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[
"Some Properties of the Higher Spin Laplace Operator",
"Some Properties of the Higher Spin Laplace Operator"
] |
[
"Chao Ding \nDepartment of Mathematics\nUniversity of Arkansas\n72701FayettevilleARUSA\n",
"John Ryan \nDepartment of Mathematics\nUniversity of Arkansas\n72701FayettevilleARUSA\n"
] |
[
"Department of Mathematics\nUniversity of Arkansas\n72701FayettevilleARUSA",
"Department of Mathematics\nUniversity of Arkansas\n72701FayettevilleARUSA"
] |
[] |
The higher spin Laplace operator has been constructed recently as the generalization of the Laplacian in higher spin theory. This acts on functions taking values in arbitrary irreducible representations of the Spin group. In this paper, we first provide a decomposition of the higher spin Laplace operator in terms of Rrita-Schwinger operators. With such a decomposition, a connection between the fundamental solutions for the higher spin Laplace operator and the fundamental solutions for the Rarita-Schwinger operators is provided. Further, we show that the two components in this decomposition are conformally invariant differential operators. An alternative proof for the conformally invariance property is also pointed out, which can be connected to Knapp-Stein intertwining operators. Last but not least, we establish a Borel-Pompeiu type formula for the higher spin Laplace operator. As an application, we give a Green type integral formula.
|
10.1090/tran/7404
|
[
"https://arxiv.org/pdf/1612.07384v1.pdf"
] | 119,266,925 |
1612.07384
|
9507e1d33996cad5c1998f428c5b5a15e516b114
|
Some Properties of the Higher Spin Laplace Operator
Chao Ding
Department of Mathematics
University of Arkansas
72701FayettevilleARUSA
John Ryan
Department of Mathematics
University of Arkansas
72701FayettevilleARUSA
Some Properties of the Higher Spin Laplace Operator
Rarita-Schwinger operatorsHigher spin Laplace operatorStokes' Theo- remConformally invariantBorel-Pompeiu type formulaGreen type integral formulaKnapp-Stein intertwining opreator AMS subject classification: Primary 30Gxx42Bxx46F1253Bxx58Jxx
The higher spin Laplace operator has been constructed recently as the generalization of the Laplacian in higher spin theory. This acts on functions taking values in arbitrary irreducible representations of the Spin group. In this paper, we first provide a decomposition of the higher spin Laplace operator in terms of Rrita-Schwinger operators. With such a decomposition, a connection between the fundamental solutions for the higher spin Laplace operator and the fundamental solutions for the Rarita-Schwinger operators is provided. Further, we show that the two components in this decomposition are conformally invariant differential operators. An alternative proof for the conformally invariance property is also pointed out, which can be connected to Knapp-Stein intertwining operators. Last but not least, we establish a Borel-Pompeiu type formula for the higher spin Laplace operator. As an application, we give a Green type integral formula.
Introduction
Classical Clifford analysis (see [4,7]) is the study of Dirac type operators and theory of monogenic functions (null solutions of the Dirac operator). It is also well known that ( [4,7]) solutions for the Dirac equation, D x f (x) = 0, in m-dimensional Euclidean space R m can be described by a Cauchy integral formula. Here, D x stands for the Euclidean Dirac operator m i=1 e i ∂ ∂x i , and the e i 's are the generators of a real Clifford algebra Cl m . As D 2 x = −∆ x , the negative Laplacian over R m , it has been shown that ( [16]) Green's formula for harmonic functions can be modified via Clifford algebras to more closely resemble a Cauchy integral formula. Further, many authors have also studied Cauchy integral formulas for modified Dirac equations. For instance, in [22], Xu studied solutions to the inhomogeneous Dirac equation (D x + λ)f (x) = 0, with λ ∈ C, which also possess a Cauchy integral formula. In [17], Ryan provided Cauchy kernels and Cauchy-Green type integral formulas for solutions to each polynomial equation (D n x + n−1 k=1 b k D k x )f (x) = 0, with b k ∈ C.
In the past few decades, many authors ( [1,5,8,9,10,11]) have been working on generalizations of classical Clifford analysis to the so called higher spin theory. This investigates higher spin differential operators acting on functions on R m , taking values in arbitrary irreducible representations of the Spin group. In Clifford analysis, these irreducible representations are traditionally constructed as homogeneous polynomial spaces satisfying certain differential equations, see [12] for more details. The first order conformally invariant differential operator, named as a Rarita-Schwinger operator, was first studied systematically in [5] and revisited in [10] with a different approach. In both papers, fundamental solutions and integral formulas, such as Borel-Pompeiu formula and Cauchy integral formula, are established. Stein and Weiss ( [21]) introduced Stein Weiss type gradients (also called Stein Weiss operators) as projections of the gradient operator with the language of representation theory. It turns out that the Dirac operator and the Rarita-Schwinger operator can both be reconstructed as Stein Weiss type gradients, see [8,20,21] for more details. In [1,11], the second order conformally invariant differential operator, named as a generalized Maxwell operator or the higher spin Laplace operator, and its fundamental solution were discovered. In [9], the authors completed the construction of arbitrary order conformally invariant differential operators in higher spin spaces as well as their fundamental solutions.
However, we have not found any results of integral formulas for these higher spin conformally invariant differential operators, except for the Rarita-Schwinger operator. In this paper, our purpose is to establish a Borel-Pompeiu type formula for the higher spin Laplace operator. The technique used here is motivated by the proof of the Green integral formula for harmonic functions via Clifford analysis. This suggests us to find the connection between the higher spin Laplace operator and the Rarita-Schwinger operators. Then we can investigate a Borel-Pompeiu type formula for the higher spin Laplace operator with the help of Stokes' theorem for the Rarita-Schwinger operators ( [5,8]). A Green type integral formula is provided at the end of this paper. This paper is organized as follows. In Section 2, we introduce the notion of Clifford analysis and some of the standard facts in higher spin theory. For instance, we review Rairta-Schwinger operators and the higher spin Laplace operator with their fundamental solutions. Further, intertwining operators for the higher spin Laplace operator are also provided for later use. Section 3 is devoted to a brief summary of Stokes' theorem for the Rarita-Schwinger operators. This is one of the main tools for establishing integral formulas for the higher spin Laplace operator. Section 4 establishes the relation between the higher spin Laplace operator with the Rarita-Schwinger operators, this allows us to apply Stokes' theorem for the Rarita-Schwinger operators in the later argument. In Sec-tion 5, we also point out the connection between the fundamental solution of the higher spin Laplace operator and the fundamental solutions of two Rarita-Schwinger type operators, which turn out to be critical in the argument for the Borel-Pompeiu type formula in Section 7. Section 6 demonstrates that the decomposition for the higher spin Laplace operator, which is obtained in Section 4, provides us two second order conformally invariant differential operators. A straightforward proof is provided, and we also point out that there is an alternative proof. This needs the help of convolution type operators and the fundamental solutions for the higher spin Laplace operator. Similar argument can be found in Section 4.1 in [9]. Such convolution type operators can also be realized as Knapp-Stein intertwining operators with the principal series representations induced by the polynomial representations of the Spin group. However, this is beyond the scope of this paper. See [6,13] for more details.
With all the preparations that have been made, we establish a Borel-Pompeiu type formula for the higher spin Laplace operator in Section 7. As an application, a Green type integral formula is stated at the end.
Preliminaries
Let {e 1 , e 2 , · · · , e m } be an orthonormal basis for the m-dimensional Euclidean space R m . The real Clifford algebra is generated by these basis elements with the defining relations
e i e j + e j e i = −2δ ij , 1 ≤ i, j ≤ m,
where δ ij is the Kronecker delta function. An arbitrary element of the basis of the Clifford algebra can be written as e A = e j 1 · · · e jr , where A = {j 1 , · · · , j r } ⊂ {1, 2, · · · , m} and 1 ≤ j 1 < j 2 < · · · < j r ≤ m. Hence for any element a ∈ Cl m , we have a = A a A e A , where a A ∈ R. The complex Clifford algebra Cl m (C) is defined as the complexification of the real Clifford algebra
Cl m (C) = Cl m ⊗ R C.
We consider real Clifford algebra Cl m throughout this subsection, but in the rest of the paper we consider the complex Clifford algebra Cl m (C) unless otherwise specified. For a = A a A e A ∈ Cl m , we will need the following anti-involutions:
• Reversion:ã = A (−1) |A|(|A|−1)/2 a A e A ,
where |A| is the cardinality of A. In particular, e j 1 · · · e jr = e jr · · · e j 1 . Alsoãb =bã for a, b ∈ Cl m .
• Clifford conjugation:ā
= A (−1) |A|(|A|+1)/2 a A e A ,
satisfying e j 1 · · · e jr = (−1) r e jr · · · e j 1 . Also ab =bā for a, b ∈ Cl m .
The Pin and Spin groups play an important role in Clifford analysis. The Pin group can be defined as P in(m) = {a ∈ Cl m : a = y 1 y 2 . . . y p , y 1 , . . . , y p ∈ S m−1 , p ∈ N},
where S m−1 is the unit sphere in R m . P in(m) is clearly a multiplicative group in Cl m . Now suppose a ∈ S m−1 ⊆ R m . If we consider axa, we may decompose
x = x a + x a⊥ ,
where x a is the projection of x onto a and x a⊥ is the remainder part perpendicular to a. Hence x a is a scalar multiple of a and we have
axa = ax a a + ax a⊥ a = −x a + x a⊥ .
So the action axa describes a reflection of x in the direction of a. By the Cartan-Dieudonné Theorem each O ∈ O(m) is the composition of a finite number of reflections. If a = y 1 · · · y p ∈ P in(m), we haveã := y p · · · y 1 and observe axã = O a (x) for some O a ∈ O(m).
Choosing y 1 , . . . , y p arbitrarily in S m−1 , we have the group homomorphism
θ : P in(m) −→ O(m) : a → O a ,
with a = y 1 · · · y p and O a x = axã is surjective. Further −ax(−ã) = axã, so 1, −1 ∈ Ker(θ). In fact Ker(θ) = {1, −1}. See [15]. The Spin group is defined as Spin(m) = {a ∈ Cl m : a = y 1 y 2 . . . y 2p , y 1 , . . . , y 2p ∈ S m−1 , p ∈ N} and it is a subgroup of P in(m). There is a group homomorphism
θ : Spin(m) −→ SO(m)
that is surjective with kernel {1, −1} and defined by the above group homomorphism for P in(m). Thus Spin(m) is the double cover of SO(m). See [15] for more details.
The Dirac operator in R m is defined to be
D x := m i=1 e i ∂ x i . Note D 2 x = −∆ x , where ∆ x is the Laplacian in R m . A Cl m -valued function f (x) defined on a domain U in R m is left monogenic if D x f (x) = 0.
Sometimes, we will consider the Dirac operator D u in a vector u rather than x.
Let M k denote the space of Cl m -valued monogenic polynomials homogeneous of degree k. Note that if h k (u) ∈ H k , the space of complex valued harmonic polynomials homoge-
neous of degree k, then D u h k (u) ∈ M k−1 , but D u up k−1 (u) = (−m − 2k + 2)p k−1 u, where p k−1 (u) ∈ M k−1 . Hence, H k = M k ⊕ uM k−1 , h k = p k + up k−1 .(1)
This is an Almansi-Fischer decomposition of H k [10]. In this Almansi-Fischer decomposition, we have P + k and P − k as the projection maps
P + k = 1 + uD u m + 2k − 2 : H k −→ M k ,(2)P − k = I − P + k = −uD u m + 2k − 2 : H k −→ uM k−1 . (3) Suppose U is a domain in R m . Consider a differentiable function f : U × R m −→ Cl m such that, for each x ∈ U , f (x, u) is a left monogenic polynomial homogeneous of degree k in u.
Then the Rarita-Schwinger operator [5,10] is defined by
R k = P + k D x : C ∞ (R m , M k ) −→ C ∞ (R m , M k )
. We also need the following three more Rarita-Schwinger type operators.
The twistor operator:
T k = P + k D x : C ∞ (R m , uM k−1 ) −→ C ∞ (R m , M k ), The dual twistor operator: T * k = P − k D x : C ∞ (R m , M k ) −→ C ∞ (R m , uM k−1 ), The remaining operator: Q k = P − k D x : C ∞ (R m , uM k−1 ) −→ C ∞ (R m , uM k−1 )
. More details can be found in [5,10]. Let Z 1 k (u, v) be the reproducing kernel for M k , which satisfies
f (v) = S m−1 Z 1 k (u, v)f (u)dS(u), f or all f (v) ∈ M k .
Then the fundamental solution for R k ( [10]) is
E k (x, u, v) = 1 ω m a k x ||x|| m Z 1 k ( xux ||x|| 2 , v), where the constant a k is m − 2 m + 2k − 2
and ω m is the area of the m-dimensional unit sphere.
Similarly, we have the fundamental solution for Q k ( [14]) as follows.
F k (x, u, v) = −1 ω m a k u x ||x|| m Z 1 k−1 ( xux ||x|| 2 , v)v.
The higher spin Laplace operator is constructed in [1], and is defined as follows.
D 2 = ∆ x − 4 u, D x D u , D x m + 2k − 2 + 4||u|| 2 D u , D x 2 (m + 2k − 2)(m + 2k − 4) .
where , is the standard inner product in Euclidean space. The fundamental solution for D 2 is also provided in the same reference. We denote it by
H k (x, u, v) = (m + 2k − 4)Γ( m 2 − 1) 4(4 − m)π m 2 ||x|| 2−m Z 2 k ( xux ||x|| 2 , v), where Z 2 k (u, v)
is the reproducing kernel for H k and satisfies
f (v) = S m−1 Z 2 k (u, v)f (u)dS(u), f or all f (v) ∈ H k .
The result of intertwining operators for D 2 is stated as follows for later use.
Theorem 1. [1, 9] Let f (x, u) ∈ C ∞ (R m , H k ), x = ϕ(x) = (ax+b)(cx+d) −1 is a Möbius transformation and u = (cx + d)u( cx + d) ||cx + d|| 2 . Then we have J −1 −2 D 2,x,u J 2 (ϕ, x)f (ϕ(x), (cx + d)u( cx + d) ||cx + d|| 2 ) = D 2,x ,u f (x , u ).
where J −2 (ϕ, x) = ||cx + d|| −m−2 and J 2 (ϕ, x) = ||cx + d|| 2−m are called intertwining operators for D 2 . The lower index in D 2,x,u stands for the variables of the operator D 2 .
To conclude this section, we point out that, if D u f (u) = 0 thenf (u)D u = −f (u)D u = 0. So we can talk of right monogenic functions, right monogenic polynomials with homogeneity of degree k, right Almansi-Fischer decomposition for H k , etc. In other words, for all the results we introduced above, we have their analogues for right monogenic functions.
Stokes' theorem for the Rarita-Schwinger type operators
For the convenience of the reader, we review Stokes' theorem for the Rarita-Schwinger type toperators as follows. More details can be found in [5,8].
Theorem 2 ([10]). (Stokes' theorem for R k )
Let Ω and Ω be domains in R m and suppose the closure of Ω lies in Ω . Further suppose the closure of Ω is compact and ∂Ω is piecewise smooth. Let f, g ∈ C 1 (Ω , M k ).
Then
Ω (g(x, u)R k , f (x, u)) u + (g(x, u), R k f (x, u)) u dx m = ∂Ω (g(x, u), dσ x f (x, u)) u .
where
dσ x = n(x)dσ(x), dσ(x) is the area element. (P (u), Q(u)) u = S m−1 P (u)Q(u)dS(u)
is the inner product for any pair of Cl m -valued polynomials.
Comparing with the version in [10], we shortened the identity by missing P + k in the last two equations on purpose. This can also be found in the proof of this theorem in [10]. We also want to clarify that, in the Stokes' theorem above, for g(x, u)R k , the R k is the right Rarita-Schwinger operator. Since it is positioned on the right hand side, there should be no confusion for this. The reader will see more such terms in the rest of this paper.
Theorem 3 ([8]). (Stokes' Theorem for T k and T * k )
Let Ω and Ω be defined as above.
Then for f ∈ C 1 (R m , M k ) and g ∈ C 1 (R m , uM k−1 ), we have ∂Ω g(x, u), dσ x f (x, u) u = Ω g(x, u)T k , f (x, u) u dx m + Ω g(x, u), T * k f (x, u) u dx m . Theorem 4. ([14])(Stokes' theorem for Q k )
Let Ω and Ω be defined as above. Then for f, g ∈ C 1 (R m , uM k−1 ), we have
Ω (g(x, u)Q k , f (x, u)) u + (g(x, u), Q k f (x, u)) u dx m = ∂Ω (g(x, u), dσ x f (x, u)) u .
We also missed P − k in the last equation for convenience. See [14].
Proposition 1. Let P + k and P − k be the projection maps defined in (2) and (3). Then the higher spin Laplace operator D 2 can be written as follows.
D 2 = −R 2 k P + k + 2T * k R k P + k m + 2k − 4 − 2T k Q k P − k m + 2k − 4 − (m + 2k)Q 2 k P − k m + 2k − 4 ,(4)= −R 2 k P + k + 2R k T k P − k m + 2k − 4 − 2Q k T * k P + k m + 2k − 4 − (m + 2k)Q 2 k P − k m + 2k − 4 ,(5)when it acts on a function f (x, u) ∈ C ∞ (R m , H k ).
Here, we first prove the equation (4). To accomplish this, we need the following lemma.
p 1 = 1 + uD u m + 2k − 2 : H k −→ M k , p 0 = −D u m + 2k − 2 : H k −→ M k−1 .
Then for any f (x, u) ∈ C ∞ (R m , H k ), we have
D 2 = −R 2 k p 1 + 4u D u , D x R k p 1 (m + 2k − 2)(m + 2k − 4) p 1 −uR 2 k−1 p 0 − 4 m + 2k − 2 u, D x − ||u|| 2 D u , D x m + 2k − 4 R k−1 p 0 .(6)
Here, we remind the reader that the projection map p 1 is the same as P + k , but p 0 is different from P − k . Indeed, P − k = up 0 . Now, we can start verifying equation (4). Proof. From the expression of D 2 in Lemma 1, we notice that, for any f (x, u) ∈ C ∞ (R m , H k ), we have R k p 1 f (x, u) ∈ C ∞ (R m , M k ). It is easy to see that
u D u , D x R k p 1 f (x, u) = − uD u D x + uD x D u 2 R k p 1 f (x, u) = − uD u D x 2 R k p 1 f (x, u).
The last equation comes from D u R k p 1 f (x, u) = 0. Hence, the first two terms of (6) become
−R 2 k p 1 + 4u D u , D x R k (m + 2k − 2)(m + 2k − 4) p 1 = −R 2 k P + k + 2T * k R k P + k m + 2k − 4 ,
which are exactly the first two terms in (4).
On the other hand, Lemma 1 tells that the last two terms in (4) come from D 2 acting on P H k ). Next, we show that D 2 is a linear combination of Q 2 k and T k Q k when acting on functions in C ∞ (R m , uM k−1 ). This makes sense, since Q 2 k and T k Q k are the two "paths", which start from C ∞ (R m , uM k−1 ) and end in C ∞ (R m , H k ). See the following diagram.
− k f (x, u) ∈ C ∞ (R m , uM k−1 ), for f (x, u) ∈ C ∞ (R m ,C ∞ (R m , H k ) C ∞ (R m , H k ) C ∞ (R m , M k ) C ∞ (R m , M k ) C ∞ (R m , M k ) C ∞ (R m , uM k−1 ) C ∞ (R m , uM k−1 ) C ∞ (R m , uM k−1 ) || || ⊕ ⊕ Q k Q k T k
Hence, we calculate D 2 , Q 2 k and T k Q k acting on C ∞ (R m , uM k−1 ), respectively. For our convenience, we assume P − k f (x, u) = ug(x, u) for some g(x, u) ∈ C ∞ (R m , M k−1 ). Now, we calculate D 2 u and keep in mind that it acts on a function g(x, u) ∈ M k−1 with respect to the variable u. In other words, in the calculation below, any operator with D u on the right hand side vanishes.
D 2 u = ∆ x − 4 m + 2k − 2 u, D x − ||u|| 2 D u , D x 2 m + 2k − 4 D u , D x u = u∆ x − 4 m + 2k − 2 u, D x − ||u|| 2 D u , D x m + 2k − 4 u D u , D x + D x = u∆ x − 4u u, D x D u , D x m + 2k − 2 − 4 u, D x D x m + 2k − 2 + 4u||u|| 2 D u , D x 2 (m + 2k − 2)(m + 2k − 4) + 8||u|| 2 D u , D x D x (m + 2k − 2)(m + 2k − 4) .(7)
On the other hand, since
D x u = −uD x − 2 u, D x , D u u = −m − 2E u − uD u , we have Q 2 k u = uD u D x uD u D x (m + 2k − 2) 2 u = uD u D x uD u (−uD x − 2 u, D x ) (m + 2k − 2) 2 = 1 (m + 2k − 2) 2 − uD u D x u(−m − uD u − 2E u )D x − 2uD u D x u( u, D x D u + D x ) , where E u = m i=1 u i ∂ ∂u i is the Euler operator. Since E u D x g(x, u) = (k − 1)D x g(x, u) and D u g(x, u) = 0 for g(x, u) ∈ C ∞ (R m , M k−1 ). The equation above becomes m (m + 2k − 2) 2 uD u D x uD x − uD u D x ||u|| 2 D u D x (m + 2k − 2) 2 + 2uD u D x uE u D x (m + 2k − 2) 2 − 2uD u D x uD x (m + 2k − 2) 2 = m + 2k − 4 m + 2k − 2 uD u (−uD x − 2 u, D x )D x − u(2u + ||u|| 2 D u )D x D u D x (m + 2k − 2) 2 .
The last equation comes from combining all uD u D x uD x terms with the fact that E u D x = (k − 1)D x . Applying similar identities that we just used again, we get
− m + 2k − 4 (m + 2k − 2) 2 u(−m − uD u − 2E u )D 2 x − 2(m + 2k − 4) (m + 2k − 2) 2 u(D x + u, D x D u )D x − 2u 2 D x D u D x (m + 2k − 2) 2 − u||u|| 2 (m + 2k − 2) 2 (−D x D u − 2 D u , D x )(−2 D u , D x ).
Simplifying the previous equation, we have
Q 2 k u = (m + 2k − 4) 2 (m + 2k − 2) 2 uD 2 x + 4(m + 2k − 4) (m + 2k − 2) 2 u u, D x D u , D x + 4u 2 D x D u , D x (m + 2k − 2) 2 − 4u||u|| 2 D u , D x 2 (m + 2k − 2) 2 .(8)
Similarly, we have the details for rewriting T k Q k u as follows.
Since
D x u = −uD x − 2 u, D x , we have T k Q k u = (D x + uD u D x m + 2k − 2 )(− uD u D x m + 2k − 2 )u = −D x uD u D x m + 2k − 2 u − Q 2 k u = −1 m + 2k − 2 (−uD x − 2 u, D x )D u (−uD x − 2 u, D x ) − Q 2 k u = − uD x D u uD x m + 2k − 2 − 2 u, D x D u uD x m + 2k − 2 − 2uD x D u u, D x m + 2k − 2 − 4 u, D x D u u, D x m + 2k − 2 − Q 2 k u.
As D u u = −m − 2E u − uD u and D u u, D x = D x + u, D x , the equation above becomes
− uD x (−m − 2E u − uD u )D x m + 2k − 2 − 2 u, D x (−m − 2E u − uD u )D x m + 2k − 2 − 2uD x ( u, D x D u + D x ) m + 2k − 2 − 4 u, D x ( u, D x D u + D x ) m + 2k − 2 − Q 2 k u.
Recall that this operator acts on g(x, u) ∈ C ∞ (R m , M k−1 ), which means
E u D x g(x, u) = (k − 1)D x g(x, u), uD x u, D x D u g(x, u) = 0 and u, D x 2 D u g(x, u) = 0. Hence, with D u D x = −D x D u − 2 D u , D x , we get T k Q k u = uD x uD u D x m + 2k − 2 + uD 2 x + 2 u, D x D x − 4u u, D x D u , D x m + 2k − 2 − 2uD 2 x m + 2k − 2 − 4 u, D x D x m + 2k − 2 − Q 2 k u = u(−uD x − 2 u, D x )(−D x D u − 2 D u , D x ) m + 2k − 2 + m + 2k − 4 m + 2k − 2 uD 2 x + 2(m + 2k − 4) m + 2k − 2 u, D x D x − 4u u, D x D u , D x m + 2k − 2 − Q 2 k u Since u(−uD x − 2 u, D x )D x D u g(x, u) = 0 for g(x, u) ∈ C ∞ (R m , M k−1 ), the previous equation becomes −2||u|| 2 D u , D x D x + 4u u, D x D u , D x m + 2k − 2 + m + 2k − 4 m + 2k − 2 uD 2 x + 2(m + 2k − 4) m + 2k − 2 u, D x D x − 4u u, D x D u , D x m + 2k − 2 − Q 2 k u.
Simplifying the equation above, we have
T k Q k u = − 2||u|| 2 D u , D x D x m + 2k − 2 + m + 2k − 4 m + 2k − 2 uD 2 x + 2(m + 2k − 4) m + 2k − 2 u, D x D x − Q 2 k u. (9)
Combining the equations (7), (8) and (9), a straightforward verification shows that
D 2 u = − 2T k Q k u m + 2k − 4 − m + 2k m + 2k − 4 Q 2 k u.
This completes the proof of equation (4). In [5], the authors point out that
T k Q k f (x, u) = −R k T k f (x, u), if f (x, u) ∈ C ∞ (R m , uM k−1 ); T * k R k f (x, u) = −Q k T * k f (x, u), if f (x, u) ∈ C ∞ (R m , M k )
. Then we can obtain (5) from (4) immediately.
Connection between fundamental solution of D 2
and fundamental solutions of R k and Q k
In equation (5), we have
D 2 = −R 2 k P + k + 2R k T k P − k m + 2k − 4 − 2Q k T * k P + k m + 2k − 4 − (m + 2k)Q 2 k P − k m + 2k − 4 = R k (−R k P + k + 2T k P − k m + 2k − 4 ) + Q k (− 2T * k P + k m + 2k − 4 − (m + 2k)Q k P − k m + 2k − 4 ). We let A k = −R k P + k + 2T k P − k m + 2k − 4 and B k = − 2T * k P + k m + 2k − 4 − (m + 2k)Q k P − k m + 2k − 4 for conve- nience, then we have D 2 = R k A k + Q k B k .
The main result of this section is the following.
Proposition 2. Let E k (x, u, v), F k (x, u, v)
and H k (x, u, v) be the fundamental solutions for R k , Q k and D 2 respectively (see Section 2). We have
H k (x, u, v)A k,r = E k (x, u, v), H k (x, u, v)B k,r = F k (x, u, v),
where A k,r and B k,r act from the right hand side of H k (x, u, v).
To prove the proposition above, we need the following technical lemma.
Lemma 2.
[10] Suppose p k is a left monogenic polynomial homogeneous of degree k and p k−1 is a right monogenic polynomial homogeneous of degree k − 1 then
(p k−1 (u)u, p k (u)) u = S m−1 p k−1 (u)up k (u)dS(u) = 0.
Now, we start proving the previous proposition.
Proof. Since H k (x, u, v) is the fundamental solution for D 2 , then we have
H k (x − y, u, v)D 2 = δ(y)Z 2 k (u, v),
where Z 2 k (u, v) is the reproducing kernel for H k . In other words, for any function
f (x, u) ∈ C ∞ (R m , H k ) with compact support with respect to the variable x, denoted by C ∞ c (R m , H k ), we have f (y, v) = R m (H k (x − y, u, v)D 2 , f (x, u)) u = R m (H k (x − y, u, v)(A k,r R k + B k,r Q k ), P + k f (x, u) + P − k f (x, u)) u . (10)
From the expression of A k,r , we observe that
H k (x − y, u, v)A k,r ∈ C ∞ (R m , M k,r ), H k (x − y, u, v)A k,r R k ∈ C ∞ (R m , M k,r ),
where M k,r stands for right monogenic polynomial space homogeneous of degree k. Similarly, we know that H k (x − y, u, v)B k,r Q k ∈ C ∞ (R m , M k−1,r u). In the meantime, we know that P + k f ∈ C ∞ (R m , M k ) and P − k f ∈ C ∞ (R m , uM k−1 ). Hence, with the help of Lemma 2, we have
(H k (x − y, u, v)A k,r R k , P − k f (x, u)) u = (H k (x − y, u, v)B k,r Q k , P + k f (x, u)) u = 0. (11)
Therefore, equation (10) tells us that
R m (H k (x − y, u, v)A k,r R k , P + k f (x, u)) u + R m (H k (x − y, u, v)B k,r Q k , P − k f (x, u)) u = f (y, v) = P + k f (y, v) + P − k f (y, v)
,
for any f (y, v) ∈ C ∞ c (R m , H k ). In particular, for any f (y, v) ∈ C ∞ c (R m , M k ), which means P − k f = 0, we have R m (H k (x − y, u, v)A k,r R k , P + k f (x, u)) u = P + k f (y, v).
This implies that
H k (x − y, u, v)A k,r R k = δ(y)Z 1 k (u, v), where Z 1 k (u, v)
is the reproducing kernel for M k . However, from the definition of fundamental solutions for a differential operator, this also implies that H k (x − y, u, v)A k,r must be the fundamental solution for R k , i.e., y, u, v).
H k (x − y, u, v)A k = E k (x −Similarly, if f (y, v) ∈ C ∞ c (R m , uM k−1 ), which means P + k f = 0, then we have R m (H k (x − y, u, v)B k,r Q k , P − k f (x, u)) u = P − k f (y, v),(12)which implies H k (x − y, u, v)B k,r Q k = δ(x − y)Z 1 k−1 (u, v)u.
We remind the reader that the "u" on the right hand side is cancelled by the "u" in P
− k f (x, u) ∈ C ∞ (R m , uM k−1 ). Therefore, H k (x − y, u, v)B k,r = F k (x − y, u, v),
which completes the proof.
Conformally invariant property for R k A k and Q k B k
We notice that in the decomposition of
D 2 = R k A k + Q k B k , R k A k : C ∞ (R m , H k ) −→ C ∞ (R m , M k ),(13)Q k B k : C ∞ (R m , H k ) −→ C ∞ (R m , uM k−1 ).(14)
In this section, we will show that R k A k and Q k B k are both conformally invariant. More specifically,
Theorem 5. Let f (x, u) ∈ C ∞ (R m , H k ), x = ϕ(x) = (ax + b)(cx + d) −1 is a Möbius transformation and u = (cx + d)u( cx + d) ||cx + d|| 2 . Then we have J −1 −2 (ϕ, x)(R k A k ) x,u J 2 (ϕ, x)f (ϕ(x), (cx + d)u( cx + d) ||cx + d|| 2 ) = (R k A k ) x ,u f (x , u ), J −1 −2 (ϕ, x)(Q k B k ) x,u J 2 (ϕ, x)f (ϕ(x), (cx + d)u( cx + d) ||cx + d|| 2 ) = (Q k B k ) x ,u f (x , u ),
where J −2 (ϕ, x) and J 2 (ϕ, x) are defined in Theorem 1.
Proof. From Theorem 1, for f (x, u) ∈ C ∞ (R m , H k ), we have
J −1 −2 (ϕ, x)D 2,x,u J 2 (ϕ, x)f (ϕ(x), (cx + d)u( cx + d) ||cx + d|| 2 ) = D 2,x ,u f (x , u ). Since D 2 = R k A k + Q k B k , the previous equation becomes J −1 −2 (ϕ, x)(R k A k ) x,u J 2 (ϕ, x)f (ϕ(x), (cx + d)u( cx + d) ||cx + d|| 2 ) +J −1 −2 (ϕ, x)(Q k B k ) x,u J 2 (ϕ, x)f (ϕ(x), (cx + d)u( cx + d) ||cx + d|| 2 ) = (R k A k ) x ,u f (x , u ) + (Q k B k ) x ,u f (x , u ).(15)
From Theorem 1, we know that
J 2 (ϕ, x)f (ϕ(x), (cx + d)u( cx + d) ||cx + d|| 2 ) ∈ C ∞ (R m , H k ).
Then with the help of (13) and (14), we obtain
J −1 −2 (ϕ, x)(R k A k ) x,u J 2 (ϕ, x)f (ϕ(x), (cx + d)u( cx + d) ||cx + d|| 2 ) ∈ C ∞ (R m , M k ), J −1 −2 (ϕ, x)(Q k B k ) x,u J 2 (ϕ, x)f (ϕ(x), (cx + d)u( cx + d) ||cx + d|| 2 ) ∈ C ∞ (R m , uM k−1 ), (R k A k ) x ,u f (x , u ) ∈ C ∞ (R m , M k ), (Q k B k ) x ,u f (x , u ) ∈ C ∞ (R m , uM k−1 ).
Meanwhile, from the Almansi-Fischer decomposition of H k (see (1)), we know that
C ∞ (R m , M k ) ∩ C ∞ (R m , uM k−1 ) = ∅.
Therefore, in equation (15), the functions on both sides, which belong to the same function space, are equal to each other. In other words,
J −1 −2 (ϕ, x)(R k A k ) x,u J 2 (ϕ, x)f (ϕ(x), (cx + d)u( cx + d) ||cx + d|| 2 ) = (R k A k ) x ,u f (x , u ), J −1 −2 (ϕ, x)(Q k B k ) x,u J 2 (ϕ, x)f (ϕ(x), (cx + d)u( cx + d) ||cx + d|| 2 ) = (Q k B k ) x ,u f (x , u ),
which completes the proof.
We point out that there is an alternative approach for the conformally invariant property for R k A k and Q k B k . We notice that equation (11) and (12) provide us integrals to reproduce P + k f (y, v) and P − k f (y, v). The integrals of the form
R m E k (x − y, u, v), f (x, u) u
are called convolution type operators generated by the fundamental solution E k (x, u, v) in [9]. These can be considered as Knapp-Stein intertwining operators with respect to a principal series representation of the Spin group. See [6,13] for more details. Further, we have already seen the intertwining operators for D 2 in Theorem 1. With similar argument as in Section 4.1 in [9], we can also have Theorem 5.
7 Integral formulas for the higher spin Laplace operator
With the decomposition of D 2 obtained in Section 4 and the Stokes' theorem for the Rarita-Schwinger operators, we will establish a Borel-Pompeiu type formula for the higher spin Laplace operator in this section.
Theorem 6. (Borel-Pompeiu type formula)
Let Ω and Ω be domains in R m and suppose the closure of Ω lies in Ω . Further suppose the closure of Ω is compact and ∂Ω is piecewise smooth. Let f (x, u) ∈ C ∞ (R m , H k ) and y ∈ Ω, then we have
∂Ω H k (x − y, u, v)(P + k − 2P − k m + 2k − 4 ), dσ x R k P + k f (x, u) u + ∂Ω E k (x − y, u, v), dσ x P + k f (x, u) u + ∂Ω H k (x − y, u, v)( 2P + k m + 2k − 4 + m + 2k m + 2k − 4 P − k ), dσ x Q k P − k f (x, u) u + ∂Ω F k (x − y, u, v), dσ x P − k f (x, u) u = − Ω H k (x − y, u, v), D 2 f (x, u) u dx m + f (y, v),
where dx m = dx 1 ∧ · · · ∧ dx m , dσ x = n(x)dσ(x), σ is scalar Lebesgue measure on ∂Ω and n(x) is unit outer normal vector to ∂Ω.
Before we prove the theorem above, we remind the reader that, in the Stokes' Theorem for the Rarita-Schwinger type operators in Section 3, the function spaces in these theorems are different. More specifically, for instance, the Stokes' theorem for R k requires that f (x, u), g(x, u) ∈ C 1 (R m , M k ) but the Stokes' theorem for Q k requires that f (x, u), g(x, u) ∈ C 1 (R m , uM k−1 ). However, in the theorem above, we notice
H k (x − y, u, v), f (x, u) ∈ C ∞ (R m , H k ).
Hence, in order to apply the Stokes' theorem for the Rarita-Schwinger type operators, we have to break them into C ∞ (R m , M k ) and C ∞ (R m , uM k−1 ) via the Almansi-Fischer decomposition for H k . That is why P + k and P − k are involved in the theorem. Proof. We first consider the first two integrals on the left hand side. From Proposition 2, we have
E k (x, u, v) = H k (x, u, v)A k,r , where A k,r = −P + k R k + 2P − k T k m + 2k − 4 .
For convenience, we let
E k = E k (x − y, u, v), F k = F k (x − y − u − v), H k = H k (x − y, u, v),
unless it is necessary to specify their dependence on the variables. Hence, the first two integrals can be rewritten as follows.
+ = − S m−1 S m−1 rZ 2 k (ζuζ, v)P + k ζR k P + k f (y + rζ, u)dS(u)dS(ζ),
where r m−1 above comes from the Jacobian of the change of variable. Since f (x, u) ∈ C ∞ (R m , H k ) and Z 2 k (u, v) is the reproducing kernel of H k , then Z 2 k (ζuζ, v)P + k ζR k P + k f (y + rζ, u) is bounded for ζ ∈ S m−1 and u ∈ S m−1 . Therefore, the previous integral goes to zero when r goes to zero.
On the other hand, from Lemma 2, we observe that
H k P − k T k ⊥ T * k R k P + k f (x, u) with respect to ( , ) u . Therefore, equation (16) becomes = ∂Br H k (−P + k R k + 2 m + 2k − 4 P − k T k ), dσ x P + k f (x, u) u + B c r H k P + k + H k P − k , (R 2 k − 2 m + 2k − 4 T * k R k )P + k f (x, u) u + B c r H k (−P + k R 2 k + 2 m + 2k − 4 P − k T k R k ), P + k f (x, u) u = ∂Br H k A k,r , dσ x P + k f (x, u) u + B c r H k , (R 2 k − 2 m + 2k − 4 T * k R k )P + k f (x, u) u + B c r H k A k,r R k , P + k f (x, u) u = ∂Br E k , dσ x P + k f (x, u) u + B c r H k , (R 2 k P + k − 2 m + 2k − 4 T * k R k P + k )f (x, u) u .(17)
The last equation comes from
H k (x − y, u, v)A k,r R k = E k (x − y, u, v)R k = 0, f or x ∈ B c r .
Similar argument applies for the last two integrals on the left hand side in Theorem 6. With
F k (x − y, u, v) = H k (x − y, u, v)B k,r , where B k,r = − 2P + k T * k m + 2k − 4 − (m + 2k)P − k Q k m + 2k − 4 ,
the last two integrals in Theorem 6 can be rewritten as follows.
∂Ω H k ( 2P + k m + 2k − 4 + m + 2k m + 2k − 4 P − k ), dσ x Q k P − k f (x, u) u + ∂Ω F k , dσ x P − k f (x, u) u =
when r approaches zero. From Lemma 2, we have when r approaches zero. If (20) holds, then the previous equation becomes
= P + k f (y, v) + P − k f (y, v) − B c r H k , D 2 f (x, u) u = f (y, v) − Ω H k , D 2 f (x, u) u ,
which completes the proof of the theorem. The last equation comes from Br H k , D 2 f (x, u) u −→ 0, when r approaches zero because of the homogeneity of x − y in H k (x − y, u, v).
Now, we prove (20). To accomplish this, we need the following lemma. where a k = m − 2 m + 2k − 2 .
We only provide the details for the first equation of (20), similar argument also applies for the second equation. This argument can also be found in the proof of Theorem 7 in [10]. We rewrite
∂Br E k , dσ x P + k f (x, u) u = ∂Br E k , dσ x P + k f (y, u) u + ∂Br E k , dσ x P + k [f (x, u) − f (y, u)] u
Since the second integral on the right hand side tends to zero as r goes to zero because of the continuity of f (x, u) with respect to the variable x, we only need to deal with the first integral. We will need the property that
E k (x, u, v) = 1 ω m a k x ||x|| m Z 1 k ( xux ||x|| 2 , v) = 1 ω m a k Z 1 k (u, xvx ||x|| 2 )
x ||x|| m , more details can be found in [10]. Hence, the first integral becomes
∂Br E k , dσ x P + k f (y, u) u = ∂Br S m−1 1 ω m a k Z 1 k (u, (x − y)v(x − y) ||x − y|| 2 )
x − y ||x − y|| m n(x)P + k f (y, u)dS(u)dσ(x)
where n(x) is the unit outer normal vector and dσ(x) is the scalar measure on ∂B r . Now n(x) here is y − x ||x − y|| . Hence the previous integral becomes
∂Br S m−1 1 ω m a k Z 1 k (u, (x − y)v(x − y) ||x − y|| 2 ) x − y ||x − y|| m y − x ||x − y|| P + k f (y, u)dS(u)dσ(x) = ∂Br 1 r m−1 S m−1 1 ω m a k Z 1 k (u, (x − y)v(x − y) ||x − y|| 2 )P + k f (y, u)dS(u)dσ(x)
By Lemma 3, this integral is qual to
S m−1 1 ω m a k Z 1 k (u, v)P + k f (y, u)dS(u) = P + k f (y, v),
which completes the proof for (20). Similar argument applies for the second equation of (20) with the help of
F k (x, u, v) = −1 ω m a k u x ||x|| m Z 1 k−1 ( xux ||x|| 2 , v)v = −1 ω m a k uZ 1 k−1 (u, xvx ||x|| 2 )
x ||x|| m v.
As an application of the previous theorem, we have a Green type integral formula for the higher spin Laplace operator D 2 immediately, when D 2 f (x, u) = 0.
Theorem 7. (Green type integral formula)
Suppose Ω and Ω are defined as in the previous theorem. Let f (x, u) ∈ C ∞ (R m , H k ), y ∈ Ω and D 2 f (x, u) = 0, then we have
f (y, v) = ∂Ω H k (x − y, u, v)(P + k − 2P − k m + 2k − 4 ), dσ x R k P + k f (x, u) u + ∂Ω E k (x − y, u, v), dσ x P + k f (x, u) u + ∂Ω H k (x − y, u, v)( 2P + k m + 2k − 4 + m + 2k m + 2k − 4 P − k ), dσ x Q k P − k f (x, u) u + ∂Ω F k (x − y, u, v), dσ x P − k f (x, u) u .
Lemma 1 .
1([1]) Let p 1 and p 0 be the projection maps in the Almansi-Fischer decomposition of H k defined as
Lemma 3 .
3[10] Suppose h k : R m −→ Cl m is a harmonic polynomial homogeneous of degree k and m > 2. Suppose u ∈ S (xux)dS(x) = a k h k (u),
Connection between D 2 and the Rarita-Schwinger operatorsIn this section, we will rewrite the higher spin Laplace operator D 2 in terms of the Rarita-Schwinger operators. This allows us to apply the Stokes' theorem for the Rarita-Schwinger operators in the last section.
Let B r = {x : ||x − y|| < r} ⊂ Ω, with some r > 0, and B c r = Ω\B r . Then we apply the Stokes' theorem for T k to the second integral and the Stokes' theorem for R k to the other three integrals. The previous equation becomesWith four integrals cancelled above, it becomesRecall thatand P + k and P − k are independent with respect to the variable x. Hence, from the homo-when r approaches zero. Here, we give the details for the first integral approaching zero, the second can be derived from similar arguments. For convenience, we ignore the constant in H k (x − y, u, v). We also remind the reader that we will see similar statements later in this section.Here, the outer normal vector n(Applying the Stokes' theorem for Q k and T k to the equation above, we haveWith four integrals cancelled above, the previous equation becomesSimilarly, from the homogeneity of x − y in H k (x − y, u, v), we have, with respect to ( , ) u . Therefore, equation(18)The last equation comes from(17)and(19), we have the left hand side in Theorem 6 is equal toThe last equation comes from (4). That is,Now, we state that ∂Br E k , dσ x P + k f (x, u) u = P + k f (y, v),
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"Understanding the Forward Muon Deficit in Coherent Pion Production",
"Understanding the Forward Muon Deficit in Coherent Pion Production"
] |
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"L M Sehgal \nInstitute of Theoretical Physics (E)\nRWTH Aachen\n52056AachenGermany\n"
] |
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"Institute of Theoretical Physics (E)\nRWTH Aachen\n52056AachenGermany"
] |
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For any inelastic process v ℓ + I → ℓ − + F with m ℓ = 0, the cross section at θ ℓ = 0 is given by Adler's PCAC theorem. Inclusion of the lepton mass has a dynamical effect ("PCAC-screening") caused by interference of spin-zero (π + ) and spin-one exchanges. This effect may be relevant to the forward suppression reported in recent experiments.
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10.1063/1.2834469
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"https://arxiv.org/pdf/0709.4450v1.pdf"
] | 118,028,498 |
0709.4450
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7dda8f4e66c7ae4394333dcb46d9d09812c576d5
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Understanding the Forward Muon Deficit in Coherent Pion Production
27 Sep 2007
L M Sehgal
Institute of Theoretical Physics (E)
RWTH Aachen
52056AachenGermany
Understanding the Forward Muon Deficit in Coherent Pion Production
27 Sep 2007
For any inelastic process v ℓ + I → ℓ − + F with m ℓ = 0, the cross section at θ ℓ = 0 is given by Adler's PCAC theorem. Inclusion of the lepton mass has a dynamical effect ("PCAC-screening") caused by interference of spin-zero (π + ) and spin-one exchanges. This effect may be relevant to the forward suppression reported in recent experiments.
Recent experiments with low energy neutrino beams suggest that in inelastic CC events, there are fewer muons coming out at small angles than expected. (For a review of the data, see the talk of Bonnie Fleming) [1]. he evidence comes principally from two-track events that are interpretable as ν µ + (p, n) → µ − + (p, n) + π + with nucleon undetected, or coherent π + production ν µ + Nucleus → µ − + π + + Nucleus. In particular, the K2K experiment (E ν ≈ 1.3GeV ) has reported a deficit at low Q 2 (Q 2 < 0.1GeV 2 ) which they interpret as a suppression of coherent π + , obtaining an upper limit σ (coh π + )/σ (CC) < 0.6% [2], compared with a theoretically expected value of 2% [3].
The deficit is puzzling since there appears to be evidence for NC coherent π 0 production ν µ + Nucleus → ν µ + π 0 + Nucleus [4] at roughly the expected rate σ (coh π 0 )/σ (CC) ≈ 1 %, and a ratio σ (coh π + )/σ (coh π 0 ) = 2 is expected from fairly general isospin considerations.
This situation has prompted us to ask whether the deficit could be a dynamical effect caused by the nonzero mass of the muon in the CC channel, absent in the NC process. We recall that in any inelastic CC reaction ν µ + I → µ − + F, F = I, the cross section in the forward scattering configuration for m ℓ = 0 is predicted by Adler's PCAC theorem
[5] dσ dxdy θ =0 = G 2 ME ν π 2 f 2 π (1 − y)σ (π + + I → F)| E π =E ν y(1)
For non-forward scattering, this result is expected to be modified by a slowly-varying factor
(1 + Q 2 /M 2 A ) −2 , where M A ≈ 1GeV
is the typical mass of the spin-one (1 ++ ) mesons mediating the process. If the muon mass is not neglected, however, the process receives an additional contribution due to the exchange of a spin-zero π + meson. It was shown by Adler [5] that the forward theorem is modified by a multiplicative factor, which may be written as [6] C Adler = 1 − 1 2
Q 2 min Q 2 + m 2 π 2 + 1 4 y Q 2 min (Q 2 − Q 2 min ) Q 2 + m 2 π ) 2 (2) where Q 2 min = m 2 ℓ y 1 − y(3)
This correction is valid for small angles, and contains the important terms in which the factor m 2 µ is accompanied by the pion propagator 1/(Q 2 + m 2 π ). The factor C Adler has non trivial consequences for all inelastic cross sections at small angles. For forward scattering, in particular,
C Adler (θ = 0) = 1 − 1 2 Q 2 min Q 2 min + m 2 π 2 (4)
The minus sign within parentheses indicates that the effect of pion-exchange is a destructive interference. Taking an average value y ≈ 1/2, the forward suppression factor is
C Adler (θ = 0, y = 1/2) = 1 − 1 2 m 2 µ m 2 µ + m 2 π 2 = 70%!(5)
We have investigated [6], the consequences of this screening effect in the coherent process ν µ + C 12 → µ − + π + + C 12 using the model described in [3]. The effects on dσ /d cosθ µ and dσ /dQ 2 are shown in Fig. 1 and Fig. 2, and exhibit a forward muon deficit. Note that a comparison of the m µ = 0 and m µ = 0 cases is essentially a comparison of ν µ and ν e scattering, and that a "muon deficit" could equally be regarded as an "electron excess".
In applying the above suppression mechanism to the K2K data, our analysis [6] indicates that the coherent π + signal in the domain Q 2 < 0.1GeV 2 is suppressed by a factor C coh ≈ 0.77. We have also estimated the incoherent resonant background, using the resonance model [7], and obtain a suppression factor C res ≈ 0.85. These results allow a reinterpretation of the K2K deficit in the interval Q 2 < 0.1GeV 2 , and reduce the discrepancy to about 2σ . A detailed discussion of muon mass effects will appear in [8].
ACKNOWLEDGMENTSI thank Chris Berger for help in preparing the figures. Thanks to Jorge Morfin and the organizers of NuInt07 for the invitation to present these ideas at their Workshop.
B Fleming, Mini Boonethese Proceedings. B. Fleming (Mini Boone), these Proceedings
. M Hasegawa, K2KPhys. Rev. Lett. 95252301M. Hasegawa et al (K2K), Phys. Rev. Lett. 95, 252301 (2005)
. D Rein, L M Sehgal, Nucl. Phys., B. 22329D. Rein and L. M. Sehgal, Nucl. Phys., B 223, 29 (1983)
. S Nakayama, K2K)Phys. Lett.B. 619255S. Nakayama (K2K), Phys. Lett.B 619, 255, (2005);
J L Link ; S, MiniBooneAdler, MiniBoonethese Proceedings 5. 135963J. Link (MiniBoone), these Proceedings 5. S. L. Adler, Phys. Rev. 135, B963 (1964);
. arXiv:hep-ph/0505177Ann. Phys. 5089Ann. Phys. 50, 89 (1968); arXiv:hep-ph/0505177.
. D Rein, L M Sehgal, arXiv:hep-ph/0606185D. Rein and L.M. Sehgal, arXiv: hep-ph / 0606185
. D Rein, L M Sehgal, Ann. Phys. 13379D. Rein and L.M. Sehgal, Ann. Phys. 133 79, (1981)
. Ch, L M Berger, Sehgal, in preparationCh. Berger and L. M. Sehgal, in preparation
|
[] |
[
"Volatility-inspired σ-LSTM cell",
"Volatility-inspired σ-LSTM cell",
"Volatility-inspired σ-LSTM cell",
"Volatility-inspired σ-LSTM cell"
] |
[
"German Rodikov ",
"Nino Antulov-Fantulin ",
"German Rodikov ",
"Nino Antulov-Fantulin "
] |
[] |
[] |
Volatility models of price fluctuations are well studied in the econometrics literature, with more than 50 years of theoretical and empirical findings. The recent advancements in neural networks (NN) in the deep learning field have naturally offered novel econometric modeling tools. However, there is still a lack of explainability and stylized knowledge about volatility modeling with neural networks; the use of stylized facts could help improve the performance of the NN for the volatility prediction task. In this paper, we investigate how the knowledge about the "physics" of the volatility process can be used as an inductive bias to design or constrain a cell state of long shortterm memory (LSTM) for volatility forecasting. We introduce a new type of σ-LSTM cell with a stochastic processing layer, design its learning mechanism and show good out-of-sample forecasting performance.
| null |
[
"https://arxiv.org/pdf/2205.07022v1.pdf"
] | 248,811,457 |
2205.07022
|
a45558e7966d920a70e4811e7a1fb749e587e012
|
Volatility-inspired σ-LSTM cell
14 May 2022
German Rodikov
Nino Antulov-Fantulin
Volatility-inspired σ-LSTM cell
14 May 2022(Dated: May 17, 2022)
Volatility models of price fluctuations are well studied in the econometrics literature, with more than 50 years of theoretical and empirical findings. The recent advancements in neural networks (NN) in the deep learning field have naturally offered novel econometric modeling tools. However, there is still a lack of explainability and stylized knowledge about volatility modeling with neural networks; the use of stylized facts could help improve the performance of the NN for the volatility prediction task. In this paper, we investigate how the knowledge about the "physics" of the volatility process can be used as an inductive bias to design or constrain a cell state of long shortterm memory (LSTM) for volatility forecasting. We introduce a new type of σ-LSTM cell with a stochastic processing layer, design its learning mechanism and show good out-of-sample forecasting performance.
I. INTRODUCTION
The structure of noise or errors in regression models is usually subtle and taken as an ansatz to use different mathematical frameworks. E.g. in the case of linear regression models y = Xβ + ǫ, where y ∈ R n represent response variable, unobservable parameters β ∈ R K and non-random explanatory variable X ∈ R n×K , while ǫ represents the noise or error. When the errors ǫ i are homoscedastic i.e. V ar[ǫ i ] = σ 2 and serially uncorrelated i.e. Cov[ǫ i , ǫ j ] = 0 by the Gauss-Markov theorem [1] the ordinary least squares is having the lowest sampling variance within the class of linear unbiased estimators. In econometrics, special interpretation and attention are given to the error structure [2].
In this paper, we focus on the problem of volatility of asset returns, which are well known to be heteroscedastic in nature. Volatility is associated with the risk and amplitude of price fluctuations.
Some models characterize the volatility from a conditional process perspective, for example, the autoregressive conditional heteroskedasticity (ARCH) model [3] and the generalized autoregressive conditional heteroskedasticity (GARCH) model [4]. The idea of the conditional process approach is the possibility of using a conditional variance that varies over time while the unconditional variance remains relatively constant.
On the other hand, researchers have recently focused on using realized volatility (RV) to build forecasting models due to the wide availability of high-frequency financial data. For example, the Heterogeneous Autoregression (HAR-RV) model [5] is widely used in the literature due to consistently good predictive performance and simple methods to estimate it.
Recently, different recurrent NN units like LSTM [6], GRU [7], SRU [8] have demonstrated high performance on forecasting tasks for time-series data. More specifically, LSTM is the particular architecture that design improves the model's performance overall and especially in the volatility prediction task [9,10].
Gated recurrent units (GRU) [7] is an enhanced LSTM architecture that improves the fitting process by eliminating the cell state. In addition, the Statistical Recurrent Unit (SRU) [8] was introduced, which can infer longterm dependencies from data by using simple moving averages of summary statistics and has multiple proxies of the past with simple linear combinations.
Our work investigated and analyzed how NN can learn to capture the temporal structure of realized volatility. We are interested in how we could add the structure of long and short-term volatility effects to the LSTM. We introduce a modified LSTM cell that we call σ-LSTM to match these needs. For this reason, we extend the equation system of the LSTM cell, which reflects the inductive bias of GARCH-like structure and HAR-RV effects of volatility, which allows easier learning of volatility by maximum likelihood estimation.
Recently, several studies [11,12] have explored the heteroskedasticity of returns with recurrent NN architectures, but not by means of modified LSTM cell. In [11], authors have proposed the RECH model, where ω-constant of the GARCH process is modeled by a particular RNN model.
In [12], authors have proposed the combination of the Stochastic Volatility (SV) model and Statistical Recurrent Unit (SRU). The idea is that the SRU captures the long-term memory effects and auto-dependence of the volatility. However, the SRU is modeling the deterministic dynamics of the hidden states in the SR-SV model.
We investigated the original Long short-term memory cell, proposed σ-LSTM with a particular loss function for realized volatility forecasting tasks, and compared the predictive ability with widely used HAR-RV, GARCH(1,1) models.
The remaining paper is organized as follows. Section II provides mathematical motivation and a formal definition of σ-LSTM cell. Section III describes how the experiment and its results. Finally, in Section IV, we provide a conclusion.
II. METHODOLOGY
One important class of econometric models are GARCH family models [3,4]:
r t = µ t + σ t ε t , σ t ε t ∼ N (0, σ 2 t )(1)
where the conditional variance [3,4] has the autoregressive structure:
σ 2 t = ω + α * r 2 t−1 + β * σ 2 t−1 .(2)
Number extensions with different functional dependence (see Table I) have been proposed like eGARCH [13], cGARCH [14], TGARCH [15], GJR-GARCH [16] and others.
) = ω + α ε t−1 σ t−1 − E ε t−1 σ t−1 + δ ε t−1 σ t−1 + β ln(σ 2 t−1 ) cGARCH σ 2 t = qt + α(ε 2 t−1 − qt−1) + β(σ 2 t−1 − qt−1) qt = ω + ρqt−1 + θ(ε 2 t−1 − σ 2 t−1 ) GJR-GARCH σ 2 t = ω + (α + γIt−1) ε 2 t−1 + βσ 2 t−1 It−1{ 0 if rt−1 ≥ µ 1 if rt−1 < µ TGARCH σt = ω + αεt−1 + βσt−1 + φεt−11 [εt−1<0]
The heterogeneous Autoregression Realized Volatility (HAR-RV) model introduced by [5] assumes that agents' behavior in financial markets, which differ in their perception of volatility depending on their investment horizons and are divided into short-term, medium-term, and long-term. Heterogeneous structures in financial markets are based on the heterogeneous market hypothesis presented by [17]. Participants' decisions refer to different time horizons that perceive and respond to different types of volatility. A memory of each component decreases with a particular time constant.
The HAR-RV model is an additive cascade of partial volatilities generated at different time horizons that follows an autoregressive process [5]. The HAR-RV approach is one more stable and accurate estimate for Realized Volatility [18] at the 3 different horizons, where RV
σ (m) t+1m = c (m) + φ (m) RV (m) t +ω (m) t+1m σ (w) t+1w = c (w) + φ (w) RV (w) t + γ (w) E t σ (m) t+1m +ω (w) t+1w ′ σ (d) t+1d = c (d) + φ (d) RV (d) t + γ (d) E t σ (w) t+1w +ω (d) t+1d (3) where c (m) -the constant andω (m)
t+1m is an innovation that is simultaneously and consistently independent with a mean zero for monthly aggregation, and φ represents the wight in a particular cascade.
Motivated by GARCH structure and long and shortterm volatility effect of HAR-RV model, we propose to model the return assets r t = φ(r t−1 , ..., r t−p ), where φ(.) is a differentiable non-linear function. In particular a modified long short-term memory (LSTM) cell [6] that should capture long and short-term volatility. The inputs to out modified LSTM cell x t = r t are directly returns and the outputs arer t andσ 2 t . The cell has directly the hidden representation h t for short-term memory and long-term C t volatility memory component. The updates rules of σ−LSTM are the following:
f t = σ (W f · [h t−1 , x t ] + b f ) ,(4)i t = σ (W i · [h t−1 , x t ] + b i ) ,(5)C t = tanh (W C · [h t−1 , x t ] + b C ) ,(6)C t = f t * C t−1 + i t * C t ,(7)o t = N (0, W o [C 2 t ]),(8)h t = o t * φ (C t ) .(9)
Finally, the output returnr t and estimated volatilityσ t is:r
t = W h · h t(10)σ 2 t = C t 2 ,(11)
where . is the mean operator and bothr t andσ 2 t are scalar values.
We implement the custom loss function as the likelihood of observed returns with estimated volatilities.
L = m t=1 − ln σ 2 t − r 2 t σ 2 t .(12)
III. EXPERIMENTS & RESULTS
This study investigates how proposed σ-LSTM could estimate and predict realized volatility on different market structures, particularly stocks, indexes, and cryptocurrency data. We consider Apple inc. stock, the S&P 500 index, and Bitcoin-USD.
We calculate RV based on minutes-based price observations for daily aggregation. Returns are calculated on the daily close price. As a best practice, we divided the dataset into three parts: training, validation, and test. The validation and the test sample are equivalent to 200 points.
Mean Squared Error (MSE) measures averaged squared difference values between the predictions and the target. The power of 2 in this metric prevents neutralizing positive and negative deviations, which minimizes the distance between actual and calculated values. Root Mean Squared Error (RMSE) is the square root of MSE. The square root is introduced to scale error is the same as the target scale.
To find the best configuration of NN is necessary to conduct multiple experiments with different hyperparameters [19]. We have results for training launches and results for the validation dataset; the next step is to select promising hyperparameters RMSE metrics appropriately. Standardization is highly recommended before training RNNs and can improve the efficiency of training models. We normalized input data from 0 to 1 by min-max scale.
In our study, we ask the following questions. First, how could the proposed σ-LSTM cell with GARCH-like structure and long and short-term volatility effect of the HAR-RV model capture the long and short-term volatility effects?
We performed standard accuracy measures for the one-step-ahead prediction using RMSE metrics for 200 data points, Table III. As a result, σ-LSTM shows the best performance for RMSE for the out-of-sample result for the S&P 500 index and Apple Inc. stock data sets. However, it should be noted that in the case of cryptocurrency, the prediction error of HAR-RV was at the same level as σ-LSTM. In our experiment, C t of the original LSTM does not provide sufficient results.
IV. CONCLUSION
This work introduces a special σ-LSTM cell to investigate whether the use of stylized facts or "physicsinformed" inductive bias [20] i.e., GARCH and HAR-RV volatility structure could help to improve the performance of the NN for the volatility prediction task. We do not use the Recurrent LSTM unit as a black box but rather design a sub-component to represent a long-short volatility memory and a stochastic part.
We add particular loss functions for the σ-LSTM. As a result, we show that σ-LSTM could outperform wellknown models in this field, such as a strong baseline HAR-RV and regular LSTM cell. We will investigate more advanced loss functions in future work that could allow faster learning convergence and accuracy.
are respectively the daily, weekly, and monthly observed realized volatilities.
TABLE I .
IGARCH familyeGARCH
ln(σ 2
t
TABLE II .
IIDescription of the data
Type of asset
Name
Price points RV points
Index
S&P500
2 821 368
3803
Stock
Apple Inc.
2 466 466
3803
Cryptocurrency Bitcoin-USD
3 613 769
3375
TABLE III .
IIIS&P 500, Apple Inc. stock and Bitcoin-USD out-of-sample tests of forecasting accuracyData Set
Model
RMSE
S&P 500 Index
GARCH (1,1)
0.00405
HAR-RV
0.00359
LSTM
0.00805
σ-LSTM
0,00351
Apple Inc. stock
GARCH (1,1)
0.00648
HAR-RV
0.00561
LSTM
0.00752
σ-LSTM
0.00560
Bitcoin-USD
GARCH (1,1)
0.01641
HAR-RV
0.01537
LSTM
0.02286
σ-LSTM
0.01542
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|
[] |
[
"Massive spinons in S = 1/2 spin chains: spinon-pair operator representation",
"Massive spinons in S = 1/2 spin chains: spinon-pair operator representation"
] |
[
"Mohsen Hafez-Torbati \nLehrstuhl für Theoretische Physik I\nTechnische Universität Dortmund\nOtto-Hahn-Straße 444221DortmundGermany\n",
"Götz S Uhrig \nLehrstuhl für Theoretische Physik I\nTechnische Universität Dortmund\nOtto-Hahn-Straße 444221DortmundGermany\n"
] |
[
"Lehrstuhl für Theoretische Physik I\nTechnische Universität Dortmund\nOtto-Hahn-Straße 444221DortmundGermany",
"Lehrstuhl für Theoretische Physik I\nTechnische Universität Dortmund\nOtto-Hahn-Straße 444221DortmundGermany"
] |
[] |
Spinons are among the generic excitations in one-dimensional spin systems; they can be massless or massive. The quantitative description of massive spinons poses a considerable challenge in spite of various variational approaches. We show that a representation in terms of hopping and Bogoliubov spinon processes, which we call "spinon-pair" operators, and their combination is possible. We refer to such a representation as second quantized form. Neglecting terms which change the number of spinons yields the variational results. Treating the bilinear and quartic terms by continuous unitary transformations leads to considerably improved results. Thus, we provide the proof-of-principle that systems displaying massive spinons as elementary excitations can be treated in second quantization based on spinon-pair representation.
|
10.1103/physrevb.95.155136
|
[
"https://arxiv.org/pdf/1612.07817v2.pdf"
] | 119,046,583 |
1612.07817
|
291d9c0615b8e365e3277ba88cddb029b6aa5851
|
Massive spinons in S = 1/2 spin chains: spinon-pair operator representation
28 May 2017
Mohsen Hafez-Torbati
Lehrstuhl für Theoretische Physik I
Technische Universität Dortmund
Otto-Hahn-Straße 444221DortmundGermany
Götz S Uhrig
Lehrstuhl für Theoretische Physik I
Technische Universität Dortmund
Otto-Hahn-Straße 444221DortmundGermany
Massive spinons in S = 1/2 spin chains: spinon-pair operator representation
28 May 2017(Dated: October 7, 2018)
Spinons are among the generic excitations in one-dimensional spin systems; they can be massless or massive. The quantitative description of massive spinons poses a considerable challenge in spite of various variational approaches. We show that a representation in terms of hopping and Bogoliubov spinon processes, which we call "spinon-pair" operators, and their combination is possible. We refer to such a representation as second quantized form. Neglecting terms which change the number of spinons yields the variational results. Treating the bilinear and quartic terms by continuous unitary transformations leads to considerably improved results. Thus, we provide the proof-of-principle that systems displaying massive spinons as elementary excitations can be treated in second quantization based on spinon-pair representation.
I. INTRODUCTION
Quantum magnets constitute a flourishing field of research. In particular, the search for unconventional excitations and their quantitative understanding represents an important issue of current interest.
The elementary excitations which are known best are spin waves or magnons. They appear as massless Goldstone bosons in long-ranged ordered magnets such as ferromagnets or antiferromagnets. Their effect of the total spin of the system is integer, i.e., they change the total magnetization or the sublattice magnetization by one 1,2 . Henceforth, we set Planck's constant to unity for the sake of simplicity. Another class of integer excitations are triplons, i.e., gapped dressed particles with S = 1 as they appear in valence bond solids, for instance in all models resulting from coupling spin dimers of S = 1/2 in one dimension, see e.g. Refs. 3-6, in two dimensions, see e.g. Refs. 3, 7-9, and in three dimensions, see e.g. Refs. 10-14. But in particular in low-dimensional systems, fractionalization may occur. This means that the integer excitations decompose into several, mostly two fractional excitations. The famous example are the S = 1/2 spinons in the nearest-neighbor Heisenberg chain [15][16][17][18] with the Hamiltonian
H := J 1 i S i · S i+1 + J 2 i S i · S i+2 ,(1)
where S i defines the spin S = 1/2 operator at site i and the sum runs over the sites of a chain. The couplings J 1 and J 2 control the interaction strengths between nearest neighbor (NN) and next-nearest neighbor (NNN) sites, respectively. The NN case is given by J 2 = 0. For later use, we define the relative coupling α := J 2 /J 1 which is a measure of the degree of frustration. Another analytically solvable case of massless spinons is realized in the Haldane-Shastry model 19,20 . The Hamiltonian of this model is related to the one in Eq. (1), but for certain long-range couplings J n . The concept of massless spinons is used to develop effective or approximate descriptions of a multitude of systems, even if the microscopic Hamiltonian does not match perfectly 21 , but it can provide a starting point for perturbative inclusion of inter-chain couplings [22][23][24] . Clearly, massless spinons represent an intensive field of research.
In addition, current research addresses massive spinons, i.e., spinons of which the creation requires a finite amount of energy. In one dimension, strongly frustrated chains such as given by the Hamiltonian (1) for α 0.241 display such elementary excitations [25][26][27] . Four-spin and six-spin interaction terms can be considered as well 28,29 which lead to spontaneous dimerization even without frustration. In two dimensions, systems such as kagomé lattices are prone to be governed by fractional massive spinons [30][31][32][33][34] Even in three dimensions, fractionalization takes place leading to magnetic monopoles [35][36][37] . But these occur in highly anisotropic spin models which marks an important difference to the spinons mentioned above in one and two dimensions.
Given the great interest in spinons and the fact that massive spinons are less well understood than there massless counterparts we study massive spinons in the present article. We start from the description introduced by Shastry and Sutherland 25 in a second quantized form. Conceptually, we extend the description of Shastry and Sutherland to general chains. As a proof-of-principle we will consider the frustrated spin chain in (1) for arbitrary frustration α. We do not attempt to define the creation or annihilation of single spinons. Instead, we introduce spinon-pair operators which denote bilinear processes involving spinons, i.e., hopping of spinons or pairwise creation or annihilation of them. In addition, we keep track of the interactions of two spinons and of the decay of one spinon into three. We show that it is indeed possible to systematically define the Hamiltonion in terms of spinon-pairs.
In a second step, we analyze the obtained second quantized Hamiltonian by continuous unitary transformations to extract the physical relevant properties. The processes changing the number of spinons are rotated away in this fashion. But their renormalizing effects on the physical properties are retained, at least on the level of our approximations. The physical properties comprise the effective spinon dispersion and the value of the spin gap in particular. In this way, we show that a second quantized description of massive spinons is possible on the proof-of-principle level.
The results obtained are considerably improved over the variational results. This illustrates the potential behind the idea to formulate microscopic Hamiltonian in terms of their elementary excitations. Of course, this route requires to know what these quantities are. Often, however, this is the case. Thus, we believe that the approach pursued here can be transferred to many other physical systems as well.
The article proceeds as follows. In Sect. II we introduce a complete spinon basis for the subspaces with total spin S t = 0, S t = 1 2 , and S t = 1. Subsequently, in Sect. III the states in this basis are orthonormalized. This allows us to introduce spinon-pair operators for second quantization in Sect. IV. This formalism is applied to the frustrated Heisenberg chain with nearest-neighbor and next-nearest neighbor interactions in Sect. V. The employed method of continuous unitary transformations is introduced in Sect. VI while the final results are presented in Sect. VII. The conclusions and the outlook terminate the paper in Sect. VIII.
II. SPINON BASIS
Here we introduce a basis for spinon states in chains consisting of localized spins S = 1/2. We distinguish between vacua and states with various numbers of spinons. This does not imply that the vacua are ground states of the spin chains studied finally. The first aim of this section and the subsequent two sections is to express the Hamiltonian in terms of spinons. Then a continuous unitary transformation is applied to obtain quantitative results.
For a chain with an even number of sites, the 0-spinon state (spinon vacuum) is defined such that each spin at a site forms a singlet with its neighboring site. For a chain of length L, the 0-spinon state is given by
|0 := L/4−1 n=−L/4 [2n, 2n + 1] =: | (2) where [i, i + d] := i i+d | := 1 √ 2 |↑ i |↓ i+d − |↓ i |↑ i+d
(3) depicts the singlet state between lattice positions i and i + d. The states |↑ i and |↓ i are eigenvectors of the S z i operator with eigenvalues +1/2 and −1/2, respectively. Planck's constant is set to unity.
For periodic boundary condition (PBC) there are two vacua which differ by a translation by one site. For open boundary condition (OBC) these two states constitute the vacuum and a 2-spinon state with spinons at both boundary points. The spinon vacuum for OBC is a product of singlet; it is the same as the well-known Majumdar-Ghosh (MG) state which represents the exact ground state of the J 1 -J 2 Heisenberg chain, see Eq.
(1), for J 2 = J 1 /2 > 0 38,39 . This spinon vacuum can also be seen as a short-ranged "resonating valence-bond" (RVB) state 40,41 defined on a chain. Although spinons as spin-1 2 quasiparticles always appear in pairs for given even chain size, understanding the dynamics of a single spinon is necessary to describe deconfined spinon pairs 25 . This parallels fermionic systems with a fixed number of particles.
A spinon is defined as the domain-wall separating two possible vacua 25 . The 1-spinon state |φ σ i with the spin index σ =↑, ↓ is given by denotes a triplet bond with the flavor p between sites j and j + d 3 .
The overlap between the spinon vacuum (2) and the singlet 2-spinon state (6) is given by
0|φ s i,i+d = − 1 2 d−1 2 .(9)
We also have the following overlap between singlet 2spinon states
φ s j,j+d2 |φ s i,i+d1 = − 1 2 d 1 +d 2 2 −1 , n > d 1 − 1 2 d 2 −d 1 2 +n , d 1 − d 2 ≤ n < d 1 − 1 2 d 1 −d 2 2 , n ≤ d 1 − d 2 (10) where n := j − i ≥ 0.
For triplet 2-spinon states, the same relation holds, except that for n > d 1 the result is zero.
How can we construct states which contain more than two spinons? In this manuscript, we focus on the subspaces with the total spin S tot = 0 (L even), S tot = 1 2 (L odd), and S tot = 1 (L even). For these cases, we show that a complete spinon basis can be systematically constructed. The systematic construction is important for the second quantization process that we want to introduce in the sequel.
By introducing longer ranged singlets in the spinon vacuum we create various states with total spin zero. But these states are not all linearly independent. This is so because "crossed" singlets can be expressed in terms of "nested" singlets and "distinct" singlets according to
| = | + | .(11)
This relation is valid for two arbitrary singlet bonds on arbitrary sites in the sequence of the chain. By recursive application of Eq. (11) any crossed singlet can be reexpressed in terms of nested and distinct singlets. One can choose any two groups from the three groups "distinct", "nested", and "crossed" singlets to have a complete basis spanning the S tot = 0 Hilbert space. Here, we adopt singlets of the type "distinct" and "nested" as they are already used in the definition of the 0-spinon and 2-spinon states. Thus, this choice appears to be the most suitable to define spinon-pair operators. We notice that the states with crossed singlets do not necessarily contain a well-defined number of spinons. Similar to the case of the S tot = 0 subspace, by introducing singlet spinon pairs above 1-spinon states we can construct 3-spinon, 5-spinon, and higher spinon states with the total spin S tot = 1 2 . The relation
| = | + |(12)
makes the basis of all such states overcomplete. The positions in (12) are assumed to be arbitrary ones along the chain. Any state with a nested spinon can be expanded in terms of states in which the spinon is either before or beyond the singlet bond. In order to avoid overcounting, we restrict the S tot = 1 2 subspace such that no single spinon occurs inside a singlet bond.
The states with S tot = 1 can be generated by replacing one of the singlet bonds of a state in the S tot = 0 subspace by a triplet bond. The identity
| = | + |(13)
together with Eq. (11) justify why no crossed bond needs to be considered in the S tot = 1 subspace. In Eq. (13) we omitted the flavor label p from the triplet bonds because the relation is valid for any fixed value of p. In addition to (13), we find
| = | + | + | ,(14)
which is a relation between distinct and nested singlet and triplet bonds. We notice that (13) and (14) are two independent equations. By recursive application of Eq. (14) one can eliminate all nested triplet bonds. This means that in the construction of S tot = 1 subspace, no nested triplet bond needs to be considered. Alternatively, one may decide to use Eq. (14) to consider only nearestneighbor (NN) triplet bonds in the S tot = 1 subspace. But this leads to an overcomplete basis for the S tot = 1 sector and further restriction is required which would complicate the subsequent treatment. This can be seen by inspecting the states on a 6-site cluster. In Fig. 1 we represent the S tot = 0 and the S tot = 1 spinon states of a 6-site cluster. One can also check that 14 spinon states with S tot = 0 and 28 (×3) spinon states with S tot = 1 on an 8-site cluster can be successfully generated spanning the respective Hilbert subspaces.
III. ORTHONORMALIZATION
The spinon basis introduced in the previous section is complete, but not orthonormal. In this section, we orthonormalize the spinon basis so that the basis can be used to define second quantized operators in the following.
We fix the vacuum as it is defined in Eq. (2). To make the 1-spinon states orthonormal to each other we employ the ansatz
|Φ σ i = α 1 |φ σ i + α 2 |φ σ i−2(15)
and determine the coefficients α 1 and α 2 such that Φ σ i |Φ σ ′ j = δ i,j δ σ,σ ′ holds. The ansatz (15) for the orthonormal 1-spinon states is not unique but we did not find a simpler one, i.e., we think there is no other ansatz involving less sites. Using Eq. (5), we find two solutions
|Φ σ,r i = 1 √ 3 2 |φ σ i + |φ σ i+2 (16a) |Φ σ,l i = 1 √ 3 2 |φ σ i + |φ σ i−2 (16b)
which we call orthonormal "right" and "left" 1-spinon states, respectively, and denote by the appropriate superscripts r or l.
One can choose any of these two solutions to form an orthonormal spinon basis 43 . We stress that in the orthonormal 1-spinon state the spinon is not localized at a single lattice site, but it has an extension of two lattice spacings to the right, |Φ σ,r i , or to the left, |Φ σ,l i . In the following, we use the 1-spinon state "left" and drop the index l to lighten the notation:
|Φ σ i := |Φ σ,l i .
The orthonormal many-spinon states with larger distances between the spinons are constructed from the direct product of orthonormal 1-spinon states. Since each orthonormal spinon extends over two sites, this construction fails if the distance between spinons is 1. In this case the state is "distorted" and we need to perform an explicit orthonormalization to define it properly. The orthonormal singlet and triplet 2-spinon states for d ≥ 3 are given by
|Φ s i,i+d := σσ ′ χ s σσ ′ |Φ σ i ⊗ |Φ σ ′ i+d = 1 3 4 |φ s i,i+d +2 |φ s i−2,i+d + 2 |φ s i,i+d−2 + |φ s i−2,i+d−2 , (17a) |Φ t,p i,i+d := σσ ′ χ t,p σσ ′ |Φ σ i ⊗ |Φ σ ′ i+d = 1 3 4 |φ t,p i,i+d +2 |φ t,p i−2,i+d + 2 |φ t,p i,i+d−2 + |φ t,p i−2,i+d−2 ,(17b)
where χ s σσ ′ and χ t,p σσ ′ are the Clebsch-Gordan coefficients for singlet and triplet states. The orthonormal triplet state with d = 1 is such a distorted state. To construct it, we start from
|Φ t,p i,i+1 := 1 N α 1 |φ t,p i,i+1 + α 2 |φ t,p i−2,i+1 + |φ t,p i−2,i−1 .(18)
This ansatz is motivated by the extension of each orthonormal spinon by two sites to the left. Requiring that (18) is orthonormal to the triplet 2-spinon states leads to the non-trivial equations
Φ t,p i−d+1,i+1 |Φ t,p i,i+1 = 0 =⇒ 2α 1 − 3α 2 = 0 (19a) Φ t,p i−d−1,i−1 |Φ t,p i,i+1 = 0 =⇒ α 2 − 2 = 0 (19b)
with d ≥ 3, and a normalization condition for N . One obtains
|Φ t,p i,i+1 = 1 √ 6 3 |φ t,p i,i+1 + 2 |φ t,p i−2,i+1 + |φ t,p i−2,i−1 .(20)
The orthonormal 3-spinon states are reported in Appendix A. We notice that the orthonormal n-spinon states can always be constructed from states with the same or a smaller number of spinons.
The many-spinon states constructed from the direct product of orthonormal 1-spinon states are not fully orthogonal. Some finite overlaps occur in the 3-and higher spinon subspaces. A finite overlap occurs if the positions of orthonormal spinons match. For instance, in the 3spinon sector we find
Φ σ i Φ s i+n,i+n+d |Φ s i,i+n Φ σ i+n+d = − 1 2 ; n ≥ 3. (21)
There is no such overlap in the 1-spinon and 2-spinon subspaces because after fixing the positions of the spinons there is only one possibilty to form states with specific total spin and flavor. But in the 3-spinon subspace there are two possibilities, see the bra and the ket states in (21). This finite overlap is not a serious issue because subspaces with different numbers of spinons are mutually orthogonal. But to define the proper interactions in second quantization one has to carefully take the finite overlaps into account, see Appendix D.
All spinon states can be expanded in orthonormal states. For the 1-spinon state (4) and the singlet 2-spinon state (6) we find
|φ σ i = √ 3 2 m≥0 − 1 2 m 2 |Φ σ i−m , (22a) |φ s i,i+d = − 1 2 |φ s i,i+d−2 + 3 4 m≥0 − 1 2 m 2 |Φ s i−m,i+d .(22b)
with m even. These relations are obtained by reversing Eqs. (16b) and (17a). Eq. (22b) is to be used in a recursive way starting from d = 3. We notice that |φ s i,i+1 := |0 . A local spinon state represented in terms of orthonormal states becomes extended over the whole chain with a prefactor decreasing exponentially for increasing distance. Generally, in the expansion of a mspinon state orthonormal states appear with m or less spinons.
IV. SPINON-PAIR OPERATORS
A. Definition
Any attempt to define single spinon creation or annihilation operators runs into severe problems. Typically, one has to work in a larger Hilbert space, for instance enlarge it artificially, and complement the description by a severe constraint so that the accessible Hilbert space is again the physical one 44 .
In the present article, we want to follow a different route. We refrain from defining a single creation or annihilation event, but define spinon operators for pairs of spinons in a rather straightforward manner. No severe constraints are required to deal with the physical Hilbert space because creation and annihilation of spinons always happen in pairs. Spinon hopping does not alter the number of spinons so that it can also be expressed by a second quantized operator which addresses the hopping process as a whole.
There is an important point that has to be clarified before defining the spinon-pair operators. Considering a segment of the chain, there can be two local vacua |0 i,i+d and |0 i,i+d given by
|0 i,i+d := i i+d | ,(23a)|0 i,i+d := i i+d | .(23b)
Creation of two spinons at positions i and i + d from |0 i,i+d changes the state on the chain between sites i and i + d. However, a two-spinon creation from |0 i,i+d corresponds to a change in the state of the chain before site i and beyond site i + d. We call |0 i,i+d "right" local vacuum and |0 i,i+d "wrong" local vacuum. The singlet operator S † i,i+d with d ≥ 3 is defined to create two orthonormal spinons at sites i and i + d with total spin zero if the state on the chain between sites i and i + d is the right local vacuum |0 i,i+d . This operator can be expressed as
S † i,i+d : |0 i,i+d −→ i i+d | (24)
where the states before i and beyond i + d are supposed to be arbitrary; they are not changed. The orthonormal spinons at positions i and i+d are indicated by empty circles. The result of S † i,i+d is zero if there is any other state different than |0 i,i+d between sites i and i + d. Hence, the singlet operator is defined only over odd distances d. We define S † i,i+1 := 0. Depending on the state on the chain sites i − 2, i − 1, i + 1, and i + 2, namely if they are occupied by spinons or not, the final state is distorted or not, as defined above. It is apparent from this definition that the singlet operator S ( †) i,i+d is a string operator of which the action depends on the state between sites i − 2 to i + d + 2. Moreover, we notice that there is no need to introduce additional operators to create distorted states at small distances. Each state from the S tot = 0 (S tot = 1 2 ) subspace can be generated by applying the singlet operators on the spinon vacuum (a 1-spinon state).
The order of singlet operators does not matter in the creation of distinct singlets. However, to create nested singlets one has to start from the outermost singlet. This implies that two creation (annihilation) singlet operators do not necessarily commute.
We define the triplet operator T p † i,i+d with d ≥ 1 which creates a triplet bond with flavor p between sites i and i + d from the right local vacuum |0 i,i+d . Similar to the singlet operator, the result is zero if there is any state different than |0 i,i+d between i and i + d. In addition, the result is zero if the action of the triplet operator leads to the creation of a nested triplet bond. This means that the application of the triplet operator depends on the state on the whole chain and not only on the state between sites i and i + d. This global feature of the triplet operator limits its general applicability. However, one can approximate it by its leading local contribution. The term "leading" refers to an expansion in terms with non-zero action on an increasing number of spinons.
First, we notice that T p † i,i+d with i odd (or even, depending on how we label the chain sites) always leads to creation of nested triplet bonds and hence it is zero. The action of T p † i,i+d with i even can still lead to nested triplet bonds, but this requires at least four spinons (two nested singlets) in the system. Hence the leading contribution of the triplet operator can be described by the local action
T p † i,i+d : |0 i,i+d −→ p i i+d |(25)
for i even and zero for i odd. In this argument, we suppose arbitrary states on the chain before site i and beyond site i + d. The whole S tot = 1 subspace can be generated by applying the singlet (24) and the triplet (25) operators to the vacuum. There will be some redundant states, i.e., some overcounting occurs. But this will only happen where at least 6 spinons are present due to the approximation (25). Hence we accept this degree of overcounting because it matters only on the hexatic level of operators. At low densities of spinons it is irrelevant. The hopping operator H σσ ′ i,i+d annihilates an orthonormal spinon with spin σ at position i and creates an orthonormal spinon with spin σ ′ at position i + d if there is the right local vacuum between i + 1 and i + d; otherwise the result is zero. This restricts the hopping distance d to even values. One notices that the crossing of spinons is prohibited, i.e., the result is zero if there is any spinon between sites i and i + d. We obtain
H ↑↑ i,i+d : | i i+d −→ | i i+d(26)
and similarly for other values σ and σ ′ . Instead of introducing hopping operators H σσ ′ i,i+d with specific spin indices σ and σ ′ it turns out to be more convenient to define
H j,j+d := H ↑↑ j,j+d + H ↓↓ j,j+d ,(27a)H x j,j+d := H ↑↓ j,j+d + H ↓↑ j,j+d ,(27b)H y j,j+d := i H ↑↓ j,j+d − H ↓↑ j,j+d ,(27c)H z j,j+d := H ↑↑ j,j+d − H ↓↓ j,j+d .(27d)
Note that in a SU(2) invariant model no hopping with spin flips such as H ↓↑ j,j+d will occur. But even in such model, the hopping operators with spin flips appear in products of intermediate calculations, see below.
The action of the neutral hopping operator (27a) on a singlet (triplet) bond is always a singlet (triplet) bond. The flavor hopping operators H p j,j+d with p = x, y, z acting on a singlet bond change it into a triplet bond with flavor p and vice versa. We note that the action of the neutral hopping operator H j,j+d only depends on the state between sites j and j + d. The actions of the flavor hopping operators depend on the state on the whole chains because they could replace a nested singlet bond with a triplet bond which is not allowed.
We call the singlet operators, the triplet operators, and the hopping operators bilinear although they are basically string operators. We do so because by analogy to a fermionic or bosonic Hamiltonian. The key property is that they are characterized by their action at two sites on the chain.
The quartic interactions are given by the normalordered product of two bilinear operators. To describe spinon interactions up to the quartic level in a SU(2)symmetric Hamiltonian only the singlet operators, the triplet operators, and the neutral hopping operators are required. The polarized hopping operators can only appear on higher levels of interactions or in the intermediate steps of the calculations, for instance, as the results of commutators.
B. Algebra
The next point to address is the commutation of different spinon-pair operators. Explicitly, we are interested in the normal-ordered form of S † , H , S † , S , S † , S † , S † , T , and [H, H]. By normal-ordering we mean that we sort the effect of the commutators according to the number of spinons needed for the term to become active, i.e., to have a non-trivial effect. The commutators are calculated by inspecting their effects on arbitrary states.
S † i,i+d , H j,j+d ′ = S † i,i+d H j,j+d ′ − H j,j+d ′ S † i,i+d = 0 ; = 0 − S † i,i+d+d ′ ; = S † i,i+d H j,j+d ′ − 0 ; = 0 − 0 = 0 ; = 0 − 0 = 0 ; = 0 − P i,i+d ′ −1 S † i+d ′ ,i+d ; = 0 − 0 = 0 ; = 0 − 0 = 0 ; = 0 ; = S † i,i+d H j,j+d ′ − 0 ; = S † i,i+d H j,j+d ′ − 0 ; = 0 − 0 = 0 ; Figure 2. (Color online) Analysis of the commutator [S † i,i+d , H j,j+d ′ ] in different cases.
The double-headed arrow creates two spinons at the two ends (i and i + d) and denotes the singlet operator S † i,i+d . The hopping operator H j,j+d ′ is shown by a dashed arrow indicating that a spinon hops from site j to j + d ′ .
Let us start with the commutator [S
† i,i+d , H j,j+d ′ ].
We restrict d ′ to positive values; negative values will be discussed below. Different cases are schematically distinguished in Fig. 2. The singlet operator S † i,i+d in this figure is represented by a double-headed arrow; two spinons are created at the two tips, i and i+d. The hopping operator H j,j+d ′ is depicted by a dashed arrow which specifies the hopping process from j to j + d ′ .
If j < i and j + d ′ ≤ i + d or if j > i + d the result will be zero considering the properties of the singlet operators and the hopping operators. The result will be also zero the second part of the commutator vanishes and we obtain [S
if i < j, j + d ′ ≤ i + d. If j < i + d and j + d ′ > i + d,† i,i+d , H j,j+d ′ ] = S † i,i+d H j,j+d ′ . For j = i+d, the first part vanishes S † i,i+d H j,j+d ′ = 0 and the commutator equals −S † i,i+d+d ′ . Finally, if j = i and j + d ′ < i + d, the commutator simplifies to −P i,i+d ′ −1 S † i+d ′ ,i+d .
The projection operator P i,i+d is defined to be identity if there is the right local vacuum (23a) between sites i and i + d and zero otherwise. We define P i,i−1 := 1. Combining everything, we obtain
S † i,i+d , H j,j+d ′ = −S † i,i+d+d ′ δ i+d,j − P i,i+d ′ −1 S † i+d ′ ,i+d θ(d − d ′ )δ i,j + S † i,i+d H j,j+d ′ θ(i + d − j)θ(j + d ′ − i − d),(28)
where the step function θ(x) is 1 for x > 0 and zero for x ≤ 0. It is instructive to compare Eq. (28) to its counterpart for hardcore bosons. For the hardcore boson b
( †) i acting on site i one has b † i b † i+d , b † j+d ′ b j = − b † i b † i+d+d ′ (1 − b † i+d b i+d )δ i+d,j − (1 − b † i b i )b † i+d ′ b † i+d δ i,j + b † i b † i+d b † j+d ′ b j (δ i+d,j + δ i,j ),(29)
where the local projection operator (1 − b † i b i ) guarantees that site i is empty. We compare the right hand side of the two Eqs. (28) and (29) term by term. The first term in Eq. (28) is similar to the one of (29) except that the projection operator is absorbed in the definition of the singlet operator because it occurs between sites i and i + d + d ′ . The local projection operator (1 − b † i b i ) in the second term of (29) is replaced by P i,i+d ′ −1 in (28). In addition, the step function θ(d− d ′ ) in the second term of (28) reflects the fact that spinons cannot cross each other while bosons can. The step functions in the third term of (28) instead of the local delta functions in (29) also stem from the fact that spinons cannot pass each other on the chain.
Similarly, the commutator [
S † i,i+d , H j+d ′ ,j ] is analyzed with d ′ ≥ 0 leading to S † i,i+d , H j+d ′ ,j = −S † i−d ′ ,i+d δ i,j+d ′ − S † i,i+d−d ′ P i+d−d ′ +1,i+d θ(d − d ′ )δ i+d,j+d ′ + S † i,i+d H j+d ′ ,j θ(i − j)θ(j + d ′ − i).(30)
The commutator of creation and annihilation singlet op-erators is given by
S † i,i+d , S j,j+d ′ = −P i,i+d δ i,j δ d,d ′ + 1 2 H i+d+d ′ ,i δ j,i+d + H i−d ′ ,i+d δ j,i−d ′ + S † i,i+d S j,j+d ′ θ(i + d − j)θ(j + d ′ − i). (31)
The leading contribution of this commutator is the projection operator P i,i+d which would reduce to the identity if the operators were normal fermions or bosons. The prefactor 1 /2 in the second line results from the normalization of singlet states. One can also check that the commutator between the creation singlet operator and the annihilation triplet operator reads
S † i,i+d , T p j,j+d ′ = 1 2 δ i,j+d ′ H p i−d ′ ,i+d − δ j,i+d H p i+d+d ′ ,i + S † i,i+d T p j,j+d ′ θ(i + d − j)θ(j + d ′ − i).(32)
Other useful commutators are provided in App. B.
C. Projection Operator
In practical calculation, the projection operator has to be expressed in terms of the singlet operators, the triplet operators, and the hopping operators. The projection operator P i,i+d is zero if there is a spinon at or between sites i and i + d. In addition, the projection returns zero if there is the wrong local vacuum (23b) between i and i + d. The former property can be simply captured by
i+d j=i (1 − H j,j )(33)
which vanishes if there exists a spinon at or between i and i + d. The latter property, however, which requires distinguishing between the right (23a) and the wrong (23b) local vacua, is not a local feature. This makes it difficult to find a representation for the projection operator. Nevertheless, the projection operator never appears alone in our calculations. It either occurs in a sum over chain sites or it is multiplied with other operators, see the following. In these cases, one can find the leading contributions of the expression by applying it to the first subspaces containing only a few spinons.
We start with the operator P d := i P i,i+d where the sum i runs over the chain sites. To find the leading contributions of P d we consider the ansatz
P d = C 0 i 1+ C 1 i H i,i + i d ′ C 2 (d ′ ) S † i,i+d ′ S i,i+d ′ + p=x,y,z T p † i,i+d ′ T p i,i+d ′ + · · ·(34)
where "· · · " stands for 3− and higher spinon interaction terms. The prefactor C 0 is determined by applying the relation (34) to the spinon vacuum. We obtain 0| P d |0 = L 2 which yields C 0 = 1 2 . To calculate the prefactor C 1 one needs to apply P d to the 1-spinon state. We
have Φ σ i | P d |Φ σ i = ( L 2 − d 2 ) which leads to C 1 = − d 2 .
To find the interaction potential C 2 (d ′ ), one needs to com-
pute Φ s i,i+d ′ | P d |Φ s i,i+d ′ . We identify C 2 (d ′ ) = d−d ′ 2
for d ′ ≤ d and zero otherwise. Therefore, the final result reads
P d = 1 2 i 1− d 2 i H i,i + i d ′ ≤d d − d ′ 2 S † i,i+d ′ S i,i+d ′ + p=x,y,z T p † i,i+d ′ T p i,i+d ′ + · · · .(35)
It is remarkable that the prefactor of the spinon density operator H i,i is proportional to the distance d. The twospinon interaction potential in P d decreases linearly with increasing distance between the spinons and vanishes at the maximum distance d.
The product of the projection operator and the singlet operator also appears in the commutators, see for example Eqs. (28) and (30). We consider S † i+de,i+de+d ′ P i,i+d . Here and in the following we use the subscripts "e" and "o" to indicate even and odd numbers. We observe that
S † i+de,i+de+d ′ P i,i+d |0 = S † i+de,i+de+d ′ |0(36)
because the singlet operator guarantees the existence of the right local vacuum between i and i + d. To study the effect of the product on 1-spinon states we distinguish three cases: (i) d e ≥ 0 and d e + d ′ ≥ d (ii) d e ≥ 0 and d e + d ′ < d (iii) d e < 0. We analyze case (i) explicitly; the other cases can be treated in the same way. The action of S † i+de,i+de+d ′ P i,i+d on the 1-spinon state |Φ σ j is given by
S † i+de,i+de+d ′ P i,i+d |Φ σ j = 1 − θ(j − i + 1)θ(i + d e − j) × S † i+de,i+de+d ′ |Φ σ j .(37)
We notice that even for i+d < j < i+d e the result is zero because for (j − i) odd the wrong local vacuum appears between i and i + d and for (j − i) even the wrong local vacuum appears between i+d e and i+d e +d ′ . Therefore, we obtain
S † i+de,i+de+d ′ P i,i+d = S † i+de,i+de+d ′ 1 − de−1 no=1 H i+n o ,i+n o + · · · ; {d e ≥ 0 and d e + d ′ ≥ d},(38)
where "· · · " involves 2-and higher spinon interactions which we neglect. The sum over n o is limited to odd numbers because for even numbers the wrong local vacuum occurs between i+d e and i+d e +d ′ and the vanishing is guaranteed by the application of the singlet operator S † i+de,i+de+d ′ . One can analyze cases (ii) and (iii) to obtain relations analogous to (38). Combining the three equations and after some simplifications we obtain
S † i+de,i+de+d ′ P i,i+d = S † i+de,i+de+d ′ 1 − ne H i+n e ,i+n e θ(d−n e )θ(n e −d e −d ′ ) − no H i+n o ,i+n o θ(d e −n o )θ(n o ) + · · · (39)
which is valid up to quartic level in spinon creation and annihilation operators. We also inspect the product of the neutral hopping operator with the projection operator: P i+de+n,i+de+n+d H i,i+de where n is odd for n > 0 and even for n < 0. The expression can be analyzed by applying it to the 1-and 2-spinon sectors similar to the above discussion. The final result reads
P i+de+n,i+de+n+d H i,i+de = θ(n) H i,i+de − mo≥1 θ(m o +d e )θ(n+d−m o ) S † i+de,i+de+mo S i,i+de+mo + p=x,y,z T p † i+de,i+de+mo T p i,i+de+mo + θ(−n−d) H i,i+de − mo≥1 θ(m o +d e )θ(−n−d e −m o ) S † i−mo,i+de S i−mo,i + p=x,y,z T p † i−mo,i+de T p i−mo,i + · · ·(40)
where 3-and higher spinon interaction terms are ignored. Before closing this section, we state that the product of two neutral hopping operators is not a valid representation for 2-spinon interactions. In fact, 2-spinon interactions are always described in terms of singlet and triplet operators. Wherever the product of two neutral hopping operators appears in the calculations it has be expanded in terms of hopping operators, singlet operators and triplet operators as we did above in the expansion of the projection operator. Further useful expansions can be found in App. C.
V. FRUSTRATED HEISENBERG CHAIN
As an example of the application of the spinon-pair operator representation we study the frustrated Heisenberg chain. Its Hamiltonian has been given in Eq. (1). For a chain with an even number of sites, the exact ground state at α = 1 2 , known as Majumar-Ghosh point, is given by the fully dimerized state (2) 38,39 . Upon decreasing the degree of frustration below α = 1 2 the spontaneously dimerized phase undergoes a second order phase transition to a Mott insulator with quasi longrange magnetic order (spin liquid) at α c = 0.241167 45,46 . In both, the dimerized and the spin liquid phase, the elementary excitations (quasiparticles) are known to be spinons 15,25 . The minimum of the spinon dispersion for chains with an odd number of sites moves from commensurate to incommensurate momenta beyond the Lifshitz point α l = 0.538(1) 47 . For systems with an even number of sites, the Lifshitz point takes place at smaller frustration α l = 0.52036(6) 48 .
To express the Hamiltonian (1) in terms of spinon-pair operators, we start from the following general ansatz
H =A 0:0 H 0:0 + A 1:1 (0)H 1:1 (0) + de≥2 A 1:1 (d e )H 1:1 (d e ) + do≥3 A 2:0 (d o )(H 2:0 (d o ) + h.c.) + do≥3 de n A 3:1 (d e , n, d o )(H 3:1 (d e , n, d o ) + h.c.) + do≥3 d ′ o ≥3 de A s 2:2 (d o , d e , d ′ o )H s 2:2 (d o , d e , d ′ o ) + do≥3 d ′ o ≥3 de A t 2:2 (d o , d e , d ′ o )H t 2:2 (d o , d e , d ′ o ) + · · ·(41)
where we defined
H 2:0 (d o ) := i S † i,i+do(42d)H 3:1 (d e , n, d o ) := i S † i+de+n,i+de+n+do H i,i+de (42e) H s 2:2 (d o , d e , d ′ o ) := i S † i+de,i+de+d ′ o S i,i+do (42f) H t 2:2 (d o , d e , d ′ o ) := i p=x,y,z T † p i+de,i+de+d ′ o T p i,i+do (42g)
The "· · · " in Eq. (41) denotes 3-and higher spinon interactions which may occur. But we neglect such contributions here, i.e., we develop an approach which is valid for low densities of spinons, but which may fail in regimes where their density is higher. The so far unkown prefactors A in the ansatz (41) are determined from the matrix elements of the Hamiltonian calculated in the orthonormal basis. The following relations are useful to analyze the action of (1) on spinon states
S (1,x) · S (2,y) ∓ 1 4 | (1,l) (1,r) (2,l) (2,r) = ± 1 2 | (1,l) (1,r) (2,l) (2,r) (43a) S (1,x) · S (2) ∓ 1 4 | (1,l) (1,r) (2) = ± 1 2 | (1,l) (1,r) (2) (43b)
where x, y ∈ {l, r} and accordingly the spin operators S (1,x) and S (2,y) act on the lattice positions in the ket states in (43). The upper sign in Eq. (43a) holds if x and y are different and the lower sign holds if they are the same. In Eq. (43b), the upper sign holds for x = r and the lower sign for x = l. In addition to Eqs. (22), we use the following expansions for the 3-spinon states
|φ σ i φ s i+n,i+n+d = √ 3 2 − 1 2 d−1 2 me≥0 − 1 2 me 2 |Φ σ i−me + √ 3 2 − 1 2 n+1 2 d−3 me≥0 − 1 2 me 2 |Φ σ i+n+d−me + · · · , (44a) |φ s i,i+d φ σ i+d+n = √ 3 2 − 1 2 n+1 2 me≥0 − 1 2 me 2 |Φ σ i−me + √ 3 2 − 1 2 d−1 2 n−1 me≥0 − 1 2 me 2 |Φ σ i+n+d−me + · · · , (44b)
where "· · · " stands for orthonormal 3-spinon states. The prefactors A 0:0 and A 2:0 (d) can be computed by applying the Hamiltonian (1) to the vacuum (2) and comparing the result with the application of the ansatz (41) to the vacuum. In this way, we obtain
H J 1 |0 = − 3L 8 |0 + α 2 i∈even 1 2 |0 + |φ s i,i+3 = − 3L 8 |0 + 3 α 2 do≥3 i∈even − 1 2 do+1 2 |Φ s i,i+do(45)
where α := 1 − 2α. The second equality is derived using Eq. (22b) for d = 3. The sum over i runs over the even sites supposing that the first lattice site in the spinon vacuum (2) is even. For the other degenerate vacuum one sums over the odd sites. We should not, however, restrict the sum over i in Eq. (42d) to even (or odd) values only. Note, that both kinds of vacuum can be present on distinct pieces of the chain, separated by a spinon or any odd number of spinons as domain-walls.
Thus it is important to keep the processes creating nested singlets. From Eq. (45), we identify
A 0:0 = − 3 8 J 1 ,(46a)A 2:0 (d) = 3 α 2 − 1 2 d+1 2 J 1 . (46b)
This result shows that the spin Hamiltonian (1) is not fully local represented in spinon-pair operators. But the Bogoliubov prefactors (46b) decrease exponentially upon increasing the distance d. Moreover, we see that there is no term linking the 0-spinon state to 4-spinon states. Such terms appear upon inclusion of longer-range (beyond NNN sites) spin-spin interactions in (1). At the MG point α = 1 2 , all the Bogoliubov terms vanish indicating that the spinon vacuum (2) is an exact state of the system as one knows previously 39 .
To compute the hopping prefactors A 1:1 (d) we apply the Hamiltonian (1) to the 1-spinon state |Φ σ i and compare it with the action of the ansatz (41) on the same state. Some lengthy calculations yield
H J 1 |Φ σ i = + − 3 8 L + 5 + 2 α 8 |Φ σ i + 2 + α 8 |Φ σ i+2 + |Φ σ i−2 − 3 α 4 de≥4 − 1 2 de 2 |Φ σ i+de + |Φ σ i−de + · · · ,(47)
where "· · · " denotes the 3-spinon contributions orthonormal to the 1-spinon subspace which we neglect. To derive this relation we employed Eqs. (44) for d = 3. No term linking the 1-spinon to the 5-spinon sector is produced. After subtracting the contribution A 0:0 , the hopping prefactors read
A 1:1 (0) = 5 + 2 α 8 J 1 ,(48a)A 1:1 (2) = 2 + α 8 J 1 ,(48b)A 1:1 (d e ) = − 3 α 4 − 1 2 d e 2 J 1 ; d e ≥ 4.(48c)
At the MG point, α = 0, only hopping over two sites is present. However, there are long-range hopping processes present away from the MG point. The hopping prefactors decay exponentially with distance. It is more cumbersome to calculate the interaction coefficients A 3:1 , A s 2:2 , and A t 2:2 , analytically. The linked cluster expansion theorem allows us to determine the irreducible matrix elements A in Eq. (41) on finite, but large enough clusters, in the thermodynamic limit. We refer the reader to Ref. 49 for details. We notice that the interaction coefficients can be computed exactly on finite clusters due to the locality of the spin Hamiltonian (1).
We implemented a program to compute the Hamiltonian matrix elements between orthonormal spinon states. In the choice of the size of the cluster, one must keep in mind that each orthonormal spinon is extended over two sites to the left. Careful attention is also required in identifying the interaction coefficients A 3:1 because of the finite overlap (21) between orthonormal 3-spinon states. The interaction coefficients are calculated and reported in App. D. Even at the MG point the spin Hamiltonian (1) is not local in the spinon-pair operator representation, but the non-local terms fall off quickly with distance.
VI. CONTINUOUS UNITARY TRANSFORMATIONS
We use continuous unitary transformations (CUT) to analyze the Hamiltonian (41). We briefly introduce the CUT method of which the first versions were suggested over 20 years ago 50 . The CUT method maps a given initial Hamiltonian H to a final effective one by a unitary transformation which is parametrized by an auxiliary parameter ℓ 51 . The transformation is such that at ℓ = 0 the transformed Hamiltonian H(ℓ) equals the initial Hamiltonian H and at ℓ = ∞ the desired effective Hamiltonian is reached. The Hamiltonian during the flow is given by
∂ ℓ H(ℓ) = [η(ℓ), H(ℓ)](49)
which is called the flow equation. The anti-hermitian operator η(ℓ), called generator, determines the essence of the transformation. One can use the Wegner generator 50 , the particle-conserving (pc) generator 5,52,53 , or various reduced generators 54 to fully or partially rotate away the off-diagonal elements. By writing the initial Hamiltonian in a quasiparticle (QP) representation and eliminating the terms which change the number of QPs in the system, one can map a many-particle problem to a few-particle problem using the CUT method. There are different approximations to truncate the flow equation (49) so that this systems of differential equations is closed. One may achieve perturbative 5,55 or renormalized effective Hamiltonians 55,56 . The method is applied to describe systems with elementary excitations such as triplons with spin S = 1 57 , magnons 58 , electrons and holes 59,60 and so on.
In the following, we employ the pc generator and keep in the transformed Hamiltonian H(ℓ) terms which are at most quartic. This truncation can be justified by two arguments. The first argument refers to the density of spinons because the hexatic terms need at least three spinons to become active. Thus at low concentrations of spinons the bilinear terms acting on single spinons and the quartic terms acting on pairs of spinons are the most important ones. The second argument refers to the scaling of the terms. For fermionic and bosonic systems in one dimension it is known, that the bilinear and the quartic terms have the lowest scaling dimension 58,61 . If the bilinear terms are without mass term they are of equal scaling dimension, hence of equal importance. Hexatic and higher terms are irrelevant in the scaling sense.
In the bilinear approximation, the terms (42a) to (42d) define the operator basis. In the quartic approximation, we consider the terms (42a) to (42f). For technical simplicity, we neglect to include the triplet 2-spinon interaction (42g) because its treatment in the present formalism breaks the translational symmetry of the lattice. Such a translational symmetry breaking is expected since we have implicitly chosen one of the two degenerate vacua. One notices that no term linking the 0-spinon sector to the 4-spinon sector appears during the flow because the pc generator preserves the band-diagonal structure of the initial Hamiltonian 5,53 .
To determine the set of flow equations we need to calculate the commutators between the operators of the basis (42) 55 . The necessary relations for the commutators up to quartic level are provided in Sect. IV and in Apps. B and C. While the flow equations up to bilinear level can be easily obtained it is a very tedious task to obtain the quartic contributions. This motivates us to implement a program for subsequent applications.
To integrate the derived flow equations we limit the range of processes in real space such that the distance between initial spinons, the distance between final spinons, and the maximum distance between initial and final spinons is not larger than a maximum distance, d max . The flow equations are integrated numerically with the initial conditions (46), (48), (D9), and (D10).
VII. RESULTS
Before presenting the CUT results let us consider the simplest approximation, i.e., neglecting all off-diagonal terms, i.e., all terms which change the number of spinons in (41). This approximation is expected to be most accurate at and near the MG point where the Bogoliubov terms (46b) vanish. But we stress that it is an approximation even at the MG point because terms exist which link the 1-spinon states to the 3-spinons states.
The approximate 1-spinon dispersion neglecting all spinon-number changing terms reads
ω(k) = A 1:1 (0) + 2 de=2 A 1:1 (d e ) cos(d e k) = α 4 (5 + 4 cos(2k)) J 1 + α 3 8 + 4 + 5 cos(2k) 5 + 4 cos(2k) J 1 ,(50)
based on Eqs. (48). This simple approximation recovers the variational results obtained by Brehmer et al. 27,62 .
The formalism presented in this article allows us to systematically improve these variational results by rotating away the off-diagonal elements using the CUT or by other approaches. We define the spin gap ∆ s as twice the minimum energy of the spinon dispersion. Thus, the spin gap equals with the minimum of the 2-spinon continuum. We notice that the spinons are asymptotically free although they cannot pass each other. The spin gap ∆ s versus the frustration degree α is depicted in Fig. 3. The variational result (50) and the CUT results in the bilinear and quartic approximation for different maximum distances, d max , are shown. The density-matrix renormalization group (DMRG) results by Chitra et al. 63 and by White and Affleck 64 are included for comparision. Fig. 3 shows that the spurious transition point predicted by the variational result at α ≈ 0.87 is shifted in CUT on bilinear level to α ≈ 1.47 making the spontaneously dimerized phase more stable as expected from the DMRG results. Both the variational result and the result from bilinear CUT vanish almost linearly at α ≈ 0.43. They do not show evidence of the exponentially slow vanishing gap at the Berezinskii-Kosterlitz-Thouless (BKT) transition at α ≈ 0.241.
The CUT results on the quartic level in Fig. 3 nicely capture the qualitatively behavior of the spin gap predicted by White and Affleck 64 . The agreement between the two approaches is quantitative near the MG point. Lack of convergence of the flow equations prevents us to reach beyond α = 1.26 for d max = 20 and beyond α = 1.14 for d max = 25. The CUT on quartic level predicts the transition to the quasi-long-range spin liquid phase at α ≈ 0.34. This value is larger than the established critical value α c ≈ 0.241. We believe that this deviation is due to the very slow vanishing of the spin gap near the BKT transition point 65 We denote by k * the momentum at which the minimum of the spinon dispersion occurs. For weak frustration it occurs at π/2. Fig. 4 denotes the deviation ( π 2 − k * ) versus the frustration α. The minimum of the spinon dispersion moves from k = π 2 to an incommensurate value beyond the Lifshitz transition point α l . We find α l = 0.539 for d max = 20 and α l = 0.540 for d max = 25 and d max = 30. This finding agrees quantitatively with the DMRG prediction α l = 0.538(1) by Deschner and Sørensen 47 . However, it is difficult to capture the behavior of k * for α > 0.538(1) by DMRG due to a plethora of level crossings 47 . We expect ( π 2 − k * ) → π 4 in the limit α → ∞ because the system approaches to weakly coupled penetrating Heisenberg chains with lattice constant 2a so that the minimum occurs for a = 1 at π 4 . In Fig. 5 we plot the low-energy spectrum of the J 1 -J 2 Heisenberg model (1) with (a) an odd and (b) an even number of sites. The value of the frustration α is set to 0.8. Fig. 5a shows the 1-spinon dispersion and the 3-spinon continuum while Fig. 5b shows the 2-spinon continuum. We used the spinon dispersion for d max = 30 to construct the 2-and the 3-spinon continua relying on energy and momentum conservation. The numerical results for d max = 25 are essentially the same. The 1spinon dispersions obtained from different values of d max deviate noticeably only inside the 3-spinon continuum where a clear distinction between 1-spinon and 3-spinon states is not possible. One can extend the CUT approach to describe the decay of the 1-spinon dispersion into the 3-spinon continuum in Fig. 5a 54 . But such an analysis is beyond the scope of the present article.
We recall that the triplet 2-spinon interactions (42g) are neglected in the present treatment. These terms are present even in the initial Hamiltonian, see Eq. (D11), and could make the CUT results on the quartic level even more accurate.
VIII. CONCLUSIONS AND OUTLOOK
The aim of the present paper was to provide the proofof-principle that models in terms of massive spinons can be formulated for generic Hamiltonian, not only at special points such as the Majumdar-Ghosh point. This aim has been successfully realized. The formulation can be achieved in second quantization in the sense that the resulting effective Hamiltonian also applies to finite densities of spinons. The asset of such a formulation is that the subsequent treatment can employ all methods known for such problems.
We have constructed an orthonormal spinon basis and introduced string spinon-pair operators which can capture the fractional nature of the spinon excitation and describe different spinon processes. By applying the spinonpair operators on the spinon vacuum (1-spinon states) one can fully construct the S = 0 and the S = 1 Hilbert spaces of an even (odd) size lattice. This enables us to write spin Hamiltonians in spinon-pair representation in second quantization. Our representation is valid for low densities of spinons because it treats processes involving single spinons and pairs of them exactly into account. Only on the level of three or more spinons processes are neglected.
Here we used continuous unitary transformations to analyze the second quantized Hamiltonians obtained in the first step. We showed that processes which change the number of spinons can be systematically treated by this approach. They can be eliminated ("rotated away") while their renormalization of the properties of the elementary spinons is kept. In this way, we obtained results for the spinon dispersion including gap and incommensurability for the frustrated spin chain in a wide range of frustration, i.e., not only for the Majumdar-Ghosh point where the next-nearest neighbor interaction takes half the value of the nearest-neighbor interaction. The results are significantly improved over a purely variational treatment. This illustrates the potential of the pursued approach.
Further research to establish effective models in terms of their elementary excitations is called for. One promising route of research consists in passing from treatments in real space to treatments in momentum space. Another current challenge is to transfer the presented ideas from one to higher dimensions.
ACKNOWLEDGMENTS
We are grateful to the Deutsche Forschungsgemeinschaft and the Russian Foundation of Basic Research for support through the ICRC TRR 160.
|Φ σ i Φ s i+d1,i+d1+d2 = |Φ σ i ⊗ |Φ s i+d1,i+d1+d2 ,(A1a)|Φ s i,i+d2 Φ σ i+d1+d2 = |Φ s i,i+d2 ⊗ |Φ σ i+d1+d2 ,(A1b)
where the orthonormal 1-spinon state |Φ σ i is defined by Eq. (16b) and the orthonormal singlet 2-spinon state |Φ s i,i+d by Eq. (17a). If d 1 = 1, the orthonormal 3-spinon states
|Φ σ i Φ s i+1,i+1+d = i i+1 i+d+1 | (A2a)
and
|Φ s i,i+d Φ σ i+d+1 = i i+d i+d+1 | (A2b)
are distorted. The empty circles indicate orthonormal spinons. We require orthonormality to derive these states. It is a subtle issue to identify the initial form of the distorted 3-spinon states. But it can be found in a systematic way. Each orthonormal spinon in Eqs. (A2) has an extension of two sites to the left. This implies that 7 sites of the lattice are involved overall in 3-spinon states, see the dashed boxes in Eqs. (A2). On these 7 sites there are 12 states with S tot = 1 2 . If d > 3 in Eqs. (A2), 2 of these 12 states are 5-spinon states which do not contribute to the orthonormal 3-spinon states. We expect the same relation to be valid also for d = 3. Hence, we arrive at the following ansatz with 10 parameters for the distorted 3-spinon states
|Φ σ i Φ s i+1,i+d+1 = 1 N 1 |φ σ i φ s i+1,i+d+1 + α 1 |φ σ i φ s i+1,i+d−1 + α 2 |φ σ i−2 φ s i+1,i+d+1 + α 3 |φ σ i−2 φ s i+1,i+d−1 + α 4 |φ σ i−2 φ s i−1,i+d+1 + α 5 |φ σ i−2 φ s i−1,i+d−1 + α 6 |φ σ i+d+1 + α 7 |φ σ i+d−1 + α 8 |φ s i−2,i+1 φ σ i+d+1 + α 9 |φ s i−2,i+1 φ σ i+d−1 ,(A3a)|Φ s i,i+d Φ σ i+d+1 = 1 N 2 |φ s i,i+d φ σ i+d+1 + β 1 |φ s i−2,i+d φ σ i+d+1 + β 2 |φ s i,i+d−2 φ σ i+d+1 + β 3 |φ s i−2,i+d−2 φ σ i+d+1 + β 4 |φ s i,i+d−2 φ σ i+d−1 + β 5 |φ s i−2,i+d−2 φ σ i+d−1 + β 6 |φ σ i + β 7 |φ σ i−2 + β 8 |φ σ i φ s i+d−2,i+d + β 9 |φ σ i−2 φ s i+d−2,i+d .(A3b)
The unknown prefactors are determined in an orthonormalization process similar to the Gram-Schmidt algorithm. We start with the state d = 3 in Eq. (A3a) and make it orthogonal to the 1-spinon states (16b) and to the 3-spinon states (A1). It is too tedious a task to calculate the overlaps by hand. Hence we implemented a C++ program for this purpose. In this way, we obtained the eight independent equations
Φ σ i−2 |Φ σ i Φ s i+1,i+4 = 0 =⇒ −4 α 1 − 4 α 2 + 8 α 3 + 2 α 4 − 4 α 5 − 1 α 6 + 2 α 7 + 2 α 8 − 4 α 9 + 2 = 0 (A4a) Φ σ i |Φ σ i Φ s i+1,i+4 = 0 =⇒ +4 α 1 + 1 α 6 − 2 α 7 − 2 = 0 (A4b) Φ σ i+2 |Φ σ i Φ s i+1,i+4 = 0 =⇒ +1 α 4 − 2 α 5 − 2 α 6 + 4 α 7 + 1 α 8 − 2 α 9 = 0 (A4c) Φ σ i+4 |Φ σ i Φ s i+1,i+4 = 0 =⇒ +1 α 2 − 2 α 4 + 4 α 6 − 2 α 8 − 2 = 0 (A4d) Φ σ i−4 Φ s i−1,i+2 |Φ σ i Φ s i+1,i+4 = 0 =⇒ +3 α 4 − 6 α 5 − 1 α 8 + 2 α 9 = 0 (A4e) Φ σ i−2 Φ s i+1,i+4 |Φ σ i Φ s i+1,i+4 = 0 =⇒ +2 α 2 − 1 α 8 − 1 = 0 (A4f) Φ σ i−4 Φ s i−1,i+4 |Φ σ i Φ s i+1,i+4 = 0 =⇒ +1 α 2 − 2 α 4 = 0 (A4g) Φ s i−2,i+1 Φ σ i+4 |Φ σ i Φ s i+1,i+4 = 0 =⇒ +1 α 2 − 2 α 8 = 0.(A4h)
Fulfilling the single equation (A4a) is sufficient to make the state |Φ σ i Φ s i+1,i+4 orthogonal to all 1-spinon states |Φ σ j with j ≤ i − 2. The 1-spinon states |Φ σ j with j ≥ i + 6 are trivially orthogonal to |Φ σ i Φ s i+1,i+4 leading to no condition for the prefactors α 1 · · · α 9 . The overlaps Φ σ i−1−d Φ s i−1,i+2 |Φ σ i Φ s i+1,i+4 for d = 3, 5, · · · all vanish based on Eq. (A4e). Similarly each of the Eqs. (A4f), (A4g), and (A4h) orthogonalize the distorted state |Φ σ i Φ s i+1,i+4 to a class of orthonormal 3-spinon states.
+ θ(j − i)θ(i + d − j)θ(j + d ′ − i − d)H i,i+d H j,j+d ′ − i → j d → d ′ (B1a) [H i+d,i , H j,j+d ′ ] =δ i+d,j+d ′ θ(d − d ′ + 1)P i+d−d ′ +1,i+d + θ(d ′ − d)P i+1,i+d H i+d−d ′ ,i − δ i,j θ(d ′ − d + 1)P i,i+d−1 + θ(d − d ′ )P i,i+d ′ −1 H i+d,i+d ′ (B1b)
with d and d ′ positive. We defined P i,i−1 = 1. If the hopping operator brings two spinons in the singlet state to nearest-neighbor sites the result is zero. This explains why the second contribution in Eq. (B1a) can lead to a finite value. These commutators are required only up to bilinear level due to the approximation that we employ in this paper. Hence, Eqs. (B1) simplify to
[H i,i+d , H j,j+d ′ ] = (δ i,j+d ′ H i−d ′ ,i+d − δ j,i+d H i,i+d+d ′ ) + · · · (B2)
with d ′ ≥ 0 and arbitrary d. Hermitian conjugation yields the commutator for d ′ ≤ 0. Two creation singlet operators do not necessarily commute; we find
S † i,i+d , S † j,j+d ′ = S † i,i+d S † j,j+d ′ θ(i − j)θ(j + d ′ − i − d) − S † j,j+d ′ S † i,i+d θ(j − i)θ(i + d − j − d ′ ).(B3)
This reflects the fact that in the creation of nested singlets the order of the singlet operators matters.
Appendix C: Expansion of products of hopping operators
The product of two neutral hopping operators can be expanded as p=x,y,z T p † i+n+d ′ ,i+d T p i+n,i + · · · (C1) where 3-and higher spinon interactions are neglected.
H i+n,i+n+d ′ H i,i+d = δ n,d H i,i+d+d ′ − mo≥1 θ(−m o + d)θ(m o − d − d ′ ) S † i+d+d ′ ,i+m o +2 S i,i+m o +2 + p=x,y,z T p † i+d+d ′ ,i+m o T p i,i+m o
Appendix D: Interaction coefficients
In order to compute the interaction coefficients A 3:1 one has to take into account the possible overlap (21) between orthonormal 3-spinon states. For each S tot = 1 2 orthonormal 3-spinon state |Φ σ i Φ s i+n,i+n+d we define a dual state given by
|Φ σ i Φ s i+n,i+n+d (d) := |Φ s i,i+n Φ σ i+n+d if n > 0 |Φ σ i+n Φ s i+n+d,i if n < −d (D1)
Each orthonormal 3-spinon state is orthogonal to all other states except to its dual state. The overlap is given by
Φ σ i Φ s i+n,i+n+d |Φ σ i Φ s i+n,i+n+d (d) = − 1 2 .(D2)
We notice that there is no such finite overlap for the distorted 3-spinon states (A3). Hence they do not have a dual state. We define the Hamiltonian matrix element
Figure 1 .
1(Color online) The spinon states with (a) Stot = 0 and (b) Stot = 1 (b) on a 6-site cluster. Each triplet bond (dashed line) can have the flavor p = x, y, z. The Stot = 1 states are constructed from Stot = 0 states by replacing one singlet bond with a triplet bond; no nested triplet bond is allowed as explained in the main text.
The variable n in the third line of Eq. (41) takes odd values if n > 0 and takes even values if n < −d o . We notice that H 3:1 (d e , n, d o ) = 0 for −d o ≤ n ≤ 0. The term H p:q in Eqs. (42) is bilinear if p + q = 2 and it is quartic if p + q = 4.
Figure 3 .
3(Color online) The spin gap ∆s versus the frustration α. The various CUT truncation schemes are compared to the variational results by Brehmer et al. 27 and to the DMRG results by Chitra et al. 63 and White and Affleck 64 .
which is only detectable if extremely long-range processes are kept track of reli-
Figure 4 .
4(Color online) The momentum ( π 2 − k * ) versus the frustration α; k * is the momentum at which the minimum of the spinon dispersion is located.
Figure 5 .
5(Color online) The low-energy spectrum of the J1-J2 Heisenberg model (1) with (a) an odd and (b) with an even number of sites at frustration α = 0.8. Panel (a) displays the 1-spinon dispersion and the 3-spinon continuum and panel (b) denotes the 2-spinon continuum. The continua are constructed by energy and momentum conservation from the spinon dispersion with dmax = 30. The results for dmax = 25 are the same within line width.
o − d)θ(m o + d + d ′ ) S † i−m o −2,i+d+d ′ S θ(n+d ′ −d−2)S † i+d,i+n+d ′ S i,i+n + θ(+n)θ(n−d)θ(n+d ′ −d) d ′ −2)S † i+n+d ′ ,i+d S i+n,i + θ(−n)θ(d−n)θ(d−n−d ′ )
Appendix A: Orthonormal 3-spinon statesThe orthonormal S tot = 1 2 3-spinon states |Φ σ i Φ s i+d1,i+d1+d2 and |Φ s i,i+d2 Φ σ i+d1+d2 for d 1 ≥ 3 are given by
C 3:1 (d e , n, d o ) := Φ σ i+de Φ s i+de+n,i+de+n+do | H |Φ σ (d e , n, d o ) = C 3:1 (d e + n + d o , −n − d o , n) if n > 0 C 3:1 (d e + n, d o , −n − d o ) if n + d o < 0. (D5)Similarly, we define the dual of the interaction coefficientA 3:1 (d e , n, d o ) as (d e + n + d o , −n − d o , n) if n > 0 A 3:1 (d e + n, d o , −n − d o )Using the relation(41), the Hamiltonian matrix elements C 3:1 (d e , n, d o ) and C (d o , d e , d ′ o ) for specific d o , d e , and d ′ o can also be calculated on finite clusters 49 . From the analysis of the coefficients we deduceδ d o −3,de (1 − 3 α) − δ d o −1,de (1 + δ d o −1,de ; d ′ o = 5, 7, · · · . (D10b)where we supposed d e ≥ 0. The triplet channel interactions are given byδ do,1 + δ d ′ δ do−1,de − δ do+1,de + 3 − √ 6 3 δ do,1 δ de,2 ; d e ≥ 2 , d ′ o ≥ 5 . (D11d)The prefactors As 2:2 (d o , d e , d ′ o ) and A t 2:2 (d o , d e , d ′ o ) with d e < 0 can be calculated from the symmetry relation A 2:2 (d o , d e , d ′ o ) = A 2:2 (d ′ o , −d e , d o ). (D12)i
(D3)
and its corresponding dual
C
(d)
3:1 (d e , n, d o ) := (d) Φ σ
i+de Φ s
i+de+n,i+de+n+do | H |Φ σ
i .
(D4)
One has the relation
C
(d)
3:1 A
(d)
3:1 (d e , n, d o ) :=
A 3:1 if n + d o < 0
(D6)
which links to the 1-spinon state |Φ σ
i to |Φ σ
i+de Φ s
i+de+n,i+de+n+do
(d) .
The interaction coefficients A s
2:2 (d o , d e , d ′
o ) and A t
2:2 A s
2:2 (d o , d e , 3) =
5 − 7 α
24
δ do,3 δ de,0 + −
1
2
do+3
2
α
2
) ,
(D10a)
A s
2:2 (d o , d e , d ′
o ) = −
3 α
2
−
1
2
do+d ′
o
2
A t
2:2 (d o , 0, d ′
o ) =
23α − 10 + 4
√
6(1 − 2α)
6
δ do,1 δ d ′
o ,1 +
α
2
4
√
6
9
− 1
δ do,1 δ d ′
o ,3 + δ do,3 δ d ′
o ,1 +
(α − 1)
18
δ do,3 δ d ′
o ,3
+
(1 − 2α)
2
4
√
6
3
− 3
−
1
2
do+d ′
o
2
o ,1
(D11a)
A t
2:2 (d o , d e , 1) = +
(2 −
√
6)α
6
δ do,1 δ de,2 +
α
2
4
√
6
9
− 1 δ do,3 δ de,2 −
√
6α
3
−
1
2
de
2
δ do+1,de
+
3(1 − 2α)
4
+
(3α − 2)
√
6
6
−
1
2
de
2
δ do−1,de ; d e ≥ 2
(D11b)
A t
2:2 (d o , d e , 3) = + −
1
2
de
2 +2
(1 − 2α)δ do+1,de +
3 − 2α
6
δ do−1,de +
3α − 1
3
δ do−3,de
+ δ do,1 δ de,2
3 −
√
6
24
(1 − 2α) ; d e ≥ 2
(D11c)
A t
2:2 (d o , d e , d ′
o ) =
(1 − 2α)
2
−
1
2
do+d ′
o
2
From Eqs. (A4) we find α 1 = 4 9 + α 9 3 ; α 2 = 2 3 ; α 3 = 2 9 + 2 α 9 3 ; α 4 = 1 3 ; α 5 = 1 9 + α 9 3 ; α 6 = 2 3 ; α 7 = 2 9 + 2 α 9 3 ; α 8 = 1 3 ;(A5)where α 9 is left undetermined. In addition to the relations (A5), the orthogonalization process of the distorted state (A3a) with d = 5 requires to satisfyleading towhere N 1 is the normalizing prefactor. We checked that the states (A3a) for different i and d are also orthogonal. The prefactors β 1 · · · β 9 in Eq. (A3b) can be computed similarly. We orthogonalize the distorted state (A3b) to the 1-spinon state (16b) and to the 3-spinon states (A1) and (A3a). For d = 3, this yields the eight independent equationsleading to β 1 = 3 β 9 ; β 2 = 2 3 ; β 3 = 2 β 9 ; β 4 = 1 3 ; β 5 = β 9 ; β 6 = 2 3 ; β 7 = 2 β 9 ; β 8 = 1 3 .(A9)The prefactor β 9 remains unspecified. The orthogonalization process for the state (A3b) with d = 5 leads to the same results as (A9). However, one still needs to orthogonalize the distorted states (A3b) for different i and d. We find the equationwhere we used the relations (A9). This gives us the two solutions β 9 = 1 6 and 2 3 . We choose the solution β 9 = 1 6 as it indicates that the spinon at position i is of type left. The other solution β 9 = 2 3 corresponds to a spinon of type right at i − 2. Hence, we obtainThis concludes the construction of the distorted orthonormal 3-spinon states (A3a) and (A3b).Appendix B: CommutatorsThe commutators between two neutral hopping operators are given by[H i,i+d , H j,j+d ′ ] =(δ i,j+d ′ H i−d ′ ,i+d − δ j,i+d H i,i+d+d ′ ) + (δ i+d,j−1 + δ i−1,j+d ′ )(H i,i+d H j,j+d ′ − H j,j+d ′ H i,i+d )
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|
[] |
[
"Resummed Wentzel-Kramers-Brillouin Series: Quantization and Physical Interpretation",
"Resummed Wentzel-Kramers-Brillouin Series: Quantization and Physical Interpretation",
"Resummed Wentzel-Kramers-Brillouin Series: Quantization and Physical Interpretation",
"Resummed Wentzel-Kramers-Brillouin Series: Quantization and Physical Interpretation"
] |
[
"B Tripathi \nDepartment of Physics\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA\n",
"B Tripathi \nDepartment of Physics\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA\n"
] |
[
"Department of Physics\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA",
"Department of Physics\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA"
] |
[] |
The Wentzel-Kramers-Brillouin (WKB) perturbative series, a widely used technique for solving linear waves, is typically divergent and at best, asymptotic, thus impeding predictions beyond the first few leading-order effects. Here, we report a closed-form formula that exactly resums the perturbative WKB series to all-orders for two turning point problem. The formula is elegantly interpreted as the action evaluated using the product of spatially-varying wavenumber and a coefficient related to the wave transmissivity; unit transmissivity yields the Bohr-Sommerfeld quantization.
|
10.1103/physrevd.105.036010
|
[
"https://arxiv.org/pdf/2006.01434v3.pdf"
] | 238,634,819 |
2006.01434
|
76c4a1a030e2bc274a302c790a1a47fd9082b9ee
|
Resummed Wentzel-Kramers-Brillouin Series: Quantization and Physical Interpretation
22 Feb 2022
B Tripathi
Department of Physics
University of Wisconsin-Madison
53706MadisonWisconsinUSA
Resummed Wentzel-Kramers-Brillouin Series: Quantization and Physical Interpretation
22 Feb 2022(Dated: February 23, 2022)
The Wentzel-Kramers-Brillouin (WKB) perturbative series, a widely used technique for solving linear waves, is typically divergent and at best, asymptotic, thus impeding predictions beyond the first few leading-order effects. Here, we report a closed-form formula that exactly resums the perturbative WKB series to all-orders for two turning point problem. The formula is elegantly interpreted as the action evaluated using the product of spatially-varying wavenumber and a coefficient related to the wave transmissivity; unit transmissivity yields the Bohr-Sommerfeld quantization.
I. INTRODUCTION
Linear waves are ubiquitous in the world of physics with applications ranging from quantum mechanics [1] to electromagnetism, fluid dynamics, and astrophysics [2]. The properties of these waves are encoded in their dispersion relations, which reveal the nature of the medium they traverse, as well as their generating source [2,3] (e.g., gravitational waves). This problem of obtaining the dispersion relation is traditionally addressed with the Wentzel-Kramers-Brillouin (WKB) perturbative series [4], which however is typically divergent and at best, asymptotic [1,5]. It thus presents challenges to predict phenomena beyond the leading-order effects. Obtaining an expression for this series to all-orders in perturbation theory, preferably in a closed-form, is therefore desirable. Despite its usefulness in finding previously unknown physical interpretations of fully quantized wave, it has remained elusive. Here, we will be accomplish both the tasks of finding a closed-form formula and assigning a physical meaning to it.
In the quest for developing a closed-form quantization condition (dispersion relation) for linear waves, several insightful but hitherto-unsuccessful attempts have been undertaken through various ways like investigating structures of higher order expressions in the WKB series [6,7], utilization of supersymmetric WKB method [8,9], complex WKB method [10], and phase integral method [11]. We present here, a simple and insightful method to achieve this aim successfully.
The unexpected simplicity (both in mathematical structure and geometric-optical interpretation) of the closed-form formula reported here, arising out of the unwieldy WKB series, is what we believe to be the most striking about this work.
Our principal result is that the one-dimensional wave equation 1 (or Schrödinger equation) (2) where T (z) = τ 2 (z) is the wave-traversing medium's transmissivity of a layer of width 1/k(z) given as [12]
ǫ 2 d 2 ψ(x) dx 2 = Q (x) ψ(x), ψ (±∞) = 0,(1)τ (z) = 1 − 1 2 d(k −1 ) dz 2 = 1 + S ′ 1 S ′ 0 2 .(3)
The contour Γ encircles the two turning points in anticlockwise direction. (S ′ 0 and S ′ 1 are explained immediately below, but presented here due to their elegant appearances.) Unit transmissivity reproduces the commonly known leading-order WKB approximation.
II. CONVENTIONAL WKB
To begin with, consider the traditional transformation that is applied to the Schrödinger equation (1),
ψ(z, ǫ) = exp 1 ǫ S(z, ǫ) ,(4a)or, S ′ (z, ǫ) = ǫ ψ(z, ǫ) dψ(z, ǫ) dz ,(4b)
to obtain the Riccati equation: (S ′ ) 2 + ǫS ′′ = Q(z). The quantization condition for energy eigenvalue E in Eq. (1) is given in terms of the WKB eigenfunction's exponent in Eq. (4a) as 2 :
1 ǫ Γ S ′ (z, ǫ)dz = K · 2πi.(5)
This equation, although exact, is not useful unless we know what S ′ (z, ǫ) is [or be able to solve the above Riccati equation or Eq. (1) exactly]. We, therefore, proceed with the perturbative method to compute S ′ (z, ǫ) as
S ′ (z, ǫ) = ∞ n=0 ǫ n S ′ n (z).(6)
Substituting this ansatz in the aforementioned Riccati equation and equating like-powers of ǫ, one finds the S ′ n (z) to obey the recurrence relation,
S ′ 0 (z) = Q(z) = ik(z),(7)S ′ 1 (z) = − 1 2 d dz ln[S ′ 0 (z)],(8)2S ′ 0 S ′ n + n−1 j=1 S ′ j S ′ n−j + S ′′ n−1 = 0, (n ≥ 2).(9)
Eq. (5) thus, to all-orders in perturbation theory, is (it was first written in this form by Dunham [13]):
1 2iǫ Γ ∞ n=0 ǫ n S ′ n (z) dz = Kπ, (K = 0, 1, 2, ...).
(10) This series on the left hand side (LHS) is now to be summed up. However, it is typically a divergent asymptotic series [14,15]. One way to circumvent this challenge is to employ the Borel summation technique and assign a physical meaning to such a series. When employed, the analytic continuation of the Borel transform, however, presents another difficulty -singularities on the integration contour (see, e.g., [16][17][18][19]). Avoiding such with contour deformation yields ambiguous imaginary terms that plague the energy eigenvalues E. Significant progresses have been made to address such problems and more recently via exact WKB and uniform WKB methods, following advances of resurgence theory, developed by Ecalle and others in the 1980s [10,[20][21][22]. In such works, the ambiguous imaginary terms arising from the Borel summation are made to cancel each other systematically to all-orders by considering a "resurgent transseries" for the energy eigenvalues [22], as opposed to a perturbative series for it as we have done here. Such a path although very insightful and useful will not be pursued here as we wish to present an alternative simpler way to resum the diverging series of Eq. (10), for a class of potentials, and assign a physical meaning to the resummed series.
Eq. (4a) along a contour Γ such that it encloses, for the K th energy level, all K zeros of the eigenfunction on the real axis, between the classical turning points. This leads to Γ S ′ (z, ǫ)dz = Note that, in Eq. (10), the term of first order in ǫ [i.e., S ′ 1 (z)] can be integrated exactly [4]:
1 2i Γ dzS ′ 1 (z) = − 1 8i Γ dz d dz ln[Q(z)] = − 1 8i ln Q(z) evaluated once around contour Γ = − 1 8i (2 · 2πi) = − π 2 ,(11)
where evaluating the logarithmic function around the contour Γ, enclosing the two turning points of Q(z) yields 4πi.
This total contribution of −π/2 on the LHS of Eq. (10) correctly accounts for the zero-point energy of the simple harmonic oscillator. The series in Eq. (10), truncated at the first order, is the Bohr-Sommerfeld quantization relation [1]. It has been considered as an exceptional case that all other higher order terms for the simple harmonic oscillator turn out to be zeros. However, in general, this is not the case. Fröman and Fröman [11] have shown that all other higher odd -order terms in the WKB series, Eq. (10), can be written as exact derivatives, regardless of the type of potential, which upon contour integrating yield zeros. Setting ǫ = 1, we can, therefore, rewrite Eq. (10) as
1 2i Γ ∞ n=0 S ′ 2n (z) dz = K + 1 2 π, (K = 0, 1, 2, ...).
(12) Several attempts have been undertaken in the past [6,23] to infer the general expression for S ′ 2n with expectations of summing up the series afterwards. It, however, has turned out to be, heretofore, insurmountable. It is the objective of this work to present such a summation in an exact manner for an arbitrary potential with two turning points. (Note that such a route has been possible only for a very few special kinds of potentials, for e.g., the Eckart and the Morse potentials [6,23]). Next, we outline our method of summing up the WKB series up to all-orders and interpret its physical meaning thereafter.
Let us recast Eq. (12) as
1 2i Γ S ′ 0 (z) · ∞ n=0 T 2n (z) dz = K + 1 2 π,(13)
where T 2n (z) = S ′ 2n (z) /S ′ 0 (z) and the summation over T 2n will be achieved below. Introduce an economical notation L(z) ≡ 1/S ′ 0 (z). (L(z) can be regarded as having the dimension of length, found using Eqs. (7) and (1), and so does D −1 where D ≡ d/dz.) Dividing both sides of Eq. (9) by (S ′ 0 ) 2 to rewrite it in these new notations of T, D, and L, we find,
2T n + n−1 j=1 T j T n−j + L 2 d dz S ′ n−1 S ′ 0 · 1 L = 0,(14)2T n = − n−1 j=1 T j T n−j − LDT n−1 + T n−1 DL.(15)
III. PATTERN SEARCHING CAMPAIGN
Note T 0 = 1 and T 1 = DL/2. This allows to cast Eq. (15) finally in a neat way as
T n = T n−1 T 1 − 1 2 n−1 j=1 T j T n−j − 1 2 LDT n−1 , (n ≥ 2).(16)
We provide below the expressions for T n . 3 Notice the appearance of L below,
T 2 = T 2 1 2 − L × DT 1 2 ,(17a)T 3 = − L × DT 2 2 ,(17b)T 4 = − T 4 1 8 − L × DT 3 2 − T 2 1 · DT 1 4 + LDT 1 · DT 1 8 ,(17c)T 5 = − L × DT 4 2 − T 2 · DT 2 2 ,(17d)T 6 = T 6 1 16 − L × [...] ,(17e)
where ellipsis with a square bracket, [...], represents a collection of functions of lower order T n . Note that all terms of odd order (in n) of T n necessarily begin with L because these expressions when substituted in Eq. (13) yield cancellation of such L with L in the denominator of S ′ 0 = 1/L. The remaining part of the integrand can be shown to be (the sum of) the product of exact derivatives or expressions that can be changed into them [25]. Such integrand with the product of exact derivatives is trivially zero upon contour integrating [26] because they are single-valued functions for the defined contour path (no logarithmic derivatives are involved here as L in the denominator of S ′ 0 = 1/L has been cancelled out). This is tantamount to stating that all odd-order terms of T n contribute to the wavefunction's amplitude (i.e., they do not play role in the quantization condition [6]) and T n 's even-order terms modulate the phase of the wavefunction -thus the quantization condition involves thereof. To reiterate, for n ≥ 1, T odd order = T 2n+1 = −L × [...] and T even order = T 2n = ...T 2n 1 −L×[...]. After explicitly computing higher order terms (e.g., T 8 = 5T 8 1 /128 − L × [...]), we recognize a completely unexpected but instructive pattern, based upon which we propose the following hypothesis and prove it subsequently. 4
IV. INDUCTIVE HYPOTHESIS
Proposition: For any n ∈ N,
P(n): T 2n = n − 3 2 − 3 2 T 1 i 2n − L × [...] ,(18)where n− 3 2 − 3 2
is the binomial coefficient and
i 2 = −1.
This statement is not challenging to prove using the principle of mathematical induction (see section A in the Appendix for the proof).
Although not immediately obvious, the inductive hypothesis, presented in the precise form as above, has paramount consequence. In the proposition, in Eq. (18), the first term of T 2n on the right hand side lacks L in front of it unlike the second term and hence, when substituted in Eq. (13), it yields a function that does not vanish upon doing the contour integration (as terms like
Γ dzS ′ 0 · T 2n 1 ∼ Γ dz(DL) 2n
/L contribute to a logarithmic derivative of L and thus the contour encloses poles of a logarithmic function, which are the zeros of Q(z)). In contrast thereof, the second term of T 2n that begins with L, written as L × [...], contributes exactly zero upon contour integrating as the cancellation of this L with S ′ 0 = 1/L in Eq. (13) modifies the WKB integrand into (a sum of) the product of exact derivatives (importantly, without any logarithmic derivative) [25]. Resulting product of exact derivatives, lacking logarithmic term, amount to zero upon contour integrating; the reasoning here can follow the same as was aforementioned for the odd-order terms in the WKB series [26,27]. Thus this campaign of searching terms which begin with L and others which don't is unexpectedly helpful. We shall, therefore, deal with only the power series in T 1 in Eq. (13). By straightforward summation of this special series up to all-orders in perturbation theory, we obtain a closed-form expression for Eq. (13) as presented below in Eq. (19).
K + 1 2 π = 1 2i Γ dz L 1 + ∞ n=1,2,.. n − 3 2 − 3 2 DL 2i 2n = 1 2i Γ dz L (z) 1 + DL 2 2 = 1 2 Γ k(z) · 1 + S ′ 1 S ′ 0 2 dz = 1 2 Γ k(z) · 1 − 1 2 d(k −1 ) dz 2 dz.(19)
We emphasize that our summation in Eq. (19) involves a power series, which is also the case in all the special problems for which the WKB series has been summed up exactly [23,28]. We believe this to be the reason why the first few terms of the WKB expansion often approximate the correct eigenvalues despite the full series being divergent (this is, in part, also an answer to why the WKB expansion is asymptotic).
We are also able to physically interpret the expressions involved on the last line of Eq. (19) using the notion of geometric optics; however, the equation in its entirety is non-trivial to elucidate, possibly owing to the quantum effects embodied in all-orders (in perturbation theory).
V. GEOMETRIC-OPTICAL MEANING
We borrow here the illustration by Bremmer [12] where the author demonstrates, mutatis mutandis, the similarity of each order of WKB series (consider its n th order) to the transmitted waves in an infinitesimally-discretized inhomogeneous medium that undergo n-number of reflections. At each reflection, the waves change their direction by 180 • . Thus a wave that begins from a point far to the left gets continually transmitted to the right while suffering reflection at each discretized boundary (and consider for now only the directly transmitted waves -not doubly or quadruply or even number of multiply reflected ones that also can eventually transmit to the right). Such resulting wavefunction at the rightmost end yields the 1 st order WKB approximation [12]. Each of the above-mentioned reflected waves can undergo further reflection(s) and keep continually transmitting to the right. The more the number of reflections they suffer before they arrive to the rightmost end, the higher they belong to in the order of WKB expansion. Referring readers to the original paper [12] for additional interesting details, we present now heuristically how Bremmer arrives at the reflection coefficient of a layer of width 1/k(z). The well-known reflection coefficient in one-dimension is (Bremmer [12] uses R notation, which we shall reserve for the reflectivity to avoid potential confusions),
r(z) = k s − k s+1 k s + k s+1 ≈ k s − k s+1 2k s ≈ − dk/dξ 2k ,(20)
where s is the layer-number in the infinitesimally discretized inhomogeneous medium (within which k s remains constant) and ξ is proportional to the number of wavelengths over which the wavelength (or k) changes appreciably (from k s to k s+1 ). Following Bremmer [12], dξ = k(z)dz. Therefore, the transmissivity of a layer of width 1/k(z) is
T (z) = 1 − r 2 (z) = 1 − − dk/dξ 2k 2 = 1 − 1 2 d(k −1 ) dz 2 ,(21)
which is exactly what appears on the last line of Eq. (19).
Note that the wavenumber, obtained via resummation of WKB series to all-orders (i.e., integrand in Eq. (19)), vanishes even before we reach the classical turning points. Interestingly, for all potentials, it vanishes exactly at the locations where p cl (x) · x = ℏ/2 where p cl = 2m[E − V (x)] (cf. with the Heisenberg uncertainty principle).
VI. APPLICATIONS
Our novel closed-form quantization condition yields exact energy eigenvalues to potentials with two turning points. We demonstrate the efficacy of our formula in an example below and present other cases like simple harmonic oscillator, 3-dimensional harmonic oscillator, Coulomb potential, Eckart potential, and Morse potential in section D of the Appendix. We also find in cases of 3-dimensional spherically symmetric potentials, the Langer-correction factor [29,30] appears naturally upon performing the contour integration of our formula that resums all-orders perturbative effects. This corroborates the previous claim that the Langer-modification comes from the higher order corrections in the WKB series [28].
Consider an asymmetric Rosen-Morse potential, V (x), with ℏ 2 = 2m: V (x) = −U 0 sech 2 (x/a) + U 1 tanh (x/a). Let z be a complex variable such that z = tanh (x/a). Then, z a and z b are the two classical turning points, sat-
isfying z a + z b = −U 1 /U 0 ; z a z b = − (E + U 0 ) /U 0 . Using Eq. (2), 1 2i Γ 16 [V (z) − E] 3 + [V ′ (z)] 2 4 [V (z) − E] dz = K + 1 2 π,(22)
where K = 0, 1, 2, ... represent different energy levels.
The poles of the integrand are at z = z a , z b , 1, −1, and ∞. So, calculating residue at each pole with a proper principal value yields
− 1 4 + 1 4 − a (z a − 1)(z b − 1)U 0 2 − a (z a + 1)(z b + 1)U 0 2 + √ 1 + 4a 2 U 0 2 = K + 1 2 ,(23)∴ 1 2 −E − U 1 U 0 + 1 2 −E + U 1 U 0 = − 1 a √ U 0 K + 1 2 + √ 1 + 4a 2 U 0 2a √ U 0 ,(24)
which agrees with Ma & Xu [31] and the individual terms directly manifest from the residues at poles whose locations are precisely predicted by Eq. (22). At last, we remark that, for reasons not fully understood, the proposed quantization relation works only for two turning point problems. Nevertheless, understanding the nature of the exactly quantized action in two turning point problems, considered here, is likely to benefit the extension of our geometric-optically interpretable equation to problems with multiple turning points. Such interesting extensions will be investigated in a future study. Our proposed quantization condition might also engender rethinking of quantization in higher dimensions in terms of geometry [32].
VII. CONCLUSION
This article presents an exact closed-form quantization relation by summing up the WKB series to all-orders in perturbation theory for arbitrary one-dimensional potentials having two turning points. The new resummation procedure utilized herein reveals an unexpectedly simple pattern in the general term of the WKB series, leading to an inductive hypothesis, which we are able to prove by the principle of mathematical induction. The presented formula is then physically interpreted as the action of a wave with wavenumber corrected by a factor related to the wave transmissivity. Unit transmissivity recovers the Bohr-Sommerfeld quantization relation. This closedform expression for the quantization might also be useful in problems with more than two turning points where non-perturbative effects that give rise to tunneling phenomena come into play, i.e., spectral curves with non-zero genus [36]. For such problems, some of the neglected terms in the series resummation appear to be necessary. In the light of resurgent perturbative/nonperturbative relations [33,34,36], collecting such terms seem interesting and further investigation is merited. This will, however, be left for the future as it is beyond the objective of the present article. Arguably the most important advancement through this work is the discovery of an elegant and physically-interpretable equation emerging from a myriad of complicated terms in the WKB series that become increasingly unmanageable at each higher order of the series. It is gratifying to find that the spectral problem with genus-0 spectral curve (i.e., with two turning points and no tunneling phenomenon) can be reduced to an exact equation as simple and economical as Eq. (2). Analyzing this equation might lead to a deeper understanding of quantum geometry [36].
ACKNOWLEDGMENTS
I am pleased to thank Dhrubaditya Mitra for enkindling interest in this work and for offering discussions and guidance; it is difficult to imagine NORDITA (Sweden) without his friendly teaching style and advocacy for perturbation methods. I am also indebted to Michael V. Berry and Carl M. Bender for their appraisals of this work that greatly improved the manuscript. Thanks are also due to Paul W. Terry, MJ Pueschel, and Dibyendu Nandi for their valuable feedback and suggestions. Correspondences with MithatÜnsal and Luca Salasnich helped to find connections of this work with others. I also acknowledge support from the Van Vleck Fellowship in Physics at UW-Madison and the conducive environment herein.
VIII. APPENDIX
A. Proof by the principle of mathematical induction
Below, we present the proof for the proposition put forth for T 2n in Eq. (18). The statement, P(n), is trivially satisfied for n = 1 [cf. Eq. (18) with Eq. (17a)]. Now, we assume it to be valid for an arbitrary n and prove that it implies the proposition, in Eq. (18), is true for n + 1 as well [i.e., P(n + 1) is true]. We begin with the WKB recurrence relation, Eq. (16),
T 2(n+1) = − 1 2 2(n+1)−2 j=2 T j T 2(n+1)−j − LDT 2(n+1)−1 2 (25a) = − 2n j=2,4,... even T j T 2n+2−j 2 − 2n−1 j=3,5,... odd T j T 2n+2−j 2 − LDT 2n+1 2 (25b) = − n j/2=1,2,... j 2 − 3 2 − 3 2 n + 1 − j 2 − 3 2 T 1 i 2n+2 − L × [...] − 2n−1 j=3,5,... odd T j T 2n+2−j 2 − LDT 2n+1 2 (25c) = − n j/2=1,2,... j 2 − 3 2 − 3 2 n + 1 − j 2 − 3 2 T 1 i 2n+2 − L × [...] − L × [...] − LDT 2n+1 2 [∵ T odd begins with L] (25d) = n + 1 − 3 2 − 3 2 T 1 i 2(n+1) − L × [...](25e)
This proves the proposition.
B. Tn beginning with L convertible to a product of exact derivatives
Here, we detail the recipe of casting any expression in T n that begins with L into a product of exact derivatives (without a logarithmic derivative) [25]. Consider the term in T n that begins with L:
Γ dzS ′ 0 T n = Γ dzS ′ 0 L × [...] = Γ dz × 1 L × L × [...] = Γ dz × 1 L × L × [(D p1 L q1 D p2 L q2 ...) × ... × (D pj−1 L qj−1 D pj L qj ...)] = Γ dz [(D p1 L q1 D p2 L q2 ...) × ... × (D pj−1 L qj−1 D pj L qj ...)] ,(26)
where D p1 L q1 D p2 L q2 ... represents D p1 (L q1 D p2 (L q2 ...)); p 1 , q 1 , p 2 , q 2 , ..., p j , q j , ... are all integers in between (and including) 0 and n, satisfying the constraints: p 1 +p 2 +...+p j +... = n (for n number of D's) and q 1 +q 2 +...+q j +... = n − 1 (for n − 1 number of L's); and one extra L lies in the very beginning of T n , which has gotten cancelled with S ′ 0 = 1/L. This makes T n to have n number of D's and L's, as argued in Ref. [24]. If all p 1 , p 2 , ..., p j , ... are equal to or greater than 1, this integrand is already a product of exact derivatives (without a logarithmic derivative as S ′ 0 = 1/L has already been cancelled with L of T n ). If any of p 1 , p 2 , ..., p j , .. is zero, the integrand in Eq. (26), using chain rule, becomes (say p 1 = 0):
integrand = D 0 L q1 D p2 L q2 ... × ... × (D pj−1 L qj−1 D pj L qj ...) = (L q1 D p2 L q2 ...) × ... × (D pj−1 L qj−1 D pj L qj ...) = D [(L q1 D p2 L q2 ...) × ... × (D pj−1 L qj−1 D pj L qj ...)] − D [(L q1 D p2 L q2 ...) × ...] × D pj−1−1 L qj−1 D pj L qj ... ,(27)
thus turning the integrand into (a product of) exact derivatives. This process can be repeated if p j−1 − 1 also happens to be 0 (and if several L multiplies each other, its exponent can be raised to abridge this procedure). It should be emphasized that it is guaranteed to find such a transformation as there are exactly the same number of D's and L's in any T n [24]. In this regard, searching a general expression for T 2n as stated in the proposition in Eq. (18) is surprisingly helpful.
C. Product of exact derivatives yields zero
We demonstrate here that the product of exact derivatives (without a logarithmic function) amounts to zero on contour integrating, expounding on Ref. [26]. Let us consider two functions f and g (without a logarithm). Then, Γ dz df /dz = 0. Now, using the Cauchy integral formula to represent the exact derivatives and thereafter employing the partial fraction decomposition,
Γ dz df dz dg dz = Γ dz df dz dg dz (28a) = Γ dz 1 2πi γ1 f (u) (u − z) 1+1 du 1 2πi γ2 g(v) (v − z) 1+1 dv [γ 1 & γ 2 enclose the pole at z] (28b) = − 1 4π 2 γ1 γ2 dudvf (u)g(v) Γ dz 1 (u − z)(v − z) 2 (28c) = − 1 4π 2 γ1 γ2 dudvf (u)g(v) Γ dz 1 v − u 1 u − z − 1 v − z 2 (28d) = − 1 4π 2 γ1 γ2 dudv f (u)g(v) (v − u) 2 Γ dz 1 u − z − 1 v − z 2 (28e) = − 1 4π 2 γ1 γ2 dudv f (u)g(v) (v − u) 2 Γ dz 1 (u − z) 2 − 2 Γ dz 1 (u − z) (v − z) + Γ dz 1 (v − z) 2 (28f) = − 1 4π 2 γ1 γ2 dudv f (u)g(v) (v − u) 2 0 − 2 Γ dz 1 (u − z) (v − z) + 0 (28g) = 1 2π 2 γ1 γ2 dudv f (u)g(v) (v − u) 2 Γ dz 1 v − u 1 u − z − 1 v − z (28h) = 1 2π 2 γ1 γ2 dudv f (u)g(v) (v − u) 3 Γ dz 1 u − z − Γ dz 1 v − z (28i) = 1 2π 2 γ1 γ2 dudv f (u)g(v) (v − u) 3 [−2πi + 2πi] (28j) = 0. (28k)
It is straightforward to prove, in the similar manner, that the product of three (or more) exact derivatives ((without a logarithmic function) under contour integration is also zero. We are thus left with a power series in (T 1 ) 2n , which we sum up in Eq. (19).
D. More Examples
Let us now test the validity of our novel formula. We find that this formula gives the exact energy eigenvalues E for all the following potentials and many more which are not listed here (but all having exactly two turning points). We choose, without the loss of generality, ℏ 2 = 2m in all of the following calculations.
I. Simple Harmonic Oscillator
Even though we already know that the leading-order WKB is exact for simple harmonic oscillator, it is desirable to test if the extra factor in the integrand of Eq. (22) would cause any deviation from the correct eigenvalues. Consider,
V (z) = z 2 ,(29a)V − E = z 2 − E, (29b) V ′ (z) = 2z.(29c)
From Eq. (22),
1 2i 16 (V − E) 3 + (V ′ ) 2 4 (V − E) dz = K + 1 2 π,(30)
which leads to
1 2i 16 (z 2 − E) 3 + (2z) 2 4 (z 2 − E) dz = K + 1 2 π.
(31) The poles of the integrand are at z = √ E, − √ E, and ∞. So, calculating residues at each of the poles respectively,
1 4 − 1 4 + E 2 = K + 1 2 ,(32)∴ E = 2 K + 1 2 .(33)
II.
3-D harmonic oscillator
Consider the potential,
V (r) = r 2 + b r 2 + l(l + 1) r 2 ,(34a)V − E = r 2 + b r 2 + l(l + 1) r 2 − E. (34b) Let u = r 2 ; u ′ = 2r = 2 √ u. ∴ V − E = 1 u u 2 − Eu + {b + l (l + 1)} (35a) = 1 u (u − u a ) (u − u b ) ,(35b)
with
u a,b = E 2 ± E 2 2 − {b + l (l + 1)}, (36a) u a + u b = E, (36b) u a u b = b + l (l + 1) .(36c)
Now,
V ′ = dV du u ′ (37a) = 2u − u a − u b u − (u − u a )(u − u b ) u 2 2 √ u. (37b)
Using Eq. (22),
1 2i 16 (V − E) 3 + (V ′ ) 2 4 (V − E) dz = K + 1 2 π. (38)
The poles of the integrand are at u = u a , u b , 0, and ∞. So, calculating residues at each of the poles respectively,
1 4 − 1 4 − 1 4 √ 1 + 4u a u b + u a + u b 4 = K + 1 2 ,(39)∴ E = 2 2K + 1 + l + 1 2 2 + b . (40)
This solution agrees with Rosenzweig & Krieger [35].
Note that the correct Langer correction factor, shown in the bold typeset, emerges naturally from our all-orders resummed WKB series, which Langer proposed to replace l(l + 1) by (l + 1/2) 2 to obtain the correct eigenvalues.
III. Coulomb potential
Now, assume the Coulomb potential,
V (r) = − V 0 r + b r 2 + l(l + 1) r 2 ,(41a)V − E = 1 r 2 −V 0 r + b + l(l + 1) − Er 2 (41b) = −E r 2 r 2 + V 0 E r − b + l(l + 1) E (41c) = −E r 2 (r − r a ) (r − r b ) ,(41d)
with
r a,b = − V 0 2E ± V 0 2E 2 + b + l (l + 1) E , (42a) r a + r b = −V 0 E ,(42b)r a r b = − b + l (l + 1) E .(42c)
Now,
V ′ = 2E(r − r a )(r − r b ) r 3 − E r 2 (2r − r a − r b ) .(43)
We use Eq. (22) below,
1 2i 16 (V − E) 3 + (V ′ ) 2 4 (V − E) dz = K + 1 2 π. (44)
The poles of the integrand are at r = r a , r b , 0, and ∞. So, calculating residues at each of the poles with proper principal value,
− 1 4 + 1 4 − 1 2 1 − 4Er a r b + √ −E 2 (r a + r b ) = K + 1 2 ,(45)∴ E = −V 2 0 4 K + 1/2+ b + l + 1 2 2 2 .(46)
This solution agrees with Rosenzweig & Krieger [35].
Notice again that the correct Langer correction factor, shown in the bold typeset, emerges naturally from our all-orders resummed WKB series.
IV. Eckart potential
Let us consider the Eckart potential,
V (x) = −λe −αx 1 − e −αx + be −αx (1 − e −αx ) 2 .(47)
Using a transformation, u = e αx − 1; u ′ = α(u + 1); we write,
V − E = −λ u + b(u + 1) u 2 − E (48a) = −E u 2 u 2 + λ − b E u − b E (48b) = −E u 2 (u − u a )(u − u b ),(48c)
with
u a,b = b − λ 2E ± b − λ 2E 2 + b E , (49a) u a + u b = b − λ E , (49b) u a u b = − b E .(49c)
Now,
V ′ = dV du u ′ (50a) = 2E u 3 (u − u a )(u − u b ) − E u 2 (2u − u a − u b ) α(u + 1).(50b)
Using Eq. (22),
1 2i 16 (V − E) 3 + (V ′ ) 2 4 (V − E) dz = K + 1 2 π. (51)
The poles of the integrand are at u = u a , u b , 0, −1, and ∞.
So, calculating residues at each of the poles with proper principal value,
− 1 4 + 1 4 − 1 2α α 2 − 4Eu a u b + 1 α −E(1 + u a )(1 + u b ) − √ −E α = K + 1 2 ,(52)∴ − 1 2 1 + 4b α 2 + √ λ − E α − √ −E α = K + 1 2(53)
This solution agrees with Romanovski & Robnik [23].
V. Morse potential
Finally, consider the potential of the form,
V (x) = Ae −2αx − Be −αx , (54a) V − E = Ae −2αx − Be −αx − E.(54b)
Let u = e αx , u ′ = αu, which leads us to,
∴ V − E = 1 u 2 A − Bu − Eu 2 (55a) = −E u 2 u 2 + B E u − A E (55b) = −E u 2 (u − u a )(u − u b ),(55c)
with
u a,b = − B 2E ± B 2E 2 + A E , (56a) u a + u b = −B E , (56b) u a u b = −A E .(56c)
Now,
V ′ = dV du u ′ (57a) = 2E u 3 (u − u a )(u − u b ) − E u 2 (2u − u a − u b ) αu.(57b)
Employing Eq. (22),
1 2i 16 (V − E) 3 + (V ′ ) 2 4 (V − E) dz = K + 1 2 π. (58)
The poles of the integrand are at u = u a , u b , 0, and ∞. So, calculating residues at each of the poles with proper principal value,
−1 4 + 1 4 + 1 2α −E u a u b (u a + u b )− √ −E α = K + 1 2 ,(59)∴ B 2α √ A − √ −E α = K + 1 2 .(60)
This solution agrees with Romanovski & Robnik [23].
* [email protected] The book-keeping parameter ǫ can be set equal to 1 at the outset or at the end of the perturbative calculations.with Q (x) = −k 2 (x) = 2m[V (z) − E]/ℏ 2 where k(z) is the local wavenumber, V (z)is the potential, E is the energy eigenvalue, m and ℏ are the mass and reduced Planck's constant respectively, for the case of two turning points [locations where Q(z) = 0 with z being a complex variable] has an exact closed-form quantization condition Γ k(z) · τ (z)dz =
d
lnψ(z) dz dz = ǫ lnψ(z) evaluated once around Γ = K · 2πiǫ.
One way to derive this is by integrating dS/dz = S ′ (z, ǫ) in
It is worth highlighting that Tn is a dimensionless function as it has exactly the same number of D's and L's; see Ref.[24].4 We are very grateful to Michael V. Berry for questions that prompted us to propose this hypothesis.
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This can be confirmed via dimensional arguments (as D's and L's have the dimensions of each other's inverse and Tn is a dimensionless quantity with n number of derivatives). One can also observe this via Eq. (16) by noticing the appearances of D and L. Note that L's and D's appear exactly n number of times in any Tn as it is constructed to be so in Eq. with T0 = 1 and T1 = DL/2Note that L's and D's appear exactly n number of times in any Tn as it is constructed to be so in Eq. (13). This can be confirmed via dimensional arguments (as D's and L's have the dimensions of each other's inverse and Tn is a dimensionless quantity with n number of derivatives). One can also observe this via Eq. (16) by noticing the appearances of D and L, with T0 = 1 and T1 = DL/2.
This implies, non-trivially, that any expression in Tn that begins with L can be changed into a product of exact derivatives (without a logarithmic derivative. has exactly n number of D's and L's each as argued in Ref. It is because the L that Tn begins with, cancels the factor S ′ 0 = 1/L, the latter of which is responsible for a logarithmic derivative. After this cancellation, all L's appear in the numerator (D's always manifest in the numerator). For more explicit details, see section B in the AppendixAny Tn in Eq. (13), by construct, has exactly n number of D's and L's each as argued in Ref. [24]. This implies, non-trivially, that any expression in Tn that begins with L can be changed into a product of exact derivatives (without a logarithmic derivative). It is because the L that Tn begins with, cancels the factor S ′ 0 = 1/L, the latter of which is responsible for a logarithmic derivative. After this cancellation, all L's appear in the numerator (D's always manifest in the numerator). For more explicit details, see section B in the Appendix.
A heuristic proof can be obtained using the Cauchy integral formula (CIF) to represent each of the exact derivatives and switching the WKB contour integration's order with the contour integrations issued by the CIF. Detailed calculations are presented in section C of the AppendixA heuristic proof can be obtained using the Cauchy inte- gral formula (CIF) to represent each of the exact deriva- tives and switching the WKB contour integration's order with the contour integrations issued by the CIF. Detailed calculations are presented in section C of the Appendix.
A more rigorous analysis is likely to help extend our formula to problems with more than two turning points. A more rigorous analysis is likely to help extend our for- mula to problems with more than two turning points.
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Exact Quantization Conditions. C & J B Rosenzweig, Krieger, J. Math. Phys. 9849C. Rosenzweig & J.B. Krieger, Exact Quantization Con- ditions, J. Math. Phys 9, 849 (1968).
Quantum geometry of resurgent perturbative/nonperturbative relations. G Basar, G V Dunne, & M Ünsal, J. High Energy Phys. 201787G. Basar, G.V. Dunne, & M.Ünsal, Quantum geometry of resurgent perturbative/nonperturbative relations, J. High Energy Phys. 2017, 87 (2017).
|
[] |
[
"Transitivity of Commutativity for Linear Time-Varying Analog Systems",
"Transitivity of Commutativity for Linear Time-Varying Analog Systems"
] |
[
"Mehmet Emir \nDepartment of Mathematics\nOndokuz Mayis University\n55139Atakum, SamsunTurkey\n",
"Koksal \nDepartment of Mathematics\nOndokuz Mayis University\n55139Atakum, SamsunTurkey\n"
] |
[
"Department of Mathematics\nOndokuz Mayis University\n55139Atakum, SamsunTurkey",
"Department of Mathematics\nOndokuz Mayis University\n55139Atakum, SamsunTurkey"
] |
[] |
In this contribution, the transitivity property of commutative first-order linear timevarying systems is investigated with and without initial conditions. It is proven that transitivity property of first-order systems holds with and without initial conditions. On the base of impulse response function, transitivity of commutation property is formulated for any triplet of commutative linear time-varying relaxed systems. Transitivity proves are given for some special combinations of first and second-order linear time-varying systems which are initially relaxed.
| null |
[
"https://arxiv.org/pdf/1709.04477v1.pdf"
] | 10,118,688 |
1709.04477
|
7568ce251cc5971818d86b74ef1bbe74ab2d655e
|
Transitivity of Commutativity for Linear Time-Varying Analog Systems
Mehmet Emir
Department of Mathematics
Ondokuz Mayis University
55139Atakum, SamsunTurkey
Koksal
Department of Mathematics
Ondokuz Mayis University
55139Atakum, SamsunTurkey
Transitivity of Commutativity for Linear Time-Varying Analog Systems
1CommutativityDifferential equationsInitial conditionsTime-varying systemslinear systemsImpulse response
In this contribution, the transitivity property of commutative first-order linear timevarying systems is investigated with and without initial conditions. It is proven that transitivity property of first-order systems holds with and without initial conditions. On the base of impulse response function, transitivity of commutation property is formulated for any triplet of commutative linear time-varying relaxed systems. Transitivity proves are given for some special combinations of first and second-order linear time-varying systems which are initially relaxed.
Introduction
As the main branches of applied mathematics, differential (and integral) equations arise in many areas of sciences and engineering including acoustics, electromagnetic, electrodynamics, fluid dynamics etc. There is a great deal of papers on the theory, technique and applications of differential equations. Especially, they are used as a major tool in order to achieve many developments in real engineering problems by modelling, analyzing and solving naturel problems. For example, an interdisciplinary branch of applied mathematics and electricelectronics engineering, they play a pioneering role in system and control theory, that deal with the behavior of dynamical systems with inputs, and how their behavior is modified by different combinations such as cascade and feedback connections which is. When the cascade connection in system design is considered, the commutativity concept places a prominent role to improve different system performances. When two systems of this type are interconnected sequentially so that the output of the former feeds the input of the later, it is said that they are connected in cascade [1]. If the order of connection does not affect the input-output relation of the combined system or , it is said that systems and are commutative.
Figure 1:
Cascade connection of the differential system and If the combined system has an overall input-output relation invariant with the sequence of connection, it is said that these systems are commutative [2]. In [2], J. E. Marshall has proven that "for commutativity, either both systems are time-invariant or both systems are timevarying". In addition, he has revealed the commutativity conditions of first-order systems.
Later Marshall's work, a great deal of researches has been done on commutativity. In [3][4][5], the necessary and sufficient conditions for commutativity of second-order systems are presented. Koksal has presented the general commutativity conditions for time-varying systems of any order and reformulated the previous results obtained for seccond-order systems in the format of general conditions [6]. In this work, the general conditions are used to show that any system with constant forward and feedback gains is commutative with the system itself, which is an important fact for the feedback control theory.
In 1985, Koksal prepared a technical report which is a survey on commutativity [7]. In this report, an iterative formula is derived and an explicit formula is given for the entries of the coefficient matrix expressing the first set of commutativity conditions and the second set of commutativity conditions, respectively. Hence, the theorem stating these conditions is formally proved. Morover, explicit commutativity results for fourth-order systems are obtained. Finally, commutativity of Euler's system is proved.
The content of the published but undistributed work [7] can be found in the exhaustive journal paper of M. Koksal introduced the basic fundamentals of the subject [8]. His paper covers almost all the previous results except the ones related with initial conditions and sensitivity.
This paper is the first tutorial paper that has appeared in the literature.
More than one decade no publication had been appeared in the literature until the work in 2011 [9]. This reference is the second basic journal publication after the first appeared in 1988 [8].
In [9], another generic paper by the same author has presented explicit commutativity conditions of fifth-order systems in addition to reviews of commutativity of systems with nonzero initial conditions, commutativity and system disturbance, commutativity of Euler systems.
Research on commutativity has not confined to analog systems only; there has been some literature on the subject for discrete-time systems as well [10,11]. Hence, the research is continuing on both digital and analog area [12]. In [12], explicit results for finding all the second order commutative pairs of a first-order linear time-varying system have been given and the derived theoretical results have been verified by an example. About the commutativity of continuous time linear time-varying systems, [13] has been the last paper appearing in the literature; it deals with driving necessary and sufficiently conditions for the decomposition of a second order linear time-varying system into two cascade connected commutative first-order linear time-varying subsystems.
After a short introduction of the literature in this section, the transitivity property of commutativity is introduced in Section 2. Transitivity property of commutative first-order systems with and without initial conditions is studied in Section 3. In Section 4, transitivity property of commutativity for relaxed systems of any order is formulated on the base of impulse response function. Section 5 covers the verification of the general results formulated in Section 4 on the base of impulse response function for first-order systems. Section 6
illustrates the results presented in Sections 4 and 5. Section 7 includes transitivity proves for some combinations of first and second-order relaxed linear time-varying systems. Finally, the paper finishes with conclusions and future work which compose the last section, Section 8.
Transitivity Property of Commutativity
Logically or mathematically, transitivity is a property of a binary relation such that whenever one element is related to a second element, and the second element is related to a third element, then the first element is also related to the third element.
Let , , be dynamical systems described by linear time-varying differential equations of the form (1) with their special input-output variables, orders of complexity, and coefficients. If and are commutative among themselves, further, and are also commutative amoung themselves, what can it be said about the commutativity of systems and . If and are also commutative among themselves, this property is called transitivity property of commutativity, in other words commutativity is a transitive relation.
For commutative systems , , satisfiying transitivity, all the triplets , , , , , yield the same functionally equivalent cascade connected system and hence the same input-output property. The preference of an individual system connection depends on its relative performance characteristics such as sensitivity, disturbance, robustness, and etc., with respect to the others.
It is well-known (or trivial to show) that linear scalar systems, that is systems of order 0, are always commutative among themselves whether time-invariant or time-varying; and transitivity always hold for such systems. Further, initially relaxed time-invariant systems are always commutative. When initial conditions exist, which is valid for non-scalar (1st or higherorder) systems, the commutativity of such systems (even including a scalar system) is not automatic and requires some conditions. Hence, the following discussions are devoted mainly to commutativity of systems at least one of them is of order one or higher.
Transitivity for First-order Systems
Let , , be first-order linear time-varying systems defined by a differential equation of the form (1). Let 1 , 0 ; 1 , 0 ; and 1 , 0 are time-varying (in general) coefficients of these systems. Assume also that ( , ) and ( , ) are commutative pairs.
The first commutativity conditions [7][8][9] for ( , ) and ( , ) yields
[ 1 0 ] = [ 1 0 0 1 ] [ 1 0 ],(2a)[ 1 0 ] = [ 1 0 0 1 ] [ ℓ 1 ℓ 0 ],(2b)
respectively; where 1 , 0 , ℓ 1 , ℓ 0 are some constants.
Inserting the values of 1 , 0 as expressed in (2a) into Eq. (2b) and rearranging, we obtain
[ 1 0 ] = [ 1 0 0 1 ] [ 1 ℓ 1 0 ℓ 1 + ℓ 0 ]. (2c)
This equation already satisfies the first commutativity condition for Systems and ; hence, and are commutative pairs under zero initial conditions.
For the case of existence of non-zero initial conditions, the second commutativity condition [9] for ( , ) and ( , ) yields
( 0 ) = ( 0 ) ≠ 0,(3a)1 − 0 ( 0 ) 1 ( 0 ) = 1 − 0 ( 0 ) 1 ( 0 ) ; (3b) ( 0 ) = ( 0 ) ≠ 0,(4a)1 − 0 ( 0 ) 1 ( 0 ) = 1 − 0 ( 0 ) 1 ( 0 ) ,(4b)
respectively. Hence, Eqs. (3) and (4) lead to write
( 0 ) = ( 0 ) ≠ 0,(5a)1 − 0 ( 0 ) 1 ( 0 ) = 1 − 0 ( 0 ) 1 ( 0 ) . (5b)
The result in Eq. (5) clearly shows the satisfaction of the second commutativity condition for Subsystems and . Hence, and are also commutative when initial conditions exist, and transitivity property is valid for non-relaxed first-order linear time-varying systems as well.
In fact, Eq. (3b) is not satisfied for all systems and satisfying commutativity conditions (2a) under relaxed conditions. Using (2a) we see that Eq. (3b) requires
1 − 0 ( 0 ) 1 ( 0 ) = 1 − 0 ( 0 ) 1 ( 0 ) = 1 − 1 0 ( 0 ) − 0 1 1 ( 0 ) = 1 − 0 1 − 0 ( 0 ) 1 ( 0 ) ;(6a)
Or equivalently
0 = 1 − 1 . (6b)
Similarly, using (2b), we see that Eq. (4b) requires
1 − 0 ( 0 ) 1 ( 0 ) = 1 − 0 ( 0 ) 1 ( 0 ) = 1 − ℓ 1 0 ( 0 ) − ℓ 0 ℓ 1 1 ( 0 ) = 1 − ℓ 0 ℓ 1 − 0 ( 0 ) 1 ( 0 ) ,(7a)ℓ 0 = 1 − ℓ 1 . (7b)
Using (2c) with (6b) and (7b), we can easily show that (5b) is satisfied; That is:
1 − 0 ( 0 ) 1 ( 0 ) = 1 − [ 1 ℓ 1 0 ( 0 ) + 0 ℓ 1 + ℓ 0 ] 1 ℓ 1 1 ( 0 ) = 1 − 0 ℓ 1 − ℓ 0 1 ℓ 1 − 0 ( 0 ) 1 ( 0 ) = 1 − (1 − 1 )ℓ 1 − (1 − ℓ 1 ) 1 ℓ 1 − 0 ( 0 ) 1 ( 0 ) = 1 − 0 ( 0 ) 1 ( 0 ) ,(8)
which is exactly Eq. (5b); hence, and are commutative under non-zero conditions as well if the constants 1 , ℓ 1 , 0 , , ℓ 0 satisfy (6b) and (7b). These are explicit commutativity conditions under non-zero initial states equivalently replacing by (3b) and (4b) in addition to (2a) and (2b) which are necessary and sufficient for relaxed case. With these conditions 5b is also satisfied, so that transitivity is valid.
We state the results obtained so far by two theorems: Note that this theorem is the first theorem expressing the commutativity conditions for unrelaxed systems by using the arbitrary constants relating the coefficients of two systems for their commutativity in relaxed conditions. Note also that Eqs. (6b) and (7b) are independent of the initial time. So, commutativity does not depend on the initial time 0 for first-order nonrelaxed systems.
Theorem 2:
The transitivity property of commutativity for first-order linear time-varying systems is always valid for both relaxed and un-relaxed commutative systems that is for systems with zero and non-zero initial conditions.
Transitivity for High-order Systems
In this section, higher-order linear time-varying systems with zero initial conditions are considered; scalar (0-order) and first-order systems are also included in the scope. To prove the transitivity property of commutativity, the impulse response function is used. The results are general in the sense the subsystems of each commutative pair may have a different order than its partner.
Let be a liear time-varying initially relaxed system having an impulse response ℎ ( , ); by definition this is the response to a unit impulse occurring at time and observed at time ≥ [14]. And the response of for an arbitrary input ( ) is expressed by superposition integral
( ) = ∫ ℎ ( , ) ( ) 0 . (9a)
Consider now the cascade connection of and as shown in When we consider a scalar system From these equations, it is obvious that a scalar system that can be commutative with a 1st or higher order system must be a time-invariant system; that is 0 ( ) = 0 ( 0 ) = for all ≥ 0 . In other words, a scalar time-varying system cannot commutative with any order of higher order (1st, 2nd …) time So, that and are commutative. This proves the transitivity proper of higher-order systems under the set assumptions.
Verifications for first-order systems
This section verifies the transitivity property of first-order linear time-varying systems in their relaxed modes as presented in Section 3 by using the general results obtained for systems of any order in Section 4.
Consider a first-order linear time-varying system defined by
1 ( ) ( ) + 0 ( ) ( ) = ( ),(20)
with zero initial conditions. The impulse response ℎ ( , 0 ) of this system to an impulse ( − 0 ) occurring at = 0 is obviously expressed as [14,15] ℎ ( , 0 ) = 1
1 ( 0 ) ∫ − 0 ( ) 1 ( ) 0 = 1 1 ( 0 ) ( )− ( 0 ) , (21a) where ( ) = ∫ − 0 ( ) 1 ( ) ,(21b)( ) = − 0 ( ) 1 ( ) . (21c)
Hence, for Systems and of the same type of , it can be written that ℎ ( , 0 ) = 1
1 ( 0 ) ∫ − 0 ( ) 1 ( ) 0 = 1 1 ( 0 ) ( )− ( 0 ) , (22a) ( ) = ∫ − 0 ( ) 1 ( ) , (22b) ( ) = − 0 ( ) 1 ( ) ,(22c)
Example
This example is given for illustration of the results presented in Sections 4 and 5. Consider system describe by
( + 1)́+ ( + 2) = , −1 < 0 ≤ ,(32)
where 1 ( ) = + 1, 0 ( ) = + 2; ( ) for is computed by using (21b) as
( ) = ∫ − + 2 + 1 = − − ln( + 1).(33)
Then, by (21a), its impulse response is computed as
where 1 ( ) = 2( + 1), 0 ( ) = 2 + 5 which can be obtained from 1 and 0 by using Eq. This is the same impulse response expressed in (41a). Hence, it is verified that ( , ) is a commutative pair.
Transitivity for Some 1 st and 2 nd Order Relaxed Systems
Let , , be linear time-varying systems of orders 2, 1, 1; respectively. Let they be represented by the general form of (1) with coefficients 2 , 1 , 0 ; 1 , 0 ; 1 , 0 ; inputs by
It is obviously true that this equation is in the form of (43). So is among the first-order commutative pairs of . Hence, it has been proven that if ( , ) and ( , ) are commutative pairs so is ( , ). Hence, transitivity is valid.
Let's get one step up and consider transitivity property between two second order and one firstorder systems. Let , , be 2 nd , 1 st , 2 nd order systems, respectively, with coefficient To prove the commutativity of and , it is sufficient to show that
[ 1 0 ] = [
Conclusions
Transitivity property of commutativity is defined for commutative linear time-varying systems of any order. The transitivity results are presented for first-order systems with and without initial conditions. The study is carried on two approaches; one is based on use of impulse response function and the other depends on use of general conditions set by the author and others for commutativity of linear time-varying systems.
) represents input-output pair of the system at any time ∈ , ( ) are time-varying coefficients with ( ) ≠ 0, ≥ 0 is the oder of system; and ( ) ( 0 ) ∈ , = 0, 1, ⋯ , − 1 are the initial conditions at the initial time 0 ∈ .
Fig. 2 Figure 2 :
22and denote this connection by . Smilar to Eq. (9a), for the output of we write Cascade This is the response of for any input ( ) applied for ≥ 0 .To find the impulse response ℎ ( , 0 ) of the connection , we substitute ( ) = ( − 0 ) in Eq. (9d) and arrive second equality results from the property of impulse function. In a similar result, the impulse response ℎ ( , ) of the connection can be written as Since equal impulse responses yield equal input-output pairs, for commutativity of and it must be true that In fact, either of Eqs. (11b) or Eq. (11c) express necessary and sufficient conditions in terms of impulse responses for relaxed systems and to be commutative. On the other hand, for commutativity of and under zero inintial conditions, the neccessary and suffifcient condition can be obtained from (11) by changing → and → , For the proof of the commutativity of and , that is the transitivity property of commutativity, In the light of the presence of equalities in (11b) and (12b), the work left for the proof of transitivity is to show the validity of (13b); if so then (13a) holds. This process will end with the proof of transitivity of linear time-varying relaxed systems of higher orders.It worth's to remark that Eqs. (10a, b) are first used by J. E. Marshall to prove his assertion mentioned in Introduction (Eqs.(2,3) in[2]). Later, they are used by the author for studying commutativity of Euler systems (Eqs. (34a, b) in[7]).
-varying system. The only commutative pairs of it are the scalar systems. Though commutativity of a constant gain system (scalar time-invariant system) is always valid with time-varying system of any order, if the initial conditions exist the commutativity is possible only if the constant gain is 1; that is System is an identity. The validation of transitivity using the equivalence of impulse responses ℎ ( , 0 ) and ℎ ( , 0 ) for first-order systems are considered in Section 5. But the extension of this result for systems at least one is 2nd order or higher is not straight forward and appears to be an unsolved problem. Instead of jumping this problem fully, we consider the special case that transitivity holds conditionally. Hence, the following discussion is not restricted by order limitation. Let , , be linear time-varying systems with impulse responses ℎ , Eq. (16a) implies that Eqs. (11a, b, c) are satisfied, hence ( , ) is a comutative pair. In a similar manner (16b) implies that ( , ) is also a commutative pair. Multiplying (16b) by ℎ ( , 0 ) and inserting the equivalence of ℎ ( , )ℎ ( , 0 ) as obtained from (16a), we find ℎ ( , )ℎ ( , 0 )ℎ ( , 0 ) − ℎ ( , )ℎ ( , 0 )ℎ ( , 0 ) = 0, (17 ) ℎ ( , )ℎ ( , 0 )ℎ ( , 0 ) − ℎ ( , 0 )ℎ ( , )ℎ ( , 0 )Comparing with Eqs. (10a) and (10b), the first integral in (18b) is the impulse response of and the second one is that of ; hence,
For commutativity of and , consider the verification of Eq. (11a) (or equivalently Eqs. (11b, c)). Using the relations in Eq. (2a) between the coefficients of and in Eq.Same property holds for the remaining first-order systems and as well. Another relation we need in the future proves is arrived as follows: the same impulse response as seen in Eqs.(28)and(29),respectively; so and are commutative.Remark 1: Although Eq. (11c), a property of impulse response of the commutative systems and , seems to mislead to the conclusion of the integrant being zero; that is, This is strictly wrong. In fact, using (24b) for ℎ ( , 0 ), and (26) later, we have impossible for an arbitrary function 1 (•) And for general values of 0 ≤ ≤ . in Eqs. (25-29) complete the verification of transitivity for initially relaxed first-order linear time-varying systems by using impulse responses.
(2a) with 1 = 2, 0 = 1 ;
1hence, ( , ) is a commutative pair. Using (22b), ( ) Obviously, these integrands are not identical and Eq. (30) in connection to Remark 1 is not satisfied for the general values of 0 ≤ ≤ ∈ . The impulse response of and are computed as in(10a)and(10b), that is the integral of the integrands in (38a) and (38b),respectively. The results are as follows:Using (23a), the impulse response of is now obtained asThe impulse response of the connection is computed by Eq. (10a) modified by replacing by , and using Eqs. (34) and (40b) for and ; we obtain Note that the impulse response of the connection can be obtained similarly by using Eq.(10a) by replacing by and using Eqs. (40b) and (34) for and ; the result is
On the other hand, it is true that has 1 st or lower order commutative pairs , if[8]
1 and ℓ 0 are arbitrary constants. Inserting the values of 1 and 0 from Eq. (43),) into Eq. (44), and rearranging,
Inserting the values of 2 ,̇2,̈2, 1 ,̇1 in Eq. (49) into Eq. (48a) and simplifying, we obtain
similar equation to (42b), where 2 , 1 , 0 are some constants. It is true that, the coefficients of can be expressed as in (51) by proper choice of constants 2 , 1 , 0 ; in fact, inserting in coefficients in Eq. (49) and coefficient in Eq. (50) into Eq. (51), all constants. Hence, Eq. (51) together with Eq. (42) imply that is a second-order commutative pair of , that is and are also commutative. This proves the transitivity property of ( , ) and ( , ) to ( , ).We express the result by a theorem: Theorem 3: The commutativity property between three subsystems of which at most 2 of them second order and the other(s) are first-order always satisfy the transitivity property under zero initial conditions.
Theorem 1 :
1The necessary and sufficient conditions that two first-order linear time-varying systems and described by differential equations of the form (1) are commutative under zero initial conditions are that i. The coefficients of System must be expressible in terms of the coefficients of those of as in Eq. (2a) where 1 ≠ 0 and 0 are arbitrary constants. ii. Further, Systems and are commutative under arbitrary non-zero initial conditions as well if and only if Condition (i) is satisfied with 0 = 1 − 1 (that is 1 ≠ 0 and 0 cannot be chosen arbitrarily), and their non-zero initial conditions must be equal.
, , ;
,output by , , . It is true that has 2 nd or lower order commutative pairs if and only if[8][9][10] − 2
0.5
[ 0 −
4 1
2 + 3̇2 2 − 8 1̇2 + 81 2 − 4 2̈2
16 2
] 1 = 0,
is also valid since ( , ) is a commutative pair. Further, (42a) is valid and the similar equationSince is of order 1, 1 in (43) and ℓ 1 in (46) are not zero; hence (42a) and (47) yield2
0.5
0
2
−0.5 (2 1 −2)
4
1
] [
ℓ 1
ℓ 0
]
(46)
for is written as
− 2
0.5
[ 0 −
4 1
2 + 3̇2 2 − 8 12 + 81 2 − 4 22
16 2
] ℓ 1 = 0.
(47)
0 = [ 0 +
4 1
2 + 3̇2 2 − 8 1̇2 + 8̇1 2 − 4 2̈2
16 2
],
(48a)
0 = [ 0 +
4 1
2 + 3̇2 2 − 8 12 + 81 2 − 4 22
( ) .(24a)Using this result in Eq. (22a), we obtain ℎ ( , 0 ) = 11 1 ( 0 ) ( )− 0 1 ∫ 1 1 ( ) − ( 0 )+ 0 1 ∫ 1 1 ( 0 ) 0 , = 1 1 1 1 ( 0 ) ( )− ( 0 ) ( 0 )− ( ) = 1 1 ( 0 )− ( ) ℎ ( , 0 ),(24b)where( ) = 0 1 ∫ 1 1 ( ) .(24c)By using (15b) in (7a), we obtain
, 1 , 0 ; 1 , 0 , 2 , 1 , 0 . Assume ( , ) and ( , ) are commutative pairs and show ( , ) is also commutative. Eq. (43) is valid since ( , ) is a commutative pair. Similar equation
One illustrative example is included to show the validity of the results. However, general transitivity property of commutative high order systems have not been proved or disproved and this remains as an unsolved problem yet.
R Boylestad, L Nashelsky, Electronic Devices and Circuit Theory. New JerseyPrentice HallR. Boylestad and L. Nashelsky, Electronic Devices and Circuit Theory, Prentice Hall, New Jersey, 2013.
Commutativity of time varying systems. E , Electronics Letters. 13E. Marshal, Commutativity of time varying systems, Electronics Letters, 13 (1977) 539-540.
Commutativity of second order time-varying systems. M , International Journal of Control. 36M. Koksal, Commutativity of second order time-varying systems, International Journal of Control, 36 (1982) 541-544.
Comments on 'Commutativity of second-order time-varying systems. S V Salehi, International Journal of Control. 37S. V. Salehi, Comments on 'Commutativity of second-order time-varying systems', International Journal of Control, 37 (1983) 1195-1196.
Corrections on 'Commutativity of second-order time-varying systems. M , International Journal of Control. 38M. Koksal, Corrections on 'Commutativity of second-order time-varying systems', International Journal of Control, 38 (1983) 273-274.
M , General conditions for the commutativity of time-varying systems, IASTED International Conference on Telecommunication and Control (TELCONi84). M. Koksal, General conditions for the commutativity of time-varying systems, IASTED International Conference on Telecommunication and Control (TELCONi84), 1984, pp. 223-225.
M , no: GEEE CAS-85/1A Survey on the Commutativity of time-varying systems, METU, Gaziantep Engineering Faculty. Technical. ReportM. Koksal, A Survey on the Commutativity of time-varying systems, METU, Gaziantep Engineering Faculty, Technical. Report no: GEEE CAS-85/1, 1985.
An exhaustive study on the commutativity of time-varying systems. M , International Journal of Control. 47M. Koksal, An exhaustive study on the commutativity of time-varying systems, International Journal of Control, 47 (1988) 1521-1537.
Commutativity of linear time-varying differential systems with non-zero initial conditions: A review and some new extensions. M Koksal, M E , Mathematical Problems in Engineering. M. Koksal and M. E. Koksal, Commutativity of linear time-varying differential systems with non-zero initial conditions: A review and some new extensions, Mathematical Problems in Engineering, 2011 (2011) 1-25.
Commutativity of cascade connected discrete time linear time-varying systems. M E Koksal, M , Automatic Control National Meeting TOK'2013. M. E. Koksal and M. Koksal, Commutativity of cascade connected discrete time linear time-varying systems, 2013 Automatic Control National Meeting TOK'2013, (2013) p.1128-1131.
Commutativity of cascade connected discrete-time linear time-varying systems. M Koksal, M E , Transactions of the Institute of Measurement and Control. 37M. Koksal and M. E. Koksal, Commutativity of cascade connected discrete-time linear time-varying systems, Transactions of the Institute of Measurement and Control, 37 (2015) 615-622.
The Second order commutative pairs of a first-order linear time-varying system. M E , Applied Mathematics and Information Sciences. 9M. E. Koksal, The Second order commutative pairs of a first-order linear time-varying system, Applied Mathematics and Information Sciences, 9 (2015) 1-6.
Decomposition of a second-order linear time-varying differential system as the series connection of two first-order commutative pairs. M E , Open Mathematics. 14M. E. Koksal, Decomposition of a second-order linear time-varying differential system as the series connection of two first-order commutative pairs, Open Mathematics, 14 (2016) 693-704.
Notes For A Second Course On Linear Systems. C A Desoer, Van Nostrand RheinholdNew YorkC. A. Desoer, Notes For A Second Course On Linear Systems, Van Nostrand Rheinhold, New York, 1970.
Chi-Tsong Chen, Linear System Theory and Design. New York OxfordOxford University PressChi-Tsong Chen, Linear System Theory and Design, New York Oxford, Oxford University Press, 1999.
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[] |
[
"DeepMoCap: Deep Optical Motion Capture Using Multiple Depth Sensors and Retro-Reflectors",
"DeepMoCap: Deep Optical Motion Capture Using Multiple Depth Sensors and Retro-Reflectors"
] |
[
"Anargyros Chatzitofis \nCentre for Research and Technology Hellas\nInformation Technologies Institute\n6th km Charilaou-Thermi57001Thermi, ThessalonikiGreece\n\nSchool of Electrical and Computer Engineering\nNational Technical University of Athens\nZografou Campus, Iroon Polytechniou 915780Zografou, AthensGreece\n",
"Dimitrios Zarpalas [email protected]. \nCentre for Research and Technology Hellas\nInformation Technologies Institute\n6th km Charilaou-Thermi57001Thermi, ThessalonikiGreece\n",
"Stefanos Kollias [email protected] \nSchool of Electrical and Computer Engineering\nNational Technical University of Athens\nZografou Campus, Iroon Polytechniou 915780Zografou, AthensGreece\n\nSchool of Computer Science\nUniversity of Lincoln\nLN67TSBrayfordUK\n",
"Petros Daras [email protected]. \nCentre for Research and Technology Hellas\nInformation Technologies Institute\n6th km Charilaou-Thermi57001Thermi, ThessalonikiGreece\n"
] |
[
"Centre for Research and Technology Hellas\nInformation Technologies Institute\n6th km Charilaou-Thermi57001Thermi, ThessalonikiGreece",
"School of Electrical and Computer Engineering\nNational Technical University of Athens\nZografou Campus, Iroon Polytechniou 915780Zografou, AthensGreece",
"Centre for Research and Technology Hellas\nInformation Technologies Institute\n6th km Charilaou-Thermi57001Thermi, ThessalonikiGreece",
"School of Electrical and Computer Engineering\nNational Technical University of Athens\nZografou Campus, Iroon Polytechniou 915780Zografou, AthensGreece",
"School of Computer Science\nUniversity of Lincoln\nLN67TSBrayfordUK",
"Centre for Research and Technology Hellas\nInformation Technologies Institute\n6th km Charilaou-Thermi57001Thermi, ThessalonikiGreece"
] |
[] |
In this paper, a marker-based, single-person optical motion capture method (DeepMoCap) is proposed using multiple spatio-temporally aligned infrared-depth sensors and retro-reflective straps and patches (reflectors). DeepMoCap explores motion capture by automatically localizing and labeling reflectors on depth images and, subsequently, on 3D space. Introducing a non-parametric representation to encode the temporal correlation among pairs of colorized depthmaps and 3D optical flow frames, a multi-stage Fully Convolutional Network (FCN) architecture is proposed to jointly learn reflector locations and their temporal dependency among sequential frames. The extracted reflector 2D locations are spatially mapped in 3D space, resulting in robust 3D optical data extraction. The subject's motion is efficiently captured by applying a template-based fitting technique on the extracted optical data. Two datasets have been created and made publicly available for evaluation purposes; one comprising multi-view depth and 3D optical flow annotated images (DMC2.5D), and a second, consisting of spatio-temporally aligned multi-view depth images along with skeleton, inertial and ground truth MoCap data (DMC3D). The FCN model outperforms its competitors on the DMC2.5D dataset using 2D Percentage of Correct Keypoints (PCK) metric, while the motion capture outcome is evaluated against RGB-D and inertial data fusion approaches on DMC3D, outperforming the next best method by 4.5% in total 3D PCK accuracy.
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10.3390/s19020282
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2110.07283
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fe2ce84de0487da5f9d4b27010a087d100572881
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DeepMoCap: Deep Optical Motion Capture Using Multiple Depth Sensors and Retro-Reflectors
Anargyros Chatzitofis
Centre for Research and Technology Hellas
Information Technologies Institute
6th km Charilaou-Thermi57001Thermi, ThessalonikiGreece
School of Electrical and Computer Engineering
National Technical University of Athens
Zografou Campus, Iroon Polytechniou 915780Zografou, AthensGreece
Dimitrios Zarpalas [email protected].
Centre for Research and Technology Hellas
Information Technologies Institute
6th km Charilaou-Thermi57001Thermi, ThessalonikiGreece
Stefanos Kollias [email protected]
School of Electrical and Computer Engineering
National Technical University of Athens
Zografou Campus, Iroon Polytechniou 915780Zografou, AthensGreece
School of Computer Science
University of Lincoln
LN67TSBrayfordUK
Petros Daras [email protected].
Centre for Research and Technology Hellas
Information Technologies Institute
6th km Charilaou-Thermi57001Thermi, ThessalonikiGreece
DeepMoCap: Deep Optical Motion Capture Using Multiple Depth Sensors and Retro-Reflectors
10.3390/s19020282Received: 13 December 2018; Accepted: 7 January 2019; Published: 11 January 2019sensors Articlemotion capturedeep learningretro-reflectorsretro-reflective markersmultiple depth sensorslow-costdeep mocapdepth data3D data3D visionoptical mocapmarker-based mocap
In this paper, a marker-based, single-person optical motion capture method (DeepMoCap) is proposed using multiple spatio-temporally aligned infrared-depth sensors and retro-reflective straps and patches (reflectors). DeepMoCap explores motion capture by automatically localizing and labeling reflectors on depth images and, subsequently, on 3D space. Introducing a non-parametric representation to encode the temporal correlation among pairs of colorized depthmaps and 3D optical flow frames, a multi-stage Fully Convolutional Network (FCN) architecture is proposed to jointly learn reflector locations and their temporal dependency among sequential frames. The extracted reflector 2D locations are spatially mapped in 3D space, resulting in robust 3D optical data extraction. The subject's motion is efficiently captured by applying a template-based fitting technique on the extracted optical data. Two datasets have been created and made publicly available for evaluation purposes; one comprising multi-view depth and 3D optical flow annotated images (DMC2.5D), and a second, consisting of spatio-temporally aligned multi-view depth images along with skeleton, inertial and ground truth MoCap data (DMC3D). The FCN model outperforms its competitors on the DMC2.5D dataset using 2D Percentage of Correct Keypoints (PCK) metric, while the motion capture outcome is evaluated against RGB-D and inertial data fusion approaches on DMC3D, outperforming the next best method by 4.5% in total 3D PCK accuracy.
Introduction
Human pose tracking, also known as motion capture (MoCap), has been studied for decades and is still a very active and challenging research topic. MoCap is widely used in industries such as gaming, virtual/augmented reality, film making and computer graphics animation, among others, as a means to provide body (and/or facial) motion data for virtual character animation, humanoid robot motion control, computer interaction, and more. To date, a specialized computer vision and marker-based MoCap technique, called Optical Motion Capture [1], constitutes the gold-standard for accurate and robust motion capture [2]. Optical MoCap solutions [3-5] employ multiple optical sensors and passive or active markers (passive markers are coated with retro-reflective material to reflect light, while active markers are powered to emit it; for passive marker-based MoCap systems, IR emitters are also used to cast IR light on the markers) placed on the body of the subject to be captured. The 3D positions of the markers are extracted by intersecting the projections of two or more spatio-temporally aligned optical sensors. These solutions precisely capture the body movements, i.e., the body joint 3D positions and orientations per frame in high frequency (ranging from 100 to 480 Hz).
In the last decade, professional optical MoCap technologies have seen rapid development due to the high demand of the industry and the strong presence of powerful game engines [6][7][8], allowing for immediate and easy consumption of motion capture data. However, difficulties in using traditional optical MoCap solutions still exist. Purchasing a professional optical MoCap system is extremely expensive, while the equipment is cumbersome and sensitive. With respect to its setup, several steps should be carefully followed, ideally by a technical expert, to appropriately setup the required hardware/software and to rigidly install the optical MoCap cameras on walls or other static objects [1,9]. In addition, time-expensive and non-trivial post-processing is required for optical data cleaning and MoCap data production [10]. To this end, there still exists an imperative need for robust MoCap methods that overcome the aforementioned barriers.
In this paper, a low-cost, fast motion capture method is proposed, namely DeepMoCap, approaching marker-based optical MoCap by combining Infrared-Depth (IR-D) imaging, retro-reflective materials (similarly to passive markers usage) and fully convolutional neural networks (FCN). In particular, DeepMoCap deals with single-person, marker-based motion capture using a set of retro-reflective straps and patches (reflector-set: a set of retro-reflective straps and patches, called reflectors for the sake of simplicity) from off-the-shelf materials (retro-reflective tape) and relying on the feed of multiple spatio-temporally aligned IR-D sensors. Placing reflectors on and IR-D sensors around the subject, the body movements are fully captured, overcoming one-side view limitations such as partial occlusions or corrupted image data. The rationale behind using reflectors is the exploitation of the intense reflections they provoke to the IR streams of the IR-D sensors [11,12], enabling their detection on the depth images. FCN [13], instead of using computationally expensive fully connected layers, are applied on the multi-view IR-D captured data, resulting in reflector 2D localization and labeling. Spatially mapping and aligning the detected 2D points to 3D Cartesian coordinates with the use of depth data and intrinsic and extrinsic IR-D camera parameters, enables frame-based 3D optical data extraction. Finally, the subject's motion is captured by fitting an articulated template model to the sequentially extracted 3D optical data.
The main contributions of the proposed method are summarized as follows:
• A low-cost, robust and fast optical motion capture framework is introduced, using multiple IR-D sensors and retro-reflectors. Contrary to the gold-standard marker-based solutions, the proposed setup is flexible and simple, the required equipment is low-cost, the 3D optical data are automatically labeled and the motion capture is immediate, without the need for post-processing. •
To the best of our knowledge, DeepMoCap is the first approach that employs fully convolutional neural networks for automatic 3D optical data localization and identification based on IR-D imaging. This process, denoted as "2D reflector-set estimation", replaces the manual marker labeling and tracking tasks required in traditional optical MoCap. •
The convolutional pose machines (CPM) architecture proposed in [14] has been extended, inserting the notion of time by adding a second 3D optical flow input stream and using 2D Vector Fields [14] in a temporal manner. • A pair of datasets consisting of (i) multi-view colorized depth and 3D optical flow annotated images and (ii) spatio-temporally aligned multi-view depthmaps along with Kinect skeleton, inertial and ground truth MoCap [5] data, have been created and made publicly available (https: //vcl.iti.gr/deepmocap/dataset).
The remainder of this paper is organized as follows: Section 2 overviews related work; Section 3 explains in detail the proposed method for 2D reflector-set estimation and motion capture; Section 4 presents the published datasets; Section 5 gives and describes the experimental frameworks and results; finally, Section 6 concludes the paper and discusses future work.
Related Work
The motion capture research field consists of a large variety of research approaches. These approaches are marker-based or marker-less, while the input they are applied on is acquired using RGB or RGB-D IR/stereo cameras, optical motion capture or inertial among other sensors. Moreover, they result in single-or multi-person 2D/3D motion data outcome, performing in real-time, close to real-time or offline. The large variance of the MoCap methods resulting from the multiple potential combinations of the above led us to classify and discuss them according to an "Input-Output" perspective. At first, approaches that consume 2D data and yield 2D and 3D motion capture outcome are presented. These methods are highly relevant to the present work since, in a similar fashion, the proposed FCN approaches 2D reflector-set estimation by predicting heat maps for each reflector location and optical flow. Subsequently, 3D motion capture methods that acquire and process 2.5D or 3D data from multiple RGB-D cameras similarly to the proposed setup are discussed. Finally, methods that fuse RGB-D with inertial data for 3D motion capture are presented, including one of the methods that are compared against DeepMoCap in the experimental evaluation.
2D Input-2D Output: Intense research effort has been devoted to the 2D pose recovery task for MoCap, providing efficient methods being effective in challenging and "in-the-wild" datasets. Pose machines architectures for efficient articulated pose estimation [15] were recently introduced, employing implicit learning of long-range dependencies between image and multi-part cues. Later on, multi-stage pose machines were extended to CPM [14,16,17] by combining pose machine rationale and FCN, allowing for learning feature representations for both image and spatial context directly from image data. At each stage, the input image feature maps and the outcome given by the previous stage, i.e., confidence maps and 2D vector fields, are used as input refining the predictions over successive stages with intermediate supervision. Beyond discrete 2D pose recovery, 2D pose tracking approaches have been introduced imposing the sequential geometric consistency by capturing the temporal correlation among frames and handling severe image quality degradation (e.g., motion blur or occlusions). In [18], the authors extend CPM by incorporating a spatio-temporal relation model and proposing a new deep structured architecture. Allowing for end-to-end training of body part regressors and spatio-temporal relation models in a unified framework, this model improves generalization capabilities by spatio-temporally regularizing the learning process. Moreover, the optical flow computed for sequential frames is taken into account by introducing a flow warping layer that temporally propagates joint prediction heat maps. Luo et al. [19] also extend CPM to capture the spatio-temporal relation between sequential frames. The multi-stage FCN of CPM has been re-written as a Recurrent Neural Network (RNN), also adopting Long Short-Term Memory (LSTM) units between sequential frames, to effectively learn the temporal dependencies. This architecture, called LSTM Pose Machines, captures the geometric relations of the joints in time, increasing motion capture stability.
2D Input-3D Output: During the last years, computer vision researchers approach 3D pose recovery on single-view RGB data [20][21][22][23] for 3D MoCap. In [24], a MoCap framework is introduced, realizing 3D pose recovery, that consists of a synthesis between discriminative image-based and 3D pose reconstruction approaches. This framework combines image-based 2D part location estimates and model-based 3D pose reconstruction, so that they can benefit from each other. Furthermore, to improve the robustness of the approach against person detection errors, occlusions, and reconstruction ambiguities, temporal filtering is imposed on the 3D MoCap task. Similarly to CPM, 2D keypoint confidence maps representing the positional uncertainty are generated with a FCN. The generated maps are combined with a sparse model of 3D human pose within an Expectation-Maximization framework to recover the 3D motion data. In [25], a real-time method that estimates temporally consistent global 3D pose for MoCap from one single-view RGB video is presented, extending top performing single-view RGB convolutional neural network (CNN) methods for MoCap [20,23]. For best quality at real-time frame rates, a shallower variant is extended to a novel fully convolutional formulation, enabling higher accuracy in 2D and 3D pose regression. Moreover, CNN-based joint position regression is combined with an efficient optimization step for 3D skeleton fitting in a temporally stable way, yielding MoCap data. In [26], an existing "in-the-wild" dataset of images with 2D pose annotations is augmented by applying image-based synthesis and 3D MoCap data. To this end, a new synthetic dataset with a large number of new "in-the-wild" images is created, providing the corresponding 3D pose annotations. On top of that, the synthetic dataset is used to train an end-to-end CNN architecture for motion capture. The proposed CNN clusters a 3D pose to K pose classes on per-frame basis, while a K−way CNN classifier returns a distribution over probable pose classes.
2.5D / 3D Input-3D Output: With respect to 2.5D/3D data acquisition and 3D MoCap, particular reference should be made to the Microsoft Kinect sensor [27], beyond its discontinuation, since it was the first low-cost RGB-D sensor for depth estimation and 3D motion capture, leading to a massive release of MoCap approaches that use the Kinect streams or are compared to Kinect motion capture [28][29][30][31]. This sensor triggered the massive production of low-cost RGB-D cameras, allowing a wide community of researchers to study on RGB-D imaging and, subsequently, resulting in a plethora of efficient MoCap approaches applied on 2.5D/3D data [32][33][34][35][36]. In [37], a multi-view and real-time method for multi-person motion capture is presented. Similarly to the proposed setup, multiple spatially aligned RGB-D cameras are placed to the scene. Multi-person motion capture is achieved by fusing single-view 2D pose estimates from CPM, as proposed in [14,16], extending them to 3D by means of depth information. Shafaei et al. [38] use multiple externally calibrated RGB-D cameras for 3D MoCap, splitting the multi-view pose estimation task into (i) dense classification, (ii) view aggregation, and (iii) pose estimation steps. Applying recent image segmentation techniques to depth data and using curriculum learning, a CNN is trained on purely synthetic data. The body parts are accurately localized without requiring an explicit shape model or any other a priori knowledge. The body joint locations are then recovered by combining evidence from multiple views in real-time, treating the problem of pose estimation for MoCap as a linear regression. In [39], a template-based fitting to point-cloud motion capture method is proposed using multiple depth cameras to capture the full body motion data, overcoming self-occlusions. A skeleton model consisting of articulated ellipsoids equipped with spherical harmonics encoded displacement and normal functions is used to estimate the 3D pose of the subject. Inertial (+2.5D) Input-3D Output: Inertial data [40][41][42][43], as well as their fusion with 2.5D data from RGB-D cameras, are also used to capture the human motion. In [44], Kinect for Xbox One [27] skeleton tracking is fused with inertial data for motion capture. In particular, inertial sensors are placed on the limbs and the torso of the subject to provide body bone rotational information by applying orientation filtering on inertial data. Initially, using Kinect, the lengths of the bones and the rotational offset between the Kinect and inertial sensors coordinate systems are estimated. Then, the bones hierarchically follow the inertial sensor rotational movements, while the Kinect camera provides the root 3D position. In a similar vain, a light-weight, robust method [45] for real time motion and performance capture is introduced using one single depth camera and inertial sensors. Considering that body movements follow articulated structures, this approach captures the motion by constructing an energy function to constrain the orientations of the bones using the orientation measurements of their corresponding inertial sensors. In [46], inertial motion capture is achieved on the basis of a very sparse inertial sensor setup, i.e., two units placed on the wrists and one on the lower trunk, and ground contact information. Detecting and identifying ground contact from the lower trunk sensor signals and combining this information with a fast database look-up enables data-driven motion reconstruction.
Despite the appearance of the aforementioned methods, traditional marker-based optical MoCap still remains the top option for robust and efficient motion capture. That is due to the stability of the marker-based optical data extraction and the deterministic way of motion tracking. To this end, the proposed method approaches marker-based optical motion capture, however overcoming restrictions of traditional marker-based optical MoCap solutions by:
• using off-the-shelf retro-reflective straps and patches to replace the spherical retro-reflective markers, which are sensitive due to potential falling off;
• automatically localizing and labeling the reflectors on a per-frame basis without the need for manual marker labeling and tracking; • extracting the 3D optical data by means of the IR-D sensor depth.
Taking the above into consideration, DeepMoCap constitutes an alternative, low-cost and flexible marker-based optical motion capture method that results in high quality and robust motion capture outcome.
Proposed Motion Capture Method
DeepMoCap constitutes an online, close to real-time marker-based approach that consumes multi-view IR-D data and results in single-person 3D motion capture. The pipeline of the proposed method, depicted in Figure 1, is summarized as follows: (1), spatio-temporally aligned IR-D streams are acquired and processed (2) to feed the FCN with colorized depth and 3D optical flow images (3). The FCN outcome, i.e., the multi-view 2D reflector-set estimates, is fused to extract the 3D optical data (4) and, finally, yield the subject's 3D pose for motion capture (5).
1. A set of retro-reflective straps and patches is placed on the subject's body. 2. Placing multiple calibrated and synchronized IR-D sensors around the subject to fully capture the body movements, IR-D raw data are acquired and processed, giving multi-view pairs of colorized depth and 3D optical flow. 3. Each pair is fed to a FCN model, resulting in 2D reflector-set estimation per view. 4. The reflector-set estimates are spatially mapped in 3D space using the depth data and the intrinsic and extrinsic calibration camera parameters. The resulting 3D point sets are fused, resulting in 3D optical data outcome.
5.
A template-based articulated structure is registered and fitted to the subject's body. 3D motion capture is achieved by applying forward kinematics to this structure based on the extracted 3D optical data.
The pipeline steps are accordingly described in the following sections.
Reflector-Set Placement
The reflector-set placement has been designed to provide robust and highly informative motion capture data, i.e., capturing large number of degrees of freedom (DoFs). The selected placement is shown in Figure 2, consisting of a set of 26 reflectors R i ∈ {R 1 , . . . , R 26 }, 16 patches and 10 straps, enabling motion capture by fitting an articulated body structure of 40 DoFs. The use of both straps and patches has been chosen due to the fact that the straps are 360°-visible on cylindrical body parts (i.e., limbs), while patches have been used on the body parts where strap placement is not feasible, i.e., torso, head and hands. Aiming to highlight the distinction between the front and the back side of the body, the reflective patches are not symmetrically placed. On the front side, two reflector patches are placed on the head, two on the chest and one on the spine middle, while on the back side, one is placed on the head, one on the back and one on the spine middle. The retro-reflective material used to create the reflector-set is the off-the-shelf 2-inch reflective tape used in protective clothing [47]. Following carefully the matching between the reflectors and the body-parts of the subject as depicted in Figure 2, the reflector-set placement is a fast procedure (it lasts approximately 2 min), since the sticky straps and patches are effortlessly placed, not requiring high placement precision (it is enough to be approximately placed to the body part locations shown in Figure 2).
Raw Data Acquisition and Processing
After the reflector-set placement, the subject is ready to be captured. Let us consider the use of N IR-D devices, thus, N is also the number of views v ∈ {1, . . . , N}. Using the multi-Kinect for Xbox One capturing setup proposed in [48], spatio-temporally aligned multi-view IR-D data are acquired. All reflector regions have distinguishable pixel values on IR images I v IR (Figure 3a), thus, applying binary hard-thresholding, the binary mask I v IR m of the reflectors is extracted (Figure 3b). The corresponding regions on the raw depth images I v D (Figure 3c) have zero values due to the retro-reflections. The multi-view IR-D raw data are processed before feeding the FCN. On the one hand, the IR-D frames are jointly processed in order to compute the 3D optical flow I v F of the body movements. In the present work, the primal-dual algorithm proposed in [49] is considered due to its demonstrated efficiency on relevant computer vision tasks, such as interaction-based RGB-D object recognition [50] and 3D action recognition [51]. In particular, the 3D motion vectors between two pairs of IR-D images and their magnitude are computed. The 3D flow and its magnitude are then colorized by normalizing each axis values and transforming the 3D motion vectors into a three-channel image. On the other hand, the depth images I v D are colorized applying JET color map conversion. Finally, the reflector mask I v IR m is subtracted from the colorized depth images, resulting in colorized depth with reflector black regions, I v CD , facilitating the detection of the reflectors. The colorization step for both streams is required in order to allow the usage of the proposed FCN, initialized by the first 10 layers of VGG-19 [52]. An example of the processed multi-view outcome is shown in Figure 4.
2D Reflector-Set Estimation Using FCN
The major challenge of the proposed method is the efficient localization and identification task of the reflectors placed on the subject's body. Studying the recent literature in 2D localization on RGB images, the efficiency of deep neural networks in complex tasks such as articulated 2D pose estimation is remarkable and, therefore, considered appropriate for the present challenge. To this end, a deep learning approach is introduced extending the multi-stage CPM architecture in order to localize and identify the reflectors on the body. In particular, a multi-stage fully convolutional network is trained to directly operate on intermediate confidence maps and optical flow 2D vector fields, instead of Part Affinity Fields (PAF) between 2D keypoints, implicitly learning image-dependent spatial models of the reflectors locations among sequential frames.
Despite the similarities between 2D pose and reflector-set estimation, both being solvers of 2D localization problems on color images, there exist noteworthy differences. Cao et al. [14] efficiently address the problem of 2D pose estimation in large, "in-the-wild" RGB datasets [53], resulting in accurate estimates in a variety of data showing multiple people in different environmental and lightning conditions. In contrast, the reflector-set estimation is applied in more "controlled" conditions; (i) the input depth data lie within a narrow range, (ii) the reflector regions have clearly distinguishable pixel values on IR images, (iii) there is only one subject to be captured and, in most cases, (iv) the subject is acting at the center of the scene. On the other hand, the reflector-set estimation task is more complicated with respect to (i) the estimation of a larger number of reflectors in comparison with the keypoints detected by CPM approaches and (ii) the fact that the reflector patches are one-side visible. For instance, the reflectors R 15 and R 14 are both placed on the right shoulder, but on the front and the back side of the body respectively, while the right shoulder keypoint in CPM 2D poses is unique for all views.
The overall FCN method is illustrated in Figure 5. A pair of images, the colorized depth I v CD and the corresponding 3D optical flow I v F , are given as input. A FCN simultaneously predicts a set of 2D confidence maps S of the reflector locations and a set of 2D vector fields L; the latter corresponds to the optical flow fields (OFFs) from the previous frame to the next one, encoding the temporal correlation between sequential frames. Both sets contain R = 26 elements, one per reflector
R i ∈ {R 1 , . . . , R 26 }, the set S = (S R 1 , S R 2 , ..., S R 26 ), S R i ∈ R w×h ,
where w and h are the width and height of the input images respectively, and the set
L = (L R 1 , L R 2 , . . . , L R 26 ), where L R i ∈ R 2×w×h .
Finally, a greedy inference step is applied on the extracted confidence maps and OFFs, resulting in 2D reflector-set estimation. The FCN architecture, shown in Figure 6, introduces a new two-stream, two-branch, multi-stage CPM-based approach which consumes colorized depth I v CD and 3D optical flow I v F images. Both input streams are separately processed by a convolutional network of 10 layers (first 10 layers of VGG-19 [52]), generating two sets of feature maps F D and F OF , correspondingly. Sequentially, an early stage fusion takes place, concatenating the feature maps of the two streams, F = F OF ⊕ F D . Let us denote t the stage of the network. At the first stage (t = 1), the fused feature set F is given to both branches producing confidence maps, S t = ρ t (F), and 2D vector fields, L t = ϕ t (F), where ρ t and ϕ t denote the inference of each FCN branch. For all the subsequent T − 1 stages, where T denotes the total number of stages, the predictions from both branches in the previous stage, along with the features set F, are fused and used to produce refined predictions based on:
S t = ρ t (F, S t−1 , L t−1 ), t ∈ {2, . . . , T} L t = ϕ t (F, S t−1 , L t−1 ), t ∈ {2, . . . , T}(1)
where the number of stages T is equal to 6, experimentally set by evaluating the results on the validation dataset for T = 3 and T = 6 stages, as proposed in [14,16], respectively. At the end of each stage, two L 2 loss functions, L t S and L t L , between the predictions and the ground truth are applied to guide the network branches to predict confidence maps and OFFs, respectively. At stage t, for a 2D location p = (x, y), p ∈ R 2 , the loss functions are given by:
L t S = R ∑ r=1 ∑ p ||S t r (p) − S * r (p)|| 2 2 L t L = R ∑ r=1 ∑ p ||L t r (p) − L * r (p)|| 2 2(2)
In that way, the vanishing gradient problem is addressed by the intermediate supervision at each stage, replenishing the gradient periodically. The overall loss function L of the network is given by: (4)) and vector fields L * R i (Equation (5)), respectively. Confidence maps: Each confidence map is a 2D representation of the belief that a reflector occurs at each pixel location. The proposed method performs single person motion capture, therefore, a single peak should exist in each confidence map. Let x R i , f ∈ R 2 be the ground truth 2D location of the reflector R i on the image, at frame f . For every 2D location p ∈ R 2 , the ground truth value of S * R i , f is given by:
L = T ∑ t=1 (L t S + L t L ) (3)S * R i , f (p) = exp(− ||p − x R i , f || 2 2 σ 2 )(4)
where σ controls the spread of the peak. At test time, non-maximum suppression is applied on the predicted confidence maps to localize the reflectors, assigning the confidence map peak value to each reflector prediction confidence, E S R i , f . Optical Flow Fields: In this work, the feature representation of 2D vector fields proposed in [14], is used in a temporal manner. Preserving both 2D location and orientation information across a region, a 2D vector field for each reflector is defined by connecting the reflector 2D locations between f − 1 and f frames. Let x R i , f −1 , x R i , f ∈ R 2 be the ground truth 2D locations of the reflector R i at frame f − 1 and f , respectively. The ground truth value for every 2D location p ∈ R 2 of L * R i , f is given by:
L * R i , f (p) = v, i f p on optical f low f ield 0, otherwise v = x R i , f − x R i , f −1 ||x R i , f − x R i , f −1 || 2 (5)
The set of points that belong to the optical flow field includes the points within a distance threshold from the line segment between the reflector 2D locations, given by:
0 ≤ v · (p − x R i , f −1 ) ≤ d R i |v ⊥ · (p − x R i , f −1 )| ≤ σ R i (6)
where σ R i is the width of the field in pixels, d R i is the euclidean distance of the R i reflector 2D locations between sequential frames in pixels, i.e.,
d R i = ||x R i ,t − x R i ,t−1 || 2 ,
and v ⊥ is a vector perpendicular to v. As an example, the 2D optical flow field for the reflector R 13 is illustrated in Figure 5. During inference, the optical flow, encoding the temporal correlations, is measured by computing the line integral over the corresponding optical flow field along the line segment connecting the candidate reflector locations between two sequential frames. Let r R i , f and r R i , f −1 be the predicted locations for the reflector R i at the current frame f and the previous one f − 1, correspondingly. The predicted optical flow field L R i , f is sampled along the line segment to measure the temporal correlation confidence between the predicted reflector positions in time by:
E L R i , f = 1 0 L R i , f (p(u)) · r R i , f − r R i , f −1 ||r R i , f − r R i , f −1 || 2 du (7)
where p(u) interpolates the reflector positions r R i , f and r R i , f −1 between sequential frames, as given by:
p(u) = (1 − u) · r R i , f −1 + u · r R i , f(8)
In other words, the integral is approximated by sampling and summing uniformly spaced values of u.
Greedy Inference: Finally, a greedy inference step is introduced, taking into consideration the temporal correlations between temporally sequential 2D reflector estimates. The confidence values E S R i , f and E L R i , f given by the confidence maps and the optical flow fields correspondingly, are summarized in a weighted manner, in order to give the fused confidence E R i , f for each reflector estimate. In detail, the major component of
E R i , f is E S R i , f , however, we weight the confidence E L R i , f based on a w L R i , f = (1 − E S R i , f )
factor that increases when E S R i , f decreases as:
E R i , f = E S R i , f + w L R i , f · E L R i , f(9)
In that way, when a confidence map prediction results in low confidence E S R i , f estimates, the total confidence E R i , f is strongly affected by the optical flow confidence, if high. The final outcome of the 2D reflector-set estimation is given by applying non-maximum suppression on the reflector predictions based on the total confidence E R i , f .
3D Optical Data
2D-to-3D Spatial Mapping
Given the reflector-set 2D locations on the depth image I v D , a 3D spatial mapping technique is applied to precisely extract the corresponding 3D positions. Considering that the reflector locations are given exclusively when the reflectors are clearly visible, a reflector estimate is considered valid only if it belongs to a region of more than b min black pixels in I v CD , otherwise it is dropped. The minimum accepted size in pixels for a region was set b min = 5, since the same region size was used for the annotation of the data.
In Figure 7, two of the potential cases with respect to the reflector spatial mapping are shown. In the first case (Figure 7a), the simple and most common one, E 0 ∈ I v CD is the detected region for a reflector R i ∈ {R 1 , . . . , R 26 }. Retrieving a pixel set P v R i of the E 0 region contour in I v CD (red pixels in Figure 7a), and mapping its points to I v D , the corresponding raw depth values of P v R i are given. Using only the non-zero depth values of P v R i , the median value d R i is considered the distance of the reflector R i from the sensor view v. The second case is illustrated in Figure 7b, where two or more (although not usual) reflectors belong to the same region E 1 . Examining the overlapping between the reflector areas, i.e., when n > 1 reflectors share the same black region, the pixels of the contour are clustered in n clusters, based on the 2D pixel coordinates and the depth values, and then mapped to the corresponding reflectors. Subsequently, d R i is determined for each reflector R i based on the clustered pixel set.
After one-to-one mapping between reflectors and regions, the central 2D point p R i of each E R i region is spatially mapped to 3D coordinates using the depth distance value d R i and the intrinsic parameters of the corresponding IR-D sensor, giving the 3D position P v R i of the reflector R i from viewpoint v.
3D Point Sets Fusion
Using the extrinsic calibration parameters of each sensor, the extracted 3D positions are spatially aligned to a global 3D-coordinate system, as shown in Figure 8. For the reflective patches, which are one-side visible to the sensors, the same retro-reflective region is captured by all IR-D sensors and, therefore, the 3D mapping yields slightly different estimates. To this end, the 3D points P v R i for a patch reflector R i for all views v are fused to one single 3D point P R i , taking into account the confidence value E v R i of the FCN reflector estimation. The 3D point P R i is estimated as the weighted central position of the 3D points from all views by: For the reflective straps, since different parts of the reflectors are visible to each sensor, the 3D points are estimated in different 3D locations around the part of the limb where the strap is placed on. In this case, the desired 3D point is approximately located to the center of the "circle" where these 3D points lie on, extracted by the method presented in Appendix A.
w v R i = E v R i / N ∑ v=1 E v R i P R i = 1 N N ∑ r=1 w R i P v R i(10)
The full set of the global retro-reflector 3D positions per multi-view group frame f , i.e., the extracted 3D optical data, is denoted as P f , while a total confidence value C R i f for each R i reflector is considered as the average value of the reflector estimation confidence
E v R i for each v−view, v ∈ {1, . . . .N}, C R i f = 1 N ∑ N v=1 E v R i .
To refine the quality and stability of the 3D point estimates in time, Kalman filtering [54] is applied to the extracted 3D optical data.
Motion Capture
The final stage of the proposed method targets at 3D motion capture based on 3D optical data. At this point, the extracted 3D optical data P f are mapped to a relative motion representation consisting of joint 3D positions and orientations. Similarly to [55,56], a body structure calibration technique is proposed, adapting an articulated humanoid template model to the real body structure of the subject. Subsequently, the calibrated articulated body is jointly moved by applying forward kinematics.
The proposed articulated body structure consists of 20 joints, j ∈ J, including D DoF = 40 DoFs. It contains L i ∈ L, i ∈ {0, . . . , 6}, hierarchical joint levels and the bones of the structure are registered to particular reflectors. To this end, a reflector subset R S j ⊂ {R 1 , . . . , R 26 }, j ∈ J, moves the body joint j ∈ J according to the body joint hierarchy. The correspondence between the joints and the reflectors is depicted in Figure 9a, while the reflector-to-body part mapping is illustrated in Figure 9b.
Initially, the template is coarsely scaled based on the first batch of optical data frames P f . Then, given the 3D positions P j , j ∈ J per frame, a per-bone alignment process is performed to precisely fit the body parts of the template to the subject's real body lengths. This step is sequentially performed to the bones following the joint hierarchy levels, from L 0 to L 6 . More specifically, after fitting the template to the optical data, the body root 3D position P H IPS with hierarchical level L 0 is given, enabling the sequential estimation of the rest of the bone lengths. Based on the assumption that the bone lengths are constant (rigid bones) and using exclusively high-confident (C R i f > 0.6, experimentally set) optical data P f , a random particle generation step is applied per level in L − 1 phases. More specifically, a set G j of G = 500 particles (experimentally set) is generated around the j-joint location P j given by the spatially aligned template placement using P f . After the particle generation, the G j particles relatively follow the optical data applying forward kinematics. The particle g j ∈ G j that moves more rigidly between P j l and P j l+1 , is considered as the closest one to the real relative position of joint j l+1 . The objective function that estimates this particle is given by:
D 0 = ||P j l ,0 − P g j l+1 ,0 || 2 + ||P j l+1,0 − P g j l+1 ,0 || 2 arg min g j l+1 D(g j l+1 ) = 1 F F ∑ f =1 (D 0 − ||P j l , f − P g j l+1 , f || 2 + ||P j l+1 , f − P g j l+1 , f || 2 )(11)
where D 0 denotes the sum of the 3D euclidean distances at the initial frame of the (l) − (l + 1) level alignment between the 3D position of the particle g j l+1 and the joints j l and j l+1 , as given by the latest template fitting, P g j l+1 , f denotes the 3D position of the particle g j l+1 at frame f ∈ {1, . . . , F}, where F is the total number for a window of frames used to align a body part of level l + 1. In a similar fashion, the next level joints and the corresponding bones are calibrated. Angular body part movements, especially elbow and knee flexions, enable faster and more efficient convergence of the per-bone alignment process.
Body Joint, j ∈ J Level L i DoFs Subset R S j S j (#) Retro-reflectors Hips L 0 6 R S 0 4 {R 1 , R 8 , R 19 , R 23 } Spinebase L 1 3 R S 1 2 {R 1 , R 8 } Neck L 2 3 R S 2 3 {R 2 , R 3 , R 7 } Head L 3 - R S 3 3 {R 4 , R 5 , R 6 } Left Shoulder L 3 3 R S 4 2 {R 9 , R 10 } Left Elbow L 4 1 R S 5 1 {R 11 } Left Wrist L 5 3 R S 6 1 {R 12 } Left Hand L 6 - R S 7 1 {R 13 } Right Shoulder L 3 3 R S 8 2 {R 14 , R 15 } Right Elbow L 4 1 R S 9 1 {R 16 } Right Wrist L 5 3 R S 10 1 {R 17 } Right Hand L 6 - R S 11 1 {R 18 } Left Hip L 1 3 R S 12 1 {R 19 } Left Knee L 2 1 R S 13 1 {R 20 } Left Ankle L 3 3 R S 14 1 {R 21 } Left Foot L 4 - R S 15 1 {R 22 } Right Hip L 1 3 R S 16 1 {R 23 } Right Knee L 2 1 R S 17 1 {R 24 } Right Ankle L 3 3 R S 18 1 {R 25 } Right Foot L 4 - R S 19 1 {R 26 } (a) (b)
Evaluation Datasets
Two public datasets have been created (https://vcl.iti.gr/deepmocap/dataset) consisting of subjects wearing retro-reflectors on their bodies. These datasets are exploited for: (i) motion capture evaluation in comparison with recent MoCap methods and ground truth and (ii) 2D reflector-set estimation FCN training and testing.
DMC3D Dataset
The DMC3D dataset consists of multi-view depth and skeleton data as well as inertial and ground truth motion capture data. Specifically, 3 Kinect for Xbox One sensors were used to capture the depth and Kinect skeleton data along with 9 XSens MT [57] inertial measurement units (IMU) to enable the comparison between the proposed method and inertial MoCap approaches based on [44]. Furthermore, a PhaseSpace Impulse X2 [5] solution was used to capture ground truth MoCap data. PhaseSpace Impulse X2 is an optical marker-based MoCap system considered appropriate for the scope of this dataset due to the usage of active instead of passive retro-reflective markers that would have interfered with the retro-reflectors. The preparation of the DMC3D dataset required the spatio-temporal alignment of the modalities (Kinect, PhaseSpace, XSens MTs). The setup [48] used for the Kinect recordings provides spatio-temporally aligned depth and skeleton frames. For the Kinect -IMU synchronization, a global clock was used to record depth and inertial data with common timestamps. Additionally, given the timestamps and using the robust recording frequency of PhaseSpace Impulse X2 as reference clock, the spatio-temporal alignment of the ground truth data was manually achieved.
With respect to the amount and the variety of data, 10 subjects, 2 females and 8 males, wearing retro-reflectors, inertial sensors and active markers (LEDs) on the body, were recorded performing 15 physical exercises, presented in Table 1. The full dataset contains more than 80 × 10 3 three-view depth and skeleton frames, the extrinsic calibration parameters and the corresponding inertial and MoCap data.
DMC2.5D Dataset
A second set comprising 2.5D data (DMC2.5D Dataset) was captured in order to train and test the proposed FCN. Using the recorded IR-D and MoCap data, colorized depth and 3D optical flow data pairs per view were created, as described in Section 3.2. The samples were randomly selected from 8 of the 10 subjects, excluding 2 of them to use them for the evaluation of the MoCap. More specifically, 25 × 10 3 single-view pair samples were annotated with over 300 × 10 3 total keypoints (i.e., reflector 2D locations of current and previous frames on the image), trying to cover a variety of poses and movements in the scene. 20 × 10 3 , 3 × 10 3 and 2 × 10 3 samples were used for training, validation and testing the FCN model, respectively. The annotation was realized by applying image processing and 3D vision techniques on the IR-D and MoCap data. In particular, applying blob detection on the IR binary image I v IR m yielded the 2D locations of the reflectors, while then, the corresponding 3D positions were estimated by applying 3D spatially mapping (Section 3.4.1). Finally, the reflectors were labeled by comparing the euclidean 3D distance per frame between the extracted 3D positions and the joint 3D positions of the ground truth data. However, the automatic labeling was erroneous in the cases that the reflector regions were merged (Figure 10) or the poses where complex. The complex pose issues occurred due to the positional offset between the reflector and the joint 3D positions, confusing the labeling process. Thus, further processing was required in order to manually refine the dataset.
Evaluation
For the evaluation of the proposed method, two types of experiments were conducted, presented and discussed in this section. The first experiment concerns the evaluation of the 2D reflector-set estimation on the DMC2.5D dataset, while, in the second one, the motion capture results are compared against robust MoCap solutions and ground truth on the DMC3D dataset. At first, FCN architectures are evaluated, highlighting the outperformance of the proposed FCN model. Accurate 2D reflector-set estimation eliminates the errors in 3D optical data extraction and, consequently, to the final motion capture outcome. Thus, afterwards, applying the proposed FCN model to the DMC3D testing dataset, 3D optical data from multi-view sequences are extracted and accordingly used for motion capture.
2D Reflector-Set Estimation on DMC2.5D
FCN Implementation
With respect to the training of the proposed FCN, data augmentation takes place randomly to increase variation of input by scaling and rotating the data. For each batch of frames being fed into the network per iteration, the transformation is consistent, thus, lower batch size results in higher variation between the iterations. Both input images are randomly scaled by f s ∈ [0.6, 1.1], randomly rotated by f θ ∈ [−40°, 40°] and flipped with binary randomness. Finally, all images are cropped to 368 × 368 resolution size, also setting the subject bodies at the image center. Regarding the method parameterization, the stages are equal to T = 6, using stochastic gradient descent with momentum α m = 0.9 and weight decay w d = 5 × 10 −4 to optimize the learning process. The batch size is equal to 10, while the initial learning rate is l r = 2 × 10 −5 and drops by f g = 3.33 × 10 −1 every 30 × 10 3 iterations.
Experimental Framework
The introduced FCN architecture approaches 2D-reflector-set estimation, a similar, yet different task in comparison with 2D pose estimation. Aiming to evaluate the present FCN approach and the introduced extension with respect to the temporal correlations between the reflector 2D positions from frame-to-frame, existing methods for keypoint are adapted and trained to address the present challenge. In detail, the methods by Wei et al. [16] and Cao et al. [14], included in OpenPose (https: //github.com/CMU-Perceptual-Computing-Lab/openpose) library, have been adapted and trained with the DMC2.5D dataset.
For the adaptation, the body parts have been replaced by the reflectors, while it is worth noting that the Part Affinity Fields (PAFs) have been altered due to the difference of the reflector sub-set placement between the front and the back side of the body. The adapted association between the reflectors is illustrated in Figure 11. Moreover, since the proposed method has been developed for single-person motion capture, even though PAFs branch contributes to the final feature space for the confidence map prediction, its output is not taken into account for the final reflector-set estimation.
Finally, a two-branch colorized depth and 3D optical flow two-stream approach similar to [14] is evaluated ([14] + 3D OF), showing remarkable results. Figure 11. The red arrows illustrate the directional associations between the reflectors to adapt the Part Affinity Fields, as proposed in [14]. The orange and blue colored labels indicate the reflective straps and patches, respectively.
With respect to the evaluation metrics, the proposed FCN is evaluated on the DMC2.5D dataset measuring Average Precision (AP) per reflector and mean Average Precision (mAP) for the entire set, based on Percentage of Correct Keypoints (PCK) [58] thresholds, i.e., a prediction is considered true if it falls within a pixel region around the ground-truth keypoint. This region is defined by multiplying the width and height of the bounding box that contains the subject by a factor α that controls the relative threshold for correctness consideration.
Setting α = 0.05, the validation set of the DMC2.5D dataset was used to indicate the optimum minimum confidence threshold c min for the highest mAP per method, aiming at fair comparison between them. c min corresponds to the minimum threshold of confidence for a reflector prediction to be considered as valid, i.e., E R i , f > c min . The results are presented in Figure 12, showing the method mAP against confidence threshold. Maximum mAP on the validation set was achieved for c min = 0.4 for all methods, thus, considered optimum for the experiments on the DMC2.5D testing set.
Results and Discussion
Evaluating the AP results per reflector, shown in Table 2, the efficiency of the methods in reflector-set estimation is perceived. The proposed FCN method outperforms the rest of the methods for the 80.7% of the reflectors (i.e., 21 out of 26). In particular, the AP is improved for the end-reflectors, i.e., the reflectors placed on the hands and the feet (R 13 , R 18 , R 22 , R 26 ), and their linked reflectors, i.e., the wrists and the ankles (R 12 , R 17 , R 21 , R 25 ), which are placed on the body parts with the highest moving capability and, therefore, the most rapid movements. From these results, we conclude that the temporal information implicitly encoded in the proposed FCN model improves the distinction among these reflectors, while for the reflectors that the AP is slightly lower (R 8 , R 10 , R 13 , R 14 , R 20 ), we can assume that the predicted optical flow was not accurate or informative enough to boost the prediction confidence and, therefore, the accuracy of the estimates. However, for all the respective methods, the prediction of the end-reflectors is weak in comparison with the rest of the reflectors. That is probably due to the fact that these reflectors are not often visible and are closely placed to their linked reflectors. To highlight this difference, mAP results are presented in Table 3 with and without the end-reflectors. Table 3. mAP for PCK with α = 0.05, with and without end-reflectors.
Method
Total Total (without End-Reflectors) [16] 92.16% 95.27% [14] 92 The proposed approach outperforms the competitive methods, presenting an absolute increase of mAP equal to 1.47%, 0.94% and 0.89% with end-reflectors, and 1.5%, 1.16% and 0.9% without them, in comparison with [14,16] and [14] + 3D OF, respectively. It is worth noting that the two-stream approach which takes into account the 3D optical flow ([14] + 3D OF) achieves higher mAP than [14,16], meaning that the temporal information given by the 3D optical flow stream is encoded in the feature space of the model, resulting in higher localization accuracy.
Finally, before feeding the motion capture method with the 2D reflector-set estimates, a filtering process is applied based on two fundamental considerations of the task. At first, the reflectors are detected only when visible; (i) if the region where a detector is located does not belong to a specific color (black) region of size greater than or equal to b min = 5 pixels, this estimate is discarded, and (ii) when more than one reflectors are detected on the same location (absolute distance less than 3 pixels, experimentally set), the reflectors with lower confidence are dropped. Secondly, the reflectors are unique on an image since we approach single-person MoCap; if more than one reflector estimates of the same reflector are detected, only the one with the highest confidence is considered valid.
The AP results per reflector after filtering are shown in Table 4. As shown, the results for all reflectors and for all methods are equal or greater than the corresponding results before filtering. At this experiment, the proposed FCN outperforms the rest of the methods at 14 of 26 (53.84%) of the reflectors. The mAP results for the total set of reflectors after filtering, with and without the end-reflectors, are presented in Table 5, all showing higher accuracy than the corresponding values before filtering. The proposed method outperforms [14,16] and [14] + 3D OF by presenting an absolute increment equal to 1.25%, 0.49% and 0.37% with end-reflectors, and 1.06%, 0.47% and 0.61% without them, respectively. Table 5. mAP for PCK with α = 0.05, with and without end-reflectors, after filtering.
Method
Total Total (without End-Reflectors) [16] 93.57% 96.41% [14] 94 In that way, the outcome of the 2D reflector-set estimation allows us to detect the 2D locations of the reflectors and, subsequently, the corresponding 3D optical data in a global coordinate system. Qualitative results of the proposed FCN outcome on sequential input data are illustrated in Figure 13, while more qualitative results have been made publicly available (https://vcl.iti.gr/deepmocap). Figure 13. Visualization of the proposed FCN outcome overlayed on sequential multi-view depth frames. Five multi-view sequential frames, from frame f − 10 to f + 10 with frame step equal to 5, are horizontally presented.
Motion Capture on DMC3D
Experimental Framework
After the introduction of an efficient FCN model for 2D reflector-set estimation, the final MoCap outcome is evaluated. Applying the proposed qualified FCN model to multi-view sequences, the 3D optical data are extracted and fed to the MoCap proposed method. For these experiments, several sequences of approximately 6 × 10 3 frames in total were selected from 2 subjects that were excluded from the dataset used to train the FCN model. Considering the ground truth data of the DCM3D dataset, i.e., the motion data captured with PhaseSpace Impulse X2 [5], the motion capture outcome is compared against the Kinect for Xbox One skeleton tracking with the highest quality, an inertial MoCap approach that fuses Kinect skeleton and inertial data from 9 IMU (Fusion) [44], and a second robust inertial method in a similar fashion as [44] (Fusion++) that fuses ground truth data for initialization and root positioning instead of Kinect skeleton tracking. It is worth noting that jerky skeleton estimates of Kinect skeleton tracking that cause highly erroneous root 3D position estimates have been excluded from the testing sequences, keeping only estimates meaningful for comparison.
Inertial MoCap methods were considered appropriate for a fair comparison due to their robust way of capturing, independent from self-occlusions. Multiple RGB-D-or 3D-based MoCap approaches were not taken into account due to the missing parts of depth and, therefore, missing 3D data on the body parts of the subject where the reflectors were placed on, resulting in unfair comparison. Finally, 3D MoCap methods from monocular RGB considered out of scope due to one-view and less-informative input, while motion capture methods from multiple RGB sources were not considered equal for comparison due to the blurry images of RGB streams on rapid movements.
Regarding the evaluation metrics, DeepMoCap is evaluated on the DMC3D dataset using Mean Average Error (MAE), Root Mean Squared Error (RMSE) and 3D PCK @ a 3D = 20 cm metrics for the 3D euclidean distance between the outcome of the methods and the ground truth on 12 joints spanned by shoulders, elbows, wrists, hips, knees and ankles. In 3D PCK, an estimate is considered correct when the 3D euclidean distance is less than a 3D .
Results and Discussion
The total results of the comparison between the MoCap methods are presented in Table 6, showing the outperformance of DeepMoCap in comparison with the rest of the methods. The total MAE, RMSE and 3D PCK for all sequences are 9.02 cm, 10.06 cm and 92.25%, respectively, showing the best results of all experimental methods. Fusion++ [44], Fusion [44] and Best Kinect [27] follow the proposed method presenting 88.75%, 85.93% and 83.37% in 3D PCK accuracy, respectively. Additionally, it is worth mentioning that the proposed method presents lower than 10 cm total mean average error (MAE). In Table 7, the 3D PCK accuracy results are shown per exercise, giving evidence with respect to the strengths and the weaknesses of the methods based on the body movement type variations. DeepMoCap is qualified outperforming the rest of the methods at 12 of the 15 exercises in total (80%), efficiently capturing most of them. In detail, Walking on the spot, a simple and slow motion, is efficiently captured from all the methods. Elbow and Knee flexion exercises, which are simple rotational joint movements, are captured with high precision by all methods, especially from inertial MoCap approaches. It is worthnoting that DeepMoCap presents lower accuracy for Side-steps exercise than Fusion++ probably due to the hand placement on the reflectors of the hips resulting in merged reflector regions, which complicated the detection and identification of the involved reflectors. However, in more complex exercises as Butt kicks left-right and Forward lunge left-right where there are occlusions for Kinect and body stretching for inertial sensors placed on the body, DeepMoCap presents approximately 5% higher absolute 3D PCK accuracy than Fusion++, which follows. For Jumping jacks, which is a fast and complex exercise where all body parts are fully involved, DeepMoCap achieves 96.05% 3D PCK accuracy followed by Best Kinect [27], while inertial MoCap approaches fail to properly capture the shoulders 3D positions due to rigid body movement of the torso, showing lower accuracy in such exercises. For Alternate side reaches and Kick-box kicking, which are the most challenging exercises, the 3D PCK accuracy of the proposed method is 4.84% and 8.33% higher in comparison with the second best method (Fusion++), respectively. Furthermore, it should be noted that all exercises are captured by DeepMoCap in 3D PCK accuracy higher than 82.53%, showing low variation between different types of body movements. In the plot presented in Figure 14, the total 3D PCK accuracy is given against a 3D threshold values. DeepMoCap shows higher efficiency for all a 3D , showing high 3D PCK accuracy from low threshold values (e.g., 32.25% and 63.27% for α 3D = 5 cm and α 3D = 10 cm, respectively), in comparison with the next best method that presents 16.38% and 54.36%, respectively. Given the fact that joint positioning varies between different motion capture approaches resulting in the existence of a positional offset between the estimated 3D joint positions, we conclude that DeepMoCap shows high efficiency by presenting 32.25% of the estimates on average to be closer than 5 cm from ground truth.
In Table 8, the euclidean MAE and RMSE are presented per joint. It can be observed that the proposed method has the lowest errors for 9 of 12 (75%) joints for MAE and RMSE. For Shoulders and Right Elbow, Fusion++ [44] shows slightly better results than DeepMoCap probably due to better skeleton structure positioning. The lower body joints (hips, knees and feet) are captured presenting 6.05 cm and 7.08 cm mean average and root mean squared errors, respectively. 3D PCK accuracy against α 3D threshold in cm.
Best Kinect [27] Fusion [44] Fusion++ [44] Proposed Figure 14. Comparative evaluation of the motion capture methods using total 3D PCK results in different α 3D threshold values in cm. Qualitative results depicting the 3D outcome of the proposed approach are presented in Figure 15. In particular, the multi-view input overlayed with the reflector-set estimates and the corresponding 3D motion capture along with optical data results are illustrated. As shown, DeepMoCap approaches motion capture similarly to the way traditional optical MoCap solutions work, in a more flexible and low-cost manner though. More qualitative results are publicly available (https://vcl.iti.gr/ deepmocap).
Performance Analysis
The runtime performance analysis was conducted by measuring the total time required to capture the motion data from the testing sequences. For approximately 6 × 10 3 three-view pairs of colorized depth and 3D optical flow frames, thus 18 × 10 3 single-view pairs, raw data pre-processing lasted 1796 s (∼100 ms per sample), while FCN model prediction required 3136 s (∼174 ms per sample). Thus, the proposed method achieves 2D reflector-set estimation at approximately 6 frames per second with ∼100 ms latency, while the motion tracking from optical data is real-time, requiring less than 10 ms. With respect to the input, the original frame size is 424 × 512, re-sized to 368 × 444 during testing to fit in GPU memory. Thus, DeepMoCap performs motion capture at approximately 2 Hz for 3-view input of 368 × 444, while the performance complexity against number of views, i.e., input image pairs, is O(n). The runtime analysis was performed on a desktop machine equipped with one NVIDIA GeForce Titan X GPU. Code (https://github.com/tofis/deepmocap) and dataset tools of the proposed method are publicly available to encourage further research in the area.
Conclusions
In the present work, a deep marker-based optical motion capture method is introduced, using multiple IR-D sensors and retro-reflectors. DeepMoCap constitutes a robust, fast and flexible approach that automatically extracts labeled 3D optical data and performs immediate motion capture without the need for post-processing. For this purpose, a novel two-stream, multi-stage CPM-based FCN is proposed that introduces a non-parametric representation to encode the temporal correlation among pairs of colorized depthmaps and 3D optical flow frames, resulting in retro-reflector 2D localization and labeling. This step enables the 3D optical data extraction from multiple spatio-temporally aligned IR-D sensors and, subsequently, motion capture. For research and evaluation purposes, two new datasets were created and made publicly available. The proposed method was evaluated with respect to the 2D reflector-set estimation and the motion capture accuracy on these datasets, outperforming recent and robust methods in both tasks. Taking into consideration this comparative evaluation, we conclude that the joint usage of traditional marker-based optical MoCap rationale and recent deep learning advancements in conjunction with 2.5D and 3D vision techniques can significantly contribute to the MoCap field, introducing a new way of approaching the task.
With respect to the limitations, the side-view capturing and the highly complex body poses that occlude or merge reflectors on the image views constitute the main barriers. These limitations can be eliminated by increasing the number of IR-D sensors around the subject, however, increasing the cost and complexity of the method. Next steps of this research would include the study of recent deep learning approaches in 3D pose recovery and motion capture, investigating key features that will allow us to address main MoCap challenges such as real-time performance, efficient multi-person capturing, in outdoor environments, with more degrees of freedom of the body to be captured and higher accuracy. Acknowledgments: This work was supported by the EU funded project VRTogether H2020 under the grant agreement 762111. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan X GPU used for this research.
Conflicts of Interest:
The authors declare no conflict of interest.
Appendix A
Strap Retro-Reflector Point-Set Fusion
b = n v i R i • n v j R i c = n v i R i • (P v i R i − P v j R i ) s = (b · f − c)/d P v i,j R i = P v i R i + s · n v i R i d = 1 − b 2 f = n v j R i • (P v i R i − P v j R i ) t = ( f − c · b)/d P v j,i R i = P v j R i + t · n v j R i (A1)
Applying Equation (10) for all N R i = 2 · m R i extracted 3D points, where m is the total number of pairs, the 3D position of the reflective strap R i is estimated to the center of the body part. Even when only one sensor captures a strap reflector, if the template-based structure is calibrated, the 3D position can be estimated using the normal vector and the body part (limb) radius.
Figure 1 .
1DeepMoCap pipeline overview. After the placement of the reflectors on the subject's body
Figure 2 .
2Proposed reflector-set placement. Reflective straps (orange) and patches (blue) placement on the subject's body.
Figure 3 .
3Raw depth and infrared data.
Figure 4 .
4Multiview two-stream input. (Up) Colorized depth, mask subtracted. (Down) Colorized 3D optical flow.
Figure 5 .
5Overall 2D reflector-set estimation from confidence maps and optical flow 2D vector fields. The 2D location for R 13 reflector is estimated taking into account its predicted heat map and optical flow between the predictions r 13, f −1 and r 13, f .3.3.1. FCN Architecture
Figure 6 .
6DeepMoCap two-stream, multi-stage FCN architecture. The outcome of each stage t ∈ {2, . . . , T} and the feature set F are fused and given as input to the next stage.3.3.2. Confidence Maps and Optical Flow Fields
L t S and L t L are evaluated during training by generating the ground truth confidence map S *
Figure 7 .
7Contour detection (red pixels) of the reflector regions for depth estimation and 3D mapping. (a) Contour of single reflector region, E 0 . (b) Contour of multi-reflector region, E 1 .
Figure 8 .
8Multiple 2D reflector-set estimates (left) are spatially mapped in a global 3D Cartesian coordinate system, resulting in 3D optical data extraction (right).
Figure 9 .
9(a) Template model joints, hierarchical level, DoFs and correspondence to the reflector subsets. (b) Reflector mapping to body structure body parts.
Figure 10 .
10Reflector annotation using blob tracking on IR binary mask, visualized though on I v IR IR data for the sake of better understanding. Erroneous estimates occur when reflector regions are merged.
Figure 12 .
12Mean Average Precision based on Percentage of Correct Keypoints thresholds (a = 0.05) against confidence threshold, mAP(c min ).
Figure 15 .
15Five samples of the method results are illustrated in rows. At the left side of the figure, the multi-view input along with the FCN reflector-set estimates are presented, while, at the right side, their corresponding 3D optical and motion capture outcomes are shown.
Author
Contributions: Conceptualization, ALL; methodology, A.C.; software, A.C.; validation, ALL; formal analysis, A.C.; investigation, A.C.; resources, A.C., D.Z., P.D.; data curation, A.C.; writing-original draft preparation, A.C.; writing-review and editing, D.Z., S.K., P.D.; visualization, A.C.; supervision, D.Z., S.K., P.D.; funding acquisition, D.Z., P.D. The research in this work was mainly conducted by A.C., PhD candidate in National Technical University of Athens and Research Associate in Centre for Research and Technology -Hellas. The research was supervised by D.Z., S.K. and P.D. They provided ideas, inputs and feedback to improve the quality of the present work.
Funding:
This research and the APC were funded by VRTogether, Horizon 2020 Framework Programme, grant number 762111.
3D normal vectors of the reflector regions for v i and v j views, v i , v j ∈ {1, . . . .N}, respectively. These vectors are defined as the normal vectors of the 3D points given by mapping the corresponding pixel sets P a view pair v i − v j are given by applying the equations:
Table 1 .
1Data captured per subject in the DMC3D dataset.Physical Exercise
# of Repetitions # of Frames
Type
Walking on the spot
10-20
200-300
Free
Single arm raise
10-20
300-500
Bilateral
Elbow flexion
10-20
300-500
Bilateral
Knee flexion
10-20
300-500
Bilateral
Closing arms above head
6-12
200-300
Free
Side steps
6-12
300-500
Bilateral
Jumping jack
6-12
200-300
Free
Butt kicks left-right
6-12
300-500
Bilateral
Forward lunge left-right
4-10
300-500
Bilateral
Classic squat
6-12
200-300
Free
Side step + knee-elbow
6-12
300-500
Bilateral
Side reaches
6-12
300-500
Bilateral
Side jumps
6-12
300-500
Bilateral
Alternate side reaches
6-12
300-500
Bilateral
Kick-box kicking
2-6
200-300
Free
Table 2 .
2AP for PCK with α = 0.05, for each of the 26 reflectors. The proposed FCN method outperforms the rest of the methods for the 80.7% of the reflectors (i.e., 21 out of 26).%
R01
R02
R03
R04
R05
R06
R07
R08
R09
R10
R11
R12
R13
[16]
96.81 96.11 99.22 95.06 90.98 85.26 98.78 99.76
95.25 94.70 96.99 93.33 85.92
[14]
96.95 95.36 98.77 96.69
91.08 85.26 98.78 99.51 96.00 95.89 97.15 93.79 87.64
[14] + 3D OF 96.81 96.61 99.45 94.85 89.98 85.53 98.78 99.45 96.00
96.27
97.25 93.49
87.66
Proposed
98.10 97.31 99.48 97.35 91.36 86.20 99.00 98.27
96.64
95.32
97.81 95.19
87.13
%
R14
R15
R16
R17
R18
R19
R20
R21
R22
R23
R24
R25
R26
[16]
83.42 93.21 98.24 96.25
75.23
98.39 98.38 94.52 74.28 94.88 99.30 97.18
64.73
[14]
84.58 92.88 98.66 95.95
76.19
98.39
98.44
94.88 74.06 96.99 99.52 97.87
71.18
[14] + 3D OF
86.33
94.11 98.44 95.92
72.29
98.40 98.42 95.14 78.68 96.53 99.21 97.76
70.56
Proposed
85.61 94.79 98.81 97.50
77.17 99.30
98.18
96.61 79.23 97.93 100.0 98.73 73.96
Table 4 .
4AP for PCK with α = 0.05, for each of the 26 reflectors, after filtering.%
R01
R02
R03
R04
R05
R06
R07
R08
R09
R10
R11
R12
R13
[16]
96.95 96.39 99.45 97.30 91.74 86.81 100.0 100.0
96.62
97.46
98.15 94.16 90.50
[14]
96.95 96.19 98.77 98.31 93.98
86.81 100.0 99.76 96.87
97.46
98.55 94.64 90.79
[14] + 3D OF 96.81 96.88 99.45 97.30 91.72 86.81 100.0 99.70 96.87
97.46
98.55 94.05
91.67
Proposed
98.52 98.61 99.81 98.65 92.58
87.35
100.0
100.0 98.90
96.59
100.0 95.50
89.60
%
R14
R15
R16
R17
R18
R19
R20
R21
R22
R23
R24
R25
R26
[16]
83.78 95.93 99.03 97.19 78.81 98.60 99.41 96.59 74.08 97.59 99.40 98.36 68.64
[14]
90.19
96.40
99.32 97.17
82.82
98.60 99.37
97.25
75.66
99.09
99.61 98.60 69.52
[14] + 3D OF
90.19 96.75 99.14 98.30 78.87 98.50
99.50
96.83
81.69
98.25 99.30 98.52 72.68
Proposed
88.90 95.10 99.13 97.82
78.20
99.63
98.50 96.93 79.49 98.25
100.0 99.05 78.21
Table 6 .
6Comparative evaluation of the motion capture results of the respective methods, presenting total MAE, RMSE and 3D PCK (α 3D = 20 cm) metrics.Method
MAE (cm) RMSE (cm) 3D PCK (a = 20 cm) [58]
Best Kinect [27]
15.35
16.06
82.03%
Fusion [44]
12.31
12.91
85.93%
Fusion++ [44]
10.66
11.30
88.75%
Proposed
9.02
10.06
92.25%
Table 7 .
7Comparative evaluation per exercise using 3D PCK, α 3D = 20 cm metric.Exercise
Best Kinect [27] Fusion [44] Fusion++ [44] Proposed
Walking on the spot
96.60%
100.00%
97.54%
100.00%
Single arm raise
93.57%
96.19%
97.38%
100.00%
Elbow flexion
91.12%
100.00%
100.00%
97.40%
Knee flexion
88.36%
94.11%
100.00%
98.80%
Closing arms above head
82.48%
80.08%
83.33%
88.62%
Side steps
85.00%
88.33%
93.33%
87.50%
Jumping jack
95.48%
84.18%
87.57%
96.05%
Butt kicks left-right
81.87%
80.99%
86.26%
90.94%
Forward lunge left-right
57.31%
87.93%
86.05%
92.01%
Classic squat
59.60%
78.67%
83.05%
90.40%
Side step + knee-elbow
77.78%
80.25%
81.94%
89.81%
Side reaches
89.24%
84.55%
87.88%
91.52%
Side jumps
90.00%
89.31%
92.78%
93.47%
Alternate side reaches
68.01%
74.19%
77.69%
82.53%
Kick-box kicking
74.07%
70.14%
76.39%
84.72%
Table 8 .
8Experimental results of the respective motion capture methods using (MAE) and (RMSE) metrics per joint (in cm).Joint
Best Kinect [27]
Fusion [44]
Fusion++ [44]
Proposed
MAE
RMSE
MAE RMSE MAE RMSE MAE RMSE
Left Shoulder
12.83
13.25
8.16
8.37
7.89
8.53
11.41
12.63
Right Shoulder 15.59
15.90
9.71
10.28
8.62
9.19
11.09
11.76
Left Elbow
16.04
17.45
16.46
17.17
15.90
16.60
13.25
14.84
Right Elbow
19.37
19.67
11.61
12.61
10.88
11.68
12.25
13.36
Left Hand
16.01
17.95
14.52
15.87
13.53
14.44
11.94
12.58
Right Hand
21.24
21.55
13.10
14.24
12.42
13.29
12.04
13.05
Left Hip
8.63
8.82
9.99
10.20
6.33
6.45
4.18
4.69
Right Hip
10.89
11.16
10.59
10.81
5.94
6.11
4.53
4.99
Left Knee
10.79
11.73
12.55
13.10
9.97
10.45
5.12
5.82
Right Knee
15.13
15.99
12.17
12.64
8.92
9.45
7.24
8.16
Left Foot
17.74
18.34
16.35
17.11
15.57
16.40
7.00
8.82
Right Foot
19.91
20.86
12.48
12.53
11.93
12.97
8.24
10.00
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[
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"Alberto Olmo \nArizona State University\n\n",
"Sailik Sengupta \nArizona State University\n\n",
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In this paper, we show that popular Generative Adversarial Networks (GANs) exacerbate biases along the axes of gender and skin tone when given a skewed distribution of face-shots. While practitioners celebrate synthetic data generation using GANs as an economical way to augment data for training data-hungry machine learning models, it is unclear whether they recognize the perils of such techniques when applied to real world datasets biased along latent dimensions. Specifically, we show that (1) traditional GANs further skew the distribution of a dataset consisting of engineering faculty headshots, generating minority modes less often and of worse quality and (2) image-to-image translation (conditional) GANs also exacerbate biases by lightening skin color of non-white faces and transforming female facial features to be masculine when generating faces of engineering professors. Thus, our study is meant to serve as a cautionary tale.
|
10.1016/j.artint.2021.103652
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"https://arxiv.org/pdf/2001.09528v3.pdf"
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2001.09528
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2b15d1a1b354573b3ce23b18991d15c850a8546b
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IMPERFECT IMAGANATION: IMPLICATIONS OF GANS EXACERBATING BIASES ON FACIAL DATA
Niharika Jain
Arizona State University
Alberto Olmo
Arizona State University
Sailik Sengupta
Arizona State University
Lydia Manikonda
Rensselaer Polytechnic Institute
Subbarao Kambhampati
Arizona State University
IMPERFECT IMAGANATION: IMPLICATIONS OF GANS EXACERBATING BIASES ON FACIAL DATA
Accepted for publication in the ICLR 2021 Workshop on Synthetic Data Generation -Quality, Privacy, Bias
In this paper, we show that popular Generative Adversarial Networks (GANs) exacerbate biases along the axes of gender and skin tone when given a skewed distribution of face-shots. While practitioners celebrate synthetic data generation using GANs as an economical way to augment data for training data-hungry machine learning models, it is unclear whether they recognize the perils of such techniques when applied to real world datasets biased along latent dimensions. Specifically, we show that (1) traditional GANs further skew the distribution of a dataset consisting of engineering faculty headshots, generating minority modes less often and of worse quality and (2) image-to-image translation (conditional) GANs also exacerbate biases by lightening skin color of non-white faces and transforming female facial features to be masculine when generating faces of engineering professors. Thus, our study is meant to serve as a cautionary tale.
INTRODUCTION
The use of Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) has grown significantly and due to data-demand of deep learning models, when faced with sparse data (owing to paywalls, privacy concerns, etc.) practitioners often turn to promising data augmentation solutions. While earlier computer vision works focused on performing affine transformations to existing samples (O'Gorman & Kasturi, 1995;Bloice et al., 2019), using GANs for synthetic data generation has recently become popular (Teich, 2019;Nisselson, 2018). GANs generate such data by approximating the original distribution with a limited training set and create examples that appear novel. These examples give a (false) sense of sampling unseen data from the same underlying distribution as the original training data, making GANs a seemingly perfect candidate for data augmentation. We note that even this best-case scenario would be a territory for practitioners to tread lightly; GANgenerated data for augmentation would only propagate the existing biases of the real-world data. Owing to theoretical limitations of GANs (Arora et al., 2018), we show a grim reality: the generated data learns a distribution shifted from that of the real world, one which exacerbates these biases and disproportionately underrepresents those already in the minority, both in number and quality. This poses serious ethical implications on any downstream tasks trained on a synthetically-augmented dataset, especially when biases exist along protected or embargoed attributes.
ARCHITECTURE AND APPROACH
Mode Collapse GANs are known to estimate an equilibrium of a minimax game played by a generator network G and discriminator network D. While D, a binary classifier, learns to discriminate between images that come from a real-world data distribution p data and those that do not, G learns to generate images from p GAN and fool D into classifying them as coming from p data . In the presence of infinite training data, computation time and network capacity for the generator and the discriminator, this process ensures that the p GAN distribution generated by G converges to that of the training data p data (Goodfellow et al., 2014). In reality, GAN-generated distributions are not nearly as diverse as their training distributions (Arora et al., 2018;2017) and the support (i.e. possible feature combinations of the generated data) is only representative of a small subset of what one would expect to see when sampling data from the real distribution. The support size of the generated images is constrained by the capacity of D. G collapses because the set of noise inputs that would correspond to some minority mode in the image space has (by definition) a low probability of being seen by D. As G only optimizes its own weights over the feedback from D, it rarely learns to generate these modes (Che et al., 2017).
There are several related works- Zhao et al. (2018) studies GANs' bias and generalization to unseen modes without discussing the problem of GANs collapsing to existing modes. While mode collapse is a well-studied phenomenon (Grnarova et al., 2018;Goodfellow, 2016;Che et al., 2017;Arora et al., 2017) and several GAN variants have been developed to alleviate its effects (Metz et al., 2017;Srivastava et al., 2017;Arjovsky et al., 2017;Miyato et al., 2018;Tolstikhin et al., 2017;Karras et al., 2019a), a distinction is rarely made between uniform and non-uniform training datasets. On these lines, Mishra et al. (2018) empirically shows that the divergence between p data and p GAN does indeed worsen as the training data is more skewed, however using four scalar metrics which do not offer much insight on how the distributions differ. For a dataset that is biased along latent axes (e.g. gender and skin color), we hypothesize G (we try several GAN variants) collapses to modes in the majority groups (e.g. masculine and white faces) amplifying biases that exist in the original data.
Data Collection and Processing
To test our hypothesis, we construct a dataset of faces of engineering professors from U.S. universities that are (1) listed in the top 47 of US News' most recent "Best Engineering Schools" and (2) had public access to faculty directories with images. The data exhibits bias along the latent dimensions of gender and race and thus, is an appropriate test-bed to study the amplification of bias in GAN-based data generation. We gather a total of 17, 245 engineering faculty 64 × 64-pixel headshots using an unsupervised face detector (Dalal & Triggs, 2005).
EXPERIMENT AND RESULTS
To explore the diversity of p GAN we test the performance on three GANs (1) DCGAN (Radford et al., 2016): the most common GAN used by practitioners due to its minimal requirements for compute power and off-the-shelf availability (carpedm20, 2015), (2) ProGAN (Karras et al., 2019a): a state-of-the-art GAN for sample quality and known to addresses the mode-collapse problem and to overcome the quality-variance tradeoff (Karras et al., 2019b;2020), and (3) CycleGAN : the most well-known image-to-image translation GAN, which transforms an image from one domain to another by minimizing cycle-consistency and identity losses. We show experiments on two other GAN architectures designed to address mode collapse -Wasserstein GAN (Arjovsky et al., 2017) and AdaGAN (Tolstikhin et al., 2017) -in the appendix.
IMAGINING ENGINEERS FROM SCRATCH
We assess the data from the GAN variants -DCGAN and ProGAN -by asking humans to annotate images from the original and generated datasets along the dimensions of race and gender. To account for variance in model training, we generate 50 images from three seeds where each seed trains the DCGAN and the ProGAN for 50 epochs. We conduct 4 seven-minute human study tasks in a Accepted for publication in the ICLR 2021 Workshop on Synthetic Data Generation -Quality, Privacy, Bias between-subject design fashion (each annotator saw images belonging to only one set) and leveraged data from 234 master Turkers on Amazon's MTurk platform. Each worker performed the tasks:
[T1(a/b)] Human subjects were asked to select the most appropriate option for an image x sampled from [T1a] p data and [T1b] G(z) with the following options: (1) face mostly has masculine features, (2) face mostly has feminine features, and (3) neither of the above is true.
[T2(a/b)] Human subjects were asked to select the most appropriate option for an image x sampled from [T2a] p data and [T2b] G(z) from the list of following options: (1) skin color is non-white, (2) skin color is white, and (3) can't tell.
We presented each annotator with 52 images-50 from the original/generated data and two high quality trivial images with known labels for gender and skin color. This helped us prune 18 bad datapoints. We had 30 valid data points for all generated datasets and 25 for the original distribution. We considered majority-voting to categorize an image as belonging to a class. Figure 5 in the appendix contains the resulting charts.
RESULTS
We plot the results for T1a and T1b in Figure 1 (left) and find that (1) both DCGAN and ProGAN penalize the original 20% of images with mostly feminine features being DCGAN the most penalizing, reducing the percentage to 6.67%. A one-tailed two-proportion z-test yields a p-value of 0.0032 confirming the amplification of bias across the latent dimension of gender for DCGAN and (2) for tasks T2a and T2b (Figure 1, right) the proportion of non-white faces decreased from 24% in the original dataset to 1.33% in the DCGAN-generated dataset and to 11.33% for ProGAN. The p-value obtained (2.7 × 10 −8 for DCGAN and 1.05 × 10 −3 for ProGAN) show strong statistical significance as both GANs collapse along the latent dimension of race, biasing the synthetic faces toward lighter skin tones. Note that while ProGAN did not collapse along the axis of gender, it was not immune to collapsing along the axis of other protected features (eg. skin color). We notice that the synthetic data not only propagates but exacerbates those biases against minority populations.
Quality and Confidence Metrics
We measure the consensus among Turkers by the amount of votes needed to classify each image in the axes of gender and color. For DCGAN, we find that the proportion of images labelled as non-white and female decreases as the voting threshold increases. This is indicative of a higher level of agreement between participants and shows that the quality of generated images for the minority classes is worse than that of the majority classes. ProGAN does not exhibit this disparity in quality across gender, but it produces lower quality for non-white faces than white ones.
IMAGINING ENGINEERING COUNTERPARTS
As image-to-image translation GANs' output distributions conditioned on the input, our intuition was that they may be less susceptible to exacerbating biases. For instance, in our task where gender is a latent feature and feminine faces are underrepresented, a GAN, provided with the input image of a female, would have to actively convert it into a male one. Unfortunately, it is known that even these conditional GAN variants are not immune to mode collapse (Ma et al., 2018). However, how conditional variants of GANs react to sensitive social features such as race and gender remains an open question.
Accepted for publication in the ICLR 2021 Workshop on Synthetic Data Generation -Quality, Privacy, Bias
To study this, we train a CycleGAN to stylize faces of non-engineering professors to look like engineering faculty. Thus, our target/output domain consists of the engineering faculty face dataset leveraged in the previous experiment and our input domain is the CelebA dataset (Liu et al., 2015) consisting of over 200,000 annotated images of celebrities. As our dataset consists of only 16,500 images, we randomly sample 16,500 faces from the CelebA dataset for training. We then create a held-out test set from CelebA in which we have 100 images for each of the four categories-white, non-white, male, and female.
In Figure 2, we showcase the transformation of celebrity faces that are representative of the minority categories (i.e. non-white, female) in the engineering professors dataset. While we see that the GAN learns to add glasses or creating smiling expressions, not all the modifications learned are socially harmless, we also notice that it lightens the skin tone of non-white celebrities and imparts masculine aspects to the faces of female celebrities. While it is reasonable to expect that a GAN might perpetuate and exacerbate biases along any arbitrary dimension where there exists a skew in the training set, we stress that this kind of innocuous bias is not our focus. Machine learning systems are designed to find correlations to recognize patterns, but this correlation-seeking becomes problematic for social features when models perpetuate and exacerbate biases for minority groups who have faced systemic disadvantage or discrimination. Before concluding, we highlight a casestudy where such models are having adverse real-world impact.
REAL-WORLD APPLICATIONS AND CONCLUSION
While our experiments meant to serve as example, the bias-exacerbation consequences of mode collapse in GANs can be seen in real-world applications. Snapchat, a popular image-sharing platform, has recently taken advantage of the image-to-image translation capabilities of conditional GANs such as CycleGAN for their "My Twin" lens, according to several sources (Yanjia Li, 2020;Magazine, 2019;Jang, 1970;red, 2019). We show that this presumably conditional-GAN-based technology reacts to the sensitive features this work discusses. When applying this lens to a female face, the GAN should ideally make no changes, but when used on women of color, it lightens skin tone, though this is not the case for white women using the same filter. While we have not performed a comprehensive study, the observations and claims open an intriguing research problem (Baeza-Yates, 2016). Examples of the lightened complexions on women of color and white women can be seen in Figure 6 in the appendix.
The implications of using a biased facial dataset augmented via GANs for a downstream task could be severe. The use of machine learning models on facial data is already prevalent in critical decisionmaking scenarios such as employment (Hymas, 2019), healthcare (Bahrampour, 2014), education (Kaur & Marco, 2019), criminal justice (Harwell, 2019), as well as security innovations like deepfake detection (Dolhansky et al., 2019). It is of clear ethical import that we ensure our training sets and models are fair and diverse with respect to sensitive features. At the very least, they ought not to rig the system against already underrepresented minorities.
GANs have proven to create less diverse distributions than the original they are trained on, but the implications of mode collapse remain unclear in scenarios where the training distribution p data is biased toward certain feature values (eg. males) along a latent feature (eg. gender). To study this, we empirically show how GANs trained on a demographic already skewed toward white and male faces exacerbate social biases in the generated distribution p GAN . In our setting, mode collapse occurs on a majority latent mode of the original data and causes a severe under-representation of feminine facial features and non-white skin tones in the generated dataset. We also demonstrate that this perpetuation of biases against female and non-white features occurs in image-to-image translation GANs, first stylizing celebrities' faces to look like those of engineering professors, and next by conducting a case study on Snapchat's "My Twin" lens. Beyond implications about social issues, this work should serve as a general caution against using GAN-based data augmentation techniques to alleviate problems arising from sparse or unbalanced datasets for any downstream task. There seems to exist a false sense of security that GANs can generate novel data samples which pick the expected semantic features relating to the defect, and place them in previously unseen settings. In actuality, the augmented data might be underrepresenting or compromising image quality for some crucial feature of the real-world data.
Accepted for publication in the ICLR 2021 Workshop on Synthetic Data Generation -Quality, Privacy, Bias A APPENDIX In addition to the variants mentioned in the paper (DCGAN and ProGAN), we investigate the performance of another two GANs which claim to reduce mode collapse: Wasserstein GANs or WGAN (Arjovsky et al., 2017) andAdaGAN Tolstikhin et al. (2017).
A.1 STUDIES WITH COMMERCIALLY AVAILABLE GENDER CLASSIFICATION SYSTEMS
As a less subjective approach to labelling, we use Microsoft Azure's Face API for classifying 5000 images from the training set and 5000 generated by the three different variants of GAN: the popular DCGAN and two others that attempt to address the problem of mode collapse, AdaGAN and Pro-GAN (we omit WGAN due to poor image quality). To ensure our results are not specific to a single generation, we obtain 5000 images by sampling from three runs of each GAN with different random seeds for weight initialization. We show 100 images randomly sampled from these 5000 obtained from each GAN variant in Figure 3.
In Figure 4, we show the percentage of images classified as female, male and "can't" tell by Microsoft's AI tool. We perform a one-tailed two-proportion z-test on the original and generated distributions to assess the null hypothesis that the proportion of feminine features in the synthetic distribution for all GAN variants, is same as in the original. In the initial dataset, 16.5% are labelled as Face has mostly feminine features | Face has mostly masculine features | Can't tell Figure 4: The percentage of faces classified as female, male and can't tell by Microsoft Azure's Face API decreases from 16.5% in the original dataset significantly in the synthetically generated datasets across several GAN variants that are popular or attempt to address the mode collapse problem.
females while 80.4% are labelled as male, clearly indicating an original bias towards males. Further, DCGAN exacerbates it significantly (with a p-value of 3.2×10 −6 ), bringing down the percentage of females in the generated set to 11.8%. The quality of images generated by AdaGAN is significantly worse than the ones produced by all other variants as indicated by the spike in the number of images, from 3.1% in the original to 11.8% (with a p-value of 3.0 × 10 −53 ). Surprisingly, regardless of the poor quality, there is a significant increase in the number of generated images that are classified as male (from 80.4% in the original data to 84.3%) while the number of generated images that are classified as females has a substantial drop (from 16.5% to 3.9%). This also highlights that many of the other GAN variants that seek to address mode-collapse, have proven to be worse than AdaGAN (Lala et al., 2018); such as WGAN (Arjovsky et al., 2017), VEEGAN (Srivastava et al., 2017) or Unrolled GAN (Metz et al., 2017) and either affect the quality of generated images, exacerbate the biases over latent features such as gender, or both. On the other hand, the more recent architecture ProGAN, clearly outperforms both the popular DCGAN and AdaGAN by reducing the exacerbation of bias and improving image quality. It only decreases the percentage of females in its generated set by 1.7%, even though this is a significant exacerbation of bias along the latent dimension of gender (with a p-value of 0.09008). Our results show that popular and state-of-the-art GAN variants paint an optimistic picture of this technology for data-augmentation while suffering from the exacerbation of biases along latent dimensions.
A.2 CONFIDENCE METRICS FOR DCGAN AND PROGAN
We measure the confidence of the annotators for the synthetic datasets by plotting how each headshot is classified when the threshold varies from 1 to 15 and show it in Figure 5. The x-axis represents the number of votes required to classify an image to each particular class (i.e. male, female, "can't tell"), and the y-axis represents the proportion of images that are classified considering that voting threshold. This is a metric for consensus among the group of annotators that the images belong to a certain class and demonstrates their annotating confidence; the less variability through y over each threshold, the more confidence the workers show. On the original data, roughly the same proportion of images are classified as male, female, white, and nonwhite irrespective of the number of Turkers needed to vote. In other words, the Turkers are confident about which faces are male, female, white, and non-white. This is not the case for the synthetic distributions. For DCGAN, the proportion of images that are marked as female or non-white significantly drops as it requires more Turkers to vote for that label, and they only lose confidence over the images depicting the minority gender and race; the proportion of images marked as male or white does not drop as the voting threshold increases. For ProGAN's images, the Turkers are confident about male, female, and white faces, but not about non-white faces.
Accepted for publication in the ICLR 2021 Workshop on Synthetic Data Generation -Quality, Privacy, Bias Figure 5: Human annotator agreements on skin color and gender between professor headshots from the original and synthetic (generated by DCGAN and ProGAN) distributions. The number of images labeled as masculine, feminine or neither, changes as the threshold number of votes required to categorize an image into a particular category increases from 1 to 15.
A.3 SNAPCHAT CASE STUDY
Image-to-image translation GANs, such as pix2pix or CycleGAN adjust colors and textures in an already-existing image from some input domain to map it to another class. Normally, the input and target domains are closely related and the mapping can be achieved by changing the geometries minimally. Some examples of successful applications for image-to-image translation are conversion of horses to zebras, street photographs to their semantic segmentation, aerial photos to Google maps, and summer landscapes to winter landscapes. CycleGAN is the most popular off-the-shelf GAN variant used by machine learning practitioners today, as measured by the number of stars on the most-used GitHub repositories for this model (junyanz, 2018; 2017), and has also, predictably, been a popular choice for synthetic data augmentation (Hiasa et al., 2018;Sandfort et al., 2019;Huang et al., 2018). Just as with the unconditional variants, our motivation is to explore if and how the diversity of the generated distribution p GAN differs from the training distribution p data . Figure 6: Faces of women of color (left six columns) and white women (right six columns) before and after using Snapchat's female gender face lens, top and bottom respectively. The sections used for the skin-color machine analysis are highlighted in white.
To assess how the skin color changed between pairs of images objectively, we crop a section of the face under the eyes and above the tip of the nose, spanning both cheeks, and find its average pixel value, then we map the RGB vector, using L2-norm distance, to the closest standard shade in the L'Oréal skin color chart 1 . While not considering skin warmth, only skin lightness, we show that the lens lightens non-white faces by one shade consistently for five faces and produces no effect for one Accepted for publication in the ICLR 2021 Workshop on Synthetic Data Generation -Quality, Privacy, Bias of them. On the other hand, it performed randomly for white faces in our example, lightening two by one shade, darkening two by one shade, and not affecting two. A potential cause of lightening skin tones in women of color is that a GAN used by the face lens collapses all inputs in a region of the image space to output lighter colors. However, more rigorous studies should be performed to make certain claims. Our case study offers initial support for the narrative of Snapchat's beautification face lenses lightening skin tones for people of color.
A.4 DOWNSTREAM TASKS AND VULNERABLE COMMUNITIES
The glaring ethical problem with automated, machine learning-powered tools is that they are "most often used on people towards whom they exhibit the most bias," and that the errors arising from bias "can be much more costly for those in marginalized communities than other groups" (Gebru, 2019). Classification tasks in the real world suffer from this dilemma. In criminal justice, automated tools predict recidivism risk in a system which disproportionately punishes Black and Hispanic people. It is unfortunate yet unsurprising, then, that the risk assessment software used in state criminal justice systems is biased against Black people (Angwin & Larson, 2016). This classification system is input over 137 features -not including race -and disproportionately classifies Black defendants as medium or high risk. In employment, automated tools predict candidate performance and fit in industries which are already male-dominated. Further, a hiring system designed by Amazon in 2018 faced public backlash when it was found to discriminate against female candidates by penalizing résumés which included participation in women's organizations. The classification system scraped résumés of candidates from the past ten years and was never given gender as an input feature. Classifiers are not the only automated tool who would use data generated by GANs. In 2020, PULSE (Menon et al., 2020), a face "depixelizer," received widespread backlash on social media (especially from world-famous contributors to the field of AI ethics) because it was shown to upsample images of non-white faces to have Caucasian features. The authors of this paper perform several studies in response and conclude that the biases in PULSE derive directly from the biased performance of the GAN from which it receives generated data. The major takeaway of all discussions mentioned here is that the data bias problem cannot be reduced solely to the dataset used. It seems that popular automated data generation tools, namely GANs, will not merely perpetuate the patterns found in the data (the theoretical ideal for the technologies), but rather amplify them. The question is, then, how we can regulate the societal applications for which these known flawed systems are used.
Figure 1 :
1Distribution of human classifications on gender and skin color.
Figure 2 :
2Illustrative test set of transformations on non-white (two rows on the left) and female celebrities (two rows on the right). Original and stylized images are one atop another respectively.
Figure 3 :
3Images of professors generated by popular GAN architectures trained on our engineering professors dataset; WGAN, AdaGAN, and ProGAN attempt to address the mode-collapse problem.
1
arXiv:2001.09528v3 [cs.LG] 16 Jun 2021 Accepted for publication in the ICLR 2021 Workshop on Synthetic Data Generation -Quality, Privacy, Bias
Accepted for publication in the ICLR 2021 Workshop on Synthetic Data Generation -Quality, Privacy, Bias
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"https://github.com/carpedm20/DCGAN-",
"https://github.com/junyanz/CycleGAN,",
"https://github.com/junyanz/pytorch-CycleGAN-and-"
] |
[
"Probing charged impurities in suspended graphene using Raman spectroscopy",
"Probing charged impurities in suspended graphene using Raman spectroscopy"
] |
[
"Zhen Hua Ni ",
"Ting Yu ",
"Zhi Qiang Luo ",
"Ying Ying Wang ",
"Lei Liu ",
"Choun Pei Wong ",
"Jianmin Miao \nMicromachines Centre\nSchool of Mechanical and Aerospace Engineering\nNanyang Technological University\n50 Nanyang Avenue639798Singapore\n",
"Wei Huang \nJiangsu Key Lab for Organic Electronics & Information Displays\nNanjing University of Posts and Telecommunications\n9 Wenyuan Road210046NanjingChina\n",
"Ze Xiang Shen ",
"\nDivision of Physics and Applied Physics\nSchool of Physical and Mathematical Sciences\nNanyang Technological University\n21 Nanyang link637371Singapore\n"
] |
[
"Micromachines Centre\nSchool of Mechanical and Aerospace Engineering\nNanyang Technological University\n50 Nanyang Avenue639798Singapore",
"Jiangsu Key Lab for Organic Electronics & Information Displays\nNanjing University of Posts and Telecommunications\n9 Wenyuan Road210046NanjingChina",
"Division of Physics and Applied Physics\nSchool of Physical and Mathematical Sciences\nNanyang Technological University\n21 Nanyang link637371Singapore"
] |
[] |
Charged impurity (CI) scattering is one of the dominant factors that affect the carrier mobility in graphene. In this paper, we use Raman spectroscopy to probe the charged impurities in suspended graphene. We find that the 2D band intensity is very sensitive to the CI concentration in graphene, while the G band intensity is not affected.The intensity ratio between the 2D and G bands, I 2D /I G , of suspended graphene is much stronger compared to that of non-suspended graphene, due to the extremely low CI concentration in the former. This finding is consistent with the ultra-high carrier mobility in suspended graphene observed in recent transport measurements. Our results also suggest that at low CI concentrations that are critical for device applications, the I 2D /I G ratio is a better criterion in selecting high quality single layer graphene samples than is the G band blue shift.
|
10.1021/nn900130g
|
[
"https://export.arxiv.org/pdf/0812.4169v3.pdf"
] | 18,997,150 |
0812.4169
|
1c95626517b909ea7a6df3ffe1c2bbfff52fa71c
|
Probing charged impurities in suspended graphene using Raman spectroscopy
Zhen Hua Ni
Ting Yu
Zhi Qiang Luo
Ying Ying Wang
Lei Liu
Choun Pei Wong
Jianmin Miao
Micromachines Centre
School of Mechanical and Aerospace Engineering
Nanyang Technological University
50 Nanyang Avenue639798Singapore
Wei Huang
Jiangsu Key Lab for Organic Electronics & Information Displays
Nanjing University of Posts and Telecommunications
9 Wenyuan Road210046NanjingChina
Ze Xiang Shen
Division of Physics and Applied Physics
School of Physical and Mathematical Sciences
Nanyang Technological University
21 Nanyang link637371Singapore
Probing charged impurities in suspended graphene using Raman spectroscopy
1Suspended graphenecharged impuritiesRamanmobilityscattering rate
Charged impurity (CI) scattering is one of the dominant factors that affect the carrier mobility in graphene. In this paper, we use Raman spectroscopy to probe the charged impurities in suspended graphene. We find that the 2D band intensity is very sensitive to the CI concentration in graphene, while the G band intensity is not affected.The intensity ratio between the 2D and G bands, I 2D /I G , of suspended graphene is much stronger compared to that of non-suspended graphene, due to the extremely low CI concentration in the former. This finding is consistent with the ultra-high carrier mobility in suspended graphene observed in recent transport measurements. Our results also suggest that at low CI concentrations that are critical for device applications, the I 2D /I G ratio is a better criterion in selecting high quality single layer graphene samples than is the G band blue shift.
The extremely high carrier mobility makes graphene a promising candidate for future electronic devices. 1 In practice, however, the carrier mobility of graphene varies from piece to piece 2,3,4 due to the different levels of charge impurity (CI) scattering present. 5,6 For example, the electron mobility of graphene can vary from 1×10 3 to 2×10 4 cm 2 /Vs on SiO 2 /Si substrate, which corresponds to a CI concentration range between 1.5×10 12 and 1×10 11 cm -2 . 4 It has been predicted that the carrier mobility of graphene can reach the ballistic limit of ~2×10 6 cm 2 /Vs if the CI concentration can be decreased to ~10 10 cm -2 . 5 In addition to the charged dopants from molecular adsorption and photoresist residues, 7 the substrate is another major source for charged impurities. Recent transport measurements on suspended graphene (SG) have revealed that the mobility of graphene can be dramatically enhanced to ~2×10 5 cm 2 /Vs. 8,9 Such an enhancement is thought to be due to the absence of long range scattering from the random charged impurities in the substrate. 8 The experimental investigation of charged impurities in SG as well as comparison with those in non-suspended graphene (NSG) would therefore be desirable.
Raman spectroscopy has been widely applied in the study of graphene. [10][11][12][13][14][15][16][17][18] It can be used for determining graphene thickness, 10 monitoring dopant concentration, 11 measuring strain 12,13,14 and for probing the electronic structure of graphene and multilayer graphene. 15 In this study, we compare the Raman spectra of SG and NSG and find that the 2D band intensity of SG is much stronger. This is attributed to the extremely low CI concentration in SG (<10 11 cm -2 ). A detailed study on many pieces of single layer graphene (SLG) suggests that at low CI concentrations that are critical for device applications, the intensity ratio between Raman 2D and G bands is a sensitive indicator of the level of charged impurities present.
RESULTS AND DISCUSSION
The process of fabrication of the SG samples is shown schematically in Figure 1. with a hole diameter of ~8 μm. The sample contains graphene sheets of different thicknesses. SLG was distinguished from 3-layer graphene from the width of the 2D Raman band. The former has a width of ~30 cm -1 while the latter has a width of ~57 cm -1 , 10,19 which can also be seen from the Raman imaging constructed using the 2D bandwidth in Figure 2b. Part of the SLG is suspended over the hole while the remaining part is supported by the SiO 2 /Si substrate. Hence our SG and NSG come from the same piece of SLG. Figures 2c and 2d show the Raman intensity mapping using the G and 2D bands, respectively. The dashed blue circles in the Raman imaging indicate the hole, i.e. the SG area. As we have shown, the Raman intensity for all the Raman bands from the NSG sample is enhanced as a result of the interference effect. 20 This explains the stronger G band Raman signal observed for the NSG sample (about twice the intensity compared to that of SG, as shown in Figure 2c). However, this is not the case for the 2D band intensity.
The 2D Raman band intensity (Figure 2d) for the SG sample is stronger instead of weaker than that of the NSG sample. The difference is more clearly shown in the Raman image in Figure 2f, which is constructed using the I 2D /I G ratio. It can be seen that the I 2D /I G ratio varies significantly from 8.7 for SG to 3.9 for NSG. Figures 2g-2i show the Raman images of the I 2D /I G ratio of three more samples. Similarly, the I 2D /I G ratios of SG are much higher than those of NSG. We will explain this phenomenon later by considering the electron scattering in graphene. The samples used in this work were of high quality as indicated by the absence of an obvious disorder-induced D band in the Raman spectra of SG and NSG in Figure 2e.
It would be interesting to check whether there is any strain in SG. 14,21 To investigate this, the G band frequencies from different pieces of SG and NSG were recorded and the results are shown in Table 1. As we know, the frequency of the G band is very sensitive to strain. It red-shifts with a coefficient of 10 to 15 cm -1 /%strain due to the phonon deformation caused by the change in lattice constant. 14,22 However, of the five SLG samples we studied, the G band frequencies of the SG are the same as those of the NSG, within an experimental error of ~1 cm -1 . The 2D band frequencies of the SG and NSG are also similar (results are not shown). This suggests that the strain in SG is negligible, which is consistent with the results of Berciaud et al. 23 Pereira et al. 21 suggested a method to open a transport bandgap in graphene by introducing local strain in it, which may be realized by placing graphene on local structures of substrates. From our results, it seems that noticeable strain (i.e. more than 1%) is not easily induced in graphene by simply placing it on local rough structures such as holes. This is reasonable as graphene is believed to be very stiff. 24,25 One way to introduce a noticeable strain may be to anneal the SG sample, so that the graphene sheet can deform greatly at the edge of the holes. However, this is not within the scope of this work. In addition to the G band frequency, Table 1 also provides the 2D band width of SG and NSG. It can be seen that the 2D band of SG is much sharper than that of NSG. Such band narrowing is universal for all the samples we tested.
Next, we will focus on the abnormal change in the G and 2D band intensities of SG. The integrated intensity ratios of SG and NSG (I SG /I NSG ) for different Raman bands are shown in Figure 3. The I SG /I NSG of the G band centers at around 0.5, while that of 2D band has a much larger spread, which varies from 0.7 to 1.4. Our previous studies showed that the Raman intensity is strongly dependent on the interference of the laser and the Raman signals. 20 The Raman intensity of NSG (i.e. graphene on a 285 nm SiO 2 film on Si substrate) is greatly increased because of the substrate interference enhancement.
The Raman intensity of the SG is also high because the optical constant n (n air =1) on both sides of graphene is smaller than that of graphene, which makes the interference and multiple reflections of the laser and Raman signal very efficient. 20 The calculated Raman intensity ratio between SG and NSG (on a 285 nm SiO 2 /Si substrate) under 532 nm excitation is ~ 0.51, as indicated by the blue line in Figure 3. This value is very close to the I SG /I NSG ratio of the G band. This suggests that the decrease in the G band intensity for SG is only due to different interference and multiple reflection conditions. On the other hand, the I SG /I NSG ratio for the 2D band is much larger than the calculated value of 0.51. There must be factors in addition to interference and multiple reflections that contribute to such a discrepancy for the 2D band. Furthermore, such factors only affect the 2D band but not the G band.
The above phenomena can be understood by considering electron scattering in graphene. 5 The 2D band is a two-phonon Raman band which comes from the TO phonons around the K point of the Brillouin zone. It is active by the double resonance process which is described as follows: 26
∑ + − + − + − 〉 〉〈 〉〈 〉〈 〈 − ∧ − ∧ − ∧ − ∧ 2 , 1 , 0 2 1 0 2 2 1 1 0 0 ) 2 )( 2 )( 2 ( | | | | | | | | S S S i i i em e ph e ph e em e i E E i E E i E E f H s s H s s H s s H i M γ γ γ (1) where 〉 i | and 〉 f |
Here, v is the Fermi velocity, a is the lattice constant of graphene, M is the mass of the carbon atom, and F K is the coupling constant. ω in and ω K are the frequencies of the incident laser and the 2D phonon at around the K point, respectively. It is clear that I 2D is proportional to 2 1 γ , where 2γ is the electron or hole inelastic scattering rate as mentioned above. As the amount of charged impurities (i.e. the random charged impurities in the substrate) increases, the carrier density in graphene will also increase. 4, 6 Therefore, the probability of electron-electron collisions and the inelastic scattering rate 2γ also increases. According to equation (2), it is obvious that the 2D band intensity will decrease for NSG due to the charged impurities in the SiO 2 substrate. As a result, the I SG /I NSG ratio of the 2D band will increase. Previous theoretical studies 5,29 have revealed that CI scattering from the substrate is one of the major factors that changes the electron mobility of graphene. It has also been observed in transport measurements of SG that the mobility is greatly enhanced due to the absence of long range scattering of electrons or holes with substrate charged impurities. 8 Here, our Raman measurements on SG and NSG provide another evidence for the existence of substrate charged impurities.
On the other hand, the effect of substrate charged impurities on the G band intensity should be very weak. The G band originates from the E 2g phonon, which has a wave vector of zero. Thus, the Raman process for the G band can be satisfied even under non-resonant conditions. As a result, the intensity of G band is expected to be insensitive to most of the external factors, such as polarization, carrier concentration and so on. 27 The effect of substrate charged impurities on the G band intensity hence can be ignored. This is consistent with our observation for SG and NSG. Accordingly, the intensity ratio of the 2D band to the G band, I 2D /I G , would be a good indication of the amount of charge impurities in graphene. Previous studies on SLG samples have revealed an overall decrease of I 2D /I G when the amount of charged impurities in graphene increases. 7 This is further support for our argument. In previous studies, the blue shift in the G band was used as a direct indication of doping or the presence of charged impurities. 11,30 However, we did not observe any obvious blue shift for the G band frequency on NSG with respect to that of SG (Table 1). This is because the NSG samples in Table 1 are only lightly doped, as indicated by their G band frequencies (~1580 cm -1 ). The blue shift in the G band at such low CI concentrations (<10 12 cm -2 ) is only ~1 cm -1 , according to the results of gated-tuned Raman spectroscopy of graphene. 11,30 According to our results, the change in the I 2D /I G ratio is more sensitive to the presence of charged impurities than is the shift of the G band frequency when graphene is lightly doped. We therefore propose that the I 2D /I G intensity ratio is a more effective criterion for the selection of intrinsic SLG sample at low impurity concentration levels (<10 12 cm -2 ) for device applications. The squares in Figure 4 show the I 2D /I G ratios of tens of SLG samples (SG and NSG) at different CI concentrations. The CI concentrations in NSG are estimated from the G band blue shift. 11 The CI concentration in SG is estimated to be 10 10 -10 11 cm -2 . 4,8,29 The higher the I 2D /I G ratio, the lower the CI concentration in graphene. Moreover, the change in the I 2D /I G ratio is more sensitive at low concentration levels. For comparison, the relation between the G band blue shift and CI concentration is also presented in Figure 4. Such a relation is obtained from the results of Raman spectroscopy of graphene with carrier concentrations tuned by gate voltage. 11 It is obvious that at low impurity concentrations (<10 12 cm -2 ), the blue shift in the G band is very small and is not easily distinguished considering the experimental error. Finally, care must be taken when directly comparing the I 2D /I G ratio obtained by different excitation lasers, because this value is also affected by the excitation energy. 31
CONCLUSION
In summary, Raman spectroscopy and imaging were used to study SG and NSG samples. The G band intensity of SG is found to be weaker than that of NSG, due to the substrate interference effect. On the other hand, the 2D band intensity of SG is much stronger than that of NSG due to the absence of substrate charged impurities in SG. This finding is consistent with the ultra-high mobility in suspended graphene observed in recent transport measurements. Our results also suggest that at low CI concentrations (<10 12 cm -2 ), the intensity of the 2D band (or I 2D /I G ) is more sensitive to the presence of charged impurities than is the blue shift of the G band. 11 We therefore propose that the I 2D /I G ratio can be used as a good criterion for selecting intrinsic single graphene samples for device application, where higher I 2D /I G indicates a lower CI concentration and hence a higher carrier mobility.
EXPERIMENTAL AND CALCULATION SECTION
Raman spectroscopy and imaging
Raman imaging /spectroscopy were carried out using a WITEC CRM200 Raman system with 532 nm (2.33 eV) excitation. The laser power at the sample was kept below 0.5 mW to avoid laser induced heating. 18
Raman intensity calculation
The Raman intensity of NSG considering the interference of laser light and Raman signal is calculated by the following formulae: 20 where t is the total amplitude of the electric field at a certain depth y and α is a factor considering the multi-reflection of scattered Raman light in graphene at the interface of graphene/air and graphene/(SiO 2 on Si). The blue line is the calculated value ( ~0.51) using the interference and multiple reflection model. 20 The results clearly indicate that while the G band intensity ratio I SG /I NSG follow the calculated value well, the 2D band intensity ratio I SG /I NSG does not. blue and red squares for NSG and SG, respectively. The solid line is a guide for the eye. For comparison, the relation between the G band blueshift and CI concentration is also presented: black and purple triangles for NSG and SG, respectively.
∫ Δ ⋅ = 1 0 2 d y t I α(3)⋅ ⋅ ⋅ − − ⋅ ⋅ − − ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ ′ + ⋅ ⋅ ′ ⋅ + ⋅ ⋅ = β λ π β λ π β(4)
First, an SiO 2 /
2Si substrate, which consists of a 285 nm-thick SiO 2 film on single crystal Si wafer, was spin-coated with ~10 μm thick photoresist (Figure 1a). Photolithography was then used to pattern holes into the photoresist (Figure 1b). After deep reactive-ion etching (DRIE) of the areas unprotected by the photoresist and subsequent removal of the photoresist, SiO 2 /Si substrate with periodic structures were obtained (Figure 1c). The diameter of the holes (typically between 3 to 8 μm) depends on the original feature size on the photolithographic mask, while the depth of the holes depends on the duration of the DRIE. Graphene samples were prepared on the patterned substrates using the micromechanical cleavage technique (Figure 1d). 19 The probability of finding graphene sheets coving the holes is quite high because of the high concentration of holes. This makes the preparation of SG easy and efficient. As examples, Figures 1e, f and g show the typical optical images of three SG samples.
Figure 2a
2ashows the optical image of a graphene sample on a patterned substrate,
1) an excitation photon creates an electron-hole pair with similar energy at wave vector k. 2) electron-phonon scattering occurs with an exchanged momentum of q. 3) electron-phonon scattering takes place with an exchanged momentum -q, with reverse direction. 4) the electron-hole pair recombines. The matrix element of the process can be schematically represented as:27
are the initial and final states of the process, and S 0 , S 1 , S 2 are the intermediate states where an electron-hole pair is created. E i and E 0 …E 2 are the energies of these states and 2γ is the inverse lifetime of the electron or hole due to collisions or scattering. 2γ is also known as the inelastic scattering rate.
graphite and λ is the excitation wavelength) is a measure of the absorption in the graphene layers. reflection coefficients at the interface of air/graphene, graphene/SiO 2 and
Figure 1 .
1(a)-(d). Schematic diagrams for the preparation of suspended graphene. (a) A layer of photo-resist (10 μm thick) was deposited on the 285 nm SiO 2 /Si substrate. (b)Photolithography was then used to pattern the photo-resist with 10 μm holes. (c)DRIE was used to etch the unprotected SiO 2 and Si. (d) Finally, suspended graphene was prepared on the patterned substrate. Figures (e) to (g) show three graphene samples with areas that are suspended.
Figure 2 .
2(a) Optical image of a graphene sheet on a patterned substrate covering a hole. (b) Raman imaging using the 2D band width. The dark strip with a 2D band width of ~30 cm -1 is SLG. The bright area with 2D width of ~57 cm -1 is three-layer graphene. (c) and (d) are the Raman imaging of G and 2D band intensity, respectively. (e) Raman spectra of SG and NSG taken from the red and blue dots in figure (d), respectively. (f) Raman imaging of the I 2D /I G ratio. (g)-(i) Raman images of I 2D /I G ratio of three more samples. The I 2D /I G ratios of SG are much higher than those of NSG. The scale bars in Raman images are 2 μm.
Figure 3 .
3The G and 2D band integrated intensity ratio of suspended and non-suspended graphene.
Figure 4 .
4The G and 2D band integrated intensity ratio of SLG with different CI concentration:
SiO 2 /Si.indices of air, graphite, SiO 2 , and Si at 532 nm, respectively. 33 d 1 =0.335 nm is the thickness of single layer graphene, d 2 =285 nm is the thickness of SiO 2 and the Si substrate is considered as semi-infinite. =1. The calculated Raman intensity ratio of SG and NSG, I SG /I NSG , is ~0.51.Table 1The G band frequency of SG and NSG from five different samples. The 2D band widths of SG and NSG are also presented.λ
π
2
,
1
2
,
1
2
,
12
d
n
fi
⋅
⋅
=
are the phase differences when light passes through graphene
and SiO 2 , respectively. 0
n =1, 1
n =2.6-1.3i, 2
n =1.46, 3
n =4.15-0.044i, are refractive
The Raman intensity of SG is calculated by simply changing 2
n and 3
n to the
refractive index of air 0
n Samples
G frequency (cm -1 )
2D width (cm -1 )
SG
NSG
SG
NSG
1
1578.7 + 1.3
1577.6 + 1.2
28.1 + 1.6
31.7 + 1.8
2
1579.6 + 0.8
1580.2 + 0.6
29.5 + 1.1
35.4 + 1.7
3
1580.9 + 0.9
1581.1 + 0.4
28.0 + 0.7
31.8 + 0.9
4
1579.2 + 1.7
1580.8 + 1.3
27.6 + 2.0
29.5 + 1.8
5
1582.6 + 1.2
1581.4 + 0.8
26.1 + 1.3
31.6 + 1.4
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[
"Fixed points of endomorphisms and relations between metrics in preGarside monoids",
"Fixed points of endomorphisms and relations between metrics in preGarside monoids"
] |
[
"Oussama Ajbal "
] |
[] |
[] |
In[10], it is proved that the fixed points submonoid and the periodic points submonoid of a trace monoid endomorphism are always finitely generated. We show that for finitely generated left preGarside monoids, that includs finitely generated preGarside monoids, Garside monoids and Artin monoids, the fixed and periodic points submonoids of any endomorphism are also finitely generated left preGarside monoids under some condition, and in the case of Artin monoids, these submonoids are always Artin monoids too. We also prove algebraically some inequalities, equivalences and non-equivalences between three metrics in finitely generated preGarside monoids, and especially in trace monoids and Garside monoids.
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"https://arxiv.org/pdf/1404.5440v1.pdf"
] | 119,325,207 |
1404.5440
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c4306292dee4e272827a58e909735e1248a8242c
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Fixed points of endomorphisms and relations between metrics in preGarside monoids
22 Apr 2014 November, 2013
Oussama Ajbal
Fixed points of endomorphisms and relations between metrics in preGarside monoids
22 Apr 2014 November, 2013arXiv:1404.5440v1 [math.GR]
In[10], it is proved that the fixed points submonoid and the periodic points submonoid of a trace monoid endomorphism are always finitely generated. We show that for finitely generated left preGarside monoids, that includs finitely generated preGarside monoids, Garside monoids and Artin monoids, the fixed and periodic points submonoids of any endomorphism are also finitely generated left preGarside monoids under some condition, and in the case of Artin monoids, these submonoids are always Artin monoids too. We also prove algebraically some inequalities, equivalences and non-equivalences between three metrics in finitely generated preGarside monoids, and especially in trace monoids and Garside monoids.
Introduction
Trace monoids, or equivalently partially commutative monoids, are monoids of a particular interest and have been widly studied. In particular, they are Artin Monoids. This explains why they are also called Right Angle Artin monoids (RAAM for short). In [10,11], the authors consider endomorphisms of trace monoids. They study the submonoid of fixed points of such an endomorphism and prove that it is finitely generated. They also caracterise those endomorphisms that are contractions relatively to some natural distance on trace monoids. In [4], Crisp obtained similar results for Artin monoids. He proved that the submonoid of fixed points is even finitely presented, but only in the special case of an isomorphism. Here, we aim to unified and extend both results to all Artin monoids and, more generally, to the larger classes of preGarside monoids.
Given a monoid M , we denote by End(M ) the endomorphism monoid of M . For ϕ ∈ End(M ), we say that x ∈ M is a fixed point of ϕ if ϕ(x) = x. If ϕ n (x) = x for some n ≥ 1, we say that x is a periodic point of ϕ. Let Fix(ϕ) (respectively Per(ϕ)) denote the submonoid of all fixed points (respectively periodic points) of ϕ. Clearly,
Per(ϕ) = n≥1 Fix(ϕ n ).
Our First result is the following:
Theorem 0.1.
(i) If M is a finitely generated left preGarside monoid, with an additive and homogeneous norm ν, and ϕ is in End(M ) such that the morphism π is well-defined, then Fix(ϕ) and Per(ϕ) are also finitely generated left preGarside monoids.
(ii) if M is an Artin monoid, and ϕ is in End(M ), then Fix(ϕ) and Per(ϕ) are also Artin monoids.
Similar result holds if one considers right preGarside monoids.
Two distances d 2 and d 3 on trace monoids have been introduced in [2] and [8]. They were proved to be uniformaly equivalent in [7]. In [10], the authors caracterise those endomorphisms that are contractions relatively to d 2 . We will prove that in the general context of Artin monoids, distance d 2 and d 3 are no more uniformaly equivalent in general. Indeed, distance d 2 should be replace by an alternative one, that we denote by d 1 . We prove that d 1 is larger than the other two in the general case, and we will also prove that d 1 and d 3 are uniformly equivalent in the case of Artin monoids of spherical type.
Finally, replacing d 2 by d 1 , we extend [10,Theorem 4.1], to all Artin monoids in Theorem 4.2 (we refer to the next sections for notations).
Theorem 0.2. Let M be an Artin monoid and ϕ be in End(M ).
(i) ϕ is a contraction with respect to d 1 ;
(ii) for all u, v ∈ M red , α(uv) = u ⇒ α(ϕ(uv)) = α(ϕ(u)).
Preliminaries
We start in this section by defining the monoids we will work with in this paper, namely left preGarside monoids and Artin monoids.
Consider a monoid M . It is said to be cancellative if, for all a, b, c, d ∈ M , the equality cad = cbd imposes a = b. An element b is called a factor of an element a if we can write a = cbd in M . We denote by Div(a) the set of factors of a. We denote by left divisibility in M (that is, for a, b ∈ M , we have a b when there exists c ∈ M such that b = ac). Right divisibility (defined similarly : b a when there exists c ∈ M such that b = ca) will rarely be used in this paper, so divisibility in M will simply mean left divisibility. When a is a left divisor of b in M , we say that b is a right multiple of a. An element a is said to be balanced if its sets of right-divisors and of left-divisors are equal, which in this case have to be equal to Div(a).
We say that M is atomic if there exists a mapping ν : M → N, called a norm, satisfying ν(a) > 0 for a = 1 and ν(ab) ≥ ν(a) + ν(b) for all a, b ∈ M . Note that the existence of such a mapping implies that the relations and are partial orders on M . When ν(ab) = ν(a) + ν(b) for all a, b ∈ M , we say that ν is additive. An atom in a monoid is an element a ∈ M satisfying: a = bc ⇒ b = 1 or c = 1 for all b, c ∈ M . We denote by S(M ) the set of atoms of M . Note that in an atomic monoid M , the set S(M ) has to be a generating set, and that any generating set of M contains S(M ). In particular, M is finitely generated if and only if S(M ) is finite. In an atomic monoid M , if ν(a) = ν(b) for all a, b ∈ S(M ), we say that ν is homogeneous.
A monoid M is said to be a left preGarside monoid if (a L ) it is atomic and left cancellative (ca = cb ⇒ a = b);
(b L ) for all a, b ∈ M , if the set {c ∈ M | a
c and b c} is nonempty, then it has a least element, denoted by a ∨ L b or a ∨ b.
It is said to be a right preGarside monoid if (a R ) it is atomic and right cancellative (ac = bc ⇒ a = b);
(b R ) for all a, b ∈ M , if the set {c ∈ M | c
a and c b} is nonempty, then it has a least element, denoted by a ∨ R b.
And it is said to be a preGarside monoid if it is both left preGarside and right preGarside monoid.
A Garside element of a preGarside monoid is a balanced element whose set of factors generates the whole monoid. When such an element exists, we say that the monoid is a Garside monoid.
Given a non empty finite set S, a Coxeter matrix is a symmetric matrix (m ab ) a,b∈S with entries in {1, 2, · · · , ∞}, such that m aa = 1 and m ab ≥ 2, for a = b. A Coxeter system associated to a Coxeter matrix (m ab ) a,b∈S is a pair (W, S), where W is the group with presentation
W = S | (ab) m ab = 1; m ab = ∞ .
The corresponding Artin monoid M is the monoid with presentation
M = S | [a, b m ab = [b, a m ab ; m ab = ∞ + .
where [a, b m denotes the alternating product aba · · · containing m terms. Since the defined Artin relations are homogeneous, M has a natural length function ℓ S compatible with the product. In [3], it is shown that every finite subset of M has a greatest common left divisor (gcd), and a greatest common right divisor (gcrd). It is also shown that a finite subset of S has a least common right multiple (lcm) if and only if it has a common right multiple, and that in that case, the least common right multiple and least common left multiple are equal. For a subset T ⊆ S, we denote its lcm by ∆(T ) when it exists.
The submonoids of fixed points and of periodic points
In general, a fixed points submonoid or any submonoid of a finitely generated monoid is not necessarily finitely generated.
Example 2.1. Consider the cancellative monoid M = a, e, b | ae = ea, ebe = b + and the endomorphism ϕ such that ϕ(a) = ae, ϕ(e) = e and ϕ(b) = b. Clearly, a n ba n ∈ Fix(ϕ) for every n ∈ N. And one can show that a n ba n is not decomposable in Fix(ϕ), which means that Fix(ϕ) is not finitely generated.
In this section, we will show that for a finitely generated left preGarside monoid M , and an endomorphism ϕ ∈ End(M ), the submonoids Fix(ϕ) and Per(ϕ) are also finitely generated left preGarside monoids under some condition (Theorems 2.3 and 2.13). And that, in the particular case of Artin monoids, Fix(ϕ) and Per(ϕ) are not only finitely generated, but even finitely presented (Proposition 2.15).
For a monoid M , finitely generated by S, and an endomorphism ϕ ∈ End(M ), we define n ϕ = max{k ∈ N * | ∃s ∈ S such that ϕ k (s) = 1 and
ϕ k−1 (s) = 1} if 1 ∈ ϕ(S) 1 if 1 / ∈ ϕ(S) .
Given a non empty set X, we denote by X * the set of all finite words x 1 · · · x n over the elements of X, that we call letters, and by ε the empty word in X * . Assume M is a monoid generated by a set S, and let T ⊆ S. Let us denote by π * T : S * → T * the forgetting morphism of monoids defined by π * T (t) = t for t ∈ T , and by π * T (t) = ε for t ∈ S \ T . In the sequel, when it is well-defined, we denote by π T : M → M the morphism of monoids induced by π * T .
Example 2.2. For M = s, t | sts = tst + , with S = {s, t} and T = {t}, we have π * T (sts) = t and π * T (tst) = tt. Or in M , one has t = t 2 , then π T is not well-defined. But for M = s, t | stst = tsts + with similar S and T , the morphism π T is well-defined, since π T (stst) = π T (tsts) = t 2 .
The submonoid of fixed points
For this subsection, let M be a finitely generated left preGarside monoid, equiped with an additive and homogeneous norm ν, and ϕ be in End(M ). Set
S = S(M ), S 0 = S ∩ Per(ϕ), S 1 = S ∩ (ϕ nϕ ) −1 {1}, S 2 = S \ S 1 ,
p = |S|!, and π := π S 2 when it is well-defined, which is always the case when 1 / ∈ ϕ(S).
Theorem 2.3. Let M be a finitely generated left preGarside monoid, with an additive and homogeneous norm ν, ϕ be in End(M ), and π as above. If the morphism π is well-defined, then Fix(ϕ) is a finitely generated left preGarside monoid.
The proof of this theorem is in the spirit of [10, Theorem 3.1], where the particular case of trace monoids was considered. We will start though by proving some lemmas. Note that in the proof of the next lemma, where ϕ| S is a permutation (i.e. ϕ is an automorphism), we do not need the additivity or the homogeneity of the norm ν. In the case of preGarside monoids, this result is shown in [1, Proposition 2.26].
Lemma 2.4. If the restriction of ϕ to S is a permutation, then Fix(ϕ) is a finitely generated left preGarside monoid.
Proof. Since Fix(ϕ) ⊆ M , it is clear that property (a L ) holds for Fix(ϕ). Let a, b ∈ Fix(ϕ) be such that the set {c ∈ Fix(ϕ) | a c and b c} is nonempty, and set δ = a ∨ b their (left) lcm in M . We have ϕ(a) = a ϕ(δ) and ϕ(b) = b ϕ(δ), then δ ϕ(δ). Thus δ ϕ(δ) · · · ϕ p (δ), and so ν(δ) ≤ ν(ϕ(δ)) ≤ · · · ≤ ν(ϕ p (δ)). Or ϕ| S is a permutation, and p = |S|!, then ϕ p = Id M . Therefore ν(δ) = ν(ϕ(δ)), with δ ϕ(δ). Thus, we have δ = ϕ(δ), and so δ ∈ Fix(ϕ). Let c ∈ Fix(ϕ) such that a c and b c. Write c = δδ ′ with δ ′ ∈ M . We have c, δ ∈ Fix(ϕ), then, by cancellativity, δ ′ ∈ Fix(ϕ). Hence property (b L ) holds, and so Fix(ϕ) is a left preGarside monoid.
Let Σ be the set of all ϕ-orbits B in S that have a right common multiple (and therefore a least right common multiple ∆(B) ([6, Lemma 2.1])). We claim that
Fix(ϕ) = ∆(B), B ∈ Σ + .(1)
Let B ∈ Σ, and b ∈ B. Since B is a ϕ-orbit, then there exists a ∈ B such that b = ϕ(a). But a ∆(B), so b = ϕ(a) ϕ(∆(B)). Thus b ϕ(∆(B)) for every b ∈ B, and then ∆(B) ϕ(∆(B)). Therefore, as above, we have ∆(B) = ϕ(∆(B)). Hence ∆(B), B ∈ Σ + ⊆ Fix(ϕ). Conversely, let u ∈ Fix(ϕ) such that u = 1, and let s ∈ S such that s u; then u is left divisible by all the elements in the ϕ-orbit B 1 of s, so is left divisible by their lcm u 1 := ∆(B 1 ) (which exists). Let v ∈ M such that u = u 1 v. Since u, u 1 ∈ Fix(ϕ), we have by cancellativity, v ∈ Fix(ϕ). By induction on ν(u), we get that v ∈ ∆(B) | B ∈ Σ + . Thus u ∈ ∆(B) | B ∈ Σ + and so (1) holds. The set S is finite, then so is Σ. Therefore Fix(ϕ) is finitely generated. i. Let M = s, t, u | st = ts, sus = usu, tut = utu + , and ϕ ∈ End(M ) such that ϕ(u) = u, ϕ(t) = s and ϕ(s) = t. The monoid M is an Artin monoid, so it satisfies all properties of preGarside monoids by [3]. Then we have Σ = {{u}, {s, t}}, and Fix(ϕ) = u, st + .
ii. Let M = a 1 , b 1 , a 2 , b 2 | a 1 b 1 a 1 = b 2 1 , a 2 b 2 a 2 = b 2 2 , a 1 a 2 = a 2 a 1 , b 1 b 2 = b 2 b 1 , a 1 b 2 = b 2 a 1 , b 1 a 2 = a 2 b 1 + , and ϕ ∈ End(M ) such that ϕ(a 1 ) = a 2 , ϕ(a 2 ) = a 1 , ϕ(b 1 ) = b 2 and ϕ(b 2 ) = b 1 . The monoid M is a preGarside monoid, as a direct product M = a 1 , b 1 | a 1 b 1 a 1 = b 2 1 + × a 2 , b 2 | a 2 b 2 a 2 = b 2 2 + of two preGarside monoids (see [5]). Then we have Σ = {{a 1 , a 2 }, {b 1 , b 2 }}, and Fix(ϕ) = a 1 a 2 , b 1 b 2 + .
Lemma 2.6. If 1 / ∈ ϕ(S), then S 0 = S ∩ Fix(ϕ p ) = S ∩ ϕ p (S),(2)
and
ϕ(S 0 ) = S 0 .(3)
Proof. Recall that S 0 = S ∩ Per(ϕ). Let a ∈ S 0 and m = min{k ∈ N * /ϕ k (a) = a}. Since 1 / ∈ ϕ(S) and ν is additive and homogeneous, we have ν(ϕ(u)) ≥ ν(u) for every u ∈ M . Hence ϕ n (a) ∈ S for every n ∈ N. Assume that there exist 0 < i < j ≤ m such that ϕ i (a) = ϕ j (a). So ϕ m−j • ϕ i (a) = ϕ m (a). Then ϕ m−(j−i) (a) = a and m − (j − i) < m, which contradicts the definition of m. Thus m = #{a, ϕ(a), · · · , ϕ m−1 (a)} ≤ |S|, and then m divides p. Hence ϕ p (a) = a. So S 0 ⊆ S ∩ Fix(ϕ p ), and we clearly have S ∩ Fix(ϕ p ) ⊆ S ∩ ϕ p (S).
Let a ∈ S ∩ ϕ p (S) and b ∈ S such that ϕ p (b) = a. Since 1 / ∈ ϕ(S), using that ν is additive and homogeneous, and ϕ p (b) = a, we get {b, ϕ(b), · · · , ϕ p (b)} ⊆ S. The inequality |S| < p + 1 yields ϕ i (b) = ϕ j (b) for some 0 ≤ i < j ≤ p. By composing with ϕ p−i , we get a = ϕ j−i (a). Then a ∈ Per(ϕ) ∩ S = S 0 , and therefore (2) holds.
Let a ∈ S 0 . As before, we have {a, ϕ(a), . . . , ϕ p−1 (a)} ⊆ S 0 = S ∩ Fix(ϕ p ). On the one hand, we have ϕ(a) ∈ S 0 , then ϕ(S 0 ) ⊆ S 0 . On the other hand, we have a = ϕ(ϕ p−1 (a)) and ϕ p−1 (a) ∈ S 0 , then ϕ(S 0 ) ⊇ S 0 , and so (3) holds.
Let M 0 , M 1 and M 2 , be the submonoids of M generated by S 0 , S 1 and S 2 respectively. By definition of S 1 and atomicity of M , note that ϕ nϕ (M 1 ) = {1} and ϕ(M 1 ) ⊆ M 1 , which we will be using more than once.
Lemma 2.7. If 1 / ∈ ϕ(S), then M 0 = Fix(ϕ p ),
it is a finitely generated left preGarside monoid, and the restriction of ν to M 0 is additive and homogeneous.
Proof. The order of every periodic point of S divides p = |S|!, then M 0 ⊆ Fix(ϕ p ). Let s 1 , . . . , s n ∈ S such that ϕ p (s 1 · · · s n ) = s 1 · · · s n . In view of the homogeneity and additivity of ν, and the fact that 1 / ∈ ϕ(S), we have ϕ(s i ) ∈ S for all i. By (2), S 0 = S ∩ ϕ p (S). Then ϕ(s 1 · · · s n ) ∈ S 0 + , which means that s 1 · · · s n ∈ M 0 . Thus we have the equality M 0 = Fix(ϕ p ). The submonoid M 0 is finitely generated by definition, and it is atomic and left cancellative because M 0 ⊆ M . The restriction ν| M 0 of ν is additive, and since S(M 0 ) = S 0 , it is also homogeneous. Let a, b lie in M 0 such that the set Γ 0 = {c ∈ M 0 | a c and b c} is nonempty, and set δ = a ∨ b their lcm in M . Set ψ = ϕ p . Since S 0 = S ∩ Fix(ψ), we have ψ| M 0 = Id M 0 . Let c lie in Γ 0 and a ′ , a ′′ belong to M such that c = aa ′ and δ = aa ′′ . One has a, c ∈ Fix(ψ), then a ′ = ψ(a ′ ) by cancellativity. We have δ c, then a ′′ a ′ , and so ψ n (a ′′ ) a ′ for all n ∈ N. The sequence of integers (ν(ψ n (a ′′ ))) n∈N is increasing because of the additivity and homogeneity of ν and the fact that 1 / ∈ ϕ(S). On the other hand, it is bounded by ν(a ′ ) because ψ n (a ′′ ) a ′ for all n. Thus (ν(ψ n (a ′′ ))) n∈N is stationary from some rank m a ∈ N * . Write ψ ma (a ′′ ) = s 1 · · · s r with s 1 , . . . , s r ∈ S. Since ν(ψ ma (a ′′ )) = ν(ψ ma+1 (a ′′ )), then
ψ(s i ) ∈ S for all i. So {s i , ϕ(s i ), . . . , ϕ p (s i )} ⊆ S, and therefore ψ(s i ) ∈ S 0 for all i. Thus ψ ma+1 (a ′′ ) ∈ M 0 . Similarly, for b ′ , b ′′ ∈ M such that c = bb ′ and δ = bb ′′ , we have some rank m b ∈ N * such that ψ m b +1 (b ′′ ) is in M 0 . The inclusion M 0 ⊆ Fix(ψ) yields ψ ma+1 (a ′′ ) = ψ ma+m b (a ′′ ) and ψ m b +1 (b ′′ ) = ψ m b +ma (b ′′ ). We have ψ ma+m b (δ) = aψ ma+1 (a ′′ ) = bψ m b +1 (b ′′ ), then ψ ma+m b (δ) ∈ Γ 0 . If c ∈ Γ 0 , then δ c, and so ψ ma+m b (δ) ψ ma+m b (c) = c. Write c = ψ ma+m b (δ)c ′ with c ′ ∈ M . Both c and ψ ma+m b (δ) are in Fix(ψ), then by cancellativity, ψ(c ′ ) = c ′ . Thus, c ′ ∈ M 0 and ψ ma+m b (δ) is the least element of Γ 0 . Whence property (b L ) holds, so M 0 is a left preGarside monoid.
Lemma 2.8. If π is well-defined, then M 2 is a finitely generated left preGarside monoid, and the restriction of ν to M 2 is additive and homogeneous.
Proof. As in the previous proof, the submonoid M 2 is atomic, left cancellative and finitely generated, and the restriction ν| M 2 is additive and homogeneous. Let a, b lie in M 2 such that the set Γ 2 = {c ∈ M 2 | a c and b c} is nonempty, and set δ = a ∨ b their lcm in M . Let c ∈ Γ 2 and a ′ , a ′′ ∈ M such that c = aa ′ and δ = aa ′′ . We have δ c, then a ′′ a ′ . So write a ′ = a ′′â withâ ∈ M . By the homogeneity and additivity of ν, for all u ∈ M we have ν(π(u)) ≤ ν(u), and ν(π(u)) = ν(u) ⇔ π(u) = u ⇔ u ∈ M 2 . We have ν(c) = ν(a) + ν(a ′′ ) + ν(â) and ν(π(c)) = ν(π(a)) + ν(π(a ′′ )) + ν(π(â)). But a and c belong to M 2 , then ν(a ′′ ) + ν(â) = ν(π(a ′′ )) + ν(π(â)).
So ν(π(a ′′ )) = ν(a ′′ ) because ν(π(u)) ≤ ν(u) for all u ∈ M . Then a ′′ ∈ M 2 . Similarly, there is b ′′ in M 2 such that δ = bb ′′ . Thus δ is the least element of Γ 2 , whence property (b L ). Therefore M 2 is a left preGarside monoid.
Lemma 2.9. Let N 1 be a left preGarside monoid, and N 2 a monoid. Assume there exists a morphism f :
N 1 → N 2 that is a retraction. Then N 2 is left preGarside.
Proof. The morphism f is a retraction, then we have a section g :
N 2 → N 1 such that f • g = Id N 2 . Thus N 2 embeds in N 1 . So N 2 is atomic and left cancellative. Let a, b ∈ N 2 such that the set Λ = {c ∈ N 2 | a c and b c} is nonempty. Set δ = g(a) ∨ g(b) the least common right multiple of g(a) and g(b) in N 1 , and write δ = g(a)a ′ = g(b)b ′ with a ′ , b ′ ∈ N 1 . Thus f (δ) = af (a ′ ) = bf (b ′ ). Let c ∈ Λ and c ′ ∈ N 1 such that g(c) = δc ′ . Then c = f (δ)f (c ′ ), and so f (δ) divides c in N 2 . Hence, f (δ) is the least element of Λ, and therefore N 2 is left preGarside. Lemma 2.10. For every d ∈ N * , we have (π • ϕ) d = π • ϕ d .
Proof. If 1 / ∈ ϕ(S), then S 1 = ∅ and π = Id M . Assume 1 ∈ ϕ(S). We show the result by induction on d. The case d = 1 being trivial, assume d > 1 and the result holds for smaller integers. It suffices to check the equality on the generators. By definition of
S 1 , if a ∈ S 1 , then ϕ d (a) ∈ M 1 = S 1 + because ϕ(M 1 ) ⊆ M 1 . Thus, (π • ϕ) d (a) = π • ϕ d (a) = 1.
If a ∈ S 2 , write ϕ(a) = u 0 a 1 u 1 · · · a k u k , with a 1 , . . . , a k ∈ S 2 and u 0 , . . . , u k ∈ M 1 . On the one hand, we have
π • ϕ d (a) = π • ϕ d−1 (u 0 a 1 u 1 · · · a k u k ) = π • ϕ d−1 (a 1 · · · a k ) in view of ϕ d−1 (u i ) ∈ M 1 for every i.
On the other hand, and by the induction hypothesis,
(π • ϕ) d (a) = (π • ϕ) d−1 • π • ϕ(a) = π • ϕ d−1 • π(u 0 a 1 u 1 · · · a k u k ) = π • ϕ d−1 (a 1 · · · a k ). Thus (π • ϕ) d (a) = π • ϕ d (a) for every d ∈ N * .
In view of the previous lemmas, we may now prove Theorem 2.3 in two parts, depending on whether 1 lies in ϕ(S) or not.
Proof of Theorem 2.3. Case I: 1 / ∈ ϕ(S). By the equality (3) in Lemma 2.6, the morphism ϕ restricts to an endomorphism ϕ 0 of M 0 = S 0 + . We show that
Fix(ϕ) = Fix(ϕ 0 ).(4)
It is immediate that Fix(ϕ 0 ) = Fix(ϕ) ∩ M 0 , so it suffices to show that Fix(ϕ) ⊆ M 0 . Let u = a 1 · · · a k belong to Fix(ϕ), with a 1 , . . . , a k ∈ S. Then a 1 · · · a k = u = ϕ p (u) = ϕ p (a 1 ) · · · ϕ p (a k ). Since 1 / ∈ ϕ(S), and ν is additive and homogeneous, we have ν(ϕ p (a i )) = ν(a i ) and so ϕ p (a i ) ∈ S for all i. By Lemma 2.6, we have S 0 = S ∩ ϕ p (S), so ϕ p (a i ) ∈ S 0 for all i, and then u = ϕ p (a 1 ) · · · ϕ p (a k ) ∈ M 0 . Therefore, Fix(ϕ) ⊆ M 0 , and so Fix(ϕ) = Fix(ϕ 0 ). Now ϕ 0 | S 0 is a permutation, and by Lemma 2.7, M 0 is a finitely generated left preGarside monoid. Then by Lemma 2.4 (where in the case of a permutation, the norm of M 0 does not have to be additive or homogeneous), Fix(ϕ 0 ), and therefore Fix(ϕ), is a finitely generated left preGarside monoid.
Case II: 1 ∈ ϕ(S). Denote n = n ϕ , and recall that π = π S 2 . Consider the morphism ϕ 2 = (π • ϕ)| M 2 that is clearly in End(M 2 ). We have 1 / ∈ ϕ 2 (S 2 ). Indeed, if ϕ 2 (s) = π(ϕ(s)) = 1 for some s ∈ S 2 , then ϕ(s) ∈ M 1 , which means that ϕ n (ϕ(s)) = ϕ n+1 (s) = 1. But since n = max{k ∈ N * | ∃s ∈ S such that ϕ k (s) = 1 and ϕ k−1 (s) = 1}, then ϕ n (s) = 1, which contradicts the fact that s ∈ S 2 . Thus, by Lemma 2.8 and Case I, Fix(ϕ 2 ) is a finitely generated left preGarside monoid.
We claim that Fix(ϕ) = ϕ n (Fix(ϕ 2 )).
As seen before, we have ϕ n (M 1 ) = {1} and ϕ(M 1 ) ⊆ M 1 . Let u ∈ Fix(ϕ). We may factor u = u 0 a 1 u 1 · · · a k u k , with a 1 , . . . , a k ∈ S 2 and u 0 , . . . , u k ∈ M 1 . It follows that u = ϕ n (u) = ϕ n (a 1 a 2 · · · a k ).
For every i, we have π • ϕ(u i ) = 1, in view of ϕ(M 1 ) ⊆ M 1 . Now a 1 a 2 · · · a k ∈ M 2 , and ϕ 2 (a 1 a 2 · · · a k ) = π • ϕ(a 1 a 2 · · · a k ) = π • ϕ(u 0 a 1 u 1 · · · a k u k ) = π(u) = a 1 a 2 · · · a k .
Hence a 1 a 2 · · · a k ∈ Fix(ϕ 2 ), and so u = ϕ n (a 1 a 2 · · · a k ) ∈ ϕ n (Fix(ϕ 2 )). Thus Fix(ϕ) ⊆ ϕ n (Fix(ϕ 2 )).
Conversely, let v = a 1 a 2 · · · a k ∈ Fix(ϕ 2 ), with a 1 , . . . , a k ∈ S 2 . Clearly, ϕ n • π = ϕ n .(6)
Hence
v = ϕ 2 (v) = π • ϕ(v) yields ϕ(ϕ n (v)) = ϕ n • ϕ(v) = ϕ n • π • ϕ(v) = ϕ n (π • ϕ(v)) = ϕ n (v) and so ϕ n (v) ∈ Fix(ϕ). Thus ϕ n (Fix(ϕ 2 )) ⊆ Fix(ϕ) and so Fix(ϕ) = ϕ n (Fix(ϕ 2 )).
In view of (5), we have a morphism f := ϕ n | Fix(ϕ 2 ) : Fix(ϕ 2 ) → Fix(ϕ). Let u ∈ Fix(ϕ). By Lemma 2.10, we get ϕ 2 (π(u)) = π •ϕ•π(u) = π •ϕ•π •ϕ(u) = π •ϕ 2 (u) = π(u). Then π(Fix(ϕ)) ⊆ Fix(ϕ 2 ), and so we have another morphism g := π| Fix(ϕ) : Fix(ϕ) → Fix(ϕ 2 ). In view of (6), one has ϕ n • π(u) = ϕ n (u) = u for every u ∈ Fix(ϕ), and then f • g = Id| Fix(ϕ) . So the morphism f is a retraction with section g. We established that Fix(ϕ 2 ) is a left preGarside monoid. So by Lemma 2.9, Fix(ϕ) is also a left preGarside monoid. The submonoid Fix(ϕ 2 ) is finitely generated, then so is Fix(ϕ), in view of (5).
We will see in the proof of Proposition 2.15 that the equality (5) induces an isomorphism between Fix(ϕ) and Fix(ϕ 2 ).
Example 2.11. Let M = a, b, c | abab = baba, ac = ca + , and ϕ ∈ End(M ) such that ϕ(a) = b, ϕ(b) = a and ϕ(c) = 1, with ν additive and ν(a) = ν(b) = ν(c) = 1. By using the notations above, we have n = 1, S 1 = {c}, S 2 = {a, b}, M 2 = a, b | abab = baba + and ϕ 2 ∈ End(M 2 ), such that ϕ 2 (a) = b and ϕ 2 (b) = a. Then one has Fix(ϕ 2 ) = abab + , and Fix(ϕ) = ϕ(Fix(ϕ 2 )) = Fix(ϕ 2 ) = abab + .
The submonoid of periodic points
As in the previous subsection, we consider a finitely generated left preGarside monoid M , equiped with an additive and homogeneous norm ν, and we fix ϕ ∈ End(M ). We also set n = n ϕ , S = S(M ), S 0 = S ∩ Per(ϕ), S 1 = S ∩ (ϕ n ) −1 {1}, S 2 = S \ S 1 , p = |S|!, and π := π S 2 when it is well-defined.
Proposition 2.12. If the morphism π is well-defined, then we have
Per(ϕ) = Fix(ϕ pn ).
The proof of this proposition is also in the spirit of [10,Theorem 3.2], where the particular case of trace monoids was considered.
Proof. Case I: 1 / ∈ ϕ(S). By definition, n = 1 in this case. We will use induction on |S|. The case |S| = 0 being trivial, assume that |S| > 0 and the result holds for smaller sets.
We may assume S 0 S, otherwise ϕ| S would be a permutation, and since the order of ϕ| S must divide the order of the symmetric group on S, which is p, we would get (ϕ| S ) p = Id S and therefore ϕ p = Id M , yielding Fix(ϕ p ) = M = Per(ϕ).
For every r ∈ N * , if we replace ϕ by ϕ r , then S 0 remains the same in view of Per(ϕ) = Per(ϕ r ), and so does M 0 . On the other hand, by (3), we restrict ϕ to ϕ 0 = ϕ| M 0 , and we have ϕ
r | M 0 = (ϕ| M 0 ) r = ϕ r 0 . Hence Fix(ϕ r ) = Fix(ϕ r 0 )(7)
by applying (4) to ϕ r . By the induction hypothesis and Lemma 2.7, we have Per(ϕ 0 ) = Fix(ϕ
|S 0 |! 0 ). Since |S 0 |! divides p, we get Per(ϕ 0 ) = Fix(ϕ |S 0 |! 0 ) ⊆ Fix(ϕ p 0 ) ⊆ Per(ϕ 0 ) and so Per(ϕ 0 ) = Fix(ϕ p 0 )
. Together with (7), this yields
Per(ϕ) = ∪ r≥1 Fix(ϕ r ) = ∪ r≥1 Fix(ϕ r 0 ) = Per(ϕ 0 ) = Fix(ϕ p 0 ) = Fix(ϕ p )
as required.
Case II: 1 ∈ ϕ(S). By definition, we have Per(ϕ) ⊇ Fix(ϕ pn ). Conversely, let u ∈ Per(ϕ), say u ∈ Fix(ϕ r ). We may factor u = u 0 a 1 u 1 · · · a k u k , with a 1 , . . . , a k ∈ S 2 and u 0 , . . . , u k ∈ M 1 . It follows that u = ϕ rn (u) = ϕ rn (a 1 a 2 · · · a k ). Now a 1 a 2 · · · a k ∈ M 2 , and Lemma 2.10 yields a 1 a 2 · · · a k = π • ϕ rn (a 1 a 2 · · · a k ) = (π•ϕ) rn (a 1 a 2 · · · a k ). Consequently a 1 a 2 · · · a k belongs to Fix(ϕ rn 2 ) ⊆ Per(ϕ 2 ). As in the proof of Theorem 2.3, we have 1 / ∈ ϕ 2 (S 2 ). Thus, by Lemma 2.8 and Case I, we have Per(ϕ 2 ) = Fix(ϕ
|S 2 |! 2
). We get a 1 a 2 · · · a k ∈ Fix(ϕ
|S 2 |! 2 ) ⊆ Fix(ϕ pn
2 ), and so a 1 a 2 · · · a k = π • ϕ pn (a 1 a 2 · · · a k ) in view of Lemma 2.10. Hence ϕ pn (u) = ϕ pn (a 1 a 2 · · · a k ) = v 0 a 1 v 1 · · · a k v k for some v 0 , v 1 , . . . , v k in M 1 . Thus ϕ 2pn (u) = ϕ pn • π • ϕ pn (u) = ϕ pn • π(v 0 a 1 v 1 · · · a k v k ) = ϕ pn • π(u 0 a 1 u 1 · · · a k u k ) = ϕ pn (u).
Since ϕ r (u) = u, this yields to u = ϕ r (u) = ϕ 2r (u) = · · · = ϕ pnr (u) = ϕ pn(r−1) (u) = · · · = ϕ pn (u). Therefore Per(ϕ) = Fix(ϕ pn ).
Theorem 2.13. If the morphism π is well-defined, then Per(ϕ) is also a finitely generated left preGarside monoid.
Proof. In Proposition 2.12, we showed that
Per(ϕ) = Fix(ψ), where ψ = ϕ pn . Denote S 1 (ϕ) = S 1 , S 1 (ψ) = S ∩ (ψ n ψ ) −1 {1}, S 2 (ϕ) = S 2 , S 2 (ψ) = S \ S 1 (ψ)
, π(ϕ) = π and π(ψ) = π S 2 (ψ) . We have S 1 (ϕ) = {s ∈ S | ∃k ∈ N such that ϕ k (s) = 1} = {s ∈ S | ∃k ∈ N such that ψ k (s) = 1} = S 1 (ψ). Then S 2 (ψ) = S 2 (ϕ), and so π(ψ) = π(ϕ) = π. Thus we can apply Theorem 2.3 to ψ, which means that Fix(ψ), and therefore Per(ϕ), is a finitely generated left preGarside monoid.
The case of Artin monoids
A symmetry of an Artin group A generated by S, is an endomorphism ϕ of A such that ϕ |S is a permutation. In [4,Lemma 10] and [9,Corollary 4.4], it is shown that, given a group G of symmetries of an Artin group A, the submonoid of elements fixed by G, is isomorphic to another Artin monoid. In particular, given an Artin monoid M generated by S, and ϕ ∈ End(M ) such that ϕ |S is a permutation (i.e. ϕ ∈ Aut(M )), the submonoid Fix(ϕ) is also an Artin monoid. Below, we will show that this is also the case for Per(ϕ), and for every ϕ ∈ End(M ).
Let M = S | [a, b m ab = [b, a m ab ; m ab = ∞ + be an Artin monoid, and ϕ be in End(M ). By [3], Artin monoids satisfy all properties of preGarside monoids. The set of atoms S(M ) of M is S, and the length ℓ S is an additive and homogeneous norm over M . Thus, we can apply the results from the previous subsections. As before, set n = n ϕ , S 0 = S ∩ Per(ϕ), S 1 = S ∩ (ϕ n ) −1 {1}, S 2 = S \ S 1 , p = |S|!, and π := π S 2 when it is well-defined. It is known that the submonoids M 0 = S 0 + , M 1 = S 1 + and M 2 = S 2 + are Artin monoids too.
Lemma 2.14. The morphism π is well-defined.
Proof. If 1 / ∈ ϕ(S), then S 2 = S and π = Id M . Suppose 1 ∈ ϕ(S). It suffices to verify that π([a, b m ab ) = π([b, a m ab ) for all m ab = ∞. Let a, b ∈ S such that m ab = ∞. If m ab is even, or if a and b are both in S 1 or in S 2 , the equality holds trivially. Suppose we have m ab = 2k + 1 for some a ∈ S 1 , b ∈ S, and k > 0. Then ϕ n ([a, b m ab ) = (ϕ n (b)) k and ϕ n ([b, a m ab ) = (ϕ n (b)) k+1 . Thus, by cancellativity, ϕ n (b) = 1, so b ∈ S 1 and we are done as remarked above.
Proposition 2.15. Let M be an Artin monoid, and ϕ be in End(M ). Then the submonoids Fix(ϕ) and
Per(ϕ) are also Artin monoids.
Proof. Assume first 1 / ∈ ϕ(S). In the proof of Theorem 2.3, we showed that Fix(ϕ) = Fix(ϕ 0 ), with ϕ 0 ∈ End(M 0 ) and ϕ| S 0 is a permutation. Then by [4, Lemma 10], Fix(ϕ 0 ), and therefore Fix(ϕ), is an Artin monoid. In this case, n = 1, and by Proposition 2.12, we have Per(ϕ) = Fix(ϕ p ). Since 1 / ∈ ϕ(S), then ℓ S (ϕ(u)) ≥ ℓ S (u) for all u ∈ M , and so 1 / ∈ ϕ p (S). Thus, Fix(ϕ p ), and therefore Per(ϕ), is again an Artin monoid. Assume now 1 ∈ ϕ(S). In the proof of Theorem 2.3, we showed that Fix(ϕ) = ϕ n (Fix(ϕ 2 )), with ϕ 2 ∈ End(M 2 ) and 1 / ∈ ϕ 2 (S 2 ). Let u, v ∈ Fix(ϕ 2 ) such that ϕ n (u) = ϕ n (v). Then π • ϕ n (u) = π • ϕ n (v), and so, by Lemma 2.10, (π • ϕ) n (u) = (π • ϕ) n (v). Thus u = ϕ n 2 (u) = ϕ n 2 (v) = v. Hence, the morphism ϕ n | Fix(ϕ 2 ) : Fix(ϕ 2 ) → Fix(ϕ) is not only surjective, but also injective. Therefore, Fix(ϕ) is isomorphic to Fix(ϕ 2 ). By Case I, Fix(ϕ 2 ) is an Artin monoid, then so is Fix(ϕ). By Proposition 2.12, we have Per(ϕ) = Fix(ϕ pn ). Since 1 ∈ ϕ(S), one has 1 ∈ ϕ pn (S). Thus Fix(ϕ pn ), and so Per(ϕ), is an Artin monoid.
Inequalities and some equivalences between metrics
The purpose of this section is to define three metrics d 1 , d 2 and d 3 in finitely generated preGarside monoids, to compare them in general, and in the particular cases of trace monoids and Garside monoids.
Metrics and normal forms
In order to define our three metrics, we start by introducing the following general framework. Recall that given a non empty set X, we denote by X * the set of all finite words over X. Henceforth, these words will be denoted as tuples, to avoid any confusion with the monoids elements. Let M be a monoid, X be a non empty set, and ι : M ֒→ X * be an injective map. For u, v ∈ M with ι(u) = (u 1 , . . . , u n ) and ι(v) = (v 1 , . . . , v m ), we define
r(u, v) = max{k ≥ 0 | u 1 = v 1 , . . . , u k = v k } if u = v ∞ if u = v .
The metric d over M , associated to ι, is defined, for all u, v ∈ M , by
d(u, v) = 2 −r(u,v) .
When ι(u) = (u 1 , . . . , u n ) for some u ∈ M , then for all k ≤ n, we denote ι [k] (u) = (u 1 , . . . , u k ) ∈ X * .
Let M be a finitely generated preGarside monoid. For each metric d i over M , we will define X i , ι i and r i as above. The set X 1 for the first distance d 1 is defined in [1], where it is denoted by P ; the subset of M with a preGarside structure. It contains the finite set of atoms S = S(M ), and whenever it contains an element, it also contains all its left and right divisors ([1, Proposition 2.4]). We will denote it by M red , since in the case of an Artin monoid, it is just the set of reduced elements, that we will recall bellow. The properties of M red shown in [9] for Artin monoids, hold in finitely generated preGarside monoids with the same proofs, as stated in [1]. Namely ([1, Proposition 2.12]), there is a unique function α : M → M red which induces the identity on M red , and satisfies
α(uv) = α(uα(v)),(8)
for all u, v ∈ M . Further, α(u) is the unique maximal element (for ) in the set {v ∈ M red | v u}.
Let M = S | [a, b m ab = [b,
a m ab ; m ab = ∞ + be an Artin monoid, whose natural length function is denoted, as in the preliminaries, by ℓ S . And let W = S | (ab) m ab = 1; m ab = ∞ be the corresponding Coxeter group. There is also a length function on W (see [9]), which we denote also by ℓ S . It is known that two minimal expressions of an element of W are equivalent by using Artin relations only. The length of an element is defined by the length of any of its minimal expressions as products of elements of S. This implies that the induced quotient map from M to W has a canonical section (as a map of sets), whose image M red consists of those elements of M which have the same length as their image in W . Let M be a finitely generated preGarside monoid. To every element of M , can be associated a (left) normal form (n.f ), that is called the (left) greedy normal form, and defined as follows. To 1 M , we associate the empty sequence. And for u ∈ M \ {1} and u 1 , . . . , u n ∈ M red , we say that u = u 1 · · · u n is in normal form (n.f), if and only if no u i is equal to 1 and for any i we have u i = α(u i · · · u n ). In view of (8), the normality of a form can be seen locally ([1, Proposition 2.21]): u 1 · · · u k is a normal form if and only if u i u i+1 is for all i. This implies that any segment u i · · · u j of a normal form is normal. For u = u 1 · · · u n (n.f), we define ι 1 (u) = (u 1 , . . . , u n ), and denote n = |u| 1 . Let u, v ∈ M with ι 1 (u) = (u 1 , . . . , u n ) and ι 1 (v) = (v 1 , . . . , v m ). We define r 1 (u, v) exactly as r(u, v) above. Using the convention 2 −∞ = 0, the metric d 1 is defined by
d 1 (u, v) = 2 −r 1 (u,v) .
Another important normal form ι 2 over M , that we call the Foata normal form, is defined as follows. Let (u 1 , . . . , u n ) ∈ X * 2 such that u = u 1 · · · u n and u i = ∆({s ∈ S | s u i · · · u n }). When ι 2 (u) = (u 1 , . . . , u n ), we denote n = |u| 2 . And similarly, the metric d 2 associated to ι 2 , is known as the FNF metric, and defined in [2], for all u, v ∈ M , by
d 2 (u, v) = 2 −r 2 (u,v) .
When the monoid M is equiped with an additive and homogeneous norm ν, we can assume that ν(s) = 1 for all s in S, call this norm the length over S, and denote it by ℓ S . In this case, and in addition to d 1 and d 2 , there is a third and useful metric, decribed in [10] for the particular case of trace monoids, that we will denote by d 3 . Given u, v ∈ M , we say that v is a prefix of u, when v left-divides u. For every n ∈ N, denote by Pref n (u) the set of all prefixes of u of length n. Let Note that in this case, for u, v ∈ M , we have r 3 (u, v) = max{n ∈ N | Pref n (u) = Pref n (v)}, because for
1 ≤ n ≤ ℓ S (u), Pref n (u) = Pref n (v) ⇔ Pref k (u) = Pref k (v), ∀1 ≤ k ≤ n.
Relations between d 1 , d 2 and d 3
In this subsection, we will compare the first distance d 1 with the other two for a finitely generated pre-Garside monoid M , equiped with a length ℓ S . We start with d 1 and d 3 .
Lemma 3.1. Let u lie in M . Set ι 1 (u) = (u 1 , . . . , u n ). Then, for 1 ≤ k ≤ n, we have
Pref k (u) = Pref k (u 1 · · · u k ).
Proof. It suffices to show that if v u with ℓ S (v) = k, then v u 1 · · · u k , which we do by induction on k. Assume first k = 1.
, v ′ left divides u 1 · · · u k−1 . Write u 1 · · · u k−1 = v ′ v ′′ with v ′′ ∈ M . Since v ′ sw = u 1 · · · u n = v ′ v ′′ u k · · · u n , we have s v ′′ u k · · · u n . But s ∈ M red , therefore, s α(v ′′ u k · · · u n ). By (8), α(v ′′ u k · · · u n ) = α(v ′′ α(u k · · · u n )) = α(v ′′ u k ), so s v ′′ u k . Hence, v = v ′ s v ′ v ′′ u k = u 1 · · · u k .
Proposition 3.2. Let M be a finitely generated preGarside monoid, equiped with a length ℓ S . Then we have
d 3 ≤ d 1 .
Proof. Consider u and v distinct in M . Set ι 1 (u) = (u 1 , . . . , u n ) and
ι 1 (v) = (v 1 , . . . , v m ). If r 1 (u, v) = 0, then r 3 (u, v) ≥ r 1 (u, v), so d 3 (u, v) ≤ d 1 (u, v). Otherwise, for 1 ≤ k ≤ r 1 (u, v), by Lemma 3.1 we have Pref k (u) = Pref k (u 1 · · · u k ) = Pref k (v 1 · · · v k ) = Pref k (v). Thus, r 3 (u, v) ≥ r 1 (u, v) and so d 3 (u, v) ≤ d 1 (u, v).
We turn now to d 1 and d 2 . Note that the existence of a length ℓ S is only necessary for d 3 , and we do not need it to compare d 1 and d 2 . The inclusion X 2 ⊆ M red , deduced from [1, Proposition 2.19], will be useful for us.
i ≤ n, we have u ′ 1 · · · u ′ i u 1 · · · u i . ii. For i ≤ n and j ≤ m, if u ′ 1 · · · u ′ j u 1 · · · u i , then ι [j] 2 (u 1 · · · u i ) = (u ′ 1 , . . . , u ′ j ).
Proof.
i. The fact that n ≤ m is a consequence of [9, Proposition 4.8], also true for preGarside monoids, as stated in [1]. On the other hand, by using X 2 ⊆ M red and the same proof as [9, Proposition 4.10], we get u ′ 1 · · · u ′ i u 1 · · · u i for all i ≤ n.
ii. Set i ≤ n. Assume u ′ 1 · · · u ′ j u 1 · · · u i . We prove that ι 2 (u 1 · · · u i ) = (u ′ 1 , . . . , u ′ j ) by induction on j. Since ι [1] 2 (u) = (u ′ 1 ) and u ′ 1 u 1 · · · u i u, we have ι [1] 2 (u 1 · · · u i ) = (u ′ 1 ). Hence, the property is true for j = 1. Assume j ≥ 2 plus the induction hypothesis. Write u 1 · · · u i = u ′ 1 · · · u ′ j v with v ∈ M . By the induction hypothesis, we have ι 2 (u 1 · · · u i ) = (u ′ 1 , . . . , u ′ j−1 , ι [1] 2 (u ′ j v)). On the other hand, ι [1] 2 (u ′ j · · · u ′ m ) = (u ′ j ) and u ′ j · · · u ′ m = u ′ j xu i+1 · · · u n . This imposes ι
[1] 2 (u ′ j x) = (u ′ j ). Therefore ι [j]
Proposition 3.4. Let M be a finitely generated preGarside monoid. Then we have
d 2 ≤ d 1 .
Proof. Let u, v be in M and distinct. Set ι 1 (u) = (u 1 , . . . , u n ), ι 1 (v) = (v 1 , . . . , v n ′ ), ι 2 (u) = (u ′ 1 , . . . , u ′ m ) and ι 2 (v) = (v ′ 1 , . . . , v ′ m ′ ). By Lemma 3.3, n ≤ m and n ′ ≤ m ′ . If m = r 2 (u, v) or m ′ = r 2 (u, v), then 1 (u, v). So assume m < r 2 (u, v) and m ′ < r 2 (u, v) and set k = r 2 (u, v). By assumption u ′ k+1 = v ′ k+1 . We can therefore assume without restriction that u ′ k+1 does not left divide v ′ k+1 . By Lemma 3.3 i. and ii., ι
r 1 (u, v) ≤ min(n, n ′ ) ≤ min(m, m ′ ) = r 2 (u, v). Therefore d 2 (u, v) ≤ d[k+1] 2 (v 1 · · · v k+1 ) = (v ′ 1 , . . . , v ′ k+1 ). Since ι [k+1] 2 (u ′ 1 · · · u ′ k+1 ) = (u ′ 1 , . . . , u ′ k+1 ) and u ′ k+1 does not left divide v ′ k+1 , it follows from Lemma 3.3 that u ′ 1 · · · u ′ k+1 does not left divide v 1 · · · v k+1
. But on the other hand, u ′ 1 · · · u ′ k+1 left divides u 1 · · · u k+1 by Lemma 3.3 i.. Thus, v 1 · · · v k+1 = u 1 · · · u k+1 and r 1 (u, v) ≤ k. Hence
r 1 (u, v) ≤ r 2 (u, v) and d 2 (u, v) ≤ d 1 (u, v).∀ε > 0, ∃δ > 0, ∀x, y ∈ X 1 : (d(x, y) < δ ⇒ d ′ (ϕ(x), ϕ(y)) < ε).
If the identity mappings between (X, d) and (X, d ′ ) are uniformly continuous, we say that the metrics d and d ′ are uniformly equivalent. It is immediate that two equivalent metrics are consequently uniformly equivalent Below, we will show that in Garside monoids, d 1 is uniformly equivalent to d 3 , and in trace monoids, d 2 and d 3 are uniformly equivalent. However, these metrics are not equivalent, nor uniformly equivalent in general. Here are some examples to illustrate that. i. In an Artin monoid M with m ab = ∞ for some a, b ∈ S, the metric d 1 is not uniformly equivalent (and so not equivalent) to d 2 , nor to d 3 . Indeed, write u n = (ab) n and v n = (ab) n+1 . By definition of the metrics, we have r 2 (u n , v n ) = r 3 (u n , v n ) = 2n, and since u n , v n ∈ M red , then r 1 (u n , v n ) = 0. Thus we have d 1 (u n , v n ) = 1 for all n, and lim n→∞ d 2 (u n , v n ) = lim n→∞ d 3 (u n , v n ) = lim n→∞ 2 −2n = 0. So d 1 cannot be uniformly equivalent to d 2 or d 3 .
ii. The metrics d 2 and d 3 are not equivalent in general. Indeed, Let M be an Artin monoid, with a, b, c ∈ S such that m ab = 2 and m ac , m bc ≥ 3. Consider u n = (ab) n and v n = (ab) n c. Then r 2 (u n , v n ) = n and r 3 (u n , v n ) = 2n. We have
∃C > 0, d 2 ≤ Cd 3 ⇔ ∃C > 0, −r 2 ≤ log 2 (C) − r 3 ⇔ ∃C ′ ∈ R, r 3 ≤ r 2 + C ′ .
But in our example, r 3 = 2n and r 2 = n, so there is no C ′ in R such that r 3 ≤ r 2 + C ′ . Therefore, d 2 and d 3 cannot be equivalent.
iii. In a non-abelian Artin monoid M , the metric d 1 is not equivalent to d 2 , nor to d 3 . Indeed, let a, b ∈ S such that m ab ≥ 3. Consider u n = (abba) n and v n = (abba) n+1 . Then r 1 (u n , v n ) = 2n and r 2 (u n , v n ) = r 3 (u n , v n ) = 4n. Or, as in ii., the existence of some C > 0 such that d 1 ≤ Cd 2 and d 1 ≤ Cd 3 , means there is a C ′ in R such that r 3 ≤ r 1 + C ′ and r 2 ≤ r 1 + C ′ , which is impossible for our example.
The case of trace monoids
In this subsection, we focus on right angled Artin monoids (RAAM). A RAAM, or a trace monoid, is an iii. if M is the free abelian monoid,
Artin monoid M = S | [a, b m ab = [b,d 1 = d 2 = d 3 .
Indeed, point i. was already proved in [7] by a topological argument. We provide an algebraic one. Let us start with the following remark :
Remark 3.8. If M ≃ F + p is the free monoid, i.e. m ab = ∞ for all a, b ∈ S, then d 2 = d 3 . Proof. Let u, v lie in M . Set ι 2 (u) = (u 1 , . . . , u n ) and ι 2 (v) = (v 1 , . . . , v m ). For all a, b ∈ S, m ab = ∞. Then {s ∈ S | s u i · · · u n } = {u i }, {s ∈ S | s v i · · · v m } = {v i }, Pref i (u) = {u 1 · · · u i } and Pref i (v) = {v 1 · · · v i } for every i ≤ min{n, m}. Hence, r 2 (u, v) = r 3 (u, v) and therefore d 2 (u, v) = d 3 (u, v).
For the remaining of the section, we fix a trace monoid M = S | ab = ba; m ab = ∞ + , and set p = |S|. For every u ∈ M , let ξ(u) denote the support of u, i.e. the set of atoms (elements of S) occurring in any expression of u.
Lemma 3.9. Let u, v ∈ M such that v u. Set ι 2 (u) = (u 1 , . . . , u n ) and ι 2 (v) = (v 1 , . . . , v k )
. Then k ≤ n, and for all i ≤ k, we have v 1 · · · v i u 1 · · · u i .
We may now prove our proposition.
Proof of Proposition 3.7.
i. Gathering Lemmas 3.10 and 3.11, we get immediately that d 2 and d 3 are uniformly equivalent.
ii. Example 3.6 i.
iii. Assume M ≃ N p is the free abelian monoid, i.e. m ab = 2 for all a, b ∈ S. Let u, v be in M . Set ι 1 (u) = (u 1 , . . . , u n ) and ι 1 (v) = (v 1 , . . . , v m ). Since ab = ba for all a, b ∈ S, we have
M red = {∆(T ) | T ⊆ S} = X 2 . So, for every w ∈ M , ι 1 (w) = ι 2 (w). Thus r 1 (u, v) = r 2 (u, v) and d 1 (u, v) = d 2 (u, v). If ξ(u) = ξ(v), then r 1 (u, v) = r 2 (u, v) = r 3 (u, v) = 0, and so d 1 (u, v) = d 2 (u, v) = d 3 (u, v). If ξ(u) = ξ(v) = {s 1 , . . . , s k } and u = v, write u = s f 1 1 · · · s f k k and v = s g 1 1 · · · s g k k . The set T = {f i , g i | f i = g i } is non empty because u = v. Let q = min(T). We may suppose q = f j for some 1 ≤ j ≤ k. Thus, for all i ≤ q, we have u i = v i , s j ⊀ u q+1 and s j v q+1 . Therefore r 1 (u, v) = r 2 (u, v) = q. On the one hand, s q+1 j v and s q+1 j ⊀ u, therefore Pref q+1 (u) = Pref q+1 (v).
On the other hand, let w ∈ M such that w u and ℓ S (w) = q. Then we can write w
= s h 1 1 · · · s h k k with h 1 + · · · + h k = q. Since w u, for 1 ≤ i ≤ k, we have h i ≤ f i . If f i = g i , then h i ≤ g i . And if f i = g i , since q = min(T) and h i ≤ q, one has h i ≤ g i . So for all 1 ≤ i ≤ k, we have h i ≤ g i , which means that w v. Similarly, if w v with ℓ S (w) = q, then w u as well. Hence, Pref q (u) = Pref q (v), and therefore r 3 (u, v) = q. Thus, d 1 (u, v) = d 2 (u, v) = d 3 (u, v).
The case of Garside monoids
In this subsection, we show that in a finitely generated Garside monoid, equiped with a length ℓ S , the metrics d 1 and d 3 are uniformly equivalent. Recall that a Garside monoid is a preGarside monoid containing a Garside element, i.e. a balanced element whose set of divisors generates the whole monoid. let M be a Garside monoid with a Garside element ∆. One of the Garside element important properties is that for all u ∈ M , α(u) is the greatest common (left) divisor of u and ∆, denoted by u ∧ ∆. In other words, we have u ∈ M red ⇔ u ∆. Proposition 3.12. Let M be a finitely generated Garside monoid, equiped with a length ℓ S . Set ℓ = ℓ S (∆).
We have d ℓ 1 ≤ 2 ℓ d 3 . Proof. Let u, v be distinct in M . Set ι 1 (u) = (u 1 , . . . , u n ), ι 1 (v) = (v 1 , . . . , v m ), r 1 = r 1 (u, v), and r 3 = r 3 (u, v). If u = 1 or v = 1, then r 1 = r 3 = 0, so (d 1 (u, v)) ℓ ≤ 2 ℓ d 3 (u, v). If r 1 = n or r 1 = m, then u v or v u, and so r 3 ≤ r 1 ℓ because ℓ S (w) ≤ ℓ for all w ∈ M red . Thus (d 1 (u, v)) ℓ ≤ d 3 (u, v) ≤ 2 ℓ d 3 (u, v). Otherwise, suppose u 1 · · · u r 1 +1 v and v 1 · · · v r 1 +1 u. Then u r 1 +1 v r 1 +1 · · · v m and v r 1 +1
u r 1 +1 · · · u n . Thus, by definition of the greedy normal form, u r 1 +1 v r 1 +1 and v r 1 +1 u r 1 +1 . So u r 1 +1 = v r 1 +1 , which contradicts the definition of r 1 (u, v). Therefore, one has either u 1 · · · u r 1 +1 ⊀ v or v 1 · · · v r 1 +1 ⊀ u. Either way, we have r 3 ≤ max{ℓ S (u 1 · · · u r 1 +1 ), ℓ S (v 1 · · · v r 1 +1 )} ≤ (r 1 + 1)ℓ. Hence,
(d 1 (u, v)) ℓ ≤ 2 ℓ d 3 (u, v).
Gathering propositions 3.2 and 3.12, we get : Let M = S | [a, b m ab = [b, a m ab ; m ab = ∞ + be an Artin monoid, and ϕ be in End(M ). As shown in [10], the metric space (M, d 1 ) admits a completion ( M , d 1 ) defined as follows. Let ∂M consist of all infinite sequences of the form u 1 u 2 · · · , such that u i ∈ M red for all i, and u 1 · · · u n is a (greedy) normal form for all n ∈ N. We have M = M ∪ ∂M .
The metric d 1 extends to M in the obvious way, and it is easy to check that ( M , d 1 ) is complete: given a Cauchy sequence (U n ) n with U n = u n1 u n2 · · · , it follows easily that each sequence (u nk ) k is stationary with limit, say, u k , and we get u 1 u 2 · · · = lim n→∞ U n . Since u 1 · · · u n ∈ M and it is in a normal form for all n, and u 1 u 2 · · · = lim n→∞ u 1 · · · u n , then ( M , d 1 ) is indeed the completion of (M, d 1 ). We may refer to ∂M as the boundary of M .
Assume that ϕ is uniformly continuous with respect to d 1 . Since ( M , d 1 ) is the completion of (M, d 1 ), ϕ admits a unique continuous extension Φ to ( M , d 1 ). By continuity, we must have Φ(X) = lim n→∞ ϕ(u n ) whenever X ∈ ∂M and (u n ) n is a sequence on M satisfying X = lim n→∞ u n .
Lemma 4.1. The following properties are equivalent:
(i) for all u, v ∈ M , α(uv) = α(u) ⇒ α(ϕ(uv)) = α(ϕ(u)); (ii) for all u, v ∈ M , α(uv) = u ⇒ α(ϕ(uv)) = α(ϕ(u));
(iii) for all u, v ∈ M red , α(uv) = u ⇒ α(ϕ(uv)) = α(ϕ(u));
(iv) for all u ∈ M , α(ϕ(u)) = α(ϕ(α(u))).
Proof. We prove that (ii) ⇒ (iv) ⇒ (i) ⇒ (iii) ⇒ (ii). Let u, v be in M . Set ι 1 (u) = (u 1 , . . . , u n ) the greedy normal form of u. Assume (ii) holds. Then α(u 1 (u 2 · · · u n )) = u 1 and, by (ii), α(ϕ(u 1 · · · u n )) = α(ϕ(u 1 )) = α(ϕ(α(u))). Thus (iv) holds. Assume (iv) holds and α(uv) = α(u). Then by (iv), α(ϕ(uv)) = α(ϕ(α(uv))) = α(ϕ(α(u))) = α(ϕ(u)) and (i) holds. Assume now (i). If u ∈ M red then α(u) = u, so α(uv) = u implies α(uv) = α(u), which in turn implies α(ϕ(uv)) = α(ϕ(u)) by (i). So (iii) holds. Assume finally (iii) and assume α(uv) = u. In particular u ∈ M red . We prove that α(ϕ(uv)) = α(ϕ(u)) by induction on |v| 1 = k. If |v| 1 = 1, then v ∈ M red and the result holds by (iii). Assume |v| 1 > 1 plus the induction hypothesis. Set k = |v| 1 and let ι 1 (v) = (v 1 , . . . , v m ) be the greedy normal form of v. In view of (8), we have α(ϕ(uv)) = α(ϕ(u)α(ϕ(v))). Now, we have α(v) = α(v 1 · · · v k ) = v 1 .
Since |v 2 · · · v k | 1 = k − 1 < k, then by the induction hypothesis, α(ϕ(v 1 · · · v k )) = α(ϕ(v 1 )). Hence α(ϕ(uv)) = α(ϕ(u)α(ϕ(v 1 ))) = α(ϕ(u)ϕ(v 1 )) = α(ϕ(uv 1 )). We also have α(uv 1 ) = α(uα(v)) = α(uv) = u and |v 1 | 1 = 1 < k. Then, by the case k = 1, we get α(ϕ(uv 1 )) = α(ϕ(u)). Thus α(ϕ(uv)) = α(ϕ(u)) and (ii) holds.
Recall that a mapping ϕ : (X, d) → (X, d) on a metric space is called a contraction with respect to d, if d(ϕ(u), ϕ(v)) ≤ d(u, v) for all u, v ∈ X. (i) ϕ is uniformly continuous, and Φ is a contraction with respect to d 1 ;
(ii) ϕ is a contraction with respect to d 1 ;
(iii) for all u, v ∈ M red , α(uv) = u ⇒ α(ϕ(uv)) = α(ϕ(u));
(iv) for all u ∈ M , α(ϕ(u)) = α(ϕ(α(u))).
Furthermore, in these cases, if u = u 1 u 2 · · · ∈ M and Φ(u) = U 1 U 2 · · · , then for all m ∈ N * with m ≤ |u| 1 , one has ι 1 (ϕ(u 1 · · · u m )) = (U 1 , . . . , U m , . . . ).
Proof. The equivalence (i) ⇔ (ii) is clear, and (iii) ⇔ (iv) follows from Lemma 4.1. Assume (ii). Let u, v be in M red such that α(uv) = u. Then d 1 (uv, u) = 1 2 , and by (ii), d 1 (ϕ(uv), ϕ(u)) ≤ 1 2 . Thus α(ϕ(uv)) = α(ϕ(u)). So (iii) holds. Conversely, assume (iii). Let u belong to M . Set ι 1 (u) = (u 1 , . . . , u n ) and ι 1 (ϕ(u)) = (U 1 , . . . , U N ). We prove by induction on k that for k ∈ {1, . . . , n}, one has ι 1 (ϕ(u 1 · · · u k )) = (U 1 , . . . , U k , . . . ). In particular n ≤ N . For k = 1, the result holds by (iv). So assume k ≥ 2 plus the induction hypothesis. By the induction hypothesis, we can write ϕ(u 1 · · · u k−1 ) = U 1 · · · U k−1 Z, with Z ∈ M . Since ϕ(u 1 · · · u n ) = U 1 · · · U N , it follows by cancellativity that Zϕ(u k · · · u n ) = U k · · · U N . Therefore, α(Zϕ(u k · · · u n )) = U k . But, using (8) and (iv), we have α(Zϕ(u k · · · u n )) = α(Zα(ϕ(u k · · · u n ))) = α(Zα(ϕ(u k ))). In particular, U k left divides Zα(ϕ(u k )). Hence, U 1 · · · U k left divides ϕ(u 1 · · · u k ). This imposes by definition of the greedy normal form that ι 1 (ϕ(u 1 · · · u k )) = (U 1 , . . . , U k , . . . ), which proves the induction step. Now let v belong to M . Set ι 1 (v) = (v 1 , . . . , v p ) and ι 1 (ϕ(v)) = (V 1 , . . . , V P ). Assume d 1 (u, v) = 2 −k . Then u 1 = v 1 , . . . , u k = v k . It follows from the above result that ι 1 (ϕ(u 1 · · · u k )) = (U 1 , . . . , U k , . . . ) = (V 1 , . . . , V k , . . . ). Thus U 1 = V 1 , . . . , U k = V k and d 1 (ϕ(u), ϕ(v)) ≤ 2 −k . So ϕ is a contraction. Hence, (ii) holds. Finally, assume (i). Let u lie in M . Say u = u 1 u 2 · · · . Set Φ(u) = U 1 U 2 · · · ∈ M . Let m be in N * . Then d 1 (u 1 · · · u m , u) = 2 −m . So by (i), we have d 1 (Φ(u 1 · · · u m ), Φ(u)) ≤ 2 −m . Hence, ι 1 (ϕ(u 1 · · · u m )) = ι 1 (Φ(u 1 · · · u m )) = (U 1 , . . . , U m , . . . ).
The following example illustrates that the equivalence between (ii) and (iii) in [10, Theorem 4.1] is not true for all Artin monoïds with respect to d 2 , which is why we used d 1 to extend [10, Theorem 4.1] into our Theorem 4.2.
Example 4.3. Let M = s, t | ststststs = tstststst + , and ϕ ∈ End(M ) such that ϕ(s) = sts and ϕ(t) = tst. Set ∆ = ststststs, and define α 2 (u) by ι [1] 2 (u) = (α 2 (u)) for u in M . We have X 2 = {s, t, ∆}, and {(u, v) ∈ X 2 2 | α 2 (uv) = u} = {(s, t), (t, s), (s, s), (t, t), (∆, s), (∆, t), (∆, ∆)}. Then, for all u, v ∈ X 2 , we have α 2 (uv) = u ⇒ α 2 (ϕ(uv)) = α 2 (ϕ(u)). However, the morphism ϕ is not a contraction with respect to d 2 , since d 2 (ϕ(s), ϕ(sts)) > d 2 (s, sts).
X 2 = {u ∈ M | ∃T ⊆ S, u = ∆(T )}, where S = S(M ) and ∆(T ) is the least right common multiple of the elements of T , which exists if and only if there is a right common multiple ([6, Lemma 2.1]). For u ∈ M \ {1}, there exists a unique ι 2 (u) =
X 3 = P(M ) be the set of all parts of M . For u ∈ M \ {1}, set ι 3 (u) = (u 1 , . . . , u n ) with n = ℓ S (u) and u i = Pref i (u) for all i. And set ι 3 (1) = ({1}). Then the metric d 3 , known as the prefix metric, is defined in [8] as above, for all u, v ∈ M , by d 3 (u, v) = 2 −r 3 (u,v) .
If ℓ S (v) = 1, then v ∈ S ⊆ M red . So, by definition of the greedy normal form, v α(u) = u 1 . Assume now k > 1 plus the induction hypothesis. Consider v in M such that v u and ℓ S (v) = k. Write v = v ′ s and u = vw with s in S and w in M . By the induction hypothesis
Lemma 3 . 3 .
33Let u be in M . Set ι 1 (u) = (u 1 , . . . , u n ) and ι 2 (u) = (u ′ 1 , . . . , u ′ m ). Then i. n ≤ m, and for all
Definition 3.5. A mapping ϕ : (X, d) → (X ′ , d ′ )between metric spaces is uniformly continuous if
Theorem 3 . 13 .
313In a finitely generated Garside monoid, equiped with a length ℓ S , the metrics d 1 and d 3 are uniformly equivalent.Question : Are d 1 and d 2 uniformly equivalent in Garside monoids ?
Theorem 4 . 2 .
42The following properties are equivalent:
a m ab ; m ab = ∞ + , where m ab ∈ {2, ∞} for all a, b ∈ S. Our objective is to obtain a complete comparison of d 1 , d 2 and d 3 for trace monoids. Here we prove :Proposition 3.7. Assume M is a trace monoid. Then
i. d 2 and d 3 are uniformly equivalent.
ii. if M is not free abelian, d 1 is not uniformly equivalent to d 2 , nor to d 3 .
4 Contractability of endomorphisms of Artin monoidsThe aim of this section is to extend[10, Theorem 4.1] to all Artin monoids. However, it is easy to verify that the assertions stated in [10, Theorem 4.1] can be not equivalent in an Artin monoids (see Example 4.3 below). So [10, Theorem 4.1] can not be directly extended. Actually, in the general case, the metric d 1 appears as more natural than d 2 , mainly because of Property (8). Moreover, one can verify that Property (iii) of Lemma 4.1 is the exact translation of Property (14) in [10, Theorem 4.1(iii)] when replacing d 2 by d 1 . So Theorem 4.2 provided a convenient generalisation of [10, Theorem 4.1] to the context of Artin monoids.
(u 1 · · · u i ) = (u ′ 1 , . . . , u ′ j ).
Proof. Let w be in M and s be in S. Assume ι 2 (w) = (w 1 , . . . , w l ), and set ι 2 (ws) = (w ′ 1 , . . . , w ′ l ′ ). By[12], ι 2 (ws) can be obtained in the following way. If s / ∈ ξ(w) and sw = ws, then ι 2 (ws) = (w 1 s, . . . , w l ) and |ws| 2 = l. If s ∈ ξ(w l ) or sw l = w l s, then ι 2 (ws) = (w 1 , . . . , w l , s) and |ws| 2 = l + 1. Otherwise, set j 0 = min{j ∈ {1, . . . , l}; s / ∈ ξ(w j · · · w l ) and sw j · · · w l = w j · · · w l s}. We have j 0 < l, ι 2 (ws) = (w 1 , . . . , w j 0 s, . . . , w l ) and |ws| 2 = l. In all cases, l ′ ≥ l and w 1 · · · w i w ′ 1 · · · w ′ i for all i ≤ l. Now we can write u = vs 1 · · · s m with s 1 , . . . , s m in S, and apply the above argument to all the pairs (vs 1 · · · s i−1 , vs 1 · · · s i ) to conclude.Proof. Let u, v be distinct in M . Set ι 2 (u) = (u 1 , . . . , u n ) andLemma 3.11. We have d p 2 ≤ 2 p d 3 . Proof. Let u, v be in M . Set ι 2 (u) = (u 1 , . . . , u n ) and ι 2 (v) = (v 1 , . . . , v m ), and denote k = r 2 (u, v). Since we have m st ∈ {2, ∞} for all s, t ∈ S, then u i = ∆{s ∈ S | s u i · · · u n } = {s ∈ S | s u i · · · u n } and ℓ S (u i ) ≤ p for all i. Assuming that u = v and n > k, we have u 1 · · · u k+1 ⊀ v. So r 3 (u, v) ≤ ℓ S (u 1 · · · u k+1 ) ≤ p(k + 1) = pr 2 (u, v) + p. Thus, −pr 2 (u, v) ≤ p − r 3 (u, v). And therefore, (d 2 (u, v)) p ≤ 2 p d 3 (u, v).
Springer theory in braids groups and the Birman-Ko-Lee monoid. D Bessis, F Digne, J Michel, Pacific J. Math. 205Bessis, D., Digne, F., and Michel, J. Springer theory in braids groups and the Birman-Ko-Lee monoid. Pacific J. Math. 205 (2002), 287-309.
About infinite traces. P Bonizzoni, G Mauri, G Pighizzini, Proceding of the ASMICS Worshops on Partially Commutative Monoids. V. Diekerteding of the ASMICS Worshops on Partially Commutative Monoids9002Bonizzoni, P., Mauri, G. and Pighizzini, G. About infinite traces, in: V. Diekert (ed.), Proceding of the ASMICS Worshops on Partially Commutative Monoids. Tech. Rep. TUM-I 9002 (1990), 1-10.
. E Brieskorn, K Saito, Artin Gruppen und Coxeter Gruppen. Invent. Math. 17Brieskorn, E., and Saito, K. Artin Gruppen und Coxeter Gruppen. Invent. Math. 17 (1972), 245-271.
Symmetrical subgroups of Artin groups. J Crisp, Adv. in Math. 152Crisp, J. Symmetrical subgroups of Artin groups. Adv. in Math. 152 (2000), 159-177.
Gaussian groups and Garside groups, two generalisations of Artin groups. P Dehornoy, Paris , L , Proc. London Math Soc. 79Dehornoy, P., and Paris, L. Gaussian groups and Garside groups, two generalisations of Artin groups. Proc. London Math Soc. 79, 3 (1999), 569-604.
Pregarside monoids and groups, parabolicity, amalgamation, and FC property. E Godelle, Paris , L , I.J.A.C. 23Godelle, E., and Paris, L. Pregarside monoids and groups, parabolicity, amalgamation, and FC property. I.J.A.C 23 (2013), 1431-1467.
The topology of mazurkiewicz traces. R Kummetz, D Kuske, Theoret. Comp. Sci. 305Kummetz, R., and Kuske, D. The topology of mazurkiewicz traces. Theoret. Comp. Sci. 305 (2003), 237-258.
A metric for traces. M Kwiatkowska, Z , Information processing Letters. 35Kwiatkowska, M., Z. A metric for traces. Information processing Letters 35 (1990), 129-135.
A note on braid monoids. J Michel, J. of Algebra. 215Michel, J. A note on braid monoids. J. of Algebra 215 (1999), 366-377.
E Rodaro, P V Silva, arXiv:1211.4517v1Fixed points of endomorphisms of trace monoids. Rodaro, E., and Silva, P. V. Fixed points of endomorphisms of trace monoids. arXiv:1211.4517v1 .
Fixed points of endomorphisms of trace monoids. E Rodaro, P V Silva, DOI10.1007/s00233-013-9553-0Semigroup Forum November. Rodaro, E., and Silva, P. V. Fixed points of endomorphisms of trace monoids. Semigroup Forum November (2013), DOI 10.1007/s00233-013-9553-0.
Graph groups are biautomatic. L Van Wyk, J. Pure Appl. Algebra. 94Van Wyk, L. Graph groups are biautomatic. J. Pure Appl. Algebra 94 (1994), 341-352.
|
[] |
[
"Holographic Matter : Deconfined String at Criticality",
"Holographic Matter : Deconfined String at Criticality"
] |
[
"Sung-Sik Lee \nDepartment of Physics & Astronomy\nMcMaster University\n1280 Main St. WL8S 4M1HamiltonONCanada\n\nPerimeter Institute for Theoretical Physics\n31 Caroline St. NN2L 2Y5WaterlooONCanada\n"
] |
[
"Department of Physics & Astronomy\nMcMaster University\n1280 Main St. WL8S 4M1HamiltonONCanada",
"Perimeter Institute for Theoretical Physics\n31 Caroline St. NN2L 2Y5WaterlooONCanada"
] |
[] |
We derive a holographic dual for a gauged matrix model in general dimensions from a first-principle construction. The dual theory is shown to be a closed string field theory which includes a compact twoform gauge field coupled with closed strings in one higher dimensional space. Possible phases of the matrix model are discussed in the holographic description. Besides the confinement phase and the IR free deconfinement phase, there can be two different classes of critical states. The first class describes holographic critical states where strings are deconfined in the bulk. The second class describes non-holographic critical states where strings are confined due to proliferation of topological defects for the two-form gauge field. This implies that the critical states of the matrix model which admit holographic descriptions with deconfined string in the bulk form novel universality classes with non-trivial quantum orders which make the holographic critical states qualitatively distinct from the non-holographic critical states. The signatures of the non-trivial quantum orders in the holographic states are discussed. Finally, we discuss a possibility that open strings emerge as fractionalized excitations of closed strings along with an emergent one-form gauge field in the bulk.
|
10.1016/j.nuclphysb.2012.04.023
|
[
"https://arxiv.org/pdf/1108.2253v2.pdf"
] | 119,254,533 |
1108.2253
|
01b2f5b794f1da0ccf692c0a66c11597dc9448c3
|
Holographic Matter : Deconfined String at Criticality
31 Dec 2011 (Dated: February 1, 2013)
Sung-Sik Lee
Department of Physics & Astronomy
McMaster University
1280 Main St. WL8S 4M1HamiltonONCanada
Perimeter Institute for Theoretical Physics
31 Caroline St. NN2L 2Y5WaterlooONCanada
Holographic Matter : Deconfined String at Criticality
31 Dec 2011 (Dated: February 1, 2013)
We derive a holographic dual for a gauged matrix model in general dimensions from a first-principle construction. The dual theory is shown to be a closed string field theory which includes a compact twoform gauge field coupled with closed strings in one higher dimensional space. Possible phases of the matrix model are discussed in the holographic description. Besides the confinement phase and the IR free deconfinement phase, there can be two different classes of critical states. The first class describes holographic critical states where strings are deconfined in the bulk. The second class describes non-holographic critical states where strings are confined due to proliferation of topological defects for the two-form gauge field. This implies that the critical states of the matrix model which admit holographic descriptions with deconfined string in the bulk form novel universality classes with non-trivial quantum orders which make the holographic critical states qualitatively distinct from the non-holographic critical states. The signatures of the non-trivial quantum orders in the holographic states are discussed. Finally, we discuss a possibility that open strings emerge as fractionalized excitations of closed strings along with an emergent one-form gauge field in the bulk.
I. INTRODUCTION
Extracting dynamical information on strongly interacting critical states of matter is in general a hard problem in theoretical physics. Fortunately, there are classes of strongly coupled quantum field theories whose non-perturbative dynamics can be accessed through dual descriptions which become weakly coupled when the number of degrees of freedom is large.
One such dual description that has been extensively studied in condensed matter physics is the so-called slave-particle formulation [1][2][3]. In this theory, a gauge redundancy is introduced in order to take into account dynamical constraints imposed by strong interactions. Unphysical states introduced in the redundant description is projected out by a dynamical gauge field. In the large N limit, where N is the number of flavor degrees of freedom, the dynamical gauge field becomes weakly coupled and emerges as a low energy collective excitation of the system.
The slave-particle theory may be viewed as a mere change of variables which allows one to compute dynamical properties conveniently, which could have been computed using a different set of variables albeit more complicated. However, the real power of the mathematical reformulation lies in the fact that it allows one to classify various novel phases of matter beyond the symmetry breaking scheme [4]. In particular, those phases that support emergent gauge boson possess subtle quantum orders that make them qualitatively distinct from the conventional phases. Because of the non-trivial quantum orders, the phases with an emergent (deconfined) gauge boson can not be smoothly connected to the conventional phases. Signatures of the non-trivial quantum order include fractionalized excitations and protected gapless excitations (or ground state degeneracy on a space with a non-trivial topology).
The gauge-string correspondence is another type of duality [5][6][7]. According to the duality, a class of D-dimensional quantum field theories is dual to a (D + 1)-dimensional string theory.
The question we would like to address in this paper is : Do those phases that admit holographic descriptions in one higher dimensional space possess non-trivial quantum orders ? If so, what we call holographic states that can be described in one higher dimensional space can not be smoothly connected to the conventional non-holographic states. We claim that the answer to this question is 'yes'. The signatures of the non-trivial quantum order in holographic phases are the emergent space with an extra dimension, deconfined strings and the existence of an operator whose scaling dimension is protected from acquiring a large quantum correction at strong coupling in the large N limit, even though the operator is not protected by any microscopic symmetry of the model. The paper is organized in the following way. In Sec. II, we start by reviewing the slave-particle theory with an emphasis on quantum order in fractionalized phases. In Sec. III, we introduce a gauged matrix model which will be the focus of the rest of the paper. The model is general enough to include the U(N) gauge theory. In Sec. IV, through a first-principle derivation, we
show that the matrix model in general dimensions is holographically dual to a closed string field theory in one-higher dimensional spaces. In Sec. V, it is shown that the partition function of the original matrix model can be interpreted as a transition amplitude between quantum many-loop states in the holographic description. In Sec. VI, we show that the holographic description has a gauge redundancy, and strings are coupled with a compact two-form gauge field in the bulk.
Because of the compact nature of the two-form gauge field, topological defects for the two-form gauge field are allowed. In Sec. VII, we discuss possible states of the matrix model. Different states are characterized by different dynamics of topological defects in the bulk. If topological defects are gapped, strings are deconfined in the bulk, and the holographic state is stable. On the other hand, if topological defects are condensed, strings are confined, and the bulk description is not useful anymore. Suppressed topological defect in the holographic phase is responsible for a non-trivial quantum order which protects the scaling dimension of the phase mode of Wilson loop operators from acquiring a large quantum correction at strong coupling in the large N limit.
We discuss the differences between the holographic and non-holographic states. The holographic critical phases can be divided further into two different classes. In the first case, there exist only closed strings in the bulk. In the second case, there are both closed and open strings, where open strings emerge as fractionalized collective excitations of closed strings. The latter state has a yet another quantum order which supports an emergent one-form gauge field in the bulk. Finally, we close with speculative discussions on a possible phase diagram, a world sheet description of deconfined strings, and a continuum limit.
The present construction is beyond the level of identifying the equations of motion in the bulk with the beta function of the boundary theory. We construct a full quantum theory of string in the bulk that is dual to the boundary theory. The construction of the dual theory makes use of the fact that loop variables associated with Wilson loops become classical objects in the planar limit of matrix models [8][9][10][11]. The current construction of the string field theory is directly based on the earlier works [12,13]. Compared to the the previous work on the U(N) gauge theory [13], the present construction has two major improvements. First, the extra dimension generated out of the renormalization group flow is continuous, while the earlier construction produces a discrete extra dimension. The infinitesimally small parameter associated with a continuously increasing length scale allows one to write the bulk action in a compact form in this formalism. As a result, one can readily take a continuum limit starting from a boundary theory defined on a lattice. Second, the earlier construction involves infinitely many loop fields in the bulk associated with multi-trace operators, which makes the theory highly redundant. In the present construction, the relation between single-trace operators and multi-trace operators are explicitly implemented. As a result, the dual theory can be written only in terms of the loop fields for single-trace operators. Because of these improvements, the dual theory takes a much simpler form, and this transparency allows one to uncover deeper structures in the theory.
There also exist alternative approaches to derive holographic duals for general quantum field theories [14][15][16][17][18][19][20]. All these constructions including the present one are based on the notion that the extra dimension in the holographic description is related to the length scale in the renormalization group flow [21][22][23][24][25].
II. QUANTUM ORDER IN FRACTIONALIZED PHASE
In this section we review some of the key features of the slave-particle theory[1-3] using a pedagogical model introduced in Ref. [26]. We consider a model defined on the four-dimensional (1)
Here θ ab i 's describe phase fluctuations of boson fields defined at site i. Each boson carries one flavor index a and one anti-flavor index b with a, b = 1, 2, ..., N. < i, j > represents nearest neighbor bonds of the lattice. We assume that the phases satisfy the constraints θ ab = −θ ba [51].
With the constraints, there are N(N − 1)/2 independent boson fields per site. The theory has U(1) N −1 global symmetry under which the boson fields transform as θ ab i → θ ab i + ϕ a − ϕ b . In the weak coupling limit (K << 1), the model describes weakly coupled bosons. As the strength of the kinetic term t is increased, there is a phase transition from the disordered phase to the bose condensed phase. In the disordered phase, all excitations are gapped. In the condensed phase, there are (N − 1) Goldstone modes. (At the special point of K = 0, there are N(N − 1)/2
Goldstone modes due to the enhanced symmetry).
In the strong coupling limit (K >> 1), the large potential energy imposes an additional set of dynamical constraints, θ ab i + θ bc i + θ ca i = 0 which is solved by a decomposition,
θ ab i = φ a i − φ b i .(2)
Here φ a i 's are boson fields which parameterize the low energy manifold. Note that these fields carry only one flavor quantum number contrary to the original boson fields. The new bosons are called slave-particles (or partons). The low energy effective action for the slave-particles becomes
S = −t <i,j> a e i(φ a i −φ a j ) b e −i(φ b i −φ b j ) .(3)
Note that this theory has a U(1) gauge symmetry,
φ a i → φ a i + ϕ i .(4)
This is due to the U(1) redundancy introduced in the decomposition in Eq. (2). Because of the gauge symmetry, the slave-particles can not hop by themselves. However, these particles can move in space by exchanging their positions with other particles. For example, in Eq. (3), the particle with flavor a can hop from site j to i as the particle with flavor b hops from i to j. In this sense, they can move only through the help of other slave-particles. One can introduce a collective hopping
field χ ij ≡ b e −i(φ b i −φ b j )
to characterize the amplitude of this mutual hopping. If we use this collective field, Eq. (3) can be written as
S = −t <i,j>,a χ ij e i(φ a i −φ a j ) .(5)
The magnitude of the collective field characterizes the strength of hopping, and the phase plays the role of the U(1) gauge field to which the slave-particles are coupled electrically. This mapping from Eq. (3) to the U(1) gauge theory can be made more rigorous, by using the Hubbard-Stratonovich transformation [26]. Although the gauge field does not have the usual Maxwell's term, the kinetic energy is generated once high energy modes of the boson fields are integrated out, which renormalizes the gauge coupling from infinity to g 2 ∼ 1/N. It is clear that slave-particles can propagate coherently in space only when the hopping field is 'condensed', and provides a smooth background. Since the hopping field is not a gauge invariant quantity, we need to be careful when we say that the hopping field is condensed. This notion can be sharply characterized by examining dynamics of topological defect.
Because the U(1) gauge field is compact, monopole is allowed as a topological defect in the theory. The mass of monopole is O(N) for a large N. Whether the slave-particles arise as low energy excitations of the theory depends on the dynamics of monopole. One can consider the following three different phases.
Confining phase
For a small N and small t, monopoles are light, and slave-particles are heavy. If monopoles are condensed, strong fluctuations of the phase mode of the hopping field confine the slaveparticles. Only gauge neutral composite particles, which are nothing but the original bosons in Eq.
(1), appear as low energy excitations. In this phase, all excitations are gapped. This phase is adiabatically connected to the disordered phase in the weak coupling limit.
Higgs phase
This is the phase which is electromagnetically dual to the confining phase. The slaveparticles are condensed when t is large. As a result of the condensation of charged fields, monopoles and anti-monopoles are connected by vortex lines which produce a linearly increasing potential : monopoles and anti-monopoles are confined. One slave-particle is eaten by the massive U(1) gauge boson, and (N − 1) gapless bosons are left. These modes are the Goldstone modes. This phase is smoothly connected to the bose condensed phase in the weak coupling limit.
Fractionalized (Coulomb) phase
For a large N, the mass of monopole is large. When both the slave-particles and monopoles are gapped, the Coulomb phase is realized. In this phase, slave-particles are deconfined, and arise as (gapped) excitations of the system. They are fractionalized modes because they carry only half the flavor quantum number of the original bosons. Moreover, the U(1) gauge field arises as a gapless excitation. It is noted that the gapless excitation in this phase is not a Goldstone mode. It is not protected by any microscopic symmetry. Saying that there is a gapless gauge boson in a gauge theory may sound trivial. However, we have to remember that the gauge boson is nothing but a collective excitation of the original boson fields. The existence of a collective excitation which remains gapless without a fine tuning is actually something remarkable : someone who does not use the language of gauge theory would find the origin of the gapless collective excitation mysterious. It turns out that the gapless mode is protected by a subtle order which is not characterized by any symmetry breaking scheme. This order, dubbed as quantum order [4], is associated with suppression of topological excitation, monopole in the long distance limit. Formally, this order can be expressed as the emergence of the Bianchi identity dF = 0 in the long distance limit,
where F is the field strength for the emergent gauge field. The key features of the non-trivial quantum order is the presence of the fractionalized excitations and the emergent gauge field.
Note that slave-particles are not gauge invariant objects. However, φ a 's become 'classical'
in the large N limit where non-perturbative fluctuations of the hopping field are suppressed.
In this regard, fractionalization is associated with the emergence of an 'internal' space. discuss about a matrix model and its possible phases. We will draw a close analogy between the quantum order present in the Coulomb phase of the boson model and a quantum order present in the holographic phase of the matrix model. We will see that the holographic phase has a distinct quantum order associated with the emergence of an 'external' space.
III. MATRIX MODEL
We start with a matrix model defined on the D-dimensional Euclidean hypercubic lattice,
Z = dU e −S[U ](6)
with the action,
S[U] = NM 2 <i,j> tr(U † ij U ij ) + N 2 V 1 N W C .(7)
Here i, j are site indices in the lattice with lattice spacing a, and U ij = U † ji is N × N complex matrix defined on the nearest neighbor bond < i, j >. W C is Wilson line defined on the closed oriented loop C,
W C = tr <ij>∈C U ij ,(8)
where the product is ordered along the path. V [W C /N] is a function of Wilson loop operators,
V = − ∞ n=1 N −n {C 1 ,..,Cn} J {C 1 ,..,Cn} n k=1 W C k(9): U ij → V † i U ij V j .
Eq. (6) may be viewed as the partition function for a (D − 1)-dimensional quantum matrix model in the imaginary time formalism.
To see that this model includes the usual U(N) gauge theory, we consider the following quartic action in Eq. (7) as an example,
N 2 V = <i,j> −NM 2 0 tr(U † ij U ij ) + Nv tr(U † ij U ij U † ij U ij ) + v ′ tr(U † ij U ij ) 2 −NJ W ,(10)
where represents unit plaquettes on the lattice. Here M 2 0 > 0, v > 0, v ′ > 0. We assume that v ′ is sufficiently large compared to J. The relative magnitude of M and M 0 determines the shape of the potential for the matrix field. For small M 0 , U ij = 0 is the minimum, and the system is fully gapped. For large M 0 , the low energy manifold is spanned by the matrices that satisfy
U ij U † ij = uI with u ∼ M 2 0 −M 2 2(v+v ′ )
. In this case, the low energy effective theory becomes the U(N) lattice gauge theory with the 't Hooft coupling λ ∼ (Ju 4 ) −1 . This theory can be viewed as a 'linear sigma model' for the U(N) gauge theory. Presumably, the gapped phase in the small M 0 limit is smoothly connected to the confinement phase of the gauge theory. As M 0 is increased further, the system can go through a phase transition to the deconfinement phase at a critical coupling M c 0 , depending on the dimension. If the phase transition is continuous, we can take the continuum limit by taking a → 0 and M 0 → M c 0 such that the confining scale is fixed.
IV. GENERAL CONSTRUCTION
In this section, we construct a holographic dual for the matrix model in Eq. (7) with general potential V in general dimensions. We will follow the idea introduced in Ref. [12] where coupling constants are lifted to dynamical fields in the bulk space where the extra dimension corresponds to the length scale of the renormalization group flow. In the presence of multi-trace operators, this formalism becomes rather complicated [13] because one has to introduce independent fields for infinitely many multi-trace operators that are generated along the renormalization group flow.
This issue is present even though multi-trace operators are not turned on initially, because they are generated at low energy scales in any case. To avoid this complication, here we express multi-trace operators in terms of single-trace one, by introducing a complex auxiliary field φ C for each loop
C (see Appendix A), Z = dUdφ (0) C dφ (0) * C e −S 1 ,(11)
where
S 1 = NM 2 tr(U † ij U ij ) + N 2 φ (0) C (φ (0) * C − W C /N) + N 2 V [φ (0) * C ].(12)
Here we dropped a multiplicative numerical factor in the partition function, which is not important.
It is noted that Z is well defined although S 1 is not bounded from below as a function of φ (0) C and φ (0) * C . This is because S 1 is complex and contributions from large negative S 1 is canceled because of rapid oscillation in phase. The repeated indices ij and C are understood to be summed over nearest neighbor links and closed loops, respectively.
To perform a real space renormalization group [12,27,28], an auxiliary matrix fieldŨ ij is introduced in each link,
Z = (N 1/2 µ) N 2 N l dφ (0) C dφ (0) * C dUdŨ e −S 2 ,(13)
where
S 2 = S U V [φ (0) * C , φ (0) C ] − Nφ (0) C W C +NM 2 tr(U † ij U ij ) + Nµ 2 tr(Ũ † ijŨ ij ).(14)
Here N l is the number of links in the lattice, and
S U V [φ (0) * C , φ (0) C ] = N 2 φ (0) C φ (0) * C + V [φ (0) * C ](15)
is an action for φ (0) C . We change the variables as
U ij = e −αdz (u ij +ũ ij ), U ij = e −αdz (Au ij + Bũ ij ),(16)
where α is a positive constant, dz is an infinitesimally small parameter, and
A = − M 2 mµ , B = m µ ,(17)
with
m 2 = M 2 e 2αdz − 1 .(18)
In terms of the new variables, the partition function becomes
Z = (N 1/2 m) N 2 N l dφ (0) C dφ (0) * C dudũ e −S 3 ,(19)
where
S 3 = S U V [φ (0) * C , φ (0) C ] − Nφ (0) ′ C W ′ C +N M 2 tr(u † ij u ij ) + m 2 tr(ũ † ijũ ij ) .(20)
Here W ′ C = tr <ij>∈C (u ij +ũ ij ) , and φ
(0) ′ C = e −αdzL C φ (0) C ,
where L C is the length of the loop C. The fieldũ ij with the large mass m has taken away a small amount of quantum fluctuations from the original field U ij , which leaves an action for u ij with smaller couplings φ (0) ′ C . Therefore, we can interpret u ij 's as low energy fields andũ ij 's as high energy fields.
Fluctuations ofũ ij renormalize the (dynamical) couplings for the low energy field u ij . Integrating overũ ij , we obtain
Z = dφ (0) C dφ (0) * C du e −S 4(21)
to the linear order in dz, where
S 4 = S U V [φ (0) * C , φ (0) C ] − Nφ (0) ′ C w C − 1 2m 2 F ij [C 1 , C 2 ]φ (0) ′ [C 1 +C 2 ] ij w C 1 w C 2 − N 2m 2 G ij [C 1 , C 2 ]φ (0) ′ C 1 φ (0) ′ C 2 w (C 1 +C 2 ) ij +NM 2 tr(u † ij u ij )(22)
with w C = tr <ij>∈C u ij . In the third and the fourth terms, ij runs over all nearest neighbor links, and C 1 , C 2 are understood to run over all possible loops including null loops with the convention φ ∅ i = 0, φ * ∅ i = 1 and w ∅ i = 1 for null loops, where ∅ i refers to the null loop at site i. Here we regard null loops at different sites as different loops. By this, we can keep the combinatorics simpler. In the third term, F ij [C 1 , C 2 ] is a form factor that tells whether or not two loops C 1 and C 2 are 'nearest neighbors' : F ij [C 1 , C 2 ] = 1 if C 1 and C 2 can be merged into one loop by adding the link ij and rejoining the loops, and F ij [C 1 , C 2 ] = 0 otherwise. [C 1 , C 2 ] ij denotes the loop that is made of C 1 and C 2 with the addition of the link ij. When both C 1 and C 2 are non-trivial loops, the third term describes a process where a loop splits into two loops ( Fig. 1 (a)). When one of the two loops is a null loop, it describes a process where a loop becomes shorter by eliminating a self-retracting link ( Fig. 1 (b)). When both are null loops, it describes a self-retracting link disappearing ( Fig. 1 (c)). In the fourth term, G ij [C 1 , C 2 ] is a form factor that tells whether or not two loops C 1 and C 2 are sharing the link ij : G ij [C 1 , C 2 ] = 1 if C 1 and C 2 can be merged into one loop by removing the shared link ij, and G ij [C 1 , C 2 ] = 0 otherwise. (C 1 , C 2 ) ij denotes the loop that is made by merging C 1 and C 2 by removing the shared link ij. The fourth term describes a process where two loops merge into one loop ( Fig. 1 (d)). In the small dz limit, 1/m 2 ∼ O(dz), and we can replace φ
(0) ′ C with φ (0)
C in the third and fourth terms of the action to the linear order in dz.
Note that double trace operators are generated for u ij . Another set of auxiliary fields is introduced to express the double-trace operator in terms of single-trace operators as
Z = dφ (0) C dφ (0) * C dφ (1) C dφ (1) * C du e −S 5 ,(23)
where
S 5 = S U V [φ (0) * C , φ (0) C ] +N 2 φ (1) C (φ (1) * C − w C /N) −N 2 φ (0) ′ C φ (1) * C − N 2 2m 2 F ij [C 1 , C 2 ]φ (0) [C 1 +C 2 ] ij φ (1) * C 1 φ (1) * C 2 + G ij [C 1 , C 2 ]φ (0) C 1 φ (0) C 2 φ (1) * (C 1 +C 2 ) ij +NM 2 tr(u † ij u ij ) = S U V [φ (0) * C , φ (0) C ] +N 2 φ (1) * C (φ (1) C − φ (0) ′ C ) − N 2 αdz M 2 F ij [C 1 , C 2 ]φ (0) [C 1 +C 2 ] ij φ (1) * C 1 φ (1) * C 2 + G ij [C 1 , C 2 ]φ (0) C 1 φ (0) C 2 φ (1) * (C 1 +C 2 ) ij −Nφ (1) C w C + NM 2 tr(u † ij u ij ).(24)
If we repeatedly apply the steps in Eqs. (13) - (24) to the last line of Eq. (24) R times, we obtain
Z = R l=0 dφ (l) C dφ (l) * C du e −S 6 ,(25)
where
S 6 = S U V [φ (0) * C , φ (0) C ] +N 2 R l=1 φ (l) * C (φ (l) C − φ (l−1) C + αL C dzφ (l−1) C ) − αdz M 2 F ij [C 1 , C 2 ]φ (l−1) [C 1 +C 2 ] ij φ (l) * C 1 φ (l) * C 2 + G ij [C 1 , C 2 ]φ (l−1) C 1 φ (l−1) C 2 φ (l) * (C 1 +C 2 ) ij −Nφ (R) C w C + NM 2 tr(u † ij u ij ).(26)
What is the physical meaning of the auxiliary fields ? In the last line of Eq. (26), we note that
φ (R)
C acts as a source for the low energy matrix field at scale e −Rdz . The key difference from the standard renormalization group procedure is that the source fields are dynamical fields rather than fixed constants at each scale [12]. On the other hand, the equation of motion for φ
(R) C implies that < φ (R) * C >= 1 N < w C > .(27)
Therefore, the conjugate field φ (R) * C describes the Wilson loop operator. As we will see below, φ C and φ * C are conjugate fields which satisfy a non-trivial commutation relation : sources and operators are conjugate to each other. Finally, we integrate out u to obtain
Z = R l=0 dφ (l) C dφ (l) * C e −S 7 ,(28)
where
S 7 = S U V [φ (0) * C , φ (0) C ] +N 2 R l=1 φ (l) * C (φ (l) C − φ (l−1) C + αL C dzφ (l−1) C ) − αdz M 2 F ij [C 1 , C 2 ]φ (l−1) [C 1 +C 2 ] ij φ (l) * C 1 φ (l) * C 2 + G ij [C 1 , C 2 ]φ (l−1) C 1 φ (l−1) C 2 φ (l) * (C 1 +C 2 ) ij +S IR [φ (R) C ].(29)
Here S IR is the effective potential given by
S IR [φ (R) C ] = − ln du e −N M 2 tr(u † ij u ij )+N φ (R) C w C .(30)
For a future use, we define
V ′ [φ (R) C ] ≡ 1 N 2 S IR [φ (R) C ],(31)
which can be computed using the strong coupling expansion,
V ′ [φ (R) C ] = −φ (R) C 1 M −L C 1δ C 1 ,0 − 1 2 φ (R) C 1 φ (R) C 2 M − 2 i=1 L C iδ C 1 +C 2 ,0 − 1 6 φ (R) C 1 φ (R) C 2 φ (R) C 3 M − 3 i=1 L C iδ C 1 +C 2 +C 3 ,0 − ....(32)
Here the delta function is defined asδ
C,0 ≡ <i,j> δ Q ij [C],0 ,(33)
where Q ij [C] is the U(1) charge defined on link ij associated with the flux of loop C [13]. If the loop C passes through the link ij from i to j (from j to i) n times, Q ij [C] = n(−n). The first, second and third terms are from a self retracting loop ( Fig.2 (a)), two loops ( Fig.2 (b)) and three loops ( Fig.2 (c)), respectively. Higher order terms can be obtained similarly. Now we take dz → 0 and R → ∞ limits with β ≡ Rdz fixed. Then, the partition function is written as
Z = Dφ C Dφ * C e −(S bulk [φ * C (z),φ C (z)]+S U V [φ * C (0),φ C (0)]+S IR [φ C (β)]) ,(34)
where
S bulk = N 2 β 0 dz φ * C ∂ z φ C + αL C φ * C φ C − α M 2 F ij [C 1 , C 2 ]φ * C 1 φ * C 2 φ [C 1 +C 2 ] ij + G ij [C 1 , C 2 ]φ * (C 1 +C 2 ) ij φ C 1 φ C 2 .(35)
Since the partition function is independent of β, we can take β → ∞. From now on, we will interpret the scale parameter z as an imaginary 'time'. The dual description becomes a (D + 1)dimensional field theory of closed loop. Although the action is written in terms of continuous z, one should go back to the discrete version whenever there is an ambiguity, e.g., when extracting boundary conditions by taking variations with respect to boundary fields.
As is the case for matrix models, there are two important parameters that are independent with each other. The first is 1 N 2 which controls the strength of quantum fluctuations of the loop fields : the whole action including the boundary actions scales as N 2 . The second is the 't Hooft coupling.
V. HAMILTONIAN PICTURE
A. Partition function as a transition amplitude between many-body loop states
The partition function can be viewed as an imaginary-time transition amplitude between manybody loop states. To see this, we will use a rescaled loop variable in this sub-section,
Φ C ≡ Nφ C .(36)
The bulk action in the new variable becomes
S bulk = ∞ 0 dz Φ * C ∂ z Φ C + αL C Φ * C Φ C − α NM 2 F ij [C 1 , C 2 ]Φ * C 1 Φ * C 2 Φ [C 1 +C 2 ] ij + G ij [C 1 , C 2 ]Φ * (C 1 +C 2 ) ij Φ C 1 Φ C 2 . (37)
The action has the form for canonical bosonic fields, where Φ C (Φ * C ) corresponds to the coherent field associated with the annihilation (creation) operator defined in the space of closed loops. The annihilation and creation operators a C , a † C satisfy the standard commutation relation
a C , a † C ′ = δ C,C ′ ,(38)
where δ C,C ′ is a Kronecker-delta function defined in the space of loops. Then the partition function can be written as an imaginary-time transition amplitude,
Z = lim β→∞ < Ψ f |e −βH |Ψ i >,(39)
between the initial (UV) state at z = 0,
|Ψ i > = dΦ * C dΦ C Ψ i [Φ * C , Φ C ]|Φ C >,(40)
with
Ψ i [Φ * C , Φ C ] = e −Φ * C Φ C −N 2 V [Φ * C /N ] ,(41)
and the final (IR) state at z = ∞,
|Ψ f > = dΦ * C dΦ C Ψ f [Φ * C , Φ C ]|Φ C >,(42)with Ψ f [Φ * C , Φ C ] = e −Φ * C Φ C −N 2 V ′ [Φ * C /N ] .(43)
Here
Ψ i [Φ * C , Φ C ] and Ψ f [Φ * C , Φ C ]
are the wavefunctions of loops written in the coherent state basis,
|Φ C >= e Φ C a † C |0 >,(44)
where |0 > is the vacuum in the Fock space of loops : a C |0 >= 0 for all a C . (For the derivation of Eqs. (41) and (43), see Appendix. B). The bulk Hamiltonian is given by
H = αL C a † C a C − α NM 2 F ij [C 1 , C 2 ]a † C 1 a † C 2 a [C 1 +C 2 ] ij + G ij [C 1 , C 2 ]a † (C 1 +C 2 ) ij a C 1 a C 2 . (45)
The first term in the Hamiltonian describes a tension of closed loops. The second and the third terms are the interaction terms which describe the processes where one loop splits into two loops, and two loops merge into one loop, respectively, as is shown in Fig. 1. We use the convention a ∅ i = 0, a † ∅ i = 1 for null loops. Similar loop Hamiltonians that describe joining and splitting processes of loops were considered in matrix models [30,31]. Hamiltonian that governs the quantum dynamics of loops along the scale z which is interpreted as an imaginary time. It is noted that the Hamiltonian is not Hermitian. Due to the cubic interaction term, the Hamiltonian is unbounded from below. However, the transition amplitude in Eq. (39) is well defined because eigenvalues of the Hamiltonian are complex. Eigenvalues with a large negative real part in general come with a large imaginary part, and their contributions cancel with each other due to oscillation in phase. Second, the bulk Hamiltonian is universal, and it is independent of the details of the matrix model. All informations pertaining to the specifics of the matrix model are encoded in the initial wavefunction at z = 0. Third, the strength of the interaction between loops is order of 1/N, and loops are weakly interacting in the large N limit. Therefore, the theory becomes classical in the large N limit. Fourth, H does not have any hopping term such as a † C 1 a C 2 with different C 1 and C 2 . This fact will become important for gauge symmetry, which will be discussed in Sec. V. For earlier works on string field theories formulated without quadratic action, see Ref. [32,33].
The fact that the partition function is independent of β has a remarkable consequence. By taking the derivative of Eq. (39) (for a finite β) with respect to β, we obtain
0 = < Ψ f |e −βH H|Ψ i > .(46)
Since physical states are singlets of H, the Hamiltonian can be viewed as a generator of a 'gauge transformation'. The gauge transformation corresponds to a reparameterization of z. It is based on the fact that one can choose different speed of renormalization group flows at different scales without affecting the physics. By choosing the parameter α to be z-dependent, the reparameterization symmetry can be made explicit [12]. Here α(z) becomes the lapse function. Reparameterizations of z form a subgroup of the full diffeomorphism in the (D + 1)-dimensional space. It would be interesting to formulate the theory where the full diffeomorphism can be made explicit in the bulk.
Here we proceed with the present formalism where we choose specific time slices along the z direction. is shown in Fig. 4 (b). As it will be shown in Sec. VII. A, loop fields for self-retracting loops have non-zero vacuum expectation values in the bulk. Therefore, a pair of loop and anti-loop can be created out of vacuum. This means that two loops with the opposite orientations act as particle and anti-particle in a relativistic field theory. Finally, those loops emitted at the UV boundary are absorbed at the IR boundary. In this sense, the UV boundary is a source of loops, and the IR boundary is a sink.
< W C >= 1 N ∂ ln Z ∂J C J C =0 = lim β→∞ < Ψ f |e −βH a † C |Ψ i > < Ψ f |e −βH |Ψ i > .(47)
If M is large, loops propagate independently in the bulk. To the zeroth order in 1/M, the loop C propagate to the sink along the straight path. However, this configuration vanishes as e −αβL C in the large β limit because of the tension. In order for the expectation value to survive, the large loop C should absorb other smaller loops from the condensate to disappear before it reaches the IR boundary. Then the evolution of the Wilson loop forms a world-sheet in the bulk. One such configuration is shown in Fig. 5. Then the expectation value is given by the sum over all worldsheets of the Wilson loop.
Since the interaction between loops is O(1/N), loops become classical in the large N limit.
This implies factorization of Wilson loop operators in the large N limit,
n k=1 W C k = n k=1 W C k + O(N n−2 ).(48)
VI. GAUGE SYMMETRY
The absence of the hopping term in the Hamiltonian has a deep origin : the loop field theory has a gauge symmetry. Note that this gauge symmetry is not related to the U(N) gauge symmetry of the original matrix model. Loop fields are singlets for the U(N) gauge symmetry. In this section, we examine the consequences of the new gauge symmetry carefully. From now on, we return to
the unscaled loop variable φ C ≡ Φ C N . The bulk action in Eq. (35) is invariant under the time-independent transformation generated by Q ij at each link φ C → e iθµ(i)Q ii+µ [C] φ C ,(49)
where i is summed over all sites, µ is summed over D directions of nearest neighbor links, and θ µ (i) is a time-independent angle defined on the link < i, i+µ >. The IR boundary action respects the symmetry, but the UV action does not. This is because the UV potential
V [φ * C ] = − ∞ n=1 {C 1 ,..,Cn} J C 1 ,C 2 ,..,Cn n k=1 φ * C k(50)
includes sources J C 1 ,C 2 ,..,Cn which explicitly break the symmetry. It is useful to view J C 1 ,C 2 ,..,Cn as an expectation value of another dynamical loop field. Then, the full theory is invariant if we allow the UV source to transform as
J C 1 ,C 2 ,..,Cn → e iθµ(i) n k=1 Q ii+µ [C k ] J C 1 ,C 2 ,..,Cn .(51)
This time-independent symmetry can be lifted to a full space-time gauge symmetry by introducing temporal components of a two-form gauge field B M N in the bulk with M, N = z, 1, 2, ..., D,
S bulk = N 2 β 0 dz φ * C ∂ z + iQ ii+µ [C]B µz (i, z) φ C + αL C φ * C φ C − α M 2 F ij [C 1 , C 2 ]φ * C 1 φ * C 2 φ [C 1 +C 2 ] ij + G ij [C 1 , C 2 ]φ * (C 1 +C 2 ) ij φ C 1 φ C 2 ,(52)
where B µz (i) with µ = 1, 2, .., D are the temporal components of the two-form gauge field defined at each spatial link. This two-form gauge field is the Kalb-Ramond gauge field [29]. Now the full theory is invariant under the space-time dependent gauge transformation with
φ C (z) → e iθµ(i,z)Q ii+µ [C] φ C (z), B µz (i, z) → B µz (i, z) + θ z (i + µ, z) − θ z (i, z) − (∂ z θ µ (i, z)),(53)
where θ z (i, z) is a temporal gauge parameter defined at each site. This is the discrete version of the usual gauge transformation for the two-form field, 6 shows a loop changing its shape by absorbing two small loops. Therefore, loops can propagate with the help of other loops. If loop fields are 'condensed', whose precise meaning will become clear in a moment, the condensate provides a coherent background on which other loops can propagate. Loops propagate 'on the shoulders of other loops' to explore the bulk space. This is analogous to the the slave-particle theory discussed in Sec. II. One difference is that loop fields themselves play the role of 'hopping fields' for other loops, while in slave-particle theory the hopping field is a bi-linear of slave-particle fields. The difference originates from the fact that loops are extended objects while slave-particles are point objects. Only when the condensates of loop fields are 'coherent', the bulk space is regarded as a well defined extended space by loops. Otherwise, loops are more or less localized in space. In this sense, an extended space emerges in the bulk as a dynamical feature of a phase where loop fields form coherent condensates.
B M N → B M N + ∂ M θ N − ∂ N θ M .θ µ (i, z) = z 0 dz ′ B µz (i, z ′ )(54)
When do loop fields become coherent ? To make this notion more precise, we first note that the phase modes of complex loop fields φ C = |φ C |e ib C play the role of the spatial components of the two-form gauge field. To see this, suppose that the loop field has a background value < φ C >.
Then the cubic interaction generates a quadratic hopping term,
− α < φ C > M 2 a † C+C ′ a C ′ .(55)
The amplitude of < φ C > is the strength of the hopping, and the phase determines the geometric phase acquired when the loop C ′ hops to C ′ + C. Therefore b C plays the role of the spatial components of the two-form gauge field to which loops are electrically coupled. Note that the two-form gauge field is also a part of dynamical loop fields. We identify
b C = A C B,(56)
where B µν is the spatial components of the two-form gauge field and the integration is over an area A C enclosed by the loop C. Let us focus on the loops with unit plaquettes in which case we take A C as the surface spanned by the unit plaquette.
Although the two-form gauge field does not have the bare action, it acquires the kinetic energy from quantum fluctuations. This is similar to the way that the Maxwell's term is dynamically generated for the auxiliary gauge field in the slave-particle theory as discussed in Sec. II. The gauge coupling for the two-form gauge field is renormalized to O(1/N 2 ). This can be understood by integrating out 'heavy' loop fields to obtain an effective action for 'light' loop fields in the bulk.
It is easiest to see the generation of the kinetic energy in the large M limit, where we can use 1/M as an expansion parameter. The 'mass' of a loop field is proportional to the length of the loop because of the tension. We integrate out loop fields with L > 4 and obtain an action for the loop fields with L ≤ 4. In particular, we focus on the effective action for the shortest non-self-retracting loops with L C = 4 whose phase modes can be viewed as the spatial components of the two-form gauge field on unit plaquettes. For simplicity, we choose the temporal gauge with B µz = 0 and the scale of z to set α = 1.
Let us consider three vertices in Eq. (52). Each vertex has the form N 2 M φ * C i φ C i1 φ C i2 with i = 1, 2, 3, where C i 's have length 6 and C il 's with l = 1, 2 have length 4 as is represented in Fig. 7 (a). They describe the processes where a loop on a unit plaquette with sides µ, ν merges with a loop on a unit plaquette with sides ν, λ to form a loop with with length 6. Here we interpret φ C i 's as heavy fields and φ C il 's with l = 1, 2 as light fields. In particular, the phase modes of φ C il represents the two-form gauge field defined on each plaquette. Now we integrate out the heavy loop fields using the quadratic action. Because this quadratic action has the local U(1) symmetry in the loop space, φ C → e iϕ C φ C , we need to introduce a series of vertices in order to saturate φ * C with φ C and obtain a non-vanishing result. A minimum path to saturate all heavy fields is shown in Fig. 7 (b). In the first step, we add a vertex of the type N 2
M 2 φ * (C 1 +C 2 ) ij φ C 1 φ C 2 , where (C 1 + C 2 )
ij is a loop that results from merging C 1 and C 2 by removing one shared bond. Integrating out φ C 1 and φ C 2 , we obtain the loop fields in the second configuration in Fig. 7 (b). In the second step, we use
a vertex N 2 M 2 φ * (C 1 +C 2 ) ′ ij φ (C 1 +C 2 ) ij ,S ef f ∼ − dz cubes N 2 M 24 3 i=1 φ C i1 φ C i2 + 3 i=1 φC i1 φC i2 ,(57)
where the summation is over all cubes in the D-dimensional lattice. The second term is from the same process for the anti-loops. The loop field for the anti-loop, φC is in priori independent of φ C .
However they are dynamically mixed. Because of pair-annihilation and pair-creation processes of loops and anti-loops as is shown in Fig. 4, the effective action should include terms of the form, φ C φC and φ * C φ * C . As a result, the phase modes of φ C and φC are locked. Only the anti-symmetric mode with bC = −b C remains gapless in the presence of mixing. If φ C il 's have finite amplitude φ 0 , this gives the standard 'magnetic' term for the two-form gauge field defined on each cube of the lattice
− 1 g 2 KR dz cubes cos a 3 (∆ µ B νλ + ∆ ν B λµ + ∆ λ B µν ) ,(58)
where we use the fact that b = a 2 B µν for a unit plaquette with sides µ, ν. The finite derivative is
defined as ∆ µ B νλ ≡ B νλ (x+x µ )−B νλ (x) a . Here g 2 KR ∼ N 2 M 24 φ 6 0 −1
is the renormalized coupling for the Kalb-Ramond (KR) two-form gauge field. Now we turn our attention to the 'electric' term which involves the time-derivative of the gauge field. We consider the quadratic action for the loop fields on unit plaquettes,
N 2 [φ * C (∂ z + L C )φ C + φ * C (∂ z + LC)φC] .(59)
If we integrate out the amplitude fluctuations of the loop fields, the time derivative term will be generated for b C . Because of the dynamical constraint bC = −b C caused by the mixing between φ C and φC, the linear time derivative term is canceled, leading to the second derivative term,
N 2 a 4 dz (∂ z B µν ) 2(60)
for each plaquette. Eqs. (58) and (60) represent the full kinetic energy term for the two-form gauge field in the temporal gauge.
Because of the gauge symmetry, the mass term is not allowed for the two-form gauge field in the bulk. However, the gauge symmetry does not automatically imply that the two-form field arises as a massless excitation in the bulk. This is because of the compactness of the phase mode
: b C ∼ b C + 2π,dH(x) = 2π dξ D−3 δ D+1 (x − ξ),(61)
where ξ is the coordinate of the NS-brane embedded in the (D + 1)-dimensional space, and dξ D−3 is the oriented volume element of the brane. This is illustrated in Fig. 8. Note that the Dirac quantization condition between the charge carried by loop fields, which is set to be 1 as can be
b 12 (x, z) = 2πδ x 1 ,x 1 NS δ x 2 ,x 2 NS Θ(x 3 − x 3 N S )Θ(z − z N S ),(62)
where the phases of the loop fields on x 1 − x 2 plaquettes are 2π along a semi-infinite line in the 3-dimensional space for z > z N S , and the phases are zero, otherwise. Since b C ≡ b C + 2π, this configuration is equivalent to the trivial configuration where b C = 0 everywhere. However, we can view this configuration as a topological defect with a non-trivial three-form flux on a cube, is deformed smoothly, b
H 123 = 2πδ 3 (x − x N S )Θ(z − z N′ C = b C + δb C .
Under a smooth deformation, the flux is smeared out over an extended region in the space, while the net flux 2π does not change. Now the flux is visible by loop fields. This is illustrated in Fig. 9. As z is increased further, the flux can merge back into one cube and disappear into the vacuum through the inverse process. This describes a pair of instanton and anti-instanton as is shown in Fig. 10. In higher dimensions, NS D−4 -branes are extended objects. In (D + 1)-dimensional bulk, we can think of Fig. 10
VII. EMERGENT SPACE AND QUANTUM ORDER IN HOLOGRAPHIC PHASES
In this section, we will discuss various phases that the matrix model can have, by focusing on the behavior of NS D−4 -branes. In particular, we will see that the dynamics of string excitations around a saddle-point configuration of the loop fields is determined by the fate of NS D−4 -branes in the bulk. In order to discuss about this issue systematically, we first turn to the saddle point equations.
A. Saddle point solution
The saddle point configuration of loop fields is determined from the equation of motion.
∂ z φ C = −L C φ C + 1 M 2 2F ij [C, C 1 ]φ * C 1 φ [C+C 1 ] ij + G ij [C 1 , C 2 ]φ C 1 φ C 2 δ (C 1 +C 2 ) ij ,C , (63) −∂ z φ * C = −L C φ * C + 1 M 2 F ij [C 1 , C 2 ]φ * C 1 φ * C 2 δ [C 1 +C 2 ] ij ,C + 2G ij [C 1 , C 2 ]φ * (C+C 1 ) ij φ C 1 .(64)
These equations are supplemented by two sets of boundary conditions. It is more convenient to use the action with discrete time step dz to isolate boundary fields from bulk fields. The UV boundary condition is obtained from Eq. (12),
∂S 1 ∂φ (0) * C = N 2 φ (0) C + ∂V [φ (0) * C ] ∂φ (0) * C = 0,(65)
and the IR boundary condition from Eq. (29),
∂S 7 ∂φ (R) C = N 2 φ (R) * C + ∂V ′ [φ (R) C ] ∂φ (R) C = 0.(66)
When the UV potential includes only single-trace operators, V = −J C φ (0) * C , Eq. (65) leads to the standard Dirichlet boundary condition for the source field : φ (0) C = J C . For more general non-linear UV potential, it becomes a mixed boundary condition. This is consistent with the prescription for the UV boundary condition in the presence of multi-trace deformations in the standard AdS/CFT dictionary [34]. The IR boundary condition is a mixed one because V ′ is in general non-linear.
For self-retracting loops, V ′ also contains terms that are linear in loop fields as is shown in the first term in Eq. (32). Eq. (66) then implies that φ * C = 0 at the IR boundary for self-retracting loops. As will be shown in the next paragraph, this means that loop fields for self-retracting loops have non-zero expectation values at all z in the bulk. This, in turn, generates non-zero vacuum expectation values of the source fields φ C for self-retracting loops. As was discussed in Fig. 4, self-retracting loops can turn into a loop/anti-loop pair through an interaction.
In general, the saddle point configuration is z-dependent, and φ C (z) is not necessarily the complex conjugate of φ * C (z). One should treat φ C and φ * C as two independent fields. Then the equations of motion can be viewed as a set of Hamiltonian equations in the phase space of {φ C , φ * C }. Although Eqs. (63) and (64) are coupled equations for φ C and φ * C , one can eliminate φ * C in favor of φ C . We first note that the partition function and all observables including vacuum expectation values of Wilson loop operators represented by φ * C (0) are independent of how we choose β in Eqs. (34) and (35). This means that the saddle point solution φ C (z) and φ * C (z) for z < β is independent of β. Since we could have put β anywhere, φ C and φ * C should satisfy the IR boundary condition at any z,
φ * C (z) = − ∂V ′ [φ C (z)] ∂φ C (z) .(67)
This is illustrated in Fig. 11. The fact that one can put the IR boundary at any z has an interesting implication on the role of the IR boundary. Usually, one can associate a boundary condition with a physical object located at the boundary. However, Eq. (67) is special in the sense that an observer at z < β can not 'feel' the presence of a physically identifiable object at z = β. Suppose one stops the renormalization group procedure at z = β and impose Eq. (67) at the IR boundary. If a UV observer sends a wave toward the IR region, the reflected wave from the IR region is exactly the same as the reflected wave one would observe in the space which is extended to z = ∞ without an boundary. In this sense, the IR boundary is not a physical boundary : one can always trade the IR boundary with the space where z is extended to infinity.
Using Eq. (67), we can write a set of first order differential equations for the source field only,
∂ z φ C = −L C φ C + 1 M 2 −2F ij [C, C 1 ] ∂V ′ [φ C 1 ] ∂φ C 1 φ [C+C 1 ] ij + G ij [C 1 , C 2 ]φ C 1 φ C 2 δ (C 1 +C 2 ) ij ,C .(68)
Once φ C (z) is solved using the UV boundary condition in Eq. (65), the conjugate field is readily
B. Fluctuations near the saddle point
Fluctuations near the saddle point configurationφ C (z) describes dynamical string in the bulk,
φ C (z) =φ C (z) + χ C (z),(69)
where χ C describes small fluctuations around the saddle point. We call χ C string field to distinguish it from the loop field φ C . The dynamics of string is governed by the action,
S bulk = N 2 β 0 dz χ * C ∂ z χ C + L C χ * C χ C − 1 M 2 F ij [C 1 , C 2 ]φ [C 1 +C 2 ] ij (z)χ * C 1 χ * C 2 + 2F ij [C 1 , C 2 ]φ * C 1 (z)χ [C 1 +C 2 ] ij χ * C 2 + G ij [C 1 , C 2 ]φ * (C 1 +C 2 ) ij (z)χ C 1 χ C 2 + 2G ij [C 1 , C 2 ]φ C 1 (z)χ C 2 χ * (C 1 +C 2 ) ij − 1 M 2 F ij [C 1 , C 2 ]χ [C 1 +C 2 ] ij χ * C 1 χ * C 2 + G ij [C 1 , C 2 ]χ C 1 χ C 2 χ * (C 1 +C 2 ) ij .(70)
Here χ C (z) = χ r C (z) + iχ i C (z) is a complex field. Following the standard method of the steepest descent, the contours of the real and imaginary parts of the complex fields are chosen so that the real part of the eigenvalues of the quadratic action becomes maximum along the deformed contours [35,36]. Note that the string fields χ C acquire the hopping term through non-zero condensates of loop fields. It also has the terms that describe pair creation/annihilation of two closed strings.
C. Possible phases of the matrix model
In this section, we describe possible states of the matrix model using the holographic description. In particular, we will see that strings have different dynamics depending on the behavior of NS D−4 -branes. One observable that is useful in distinguishing different states is the correlation function between Wilson loop operators. In particular, we focus on the correlation function of phase fluctuations of Wilson loop operators,
F (C, C ′ ) = δb C δb C ′ ,(71)where δb C = b C − < b C >.
In the bulk description, this correlation function corresponds to a two-point string-string correlation function. This object is of particular interest because the string state that corresponds to the phase mode describes the two-form gauge field in the bulk.
Confinement phase
For non-self-retracting loops, V ′ is quadratic or of higher order in the loop fields. Fig. 12 (a). This leads to an exponentially decaying correlation function for the Wilson loop operators. In the confinement phase, the bulk geometry ends at a finite scale due to the proliferation of NS D−4 -branes. This is reminiscent of the idea that geometry can get truncated by tachyon condensation [40]. deep inside the bulk. In the following, we will discuss the two scenarios in more detail.
Non-holographic critical phase
Here we discuss the first case which is likely to be realized when N is small. strings is predominantly along the D space dimensions as is illustrated in Fig. 12 (b).
Holographic critical phase I : deconfined closed string
The second scenario is qualitatively different from the previous one. In this phase, loop fields matrix theory. We identify this as the singleton mode localized at the boundary [7,44,45].
It is noted that the symmetry breaking source J {C 1 ,..,Cn} at the UV boundary does not necessarily open up a gap for the phase modes of loop fields in the bulk. This is because the symmetry breaking source is only at the UV boundary, but not in the bulk. This is analogous to the case where one applies a symmetry breaking field at the boundary of a system where a global symmetry is spontaneously broken in the bulk. Although the boundary field determines the direction of the symmetry breaking in the whole system, the Goldstone mode in the bulk survives in the thermodynamic limit.
The dynamics of the D-dimensional matrix model in the long distance limit is governed by strings that propagate in the (D + 1)-dimensional space. We call this phase holographic phase.
The hallmarks of the holographic phase are deconfined strings that propagate in the bulk with the extra dimension and the emergence of the Bianchi identity dH = 0 for the two-form gauge field in the long distance limit. It is emphasized that these features are not protected by any symmetry of the original matrix model. They are dynamical properties which emerge only in the holographic phase. The quantum order present in the holographic phase is analogous to the quantum order associated with the emergent Bianchi identity in the fractionalized phase of the slave-particle theory as discussed in Sec. II.
In the holographic critical phase, strings can propagate deep inside the bulk as is shown in Fig. 12 (c). The correlation function shows a power-law decay through a classical trajectory that is extended to the bulk. Because of the gauge symmetry and the non-trivial quantum order, we expect that the scaling dimension of the phase mode of Wilson loop operators will be protected accordingly. To determine the scaling dimension, one has to first solve the loop equations in the bulk and find the string Green's function in the background determined by the loop fields. We defer an explicit calculation for future studies.
In the four-dimensional N = 4 super Yang-Mills theory, the phase mode of Wilson loop operators has the scaling dimension 6 [47]. The reason why it is not 2, which is the expected scaling dimension for massless two-form fields in D = 4 [7], is that a Chern-Simons coupling generates a mass for the two-form gauge field through the mixing with the Ramond-Ramond fields. However, the scaling dimension is still protected from acquiring a large quantum correction in the large N limit. Such protection of scaling dimension is often attributed to supersymmetry. However, the non-trivial quantum order will protect the scaling dimension of the phase mode from acquiring a large quantum correction even in non-supersymmetric holographic critical phases in the large N limit. This is true whether or not the two-form gauge field becomes massive through a Chern-Simons coupling with Ramond-Ramond fields. This is because the string theory becomes classical in the large N limit, and the coefficient of the Chern-Simons term is quantized. Therefore the mass of the two-form gauge field can not become large even when other string modes become very massive at strong coupling of the boundary matrix model. This, in turn, implies that the scaling dimension remains small for the phase mode of Wilson loop operators. This is in sharp contrast to the non-holographic critical phase where it is expected that the operator generally receives a large quantum correction at strong coupling.
Deconfinement phase
Strictly speaking, the critical phases discussed in the previous two sections are kinds of deconfinement phases. Here we use the term 'deconfinement phase' in a narrower meaning, that is, free theory in the IR limit. If the sources at the boundary are very large, the second and third terms in Eq. (68) dominate, and the source fields will grow as z increases for D > 4 in which case the boundary matrix model is expected to flow into IR free gauge theory in the weak coupling (large J {C 1 ,..,Cn} ) limit. As the amplitudes of the loop fields become larger, large loops are generated through the joining processes. As a result, loop fields with all sizes are condensed in the bulk.
Then strings propagating in the bulk become highly non-local because strings can hop from one configuration into another configuration which is very different from the initial one. This is a string condensed phase. In this phase, the two-form gauge field acquires a mass and NS D−4 -brane is confined due to the Higgs mechanism [43,46].
In the deconfinement phase, a string emitted from the boundary becomes very large in the bulk and lose its identity as a closed string. The critical fluctuations are mediated by highly non-local fluctuations in the bulk. In this phase, the locality is lost in the bulk.
The deconfinement phase can be viewed as an extreme limit of the holographic critical phase In non-supersymmetric theories, it is expected to be harder to stabilize a theory at an arbitrary gauge coupling. Most likely, we expect that the holographic critical phase will arise as a multi-critical point between the confinement phase and the deconfinement phase for a sufficiently large N.
D. A mean-field description
Some features of the phases discussed in the previous section can be easily understood if we focus on a subspace within the space of loop fields {φ C }. We focus on the mean-field Ansatz [43,46] where a loop field is represented by a product of link fields along the loop,
φ C (z) = (i,i+µ)∈C ξ µ (i, z).(72)
Here ξ µ (i, z) = ξ * −µ (i + µ, z) is a complex scalar field defined on the link i, i + µ. This is a huge simplification where we reduce the space spanned by functions defined on loop space into the space spanned by functions defined on the links. Under the gauge transformation, the link variables transform as
ξ µ (i, z) → e iθµ(i,z) ξ µ (i, z).(73)
Therefore, these link fields should be charged with respect to the two-form gauge field. A minimal action for the link field that has the same symmetry as the original loop model is an Abelian-Higgs model [43] for the link field,
S bulk = −t ξ M (i)ξ N (i + M)ξ * M (i + N)ξ * N (i)e −iB M N (i) + c.c. + i,M V (|ξ M (i)| 2 ) − 1 g 2 KR cubes cos(∆ L B M N + ∆ M B N L + ∆ N B LM ).(74)
Here we discretize the z direction, and the action is written in a (D + 1)-dimensional lattice. The box represents sum over all plaquettes including temporal plaquettes. The link fields ξ z along the temporal directions can be viewed as an auxiliary field that is introduced to keep the two-form gauge symmetry. One can use a continuum description as well [46]. If the link fields are condensed, the two-form gauge field acquires a mass due to the Higgs mechanism [43,46]. Note that loop fields with arbitrarily large size acquires expectation values in the Higgs phase because loop fields are just products of link fields. This corresponds to the deconfinement phase of the boundary matrix model.
If the link field is gapped and the two-form gauge coupling is small, the theory can be in the Coulomb phase. In this phase, closed strings are deconfined and the two-form gauge field arises as a light mode in the bulk. This corresponds to the holographic critical phase. Note that the loop fields can have finite expectation values even though link fields are gapped in this phase.
E. Holographic critical phase II : deconfined open string
The mean field description discussed in the previous section allows one to understand a yet another new phase of the matrix model. To see this, we first note that the decomposition in Eq.
(72) has a U(1) gauge redundancy,
ξ M (i) → e i(γ i+M −γ i ) ξ M (i),(75)
where γ i is a U(1) phase defined on each site on the bulk space. Because of this U(1) redundancy, the link field can not have a quadratic hopping term. This is similar to the U(1) gauge redundancy present in the slave-particle theory discussed in Sec. II. One can decouple the quartic term for the link fields using the Hubbard-Stratonovich transformation. The resulting action should include a dynamical compact U(1) gauge field, Under the two-form gauge transformation in Eq. (73), the U(1) gauge field transforms as
S ′ bulk = −t i,M =N ξ * M (i + N)ξ M (i)e −i[B M N (i)−A N (i+M )+A N (i)] + c.c. −t i,M =N ξ * N (i)ξ M (i)e −i[B M N (i)−A N (i+M )+A M (i+N )] + c.c. −t i,M =N ξ N (i + M)ξ M (i)e −i[B M N (i)+A M (i+N )+A N (i)] + c.c. + i,M V (|ξ M (i)| 2 ) − 1 g 2 KR cubes cos(∆ L B M N + ∆ M B N L + ∆ N B LM ).(76)A M (i) → A M (i) + θ M (i).(77)
The first term in Eq. (76) describes a link parallel to the direction M hops along the direction N as is shown in Fig. 13 (a), and the second and third terms describe hoppings where the direction of a link changes as in Fig. 13 (b) and (c).
φ C (z) = tr (i,i+µ)∈C Ξ µ (i, z) ,(78)
where Ξ µ (i, z) = Ξ † −µ (i + µ, z) is aÑ ×Ñ complex matrix field defined on the link < i, i + µ > in the bulk. This decomposition has the U(Ñ ) gauge redundancy,
Ξ µ (i, z) → V † i (z)Ξ µ (i, z)V i+µ (z).(79)
Therefore the link fields now have to be coupled with a dynamical U(Ñ ) gauge field in the bulk.
S ′′ bulk = −t i,M =N tr Ξ † M (i + N)U i+N,i Ξ M (i)U i+M,i+M +N e −iB M N (i) + c.c. −t i,M =N tr Ξ † N (i)Ξ M (i)U i+M,i+M +N U i+M +N,i+N e −iB M N (i) + c.c. −t i,M =N tr Ξ N (i + M)Ξ M (i)U i+M +N,i+N U i+N,i e −iB M N (i) + c.c. + i,M V (|Ξ M (i)| 2 ) − 1 g 2 KR cubes cos(∆ L B M N + ∆ M B N L + ∆ N B LM ).(80)
Here In summary, we showed that a D-dimensional gauged matrix model can be mapped into a closed string field theory in (D + 1)-dimensional space. The string field in the bulk is coupled with a compact two-form gauge field which is also a part of the string field. Holographic states with deconfined string in the bulk are stable only when topological defects for the two-form gauge field are suppressed in the bulk, which is likely to be realized for a sufficiently large N. Table. II.
U i,i+M = e iτ a A
Although many structures on the holographic description have been learned from general considerations, it is desirable to obtain explicit solutions to the saddle point equation. In principle, one has to solve a set of infinitely many coupled differential equations. In the future, it will be interesting to simplify these equations by focusing on light modes. Finally, we close with discussions on a speculative phase diagram of the matrix model, a world-sheet description of deconfined strings in the holographic phases, and a continuum limit.
A. A schematic phase diagram
It may be difficult to find a specific microscopic model which realizes each phase discussed in this paper. However, one may still guess a possible phase diagram. For D ≤ 4, it is believed that the present matrix model is always in the confinement phase. In these low dimensional cases, one may have to introduce more degrees of freedom (fermions or fundamental matters) to stabilize critical phases. Here we focus on the pure bosonic matrix model in D > 4. In Fig. 15, we show a speculative phase diagram. In the strong coupling limit, the matrix model is in the confinement phase. As the gauge coupling is weakened, there is a phase transition into the IR free It is of note that the structure of the proposed phase diagram is reminiscent of known examples where systems flow into novel universality classes at interacting critical points. For example, in the two-dimensional Z N clock model with N greater than a critical value, the critical point between the disordered phase and the ordered phase has an emergent U(1) symmetry [48]. More recently, it has been proposed that the critical point between an antiferromagnetic state and a valence bond state in 2+1 dimensions can possess a non-trivial quantum order which supports an emergent gauge boson and fractionalized excitations [49].
B. World sheet description of deconfined string
In order to make a contact with the traditional first quantization formulation of string theory, it will be useful to have a world-sheet description of deconfined strings in the holographic phases.
Here we focus on closed string. The generalization to open string is straightforward. Note that the hopping integral from loop C 1 to C 1 + C 2 is determined by the loop fieldφ C 2 = |φ C 2 |e ib C 2 which is complex. The amplitude |φ C 2 | determines the strength of the hopping, and defines the notion of 'distance' between the two loops. The distance between two loops, in turn, defines the metric of the space in which loops are defined. In this sense, condensates of loop fields determine the metric of the space in which closed loops propagate. The U(1) phase b C 2 corresponds to the background two-form field to which closed strings are electrically coupled. This can be made more intuitive if we use a world-sheet representation. Let us consider the the quadratic Hamiltonian that includes the tension and the hopping terms,
H 0 (z) = L C χ † C χ C − 1 M 2 2φ * C 2 (z)χ † C 1 χ C 1 +C 2 + 2φ C 2 (z)χ † C 1 +C 2 χ C 1 .(81)
Here we suppressed the form factors F ij (C 1 , C 2 ) and G ij (C 1 , C 2 ). For this discussion, we ignore a possible deformation of the path integral of the string fields. We consider the single loop propagator given by g(C 2 , z 2 ; C 1 , z 1 ) = < C 2 |e
− z 2 z 1 H 0 (z)dz |C 1 >,(82)
where |C >= χ † C |0 > is a single string state. We can 're-discretize' the imaginary time z into small steps with size ǫ to write g(C 2 , z 2 ; C 1 , z 1 ) = all world sheets
e −S 1 W S +iS 2 W S ,(83)
where
S 1 W S = A z − i ln(2ǫ|φ C i (z i )|/M 2 ),(84)
where A z is the area of the world sheet whose faces are tangential to the z direction, and the second term in S 1 W S includes contributions
2ǫ|φ C i (z i )| M 2
for each hopping mediated by loop field φ C i at time z i . The second term is associated with the parts of the world sheet that are perpendicular to the z direction. This is illustrated in Fig. 16. One can view S 1 SW as the Nambu-Goto action provided that the area associated with a loop C at time z is taken to be A C = − ln(2ǫ|φ C (z i )|/M 2 ). The areas associated with loops in turn would determine the spatial metric of the space. The imaginary part of the action,
S 2 W S = world sheet B(85)
is simply the Berry phase associated with the phase of the background loop fields in the temporal gauge. Since both the phase and amplitude modes of loop fields are dynamical, not only the two-form gauge field but also a metric field should arise as a dynamical degree of freedom.
A typical configuration of vacuum fluctuation in the full interacting string theory are shown in third scale L is the scale over which loop fields change appreciably in the bulk. Roughly, the last one determines the 'curvature' of the bulk space in which strings propagate. One can take the continuum limit by tuning the system such that all these length scales are fixed in the a → 0 limit.
If these scales satisfy L >> l s >> l N S D−4 , strings propagate in a weakly curved background.
It is expected that a continuum description of string theory emerges in this limit. Since graviton has the same mass as the two-form gauge field in the continuum limit, graviton may emerge as a massless mode along with the two-form gauge field in the holographic phase. It will be interesting to see how the resulting theory in the continuum limit compares with the existing formulation of the closed string field theory [50].
X. APPENDIX A
We prove the identity
I = dφ * dφ e −φ(φ * −A * ) f (φ * ) = πf (A * )(86)
for any analytic function f (x). We define a new variable ξ = φ − A to write
C |e −ǫH |Φ
(1)
C > e −Φ (1) * C Φ
(1)
C < Φ (1) C |e −ǫH |Ψ i >,(89)
where we use the identity
dΦ (l) * C dΦ (l) C |Φ (l) C > e −Φ (l) * C Φ (l) C < Φ (l) C | = 1.(90)
For (40) and (42), the transition amplitude becomes
Z = R l=0 dΦ (l) * C dΦ (l) C Ψ * f [Φ (R) * C , Φ (R) C ]Ψ i [Φ (0) * C , Φ (0) C ] e −S ,(91)
where
S = −Φ (R) * C Φ (R) C + R l=1 Φ (l) * C (Φ (l) C − Φ (l−1) C ) + ǫH(Φ (l) * C , Φ (l−1) C ) .(92)
We take R → ∞ limit and equate Eq. (92) with Eq. (34) to identify
Ψ i [Φ * C , Φ C ] = e −S U V [Φ * C /N,Φ C /N ] , Ψ * f [Φ * C , Φ C ] = e −Φ * C Φ C −S IR [Φ C /N ] .(93)
with the boundary conditions in Eqs. (65) and (66). Especially, the IR boundary condition implies φ * C (∞) = 0, which then implies φ * C (z) = 0. This makes sense physically because the source for
with a self-retracting link splits into two loops (a), becomes shorter (b), or disappears (c), as the matrix fieldũ ij on the link is integrated out. In (d), two loops which share a link merge into one.
FIG. 2 :
2Examples of the diagrams that contribute to the IR effective potential to the leading order in 1/N . Every link should be paired with another link with the opposite orientation.
In this theory, there is no unique 't Hooft coupling. Instead there is a set of couplings defined in the space of loops, J {C 1 ,..,Cn} which scales as the inverse of the 't Hooft coupling. Since we could have scaled out M 2 by redefining U ij = U ′ ij /M in Eq. (7), the theory depends only on the combination J {C 1 ,..,Cn} M − i L C . The small J {C 1 ,..,Cn} limit is equivalent to the large M limit, which corresponds to the strong coupling limit of the matrix model where one expects to have the confinement phase. The set of J {C 1 ,..,Cn} 's sets the magnitudes of loop fields in the bulk. We will see that background loop fields, in turn, control the size of strings which describe small fluctuations of the loop fields.
This is an exact mapping between the D-dimensional matrix model ((D − 1)-dimensional quantum matrix model) and the (D + 1)-dimensional loop model (or D-dimensional quantum loop model). Several remarks are in order. First, the Hamiltonian in Eq. (45) is a many-body
FIG. 3 :FIG. 4 :
34Loops are emitted at the UV boundary and propagate to the IR boundary. Since there is no hopping term in the Hamiltonian, loops can not move. Instead, they can join or split following the processes shown inFig. 1. Two loops with opposite orientations can get pair-annihilated through multiple interactions.The physical picture for the transition amplitude is the following. At the UV boundary (z = 0), a condensate of loops are emitted and propagate in z under the evolution governed by H. The amplitude of the condensate is < Φ C >∼ O(N). This can be seen from the fact that the action for the unscaled loop fields φ C has N 2 as an overall prefactor, which implies < φ C >∼ O(1). Loops can join and split through the interactions as is illustrated inFig. 3. A loop C and its anti-loop C, the loop with the opposite orientation, can get pair-annihilated through a series of interactions as is shown inFig. 4 (a). Moreover, a self-retracting loop can become a loop and an anti-loop as (a) A process where a loop and its anti-loop get annihilated in pair through a series of interactions.(b) A self-retracting loop in the vacuum can split into a loop and an anti-loop.
sheet formed by multiple processes where a large loop absorbs many small loops at different stages to change its shape to disappear before it reach the IR boundary. Now let us consider a Wilson loop operator for a loop C which is much larger than the size of Wilson loops for which sources are turned on at the UV boundary. The expectation value of the Wilson loop operator is given by the one-point function,
with θ z (i, z) = 0 in Eq. (53). The temporal components can be completely gauged away because they are pure gauge degrees of freedom in the presence of boundaries. This is in contrast to the case with the periodic boundary condition, where the time independent component of the temporal gauge field can not be gauged away. As a result of the gauge symmetry, there is no quadratic hopping term for loops in the Hamiltonian. However, this does not necessarily mean that loops are always localized in space. Loops can change their shapes and move in space by absorbing or emitting other loops. For example, z FIG. 6: Loops can change their shapes and move in space like amoebas, by absorbing or emitting small loops.Fig.
FIG. 7 :
7and integrate out φ (C 1 +C 2 ) ij . In this step, the merged loop in the first step become a shorter loop (C 1 + C 2 ) ′ ij by eliminating one self-retracting link. Kinetic energy for the two-form gauge field generated by heavy loop fields.(a) Three vertices each of which involves one loop with length 6 and two loops with length 4 can generate a term for the six loops with length 4, once the loop fields with length 6 are integrated out. The resulting term for the small loops becomes the kinetic energy for the two-form gauge field. (b) In integrating out the loop fields with length 6 in (a), one has to introduce a series of nine vertices. Each step depicts a process of adding a new vertex and integrating out one or two loop fields : two fields for the first and the fourth steps, and one loop field for the other steps.steps can be understood in a similar way. In total, nine vertices and eleven loop propagators are needed. (Note that each of the first and fourth steps introduces two propagators because two loop fields are integrated out in those steps, while all the others involve only one propagator.three vertices, we obtain an action for the light loop fields,
FIG. 8 :
8An N S D−4 -brane in the (D+1)-dimensional space. Around an N S D−4 -brane, there is a net flux of 2π for the three-form flux : S 3 H = 2π. When the tension of the N S D−4 -brane is positive, N S D−4 -branes generated out of quantum fluctuations remain small. When the tension becomes negative, N S D−4 -branes are proliferated in the bulk.
which allows for a topological defect to exist as a magnetic excitation of the gauge field. In the presence of topological defects, the field strength H = dB does not satisfy the Bianchi identity dH = 0. In (D + 1)-dimensional space-time, the topological defect which carries a magnetic charge is a (D − 4)-brane, which is a (D − 3)-dimensional object in space-time.We call this object NS D−4 brane[52]. Around the NS D−4 -brane, there is a net 2π flux for the three-form flux,
FIG. 9 :
9seen from Eq.(49), and the charge of the NS D−4 -brane is automatically satisfied. This follows from the fact that the phase 2π on a unit plaquette is invisible to loop fields. In D > 4, this is a brane extended along (D − 4)-directions in space at a given time slice with fixed z. In D = 4, this is a point-like particle. In D = 3, this is an instanton which is localized both in space and time.Whether loop fields provide a coherent background for other loops is determined by dynamics ofNS D−4 -branes. An N S D−4 -brane can be viewed as an instanton in D = 3 where one inserts a source of 2π magnetic charge to the theory at a given z N S . (a) For z < z N S , the phases of loop fields are zero. (b) At z = z N S , the phases of loop fields are 2π along a semi-infinite line creating a cube which contains the 3-form flux of 2π : H 123 = ∆ 1 B 23 + ∆ 2 B 31 + ∆ 3 B 12 = 2π a 3 δ x,x NS . (c) For z > z N S the configuration in (b) is smoothly deformed and the flux is smeared over a region which contains the net flux 2π. The size of red dots represents the amount of 3-form flux contained in each cube. The physical nature of NS D−4 -brane can be most easily understood in D = 3 where NS D−4brane is an instanton localized at a point in the four-dimensional bulk. We start with a configuration of the loop fields φ for unit plaquettes with
S ). This means that there is a source of magnetic flux for the two-form gauge field localized at a point in the bulk, dH = 2πδ 3 (x − x N S )δ(z − z N S ). Since 2πmagnetic flux is concentrated at one cube, the topological defect is trivial. 10: A pair of instanton and anti-instanton in the four-dimensional bulk for D = 3.
as a configuration in a slice at a fixed x 5 , ..., x D+1 , where the NS D−4 -brane is extended along the (D − 3) directions. They can be wrapped into compact objects as is shown in Fig. 8, which in a sense describe bound states of NS D−4 -brane and anti-NS D−4 -brane. If the size of the wrapped NS D−4 -branes become infinite, NS D−4 -brane and anti-NS D−4 branes become unbound. The tension of the NS D−4 -brane is proportional to N 2 because NS D−4 -brane is a topological defect of the two-form gauge field with the coupling proportional to 1/N 2 . Here we are using the term 'tension' in a loose sense. In D > 4, it literally means the tension of NS D−4 -branes. In D = 4, it refers to the mass of 'NS D−4 -particle'. In D = 3, it refers to the action of 'NS D−4instanton'. For a sufficiently large N, we expect that NS D−4 -branes are gapped. In this case, NS D−4 -branes will be wrapped into compact objects with a finite size in the vacuum. For a small N, the bare tension of NS D−4 -brane is small, and quantum fluctuations can renormalize the tension into a negative value. Then NS D−4 -branes are condensed, and extended NS D−4 -branes fill the space in the bulk. It is also possible that NS D−4 -branes always condense for any finite N in low dimensions. We will discuss the consequences of different dynamics of NS D−4 -branes in the following section.
11: A schematic profile of a loop field φ C and the conjugate field φ * C in a deep confinement phase. φ C and φ * C satisfy the boundary condition given by Eq. (67) at any z in the bulk.
determined from Eq. (67). One can check that the source and the conjugate field satisfy Eq. (67) at all z through an explicit calculation perturbatively in 1/M (see Appendix C). It is tempting to interpret Eq. (68) as the beta function of the sources for Wilson loop operators. However, there is an important caveat for this interpretation, which comes from the fact that this is the saddle point equation of the quantum theory for dynamical loop fields. The saddle point equation is expected to be valid only when quantum fluctuations are weak for a sufficiently large N. For a small N, one can still have a well-defined beta function under the usual renormalization group flow[27, 28]. However, the beta function can not be directly identified with the saddle point equation of the loop fields if the saddle point solution becomes unstable by strong quantum fluctuations. As we will see in the following sections, non-perturbative fluctuations can invalidate the holographic description for small N.
FIG. 12 :
12For small φ C (0), the first term on the right hand side of Eq. (68) dominates. As a result, φ C (z) The Wilson loop correlation function in different phases. (a) In the confinement phase, N S D−4branes are condensed in the bulk, and the string emitted at the boundary stays near the boundary. Due to weak fluctuations of string world sheet, the string propagates in a straight path. (b) In the non-holographic critical phase, N S D−4 -branes remain condensed in the IR region. However, large amplitudes of loop fields in the UV region make the string world sheet highly fluctuating along the D dimensions. As a result, the correlation function decays algebraically. (c) In the holographic critical phase, loop fields acquire finite expectation values everywhere in the bulk, and N S D−4 -brane is gapped. Strings can propagate deep inside the bulk, and the correlation function shows a power-law behavior with a scaling dimension determined by the mass of the string. (d) In the deconfinement phase, loop fields with infinitely large loops acquire non-zero expectation value in the bulk. This causes highly non-local fluctuations of the world sheet of a string in the bulk. exponentially in z. Because larger loops for which sources are not turned on at the UV boundary are generated out of many small loops, amplitudes with larger loops decay exponentially with the area enclosed by the loop. This corresponds to the confinement phase of the matrix model. In the confinement phase, one can define a cross-over scale z * beyond which loop fields have negligible amplitudes.Now let us consider dynamics of strings in the IR (z > z * ) and the UV (z < z * ) regions separately. In the deep IR region, amplitudes of non-self-retracting loop fields are exponentially small. This means that loops that are emitted from the UV boundary rarely reach the IR region. In this region, phase fluctuations of loop fields do not have a stiffness, which leads to a condensation of NS D−4 -branes. As a result, strings are subject to the strongly fluctuating two-form gauge field. In this region, strings are confined, and the two-form gauge field is gapped[37,38]. For a more systematic discussion on possible phases of anti-symmetric fields, see Ref.[39].In the deep IR region, strings can not propagate by themselves; only charge neutral bound states of string and anti-string can propagate. On the other hand, loop fields have significant amplitudes in the UV region. The source field J {C 1 ,..,Cn} plays the role of a symmetry-breaking field at the UV boundary. As a result, phase fluctuations of loop fields are small, and NS D−4 -branes are suppressed in the UV region. Because loop fields are coherent near the UV boundary, strings are deconfined in this region. There is a domain wall that separates the IR region with condensed NS D−4 -brane and the UV region without NS D−4 -brane. In the confinement phase, a string that is emitted from the boundary can not penetrate through the wall of condensed NS D−4 -brane. Therefore it stays within the UV region. Deep inside the confinement phase with small J {C 1 ,..,Cn} , the condensates of loop fields are small. As a result, the hopping amplitudes of strings are small, and fluctuations of string world sheet is small. For the correlation function in Eq. (71), the strings inserted at the UV boundary are connected through a minimum number of hoppings, forming a straight path as is shown in
As J {C 1 ,..,Cn} is dialed up, amplitudes of loop fields in the bulk increase. Accordingly the cross-over scale z * increases. At the same time, fluctuations of string world sheet increase as the amplitudes of loop fields become larger. Suppose the system becomes critical either by fine tuning or dynamical tuning. In the case of fine tuning, one may have to tune more than one microscopic parameters to reach a critical point. For the following discussions which focus on physical properties of the critical states, it is not important whether those states are realized as phases or critical points. So we will use the term 'critical phase' in a broad sense to include not only critical phases realized within a finite region in the parameter space of a microscopic model but also critical states realized at critical points by fine tuning. Logically, there exist at least two different scenarios via a criticality is achieved. In the first scenario, NS D−4 -branes remain condensed in the IR region with a finite z * . But loop fields acquire large amplitudes in the UV region so that strings are delocalized along the D-directions, mediating critical correlations between operators inserted on the UV boundary. In the second scenario, the cross-over scale diverges and NS D−4 -branes are suppressed out in the bulk. In this case, strings can propagate
For a small N, NS D−4 -branes are 'light'. Even though loop fields have finite amplitudes in the bulk, quantum fluctuations may destabilize the saddle-point solution. Indeed this is what always occurs in the pure 2-form gauge theory in the flat four dimensional space[37, 41, 42]. If the mass or tension of NS D−4 -brane becomes negative, NS D−4 -branes are condensed. Once NS D−4 -branes are proliferated, the two-form gauge field acquires a mass gap, and strings are confined in the bulk, as is the case in the confinement phase. The difference from the confinement phase is that the boundary theory is critical. The boundary theory can be critical although strings are confined deep inside the bulk because critical correlation between boundary fields can be mediated by strings that propagate near the UV region. It is noted that 'confinement' in the boundary matrix model and 'confinement' of strings in the bulk are not the same thing. In this phase, strings are confined in the bulk, but the matrix model is not in the confinement phase. We call this non-holographic critical phase. In this critical phase, a string emitted from the UV boundary no longer takes the straight path because of large amplitudes of background loop fields in the UV region. Rather, the world sheet of string strongly fluctuates, and the correlation function can decay in a power law because of delocalized strings. However, strings are still localized within the UV region along the z direction due to the condensation of NS D−4 -branes in the IR region. The fluctuations of the world sheets of
develop non-zero amplitudes in the IR region. For a sufficiently large N, NS D−4 -branes are suppressed, and the saddle-point solution is stable against non-perturbative fluctuations of the two-form gauge field. If NS D−4 -branes are suppressed, the compactness of the gauge field is unimportant at long distances. Strings are deconfined in the bulk because loop fields provide a background in which strings can propagate coherently. Note that strings are still coupled with the dynamical two-form gauge field, but the gauge field is no longer confining in this phase. This is analogous to the Coulomb phase discussed in Sec. II. Thanks to the coherent background loop fields, closed strings can explore the extended space in the bulk. The bulk space is not a gauge invariant object. However, it assumes a 'classical identity' in the large N limit where fluctuations of loop fields are suppressed. In this sense, the bulk space emerges in the holographic phase, but not in the non-holographic phase. Because of the source that explicitly breaks the gauge symmetry, the transformation B M N → B M N + ∂ M θ N − ∂ N θ M , is not a symmetry at the UV boundary. As a result, the U(1) mode θ M becomes a physical mode. This means that there is a U(1) gauge field localized at the UV boundary in the dual theory. This U(1) mode originates from the Abelian component of the U(N)
discussed in the previous section. Even in the holographic critical phase, some loop fields are condensed in the bulk, as is the case in the deconfinement phase. The difference is that only small loops are condensed in the holographic critical phase while loops with all sizes are condensed in the deconfinement phase. Note that condensations of small loops do not generate a mass gap for the two-form gauge field. This is because closed strings with finite sizes as point-like particles are coupled only with the field strength tensor of the two-form gauge field. In certain models, one can in principle change microscopic parameters to tune the size of condensed loops, smoothly interpolating between the holographic critical phase and the deconfinement phase. This basically controls the size of strings in the bulk. The N = 4 SU(N) gauge theory in four dimensions is believed to be in this class : for a sufficiently large N, one can smoothly tune the 't Hooft coupling λ from a large value to zero without going through a phase transition. The one parameter family of the critical theories form a line of fixed points. Here, λ = 0 is a special point where the size of string diverges.
The phase structure of this model is very similar to the one for the Abelian-Higgs model for scalar fields discussed in Sec. II. If NS D−4 -branes are condensed in the bulk, the link fields are confined. This is what happens in the confinement phase and the non-holographic critical phase discussed in the previous section. The bulk physics alone can not distinguish the confinement phase and the non-holographic critical phase.
Under the gauge transformation in Eq. (75), the U(1) gauge field transforms as usual,A M (i) → A M (i) + γ i+M − γ i ,and the two-form gauge field is invariant. Eq. (75) implies that the link field ξ M (i) carries U(1) charge +1 on one end at site i + M and charge −1 on the other end at site i.
FIG. 13 :
13The figures in (a), (b) and (c) show a 'particle' denoted as a thick line defined on the link < i, i + M > hops to the link < i + N, i + M + N >, < i, i + N > and < i + M + N, i + M >, respectively. The dashed lines represent the paths along which the end points of the link follow to which the U(1) gauge field is electrically coupled.Although the bare coupling for the U(1) gauge field is infinite, the kinetic term will be generatedonce high energy fluctuations of the link fields are integrated out, renormalizing the gauge coupling to a finite value. Suppose we are in the holographic critical phase where the link fields are gapped and the two-form gauge field is in the deconfinement phase. The U(1) gauge field can be either in the confinement phase or in the deconfinement phase. If the U(1) gauge field is confining, all open links are joined with each other to form closed loops because there is a linearly confining force between open ends. In this phase, only closed strings are allowed. This is the holographic critical phase I discussed in Sec. VII.C.3. On the other hand, if the U(1) gauge field is in the deconfinement phase, closed strings can get fractionalized into open strings, and open strings arise as deconfined excitations. Closed strings can still exist as a bound state of open strings. In this phase, there is an emergent U(1) gauge field in addition to the two-form gauge field. The twoform gauge field is coupled with the world-sheet of strings and the U(1) gauge field is coupled with boundaries of open strings. Here the gapless U(1) gauge field is protected by a quantum order associated with the emergent Bianchi identity dF = 0 for the U(1) gauge field in the bulk. Open strings are fractionalized excitations of closed strings. The way open strings and the U(1) gauge field arise as collective excitations of closed string fields is very similar to the way slave-particles emerge as fractionalized excitations along with the emergent U(1) gauge boson in the slave-particle theory discussed in Sec. II. One can go one step further to obtain a different gauge group for open strings. For this we introduce a larger gauge 14: (a) In the confinement phase for the one-form gauge field in the bulk, link fields are always joined to form closed loops, and there exist only closed strings as finite energy states. (b) In the deconfinement phase, closed strings can be fractionalized into open strings. The end points of open strings carry gauge charge for the emergent gauge field in the bulk. redundancy in Eq. (72),
a M (i) , and A a M (i) is U(Ñ ) gauge field defined on the link < i, i + M >. It is noted that the U(Ñ ) gauge field that emerges in the bulk is different from the original U(N) gauge field of the boundary matrix model. In this description, end points of the link fields carry fundamental and anti-fundamental charges for the dynamical U(Ñ ) gauge field. If the U(Ñ ) gauge field in the bulk is in the deconfinement phase, open strings with the U(Ñ ) Chan-Paton factor emerge as collective excitations of the closed string fields. Although one can choose a description with an arbitrary gauge redundancy, the gauge group is determined by dynamics in the end. Holographic theories with different gauge groups for open strings in the bulk describe different states of the matrix model. In the T-dual description, configurations with background gauge fields describe D-branes. It is interesting to note that D-branes can emerge as non-perturbative excitations in the closed string field theory. VIII. DISCUSSION
The holographic states are in different universality classes from the non-holographic states where strings are confined in the bulk due to condensed topological defects. We also discussed a holographic critical state where closed strings get fractionalized into open strings. In this state, there are both closed and open strings along with the two-form and one-form gauge fields in the bulk. The nontrivial quantum order present in the holographic phases is responsible for the existence of operators whose scaling dimensions are protected, which otherwise would have received a large quantum correction at strong coupling. The possible phases of the matrix model are summarized in
deconfinement phase. If the phase transition is continuous, there can be two different universality classes for the critical points. For small N, NS D−4 -branes are condensed and strings are confined in the bulk. At this non-holographic critical point, the scaling dimension of Wilson loop operators generally receive a large quantum correction. For N greater than a critical value, NS D−4 -branes are suppressed, and strings are deconfined in the bulk. The quantum order protects the two-form gauge field from acquiring a large mass in the large N limit, which in turn protects the scaling dimension of the phase fluctuations of Wilson loop operators even at large 't Hooft couplings. As was discussed in Sec. VII. E, there exist two different kinds of holographic critical points. In the first case, there are only closed string excitations in the bulk. In the second case, there are both closed and open string excitations along with the emergent gauge field and the two-form gauge field. We expect that it is easier to stabilize the state with both closed and open strings for N'15: A proposed phase diagram for the pure bosonic gauged matrix model in D > 4. Here λ represents a set of 't Hooft couplings associated with various Wilson loops in the matrix model. Moving along the direction of λ in this figure may mean tuning more than one microscopic parameters at the same time. The critical points, in general, represent multi-critical points. Therefore the shape of the phase boundary in the figure should not be taken seriously. The actual phase boundary is not likely to be a straight line in the multi-dimensional space of microscopic couplings. In the strong coupling limit (large λ), the theory is in the confinement phase with the exponentially decaying Wilson loop correlation function. As the coupling becomes smaller, the model goes through a phase transition to the IR free deconfinement phase. For small N , N S D−4 -branes remain condensed at the critical point. Strings are confined, and the two-form gauge field is massive in the bulk. Critical correlations are mediated by the D-dimensional fluctuations of strings near the UV boundary. Scaling dimensions generally receive a large quantum correction at strong couplings. This is the non-holographic critical point. For large N , N S D−4 -brane is gapped and the two-form gauge field remains light even at strong coupling. Low energy physics is governed by deconfined strings that propagate deep inside the bulk. In this holographic critical point, critical correlations of the matrix model is mediated by (D + 1)-dimensional fluctuations of strings. The scaling dimension of the operator associated with the two-form gauge field is protected from acquiring a large quantum correction due to the quantum order associated with the dynamical suppression of N S D−4 -branes. The holographic critical point can be either in the state with only closed strings, or in the state with both closed and open strings.are large but not too large. In the large N limit, closed strings are free, and there is no dynamical reason why they decay into open strings. For N large enough to suppress NS D−4 -brane, but still small enough to support strong interactions between closed strings, closed strings may decay into open strings. We believe that further studies are needed to understand this phenomenon more systematically.
FIG. 16 :
16The world sheet action has two contributions : A z is the area of the surface which is parallel to z, and A C is the contribution from hoppings mediated by condensates of loop fields.
Fig. 17 .
17For every vertex where closed loops join or split, there is a factor of 1/N. For a large N, the theory describes weakly interacting strings propagating in the time-dependent background.z=0 z FIG. 17: A snap shot of a vacuum fluctuation in the holographic critical phase with deconfined closed string. C. Continuum limit In the holographic phase, there are three important length scales. The first scale is associated with the tension µ N S D−4 of NS D−4 -brane, l N S D−4 ∼ 1/(µ N S D−4 ) D−3 . The second scale is the 'string scale' l s which corresponds to the typical size of closed string excitation in the bulk. The
Myers and Subir Sachdev for helpful comments and discussions. This research was supported in part by the Natural Sciences and Engineering Research Council of Canada and the Early Research Award from the Ontario Ministry of Research and Innovation. Research at the Perimeter Institute is supported in part by the Government of Canada through Industry Canada, and by the Province of Ontario through the Ministry of Research and Information.
IC
= dξ * dξ e −|ξ| 2 −Aξ * f (ξ * + A * * dξ e −|ξ| 2 (−Aξ * ) m f (n) (A * )ξ * n ,(87)where we Taylor expanded e Aξ * and f (ξ * + A * ). Here only m = n = 0 component survives in the angular integration of ξ = |ξ|e iθ and we have I = πf (A * ). (88) XI. APPENDIX B Here we show that Eq. (39) with Eqs. (40) -(43) is equivalent to Eq. (34). The imaginary time β is divided into R pieces, < Ψ f |e −ǫH |Φ (R−1) C > e −Φ (R−1
section, we solve the loop equations perturbative in 1/M to the lowest non-trivial order for small loops. Let us first consider the saddle point solution in the large M limit, which corresponds to the deep inside the confinement phase of the gauge theory. For M = ∞, we have simple solution,φ C (z) = φ C (0)e −L C z , φ * C (z) = φ * C (0)e L C z
TABLE I :
Islave-particle monopole
low energy excitations
Confining phase confined condensed
θ ab
Coulomb phase deconfined
gapped φ a , monopole, gauge boson
Higgs phase
condensed confined
Goldstone bosons
Table . I
.summarizes the physics in each phase of the boson model. Now, we switch gear to
which is, in general, non-linear in the presence of multi-trace operators. Here J {C 1 ,..,Cn} 's are loop dependent coupling constants. This theory is invariant under the U(N) gauge transformation
Note that introducing the temporal components of the two-form gauge field into the theory doesn't do anything except for making the gauge symmetry more explicit. This can be understood from the fact that one can reproduce the original action in Eq. (35) by choosing the temporal gauge with B µz = 0. This can be done by choosing
TABLE II :
IIclosed string open string NS-brane bulk excitations Holographic deconfined deconfined gapped closed & open string (B M N , A M ), critical phase II NS-brane, D-brane Deconfinement (IR free) phase condensed condensed confined non-local stringConfinement phase
confined
confined condensed
×
Non-holographic
confined
confined condensed
×
critical phase
Holographic
deconfined confined
gapped
closed string (B M N ),
critical phase I
NS-brane
Wilson loops decreases exponentially with z, and the expectation values of Wilson loop operators vanish in the strong coupling limit of the gauge theory. For a large but finite M, the loop equations can be solved perturbatively in 1/M.To illustrate the idea, we solve the loop equation for a simple model which include only the single trace operator for the unit plaquette, V = J φ * . In this model, we solve φ * to the lowest order in 1/M. Keeping the first four shortest loops that contribute to the evolution of φ * C of unit plaquette, we write the saddle point equations,The symbols are defined inFig. 18. We consider solutions with the translational and discrete rotational symmetries of the lattice, and suppress the site index for the loop fields. The equations for φ * ⊓ , φ * and φ are similar to the one for φ * ⊔ . Similarly, the equation for φ * ⌋ (φ | ) is similar to the one for φ * ⌊ (φ − ). From Eqs.(32)and(66), we read the boundary conditions,Using the zero-th order solution for φ¯ = J¯ e −4z , we obtainThis implies that the expectation value of the Wilson loop operator for the unit plaquette become J¯ /M 8 as expected in the large M limit. The difference from the case with M = ∞ is that φ * C is now non-zero at z = 0 because Wilson loops have non-zero expectation values. As expected, φ C and φ * C satisfy Eq. (67) to the leading order in 1/M at all z.[1] P. W. Anderson, Science 235 (1987) 1196 ; P. Fazekas and P. W. Anderson, Philos. Mag.30 (1974)432.
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This model including the constraints can arise as an effective theory for exciton bose condensates in a multi-band insulator[26]. But we treat this model as our 'microscopic model' for the following discussion. This model including the constraints can arise as an effective theory for exciton bose condensates in a multi-band insulator[26]. But we treat this model as our 'microscopic model' for the following discussion.
This name has been borrowed from the NS 5 brane which is the magnetically charged object for the Kalb-Ramond two-form gauge field in the ten dimensional superstring theory. This name has been borrowed from the NS 5 brane which is the magnetically charged object for the Kalb-Ramond two-form gauge field in the ten dimensional superstring theory.
|
[] |
[
"Caffeinated FPGAs: FPGA Framework For Convolutional Neural Networks",
"Caffeinated FPGAs: FPGA Framework For Convolutional Neural Networks"
] |
[
"Roberto Dicecco [email protected] ",
"Griffin Lacey [email protected] ",
"Jasmina Vasiljevic [email protected] ",
"Paul Chow ",
"Graham Taylor [email protected] ",
"Shawki Areibi [email protected] ",
"\nDepartment of Electrical and Computer Engineering\nUniversity of Toronto\nOntarioCanada\n",
"\nUniversity of Guelph\nOntarioCanada\n"
] |
[
"Department of Electrical and Computer Engineering\nUniversity of Toronto\nOntarioCanada",
"University of Guelph\nOntarioCanada"
] |
[] |
Convolutional Neural Networks (CNNs) have gained significant traction in the field of machine learning, particularly due to their high accuracy in visual recognition. Recent works have pushed the performance of GPU implementations of CNNs to significantly improve their classification and training times. With these improvements, many frameworks have become available for implementing CNNs on both CPUs and GPUs, with no support for FPGA implementations. In this work we present a modified version of the popular CNN framework Caffe, with FPGA support. This allows for classification using CNN models and specialized FPGA implementations with the flexibility of reprogramming the device when necessary, seamless memory transactions between host and device, simple-to-use test benches, and the ability to create pipelined layer implementations. To validate the framework, we use the Xilinx SDAccel environment to implement an FPGA-based Winograd convolution engine and show that the FPGA layer can be used alongside other layers running on a host processor to run several popular CNNs (AlexNet, GoogleNet, VGG A, Overfeat). The results show that our framework achieves 50 GFLOPS across 3 × 3 convolutions in the benchmarks. This is achieved within a practical framework, which will aid in future development of FPGA-based CNNs.
|
10.1109/fpt.2016.7929549
|
[
"https://arxiv.org/pdf/1609.09671v1.pdf"
] | 18,372,711 |
1609.09671
|
eb95f8cb5f3c3678e83dddbf29063b5021440fb9
|
Caffeinated FPGAs: FPGA Framework For Convolutional Neural Networks
30 Sep 2016
Roberto Dicecco [email protected]
Griffin Lacey [email protected]
Jasmina Vasiljevic [email protected]
Paul Chow
Graham Taylor [email protected]
Shawki Areibi [email protected]
Department of Electrical and Computer Engineering
University of Toronto
OntarioCanada
University of Guelph
OntarioCanada
Caffeinated FPGAs: FPGA Framework For Convolutional Neural Networks
30 Sep 2016
Convolutional Neural Networks (CNNs) have gained significant traction in the field of machine learning, particularly due to their high accuracy in visual recognition. Recent works have pushed the performance of GPU implementations of CNNs to significantly improve their classification and training times. With these improvements, many frameworks have become available for implementing CNNs on both CPUs and GPUs, with no support for FPGA implementations. In this work we present a modified version of the popular CNN framework Caffe, with FPGA support. This allows for classification using CNN models and specialized FPGA implementations with the flexibility of reprogramming the device when necessary, seamless memory transactions between host and device, simple-to-use test benches, and the ability to create pipelined layer implementations. To validate the framework, we use the Xilinx SDAccel environment to implement an FPGA-based Winograd convolution engine and show that the FPGA layer can be used alongside other layers running on a host processor to run several popular CNNs (AlexNet, GoogleNet, VGG A, Overfeat). The results show that our framework achieves 50 GFLOPS across 3 × 3 convolutions in the benchmarks. This is achieved within a practical framework, which will aid in future development of FPGA-based CNNs.
I. INTRODUCTION
Convolutional Neural Networks (CNNs) are highly accurate deep learning networks inspired by the mammalian visual cortex. A number of works explored the implementation of CNNs on FPGAs [1]- [3] to take advantage of their lowpower, customizable and programmable fabric. While FPGA implementations show promise in efficiently computing CNNs, they lack the infrastructure available for both CPUs and GPUs. This makes FPGAs inaccessible to deep learning scientists. There are many frameworks for CNN implementations, most of which provide support for CPU, GPU or the option of both [4]. These frameworks allow the programmer to launch any CNN model, and contain comprehensive tests for both layer-based and system-based executions [4]. However, none of the prominent CNN frameworks provide support for FPGA implementations. As a result, to implement a CNN on the FPGA, the designer has to manually design the implementation for each model, as well as test for correctness and optimize for performance, essentially rebuilding from scratch, rather than taking advantage of existing work.
CNNs are very computationally intensive with most of the computation in the convolution layers. The convolution layers require a large number of multiply-accumulate operations. The large computational complexity motivates efforts to reduce the number of required operations. One approach is to use the FFT for convolution because in the frequency domain, the convolution becomes multiplication of each transformed input with the corresponding transformed filter coefficients, resulting in a compute reduction and speedup [5]. An alternative approach uses the Winograd minimal filtering algorithm to take advantage of the overlapping computations between adjacent convolution windows [6], [7]. In this work we implement and optimize the Winograd algorithm on an FPGA within the Caffe framework [4].
This paper makes the following contributions:
• We present an adaptation of the Caffe CNN framework with support for the Xilinx FPGA SDAccel environment. This adaptation allows us to launch CNN classification on CPU-FPGA-based systems.
• We describe a modification to the Winograd convolution algorithm to further reduce DSP utilization for FPGAbased implementations.
• We implement the Winograd convolution algorithm targeting any 3 × 3 convolution layer with unity stride and benchmark it across several popular CNNs. Results show that the architecture achieves approximately 50 GFLOPS across the 3×3 convolution layers of the benchmark suite, while using 83.2% of the available SDAccel resources in a Xilinx Virtex 7 XC7VX690T-2.
• Finally, the software and hardware implementation details have been made open-source and can be found at the following link: https://github.com/dicecco1/fpga caffe. This work is organized as follows: Section II provides background information on CNNs and the Xilinx SDAccel OpenCL framework. Section III discusses the Winograd convolution algorithm and FPGA implementation. Section IV details the features included in the FPGA Caffe framework and Section V shows the area utilization and performance results of the Winograd convolution engine within the FPGA Caffe framework. Section VI reviews related work and compares this work to other recent FPGA implementations. Finally, Section VII discusses future work related to the FPGA Caffe framework, and Section VIII concludes the paper.
II. BACKGROUND
The following subsections detail the necessary background information regarding CNNs and the Xilinx SDAccel OpenCL development environment.
A. Convolutional Neural Networks
CNNs are a popular type of supervised machine learning algorithm. Similar to other machine learning algorithms, CNNs can be trained using back propagation to learn complex representations useful for many applications. CNNs are commonly used for performing object recognition in pixel-based input. A popular CNN model such as AlexNet [8] can be used to classify up to 1000 different objects in images with high accuracy.
These networks have two different modes of operation: training and inference. In the case of object recognition, training involves feeding a large number of human-annotated images into the network. These images are used by the CNN to repeatedly update the model's weights and biases such that it can learn how to recognize the objects in the humanannotated images. Classification uses the trained CNN model and presents it images that the model has never seen to attempt to predict what the object in the image is.
B. CNN Layers
AlexNet was one of the first successful ImageNet submissions employing CNNs and it consists of several layers: Convolution, Max Pooling, Fully Connected (FC), Rectified Linear Unit (ReLU), and Local Response Normalization (LRN) [8]. In many of the top performing CNNs, Convolution, Pooling, Fully Connected, and ReLU layers are typically used, while LRN and other layers are sometimes used depending on the model [8]- [11]. The general structure of a CNN usually consists of stacks of convolution layers with ReLU activations followed by a pooling layer. In recognition applications, fully connected layers are used towards the output to reduce spatially organized information into a decision. Convolution layers represent the majority of the computation for a CNN, with extreme cases requiring upwards of 30 GFLOPs of computation [9]. The ReLU layer is an activation function used to introduce a non-linearity into the network and can be described by y = max(0, x), which is applied to every data point of the input. Pooling is a simple reduction operation, such as max or average, applied to local regions of the input using a sliding window approach. FC layers consist of dense connections between neurons and usually contain the majority of the weights in the network. The computation of FC layers corresponds to a matrix multiplication followed by the addition of an optional bias parameter to each output.
C. Parallelism Strategies
Given the large computational requirements and inherent parallelism of neural networks, these architectures are ideal for hardware accelerators. Popular parallelism strategies can be reduced to three main categories. Data Parallelism -splitting the data across different execution threads, but using the same model. For the pixel-based input (e.g. images) natural to CNNs, data parallelism is inherent given the independence of individual and local groups of pixels. Fine-grained data parallelism can be applied using operations applied concurrently to all pixels, while coarsegrained data parallelism can be applied by processing "minibatches" of hundreds or thousands of input images during training. Model parallelism -splitting the model across different execution threads, but using the same data. This strategy offers several advantages, such as being able to accommodate large neural network sizes by splitting the weights across hardware accelerators, and employing a type of efficient model averaging during training. Pipeline Parallelism -operating different dependent steps of computation concurrently on different threads, so that output from one step is streamed as input to the next, while execution of steps is overlapping. The feed-forward computation of CNNs are well suited for pipeline parallelism, so hardware that can exploit deep pipeline parallelism (e.g. FPGAs) can offer an advantage.
D. SDAccel OpenCL FPGA Programming Model
Using the SDAccel OpenCL environment to perform computations on an FPGA involves both host and kernel code. The host code is used for programming the FPGA, passing data between the host's memory and the FPGA's global memory, and launching the kernel on the FPGA. The FPGA is segmented into two regions, the programmable region and the static region. The static region is programmed upon power-up and it contains the interfaces to global memory and PCIe. The programmable region contains the kernel, the computation to be accelerated. The kernel code is synthesized into hardware and configured into the programmable region of the FPGA. The synthesized kernel can contain one or more compute units (CUs), where a CU corresponds to the hardware unit responsible for the required computation. One approach to increasing parallelism in the host code is to instantiate multiple CUs as shown in Fig. 1, with each CU handling an equally sized portion of the problem [12]. The portion of the problem handled by a single CU is referred to as the local work group size, while the size of the overall task to be completed is referred to as the global work group size. Each CU has its own local memory that is only accessible to the CU while all CUs share the global off-chip memory of the platform.
We have integrated SDaccel into the Caffe framework and used it to develop a convolution kernel using the Winograd transform.
III. WINOGRAD CONVOLUTION
Winograd convolution exploits the Winograd minimal filtering algorithm [6]. This approach has been shown to reduce the amount of required floating point operations [7]. The sections below provide an overview for the Winograd convolution algorithm and discusses the implementation details of the FPGA-based Winograd convolution engine.
A. Winograd Convolution Algorithm
The Winograd convolution algorithm output is referred to as F (m×m, r ×r). In this expression m×m refers to the output tile size for a given input tile, meaning that m × m output The filter size in this case is r × r. For a given filter size, many different values of m can be chosen, which changes the computational complexity of F (m × m, r × r), but does not impact the overall result. In this work, we implement a
F (2 × 2, 3 × 3) Winograd algorithm. In the case of 3 × 3 convolutions, F (2 × 2, 3 × 3)
has been shown to provide significant performance gains for GPU implementations in [7], though larger tile sizes may produce additional gains. This work targets F (2 × 2, 3 × 3), mainly due to its simplicity of implementation however future work may explore other tile sizes as well. The equations for Winograd convolution are shown in Equations 1 to 3. Equation 1 shows how the filter transformation is calculated.
U = GgG T (1) Where U is a (m + r − 1) × (m + r − 1) transformed filter;
g is an r × r filter; G is an (m + r − 1) × r transform matrix, defined by the Winograd algorithm.
The filter values (g) are known at compile time and remain constant during run-time. Therefore, to save resources during run-time, the Winograd transformation for the filter values, shown in Equation 1, can be executed at compile time, on the CPU. This approach saves FPGA resources. However, pre-computing filter values increases the memory storage requirement. For direct convolution, the 3 × 3 filters require C × 3 × 3 storage, where C is the number of input channels. For Winograd-based convolution, after the transformation, the 3×3 filter is transformed into a 4×4 matrix, requiring C×4×4. Therefore the storage requirement is increased by 33%. Equation 2 shows how the input transformation is calculated.
V = B T dB (2) Where V is an (m + r − 1) × (m + r − 1) transformed data tile; d is an (m + r − 1) × (m + r − 1) input tile; B is an (m + r − 1) × (m + r − 1) transform matrix,
defined by the Winograd algorithm.
For F (2 × 2, 3 × 3) input tile (d) is 4 × 4 and is generated by a sliding window across the 2-D input feature data. Shown in Fig. 2, the d window slides horizontally across the input data, with a stride of two. After the Winograd transformation, the 4 × 4 V tiles are stored back into memory. Because the input tiles contain overlaps of the input data, four times more memory storage is required. Equation 3 shows how the pre-computed transformed filter U and the run-time transformed input data V are used to calculate the final output, a 2 × 2 tile Y . Each Y tile corresponds to a 2 × 2 non-overlapping subsection of the overall convolution output.
Y = F (m × m, r × r) = A T [U ⊙ V ]A (3) Where Y is an m × m output tile; ⊙ is an element wise multiplication; A is an (m + r − 1) × m transform matrix, defined by the Winograd algorithm. INPUT DATA 4 4 stride = 2 d -input tile
B. FPGA Winograd Convolution
To validate the FPGA Caffe framework, efforts were primarily focused on implementing the Winograd algorithm for convolution (F (2×2, 3×3)) discussed in Section III-A, though other less optimized implementations of the layers discussed in Section II-B have been created as well. The architecture created for the Winograd algorithm on the FPGA can be separated into three stages of operation: input, compute, and output.
Input Stage -used to move the input frames from off-chip SDRAM memory to the on-chip BRAMs. A portion of the input frame is burst read into a temporary buffer through an AXI request and moved into tiles of four rows by two columns stored in BRAM. The tiles in this case deviate from the algorithm discussed in Section III-A as data is only replicated in overlapping rows of tiles rather than both overlapping rows and columns to reduce the memory overhead of storing input tiles. This tiling method is similar to the method in Fig. 2, however while the stride stays the same, the tile width is two resulting in no overlaps. To facilitate the compute stage and output stages, the number of tiles per row is set to be a multiple of eight, with padding added as required. After tiling is completed, a portion of the Winograd input transformation shown in Equation 4 is applied to every column of the 4 × 2 tiles.
O[0, j] = I[0, j] − I[2, j] O[1, j] = I[1, j] + I[2, j] O[2, j] = I[2, j] − I[1, j] O[3, j] = I[1, j] − I[3, j](4)
Where I[n, m] is the input of the partial transform at (n, m); O[n, m] is the output of the partial transform at (n, m).
This partial transformation is replicated eight times in the input stage such that eight columns of the input tiles can be processed per cycle to reduce the overhead that this preprocessing causes.
The full input transformation in Equation 2 requires a total of eight instances of Equation 4 per 4 × 4 tile, with four instances being applied to the columns, and four instances being applied to the rows of the tile (indices of Equation 4 are swapped for row calculations) as shown in Fig. 3. This results in 32 floating point additions and 64 DSPs per input tile transformation instantiation. Exploiting the fact that each input tile overlaps every two rows and every two columns with its neighbor, savings can be achieved by first precomputing either all of the column wise instances of Equation 4 or all of the row wise instances rather than computing the full transform for each tile. This reduces the number of partial transforms required for either the column wise or row wise instances from four to two per tile, which reduces the number of floating point additions to 24 from 32 per input tile transformation (with potentially one additional set of column transformations for the edge of each row). Compute Stage -where the bulk of the processing time is spent in the architecture. Following the input stage, each 4x2 tile and its neighbor are fed into a pipelined processing element that handles all subsequent computations required by the algorithm. The processing element completes the remaining set of partial transforms required after the input stage, performs the element wise multiplication between the input and the weights, computes the output transform, and accumulates the result within an output buffer. This process is repeated with different weights per output feature map until all of the output feature maps have been computed. Per output feature map this requires C × P × Q + D clock cycles, where C is the number of input channels, P is the number of output tiles per column, Q is the number of output tiles per row, and D is the number of cycles required to fill the pipeline. To reduce the cycles required per output feature map, the processing element is replicated four times, allowing this stage to effectively be completed four times faster (assuming that D is small).
Output Stage -handles transferring the output frame back into the off-chip DRR memory. First the results are gathered from the partial result buffers of the compute stage to an output buffer. Then the output is burst written to the DDR through AXI.
To further improve the performance of the engine, it has been replicated such that there are two CUs rather than one. Each CU handles a separate input image to exploit coarsegrained data parallelism. The performance of the engine is directly proportional to how many CUs can be replicated, with the execution time being dictated by the equation T = (C × N )/(F × #CU s), where F is the operating frequency, C is the number of clock cycles, N is the number of images, and T is the total latency.
IV. FPGA CAFFE FRAMEWORK
The Caffe framework [4] is used to describe CNNs based on predefined layer implementations with CPU and GPU backends. In this section we describe our approach to augmenting the Caffe framework to enable CNN classification using FPGAs. The discussion below will detail how memory transfers between the device and host are handled, how FPGA test benches may be used within the framework, and several FPGA specific layer implementations. The layers include a custom layer for reprogramming the FPGA and pipeline layers for fused layer implementations.
A. Caffe Model Description
The infrastructure in Caffe allows for simple description of common layers used in CNNs and provides several implementations of existing high performance CNNs as well. Each layer in Caffe corresponds to a set of computations required by a given CNN model, allowing for modular CNN implementations. Caffe also allows for networks to be defined without modification of source code by providing model definitions through the Protocol Buffer Language [4]. This allows for networks to be constructed through a file describing which layers are required and their respective ordering.
The model description format in Caffe has been augmented in this work to support additional features described in the sections below. Namely, in the FPGA Caffe framework a program layer can be specified in any position of the network to force the FPGA to be reprogrammed, pipelined layers can be specified if a fused layer is required, and the Winograd
B. OpenCL Brew
In Caffe a Brew is referred to as a mode of operation that determines the target architecture on which CNN classification or training is executed. The original Brews are CPU or GPU, with the CPU Brew containing the C++ infrastructure required to define layers using a CPU, and the GPU Brew providing similar features but for NVIDIA GPUs using CUDA and cuDNN [4], [13]. For each Brew, Caffe contains test cases available for every layer, allowing for fast determination of functional correctness and benchmarking.
This work extends the baseline Caffe framework to include the OCL (OpenCL) Brew, which provides support for Xilinx FPGA-based CNNs and could easily be adapted to target Altera's OpenCL programming environment as well. The user can choose between the different Brews by building the framework using the corresponding Makefile flags and changing the Brew to OCL. Fig. 4 shows an overview of the augmented system with the OCL Brew, where inputs and outputs are the same as in the CPU and GPU Brews, but the underlying hardware of the system is comprised of the CPU for host code and the FPGA for layer computations.
To perform a forward pass (inference) using the OCL Brew, we added an API call: forward ocl(). The forward ocl() API call is used as the forward operator on the condition that the Brew is OCL and the function is defined, otherwise it defaults to the forward cpu() call as in the baseline Caffe implementation.
C. OpenCL Memory Management and Synchronization
Data in Caffe is represented as a four-dimensional flattened array, with allocation, resizing, and synchronization between CPU and GPU resources abstracted from its usage [4]. The memory management API in Caffe handles synchronization between the host and GPU devices such that memory is only transferred back to the host when necessary. To accomplish this, the state of the memory is stored as either HEAD AT GPU, HEAD AT CPU, or SYNCED which is verified upon accessing the data. If the state of the data is HEAD AT GPU and the host requests the data, a data transfer from the device to the host will be issued and the state will change to SYNCED. Support for memory synchronization between the host and the FPGA in the FPGA Caffe framework builds on the memory synchronization features described above. To accomplish similar functionality, OpenCL APIs are used with an additional object corresponding to the FPGA device memory object for each data structure. When data is passed from the host to the FPGA, the state of the memory changes to HEAD AT OCL such that on subsequent accesses it will either stay in the device memory or be transferred back to host memory. If the data is required by the host, a memory transfer will be issued from the device to the host and the state of the memory will change to SYNCED. To access the FPGA memory object, calls to either mutable ocl data() for modifying data (layer output data) or ocl data() for static data (layer input data, such as weights), are required. These two functions were added to Caffe to handle both the creation and synchronization of the device and host memory while maintaining transparency of memory manipulation as in the baseline Caffe implementation.
D. FPGA Testbenches
Testing a given layer in the FPGA framework can be accomplished in two different ways depending on the stage of development. Layers can be tested using individual test cases through the test framework provided in Caffe. Alternatively they can also be tested through the use of standalone host code by invoking only the host code required to launch the kernel. In either case, the layer can be tested using a hardware implementation or using software emulation based implementations created in the Xilinx SDAccel environment.
The baseline Caffe framework has a number of tests that are available for each layer of the system [4]. Each test can be made into a test for the FPGA implementations by changing the Brew to OCL and modifying parameters to suit a given layer. These tests allow for larger scale testing to verify that the layer has been integrated properly within the Caffe framework. Aside from providing breadth to the test suite, this also allows for fast prototyping of layers through the use of software emulated layers provided by the capabilities of SDAccel [12].
E. Kernel Compilation
Compiling kernels for CPUs and GPUs amounts to compiling programs into instructions that program the hardware, whereas compiling for FPGAs involves synthesizing full circuits. As a result, the overhead of compilation for FPGAs (hours) is much greater than that of CPUs and GPUs (milliseconds), and so runtime compilation of FPGA kernels is not possible. We deal with this problem by employing an offline compilation strategy, where deep learning practitioners can make use of precompiled binaries at run-time.
F. XCLProgram Layer
Though FPGA Caffe makes use of offline binary compilation, there is still significant overhead from programming the FPGA (100-300 ms) compared to the CPU and GPU (0.001-0.005 ms). This programming overhead conflates the measured execution time for a layer in the Caffe benchmarking functionality. We introduce a new layer, the XCLProgram layer, as a method for giving the user greater control over how the FPGA is programmed, as well as the ability to separately benchmark the execution time of each layer and programming overhead. The XCLProgram layer as input receives a pointer to the FPGA binary file, as well as the kernel name.
G. Pipelined Layers
In the GPU-based approach native to Caffe, modularity is enforced layer-wise, meaning before each layer is executed the GPU is programmed with the appropriate kernel and memory is synchronized with the host. In FPGA Caffe this becomes a bottleneck given the large overhead of programming the FPGA, and requiring such frequent memory synchronization with the host is much more expensive on FPGAs compared to high memory bandwidth GPUs. Additionally, this modularity used in Caffe limits parallelism strategies to within each kernel. To address these issues, we introduce a new layer type in FPGA Caffe called pipeline layers. Facilitated by XCLProgram layers, pipeline layers package multiple kernels into a single binary, with kernel-kernel communication occurring through local memory structures on the FPGA (i.e. FIFO). Pipeline layers reduce the number of times the FPGA is programmed, and eliminate the need to synchronize memory with the host between every layer. Most importantly, pipeline layers allow pipeline parallelism strategies across layers, increasing throughput by allowing multiple layers to execute concurrently. While the use of pipeline layers violates some modularity assumptions of Caffe, we argue that this is practical given that combinations of layer groups are very predictable in practice (e.g. convolution, ReLU, pooling).
V. RESULTS
This section describes our study of the Winograd Convolution Algorithm and the results gathered from using FPGA Caffe to implement the 3×3 convolution layer using the Winograd Convolution Algorithm. The platform we use includes an Alpha-Data ADM-PCIE-7V3 card with a Xilinx Virtex 7 XC7VX690T-2 running at 200 MHz and an Intel Xeon CPU E5-2620 running at 2.0 GHz for the host application code. The Xilinx Virtex 7 is contained within a server that has been virtualized to support virtual machines (VM) and is connected through PCIe. The VM in use has 8GB RAM and four cores. An Intel i7-4770k running at 3.5 GHz was used for CPU comparison and an nVidia Quadro K620 for GPU comparisons. The Xilinx SDAccel version number is 2015.1.3, CUDA version number is 7.5, cuDNN version number is 4.0 [13] and the CPU host code uses OpenBLAS [14] with eight threads enabled. The CPU, GPU, and FPGA implementations all use single precision floating point as their data representation.
A. Winograd Resource Utilization
The resource utilization post place and route is shown in Table I. The highest utilization post place and route for both CUs is the LUT utilization at 83.2% of the SDAccel region's available LUTs. The utilization post place and route accounts for additional resources required to integrate into the SDAccel framework, which drives the significant LUT utilization in comparison to other resources on the device. Given the utilization of the device resources, further instances of the CU could theoretically be added, though the limits on the available resources in the SDAccel region of the FPGA shown in Table I makes it impossible to place more than two CUs. This in turn limits the potential performance of the architecture within the SDAccel environment because of the overhead of the static region and the lack of resources available in the reconfigurable region.
To quantify the DSP savings from using the input stage column transformation discussed in Section III-B we consider three separate cases. The first case is that the full input transform is pre-computed for each four by four tile before sending the data to the processing elements of the compute stage, with only one instance of the full input transformation in Equation 2. The second case is that the full input transformation in Equation 2 is computed within each processing element of the compute stage using one full input transformation per processing element. Finally, the last case is the one discussed in Section III-B, in which a column-wise partial transformation is computed for all input tile columns, using eight partial transformations to reduce computation overhead, with the remaining partial transformations computed within the processing elements using four partial transformations per processing element. Table II shows the resource utilization of the three cases post place and route. Between cases 2 and 3 there is a DSP savings of 61 units and a decrease in LUT usage. This is slightly less than what is anticipated in Section III-B, though the difference can be attributed to fewer addressing calculations being required in case 2 due to the elimination of the partial transformations in the input stage. When comparing with case 1, the DSP utilization is approximately 20% less than case 3, however the BRAM utilization has increased by 33% due to the storing of replicated data in overlapping tiles. While the LUT usage and DSP utilization is better in case 1, the BRAM utilization makes it difficult to place and route more than 1 CU, as the BRAM utilization for two would require most of the available BRAM.
B. FPGA Caffe Benchmark Results
To evaluate the Winograd convolution engine within the FPGA Caffe framework, a set of benchmark CNNs is required to view its performance across varying workloads (number of output feature maps and output sizes). The benchmark suite that we use is adopted from the Soumith Chintala convnetbenchmarks [15], which is composed of previous ImageNet winners including: AlexNet [8], VGG A [9], Overfeat [10], and GoogleNet [11]. Due to the RAM size of the virtual machine used for the host code, the batch size for each benchmark is reduced by half to fit within the host VM. Table III shows the performance of the system in comparison to both CPU and GPU implementations of the 3 × 3 convolution layers of each CNN. To calculate the GFLOPS of the Winograd convolution engine, the number of floatingpoint operations is taken to be the same as direct convolution, as is the case in [7], which is considered to be the effective GFLOPS. Comparing the geometric averages in Table III, the Winograd convolution engine performs approximately 2.1 times slower than the CPU implementation and 9.4 times slower than the GPU implementation.
VI. RELATED WORK
There exist several frameworks for implementing CNNs depending on the targeted platform and programming language, with many of the popular frameworks discussed in [4]. These frameworks typically support a number of programming languages including: C/C++, Matlab, Python, etc. With respect to Caffe, there is an independent effort underway to add OpenCL support [16], though this support is meant primarily for exposing more GPUs to Caffe rather than FPGAs. The work in [16] provides additional functionality to support AMD devices through OpenCL and provides some similar features to those explored in this work. However, these works are differentiated in that the OpenCL implementations are abstracted away from being GPUs in this work through a separate Brew for OpenCL to allow for implementations across many different compute engines. Additionally, given the non-standard development model currently supported in FPGA OpenCL tools, frameworks that support standard OpenCL will not work with FPGAs without significant framework modification. This is the case primarily because the FPGA OpenCL tools require offline compilation of kernels and vendor-specific attributes to achieve suitable performance. The framework introduced in this paper allows for the support of FPGAs through layerspecific implementations. These include the ability to specify when to program the FPGA, the addition of pipelined layers, and precompiled FPGA-specific layer implementations. Table IV shows the performance of this work compared to several recent FPGA works. The highest performing implementations are Qiu [3] and Suda [2], which is enabled by their use of fixed point representations. The work of Zhang [1] is approximately 1.2-fold higher performance than this work while using the same data representation, though as expected our DSP utilization is significantly lower while achieving comparable throughput. While the performance of this implementation is lower, it does not require precision analysis prior to usage and it does not need to be resynthesized for new work loads as is the case in all of the prior works.
VII. FUTURE WORK
Future work related to this framework will contribute to both performance and usability. First, completing the implementation of back propagation will ensure FPGAs can be used for both classification and training, where hardware acceleration is especially important. Given the modularity between the solver, network, and layer in Caffe, only the layers need to be modified to accommodate backward propagation. The structure of the network that defines the collection of layers, as well as the operation of the solver that calls the backward methods to generate gradients and perform a weight update, is already in place.
As well, experimenting with reduced-precision implementations of common layers is an important next step. Recently, GPUs have started to support half precision based implementations of CNNs [7], [13] and many of the FPGA works have been focused on reduced precision implementations [2], [3]. Half precision or fixed-point implementations both would offer significant area gains for FPGA implementations as shown in [2], [3], as a floating-point multiplication requires three DSP units and additional LUTs and flip-flops for current Xilinx FPGAs, while half precision or fixed-point multiplications require only one DSP unit and significantly fewer LUTs and flip-flops. This would allow for the replication of both processing elements and CUs to improve performance.
Finally, multi-FPGA-based parallelism strategies are crucially important in scaling up to accommodate larger data and model sizes. Current solutions involve using GPU clusters with Infiniband interconnects and MPI, which allow fast node to node data transfer and increase parallelism capabilities [17]. The use of FPGAs are attractive in this domain given the flexibility and high performance/watt, and FPGAs can benefit from much of the work being done investigating multi-GPU parallelism strategies. 470.17 50 a GPU memory could only fit 8 images, so the model was run with 8 images and execution time was multiplied by 4 to get a performance estimate. b GPU memory could only fit 32 images, so the model was run with 32 images and execution time was multiplied by 2 to get a performance estimate.
VIII. CONCLUSION
In this work we presented a framework for implementing CNNs using FPGAs based on the Caffe CNN framework. The framework allows for transparent support for individual FPGA implementations of layers for testing and verification. This framework was validated by implementing the Winograd convolution layer and testing it across several CNNs. The results show that with 83.2% of the available SDAccel resources we are able to achieve 50 GFLOPs across the 3 × 3 convolution layers of four different CNNs. While this does not improve upon current implementations in terms of performance, it demonstrates the capabilities of the framework, which allows for further work that could lead to higher performance.
Fig. 1 .
1SDAccel Platform values are produced for every instance of F (m × m, r × r).
Fig. 2 .
2Winograd Input Tile Stencil
Fig. 3. Winograd Forward Transform
Fig. 4 .
4High-Level View of the Brew Options in Caffe convolution engine discussed in Section III-B can be specified when needed.
TABLE I SINGLE
IPRECISION WINOGRAD CONVOLUTION ENGINE RESOURCE
UTILIZATION POST PLACE AND ROUTE
-
FF
LUT
DSP
BRAM
(18Kb)
XC7VX690T
Total Resources (A)
866,400
433,200
3,600
2,940
SDAccel
Region (B)
551,040
275,520
2,376
1,940
Winograd
Conv. Engine (C)
253,873
229,226
1,307
1,188
SDAccel
Utilization (C/B)
46.7%
83.2%
55%
61.2%
Total
Device Utilization (C/A)
29.3%
52.9%
36.3%
40.4%
TABLE II RESOURCE
IIUTILIZATION FOR DIFFERENT WINOGRAD STRATEGIES WITH
ONE CU
Layer
FF
LUT
DSP
BRAM
(18Kb)
Case 1: Full
Pre-Transform
143,965
127,989
523
914
Case 2: Full
Transform in PE
166,905
153,887
715
688
Case 3: Partial
Transform in PE
158,266
145,892
654
688
TABLE III CPU
III, GPU, AND FPGA 3×3 CONVOLUTION BENCHMARK RESULTSNetwork
CPU Run
Time (ms)
GPU Run
Time (ms)
FPGA Run
Time (ms)
CPU GFLOPS
GPU GFLOPS
FPGA GFLOPS
AlexNet (64 Images)
492
261.2
1,010
94
177.1
45.8
VGG A (32 Images)
4,310
745.14 a
8,713
111.2
642.9
55
Overfeat (64 Images)
2,030
387.1
4,781
139.2
730.2
59.1
GoogleNet (64 Images)
1,506
209.66 b
2,937
81.8
587.8
42
Geometric Average
1,595.6
354.51
3,333.9
104.46
TABLE IV COMPARISON
IVBETWEEN EXISTING FPGA WORKSMetric
Zhang [1]
Suda [2]
Qiu [3]
This Work
Clock Frequency (MHz)
100
120
150
200
Precision
32 bit float
8-16-bit fixed
16-bit fixed
32 bit float
FPGA Version
Virtex 7 VX485T
Stratix-V GSD8
Zynq XC7Z045
Virtex 7 XC7VX690T-2
DSP Utilization
2,240
(Not specified)
780
1,307
Host Connection
Microblaze, on chip host
PCIe
ARM Cortex-A9 Processor, on chip host
PCIe
GFLOPS/GOPS
61.62
136.5
187.8
50
ACKNOWLEDGMENTThe authors would like to thank Xilinx, CMC and em-SYSCAN, NSERC, and CFI for the funding and resources provided for this work.
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|
[
"https://github.com/dicecco1/fpga"
] |
[
"KREIN FORMULA AND S-MATRIX FOR EUCLIDEAN SURFACES WITH CONICAL SINGULARITIES",
"KREIN FORMULA AND S-MATRIX FOR EUCLIDEAN SURFACES WITH CONICAL SINGULARITIES"
] |
[
"Luc Hillairet ",
"Alexey Kokotov "
] |
[] |
[] |
Using the Krein formula for the difference of the resolvents of two self-adjoint extensions of a symmetric operator with finite deficiency indices, we establish a comparison formula for ζ-regularized determinants of two self-adjoint extensions of the Laplace operator on a Euclidean surface with conical singularities (E. s. c. s.). The ratio of two determinants is expressed through the value S(0) of the S-matrix, S(λ), of the surface. We study the asymptotic behavior of the S-matrix, give an explicit expression for S(0) relating it to the Bergman projective connection on the underlying compact Riemann surface and derive variational formulas for S(λ) with respect to coordinates on the moduli space of E. s. c. s. with trivial holonomy.
|
10.1007/s12220-012-9295-3
|
[
"https://arxiv.org/pdf/1011.5034v2.pdf"
] | 36,327,362 |
1011.5034
|
68809a9ae6224bbb56be7222bb0abc6504c9782c
|
KREIN FORMULA AND S-MATRIX FOR EUCLIDEAN SURFACES WITH CONICAL SINGULARITIES
28 Feb 2012
Luc Hillairet
Alexey Kokotov
KREIN FORMULA AND S-MATRIX FOR EUCLIDEAN SURFACES WITH CONICAL SINGULARITIES
28 Feb 2012
Using the Krein formula for the difference of the resolvents of two self-adjoint extensions of a symmetric operator with finite deficiency indices, we establish a comparison formula for ζ-regularized determinants of two self-adjoint extensions of the Laplace operator on a Euclidean surface with conical singularities (E. s. c. s.). The ratio of two determinants is expressed through the value S(0) of the S-matrix, S(λ), of the surface. We study the asymptotic behavior of the S-matrix, give an explicit expression for S(0) relating it to the Bergman projective connection on the underlying compact Riemann surface and derive variational formulas for S(λ) with respect to coordinates on the moduli space of E. s. c. s. with trivial holonomy.
Introduction
Spectral geometry aims at understanding the relations between the spectrum of some Laplace operator in a given geometrical setting and geometric properties of the latter. Polygons and polyhedra are among the simplest shapes one can consider and one could hope in this setting for a better understanding. This leads naturally to study the spectral geometry of Euclidean surfaces with conical singularities. Another motivation is the spectral theory of translation surfaces for which the geometric picture has many interesting developments (see [30] for instance).
One peculiarity of Laplacians on manifolds with conical points is that, due to the presence of conical points, a choice has to be made in order to get a self-adjoint operator. In this paper, we are interested in understanding how this choice affects several spectral quantities such as the resolvent and the zetaregularized determinant. Depending on the self-adjoint extension, this zeta-regularization procedure is not as straightforward as usual because of unusual behavior of the zeta function but it is still possible to define such a regularization (see [17,12] and section 5.3) and we will prove a comparison formula for these determinants.
Comparison formulas for regularized determinants for conical manifolds were first found in [22] using a surgery formulaà la BFK (see [5]) and in [17] using a contour integral method based on a secular equation that defines the spectrum. One of our motivations was to understand how the comparison formulas for different self-adjoint extensions from [22] read in the case of Euclidean surfaces with conical singularities and whether it is possible to express the determinants of the non Friedrichs selfadjoint extensions of the Laplacian on these surfaces through holomorphic invariants of the underlying Riemann surface (as it was done in [19] for the determinant of the Friedrichs extension). Indeed, Euclidean surfaces with conical singularities are our primary interest and we will restrict to this setting although many statements still make sense for more general conical manifolds.
It turns out that the geometric interpretation of the formulas obtained in [22] and [17] is not that straightforward and we have found it more convenient to establish the comparison formula for determinants using the Krein formula for the difference of resolvents of two self-adjoint extensions of a symmetric operator. We observe that the trace of the difference of two resolvents admits a nice representation through the so-called S-matrix of a Euclidean surface with conical singularities (E. s. c. s.) X. The latter matrix, or, more precisely, the meromorphic family of matrices S(λ) is in some sense a characteristic feature of X. Indeed, we believe that some of the geometry of X (such as for instance the lengths of saddle-connections between conical points-see Remark 4.3) is encoded in S(λ) although it seems quite difficult to retrieve this kind of information. We should also remark that this S-matrix allows to write down a secular equation that can then be treated using the approach of [17] so that what we propose here may be seen as a geometric interpretation for the latter method. The comparison with [22] is less straightforward, it relies in interpreting the S-matrix as some kind of limiting Dirichlet-to-Neumann operator on a circle around the conical point when the radius of that circle goes to 0. It can be noted here that, in contrast with [22] no extra condition is needed to obtain our formula.
We will thus prove the following theorem. The notion of regular self-adjoint extensions will be introduced in definition 5.2, section 5 and, for these self-adjoint extensions, the expression P + QS(0) makes sense (see remark 5.5).
Theorem 1. On a compact E.s.c.s. X, let S(λ) be the S-matrix and ∆ F be the Friedrichs extension.
Let P and Q be matrices that define a regular self-adjoint extension ∆ L , and define D(λ) := det(P + QS(λ)).
Let d be the dimension of ker(P + QS(0)) and let D * (0) := lim λ→0 (−λ) −d D(λ). There exist α 0 and Γ such that the asymptotic expansion of D(−|λ|) as λ goes to ∞ is ln D(−|λ|) := α 0 ln(|λ|) + Γ + o(1).
The following identity then holds :
det * ζ (∆ L ) = exp(−Γ)D * (0)det * ζ (∆ F ), in which det *
ζ is the modified zeta-regularized determinant (see definition 5.14.) To fulfil our second aim we then need to understand more explicitly what kind of geometric information is encoded in the family S(λ). We focus on the limiting behavior when the spectral parameter goes to 0 since this is the regime that comes up in the comparison formula. We will prove that most of the matrix elements in this limit have an interpretation through values of the Bergman projective connection and the basic holomorphic differentials taken at the conical point in the corresponding distinguished holomorphic local parameter (see section 4.3). Since we expect translation surfaces to have particular and interesting features, we will also say a word on the S-matrix on these special kind of surfaces. Namely, we will derive variational formulas for the S-matrix when it is differentiated with respect to moduli parameters. These results answer most of the questions which motivated our study.
Organization of the paper. In the small second section we will recall the basic facts about Euclidean surfaces with conical singularities. We will in particular recall that these can be viewed as Riemann surfaces with flat conformal conical metric.
In section 3 we recall some basic properties of the Friedrichs Laplace operator on E. s. c. s., and introduce the object of our primary interest -the S-matrix; we also derive here the standard formula for the derivative of the S-matrix with respect to λ.
In section 4 we study the asymptotic behavior of S(λ) as λ goes to −∞ and find the geometric interpretation of S(0). We also also apply the variational formulas of [19] to obtain the variations of S(λ) with respect to moduli parameters on translation surfaces.
In section 5 we study various self-adjoint extensions of the Laplace operator on E. s. c. s. and prove the comparison formula for their ζ-regularized determinants.
We acknowledge useful conversations with G. Carron and with D. Korotkin whose advice in particular helped us to simplify some constructions from section 4.2.
2. Euclidean surfaces with conical singularities 2.1. Euclidean surfaces with conical singularities as Riemann surfaces with conformal flat conical metrics. A Euclidean surface with conical singularities (E. s. c. s.) is a compact (orientable) surface glued from Euclidean triangles. One can take as an example of such a surface the boundary of a connected but not necessarily simply connected polyhedron in R 3 .
When two triangles are glued together and after rotating one of the triangles around the common edge we observe that the intrinsic geometry of the surface is locally that of the plane. There, the surface actually is smooth and equipped with a smooth Euclidean metric. At a vertex p where k triangles with angles ϑ 1 , . . . , ϑ k are glued together, the surface is locally isometric to a neighbourhood of the tip of the Euclidean cone of total angle θ p = ϑ 1 + · · · + ϑ k . The surface X is thus equipped with a Euclidean metric that is smooth except at the vertices p for which θ p = 2π.
It follows for instance from [28] that X can be provided with a complex analytic structure becoming a compact Riemann surfaceX; moreover, the usual Euclidean metric on X gives rise to a flat conformal (i. e. defining the same complex structure) metric onX. Abusing notations slightly, from now on we won't make any difference between X andX.
On the other hand, consider a flat conformal metric m with conical singularities on a Riemann surface X. In a vicinity of a conical point p, m can be written as
m = |g(z)||z| 2b |dz| 2 ,
where z is a holomorphic local parameter near p, z(p) = 0, b > −1 and g(z) is a holomorphic function of the local parameter such that g(0) = 0.
It is shown in [28] that one can choose a holomorphic change of variables z = z(ζ) such that
(2.1) |g(z(ζ))||z(ζ)| 2b |z ′ (ζ)| 2 = |ζ| 2b
and, therefore,
(2.2) m = |ζ| 2b |dζ| 2
in the local parameter ζ. This means that the Riemannian surface (X, m) near p is isometric to the standard Euclidean cone of angle 2π(b + 1). Troyanov [28] showed that the Riemannian manifold (X, m) can be triangulated in such a way that all the conical points will be among the vertices of the triangulation meaning thus that (X, m) is an E. s. c. s. Definition 1. Let X be a compact Riemann surface with conformal flat conical metric (i. e. a E. s. c. s.) and let p ∈ X be a conical point. Then any holomorphic local parameter ζ in which the metric takes the form (2.2) is called distinguished.
Notation : We will denote by P the set of conical points and by X 0 := X \ P the complement of P in X. We set M := Card(P ) the number of conical points. At each p ∈ P, the total cone angle is denoted by θ p .
2.1.1. Translation and half-translation surfaces. A translation (resp. half-translation) surface is a E. s. c. s. that has trivial holonomy (resp. holonomy group Z 2 ). These are important examples of E.s.c.s. with very nice geometric properties (see [30] for a survey on these).
Translation surfaces are Riemann surfaces X that are equipped with a conformal flat conical metric given by the modulus square, m = |ω| 2 , of a holomorphic 1-form (an Abelian differential) ω. If P is a zero of ω of multiplicity k then p is a conical point of the translation surface X with conical angle 2π(k+1). The moduli space H g of pairs (X, ω) (where X is a compact Riemann surface of genus g ≥ 1, ω is a holomorphic 1-form on X) is stratified according to the multiplicities of the zeros of the 1-form ω. Denote by H g (k 1 , . . . , k M ) the stratum consisting of pairs (X, ω), where ω has M zeros, p 1 , . . . , p M of multiplicities k 1 , . . . , k M (according to Riemann-Roch theorem one has k 1 + · · · + k M = 2g − 2). The stratum H g (k 1 , . . . , k M ) is a complex orbifold of dimension 2g + M − 1.
Let (X, ω) ∈ H g (k 1 , . . . , k M ). Choose a canonical basis of cycles {a α , b α } on the Riemann surface X and take M − 1 contours γ k , k = 2, . . . , M on X connecting p 1 with p 2 , . . . , p M The local coordinates on H g (k 1 , . . . , k M ) (which are called Kontsevich-Zorich homological coordinates, see [20]) are given by the following integrals:
A α = aα ω; α = 1, . . . , g, B α = bα ω; α = 1, . . . , g, z k = γ k ω; k = 2, . . . , M − 1 .
A half-translation surface is a compact Riemann surface with flat conical metric m = |q|, where q is a meromorphic quadratic differential with at most simple poles.
Example 2.1. Consider the Riemann sphere CP 1 with metric
|z| 2 |dz| 2 6 k=1 |z − z k | , where z k ∈ C, z k = 0 and z i = z k if i = k.
This is a half-translation surface with 7 conical points 0, z 1 , . . . , z 6 . The conical angle at 0 is 4π, the conical angles at each point z k are equal to π.
Such a surface can be viewed by considering a Euclidean pair of pants (with one 4π singularity) and by sewing each leg and the waist with itself (thus creating the six π singularities).
The Friedrichs Laplacian and the S-matrix
Let X be a compact E.s.c.s.. In this section we will recall the definition of the Friedrichs Laplacian associated with the (singular) metric and define the so-called S-matrix. We will then collect several properties of this matrix.
We denote by ∆ the minimal closed extension of the Euclidean Laplacian defined on C ∞ 0 (X 0 ), and by ∆ * its adjoint with respect to the Euclidean L 2 scalar product
u, v := X uv dx.
Near each conical point p, any u ∈ dom(∆ * ) has the following asymptotic behavior in polar coordinates (r, θ) (see, e. g., [23], [22], [24] or [19]) :
(3.1) u(r, θ) = 2θ p a + 0 + a − 0 ln(r) + ν 2|ν|θ p a + ν r |ν| + a − ν r −|ν| exp(iνθ) + u 0 ,
where ν ranges over N p := 2π θp · k, | k ∈ Z\{0}, |k| < θp 2π , and u 0 ∈ dom(∆).
Notation : We will denote by N = ∪ p∈P N p , and we will abusively still denote by ν an element of N. Choosing an element ν of N thus amounts to choosing a conical point p and then some ν in N p . Unless needed we will omit the reference to p. The square roots prefactor in (3.1) are just normalization constants. We will denote these constants by C 0 := 2θ p and C ν := 2|ν|θ p (we recall that since ν implicitly depends on p, so does C ν ).
In the distinguished local parameter ζ near p we have, for ν = 2π θp · k
ζ k = r ν exp(iνθ) = r |ν| exp(iνθ) if ν > 0, r −|ν| exp(iνθ) if ν < 0. (3.2) ζ −k = r −ν exp(iνθ) = r |ν| exp(iνθ) if ν < 0, r −|ν| exp(iνθ) if ν > 0. (3.3)
Thus the asymptotic expansion (3.1) may also be written
(3.4) u(ζ,ζ) = C 0 a + 0 + a − 0 ln(|ζ|) + θp 2π −1 k=1 C k 2π θp a + k ζ k + a − kζ −k + a + −kζ k + a − −k ζ −k + u 0 .
A straightforward application of Green's formula (combined with the choice of the normalization constants C 0 , C ν ) then implies that, for any u, v in dom(∆ * ),
∆ * u, v − u, ∆ * v = p∈P a + 0 · b − 0 − a − 0 · b + 0 + ν∈Np a + ν · b − ν − a − ν · b + ν (3.5)
where the a ± ν are the coefficients in the expansion of u and the b ± ν those in the expansion of v. Setting G(u, v) := ∆ * u, v − u, ∆ * v we define a Hermitian symplectic form on dom(∆ * )/dom(∆) whose lagrangian subspaces parametrize the self-adjoint extensions of ∆.
3.1. The Friedrichs extension. For any u ∈ dom(∆) a straightforward integration by parts gives ∆u, u = X |∇u| 2 dx so that the Friedrichs procedure (see [4] section 10.3 or [26] theorem X.23) provides us with a selfadjoint extension that we denote by ∆ F . Since a function u in dom(∆ F ) is characterized by ∇u ∈ L 2 (X), we obtain the following lemma. Remark 1. This definition of H s is not completely standard. In particular, because of the conical singularities, for m > 1 the following inclusion is strict (see [11] for a much more detailed discussion about this fact) :
{u ∈ L 2 | ∀|α| ≤ m, ∂ α u ∈ L 2 } ⊂ H m .
By standard spectral theory, the resolvent of ∆ F defines a continuous operator from H s to H s+2 . We also recall that since X is compact, the Rellich-type injection theorem from [8] implies that ∆ F has compact resolvent so that the spectrum is non-negative and discrete.
3.2. The S-matrix. We will now define a matrix associated to the flat structure and to the choice of the Friedrichs extension.
First, for any ν, we fix F ν = C ν r −|ν| exp(iνθ)ρ(r) where ρ is some fixed cut-off function that is identically 1 near the corresponding conical point p.
We define Λ ν to be the linear functional on H 2 satisfying
(3.6) ∀u ∈ H 2 , Λ ν (u) = G(u, F ν ).
We have the following lemma.
Lemma 3.2. The linear functional Λ ν is continuous on H 2 and
∀u ∈ H 2 , Λ ν (u) = a + ν
where a + ν is the coefficient in the expansion (3.1) of u near p. Proof. The fact that F ν ∈ dom(∆ * ) implies that Λ ν is indeed continuous. The second statement follows from the respective asymptotic behaviors of F ν and u near p.
Remark 3.3. The preceding lemma in particular implies that the linear functional Λ ν doesn't depend on the choice of the cut-off function ρ.
For λ ∈ C \ [0, ∞), we set G ν (· ; λ) := (∆ F − λ) −1 Λ ν . Since Λ ν is in H −2 , G ν is in L 2 , and for any u ∈ H 2 , we have (3.7) Λ ν (u) = (∆ F − λ)u, G ν (· ; λ) .
Since the resolvent is analytic in λ, G ν (· ; λ) defines an analytic family of L 2 functions. Observe that the latter equation is equivalent to
(∆ * − λ)G ν (·; λ) = 0,
so that G ν (·; λ) ∈ dom(∆ * ). Moreover, by testing against an appropriate u ∈ H 2 we can compute the coefficients a − µ of G ν . This yields a − µ = δ µν (where δ is the Kronecker symbol). The following proposition gives a formula for G ν
Proposition 3.4. For any λ ∈ C \ [0, ∞), set f ν (· ; λ) := (∆ * − λ)F ν and g ν (· ; λ) := − (∆ F − λ) −1 f ν (·; λ). Then g ν (· ; λ) is an analytic family in H 2 and G ν (· ; λ) = F ν (·) + g ν (· ; λ).
Proof. Computation shows that f ν is in L 2 (X) which yields that g ν is in H 2 since λ is in the resolvent set of ∆ F . Since f ν and the resolvent depend analytically on λ so does g ν . By construction,
(∆ * − λ) (F ν + g ν ) = 0 and all the a − µ coefficients of G ν − (F ν + g ν ) vanish
. This means that the latter function is in H 2 and thus is 0 since λ is in the resolvent set.
Example 3.5. Let us consider the complete cone [0, ∞) × R/αZ. Using separation of variables we have that G ν (r, θ ; λ) = k(r) exp(iνθ). For ν = 0, by definition k is the unique solution to
−k ′′ − 1 r k ′ + ν 2 r 2 − λ k = 0,
which is L 2 (rdr) and asymptotic to C ν r −|ν| near 0. Thus k is proportional to K ν ( √ −λ r) where K ν is Bessel-MacDonald function (see [25] for instance). For ν = 0, the singular behavior is logarithmic but k(r) still is proportional to K 0 ( √ −λ r) Definition 3.6 (The S-matrix). We define the S-matrix S(λ) by
(3.8) S µν (λ) = Λ µ (g ν (· ; λ)). Remark 3.7. Alternatively, S µν (λ) is the a + µ coefficient of g ν (· ; λ). It is also the a + µ coefficient of G ν (· ; λ).
Observe that the entries of the S-matrix are numbered by non-integer numbers.
Using (3.7), we have the following alternative expression
S µν (λ) = (∆ F − λ) g ν (· ; λ), G µ (· ; λ) = f ν (· ; λ), G µ (· ; λ)
It follows from the analyticity of g ν that S(λ) is analytic on C \ [0; ∞).
Example 3.8. We define S α (λ) to be the S-matrix of the cone of angle α. According to example 3.5, S α (λ) is diagonal. Moreover, the asymptotic expansion of Bessel-Macdonald functions near 0 is
K 0 (z) = − ln(z) + ln(2) − γ + o(1), K |ν| (z) = π 2 sin(|ν|π) z −|ν| 2 −|ν| Γ(1 − |ν|) − z |ν| 2 |ν| Γ(1 + |ν|) + O(z 2−|ν| )
where Γ is Euler gamma function and γ Euler's constant (see for instance [25]). This yields
[S α (λ)] 00 = ln( √ −λ) − (ln(2) − γ) , [S α (λ)] νν = − Γ(1 − |ν|)(−λ) |ν| 2 2|ν| Γ(1 + |ν|) .
The interpretation of S(λ) is given by the following lemma.
Lemma 3.9. For any λ ∈ C \ [0, ∞) and any F ∈ ker(∆ * − λ). Denote by A ± (F ) the vector consisting of all the coefficients a − ν (resp. a + ν ) of F. Then we have
A + = S(λ)A − .
Remark 3.10. Interpreting A − as some kind of incoming data and A + as the outgoing data justifies the interpretation of the S-matrix as a scattering matrix.
Proof. SetF := ν a − ν G ν (· ; λ) then F −F is in dom(ker(∆ * − λ)). Since all the a − ν vanish, F −F actually is in dom(∆ F ). This implies F =F since λ is in the resolvent set of ∆ F . Writing each G ν = F ν + g ν , we obtain : a + µ = Λ µ ( ν a − ν g ν ) = ν S(λ) µν a − ν .
Remark 3.11. Until now we haven't used the fact that the underlying metric actually is Euclidean with conical singularities. The preceding construction is fairly general and can be made on any manifold with conical singularities. Actually, it can be done in an abstract manner for any symmetric operator with (equal) finite deficiency indices (compare with section 13.4 of [14]).
Before coming to the main aim of this paper, which is to understand how much geometric information is contained in the S-matrix, we derive first two basic properties of S µν (λ).
3.3.
Derivative of the S-matrix. In this section a dot will mean differentiation with respect to λ, and we prove the following lemma.
Lemma 3.12. On C \ [0, ∞), we have (3.9)Ṡ µν = G ν (· ; λ), G µ (· ; λ) .
Proof. We start from the relation
(∆ F − λ) g ν (· ; λ) = −∆ * F ν (·) + λF ν (·),
that we differentiate with respect to λ. Since F ν doesn't depend on λ and g ν is analytic in
H 2 we obtain (∆ F − λ)ġ ν (· ; λ) = F ν (·) + g ν (· ; λ) = G ν (· ; λ). This givesṠ (λ) µν = Λ µ (ġ ν (· ; λ)) = Λ µ (∆ F − λ) −1 G ν (· ; λ) = G ν (· ; λ), G µ (· ; λ) ,
where we have used (3.7) for the last identity.
3.4.
Relation with the resolvent kernel. Denote by R(x, x ′ ; λ) the resolvent kernel of the Friedrichs extension ∆ F . Fix x ′ ∈ X 0 . As a function of the first argument, R(·, x ′ ; λ) is locally in H 2 near each conical point p. Thus according to (3.1), there exists a collection a + ν (x ′ ; λ) such that, in the neighbourhood of p we have the following asymptotic expansion :
(3.10) R(r exp(iθ), x ′ ; λ) = ν∈Np C ν a + ν (x ′ ; λ)r |ν| exp(iνθ) + r 0 with r 0 ∈ C ∞ 0 (X 0 ) H 2 .
Using (3.6), we see that a + ν (x ′ ; λ) = G(R(· , x ′ ; λ), F ν ) and thus, the former expansion may be differentiated with respect to x ′ in any compact set of X 0 .
The following proposition makes the relation between a + ν (x ′ ; λ) and G ν (x ′ ; λ) more explicit. Proposition 3.13. For any x ′ ∈ X 0 , we have
(3.11) G ν (x ′ ; λ) = a + ν (x ′ ; λ) where a + ν (x ′ ; λ) is the previously described coefficient in the asymptotic expansion of R(·, x ′ ; λ) near p. In other words, G ν (x ′ ; λ) is obtained by selecting in the resolvent kernel R(x, x ′ ; λ) some particular term in the asymptotic behavior x → p. Using R(x ′ , x ; λ) = R(x, x ′ ; λ)
there are similar statements when we fix x and let x ′ tends to p.
Proof. Denote by ∆ 1 the Euclidean Laplace operator on C ∞ 0 (X \ (P ∪ {x ′ })). This operator fits in the general theory described in section 3 by considering that x ′ actually is the vertex of a cone of angle 2π. In particular, Green's formula (3.1) is still valid provided we take into account log singularities at x ′ . The resolvent kernel R(·, x ′ ; λ) and G ν (·; λ) both belong to dom(∆ * 1 ). the singularities of R are described by the functions a + ν near the conical points and R has a log singularity near x ′ whereas G ν is smooth near x ′ and its singular behavior near the conical points G ν is prescribed by (3.10). Green's formula thus yields :
(∆ * 1 − λ)R(·, x ′ ; λ), G ν (· λ) − R(·, x ′ ; λ), (∆ * 1 − λ)G ν (x ′ ; λ) = G ν (x ′ ; λ) − a + ν (x ′ ; λ). Since (∆ * 1 − λ)R(·, x ′ ; λ) = 0 = (∆ * 1 − λ)G ν (x ′ ; λ), we obtain G ν (x ′ ; λ) = a + ν (x ′ ; λ)
. We now use the fact that G ν (x; λ) is analytic for λ ∈ C\[0, ∞) and real for real (and negative) λ. Thus by analytic continuation
G ν (x ′ ; λ) = G ν (x ′ , λ). 4. The S-matrix of E.s.c.s.
In this section we try to understand what kind of geometric information is encoded in the S-matrix of a Euclidean surface with conical singularities. We begin by studying the asymptotic behavior of S(λ) as λ goes to −∞.
4.1. S(−|λ|) for large λ.
It is a general fact that the behavior of the resolvent kernel when λ goes to −∞ is a local quantity. This is confirmed by the following lemma.
Lemma 4.1. When λ goes to ∞ then [S(−|λ|)] µν = O(|λ| −∞ ),
if µ and ν do not correspond to the same conical point. When µ and ν correspond to the same conical point p of angle α then we have
[S(−|λ|)] µν = [S α (−|λ|)] µν + O(|λ| −∞ ),
where S α denotes the S-matrix on the infinite cone of total angle α. Moreover both identities may be differentiated with respect to λ.
Proof. We use the representation of the resolvent kernel using the heat kernel (that we denote here by P(t, x, x ′ )) :
(4.1) R(x, x ′ ; −|λ|) = ∞ 0 exp(−t|λ|)P(t, x, x ′ ) dt.
We now use a standard construction of a parametrix for the heat kernel (see [7] for instance). We first enumerate the set of conical points writing P := {p i , 1 ≤ p i ≤ M }. Then, for each p i we choosẽ χ i and χ i two smooth cut-off functions such that supp(χ i ) ⊂ {χ i = 1}, χ i is identically 1 near p and X is isometric to a neighbourhood of the tip of the cone of angle θ p i on the support ofχ i . We complete the collections (χ i ) i≤M and (χ i ) i≤M to (χ i ) i≤M , (χ i ) i≤M in such a way that (χ i ) i≤M is a partition of unity,χ i is identically 1 on the support of χ i and, for M < i ≤M , X is isometric to a neighbourhood of the origin in R 2 on the support ofχ i . We also set P i to be the heat kernel on the cone corresponding to p i if i ≤ M and on the plane otherwise and definẽ
P(t, x, x ′ ) =M i=1χ i (x)P i (t, x, x ′ )χ i (x).
Using Duhamel's principle and the fact that P i fastly decays away of the diagonal (see eq (1.1) of [7]) yields thatP(t) − P(t) maps L 2 into H s for any s, and
P (t) − P(t) L 2 →H s = O(t ∞ )
when t goes to 0, so thatP is a parametrix for the heat kernel.
Inserting into (4.1) and integrating against f ν we obtain
g ν (x; −|λ|) =χ i (x) ∞ 0 X P i (t, x, x ′ )f ν (x ′ ; −|λ|)dS(x ′ )dt + r λ (x),
where the remainder r λ ∈ H 2 and r λ H 2 = O(|λ| −∞ ) and the index i corresponds to the conical point corresponding to ν. The first statement follows. The second also follows by remarking that F ν , f ν and Λ ν can also be seen as living on the cone and that the latter equation is also valid on the complete cone. Differentiating with respect to λ amounts to replace P by ∆ F P and we can use the same argument.
Using example 3.8 we obtain the following proposition as a corollary.
Proposition 4.2. When λ goes to ∞ we have
[S(−|λ|)] µν = O(|λ| −∞ ) if µ = ν, [S(−|λ|)] νν = − Γ(1−|ν|) 2 2|ν| Γ(1+|ν|) · |λ| |ν| + O(|λ| −∞ ), if ν = 0, [S(−|λ|)] 00 = 1 2 ln(|λ|) − (ln(2) − γ) + O(|λ| −∞ ). Remark 4.3.
It would be interesting to study the asymptotic behavior of S(±i|λ|). It is then expected to see contributions of periodic diffractive orbits (compare with [16]).
4.2.
Explicit formulas for S(0). In this subsection we will show that for ν = 0 the coefficient S µν (λ) is continuous at λ = 0 and may be expressed using standard objects of the Riemannian surface X.
Recall that, in the distinguished local parameter ζ near some conical point P the asymptotic expansion was given in (3.4). It follows that we have
F ν (ζ, ζ) ∼ C ν ζ −k k > 0, F ν (ζ, ζ) ∼ C ν ζ k k < 0,
where, as usual ν and k are related by the relation ν = 2π θp · k. We first prove the following lemma.
Lemma 4.4. If ν = 0 then G ν (·; λ) is continuous at λ = 0 and G ν (·; 0) is a harmonic L 2 function on X such that
G ν (ζ, ζ; 0) = ζ −k + O(1) k > 0, G ν (ζ, ζ; 0) = ζ k + O(1) k < 0.
Proof. Recall that we have set G ν (·; λ) = F ν + g ν (·; λ), where g ν (·; λ) is the unique solution to
(∆ F − λ) g ν (·; λ) = − (∆ * − λ) F ν .
Since X (∆ * − λ) F ν dx = 0 the continuity at 0 follows from the fact that the ker(∆ F ) consists only in the constant function. By continuity we obtain G ν (· ; 0) is a solution to ∆ * G ν (· ; 0) = 0 and, therefore, G ν (· ; 0) is harmonic on X 0 .
Remark 4.5. Let ζ be denoting the distinguished local parameter near a fixed p ∈ P . The problems
(4.2) ∆U k = 0 on X \ P U k ∼ ζ −k + O(1), as ζ → 0 for 0 < k < θp 2π and (4.3) ∆U k = 0 on X \ P U k ∼ ζ k + O(1), as ζ → 0
for − θp 2π < k < 0 have solutions only up to an additive constant. On the other hand, the problem
∆u = 0 on X \ P u ∼ log r + O(1), as ζ → 0
doesn't have a solution. Thus the behaviour of the coefficients S 0ν (λ) and S µ0 (λ) may not even be properly defined for λ = 0. When writing S(0) we will implicitly assume that only the coefficients S µν with nonzero µ and ν are considered (see also remark 5.5).
In the next subsection we construct solutions to the problems (4.2, 4.3) since they give the functions G ν (· ; 0) from which the coefficients S µν can be computed (for nonzero µ and ν).
4.3.
Special solutions and an explicit expression for S(0). Choose a canonical basis of cycles, {a α , b α } on the Riemann surface X and let {v α } α=1,...,g be the corresponding basis of holomorphic normalized differentials. Let B be the matrix of b-periods of X.
We have the following proposition.
Proposition 4.6. Fixing P a conical point and k ∈ N, there exist Ω k and Σ k such that (1) Ω k and Σ k are meromorphic differentials of the second kind on X with only one pole of order k + 1 at P. (2) In the distinguished local parameter near P , they satisfy
(4.4) Ω k (ζ) = − k ζ k+1 dζ + O(1) Σ k (ζ) = − ik ζ k+1 dζ + O(1). (3)
All the a and b-periods of Ω k (P, ·) and Σ k (P, ·) are purely imaginary.
Proof. Let ω(·, ·) be the canonical meromorphic bidifferential on the Riemann surface X (see [9], p. 3), for which the following asymptotic expansion holds
ω(ζ(Q 1 ), ζ(Q 2 )) dζ(Q 1 )dζ(Q 2 ) = 1 (ζ(Q 1 ) − ζ(Q 2 )) 2 + 1 6 S B (ζ(Q 2 )) + o(1) as Q 1 → Q 2 ,
where S B is the Bergman projective connection. Moreover, ω is normalized in such a way that
(4.5) aα ω(·, ζ) dζ ζ=0 = 0 bα ω(·, ζ) dζ ζ=0 = 2πi v α (ζ) dζ ζ=0 ,
for α = 1, · · · , g. Let (c α ) α=1···g be coefficients to be chosen later and consider the meromorphic differential
(4.6) − ω(·, ζ) dζ ζ=0 + g α=1 c α v α .
We want to choose c α in (4.6) so that all the a-and b-periods of this differential are purely imaginary. The vanishing of the real parts of all a-periods implies that all the constants c α are purely imaginary. The vanishing of the real part of the period over the cycle b β then gives :
Re b β c α v α = Re b β ω(·, ζ) dζ ζ=0 .
Using the fact that the c α are known to be purely imaginary and the normalization of ω recalled in (4.5) we obtain the following system of equations :
(4.7) g α=1 [Im B] βα c α = 2πiIm v β dζ ζ=0
Since Im (B) is invertible, this uniquely determines c α . In order to get Σ 1 we apply the same method searching coefficients c α so that the meromorphic differential
−i ω(·, ζ) dζ + g α=1 c α v α
has purely imaginary periods. The system of equations we obtain is similar to (4.7) except that
Im v β dζ ζ=0 is replaced by Re v β dζ ζ=0
. It still has a solution using the same invertibility of Im (B).
To get Ω k and Σ k with an arbitrary k ≥ 1 we repeat the same construction taking the first term in (4.6) to be
(−1) k (k − 1)! d dζ k−1 ω(·, ζ) dζ ζ=0 .
We will obtain an equation similar to (4.7) so that eventually, the existence result thus follows from the existence of ω and the fact that the matrix Im (B) is invertible.
This proposition gives the following corollary.
Corollary 4.7.
Let Ω k and Σ k be defined by the preceding proposition, then the following formula defines a function f k which is harmonic in X \ {P } :
(4.8) f k (Q) = Re Q P 0 Ω k − iRe Q P 0 Σ k .
Moreover, in the distinguished local parameter near P, f k admits the following asymptotic behavior :
f k (ζ) = 1 ζ k + O(1).
Proof. Since all the a− and b− periods of Ω and Σ are purely imaginary, f k is indeed well-defined on X. The remaining statements follow from the construction.
By considering C ν f k or C ν f k we obtain the functions G ν (·; 0) up to an additive constant. This additive constant is harmless when computing the matrix elements S µν (0).
Examples.
(1) A conical point of angle 2π < β ≤ 4π on a Euclidean surface of genus ≥ 1. In this case one has n = 1. The proposition 4.6 combined with the asymptotics of ω yield ζ P 0
Ω 1 (P, ·) = 1 ζ + c 0 + − 1 6 S B (ζ) ζ=0 + 2πi g α=1,β=1 ((Im B) −1 ) αβ Im v β (ζ) dζ ζ=0 v α (ζ) dζ ζ=0 ζ + O(ζ 2 ) (4.9)
with some constant c 0 , and ζ P 0
Σ 1 (P, ·) = i ζ + d 0 + − i 6 S B (ζ) ζ=0 + 2πi g α=1,β=1 ((Im B) −1 ) αβ Re v β (ζ) dζ ζ=0 v α (ζ) dζ ζ=0 ζ + O(ζ 2 ) (4.10)
with some constant d 0 .
Denoting the expressions in square brackets in (4.9) and (4.10) by A and B respectively, one gets the asymptotics
f 1 (ζ, ζ) = 1 ζ + const + A − iB 2 ζ + A − iB 2 ζ + O(|ζ| 2 )
and, therefore,
(4.11) S p (0) = * * * * A−iB 2 A−iB 2 * A+iB 2 A+iB 2 ,
where the index p means that we have written down only the coefficients of S(0) that corresponds to indices ν ∈ N p (2) A Euclidean sphere with one 4π singularity and six π singularities. Consider the surface of example 2.1 i.e. the Riemann sphere with metric
|z| 2 |dz| 2 6 k=1 |z − z k | .
We consider the part of the S-matrix with non-zero indices µ and ν. We thus only have to consider the asymptotic behavior near 0 and compute the coefficients S 1 The distinguished local parameter ζ in a vicinity of the conical point z = 0 is given by
ζ(z) = z 0 w dw 6 k=1 (w − z k ) 1/2 .
The special solution f 1 is now not only harmonic but even holomorphic in CP 1 \ 0 and is nothing but the function A/z with some constant A.
One has
A z = 1 ζ + const + S 1 2 1 2 (0)ζ + O(ζ 2 ), Therefore, A = dz dζ ζ=0
and a simple calculation shows that
S 1 2 1 2 (0) = − 1 6 z ′′′ (ζ)z ′ (ζ) − 3 2 (z ′′ (ζ)) 2 (z ′ (ζ) 2 ζ=0 = − 1 6 {z, ζ}| ζ=0 ,
where {z, ζ} is the Schwarzian derivative. One has also S −
Θ µν = [G µ (z; λ)] zz G ν (z; λ)dz + [G µ (z; λ)] z [G ν (z; λ)] z dz ,
Then the variational formulas hold:
(4.12) ∂S µν (λ) ∂A α = 2i bα Θ µν ; α = 1, . . . , g, (4.13) ∂S µν (λ) ∂B α = −2i aα Θ µν ; α = 1, . . . , g, (4.14) ∂S µν (λ) ∂z k = 2i p k Θ µν ; k = 2, . . . , M,
where the integrals in (4.14) are taken over some small contours encircling conical points p k .
Proof. The method of proof follows closely [19]. We will prove only the variational formulas with respect to coordinates A α since the other formulas can be established similarly.
According to [19] (Proposition 2, p. 84) one has (4.15) ∂ Aα R(x, y; λ) = 2i bα R(x, z; λ)R zz (y, z; λ)dz + R z (x, z; λ)R z (y, z; λ) dz .
(Here R(x, y; λ) stands for the resolvent kernel of the Friedrichs extension; one has R zz (x, z; λ) = and Lemma 7 on page 88 of [19] reads as
(4.17) ∂ Aα X Φ(x, x; moduli)dx = X ∂ Aα Φ(x, x, moduli)dx + i 2 bα Φ(x, x, moduli)dx .
The cycle b α does not intersect the support of F ν and the terms F ν and (∆ − λ)F ν are moduli independent, therefore,
∂ Aα G ν (x ; λ) = ∂ Aα (F ν + g ν ) = ∂ Aα g ν (x ; λ).
Using (4.16) and (4.17), we obtain
∂ Aα G ν (x ; λ) = 2i X dy[(∆ − λ)F ν (y)] bα {R(x, z; λ)R zz (y, z; λ)dz + R z (x, z; λ)R z (y, z; λ)dz} = 2i bα R(x, z; λ) X λ 4 R(y, z; λ)(∆ − λ)F ν (y)dy dz + 2i bα R z (x, z; λ) X R z (y, z; λ)(∆ − λ)F ν (y)dy dz = 2i bα R zz (x, z; λ)g ν (z; λ)dz + R z (x, z; λ)[g ν (z; λ)] z dz = 2i bα R zz (x, z; λ)G ν (z; λ)dz + R z (x, z; λ)[G ν (z; λ)] z dz
We finally obtain
∂ Aα g ν (λ, x) = 2i bα R zz (x, z; λ)G ν (z; λ)dz + R z (x, z; λ)[G ν (z; λ)] z dz.
Using proposition 3.13 and equation (3.10) to identify the behavior near the conical points of the different terms we obtain
∂ Aα S µν = 2i bα a + µ (z; λ) zz G ν (z; λ)dz + a + µ (z; λ) z [G ν (z; λ)] z dz.
Using proposition 3.13, this gives the result.
Krein's formula and relative determinants
There are several ways of defining determinants of operators acting on an infinite dimensional space. We recall the following two basic constructions : first a perturbative determinant when the operator is a trace-class perturbation of the identity, and second zeta-regularization which is used in particular for Laplacian-like operators.
Both these approaches can also be used to define relative determinants when comparing two operators H 0 and H 1 in which one is thought to be a perturbation of the other. Krein's formula is a classical tool in this setting and usually applies when the difference f (H 1 )−f (H 0 ) is trace-class for some simple function f . In that case it is possible to define a relative perturbative determinant (see [29]). This approach applies well to the case when H 0 and H 1 are different self-adjoint extensions of a symmetric operator that has finite deficiency indices. Indeed, in that case the difference of the resolvents is a finite-rank operator and, moreover, the perturbative determinant is actually the determinant of a finite dimensional matrix.
We will thus adapt these techniques to our setting. The method is clearly identified in the literature (see [29] and also [5]) and the main task here consists in identifying the perturbative determinant in terms of the boundary condition and the S-matrix.
Once this is done, we will use this determinant to define a zeta-regularization and compare the determinants that are obtained this way.
Remark 5.1. We insist here that we will actually use the perturbative determinant to show that zetaregularization is possible and then to compare the two definitions of determinants. In particular, all the issues that are relative to zeta-regularization may be expressed using the perturbative determinant (when the latter can be defined).
5.1.
Krein's formula and perturbative determinant. One convenient way of parametrizing the self-adjoint extensions of ∆ is by using two matrices P and Q in the following way (see [21]).
We first construct two vectors A ± that collect the coefficients a ± ν . We organize these coefficients so that the first n p 1 entries correspond to the first conical point p 1 then we put the data corresponding to the second conical points and so on.
A lagrangian subspace L in dom(∆ * )/dom(∆) can be parametrized by a system of linear equations of the following form :
P A − + QA + = 0,
where P and Q are square matrices satisfying rank(P, Q) is maximal and P * Q is self-adjoint. We fix two such matrices and denote by ∆ L the corresponding self-adjoint extensions. It is possible to find a basis in which the n × 2n matrix (P Q) has the following block-decomposition ( [21]) :
(5.1) P 2 P 3 Q 1 0 0 P 1 0 0 ,
in which P 1 and Q 1 are invertible and L := Q −1 1 P 2 is self-adjoint. Definition 5.2. We will call an extension ∆ L regular if functions in dom(∆ L ) are not allowed to have logarithmic singularities. Equivalently, ∆ L is regular if and only if for any u ∈ dom(∆ L ), and any conical point p, the coefficient a − p,0 of u vanishes. The following observation (based on the classical Krein formula) is the key technical result of the present paper. Proposition 5.3. For any λ ∈ C \ (spec(∆ F ) ∪ spec(∆ L )) the following identity holds :
Tr (∆ L − λ) −1 − (∆ F − λ) −1 = −Tr (P + QS(λ)) −1 QṠ(λ) ,
where the dot indicates derivation with respect to λ.
Proof. Let λ be in the union of the resolvent sets of ∆ F and ∆ L , and let f be in L 2 . We search a matrix X = [x µν ] such that we have the following Krein formula (see, e. g., [4] or [1], Theorem A.3)
(5.2) (∆ L − λ) −1 f = (∆ F − λ) −1 (f ) + µ,ν x µν G µ (· ; λ)Λ ν (∆ F − λ) −1 (f ) .
We denote by u = (∆ F − λ) −1 (f ) and we compute the vectors A ± of the right-hand side
a − µ = ν x µν Λ ν (u), a + µ ′ = Λ µ ′ (u) + µ,ν x µν [S(λ)] µ ′ µ Λ ν (u).
Denoting by Λ the vector Λ ν (u) we thus have
A − = XΛ, A + = (I + S(λ)X)Λ.
Plugging into the self-adjoint condition we obtain that the following relation is satisfied.
[P X + Q(I + S(λ)X)] · Λ = 0.
Using the block decomposition (5.1), we see that
P + QS(λ) = P 2 + Q 1 S(λ) * 0 P 1
Since λ is in both resolvent sets, Λ is arbitrary and the preceding system always has a solution. We obtain that (P 2 + Q 1 S(λ)) must be invertible and hence P + QS(λ). Finally, we obtain
X = −(P + QS(λ)) −1 Q.
Denoting by Π µν (λ) the (rank one) operator defined from H 2 into L 2 by
Π µν (λ)(u) = G µ (· ; λ)Λ ν (u), equation (5.
2) may be rewritten :
(∆ L − λ) −1 − (∆ F − λ) −1 = µ,ν x µν Π µν (λ) • (∆ F − λ) −1
Observe that the right-hand side is finite rank so that we can trace this equation and obtain
Tr (∆ L − λ) −1 − (∆ F − λ) −1 ) = µ,ν x µν Tr Π µν (λ) • (∆ F − λ) −1 .
Using lemma 5.4 below and lemma 3.12 we obtain
Tr (∆ L − λ) −1 − (∆ F − λ) −1 ) = µ,ν x µν G µ (· ; λ), G ν (· ; λ) = µν x µν [Ṡ(λ)] νµ = −Tr (P + QS(λ)) −1 QṠ(λ) .
It remains to prove the following lemma.
Lemma 5.4. The trace of the rank one operator Π µν (λ) • (∆ F − λ) −1 is given by
Tr Π µν (λ) • (∆ F − λ) −1 = G µ (· ; λ), G ν (· ; λ)
Proof. Let e n be an orthonormal basis of L 2 then Π µν (λ) • (∆ F − λ) −1 e n , e n = G µ (· ; λ), e n · Λ ν (∆ F − λ) −1 e n = G µ (· ; λ), e n · e n , G ν (· ; λ) Summing over n and using Parseval's identity gives the lemma.
We may now define D on the union of the resolvent sets of ∆ F and ∆ L by (5.3) D(λ) = det (P + QS(λ)) .
Remark 5.5. When the extension is regular the matrix P + QS(λ) doesn't involve the coefficients S µν whenever µ or ν is 0 (because these are multiplied by a zero entry of Q). Hence the matrix P + QS(0) makes perfect sense and can be computed using the results of section 4.2.
The preceding proposition gives
(5.4) Tr (∆ L − λ) −1 − (∆ F − λ) −1 = − D ′ (λ) D(λ)
This implies that D ′ D extends to a meromorphic function with poles that correspond to eigenvalues of ∆ L and ∆ F and with residues dim(ker(∆ L − λ)) − dim(ker(∆ F − λ)).
Since D ′ D is the logarithmic derivative of D, it is convenient to give a name to ln(D). We thus denote by Ω ⊂ C the set obtained by removing a downward vertical cut starting at each eigenvalue of ∆ F and ∆ L i.e.
Ω = C \ {λ − it, λ ∈ spec(∆ F ) ∪ spec(∆ L ), t ∈ (−∞, 0] } ,
and, on Ω, we define the functionξ byξ(λ) := − 1 2iπ ln (det(P + QS(λ))) . Observe that on Ω we have, by definition,
(5.5) D(λ) = exp −2iπξ(λ) .
The functionξ is intimately related to the spectral shift function ξ (see [29,13]). Although the latter is usually used in settings with continuous spectrum, it is possible to define it even when H 0 and H 1 have pure point spectrum. In the latter case, it follows from the definitions that ξ is the step-function :
ξ(t) := N 1 (t) − N 0 (t) where N i is the counting function associated with H i .
It follows from our definition ofξ that the function ξ defined on R by
ξ(t) := − 1 2π Arg(D(t)) = Reξ(t)
is a step function with jumps located at the eigenvalues of ∆ F and ∆ L . Moreover the jumps are exactly the differences dim(ker(∆ L −λ))−dim(ker(∆ F −λ)). We thus obtain the spectral shift function of ∆ F and ∆ L . (compare with [29] Thm 1 p. 272).
In our setting Birman-Krein formula would be (5.5) (compare with [29] p.272) and would follow, in our case, from our definitions. In the next subsection we will prove that, using D, one may define a determinant of ∆ L via zeta-regularization and then establish the relation :
(5.6) det ζ (∆ L − λ) = C 0 · D(λ)det ζ (∆ F − λ),
in which C 0 is some constant that we will also determine. In particular, we will now prove that D allows us to recover the 'exotic' features of the zeta function associated with ∆ L . This unusual behavior has been extensively studied in [17] in a setting very close to ours and in [12] in greater generality. Our main contribution here is the interpretation of D using S-matrix that, in some sense, gives a geometrical interpretation to the 'secular equation' method of [17].
Comparing determinants.
The procedure here is not as straightforward as usual because of unusual behavior of the zeta function near s = 0. In particular, ζ(s, ∆ L ) will admit a analytic continuation that is regular at 0 only if L is regular (though with possible unusual poles). This unusual behavior as we just mentioned has been extensively studied in literature (see [12,17,22]); from our point of view, it is linked with the asymptotic behavior of D(λ) for large negative λ. We thus begin by deriving this asymptotic expansion. 5.2.1. D(λ) for large negative λ. The analysis that follows is closely related to the one performed in [17]. This is not surprising since the asymptotic regime λ goes to −∞ is local. In particular, the function D(−|λ|) := det(P + QS(−|λ|)) on a cone has to be compared to the function F (i |λ|) in [17].
We first use prop. 4.2 and consider all possible sums of the exponents ν i that appear in this proposition. This gives us a collection of numbers that we order and denote by α 0 > α 1 > · · · > α k > · · · . Expanding now the determinant, and ordering the terms, we get
D(−|λ|) = f inite a kl |λ| α k (ln |λ|) l + O(|λ| −∞ ).
By definition, there are no logarithm in the expansion corresponding to a regular self-adjoint extension, therefore, in that case, the expansion reads :
D(−|λ|) = f inite a k |λ| α k + O(|λ| −∞ ).
We set l k the largest integer l such that |λ| α k (ln |λ|) l appears in the expansion and we set β k = α 0 −α k we have
D(−|λ|) = a 0l 0 |λ| α 0 (ln |λ|) l 0 1 + l≥1 a 0l (ln |λ|) −l + β k >0 l 0 −l k a kl |λ| −β k (ln |λ|) l + O(|λ| −∞ ) We denote by F (λ) = 1 + l≥1 a 0l (ln |λ|) −l + β k >0 l k −l k a kl |λ| −β k (ln |λ|) l + O(|λ| −∞ )
Taking the logarithmic derivative, we obtain
− D ′ (−|λ|) D(−|λ|) = 2iπξ ′ (−|λ|) = α 0 |λ| + l 0 |λ| ln |λ| + F ′ (λ) F (λ) .
By inspection we find
F ′ (λ) F (λ) = O |λ| −β 1 −1 regular case O |λ| −1 (ln |λ|) −2 otherwise.
Lemma 5.6.
(1) In the regular case, there exist three positive numbers α 0 , β 1 and M such that the estimate
(5.7) 2iπξ ′ (−|λ|) − α 0 |λ| ≤ M |λ| −β 1 −1 ,
holds for λ large enough.
(2) In the other cases, there exist two positive real numbers α 0 and β 1 , a non-negative integer l 0 and a constant M such that the estimate
(5.8) 2iπξ ′ (−|λ|) − α 0 |λ| − l 0 |λ| ln |λ| ≤ M · |λ| −1 (ln |λ|) −2
holds for |λ| large enough.
In the regular case, for any C > 0, define h C (s) for Re(s) large enough by
(5.9) h C (s) = 2iπ ∞ C λ −sξ′ (−|λ|) dλ − α 0 s exp(−s ln(C)).
The estimates of the previous lemma imply the following corollary. We restrict to the regular case although similar statements are valid in the non-regular case (with extra logarithmic singularities -see [17]).
Proposition 5.7. For a regular extension, the function h C extends to a holomorphic function on {Re(s) ≥ −β 1 } .
Proof. We have ∞ C λ −s 2iπξ ′ (−|λ|)dλ = ∞ C λ −s 2iπξ ′ (−|λ|) − α 0 λ dλ + ∞ C λ −s α 0 λ dλ.
The second integral on the right-hand side is computed directly :
∞ C λ −s α 0 λ dλ = α 0 s exp(−s ln C),
so that h C actually represents the first integral. Lemma 5.6 then gives that h C extends to a holomorphic function on Res > −β 1 .
Zeta-regularization.
For any A and any C that is large enough, for anyλ ∈ Ω such that Re(λ) > A we choose a cut cλ ⊂ Ω that starts from −∞ + i0 and that ends atλ. We may choose it in such a way that it begins with the interval (−∞, −C]. For anyλ and any s ∈ C, the function λ → (λ −λ) −s , which is well defined when λ −λ is a positive real number, extends to a holomorphic function on the complement of the cut cλ. Moreover, when λ goes to the cut cλ from above or from below, we have the following jump condition For λ on cλ, we define (λ −λ) −s 0 to be this common limit. Let A + be any number greater than A that is neither an eigenvalue of ∆ F nor of ∆ L . Define a contour γ that avoids cλ and that consists in one part that encloses the half-line {x ≥ A + } and then small circles that enclose the eigenvalues of ∆ L and ∆ F that are smaller than A + .
For Re(s) > 1 we have
ζ(s, ∆ L −λ) = 1 2iπ Tr γ (λ −λ) −s (∆ L − λ) −1 dλ , = 1 2iπ
Tr
cλ ,ε (λ −λ) −s (∆ L − λ) −1 dλ ,
in which cλ ,ε denotes the contour obtained by following cλ at a distance ε. The second identity comes from Cauchy integral formula since, when Re(s) > 1 the contribution of a large circle centered atλ tends to zero when the radius grows to infinity. The same formulas are true for ∆ F and using the fact that (∆ L − λ) −s and (∆ F − λ) −s are trace class for Res > 1, we can exchange the contour integration and the trace operation to obtain
ζ(s, ∆ L −λ) − ζ(s, ∆ F −λ) = 1 2iπ cλ ,ε (λ −λ) −s Tr (∆ L − λ) −1 − (∆ F − λ) −1 dλ
Using prop. 5.3 and the definition ofξ we obtain
ζ(s, ∆ L −λ) − ζ(s, ∆ F −λ) = c λ,ε (λ −λ) −sξ′ (λ)dλ.
We rewrite the right-hand side in the following form : ζ 1 (s) + ζ 2 (s) where ζ 1 corresponds to the part of the contour c λ,ε that is in the half-plane {Reλ ≤ −C} , and ζ 2 is the remaining part of that contour.
The function ζ 2 extends to an entire function of s and for Re(s) < 1 we may let ε go to 0, giving ∀s, Re(s) < 1, ζ 2 (s) = 2i sin(πs)
λ −C (λ −λ) −s 0ξ ′ (λ)dλ,
where the integral is along the part of the cut cλ that belongs to the half-plane {Re(λ) > −C} .
For ζ 1 we may first let ε go to 0 and obtain :
ζ 1 (s) = 2i sin(πs) −C −∞ (λ −λ) −s 0ξ ′ (λ)dλ.
We make a further reduction by using the following technical lemma.
Lemma 5.8. On C × {|z| < 1}, we define ρ(s, z) = (1 − z) −s − 1.
For any r < 1, and any R > 0, the following holds for any |z| ≤ r, and any |s| ≤ R
(5.10) |ρ(s, z)| ≤ exp Rr 1−r 1 − r · |s| · |z|.
Proof. We start from
ρ(s, z) = k≥1 (−s) k [ln(1 − z)] k k! .
By integration we have | ln(1 − z)| ≤ 1 1−r |z| so that
|ρ(s, z)| ≤ exp |s||z| 1 − r − 1 = |s||z| 1−r 0 exp(v) dv.
The claim then follows.
For Re(λ) ≤ −C, there exists some r < 1 such that λ λ ≤ r. We can thus write
(λ −λ) −s = λ −s 1 + ρ s,λ λ
for any λ such that Re(λ) ≤ −C and λ / ∈ (−∞, −C). Fix some R, For s such that Re(s) > 0 and |s| ≤ R, using the bound in Lemma 5.8 we may let ε go to zero and write Using Lemma 5.8 and Lemma 5.6 we find that, for any extension (regular or not)R C (·, λ) can be analytically continued to Re(s) > −1, and thatR C (0) = 0.
Adding up ζ 1 and ζ 2 we obtain the following proposition.
Proposition 5.9. For any extension, the function R C (s,λ) which is defined for s large by
R C (s,λ) = ζ(s, ∆ L −λ) − ζ(s, ∆ F −λ) − 2i sin(πs) −C −∞ |λ| −sξ′ (λ) dλ − ζ 2 (s)
can be analytically continued to Re(s) > −1 and R C (s,λ) vanishes at least at second order at s = 0.
Proof. By inspection and using the definitions of the different functions that appear in the expression of R C we find that R C (s,λ) = 2i sin(πs)R C (s,λ).
Using the bounds given by Lemmas 5.6 and 5.8 we find a constantC such that
∀λ < −C, |λ| −sξ′ (λ)ρ s,λ λ ≤C|s| · |λ| −Re(s)−2 ,
whereC depends on C,λ and is uniform for |s| ≤ R. The claim follows
In particular, in the regular case, we obtain the following corollary (compare with [23]) Proof. The zeta regularization of the Friedrichs extension is well-known and well studied starting from the small-time asymptotics of the heat kernel (obtained for instance from [7]). The function (s − 1)ζ(∆ F −λ) is thus known to extends holomorphically to C (see [2,3,17,19]). Moreover the preceding proposition yields that (s−1)ζ(s, ∆ L −λ) = (s−1)· ζ(s, ∆ F −λ) + sin(πs) π h C (s) + α 0 s exp(−s ln C) + ζ 2 (s) + R C (s,λ) .
The statement thus follows by examining the analytic continuation of each individual term.
Remark 5.11. By evaluating everything at s = 0 we obtain
ζ(0, ∆ L −λ) = ζ(0, ∆ F −λ) + α 0 .
In the regular case, we can thus define the regularized zeta determinant by the usual formula det ζ (∆ L −λ) = exp −ζ ′ (0, ∆ L −λ) , and we obtain the following theorem. Then, for anyλ ∈ Ω we have (5.12) det ζ (∆ L −λ) = e −Γ · D(λ) · det ζ (∆ F −λ).
Proof. According to the preceding proposition, we have ζ ′ (0, ∆ L −λ) − ζ ′ (0, ∆ F −λ) = ζ ′ 2 (0) + h C (0) − α 0 ln(C). We compute ζ ′ 2 (0) = 2iπ ξ (λ) −ξ(−C) . Combining the two we find ζ ′ (0, ∆ L −λ) − ζ ′ (0, ∆ F −λ) = 2iπξ(λ) − 2iπξ(−C) + h C (0) − α 0 ln(C) = 2iπξ(λ) + ln (D(−C)) − α 0 ln(C) + h C (0)
This implies the result with Γ replaced by the quantity ln(D(−C)) − α 0 ln(C) + h C (0) (which proves in particular that the latter doesn't depend on C large enough). When we let C go to infinity, on the one hand ln(D(−C)) − α 0 ln(C) goes to Γ, and on the other hand, since
h C (0) = ∞ C 2iπξ ′ (−|λ|) − α 0 λ dλ
and the function inside the integral is L 1 , h C (0) goes to 0. This finishes proving the theorem.
Remark 5.12. As soon as h C allows the definition of the relative zeta determinant of ∆ L −λ and ∆ F −λ, then, using theorem 2 and differentiating with respect to λ, we recover a well-known fact of this theory :
∂λ ln det(∆ L −λ) − ln det(∆ F −λ) = 2iπξ ′ (λ).
(compare with [10, 15, 6])
Remark 5.13. For non-regular extensions, it is still possible to analytically continue ζ to Res > 0 and to define a zeta-regularized determinant by picking some coefficient in the asymptotic expansion of ζ(s, ∆ L −λ) near 0 (see [17]). Note however, that the limitλ → 0 will be problematic.
5.4.
Proof of theorem 1. In order to get the theorem of the introduction, we now letλ go to 0. We thus modify the zeta-regularized determinant in order to exclude the eigenvalue 0. Define by δ L (resp. δ F ) the dimensions of ker(∆ L ) (resp. ker(∆ F )). Equation ( Using this definition for ∆ L and ∆ F , and plugging into (5.12), the powers of −λ cancel out and we may letλ go to zero. We thus obtain the theorem in the introduction (Thm. 1).
When d = 0, the prefactor D * (0) may be computed using the method of section 4.3. When d > 0 then this method has to be refined to compute more terms in the Taylor expansion of S(λ) at λ = 0. In the following example, we will pay special attention to addressing the question of the kernel of P + QS(0).
5.5.
On the Euclidean sphere with one 4π and six π singularities. We consider the Euclidean sphere with six π singularities and one 4π conical point. We define
A ± = a ± 0 a ± − 1 2 a ± + 1 2 A ±
where a ± i , i = − 1 2 , 0, 1 2 correspond to the 4π singularity andà ± are the coefficients corresponding to the remaining six π singularities. Recall that for each of the latter there are only two coefficients a ± 0 . A regular extension thus relates only the coefficients a ± ± 1 2 We define P θ and Q θ by This choice defines a regular self-adjoint extension (which is, moreover invariant under complex conjugation). We have D(λ) = det(P + QS(λ)) = det(cos θI 2 + sin θS(λ)), whereS is the 2 × 2 matrix obtained from S by erasing the first row and column (that correspond to a ± 0 ) and all the rows and columns corresponding toà ± According to proposition 4.2, when θ = 0, π, the asymptotic expansion of D is given by since |ν| = 1 2 . Finally, we obtain that, for any θ = 0 , π such that −cotan(θ) isn't an eigenvalue ofS(0) the following holds : det * ζ (∆ L ) = det(cos θI 2 + sin θS(0)) sin 2 θ · det * ζ (∆ F ).
Lemma 3. 1 .s 2 F
12The lagrangian subspace in dom(∆ * )/dom(∆) that corresponds to the Friedrichs extension isa − ν = 0 .Definition 2. We denote by H s := dom(∆ ) the scale of Sobolev spaces associated with it. In particular we set dom(∆ F ) := H 2 .
very symmetric case where the z k form a regular hexagon, the computation yields that z = c · ζ(1 + O(ζ 6 )) so that S . S-matrix as a function on the moduli space of holomorphic differentials: variational formulas. Let (X, ω) ∈ H g (k 1 , . . . , k M ) and let S(λ) be the S-matrix corresponding to a conical point of the translation surface (X, |ω| 2 ) (i. e. one of the zeros of the holomorphic one-form ω). Here we derive the variational formulas for S(λ) with respect to Kontsevich-Zorich homological coordinates on H g (k 1 , . . . , k M ).
Proposition 4 . 8 .
48Let z(p) = p ω. Introduce the following (closed) (1-1)-form on X 0 :
λ 4 R
4(x, z; λ). ) On the other hand, by definition of g ν we have (4.16) g ν (x ; λ) = − X [R(x, y ; λ)(∆ − λ)F ν (y)]dy;
lim λ↓cλ exp(−iπs)(λ −λ) −s = lim λ↑cλ exp(iπs)(λ −λ) −s .
|λ| −sξ′ (λ) dλ + 2i sin(πs)R C (s,λ) whereR C (s,λ) = −C −∞ |λ| −sξ′ (λ)ρ s,λ λ dλ.
Corollary 5 . 10 .
510If L defines a regular extension then (s − 1)ζ(s, ∆ L −λ) extends to a holomorphic function on Re(s) > −β 1 .
Theorem 2 .
2Let L define a regular extension and set Γ to be (5.11) Γ = lim λ→∞ ln (D(−|λ|)) − α 0 ln(−|λ|).
5.4) implies that 0 is a pole of D ′ D with residue d := δ L − δ F so that we can define D * (0)D * (0) := lim λ→0 D(λ)(−λ) −(δ L −δ F ) .On the other hand, we define the modified zeta function byζ * (s, ∆ F −λ) = ζ(s, ∆ F −λ) − δ F (−λ) −s and the corresponding modified determinant.Definition 5.14. Let L be defining a regular extension (or L = F ), the modified zeta determinant of ∆ L is defined by det * ζ (∆ L ) = lim λ→0 (−λ) −δ L det ζ (∆ L −λ)
(|λ|) + ln [sin θ] 2 + O(1),
Acknowledgements. The research of LH was partly supported by the ANR programs NONaa and Teichmüller. The research of AK was supported by NSERC. AK thanks Hausdorff Research Institute for Mathematics (Bonn) and Laboratoire de Mathématiques Jean Leray (Nantes) for hospitality. AK also thanks the MATPYL program for supporting his coming and stay in Nantes where this research began.
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E-mail address: [email protected]. E-mail address: [email protected]
. France E-mail address: [email protected] Department of Mathematics and Statistics. 92208UMR CNRS 6629-Université de Nantes, 2 rue de la Houssinière ; Concordia University, 1455 de Maisonneuve BlvdNantes CedexUMR CNRS 6629-Université de Nantes, 2 rue de la Houssinière,, BP 92 208, F-44 322 Nantes Cedex 3, France E-mail address: [email protected] Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8 Canada
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[
"Application of the Theory of Linear Singular Integral Equations to a Riemann Hilbert Problem for a New Expression of Chandrasekhar's H- function in Radiative Transfer",
"Application of the Theory of Linear Singular Integral Equations to a Riemann Hilbert Problem for a New Expression of Chandrasekhar's H- function in Radiative Transfer"
] |
[
"Rabindra Nath Das :[email protected] \nVisiting Faculty Member\nDepartment of Mathematics\nHeritage Institute of Technology\nChowbaga Road, Anandapur, P.O. East Kolkata Township, Kolkata-700 107, West BengalIndia\n"
] |
[
"Visiting Faculty Member\nDepartment of Mathematics\nHeritage Institute of Technology\nChowbaga Road, Anandapur, P.O. East Kolkata Township, Kolkata-700 107, West BengalIndia"
] |
[] |
The linear non homogeneous singular integral equation( LNSIE) derived from the non-linear non homogeneous integral equation (NNIE) of Chandrasekhar's H -functions is considered here to develop a new form of Hfunctions .The Plemelj's formulae are applied to that equation to determine a new linear non homogeneous integral equation (LNIE) for H-functions in complex plane. The analytic properties of this new linear integral equation are assessed and compared with the known linear integral equations satisfied by H-functions. The Cauchy integral formulae in complex plane are used to obtain this form of Hfunctions not dependent on H-function in the integral. This new form of Hfunction is represented as a simple integral in terms of known functions both for conservative and non-conservative cases. This is identical with the form of Hfunctions derived by this author by application of Wiener -Hopf technique to LNIE . The equivalence of application of the theory of linear singular integral equation in Riemann Hilbert Problem and of the technique of Wiener-Hopf in linear integral equation in representing the H -functions is therefore established . This new form may be used for solving the problems of radiative transfer in anisotropic and non coherent scattering using the method Laplace transform and Wiener-Hopf technique.
| null |
[
"https://arxiv.org/pdf/astro-ph/0701457v1.pdf"
] | 117,066,746 |
astro-ph/0701457
|
6f707c3e65dbb4be22aff23680bb9054338532d5
|
Application of the Theory of Linear Singular Integral Equations to a Riemann Hilbert Problem for a New Expression of Chandrasekhar's H- function in Radiative Transfer
Rabindra Nath Das :[email protected]
Visiting Faculty Member
Department of Mathematics
Heritage Institute of Technology
Chowbaga Road, Anandapur, P.O. East Kolkata Township, Kolkata-700 107, West BengalIndia
Application of the Theory of Linear Singular Integral Equations to a Riemann Hilbert Problem for a New Expression of Chandrasekhar's H- function in Radiative Transfer
1 1Radiative transferSingular integral equationRiemann Hilbert ProblemCauchy Integral formulae *Permanent address: KB-9Flat -7Sector IIISaltlake City (Bidhan nagar)Kolkata-700098West BengalIndia
The linear non homogeneous singular integral equation( LNSIE) derived from the non-linear non homogeneous integral equation (NNIE) of Chandrasekhar's H -functions is considered here to develop a new form of Hfunctions .The Plemelj's formulae are applied to that equation to determine a new linear non homogeneous integral equation (LNIE) for H-functions in complex plane. The analytic properties of this new linear integral equation are assessed and compared with the known linear integral equations satisfied by H-functions. The Cauchy integral formulae in complex plane are used to obtain this form of Hfunctions not dependent on H-function in the integral. This new form of Hfunction is represented as a simple integral in terms of known functions both for conservative and non-conservative cases. This is identical with the form of Hfunctions derived by this author by application of Wiener -Hopf technique to LNIE . The equivalence of application of the theory of linear singular integral equation in Riemann Hilbert Problem and of the technique of Wiener-Hopf in linear integral equation in representing the H -functions is therefore established . This new form may be used for solving the problems of radiative transfer in anisotropic and non coherent scattering using the method Laplace transform and Wiener-Hopf technique.
. Introduction :
The H-functions, extensively developed by Chandrasekhar [1] play an important role in the study of radiative transfer in homogeneous plane parallel scattering atmosphere of finite and infinite thickness .. The detailed application of the mathematical theory to the equations of radiative transfer or neutron transport depends , however upon the availability of the H-function in sufficient closed form. This requirement has been partly met by Chandrasekhar [1] . Mathematical framework and the numerical evaluation of these functions have been made by him extensively by his own method. There is still a need for further detailed theoretical and numerical work on these functions , particularly to have a closed form by application of some other methods . Accordingly , we have re -examined the question of mathematical formulation of a new form of the H-functions from a different angle. The object of this is to providing current and future requirements of numerical value of these functions in tabular form for the case of isotropic scattering for any value of particle albedo both in conservative and non conservative cases of physical interest . The work presented in this paper is an extensive formulation of mathematical theory to build up a new form of the H-functions for atmosphere with coherent scattering.
The mathematical form of H functions presented by different authors Chandrasekhar [1] , Kourganoff [2] , Busbridge [3] , Fox [4] , Dasgupta [5], [6] , Das [7] , by different methods are stated to be exact for numerical integration with some suitable constraints.
We believe that the forms of H-functions derived so far, as a solution of NNIE or LNIE for numerical computation are in fact dependent on these H-function itself within the integral. However ,we believe that those are worthwhile to frame a basis for a new functional representation of H -functions as a solution of the LNIE of H-functions .
The forms of H-functions derived by Fox [4], Zelazny [8] , Mullikin[9] , Zweifel [10],Ivanov [11] ,Siewert [12], Das [13] are considered to be important from theoretical and numerical point of view . Those solutions are not dependent on the H-function itself.
The forms of H -functions derived by Siewert [14] , Garcia and Siewert [15],Sulties and Hill [16] , Barichello and Siewert [17 ,18] , Bergeat and Rutily [ 19,20] in terms of new functions from different standpoints are still considered to be useful for numerical and theoretical point of view.
Further more we believe that the result obtained by this method of application of the theory of singular integral equation in this work to a LNSIE of H functions for solving a Riemann Hilbert problem is a new and sufficiently different from others in the literature of H-functions to warrant its communication .
Consequently we hope that this functional representation of H -functions extends the result of Fox [4] for better understanding of the method used by him . We hope that these results identify the equivalence of the application of Wiener-Hopf technique to LNIE of H-functions Das [13] and the application of theory of linear singular integral equation to LNSIE of H-functions so as to have the same form of H -function by two different approaches . The existence and uniqueness of those functions are achieved already in Das [13]. It is therefore shown that it is now safe to handle the LNSIE or LNIE of H-functions with due respect to the concluding comment of Busbridge [21].
Mathematical analysis :
The integro-differential equations of radiative transfer have been solved by different authors to obtain the intensity of radiation at any optical depth. and at the boundary. They finally derived solutions in terms of Chandrasekhar [1] Hfunctions . The H -functions satisfy NNIE as
H(x)= 1 + x H(x) 1 0 U(u) H(u) d u / (u + x) , 0≤ x ≤1 ,(1)
where U(u) is a known characteristic function . In physical context to solve equation (1), certain restrictions on U(u) are necessary :
i) U 0 = 1 0 U (u) d u = ½ (2)
where U 0 = ½ refers to conservative case and U 0 < ½ refers to non conservative case ;
ii) U(u) is continuous in the interval (0,1) and satisfy Holder conditions , viz . ,
I U(u 2 ) -U(u 1 ) I < K I u 2 -u 1 I α(3 )
where u 2 , u 1 lie in the interval (0,1) ( u 1 is in the neighbourhood of u 2 ) , where K and α are constants and 0 < α ≤ 1 .
If z = x +iy , a complex variable in the complex plane cut along (-1,1) then Chandrasekhar [1] proved that ( H (z) H (-z) ) -1 = 1 -2 z 2 1 0 U(u) d u / ( z 2 -u 2 ) = T(z) , (4) and he determined H -function to satisfy the integral solution as
i ∞ log H(z) = ( 2πi) -1 ∫ z log ( T(w) ) d w / ( w 2 -z 2 ) (5) -i ∞
where T(z) has the properties :
i)T( z) is analytic in the complex z plane cut along (-1,1); ii) it is an even function of z ;iii) it has two logarithmic branch points at -1 and +1 ; iv) T(z) 1 as 1 z 1 0 , v) it has only two simple zeros at infinity when U 0 = ½ ; vi) only two real simple zeros at z = +1/k and z = -1/k where k is real , 0< k ≤ 1 when U 0 < ½ ;vii) T(z) -C/ z 2 as z ∞ when U 0 = ½ , C = 2 U 2 , a real positive constant ;viii)T (z) D = 1-2 U 0 as z ∞ when U 0 < 1/2 , D is a real positive constant ; ix) it is bounded on the entire imaginary x)T ( ∞) = D = 1-2 U 0 when U 0 <1/2
= ( 1 - 1 0 H(x) U (x) dx ) 2 = 1/ ( H ( ∞ ) ) 2 where U r = 1 0 u r U(u) d u, r = 1,2,3….
5 where P represents the Cauchy Principal value of the integral and .
T 0 (t) = 1 -t 1 0 U (x) d x / (x + t) + t 1 0 ( U(x)-U(t)) dx /( x -t)+ t U(t) ln( ( 1-t ) / t ) .(13)
In this paper, we shall consider the LNSI Eq (12) of Hfunctions to solve it by the theory of linear singular integral equation fully described in the standard work of Muskhelishvilli [22] associated with the theory of analytic continuation and Cauchy theory on analytic function in complex plane . We derive a unique solution which is obtained by Das [23] by using Wiener Hopf technique along with theory of analytic continuation and Cauchy theory of analytic function in complex plane . We shall show the equivalence of Riemann Hilbert Theory for solving LNSIE and the Wiener Hopf technique for solving LNIE so far as it relates to determination of a new form for H-functions Let us consider the function
Φ(z)= ψ(t )d t / ( t -z) ,(14)
L where L is some contour in the complex z plane with a cut along (0,1) and z does not lie on L . Here Φ (z) be an analytic function of z in the complex plane cut along (0,1) and be O ( 1/z) when z ∞ . If z is on L , then the integral for Φ(z) is taken to be a Cauchy principal value and the curve L becomes a line of discontinuity for the function Φ(z)., The function Φ(z ) must pass through a discontinuity when z crosses L. Let A and B be the end points of the arc L at z = 0 and z = 1 respectively .
We want to determine the value of Φ (z) as z t 0 from both sides of the contour where t 0 lie on L and t 0 is not an end point of L. We define the region D + as the region to the left of the contour( if we look to the left moving counter clockwise along the contour) and Das the region to the right .
.Let us define Φ + or -(t 0 ) = lim Φ(z) , z belongs to D + or - (15) z t 0
We also define Cauchy principal value integral
Φ (t 0 ) = P ∫ ψ(t) d t / ( t -t 0 ) (16) L = lim ∫ ψ (t) d t / ( t -t 0 ) (17) є 0 L-L є
where L є is that part of L contained within a circle of radius є with center at t= t 0 and where L -L є is that part of L contained without it .
We assume ψ(t) is analytic at t = t 0 and continuous every where, hence by analytic continuation ψ(t) is analytic in a small neighbourhood of t 0 and this continuation can be extended to the whole complex plane.
We therefore using the definition in equations ( 16 ) and (17)
Φ + (t 0 ) = lim Φ (z) , z lie within D + z t 0 є0 = Φ (t 0 ) + i π ψ (t 0 ) ( 18) Φ -(t 0 ) = lim Φ (z) , z lie within D - z t 0 є0 Φ (t 0 ) = Φ (t 0 ) -iπ ψ (t 0 )( 19)
Equation ( 18 ) and (19) are referred to as the Plemelj formulae .They may be equivalently written as
+ (t 0 ) -Φ -(t 0 ) = 2πi ψ (t 0 ) (20) Φ + (t 0 ) + Φ -(t 0 ) = 2 P ∫ ψ(t) d t / ( t -t 0 )(21)
The end points of the contour L cause considerable difficulty . A complicated analysis in Muskhelishvilli (1953) shows that at the end points t A and t B of L , Φ(t) will behave as
1 Φ (t) 1 ≤ K A / 1 t -t A 1 α A , 0 ≤ α A < 1 (22) 1 Φ (t) 1 ≤ K B / 1 t -t B 1 α B , 0 ≤ α B < 1(23)
where K A , K B , α A and α B are constants .
Determination of a new LNIE for H-functions :
we shall now introduce a function Y(z) of the complex variable z
Y( z) = 1 0 U(t) H(t) d t / ( t -z )(24)
which has the properties :
i)Y(z) is analytic in the complex plane cut along (0,1) ; ii) Y(z) = O (1/z) when z ∞ ; iii)1 Y (z) 1 ≤ K 0 / 1 z 1 α 0 , 0 ≤ α 0 < 1 ; iv) 1 Y (z) 1 ≤ K 1 / 1 1-z 1 α 1 , 0 ≤ α 1 < 1; v) U(t) H(t)
is continuous in the interval (0,1) and satisfy equivalent Holder conditions stated in equation (3) and K 0 , K 1 , α 0 and α 1 are constants .
We shall now apply the Plemelj's formulae (20 ) and (21) to the equation (12) to have
Y + (x) = g(x)Y -(x) + 2πi U(x) / T -(x) , 0≤ x ≤1 (25)
where Y + (x) , Y -(x) and g(x) are complex numbers and g(x) is related to
T + (x)= T 0 (x) + i π x U(x) (26) T -(x) = T 0 (x) -i π x U(x) (27) g(x) = (T 0 (x) + i π x U(x) ) / ( T 0 (x) -i π x U(x) ) = T + (x)/T -(x)(28)
where T 0 (x) is given by equation ( 13 ).
From equation (25) we have to determine the unknown function Y(z) for being analytic in the complex plane cut along (0,1) having properties outlined above and g(x) and U(x) are known functions. This is known as non homogeneous Riemann Hilbert problem. 8 We shall first determine the solution of homogeneous Riemann Hilbert problem obtained from equation (25) .
It is that of finding a new analytic function X(z) for which
X + (x) = g(x) X -(x) , 0≤x≤1(29)
where X(z) is to satisfy the following properties :
i) X(z) is analytic in the complex plane cut along (0,1) ; ii) X(z) is not zero for all z in complex plane cut along (0 ,1) ; iii) O(1/z) when z ∞ ;iv) 1 X (z) 1 ≤ K 2 / 1 z 1 α 2 , 0 ≤ α 2 < 1 ;and v)1 X (z) 1 ≤ K 3 / 1 1 -z 1 α 3 , 0 ≤ α 3 < 1 , where K 2 , K 3 , α 2 and α 3 are constants .
We now assume that for 0 ≤ x ≤ 1, U(x) is real , positive , single valued and satisfy Holder condition in (0,1) . We also assume that T 0 (x) is not equal to zero both in the conservative case U 0 = ½ and in non conservative cases U 0 < ½ . As T 0 (x) is dependent on U(x) it can be proved that T 0 (x) is real ,one valued and satisfy Holder conditions in (0,1) and modulus of ( T 0 (x) +i π x U(x) ) , ( T 0 (x) -i π x U(x) ) are not equal to zero, T 0 (0) = 1 and T 0 (1) -∞ as z 1 from within the interval (0,1) .
Taking logarithm to equation (29)and using (28) we find that
log( X + (x) ) -log (X -(x)) =log g(x) = 2 i θ (x)(30)
where tan θ(x) = π x U (x) / T 0 (x) ; -π/2 < θ (x) < π/2 (31)
θ(0)=0,(32)
and θ(x) is assumed single valued .
We shall evaluate X(z).Using Plemelj's formulae (20) to equation (30) we get
log X(z) = (2πi) -1 1 0 log g(u) du /(u -z) + P n (z)(33)
where P n (z) is an arbitrary polynomial in z of degree n. in the complex plane cut along (0,1) and it is continuous in the complex plane across a cut along (0,1) because it is analytic there. Thus equation (33) does also satisfy the first of Plemelj's formulae (20).
We set
L(z) = (2πi) -1 1 0 log g(u) d u / ( u -z)(34)
Equation (33) with equation (34) then yields
X(z) = X 0 (z) exp ( L(z) )(35)
where X 0 (z) = exp ( P n (z) ) is analytic for all z in the complex plane and is to be determined such that X(z) given by equation (35) satisfy all properties outlined above and the use of the Plemelj's formulae is not invalidated .
We have to determine the value of X 0 (z) at the end points: Equation (34) may be written as
L(z) = (2πi ) -1 log g(z) 1 0 d u /( u-z) + (2πi) -1 1 0 ( log g(u) -log g(z) ) d u / (u -z)(36)
At the end point z = 0 , equation (36) can be written as
L(z) = = -(log g(z) / 2πi ) log 1z1 + q 0 (z)(37)
where q 0 (z) incorporates the second integral of equation (36) and the value of first term at z = 1 . Here q 0 (z) is clearly a bounded function .We therefore find that , in the neighborhood of the end point at z= 0 ,the behavior of X(z) is dominated by
X(z) X 0 (z) 1z1 -(log g(z) / 2πi )(38)
Near the end point z=0 for the Plemelj' formulae to be valid , X 0 (z) and X(z) must behave in the neighborhood of z = 0 as
X 0 (z) 1z1 m 0 (39) X(z) 1z1 (m 0 -(log g(z) / 2πi ) )(40)
We must have the constant m 0 to satisfy there 0 ≤ Real ( log g(z) / 2πi ) -m 0 < 1
As X 0 (z) will be analytic and continuous across the cut in complex z plane along (0,1) we must have using eqs.
where q 1 (z) incorporates the second integral of equation (36) and the value of first term at z = 0 . Here q 1 (z) is clearly a bounded function . Hence, in the neighborhood of the end point at z= 1 ,the behavior of X(z) will be dominated by X 0 (z) as
X(z) X 0 (z) 11-z1 (log g(z) / 2πi )(44)
and for the Plemelj' formulae to be valid there , X 0 (z) and X(z) must also behave in the neighborhood of z = 1 as
X 0 (z ) 1 1-z1 m 1 (45) X(z) 11-z1 (m 1 + (log g(z) / 2πi ) )(46)
We must have the constant m 1 to satisfy there 0 ≤ -Real ( log g(z) / 2πi ) -m 1 < 1
As X 0 (z) will be analytic and continuous across the cut in complex z plane along (0,1) hence using eq.(30)
m 1 = -Real ( log g(1) / 2πi) = -θ(1 ) / π(48)
But we know that θ(1) = r π , r = 1,2,3 etc (49) so m 1 = -1 for r =1 = ½ the zeros of T(z) in the complex plane cut along(-1,1) .Hence on consideration of the properties of X 0 (z) at both end points we have X 0 (z) = (1-z) -1 (50) .
Hence equation (35)
= (1-z ) -1 exp ( π -1 1 0 θ(u) d u / ( u -z) )(52)
where θ(u) is given by using equation (31) as
θ(u) = tan -1 ( π u U(u) / T 0 (u) )(54)
We shall determine a solution of non homogeneous Hilbert problem. The equation (25) may be changed by first writing
X + (u) = g(u) X -(u) , 0<u<1(55)
when u lie on cut along (0,1),X + (u),g(u) and X -(u)are all nonzero .Substitution of equation (55) in equation (25) we get
Y + (u) / X + (u)-Y -(u) / X -(u) = 2πi U(u) / X + (u) T -(u) ,(56)
when u lie in the interval (0,1). Using Plemelj's formulae in equation (20) to equation (56) we get for z in complex plane cut along (0,1),
Y(z) = X(z) 1 0 U(u) d u / (X + (u) T -(u) ( u -z) ) + P m (z) X(z)(57)
where P m (z) is an arbitrary polynomial in z of degree m and is continuous in the complex plane across a cut along (0,1 ).
Thus equation (57) does satisfy the first of Plemelj's formulae as in. equation (20) The arbitrariness of P m (z) is removed usually by examining the behaviour of Y(z) and X(z) at infinity and /or the end points of the cut along (0,1) .
Knowledge of X (z) then completes the solution for Y (z). Y(z) will then be the sum of solution of homogeneous Hilbert problem in equation (29) and the particular solution of the equation (56).
We now return our attention to the evaluation of the particular solution of equation (56)for Y(z) from equation (57) and in particular , to the determination of the polynomial P m (z) . Since X(z) is equivalent to exp(L(z) ) , the term appearing on the RHS of equation (57) is a polynomial of degree m . However, if we use the properties of X(z) and Y(z) when z ∞, we must have P m (z) = 0 (58)
Hence we get the particular solution Y p (z) of equation (56)in complex z plane cut along (0,1)) for Y(z) as Y p (z) = X(z) 1 0 U(u) d u / (X + (u) T -(z) ( u -z) ) (59).
We shall now represent Y p (z) in terms of equation X(z) in the complex z plane cut along (0,1) by using Cauchy's integral theorems
We set
V(z) =(2πi) -1 1 0 U(u)du /(X + (u)T -(u) (u-z) )(60)
where V(z) is analytic in the complex z plane cut along (0 , 1).
We consider two contours as follows:
F(z) = (2πi) -1 1 C F(u) d u / (u-z) )(61)
where C 1 = L 1 U L 2 . Here contour L 1 is a sufficiently large in the complex z plane to contain the cut along (0,1) within and L 2 is a circle with center at the origin and of very large radius R and L 1 lies interior to this L 2 and z lies outside L 1 but inside L 2 . Both L 1 and L 2 are taken in counter clock wise sense. The function X(z) and 1/X(z) are analytic and nonzero in the annulus C 1 . We shall now apply the Cauchy Integral theorem on the function F(z) =1/ z X(z) around C 1 to obtain
where the superscript '+' denotes the value from above the cut along (0,1) and the superscript '-' denotes the value from below the cut along (0,1) of the respective functions in the complex z plane and
L + (u) = ( π -1 P 1 0 θ(t) d t / ( t -u) + i θ (u) ) (66) L -(u) = ( π -1 P 1 0 θ(t) d t / ( t -u) -i θ (u) ) ( 67)
Using eqs. (66) & (67) in equations (64) and (65) we can write that 1/X + (u) -1/X -(u) = ( 1 -exp( 2iθ(u) ) / X + (u) (68).
Using equation (29 ) and ( 30 ) To make the contour L 1 to be well defined we shall shrink the contour L 1 to i) a circle C 0 around the origin of the complex plane in counter clockwise sense such that w= r exp (i α), 0≤ α ≤ 2π ; ii) a line CD below the cut along (0,1) from r to 1-r where w = u, real , u varies from 0 to 1; iii) a circle C 2 counter clockwise sense around w=1 such that w = 1 + r exp(i β ) , -π ≤ β≤ π and iv) a line BA above the cut along (0,1) from 1-r to r on which w = u , real , u varies from 1 to 0 . . The value of the integral , on the circle C 0 , in the limit r0 becomes
(2πi) -1 0 C dw/(X(w)w(w-z) )= -( z X(0) ) -1 (71)
The value of the integral , on the circle C 2 , in the limit r0 becomes
(2πi) -1 2 C d w / ( X(w) w (w-z) ) = 0 (72)
The value of the integral ,on the line CD , in the limit r0 becomes
(2πi) -1 CD d w / ( X(w) w (w-z)) = (2πi) -1 1 0 d u / ( X -(u) u (u-z) ) (73)
The value of the integral , on the line BA , in the limit r0 becomes
(2πi) -1 BA d w / ( X(w) w (w-z)) = (2πi) -1 0 1 d u / ( X + (u) u (u-z) ) (74)
Hence the integral , on the contour L 1 in eq(62), becomes using eqs.(70-74)
(2πi) -1 1 L d w /(X(w) w (w-z) ) = -(2πi) -1 1 0 (1/ X + (u) -1/X -(u) )d u /(u (u-z) ) -1 / ( z X(0) )(75)= 1 0 U (u) d u /( X + (u)T -(u) (u-z) ) -1 / ( z X(0) )(76)
The integral , on the contour L 2 , when R ∞ becomes
(2πi) -1 2 L d w /(X(w) w (w-z) ) = -1 (77 )
as z X(z) = -1 when z ∞
Hence equation (60) with equations (62), (76) and (77) gives V(z) = ( z X(0) ) -1 -1 -(z X(z)) -1 (79)
Hence the particular solution Y p (z) of equation ( 56) gives Y p (z) = X(z) V(z) = X(z) / (z X(0) ) -X(z) -1/ z (80)
Hence the general solution of the equation (25) in the complex z plane , is
Y(z) = A X(z) + Y p (z)(81)
where A is a constant yet to be determined by using the pole of Y(z), if any, at the zero of T(z) and X(z) , Y p (z) are determined by equations (51) and (80) respectively .This Y(z) in Equation (81) will help in representing 1/H(-z) in equation (8).
We shall now show that H(z) in equation (8) is continuous across the cut along (0,1).. Let H(x) be any real valued solution of NNIE at equation ( 1 ) for 0≤ x≤ 1 then it can be proved that H(z) defines in equation (8) is the meromorphic extension of H(x) , 0 ≤ x≤ 1 to the complex domain 1z1>0 . In addition H(x) will satisfy the LNSIE given by equation ( 12 ) .By using the Plemelj's first formulae as in equations (18) & (19) to equation (8) when z approaching the cut along (0,1) from above and below respectively :
T + (x) H + (x) = 1 + x P 1 0 H(t) U(t) dt / (t-x) + i π H(x) U(x), 0≤ x≤ 1 (82) T -(x) H -(x) = 1 + x P 1 0 H(t) U(t) dt / (t-x) -i π H(x) U(x), 0≤ x≤ 1 (83)
where superscript '+' and '-' denotes the value as z approaches from above and below to the cut along (0,1) . Using the values of T + (x) and T -(x) from equations (26) , (27) to equations (82) and (83) and using the equation (12) therein we get
H + (x) = H(x) = H -(x)(84)
This shows that H(z) defined by equation (8) is continuous across the cut along (0,1) and real valued on (0,1) Hence it indeed defines a meromorphic extension of H(x) at least in the region 0 <Re(z) <1 , Im(z) > 0 and Im (z) < 0 and T(1) 0 ,and it can be said that z =1 is a removable singularity. We shall now frame a new LNIE from the theory of linear singular integral equations. Using equations (24), we get ,
(T + (x) H(x) -x A X + (x) -x Y p + (x)) -( T -(x) H(x -x X -(x) -x Y p - (x ) ) = 0(85)
Using first of Plemelj's formulae as in equation (20) to equation (85) we get
T(z) H(z) -z A X(z) -zYp(z) = Pp(z)(86)
where P p (z) an arbitrary polynomial is to be determined from the properties of X(z) , Yp(z) ,T(z) and H(z) at the origin..
Since H(z) and T(z) both 1 when z0 , equation (86) gives
Pp (z) = 1(87)
Equation (86) with equation (87) gives a new representation of LNIE in the complex z plane cut along (0,1)from the theory of linear singular integral equation and by analytic continuation to complex plane cut along (-1,1) to T(z) H(z) = 1+ z A X(z) + z Y p (z) (88) .
Determination of constant :
Equation (88) with equation (8) and (81) gives
1/H(-z) = 1+ z A X(z) + z Y p (z) (89) = 1 + z Y(z)(90)
We have now to determine the constant A in equation (88).We know that T(z) has two zeros at z = 1/k or -1/k , 0≤ k ≤1 . 1/H(-z) has a zero at z=1/k . Equation (89) , on substitution of z = 1/k , gives
1+ A X(1/k) / k + Y p (1/k) / k =0 (91).
Using equation (80) for z =1/k we get
Y p (1/k) = k ( X(1/k)/X(0) -X(1/k)/k -1) (92).
Equation (91) and (92) we get
1+ A X(1/k) /k + X(1/k) / X(0) -X(1/k) /k -1 =0(93)
This equation (93) on simplification gives
A = 1 -k / X(0)(94)
We shall have to determine X(0). Substituting z=0 in equation ( 53 ) we get
X(0)=exp( π -1 1 0 θ(t)d t/t )(95)
We can determine other form of A for a new form of X(0). In equation (24)
where X(z) is given by equation (53) Hence in the complex z plane cut along (-1,0) we get
H(z) = k /( X(-z)( 1 + k z ) ( D) ½ ).(104)
Using the values of X(z) and X(-z) in equations (103) and (104) we These are the same form derived by Das [13] using Wiener Hopf technique to linear non homogeneous integral equation in Eq.(8).
New form of H-function in Cauchy Principal value sense:
We have to find a new form for H(x) in Cauchy's principal value sense by application of Plemelj's formula.We know from equation ( 81 ) using eq.(94) that
Y(z) = X(z) (1-kz) / ( z X(0) )(107)
Using first of the Plemelj's formulae (20) to equation (107) we get
H(x) = ( Y + (x) -Y -(x) ) / ( 2πi U(x) )(108)
= ( 1-k x ) ( X + (x) -X -(x) ) / (2πi x U(x) X(0) ) .
(109) But X + (x) -X -(x) = 2i sin ( θ(x) ) exp( P π -1 1 0 θ(t) d t / (t -x) ) / (1-x) ,(110) sin (θ(x) ) = π x U(x) /( T 0 (x) + π 2 x 2 U 2 (x) ) 1/2 (111)
X(0) = k D -1/2 (112) D = ( 1 -2 U o )(113)
Equation (109) with equations (110-113) gives
H(x) = (1 -k x) D 1/2 exp( -π -1 P 1 0 θ(t) dt / (t -x) ) X ( T 0 (x) + π 2 x 2 U 2 (x) ) -1/ 2 ( k (1-x) ) -1(114)
This is the new explicit form of H(x) different from Fox's [4]solution . The integral is Cauchy principal Value sense.
Removal of Cauchy Principal Value sense from H-function:
We shall now show that the representation of H(x) in eq.(114) is nothing but the representation of H(x) in equation .(105). It can be done by way of removal of singular part from equation (114).
Let M(x) = π -1 P 1 0 θ(t) dt / (t -x) , 0 x 1 (115) N(x) = π -1 1 0 θ(t) dt / (t + x ) ,0 x 1(116)
If z approaches from above to the cut along (0,1) to x , 0≤x≤1 we get
H + (x) H + (-x) = 1/ T + (x)(117)
Here H + (x) = H(x) as H(x ) is continuous across the cut along (0,1) but H + (-x) is not continuous there . Therefore we get H + (-x) , H + (x) from equation (106)
T + (x) = T 0 (x) + i π x U(x) (120)
Using equations (118-120) in equation (117) we get ( 1 -x 2 ) k 2 D -1 ( 1 -k 2 x 2 ) -1 exp(-M(x) -N(x) ) ( cos θ(x) -i sin θ(x) )
= ( T 0 (x) -iπ x U(x) }/ ( T 0 (x) + π 2 x 2 U 2 (x) )(121)
Equating the real and imaginary parts of equation (141) we get
( 1 -x 2 ) k 2 D -1 ( 1 -k 2 x 2 ) -1 exp(-M(x) -N(x) ) cos θ(x) = T 0 (x) / ( T 0 (x) + π 2 x 2 U 2 (x) )(122)
( 1 -x 2 ) k 2 D -1 ( 1 -k 2 x 2 ) -1 exp(-M(x) -N(x) ) sin θ(x)
= π x U(x) }/ ( T 0 (x) + π 2 x 2 U 2 (x) ) .
But using equations (111), (115) and (116) in equation (123) we get
( 1 -x 2 ) k 2 D -1 ( 1 -k 2 x 2 ) -1 exp(-M(x) -N(x) ) = 1 / ( T 0 (x) + π 2 x 2 U 2 (x) ) 1/2 .(124)
Now we use equation (124) to equation (114) in order to eliminate the Principal part of the integral in equation (114) and we get the same form of H(x) ( as in equation (105) ) with constraints (9) & (10) in non conservative cases and constraint (11) in conservative cases
Using equations (101) and (95) to equation (105) we get the form outlined by Mullikin [9].in non conservative cases:
H(z) = (1+z) exp ( π -1 1 0 θ(u) d u / u( u + z) ) / ( 1 + k z ). (125)
Determination of H-function in conservative cases :
From equation (105) we can derive the representation of H(z) for conservative cases . In conservative cases U 0 =1/2 , T(z) will have zeros at infinity . Hence k = 0. When k=0, U 0 =1/2 , D = 0 by equation( 113 ) . Hence when k0 , k D -1/2 ( 2 U 2 ) -1/2 , in conservative case , H(z) in equation (105) will take the form H(z) = ( 2 U 2 ) -1/2 (1+z) exp ( -π -1 1 0 θ(u) d u / ( u + z) ) . (126) with constraint in equation (11) .
Admissible solutions:
Chandrasekhar [1], Busbridge [3] ,Dasgupta [23] , Abhyankar [24] proved that , in non conservative cases , these two constraints(9) &(10) will provide two different unique solutions but in conservative cases those two solutions will be identical to form one unique solution with constraint (11) . In non conservative cases , they also derived the form of constraints from equations (9)
We therefore, resolve that, in non conservative cases, equation (105) for H(z) with constraints (9) or (127) will provide meaningful unique solution of equation (12) of physical interest and the other unique solution H 1 (z) will be defined by equation (129) with H(z) from equation (105) and with constraints (10) or (128).
10.Conclusion :
The basic approach in this paper is to place a new method to extend the only available solution of Fox [4] for the H-functions from LNSIE as solution of a Riemann Hilbert problem. The representation of H-functions obtained from LNSIE and from LNIE has become the same . The equivalence between application of the theory of linear singular integral equation and application of Wiener-Hopf technique to the linear integral equations is proved to be true so far it relates to the H-functions .The numerical evaluation of these H-functions from this new form is awaiting for communication . This new method may be applied in anisotropic line transfer problems in non coherent scattering , in problems of multiple scattering and also in time dependent problems of radiative transfer to determine the H -function related to those problems .
U(u) H(u) d u / ( u -x )
end point z = 1 , equation (36) can be written as L(z) = = (log g(z) / 2πi ) log 11-z1 + q 1 (z)
the solution of homogeneous Hilbert problem mentioned in equation (29) in the complex z plane cut along (0,1) .This can be written in explicit form when z is in complex in z plane cut along(0,1) using eqs.(34)
X -(u) = ( 1 -u) exp ( -L -(u) )
( 8 )
8,(26),(27),(81) ,(82) ,(83)& (84) in equation
U (x) H (x) d x = 1 -( D) Determination of new form of H-function : We shall now determine the new form of H(z).On substitution of the values of A , X(0) and Y p (z) from eqs.(94),(101) and(80)respectivelyin equation (89)we get in the complex plane cut along (
get the new form of H(z) and H(-z) as H(z) = k ( D) -½ (1+z) exp ( -π
(x) = (1-x ) k ( D) -1/2 exp ( -N(x) ) / (1+kx)
Acknowledgement:I express my sincere thanks to the Department of Mathematics , Heritage Institute of Technology , Anandpur , West Bengal , India for the support extended.References :
. S Chandrasekhar, Transfer, Dover , New YorkChandrasekhar , S . Radiative transfer , Dover , New York , 1950.
Basic methods in transfer problems. V A Kourganoff, Clarendon PressOxford.Kourganoff,V.A. Basic methods in transfer problems. Clarendon Press, Oxford., 1952.
The mathematics of radiative transfer. I W Busbridge, Cambridge Tracts. 50Busbridge,I.W. The mathematics of radiative transfer , Cambridge Tracts, No 50 , Cambridge, 1960.
A solution of Chandrasekhar's integral equation. S C Fox, Trans. Amer. Math. Soc. 99Fox ,S.C. A solution of Chandrasekhar's integral equation , , Trans. Amer. Math. Soc. 1961, 99 , 285-291.
On H-functions of radiative transfer. S R Dasagupta, Astrophysics and space science. 30Dasagupta ,S.R.On H-functions of radiative transfer , Astrophysics and space science ,1974 , 30,327-342.
A new representation of the H-function of radiative transfer. S R Dasagupta, Astrophysics and space science. 50Dasagupta ,S.R.A new representation of the H-function of radiative transfer , Astrophysics and space science ,1977,50, 187-203. .
Exact solution of the transport equation for radiative transfer with scattering albedo w 0 <1 using Laplace transform and Wiener hopf technique and a expression of H-function. R N Das, Astrophysics and space science. 60Das ,R.N. Exact solution of the transport equation for radiative transfer with scattering albedo w 0 <1 using Laplace transform and Wiener hopf technique and a expression of H-function, Astrophysics and space science ,60 ,(1979), 49-58.
Exact solution of a critical problem for a slab. R Zelazny, J. Math.Phys. 2Zelazny,R. Exact solution of a critical problem for a slab , J. Math.Phys. , . 1961, 2,538-542.
Chandrasekhar's X and Y equations ( 1 ). T W Mullikin, Trans. Amer. Math. Soc. 113Mullikin , T.W.Chandrasekhar's X and Y equations ( 1 )Trans. Amer. Math. Soc. 1964,113 , 316-332.
Recent applications of neutron transport theory , Micigan Memorial phoenix project report, university of Michigan. P F Zweifel, P F , P.F. Zweifel ,P.F. Recent applications of neutron transport theory , Micigan Memorial phoenix project report, university of Michigan, 1964.
Transfer of radiation in spectral lines. V V Ivanov, National Bureau of Standards special publication. 385Ivanov ,V.V. Transfer of radiation in spectral lines, U.S Department of commerce, National Bureau of Standards special publication 385, 1973.
On computing eigenvalues in radiative transfer. C E Siewert, J. Math. Phys. 21Siewert,C.E. On computing eigenvalues in radiative transfer, J. Math. Phys. 1980,21,2468-2470. .
Application of the theory of Linear singular Integral equation to a Riemann Hilbert Problem for a new representation of Chandrasekhar's H-in radiative transfer. R N Das, JMAA-06-2670Journal of Mathematical Analysis and Applications. submitted in 2006 for publication to theDas ,R.N. Application of the theory of Linear singular Integral equation to a Riemann Hilbert Problem for a new representation of Chandrasekhar's H-in radiative transfer . submitted in 2006 for publication to the Journal of Mathematical Analysis and Applications .(JMAA-06-2670 ).
. . C Siewert, Journal of Mathematical Physics. Siewert.C.E , Journal of Mathematical Physics , 1980, 21, 2468.
. R D Garcia, C E Siewert, Transport Theory and Statistical Physics. 437Garcia , R.D.M and Siewert , C.E . Transport Theory and Statistical Physics ,1987, 14 , 437.
. J Sulties, T H Hill, Nuclear Science and Engineering. 53Sulties ,J.K and Hill,T.H . Nuclear Science and Engineering , 1976, 59, 53.
. L Barichello, C E Siewert, J of Quantitative Spectroscopy and Radiative Transfer. 261Barichello,L.B and Siewert , C.E . J of Quantitative Spectroscopy and Radiative Transfer , 1998, 60, 261.
. L Barichello, C E Siewert, J of Quantitative Spectroscopy and Radiative Transfer. 62665Barichello,L.B and Siewert , C.E . J of Quantitative Spectroscopy and Radiative Transfer , 1999, 62, 665.
Auxiliary functions for Radiative transfer problems in plane parallel geometry -I -the infinite medium. Bergeat, B Rutily, J of Quantitative Spectroscopy and Radiative Transfer. 52Bergeat ,J and Rutily ,B. Auxiliary functions for Radiative transfer problems in plane parallel geometry -I -the infinite medium , J of Quantitative Spectroscopy and Radiative Transfer , 1996, 52, 857-886.
Auxiliary functions for Radiative transfer problems in plane parallel geometry -II -the semi-infinite medium. Bergeat, B Rutily, J of Quantitative Spectroscopy and Radiative Transfer. 60Bergeat ,J and Rutily ,B. Auxiliary functions for Radiative transfer problems in plane parallel geometry -II -the semi-infinite medium , J of Quantitative Spectroscopy and Radiative Transfer , 1998, 60, 1033-1051.
On solutions of Chandrasekhar's Integral equations. I W Busbridge, Trans. Amer. Math. Soc. 105Busbridge, I.W. On solutions of Chandrasekhar's Integral equations, Trans. Amer. Math. Soc. 1962 , 105 ,112-117.
Singular integral equations. N I Muskhelishvili, Noordhoff, GroningenMuskhelishvili,N.I. Singular integral equations , Noordhoff, Groningen,1953.
Doctoral thesis, Radiation problems in Astrophysics. S R Dasgupta, Calcutta UniversityDasgupta,S.R. Doctoral thesis, Radiation problems in Astrophysics, Calcutta University ,1962.
On the solution H 1 (u) of Ambarzumian -Chandrasekhar H-equation. K Abhyankar, A L Fymat, J. Quant. Spectrosc.Radiat.Transfer. 9Abhyankar,K.D and Fymat ,A.L. On the solution H 1 (u) of Ambarzumian - Chandrasekhar H-equation, J. Quant. Spectrosc.Radiat.Transfer, 1969, 9 , 1563- 1566. -----------------------------------------------
|
[] |
[
"EMBEDDING LARGE SUBGRAPHS INTO DENSE GRAPHS",
"EMBEDDING LARGE SUBGRAPHS INTO DENSE GRAPHS"
] |
[
"Daniela Kühn ",
"Deryk Osthus "
] |
[] |
[] |
What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F -packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F . It is unlikely that there is a characterization of all graphs G which contain a perfect F -packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F -packing.The Regularity lemma of Szemerédi and the Blow-up lemma of Komlós, Sárközy and Szemerédi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F -packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved.
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10.1017/cbo9781107325975.007
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[
"https://arxiv.org/pdf/0901.3541v1.pdf"
] | 16,952,340 |
0901.3541
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db2d5dae9a0799ff4377f2a993052e49e7f833e3
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EMBEDDING LARGE SUBGRAPHS INTO DENSE GRAPHS
22 Jan 2009
Daniela Kühn
Deryk Osthus
EMBEDDING LARGE SUBGRAPHS INTO DENSE GRAPHS
22 Jan 2009
What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F -packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F . It is unlikely that there is a characterization of all graphs G which contain a perfect F -packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F -packing.The Regularity lemma of Szemerédi and the Blow-up lemma of Komlós, Sárközy and Szemerédi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F -packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved.
Introduction, overview and basic notation
In this survey, we study the question of when a graph G contains some given large or spanning graph H as a subgraph. Many important problems can be phrased in this way: one example is Dirac's theorem, which states that every graph G on n ≥ 3 vertices with minimum degree at least n/2 contains a Hamilton cycle. Another example is Tutte's theorem on perfect matchings which gives a characterization of all those graphs which contain a perfect matching (so H corresponds to a perfect matching in this case). A result which gives a complete characterization of all those graphs G which contain H (as in the case of Tutte's theorem) is of course much more desirable than a sufficient condition (as in the case of Dirac's theorem). However, for most H that we consider, it is unlikely that such a characterization exists as the corresponding decision problems are usually NP-complete. So it is natural to seek simple sufficient conditions. Here we will focus mostly on degree conditions. This means that G will usually be a dense graph and that we have to restrict H to be rather sparse in order to get interesting results. We will survey the following topics:
• a generalization of the matching problem, which is called the F -packing or F -tiling problem (here the aim is to cover the vertices of G with disjoint copies of a fixed graph F instead of disjoint edges); • Hamilton cycles (and generalizations) in graphs, directed graphs and hypergraphs;
• large subtrees of graphs;
• arbitrary subgraphs H of bounded degree;
• Ramsey numbers of sparse graphs.
A large part of the progress in the above areas is due to the Regularity lemma of Szemerédi [125] and the Blow-up lemma of Komlós, Sárközy and Szemerédi [83]. Roughly speaking, the former states that one can decompose an arbitrary large dense graph into a bounded number of random-like graphs. The latter is a powerful tool for embedding spanning subgraphs H into such random-like graphs. In the final section we give a formal statement of these results and describe in detail an application to a special case of the F -packing problem. We hope that readers who are unfamiliar with these tools will find this a useful guide to how they can be applied.
There are related surveys in the area by Komlós and Simonovits [89] (some minor updates were added later in [88]) and by Komlós [80]. However, much has happened since these were written and the emphasis is different in each case. So we hope that the current survey will be a useful complement and update to these. In particular, as the title indicates, our focus is mainly on embedding large subgraphs and we will ignore other aspects of regularity/quasi-randomness. There is also a recent survey on F -packings (and so-called F -decompositions) by Yuster [130], which is written from a computational perspective.
2. Packing small subgraphs in graphs 2.1. F -packings in graphs of large minimum degree. Given two graphs F and G, an F -packing in G is a collection of vertex-disjoint copies of F in G. (Alternatively, this is often called an F -tiling.) F -packings are natural generalizations of graph matchings (which correspond to the case when F consists of a single edge). An F -packing in G is called perfect if it covers all vertices of G. In this case, we also say that G contains an F -factor or a perfect F -matching. If F has a component which contains at least 3 vertices then the question whether G has a perfect F -packing is difficult from both a structural and algorithmic point of view: Tutte's theorem characterizes those graphs which have a perfect F -packing if F is an edge but for other connected graphs F no such characterization is known. Moreover, Hell and Kirkpatrick [61] showed that the decision problem of whether a graph G has a perfect F -packing is NP-complete if and only if F has a component which contains at least 3 vertices. So as mentioned earlier, this means that it makes sense to search for degree conditions which ensure the existence of a perfect F -packing. The fundamental result in the area is the Hajnal-Szemerédi theorem: Theorem 1. (Hajnal and Szemerédi [55]) Every graph whose order n is divisible by r and whose minimum degree is at least (1 − 1/r)n contains a perfect K r -packing.
The minimum degree condition is easily seen to be best possible. (The case when r = 3 was proved earlier by Corrádi and Hajnal [30].) The result is often phrased in terms of colourings: any graph G whose order is divisible by k and with ∆(G) ≤ k − 1 has an equitable k-colouring, i.e. a colouring with colour classes of equal size. (So k := n/r here.) Theorem 1 raises the question of what minimum degree condition forces a perfect F -packing for arbitrary graphs F . The following result gives a general bound.
Theorem 2. (Komlós, Sárközy and Szemerédi [86]) For every graph F there exists a constant C = C(F ) such that every graph G whose order n is divisible by |F | and whose minimum degree is at least (1 − 1/χ(F ))n + C contains a perfect F -packing.
This confirmed a conjecture of Alon and Yuster [9], who had obtained the above result with an additional error term of εn in the minimum degree condition. As observed in [9], there are graphs F for which the above constant C cannot be omitted completely (e.g. F = K s,s where s ≥ 3 and s is odd). Thus one might think that this settles the question of which minimum degree guarantees a perfect F -packing. However, we shall see that this is not the case. There are graphs F for which the bound on the minimum degree can be improved significantly: we can often replace χ(F ) by a smaller parameter. For a detailed statement of this, we define the critical chromatic number χ cr (F ) of a graph F as
χ cr (F ) := (χ(F ) − 1) |F | |F | − σ(F ) ,
where σ(F ) denotes the minimum size of the smallest colour class in an optimal colouring of F . (We say that a colouring of F is optimal if it uses exactly χ(F ) colours.) So for instance a k-cycle C k with k odd has χ cr (C k ) = 2+2/(k−1). Note that χ cr (F ) always satisfies χ(F ) − 1 < χ cr (F ) ≤ χ(F ) and equals χ(F ) if and only if for every optimal colouring of F all the colour classes have equal size. The critical chromatic number was introduced by Komlós [81]. He (and independently Alon and Fischer [8]) observed that for any graph F it can be used to give a lower bound on the minimum degree that guarantees a perfect F -packing.
Proposition 3. For every graph F and every integer n that is divisible by |F | there exists a graph G of order n and minimum degree ⌈(1 − 1/χ cr (F ))n⌉ − 1 which does not contain a perfect F -packing.
Given a graph F , the graph G in the proposition is constructed as follows: write k := χ(F ) and let ℓ ∈ N be arbitrary. G is a complete k-partite graph with vertex classes V 1 , . . . , V k , where |V 1 | = σ(F )ℓ − 1, n = ℓ|F | and the sizes of V 2 , . . . , V k are as equal as possible. Then any perfect F -packing would consist of ℓ copies of F . On the other hand, each such copy would contain at least σ(F ) vertices in V 1 , which is impossible.
Komlós also showed that the critical chromatic number is the parameter which governs the existence of almost perfect packings in graphs of large minimum degree. (More generally, he also determined the minimum degree which ensures that a given fraction of vertices is covered.) Theorem 4. (Komlós [81]) For every graph F and every γ > 0 there exists an integer n 0 = n 0 (γ, F ) such that every graph G of order n ≥ n 0 and minimum degree at least (1 − 1/χ cr (F ))n contains an F -packing which covers all but at most γn vertices of G.
By making V 1 slightly smaller in the previous example, it is easy to see that the minimum degree bound in Theorem 4 is also best possible. Confirming a conjecture of Komlós [81], Shokoufandeh and Zhao [121,122] subsequently proved that the number of uncovered vertices can be reduced to a constant depending only on F .
We [96] proved that for any graph F , either its critical chromatic number or its chromatic number is the relevant parameter which governs the existence of perfect packings in graphs of large minimum degree. The classification depends on a parameter which we call the highest common factor of F . This is defined as follows for non-bipartite graphs F . Given an optimal colouring c of F , let x 1 ≤ x 2 ≤ · · · ≤ x ℓ denote the sizes of the colour classes of c. Put
D(c) := {x i+1 − x i | i = 1, . . . , ℓ − 1}.
Let D(F ) denote the union of all the sets D(c) taken over all optimal colourings c. We denote by hcf(F ) the highest common factor of all integers in D(F ). If D(F ) = {0} we set hcf(F ) := ∞. Note that if all the optimal colourings of F have the property that all colour classes have equal size, then D(F ) = {0} and so hcf(F ) = 1 in this case. In particular, if χ cr (F ) = χ(F ), then hcf(F ) = 1. So for example, odd cycles of length at least 5 have hcf = 1 whereas complete graphs have hcf = 1.
The definition can be extended to bipartite graphs F . For connected bipartite graphs, we always have hcf(F ) = 1, but for disconnected bipartite graphs the definition also takes into account the relative sizes of the components of F (see [96]).
We proved that in Theorem 2 one can replace the chromatic number by the critical chromatic number if hcf(F ) = 1. (A much simpler proof of a weaker result can be found in [93].) [96]) Suppose that F is a graph with hcf(F ) = 1. Then there exists a constant C = C(F ) such that every graph G whose order n is divisible by |F | and whose minimum degree is at least (1 − 1/χ cr (F ))n + C contains a perfect F -packing.
Theorem 5. (Kühn and Osthus
Note that Proposition 3 shows that the result is best possible up to the value of the constant C. A simple modification of the examples in [8,81] shows that there are graphs F for which the constant C cannot be omitted entirely. Moreover, it turns out that Theorem 2 is already best possible up to the value of the constant C if hcf(F ) = 1. To see this, for simplicity assume that k := χ(F ) ≥ 3 and n = kℓ|F | for some ℓ ∈ N and let G be a complete k-partite graph with vertex classes V 1 , . . . , V k , where |V 1 | := ℓ|F | − 1, |V 2 | := ℓ|F | + 1 and |V i | = ℓ|F | for i ≥ 3. Consider any F -packing F 1 , . . . , F t in G. Let G i be the graph obtained from G by removing F 1 , . . . , F i . So G = G 0 . If t := hcf(F ) = 1, then the vertex classes V 1 i of G 1 still have property that |V 1 1 | − |V 1 k | ≡ 0 modulo t. More generally, this property is preserved for all G i , so the original F -packing cannot cover all the vertices in V 1 ∪ V k .
One can now combine Theorems 2 and 5 (and the corresponding lower bounds which are discussed in detail in [96]) to obtain a complete answer to the question of which minimum degree forces a perfect F -packing (up to an additive constant). For this, let
χ * (F ) := χ cr (F ) if hcf(F ) = 1; χ(F ) otherwise.
Also let δ(F, n) denote the smallest integer k such that every graph G whose order n is divisible by |F | and with δ(G) ≥ k contains a perfect F -packing.
Theorem 6. (Kühn and Osthus [96]) For every graph F there exists a constant C = C(F ) such that
1 − 1 χ * (F ) n − 1 ≤ δ(F, n) ≤ 1 − 1 χ * (F ) n + C.
The constant C appearing in Theorems 5 and 6 is rather large since it is related to the number of partition classes (clusters) obtained by the Regularity lemma. It would be interesting to know whether one can take e.g. C = |F | (this holds for large n in Theorem 2). Another open problem is to characterize all those graphs F for which δ(F, n) = ⌈(1 − 1/χ * (F ))n⌉. This is known to be the case for complete graphs by Theorem 1 and all graphs with at most 4 vertices (see Kawarabayashi [71] for a proof of the case when F is a K 4 minus an edge and a discussion of the other cases). If n is large, this is also known to hold for cycles (this follows from Theorem 32 below) and for the case when F is a complete graph minus an edge [29] (the latter was conjectured in [71]).
2.2.
Ore-type degree conditions. Recently, a simple proof (based on an inductive argument) of the Hajnal-Szemerédi theorem was found by Kierstead and Kostochka [78]. Using similar methods, they subsequently strengthened this to an Ore-type condition [79]: Theorem 7. (Kierstead and Kostochka [79]) Let G be a graph whose order n is divisible by r. If d(x) + d(y) ≥ 2(1 − 1/r)n − 1 for all pairs x = y of nonadjacent vertices, then G has a perfect K r -packing.
Equivalently, if a graph G whose order is divisible by k satisfies d(x) + d(y) ≤ 2k − 1 for every edge xy, then G has an equitable k-colouring. (So k := n/r.) Recently, together with Treglown [99], we proved an Ore-type analogue of Theorem 6 (but with a linear error term εn instead of the additive constant C). The result in this case turns out to be genuinely different: again, there are some graphs F for which the degree condition depends on χ(F ) and some for which it depends on χ cr (F ). However, there are also graphs F for which it depends on a parameter which lies strictly between χ cr (F ) and χ(F ). This parameter in turn depends on how many additional colours are necessary to extend colourings of neighbourhoods of certain vertices of F to a colouring of F . It is an open question whether the linear error term in [99] can be reduced to a constant one. ∈ W of the number of neighbours of x in W . The obvious question is what value of δ ′ (G) ensures that G has a perfect K r -packing. The following (surprisingly difficult) conjecture is implicit in [101]. Fischer [39] originally made a stronger conjecture which did not include the 'exceptional' graph Γ r,n defined below.
Conjecture 8. Suppose that r ≥ 2 and that G is an r-partite graph with vertex classes of size n. If δ ′ (G) ≥ (1 − 1/r)n, then G has a perfect K r -packing unless both r and n are odd and G = Γ r,n .
To define the graph Γ r,n , we first construct a graph Γ r : its vertices are labelled g ij with 1 ≤ i, j ≤ r. We have an edge between g ij and g i ′ j ′ if i = i ′ , j = j ′ and j ≤ r − 2 or j ′ ≤ r − 2. We also have an edge if i = i ′ and we have either j = j ′ = r − 1 or j = j ′ = r (see Fig. 1). Γ r,n is then obtained from Γ r by replacing each vertex with an independent set of size n/r and replacing each edge with a complete bipartite graph.
To see that Γ r,n has no perfect K r -packing when both r and n are odd, let W ℓ denote the set of vertices of Γ r,n which correspond to a vertex of Γ r with j = ℓ. Note that every copy of K r which covers a vertex in W 1 ∪ · · · ∪ W r−2 has to contain at least 2 vertices in W r−1 or at least 2 vertices in W r . So in order to cover all vertices in W 1 ∪ · · · ∪ W r−2 we can only use copies of K r which contain exactly 2 vertices in W r−1 or exactly 2 vertices in W r . But since |W r−1 | = |W r | = n is odd this means that it is impossible to cover all vertices of Γ r,n with vertex-disjoint copies of K r . (Note that the argument uses only that n is odd, but we cannot have that n is odd and r is even.)
A much simpler example which works for all r and n but which gives a weaker bound when r and n are odd is obtained as follows: choose a set A which has less than (1 − 1/r)n vertices in each vertex class and include all edges which have at least one endpoint in A. For large n, the case r = 3 of Conjecture 8 was solved by Magyar and Martin [101] and the case r = 4 by Martin and Szemerédi [102], both using the Regularity lemma (the case r = 2 is elementary). Johansson [67] had earlier proved an approximate version of the case r = 3. Csaba and Mydlarz [33] proved a result which implies that Conjecture 8 holds approximately when r is large (and n large compared to r). Generalizations to packings of arbitrary graphs were considered in [63,103,132]. A variant of the problem (where one considers usual minimum degree δ(G)) was considered by Johansson, Johansson and Markström [68]. They solved the case r = 3 and gave bounds for the case r > 3. This problem is related to bounding the so-called 'strong chromatic number'.
2.4.
Hypergraphs. (Perfect) F -packings have also been investigated for the case when F is a uniform hypergraph. Unsurprisingly, the hypergraph problem turns out to be much more difficult than the graph problem. There are two natural notions of (minimum) degree of the 'dense' hypergraph G. Firstly, one can consider the vertex degree. Secondly, given an r-uniform hypergraph G and an (r − 1)tuple W of vertices in G, the degree of W is defined to be the number of hyperedges which contain W . This notion of degree is called collective degree or co-degree. In contrast to the graph case, even the minimum collective degree which ensures a perfect matching (i.e. when F consists of a single edge) is not easy to determine. Rödl, Ruciński and Szemerédi [118] gave a precise solution to this problem, the answer turns out to be close to n/2. This improved bounds of [94,115]. An rpartite version (which is best possible for infinitely many values of n) was proved by Aharoni, Georgakopoulos and Sprüssel [3]. The minimum vertex degree which forces the existence of a perfect matching is unknown. It is natural to make the following conjecture (a related r-partite version is conjectured in [3]).
Conjecture 9.
For all integers r and all ε > 0 there is an integer n 0 = n 0 (r, ε) so that the following holds for all n ≥ n 0 which are divisible by r: if G is an r-uniform hypergraph on n vertices whose minimum vertex degree is at least
(1 − (1 − 1/r) r−1 + ε) n r − 1 ,
then G has a perfect matching.
The following construction gives a corresponding lower bound: let V be a set of n vertices and let A ⊆ V be a set of less than n/r vertices and include as hyperedges all r-tuples with at least one vertex in A. The case r = 3 of the conjecture was proved recently by Han, Person and Schacht [56].
A hypergraph analogue of Theorem 6 currently seems out of reach. So far, the only hypergraph F (apart from the single edge) for which the approximate minimum collective degree which forces a perfect F -packing has been determined is the 3-uniform hypergraph with 4 vertices and 2 edges [95]. Pikhurko [113] gave bounds on the minimum collective degree which forces the complete 3-uniform hypergraph on 4 vertices. In the same paper, he also shows that if ℓ ≥ r/2 and G is an r-uniform hypergraph where every ℓ-tuple of vertices is contained in at least (1/2 + o(1)) n r−ℓ hyperedges, then G has a perfect matching, which is best possible up to the o(1)-term. This result is rather surprising in view of the fact that Conjecture 9 (which corresponds to the case when ℓ = 1) has a rather different form. Further results on this question are also proved in [56].
Trees
One of the earliest applications of the Blow-up lemma was the solution by Komlós, Sárközy and Szemerédi [82] of a conjecture of Bollobás on the existence of given bounded degree spanning trees. The authors later relaxed the condition of bounded degree to obtain the following result.
Theorem 10. (Komlós, Sárközy and Szemerédi [87]) For any γ > 0 there exist constants c > 0 and n 0 with the following properties. If n ≥ n 0 , T is a tree of order n with ∆(T ) ≤ cn/ log n, and G is a graph of order n with δ(G) ≥ (1/2+γ)n, then T is a subgraph of G.
The condition ∆(T ) ≤ cn/ log n is best possible up to the value of c. (The example given in [87] to show this is a random graph G with edge probability 0.9 and a tree of depth 2 whose root has degree close to log n.)
It is an easy exercise to see that every graph of minimum degree at least k contains any tree with k edges. The following classical conjecture would imply that we can replace the minimum degree condition by one on the average degree. [37]) Every graph of average degree greater than k − 1 contains any tree with k edges. This is trivially true for stars. (On the other hand, stars also show that the bound is best possible in general.) It is also trivial if one assumes an extra factor of 2 in the average degree. It has been proved for some special classes of trees, most notably those of diameter at most 4 [105]. The conjecture is also true for 'locally sparse' graphs -see Sudakov and Vondrak [124] for a discussion of this.
Conjecture 11. (Erdős and Sós
The following result proves (for large n) a related conjecture of Loebl. An approximate version was proved earlier by Ajtai, Komlós and Szemerédi [5].
Theorem 12. (Zhao [131]) There is an integer n 0 so that every graph G on n ≥ n 0 vertices which has at least n/2 vertices of degree at least n/2 contains all trees with at most n/2 edges. This would be generalized by the following conjecture.
Conjecture 13. (Komlós and Sós)
Every graph G on n vertices which has at least n/2 vertices of degree at least k contains all trees with k edges.
Again, the conjecture is trivially true (and best possible) for stars. Piguet and Stein [111] proved an approximate version for the case when k is linear in n and n is large. Cooley [26] as well as Hladký and Piguet [62] proved an exact version for this case. All of these proofs are based on the Regularity lemma. As with Conjecture 11, there are several results on special cases which are not based on the Regularity lemma. For instance, Piguet and Stein proved it for trees of diameter at most 5 [112].
Hamilton cycles
4.1. Classical results for graphs and digraphs. As mentioned in the introduction, the decision problem of whether a graph has a Hamilton cycle is NPcomplete, so it makes sense to ask for degree conditions which ensure that a graph has a Hamilton cycle. One such result is the classical theorem of Dirac.
Theorem 14. (Dirac [36]) Every graph on n ≥ 3 vertices with minimum degree at least n/2 contains a Hamilton cycle.
For an analogue in directed graphs it is natural to consider the minimum semidegree δ 0 (G) of a digraph G, which is the minimum of its minimum outdegree δ + (G) and its minimum indegree δ − (G). (Here a directed graph may have two edges between a pair of vertices, but in this case their directions must be opposite.) The corresponding result is a theorem of Ghouila-Houri [45].
Theorem 15. (Ghouila-Houri [45]) Every digraph on n vertices with minimum semidegree at least n/2 contains a Hamilton cycle.
In fact, Ghouila-Houri proved the stronger result that every strongly connected digraph of order n where every vertex has total degree at least n has a Hamilton cycle. (When referring to paths and cycles in directed graphs we always mean that these are directed, without mentioning this explicitly.) All of the above degree conditions are best possible. Theorems 14 and 15 were generalized to a degree condition on pairs of vertices for graphs as well as digraphs:
Theorem 16. (Ore [110]) Suppose that G is a graph with n ≥ 3 vertices such that every pair x = y of nonadjacent vertices satisfies d(x) + d(y) ≥ n. Then G has a Hamilton cycle.
Theorem 17. (Woodall [129]) Let G be a strongly connected digraph on n ≥ 2 vertices. If d + (x) + d − (y) ≥ n for every pair x = y of vertices for which there is no edge from x to y, then G has a Hamilton cycle.
There are many generalizations of these results. The survey [46] gives an overview for undirected graphs and the monograph [10] gives a discussion of directed versions. Below, we describe some recent progress on degree conditions for Hamilton cycles, much of which is based on the Regularity lemma.
4.2.
Hamilton cycles in oriented graphs. Thomassen [127] raised the natural question of determining the minimum semidegree that forces a Hamilton cycle in an oriented graph (i.e. in a directed graph that can be obtained from a simple undirected graph by orienting its edges). Thomassen initially believed that the correct minimum semidegree bound should be n/3 (this bound is obtained by considering a 'blow-up' of an oriented triangle). However, Häggkvist [52] later gave the following construction which gives a lower bound of ⌈(3n − 4)/8⌉ − 1 (see bipartite graph) which is as regular as possible. We also add all edges from A to B, from B to C, from C to D and from D to A. Since every path which joins two vertices in D has to pass through B, it follows that every cycle contains at least as many vertices from B as it contains from D. As |D| > |B| this means that one cannot cover all the vertices of G by disjoint cycles. This construction can be extended to arbitrary n (see [74]). The following result exactly matches this bound and improves earlier ones of several authors, e.g. [52,54,76]. The proof of this result is based on some ideas in [76]. Häggkvist [52] also made the following conjecture which is closely related to Theorem 18. Given an oriented graph G, let δ(G) denote the minimum degree of G (i.e. the minimum number of edges incident to a vertex) and set δ * (G) := δ(G) + δ + (G) + δ − (G). [52]) Every oriented graph G on n vertices with δ * (G) > (3n − 3)/2 contains a Hamilton cycle.
Conjecture 19. (Häggkvist
(Note that this conjecture does not quite imply Theorem 18 as it results in a marginally greater minimum semidegree condition.) In [76], Conjecture 19 was verified approximately, i.e. if δ * (G) ≥ (3/2 + o(1))n, then G has a Hamilton cycle (note this implies an approximate version of Theorem 18). The same methods also yield an approximate version of Theorem 17 for oriented graphs.
Theorem 20. (Kelly, Kühn and Osthus [76]) For every α > 0 there exists an integer n 0 = n 0 (α) such that every oriented graph G of order n ≥ n 0 with d + (x) + d − (y) ≥ (3/4 + α)n whenever G does not contain an edge from x to y contains a Hamilton cycle.
The above construction of Häggkvist shows that the bound is best possible up to the term αn. It would be interesting to obtain an exact version of this result.
Note that Theorem 18 implies that every sufficiently large regular tournament on n vertices contains at least n/8 edge-disjoint Hamilton cycles. (To verify this, note that in a regular tournament, all in-and outdegrees are equal to (n − 1)/2. We can then greedily remove Hamilton cycles as long as the degrees satisfy the condition in Theorem 18.) It is the best bound so far towards the following conjecture of Kelly (see e.g. [10]). A result of Frieze and Krivelevich [43] states that every dense ε-regular digraph contains a collection of edge-disjoint Hamilton cycles which covers almost all of its edges. This implies that the same holds for almost every tournament. Together with a lower bound by McKay [104] on the number of regular tournaments, it is easy to see that the above result in [43] also implies that almost every regular tournament contains a collection of edge-disjoint Hamilton cycles which covers almost all of its edges. Thomassen made the following conjecture which replaces the assumption of regularity by high connectivity. [128]) For every k ≥ 2 there is an integer f (k) so that every strongly f (k)-connected tournament has k edge-disjoint Hamilton cycles.
Conjecture 22. (Thomassen
The following conjecture of Jackson is also closely related to Theorem 18 -it would imply a much better degree condition for regular oriented graphs. [66]) For d > 2, every d-regular oriented graph G on n ≤ 4d + 1 vertices is Hamiltonian.
Conjecture 23. (Jackson
The disjoint union of two regular tournaments on n/2 vertices shows that this would be best possible. An undirected analogue of Conjecture 23 was proved by Jackson [65]. It is easy to see that every tournament on n vertices with minimum semidegree at least n/4 has a Hamilton cycle. In fact, for tournaments T of large order n with minimum semidegree at least n/4 + εn, Bollobás and Häggkvist [18] proved the stronger result that (for fixed k) T even contains the kth power of a Hamilton cycle. It would be interesting to find corresponding degree conditions which ensure this for arbitrary digraphs and for oriented graphs.
4.3.
Degree sequences forcing Hamilton cycles in directed graphs. For undirected graphs, Dirac's theorem is generalized by Chvátal's theorem [22] that characterizes all those degree sequences which ensure the existence of a Hamilton cycle in a graph: suppose that the degrees of the graph are d 1 ≤ · · · ≤ d n . If n ≥ 3 and d i ≥ i + 1 or d n−i ≥ n − i for all i < n/2 then G is Hamiltonian. This condition on the degree sequence is best possible in the sense that for any degree sequence violating this condition there is a corresponding graph with no Hamilton cycle. Nash-Williams [109] raised the question of a digraph analogue of Chvátal's theorem quite soon after the latter was proved: for a digraph G it is natural to consider both its outdegree sequence d + 1 , . . . , d + n and its indegree sequence d − 1 , . . . , d − n . Throughout, we take the convention that d + 1 ≤ · · · ≤ d + n and d − 1 ≤ · · · ≤ d − n without mentioning this explicitly. Note that the terms d + i and d − i do not necessarily correspond to the degree of the same vertex of G.
Conjecture 24 (Nash-Williams [109]). Suppose that G is a strongly connected digraph on n ≥ 3 vertices such that for all i < n/2
(i) d + i ≥ i + 1 or d − n−i ≥ n − i, (ii) d − i ≥ i + 1 or d + n−i ≥ n − i. Then G contains a Hamilton cycle.
It is even an open problem whether the conditions imply the existence of a cycle through any pair of given vertices (see [12]). It is easy to see that one cannot omit the condition that G is strongly connected. The following example (which is a straightforward generalization of the corresponding undirected example) shows that the degree condition in Conjecture 24 would be best possible in the sense that for all n ≥ 3 and all k < n/2 there is a non-Hamiltonian strongly connected digraph G on n vertices which satisfies the degree conditions except that d + k , d − k ≥ k + 1 are replaced by d + k , d − k ≥ k in the kth pair of conditions. To see this, take an independent set I of size k < n/2 and a complete digraph K of order n − k. Pick a set X of k vertices of K and add all possible edges (in both directions) between I and X. The digraph G thus obtained is strongly connected, not Hamiltonian and is both the out-and indegree sequence of G. In contrast to the undirected case there exist examples with a similar degree sequence to the above but whose structure is quite different (see [98]). In [98], the following approximate version of Conjecture 24 for large digraphs was proved.
Theorem 25 (Kühn, Osthus and Treglown [98]). For every α > 0 there exists an integer n 0 = n 0 (α) such that the following holds. Suppose G is a digraph on n ≥ n 0 vertices such that for all i < n/2
• d + i ≥ i + αn or d − n−i−αn ≥ n − i, • d − i ≥ i + αn or d + n−i−αn ≥ n − i.
Then G contains a Hamilton cycle.
Theorem 25 was derived from a result in [74] on the existence of a Hamilton cycle in an oriented graph satisfying a 'robust' expansion property.
The following weakening of Conjecture 24 was posed earlier by Nash-Williams [108]. It would yield a digraph analogue of Pósa's theorem which states that a graph G on n ≥ 3 vertices has a Hamilton cycle if its degree sequence d 1 ≤ · · · ≤ d n satisfies d i ≥ i + 1 for all i < (n − 1)/2 and if additionally d ⌈n/2⌉ ≥ ⌈n/2⌉ when n is odd [114]. Note that Pósa's theorem is much stronger than Dirac's theorem but is a special case of Chvátal's theorem.
Conjecture 26 (Nash-Williams [108]). Let G be a digraph on n ≥ 3 vertices such that d + i , d − i ≥ i + 1 for all i < (n − 1)/2 and such that additionally d + ⌈n/2⌉ , d − ⌈n/2⌉ ≥ ⌈n/2⌉ when n is odd. Then G contains a Hamilton cycle.
The previous example shows the degree condition would be best possible in the same sense as described there. The assumption of strong connectivity is not necessary in Conjecture 26, as it follows from the degree conditions. Theorem 25 immediately implies an approximate version of Conjecture 26.
It turns out that the conditions of Theorem 25 even guarantee the digraph G to be pancyclic, i.e. G contains a cycle of length t for all t = 2, . . . , n. Thomassen [126] as well as Häggkvist and Thomassen [53] gave degree conditions which imply that every digraph with minimum semidegree > n/2 is pancyclic. The latter bound can also be deduced directly from Theorem 15. The complete bipartite digraph whose vertex class sizes are as equal as possible shows that the bound is best possible. For oriented graphs the minimum semidegree threshold which guarantees pancyclicity turns out to be (3n − 4)/8 (see [77]).
Powers of Hamilton cycles in graphs.
The following result is a common extension (for large n) of Dirac's theorem and the Hajnal-Szemerédi theorem. It was originally conjectured (for all n) by Seymour. [85]) For every k ≥ 1 there is an integer n 0 so that every graph G on n ≥ n 0 vertices and with δ(G) ≥ k k+1 n contains the kth power of a Hamilton cycle.
Theorem 27. (Komlós, Sárközy and Szemerédi
Complete (k + 1)-partite graphs whose vertex classes have almost (but not exactly) equal size show that the minimum degree bound is best possible. Previous to this a large number of partial results had been proved (see e.g. [100] for a history of the problem). Very recently, Levitt, Sarközy and Szemerédi [100] gave a proof of the case k = 2 which avoids the use of the Regularity lemma, resulting in a much better bound on n 0 . Their proof is based on a technique introduced by Rödl, Ruciński and Szemerédi [117] for hypergraphs. The idea of this method (as applied in [100]) is first to find an 'absorbing' path P 2 : roughly, P 2 is the second power of a path P which, given any vertex x, has the property that x can be inserted into P so that P ∪ x still induces the second power of a path. The proof of the existence of P 2 is heavily based on probabilistic arguments. Then one finds the second power Q 2 of a path which is almost spanning in G − P 2 . One can achieve this by repeated applications of the Erdős-Stone theorem. One then connects up Q 2 and P 2 into the second power of a cycle and finally uses the absorbing property of P 2 to incorporate the vertices left over so far.
Hamilton cycles in hypergraphs.
It is natural to ask whether one can generalize Dirac's theorem to uniform hypergraphs. There are several possible notions of a hypergraph cycle. One generalization of the definition of a cycle in a graph is the following one. An r-uniform hypergraph C is a cycle of order n if there a exists a cyclic ordering v 1 , . . . , v n of its n vertices such that every consecutive pair v i v i+1 lies in a hyperedge of C and such that every hyperedge of C consists of consecutive vertices. Thus the cyclic ordering of the vertices of C induces a cyclic ordering of its hyperedges. A cycle is tight if every r consecutive vertices form a hyperedge. A cycle of order n is loose if all pairs of consecutive edges (except possibly one pair) have exactly one vertex in common. (So every tight cycle contains a spanning loose cycle but a cycle might not necessarily contain a spanning loose cycle.) There is also the even more general notion of a Berge-cycle, which consists of a sequence of vertices where each pair of consecutive vertices is contained in a common hyperedge. 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 tight cycle cycle 0 0 1 1 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 loose cycle A Hamilton cycle of a uniform hypergraph G is a subhypergraph of G which is a cycle containing all its vertices. Theorem 28 gives an analogue of Dirac's theorem for tight hypergraph cycles, while Theorem 29 gives an analogue for 3-uniform (loose) cycles.
Theorem 28. (Rödl, Ruciński and Szemerédi [117]) For all r ∈ N and α > 0 there is an integer n 0 = n 0 (r, α) such that every r-uniform hypergraph G with n ≥ n 0 vertices and minimum degree at least n/2 + αn contains a tight Hamilton cycle. [57]; Keevash, Kühn, Mycroft and Osthus [73]) For all r ∈ N and α > 0 there is an integer n 0 = n 0 (α) such that every r-uniform hypergraph G with n ≥ n 0 vertices and minimum degree at least n/(2r − 2) + αn contains a loose Hamilton cycle.
Theorem 29. (Han and Schacht
Both results are best possible up to the error term αn. In fact, if the minimum degree is less than ⌈n/(2r−2)⌉, then we cannot even guarantee any Hamilton cycle in an r-uniform hypergraph. The case r = 3 of Theorems 28 and 29 was proved earlier in [116] and [95] respectively. The result in [57] also covers the notion of an r-uniform ℓ-cycle for ℓ < r/2 (here we ask for consecutive edges to intersect in precisely ℓ vertices). Hamiltonian Berge-cycles were considered by Bermond et al. [11].
Bounded degree spanning subgraphs
Bollobás and Eldridge [17] as well as Catlin [21] made the following very general conjecture on embedding graphs. If true, this conjecture would be a far-reaching generalization of the Hajnal-Szemerédi theorem (Theorem 1). [17], Catlin [21]). If G is a graph on n vertices with δ(G) ≥ ∆n−1 ∆+1 , then G contains any graph H on n vertices with maximum degree at most ∆.
Conjecture 30 (Bollobás and Eldridge
The conjecture has been proved for graphs H of maximum degree at most 2 [4,7] and for large graphs of maximum degree at most 3 [34]. Recently, Csaba [31] proved it for bipartite graphs H of arbitrary maximum degree ∆, provided the order of H is sufficiently large compared to ∆. In many applications of the Blowup lemma, the graph H is embedded into G by splitting H up into several suitable parts and applying the Blow-up lemma to each of these parts (see e.g. the example in Section 7). It is not clear how to achieve this for H as in Conjecture 30, as H may be an 'expander'. So the proofs in [31,34] rely on a variant of the Blowup lemma which is suitable for embedding such 'expander graphs'. Also, Kaul, Kostochka and Yu [70] showed (without using the Regularity lemma) that the conjecture holds if we increase the minimum degree condition to ∆n+2n/5−1
∆+1
. Theorem 2 suggests that one might replace ∆ in Conjecture 30 with χ(H) − 1, resulting in a smaller minimum degree bound for some graphs H. This is far from being true in general (e.g. let H be a 3-regular bipartite expander and let G be the union of two cliques which have equal size and are almost disjoint). However, Bollobás and Komlós conjectured that this does turn out to be true if we restrict our attention to a certain class of 'non-expanding' graphs. This conjecture was recently confirmed in [15]. The bipartite case was proved earlier by Abbasi [1]. [15]) For every γ > 0 and all integers r ≥ 2 and ∆, there exist β > 0 and n 0 with the following property. Every graph G of order n ≥ n 0 and minimum degree at least (1 − 1/r + γ)n contains every r-chromatic graph H of order n, maximum degree at most ∆ and bandwidth at most βn as a subgraph.
Theorem 31. (Böttcher, Schacht and Taraz
Here the bandwidth of a graph H is the smallest integer b for which there exists an enumeration v 1 , . . . , v |H| of the vertices of H such that every edge v i v j of H satisfies |i − j| ≤ b. Note that kth powers of cycles have bandwidth 2k, so Theorem 31 implies an approximate version of Theorem 27. (Actually, this is only the case if n is a multiple of k + 1, as otherwise the kth power of a Hamilton cycle fails to be (k + 1)-colourable. But [15] contains a more general result which allows for a small number of vertices of colour k + 2.) A further class of graphs having small bandwidth and bounded degree are planar graphs with bounded degree [13]. (See [92,97] for further results on embedding planar graphs in graphs of large minimum degree.) Note that the discussion in Section 2 implies that the minimum degree bound in Theorem 31 is approximately best possible for certain graphs H but not for all graphs. Abbasi [2] showed that there are graphs H for which the linear error term γn in Theorem 31 is necessary. One might think that one could reduce the error term to a constant for graphs of bounded bandwidth. However, this turns out to be incorrect. (We grateful to Peter Allen for pointing this out to us.)
Alternatively, one can try to replace the bandwidth assumption in Theorem 31 with a less restrictive parameter. For instance, Csaba [32] gave a minimum degree condition on G which guarantees a copy of a 'well-separated' graph H in G. Here a graph with n vertices is α-separable if there is a set S of vertices of size at most αn so that all components of H − S have size at most αn. It is easy to see that every graph with n vertices and bandwidth at most βn is √ β-separable. (Moreover large trees are α-separable for α → 0 but need not have small bandwidth, so considering separability is less restrictive than bandwidth.)
Here is another common generalization of Dirac's theorem and the triangle case of Theorem 1 (i.e. the Corrádi-Hajnal theorem). It proves a conjecture by El-Zahar (actually, El-Zahar made the conjecture for all values of n, this is still open).
Theorem 32. (Abbasi [1]) There exists an integer n 0 so that the following holds. Suppose that G is a graph on n ≥ n 0 vertices and n 1 , . . . , n k ≥ 3 are so that
k i=1 n i = n and δ(G) ≥ k i=1 ⌈n i /2⌉.
Then G has k vertex-disjoint cycles whose lengths are n 1 , . . . , n k .
Note that k i=1 ⌈n i /2⌉ = k i=1 (1 − 1/χ cr (C i ))n i ,
where C i denotes a cycle of length n i . This suggests the following more general question (which was raised by Komlós [81]): Given t ∈ N, does there exists an n 0 = n 0 (t) such that whenever H 1 , . . . , H k are graphs which each have at most t vertices and which together have n ≥ n 0 vertices and whenever G is a graph on n vertices with minimum degree at least i (1 − 1/χ cr (H i ))|H i |, then there is a set of vertex-disjoint copies of H 1 , . . . , H k in G? In this form, the question has a negative answer by (the lower bound in) Theorem 6, but it would be interesting to find a common generalization of Theorems 6 and 32.
It is also natural to ask corresponding questions for oriented and directed graphs. As in the case of Hamilton cycles, the questions appear much harder than in the undirected case and again much less is known. Keevash and Sudakov [75] recently obtained the following result which can be viewed as an oriented version of the ∆ = 2 case of Conjecture 30. [75]) There exist constants c, C and an integer n 0 so that whenever G is an oriented graph on n ≥ n 0 vertices with minimum semidegree at least (1/2 − c)n and whenever n 1 , . . . , n t are so that t i=1 n i ≤ n − C, then G contains disjoint cycles of length n 1 , . . . , n t . In the case of triangles (i.e. when all the n i = 3), they show that one can choose C = 3 (one cannot take C = 0). [75] also contains a discussion of related open questions for tournaments and directed graphs. Similar questions were also raised earlier by Song [123]. For instance, given t, what is the smallest integer f (t) so that all but a finite number of f (t)-connected tournaments T satisfy the following: Let n be the number of vertices of T and let t i=1 n i = n. Then T contains disjoint cycles of length n 1 , . . . , n t .
Theorem 33. (Keevash and Sudakov
Ramsey Theory
The Regularity lemma can often be used to show that the Ramsey numbers of sparse graphs H are small. (The Ramsey number R(H) of H is the smallest N ∈ N such that for every 2-colouring of the complete graph on N vertices one can find a monochromatic copy of H.) In fact, the first result which demonstrated the use of the Regularity lemma in extremal graph theory was the following result of Chvátal, Rödl, Szemerédi and Trotter [23], which states that graphs of bounded degree have linear Ramsey numbers: Theorem 34. (Chvátal, Rödl, Szemerédi and Trotter [23]) For all ∆ ∈ N there is a constant C = C(∆) so that every graph H with maximum degree ∆(H) ≤ ∆ and n vertices satisfies R(H) ≤ Cn.
The constant C arising from the original proof (based on the Regularity lemma) is quite large. The bound was improved in a series of papers. Recently, Fox and Sudakov [40] showed that R(H) ≤ 2 4χ(H)∆ n (the bipartite case was also proved independently by Conlon [24]). For bipartite graphs, a construction from [49] shows that this bound is best possible apart from the value of the absolute constant 4 · 2 appearing in the exponent.
Theorem 34 was recently generalized to hypergraphs [27,28,106,64] using hypergraph versions of the Regularity lemma. Subsequently, Conlon, Fox and Sudakov [25] gave a shorter proof which gives a better constant and does not rely on the Regularity lemma.
One of the most famous conjectures in Ramsey theory is the Burr-Erdős conjecture on d-degenerate graphs, which generalizes Theorem 34. Here a graph G is d-degenerate if every subgraph has a vertex of degree at most d. In other words, G has no 'dense' subgraphs. [19]) For every d there is a constant C = C(d) so that every d-degenerate graph H on n vertices satisfies R(H) ≤ Cn.
Conjecture 35. (Burr and Erdős
It has been proved in many special cases (see e.g. the introduction of [41] for a recent overview). Also, Kostochka and Sudakov [91] proved that it is 'approximately' true: The exponent '2d/(2d + 1)' of the logarithm was improved to '1/2' in [41]. All the results in [24,40,41,91] rely on variants of the same probabilistic argument, which was first applied to special cases of Conjecture 35 in [90]. To give an idea of this beautiful argument, we use a simple version to give a proof of the following density result (which is implicit in several of the above papers): it implies that bipartite graphs H whose maximum degree is logarithmic in their order have polynomial Ramsey numbers. An immediate corollary is that the Ramsey number of a d-dimensional cube Q d is polynomial in its number n = 2 d of vertices (this fact was first observed in [120] based on an argument similar to that in [90]). The best current bound of R(Q d ) ≤ d2 2d+5 is given in [40]. Burr and Erdős [19] conjectured that the bound should actually be linear in n = 2 d . Note that the proof immediately shows that the bound on the maximum degree of H can be relaxed: all we need is the property that every subgraph of H has a vertex b ∈ B ′ of low degree. In the proof of (the bipartite case) of Theorem 36, this was exploited as follows: roughly speaking one carries out the above argument twice (of course with different parameters than the above). The first time we consider a random subset S ⊆ B and the second time we consider a smaller random subset S ′ ⊆ T .
For some types of sparse graphs H, one can give even more precise estimates for R(H) than the ones which follow from the above results. For instance, Theorem 12 has an immediate application to the Ramsey number of trees.
Corollary 38.
There is an integer n 0 so that if T n is a tree on n ≥ n 0 vertices then R(T n ) ≤ 2n − 2.
Indeed, to derive Corollary 38 from Theorem 12, consider a 2-colouring of a complete graph K 2n−2 on 2n − 2 vertices, yielding a red graph G r and a blue graph G b . Order the vertices x i according to their degree (in ascending order) in G r . If x n−1 has degree at least n − 1 in G r , then we can apply Theorem 12 to find a red copy of T in G r . If not, we can apply it to find a blue copy of T in G b . For even n, the bound is best possible (let T be a star and let G b and G r be regular of the same degree) and proves a conjecture of Burr and Erdős [20]. For odd n, they conjectured that the answer is 2n − 3. Similarly, the Komlós-Sós conjecture (Conjecture 13) would imply that R(T n , T m ) ≤ n+m−2, where T n and T m are trees on n and m vertices respectively. Of course, Corollary 38 is not best possible for every tree. For instance, in the case when the tree is a path, Gerencsér and Gyarfas [44] showed that R(P n , P n ) = ⌊(3n − 2)/2⌋. Further recent results on Ramsey numbers of paths and cycles (many of which rely on the Regularity lemma) can be found e.g. in [51,38]. Hypergraph versions (i.e. Ramsey numbers of tight cycles, loose cycles and Berge-cycles) were considered e.g. in [58,59,50].
A sample application of the Regularity and Blow-up lemma
In order to illustrate the details of the Regularity method for those not familiar with it, we now prove Theorem 2 for the case when H := C 4 and when we replace the constant C in the minimum degree condition with a linear error term.
Theorem 39. For every 0 < η < 1/2 there exists an integer n 0 such that every graph G whose order n ≥ n 0 is divisible by 4 and whose minimum degree is at least n/2 + ηn contains a perfect C 4 -packing. We also write d(A, B) if this is unambiguous. Given ε > 0, we say that G is ε-regular if for all sets X ⊆ A and Y ⊆ B with |X| ≥ ε|A| and |Y | ≥ ε|B| we have |d(A, B) − d(X, Y )| < ε. Given d ∈ [0, 1), we say that G is (ε, d)-superregular if all sets X ⊆ A and Y ⊆ B with |X| ≥ ε|A| and |Y | ≥ ε|B| satisfy d(X, Y ) > d and, furthermore, if d G (a) > d|B| for all a ∈ A and d G (b) > d|A| for all b ∈ B. Moreover, we will denote the neighbourhood of a vertex x in a graph G by N G (x). Given disjoint sets A and B of vertices of G, we write (A, B) G for the bipartite subgraph of G whose vertex classes are A and B and whose edges are all the edges of G between A and B.
Szemerédi's Regularity lemma [125] states that one can partition the vertices of every large graph into a bounded number 'clusters' so that most of the pairs of clusters induce ε-regular bipartite graphs. Proofs are also included in [16] and [35]. Algorithmic proofs of the Regularity lemma were given in [6,42]. There are also several versions for hypergraphs (in fact, all the results in Section 4.5 are based on some hypergraph version of the Regularity lemma). The first so-called 'strong' versions for r-uniform hypergraphs were proved in [48] and [107,119].
Lemma 40 (Szemerédi [125]). For all ε > 0 and all integers k 0 there is an N = N (ε, k 0 ) such that for every graph G on n ≥ N vertices there exists a partition of V (G) into V 0 , V 1 , . . . , V k such that the following holds:
• k 0 ≤ k ≤ N and |V 0 | ≤ εn, • |V 1 | = · · · = |V k | =: m, • for all but εk 2 pairs 1 ≤ i < j ≤ k the graph (V i , V j ) G is ε-regular.
Unfortunately, the constant N appearing in the lemma is very large, Gowers [47] showed that it has at least a tower-type dependency on ε. We will use the following degree form of Szemerédi's Regularity lemma which can be easily derived from Lemma 40.
Lemma 41 (Degree form of the Regularity lemma). For all ε > 0 and all integers k 0 there is an N = N (ε, k 0 ) such that for every number d ∈ [0, 1) and for every graph G on n ≥ N vertices there exist a partition of V (G) into V 0 , V 1 , . . . , V k and a spanning subgraph G ′ of G such that the following holds:
• k 0 ≤ k ≤ N and |V 0 | ≤ εn, • |V 1 | = · · · = |V k | =: m, • d G ′ (x) > d G (x) − (d + ε)n for all vertices x ∈ G, • for all i ≥ 1 the graph G ′ [V i ] is empty, • for all 1 ≤ i < j ≤ k the graph (V i , V j ) G ′ is ε-regular and has density either 0 or > d.
The sets V i (i ≥ 1) are called clusters, V 0 is called the exceptional set and G ′ is called the pure graph.
Sketch of proof of Lemma 41
To obtain a partition as in Lemma 41, apply Lemma 40 with parameters d, ε ′ , k ′ 0 satisfying 1/k ′ 0 , ε ′ ≪ ε, d, 1/k 0 to obtain clusters V ′ 1 , . . . , V ′ k ′ and an exceptional set V ′ 0 . (Here a ≪ b < 1 means that there is an increasing function f such that all the calculations in the argument work as long as a ≤ f (b).) Let m ′ := |V ′ 1 | = · · · = |V ′ k ′ |. Now delete all edges between pairs of clusters which are not ε ′ -regular and move any vertices into V ′ 0 which were incident to at least εn/10 (say) of these deleted edges. Secondly, delete all (remaining) edges between pairs of clusters whose density is at most d + ε ′ . Consider such a pair (V ′ i , V ′ j ) of clusters. For every vertex x ∈ V ′ i which has more than (d + 2ε ′ )m ′ neighbours in V ′ j mark all but (d + 2ε ′ )m ′ edges between x and V ′ j . Do the same for the vertices in V ′ j and more generally for all pairs of clusters of density at most d + ε ′ . It is easy to check that in total this yields at most ε ′ n 2 marked edges. Move all vertices into V ′ 0 which are incident to at least εn/10 of the marked edges. Thirdly, delete any edges within the clusters. Finally, we need to make sure that the clusters have equal size again (as we may have lost this property during the deletion process). This can be done by splitting up the clusters into smaller subclusters (which contain almost all the vertices and have equal size) and moving a small number of further vertices into V ′ 0 . A straightforward calculation shows that the new exceptional set V 0 has size at most εn as required.
The reduced graph R is the graph whose vertices are 1, . . . , k and in which i is joined to j whenever the bipartite subgraph (V i , V j ) G ′ of G ′ induced by V i and V j is ε-regular and has density > d. Thus ij is an edge of R if and only if G ′ has an edge between V i and V j . Roughly speaking, the following result states that R almost 'inherits' the minimum degree of G.
Proposition 42. If 0 < 2ε ≤ d ≤ c/2 and δ(G) ≥ cn then δ(R) ≥ (c − 2d)|R|.
Proof. Consider any vertex i of R and pick x ∈ V i . Then every neighbour of
x in G ′ lies in V 0 ∪ j∈N R (i) V j . Thus (c − (d + ε))n ≤ d G ′ (x) ≤ d R (i)m + εn and so d R (i) ≥ (c − 2d)n/m ≥ (c − 2d)|R| as required.
The proof of Proposition 42 is a point where it is important that R was defined using the graph G ′ obtained from Lemma 41 and not using the partition given by Lemma 40. In our proof of Theorem 39 the reduced graph R will contain a Hamilton path P . Recall that every edge ij of P ⊆ R corresponds to the ε-regular bipartite subgraph (V i , V j ) G ′ of G ′ having density > d. The next result shows that by removing a small number of vertices from each cluster (which will be added to the exceptional set V 0 ) we can guarantee that the edges of P even correspond to superregular pairs. Proposition 43. Suppose that 4ε < d ≤ 1 and that P is a Hamilton path in R. Then every cluster V i contains a subcluster V ′ i ⊆ V i of size m − 2εm such that (V ′ i , V ′ j ) G ′ is (2ε, d − 3ε)-superregular for every edge ij ∈ P . Proof. We may assume that P = 1 . . . k. Given any i < k, the definition of regularity implies that there are at most εm vertices x ∈ V i such that |N G ′ (x) ∩ V i+1 | ≤ (d − ε)m. Similarly, for each i > 1 there are at most εm vertices x ∈ V i such that |N G ′ (x) ∩ V i−1 | ≤ (d − ε)m. Let V ′ i be a subset of size m − 2εm of V i which contains none of the above vertices (for all i = 1, . . . , k). Then V ′ 1 , . . . , V ′ k are as required.
Of course, in Proposition 43 it is not important that P is a Hamilton path. One can prove an analogue whenever P is a subgraph of R of bounded maximum degree. We will also use the following special case of the Blow-up lemma of Komlós, Sárközy and Szemerédi [83]. It implies that dense superregular pairs behave like complete bipartite graphs with respect to containing bounded degree graphs as subgraphs, i.e. if the superregular pair has vertex classes V i and V j then any bounded degree bipartite graph on these vertex classes is a subgraph of this superregular pair. An algorithmic version of the Blow-up lemma was proved by the same authors in [84]. A hypergraph version was recently proved by Keevash [72].
Lemma 44 (Blow-up lemma, bipartite case). Given d > 0 and ∆ ∈ N, there is a positive constant ε 0 = ε 0 (d, ∆) such that the following holds for every ε < ε 0 . Given m ∈ N, let G * be an (ε, d)-superregular bipartite graph with vertex classes of size m. Then G * contains a copy of every subgraph H of K m,m with ∆(H) ≤ ∆.
Proof of Theorem 39
We choose further positive constants ε and d as well as n 0 ∈ N such that 1/n 0 ≪ ε ≪ d ≪ η < 1/2. (In order to simplify the exposition we will not determine these constants explicitly.) We start by applying the degree form of the Regularity lemma (Lemma 41) with parameters ε, d and k 0 := 1/ε to G to obtain clusters V 1 , . . . , V k , an exceptional set V 0 , a pure graph G ′ and a reduced graph R. Thus k := |R| and (1) δ(R) ≥ (1/2 + η − 2d)k ≥ (1 + η)k/2 by Proposition 42. So R contains a Hamilton path P (this follows e.g. from Dirac's theorem). By relabelling if necessary we may assume that P = 1 . . . k.
Apply Proposition 43 to obtain subclusters V ′ i ⊆ V i of size m − 2εm =: m ′ such that for every edge i(i + 1) ∈ P the bipartite subgraph (V ′ i , V ′ i+1 ) G ′ of G ′ induced by V ′ i and V ′ i+1 is (2ε, d/2)-superegular. Note that the definition of ε-regularity implies that (V ′ i , V ′ j ) G ′ is still 2ε-regular of density at least d−ε ≥ d/2 whenever ij is an edge of R. We add all those vertices of G that are not contained in some V ′ i to the exceptional set V 0 . Moreover, if k is odd then we also add all the vertices in V ′ k to V 0 . We still denote the reduced graph by R, its number of vertices by k and the exceptional set by V 0 . Thus |V 0 | ≤ εn + 2εn + m ≤ 4εn.
Let M denote the perfect matching in P . So M consists of the edges 12, 34, . . . , (k− 1)k. The Blow-up lemma would imply that for every odd i the bipartite graph (V ′ i , V ′ i+1 ) G ′ contains a perfect C 4 -packing, provided that 2 divides m ′ . So we have already proved that G contains a C 4 -packing covering almost all of its vertices (this can also be easily proved without the Regularity lemma). In order to obtain a perfect C 4 -packing, we have to incorporate the exceptional vertices.
To make it simpler to deal with divisibility issues later on, for every odd i we will now choose a set X i of 7 vertices of G which we can put in any of V ′ i and V ′ i+1 without destroying the superregularity of (V ′ i , V ′ i+1 ) G ′ . More precisely, (1) implies that the vertices i and i + 1 of R have a common neighbour, j say. Recall that both (V ′ i , V ′ j ) G ′ and (V ′ i+1 , V ′ j ) G ′ are 2ε-regular and have density at least d/2. So almost all vertices in V ′ j have at least (d/2−2ε)m ′ neighbours in both V ′ i and V ′ i+1 . Let X i ⊆ V ′ j be a set of 7 such vertices. Clearly, we may choose the sets X i disjoint for distinct odd i. Remove all the vertices in X 1 ∪ X 3 ∪ · · · ∪ X k−1 =: X from the clusters they belong to. By removing at most |X|k ≤ 7k 2 further vertices and adding them to the exceptional set we may assume that the subclusters V ′′ i ⊆ V ′ i thus obtained satisfy |V ′′ 1 | = · · · = |V ′′ k | =: m ′′ . (The vertices in X are not added to V 0 .) Note that we now have |V 0 | ≤ 4εn + 7k 2 ≤ 5εn.
Consider any vertex x ∈ V 0 . Call an odd i good for x if x has at least η 2 m ′′ neighbours in both V ′′ i and V ′′ i+1 (in the graph G ′ ). Then the number g x of good indices satisfies (1/2+η/2)n ≤ d G ′ (x)−|V 0 |−|X| ≤ 2g x m ′′ +(k/2−g x )(1+η 2 )m ′′ ≤ 2g x m ′′ +(1+η 2 )n/2,
Acknowledgment
We would like to thank Demetres Christofides, Nikolaos Fountoulakis and Andrew Treglown for their comments on an earlier version of this manuscript.
Figure 1 .
1The graph Γ 3 = Γ 3,1 in Conjecture 8 2.3. r-partite versions. Also, it is natural to consider r-partite versions of the Hajnal-Szemerédi theorem. For this, given an r-partite graph G, let δ ′ (G) denote the minimum over all vertex classes W of G and all vertices x /
Fig. 2 )Figure 2 .
22. For n of the form n = 4m+3 where m is odd, we construct G on n vertices as follows. Partition the vertices into 4 parts A, B, C, D, with |A| = |C| = m, |B| = m + 1 and |D| = m + 2. Each of A and C spans a regular tournament, B and D are joined by a bipartite tournament (i.e. an orientation of the complete An extremal example for Theorem 18
Theorem 18. (Keevash, Kühn and Osthus[74]) There exists an integer n 0 so that any oriented graph G on n ≥ n 0 vertices with minimum semidegree δ 0 (G)
Conjecture 21. (Kelly) Every regular tournament on n vertices can be partitioned into (n − 1)/2 edge-disjoint Hamilton cycles.
− 1 − k, . . . , n − 1 − k n−2k times , n − 1, . . . , n − 1 k times
Theorem 36.(Kostochka and Sudakov [91]) For every d there is a constant C = C(d) so that every d-degenerate graph H on n vertices satisfies R(H) ≤ 2 C(log n) 2d/(2d+1) n.
(The logarithms in the statement and the proof are binary.) Theorem 37. Suppose that H = (A ′ , B ′ , E ′ ) is a bipartite graph on n ≥ 2 vertices and ∆(H) ≤ log n. Suppose that m ≥ n 8 . Then every bipartite graph G = (A, B, E) with |A| = |B| = m and at least m 2 /8 edges contains a copy of H. In particular, R(H) ≤ 2n 8 .
= n 2 .
2Proof. Write ∆ := log n. Let b 1 , . . . , b s be a sequence of s := 2∆ not necessarily distinct vertices of B, chosen uniformly and independently at random and write S := {b 1 , . . . , b s }. Let N (S) denote the set of common neighbours of vertices in S. Clearly, S ⊆ N (a) for every a ∈ N (S). So Jensen's inequality implies that E(|N (S)We say that a subset W ⊆ A is bad if it has size ∆ and its common neighbourhood N (W ) satisfies |N (W )| < n. Now let Z denote the number of bad subsets W of N (S). Note that the probability that a given set W ⊆ A lies in N (S) equals (|N (W )|/m) s (since the probability that it lies in the neighbourhood of a fixed vertex b ∈ B is |N (W )(|N (S)|−Z) ≥ n 2 −1 ≥ n and hence there is a choice of S with |N (S)|−Z ≥ n. By definition, we can delete a vertex from every bad W contained in N (S) to obtain a set T ⊆ N (S) with |T | ≥ n so that every subset W ⊆ T with |W | = ∆ satisfies |N (W )| ≥ n. Clearly we can now embed H: first embed A ′ arbitrarily into T and then embed the vertices of B ′ one by one into B, using the property that T has no bad subset.The bound on R(H) can be derived as follows: consider any 2-colouring of the complete graph on 2n 8 vertices. Partition its vertices arbitrarily into two sets A and B of size n 8 and then apply the main statement to the subgraph of G induced by the colour class having the most edges between A and B.
(
Note that Theorem 39 also follows from Theorems 31 and 32.) We start with the formal definition of ε-regularity. The density of a bipartite graph G = (A, B) with vertex classes A and B is d G (A, B) := e G (A, B) |A||B| .
which shows that g x ≥ ηk/8 = η|M |/4. Since |V 0 |/( √ εm ′′ ) ≤ η|M |/4, this implies that we can assign each x ∈ V 0 to an odd index i which is good for x in such a way that to each odd i we assign at most √ εm ′′ exceptional vertices. Now consider any matching edge i(i + 1) ∈ M . Add each exceptional vertex assigned to i to V ′ i or V ′ i+1 so that the sizes of the sets V * i ⊇ V ′′ i and V * i+1 ⊇ V ′′ i+1 obtained in this way differ by at most 1. It is easy to check that the bipartite subgraph (Since the vertices in X i can be added to any of V * i and V * i+1 without destroying the superregularity of (V * i , V * i+1 ) G ′ , we could now apply the Blow-up lemma to find a C 4 -packing ofwhich covers all but at most 3 vertices (and so altogether these packings would form a C 4 -packing of G covering all but at most 3k vertices of G). To ensure the existence of a perfect C 4 -packing, we need to make |V * i ∪ V * i+1 ∪ X i | divisible by 4 for every odd i. We will do this for every i = 1, 3, . . . , k − 1 in turn by shifting the remainders mod 4 along the path P . More precisely, suppose that |V *Remove the vertices in these copies from the clusters they belong to and still denote the subclusters thus obtained by V * i . (Each such copy of C 4 can be found greedily using that bothhave density at least d/8. Indeed, to find the first copy, pick any vertexThe regularity of (V * 2 , V * 3 ) G ′ implies that almost all vertices in V * 2 can play the role of x. The regularity of (V * 3 , V * 4 ) G ′ now implies that its bipartite subgraph induced by the neighbourhood of x in V * 3 and by V * 4 has density at least d/8 − 2 √ ε. So there are many vertices y ∈ V * 4 which have at least 2 neighbours in N G ′ (x) ∩ V * 3 . Then x and y together with 2 such neighbours form a copy of C 4 .) Now |V * 1 ∪ V * 2 ∪ X 1 | is divisible by 4. Similarly, by removing at most 3 further copies of C 4 , each having 1 vertex in V * 4 , 2 vertices in V * 5 and 1 vertex in V * 6 we can achieve that |V * 3 ∪ V * 4 ∪ X 3 | is divisible by 4. Since n = |G| is divisible by 4 we can continue in this way to achieve that |V * i ∪ V * i+1 ∪ X i | divisible by 4 for every odd i. Recall that before we took out all these copies of C 4 , for every odd i the sizes of V * i and V * i+1 differed by at most 1. Thus now these sizes differ (crudely) by at most 7. But every vertex x ∈ X i can be added to both V * i and V * i+1 without destroying the superregularity. Add the vertices from X i to V * i and V * i+1 in such a way that the sets V ⋄ i ⊇ V * i and V ⋄ i+1 ⊇ V * i+1 thus obtained have equal size. (This size must be even since)-superregular. Thus we can apply the Blow-up lemma (Lemma 44) to obtain a perfect C 4 -packing in (V ⋄ i , V ⋄ i+1 ) G ′ . The union of all these packings (over all odd i) together with the C 4 's we have chosen before form a perfect C 4 -packing of G.
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|
[] |
[
"QUIVER W-ALGEBRAS",
"QUIVER W-ALGEBRAS"
] |
[
"Taro Kimura ",
"Vasily Pestun "
] |
[] |
[] |
For a quiver with weighted arrows we define gauge-theory K-theoretic Walgebra generalizing the definition of Shiraishi et al., and Frenkel and Reshetikhin. In particular, we show that the qq-character construction of gauge theory presented by Nekrasov is isomorphic to the definition of the W-algebra in the operator formalism as a commutant of screening charges in the free field representation. Besides, we allow arbitrary quiver and expect interesting applications to representation theory of generalized Borcherds-Kac-Moody Lie algebras, their quantum affinizations and associated W-algebras.
|
10.1007/s11005-018-1072-1
|
[
"https://arxiv.org/pdf/1512.08533v4.pdf"
] | 119,632,607 |
1512.08533
|
dc1ebd4159224cde509017ac7b74c161e9285262
|
QUIVER W-ALGEBRAS
28 Aug 2016
Taro Kimura
Vasily Pestun
QUIVER W-ALGEBRAS
28 Aug 2016
For a quiver with weighted arrows we define gauge-theory K-theoretic Walgebra generalizing the definition of Shiraishi et al., and Frenkel and Reshetikhin. In particular, we show that the qq-character construction of gauge theory presented by Nekrasov is isomorphic to the definition of the W-algebra in the operator formalism as a commutant of screening charges in the free field representation. Besides, we allow arbitrary quiver and expect interesting applications to representation theory of generalized Borcherds-Kac-Moody Lie algebras, their quantum affinizations and associated W-algebras.
Introduction
Let Γ be a quiver with µ-weighted arrows µ : Γ 1 → C × . We construct two-parametric algebra W q 1 ,q 2 (Γ) from the equivariant K-theory on the moduli space of Γ-quiver sheaves on C 2 q 1 ,q 2 . If a quiver is simply-laced Dynkin graph, our construction agrees with Frenkel-Reshetikhin definition [1] of W q 1 ,q 2 (g Γ ) on the one hand. On the other hand, it explains that certain observables of the gauge theory, coming from the q 2 -lift of the gauge-theory construction of q 1 -characters in [2], described in details and called q 1 q 2 -characters in [3], can be promoted to operator valued currents that form non-commutative associative current algebra W q 1 ,q 2 (g Γ ).
In our constructions the q 1 q 2 -character currents are valued in the algebra of differential operators in higher times t i,1 , . . . , t i,∞ of the gauge theory [4]. We show that the algebra of these differential operators is equivalent to the Heisenberg algebra of q 1 q 2 -bosons used by Shiraishi et al. [5] and Frenkel-Reshetikhin [1,6] to define W q 1 ,q 2 (g Γ ) algebra. Specializing to g Γ of ADE type (in this case Γ has no loops and hence µ-parameters represent necessarily the trivial class in H 1 (Γ, C × ) and hence are gauged away) we show that the pole cancellation construction of [2] and [3] developed from the cut cancellation construction of [7], is isomorphic to the definition of W q 1 ,q 2 (g Γ ) algebra in [1] as the commutant of screening charges, hence explaining the isomorphism between the gauge theory construction presented in [3] and the algebraic contruction of [1].
The gauge theory definition of W q 1 ,q 2 (Γ) algebra is symmetric in exchange 1 ↔ 2. However, for the free field realization there is a choice between (12) and (21). The equivalence between the two realizations, transparent from the gauge theory, leads to 'quantum q-geometric' Langlands equivalence. The duality of W-algebras and the connection with geometric Langlands was first found in [8]. The 'quantum q-geometric' Langlands [1,9] degenerates to the (CFT) 'quantum geometric' Langlands duality β ↔ β −1 in the limit q 1 = e ǫ 1 , q 2 = e ǫ 2 with ǫ 1 , ǫ 2 → 0 and β = −ǫ 1 /ǫ 2 , and further down in the limit ǫ 1 = , ǫ 2 = 0 to the 'geometric' Langlands duality [10][11][12].
For a survey of duality of W-algebras and its connection with the geometric Langlands program see [1] and [13] section 8.6.
In the language of complex integrable systems, and in the reverse order, the 'geometric' Langlands duality (ǫ 1 = , ǫ 2 = 0) is T-duality along the fibers of the phase space of Hitchin integrable system. The K-theory lift to q 1 = e , q 2 = 1 converts Hitchin integrable system on flat curve C to group valued Hitchin integrable system on C [14], equivalently to the integrable system of periodic monopoles on C × S 1 [7]. In the limit q 2 = 1 there is isomorphism, found in [15], between the algebra W q 1 ,q 2 (g Γ ), which turns into commutative algebra, and the K-theory ring of the category of certain representations of the quantum loop group U q 1 (Lg Γ ) W q 1 ,1 (g Γ ) ≃ K(Rep U q 1 (Lg)) (1.1) The character of the R-matrix maps the elements of W q 1 ,1 (g Γ ) to the commuting Hamiltonians of quantum integrable system [2,15].
The geometric realization of U q 1 (Lg Γ ) and of q 1 -characters in W q 1 ,1 (g Γ ) was obtained by Nakajima [16] after Ringel [17], Lusztig [18], Ginzburg-Vasserot [19] from the equivariant K-theory of the C × q 1 -equivariant cotangent bundle of the moduli space T * q 1 M(Γ, CMod) of Γquiver representations in the category of vector spaces C-Mod. To see the q 2 -parameter one needs to consider the central extension of U q 1 (Lg Γ ) to U q 1 (ĝ Γ ) (quantum Drinfeld affinization of g Γ ). The central extension is missing in Nakajima's construction which concerns only the specialization of U q 1 (ĝ Γ ) by the trivial center to U q 1 (Lg Γ ). 1 Compared to Nakajima, we replace a point by a complex variety S and replace CMod by O S Mod, so that we consider equivariant K-theory on the moduli space M(Γ, O S Mod) of Γ-quiver representations in the category of coherent sheaves on a complex variety S. For S = C q 1 ,q 2 we recover W q 1 ,q 2 (g Γ ) from K-theory on M(Γ, O Cq 1 ,q 2 Mod). As proposed by Nekrasov in [3] using complex 4-dimensional setup this should be equivalent to considering the K-theory on the q 2 -twisted fiber-parity-inversed total space of the tangent bundle to Nakajima's quiver variety ΠT q 2 T * q 1 q 2 M(Γ, CMod). We expect that K-theory definition of quiver W-algebra W (Γ, C q 1 ,q 2 ) can be given a more geometric sense in the more general situation when C q 1 ,q 2 is replaced by a generic complex variety S factorized into a product S = S 1 × S 2 and we expect that the 1 ↔ 2 duality will be lifted to higher Langlands duality. The relation to cohomological Hall algebra of quiver [21] remains to be clarified. This paper takes equivariant K-theory as example of generalized cohomological theory corresponding to the supersymmetric 5d theory reduced on S 1 . However, all constructions 1 The parameters (q 2 ) are called by (q, t) in [1,5,9]. However, the parameter t in Nakajima's (q, t)-characters [20], which grades the cohomological degree, has different meaning from the present q 2 remain intact if equivariant K-theory is replaced by the ordinary equivariant cohomology (4d theory) or by equivariant elliptic cohomology (6d theory reduced on elliptic curve). Consequently, the geometric construction of K-theoretic W-algebra can be scaled to its Yangian version [1,15,22] using cohomology and lifted to the elliptic version [23,24] using elliptic cohomology.
For A r -quivers the defining relation of the present note between gauge theory and W (A r )algebra after the 90 • brane rotation (the exchange between the rank of the gauge group in the quiver nodes and the rank of the quiver [25], equivalently Nahm transform, equivalently fiberbase duality) implies the AGT duality of [26][27][28]. The invariance under the brane rotation of the gauge theory partition function is clear from the formalism of refined topological vertex [29][30][31] and was explicitly checked in [32].
We do not restrict Γ to be quiver of finite Dynkin type and consequently expect interesting applications to representation theory of (quantum affinization of) generalized Borcherds-Kac-Moody Lie algebras, such as E 11 symmetry prominently appearing in M-theory or Borcherds Monster Lie algebra for Conway-Norton moonshine. The affine and hyperbolic quivers generate new W-algebras describing affine (such as sinh-Gordon) and hyperbolic quantum 2d Toda models.
Also it would be interesting to interpret the higher times and the meaning of the presented W (Γ)-symmetry in the context of topological string on toric CY realization of the gauge theory partition function for ADE and affine ADE quivers [33].
Note. The origins of the q-Virasoro symmetry the case of single node quiver Γ = A 1 = • to the Vir q 1 ,q 2 = W q 1 ,q 2 (A 1 ) algebra can be traced to Eynard's q-deformed single matrix model [34][35][36]. Elliptic version of matrix model is discussed in [37]. It would be interesting to explore Γ-quiver matrix models beyond Dynkin graphs of finite and affine type [38].
The regularity of qq-characters was explained by Nekrasov in the talk in Strings 2014 and multiple other talks. In [39] the regularity for a linear quiver was interpreted in the language of the quantum toroidal algebra U q 1 ,q 2 (ĝl 1 ) which by Nakajima's construction [40][41][42][43][44][45][46] acts on the instanton moduli spaces on C 2 for each individual node.
The relation between quiver gauge theories and W-algebras in terms of Toda conformal blocks for finite ADE quivers also appeared in [47].
2.1. Quiver. Let Γ be a quiver with the set of nodes Γ 0 and the set of arrows Γ 1 . By i, j ∈ Γ 0 we label the nodes, and by e : i → j we denote an arrow e from the source i = s(e) to the target j = t(e). We allow loops and multiple arrows.
Cartan matrix and Kac
-Moody algebra. A quiver Γ defines |Γ 0 | ×|Γ 0 | matrix (c ij ) c ij = 2 − #(e : i → j) − #(e : j → i) (2.1)
that is called quiver Cartan matrix c. By definition, the quiver Cartan matrix c is symmetric. If there are no single node loops, all diagonal entries of the quiver Cartan matrix are equal to 2 and such Cartan matrix defines Kac-Moody algebra g Γ with Dynkin graph Γ. Let Coh(S) Γ = Rep(Γ, Coh(S)) be the category of representations of quiver Γ in Coh(Q). We call Rep(Γ, Coh(S)) by Γ-quiver gauge theory on S: each node i is sent to a sheaf Y i on S and each arrow e : i → j is sent to an element of
Hom O S (Y i , Y j ).
In the context of N = 2 gauge theories, a sheaf Y i represents gauge connection in the i-th vector multiplet, and an element in
Hom O S (Y i , Y j ) represents field in the i → j bi- fundamental hypermultiplet. 2.4. Moduli space. Let M(Γ, S) = Coh(S) Γ /Aut(Coh(S) Γ ) (2.2)
be the moduli space of Γ-quiver sheaves on S.
Let γ = ch Y denote the Chern character of the collection Y = (Y i ) i∈Γ 0 so that γ = (γ i ) i∈Γ 0 with γ i = ch Y i ∈ H • (S). The Chern character γ i characterizes the topological class of sheaf Y i .
The total moduli space M(Γ, S) of Γ-quiver sheaves on S is a disjoint union over topological sectors
M(Γ, S) = γ M(Γ, S) γ (2.3)
Algebraically, the moduli space M(Γ, S) is a derived stack with virtual tangent bundle T M(Γ, S) at Y given by The total partition function is the sum over the charges
T Y M(Γ, S) = Coh(S) Γ (Y, Y)[1] (2.4) More explicitly T Y M(Γ, S) • = (i e →j)∈Γ 1 Ext • O S (Y i , Y j ) ⊕ i∈Γ 0 Ext •+1 O S (Y i , Y i ) (2.5) 2.5. Universal sheaf. Let Y = (Ŷ i ) i∈ΓZ T (Γ, S) = γ q γ Z T (Γ, S) γ (2.7)
This partition function in the context of N = 2 gauge theories is known under the name K-theoretic Nekrasov partition function, or the partition function of the 5d quiver gauge theory reduced on S 1 [49] where q is the coupling constant.
We can write the partition function using the notation of the derived pushforward π ! = (−1) i R i π * for the projection (integration) map π : M(Γ, S) → point
Z T (Γ, S) = ch T π ! q γ (2.8)
By definition, the partition function Z T,γ , being a character of a virtual representation in Rep(T), can be evaluated on an element t ∈ T. In the context of Nekrasov's partition function the element t comprises all equivariant parameters.
2.8. Fundamental matter. Since quiver Γ is arbitrary, unlike [2,7] where Γ was of finite or affine type, in the present formalism (anti) fundamental matter for a node i is treated simply as bi-fundamental arrow between the node i and another frozen node which is represented in constant sheaves with gauge coupling constant q turned off.
Y i = i * oŶ i (2.9) Let Y [p]
i be the p-th Adams operation applied to Y i . The sheaves (Y [p] i ) i∈Γ 0 ,p∈Z ≥0 generate the ring of observables (using the direct sum modulo equivalence from exact sequences as the addition and the tensor product as the multiplication) which is a subring in the T-equivariant K-theory of sheaves on M(Γ, S).
2.10.
Extended partition function. We fix quiver Γ and the space-time S and drop the symbols from the notations.
Associated to the local observables (Y [p] ) i∈Γ 0 ,p∈Z ≥0 introduce parameters, called higher times t = (t i,p ) i∈Γ 0 ,p∈Z ≥1 and Chern-Simons levels (κ i ) i∈Γ 0 . We treat higher times t i,p as the conjugate variable to Y [p] i in the sense of the generating function [4]
Z T (t) = ch T q γ π ! i∈Γ 0 [detŶ i ] κ i exp( ∞ p=1 t i,p Y [p] i ) (2.10) 2.11. Localization. Suppose that space of the T-fixed points in M is a discrete set of points M T with inclusion i T : M T ֒→ M .
Then the generating function (2.10) can be computed by localization formula 2 :
Z T (t) = M T q γ exp ∞ p=1 1 p ch T (T ∨ M T M) [p] i∈Γ 0 ch T [det i * TŶ ] κ i exp ∞ p=1 t i,p ch T (i * T Y i ) [p]
(2.11)
3. Quiver gauge theory on C 2
In this section we specialize to the space-time S = C 2 with marked point o ∈ S and the natural action of complex group GL(2) on S by with fixed point o.
Automorphism group GL(Q).
We denote by Q the fiber of the cotangent bundle to
S at o Q = T ∨ o S (3.1)
Then Q is the defining module for the group of its automorphisms GL(Q). We split Q = Q 1 ⊕ Q 2 with respect to Cartan torus
T Q ≃ GL(Q 1 ) × GL(Q 2 ) ⊂ GL(Q) (3.2)
and define q 1 , q 2 to be corresponding characters
q 1 = ch Q 1 , q 2 = ch Q 2 , Q = ch T Q Q = q 1 + q 2 , q = q 1 q 2 = ch Λ 2 Q (3.3)
Remark. The parameters (q 1 , q 2 ) are exponentiated ǫ-parameters of the gauge theory [7,50].
Automorphism groups GL(N) and GL(M). Let
O(S) = C[z 1 , z 2 ]
, the ring of polynomials in two variables, be the coordinate ring of S = C 2 . The space of sections of a coherent sheaf on S = C 2 is a O(S)-module. Then Γ-quiver gauge theory on S is identified with representation of Γ in O(S)-modules. We can take C 2 ≃ P 2 \ P 1 ∞ and fix framing at
the P 1 ∞ . For a Γ-quiver sheaf Y, let n ∈ Z Γ 0 >0 denote the rank and let k ∈ Z Γ 0 ≥0 denote the instanton charge n(Y) = ch 0 Y, k(Y) = − ch 2 Y (3.4)
Let N = (N i ) i∈Γ 0 be the framing space associated to Γ 0 part of quiver (nodes). To each node i we associate the framing N i ≃ C n i for the respective sheaf Y i on S. Let GL(N) = i∈Γ 0 GL(N i ) be the respective group of automorphisms and let T N be a Cartan torus of GL(N). Let N be the character of N Remark. The parameters µ e are exponentiated mass parameters of the bifundamental fields e : i → j of the gauge theory. If we assign a multiplicity m e to an arrow e ∈ Γ 1 , then the mass-twisting space is M e ≃ C me and its character M = ch T M M = µ 1 + . . . µ me (3.7) Since in our formalism GL(M e ) is reduced to its Cartan torus T Me = (C × ) me , the formalism where an arrow e : i → j is assigned a multiplicity m e is equivalent to the formalism where this arrow is replaced by m e individual arrows i → j.
N = ch T N N = ν 1 + · · · + ν n (3.5) Remark. The parameters (ν i,α ) i∈Γ 0 ,
3.4.
Fundamental matter as background of higher times. Alternatively, fundamental matter can be realized as a background in higher times. To add fundamental multiplet with mass µ to node i it is sufficiently to additively modify the times to
t i,p → t i,p + 1 p q p (1 − q p 1 )(1 − q p 2 ) µ −p (3.9)
To simplify presentation we don't keep track of the fundamental matter since it is a particular case of the higher times theory. In the operator formalism, on the other hand, this shift is imposed by additional vertex operators introduced in Sec. 3.21.
3.5. Localization in quiver theory on C 2 . The localization formula (2.11) to the T-fixed point set M T in the moduli space of Γ-quiver sheaves on S = C 2 can be explicitly computed [49][50][51].
The T-fixed sheaves split into the direct sum of 1-dimensional T-fixed ideal sheaves, which are classified as
T Q -fixed ideals in O(S) ≃ C[z 1 , z 2 ] where the fixed point o ∈ C 2 is the origin o = (0, 0). A T Q -fixed ideal in C[z 1 , z 2 ] of ch 2 = −k is labelled by a partition (λ) = λ 1 ≥ λ 2 ≥ · · · ≥ 0 ≥ 0 . . . of total size |λ| = ∞ i=1 λ i = k. Each box s = (s 1 , s 2 ) in the partition λ with s 1 ∈ [1 . . . ∞] and s 2 ∈ [1 . . . λ s 1 ] is associated to the monomial z s 1 −1 1 z s 2 −1 2 .
The ideal I λ ⊂ O(S) = I ∅ is O(S)-generated by all monomials outside of the partition λ.
Let K λ = I ∅ /I λ be generated by the monomials in the partition λ.
A T N × T Q -fixed O(S)-module Y S =Ŷ(S) of rank n splits into direct sum of T Q -fixed ideals Y S = ⊕ α∈[1...n] I λα ⊗ N α (3.10) and let Y ≡ Y o = i * o Y S . Then we have in K-theory by localization to i o : o ֒→ S [Y S ] = [Y o ]/[ΛQ] (3.11)
where ΛQ = i (−1) i Λ i Q and the division is in formal series. Then
[Y o ] = [N] − [ΛQ][K] (3.12) 3.6. Cotangent moduli space. From (2.5) we find the K-theory class [T ∨ Y M] at T-fixed pointŶ ∈ M T [T ∨ Y M] = 1 [ΛQ ∨ ] (i e →j)∈Γ 1 [M ∨ e ][Y o ] i [Y ∨ o ] j − i∈Γ 0 [Y o ] i [Y ∨ o ] i (3.13)
3.7. Two commutative reductions. Since the space-time S is a product S = S 1 × S 2 the reduction from Y S to Y o can be done in two steps in two ways, either first project along S 2 and then along S 1 (left path) or first project along S 1 and then along S 2 (right path)
[Y S ] [X] := [Y S 1 ] [Y S 2 ] =: [X] [Y o ] ·[ΛQ] ·[ΛQ2] ·[ΛQ1] ·[ΛQ1] ·[ΛQ2]
(3.14)
so that it holds
[Y o ] = [ΛQ 1 ][X], [Y o ] = [ΛQ 2 ][X] (3.15)
3.8. Quantum q-geometric Langlands Duality. The exchange 1 ↔ 2 in the above diagram leads to the quantum q-geometric Langlands duality q 1 ↔ q 2 . See the section 1 for references.
3.9. Intermediate reduction. The class [T ∨ Y M] at fixed pointŶ ∈ M T in the equation (3.13) of the partition function can be expressed in terms of [X] ≡ [Y S 1 ] [T ∨ Y M] = [ΛQ 1 ] [ΛQ ∨ 2 ] − (i,j)∈Γ 0 ×Γ 0 [X ∨ ] i c + ij [X] j (3X i = {x i,α,s 1 } α∈[1...n i ],s 1 ∈[1...∞] , X = ⊔ i∈Γ 0 X i (3.19)
be the set of characters of the monomials associated to boxes (s 1 , λ s 1 + 1) that generate (Y S ) as O(S 2 )-module so that
x i,α,s 1 = ν i,α q s 1 −1 1 q λs 1 2 (3.20) and X i = x∈X i x. Let i : X → Γ 0 be the node label so that i(x) = i for x ∈ X i .
3.11. The partition function. In terms of x-variables the extended partition function (2.11) is
Z T (t) = X ∈M T exp − (x L ,x R )∈X ×X ∞ p=1 1 p 1 − q p 1 1 − q −p 2 (c + i(x L ),i(x R ) ) [p] x −p L x p R × exp x∈X − κ i(x) 2 (log q 2 x − 1) log q 2 x + log q i(x) log q 2 x x + ∞ p=1 (1 − q p 1 )t i(x),p x p (3.21) wherex i,α,s 1 = ν i,α q s 1 −1 1
denotes ground configuration of the empty partition λ = 0, so that the log q 2 x x counts the size of the partition λ equal to the instanton charge k. Remark. In the limit q 2 → 1 the extended partition function is dominated by the critical set X crit determined in [2] and the variables x ∈ X crit satisfy the Bethe ansatz equations.
3.12. Reflection of the index. Let X [p] be p-th Adams operation applied to an object X. Then the following reflection equation holds
exp( ∞ p=1 1 p [(X ∨ ) [p] ]) = (−1) rk X [det X] exp( ∞ p=1 1 p [X [p] ]) (3.22)
3.13. The ordered partition function. Pick an order ≻ on the set X . For example, an order can be chosen by taking |q 1 | ≪ |q −1 2 | < 1 and |ν| ≃ |µ| ≃ 1. Then define
x L ≻ x R if |x L | > |x R |.
The sum over all pairs (x L , x R ) ∈ X × X in the partition function (3.21) can be transformed to the sum over pairs (x L ≻ x R ), over pairs (x L ≺ x R ) and the diagonal pairs (x L = x R ). The diagonal part gives (q, ν, µ, t)-independent factor that we omit. The sum over pairs (x L ≻ x R ) and (x L ≺ x R ) can be combined together using the reflection equation
(3.22) Z T (t) = X ∈M T exp (x L ≻x R )∈Λ 2 X −(c + i(x R ),i(x L ) ) [0] β log x R x L − ∞ p=1 1 p 1 − q p 1 1 − q −p 2 (c i(x L ),i(x R ) ) [p] x p R x L p × exp x∈X − κ i(x) 2 (log q 2 x − 1) log q 2 x + log q i(x) log q 2 x x + ∞ p=1 (1 − q p 1 )t i(x),p x p (3.23)
where β = − log q 1 log q 2 = − ǫ 1 ǫ 2 and the mass deformed Cartan matrix is .24) 3.14. Extended partition function is a state. The exponentiated sum over the pairs (x L ≻ x R ) in the equation (
c ij = c + ij + c − ij , c − ij = q −1 (c + ji ) ∨ , c ij = (δ ji − e:j→i µ −1 e ) + (q −1 δ ji − e:i→j q −1 µ e ) (3|Z T = X ∈M T ≻ x∈X S i(x),x |1 (3.27)
where ≻ denotes the ≻-ordered product over x ∈ X of the vertex operators
S i,x =: exp p>0 s i,−p x p + s i,0 log x +s i,0 + p>0 s i,p x −p : (3.28)
Here the free field modes or oscillators are
s i,−p p>0 = (1 − q p 1 )t i,p , s i,0 = t i,0 , s i,p p>0 = − 1 p 1 1 − q −p 2 c [p] ij ∂ j,p (3.29) with commutation relations [s i,p , s j,p ′ ] = −δ p+p ′ ,0 1 p 1 − q p 1 1 − q −p 2 c [p] ij , p > 0 (3.30)
The conjugate zero modes i,0 satisfies
[s i,0 , s j,p ] = −βδ 0,p c [0] ij (3.31)
The normal product notation : e A 1 e A 2 :, where operators A 1 , A 2 are linear in the free fields, means that all operators (s i,p ) p≤0 are placed to the left of (s i,p ) p>0 ands i,p .
κ i = −(c − ij ) [0] n i , log q 2 q i = β + t i,0 + (c − ij ) [log q 2 ] n j − (c − ij ) [0] log q 2 ((−1) n j ν j ) (3.32) where (c − ij ) [log q 2 ] = δ ij log q 2 q −1 − e:i→j log q 2 (q −1 µ e ) (3.33)
3.16. Screening charges. The configuration sets X ∈ M T are described by the partitions, which are explicitly collections of constrained sequences
(λ i,α,1 ≥ λ i,α,2 ≥ · · · ≥ 0 = 0 = 0 = . . . ) i∈Γ 0 ,α∈[1...n i ](3.34)
Let X 0 be the ground configuration with all λ i,α, * = 0.
Let Z X 0 be the set of collections of arbitrary integer sequences terminating by zeroes
(λ i,α,s 1 ? λ i,α,s 2 ? . . . ? = 0 = 0 = . . . ) i∈Γ 0 ,α∈[1...n i ] (3.35)
Then M T ⊂ Z X 0 . It turns out that the summation over M T in (3.27) can be extended to the whole Z X 0 ≥0 without changing the result because
≻ x∈X S i(x),x |1 = 0 if X ∈ Z X 0 but X / ∈ M T (3.36)
due to the zero factors in the normal ordering product of vertex operators S i,x for the sequences (x i,α,
s 1 = ν i,α q s 1 −1 1 q λ i,α,s 1 2 )
where λ does not satisfy the constraint (3.34).
Therefore
|Z T = X ∈Z X 0 ≻ x∈X S i(x),x |1 (3.37)
For every pointx i,α,s 1 = ν i,α q s 1 −1 1 in the ground configuration X 0 define the operator called screening charge
S i,x = s 2 ∈Z S i,q s 2 2x (3.38)
Then the state |Z T is obtained by applying to the vacuum the ordered product of S i operators
|Z T = ≻ x∈X 0 S i(x),x |1 (3.39)
The partition function of plain, not t-extended theory, can be interpreted as the projection
1|Z T = 1| ≻ x∈X 0 S i(x),x |1 (3.40)
3.17. The Ward identities. The sum representation of the operator S i,x in (3.38) is explicitly invariant under the Z-translational symmetry s 2 → s 2 + Z (change of variables). Hence the representation of the partition function (3.39) is invariant under the Z X 0 symmetry that shifts the summation variables s 2 for eachx i,α,s 1 . In the S space-time picture the variation s 2 → s 2 + 1 amounts to the z 2 multiplicative change of variables in the z 1 , z 2 -mode expansion φ i,α,s 1 z s 1 1 → z 2 φ i,α,s 1 z s 1 1 where φ is in the sheaf Y. The shift s 2 → s 2 + 1 adds one box to the partition, or equivalently one instanton to the gauge field on the space-time.
3.18. The Y-operators. In [2,7] the (Y i,x ) i∈Γ 0 observables were introduced in the K-theory of the moduli space M T of the quiver gauge theory
Y i,x := exp − ∞ p=1 x −p p Y [p] i (3.41)
The expectation value of the observable Y i,x in the plain (not t-extended) theory is computed by the pushforward integration over the moduli space M T (2.10)
Y i,x := ch T π ! q γ Y i,x (3.42)
It is natural to lift the Y i,x observables to the t-extended theory by giving them the operator definition:
Y i,x = qρ i 1 : exp p>0 y i,−p x p + y i,0 + p>0 y i,p x −p : (3.43) whereρ i := j∈Γ 0c [0]
ij are components of the Weyl vector in the basis of simple roots. If the quiver is the affine type, we putρ i = 0.
The operator Y i,x is an element of the Heisenberg algebra H. The oscillators y i,p are expressed in terms of t i,p and ∂ i,p
(p > 0) y i,−p = (1 − q p 1 )(1 − q p 2 )(c [−p] ) ij t j,p , y i,0 = −c [0] ij t j,0 log q 2 y i,p = − 1 p ∂ i,p (3.44)
or equivalently terms of the free field s i,p
y i,p p =0 = (1 − q −p 2 )c [p] ij s j,p , y i,0 = (log q −1 2 )c [0] ij s j,0 (3.45)
wherec ij is the inverse to the mass-deformed Cartan matrix c ij defined in (3.24). The definition (3.44) and the definition (2.10) imply
Y i,x = 1|Y i,x |Z T (3.46)
3.19. The OPE of Y and S. The commutation relations between y i,p and s j,p ′ are
[y i,p , s j,p ′ ] = − 1 p (1 − q p 1 )δ p+p ′ ,0 δ ij , [s i,0 , y j,0 ] = −δ ij log q 1 (3.47)
Then (3.46) can be also seen from the commutation relations (3.47) and normal ordering because at |x| > |x ′ | we have
Y i,x S i,x ′ = 1 − x ′ /x 1 − q 1 x ′ /x : Y i,x S i,x ′ : Y i,x S j,x ′ = : Y i,x S j,x ′ : i = j (3.48)
Therefore at each fixed point configuration X ∈ M T
1|Y i,x ≻ x ′ ∈X S i(x),x ′ |1 = qρ i 1 x ′ ∈X i 1 − x ′ /x 1 − q 1 x ′ /x 1| ≻ x ′ ∈X S i(x),x ′ |1 (3.49)
like in the definition that was given in [2]. The observable Y i,x is not regular in the C × x because of the possible poles at points x = q 1 x ′ .
3.20. The commutator of Y and S. The commutation relations (3.47) also imply for |x ′ | > |x|
S i,x ′ Y i,x = q −1 1 1 − x/x ′ 1 − q −1 1 x/x ′ : S i,x ′ Y i,x :
(3.50) Therefore (3.48)(3.50) imply the non-zero radial-ordered commutator
[Y i,x , S i,x ′ ] = (1 − q −1 1 )δ(q 1 x ′ x ) : S i,x ′ Y i,x : (3.51)
where by definition δ(z) = n∈Z z n .
The fact that observable Y i,x has singularities at x = q 1 x ′ in (3.50) is equivalent to the presence of the δ(q 1 x ′ /x) in the radial ordered commutator between Y i,x and S i,x ′ . This is a general statement implied by Cauchy integral formula and familiar from the formalism of radial quantization in CFT.
3.21.
The V-operators. We introduce another kind of vertex operator to reproduce the fundamental matter contribution in gauge theory. As explained before, this contribution is given by shift of the time variables (3.9), which can be implemented by the operator
V i,x = exp p =0 v i,p x −p .
(3.52)
The corresponding free field is explicitly written
(p > 0) v i,−p = − c [−p] ij t j,p v i,p = 1 p 1 (1 − q p 1 )(1 − q p 2 ) ∂ i,p . (3.53)
We remark a simple relation to the y-operators (3.44)
v i,p = − 1 (1 − q p 1 )(1 − q p 2 ) y i,p . (3.54)
Then the OPE of V and S operators are given by
V i,x S i,x ′ = x ′ x ; q 2 −1 ∞ : V i,x S i,x ′ : , S i,x ′ V i,x = q 2 x x ′ ; q 2 ∞ : V i,x S i,x ′ : (3.55)
corresponding to the fundamental and antifundamental hypermultiplet contributions. Thus the extended partition function in the presence of (anti)fundamental matters is obtained by inserting the V-operators
|Z T = x∈X f V i(x),x ≻ x∈X 0 S i(x),x x∈X f V i(x),x |1 (3.56) where X f = {µ i,f } i∈Γ 0 ,f ∈[1...n f i ] andX f = {μ i,f } i∈Γ 0 ,f ∈[1.
..ñ f i ] are sets of fundamental and antifundamental mass parameters. This V-operator creates a singularity on the curve at x = µ i,f . Then the plain partition function (t = 0) is given as a correlator as shown in (3.40),
Z T (t = 0) = 1|Z T = 1| x∈X f V i(x),x ≻ x∈X 0 S i(x),x x∈X f V i(x),x |1 .
(3.57)
W-algebra
Here we describe the construction of regular observables T of the extended gauge theory and explain isomoprhism with Shiraishi et al. [5] and Frenkel-Reshetikhin [1] definition of W q 1 ,q 2 algebra as commutant of screening charges in the Heisenberg algebra H, and define K-theoretical quiver W-algebra for S = C q 1 ,q 2 . 4.1. Pole cancellation in T : A 1 -example. Consider simplest quiver Γ = A 1 for example. In [2] in the study of the q 2 = 1 limit of the gauge theory partition function, motivated by cut-crossing story of [7], it was suggested to consider the observable 3
T 1,x = Y 1,x + Y 1,q −1 x (4.1)
for its virtue of being regular function in C × x . This is the simplest example of q-character representing the T -matrix of Baxter coming from U q (ŝl 2 )-integrable system and Baxter equation.
In fact, the same observable T 1,x remains regular function of x for generic q 2 . Indeed, in the operator formalism we find
Y 1,x S 1,x ′ = 1 − x ′ /x 1 − q 1 x ′ /x : Y 1,x S 1,x ′ : Y −1 1,xq −1 S 1,x ′′ = 1 − qq 1 x ′′ /x 1 − qx ′′ /x : Y −1 1,xq −1 S 1,x ′′ : (4.2)
so the potential singularity in the first line is for x ′ = q −1 1 x and in the second line for x ′′ = q −1 x. Therefore, the two singularities have chance to cancel at x ′ = q 2 x ′′ . Recall, that the state |Z T is obtained with the sums (3.38) and there is internal symmetry for the shift of the summation indexing variable (see Ward identity in Sec. 3.17) so that for every term S 1,x ′′ there is a term with S 1,x ′ with x ′ = q 2 x ′′ .
Indeed, we find for the first term the normal ordered expression
: Y 1,x S 1,xq −1 1 : = : exp − 1 2 s 1,0 log q 2 + s 1,0 log(xq −1 1 ) +s 1,0 + p =0 (q p 1 + 1 − q −p 2 1 + q −p )s 1,p x −p :
(4.3) and for the second term the normal ordered expression
q 1 : Y −1 1,xq −1 S 1,xq −1 : = : exp + 1 2 s 1,0 log q 2 + s 1,0 log(xq −1 ) +s 1,0 + p =0 (q p − 1 − q −p 2 1 + q −p q p )s 1,p x −p :
(4.4) which are exactly identical. The respective residues in the prefactors (4.2) are (1 − q −1 1 ) and (q −1 1 − 1) which respectively cancel each other. This computation proves regularity in x ∈ C × of the state |Z T of higher t-extended gauge theory in A 1 example ∂xT 1,x |Z T = 0 (4.5) 3 We adopted the normalizations and the zero modes to the conventions of the present paper in which the T -observables have the simplest canonical form.
4.2.
Commutator of T and S vanishing: A 1 -example. An exactly equivalent presentation of the regularity of T 1,x is the statement that
[T 1,x , S 1,x ′ ] = 0 (4.6)
where S 1,x ′ is the screening charge (3.38) defined as the summation over the q Z 2 shifts. Indeed,
[Y 1,x , S 1,x ′ ] = (1 − q −1 1 )δ(q 1 x ′ x ) : Y 1,x S 1,x ′ : [Y −1 1,xq −1 , S 1,x ′ ] = (q −1 1 − 1)δ(q x ′ x ) : Y 1,xq −1 S 1,x ′ : (4.7)
The total sum is q 2 -difference which cancels after summation over q Z 2 shifts entering definition of screening charge (3.38). This is the consequence of the Ward identity in Sec. 3.17.
4.3. W-algebra of A 1 -quiver. Consequently the operator T 1,x can be moved in the position in the radial-ordered operator-state presentation of the extended gauge theory partition state (3.39)
T 1,x |1 = T 1,x S 1,x ′ S 1,x ′′ . . . |1 = S 1,x ′ T 1,x S 1,x ′′ . . . |1 = S 1,x ′ S 1,x ′′ T 1,x . . . |1 (4.8)
The operators S i,x ′ can be thought as exponentiated Hamiltonians of the q 1 , q 2 -deformed CFT. The commutant of the Hamiltonians is the conserved current T 1,x which is regular
∂xT 1,x = 0 (4.9)
Consequently, T 1,x has well defined, time-radial independent, modes
T 1,x = p∈Z T 1,[p] x −p (4.10)
We define algebra W q 1 ,q 2 (A 1 ) to be the subalgebra in H generated by the modes of the conserved current T 1,x . This definition is in the exact agreement with Shiraishi et al. [5] and Frenkel-Reshetikhin [1]. 4.4. W-algebra of quiver: definition. The definition of the state |Z T (3.39) implies that the current T i,x is regular ∂xT i,x |Z T = 0 (4.11) if it commutes with all screening operators:
[T i,x , S j,x ′ ] = 0 j ∈ Γ 0 , x ′ ∈ X j (4.12)
This explains isomorphism between the gauge theoretic construction of q 1 q 2 -characters [2,3] and definition of W q 1 ,q 2 -algebras [1,5,9] as the algebra generated by currents T i,x which are defined as commutants of screening charges S j in the vertex operator algebra defined by the free fields from Heisenberg algebra H and expressed as
T i,x = Y i,x + . . . (4.13)
We define in the same way the W-algebra W (Γ, S) for generic quiver Γ with generalized even symmetric Borcherds-Kac-Moody-Cartan matrix, mass deformed by µ : Γ 1 → C × , as in equation (3.24), and for S = C q 1 ,q 2 as the algebra generated by currents T i,x commuting with all screening charges (S j,x ) j∈Γ 0 , or equivalently, regular on the higher times extended gauge theory state (4.11). We expect to generalize the definition for more general and possibly higher dimensional varieties S.
Examples
We consider a few examples to illustrate the equivalence between gauge-theory formalism [2,3,7] and the operator formalism [1,5,9,15].
5.1.
Commutator of T and S vanishing: general quiver, local reflection. Suppose that there is no edge loop from a node i to itself and consider
T i,x = Y i,x + : Y −1 i,q −1 x e:i→j Y j,µ −1 e x e:j→i Y j,q −1 µex : + . . . (5.1)
The vanishing of commutator
[T i,x , S i,x ′ ] = 0 (5.2)
follows from (3.51) and the relation
q 1 : Y −1 i,q −1 x e:i→j Y j,µ −1 e x e:j→i Y j,q −1 µex S i,q −1 x : = : Y i,x S i,q −1 1 x : (5.3)
Indeed, this relation is equivalent to
q −1 1 : Y i,x Y i,q −1 x e:i→jc [0] ijc [0] jk log q −1 2 s k,0 + p =0 (1−q −p 2 )q p c [p] ijc [p] jk s k,p x −p = (log(xq −1 )−log(xq −1 1 ))s i,0 + p =0
(q p −q p 1 )s i,p x −p (5.5) thanks to the definition of the µ-dependent Cartan matrix c ij (3.24) and its inversec ij so that c
[p] ijc [p] jk = δ ij . This leads to : Y −1 i,q −1 x e:i→j Y j,µ −1 e x e:j→i Y j,q −1 µex : , S i,x ′ = (q −1 1 − 1)δ(q x ′ x ) : Y 1,xq −1 S 1,x ′ : (5.6)
and therefore (5.2) holds at the level of the first two terms.
The second term contains the Y j fields for the nodes j linked to the node i. This term might give potential singularities, or, equivalently, δ-functions in the commutators associated to the S j operators. Then one needs to continue to apply Weyl reflections to generate terms which cancel the singularities.
The algebraic structure is associated to highest weight Verma module of the generalized Borcherds-Kac-Moody algebra g Γ .
If Γ is of finite Dynkin type the process terminates, the associated Verma module is finitedimensional. The finite-dimensional case was studied in details in [9].
More generally, for infinite-dimensional Verma module, the recursive algorithm is also applicable which builds a tree starting from the root node Y i,x . The vertices of the tree are monomials in the T i,x current, and the edges are colored by the nodes i of the quiver. Two monomials are linked by edge of color i if they are related by the local reflection move (5.1). The algorithm can be computerized.
Alternatively, Nekrasov presented closed formula [3] which express the q 1 q 2 -characters in terms of geometry of Nakajima's quiver variety [16,40]. This formula can be thought as q 2 -deformation of original Nakajima's construction of q-characters of U q (Lg Γ ) from the qequivariant K-theory on the quiver variety M Nak [20]. The formula in [3] amounts to replacing Euler characteristic of T * q M w,v by the q 2 -equivariant Euler class of the tangent bundle to T * q M w,v so effectively to the integration over ΠT q 2 T * q M w,v . Here w : Γ 0 → Z labels the components of the highest weight in the basis of fundamental weights, and v : Γ 0 → Z labels the components of a positive root in the basis of simple roots which is added to the highest weight to get a weight at the level i∈Γ 0 v i in the Verma module.
w,v = T * q M w,v
5.2.
Higher weight currents. Conjecturally, quiver W-algebra is completely generated by the fundamental currents
T i,x = Y i,x + . . . (5.7)
However, higher weight currents T w w,x can be defined where to each node i we assign vector space W of dimension w and character
W i = w i ω=1 w i,ω w i,ω ∈ C × (5.8)
with the first term
T w w,x = : i∈Γ 0 w i ω=1 Y i,xw i,ω : + . . . (5.9)
In the finite-dimensional and irreducible modules of higher weights can be found in the tensor product of the i-fundamental modules with weights w i = 1, w j =i = 0. In the not qdeformed case, usually the tensor product of fundamental modules decomposes into several irreducible components. For example, for sl 2 we have C 2 ⊗ C 2 = C 3 ⊕ C 1 . This does not hold after q-deformation. For generic weights w the tensor product is irreducible.
5.3.
Higher weight current in the A 1 example. In the example of A 1 quiver the higher weight current T w w,x with w 1 ∈ Z >0 for generic weights (w 1,1 , . . . , w 1,w 1 ) contains 2 w terms [3] coming from the cohomologies of Nakajima's quiver variety which in this case are ∐ v≤w T * Gr(w, v).
This higher weight character current T w w,x is elementary to compute from the free-field formalism and normal ordering given the fundamental current T 1,x in equation (4.1)
Consider the product
T 1,w 1 x T 1,w 2 x = (Y 1,xw 1 + Y −1 1,q −1 w 1 x )(Y 1,xw 2 + Y −1 1,q −1 w 2 x ) (5.10)
The normal ordering is computed using the commutator from (3.44) [y i,p , y j,−p ]
p>0 = − 1 p (1 − q p 1 )(1 − q p 2 )c [−p] ji (5.11)
with the result
T 1,w 1 x T 1,w 2 x = f (w 2 /w 1 ) −1 : Y 1,xw 1 Y 1,xw 2 : + S(w 1 /w 2 ) : Y 1,xw 1 Y −1 1,q −1 w 2 x : +S(w 2 /w 1 ) : Y −1 1,xw 1 q −1 Y 1,xw 2 : + : Y −1 1,xw 1 q −1 Y −1 1,q −1 w 2 x : (5.12)
where the scalar prefactor
f (w) = exp ∞ p=1 1 p (1 − q p 1 )(1 − q p 2 ) 1 + q p w p (5.13)
is in agreement with the function f (w) generating the commutation relations for W q 1 ,q 2 (A 1 ) current T 1 (x) in Shiraishi et al. [5] and the permutation factor S(u) is in agreement with formulae for higher qq-characters in [3]
S(w) = (1 − q 1 w)(1 − q 2 w) (1 − qw)(1 − w) (5.14)
which comes from the equivariant Euler characteristic (or its K-theory version) of the fixed point in ΠT q 2 T * q 1 P 1 where T * q 1 P 1 = M Nak w=2,v=1 . This relation leads to
f w 2 w 1 T 1,w 1 x T 1,w 2 x − f w 1 w 2 T 1,w 1 x T 1,w 2 x = (1 − q 1 )(1 − q 2 ) 1 − q δ q w 1 w 2 − δ q w 2 w 1 ,(5.15)
which determines the algebraic relation for the modes T 1, [p] . We remark f (w)f (qw) = S(w).
The degree w current is similarly computed
T [w] 1,x = : Y 1,w 1 x Y 1,w 2 x · · · Y 1,wnx : + · · · = I∪J={1...n} i∈I,j∈J S( w i w j ) : i∈I Y 1,w i x j∈J Y −1 1,w j xq −1 : (5.16)
in agreement with [3]. The S factor becomes trivial in the limit q 2 → 1 and the ordinary formulae for q-character is recovered [2,15].
5.4.
Degeneration and derivatives. By definition, vertex operator algebra involves expressions in fields and their derivatives. Hence we shall expect appearance of the derivatives when two vertex operators fuse.
So consider slightly more general situation of W-algebra currents with local structure
: Y i,x Y i,ux : + S(u) : Y i,ux Y i,xq −1 e:i→j Y j,µ −1 e x e:j→i Y j,µeq −1 x : + S(u −1 ) : Y i,x Y i,uxq −1 e:i→j Y j,uµ −1 e x e:j→i Y j,uµeq −1 x : + : Y −1 i,xq −1 Y −1 i,uxq −1 e:i→j Y j,µ −1 e x Y j,uµ −1 e x e:j→i Y j,µeq −1 x Y j,uµeq −1 x : (5.17)
Taking the collision limit u → 1, this yields a derivative term
Y 2 i,x + : Y i,x Y i,xq −1 e:i→j Y j,µ −1 e x e:j→i Y j,µeq −1 x × c(q 1 , q 2 ) − (1 − q 1 )(1 − q 2 ) 1 − q x∂ x log Y i,x Y i,xq −1 e:i→j Y j,µ −1 e x e:j→i Y j,µeq −1 x : + : Y −2 i,xq −1 e:i→j Y j,µ −1 e x e:j→i Y j,µeq −1 x 2 : (5.18)
where the coefficient c(q 1 , q 2 ) is determined by
c(q 1 , q 2 ) = lim u→1 S(u) + S(u −1 ) = 1 − 6q 1 q 2 + q 2 1 q 2 2 + (1 + q 1 q 2 )(q 1 + q 2 ) (1 − q 1 q 2 ) 2 q 1,2 →1 −→ 2 .
(5.19) 5.5. Edge loop:Â 0 -example. Consider an example of a single node with a loop edge. This corresponds to N = 2 * theory in 4d. Let n be the gauge group rank and µ be the (exponentiated) adjoint mass. The Cartan matrix is (0) and the mass deformed Cartan matrix is
c = 1 + q −1 − µ −1 − q −1 µ (5.20)
The quantum affinization of the respective algebra by Nakajima's quiver construction is U q,µ (Lĝl 1 ) [43][44][45] with q-character given by the sum over all partitions [2]. Here we consider q 2 -deformation to recover W-algebra of 0 -quiver. We need the commutation relation for the oscillator (3.44) [y 1,p , y 1,
p ′ ] = −δ p+p ′ ,0 1 p (1 − q p 1 )(1 − q p 2 ) (1 − µ p )(1 − q p µ −p ) (5.21)
Using this oscillator, we construct W q 1 ,q 2 (g Γ ) algebra associated with the affine quiver Γ =Â 0 . In this case, the local pole cancellation structure is
Y 1,x + S(µ −1 ) : Y −1 1,q −1 x Y 1,µ −1 x Y 1,T 1,x = Y 1,x + S(µ −1 ) : Y −1 1,q −1 x Y 1,µ −1 x Y 1,µq −1 x : + · · · = λZ λ : s∈∂ + λ Y 1,qx/x(s) s∈∂ − λ Y −1 1,x/x(s) : (5.23)
where ∂ + λ and ∂ − λ are the outer and inner boundary of the partition λ, and we definẽ
x(s) = (µ −1 q) s 1 −1 µ s 2 −1 q (5.24)
The combinatorial weightZ λ obeys
Z λ ′ Z λ = − (1 − µ −1 q 1 )(1 − µ −1 q 2 ) (1 − q 1 )(1 − q −1 2 )Ỹ q 1 xỸq 2 x Y ′ qxỸ x x=x k (5.25)
where λ ′ is the shifted partition λ k → λ k + 1, and we define the "dual" functionỸ x
Y x = k=1 1 −x k /x 1 −q 1xk /x (5.26)
with the "dual" parameters
q 1 = µ −1 q ,q 2 = µ ,μ = q 2 ,x k =x(k, λ k + 1) (5.27)
HereỸ ′ is evaluated with the shifted configuration λ ′ . Although this dual function also has poles, such a singularity is cancelled in the following combinatioñ
Y x + (1 −μ −1q 1 )(1 −μ −1q 2 ) (1 −μ −1q )(1 −μ −1 )Ỹ −1 q −1 xỸμ −1 xỸμqx (5.28)
This expression is equivalent to the original one (5.22) in particular for the rank one theory. Again, operator formalism of W-algebra is equivalent to q 2 -deformation of Nakajima's construction [3].
5.6. W-algebra of hyperbolic quiver example. We consider examples of the hyperbolic quiver, where the determinant of the corresponding Cartan matrix is negative. The simplest example is the quiver having a single node with two loop edges:
• c = −(2) (5.29)
Let µ 1,2 be the mass parameter associated with the edges, and the mass deformed Cartan matrix is given by
c = 1 + q −1 − µ −1 1 − µ 1 q −1 − µ −1 2 − µ 2 q −1 (5.30)
Since the local pole cancellation occurs in the following combination Y 1,x + S(µ −1 1 )S(µ −1 2 ) : Y −1 1,q −1 x Y 1,µ −1 1 x Y 1,µ 1 q −1 x Y 1,µ −1 2 x Y 1,µ 2 q −1 x : (5.31) the first few terms of the holomorphic current are given by
T 1,x = Y 1,x + S(µ −1 1 )S(µ −1 2 ) : Y −1 1,q −1 x Y 1,µ −1 1 x Y 1,µ 1 q −1 x Y 1,µ −1 2 x Y 1,µ 2 q −1 x : + S(µ −1 1 )S(µ −1 2 ) 2
S(µ 2 1 q −1 )S(µ 1 µ −1 2 )S(µ 1 µ 2 q −1 ) :
Y 1,µ −2 1 x Y 1,µ 1 q −1 x Y 1,µ −1 1 µ −1 2 x Y 1,µ −1 1 µ 2 q −1 x Y 1,µ −1 2 x Y 1,µ 2 q −1 x Y 1,µ −1 1 q −1 x : + S(µ −2 1 q)S(µ −1 1 µ −1 2 q)S(µ −1 1 µ 2 ) : Y 1,µ −1 1 x Y 1,µ 2 1 q −2 x Y 1,µ 1 µ −1 2 x Y 1,µ 1 µ 2 q −1 x Y 1,µ −1 2 x Y 1,µ 2 q −1 x Y 1,µ 1 q −2 x :
+ (1 ↔ 2) + · · · (5.32)
We can see a cancellation of factors, which is similar to 0 theory, and thus there is no colliding term, e.g. Y 2 1, * , in a numerator. Next example is a rank two quiver with three arrows:
• • c = 2 −3 −3 2 (5.33)
Let us assign three mass parameters µ 1,2,3 to the arrows, and then the local cancellation is
Y 1,x + : Y 2,µ −1 1 x Y 2,µ −1 2 x Y 2,µ −1 3 x Y 1,q −1 x : Y 2,x + : Y 1,µ 1 q −1 x Y 1,µ 2 q −1 x Y 1,µ 3 q −1 x Y 2,q −1 x : (5.34)
The holomorphic current becomes
T 1,x = Y 1,x + : Y 2,µ −1 1 x Y 2,µ −1 2 x Y 2,µ −1 3 x Y 1,q −1 x : + S(µ 1 µ −2 2 )S(µ 1 µ −1 3 ) : Y 1,µ −1 1 µ 2 q −1 Y 1,µ −1 1 µ 3 q −1 Y 2,µ −1 2 x Y 2,µ −1 3 x Y 2,µ −1 1 q −1 x : +S(µ 2 µ −2 1 )S(µ 2 µ −1 3 ) : Y 1,µ −1 2 µ 1 q −1 Y 1,µ −1 2 µ 3 q −1 Y 2,µ −1 1 x Y 2,µ −1 3 x Y 2,µ −1 2 q −1 x
: +S(µ 3 µ −2 1 )S(µ 3 µ −1 1 ) :
Y 1,µ −1 3 µ 1 q −1 Y 1,µ −1 3 µ 2 q −1 Y 2,µ −1 1 x Y 2,µ −1 2 x Y 2,µ −1 3 q −1 x
: + · · · (5.35)
The other current T 2,x is obtained in the same way. Similarly it is expected that there is no collision term in these holomorphic currents.
6. Applications 6.1. Toda scaling limit. In the scaling limit q 1 → 1, q 2 → 1, q i → 1, and log q 2 q 1 , log q 2 q i are finite, the free field commutation relations ( where φ i (x) is the free boson that takes value in the Cartan of g Γ with canonical commutation relations defined by the bilinear form with matrix (c ij ) in the basis of simple roots. Hence S i (x) are vertex primary operators of Kac-Moody g Γ -Toda field theory on punctured disc C × x . In the same scaling limit we find from (3.45) that the field y i (x) = −ǫ 2 bc ij ∂φ j (x) is also primary.
For example, in the ǫ 2 -expansion of T 1 (x) for A 1 -quiver T 1 (x) = : e y(x) : + : e −y(xq −1 ) : = 2 + 1 4
ǫ 2 b 2 ((∂φ) 2 − (b + b −1 )∂ 2 φ) + . . . (6.3)
we find the stress-energy Virasoro current of the free field φ with background charge and the central charge c = 1 + 6(b + b −1 ) 2 (6.4) 6.2. Affine type. If g Γ is of affine type, the g Γ -Toda is affine Toda. For example, the scaling limit of the W-algebra defined by the quiver
• • c = 2 −2 −2 2 (6.5) with g Γ = A(1)
1 describes quantum sin(h)-Gordon theory on punctured disc C × x .
6.3. Nahm transform. The g Γ -Toda theory specializes to the finite Toda if g Γ is of finite type. For sl r -quiver with n colors at each node the sl r -Toda is Nahm dual to the sl n -Toda proposed in [26,27].
2. 3 .
3Quiver sheaves. Choose the space-time to be a complex variety S with structure sheaf O S , and let Coh(S) denote the category of coherent sheaves on S (the category O Q Mod of O S -modules).
0 denote the universal sheaf over M(Γ, S) × S that is associated to the family of sheaves Y i on S parametrized by M(Γ, S).
2. 6 .
6Equivariant version. Suppose we are given an equivariant action of a complex group T on the sheaves Coh(S). Then quiver gauge theory can be defined T-equivariantly. In particular, group T acts on the moduli space M(Γ, S) of T-equivariant Γ-quiver sheaves on S. 2.7. Partition function. Define partition function Z T (Γ, S) γ in topological sector γ be the T-equivariant index (holomorphic equivariant Euler characteristic) of the structure sheaf on the moduli space of Γ-quiver sheaves on S of charge γ Z T (Γ, S) γ = n∈Z (−1) n ch T H n (M(Γ, S) γ , O M(Γ,S)γ ) (2.6)
2. 9 .
9Local observables. Let o ∈ S be a T-invariant point on space-time S and let i o : o → S be the inclusion map that naturally induces i o : M(Γ, S) → M(Γ, S) × S.
We define observable sheaves (Y i ) i∈Γ 0 over the moduli space M(Γ, S) as the pullback of the universal sheaf (Ŷ i ) i∈Γ 0 from M(Γ, S) × S to M(Γ, S) by the inclusion i o
α∈[1...n i ] are the exponentiated Coloumb parameters of the gauge theory. Let M = (M e ) e∈Γ 1 be the framing space associated to Γ 1 part of quiver (arrows). To each individual arrow e we associate 1-dimensional mass-twisting space M e ≃ C. Let GL(M) = e∈Γ 1 GL(M e ) be the respective group of automorphisms and let T M be a Cartan torus in GL(M). Let M be the character of M M = ch T M M = µ (3.6)
3. 3 .
3Complete group of equivariance. For Γ-quiver gauge theory on S = C 2 we denote by T = T Q × T N × T M (3.8) the Cartan torus in the automorphism group of the moduli space M(S, Γ).
expresses the defining relation(3.45) between Y i,x and S j,x in the exponentiated form: the field y i (x) (of the Cartan weight type) is the q 2 -derivative of the field s i (x) (of the Cartan root type). Namely, the relation (5.4) is the identity
was studied in details in section 4.1 of [1]. In terms of the parameter b 2 = −β the vertex operator (3.28) can be written as S i (x) =: e bφ i (x) : (6.2)
.16) and the K-theory valued half Cartan matrix c + ij defined as [c + ij ] := δ ij − The set of eigenvalues. Then the Chern characters X = ch X at T-fixed point λ can be explicitly described. Lete:j→i
[M ∨
e ]
(3.17)
with Chern character
ch[c +
ij ] = δ ji −
e:j→i
µ −1
e
(3.18)
3.10.
The elements of the Fock space ch Rep T [[t]] are formal t-series valued in the ring of Tcharacters. The t-constants are lowest-weight states (vacua); they are annihilated by all lowering operators ∂ i,p . A state in the Fock space ch Rep T [[t]] can be obtained by an action of an operator in the algebra H on the vacuum |1 3.15. Free bosons and vertex operators. The state |Z T can be presented as3.23) suggests a natural way to present the extended partition
function Z T (t) as a state |Z T in the infinite-dimensional T-character valued Fock space
ch Rep T [[t]]. The Fock space ch Rep T [[t]] is Verma module for the Heisenberg algebra H gen-
erated by the operators (∂ i,p ) i∈Γ 0 ,p∈[0...∞] and t = (t i,p ) i∈Γ 0 ,p∈[0...∞] over ch Rep T with canonical
commutators
[∂ i,m , t j,n ] = δ ij δ mn
(3.25)
where
t i,0 = log q 2 q i
(3.26)
imply that the operator-state representation of the partition function (3.27) is equivalent to the quiver gauge theory definition (3.23) if gauge theory couplings κ i and q i are evaluated asThe relations (3.31) and (3.30), and the relation e A 1 e A 2 = e [A 1 ,A 2 ] e A 2 e A 1 for central [A 1 , A 2 ],
and the holomorphic current can be characterized by a single partition[2] µq −1 x :
(5.22)
Lefshetz -Grothendieck-Hirzebruch-Riemann-Roch -Atiyah-Singer formula
Taro Kimura, Keio University, JapanVasily Pestun, IHES, France
Acknowledgements. We thank for discussions and comments Alexey Sevastyanov, Edward Frenkel, Nikita Nekrasov and Samson Shatashvili. VP acknowledges grant RFBR 15-01-04217. TK is grateful to Institut des HautesÉtudes Scientifiques for hospitality where a part of this work has been done. The work of TK was supported in part by JSPS Grant-in-Aid for Scientific Research (No. 13J04302) from MEXT of Japan.Quiver gauge theoryIn this section we define the extended partition function Z of quiver gauge theories[4,48,49].
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[
"MONIC MODULES AND SEMI-GORENSTEIN-PROJECTIVE MODULES",
"MONIC MODULES AND SEMI-GORENSTEIN-PROJECTIVE MODULES"
] |
[
"P U Zhang \nSCHOOL OF MATHEMATICAL SCIENCES\nSHANGHAI JIAO TONG UNIVERSITY\n200240SHANGHAIP. R. CHINA\n"
] |
[
"SCHOOL OF MATHEMATICAL SCIENCES\nSHANGHAI JIAO TONG UNIVERSITY\n200240SHANGHAIP. R. CHINA"
] |
[] |
The category gp(Λ) of Gorenstein-projective modules over tensor algebra Λ = A ⊗ k B can be described as the monomorphism category mon(B, gp(A)) of B over gp(A). In particular, Gorensteinprojective Λ-modules are monic. In this paper, we find the similar relation between semi-Gorensteinprojective Λ-modules and A-modules, via monic modules, namely, mon(B, ⊥ A) = mon(B, A) ∩ ⊥ Λ.Using this, it is proved that if A is weakly Gorenstein, then Λ is weakly Gorenstein if and only each semi-Gorenstein-projective Λ-modules are monic; and that if B = kQ with Q a finite acyclic quiver, then Λ is weakly Gorenstein if and only if A is weakly Gorenstein. However, this relation itself does not answer the question whether there exist double semi-Gorenstein-projective Λ-modules which are not monic. Using the recent discovered examples of double semi-Gorenstein-projective A-modules which are not torsionless, we positively answer this question, by explicitly constructing a class of double semi-Gorenstein-projective T 2 (A)-modules with one parameter such that they are not monic, and hence not torsionless. The corresponding results are obtained also for the monic modules and semi-Gorensteinprojective modules over the triangular matrix algebras given by bimodules. MSC 2020: Primary: 16G10; 16G50; secondary: 16E05; 16E65. Key words and phrases: monic module, monomorphism category, (double) semi-Gorenstein-projective module, Gorenstein-projective module, the canonical map, torsionless module, reflexive module, (left) weakly Gorenstein algebra, semi-Gorenstein-projective-free algebra PU ZHANG Let M * denote the A-dual Hom A (M, A) of M . Denote by φ M : M −→ M * * the canonical A-map, defined by φ M (m)(f ) = f (m) for m ∈ M and f ∈ M * . A module M is torsionless if it is a submodule of a projective module, or, equivalently, φ M is a monomorphism; and M is reflexive if φ M is an isomorphism. A module M is semi-Gorenstein-projective if M ∈ ⊥ A; and M will be called double semi-Gorensteinprojective, if both M and M * are semi-Gorenstein-projective. By definition, a Gorenstein-projective module is double semi-Gorenstein-projective and reflexive. This is introduced by Auslander and Bridger [AB], under the name of modules of G-dimension zero, and it is equivalent to the definition in terms of complete projective resolution given by Enochs and Jenda ([EJ1], [EJ2]). For the equivalence we refer to [AM, p.398] (where it is called a total reflexive module) and [Chr, Theorem 4.2.6]. Denote by gp(A) the full subcategory of A-mod of Gorenstein-projective modules. Thus add(A) ⊆ gp(A) ⊆ ⊥ A. 1.2. Avramov and Martsinkovsky [AM, p.398] has proposed the independence problem of the total reflexivity. In fact, the known examples of semi-Gorenstein-projective modules which are not Gorensteinprojective are few and complicated. The first examples of reflexive semi-Gorenstein-projective modules which are not Gorenstein-projective, and the first examples of reflexive modules M with M * semi-Gorenstein-projective such that M are not semi-Gorenstein-projective, are discovered by Jorgensen and Şega [JS]; and the first examples of double semi-Gorenstein-projective modules which are not torsionless, are recently founded in [RZ2, RZ3]. Putting together, this solves the independence problem of the total reflexivity. Note that the first examples of semi-Gorenstein-projective modules which are not Gorensteinprojective over noncommutative algebras, are presented by Marczinzik [M2].
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10.1016/j.jpaa.2022.107181
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"https://export.arxiv.org/pdf/2208.05690v1.pdf"
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2208.05690
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71785d008dfd5ddf92b5abb84509dad2c60e9ce0
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MONIC MODULES AND SEMI-GORENSTEIN-PROJECTIVE MODULES
11 Aug 2022
P U Zhang
SCHOOL OF MATHEMATICAL SCIENCES
SHANGHAI JIAO TONG UNIVERSITY
200240SHANGHAIP. R. CHINA
MONIC MODULES AND SEMI-GORENSTEIN-PROJECTIVE MODULES
11 Aug 2022arXiv:2208.05690v1 [math.RT]and phrases: monic modulemonomorphism category(double) semi-Gorenstein-projective moduleGorenstein-projective modulethe canonical maptorsionless modulereflexive module(left) weakly Gorenstein algebrasemi-Gorenstein-projective-free algebra
The category gp(Λ) of Gorenstein-projective modules over tensor algebra Λ = A ⊗ k B can be described as the monomorphism category mon(B, gp(A)) of B over gp(A). In particular, Gorensteinprojective Λ-modules are monic. In this paper, we find the similar relation between semi-Gorensteinprojective Λ-modules and A-modules, via monic modules, namely, mon(B, ⊥ A) = mon(B, A) ∩ ⊥ Λ.Using this, it is proved that if A is weakly Gorenstein, then Λ is weakly Gorenstein if and only each semi-Gorenstein-projective Λ-modules are monic; and that if B = kQ with Q a finite acyclic quiver, then Λ is weakly Gorenstein if and only if A is weakly Gorenstein. However, this relation itself does not answer the question whether there exist double semi-Gorenstein-projective Λ-modules which are not monic. Using the recent discovered examples of double semi-Gorenstein-projective A-modules which are not torsionless, we positively answer this question, by explicitly constructing a class of double semi-Gorenstein-projective T 2 (A)-modules with one parameter such that they are not monic, and hence not torsionless. The corresponding results are obtained also for the monic modules and semi-Gorensteinprojective modules over the triangular matrix algebras given by bimodules. MSC 2020: Primary: 16G10; 16G50; secondary: 16E05; 16E65. Key words and phrases: monic module, monomorphism category, (double) semi-Gorenstein-projective module, Gorenstein-projective module, the canonical map, torsionless module, reflexive module, (left) weakly Gorenstein algebra, semi-Gorenstein-projective-free algebra PU ZHANG Let M * denote the A-dual Hom A (M, A) of M . Denote by φ M : M −→ M * * the canonical A-map, defined by φ M (m)(f ) = f (m) for m ∈ M and f ∈ M * . A module M is torsionless if it is a submodule of a projective module, or, equivalently, φ M is a monomorphism; and M is reflexive if φ M is an isomorphism. A module M is semi-Gorenstein-projective if M ∈ ⊥ A; and M will be called double semi-Gorensteinprojective, if both M and M * are semi-Gorenstein-projective. By definition, a Gorenstein-projective module is double semi-Gorenstein-projective and reflexive. This is introduced by Auslander and Bridger [AB], under the name of modules of G-dimension zero, and it is equivalent to the definition in terms of complete projective resolution given by Enochs and Jenda ([EJ1], [EJ2]). For the equivalence we refer to [AM, p.398] (where it is called a total reflexive module) and [Chr, Theorem 4.2.6]. Denote by gp(A) the full subcategory of A-mod of Gorenstein-projective modules. Thus add(A) ⊆ gp(A) ⊆ ⊥ A. 1.2. Avramov and Martsinkovsky [AM, p.398] has proposed the independence problem of the total reflexivity. In fact, the known examples of semi-Gorenstein-projective modules which are not Gorensteinprojective are few and complicated. The first examples of reflexive semi-Gorenstein-projective modules which are not Gorenstein-projective, and the first examples of reflexive modules M with M * semi-Gorenstein-projective such that M are not semi-Gorenstein-projective, are discovered by Jorgensen and Şega [JS]; and the first examples of double semi-Gorenstein-projective modules which are not torsionless, are recently founded in [RZ2, RZ3]. Putting together, this solves the independence problem of the total reflexivity. Note that the first examples of semi-Gorenstein-projective modules which are not Gorensteinprojective over noncommutative algebras, are presented by Marczinzik [M2].
Introduction
Monic modules, defined on tensor products Λ = A ⊗ k B, or on matrix algebras Λ = ( A M 0 B ) of bimodule modules M , built a bridge between Gorenstein-projective Λ-modules and Gorenstein-projective A-modules (Theorems 2.6 and 2.13). In particular, Gorenstein-projective Λ-modules are monic, in the both cases. This paper is to show that they also play an important role in the study of semi-Gorensteinprojective modules. In the both cases, we will give sufficient and necessary conditions such that Λ is weakly Gorenstein, and positively answer the question whether there exist double semi-Gorensteinprojective Λ-modules which are not monic, and hence not torsionless, and not Goresnstein-projective.
1.1. Let A be an Artin algebra. All the modules in this paper are finitely generated, and we start from left modules. Let A-mod be the category of left A-modules. For M ∈ A-mod, denote by add(M ) the full subcategory of A-mod of direct summands of a direct sum of copies of M ; by ⊥ M the full subcategory of A-mod of modules X with Ext i A (X, M ) = 0 for i ≥ 1; and by M ⊥ the full subcategory of A-mod of modules X with Ext i A (M, X) = 0 for i ≥ 1.
Supported by National Natural Science Foundation of China, Grant No. 12131015, 11971304. [email protected].
Let
A and B be finite-dimensional algebras over field k. Since Cartan-Eilenberg [CE], modules over tensor algebra Λ = A ⊗ k B have got interest. They are complicated in the sense that Λ-mod can not be controlled by U ⊗ k V with U ∈ A-mod and V ∈ B-mod. However, if B is given by a bound quiver (Q, I), one can study Λ-module by taking the advantage of the representations of quivers over algebra A ([ , [S2-S5], [KLM1,KLM2], [LZ1,LZ2], [RZ1], [ZX]), i.e., any Λ-module can be identified with a representation (X i , X α , i ∈ Q 0 , α ∈ Q 1 ) of (Q, I) over A, where each X i ∈ A-mod, and each X α : X s(α) −→ X e(α) is an A-map, such that X α 's satisfy all the relations which generate I.
When Q is finite acyclic and I is generated by monomial relations, this identification permits us to define monic Λ-modules and monomorphism category mon(B, C ) = mon(Q, I, C ) ( [LZ1,LZ2], [ZX]), for any additive full subcategory C of A-mod. This definition is combinatorial and constructive, and it admits a homological interpretation. In general, there is no longer the corresponding combinatorial definition of a monic module, but this homological interpretation still makes sense, and it is taken as the definition of the monomorphism category mon(B, C ) by Hu, Luo, Xiong and Zhou in [HLXZ]. See Subsection 2.1.
The study of the monomorphism categories can be traced to G. Birkhoff [Bir]. When B is the path algebra of quiver A n with linear orientation, i.e., B = T n (k) = and the monomorphism category mon(B, A) = mon(B, A-mod) is exactly the submodule category S n (A) ( ), or called the filtered chain category ([S1 -S5]). They have Auslander-Reiten sequences ( [RS2]) and the RSS equivalence ( [ZX]). Simson ([S2]- [S5]) has studied their representation type. By Kussin,Lenzing and Meltzer [KLM1,KLM2] and Chen [Chen1], they are related to the singularity theory.
1.4. An important application of monomorphism categories is that they can describe Gorenstein-projective modules as gp(Λ) = mon(B, gp(A)) (cf. Theorem 2.6 below). Thus, Gorenstein-projective Λ-modules are monic Λ-modules over Gorenstein-projective A-modules. If B is given by a finite acyclic quiver and monomial relations, by the combinatorial definition of monic modules, this gives in practice a reductive construction of Gorenstein-projective Λ-modules.
Question 1. Is there the similar relation between semi-Gorenstein-projective Λ-modules and Amodules?
Theorem 1.1. Let A and B be finite-dimensional k-algebras with gl.dimB < ∞, and Λ = A ⊗ k B. Then
mon(B, ⊥ A) = mon(B, A) ∩ ⊥ Λ.
Moreover, if inj.dim A A < ∞, then ⊥ Λ = mon(B, ⊥ A).
Theorem 1.1 will be proved in Section 3, as a special case of Theorem 3.1.
An Artin algebra
A is Gorenstein, if inj.dim A A < ∞ and inj.dimA A < ∞. An Artin algebra A is left weakly Gorenstein ( [M1], [RZ2]), if any left semi-Gorenstein-projective A-module is Gorenstein-projective, i.e., ⊥ A = gp (A). It is open whether a left weakly Gorenstein algebra is right weakly Gorenstein ( [M1, § 5], [RZ2,9.3]). However, if no confusions caused, we will omit the word "left".
By Enochs and Jenda [EJ2,11.5.3], Gorenstein algebras are weakly Gorenstein. By Yoshino [Y1,Theorem 5.5] and Beligiananis [Bel2,Corollary 5.11], if ⊥ A is of finite type, then A is weakly Gorenstein. By Marczinzik [M1,Theorem 3.5(3)], torsinless finite algebras are weakly Gorenstein. For more information on weakly Gorenstein algebras we refer to [Bel1,Bel2], [M1], and [RZ2,3.6]. (ii) Let A and B be finite-dimensional k-algebras with gl.dimB < ∞. When the tensor product A⊗ k B is weakly Gorenstein?
It turns out that, in the both cases, monic modules will play a crucial role. (2) If proj.dimM B < ∞ and B is a Gorenstein algebra, then Λ is weakly Gorenstein if and only if each semi-Gorenstein-projective Λ-module is monic respect to bimodule M and A is weakly Gorenstein.
( (1) Assume that gl.dimB < ∞ and Λ = A ⊗ k B. If Λ is weakly Gorenstein, then so is A. Conversely, if A is weakly Gorenstein, then a semi-Gorenstein-projective Λ-module M is Gorenstein-projective if and only if M is monic.
Thus, if A is weakly Gorenstein, then Λ is weakly Gorenstein if and only if each semi-Gorensteinprojective Λ-module is monic, or equivalently, ⊥ Λ = mon(B, ⊥ A).
(2) Let Q be a finite acyclic quiver. Then A ⊗ k kQ is weakly Gorenstein if and only if A is weakly Gorenstein.
The assumption gl.dimB < ∞ holds automatically if B is given by a bound acyclic quiver. Theorems 1.3 is the combination of Propositions 4.7 and 4.8.
1.6. An Artin algebra A will be called left semi-Groenstein-projective-free, or in short, lsgp-free, provided that each left semi-Gorenstein-projective A-module is a projective module, i.e., ⊥ A = add (A). We do not know whether a lsgp-free algebra is right semi-Groenstein-projective-free.
Recall that
⊥ Λ = gp(Λ) = { M ⊗ B P P Id M ⊗ B P ⊕ G 0 | P ∈ add(B), G ∈ ⊥ A = gp(A)}
and ⊥ Λ = add(Λ) if and only if ⊥ A = add(A).
(2) Let I be an admissible ideal of kQ, and Λ = A ⊗ k kQ/I. Then ⊥ Λ = add(Λ) if and only if ⊥ A = add (A).
Theorems 1.4 is the combination of Propositions 4.9 and 4.11.
1.7. If Λ = A ⊗ k B with gl.dim.B < ∞, or if Λ = ( A M 0 B )
, then Gorenstein-projective Λ-modules are always monic (cf. Theorems 2.6 and 1.1). Is this true for semi-Gorenstein-projective Λ-modules? One may ask a stronger question:
Question 3. In the both cases, whether there exist double semi-Gorenstein-projective Λ-modules which are not monic?
The positive answer will in particular gives double semi-Gorenstein-projective modules which are not Gorenstein-projective. As mentioned in Subsection 1.2, this is highly nontrivial.
To answer Question 3, we consider Λ
= A ⊗ k k(• −→ •) = ( A A 0 A ) = T 2 (A). Any left Λ-module M can be identified with a triple ( X Y ) ϕ , where ϕ : Y −→ X is a left A-map.
Thus, one has the exact sequence of left A-modules Y ϕ −→ X π −→ Cokerϕ −→ 0, and there is a unique A-map β : Cokerπ * −→ Y * , such that the diagram with exact rows
0 / / (Cokerϕ) * π * / / X * p / / Cokerπ * β ✤ ✤ ✤ / / 0 0 / / (Cokerϕ) * π * / / X * ϕ * / / Y * . commutes, where p is the canonical A-epimorphism. So one has the left A-map β * : Y * * −→ (Cokerπ * ) * , and the composition β * φ Y : Y −→ (Cokerπ * ) * , where φ Y : Y −→ Y * * is the canonical map.
Theorem 1.5. Let A be an Artin algebra,
Λ = T 2 (A) = ( A A 0 A ) , and ( X Y ) ϕ a left Λ-module. Then (1) There is a left Λ-module isomorphism ( X Y ) * * ϕ ∼ = X * * (Cokerπ * ) * p * , where p * : (Cokerπ * ) * −→ X * * is the A-monomorphism induced by p : X * −→ (Cokerϕ) * .
Taking this isomorphism as identity, then the canonical
Λ-map φ ( X Y ) ϕ : X Y ϕ −→ X * * (Cokerπ * ) * p * is given by φ ( X Y ) ϕ = φX β * φY .
(2) ( X Y ) ϕ is torsionless and double semi-Gorenstein-projective if and only if ( X Y ) ϕ is monic, X, Y , and Cokerϕ are double semi-Gorenstein-projective, and X and Y are torsionless.
(3) ( X Y ) ϕ is double semi-Gorenstein-projective with epimorphism φ ( X Y ) ϕ
if and only if ϕ * : X * −→ Y * is an epimorphism, X and Y are double semi-Gorenstein-projective, (Cokerϕ) * is semi-Gorensteinprojective, and φ X and φ Y are epimorphisms.
Theorem 1.5(1) is a summary of Lemma 5.3 and Proposition 5.4; and Theorem 1.5(2) and (3) will be clear after Proposition 5.7.
As remarked in [RZ4,3.1], up to now, all the known examples have the following property:
Double semi-Gorenstein-projective modules M such that φ M is a monomorphism (an epimorphism, respectively) are Gorenstein-projective.
The following result shows that this property is preserved under the T 2 -extensions.
Theorem 1.6. Let A be an Artin algebra and Λ = T 2 (A) = ( A A 0 A ) . Then (1) Any torsionless and double semi-Gorenstein-projective A-module is Gorenstein-projective if and only if any torsionless and double semi-Gorenstein-projective Λ-module is Gorenstein-projective.
(2)
Any double semi-Gorenstein-projective A-module L with φ L an epimorphism is Gorensteinprojective if and only if any double semi-Gorenstein-projective Λ-module M with φ M an epimorphism is Gorenstein-projective.
Theorem 1.6 will be proved in Subsection 5.9.
1.8. The following result positively answers Question 3, and gives a construction of double semi-Gorenstein-projective T 2 (A)-modules which are not monic.
Theorem 1.7. Suppose that Y is a double semi-Gorenstein-projective A-module which is not torsionless. Let ϕ : Y −→ P be a left add(A)-approximation of Y . Then P Y ϕ is a double semi-Gorenstein-projective T 2 (A)-module which is not monic. In particular, P Y ϕ is not torsionless.
Theorem 1.7 will be proved in Subsection 6.1. Using the algebra A in [RZ2] and the A-modules M (1, −q, c) in [RZ3], by Theorem 1.7, we obtain a class of double semi-Gorenstein-projective T 2 (A)modules with parameter c as
X(c) := A A M(1,−q,c) f1
such that X(c) not monic, and hence not torsionless; moreover, all the canonical maps φ X(c) : X(c) −→ X(c) * * are neither monomorphisms nor epimorphisms, and X(c) * * are not semi-Gorenstein-projective. See Proposition 6.2. (1) A left Λ-module X is monic, if Tor Λ i (A ⊗ k V, X) = 0 for all i ≥ 1 and for all right B-modules V . Denote by mon(B, A) the full subcategory of Λ-mod consisting of monic modules, which is called the monomorphism category of B over A.
(2) Let C be an additive full subcategory of A-mod. An object X ∈ mon(B, A) is a monic module over C , if (A ⊗ k V ) ⊗ Λ X ∈ C for all right B-module V .
Denote by mon(B, C ) the full subcategory of mon(B, A) of monic modules over C , which is called the monomorphism category of B over C .
mon(B, A) = {X ∈ Λ-mod | Tor Λ i (A ⊗ k D(S), X) = 0, ∀ i ≥ 1, ∀ simple left B-module S} = ⊥ (D(A A ) ⊗ k B).
Example 2.3. (1) If B is the path algebra of the quiver A n (n ≥ 2) with linear orientation, then
B = T n (k) = k k ··· k 0 k ··· k . . . . . . . . . . . . 0 0 ··· k , Λ = A ⊗ k B = A A ··· A 0 A ··· A . . . . . . . . . . . . 0 0 ··· A = T n (A)
, and mon(B, A) turns out to be
S n (A) = { X1 . . . Xn (ϕi) ∈ T n (A)-mod | ϕ i : X i+1 −→ X i is a monomorphism, ∀ 1 ≤ i ≤ n − 1}.
This submodule category has been studied in [A], [S1-S5], , [Z1].
(2) If B is the path algebra kQ, where Q = (Q 0 , Q 1 , s, e) is a finite acyclic quiver, then a monical Λ-module has been defined in [LZ1] as a representation (X i , X α , i ∈ Q 0 , α ∈ Q 1 ) of Q over A, such that for each i ∈ Q 0 the A-map
(X α ) α∈Q1 e(α)=i : α∈Q1 e(α)=i X s(α) −→ X i is a monomorphism.
For any additive full subcategory C of A-mod, a monic Λ-module over C has been defined in [ZX,2.1], as a monic Λ-module (X i , X α , i ∈ Q 0 , α ∈ Q 1 ) satisfying
X i /Im(X α ) α∈Q1 e(α)=i ∈ C , ∀ i ∈ Q 0 .
(3) If B = kQ/I with I generated by monomial relations, then mon(B, C ) has also been defined combinatorially. For details see [LZ2] and [ZX].
In all these monomorphism categories defined via quivers, "monomorphisms" are visible, and they also admit the homological description in Definition 2.1 ([Z1, Theorem 3.1], [LZ2,2.1], [ZX,Theorem 2.6]).
Lemma 2.4. Let Λ = A ⊗ k kQ, where Q is a finite acyclic quiver. Then torsionless Λ-modules are monic.
Proof. Let X = (X i , X α ) be a torsionless Λ-module. Then X is a submodule of a projective Λ-module, which is of the form P ⊗ k L, where P is a projective left A-module, and L = (
L i , L α ) is a projective left kQ-module. Thus there is a monomorphism (f i ) i∈Q0 : (X i , X α ) −→ (P ⊗ k L(i), Id P ⊗ k L α ) of Λ-modules. Hence, for each i ∈ Q 0 , the diagram of A-maps α∈Q1,e(α)=i X s(α) _ α∈Q 1 ,e(α)=i f s(α) (Xα) α∈Q 1 ,e(α)=i / / X i _ fi α∈Q1,e(α)=i P ⊗ k L s(α) (IdP ⊗ k Lα) α∈Q 1 ,e(α)=i / / P ⊗ k L i
commutes. Since both α∈Q1,e(α)=i f s(α) and (Id P ⊗ k L α ) α∈Q1,e(α)=i are monomorphisms, it follows that (X α ) α∈Q1,e(α)=i is a monomorphism, i.e., X is a monic Λ-module.
Remark 2.5. Lemma 2.4 is not true for Λ = A ⊗ k (kQ/I), even if I is generated by monomial relations.
For example, if A = k, Q = 3 α −→ 2 β −→ 1, and I = βα , then the simple module S(2) = radP (3) is a torsionless (kQ/I)-module, but it is not monic.
2.2. Gorenstein-projective modules over tensor algebras. The relationship between Gorensteinprojective modules over Λ = A ⊗ k B and monomorphism categories of B over A is Theorem 2.6. ( [HLXZ,Theorem 4.5]) Let A and B be finite-dimensional k-algebras with gl.dimB < ∞, and Λ = A ⊗ k B. Then gp(Λ) = mon(B, gp(A)). In particular, a Gorenstein-projective Λ-module is monic.
Theorem 2.6 is proved for Λ = T n (A) = A ⊗ k T n (k) with A Gorenstein in [Z1,Corollary 4.1(ii)]; it is proved for B = kQ in [LZ1,Theorem 5.1], and for B = kQ/I in [LZ2,Theorem 4.1], where Q is any finite acyclic quiver, and I is generated by monomial relations. In all these cases, since mon(B, gp (A)) are defined via the combinatorics of quivers, Theorem 2.6 provides in practice an inductive construction of Gorenstein-projective modules.
2.3. Monic modules respect to bimodules. Let A and B be Artin algebras, and M an A-B-bimodule such that Λ = ( A M 0 B ) is an Artin algebra. This is equivalent to say that A and B are Artin R-algebra, and M is finitely generated over R which acts centrally on M , where R is a commutative Artin ring ( [ARS,Proposition 2.1,p.72]
). Any left Λ-module is identified with a triple X Y ϕ , where X is a left A-module, Y is a left B-module, and ϕ : M ⊗ B Y −→ X is a left A-map. Definition 2.7. ([XZZ, 2.1]) Let Λ = ( A M 0 B ) be an Artin algebra. A Λ-module X Y ϕ is monic respect to bimodule A M B , provided that ϕ : M ⊗ B Y −→ X is a monomorphism.if Λ = A ⊗ k B = A ′ M 0 B ′ , mon(B, A) = M(A ′ , M, B ′ ) in general. For example, consider T n (A) = A⊗ k T n (k). A T n (A)-module X = X1 . . . Xn (ϕi) is a monic T n (A)-module if and only if ϕ i : X i+1 −→ X i is a monomorphism for all 1 ≤ i ≤ n − 1. Thus mon(T n (k), A) = S n (A).
On the other hand,
T n (A) = Tn−1(A) Mn−1 0 A for n ≥ 2, where M n−1 = A . . . A (n − 1 rows) is a T n−1 (A)-A-bimodule, and X is a monic T n (A)-module respect to bimodule M n−1 if and only if ϕ i · · · ϕ n−1 : X n −→ X i is a monomorphism for all 1 ≤ i ≤ n − 1. Thus, a T n (A)-module X is a monic T n (A)-module if and only if X1 . . . Xm (ϕi) is a monic T m (A)-module respect to bimodule M m−1 for all 2 ≤ m ≤ n, where T m (A) = Tm−1(A) Mm−1 0 A , and M m−1 = A . . . A
(m − 1 rows); and a monic T n (A)-module X respect to M n−1 is not necessarily a monic T n (A)-module.
In some sense, M(T n−1 (A), M n−1 , A) can be view as the local version of mon(T n (k), A). Proof. Let L = X Y ϕ be a torsionless Λ-module. Then L is a submodule of a projective Λ-module, which is of the
For example, let n ≥ 3. Consider T n (A)-module X = A . . . A A⊕A Xn=A (ϕi) , where ϕ n−1 = Id A 0 : A −→ A ⊕ A, ϕ n−2 = (Id A , 0) : A ⊕ A −→ A and ϕ i = Id A : X i+1 = A −→ A = X i for all 1 ≤ i ≤ n − 3. Then X / ∈ mon(T n (k), A), but X ∈ M(T n−1 (A), M n−1 , A).form P 0 ⊕ M⊗B Q Q Id M ⊗ B Q , where P is a projective left A-module, and Q is a projective left B-module. Thus, there is a monomorphism ( f 1 f 2 ) g : X Y ϕ −→ P ⊕M⊗B Q Q ( 0 1 ) . Since M B is projective, Id M ⊗ B g is a monomorphism. By the commutative diagram M ⊗ B Y _ IdM ⊗B g ϕ / / X _ ( f 1 f 2 ) M ⊗ B Q ( 0 1 ) / / P ⊕ M ⊗ B Q ϕ is a monomorphism, i.e., L is a monic Λ-module. Remark 2.10. Lemma 2.9 is not true if M B is not projective. For example, let Λ = k M 0 A , where A is the path algebra k(2 −→ 1), k M A = D(Ae 1 ) = Hom k (Ae 1 , k) is a k-A-bimodule. Since M e 2 = 0, M ⊗ A Ae 2 = M e 2 ⊗ A e 2 = 0. Thus 0 Ae2 is a left projective Λ-module. Let σ : Ae 1 −→ Ae 2 be the embedding. Then 0 σ : 0 Ae1 −→ 0 Ae2 is a Λ-monomorphism, and hence 0 Ae1 is a torsionless Λ- module. But since M ⊗ A Ae 1 = M ⊗ A e 1 Ae 1 = k = 0, 0 Ae1 is not monic respect to bimodule k M A .
(Semi-)Gorenstein-projective modules over triangular matrix algebras of bimodules.
For an Artin algebra B, let D denote the duality of B ([ARS, p.37]).
Let A and B be Artin algebras, M an A-B-bimodule such that Λ = ( A M 0 B )
is an Artin algebra. Under suitable conditions, semi-Gorenstein-projective Λ-modules can be described as follows.
Theorem 2.11. ([XZ, Theorem 1.1]) Assume that proj.dim A M < ∞ and D(M B ) ∈ ( ⊥ ( B B)) ⊥ . Then a Λ-module ( X Y ) ϕ ∈ ⊥ Λ if and only if Y ∈ ⊥ B, the left A-map ϕ : M ⊗ B Y −→ X induces isomorphisms Ext i A (X, A) ∼ = Ext i A (M ⊗ B Y, A) for all i ≥ 1, and ϕ * : X * −→ (M ⊗ B Y ) * is a right A-epimorphism. An A-B-bimodule M is compatible ([Z2, Definition 1.1])
, if the following two conditions hold:
If Q • is an exact sequence of projective B-modules, then M ⊗ B Q • is exact; and
If P • is a complete A-projective resolution, then Hom A (P • , M ) is exact.
Lemma 2.12. ([Z2, Proposition 1.3(1)]) Let M be an A-B-bimodule. If proj.dim A M < ∞ and proj.dimM B < ∞, then M is compatible.
Under the condition of compatible bimodule, Gorenstein-projective Λ-modules can be described as follows. In particular, again, Gorenstein-projective Λ-modules are monic, but in the sense of respect to bimodule A M B (compare Theorem 2.6).
Theorem 2.13. ([Z2, Theorem 1.4]) Assume that M is a compatible A-B-bimodule. Then ( X Y ) φ ∈ gp(Λ) if and only if ϕ : M ⊗ B Y −→ X is a monomorphism, Coker ϕ ∈ gp(A), and Y ∈ gp(B).
If ths is the case,
X ∈ gp(A) if and only if M ⊗ B Y ∈ gp(A).
Corollary 2.14. Let A be an Artin algebra, and
Λ = T 2 (A) = ( A A 0 A ). Then (1) ⊥ Λ = {( X Y ) ϕ ∈ Λ-mod | X ∈ ⊥ A, Y ∈ ⊥ A, ϕ * : X * −→ Y * is epic}. (2) gp(Λ) = {( X Y ) ϕ ∈ Λ-mod | ϕ : Y −→ X is monic, Coker ϕ ∈ gp(A), Y ∈ gp(A)} = {( X Y ) ϕ ∈ Λ-mod | ϕ : Y −→ X is monic, Coker ϕ ∈ gp(A), Y ∈ gp(A), X ∈ gp(A)}.
Monomorphism categories over perpendicular categories
Let A and B be finite-dimensional k-algebras, and Λ = A ⊗ k B. A relation between semi-Gorensteinprojective Λ-modules and semi-Gorenstein-projective A-modules is contained in the following general result.
Theorem 3.1. Let A and B be finite-dimensional k-algebra with gl.dimB < ∞, T an A-module, and Λ = A ⊗ k B. Then mon(B, ⊥ T ) = mon(B, A) ∩ ⊥ (T ⊗ k B). Moreover, if there is an exact sequence of left A-modules 0 −→ T m −→ · · · −→ T 0 −→ D(A A ) −→ 0 with each T j ∈ add(T ), then mon(B, ⊥ T ) = ⊥ (T ⊗ k B).
In particular, there holds mon(B,
⊥ A) = mon(B, A) ∩ ⊥ Λ; and if inj.dim A A < ∞, then ⊥ Λ = mon(B, ⊥ A). Proof. Let X ∈ mon(B, A). Since by definition mon(B, ⊥ T ) ⊆ mon(B, A), it follows that, in order to prove mon(B, ⊥ T ) = mon(B, A) ∩ ⊥ (T ⊗ k B), it suffices to prove that X ∈ ⊥ (T ⊗ k B) if and only if X ∈ mon(B, ⊥ T ), i.e., (A ⊗ k V ) ⊗ Λ X ∈ ⊥ T for all right B-modules V .
Take a Λ-projective resolution
P • : · · · −→ P 1 −→ P 0 −→ X −→ 0. Claim 1: X ∈ ⊥ (T ⊗ k B) if and only if the complex Hom Λ (P • , Hom k (S ′ , T )) is exact, for each right simple B-module S ′ . Since gl.dimB < ∞, it is clear that ⊥ (T ⊗ k B) = S ⊥ (T ⊗ k S),
where S ranges over all the left simple B-modules. To use the Tensor-Hom adjoint pair later, we write a left simple B-module S as D
(S ′ ), where S ′ is a right simple B-module. Thus, ⊥ (T ⊗ k B) = S ′ ⊥ (T ⊗ k D(S ′ )), where S ′ ranges over all the right simple B-modules. Therefore, X ∈ ⊥ (T ⊗ k B) if and only if Hom Λ (P • , T ⊗ k D(S ′ )) is exact, for each right simple B-module S ′ . Note that the canonical k-linear isomorphism T ⊗ k D(S ′ ) −→ Hom k (S ′ , T ), t ⊗ f → "s ′ → f (s ′ )t", ∀ t ∈ T, f ∈ D(S ′ ), s ′ ∈ S ′ is a left Λ-isomorphism. Thus, X ∈ ⊥ (T ⊗ k B) if and only if Hom Λ (P • , Hom k (S ′ , T )) is exact, for each right simple B-module S ′ . Claim 2: (A ⊗ k V ) ⊗ Λ X ∈ ⊥ T for all right B-modules V if and only if the complex Hom A ((A ⊗ k S ′ ) ⊗ Λ P • , T )
is exact for each right simple B-module S ′ .
By assumption X ∈ mon(B, A), i.e., Tor Λ i (A ⊗ k V, X) = 0 for all i ≥ 1 and for all right B-modules V . It follows that the functor
(A ⊗ k −) ⊗ Λ X : modB −→ A-mod is an exact functor. As a consequence, (A ⊗ k V ) ⊗ Λ X ∈ ⊥ T for all right B-modules V if and only if (A ⊗ k S ′ ) ⊗ Λ X ∈ ⊥ T for each right simple B-module S ′ , since ⊥ T is extension closed. Since Tor Λ i (A ⊗ k S ′ , X) = 0 for all i ≥ 1, it follows that (A ⊗ k S ′ ) ⊗ Λ P • : · · · −→ (A ⊗ k S ′ ) ⊗ Λ P 1 −→ · · · −→ (A ⊗ k S ′ ) ⊗ Λ P 0 −→ (A ⊗ k S ′ ) ⊗ Λ X −→ 0 is an exact sequence of left A-modules. Since each P i is a projective left Λ-module, each (A ⊗ k S ′ ) ⊗ Λ P i ∈ add(A ⊗ k S ′ ). Thus each (A ⊗ k S ′ ) ⊗ Λ P i is projective as a left A-module, and hence (A ⊗ k S ′ ) ⊗ Λ P • is an A-projective resolution of left A-module (A ⊗ k S ′ ) ⊗ Λ X, for each right simple B-module S ′ . Therefore, (A ⊗ k S ′ ) ⊗ Λ X ∈ ⊥ T for each right simple B-module S ′ if and only if Hom A ((A ⊗ k S ′ ) ⊗ Λ P • , T ) is exact for each right simple B-module S ′ .
Claim 3: There is an isomorphism of complexes
Hom A ((A ⊗ k S ′ ) ⊗ Λ P • , T ) ∼ = Hom Λ (P • , Hom k (S ′ , T )) for each right simple B-module S ′ .
Applying the Tensor-Hom adjoint pair (
(A ⊗ k S ′ ) ⊗ Λ −, Hom A (A ⊗ k S ′ , −)
) between Λ-mod and A-mod, one has the following isomorphism of complexes of k-spaces
Hom A ((A ⊗ k S ′ ) ⊗ Λ P • , T ) ∼ = Hom Λ (P • , Hom A (A ⊗ k S ′ , T )).
Applying the adjoint pair (A ⊗ k −, Hom A (A, −)) between k-mod and A-mod, one has the isomorphisms of k-spaces
Hom A (A ⊗ k S ′ , T ) ∼ = Hom k (S ′ , Hom A (A, T )) ∼ = Hom k (S ′ , T ),
which is clearly also an isomorphism of left Λ-modules. All together we get an isomorphism of complexes
Hom A ((A ⊗ k S ′ ) ⊗ Λ P • , T ) ∼ = Hom Λ (P • , Hom k (S ′ , T )) for each right simple B-module S ′ .
It follows from Claim 1, Claim 2 and Claim 3 that ) is an Artin algebra. We will give various conditions for Λ being a left weakly Gorenstein algebra, i.e., ⊥ Λ = gp(Λ).
X ∈ ⊥ (T ⊗ k B) if and only if (A⊗ k V )⊗ Λ X ∈ ⊥ T for all right B-module V . This proves mon(B, ⊥ T ) = mon(B, A) ∩ ⊥ (T ⊗ k B). Finally, assume that there is an exact sequence 0 −→ T m −→ · · · −→ T 0 −→ D(A A ) −→ 0 with each T j ∈ add(T ). To show mon(B, ⊥ T ) = ⊥ (T ⊗ k B), it suffices to show ⊥ (T ⊗ k B) ⊆ mon(B, A). By Lemma 2.2, mon(B, A) = ⊥ (D(A A ) ⊗ k B). Thus, it suffices to show ⊥ (T ⊗ k B) ⊆ ⊥ (D(A A ) ⊗ k B). This follows from the exact sequence 0 −→ T m ⊗ k B −→ · · · −→ T 0 ⊗ k B −→ D(A A ) ⊗ k B −→ 0 with each T j ⊗ k B ∈ add(T ⊗ k B).Proposition 4.1. Assume that proj.dim A M < ∞, proj.dimM B < ∞, and D(M B ) ∈ ( ⊥ ( B B)) ⊥ . Then Λ = ( A M 0 B )
is weakly Gorenstein if and only if each semi-Gorenstein-projective Λ-module is monic respect to bimodule A M B , and A and B are weakly Gorenstein.
Proof. Since proj.dim A M < ∞ and proj.dimM B < ∞, the A-B-bimodule M is compatible (cf. Lemma 2.12). Thus, under the assumptions, one can apply Theorems 2.11 and 2.13.
Assume that each semi-Gorenstein-projective Λ-module is monic respect to bimodule A M B , and that A and B are weakly Gorenstein. Let X Y ϕ ∈ ⊥ Λ. We need to prove X Y ϕ ∈ gp(Λ). By the assumption, ϕ : M ⊗ B Y −→ X is a monomorphism; together with Theorem 2.11, one gets the conclusions: (A), since by assumption A is weakly Gorenstein. Thus, we get the following:
• ϕ : M ⊗ B Y −→ X is a monomorphism; • Y ∈ ⊥ B, and hence Y ∈ gp(B) (since by assumption B is weakly Gorenstein); • ϕ induces isomorphisms Ext i A (X, A) ∼ = Ext i A (M ⊗ B Y, A) for all i ≥ 1; • ϕ * : X * −→ (M ⊗ B Y ) * is a right A-epimorphism. Applying Hom A (−, A) to the exact sequence 0 −→ M ⊗ B Y ϕ −→ X −→ Coker ϕ −→ 0, since ϕ * : X * −→ (M ⊗ B Y ) * is an epimorphism and ϕ induces isomorphisms Ext i A (X, A) ∼ = Ext i A (M ⊗ B Y, A) for all i ≥ 1, it follows that Coker ϕ ∈ ⊥ A. Hence Coker ϕ ∈ gp• ϕ : M ⊗ B Y −→ X is a monomorphism; • Coker ϕ ∈ gp(A); and • Y ∈ gp(B).
Applying Theorem 2.13, one gets X Y ϕ ∈ gp(Λ). This proves the "if" part.
Conversely, assume that Λ is weakly Gorenstein. Thus, any semi-Gorenstein-projective Λ-module is Gorenstein-projective, and hence it is monic respect to bimodule A M B , by Theorem 2.13. It remains to prove that A and B are weakly Gorenstein. Let X ∈ ⊥ A. Applying Theorem 2.11 one knows X 0 ∈ ⊥ Λ, thus X 0 ∈ gp(Λ) by the assumption, and then by Theorem 2.13 one has X ∈ gp (A). This proves that A is weakly Gorenstein.
Similarly, let Y ∈ ⊥ B. Proof. Since by assumption B is a Gorenstein algebra, it follows that ⊥ ( B B) = gp(B). Recall that for a Gorenstein algebra B, (gp(B), p(B) <∞ ) is a cotorsion pair (see e.g., [H], [EJ2], [BR]), where p(B) <∞ is the full subcategory of B-mod consisting of modules of finite projective dimension. Thus
( ⊥ ( B B)) ⊥ = gp(B) ⊥ = p(B) <∞ .
Since by assumption proj.dimM B < ∞, It follows that inj.dimD(M B ) < ∞. Since B is Gorenstein, it follows that proj.dimD(M B ) < ∞, i.e., D(M B ) ∈ p(B) <∞ = ( ⊥ ( B B)) ⊥ . Thus, the assertion follows from Proposition 4.1.
Taking B to be a field k in Proposition 4.2, we get Corollary 4.3. Let A be a finite-dimensional k-algebra.
(1) Let M be a finite-dimensional A-module. Assume that proj.dim A M < ∞. Then Λ = A M 0 k is weakly Gorenstein if and only if each semi-Gorenstein-projective Λ-module is monic respect to bimodule A M k and A is weakly Gorenstein.
(2) Let P be a finite-dimensional projective left A-module, and Λ = A P 0 k . Then
⊥ Λ = { G 0 ⊕ P ⊗ k V V IdP ⊗ k V | G ∈ ⊥ A, V ∈ k-mod} gp(Λ) = { G 0 ⊕ P ⊗ k V V Id P ⊗ k V | G ∈ gp(A), V ∈ k-mod}
and Λ is weakly Gorenstein if and only if A is weakly Gorenstein.
Proof.
(2) Let X V ϕ ∈ ⊥ Λ. By Theorem 2.11, one has • ϕ :
P ⊕dimV −→ X induces isomorphisms Ext i A (X, A) ∼ = Ext i A (P ⊕dimV , A) = 0, ∀ i ≥ 1; and • ϕ * : X * −→ (P * ) ⊕dimV is a right A-epimorphism.
Thus X ∈ ⊥ A, and ϕ * is a splitting epimorphism. Hence ϕ * * is a splitting monomorphism. By the commutative diagram
P ⊕dimV ∼ = ϕ / / X φX (P * * ) ⊕dimV ϕ * * / / X * *
one sees that ϕ is a splitting monomorphism. Thus X = G ⊕ P ⊕dimV where G is semi-Gorensteinprojective, and hence X V ϕ
∼ = G 0 ⊕ P ⊗ k V V Id . This proves ⊥ Λ = { G 0 ⊕ P ⊗ k V V Id P ⊗ k V | G ∈ ⊥ A, V ∈ k-mod}. Since P ⊗ k V V
Id is a projective Λ-module and G 0 is a Gorenstein-projective Λ-module if and only if G a Gorenstein-projective A-module, it follows that gp( Proof. Since M B is projective, D(M B ) is an injective B-module, and hence D(M B ) ∈ ( ⊥ B) ⊥ . Thus, the assumption that proj.dim. A M < ∞ and M B is projective guarantee that the conditions of Proposition 4.1 are satisfied. By Proposition 4.1, it suffices to prove that if A and B are weakly Gorenstein, then any semi-Gorenstein-projective Λ-module X Y ϕ is monic respect to bimodule A M B . Applying Theorem 2.11 to X Y ϕ ∈ ⊥ Λ, one gets the following conclusions: • Y ∈ ⊥ B, and hence Y ∈ gp(B) (since by assumption B is weakly Gorenstein);
Λ) = { G 0 ⊕ P ⊗ k V V IdP ⊗ k V | G ∈ gp(A), V ∈ k-mod}.• ϕ : M ⊗ B Y −→ X induces isomorphisms Ext i A (X, A) ∼ = Ext i A (M ⊗ B Y, A), ∀ i ≥ 1; and • ϕ * : X * −→ (M ⊗ B Y ) * is a right A-epimorphism. Since Y ∈ gp(B), B Y is a submodule of some projective B-module B P . Since M B is projective, it follows that A (M ⊗ B Y ) is a submodule of A (M ⊗ B P ). Since B P is projective, A (M ⊗ B P ) ∈ add( A M ). Since by assumption A M is torsionless, it follows that M ⊗ B P is a torsionless left A-module, and hence M ⊗ B Y is a torsionless left A-module. Thus, the canonical map φ M⊗B Y : M ⊗ B Y −→ (M ⊗ B Y ) * * is a monomorphism. Since ϕ * : X * −→ (M ⊗ B Y ) * is an epimorphism, it follows that ϕ * * : (M ⊗ B Y ) * * −→ X * * is a monomorphism. From the commutative diagram with monomorphism φ M⊗B Y M ⊗ B Y _ φM⊗ B Y ϕ / / X φX (M ⊗ B Y ) * * ϕ * * / / X * * one sees that ϕ : M ⊗ B Y −→ X is a monomorphism, i.e., X Y ϕ is monic respect to bimodule A M B .
This completes the proof. (ii) If M is a semi-Gorenstein-projective A-module which is not Gorenstein-projective, then M ⊗ k B is a semi-Gorenstein-projective Λ-module which is not Gorenstein-projective. is an injective map.
Proof. (i) Assume that M ∈ C . For any right B-module V , taking a B-projective resolution
P • : · · · −→ P 1 −→ P 0 −→ V −→ 0 of V , one has a projective resolution A ⊗ k P • of right Λ-module A ⊗ k V . By the isomorphisms (A ⊗ k V ) ⊗ Λ (M ⊗ k B) ∼ = (A ⊗ A M ) ⊗ k (V ⊗ B B) ∼ = M ⊗ k V one sees that there is an isomorphism of complexes (A ⊗ k P • ) ⊗ Λ (M ⊗ k B) ∼ = M ⊗ k P • and hence Tor Λ i (A ⊗ k V, M ⊗ k B) ∼ = Tor k i (M, V ) = 0. This shows M ⊗ k B ∈ mon(B, A). Further, by (A ⊗ k V ) ⊗ Λ (M ⊗ k B) ∼ = M ⊗ k V ∈ C , one gets M ⊗ k B ∈ mon(B, C ). Conversely, if M ⊗ k B ∈ mon(B, C ), then by definition (A ⊗ k B) ⊗ Λ (M ⊗ k B) ∼ = M ⊗ k B ∈ C .
Since C is closed under direct summands, it follows that M ∈ C .
(ii) Assume that M ∈ ⊥ A and M / ∈ gp(A). By (i), M ⊗ k B ∈ mon(B, ⊥ A) ⊆ ⊥ Λ, where the inclusion follows from Theorem 3.1. Again by (i), M ⊗ k B / ∈ mon(B, gp(A)) = gp(Λ), where the equality follows from Theorem 2.6.
(iii) Assume that C 1 and C 2 are additive full subcategories of A-mod closed under direct summands, such that mon(B, C 1 ) = mon(B, C 2 ). We need to prove C 1 = C 2 . Let M ∈ C 1 . By (i), M ⊗ k B ∈ mon(B, C 1 ). Thus M ⊗ k B ∈ mon(B, C 2 ). Again by (i), M ∈ C 2 . This completes the proof.
Proposition 4.7. If Λ is weakly Gorenstein, then so is A. Conversely, if A is weakly Gorenstein, then a semi-Gorenstein-projective Λ-module is Gorenstein-projective if and only if it is monic.
Thus, if A is weakly Gorenstein, then Λ is weakly Gorenstein if and only if each semi-Gorensteinprojective Λ-module is monic, or equivalently, ⊥ Λ = mon(B, ⊥ A).
Proof. If Λ is weakly Gorenstein, then A is weakly Gorenstein, by Lemma 4.6(ii).
Assume that A is weakly Gorenstein and M is a semi-Gorenstein-projective Λ-module. If M is Gorenstein-projective, then M is monic, by Theorem 2.6. If M is monic, then by Theorem 3.1 and Theorem 2.6 one has M ∈ mon(B, A) ∩ ⊥ Λ = mon(B, ⊥ A) = mon(B, gp(A)) = gp(Λ). Proof. By Proposition 4.7, it remains to prove the "if" part. Assume that A is weakly Gorenstein. We will prove that Λ = A ⊗ k kQ is weakly Gorenstein, by using induction on |Q 0 |. If |Q 0 | = 1, then Λ = A is weakly Gorenstein, by the assumption.
Assume that |Q 0 | ≥ 2. We write the conjunction of paths of Q from left to right. Since Q is an acyclic quiver, Q has a source vertex, say, n, and then
kQ = kQ ′ radP (n) 0 k
where Q ′ is the subquiver of Q by deleting the source vertex n, and P (n) = kQe n . Then radP (n) is a kQ ′ -k-bimodule. Thus
Λ = A ⊗ k kQ = A⊗ k kQ ′ M 0 A where M = A ⊗ k radP (n) is an (A ⊗ k kQ ′ )-A-bimodule.
Since Q is acyclic, so is Q ′ . Hence radP (n) is a projective left kQ ′ -module. Thus M = A ⊗ k radP (n) is a projective left (A ⊗ k kQ ′ )-module, and also M = A ⊗ k radP (n) is a projective right A-module.
Since |Q ′ 0 | = |Q 0 | − 1, by induction A ⊗ kQ ′ is weakly Gorenstein. Applying Proposition 4.4 to A⊗kQ ′ M 0 A = Λ, one sees that Λ is weakly Gorenstein.
4.3.
Semi-Groenstein-projective-free algebras. Replacing the condition that M B is projective in Proposition 4.4 by " ⊥ B = add(B) and proj.dimM B < ∞", we then get the following result on lsgp-free algebras.
11 to X Y ϕ ∈ ⊥ Λ one gets that Y ∈ ⊥ B, that ϕ : M ⊗ B Y −→ X induces isomorphisms Ext i A (X, A) ∼ = Ext i A (M ⊗ B Y, A) for all i ≥ 1, and that ϕ * : X * −→ (M ⊗ B Y ) * is a right A-epimorphism. Since Y ∈ ⊥ B = add(B), B Y is projective. Thus A (M ⊗ B Y ) ∈ add( A M ). Since by assumption A M is torsionless, it follows that A (M ⊗ B Y ) is torsionless. Thus, the canonical map φ M⊗B Y : M ⊗ B Y −→ (M ⊗ B Y ) * * is a monomorphism.
By the same argument as in the proof of Proposition 4.4 one concludes that X Y ϕ is monic respect to bimodule A M B . Now, assume in addition that A M is projective. Continuing the argument above, one knows that
A (M ⊗ B Y ) is projective, thus, φ M⊗B Y : M ⊗ B Y −→ (M ⊗ B Y ) * * is an isomorphism. By Ext i A (X, A) ∼ = Ext i A (M ⊗ B Y, A)
= 0 for all i ≥ 1, one has X ∈ ⊥ A. Since by assumption A is weakly Gorenstein, X ∈ gp(A), and hence φ X : X −→ X * * is an isomorphism. Since ϕ * : X * −→ (M ⊗ B Y ) * is an epimorphism and (M ⊗ B Y ) * is a right projective A-module, it follows that ϕ * is a splitting epimorphism, and hence ϕ * * : (M ⊗ B Y ) * * −→ X * * is a splitting monomorphism. From the commutative diagram
M ⊗ B Y φM⊗ B Y ∼ = ϕ / / X φX ∼ = (M ⊗ B Y ) * * ϕ * * / / X * * one sees that ϕ : M ⊗ B Y −→ X is also a splitting monomorphism. Thus X ∼ = (M ⊗ B Y ) ⊕ X ′ for some X ′ ∈ gp(A) and X Y ϕ = M⊗B Y Y IdM⊗ B Y ⊕ X ′ 0 . Since A M is projective and proj.dimM B < ∞, the A-B-bimodule M is compatible (cf. Lemma 2.12). By Theorem 2.13, X ′ 0 ϕ ∈ gp(Λ), and hence X Y ϕ = M⊗B Y Y Id M ⊗ B Y ⊕ X ′ 0 ∈ gp(Λ). This proves ⊥ Λ = gp(Λ) = { M ⊗ B P P IdM⊗ B P ⊕ G 0 | P ∈ add(B), G ∈ ⊥ A = gp(A)}
and from which one sees that ⊥ Λ = add(Λ) if and only if ⊥ A = add (A). Proof. Assume that ⊥ A = add (A). We will prove ⊥ Λ = add(Λ), again by using induction on |Q 0 |. If |Q 0 | = 1, then Λ = A, thus the assertion holds, by the assumption ⊥ A = add (A).
Assume that |Q 0 | ≥ 2. Similar as in the proof of Proposition 4.8, we write Λ as a triangular matrix algebra. However, in order to apply Proposition 4.9, this time we need to use the subquiver Q ′ of Q by deleting a sink vertex, say, 1, and the corresponding algebra kQ ′ /I ′ . Then
kQ/I = k rad(e1kQ/I) 0 kQ ′ /I ′ where rad(e 1 kQ/I) is a k-(kQ ′ /I ′ )-bimodule. Thus Λ = A ⊗ k (kQ/I) = A M 0 Λ ′ where Λ ′ = A ⊗ k (kQ ′ /I ′ ), M = A ⊗ k rad(e 1 kQ/I) is a A-Λ ′ -bimodule. Since |Q 0 | = |Q 0 | − 1, by induction one gets ⊥ Λ ′ = add(Λ ′ ).
Since proj.dim.rad(e 1 kQ/I) kQ ′ /I ′ < ∞, it follows that
proj.dim.M Λ ′ < ∞.
Also, A M is projective. Since we already known ⊥ Λ ′ = add(Λ ′ ) by induction, thus we can apply Proposition 4.9 to Λ = A ⊗ k (kQ/I)
= A M 0 Λ ′ to get ⊥ Λ = gp(Λ) = { M ⊗ Λ ′ P P IdM⊗ Λ ′ P ⊕ G 0 | P ∈ add(Λ ′ ), G ∈ ⊥ A}.
Since G ∈ ⊥ A = add (A), it follows that G 0 ∈ add(Λ), and hence ⊥ Λ = add(Λ). Conversely, assume that ⊥ Λ = add(Λ). Let X be an indecomposable A-module with X ∈ ⊥ A. For any indecomposable projective (kQ/I)-module P , by the Cartan-Eilenberg isomorphism ( [CE,Thm. 3.1,p.209,p.205]) one has Ext i Λ (X ⊗ k P, A ⊗ k kQ/I) = p+q=i (Ext p A (X, A) ⊗ k Ext q kQ/I (P, kQ/I)) = 0, ∀ i ≥ 1.
So X ⊗ k P ∈ ⊥ Λ = add(Λ), and hence X ∈ add(A).
Example and Problem. Let
A be the algebra given by quiver 2 β α / / 1 and relations β 2 , αβ. ; ; ① ① ① ① with indecomposable projective modules P (1) = 1 and P (2) = 2 2 1 , and indecomposable injective modules I(1) = 2 1 and I(2) = 2 2 . Since
The Auslander-Reiten quiver of
A is 2 % % ❑ ❑ ❑ ❑ 2 1 # # ❋ ❋ ❋ ❋ 2 2 1 % % ▲ ▲ ▲ ▲
Ext 1 A (2, 2 2 1 ) = 0, Ext 1 A ( 2 2 , 1) = 0, Ext 2 A ( 2 1 , 1) = Ext 1 A ( 2 1 , 2) = 0 one sees that A is lsgp-free, i.e., ⊥ A = add (A). Note that A is not Gorenstein. By Proposition 4.11, ⊥ (A ⊗ k kQ/I) = gp(A ⊗ k kQ/I) = add(A ⊗ k kQ/I), for any finite acyclic quiver Q and any admissible ideal I.
Problem 1.
Are there a left weakly Gorenstein algebra A, a finite acyclic quiver Q, and an admissible ideal I of kQ, such that A ⊗ k kQ/I is not left weakly Gorenstein, or equivalently, such that there is a semi-Gorenstein-projective (A ⊗ k kQ/I)-module which is not monic?
This problem is different from Question 4. Such an algebra A (if there exists) is not Gorenstein (otherwise, A ⊗ k kQ/I is Gorenstein, by [AR,Proposition 2.2]); such an I = 0, by Proposition 4.8; and also ⊥ A = add(A), by Proposition 4.11.
Canonical maps of modules over T 2 (A)
Let A be an Artin algebra, Λ = T 2 (A) = ( A A 0 A ) , and M a Λ-module. We will give a sufficient and necessary condition, such that the canonical Λ-map φ M : M −→ M * * is a monomorphism (an epimorphism, and reflexive, respectively); and we will give a sufficient and necessary condition such that M is double semi-Gorenstein-projective with φ M a monomorphism (an epimorphism, respectively).
Recall that a left Λ-module
M is identified with the triple ( X Y ) ϕ , where ϕ : Y −→ X is a left A- map; and a right Λ-module is identified with a triple (U, V ) ψ , where ψ : U −→ V is a right A-map.
Using the identifications, we will determine the right Λ-module M * = Hom Λ (M, Λ Λ), the left Λ-module
M * * = Hom Λ (M * , Λ Λ ), and φ M : M −→ M * * . 5.1. The Λ-dual of a left Λ-module. For a left Λ-module ( X Y ) ϕ , we will determine the right Λ-module ( X Y ) * ϕ = Hom Λ (( X Y ) ϕ , Λ Λ). As a left Λ-module, Λ Λ = ( A 0 ) ⊕ ( A A ) IdA = A⊕A A ( 0 Id A ) . Thus, any Λ-map f ∈ ( X Y ) * ϕ = Hom Λ (( X Y ) ϕ , A⊕A A ( 0 Id A ) ) is of the form ( α1 α2 ) β , where α 1 , α 2 ∈ X * = Hom A (X, A), β ∈ Y * , such that the square Y β ϕ / / X ( α 1 α 2 ) A ( 0 1 ) / / A ⊕ A commutes. So α 1 ϕ = 0, β = α 2 ϕ. Thus, there is a unique g ∈ (Cokerϕ) * = Hom A (Cokerϕ, A) such that α 1 = gπ, where π : X −→ Cokerϕ is the canonical A-epimorphism. Lemma 5.1. Let ( X Y ) ϕ be a left Λ-module with ϕ : Y −→ X a left A-map. Then (i) Any f ∈ ( X Y ) * ϕ is of the form ( gπ α2 ) α2ϕ , where g ∈ (Cokerϕ) * , π : X −→ Cokerϕ is the canonical A-epimorphism, and α 2 ∈ X * . (ii) There is a unique right Λ-module isomorphism h : ( X Y ) * ϕ ∼ = ((Cokerϕ) * , X * ) π * , given by f = ( gπ α2 ) α2ϕ → (g, α 2 )
where π * : (Cokerϕ) * −→ X * is the right A-monomorphism induced by π.
Proof. (ii) We claim that h is a right Λ-map, i.e.,
h(f ( a1 a2 0 a3 )) = h(f ) ( a1 a2 0 a3 ) , ∀ f = ( gπ α2 ) α2ϕ ∈ ( X Y ) * ϕ , ∀ ( a1 a2 0 a3 ) ∈ Λ.
In fact, for any x y ∈ X Y ϕ , since πϕ = 0, one has (f ( a1 a2 0 a3 ))
x y = (f x y ) ( a1 a2 0 a3 ) = (gπ)(x) α2(x) 0 α2(ϕ(y)) ( a1 a2 0 a3 ) = (gπ)(x)a1 (gπ)(x)a2+α2(x)a3 0 α2(ϕ(y))a3 = ((ga1)π)(x) ((ga2)π+α2a3)(x) 0 ((ga2)π+α2a3)(ϕ(y)) = (ga1)π (ga2)π+α2a3 ((ga2)π+α2a3)ϕ x y Thus f ( a1 a2 0 a3 ) = (ga1)π (ga2)π+α2a3 ((ga2)π+α2a3)ϕ
, and hence h(f ( a1 a2 0 a3 )) = (ga 1 , (ga 2 )π + α 2 a 3 ).
One the other hand, by the right Λ-module structure of ((Cokerϕ) * , X * ) π * , one has h(f ) ( a1 a2 0 a3 ) = (g, α 2 ) ( a1 a2 0 a3 ) = (ga 1 , π * (ga 2 ) + α 2 a 3 ) = (ga 1 , (ga 2 )π + α 2 a 3 ).
This proves the claim.
Since the map
((Cokerϕ) * , X * ) π * −→ ( X Y ) * ϕ , (g, α 2 ) → f = ( gπ α2 ) α2ϕ
is the inverse of h, h is a right Λ-isomorphism.
5.2.
The Λ-dual of a right Λ-module. Similarly, one can determine the Λ-dual of a right Λ-module
(U, V ) ψ , where ψ : U −→ V is a right A-map. As a right Λ-module, Λ Λ = (A, A) IdA ⊕ (0, A) = (A, A ⊕ A) ( Id A 0 ) . So any Λ-map f ∈ (U, V ) * ψ = Hom Λ ((U, V ) ψ , (A, A ⊕ A) ( Id A 0 ) ) is of the form (α, β1 β2 ), where α ∈ U * = Hom A (U, A A ), β 1 , β 2 ∈ V * , such that U α ψ / / V ( β 1 β 2 ) A ( 1 0 ) / / A ⊕ A
commutes. Thus α = β 1 ψ, β 2 ψ = 0. Hence, there is a unique g ∈ (Cokerψ) * = Hom A (Cokerψ, A A ) such that β 2 = gπ, where π : V −→ Cokerψ is the canonical A-map. By the similar argument one has
Lemma 5.2. Let (U, V ) ψ be a right Λ-module with ψ : U −→ V a right A-map. Then (i) Any f ∈ (U, V ) * ψ is of the form (β 1 ψ, β1 gπ ), where β 1 ∈ V * , g ∈ (Cokerψ) * , and π : V −→ Cokerψ is the canonical A-epimorphism. (ii) There is a unique left Λ-module isomorphism h ′ : (U, V ) * ψ ∼ = V * (Cokerψ) * π *
, given by
f = (β 1 ψ, β1 gπ ) → β 1 g
where π * : (Cokerψ) * −→ V * is the left A-monomorphism induced by π.
The left Λ-module
X Y * * ϕ .
For any left Λ-module ( X Y ) ϕ with left A-map ϕ : Y −→ X, by Lemma 5.1, one has the right module isomorphism h :
X Y * ϕ ∼ = ((Cokerϕ) * , X * ) π * , f = ( gπ α2 ) α2ϕ → (g, α 2 ) (5.1)
where π * : (Cokerϕ) * −→ X * is the right A-monomorphism induced by π : X −→ Cokerϕ. Applying Lemma 5.2 to ((Cokerϕ) * , X * ) π * , we then get Lemma 5.3. (i) Any f ∈ ((Cokerϕ) * , X * ) * π * is of the form (β 1 π * , β1 gp ), where β 1 ∈ X * * , g ∈ (Cokerπ * ) * , and p : X * −→ Cokerπ * is the canonical A-epimorphism.
(ii) There is a unique left Λ-module isomorphism h :
X * * (Cokerπ * ) * p * ∼ = ( X Y ) * * ϕ , given by β 1 g → h * ((β 1 π * , β1 gp )) where p * : (Cokerπ * ) * −→ X * * is the A-monomorphism induced by p, h : X Y * ϕ ∼ = ((Cokerϕ) * , X * ) π *
is given in (5.1), and h * :
((Cokerϕ) * , X * ) * π * −→ X Y * * ϕ is induced by h. 5.4. The canonical Λ-map φ ( X Y ) ϕ : X Y ϕ −→ X Y * * ϕ . For a left Λ-module ( X Y ) ϕ , one has an exact sequence Y ϕ −→ X π −→ Cokerϕ −→ 0 of left A-modules. Applying Hom A (−, A A), one gets an exact sequence of right A-modules 0 −→ (Cokerϕ) * π * −→ X * ϕ * −→ Y * and the exact sequence 0 −→ (Cokerϕ) * π * −→ X * p −→ Cokerπ * −→ 0.
Thus, there is a unique A-map β : Cokerπ * −→ Y * such that the diagram
0 / / (Cokerϕ) * π * / / X * p / / Cokerπ * β ✤ ✤ ✤ / / 0 0 / / (Cokerϕ) * π * / / X * ϕ * / / Y * .
(5.2) commutes, i.e., ϕ * = βp. Thus, ϕ * is an epimorphism if and only if so is β, and if and only if β is an isomorphism. So one has the A-map β * : Y * * −→ (Cokerπ * ) * . Consider the composition
β * φ Y : Y −→ (Cokerπ * ) * where φ Y : Y −→ Y * * is the canonical map.
By the definition of φ Y and β * , one knows that β * φ Y : Y −→ (Cokerπ * ) * is given by y → "g → (β(g))(y)", ∀ g ∈ Cokerπ *
i.e., ((β * φ Y )(y))(g) = (β(g))(y), ∀ y ∈ Y.
Proposition 5.4. For any left Λ-module ( X Y ) ϕ with left A-map ϕ : Y −→ X, with the notations above one has
(i) φX β * φY : X Y ϕ −→ X * * (Cokerπ * ) * p * is left Λ-map,
where φ X : X −→ X * * and φ Y : Y −→ Y * * are the canonical A-maps, β : Cokerπ * −→ Y * is the canonical A-map such that ϕ * = βp, and β * : Y * * −→ (Cokerπ * ) * is induced by β.
(ii) The canonical Λ-map φ ( X Y ) ϕ : X Y ϕ −→ X Y * * ϕ is given by φ ( X Y ) ϕ = h • φ X β * φ Y .
where h :
X * * (Cokerπ * ) * p * −→ X Y * * ϕ is the isomorphism given in Lemma 5.3.
Proof. (i) One needs to prove the diagram
Y ϕ / / β * φY X φX (Cokerπ * ) * p * / / X * * commutes, i.e., p * β * φ Y = φ X ϕ.
In fact, since ϕ * = βp, one has ϕ * * = p * β * . By the functorial property of the canonical map φ X : X −→ X * * one has the commutative diagram
Y ϕ / / φY X φX Y * * ϕ * * / / X * * . It follows that p * β * φ Y = ϕ * * φ Y = φ X ϕ. (ii) We need to prove φ ( X Y ) ϕ x y = h( φX β * φY x y ), ∀ x y ∈ X Y ϕ . For this, let f ∈ X Y * ϕ . By Lemma 5.1(i), f = ( gπ α2 ) α2ϕ ,
where g ∈ (Cokerϕ) * , π : X −→ Cokerϕ is the canonical A-epimorphism, and α 2 ∈ X * . By the definition of φ ( X Y ) ϕ one has
φ ( X Y ) ϕ x y (f ) = f x y = ( gπ α2 ) α2ϕ x y = (gπ)(x) α2(x) 0 α2(ϕ(y)) ∈ Λ.
On the other hand, by the definitions of β * φ Y and h one has
h( φ X β * φ Y x y )(f ) = h( φ X (x) β * φ Y (y) )(f ) = h * ((φ X (x)π * , φX (x) (β * φY (y))p ))(f ) = (φ X (x)π * , φX (x) (β * φY (y))p )(h(f )) = (φ X (x)π * , φX (x) (β * φY (y))p )(g, α 2 ) = φX (x)(π * (g)) φX (x)(α2) 0 β * φY (y)(p(α2)) = π * (g)(x) α2(x) 0 φY (y)(β(p(α2))) = π * (g)(x) α2(x) 0 β(p(α2))(y) = g(π(x)) α2(x) 0 ϕ * (α2)(y) = (gπ)(x) α2(x) 0
α2(ϕ(y)) ∈ Λ. This completes the proof. 5.5. Torsionless Λ-modules and reflexive Λ-modules.
Corollary 5.5. Let ( X Y ) ϕ be a left Λ-module, where ϕ : Y −→ X is a left A-map. Then (i) ( X Y ) ϕ is a torsionless Λ-module if and only if it is monic (i.e., ϕ is a monomorphism), X and Y are torsionless A-modules.
(ii) φ ( X Y ) ϕ
is a Λ-epimorphism if and only if φ X and β * φ Y : Y −→ (Cokerπ * ) * are A-epimorphisms.
(iii) ( X Y ) ϕ is a reflexive Λ-module if and only if ϕ is a monomorphism, X is reflexive A-module, and β * φ Y is an isomorphism.
Proof. By Proposition 5.4, φ ( X Y ) ϕ : X Y ϕ −→ X Y * * ϕ is given by φ ( X Y ) ϕ = h • φ X β * φ Y
where h :
X * * (Cokerπ * ) * p * −→ X Y * * ϕ is an isomorphism. Since φX β * φY : X Y ϕ −→ X * * (Cokerπ * ) * p * is a left Λ-map, the diagram Y ϕ / / β * φY X φX (Cokerπ * ) * p * / / X * * commutes, i.e., p * β * φ Y = φ X ϕ. (i) Assume that ( X Y ) ϕ is a torsionless Λ-module, i.e., φ ( X Y ) ϕ : X Y ϕ −→ X Y * *
ϕ is a Λ-monomorphism. Thus φ X and β * φ Y are monomorphisms, in particular φ Y is a monomorphism, so X and Y are torsionless.
Since p * β * φ Y = φ X ϕ and p * : (Cokerπ * ) * −→ X * * is a monomorphism, it follows that φ X ϕ is a monomorphism, and hence ϕ is a monomorphism, i.e., ( X Y ) ϕ is monic. Conversely, assume that ( X Y ) ϕ is monic, X and Y are torsionless A-modules, i.e., ϕ is a monomorphism, φ X and φ Y are monomorphisms. By
p * β * φ Y = φ X ϕ, p * β * φ Y is a monomorphism, and hence β * φ Y is a monomorphism. Thus φ ( X Y ) ϕ = h • φX β * φY is a monomorphism, i.e., ( X Y ) ϕ is a torsionless Λ-module.
The assertion (ii) a direct consequence of the formula φ ( X Y ) ϕ = h • φX β * φY .
The assertion (iii) a direct consequence of (i) and φ ( X Y ) ϕ = h • φX β * φY . 5.6. Double semi-Gorenstein-projective Λ-modules. For a left Λ-module ( X Y ) ϕ with ϕ : Y −→ X a left A-map, by Corollary 2.14(i), ( X Y ) ϕ ∈ ⊥ A if and only if the following conditions (1)-(3) hold: (1) X ∈ ⊥ A;
(2) Y ∈ ⊥ A;
(3) ϕ * : X * −→ Y * is an epimorphism.
By Lemma 5.1, X Y * ϕ ∼ = ((Cokerϕ) * , X * ) π * as right Λ-modules, where π * : (Cokerϕ) * −→ X * is the right A-monomorphism induced by π : X −→ Cokerϕ. Thus, by the right module version of Corollary 2.14(i), ( X Y ) * ϕ ∈ ⊥ A if and only if the following conditions (4)-(6) hold: (4) (Cokerϕ) * ∈ ⊥ A; (5) X * ∈ ⊥ A; (6) π * * : X * * −→ (Cokerϕ) * * is an epimorphism. Proof. It remains to show that the conditions (1) - (6) imply the conditions (7) and (8).
Lemma 5.6. Let ( X Y ) ϕ be a left Λ-module, where ϕ : Y −→ X is a left A-map. Then ( X Y ) ϕ
Assume that the conditions (1) -(6) hold. Since ϕ * : X * −→ Y * is an epimorphism, β : Cokerπ * −→ Y * is an isomorphism (cf. the diagram (5.2)). Applying Hom A (−, A A) to the exact sequence 0 −→ (Cokerϕ) * π * −→ X * ϕ * −→ Y * −→ 0, by the assumption that π * * : X * * −→ (Cokerϕ) * * is an epimorphism and by the assumptions X * ∈ ⊥ A and (Cokerϕ) * ∈ ⊥ A, one sees that Y * ∈ ⊥ A.
5.7.
A double semi-Gorenstein-projective Λ-module M with φ M monomorphism or epimorphism.
Proposition 5.7. Let ( X Y ) ϕ be a left Λ-module with left A-map ϕ : Y −→ X. Then (i) ( X Y ) ϕ is torsionless and double semi-Gorenstein-projective if and only if ( X Y ) ϕ is monic (i.e. ϕ is a monomorphism), X, Y , and Cokerϕ are double semi-Gorenstein-projective, and X and Y are torsionless.
(ii) ( X Y ) ϕ is double semi-Gorenstein-projective with epimorphism φ ( X Y ) ϕ
if and only if the following conditions are satisfied:
• ϕ * : X * −→ Y * is an epimorphism;
• All the five modules X, Y, X * , Y * , (Cokerϕ) * are semi-Gorenstein-projective;
• φ X and φ Y are epimorphisms.
(iii) (Corollary 2.14) ( X Y ) ϕ is Gorenstein-projective if and only if ϕ is a monomorphism, Y and Cokerϕ are Gorenstein-projective. If this is the case, then X is Gorenstein-projective.
Proof. (i) Assume that ( X Y ) ϕ is torsionless and double semi-Gorenstein-projective. By Corollary 5.5(i), ϕ is a monomorphism, and X and Y are torsionless. By Lemma 5.6, all the conditions (1)-(8) hold. Applying Hom A (−, A A) to the exact sequence 0 −→ Y ϕ −→ X π −→ Cokerϕ −→ 0, since ϕ * : X * −→ Y * is an epimorphism, and since X and Y are semi-Gorenstein-projective, it follows that Cokerϕ is semi-Gorenstein-projective.
Conversely, assume that ϕ is a monomorphism, X, Y and Cokerϕ are double semi-Gorensteinprojective, and that X and Y are torsionless. By Corollary 5.5(i), ( X Y ) ϕ is torsionless. Again applying
Hom A (−, A A) to the exact sequence 0 −→ Y ϕ −→ X π −→ Cokerϕ −→ 0, since Cokerϕ is semi-Gorenstein- projective, ϕ * : X * −→ Y * is an epimorphism and 0 −→ (Cokerϕ) * π * −→ X * ϕ * −→ Y * −→ 0 is an exact sequence.
Since Y * is semi-Gorenstein-projective, π * * : X * * −→ (Cokerϕ) * * is an epimorphism. Thus, all the conditions (1)-(6) hold. By Lemma 5.6, ( X Y ) ϕ is double semi-Gorenstein-projective.
(ii) Assume that ( X Y ) ϕ is double semi-Gorenstein-projective and φ ( X Y ) ϕ is an epimorphism. By Lemma 5.6, all the conditions (1)-(8) are satisfied. By Corollary 5.5(ii), φ X and β * φ Y are epimorphisms, where β : Cokerπ * −→ Y * is the canonical A-map such that ϕ * = βp, p : X * −→ Cokerπ * is the canonical A-epimorphism, and β * : Y * * −→ (Cokerπ * ) * is induced by β. It remains to show that φ Y is an epimorphism. In fact, by Condition (8), β * is an isomorphism, hence φ Y is an epimorphism.
Conversely, assume that ϕ * : X * −→ Y * is an epimorphism, all the five modules X, Y , X * , Y * , (Cokerϕ) * are semi-Gorenstein-projective, and that φ X and φ Y are epimorphisms. Since ϕ * is an epimorphism, β : Cokerπ * −→ Y * is an isomorphism (cf. Subsection 5.4), and hence β * is an isomorphism. Thus β * φ Y is an epimorphism. By Corollary 5.5(ii), φ ( X Y ) ϕ is an epimorphism.
Applying Hom
A (−, A A) to the exact sequence Y ϕ −→ X π −→ Cokerϕ −→ 0, since ϕ * : X * −→ Y * is an epimorphism, it follows that 0 −→ (Cokerϕ) * π * −→ X * ϕ * −→ Y * −→ 0 is an exact sequence.
Since Y * is semi-Gorenstein-projective, π * * : X * * −→ (Cokerϕ) * * is an epimorphism. Thus, all the conditions (1)-(6) hold. By Lemma 5.6, ( X Y ) ϕ is double semi-Gorenstein-projective. (iii) This is just Corollary 2.14. We rewrite here, because in the setting of (i) and (ii), it admits a simple proof. The "if" part follows from (i) and (ii) and the fact that Gorenstein-projective modules are closed under extensions.
Assume that ( X Y ) ϕ is Gorenstein-projective. Then by (i) and (ii), ϕ is a monomorphism, X and Y are Gorenstein-projective, and Cokerϕ is double semi-Gorenstein-projective. Moreover, the diagram
0 / / Y ϕ / / φY ∼ = X π / / φX ∼ = Cokerϕ φ Cokerϕ / / 0 0 / / Y * * ϕ * * / / X * * π * * / / (Cokerϕ) * * / / 0 (5.3)
commutes with exact rows. So φ Cokerϕ is an isomorphism, and thus Cokerϕ is Gorenstein-projective. (1) Assume that any torsionless and double semi-Gorenstein-projective Amodule is Gorenstein-projective. Let M = ( X Y ) ϕ be a torsionless and double semi-Gorenstein-projective Λ-module. We need to show that M is Gorenstein-projective.
By Proposition 5.7(i), ϕ is a monomorphism, X, Y , and Cokerϕ are double semi-Gorenstein-projective, and X and Y are torsionless. By the assumption, X and Y are Gorenstein-projective.
By Lemma 5.6, ϕ * and π * * are epimorphisms. Thus, one again has the commutative diagram (5.3) with exact rows, from which one knows that φ Cokerϕ is also an isomorphism, and hence Cokerϕ is Gorensteinprojective. Thus, M is Gorenstein-projective, by Proposition 5.7(iii).
Conversely, assume that any torsionless and double semi-Gorenstein-projective Λ-module is Gorensteinprojective. Let L be a torsionless and double semi-Gorenstein-projective A-module. We need to prove that L is Gorenstein-projective.
Since L is torsionless, a left add(A)-approximation ϕ : L −→ P of L is a monomorphism, where P is a projective A-module. Since both P and L are semi-Gorenstein-projective and ϕ is a left add(A)approximation, it follows that Cokerϕ is also semi-Gorenstein-projective. Consider the Λ-module ( P L ) ϕ . Since P * and L * are semi-Gorenstein-projective and is an exact sequence, it follows that (Cokerϕ) * is also semi-Gorenstein-projective. Thus, by Proposition 5.7(i), ( P L ) ϕ is a torsionless and double semi-Gorenstein-projective Λ-module. By the assumption, ( P L ) ϕ is Gorenstein-projective. Hence L is Gorenstein-projective, by Proposition 5.7(iii).
(2) Assume that any double semi-Gorenstein-projective A-module L with φ L an epimorphism is Gorenstein-projective. Let M = ( X Y ) ϕ be a double semi-Gorenstein-projective Λ-module such that φ M is an epimorphism. We need to prove that M is Gorenstein-projective.
By Proposition 5.7(ii), ϕ * : X * −→ Y * is an epimorphism, all the five modules X, Y , X * , Y * , (Cokerϕ) * are semi-Gorenstein-projective, and φ X and φ Y are epimorphisms. By the assumption, X and Y are Gorenstein-projective, in particular, φ X and φ Y are isomorphisms.
We claim that ϕ : Y −→ X is a monomorphism and Cokerϕ is reflexive. In fact, applying
Hom A (−, A) to Y ϕ −→ X π −→ Cokerϕ −→ 0, since ϕ * : X * −→ Y * is an epimorphism, it follows that 0 −→ (Cokerϕ) * π * −→ X * ϕ * −→ Y * −→ 0 is an exact sequence. Since Y * is semi-Gorenstein-projective, 0 −→ Y * * ϕ * * −→ X * * π * * −→ (Cokerϕ) * * −→ 0
is an exact sequence. Thus, by the functorial property of φ one has the commutative diagram
Y ϕ / / φY ∼ = X π / / φX ∼ = Cokerϕ φ Cokerϕ / / 0 0 / / Y * * ϕ * * / / X * * π * * / / (Cokerϕ) * * / / 0
with exact rows. Since both φ Y and ϕ * * are monomorphisms, ϕ is a monomorphism. Also, this commutative diagram shows that φ Cokerϕ is an isomorphism, i.e., Cokerϕ is reflexive. This proves the claim. Applying Hom A (−, A A) to the exact sequence 0 −→ Y ϕ −→ X π −→ Cokerϕ −→ 0, since ϕ * : X * −→ Y * is an epimorphism and X and Y are semi-Gorenstein-projective, it follows that Cokerϕ is also semi-Gorenstein-projective. So, Cokerϕ is double semi-Gorenstein-projective and reflexive, i.e., Cokerϕ is Gorenstein-projective. By Proposition 5.7(iii), M = ( X Y ) ϕ is Gorenstein-projective.
Conversely, assume that any double semi-Gorenstein-projective Λ-module M with φ M an epimorphism is Gorenstein-projective. Let L be a double semi-Gorenstein-projective A-module such that φ L is an epimorphism. We need to show that L is Gorenstein-projective.
Take a left add(A)-approximation ϕ : L −→ P of L. Applying Hom A (−, A A) to the exact sequence L ϕ −→ P π −→ Cokerϕ −→ 0, since ϕ is left add(A)-approximation, ϕ * : P * −→ L * is an epimorphism and 0 −→ (Cokerϕ) * π * −→ P * ϕ * −→ L * −→ 0
is an exact sequence. Since L * and P * are semi-Gorenstein-projective, so is (Cokerϕ) * . Thus, by Proposition 5.7(ii), ( P L ) ϕ is a double semi-Gorenstein-projective Λ-module such that φ ( P L ) ϕ is an epimorphism.
By the assumption, ( P L ) ϕ is Gorenstein-projective. Hence by Proposition 5.7(iii), L is Gorensteinprojective. 6. Double semi-Gorenstein-projective modules which are not monic 6.1. Proof of Theorem 1.7. Since by assumption Y is not torsionless and ϕ : Y −→ P is a left add(A)approximation of Y , it follows that ϕ is not a monomorphism. Thus P Y ϕ is not a monic T 2 (A)-module. By Corollary 5.5(i), P Y ϕ is not torsionless.
Apply Hom A (−, A A) to the exact sequence Y ϕ −→ P π −→ Cokerϕ −→ 0. Since ϕ is a left add(A)approximation of Y , ϕ * : P * −→ Y * is an epimorphism and
0 −→ (Cokerϕ) * π * −→ P * ϕ * −→ Y * −→ 0
is an exact sequence. Since Y * and P * are semi-Gorenstein-projective, it follows that (Cokerϕ) * is semi-Gorenstein-projective and π * * : P * * −→ (Cokerϕ) * * is an epimorphism. Thus, all the conditions (1) - (6) in Subsection 5.6 are satisfied. By Lemma 5.6, ( P Y ) ϕ is a double semi-Gorenstein-projective T 2 (A)-module. is not semi-Gorenstein-projective, where ι : A(x − y) −→ A is the embedding. In particular, X(c) * is not Gorenstein-projective.
(ii) If one identifies X(c) * * with where r x−y is the right multiplication by x − y. Thus, φ X(c) is neither a monomorphism nor an epimorphism.
Proof. It remains to prove (i) and (ii).
(i) Note that there are isomorphisms (Cokerf 1 ) * = (A/A(x − y)) * ∼ = (x − q −1 y)A ∼ = A/(x − q −2 y)A as right A-modules, where the first isomorphism is given by g → g(1), and the second isomorphism is given by x − q −1 y →1. We stress that Note that (Cokerσ) * = (A/(x − q −1 y)A) * ∼ = A(x − y)A as left A-modules, with the isomorphism given by g → g(1). By Lemma 5.3(ii), there is a left Λ-module isomorphism Note that A(x−y)A = A(x−y)⊕kzx is decomposable left A-module of dimension 3. By [RZ3, Theorem 1.5], A(x − y)A not semi-Gorenstein-projective. It follows from Corollary 2.14(1) that A A(x−y)A ι is not semi-Gorenstein-projective, and hence X(c) * * is not semi-Gorenstein-projective. In particular, X(c) * is not Gorenstein-projective.
(ii) To get φ X(c) , we apply Proposition 5.4 to X(c) = AA M(1,−q,c) f1
. It is clear that φ A = Id A .
Since X(c) is double semi-Gorenstein-projective, it follows from Lemma 5.6 that the canonical A-map β : Cokerσ −→ M (1, −q, c) * appeared in Proposition 5.4, is an isomorphism. Without loss of generality, one may regarded β as the identity. Note that
M (1, −q, c) * * ∼ = M ′ (1, −q −1 , 0) * ∼ = A(x − y)A
where the first isomorphism is given in Lemma 6.1, and the second isomorphism is given by g → g(1). By Proposition 5.4(ii), if we identify X(c) * * with Thus, φ M(1,−q,c) is neither a monomorphism nor an epimorphism, and hence φ X(c) is neither a monomorphism nor an epimorphism.
, then Λ = A ⊗ k B = T n (A)
Question 2 .
2(i) Let A and B be Artin algebras, M an A-B-bimodule such that Λ = ( A M 0 B ) is an Artin algebra. When Λ is weakly Gorenstein?
Theorem 1. 2 .
2Let A and B be Artin algebras, M an A-B-bimodule with proj.dim A M < ∞, and Λ = ( A M 0 B ). (1) If proj.dimM B < ∞ and D(M B ) ∈ ( ⊥ B) ⊥ , then Λ is weakly Gorenstein if and only if each semi-Gorenstein-projective Λ-module is monic respect to bimodule M , and A and B are weakly Gorenstein.
2 .
2Preliminaries: Monic modules with relations to Gorenstein-projective modules 2.1. Monic modules over tensor algebras. Definition 2.1. ([HLXZ, 3.1]) Let A and B be finite-dimensional k-algebras, and Λ = A ⊗ k B.
Lemma 2.2. ([HLXZ, Lemma 3.2(7)]; [ZX, Theorem 2.6(1)]) One has
Denote by M(A, M, B) the full subcategory of Λ-mod of monic Λ-modules respect to bimodule A M B , which is called the monomorphism category respect to bimodule A M B . Example 2.8. The monomorphism category mon(B, A) and the monomorphism category M(A, M, B) are in different setting. Even
Lemma 2 . 9 .
29Let Λ = ( A M 0 B ) be an Artin algebra, where M B is projective. Then torsionless Λ-modules are monic respect to bimodule A M B .
This completes the proof. 4. Weakly Gorenstein algebras: Proof of Theorems 1.2, 1.3, and 1.4 4.1. When triangular matrix algebras of bimodules are weakly Gorenstein? Let A and B be Artin algebras, M an A-B-bimodule such that Λ = ( A M 0 B
Remark 4. 5 .
5The "only if" part in Proposition 4.4 does not need the condition that A M is torsionless.4.2.When tensor algebras are weakly Gorenstein? Let A and B be finite-dimensional k-algebras with gl.dimB < ∞, and Λ = A ⊗ k B. We first look at some properties of a map mon(B, −).
Lemma 4. 6 .
6Let A and B be finite-dimensional k-algebra, and Λ = A ⊗ k B.
( i )
iLet C be an additive full subcategory of A-mod closed under direct summands, and M ∈ A-mod. Then M ⊗ k B ∈ mon(B, C ) if and only if M ∈ C .
( iii )
iiiLet Ω (respectively, Γ) be the set of additive full subcategories of A-mod (respectively, Λ-mod) closed under direct summands. Then the map mon(B, −) : Ω −→ Γ, C → mon(B, C )
Proposition 4 . 8 .
48Let Q be a finite acyclic quiver. Then A ⊗ k kQ is weakly Gorenstein if and only if A is weakly Gorenstein.In particular, T n (Aweakly Gorenstein if and only if A is weakly Gorenstein.
Proposition 4. 9 .
9Assume that ⊥ B = add(B) with proj.dimM B < ∞, and that A M is torsionless withproj.dim A M < ∞. Then Λ = ( A M 0 B ) is left weakly Gorenstein if and only if A is left weakly Gorenstein. Moreover, if in addition A M is projective, then ⊥ Λ = gp(Λ) = { M ⊗ B P P IdM⊗ B P ⊕ G 0 | P ∈ add(B), G ∈ ⊥ A = gp(A)} and ⊥ Λ = add(Λ) if and only if ⊥ A = add(A). Proof. Since ⊥ B = add(B), it follows that ( ⊥ B) ⊥ = B-mod, and hence D(M B ) ∈ ( ⊥ B) ⊥ . So, the conditions of Proposition 4.1 are satisfied. To prove the first assertion, by Proposition 4.1, it suffices to prove that if A is left weakly Gorenstein, then any semi-Gorenstein-projective Λ-module X Y ϕ is monic respect to bimodule A M B . In fact, applying Theorem 2.
Remark 4 . 10 .
410The "only if" part in Proposition 4.9 does not need the conditions that ⊥ B = add(B) and A M is torsionless.Proposition 4.11. Let Q be a finite acyclic quiver, I an admissible ideal of kQ, and Λ = A ⊗ k kQ/I. Then ⊥ Λ = add(Λ) if and only if ⊥ A = add(A).
is double semi-Gorenstein-projective if and only if the conditions (1) − (6) above hold, and if and only if the conditions (1) − (8) hold, where (7) Y * ∈ ⊥ A; (8) The canonical A-map β : Cokerπ * −→ Y * is an isomorphism.
Problem 3 .
3Is there a double semi-Gorenstein-projective module M with φ M an epimorphism such that M is not semi-Gorenstein-projective? Theorem 1.6 is a result in this direction. 5.9. Proof of Theorem 1.6.
Hence f ( 1 )
1∈ (x − y)A and M (1, −q, c) * has a k-basis f 1 , f 2 , f 3 , where f i : M (1, −q, c) −→ A is the left A-map given byf 1 (1) = x − y, f 2 (1) = yx, f 3 (1) = zx. Therefore M (1, −q, c) * = f 1 A and f 1 : M (1, −q, c) −→ Ais a left add(A)-approximation of M (1, −q, c). Applying Theorem 1.7 one gets Proposition 6.2. For all c ∈ k, the T 2 f 1 : M (1, −q, c) −→ A is the left A-map given by f 1 (1) = x − y, are double semi-Gorensteinprojective, but not monic, and hence not torsionless.Moreover, one has(i) X(c) * ∼ = ((x − q −1 y)A, A A ) σ , where σ : (x − q −1 y)A −→ A is the embedding; X(c)
y)A ι , then the canonical Λ-map φ X(c) : X(c) −→ X(c)
A
/A(x − y) ≇ M (1, −1, 0) = A/[A(x − y) + socA] and that (A/A(x − y)) * ∼ = (x − q −1 y)A ∼ = M (1, −1, 0) * . By Lemma 5.1(ii), there is a right Λ-module isomorphism h : X(c) * ∼ = ((x − q −1 y)A, A A ) σ where σ : (x − q −1 y)A −→ A is the embedding.
∼
= X(c) * * where ι : A(x − y)A −→ A is the embedding.
it follows that φ M(1,−q,c) : M (1, −q, c) −→ A(x − y)A is just the right multiplication r x−y .
3 )
3If A M is torsionless and M B is projective, then Λ is weakly Gorenstein if and only if A and B are weakly Gorenstein. In particular, if A M and M B are projective, then Λ is weakly Gorenstein if and only if A and B are weakly Gorenstein. Theorem 1.2 is the combination of Propositions 4.1, 4.2 and 4.4. Theorem 1.3. Let A and B be finite-dimensional k-algebras.
A is left CM-free ([Chen 2]) if gp(A) = add(A). Thus, A is lsgp-free if and only if A is left CM-free and left weakly Gorenstein. It is open whether a left CM-free algebra is left weakly Gorenstein (or equivalently, lsgp-free). See [RZ2, 9.2]. Many algebras are lsgp-free. For example, this is the case if gl.dim.A < ∞. There are also non Gorenstein algebras A (thus, gl.dim.A = ∞) which are lsgp-free. Theorem 1.4. (1) Assume that ⊥ B = add(B) with proj.dimM B < ∞ and that A M is torsionless with proj.dim A M < ∞. Then Λ = ( A M 0 B ) is weakly Gorenstein if and only if A is weakly Gorenstein. Moreover, if in addition A M is projective, then
By Theorem 2.11 one knows M⊗B Y Assume that proj.dim A M < ∞, proj.dimM B < ∞, and that B is a Gorenstein algebra. Then Λ = ( A M 0 B ) is weakly Gorenstein if and only if each semi-Gorenstein-projective Λ-module is monic respect to bimodule A M B and A is weakly Gorenstein.Y
IdM⊗ B Y
∈ ⊥ Λ, and hence M⊗B Y
Y
IdM⊗ B Y
∈
gp(Λ). Then by Theorem 2.13, Y ∈ gp(B). This proves that B is weakly Gorenstein.
Taking B to be a Gorenstein algebra in Proposition 4.1, we get
Proposition 4.2.
Therefore Λ is weakly Gorenstein if and only if A is weakly Gorenstein. Proposition 4.4. Assume that A M is torsionless with proj.dim A M < ∞ and that M B is projective. Then Λ = ( A M 0 B ) is weakly Gorenstein if and only if A and B are weakly Gorenstein. In particular, if A M and M B are projective, then Λ is weakly Gorenstein if and only if A and B are weakly Gorenstein.
5.8. Problems. As remarked in[RZ4, 3.1], all known examples of double semi-Gorenstein-projective modules M such that φ M is a monomorphism (an epimorphism, respectively) are Gorenstein-projective.Problem 2. Is there a torsionless and double semi-Gorenstein-projective module M such that M is not Gorenstein-projective?
−→ (Cokerϕ) * −→ P * ϕ * −→ L * −→ 0
Acknowledgement: The author sincerely thanks Claus Michael Ringel for his helpful discussions and comments, and the anonymous referee for suggestions on the presentation of the paper.
A = k x, y, z / x 2 , y 2 , z 2 , yz, xy + qyx, xz − zx, zy − zx where q is an non-zero element in field k, and q is of multiplicative order ∞. Then A is a short local algebra of wild representation type. class of double semi-Gorenstein-projective modules which are not torsionless. From now on, A is the algebra Λ(q), which has been studied in. RZ2, RZ3], i.e.. with a basis 1, x, y, z, yx, zx, and with Hilbert type (|J/J 2 |, |J 2 |) = (3, 2), where J is the Jacobson radical of A. For the studies on short local algebras, we refer to e.g. [L], [Y2], [CV], [AIS], [RZ5, RZ62. A class of double semi-Gorenstein-projective modules which are not torsionless. From now on, A is the algebra Λ(q), which has been studied in [RZ2, RZ3], i.e., A = k x, y, z / x 2 , y 2 , z 2 , yz, xy + qyx, xz − zx, zy − zx where q is an non-zero element in field k, and q is of multiplicative order ∞. Then A is a short local algebra of wild representation type, with a basis 1, x, y, z, yx, zx, and with Hilbert type (|J/J 2 |, |J 2 |) = (3, 2), where J is the Jacobson radical of A. For the studies on short local algebras, we refer to e.g. [L], [Y2], [CV], [AIS], [RZ5, RZ6].
c) ranges over k 3 \ {0}, left A-modules M. = A A/[A(ax + by + cz) + socA] = A1When (a, b, c) ranges over k 3 \ {0}, left A-modules M (a, b, c) := A A/[A(ax + by + cz) + socA] = A1
They are (A/J 2 )-modules. Since A/J 2 is commutative, D(M (a, b, c)) = Hom k (M (a, b, c), k) are also left (A/J 2 )-modules, and hence left A-modules. give all the 3-dimensional local A-modules. RZ3, Proposition A.1] asserts that M (a, b, c) and D(M (a, b, c)) give all the indecomposable left A-modules of dimension 3give all the 3-dimensional local A-modules. They are (A/J 2 )-modules. Since A/J 2 is commutative, D(M (a, b, c)) = Hom k (M (a, b, c), k) are also left (A/J 2 )-modules, and hence left A-modules. [RZ3, Proposition A.1] asserts that M (a, b, c) and D(M (a, b, c)) give all the indecomposable left A-modules of dimension 3.
k 3 \ {0}, we also consider right A-modules M ′ (a, b, c) := A/[(ax + by + cz)A + socA. 1For (a, b, c) ∈ k 3 \ {0}, we also consider right A-modules M ′ (a, b, c) := A/[(ax + by + cz)A + socA] =1A.
An indecomposable A-module M of dimension at most 3 is double semi-Gorenstein-projective which are not torsionless if and only if M ∼ = M (1, −q, c) for some c ∈ k. Moreover, M (1, −q, c) * ∼ = (x − y)A ∼ = A/(x − q −1 y)A = M ′ (1, −q −1 , 0) where the first isomorphism is given by f →. Lemma 6.1. ([RZ3, 1.7. and the second isomorphism is given by x − y →1Lemma 6.1. ([RZ3, 1.7]) An indecomposable A-module M of dimension at most 3 is double semi- Gorenstein-projective which are not torsionless if and only if M ∼ = M (1, −q, c) for some c ∈ k. Moreover, M (1, −q, c) * ∼ = (x − y)A ∼ = A/(x − q −1 y)A = M ′ (1, −q −1 , 0) where the first isomorphism is given by f → f (1), and the second isomorphism is given by x − y →1.
In order to apply Theorem 1.7 to get a family of double semi-Gorenstein-projective T 2 (A)-modules which are not monic, we look for a left add(A)-approximation of M (1, −q, c) = A/A(x − qy + cz) = A1. Any f ∈ M (1, −q, c) * = Hom A (A1, A) is the right multiplication by f (1). A class of double semi-Gorenstein-projective T 2 (A)-modules which are not monic. A class of double semi-Gorenstein-projective T 2 (A)-modules which are not monic. In order to apply Theorem 1.7 to get a family of double semi-Gorenstein-projective T 2 (A)-modules which are not monic, we look for a left add(A)-approximation of M (1, −q, c) = A/A(x − qy + cz) = A1. Any f ∈ M (1, −q, c) * = Hom A (A1, A) is the right multiplication by f (1). Since Imf ∈ J, f (1) =
yx + c 5 zx with c i ∈ k, such that (x − qy + cz)f (1) = 0. Thus c 1 + c 2 = 0. c 3 = 0. Referencesx + c 2 y + c 3 z + c 4 yx + c 5 zx with c i ∈ k, such that (x − qy + cz)f (1) = 0. Thus c 1 + c 2 = 0, c 3 = 0. References
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[
"Shoresh Soltani \nInstitute of Physics and Applied Physics\nYonsei University\n120-749SeoulRepublic of Korea\n\nCenter for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulRepublic of Korea\n",
"Soohyun Cho \nInstitute of Physics and Applied Physics\nYonsei University\n120-749SeoulRepublic of Korea\n\nCenter for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulRepublic of Korea\n",
"Hanyoung Ryu \nCenter for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulRepublic of Korea\n\nDepartment of Physics and Astronomy\nSeoul National University (SNU)\n08826SeoulRepublic of Korea\n",
"Garam Han ",
"Beomyoung Kim \nCenter for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulRepublic of Korea\n\nDepartment of Physics and Astronomy\nSeoul National University (SNU)\n08826SeoulRepublic of Korea\n\nAdvanced Light Source\nLawrence Berkeley National Laboratory\n94720BerkeleyCAUSA\n",
"Dongjoon Song \nElectronic and Photonics Research Institute\nNational Institute of Advanced Industrial Science and Technology (AIST)\n305-8568TsukubaJapan\n",
"Timur K Kim \nDiamond Light Source\nHarwell Campus\nOX11 0DEDidcotUnited Kingdom\n",
"Moritz Hoesch \nDiamond Light Source\nHarwell Campus\nOX11 0DEDidcotUnited Kingdom\n",
"Changyoung Kim \nCenter for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulRepublic of Korea\n\nDepartment of Physics and Astronomy\nSeoul National University (SNU)\n08826SeoulRepublic of Korea\n"
] |
[
"Institute of Physics and Applied Physics\nYonsei University\n120-749SeoulRepublic of Korea",
"Center for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulRepublic of Korea",
"Institute of Physics and Applied Physics\nYonsei University\n120-749SeoulRepublic of Korea",
"Center for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulRepublic of Korea",
"Center for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulRepublic of Korea",
"Department of Physics and Astronomy\nSeoul National University (SNU)\n08826SeoulRepublic of Korea",
"Center for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulRepublic of Korea",
"Department of Physics and Astronomy\nSeoul National University (SNU)\n08826SeoulRepublic of Korea",
"Advanced Light Source\nLawrence Berkeley National Laboratory\n94720BerkeleyCAUSA",
"Electronic and Photonics Research Institute\nNational Institute of Advanced Industrial Science and Technology (AIST)\n305-8568TsukubaJapan",
"Diamond Light Source\nHarwell Campus\nOX11 0DEDidcotUnited Kingdom",
"Diamond Light Source\nHarwell Campus\nOX11 0DEDidcotUnited Kingdom",
"Center for Correlated Electron Systems\nInstitute for Basic Science (IBS)\n08826SeoulRepublic of Korea",
"Department of Physics and Astronomy\nSeoul National University (SNU)\n08826SeoulRepublic of Korea"
] |
[] |
We have performed angle resolved photoemission spectroscopy (ARPES) experiments on the surface states of SrTiO3(001) using linearly and circularly polarized light to investigate the subband structures of out-of-plane d xz/yz orbitals and chiral orbital angular momentum (OAM). The data taken in the first Brillouin zone reveal new subbands for d xz/yz orbitals with Fermi wave vectors of 0.25 and 0.45Å −1 in addition to the previously reported ones. As a result, there are at least two subbands for all the Ti 3d t2g orbitals. Our circular dichroism ARPES data is suggestive of a chiral OAM structure in the surface states and may provide clues to the origin of the linear Rashba-like surface band splitting.
|
10.1103/physrevb.95.125103
|
[
"https://arxiv.org/pdf/1706.05488v1.pdf"
] | 55,956,156 |
1706.05488
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7937e8e8c0609c45cfd1e1b0797cf85e892e2423
|
17 Jun 2017
Shoresh Soltani
Institute of Physics and Applied Physics
Yonsei University
120-749SeoulRepublic of Korea
Center for Correlated Electron Systems
Institute for Basic Science (IBS)
08826SeoulRepublic of Korea
Soohyun Cho
Institute of Physics and Applied Physics
Yonsei University
120-749SeoulRepublic of Korea
Center for Correlated Electron Systems
Institute for Basic Science (IBS)
08826SeoulRepublic of Korea
Hanyoung Ryu
Center for Correlated Electron Systems
Institute for Basic Science (IBS)
08826SeoulRepublic of Korea
Department of Physics and Astronomy
Seoul National University (SNU)
08826SeoulRepublic of Korea
Garam Han
Beomyoung Kim
Center for Correlated Electron Systems
Institute for Basic Science (IBS)
08826SeoulRepublic of Korea
Department of Physics and Astronomy
Seoul National University (SNU)
08826SeoulRepublic of Korea
Advanced Light Source
Lawrence Berkeley National Laboratory
94720BerkeleyCAUSA
Dongjoon Song
Electronic and Photonics Research Institute
National Institute of Advanced Industrial Science and Technology (AIST)
305-8568TsukubaJapan
Timur K Kim
Diamond Light Source
Harwell Campus
OX11 0DEDidcotUnited Kingdom
Moritz Hoesch
Diamond Light Source
Harwell Campus
OX11 0DEDidcotUnited Kingdom
Changyoung Kim
Center for Correlated Electron Systems
Institute for Basic Science (IBS)
08826SeoulRepublic of Korea
Department of Physics and Astronomy
Seoul National University (SNU)
08826SeoulRepublic of Korea
17 Jun 2017(Dated: March 20, 2018)d xz/yz Orbital Subband Structures and Chiral Orbital Angular Momentum in the (001) Surface States of SrTiO 3 PACS number(s): 79.60.-i, 79.60.Bm, 73.20.At, 73.21.Fg
We have performed angle resolved photoemission spectroscopy (ARPES) experiments on the surface states of SrTiO3(001) using linearly and circularly polarized light to investigate the subband structures of out-of-plane d xz/yz orbitals and chiral orbital angular momentum (OAM). The data taken in the first Brillouin zone reveal new subbands for d xz/yz orbitals with Fermi wave vectors of 0.25 and 0.45Å −1 in addition to the previously reported ones. As a result, there are at least two subbands for all the Ti 3d t2g orbitals. Our circular dichroism ARPES data is suggestive of a chiral OAM structure in the surface states and may provide clues to the origin of the linear Rashba-like surface band splitting.
I. INTRODUCTION
Studying two dimensional electron gases (2DEGs) in the surfaces and interfaces of transitional metal oxides has been an interesting topic during the past decades due to the intriguing properties of confined electronic states. In particular, the creation and control of a 2DEG on surfaces of SrTiO 3 (STO) and in STO/Al 2 O 3 interfaces has ignited extensive research 1,2 . The confined electronic states show unique and interesting properties including superconductivity, magnetism, multiferroicity and enhanced Seebeck coefficients [3][4][5][6][7] . Such novel phenomena make STO important in oxide electronics applications. To have deeper understanding of these phenomena and fabricate practical devices, we also need realistic models and experimental observations of confined states within a narrow surface potential. All these make studies of metallic states on STO important in the field of solid state physics.
Several theoretical and experimental studies of the surface states of STO have been performed so far [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] . Especially, exploiting its surface sensitivity, ARPES has been used to directly measure the band structures of STO metallic states. For instance, subband structures and their orbital characters have been investigated 11 . Other than ARPES studies, it is known from transport measurements that each subband has a small Rashba-like splitting with linear and non-linear terms. 26 While the band dispersion and the origin of 2DEG at the surface seem to be well studied, there are still issues to be resolved. For example, the origin of the surface band splitting deduced from the transport experiments is not settled 26 . Different approaches have been used to explain it within the standard Rashba or unconventional Rashba models 12,17,18,[27][28][29][30] . Zhong et. al. suggested that the spin orbit coupling (SOC) effect at the crossing point of the d xy and d yz (or d zx ) bands can result in a Rashba spin splitting with a cubic term. 27 However, the Rashba effect of t 2g bands was phenomenologically treated which, for example, cannot explain the complex spin or orbital angular momentum structures 12 . On the other hand, Kim et. al. 28 used an approach based on the orbital Rashba effect model 31,32 and claimed that the approach not only explains the linear and cubic momentum terms but also predicts the chiral spin and orbital angular momentum. In addition, a recent spin-ARPES study 29 shows the existence of a giant spin splitting (100 meV) and suggested a magnetic order on the surface within a non-Rashba picture while another spin-ARPES measurement 30 rejects the existence of such spin splitting and treats it as an unconventional Rashba splitting. These raise new questions on the origin of the band splitting and possibility for time-reversal symmetry breaking. As for the band structure, previous data show only one elliptical Fermi surface for d xz and d yz orbitals while two circular Fermi surfaces exist for the d xy orbital.
Addressing the above mentioned issues needs high quality data as well as a different approach. As for the experimental side, we take data in a different Brillouin zone (BZ) with all possible combinations of light polarizations to look for any missing bands. Meanwhile, our strategy is look for local orbital angular momentum (OAM) which could play a role in the surface band splitting. In that case, the existence of OAM in the surface states, which could be indirectly probed by circular dichroism (CD) ARPES experiment, may be used to explain the linear term in the surface band splitting within a Rashba-like model 32 . Our experimental results reveal additional subbands for d xz/yz and are suggestive of a chiral OAM structure, providing possible clues for the Rashba-like splitting effect.
II. METHODS
Single crystals of lightly Nb doped (0.05 weight %) SrTiO 3 (MTI, USA) were cut into 5 × 2 × 0.5 mm 3 pieces and mounted on a custom designed sample holder with a scratch line for cleavage ( Fig. 1(a),(b)). Samples were cleaved in situ and measured at 18 K. Flat and shiny surfaces with areas > 1 × 0.2 mm 2 as required for the ARPES experiment were obtained ( Fig. 1(c)). The data presented in this paper was taken at the beam line I05 of the Diamond Light Source in the United Kingdom. Preliminary experiments were performed at beamline 4A1 at Pohang Light Source, BL21B1 at National Synchrotron Radiation Research Center, and I3 at MAX IV Laboratory. 51 eV photons with linear and circular polarizations were used. The experimental chamber is equipped with a Scienta R4000 analyzer with a vertical slit. The base pressure in the measurement chamber was better than 1.2 × 10 −10 mbar. The energy and angular resolutions were 15 meV and 0.25 • , respectively. The experimental geometry is schematically shown in Fig. 1(d). When we have xz-plane as the mirror plane and linear vertical (LV) polarized light ( Fig. 1(d)), we expect to see only the d xy and d yz bands. For the linear horizontal (LH) case, we expect to see d xz band only. For a Fermi surface map, θ ( Fig. 1(d))) changes as much as 16 degrees, which results in a slight change in the polarization. The effect of such polarization change is estimated to be about 2% which is not significant at the qualitative level of our discussion.
III. SUBBAND STRUCTURE OF d xz/yz ORBITALS Figure 2 summarizes our ARPES data from STO (001) surface taken in the first BZ with all possible polarizations which are necessary to discern all the bands. We first discuss the Fermi surface map data in Fig. 2(a) and (b). Looking at the LV data along the k x direction in Fig. 2(a), we find at least two elliptical Fermi surfaces with semi-major axes of 0.45 and 0.25Å −1 and semi-minor axes of 0.11 and 0.07Å −1 (shown by number 3 and 3 ′ ). Considering the light polarization and shape, the two elliptical Fermi surfaces are from the d yz orbital and we will call them outer and inner subbands, respectively. On the other hand, the data taken with LH shows that the two Fermi surfaces are suppressed along the k x direction while they appear along the k y direction. These observations reveal that the two elliptical Fermi surfaces are indeed from the d yz orbital. The data taken with circularly polarized light in Fig. 2 polarized light) shows all the subbands even though they are not clearly discerned. (number 3 ′ and 3), respectively. Considering the selection rules, we attribute these to d yz subbands. The peaks in the MDC curve suggest that there are two other bands with k F =0.05 and 0.21Å −1 . The first one belongs to d yz or d xz orbital, but from our polarization dependence data we can not say which one is and label it as number 4. The band with k F =0.21Å Figure 2(d) shows data along the k x direction. The data taken with LV provides more evidence for the subband structure for d yz orbital. The LV data on the left hand side shows two heavy bands from d yz . Their band minima are located at 45 and 100 meV with k F =0.25 and 0.45Å −1 (number 3 ′ and 3), respectively. The band from the inner elliptical band (3 ′ ) is indicated by a red arrow. In the data taken with LH polarization depicted in Fig. 2(d), these two heavy bands are suppressed while the d xz intensity (1) is relatively strong as expected from the polarization dependence. Figure 2(e) is the schematic Fermi surface determined based on our ARPES results. There are two elliptical Fermi surfaces for d xz/yz . Note that the dashed green ellipsoid (1 ′ ) is missing in our ARPES data but we expect it based on the symmetry consideration. With the newly found Fermi surface topology, we estimate the carrier density from the area enclosed by the Fermi surfaces to be n 2D =A F /2π 2 ≈3×10 14 cm −2 . Shown in Fig. 2(f) is the schematic summary of the band structure. The subband structure for d yz orbital deduced from Fig. 2(a) and (d) is drawn with blue lines. The effective mass for the d yz bands in the k x direction is found to be m * ≈ 10m e consistent with previous reports 2, 11,15 . As mentioned above, a discernible inner d xz subband was not observed in the LH data in Figs. 2(a) and (d). We therefore plot the expected but missing inner band with dashed green parabola. The effective mass for the observed d xz band in the k x direction is estimated to be m * ≈0.6m e . d xy subbands obtained based on LH and RC+LC data are shown with red lines (number 2 and 2 ′ ) with the energy minima located at 130 and 240 meV. We wish to emphasize two points here. First, the actual Fermi surface can be more complicated than the one illustrated in our simple schematic figure as we did not consider possible hybridization between different orbitals at the crossing points due to distorted crystal structure and spin-orbit coupling. Such effects cannot be resolved in the experimental data due to the limitated resolution. Second, there is a structural phase transition from cubic to tetragonal at 110 K, 11,25,[34][35][36] and to orthorhombic below 65 K 37 . As a result, the degeneracy between d xz and d yz bands is lifted at the measurement temperature (18 K) and the effect appears as the non-degeneracy in the band dispersion at the Γ-point in Fig. 2(f).
LH LV Z F E 0 V − STO dxz/yz dxy W 0 + (g) -0.3 -0.2 -0.1 0 E-E F (eV) -0.4 -0.2 0 0.2 0.4 k y (Å -1 ) (c) (d) Γ Y Y LV Γ X X -0.4 -0.2 0 0.2 0.4 k x (Å -1 ) LH LV LH dxy dyz dxz (f) k x (Å -1 ) k y (Å -1 ) k y (Å -1 ) k x (Å -1 ) -0.2 0 0.2 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Γ X X -0.3 -0.2 -0.1 0 E-E F (eV) -0.4 -0.2 0 0.2 0.4 0.1 0.2 k x (Å -1 ) 1 2 3 3ʹ 4 1 2 3 3ʹ 1 1 2 3 3ʹ 3ʹ 3 2 1 3 2ʹ 2ʹ 1ʹ 1ʹ 3 3ʹ
Comparing our data with previously published data 2,11,12 , one may ask why only a single band has been observed for d xz/yz before. Our data is collected in the first BZ where only the circular subbands from d xy orbital were observable. One possible explanation is that experimental conditions including photon energy, light polarization, and experimental geometry allowed us to observe the extra band. For example, the orientation of the d xz/yz orbitals inside surface potential well ( Fig. 2(g)) means more three dimensional (3D) states and possibly a weak k z dispersion as it was suggested previously 15 . Our photon energy dependent data indeed show a weak but discernible effect (not shown). In such a case, the selection of the photon energy could greatly affect the cross section. Another possibility is that the second d xz/yz band was not formed in other cases due to the finite depth of the potential well. However, the fact that the d xy band bottom is similar to the previously reported value suggests that the quantum well is similar to previous cases. This makes the latter explanation less likely.
IV. CHIRAL OAM STRUCTURE
With all subbands characterized in the previous section, we shift our focus to the study of the surface band splitting mechanism. As mentioned before, different approaches such as unconventional Rashba model were used to explain the surface band splitting. Unfortunately, the split bands and spin structure are not expected to be observed by ARPES due to the small Rashba parameter of STO 12 . On the other hand, it was argued that, in the presence of inversion symmetry breaking, the multiorbital bands can lead to chiral OAM structures. 28 In that approach, a net OAM, which is defined as the sum of two OAM vectors of inner and outer Rashba split bands, is expected. Noting the suggestion that useful information on the OAM can be obtained from CD-ARPES sig-nal, 31 CD-ARPES on the STO surface state may shed light on the issue.
Before discussing the usefulness of the CD-ARPES technique, we discuss the origin of the CD-ARPES signal as there are different views on the origin of the CD signal 32,[38][39][40] . CD signal refers to the difference in the reaction to right and left circularly polarized light. The difference may come from experimental geometry [38][39][40] and OAM in the initial 32 and/or final states 41,42 . To estimate the contribution from experimental geometry, we took CD-ARPES data on a polycrystalline gold sample with the same experimental condition. As polycrystalline gold is expected to have no OAM, any observed signal would be from experimental geometry. The normalized CD (NCD) from gold sample, defined as NCD=(RC-LC)/(RC+LC), shows a weak signal less than 5% which is much smaller than the typical NCD value from STO of about 60%. In addition, the complicated CD pattern in Figs. 3(c) and (f) also shows that it is not from the geometrical effect for which a simple asymmetric CD pattern is expected. On the other hand, with the relatively high photon energy used in the experiment, the final state is expected to be close to a free electron-like state. Therefore, we assume that the major part of the CD pattern is determined by the OAM of initial states, and discuss the CD-ARPES within this interpretation.
We plot our CD-ARPES results in Fig. 3. Figures 3(a) and (b) show high symmetry cuts along the Γ-Y direction with RC and LC polarized light, respectively. It is clear that the RC data has higher intensity on the right while the LC data shows an opposite behavior. The CD signal defined as CD=RC-LC is shown in Fig. 3(c). The sign change in the CD signal may be attributed to the reversal of the OAM direction. An OAM pattern consistent with the CD data is marked in the figure. We point out that the pattern with OAM direction reversed is also consistent with the CD data. Fermi surface maps for CD-ARPES study are depicted in Figs. 3(d) -(f). Due to the complicated band structure, a clear identification of the bands can be made only for two large elliptical d xz/yz and two circular d xy Fermi surfaces in the CD Fermi surface map in Fig. 3(f). Figure 3(g) shows an OAM pattern on the Fermi surface that is consistent with the CD-ARPES data. To deduce the OAM pattern, we use the following simple rules. First, we take the positive (blue color) and negative (red color) signal in Fig. 3(f) as rightward and leftward OAM vectors, respectively. Second, any OAM pattern should preserve the fourfold symmetry of the Fermi surface. As for the magnitude of the OAM vectors, we note that at the crossing points where multiorbital bands exist the magnitude of the OAM is larger due to the enhanced spin orbit coupling effect. 12,28 Finally, near the k y =0 line which coincides with the mirror plane, CD is expected to be very weak. In that case, we have applied the symmetry rule to deduce OAM vectors. Here, we point out that the OAM texture we have shown in Fig. 3(g) is not unique. For example, the pattern with reversed OAM vectors is still valid. Therefore, the OAM texture in Fig. 3(g) is not a unique solution but a consistent one. However, it is sufficient for the discussion to follow.
(a) (b) (c) (d) (e) (f) (g) + - 0 0 + Γ Y Y Γ Y Y Γ Y Y
Our observation suggest that in-plane OAM vectors for d xz/yz and d xy orbitals have opposite chiralities while the chiralities for the two d xy subbands are the same. The non-zero OAM implies that the OAM of the two Rashba bands do not cancel each other. This matches theoretical reports 12,28 that predict the same OAM directions for the Rashba bands near crossing points (shown with dashed red circles in Fig. 3(f) and (g)) due to orbital mixing. Far from crossing points, Ref. 12 predicts OAM vectors of the d xz/yz Rashba bands to have opposite directions while Ref. 28 suggests for the same OAM directions for the d xz/yz Rashba bands. In this sense, our non-zero CD-signal for d xz/yz bands is in a better agree-ment with predictions of Ref. 28 . On the other hand, the k cubic term due to multiorbital effects 27 is much smaller than the linear term 26,28 and thus cannot be detected in our CD-ARPES. Within the interpretation discussed so far, we should see more pronounced signal near crossing points and weaker CD signal away from these points as seen in Fig. 3(f). The observation of OAM also suggests that the Rashba-related band splitting can be understood within the so-called orbital Rashba effect 28,31,32 rather than the conventional Rashba model 43,44 .
Our results could shed light on the recent spin-ARPES measurements on the surface states of STO. 29,30 Based on their experimental observations, authors of Ref. 29 suggested that a giant non-Rashba type splitting of 100 meV with opposite spin chiralities in the d xy subbands exists. On the other hand, a more recent spin-ARPES measure-ment 30 shows no such giant spin splitting. While CD-ARPES is not a spin sensitive technique and these inconsistent results should be further resolved experimentally and theoretically, we believe our result may have some implication on the issue. First of all, our CD-ARPES data, being consistent with the predictions of Ref. 28 , suggest the same OAM direction for the two Rashba bands, which in turn tells us that the spins in the two Rashba bands are pointing oppositely. Therefore, the net spin from a d xy subband should be zero, consistent with the result reported in Ref. 30 . Why are there two conflicting results then? A possible solution comes from the earlier suggestion that, in systems with chiral OAM texture, the observed spin polarization of surface states could be strongly affected by the light polarization or experimental geometry [45][46][47] . While the net spin is zero, the measured spin polarization can be non-zero if the two spin signals from the two bands are affected differently by the polarization of the light or experimental geometry. However, we should point out that if the observed nonzero spin comes from the above mentioned effect, we expect similar effects for the two d xy subbnads. Therefore, within our interpretation, it is still hard to explain the fact that the two d xy subbands show opposite chiralities while our data show the same OAM chiralities.
V. CONCLUSION
In summary, we have performed ARPES measurements on the 2DEG on the surface of STO single crystal using linearly and circularly polarized light. Our measurements in the first BZ using LV polarized light reveals subbands for out of plane d xz orbital with k F =0.25 and 0.45Å −1 , which implies d yz orbital also has a subband considering the four fold symmetry. Therefore, subbands exist not only for d xy but also for d yz/zx orbitals. In addition, CD results suggest for a chiral OAM texture in momentum space. It supports the theoretical predictions 28 of same OAM directions for the two Rashba bands near crossing points where multiorbital bands exist, which implies a better consistency with an orbital Rashba 28,31,32 than a conventional Rashba model 40,43 .
ACKNOWLEDGMENT
We are grateful for helpful discussions with Jung Hoon Han and Young Jun Chang. We thank Wonshik Kyung for his role in preliminary experiments. The work was supported by IBS-R009-G2. S.S., S.C., H.Y. and G. H. acknowledge were supported by Yonsei university, BK21 program. The data presented in this paper was taken at beamline I05, Diamond Light Source under proposal SI11445.
FIG. 1 .
1(b) (sum of data taken with right and left circularly (RC and LC) (a) Crystal structure of SrTiO3 at room temperature. (b) Schematic of the sample holder. Force F is exerted normal to the sample side-surface for cleaving. (c) Microscope image of a cleaved (001) surface (0.5 × 5 mm 2 ). (d) Schematic diagram for the experimental geometry; The analyzer slit is in the vertical direction; LV and LH show the case for vertical and horizontal polarization of the light with α =50 • ; Synchrotron beam (red line) with the energy hν comes in in the xz-plane; Photoelectrons (shown by a red sphere) are collected by the analyzer with the slit in the vertical direction. For Fermi surface mapping, θ is rotated.
Figure 2 (
2c) shows high-symmetry cuts along the k y direction (vertical dash-dot line in Fig. 2(a)) taken with LV (left) and LH (right) polarized light. Fitted momentum distribution curves (MDC) at the Fermi level are shown at the top for two polarizations. The numbered grey vertical lines mark the momentum positions where we have peaks in the MDC or Fermi level crossing. The LV data reveals a few heavy and light bands. The heavy band (number 1) is characterized as d xz with the band minimum located at 80 meV and Fermi wave vector (k F ) of 0.45Å −1 . The suppressed intensity of this band is due to the polarization dependence of orbitals 33 . In addition, there are at least two light bands with their minima located at 45 and 100 meV with k F =0.07 and 0.11Å −1
− 1
1(shown by number 2) is the outer subband of d xy orbital. Right-hand side of Fig. 2(c) shows at least three Fermi level crossings. The heavy band from d xz (number 1), one light band from
FIG. 2 .
2Fermi surface map of STO(001) surface taken with (a) LV and LH polarized light, and (b) sum of the data taken with RC and LC polarizations. (c) The cut along the ky direction taken with LV and LH polarizations. Fitted MDC at the Fermi level case is plotted at the top with a black curve. The numbered grey vertical lines indicate peak positions in the MDCs. Primed numbers show inner subbands. (d) Data along the kx direction with two linear polarizations. The inner subband for dyz orbital is indicated by vertical red arrow. (e) Schematic diagrams for the Fermi surface extracted from the ARPES. The dashed green ellipsoid (1 ′ ) is expected based on the symmetry consideration but was not clearly observed in the data. (f) Schematic band dispersions along the kx direction. (g) Schematic of the confining potential on the (001) surface and illustration of the energy positions of the bands. In-plane dxy orbital are located at the bottom of the potential well while out-of-plane d xz/yz orbitals have a higher energy with the extended orbitals along the z direction.
d xy orbital (number 2) and one located at k F =0.11Å−1 (number 3) from d yz orbital.
FIG. 3 .
3Y-Γ-Y cut taken with (a) RC polarized light and (b) LC polarized light. Dashed gray lines represent the Fermi energy. (c) CD signal which is defined as the difference between RC and LC. A possible configuration of the OAM direction near Fermi energy is shown. Fermi surface maps of (d) RC polarized light, (e) LC polarized light, and (f) CD signal. Red dashed circle indicates the crossing points where multiorbital bands exist due to hybridization. (g) A possible configuration of the chiral OAM structure based on the CD signal.
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|
[] |
[
"Achieving Representative Data via Convex Hull Feasibility Sampling Algorithms",
"Achieving Representative Data via Convex Hull Feasibility Sampling Algorithms"
] |
[
"Laura Niss \nUniversity of Michigan\n\n",
"Yuekai Sun \nUniversity of Michigan\n\n",
"Ambuj Tewari \nUniversity of Michigan\n\n"
] |
[
"University of Michigan\n",
"University of Michigan\n",
"University of Michigan\n"
] |
[] |
Sampling biases in training data are a major source of algorithmic biases in machine learning systems. Although there are many methods that attempt to mitigate such algorithmic biases during training, the most direct and obvious way is simply collecting more representative training data. In this paper, we consider the task of assembling a training dataset in which minority groups are adequately represented from a given set of data sources. In essence, this is an adaptive sampling problem to determine if a given point lies in the convex hull of the means from a set of unknown distributions. We present adaptive sampling methods to determine, with high confidence, whether it is possible to assemble a representative dataset from the given data sources. We also demonstrate the efficacy of our policies in simulations in the Bernoulli and a multinomial setting.
|
10.48550/arxiv.2204.06664
|
[
"https://arxiv.org/pdf/2204.06664v1.pdf"
] | 248,178,256 |
2204.06664
|
a7a8ad822d6917e3950a4bd3d6fbf358d3a0de76
|
Achieving Representative Data via Convex Hull Feasibility Sampling Algorithms
Laura Niss
University of Michigan
Yuekai Sun
University of Michigan
Ambuj Tewari
University of Michigan
Achieving Representative Data via Convex Hull Feasibility Sampling Algorithms
Sampling biases in training data are a major source of algorithmic biases in machine learning systems. Although there are many methods that attempt to mitigate such algorithmic biases during training, the most direct and obvious way is simply collecting more representative training data. In this paper, we consider the task of assembling a training dataset in which minority groups are adequately represented from a given set of data sources. In essence, this is an adaptive sampling problem to determine if a given point lies in the convex hull of the means from a set of unknown distributions. We present adaptive sampling methods to determine, with high confidence, whether it is possible to assemble a representative dataset from the given data sources. We also demonstrate the efficacy of our policies in simulations in the Bernoulli and a multinomial setting.
Introduction
Implementing algorithmic fairness in practice is a difficult task because most data science pipelines consists of many steps (e.g. data collection, data cleaning, training and post-processing), and any of these steps can affect the fairness of the outcome. Thus implementing algorithmic fairness in practice is generally non-trivial. Representation bias is a known issue when training ML models (Hashimoto et al., 2018;Rolf et al., 2021). This bias represent a lack of or minimal data from a subgroup of the desired population that can negatively impact the algorithmic outcomes. Unlike historical bias which is inherent in the data (Julia Angwin, 2016), representation bias can be alleviated through intentional data collection. When queried about ways individuals have attempted to address fairness, many cited more data collection as a first approach (Holstein et al., 2019). While this is possible in settings where group membership and, when applicable, outcome labels are known and can be directly sampled, there are circumstances where data collection comes from sources with unknown distributions of attributes.
An example of this is given in Holstein et al., 2019. Here they describe a company that wishes to automate essay scoring whose current iteration has unfair outcomes for a minority group. Their algorithm is scoring these minority students on average lower than a human specialist. They desire more high scoring essays from minority students to improve their scoring accuracy within that group. Because they do not know the distribution of these students at the schools they are collecting essays from, they do not have an efficient strategy to collect those needed samples, or know if it is possible to collect a data set with their desired distribution.
An approach to this problem would be to have a sampling policy to determine if there exists a distribution across schools that would produce a data set with the desired proportion of high scoring essay from the minority group. The goal of this iterative sampling policy would be to make this determination using a minimum number of samples. Once this feasibility is known, one can either sample accordingly or seek out other sources.
There are a myriad of strategies now published that are methods to improve fairness at the post-data collection stages (Dwork et al., 2012;Friedler et al., 2019). These training strategies and post-processing strategies will improve fairness outcomes, but there is a limit to improvement before impacting accuracy. It is always preferred in any machine learning application to start with the best data one can access. This highlights another benefit collecting fair data over post-collection strategies. If fairness is truly a concern, it must also be recognized that data collected today will be used for a different purpose tomorrow. By considering how to curate fair data in isolation, this can impact fairness outcomes regardless of the way data is used apart from its original purpose.
For example, consider that, in general, different definitions of fairness cannot be simultaneously satisfied except for certain possibly unattainable scenarios (Kleinberg, Mullainathan, and Raghavan, 2017;Pleiss et al., 2017). Collecting data to achieve one type of fairness when trained with a particular algorithm gives no guarantee for outcomes of other measures of fairness. If the measurement for fairness changes over the life of a project, the data is no longer optimal. Aiming for fair representation from the onset will mitigate some of these problems. Additionally, equal representation is one of the scenarios that can produce fair outcomes in relation to calibration and equalized odds, something that lopsided data cannot achieve.
This work aims to provide a sampling method that tests whether a curator can create a fair data set from available sources, where "fair" is defined in terms of a predefined proportion of group memberships. To the best of our knowledge, similar work in this area of fair sampling assumes a fair data set is achievable. This work focuses on testing that assumption. Considering the cost of collecting data, the goal will be to determine the feasibility of these sources with a minimum number of samples. When collecting data, if one can sample any protected attribute any number of times, it is simple to create training data that is consistent with some notion of fairness, such as equal proportions of protected attributes. In this paper, we consider the scenario where the sampling sources have unknown distributions of attributes and the curator has defined a "balanced" set in regards to the desired proportions of the training data. That is, data can be sampled from different sources (such as polling in different cities) but knowledge of the distributions of data from those sources is unknown. This problem setting is described in full in section 2.
Aside from collecting fair data for training, this method could also be used when fair sampling is the end for. For example, advertising community services with a desired outcome of equal men and women using those services. Different advertising strategies would reach different populations. A practitioner would want to know as quickly as possible whether their selected strategies can achieve their desired distribution, and if so what combination of strategies would do this.
Contributions We introduce the convex hull feasibility problem. In the Bernoulli setting, we give a lower bound on the expected sample size in the infeasible case and an oracle lower bound of the expected sample size in the feasible case. We define the direction of greatest uncertainty and present three policies that use this direction, along with a naive Uniform policy. Using high-probability upper bounds, we prove that one policy, Lower Upper Confidence Bound (LUCB) Mean is superior to Uniform. We define the Multinomial version of the problem along with adjusted algorithms, and using simulations show the performance of our three policies outperform Uniform in the Bernoulli setting and the Multinomial setting with three dimensions.
Related Work
Fair Sampling
To the best of our knowledge, the first work to address data collection as a part of bias mitigation is Abernethy et al., 2020. Here the goal is to optimize over both a loss function for accuracy and a loss function for fairness. They assume an infinite availability of group labeled data, and at every iteration of sampling they choose the sample which will either minimize the accuracy loss or minimize the fairness loss. The choice of which loss to minimize at every time point is determined by a Bernoulli variable with probability p, where p is a parameter chosen beforehand. When a sample is chosen to increase fairness, a sample is drawn from the group which currently has the worst loss performance. Otherwise a sample is chosen randomly. The intuition in both cases is that more training samples will improve performance, either overall performance when sampling at random or a specific group's performance when sampling to improve fairness. A similar framework is presented in Tae and Whang, 2021, where the groupings are predefined slices of a current data set, and the goal is to obtain additional samples within a budget so as to maximize average accuracy as well as minimize the average difference between the accuracy of each slice and that of the total data. Their sampling method relies on estimating learning curves and allocating the sampling budget to slices that will have maximum impact on accuracy and fairness.
Along this vein of work is Shekhar et al., 2021. Their goal identify a minimax optimal classifier across the sampling proportion of protected attributes and the loss of the worst performing group. Given a function class F, loss l, and protected attributes z ∈ Z, they propose an adaptive sampling policy that identifies the worst performing group z and dedicates a larger proportion of the sampling budget to that group.
In Asudeh, Jin, and Jagadish, 2019, they forgo optimization for a particular learning algorithm and focus on the coverage of features within the data. They define the set maximum uncovered patterns (MUP), which aims to identifying feature combinations that fail to meet predefined threshold counts. In addition to providing several algorithms to identify the set of MUP, they provide a greedy algorithm to sample additions data whose feature patterns are MUP until all meet the required sampling threshold. The work closest to ours is presented in Nargesian, Asudeh, and Jagadish, 2021, where the goal is to collect a data set of a given size consisting of a desired count from each defined group. Here they assume a priori that the desired counts are feasible, and if minimums are not achieved they propose oversampling until minority group counts are met and removing excess majority samples. In addition to results for when the sampling distributions are known, they tackle the unknown distribution model with a multi-armed bandit strategy. They propose a reward function that depends on the true distribution of a group (such as from census population data), with the intuition being if a sample is from a group with a high proportion in the population then the reward is low and if from a minority group the reward should be high. Using a UCB type strategy with this reward function presents a sampling strategy that aims to sample from the distribution with the largest proportion of the minority group. Our work differs substantially by focusing on the feasibility of the desired proportions, and frames the problem through use of a convex-hull composed of points defined by a confidence region.
There are several other frameworks around obtaining a fair data set. For example, an active learning application is presented in Anahideh, Asudeh, and Thirumuruganathan, 2021, where the goal is to sequentially select which points to label so as to balance model accuracy along with a predetermined notion of fairness. Data augmentation with synthetic points has also been explored Sharma et al., 2020.
Bandit Pure Exploration
The feasibility problem is closely related to the pure exploration multi-armed bandit problem. In pure exploration a learner has k actions with unknown means and the goal is to identify the action or subset of actions with the largest mean from the fewest samples. There are two settings in this problem, fixed-confidence and fixed-budget. In the fixed-confidence setting, a policy aims to minimize the sample complexity while guaranteeing the outcome of a policy is correct with some minimum predetermined probability. In the fixed-budget setting, a policy, given a predetermined sample size, aims to provide the largest confidence with which the largest means are correctly identified.
To see the connection to our feasibility problem to the fixed-confidence setting, consider the two class case, which reduces to identifying if there exists p i ≤ x ≤ p j . Here p 1 , . . . , p k are the k unknown means and the desired mean x encodes our definition of a balanced data set. Then by determining if x is or isn't the maximum or minimum mean with some probability 1 − δ we determine whether we correctly identify feasibility with probability 1 − δ.
The PAC pure-exploration setting was first presented in Even-dar, Mannor, and Mansour, 2002 for identifying the top action with a fixed confidence. Their successive elimination algorithm relies on uniformly sampling actions from a decreasing set, removing actions from the set as they are determined to be lower than the top action with high confidence. Another set of policies uses lower upper confidence bounds on the means of the actions (Gabillon, Ghavamzadeh, and Lazaric, 2012;Jamieson et al., 2014;Kalyanakrishnan et al., 2012;Kaufmann and Kalyanakrishnan, 2013). A lower bound on the expected sample complexity for Bernoulli rewards is presented in Mannor and Tsitsiklis, 2004, where they provide worst case and gap dependent bounds. This is expanded upon by Garivier and Kaufmann, 2016, who provide a lower bound on sample complexity for one parameter exponential families and a policy with a asymptotically matching upper bound.
Probabilistic Hyperplane Separability
The fields of computational geometry and computer science are not new to the problems of convex hull feasibility and hyperplane separability with probabilistic points. Though the underlying data assumptions are not quite matched to the convex-hull feasibility problem we present in this paper, there are significant similarities that may ultimately be used in future research and we would be remiss not to point them out. The goal of these papers is typically to provide an algorithm identifying separability or the probability of separability that minimizes run time complexity.
We briefly characterize three variations of these problems that are similar to ours. The first is that which considers the probability of linear separability between two sets of points A and B which are drawn from sets A and B, as in Fink et al., 2017. The second variation considers n labeled points from sets A and B, each with a known uncertainty region. The question then is to determine separability of sets of uncertainty regions, as seen in Sheikhi et al., 2017. Finally, there is the problem formulation where there are n points, with the value of each point i having a probability distribution over a discrete set s i with the goal to find the probability a set O lies within the probabilistic convex hull (Yan et al., 2015).
General Problem Definition
The fixed-confidence -convex hull feasibility problem is defined as follows. Each of k distributions, which we will hereto refer to as actions, are independently belong to some known family P with unknown means µ i in dimension d. We are given a known variable x ∈ R d and a relaxation of ≥ 0 and define x as the open set {y : ||y − x|| < }, with x = x when = 0. We define the feasible case as when there exists some y ∈ x that lies in the convex hull of {µ 1 , ..., µ k } and the infeasible case as when the set x lies outside of the convex hull of {µ 1 , ..., µ k }. We include the relaxation of x with because it may not be necessary to achieve exact feasibility.
If the µ i 's are known, then it is possible to determine whether x is in the convex hull of the µ i 's by solving a linear optimization problem. Instead, we consider the setting in which the µ i 's are unknown, but we can (actively) observe noisy versions of the µ i 's. The goal is to give a determination of the feasibility of x with a predetermined confidence while minimizing the number of times the actions are sampled.
In the fairness setting, the dimension d represents the number of groups defined by the protected attribute labels that the curator wishes to balance on. For example d = 2 could represent the groupings of 'men' and 'women'. The points µ i 's correspond to data sources: the j-th component of µ i is the fraction of samples from the j-th group in samples from the i-th data source. The components of the query point x correspond to the desired fractions of samples from each group in the data set. The convex hull feasibility problem is thus equivalent to determining whether there is a set of weights w i such that drawing w i fraction of samples from the i-th data source will lead to a data set with the desired fractions of samples from each group.
Feasibility and Infeasbility
Given i ∈ [k], µ i ∈ R d , x ∈ R d and ≥ 0, we first state the feasible and infeasible cases more formally.
Definition 2.1 (Infeasible Case). The problem is (x, )-infeasible if there exists some separating hyperplane between x and the µ i .
∃a ∈ R d such that ∀i ∈ [k], x ∈ x (µ i − x) T a < 0.
Definition 2.2 (Feasible Case). The problem is (x, )-feasible if there exists a convex combination that expresses some y ∈ x in terms of the µ i 's:
∃λ ∈ ∆ k−1 such that y = k i=1 λ i µ i . where ∆ k−1 is the (k − 1)-dimensional probability simplex in R k .
Because the µ i are unknown, we must rely on confidence regions to inform a decision of whether the underlying case is feasible or infeasible. If each confidence region R i contains µ i with probability at least 1 − δ k then we can make a high-confidence decision on the underlying case.
Definition 2.3 (1−δ Confident Infeasible).
There exists a separating hyperplane between the set x and the confidence regions for all actions.
∃a ∈ R d such that ∀i ∈ [k], y ∈ x , v i ∈ R i , we have that (v i − y) T a < 0. Definition 2.4 (1 − δ Confident Feasible)
. For all sets consisting of a point from each confidence region, there exists a point in x within their convex hull.
∀v i ∈ R i , i ∈ [k], ∃λ ∈ ∆ k−1 , y ∈ x such that y = k i=1 λ i v i .
Sampling Policy
A sampling policy π is a mapping of the history of all samples drawn up to the current time to the choice of which action to sample next and the termination of the algorithm. When a policy terminates, it outputs a result of either feasible or infeasible. Let τ represent the stopping time of a policy, and I(π, δ) ∈ {feasible, infeasible} be the indicator function of the output for policy π given confidence 1 − δ.
Definition 2.5 (Sound Policy). Given some δ, We call a policy (1 − δ)-sound if the expected value of the stopping time is finite and if with probability at least 1 − δ the policy selects the correct underlying case,
E[τ ] < ∞ P (I(π, δ) = f easible|f easible) ≥ 1 − δ, P (I(π, δ) = inf easible|inf easible) ≥ 1 − δ
Bernoulli Feasibility Sampling
We focus on the case where there are two protected categories (d = 2). In this case the µ i lie in the 2-dimensional simplex and convex-hull feasibility simplifies into testing in 1-dimension with Bernoulli means. This setting maps onto the scenario with two groups labels, {0, 1}, with x ∈ [0, 1] representing the desired proportion of samples from group 1 and the probability of sampling group 1 from action i is p i . For our theoretical analysis, we assume without loss of generality that p 1 ≥ p 2 ≥ ... ≥ p k .
Sample Complexity Lower Bounds
We will take inspiration from the pure exploration bandit literature and give a lower bound on the expected value of the stopping time τ as a measure of sample complexity in the Bernoulli setting. The multi-armed bandit best arm identification problem and the Bernoulli convex hull feasibility problem share certain similarities pointing towards similar techniques, but significant differences prevent direct application. In the best arm identification problem, to determine the best action with high confidence, all sub-optimal actions must be sampled to some extent to rule them sub-optimal. This remains true in our problem when the problem instance is infeasible, as all actions must be sampled sufficiently to determine them separable from our set of interest x . If the instance is feasible, the relation of the "sub-optimal" actions to each other or x becomes irrelevant. For example, if two actions are sampled such they are determined with high confidence to be above and below x respectively, sampling from the other actions provides no additional information about the feasibility or infeasibility of the problem. Additionally, there may be multiple sets of actions whose convex hull is feasible.
The possibility of multiple optimal subsets of actions presents a difficulty in determining a lower bound for feasible instances since for any (1 − δ)-sound policy, it may not have sampled all actions and there may be multiple sets of actions that would trigger termination with the correct outcome. Therefore, for a specific feasible instance, it becomes difficult to give an expected lower bound for each action, except for the case when the playable actions comprise a unique feasible set.
Considering this, we give a looser oracle lower bound for the feasible case. Here, the oracle knows the optimal subset(s) of actions but does not know their means. The oracle lower bound then is the minimum expected sample complexity when only actions in an optimal subset are played. Note that the oracle lower bound is still a valid lower bound since we are only giving the learner more information about the problem. However, the true lower bound might be much higher than our oracle lower bound.
Notation
: Let E f (x, ), E if (x, )
represent the set of feasible and infeasible instances of {p 1 , ..., p k } for given (x, ), respectively. Where a feasible instance represents a vector {p 1 , ..., p k } whose convex hull contains a point in x , and and infeasible instance is otherwise. For any feasible problem instance ν ∈ E f , let
Ω = {J ⊆ [k]|{p i } i∈J is (x, ) feasible}
be the set of all subsets of actions whose means are (x, )-feasible. Then we define the optimal subset of actions, J * , as the subset(s) that is furthest from any infeasible instance, J * = argmax J ∈Ω min ν ∈E if i∈J D(ν i , ν i ), where D is the Kullback-Leibler divergence. There are two feasible cases, either only one source is feasible or two sources are a feasible set, so |J * | ∈ {1, 2}. When analysis differs for these cases and |J * | = 1 then we write J * = {l * }, else we write J * = {1, k}, as in this case the optimal subset consists of the sources with the largest and smallest mean, p 1 and p k .
Theorem 1 (Oracle Feasible case). For a problem instance ν that is (x, )-feasible, for any (1−δ)-sound deterministic policy with d = 2, δ < 1/2,
E ν [τ ] ≥ max D(p l * |x − ) −1 , D(p l * |x + ) −1 1 2 log 1 4δ J * = {l * } 1 D(p1|x− ) + 1 D(p k |x+ ) 1 2 log 1 4δ J * = {1, k},
Theorem 2 (Infeasible case). For a problem instance ν that is (x, )-infeasible, for any (1 − δ)-sound deterministic policy with d = 2, δ < 1/2,
E ν [τ ] ≥ k i=1 max D(p i |x − ) −1 , D(p i |x + ) −1 1 2 log( 1 4δ )
Lower bound proofs can be found in section A.1.
Sampling Policies
We present four sampling policies, a naive Uniform policy as a baseline along with Lower Upper Confidence Bound (LUCB) Mean, LUCB Ratio and Beta Thompson Sampling. We give high probability upper bounds for Uniform and LUCB Mean, and empirical evidence that LUCB Mean, LUCB Ratio, and Beta TS significantly outperform Uniform. Notation: Let B(n, δ) be a confidence margin dependent upon sample size n and confidence parameter δ such that
∞ n=1 P (|p i −p i (n)| > B (n, δ)) < δ k . (3.1)
We write B i (t) to represent the confidence margin for action i given its sample size at time t when δ is implied. Letp i (t) be the estimated mean of action i at time t, and R i (t) = {y :p i (t) − B i (t) ≤ y ≤p i (t) + B i (t)} be the confidence region of action i at time t. We use a t to specify the action chosen at time t and n i (t) the number of times action i has been chosen at time t. Each policy follows the same stopping rules for termination.
We next define the direction of greatest uncertainty, which is used to determine termination and in our sampling policies for action selection. This measure aims to capture which direction away from x, we are least certain an action mean lies on.
Definition 3.1 (Direction of greatest uncertainty). Given a confidence margin B i and mean estimatep i , the direction of greatest uncertainty u ∈ {1, −1} is defined as,
u = argmin u∈{1,−1} max i∈[k] u(p i − x) − B i .
The intuition behind this definition is that it identifies the direction from x we are furthest from determining a mean exists in that direction. For example, if x = .5, and there are two confidence regions (.48, .9) and (.49, .8), then the closest lower bound in direction u = −1 is .8, and the closest lower bound in direction u = 1 is .48. The decision boundary that implies a mean lies below x is further from x than a decision boundary that implies a mean lies above it, so our direction of greatest uncertainty is u = −1 and we should sample actions that we have a higher belief are below x.
All the policies presented follow the same stopping rules.
Stopping Rules: If one of the following criteria are met, the policy terminates,
1.
Feasible: x is not separable from any subset consisting of a point from each of the confidence regions.
min u∈{−1,1} max i∈[k] (p i − x)u − B i (t) > − 2. Infeasible: x is separable from all confidence regions. min u∈{−1,1} max i∈[k] (p i − x)u + B i (t) < −
Where stopping rule 1 states there is a mean whose confidence interval lies above x − and one whose confidence intervals lies below x + . The same confidence interval may satisfy both of these conditions. Intuitively, stopping rule 1 says that if the true means lie in their respective confidence intervals, then no matter their value, a point in x lies in their convex hull.
Uniform
This simple policy samples from the active actions and chooses the action with the least samples, leading to uniform sample sizes across active actions. Active actions at time t are those whose confidence regions at time (t − 1) contain a boundary point of x . The policy is given in algorithm 1.
Algorithm 1: Uniform Bernoulli input: Number of actions k, confidence 1 − δ, x, . Sample from each source once. while Stop = False do Update active actions A t = {i : ∃y ∈ ∂x , y ∈ R i (t)}. a t+1 = argmin i∈At n i (t) end
LUCB Mean
This policy is based on the idea of sampling the active action with the confidence boundary furthest from x in the direction of greatest uncertainty, as given in definition 3.1. Given this direction, we exploit the action whose confidence bound is furthest from x. The policy is given in algorithm 2.
Algorithm 2: LUCB Mean Bernoulli input: Number of actions k, confidence 1 − δ, x, . Sample from each source once.
while Stop = False do u t = argmin u∈{1,−1} max i∈[k] u(p i (t) − x) − B i (t) a t+1 = argmax i∈[k] u t (p i (t) − x) + B i (t) end
LUCB Ratio
Using definition 3.1 to define the direction of greatest uncertainty, the intuition of this policy is to sample from the active action whose confidence region has the largest proportion of area on the side of x in this direction. It is possible that two actions have the same confidence ratio, at which point exploring the less sampled action provides more information. To account for this, we scale the confidence ratio by 1 √ ni . The policy is given in algorithm 3.
Thompson Sampling
This probabilistic algorithm is a standard choice in the bandit literature. With few changes we adjust it to the convex hull feasibility problem. Again we use the direction of greatest uncertainty, sample a mean from the posterior of each action, and play the action with the mean furthest from x in the given direction. The policy is given in line 4 where r i (t) are the number of success drawn from action i at time t.
Algorithm 3: LUCB Ratio Bernoulli input: Number of actions k, confidence 1 − δ, x, . Sample from each source once.
while Stop = False do u t = argmin u∈{1,−1} max i∈[k] u(p i (t) − x) − B i (t) a t+1 = argmax i∈[k] 1 √ ni ut(pi(t)−x)+Bi(t) ut(x−pi(t))+Bi(t) end
Algorithm 4: Beta Thompson Sampling input: Number of actions k, confidence 1 − δ, x, Sample from each source once.
while Stop = False do Update posteriors π i (t) = Beta(1 + r i (t), 1 + (n i (t) − r i (t))) u t = argmin u∈{1,−1} max i∈[k] u(p i (t) − x) − B i (t) Samplep i (t) from posterior π i (t) for all i ∈ [k] a t+1 = argmax i∈[k] u t (p i (t) − x) end
Sample Complexity Upper Bounds
For both Uniform and LUCB Mean policies we give high probability upper bounds on the sample complexity. These policies sample all actions in relation to the optimal feasible subset, and thus allow a simple bounding on the complexity of each action. In section 5, we show that Beta TS, LUCB Ratio, and LUCB Mean outperform Uniform empirically.
Notation:We define ∆ max i to be the maximum distance from p i to a boundary of x and define ∆ min i to be the minimum distance from p i to a boundary of x . Let s max i be the minimum integer solution to ∆ max i > 2B(s max i , δ) and similarly for s min i . Therefore we have that with probability at least 1 − δ/k, when action i is sampled s max i (s min i ) times, ∆ max i (∆ min i ) will not be contained in its confidence region. A visualization of the gap relationship is show in figure 1. We additionally define ∆ i,j = |p i − p j | and s i,j and the smallest integer such that ∆ i,j > 2B(s i,j ).
For the feasible Bernoulli setting, the optimal subset J * will consist of one action, J * = {l * }, or two actions, J * = {1, k}. In the following theorems we include the general case for any B(n, δ) that satisfies equation (3.1) and
where s depends on choice of B(n, δ), as well as for the case with B(n, δ) = 1 2n log(n 2 5k 3δ ), which shows the gap dependencies clearly.
Theorem 3 (Uniform Complexity). Let j * = argmax i∈{1,k} s max i . Assume B(n, δ) satisfies equation (3.1). When the underlying case is feasible, the sample complexity of Uniform is bounded above by and for B(n, δ) = 1 2n log(n 2 5k 3δ ),
τ ≤ O k i=1 min s max j * , s min i J * = {1, k} O k i=1 min s min l * , s min i J * = {l * } ( ) p i x ∆ max i ∆ min iτ ≤ O k i=1 min 1 (∆ max j * ) 2 , 1 (∆ min i ) 2 J * = {1, k} O k i=1 min 1 (∆ min l * ) 2 , 1 (∆ min i ) 2 J * = {l * } with probability at least 1 − δ.
When the cases is infeasible, the sample complexity of Uniform is bounded above by
τ ≤ O k i=1 s min i τ ≤ O k i=1 1 (∆ min i ) 2
For B(n, δ) = 1 2n log(n 2 5k 3δ )
With probability at least 1 − δ.
Theorem 4 (LUCB Mean Complexity). Let j * = argmax i∈{1,k} s max i and i * = argmin i∈{1,k} s max i . Assume B(n, δ) satisfies equation (3.1). When the underlying is feasible, the sample complexity of LUCB Mean is bounded above by
τ ≤ O i:∆ i,j * ≤∆ max j * s max j * + i:∆ i,j * >∆ max j * max (s i,j * , s max i * ) ∃i, j, p i < x < p j O k i=1 min s i,j * , s min j * otherwise and for B(n, δ) = 1 2n log(n 2 5k 3δ ) τ ≤ O i:∆ i,j * ≤∆ max j * 1 (∆ max j * ) 2 + i:∆ i,j * >∆ max j * max 1 ∆ 2 i,j * , 1 (∆ max i * ) 2 ∃i, j, p i < x < p j O k i=1 min 1 ∆ 2 i,j * , 1 (∆ min j * ) 2
otherwise with probability at last 1 − δ.
When the underlying case is infeasible, the sample complexity of LUCB Mean is bounded above by,
τ ≤ O k i=1 s min i τ ≤ O k i=1 1 (∆ min i ) 2
For B(n, δ) = 1 2n log(n 2 5k 3δ )
With probability at least 1 − δ.
This shows that for any problem instance, the worst case sample complexity is lower using the LUCB Mean policy compared to the Uniform policy, since min(s max j * , s min l ) ≥ max(s l,j * , s max i * ) for all l ∈ [k]. We leave details of this to section A.3.
Intuitively, theorem 3 says that all actions are sampled as many times as the most sampled optimal arm or until it's confidence region is disjoint from x . For theorem 4, the bounds are more complicated because it depends upon how close the action mean is to p j * and the instance setting. Generally speaking, the bounds describe a relationship between the relative distance of an action's mean and the most sampled optimal action's mean, an action's mean and a boundary point in x , and the worst case behavior of the action's confidence region. These particular's are detailed in the proof.
Multinomial Feasibility Sampling
By expanding our definition of the direction of greatest uncertainty and our stopping rules, we can modify each policy to work in higher dimensions.
Feasibility and Infeasibility Checks
Recall the definition of 1 − δ Confident Feasible (definition 2.4). If we assume = 0, an equivalent definition would be ∀u, ||u|| = 1, ∃R i such that (q i − (x + w)) T u > 0 ∀q i ∈ R i which states that x is not separable from any subset of points constructed from the confidence regions. Alternatively, we may say that for all unit vectors u, max i∈[k] min q∈Ri (q i − x) T u > 0. If we limit the confidence regions to be balls with radius B, then we can simplify to say x is feasible if
min u:||u||=1 max i∈[k] (p i − x) T u − B i > 0.
Now considering > 0, we would need to show there exists some point (x + w) ∈ x , ||w|| < ,
min u:||u||=1 max i∈[k] (p i − (x + w)) T u − B i > 0
We have that for some λ ∈ (0, 1),
min u:||u||=1 max i∈[k] (p i − x) T u − B i > −λ min u:||u||=1 max i∈[k] (p i − x) T u − B i − w T u > −λ − w T u min u:||u||=1 max i∈[k] (p i − (x + w)) T u − B i − w T a > 0
Where in the last line we have that since u is a unit vector and the length of w is bounded by , we can pick w, λ ∈ (0, 1) such that w T u = −λ . Therefore a feasibility check becomes if, given some λ ∈ (0, 1),
min u:||u||=1 max i∈[k] (p i − (x + w)) T u − B i > 0.
In a similar fashion, an infeasibility check would be if there exists a unit vector a such that,
min u:||u||=1 max i∈[k] (p i − x) T u + B i < − .
Sampling Policies
Using the above formulation for checking feasibility lends itself to defining the direction of greatest uncertainty in any dimension.
Definition 4.1 (Direction of greatest uncertainty). Given a confidence margin B i and mean estimatep i , the direction of greatest uncertainty a is defined as,
u = argmin u: ||u||=1 max i∈[k] (p i − x) T u − B i .
Unfortunately, finding the direction of greatest uncertainty for d ≥ 3, and thus also checking feasibility, is a nonconvex problem, so we cannot obtain the optimal solution. One obvious workaround to this is simply doing a grid search over some subset of points on the unit ball. This is the approach we take.
Let G be some subset of the unit ball in dimension d which will be the directions we search over, and let λ ∈ (0, 1) be a parameter.
Stopping Rules: If one of the following criteria are met, the policy terminates,
1.
x is not separable from the confidence balls in any direction u ∈ G.
min u∈G max i∈[k] (p i (t) − x) T u − B i (t) > −λ 2.
x is separable from all confidence balls.
min u∈G max i∈[k] (p i (t) − x) T u + B i (t) < −
Our sampling algorithms do not change significantly to accommodate higher dimensions. The Uniform policy no longer has an active action set, and the other policies use the updated definition of direction of greatest uncertainty and vector dot products instead of scalar multiplication. The policies are given in algorithm 5 (Uniform), algorithm 6 (LUCB Mean), algorithm 7 (LUCB Ratio), and algorithm 8 (Dirichlet Thompson sampling). Sample from each source once.
while Stop = False do u t = argmin u∈G umax i∈[k] (p i − x) T u − B i (t) a t+1 = argmax i∈[k] (p i (t) − x) T u t + B i (t) end
Algorithm 7: Confidence Ratio Sampling input: Number of actions K, confidence 1 − δ, unit vectors G. Sample from each source once.
while Stop = False do u t = argmin u∈G max i∈[k] (p i − x) T u − B i (t) a t+1 = argmax i∈[k] 1 √ ni (pi(t)−x) T ut+Bi(t) (x−pi(t)) T ut+Bi(t) end
Simulations
We compare the average sample size till termination of our three policies against the naive uniform sampling method.
Setup
We run our policies in the Bernoulli setting (which correlates to both d = 1 and d = 2) and the Multinomial setting with d = 3. Each graph shows the average sample size at termination for the four policies when averaged over 30 trials using B(n, δ) = 1 2n log(n 2 5k 3δ ). In all trials, we set δ = .01, k = 10, = 0.1, and set λ = .99 when d = 3. In the Multinomial setting, we use a grid search over 300 points on the unit sphere. For the Bernoulli setting we run scenarios Algorithm 8: Dirichlet Thompson Sampling input: Number of actions K, confidence 1 − δ, priors π i , unit vectors G Sample from each source once.
while Stop = False do u t = argmin a∈G max i∈[k] (p i − x) T u − B i (t) Sample p i (t) from posterior π i (t) for all i ∈ [k] a t+1 = argmax i∈[k] (p i (t) − x) T u t end Solve for t in plug optimization. If t ≥ 0, out = feasible, else if t < 0 out = infeasible.
for |J * | ∈ {1, 2} and for the Multinomial setting |J * | ∈ {1, 2, 3}. In each of these settings we further consider two cases, when J * is unique and when it is not. When the J * is unique, it is an element of the set of optimal subsets when J * is not unique. Therefore the oracle lower bound is the same for unique and non-unique cases and we can compare the two outcomes when all other parameters are fixed. The setting for each scenario is listed in the caption.
The desired values is in the Bernoulli case x = .5, and x = (.33, .33, .33) in the Multinomial case. The means used in each setting are listed in tables 1 and 2, and were chosen as a general representation of several different scenarios.
Results
When the average sample size of the Uniform policy is substantially larger than that of the best performing policy, the y-axis has a break point to indicate a change in the scale. Figures 2 and 3 show results for Bernoulli sampling and figures 4 to 6 show resuts for the Multinomial sampling with d = 3. It is clear that the Uniform sampling policy performs the worst in all cases, and is improved upon by all other policies presented in this paper. It is clearly seen, and somewhat surprising, that there is a large relative difference in performance of LUCB Ratio and Thompson sampling between the Bernoulli and Multinomial setting.
In the Multinomial setting Dirichlet Thompson sampling has superior performance, while in the Bernoulli setting LUCB Ratio has the best performance, except when there in one unique optimal action, as seen in figure 2a. Here Beta Thompson sampling (Beta TS) outperforms the other policies. We speculate that in this particular Bernoulli setting, Beta TS this may be because this case is most similar to the standard multi-armed bandit problem, which aims to select the action with the highest mean as often as possible. The multi-armed bandit Beta TS policy is one of the simplest and most effective policies in practice (Chapelle and Li, 2011). Looking at figure 2, when J * is unique it requires fewer sample sizes on average for each policy than when J * is not unique. This relationship reversed in figure 3. This example shows that uniqueness of J * in the Bernoulli setting does not imply a simpler problem. This is similarly seen in the Multinomial setting. We see in figure 4 that Dirichlet TS and LUCB Ratio perform better in the unique optimal subset setting, and there is no difference for LUCB Mean and Uniform. Whereas in figure 5, all but LUCB Ratio perform better in the non-unique optimal subset setting.
We see in both the Bernoulli and Multinomial setting that the larger the optimal subset, the fewer average samples before termination. This is because when |J * | < d the optimal subsets must be sampled until B(n, δ) ≈ to ensure there is a mean either on both sides of x or that a confidence region is fully contained in x in all directions.
In practice, results will be dependent upon the underlying truth, as can be inferred by the Oracle average sample complexity lower bound and the high probably sample complexity upper bounds given in this work. These simulations give evidence of the magnitude of improvement using an adaptive sampling method over the naive uniform method. Depending on the setting, average sample size can be reasonably small, as seen in figures 3a and 3b. In the Multinomial setting, average sample sizes are in the thousands. The practicality of this method can be seen to depend upon the true distributions, sampling budget, and parameter values.
Summary and Discussion
We introduce the convex hull feasibility problem in the context of fair data collection. In the Bernoulli setting, we give a lower bound on the expected sample complexity in the (x, )-infeasible instance and an oracle lower bound on the expected sample complexity in the (x, )-feasible instance. We introduce four sampling policies for the Bernoulli setting, Uniform, LUCB Mean, LUCB Ratio, and Beta TS and give high probability upper bounds on sample complexity for the Uniform and LUCB Mean policies. We give the adaptation of the Binomial policies to the Multinomial case. Through simulation, we show LUCB Mean, LUCB Ratio, and the Thompson sampling policies significantly outperform Uniform in the Bernoulli and Multinomial setting. Under our simulation scenarios, we see that LUCB Ratio is typically the best performing policy in our Bernoulli settings, while Dirichlet TS is the best performing policy in our Multinomial settings. We discuss that the practicality of implementation is dependent upon the underlying distributions, sampling budget, and chosen parameters. Large sampling budgets would enable this method practical under most settings, whereas with small sampling budgets this method would only be practical if there was a strong prior that the underlying distributions has a small oracle lower bound. While this work focused on Bernoulli and Multinomial convex hull feasibility sampling, the general problem is applicable when points are drawn from any distribution for which one can construct a confidence region that satisfies equation (3.1).
There are some limitations within this work. Notably, we were only able to give an oracle lower bound on the expected sample complexity in the feasible case. A true lower bound would allow for better comparison of a policy's theoretical performance. Addtionally, we provide theoretical results in the Bernoulli settings, but not the Multinomial setting.
The work in this paper is somewhat analogous to multi-armed bandit best arm identification with fixed-confidence. Another approach seen in the best arm identification literature is the fixed-budget setting, which could also be applied to the convex hull feasibility problem. If given a set of samples, the confidence regions can be such that they do not meet the definition of either (1 − δ)-confident feasible or (1 − δ)-confident infeasible. In this case we could ask instead what is the probability of feasibility or infeasible given the current sample, or if adaptively sampling, what is the highest probably of a correct decision when sampling with a budget. One application of this could be to check Pareto frontier feasibility, where if given noisy gradients, we ask what is the probability all groups can be improved versus the probability that improving some groups may harm others.
A Proofs
A.1 Lower Bounds
Proof of theorem 1. Let ν be the feasible instance. With the optimal set known, to check feasibility we only need to check the relationship between the mean of each action mean and the set x . If the set J * consists of only one point, it must be sampled enough to determine it lies within x . If J * = {1,k}, we must sample to determine that one of the means lies above x − and one lies below x + .
We start with the case where J * = {i * } ∈ {1, k}. Since this is a feasible set, it must be that |p i − x| < . The closest infeasible case is the boundary of x closest to p i . The KL divergence from this infeasible case is given by
min (D(p i |x − ), D(p i |x + )) Lett i = max D(p i |x − ) −1 , D(p i |x + ) −1 1 2 log( 1 4δ )
. We let N i be the random variable representing the number of times action i was sampled when the policy terminates. We will use a proof by contradiction similar to that presented by Mannor and Tsitsiklis, 2004 along with a divergence decomposition Lattimore and Szepesvári, 2020, Lemma 15.1.
Assume E[N i * ] ≤t i * . Let O ∈ {f easible, inf easible} be the output of a policy, and define event B = {O = f easible}. Then by definition of a 1 − δ-sound policy, P ν (B) ≥ 1 − δ ≥ 1/2 for ν ∈ E f . Without loss of generality, assume x − < p i * ≤ x. Then the closest infeasible case would be p i * = x − . We will call H 0 : p i * = p i * , H 1 : p i * = x − . We get that,
P 1 (B) = E 1 [1{B}] = E 0 L 1 L 0 1{B} = E 0 L 1 L 0 |B P 0 (B) = E 0 exp − log L 0 L 1 |B P 0 (B) ≥ exp −E 0 log L 0 L 1 |B P 0 (B) = exp {−E 0 [N i |B] D(p i * , x − )} P 0 (B) ≥ exp −2t i * D(p i * , x − ) P 0 (B) = 4δP 0 (B) > δ
which contradicts that the policy is 1 − δ sound under hypothesis H 1 , which means that
E 0 [N i * ] ≥ max D(p * |x − ) −1 , D(p i * |x + ) −1 1 2 log 1 4δ .
For J * = {1, k}, we define H 0 : p 1 = p 1 , p k = p k and H 1 : P 1 = x − or p k = p + . Because p 1 ≥ p k by definition, if either p 1 = x − or p k = x + then the problem is infeasible. By settingt i = max D(p i |x − ) −1 , D(p i |x + ) −1 1 2 log( 1 4δ ) and following the same method as above for actions i ∈ {1, k}, get E 0 [T i ] ≥ min D(p i |x − ) −1 , D(p i |x + ) −1 1 2 log 1 4δ .
Proof sketch of theorem 2. The proof for the infeasible lower bound follows closely to that of the feasible case, therefore we provide a brief proof outline. Because all means must lie outside x , the closest feasible case is the boundary of x nearest the means. To determine infeasibility, all actions must be sampled sufficiently to reject this boundary. Without loss of generality, if we assume p i < x − , then the closest boundary would be x − . Setting H i : p i = p i , H 1 : p i = x − for all i,t i = max D(p i |x − ) −1 , D(p i |x + ) −1 1 2 log( 1 4δ ) and following the methods from the feasible case, we get the desired result.
A.2 Upper Bounds
Proof of theorem 3. Given some δ and B(n, δ) that satisfies equation (3.1), let event E be the event that all confidences regions contain their mean, E = {∀i ∈ [k], n ∈ N,p i (n) − B(n, δ) ≤ p i ≤p i (n) + B(n, δ)}. Under event E, each action i will become inactive at or before being sampled s min i times. We start with the feasible cases. When where J * = {i * } and under event E, action i * can be sampled at most s min i * times before the policy will terminate due to stopping rule 1. Thus the bound on the sample size of each action is the minimum of the sample size it is guaranteed to become inactive under E, which is s min i , and the sample size of s min i * when the policy terminates. Since event E happens with probability at least 1 − δ, this concludes the proof when the optimal subset is one action.
In the case where J * = {1, k}, under event E the policy will terminate due to stopping rule 1, which will happen when action 1 is sampled s max 1 times and action k is sampled s max k times. Again, under E an action becomes inactive when sampled at most s min i times, this gives that each action is sampled at most s min i , max (s max 1 , s max k ) times. Again, event E happens with probability at least 1 − δ.
When the problem is infeasible, each action will be sampled until its confidence region is disjoint form x . Under event E, this sample size is bounded above by s min i for all i.
Proof of theorem 4. We Start with the feasible case where there exists a mean on both sides of x. Let event E be the event that all confidences regions contain their mean, E = {∀i ∈ [k], n ∈ N,p i (n)−B(n, δ) ≤ p i ≤p i (n)+B(n, δ)}. Without loss of generality, let actions i, j be the action that triggers termination at time τ . Define the sample size of action l at time t as N l (t). Assume without loss of generality that j * = k, and p j < x, thus i * = 1 and p i > x. If j = k, then under event E, N j (τ ) ≤ s max k . If j = k it must be that,
p j (N j (τ ) − 1) − B(N j (τ ) − 1) ≤ p k by Ê p j (N j (τ ) − 1) + B(N j (τ ) − 1) ≥ x + definition of τ Thereforep j (N j (τ ) − 1) + B(N j (τ ) − 1) ≥ x + 2B(N j (τ ) − 1) ≥ x + − (p j (N j (τ ) − 1) − B(N j (τ ) − 1)) ≥ x + − p k = ∆ max k > 2B(s max k ) since B is a decreasing function, N j (τ ) − 1 < s max k =⇒ N j (τ ) ≤ s max k .
Similarly for action i we have that N j (τ ) ≤ s max 1 For non-terminating actions we have that under E, p l (N l (τ ) − 1) − B(N l (τ ) − 1) ≤ p kpl (N l (τ ) − 1) + B(N l (τ ) − 1) ≥ max(x + , p l ) which implies that 2B(N l (τ ) − 1) ≥ max(|p k − (x + )|, |p l − p k |) = max(∆ max k , ∆ l,j ) > max(2B(s max k ), 2B(s k,l )) giving N l (τ ) ≤ min(s max k , s k,l ) and similarly N l (τ ) ≤ min(s max 1 , s 1,l ). To meet both these bounds, it must be that N l (τ ) ≤ max(min(s max 1 , s 1,l ), min(s max k , s k,l )) = max(min(s max i * , s l,i * ), min(s max j * , s l,j * )).
Figure 1 :
1Visualization of ∆ max i , ∆ min i for some p i given x, .
Algorithm 5 :
5Uniform input: Number of actions k, confidence 1 − δ, x, . Sample from each source once. while Stop = False do a t+1 = argmin i∈[k] n i (t) end Algorithm 6: LUCB Sampling input: Number of actions K, confidence 1 − δ, unit vectors G. fix : A = [k]
) J * not unique.
Figure 2 :
2Average stopping time in Bernoulli setting, d = 1, |J * | = 1, k = 10. (a) J * unique. (b) J * not unique.
Figure 3 :
3Average stopping time in Bernoulli setting, d = 1, |J * | = 2, k = 10.
Figure 4 :
4Average stopping time in Multinomial setting, d = 3, |J * | = 1, k = 10. (a) J * unique. (b) J * not unique.
Figure 5 :
5Average stopping time in Multinomial setting, d = 3, |J * | = 2, k = 10.(a) J * unique. (b) J * not unique.
Figure 6 :
6Average stopping time in Multinomial setting, d = 3, |J * | = 3, k = 10.
Table 1 :
1Bernoulli Mean ValuesBernoulli
|J * | = 1 |J * | = 2
Optimal
.5
.3, .7
Non-optimal .48, .52
.48, .52
Table 2 :
2Multinomial Mean VectorsMultinomial |J * | = 1
|J * | = 2
|J * | = 3
Optimal
(.33, .33, .33)
(.1, .57, .33)
(.57, .1, .33)
(.2, .1, .7)
(.7, .2, .1)
(.1, .7, .2)
Non-optimal (0, 0, .1)
(.2, .47, .33)
(.47, .2, .33)
(.33, .33, .34)
(.33, .34, .33)
(.34, .33, .33)
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|
[] |
[
"Neuroevolutionary Feature Representations for Causal Inference *",
"Neuroevolutionary Feature Representations for Causal Inference *"
] |
[
"Michael C Burkhart \nUniversity of Cambridge\nCambridgeU.K\n",
"Gabriel Ruiz [email protected] \nUCLA\nLos AngelesCA, U.S.A\n"
] |
[
"University of Cambridge\nCambridgeU.K",
"UCLA\nLos AngelesCA, U.S.A"
] |
[] |
Within the field of causal inference, we consider the problem of estimating heterogeneous treatment effects from data. We propose and validate a novel approach for learning feature representations to aid the estimation of the conditional average treatment effect or cate. Our method focuses on an intermediate layer in a neural network trained to predict the outcome from the features. In contrast to previous approaches that encourage the distribution of representations to be treatment-invariant, we leverage a genetic algorithm that optimizes over representations useful for predicting the outcome to select those less useful for predicting the treatment. This allows us to retain information within the features useful for predicting outcome even if that information may be related to treatment assignment. We validate our method on synthetic examples and illustrate its use on a real life dataset.
|
10.48550/arxiv.2205.10541
|
[
"https://arxiv.org/pdf/2205.10541v1.pdf"
] | 248,987,304 |
2205.10541
|
af3899c143ea6673c4e26d7000b0d3fc6130e5cf
|
Neuroevolutionary Feature Representations for Causal Inference *
Michael C Burkhart
University of Cambridge
CambridgeU.K
Gabriel Ruiz [email protected]
UCLA
Los AngelesCA, U.S.A
Neuroevolutionary Feature Representations for Causal Inference *
causal inferenceheterogeneous treatment effectsfeature represen- tationsneuroevolutionary algorithmscounterfactual inference
Within the field of causal inference, we consider the problem of estimating heterogeneous treatment effects from data. We propose and validate a novel approach for learning feature representations to aid the estimation of the conditional average treatment effect or cate. Our method focuses on an intermediate layer in a neural network trained to predict the outcome from the features. In contrast to previous approaches that encourage the distribution of representations to be treatment-invariant, we leverage a genetic algorithm that optimizes over representations useful for predicting the outcome to select those less useful for predicting the treatment. This allows us to retain information within the features useful for predicting outcome even if that information may be related to treatment assignment. We validate our method on synthetic examples and illustrate its use on a real life dataset.
Introduction
In this note, we aim to engineer feature representations to aid in the estimation of heterogeneous treatment effects. Specifically, we consider the following graphical model X
W Y ← → ← → ← →(1)
where X ∈ R d denotes a vector of features, W ∈ {0, 1} represents a boolean treatment, and Y ∈ R denotes the outcome. Suppose (X i , W i , Y i ) for i = 1, . . . , n are i.i.d. samples from a distribution P respecting the graph (1). Within the potential outcomes framework [20,27], we let Y i (0) denote the potential outcome if W i were set to 0 and Y i (1) denote the potential outcome if W i were set to 1.
We wish to estimate the conditional average treatment effect (cate) defined by
τ (x) = E[Y (1) − Y (0)|X = x].(2)
We impose standard assumptions that the treatment assignment is unconfounded, meaning that
{Y i (0), Y i (1)} ⊥ W i | X i ,
for all i, and random in the sense that
< P (W i = 1|X i = x i ) < 1 −
for all i, some > 0, and all x i ∈ R d in the support of X i . These assumptions are jointly known as strong ignorability [26] and prove sufficient for the cate to be identifiable. Under these assumptions, there exist well-established methods to estimate the cate from observed samples (see section 2.1 for a discussion) that then allow us to predict the expected individualized impact of an intervention for novel examples using only their features [22]. Viewing these approaches as black box estimators, we aim to learn a mapping Φ : R d → R m such that the estimate of the cate learned from the transformed training data (Φ(X i ), W i , Y i ) is more accurate than an estimate learned on the original samples (X i , W i , Y i ). In particular, we desire a function Φ yielding a corresponding representation Φ(X) such that 1. Φ(X) is as useful as X for estimating Y , and 2. among such representations, Φ(X) is least useful for estimating W .
In this way, we hope to produce a new set of features Φ(X) that retain all information relevant for predicting the outcome, but are less related to treatment assignment. We propose learning Φ as an intermediate layer in a neural network estimating a functional relationship of Y given X. We apply a genetic algorithm [9] to a population of such mappings to evolve and select the one for which the associated representation Φ(X) is least useful for approximating W . Feature representations are commonly used in machine learning to aid in the training of supervised models [2] and have been previously demonstrated to aid in causal modeling. Johansson, et al. [11,29] viewed counterfactual inference on observational data as a covariate shift problem and learned neural network-based representations designed to produce similar empirical distributions among the treatment and control populations, namely [16] and Yao, et al. [34] developed representations in a related vein designed to preserve local similarity. We generally agree with Zhang et al.'s [36] recent argument that domain invariance often removes too much information from the features for causal inference. † In contrast to most previous approaches, we develop a feature representation that attempts to preserve information useful for predicting the treatment effect if it is also useful for predicting the outcome. † Zhao et al. [37] make this argument in a more general setting.
{Φ(X i )} Wi=1 and {Φ(X i )} Wi=0 . Li & Fu
Outline. We proceed as follows. In the next section, we describe methods for learning the cate from observational data and introduce genetic algorithms. In section 3, we describe our methodology in full. We then validate our method on artificial data in section 4 and on a publicly available experimental dataset in section 5, before concluding in section 6.
Related work
In the first part of this section, we discuss standard methods for learning the cate function from data. We will subsequently use these to test our proposed feature engineering methods in section 4. In the second part, we briefly outline evolutionary algorithms for training neural networks, commonly called neuroevolutionary methods.
Meta-learners
We adopt the standard assumptions of unconfoundedness and the random assignment of treatment effects that together constitute strong ignorability. Given i.i.d. samples from a distribution P respecting (1) and these assumptions, there exist numerous meta-learning approaches that leverage an arbitrary regression framework (e.g., random forests, neural networks, linear regression models) to estimate the cate that we now describe.
S-learner. The S-learner (single-learner) uses a standard supervised learner (regression model) to estimate µ(x, w) = E[Y |X = x, W = w] from observation data and then predictsτ S (x) =μ(x, 1) −μ(x, 0) where we use the standard hat notation to denote estimated versions of the underlying functions.
T-learner. The T-learner (two-learner) estimates
µ 1 (x) = E[Y (1)|X = x] from observed treatment data {(X i , Y i )} Wi=1 and µ 0 (x) = E[Y (0)|X = x] from observed control data {(X i , Y i )} Wi=0 , and then predictŝ τ T (x) =μ 1 (x) −μ 0 (x).
X-learner. The X-learner [14] estimates µ 1 and µ 0 as in the T-learner, and then predicts the contrapositive outcome for each training point. Next, the algorithm estimates
τ 1 (x) = E[D 1 i | X = x] on {(X i ,D 1 i )} Wi=1 whereD 1 i = Y i −μ 0 (X i ) and τ 0 (x) = E[D 0 i | X = x] on {(X i ,D 0 i )} Wi=0 whereD 0 i =μ 1 (X i ) − Y i . The X-learner then predictŝ τ X (x) = g(x)τ 0 (x) + (1 − g(x))τ 1 (x)
where g : R d → [0, 1] is a weight function. The creators of the X-learner remark that the treatment propensity function (3) often works well for g, as do the constant functions 1 and 0. In our implementation, we use g(x) ≡ 1/2.
Concerning this method, we note that it is also possible to directly estimate τ from
{(X i , Y i −μ 0 (X i ))} Wi=1 ∪ {(X i ,μ 1 (X i ) − Y i )} Wi=0
or, usingμ(x, w) from the S-learner approach, with
{(X i , Y i −μ(X i , 0))} Wi=1 ∪ {(X i ,μ(X i , 1) − Y i )} Wi=0 .
We find that these alternate approaches work well in practice and obviate the need to estimate or fix g.
R-learner.
Within the setting of the graphical model (1), we define the treatment propensity (sometimes called the propensity score) as
e(x) = P (W = 1|X = x)(3)
and the conditional mean outcome as
m(x) = E[Y |X = x].(4)
The R-learner [21] leverages Robinson's decomposition [25] that led to Robin's reformulation [24] of the cate function as the solution to the optimization problem
τ (·) = arg min τ E (X,W,Y )∼P Y − m(X) − W − e(X) τ (X) 2(5)
in terms of the treatment propensity (3) and the conditional mean outcome (4). In practice, a regularized, empirical version of (5) is minimized via a two-step process: (1) cross-validated estimatesm andê are obtained for m and e, respectively, and then (2) the empirical loss is evaluated using folds of the data not used for estimatingm andê, and then minimized. The authors Nie & Wager note that the structure of the loss function eliminates correlations between m and e while allowing one to separately specify the form of τ through the choice of optimization method. In this paper, the only R-learner we use is the causal forest as implemented with generalized random forests [1] using the default options, including honest splitting [33].
Genetic and neuroevolutionary algorithms
Holland introduced genetic algorithms [9] as a nature-inspired approach to optimization. Generally speaking, these algorithms produce successive generations of candidate solutions. New generations are formed by selecting the fittest members from the previous generation and performing cross-over and/or mutation operations on them to produce new offspring candidates. Evolutionary algorithms encompass extensions and generalizations to this approach including memetic algorithms [18] that perform local refinements, genetic programming [7] that acts on programs represented as trees, and evolutionary programming [6] and strategies [23,28] that operate on more general representations. When such methods are applied specifically to the design and training of neural networks, they are commonly known as neuroevolutionary algorithms. See Stanley et al. [31] for a comprehensive survey. In the next section, we describe a specific neuroevolutionary strategy for feature engineering.
Methodology
In this section, we describe how we form our feature mapping Φ : R d → R m . To generate a single candidate solution, we train a shallow neural network to predict Y from X and extract an intermediate layer of this network. Each candidate map created in this way should yield a representation as functionally useful for predicting Y as X is. We then iteratively evolve cohorts of parameter sets for such maps to create a representation that carries the least amount of useful information for predicting the treatment W .
Candidate solutions
We consider neural networks f Θ :
R d → R of the form f Θ (x) = M 2 · a(M 1 · x + b 1 ) + b 2(6)
where M 1 ∈ R m×d , M 2 ∈ R 1×m are real-valued matrices (often called weights), b 1 ∈ R m and b 2 ∈ R 1 are vectors (often called biases), and a is a nonlinear activation function applied component-wise. We let Θ = (M 1 , M 2 , b 1 , b 2 ) denote the parameters for f Θ . Though f Θ is decidedly not a deep neural network, we note that, as a neural network with a single hidden layer, it remains a universal function approximator in the sense of Hornik et al. [10]. Optimizing the network (6) in order to best predict Y from X seeks the solution
Θ * = arg min Θ E |Y − f Θ (X)| 2 .(7)
Given parameters Θ for the network (6), we let Φ Θ : R d → R m given by
Φ Θ (x) = a(M 1 · x + b 1 )(8)
denote the output of the hidden layer. We restrict to candidate feature mappings of this form. As these mappings are completely characterized by their associated parameters, we define a fitness function and evolutionary algorithm directly in terms of parameter sets Θ in the following subsections.
Fitness function
For parameters Θ near the optimum (7), we note that Φ Θ (X) should be approximately as useful as X for learning a functional relationship with Y . However, for some values of Θ, the mapped features Φ Θ (X) may carry information useful for predicting W , and for this reason we consider a network g Ψ,Θ :
R d → [0, 1] given by g Ψ,Θ (x) = σ(M 4 · a(M 3 · Φ Θ (x) + b 3 ) + b 4 )(9)
where M 3 ∈ R k×m and M 4 ∈ R 1×k are weights, b 3 ∈ R k and b 4 ∈ R are biases, a is a nonlinear activation function applied component-wise, ‡ and σ(x) = (1 + exp(−x)) −1 denotes the sigmoidal activation function. In this case, Ψ = (M 3 , M 4 , b 3 , b 4 ) denotes the collection of tunable parameters. We define the fitness of a parameter set Θ to be
µ(Θ) = min Ψ E |W − g Ψ,Θ (X)| 2 .(10)
In this way, we express a preference for representations Φ Θ (X) that are less useful for predicting W . For a schematic of these architectures, please see Figure 1. Figure 1: In this schematic, the arrows connecting the X i to the Φ(X) j represent the map Φ Θ as in (8); these, in addition to the ones joining the Φ(X) j to Y , represent f Θ given explicitly in (6); and the original arrows along with the dashed arrows represent g Ψ,Θ as in (9).
T X 1 X 2 X 3 X d Φ(X) 1 Φ(X) 2 Φ(X) m Y
Evolutionary algorithm
We now describe a method to generate and evolve a cohort of candidate parameter sets Θ intended to seek a parameter set Θ * such that Φ Θ * (X) is nearly as useful for predicting Y as X is and, among such representations,
Φ Θ * (X) is least useful for predicting W . Given training data (X i , W i , Y i ) ∼ i.i.d. P for 1 ≤ i ≤ n,
we first partition the data into training and validation sets. We form an initial cohort of c candidates independently as follows. For 1 ≤ j ≤ c, we randomly instantiate Θ j using Glorot normal initialization [8] for the weights and zeros for the biases and then apply batch-based gradient descent on the training set to seek the solution to (7). In particular, we use the Adam optimizer [13] that maintains parameter-specific learning rates [5, cf. AdaGrad] and allows these rates to sometimes increase [35, ‡ though not necessarily the same as the one used in (6) and (8) cf. Adadelta] by adapting them using the first two moments from recent gradient updates. We use Tikhonov regularization [32] for the weights and apply a dropout layer [30] after the a(x) = tanh(x) activation function § to prevent overfitting.
For each constituent Θ j in the cohort, we then initialize and train a network g Ψ,Θj as in (9) to seek Ψ j = arg min Ψ E W − g Ψ,Θj (X) 2 on the training set and then evaluate E W − g Ψj ,Θj (X) 2 empirically on the validation set to estimate µ(Θ j ). We then use the fittest members of the current cohort to form a new cohort as follows. For each of the 2 pairings, we apply Montana and Davis's node-based crossover [17] method to the parameters M 1 and b 1 that we use to form Φ. This amounts to forming a new Φ by randomly selecting one of the two parents and using that parent's mapping for each coordinate. Thus, the new M 1 and b 1 are selected in a row-wise manner from the corresponding rows of the parents, and then the new M 2 and b 2 are randomly initialized and a few steps of optimization are performed to form the offspring candidate. The next generation then consists of the best performing candidate from the previous generation, 2 candidates formed by crossing the best candidates of the previous generation, and c − 1 − 2 entirely new candidates generated from scratch. We summarize our approach using pseudo-code in Algorithm 1. All computation of the valuation function is done by training a network of the form (9) to minimize |W − g Ψ,Θ (X)| 2 on training batches and then approximating (10) by taking the empirical mean on the validation set.
Remark on linearity
Due to our choice of representation Φ in (8), after training the network (6) to optimize (7), we expect the relationship between the learned features Φ(X) and the outcome Y to be approximately linear. In particular, we will have Y ≈ M 2 · Φ(X) for M 2 as given in (6). For this reason, the causal meta-learners trained using a linear regression base learner may benefit more extensively from using the transformed features instead of the original features, especially in cases where the relationship between the original features and outcomes is not well-approximated as linear. We provide specific examples in the next section involving meta-learners trained with ridge regression.
Remark on our assumptions
In order to use the represented features Φ(X i ) in place of the original features X i when learning the cate, we require that strong ignorability holds for the transformed dataset (Φ(X i ), W i , Y i ), i = 1, . . . , n. One sufficient, though generally not necessary, assumption that would imply strong ignorability is for Φ to be invertible on the support of X [29, assumption 1]. Unconfoundedness would also be guaranteed if Φ(X) satisfied the backdoor condition with respect to (W, Y ) [22, section 3.3.1]. § We tested rectified [19] and exponential [3] linear unit activation functions for a in ΦΘ but noticed only minor differences in subsequent performance of the causal forest. (1); positive integer parameters: c cohort size, number of members involved in forming the next generation, g number of generations, m dimensionality of the latent representation, and k dimensionality of the final hidden layer in (9) Result: parameterized function ΦΘ * : R d → R m such that models for the cate learned using the transformed training data {(Φ(Xi), Wi, Yi)}i∈T perform better than those learned on the original dataset T for j = 1, . . . , c do Optimize a parameter set Θj to seek (7) on training batches from a random initialization; end Form the first generation G1 = {Θj}j=1,...,c; for t = 2, . . . , g do #form the next generation
Initialize new generation Gt = {arg max Θ∈G t−1 µ(Θ)} with the best-performing candidate from the previous generation; for unique pairs {Θj, Θ k } formed from the top candidates from Gt−1 do #form new candidates using crossover Initialize M1 ∈ R m×d and b1 ∈ R m as the M1 and b1 from Θj; for κ = 1, . . . , m do Let ξ ∼ Bernoulli(1/2); if ξ = 1 then replace the κth row of M1 and the κth component of b1 with those from Θ k ; end Randomly initialize M2 and b2 and take optimization steps towards the solution to (7); Add Θ = (M1, M2, b1, b2) to Gt end while |Gt| < c do Optimize a parameter set Θ to seek (7) on training batches from a random initialization; Add Θ to Gt end end Let Θ * = arg max Θ∈Gg µ(Θ); return ΦΘ * as in (8) Algorithm 1: Neuroevolutionary Feature Engineering for Causal Inference
Ablation study on generated data
Due to the fundamental challenge of causal inference (namely, that the counterfactual outcome cannot be observed, even in controlled experiments), it is common practice to compare approaches to cate estimation on artificially generated datasets that allow the cate to be calculated directly for evaluative purposes. In this section, we perform experiments using two such data generation mechanisms from Nie & Wager's paper [21] that we now describe. ‡ Both setups provide a joint distribution satisfying the graph (1). For the vector-valued random variable X ∈ R d , we let X ij denote jth component (1 ≤ j ≤ d) of the ith sample (1 ≤ i ≤ n). The specifics for both setups are given as follows.
Setup A. For σ > 0 and an integer d > 0, we let
X i ∼ i.i.d. Uniform([0, 1] d ) and W i | X i ∼ Bernoulli(e(X i )),
where e(X i ) = max{0.1, min{sin(πX i1 X i2 ), 0.9}} and
Y i | X i , W i ∼ N b(X i ) + (W i − 0.5)τ (X i ), σ 2 , where b(X i ) = sin(πX i1 X i2 ) + 2(X i3 − 0.5) 2 + X i4 + 0.5X i5 and τ (X i ) = (X i1 + X i2 )/2.
In this paper, we let d = 24, n = 200, and σ = 1.
Setup C. For σ > 0 and an integer d > 0, we let
X i ∼ i.i.d. N d ( 0, I d×d ) and W i | X i ∼ Bernoulli(e(X i )),
where in this case e(X i ) = (1 + exp(X i2 + X i3 )) −1 and
Y i | X i , W i ∼ N b(X i ) + (W i − 0.5)τ (X i ), σ 2 ,
where now b(X i ) = 2 log(1+exp(X i1 +X i2 +X i3 )) and τ (X i ) = 1. In our example, we let d = 12, n = 500, and σ = 1.
Comparison methodology
For each data generation method, we ran 100 independent trials. Within each trial, we simulated a dataset of size n and randomly partitioned it into training, validation, and testing subsets at a 70%-15%-15% rate. We trained causal inference methods on the training set, using the validation data to aid the training of some base estimators for the meta-learners, and predicted on the test dataset. We then developed a feature map using the training and validation data as described in the previous section, applied this map to all features, and repeated the training and testing process using the new features.
To determine the impact of the fitness selection process, we also learned a feature transformation that did not make use of the fitness function at all. In effect, it simply generated a single candidate mapping and used it to transform all the features (without any cross-over or further mutation). Features developed in this way are described as "no fitness" in the tables that follow.
We compared the causal forest with default options (as found in R's grf package), and the S-, T-, and X-learners as described in section 2.1 with two base learners. The first base learner is LightGBM [12], a boosted random forest algorithm that introduced novel techniques for sampling and feature bundling. The second is a cross-validated ridge regression model (as found in scikit-learn) that performs multiple linear regression with an L 2 normalization on the weights. ‡ Nie & Wager's paper included four setups, namely A-D; however setup B modeled a controlled randomized trial and setup D had unrelated treatment and control arms.
Results
We report results for setup A in Table 1 and results for setup C in Table 2. For both setups, we consider a paired t-test for equal means against a two-sided alternative (as implemented in Python's scipy package). For setup A, we find that the improvement in MSE from using the transformed features in place of the original features corresponds to a statistically significant difference for the following learners: the causal forest (p < 0.001), the S-learner with ridge regression (p < 0.001), the T-learner with both LightGBM (p < 0.001) and ridge regression (p < 0.001), and the X-learner with both LightGBM (p < 0.001) and ridge regression (p < 0.001). For setup C, we again find significant differences for the causal forest (p = 0.023), S-learner with LightGBM (p = 0.003), T-learner with ridge regression (p < 0.001) and X-learner with ridge regression (p < 0.001).
In summary, we find that our feature transformation method improves the performance of multiple standard estimators for the cate under two data generation models.
Application to econometric data
In this section, we apply our feature engineering method to the LaLonde dataset [15,4] chronicling the results of an experimental study on temporary employment opportunities. The dataset contains information from 445 participants who were randomly assigned to either an experimental group that received a temporary job and career counseling or to a control group that received no assistance. Features include age and education (in years), earnings in 1974 (in $, prior to treatment), and indicators for African-American heritage, Hispanic-American heritage, marital status, and possession of a high school diploma. We consider the outcome of earnings in 1978 (in $, after treatment).
We cannot determine true average treatment effects based on individual-level characteristics (i.e. the true cate values) for real life experimental data as we can with the synthetic examples of the previous section. Instead, we evaluate performance by comparing the average realized treatment effect and average predicted treatment effect within bins formed by sorting study participants according to predicted treatment effect as demonstrated in Figure 2. Applying the causal forest predictor to the original features results in a root mean square difference between the average predicted and realized treatment effects of 4729.51. If the transformed features are used instead, this discrepancy improves to 3114.82.
From a practical perspective, one may learn the cate in order to select a subset of people for whom a given intervention has an expected net benefit (and then deliver that intervention only to persons predicted to benefit from it). When we focus on the 20% of people predicted to benefit most from this treatment, we find that the estimated realized benefit for those chosen using the transformed features ($4732.89) is much greater than the benefit for those chosen using the original feature set ($816.92). This can be seen visually in Figure 2 by comparing the estimated realized average treatment effect for bin #5 (the rightmost bin) in both plots. Table 1: Mean Squared Error (MSE) over 100 independent trials run using setup A. Algorithm 1 was run with parameters: cohort size c = 4, progenitors = 2, number of cohorts g = 5, representation dimensionality m = 20, and fitness function parameter k = 10.
Conclusions
Causal inference, especially on real life datasets, poses significant challenges but offers a crucial avenue for predicting the impact of potential interventions. Learned feature representations help us to better infer the conditional average treatment effect, improving our ability to individually tailor predictions and target subsets of the general population. In this paper, we propose and validate a novel representation-based method that uses a neuroevolutionary approach to remove information from features irrelevant for predicting the outcome. We demonstrate that this method can yield improved estimates for heterogeneous treatment effects on standard synthetic examples and illustrate its use on a real life dataset. We believe that representational learning is particularly well-suited for removing extraneous information in causal models and we anticipate future research in this area. Table 2: Mean Squared Error (MSE) over 100 independent trials run using setup C. Algorithm 1 was run with c = 4, = 2, g = 2, m = 20, and k = 10. Figure 2: We plot estimated realized and predicted average treatment effects versus the quintiles of predicted treatment effect (5 bins) for a causal forest using (left) the initial features and (right) the features transformed using our method. In order to estimate realized treatment effect within each bin, we take the difference between the average outcome for persons randomly assigned to the experimental group and the average outcome for persons randomly assigned to the control group.
Data: training T = {(Xi, Wi, Yi)}i∈T and validation V = {(Xi, Wi, Yi)}i∈V datasets drawn i.i.d. from P respecting the graphical model
Acknowledgements. We would like to thank Binjie Lai, Yi-Hong Kuo, and Xiang Wu for their support and feedback. We are also grateful to the anonymous reviewers for their insights and suggestions.
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"Metric Temporal Logic for Ontology-Based Data Access over Log Data",
"Metric Temporal Logic for Ontology-Based Data Access over Log Data"
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"D Calvanese \nFaculty of Computer Science Department of Computer Science\nItaly Birkbeck\nFree University of Bozen-Bolzano\nUniversity of London\nUK\n",
"E Güzel Kalaycı \nFaculty of Computer Science Department of Computer Science\nItaly Birkbeck\nFree University of Bozen-Bolzano\nUniversity of London\nUK\n",
"V Ryzhikov \nFaculty of Computer Science Department of Computer Science\nItaly Birkbeck\nFree University of Bozen-Bolzano\nUniversity of London\nUK\n",
"G Xiao [email protected]@dcs.bbk.ac.uk \nFaculty of Computer Science Department of Computer Science\nItaly Birkbeck\nFree University of Bozen-Bolzano\nUniversity of London\nUK\n",
"M Zakharyaschev \nFaculty of Computer Science Department of Computer Science\nItaly Birkbeck\nFree University of Bozen-Bolzano\nUniversity of London\nUK\n"
] |
[
"Faculty of Computer Science Department of Computer Science\nItaly Birkbeck\nFree University of Bozen-Bolzano\nUniversity of London\nUK",
"Faculty of Computer Science Department of Computer Science\nItaly Birkbeck\nFree University of Bozen-Bolzano\nUniversity of London\nUK",
"Faculty of Computer Science Department of Computer Science\nItaly Birkbeck\nFree University of Bozen-Bolzano\nUniversity of London\nUK",
"Faculty of Computer Science Department of Computer Science\nItaly Birkbeck\nFree University of Bozen-Bolzano\nUniversity of London\nUK",
"Faculty of Computer Science Department of Computer Science\nItaly Birkbeck\nFree University of Bozen-Bolzano\nUniversity of London\nUK"
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We present a new metric temporal logic HornMTL over dense time and its datalog extension datalogMTL. The use of datalogMTL is demonstrated in the context of ontology-based data access over meteorological data. We show decidability of answering ontology-mediated queries for a practically relevant non-recursive fragment of datalogMTL. Finally, we discuss directions of the future work, including the potential use-cases in analyzing log data of engines and devices.
| null |
[
"https://arxiv.org/pdf/1701.00976v1.pdf"
] | 16,591,967 |
1701.00976
|
682c580ec7f8e5328794128ae356fc0327fd5bd7
|
Metric Temporal Logic for Ontology-Based Data Access over Log Data
4 Jan 2017
D Calvanese
Faculty of Computer Science Department of Computer Science
Italy Birkbeck
Free University of Bozen-Bolzano
University of London
UK
E Güzel Kalaycı
Faculty of Computer Science Department of Computer Science
Italy Birkbeck
Free University of Bozen-Bolzano
University of London
UK
V Ryzhikov
Faculty of Computer Science Department of Computer Science
Italy Birkbeck
Free University of Bozen-Bolzano
University of London
UK
G Xiao [email protected]@dcs.bbk.ac.uk
Faculty of Computer Science Department of Computer Science
Italy Birkbeck
Free University of Bozen-Bolzano
University of London
UK
M Zakharyaschev
Faculty of Computer Science Department of Computer Science
Italy Birkbeck
Free University of Bozen-Bolzano
University of London
UK
Metric Temporal Logic for Ontology-Based Data Access over Log Data
4 Jan 2017
We present a new metric temporal logic HornMTL over dense time and its datalog extension datalogMTL. The use of datalogMTL is demonstrated in the context of ontology-based data access over meteorological data. We show decidability of answering ontology-mediated queries for a practically relevant non-recursive fragment of datalogMTL. Finally, we discuss directions of the future work, including the potential use-cases in analyzing log data of engines and devices.
Introduction
The aim of ontology-based data access (OBDA) (Poggi et al. 2008) is, on one hand, to represent the information from various heterogeneous data sources in a unified and conceptually transparent way by means of mappings. On the other hand, the ontology language allows one to define concepts in terms of other concepts, and thereby represent frequently used query patterns as reusable concepts. The end-user, in that case, can obtain the required information by means of simple conceptual queries and is not required to know neither the structure of the source data nor the definitions of the concepts he is using.
Due to up-to-date requirements of industry (see, e.g., (Kharlamov et al. 2014)) the OBDA approach is being actively adopted in the context of the temporal data of streams and logs. Initially, only the classical non-temporal ontology languages were considered to mediate the access to temporal data (Gutiérrez- Basulto and Klarman 2012;Ozcep et al. 2013;Baader, Borgwardt, and Lippmann 2013;Klarman and Meyer 2014).
Later, the ontology languages with temporalized concepts were studied in this context (Artale et al. 2015;Basulto, Jung, and Kontchakov 2016). Such concepts are defined by means of linear temporal logic (LTL); for example, the axiom Hurricane ← HurricaneForceWind ∧ X − HurricaneForceWind defines a hurricane as hurricane force wind lasting for 1 hour (X − is the previous time LTL operator). One easily notices that this definition works only if the temporal data arrives strictly in hourly periods, such as 13:21, 14:21, etc. If these periods are smaller and have a fixed length, the definition above can still be adjusted by using the conjunction of the form HFW ∧ X − HFW ∧ X − X − HFW ∧ . . . . However, first, having the data with fixed-period timestamps is not always a realistic assumption, and, second, doing the adjustment above contradicts the OBDA philosophy, where the ontology user is not required to have knowledge of the structure of the data sources. Therefore, the following definition would be more natural Hurricane ← ⊟ 1h 0 HurricaneForceWind, where ⊟ 1h 0 is a metric temporal operator during the previous hour.
The logic required to express such statements is a kind of metric temporal logic or modal logic of metric spaces; see (Koymans 1990;Kurucz, Wolter, and Zakharyaschev 2005) for surveys and further references.
In this paper, we introduce a metric temporal logic HornMTL with the operator ⊟ ✁d ✄e , where ✄ is either > or (and similarly for ✁) and e, d are time distances, its future analogue ⊞ ✁d ✄e , as well as their duals ✁d ✄e and ✁d ✄e . We interpret this logic over a dense temporal domain. The reason for not considering a discrete domain is that we want to abstract from the granularities of time (periods of timestamps) in the data sources. In our logic, we allow the statements of the form P @ι, where ι is an interval specified by a pair of time instants, to represent the conceptualized temporal data. The meaning of, say, P @(t 1 , t 2 ] is that P holds at all times t between t 1 (not including it) and t 2 (including it). We assume that we can convert data from any source with timestamped tuples to this format by means of mappings. For example, if a source contains the information of temperature measurements taken every hour, such as 13:21: -1 • C, 14:21: 2 • C, 15:21: -1 • C, etc., we can conceptualize them as the statements PositiveTemp@(13:21, 14:21], etc. Note that whether to include the ends of intervals or not, as well as whether to consider 2 • C to be the case in the hour preceding or following 14:21, is the choice of the mapping designer. We then extend HornMTL to datalogMTL that also allows for standard Datalog reasoning about objects of the application domain (weather stations, cities, sensors, etc.).
We present a few preliminary results on datalogMTL. First, we describe a use-case of OBDA over meteorological data with SQL mappings to a large real-world weather database and datalogMTL as an ontology language. Second, we develop an ontology-mediated query answering algorithm for a non-recursive fragment datalogMTL ✷ nr of datalogMTL. Finally, we report some preliminary evaluation results showing the feasibility of our approach.
HornMTL and datalogMTL
Syntax. We consider a propositional temporal logic HornMTL with the set of propositional variables P 0 , P 1 , . . . over the temporal domain T isomorphic to (R, ) with 0 and arithmetic operations +, −. That is, we assume dense time. Let int(T) be the set of (non-empty) intervals on T, which are of the form [t 1 , t 2 ], [t 1 , t 2 ), (t 1 , t 2 ], and (t 1 , t 2 ), where t i ∈ T∪{−∞, ∞}, is either ( or [, and is either ) or ]. (We do not distinguish between the intervals t 1 , ∞] and t 1 , ∞), consider ∞, ∞ to be empty, and analogously for −∞. We also assume that ≤ is defined on T ∪ {−∞, ∞} and +, − are defined on pairs of elements from T and {−∞, ∞}, in a standard way.) Define a data instance D as a non-empty finite set of data assertions (or facts) of the form:
P i @ι,
where P i is a propositional variable and ι ∈ int(T).
We use the temporal operators of the form:
-⊞ ✁d ✄e (always between e and d in the future), -⊟ ✁d ✄e (always between e and d in the past),
-✁d ✄e (sometime between e and d in the future), -✁d ✄e (sometime between e and d in the past), where ✁ is either < or , e, d are distances, that is, positive elements of T, and ✄ is either > or . Thus, e.g., ⊞ <d e expresses 'always between e and d in the future including e and excluding d' and similarly for the other operators. We also impose the following consistency requirement on every operator O ✁d ✄e (henceforth we assume O ∈ {⊞, ⊟, , }, ✷ ∈ {⊞, ⊟}, and ✸ ∈ { , }):
-there exists t ∈ T such that t ✄ e and t ✁ d.
Propositional literals are defined by the following grammar:
λ ::= P i | O ✁d ✄e λ.
An ontology, Ø, is a finite set of axioms of the form:
λ ← λ 1 ∧ · · · ∧ λ k , ⊥ ← λ 1 ∧ · · · ∧ λ k .(1)
A knowledge base (KB) is a pair (Ø, D).
Semantics.
Consider an interpretation M = (T, · M ) such that P M i ⊆ T for each propositional variable P i and write M, t |= P i when t ∈ P M i for t ∈ T. As usual, it is assumed that M, t |= ⊥ for all t ∈ T. We extend the definition of |= to λ as follows:
M, t |= ⊞ ✁d ✄e λ iff M, t ′ |= λ for all t ′ such that t ′ − t ✄ e and t ′ − t ✁ d, (2) M, t |= ⊟ ✁d ✄e λ iff M, t ′ |= λ for all t ′ such that t − t ′ ✄ e and t − t ′ ✁ d, (3) M, t |= ✁d ✄e λ iff M, t ′ |= λ for some t ′ such that t ′ − t ✄ e and t ′ − t ✁ d, (4) M, t |= ✁d ✄e λ iff M, t ′ |= λ for some t ′ such that t − t ′ ✄ e and t − t ′ ✁ d. (5) We say that M satisfies a data assertion P @ι if M, t |= P for all t ∈ ι. We say that M satisfies an ontology axiom λ ← λ 1 ∧· · ·∧λ k (respectively, ⊥ ← λ 1 ∧· · ·∧λ k ), if M, t |= λ i , for all i = 1, . . . , k, imply M, t |= λ (resp., M, t |= ⊥), for every t ∈ T.
Thus, the ontology axioms are global. We say that M satisfies a data instance D (resp., ontology Ø) if it satisfies each statement in it. Finally, we say that M satisfies a knowledge base (Ø, D) and write M |= (Ø, D) if M satisfies both Ø and D.
Our main reasoning problem is query answering. Define an atomic query (AQ) as an expression P @δ, where P is a proposition and δ is an interval variable. An ontology Ø and an AQ P @δ constitute an ontology-mediated query
(OMQ) Q(δ) = (Ø, P @δ). A certain answer to Q(δ) over D is any interval ι ∈ int(T) such that M |= (Ø, D) implies M, t |= P for all t ∈ ι.
HornMTL ✷ fragment. We consider one important fragment HornMTL ✷ of HornMTL, where the operators ✁d ✄e and ✁d ✄e are disallowed in the heads of the rules. Note that each HornMTL ✷ KB can be converted to KB that has ⊞ ✁d ✄e and ⊟ ✁d ✄e operators only, and the original KB is a conservative extension of it. For example, an axiom R ← P ∧ ✁d ✄e Q can be replaced by the pair of axioms R ← P ∧ Q ′ and ⊞ ✁d ✄e Q ′ ← Q. Finally, we consider a non-recursive fragment HornMTL ✷ nr of HornMTL ✷ by adopting the simplest definition of non-recursivivity: consider the relation ≺ on the symbols of Ø defined as P ≺ Q iff there is an axiom in Ø, where P occurs in the head and Q in the body (P depends on Q). We require that P ≺ * P for no symbol P in Ø, where ≺ * is a transitive closure of ≺.
datalogMTL. Consider the predicate symbols P 0 , P 1 , . . . , each of some arity m ≥ 0, and a set of object variables x 0 , x 1 , . . . . Data instances D here contain assertions P (c)@ι, where P is an m-ary predicate symbol, c an mtuple of individual constants, and ι ∈ int(T). This assertion says that P (c) is true at ι. We denote by ind(D) the set of all individual constants in D. An ontology Ø is a finite set of axioms of the form (1) with the literals λ defined by the grammar:
constants. We also impose other standard datalog restrictions on our programs, and forbid (in)equality predicates in the heads. We call the predicates occurring in D extensional and those occurring in the head of the axioms of Ø intentional. An interpretation, M, is based on the domain ∆ = ind(D) (for the individual variables and constants) and T. For any m-ary predicate P , m-tuple c from ∆ and t ∈ T, M specifies whether P is true on c at t, in which case we write M, t |= P (c). Let ν be an assignment of elements of ∆ to individual terms (we adopt the standard name assumption: ν(c) = c, for every individual constant c). We set:
M, t |= ν τ = τ ′ iff ν(x 0 ) = ν(x 1 ), M, t |= ν τ = τ ′ iff ν(x 0 ) = ν(x 1 ), M, t |= ν P (x) iff M, t |= ν P (ν(x)),
and use inductively the formulas (2)-(5) with |= ν instead of |= for the cases O ✁d ✄e λ. We say M satisfies an ontology axiom λ ← λ 1 ∧ · · · ∧ λ k (respectively, ⊥ ← λ 1 ∧ · · · ∧ λ k ), if M, t |= ν λ i for each i implies M, t |= ν λ (resp., M, t |= ν ⊥), for every t ∈ T and assignment ν. Finally, M satisfies a data assertion P (c)@ι if M, t |= P (c) for each t ∈ ι, and M |= (Ø, D) is defined straightforwardly.
AQs are defined as P (x)@δ, where P is a predicate symbol of arity m, and δ is an interval variable. An ontologymediated query is defined Q(x, δ) = (Ø, P (x)@δ). A certain answer to Q(x, δ) over D is any pair (c, ι), such that c = ν(x) for some ν, and M |= (Ø, D) implies M, t |= P (c) for all t ∈ ι.
Note that HornMTL is a fragment of datalogMTL (where all predicates have arity 0). We also consider the fragments datalogMTL ✷ and datalogMTL ✷ nr defined with the same syntactic restrictions as HornMTL ✷ and HornMTL ✷ nr .
Weather Use Case
Our OBDA approach can be used to analyze meteorological data through ontology-mediated queries. We can conceptualize this raw data by means of the SQL mappings. For example, to extract the data for the extensional predicate Precipitation(x)@ t 1 , t 2 (with the meaning precipitation occurs at x during t 1 , t 2 ), we can use the following SQL query:
SELECT ID AS x, lag(TIME) over (partition by ID order by TIME) AS t 1 , TIME AS t 2 , "(" AS , "]" AS FROM Weather WHERE P01I > lag(P01l) over(partition by ID order by TIME)
That is, we extract the intervals of the shape (t 1 , t 2 ], where t 1 and t 2 are the two next timestamps for a given station. The ends of the interval are chosen to reflect the fact that, e.g., the precipitation is measured accumulatively and the device produces the output in the end of the measurement interval. Analogously to Precipitation, we populate by the data the other extensional predicates, such as PositiveTemp (temperature well above 0 • C), HurricaneForceWind (wind with the speed above 118 km/h), TempAbove24 and TempAbove41 (temperature above 24 and 41 • C). Consider the ontology containing the axioms:
Rain(x) ← PositiveTemp(x) ∧ Precipitation(x), ⊟ 1h 0 Hurricane(x) ← ⊟ 1h 0 HurricaneForceWind(x), ⊟ 24h 0 ExcessiveHeat(x) ← ⊟ 24h 0 TempAbove24(x)∧ 24h 0 TempAbove41(x),
The second axiom is already discussed in the introduction (here we use a slightly modified version to say that hurricane holds also at the time point, when the hurricane force wind begins), whereas the last axiom formalizes the definition of the situation when an excessive heat warning should be issued according to the US Weather Forecast Offices (24 hours with the minimal temperature above 24 • C and the maximal above 41 • C).
We can also populate the binary predicate LocationOf(x, y)@ t 1 , t 2 by using:
SELECT COUNTY AS x, ID AS y, −∞ AS t 1 , ∞ AS t 2 , "(" AS , ")" AS FROM Metadata
Note that we assume that LocationOf holds between a county and a station globally. It is now possible to define:
HurricaneAffectedCounty(x) ← LocationOf(x, y) ∧ Hurricane(y), SpreadRainCounty(x) ← LocationOf(x, y)∧ LocationOf(x, z) ∧ (y = z) ∧ Rain(y) ∧ Rain(z).
Query Answering in datalogMTL ✷
nr
In this section we first present an algorithm for computing certain answers to an HornMTL ✷ nr OMQ Q(δ) = (Ø, P @δ) over D.
Normal form for HornMTL ✷ nr . Our procedure works on the ontology Ø containing only the clauses of the shape:
P ← Q ∧ R, ⊥ ← Q ∧ R, ⊞ ✁d ✄e P ← Q, ⊟ ✁d ✄e P ← Q, P ← ⊞ ✁d ✄e Q, P ← ⊟ ✁d ✄e Q
It is an easy exercise to verify that every HornMTL ✷ nr can be brought to the normal form by performing the following operations:
-Substitute the axioms of the shape λ ← λ 1 ∧ · · · ∧ λ k for k ≥ 3 by k − 1 axioms with binary conjunctions using fresh symbols. Analogously for the axioms with ⊥ in the head.
-Remove ✸ ✁d ✄e λ literals in the body of the axioms as sketched in Preliminaries.
-Remove the nested modalities ✷ ✁d ✄e λ by substituting them for ✷ ✁d ✄e P λ , for a fresh symbols P λ , and adding: -P λ ← λ, if ✷ ✁d ✄e λ occurred in the body of the axiom, -λ ← P λ , if ✷ ✁d ✄e λ occurred in the head of the axiom. -Remove the axioms of the shape λ 0 ← λ 1 ∧ λ 2 , if λ i = ✷ ✁d ✄e P for some 0 ≤ i ≤ 2, as described in the previous step. Analogously for the axioms with ⊥ in the head.
It can be readily verified that the resulting ontology in the normal form is in HornMTL ✷ nr .
Algorithm. We first assume that the facts of D are stored in the tables of the shape P * i (t 1 , t 2 , , ), where t 1 , t 2 ∈ T, is either ( or [, and is either ) or ]. E.g., for D =
{P i @(t 1 , t 2 ], P i @[t ′ 1 , t ′ 2 ]} we produce the table P * i with two tuples { t 1 , t 2 , (, ] , t ′ 1 , t ′ 2 , [, ] }.
Consider an intentional symbol P and assume that for all Q such that P ≺ Q the tables Q * are computed. Consider now the cases: P ← Q ∧ R. Then P * is computed as the minimal table satisfying the condition:
Q * t 1 ,t 2 , , ∧ R * t ′ 1 , t ′ 2 , ′ , ′ ∧ ints t 1 , t 2 , , , t ′ 1 , t ′ 2 , ′ , ′ → P * t ′′ 1 , t ′′ 2 , ′′ , ′′ , where ints(t 1 , t 2 , , , t ′ 1 , t ′ 2 , ′ , ′ ) is ⊤ if t 1 , t 2 ∩ ′ t ′ 1 , t ′ 2 ′ = ∅ (the intervals intersect), otherwise it is ⊥, and ′′ t ′′ 1 , t ′′ 2 ′′ = t 1 , t 2 ∩ ′ t ′ 1 , t ′ 2
′ (the result of the intersection). Note that P * is computed as a temporal join (Gao et al. 2005) of Q * and R * . We also create a table ⊥ * for the axioms ⊥ ← Q ∧ R.
⊞ ⊞ ⊞ ✁d ✄e P ← Q. Then P * is computed as a minimal table satisfying:
Q * t 1 , t 2 , , → P * t 1 + e, t 2 + d, ed , ✄ , ed , ✁
where the edge function ed( , ✄) returns [, if is [ and ✄ is , and (, otherwise. Then ed( , ✁) is defined symmetrically. For example, if Q * = { t 1 , t 2 , (, ] } and the axiom is ⊞ <d e P ← Q, then P * = { t 1 + e, t 2 + d, (, ) }. The axiom ⊟ ✁d ✄e P ← Q is handled analogously. P ← ⊞ ⊞ ⊞ ✁d ✄e Q. Consider the following example: let Q * = { t 1 , t 2 , (, ] , t 2 , t 3 , (, ) } and the axiom P ← ⊟ <d e Q such that d−e < t 3 −t 1 . Then, according to the semantics, P * = { t 1 − e, t 3 − d, (, ] }. In order to compute P * correctly we need to consider the concatenation of the intervals (t 1 , t 2 ]
and (t 2 , t 3 ). To compute P * in general we first produce a closure Q ′ of Q * as the minimal table satisfying:
Q * t 1 , t 2 , , → Q ′ t 1 , t 2 , , , Q * t 1 , t 2 , , ∧ Q ′ t ′ 1 , t ′ 2 , ′ , ′ ∧ (t ′ 2 ≤ t 2 )∧ ints t 1 , t 2 , , , t ′ 1 , t ′ 2 , ′ , ′ → Q ′ t ′ 1 , t 2 , ′ , .
After that P * can be obtained by:
Q ′ t 1 , t 2 , , ∧ fit t 1 , t 2 , , , e, d, ✄, ✁ → P * t 1 − e, t 2 − d, de , ✄ , de , ✁ ,
where fit t 1 , t 2 , , , e, d, ✄, ✁ is ⊤, if there exists t ∈ T such that {t + t ′ | t ′ ✄ e and t ′ ✁ d} ⊆ t 1 , t 2 , and ⊥ otherwise. Essentially, fit holds if the segment {t ′ | t ′ ✄ e and t ′ ✁ d} can be shifted so that it fits inside t 1 , t 2 . Finally, another edge function de is needed to compute the ends of the resulting interval. Here de , ✄ is [, if either is ( and ✄ is >, or is [ and ✄ is ; otherwise de , ✄ is (. The definition of de , ✁ is symmetric. The axiom ⊟ ✁d ✄e P ← Q is handled analogously. Observe that the computation of Q ′ requires recursion.
Clearly, when P occurs in the head of several axioms, the table P * is taken equal to the union of the tables computed above. In fact, for every symbol P in Ø the algorithm computes P * that, for a consistent KB (Ø, D), satisfies:
• for every t ∈ T, there exists a certain answer ι to OMQ Q(δ) = (Ø, P @δ) over D such that t ∈ ι iff there exists a tuple t 1 , t 2 , , in P * such that t ∈ t 1 , t 2 .
This correctness follows directly from the semantics of HornMTL ✷ nr . Then, if the table ⊥ * is empty, as an output of the OMQ Q(δ) = (Ø, G@δ) over D we produce the table G * (otherwise, we return G * with one special tuple −∞, ∞, (, ) as (Ø, D) is inconsistent). Clearly, the correctness above guarantees that G * represents the set of all certain answers.
One can extend the approach presented above to OMQ answering in datalogMTL ✷ nr . Indeed, it is possible to convert an arbitrary datalogMTL ✷ nr ontology to the one in the normal form similar to that used above. The tables P * need to contain the tuples of the shape c 1 , . . . , c m , t 1 , t 2 , , , where m is the arity of P . The rules for processing the temporal axioms essentially remain the same. The rules for computing the conjunctions (joins) need to be adjusted to correctly handle the individual arguments of the predicates.
Discussion and Future Work
Initial Experiments. We made experiments to evaluate the performance of the proposed algorithm on the Hurricane(x)@δ and ExcessiveHeat(x)@δ OMQs with the ontology from the weather use case. We implemented the algorithm of the previous section, for a given OMQ, as an SQL query using WITH clause and the RECURSIVE operator. That is, the intermediate tables of the algorithm are defined as a sequence of virtual SQL tables. The configuration of the computer that was used for the experiments is Intel Core i5 @ 2.7 GHz, 8 GB RAM with 1867 MHz DDR3 and OS X El Capitan operating system in version 10.11.4. The weather data is stored in 64 bit PostgreSQL version 9.4.5. We ran the queries over a table including 140 881 rows. It took 3 199 ms for Hurricane and 481 876 ms for ExcessiveHeat to retrieve the results. We interpret this outcome as a positive indication of the feasibility of our approach: even a straightforward implementation appears to work. We foresee the following three directions of the future work: New Use Cases. Our language is capable of expressing complex patterns of events that are of interest for such purposes as diagnostics of engines or devices. The axiom
SmoothShutDown ← IdleRPM∧⊟ <15min >0 IntermRPM∧ 25min 15min RunningRPM, for instance, describes the event of smooth shutdown of an engine as being in an idle state after having intermediate speed (RpM) for 15 minutes and having a running speed before that (not further than 25 minutes). The axiom:
ConsHighVibration ← ⊟ 50sec 0 10sec >0
HighVibration describes consistent high vibration as high vibration occurring every 10 seconds during a minute. Our OBDA approach seems to be able to capture many industrial use-cases. In the future, we plan to investigate such potential applications.
Open Theoretical Problems. At the moment, we do not know whether OMQ answering in HornMTL is decidable. In fact, this question is open even for the fragment HornMTL ✷ . We plan to obtain complexity results for those languages, and we are particularly interested in data complexity (that is, the complexity in the size of D when Q(δ) is assumed to be fixed). It is also important to understand how the complexity results for HornMTL carry over to datalogMTL. To achieve our goal, we plan to study various techniques developed in the area of metric temporal logics (Ouaknine and Worrell 2005;Ouaknine and Worrell 2008;Hirshfeld and Rabinovich 2005) and modal logics over metric spaces (Kutz et al. 2003;Sheremet, Wolter, and Zakharyaschev 2010;.
Implementation and Optimizations.
The proposed query answering algorithm for datalogMTL ✷ nr clearly allows for optimizations. For example, computing the transitive closure of the table Q * when processing the axiom P ← ⊞ ✁d ✄e Q seems to be avoidable. Moreover, our algorithm does not make any assumption regarding the temporal ordering of the tuples. If such a realistic assumption is made, we may be able to develop more efficient algorithms, in particular, by using indexes on timestamps.
The MesoWest 1 project makes publicly available historical records of the weather stations across the US. This data is available in the relational tables Weather containing the following fields: ID. Station ID. Example: KHYS.TIME. Timestamp. Example: 11-11-2015 8:55 CST.
TMP. Temperature. Example: 15.6 • C.
SKNT. Wind Speed. Example: 9.2 km/h.
P01I. Precipitation in 1 hour. Example: 0.09 cm.
Moreover, there are metadata tables Metadata containing,
in particular, location information of stations in the fields:
ID. Station ID. Example: KHYS.
COUNTY. Example: Ellis.
STATE. Example: Kansas.
λ ::= (τ = τ ′ ) | (τ = τ ′ ) | P (x) | O ✁d ✄e λ,where P is a predicate symbol of arity m, x is a vector of m variables, and τ, τ ′ are individual terms, i.e., variables or
http://mesowest.utah.edu/
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A modal logic framework for reasoning about comparative distances and topology. [ Özcep, Ann. Pure Appl. Logic. 1614EUTechnical reportJ. Symb. Log.[Özcep et al. 2013]Özcep, O.; Möller, R.; Neuenstadt, C.; Zheleznyakov, D.; and Kharlamov, E. 2013. A semantics for temporal and stream-based query answering in an OBDA context. Technical report, Deliverable D5.1, FP7-318338, EU. [Poggi et al. 2008] Poggi, A.; Lembo, D.; Calvanese, D.; De Giacomo, G.; Lenzerini, M.; and Rosati, R. 2008. Link- ing data to ontologies. J. on Data Semantics X:133-173. [Sheremet, Wolter, and Zakharyaschev 2010] Sheremet, M.; Wolter, F.; and Zakharyaschev, M. 2010. A modal logic framework for reasoning about comparative distances and topology. Ann. Pure Appl. Logic 161(4):534-559. [Wolter and Zakharyaschev 2005] Wolter, F., and Za- kharyaschev, M. 2005. A logic for metric and topology. J. Symb. Log. 70(3):795-828.
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[
"Phase Diagram for Magnetic Reconnection in Heliophysical, Astrophysical and Laboratory Plasmas",
"Phase Diagram for Magnetic Reconnection in Heliophysical, Astrophysical and Laboratory Plasmas"
] |
[
"Hantao Ji \nCenter for Magnetic Self-Organization\nPlasma Physics Laboratory\nPrinceton University\n08543Princeton, PrincetonNew Jersey\n",
"William Daughton \nLos Alamos National Laboratory\n87545Los AlamosNew Mexico\n"
] |
[
"Center for Magnetic Self-Organization\nPlasma Physics Laboratory\nPrinceton University\n08543Princeton, PrincetonNew Jersey",
"Los Alamos National Laboratory\n87545Los AlamosNew Mexico"
] |
[] |
Recent progress in understanding the physics of magnetic reconnection is conveniently summarized in terms of a phase diagram which organizes the essential dynamics for a wide variety of applications in heliophysics, laboratory and astrophysics. The two key dimensionless parameters arXiv:1109.0756v1 [astro-ph.IM]
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10.1063/1.3647505
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[
"https://arxiv.org/pdf/1109.0756v1.pdf"
] | 118,885,619 |
1109.0756
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fbd1590815b629df1a2936d58d20eb943d1a41b0
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Phase Diagram for Magnetic Reconnection in Heliophysical, Astrophysical and Laboratory Plasmas
4 Sep 2011
Hantao Ji
Center for Magnetic Self-Organization
Plasma Physics Laboratory
Princeton University
08543Princeton, PrincetonNew Jersey
William Daughton
Los Alamos National Laboratory
87545Los AlamosNew Mexico
Phase Diagram for Magnetic Reconnection in Heliophysical, Astrophysical and Laboratory Plasmas
4 Sep 2011(Dated: September 6, 2011)
Recent progress in understanding the physics of magnetic reconnection is conveniently summarized in terms of a phase diagram which organizes the essential dynamics for a wide variety of applications in heliophysics, laboratory and astrophysics. The two key dimensionless parameters arXiv:1109.0756v1 [astro-ph.IM]
are the Lundquist number and the macrosopic system size in units of the ion sound gyroradius.
In addition to the conventional single X-line collisional and collisionless phases, multiple X-line reconnection phases arise due to the presence of the plasmoid instability either in collisional and collisionless current sheets. In particular, there exists a unique phase termed "multiple X-line hybrid phase" where a hierarchy of collisional islands or plasmoids is terminated by a collisionless current sheet, resulting in a rapid coupling between the macroscopic and kinetic scales and a mixture of collisional and collisionless dynamics. The new phases involving multiple X-lines and collisionless physics may be important for the emerging applications of magnetic reconnection to accelerate charged particles beyond their thermal speeds. A large number of heliophysical and astrophysical plasmas are surveyed and grouped in the phase diagram: Earth's magnetosphere, solar plasmas (chromosphere, corona, wind and tachocline), galactic plasmas (molecular clouds, interstellar media, accretion disks and their coronae, Crab nebula, Sgr A*, gamma ray bursts, magnetars), extragalactic plasmas (Active Galactic Nuclei disks and their coronae, galaxy clusters, radio lobes, and extragalactic jets). Significance of laboratory experiments, including a next generation reconnection experiment, is also discussed.
PACS numbers:
I. COLLISIONAL AND COLLISIONLESS RECONNECTION
It has been a long held view that magnetic reconnection is primarily characterized by plasma collisionality. This is evidenced by the common uses of the resistive magnetohydrodynamic (MHD) models, which is parameterized solely by the dimensionless Lundquist number,
S ≡ µ 0 L CS V A η ,(1)V R V A = 1 √ S ,(2)
where V R is the reconnection inflow speed. When collisions are sufficiently infrequent or S is sufficiently large, physics beyond resistive MHD becomes crucial 3 , leading to a fast reconnection rate nearly independent of S. A large body of the work in the past decades, therefore, has focused on reconnection either in collisional or collisionless limit as summarized by recent reviews 4,5 .
The collisional MHD description provides a good description of magnetic reconnection for plasmas in which all the resistive layers remain larger than the relevant ion kinetic scale.
For example, without a guide field (i.e. anti-parallel reconnection), the transition between collisional and collisionless reconnection occurs 6-10 when the current sheet half thickness predicted by the Sweet-Parker model approaches
δ SP ≡ L CS √ S = d i(3)
where d i ≡ c/ω pi is the ion skin depth. By properly varying both L (and thus L CS ) and
η (through changing e.g. electron temperature), S and d i can be kept constant while the relative magnitude of δ SP to d i can be reversed, leading to dramatic differences in the structure of the reconnection layer along with clear changes in the magnitude and scaling of the reconnection rate. This qualitative change can be characterized by the effective plasma size which is defined by λ ≡ L d i (4) so that the second equality in Eq.(3) can be written as
S = 2 λ 2 .(5)
In the case of a finite guide field, the transition occurs [11][12][13][14] when δ SP = ρ s where ρ s ≡
(T i + T e )m i /q i B T isλ ≡ L ρ s .(6)
Thus, the collisional and collisionless reconnection phases are distinguished in the parameter space of (λ, S). This is illustrated as the black line in the phase diagram in Fig. 1 electrons) simulations have revealed significant discrepancies in the expected structure of the discontinuities due to the strong ion temperature anisotropy that is naturally generated in these configurations 24,25 .
There is a growing body of work that suggests it may be necessary to move beyond these steady-state models in order to understand the dynamics of magnetic reconnection in largescale collisional plasmas. In particular, for sufficiently high Lundquist numbers, resistive MHD simulations feature highly elongated layers which breakup into muliple X-lines sepa-
S c ∼ 10 4 ,(7)
shown by the green line in Fig. 1. However, the precise value of S c probably depends on the level of pre-existing fluctuations in the plasma 30,44,50,51 . We note that the green line is necessarily stopped at the low λ end defined by Eq.(5) due to the invalidity of MHD models in the collisionless phase.
N 1 − islands N 2 − islands S 1 = (L 1 /δ 1 ) 2 S 2 = (L 2 /δ 2 ) 2 2L 2 = 2L 1 /N 1 1 st Level FIG. 2:
In the regimes of high Lundquist number and large plasma size, the plasmoid instability gives rise to a hierarchy of interacting current sheets and islands. The above sketch gives the notation used here for describing this hierarchy. physics, which we denote as the multiple X-line hybrid phase.
The boundary line in the (λ, S) space between these two sub-phases depends on the detailed physics of unstable current sheets. The main uncertainty originates from the question of how many islands, on the average, remain within the unstable current sheet at any given time. According to linear analytic theory 40,52 , the number of secondary islands scales as
N ∼ S S c α ,(8)hierarchy is N 2 = (S 2 /S c ) α where S 2 = (L 2 /δ 2 ) 2 is the Lundquist number of the new sheets,
assuming the same reconnection magnetic field strength upstream. Therefore, the jth level quantities in the hierarchy are related to the (j-1)th level quantities by
L j = L j−1 N j−1 (9) S j = S j−1 N j−1 (10) δ j ≡ L j S j = L j−1 S j−1 1 N j−1 = δ j−1 N j−1 ,(11)
where the number of islands in the jth level, N j , is given by a recursion relation,
N j = S j S c α = S j−1 S c α S j S j−1 α = N (1−α) j−1 = N (1−α) 2 j−2 = ... = N (1−α) j−1 1 .(12)
If we terminate the hierarchy at the jth level, then the total number of islands in the system N j corresponds to the product of the islands in all the levels
N j ≡ N 1 N 2 N 3 ... N j = N 1 N (1−α) 1 N (1−α) 2 1 ... N (1−α) j−1 1 ≡ N β j 1 = S 1 S c αβ j(13)
where
β j ≡ j−1 n=0 (1 − α) n = 1 − (1 − α) j α .(14)
Note that as the hierarchy becomes deeper j 1 it converges towards β ∞ = α −1 . We can now conveniently express the scaling for the total number of islands up through jth level in terms of the global Lundquist number
N j = S 1 S c 1−(1−α) j ,(15)
while the Lundquist number of the new current sheets at the jth level is
S j = S j−1 N j−1 = S j−2 N j−1 N j−2 = ... = S 1 N j−1 N j−2 ...N 1 = S 1 N j−1 = S (1−α) j−1 1 S 1−(1−α) j−1 c .(16)
Notice that for α < 1 this result implies S j > S c for any finite level in the hierarchy, which implies the new levels are always unstable to the plasmoid instability! Strangely enough, this implies the hierarchy has infinite depth for α < 1, and only terminates in the limit j → ∞ where N ∞ = S 1 /S c and S ∞ = S c . However, our basic scaling assumption for the number of islands at each level N j = (S j /S c ) α becomes invalid when N j decreases towards unity. To make a reasonable estimate for the number of levels in the hierarchy, one should consider a cutoff N j ∼ N min below which continuous scaling arguments are meaningless. For example, by setting N j ≥ N min in Eq. (12) as a cutoff, the maximum level in the hierarchy is
j max = 1 + ln(ln(N min )) − ln(α ln(S 1 /S c )) ln(1 − α) ,(17)
which diverges logarithmically as N min → 1. Picking some reasonable cutoff N min ∼ 2 and assuming α = 3/8 will terminate the hierarchy fairly quickly (j max = 2 → 8) for most any conceivable Lundquist numbers (see Sec. VI).
Therefore, the scaling of the total number of islands in the hierarchy [Eq. (15)
B. Transition to multiple X-line collisionless reconnection
There is a second way to terminate the downward progression in the hierarchy before reaching the maximum level estimated by Eq. (17). As discussed in Sec. I, this occurs when the thickness of a current sheet at a level j (≥ 1), given by Eq. (11), approaches the ion log (S) sound radius to trigger collisionless reconnection:
δ j = δ j−1 N j−1 = δ j−2 N j−1 N j−2 = ... = δ 1 N j−1 N j−2 ...N 1 = δ 1 N j−1 = ρ s .(18)
Using Eq. (15), this can be expressed in terms of the critical global Lundquist number, S = S 1 , as a function of plasma size, λ, as
S = S (1−c)/2 c ( λ) 1+c ,(19)
where
c = (1 − α) j−1 2 − (1 − α) j−1
which vanishes in the limits of α = 1 or j = ∞ for 0 < α < 1. The blue line in Fig. 1 shows the transition boundary in the (λ, S) space from multiple X-line collisional phase to multiple X-line hybrid phase in these limiting cases,
S = S c λ(20)
where S c = 10 4 and = 1/2, consistent with the previous heuristic arguments 51,54 .
For more realistic scenarios with α < 1 and a finite number of levels, the appxoimate transition boundary can also be estimated. For any given S(> S c ), the deepest level, j max , in the hierarchy is given by Eq. (17). Then using this j max , the maximum λ for current sheets at the deepest level in the hiearchy to remain collisional is determined by Eq. (19). Examples for α = 3/8 and α = 0.8 are shown in Fig. 3 as a dotted line and a dashed line, respectively.
The steps in these lines correspond to the increases in j max , but overall they are only slightly above the limiting cases of c = 0. Lastly, we note that the physics across this boundary
λ = λ c ∼ 50,(21)
which is shown as the orange line in Fig. 1.
In principle, multiple X-line collisionless reconnection can also occur in the hybrid phase if the effective plasma size is larger than λ c at the hierarchy level when the current sheet thickness reaches ρ s . But it turns out that the condition for such transition is almost identical to Eq. (20) as shown below. The effective plasma size at the jth level in the hierarchy, λ j , is given by
λ j = L j ρ s = L 1 ρ s 1 N j−1 = λ S c S 1−(1−α) j−1 .(22)
Equating this to λ c yields
λ λ c = S S c 1−(1−α) j−1(23)
V. ELECTRON RUNAWAY CONDITION
In real plasmas, the whole concept of collisional resistivity (and thus Lundquist number) is restricted to parameter regimes in which the reconnection electric field is small in comparison to the Dreicer runaway limit 59 given by
E D ≡ √ m e T e ν ei e ,(24)
where ν ei is electron-ion collision frequency. To put the phase diagram into better perspectives with regard to its previous version 60 , it is important to understand where runaway conditions are unavoidable and how these boundaries correlate with kinetic-scale transitions already discussed. For a Sweet-Parker reconnecting current sheet, the reconnection electric field is given by
E R = ηj = m e ν ei e 2 n B R µ 0 δ SP(25)
where η is classical resistivity, j is the peak current density, and B R is the reconnecting field component. Using the relation for the plasma β
β = 2 ρ s d i 2(26)
where T e = T i is assumed, we have
E R E D = 2 √ 2 β B R B T m e m i ρ s δ SP .(27)
For the single X-line reconnection phase, δ SP is given by Eq. (3), and E R /E D = 1 leads to
S = β 2 128 B T B R 2 m i m e λ 2(28)
where B T is the total magnetic field. Assuming β = 0.01, B T /B R = 10, m i /m e = 1836, the above equation becomes
S = 0.14λ 2(29)
which is below the boundary line defined by Eq.(5) but within only a factor of 2 when = 1/2. Therefore, the electron runaway condition is well coincident with the boundary line between collisional and collisionless phases.
For the multiple X-line reconnection phase, we can use Eq.(18) with α = 1 or j = ∞ for simplicity, yielding
S = β 8 √ 2 B T B R m i m e S c λ.(30)
Using the same example parameters as above, we have
S = 0.38 S c λ(31)
which is again within a factor of 2 from Eq.(20) when c = 0 and = 1/2. Red lines in we have largely avoided the onset question (i.e. how reconnection gets started) which is likely very different for the various parameter regimes. In particular, the detailed properties of tearing instabilities in various regions of parameter space may play some role in the onset phase of reconnection, but this subject is beyond the focus of the present paper. As another example, the structure of the large-scale initial condition can also influence the reconnection dynamics, including features such as asymmetric current layers and velocity shear. There has been some work on both of these issues, but many uncertainties remain. Below we specifically discuss external drive dependence and three-dimensional effects, followed by the discussions on the locations of various plasmas in the phase diagram.
S λ = µ 0 ρ s V A η ,(32)
A. Dependence of external drive
One might consider that the reconnection process should depend on the external drive or on how much free energy available in the system for reconnection. This is especially relevant when reconnection is modeled by a local box around the diffusion region. In real applications, the boundary condition may significantly influence the reconnection process within. However, we point out that the definition of the Lundquist number given by Eq. (1) already takes into account of the available free energy in the system: the half length of the reconnecting current, L CS , which is taken as a fraction of the system size, L, is used. If the system is completely relaxed without free energy for reconnection, then L CS = 0 even L can be very large. Having L CS on the order of L implies that the available free energy is near its maximum. One can imagine a time evolution when the system is driven from its completely relaxed state with L CS = 0 to L CS L/2 to reach a maximum S, and the dependence on the drive is actually reflected in the magnitude of S already. If the free energy is less than its maximum in a given system, S should be less than its maximum value even L is still same.
B. Influence of realistic three-dimensional dynamics
The ideas leading to the phase diagram in Fig. 1 are largely based on two-dimensional (2D) models and simulations of reconnection. At present time, very little is known regarding how reconnection will proceed in large three-dimensional (3D) systems -either in fluid or kinetic parameter regimes. To begin with, the whole idea of magnetic islands relies upon a high degree of symmetry, which can be achieved in laboratory plasmas (and 2D simulations)
but is unlikely to occur in space and astrophysical plasmas. Instead, extended flux ropes are the natural 3D extension of magnetic islands, and the manner in which these can form and interact is much more complicated than in 2D models. This is true of both the primary flux ropes which may form due to tearing instabilities, and also secondary flux ropes which can form in the new current sheets that arise from the nonlinear evolution of the primary flux ropes. The advent of petascale computers over the last few years is permitting 3D kinetic simulations 65 to explore these ideas for guide field reconnection geometries, where tearing modes can be localized at resonant surfaces across the initial current sheet. These initial simulations together with Vlasov theory have demonstrated that the spectrum of oblique tearing modes within ion-scale layers is simpler than previously thought 66 , but the resulting flux rope dynamics is still quite rich. Furthermore, the nonlinear development of the primary flux ropes produces intense electron-scale current sheets near the active xlines and along the separatrices. As illustrated in Fig. 4, in 3D simulations these layers are unstable to the formation of secondary flux ropes over a broad range of oblique angles. The continual formation and interaction of these flux ropes gives rise to a turbulent evolution that is significantly different than 2D models. However, the full implication of these results will take years to sort out; researchers are just beginning to scratch the surface.
In addition to these fundamental issues associated with island (flux rope) formation, there are a wide range of other processes to consider in 3D which may potentially influence the dynamics of magnetic reconnection. These processes include the lower-hybrid drift instability, driven by the strong diamagnetic currents, streaming instabilities, modes driven by either electron or ion velocity shear, and a range of kinetic instabilities driven by temperature anisotropy. Even in MHD, influence of a pre-exiting turbulence on reconnection remains an outstanding issue in both 2D 50 and 3D 67 studies. Huge challenges remain in understanding the role these various process play in reconnection, and how they might change the phase diagram in Fig. 1. One of the most long-standing ideas is that instabilities may modify the dissipation physics within electron-scale regions. However, there are other possibilities to consider including non-linear couplings between electron and ion-scale features, or the possibility that these instabilities may seed the formation of new flux ropes.
C. Reconnection in heliophysical, astrophysical and laboratory plasmas
Despite these rather serious caveats discussed in the previous section, it is interesting to place plasmas from laboratory, heliophysics and astrophysics in the phase diagram. In Fig. 1, some heliophysical and laboratory example plasmas are shown. In this section, results from a more extensive survey of astrophysical plasmas are summarized in Table I and Fig. 5 with references from which typical plasma parameters were taken. In general, these parameters are associated with large uncertainties due to limited measurements available from these distant plasmas and crude models used for the estimation. Extreme astrophysical conditions 92 , such as special relativity and radiation, are not taken into account here since these effects on collisionless plasmoid instabilities is just beginning to be explored numerically 101,102 . On the log-scales as in Fig. 5, however, even an order of magnitude of the uncertainty does not change the location of these plasmas by much in the phase diagram.
The cases shown in the phase diagram can be roughly grouped into three groups. The first group includes high temperature fusion plasmas and Earth's magnetospheric plasmas. These plasmas are completely in the collisionless phases, either with a single X-line or multiple Xlines, depending on whether the plasma effective sizes, λ, are larger than the critical λ c . The plasma for the Sgr A* flares may also belong to this group.
The second group of plasmas cluster along the black line separating multiple X-line collisionless phases and multiple X-line hybrid phase. It spans over huge ranges from solar Therefore, the self regulation arguments for the collisionality mentioned above do not seem to hold at the large S and λ phases, but much work still remains to be done. The accumulation of energetic particle populations is suggested 57 as another player in the self regulation process of reconnection rate in the multiple X-line collisionless phase. Energetic particle populations should be regulated also by finite collisions in the hybrid phase, but detailed dynamical processes need to be investigated.
The third and final group of plasmas shown in Fig. 5 occupy much of the multiple Xline collisional phase: accretion disk interiors, solar chromosphere and tachocline, molecular clouds, gamma ray bursts and magnetar flares. It could be argued that they form a line slightly below but along the boundary (blue line) between the multiple X-line collisional and hybrid phases. It is conceivable that the self-regulation arguments for collisionality 103,104 could be applied here since collisional reconnection dominates at the deepest level of the hierarchy on the one side of the boundary while collisionless reconnection dominates on the other side. In fact, it has been suggested through Hall MHD simulations 48 that reconnection in the multiple X-line collisional phase is much slower than that in the hybrid phase although it is much faster than the single X-line collisional (Sweet-Parker) rate. However, the reconnection rate is not so different: collisionless rates are around 0.1 and while the collisional rates are around 1/ √ S c ∼ 0.01 at the deepest level of the hierarchy. A key question here is what determines the overall reconnection rate in a hierarchy of islands and whether it is indeed dominated by the reconnection process at the deepest level 49 or by the reconnection process at all levels in an integrated way.
There are two special cases which do not belong to either of the above three groups: protostellar disks and interstellar media. Protostellar disks have lowest S among the objects we surveyed and are located between the single X-line and multiple X-line collisional phase.
Interstellar media are right in the middle of the hybrid phase, and probably both collisional MHD physics and collisionless physics are important in charactering reconnection processes there as a part of the galactic dynamo.
Lastly, we note that currently there are no laboratory experiments which can be used to study all of these new phases of magnetic reconnection. Laboratory experiments have been playing important roles in the reconnection research: confirming some leading theoretical or numerical models such as Sweet-Parker 18 and collisionless reconnection models 105 while challenging others such as Petschek model; benchmarking state-of-the-art numerical simulations 63,106,107 ; discovering 3D phenomena [108][109][110][111] ; studying flux rope dynamics 71,72 , to name a few. As mentioned above, the main research tool on physics of new reconnection phases is numerical simulations using either full particle, Hall MHD, or resistivity MHD codes.
Existing experiments, such as MRX, do not have accesses to these new phases which are important for the emerging themes of particle acceleration by magnetic reconnection 56,57 .
While numerical simulations, coupled closely with analytic theory, will continue to be a major player at this front, a next generation reconnection experiment (NGRX) based on the MRX concept is considered as a candidate for such a laboratory experiment 80 . The parameter ranges for both MRX and NGRX are also indicated in the phase diagram for their coverages.
as a starting point of the discussion for magnetic reconnection. In Eq.(1), L CS is the half length of the reconnecting current sheet, and can be taken as L CS = L where L is the plasma size and 0 ≤ ≤ 1/2 (the choices of are discussed in Sec.VI.A.). V A is the Alfvén velocity based on the reconnecting magnetic field component and η is the plasma resistivity due to Coulomb collisions. The well-known Sweet-Parker model 1,2 predicts reconnection rates as an explicit function of S,
rated by magnetic islands (or plasmoids)16,[26][27][28][29][30] . These multiple-X line models are inherently time dependent, often generating impulsive reconnection consistent with observations such as Flux Transfer Events (FTE)31 . The plasmoid-like structures are also observed in Earth's distant magnetotail during substorms 32 and in the current sheet during solar Coronal Mass Ejections (CME)33 . Although the multiple-X line models were also applied to explain these observed plasmoids in the magnetotail[34][35][36] or on the solar surface 37-39 , they did not receive much attention until recent theory40 and numerical simulations 41-47 offered detailed predictions concerning the break-up of Sweet-Parker layers to the so-called plasmoid instability, which produces numerous secondary magnetic islands. Although the time-averaged rates can be still different 45,48,49 depending on the detailed divisions within (see Sec. III below), all of them are definitely much faster than the Sweet-Parker rate [Eq.(2)]. Thus, the multiple X-line reconnection, associated with the plasmoid instability, constitutes a new reconnection phase within the collisional reconnection regime. Recent MHD simulations indicate that the critical Lundquist number, S c , for the onset of the plasmoid instability is approximately
III. MULTIPLE X-LINE COLLISIONAL AND COLLISIONLESS RECONNEC-TIONUntil quite recently, the boundary between collisional and collisionless reconnection was thought to be given by Eq.(5). This may not be true anymore when the current sheet is unstable to the plasmoid instability, forming thinner current sheets which may be further subject to new plasmoid instability leading to yet thinner current sheets in a hierarchical fashion as proposed by Shibata and Tanuma 39 . Eventually, these new current sheets can approach the ion kinetic scales triggering collisionless reconnection as recently demonstrated by full kinetic simulations with a Fokker-Planck treatment of Coulomb collisions 44 . Therefore, the multiple X-line phase can be further divided into a phase involving only collisional physics (i.e. purely resistive MHD) and a phase involving both collisional and collisionless
where α = 3/ 8 .
8This linear prediction has been carefully verified in simulations designed to study the initial breakup of the Sweet-Parker layers43,51 . However, nonlinearly many more islands are observed in the simulations44,45,51,53 corresponding to scaling parameters in the range α = 0.6 → 1. A possible explanation for this discrepancy is that, in these nonlinear simulations, the break-up of the original Sweet-Parker layer leads to new current sheets between the islands which are also unstable to the same plasmoid instability as illustrated inFig. 2, and the islands on more than one level in the hierarchy were counted (see below for more discussions).At the present time, it appears that there are only two ways to terminate the downward progression in this hierarchy: (1) either the local Lundquist number of the new current sheets falls below the critical value for the plasmoid instability or (2) the new current sheets approach the ion ρ s scale where collisionless effects dominate. One can make quantitative predictions regarding these two possible outcomes with just a few simple assumptions.A. Multiple X-line collisional reconnection As illustrated in Fig. 2, we start by defining the half-length and thickness of the top level Sweet-Parker by L 1 ≡ L CS and δ 1 ≡ δ SP [Eq.(3)], corresponding to a macroscopic Lundquist number of S 1 ≡ S [Eq.(1)]: S 1 = (L 1 /δ 1 ) 2 . At this top level, the development of the plasmoid instability gives rise to N 1 = (S 1 /S c ) α islands, which breaks the original layer into new sheets with length L 2 = L 1 /N 1 . We assume these new layers are governed by the Sweet-Parker scaling relationships and are also susceptible to the plasmoid instability in the same manner. Then, the number of islands generated within the second level of the
] depends on the maximum number of levels, j max [Eq.(17)], and the island number scaling power index from one level to the next level, α. Applying this estimate to the reported numerical MHD simulations 51 yields j max ∼ 3, using α = 3/8. This leads to the predicted scaling ofN jmax ∼ (S 1 /S c ) 0.76 . However, the linear scaling of α = 3/8 does not necessarily apply in the hierarchy model where the nonlinear evolution of islands at one level is required to generate new current sheets for the islands at the next level. Using α = 0.8, for example, leads to j max ∼ 2 and N jmax ∼ (S 1 /S c ) 0.96 . These scalings are not very far from the reported linear scaling of S ∼1 given the large uncertainties that still exist (see Fig. 5 in Ref. 51). As the hiearchy becomes increasingly deep, the precise value of α no longer matters and the result approaches the linear scaling of S as evident from Eq.(15), consistent with earlier heuristic arguments 44,49,51 .
FIG. 3 :
3The phase diagram in a smaller parameter space to show dotted and dashed lines better (see texts in Sec. III). Other symbols are same as inFig.1.
is vastly different: on the collisional side the reconnection is completely determined by collisional MHD physics while on the hybrid side, both collisional and collsionless physics is important. This is consistent with the boundary determined by electron runaway conditions which also indicate the change in the required physics (see Sec.V later). The reconnection rate, on the other hand, is given by the Sweet-Parker rate of S on the hybrid side, the reconnection is faster, but not by a large amount, at the collisionless rate of 0.01 − 0.1.IV. SINGLE AND MULTIPLE X-LINE COLLISIONLESS RECONNECTIONThe last part in our phase diagram concerns the fact that the single X-line current sheet in the collisionless phase [defined by Eq.(5)] may be also subject to secondary collisionless tearing instability. Unlike the MHD counterpart, we are not aware of any analytic work in this area although there have been numerical demonstrations[55][56][57] and some observational evidence 58 . In the collisionless limit, sufficiently large kinetic simulations suggest 55 that the critical size for the secondary island formation in the extended current sheet as a result of nonlinear evolution is,
which reduces to S = (S c /λ c )λ in the limiting cases of α = 1 or j = ∞. This condition is different from Eq. (20) by only a factor of 2 when S c = 10 4 and λ c = 50. Thus, the parameter space for the multiple X-line collisionless reconnection is very limited within the hybrid phase.
Fig. 1
1illustrate the boundaries for electron runaway conditions, which separate collisional and collisionless reconnection phases for both single X-line and multiple X-line geometries.The significance of the red lines in the phase diagram is that they separate the regime where reconnection can be described by collisional physics alone from the regime where collisionless physics is required. The alignment of red lines with either the black line or blue line is consistent with the transitions from collisional reconnection to collisionless reconnection, regardless whether it takes the form of the single X-line or multiple X-lines.Besides MHD models and fully kinetic models, Hall MHD models and hybrid models (fluid electrons and kinetic ions) are often used to study the transitions between collisional and collisionless phases3,6,7,9,10,12,13,48,54,61 . As demonstrated by the comparative studies between different models 3 , Hall MHD models and hybrid models can capture the qualitative boundaries between these two phases. However, the coincidence of electron runaway conditions with the transition boundaries between collisional reconnection and collisionless reconnection raises questions on the suitability of these fluid models when they are used to study detailed dynamics near the transitions 7,13,48,54,61 . The detailed electron kinetic dynamics become important in these regimes but are not yet accurately treated in fluid models. In particular, the transition into the multiple X-line hybrid phase unavoidably leads to runaway electric fields (E > E D ) as illustrated by the red line inFig. 1. Fully kinetic simulationsincluding the collision operator have demonstrated 62,63 that the mechanism breaking the frozen-in condition changes rapidly across this transition, from ordinary resistivity in the sub-Dreicer collisional limit (E E D ) to off-diagonal terms in the electron pressure tensor for the runaway regime. Once this transition to runway electric fields has occurred, it is unlikely to be reversed as suggested by Hall MHD models 7,54 until fast reconnection has depleted the available flux. Indeed, large-scale collisional kinetic simulations44 have demonstrated that resulting electron layers in this runaway regime can become highly extended and are unstable to secondary magnetic islands in a manner similar to previous collisionless simulations55,64 . Further insights emerge when we divide S by λ, yielding
which is simply the Lundquist number based on ρ s . It has been suggested that there exists a critical value of S/λ ∼ 50 where the dynamics can revert from Hall dynamics (kinetic) back into the Sweet-Parker regime 7,54 . However, notice that Eq.(31) implies that electron runaway will occur for S/λ > 40 beyond which simple resistivity models are known to break down. It remains an outstanding challenge to properly model this dynamics within two-fluid approaches, but comparative studies between these different models should be useful to provide guidance on reliable two-fluid models which can be practically used for the detailed investigations of the phase diagram at large S and λ values.VI. DISCUSSIONWhile the simple S − λ diagram conveniently summarizes much of the present knowledge regarding the dynamics of magnetic reconnection, there are a variety of other factors that may significantly influence reconnection which have not yet been discussed. For example,
FIG. 4 :
4Results from a kinetic simulation 65 of guide field reconnection showing the formation and interaction of flux ropes as illustrated by an isosurface of the particle density colored by the magnitude of the current density along with sample magnetic field lines (yellow). Simulation parameters are m i /m e = 100 with the initial guide field equal to the reconnecting field. The domain size is 70d i × 70d i × 35d i corresponding to 2048 × 2048 × 1024 cells and ∼ 10 12 particles.
3
, B R = 0.1B T , (3) equal electron and ion temperatures, T e = T i , and (4) ions are protons. We note that there are opinions that the plasmas in Crab pulsar wind and radio lobes are nonthermal so that temperature may not be a good description68,69 . There are some laboratory experiments which are not listed: flux rope experiments[70][71][72] with S ∼ λ ∼ 10 1 , and plasma merging experiments 73,74 with S = 10 2 − 10 3 and λ = 10 1 − 10 2 .location plasma size(m) T e (eV) n e (m −3 ) B T (Telsa) 10 1 1 × 10 1 Al +13 , B R = B T 10 6 6.2 × 10 1 T i = 350eV, D + , B R = 0.05B T 10 8 2.3 × 10 2 T i = 36keV, D + , B R = 0.01B T ITER 79 4 2 × 10 4 1 × 10 20 5.3 6 × 10 8 5 × 10 2 D + , B R = 0.01B T 10 5 1.2 × 10 3 = 1/4, T i = T e /2, B R = 0.2B T Magnetopause 81 6 × 10 7 300 1 × 10 7 5 × 10 −8 6 × 10 13 9 × 10 2 B R = B T , (p.267) Magnetotail 81 6 × 10 8 600 3 × 10 5 2 × 10 −8 4 × 10 15 1.3 × 10 3 B R = B T , T i = 4.2keV, (p.233) Solar Solar Wind 81 2 × 10 10 10 7 × 10 6 7 × 10 −9 3 × 10 12 2 × 10 5 (p.92) System Solar Corona 81 1 × 10 7 200 1 × 10 15 2 × 10 −2 1 × 10 13 4 × 10 7 (p.79) Solar Chromosphere 82 1 × 10 7 0.5 1 × 10 17 2 × 10 −2 1 × 10 8 3 × 10 8 neutral particle effects are weak 82 Solar Tachocline 83,84 1 × 10 7 200 1 × 10 29 1 1 × 10 9 5 × 10 10 Protostellar Disks 85 9 × 10 9 3 × 10 −2 6 × 10 8 2 × 10 −5 8 × 10 3 1 × 10 9 L = 2h(R=1AU), e-n collisions included 82 , Mg + X-ray Binary Disks 86,87 4 × 10 4 75 1 × 10 27 36 3 × 10 7 9 × 10 8 M = 10M , L = 2h(R = 10 2 R S ), × 10 4 5 × 10 5 1 × 10 24 1 × 10 4 1 × 10 16 9 × 10 7 M = 10M , R = R S , T i = (m p /m e )T e , η Compton included 88 Galaxy Crab Nebula Flares 89-91 1 × 10 14 130 10 6 10 −7 5 × 10 20 2 × 10 11 pair plasma, T from B 2 R /2µ 0 = 2nT Gamma Ray Bursts 92 10 4 3 × 10 5 2 × 10 35 4 × 10 9 6 × 10 17 2 × 10 16 pair plasma Magnetar Flares 92,93 10 4 5 × 10 5 10 41 2 × 10 11 6 × 10 16 5 × 10 17 pair plasma, SGR 1806-20 Sgr A* Flares 94,95 2 × 10 11 7 × 10 6 10 13 10 −3 2 × 10 24 5 × 10 8 L = 2R = 20R S Molecular Clouds 96,97 3 × 10 16 10 −3 10 9 2 × 10 −9 1 × 10 11 7 × 10 12 neutral particle effects included 82 , HCO + Interstellar Media 96,97 5 × 10 19 1 10 5 5 × 10 −10 2 × 10 20 1 × 10 14 L=magnetic field scale height AGN Disks 86,87,98 2 × 10 11 24 8 × 10 23 0.5 2 × 10 13 1 × 10 14 M = 10 8 M , L = 2h(R = 10 2 R S ), α = 10 −2 ,Ṁ = 10 26 g/s Extra-AGN Disk Coronae 88 3 × 10 11 5 × 10 5 1 × 10 17 4 10 23 3 × 10 11 M = 10 8 M , R = R S , T i = (m p /m e )T e , η Compton included 88 galactic Radio Lobes 69 3 × 10 19 100 1 5 × 10 −10 2 × 10 25 8 × 10 12 Extragalactic Jets 99 3 × 10 19 10 4 3 × 10 1 10 −7 6 × 10 29 1 × 10 14 3C 303 Galaxy Clusters 100 6 × 10 18 5 × 10 3 4 × 10 4 2 × 10 −9 2 × 10 25 6 × 10 11 A1835
ion sound gyroradius, T e and T i are electron and ion temperatures, B T is the total magnetic field including both the reconnecting and guide components, and m i and q i are the ion mass and charge. In the case of anti-parallel reconnection with upstream plasma β up 1, ρ s will be equal to d i by the virtue of the force balance across the current sheet, if the reconnecting magnetic field and the temperatures at the current sheet center are used to calculate ρ s . (We note that when β up 1, the transition scale for δ SP is less clear since ρ s ( β up d i ) is separated from d i .) Therefore, the boundary between collisional and collisionless reconnection is defined by Eq.(5) regardless of the presence of a guide field when the definition of plasma effective size is modified to
assuming = 1/2. We note that the term "collisionless reconnection" is used in this paper for the reconnection process dominated by the effects beyond collisional MHD, such as two-fluid effects, ion and electron kinetic effects. Among these, the electron kinetic effects should become important in a similar parameter space defined by the black line as shall be discussed in Sec.V.II. SINGLE AND MULTIPLE X-LINE COLLISIONAL RECONNECTIONFor plasmas larger than those specified by Eq.(5), it would appear that the collisional MHD description might be valid despite the large S. It shall become clear later in Sec.III.B., however, collisional models are not sufficient for describing reconnection in these regimes.FIG. 1: A phase diagram for magnetic reconnection in two dimensions. If either S or the normalized size, λ, is small, reconnection with a single X-line occurs in collisional or in collisionless phases. When both S and λ are sufficiently large, three new multiple X-line phases appear with magnetic islands. The dynamics of new current sheets between these islands are determined either by collisional physics or by collisionless physics (see Sec.III and Sec.IV.). The conditions for electron runaway are shown as red lines (see Sec. V). The locations for reconnection in Earth's magnetosphere, solar corona, solar chromosphere, and solar tachocline are also shown. The existing experiments, such as Magnetic Reconnection Experiment (MRX), do not have accesses to these new phases. A next generation reconnection experiment (NGRX) is required for such accesses to these new phases directly relevant to reconnection in heliophysical and astrophysical plasmas. Sweet-Parker model 1,2 and the Petschek model 15 , both of which, however, are unsatisfactory. The Sweet-Parker model has been verified numerically 16,17 and experimentally 18 at relatively small values of S, but it predicts reconnection rates too slow to be consistent with observations of larger S plasmas. On the other hand, the Petschek model, invokingDiscussions on collisional reconnection have been long dominated by debates between the
log (S)
log (λ)
0
5
10
15
0
2
4
6
8
12
10
NGRX NGRX
MRX MRX
Magnetosphere Magnetosphere
Solar Solar
Corona Corona
Solar Solar
Tachocline Tachocline
Solar Solar
Chromosphere Chromosphere
Single
X-line
collisionless
Multiple X-line
collisionless
M
u l t i p l e X -l i n e
h y b r i d
Multiple X-line
collisional
Single X-line collisional
S=S c
S=
λ
=
λ
√S
2
2
c
S=λ /4
2
λ=λ c
slow-mode shocks, predicts rates consistent with observations but it requires a localized re-
sistivity enhancement in simulations 19-22 and has not yet been verified experimentally. The
origin of the localized resistivity enhancement is hypothesized to be kinetic in nature, but
the underlying mechanisms still remain illusive. While signatures of slow-mode shocks have
been reported in the Earth's distant magnetotail 23 , large-scale hybrid (kinetic ions and fluid
TABLE
Laser LaserFIG. 5: Various laboratory, heliophysical and astrophysical plasmas, in which magnetic reconnection is believed to occur, are shown in the phase diagram. Other symbols are same as in Fig.1. See the text andTable I for details.corona, accretion disk coronae, Crab nebula flare, to galaxy clusters, radio lobes, and extragalactic jets. When S and λ are both small, this same line separates single X-line collisional phase and collisionless phase. With a single X line, the reconnection in the collisional phase was known to be much slower than in the collisionless phase. The plasma collisionality was argued 103 to regulate itself so that the plasma always stay near the marginal collisionality, based on the reasoning that fast reconnection should effectively release magnetic energy evaporating nearby dense neutral gases (such as in the solar chromosphere) to increase density and collisionality until reconnection slows to a collisional rate. An alternative model was also proposed 104 based on self-regulation of electron temperature to maintain marginal collisionality through a similar but different reasoning: higher temperature lowers collisionality and fastens reconnection, and thus depletes quickly available magnetic energy and eventually slows reconnection and cools the plasma while lower temperature increases collisionality and slows reconnection, and thus accumulate magnetic energy and eventually trigger faster reconnection and heat the plasma.However, at large S and λ values for all plasmas in the second group, the marginality black line now separates multiple X-line collisionless phases and multiple X-line hybrid phase in the phase diagram. Now there is numerical evidence 44,48 that the reconnection rates in multiple X-line hybrid phase are as fast as the single X-line collisionless rate, consistent with the theoretical argument 49 that the global reconnection rate is determined by a dominant reconnection site in the island hierarchy which should be collisionless in the hybrid phase.TFTR TFTR
ITER ITER
Magnetotail Magnetotail
Magnetopause Magnetopause
Solar Solar
Wind Wind
Solar Solar
Corona Corona
X-ray Binary
X-ray Binary
Disk Corona
Disk Corona
Crab Nebula
Crab Nebula
Flare Flare
Sgr A*
Sgr A*
Flare Flare
AGN Disk
AGN Disk
Corona Corona
Galaxy Galaxy
Cluster Cluster
Radio Radio
Lobe Lobe
Extragalactic Extragalactic
Jet Jet
Interstellar Interstellar
Medium Medium
γ-ray Burst
-ray Burst
Magnetar Magnetar
AGN AGN
Disk Disk
Molecular Molecular
Cloud Cloud
Solar Solar
Tachocline Tachocline
Protostellar Protostellar
Disk Disk
Solar Chromosphere
Solar Chromosphere
X-ray Binary Disk
X-ray Binary Disk
Multiple X-line
collisionless
Single X-line collisional
Multiple X-line
collisional
M
u l t i p l e X -l i n e
h y b r i d
Single
X-line
collisionless
λ=λ c
S=λ /4
2
S=
λ
=
λ
√S
2 2
c
S=S c
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[] |
[
"Algebraic Connectivity Under Site Percolation in Finite Weighted Graphs",
"Algebraic Connectivity Under Site Percolation in Finite Weighted Graphs"
] |
[
"Sohail Bahmani ",
"Senior Member, IEEEJustin Romberg ",
"Prasad Tetali "
] |
[] |
[] |
We study the behavior of algebraic connectivity in a weighted graph that is subject to site percolation, random deletion of the vertices. Using a refined concentration inequality for random matrices we show in our main theorem that the (augmented) Laplacian of the percolated graph concentrates around its expectation. This concentration bound then provides a lower bound on the algebraic connectivity of the percolated graph. As a special case for (n, d, λ)-graphs (i.e., d-regular graphs on n vertices with non-trivial eigenvalues less than λ in magnitude) our result shows that, with high probability, the graph remains connected under a homogeneous site percolation with survival probability p ≥ 1 − C1n −C 2 /d with C1 and C2 depending only on λ/d.Index Termssite percolation, algebraic connectivity, matrix concentration inequalities ! • S. Bahmani and J. Romberg are with the
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10.1109/tnse.2017.2757762
|
[
"https://arxiv.org/pdf/1612.05986v2.pdf"
] | 80,241 |
1612.05986
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cedb7cef73f023a6d23b01511f223b6007f527cc
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Algebraic Connectivity Under Site Percolation in Finite Weighted Graphs
Sohail Bahmani
Senior Member, IEEEJustin Romberg
Prasad Tetali
Algebraic Connectivity Under Site Percolation in Finite Weighted Graphs
1Index Terms site percolationalgebraic connectivitymatrix concentration inequalities !
We study the behavior of algebraic connectivity in a weighted graph that is subject to site percolation, random deletion of the vertices. Using a refined concentration inequality for random matrices we show in our main theorem that the (augmented) Laplacian of the percolated graph concentrates around its expectation. This concentration bound then provides a lower bound on the algebraic connectivity of the percolated graph. As a special case for (n, d, λ)-graphs (i.e., d-regular graphs on n vertices with non-trivial eigenvalues less than λ in magnitude) our result shows that, with high probability, the graph remains connected under a homogeneous site percolation with survival probability p ≥ 1 − C1n −C 2 /d with C1 and C2 depending only on λ/d.Index Termssite percolation, algebraic connectivity, matrix concentration inequalities ! • S. Bahmani and J. Romberg are with the
Introduction
C onsider a connected weighted graph G = (V = [n] , E) with (non-negative) edge weights {w i,j } 1≤i,j≤n and no self-loop (i.e., w i,i = 0 for all 1 ≤ i ≤ n) and suppose that each vertex i of G is deleted independently with probability 1 − p i . These types of random graph models can describe certain phenomena in random media and are studied under percolation theory [6] in mathematics and statistical physics. The process of vertex deletion, as described above, is usually referred to as site percolation whereas bond percolation refers to the process of random deletion (or addition) of the edges of a graph. In this paper we establish a lower bound on algebraic connectivity of the surviving subgraph in the described site percolation model. The algebraic connectivity of a graph G is a = λ 2 (L), the second smallest eigenvalue of the graph Laplacian
L def = 1≤i<j≤n w i,j (e i − e j )(e i − e j ) T ,
where e i 's are canonical basis vectors. Algebraic connectivity and its analog for normalized Laplacians are important because they provide a bound on isoperimetric constants of graphs through Cheeger's inequality [see e.g., 13] and they are critical in approximation of the mixing rate of continuous-time Markov chains [12;15,Ch. 20].
Properties such as connectivity, spectral gap, and emergence of a giant component (i.e., a connected component with Ω(n) vertices) have received more attention and are better understood for bond percolation models compared to site percolation models. Perhaps, the main reason is that in bond percolation edges are removed independently whereas in site percolation edge deletions are dependent since they share a common vertex which lead to more intricate behavior.
In this paper we focus on algebraic connectivity of the surviving subgraph of a weighted graph under (inhomogeneous) site percolation. Using a delicate matrix concentration inequality (Proposition 6), in our main result (Theorem 1) we show that the "augmented" Laplacian concentrates around its expectation. This result allows us to find a non-trivial lower bound on the algebraic connectivity in a straightforward way. For concreteness, we also apply the general result of the Theorem 1 to obtain a threshold for connectivity in the special case of (n, d, λ)-graphs under uniform site percolation. In particular, Corollary 3 below shows that if the vertices of an (n, d, λ)-graph are removed independently with probability 1 − p then, with high probability, the surviving graph is connected if
p ≥ 1 − n − O( 1 d )
with the hidden constants depending only on λ d .
Related work
In [3] the bond percolation model with a uniform edge survival probability p is studied. With d i denoting the degree of vertex i, it is shown in [3] that asymptotically almost surely a giant component survives
(or not) if p > (1 + ) i d i i d 2 i (or p < (1 − ) i d i i d 2 i
). Furthermore, the spectral gap under bond percolation is studied in [2] and [14], where the latter established a sharper bound by means of concentration inequalities for random matrices.
A more relevant problem to our work is the problem of network (un)reliability [4] where the goal is to estimate the probability that a percolated graph remains connected. Under the bond percolation model, [8] proposes a method to approximate the network reliability through a fully polynomial-time approximation scheme. Approximation algorithms for the same problem with better computational complexity were proposed later in [7] and [9]. The site percolation model for random d-regular graphs is analyzed in [5]. Specifically, [5] shows that, with high probability, for vertex deletion probability of the form n −γ , the surviving subgraph has a giant component of order n − o(n) that is an expander graph and, if γ ≥ 1 d−1 , then it is connected as well. This result was later improved and generalized in [1]. Recall that an (n, d, λ)-graph is a d-regular graph of order n with the non-trivial eigenvalue less than λ in magnitude. A phase transition for site percolation on such (n, d, λ)-graphs is established in [11]. In particular, the mentioned paper shows that if the vertex survival probability is p = 1− d , then with high probability, the connected components of the surviving subgraph have O (log n) vertices; whereas if p = 1+ d , d = o(n), and λ d is relatively small, then with high probability a giant component with Ω( n d ) vertices survives. Our main result, Theorem 1, relies on a refined concentration inequality stated in Proposition 6 for random Bernoulli matrices and, consequently, is distinct from most of the previous work mentioned above which rely on combinatorial arguments. We also apply our general result to the special case of (n, d, λ)-graphs (Corollary 3), and reproduce bounds comparable to those established in [1; 5]. In particular, [1,Proposition 3.5] shows that any (n, d, λ)-graph G with d ≥ 3 and λ ≤ 2
√ d − 1 + 1 40 , that is also "locally sparse" in the sense that max H⊆G , |V (H)|≤d+29 |E(H)| |V (H)| ≤ 1 ,
with high probability, remains connected under a homogeneous site percolation with survival probability
p > 1 − n − 1 d .
Similarly, our result in Corollary 3, guarantees that with probability ≥ 1 − 4 n , any (n, d, λ)-graph remains connected under a homogeneous site percolation with survival probability p ≥ 1 − C 1 n − 2C 2 d . The constants C 1 and C 2 depend only on λ d ; their exact form is provided in the proof of Corollary 3. If we have λ d = 1 − for some ∈ (0, 1), then − log C 1 = O (1 + 2 ) 2 and C 2 = O −4 both of which are decreasing in . These quantities can be fairly large for small values of , which implies that our required lower bound on p would be stricter than that of [1]. However, our analysis does not explicitly assume a bound on λ or local sparsity as in [1]. The fact that Corollary 3 leads to suboptimal constants compared to [1, Proposition 3.5] is not surprising; Corollary 3 is an application of a very general bound established in Theorem 1 to the case of (n, d, λ)-graphs.
Future directions
There are natural extensions to the connectivity problem studied in this paper that we would like to study through the lens of random matrix theory as done here. For example, an immediate question is to find a bound on the size of the giant component of the site-percolated graph. Furthermore, an interesting research direction is to study other properties of the site-percolated random graphs such as their clique number, chromatic number, etc by means of random matrix theory. While the best results might still be obtained through specifically tailored combinatorial arguments, we believe that the analysis based on algebraic methods and random matrix theory would be more robust to model errors.
Problem Setup
For 1 ≤ i ≤ n, let δ i ∼ Bernoulli(p i ) be the independent random variables that indicate whether or not the corresponding vertex survives. In order to operate on a Laplacian with fixed dimensions we interpret site percolation as removing every edge connected to the affected vertices. The Laplacian of the remaining graph G δ is then given by
L δ = 1≤i<j≤n δ i δ j w i,j (e i − e j )(e i − e j ) T ,
which also includes the vertices affected by the site percolation as isolated vertices. We need to take into account the effect of these "ghost vertices" to find a non-trivial bound for the desired algebraic connectivity which we denote by a δ . To this end, for a coefficient α ≥ 0, we introduce the augmented Laplacian given by
L δ = L δ + 1≤i≤n α (1 − δ i ) e i e T i .(1)
The Laplacian L δ and the diagonal matrix (1) are supported on the vertices that survived and the ghost vertices, respectively. Because these two vertex sets are disjoint, the eigenvalues and eigenvectors of the corresponding terms on the right-hand side of (1) constitute the eigendecomposition of L δ . We either have a δ > α or a δ ≤ α. If the latter holds, then a δ would coincide with the second smallest eigenvalue of L δ and by Weyl's eigenvalues inequality we obtain
1≤i≤n α (1 − δ i ) e i e T i ina δ = λ 2 L δ ≥ λ 2 EL δ − L δ − EL δ .
An immediate implication is that
a δ ≥ min λ 2 EL δ − L δ − EL δ , α ,(2)
holds for all α ≥ 0. Hence, we can obtain a non-trivial lower bound for a δ by studying the tail behavior of L δ − EL δ which also depends on α. The lower bound given by (2) can also be optimized with respect to α.
Main Result
Our main theorem below provides an upper bound for L δ − EL δ . To state the theorem it is necessary to introduce some notation. For each 1 ≤ i ≤ n, let
K i = 1 2 1 − 2p i log 1−p i p i(3)
denote the sub-Gaussian parameter of δ i − p i as used in the Kearns-Saul inequality (Lemma 4 below).
Compared to the bounds on the moment generating function used in the Hoeffding and the Bernstein inequalities, the parameter (3) yields tighter bounds, particularly, if p i is close to 0 or 1. This property is crucial in our analysis as non-trivial events occur if the vertex survival probabilities (i.e., p i s) are relatively close to 1. We use p = p 1 p 2 · · · p n T to denote the vector of survival probabilities and a i to denote the ith column of the adjacency matrix A. The diagonal matrix whose diagonal entries are given by a vector u is denoted by D u . The binary operation • denotes the entrywise (or Hadamard) product.
Theorem 1.
Let δ i ∼ Bernoulli(p i ) be independent random variables for 1 ≤ i ≤ n. Furthermore, with K i given by (3) define
σ 2 = i K 2 i (1 − 2p i ) 2 (a i a T i ) 2 , K = max i j w 2 i,j K 2 j 1 2 .
Then for any ε ∈ (0, 1), with probability ≥ 1 − ε we have To evaluate the bound produced using (2) and Theorem 1, we apply the result to two special problems with (n, d, λ)-graphs. First we recall the definition of these graphs. Definition 2 ((n, d, λ)-graphs). An (n, d, λ)-graph is a d-regular graph with n vertices whose adjacency matrix has no non-trivial eigenvalue with magnitude greater than λ.
L δ − EL δ ≤ 2K log 4n ε + max i α − j p j w i,j + D
Below, we assume that the vertex deletion probabilities are identical, i.e., p 1 = p 2 = . . . = p n = p. This assumption also implies that
K 1 = K 2 = . . . = K n = K = 1 2 1−2p log 1−p p
. Also we assume all the edge weights w i,j are {0, 1}-valued and effectively indicate existence of an edge in G. We need to quantify or bound λ 2 EL δ as well as the parameters σ and K. Using Theorem 1, the following corollary basically
shows that p = 1 − 4n ε −O( 1 d )
, could suffice for any (n, d, λ)-graph affected by the prescribed site percolation to remain connected with probability ≥ 1 − ε. Proof: With A and L denoting the adjacency and Laplacian matrices of the (n, d, λ)-graph G, the expected value of the augmented Laplacian under the considered site percolation would be
EL δ = p 2 L + α (1 − p) I = p 2 d + α (1 − p) I − p 2 A.
Let α = pd. It follows from the definition of the graph and the equation above that
λ 2 EL δ ≥ p 2 (d − λ) + p(1 − p)d = pd − p 2 λ .(4)
Furthermore, the parameters σ 2 and K can be expressed as
σ 2 = K 2 (1 − 2p) 2 i (a i a T i ) 2 and K = K √ d. = K 2 (1 − 2p) 2 d A 2 = K 2 (1 − 2p) 2 d 3 Finally, we have D 1 2 p•(1−p) AD 1 2 p•(1−p) = p (1 − p) A = p (1 − p) d, D p AD 1 2 p•(1−p) = p 3 2 (1 − p) 1 2 d, and max i α − j p j w i,j = 0.
We can now invoke Theorem 1 and apply the above bounds to obtain
L δ − EL δ ≤ 2K d log 4n ε + p (1 − p) d + 2p 3 2 (1 − p) 1 2 d + 9 2 K |1 − 2p|d 3 4 log 4n ε 1 4 .
Given the inequalities (2), (4), and the assumption that α = pd, we are naturally interested in values of p for which the right-hand side of the inequality above is strictly smaller than pd − p 2 λ. Specifically, we would like to find p for which we have
pd − p 2 λ > 2K d log 4n ε + p (1 − p) d + 2p 3 2 (1 − p) 1 2 d + 9 2 K |1 − 2p|d 3 4 log 4n ε 1 4 , or equivalently (1 − λ d )p > 2 log 4n ε d · K 2 1 2 + 2 p(1 − p) + 9 2p |1 − 2p| log 4n ε d · K 2 1 4 .(5)
For p ≥ 1 2 we have K ≤ . Therefore, if we parameterize p by β ≥ 1 as p = 1 − e −β 4 we
have K 2 ≤ β −4 4 . Furthermore, we can write p ≥ 1 1+β −4 ≥ 1 − β −2 , 2 p(1 − p) ≤ 2e −β 4 /2 ≤ 2β −2 , and 1 2p |1 − 2p| ≤ 1.
Therefore, to guarantee (5) it suffices to have
(1 − λ d )(1 − β −2 ) > 2 β −4 log 4n ε 4d 1 2 + 2β −2 + 9 β −4 log 4n ε 4d 1 4 = log 4n ε d 1 2 + 2 β −2 + 9 √ 2 log 4n ε d 1 4 β −1 ,
which is equivalent to
1 − λ d > log 4n ε d 1 2 + 3 − λ d β −2 + 9 √ 2 log 4n ε d 1 4 β −1 .
The inequality above holds for
β 2 > max 2 1 − λ d −1 log 4n ε d 1 2 + 3 − λ d , 81 1 − λ d −2 log 4n ε d 1 2
which, for the sake of simpler expressions, can be further relaxed to
β 4 ≥ 81 2 1 − λ d −4 log 4n ε d + 8 1 − λ d −2 3 − λ d 2 .
The desired results follows immediately by setting
C 1 = exp −8 1 − λ d −2 3 − λ d 2 and C 2 = 81 2 1 − λ d −4 .
Proof of Theorem 1
In this section we prove our main result. The lemmas and other technical tools we use are summarized below in Appendix A. Proof of Theorem 1: Splitting L δ − EL δ into the sum of diagonal and off-diagonal terms as
L δ − EL δ = i δ i ( j δ j w i,j ) − p i ( j p j w i,j ) − α(δ i − p i ) e i e T i + i<j (δ i δ j − p i p j )(e i e T j + e j e T i )
and applying triangle inequality yields
L δ − EL δ ≤ max i δ i ( j δ j w i,j ) − p i ( j p j w i,j ) − α(δ i − p i ) + i<j (δ i δ j − p i p j )w i,j (e i e T j + e j e T i ) .
Our goal is to bound the two terms on the right-hand side of the inequality above. To lighten the notation we use ξ i = δ i − p i for i = 1, 2, . . . , n and let
S 1 = max i δ i ( j δ j w i,j ) − p i ( j p j w i,j ) − α(δ i − p i ) and S 2 = i<j (δ i δ j − p i p j )w i,j (e i e T j + e j e T i ) .
It is straightforward to verify that
δ i ( j δ j w i,j ) − p i ( j p j w i,j ) − α(δ i − p i ) = δ i j (δ j − p j )w i,j − (δ i − p i )(α − j p j w i,j ) ≤ j ξ j w i,j + α − j p j w i,j
using which we obtain
S 1 ≤ max i j ξ j w i,j + α − j p j w i,j .
By Chernoff's inequality and Lemma 4, for each i we have
j ξ j w i,j ≤ 2 j w 2 i,j K 2 j 1 2 log 4n ε ,
with probability ≥ 1 − ε 2n . Then by union bound we have
S 1 ≤ max i j w 2 i,j K 2 j 1 2 log 8n ε + α − j p j w i,j ≤ 2K log 4n ε + max i α − j p j w i,j ,(6)
with probability ≥ 1 − ε 2 . Expressing S 2 in terms of ξ i s and applying the triangle inequality reveals that
S 2 = i<j (ξ i ξ j − p i ξ j − ξ i p j )w i,j (e i e T j + e j e T i ) ≤ i<j ξ i ξ j w i,j (e i e T j + e j e T i ) + i<j (p i ξ j + p j ξ i ) w i,j (e i e T j + e j e T i ) = D ξ AD ξ + 2 D p AD ξ ,(7)
with D ξ and D p respectively denoting diagonal matrices with ξ i s and p i s on their diagonals. We can write
D ξ AD ξ 2 = D ξ AD 2 ξ AD ξ ≤ D ξ AD p•(1−p) AD ξ + D ξ AD ξ•(1−2p) AD ξ ≤ D 1 2 p•(1−p) AD 2 ξ AD 1 2 p•(1−p) + AD ξ•(1−2p) A ≤ D 1 2 p•(1−p) AD p•(1−p) AD 1 2 p•(1−p) + D 1 2 p•(1−p) AD ξ•(1−2p) AD 1 2 p•(1−p) + AD ξ•(1−2p) A .(8)
We used the identity δ 2 i = δ i or equivalently
ξ 2 i = p i (1 − p i ) + ξ i (1 − 2p i )
followed by a triangle inequality to obtain the first inequality. To obtain the second inequality we simply rearranged the matrices in the first term and used the fact that |ξ i | ≤ 1 to bound the second term. Applying the identity δ 2 i = δ i again yields the third inequality. With a i denoting the ith column of the adjacency matrix A, we can invoke Proposition 6 to guarantee that with probability ≥ 1 − ε 2 we have
AD ξ•(1−2p) A = i ξ i (1 − 2p i ) a i a T i ≤ 2 i K 2 i (1 − 2p i ) 2 (a i a T i ) 2 1 2 log 4n ε = 2σ log 4n ε . where K(p) def = 1 2 1 − 2p log 1−p p .
We also use the following master tail bound for sums of independent random matrices due to [16].
P λ max n i=1 Z i ≥ t ≤ inf θ>0 e −θt tr exp n i=1
log Ee θZ i .
In particular, we combine Lemma 4 and Theorem 5 to obtain a sharper analog to the tail bounds for Rademacher series derived in [16], for general centered Bernoulli random variables. As a consequence of the use of Kearns-Saul inequality (i.e., Lemma 4), the improvement over similar bounds obtained via matrix Hoeffding or matrix Bernstein inequalities can be particularly significant if the Bernoulli random variables have means close to 0 or 1.
Proposition 6.
For i = 1, 2, . . . , n let δ i ∼ Bernoulli(p i ) be independent random variables. Furthermore, let X i be deterministic N × N self-adjoint matrices. Then with K i = K(p i ) defined as in Lemma 4, we have
P n i=1 (δ i − p i ) X i ≥ t ≤ 2N e − t 2 4σ 2 , where σ 2 = n i=1 K 2 i X 2 i .
Proof: Let Z i = (δ i − p i ) X i for i = 1, 2, . . . , n. For any real number θ we have
Ee θZ i = p i e θ(1−p i )X i + (1 − p i )e −θp i X i .
Since θX i is self-adjoint, it can be diagonalized. Therefore, by applying Lemma 4 to the eigenvalues of θX i the above equation implies that
Ee θZ i e θ 2 K 2 i X 2 i ,
where the inequality is with respect to the positive semidefinite cone. Therefore, we have tr exp
n i=1 log Ee θZ i ≤ tr exp θ 2 n i=1 K 2 i X 2 i ≤ N exp θ 2 n i=1 K 2 i X 2 i = N e θ 2 σ 2 .
Then it follows from Theorem 5 that
P λ max n i=1 Z i ≥ t ≤ inf θ>0 N e −θt+θ 2 σ 2 = N e − t 2 4σ 2 .
Replacing X i by −X i and repeating the above argument we can similarly show that
P λ min n i=1 Z i ≤ −t ≤ N e − t 2 4σ 2 .
The union bound then guarantees that
P n i=1 Z i ≥ t ≤ 2N e − t 2 4σ 2 ,
as desired.
Corollary 3 .
3Let G be an arbitrary (n, d, λ)-graph. There are positive constants C 1 and C 2 depending only on λ d such that under the site percolation model with vertex survival probability ofp ≥ 1 − C 1 4n ε −C 2 /dthe surviving subgraph of G is connected with probability ≥ 1 − ε.
Theorem 5. [16, Theorem 3.6] Consider a finite sequence {Z i } of independent, random, self-adjoint matrices. For all t ∈ R,
AcknowledgementsS. Bahmani and J. Romberg were supported in part by ONR grant N00014-11-1-0459, NSF grants CCF-1415498 and CCF-1422540, and the Packard Foundation. P. Tetali was supported in part by NSF grants DMS-1407657 and CCF-1415498.On the same event we also havewhere we used the fact that D p•(1−p) ≤ 1 4 in the second line. These probabilistic upper bounds together with(8)with probability ≥ 1 − ε 2 . Furthermore, using similar arguments as above we haveand thusIt follows from(7),(9), and (10) thatwith probability ≥ 1 − ε 2 . The desired result follows immediately using the derived bounds(6)and(11).Appendix A Auxiliary tools and technical lemmasWe use the following lemma due to[10]which provides a sharp bound for the sub-Gaussian norm of general Bernoulli random variables.Lemma 4 (Kearns-Saul inequality[10]). For p ∈ [0, 1] let δ be a Bernoulli(p) random variable. Then for all t ∈ R we have Ee t(δ−p) = pe t(1−p) + (1 − p)e −tp ≤ e (K(p) t)2,
Vertex percolation on expander graphs. S Ben-Shimon, M Krivelevich, European Journal of Combinatorics. 302S. Ben-Shimon and M. Krivelevich. Vertex percolation on expander graphs. European Journal of Combinatorics, 30(2):339-350, 2009.
The spectral gap of a random subgraph of a graph. F Chung, P Horn, Internet Math. 42-3F. Chung and P. Horn. The spectral gap of a random subgraph of a graph. Internet Math., 4(2-3): 225-244, 2007.
The giant component in a random subgraph of a given graph. F Chung, P Horn, L Lu, International Workshop on Algorithms and Models for the Web-Graph. SpringerF. Chung, P. Horn, and L. Lu. The giant component in a random subgraph of a given graph. In International Workshop on Algorithms and Models for the Web-Graph, pages 38-49. Springer, 2009.
The Combinatorics of Network Reliability. C J Colbourn, International Series of Monographs on Computer Science. 4Oxford University Press, IncC. J. Colbourn. The Combinatorics of Network Reliability, volume 4 of International Series of Monographs on Computer Science. Oxford University Press, Inc., New York, 1987.
Expansion properties of a random regular graph after random vertex deletions. C Greenhill, F B Holt, N Wormald, European Journal of Combinatorics. 295C. Greenhill, F. B. Holt, and N. Wormald. Expansion properties of a random regular graph after random vertex deletions. European Journal of Combinatorics, 29(5):1139-1150, 2008.
Percolation, volume 321 of Grundlehren der mathematischen Wissenschaften. G Grimmett, SpringerBerlin Heidelberg; Berlin, HeidelbergG. Grimmett. Percolation, volume 321 of Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, Berlin, Heidelberg, 1999.
Improved bounds and algorithms for graph cuts and network reliability. D G Harris, A Srinivasan, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '14. the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '14Philadelphia, PA, USA, 2014. SIAMD. G. Harris and A. Srinivasan. Improved bounds and algorithms for graph cuts and network reliability. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '14, pages 259-278, Philadelphia, PA, USA, 2014. SIAM.
A randomized fully polynomial time approximation scheme for the all-terminal network reliability problem. D R Karger, SIAM Review. 433D. R. Karger. A randomized fully polynomial time approximation scheme for the all-terminal network reliability problem. SIAM Review, 43(3):499-522, 2001.
A fast and simple unbiased estimator for network (un)reliability. D R Karger, Foundations of Computer Science (FOCS). IEEE 57th Annual Symposium onD. R. Karger. A fast and simple unbiased estimator for network (un)reliability. In Foundations of Computer Science (FOCS), 2016 IEEE 57th Annual Symposium on, 2016.
Large deviation methods for approximate probabilistic inference. M Kearns, L Saul, Proceedings of the Fourteenth Conference on Uncertainty in Artificial Intelligence, UAI'98. the Fourteenth Conference on Uncertainty in Artificial Intelligence, UAI'98San Francisco, CA, USAMorgan Kaufmann Publishers IncISBN 1-55860-555-XM. Kearns and L. Saul. Large deviation methods for approximate probabilistic inference. In Proceedings of the Fourteenth Conference on Uncertainty in Artificial Intelligence, UAI'98, pages 311-319, San Francisco, CA, USA, 1998. Morgan Kaufmann Publishers Inc. ISBN 1-55860-555-X.
The phase transition in site percolation on pseudo-random graphs. M Krivelevich, The Electronic Journal of Combinatorics. 231M. Krivelevich. The phase transition in site percolation on pseudo-random graphs. The Electronic Journal of Combinatorics, 23(1):1-12, 2016.
Markov chains and mixing times. D A Levin, Y Peres, E L Wilmer, American Mathematical SocietyD. A. Levin, Y. Peres, and E. L. Wilmer. Markov chains and mixing times. American Mathematical Society, 2009.
Isoperimetric numbers of graphs. B Mohar, Journal of Combinatorial Theory, Series B. 473B. Mohar. Isoperimetric numbers of graphs. Journal of Combinatorial Theory, Series B, 47(3): 274-291, 1989.
Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges. R I Oliveira, arXiv:0911.0600math.COR. I. Oliveira. Concentration of the adjacency matrix and of the Laplacian in random graphs with independent edges. arXiv:0911.0600 [math.CO], Nov. 2009.
The fastest mixing Markov process on a graph and a connection to a maximum variance unfolding problem. J Sun, S Boyd, L Xiao, P Diaconis, SIAM review. 484J. Sun, S. Boyd, L. Xiao, and P. Diaconis. The fastest mixing Markov process on a graph and a connection to a maximum variance unfolding problem. SIAM review, 48(4):681-699, 2006.
User-friendly tail bounds for sums of random matrices. J A Tropp, Foundations of Computational Mathematics. J. A. Tropp. User-friendly tail bounds for sums of random matrices. Foundations of Computational Mathematics, Aug. 2011.
|
[] |
[
"A 6bit, 1.2GSps Low-Power Flash-ADC in 0.13µm Digital CMOS",
"A 6bit, 1.2GSps Low-Power Flash-ADC in 0.13µm Digital CMOS"
] |
[
"Christoph Sandner 2mailto:[email protected] \nInfineon Technologies Austria\nDevelopment Center VillachA-9500VillachSiemensstr\n",
"Martin Clara \nInfineon Technologies Austria\nDevelopment Center VillachA-9500VillachSiemensstr\n",
"Andreas Santner \nInfineon Technologies Austria\nDevelopment Center VillachA-9500VillachSiemensstr\n",
"Thomas Hartig \nInfineon Technologies Austria\nDevelopment Center VillachA-9500VillachSiemensstr\n",
"Franz Kuttner \nInfineon Technologies Austria\nDevelopment Center VillachA-9500VillachSiemensstr\n"
] |
[
"Infineon Technologies Austria\nDevelopment Center VillachA-9500VillachSiemensstr",
"Infineon Technologies Austria\nDevelopment Center VillachA-9500VillachSiemensstr",
"Infineon Technologies Austria\nDevelopment Center VillachA-9500VillachSiemensstr",
"Infineon Technologies Austria\nDevelopment Center VillachA-9500VillachSiemensstr",
"Infineon Technologies Austria\nDevelopment Center VillachA-9500VillachSiemensstr"
] |
[] |
A 6bit flash-ADC with 1.2GSps, wide analog bandwidth and low power, realized in a standard digital 0.13µm CMOS copper technology is presented. Employing capacitive interpolation gives various advantages when designing for low power: no need for a reference resistor ladder, implicit sample-and-hold operation, no edge effects in the interpolation network (as compared to resistive interpolation), and a very low input capacitance of only 400fF, which leads to an easily drivable analog converter interface.Operating at 1.2GSps the ADC achieves an effective resolution bandwidth (ERBW) of 700MHz, while consuming 160mW of power. At 600MSps we achieve an ERBW of 600MHz with only 90mW power consumption, both from a 1.5V supply. This corresponds to outstandingFigure-of-Merit numbers (FoM) of 2.2 and 1.5pJ/convstep, respectively. The module area is 0.12mm².
|
10.1109/date.2005.2
|
[
"https://arxiv.org/pdf/0710.4838v1.pdf"
] | 10,042,806 |
0710.4838
|
7d6bf76b0c0bd58f357249949d4e8ed02d606c27
|
A 6bit, 1.2GSps Low-Power Flash-ADC in 0.13µm Digital CMOS
Christoph Sandner 2mailto:[email protected]
Infineon Technologies Austria
Development Center VillachA-9500VillachSiemensstr
Martin Clara
Infineon Technologies Austria
Development Center VillachA-9500VillachSiemensstr
Andreas Santner
Infineon Technologies Austria
Development Center VillachA-9500VillachSiemensstr
Thomas Hartig
Infineon Technologies Austria
Development Center VillachA-9500VillachSiemensstr
Franz Kuttner
Infineon Technologies Austria
Development Center VillachA-9500VillachSiemensstr
A 6bit, 1.2GSps Low-Power Flash-ADC in 0.13µm Digital CMOS
A 6bit flash-ADC with 1.2GSps, wide analog bandwidth and low power, realized in a standard digital 0.13µm CMOS copper technology is presented. Employing capacitive interpolation gives various advantages when designing for low power: no need for a reference resistor ladder, implicit sample-and-hold operation, no edge effects in the interpolation network (as compared to resistive interpolation), and a very low input capacitance of only 400fF, which leads to an easily drivable analog converter interface.Operating at 1.2GSps the ADC achieves an effective resolution bandwidth (ERBW) of 700MHz, while consuming 160mW of power. At 600MSps we achieve an ERBW of 600MHz with only 90mW power consumption, both from a 1.5V supply. This corresponds to outstandingFigure-of-Merit numbers (FoM) of 2.2 and 1.5pJ/convstep, respectively. The module area is 0.12mm².
Introduction
Flash Analog-to-Digital Converters (ADCs) are still the architecture of choice, where maximum sample rate and low to moderate resolution is required. A typical example is e.g. the read-write channel of a disk drive system, where customers often ask for the maximum sample rate that is offered by the currently available technology generation.
However, there are additional applications in the wireless area coming up, where a flash-ADC gives sufficient accuracy at the required large analog bandwidth, for example in ultra-wideband (UWB) systems. Since a lot of wireless applications are hand-held as well, this poses an important constraint to the specification of the ADC, which is power consumption. The ADC can be the dominant block in terms of power consumption for the whole analog frontend of such a system.
In this paper we describe the concept, design and measurement results of a 6bit flash-ADC, optimized for low power at sufficiently high data rates and analog bandwidth. With 1.2GSps and ERBW of 700MHz the ADC fulfils the requirements currently under discussion for multi-band OFDM UWB systems [1]. Due to the large ERBW different receiver topologies, like low-IF or zero-IF can be supported, both at Nyquist rate, and under sub-sampling operation. However, the presented ADC architecture is not limited to these applications.
Although almost all recently published high-speed full-flash-ADCs employ resistive interpolation and averaging, the capacitive interpolation structure with distributed front-end sample-and-hold seems to have advantages in terms of power and area, even at GSps-speed. The outstanding FoM numbers prove the efficiency of the implemented architecture, when compared to stateof-the-art 6bit flash ADCs.
Converter Architecture
Averaging is a well known technique to improve the linearity of a flash-type ADC beyond the matching limit of the single comparator [2]. Multiple gain-stage architectures allow the use of interpolation to reduce the number of front-end amplifiers and thus the input capacitance of the converter. By scaling the amplifiers in the analog preprocessing chain from front to back also the overall power consumption can be optimized under the given gain/bandwidth constraints. There exist basically two different strategies for implementing a flash Resistive interpolation employs averaging resistors between the outputs of adjacent amplifiers. [3]. A common problem of such architectures is the need for overrange comparators to maintain linearity at the edges of the conversion range. Special circuit techniques [4] allow reducing the number of overrange comparators, but they rely on matching the termination resistor with the output resistance of the overrange blocks. Furthermore, an external sample-and-hold circuit is almost always required, which consumes a significant fraction of the total power budget in wide-band applications.
Capacitive interpolation, on the other hand, uses a purely reactive averaging network between the outputs of adjacent amplifiers [5]. A big advantage of capacitive interpolation is that it requires neither power consuming overrange comparators nor any static averaging termination. Also, no external sample-and-hold is required, because the interpolation capacitors at each stage are readily used as sampling capacitors implementing a multistage input-offset-sampling (IOS) architecture with distributed front-end sample-and-hold.
The principle of capacitive interpolation combined with distributed front-end sample-and-hold is illustrated in Fig. 1, showing a portion of the converter input. The inverting first stage amplifiers drive the interpolating capacitive network of the second stage, here implementing an interpolation factor 2. The interpolation factor at the converter input is given by the total number of frontend amplifiers minus 1. The total input capacitance of the converter in the sampling phase is given by the sum of the front-end capacitors plus wiring parasitics.
One drawback of the capacitive interpolation structure is the capacitive divider formed by the sampling capacitor and the input capacitance of the amplifier during the amplification phase. The overall gain of each stage is given by the intrinsic gain of the amplifier and the capacitive divider ratio. In order to minimize the total input capacitance of the converter, the sampling capacitors will be chosen as small as possible, for a resolution of 6 bits ultimately limited by capacitor mismatch. For minimum gain loss the amplifiers' input devices should therefore be as small as possible. Here, IOS helps to further reduce the input devices' effective mismatch, thus allowing additional downscaling to save power and area.
Implemented Design
The block diagram of the implemented A/D-converter is shown in Fig. 2. The circuitry is fully differential, although drawn single ended for simplicity. The input signal is sampled in the first clock phase φ 1 by the front-end amplifiers forming a distributed sample-and-hold, while the subsequent stages sample the offset voltage. During the second clock phase φ 2 the reference voltage is applied to the bottom plates of the front-end sampling capacitors and the difference between input voltage and reference voltage is amplified before it is latched by the comparators at the end of this clock phase.
The capacitive load at each amplifier output and thus the bandwidth of the amplifier is linearly related to the interpolation factor. To optimize the power-bandwidth product of the amplifier, while still profiting from the averaging property, the minimum possible interpolation factor of 2 is chosen at the output of each gain stage. To reach 64 decision levels after three interpolating stages, an interpolation factor of 8 is implemented at the converter input, thus formed by 9 parallel input amplifiers. A fourth gain stage directly drives the latching comparator, leading to an 8-2-2-2-1 interpolation topology. Since the overall interpolation factor is 64, only two reference voltages are needed at the input of the 6 bit ADC, thus avoiding the silicon area and power penalty of a lowresistance reference ladder. Instead, the reference voltages are generated by capacitive voltage division during the amplification phase φ 2 (referring to single-ended representation in Fig. 1
): ( ) REFN REFP REFN i REF V V C C C V V − + + = 2 1 1 (1)
Through optimized sizing of the converter front-end, it was possible to reach a very low input capacitance of only 400fF. The subsequent gain stages are progressively scaled down by a factor of 2, thus achieving a very compact layout, since all 4 gain stages end up to have the same height (Fig. 4).
Regarding DC performance, the input referred offset is the most important specification. Since, due to minimum sizing for maximum speed, the offset of the final comparator stage is dominant, this offset must be reduced by achieving sufficient gain in the preamplifiers, thus reducing the input referred offset. The amplifier block (Fig. 3) consists of a differential pair with resistive load. PMOS switches connected between inputs and outputs provide the offset sampling. The gain of each stage is chosen to be 2.5 for achieving maximum bandwidth. The final comparator latch is shown in Fig. 5. The input voltage difference is first converted into a current difference and then fed to the cross-coupled latch, formed by transistors N2. During the sampling phase, when the amplifiers sample the analog input signal, the CLK signal is high, thus keeping the comparator outputs OUTP and OUTN at the same level. During amplification phase CLK goes low, thus releasing the outputs, and the latch can decide whether the input voltage difference was positive or negative. The use of a single NMOS clock switch between both outputs is not possible, due to the low supply voltage of 1.5V. The gm of transistors N2 must be chosen such that the desired Bit Error Rate (BER) specifications are met. Main advantages of this topology are the almost rail-to-rail output swing, and the reduced kick-back to the pre-amplifiers due to the current mirrors.
For increasing the BER the 64 outputs of the comparator row are again latched twice by differential latches, before entering a first order bubble correction stage, which also converts the thermometer code to a 1out-of-N code. Finally, this code is converted into a 6 bit Gray-code using a ROM-table based on a current steering topology optimized for high frequency operation. For test purposes the digital data is downsampled by a factor of 64, thus standard CMOS pads can be used for the digital outputs. Fig. 4 shows the layout plot of the implemented testchip. It was fabricated in a 0.13µm digital CMOS technology with standard-threshold MOS devices (no low-VT option), single poly, 4 thin copper metals, and 2 thick copper metals. The capacitors are of metal-metal sandwich type, taking advantage of both vertical and lateral capacitances.
Measurements
All measurements are done on a standard PCB with ceramic TQFP-44 package and no socket, at room temperature and nominal supply of 1.5V. Both, analog input and sampling clock are applied differentially to the chip by using baluns. Fig. 6 shows the DNL of the converter, measured at 600MSps with a full-scale 50MHz sine-wave input employing a histogram method. The measured peak DNL of 0.4LSB thus already includes dynamic effects at that sampling rate. Peak INL is measured <0.6LSB.
In Fig. 7 the maximum sampling rate of the ADC is explored for nominal bias current. The analog signal frequency is kept constant at 121MHz, while the sampling clock frequency is varied on the x-axis. The ADC shows 5bit ENOB up to a sample rate of 1.4GSps. The clock frequency limit for the digital circuitry is around 1.6GSps. Fig. 8 shows the SNR and SNDR for a sample rate of 600MSps, at reduced bias current to save power. The SNDR at 51MHz is 35.5dB, with an SFDR of 52dB and a THD of 49dB, thus proving the excellent linearity of the capacitive interpolation topology. The SNDR drops by 3dB at an ERBW of 600MHz. In Fig. 9 the same measurement is done for 1.2GSps at nominal bias current. SNDR starts at 35.8dB for low frequencies (THD is 46dB), dropping by 3dB at an ERBW of 700MHz. These measurements show that this ADC topology is very well suited for applications with wide analog bandwidth.
To do a comparison with state-of-the-art 6bit flash-ADCs, a Figure-
⋅ ⋅ = 2 2 , [pJ/convstep](2)
For our ADC we achieve a FoM of 2.2pJ/conv at 1.2GSps, and 1.5pJ/conv at 600MSps. As can be seen in Fig. 10 these are the best FoM numbers for flash ADCs ever published [4,[6][7][8][9]. Although a smaller feature size technology is used in this work, the achieved performance nevertheless proves the efficiency of the capacitive interpolation architecture with distributed sample-and-hold for flash ADCs in the GHz range.
5.
Fig. 1 :
1ADC frontend portion using capacitive interpolation ADC employing interpolation and averaging.
Fig. 2 :
2Implemented ADC Block Diagram Fig. 3: Amplifier Block Diagram
Fig. 4 Fig. 5 :
45Final Comparator LatchFig. 6: Measured DNL typ. <0.4LSB (at 600MSps)
Fig. 8 :Fig. 10 :
810Measured Comparison to state-of-the-art 6b Flash ADCsThis work:
of-Merit (FoM) is calculated[8]:ERBW
Power
FoM
DC
ENOB
Proceedings of the Design, Automation and Test in Europe Conference and Exhibition (DATE'05) 1530-1591/05 $ 20.00 IEEE
AcknowledgementsWe thank C. Kropf for layout work, P. Schreilechner for board design, P. Bogner, D. Draxelmayr and G. Knoblinger for fruitful discussions.6.
Multi-band OFDM Physical Layer Proposal. A Batra, IEEE 802.15 WPAN High Rate Alternative PHY Task Group 3a (TG3a). A. Batra et al, "Multi-band OFDM Physical Layer Proposal", IEEE 802.15 WPAN High Rate Alternative PHY Task Group 3a (TG3a), Sep. 2003, http://www.ieee802.org/15
An Embedded 240-mW 10-b 50-MS/s CMOS ADC in 1-mm². Klaas Bult, Aaron Buchwald, IEEE Journal of Solid-State Circuits. 3212Klaas Bult, Aaron Buchwald, "An Embedded 240-mW 10-b 50-MS/s CMOS ADC in 1-mm²", IEEE Journal of Solid-State Circuits, Vol. 32, No. 12, Dec. 1997, pp. 1887-1895.
A technique for reducing differential non-linearity errors in flash A/D converters. Kevin Kattmann, Jeff Barrow, IEEE International Solid-State Circuits Conference. XXXIVKevin Kattmann, Jeff Barrow; "A technique for reducing differential non-linearity errors in flash A/D converters", IEEE International Solid-State Circuits Conference, Vol. XXXIV, Feb. 1991, pp. 170 -171.
A 6-b 1.6-Gsample/s Flash ADC in 0.18-um CMOS Using Averaging Termination. Peter Scholtens, Maarten Vertregt, IEEE Journal of Solid-State Circuits. 3712Peter Scholtens, Maarten Vertregt, "A 6-b 1.6-Gsample/s Flash ADC in 0.18-um CMOS Using Averaging Termination", IEEE Journal of Solid-State Circuits, Vol. 37, No. 12, Dec. 2002, pp. 1599-1609.
A 10b 20MHz 30mW Pipelined Interpolating CMOS ADC. Keiichi Kusumoto, Akira Matsuzawa, Kenji Murata, IEEE Journal of Solid-State Circuits. 2812Keiichi Kusumoto, Akira Matsuzawa and Kenji Murata, "A 10b 20MHz 30mW Pipelined Interpolating CMOS ADC", IEEE Journal of Solid-State Circuits, Vol. 28, No. 12, December 1993, pp. 1200-1206
A 6-b 1.3-Gsample/s A/D Converter in 0.35µm CMOS. Michael Choi, Asad A Abidi, IEEE Journal of Solid-State Circuits. 3612Michael Choi, Asad A. Abidi, "A 6-b 1.3-Gsample/s A/D Converter in 0.35µm CMOS", IEEE Journal of Solid-State Circuits, Vol. 36, No. 12, Dec. 2001, pp.1847-1858.
A 6b 1.1GSample/s CMOS A/D Converter. Govert Geelen, IEEE International Solid-State Circuits Conference. 44Govert Geelen, "A 6b 1.1GSample/s CMOS A/D Converter", IEEE International Solid-State Circuits Conference, Vol. 44, Feb. 2001, pp. 128-129.
A 2GS/s 6b ADC in 0.18µm CMOS. Xicheng Jiang, Zhengyu Wang, M Frank Chang, IEEE International Solid-State Circuits Conference. 46Xicheng Jiang, Zhengyu Wang, M. Frank Chang, "A 2GS/s 6b ADC in 0.18µm CMOS", IEEE International Solid-State Circuits Conference, Vol. 46, Feb. 2003, pp. 322-323.
A 1.8-V 6-bit 1.3-GHz flash ADC in 0.25µm CMOS. Koen Uyttenhove, Michiel Steyaert, IEEE Journal of Solid-State Circuits. 3812Koen Uyttenhove, Michiel Steyaert, "A 1.8-V 6-bit 1.3- GHz flash ADC in 0.25µm CMOS", IEEE Journal of Solid- State Circuits, Vol. 38, No. 12, July 2003, pp. 1115-1122.
|
[] |
[
"A New Approach for Automatic Segmentation and Evaluation of Pigmentation Lesion by using Active Contour Model and Speeded Up Robust Features",
"A New Approach for Automatic Segmentation and Evaluation of Pigmentation Lesion by using Active Contour Model and Speeded Up Robust Features"
] |
[
"I ",
"Ms Sara Mardanisamani ",
"ProfAkram Jamshidzadeh ",
"DrMelika Farshad ",
"DrZahra Karimi ",
"ProfMehran Yazdi ",
"Mr Amirmehdi Farshad ",
"\nInternational Journal of Engineering Research & Technology (IJERT)\nDepartment of Electrical Engineering, Zarghan Branch\nDepartment of Pharmacology and Toxicology\nFaculty of Pharmacy\nIslamic Azad University\nZarghanIran\n",
"\nDepartment of Dentistry\nShiraz University of Medical Sciences\nShirazIran\n",
"\nPharmaceutical Sciences Research Center\nShiraz University of Medical Sciences\nShirazIran\n",
"\nDepartment of Communications and Electronics\nFaculty of Electrical and Computer Engineering\nShiraz University of Medical Sciences\nShirazIran\n",
"\nDepartment of Power and Control Engineering\nSchool of Electrical and Computer Engineering\nShiraz University\nShirazIran\n",
"\nShiraz University\nShirazIran\n"
] |
[
"International Journal of Engineering Research & Technology (IJERT)\nDepartment of Electrical Engineering, Zarghan Branch\nDepartment of Pharmacology and Toxicology\nFaculty of Pharmacy\nIslamic Azad University\nZarghanIran",
"Department of Dentistry\nShiraz University of Medical Sciences\nShirazIran",
"Pharmaceutical Sciences Research Center\nShiraz University of Medical Sciences\nShirazIran",
"Department of Communications and Electronics\nFaculty of Electrical and Computer Engineering\nShiraz University of Medical Sciences\nShirazIran",
"Department of Power and Control Engineering\nSchool of Electrical and Computer Engineering\nShiraz University\nShirazIran",
"Shiraz University\nShirazIran"
] |
[
"International License.)"
] |
Digital image processing techniques have wide applications in different scientific fields including the medicine. By use of image processing algorithms, physicians have been more successful in diagnosis of different diseases and have achieved much better treatment results. In this paper, we propose an automatic method for segmenting the skin lesions and extracting features that are associated to them. At this aim, a combination of Speeded-Up Robust Features (SURF) and Active Contour Model (ACM), is used. In the suggested method, at first region of skin lesion is segmented from the whole skin image, and then some features like the mean, variance, RGB and HSV parameters are extracted from the segmented region. Comparing the segmentation results, by use of Otsu thresholding, our proposed method, shows the superiority of our procedure over the Otsu theresholding method. Segmentation of the skin lesion by the proposed method and Otsu thresholding compared the results with physician's manual method. The proposed method for skin lesion segmentation, which is a combination of SURF and ACM, gives the best result. For empirical evaluation of our method, we have applied it on twenty different skin lesion images. Obtained results confirm the high performance, speed and accuracy of our method.
| null |
[
"https://arxiv.org/pdf/2101.07195v1.pdf"
] | 56,837,129 |
2101.07195
|
14e60c94a8c1925333d799fd80f98ce35aa91efd
|
A New Approach for Automatic Segmentation and Evaluation of Pigmentation Lesion by using Active Contour Model and Speeded Up Robust Features
October-2015
I
Ms Sara Mardanisamani
ProfAkram Jamshidzadeh
DrMelika Farshad
DrZahra Karimi
ProfMehran Yazdi
Mr Amirmehdi Farshad
International Journal of Engineering Research & Technology (IJERT)
Department of Electrical Engineering, Zarghan Branch
Department of Pharmacology and Toxicology
Faculty of Pharmacy
Islamic Azad University
ZarghanIran
Department of Dentistry
Shiraz University of Medical Sciences
ShirazIran
Pharmaceutical Sciences Research Center
Shiraz University of Medical Sciences
ShirazIran
Department of Communications and Electronics
Faculty of Electrical and Computer Engineering
Shiraz University of Medical Sciences
ShirazIran
Department of Power and Control Engineering
School of Electrical and Computer Engineering
Shiraz University
ShirazIran
Shiraz University
ShirazIran
A New Approach for Automatic Segmentation and Evaluation of Pigmentation Lesion by using Active Contour Model and Speeded Up Robust Features
International License.)
4October-2015(This work is licensed under a Creative Commons Attribution 4.0Pigmentation disorderSpeeded-Up Robust FeaturesActive Contour ModelImage segmentation
Digital image processing techniques have wide applications in different scientific fields including the medicine. By use of image processing algorithms, physicians have been more successful in diagnosis of different diseases and have achieved much better treatment results. In this paper, we propose an automatic method for segmenting the skin lesions and extracting features that are associated to them. At this aim, a combination of Speeded-Up Robust Features (SURF) and Active Contour Model (ACM), is used. In the suggested method, at first region of skin lesion is segmented from the whole skin image, and then some features like the mean, variance, RGB and HSV parameters are extracted from the segmented region. Comparing the segmentation results, by use of Otsu thresholding, our proposed method, shows the superiority of our procedure over the Otsu theresholding method. Segmentation of the skin lesion by the proposed method and Otsu thresholding compared the results with physician's manual method. The proposed method for skin lesion segmentation, which is a combination of SURF and ACM, gives the best result. For empirical evaluation of our method, we have applied it on twenty different skin lesion images. Obtained results confirm the high performance, speed and accuracy of our method.
INTRODUCTION
Quantification of skin color or determining the extent of pigmentation is necessary in dermatology and cosmetic science. In early 1920s, non-invasive techniques began to be used for skin pigmentation measurement, using different devices. Assuming that light intensity attenuates when remitted by the skin, these instruments were designed to enhance the performance of skin pigmentation by use of skin images [1].
Spectrophotometers and colorimeters, known as Skin reflectance measurement devices, supply the techniques for objective measurement and high inter/intra-rater reliability [2,3]. Not only the expense, but also needs for user training for these devices are the main limitations in their use. Therefore, developing more user-friendly and less costly methods would be advantageous for physicians and researchers.
In the last decades a number of computer vision based methods have been proposed to enhance skin images. These techniques are mostly used for melanoma diagnosis. Using Digital image process helps physicians in better diagnosis and treatments. In general physician could locate the lesions and describe their dermatological features by extracting the image parameters. Studies have shown that practically, using these automated systems is sufficient for diagnosis of skin color disorders or melanoma [4,5].
Numerous methods have been reported in computer vision and image processing, for Image segmentation [6][7][8]. Gabriella et al. proposed a method to detect some specific dermoscopic criteria for Melanocytic Skin Lesion which is based on blue whitish veil, the regression, and the irregular streaks techniques [9]. El-Zaart modeled a new threshold estimation method by an unsupervised learning technique with beta distribution [10].
Abbas et al. used a different approach. He rescaled all images, then reduced their artifacts by applying some filters, and finally segmented the lesion borders. This unsupervised approach uses Region-based Active Contours (RACs) for lesion border segmentation [11].
Chan kim et al. proposed a method based on L*a*b color coordinates [12] and developed a Visual Basic program to compare the pre-and post-treatment skin color. Xu et al. used another method for determination of lesion boundaries. In this method, threshold value determined for the gray-scaled images and initial lesion segmentation were achieved [8].
Using digital filters can provide efficient techniques for determining skin disorder. These methods are studied by Güçin, Patias and Altan [13].
Studying skin cancer images could be more complicated. For this matter, not only the skin color but also its texture should be used. Padmapriya Nammalwar et al. combine these two factors to present an efficient procedure for skin cancer image segmentation [14]. Fassihi et al. suggested another method for skin cancer images. In this method the segmentation is performed by morphological operators and features are extract using wavelet transforms [15].
In this paper an automatic segmentation algorithm based on SURF algorithm and Active contour model thresholding is proposed. For automatic border detection, our approach exploits two main steps: (1) finding interest point by use of the surf algorithm, (2) segmenting the skin lesions by use of the ACM. Our method is then compared with the manual segmentation results performed by an expert physician and the automatic Otsu thresholding algorithm.
II. MATERIAL AND METHODS
A. Image acquisition
The process of manual segmentation of the skin lesion images is very time consuming. It also requires significant medical expertise, and can be prone to error. In this work, 20 color images with 256×256 pixels, from the image database Dermatlas [16] are used for the experiments. We conducted the experiments on these skin images and compared the segmentation results obtained by our approach with those subjectively chosen by our physician and other methods. The examination is performed on 20 images, all in JPEG format. To conduct the experiments we used Matlab 7.1 on a dual core Pentium IV computer with 4GB Ram and 2.53 GHZ processor speed.
B. Segmentation method
Segmentation is the first step of computer-based skin lesion diagnosis. The lesion boundary provides important information for accurate diagnosis. Furthermore, the extraction of other clinical features critically depends on the accuracy of the boundary [17]. Our goal in this article is to find a contour that best approximates the perimeter of a skin lesion. In this paper, we present an active contour segmentation approach and apply it to skin lesion images. For active contour segmentation, an initial contour is needed as a first step of segmentation. At this aim we use an automatic strategy, in which the initial contour is obtained roughly by Surf algorithm.
C. Speeded Up Robust Features
For real-time visual navigation, interest point detection and description are required. SURF is one of the methods which could detect the interest point [18]. Bay et al. proposed this scale-invariant feature detector method based on Hessianmatrix [19]. Three steps involved in this method: 1-describing the basic idea of integral images, 2-obtaining approximation of the Hessian matrix and 3-computing the determinant of the Hessian matrix.
Interest Point Detection
An integral image I(x) at location x represents the sum of all pixels in the input image I and is defined as (1)
I(x) = ∑ ∑ I(i, j) j≤y j=0 i≤x i=0
(1) By utilizing the integral image, the area within a bounded region (A, B, C, D) of the original image can be computed using four places in memory and three summations. Fig. 1 shows calculating the sum of intensities for obtaining integral images.
Hessian matrix at scale σ is defined as follows:
( , ) = [ ( , ) ( , ) ( , ) ( , ) ]
(2) Figure 1: Calculating the sum of intensities inside a rectangular region of any size for obtaining integral images [19]. Interest points are obtained after calculating the Hessian matrix [18,20]. These data then used to detect the keypoints. SURF first allocates an orientation to each keypoint: A circular region around the keypoint is convolved with two Haar wavelets. Scale σ, at which the keypoint is detected, determines the size of region, wavelets and sampling steps. A circular window centered on each keypoint is used for computing the SURF descriptor [21]. These keypoints are depicted in Fig. 2. After keypoints are detected with SURF algorithm, several keypoints are randomly connected to each other so that the initial contour are created.
Active Contour Model
To segment the skin lesion by use of ACM, interest points were extracted and used to form the initial contours automatically. Active contour is defined parametrically [22,23,24].We use a type of active contour that minimizes the energy defined on contours or curves living in the domain of the image. This minimization detects specified features within an image. It is a flexible curve which can be dynamically derived to the boundary of the object of interest in the image. Fitting active contours to objects in images is an interactive process. To obtain active contour we need to have initial contour first; this could be obtained by SURF algorithm.
⃗( ) = ( ⃗( ), ⃗( ))(3)
Where (s) and y(s) are , y coordinates of pixels that pass through the contour and s is the normalized index of the control points. There are two components that describe the energy function of active contour: internal and external energy. To calculate internal energy the following formula could be used:
= ( ) | | 2 + ( ) | 2 2 | 2(4)
Where α is an adjustable constant that specifies continuity and β is adjustable constant that specifies contour curving. Sum of elastic and bending energies are considered as the internal energy. To calculate these two energies the following formulas are used: Minimized energy function is defined as follow: * = ∫{ ( ( )) + ( ( )) + ( ( ))} 1 0 (7) Where Eint, Eimage and Econ are curve's internal energy, picture's energy and external limitations, respectively [25]. Therefore, we can successfully segment the lesions by using the SURF algorithm for interest point extraction, to form the initial contours automatically. The segmentation is then completed by using the ACM procedure. The combination of these methods gives high performance. The results of using this procedure for 20 real images are provided in the experimental section.
Evaluation of the Segmented Skin Lesion
After segmenting the lesion area, some features can be extracted from that area. These features can be useful for the physicians in their diagnosis and determining the degree of an especial disorder. Physicians usually use Photoshop software to obtain these features. So the automatic access to these features can be very helpful. The most useful features which could be extracted from the segmented regions are; mean and variance of the lesion area and the healthy skin part, the pixel value of images in RGB and HSV color spaces and the histogram of image in each of these two spaces. These features are depicted in Fig. 3.
III. EXPERIMENTAL RESULTS
The images are segmented by using the active contour model and SURF method. In fact, in the images, the initial contours are obtained by the SURF algorithm. Then active contour model is used to obtain the final contours for lesions. 20 color images, with 256×256 pixels, from "Dermatlas" database are used in the experiments [16]. The obtained segmentation results were validated by an expert on this field. In Fig. 4, ten segmented images are presented: The original images (1 to 10) were processed by the proposed method and the final contours presented in the images (a to l, respectively) were obtained. All final contours were visually evaluated by an expert. He confirmed that all ill regions were successfully detected. After segmenting the skin lesion areas from the images, in our work, we extracted some features like the mean, variance, RGB and HSV parameters from the segmented regions [26] features was shown in Fig. 3. We apply the Otsu's method on each box containing a skin lesion. To do this, a four class thresholding is used. Some examples of the results obtained by applying this method are shown in Fig. 5. The results of segmenting the skin lesion, based on our method in comparison with manual physician segmentation and Otsu thresholding method are given in Table 1 and Table 2 for 20 images. The manual segmentations provided by our physician are used for performance evaluation. Table 3, shows the average recall and precision values in segmenting les for all the images. The first row in this table shows the percentage average recall and precision of segmentation, achieved in our method with respect to manual segmentation by physicians. The second row shows those results for the Otsu thresholding method with respect to the manual segmentation. Results indicate that our method have good performance in comparison with Otsu procedure. Recall and precision rates are computed here by use of relations (8) and (9), [27]. TPs indicates the skin lesion pixels correctly classified. FPs represents the healthy skin pixels classified as skin lesion pixels. TNs indicates the healthy pixels correctly classified. And finally FNs shows the skin lesion pixels classified as healthy skin pixels. The following measures have been applied to evaluate the algorithms [28]. The proposed image segmentation method uses the active contour model and SURF algorithm. As active contours always provide continuous boundaries of sub-regions, they can produce more reasonable segmentation results than traditional segmentation methods, and consequently improve the final results of image analysis. The mathematical implementation of the proposed active contour models is accomplished using level set method. By presenting contours as a level of a topological function, we can merge multiple contours into one contour, or can split a contour into multiple contours, providing a good flexibility in the use of active contours. The proposed image segmentation method in this work is successfully used for detection of lesions in real skin images. The proposed algorithm demonstrate good performance for segmenting skin lesion and extracting features such as variance and mean. These feature can help physician to diagnosis and treat skin diseases. We do this using a fully automatic and accurate approach. Many physician obtain these features with non-automatic methods such as using Photoshop software that takes a much time.
Figure 2 :
2extracting interest point by using the SURF algorithm. The original images are colorful. Interest points are shown on gray scale images.
Figure 3 .
3Top pictures: Histogram of RGB space components. Bottom pictures: Image in RGB and HSV color spaces and HSV components of the image.
Figure 4 .
4The results of segmenting the skin lesions in 10 images, by drawing the lesions' contours based on the proposed method.
Recall: The ratio between the number of skin lesion pixels correctly classified and the total number of actual skin lesion pixels (TPr) The ratio between the number of skin lesion pixels correctly classified and the total number of pixels labeled as skin lesion pixels, by the applied skin lesion segmentation method.
Figure 5 .
5Results of skin lesion segmentation by using Otsu's thresholding method in four classes.
TABLE 1 .TABLE 2 .
12COMPARING RESULT BETWEEN OUR METHOD AND OTSU THRESHOLDING ALGORITHM BY USE OF RECALL AND PRECISION FACTORS ACKNOWLEGMENT The result of this investigation was originated from the research project funded by research affairs office of Islamic Azad University, Zarghan Branch. The authors express gratitude to Islamic Azad University Zarghan Branch because of financial support. SEGMENTATION OF THE SKIN LESION BY OTSU THRESHOLDING METHOD COMPARED THE RESULTS WITH RESULTS WITH PHYSICIAN'S MANUAL METHOD BY USE OF RECALL AND PRECISION FACTORS This work is licensed under a Creative Commons Attribution 4.0 International License.)Average Precision
Percentage
Average Recall
Percentage
92
91
Our method
54
75
Otsu thresholding
method
TABLE 1 .
1SEGMENTATION OF THE SKIN LESION BY THE PROPOSED METHOD COMPARED THE RESULTS WITH PHYSICIAN'S MANUAL METHOD BY USE OF RECALL AND PRECISION FACTORS
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|
[] |
[
"Stability Aspects of Wormholes in R 2 Gravity",
"Stability Aspects of Wormholes in R 2 Gravity"
] |
[
"James B Dent \nDepartment of Physics\nUniversity of Louisiana at Lafayette\n70504LafayetteLAUSA\n",
"Damien A Easson \nDepartment of Physics & Beyond Center for Fundamental Concepts in Science\nArizona State University\n85287-1504TempeAZUSA\n",
"Thomas W Kephart \nDepartment of Physics and Astronomy\nVanderbilt University\n37235NashvilleTNUSA\n",
"Sara C White \nDepartment of Physics\nUniversity of Louisiana at Lafayette\n70504LafayetteLAUSA\n"
] |
[
"Department of Physics\nUniversity of Louisiana at Lafayette\n70504LafayetteLAUSA",
"Department of Physics & Beyond Center for Fundamental Concepts in Science\nArizona State University\n85287-1504TempeAZUSA",
"Department of Physics and Astronomy\nVanderbilt University\n37235NashvilleTNUSA",
"Department of Physics\nUniversity of Louisiana at Lafayette\n70504LafayetteLAUSA"
] |
[] |
We study radial perturbations of a wormhole in R 2 -gravity to determine regions of stability.We also investigate massive and massless particle orbits and tidal forces in this space-time for a radially infalling observer.
|
10.1142/s0218271817501176
|
[
"https://arxiv.org/pdf/1608.00589v1.pdf"
] | 119,108,623 |
1608.00589
|
f205d2170d894c74bca55bbc92e256426ba35304
|
Stability Aspects of Wormholes in R 2 Gravity
James B Dent
Department of Physics
University of Louisiana at Lafayette
70504LafayetteLAUSA
Damien A Easson
Department of Physics & Beyond Center for Fundamental Concepts in Science
Arizona State University
85287-1504TempeAZUSA
Thomas W Kephart
Department of Physics and Astronomy
Vanderbilt University
37235NashvilleTNUSA
Sara C White
Department of Physics
University of Louisiana at Lafayette
70504LafayetteLAUSA
Stability Aspects of Wormholes in R 2 Gravity
We study radial perturbations of a wormhole in R 2 -gravity to determine regions of stability.We also investigate massive and massless particle orbits and tidal forces in this space-time for a radially infalling observer.
I. INTRODUCTION
Wormholes are intriguing solutions to Einstein's theory of general relativity, though their physical existence is highly contested. Wormholes are shortcut paths between different parts of spacetime and have long served as a theoretical laboratory for exploring exotic phenomena in general relativity [2][3][4]. Wormhole solutions are found by specifying the desired spacetime through sewing together two separate spacetimes with black hole geometries. This procedure determines the type of energy-momentum tensor necessary to support such a solution and typically describes "exotic" types of mass and energy. There are longstanding discussions regarding the viability of constructing wormhole solutions in standard general relativity. Surprisingly, wormholes in a certain modification of general relativity, namely R = 0 solutions of R 2 gravity, exist without the need for exotic matter [5]. R 2 gravity has recently gained attention due to the discovery of new spherically symmetric solutions in the R = 0 regime [6], including black hole solutions in general quadratic gravity [7]. One of the intriguing features of R 2 gravity is, unlike in General Relativity, it is possible to have non-trivial Ricci tensor R µν = 0 even when R = 0.
In this paper we study stability properties of the aforementioned R 2 gravity wormhole solutions. We identify regions in parameter space which yield a stable wormhole by studying radial perturbations. In addition, we search for stable orbits around these wormholes and determine whether an astrophysical probe in an R 2 universe could possibly traverse such a wormhole without experiencing destructive tidal forces. The present work explores these questions by adapting previous stability inquiries for wormholes in standard general relativity such as those discussed in [9].
We begin in section II by reviewing the wormhole construction found in [5], and in section III we determine the junction conditions needed in f (R) gravity. In section IV we discuss the criteria for a stable wormhole and find regions in parameter space where stability holds.
Section V examines the stability of orbits around the stable wormhole construction, and section VI determines the tidal forces encountered by a radially infalling observer passing through the wormhole. We conclude in section VII.
Consider the wormhole R = 0 solution to R 2 gravity. R = 0 is a requirement for this vacuum solution. The metric is
ds 2 = −G( )dt 2 + d 2 G( ) + ( 2 + k 2 )dΩ 2(1)
where we define
2 = r 2 − k 2 .(2)
Here, l is the radius of the wormhole throat and dΩ 2 is the metric on the unit two-sphere.
The constant k sets the minimal throat radius, and the full form of G( ) can be found in the Appendix of [5]. If G( ) → 1, then → ±∞ and the space is asymptotically Minkowski.
For large we have G( ) → 1 − 2M ± + Q 2 ± and the space is similar to Reissner-Nordström.
(We shall not need the definitions of M ± and Q 2 ± , but they can be found in [5].) Therefore, allowing the throat at r = a to have time dependence, we have the relations
d dτ = a da dτ = aȧ (3) d 2 dτ 2 = aä −ȧ 2 k 2 3 (4) dG( ) dτ = dG d aȧ = G aȧ (5) From equation (1) we find dt dτ = G( ) 2 + ( 2 + k 2 )ȧ 2 G( ) = G +˙ 2 G(6)
along with the timelike Killing vector
X µ = (−G( ), 0, 0, 0)(7)
and the four-velocity
U µ = ( dt dτ ,˙ , 0, 0)(8)
The extrinsic curvature K τ τ is then given by
D Dτ (X µ U µ ) = −A˙(9)
One can perform the τ derivative, which gives
K τ τ =¨ + G ( )/2 G( ) +˙ 2(10)
It can be seen that this is equivalent to using the replacements
F (r) → G( ) (11) a →˙ (12) a →¨(13)
in the K τ τ found from the metric
ds 2 = −F (r)dt 2 + dr 2 F (r) + r 2 dΩ 2(14)
This makes sense when directly comparing the metrics given in equations (1) and (14).
Similar replacements can be made to recover K θ θ and K φ φ . We find
K θ θ = K φ φ = G +˙ 2 (15) III. f (R) GRAVITY JUNCTION CONDITIONS
We have the energy-momentum on the throat given in terms of the extrinsic curvature
8πGS ij = −κ ij + h ij κ(16)
where
8πGS ij = f − Rf (R) 2 h ij + ∇ i ∇ j f (R) − f (R)h ij(17)
and κ ij and κ are given by the jump discontinuity equations for the extrinsic curvature
κ ij = K + ij − K − ij (18) κ = K + − K −(19)
with K being the trace of the extrinsic curvature
K = K τ τ + K θ θ + K φ φ(20)
In the case of our metric (1) we have the following relations of importance
R tt = 1 2 G 2 G 2 + k 2 + G (21) R rr = − 1 2G 4k 2 G + 2 ( 2 + k 2 )G ( 2 + k 2 ) 2 + G (22) R θθ = 1 − G − G (23) R φφ = R θθ sin 2 θ (24) R = − ( 2 + k 2 )[( 2 + k 2 )G + 4 G − 2] + 2( 2 + 2k 2 )G ( 2 + k 2 ) 2 (25) f (R) = R 2 =⇒ f (R) = 2R (26) K =¨ + 1 2 G + 2G + 2˙ 2 G +˙ 2 (27) IV. STABILITY
Apparently f (R) models are all subject to the junction condition [K] = 0 [8] in order for there to be no jump discontinuity between the two spaces. In our case this becomes
= − G ( ) 2 − 2 (G( ) +˙ 2 )(28)
where we will be interested in the static case where˙ = 0. This leads to
G ( 0 ) = −4 G( 0 ) 0(29)
Defining the potential
V ( ) = −˙ 2(30)
we find
V ( 0 ) = G ( 0 ) − 20 2 0 G( 0 )(31)
Stability is then given by the condition V ( 0 ) > 0.
We can see in Figures (1-4) that regions of stability exist for certain parameters.
V. ORBITS AROUND THE WORMHOLE
Beginning with the explicit form of the metric we examine equatorial orbits with θ = π/2, where we have the geodesic equations
ds 2 = −G( )dt 2 + 1 G( ) d 2 + ( 2 + k 2 )dθ 2 + ( 2 + k 2 )sin 2 θdφ 2(32)0 = d 2 dλ 2 − G ( ) 2G( ) d dλ 2 + G ( )G( ) 2 dt dλ 2 − G( ) dφ dλ 2 (33) 0 = d 2 φ dλ 2 + 2 2 + k 2 dφ dλ d dλ (34) 0 = d 2 t dλ 2 + G ( ) G( ) d dλ dt dλ(35)
The φ equation can be written as
1 2 + k 2 d dλ ( 2 + k 2 ) dφ dλ = 0(36)
Which leads to the constant of motion
( 2 + k 2 ) dφ dλ ≡ L(37)
The t equation can be written as
d dλ G( ) dt dλ = 0(38)
This gives the constant
G( ) dt dλ = E(39)
We also know that the quantity
g µν dx µ dλ dx ν dλ(40)
is a constant along the path parameterized by λ. If dλ = dτ then this is −1 for a massive particle, and it is always equal to zero for a massless particle. Following [10] we will call this constant − , which then gives
− = −G( ) dt dλ 2 + 1 G( ) d dλ 2 + ( 2 + k 2 ) dφ dλ 2 (41) = − E 2 G( ) + 1 G( ) d dλ 2 + L 2 2 + k 2(42)
This can be written as with the effective potential defined as
1 2 d dλ 2 + V ef f ( ) = E 2 2(43)V ef f ( ) = 1 2 G( ) + G( )L 2 2 + k 2(44)
One can then find the stable orbits for V ef f = 0 and V ef f > 0.
VI. TIDAL FORCES
By looking at the difference in acceleration between two neighboring points we are able to determine the tidal forces that a radially infalling observer will experience. To do this we use the equation for the tidal acceleration previously found in [5] given as
∆aĵ = −Rĵαβρuαξβuρ = −Rĵ0k0ξk(45)
where ξ is the separation distance between the two points, measured by our infalling observer, and u is the four-velocity.
The Riemann tensors, which determine the tidal forces for the metric (1) are found in [5], and here we reproduce them in terms of the variable x
R1010 = − G 2k 2 (46) R2020 = R3030 = γ 2 xG k 2 (x 2 + 1) − v 2 γ 2 k 2 (x 2 + 1)G + 2k 2 G 2k 4 (x 2 + 1) 2(47)
In order to determine if the infalling observer will survive the journey through the wormhole, we now examine the tidal forces in the range of parameter values previously found to produce a stable wormhole configuration. As a benchmark, and following the traditional treatment found in [3], we will utilize the acceleration standard of the gravitational acceleration, g been made for the physical significance of conformal gravity [11], which is also quadratic in the curvature.) Hence the solutions and properties thereof discussed here may be relevant to more physically well motivated general theories of quadratic gravity. Here we have found stability conditions for radially perturbed wormholes in R 2 gravity with R = 0 that do not require exotic matter. Properties of stable particle orbits in these solutions then give us hints of what we may expect to find in classes of more general theories that are quadratic in curvature.
FIG. 1 :
1Contour plot of V varying G 0 and /k for the parameter values G ( ) = 1.0 and k = 1.0.
FIG. 2 :FIG. 3 :FIG. 4 :
234Contour plot of G varying G 0 and /k for the parameter values G ( ) = 1.0 and k = 1.0. Contour plot of V varying G 0 and G ( ) for the parameter values /k = 0.5 and k = 1.0. Contour plot of G varying G 0 and G ( ) for the parameter values /k = 0.5 and k = 1.0.
FIG. 5 :
5A plot of V ef f (x) as a function of x ≡ /k with L = 0.1, 0.3, 0.5, 0.7 which are the blue, red, gold, and green curves, respectively, and k = 1.0, G 0 = 1.0, and v 0 = 0.1.
Figure 5
5demonstrates there exists a region of orbital stability due to the presence of a minimum for the effective potential, and it is just on the positive x side as L is lowered towards zero (x = .05 is roughly where the minimum value occurs as L → 0). Unstable orbits are seen to exist when maxima of the effective potential are present.
( 9 .FIG. 6 :FIG. 7 :
9678m/s 2 in SI units), at the surface of the Earth. The absolute values of the Riemann tensors are translated to factors of g as shown in figures (6-9). In these figures, the parameter k set to 1.0 meter implies that the /k values give in meters. Unsurprisingly, for very small values, equivalently small wormhole throats, the tidal forces produced are immense, but decline precipitously with increasing , reaching values sustainable by humans (or human-built spacecraft) at only a few hundred km for these parameter values. This is a plot of the absolute value of R 1010 for the parameter values k = 1.0, G 0 = 0.1 (blue), G 0 = 0.2 (red), and v 0 = −0.9. This is a plot of the absolute value of R 1010 for the parameter values k = 1.0, G 0 = 0.1 (blue), G 0 = 0.2 (red), and v 0 = −0.9.
FIG. 8 :FIG. 9 :
89This is a plot of the absolute value of R 2020 for the parameter values k = 1.0, G 0 = 0.1 (blue), G 0 = 0.2 (red), and v 0 = −0.9.VII. DISCUSSION AND CONCLUSIONWhile R 2 gravity may not correspond to physical reality, it provides a testing ground where we can find analytic solutions and study their corresponding stability. (The case has This is a plot of the absolute value of R 2020 for the parameter values k = 1.0, G 0 = 0.1 (blue), G 0 = 0.2 (red), and v 0 = −0.9.
VIII. ACKNOWLEDGEMENTS J.B.D. thanks Dr. and Mrs. Sammie W. Cosper at the University of Louisiana at Lafayette, and the Louisiana Board of Regents for support. S.C.W. would like to thank Eric M. Schlegel for useful discussions.[1] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], Phys. Rev. Lett. 116, 061102 (2016) [arXiv:1602.03837 [gr-qc]]. [2] M.S. Morris, K.S. Thorne, and U. Yurtsever, Phys. Rev. Lett. 61, 1446 (1988).
. M S Morris, K S Thorne, Am. J. Phys. 56395M.S. Morris and K.S. Thorne, Am. J. Phys. 56, 395 (1988).
Lorentzian Wormholes: From Einstein to Hawking. M Visser, AIP Series in Computational and Applied Mathematical Physics. M. Visser, "Lorentzian Wormholes: From Einstein to Hawking", (1995) AIP Series in Com- putational and Applied Mathematical Physics, Woodbury, USA.
. F Duplessis, D Easson, arXiv:1506.00988Phys. Rev. D. 9243516F. Duplessis and D. Easson, Phys. Rev. D 92, 043516 (2015) [arXiv:1506.00988].
. A Kehagias, C Kounnas, D Lust, A Riotto, arXiv:1502.04192JHEP. 1505143A. Kehagias, C. Kounnas, D. Lust, and A. Riotto, JHEP 1505, 143 (2015) [arXiv:1502.04192].
. H Lu, A Perkins, C N Pope, K S Stelle, arXiv:1502.01028Phys. Rev. Lett. 114171601H. Lu, A. Perkins, C.N. Pope, and K.S. Stelle, Phys. Rev. Lett. 114, 171601 (2015) [arXiv:1502.01028].
. E F Eiroa, G Figueroa Aguirre, arXiv:1511.02806Eur. Phys. J. C. 76132gr-qcE. F. Eiroa and G. Figueroa Aguirre, Eur. Phys. J. C 76, 132 (2016) [arXiv:1511.02806 [gr-qc]].
. E Poisson, M Visser, gr-qc/9506083Phys. Rev. D. 527318E. Poisson and M. Visser, Phys. Rev. D 52, 7318 (1995) [gr-qc/9506083].
Lecture notes on general relativity. S M Carroll, gr-qc/9712019S. M. Carroll, "Lecture notes on general relativity," gr-qc/9712019.
. P D Mannheim, arXiv:1101.2186Found. Phys. 42hep-thP. D. Mannheim, Found. Phys. 42, 388 (2012) [arXiv:1101.2186 [hep-th]].
|
[] |
[
"A Tale of Two Herbig Ae stars -MWC275 and AB Aurigae: Comprehensive Models for SED and Interferometry",
"A Tale of Two Herbig Ae stars -MWC275 and AB Aurigae: Comprehensive Models for SED and Interferometry"
] |
[
"A Tannirkulam ",
"J D Monnier ",
"T J Harries \nSchool of Physics\nUniversity of Exeter\nStocker RoadEX4 4QLExeterUK\n",
"R Millan-Gabet \nMichelson Science Center\nPasadenaCAUSA\n",
"Z Zhu ",
"E Pedretti \nUniversity of St. Andrews\nScotland, UK\n",
"M Ireland \nSchool of Physics\nSydney University\n2006N. S. WAustralia\n",
"P Tuthill \nSchool of Physics\nSydney University\n2006N. S. WAustralia\n",
"T Ten Brummelaar \nCHARA\nGeorgia State University\nAtlantaGAUSA\n",
"H Mcalister \nCHARA\nGeorgia State University\nAtlantaGAUSA\n",
"C Farrington \nCHARA\nGeorgia State University\nAtlantaGAUSA\n",
"P J Goldfinger \nCHARA\nGeorgia State University\nAtlantaGAUSA\n",
"J Sturmann \nCHARA\nGeorgia State University\nAtlantaGAUSA\n",
"L Sturmann \nCHARA\nGeorgia State University\nAtlantaGAUSA\n",
"N Turner \nCHARA\nGeorgia State University\nAtlantaGAUSA\n",
"\nUniversity of Michigan\nAstronomy Dept, 500 Church Street, 1017 Dennison Bldg48109-1042Ann ArborMIUSA\n"
] |
[
"School of Physics\nUniversity of Exeter\nStocker RoadEX4 4QLExeterUK",
"Michelson Science Center\nPasadenaCAUSA",
"University of St. Andrews\nScotland, UK",
"School of Physics\nSydney University\n2006N. S. WAustralia",
"School of Physics\nSydney University\n2006N. S. WAustralia",
"CHARA\nGeorgia State University\nAtlantaGAUSA",
"CHARA\nGeorgia State University\nAtlantaGAUSA",
"CHARA\nGeorgia State University\nAtlantaGAUSA",
"CHARA\nGeorgia State University\nAtlantaGAUSA",
"CHARA\nGeorgia State University\nAtlantaGAUSA",
"CHARA\nGeorgia State University\nAtlantaGAUSA",
"CHARA\nGeorgia State University\nAtlantaGAUSA",
"University of Michigan\nAstronomy Dept, 500 Church Street, 1017 Dennison Bldg48109-1042Ann ArborMIUSA"
] |
[] |
We present comprehensive models for the Herbig Ae stars MWC275 and AB Aur that aim to explain their spectral energy distribution (from UV to millimeter) and long baseline interferometry (from near-infrared to millimeter) simultaneously. Data from the literature, combined with new mid-infrared (MIR) interferometry from the Keck Segment Tilting Experiment, are modeled using an axisymmetric Monte Carlo radiative transfer code. Models in which most of the near-infrared (NIR) emission arises from a dust rim fail to fit the NIR spectral energy distribution (SED) and sub-milli-arcsecond NIR CHARA interferometry. Following recent work, we include an additional gas emission component with similar size scale to the dust rim, inside the sublimation radius, to fit the NIR SED and long-baseline NIR interferometry on MWC275 and AB Aur. In the absence of shielding of star light by gas, we show that the gas-dust transition region in these YSOs will have to contain highly refractory dust, sublimating at ∼1850K. Despite having nearly identical structure in the thermal NIR, the outer disks of MWC275 and AB Aur differ substantially. In contrast to the AB Aur disk, MWC275 lacks small grains in the disk atmosphere capable of producing significant 10-20 µm emission beyond ∼7AU, forcing the outer regions into the "shadow" of the inner disk.
|
10.1086/592346
|
[
"https://arxiv.org/pdf/0808.1728v1.pdf"
] | 45,548 |
0808.1728
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442be3068d32523096b3c5b499a20997354930fe
|
A Tale of Two Herbig Ae stars -MWC275 and AB Aurigae: Comprehensive Models for SED and Interferometry
12 Aug 2008
A Tannirkulam
J D Monnier
T J Harries
School of Physics
University of Exeter
Stocker RoadEX4 4QLExeterUK
R Millan-Gabet
Michelson Science Center
PasadenaCAUSA
Z Zhu
E Pedretti
University of St. Andrews
Scotland, UK
M Ireland
School of Physics
Sydney University
2006N. S. WAustralia
P Tuthill
School of Physics
Sydney University
2006N. S. WAustralia
T Ten Brummelaar
CHARA
Georgia State University
AtlantaGAUSA
H Mcalister
CHARA
Georgia State University
AtlantaGAUSA
C Farrington
CHARA
Georgia State University
AtlantaGAUSA
P J Goldfinger
CHARA
Georgia State University
AtlantaGAUSA
J Sturmann
CHARA
Georgia State University
AtlantaGAUSA
L Sturmann
CHARA
Georgia State University
AtlantaGAUSA
N Turner
CHARA
Georgia State University
AtlantaGAUSA
University of Michigan
Astronomy Dept, 500 Church Street, 1017 Dennison Bldg48109-1042Ann ArborMIUSA
A Tale of Two Herbig Ae stars -MWC275 and AB Aurigae: Comprehensive Models for SED and Interferometry
12 Aug 2008Subject headings: young stellar objects -circumstellar disks -radiative transfer -Monte Carlo codes-dust sublimation -grain evolution -interferometry
We present comprehensive models for the Herbig Ae stars MWC275 and AB Aur that aim to explain their spectral energy distribution (from UV to millimeter) and long baseline interferometry (from near-infrared to millimeter) simultaneously. Data from the literature, combined with new mid-infrared (MIR) interferometry from the Keck Segment Tilting Experiment, are modeled using an axisymmetric Monte Carlo radiative transfer code. Models in which most of the near-infrared (NIR) emission arises from a dust rim fail to fit the NIR spectral energy distribution (SED) and sub-milli-arcsecond NIR CHARA interferometry. Following recent work, we include an additional gas emission component with similar size scale to the dust rim, inside the sublimation radius, to fit the NIR SED and long-baseline NIR interferometry on MWC275 and AB Aur. In the absence of shielding of star light by gas, we show that the gas-dust transition region in these YSOs will have to contain highly refractory dust, sublimating at ∼1850K. Despite having nearly identical structure in the thermal NIR, the outer disks of MWC275 and AB Aur differ substantially. In contrast to the AB Aur disk, MWC275 lacks small grains in the disk atmosphere capable of producing significant 10-20 µm emission beyond ∼7AU, forcing the outer regions into the "shadow" of the inner disk.
-2 -
Introduction
Herbig Ae (HAe) stars are pre-main-sequence stars of intermediate mass (1.5-3 solar masses). They exhibit a robust excess in emission over stellar photospheric values from nearinfrared (NIR) to the millimeter (mm) wavelengths. This excess is now attributed to the passive reprocessing of stellar light by dust in the circumstellar environment (Tuthill et al. 2001;Natta et al. 2001;Dullemond et al. 2001). The geometry of the circumstellar environment of HAe stars has been actively debated in the astronomy community over the last two decades. Some of the early workers in this field (Hillenbrand et al. 1992) showed that spectral energy distribution (SED) of HAe stars could be explained by emission from circumstellar matter in disk-like geometry. Others (Miroshnichenko et al. 1997) argued that the emission could also arise from dust in a spherical geometry around the star, proving the inadequacy of SED modeling alone in uniquely fixing the geometry of the circumstellar matter. The first observational evidence in favor of a disk geometry came from millimeter (mm) interferometry in the form of asymmetries detected (Mannings & Sargent 1997) in the mm images. Asymmetries in the NIR emission were also detected by the Palomar Test-Bed Interferometer (Eisner et al. 2003(Eisner et al. , 2004, settling the debate in support of a disk geometry for circumstellar material in Herbig Ae stars.
Most interferometric studies of HAe stars have relied on simple geometric models (Mannings & Sargent 1997;Millan-Gabet et al. 1999, 2001Eisner et al. 2003Eisner et al. , 2004Monnier et al. 2005) that explain the emission geometry of the system in only narrow wavelength ranges. This method, albeit extremely useful in elucidating some of the morphology details, is not adequate for exploring the interdependency in structure of the inner and outer parts of the disk. A number of studies (Dullemond et al. 2001;Dullemond & Dominik 2004;van Boekel et al. 2005b) have shown that the structure of the inner disk at fractions of an AU scale clearly affects the structure of the outer disk. A complete understanding of the circumstellar disk structure in HAe stars therefore requires models that simultaneously explain the SED and interferometry over a large wavelength range. Such models have begun to appear in the literature only recently (Pontoppidan et al. 2007;Kraus et al. 2008).
In this paper, we develop comprehensive disk models to explain the SED and interferometry of the HAe stars MWC275 and AB Aur. MWC275 and AB Aur are prototype pre-main-sequence stars of similar ages and spectral type with extensive circumstellar disks. Due to the availability of photometric and interferometric data over a large wavelength range, MWC275 and AB Aur are ideal candidates for testing disk models for YSOs. The extent of their circumstellar-dust disks was first measured by Mannings & Sargent (1997) to be several 100AU using the Owens Valley Radio Observatory (OWRO). Natta et al. (2004) resolved the MWC275 disk in the mm and reported a de-convolved, projected dust-disk size of 300AU×180AU. More recently, Isella et al. (2007) analyzed IRAM, SMA and VLA continuum and 12 CO, 13 COand 18 CO line data constraining the gas-disk radius to be 540AU with the gas in Keplerian rotation around the central star. Scattered light studies of MWC275 (Grady et al. 2000) and AB Aur (Grady et al. 1999;Oppenheimer et al. 2008) show the presence of arcs and rings in the circumstellar disk. Corder et al. (2005) resolved the AB Aur CO disk radius to be ∼600AU, finding strong evidence for Keplerian rotation for the bulk of the disk. Corder et al. (2005) and Lin et al. (2006) detected spiral arms in CO emission with radii of ∼150AU, while Fukagawa et al. (2004) detected similar structure in Subaru H-band scattered light images. AB Aur also has substantial envelope material on scales larger then 600 AU (Grady et al. 1999;Semenov et al. 2005;Corder et al. 2005;Lin et al. 2006).
MIR emission probes the giant planet formation region in circumstellar disks (Calvet et al. 1992;Chiang & Goldreich 1997;Dullemond et al. 2001) with the emission arising from warm dust (T > 150K). Meeus et al. (2001) and used the 10µm MIR silicate emission feature from MWC275 and AB Aur to show that dust grains in these systems had grown larger than the typical interstellar medium grain sizes. Mariñas et al. (2006) imaged AB Aur at 11.7µm and found the emission FWHM size to be 17±4 AU consistent with the flared disk models of Dullemond & Dominik (2004). In this paper, we present new 10µm measurements of AB Aur and MWC275 with the Keck Segment Tilting Experiment (Monnier et al. 2004, described in §2). In contrast to AB Aur, the MWC275 disk is unresolved by the Segment Tilting Experiment (maximum baseline of 10m), requiring the VLT Interferometer (100m baseline) to probe to its MIR structure ). These observations suggest that MWC275 disk differs considerably from AB Aur and we present a detailed comparison of the two disk structures in the discussion ( §6).
Thermal NIR emission probes hot regions (typically the inner AU) of the disk with temperatures greater than 700K. The NIR disks of MWC275 and AB Aur were first resolved with IOTA by Millan-Gabet et al. (1999, 2001 and subsequently observed at higher resolution with PTI (Eisner et al. 2004), Keck Interferometer (Monnier et al. 2005) and the CHARA interferometer array ). In Tannirkulam et al. (2008, hereafter T08) we showed that inner-disk models in which majority of the K-band emission arises in a dust rim (Dullemond et al. 2001;Isella & Natta 2005;Tannirkulam et al. 2007) fail to fit the CHARA data at milli-arcsecond resolution. We also demonstrated that the presence of additional NIR emission (presumably from hot gas) inside the dust destruction radius can help explain the CHARA data and the NIR SED. First calculations for the effects of gas on rim structure (Muzerolle et al. 2004) showed that for plausible disk parameters, presence of gas does not modify dust-rim geometry significantly. Besides a poorly understood interferometric visibility profile, MWC275 also displays as yet ill-understood NIR and MIR SED time variability (Sitko et al. 2008) which has been interpreted as variations of the inner -4disk structure. In §4 and §5 we present a detailed analysis of the NIR visibility and SED for MWC275 and AB Aur, placing constraints on the wavelength dependence of the opacity source inside the dust destruction radius.
In this study, we focus on (i) explaining the inner-disk structure and discuss important open problems and (ii) modeling the MIR emission morphology of the disks and the shape of the MIR spectrum. The paper is organized into 7 sections with §2 detailing the observations. §3 explains the disk model and the modeling strategy. §4 and §5 analyze MWC275 and AB Aur SED and visibilities in relation to the disk models. We present a discussion on our results and our conclusions in §6 and §7 respectively.
New Observations and Literature Data
The properties of the circumstellar disks in MWC275 and AB Aur have been constrained using IR and mm interferometry and SED. We include new NIR interferometry from the CHARA array, MIR interferometry from the Keck Segment Tilting Experiment and mm interferometry from the literature in our analysis. Optical and NIR photometry obtained at the MDM Observatories, and mid and far-infrared data from ISO are also included to constrain model SED. We describe the data in detail in the following paragraphs.
K-band (central wavelength of 2.13µm, ∆λ 0.3µm) interferometry on MWC275 and AB Aur, first presented in T08, was obtained using the CHARA array with the 2-beam CHARA "Classic" combiner at the back end (ten Brummelaar et al. 2005). The targets were observed with multiple baselines of CHARA at a variety of orientations in 8 runs between June 2004 and June 2007. The longest baseline observation for MWC275 was 325m (resolution 1 of 0.67 milli-arcseconds) and 320m (resolution of 0.68 milli-arcseconds) for AB Aur. The data were reduced using standard CHARA reduction software (ten Brummelaar et al. 2005) and these results were cross-checked with an independent code developed at University of Michigan. HD164031 , HD166295 and HD156365 with uniform-disk (UD) diameters of 0.83± 0.08 mas, 1.274±0.018 mas and 0.44±0.06 mas (Merand et al. 2005, and getCal -http://mscweb.ipac.caltech.edu/gcWeb/gcWeb.jsp) were used as calibrators for MWC275. AB Aur visibilities were calibrated with HD29645 (UD diameter=0.54±0.07 mas, getCal) and HD31233 (UD diameter=0.76±0.13 mas, getCal). During the reduction procedure the flux ratios between the two interferometer telescopes were carefully monitored for the source and the calibrators. Data points having a flux ratio variation greater than 10% of the mean, -5indicating large changes in instrument alignment, were discarded. Three MWC275 data points were removed by this criterion. The procedure was adopted to minimize calibration errors caused by differences in beam overlap in the source and calibrator. The relative visibility errors which include calibration errors, statistics and uncertainties in calibrator size, are at the ∼6% level, typical for CHARA Classic. In T08, we had shown the reduced data briefly in the form of visibility interferometric-baseline plots. Here we present the complete observing logs with the uv coverage (see Figs 1 and 2) and the calibrated visibilities listed in Tables 1 and 2. NIR interferometry from IOTA , PTI (Eisner et al. 2003(Eisner et al. , 2004 and the Keck Interferometer (Monnier et al. 2005) have also been included to constrain the models.
MW275 and AB Aur were observed with the Keck Segment Tilting Experiment to study their MIR emission morphology. Four subsets of Keck primary mirror segments were repointed and rephased so as to achieve four separate sparse aperture Fizeau interferometers, each with a separate pointing origin on the LWS detector (Jones & Puetter 1993). The details of the experiment and the data reduction procedure are provided in Monnier et al. (2004) and Monnier et al. (2008, in prep). The calibrated data are listed in Table 3. In addition to the Keck Segment Tilting data, we also include MWC275 MIR intereferometry from VLTI-MIDI ) in our analysis.
Millimeter interferometry from Mannings & Sargent (1997), Natta et al. (2004), Semenov et al. (2005), Lin et al. (2006) and Isella et al. (2007) are used to constrain the circumstellar disk masses and disk sizes and surface density profile. In conjunction with the interferometry, the shape of the mm SED provides information on sizes of the dust grains in the bulk of the circumstellar disks.
To constrain the SED computed from radiative transfer models we obtained optical and NIR photometry on MWC275 and AB Aur with the MDM 2.4m telescope at Kitt Peak. Due to the high brightness of our targets, we defocussed the telescope during observations to avoid camera saturation. After bias correction, flat fielding, and background subtraction, the reduced images were used to obtain source counts using ATV -an aperture photometry code (Barth 2001). Targets were calibrated using Landoldt standards (Landolt 1983). The calibrated UBVRIJHK photometry are listed in Tables 10 and 11 in the Appendix. We also include photometry for a number of other YSOs that we observed. NIR photometry from Sitko et al. (2008) , mid and far-IR SED from Meeus et al. (2001) and mm fluxes (Mannings & Sargent 1997;Natta et al. 2004;Semenov et al. 2005;Lin et al. 2006) were also used in the model analysis. modeled the 10µm spectra of a large sample of Herbig Ae stars and derived the mineralogy and typical grain sizes responsible for the emission. We use results from for fixing the composition of dust -6grains in the atmosphere of the MWC275 and AB Aur disks.
A compilation of stellar properties and circumstellar disk properties extracted from the literature is listed in Tables 4, 5, 6 and 7.
Circumstellar Disk Modeling
The circumstellar material around MWC275 and AB Aur is modeled as a passive dust disk reprocessing stellar radiation (Dullemond et al. 2001). The disk is in thermal and hydrostatic equilibrium and can be divided into 3 distinct regions (Fig 3) -
• Curved Inner Rim Sufficiently close to the star (distance depends on stellar luminosity and dust sublimation temperatures), dust in the circumstellar disk reaches sublimation temperatures and evaporates. Inside of the evaporation radius, the disk is optically thin. The truncated dust disk is frontally illuminated by the star and forms a 'rim' whose shape depends sensitively on dust properties (Isella & Natta 2005;Tannirkulam et al. 2007). The rim puffs up, traps a significant fraction of stellar photons and re-radiates predominantly in the NIR (Dullemond et al. 2001).
• Shadow Region The inner rim casts a geometric shadow on the region behind it (Dullemond et al. 2001;Dullemond & Dominik 2004), preventing it from receiving direct star light. The shadow is heated by scattered photons from the rim edge and through diffusion. The size of the shadow depends on the rim geometry, mass of dust in the outer disk and dust grain properties in the outer disk.
• Flared Disk If sufficient dust is present in the outer disk, the disk eventually emerges from the shadow and "flares". The flared disk emits radiation in the MIR and longer wavelengths.
Details of the modeling procedure and comparison to data are described below.
The Monte Carlo Transfer Code -TORUS
The calculations in this paper were performed using the torus Monte-Carlo radiativetransfer code (Harries 2000;Harries et al. 2004;Kurosawa et al. 2004;Tannirkulam et al. 2007). Radiative equilibrium is computed using Lucy's (Lucy 1999) algorithm on a twodimensional, cylindrical adaptive-mesh grid. The initial density structure for the disk calculations is based on the canonical description of the α-disk developed by Shakura & Sunyaev -7 -(1973), viz
ρ(r, z) = ρ 0 r r 0 −α exp − 1 2 z 2 h(r) 2(1)
where r is the radial distance in the mid-plane, r 0 is some characteristic radius, z is the distance perpendicular to the mid-plane, and h(r) is the scaleheight, given by
h(r) = h 0 r r 0 β(2)
with parameters of α = 2.125 and β = 1.125, giving a radial dependence of the surface density of Σ(r) ∝ r −1.0 . Once the temperature (we assume that the disk is in local thermodynamic equilibrium passively heated by the star, and gas and dust are thermally coupled) and dust sublimation (see next paragraph) structures have converged using the Lucy algorithm, the vertical disk structure is modified via the equation of hydrostatic equilibrium following a similar algorithm to that detailed by Walker et al. (2004). A self-consistent calculation for dust sublimation and disk temperature followed by a hydrostatic equilibrium calculation is repeated until the disk density structure has converged. Convergence is typically achieved in four iterations. Images and SEDs are subsequently calculated using a separate Monte Carlo algorithm based on the dust emissivities and opacities (Harries 2000).
The shape of the dust evaporation front is computed following Tannirkulam et al. (2007). The dust content is first reduced to a very low value in the computational grid for the circumstellar disk, to make each of the grid cells optically thin. Stellar photons then propagate through the disk and the temperature of grid cells is determined. Dust is added to cells that are cooler (see eqn 3 for sublimation temperature parameterization) than the sublimation temperature in small steps of τ . The step size is a τ of 10 −3 (computed at 5500Å) for the first five dust growth steps. The step size is then increased logarithmically, first to 10 −2 , then to 10 −1 and so on until a 100:1 gas to dust ratio is reached in each grid cell. The grid cell temperatures are recomputed after every dust growth step and the process is repeated until the shape of the dust sublimation region converges.
We use Kurucz (Kurucz 1970) stellar atmosphere models as input spectra for the stars. We consider a mixture of 0.1,1.3 and 50 micron warm silicates (Ossenkopf et al. 1992) and power-law opacity mm grains (Mannings & Sargent 1997;Natta et al. 2004) to model the opacity in the disk. The micron and sub-micron grain mixture is based on work by Meeus et al. (2001) and . To keep the analysis simple, the grain species are assumed to be well mixed with gas following a delta function size distribution. Dust scattering is not included in the models. Scattering does not seem to have significant impact on disk structure and interpretation of infrared interferometry for HAe stars (Dullemond & Natta 2003;Pinte et al. 2008).
During the course of our modeling and as outlined in Tannirkulam et al. (2008), we realized that an additional emission component (Fig 3), which we argue to be gas, is needed inside the dust destruction radius to explain the NIR SED and interferometry of MWC275 and AB Aur (see section 4.1.2 for discussion on gas opacity). This additional component is not treated self consistently in our modeling and is added after the dust-opacity-dominated circumstellar-disk model converges in structure. Calculations by Muzerolle et al. (2004) have shown that for parameters suitable to MWC275 and AB Aur, gas does not significantly alter the structure of the dust rim, justifying our simple treatment for the NIR emission geometry.
In sections 4.1.1 and 5.1, we demonstrate that the NIR emitting region in MWC275 and AB Aur has a radius of ∼0.2AU. Furthermore, long-baseline interferometry beyond the first visibility minimum constrains the gas and dust emission to be on similar length scales (Fig 7). The two statements together imply that in the absence of shielding of the evaporation front by gas, the mid-plane sublimation temperature in the dust rim is ∼1850K (see section 4.1.1).
Comprehensive models for SED and Interferometry
To fit the SED and visibilities of MWC275 and AB Aur we adopted the following algorithm: First, we computed models for the dust evaporation front as described in §3.1. The inner edge of the dust disk is assumed to be dominated by grains larger than 1 micron (Tuthill et al. 2001;Isella et al. 2006) and the evaporation front shape is set by the density dependence of dust sublimation temperatures (Isella & Natta 2005;Tannirkulam et al. 2007). The K-band visibilities are computed for the model and compared with data. The normalization of the dust evaporation law is then adjusted so that the model visibilities fit the visibility data before the first visibility minimum. These models fail to fit the visibility beyond the minimum and do not have sufficient emission to explain the observed NIR SED. Therefore an additional emission component has been added inside the dust sublimation radius to reconcile the model with the visibility data and NIR SED.
With the inner-rim parameters fixed, we next focus on MIR and the mm disk. Millimeter interferometry results from the literature are used to fix disk masses and sizes. The majority of the dust mass is placed in mm sized grains with a power-law opacity function (Natta et al. 2004). A small fraction (∼10%) of the dust mass is in micron and sub-micron (small) grains with their relative mass fractions based on literature results (Meeus et al. 2001;. The physical extent of small grains is constrained with MIR imaging and interferometry. The model is then allowed to run to convergence. The model SED is compared with MIR and far-infrared data, the mass of the small grain population is then adjusted and models are recomputed until a good fit to the MIR and far-infrared SED -9is obtained.
The MIR visibilities are computed for the SED-converged model and compared with the data and the spatial distribution of the small grain component is adjusted until model visibilities match with data. The only free parameters in our models are the absolute masses of the small grains, mass of the 50µm silicate grain and their spatial distribution. Each of the models is computationally expensive. To achieve fast convergence, the parameter space was varied by hand, until a good fit was found for the observable quantities.
MWC275: Analysis
MWC275 is a Herbig Ae star (refer to Table 4 for basic properties and photometry) with a total luminosity of 36L ⊙ . The large stellar luminosity, coupled with the fact that the mass accretion rate is ≤ 10 −7 M ⊙ /year (Garcia Lopez et al. 2006) allows us to ignore accretion heating and model the MWC275 circumstellar disk as a passive disk, reprocessing stellar radiation (Chiang & Goldreich 1997;Dullemond et al. 2001). For our models, we choose the MWC275 disk mass to be between 0.05-0.1 M ⊙ (Natta et al. 2004) and a surface density profile that varies radially as r −1 (Isella et al. 2007). The disk outer edge is truncated at 200AU and bulk (∼80%) of the dust mass is assumed to reside in mm grains having an opacity with a wavelength dependence of λ −1 at long wavelengths. Here, we describe in detail our modeling results for the NIR and MIR morphology of MWC275.
The Thermal NIR Disk
Visibilities
Like many other Herbig Ae stars, MWC275 shows a strong NIR excess over stellar photospheric values (Hillenbrand et al. 1992). This excess has been traditionally interpreted in terms of the dust disk being truncated by sublimation and forming a 'rim'. The rim intercepts stellar photons, re-radiatiing predominantly in the NIR (Dullemond et al. 2001;Isella & Natta 2005;Tannirkulam et al. 2007). However, in T08 we had conclusively shown that models in which all of the NIR excess arises from dust rims alone cannot explain the CHARA interferometry data on MWC275. Our arguments in T08 were necessarily brief. We present a more elaborate analysis in this section.
B ef f = B projected cos 2 (θ) + cos 2 (φ)sin 2 (θ)
where θ is the angle between the uv vector for the observation and the major axis of the disk and φ is the inclination of the disk. Under the flat disk assumption, the effective baseline correctly accounts for the change in resolution due to the disk inclination and PA (the geometry of thick disks is represented only approximately with optical depth effects and 3-D geometry of thick disks not being taken into account), allowing us to plot the visibility measurements as a function of one coordinate, simplifying presentation and analysis.
We attempt at fitting the visibilities with a curved inner-rim model (the "standard" model) where the rim curvature (variation in cylindrical radius between rim midplane and the atmosphere) is set by the density dependence of dust sublimation temperatures, taken from Pollack et al. (1994). In this model, silicate grains sublimate at a higher temperature compared to other grains and hence fix the rim location. The rim is assumed to be composed of 1.3µm grains, as larger grains do not affect the rim shape and location significantly (Isella & Natta 2005), at the same time making numerical convergence slower due to strong back-warming effects (Isella & Natta 2005;Tannirkulam et al. 2007). For silicate dust, the evaporation temperature T evp can be parameterized as
T evp = G ρ gas (r, z) 1g cm −3 γ(3)
where G = 2000 K, γ = 1.95 × 10 −2 and ρ gas is the density of gas in g cm −3 (see IN05 eq.
[16]). This parameterization, derived from a fit to sublimation temperatures recorded in the laboratory (Pollack et al. 1994), produces a dust rim with an inner edge at 0.36 AU (Fig 6a). The rim radius is too large to fit even the relatively short baseline visibility data from IOTA (Figs 4 and 5). In order to fit the data before the first visibility minimum, we had to increase the T evp normalization-G by ∼30% from 2000K to 2600K. This increases the sublimation temperature at the base of the rim from ∼1350K to ∼1800K. However, as seen in Figs 4 and 5, rim models which are sharply truncated due to dust sublimation and produce all of the NIR excess, fail to fit observations beyond the first visibility minimum. These models display bounces in visibility at long baselines (not seen in the data) because of the presence of sharp ring-like features with high spatial frequency -11components in the corresponding images, even for the smoothest rims physically plausible. In T08, we showed that the presence of a smooth emission component inside the dust destruction radius (Fig 6c) providing 56% of the total K band emission helps fit the data well (solid line in Figs 4 and 5). The NIR visibility data cannot constrain the surface brightness profile of the smooth emission component (we have adopted a constant surface brightness profile -a Uniform Disk for simplicity), but can constrain the size scale of the emission fairly robustly. Fig 7 shows a series of visibility curves where the smooth emission component is fixed to be 56% of the total emission and the radius of the Uniform Disk component is decreased by 15% successively from the initial radius of 0.23AU radius. The model image is then rescaled to maintain a good visibility fit at baselines shorter than 100m. It can be seen in Fig 7 that for Uniform Disk (UD) radii smaller than 0.19AU, the model visibilities begin to deviate significantly from the observations. Thus, the CHARA data constrains the smooth emission component to be on the same length scale as the dust sublimation rim filling the region between the disk and the central star.
SED
Fig 8 shows the NIR SED for MWC275. Besides failing to explain the NIR interferometry, the standard model also fails at producing sufficient NIR emission to explain the MWC275 SED even in its 'low' state. In T08, we had shown that binarity and source variability cannot account for the discrepancy between the standard model and data. We had argued that the presence of smooth emission inside the dust destruction radius can help explain the NIR visibility and account for the "missing" NIR flux in standard models. Opacity candidates for the smooth emission component are: (1) a dusty halo around the stars (Vinkovic et al. 2006a) and (2) gas inside the evaporation front. However, to fit the visibility data, the halo emission would have to be closer to the star than the dust destruction radius in the disk. This would require even higher dust-sublimation temperatures than the ∼ 1850K we are adopting.
The most plausible physical mechanism for the smooth emission is hot gas. The required emission levels to explain the long-baseline K-band visibility data can be obtained with optically thin gas (τ ∼ 0.15) with a temperature range of 2000K-3000K (Muzerolle et al. 2004;Eisner et al. 2007a;Eisner 2007b). Assuming that the gas has sufficient opacity to produce the difference in flux between the standard model and the observed photometry, we can place limits on the wavelength dependence of gas opacity. Fig 9 plots limits on the gas opacity (normalized at K band) such that flux from the gas component + the standard model falls within 10% of the observed photometry (we have assumed that gas does not significantly alter the geometry of the dust rim). In the next 2 paragraphs we compare theoretical gas-opacity curves with our empirically derived opacity from SED. Fig. 9 shows the wavelength dependence of molecular (Ferguson et al. 2005;Zhu et al. 2007) and free-free+free-bound (henceforth FF-BF) opacity, both good candidates for the gas emission (refer T08). At 5000K, FF-BF opacity (Ferguson et al. 2005) agrees well with the derived opacity at long wavelengths but overshoots limits shortward of 2µm. At temperatures greater than 8000K , FF-BF opacities rise quickly with wavelength producing excessive midinfrared light.
Theoretical molecular opacities compare fairly with empirical derivation between 1µm and 4µm. Beyond 4µm , theoretical molecular opacities rise rapidly with wavelength. However, the observed SED can be matched with models only if the gas opacity is flat between 4µm and 9µm. Also at 2000K and 2500K, strong opacity bands of CO and water vapor are present at 2.5µm and 5µm respectively, which have not been observed in MWC275. This suggests that if molecules were contributing to the bulk of NIR opacity, then some of the species providing opacity between 4-8 µm in Ferguson et al. (2005) and Zhu et al. (2007) are being destroyed in the vicinity of Herbig Ae star MWC275. We note that FF-BF opacities seem to better fit the empirically derived values than molecular opacity. Sitko et al. (2008) have obtained fairly dense time coverage on the NIR and MIR SED of MWC275. The NIR SED shows variability at the 20% level. During the same period, the flux in the visible shows no detectable change, indicating that that stellar luminosity remained fairly constant. Sitko et al. (2008) interpret their observations as variations in the structure of the thermal NIR disk. A variation in the NIR morphology of MWC275 was also detected in the interferometry. The NIR disk size deduced from the Keck Interferometer data (April 2003 epoch, Figs 4 and 5) is ∼20% larger than the size obtained with the CHARA data (June 2004-Aug 2006 epochs). The size determined from the S2W1 June2007 data also differs at the ∼25% level from the size obtained from earlier CHARA epochs. These variations are poorly understood and could be caused by changes in size/geometry, mass accretion rate and gas content in the inner disk. More evidence for MWC275 variability was recently reported by Wisniewski et al. (2008), who found changes in scattered light intensity between 1998 and 2003-2004.
MIR SED and Emission Morphology
van analyzed the MIR SED (Fig 10) of MWC275 in detail and showed that the SED could be reproduced well with a grain mixture of 1.5µm and 0.1µm -13silicates with mass ratio of 4:1. We use results from in fixing the small grain composition in our disk models.
As seen in Fig 10, the MWC275 SED falls sharply between 20µm and 30µm. This drop and the 10µm silicate feature can be simultaneously reproduced only if the mass fraction of the small grain dust component relative to gas beyond 7AU is less than 20% of the mass fraction inside of 7AU. If the small grain component is allowed to exist beyond 7AU, then the model far-infrared spectrum becomes much stronger than observed SED. Fig 10 shows a TORUS model SED that fits the MIR and longer wavelength spectrum of MWC275 well. In this model, 40% of the 8µm emission arises from the dust rim, with the rim contribution declining to ∼20% at 13µm. This model also fits the MIDI-VLTI MIR visibilities ) and reproduces the 0.8±0.1AU 11µm FWHM minor-axis size of MWC275 (Fig 11), naturally explaining why MWC275 is unresolved by the Keck Segment Tilting Experiment. The quality of the SED and visibility fit in the 8-15µm region is only moderate, probably due to the simple dust composition and distribution that we have assumed in the model. The initial model setup has been chosen to reproduce MWC275 mm-interferometry. Table 8 lists disk parameters for the MWC275 model and Fig 12 shows the radial distribution of the small grain fractions. The mid-plane temperature profile and the "flaring" geometry of the disk surface are shown in Fig 13. The dust rim "shadows" (Dullemond et al. 2001;Dullemond & Dominik 2004) the region of the disk between 0.3 and 1AU beyond which the disk begins to flare. The τ =1 surface drops down in scale height steeply after 6.5AU where the small grain fraction reduces sharply. Our conclusions on dust-grain distribution in the MWC275 disk are consistent with that of Sitko et al. (2008).
AB Aur: Analysis
AB Aur is a Herbig Ae star (refer to Table 5 for basic properties and photometry) with a total luminosity of 47L ⊙ (Isella et al. 2006). As in the case of MWC275, AB Aur's large stellar luminosity dominates the circumstellar disk's energy budget (Garcia Lopez et al. 2006, accretion rates ≤ 10 −7 M ⊙ /year). This allows us to ignore accretion heating and model the AB Aur circumstellar disk as a passive disk, reprocessing stellar radiation (Chiang & Goldreich 1997;Dullemond et al. 2001). For our models, we choose the AB Aur disk mass to be between 0.007-0.013 M ⊙ (Lin et al. 2006) and a surface density profile that falls radially as r −1 (Corder et al. 2005). The disk outer edge is truncated at 300AU and the bulk (∼80%) of the dust mass is assumed to reside in mm grains with an opacity that depends on wavelength as λ −1 for long wavelenghts. Here, we describe in detail our modeling results for the NIR and MIR morphology, and SED of AB Aur.
-14 -
The Thermal NIR Disk
We follow the procedure outlined in §4.1.1 to model the SED and visibilities of AB Aur. We first attempt at fitting a standard curved dust rim model (Fig 14) to the NIR visibilities. The rim is assumed to be composed of 1.3µm silicate grains and the dust evaporation temperature law is described by equation (3). This produces rim radii too large to fit baselines shorter than 100m and we had to increase the T evp normalization to 2800K from 2000K. This increases the sublimation temperature at the base of the rim from ∼1350K to ∼1950K. The dashed line in Fig 15 traces the visibility for this model and provides a good fit to the data at baselines shorter than 100m.
The dust-rim-only model produces large bounces in visibility beyond 150m and as in the case of MWC275, this bounce is not observed. We have scanned the 150m-300m baseline (Fig 15) range several times with CHARA and have failed to detect fringes, ruling out dustrim only models for the AB Aur NIR emission. The addition of a Uniform Disk of emission interior to the dust destruction radius (Fig 14c) helps fit the data well (solid line in Fig 15). The gas component also helps fit the NIR SED (Fig 16). Parameters for the 'dust rim + Uniform Disk' model are listed in Table 9. Liu et al. (2007) interpreted their result in terms of the AB Aur circumstellar environment being more complicated than a disk. Since AB Aur is well resolved by the Keck Segment Tilting Experiment (Fig 17), a disk inclination of 45 o -65 o would have produced observable size difference between the major and minor axis of the disk. We do not find evidence for this size variation in our Segment Tilting data, and hence support a face on model for the mid-infrared disk around AB Aur consistent with the NIR and mm results.
MIR SED and Emission Morphology
The MIR spectrum of AB Aur in the 10.7µm to 20µm range can be modeled well with a dust grain mixture of 1.3µm and 0.1µm silicates with equal mass fractions . In addition to the micron and sub-micron silicates, we include a 50µm silicate component to model the relatively flat spectrum of AB Aur between 35µm and 80µm .
Fig 18 shows a TORUS model SED that fits the MIR and longer wavelength spectrum of AB Aur well . In this model, ∼40% of the 8µm emission arises from the dust rim, with the rim contribution declining to ∼10% at 13µm. This model also fits the Keck Segment -15 -Tilting data visibilities reproducing the 10.5±0.7AU 10µm FWHM size of AB Aur (Fig 17). By initial design, the model fits AB Aur mm-interferometry and SED. Table 9 lists disk parameters for the AB Aur model and Fig 12 shows the radial distribution of the small grain fractions. The mid-plane temperature profile and the τ =1 surface at 5500Å are shown in Fig 13. The inner rim shadows the disk between 0.3AU and 1AU, beyond which the disk surface takes a flared geometry.
Discussion
The simultaneous modeling of the infrared and millimeter SED and interferometry of MWC275 and AB Aur allows us to address several important issues regarding the structure of their circumstellar disks. To maintain clarity in our discussion we divide the disk into two regions (i) thermal NIR region (< 0.3 AU) (ii) outer disk (between 0.3AU and the disk outer edge).
The Thermal NIR Disk
Detailed modeling ( §4.1.1 and §5.1) of the inner disk shows that models where bulk of the NIR emission arises in a dust rim truncated by sublimation fail to fit the longbaseline interferometry data and under-estimate the NIR emission by a factor of 2 relative to observations. As mentioned in T08 and demonstrated in detail in this work, the presence of a gas emission component inside the dust destruction radius can solve the interferometry and SED problem simultaneously.
This however opens up a number of new questions, namely (i) What is the geometry of the gas dust transition region? To date there has been no calculation of transition region structure that treats both gas and dust simultaneously in a self consistent manner. (ii) What are the relative contributions of accretion and stellar radiation to heating the gas? We have shown that an ad-hoc addition of an NIR emission component inside the dust destruction radius helps explain the data, but the current modeling does not shed any light on the energy budget question. (iii) What are the gas species that provide the NIR opacity? Is the opacity molecular in nature or is it from free-free and free-bound processes? If a significant portion of the NIR emission were indeed arising from molecular gas, then Fig 9 shows that theoretical gas opacities depend much more sensitively on wavelength between 4 and 10µm than what is observed. This suggests that some of the molecules providing the model opacities might be getting destroyed by the stellar UV radiation field.
-16 -In the course of modeling the MWC275 and AB Aur disks, we realized ( §4.1.1) that the observed K-band sizes could be reproduced only if the dust sublimation temperature at the base of the dust rims were increased to ∼1850K from the experimentally measured silicate evaporation temperatures of ∼1400K (Pollack et al. 1994). A simple treatment of the gas-dust transition region by Muzerolle et al. (2004) suggests that gas is not effective in modifying rim geometry. In the absence of shielding by gas, the large dust sublimation temperatures indicate that the grains in the inner disks of young stars are significantly more refractory and/or optically transparent than has been assumed in the literature. There is also the possibility that the gas gets optically thick along the mid-plane, shielding the dust from direct stellar radiation and allowing the dust rim to exist closer to the star (Monnier et al. 2005;Isella et al. 2006). Future, high resolution NIR spectroscopic studies of MWC275 and AB Aur, combined with self consistent models of the gas density and temperature structure, will help address many of the questions raised here.
MWC275 and AB Aur require gas emission to explain the their SED interferometry. In contrast, past modeling work by Isella et al. (2006) seems to suggest that dust rims alone are probably sufficient to explain the NIR data on the young stars V1295 Aql (A2 IVe) and CQ Tau (F2 IVe). A larger sample of young stars will therefore have to be observed with milliarcsecond interferometry to establish and understand trends between spectral type, stellar mass, accretion rates and the contribution of gas emission to NIR SED.
A new and exciting observational domain will be opened with the commissioning of the fringe tracker for CHARA-MIRC (Monnier et al. 2007) in the summer of 2008. This will sufficiently improve CHARA-MIRC sensitivities to combine light from 3 or more telescopes, allowing the first milli-arcsecond non-parametric imaging of MWC275 and AB Aur in the NIR. The snapshot multiple-baseline coverage will provide us a powerful tool in understanding the infrared time variability of YSO disks.
The Outer Disk
Our models for the MWC275 and AB Aur MIR interferometry and SED suggest that the outer disks of these systems are at different evolutionary stages. MWC275 10 micron size and MIR SED can only be reproduced if the disk is depleted in micron and sub-micron sized grains beyond ∼7AU ( §4.2). This meshes well with the fact that the observed 10.7µm size of MWC275 is ∼3 times smaller than AB Aur. The depletion of small grains beyond 7AU in the disk atmosphere indicates that the dust particles in MWC275 have undergone significant settling. However, the presence of the 10µm silicate feature in MWC275 implies that there is some process (like planetesimal collisions) that maintains the supply of micron -17sized grains in the inner regions of the disk.
Our models predict that the inner dust rim shadows (Dullemond & Dominik 2004) the region of the disk between 0.3AU and 1AU. The structure and size of the shadow depends sensitively on the composition of grains in the circumstellar disk and hence is an important probe of dust physics. The presence of the shadow has not been observationally confirmed yet in any YSO system, although some indirect evidence has been found in VV Ser (Pontoppidan et al. 2007).
Conclusions
We have presented the first set of comprehensive disk models for the SED and interferometry of Herbig Ae stars MWC275 and AB Aur. We have shown that 'standard' models for the dust evaporation front where the bulk of the near-infrared emission arises from a dust wall, fail to explain the near-infrared spectral energy distribution and interferometry. Standard models produce large bounces in visibility at high spatial frequency, which is not observed in the data. We have conclusively demonstrated that the presence of an additional smooth emission component (presumably hot gas) inside the dust destruction radius and on a similar size scale to the dust rim can ameliorate the situation. In the absence of shielding of star light by gas, we have established that dust grains in the gas-dust transition region will have to be highly refractory, sublimating at 1850K. The small mid-infrared size of MWC275 relative to AB Aur, shows that the dust grains in the outer disk MWC275 are significantly more evolved/settled than the grains in the AB Aur disk. We suggest that dynamical processes (like planetesimal collisions) that maintain the population of micron-sized grains producing the 10µm feature in the spectrum, are operational only in the inner 7AU of MWC275. However, in AB-Aur the small-dust producing mechanisms exist at least out to 20 AU and maybe even beyond. Tannirkulam et al. (2008). The star contributes 10% of the K-band flux and an extended envelope ) contributes 5%.
-25 - Tannirkulam et al. (2008). The star contributes 8% of the K-band flux and an extended envelope ) contributes 5%. iota data KI data chara data S2W2 " " E1W1 " " S1W1 " " E2S2 " " W1W2 "
" S2W1
PA of MWC 275 on sky IOTA data PTI data chara data E1S1 " " E2W2 " upper limits Fig. 2.-uv coverage for AB Aur. We include data from PTI (Eisner et al. 2004), IOTA and CHARA ) in our analysis. Dust-rim + gas model iota data KI data chara data S2W2 " " E1W1 " " S1W1 " " E2S2 " " W1W2 " " S2W1 iota data KI data chara data S2W2 " " E1W1 " " S1W1 " " E2S2 " " W1W2 " " S2W1 Table 4). The 'squares' and 'diamonds' are high and low state measurements from Sitko et al. (2008). The solid line is the SED produced by the 'star + dust-rim only' model in Fig 6b. The dotted line is the SED of the star. Fig. 9.-The plusses (+) represent empirically derived gas opacities from observed photometry and NIR disk models for MWC275 (see Fig 6 ). a) Top panel. The stars represent fiducial theoretical molecular absorption opacities smoothed over the photometry band for 2000K and 2500K gas respectively (Zhu et al. 2007). The opacity jump at 5µm is due to water vapor. b) Bottom Panel. The gas absorption opacity at infrared wavelengths is dominated by free-free and free-bound transitions of H − at 5000K and by hydrogen at 8000K (Ferguson et al. 2005;Zhu et al. 2007). Fig. 12.-Mass fractions of dust components relative to gas. The micron and sub-micron grain fraction in MWC275 (red solid and dotted lines) have to be reduced below 20% of their values inside of 6.5AU at larger radii to fit the SED and interferometry. The silicate-grain opacities are from Ossenkopf et al. (1992) and the relative masses of dust grains are from . Between 0.9AU and 1.6 AU, 0.1µm grains are added smoothly to avoid the formation of two distinct dust rims. Bulk of the dust mass is in mm sized grains with a power law opacity profile (Natta et al. 2004). For AB Aur, we also add a 50µm silicate component to improve SED fits between 40µm and 100mum. The dust parameters are derived assuming gas and dust are well mixed. (Table 5). The solid line is the SED produced by the 'star + dust-rim only' model in Fig 14b. The dashed line traces the stellar SED. The dotted line includes emission from gas at 2500K, assuming that the gas opacity curve derived for MWC275 (see Fig. 9) is valid for AB Aur as well. A.2. "Effective Baselines" as a tool in characterizing visibility information on MWC275.
Let B projected be the projected interferometric baseline and let V(B projected ) be the visibility for a circularly symmetric brightness distribution. For a flat disk inclined at angle φ and oriented at some PA, we plotted V(B ef f ) in Fig 5. The effective baseline B ef f is defined as -B ef f = B projected cos 2 (θ) + cos 2 (φ)sin 2 (θ)
where θ is the angle between the uv vector for the observation and the major axis of the inclined disk and φ is the inclination of the disk (0 o inclination is face on). Effective baselines account for the decrease in interferometric resolution due to the inclination of the disk in the sky. They capture the geometry of flat disks correctly, but the geometry of finitely thick disks is represented only approximately (optical depth effects and 3-D geometry of thick disks are not taken into account). Here, we argue that effective baselines are good (albeit approximate) tools for capturing details of the MWC275 disk geometry.
In order to determine the inclination angle and sky orientation of the disk, we adopted the following procedure. MC275 visibility values measured with W1W2, S2W2 and E2S2 CHARA-telescope-pairs are close to and just prior to the first minimum in the visibility curve (see Fig 5). In this region the visibility-baseline relation for the emission models in −9 degrees determined in Wassell et al. (2006). The excellent agreement in inclination and PA values for MWC275 from two independent methods strongly supports a disk model for MWC275, validating the use of "effective baselines" to plot MWC275 visibilities.
MWC275 observations allow us to clearly detect the asymmetry of the MWC275 disk (see Appendix A.2), as havinginclination=48 o ±2 o , PA=136 o ±2 o , consistent with the incli-, PA of 139 o ±15 o determined inWassell et al. (2006) and inclination of 46 o ± 4 o , PA of 128 o ±4 o determined inIsella et al. (2007). The complete visibility data along each of the baselines are presented inFig 4.Following T08, we show the data in a concise manner inFig 5 usingthe notion of an "effective baseline" -
Fig 6b showsthe synthetic K-band image for the rim with the increased normalization. The dashed line inFig 4 tracesthe visibility for this model and provides a good fit to the short baseline (< 100m) data.
Liu et al. (2007) resolved the AB Aur disk at 10.3µm using nulling interferometry and measured a disk is inclination of 45 o -65 o inconsistent with nearly face on measurements in the mm(Corder et al. 2005) and the NIR(Millan-Gabet et al. 2001;Eisner et al. 2004).
AT acknowledges contributions from Nuria Calvet, Michael Busha, Marlin Whitaker and Steve Golden. Research at the CHARA array is supported by the National Science Foundation through grants AST 06-06958 and AST 03-52723 and by the Georgia Sate University through the offices of the Dean of the College of Arts and Sciences and the Vice President for Research. This project was partially supported by NASA grant 050283 and NSF grant AST 03-52723 . This publication makes use of NASA's Astrophysics Data System Abstract Service. CHARA visibility-calibrator sizes were obtained with the fBol module of getCal, a package made available by the Michelson Science Center, California Institute of Technology (http://msc.caltech.edu). Computations were performed on the Legato-Opus -18 -Cluster Network at the University of Michigan.
Fractions of micron and sub-micron grains in the disk atmosphere c
Fig. 1 .
1-uv coverage for MWC275. We include data from KI(Monnier et al. 2005), IOTA) and CHARA) in our analysis. A position angle (measured East of North) of 136 o for MWC275 is marked in the left panel.
Fig. 3 .Fig. 4 .
34-Schematic of disk models. a)Top panel. Flared disk with a curved inner rim. b) Bottom panel. An additional "smooth" emission component (presumably gas) has been added inside the dust destruction radius to explain MWC275 and AB Aur NIR photometry and interferometry. Note: The models are not to scale. -MWC 275 visibility data and model curves. The quoted model temperatures are at the base of the dust rim. The NIR size deduced from the Keck Interferometer data (triangles) is ∼20% larger than the size obtained with the CHARA data. This variability of MWC275 is discussed in §4.1.2.
Fig. 5 .Fig. 6 .
56-MWC275 visibility vs 'Effective Baseline'. Effective Baselines are useful in presenting data along multiple uv vectors in a concise manner (under the assumption of axial symmetry). The NIR size deduced from the Keck Interferometer data (triangles) is ∼20% larger than the size obtained with the CHARA data. This variability of MWC275 is discussed in §4.-Inclined-disk models for NIR emission in MWC275. The disk has an inclination of 48 o and a PA of 136 o (North is towards the top and East is on the left). The sense of the inclination is from Grady et al. (1999) a) Top left panel. A standard curved dust-rim-only model with rim-base temperature ∼1350K. b) Top right panel. Standard curved dust-rimonly model with rim-base temperature ∼1800K. c) Bottom panel. Curved dust-rim model with gas emission (modeled as a uniform disk centered on the star) inside the dust rim to smooth out the emission profile.
Fig. 7 .Fig. 8 .
78-Constraining the size scale of the smooth emission component interior to the dust destruction radius in MWC275. The model visibilities begin to deviate significantly from the data when the radius of the smooth emission component becomes smaller than 0.19 AU. -The NIR SED for MWC275. The 'stars' are photometry points from MDM (Appendix
FigFig. 11 .
11. 10.-MWC275 SED from UV to mm.The mid, far-infrared and sub-mm data are fromMeeus et al. (2001) and references therein. The solid line traces the dust-disk model SED (see §4.2). The dotted line traces the dust-disk+smooth emission SED. The smooth component is modeled as optically-thin grey emission at 2500K. The relative contributions of star, dust and gas to the total integrated flux are 0.79, 0.16 and 0.05 respectively. -MIR image and visibilities for MWC275. The disk has an inclination of 48 o and a PA of 136 o (North is towards the top and East is on the left) a) Top panel.
Fig. 13 .Fig
13-Temperature profile and disk-surface shapes for MWC275 and AB Aur. a) Left panel. Midplane temperature profile for MWC275 (dotted line) and AB Aur (dashed line). b) Right panel. The figure shows τ =1 at 5500Å surface of the disk measured along radial lines from the central star. The y axis is the polar angle (0 is the equatorial plane) in radians. . 14.-Face-on models for NIR emission in AB Aur. a) A standard curved dust-rim-only model with rim-base temperature ∼1350K. b) Top right panel. Standard curved dust-rimonly model with rim-base temperature ∼ 1950K. c) Bottom panel. Curved dust-rim model with gas emission (modeled as a uniform disk centered on the star) added inside the dust rim in to smooth out the emission profile. The central star has been suppressed in all the panels.
Fig. 15 .
15-AB Aur visibility vs Baseline. The arrows are upper limits on the visibility. The quoted model temperatures are at the base of the dust rims.
Fig. 16 .
16-The NIR SED for AB Aur. The 'stars' are photometry points from MDM
Fig. 17 .
17-10.7µm image and visibilities for AB Aur. a) Left panel. Synthetic 10.7µm TORUS image b)Right panel. Model visibilities (solid line) compared with azimuthally averaged Keck Segment Tilting data. The model also includes 5% emission arising from an extended envelope.
Fig. 18 .
18-AB Aur SED from UV to mm. The mid, far-infrared and sub-millimeter data are fromMeeus et al. (2001) and references therein. The solid line traces the dust-disk model SED (see §5.2). The dotted line traces the dust-disk + gas model. The relative contributions of star, dust and gas to the total integrated flux are 0.67, 0.27 and 0.06 respectively.
Fig 6 can be approximated with a linear function. We calculated reduced χ 2 values for the best-fit line to the W1W2, S2W2 and E2S2 visibilities as a function of effective baseline, varying the assumed inclination and position angle of the observed disk. Fig 19 shows the reduced χ 2 surface for the fits, plotted against the assumed disk-inclination and position angle. To further illustrate the change in quality of fits as inclination and position angles are varied , Fig 20 shows the linear fits to the visibility vs effective baseline data set. As seen in Figs 19 and 20, the quality of the fits show dramatic improvement at MWC275 disk PA of 136 o ±2 o and inclination of 48 o ±2 o . These values are very close to a disk PA of 139 o ±15 o and inclination of 51 +11
FigFig. 20 .
20. 19.-Reduced χ 2 surface for the linear fits to the observed visibilities (obtained with the W1W2, S2W2 and E2S2 CHARA-telescope-pairs) as a function of effective interferometric baseline. The solid curves are reduced χ 2 contours of 5-Linear fits to the observed visibilities (obtained with the W1W2, S2W2 and E2S2 CHARA-telescope-pairs) as a function of effective interferometric baseline. The data symbols are explained inFig. 3a.
Table 1 :
1CHARA uv coverage and visibility data for MWC275. The array geometry is illustrated inFig 1,ten Brummelaar et al. (2005).UT-Date
u(m)
v(m) Telescope
Calibrated
Calibrator
of Observation
pair
Visibility
Names
2004July09
-210.61 138.79
S1W1
0.150±0.008
HD164031
-200.38 127.78
0.143±0.009
2005July22
106.91 -11.88
W1W2
0.218±0.011
HD164031
103.22 -18.38
0.227±0.009
2005July26
102.45
5.03
W1W2
0.260±0.014
HD164031
106.45 -1.61
0.241±0.011
107.18 -10.26
0.201±0.011
105.94 -13.95
0.232±0.011
2006June22
-11.99 84.94
S2W2
0.345±0.016
HD164031
-24.60 85.73
0.301±0.017
-42.30 87.77
0.203±0.013
2006June23
-301.23 -85.20
E1W1
0.0715±0.0043
HD164031
-302.93 -78.34
0.0730±0.0044
2006June23
-84.05 98.77
S2W2
0.0925±0.0041
HD164031
2006Aug23
60.15 125.48
E2S2
0.181±0.010 HD164031, HD166295
28.16 121.90
0.189±0.011
2007June17
-94.21 66.98
S2W1
0.232±0.013 HD164031, HD156365
-166.86 90.88
0.080±0.005
-184.56 102.73
0.096±0.006
-195.84 114.04
0.110±0.007
Table 2 :
2CHARA uv coverage and visibility data for AB Aur.UT-Date
u(m)
v(m) Telescope Calibrated
Calibrator
of Observation
pair
Visibility
Names
2006Aug23
212.04 237.05
E1S1
0.095±0.005 HD29645, HD31233
203.79 251.87
0.120±0.006
197.53 259.95
0.123±0.007
2006Dec14
-5.77 -325.14
E1S1
0.115±0.007 HD29645, HD31233
2006Dec15
-93.57
8.26
E2W2
0.188±0.011 HD29645, HD31233
Table 3 :
3Keck Segment Tilting Experiment baseline coverage and uv averaged visibility data for MWC275 and AB Aur.Table 5. Basic stellar properties and photometry for AB Aur.UT-Date
Baseline(m)
Calibrated
Calibrator
of Observation
Visibility
Names
MWC275
2004Sep01
3.03
0.969±0.049
v3879 Sgr
"
4.72
0.944±0.040
"
"
5.49
0.946±0.036
"
"
7.21
0.942±0.033
"
"
8.43
0.963±0.033
"
AB Aur
2004Aug30, 31 &
3.03
0.870±0.039
iota Aur
Sep01
"
4.72
0.823±0.027
"
"
5.49
0.807±0.033
"
"
7.21
0.753±0.047
"
"
8.43
0.708±0.039
"
Table 8 .
8MWC275 model-disk properties constrained by this work.Dust Disk
Inner Radius
0.22 AU a
K-band flux contribution from dust rim
29% a
Mass fractions of dust components
see Fig 12
NIR Gas Disk
Surface Brightness Profile
constant
(poorly constrained)
Outer Radius
0.22 AU a
K-band flux contribution
56% a
Temperature
> 1800K
Vertical Optical Depth
0.15 a
Gas-Opacity Profile
see Fig 9
a
Table 9 .
9AB Aur model-disk properties constrained by this work.Dust Disk
Inner Radius
0.24 AU a
K-band flux contribution from dust rim
22% a
Mass fractions of dust components
refer Fig 12
NIR Gas Disk
Surface Brightness Profile
constant
(poorly constrained)
Outer Radius
0.24 AU a
K-band flux contribution
65%
Temperature
> 1900K
Vertical Optical Depth
0.14 a
Gas-Opacity Profile
refer Fig 9
a
Synthetic 11µm TORUS image. b) Bottom left panel. Azimuthally averaged 10.7µm visibilities from the Keck Segment Tilting Experiment (Monnier et al. 2008). The 'stars' are measured values and the solid line is the model visibility. MWC275 is not resolved by Keck. c) Bottom right panel. Model visibilities compared with MIDI) data. The MIDI data was obtained at a projected baseline of ∼ 99m and a PA of 16 o , nearly aligned with disk minor axis.-37 -
Mass fractions of dust components
1
10
100
10 -5
10 -4
Radius [AU]
dust fraction
MWC275 1.3 micron silicates
" 0.1 micron silicates
AB Aur 1.3 micron silicates
" 0.1 micron silicates
Table 11 :
11: JHK Photometry. Majority of the MDM targets are YSOs. 93±0.10 6.96±0.04 5.81±0.07 01±0.05 5.98±0.05 4.78±0.05 12/17/2005 6.94±0.08 5.92±0.08 4.65±0.08Target
RA (J2000)
Dec
J
H
K
UT Date
of Observation
HD158643
17 31 25.0
−23 57 46
4.76±0.08 4.63±0.07 4.34±0.08
06/02/2006
RSOph
17 50 13.2
−06 42 29
7.94±0.08 7.17±0.07 6.77±0.08
06/04/2006
MWC275
17 56 21.3
−21 57 22
6.20±0.08 5.48±0.07 4.59±0.08
06/02/2006
HD169412
18 21 33.5
+52 54 08
7.77±0.08 7.78±0.07 7.79±0.08
06/02/2006
MWC297
18 27 39.6
−03 49 52
6.06±0.08 4.54±0.07 3.12±0.08
06/04/2006
VVSer
18 28 47.9
+00 08 40
8.60±0.08 7.37±0.07 6.20±0.08
06/02/2006
MWC614
19 11 11.3
+15 47 16
6.91±0.08 6.58± 0.07 5.88±0.08
06/03/2006
V1295Aql
20 03 02.5
+05 44 17
7.15±0.08 6.61±0.07 5.75±0.08
06/02/2006
V1685Cyg
20 20 28.3
+41 21 52
7.97±0.05 7.01±0.05 5.86±0.05
12/17/2005
7.06/03/2006
MWC342
20 23 03.6
+39 29 50
7.06/03/2006
MWC361
21 01 36.9
+68 09 48
6.12±0.05 5.58±0.05 4.77±0.05
12/17/2005
MWC1080
23 17 25.6
+60 50 43
7.38±0.05 6.04±0.05 4.68±0.05
12/17/2005
Resolution is defined as λ 2D , where λ is wavelength of observation and D is the interferometer baseline length.
This preprint was prepared with the AAS L A T E X macros v5.2.
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Owing to the potential higher energy supply efficiency and operation flexibility, integrated energy system (IES), which usually includes electric power, gas and heating/cooling systems, is considered as one of the primary forms of energy carrier in the future. However, with the increasing complexity of multiple energy devices and systems integration, IES planning is facing a significant challenge in terms of risk assessment. To this end, an energy hub (EH) planning model considering renewable energy sources (RES) and energy storage system (ESS) integration is proposed in this paper, in which the risk is measured by Conditional Value-at-Risk (CVaR). The proposed IES planning model includes two stages: 1) investment planning on equipment types and capacity (e.g., energy converters, distributed RES and ESS) and 2) optimizing the potential risk loss in operation scenarios along with confidence level and risk preference. The problem solving is accelerated by Benders Decomposition and Improved Backward Scenario Reduction Method. The numerical results illustrate the proposed method's effectiveness in balancing the potential operation risk and investment cost. Moreover, the effectiveness of reducing potential operation risk by introducing ESS and RES are also verified.
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10.1016/j.apenergy.2021.116971
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"https://arxiv.org/pdf/2104.10862v1.pdf"
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2104.10862
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5bd0727479e58920fa929385bccbd70991f81f57
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integrated energy systemenergy hub planningenergy storage systemsrenewable energy sourceconditional value-at-risk
Owing to the potential higher energy supply efficiency and operation flexibility, integrated energy system (IES), which usually includes electric power, gas and heating/cooling systems, is considered as one of the primary forms of energy carrier in the future. However, with the increasing complexity of multiple energy devices and systems integration, IES planning is facing a significant challenge in terms of risk assessment. To this end, an energy hub (EH) planning model considering renewable energy sources (RES) and energy storage system (ESS) integration is proposed in this paper, in which the risk is measured by Conditional Value-at-Risk (CVaR). The proposed IES planning model includes two stages: 1) investment planning on equipment types and capacity (e.g., energy converters, distributed RES and ESS) and 2) optimizing the potential risk loss in operation scenarios along with confidence level and risk preference. The problem solving is accelerated by Benders Decomposition and Improved Backward Scenario Reduction Method. The numerical results illustrate the proposed method's effectiveness in balancing the potential operation risk and investment cost. Moreover, the effectiveness of reducing potential operation risk by introducing ESS and RES are also verified.
The short version of this paper was presented at virtual CUE2020, OCT 10-17, 2020. This paper is a substantial extension of the short version of conference paper. This work is supported by the National Key R&D Program of China (2018YFB0905000), the National Natural Science Foundation of China (NSFC) (
A. Background and Motivation
LONG with the increasing pressure on energy crisis and environmental pollution, the integrated energy system (IES) has attracted broad interests as different energy systems can be combined to achieve a higher energy supply efficiency and flexibility [1]. Energy hub (EH) [2] concept is introduced as a tool to model IES in the project, "Vision of Future Energy Networks." An EH is a group of energy facilities where the production, conversion, storage, and consumption of different energy carriers occurs, which is a promising option for IES planning. Energy storage systems (ESS) are vital in alleviating renewable energy and load fluctuations, which can provide other services, including peak shaving, uninterruptible power supply, and energy arbitrage [3]. Moreover, it is predicted that the investment and operation costs of ESSs will become more affordable [4], which has also been proved by price data from the vendors. Correspondingly, renewable energy sources (RES) can perform an essential role by addressing fossil fuel depletion and global warming, which are critically important due to their environmental-friendly nature [5].
How to efficiently assess and reduce the potential risk when planning and operating an IES is an important and essential issue due to the complexity of IES components. To tackle this problem, extensive efforts have been made to study the optimal operation of IES considering risk. However, few works have been focused on IES planning, in which the models and methods mainly focused on the co-optimization for investment and operation strategy instead of risk management.
Besides, current research has no unified standard for IES's risk management, and the definition of risk indices is relatively rough, which is primarily based on traditional power-supply reliability indices. Therefore, further research on the risk management of IES should be performed.
B. Literature Review
Considering the differences between diverse energy systems, IES's operation and planning still faces many difficulties and needs to be further studied. Currently, several scholars have carried out related research on the operation and planning of IES. On IES operation, reference [6] proposed a mixed-integer linear programming (MILP) short-term operation model that couples power and gas networks for more flexibility and reliability. Literature [6] designed a paradigm and its operation model for interconnected EHs. Alternating direction method of multipliers (ADMM) was applied for synergistic operation of distributed IES in [8]. A coordinated regional-district operation method for IES to enhance the resilience in extreme conditions was proposed in [9]. On IES planning, an EH expansion planning model was proposed in [10], and [11] proposed a multi-stage active distribution network planning model integrated with ESS. Combining both the long-term investment and the short-term operation strategies, reference [12] presented a two-stage optimization method for a coupled IES capacity planning problem, considering economical operation and environmental issues of regional IES. In [13], an IES planning model containing an operational module that developed a steady-state optimal multi-energy flow is proposed, in which the multi-stage expansion module can also optimize the investment decisions. Meanwhile, how to decide the component capacities and operate the IES in applications were investigated in [14]- [15].
Moreover, with growing complexity, IES is facing new challenges regarding the uncertainties from different sources, e. g. RES, energy price, load demand. Stochastic optimization and robust optimization are typical methods to handle inherent uncertainties. The proposed approach in [16] adopted robust optimization to plan transmission lines to characterize the uncertainty sources pertaining to load demands and wind power productions through polyhedral uncertainty sets. To address the uncertainties of wind power output, a two-stage robust optimization model was proposed in [17] to coordinate the reactive power compensators and find a robust optimal solution. Similarly, an operation model decomposed into two subproblems representing the feasibility (security) and the optimality (economic) can be found in [18]. As for stochastic optimization, [19] presented a two-stage model for distributed energy system planning, in which Monte Carlo simulation (MCS) is applied to model energy demand and supply uncertainties. Similarly, a two-stage stochastic MILP model was proposed in [20] to invest renewable generation economically.
Conditional Value-at-Risk (CVaR) is a widely-used concept for quantifying the uncertainties and risk of portfolios in financial industry, which has been applied in power system by some scholars. It has been proved to be effective for application to maximize the profits of aggregators in [21]. It can also be applied for energy storage operation in transmission system [22] and microgrid [23], as well as quantifying risk of wind power ramps [24]. Moreover, CVaR has also been applied in economic dispatch problems in different scales. For instance, it was used to minimize the total procurement cost of conventional generation and reserve in [25]; and in [26], it was used to minimize the risk value of energy cost in day-ahead home energy management, while [27] presented a two-stage stochastic unit commitment model in transmission systems.
To conclude, we summarized the features of previous work as follows: 1) current analytical methods for addressing uncertainties in electric power system and IES such as stochastic/robust optimization [16]- [20] have been relatively mature, while the risk assessment and management considering uncertainties are somewhat roughly. 2) the applications of CVaR [21]- [27] in power systems mainly focus on operation A towards one or more uncertainties such as load variations, RES, etc. 3) CVaR has been applied mostly in power system, while study on CVaR-based IES planning is still a blank.
C. Problem Identification and Main Contributions
Based on the literature mentioned above and the essential needs in IES applications, in this paper, CVaR approach is applied to model the risk caused by uncertainties in IES operation, which could in turn benefit the IES planning by investment risk management.
Major contributions of the paper are therefore twofold: 1) A two-stage investment-operation IES planning method is proposed based on CVaR, which can measure operation risk brought by renewable generation and load variations effectively and make a trade-off between risk and cost.
With proposed model and parameters on confidence level and risk preference, the decision-makers can choose the investment strategies properly. Moreover, ESSs are included in the proposed model to improve IES's operation flexibility and proved to be cost-effective. 2) To address the computation efficiency issue brought by massive operation scenarios integrations in proposed IES planning model, we further applied an improved Backward Reduction Method in scenario reduction, cases showed the results' deviation compared with standard case and its advantages over K-means clustering. Furthermore, Benders Decomposition is also applied in the solving process to help accelerate the convergence, which outperforms the state-of-the-art commercial solver Gurobi/Cplex. The remaining parts are structured as follows: Section II presents CVaR to quantify potential risk loss, the mathematical formulation of the planning model and the solution strategy are shown in Section III. The numerical case study and analysis are performed in section IV. Conclusions are given in section V.
II. PRELIMINARIES OF CONDITIONAL VALUE AT RISK
Value-at-risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investment portfolios might lose with a given probability, given market condition, in a set period [28]. VaR was firstly proposed by J. P. Morgan in the early 1990s and has become one of the most popular methods in financial institutions to measure investment portfolios' risk. Its mathematical definition is as follows:
( , ) | ( , ) x P y f x y = (1) min | ( , ) VaR x = (2)
In (1) x is the investment decision vector representing a portfolio, and the vector y stands for uncertainties governed by a probability measure P . For a certain x , we use ( , )
x to denote the accumulative distribution function (1) for the loss (2) is the general definition of VaR, is the confidence level, which in some applications its value would usually be close to 1, e.g., 0.95
( , ) f x y . = .
The minimun can always be attained because is continuous and increasing, VaR is the unique value satisfying ( , )
x = .
In 2008, several charges about VaR in Global Association of Risk Professionals Review [29] had been proposed: One was that tail risks are non-measurable; another was that VaR is not a coherent risk measure since it violates the sub-additivity property. Conditional Value-at-Risk (CVaR), also known as Mean Excess Loss, or Tail VaR, is considered to be a more consistent measure of risk than VaR [30]. CVaR is derived by taking a weighted average of the "extreme" losses in the tail of the frequency distribution function, beyond the VaR cut-off point in Fig.1..
CVaR and its minimization formula were firstly proposed in [30], which demonstrated its numerical effectiveness including portfolio optimization through several case studies. CVaR quantifies the average loss over a specified period of unlikely scenarios beyond the confidence level. For example, a one-day 99% CVaR valued at $12 million means that the worst 1% scenario's expected loss over one day is $12 million. It was proved in [30] that as a function of x and VaR , ( , ) F x VaR is convex and continuously differentiable, which means CVaR associated with investment decision vector x , for a given confidence level , can be determined by the formula (3a)- [31] persisted the CVaR concept can be calculated in multi-scenario simulation and for general loss distributions by:
(3b): 1 ( , ) ( , ) ( ) 1 max 0, y F x VaR VaR f x y VaR p y dy where t t + + = + − − = (3a) min ( , ) CVaR F x VaR = (3b) Reference 1 1 ( , ) ( , ) 1 max 0, N s s s F x VaR VaR p f x y VaR where t t + = + = + − − = (4)
Due to the limitations of VaR in the estimation of risk and the advantages of CVaR compared with VaR, the risk management in this study is addressed by CVaR in (4). Therefore, for IES planning problem, considering the investment decisions x as investment portfolios, ( , ) III. MODEL FORMULATION Based on CVaR concept in previous section, a regional IES planning model is presented considering multi-energy device {combined cooling, heating, and power (CCHP), gas boiler (GB), air conditioner (AC), transformer (TX)}, RES {photovoltaics (PV), wind turbine (WT)}, and ESS {battery energy storage system(BESS), heating energy storage system (HESS), cooling energy storage system (CESS)}. It provides a novel approach for planning an EH with the balance between system's risk and economy.
A. Two-stage Planning Model for Investment-Operation Cooptimization
As explained in the Introduction Section, the proposed twostage model consists of two stages, in which the investment decisions are made in the first stage and the operation strategies are optimized in the second stage both from the system operators' perspective. As shown in Fig.2., the variables consist of binary variables ( ) [34] has been employed in many literatures.
B. Objective Function
The objective function without CVaR in (5) is to minimize the total cost considering annualized investment cost and a weighted summation of expectation operation cost containing trading cost, maintenance cost, and load shedding cost in one year's multiple typical daily scenarios.
The first term IC in (6) of IES denotes the investment cost with annualized amortization coefficient (7) to amortize over their lifetime. We use binary variables , ij u to represent the selection with a specific capacity j of a specific candidate device i , which is equal to 1 if the devices' candidate option is invested, and being 0 otherwise; meanwhile, this model
(1 ) , , ,(1 )1 t T T dr dr k i m n dr + == +− (7)
In (5)
In the superscript on the upper right corner, , , , g h c e represents different energy forms: natural gas, heating, cooling, and electricity, respectively. For each scenario s, Trading cost s TC in (8) (11) where
CVaR denotes the loss expectation of the detected scenarios in (11) derived from (4) with the auxiliary variable . In [30], it was proved that auxiliary variable could be approximated to VaR while CVaR reaches the optimal value. Then the final objective function considering CVaR is shown as (12),
min (1 ) obj IC OC CVaR = + − +(12)
To manage the risk in the study, the second term
2) RES constraints
,, 0 WT WT WT s t s t P z P (15) ,, 3 , ,, 3 , 0, 2 2 st in s t out WT WT st WT s t in s t r WT WT r r s t out v v or v v vS P v v v vS v v v = (16)
In (16) and (18) (15) and (17).
3) ESS constraints
Constraint (19) ensures that the ESSs cannot be charged and discharged at the same time. Charging/discharging power of ESSs are limited by the power of investment option in (20) ' , , ,
4) Energy hub constraints
According to the EH theorem [2], the energy coupling matrix equations is constructed as shown in (25). Constraints (26) and (27) limit the power of multi-energy coupling device with binary variable , ij u .
5) Additional constraints
Constraint (28) represents the shedding load should be less than the total load. Constraint (29) limits the planned RES energy capacity maximum to a certain percentage of total energy capacity in IES to avoid system instability brought by renewable power intermittence. The value of is determined by system operators based on experiences and case conditions. , ' , , , ,
6) Power balance constraints
Constraints (30)-(32) are the power balance between demand and supply. It should be noted that, the supply and demand of electric load should be strictly equal, while the heating and cooling load can be relaxed based on real-world experiences.
D. Solution Strategy
There are 106002 continous variables and 14425 integer (14420 binary) variables in 100 scenarios, massive numbers of binary variables aggregated to model due to charge/discharge variables ( ) /, ,
ch dis n st v
of ESSs. It can be inferred that as the number of typical daily scenarios increases, the computational burden will become extremely heavy. Therefore, it is necessary to apply some methods to accelerate computational speed while ensuring accuracy.
The following methods are considerable for reducing computational time consumption:
1) Apply some more efficient computing method for solving the model, like Benders' Decomposition;
2) Increase the convergence index, e.g., change gap tolerance from 0.01% to 0.1%;
3) Increase the time step in operation stage, e. g. from 1 hour to 2 hours or 4 hours.
As the number of scenarios increases, the computational burden will become extremely heavy. Therefore, it is necessary to reduce the scenario number to accelerate computational speed while ensuring accuracy. Scenario reduction techniques aggregate similar scenarios based on a particular index, currently a number of scenario reduction techniques have been proposed to make practical planning problem with massive scenarios solvable. In this work, we utilize an improved backward scenario reduction method [35], and compare the clustering results with traditional k-means clustering method [36].
The process of Improved Backward Scenario Reduction method is shown as follows:
Step Step 6: repeat Step 3-Step 5 until the scenario reduction requirements are met.
IV. CASE STUDIES
A. Basic Settings
The proposed model is applied to an EH in Fig. 3. to simulate an industrial park with electricity, heating and cooling load as well as electricity and natural gas input. Dotted lines mean they are investment options to be planned and connected. The conditions for planning, including load demand, energy price, candidate device parameter, ESS/RES information, are stated as follows [37].
1) Load Demand and Weather Condition:
The load demand is characterized by three types of load, i. e. electricity, cooling, and heating, based on an industrial park to be planned in Hebei Province, China. with 8760 hours' data in a year. The annual wind speed and light intensity are obtained from China Meteorological Data Service Centre at http://data.cma.cn/.
2) Energy Price: The gas price is 3.4 RMB/m3 and is considered constant in planning period. The hourly electricity price adopts peak-valley electricity price issued by the local government. No heating/cooling power is consumed at the input ports of the EH.
3) Candidate Device Parameters: The candidate options for energy-supply facilities are listed in Table I, with some detailed parameters illustrated in [37] due to space limit. Each multienergy device has five types from which to choose. The maximum charging/discharging power of ESS is set to 50% of the planned capacity.
4) RES Information: The PV and WT candidate module parameters are listed in Table II. The proposed model is a mixed-integer programming problem which can be solved by commercial solver such as Gurobi and Cplex. The numerical experiments of all cases are performed on a personal computer with AMD Ryzen CPU (2.10 GHz) and 20.00 GB RAM. YALMIP toolbox in MATLAB R2020b is used for modelling and Cplex 12.9.0 and Gurobi 9.0.0 optimizer for solving.
B. Analysis of investment strategies
To address different investors' risk preferences, different values of risk parameters are considered. In general, we describe investors with risk parameter 0.5 as "risk-averse investors", investors with risk parameter 0.5 as "riskseeking investors", investors with risk parameter 0.5 = as "risk-neutral investors". In the objective function (12), the decision variables of investment terms are , ,, i j m n u z z , which limit the operation bounds of devices. We explored the investment costs variation of varying confidence levels and risk parameters to analyze the investment strategy in Fig.4.. The value of the risk parameter represents the degree of investor aversion to risk, and the confidence level describes the probability threshold of the expected loss.
In Fig.4., we observed that: 1) With higher confidence level ( ) and risk parameter ( ) requirements, more investments on energy-supply facilities are tending to be made. 2) Multienergy devices and ESSs often occur in additional investments simultaneously, such as GB&HESS (①&⑤), AC&CESS(②& ⑥), WT(PV)&BESS(④⑦&⑧). 3) At the same confidence level, risk-seeking investors prefer heating and cooling facilities as additional investments due to their relatively low investment costs; as for risk-averse investors, they add electric facilities based on risk-seeking investors' investment strategies. 4) At higher confidence levels, similar investment strategies could be made in advance along with increasing risk parameters. Overall, Fig.4. can be used as an auxiliary tool to provide customized investment advice for different investors, and this is also the novelty of the model. 5. takes the shedding cooling load as an example to further illustrate the superiority of the planning method in reducing shedding load considering CVaR. As the increasing risk aversion, the quantity of shedding cooling load decreases until it is reduced to nearly zero when 0.8
, it means planning schemes will reduce adverse effects by extreme fluctuations considering load uncertainties with more investments in higher risk aversion.
C. Analysis of different coupling conditions
To further verify the model effectiveness, four cases considering different coupling conditions are designed to analyze the planning results in this paper: Case 1-plan EH with multi-energy device (CCHP, GB, AC and TX); The specific planning results are shown in Table IV. The trading cost, maintenance cost and load shedding cost in Case 2, Case 3 and Case 4 has been reduced compared to that of Case 1. This is related to the superiority of ESS and RES in O&M cost, optimizing scheduling, and energy arbitrage compared to multi-energy coupling device. It is believed this trend will become more evident with the improvement of corresponding technology. Similar trends can be found at CVaR and VaR as well, which indicates the potential loss, or "Tail VaR" (See Section II.), could be reduced with RES and ESS aggregations. Results also illustrate that CVaR can not only provide investors with auxiliary investment strategies but also can be used as a risk measurement index to express the potential risk loss of the entire system.
The hourly electricity transaction summation of all scenarios in Fig.6. can further illustrate the performance of RES and ESS in daily scheduling and the energy arbitrage with the peakvalley electricity price difference. During the valley hours (23:00-7:00), Case 3 and Case 4 are significantly higher than Case 1 and Case 2 because of BESS's charging power, thus during other periods the purchased power is lower than the above period due to discharging power. Compared with Case 1/3, the purchased electricity in Case 2/4 is always at a lower level due to local wind energy and solar energy utilization.
D. Analysis of computational speed and accuracy
In this study, a calendar day containing 24 hours is considered to be one scenario, different scenario numbers after reduction are set from 365 scenarios in one year to relieve the computational burdens. We select 10, 30, 50, 100, 200 and 300 scenarios, respectively, and these reduced scenarios are usually called typical days. Heuristic feasible solution search is turned off in Gurobi and Cplex, with optimality gap tolerance set to 0.0001. Compared time consumptions and objective with different scenario numbers in Table V. The calculation results are the same because a certain MIP gap is set among three solution methods. In that case, we could verify the acceleration effectiveness of Benders decomposition by comparing the time consumption.The results proved the effectiveness of BD to help accelerate the convergence in solving process.
We choose the result based on 8760-hour datasets (i. e. 365 typical days without scenario reduction) as a benchmark to calculate the results' deviation: a positive sign means exceeding the actual value, and a negative sign means lower than the actual value. The k-means clustering, as a well-known scenario reduction method, is used to compare with the proposed improved backward reduction method in Table VI. Table VI shows that the deviations with the benchmark are smaller using proposed improved backward reduction method, compared with k-means clustering method. The proposed method considers the Probability Distance based on Euclidean distance instead of the Within-Cluster Sum of Squares (WCSS) [38] in k-means clustering, while the effect of low-probability scenarios on the reduction result is preferentially excluded. Therefore, this method is more applicable for scenario sets with scattered probability distributions, it is also one of its advantages. Moreover, the k-means method has several disadvantages: 1) the value of K needs to be set manually, and the results obtained with different K are different; 2) it's sensitive to the initial cluster center, different cluster center selection methods will produce different clustering results; 3) it's sensitive to outliers, while the proposed method overcomes these shortcomings.
V. CONCLUSIONS
This paper presents a two-stage IES planning model considering CVaR aggregated with RES and ESS. The proposed model effectively co-optimizes the planning strategy and the operation strategy of IES, as well as the assessment on CVaR. In the proposed model, three indices on maintenance cost, trading cost, and load shedding cost are included in the objective function to optimize the annualized investment cost and operating conditions. The planning decisions, including configurations of multi-energy devices, RES and ESS module, and operation strategies for multi-energy device, RES and charging/discharging ESS, are optimized in the model. In terms of risk assessment, a CVaR based analysis is implemented in different confidence level with different risk preference.
The case studies demonstrate the effectiveness of the proposed model and illustrate the benefits of RES and ESS applications by CVaR in the IES planning and operation. ESS located on the demand side can benefit system operation by peak-valley load shifting and energy arbitrage to enhance resilience. And increasing risk parameter could benefit IES planning by eliminating load shedding caused by load and renewables' uncertainties.
Since the model is time-consuming when applied to the IES planning with hundreds of scenarios, we also propose to speed up the convergence by the improved backward reduction method and benders decomposition, which is proved to be effective, too. Further research on this topic will help balance the convergence speed and planning results' deviations.
s f x y denotes the loss in scenario s, i.e. the operation cost including electricity/gas consumption...... , and scenario 1, , sN = have probabilities s p respectively.
Fig. 1 .
1VaR and CVaR[32]
the number of RES/ESS modules, while superscript m and n are device options of RES/ESS.
in (9) is related to operational power of multi-energy device, RES and charging/ discharging mileage of ESS, which are denoted by (9a)-(9c), respectively. Load shedding cost (10) is calculated by multiplying unit cost of different types of loads shedding r with the summation of shedding load , , SHED r st L . Then we introduce the Conditional Value-at-Risk (CVaR) of
Fig. 2 .
2Structure of two-stage planning-
OC in all scenarios and corresponding expected risk loss. Different values of risk parameters between 0 and 1 are considered for the investors to address different investors' risk aversion degrees, if the value of is closer to 1, the significance of the risk namely CVaR, increases. 13) and (14) model the binary decisions and integer decisions for investments on multi-energy device and RES/ESS, which means there should be at least one device of CCHP and TX constructed to complete the EH and the number of RES/ESS module should be positive.
S
imply PV module area, conversion efficiency of MPPT (Maximum Power Point Tracking) and PV panels, denote air density, wind speed, light intensity and solar tilt angle depend on weather conditions. (16) and (18) are general output power formulas of wind turbine and PV for one module, which become the outputs upper limitation through multiplying with number of modules WT z and PV z in
the charge/discharge states, while M is a large number used in Big-M method. Constraint (22) represents the SOC (state of charge) is limited by the number of ESSs module and energy capacity for one module. Constraint (23) denote the relationship between charge/discharge power and SOC (state of charge). Constraint (24) ensures the initial and final values of SOC in a scenario are the same.
1 :
1suppose D is the initial set of scenarios, J is the detected set of scenarios. Compute Kantorovich Distance (KD) for each pair of scenarios in D to form the Kantorovich Distance Matrix (KDM), KD for i and j with nearest scenario k and mark it; Step 3: compute the Probabilistic Distance (PD) for each scenario i PD among all scenarios in D and corresponding scenario l , add l to J and delete it in D; Step 5: find the marked nearest scenario of l in Step 2, add the probability of l to the marked nearest scenario;
Fig. 3 .
3Energy hub model studied in the case
Fig. 5 .
5Hourly shedding cold load in different risk parameter Case 2-plan EH with multi-energy device (CCHP, GB, AC, TX) and RES (WT, PV); Case 3-plan EH with multi-energy device (CCHP, GB, AC and TX) and ESS (BESS, HESS, CESS); Case 4-plan EH with multi-energy device (CCHP, GB, AC and TX), RES (WT, PV) and ESS (BESS, HESS, CESS).
second-stage problem. The capacity of multi-energy device, RES, as well as the capacity and maximum charging/discharging power of ESS resulting from the firststage problem, thereby affects the decision variables evaluated in the second-stage problem. To solve the two-stage optimization model, Benders Decomposition (BD) technique[33]-n
i j
m
SHED r
s t
s t
s t
s t
p
p p L
parameterize the
TC MC LC denote trading cost, maintenance cost, and load shedding cost in scenario s , respectively. Operation cost is the probabilistic weighted summation of scenarios, more scenarios account for a more accurate calculation result.,
,,
s
s
s
,
e e
g g
s
t s t
s
t
TC
pr P
pr P
=+
(8)
,,
,,
e
i j e
s t
s t
ij
PP =
(8a)
,,
,
g
i j g
s
s t
t
i
j
PP =
(8b)
+
i i
m m
n n
s
s
s
s
i
m
n
MC
Q
Q
Q
=+
(9)
,
,
i
i j
s
s t
jt
QP =
(9a)
,
mm
s
s t
t
QP =
(9b)
,,
,,
()
n
ch n
dis n
s
s t
s t
t
Q
P
P
=+
(9c)
,
,
,,
r
SHED r
s
s t
r e c h
t
LC
L
=
Table I
IMULTI-ENERGY DEVICE AND ESS CANDIDATE PARAMETERSCandidate options
Investment cost
(10 4 RMB/MW)
Maintenance cost
(RMB/MWh)
CCHP
5
900
1.5
GB
5
80
10
AC
5
150
2
TX
5
30
10
BESS
1 MWh per module
90
90
HESS
1 MWh per module
9
9
CESS
1 MWh per module
19
19
Table II
IIRES CANDIDATE PARAMETERSFig. 4. Investment strategies and changes in different confidence level and risk parameterWT Module
PV Module
Investment cost
(10 4 RMB/MW)
350
600
Maintenance cost
(RMB/MWh)
120
625
Conversion Efficiency
0.3
0.2
Other parameters
Vin (m/s):2.5
Vr (m/s):12
Vout (m/s):25
MPPT
conversion efficiency:0.97
blade length (m):40
solar inclination:38°
air density: 1.29kg/m 3
Table III
IIICOMPARISON OF DIFFERENT RISK PREFERENCEScenario=100,=
=
=
=
Planned results
CCHP
5MW
5MW
5MW
GB
20MW
20MW
25MW
AC
31MW
36MW
36MW
TX
20MW
20MW
20MW
BESS
15MWh
14MWh
14MWh
HESS
16MWh
42MWh
13MWh
CESS
13MWh
19MWh
21MWh
WT
6MW
6MW
6MW
PV
4120m 2
4120m 2
4120m 2
Investment cost (✕10 4 RMB)
25813.99
27870.57
28584.03
Trading cost (✕10 4 RMB)
14558.26
14578.45
14569.71
Maintenance cost (✕10 4 RMB)
165.13
165.56
165.54
Load shedding cost (✕104RMB)
545.13
91.18
33.04
VaR(✕10 4 RMB)
18142.76
18134.64
18104.16
CVaR(✕10 4 RMB)
28530.08
20013.27
18898.59
Total cost (✕10 4 RMB)
42408.67
45294.80
47069.59
Table IV
IVCOMPARISON OF DIFFERENT COUPLING CONDITIONS Scenario=100,= =Case1
Case2
Case3
Case4
Planned
results
CCHP
5MW
5MW
5MW
5MW
GB
25MW
25MW
25MW
20MW
AC
38.5MW
38.5MW
36MW
36MW
TX
20MW
20MW
20MW
20MW
BESS
-
-
1MWh
14MWh
HESS
-
-
42MWh
16MWh
CESS
-
-
17MWh
8MWh
WT
-
6
-
42MW
PV
-
4120m 2
-
4120m 2
Investment cost (✕10 4 RMB)
26739.25
27366.34
26458.72
27870.57
Trading cost (✕10 4 RMB)
17948.96
15451.78
17810.78
14578.45
Maintenance cost (✕10 4 RMB)
191.07
161.74
192.20
165.56
Load shedding cost (✕10 4 RMB)
151.46
151.46
107.48
91.18
VaR(✕10 4 RMB)
20956.41
18896.09
20927.49
18134.64
CVaR(✕10 4 RMB)
24113.34
21788.36
23394.08
20013.27
Total cost (✕10 4 RMB)
47941.66
46143.01
47210.98
45294.80
Table V
VCOMPARISON OF COMPUTATION TIME AND SOLUTION① K-means clustering method (Euclidean Distance) ② Improved backward scenario reduction methodIN DIFFERENT SCENARIOS
Scenario
(= =)
Method
Time (s)
Objective
(✕10 4 RMB)
10
Gurobi
12.32
40769.85
Cplex
11.26
40769.85
Benders
1.98
40769.85
30
Gurobi
50.40
42934.94
Cplex
65.42
42934.94
Benders
11.91
42934.94
50
Gurobi
55.76
43791.01
Cplex
54.12
43791.01
Benders
15.25
43791.01
100
Gurobi
319.44
44366.26
Cplex
287.84
44366.26
Benders
49.30
44366.26
200
Gurobi
815.84
44927.91
Cplex
945.60
44927.91
Benders
382.92
44927.91
300
Gurobi
2022.94
45296.15
Cplex
1547.94
45296.15
Benders
623.13
45296.15
365
(Without reduction)
Gurobi
2479.88
45294.80
Cplex
2753.12
45294.80
Benders
1132.61
45294.80
Table VI
COMPARISON OF CALCULATION DEVIATION
Scenario
(= =)
IC
TC
MC
LC
CVaR
Total
Cost
10
①
-15.39%
-5.28%
+16.75%
-
-25.52%
-13.90%
②
-12.16%
-3.20%
+12.12%
-
-20.20%
-9.99%
30
①
-11.35%
-3.77%
+5.41%
-97.92%
-21.23%
-6.74%
②
-9.08%
-2.65%
+4.34%
-82.45%
-18.34%
-5.21%
50
①
-10.72%
-2.12%
-3.86%
-50.84%
-15.86%
-5.44%
②
-8.76%
-1.28%
-3.12%
-38.11%
-13.12%
-3.32%
100
①
-5.72‰
-9.25‰
-2.84%
-15.27%
-12.46%
-3.82%
②
-4.44‰
-2.74‰
-1.75%
-10.21%
-9.44%
-2.05%
200
①
-5.75‰
+2.68‰
-2.71%
-8.14%
-5.69%
-1.84%
②
-4.74‰
-0.25‰
-2.18%
-5.23%
-2.33%
-0.81%
300
①
+0.17‰
+0.67‰
-2.35%
-5.78%
-2.81%
-0.05‰
②
-0.04‰
-0.03‰
-7.78‰
-3.90%
+9.07‰
+0.03‰
365
0
0
0
0
0
0
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. 10.6084/m9.figshare.14188262.v1URLhttps:/figshare.com/s/013bd6d0bc5e4d44effehttps://doi.org/10.6084/m9.figshare.14188262.v1 URL https://figshare.com/s/013bd6d0bc5e4d44effe
On some significance tests in cluster analysis. H H Bock, 10.1007/BF019080Journal of Classification. 2Bock, H.H. On some significance tests in cluster analysis. Journal of Classification 2, 77-108 (1985). https://doi.org/10.1007/BF019080
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[
"Hall anomaly by vacancies in pinned lattice of vortices: A quantitative analysis on the thin-film data of BSCCO",
"Hall anomaly by vacancies in pinned lattice of vortices: A quantitative analysis on the thin-film data of BSCCO",
"Hall anomaly by vacancies in pinned lattice of vortices: A quantitative analysis on the thin-film data of BSCCO",
"Hall anomaly by vacancies in pinned lattice of vortices: A quantitative analysis on the thin-film data of BSCCO"
] |
[
"Ruonan Guo \nShanghai Center for Quantitative Life Sciences & Physics Department\nShanghai University\n200444ShanghaiChina\n\nShanghai Key Laboratory of High Temperature Superconductors\nShanghai University\n200444ShanghaiChina\n",
"Yong-Cong Chen *[email protected]†[email protected]:2206.03384v1[cond-mat.supr-con] \nShanghai Center for Quantitative Life Sciences & Physics Department\nShanghai University\n200444ShanghaiChina\n",
"Ping Ao \nShanghai Center for Quantitative Life Sciences & Physics Department\nShanghai University\n200444ShanghaiChina\n",
"Ruonan Guo \nShanghai Center for Quantitative Life Sciences & Physics Department\nShanghai University\n200444ShanghaiChina\n\nShanghai Key Laboratory of High Temperature Superconductors\nShanghai University\n200444ShanghaiChina\n",
"Yong-Cong Chen *[email protected]†[email protected]:2206.03384v1[cond-mat.supr-con] \nShanghai Center for Quantitative Life Sciences & Physics Department\nShanghai University\n200444ShanghaiChina\n",
"Ping Ao \nShanghai Center for Quantitative Life Sciences & Physics Department\nShanghai University\n200444ShanghaiChina\n"
] |
[
"Shanghai Center for Quantitative Life Sciences & Physics Department\nShanghai University\n200444ShanghaiChina",
"Shanghai Key Laboratory of High Temperature Superconductors\nShanghai University\n200444ShanghaiChina",
"Shanghai Center for Quantitative Life Sciences & Physics Department\nShanghai University\n200444ShanghaiChina",
"Shanghai Center for Quantitative Life Sciences & Physics Department\nShanghai University\n200444ShanghaiChina",
"Shanghai Center for Quantitative Life Sciences & Physics Department\nShanghai University\n200444ShanghaiChina",
"Shanghai Key Laboratory of High Temperature Superconductors\nShanghai University\n200444ShanghaiChina",
"Shanghai Center for Quantitative Life Sciences & Physics Department\nShanghai University\n200444ShanghaiChina",
"Shanghai Center for Quantitative Life Sciences & Physics Department\nShanghai University\n200444ShanghaiChina"
] |
[] |
Hall anomaly, as appears in the mixed-state Hall resistivity of type-II superconductors, has had numerous theories but yet a consensus on its origin. In this work, we conducted a quantitative analysis of the magnetotransport measurements on BSCCO thin films by Zhao et al. [Phys. Rev. Lett. 122, 247001 (2019)] and validate a previously proposed vacancy mechanism [cf. J. Phys. Condens. Matter. 10, L677 (1998)] with many-body vortex correlations for the phenomenon. The model attributes the Hall anomaly to the motion of vacancies in pinned fragments of vortex lattice. Its validity is first examined by an exploration on the vortex states near the Kosterlitz-Thouless transition on the vortex crystal. Comparisons are then carried out between the measured activation energies with the calculated creation energy of the vortex-anti-vortex pair and the vacancy energy on the flux-line lattice, with no adjustable parameter. Our analysis elucidates the theoretical basis and prerequisites of the vacancy model. In particular, the vacancy activation energies are an order of magnitude smaller than that of a sole vortex line. The proposed mechanism may provide a macro-theoretical framework for other studies.Abstract This Supplementary Material (SM) covers additional computational, comparison works used or referred in the main article published in [1].
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10.1103/physrevb.106.104507
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[
"https://export.arxiv.org/pdf/2206.03384v1.pdf"
] | 249,431,771 |
2206.03384
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12a154fede8b853296f473f79fbdc02c248a3a52
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Hall anomaly by vacancies in pinned lattice of vortices: A quantitative analysis on the thin-film data of BSCCO
7 Jun 2022
Ruonan Guo
Shanghai Center for Quantitative Life Sciences & Physics Department
Shanghai University
200444ShanghaiChina
Shanghai Key Laboratory of High Temperature Superconductors
Shanghai University
200444ShanghaiChina
Yong-Cong Chen *[email protected]†[email protected]:2206.03384v1[cond-mat.supr-con]
Shanghai Center for Quantitative Life Sciences & Physics Department
Shanghai University
200444ShanghaiChina
Ping Ao
Shanghai Center for Quantitative Life Sciences & Physics Department
Shanghai University
200444ShanghaiChina
Hall anomaly by vacancies in pinned lattice of vortices: A quantitative analysis on the thin-film data of BSCCO
7 Jun 2022High-temperature superconductorBSCCO thin filmHall anomalyVacancy and flux-line latticeVortex many-body effectActivation energyKosterlitz-Thouless transition
Hall anomaly, as appears in the mixed-state Hall resistivity of type-II superconductors, has had numerous theories but yet a consensus on its origin. In this work, we conducted a quantitative analysis of the magnetotransport measurements on BSCCO thin films by Zhao et al. [Phys. Rev. Lett. 122, 247001 (2019)] and validate a previously proposed vacancy mechanism [cf. J. Phys. Condens. Matter. 10, L677 (1998)] with many-body vortex correlations for the phenomenon. The model attributes the Hall anomaly to the motion of vacancies in pinned fragments of vortex lattice. Its validity is first examined by an exploration on the vortex states near the Kosterlitz-Thouless transition on the vortex crystal. Comparisons are then carried out between the measured activation energies with the calculated creation energy of the vortex-anti-vortex pair and the vacancy energy on the flux-line lattice, with no adjustable parameter. Our analysis elucidates the theoretical basis and prerequisites of the vacancy model. In particular, the vacancy activation energies are an order of magnitude smaller than that of a sole vortex line. The proposed mechanism may provide a macro-theoretical framework for other studies.Abstract This Supplementary Material (SM) covers additional computational, comparison works used or referred in the main article published in [1].
I. INTRODUCTION
The Hall anomaly in the mixed-state Hall resistivity of superconductors, i.e. the sign reversal of the Hall resistivity below the superconducting transition temperature and in the presence of flux-line vortices, was discovered as early as in the 1950s [1,2]. Much attention has since been set on its physical origin. Prior to the discovery, Onsager [3] had established the framework of the vortex theory based on fluid dynamics in 1949. A modern version of his work, the equation for a j th vortex of unit length in a superconductor takes the same Langevin equation as a charged particle in the presence of a magnetic field [4],
mr j = q n s 2 h (v s,t −ṙ j ) ×ẑ − ηṙ j + F p + f ,(1)
where the overhead of dots stands for time derivatives. The unit length vortex at r j has an effective mass m, subject to a pinning force F p , a fluctuating force f , viscosity η, and moves in a background of a superfluid with total velocity v s,t (which includes contributions from all other vortices). For the parameters in Eq. (1), q = ±1 indicates the vorticity (under the usual right-hand rules), h is the Planck constant, n s the superfluid carrier density, and z the unit vector in the direction of the magnetic field. The term with the velocityṙ j at the right-hand side is also known as the Magnus force. Note that CGS units are assumed throughout this work, in line with the majority of work in the literature.
With Eq. (1), two idealized pictures can be drawn. Take e.g. q = +1, Fig. 1(a) shows that in the absence of the pinning force F p and the frictional force (η = 0; F p = 0), the vortex velocityṙ j matches that of superfluid in both direction and magnitude. In other words, for an average charge e in the vortex, the Magnus Force (due to the electric field E generated by moving vortex), i.e. eE cancels the Lorentz force (e/c)v s × H, leading to the same Hall effect as in a normal metal. One has no reason to expect a sign reversal on the Hall resistance R xy below the superconducting transition temperature T c . Fig. 1(b) presents an alternative scenario of an extreme situation: The vortex is firmly trapped by a strong pinning force F p such that there is no Magnus force. Moreover, the pinning force is opposite to the Lorentz force and there will be no change of sign on the Hall resistance R xy either.
These scenarios raise an apparent paradox between conceptual reasoning and experimental observation.
To address the abnormal sign reversal, several groups [5][6][7][8] have proposed a phenomenological model, which adds an adjustable parameter α directly in front of the Magnus force.
r j = v s . (b)
The vortex is firmly trapped by strong pinning force F p such thatṙ j = 0.
This modifies the vortex dynamic equation to
mr j = q n s 2 h(v s,t − αṙ j ) ×ẑ − ηṙ j + F p + f .(2)
In a steady-state where the vortex is subject to the friction force only, the Hall resistivity ρ xy (0) and the longitudinal resistivity ρ xx (0) can be obtained respectively
ρ xx (0) = − ηqhH ec (α 2 q 2 n s 2 h 2 + 4η 2 ) ,(3)ρ xy (0) = αn s q 2 h 2 H 2ec (α 2 q 2 n s 2 h 2 + 4η 2 ) .(4)
This allows the sign reversal of the Hall resistivity ρ xy (0) when α < 0, though it is unclear how α can be so adjusted. Furthermore, there should also be resistivity coming from the normal component of electrons at finite temperatures. The latter follows a parallel connection to the superconducting circuit so that we have
ρ xx (T ) = − ηη qhH e [cη α 2 q 2 n 2 s (T )h 2 − ehηqH(n s (0) − n s (T )) + 4cη η 2 ] ,(5)ρ xy (T ) = αHn s (T )q 2 h 2 ce [2α 2 q 2 n 2 s (T )h 2 + αn s (T )q 2 h 2 (n s (0) − n s (T )) + 8η 2 ] .(6)
Here n s (T ) is the density of superfluid electron at temperature T and the density of normal electron can be written as n s (0) − n s (T ). The change in the sign of α has to overcome the normal-state electron contribution to achieve the sign reversal on ρ xy (T ). The details are presented in Part A of the Supplemental Material (SM) to this work [9].
The adjustable parameter α itself has not had a consistent origin in the approach. Hall and Vinen [10,11] in 1956 assumed a big drag force of the same as the Magnus force.
Nozieres and Vinen [5], Hagen et al. [6,7] further developed a theoretical explanation for the existence of such a force. In contrast, Wang and Ting [8] argued that the parameter α should be zero. They based their theory on the well-known normal-core model for flux lines in extreme type-II superconductors and correctly took into account the backflow current due to pinning forces, which concluded in no Magnus force. As a result, there can be a small α with either sign, leading to a signing inversion of the Hall coefficient under certain circumstances. However, Ao and Thouless [12] in 1993 found that α ≡ 1 at zero temperature.
And three years later, Thouless, Ao and Niu [13] in an atomically thin high-temperature superconductor by a Harvard group [20] in 2019 drastically extended the region which displays the Hall sign reversal.
In this work, we present an in-depth analysis of the activation energies deduced from the experimental data in [20]. It is then compared to the predicted activation energy of independent vortices and the energy of vortex many-body correlation under flux-line lattices. In In this section, we will first review some basics of the vortex lattice state near the KT transition which melts the crystal. Then we discuss the nature of the KT transition from the thermodynamic perspective and estimate the theoretical value of the transition temperature T KT of the BSCCO thin film. Finally, the question of whether the concept and presence of vacancies still apply above T KT is addressed.
A. KT transition of molten crystals
The melting transition of most solid materials at the present has not been well understood as there is a lack of theories explaining the transition on the microscopic scale. Furthermore, the mechanism of melting depends on the interaction details between the constituents forming a crystal lattice. In particular various defects which reduce the translational order of the crystal play a major role. It should be noted that in two dimensions, only edge dislocations and not the screw ones are important in the melting transition. The core energy of the dislocations upon which these effects can form must be sufficiently low for their spontaneous appearance. The specific analysis of the dislocation formation energy in a 2D BSCCO film will be presented in the subsequent section.
These dislocations in question are topological point defects, which implies that a single one cannot be created isolated by an affine transformation without cutting the hexagonal crystal up to infinity (i.e. up to its borders). Hence they must be created in pairs with antiparallel Burgers vectors. When a large number of dislocations were e.g. thermally excited, the discrete translational order of the crystal would be destroyed. Simultaneously, the shear modulus and Young's modulus would disappear, signaling the starting of the molten transition from a solid to a fluid phase.
However, it is possible that the orientational order is not yet fully destroyed (as indicated by lattice lines in one direction) and one finds -very similar to liquid crystals -a fluid phase with typically a six-folded direction field. This so-called hexatic phase still has an orientational stiffness. Such an anisotropic fluid phase can appear when the dislocations dissociate into isolated five-folded and seven-folded disclinations [21]. This two-step melting phenomenon is described within the so-called Kosterlitz-Thouless-Halperin-Nelson-Youngtheory (KTHNY theory), based on two separate transitions of the Kosterlitz-Thouless-type.
In 2010, Urs Gasser et al. [21] presented the first conclusive evidence for the existence of the hexatic phase and two continuous phase transitions in 2D melts in a colloidal model system with repulsive magnetic dipole-dipole interaction.
B. Thermodynamics of KT transition
The topological phase transition in a 2D superfluid was predicted by Berezinskii [22] and Kosterlitz and Thouless [23] and elaborated by Halperin and Nelson [24,25]. A simple thermodynamic argument allows us to understand the intrinsic quality of the KT transition [26]. The Helmholtz free energy is given by the difference between the energy E and the entropy of a dislocation S multiplied by the temperature T ,
F = E − T S.(7)
The energy E is given by Eq. (15), which is contributed by a dislocation pair over a large distance. For convenience, we rewrite it as
E = dε 0 2 √ 3π ln L a .(8)
6
The entropy can be estimated from the number of places the dislocation can be positioned, namely on each of the ∼ L 2 plaquette of the lattice, i.e.,
S = k B ln L 2 a 2 .(9)
Accordingly, the free energy is given by
F = d ε 0 2 √ 3π − 2k B T ln L a .(10)
Evidently, there exists a temperature above which a vast number of dislocations are preferred. Such a transition is of topological nature and is referred to as the KT transition.
The transition temperature T KT could be expressed as
T KT = 1 4 √ 3πk B Φ 0 4πλ 2 d ∼ = 57.3K.(11)
Ideally at T < T KT , thermally excited dislocations form pairs of close compact, namely a dislocation pair. But they are spontaneously separated when T > T KT . The KT transition temperature T KT measured by a Harvard group is 60 K [20], which validates the effective thickness and the London penetration depth chosen below for our analysis. temperature [27]. An urgent question is whether the Hall anomaly mainly appears in the vortex-liquid regime. Scilicet fades away as the vortex liquid freezes into a solid-state crystal.
However, there is no evidence for such a solid-liquid phase transition in the BSCCO film.
As the transition would be a first-order phase transition in which observation of latent heat was to be expected, the resistance of the BSCCO film should measure an abrupt change at the transition. More crucially Kosterlitz-Thouless transition is classified as a topological phase transition, the third type of phase transition. Experientially, all curves intercepted by the isotherm of T KT = 60 K are smooth, with no sign of jump around the node.
According to the generic nature of phase transitions in two-dimensional materials [28][29][30], we can reasonably deduct the phase diagram as shown in Fig. 2. In the illustration, T KT is This sets the prerequisites for the existence of vacancies and the applicability of the vacancy model proposed by Ao et. al. [4].
III. HALL ANOMALY BY VACANCIES OF THE VORTEX LATTICE
In this section, we intend to quantitatively demonstrate that the Hall anomaly is in full concert with the theoretical basis of the vacancy hypothesis. Namely, it is due to the movement of vacancies, a direct result of many-body vortex interaction and the origin of the Hall anomaly. First, the dynamics of vacancies in a vortex lattice are discussed in detail.
Then some results are applied to fit the experimental data by the Harvard group [20] with no adjustable parameters. The vacancy activation/formation energy under a diverse set of magnetic fields is found to be the same order of magnitude as theoretical predictions. In particular, both theoretical and experimental results show that the energy of an independent vortex line or a vortex-anti-vortex pair is an order of magnitude larger than the vacancy energy.
A. Properties of vacancies in a pinned vortex lattice
In 1993, Thouless and Ao [12] proved that the existence of Magnus force is a universal
e d (r) ∼ = da 2 c 66 4π ln r a ,(12)
where a = (2Φ 0 / √ 3B) is the lattice spacing, r a is the distance between two dislocations, and c 66 is the shear modulus of the FLL, cf. [33]. For uniform distortions the elastic moduli [32] of a triangular FLL reads
c 66 ≈ Bφ 0 16πλ 2 µ 0 1 − 1 2κ 2 1 − b 2 1 − 0.58b + 0.29b 2 .(13)
Here λ is then a uniaxial elastic medium similar to that of an isotropic and bulk superconductor.
In the limit of large κ and relatively small magnetic field, we have
c 66 ≈ BΦ 0 16πλ 2 µ 0 .(14)
Using Eqs. (12)- (14) we can re-express the energy for a dislocation pair as
e d (r) = d ε 0 2 √ 3π ln r a .(15)
Here the major variable ε 0 ≡ (Φ 0 /4πλ) 2 defines vortex creation energy per unit-length, (d ε 0 ) then sets the scale for both the vortex-vortex and strong pinning interactions [4]. The energy scale (ε 0 /2 √ 3π) for the dislocation pair is about ten times smaller than ε 0 , it is energetically favourable to have close-distance dislocation pairs in the lattice as carriers of transverse current.
Thus at temperature k B T (d ε 0 ) we can ignore the contribution to the current from the vortices hopping out of pinning as well as thermal activation of vortex-antivortex pairs. This is because the entire vortex lattice, formed via inter-vortex interactions should be effectively pinned down. Instead, we should look into the vacancies and interstitials which can be viewed as the smallest dislocation pairs [34]. The vacancy formation energy ε v per unit length can be estimated [4] by setting r ∼ 2a in e d (r) together with an extra factor
ε v ∼ ε 0 ln 2 2 √ 3π a ξ .(16)
Note that in Ao's initial valuations, the effect of the magnetic field on the energy barrier in the activation process has been ignored. It is anticipated that such an effect should relate to the ratio of the lattice constant as a function of a magnetic field to the coherence length ξ in some way. In this work, we attempt to take the effect into account with the simplest multiplication factor (a/ξ).
B. Vacancy activation energy under magnetic field
We can now analyze the experimental data from the SM of [20] with the Arrhenius empirical formula whose validity in solid-state kinetics has been illustrated in e.g. [35].
Taking that the dominant source of resistance R is the thermal activation of the dissipative vacancies in the superconductor film, we have
R = A exp (E a /k B T ) ⇒ log 10 R(T ) = log 10 A − (log 10 e)[E a (T )/k B T ],(17)
where A is a pre-factor for the exponential term, and T is the temperature with k B the usual Boltzmann constant. Both A and the activation energy E a can be themselves temperaturedependent. Now in the GL analysis, the penetration depth λ(T ) is given by
λ(T ) = λ(0)/[1 − (T /T c )] 1/2 .(18)
The length scale is greatly affected by temperature, while its dependence on the magnetic field is negligible [36]. Substituting λ(T ) into the vacancy formation energy Eq. (16), we get
E a (T ) = d ln 2 2 √ 3π a ξ Φ 0 4πλ(T ) 2 .(19)
Observe that the fitting requires the magnetic field B to be in the middle of the lower critical magnetic field H c1 and the upper critical magnetic field H c2 . Although for thicker films the thickness in Eq. (17) depends on various parameters such as magnetic field, pinning, temperature and anisotropy. The case here is relatively simple. We take d to be 50% of the physical thickness of the thin film, which is based on the observation that the ratio roughly corresponds to the "superconductive" portion of the material (along the c-axis).
To proceed further, we set λ(0) = 2690Å [37] for BSCCO-2212, which can be compared to other values from i) reversible magnetization, λ ≈ 2100Å [38]; ii) uSR, λ ≈ 1800Å [39];
and iii) lower critical field measurements, λ ≈ 2700Å [40]. [20] and the open symbols and dash lines from Eq. (17). In these dashed lines, there is an external magnetic field, the activation energy is governed by the vacancy motion, which is determined by Eq. (19). In the absence of an external magnetic field, the activation energy is caused by the independent vortex motion, which depends on Eq. (21).
C. Vortex-anti-vortex pairs in the absence of a magnetic field
In the zero-field case, the activation E a (T ) is in the order of an independent vortex energy ε in , which reads [26]
ε in = d Φ 0 4πλ(T ) 2 ln κ.(20)
For a vortex-anti-vortex pair, the distant current contribution cancels out, hence the item ln κ should drop out. Therefore, for an impact pair, the creation energy can be taken as [20]. The outcome with no further variable again is shown in Fig. 3, with the calculated average formation energy of vortex-antivortex pair in Table I. Both the field-dependent and zero-field predictions are in excellent agreement with experimental measurements. They are further discussed below.
E a (T ) = 2d ε 0 .(21)
D. A comparison of vacancy and vortex-pair activation energies
Eq. (17) states that the slopes of the curves in Fig 7 of SM in [20] match to E a /k B .
Their average values are tabulated in Table I. Observe that the activation energy at zero field B = 0 is an order of magnitude larger than that at B > 0, which represents one of the Primary outcomes of the present paper. For B = 0 there are no vacancies and the activation energy is at the same order of magnitude as the independent vortex energy.
On the other hand, the prime contribution to the activation energy comes from vacancies for B = 0, the influence of the independent vortex is very small and can be ignored beyond the likely melting temperature T m , cf. Fig. 2. Keeping the same temperature, the films under a magnetic field have much larger resistance than they have under a zero field, cf. Fig 7 of SM in [20]. The corresponding activation energies at B = 0 T are all in the same order of magnitude as the vacancy formation energies in Table 1.
By quantitative comparisons between their experimental and the theoretical values, we conclude that the Hall anomaly is a consequence of many-body vortex interactions, which further conforms to the theoretical analysis of the vacancy model. In particular, according to the above analysis and Table I, the vortex-anti-vortex energy is an order of magnitude larger than the vacancy energy. The latter constitutes one of our primary arguments.
IV. DISCUSSIONS
The current microscopic understandings of the Hall anomaly in the 3UC BSCCO crystal may be classified into two opposite physical models, which are based on disparate theoretical approaches and give contradictory interpretations. The Harvard group [4] fabricated a few unit-cell (UC) thick BSCCO and detected the reversal depicted by vortex dynamics in this system. It laid the foundation to differentiate diverse models experimentally. The group [20] has made a theoretical explanation based on the individual vortex dynamic model [43].
Inversely, Ao proposed a multi-body correlation model [4] in which vacancy movements on pinned vortex lattice are the primary mechanism for the phenomenon.
In this paper we elaborate, via quantitative analysis, on the three main aspects of the The extreme scenario of a strong pining force F p is discussed in the main article [1], in this section with no pinning force F p = 0.
We consider a two-fluid model of a superconductor placed at fixed temperature T with a current density J, the normal circuit follows a parallel connection to the superconducting circuit so that For zero degrees, the total carriers become superfluid electrons, i.e., the normal electron density n n (0) = 0, hence σ N (0) = 0. Then, the film conductivity is the superconducting contribution σ S (0)
J = J S (T ) + J N (T ),(S1)σ(0) = σ S (0).(S9)
In a steady-state where the vortex dynamics analysis is shown in Fig. S1(a), and the dynamic equation is
q n s (0) 2 h(v s,t (0) − αṙ j (0)) ×ẑ − ηṙ j (0) = 0.(S10)
The physical significance of these symbols is detailed in the main article [1]. The superfluid electron moves with a velocity v s,t (0) = J/2en s (0) under the applied transport current J, where n s (0) is the density of superfluid electron at zero temperature. For calculation purposes, according to Fig. S1(a), with the direction parallel and perpendicular to the superfluid electron motion as the base vector, the vector Eq. (S10) decomposed into two scalar equations by −q n s (0) 2 hv s,t (0) + αq n s (0) 2 hṙ jxx (0) + ηṙ jxy (0) = 0, (S11)
αq n s (0) 2 hṙ jxy (0) − ηṙ jxx (0) = 0.(S12)
Hereṙ jxx (0) andṙ jxy (0) denote the components of the vortex velocity parallel and perpendicular to the direction of superfluid electron, respectively. The macroscopic electric field is equal to E = −ṙ j × H/c [8] (satisfies the usual right-hand rules), one finds the longitudinal E S xx (0) and Hall E S xy (0) electric fields
E S xx (0) = −ṙ jxy (0)H/c,(S13)E S xy (0) =ṙ jxx (0)H/c.(S14)
Thus the longitudinal σ xx (0) and Hall resistivities σ xy (0) are immediately obtained
σ xx (0) = − ec(α 2 q 2 n 2 s (0)h 2 + 4η 2 ) ηqhH ,(S15)
σ xy (0) = 2ec(α 2 q 2 n 2 s (0)h 2 + 4η 2 ) αn s (0)q 2 h 2 H , (S16) then we have the longitudinal ρ xx (0) and Hall resistivities ρ xy (0)
ρ xx (0) = − ηqhH ec(α 2 q 2 n 2 s (0)h 2 + 4η 2 ) ,(S17)ρ xy (0) = αn s (0)Hq 2 h 2 2ec(α 2 q 2 n 2 s (0)h 2 + 4η 2 )
.
(S18)
This allows the sign reversal of the Hall resistivity ρ xy (0) when α < 0.
We now proceed to discuss the low-temperature (0 < T < T c ) scenario. There should be resistivity coming from the normal and superfluid components. In terms of Eq. (S17) and
Eq. (S18), the superfluid contribution at T is
σ xx (T ) = − ec(α 2 q 2 n 2 s (T )h 2 + 4η 2 ) ηqhH ,(S19)
σ xy (T ) = 2ec(α 2 q 2 n 2
s (T )h 2 + 4η 2 ) αn s (T )q 2 h 2 H ,(S20)
Under Eq. (S3) and Eq. (S4), we get the normal current density J N (T ). The normal electron moves with an average velocity v n = J N (T )/en n (T ), with the dynamic analysis is illustrated in Fig. S1(b), and the steady-state equation is
e c v n × H + eE − η v n = 0. (S21)
The term −η v n is the frictional force on the normal electron. In line with the procedure in Eqs. (S10)-(S12), the vector Eq. (S21) decomposed into .
The change in the sign of α has to overcome the normal electron contribution to achieve the sign reversal on Hall resistivity ρ xy (T ).
B. A POWER-LAW BEHAVIOR WITH v = 2
As pointed out in the main article [1], our analysis elucidates the theoretical basis and prerequisites of the vacancy model. Now we examine its prediction. In the low-temperature limit, both longitudinal and Hall resistivities vanish exponentially [9] ρ xy ∝ ρ v xx .
Introduce the Hall conductivity,
σ xy = ρ xy ρ 2 xx + ρ 2 xy .(S31)
Whereas ρ xy ρ xx ρ 0 in real superconductors, ρ 0 is the resistivity at room temperature.
The Hall conductivity term (S31) becomes
σ xy = ρ xy /ρ 2 xx .(S32)
The scaling relation reads after some manipulations of Eq. (S30) and Eq. (S32),
σ xy ∝ ρ −(2−v) xx .
(S33)
The model predicts the power
v = 2a v + b v a v + b v (S34)
varying between 1 and 2, depending on the detail of a sample which determines the numerical factors a v and b v . If all vacancies produced by pinnings, the b v = 0 and v = 2. Consider that the longitudinal resistivity obeys ρ xx (T ) = ρ 0 e −Ea/kT , where ρ 0 is a constant. Thus, the Hall conductivity is independent of temperature when v = 2, in agreement with Fig. 1 of [10]. In the illustration, the Hall conductivity is temperature independent at the low temperature, especially at a fixed magnetic field B = 6T . On BSCCO films, the work of Zhao el al. [10] revealed such a power-law behavior with v = 2. Data from other laboratories [11][12][13][14][15] also admit 1 < v < 2.
FIG. 1 .
1Analysis of vortex dynamics in two ideal situations. In the picture E is the electric field generated by moving vortex and H is the applied magnetic field. (a) With no pinning force F p = 0 and no frictional force (η = 0; f = 0), the vortex moves at the same velocity as the superfluiḋ
the next section, we examine whether the abnormal Hall effect on the BSCCO film meets the pre-requisites of the vacancy model by analyzing the state of the vortices near the KT transition temperature. In section III, we first review the fundamentals of our methodology, namely the pinning and dynamics of vacancies in a vortex lattice. The core concept presented follows throughout the entire section. The experimental data are extracted and compared with the theoretical calculations of activation energies of carriers in the BSCCO film, under varying magnetic fields. The excellent agreement between them elucidates the conformation of the Hall anomaly to the vacancy model. Some concluding remarks and possible connections to other works and future directions are discussed in the final section. II. VORTEX STATES AROUND THE KOSTERLITZ-THOULESS TRANSITION To validate that the Hall anomaly complies with the pre-requisites of the vacancy model, it is crucial to clarify the states of vortex crystal near the Kosterlitz-Thouless (KT) transition.
C. The presence of vacancies above T KT Let us for the moment assume a vortex solid-liquid phase transition near (the first) T KT , around 60 K. Then the Arrhenius behavior of longitudinal resistance places the Hall sign reversal within the thermally activated flux flow regime above the vortex lattice melting
FIG. 2 .
2Schematic phase diagram of the superconducting gap vs. temperature. The solid curve represents the relationship between the order parameter and temperature of the type-II superconductor under a constant external magnetic field. When T KT < T < T m , the vortex lattice possesses quasi-long order. Three insets show the states of the vortex lattice at diverse temperature intervals.the lowest with a second melting temperature T m for the solid-liquid phase transition below the superconducting transition temperature T c . For T KT < T < T m , there should exist local fragments of vortex lattice, and in each of them long-range order out to be preserved.
essence of superconductor vortex line. In 1998, Ao [4] further established a set of processes leading to the Hall effect as a result of moving vacancies in a background of pinned vortex lattice(s). It manifests in particular that neither a modification on the vortex equation nor an assumption of two types of carriers is necessary. One only needs to study the vortex dynamics equation proposed by Niu, Ao and Thouless [31] in 1994. To recite the theory, we turn to a crucial quantitative result regarding the motion of vacancies in a pinned vortex lattice used in the subsequent analysis, namely the vacancy formation energy in a flux line lattice.First we look at the energy scale of dislocations in the lattice. In a type-II superconductor with mixed states, the many-body correlation between the vortices and the pinning forces usually cannot be ignored. On a two-dimensional flux line lattice (FLL) of a thin film of thickness d, spontaneous nucleation of a pair of edge dislocations costs an energy[26,32]
2 = (m * c 2 /8πρ s e 2 ) is the London penetration depth (with the effective mass m * and superfluid density ρ s of the underlying carriers of charge 2e), Φ 0 = (hc/2 |e|) is the flux quantum of a Cooper pair, κ is the GL parameter and b = H/H c2 (with the applied magnetic field strength H the upper critical field H c2 of the superconductor). The 2-d FLL
FIG. 3 .
3The superconducting transition temperature T c = 89 K and log 10 A ≈ 2 are read off from the experimental figure [20] for the BSCCO-2212 film. Other experimental parameters include the effective film thickness = 50% of a 3-layer film d = 1.5U C = 2 × 1.5 × 15.35 × 10 −8 cm (The half height a unit cell in Bi-2212 is 15.35Å [41]), the GL parameter κ = 86 [42], the coherence length ξ = λ(T )/κ, together with the flux quantum of Cooper pair Φ 0 = hc/2 |e| = 2.07 × 10 −7 G· cm 2 and the Boltzmann constant k B = 1.38 × 10 −16 erg/K. All quantities are in CGS units. We then apply Eq. (17) to Fig 7 of SM in[20]. The result that requires no extra fitting parameter is shown inFig. 3. The theoretical values of the average energy of vacancy formation under diverse magnetic fields are presented inTable I. Both display excellent agreement between theory and experiment, with no adjustable parameters. An Arrhenius plot of the resistance vs. temperature with the solid lines and symbols from the experimental data
. The energy of the 3UC BSCCO film on various magnetic fields. The experimental value is the activation energy of the film, which is gotten by Eq. (17). The theoretical values of the film at B = 0 T and B = 0 T are determined by Eq. (21) and Eq. (19), respectively.
Ao's model. Firstly, the precondition and theoretical basis of the Hall anomaly warrant the vacancy model in a pinned vortex lattice or fragments of the lattice. Secondly, both theory and experiment reveal that the energy of the vortex-anti-vortex pair is about an order of magnitude higher than the vacancy energy. Last but not least, distinct theoretical models of the Hall anomaly can be quantitatively distinguished via experiment. Particularly, our theoretical predictions have no adjustable parameters. As discussed in Part E of SM[9], several predictions of the vacancy model explain the Hall anomaly of BSCCO in the mixed state better than some other models.We also look forward to providing a macro-theoretical framework for related topics in future works. For example, Yang et al.[44] observed linear-in-temperature and linear-inmagnetic field resistance on nano-patterned YBCO film arrays over extended temperature and magnetic field. Meanwhile, the low-field magneto-resistance oscillates with a perioddictated by the superconducting flux quantum. It is possible that the unexpected signatures may be explained by a pinned vortex lattice model in a similar consideration. In particular, the plasticity of a vortex lattice pinned by the periodic nano holes can lead to diminishing of super concurrent as a result of free-energy minimisation. Further exploration will be presented elsewhere. THE RESISTIVITY SOLUTIONS IN A PHENOMENOLOGICAL MODELThe phenomenological model, which attempts to explain the Hall anomaly by adding an adjustable parameter α in front of the Magnus force, was first proposed by Hall and Vinen[2, 3] in 1956 and later developed by Nozieres and Vinen[4], Hagen et al.[5, 6]. This model makes a slight alteration on the vortex dynamics equation proposed by Onsager[7] in 1949, as in Eq. (??). To describe its theoretical basis, one has to derive both longitudinal ρ xx and Hall ρ xy resistivities at zero degrees and low temperatures (T < T c), starting from Eq. (??).
E
(T ) = E S (T ) = E N (T ). (S2) Here the electric field of superfluid electrons E S (T ), normal electrons E N (T ), and the film E(T ) is the same, while J is the sum of the superconducting J S (T ) and the normal J N (T ) current density. And J S (T ) = σ S (T )E S (T ), (S3) J N (T ) = σ N (T )E N (T ). (S4) σ N (T ) and σ S (T ) are the conductivity of the normal and superconducting circuit respectively, leading us to the combined conductivity σ(T ) = σ N (T ) + σ S (T ). (T ), ρ N (T ) and ρ S (T ) are, respectively, the film, normal electrons and superfluid electrons resistivity. In a true superconductor, the resistivity and conductivity are determined by a matrix, and the non-diagonal elements are antisymmetric. S1. With no pinning force F p = 0, the dynamic analysis of vortex and normal electron at temperature T . In the picture E is the electric field generated by moving vortex and H is the applied magnetic field. (a) The moving vortex generates resistivity at temperature T (0 T < T c ). (b) Partial resistivity comes from the normal component of electrons at temperature T (0 < T T c ).
N xx (T ) and E N xy (T ) present the longitudinal and Hall electric fields of the normal electrons, separately. The solutions of Eq. (S19-S23) are the normal state longitudinal conductivity σ N xx (T ) and Hall conductivity σ N xy (T ) σ N xx (T ) = e 2 n n (T )/η , (S24) σ N xy (T ) = ecn n (T )/H. (S25) Here the density of normal electron n n (T ) can be written as n s (0) − n s (T ). Since the superconducting circuit is connected in parallel with the normal circuit, the longitudinal σ xx (T ) and Hall σ xy (T ) conductivity of the film read σ xx (T ) = − e [cη α 2 q 2 n 2 s (T )h 2 − ehηqH(n s (0) − n s (T )) + 4cη η 2 ] ηη qhH , (S26)σ xy (T ) = ce [2α 2 q 2 n 2 s (T )h 2 + αn s (T )q 2 h 2 (n s (0) − n s (T )) + 8η 2 ] αHn s (T )q 2 h 2 , (S27) therefore ρ xx (T ) = − ηη qhH e [cη α 2 q 2 n 2 s (T )h 2 − ehηqH(n s (0) − n s (T )) + 4cη η 2 ] ,(S28)ρ xy (T ) = αHn s (T )q 2 h 2 ce [2α 2 q 2 n 2 s (T )h 2 + αn s (T )q 2 h 2 (n s (0) − n s (T )) + 8η 2 ]
extended the result to low but finite temperature. Additionally, from the Ginzburg-Landau theory, another phenomenological model was proposed by Xu et al. [14] in 2002 for the Hall anomaly in High-temperature superconductors. And in 2004, Ghenim et al. [15] also suggested a model for conventional superconductors.Despite that the Hall anomaly attracted numerous theoretical studies, a lack of advancement in experimental techniques has hindered a consensus on its origin. But recently, there has been a steady improvement in the situation over the last two decades. In 1996, Zhu et al.[16] designed a mechanical experiment to directly measure the total transverse force on moving vortices in a type II superconductor for the first time, and their result is consistentwith the Ao-Thouless theory [12]. In 2001, Zhu and Nyeanchi [17] demonstrated that cer-
tain features of the Kosterlitz-Thouless transition of a vortex lattice are preserved near the
superconducting transition temperature. In 2019, Chen's team [18] developed a fabrication
process which can produce intrinsic monolayer crystals of BSCCO. In 2021, Richter et al.
[19] measured the resistivity, Hall effect, and anisotropic superconducting coherence lengths
in HBCCO thin films with morphological variations. Excellent measurement of Hall effects
Substituting the above formula along with parameters taken for the other estimates to Eq. (17), we can compare the result to Fig 7 of SM in
One of us (NG) is grateful to Prof. Chuanbing Cai for useful discussions on the research. This work was supported in part by the National Natural Science Foundation of China No. 16Z103060007 (PA). One of us (NG) is grateful to Prof. Chuanbing Cai for useful discussions on the research. This work was supported in part by the National Natural Science Foundation of China No. 16Z103060007 (PA).
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|
[] |
[
"Fully-dynamic α + 2 Arboricity Decompositions and Implicit Colouring",
"Fully-dynamic α + 2 Arboricity Decompositions and Implicit Colouring"
] |
[
"Aleksander B G Christiansen \nTechnical University of Denmark\nLyngbyDenmark\n",
"Eva Rotenberg \nTechnical University of Denmark\nLyngbyDenmark\n"
] |
[
"Technical University of Denmark\nLyngbyDenmark",
"Technical University of Denmark\nLyngbyDenmark"
] |
[] |
In the implicit dynamic colouring problem, the task is to maintain a representation of a proper colouring as a dynamic graph is subject to insertions and deletions of edges, while facilitating interspersed queries to the colours of vertices. The goal is to use few colours, while still efficiently handling edge-updates and responding to colour-queries. For an n-vertex dynamic graph of arboricity α, we present an algorithm that maintains an implicit vertex colouring with 4·2 α colours, in amortised poly-log n update time, and with O(α log n) worst-case query time. The previous best implicit dynamic colouring algorithm uses 2 40α colours, and has a more efficient update time of O(log 3 n) and the same query time of O(α log n)[25].For graphs undergoing arboricity α preserving updates, we give a fully-dynamic α + 2 arboricity decomposition in poly(log n, α) time, which matches the number of forests in the best near-linear static algorithm by Blumenstock and Fischer [12] who obtain α + 2 forests in near-linear time.Our construction goes via dynamic bounded out-degree orientations, where we present a fullydynamic explicit, deterministic, worst-case algorithm for (1+ε)α +2 bounded out-degree orientation with update time O(ε −6 α 2 log 3 n). The state-of-the-art explicit, deterministic, worst-case algorithm for bounded out-degree orientations maintains a β · α + log β n out-orientation in O(β 2 α 2 + βα log β n) time[28].algorithms, the state-of-the-art algorithms spend time O(m 10/7 )[32]or O(m √ n) [31] for the out-orientation problem, andÕ(m 3/2 ) for the arboricity decomposition problem [19, 20]. Even statically computing an α+1 out-orientation[29]resp. an α+2 arboricity decomposition [12] takesÕ(m) time. In the dynamic case, the out-orientation with the lowest bound on the out-degree with O(poly(log n, α)) update time seem to be the algorithm of Brodal & Fagerberg [13] that achieves 2(α max + 1) out-degree. In [13], it is also noted that determining exactly the complexity of maintaining a d out-orientation for d ∈ [α, 2α] is a 'theoretically interesting direction for further research'. We make some progress in this direction by showing how to maintain a (1 + ε)α + 2 out-orientation with poly(log n, α, ε −1 ) update time. Thus, if α is a constant, we may carefully choose ε to obtain a polylogarithmic α + 2 out-orientation.ResultsLet G be a dynamic graph with n vertices undergoing insertion and deletions of edges, and let α be the current arboricity of the graph; that is α might change, when edges are inserted and deleted. If we at all times have an upper bound α max on α, we say that G is undergoing an α max preserving sequence of updates. We have the following: Theorem 1. For 1 > ε > 0, there exists a fully-dynamic algorithm maintaining an explicit ((1 + ε)α + 2)-bounded out-degree orientation with worst-case insertion time O(log 3 n · α 2 /ε 6 ) and worst-case deletion time O(log 3 n · α/ε 4 ) Using pseudoforest decompositions, we obtain a fully dynamic, implicit colouring algorithm:Corollary 2. Given a dynamic graph with n vertices, there exists a fully dynamic algorithm that maintains an implicit 2 · 3 (1+ε)α colouring with an amortized update time of O(log 4 n · α 2 /ε 6 ) and a query time of O(α log n).By moving edges between pseudoforests, we can turn the pseudoforest decomposition into a forest decomposition. This also gives a colouring algorithm using fewer colours.Theorem 3. Given an initially empty and dynamic graph undergoing an arboricity α max preserving sequence of updates, there exists an algorithm maintaining a (1+ε)α +2 arboricity decomposition with an amortized update time of O(poly(log n, α max , ε −1 )). In particular, setting ε < α −1 max yields α + 2 forests with an amortized update time of O(poly(log n, α max )).Corollary 4. Given a dynamic graph with n vertices, there exists a fully dynamic algorithm that maintains an implicit 4 · 2 α colouring with an amortized update of O(polylog n) and a query time of O(α log n).Finally, we modify an algorithm of Brodal & Fagerberg [13] so that it maintains an acyclic out-orientation.Theorem 5. Given an initially empty and dynamic graph G undergoing an arboricity α max preserving sequence of insertions and deletions, there exists an algorithm maintaining an acyclic (2α max + 1) out-degree orientation with an amortized insertion cost of O(α 2 max ), and an amortized deletion cost of O(α 2 max log n).Paper outline We recall related work below. In Section 2 we recall preliminaries and and show how to explicitly 2-orient dynamic forests as a warm.up; Section 3 outlines the proof of Theorem 1 and Corollary 2; in Section 4, we outline how to show Theorem 3 and Corollary 4; and Section 5 is dedicated to Theorem 5. Finally, in Section 6 we give the missing details in the proof of Corollaries 4 and 2.
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[
"https://arxiv.org/pdf/2203.06039v1.pdf"
] | 247,411,246 |
2203.06039
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454c00d6938bbdd7e4815b214378c296ca5f7d00
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Fully-dynamic α + 2 Arboricity Decompositions and Implicit Colouring
Aleksander B G Christiansen
Technical University of Denmark
LyngbyDenmark
Eva Rotenberg
Technical University of Denmark
LyngbyDenmark
Fully-dynamic α + 2 Arboricity Decompositions and Implicit Colouring
2012 ACM Subject Classification Theory of computation → Dynamic graph algorithms Keywords and phrases Dynamic graphsbounded arboricitygraph colouringdata structures
In the implicit dynamic colouring problem, the task is to maintain a representation of a proper colouring as a dynamic graph is subject to insertions and deletions of edges, while facilitating interspersed queries to the colours of vertices. The goal is to use few colours, while still efficiently handling edge-updates and responding to colour-queries. For an n-vertex dynamic graph of arboricity α, we present an algorithm that maintains an implicit vertex colouring with 4·2 α colours, in amortised poly-log n update time, and with O(α log n) worst-case query time. The previous best implicit dynamic colouring algorithm uses 2 40α colours, and has a more efficient update time of O(log 3 n) and the same query time of O(α log n)[25].For graphs undergoing arboricity α preserving updates, we give a fully-dynamic α + 2 arboricity decomposition in poly(log n, α) time, which matches the number of forests in the best near-linear static algorithm by Blumenstock and Fischer [12] who obtain α + 2 forests in near-linear time.Our construction goes via dynamic bounded out-degree orientations, where we present a fullydynamic explicit, deterministic, worst-case algorithm for (1+ε)α +2 bounded out-degree orientation with update time O(ε −6 α 2 log 3 n). The state-of-the-art explicit, deterministic, worst-case algorithm for bounded out-degree orientations maintains a β · α + log β n out-orientation in O(β 2 α 2 + βα log β n) time[28].algorithms, the state-of-the-art algorithms spend time O(m 10/7 )[32]or O(m √ n) [31] for the out-orientation problem, andÕ(m 3/2 ) for the arboricity decomposition problem [19, 20]. Even statically computing an α+1 out-orientation[29]resp. an α+2 arboricity decomposition [12] takesÕ(m) time. In the dynamic case, the out-orientation with the lowest bound on the out-degree with O(poly(log n, α)) update time seem to be the algorithm of Brodal & Fagerberg [13] that achieves 2(α max + 1) out-degree. In [13], it is also noted that determining exactly the complexity of maintaining a d out-orientation for d ∈ [α, 2α] is a 'theoretically interesting direction for further research'. We make some progress in this direction by showing how to maintain a (1 + ε)α + 2 out-orientation with poly(log n, α, ε −1 ) update time. Thus, if α is a constant, we may carefully choose ε to obtain a polylogarithmic α + 2 out-orientation.ResultsLet G be a dynamic graph with n vertices undergoing insertion and deletions of edges, and let α be the current arboricity of the graph; that is α might change, when edges are inserted and deleted. If we at all times have an upper bound α max on α, we say that G is undergoing an α max preserving sequence of updates. We have the following: Theorem 1. For 1 > ε > 0, there exists a fully-dynamic algorithm maintaining an explicit ((1 + ε)α + 2)-bounded out-degree orientation with worst-case insertion time O(log 3 n · α 2 /ε 6 ) and worst-case deletion time O(log 3 n · α/ε 4 ) Using pseudoforest decompositions, we obtain a fully dynamic, implicit colouring algorithm:Corollary 2. Given a dynamic graph with n vertices, there exists a fully dynamic algorithm that maintains an implicit 2 · 3 (1+ε)α colouring with an amortized update time of O(log 4 n · α 2 /ε 6 ) and a query time of O(α log n).By moving edges between pseudoforests, we can turn the pseudoforest decomposition into a forest decomposition. This also gives a colouring algorithm using fewer colours.Theorem 3. Given an initially empty and dynamic graph undergoing an arboricity α max preserving sequence of updates, there exists an algorithm maintaining a (1+ε)α +2 arboricity decomposition with an amortized update time of O(poly(log n, α max , ε −1 )). In particular, setting ε < α −1 max yields α + 2 forests with an amortized update time of O(poly(log n, α max )).Corollary 4. Given a dynamic graph with n vertices, there exists a fully dynamic algorithm that maintains an implicit 4 · 2 α colouring with an amortized update of O(polylog n) and a query time of O(α log n).Finally, we modify an algorithm of Brodal & Fagerberg [13] so that it maintains an acyclic out-orientation.Theorem 5. Given an initially empty and dynamic graph G undergoing an arboricity α max preserving sequence of insertions and deletions, there exists an algorithm maintaining an acyclic (2α max + 1) out-degree orientation with an amortized insertion cost of O(α 2 max ), and an amortized deletion cost of O(α 2 max log n).Paper outline We recall related work below. In Section 2 we recall preliminaries and and show how to explicitly 2-orient dynamic forests as a warm.up; Section 3 outlines the proof of Theorem 1 and Corollary 2; in Section 4, we outline how to show Theorem 3 and Corollary 4; and Section 5 is dedicated to Theorem 5. Finally, in Section 6 we give the missing details in the proof of Corollaries 4 and 2.
Introduction
Graph colouring is a well-studied problem in computer science and discrete mathematics and has many applications such as planar routing and network optimization [15]. A proper colouring of a graph G = (V, E) on n vertices is an assignment of colours to each vertex in V (G) such that no neighbours receive the same colour. We are interested in minimising the number of colours used. The minimum number of colours that can be used to properly colour G, is called the chromatic number of G. It is NP-hard to even approximate the chromatic number to within a factor of n 1−ε for all ε > 0 [41,27], but colourings with respect to certain parameters can be efficiently computed. For instance, it is well known that if a graph is uniformly sparse in the sense that we can decompose it into k forests, then we can efficiently compute a colouring: the sparsity of the graph ensures that every subgraph has a vertex of degree at most 2k − 1, allowing us to compute a 2k colouring of the graph in linear time by colouring the vertices in a clever order. The minimum number of forests that the graph can be decomposed into is called the arboricity of G. In the past decades, much work has gone into the study of dynamic algorithms that are able to efficiently update a solution, as the problem undergoes updates. A general question about dynamic problems is: which (near-) linear-time solvable problems have polylogarithmic updatable solutions?
We study the problem of maintaining a proper colouring of a dynamic graph with bounded arboricity. This class of graphs encompasses, for instance, dynamic planar graphs where α ≤ 3. Here, the graph undergoes changes in the form of insertions and deletions of edges and one needs to maintain a proper colouring of the vertices with fast update times. We distinguish between two scenarios: one where, as is the case for dynamic planar graphs, we have access to an upper bound on the arboricity α max throughout all updates, and one where we do not. Note that due to insights presented in [37], we can often turn an algorithm for the first scenario into an algorithm for the second by scheduling updates to O(log n) (partial) copies of the graph, thus incurring only an O(log n) overhead in the update time.
Barba et al. [6] showed that one cannot hope to maintain a proper, explicit vertex-colouring of a dynamic forest with a constant number of colours in poly-logarithmic update time. Consequently, we cannot maintain explicit colourings where the number of colours depend entirely on α with poly-logarithmic update time -even if we know an upper bound on α. This motivated Henzinger et al. [25] to initiate the study of implicit colourings. Here, instead of storing the colours of vertices explicitly in memory, a queryable data structure is provided which after some computations returns the colour of a vertex. If one queries the colours of two neighbouring vertices between updates, the returned colours must differ. Now, we can circumvent the lower bound by using known data structures for maintaining information in dynamic forest to 2-colour dynamic forests in poly-logarithmic update time. Henzinger et al. [25] use this to colour graphs via an arboricity decomposition i.e. a decomposition of the graph into forests. They present a dynamic algorithm that maintains an implicit proper 2 O(α) -colouring of a dynamic graph G with arboricity α. Their algorithm adapts to α, but in return it hides a constant (around 40) in the asymptotic notation. Even if one has an upper bound α max on α, the currently best obtainable colouring uses 2 4(αmax+1) colours by combining the arboricity decomposition algorithm from Henzinger et al. [25] with an algorithm of Brodal & Fagerberg [13] that maintains a 2(α max + 1) bounded out-degree orientation. Both of these algorithms use a lot of colours. Even for planar graphs with arboricity at most 3, 2 16 > 60.000 colours are used. This is quite far from 4 colours, which is always sufficient [3,36], or the 5 colouring that can be computed in linear time [33,14,18].
Dynamic arboricity decompositions: Both colouring algorithms go via dynamic α -bounded out-orientations. Here, the goal is to orient the edges of the graph while keeping out-degrees low. These are then turned into dynamic 2α -arboricity decompositions. By 2-colouring each forest, such a decomposition yields a 2 2α colouring. Thus the lower α is, the fewer colours we use. There has been a lot of work on maintaining dynamic low out-orientations [13,8,28,24,39], and much of this work aim to improve update complexity by relaxing the allowed out-degree. Motivated by implicit colourings, we provide a different trade-off, providing a lower α value within polylog(n, α) update time. Specifically, a (1 + ε)α + 2 dynamic out-orientation with O(log 3 (n)α 2 /ε 6 ) update-time adaptive to α, and an α + 2 dynamic arboricity decomposition with O(poly(log n, α max )) update time, when we have an upper bound α max on the arboricity. Our algorithm maintaining the arboricity decomposition matches the number of forests obtained by the best static algorithm running in near-linear time [12].
These algorithms may also be interesting as they go below the 2α barrier on out-edges and forests respectively. In the static case there exist simple and elegant algorithms computing 2α − 1 out-orientations and arboricity decompositions in linear time [4,17]. For exact Reference Out-degree Update time α Brodal & Fagerberg [13] 2(α + 1) O(α + log n) am. fixed Kopelowitz et al. [28] βα + log β n O(β 2 α 2 + βα log n) adaptive He et al. [24] O(α √ log n) O( √ log n) am. fixed Berglin & Brodal [8] O(α + log n) O(log n) adaptive Henzinger et al. [25] 40α O(log 2 n) am. adaptive Kowalik [30].
O(α log n) O(1) am. fixed New (Thm. 1) (1 + ε)α + 2 O(log 5 (n)α 2 /ε 6 )) adaptive Table 1 Different dynamic algorithms for maintaining out-orientations.
Related Work
Dynamic colouring: Barba et al. [6] give algorithms for the dynamic recolouring problem, and show that c-colouring a dynamic forests incurs Ω(n 1 c(c−1) ) recolourings per update. Solomon & Wein give improved trade-offs between update time and recolourings and give a deterministic dynamic colouring algorithm parametrized by the arboricity α, using O(α 2 log n) colours with O(1) amortized update time [39]. Henzinger et al. [25] introduced the study of implicit colouring of sparse graphs in order to circumvent the explicit lower bound of Barba et al. [6]; they maintain an implicit colouring using 2 O(α) colours, with O(log 3 n) update time and O(α log n) query-time. Bhattacharya et al. [9] studied the dynamic colouring problem parameterized by the maximum degree ∆, presenting a (1 + o(1))∆ colouring algorithm with O(polylog(∆)) update time, and a randomized ∆ + 1 colouring with expected amortized update time O(log ∆). This randomized result was subsequently improved independently by Bhattacharya et al. [10] and Henzinger & Peng [26] achieving O(1) amortized update time (respectively, w.h.p. and expected).
Bounded out-degree orientations: Much of the work with respect to bounded out-degree orientations has gone into either 1) statically computing bounded out-degree orientations with the minimum (or close to it) out-degree [19, 35,11,1], or 2) dynamically maintaining bounded out-degree orientations with efficient updates [13, 28,39,30,8], but allowing weaker guarantees on the minimum out-degree (see Table 1 for an overview). The current stateof-the-art for exact, static algorithms have running times O(m 10/7 ) [32] and O(m √ n) [31]. Kowalik also gave an algorithm computing a (1 + ε)α out-orientation inÕ(m log n) time.
Arboricity decompositions: A lot of work has been put into producing efficient static algorithms for computing arboricity decompositions [19,20,16,35] (see [12] for an overview). The fastest static algorithm runs inÕ(m 3/2 ) time [19,20]. Also approximation algorithms have been studied in the static case. There exists a linear-time 2-approximation algorithm [4,17]. Furthermore, Blumenstock & Fischer provide an algorithm computing a (1 + ε)α + 1 arboricity decomposition inÕ(m log n) time. Bannerjee et al. [5] provide anÕ(m) dynamic algorithm maintaining the exact arboricity α of a dynamic graph, and show a lower bound of Ω(log n) for dynamically maintaining arboricity. Henzinger et al. [25] provide a dynamic algorithm for maintaining a 2α arboricity decomposition, given access to any black box dynamic α out-degree orientation algorithm (See Table 2).
Other related work: Motivated by the problem of finding a densest subgraph, Sawlani & Wang [37] gave an (implicit) dynamic approximation algorithm for maintaining a (1 + ε)ρ fractional out-degree orientation, where ρ is the maximum subgraph density. In order to Table 2 Overview of dynamic algorithms for maintaining arboricity decompositions. Note that applying Lemma 27 to Theorem 5 gives an arboricity decomposition, since the orientation in Theorem 5 is acyclic.
tune the parameters in the algorithm, they use multiple (partial) copies of the same graph, where each copy has a different estimate of the maximum density of the graph.
Computing near optimal out-orientations and arboricity decompositions has also been studied from a distributed point of view. Barenboim & Elkin gave a (2 + ε)-approximation in [7]. This has since then been improved to (1 + ε)-approximations [22,23,40,21].
Preliminaries & Warm-up
Nash-Williams showed that the arboricity α of a graph G satisfies α = max J⊂G |E(J)| |V (J)|−1 [34]. A closely related sparsity measure is the maximum (subgraph) density ρ defined as ρ = max J⊂G |E(J)| |V (J)| . Note that α and ρ are numerically very close.
Explicit 2-out orientation of dynamic forests
We begin by considering the simpler problem of orienting the edges of dynamic forests so as to minimise the maximum out-degree of vertices. The edges of a forest can be oriented in such a way that the maximum out-degree of a vertex is 1. Indeed, we root every tree in the forest arbitrarily and orient all edges towards the roots. It is well-known that we can maintain an implicit representation of such a 1-orientation in a dynamic forest using data structures for maintaining information in a dynamic forest as e.g. top trees [2] or Sleator and Tarjans dynamic trees [38]. The representation is implicit in the sense that in order to determine the out-edge of a vertex, the dynamic algorithm has to perform some computations. This is achieved by maintaining the dynamic forests using for instance top trees [2]. Each top tree is then arbitrarily rooted, and one can determine an out-edge, if it exists, of a vertex v by finding the first edge on the unique v-to-root-path. This solution has worst case query and update time of O(log n).
A natural next question is if we can maintain such an orientation explicitly i.e. in such a way that we at all times explicitly store which way an edge is oriented, whilst still achieving logarithmic update times. In turns out, we cannot dynamically maintain such a 1-out orientation in logarithmic time, not even for a dynamic set of paths. Indeed, consider two 1-out oriented paths P 1 , P 2 both of length k. P i has exactly one vertex of out-degree 0 that all edges are oriented towards, and this vertex is of distance at least k/2 from one of the endpoints of P i , say x i . Consequently, adding the edge x 1 x 2 forces the reorientation of at least k/2 edges. Furthermore, this is repeatable: deleting this edge again, determining new vertices x 1 , x 2 and adding the edge between them, again forces the algorithm to reorient at least k/2 edges. Hence, by setting k = n/2 we find that there exists an n-vertex dynamic set of paths together with a sequence of O(1) moves forcing any algorithm maintaining a 1-out orientation of the paths to perform Ω(n) reorientations for either an insertion or a deletion.
So in the explicit version of the problem, we have to settle for a 2-out orientation. In this setting, it is straight-forward to solve the problem for dynamic paths: one can orient edges arbitrarily and every vertex will still have out-degree at most two. Note that in particular the endpoints of paths have out-degree at most one. By dynamically maintaining a decomposition of a dynamic forest into a set of paths together with a set of edges going between the paths such that every such edge is assigned to a unique endpoint of a path, we can extend the above solution to dynamic forests. Indeed, we orient each path arbitrarily, and orient the inter-path edges away from the unique endpoint the edges were assigned. Since these endpoints are endpoints of paths, no vertex receives out-degree more than two.
In order to maintain this decomposition dynamically, one can maintain the well-known heavy-light decomposition of each tree using dynamic trees [38]. For a rooted-tree T , we have the notion of parents and children of the vertices. The parent of v is the first vertex from v on the v-to-root path in T . The children of v are all neighbours of v that are not the parent of v. A heavy child w of v is then a child of v such that the sub-tree of T rooted a w contains more than half of the vertices of the sub-tree rooted at v. The heavy children in T induces heavy edges going from a vertex to its heavy child, and light edges going from a vertex to a non-heavy child. Every root-to-leaf path then contains at most O(log n) light edges. The heavy edges form the desired paths, and the light edges can be assigned to the endpoint that is a child of the other endpoint.
Sleator and Tarjan [38] showed how to maintain such a heavy-light decomposition in O(log n) worst case update time. The algorithm maintains a decomposition of the edges into solid and dashed edges and at the end of each update the solid edges coincide exactly with the heavy edges, and the dashed edges with the light edges. With O(1) overhead we can orient every solid edge arbitrarily and every dashed edge from the child endpoint to the parent endpoint. An update changes at most O(log n) solid edges to become dashed edges, and so the update time is still O(log n).
To argue correctness, we need to argue that at the end of each update every dashed edge is oriented from the child towards the parent. Whenever a solid edge is turned into a dashed edge, the orientation is chosen such that this invariant is ensured. Therefore, the only problematic case might be if a dashed edge has the parent-child relationship of its endpoints switched, without the edge having been a solid edge in the meantime. However, the only way a dashed edge e has the parent-child relationship of its endpoints switched, is if the root r 1 of a tree T is switched to be r 2 such that e is on the unique path between r 1 and r 2 in T . For the root to be switched to r 2 , every edge on the path from r 1 to r 2 is turned solid, hence e is turned solid, and so the problematic case never occurs. As such, we have shown: Lemma 6. There exists a fully-dynamic algorithm maintaining an explicit 2-out orientation of an n-vertex dynamic forest with O(log n) worst-case update time.
Fractional Out-degree Orientations We will obtain a low-bounded out-degree orientation in the general case by deterministically rounding, what we shall refer to as a fractional out-degree orientation. Here, the orientation problem is relaxed so that the edges are allowed to be assigned partially to each end-point, and the goal is to compute an orientation such that the maximum total load assigned to a vertex is minimized. A formal definition is as follows:
Definition 7. A fractional α -bounded out-degree orientation O of a graph G1. ∀e = uv ∈ E(G): X u e + X v e = 1 2. ∀v ∈ V (G): s(v) = e:v∈e X v e ≤ α
If furthermore X u e ,X v e ∈ γ −1 · Z for all e ∈ E(G), we say that O is a (γ, α )-orientation.
In particular, an α -bounded out-degree orientation is just a (1, α )-orientation. We think of s(v) as the load on vertex v, and α as an upper bound on the allowed vertex load. The γ-parameter underlines the fact that we wish to discretise the fractional loads on edges to rational loads. If one does so in a symmetric manner for each edge, one can view a (γ, α )-orientation of a graph G as a (1, γα)-orientation of G γ , where we define G γ to be G, where every edge is replaced by γ copies. For an edge e = uv ∈ E(G), we denote by B e the bundle of γ edges representing e in G γ . If G γ is oriented, we denote by B u e the bundle of edges oriented u → v. Since the copies of e in B e are identical, we only care about the size of B u e , and not which copies of e it contains. Hence:
Observation 8. For a graph G, there is a natural bijection (up to symmetry) between (1, γ · α ) orientations of G γ and (γ, α )-orientations of G.
In light of this observation, we shall use these two descriptions interchangeably, and in some cases we shall refer to the same orientation as being both a (1, γα )-orientation of G γ and a (γ, α )-orientation of G. We follow the approaches of Sawlani & Wang [37] and Kopelowitz et al. [28], so we repeat the following:
Definition 9 ([37]). Given a (1, α )-orientation of a graph G γ , we say that an edge u → v ∈ E(G) is η-valid if s(u) − s(v) ≤ η and η-invalid otherwise. If also s(v) − s(u) ≤ η, we say that e = uv ∈ G is doubly η-valid. Furthermore, if s(v) − s(u) ≤ −η/2 we say that e is an η-tight out-edge of u and an η-tight in-edge of v. Note that if u → v is η-invalid, then s(u) − s(v) > η and so −η > s(v) − s(u), so uv is η-tight.
Also note that η will only differ from 1 in Appendix B. Lemma 11 (Implicit in [37]). Inserting an η-valid edge oriented u → v and reorienting a maximal η-tight chain from u will η-invalidate no η-valid edges.
Deleting an edge oriented u → v and reorienting a maximal η-tight chain to u will η-invalidate no η-valid edges.
Proof. Inserting the edge increases s(w) by one for some w. This will never η-invalidate any in-edges of w, but it might invalidate an out-edge of w. Suppose the edge w → w is invalidated if s(w) increases by 1. Then before s(w) increases, we must have s(w) ≥ s(w ) + η and so the edge ww is tight, contradicting that we reorient a maximal η-tight chain from u. The other statement is similar.
Remark 12. Note that a maximal η-tight chain has length at most 2· maxv s(v) η . Indeed, each time we follow an η-tight out-edge the load on the vertex increases by at least η/2.
If every edge is η-valid, Sawlani & Wang say that the orientation is locally η-stable. Kopelowitz et al. show the following guarantees for locally 1-stable orientations, where we, for ease of notation, define ∆ + := (1 + ε)αγ + log (1+ε) (n):
Lemma 13 (Implicit in [28]). If every edge in G γ is 1-valid, then max v s(v) ≤ ∆ + .
Proof. The proof is completely synchronous to that of Theorem 2.2 in [28], but we present it here for completeness.
Assume every edge is 1-valid and suppose for contradiction that d + (v) > ∆ + in G γ . Consider the i'th distance class of v in G γ , V i , i.e. V i is the set of vertices reachable from v via directed paths of length no more than i. Then for 1 ≤ i ≤ log (1+ε) n and for every vertex w ∈ V i , we have:
d + (w) ≥ d + (v) − i ≥ ∆ + − i ≥ (1 + ε)αγ
Now, it follows by induction on i that |V i | ≥ (1 + ε) i for all 1 ≤ i ≤ log (1+ε) n. Indeed, by the Theorem of Nash-Williams [34] we have α(V 1 ) ≤ γα(G), and so |V 1 | > ∆ + /(γα) ≥ (1 + ε). For the induction step, note that the number of out-edges of vertices in V i−1 is at least |V i−1 | · (1 + ε)αγ. Now, the induction step follows by applying the Theorem of Nash-Williams [34]
to get that α(V i ) ≤ α(G)γ and that |V i | − 1 ≥ (1 + ε)|V i−1 | > (1 + ε) i .
Implicit orientations
We are interested in maintaining a fractional out-degree orientation in which the fractional orientation of edges allow us to 'round' the fractional orientation to a low out-degree orientation. We are interested in two properties: first of all the maximum load of a vertex should be low, and second of all many of the edges should have either X u e or X v e close to 1, so that a naive rounding strategy does not increase the load of a vertex by much. By Lemma 13, if we ensure that the orientation is locally 1-stable, then we get an upper bound on the maximum vertex load. In order to ensure the second property, we redistribute load along cycles without breaking local stability. Our algorithm has two phases. A phase for inserting/deleting edges in a manner that η-invalidates no edges, thus ensuring the first property, and a second phase for redistributing load along edges in order to ensure that the orientation also has the second property. In order for these two phases to work (somewhat) independently, we think of each phase as having implicit access to the orientation; that is the insertion/deletion algorithm might have to pay a query cost in order to identify the precise fractional load of an edge or neighbourhood of a vertex.
Definition 14.
An algorithm on an n vertex dynamic and oriented graph has implicit (|L|, q(n)) access to an orientation, if it has access to: Implicitly Accessing Orientations In this section, we outline how to modify the algorithm of Kopelowitz et al. [28] to run on G γ and to support implicit access to the orientation. We describe the modifications here and give pseudocode Appendix A. The ideas presented here are not new; they arise in [28] and [37], but we present them for completeness. We think of the algorithm as being run on G γ for some γ to be specified later. We think of each edge e ∈ G as γ copies in G γ , but in practice we only store e along with counters |B u e |, |B v e | denoting the number of copies oriented in each direction. Now, we wish to run the algorithm from [28] in order to insert/delete each copy of an edge one-by-one. This algorithm inserts/deletes a copy of an edge in G γ using Lemma 11 with η = 1. We identify a tight chain from u by continuously looking at all out-neighbours and following tight out-edges, until the chain becomes maximal. We use max-heaps, stored at each vertex, to identify maximally tight chains to u. Since we only have implicit access to the orientation, we have to first process the list of possible changes to in-and out-neighbours before trying to identify the next tight edge. Furthermore, when we reorient said chains, we have to access the fractional load of each edge on the chain, before we can change it. Hence, we have:
Theorem 15 (implicit in [28]). Given implicit (|L|, q(n)) access to an orientation with max v s(v) ≤ ∆ + , there exists an algorithm that can insert and delete edges from the orientation without creating any new 1-invalid edges. The algorithm has worst-case insertion time of O(γ · ∆ + (∆ + + log(n)(|L| + 1) + q(n)))) and a worst case deletion time of O(γ · ∆ + (log(n)(|L| + 1) + q(n)))).
Proof. See Appendix A.
Remark 16. Each insertion/deletion of a copy of an edge in G γ with max v s(v) ≤ ∆ + changes the load of at most O(∆ + ) edges. Indeed, we only change the load of edges on tight chains (and potentially one new edge), so the statement follows from Remark 12.
Scheduling Updates Some of our algorithms need upper bounds on the arboricity to provide the ensured guarantees. This is, however, not as limiting a factor as one might think, if we are willing to settle for implicit algorithms. In this section we describe how to use the algorithm of Sawlani & Wang [37] to schedule updates to O(log n) different copies of a graph such that each copy satisfies different density constraints. Here, we describe the main ideas behind the algorithm, and in Appendix B, we paraphrase the ideas in more details.
Sawlani & Wang [37] maintain a fractional out-orientation of a graph G by using an algorithm similar to Theorem 15 to insert and delete edges in G γ . By allowing η to scale with the maximum density ρ of G, they are able to make the update time independent of the actual value of ρ, provided that they have accurate estimates of ρ. By using log n copies of G -each with different estimates ρ est of ρ, they are able to at all times keep the copy where ρ est ≤ max v s(v) < 2ρ est fully updated. They call this copy the active copy. Similar to Remark 12 they observe that one can safely insert an edge uv in a copy where at least one of s(u) or s(v) is below 2ρ est . If, however, this is not the case, one cannot afford to update the copy. Sawlani & Wang resolve this issue by scheduling the updates so that they are only performed, when we can afford to do them. We can use this algorithm as a scheduler for our algorithms: We also run our algorithm on log n copies of G. Whenever the algorithm from Theorem 1.1 in [37] has fully inserted or deleted an edge in a copy, we insert or delete the edge in our corresponding copy. Whenever our algorithm is queried, we then use the structure from the currently active copy to answer the query. Hence, we have:
Theorem 17 (Implicit in [37] as Theorem 1.1). There exists a fully dynamic algorithm for scheduling updates that at all times maintains a pointer to a fully-updated copy with estimate
ρ est where (1 − ε)ρ est /2 ≤ α(G) < 4ρ est .
Furthermore, the updates are scheduled such that a copy G with estimate ρ satisfies α(G ) ≤ 4ρ . The algorithm has amortised O(log 4 (n)/ε 6 ) update times. that distribute their loads somewhat equally between both endpoints, form a forest. This property is nice, if we want to transform our orientation to a bounded out-degree orientation, since all edges outside of the forest almost already have decided on an orientation, and we can 2-orient trees using Lemma 6. Definition 18 formalises this idea:
Definition 18. Let O be a (γ, α )-orientation of a graph G. Then H is a (δ, µ)-refinement of G wrt. O if: 1. V (H) = V (G) 2. For all e = uv ∈ E(G) : X u e , X v e ∈ (δ, 1 − δ) implies that e ∈ H. 3. If e ∈ H, then X u e , X v e ∈ [δ − µ, 1 − δ + µ]
The basic idea behind our algorithm is to maintain a refinement that is a forest. Whenever a cycle C occurs in the refinement, we can redistribute the fractional loads along the cycle so as to not change s(v) for any v ∈ C, but such that an edge of C does not satisfy condition 2. in Definition 18. Thus we can remove this edge and a again obtain an acyclic refinement with respect to this new orientation. Hence, we have the following observation:
Observation 19. Suppose 1 > δ > γ −1 + µ ≥ 2γ −1 > 0. Let H be (δ, µ)-refinement of a graph G wrt. some (γ, α )-orientation O of G. Then there exists a (δ, µ)-refinement of G, say H , wrt. some (γ, α )-orientation O of G, such that H is a forest. Proof. Suppose H is not one such (δ, µ)-refinement i.e. H is not a forest. Let C = v 0 , v 1 , . . . , v c , v 0 be a cycle in H. Set l(C) = min i {min{X vi vivi+1 , 1 − X vi vivi+1 }}
Now if l(C) ≤ δ, we can just remove the edge minimizing l(C) from H, so suppose this is not the case.
If X vi vivi+1 = l(C) for some i, we set X vj vj vj+1 = X vj vj vj+1 − (l(C) − δ + µ) and X vj+1 vj vj+1 = X vj+1 vj vj+1 + (l(C) − δ + µ) for all j. Otherwise, 1 − X vj vivi+1 = l(C) for some i, and we set X vj vj vj+1 = X vj vj vj+1 + (l(C) − δ + µ) and X vj+1 vj vj+1 = X vj+1 vj vj+1 − (l(C) − δ + µ) for all j.
Observe that this change of the fractional orientations on the cycles does not violate the 2 conditions in Definition 7, so this yields a new (γ, α )-orientationÔ of G. Furthermore, H is still a (δ, µ)-refinement of G. Indeed, we only change the fractional load of edges in C, and for these edges, after the changes, we have:
X vj vj vj+1 ≥ δ − µ X vj vj vj+1 ≤ 1 − δ + µ Furthermore,
for the i minimizing l(C) one of the inequalities holds with equality, allowing us to remove it from H and obtain a new, smaller (δ, µ)-refinement of G wrt. the new orientation. Continuing like this eventually turns H in to a forest.
Note that if every edge of a graph G is η-valid, then an edge e = uv ∈ G can only distribute its load somewhat evenly between u and v, if s(u) and s(v) are approximately the same. This implies that e is actually doubly η-valid:
Observation 20. If every edge in G γ is η-valid, then every edge of a (δ, µ)-refinement H of G with 1 > δ > γ −1 + µ ≥ 2γ −1 is doubly η-valid. Proof. If e = uv ∈ H, then X u e , X v e ∈ [δ − µ, 1 − δ + µ] so X u e , X v e > 0. Hence, in G γ we have |B u e |,
|B v e | > 0, and so there is at least one edge oriented u → v and at least one edge oriented v → u in G γ . Since all edges are η-valid, the statement follows.
Since the redistribution of fractional load of edges along a cycle does not change the load s(v) of any vertex v, performing the redistribution from Observation 19 η-invalidates no edges.
The algorithm
As outlined earlier, our algorithm has 2 phases. In the first phase, we will insert and delete edges without η-invalidating any edges. We do this using the algorithm from Section 2. In the second phase, we examine all of the edges, whose fractional load was altered in phase 1. These edges might need to enter or exit H, depending on their new load. If such an insertion in H creates a unique cycle, we remove it as described in Observation 19. More precisely, the algorithm works as follows:
1. Insert (delete) γ copies of e into G γ one at a time using a phase I algorithm from Section 2. Whenever a copy of an edge f ∈ E(G) is reoriented in phase I, we push f onto a stack Q. If e is deleted in G, we also remove e from H.
2.
When all γ copies of e are inserted, we set g = uv = pop(Q) and update H as follows until Q is empty:
If g ∈ H and X u g ∈ (δ, 1 − δ), we update the weight of g in H to match that of G γ . If g ∈ H and X u g / ∈ (δ, 1 − δ), we remove g from H. If g / ∈ H and X u g ∈ (δ, 1 − δ), we push g onto a new stack S. If g / ∈ H and X u g / ∈ (δ, 1 − δ), we do nothing.
3.
After processing all of Q, H together with the edges in S form a (δ, µ)-refinement of G. We now process each edge h = uv ∈ S as follows:
If u, v are not in the same tree in H, we insert h into H. Otherwise, u, v are in a unique cycle C in H. We update the weights along C, locate an edge wz along C with X w wz , X z wz / ∈ (δ, 1 − δ) and remove it from H. If uv = wz, we insert uv into H.
Finally, we update B w wz , B z wz in G − H to match the weights wz had in H.
Since only edges from Q can enter S, we have the following Observation:
Observation 21. Let S max and Q max denote the maximum size of the stacks above during an insertion or a deletion. Then we have S max ≤ Q max ≤ T , where T is the total number of edges whose fractional orientation are altered during an insertion or a deletion.
Furthermore, Observations 19 and 20 and Theorem 15 imply the invariants:
Invariant 22. Under the orientation induced by H for edges in E(H) and by
G − H for edges in E(G − H), every edge in E(G) is η-valid.
Invariant 23. H is both a (δ, µ)-refinement and a forest.
Implementing updates
Since we maintain the invariant that H is a forest, we can use data structures for maintaining information in fully dynamic forests to store and update H:
Lemma 24 (Implicit in [2]). Let F be a dynamic forest in which every edge e = wz is assigned a pair of variables X w e , X z e ∈ [0, 1] s.t. X w e + X z e = 1. Then there exists a data structure supporting the following operations, all in O(log |F |)-time:
link(u, v, X u uv , X v uv ): Add the edge uv to F and set X u uv , X v uv = 1 − X u uv as indicated. cut(u, v): Remove the edge uv from F . connected(u, v): Return true if u, v are in the same tree, and false otherwise. add_weight(u, v, x): For all edges wz on the path u . . . wz . . . v between u and v in F , set X w wz = X w wz + x and X z wz = X z wz − x. min_weight(u, v): Return the minimum X w wz s.t. wz is on the path u . . . wz . . . v in F . max_weight(u, v)): Return the maximum X w wz s.t. wz is on the path u . . . wz . . . v in F . Proof. See Appendix C.
Note that using non-local search as described in [2], one can also locate the edges of minimum/maximum weight in O(log |F |)-time. The Lemma also shows that we can process an edge in Q in O(log n)-time. To process an edge uv ∈ S creating a cycle C in the (δ, µ) refinement H, we do as follows. Depending on the argument minimizing l(
C) = µ + min{min_weight(u, v) − δ, 1 − δ − max_weight(u, v), X u uv − δ, 1 − δ − X v uv },
we either add or subtract l(C) to every edge in C. We determine and remove the edge that minimized l(C) from H. Thus we can process an edge in S in O(log n)-time. Finally, if µ > γ −1 then every edge in S ∪ H has at least one copy in G γ pointing in each direction both before and after the inversion of a cycle. Hence, no vertex receives any new in-nor out-neighbours. Since inverting a cycle does not change the load of any vertex, we need not update any priority queues for the insertion/deletion algorithm. Hence, we do not have to return any lists and so |L| = q(n) = 0. Theorem (Theorem 1). For 1 > ε > 0, there exists a fully-dynamic algorithm maintaining an explicit ((1 + ε)α + 2)-bounded out-degree orientation with worst-case insertion time O(log 3 n · α 2 /ε 6 ) and worst-case deletion time O(log 3 n · α/ε 4 ).
Conclusions
Theorem 26. Suppose 1 > δ > γ −1 + µ > 2γ −1 > 0, ε > 0. Then, there exists a dynamic algorithm that maintains a (γ, (1 + ε)α + γ −1 log (1+ε) n)-orientation of a dynamic graph G with arboricity α as well as a (δ, µ)-refinement H of G wrt. this orientation such that H is a forest. The fractional orientation of an edge can be computed in time O(log n), insertion takes worst-case O(γ · (∆ + ) 2 ) time and deletion takes worst-case O(γ · ∆ + · log (n)) time.
Proof. Using Theorem 26 with ε = ε/20, γ = log n/ε 2 , δ = 2γ −1 and µ = γ −1 , we can with only O(1) overhead deterministically round any edge in G − H to point away from the vertex to which it assigns the highest load. This gives an out-degree upper bounded by
((1 + ε )α + γ −1 log 1+ε (n))/(1 − 2γ −1 ) ≤ ((1 + ε )α + γ −1 ε −1 log(n)) · (1 + 4γ −1 ) ≤ α(1 + ε + γ −1 ε −1 log(n) + 4γ −1 + 4γ −1 ε + 4γ −2 ε −1 log(n)) ≤ α(1 + 2ε + 4ε 2 + 4ε 3 + 4ε 5 ) ≤ α(1 + ε)
where we used the fact that for r ≤ 1/2, we have 1/(1 − r) ≤ 1 + 2r. Note that this is an upper bound on the out-degree, so the actual out-degree is at most the floor of this expression. Finally, we also run the algorithm from Lemma 6 to 2-orient H. This gives at most 2 extra out-edges per vertex, and takes time O(log(n) log(n)/ε −2 · α log n) = O(log 3 (n)αε −2 ), since we have at most O(γ∆ + ) insertions and deletions into H per insertion into G, and we can insert each such edge into the rounding scheme on H spending O(log(n))-time.
By naively rounding G − H in Theorem 26 (for specific values of parameters), and splitting the out-orientation using Lemma 27, we get an algorithm for dynamically maintaining a decomposition into (1 + ε)α pseudoforests and a single forest. Applying the colouring techniques described in Section 6, then yields Corollary 2.
Forests
We begin this section by outlining the main ideas for turning a dynamic low out-orientation into a dynamic low arboricity decomposition. Given a dynamic α -bounded out-degree orientation, one can, with very little overhead, split it into α 1-bounded out-degree orientations using a (slight modification) of an algorithm by Henzinger et al. [25]. Now, given this dynamic pseudoforest partition, we wish to apply the ideas of Blumenstock & Fischer [12] in order to turn the α pseudoforests into α + 1 forests. The main technical challenge of making this process dynamic is the following: the algorithm from [25] relies heavily on each vertex having out-degree no more than 1 in each pseudoforest. However, the approach of Blumenstock & Fischer [12] is to move edges between pseudoforests, showing no regards as to why an edge was placed in a pseudoforest to begin with. Hence, if one naively applies this approach on top of the pseudoforest partition, one could potentially ruin the invariant that every vertex has out-degree no more than 1 in each pseudoforest, causing the algorithm of Henzinger et al. [25] to fail. We tackle this problem in steps. First, we show that if we were somehow able to invert the orientations of cycles, then we can make the moves of Blumenstock & Fischer's approach faithful to the degree condition of the pseudoforest algorithm of Henzinger et al. [25]. If we invert orientations along cycles in the pseudoforests, the out-degree of no vertex in the pseudoforests is changed. However, if we wish to perform these operations, we will have to do it in a manner that still allows us to maintain the underlying α -bounded out-degree orientation. If the cycles are doubly η-valid, we invert the cycles using Lemma 24. We do as in Section 3, but this time we add or subtract 1 − δ along the cycles. This ensures that every edge on the cycle now prefers the other endpoint, and so is naively rounded to the opposite direction without ending in H. The problem is that we have no guarantee that all edges are doubly η-valid. If an edge is only singly η-valid, then redistributing the load along a cycle containing this edge causes the edge to become invalid. However, by Lemma 11, we can delete such invalid edges and reinsert them again to restore the invariant that all edges are η-valid.
Ideas of Henzinger et al. and Blumenstock & Fischer
Given an α -bounded out-degree orientation, one can split it into α pseudoforests by partitioning the edges such that each vertex has out-degree at most one in each partition. Then every connected component P C in a partition is a pseudoforest. Indeed, |E(P C )| ≤ |P C | since every vertex has out-degree at most one. Hence, there can be at most one cycle in P C . This idea is implicit in an algorithm of Henzinger et al. [25]. Note that we can store each pseudotree as a top tree with one extra edge with only O(log n) overhead per operation.
Lemma 27 (Implicit in [25]). Given black box access to an algorithm maintaining an α -bounded out-degree with update time T (n), there exist an algorithm maintaining an α pseudo-forest decomposition with update time O(T (n)). Moving edges between pseudoforests. Before performing the swap, we reorient a cycle so that the swapped edges both are out-edges of their common endpoint These Lemmas motivate the following approach: use Lemma 28 to ensure that the surplus graph is always colourful. Next use Lemma 28 to remove any cycles from the surplus graph.
Proof. See
Our algorithm for maintaining dynamic arboricity decompositions
Assume that we have an upper bound α max on the arboricity throughout the entire update sequence. The algorithm works roughly as follows:
1. Run the algorithm from Theorem 26. If a non-doubly valid cycle is reoriented in this process, we remove the singly-valid edges from the pseudoforests and add them to R. This ensures that the two edges from the same matching that we were trying to separate into two different components, are indeed separated. We will later bound the total number of edges pushed to R. If, on the other hand, the component is colourful, e may sit in a cyclic component. Then we apply Lemma 29 to remove the unique cycle. This may create a new non-colourful component, which we handle as before.
In the following, we describe the necessary data structures and sub-routines needed to perform these operations. Note that in order to show Lemma 30 in Section 4.3, we use ideas from Sections 4.5 and 4.6, so the proof of Lemma 30 is deferred to Section 4.6.
Operations on the surplus graph
In this section, we assume all cycles are doubly η-valid. In section 4.5, we handle cases where this is not the case. Assuming that G[M ] is both colourful and acyclic, we can insert an edge in G[M ] and restore these invariants by performing switches according to Lemmas 28 and 29. Indeed, after inserting an edge, we can run, for example, a DFS on the component in G[M ] to determine if it is colourful. If it is not, we locate a path e 1 , . . . , e k such that e 1 , e k ∈ M i . Then we apply Lemma 28 to e i−1 and e i beginning with i = k, until an edge from M i is removed from G[M ]. Note that this is certain to happen when e 1 and e 3 belong to the same pseudoforest. We continue locating and handling paths until the component becomes colourful. If the component is colourful, but not acyclic, we choose a vertex v on the cycle and apply Lemma 29 to determine a pseudoforest represented in the component in which v is connected to no other vertex in the cyclic component. Then we determine a path in the surplus graph between an edge in said pseudoforest and v. Now we move the edge in this pseudoforest to v using Lemma 28. If the edge is removed from G[M ] or the path is disconnected, we repeat the process. When such an edge is incident to v, we switch it with an edge on the cycle. Finally, we replace it in M with the unique other edge incident to v in the pseudotree that put it in M . Now G[M ] is acyclic, but it may not be colourful. If this it the case, we repeat the arguments above until it becomes colourful. Note that these moves never create a cycle.
Lemma 30. After inserting an edge into G[M ]
, we can restore acyclicity and colourfulness in O(α 3 log 2 (n)) time.
Recovering neighbours
For each vertex, we will lazily maintain which of its out-edges belong to which pseudoforest. This costs only O(1) overhead, when actually moving said edges. However, whenever we invert a cycle, these edges may change. Since the cycles can be long, we can only afford to update this information lazily, whenever the insertion/deletion algorithm determines the new out-neighbours of a vertex. When this happens, we say the vertex is accessed. Whenever an edge has its fractional load changed via a cycle inversion, it is always changed by the same amount. Hence, we make the following Observation:
Observation 31. Between two accesses of a vertex v, the only possible new in-neighbours are the edges which were out-neighbours at the last access of v, and the only new out-neighbours are the vertices that are out-neighbours at the current access of v.
Proof. Consider an edge oriented u → v at two consecutive accesses of v. Then, the fractional orientation of uv is the same as the last time v was accessed. Indeed, the fractional orientation can only change, if either the edge is on a maximally tight chain being reoriented, in which case v was accessed, or if uv is on a cycle being inverted. Every time, this happens the orientation of uv changes. Hence, uv has been inverted exactly the same amount of times in both directions. Since each inversion changes the fractional orientation by the same amount, it follows that the fractional orientation of uv has not changed.
The same argument also shows that the edges, whose fractional orientation have changed, are exactly the edges that have had there orientation changed since the last time v was accessed. These edges are precisely the old and the new out-edges.
Thus, we can recover exactly which incident edges might have changed in-and/or outneighbour status from v, since the last time v was accessed by the insertion/deletion algorithm. To do so, we maintain that each top tree is rooted in the unique vertex, which has out-degree 0, when the underlying orientation is restricted to the tree. This ensures that we can recover v's unique out-edge in a pseudoforest by finding the first edge on the unique v-to-root path in the top tree. We maintain this information as follows:
When we link(u, v) with an edge oriented u → v, we set the root of the new tree to be that of the tree containing v. When we cut(u, v) with an edge oriented u → v, we set the root of the tree containing u to be u and that containing v to be the same as the old tree. When we invert the orientation along a cycle originally oriented u → v → · · · → u, we change the root from v to u. When we perform a Lemma 29 swap, we also update the root accordingly.
Note that each update is accompanied by an operation costing O(log n) time, so the overhead for maintaining this information is only O (1). With this information, we can recover the old out-neighbours as the stored out-neighbours, and the new out-neighbours by taking the first edge on the path from v to the root. Hence, we have shown:
Non doubly η-valid cycles
If a cycle is not doubly η-invalid, we still switch the orientation as before, but now we have to fix invalid edges. Assuming we know which edges have become η-invalidated, we fix them as follows: For every invalid edge, we first remove the edge from the pseudoforest it resides in. This has two consequences. Firstly, the algorithm from Lemma 27 might move O(1) edges between pseudoforests, and secondly, we also have to move an edge from the surplus graph back down as a normal edge in the pseudoforest it comes from. All of the (re)moved edges are pushed to the queue R. Then, we delete all invalid copies of edges in G γ , and reinsert them. Now, all edges are valid again, and so we continue processing edges in R as described in Section 4.2. If an edge now belongs to H, we do not insert it into any pseudoforest.
It is important to note that the second consequence i.e. that we remove an edge from G[M ], either makes a Lemma 28 switch successful by removing one of the edges from M i , or it removes an edge on one of the at most two paths between edges in M i . In this case, we try to locate a second path, and handle it as before. This happens at most once: the component has at most one cycle, and hence at most two paths between two vertices. We ascribe the cost of deleting and reinserting invalid edges to the potential in Lemma 33 that bounds the total number of copies of edges that are inserted into G γ . This cost is not ascribed to the algorithm maintaining G[M ]. Set ∆ + max = (1 + ε)α max γ + log (1+ε) n, we have:
Lemma 33. The total amount of insertions and deletions performed by the insertion/deletion algorithm over the entire update sequence can be upper bounded by
O( γ∆ + max η (∆ + max · i + d))
Proof. We use the following potential to bound the number of insertions/deletions performed:
Φ(O i ) = v:s(v)≥1 (∆ + max − s(v)) 2
Note that by Lemma 13 max s(v) ≤ ∆ + ≤ ∆ + max , so an insertion or a deletion of a single copy of an edge in G γ increases the potential by O((∆ + max ) 2 ) and O(∆ + max ) respectively, since each operation increments/decrements the load of at most one vertex each. Hence, we have that the total sum of Φ increases over the entire update sequence, ∆(Φ), satisfies:
∆(Φ) ≤ O(i · γ(∆ + max ) 2 + d · γ∆ + max )
Deleting an η-invalid edge and reinserting it again, decreases the out-degree of a vertex of degree, say δ, and increases the out-degree of a vertex of degree ≤ δ − η − 1. Hence, for the low degree vertex the potential decreases by at least
(∆ + max − δ + η + 1) 2 − (∆ + max − δ + η) 2 = 1 + 2 · (∆ + max − δ + η)
For the high-degree vertex the potential is increased by at most:
(∆ + max − δ + 1) 2 − (∆ + max − δ) 2 = 1 + 2 · (∆ + max − δ)
Hence, in total the potential decreases by 2η units. Therefore, we can reorient at most O( γ∆ + max η (∆ + max · i + d)) η-invalid edges. Each reorientation takes two updates -one insertion and one deletion. Thus, we arrive at the stated bound.
Lemma 33 allows us to bound the total number of edges moved between pseudoforests:
Lemma 34. We move at most O( γ(∆ + max ) 3 η (i + d
)) edges between pseudoforests.
Proof. Each time an edge has its load changed by the insertion/deletion algorithm, it might cause an edge to have its orientation changed, and hence end up in a new pseudo forest. This might cause the algorithm from Lemma 27 to move O(1) other edges between pseudoforests, and consequently these edges may also end up in the surplus graph. Since each insertion/deletion changes the fractional orientation of at most ∆ + edges, the Lemma follows from Lemma 33.
Note that this implies that the total no. of insertions into R is O(
γ(∆ + max ) 3 η (i + d)).
Locating singly η-valid edges
When we are accessing an edge, we can check if it is doubly η-valid or not (this information depends only on the load on the endpoints), and maintain this information in a dynamic forest using just 1 bit of information per edge. This allows us to later locate these edges using non-local searches in top trees. However, when edges go between being singly η-valid and doubly η-valid through operations not accessing said edge, we are not able to maintain this information. This can happen in two ways: either 1) a vertex has its load lowered causing an in-going edge to now become doubly η-valid or an outgoing edge to become singly η-valid or 2) a vertex has its load increased causing similar issues. We say an edge is clean if we updated the validity bit of an edge, the last time the out-degree of an endpoint of the edge was altered. Otherwise, we say it is dirty. Now if all edges on a cycle are clean, we can use top trees to direct searches for the edges that become invalidated by inverting the cycle. We maintain a heavy-light decomposition of every forest using dynamic st-trees [38] to help us ensure that we can clean all edges in a cycle in time O(log 2 n). The idea is to maintain the invariant that all heavy edges are clean. Now we can clean a cycle by cleaning the at most O(log n) light edges on said cycle. In order to realise this invariant, whenever the degree of a vertex is changed, we need to update all of its incident heavy edges in all of the heavy-light decompositions. Since a vertex is incident to at most 2 heavy edges in each forest, we have to update O(α) heavy edges. The following holds:
Observation 35. We can locate singly η-valid edges on a clean cycle in time O(log n) per edge, if we spend O(log n) overhead updating the bit indicating double validity.
Proof. With O(log n) overhead, we can update a top tree with this information, so that we can locate singly valid edges in time O(log n) per singly valid edge using non-local searches [2].
Observation 36. We can insert and delete edges in the heavy-light decomposition in worst case O(log 2 n) time.
Proof. We maintain the heavy-light decomposition using dynamic st-trees [38]. An insertion (or a deletion) creates at most O(log n) new heavy edges. By Observation 35, we have to pay O(log n) overhead to clean each edge. O(log 2 n).
Lemma 37. We can check if a cycle is doubly valid in time
Proof. The algorithm maintains the invariant that all heavy edges are clean, hence the only dirty edges are light edges. There are at most log n such edges on any root-to-leaf path, and so at most 2 log n such edges on the path between any two vertices in the tree. We can locate each light edge in time O(1) per edge. Indeed, we begin at u resp. v and chase pointers to the end of heavy paths on the way to the root.
Proof. A component V C has size at most O(α), so we can determine colourfulness and locate relevant paths and cycles using for example a DFS in O(α) time.
To locate a Lemma 29 switch for a vertex v on the cycle, we check for each i such that Proof. Before performing a switch we determine in O(log 2 n) time, using Lemma 37, if the cycles are doubly valid. If a cycle is singly valid, we still achieve our goal by Section 4.5. We ascribe the cost of fixing singly valid edges to the potential in Lemma 33. Finally, we can reorient a cycle by inverting it with x = 1 − δ using Lemma 24 in O(log n) time. This ensures that every edge on the cycle now prefers the other endpoint, and so is naively rounded to the opposite direction. Also note that this ensures that no edge on the cycle ends up in H.
M i ∩ V C = ∅ whether N i (v) ∩ V C is
If we end up with a non-colourful component, by Observation 38 and 39 we spend O(α log 2 n) time locating a path and performing switches, since such a path has length at most O(α). We possibly need to do this for every pair of edges belonging to the same pseudoforest in said component. In the worst case, the inserted edge connects to colourful components. Thus there are at most 2 edges from the same pseudoforest, and therefore at most one pair per pseudoforest. Hence, there are at most O(α) such pairs, and we can handle each pair in O(α log 2 n) time by above.
If we end up with a cyclic component, we spend time O(α 2 log n) locating a Lemma 29 switch. In order to perform it, we do at most O(α) Lemma 28 switches. We fail this process at most O(α) times, since each time we fail the size of the component in the surplus graph decreases. The component has size O(α), since the graph was colourful before the insertion. When the process is successful, we can swap the edge in G [M ] in O(log n) time. Finally, this might yield an even bigger non-colourful, but acyclic component that we handle as before. Correctness follows since no switches can create any new cycles.
Conclusion
Theorem 3 follows from Lemma 40 below. We prove Corollary 4 in Section 6.
Lemma 40. Consider a sequence of updates with i insertions and d deletions.
1. The insertion/deletion algorithm spends O(log 6 (n) · α 4 max · ε −12 (i + d)) time to update the fractional out-degree orientation and the refinement. Now we can show Theorem 3. We recall it below:
Theorem (Theorem 3). Given an initially empty and dynamic graph undergoing an arboricity α max preserving sequence of updates, there exists an algorithm maintaining a (1 + ε)α + 2 arboricity decomposition with an amortized update time of O(poly(log n, α max , ε −1 )). In particular, setting ε < α −1 max yields α + 2 forests with an amortized update time of O(poly(log n, α max )).
Proof. Apply the algorithm from Theorem 26 with ε = ε/10,γ = log n/ε 2 , δ = 2γ −1 and µ = γ −1 as in the proof of Theorem 1 in order to insert/delete edges and maintain the refinements. Now Lemma 40 shows that the amortised cost of running the algorithm from Section 4 is O(log 6 (n) · α 6 max · ε −12 ).
We have shown how to maintain an α + 2 arboricity decomposition of a fully dynamic graph as it undergoes an arboricity α preserving sequence of updates in poly(log n, α) time per update. We have also shown how to maintain an (1 + ε)α + 2 out-orientation of a fully dynamic graph in poly(log n, α) time per update. These algorithms are the first dynamic algorithms to go below 2α forests and out-edges, respectively, and the number of forests matches the best near-linear static algorithm by Blumenstock and Fischer [12]. We apply these algorithms to get new trade-offs for implicit colouring algorithms for bounded arboricity graphs. In particular, we maintain 4 · 2 α and 2 · 3 α implicit colourings in poly(log n, α) time per update. This improves upon the 2 40α colours of the previous most colour-efficient algorithm maintaining poly(log n, α) update time [25]. In particular, this reduces the number of colours for planar graphs from 2 120 to 32. An interesting direction for future work is to see, if one can reduce the number of forests even further in the static case, while still achieving near-linear running time. Also, even though our algorithms use few colours and forests, the update times contain quite high polynomials in both log n and α. Is it possible to get more efficient update times without using more forests? Finally, for constant α, we get α + 2 out-edges. Brodal & Fagerberg [13] showed that one cannot get α out-edges with faster than Ω(n) update time (even amortised). The question remains, can one get α + 1?
Acyclic Orientations and Arboricity Decompositions
In this section, we describe the algorithm in Theorem 5. We modify an algorithm by Brodal & Fagerberg [13]. Specifically, we change how an edge is inserted. The algorithm maintains a list of out-edges out(u) for each vertex u ∈ G. An edge e is in out(u) if and only if e ∈ E(G) and e is oriented away from u. As a result d + (u) = |out(u)|. All of these lists are initialized to be empty. The algorithm ensures that the maximum out-degree of the vertices in G is d for some constant d > 2α to be specified later. The algorithm handles deletions and insertions in the following way: Deletion: If e incident to x, y is deleted, we search out(x) and out(y) for e, and delete it. Insertion: When an edge e is inserted, an arbitrary endpoint u of e is chosen, and e is added to out(u). Now every edge in out(u) is oriented in the other direction (also e) i.e. we delete f = (u, v) from out(u) and add f to out(v) instead for all edges f ∈ out(u). Now u has out-degree at most d, but the reorientations of an edge f = (u, v) might increase |out(v)| above d. The algorithm then proceeds by reorienting all out-edges out of v. It continues this process until all vertices have out-degree at most d.
Note that this process terminates: Since G has arboricity α, it has an orientation O such that the maximum out-degree in G is α. Call an edge in E(G) good, if it is oriented the same way by both the algorithm and O and bad if it isn't. Now, inserting e could, in the worst case, make the algorithm change the orientation of α good edges. However from here on, every vertex whose edges are reoriented will increase the total number of good edges by at least d − 2α ≥ 1, so the process terminates. The algorithm differs from the one presented in [13] in only one way. When an edge e = uv is inserted, we always turn u into a sink. In [13], this only happens if u's out-degree increases above d. This small modification ensures no cycles are created: when an edge is inserted, one of its endpoints is turned into a sink, and so this edge is in no cycle. Also, turning a vertex into a sink does not create any cycles.
Correctness
The algorithm clearly maintains a d-bounded out-degree orientation. Hence, correctness of the algorithm follows by showing Lemma 41: Lemma 41. Let G s be G after the first s updates. Then the orientation O s maintained by the algorithm is acyclic.
Proof. We will conceptually attach update time-stamps to all vertices: Assume we run the algorithm on an initially empty and dynamic graph G undergoing an arboricity α preserving sequence of insertions and deletions. After the first s updates, the algorithm will have reoriented the out-edges out of some vertices. Suppose v s 1 , . . . , v s k is the order in which this is done. That is for all i the out-edges out of v s i were reoriented by the algorithm before the out-edges out of v s i+1 were reoriented. After the first s updates, we conceptually attach a time
-stamp t s v to each vertex v ∈ V (G) equal to the maximum index t such that v s t = v. If v ∈ v s 1 , . . . , v s k , we define t s v = −1.
Claim 42. Let G s be G after the first s updates. Suppose e = uv ∈ E(G s ) is oriented from u towards v in the orientation of G s maintained by the algorithm. Then t s u < t s v , that is the edges out of u were last reoriented, before the edges out of v were reoriented.
Proof. Note that t s v > −1. Indeed, an edge cannot be oriented towards a vertex without that vertex having had its out-edges reoriented. Since all time-stamps except for −1 are unique, we cannot have t s u = t s v . Suppose for contradiction that t s u > t s v . Since e = uv only can have its orientation changed if either u or v has the orientation of all its out-edges flipped, the last time that the orientation of e could have changed, was at time-step t s u . But then e cannot point away from u, since u has out-degree 0 after all its out-edges are reoriented.
Suppose now that there is a directed cycle C = v 0 , v 1 , . . . , v c , v 0 in G s , with edges oriented from v i to v i+1 for i = 0, . . . , c − 1 and from v c towards v 0 . Then by Claim 42, we must have that t s vi < t s vi+1 for all i = 1, . . . , c − 1. But by Claim 42, we also have t s vc < t s v0 , and so we arrive at the contradiction t s v0 < t s vc < t s v0 .
Analysis
The analysis of the algorithm is synchronous to that in [13]: we just need to show a modified version of Lemma 1 in [13]. This reduces the problem to that of describing an offline reorientation scheme using few reorientations throughout the updates. Here, we use the scheme presented as Lemma 3 in [13]. The idea behind this scheme is the following: If a vertex v has out-degree δ > α, then it must have a neighbour, reachable by following only O(log δ/α (n)) out-edges, with out-degree no more than α. Indeed, if the t-neighbourhood of v does not contain such a vertex, then induction and the Theorem of Nash-Williams [38] implies that the (t + 1)-neighbourhood must have size at least (δ/α) t+1 , and so the O(log δ/α (n))-neighbourhood must contain such a vertex. The reorientation scheme is then to accommodate insertion of an out-edge of u by lowering the out-degree of u via reorienting edges along such short paths. Since an edge must be inserted before it can be deleted, the reorientation scheme is run on the update sequence played in reverse in order to push the update cost unto delete operations, and we end up with a scheme achieving out-degree ≤ δ using r = c · log δ/α n reorientations with c being the number of deletes performed. The modified version of the reduction to the existence of such a reorientation scheme is as follows.
Lemma 43. Assume we are given an initially empty and dynamic graph G undergoing an arboricity α preserving sequence σ of insertions and deletions. Suppose there are i insertions in σ. If there exists a sequence of δ-oriented graphsĜ 1 , . . .Ĝ |σ| using at most r reorientations between them in total, then the algorithm performs at most
(δ · i + r)( d + 1 d + 1 − 2δ ) reorientations in total.
Proof. The proof is synchronous to that of Lemma 2 in [13]. We count how many times we can possibly turn a good edge bad, and then use this to bound the number of edges that are reoriented. We do this by comparing the orientation of G j with that ofĜ j . We say an edge is bad, if it is oriented differently in G j andĜ j . Otherwise, we call it good. Doing this, we get a non-negative potential function
Φ(G j ) = the no. of bad edges in E(G j )
Initially, there are no bad edges, since we begin with an empty graph. Furthermore, Φ is increased by at most δ for each of the i edge insertions and 1 for each of the r reorientations. Indeed, when we reorient all out-edges, we create at most δ bad edges. Thus we lose at least d + 1 − 2δ ≥ 1 bad edges every time the out-edges out of an out-degree d + 1 vertex are reoriented. Consequently, we can perform no more than
(δ · i + r) d + 1 − 2δ
such operations. Each such operation turns at most δ good edges bad, so the total number of times these operations turn a good edge bad is upper bounded by
δ(δ · i + r) d + 1 − 2δ
In particular, the number of bad edges the operation can turn good is at most δi + r + δ(δ·i+r) d+1−2δ . Adding these numbers together, we see that the algorithm performs at most
δ · i + r + 2δ(δ · i + r) d + 1 − 2δ = (δ · i + r)(1 + 2δ d + 1 − 2δ ) = (δ · i + r)( d + 1 d + 1 − 2δ )
reorientations, since each reorientation either turns a bad edge good or a good edge bad.
Combining the scheme from [13] with Lemma 41 yields the following for which specifying parameters yields Theorem 5: Applying Lemma 43, we find that the algorithm performs at most (δ·i+c· log δ α (n) ) d+1 d+1−2δ reorientations in total. As such we retrieve the stated update costs.
Theorem
Setting δ = α max + 1 and d = 2δ in the above Theorem, we arrive at Theorem 5.
Dynamic Colouring
In this section, we discuss how to turn dynamic arboricity decomposition algorithms and dynamic out-orientation algorithms into implicit dynamic colouring algorithms, and use this to deduce Corollaries 2 and 4. For the dynamic arbroricity decomposition algorithms, we do exactly as in [25]. Here, Henzinger et al. maintain each forest in a data structure for maintaining information in dynamic forests, more specifically they use top trees [2]. Now they root each forest arbitrarily and colour the forests by the parity of the distance to the root. Finally, in order to obtain a proper colouring of the whole graph, they return the product colouring over all of the forests in the arboricity decomposition. It costs O(log n) overhead to maintain each forest as a top forest, and so they arrive at Lemma 45 ([25]). Given blackbox access to a dynamic algorithm maintaining an α arboricity decomposition of a graph G with update time T (n), there exists a dynamic algorithm maintaining an implicit 2 α colouring of G with update time O(T (n) log n) and query-time O(α log n).
In order to use dynamic out-orientations, they show how to turn a dynamic algorithm maintaining an α -bounded out-degree orientation into a dynamic algorithm maintaining a 2α -arboricity decomposition. Finally, they apply Lemma 45 to get a 2 2α = 4 α colouring.
We use the following strategy instead. We use Lemma 27 to turn the out-orientation into a pseudoarboricity decomposition. Then we maintain each pseudoforest as a forest and a matching. Finally, we root each pseudotree in an arbitrary vertex on the unicycle (as we did in Section 4.4). Now with O(log n) overhead, we can 3 colour each pseudoforest. Indeed, we store each pseudotree as a rooted top tree plus an edge such that one of the endpoints of the edge not in the top tree is the root of the top tree. Then we can colour the root a unique colour and all other vertices by the parity of the distance to the root as before. Finally, we form the final colouring as the product colouring over all pseudoforests. This colouring can again be queried in time O(α log n).
Using these strategies for colouring, we can now deduce Corollaries 4 and 2.
Corollary (Corollary 4). Given a dynamic graph with n vertices, there exists a fully dynamic algorithm that maintains an implicit 4 · 2 α colouring with an amortized update of O(polylog n) and a query time of O(α log n).
Proof. We schedule the updates on O(log n) copies of G using Theorem 17. On each copy we apply the algorithm of Theorem 3 with α max = 4ρ est and ε < α −1 max combined with Lemma 45. In the remaining copies, we do nothing. In order to answer a colouring query, we query the currently active copy. If a copy with ρ est > log n is active, we just colour each vertex by their vertex id, since then α(G) ≥ ρ est ≥ log n, and so 2 α(G) ≥ n.
Corollary (Corollary 2). Given a dynamic graph with n vertices, there exists a fully dynamic algorithm that maintains an implicit 2 · 3 (1+ε)α colouring with an amortized update time of O(log 4 n · α 2 /ε 6 ) and a query time of O(α log n).
Proof. Apply the algorithm from Theorem 26 with ε = ε/20,γ = log n/ε 2 , δ = 2γ −1 and µ = γ −1 as in the proof of Theorem 1. Naively round each edge in G − H to get a forest and a (1 + ε)α out-orientation. Apply Lemma 27 to get a pseudoforest decomposition of G − H. Colour G − H with the strategy for colouring pseudoforests discussed above, and colour H with the strategy from Lemma 45. Finally, take the product colouring.
A Missing details of Section 2
Theorem (Theorem 15,implicit in [28]). Given implicit (|L|, q(n)) access to an orientation with max v s(v) ≤ ∆ + , there exists an algorithm that can insert and delete edges from the orientation without creating any new 1-invalid edges. The algorithm has worst-case insertion time of O(γ · ∆ + (∆ + + log(n)(|L| + 1) + q(n)))) and a worst case deletion time of O(γ · ∆ + (log(n)(|L| + 1) + q(n)))).
We provide pseudo-code for the algorithm in Theorem 15: We store the following global data structure.
Edges : Balanced binary search tree sorted by edge id. Initially empty.
Each vertex u stores the following data structures: s(u), initialized to 0. OutN brs u : Balanced binary search tree containing out-neighbours of u sorted by vertex id. Initially empty InN brs u : Max heap containing in-neighbours indexed using s(v). Initially empty.
We update them using the operations in Algorithms 1, 2 and 3. Theorem 15 now follows from the following two Lemmas:
Lemma 46 (Implicit in [28]). We can insert a copy of an edge into G γ in time O(∆ + · (∆ + + |L| log n + q(n))) Proof. For each vertex on the chain, we have to process the neighbourhood changes in time O(|L| · log n + q(n)), look at all out-neighbours in time O(∆ + ) and access and update an out-going edge in time O(log n). By Remark 12, the chain has at most ∆ + vertices.
Lemma 47 (Implicit in [28]). We can delete a copy of an edge into G γ in time O(∆ + · (log n + |L| log n + q(n))) Proof. For each vertex on the chain, we have to process the neighbourhood changes in time O(|L| · log n + q(n)), extract the maximum from a max-heap, and access and update an in-going edge in time O(log n). By Remark 12, the chain has at most ∆ + vertices.
if v / ∈ OutN brs u then add v to OutN brs u u, v ← v, u B
Scheduling updates
In this Appendix, we paraphrase the algorithms and ideas of Sawlani & Wang. To be consistent, we use the terminology introduced in Section 2. In particular, we consider out-orientations instead of the symmetric problem of orienting the edges in order to provide guarantees on the number of in-edges for each vertex. We also make similar modifications to the algorithm in Lemma 3.6 in [37], as we did to the algorithm of Kopelowitz et al. [28] in Section 2, and show that this algorithm can be used to maintain implicit approximate outorientations in dense graphs with polylogarithmic in n update time by using the techniques of Section 3.
A key insight in [37] is to allow η to scale with an estimate of the maximum density. Combined with the following guarantee for locally η-stable orientations and Remark 12, this reduces the length of maximal tight chains to O(γ), which means that the update times become independent of α(G).
Theorem 48 ([37] Theorem 3.1 and Corollary 3.2).
If O is a locally η-stable and α -bounded fractional out-degree orientation of a graph G, then for ρ est ≤ α = max v s(v) ≤ 2ρ est it holds that:
1 − 4 η log n/ρ est α ≤ ρ(G) ≤ α(G)
However, in order to use this guarantee, one needs an estimate of the maximum density. Sawlani and Wang overcome this by maintaining O(log n) copies of G -each with a different estimate of ρ. Then there will always be a copy with the guarantees of Theorem 49, provided of course that the algorithm is fully updated.
Furthermore, since η now scales with ρ, we cannot afford to look at all out-neighbours, as we do during insertions in Theorem 15. In order to overcome this, Sawlani and Wang lazily update different parts of the out neighbourhood each time a vertex' load is changed. Since a vertex's load now can be increased multiple times, before a tight edge becomes invalid, one is always able to identify an edge as tight, before it becomes invalid. We have the following Theorem:
Theorem 49 (Implicit in [37]). Given implicit (|L|, q(n)) access to an orientation with max v s(v) ≤ ρ max where ρ max is an upper bound on the maximum density, there exists an algorithm that can insert and delete edges from the orientation without creating any new ε 2 ·ρmax 16 log n -invalid edges. The algorithm has worst-case insertion and deletion time O(ε −6 log 3 (n)(1 + log(n)|L| + q(n))) Before we give a proof, we note that by using Theorem 49 for updates and Theorem 17 to schedule the updates in the proof of Theorem 26, we obtain: Theorem 50. Suppose γ = 128 log n/ε 2 and that 1 > δ > γ −1 + µ > 2γ −1 > 0, ε > 0. Then, there exists a dynamic algorithm that implicitly maintains (γ, (1 + ε)α)-orientation of a dynamic graph G with arboricity α as well as a (δ, µ)-refinement H of G wrt. this orientation such that H is a forest. Insertion and deletion take amortised O(ε −6 log 4 n) time.
In each copy, we can round H and G − H as in the proof of Theorem 1 in order to obtain an implicit out-orientation with update times independent of α.
Proof of Theorem 49
For the algorithm proving Theorem 49, we store the following global data structure.
Edges : Balanced binary search tree sorted by edge id. Initially empty.
Each vertex u stores the following data structures:
s(u), initialized to 0.
OutN brs u : Balanced binary search tree containing out-neighbours of u sorted by vertex id. Initially empty InN brs u : Max heap containing in-neighbours indexed using s v . Initially empty.
We update them using the operations in algorithms 4, 5 and 6. Assuming we know ρ max , n and ε, we set η = 2 ρmax γ and γ = 128 log (n)/ε 2 .
Correctness Correctness follows from arguments symmetric to those given in Lemma 3.7 in [37]. We have to argue that the orientation at all times is locally η-stable. In order to do so, we observe that to break the local stability, we have to, at some point, increment the load of a vertex with a critically tight in-neighbour or decrement the load of a vertex with a critically tight out-neighbour. We always check the edges to neighbours returned from the list, and the remaining neighbours are updated sufficiently often (see Lemma 3.7 in [37]), and so this never happens.
Analysis We can insert a copy of an edge in time O(γ(γ + |L| log n)). Indeed, by Remark 12 a maximal chain has length no more than 2ρ max /η = O(γ). At each vertex in the chain we spend O(γ) time checking for tight out-neighbours and O(|L| log n) time retrieving and checking the lists. Similarly for delete. Hence, we arrive at Theorem 49.
The scheduling algorithm Here we describe the ideas from [37] needed for the algorithm in Theorem 17:
We will maintain O(log n) copies of G γ some of which are only partially updated. In each copy, we maintain a different estimate ρ est of the maximum density. Based on this estimate we set ρ max = 2ρ est . We set γ = 128 log (n)/ε 2 , and in the i'th copy, we set ρ est = 2 i−2 γ. In copy i, we initialize the data structures from Theorem 49 and set η i = 2 ρ max i γ . Furthermore, for each copy G γ i , we initialize an empty, sorted list of edges pending i using two BST (sorted by each endpoint of the edge). Finally, we maintain a counter active, indicating that G γ active is the currently active copy. Whenever an edge is inserted or deleted in G, we update as follows.
Insertion: To insert a copy of an edge uv, we insert it into all copies i for which ρ max i is greater than ρ max active . These insertions all take poly(log n, ε −1 ) time by Theorem 49. If the copy with i = active + 1 satisfiesρ active+1 = max v s active+1 (v) ≥ ρ max active , we make G γ active+1 the active copy and increment active. Otherwise, we also insert uv in the active copy.
For the remaining copies of G γ , we cannot necessarily afford to insert the edge. We need a bound on the length of the maximal tight chains to achieve the update guarantees of Theorem 49. By Remark 12 this is upper bounded by 2· maxw s(w) η for a chain beginning or ending at w. Therefore, we only insert uv in copy i if at least one of u or v has load s strictly smaller than ρ max i . We can always afford to do this, since this only increases the load of a vertex with load at most s . Otherwise, we add the edge to pending i .
Deletion:
To delete a copy of an edge uv, we delete it in all copies i for which ρ max is greater thanρ = max v∈G γ active s(v). If the active copy satisfies (1 − ε)ρ < ρ est active , we make G active−1 the active copy. These deletions all take poly(log n, ε −1 ) time by Theorem 49.
For the remaining copies of G γ , we do as follows: If uv ∈ pending i , we just remove it. Otherwise, delete a copy of uv in G γ i , and locate the unique vertex w whose loads was decremented. Now if w is incident to an edge in pending i , we insert one such edge into the copy and remove the corresponding edge from pending i . We can continue doing this until the load of w again becomes too high.
Analysis:
We repeat the arguments of Sawlani and Wang [37]. Observe first that ρ active est < ρ ≤ 2ρ active est . Indeed, after each insertion, we check whether this is the case and update accordingly. Similarly for delete.
Secondly, pending active is always empty. Indeed, if uv is put in pending i at some point during the algorithm, then at this point in time s i (v), s i (u) = ρ max i . Hence, G γ i can only become active if the load of either u or v is decreased, meaning that all edges incident to this vertex in pending i has been inserted. In particular, uv is no longer in pending i . Since we only insert an edge from pending i once, a copy of an edge is only inserted O(log n)-times in total. Sawlani and Wang state that we only have to remove one incident edge from pending i per decrement, which makes the algorithm a worst-case algorithm (see [37] Theorem 1.1). Now, we are ready to show Theorem 17. We restate it for convenience:
Theorem (Theorem 17, implicit in [37] as Theorem 1.1). There exists a fully dynamic algorithm for scheduling updates that at all times maintains a pointer to a fully-updated copy with estimate ρ est where (1 − ε)ρ est /2 ≤ α(G) < 4ρ est . Furthermore, the updates are scheduled such that a copy G with estimate ρ satisfies α(G ) ≤ 4ρ . The algorithm has amortised update times O(log 4 (n)/ε 6 ).
Proof. We use the algorithm from Theorem 1.1 in [37] paraphrased above. This algorithm maintains O(log n) different (partial) copies of G γ . For each copy, we maintain a (partial) copy of G. Whenever this algorithm has removed all γ copies of an edge from pending i , we insert the edge into the corresponding copy of G.
Sawlani & Wang show that the active copy is always fully updated and thatρ = max v s(v) satisfies thatρ/2 ≤ ρ active est <ρ. Theorem 48 (Corollary 3.2 in [37]) then implies that ρ(G) satisfies (1 − ε)ρ ≤ ρ ≤ρ. Since ρ(G) ≤ α(G) ≤ 2ρ(G) when n ≥ 2, we get the bounds for the active copy.
For a non-active copy G with maximum density estimate ρ , we need to show that α(G ) ≤ 4ρ . The algorithm only inserts an edge uv in G γ if one of s(u) and s(v) is strictly smaller than 2ρ . Inserting this edge increases s(w) by one for exactly one vertex w at the end of a tight chain. Since Sawlani & Wang reorient a tight chain incident to the vertex out of u and v with the lowest load, this never increases the load of any vertex above 2ρ . In particular, since ρ(G ) ≤ max w∈G γ s(w) ≤ 2ρ by Theorem 48 (Corollary 3.2 in [37]), and since α(G ) ≤ 2ρ(G ) for n ≥ 2, we get the desired bound on the arboricity of the copy.
C
Missing proofs from Section 3 Lemma (Lemma 24, implicit in [2]). Let F be a dynamic forest in which every edge e = wz is assigned a pair of variables X w e , X z e ∈ [0, 1] s.t. X w e + X z e = 1. Then there exists a data structure supporting the following operations, all in O(log |F |)-time:
link(u, v, X u uv , X v uv ): Add the edge uv to F and set X u uv , X v uv = 1 − X u uv as indicated. Proof. [2] shows how to support the first 3 operations. The remaining operations also follow from [2], but we give a proof for completeness: Each path-cluster C maintains the variables min_weight(C), max_weight(C), extra(C) and head(C) ∈ ∂C. When a leaf-cluster C with cluster path π(C) = uv is created, we
D Missing proofs of Section 4
Lemma (Lemma 27, Implicit in [25]). Given black box access to an algorithm maintaining an α -bounded out-degree with update time T (n), there exist an algorithm maintaining an α pseudo-forest decomposition with update time O(T (n)).
Proof. We describe the algorithm below for completeness and stress that it is almost identical to one presented in [25]. The algorithm runs the black box algorithm maintaining an αbounded out-degree orientation. Then, with constant overhead, it maintains n pseudoforests P 0 , P 1 , . . . , P n−1 , but only the first α pseudoforests are non-empty. We will conceptually maintain an array A v of bits of length n for each v, with A v (i) indicating the out-degree of v in F i . The algorithm will preserve the following invariants for every v ∈ V (G): The algorithm updates in the following way: Suppose an edge e is inserted/deleted from the graph. Then e is inserted into the algorithm maintaining the out-orientation. Then three things might happen: A) the orientation algorithm might reorient the orientation of an existing edge, B) a new oriented edge might be inserted into the graph and finally C) an oriented edge might be deleted.
Consider the case 1) firstly. Suppose u → v is reoriented, so that it now has direction v → u. Then we update as follows:
1. delete u → v from the forest where e is currently residing. Set A u (i) = 0 and update s u = i accordingly. 2. If s u ≥ l u , all invariants are still maintained for u, and all we have to do is update l u = l u − 1. If not, then we will move the out-edge of u, u → w, residing in F lu to F su Then we update s u = l u and l u = l u − 1, and now all of u's invariants are restored again.
3.
Insert v → u into F lv . Then update A(l v + 1) = 1, l v = l v + 1 and s v = s v + 1, again restoring all of the invariants.
In case B), where we are inserting an edge, we perform only step 3, and in case C), where an edge is deleted, we perform steps 1 and 2.
Analysis of the algorithm The update procedure clearly maintains the claimed invariants.
The following Lemma establishes correctness.
Claim 53 (Implicit in [25]). After each update P 0 , P 1 , . . . , P d−1 form a pseudoarboricity decomposition of the graph G.
Proof. Indeed by Invariant 51, E(G) ⊂ α −1 j=0 E(P j ) so the pseudoforests cover all of the edges of G. Furthermore, every v ∈ P i has out-degree at most 1, and so any cycle in P i must be directed. Since by Invariant 51, |E(P C )| ≤ |P C | for any connected component P C ⊂ P i , we must have that if P C is cyclic, then equality holds. Deleting an edge e from a cycle in such a cyclic and connected component ensures that |E(P C − e)| = |P C | − 1 i.e. P C − e is a tree. Thus P C is a pseudotree.
Definition 10 ([37] Def. 3.5). A maximal η-tight chain from v is a path of η-tight edges v 0 v 1 , . . . v k−1 v k , such that v 0 = v and v k has no η-tight out-edges. A maximal η-tight chain to v is a path of η-tight edges v 0 v 1 , . . . v k−1 v k , such that v k = v and v 0 has no η-tight in-edges.
Figure 1
1Inverting a cycle. We can reorient copies of edges forming a directed cycle without changing the load of any vertex.
Observation 25 .
25We can access and change the fractional load of e ∈ H in time O(log n). We can do the same for e ∈ G − H in O(1) time, since these loads are not stored in top trees.
Proof. Apply Theorem 15 for insertion/deletion. Note that |L| = 0 and q(n) = 0. The time spent repairing H after each insertion/deletion is in O(γ∆ + log n) by Remark 16 and Observations 21 and 25, since we can process an edge from both Q and S in O(log n) time. Finally, Observation 20 and the Invariants 22 and 23 show correctness of the algorithm. Now tuning the parameters of Theorem 26, rounding edges in G − H and 2-orienting H yields Theorem 1. We restate Theorem 1 for convenience:
Appendix D. Using the ideas of Blumenstock & Fischer [12], we can represent a pseudoforest P by a pair (F, M ) s.t. F is a forest and M is matching, by adding exactly one edge from each cycle in P to M . Similarly, we can represent a partition of E(G) into pseudoforest (P 1 , . . . , P k ) by a pair (F, M ) s.t. F = ∪F i and M = ∪M i and (F i , M i ) represents P i for all i.In order to ensure the guarantees of Lemma 27, we need to maintain the invariant that every vertex has out-degree at most one in every pseudoforest. If this is the case, we say that the partition is faithful to the underlying orientation.Blumenstock & Fischer [12] perform operations on G[M ] in order to turn it into a forest. They call G[M ] the surplus graph. Some of the operations, they perform, are described in the following Lemma: Lemma 28 (Implicit in [12]). Let (F, M ) be a faithful representation of a pseudoforest partition of a simple graph G equipped with an α -bounded out-degree orientation. If uv ∈ M i and vw ∈ M j with i = j, then there exists an α -bounded out-degree orientation with respect to which the partition gained by swapping P i ← P i ∪ {vw} − {uv} and P j ← P j ∪ {uv} − {vw} yields a faithful partition, and uv ∈ M j resp. vw ∈ M i iff. uv resp. vw are on the uni-cycle in their new pseudoforests.Furthermore, if wx ∈ M i for some x, then vw is not on a uni-cycle in P i .
Lemma 29 ([12]). Suppose J is a colourful component of the surplus graph G[M ] of a graph G. Then for all v ∈ J there exists an index i s.t. N Fi (v) ∩ J = ∅ and J ∩ M i = ∅.
Figure 2
2Figure 2 Moving edges between pseudoforests. Before performing the swap, we reorient a cycle so that the swapped edges both are out-edges of their common endpoint
2 .
2Naively round the orientation of each edge in G − H. 3. Split the rounded out-degree orientation on G − H into pseudoforests. 4. Whenever an edge enters or moves between pseudoforests, we push it to a queue R. We process each edge in e ∈ R as follows: Put e into a pseudoforest. If e completes a cycle in a pseudoforest add it to G[M ]. When e enters G[M ], we determine if it sits in a colourful component. If it doesn't, we apply Lemma 28 until all components in G[M ] are colourful.
Lemma 32 .
32We can supply each vertex with a query returning a list L of neighbours which might have their status changed in time O(α log n). Furthermore, |L| = O(α).
Now we clean each light edge in time O(log n) by Observation 35. Finally, we can search for a singly valid edge in time O(log n) using Observation 35. If no edge is returned, we conclude that the cycle is doubly valid. Finally, we can show Lemma 30 (we recall it below): Lemma (Lemma 30). After inserting an edge into G[M ], we can restore acyclicity and colourfulness in O(α 3 log 2 (n)) time. Proof. We begin by showing the following two observations: Observation 38. We can locate a chain of Lemma 28 switches in time O(α), and a Lemma 29 switch in time O(α 2 log n).
empty. We check an i in time O(α log n) by checking whether v and u ∈ V c are neighbours in P i by using the top trees. Since we check at most O(α) different i's, the statement follows. Observation 39. We can perform a Lemma 28 switch in O(log 2 n)-time.
2 .
2The algorithm maintaining the pseudoforests spends O(log 6 (n) · α 3 max · ε −8 (i + d)) time. 3. The algorithm maintaining the surplus graph spends O(log 6 (n) · α 6 max · ε −8 (i + d)) time.Proof. 1. By Lemma 32, |L| = O(α) and q(n) = O(α log n). Furthermore, we spend time O(α log 2 n) updating heavy-edges per vertex accessed per insertion/deletion. By Lemma 33, we perform at most O( γ∆ + max η (∆ + · i + d)) updates and so setting γ = Θ(log n/ε −2 ), η = 1 and applying Theorem 26 yields the result. 2. We spend overhead O(log 2 n) for each such operation, and by Lemma 34 we perform at most O( γ(∆ + max ) 3 η (i + d)) of these operations. Thus the claimed follows by inserting parameters. 3. An edge can only end up in a surplus graph, if it inserted into a new pseudoforest, so the statement follows from Lemmas 34 and 30.
44 .
44Given an initially empty and dynamic graph G undergoing an arboricity α preserving sequence σ of insertions and deletions (say i insertions and c deletions), the algorithm with parameters d/2 ≥ δ > α will have an amortized insertion cost of O(δ · d+1 d+1−2δ ), and an amortized deletion cost of O(d + log δ α (n) d+1 d+1−2δ ). Proof. For any δ > α the reorientation scheme from [13] gives δ-orientations O j such that O j is a δ-bounded out-degree orientation of G j for all j = 1, . . . |σ| such that the total number of reorientations is at most log δ α (n)) times the number of deletions. Each reorientation can be done in O(1) time per reorientation, since we reorient the entire out-degree list. For deletions, we incur an extra cost of O(d) to locate the deleted edge in the out-list.
Figure 3
3Different choices of head(A), head(B) and head(C). The head of C is indicated by the filled vertex. The arrows point towards the heads of clusters A and B. We update the information differently depending on how head(B) relates to head(C). cut(u, v): Remove the edge uv from F . connected(u, v): Return true if u, v are in the same tree, and false otherwise. add_weight(u, v, x): For all edges wz on the path u . . . wz . . . v between u and v in F , set X w wz = X w wz + x and X z wz = X z wz − x. min_weight(u, v): Return the minimum X w wz s.t. wz is on the path u . . . wz . . . v in F . max_weight(u, v)): Return the maximum X w wz s.t. wz is on the path u . . . wz . . . v in F .
initialize min_weight(C) = max_weight(C) = X u uv , extra(C) = 0, head(C) = v. When a path-cluster C = A ∪ B with two cluster children is created, we set head(C) to head(A) if possible. Otherwise, we set head(C) = ∂C ∩ ∂B. If head(C) = head(A) and head(B) ∈ ∂A or if head(C) = head(B) and head(B) / ∈ ∂(A) (Case 1), we update: min_weight(C) = min{min_weight(A), min_weight(B)} max_weight(C) = min{max_weight(A), max_weight(B)} Otherwise (Case 2), we update: min_weight(C) = min{min_weight(A), 1 − max_weight(B)} max_weight(C) = min{max_weight(A), 1 − min_weight(B)} Finally, we set extra(C) = 0. When a path-cluster C = A ∪ B with two cluster children is split, we set: extra(A) = extra(A) + extra(C) min_weight(A) = min_weight(A) + extra(C) max_weight(A) = max_weight(A) + extra(C) If head(C) = head(A) and head(B) ∈ ∂A or if head(C) = head(B) and head(B) / ∈ ∂A (Case 1), we set; extra(B) = extra(B) + extra(C) min_weight(B) = min_weight(B) + extra(C) max_weight(B) = max_weight(B) + extra(C) Otherwise (Case 2), we set: extra(B) = extra(B) − extra(C) min_weight(B) = min_weight(B) − extra(C) max_weight(B) = max_weight(B) − extra(C) Note that only Case 1 and Case 2 can occur due to the way we create the joined clusters. To add_weight(u, v, x), we set C = expose(u, v) and add x to min_weight(C), max_weight(C) and extra(C) if head(C) = v. Otherwise, we subtract x from min_weight(C), max_weight(C) and extra(C). To do max_weight(u, v), we again set C = expose(u, v), and return max_weight(C) if head(C) = v and 1 − min_weight(C) otherwise. min_weight(u, v) is similar.
Invariant 51 .
51For all i ∈ {0, . . . , d + (v) − 1}, P i , will contain an out-edge of v.Invariant 52. For all i ∈ {0, . . . , n − 1}, we have that A v (i) = 1 iff. N + (v) ∩ P i = ∅.Finally, we maintain an index s v for each v indicating the smallest i s.t. A v (i) = 0, and an index l v indicating the largest i s.t. A v (i) = 1. Note that when the above invariants are maintained, we necessarily have s v > l v .
1 .
1Operations for querying and changing fractional loads of edges in O(log n) time. 2. A query that returns a list containing a superset of all neighbours of a vertex that have changed status as in-or out-neighbour, since the last time the query was called on this vertex. The list should have length ≤ |L| and the query should run in O(q(n)) time.
Proof. See [12] Lemma 2. To see that we can modify the orientation to accommodate the swaps, note that we can always reverse the direction of at most two cycles, without changing the out-degree of any vertex, such that both of the edges swapped are out-edges of v. Now swapping the two out-edges ensures that the partition stays faithful to the orientation.Following Blumenstock & Fischer we note that if e 1 , . . . , e k is a path in a surplus graph
G[M ] such that e 1 and e k belong to the same matching M i , then we can use the moves from
Lemma 28 to restore colourfulness (see [12] Lemma 3). The key is that we can move the
other edge in M i towards e 1 , and then after O(k) switches, we are sure to end up in the
furthermore part of Lemma 28. If a surplus graph contains no such paths, Blumenstock &
Fischer say it is a colourful surplus graph. They show the following Lemma:
Proof. See Appendix B.3 Dynamic Maintenance of RefinementsIn this section, we work towards the second goal: an orientation where many edges assign most of their load to one endpoint. To realise this, we introduce refinements of fractional orientations and show how to dynamically maintain them. The basic idea is to maintain an orientation such that the fractional load of each vertex is small and such that the edges,
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|
[] |
[
"Gradient solitons on statistical manifolds",
"Gradient solitons on statistical manifolds"
] |
[
"Adara M Blaga ",
"Bang-Yen Chen "
] |
[] |
[] |
We provide necessary and sufficient conditions for some particular couples (g, ∇) of pseudo-Riemannian metrics and affine connections to be statistical structures if we have gradient almost Einstein, almost Ricci, almost Yamabe solitons, or a more general type of solitons on the manifold. In particular cases, we establish a formula for the volume of the manifold and give a lower and an upper bound for the norm of the Ricci curvature tensor field. 2020 Mathematics Subject Classification. 35C08, 35Q51, 53B05.
|
10.1016/j.geomphys.2021.104195
|
[
"https://arxiv.org/pdf/2005.13470v5.pdf"
] | 218,900,882 |
2005.13470
|
b45011667525649753626d8475007f540ab2c98f
|
Gradient solitons on statistical manifolds
9 Mar 2021
Adara M Blaga
Bang-Yen Chen
Gradient solitons on statistical manifolds
9 Mar 2021
We provide necessary and sufficient conditions for some particular couples (g, ∇) of pseudo-Riemannian metrics and affine connections to be statistical structures if we have gradient almost Einstein, almost Ricci, almost Yamabe solitons, or a more general type of solitons on the manifold. In particular cases, we establish a formula for the volume of the manifold and give a lower and an upper bound for the norm of the Ricci curvature tensor field. 2020 Mathematics Subject Classification. 35C08, 35Q51, 53B05.
Introduction
Information geometry was firstly studied by Amari [1] treating the properties of the geometrical structures, such as Riemannian metrics and affine connections, that naturally arise on a space of probability distributions. In this way, statistical structures provide a link between information geometry and affine geometry. Such a Riemannian metric is the Fisher information metric defined on the manifold of probability distributions [13]. According to Chentson's theorem, up to rescaling, the Fisher information metric on statistical models is the only Riemannian metric that is invariant under sufficient statistics [2].
In differential geometry, a statistical structure on a smooth manifold M consists of a pseudo-Riemannian metric g and a torsion-free affine connection ∇ such that ∇g is a Codazzi tensor field. For such a pair (g, ∇), the dual connection ∇ * of ∇ with respect to g is uniquely defined by X(g(Y, Z)) = g(∇ X Y, Z) + g(Y, ∇ * X Z), for X, Y , Z ∈ X(M). The pair (∇, ∇ * ) is said to be a dualistic structure which plays an important role in statistics. A particular statistical structure when the curvature tensor field of ∇ vanishes is the Hessian structure [12]. In this case, also the curvature tensor field of the dual connection ∇ * vanishes and the manifold is called dually flat.
We extend the notion of statistical structure in the following ways: i) if h is a symmetric (0, 2)-tensor field and ∇ is a torsion-free affine connection, we call (h, ∇) a nearly statistical structure on M if
(∇ X h)(Y, Z) = (∇ Y h)(X, Z),
for any X, Y , Z ∈ X(M);
ii) if h is a (0, 2)-tensor field and ∇ is an affine connection, we call (h, ∇) a quasi-
statistical structure on M if d ∇ h = 0 [8], where d ∇ is defined by (d ∇ h)(X, Y, Z) := (∇ X h)(Y, Z) − (∇ Y h)(X, Z) + h(T ∇ (X, Y ), Z),
for any X, Y , Z ∈ X(M).
Regarded as stationary solutions of a geometric flow, the notion of solitons can be generalized in some very natural ways. One question is that if some of these generalizations are coming from certain particular flows, what geometrical and topological properties of the manifold shall reveal?
In the context of statistical geometry, we will investigate in this article the consequences of the existence of different kind of solitons; such as Ricci, Einstein, Yamabe or a more general type defined by an affine connection, with a special view towards curvature. By means of an arbitrary 1-form, we consider a statistical structure, which is equiaffine if the 1-form is exact, and study some geometrical and topological properties of the solitons defined by it. Precisely, we provide a lower and an upper bound for the Ricci curvature tensor's norm, and in the compact case, using the classical Bochner formula, we determine a relation for the volume of the manifold. It is known that the Ricci tensor is the component of the curvature tensor of spacetime, related to the matter content of the universe via Einstein's field equation, its lower bounds allow us to deduce global geometrical properties of the manifold.
Solitons and statistical structures
Consider a pseudo-Riemannian manifold (M, g) and let ∇ g be the Levi-Civita connection of g. We denote by Q the Ricci operator defined by g(QX, Y ) := Ric(X, Y ), where Ric is the Ricci tensor of (M, g). If the Ricci tensor is non-degenerate, then it is a pseudo-Riemannian metric and we denote by ∇ Ric the Levi-Civita connection of Ric.
From Koszul's formula, we deduce:
2g(∇ Ric X Y − ∇ g X Y, QZ) = g((∇ g Y Q)X, Z) + g((∇ g X Q)Y, Z) − g((∇ g Z Q)X, Y ), for any X, Y , Z ∈ X(M).
Remark the following facts [5]:
i) (Ric, ∇ g ) is a statistical structure if and only if (∇ g X Q)Y = (∇ g Y Q)X, for any X, Y ∈ X(M). Moreover, if the Ricci operator Q is a Codazzi tensor, then QT = 1 2 ∇ g Q, where T := ∇ Ric − ∇ g ; ii) if (g, ∇ Ric ) is a statistical structure, then g(X, T (Y, Z)) = g(Y, T (X, Z)), for any X, Y , Z ∈ X(M), where T := ∇ Ric − ∇ g ; iii) if (g, ∇)
is a quasi-statistical structure, then (Ric, ∇) is a quasi-statistical structure if and only if g((∇ X Q)Z, Y ) = g((∇ Y Q)Z, X), for any X, Y , Z ∈ X(M).
Let f be a smooth function on M. If the Hessian of f , denoted by Hess(f ), is nondegenerate and of constant signature, then Hess(f ) is a pseudo-Riemannian metric. A nice geometrical interpretation of Hessian metrics has recently appeared in mirror symmetry [6], their practical importance being also shown in [3].
If we denote by ∇ Hess(f ) the Levi-Civita connection of Hess(f ), then it follows from Koszul's formula that
2g(∇ Hess(f ) X Y − ∇ g X Y, ∇ g Z ∇f ) = g((∇ g ) 2 X,Y ∇f, Z) + g((∇ g ) 2 Y,Z ∇f, X) − g((∇ g ) 2 Z,X ∇f, Y ), for any X, Y , Z ∈ X(M), where ∇f denotes the gradient of f and (∇ g ) 2 X,Y := ∇ g X ∇ g Y − ∇ g ∇ g
X Y , and we prove: Theorem 2.1. (Hess(f ), ∇ g ) is a statistical structure on M if and only if the radial curvature vanishes, i.e. R ∇ g (X, Y )∇f = 0, for any X, Y ∈ X(M).
Proof. d ∇ g Hess(f ) = 0 is equivalent to (∇ g X Hess(f ))(Y, Z) = (∇ g Y Hess(f ))(X, Z), for any X, Y , Z ∈ X(M), which gives X(g(∇ g Y ∇f, Z)) − g(∇ g ∇ g X Y ∇f, Z) − g(Y, ∇ g ∇ g X Z ∇f ) = = Y (g(∇ g X ∇f, Z)) − g(∇ g ∇ g Y X ∇f, Z) − g(X, ∇ g ∇ g Y Z ∇f ) ⇐⇒ X(g(∇ g Y ∇f, Z)) − g(∇ g ∇ g X Y ∇f, Z) − g(∇ g X Z, ∇ g Y ∇f ) = = Y (g(∇ g X ∇f, Z)) − g(∇ g ∇ g Y X ∇f, Z) − g(∇ g Y Z, ∇ g X ∇f ) ⇐⇒ g(∇ g X ∇ g Y ∇f, Z) − g(∇ g ∇ g X Y ∇f, Z) = g(∇ g Y ∇ g X ∇f, Z) − g(∇ g ∇ g Y X ∇f, Z) ⇐⇒ g((∇ g ) 2 X,Y ∇f − (∇ g ) 2 Y,X ∇f, Z) = 0 which is equivalent to R ∇ g (X, Y )∇f = 0.
Note that the radial curvature was introduced by Klingenberg [7] in the context of algebraic topology, to prove a homotopy sphere theorem.
Theorem 2.2. If (g, ∇) is a statistical structure, then (Hess(f ), ∇) is a statistical structure if and only if R ∇ (X, Y )∇f = 0, for any X, Y ∈ X(M). Proof. d ∇ Hess(f ) = 0 is equivalent to (∇ X Hess(f ))(Y, Z) − (∇ Y Hess(f ))(X, Z) + Hess(f )(T ∇ (X, Y ), Z) = 0,
for any X, Y , Z ∈ X(M), which gives
X(g(∇ Y ∇f, Z)) − g(∇ ∇ X Y ∇f, Z) − g(Y, ∇ ∇ X Z ∇f ) − Y (g(∇ X ∇f, Z))+ +g(∇ ∇ Y X ∇f, Z) + g(X, ∇ ∇ Y Z ∇f ) + g(T ∇ (X, Y ), ∇ Z ∇f ) = 0.
Since ∇ is torsion-free, we can express its curvature in terms of the second order derivatives, namely
R ∇ (X, Y ) = ∇ 2 X,Y − ∇ 2 Y,X , where ∇ 2 X,Y := ∇ X ∇ Y − ∇ ∇ X Y
and the above equation becomes:
(∇ X g)(∇ Y ∇f, Z) − (∇ Y g)(∇ X ∇f, Z) + g(R ∇ (X, Y )∇f, Z) = 0 which, from d ∇ g = 0, is equivalent to R ∇ (X, Y )∇f = 0.
Next, we shall relate statistical structures to gradient solitons (see also [5]). Recall that, for a pseudo-Riemannian metric g and two smooth functions f and λ, the triple (g, f, λ) is called: i) a gradient almost Ricci soliton if
Hess(f ) + Ric = λg,
where Hess(f ) is the Hessian of f and Ric is the Ricci tensor of g;
ii) a gradient almost Einstein soliton if
Hess(f ) + Ric = λ + scal 2 g,
where scal is the scalar curvature of (M, g);
iii) a gradient almost Yamabe soliton if
Hess(f ) = (λ − scal)g.
In particular, if λ is a constant, then we drop the adjective "almost" from the previous definitions and call the solitons the gradient Ricci, gradient Einstein and gradient Yamabe, respectively.
Taking now the covariant derivative in the soliton equations, we obtain respectively:
(∇ g X Hess(f ))(Y, Z) + (∇ g X Ric)(Y, Z) = X(λ)g(Y, Z), (∇ g X Hess(f ))(Y, Z) + (∇ g X Ric)(Y, Z) = X λ + scal 2 g(Y, Z), (∇ g X Hess(f ))(Y, Z) = X(λ − scal)g(Y, Z)
, for any X, Y , Z ∈ X(M) and we can state: iii) If (g, f, λ) defines a gradient Yamabe soliton and M is of constant scalar curvature, then (Hess(f ), ∇ g ) is a statistical structure on M.
Proposition 2.3. i) If (g, f, λ)
We deduce the followings:
Proposition 2.4.
If (g, f, λ) defines a gradient almost Einstein soliton on the smooth manifold M with non-degenerate Ricci tensor, then (Ric, ∇ g ) is a statistical structure on M if and only if
(1) g(R ∇ g (X, Y )∇f, Z) = X λ + scal 2 g(Y, Z) − Y λ + scal 2 g(X, Z),
for any X, Y , Z ∈ X(M).
Proposition 2.5. If (g, f, λ) defines a gradient almost Ricci soliton on the smooth manifold M with non-degenerate Ricci tensor, then (Ric, ∇ g ) is a statistical structure on M if and only if
(2) g(R ∇ g (X, Y )∇f, Z) = X(λ)g(Y, Z) − Y (λ)g(X, Z),
for any X, Y , Z ∈ X(M).
Proposition 2.6. If (g, ∇) is a statistical structure on the smooth manifold M and (g, f, λ) defines a gradient almost Einstein soliton on M with non-degenerate Ricci tensor, then (Ric, ∇) is a statistical structure on M if and only if
(3) g(∇ 2 X,Z ∇f, Y ) − g(∇ 2 Y,Z ∇f, X) = X λ + scal 2 g(Y, Z) − Y λ + scal 2 g(X, Z), for any X, Y , Z ∈ X(M), where ∇ 2 X,Y Z := ∇ X ∇ Y Z − ∇ ∇ X Y Z.
From the soliton equations, we deduce respectively the followings:
∇ g ξ + Q = λI, ∇ g ξ + Q = λ + scal 2 I, ∇ g ξ = (λ − scal)I,
where Q stands for the Ricci operator and ξ := ∇f . These lead to a more general notion of soliton, precisely we consider an almost (∇, J, ξ, λ)-soliton on M as a data (∇, J, ξ, λ) which satisfy the equation:
(4) ∇ξ + J = λI,
where ∇ is an affine connection, J is a (1, 1)-tensor field, ξ is a vector field and λ is a smooth function on M.
A straightforward computation gives:
Lemma 2.7. If (∇, J, ξ, λ) defines an almost (∇, J, ξ)-soliton on the pseudo-Riemannian manifold (M, g), then the 2-form Ω := g(J·, ·) is symmetric if and only if the endomorphism ∇ξ is self-adjoint with respect to g, i.e.
g(∇ X ξ, Y ) = g(X, ∇ Y ξ),
for any X, Y ∈ X(M).
if (∇ X g)(JY, Z) − (∇ Y g)(JX, Z) = g((∇ Y J)X − (∇ X J)Y, Z).
In particular, Ω is a Codazzi tensor field, i.e. (∇ g X Ω)(Y, Z) = (∇ g Y Ω)(X, Z), if and only if J is a Codazzi tensor field, i.e. (∇ g X J)Y = (∇ g Y J)X.
Remark 2.9. If Ω is a Codazzi tensor field and J is a Killing tensor field (i.e. (∇ g X J)X = 0, for any X ∈ X(M)), then J is ∇ g -parallel.
As particular cases, we deduce from [5] the followings:
g(R ∇ (X, Y )ξ, Z) = (∇ X g)(JY, Z) − (∇ Y g)(JX, Z) + g(X(λ)Y − Y (λ)X, Z),
for any X, Y , Z ∈ X(M).
Corollary 2.11. If (∇ g , J, ξ, λ) defines an almost (∇ g , J, ξ)-soliton on a pseudo-Riemannian manifold (M, g) and Ω := g(J·, ·) is symmetric, then (Ω, ∇ g ) is a nearly statistical structure on M if and only if R ∇ g (·, ·)ξ = dλ ⊗ I − I ⊗ dλ.
Assume now ξ = ∇f and from the soliton equation (4) we get:
Hess ∇ (f ) + Ω = λg, hence, ∇ g X Hess ∇ (f ) + ∇ g X Ω = X(λ)g, for any X ∈ X(M).
Corollary 2.12. Let (∇ g , J, ξ, λ) define an almost (∇ g , J, ξ)-soliton on the pseudo-Riemannian manifold (M, g) with λ a constant and ξ = ∇f . Then the following statements are equivalent: i) (Ω, ∇ g ) is a nearly statistical structure on M;
ii) R ∇ g (X, Y )∇f = 0, for any X, Y ∈ X(M);
iii) (Hess ∇ (f ), ∇ g ) is a nearly statistical structure on M.
Connections defined by 1-forms and solitons
Inspired by the property of projectively equivalence of connections, given an arbitrary 1-form η on the pseudo-Riemannian manifold (M, g), we consider the affine connection:
∇ η := ∇ g + η ⊗ I + I ⊗ η + g ⊗ ξ,
where ∇ g is the Levi-Civita connection of g and ξ is the g-dual vector field of η (i.e. η = i ξ g). We get:
T ∇ η = 0, d ∇ η g = 0.
Hence, we have Proposition 3.1. For any 1-form η on the pseudo-Riemannian manifold (M, g), the pair (g, ∇ η ) is a statistical structure on M and ∇ −η is the dual connection of ∇ η .
In particular, ∇ η ξ ξ = ∇ g ξ ξ + 3|ξ| 2 g ξ, therefore, ξ is a geodesic vector field for ∇ η if and only if ∇ g ξ ξ = −3|ξ| 2 g ξ. Moreover: i) ξ is ∇ η -parallel if and only if (∇ g , J := 2η ⊗ ξ, ξ, λ := |ξ| 2 g ) is a soliton; ii) η is ∇ η -parallel if and only if (∇ g , J := −2η ⊗ ξ, ξ, λ := −|ξ| 2 g ) is a soliton. The curvature of the connection ∇ η is given by:
(R ∇ η − R ∇ g )(X, Y )Z = [g(Y, ∇ g X ξ) − g(X, ∇ g Y ξ)]Z+ + [η(Y )η(Z) + g(Y, Z)|ξ| 2 g − g(Z, ∇ g Y ξ)]X + g(Y, Z)∇ g X ξ − g(X, Z)∇ g Y ξ− − [η(X)η(Z) + g(X, Z)|ξ| 2 g − g(Z, ∇ g X ξ)]Y + [η(X)g(Y, Z) − η(Y )g(X, Z)]ξ,
and we deduce that, if ξ is a g-null and ∇ g -parallel vector field (i.e. η(ξ) = 0 and ∇ g ξ = 0), condition appearing in Walker manifolds [14], then
(R ∇ η − R ∇ g )(X, Y )Z ∈ ker η,
for any X, Y , Z ∈ X(M).
Example 3.2. Let (M, ϕ, ξ, η, g) be a Kenmotsu manifold and let ∇ η be the affine connection defined by the structure, ∇ η := ∇ g +η ⊗I +I ⊗η +g ⊗ξ. Since ∇ g ξ = I −η ⊗ξ, we get ∇ η ξ = 2I + η ⊗ ξ, and (∇ η , J := −η ⊗ ξ, λ = 2) is a soliton on M.
The divergence operator with respect to ∇ df is given by: div (g,∇ df ) = div (g,∇ g ) +(n + 2)df.
In the compact case, it follows from the divergence theorem that M div (g,∇ df ) (X)dµ g = (n + 2) M g(grad g (f ), X)dµ g , for any X ∈ X(M). Moreover, if | grad g (f )| g is constant, then:
vol(M) = 1 (n + 2)| grad g (f )| 2 g M div (g,∇ df ) (grad g (f ))dµ g .
Denote by ∆ g := div (g,∇ g ) • grad g and ∆ η := div (g,∇ η ) • grad g the corresponding Laplace operators. Then:
∆ df (f ) = ∆ g (f ) + (n + 2)g(grad g (f ), grad g (f )), for any smooth functionf on M.
Note that iff is harmonic for ∆ df , thenf is also harmonic for ∆ g if and only if the vector fields grad g (f ) and grad g (f ) are g-orthogonal.
In particular, we have
∆ df (f ) = ∆ g (f ) + (n + 2)| grad g (f )| 2 g ,
hence:
i) if f is harmonic for ∆ g , then it is a subharmonic function for ∆ df (i.e. ∆ df (f ) ≥ 0) provided | grad g (f )| 2 g ≥ 0; ii) if f is harmonic for ∆ df and (M, g) is a compact Riemannian manifold, then f is locally constant.
Also, for any smooth function f ∈ C ∞ (M), if we denote by Hess g (f ) and Hess η (f ) the Hessian tensor fields with respect to ∇ g and ∇ η , then we have:
Hess η (f )(X, Y ) := g(∇ η X grad g (f ), Y ) = = Hess g (f )(X, Y ) + η(grad g (f ))g(X, Y ) + η(X)df (Y ) + η(Y )df (X)
and by tracing this relation we find ∆ η (f ) = ∆ g (f ) + (n + 2)η(grad g (f )), where n = dim(M). In particular, if ξ = grad g (f ), then η = df and we obtain:
(5)
Hess df (f ) = Hess g (f ) + | grad g (f )| 2 g g + 2df ⊗ df.
Recall that a pseudo-Riemannian manifold (M, g) with a pair of dual connections (∇, ∇ * ) is called conjugate Ricci-symmetric [9] if Ric ∇ = Ric ∇ * .
If we denote by Ric g and Ric η the Ricci tensors for ∇ g and ∇ η , then from the curvature relation we obtain that the Ricci curvature of ∇ η satisfies
Ric η (Y, Z) = Ric g (Y, Z) + g(Y, Z){n|ξ| 2 g + div (g,∇ g ) (ξ)}+ + (n − 2)η(Y )η(Z) + g(Y, ∇ g Z ξ) − (n + 1)g(Z, ∇ g Y ξ),
where n = dim(M) and we can state:
Proposition 3.3. (M, g, ∇ η , ∇ −η ) is a conjugate Ricci-symmetric manifold.
An affine connection on M is called equiaffine [10] if it admits a parallel volume form on M. It is known that [10] the necessary and sufficient condition for a torsion-free affine connection to be equiaffine is that the Ricci tensor is symmetric.
Since ∇ η is torsion-free, we get:
Proposition 3.4. ∇ η is an equiaffine connection on M if and only if the endomorphism ∇ g ξ is self-adjoint with respect to g, i.e.
g(∇ g X ξ, Y ) = g(X, ∇ g Y ξ),
for any X, Y ∈ X(M).
In particular, if ξ = grad g (f ), then η = df and we obtain: (6) Ric df = Ric g +{n| grad g (f )| 2 g + ∆ g (f )}g + (n − 2)df ⊗ df − n Hess g (f ).
Hence we have
Corollary 3.5. ∇ df is an equiaffine connection on M.
Taking the trace in the previous relation, we get:
scal (g,∇ df ) = scal (g,∇ g ) +(n − 1)(n + 2)| grad g (f )| 2 g , which implies scal (g,∇ df ) ≥ scal (g,∇ g ) provided | grad g (f )| 2 g ≥ 0.
If we denote by Q g and Q df the Ricci operators defined by g(Q g X, Y ) := Ric g (X, Y ) and g(Q df X, Y ) := Ric df (X, Y ), X, Y ∈ X(M), then:
Q df = Q g + {n| grad g (f )| 2 g + ∆ g (f )}I + (n − 2)df ⊗ grad g (f ) − n∇ g grad g (f )
and by direct computations, we obtain: Proposition 3.6. Let (M, g) be an n-dimensional pseudo-Riemannian manifold, ξ = grad g (f ) and η = df . Then: i) (∇ η , Q g , ξ, λ) is a gradient almost soliton if and only if (∇ g , Q g + 2η ⊗ ξ, λ − |ξ| 2 g ) is a gradient almost soliton;
ii) (∇ g , Q η , ξ, λ) is a gradient almost soliton if and only if (∇ g , 1
1−n {Q g + (n − 2)η ⊗ ξ}, 1 1−n {λ − n|ξ| 2 g − ∆ g (f )})
is a gradient almost soliton;
iii) (∇ η , Q η , ξ, λ) is a gradient almost soliton if and only if (∇ g , 1 1−n (Q g +nη⊗ξ), 1 1−n {λ− (n + 1)|ξ| 2 g − ∆ g (f )}) is a gradient almost soliton.
Now, we shall relate the previously considered types of solitons to almost Ricci and almost η-Ricci solitons [4].
Proposition 3.7. Let (M, g) be an n-dimensional pseudo-Riemannian manifold, ξ = grad g (f ) and η = df . Then we have: i) (∇ η , Q g , ξ, λ) is a gradient almost soliton if and only if (g, ξ, λ−|ξ| 2 g , −2) is a gradient almost η-Ricci soliton.
ii) If ∇ g ξ = η ⊗ ξ, then:
(ii.1) (∇ g , Q η , ξ, λ) is a gradient almost soliton of M if and only if (g, ξ, λ − (n + 1)|ξ| 2 g , 2) is a gradient almost η-Ricci soliton; in this case, scal (g,∇ η ) = nλ−|ξ| 2 g ; (ii.2) (∇ η , Q η , ξ, λ) is a gradient almost soliton of M if and only if (g, ξ, λ−(n+2)|ξ| 2 g ) is a gradient almost Ricci soliton; in this case, scal (g,∇ g ) = nλ − (n + 1) 2 |ξ| 2 g .
Proof. i) ∇ η ξ + Q g = λI is equivalent to ∇ g ξ + Q g = (λ − |ξ| 2 g )I − 2η ⊗ ξ. ii) From hypotheses we get ∆ g (f ) = |ξ| 2 g . For (ii.1), by taking the trace in − Hess g (f ) + Ric g = {λ − (n + 1)|ξ| 2 g }g, we obtain scal (g,∇ g ) = nλ − (n 2 + n − 1)|ξ| 2 g , therefore, scal (g,∇ η ) = nλ − |ξ| 2 g . By a similar proof we get the conclusion (ii.2).
We shall further derive a formula for the volume of M whenever it admits an almost soliton.
By computing the scalar product with respect to g, we find:
Ric df , df ⊗ df g = Ric g (grad g (f ), grad g (f )) − n Hess g (f )(grad g (f ), grad g (f ))+ +| grad g (f )| 2 g ∆ g (f ) + 2(n − 1)| grad g (f )| 4 g and using the classical Bochner formula, we obtain:
Ric df , df ⊗ df g = 1 2 ∆ g (| grad g (f )| 2 g ) − | Hess g (f )| 2 g − grad g (f )(∆ g (f ))− −n Hess g (f )(grad g (f ), grad g (f )) + | grad g (f )| 2 g ∆ g (f ) + 2(n − 1)| grad g (f )| 4 g and we can state: grad g (f )(∆ g (f ))dµ g + M Ric df , df ⊗ df g dµ g .
Also, the Bochner formula can be written in terms of Ric df and Hess df (f ) as follows: 1 2 ∆ g (| grad g (f )| 2 g ) = | Hess df (f )| 2 g + Ric df (grad g (f ), grad g (f )) + grad g (f )(∆ g (f ))− −3(n + 2)| grad g (f )| 4 g − 3| grad g (f )| 2 g ∆ g (f ) + n − 4 2 grad g (f )(| grad g (f )| 2 g ) = = | Hess df (f )| 2 g + Ric df (grad g (f ), grad g (f )) + grad g (f )(∆ df (f ))− −3| grad g (f )| 2 g ∆ df (f ) − n + 8 2 grad g (f )(| grad g (f )| 2 g ) and we can state: grad g (f )(∆ df (f ))dµ g + M Ric df (grad g (f ), grad g (f ))dµ g .
defines a gradient Ricci soliton, then (Hess(f ), ∇ g ) is a statistical structure on M if and only if (Ric, ∇ g ) is a statistical structure on M. ii) If (g, f, λ) defines a gradient Einstein soliton and M is of constant scalar curvature, then (Hess(f ), ∇ g ) is a statistical structure on M if and only if (Ric, ∇ g ) is a statistical structure on M.
]
The 2-form Ω := g(J·, ·) satisfies (∇ X Ω)(Y, Z) = (∇ Y Ω)(X, Z) if and only
Proposition 2 . 10 .
210Let (∇, J, ξ, λ) define an almost (∇, J, ξ)-soliton on a pseudo-Riemannian manifold (M, g). If Ω := g(J·, ·) is symmetric and ∇ is torsion-free, then (Ω, ∇) is a nearly statistical structure on M if and only if
Proposition 3. 8 .
8Let (M, g) be a compact n-dimensional pseudo-Riemannian manifold, f a smooth function on M such that | grad g (f )| g is constant. Then:
Proposition 3 . 9 .
39Let (M, g) be a compact n-dimensional pseudo-Riemannian manifold, f a smooth function on M such that | grad g (f )| g is constant. Then:
Remark 3.10. Notice that, under the same hypotheses, we have:grad g (f )(∆ g (f ))dµ g + M Ric g (grad g (f ), grad g (f ))dµ g = 0.Proposition 3.11. Let (M, g) be a compact n-dimensional pseudo-Riemannian manifold, f a smooth function on M such that | grad g (f )| g is constant. If (∇ g , Q df , grad g (f ), λ) is a gradient almost soliton, then:Hess g (f ) + Ric df = λg.Then:Also, replacing Ric df from (6) in(7)and taking the trace with respect to g, we get:Now, by applying grad g (f ) to the previous relation and using Proposition 3.8, we obtain the conclusion.Remark 3.12. Under the same hypotheses, we have:provided | grad g (f )| 2 g ≥ 0 and f is a harmonic function for ∆ g ;ii) if λ is a constant and λ = 2(n − 1)| grad g (f )| 2 g , thenMoreover, if f is a harmonic function for ∆ g , thenHence, in the Riemannian case, λ < 2(n − 1)| grad g (f )| 2 g , thereforeProposition 3.13. Let (M, g) be an n-dimensional pseudo-Riemannian manifold and f a smooth function on M. If (∇ g , Q df , grad g (f ), λ) is a gradient almost soliton, then:Proof. This proposition follows by computing | Hess g (f )| 2 g fromas well as the fact that the conditions to exist a solution (in λ) give precisely the double inequality from the conclusion.Proposition 3.14. Let (M, g) be a compact n-dimensional pseudo-Riemannian manifold, f a smooth function on M such that | grad g (f )| g is constant. If (∇ df , Q g , grad g (f ), λ) is a gradient soliton, then:Proof. Indeed, ∇ df grad g (f ) + Q g = λI implies(8)Hess df (f ) + Ric g = λg.Then:Also, replacing Hess df (f ) from (5) in(8)and taking the trace with respect to g, we get:∆ g (f ) + scal (g,∇ g ) +(n + 2)| grad g (f )| 2 g = nλ.Now, by applying grad g (f ) to the previous relation and using Proposition 3.8, we obtain the conclusion.Proposition 3.15. Let (M, g) be an n-dimensional pseudo-Riemannian manifold and f a smooth function on M. If (∇ df , Q g , grad g (f ), λ) is a gradient almost soliton, then:n (scal (g,∇ g ) ) 2 + 4 n | grad g (f )| 2 g scal (g,∇ g ) − −4 Ric g (grad g (f ), grad g (f )).Proof. This proposition follows by computing | Hess g (f )| 2 g fromHessas well as the fact that the conditions to exist a solution (in λ) give precisely the double inequality from the conclusion.Remark 3.16. If | grad g (f )| g = 1, then the double equality from the previous Proposition impliesIn this case, from the soliton equation(8), we get Ric g (grad g (f ), grad g (f )) = λ − 3 and ∆ g (f ) = nλ − (n + 2) − scal (g,∇ g ) .Replacing the last two expressions in the first one and asking for the equation of order two in λ to have solution, we get n 2 (scal (g,∇ g ) ) 2 ≤ 0, hence the manifold is of zero scalar curvature. Moreover, if f is a harmonic function for ∆ g , then λ = n+2 n and Ric g (grad g (f ), grad g (f )) = − 2(n−1) n < 0. Note that Petersen[11]called f a distance function if it is a solution of the Hamilton-Jacobi equation | grad g (f )| 2 g = 1, which he has used in his book.
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[
"A FOURIER TRANSFORM FOR THE QUANTUM TODA LATTICE",
"A FOURIER TRANSFORM FOR THE QUANTUM TODA LATTICE"
] |
[
"Gus Lonergan "
] |
[] |
[] |
We introduce an algebraic Fourier transform for the quantum Toda lattice.
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10.1007/s00029-018-0419-x
|
[
"https://arxiv.org/pdf/1706.05344v1.pdf"
] | 119,698,008 |
1706.05344
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f1bba834e4beccc78d67c93b641117229ff8474f
|
A FOURIER TRANSFORM FOR THE QUANTUM TODA LATTICE
16 Jun 2017
Gus Lonergan
A FOURIER TRANSFORM FOR THE QUANTUM TODA LATTICE
16 Jun 2017
We introduce an algebraic Fourier transform for the quantum Toda lattice.
1. Introduction 1.1. The Toda lattice. Following [1], let G be a complex reductive algebraic group and denote by T oda 1 (G) the partially compactified quantum Toda lattice of G. By definition, this is the two-sided 1 quantum Hamiltonian reduction of D(G) with respect to a generic character ψ of a maximal unipotent subgroup N . In a formula, we have T oda 1 (G) = (D(G)/((l − ψ)(n) + (r − ψ)(n)) D(G)) N ×N where l, resp. r are the embeddings of g in D(G) as left-, resp. right-invariant vector fields. Here g = Lie(G), n = Lie(N ) (and this pattern will continue). It is naturally a Hopf algebroid over O(g * //G). A different choice of N, ψ gives a canonically isomorphic Hopf algebroid (justifying the definite article). Kostant's classic result gives a canonical isomorphism between T oda 1 (G) and the quantum Hamiltonian reduction of D(G) with respect to the trivial character of G itself, acting adjointly.
The order filtration on D(G) induces 2 a filtration on T oda 1 (G). The base O(g * //G), viewed as a subalgebra of T oda 1 (G), is canonically isomorphic to its associated graded and one thus obtains an associated graded Hopf algebroid (over the same base). This is the partially compactified Toda lattice and will be denoted 1 The epithet 'two-sided' refers to the use of both the left-and the right-regular actions of N on G.
2 This is true for the presentation as D(G)// triv G. For the presentation as N ψ \\ D(G)// ψ N one needs to adjust this filtration by the ρ ∨ -weight. 1 T oda 0 (G). In fact, it is a commutative Hopf algebra over O(g * //G), and its corresponding group scheme is canonically identified with the flat abelian group scheme z G ψ+n ⊥ /N over (ψ+n ⊥ )/N ∼ = g * //G 3 . Here n ⊥ denotes the orthogonal complement to n in g * . In this classical setting, Kostant's result identifies this group scheme with the adjoint quotient z G g * //G. Let us now fix a maximal torus T of the normalizer B of N . We obtain an opposite subgroup N − to N , and a splitting g * = n * × t * × n * − . We can therefore form the subscheme ψ + t * ⊂ ψ + n ⊥ . This fits into a commutative diagram
ψ + t * ∼ − → t * ↓ ↓ g * //G ∼ − → t * //W
and consequently we have an isomorphism z G ψ+t * ∼ = t * × g * //G (z G ψ+n ⊥ /N ). In other words, there exists an action of W on z G ψ+t * , compatible with its usual action on the base t * , and the geometric quotient is the spectrum of the partially compactified Toda lattice. Noting that ψ + t * is contained in (b * ) reg , we see that z G ψ+t * = z B ψ+t * . The resulting projection z G ψ+t * → (ψ + t * ) × T is an isomorphism over ψ + (t * ) reg . This map is W -equivariant, where W acts 'diagonally' on (ψ + t * ) × T ∼ = t * ×T . All this goes to show that we have a map Spec(T oda 0 (G)) → (T * T )//W which is an isomorphism generically over the base t * //W . This classical picture is quantizes in the natural manner, and in particular we have a map D(T ) W → T oda 1 (G) (1.1) of Hopf algebroids over t * //W , which is generically an isomorphism.
1.2. The Fourier transform. One is interested in understanding modules for T oda 1 (G). Restricting along 1.1, such a thing becomes a module for D(T ) W . Since D(T ) W is Morita equivalent to D(T )# C W , this is the same thing as a W -equivariant D-module on T . There is a well-known equivalence D(T ) − mod ∼ = QCoh X • (T ) (t * ) (1.2) and likewise
D(T ) W − mod ∼ = QCoh W af f (t * ) (1.3)
where X • (T ) denotes the character lattice of T and W af f = X • (T )#W is the partially-extended affine Weyl group 4 . This equivalence may be regarded as an algebraic incarnation of the Fourier transform, but it is completely trivial when one writes out the definition of the categories to be related.
A natural question arises: is there a similar kind of 'algebraic Fourier transform' for T oda 1 (G)? That is the subject of this paper. In fact, we have 3 In fact, z G ψ+n ⊥ is itself a flat abelian group scheme over ψ + n ⊥ , on which base N acts freely, justifying the notation '/N ' rather than '//N '. It is customary to trivialize the N -torsor ψ + n ⊥ → g * //G. The resulting section κ is called 'the' Kostant slice, and we obtain Spec(T oda 0 (G)) ∼ = z G κ . 4 As opposed to the fully extended affine Weyl group, which is usually defined to be the group obtained in this manner starting from the universal cover of G. The affine Weyl group, W af f , is the group obtained in this manner starting from the adjoint quotient of G. Theorem 1.1. There exists an equivalence of categories T oda 1 (G) − mod ∼ = QCoh W af f (t * ) " I " (1.4) which is compatible with 1.3 in the natural way.
Here the category QCoh W af f (t * ) " I " denotes the full subcategory of QCoh W af f (t * ) whose objects are all those with trivial derived isotropy for W af f . The precise meaning of this will be spelled out in the main body of the paper (see Proposition 4.4). An alternative formulation is as follows:
Theorem 1.2.
There exists an equivalence of categories (1.5) which is compatible with 1.3 in the natural way.
T oda 1 (G) − mod ∼ = QCoh W af f (t * ) f pd
Here the category QCoh W af f (t * ) f pd denotes the full subcategory of QCoh W af f (t * ) whose objects are all those whose Γ-equivariant structure descends to t * //Γ, for every finite parabolic subgroup Γ ⊂ W af f ⊂ W af f .
The Affine Grassmannian
For background material on the affine Grassmannian, see [5] [6]. For algebraic groups, see [2]. The material of paragraphs 2.1-2.5, 2.7, 2.9 is borrowed from these.
2.1. Let G ∨ be the complex algebraic group which is Langlands dual to G and has maximal torus T ∨ . Let Gr denote the affine Grassmannian for the G ∨ . This is a certain projective ind-scheme whose C-valued points are
G ∨ (K)/G ∨ (O), where K = C((t)), O = C[[t]]
. The translation action of G ∨ (O) has finite dimensional orbits, whose closures (the so-called 'spherical Schubert varieties') give the indscheme structure. The cocharacter lattice X • (T ∨ ) embeds in Gr as its T ∨ -fixed point subset, and each G ∨ (O)-orbits contains a unique W -orbit in X • (T ∨ ). The group G m also acts, by the 'loop rotation' local automorphisms of K, and this action fixes the T ∨ -fixed points and preserves the spherical Schubert varieties.
2.2. We will be interested the (complex) cohomology and homology of Gr 5 . By definition, the cohomology H • (Gr) of Gr is the cofiltered system of the cohomologies of the spherical Schubert varieties, and the homology H • (Gr) of Gr is the filtered system of the homologies of the spherical Schubert varieties. The transition morphisms in H • (Gr) are all surjective, and those in H • (Gr) are all injective. Forgetting the ind-scheme structure on Gr, one obtains the topological space Gr top 6 . We then have
H • (Gr top ) = colim − −− → H • (Gr). However, lim ← − H • (Gr)
is some completion of H • (Gr top ) (and so bigger than it). Since each spherical Schubert variety is compact, its homology is equal to its Borel-Moore homology, i.e. its cohomology with coefficients in its dualizing complex. Gluing together these dualizing complexes by the !-pullbacks, one may think of H • (Gr) rather literally as its cohomology with coefficients in its dualizing complex. 5 Strictly speaking, we are working here with the ind-analytic space Gr an . 6 Again, strictly speaking this is the filtered colimit in the category of topological spaces of the filtered system of analytic spaces Gr an .
2.3.
We fix a Borel subgroup B ∨ of G ∨ containing T ∨ and write I ∨ = G ∨ (O) × G ∨ B ∨ for the corresponding Iwahori subgroup of G ∨ (O). The orbits of I ∨ on Gr form a complex cell decomposition, compatible with the stratification by spherical Schubert varieties. It follows that Gr is equivariantly formal (for, say the T ∨ -action). We then define H • T ∨ (Gr) as the cofiltered system of the T ∨ -equivariant cohomologies of the spherical Schubert varieties, and H T ∨ • (Gr) as the filtered system of the T ∨ -equivariant cohomologies of the spherical Schubert varieties with coefficients in their respective (canonically equivariant) dualizing complexes 7 . By equivariant formality (and finite-dimensionality of the spherical Schubert varieties), H T ∨ • (Gr) and
H • T ∨ (Gr) are dual over H • T ∨ (pt) = O(t * )
. Similarly we define the G ∨ -equivariant homology and cohomology; it is a standard consequence of equivariant formality that
H • T ∨ (Gr) = π * H • G ∨ (Gr) and H T ∨ • (Gr) = π * H G ∨ • (Gr) where π : t * → t * //W = Spec(H • G ∨ (pt))
is the natural projection. Inversely, W acts naturally on H • T ∨ (Gr) (resp. H T ∨ • (Gr)), and one obtains π * H • G ∨ (Gr) (resp. π * H G ∨ • (Gr)) from them by taking invariants. All of the above goes through when one introduces the additional 'loop rotation' factor of equivariance (and the corresponding additional A 1 factor in the base). As in the non-equivariant case, the transition maps in cohomology are all surjective, while those in homology are all injective, and the analogous remarks comparing their (co-) limits to the equivariant (co-) cohomology of Gr top hold.
2.4. Since G ∨ is the maximal reductive quotient of G ∨ (O), we may replace G ∨ by G ∨ (O) as a factor of equivariance. That is to say, we have natural isomorphisms
H • G ∨ (Gr) ∼ = H • G ∨ (O) (Gr) and H • G ∨ ×Gm (Gr) ∼ = H • G ∨ (O)⋊Gm (Gr)
where the right-hand sides are defined just as before (recall that the spherical Schubert varieties are G ∨ (O)-equivariant by definition). We have the analogous isomorphisms in homology. In the same fashion, one may replace T ∨ by I ∨ .
2.5. Putting the ind-scheme structure of Gr together with the pro-scheme structure of G ∨ (O), we may regard G ∨ (K) as a pro-ind-scheme. With some rearranging, it may be viewed as an ind-pro-scheme, admitting the G ∨ (O) × G ∨ (O)-orbits as a final family of sub-pro-schemes. We have
H • G ∨ (O) (Gr) = H • G ∨ (O)×G ∨ (O) (G(K)) H • G ∨ (O)⋊Gm (Gr) = H • (G ∨ (O)×G ∨ (O)
)⋊Gm (G(K)) and so on. This symmetric presentation makes it clear that in both cases we have an extra action of O(t * //W ). It happens that the two O(t * //W )-module structures differ in the presence of loop-rotation equivariance, and coincide without it. 7 As before, we like to think of this as the equivariant cohomology of Gr with coefficients in its dualizing complex.
A remark on equivariant homology. The description of H
G ∨ (O)⋊Gm • (Gr) as the O(t * //W ×A 1 )-linear dual to H • G ∨ (O)⋊Gm (Gr) = H • (G ∨ (O)×G ∨ (O)
)⋊Gm (G(K)) (using the 'left-hand' action of O(t * //W )) appears asymmetrical. However, essentially because G ∨ (O) is (pro-) smooth, the * -pullback to G ∨ (K) of the dualizing complex on Gr descends to the dualizing complex on G ∨ (O)\G ∨ (K). One can then give a symmetric description of H
G ∨ (O)⋊Gm • (Gr) as the (G ∨ (O) × G ∨ (O)) ⋊ G m - equivariant cohomology of G ∨ (K)
with coefficients in this pullback 8 . In particular this gives a canonical isomorphism between the two 9 . This line of reasoning has an interesting application for finite flag varieties. Namely, let P and Q be two parabolic subgroups of G. Since P, Q are smooth and P \G, G/Q are compact, the P × Q-equivariant cohomology of the dualizing complex on G is at once the H • P (pt) = O(t //W P )-linear dual of H • P (G/Q), shifted by 2 dim Q, and the H • Q (pt) = O(t //W Q )-linear dual of H • Q (P \G), shifted by 2 dim P . The consequence in algebraic geometry is that, whenever P ⊂ Q, we have the canonical isomorphism of functors
O(t * //W × A 1 )-linear duals to H • G ∨ (O)⋊Gm (Gr)(π P Q ) ! (2 dim P ) ∼ = (π P Q ) * (2 dim Q) : QCoh Gm (t //W Q ) → QCoh Gm (t //W P ) exactly mirroring the topological statement that (π P Q ) ! [2 dim P ] ∼ = (π P Q ) * [2 dim Q] : D b (BQ) → D b (BP )
. Here π P Q stands simultaneously (and very abusively) for the natural maps t //W P → t //W Q and BP → BQ. (Gr) are Hopf algebroids over t * //W ×A 1 . They are supported on the diagonal t * //W × t * //W × A 1 ⊂ (t * //W × A 1 ) 2 , and thus we may think of them as an A 1 -family of Hopf algebroids over t * //W . The same is true for their (co-) limits discussed in 2.2. Of course some care has to be taken to say what it means for a pro-or ind-object to be a Hopf algebroid. For our purposes, it happens that the pro-object H • G ∨ (O)⋊Gm (Gr) is a pro-algebra (but not a pro-coalgebra), while the ind-object H G ∨ (O)⋊Gm • (Gr) is an ind-coalgebra (but not an ind-algebra). We do not know whether these restrictions are critical to have a good general theory, but they are very natural in the topological setting.
The pro-object Hopf algebroid H • G ∨ (O)⋊Gm (Gr) is commutative. Equivalently, its spectrum is a groupoid ind-scheme. In a sense which will be made precise, the (A 1 -families of) representations of this groupoid over t * //W are the same as comodules for H • G ∨ (O)⋊Gm (Gr), which in turn (by pro-finite flatness) are the same as modules for the dual algebra H 2.8. Following [3] (and, originally, [5]), the following is a consequence of the geometric Satake equivalence (whose details we need not recall here):
Theorem 2.1. There are natural isomorphisms H G ∨ ⋊Gm • (Gr top ) ∼ = T oda (G)
and
H • G ∨ ⋊Gm (Gr top ) ∼ = O(π 1 (G ∨ ) × N t * //W (t * //W ) 2 ) of Hopf algebroids over t * //W × A 1 .
Here T oda (G) is the Rees construction of T oda 1 (G) with respect to the filtration of 1.1, and is the parameter of A 1 . Also N X Y denotes the 'deformation to the normal cone' of X in Y whenever X ⊂ Y is a closed subscheme. It is a flat G mequivariant A 1 -scheme whose restriction to
A 1 −{0} is Y × (A 1 −{0})
and whose 0-fiber is the normal cone of X in Y . The groupoid scheme structure comes from the trivial groupoid structure on (t * //W ) 2 (with t * //W its subgroup).
Recall that H G ∨ ⋊Gm
• (Gr top ) = colim − −− → H G ∨ ⋊Gm •
(Gr); thus the data provided by H G ∨ ⋊Gm • (Gr) is nothing more than a filtration of T oda (G), the so called spherical Schubert filtration. A module for T oda (G) is therefore precisely a trivially filtered module for H G ∨ ⋊Gm • (Gr). These are equivalent to trivially cofiltered comodules for H • G ∨ ⋊Gm (Gr), and one obtains:
Corollary 2.2.
There is an equivalence of categories
T oda (G) − mod ∼ = Rep t * //W ×A 1 Spec(H • G ∨ ⋊Gm (Gr)) whence T oda 1 (G) − mod ∼ = Rep t * //W Spec(H • G ∨ ⋊Gm (Gr)) =1 . Here Spec(H • G ∨ ⋊Gm (Gr)
) is a certain sub-groupoid-ind-scheme of N t * //W (t * //W ) 2 , which the remainder of this paper is aimed at understanding.
2.9. The affine flag variety. The affine flag variety, denoted F l, is a certain G ∨ /B ∨ -bundle over Gr whose C-points are G ∨ (K)/I ∨ . We define the A 1 -families of Hopf algebroids over t * :
H • I ∨ ⋊Gm (F l) H I ∨ ⋊Gm • (F l)
in essentially the same way as for Gr 10 . They are respectively pro-finite flat and ind-finite flat objects with respect to both O(t * × A 1 )-module structures, over which they are also dual. Again H • I ∨ ⋊Gm (F l) is commutative, and so its spectrum is an A 1 -family of groupoid ind-schemes over t * . It is related to Spec(H • G ∨ ⋊Gm (Gr)) in the following way:
Spec(H • I ∨ ⋊Gm (F l)) ∼ = t * × t * //W Spec(H • G ∨ ⋊Gm (Gr)) × t * //W t * , i.e.
it is obtained from Spec(H • G ∨ ⋊Gm (Gr)) by applying the natural pullback functor for ind-schemes along π : t * → t * //W .
We will see that the representations of these two groupoids are equivalent. Let T denote the pullback groupoid evaluated at = 1. It is the study of this groupoid which will eventually yield Theorem 1.1.
2.10. Localization. The localization theorem of [4] allows us to describe T precisely. Indeed they show that the restriction map
H • I ∨ ⋊Gm (F l) ∼ = H • T ∨ ⋊Gm (F l) → H • T ∨ ⋊Gm (F l T ∨ ×Gm )
is generically 11 an isomorphism (and so in particular injective, since the left hand side is torsion-free over t * × A 1 ). The same holds when we set = 1, by gradedness. The T ∨ × G m -fixed point set of F l is identified with W af f , and the spectrum of its equivariant cohomology is the ind-scheme
γ∈ W af f Γ γ .
Here Γ γ = {(x, y, ∈ t * × t * × A 1 )|x = γ (y)} and γ denotes the operator obtained from γ by dilating its translational part by . This is the same as the transformation groupoid
W af f #(t * × A 1 )
where W af f now is regarded as a group ind-scheme which acts on t * × A 1 by γ → γ . In particular setting = 1 we have
Spec(H • T ∨ ⋊Gm (F l T ∨ ×Gm )) =1 ∼ = W af f # t * . Representations over t * of this groupoid are by definition precisely W af f -equivariant quasicoherent sheaves on t * . Identifying H T ∨ ⋊Gm • (F l T ∨ ×Gm ) =1 with D(T )# C W
gives the algebraic Fourier transform of 1.3. Therefore it is natural to regard the category of representations of T as the 'Fourier transform description' of the category of modules for T oda 1 (G).
Since
H • T ∨ ⋊Gm (F l) =1 → H • T ∨ ⋊Gm (F l T ∨ ×Gm ) =1 is injective, and O(π 1 (G ∨ ) × t * × t * ) ∼ = H • T ∨ ⋊Gm (F l top ) =1 → H • T ∨ ⋊Gm (F l) =1
is surjective in every piece of the right-hand pro-object, we obtain:
Lemma 2.3. T is the image of the morphism W af f # t * (p•π1,i) − −−−− → π 1 (G ∨ ) × t * × t * .
Here p :
W af f → π 1 (G ∨ ) is the quotient map (with kernel W af f ) and i : W af f # t * → t * × t * is the closed embedding.
The idea now is that this image is in some reasonable sense the quotient of W af f # t * by the isotropy (i.e. maximal) subgroup of W af f # t * -as for instance can be seen on closed points -and thus one expects that representations of T are W af f -equivariant quasicoherent sheaves on t * with trivial isotropy in W af f . However, the fact that the isotropy subgroup is not flat over t * causes difficulties, and in fact this is why we must introduce the notion of 'derived isotropy'. We deal with these issues in the following section.
We make one final remark. The localization theorem of [4] also includes a de-
scription of the image of H • T ∨ ⋊Gm (F l) → H • T ∨ ⋊Gm (F l T ∨ ×Gm ).
We have of course already described this (via the calculation of H • T ∨ ⋊Gm (F l top )), but the result of [4] provides some crucial and not obvious additional information (see Section 5).
Groupoids and descent
3.1. This section contains a number of general lemmas on groupoid schemes which would suffice to prove Theorem 1.1 if T were a scheme rather than an ind-scheme. In fact, the arguments carry over more or less directly to the case of ind-schemes, but for readability we have chosen to leave the thorough treatment of ind-schemes to the following section.
Let G ⇒ X be an algebraic groupoid over the scheme X. We denote by s, t the two maps to X (heads and tails). Expressions G× X , × X G denote the Cartesian product using respectively s, t, and likewise expressions
O(G)⊗ O(X) , ⊗ O(X) O(G)
denote the tensor product using respectively s # , t # .
3.2.
A lemma on descent. We make the important assumption that t is flat. In that case, the category Rep X (G) of right O(G)-comodules is abelian. We have:
Lemma 3.1 (faithfully flat descent). Suppose f : Y → X is faithfully flat (or more generally that it induces a universally injective map O(X) → O(Y ) of O(X)- modules).
Then the functor
f * : Rep X (G) → Rep Y (Y × X G × X Y ) (3.1)
is an equivalence of (monoidal) categories.
Remark 3.2. The usual statement of faithfully flat descent is the special case where G is the trivial groupoid.
Proof. This is an instructive application of Barr-Beck. Consider the composition
Rep X (G) Res G X −−−→ QCoh(X) f * −→ QCoh(Y ). (3.2) Then • f * • Res G X admits a right adjoint, namely Coind G X • f * , where Coind G X = (−) ⊗ O(X) O(G) is the right adjoint to Res G X . • f * • Res G X reflects
isomorphisms, since f * does (f being faithfully flat), and Res G X does (being an exact faithful functor between abelian categories). • Rep X (G) has, and f * • Res G X preserves, equalizers of f * • Res G X -split equalizers. This is because f * has this property (by hypothesis) and Res G X has the stronger property that Rep X (G) has, and Res G X preserves, equalizers of Res G X -equalizers (being an exact functor between abelian categories). Consequently f * • Res G X induces an equivalence between Rep X (G) and the category of
f * • Res G X • Coind G X • f * -comodules in QCoh(Y ). The comonad in question is nothing more than (−) ⊗ O(X) O(G) ⊗ O(X) O(Y ) ∼ = (−) ⊗ O(Y ) O(Y × X G × X Y )
with the comonad structure given by the groupoid structure on Y × X G × X Y .
Adjacency and isotropy. Let C be the coequalizer (in affine schemes) of
G ⇒ X; that is, O(C) = {x ∈ O(X)|s # (x) = t # (x)} (a subalgebra of O(X)).
The adjacency groupoid T is defined by setting O(T ) to be the subalgebra of O(G) generated by the images of s # , t # . This is naturally an algebraic groupoid over X and we have the maps G → T → X × C X of algebraic groupoids over X. We also have the isotropy subgroup I := G × T X of G (here X → T is the identity section). Let us write M for the kernel of the surjective map O(G) → O(I); M is nothing more than the ideal generated by (s # − t # )(O(X)).
Heuristically one thinks of G as a space of arrows with heads and tails in X, which are composable in the natural way (and satisfy the groupoid axioms); the relation of being connected by an arrow is an equivalence relation on X. Then T is the adjacency groupoid of the equivalence relation, and I is the subgroup of G consisting of all arrows whose head and tail coincide. In some cases C is the space of equivalence classes: for instance in the case of a transformation groupoid G = X × Γ, C is the GIT quotient and if it is also a geometric quotient then this condition is satisfied. In that case, one might hope 12 that T → X × C X is an isomorphism. Naively one expects that formulas such as 'T = G/I' and 'Rep X (T ) = Rep X (G) I ' to hold. Here Rep X (G) I denotes the full subcategory of Rep X (G) consisting of objects with trivial I-action. However, this simply isn't true in general. Nonetheless we shall demonstrate some appropriate replacement, given the following hypothesis:
Hypothesis 3.3.
(1) We assume that G, T are both flat over X, with respect to both the heads and tails maps. (2) We assume that G → T is the coequalizer of G × T G = G × X I ⇒ G. In light of the first hypothesis, on the level of functions, this is the statement that the images of s # , t # in O(G) together generate the entire subalgebra
{x ∈ O(G)|∆(x) − x ⊗ 1 ∈ O(G) ⊗ O(X) M }.
3.4. Behaviour on flat objects. Consider the functor of restriction F :
Rep X (T ) → Rep X (G) I .
Lemma 3.4. F induces an equivalence between the full subcategories of objects which are flat as quasicoherent sheaves on X.
Proof. To avoid excessive notation, the symbol ⊗ will denote ⊗ O(X) unless given some other subscript. The main point is the essential surjectivity. Let V be an object in the target category. By definition, this is a flat
O(X)-module V together with an O(G)- comodule structure m : V → V ⊗ O(G) (3.3) such that the composition V m − → V ⊗ O(G) → V ⊗ O(I) (3.4)
12 For instance, this holds when X is the reflection representation of a finite complex reflection group Γ and G = Γ#X; see 5.3 to deduce a proof.
coincides with id ⊗ 1. Consider the diagram V ? − → V ⊗ O(G) p − → V ⊗ O(I) ↓ * ↓ m⊗id V ⊗ O(G) ⊗ O(G) q − → V ⊗ O(G) ⊗ O(I) (3.5)
where the horizontal maps labelled p, q are the obvious quotient maps. We perform the following diagram chase:
q • (id ⊗ ∆) • m = q • (m ⊗ id) • m = (m ⊗ id) • p • m = (m ⊗ id) • p • (id ⊗ 1) = q • (m ⊗ id) • (id ⊗ 1) = q • (id ⊗ id ⊗ 1) • m. Thus m lands inside the equalizer of V ⊗ O(G) ⇒ V ⊗ O(G) ⊗ O(I), the parallel morphisms being q • (id ⊗ ∆) and q • (id ⊗ id ⊗ 1). This equalizer is the kernel of V ⊗ O(G) q•(id⊗(∆−id⊗1)) −−−−−−−−−−−→ V ⊗ O(G) ⊗ O(I), (3.6) which since V is flat is precisely V ⊗ O(T ).
Thus we have the unique factorization
V → V ⊗ O(T ) → V ⊗ O(G) (3.7) of m. That this is an O(T )-comodule structure on V follows from the fact that V ⊗ O(T ) ⊗ O(T ) → V ⊗ O(G) ⊗ O(G) is injective. This is because O(T ) → O(G) is injective and every O(X)-module in sight is flat.
Next, F is of course faithful, being the identity on underlying O(X)-modules.
Finally, F is full on its flat O(X)-modules. Indeed if V , V ′ are any two such, and V → V ′ is a morphism of O(G)-comodules then consider the diagram V → V ⊗ O(T ) → V ⊗ O(G) ↓ ↓ ↓ V ′ → V ′ ⊗ O(T ) → V ′ ⊗ O(G).
(3.8)
The outer and rightmost squares are both commutative, and by flatness of V ′ the lower horizontal arrow in the rightmost square is injective. It follows that the leftmost square is commutative, as required.
A closing remark: the proof shows that F :
Hom(V, V ′ ) → Hom(F V, F V ′ )
is an isomorphism as long as V ′ is a flat O(X)-module (V may be arbitrary).
3.5. Fullness. In fact, F is full under some additional hypotheses which we will describe. First, we will require: Hypothesis 3.5. T , G are finite over X (with respect to both s and t).
It follows that for any objects
V, V ′ in Rep X (T ), the space Hom O(X) (V, V ′ )
is also an object of Rep X (T ), and Hom T (V, V ′ ) coincides with its maximal invariant submodule:
Hom T (V, V ′ ) = Hom O(X) (V, V ′ ) T = {f ∈ Hom O(X) (V, V ′ )|m(f ) = f ⊗ 1} (3.9) = m −1 (Hom O(X) (V, V ′ ) ⊗ 1). (3.10) (Likewise for G).
To see this, consider the following diagram:
V m − → V ⊗ O(T ) ↓ f ↓ f ⊗id V ′ m − → V ′ ⊗ O(T ) m⊗id − −−− → V ′ ⊗ O(T ) ⊗ O(T ) id⊗id⊗S − −−−−− → id⊗mult −−−−−→ V ′ ⊗ O(T ). (3.11) The high road is a map of O(X)-modules from V to V ′ ⊗ O(X) O(T ) with its O(X)- structure coming from V ′ (or equivalently t # )
; this is different from the one we have been using until now. By the finiteness hypothesis this is the same as an
element of Hom O(X) (V, V ′ ) ⊗ O(X) O(T ). One checks that the map so constructed from Hom O(X) (V, V ′ ) to Hom O(X) (V, V ′ ) ⊗ O(X) O(T ) is O(X)-linear (in the usual sense) and makes Hom O(X) (V, V ′ ) into a representation of T .
The condition that f is invariant is the condition that the high road of the diagram is equal to f ⊗1. This is equal to the low road. Since the tail of the diagram is an isomorphism (in fact, an involution!) it is equivalent to the condition that the square is commutative.
Consequently, F is full if and only if F 'reflects invariants': the natural map
V T → V G is an isomorphism (for all V ).
3.6. Reflection of invariants. We will now give some conditions which guarantee that F reflects invariants independently of any earlier hypothesis (of course finite flatness is required to deduce fullness from this). For instance, one such condition is:
Hypothesis 3.6. The equalizer O(C) → O(X) ⇒ O(G) is split, with a (t # ) O(X)- linear section β of t # .
Indeed, in that case one may consider the composition α
V m − → V ⊗ O(T ) id⊗f − −− → V ⊗ O(G) id⊗β − −− → V. (3.12)
Note that the condition that β be (t # ) O(X)-linear is necessary for this to be welldefined. Certainly α is the identity on V G , and so in particular on V T . We claim moreover that α is a section of the inclusion V T → V ; we have
m • α = m • (id ⊗ β) • (id ⊗ f ) • m (3.13) = (id ⊗ id ⊗ β) • (m ⊗ id) • (id ⊗ f ) • m (3.14) = (id ⊗ id ⊗ β) • (id ⊗ id ⊗ f ) • (m ⊗ id) • m (3.15) = (id ⊗ id ⊗ β) • (id ⊗ id ⊗ f ) • (id ⊗ ∆) • m. (3.16) Since O(T ) is generated by s # (O(X)) and t # (O(X)), we have m(V ) ⊂ V ⊗ s # (O(X)) ⊂ V ⊗ O(T ). By linearity of ∆, we have therefore (id ⊗ ∆) • m(V ) ⊂ V ⊗ 1 ⊗ s # (O(X)). Since β • s # (O(X)) ⊂ O(C), we get finally m • α(V ) ⊂ V ⊗ O(C) = V ⊗ 1 ⊂ V ⊗ O(T ). We conclude as follows: suppose v ∈ V G . Then v = α(v) ∈ V T , as required.
It may happen that O(X) ⇒ O(G) does not admit a split equalizer globally, but does locally. Thus we make:
(t # ) O(X)-linear section of t # x : O(X x ) → O(G (x,X) ) (denoted by β x ) such that s # x • β x • s # and t # x • β x • s # coincide in O(G (x,x) ) (equivalently, in O(T (x,x) )).
Remark 3.8. This is satisfied if, for each x, the groupoid G (x,x) over X x satisfies Hypothesis 3.6. Hypothesis 3.7 seems to be weaker in general.
We run through the previous argument, starting from the composition α x given by
V m − → V ⊗ O(T ) id⊗f − −− → V ⊗ O(G) id⊗βx − −−− → V x . (3.17)
We have that α x coincides with the localization map when restricted to V G . Also arguing as before we have that
m x • α x = (id ⊗ id ⊗ β x ) • (id ⊗ id ⊗ f ) • (id ⊗ ∆) • m, (3.18)
where m x denotes the comultiplication map V x → V ⊗ O(T (X,x) ). We conclude as before that m x • α x (V ) ⊂ V x ⊗ 1 after passing to V ⊗ O (T (x,x) ). Since T (x,x) is a groupoid over X x , and V x its representation, it must be that m ,x) ). Since this is true for all x, we get finally that
x • α x (v) = v ⊗ 1 in V ⊗ O(T (x,x) ) = V x ⊗ O(Xx) O(T (x,x) ) for any v ∈ V . So for v ∈ V G , we have m(v) = m x •α x (v) = v ⊗1 in V ⊗O(T (xm(v) = v ⊗ 1 in V ⊗ O(T U ) for some open neighborhood U of the diagonal X ⊂ X 2 , whenever v ∈ V G .
To extend this equality over X 2 , we will need some further hypothesis. There are probably several options, but here is a natural choice: Hypothesis 3.9. There exist closed subschemes R i of T such that:
(1) The projection map G × T R i → R i induces a universally injective map of O(X)-modules for each i, and (2) T U and the various R i generate T .
Here the second condition means precisely that the multiplication maps A 1 × X . . . × X A n → T , where n ranges from 1 to ∞ and the A j range over T U and the various R i (allowing repeats), induce jointly universally injective maps of O(X)modules.
The first condition guarantees that the map
V ⊗ O(R) → V ⊗ O(G × T R)Hom P QCoh(X) ((V j ) j∈J , (W k ) k∈K ) = lim ← − k colim − −− → j Hom QCoh(X) (V j , W k ). (4.1)
An equivalent, and useful, way to think about this is as follows. A morphism (V j ) j∈J → (W k ) k∈K consists of the following data:
(1) For each k ∈ K, a cofinal subsystem S(k) of J, satisfying S(k) ⊂ S(k ′ ) whenever k → k ′ ;
(2) For each j ∈ S(k), a morphism f k j : V j → W k , such that the diagram
V j → W k ↓ ↓ V j ′ → W k ′ (4.2)
commutes whenever it exists.
We take such data up to equivalence; two such data (S, f ), (S ′ , f ′ ) are equivalent if for every k, every j ∈ S(k), j ′ ∈ S ′ (k), and every lower (upper?) bound j ′′ of j, j ′ , the diagram
V j ′′ − → V j ↓ ↓ f k j V j ′ (f ′ ) k j ′ −−−→ W k (4.3)
commutes. It is enough to check for k in some cofinal subsystem of K, for (j, j ′ ) in some cofinal subsystem of S(k) × S ′ (k), and for j ′′ being any one (rather than all) lower bound(s) of j, j ′ . Yet another way to think of this is as follows: we view lim ← −j V j , lim ← −k V k as topological O(X)-modules (with the pro-discrete topology) and then Hom P QCoh(X) ((V j ), (W k )) is none other than the set of continuous morphisms between these topologized limits. For this, the countability is essential: a countable cofiltered system (V j ) admits a cofinal inverse (i.e. ordered as N) subsystem, and consequently each map lim ← −j V j → V j is surjective if the transition morphisms are. To present a projectively discrete topological O(X)-module as the limit of an object of P QCoh(X) is to give a countable cofinal subsystem of its lattice of open submodules. However we will not usually think of P QCoh(X) in this way, preferring to reserve the notation lim ← − for the functor lim ← − : P QCoh(X) → QCoh(X).
(4.4)
We note that lim ← − is right adjoint to the functor QCoh(X) → P QCoh(X) which takes a quasi-coherent sheaf to the corresponding single-object cofiltered system. lim ← − is faithful.
Perhaps the most useful way to think of this is given by the following:
Lemma 4.1.
(1) Every object of P QCoh(X) is isomorphic to an inverse (i.e. ordered as N) system;
(2) Let V be an object of P QCoh(X), let (W k ) k∈N be an inverse system in P QCoh(X), and let V → W be a morphism; then there exists an isomorphism (U i ) i∈N → V such that the composition (U i ) → V → (W k ) is equivalent to a map of inverse systems in the traditional sense. In other words, writing (f, S) for said composition, we may take S(k) = [k, ∞) for all k. In other other words, the composition (U i ) → V → (W k ) is equivalent to a surjective inverse system of morphisms.
It follows of course that any sequence in P QCoh(X) is isomorphic to an inverse system of sequences.
P QCoh(X) is not abelian, but it is exact. First we describe the admissible sequences:
Lemma 4.2. Let 0 → U → V → W → 0 be a sequence in P QCoh(X) which is isomorphic to an inverse system of of short exact sequences.
Then 0 → lim ← − U → lim ← − V → lim ← − W → 0 is exact.
We call such sequences Mittag-Leffler. To justify the name, observe first that given
a morphism V → W in P QCoh(X), presented as (V i ) (S,f ) − −− → (W j )
say, the property that every extant f j i is surjective is independent of the presentation. These are precisely the epimorphisms (epis) in P QCoh(X). Next, we observe that an epimorphism V → W may be extended to a Mittag-Leffler sequence U → V → W if and only if for every (equivalently, some) presentation of V → W as a surjective inverse system of morphisms V i → W i , the resulting inverse system ker(V i → W i ) satisfies the Mittag-Leffler condition. (In that case, to construct U we take the stabilization of the pointwise kernel of any presentation of V → W as a surjective inverse system of morphisms). We note also that monomorphisms are the same as morphisms which give injections in the limit.
4.4. The class of Mittag-Leffler sequences is an exact structure for P QCoh(X) (i.e. P Qcoh(X) has a unique structure of exact category in which the admissible sequences are precisely Mittag-Leffler sequences). This is a simple excersise in diagram chasing. P QCoh(X) has all cokernels, and every cokernel map is admissible. P QCoh(X) also has all kernels, and every kernel map is admissible. If f : V → W has kernel K and cokernel C then the map coim(f ) := coker(K → V ) → ker(W → C) =: im(f ) is always both epi and mono, but it is not always an isomorphism. It is an isomorphism if for some (equivalently any) presentation of f : V → W as a surjective inverse system of morphisms V i → W i , the inverse system ker(V i → W i ) satisfies the Mittag-Leffler condition (but not conversely!); in that case K is the stabilization of ker(V i → W i ). We call such f admissible. 4.5. If X → Y then we have the pushforward functor P QCoh(X) → P QCoh(Y ). This functor preserves kernels and cokernels, kills no objects, and reflects isomorphisms. Consequently it preserves and reflects monos, epis, and admissible morphisms. In particular it preserves and reflects Mittag-Leffler sequences. 4.6. We say an object of P QCoh(X) is flat, coherent etc. if it is isomorphic to a (surjective countable) cofiltered system of flat, coherent etc. O(X)-modules. Sometimes (e.g. 'coherent') this property is independent of the presentation, but more usually (e.g. 'flat') it depends very much on the presentation.
4.7. P QCoh(X) is monoidal: we set (V j ) j∈J ⊗ (W k ) k∈K = (V j ⊗ W k ) (j,k)∈J×K .
The one-object cofiltered system O(X) is the unit. It is convenient to note that for inverse systems (
V j ) j∈N , (W k ) k∈N , we have (V j ) ⊗ (W k ) ∼ = (V j ⊗ W j ) j∈N .
For instace, let A be a flat coherent (i.e. finite rank projective) sheaf on X, regarded as a single-object cofiltered system in P QCoh(X). Since A is dualizable, A ⊗ (−) is a right adjoint and thus we have A ⊗ lim ← − (V ) = lim ← − (A ⊗ V ) for any object V of P QCoh(X). Consequently, if more generally A is a flat coherent object of P QCoh(X), and V is any object of P QCoh(X), we have lim
← − (A ⊗ lim ← − (V )) = lim ← − (A ⊗ V ).
The same formulas hold if V is replaced with a countable cofiltered system with not necessarily surjective morphisms, from which point (3) of the following otherwise easy lemma follows: Lemma 4.3. Let A be an object of P QCoh(X). Then
(1) A ⊗ (−) preserves cokernels, hence epis;
(2) If A is flat then A ⊗ (−) preserves also admissible morphisms and their kernels, hence Mittag-Leffler sequences and admissible monos (but not monos or kernels in general); (3) If A is flat coherent then A ⊗ (−) preserves also kernels, hence monos.
4.
8. An affine ind-scheme over X is by definition a (surjective, countable) cofiltered system of O(X)-algebras 13 . Affine ind-schemes form a category, IAf f X , where by definition a morphism in IAf f op X , between (A i ) and (B j ) say, is any morphism (S, f ) in P QCoh(X) for which every extant f j i is a ring map 14 . An affine ind-scheme is called flat, coherent etc. if it is so as an object of P QCoh(X) 15 . IAf f X inherits the monoidal structure from P QCoh(X). A groupoid ind-scheme is a groupoid in IAf f 16 . As usual, for such a thing G, we will write O(G) for the corresponding object of IAf f op X , and keep the notations ∆, η, t # , s # of the previous section. Since η, s # split each other, they are respectively admissible epi, mono. Similarly t # and ∆ are admissible mono. A (right) representation of the groupoid ind-scheme G is a quasi-coherent sheaf V on X together with a morphism
m : V → V ⊗ O(G) (4.5)
satisfying the natural comodule axioms. Representations of G form a category, denoted Rep X (G), in the obvious manner. Of course, one could make the same definition with V being an arbitrary object of P QCoh(X); however, this would apparently make it difficult for Rep X (G) to be abelian.
4.9.
Coinduction. We assume that t is finite flat. In that case, the forgetful functor Rep X (G) → QCoh(X) has the all-important right adjoint
Coind G X : QCoh(X) → Rep X (G) V → lim ← − (V ⊗ O(G)) (4.6)
where the structure of representation on lim ← − (V ⊗O(G)) is given as follows. Certainly
there is a map V ⊗O(G) id⊗∆ −−−→ V ⊗O(G)⊗O(G); taking lim ← − we get lim ← − (V ⊗O(G)) − → lim ← − (V ⊗ O(G) ⊗ O(G)) = lim ← − (lim ← − (V ⊗ O(G)) ⊗ O(G)) since O(G)
is flat coherent with respect to t # . By the adjunction property of lim ← − , this is the same as a map lim
← − (V ⊗ O(G)) → lim ← − (V ⊗ O(G)) ⊗ O(G).
Having constructed the map, it is easy to 13 This definition is more restrictive than being simply a ring object in P QCoh(X). 14 Equivalently, noting that the limit of an affine ind-scheme is naturally a ring, we see that morphisms between affine ind-schemes are exactly those morphisms in P QCoh(X) whose limit is a ring homomorphism. This shows, for instance, that the forgetful functor IAf f op X → P QCoh(X) reflects isomorphisms. To present a projectively discrete topological ring as the limit of an object of IAf f X is to give a countable cofinal subsystem of ideals in its lattice of open submodules.
15 I do not know (nor care) whether a flat affine ind-scheme may be presented as a surjective cofiltered system of flat O(X)-algebras. 16 IAf f is meant as a stack over Af f ; this is just a convenient way of saying that a groupoid ind-scheme is an object of IAf f X×X for some X with the appropriate operations between its two projections to IAf f X . see that it satisfies the comodule axioms, and that Coind G X is indeed right adjoint to the forgetful functor. 4.10. Abelian-ness. It follows from the flatness of t that Rep G X is abelian. The proof is more or less the same as in the scheme case, but we'll indicate it anyway.
Suppose U → V → W → 0 is an exact sequence in QCoh(X). If U → V lifts to Rep X (G), then V → V ⊗ O(G) → W ⊗ O(G)
factors through W , and one checks that this defines a comodule structure on W (independently of the assumption on t), and that V → W is a map of comodules, and that this is the unique way to lift
V → W to Rep G X . If instead U → V is injective and V → W lifts to Rep X (G), then consider the diagram 0 0 0 ↓ ↓ ↓ U U ⊗ O(G) − − → U ⊗ O(G) ⊗ O(G) ↓ ↓ ↓ V − → V ⊗ O(G) − → − − → V ⊗ O(G) ⊗ O(G) ↓ ↓ ↓ W − → W ⊗ O(G) − → − − → W ⊗ O(G) ⊗ O(G) ↓ ↓ ↓ 0 0 0 (4.7)
By the flatness of t, each column is Mittag-Leffler. Therefore in the limit, the columns become short exact sequences. The leftmost dashed arrow exists in the limit, and hence exists outright by adjunction (U being already quasi-coherent). Hence both dashed arrows exist outright. All necessary commutativity/equalizing properties follow from the corresponding limiting statements, since lim ← − is faithful. A similar diagram yields the counity condition, and thus one obtains the (unique) lifting of U → V to Rep X (G). From this it is a formal consequence that Rep G X is abelian, and that the forgetful functor to QCoh(X) is exact and faithful. 4.11. Descent. By Barr-Beck, one obtains that Rep X (G) is equivalent to the category of comodules in QCoh(X) for the comonad lim ← − ((−) ⊗ O(G)). In order for Lemma 3.1 to go through, one must make the additional assumption that f : Y → X is finite (as well as faithfully flat), so that the comonads
f * • Res G X • Coind G X • f * and Res Y ×G×Y Y • Coind Y ×G×Y Y on QCoh(Y ) coincide.
4.12. The remaining arguments of the previous section apply more or less verbatim. We will point out (in chronological order) the points where some extra thought is needed:
(1) C is defined the same way as before (it is a scheme). . The waffle about high roads and low roads works out the same and one obtains Equation 3.9, whose RHS is interpreted in exact categories-speak as the kernel of
Hom QCoh(X) (V, V ′ ) m−id⊗1 −−−−−→ Hom QCoh(X) (V, V ′ ) ⊗ O(T ).
Since m, id ⊗ 1 have the common section id ⊗ η, we also get Equation 3.10 (interpreted as a pullback in the obvious way). Thus again we will deduce fullness from the 'reflects invariants' property. (7) We make Hypothesis 3.6 and define α, β as before. It is still true that α is a section of the inclusion V T → V , but one must be a little more careful.
(id ⊗ β) • (id ⊗ f ) • ∆ factors through t # . Likewise the map (id ⊗ id ⊗ β) • (id ⊗ id ⊗ f ) • (id ⊗ ∆) : V ⊗ O(T ) → V ⊗ O(T ) factors through id ⊗ 1 : V → V ⊗ O(T ) as required. (8)
We make the same alternative hypothesis as Hypothesis 3.7, and make the same ,x) ) for all closed points x ∈ X. Of course it does not necessarily follow that we have the equality in V ⊗ O(T U ) for some Zariski-open neighborhood U of the diagonal X ⊂ X × X, but it does hold for U being the 'complement' of some closed ind-subscheme of X × X, which suffices. (9) From Hypothesis 3.9 onwards the argument is identical, reading 'sub-indscheme' for subscheme and 'mono' for injective.
conclusion that if v ∈ V G then m(v) = v ⊗ 1 in V ⊗ O(T (x
In summary, we have the following:
Proposition 4.4. Let G be an affine groupoid ind-scheme over the affine base scheme X, with adjacency groupoid T and isotropy subgroup I.
(1) Suppose the following conditions hold: (a) Both G, T are finite flat over X with respect to both the head and tails maps;
(b) O(G) → O(G) ⊗ O(I) is admissible and O(T ) is its kernel; (c) For every closed point x of X, there exists a (t # ) O(X)-linear section of t # x : O(X x ) → O(G (x,X) ) (denoted by β x ) such that s # x • β x • s # and t # x • β x • s # coincide in O(G (x,x) ) (equivalently, in O(T (x,x) )).
Then the functor of restriction Rep X (T ) → Rep X (G) reflects invariants in some neighborhood U of the diagonal X ⊂ X 2 .
(2) Suppose in addition that there exist closed sub-ind-schemes R i of T such that: where the latter category denotes the full subcategory of Rep X (G) consisting of those objects which admit resolutions by O(X)-flat objects which have trivial isotropy.
(a) The map G × T R i → R i
Remark 4.5. We think of the above isotropy condition as 'having trivial derived isotropy'. We expect that the better way to phrase it is to replace I by the natural groupoid ind-dga-scheme (whatever that means!), denoted " I ", at which point the condition may be literally interpreted as having trivial " I "-action. We do not pursue this here.
Proof of Theorems 1.1 and 1.2
For background material on root systems and reflection groups, as used heavily in paragraph 5.3, see [7]. Indeed by [4], Theorem 1.2.2 we have that for an equivariantly formal T -variety X with finitely many fixed points,
H • T (X) = ker(H • T (X T ) → j H • T (X j )), (5.1)
where the X j are the one-dimensional orbits of T on X and the j-component of the map is (ξ x ) x∈X T → ξ j0 − ξ j∞ (on Lie(Stab T (X j ))). In our situation, the one-dimensional T ∨ × G m -orbits on F l λ correspond (up to equivalence, i.e. repetition in the above morphism) to pairs (s, γ), where s is an affine reflection and γ is any fixed point in F l λ such that γs is also in F l λ . Since T ∨ has the same fixed point set in F l λ as T ∨ × G m , each H • T ∨ ×Gm (F l λ ) is free over A 1 . Since A 1 has homological dimension 1, one may set = 1 in Equation 5.1 (and obtain a correct formula). Then
H • T ∨ ×Gm (F l λ(s,γ) ) =1 = O(Γ γ | (t * ) s ). Writing O(G) λ := H • T ∨ ×Gm (F l λ T ∨ ) =1 , O(T ) λ := H • T ∨ ×Gm (F l λ ) =1 , we have O(G) λ = γ∈F l λ T ∨ O(Γ γ ) O(I) λ = γ∈F l λ T ∨ O(Γ γ | (t * ) γ ) O(G) λ ⊗ O(I) µ = (γ0,γ∞)∈F l λ T ∨ ×F lµ T ∨ O(Γ γ0 | (t * ) γ∞ ). (5.2)
In the limit, we have lim
← − O(G) = γ∈F l T ∨ O(Γ γ ) and lim ← − (O(G) ⊗ O(I)) = γ0,γ∞∈F l T ∨ O(Γ γ0 | (t * ) γ∞ )
. Both are equipped with the product topology, and the morphism ∆ − id ⊗ 1 between them sends (ξ γ ) γ to (ξ γ0.γ∞ | (t * ) γ∞ − ξ γ0 | (t * ) γ∞ ) (γ0,γ∞) . For each λ, the set S λ of pairs γ 0 , γ ∞ such that both γ 0 and γ 0 .γ ∞ are contained in F l λ is finite, and so one obtains the discrete
quotient (γ0,γ∞)∈S λ O(Γ γ0 | (t * ) γ∞ ) of lim ← − (O(G) ⊗ O(I))
. These quotients are cofinal in the cofiltered system of all discrete quotients, and thus may be used as a presentation. But then composition of ∆ − id ⊗ 1 with projection to the λ-piece of this presentation factors through O(G) λ , and yields the the map
γ∈F l λ T ∨ O(Γ γ ) → (γ0,γ∞)∈S λ O(Γ γ0 | (t * ) γ∞ ) (ξ γ ) γ → (ξ γ0.γ∞ | (t * ) γ∞ − ξ γ0 | (t * ) γ∞ ) (γ0,γ∞) (5.3)
If one projects the right-hand side to the product of all those factors for which γ ∞ is an affine reflection, then one obtains the map whose kernel is O(T ) λ , according to Equation 5.1. Therefore O(T ) λ is also the kernel of Equation 5.3, which is to say that O(T ) is the kernel of the admissible
morphism O(G) ∆−id⊗1 −−−−−→ O(G) ⊗ O(I), as required.
(3) Hypothesis 3.7 holds. Indeed, note that for any closed point x ∈ t * , the stabilizer W af f x is finite, and so one may define a map G (x,x) ), we get the desired result 17 . (4) Hypothesis 3.9 holds taking the closed sub-ind-schemes R i of T to be the graphs Γ γ (γ ranging over all of W af f ). Of course Γ γ is most naturally a closed sub-ind-scheme of G; it is also a closed subscheme of T , because it is a closed subscheme of t * × t * . Then the projection G × T Γ γ → Γ γ is an isomorphism, so certainly induces a universally injective map of O(X)modules. Finally, the various multiplications Γ γ × X T U → T constitute an open cover of T , so certainly induce jointly universally injective morphisms of O(X)-modules.
β x : O(G (x,X) ) = γ∈ W af f O(Γ γ (x,X) ) → O(t * x ) (ξ γ ) γ → 1 | W af f x | γ∈ W af f x π 1 * (ξ γ ). (5.4) β x is a t # -linear section of t # x . The map β x • s # : O(t * ) → O(t * x )
General case.
For general reductive G, we write T ad , G ad for the groupoids associated as in the previous paragraph to its adjoint group. W af f acts (by 'conjugation') on both T ad , G ad by groupoid automorphisms covering its natural action on t * (for both the heads and the tails map), and the map G ad → T ad is W af fequivariant. From Lemma 2.3 it follows that T , G are obtained from T ad , G ad as follows:
T = T ad × W af f W af f G = G ad × W af f W af f .
Here the symbol × W af f denotes the balanced product, where W af f acts on W af f by left translations and on T , G by right translations. It is left as an exercise to check that the right-hand expressions have natural groupoid structures and that the equalities are as groupoids 18 . It follows that the right adjoint of the restriction functor Res T T ad ,
Coind T T ad (−) := (−) ⊗ O(T ad ) O(T ) = V ⊗ O(W af f ) O( W af f ) ≃ π1(G ∨ ) V 19 ,
satisfies the equation
Res G T • Coind T T ad ∼ = Coind G G ad • Res G ad T ad .
Both Res T T ad and Res G G ad satisfy the conditions of comonadicity (their right adjoints are the coinduction functors above; they are exact functors between abelian categories which kill no objects). Writing C T , C G for the corresponding comonads, we have
C G • Res G ad T ad ∼ = Res G ad T ad • C T ;
17 With a little more care, we may take U ⊂ t * × t * to be the complement of the (ind-scheme) union of the graphs of the fixed-point-free elements of W af f . This is not necessary for what follows. 18 This is essentially the same as the original construction of the transformation groupoid
G = W af f × t * . 19
The symbols ⊗ O(T ad ) , ⊗ O(W af f ) denote the balanced tensor product (over t * ), which is to say the subspace of the tensor product on which the two 'inner' comodule structures coincide.
thus Res G ad T ad defines a functor Res G ad T ad : C T − comod → C G − comod which makes the following diagram commutative:
Rep t * (T ) ∼ − → C T − comod ↓ Res G T ↓ Res G ad T ad Rep t * (G) ∼ − → C G − comod.
That Res G ad T ad : C T − comod → C G − comod is fully faithful follows from the fact established previously that Res G ad T ad : Rep t * (T ad ) → Rep t * (G ad ) is fully faithful. The fully faithfulness of Res G ad T ad also implies that a C G -comodule which is also a representation of T ad is a C T -comodule in a unique way. Equivalently, a representation of G is the restriction of a representation of T if and only if its restricted structure of G af f -representation descends to one of T af f -representation. This shows that the objects of T oda 1 (G) are precisely W af f -equivariant quasicoherent sheaves on t * which admit W af f -equivariant resolutions by W af f -equivariant quasicoherent sheaves which are flat over t * and have trivial isotropy. On the other hand, every T oda 1 (G)-module admits a resolution by free T oda 1 (G)-modules, which correspond to W af f -equivariant quasicoherent sheaves which are free over t * and have trivial isotropy in W af f . Thus we have:
Theorem 5.1.
There is an equivalence of categories:
Rep t * (T ) ∼ − → QCoh W af f (t * ) " I " .
By definition the second category is the full subcategory of QCoh W af f (t * ) consisting of objects which admit resolutions by W af f -equivariant quasicoherent sheaves which are free over t * and have trivial isotropy in W af f . However, to check this condition it is sufficient to find a W af f -equivariant resolution by W af f -equivariant quasicoherent sheaves which are flat over t * and have trivial isotropy.
5.3.
Finite parabolic descent. In this special case, the 'derived isotropy' condition has a more familiar description. Let us denote the stabilizer in W af f of the closed point x ∈ t * by Γ x . Recall that W af f acts simply transitively on the connected components of the complement of the affine reflection hyperplanes ('hyperplanes' for short) in the real span t * R of the character lattice. These connected components are called alcoves. It follows immediately that Γ Re(x) acts sub-simply transitively on the set of alcoves containing Re(x) in their closure, so that Γ Re(x) is in particular finite. On the other hand, for any two such alcoves P, Q one may draw a path between them sufficiently close to Re(x) that the only hyperplanes it crosses pass through Re(x); we may deform this path slightly (staying in t * R ) so that it does not meet any pairwise intersections of hyperplanes, and the result is a sequence P = P 0 , . . . , P n = Q of alcoves such that P j−1 , P j share a common face for each j = 1, . . . , n, which is contained in the hyperplane corresponding to the affine root α j . Then s α j P j−1 = P j , so that s α n . . . s α 1 P 0 = P n . This shows that Γ Re(x) acts simply transitively on the alcoves containing Re(x) in their closure and is generated by reflections. Moreover, the reflection s α 1 s α 2 . . . s α n−1 s α n s α n−1 . . . s α 2 s α 1 is through some face of P ; it follows by induction on n that Γ Re(x) is generated by its reflections through faces of P . Since P is conjugate to the fundamental alcove, which is bounded by the simple root hyperplanes, it follows that Γ Re(x) is a parabolic subgroup of W af f . On the other hand, translating by −x sends the hyperplanes passing through x to certain hyperplanes passing through 0, which are still reflection hyperplanes. We have thus proved:
Lemma 5.2. The composition Γ Re(x) → W af f π − → W
is injective and realizes Γ Re(x) as a reflection subgroup of W . In fact, Γ Re(x) is realized as the Weyl subgroup corresponding to some root subsystem Φ x of the root system Φ of W 20 . Now we may calculate:
Γ x = Γ Re(x) ∩ π −1 Stab W (Im(x)) ∼ − → π(Γ Re(x) ∩ π −1 Stab W (Im(x))) = π(Γ Re(x) ) ∩ Stab W (Im(x)) = Stab π(Γ Re(x) ) (Im(x))
Let V x denote the fixed point subspace of π(Γ Re(x) ) acting on t * R . V x is complementary to the span of Φ x , so that Φ x + V x is a root system of full rank in t * +R/V x , with Weyl group π(Γ Re(x) ). We have
Γ x = Stab π(Γ Re(x) ) (Im(x)) = Stab π(Γ Re(x) ) (Im(x) + V x )
which is a parabolic subgroup of π(Γ Re(x) ) 21 . We have proved:
O(V × V //Γ V ) → O(Γ#V ),
which is an isomorphism generically over V for any finite Γ (not necessarily generated by reflections), is injective. We conclude that V × V //Γ V is the image of the natural map Γ#V → V × V , i.e. the adjacency groupoid of Γ#V . These conclusions hold also for the action of Γ x on t * , since it is conjugate to the action of a finite reflection group, under the automorphism of t * given by translating by −x. 20 Φx is not necessarily integrally closed in Φ, nor irreducible even if Φ is; see for instance what happens in type G 2 . 21 being the stabilizer of a point in the reflection representation of a Weyl group; remove the words 'affine' from the discussion at the start of this paragraph. 22 It seems likely that this is well known, but I have not been able to find a reference for it.
We will write G x for Γ x # t * and T x , I x for the resulting adjacency groupoid and isotropy subgroup.
In fact, we have the following under the multiplication map, which is isomorphic to I x (y,y) × G x (y,y) ⇒ G x (y,y) → T x (y,y) under the projection, which we have just shown to be a coequalizer diagram.
Consequently, an O(G x )-comodule (over t * ) has at most one compatible structure of O(T x )-comodule. Likewise, an O(G x ) (y,y) -comodule (over t * y ) has at most one compatible structure of O(T x ) (y,y) -comodule. We are now ready to prove 24 the following:
Theorem 5.6. An object V of QCoh W af f (t * ) has trivial derived isotropy if and only if for every finite parabolic subgroup Γ ⊂ W af f , the Γ-equivariant structure on V is descent datum for t * → t * //Γ.
Proof. That trivial derived isotropy implies descent for all finite parabolic subgroups is immediate. Conversely, assume the W af f -equivariant quasicoherent sheaf V has descent for all finite parabolic subgroups. It means that for each closed point x ∈ t * there is a unique dashed comultiplication making the diagram
V → V ⊗ O(G) ↓ V ⊗ O(T x ) → V ⊗ O(G x )
commutative. Also for any closed point y the composition V → V ⊗ O(T x ) → V ⊗ O(T x (y,y) ) is the unique comodule map making the diagram
V → V ⊗ O(G) ↓ V ⊗ O(T x
(y,y) ) → V ⊗ O(G x (y,y) ) commutative. Denote by T t * the stalk of T at the diagonal t * ⊂ t * × t * . Choose an enumeration γ 1 , γ 2 , . . . of W af f . Set S i to be the closed subscheme of T which is the union of the graphs Γ γ1 , . . . , Γ γi . These exhaust T . Write also Ω i = {γ 1 , . . . , γ i }. For any closed point (x, y) in t * × t * let us write T y→x for the union of graphs passing through (x, y). This is a torsor for T x in the sense that choosing any component of T y→x gives an isomorphism with T x ; likewise it is a torsor for T y . Similarly write Γ y→x for subset of W af f consisting of all γ such that γ(y) = x, a torsor for both Γ x and Γ y . We construct the map V → V ⊗ O(T ) as follows. for any choice γ ∈ Γ y→x 25 . To see that these glue together, it suffices to check that for every three closed points (x 1 , x 2 ), (y 1 , y 2 ), (z 1 , z 2 ) of t * × t * such that (z 1 , z 2 ) ∈ U x2→x1 i ∩ U y2→y1 i , the two resulting maps
V → V ⊗ O(T x2→x1 ) → V ⊗ O(U x1,x2 i ) → V ⊗ O((S i ) (z1,z2) ) and V → V ⊗ O(T y2→y1 ) → V ⊗ O(U y1,y2 i ) → V ⊗ O((S i ) (z1,z2) ) coincide. Now (z 1 , z 2 ) ∈ U x2→x1 i ∩ U y2→y1 i implies Γ z2→z1 ∩ Ω i ⊂ Γ x2→x1 ∩ Γ y2→y1 .
It follows that our two morphisms can be written as
V → V ⊗ O(T x2→x1 ) → V ⊗ O(T x2→x1,y2→y1,z2→z1 ) → V ⊗ O((S i ) (z1,z2) ) and V → V ⊗ O(T y2→y1 ) → V ⊗ O(T x2→x1,y2→y1,z2→z1 ) → V ⊗ O((S i ) (z1,z2) )
where T x2→x1,y2→y1,z2→z1 denotes the union of those graphs which pass through all three points (x 1 , x 2 ), (y 1 , y 2 ), (z 1 , z 2 ). This is a torsor for the adjacency group T x2,y2,z2 of the reflection group Γ x2 ∩ Γ y2 ∩ Γ z2 . These morphisms coincide, since either both are 0 (if T x2→x1,y2→y1,z2→z1 is empty) or in both cases the induced morphism V → V ⊗ O(T x2,y2,z2 ) is the unique comodule structure which restricts to the given O(G x2,y2,z2 )-comodule structure.
We have constructed the maps V → V ⊗O(S i ), which have the property that each y) ), and it follows that they are compatible as i ranges to ∞, so that we get a morphism V → V ⊗ O(T ). That this is a comodule structure can be checked on stalks, where it holds by construction. 5.4. We find it interesting to note that we may view T oda 1 (G)-mod as being made up of the various QCoh(t * //Γ), glued together along their common ramified cover t * . We do not yet know what to make of this.
composition V → V ⊗ O(S i ) → V ⊗ O((S i ) (x,y) ) factors as V → V ⊗ O(T y→x ) → V ⊗ O((S i ) (x,
2. 7 .
7Convolution. The multiplication and inversion in the group G ∨ (K) are bounded with respect to the ind-pro-structure. It follows that H • G ∨ (O)⋊Gm (Gr) and H G ∨ (O)⋊Gm •
Hypothesis 3. 7 (
7Replacement for Hypothesis 3.6). For every closed point x of X, there exists a
is injective; it follows that for any v ∈ V G , m(v) and v ⊗ 1 have the same image in V ⊗ O(R). The second condition gives that the various compositionsV ⊗ O(T ) id⊗∆ n−1 − −−−−− → V ⊗ O(T ) ⊗ . . . ⊗ O(T ) → V ⊗ O(A 1 ) ⊗ . . . ⊗ O(A n ) (3.19) are jointly injective. If v ∈ V Gthen the image of m(v) under any one of these compositions certainly coincides with v ⊗ 1 ⊗ . . . ⊗ 1, and hence m(v) = v ⊗ 1 as required. 4. Ind-schemes 4.1. In this section, we develop the theory of groupoid ind-schemes to the point where we are able to formulate appropriate replacements for the hypotheses, arguments and conclusions of the previous section. 4.2. Consider the collection of non-empty countable cofiltered systems of O(X)modules with surjective transition maps. These form an additive category, denoted P QCoh(X), where by definition
( 2 )
2O(T ) is defined as the subsystem of O(G) generated by the images of s # , t # , which is automatically surjective; it is left as an exercise that T inherits the groupoid structure from G.From the point of view of exact categories, O(T ) is the kernel of the cokernel morphism O(G) → coker(O(X × X) → O(G)); since cokernel maps are admissible, we get that O(T ) → O(G) is admissible. (3) I is defined as I = G× X×X X; on level of surjective systems, it is the system obtained by quotienting out each piece of O(G) by the ideal generated by the image of (s # −t # )O(X). Those ideals also form a surjective system, denoted M , and so we have the Mittag-Leffler sequence M → O(G) → O(I).
( 4 )
4Hypothesis 3.3 needs changing slightly. First we replace 'flat' by 'finite flat'. Next note that the equalizer of O(G) ⇒ O(G) ⊗ O(I) is equal to the kernel of the composition O(G) I) and is an ind-scheme (the forgetful functor IAf f op X → P QCoh(X) has and preserves equalizers of equalizers it creates). Moreover since M → O(G) → O(I) is Mittag-Leffler, so is O(G) ⊗ M → O(G) ⊗ O(G) → O(G) ⊗ O(I), and it follows that the equalizer in question is the Cartesian product of O(G) ↓ ∆−id⊗1 O(G) ⊗ M → O(G) ⊗ O(G).
condition that G × X I ⇒ G → T is a coequalizer in IAf f X is equivalent to the condition that O(T ) maps isomorphically to the Cartesian product of that diagram in P QCoh(X), or equivalently that it maps isomorphically to the kernel of O(G) → O(G) ⊗ O(I) in P QCoh(X). We require the additional hypothesis that this map is admissible (recall that O(T ) → O(G) is admissible). This hypothesis is equivalent to that in any presentation of diagram 4.8 as a surjective inverse systems of diagrams the pointwise pullback satisfies the Mittag-Leffler condition and O(T ) maps isomorphically to its stabilization.(5) Lemma 3.4 goes through as written if one understands 'injective' as mono, and recalling that O(T ) → O(G) is the kernel of the admissible morphism O(G) ⊗ O(I) (and these properties are preserved when tensoring with flat objects). (6) Hypothesis 3.5 has already been made. The construction of comodule structure on Hom O(X) (V, V ′ ) is the same, noting that Hom P QCoh(X) (V, V ′ ⊗ O(T )) = lim ← − Hom QCoh(X) (V, V ′ ⊗O(T ) i ) (choosing a presentation of O(T )), which equals lim ← − (Hom QCoh(X) (V, V ′ )⊗O(T ) i ), and thus Hom P QCoh(X) (V, V ′ ⊗ O(T )) is 'internalized' as Hom Qcoh(X) (V, V ′ ) ⊗ O(T )
Indeed since the map O(X×X) → O(T ) is epi, the map (id⊗β)•(id⊗f )•∆ : O(T ) → O(T ) factors through the image of t # ⊗(s # •β •s # ) : O(X ×X) → O(T ). This latter map equals t # ⊗ (t # • β • s # ) which of course factors through t # : O(X) → O(T ). This last map is an isomorphism with its image, so we see that
induces a universally injective of O(X)modules for each i, and (b) T U and the various R i generate T . Then Rep X (T ) → Rep X (G) is full. (3) The functor Rep X (T ) → Rep X (G) I is an equivalence on the full subcategories of objects which are flat over X. Noting that O(X)-flat (even projective) resolutions exist in Rep X (T ), one obtains the equivalence Rep X (T ) ∼ = Rep X (G) " I "
5. 1 .
1Adjoint case. Let us begins with the case where G is adjoint. Then π 1 (G ∨ ) is trivial, W af f = W af f and T = Spec(H • G ∨ (O)⋊Gm (F l)) is really the adjacency groupoid of G = W af f # t * . We simply check the conditions of Proposition 4.4.(1) Certainly s, t are finite flat (for T , G).
Proposition 5 . 3 .
53The stabilizer group Γ x is a finite parabolic subgroup of W af f . In particular it is generated by affine reflections passing through x 22 .Moreover, every finite parabolic subgroup of W af f arises in this way. Next, we have (paraphrasing):Theorem 5.4 (Chevalley-Shephard-Todd, [8][9]). Let V be a complex vector space and Γ be a finite subgroup of GL(V ) generated by reflections. Then O(V ) is free of finite rank over O(V //Γ).Thus in the situation of Chevalley-Shephard-Todd, we have:QCoh(V //Γ) ∼ = Rep V (V × V //Γ V )by faithfully flat descent. Also, since O(V × V //Γ V ) is free of finite rank over either copy of O(V ), it is in particular torsion-free and so the natural map
For each subscheme S i we form the open cover open subscheme of a closed subscheme of T y→x , and so we define the mapV → V ⊗ O(T y→x ) → V ⊗ O(U x,y i ). Here the map V → V ⊗ O(T y→x ) by definition equals either 0 (if T y→x is empty) or otherwise the composition V → V ⊗ O(T y ) → V ⊗ O(Γ γ ) ⊗ O(T y ) ∼ − → V ⊗ O(T y→x )24 and indeed, to formulate: in the statement of the theorem the uniqueness of a compatible descent datum is implicit.
is the composition of the averaging map with respect to W af fx
with the localiza-
tion map. Therefore s #
x • β x • s # and t #
x • β x • s # coincide on every factor
of the form O(Γ γ ) with γ ∈ W af f
x
. Noting that these are the only factors
which survive in O(
which is to be thought of as its dualizing complex, homologically shifted by −2 dim G ∨ (O).9 If one restricts to 0 ∈ A 1 , these two duals are identical and the resulting automorphism is the identity.
And as for Gr they may also be regarded as the (I ∨ × I ∨ ) ⋊ Gm-equivariant cohomology of some 'complexes' (the constant sheaf and the shifted dualizing complex) on G ∨ (K).
This means that for each closure of G ∨ (O)-orbit in F l, whose (T ∨ × Gm-) equivariant cohomologies form the pieces of the pro-object H • T ∨ ⋊Gm (F l), the restriction map from its equivariant cohomology to that of its (T ∨ × Gm-) fixed point set is an isomorphism over some non-empty Zariski-open subset of the base t * × A 1 . These open subsets do not stabilize as one exhausts F l.
Lemma 5.5. The groupoids G x (over t * ) and G x (y,y) (over t * y ) for any closed point y of t * all satisfy the conditions of Proposition 4.4.Proof. Note that:Since Γ x , Γ y are both parabolic, so is Γ x ∩ Γ y ; in particular it is generated by reflections. Therefore the adjacency groupoid of Γ x ∩ Γ y # t * is finite flat over t * (with respect to either heads or tails); it is easy to see 23 that the stalk at (y, y) of this adjacency groupoid is the same as its stalk at (y, t * ). It follows that this stalk is finite flat over t * y , and also that it coincides with the adjacency groupoid of G x (y,y) ; this gives condition (1)(a). We note that the adjacency groupoid of G x (y,y) also coincides with T x (y,y) . Conditions (1)(c) and(2)are solved in the same way as in points(3)and(4)of paragraph 5.1. This leaves condition (1)(b). Noting that the formation of T x , I x from G x commutes with taking stalks, this amounts to checking thatare coequalizer diagrams. For the second diagram, let z be any closed point of t * whose stabilizer in W af f is Γ x ∩ Γ y . Then, translating by z − y, we see that the groupoid G x (y,y) over t * y is isomorphic to the groupoid G z (z,z) over t * z . This is isomorphic to G (z,z) (over t * z ). Noting now that the formation of T , I from G commutes with taking stalks, we see that the second diagram is isomorphic to the localization at (z, z) of the big (admissible) coequalizer diagram I × G ⇒ G → T and so is indeed a coequalizer diagram.Consider now the first coequalizer diagram-to-be. To check that it is a coequalizer diagram it suffices to check at stalks of closed points. At any closed point (y, z) say, we need to show thatis a coequalizer diagram. If y, z are not conjugate under Γ x , this is vacuous; otherwise suppose γ ∈ Γ x with γ(z) = y. Then this diagram is isomorphic to23 Indeed let f be a function on t * × t * which does not vanish at (y, y). Consider the function f y := γ∈Γ x ∩Γ y γ(f ), where γ acts on the second factor of t * . This function also does not vanish at (y, y), and its restriction to the adjacency groupoid A in question coincides with the functionThus to invert f on A it suffices to invert some function which factors through π 1 as claimed.
Quantization and representation theory, in Representation theory of Lie groups. B Kostant, London Math. Soc. Lect. Note Series. 34B. Kostant, Quantization and representation theory, in Representation theory of Lie groups, London Math. Soc. Lect. Note Series 34 (1979), 287-316.
M Demazure, P Gabriel, Groupes algebriques. Tome I: Géométrie algebrique, généralités, groupes commutatifs, Masson & Cie. Editeur, ParisAmsM.Demazure, P.Gabriel, Groupes algebriques. Tome I: Géométrie algebrique, généralités, groupes commutatifs, Masson & Cie, Editeur, Paris; North-Holland Publishing Co., Ams- terdam (1970).
R Bezrukavnikov, M Finkelberg, arXiv:0707.3799Equivariant Satake category and Kostant-Whittaker reduction. R. Bezrukavnikov, M. Finkelberg, Equivariant Satake category and Kostant-Whittaker re- duction, arXiv:0707.3799 (2007).
Equivariant cohomology, Koszul duality, and the localization theorem. M Goresky, R Kottwitz, R Macpherson, Invent. Math. 131M. Goresky, R. Kottwitz, and R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998).
V Ginzburg, arXiv:alg-geom/9511007Perverse sheaves on a Loop group and Langlands' duality. V. Ginzburg, Perverse sheaves on a Loop group and Langlands' duality, arXiv:alg-geom/9511007 (1995)
Singularities, character formulas, and a q-analogue for weight multiplicities. G Lusztig, Analyse et topologie sur les espaces singuliers. 101G. Lusztig, Singularities, character formulas, and a q-analogue for weight multiplicities, in Analyse et topologie sur les espaces singuliers, Astérisque 101-102 (1982) 208-229.
Reflection groups and Coxeter groups. James E Humphreys, of Cambridge Studies in Advanced Mathematics. CambridgeCambridge University Press29James E. Humphreys. Reflection groups and Coxeter groups, volume 29 of Cambridge Stud- ies in Advanced Mathematics. Cambridge University Press, Cambridge, 1990.
Invariants of finite groups generated by reflections. C Chevalley, Amer. J. Math. 77C. Chevalley, Invariants of finite groups generated by reflections. Amer. J. Math. 77 (1955), 778-782.
Here the first morphism is the comultiplication given by the hypothesis of finite parabolic descent; the second morphism is induced by the morphism V → V ⊗ O(Γγ ) determined by the W af f -equivariant structure; the third morphism is the torsor isomorphism determined by γ. The composition is independent of γ. Here the first morphism is the comultiplication given by the hypothesis of finite parabolic descent; the second morphism is induced by the morphism V → V ⊗ O(Γγ ) determined by the W af f -equivariant structure; the third morphism is the torsor isomorphism determined by γ. The composition is independent of γ.
Finite unitary reflection groups. G C Shephard, J A Todd, Canad. J. Math. 6G. C. Shephard and J. A. Todd, Finite unitary reflection groups. Canad. J. Math. 6 (1954), 274-304.
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[] |
[
"Classification of affine matrix means",
"Classification of affine matrix means",
"Classification of affine matrix means",
"Classification of affine matrix means"
] |
[
"Miklós Pálfia ",
"Miklós Pálfia "
] |
[] |
[] |
In this article we find all possible matrix means which are points of geodesics of affinely connected manifolds. We characterize certain properties of these manifolds, decide whether they are metrizable or not. We show that the matrix means that occur in this form are exactly the matrix power means. We show that the several variable formulas of power means are the unique solutions of certain Karcher equations corresponding to affinely connected geometric structures. At the same time we construct weighted matrix means corresponding to a matrix mean and we show that the representing operator monotone functions of the weighted means form a one parameter family of functions that has a semigroup structure in the sense of Loewner. These one parameter families are induced by operator monotone functions that behave similarly as logarithm maps of affinely connected manifolds. We show that the holomorphicity of the members of these one-parameter families is controlled by the distribution of the branch points of the corresponding logarithm map.
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[
"https://arxiv.org/pdf/1208.5603v5.pdf"
] | 119,621,073 |
1208.5603
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79d30bd3450fd0a1e4c89533fa0a9cf92bcde7fc
|
Classification of affine matrix means
28 Aug 2012
Miklós Pálfia
Classification of affine matrix means
28 Aug 2012Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor)operator monotone function · starlike function · matrix mean · symmetric space · affine connection
In this article we find all possible matrix means which are points of geodesics of affinely connected manifolds. We characterize certain properties of these manifolds, decide whether they are metrizable or not. We show that the matrix means that occur in this form are exactly the matrix power means. We show that the several variable formulas of power means are the unique solutions of certain Karcher equations corresponding to affinely connected geometric structures. At the same time we construct weighted matrix means corresponding to a matrix mean and we show that the representing operator monotone functions of the weighted means form a one parameter family of functions that has a semigroup structure in the sense of Loewner. These one parameter families are induced by operator monotone functions that behave similarly as logarithm maps of affinely connected manifolds. We show that the holomorphicity of the members of these one-parameter families is controlled by the distribution of the branch points of the corresponding logarithm map.
this space, which is defined as taking middle points of the geodesic connecting two points, is the geometric mean of two positive definite matrices [6].
The other two symmetric spaces which are well known are Euclidean spaces. One of them is just the vector space of squared complex matrices. The subset P(n, C) therefore inherits its Euclidean structure and the midpoint operation is the arithmetic mean. The third case is the harmonic mean, the corresponding space is isometric to the previous one.
These affinely connected manifold structures are important and widely used in several cases. For instance the extension of the two variable geometric mean to several variables are heavily based on the Riemannian structure of the corresponding space [1,28,25]. The multivariable geometric mean or Karcher mean is defined as the center of mass This mean and the geometric structure corresponding to it is also used in practical applications when one considers averaging of symmetric tensors [2,11,21,3,4]. The Karcher mean Λ(A 1 , . . . , An) is also the unique positive definite matrix where the gradient of the function in the minimization problem (1) vanishes
n i=1 log(X −1 A i ) = 0.
This equation is called the Karcher equation. Recently Lim and Pálfia [26] found a one parameter family of multivariable matrix means called the matrix power means which are defined as the unique positive definite solution of the equation
X = 1 n n i=1 G t (X, A i )(2)
where G t (A, B) = A 1/2 A −1/2 BA −1/2 t A 1/2 is the weighted geometric mean of A, B ∈ P(n, C). The attractive property of this family is that as the defining parameter t → 0, the matrix power means converge to the Karcher mean. In this paper in Section 7 we will show that the matrix power means are actually unique solutions of Karcher equations n i=1 log X (A i ) = 0 corresponding to affine connections given on the manifold P(n, C). In other words the two-variable matrix power means give back the geodesic lines corresponding to these affine connections. Moreover in Section 8 we find all possible two-variable matrix means which occur such a way, in other words as affine matrix means, which question was raised in [32]. It turns out that these means are exactly the matrix power means. We prove also that the corresponding affine connections are
∇ Xp Yp = DY [p][Xp] − κ 2 Xpp −1 Yp + Ypp −1 Xp ,
where 0 ≤ κ ≤ 2 and the tangent space is H(n, C), the space of n-by-n hermitian matrices, at every point p ∈ P(n, C). These connections appear earlier when we construct them as prototypes of invariant affine connections in Section 6. In Section 9 among other results we show that these affine connections are non-metric in general, i.e. there exist no other Riemannian structures as in the case of the Karcher mean (1). In order to achieve this we investigate the holonomy groups and other properties of these affine connections.
During the above process we give some geometric construction which can be used over a general affinely connected space to reconstruct the logarithm (hence also the exponential) map of the corresponding affine connection from the midpoint operation m(p, q) = exp p (1/2 log p (q)) on the manifold as log p (q) = lim n→∞ m(p, q) •n − p 1 2 n where we use the notation m(p, q) •n ≡ m p, m(p, q) •(n−1) and log p (q) is the logarithm map. This is dicussed in Section 3. We apply an analogue of such a process to arbitrary two-variable matrix means in Section 4 and we obtain a corresponding logarithm map log A (B) = A 1/2 log I A −1/2 BA −1/2 A 1/2 to the matrix mean, where log I (x) is an operator monotone function. We show that log A (B), hence log I (x) directly induce a one parameter family of operator monotone functions which represent matrix means that can be thought of the weighted counterparts of the original mean. Then in Section 5 we show that taking directly such operator monotone functions log I (x) that can be prototype of logarithm maps, we obtain similar one parameter families of matrix means. We show that in both cases these families are Loewner semigroups of Pick functions which itself has a classical and rich theory [12,13,17,18,19,20]. We also show that the further extendability to greater parameter values of the one parameter family depends on the distribution of the branch points of the corresponding logarithm map. We relate this extendability property to functional equations over the upper complex half-plane of the form log I (f t (z)) = t log I (z),
where f t (z) is the representing operator monotone function of the matrix mean and log I (z) is the corresponding logarithm map. We show that if log I (z) has no branch points in the uper half-plane, then the functional equation, hence the the one parameter family f t (z) is a Pick function, i.e. an operator monotone function for all t ∈ [0, 1].
Matrix means and some constructions
Let us recall the family of matrix means [24]:
Definition 1 A two-variable function M : P(n, C) × P(n, C) → P(n, C) is called a matrix mean if In property (ii), (iii), (iv) the partial order being used is the positive definite order, i.e. A ≤ B if and only if B − A is positive semidefinite. An important consequence of these properties is [24] that every matrix mean can be uniquely represented by a normalized, operator monotone function f (t) in the following form
M (A, B) = A 1/2 f A −1/2 BA −1/2 A 1/2 .(4)
This unique f (t) is said to be the representing function of the matrix mean M (A, B). So actually matrix means are in one to one correspondence with normalized operator monotone functions, the above characterization provides an orderisomorphism between them. Normalization means that f (1) = 1. For symmetric means, i.e. for means M (A, B) = M (B, A), we have f (t) = tf (1/t) which implies that f ′ (1) = 1/2. Operator monotone functions have strong continuity properties, namely all of them are analytic functions and can be analytically continued to the upper complex half-plane. This is the consequence of the integral characterization of an operator monotone function f (t), which is given over the interval (0, ∞):
f (t) = α + βt + ∞ 0 λ λ 2 + 1 − 1 λ + t dµ(λ),(5)
where α is a real number, β ≥ 0 and µ is a unique positive measure on (0, ∞) such that
∞ 0 1 λ 2 + 1 dµ(λ) < ∞.(6)
Actually the interval (0, ∞) may be changed to an arbitrary (a, b), in this case the integral is transformed to this interval accordingly. These are the consequences of the theory of Loewner, an introduction to the theory can be found in Chapter V [5]. We will use this integral characterization at several points in the article in order to show that certain functions are analytic.
In this article we are interested in finding all possible symmetric matrix means which are also geodesic midpoint operations on smooth manifolds. Or more generally those matrix means that are arbitrary dividing points of geodesics. We will call such a matrix mean affine [32]: Definition 2 (Affine matrix mean) An affine matrix mean M : W 2 → W is a matrix mean which is also a point of an arc-length parametrized geodesic on a smooth manifold W ⊇ P(n, C) equipped with an affine connection ∇. I.e. M (A, B) = exp A (t log A (B)) for a fixed t ∈ (0, 1) and for all A, B ∈ P(n, C), B is assumed to be in the injectivity radius of the exponential map exp A (x) of the connection ∇ given at the point A. The mapping log A (x) is just the inverse of the exponential map at the point A ∈ W .
We can make some basic observations about affine matrix means. First of all note, that by (4) we have that f (X) = M (I, X), so if M (A, B) is an affine matrix mean, then f (X) is some point of a geodesic connecting X and I. Also on a smooth manifold with an affine connection if we differentiate the exponential map exp p (X) at p, then we get d exp p = Ip, where Ip is the identity transformation of the tangent space at p [15]. Therefore if we differentiate its inverse, the logarithm map log p (q) we also get d log p = Ip at p. So if we combine this with the chain rule we get that the differential of the mapping M (p, q) = exp p (t log p (q)) is dM (p, ·) = tIp. Now if we apply the above argument to an affine matrix mean M (A, B) we get the following result.
Proposition 1 Let M (A, B) := exp A (t log A (B)) be an affine matrix mean. Then f ′ (1) = t.
Proof Since P(n, C) is diffeomorphically embedded in H(n, C), therefore we can differentiate the map M (A, B) = exp A (t log A (B)) using the vector space structure of H(n, C), i.e. calculate the Fréchet differential which we denote for an arbitrary differentiable function g by
Dg[X][Y ] = lim s→0 g(X + Y s) − g(X) s (7)
at the matrix X in the direction of the matrix Y . So by (4)
for all H ∈ H(n, C) we have lim s→0 M (A, A + Hs) − M (A, A) s = lim s→0 A 1/2 M (I, I + A −1/2 HA −1/2 s) − M (I, I) s A 1/2 = = A 1/2 lim s→0 f (I + A −1/2 HA −1/2 s) − f (I) s A 1/2 = A 1/2 Df [I][A −1/2 HA −1/2 ]A 1/2 .
Since f is an operator monotone function on (0, ∞), it admits an integral characterization (5), so it can be analytically continued to the upper half-plane through the interval (0, ∞). Therefore we may differentiate a power series representation of f , that uniformly converges on an open interval which contains 1, so then we get that Df [I][K] = Df [I][I]K = f ′ (1)K for all K ∈ H(n, C). Combining this with the above we get that
lim s→0 M (A, A + Hs) − M (A, A) s = A 1/2 Df [I][I]A −1/2 HA −1/2 A 1/2 = f ′ (1)H.
Since H was arbitrary this yields that t = f ′ (1), because dM (p, ·) = tIp and also the tangent space of P(n, C) at every point can be indentified by H(n, C).
By the preceding proposition we shall focus on matrix means represented by operator monotone functions f on (0, ∞) such that f ′ (1) ∈ (0, 1). We will use the notation P(t) to denote the set of all operator monotone functions f on (0, ∞) such that f (x) > 0 for all x ∈ (0, ∞) and f (1) = 1, f ′ (1) = t. We can find the minimal and maximal elements of P(t) for all t ∈ (0, 1) easily.
Lemma 1 For all f (x) ∈ P(t) we have (1 − t) + tx −1 −1 ≤ f (x) ≤ (1 − t) + tx.(8)
Proof Since every operator monotone function is operator concave, see Chapter V [5], therefore we must have f (x) ≤ (1 − t) + tx by concavity and the normalization conditions on elements of P(t). Since the map x −1 is order reversing on positive matrices, we have that if f (x) ∈ P(t) then also f (x −1 ) −1 ∈ P(t). So again by concavity
f (x −1 ) −1 ≤ (1 − t) + tx f (x −1 ) ≥ ((1 − t) + tx) −1 f (x) ≥ (1 − t) + tx −1 −1 .
Since (1 − t) + tx −1 −1 and (1 − t) + tx are operator monotone we see that the they are the minimal and maximal elements of P(t) respectively, and also they are the representing functions of the weighted harmonic and arithmetic means. This already gives us that the minimal and maximal affine matrix means are the weighted harmonic and arithmetic means respectively, so if M (A, B) is an affine matrix mean, then
(1 − t)A −1 + tB −1 −1 ≤ M (A, B) ≤ (1 − t)A + tB.(9)
In general by the previous Lemma 1 the above inequality is true for all M (A, B) matrix means with representing operator monotone function f for which we have f ′ (1) = t. In this sense P(t) characterizes weighted matrix means. If we take this as the definition of weighted matrix means, one can compare it with the definition of weighted matrix means given in [32].
Lemma 2 All f (x) ∈ P(t) for t ∈ (0, 1) has only one fixed point in (0, ∞) which is 1 and 1 is an attractive fixed point on (0, ∞).
at the fixed point 1, so by Banach's fixed point theorem this fixed point is attractive on (ǫ, ∞), where ǫ < 1 is such that the derivative f ′ (ǫ) = 1. On (0, ǫ) the function (1 − t) + tx ≥ f (x) > x so its subsequent iterates form an increasing sequence of functions. I.e. if we start an iteration with x 0 ∈ (0, ǫ), then after finitely many iterations by f (x), xn = f (x n−1 ) will be in the interval (ǫ, 1). From there convergence to 1 follows again from Banach's fixed point theorem.
In order to advance further in the understanding of affine matrix means, we should be able to grasp more geometrical structure related to the affinely connected manifolds corresponding to affine matrix means. In the next section we will study the general situation of affinely connected manifolds given with a geodesic dividing point operation. We will see that in this case we can reconstruct the exponential map and its inverse, the logarithm map from the geodesic dividing point operation.
The reconstruction of the exponential map
In this section we reconstruct the exponential map of an arbitrary affinely connected differentiable manifold based first on its midpoint map. Without loss of generality we fix a base point p as the starting point of the geodesics. The basics of the exponential map of a manifold can be found for example in Chapter I. paragraph 6 [15].
Theorem 1 Let M be an affinely connected smooth manifold diffeomorphically embedded into a vector space V . Suppose that the midpoint map m(p, q) = exp p (1/2 log p (q)) is known in every normal neighborhood where the exponential map exp p (X) is a diffeomorphism. Then in these normal neighborhoods the inverse of the exponential map log p (q) can be fully reconstructed from the midpoint map in the form
log p (q) = lim n→∞ m(p, q) •n − p 1 2 n ,(10)
where we use the notation m(p, q) •n ≡ m p, m(p, q) •(n−1) . Proof We will use some basic properties of the differential of the exponential map to construct the inverse of it, the logarithm map. Since in small enough normal neighborhoods the exponential map is a diffeomorphism, it can be given as the inverse of the logarithm map log p (q).
By the basic properties of the exponential map we have
∂ exp p (Xt) ∂t t=0 = lim t→0 exp p (Xt) − p t = X,
where X ∈ TpM . Here we used the fact that we have an embedding into a vector space. Suppose exp p (X) = q is in the normal neighborhood. We are going to provide the limit on the right hand side of the above equation. The limit clearly exists in the normal neighborhood so
lim t→0 exp p (Xt) − p t = lim n→∞ exp p X 1 2 n − p 1 2 n = lim n→∞ m(p, q) •n − p 1 2 n .
Here we use the notation m(p, q) •n ≡ m p, m(p, q) •(n−1) . We are in a normal neighborhood so the exponential map has an inverse, the logarithm map, so the limit can be written as
X = lim t→0 exp p (Xt) − p t = lim n→∞ m(p, q) •n − p 1 2 n = log p (q).
In the above assertion we used the midpoint map to reconstruct the exponential map, but we can use arbitrary dividing point operation that yields a point, other then the ending points on the geodesic connecting two points in the normal neighborhood. This is summarized in the following proposition.
Proposition 2 Let M be an affinely connected smooth manifold diffeomorphically embedded into a vector space V . In every normal neighborhood N let γ a,b (t) denote the geodesic connecting a, b ∈ N with parametrization γ a,b (0) = a and γ a,b (1) = b. Suppose that the map m(a, b) t0 = γ a,b (t 0 ) = exp p (t 0 log p (q)) is known for a t 0 ∈ (0, 1) in every normal neighborhood N where the exponential map is a diffeomorphism and a, b ∈ N . Then in these normal neighborhoods the logarithm map can be fully reconstructed as log p (q) = lim n→∞ m(p, q) •n t0 − p t n 0 , with the notation m(p, q) •n t0 ≡ m p, m(p, q)
•(n−1) t0 t0
. We also obtain the exponential map by inverting log p (q).
We are going to use this construction in the next sections to characterize affine matrix means. 4 The exponential map of affine matrix means Based on the idea of reconstruction given by Proposition 2 we are going to formally take the limits for matrix means in P(t). The following result will show that if a matrix mean is affine then the exponential map of the corresponding smooth manifold has a special structure. We will use similarly the notation M (A, B) •n = M A, M (A, B) •(n−1) as before in the previous section.
lim n→∞ M (A, B) •n − A f ′ (1) n = A 1/2 log I A −1/2 BA −1/2 A 1/2(11)
where the limit exists and is uniform for all A, B ∈ P(n, C) and log I (x) is an operator monotone function which fulfills the functional equation
log I (f(x)) = f ′ (1) log I (x)(12)
on the interval (0, ∞).
Proof We will prove the convergence to a continuous function log I (t) in a more general setting. The operator monotonicity in the matrix mean case will be a particularization.
First of all note that by the repeated usage of (4) we can reduce the above problem to the right hand side of the following formula:
M (A, B) •n − A f ′ (1) n = A 1/2 f A −1/2 BA −1/2 •n − I f ′ (1) n A 1/2 .
From now on we will explicitly use the notation g(x) •n = g g(x) •(n−1) for arbitrary function g(t) where this notation is straightforward. Due to the above formula it is enough to prove the assertion for a single operator monotone function f (x). By operator monotonicity of f (x) this is just the special case of the problem considered for arbitrary concave, analytic functions f (x) given in the following form
lim n→∞ f (X) •n − I f ′ (1) n ,(13)
for X ∈ P(n, C). As every operator monotone function which maps (0, ∞) to (0, ∞), is analytic on (0, ∞) and has an analytic continuation to the complex upper halfplane across (0, ∞), we can consider the functional calculus for hermitian matrices in the above equations. Therefore we can further reduce the problem to the set of the positive reals by diagonalizing X and considering the convergence for every distinct diagonal element separately. For an extensive study on operator monotone functions one may refer to Chapter V in [5].
Without loss of generality we may shift the function f (x) by 1 so it is enough to show the assertion for lim
n→∞ g(x) •n g ′ (0) n , where g(x) = f (x + 1) − 1 and so g(x) •n = f (x + 1) •n − 1.
From now on we will be considering the shifted problem. At this point we must emphasize the fact that the function g must have 0 as an attractive and only fixed point on the interval of interest (−1, ∞). In the unshifted case this is equivalent to f having 1 as the only attractive fixed point on the interval (0, ∞), which is the case by Lemma 2. So we can also assume that 0 < g ′ (0) < 1. The rest of the argument will be based on the claim that the above limit of analytic functions of the form g(x) •n /g ′ (0) n is uniform Cauchy therefore the limit function exists and is continuous. First of all we have 0 as the attractive and only fixed point of g, so for arbitrary x ∈ (−1, ∞) the sequence xn = g(x) •n converges to 0. We have g(0) = 0 and by the mean value theorem we have
xn = g(x) •n = g ′ (tn)g(x) •(n−1) = n i=1 g ′ (t i )x, where t i ∈ 0, g(x) •(i−1) if x ≥ 0 or t i ∈ g(x) •(i−1) , 0 if x < 0, since g is a concave function on (−1, ∞). As xn → 0 for arbitrary x we have g ′ (t i ) → g ′ (0). Now we have to obtain a suitable upper bound on g(x) •n g ′ (0) n − g(x) •m g ′ (0) m .(14)
We argue as follows (14) can be arbitrarily small on any closed interval in (−1, ∞) by choosing a uniform m. By the continuity of g ′ (t) and xn → 0 we have g ′ (t i ) → g ′ (0) and by assumption 0 < g ′ (0) < 1, therefore there exists N and q such that for all i > N we have 0 < g ′ (t i ) ≤ q < 1. What follows here is that
g(x) •n g ′ (0) n − g(x) •m g ′ (0) m = g(x) •n − g ′ (0) n−m g(x) •m g ′ (0) n ≤ ≤ n i=m+1 g ′ (t i ) − g ′ (0) n−m m i=1 g ′ (t i ) g ′ (0) n |x| = = n i=m+1 g ′ (t i ) g ′ (0) − 1 m i=1 g ′ (t i ) g ′ (0) |x|. Now uniform convergence follows if ∞ i=1 g ′ (t i )/g ′ (0) < ∞ because then the tail ∞ i=m+1 g ′ (t i )/g ′ (0) → 1 so∃K 1 , K 2 < ∞ such that |t N | ≤ K 1 and |g ′′ (t i )| ≤ K 2 for all i > N . This yields the bound |t i | ≤ K 1 q i−N for all i > N . Considering the Taylor expansion of g ′ around 0 we get g ′ (t i ) g ′ (0) = g ′ (0) + g ′′ (t ′ i )t i g ′ (0) for 0 < t ′ i < t i . What follows from this is that ∞ i=N g ′ (t i ) g ′ (0) ≤ ∞ i=N 1 + K 1 K 2 g ′ (0) q i−N .
The infinite product on the right hand side converges because ∞ j=0 K1K2
g ′ (0) q j con- verges hence ∞ i=1 g ′ (t i )/g ′ (0) < ∞ for all x in the closed interval.
At this point we can easily establish the convergence for normalized operator monotone functions because they are concave functions by Theorem V.2.5 in [5], so f ′′ (t) ≤ 0 and they have only one fixed point which is 1. The fact that the limit is operator monotone function in this case follows from the operator monotonicity of the generating f (t).
The functional equation (12) is the consequence of the following:
log I (f(x)) = lim n→∞ f (f(x)) •n − 1 f ′ (1) n = lim n→∞ f (x) •(n+1) − 1 f ′ (1) n = lim n→∞ f ′ (1) f (x) •(n+1) − 1 f ′ (1) n+1 = f ′ (1) log I (x).
Actually the above proof works for a larger class of functions then the family of normalized operator monotone functions. The limit in (13) exists and it is a continuous function if the twicely differentiable function f (x) has 1 as the only attractive fixed point and the derivative −1 < f ′ (x) < 1. The next example shows how to calculate the limit function explicitly.
Example 1 Consider the one-parameter family of functions
fq(x) = [(1 − t) + tx q ] 1/q
for t ∈ (0, 1). These are in P(t) if and only if q ∈ [−1, 1], because for other values of q the function is not operator monotone, see exercise 4.5.11 in [6]. It is easy to see that
fq(x) •n = t n x q + n−1 k=0 t k (1 − t) 1/q = [t n x q − t n + 1] 1/q .
In this case we can easily calculate the limit function log I,fq (x) by turning the limit into a derivative:
log I,fq (x) = lim n→∞ (t n x q − t n + 1) 1/q − 1 t n = lim s→0 (sx q − s + 1) 1/q − 1 s = ∂ ∂s (sx q − s + 1) 1/q s=0 = x q − 1 q .
The limit functions indeed are operator monotone again if and only if q ∈ [−1, 1]. This family has a singularity at q = 0 but it is easy to verify that it is a removable singularity, so in fact we have
f 0 (x) = x t log I,f0 (x) = log(x),
where log(x) and x t are also well known to be operator monotone. Particularly x t as a representing function corresponds to the weighted geometric mean.
Proposition 3
The limit function log I (x) in Theorem 2 satisfies the following:
(i) log I (x) maps P(n, C) to H(n, C) injectively, (ii) 1 − x −1 ≤ log I (x) ≤ x − 1 for all x > 0, (iii)
If log I,f (x) and log I,g (x) are the corresponding limit functions for f, g ∈ P(t) such that f (x) ≤ g(x) for all x > 0, then log I,f (x) ≤ log I,g (x) for all x > 0, (iv) log I (1) = 0 and log ′ I (1) = 1.
Proof (iii): Since f (x) ≤ g(x) by monotonicity we have f (x) •n ≤ g(x) •n . From this it follows that f (x) •n − 1 f ′ (1) n ≤ g(x) •n − 1 g ′ (1) n ,
and the inequality is also preserved in the limit. (ii): By Lemma 1 we have
(1 − t) + tx −1 −1 ≤ f (x) ≤ (1 − t) + tx
where on the left hand side we have the function f −1 (x) and on the right hand side we have f 1 (x) from Example 1. In Example 1 we calculated the corresponding limit functions, so these combined with the previous property (iii) proves property (ii). (i): By property (ii) it follows that log I (x) is nonconstant on (0, ∞). Also log I (x) is operator monotone there, so it is strictly concave, therefore injective and real valued. This combined with the functional calculus for matrix functions proves the property.
(iv): log I (1) = 0 follows from (ii). Using this and (ii) again we have
1 − (1 + h) −1 h ≤ log I (1 + h) − log I (1) h ≤ (1 + h) − 1 h .
Taking the limit h → 0 we get derivatives on the left and right hand sides are 1, so also log ′ I (1) = 1.
Since log I (x) is operator monotone on (0, ∞), it is also analytic there, so it has an analytic inverse exp I (x) by Lagrange's inversion theorem, since its derivative is nonzero due to Proposition 3. It is also easy to see that exp ′ I (0) = 1 and exp I (0) = 1. By these considerations we have just arrived at the following
Proposition 4 Let f ∈ P(t). Then f (x) = exp I f ′ (1) log I (x) ,(15)
where log I (x) is the unique solution of the functional equation (12) in the class of functions that are continuous, invertible on (0, ∞), vanish at 1 and have derivative 1 at 1.
Proof The first part of the assertion follows from the analyticity and invertibility of log I (x) on (0, ∞). For the second part note that if log I (x) is an invertible solution of the functional equation (12) and also log I (1) = 0 and log ′ I (1) = 1, then its inverse exp I (x) exists, exp I (0) = 1 and exp ′
I (0) = 1. Moreover lim n→∞ f (x) •n − 1 f ′ (1) n = lim n→∞ exp I (f ′ (1) n log I (x)) − exp I (0) f ′ (1) n = lim s→0 exp I (s log I (x)) − exp I (0) s = ∂ ∂s exp I (s log I (x)) s=0 = log I (x).
The above propositions put some restrictions on the possible functions log I (x) that can occur as limits in Theorem 2. Therefore we will use the notation L to denote the set of operator monotone functions g(x) on (0, ∞) such that g(1) = 0 and g ′ (1) = 1. By Proposition 4 it is clear, that for each f ∈ P(t) we have a unique corresponding log I (x) in L. We will further say that for an arbitrary f ∈ P(t) the corresponding solution log I (x) of the functional equation (12) is the logarithm map corresponding to f (x), while its inverse exp I (x) is the exponential map corresponding to f (x).
In the following section we will go the other way around and see whether the function
f (x) = exp I (t log I (x))
is in P(t) for all log I ∈ L and t ∈ (0, 1).
Representing functions induced by logarithm maps
In the previous section we established that for every f ∈ P(t) there exists a unique function log I ∈ L such that it fulfills the functional equation (12). In order to see whether an element log I ∈ L also induces a representing function f ∈ P(t) with the generalized functional equation
log I (f(x)) = t log I (x)
for all t ∈ (0, 1), we must extend our investigations into the upper complex half-
plane H + = {z ∈ C : ℑz > 0}.
First of all let us recall Nevanlinna's representation [5] of holomorphic functions f : H + → H + . By Nevanlinna's theorem each such f can uniquely be written as
f (z) = α + βz + ∞ −∞ λz + 1 λ − z dν(λ),(16)
where α ∈ R, β ≥ 0 and ν is a positive measure with support in (−∞, ∞). It is well known that f can be extended to the lower half-plane H − = {z ∈ C : ℑz < 0} as well by Schwarz reflection f (z) for all z ∈ H − . Therefore also if this extension is by analytic continuation over an interval (a, b), then ν vanishes on the interval [5].
Similarly if ν vanishes on a real interval, then f is holomorphic on the interval as well and can be analyticly continued to the lower half-plane. Representation (16) will be useful for studying functions in L. For example Nevanlinna's representation yields that all f ∈ L can be represented as
f (z) = α + βz + 0 −∞ λz + 1 λ − z dν(λ),(17)
where α ∈ R, β ≥ 0 and ν is a positive measure with support in (−∞, 0). This is due to the required holomorphicity of f on (0, ∞). Next let us find the maximal and minimal elements in L.
Lemma 3 For all f (x) ∈ L we have 1 − x −1 ≤ f (x) ≤ x − 1.(18)
Proof Since every operator monotone function is operator concave, therefore we must have f (x) ≤ x − 1 by concavity and the normalization conditions on elements of L. Since the map x −1 and −x is order reversing on hermitian matrices, we have
that if f (x) ∈ L then also −f (x −1 ) ∈ L. So again by concavity −f (x −1 ) ≤ x − 1 f (x) ≥ 1 − x −1 . Clearly x − 1 and 1 − x −1 are also in L.
At this point let us refer again to the functional equation (12) in the previous section. By the above considerations we can generalize (12) by analytic continuation.
Proposition 5 Let f ∈ P(t). Then the function log I (x) given in Theorem 2 admits analytic continuation to H + and also to H − across (0, ∞) by relfection, moreover it fulfills the functional equation
log I (f(z)) = f ′ (1) log I (z) (19) for all z ∈ C\(−∞, 0].
Proof Since analytic continuation of f (x) and log I (x) can be performed using the integral characterizations (5) and (17) respectively, we end up with holomorphic functions living on C\(−∞, 0]. Since log I (z) is a holomorphic function, it has a meromorphic inverse exp I (z). So we have
F (z) = exp I f ′ (1) log I (z) ,
a meromorphic function that is identical to f (z) everywhere on the domain (0, ∞). Therefore by uniqueness of meromorphic and analytic continuation we must have
F (z) = f (z) everywhere on the domain C\(−∞, 0].
The above result tells us, that for a given log I ∈ L we should consider the generalized functional equation
log I (f t (z)) = t log I (z)(20)
to define a representing function f t (z) for all t ∈ (0, 1) corresponding to log I (z) which was itself obtained by analyitc continuation using representation (17). The obvious question that arizes here is whether every log I (z) in L has a corresponding f t ∈ P(t)? We need the following:
Definition 3 (Radial convexity) Let S ⊆ C be such that 0 ∈ S. We will say that S is radially convex if and only if for all z ∈ S also tz ∈ S for all t ∈ [0, 1].
Proposition 6 Let log I ∈ L. Then log I maps H + to a radially convex set in H + .
Proof Using Nevanlinna's characterization (17) we have that log I is a convex combination of functions of the form
h λ (z) = λz + 1 λ − z .
If we have h λ (z) = w for z ∈ H + , then after some calculations we get that
z = λ − (λ 2 + 1) λ + w |λ + w| 2 which means that ℑz = λ 2 + 1 |λ + w| 2 ℑw.
In other words if h λ (z) = w ∈ H + , then for all s ∈ (0, 1) there exists zs ∈ H + such that h λ (zs) = sw. Now if we consider any convex combination of such functions h λ (z), the resulting function will still have a radially convex image of H + . The reason for this is that if x i ∈ S i ⊆ H + where S i are radially convex sets, then sx i ∈ S i for all s ∈ (0, 1). Therefore if S i are the images of H + under the mappings K i h λi (z) for some K i > 0 and λ i , then the image S of H + under the function that we get as the sum of the functions K i h λi (z), is radially convex, since every element of it can be written as a sum of some x i ∈ S i . So we also have that the sum of sx i is in S too by the convexity of each S i . Therefore S must be radially convex.
Theorem 3 Let log I ∈ L. Then f t ∈ P(t) for all t ∈ (0, 1) if and only if log I (z) has no branch point in H + .
Proof First of all since log I ∈ L, it follows that log I (x) is invertible on (0, ∞) because it is nonconstant monotone increasing there, also it is invertible in a neighborhood of (0, ∞) and its inverse exp I (z) is holomorphic in that neighborhood and f t (x) ∈ (0, ∞) and meromorphic in H + .
Suppose that log I (z) has no branch point in H + . Then by the previous Proposition 6 it maps H + to a radially convex set. Since log I (z) has no branch point, it has a univalent holomorphic inverse exp I (z), so
f t (z) = exp I (t log I (z))
is a well defined holomorphic function on H + . Moreover f t (z) is real valued over (0, ∞). Since the image log I (H + ) of H + under the map log I (z) is radially convex, we have that for any s ∈ log I (H + ) also ts ∈ log I (H + ). Therefore t log I (H + ) ⊆ log I (H + ), so also exp I (t log I (H + )) ⊆ H + . Now for the only if part suppose on the contrary that log I has a branch point in H + . Then its inverse exp I has a pole at the image of the branch point under log I which means that f t (z) is not holomorphic there, but this contradicts f t ∈ P(t).
What happens if log I has a branch point in H + ? What can then be said about f t (z)?
Proposition 7 Let log I ∈ L be induced by an f t0 ∈ P(t 0 ) using Proposition 4. Then f t ∈ P(t) for all 0 < t ≤ t 0 .
Proof By Proposition 4 we have that there is no image of a branch point of log I in the domain t 0 log I (H + ) ⊆ H + , otherwise f t0 (z) would have a singularity in H + . But since for all 0 < t ≤ t 0 we have that t log I (H + ) ⊆ t 0 log I (H + ), therefore f t (z) is singularity free as well. Remark 1 In general one can assure that if for a given log I ∈ L with branch points t log I (H + ) avoids the image of the branch points (of log I ) under log I in H + , then f t ∈ P(t).
According to Theorem 3 we need to find members of L without branch points. In other words we are looking for mappings that are univalent (schlicht) holomorphic functions on H + mapping H + into itself. Such mappings are characterized by FitzGerald in the classical article [12]. (a, b). Suppose the origin is in (a, b) and f (0) = 0. A necessary and sufficient condition that f can be continued to be a univalent analytic function of H + onto a subset of itself that is radially convex with respect to the origin is that the function Note that by Weierstrass's approximation theorem we also have
η(x) = − f (x) f ′ (x) be conditionally positive definite, i.e. b a b a φ(s) η(s) − η(t) s − t φ(t)dsdtp 1/2 exp I p −1/2 Xp −1/2 p 1/2 = p exp I p −1 X p 1/2 log I p −1/2 Xp −1/2 p 1/2 = p log I p −1 X .(22)
In some cases, to ensure easier reading, similarly as in the above formulas, we will denote matrices with uppercase letters which are elements of some tangent space, while at the same time we will use lowercase letters for denoting matrices which are points of a differentiable manifold.
Construction of an invariant affine connection
Let us recall the classical symmetric space GL(n, C)/U(n, C), the cone of positive definite n × n matrices P(n, C) [9]. This is a Lie group and the K = U(n, C) isotropy group invariant inner product at the identity I is U, V = T r {U V }. The tangent space, considering the Cartan decomposition of the Lie algebra, is the space of Hermitian matrices H(n, C). The action of the isometry group GL(n, C) on this manifold is g(o) = gog * and acting with left translations we can transport the inner product to any point p on this manifold and we get the Riemannian metric U, V p = T r p −1 U p −1 V . The exponential map is just the ordinary matrix exponential at the identity. The left invariant affine connection is
∇ Xp Yp = DY [p][Xp] − 1 2 Xpp −1 Yp + Ypp −1 Xp ,(23)
here DY [p][Xp] denotes the Fréchet-differential of Y at the point p in the direction Xp. A well known property of this metric is that the midpoint map of the space m(p, q) = expp(1/2logp(q)) is just the geometric mean of two positive matrices
G(A, B) = A 1/2 A −1/2 BA −1/2 1/2 A 1/2 .(24)
The question that can be asked at this point is that are there other symmetric matrix means which correspond to symmetric spaces as midpoint maps on P(n, C)?
Two other examples are known, these are the arithmetic mean (A + B)/2 and the harmonic mean 2(A −1 +B −1 ) −1 . The symmetric spaces corresponding to these two means are Euclidean while the symmetric space corresponding to the geometric mean has nonpositive curvature. It has flat and negatively curved de Rham factors.
At this point we begin with the characterization of means that correspond to affine symmetric spaces in general. What we know at this point is that the two functions, which are of each others inverse, log I (t) and exp I (t) exist for all matrix means, as it was proved in Theorem 2.
In [14] and [15] there is an extensive study of affine connections on manifolds. A well known fact is that the affine connection on a manifold can be reconstructed by differentiating the parallel transport:
∇ Xp Yp = lim t→0 Γ 0 t (γ)Y γ(t) − Y γ(0) t ,
where γ(t) denotes an arbitrary smooth curve emanating from p in the direction Xp = ∂γ(t)/∂t| t=0 and Γ s t (γ)Y denotes the parallel transport of the vector field Y along the curve γ from γ(t) to γ(s), refer to [14,15]. The above limit does not depend on the curve itself, only on its initial direction vector and it depends on the vector field Y in an open neighborhood of p. On affine symmetric spaces the parallel transport from one point to another along the connecting geodesic is given by the differential of the geodesic symmetries with a negative sign. The geodesic symmetry is given as Sp(q) = exp p (− log p (q)).
On affine symmetric spaces this map is an affine transformation so one can conclude that
Γ 0 1 (γ)Y = − ∂S γ(1/2) (exp q (Y t)) ∂t t=0 ,(25)
where γ(t) is a geodesic connecting p = γ(0) and q = γ(1). We have already proved the following formulas for the exponential and logarithm maps at the end of the preceding section exp p (X) = p 1/2 exp I p −1/2 Xp −1/2 p 1/2 = p exp I p −1 X log p (X) = p 1/2 log I p −1/2 Xp −1/2 p 1/2 = p log I p −1 X .
The above identities already specify the geodesic symmetries with the notation S I (X) = exp I (− log I (X)) as Sp(q) = exp p (− log p (q)) = p 1/2 S I p −1/2 qp −1/2 p 1/2 = pS I p −1 q .
Now we are in position to prove the following Theorem 5 Let P(n, C) be subset of an affine symmetric space with affine geodesic symmetries given as (27). Then the invariant affine connection has the form
∇ Xp Yp = DY [p][Xp] − κ 2 Xpp −1 Yp + Ypp −1 Xp ,(28)
where κ = S ′′ I (1)/2.
Proof We are going to use (25) to obtain the connection (28). We make the assumption that the geodesic symmetries are of the form (27). The functions exp p (X) and log p (X) are of the form (26), where exp I (t) and log I (t) are analytic functions on a disk centered around 0 and 1 respectively. We also have that log I (1) = 0, exp I (0) = 1 and furthermore
∂ exp I (t) ∂t t=0 = 1.
First of all we have to differentiate the map Sp(q) given in (27) to obtain
Γ 0 1 (γ)Y = Tq→pY , where γ(t)
is a geodesic connecting p = γ(0) and q = γ(1).
∂Sp(exp q (Y t)) ∂t t=0 = ∂pS I (p −1 exp q (Y t)) ∂t t=0 = = pDS I p −1 q p −1 Y(29)
We used the fact that ∂ exp q (Y t)/∂t| t=0 = Y which is a consequence of exp ′ I (0) = 1. Now we are going to differentiate the parallel transport as given by (25) to get back the connection. We use the holomorphic functional calculus to express the Fréchet-differential in (29) as
DS I [X][U] = 1 2πi g S I (z)[zI − X] −1 U [zI − X] −1 dz.
It also easy to see that DS I [I][I] = S ′ I (1) = −1, so we may express the limit (25) by the following differential
∇ γ ′ (0) Y γ(0) = − ∂γ(t/2)DS I γ(t/2) −1 γ(t) γ(t/2) −1 Y γ(t) ∂t t=0 =
we massage this further by using the holomorphic functional calculus
= − ∂ ∂t γ(t/2) 1 2πi g S I (z)[zI − γ(t/2) −1 γ(t)] −1 γ(t/2) −1 Y γ(t) × [zI − γ(t/2) −1 γ(t)] −1 dz t=0 = − 1 2 γ ′ (0)γ(0) −1 Y γ(0) DS I [I][I]− − γ(0) 1 2πi g S I (z) [zI − I] −1 1 2 γ(0) −1 γ ′ (0)[zI − I] −1 γ(0) −1 Y γ(0) [zI − I] −1 + + [zI − I] −1 γ(0) −1 Y γ(0) [zI − I] −1 1 2 γ(0) −1 γ ′ (0)[zI − I] −1 + + [zI − I] −1 −γ(0) −1 1 2 γ ′ (0)γ(0) −1 Y γ(0) + γ(0) −1 DY [γ(0)][γ ′ (0)] × [zI − I]= − DS I [I][I] 2 γ ′ (0)γ(0) −1 Y γ(0) − − γ(0) 1 2πi g S I (z)dz (z − 1) 3 1 2 γ(0) −1 γ ′ (0)γ(0) −1 Y γ(0) − − γ(0) 1 2πi g S I (z)dz (z − 1) 3 1 2 γ(0) −1 Y γ(0) γ ′ (0)γ(0) −1 − − γ(0) 1 2πi g S I (z)dz (z − 1) 2 − 1 2 γ(0) −1 γ ′ (0)γ(0) −1 Y γ(0) + γ(0) −1 DY [γ(0)][γ ′ (0)]
at this point we use the integral representation S (j)
I (1) = j! 2πi g S I (z) (z − 1) j+1 dz
to further simplify the above.
∇ γ ′ (0) Y γ(0) = − S ′′ I (1) 4 γ ′ (0)γ(0) −1 Y γ(0) + Y γ(0) γ(0) −1 γ ′ (0) − − S ′ I (1) 2 γ ′ (0)γ(0) −1 Y γ(0) − S ′ I (1) 2 −γ ′ (0)γ(0) −1 Y γ(0) + 2DY [γ(0)][γ ′ (0)] = = −S ′ I (1)DY [γ(0)][γ ′ (0)] − S ′′ I (1) 4 γ ′ (0)γ(0) −1 Y γ(0) + Y γ(0) γ(0) −1 γ ′ (0) .
So we have that κ = S ′′ I (1)/2.
The above clearly tells us that all symmetric spaces occuring in such a way that their geodesic division maps are matrix means, have invariant affine connections in the form (28). We are going to study these connections as κ being a parameter. We will find out later for which values of κ are these spaces symmetric. Also for arbitrary real κ (28) defines an affine connection with corresponding exponential and logarithm map which are of the form (26) as we will see later. We will also determine if these connections are metric or not.
Properties of these affine connections
In this section we study the connections
∇ Xp Yp = DY [p][Xp] − κ 2 Xpp −1 Yp + Ypp −1 Xp(30)
for p ∈ P(n, C) and vector fields Xp, Yp ∈ H(n, C) on the smooth manifold P(n, C) with tangent bundle H(n, C). It is easy to see that indeed these connections are affine and analytic for real κ. We can fix a coordinate frame by taking the basis E i ∈ H(n, C), where i indices over the set of distinct hermitian matrices which have zero entries, excluding exactly the entry [E i ] kl = 1 and its transpose [E i ] lk = 1. If we equip H(n, C) with the inner product X, Y = T r {XY }, then the E i form an orthonormal basis of H(n, C). The dimension of H(n, C) is n(n+1)/2 such as the dimension of the smooth manifold P(n, C). In this coordinate frame the Christoffel symbols are given as
Γ k ij E k = − κ 2 E i p −1 E j + E j p −1 E i ,(31)
where we used the Einstein summation convention for repeated covariant and contravariant indices. Given an arbitrary connection ∇ the geodesic equations corresponding to it are given as
∇γ (t)γ (t) = 0(32)
with given initial conditions γ(0) andγ(0), for all t ∈ [0, T ). I.e. the curve γ must be parallel along itself.
(35)
Proof For the connections (30) it is easy to see that the corresponding ∇γ (t)γ (t) = 0 geodesic equations are (33).
Let us first consider the case when γ(0) = p = I =γ(0) = X. Then it is enough to solve the equation (33) for real numbers. Therefore the equation takes the form
exp ′′ I (t) = κ exp ′ I (t) 2 exp I (t) −1 .(36)
If we transform the equation to the inverse function of exp I (t) which will be the logarithm map log I (t), then we get a separable first order differential equation of the form log ′′
I (t) = −κ log ′ I (t)t −1 .
Solving the above we get the logarithm map as log I (X) =
X 1−κ −1 1−κ if κ = 1, log(X) else.
From this by inverting the above function we get the assertion for real numbers. Now we check by substitution into (33) that the curve
γ(t) = p 1/2 exp I p −1/2 Xp −1/2 t p 1/2
is also a solution of the equations (33), since the function exp I is analytic. Indeeḋ
γ(t) = Xp −1/2 exp ′ I p −1/2 Xp −1/2 t p 1/2 = p 1/2 exp ′ I p −1/2 Xp −1/2 t p −1/2 Ẍ γ(t) = Xp −1/2 exp ′′ I p −1/2 Xp −1/2 t p −1/2 X
and after substitution we get
Xp −1/2 exp ′′ I p −1/2 Xp −1/2 t p −1/2 X = κXp −1/2 exp ′ I p −1/2 Xp −1/2 t × exp I p −1/2 Xp −1/2 t −1 exp ′ I p −1/2 Xp −1/2 t p −1/2 X which is fulfilled since exp ′′ I p −1/2 Xp −1/2 t = κ exp ′ I p −1/2 Xp −1/2 t 2 exp I p −1/2 Xp −1/2 t −1
holds by the functional calculus for exp I and its derivatives and (36).
(38)
The affine matrix means which induce these affinely connected manifolds are
M t (X, Y ) = exp X (t log X (Y )) = = X 1/2 (1 − t)I + t X −1/2 Y X −1/2 1−κ 1 1−κ X 1/2 if κ = 1, X 1/2 X −1/2 Y X −1/2 t X 1/2 else,(39)
if κ ∈ [0, 2], for other values of κ the functions (39) are not matrix means.
Proof The first part of the assertion is clear, the second part follows from the fact that (39) are matrix means if and only if κ ∈ [0, 2], refer to Example 1.
Remark 2
The one-parameter family of affine matrix means (39) seems to have a singularity at κ = 1, however it is known that the singularity is removable and indeed as κ → 1 we get the matrix geometric mean as the limit. This phenomenon has already been investigated in [26]. In that paper the same one-parameter family of matrix means were considered under the name of matrix power means. If κ = 0 we get back the arithmetic mean as the midpoint operation, and the weighted arithmetic mean
A t (A, B) = (1 − t)A + tB (40)
is the geodesic line connecting A and B with respect to the metric X, Y p = T r {X * Y }. If κ = 2 we get back the harmonic mean as the midpoint operation, and the weighted harmonic mean
H t (A, B) = (1 − t)A −1 + tB −1 −1 (41)
is also a geodesic with respect to the metric X, Y p = T r p −2 Xp −2 Y . We have already mentioned that the second metric is isometric to the first one, so it is also Euclidean.
In the case when κ = 1 the midpoint is the geometric mean and the geodesics are given by the weighted geometric mean
G t (A, B) = A 1/2 A −1/2 BA −1/2 t A 1/2 .(42)
The corresponding Riemannian metric is X, Y p = T r p −1 Xp −1 Y . This manifold, which is the symmetric space GL(n, C)/U(n, C), is nonpositively curved while the other two has zero curvature.
In the paper [26] matrix power means Ps(w 1 , . . . , w k ; A 1 , . . . , A k ) are defined as the unique positive definite solution of the equations
X = k i=1 w i Gs(X, A i )(43)
where s ∈ [−1, 1], w i > 0, k i=1 w i = 1 and A i ∈ P(n, C). Existence and uniqueness of the solutions follow from the fact that the function
f (X) = k i=1 w i Gs(X, A i )
is a strict contraction for s ∈ [−1, 1], s = 0 with repsect to Thompson's part metric [26]. In the case k = 2 we get back the affine matrix means (39) with s = κ − 1 and t = w 2 .
Corollary 2 With the identification s = κ − 1, the two-variable matrix power means Ps(w 1 , w 2 ; A 1 , A 2 ) are geodesic lines, with w 2 being the arc-length parameter, of the affinely connected spaces with affine connections (30).
The arithmetic (40), harmonic (41) and geometric (42) means have nice chracterizations and extensions to several variables as the center of mass or Karcher mean of the corresponding manifolds [31,32,7,26]. I.e. Λ(w 1 , . . . , w k ; A 1 , . . . , A k ) = arg min
X∈P(n,C) n i=1 w i d 2 (X, A i ),(44)
where d(·, ·) is a Riemannian metric given as
d 2 (X, Y ) = log X (Y ), log X (Y ) X(45)
where ·, · X is one of the corresponding metrics given above for the arithmetic (40), harmonic (41) and geometric (42) means, and log X (Y ) are the corresponding logarithm maps (37) (for κ = 0, 2, 1 respectively). It is well known that in geodesically convex neighborhoods on a Riemannian manifold (44) has a unique solution [21,31]. The solution can be expressed by taking the gradient of the cost function on the right hand side of (44) [21] and then one arrives at
n i=1 w i log X (A i ) = 0.(46)
The unique solution of this equation can be expressed in closed form in the case of the arithmetic and harmonic means, since the corresponding manifolds are Euclidean. The solutions are just the multivariable weighted arithmetic k [32]. These functions are monotone in their variables with respect to the positive definite order and have some other desirable properties [26]. These two cases are well known and of less interest, however the same situation is of much more interest in the case of the geometric mean. In this case the unique solution of the minimization problem (44) cannot be expressed easily in closed form since the corresponding Riemannian manifold is no longer flat. The corresponding equation for the gradient (46) is given in the form
i=1 w i A i and harmonic means k i=1 w i A −1 i −1n i=1 w i log(X 1/2 A −1 i X 1/2 ) = 0(47)
and usually this equation is called the Karcher equation [26] and the corresponding unique positive definite solution (44) the Karcher mean. Several properties of this mean were open problems, for example its monotonicity with repsect to the positive definite order, however this and other key properties of the mean were proved by using different techniques [25,26,8]. One of the techniques given in [26] is based on the matrix power means Ps(w 1 , . . . , w k ; A 1 , . . . , A k ). These means are given as the unique positive definite solutions of (43). The following result provides a geometric characterization of matrix power means. where log X (A i ) are the logarithm maps (37) corresponding to the affine family (30) with parameter identification s = κ − 1.
Proof The defining equation (43) of matrix power means with s = 0 is equivalent to
k i=1 w i (Gs(X, A i ) − X) = 0 n i=1 w i X 1/2 X −1/2 A i X −1/2 s − I X 1/2 = 0 n i=1 w i X 1/2 X −1/2 A i X −1/2 s − I s X 1/2 = 0,
which is by (37) equivalent to
n i=1 w i log X (A i ) = 0.
The case s = 0 is just the case (47).
Remark 3 By the continuity of fixed points of pointwise continuous families of strict contractions [26], the unique solution of (48) varies continuously with resppect to the parameter s. The singularity at s = 0 is known to be removable and the limit is just the Karcher mean [26].
Now since the Karcher equations (48) admit unique positive definite solutions the obvious question arises whether there are Riemannian metrics corresponding to other values of κ? Also for what other values of κ is the manifold P(n, C) with affine connection (30) a symmetric space? The full solution of these questions requires the study of the curvature tensors and holonomy groups which is postponed to the last section. At this point we prove some other results which gets us closer to this metrizability problem of the affine connections (30). First of all we compute the parallel transport over a geodesic connecting an arbitrary point and the identity. The parallel transport of a vector Y γ(0) given in the tangent space at γ(0) with respect to the connection ∇ along the curve γ(t) is defined to be the vector field Y γ(t) which is the solution of the ODE
∇γ (t) Y = 0.
Proposition 10 Let c(t) be a geodesic with repsect to the connection (30) and c(0) = I, c(1) = p. Then the unique solution of ∇ċ (t) Y = 0 with respect to the connection (30) and the initial condition Y c(0) = Y 0 is the vector field
Y (t) = c(t) κ 2 Y 0 c(t) κ 2 . (49) Proof We have to integrate the equation ∇ c ′ (t) Y c(t) = 0. This is equivalent to DY [c(t)][c ′ (t)] − κ 2 c ′ (t)c(t) −1 Y c(t) + Y c(t) c(t) −1 c ′ (t) = 0.
We are looking for the solution Y c(t) = Y (t) in the form
Y (t) = f (c(t))Y 0 f (c(t)),
for some analytic function f (x). We have for the Fréchet-differential
DY [c(t)][c ′ (t)] = dY (t) dt = df (c(t)) dt Y 0 f (c(t)) + f (c(t))Y 0 df (c(t)) dt .
Now substituting into the equation of the parallel transport above, we get
df (c(t)) dt Y 0 f (c(t)) + f (c(t))Y 0 df (c(t)) dt = κ 2 c ′ (t)c(t) −1 Y c(t) + Y c(t) c(t) −1 c ′ (t) .
Since c(t) = exp I (t log I (p)), it has a power series expansion, as has f (x), so we have by commutativity that
κ 2 c ′ (t)c(t) −1 f (c(t)) = df (c(t)) dt = Df [c(t)][c ′ (t)] = f ′ (c)c ′ (t).
Since everything on the left and right hand side commutes with one another, we arrive at the following separable differential equation
κ 2 c −1 = f ′ (c)f(c) −1 ,
which has its solution in the form f (c) = c κ/2 .
On a Riemannian manifold the length of vectors are left invariant by parallel transports with respect to the Levi-Civita connection due to the Fundamental Theorem of Riemannian geometry [22]. It is easy to check that the connections (30) are symmetric and torsion free so any of them can possibly be a Levi-Civita connection of a Riemannian manifold. So by the above proposition we should look for the Riemannian metrics in the form
p −κ/2 Xp −κ/2 , p −κ/2 Y p −κ/2 κ (50)
for some positive definite bilinear forms ·, · κ given on the tangent space at I. In the next section we prove that all affine matrix means are actually matrix power means, i.e. we do not have to look for other connections than (30). 8 The classification of affine matrix means Due to Proposition 8 we have the exponential and logarithm map of affine matrix means in the form exp p (X) = p 1/2 exp I p −1/2 Xp −1/2 p 1/2 log p (X) = p 1/2 log I p −1/2 Xp −1/2 p 1/2 (51) for p ∈ P(n, C), where exp I (X) and log I (X) are analytic functions. The function exp I : H(n, C) → P(n, C) and log I (X) is its inverse, log ′ I (I) = I, exp ′ I (0) = I, log I (I) = 0, exp I (0) = I. Suppose that (51) represent the exponential and logarithm map of an affinely connected manifold. Then the analytic function exp I (t) is the solution of some geodesic equations
exp ′′ I (t) + Γ exp ′ I (t), exp ′ I (t), exp I (t) = 0,
where Γ (·, ·, ·) : H(n, C) × H(n, C) × P(n, C) → H(n, C) is a smooth function in all variables and linear in the first two, representing the Christoffel symbols of an affine connection. By Propostion 15 and Corollary 16 of Chapter 6 in [33] we know that connections which have the same torsion and geodesics are identical and for an arbitrary connection there is a unique connection with vanishing torsion and with the same geodesics. If we have an affine connection with non-symmetric Christoffel symbols Γ i jk , it has the same geodesics as its symmetric part
Γ i jk +Γ i kj 2
, so without loss of generality we can assume in our case that all connections are symmetric, so we will be considering mappings Γ (·, ·, ·) which are symmetric in their first two arguments.
Proposition 11 Suppose that Γ (·, ·, ·), exp I (·), exp p (·) are functions given with the above properties. Then
Γ (X, X, p) = p 1/2 Γ p −1/2 Xp −1/2 , p −1/2 Xp −1/2 , I p 1/2 (52)
for p ∈ P(n, C) and X ∈ H(n, C).
Proof Let γ(t) = exp I p −1/2 Xp −1/2 t . Since exp I is an analytic function we havė
γ(t) = p −1/2 Xp −1/2 exp ′ I p −1/2 Xp −1/2 t γ(t) = p −1/2 Xp −1/2 exp ′′ I p −1/2 Xp −1/2 t p −1/2 Xp −1/2
and other formulas hold forγ(t) andγ(t) similarly to the second part of the proof of Theorem 6. By the geodesic equations we havë
γ(t) = −Γ (γ(t),γ(t), γ(t)) Xp −1/2 exp ′′ I p −1/2 Xp −1/2 t p −1/2 X = −p 1/2 Γ p −1/2 Xp −1/2 exp ′ I p −1/2 × Xp −1/2 t , p −1/2 Xp −1/2 exp ′ I p −1/2 Xp −1/2 t , exp I p −1/2 Xp −1/2 t p 1/2 .
If we consider the geodesic equations for γ(t) = exp p (Xt) we get
Xp −1/2 exp ′′ I p −1/2 Xp −1/2 t p −1/2 X = −Γ Xp −1/2 exp ′ I p −1/2 Xp −1/2 t p 1/2 , p 1/2 exp ′ I p −1/2 Xp −1/2 t p −1/2 X, p 1/2 exp I p −1/2 Xp −1/2 t p 1/2 .
The left hand sides of the two equations above are the same so as the right hand sides. Taking t = 0 and that exp ′ I (0) = I, exp I (0) = I we get for all p ∈ P(n, C), X ∈ H(n, C) that
p 1/2 Γ p −1/2 Xp −1/2 , p −1/2 Xp −1/2 , I p 1/2 = = Γ (X, X, p) ,
which proves the assertion.
By the above result we have just reduced the problem of characterizing Γ (X, X, p) to the characterzation of Γ (X, X, I). Now we will show that Γ (X, X, p) is invariant under similarity transformations.
Proposition 12 For all p ∈ P(n, C) and X ∈ H(n, C) and invertible S we have Γ SXS −1 , SXS −1 , SpS −1 = SΓ (X, X, p) S −1 .
(53) Proof We have by the geodesic equations
X 2 exp ′′ I (Xt) = −Γ X exp ′ I (Xt), X exp ′ I (Xt), exp I (Xt) SX 2 exp ′′ I (Xt)S −1 = −SΓ X exp ′ I (Xt), X exp ′ I (Xt), exp I (Xt) S −1 .
Similarly if we consider the geodesic equations for the curve γ(t) = exp I SXS −1 t we get
SX 2 S −1 exp ′′ I (SXS −1 t) = −Γ SXS −1 exp ′ I (SXS −1 t), SXS −1 exp ′ I (SXS −1 t), exp I (SXS −1 t) SX 2 exp ′′ I (Xt)S −1 = −Γ SX exp ′ I (Xt)S −1 , SX exp ′ I (Xt)S −1 , S exp I (Xt)S −1 .
Again since the above two equations are identical we get the assertion.
By the above proposition we have for hermitian X that
Γ (X, X, I) = U Γ (D, D, I) U * ,(54)
for some diagonal D and unitary U , so it is enough to characterize Γ (X, X, I) for diagonal X.
for some real valued constant c.
Proof First we will show that Γ (I, I, I) = cI for some real constant c. Consider the case when γ(t) = exp I (λIt) for some real λ. Then by the geodesic equations for γ(t) we have
λ 2 exp ′′ I (λIt) = −Γ λ exp ′ I (λIt), λ exp ′ I (λIt), exp I (λIt) .
By linearity of Γ (·, ·, ·) in the first two variables, this is equivalent to
λ 2 exp ′′ I (λIt) = −λ 2 Γ exp ′ I (λIt), exp ′ I (λIt), exp I (λIt) .
Letting t = 0 we get cI = −Γ (I, I, I) ,
where c = exp ′′ I (0) is a real number, since exp I : H(n, C) → P(n, C) is an analytic function with real coefficients in its Taylor series.
The next step is to show that for a projection P = P 2 = P * we have Γ (P, P, I) = −cP . Consider again γ(t) = exp I (P t). Then the geodesic equations read P 2 exp ′′ I (P t) = −Γ P exp ′ I (P t), P exp ′ I (P t), exp I (P t) .
Since P 2 = P and again letting t = 0 we get
cP = −Γ (P, P, I) ,
where c is trivially the same constant as determined above for Γ (I, I, I). Now suppose that we have two mutually orthogonal projections P 1 , P 2 such that P 1 P 2 = 0. Then we have for the projection P 1 + P 2 using linearity of Γ (·, ·, ·) in the first two variables that Γ (P 1 , P 1 , I) + Γ (P 2 , P 2 , I) = −c(P 1 + P 2 ) = Γ (P 1 + P 2 , P 1 + P 2 , I) = = Γ (P 1 , P 1 , I) + Γ (P 1 , P 2 , I) + Γ (P 2 , P 1 , I) + Γ (P 2 , P 2 , I) , which yields that for mutually orthogonal projections P 1 , P 2 we get the orthogonality relation Γ (P 1 , P 2 , I) = 0.
Finally since a diagonal D can be written as D = i λ i P i for mutually orthogonal projections P i , we have
Γ (D, D, I) = Γ i λ i P i , i λ i P i , I = = i λ 2 i Γ (P i , P i , I) = − i λ 2 i cP i = = −cD 2 ,
which is what needed to be shown.
The above three theorems with the other preceeding results presented here, lead us to the concluding
Theorem 8 All affine matrix means M t (X, Y ) are of the form M (X, Y ) = X 1/2 (1 − t)I + t X −1/2 Y X −1/2 1−κ 1 1−κ X 1/2 if κ = 1, X 1/2 X −1/2 Y X −1/2 t X 1/2 if κ = 1,(56)
where 0 ≤ κ ≤ 2. The symmetric affine connections corresponding to these means are
∇ Xp Yp = DY [p][Xp] − κ 2 Xpp −1 Yp + Ypp −1 Xp .(57)
Proof By Proposition 11, 12 and Theorem 7 we have that the functions Γ (·, ·, ·) : H(n, C) × H(n, C) × P(n, C) → H(n, C) representing the Christoffel symbols are of the form
Γ (X, X, p) = −cXp −1 X.(58)
This formula determines the functions that are the symmetric parts of the possible connections, and these connections have geodesics determined by Theorem 6 in the form (56). Again by Propostion 15 and Corollary 16 of Chapter 6 in [33] we know that connections which have the same torsion and geodesics are identical and for an arbitrary connection there is a unique connection with vanishing torsion and with the same geodesics. So in other words since the connections (57) are symmetric, affine and have the same geodesics, therefore they give the sought symmetric connections for each κ if we choose c = κ. The corresponding geodesics are given in (39), and these are matrix means if and only if κ ∈ [0, 2], since the representing functions f (t) in (4) turn out to be operator monotone only in these cases due to Example 1.
The above result gives us the complete classification of affine matrix means. So now we can concetrate only on the connections (57). In the next section we solve the metrization problem of these connections. 9 The holonomy groups and metrizability of the affine family Let W be a smooth connected manifold with an affine connection ∇. The holonomy group Hp(∇) of the connection ∇ at point p ∈ W is defined to be the set of all linear automorphisms of the tangent space TpW at p induced by parallel transports along p based closed rectificable curves. If W is simply connected then Hp(∇) is known to be a Lie subgroup of End(TpW ) [27]. In case of non-simply connectedness the restricted holonomy groupĤp(∇) is defined as the normal subgroup of Hp(∇) which is induced by closed rectificable curves homotopic to zero, see Chapter II Section 4 in [22] for more detailed information. Let hp(∇) andĥp(∇) denote the Lie algebra of Hp(∇) andĤp(∇) respectively. The holonomy group Hp(∇) is known to be an invariant of the connected manifold W , since Hp(∇) is conjugate to every other Hq(∇) by parallel transports. Now suppose that the connection ∇ is real analytic. Then by Theorem 10.8 of Chapter II and Theorem 9.2 of Chapter III in [22],ĥp(∇) is generated by the successive covariant differentials ∇ r R, r = 0, 1, 2, . . . at the point p where R(X, Y ) denotes the curvature endomorphism of the connection ∇. This is a version of Ambrose-Singer's theorem of Kobayashi-Nomizu. The curvature tensor R is defined as
R(X, Y )Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [X,Y ] Z
or expressed in local coordinate system with the Christoffel symbols Γ i jk as
R i jkl = ∂Γ i lj ∂x k − ∂Γ i kj ∂x l + Γ i km Γ m lj − Γ i lm Γ m kj .(59)
Suppose now that the connection ∇ is torsion-free, i.e.
∇ X Y − ∇ Y X − [X, Y ] = 0(60)
for all vector fields X, Y or equivalently Γ i jk = Γ i kj everywhere. Then ∇ is the Levi-Civita connection of a Riemannian metric if and only if the corresponding holonomy groupĤp(∇) is a compact Lie group. More generally there exists a non-degenerate ∇ invariant bilinear form ·, · p if and only ifĤp(∇) leaves ·, · p invariant.
In [27] all possible irreducible holonomy groups of torsion-free affine connections are classified, so in principle we know what kind of groups can occur, at least in the reducible case. Again we are interested in the connections
∇ Xp Yp = DY [p][Xp] − κ 2 Xpp −1 Yp + Ypp −1 Xp .(61)
These connections are real analytic, torsion-free and the corresponding manifold P(n, C) is analytic simply connected. So to answer the question of metrizability we have to determine the holonomy groupsĤp(∇). In our case it turns out that
Γ i jk E i = − κ 2 (E j p −1 E k + E k p −1 E j ) R i jkl E i = κ 2 − κ 2 4 p p −1 E j , p −1 E k , p −1 E l ,(62)
where the E i form the standard basis of the vector space of H(n, C) and [·, ·] is the commutator. Note that the tangent space is H(n, C), so the left hand sides are in H(n, C). In order to determine which of these manifolds are symmetric spaces it is sufficient to calculate the covariant differential R s jkl;m , since it vanishes everywhere if and only if the underlying manifold is a symmetric space [15]. Given the basis E i for H(n, C) we have the identities where indices after ; denote covariant differentiation. Now we prove an analogue of Lemma 1 given in the proof of Theorem 9.2 of Chapter III [22].
R(X, Y ; A 1 , . . . , Ar)Z = i,j,k,l1,...,lr,m R i jkm;l1 ,...,lr E i X j Y k Z m A l1 1 · · · A
Theorem 9 Let the smooth connected manifold P(n, C) be equipped with real analytic connection ∇ and curvature tensor given by (62) with κ ∈ R. Then Proof The proof is based on writing R(X, Y )Z and its subsequent covariant differentials in essentially two equivalent ways. First of all note that
∂ ∂x i p −1 = D(x −1 )[p][E i ] = −p −1 E i p −1 ,(64)
so the differential operator ∂ ∂x i is equivalent to Fréchet differentiation at p in the direction of E i , also
R(X, Y )Z = κ 2 − κ 2 4 p p −1 Z, p −1 X, p −1 Y = κ 2 − κ 2 4 Z p −1 X, p −1 Y + Y p −1 , Xp −1 Z = κ 2 − κ 2 4 Zp −1 , Xp −1 , Y p −1 p.(65)
Using index-less notation and the linearity of R(X, Y ; A 1 , . . . , Ar)Z we have
R(X, Y ; A 1 , . . . , A r+1 )Z = ∇ Ar+1 (R(X, Y ; A 1 , . . . , Ar)Z) − κ 2 A r+1 p −1 R(X, Y ; A 1 , . . . , Ar)Z + R(X, Y ; A 1 , . . . , Ar)Zp −1 A r+1 − R(A r+1 p −1 X + Xp −1 A r+1 , Y ; A 1 , . . . , Ar)Z − R(X, A r+1 p −1 Y + Y p −1 A r+1 ; A 1 , . . . , Ar)Z − R(X, Y ; A 1 , . . . , Ar)(A r+1 p −1 Z + Zp −1 A r+1 ) − r i=1 R(X, Y ; A 1 , . . . , A r+1 p −1 A i + A i p −1 A r+1 , . . . , Ar)Z .(66)
Again the first term in the above equation is equivalent to
(A r+1 ) s ∂ ∂x s R(X, Y ; A 1 , . . . , Ar)Z = D(R(X, Y ; A 1 , . . . , Ar)Z)[p][A r+1 ].(67)
Claim R(X, Y ; A 1 , . . . , Ar)Z is the linear combination of terms pS, where S is some word which is a product of the terms p −1 X, p −1 Y, p −1 A 1 , . . . , p −1 Ar of the first order.
Proof (of the claim) We prove by induction. For r = 0 it clearly holds by the first equality in (65). Suppose that it holds for some r. Then by (66) it is easy to see that it holds for r + 1, due to (64), the linearity of R(X, Y ; A 1 , . . . , Ar)Z and the product rule of Fréchet differentiation.
By the claim R(X, Y ; A 1 , . . . , Ar)Z is the linear combination of terms pS, therefore by linearity, (67) and (64) we have
∇ Ar+1 (R(X, Y ; A 1 , . . . , Ar)Z) = A r+1 p −1 R(X, Y ; A 1 , . . . , Ar)Z − R(A r+1 p −1 X, Y ; A 1 , . . . , Ar)Z − R(X, A r+1 p −1 Y ; A 1 , . . . , Ar)Z − R(X, Y ; A 1 , . . . , Ar)(A r+1 p −1 Z) − r i=1 R(X, Y ; A 1 , . . . , A r+1 p −1 A i , . . . , Ar)Z.(68)
Combining the above we arrive at a version of (66):
R(X, Y ; A 1 , . . . , A r+1 )Z = = 1 − κ 2 A r+1 p −1 R(X, Y ; A 1 , . . . , Ar)Z − R(A r+1 p −1 X, Y ; A 1 , . . . , Ar)Z − R(X, A r+1 p −1 Y ; A 1 , . . . , Ar)Z − R(X, Y ; A 1 , . . . , Ar)(A r+1 p −1 Z) − r i=1 R(X, Y ; A 1 , . . . , A r+1 p −1 A i , . . . , Ar)Z − κ 2 R(X, Y ; A 1 , . . . , Ar)Zp −1 A r+1 − R(Xp −1 A r+1 , Y ; A 1 , . . . , Ar)Z − R(X, Y p −1 A r+1 ; A 1 , . . . , Ar)Z − R(X, Y ; A 1 , . . . , Ar)(Zp −1 A r+1 ) − r i=1 R(X, Y ; A 1 , . . . , A i p −1 A r+1 , . . . , Ar)Z ,
for any A, B ∈ H(n, C) and linearity of R(X, Y ; A 1 , . . . , Ar)Z, we get that
R(X, Y ; A 1 , . . . , A r+1 )Z − R(X, Y ; A 1 , . . . , A r+1 )Z = 0 = = p p −1 A r+1 , p −1 R(X, Y ; A 1 , . . . , Ar)Z − R(p[p −1 A r+1 , p −1 X], Y ; A 1 , . . . , Ar)Z − R(X, p[p −1 A r+1 , p −1 Y ]; A 1 , . . . , Ar)Z − R(X, Y ; A 1 , . . . , Ar)(p[p −1 A r+1 , p −1 Z]) − r i=1 R(X, Y ; A 1 , . . . , p[p −1 A r+1 , p −1 A i ], . . . , Ar)Z.(71)
So in particular (69) is just
R(X, Y ; A 1 , . . . , A r+1 )Z = = (1 − κ) A r+1 p −1 R(X, Y ; A 1 , . . . , Ar)Z − R(A r+1 p −1 X, Y ; A 1 , . . . , Ar)Z − R(X, A r+1 p −1 Y ; A 1 , . . . , Ar)Z − R(X, Y ; A 1 , . . . , Ar)(A r+1 p −1 Z) − r i=1 R(X, Y ; A 1 , . . . , A r+1 p −1 A i , . . . , Ar)Z = (1 − κ)p p −1 A r+1 p −1 R(X, Y ; A 1 , . . . , Ar)Z − p −1 R(A r+1 p −1 X, Y ; A 1 , . . . , Ar)Z − p −1 R(X, A r+1 p −1 Y ; A 1 , . . . , Ar)Z − p −1 R(X, Y ; A 1 , . . . , Ar)(A r+1 p −1 Z) − r i=1 p −1 R(X, Y ; A 1 , . . . , A r+1 p −1 A i , . . . , Ar)Z .
Considering again (64) and the first claim we get that the above is equivalent to
R(X, Y ; A 1 , . . . , A r+1 )Z = (1 − κ)D(R(X, Y ; A 1 , . . . , Ar))[p][A r+1 ],
which is just (63). Now again the Lie algebraĥp(∇) is generated by the endomosphisms ∇ r R. This means that the generated algebra grows as r increases and after some finitely many steps it stabilizes and taking higher covariant derivatives of R is unnecessary. Since the manifold P(n, C) is simply connected the holonomy group and the restricted holonomy group coincide, soĥp(∇) = hp(∇). By the second formula in (65) and (63) we have the following Corollary 3 The Lie algebra hp(∇) is faithfully represented over the vector space V = H(n, C) (or V = H(n, R)) with ρ : hp(∇) → End(V ) given as
ρ(W )Z = ZW + W * Z(72)
for W ∈ hp(∇) and Z ∈ H(n, C) (or Z ∈ H(n, R)).
We are in position to do a case by case analysis for different values of κ. so(n, R) denotes the Lie algebra of skew-symmetric n-by-n matrices over the real field R, su(n, C) denotes the Lie algebra of skew-hermitian matrices with vanishing trace over C, sl(n, F) denotes the Lie algebra of traceless matrices over the field F. Suppose κ = 0, 2. Then the curvature (62) of the connection vanishes, so hp(∇) is the trivial algebra.
Suppose κ = 1. Then the curvature (62) is nonzero, but is covariantly constant, all first and higher order covariant derivatives vanish due to Theorem 9. Therefore the manifold is a symmetric space that is very well known and the algebra hp(∇) by (65) is generated by elements of the form [X, Y ] where X, Y ∈ H(n, F). We have for all [X, Y ] = W ∈ hp(∇) that
W * = [X, Y ] * = −[X * , Y * ] = −[X, Y ]
where * can be replaced by the transpose T over F = R. Also since T rW = T r[X, Y ] = 0 we have hp(∇) = so(n, R) if F = R and hp(∇) = su(n, C) if F = C.
Suppose κ = 0, 1, 2. Then by Theorem 9 the higher order covariant derivatives ∇ r R (as we will see immediately) no longer vanish. Let W ∈ hp(∇). Then by (65) where X, Y ∈ H(n, F). I.e. W is given by the linear combination of commutators of some n-by-n matrices over the field F, so T rW = 0. This tells us that hp(∇) ⊆ sl(n, F).
Now we will show that the generated algebra already stabilizes for r = 1. Without loss of generality we can assume that p = I. Then we have to consider the generators of the form
G = D([p −1 X, p −1 Y ])[I][A 1 ] = −[A 1 X, Y ] − [X, A 1 Y ].(77)Let E + ik = E ik + E ki if i = k, E i i if i = k, E − ik =E ik − E ki where E ik
is the matrix with zero entries excluding the (ik) entry which is 1. Then E + ik form a basis of H(n, R) and E − ik form a basis of the vector space of skew-hermitian matrices SH(n, R) over the real field R. The vector space SH(n, C) is defined similarly over C. Note also that H(n, C) ∼ = H(n, R) ⊕ SH(n, R) and that E + ik E − lm = 0 in general. Suppose that A 1 = E + iz , X = E + ky and Y = E + ik . Then by (77)
G = −E + iz E + ky E + ik + E + ik E + iz E + ky − E + ky E + iz E + ik + E + iz E + ik E + ky .
Using that E + ik = E + ki and imposing restrictions z = k and y = i we get that
G = Ezy if z = y, Ezz − E ii if y = z.
So the matrices G of this form span the whole sl(n, R), i.e. considering (76) we have hp(∇) = sl(n, R) if F = R. Similarly if A 1 = E − iz , X = E − ky and Y = E − ik , then we get the same generator G, so hp(∇) = sl(n, R) ⊕ sl(n, R) due to E + ik E − lm = 0 if F = C; that is hp(∇) = sl(n, C) in the complex case.
By the proof of the previous Theorem 10 we see that R s jkl;m = 0 everywhere if and only if κ = 0, 1, 2. This proves the following Corollary 4 The only matrix means which are affine means corresponding to symmetric spaces are the arithmetic, harmonic and geometric means.
Since we know the holonomy groups we can decide their metrizability.
Corollary 5 The affine connections (73) are metric in the following cases:
1. n = 1, 2, κ arbitrary, F = R or C, 2. n ≥ 3, κ = 0, 1, 2, F = R or C.
Proof The case n = 1 is trivial. In [27] all irreducible holonomies of affine connections are classified and metrizability is also dicussed. The metric connections were classified by Berger long ago. The holonomy sl(2, R) is isomorphic to sp(2, R) which is metric, there exists an invariant symplectic form. Also sl(2, C) is isomorphic to so(1, 3) which is also metric, there exists an invariant pseudo-Riemannian metric with signature (1,3). This isomorphic correspondence fails in higher dimensions n ≥ 3, where the holonomies sl(n, F) (F = R or C) with representation over H(n, F) is non-metric. Remark 4 In the second case in Corollary 5 although there exists no metric structure, however by inspecting the holonomy group Hp(∇) we get that there exist totally geodesic flat submanifolds. That is if we consider the subset D(n, F) of diagonal matrices of H(n, F) in both cases F = R or C, we get a totally geodesic Euclidean submanifold and a Riemannian metric on D(n, F) is given in the form
T r p −2κ log 2 p (q)
where log p (q) is the logarithm map given in (37).
So there exist no previously unknown affine matrix mean which correpsond to a Riemannian manifold. Although we have found a previously unknown, generally non-metrizable, one parameter family of affinely connected manifolds where the points of the geodesics are matrix means, in particular matrix power means.
Riemannian manifold P(n, C) endowed with the trace metric d(A, B) = T r log(A −1 B).
(i) M (I, I) = I where I denotes the identity,(ii) if A ≤ A ′ and B ≤ B ′ , then M (A, B) ≤ M (A ′ , B ′ ), (iii) CM (A, B)C ≤ M (CAC,CBC) for all hermitian C, (iv) if An ↓ A and Bn ↓ B then M (An, Bn) ↓ M(A, B).
Theorem 2
2Let M (A, B) be a matrix mean with representing function f ∈ P(t). Then
Theorem 4 (
4FitzGerald) Suppose f (x) is a twice continuously differentiable, realvalued function with positive first derivative on
≥ 0 for
0all real continuous φ having compact support in (a, b) with η ′ (s).Summarizing the results in the previous sections from the point of view of affine matrix means we arrive at the following Proposition 8 If a matrix mean M (A, B) is affine, then the exponential map and its inverse, the logarithm map are of the following forms exp p (X) = p 1/2 exp I p −1/2 Xp −1/2 p 1/2 log p (X) = p 1/2 log I p −1/2 Xp −1/2 p 1/2(21) for p ∈ P(n, C), where exp I (X) and log I (X) are analytic functions such that exp I : H(n, C) → P(n, C) and log I (X) is its inverse and log ′ I (I) = I, exp ′ I (0) = I, log I (I) = 0, exp I (0) = I.
the fact that DS I [I][I] and [zI − I] −1 commutes with every matrix we get
Theorem 6 / 2
62The geodesic equations corresponding to the affine connections (30) arëγ = κγγ −1γ .(33)The solutions of these equations with initial conditions γ(0) = p,γ(0) = X are the folowing one parameter family of functions γ(t) = exp p (Xt) = p 1/2 exp I p −1Xp
Corollary 1
1The exponential and logarithm map for the affine connections (30) are given in the form exp p (X) = p 1/2 exp I p −1/2 Xp −1/2 p 1/2 log p (X) = p 1/2 log I p −1/2 Xp −1/2 p 1/2 ,
Proposition 9
9The matrix power means Ps(w 1 , . . . , w k ; A 1 , . . . , A k ) for s ∈ [−1, 1] are the unique positive definite solutions of the Karcher equations n i=1 w i log X (A i ) = 0 (48)
Theorem 7
7Let D be diagonal with real coefficients. Then Γ (D, D, I) = −cD 2 ,
R
(X, Y ; A 1 , . . . , Ar)Z = (1 − κ) r D(R(X, Y )Z)[p][A 1 , . . . , Ar] (63) where D(R(X, Y )Z)[p][A 1 , . . . , Ar] denotes the r-th Fréchet differential of the map R(X, Y )Z at the point p ∈ P(n, C) in the directions A i ∈ H(n, C).
Proof
κ ∈ R. Then the holonomy algebra hp(∇) is as follows:In the case of the submanifold P(n, R) with tangent space H(n, R) By the conjugate invariancy of Hp(∇) it is enough to consider the case when p = I.
κ) r D([p −1 X, p −1 Y ])[p][A 1 , . . . , Ar],
lr 1
lrR i jkm;l1 ,...,lr+1 = ∂ ∂x lr+1 R i jkm;l1 ,...,lr + Γ i slr+1 R s jkm;l1 ,...,lr jkm;l1 ,...,s,...,lr−
µ
Γ s
lµlr+1 R i
which is equivalent to R(X, Y ; A 1 , . . . , A r+1 )Z =Now we can reverse the claim and using the exactly the same argument starting with the third equality in (65) we can prove that R(X, Y ; A 1 , . . . , Ar)Z is the linear combination of terms Sp, where S is some word which is a product of the terms
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[
"Radial thermal expansion of pure and Xe-saturated bundles of single- walled carbon nanotubes at low temperatures",
"Radial thermal expansion of pure and Xe-saturated bundles of single- walled carbon nanotubes at low temperatures"
] |
[
"A V Dolbin [email protected]:74.70.wz \nVerkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine\n47 Lenin Ave61103KharkovUkraine\n",
"V B Esel'son \nVerkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine\n47 Lenin Ave61103KharkovUkraine\n",
"V G Gavrilko \nVerkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine\n47 Lenin Ave61103KharkovUkraine\n",
"V G Manzhelii \nVerkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine\n47 Lenin Ave61103KharkovUkraine\n",
"S N Popov \nVerkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine\n47 Lenin Ave61103KharkovUkraine\n",
"N A Vinnikov \nVerkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine\n47 Lenin Ave61103KharkovUkraine\n",
"N I Danilenko \nFrantsevich Institute for Problems of Materials Science\nNational Academy of Sciences of Ukraine\nKrzhizhanovsky str., 303680KyivUkraine\n",
"B Sundqvist \nDepartment of Physics\nUmea University\nSE -901 87UmeaSweden\n"
] |
[
"Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine\n47 Lenin Ave61103KharkovUkraine",
"Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine\n47 Lenin Ave61103KharkovUkraine",
"Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine\n47 Lenin Ave61103KharkovUkraine",
"Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine\n47 Lenin Ave61103KharkovUkraine",
"Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine\n47 Lenin Ave61103KharkovUkraine",
"Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine\n47 Lenin Ave61103KharkovUkraine",
"Frantsevich Institute for Problems of Materials Science\nNational Academy of Sciences of Ukraine\nKrzhizhanovsky str., 303680KyivUkraine",
"Department of Physics\nUmea University\nSE -901 87UmeaSweden"
] |
[] |
The radial thermal expansion coefficient α r of pure and Xe-saturated bundles of singlewalled carbon nanotubes has been measured in the interval 2.2-120 K. The coefficient is positive above T = 5.5 K and negative at lower temperatures. The experiment was made using a low temperature capacitance dilatometer with a sensitivity of 2·10 -9 cm and the sample was prepared by compacting a CNT powder such that the pressure applied oriented the nanotube axes perpendicular to the axis of the cylindrical sample. The data show that individual nanotubes have a negative thermal expansion while the solid compacted material has a positive expansion coefficient due to expansion of the intertube volume in the bundles. Doping the nanotubes with Xe caused a sharp increase in the magnitude of α r in the whole range of temperatures used, and a peak in the dependence α r (T) in the interval 50-65 K. A subsequent decrease in the Xe concentration lowered the peak considerably but had little effect on the thermal expansion coefficient of the sample outside the region of the peak. The features revealed have been explained qualitatively.
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10.1063/1.1542277
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"https://arxiv.org/pdf/0902.2929v2.pdf"
] | 116,907,271 |
0902.2929
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2e3be9e3fb1ddfc42c5e3c8ff766e7eca8b5d99c
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Radial thermal expansion of pure and Xe-saturated bundles of single- walled carbon nanotubes at low temperatures
Published: Fizika Nizkikh Temperatur, 2009,
A V Dolbin [email protected]:74.70.wz
Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave61103KharkovUkraine
V B Esel'son
Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave61103KharkovUkraine
V G Gavrilko
Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave61103KharkovUkraine
V G Manzhelii
Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave61103KharkovUkraine
S N Popov
Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave61103KharkovUkraine
N A Vinnikov
Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave61103KharkovUkraine
N I Danilenko
Frantsevich Institute for Problems of Materials Science
National Academy of Sciences of Ukraine
Krzhizhanovsky str., 303680KyivUkraine
B Sundqvist
Department of Physics
Umea University
SE -901 87UmeaSweden
Radial thermal expansion of pure and Xe-saturated bundles of single- walled carbon nanotubes at low temperatures
Published: Fizika Nizkikh Temperatur, 2009,10.1063/1.1542277]
The radial thermal expansion coefficient α r of pure and Xe-saturated bundles of singlewalled carbon nanotubes has been measured in the interval 2.2-120 K. The coefficient is positive above T = 5.5 K and negative at lower temperatures. The experiment was made using a low temperature capacitance dilatometer with a sensitivity of 2·10 -9 cm and the sample was prepared by compacting a CNT powder such that the pressure applied oriented the nanotube axes perpendicular to the axis of the cylindrical sample. The data show that individual nanotubes have a negative thermal expansion while the solid compacted material has a positive expansion coefficient due to expansion of the intertube volume in the bundles. Doping the nanotubes with Xe caused a sharp increase in the magnitude of α r in the whole range of temperatures used, and a peak in the dependence α r (T) in the interval 50-65 K. A subsequent decrease in the Xe concentration lowered the peak considerably but had little effect on the thermal expansion coefficient of the sample outside the region of the peak. The features revealed have been explained qualitatively.
Introduction
Since the discovery of carbon nanotubes (CNTs) in 1991 [1], this novel class of physical objects has been stimulating intense experimental and theoretical research activities. The diversity of CNT types and the problems encountered in obtaining pure CNT material in quantities needed for experimental investigations make it rather difficult to trace the basic trends in the behavior of carbon nanotubes (e.g., see the text and References in [2]). Thermal expansion is one of the least studied properties of CNTs. The currently available experimental evidence on the thermal expansion of single-walled nanotubes (SWNTs) and their bundles is confined to the region near and above room temperature, whereas low temperature data are essential for understanding the CNT dynamics. The theoretically estimated thermal expansion coefficients (TEC) of SWNTs [3][4][5][6][7][8][9] vary appreciably both in magnitude and sign.
Owing to their unique geometry, CNTs can be a basis for forming novel low-dimensional systems. For example, bundles can be used as templates to form one-dimensional chains or twodimensional surfaces consisting of condensed impurity molecules.
In recent years much experimental effort has been devoted to the study of structural and thermal properties of such systems and a number of theoretical models have been advanced to predict these properties [10][11][12][13][14][15][16][17][18][19][20][21][22][23]. However, the thermal expansion behavior of SWNT-gas impurity systems still remains obscure.
In this study the radial thermal expansion was measured on a sample consisting of bundles of single-walled nanotubes closed at the ends (c-SWNT) in the range T = 2.2-120 K and on bundles of SWNTs saturated with Xe at T = 2.2-75 K. The sorption properties of bundles of SWNTs with closed (c-SWNT) and open (o-SWNT) ends were investigated using the technique described below.
1.Radial thermal expansion of pure single-walled carbon nanotubes
1.1.Measurement technique and investigated sample
The sample for thermal expansion measurements was prepared using a procedure for ordering the SWNT axes by applying a pressure of 1 GPa, as described by Bendiab et al. [24]. These authors showed that in SWNT plates of up to 0.4 mm thickness, such a pressure aligned the CNT axes in the sample such that their average angular deviation from a plane normal to the pressure vector was ~4º.
The starting material was a CNT powder (CCVD method, Cheap Tubes, USA) which according to the manufacturer contained over 90% of SWNTs. The main characteristics are given in the The quality of the powder was confirmed by Raman analysis performed both by the supplying company and at Umeå University, Sweden. According to the manufacturer, the average outer diameter of the tubes was 1.1 nm but no information is available about the chirality distribution. From our own Raman data, obtained using four different excitation lasers with wavelengths in the range 541-830 nm, we find that the radial breathing modes indicate a wide range of tube diameters, 0.8 -2.1 nm. All samples studied show typical SWNT G-bands and only weak disorder bands. Although multi-wall tubes may also be present, judging from the spread in diameters, the Raman spectra are completely dominated by the response from single (or possibly few-) wall nanotubes. However, a small fraction of MWNTs might be invisible due to their large diameters and possibly lower Raman cross sections.
The starting SWNT powder was also investigated by high-resolution transmission electron microscopy (HRTEM) at both the Institute of Problems of Material Science, NAS of Ukraine (Fig. 1a) and at Umea University, Sweden (Fig. 1b). The pictures show that large sample fractions contain little amorphous carbon or residual catalyst. By measuring the bundle diameters we estimate that in the starting powder each bundle contains 7 to 600 SWNTs. The compacted sample used was prepared at Umea University (Sweden) by first compacting pressure-oriented (P = 1.1 GPa) SWNT plates (an individual plate was up to 0.4 mm thick), then pressing several stacked plates together at a ten percent higher pressure to form a cylinder 7.2 mm high and 10 mm in diameter with a density of 1.2 g/cm 3 . The sample was made in a special cylindrical segmented die designed for compacting CNT powder under effective pressures 0.5-2 GPa, consisting of a ring with a cylindrical channel and a conical outer surface, which was inserted into a hardened-steel cylinder supported inside a larger pressure vessel. The structure so arranged was resistant to internal stresses. The 10 mm in diameter piston was made from sintered tungsten carbide (WC). The pressures used were high enough to consolidate the powder into a solid with well oriented tubes [24], but still low enough to keep the integrity and structure of the tubes and avoid tube collapse, and Raman spectra taken on pressed plates showed no systematic changes relative to spectra taken on the pristine powder. The sample prepared by this technology has a pronounced anisotropy of properties in the directions perpendicular and parallel to the sample axis. In the direction perpendicular to the applied pressure the axes of the SWNT bundles are disordered. The compaction aligns the axes of the SWNT bundles in the plane perpendicular to the sample axis [24]. As a result the radial component of the expansion of the SWNT bundles makes a dominant contribution to the thermal expansion of the sample in the directional parallel to the sample axis. If the axial component of the thermal expansion coefficient has a magnitude comparable to that of the radial one, an angle of typically 4 o implies that the typical contribution to the total coefficient from the axial component is about 7 % of the magnitude of the radial component.
The radial thermal expansion of the sample was investigated using a capacitance dilatometer (its design and the measurement technique are described in [25]). The linear thermal expansion coefficient (LTEC) was measured in the direction of the applied compacting pressure, i.e. radially to the SWNT bundles. Prior to measurement, the gas impurities were removed from the sample by dynamic evacuation for 72 hours at 10 -3 mm Hg and room temperature. Immediately before dilatometric investigation, the measuring cell with the sample was cooled slowly (for 8 hours) down to liquid helium temperature (4.2 K) and the sample was held at this temperature for about 4 hours. The cooling and investigation were made in vacuum down to 10 -5 mm Hg.
1.2.Experimental results and discussion
The temperature dependence of the LTEC in the interval 2.2-120 K is shown in Fig. 2. The curves were obtained by least-square averaging over several series of measurement. Curve 2 was taken on the first heating of the sample from T = 2.2 K. Curve 1 data were measured in the subsequent heating-cooling process. The non-equilibrium LTECs obtained on the first heating from T = 2.2 K may account for heating-induced alignment and ordering of the bundles and the nanotubes in them, which causes a compression of bundles and, as a result, negative thermal expansion.
The equilibrium radial LTEC α r (curve 1) is positive above 5.5 K and negative at lower temperatures.
Assuming that the impurity effect is negligible, α r comprises two components α d and α g accounting for temperature-induced changes in the CNT diameters and the intertube gap. From a simple Grüneisen-type model it might be expected that α d should be similar to the in-plane thermal expansion of graphite, and thus probably small and negative well below room temperature. Because the sample is a mixture of all chiralities, the average α d should also be very similar to the average axial expansion coefficient of the tubes. The thermal expansion of a bundle should thus probably be dominated by α g , which should be similar to the out-of-plane thermal expansion of graphite or, considering the curvature, to the thermal expansion of fullerenes or linear fullerene polymers [26].
So far there has been only one study [27] in which both α d and α g were measured by the Xray diffraction method in the interval 300-950 K. At T = 300 K α r = (0.75 ± 0.25) · 10 -5 K -1 , α d = (-0.15 ± 0.2) · 10 -5 K -1 and α g = (4.2 ± 1.4) · 10 -5 K -1 . Another measurement, of α r only, by the same method [28] arrived at negative values in the whole range of measurement temperatures (200-1600 K). We are not aware of further experimental attempts to directly investigate the thermal expansion of SWNT bundles, but some experiments have been made to estimate the thermal expansion from the temperature dependence of the radial breathing Raman modes of nanotubes. Although these modes shift down rapidly with increasing temperature, indicating a large strong positive thermal expansion coefficient, it was concluded by Raravikar et al. [7] that this effect is almost completely caused by changes in intra-and intertube interactions, and that α d is very small.
It is rather problematic to compare our results with theoretical data quantitatively, mainly because the available theoretical studies are concerned with the radial and axial thermal expansion of individual CNTs. Some of them offer general conjectures on how thermal expansion can be affected by the interaction of nanotubes in a bundle (e.g., see [8]). Also, there is little agreement between the theoretical conclusions from different groups about the TEC magnitude, sign and temperature dependence, about the effect of chirality and CNT diameter upon thermal expansion, and about the correlation between the radial and axial components of the thermal expansion of nanotubes. For example, the thermal expansion is negative in a wide temperature interval (0-800 K) in [4], changes from negative magnitudes at low temperatures to positive ones at moderate and high temperatures in [8] or is positive at all the temperatures investigated in [6].
The qualitative interpretation of our results is based on the Grüneisen coefficients calculated [8] for carbon modifications -diamond, graphene, graphite and nanotubes. It is found [8] that the Grüneisen coefficients and the radial thermal expansion of CNTs are negative at relatively low temperatures, an effect caused mainly by the contribution from transverse acoustic vibrations perpendicular to the CNT surface. However, our measurements show that a negative thermal expansion coefficient exists only in a temperature interval much more narrow than found in the calculations [8]. We believe that the main reason for this is that the calculations were performed for individual nanotubes only [8]. Our sample is clearly dominated by CNT bundles (Fig. 1), and in this case additional factors contributing to the thermal expansion come into play. Firstly, there appears a positive contribution α g caused by variations of the intertube gaps with temperature. Secondly, the nanotube interaction in the bundles suppresses the negative contribution of the transverse acoustic vibrations perpendicular to the nanotube surfaces [8]. These two positive contributions to the thermal expansion of SWNT bundles decrease both the magnitude and the temperature region of the total negative thermal expansion. If we use this model and assume α d to vary slowly with temperature over a wide temperature interval we can use the data shown in Fig. 2b to estimate α d = (-4 ± 1) ⋅ 10 -8 K -1 at T = 2.2 K. Assuming further that the temperature dependent part of α at low temperatures is dominated by a positive coefficient α g , we see from Fig. 2b that a polynomial of the third order in T is a good approximation to α g (T) up to about 25 K. Although the scatter in the data is somewhat high it is clear that to get a good fit it is necessary to include one term in T 3 and one term linear (or, with a less good fit, quadratic) in T. In a Grüneisen model, the thermal expansion coefficient of a bundle is closely related to its specific heat capacity, and it is well known that the experimentally found low-temperature specific heat of nanotube bundles shows a similar behaviour above 2 K [29]. In that case the experimental behaviour c p (T) = aT + bT 3 could be fitted by an anisotropic two-band Debye model with weak coupling between tubes in the bundle by adding a contribution from the first optic branch. It thus seems quite reasonable to attribute the strongly temperature dependent positive component of the total thermal expansion to α g . The data in Fig. 2a also shows a noticeable plateau-like structure between 40 and 60 K. We point out that the intermolecular interaction in C 60 , which should be similar in magnitude to the inter-tube interaction, corresponds to an effective Debye temperature near 50-60 K which gives rise to a plateau in the specific heat in this range for both molecular and polymeric C 60 [30]. The plateau structure observed here might thus indicate the cross-over between the acoustic modes and the lowest optical/molecular 3D modes in the bundle lattice.
Xe sorption in the powder of carbon nanotubes with closed and open ends
2.1.Measurement technique and investigated samples
Carbon nanotubes (CNT) prepared by standard methods (electric-arc, laser evaporation of carbon, or CCVD method) are arranged into bundles. Inside a bundle the CNTs form a close-packed two-dimensional (2D) triangular lattice [31]. Normally, CNTs have fullerene-like semispheres at the ends (CNTs with closed ends, or c-SWNT). The final CNT product can contain large amounts of amorphous carbon, fullerenes and other carbon modifications [2,[31][32][33][34][35][36][37]. The currently used methods of cleaning CNT materials involve oxidative treatment with acid-oxidant mixtures, ozone [38], etc. They lead to partial or complete opening of the CNT ends and produce defects at the lateral surfaces.
The possible sites for sorption of gas impurity molecules in bundles of infinite, open and equal-diameter SWNTs are shown in Fig. 3. However, in practice such SWNT systems can have additional zones of impurity sorption. For example, nanotubes of different diameters form rather large channels parallel to the nanotube axes, which can be occupied by impurity molecules [39]. Besides, oxidation can produce interstices between the nanotubes in a bundle [18]. We investigated Xe sorption in c-SWNT and o-SWNT powders at T = 78-200 K. The choice of the temperature interval and the impurity was dictated by the following considerations. The interaction of gas impurities with different parts of the CNT surface is most evident at low temperatures. Owing to their geometric configuration, SWNT bundles ideally (Fig. 3) have favorable sites where sorption of impurity molecules is energy-advantageous. A number of theoretical models were proposed [39][40][41][42][43] to describe the physical sorption and dynamics of admixed gas molecules at the surface and in the interstitial channels of SWNT bundles. According to mathematical simulations [43], the inner CNT surfaces and the interstices between the neighboring tubes at the surface of SWNT bundles (the grooves -G, Fig. 3) are the most energyadvantageous sites for sorbing impurity gas molecules. Xe was used because the SWNT-Xe system is already a well-studied "model" system [44][45][46][47]. A Xe atom is too large to penetrate into the interstitial channels (IC) of close-packed bundles of identical nanotubes whose energy of binding to impurity molecules is comparable to that at the inner surface [20]. Therefore, the Xe impurity is sorbed inside a nanotube (I), in a groove between two neighboring tubes at the outer surface of a bundle (G) and at the surface of the individual tubes forming the outer surface of a bundle (S) (see Fig. 3).
To obtain the necessary information about gas impurity desorption from CNT materials, a laboratory test bench (Fig. 4) was constructed for investigating the process of Xe sorption and desorption in a CNT powder at T = 78-200 K.
The measuring cell V 1 containing a CNT sample was filled with Xe at 12 torr and cooled slowly to T = 78 K. At this temperature the xenon available in the cell was sorbed by the CNT powder and condensed on the cell walls. The cell temperature was then increased in steps of 5 K. The Xe evaporated from the cell surface and was desorbed from different sites of the SWNT bundle surface. The evaporated Xe was condensed in the vessel V 2 cooled with liquid nitrogen. When the stepwise heating brought the pressure in the V 1 -V 2 system to a constant value, the cell V 1 with the sample was separated from the vessel V 2 . The Xe condensed in the vessel V 2 was evaporated and its pressure in the system was measured with the capacitive pressure transducer 5. With the volume of the system known, we could estimate the quantity of Xe desorbed from the sample at a particular temperature. To reduce the error due to the temperature gradient over the vessel V 2 , the vessels V 2 and V 3 were minimized to the form of capillaries 1 mm in diameter. After each measurement run, Xe was recondensed from vessel V 2 to vessel V 3 .
2.2.Results and discussion
The sorption properties of the starting pure c-SWNT powder (0.0416 g) were investigated through thermo-programmed desorption (see above). Fig. 5 illustrates the temperature distribution of the desorbed impurity. The greatest quantities of Xe were desorbed at T = 125-135 K. In the case of close-packed bundles of infinite equal-diameter SWNTs (Fig. 3), the highest desorption of Xe in this temperature interval is expected from the grooves at the outer bundle surface (G) and from the interior channels of some nanotubes (I) because Xe atoms have the highest and nearly equal binding energies at these sites [47]. In our powder the desorption can be enhanced considerably by removal of Xe atoms from the axial large-diameter channels (IC). Such channels are possible in bundles of nanotubes of varying diameters. Xenon can penetrate into interior channels through defects at the ends or the lateral surfaces that can be present in some nanotubes of the starting powder. A rather small quantity of Xe was also desorbed at T = 100-105 K, which may be due to removal of the layers (S) of Xe molecules that form at the outer surface of SWNT bundles.
To open the nanotube ends, a portion (0.0705 g) of the starting powder was placed into a capsule which was then evacuated for 8 hours and heated to 450 ºC. At this temperature the capsule was filled with air for 12 min. under atmospheric pressure. According to the literature data, the ends of over 90% of CNTs can be opened through this procedure [48]. Thereafter, the capsule was evacuated again to about 10 -3 mm Hg, heated to 750 ºC and held at this temperature for an hour to remove the gaseous oxidation products. The post-treatment weighting showed a loss of ~ 5% in the powder mass.
The sorption properties of the nanotubes with the opened ends were then investigated using 8 the same thermoprogrammed desorption technique (see above). The oxidation-induced opening of the CNT ends made the inner CNT surfaces and the intertube interstice in the bundles accessible to Xe sorption [18], which enhanced the sorption capacity of the SWNT powder almost fivefold as compared to the starting material (see Fig. 5).
Radial thermal expansion of xenon-saturated single-walled carbon nanotubes. Discussion
The radial thermal expansion of Xe-saturated SWNTs was also investigated on the compacted sample used previously to measure the LTECs of pure SWNTs. The measurement technique is described in Section 1. Immediately before measurement, the cell with a pure CNT sample was evacuated at room temperature for 96 hours and then filled with Xe at 760 mm Hg. The evacuated measuring cell of the dilatometer with the sample in the Xe atmosphere was cooled to 90 K. At this temperature it was evacuated again and then cooled to liquid helium temperature. The thermal expansion was measured in vacuum down to 1·10 -5 mm Hg.
The temperature dependence of the LTEC taken on a Xe-SWNT sample in the interval 2.2-75 K is shown in Fig. 6 (curve 1). The sharp increase in the LTECs of the Xe-saturated sample (cf. curves 1, 3) can reasonably be attributed to the heavy Xe atoms affecting the transverse vibrations of the nanotubes in the direction perpendicular to their surface. At low temperatures the Grüneisen coefficients of such vibrations are negative in two-dimensional (graphene) or quasi-twodimensional (graphite, nanotubes) carbon systems [8,49] and positive in a three-dimensional carbon modification (diamond). The formation of SWNT bundles and the sorption of impurity atoms at the surface or inside the nanotubes generate three-dimensional features in the system. As a result, the negative Grüneisen coefficients of such system decrease in magnitude or become positive. The thermal expansion coefficients are expected to behave in a similar way. That is why the negative contribution to the radial thermal expansion of Xe-saturated SWNT bundles decreases and shifts towards lower temperatures (see Fig. 6; cf. curves 1, 2 and 3).
In contrast to pure CNTs the thermal expansion of Xe-saturated SWNTs is similar during the first heating and in the subsequent heating and cooling runs. It is possible that the first heating of pure SWNT bundles with xenon can make the system more rigid and its geometry insensitive to heating at low temperatures. It is interesting that the LTECs have maximum values in the interval 50-65 K, which may be a manifestation of spatial redistribution of the Xe atoms in the SWNT bundles. The simulation (by Wang-Landau algorithm) [23,50] of the potential energy for a system of SWNT bundles saturated with inert gases predicted peaks in the temperature dependence of the heat capacity at T = 50-100 K, attributed to reordering of the impurity atoms.
To test the prediction, it was necessary to remove the Xe impurity from the surface of the SWNT bundles. For this purpose, the sample was heated to T = 110 K. This temperature causes intensive desorption of Xe from the sample surface but leaves it undisturbed in the grooves of the SWNT bundles (G) and the inner interstices (I) of the nanotubes having surface defects (Fig. 5). The sample was kept at T = 110 K until the desorbed Xe was entirely removed and the pressure in the measuring cell reached ~ 1·10 -5 mm Hg. The sample was then cooled to T = 2.2 K and the thermal expansion was measured again (Fig. 6, curve 2). It is seen that the LTEC peak is much lower after Xe was removed from the SWNT bundle surfaces. However, this partial Xe desorption leaves the temperature dependence of the LTEC practically unaffected outside the interval of the peak. This suggests that the Xe atoms residing on the bundle surface have only a small effect upon the thermal expansion of SWNT bundles when the process of spatial redistribution of atoms are absent.
Conclusions
This is the first time that the temperature dependences of the radial thermal expansion coefficients α r (T) of pure and Xe-saturated SWNT bundles have been investigated experimentally at low temperatures. The measurements were made on heating and cooling the samples in the interval 2.2-120 K using a capacitance dilatometer.
The dependence α r (T) measured on the first heating showed very strong nonequilibrium effects, and in the interval 3.2-120 K it differed significantly from the well reproducible equilibrium dependences α r (T) that were found on subsequent heating and cooling runs in this measurement.
The equilibrium coefficients of the radial thermal expansion α r (Fig. 2, curve 1) are positive above 5.5 K and negative at lower temperatures. The nonequilibrium coefficients of the radial thermal expansion α r (Fig. 2, curve 2) are negative in the interval 2.2-82 K. It is assumed that the non-equilibrium α r -values measured on the first heating of the sample are due to the irreversible alignment and ordering of the bundle positions and the nanotubes in the bundles at rising temperature. As this occurs, the density of the system increases, and the thermal expansion becomes negative.
The qualitative interpretation of the equilibrium dependence α r (T) was based on the theoretical conclusions about the Grüneisen coefficients for carbon modifications [8]. The Grüneisen coefficient and the radial thermal expansion of nanotubes are negative at reasonably low temperatures [8], which is determined mainly by the contribution of the transverse acoustic vibrations perpendicular to the nanotube surfaces. However, in the experiment the temperature interval of the negative thermal expansion is much narrower in comparison with the theoretical predictions. This is most likely because the cited theory [8] investigated individual nanotubes. Additional contributions to the thermal expansion come into play in SWNT bundles. First, there is a positive contribution α g , generated by the variations of the intertube gaps with temperature. In addition, the nanotube interaction in the bundles suppresses the negative contribution from the transverse acoustic vibrations perpendicular to the nanotube surfaces [8]. These two positive contributions to the thermal expansion of the SWNT bundles decrease both the magnitude and the temperature interval of the negative thermal expansion.
The saturation of SWNT bundles with xenon brings about new features in their thermal expansion.
1) The magnitude of α r increases sharply in the whole range of temperature investigated. This is because the Xe impurity suppresses the negative contribution to the thermal expansion from the transverse acoustic vibrations perpendicular to the nanotube surfaces [8].
2) The dependence α r (T) has a peak in the interval 50-65 K, which appears to be due to the spatial redistribution of the Xe atoms over the SWNT bundle surfaces. Removal of the Xe impurity from these surfaces decreases the peak significantly but leaves the temperature dependence of the LTEC practically unchanged outside the interval of the peak. This suggests that the Xe atoms atoms located at the bundle surfaces have little effect on the thermal expansion of SWNT bundles when the processes of their spatial redistribution are inoperative.
3) For the Xe saturated material there is no non-equilibrium thermal expansion behaviour such as was observed during the first heating of the sample and attributed to irreversible alignment and ordering of the bundle positions and the nanotubes in the bundles at rising temperature. It is likely that the saturation with Xe makes the system of SWNT bundles more rigid and its geometry insensitive to heating in a low temperature interval.
Finally, the employed technique of thermoprogrammed desorption has also enabled us to measure the temperature dependence of Xe desorption from both open and closed SWNT bundles.
Fig. 1
1TEM images of the starting SWNT powder.
Fig. 2 .
2LTECs of pressure-oriented SWNT compacted sample in the direction perpendicular to the SWNT bundle axes: a) T = 2.2-120 K; b) T = 2.2-25 K (curve 1 -heating and cooling, curve 2 -first heating from T = 2.2 K).
Fig. 3 .
3Sites of possible sorption of gas impurity molecules in bundles of infinite, open and equaldiameter SWNTs.
Fig. 4 .
4Schematic view of the laboratory test bench for investigation of gas sorption-desorption in CNT samples at low temperatures. 1 -Sample of nanotubes 2, 3, 10 -Heaters 4, 11 -Temperature sensor (silicone diode DT -470) 5 -Pressure transducer (capacitance manometer MKS Baratron 627B) 6 -Gas input 7 -Digital multimeter (Keithley 2700) 8 -Temperature controller (Cryo-Con model 34) 9 -Matching device (Advantech PCI -1670)
Fig. 5 .
5Temperature distribution of Xe impurity (mole per mole and mole per gram) desorbed from powder samples of c-SWNTs (dark columns) and o-SWNTs (light columns).
Fig. 6 .
6The radial LTECs of SWNT bundles: 1 -Xe-saturated bundles, 2 -after partial Xe removal at T = 110 K, 3 -pure SWNTs (α r ) at T = 2.2-75 K (a) and 2.2-7 K (b), compacted sample.
Table . Table I .
.ICharacteristics of carbon nanotube powder given by the manufacturer.Diameter
l-2 nm
Length
5-30 µm
SWNT fraction
> 90 wt %
Amorphous carbon fraction
< 1.5 wt %
Co catalyst fraction
2,9 wt %
Specific surface
> 407 m 2 /g
Electrical conduction
> 102 S/cm
We wish to thank Prof. V.M. Loktev for valuable discussion.The authors are indebted to the Science and Technology Center of Ukraine (STCU) for the financial support of this study (project No 4266).Bibliography
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|
[] |
[
"IEEE Copyright Notice Sobi: An Interactive Social Service Robot for Long-Term Autonomy in Open Environments",
"IEEE Copyright Notice Sobi: An Interactive Social Service Robot for Long-Term Autonomy in Open Environments"
] |
[
"Marvin Stuede ",
"Konrad Westermann ",
"Moritz Schappler ",
"Svenja Spindeldreier "
] |
[] |
[
"Proceedings of the 10th European Conference on Mobile Robots (ECMR 2021)"
] |
Long-term autonomy in service robotics is a current research topic, especially for dynamic, large-scale environments that change over time. We present Sobi, a mobile service robot developed as an interactive guide for open environments, such as public places with indoor and outdoor areas. The robot will serve as a platform for environmental modeling and humanrobot interaction. Its main hardware and software components, which we freely license as a documented open source project, are presented. Another key focus is Sobi's monitoring system for long-term autonomy, which restores system components in a targeted manner in order to extend the total system lifetime without unplanned intervention. We demonstrate first results of the long-term autonomous capabilities in a 16-day indoor deployment, in which the robot patrols a total of 66.6 km with an average of 5.5 hours of travel time per weekday, charging autonomously in between. In a user study with 12 participants, we evaluate the appearance and usability of the user interface, which allows users to interactively query information about the environment and directions.
|
10.1109/ecmr50962.2021.9568838
|
[
"https://arxiv.org/pdf/2105.03242v2.pdf"
] | 234,094,218 |
2105.03242
|
306feb42b4b30191c20c378527632dcb25a1b80e
|
IEEE Copyright Notice Sobi: An Interactive Social Service Robot for Long-Term Autonomy in Open Environments
August 31st -September 3rd
Marvin Stuede
Konrad Westermann
Moritz Schappler
Svenja Spindeldreier
IEEE Copyright Notice Sobi: An Interactive Social Service Robot for Long-Term Autonomy in Open Environments
Proceedings of the 10th European Conference on Mobile Robots (ECMR 2021)
the 10th European Conference on Mobile Robots (ECMR 2021)Bonn, GermanyAugust 31st -September 3rdAccepted to be published in:
Long-term autonomy in service robotics is a current research topic, especially for dynamic, large-scale environments that change over time. We present Sobi, a mobile service robot developed as an interactive guide for open environments, such as public places with indoor and outdoor areas. The robot will serve as a platform for environmental modeling and humanrobot interaction. Its main hardware and software components, which we freely license as a documented open source project, are presented. Another key focus is Sobi's monitoring system for long-term autonomy, which restores system components in a targeted manner in order to extend the total system lifetime without unplanned intervention. We demonstrate first results of the long-term autonomous capabilities in a 16-day indoor deployment, in which the robot patrols a total of 66.6 km with an average of 5.5 hours of travel time per weekday, charging autonomously in between. In a user study with 12 participants, we evaluate the appearance and usability of the user interface, which allows users to interactively query information about the environment and directions.
I. INTRODUCTION
Social service robots can now be found performing tasks such as providing information, guiding, transport or entertainment in various contexts, e.g. retail, airports [1] or hotels [2]. Despite their commercial availability, long term autonomy (LTA) without human intervention, especially in unstructured and dynamic environments, remains a current research challenge. Long-term autonomous operations thereby pose special demands to robot design, both in terms of robust and reliant software and hardware.
In this paper we present Sobi, a social service robot for information provision and guiding in open environments (see Fig. 1). Sobi is designed for use on a campus inside and outside buildings and answers voice-and touch-based requests for e.g. directions, room plans, canteen menus or small talk and provides a guiding functionality.
The robotic tour guide and information terminal scenario is a popular use case and has been considered in various contexts in research. Early developments focused mostly on robust localization and navigation, such as the RHINO robot which was deployed for 6 days in a museum [3]. Later implementations reached cumulative operating times of several weeks. Sacarino is a service robot that was deployed for multiple weeks to provide information and do navigation tasks for guests in a hotel [4]. In the CoBot project, cumulative travelled distances of more than 1000 km were achieved with multiple robots [5]. The project fo- cused on long-term mapping, navigation and human-robot interaction (HRI), especially to proactively ask humans for help in problematic situations (e.g. operating a lift without physical manipulators). Between autonomous operations, however, manual intervention by supervisors is necessary, for example for the charging process. In [6], it is described that autonomous charging capabilities as well as deliberate requests for supervisor assistance can significantly lengthen the deployment time. The EU project STRANDS focused on the development of specific methods for environmental modeling and HRI for LTA operations. Their robots reached multiple weeks of uninterrupted autonomy in security and care contexts [7] and tour-guide scenarios [8] in subsequent works. Other projects focus on types of interaction e.g. by voice and facial recognition [9] [10], specific environments (e.g. outdoors [11] or elder care [12]), or perception and social navigation [1].
However, research is often based on the use of commercially available robots, usually supplemented by only a few hardware components, or the systems are designed for either exclusive indoor [4]- [8], [10], [12] or outdoor use [11], and the used sensors and actuators would make them unsuitable for the respective other area. In contrast, Sobi is aimed at long term autonomous use in unsupervised open spaces, multiple days and weeks indoors and frequent partially supervised deployments outdoors. Contrary to other LTA systems, we provide not only the complete robot's software as open source, but also make the system available as an open source hardware project, including 3D-model and design data, mechanical and electrical drawings and parts lists 1 .
One key aspect of LTA is monitoring system variables and executing defined recoveries in the event of a failure, which has been explored particularly for areas outside of service robotics, such as underwater or extraterrestrial applications. This can involve model-based methods that compare the system and component behavior with a nominal target [13] or data-driven approaches for outlier rejection [14]. Although various descriptions for LTA applications in service robotics note that monitoring is an important aspect [6], [7], [10], the specific methods used are not presented.
Therefore, this paper firstly introduces the main design and components of Sobi, with an emphasis on LTA aspects such as sensor and algorithmic setup for robust localization indoors and outdoors. Furthermore, a novel type of stateless reactive monitoring system based on Behavior Trees (BTs) as part of Sobi is presented, which detects faults and reacts accordingly by restarts or recoveries. In the course of the evaluation, we show first results on LTA with a 16-day deployment in which the robot patrols a total of 66.6 km within a building and notifies people of compliance to hygiene regulations. The external appearance as well as the humanrobot interface are examined in a user study, that includes verbal and touch requests for places of interest, public transport and meal plans. To summarize, the contributions of this paper are 1) a novel social service robot under free license for the use in indoor and outdoor environments, 2) a stateless reactive monitoring system based on BTs for LTA of the robot, 3) experimental evaluation of the robot in a scenario aiming at LTA and in a user study.
The remainder of this paper is structured as follows: The next section II gives on overview of the main hardware and software components of Sobi. The monitoring system is described in section III. Subsequently, section IV describes the long-term test and user study. In section V we give some remarks about the lessons learned throughout the development process. Finally, in section VI we provide a summary and an outlook on further work.
II. ROBOT DESIGN AND COMPONENTS
This section describes the design, hardware and software components of Sobi, starting with a brief overview of the core requirements:
The robot is supposed to operate in various buildings as well as in the outdoor area of the newly built campus of the Faculty of Mechanical Engineering of the Leibniz University Hannover. Within the robot's home building, fully autonomous operation should be possible by specifically using a charging station when needed. In other areas, the robot should operate autonomously as long as the batteries are charged and will then be manually transported back to the home building. It should be able to operate for several hours with robust localization and navigation in these dynamic environments, and also enable multi modal perceptual sensing for applications such as person recognition and tracking. Since the robot should look appealing, one of the development priorities is an approachable outer design, which at the same time protects the robot's components from environmental influences, and an intuitive user interface. The robot should therefore have basic splash water protection to withstand spills or brief drizzles (according to the IP21 protection class to protect against touch and vertically falling water). However, it should be noted that we used the IP rating only as a guideline and no industrial grade tests were performed.
A. Exterior Shape and Design Concept
The shape and color concept of the robot was developed in close cooperation with the Hanover University of Applied Sciences and Arts. Sobi's design is based on a futuristic appearance and clear lines, which is intended to have a friendly and inviting effect on users. The shape is based on simple geometric bodies offset with chamfers, which can be found throughout the design of the robot. The bevelled design and the use of straight lines are furthermore intended to avoid water build-up and to ease sealing of transition points. One individual design feature is the circumferential aluminum-bracket that connects the upper body to the base and was CNC-machined. Since the robot is relatively large with a height of 1.56 m and width of 0.66 m, this feature is supposed to give an impression of lightness. Humanoid features are the torso and head with eyes, which are indicated as rings, as well as movable arms and ears, which can be illuminated in color by LEDs. Flat elements (e.g. for arms and ears) were laser-cut from aluminum and ABS plastic. The outer white covers of the robot, with the exception of the base cover, are laser sintered and coated to repel water.
B. Hardware Components
The described externally visible elements of the robot are supported by a structure made of aluminum squareprofiles, which is rigidly connected to a wheeled platform. All components mentioned in this section are shown in Fig. 2. The platform (Neobotix MP-500: differential drive with one caster wheel) has an onboard computer and 2D-Lidar, which can trigger low-level emergency stops for collision avoidance. It includes two 12 V AGM batteries in series with a total capacity of 50 Ah, which we extended by two smaller batteries for a total capacity of 75 Ah. This gives a minimum time of 4-5 hours until a recharge is needed for heavy usage and over twelve hours standby time.
The embedded main computer used for controlling the robot is mounted on the platform (Vecow EVS-1010: Intel i7-7700T, 16 GB RAM, GeForce GTX1050 GPU). It has two built-in WiFi modules to be permanently connected to the internet and to provide a WiFi hotspot for external access in field operation. The robot includes the following sensors for localization, navigation and interaction:
• Xsens MTi-30 IMU/AHRS for odometry calculation • Bosch Parkpilot URF7 sonar sensors for collision avoidance (3×) • SEEED ReSpeaker v2.0 microphone array for speech recognition • Velodyne Puck 3D-Lidar for localization, people perception, and collision avoidance • Intel D435 RGBD cameras (2×, mounted frontal and dorsal) for localization and people/object perception. The cameras are connected to a Nvidia Jetson Nano inside the head, which provides the synchronized image data via Ethernet and thus reduces the load on the main computer.
Apart from the platform, the only actuators are BLDC motors to move the arms and servo motors for the ears. These are not used for physical interaction or manipulation, but for social interaction to make the appearance of the robot more natural by supporting the emotions to be displayed (e.g. wiggling ears to depict happiness or lowering ears for sadness). Besides a speaker and tablet (Fig. 3), another main component for interaction is the 64×32 px RGB LED matrix that displays still images or animations. The speaker, ears and LED panel are controlled by a Raspberry Pi, which communicates with the other computers via Ethernet.
C. Software Architecture
All four computers in Sobi run Linux (Ubuntu) and the Robot Operating System (ROS) as a framework for communication and control. The software is partially based on freely available ROS packages, some of which have been adapted for the specific robot setup, but mainly consists of self-developed solutions. The layered architecture is shown in Fig. 4. All relevant ROS messages are continuously stored to an external server running a MongoDB instance. The central control unit in the application layer is a Behavior Tree (BT), partly composed of sub-trees of the behavior layer, which can also be executed independently. Since the robot must act purely reactively on immediate requests, modelling via BTs is well suited and the use of a task scheduler is not necessary. However, recent advances in BT research would also enable more sophisticated use cases which incorporate planning capabilities for BT synthesis [15], [16].
In the following sections, the individual software functions of Sobi are presented in more detail.
1) People and Object Perception
People perception and tracking is conducted through the 3D Lidar and RGBD cameras. YOLO v3 [17] is used for people and object detection in the RGB images. The centroids of resulting bounding boxes are registered with a median filter based on the distance either with the 3D Lidar data or the depth data of the cameras. We use a modified version of [18] for 3D point cloud segmentation, extended by the approach proposed in [19] for 2D clustering to decrease the false negative rate. The aggregation and tracking is performed with the SPENCER framework [20], which enables tracking in a circular area with a radius of up to 5 m through the multimodal combination of different sensors.
2) Localization and Navigation
A basic requirement for a mobile robot is to know its own position in relation to the environment. Especially in dynamically changing environments this poses a challenge. As Sobi is designed to be used in different areas, both inside and outside of buildings, a powerful SLAM solution is one of the most important requirements. RTAB-Map (real-time appearance based mapping) [21] is used as the basic SLAM method, as it allows flexible use of different sensor types (e.g. RGBD cameras, Lidar). RTAB-Map comprises a powerful framework that can be used for long-term, large-scale and multi-session mapping. As input for the SLAM front-end we use the wheel odometry, fused with the measurement data of the IMU via an extended Kalman filter, as odometry source and both cameras and 3D Lidar as sensors. Since the different types of environment place different demands on the SLAM system, we developed a method that automatically selects predefined SLAM configurations for different environments depending on various criteria, such as distances to walls. For example, for outdoor or large-scale environments the depth information of the 3D Lidar is used instead of the cameras, Fig. 4: Layered software structure of the robotic system. White boxes indicate custom self-developed programs, dark grey boxes third-party ROS programs and light grey boxes modified or extended third-party ROS programs. and for indoor environments the maximum range for map generation is reduced [22].
One of the biggest challenges of visual SLAM methods like RTAB-Map are changes in illumination or appearance over time. To increase the number of loop closures, we added to RTAB-Map the possibility to use 3D point clouds for loop closure detection by a trained classifier of global point cloud descriptors and then register them in a multi-step process. Especially in outdoor environments and environments poor in visual features, localization could thus be significantly improved. Further information and experimental results can be found in the corresponding paper [23].
For navigation on metric maps, respectively local and global path planning, standard techniques of the ROS framework (i.e. MoveBase) are used. The 3D map is projected onto the ground plane to create an occupancy grid map that can be used with standard planners. Above this level, we use navigation on a topological map, where nodes represent either relevant locations, such as specific rooms or facilities, or waypoints between which the robot can either navigate directly or move using problem-specific planners. An example for problem-specific navigation planning is docking and undocking at the charging station. The former uses a triangular landmark that can be detected by the 2D laser scanner. It plans a Dubins path consisting of circular-and linear segments and follows the path with a pure pursuit controller [24]. These approaches are not complex in their computation and have resulted in robust docking processes in initial evaluations.
3) Human-Robot Interaction
The main task of Sobi is to provide campus-specific information as well as guiding applications. For intuitive accessibility, the human-robot interface therefore consists of speech processing as well as touch operation. Speech processing is based on a combination of Google's Speech-to-Text and Text-to-Speech services and the Natural Language Processing pipeline Dialogflow (see further information in [25]). Since WiFi coverage is available throughout the robot's area of operation, the latencies of the speech processing pipeline are short enough to allow smooth interaction. In addition, no offline backup solution is required, because the internet connection was available throughout the entire development and test period. All information can be accessed via both the speech and touch interfaces. The interface includes the following functions: display and queries of the canteen menu, public transport timetables, staff offices and room locations as well as options for small talk. By connecting the topological map of the robot with the environmental structure, it is also possible to locate nearby places (e.g. restrooms, seminar rooms or offices). The path to the requested location can then be displayed on a 3D map for destinations on the same floor (Fig. 3). A daytime-specific greeting and thus the start of an interaction occurs when a person is detected in front of the robot or the touch display is activated. To further enhance HRI, animations on the LED panel, varying colors of the LED strips, and movements of the ears are also used. Blinking eyes indicated as rings are displayed in normal operation, as well as various animations for specific situations (e.g. laughing, sad or sleeping during the charging process). When the voice input is activated by a button on the tablet, the robot's LEDs turn green, the ears move forward, and the indicated eyes widen. Unanswerable requests, on the other hand, are underlined by lowered ears and the depiction of a sad face.
III. MONITORING FOR LONG-TERM AUTONOMY
Monitoring system variables of hardware and software applications is an essential building block for achieving long-term autonomy. We therefore developed a framework for Sobi, which monitors various system variables and reacts to faults. The monitoring system consists of separate applications that monitor hardware-and ROS framework parameters. The hardware monitors include CPU, RAM and network load monitoring as well as measurement of time differences of host computers based on Network Time Protocol (NTP). ROS nodes are monitored by continuously checking if they are pingable and that essential topic publishing rates are within a tolerance band. Furthermore, we implemented monitors that continuously check whether there is a valid loop closure in the localization system and if there are error cases in the navigation system. Each monitor includes an individual warning and error range and respective messages are aggregated on one single ROS topic. The aggregated values are then used as input to an arbiter based on the Behavior Tree framework, that deterministically reacts to the different error cases. Although Behavior Trees are mainly used for sequential control of autonomous agents, their advantages in terms of reactivity and modularity combined with intuitive modelability are also applicable for system monitoring. Due to the statelessness of BT and their reactive structure, it is thus possible to react immediately to errors that occur, without the need for explicit state transition modeling as in the case of finite state machines, for example. The structure of the monitoring system is shown in Fig. 5.
The system entities (i.e. programs/topics) to monitor are organized in configurations, that can be switched autonomously or manually and are created for different use ... cases, e.g. for normal operation, charging or mapping. All unneeded programs from other configurations are terminated when a configuration change is made. Whether an error is present is determined on a monitor-specific basis via detection signals. Table I summarizes these signals together with the associated recovery reaction. Similar to [6], we follow the approach that maximum robustness may not be achieved by full autonomy alone, but by planned interventions of supervisors in case of failure. Therefore, the last resort for navigation and localization errors is a predefined request to a list of supervisors, in which the robot sends an instant message with an URL. On the linked website, a teleoperation can then be performed based on the camera views, or the current position on the map can be specified. This simple system intervention usually takes less than a minute of the operator's time, but prevents a total failure of the system. Once the problem has been fixed, this is confirmed by the operator and an all-clear is sent to the other supervisors.
Configurations
IV. EVALUATION
The evaluation of the robot consists of two parts: In the first part, we test the LTA capabilities in a continuous 16 day deployment and in the second part, we conduct a user study on human-robot interaction to evaluate the user interface and robot's appearance.
A. Long-Term Autonomy
The goal of this evaluation is to determine the robustness of the localization, navigation and monitoring system as well as the individually created algorithms, e.g. for docking, and to identify possible weaknesses. The robot's task is to permanently patrol within one floor during office hours (9 am-5 pm) on working days. Outside these hours and if necessary in between, Sobi autonomously approaches the charging station and performs the charging process. The environment mainly consists of a hallway with several offices and an entrance area. The metric and topological maps of the environment are shown in Fig. 6. Due to hygiene restrictions in place at the time of the evaluation, passers-by are not allowed to access the robot by touch and are also required to wear a face mask while in the building. The robot is therefore tasked with detecting whether people are wearing a face mask, by analyzing the image of the frontal camera every 0.5 s and using face detection. The underlying YOLO network was trained with 600 images containing both mask and no mask classes. A test set of 100 images results in an average precision of 92.9 % for the mask and 82.34 % for the no mask class. If the no mask class is detected, the robot verbally asks the person to put on a mask. Given the total number of detections in the evaluation period, the average precision can give a rough estimate about how many people were encountered directly in front of the robot. This therefore also serves as a measure of the environment dynamics, since crowded environments pose a greater challenge for longterm autonomous systems. Together with typical metrics for long-term autonomy the number of detections is summarized in Table II. In total, Sobi was undocked from the charging station for 71.2 h, of which 65.4 h were spent in motion, so that a distance of 66.6 km was covered during the evaluation period. The time in motion indicates the time during which the robot was able to perform the patrolling task and was not in an error or recovery state. With respect to an 8-hour workday of the robot, this results in an autonomy percentage (A%) of 69 %, which is an indicator of the percentage of the available time that the robot actually uses to perform its services (patrolling). In our case, this indicative parameter is mainly influenced by the average additional charging time of 2 h during the day.
An overview of the recovery behaviors performed is shown in Table III. As in [7], a recovery behavior is considered successful if no further recovery behavior needs to be performed within one minute. For the recovery behaviors for navigation, a further requirement is that no additional recovery behavior must be performed within a circle with a radius of 1 m within this time. From the table it can be seen that many problems can already be solved by waiting and going back (in case of navigation errors) or slow rotation and restarts (in case of localization errors). Requested remote access was necessary in a total of nine cases. These occurred more frequently when there were major changes to the environment and, for example, many doors were open that had been closed during the mapping process, or when closed fire doors or obstacles blocked the way.
A typical quantity for evaluating robustness is the total system lifetime (TSL), which specifies the time interval between interventions by supervisors in the event of a failure, which were not specifically requested by the robot. During the evaluation period, a total of four of such interventions were necessary, resulting in the TSL values shown. Two of the interventions were due to software errors that resulted in the signal to approach the charging station not being sent, requiring manual intervention to avoid shutdown. On one other occasion, various programs were permanently restarted within a deadlock state, making it impossible to continue the process. This deadlock situation mainly influences the 62.5 % success rate of the node restart recovery, because ten node restarts occurred in a timespan of less than three minutes. The fourth intervention was necessary when a docking attempt failed, so that the robot performed an emergency shutdown and had to be switched on again manually. A total of 28 docking attempts were made during the evaluation period, with all other 27 attempts carried out successfully. Except for this last error, all problems could be recovered via remote SSH access to the system and no hardware errors occurred. One drawback of the TSL, and also the A% is, that it does not validate how useful the services of the robot actually are for users. In the next section, we therefore present a separate study on how users perceive the robot and its info-terminal services.
B. User Study on Human-Robot Interaction
The user study is based on the Godspeed Questionnaire Series (GQS) [26] and is conducted with 12 participants (10 male, 2 female, all between 25 and 34 years old), who had no prior knowledge or direct contact with the robot before the study. The GQS measures a user's perceived sensations during a social interaction with a robot and is one of the most widely used assessment criteria in this area [27]. It describes the perceived impression of the robot via five categories and different bipolar adjective pairs on a scale from 1 to 5 using a semantic differential.
The interaction between Sobi and a user is conducted in the location shown in Fig. 6 and is divided into three independent phases. During the experiment, the robot remained in a static position and interaction was started by the users. The shown POI are visualized on the 3D map of the tablet accordingly. In the first phase, users were given specific tasks to perform. This included figuring out how to get to the nearest restroom as well as to the office of a given person. Users were not provided with any assistance, nor were they made aware that there was more than one possibility to get the requested information, using speech commands or the tablet GUI. After successfully visualizing the path to the destination on the tablet's 3D map, users should identify information about public transportation departure times. In the second phase of the study, five minutes were provided for free interaction, during which Sobi's abilities were to be learned in more detail by trying them out for themselves. In a concluding observation phase, Sobi gave some verbal closing remarks and insight about further possible emotions and robot states.
The results of the GQS are shown in Fig. 7. For the evaluation of reliability, the respective value of Cronbach's alpha represents a measure of internal consistency within a category. The results show that the robot likeability level felt during the interaction is not dependent on distinctive humanlike characteristics of the robot. Even though human-like characteristics within the anthropomorphism category can only be partially applied to Sobi, the interaction was predominantly perceived as pleasant. Sobi's appearance and behavior are positively highlighted with overall high perceived likability by all users. Sobi is perceived as a mechanical robot, but its dynamic behavior clearly distinguishes him from a static machine. By greeting a user automatically when he or she approaches as well as providing immediate verbal and visual feedback to user requests, the robot is perceived as lively and responsive. With high internal consistency within the corresponding category, the perceived intelligence of the robot was rated relatively positively with medium to high values, but continues to show potential for development. In many cases, speech input was used to search for locations, which were not always correctly comprehended by the robot or were fundamentally unknown to it. However, regardless of any failed attempts, the robot's behavior in responding to user instructions is considered reasonable. The outliers at values of 1 could be due to an inadvertent erroneous rating of the category by one person, since this person otherwise positively highlighted the interaction with Sobi as well as The consistency within a category is indicated in each case with Cronbach's α. The values for the anthropomorphism, likeability, and perceived intelligence categories are above the 0.70 threshold, indicating internal consistency. A low value in the safety category was also reported in similar research [28].
his abilities in an additional free text field that had to be answered. Other people described the operation of the robot as smooth and intuitive, but also mentioned possible scopes of knowledge that could be learned by Sobi. Overall, Sobi was perceived as a competent robot, which was able to answer most of the users' concerns. Due to its appearance as well as its active and reactive behavior, the robot was accepted as a pleasant interaction partner.
V. LESSONS LEARNED
During the development period and the evaluation runs we had the opportunity to test Sobi, identify errors and obtain feedback. We will therefore briefly discuss the most important insights and challenges of this process. The most prominent factor in the first months of development of the robot was the choice of main hardware components (i.e. the mobile base, sensors and computers). Since the selection of reliable hardware components is the basis for long-term autonomous use, we mainly use established components with maintained ROS support, full integration and standard transmission interfaces (e.g. USB or Ethernet). The signal lines are kept as short and shielded as possible, since in our experience, especially with data-intensive transmission (e.g. camera data), line losses or interference lead to errors as the operating time increases. To develop the SLAM functionality of the robot, a functional model with a simple aluminum frame structure was used early in the development process. This allowed the exact positioning of sensors to be easily varied. After these initial tests, we started working with the design team early on to conceptualize and develop the eventual design of the robot appearance. By defining this at an early stage, it was thus possible to avoid redesigns of the mechanical structure and provide a guideline for the placement of the other robot components.
The robot initially contained only the main computer and the computer in the mobile platform to run the software. However, it has proven useful to outsource individual functions to function-specific embedded computers. The execution of many hardware drivers can already lead to a considerable system load, which can be counteracted by distributing them over several computing units. Thus, for example, the camera data is read out and synchronized by the Jetson computer, so that only the compressed RGBD data has to be made available to the ROS system via Ethernet. An essential prerequisite for this is that the system clock times of all computers are synchronized with each other to avoid inconsistencies in ROS message processing. A common approach is to use Network Time Protocol (NTP), with the main computer acting as the NTP server in our case, so time synchronization is possible regardless of an internet connection.
In terms of software development and implementation for long term autonomy, we share the findings from similar efforts in the literature [6], [7], [10]. Especially the design principles that all programs should tolerate the temporary unavailability of other programs and that program restarts should transition to a clean state are the basis for the design of our software and the monitoring system. In terms of localization, our system differs from the aforementioned literature in the use of multiple RGBD cameras in conjunction with a 3D laser scanner. Since the localization programs are terminated during the charging process, it is necessary to quickly regain a valid localization estimate when resuming operation. For this purpose, we hung up a poster with many images and text directly above the charging station, so that a large number of distinctive features leads to a loop closure detection directly after undocking (see Fig. 6). It has also proven useful to cover a large field of view with the different perceptual sensors in order to prevent the robot from losing localization if, for example, people are standing around it. On the other hand, the fields of view of the cameras and the 3D laser scanner overlap, so that the depth information of the scanner can be registered with the RGB images. This is particularly advantageous for large distances, since the depth data from the cameras is only useful for distances of a few meters.
We also had to discard some original plans during the course of the project. The implementation of actuators requires a significant amount of time and therefore we did not implement a rotatable head and a tiltable tablet as originally intended. Furthermore, the plan to let the robot move autonomously between indoor and outdoor areas is prevented by door sills. These are too high (20-30mm) for the robot to pass, due to its mass of 150 kg with a wheel diameter of 260 mm. This problem arises from an incorrect assumption regarding accessibility, as the campus was still under construction during the project period.
VI. CONCLUSION AND FUTURE WORKS
In this work we presented our approach to building a long-term autonomous robot in terms of design, hardware components and software structure. The robot's monitoring system was presented and tested in a 16-day evaluation with regard to long-term autonomy. In a user study with 12 participants, the functionalities of the HRI were assessed in terms of impression and usefulness. Participants described the services offered by the robot as useful and perceived the appearance as very pleasant. During the long-term evaluation the robot was in motion for 65.4 h, traveling a total distance of 66.6 km. These first results show that the robot is suitable for long-term autonomous tasks, since the main sources of error were due to the developed software. In addition to correcting these errors, the robot will be tested in further outdoor deployments in the future to test the capabilites for long-term autonomy under even more dynamic environmental conditions. Within the deployment on our campus, further insights are to be gained through a larger number of users and recording of time-and location-dependent longterm usage patterns. This information will then be used to create data-driven models of people occurrence and usage patterns, which could then be employed to actively improve the services offered by the robot.
1Fig. 1 :
1All authors are with the Leibniz University Hannover, Institute of Mechatronic Systems, D-30823 Garbsen, Germany, [email protected] Sobi interacting with users during one of the presentations at public events (Hannover city hall).
Fig. 2 :
2Hardware components and their placement in Sobi. Fig. 3: GUI of the tablet. Places of interest (POI) can be chosen by category (top). A 3D map then shows the path from the current position (bottom).
Fig. 5 :
5Structure of the monitoring framework. Configuration dependent monitored values are processed in a BT. The composition of the tree defines the priority of error handling.
Fig. 6 :
6Environment used for validation. Dots indicate nodes and blue lines indicate edges of the topological map. The robot continuously patrols between the shown waypoints in a random fashion.
Fig. 7 :
7Results of the user study on sensations and perceptions (Godspeed Questionnaire Series) of a user during interaction with Sobi.
Operating ,
OperatingCharging , Mapping , ...Node Mon.
Topic Mon.
Navigation
Mon.
Localization
Mon.
Change Configuration
Aggregation
→
Check & Set
Configuration
Process
Monitors
?
→
?
Node
error
Topic
error
Restart
Node
→
Nav.
error
Recover
Navigation
TABLE I :
IOverview of the utilized monitors with detection signal and corresponding reaction.Name
Detection
Reaction
Node monitor
Node not pingable
Restart node
Topic monitor
Publish frequency not in
tolerance band
Restart publishing node
CPU, RAM, NTP, Network
Monitor
Not in tolerance band
Send message
Navigation monitor
No global path found
and MoveBase recoveries
failed
1) Wait and retry
2) Move backwards
3) Ask Supervisor
Localization monitor
No loop closure detected 1) Slow rotate
2) Restart localization
3) Slow rotate
4) Ask Supervisor
TABLE II :
IIMetrics of the de-
ployment. Weekend days are
excluded.
Metric
Value
Timespan
12 days
Mean time operat-
ing per day
5.9 h
Mean time moving
per day t m
5.5 h
A% = tm /8 h
69 %
Traveled distance
66.6 km
Mean TSL
56.7 h
Max TSL
90.6 h
Detected faces with
mask
983
Detected faces w/o
mask
101
TABLE III :
IIIExecuted recovery behaviors sorted by category.Type Reaction
#
success
ROS Restart node
32 62.5 %
Navigation
Wait and retry
194 70.6 %
Move backwards
41 48.8 %
Ask supervisor for
teleoperation
5
100 %
Localization
Slow rotate
76 84.2 %
Restart localiza-
tion & rotate again
12 66.6 %
Ask supervisor for
relocalization
4
100 %
https://marvinstuede.github.io/Sobi
ACKNOWLEDGMENT This work was funded by the Faculty of Mechanical Engineering of the Leibniz University Hannover. Special thanks go to the product design team of the Hannover University of Applied Sciences and Arts.
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iSocioBot: A Multimodal Interactive Social Robot. Z.-H Tan, International Journal of Social Robotics. 101Z.-H. Tan et al., "iSocioBot: A Multimodal Interactive Social Robot," International Journal of Social Robotics, vol. 10, no. 1, pp. 5-19, 2018.
|
[] |
[
"Thermo -mechanical instabilities in friction contact Senior Consultant material and damage models, probabilistics, formerly BMW Group, power train development",
"Thermo -mechanical instabilities in friction contact Senior Consultant material and damage models, probabilistics, formerly BMW Group, power train development"
] |
[
"Valentin L Popov \nDept. of System Dynamics\nPhysics of Friction\nTechnische Universität Berlin\n10623BerlinGermany\n",
"Andreas Fischersworring-Bunk \nMTU AeroEngines AG\n\n"
] |
[
"Dept. of System Dynamics\nPhysics of Friction\nTechnische Universität Berlin\n10623BerlinGermany",
"MTU AeroEngines AG\n"
] |
[] |
The phenomenon of corrugated surfaces is a known technical problem of tribological systems; considerable work has been published in the past on the aspect of rail corrugation of railway systems. Less known is a similar phenomenon observed within the cylinder-piston system of advanced automotive engines using aluminium cylinders. This paper investigates the condition leading to cylinder corrugation in the piston/cylinder system. Material investigations strongly indicate that heat in the contact is playing a major role. Using basic analytical relationships from contact mechanics, the condition required for the onset of such thermo-mechanical instabilities are investigated. Using the concept of a critical velocity it is shown that such instabilities can occur for a realistic set of parameters. A significant technical key factor is the friction coefficient.
| null |
[
"https://export.arxiv.org/pdf/1801.04193v1.pdf"
] | 119,095,087 |
1801.04193
|
e99a53337dd5672a0b8b36f17814a3dbc92455b4
|
Thermo -mechanical instabilities in friction contact Senior Consultant material and damage models, probabilistics, formerly BMW Group, power train development
Valentin L Popov
Dept. of System Dynamics
Physics of Friction
Technische Universität Berlin
10623BerlinGermany
Andreas Fischersworring-Bunk
MTU AeroEngines AG
Thermo -mechanical instabilities in friction contact Senior Consultant material and damage models, probabilistics, formerly BMW Group, power train development
1
The phenomenon of corrugated surfaces is a known technical problem of tribological systems; considerable work has been published in the past on the aspect of rail corrugation of railway systems. Less known is a similar phenomenon observed within the cylinder-piston system of advanced automotive engines using aluminium cylinders. This paper investigates the condition leading to cylinder corrugation in the piston/cylinder system. Material investigations strongly indicate that heat in the contact is playing a major role. Using basic analytical relationships from contact mechanics, the condition required for the onset of such thermo-mechanical instabilities are investigated. Using the concept of a critical velocity it is shown that such instabilities can occur for a realistic set of parameters. A significant technical key factor is the friction coefficient.
Introduction
The development of instabilities in a tribological system can lead to a permanent deformation pattern (corrugation) of the friction surface and can result in it's functional deterioration. This phenomenon is best known from the rail -wheel system documented in numerous publications [1][2][3][4][5][6], however it is known to a much lesser extent from automotive internal combustion engines.
Already since the 1970's hypereutectic aluminium-silicon alloys are used in the development of crankcases of automotive internal combustion engines as an alternative material solution. The primary Si precipitation results in a wear resistant surface making the use of grey cast iron liner technology superfluous. This results in a more compact and lightweight design. The excellent thermo-physical properties with high thermal conductivity are especially attractive for high performance engines because of a reduced thermal loading of the liner surface. With the increase of the specific power however the wear resistance is a concern. Already in the early 1990's a wear phenomena was reported specific to high performance engines: Next to the wear marks in the top centre (TC) and bottom centre (BC) crank position of the top piston ring a corrugated surface with a 'washboard pattern' was observed on the liner (see figure 1) in the contact surface between liner and piston rings. This wear can develop to a different extent and results in the most detrimental case in a total loss of the engine. In general, the wear is more pronounced in the longitudinal axis of the engine than at the thrust and anti-thrust side. Wear in the range of 1/10 mm can develop even after short engine running times. A particular characteristic is the almost constant wavelength in the wear pattern despite the significant change in the piston velocity. In general, the engineering problem is solved without a thorough understanding of the root cause. The list of the potential influence parameters includes the piston secondary motion, the piston ring design and its manufactured surface condition, the Al-Si cylinder surface condition, and the lubrication. High resolution material microstructure examination using TEM (Transmission Electron Microscopy) of the cylinder reveals that subsurface material is softened by dissolution of the hardening precipitations and also shows a low dislocation density. This change in microstructure can be attributed to the influence of high temperature pointing towards an excessive friction heat. One principal question is related to the topographic regularity of the wear pattern.
This paper tries to explain the washboard wear pattern formation with the help of thermomechanical instabilities. If two elastic bodies in contact are put into relative motion, the interrelation of the released friction energy and the local thermal expansion can result in an instability: areas with higher temperature and hence higher thermal expansion are exposed to higher normal stresses and therefore will heat up even more (see figure 2). The required conditions for the development of such an instability will be reviewed in the following.
Instability problem in a infinite system
We model a system of two bodies, one of which is rigid and has zero-heat conductance. In addition we assume no variation in transverse direction. The boundary between a stable and non-stable condition is defined by a stationary disturbance. Therefore we can assume an equilibrium condition for the elastic body including the effect of the thermal expansion [5,7],
Fehler! Es ist nicht möglich, durch die Bearbeitung von Feldfunktionen Objekte zu erstellen.
(1) and the steady heat conductance equation
T 0 ∆ =.(2)
Here u denotes the displacement vector, ν is the Poisson-contraction, T is the deviation of the temperature of it's stationary value far from the surface, γ is the thermal coefficient of expansion, and x z
∂ ∂ ∆ = + ∂ ∂
is the Laplacian-operator. The stress tensor components are given by
l i k l ik ik ik ik l k i l u u u u 2 G(1 ) 2 G(1 ) 2 T G 3 (1 2 ) 3 (1 2 ) x x x 3 x ∂ ∂ ∂ ∂ + ν + ν σ = − γ δ + δ + + − δ − ν − ν ∂ ∂ ∂ ∂ .(3)
Area of high temperature with shear modulus G. Assuming a rigid surface for the upper body (body 1) we have zero vertical displacement in the elastic body (body 2) (see Figure 3):
z u (z 0) 0 = = .
(4) Figure 3: A rigid body (1) with zero heat conductivity is in contact with an elastic continua (2). Both bodies are in relative motion to each other with a tangential velocity v.
In view of the situation in an internal combustion engine where we can encounter the thermo-mechanical instability situation, we assume that the coefficient of friction is small and that the normal stress component σ zz (e.g. from the piston ring force) is the dominating component. Under these assumptions for the mechanical equilibrium condition the tangential stress component has a negligible contribution:
xz (z 0) 0 σ = = .(5)
One solution of equations (1) and (2) subject to the boundary conditions (3) and (4) is given by the solution
( ) kz kz 0 0 T (1 ) T T coskx e , u ( 1 kz) sinkx, 0, kz coskx e 6(1 )k γ + ν = ⋅ =− − + − ⋅ − ν .(6)
(The choice of the term coskx in the solution on the coordinate x means that we are examining the development of a harmonic disturbance. Due to the linearity of the problem an arbitrary disturbance can be described by the superposition of the Fourier coefficients with different wave numbers k). Under stationary conditions the heat released at the surface must equal the heat flux from the surface (in our model assumption only in the lower body):
zz T v (z 0) z ∂ κ = −µ σ = ∂(7)
with heat conductivity κ . Therefore the critical value of the wave number is
( ) ( ) c v G 1 k 3 1 µ γ + ν = κ − ν .(8)For 1/ 3 ν = the critical wave number is c 2 v G k 3 µ γ = κ .
Temperature disturbances with smaller wave numbers than k c are non-stable.
Dynamics of an instable disturbance
We will now examine the development of an instable disturbance. For the non-stationary case we can assume that a mechanical equilibrium is achieved and hence equation (1) holds. The steady state heat conductance equation (2) however is replaced by it's unsteady state differential form
T T t ∂ = α∆ ∂ (9)
with thermal diffusivity α. Again we investigate the evolution of a disturbance which is a periodic function in direction x with wave number k.
( k ) ω = α λ −(10)
holds. A particular solution of equation (1) with regard to the displacement vector u is
1 u T 3 1 αγ + ν = ∇ ω − ν .(11)
For the temperature distribution
we obtain the solution
( ) t z T 0 2 2 1 u T k sinkx, 0, coskx e e 1 3( k ) ω λ γ + ν = − λ ⋅ − ν λ − .(13)
Any solution to the homogeneous equation
3 (1 2 ) 3 1 u divu 0 2 (1 ) 2 (1 ) − ν ∆ + ∇ = + ν + ν (14)
can be added. These in general have the form
( ) 4 kz 3 h 3B(1 ) kA u A Bz sinkx, 0, Bz coskx e k − − ν + = + − − ⋅ .(15)
The solution which satisfies the boundary conditions z u (z 0) = and zx (z 0) 0 σ = = is therefore ( )
kz z t 0 x 2 2 T (1 ) u sinkx e ke e 3 k (1 ) λ ω γ + ν = ⋅ λ − λ − − ν (16) ( ) z kz t 0 z 2 2 T (1 ) u coskx e e e 3 k (1 ) λ ω γλ + ν = − λ − − ν(17)
From the heat balance (7) we obtain the equation
( ) ( ) ( ) 2 3 1 k 3 (1 ) 2 vG 1 0 − λ κ − ν + − λκ − ν + µ γ + ν =(19)
from which we can compute the wave number and the magnification constant ω (10) as functions of λ :
2 1 c 2 1 c c 2 k 1 ( / k ) k 1 ( / k ) = λ − λ ,(20)( ) ( ) ( ) ( ) 2 c c 2 2 1 c c 2 1 / k / k k 1 / k − λ ω = λ α − λ ,(21)
or explicitly the magnification constant as a function of the wave number k :
( ) ( ) 2 1 1 2 2 2 2 1 1 c 4 4 2 1 8 k 1 8 ξ + ξ − ξ + ξ ω = α + ξ − ξ + ξ (22) where c k / k ξ = .
The dependence is shown in Figure 4. Within the range of interest for an unstable disturbance it can approximated by 2 c
1,5 (1 ) k ω ≈ ξ − ξ α .(23)
We again can observe, that all disturbances with a wave number c k k < are non-stable ( 0 ω > ), meanwhile all disturbances with a larger wave number remain stable.
/ v k L / v 1 α − − α = .(24)
Instability will therefore occur only for disturbances with a wave number smaller than 0 k , where
This estimate is looses validity if the length of the gliding contact is smaller than the wavelength 2 / k Λ = π . The domain of validity is given by the inequality relation ( ) c 2 / l k 1,5k l / L π < < .
This relation can only be satisfied if ( ) c 2 / l 1,5k l / L π < and therefore for velocities larger than
c 2 L v 2 G l κ ≈ πµ γ .(27)
However we need to emphasize that this critical velocity must be seen in the context of the used assumptions. Using characteristic material parameters of a hypereutectic aluminium alloy: a shear modulus 10 G 2.8 10 Pa ⋅ .
, a thermal expansion of , we obtain a critical velocity of roughly 35-350m/s. We can state that only in the case of relatively high friction coefficients of 0,1 µ ≈ the estimated critical velocity is in the range of real piston velocities (for high engine revolutions). Hence the friction coefficient is the most important technical parameter. However the instability can also develop at lower velocities, therefore we will examine the case of point contacts as limiting case the short contacts.
Estimation of the instability condition for a system in point contact
We will perform a coarse estimate to demonstrate that a thermo -mechanical instability is possible for point contacts as well using the simplified model shown in Figure 5. The point contact can describe e.g. the contact between the piston ring and the cylinder surface. The length of the contact in transverse direction is U L (in case of the piston ring its circumferential length), and the contact stiffness is c. The contact is moving in x-direction with velocity v. Due to a non-homogeneous heating the surface of the body has a wavy form 0 z h coskx = .
(28) Therefore the normal force will depend on the coordinate x. The periodic force component equals N 0
F cz ch coskx ∆ = = .(29)
The resulting change in friction force is computed by
R N 0 F F ch coskx ∆ = µ∆ = µ .(30)
This leads to a heterogeneous heat production on the surface. Due to it's short contact length the heat production will decay during the time t L / v ≈ between two subsequent contacts following an exponential law 2 k t e − α . Therefore only disturbances with wave vectors ( )
1/ 2 1/ 2 L k t v − − α ≈ α = (31)
will become unstable.
Summary
We have shown that the interplay of the thermal expansion, the contact mechanics and the frictional heat production in both finite and point contact conditions can result in a thermomechanical instability under certain conditions. A system with sufficiently large contact length can be analyzed by analytical means as accomplished in the first two paragraphs. Short contact lengths however will need a numerical analysis. Using a first rough analytical estimate we proved that a thermo-mechanical instability can be caused by a gliding point contact. A thorough numerical analysis is still pending. The thermo-mechanical instabilities and their particular characteristics seem to have simple physical reasons. The characteristic wave number (32) is based in principle on the propagation length of the heat during the half period of the stroke. The most important technical parameter on the critical speed is the coefficient of friction. An instability will only develop for sufficiently high sliding velocities, and results from the balance of heat production and heat conductance.
Figure 1 :
1View of a corrugated (washboard pattern) cylinder
Figure 2 :
2Areas with higher temperature bulge due to their thermal expansion, resulting in an increased friction energy dissipation and therefore an increase in local temperature. This can lead to an instability and permanent pattern formation.
Figure 4 :
4Magnification constant ω as function of the wave number k 4Instability problem in a contact of finite lengthNext, we investigate the qualitative condition for a thermo -mechanical instability in a moving contact with finite contact length. For a contact of length l, the contact time is1 t l / v = .Under this condition the amplitude of the disturbance of wave vector k will be magnified during the contact time by the factor
Figure 5 :
5Gliding point contact on a wavy surface.
The conditions for instability can be determined as follows. Half the wave length ∆ ≈µ ∆ = µ π . The penetration depth of the heat has the same order of magnitude as half the wave length x ∆ . The increase in temperature T ∆ can be estimated from the condition that the heat flux j T / x Tkv / ≈ κ∆ ∆ = κ∆ π equals the friction heat per area in square meter and second. The waviness resulting from the thermal expansionx
/ k
∆ ≈ π
will be overrun during time t
/ kv
∆ ≈ π
. During that time the heat release equals
0
0
W
ch x
ch / k
0
U
ch kv
T
L
µ
κ∆ ≈
π
(32)
equals to 0
T
h
k
γ∆ π
≈
. With this equation (32) takes the form
U
cv
L
µγ
κ ≈
. Therefore we obtain
the characteristic critical velocity
U
L
v
c
κ
≈
µγ
.
(33)
The Corrugation of Rails. The Electrician. The Corrugation of Rails. The Electrician, June 15 (1906).
Überblick über das Riffelproblem. M Fink, Jahrbuch des Eisenbahnwesens. 8Fink, M.: Überblick über das Riffelproblem. Jahrbuch des Eisenbahnwesens. 8 (1957) S. 135-148.
Short wavelength rail corrugation and non-steady-state contact mechanics. K Knothe, A Gross-Thebing, Vehicle System Dynamics. 46Knothe, K, Gross-Thebing, A.: Short wavelength rail corrugation and non-steady-state contact mechanics. Vehicle System Dynamics, 46 (2008) pp. 49-66.
Thermally induced roughness of tread-braked railway wheels. T Vernersson, Wear. 236Vernersson, T.: Thermally induced roughness of tread-braked railway wheels. Wear. 236 (1999) pp. 96-105.
Contact mechanics and friction. Physical principles and applifcations, Springer. V L Popov, Berlin, HeidelbergPopov, V.L. Contact mechanics and friction. Physical principles and applifcations, Spring- er, Berlin, Heidelberg, 2017.
Method of dimensionality reduction. V L Popov, M Heß, SpringerBerlin, HeidelbergPopov, V.L., Heß, M. Method of dimensionality reduction, Springer, Berlin, Heidelberg, 2015.
. L D Landau, E M Lifschitz, Elastizitätstheorie, Akademie VerlagBerlinLandau, L.D., Lifschitz, E.M. Elastizitätstheorie. Akademie Verlag, Berlin, 1991
|
[] |
[
"Dynamical analysis of the buildup process near resonance",
"Dynamical analysis of the buildup process near resonance"
] |
[
"Jorge Villavicencio \nCentro de Investigación Científica y de Educación Superior de Ensenada Apartado Postal\nFacultad de Ciencias\nAutónoma de Baja California Apartado Postal 1880\nUniversidad\n2732, 22800, 22800Ensenada, Baja California, Ensenada, Baja CaliforniaMéxico, México\n",
"Roberto Romo \nCentro de Investigación Científica y de Educación Superior de Ensenada Apartado Postal\nFacultad de Ciencias\nAutónoma de Baja California Apartado Postal 1880\nUniversidad\n2732, 22800, 22800Ensenada, Baja California, Ensenada, Baja CaliforniaMéxico, México\n"
] |
[
"Centro de Investigación Científica y de Educación Superior de Ensenada Apartado Postal\nFacultad de Ciencias\nAutónoma de Baja California Apartado Postal 1880\nUniversidad\n2732, 22800, 22800Ensenada, Baja California, Ensenada, Baja CaliforniaMéxico, México",
"Centro de Investigación Científica y de Educación Superior de Ensenada Apartado Postal\nFacultad de Ciencias\nAutónoma de Baja California Apartado Postal 1880\nUniversidad\n2732, 22800, 22800Ensenada, Baja California, Ensenada, Baja CaliforniaMéxico, México"
] |
[] |
The time evolution of the buildup process inside a double-barrier system for off-resonance incidence energies is studied by considering the analytic solution of the time dependent Schrödinger equation with cutoff plane wave initial conditions. We show that the buildup process exhibits invariances under arbitrary changes on the system parameters, which can be successfully described by a simple and easy-to-use one-level formula. We find that the buildup of the off-resonant probability density is characterized by an oscillatory pattern modulated by the resonant case which governs the duration of the transient regime. This is evidence that off-resonant and resonant tunneling are two correlated processes, whose transient regime is characterized by the same transient time constant of two lifetimes.
|
10.1063/1.126982
|
[
"https://export.arxiv.org/pdf/quant-ph/0011032v1.pdf"
] | 119,096,753 |
quant-ph/0011032
|
18f4d7ee3350ebbaaea782641daaff18822e4517
|
Dynamical analysis of the buildup process near resonance
Jorge Villavicencio
Centro de Investigación Científica y de Educación Superior de Ensenada Apartado Postal
Facultad de Ciencias
Autónoma de Baja California Apartado Postal 1880
Universidad
2732, 22800, 22800Ensenada, Baja California, Ensenada, Baja CaliforniaMéxico, México
Roberto Romo
Centro de Investigación Científica y de Educación Superior de Ensenada Apartado Postal
Facultad de Ciencias
Autónoma de Baja California Apartado Postal 1880
Universidad
2732, 22800, 22800Ensenada, Baja California, Ensenada, Baja CaliforniaMéxico, México
Dynamical analysis of the buildup process near resonance
(March 31, 2022)arXiv:quant-ph/0011032v1 9 Nov 2000
The time evolution of the buildup process inside a double-barrier system for off-resonance incidence energies is studied by considering the analytic solution of the time dependent Schrödinger equation with cutoff plane wave initial conditions. We show that the buildup process exhibits invariances under arbitrary changes on the system parameters, which can be successfully described by a simple and easy-to-use one-level formula. We find that the buildup of the off-resonant probability density is characterized by an oscillatory pattern modulated by the resonant case which governs the duration of the transient regime. This is evidence that off-resonant and resonant tunneling are two correlated processes, whose transient regime is characterized by the same transient time constant of two lifetimes.
The buildup process of electrons inside the quantum well of a double barrier (DB) resonant structure has been one of the most important problems under investigation since it governs the ultimate speed of high frequency tunneling devices 1,2 . Although the first theoretical efforts to estimate the relevant time scales for this mechanism were based on stationary approaches 2, 3 , it has been widely recognized that this process is of a dynamical nature 3,4 ; hence, the solution of the time-dependent Schrödinger equation provides the most reliable way to tackle this fundamental problem.
In this letter we provide a full quantum dynamical study of the buildup process at resonance and offresonance incidence energies, based on an exact analytical solution. We show that within the complexity of the process, underlying invariances can be found in the time evolution of the probability density. Important conclusions on the relevant time scale that characterizes both the resonant and off-resonant buildup are brought out from such invariances.
Our analysis deals with the analytic solution of the time-dependent Schrödinger equation for a finite range potential V (x) that vanishes outside the region 0 ≤ x ≤ L, with a cutoff plane initial condition,
Ψ (x, k; t = 0) = e ikx − e −ikx , −∞ < x ≤ 0, 0, x > 0,(1)
which refers to a perfect reflecting shutter 5 . The solution for the internal region 6 reads,
Ψ(x, k; t) = φ(x, k)M (0, k; t) − φ * (x, k)M (0, −k; t) −i ∞ n=−∞ φ n M (0, k n ; t),(2)
where φ(x) stands for the stationary solution, and φ n = 2ku n (0)u n (x)/(k 2 − k 2 n ) is given in terms of the resonant states, {u n (x)}, and the S−matrix poles, {k n }, of the system. The index n runs over the complex poles k n , distributed in the third and fourth quadrants in the complex k-plane. In the above equation the Moshinsky functions M (0, q; t) are defined in terms of the complex error function w(z), as M (0, q; t) ≡ M (y q ) = w(iy q )/2, where the argument is given by y q = −exp(−iπ/4)(m/2h) 1/2 (hq/m)t 1/2 , and q stands either for ±k or k ±n .
The formal solution for the internal region, Eq. (2), allows us to study both the spatial behavior and the time evolution of the probability density for any incidence energy E =h 2 k 2 /2m. The main ingredients of Eq. (2) are the stationary wavefunction φ(x, k), the resonance parameters {E n = ε n − iΓ n /2 =h 2 k 2 n /2m} and the corresponding resonant eigenfunctions {u n (x)}. The latter can be obtained by a straightforward calculation using the transfer matrix method adapted to the complex eigenvalue problem 7 . For systems with isolated (non-overlapping) resonances, the single-resonance term approximation to Eq. (2) gives an excellent description, except for very short times (t ≪h/Γ n ) in which one has to consider contributions from additional resonance terms. In all the numerical examples presented here the single term approximation applies.
As a first example, let us consider the DB struc-ture (system A) with parameters: barrier heights V 1 = V 2 = 0.23 eV , barrier widths b 1 = b 2 = 5.0 nm, well width ω 0 = 5.0 nm, and effective mass for the electron m = 0.067m e . The resonance parameters for the first resonant state are: energy position, ε 1 = 80.11 meV , and resonance width, Γ 1 = 1.03 meV . In Fig. 1 (a) we plot |Ψ| 2 , calculated from Eq. (2), as a function of the position x along the internal region, for specific times {t i } whose increasing values are given in the figure.
Here, the incidence energy is chosen below resonance, at E = 75.0 meV . Note that the off-resonant buildup occurs in such a way that |Ψ| 2 is found sometimes above or below the asymptotic value |φ| 2 . This behavior is dramatically different from the monotonic growth that characterizes the special case of incidence at resonance, see Fig. 4 of Ref. 8. In order to show the time dependence of the probability density for a fixed position x 0 , we plot in part (b) |Ψ(x 0 , k; t)| 2 versus t for different deviations from resonance ∆E = |E − ε 1 |. Note that |Ψ| 2 fluctuates around its asymptotic value |φ| 2 , and exhibits an oscillatory behavior not present in the resonant case, as shown in Fig. 1 (b). Up to here, we have illustrated the behavior of the offresonance buildup only for a particular potential profile. It is clear that any changes in either the incident energy or the potential profile parameters will affect the solution Ψ(x, k; t) since the relevant input to Eq. (2), namely, φ (x, k), u n (x) and E n , strongly depends on the potential parameters. For instance, let us consider two additional DB systems with potential profiles quite different from system A; the first corresponds to the symmetrical structure (system B) with parameters: barrier heights V 1 = V 2 = 0.5 eV , barrier widths b 1 = b 2 = 3.0 nm and well width ω 0 = 10.0 nm; the second corresponds to an asymmetrical structure (system C), whose parameters are: V 1 = 0.45 eV, V 2 = 0.35 eV , b 1 = 3.0 nm, b 2 = 10.0 nm and ω 0 = 8.0 nm. The resonance parameters for the first eigenstate are: ε 1 = 37.80 meV , Γ 1 = 0.12 meV (system B); and ε 1 = 51.29 meV , Γ 1 = 0.17 meV (system C). The value of the transmission peak T (ε 1 ) is unity for A and B; for system C, T (ε 1 ) < 1, since it is asymmetric. We choose here the incidence energies E = ε 1 − ∆E such that the ratio γ = T (E)/T (ε 1 ) is the same for A, B, and C. For example, if we choose E such that T (E) is 1.0 % of T (ε 1 ), from numerical inspection from a T versus E plot (not shown here), the incidence energies for A, B, and C, must be 74.97, 37.20 and 50.44 (meV ), respectively. The results of the comparison of the time evolution of |Ψ| 2 for this selection (γ = .01), are shown in Fig. 2 (a). The three curves are strongly different, as expected. Thus, the complete characterization of the buildup process for a broad range of potential geometries seems to be a too involved task; however, one of the purposes of this work is to show that despite this complex situation, the probability density has striking invariances under changes in the potential profiles. To illustrate the above let us consider a more suitable representation for the probability density i. e. |Ψ/φ| 2 as a function of τ , which is the time normalized to lifetime units. We find a striking result: all curves coincide exactly for the three different systems, see Fig. 2 (b). This result suggests the existence of an underlying invariance in the process.
In order to show the existence of such invariance, we derive a one-level formula for the normalized probability density starting from the formal solution (2). We proceed along the same lines discussed in our recent work 9 , but considering incidence energies different from resonance (E = ε n ). Following such a procedure we obtain,
|Ψ (τ ) /φ| 2 = 1 + e −τ − 2e −τ /2 cos [ω n τ ] ,(3)
where ω n = (ε n − E)/Γ n is a dimensionless frequency.
The reliability of this one-level formula is shown in a plot of Eq. (3) included in Fig. 2 (b), and we see an excellent agreement. Furthermore, note that Eq. (3) depends in general on the system parameters through the frequency ω n ; however, it can be shown straightforwardly that the condition previously imposed on the ratio T (E)/T (ε n ) guarantees the independence of Eq. (3) on the potential profile. Consider the Breit-Wigner expression for the transmission coefficient 7 ,
T (E) = Γ 0 n Γ L n (E − ε n ) 2 + Γ 2 n /4 ,(4)
where Γ 0 n and Γ L n are the partial decay widths of the system which satisfy Γ n = Γ 0 n + Γ L n . This formula of T (E) is valid for isolated non-overlapping resonances, that is Γ n ≪ |ε n − ε n±1 |, which is the case of a broad range of typical DB structures. From Eq. (4) we can easily calculate the ratio γ = T (E)/T (ε 1 ) and obtain an expression for the frequency ω 1 = γ −1 − 1 1/2 /2. As a consequence, Eq. (3) is no longer dependent on the system parameters since it only depends on γ. Note that the frequency ω 1 also measures the deviations of the incidence energy E from the resonance ε 1 in multiple numbers of Γ 1 , i.e. ∆E = ω 1 Γ 1 . In other words, different systems share the same curve of the probability density provided that deviations from resonance are the same in units of the corresponding resonance width Γ 1 . Note also that Eq. (3) is independent of the choice ±ω n , which implies that deviations above and below resonance give the same result. Since in our example γ = 0.01, we have that ω 1 ≈ 5.0; this can also be verified by computing the values of ω 1 = |∆E|/Γ 1 from the incidence energies used in Fig. 2. Note another interesting regularity of |Ψ (τ ) /φ| 2 ; the damped oscillatory behavior in Eq. (3) is modulated by the lower envelope
|Ψ (τ ) /φ| 2 = 1 − e −τ /τ0 2 ,(5)
which is exactly the capacitor-like buildup law obtained for the special case of incidence at resonance 9 , where the transient time constant τ 0 of the process is exactly two lifetimes, τ 0 = 2. A plot of Eq. (5) is included in Fig. 2 (b). This result is relevant from a physical point of view, since it is a manifestation of the subtle interplay between the incident off-resonant carriers and the quasibound state of the system: resonant and off-resonant buildup, although different processes, are not uncorrelated at all, the latter is governed by the former in the way exhibited in Fig. 2(b). As a consequence, the transient regime for both situations is characterized by the same transient time constant τ 0 . The above mentioned quantity is relevant for the design and optimization of resonant tunneling diodes; in this respect, a detailed discussion can be found in a recent work by Luryi and Zaslavsky 10 , in which the distinction between capacitive and quantum contributions to an effective time constant is analyzed.
In conclusion, the dynamics of the buildup mechanism at off-resonance incidence energies has been explored in typical DB resonant structures. We have shown that, despite the complexity that characterizes the dynamical process, the time evolution of the probability density exhibits invariances under arbitrary changes on the system parameters. From such invariances we conclude that the transient regime in both resonant and off-resonant processes is characterized by the same transient time constant of two lifetimes. Our results are valid for any DB system with isolated resonances and incident plane wave initial condition.
FIG. 1 .
1(a) The birth of |Ψ| 2 inside the structure as a function of the position x, for increasing values of time: t1 = 0.04 ps, t2 = 0.4 ps t3 = 0.8 ps, and t4 = 1.2 ps (solid lines). The stationary solution |φ| 2 (dashed line) is also included for comparison. (b) The time evolution of |Ψ| 2 at the fixed position at the center of the well, for different values of ∆E = ∆E k , where: ∆E1 = 0.6, ∆E2 = 1.1, and ∆E3 = 1.6 (meV).
FIG. 2 .
2(a) The time evolution of |Ψ| 2 in the center of the well at off-resonance incidence energy using Eq. (2) for systems A, B, and C. (b) Also from Eq. (2), shows the time evolution of |Ψ(τ )/φ| 2 as a function of the time τ given now in lifetime units; the curves of A, B and C are indistinguishable among them. The calculation using the one-level formula, Eq.(3), is also included in (b) for comparison and is indistinguishable from A, B, and C. The lower envelope calculated from Eq. (5) is also shown.
τ (lifetimes)
The authors acknowledge financial support from Conacyt, México, through Contract No. 431100-5-32082E. The authors also thank G. García-Calderón for useful discussions.
. T C L G Sollner, W D Goodhue, P E Tannenwald, C Parker, D D Peck, Appl. Phys. Lett. 43588T. C. L. G. Sollner, W. D. Goodhue, P.E. Tannenwald, C. D Parker and D. D. Peck, Appl. Phys. Lett. 43, 588 (1983).
. S Luryi, Appl. Phys. Lett. 47490S. Luryi, Appl. Phys. Lett. 47, 490 (1985).
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. M Tsuchiya, T Matsusue, H Sakaki, Phys. Rev. Lett. 592356M. Tsuchiya, T. Matsusue, and H. Sakaki, Phys. Rev. Lett. 59, 2356 (1987);
. H Yoshimura, J N Schulman, H Sakaki, Phys. Rev. Lett. 642422H. Yoshimura, J. N. Schulman, H. Sakaki, Phys. Rev. Lett. 64, 2422 (1990);
. M A Talebian, W Pötz, Appl. Phys. Lett. 691148M. A. Talebian and W. Pötz, Appl. Phys. Lett. 69, 1148 (1996).
Note that the shutter is a device that aids to visualize the initial condition and hence it is not part of the system. Note that the shutter is a device that aids to visualize the initial condition and hence it is not part of the system.
This solution was obtained by one of the authors (J. V.) as an extension of the solution for the absorbing shutter introduced by G. García-Calderón and A. Rubio. Phys. Rev. A. 553361This solution was obtained by one of the authors (J. V.) as an extension of the solution for the absorbing shutter in- troduced by G. García-Calderón and A. Rubio, Phys. Rev. A 55, 3361 (1997).
. G García-Calderón, R Romo, A Rubio, Phys. Rev. B. 5015142G. García-Calderón, R. Romo and A. Rubio, Phys. Rev. B 50, 15142 (1994).
. G García-Calderón, A Rubio, Phys. Rev. A. 553361G. García-Calderón and A. Rubio, Phys. Rev. A 55, 3361 (1997).
. R Romo, J Villavicencio, Phys. Rev. B. 602142R. Romo and J. Villavicencio, Phys. Rev. B 60, R2142 (1999).
S Luryi, A Zaslavsky, Modern Semiconductor Device Physics. S. M. SzeNew YorkWileyS. Luryi and A. Zaslavsky, in Modern Semiconductor De- vice Physics, edited by S. M. Sze (Wiley, New York, 1998), Chap. 5, pp. 253-342.
|
[] |
[
"Beam polarization effects on top-pair production at the ILC",
"Beam polarization effects on top-pair production at the ILC"
] |
[
"Nhi M U Quach \nThe Graduate University for Advanced Studies (SOKENDAI)\n240-0193HayamaKanagawaJapan\n\nHigh Energy Accelerator Research Organization (KEK)\n305-0801TsukubaIbarakiJapan\n",
"Yoshimasa Kurihara \nHigh Energy Accelerator Research Organization (KEK)\n305-0801TsukubaIbarakiJapan\n",
"Khiem H Phan \nUniversity of Science Ho Chi Minh City\n227 Nguyen Van Cu, Dist. 5Ho Chi Minh CityVietnam\n",
"Takahiro Ueda \nNikhef\nScience Park 1051098 XGAmsterdamThe Netherlands\n"
] |
[
"The Graduate University for Advanced Studies (SOKENDAI)\n240-0193HayamaKanagawaJapan",
"High Energy Accelerator Research Organization (KEK)\n305-0801TsukubaIbarakiJapan",
"High Energy Accelerator Research Organization (KEK)\n305-0801TsukubaIbarakiJapan",
"University of Science Ho Chi Minh City\n227 Nguyen Van Cu, Dist. 5Ho Chi Minh CityVietnam",
"Nikhef\nScience Park 1051098 XGAmsterdamThe Netherlands"
] |
[
"Eur. Phys. J. C"
] |
Full one-loop electroweak corrections for an e − e + → tt process associated with sequential t → bμν μ decay are discussed. At the one-loop level, the spinpolarization effects of the initial electron and positron beams are included in the total and differential cross sections. A narrow-width approximation is used to treat the top-quark production and decay while including full spin correlations between them. We observed that the radiative corrections due to the weak interaction have a large polarization dependence on both the total and the differential cross sections. Therefore, experimental observables that depend on angular distributions such as the forward-backward asymmetry of the top-production angle must be treated carefully including radiative corrections. We also observed that the energy distribution of bottom quarks is largely affected by the radiative corrections.
|
10.1140/epjc/s10052-018-5895-9
| null | 115,174,507 |
1706.03432
|
e4f944633c7c72c74b3b262c6be99f7131994e35
|
Beam polarization effects on top-pair production at the ILC
2018
Nhi M U Quach
The Graduate University for Advanced Studies (SOKENDAI)
240-0193HayamaKanagawaJapan
High Energy Accelerator Research Organization (KEK)
305-0801TsukubaIbarakiJapan
Yoshimasa Kurihara
High Energy Accelerator Research Organization (KEK)
305-0801TsukubaIbarakiJapan
Khiem H Phan
University of Science Ho Chi Minh City
227 Nguyen Van Cu, Dist. 5Ho Chi Minh CityVietnam
Takahiro Ueda
Nikhef
Science Park 1051098 XGAmsterdamThe Netherlands
Beam polarization effects on top-pair production at the ILC
Eur. Phys. J. C
78422201810.1140/epjc/s10052-018-5895-9Received: 11 June 2017 / Accepted: 14 May 2018 / Published online: 28 May 2018Regular Article -Theoretical Physics
Full one-loop electroweak corrections for an e − e + → tt process associated with sequential t → bμν μ decay are discussed. At the one-loop level, the spinpolarization effects of the initial electron and positron beams are included in the total and differential cross sections. A narrow-width approximation is used to treat the top-quark production and decay while including full spin correlations between them. We observed that the radiative corrections due to the weak interaction have a large polarization dependence on both the total and the differential cross sections. Therefore, experimental observables that depend on angular distributions such as the forward-backward asymmetry of the top-production angle must be treated carefully including radiative corrections. We also observed that the energy distribution of bottom quarks is largely affected by the radiative corrections.
Introduction
The discovery of the Higgs boson [1,2] in 2012 showed the standard theory of particle physics to be well established. Even though the standard theory can describe the microscopic nature at a subatomic level very precisely [3], it cannot be the most fundamental theory of nature because, for instance, it includes many parameters (e.g., particle masses and couplings, number of generations) that are not determined within the theory. While experiments at the Large Hadron Collider continue to search for signals beyond the standard model (BSM), none have been reported to date. 1 Besides discovering new particles, pursuing the BSM also involves precise measurements of the properties of known particles. Milestones along this direction must surely be the Higgs boson and the top quark. Because the top quark is the heaviest fermion with a mass above even the electroweak a e-mail: [email protected] symmetry-breaking scale, it is naturally expected to play a special role in the BSM. In addition, it has been pointed out that the vacuum stability of the Higgs potential depends strongly on the Higgs and top-quark masses [6]. Hence, the precise measurement of top-quark properties is crucial for understanding the stability of the universe, as well as for the search for BSM signals.
The International Linear Collider (ILC) [7], which is a proposed electron-positron colliding experiment with centerof-mass (CM) energies above 250 GeV, is being discussed intensively as a future project in high-energy physics. The main goals of ILC experiments would be a precise measurement of the Higgs and top-quark properties and searching directly for new particles. The ILC will use spin-polarized beams for both electron and positron beams [8,9] to increase its sensitivity to new physics and to improve its measurement accuracy. The design values of the beam polarization are 80% for the electron beam and 30% for the positron beam with beam energies below 1000 GeV [10]. For many processes, beam polarization is a simple way to increase the signal cross section while suppressing the background. Moreover, beam polarization allows new properties to be measure (e.g., the polarization dependence of cross sections). Detailed Monte Carlo studies have shown that the ILC would be able to measure most of the standard model parameters to within subpercent levels [11].
Because of the improved experimental accuracy intended of the ILC, theoretical predictions must be given with new level of precision. In particular, a radiative correction due to the electroweak interaction (including spin polarizations) is mandatory for such requirements. Before the discovery of the top quark, a full electroweak radiative correction was conducted for an e − e + → tt process at a lower energy [12], and it was then obtained independently for higher energies [13,14]. The same correction including radiative photon, the e − e + → ttγ process, has also been reported [15]. Higherorder corrections including photon radiation are important for the precise prediction of cross sections because the initial photon radiation affects the total cross sections significantly. However, none of previous calculations include the effect of spin polarization. Some application of polarized cross sections of this process including full O(α) electroweak corrections is reported in Ref. [16], in which polarized cross sections are obtained using the method presented here by the authors of the current report.
In the present study, we report full electroweak radiative corrections for the process e − e + → tt → bbμ + μ − ν μνμ using a narrow-width approximation for the top quarks. Spinpolarization effects are included, not only in the initial beams, but also in the full spin correlations of the production and decay of top quarks. While Born cross sections of the process e − e + → bbμ + μ − ν μνμ including all six-body final state are given in Ref. [17], an electroweak radiative correction of the tt process associated by their decay including a spin correlation is not calculated yet. On the other hand, NLO QCD corrections for on-shell tt and tt H including decays are calculated in Ref. [18]. A detailed study of the electroweak correction on the top-quark decay is also reported in Refs. [19,20].
This report is organized as follows. The calculation method is explained in Sect. 2. We use the GRACE-Loop system to calculate the cross sections. A system-checking method is also explained in Sect. 2. In Sect. 3, we show results of electroweak corrections of the total cross section as well as of the angular distribution with spin-polarized beams. The effects of radiative corrections on top-quark decay products, including a spin correlation, are also discussed using a narrow-width approximation. The contribution of an NLO-QCD correction is briefly discussed in Sect. 3. We summarize and conclude this report in Sect. 4. In Appendix A, we summarize the formulas of the NLO-QCD correction for massive quark production.
Calculation method
For precise cross-section calculations of the target process in this study, we used the GRACE-Loop system, which is an automatic system for calculating cross sections of scattering processes at one-loop level for the standard theory [21] and the minimal supersymmetric standard model [22]. This system has been used to treat electroweak processes with two, three, or four particles in the final state [23][24][25][26]. The GRACE-Loop system has the following features: (1) The renormalization of the electroweak interaction is carried out using an on-shell scheme [27,28]. (2) The infrared divergences are regulated using a fictitious photon mass λ [28]. (3) The symbolic manipulation system FORM [29] is used to handle all Dirac and tensor algebras in n spacetime dimensions. (4) GRACE generates FORTRAN source code that calls library subroutines to calculate the scattering amplitudes. (5) For loop integrations, all tensor one-loop integrals are reduced to scalar integrals using our own formalism, whereupon the integrations are performed using the packages FF [30] and LoopTools [31]. (6) Phase-space integrations are done using an adaptive Monte Carlo integration package BASES [32,33]. (7) For numerical calculations, we use quadruple precision for floating-point variables.
To treat spin polarization in loop calculations, the projection operators are applied on fermion wave functions. A spin projection of the initial beams is realized simply by multiplying the spin-projection operator P λ = 1 2 (1 + λγ 5 / p/m), where p is the four-momentum of beam particles and λ = ±1 is their helicity. Here, we assume that the initial beams comprise light fermions with no transverse momenta. The electron/positron completeness relation becomes s u( p) sūs ( p) = 1 2 (1 + λγ 5 )( / p + m). For top quarks, the spin-polarization vector can be taken as
s μ t = p t ·ŝ t m t ,ŝ t + ( p t ·ŝ t ) p t m t (E t + m t ) ,
where m t is the top-quark mass, E t is the top-quark energy, and p t is the top-quark three-momentum. The spin is projected on the direction of the top-quark three-momentum using a direction vectorŝ t = p t /| p t |. The completeness relations in this case are given as λ u( p, λ)ū( p, λ) = In GRACE, while using the R ξ -gauge in the linear gaugefixing terms, the non-linear gauge-fixing Lagrangian [21,34] is employed, namely
L G F = − 1 ξ W ∂ μ − ieα A μ − igc Wβ Z μ W μ+ + ξ W g 2 v +δ H + iκχ 3 χ + 2 − 1 2ξ Z ∂ · Z + ξ Z g 2c W v +ε H χ 3 2 − 1 2ξ A (∂ · A) 2 ,
for the sake of system checking. Here A, Z , W, χ, and H denote the wave functions of the corresponding fields, and ξ 's are gauge parameters for the linear gauge-fixing terms. The results must be independent of the non-linear gauge parameters {α,β,δ,κ,ε}. We can perform system checking numerically to confirm the correctness of the system. Before calculating cross sections, we checked for ultra-violet coefficient (C UV ) independence, photon-mass (λ) independence, and gauge invariance numerically at several randomly cho- Yukawa coupling cannot be neglected to achieve such precision. We note that the parameter dependence of the amplitude is logarithmic for C UV and λ, while it is up to quartic for the non-linear gauge parameters. In addition to the above checks, we examined the soft-photon cut-off independence: for cross sections at the one-loop level, the results must be independent of a hard-photon cut-off parameter k c . We confirmed that the integration results are self-consistent within the statistical error of numerical phase-space integrations while varying k c from 10 −4 to 10 −1 GeV.
Results and discussions
For cross-section calculations of the production process e − e + → tt and its sequential top decay, we use the input parameters listed in Table 1. The masses of the light quarks (i.e., other than the top quark) and W boson are chosen to be consistent with low-energy experiments [35]. Other particle masses (including an electron mass) are taken from recent measurements [3]. For renormalization scheme, the on-shell scheme is used, in which input parameters to determine electroweak couplings are W -boson and Z -boson masses, and the fine-structure constant. The Weinberg mixing angle is obtained using the on-shell condition,
sin 2 θ W = 1 − m 2 W /m 2 Z .
The fine-structure constant α = 1/137.0359859 is taken from the low-energy limit of Thomson scattering due to our renormalization scheme. The W -boson and Zboson widths are taken as the calculated value at tree level using the same parameters given above.
Production cross sections
We focus on CM energies above 500 GeV to avoid possible complications from large QCD corrections near the produc- Fig. 1 Examples of Feynman diagrams for e − e + → tt at tree level, with real radiation and at loop level. In our cross-section calculations, all diagrams include contributions from Goldstone bosons and lightfermion Yukawa couplings tion threshold. In the energy region, an experimental target of top-quark physics is a precise measurement of the Z -top and top-Yukawa couplings. It is reasonable to expect that information beyond the standard theory could be probed through precise measurements of the top-production form factor [16]. To extract new physics from the form-factor measurement, one has to understand precisely the effects of higher-order corrections on the measurements. For instance, a signal of the scalar top in the MSSM can be observed through the loop effect in the top-quark production [36].
For the e − e + → tt process, there are four Feynman diagrams at tree level, 16 with real-photon radiation, and 150 at the one-loop level. Typical diagrams are shown in Fig. 1.
We calculate the total cross sections as a function of CM energy of 500-1000 GeV assuming 100% left-hand polarization for electrons (e − L ) and 100% right-hand polarization for positrons (e + R ), or vice versa (e − R and e + L ). The cross sections so obtained are shown in Fig. 2 as functions of the colliding energy. As shown in the upper panels of Fig. 2, the total cross sections for the e − L e + R collision are roughly twice those for the e − R e + L collision due to the P-violation of the weak interaction.
Cross sections with realistic polarizations of the design value (e − L = 80% and e + R = 30%) can be obtained from those with 100% polarized results as follows: the left-handed polarization degree of the electron beam is defined as p e = (N L − N R )/(N L + N R ), where N L and N R are number of lefthanded and right-handed electrons in the beam, respectively. When a normalization N L + N R = 1 is used, the normalized number of left-handed and right-handed electrons can be obtained as N L = (1 + p e )/2 and N R = (1 − p e )/2, respectively. Therefore, the cross sections with left-handed
σ ( p e , p p ) = (1 + p e )(1 + p p ) 4 σ LR + (1 − p e )(1 − p p ) 4 σ RL .
where σ LR (σ RL ) are cross sections with the 100% left polarized (right polarized) electron and the 100% right polarized (left polarized) positron beams, respectively. We omit contributions involving e − L e + L and e − R e + R collisions because they yield negligible cross sections.
When design values of polarizations will be realized at the ILC, one can gain roughly 50% in total cross section compared with the non-polarized case. In addition, the total amount of electroweak corrections is smaller for the e − L e + R case than that for the e − R e + L case. For a simple evaluation of the fraction of higher-order corrections, let us introduce the ratio δ = (σ NLO − σ Tree )/σ Tree , where σ NLO and σ Tree are the total cross sections at a full O(α) correction and that at tree level, respectively. The results so obtained are summarized in Fig. 3. For instance, at a CM energy of 500 GeV, the electroweak correction of e − L e + R is − 0.8% and the electroweak correction of e − R e + L is 12%. At a CM energy of 1000 GeV, the electroweak correction of e − L e + R is 4.0%, where the electroweak correction of e − R e + L is 26%. The e − R e + L polarization has larger radiative corrections than those of the e − L e + R one. Together with the larger cross sections, one can expect smaller systematic errors for the cross-section measurement with the polarized beam than in the non-polarized case. As shown in Fig. 3, an electroweak radiative correction gives very small radiative corrections on the polarized beam with the design value. While the non-polarized cross section also has small radiative corrections, the difference between the non-polarized and design polarized cases is significant. These small corrections on the total cross sec- tions are due to the accidental cancellation among loop diagrams. This situation is suitable for new physics searches. If the top quark has anomalous couplings with gauge bosons, those signals can be observed with small systematic errors [16].
We note that the full electroweak correction reported here includes a trivial photonic correction from the initial-state photon radiation (ISR). It is well known that the ISR correction can be factorized and be improved using a higherorder re-summation [28]. The polarization asymmetry of electroweak corrections may be induced by diagrams involving W bosons [37], i.e., the diagrams shown in Fig. 1. In this report, we do not discuss the origin of the radiative-correction asymmetry in detail.
Angular distributions
The angular distribution of the top-pair production has a large forward peak, and thus it has a sizable forward-backward asymmetry that allows us to make a good test of the standard theory. However, radiative corrections may distort the angular distribution as well as the total cross sections. Angular distributions of the top-pair production with and without radiative corrections at the CM energy of 500 GeV are shown in Fig. 4 for both e − L e + R (left figure) and e − R e + L (right figure) polarizations. The ISR corrections generally flatten the forward peak because of a smearing effect of the CM system. One can see this smearing effect clearly in the e − L e + R polarization case. Even though the total correction δ is small at √ s = 500 GeV, as mentioned above, the electroweak correction modifies the angular distribution. A small cor-rection to the total cross section is caused by an accidental cancellation between negative corrections for the forward region and a positive contribution in the backward region. In contrast, the electroweak correction for the e − R e + L polarization gives positive corrections in the whole angular region, as shown in the right-hand panel in Fig. 4. In conclusion, the observed value of the forward-backward asymmetry is largely affected by the electroweak radiative corrections. Moreover, the effect of the radiative corrections depends on the spin polarization of the initial beams. Therefore, careful investigations of the forward-backward asymmetry are required.
A definition of the forward-backward asymmetry is given as follows. The forward and backward cross sections are defined as σ F = 1 0 dσ /d cos θ t d cos θ t and σ B = 0 −1 dσ /d cos θ t d cos θ t , respectively. Thus, the forwardbackward asymmetry is defined by
A FB = (σ F − σ B )/(σ F + σ B )
. The tree and electroweak-corrected values of the forward-backward asymmetry at the CM energy of 500 GeV are summarized in Table 2. For e − L e + R (e − R e + L ) polarization, the forward-backward asymmetry at tree level is 0.385 (0.467), which becomes 0.317 (0.443) after the full electroweak correction. When the design values of polarizations are assumed, the forward-backward asymmetry is determined mainly by the contribution from the e − L e + R component, as shown in the last row of Table 2.
Top-quark decay
According to the beam polarization, the produced top quarks are also polarized. The polarization degree is defined as δ pol = (σ L − σ R )/(σ L + σ R ), where σ L and σ R are the cross sections for creating the left-handed and right-handed top quark, respectively. The polarization degree depends on the CM energy, as shown in Fig. 5. At tree level, the polarization degree increases from 8.8% at 350 GeV to 67.6% at 800 GeV. At a CM energy of 350 GeV (close to the production threshold), the produced top quark moves slowly and thus its helicity state is easily flipped. In contrast, at higher energies, the particle moves much faster and the helicity is stable. That causes the difference in polarization to increase with energy, as shown in Fig. 5. The full electroweak corrections reduce the polarization degree by roughly 10% in the high-energy region. The top quark immediately decays into a bottom quark and a fermion pair. Because the angular and energy distributions of the decay products depend strongly on the top polarization, an exact treatment of the top polarization is mandatory. We discuss the top decay of t −→ bμ + ν μ at a CM energy of 500 GeV as a benchmark process. Because b-quark tagging is required to identify the top quark experimentally, precise calculation of b-quark distributions is important. The number of Feynman diagrams for the six-body final state e − e + → bbμ − μ + νν is too large, and thus a full electroweak correction is impossible using the current computing power. Instead, we have used a narrow-width approximation (NWA) for the top-quark production and decay, including the spin correlation exactly. A more sophisticate method to treat a particle production and decay consistently at a one loop order is known as the double-pole approximation. This method is developed for a W -boson pair production [38,39] at first, and later it is applied to a top-quark production [40] too. We do not employ the double-pole approximation in this study, because a simple NWA is enough to discuss an effect of electroweak corrections on a top-quark polarization. E.g., an energy distribution of decayed b-quarks is mainly determined by the top-quark polarization degree.
The branching ratio of the bμ + ν μ decay is obtained with the O(α) correction as follows: the top width at tree level is calculated to be Tree = 1.416 GeV. The full electroweak-corrected width is calculated by summing all possible decay channels of t → blν l and t → bqq as Loop = 1.421 GeV. The partial width of the decay channel to bμ + ν μ is 0.1535 GeV, thus the branching ratio of this channel is obtained as 10.8% after the O(α) correction. Here, only electroweak corrections are included. The effect of the QCD higher-order correction is known to be about − 5% (hadronic decays) and − 9% (semi leptonic decays) [20,41], and they are not included in this study. In our approximation, corrections on the top-quark width affect only on the branching ratio of some specific decay channel, and then they does not affect on any distributions. On the other hand, radiative corrections on the top-quark spin polarization largely affect on energy distributions of b-quarks.
The total cross section of N -body production including a narrow fermion resonance with mass m and width , which decays into N bodies, can be expressed as
σ = 1 f lux |M| 2 d N = 1 f lux λ M p u λ (q)ū λ (q)M d 2 (q 2 − m 2 ) 2 + m 2 2 × dq 2 2π d cos θ q dϕ q d n d N −n ,
where u λ is the spinor, q μ is the momentum (off-shell), and λ is the spin of the resonance particle. The term d n denotes an n-body phase space, and M p and M d are the product and decay amplitudes, respectively. Using an onshell approximation as q 2 ∼ q 2 0 = m 2 for the numerator, the amplitudes can be approximated byM λ p = M p u λ (q 0 ) andM λ d = M d u λ (q 0 ). Therefore, the total cross section becomes
σ 1 f lux λ M λ p 2 d cos θ q dϕ q d N −n M λ d 2 d n 1 (q 2 − m 2 ) 2 + m 2 2 dq 2 2π .
We note that the spin correlation is maintained between production and decay. Integration can be performed over the resonance masses, namely
M λ d 2 d n +∞ −∞ 1 (q 2 − m 2 ) 2 + m 2 2 dq 2 2π = 1 1 2m M λ d 2 d n ,
which gives the branching ratio of a specific decay channel. In reality, calculations are performed using the exact six-body phase space. The validity of the NWA is verified by com-paring b-quark distributions obtained by the narrow-width and the exact six-body calculations at tree level. Both results agree each other within the statistical error of Monte Carlo calculations. Since the contribution from non-resonant diagrams is negligible [17] up to the CM energies considering in this study, the NWA are precise enough. For a higher energy region than at TeV order, the contribution from non-resonant diagrams becomes important [18]. The angular and energy distributions of b-quarks are shown in Figs. 6 and 7, respectively. For the e − L e + R polarization case, the decayed b-quarks tend to be produced in the forward direction of the top-quark momentum, and in the backward direction for the e − R e + L polarization. The angular distributions of the b-quarks at tree level reflect this tendency. The electroweak corrections distort the angular distribution rather largely in the e − L e + R polarization case, as shown in the left-hand panel of Fig. 6.
In the top-quark rest frame, the b-quark energy is monochromatic (while ignoring the W -boson width). Thus, the energy distribution of the b-quarks are a reflection of their angular distribution with respect to the top-quark momentum, after the Lorentz boost due to finite top-momentum. From this point on view, the energy distribution of b-quarks can be understood intuitively. Again, the electroweak corrections distort the distribution largely for the e − L e + R case, as shown in Fig. 7.
While these effects on the decay products from the higherorder corrections are important for the precise estimation of the event acceptance, it is also important for the new physics searches. For instance, it is reported that the spin correlation between top and anti-top quarks is sensitive to the BSM [42]. The spin polarization of (anti-)top quarks is affected by the
QCD correction
We have not discussed the QCD correction so far in this report because the QCD correction for the top-pair production is independent of the beam polarization and simply modifies the total cross section while maintaining the distributions. However, the QCD correction is not small at a CM energy of 500 GeV. The formulas used here are summarized in Appendix A. While the QCD correction is expected to be α s /π 3.8% at higher energies, it still makes a contribution of 9.7% to the total cross section at a CM energy of 500 GeV. While the QCD correction gradually approaches the asymptotic value of α s /π with increase of the CM energy, as shown in Fig. 8, it still makes a large contribution around a CM energy of 500 GeV. While results including only electroweak corrections are shown in this report, more precise QCD corrections [18,43] must be included for future experimental analysis.
Summary and conclusions
In this report, we have presented full O(α) electroweak corrections for the e − e + → tt process associated with the sequential decay t → bμν μ . Calculations were performed using the GRACE-Loop system. The electroweak radiative correction was estimated typically at a level of 10% on the total cross section in the on-shell scheme for the non- L e + R polarization was roughly twice that with e − R e + L polarization at tree level, the radiative correction of the former was smaller than that of the latter. The electroweak correction with the design polarizations (e − L = 80% and e + R = 30%) was estimated to be less than 5%. Even though the electroweak correction of the total cross sections was rather small for e − L e + R polarization, the radiative corrections modified the angular distribution of the produced top quarks. The radiative corrections decreased the forward-backward asymmetry of the topquark production from 0.388 to 0.321 for the design polar-ization. We also studied the properties of top-quark decay t → bμ + ν μ including the spin correlation. Both production and decay processes were calculated with O(α) corrections and combined with using the narrow-width approximation. We observed the energy distribution of b-quarks to be largely distorted because of the radiative correction. Therefore, an event generator including radiative corrections for both production and decay with the spin correlation will be necessary for precise measurements in future ILC experiments. Because the NLO-QCD correction is still large at CM energies of 500 GeV, a precise QCD correction is also desired. The authors wish to thank Prof. J. Vermaseren and Prof. J. Fujimoto for their continuous encouragement and fruitful discussions. T.U. is supported by the ERC Advanced Grant No.320651 "HEPGAME".
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP 3 .
Appendix A: QCD correction
The detailed formulas of the NLO-QCD correction for massive quark-pair production by electroweak interaction are summarized in this appendix. In the following calculations, the standard MS renormalization scheme is used. After renormalization, a space-time dimension other than four is reinterpreted to regulate the infrared divergence as d = 4−2ε UV → 4+2ε IR with ε IR > 0. The NLO-QCD correction consists of three parts: vertex, self-energy, and real-gluon-emission corrections. The contributions of each part are given separately below.
Vertex correction
The total vertex correction is given as
= C F (I k + I 0 ) ,
where C F = 4/3 is a color factor. Each integration term is given as
I k = α s 4π −1 ε IR + L + log μ t + 1 +μ t log − 1 −μ t 1 +μ t , I 0 = α s 4π 1 ε IR 2 (2μ t + 1) μ t log − 1 −μ t 1 +μ t − 1 − 2 (2μ t + 1) μ t Sp 1 2 − 1 2μ t − Sp 1 2 + 1 2μ t + 2 μ t log − 1 −μ t 1 +μ t (−2 (6μ t + 1) + (2μ t + 1) L + 1 2 log − 1 −μ t 1 +μ t + log −μ t (μ t + 1) 2 , L = log −s μ F , μ t = − m 2 t s ,μ t = 4μ t + 1,
where μ F is the factorization energy scale, m t is the top-quark mass, and s is the momentum square of a tt-system.
Self-energy correction
The self-energy correction appears because of the renormalization scheme. The top mass that appears here must be interpreted as the MS mass:
p 2 = m 2 t = C F α s 4π −1 ε IR + (L t − 4) ,
where L t = log m 2 t /μ 2 F .
Real-emission correction
The real-emission correction is further separated into two parts: soft-gluon emission and hard-gluon emission. A threshold energy k c is introduced to separate soft and hard emissions. The soft-emission corrections are given as
R ii = C F α s 2π 1 ε IR − L k − 1 μ t log − 1 −μ t 1 +μ t , R i j = C F α s 2π −1 ε IR 2μ t + 1 μ t log − 1 −μ t 1 +μ t − 2μ t + 1 μ t L k log − 1 −μ t 1 +μ t + Sp 2 1 + 1/μ t − Sp 2 1 − 1/μ t ,
where L k = 2 log (2k c /μ F ). These formulas are obtained via an approximation in which the gluon energy is much smaller than m t . The hard-emission cross section can be calculated using the GRACE system based on the exact matrix element. We confirmed numerically that real-emission corrections are independent of k c , whose values are below 1 GeV.
Total correction
The NLO-QCD cross section σ NLO can be obtained as
σ NLO = 1 + 2 R ii + R i j + Re [ + ] σ 0 + σ g ,
where σ 0 and σ g are the Born and hard-emission cross sections, respectively. After summing up all contributions, the infrared divergence and μ F dependence disappear completely.
1 2
1(1+λγ 5 / s)( / p+m) for top quarks and λ v( p, λ)v( p, λ) = 1 2 (1 + λγ 5 / s)( / p − m) for anti-top quarks.
Fig. 2
2Total cross sections with respect to the CM energy √ s from 500 to 1000 GeV, assuming 100% of e − L and e + R for the upper-left figure, and vice versa (e − R and e + L ) for the upper-right figure. Lower-left and lower-right figures show cross sections with non-polarization and polarization with a design value (e − L = 80% and e + R = 30%, respectively). The dotted lines show the results for the tree level, while the solid lines correspond to the full one-loop electroweak correction electron polarization p e and right-handed positron polarization p p can be obtained as
Fig. 3
3Ratio of the full correction δ for various polarization conditions. From the top of the figure, the lines show e − R e + L polarization, non-polarization, design polarization, and e − L e + R polarization, in that order
Fig. 4
4Angular distributions of the production angle of top quark θ top at a CM energy of 500 GeV with e − L e + R polarization (left) and e − R e + L polarization (right). The dotted lines show tree-level results whereas the solid lines show full electroweak-corrected results
Fig. 5
5Top-quark polarization as a function of the CM energy from 300 to 800 GeV for the process e − L e + R → tt. The dotted lines show tree-level results, whereas the solid lines show full electroweak-corrected results
Fig. 6
6Angular distributions of b-quarks with e − L e + R (left) and e − R e + L (right) polarizations. Circle and square points show tree and electroweakcorrected distributions, respectively
Fig. 7
7Energy distributions of b-quarks with e − L e + R (left) and e − R e + L (right) polarizations. Circle and square points show tree and electroweakcorrected distributions, respectively electroweak radiative correction, it is important to include effects from the radiative corrections in such kind of analysis in the ILC experiments.
Fig. 8
8NLO-QCD correction of the top-pair production process. A strong coupling constant α s = 0.12 is used polarized case. While the cross section with e −
Table 1
1Input parametersu-quark mass 58.0 × 10 −3 GeV d-quark mass 58.0 × 10 −3 GeV
c-quark mass 1.5 GeV
s-quark mass 92.0 × 10 −3 GeV
t-quark mass 173.5 GeV
b-quark mass 4.7 GeV
Z -boson mass 91.187 GeV
W -boson mass 80.370 GeV
Z -boson width 2.356 GeV
W -boson width 1.993 GeV
Higgs mass
126 GeV
sen phase points. For instance, in the polarized case at a CM
energy of 500 GeV, we confirmed ultra-violet coefficient and
photon-mass independence, both with stable results over 19
digits, when the parameters C UV and λ changed by three
orders of magnitude from their nominal values. Meanwhile,
the non-linear gauge-invariance results are stable over 25
digits against changing those values. Even a light-fermion
Table 2
2Estimated values of the forward-backward asymmetry at a
CM energy of 500 GeV
e − e + → tt
A FB (Tree)
A FB (Full)
e − e +
0.410
0.359
e −
L e +
R
0.385
0.317
e −
R e +
L
0.467
0.443
e −
L (80%)e +
R (30%)
0.388
0.321
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[
"Unique continuation estimates for the Kolmogorov equation in the whole space",
"Unique continuation estimates for the Kolmogorov equation in the whole space"
] |
[
"Yubiao Zhang \nSchool of Mathematics and Statistics\nWuhan University\n430072WuhanChina\n"
] |
[
"School of Mathematics and Statistics\nWuhan University\n430072WuhanChina"
] |
[] |
We prove in this Note an observation estimate at one point in time for the Kolmogorov equation in the whole space. Such estimate implies the observability and the null controllability for the Kolmogorov equation with a control region which is sufficiently spread out throughout the whole space.RésuméInégalités de continuation unique pour l'équation de Kolmogorov dans l'espace tout entier. Nous montrons dans cette Note des inégalités d'observation traduisant la continuation unique pour l'équation de Kolmogorov définie sur l'espace tout entier.
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10.1016/j.crma.2016.01.009
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"https://arxiv.org/pdf/1509.08292v1.pdf"
] | 54,698,073 |
1509.08292
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d6c91c0505792a1d1ccd8e7407ff70f3e48935ab
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Unique continuation estimates for the Kolmogorov equation in the whole space
28 Sep 2015
Yubiao Zhang
School of Mathematics and Statistics
Wuhan University
430072WuhanChina
Unique continuation estimates for the Kolmogorov equation in the whole space
28 Sep 2015Version française abrégée
We prove in this Note an observation estimate at one point in time for the Kolmogorov equation in the whole space. Such estimate implies the observability and the null controllability for the Kolmogorov equation with a control region which is sufficiently spread out throughout the whole space.RésuméInégalités de continuation unique pour l'équation de Kolmogorov dans l'espace tout entier. Nous montrons dans cette Note des inégalités d'observation traduisant la continuation unique pour l'équation de Kolmogorov définie sur l'espace tout entier.
Introduction and the main result
Consider the following Kolmogorov equation in the whole space (
d ∈ N + ) (∂ t + v · ∇ x − ∆ v )g(t, x, v) = 0, (t, x, v) ∈ R + × R d × R d , g(0, x, v) = g 0 (x, v), (x, v) ∈ R d × R d .(1)
The well-posedness of the solution of (1) was proved in Propositions 2.1 and 2.2 in [3]. In [3], the authors considered the following definition.
Definition 1.1 (See Definition 1.1 in [3]) An open set O of R n (n ∈ N + ) is said to be an observability open set on the whole space R n if there exist δ > 0 and r > 0 such that ∀ y ∈ R n , ∃ y ′ ∈ O such that B R n (y ′ , r) ⊂ O and |y − y ′ | ≤ δ.
(2)
Here B R n (x, r) denotes an open ball in R n of radius r centered at x.
From this definition, the authors in [3] proved the following estimate: Assume that ω x ⊂ R d and ω v ⊂ R d both verifies the property (2) with n = d. Then for all T > 0, there exists C > 0 so that for each g 0 ∈ L 2 (R 2d ), the solution of (1) satisfies that
g| t=T L 2 (R 2d ) ≤ C g L 2 ((0,T )×ωx×ωv ) .(3)
In [3], the proof of (3) is based on a spectral inequality, a Carleman inequality with respect to the variable v and a decay inequality for the Fourier transform of the solution of (1) with respect to the variable x.
The geometric condition (2) plays an important role in proving (3). The authors in [3] (2) with n = 2d, we get a unique continuation estimate for the Kolmogorov equation. Such kind of estimate has been studied in [1] and [6]. Our proof combines the spectral inequality given in [3] and a decay inequality on the Fourier transform of the solution of (1) with respect to the variables x and v. The main result is as follows.
Theorem 1.2 Let ω ⊂ R 2d
be an observability open set on the whole space R 2d . Then there exists C = C(ω, d) > 0 so that for all T > 0, α ∈ (0, 1) and g 0 ∈ L 2 (R 2d ), the solution of (1) satisfies that
g| t=T L 2 (R 2d ) ≤ e C α (1+ 1 T 3 ) g| t=T 1−α L 2 (ω) g 0 α L 2 (R 2d ) .(4)
By a telescoping series method (see [6, Theorem1.1]), a direct consequence of (4) is the following observability estimate.
Corollary 1.3 Let ω ⊂ R 2d
be an observability open set on the whole space R 2d . Let T > 0 and E ⊂ (0, T ) be a measurable set of positive measure. Then there exists C obs = C(ω, d, T, E) > 0 so that for each g 0 ∈ L 2 (R 2d ), the solution of (1) verifies that
g| t=T L 2 (R 2d ) ≤ C obs E g(t, ·, ·) L 2 (ω) dt.(5)When E = (0, T ), C obs = e C(1+ 1 T 3 )
where C only depends on ω and d.
Such observability estimate implies by duality the null controllability for the Kolmogorov equation.
A spectral inequality
The following spectral inequality plays a key role to deduce the estimate (4). Heref denotes the Fourier transform of f .
C = C(ω, d) > 0 such that for all N > 0, every f ∈ L 2 (R 2d ) verifies that |ξ|≤N |f (ξ)| 2 dξ ≤ e C(1+N ) ω |ξ|≤Nf (ξ)e ixξ dξ 2 dx.(6)
We mention that, for smooth compact and connected Riemannian manifold M with metric g and boundary ∂M , the following inequality was obtained in [4]: Let ω ⊂ M be an open nonempty subset. There exists C > 0 such that the Laplace-Beltrami operator −∆ g on M satisfies that
u L 2 (M) ≤ Ce C √ λ u L 2 (ω) for all u ∈ span{e j ; λ j ≤ λ},(7)
where {λ j } and {e j } are the eigenvalues and the corresponding eigenvectors of −∆ g with the zero Dirichlet boundary condition. Based on this type of inequality (7), a similar estimate to (4) was obtained for the heat equation in a bounded domain (see [1,Theorem 6]). The strategy in this Note also works for the heat equation in the whole space. This can be compared with [5], where M is non-compact with a Ricci curvature bounded below. The author in [5] proves that, under an interpolation inequality in [5, (6) on Page 40], (2) implies the spectral inequality (6), which yields the observability for the heat equation in M .
A decay inequality
We apply the Fourier transform, with respect to the variables x and v, to Equation (1). Then we get the following equation in the corresponding frequency space
(∂ t − ξ · ∇ η + |η| 2 )ĝ(t, ξ, η) = 0, (t, ξ, η) ∈ R + × R d × R d , g(0, ξ, η) =ĝ 0 (ξ, η), (ξ, η) ∈ R d × R d .(8)
The solution of (8) has an explicit representation, which has been obtained in [2, Section 7.6, Pages 210-211]. Based on this, we get a decay estimate for the Kolmogorov equation as follows.
Proposition 3.1 There exist C > 0 and C ′ = C ′ (d) > 0 such that for all N , T > 0 and each g 0 ∈ L 2 (R 2d ), the solution of (8) verifies that
|(ξ,η)|>N |ĝ(T, ξ, η)| 2 dξdη ≤ e C ′ −CN 2 min{T,T 3 } R d x ×R d v |g 0 (x, v)| 2 dxdv.(9)
Proof. Let g be a solution of (8). One can directly compute that
g(t, ξ, η) =ĝ 0 (ξ, η + ξt) exp − |η| 2 t − η · ξt 2 − |ξ| 2 t 3 /3 , ∀ (t, ξ, η) ∈ R + × R d × R d .
This yields that for all (t, ξ, η)
∈ R + × R d × R d , |ĝ(t, ξ, η)| ≤ |ĝ 0 (ξ, η + ξt)| exp − (|η| 2 + |ξ| 2 ) min{t, t 3 }/30 .
From this, we see that for all N , T > 0,
|(ξ,η)|>N |ĝ(T, ξ, η)| 2 dξdη ≤ exp − N 2 min{T, T 3 }/15 R d ξ ×R d η |ĝ 0 (ξ, η)| 2 dξdη,
which leads to (9). This ends the proof. In this section, we first prove Theorem 1.2 by combining Theorem 2.1 and Proposition 3.1 as follows.
Proof of Theorem 1.2. Let g be the solution of Equation (1) with the initial data g 0 ∈ L 2 (R 2d ). For each N > 0, writê
g(t, ξ, η) = χ BN (ξ, η)ĝ(t, ξ, η) + χ B c N (ξ, η)ĝ(t, ξ, η), ∀ (t, ξ, η) ∈ R + × R d × R d , where χ BN and χ B c
N denote the characteristic functions of the set B N (ξ, η) ∈ R 2d ; |(ξ, η)| ≤ N and its complement, respectively. Let T > 0. We observe that for all N > 0,
(2π) d g| t=T L 2 (R 2d ) = ĝ| t=T L 2 (R 2d ) ≤ χ BNĝ | t=T L 2 (R 2d ) + χ B c Nĝ | t=T L 2 (R 2d ) .(10)
On one hand, we apply (6) to g to get the existence of a positive constant
C 1 = C 1 (ω, d) so that for all N > 0, BN |ĝ(T, ξ, η)| 2 dξdη ≤ e 2C1(N +1) ω R d ξ ×R d ηĝ (T, ξ, η)e i(x·ξ+v·η) dξdη 2 dxdv + R d x ×R d v B c Nĝ (T, ξ, η)e i(x·ξ+v·η) dξdη 2 dxdv .(11)
On the other hand, let f (ξ, η) χ B c N (ξ, η)ĝ(T, ξ, η), (ξ, η) ∈ R d ξ × R d η . It follows from the inverse Fourier transform formula that f (ξ, η)e i(x·ξ+v·η) dξdη is the inverse Fourier transform of f . Then
1 (2π) 2d R d x ×R d v B c Nĝ (T, ξ, η)e i(x·ξ+v·η) dξdη 2 dxdv = 1 (2π) 2d R d x ×R d v R d ξ ×R d η f (ξ, η)e i(x·ξ+v·η) dξdη 2 dxdv = R d ξ ×R d η |f (ξ, η)| 2 dξdη = B c N |ĝ(T, ξ, η)| 2 dξdη. (12)
Meanwhile, we apply (9) to g to obtain that there exist C 2 > 0 and C 3 = C 3 (d) > 0 so that for all N > 0,
B c N |ĝ(T, ξ, η)| 2 dξdη ≤ e 2[C3−C2N 2 min{T,T 3 }] R d x ×R d v |g 0 (x, v)| 2 dxdv.(13)
Write T 1 3 min{T, T 3 }. By the inverse Fourier transform formula, we see from (10)-(13) that for all N > 0,
g| t=T L 2 (R 2d ) ≤ e C1(N +1) g| t=T L 2 (ω) + 2e C1(N +1)+C3−C2N 2 T 1 3 g 0 L 2 (R 2d ) .(14)
Let α ∈ (0, 1). We set k(α) α/(1 − α). Then we have that for all N > 0,
C 1 N ≤ C 2 1 2k(α)C 2 T 1 3 + k(α) C 2 N 2 T 1 3 2 and C 1 N − C 2 N 2 T 1 3 ≤ C 2 1 2C 2 T 1 3 − C 2 N 2 T 1 3 2 .
These, together with (14), yield that for all ε ∈ (0, 1),
g| t=T L 2 (R 2d ) ≤ C 1 ε −k(α) g| t=T L 2 (ω) + ε g 0 L 2 (R 2d ) ,(15)
where
C 1 max e C1+ C 2 1 2k(α)C 2 T 1 3 , 2e C1+C3+ C 2 1 2C 2 T 1 3 ≤ 2e (C 1 +C 2 +C 3 ) 2 C 2 α (1+ 1 T 3 ) . Since g| t=T L 2 (R 2d ) ≤ g 0 L 2 (R 2d )
, the minimization of the right side of (15), with respect to the variable ε over R + , leads to (4). This completes the proof. ✷
We next use the telescoping series method to deduce the Corollary 1.3 from Theorem 1.2.
Proof of Corollary 1.3. Let g be the solution of Equation (1) with the initial data g 0 ∈ L 2 (R 2d ). We take α = 1/2 in (4) and then see from the Young inequality that there exists C 1 = C 1 (ω, d) > 0 so that
g| t=T ≤ 1 ε e C1(1+ 1 T 3 ) g(T, ·, ·) L 2 (ω) + ε g| t=0 , ∀ ε > 0.
Generally, for each 0 < t 1 < t 2 , we have that
g| t=t2 ≤ 1 ε e C1[1+ 1 (t 2 −t 1 ) 3 ] g(t 2 , ·, ·) L 2 (ω) + ε g| t=t1 , ∀ ε > 0.(16)
Let l be a Lebesgue density point of E. Then by [6, Proposition 2.1], we know that for each λ ∈ (1/ 6 √ 2, 1), there exists a sequence {l m } ⊂ (l, T ) so that for each m ∈ N + , l m − l = λ m−1 (l 1 − l) and 3|E ∩ (l m+1 , l m )| ≥ |l m+1 − l m |.
(17) Take a m ∈ N + and let 0 < l m+2 < l m+1 ≤ s < l m < T . Since g| t=lm ≤ g| t=s and l m+1 − l m+2 ≤ s − l m+2 , we apply (16), where t 1 = l m+2 and t 2 = s, to get that
g| t=lm ≤ 1 ε e C1[1+ 1
(l m+1 −l m+2 ) 3 ] g(s, ·, ·) L 2 (ω) + ε g| t=lm+2 , ∀ ε > 0.
By integrating both sides over E ∩ (l m+1 , l m ) in the above inequality, we know that
ε|E ∩ (l m+1 , l m )|e − C 1 (l m+1 −l m+2 ) 3 g| t=lm − ε 2 |E ∩ (l m+1 , l m )|e − C 1 (l m+1 −l m+2 ) 3 g| t=lm+2 ≤ e C1 E∩(lm+1,lm) g(s, ·, ·) L 2 (ω) ds, ∀ ε > 0.(18)
Meanwhile, we know from (17) that
3|E ∩ (l m+1 , l m )| ≥ |l m+1 − l m | ≥ e − 1 |l m+1 −lm | ≥ e − λ 3 (l 1 −l 2 ) 2 (l m+1 −l m+2 ) 3 , ∀ m ∈ N + .
Since l m − l m+2 = (1 + 1 λ )(l m+1 − l m+2 ), the above, as well as (18), yields that for all m ∈ N + and ε > 0,
εe − C 2 (lm −l m+2 ) 3 g| t=lm − ε 2 e − C 2 (lm−l m+2 ) 3 g| t=lm+2 ≤ 3e C1
E∩(lm+1,lm) g(s, ·, ·) L 2 (ω) ds,
where C 2 = (1 + 1 λ ) 3 [C 1 + λ 3 (l 1 − l 2 ) 2 ]. Let β λ 6 2λ 6 −1 (> 0) and ε = e − (β−1)C 2 (lm −l m+2 ) 3 . Since λ 2 (l m − l m+2 ) = l m+2 − l m+4 , ∀ m ∈ N + , it follows from (19) that e − βC 2 (lm−l m+2 ) 3 g| t=lm − e − βC 2 (l m+2 −l m+4 ) 3 g| t=lm+2 ≤ 3e C1 E∩(lm+1,lm) g(s, ·, ·) L 2 (ω) ds.
We deduce from this that g(s, ·, ·) L 2 (ω) ds ≤ 3e C1 E∩(l,l1) g(s, ·, ·) L 2 (ω) ds.
Since g| t=T ≤ g| t=l1 , the above implies that g| t=T ≤ 3e C1+ βC 2 (l 1 −l 3 ) 3 E g(s, ·, ·) L 2 (ω) ds.
This proves (5). Especially, when E = (0, T ), we can take l 1 = T and l 3 = T /4. We end the proof.
✷
pointed out the following fact: There exists an open set O of R 2d , which is an observability open set in the whole R 2d , and does not contain any cartesian product O 1 × O 2 , where each O 1 and O 2 are both observability open sets in the whole space R d .In this Note, when assume that ω ⊂ R 2d verifies
Let ω ⊂ R 2d be an observability open set on the whole space R 2d . Then there existsTheorem 2.1 (See Theorem 1.2 in [3])
Acknowledgements. The author gratefully thanks Professor Kim Dang Phung for discussing and his valuable suggestions. Also, the author would like to thank Can Zhang for his help.
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[
"Terahertz streaking of few-femtosecond relativistic electron beams",
"Terahertz streaking of few-femtosecond relativistic electron beams"
] |
[
"Lingrong Zhao \nSchool of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nCollaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina\n",
"Zhe Wang \nSchool of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nCollaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina\n",
"Chao Lu \nSchool of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nCollaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina\n",
"Rui Wang ",
"Cheng Hu \nSchool of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nCollaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nSchool of Physics and Astronomy\nKey Laboratory of Artificial Structures and Quantum Control (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nCollaborative Innovation Center of Advanced Microstructures\n210093NanjingChina\n",
"Peng Wang \nShanghai Institute of Optics and Fine Mechanics\nState Key Laboratory of High Field Laser Physics\nChinese Academy of Sciences\n201800ShanghaiChina\n",
"Jia Qi \nShanghai Institute of Optics and Fine Mechanics\nState Key Laboratory of High Field Laser Physics\nChinese Academy of Sciences\n201800ShanghaiChina\n",
"Tao Jiang \nSchool of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nCollaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina\n",
"Shengguang Liu \nSchool of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nCollaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina\n",
"Zhuoran Ma \nSchool of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nCollaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina\n",
"Fengfeng Qi \nSchool of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nCollaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina\n",
"Pengfei Zhu ",
"Ya Cheng \nSchool of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nCollaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nShanghai Institute of Optics and Fine Mechanics\nState Key Laboratory of High Field Laser Physics\nChinese Academy of Sciences\n201800ShanghaiChina\n\nState Key Laboratory of Precision Spectroscopy\nEast China Normal University\n200062ShanghaiChina\n",
"Zhiwen Shi \nSchool of Physics and Astronomy\nKey Laboratory of Artificial Structures and Quantum Control (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nCollaborative Innovation Center of Advanced Microstructures\n210093NanjingChina\n",
"Yanchao Shi \nScience and Technology on High Power Microwave Laboratory\nNorthwest Institute of Nuclear Technology\n710024Xi'anShanxiChina\n",
"Wei Song \nScience and Technology on High Power Microwave Laboratory\nNorthwest Institute of Nuclear Technology\n710024Xi'anShanxiChina\n",
"Xiaoxin Zhu \nScience and Technology on High Power Microwave Laboratory\nNorthwest Institute of Nuclear Technology\n710024Xi'anShanxiChina\n",
"Jiaru Shi \nDepartment of Engineering Physics\nTsinghua University\n100084BeijingChina\n",
"Yingxin Wang \nDepartment of Engineering Physics\nTsinghua University\n100084BeijingChina\n",
"Lixin Yan \nDepartment of Engineering Physics\nTsinghua University\n100084BeijingChina\n",
"Liguo Zhu \nInstitute of Fluid Physics\nAcademy of Engineering Physics\n621900MianyangSichuanChina, China\n",
"Dao Xiang \nSchool of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nCollaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nTsung-Dao Lee Institute\n200240ShanghaiChina\n",
"Jie Zhang \nSchool of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina\n\nCollaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina\n"
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"School of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Collaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"School of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Collaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"School of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Collaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"School of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Collaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"School of Physics and Astronomy\nKey Laboratory of Artificial Structures and Quantum Control (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Collaborative Innovation Center of Advanced Microstructures\n210093NanjingChina",
"Shanghai Institute of Optics and Fine Mechanics\nState Key Laboratory of High Field Laser Physics\nChinese Academy of Sciences\n201800ShanghaiChina",
"Shanghai Institute of Optics and Fine Mechanics\nState Key Laboratory of High Field Laser Physics\nChinese Academy of Sciences\n201800ShanghaiChina",
"School of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Collaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"School of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Collaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"School of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Collaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"School of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Collaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"School of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Collaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Shanghai Institute of Optics and Fine Mechanics\nState Key Laboratory of High Field Laser Physics\nChinese Academy of Sciences\n201800ShanghaiChina",
"State Key Laboratory of Precision Spectroscopy\nEast China Normal University\n200062ShanghaiChina",
"School of Physics and Astronomy\nKey Laboratory of Artificial Structures and Quantum Control (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Collaborative Innovation Center of Advanced Microstructures\n210093NanjingChina",
"Science and Technology on High Power Microwave Laboratory\nNorthwest Institute of Nuclear Technology\n710024Xi'anShanxiChina",
"Science and Technology on High Power Microwave Laboratory\nNorthwest Institute of Nuclear Technology\n710024Xi'anShanxiChina",
"Science and Technology on High Power Microwave Laboratory\nNorthwest Institute of Nuclear Technology\n710024Xi'anShanxiChina",
"Department of Engineering Physics\nTsinghua University\n100084BeijingChina",
"Department of Engineering Physics\nTsinghua University\n100084BeijingChina",
"Department of Engineering Physics\nTsinghua University\n100084BeijingChina",
"Institute of Fluid Physics\nAcademy of Engineering Physics\n621900MianyangSichuanChina, China",
"School of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Collaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Tsung-Dao Lee Institute\n200240ShanghaiChina",
"School of Physics and Astronomy\nKey Laboratory for Laser Plasmas (Ministry of Education)\nShanghai Jiao Tong University\n200240ShanghaiChina",
"Collaborative Innovation Center of IFSA (CICIFSA)\nShanghai Jiao Tong University\n200240ShanghaiChina"
] |
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Streaking of photoelectrons with optical lasers has been widely used for temporal characterization of attosecond extreme ultraviolet pulses. Recently, this technique has been adapted to characterize femtosecond x-ray pulses in free-electron lasers with the streaking imprinted by farinfrared and Terahertz (THz) pulses. Here, we report successful implementation of THz streaking for time-stamping of an ultrashort relativistic electron beam of which the energy is several orders of magnitude higher than photoelectrons. Such ability is especially important for MeV ultrafast electron diffraction (UED) applications where electron beams with a few femtosecond pulse width may be obtained with longitudinal compression while the arrival time may fluctuate at a much larger time scale. Using this laser-driven THz streaking technique, the arrival time of an ultrashort electron beam with 6 fs (rms) pulse width has been determined with 1.5 fs (rms) accuracy. Furthermore, we have proposed and demonstrated a non-invasive method for correction of the timing jitter with femtosecond accuracy through measurement of the compressed beam energy, which may allow one to advance UED towards sub-10 fs frontier far beyond the ∼100 fs (rms) jitter.2
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10.1103/physrevx.8.021061
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[
"https://arxiv.org/pdf/1805.03923v1.pdf"
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1805.03923
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3f60ef5bd8cb16dea2fa8b5036913a613a86c53c
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Terahertz streaking of few-femtosecond relativistic electron beams
10 May 2018
Lingrong Zhao
School of Physics and Astronomy
Key Laboratory for Laser Plasmas (Ministry of Education)
Shanghai Jiao Tong University
200240ShanghaiChina
Collaborative Innovation Center of IFSA (CICIFSA)
Shanghai Jiao Tong University
200240ShanghaiChina
Zhe Wang
School of Physics and Astronomy
Key Laboratory for Laser Plasmas (Ministry of Education)
Shanghai Jiao Tong University
200240ShanghaiChina
Collaborative Innovation Center of IFSA (CICIFSA)
Shanghai Jiao Tong University
200240ShanghaiChina
Chao Lu
School of Physics and Astronomy
Key Laboratory for Laser Plasmas (Ministry of Education)
Shanghai Jiao Tong University
200240ShanghaiChina
Collaborative Innovation Center of IFSA (CICIFSA)
Shanghai Jiao Tong University
200240ShanghaiChina
Rui Wang
Cheng Hu
School of Physics and Astronomy
Key Laboratory for Laser Plasmas (Ministry of Education)
Shanghai Jiao Tong University
200240ShanghaiChina
Collaborative Innovation Center of IFSA (CICIFSA)
Shanghai Jiao Tong University
200240ShanghaiChina
School of Physics and Astronomy
Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education)
Shanghai Jiao Tong University
200240ShanghaiChina
Collaborative Innovation Center of Advanced Microstructures
210093NanjingChina
Peng Wang
Shanghai Institute of Optics and Fine Mechanics
State Key Laboratory of High Field Laser Physics
Chinese Academy of Sciences
201800ShanghaiChina
Jia Qi
Shanghai Institute of Optics and Fine Mechanics
State Key Laboratory of High Field Laser Physics
Chinese Academy of Sciences
201800ShanghaiChina
Tao Jiang
School of Physics and Astronomy
Key Laboratory for Laser Plasmas (Ministry of Education)
Shanghai Jiao Tong University
200240ShanghaiChina
Collaborative Innovation Center of IFSA (CICIFSA)
Shanghai Jiao Tong University
200240ShanghaiChina
Shengguang Liu
School of Physics and Astronomy
Key Laboratory for Laser Plasmas (Ministry of Education)
Shanghai Jiao Tong University
200240ShanghaiChina
Collaborative Innovation Center of IFSA (CICIFSA)
Shanghai Jiao Tong University
200240ShanghaiChina
Zhuoran Ma
School of Physics and Astronomy
Key Laboratory for Laser Plasmas (Ministry of Education)
Shanghai Jiao Tong University
200240ShanghaiChina
Collaborative Innovation Center of IFSA (CICIFSA)
Shanghai Jiao Tong University
200240ShanghaiChina
Fengfeng Qi
School of Physics and Astronomy
Key Laboratory for Laser Plasmas (Ministry of Education)
Shanghai Jiao Tong University
200240ShanghaiChina
Collaborative Innovation Center of IFSA (CICIFSA)
Shanghai Jiao Tong University
200240ShanghaiChina
Pengfei Zhu
Ya Cheng
School of Physics and Astronomy
Key Laboratory for Laser Plasmas (Ministry of Education)
Shanghai Jiao Tong University
200240ShanghaiChina
Collaborative Innovation Center of IFSA (CICIFSA)
Shanghai Jiao Tong University
200240ShanghaiChina
Shanghai Institute of Optics and Fine Mechanics
State Key Laboratory of High Field Laser Physics
Chinese Academy of Sciences
201800ShanghaiChina
State Key Laboratory of Precision Spectroscopy
East China Normal University
200062ShanghaiChina
Zhiwen Shi
School of Physics and Astronomy
Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education)
Shanghai Jiao Tong University
200240ShanghaiChina
Collaborative Innovation Center of Advanced Microstructures
210093NanjingChina
Yanchao Shi
Science and Technology on High Power Microwave Laboratory
Northwest Institute of Nuclear Technology
710024Xi'anShanxiChina
Wei Song
Science and Technology on High Power Microwave Laboratory
Northwest Institute of Nuclear Technology
710024Xi'anShanxiChina
Xiaoxin Zhu
Science and Technology on High Power Microwave Laboratory
Northwest Institute of Nuclear Technology
710024Xi'anShanxiChina
Jiaru Shi
Department of Engineering Physics
Tsinghua University
100084BeijingChina
Yingxin Wang
Department of Engineering Physics
Tsinghua University
100084BeijingChina
Lixin Yan
Department of Engineering Physics
Tsinghua University
100084BeijingChina
Liguo Zhu
Institute of Fluid Physics
Academy of Engineering Physics
621900MianyangSichuanChina, China
Dao Xiang
School of Physics and Astronomy
Key Laboratory for Laser Plasmas (Ministry of Education)
Shanghai Jiao Tong University
200240ShanghaiChina
Collaborative Innovation Center of IFSA (CICIFSA)
Shanghai Jiao Tong University
200240ShanghaiChina
Tsung-Dao Lee Institute
200240ShanghaiChina
Jie Zhang
School of Physics and Astronomy
Key Laboratory for Laser Plasmas (Ministry of Education)
Shanghai Jiao Tong University
200240ShanghaiChina
Collaborative Innovation Center of IFSA (CICIFSA)
Shanghai Jiao Tong University
200240ShanghaiChina
Terahertz streaking of few-femtosecond relativistic electron beams
10 May 2018(Dated: May 11, 2018)1
Streaking of photoelectrons with optical lasers has been widely used for temporal characterization of attosecond extreme ultraviolet pulses. Recently, this technique has been adapted to characterize femtosecond x-ray pulses in free-electron lasers with the streaking imprinted by farinfrared and Terahertz (THz) pulses. Here, we report successful implementation of THz streaking for time-stamping of an ultrashort relativistic electron beam of which the energy is several orders of magnitude higher than photoelectrons. Such ability is especially important for MeV ultrafast electron diffraction (UED) applications where electron beams with a few femtosecond pulse width may be obtained with longitudinal compression while the arrival time may fluctuate at a much larger time scale. Using this laser-driven THz streaking technique, the arrival time of an ultrashort electron beam with 6 fs (rms) pulse width has been determined with 1.5 fs (rms) accuracy. Furthermore, we have proposed and demonstrated a non-invasive method for correction of the timing jitter with femtosecond accuracy through measurement of the compressed beam energy, which may allow one to advance UED towards sub-10 fs frontier far beyond the ∼100 fs (rms) jitter.2
I. INTRODUCTION
Ultrafast phenomena are typically studied with a pump-probe technique in which the dynamics are initiated by a pump laser and then probed by a delayed pulse [1]. Because of theÅngstrom-scale wavelength, both electrons and x-rays have been used as the probe pulses for watching atoms in motion during structural changes [2][3][4]. With the advent of ultrashort lasers, the temporal resolution in such experiments depends primarily on the pulse width and arrival time jitter of the probe pulse. Currently the brightest hard x-ray pulse is provided by free-electron lasers (FELs [5][6][7]) and with tremendous efforts devoted it is now possible to produce sub-femtosecond x-ray pulse [8,9] with its arrival time determined with femtosecond precision [10][11][12][13]. In contrast, for electron probes as in ultrafast electron diffraction (UED [14,15]), the long standing goal to deliver few femtosecond high-brightness electron beam with well-characterized arrival time still remains quite challenging.
In UED the shortest electron pulse width is mainly limited by Coulomb repulsion [16].
In the past two decades, many methods have been developed to mitigate this effect, e.g.
reducing the beam propagation length [2,17], reducing the number of electrons per pulse [18][19][20], increasing the electron beam energy [21][22][23][24][25][26][27][28], and compressing the beam with an radiofrequency (rf) buncher [29,30]. Among all these approaches, bunch compression provided the most significant advance that enabled a new class of experiments [31,32]. In this method, the elongated electron beam is first sent through an rf buncher cavity where the bunch head is decelerated to lower energy while the bunch tail is accelerated to higher energy; after passing through a drift, the bunch tail with higher energy will catch up with the bunch head, leading to longitudinally compressed isolated bunch. When the rf buncher cavity is replaced by a laser, the energy modulation at optical wavelength may lead to formation of attosecond electron bunch trains that can also be applied to a certain type of experiments [33,34].
While the rf buncher technique has been widely used in UED community and very recently a relativistic electron beam as short as 7 fs (rms) has been produced [30], it is also realized that the space charge force induced pulse broadening was solved at the cost of increasing timing jitter. This is because the phase jitter in the rf cavity leads to beam energy jitter which is further converted into timing jitter at the sample [19,35]. Similar jitter sources exist for FELs as well [10][11][12][13]. Such timing jitter, if not measured and corrected, will limit the temporal resolution in pump-probe applications to a similar level. Unfortunately, timestamping techniques developed for high-charge GeV and low-energy keV electron beams can not be easily implemented for MeV UED beams. For instance, the arrival time of a GeV beam has been measured with electro-optic sampling (EOS) technique [10,36], but it is difficult to reach high temporal resolution when applying this technique to a MeV UED facility where the beam charge is relatively low. Time stamping of keV electron beams with a laser triggered streak camera has been used to correct timing jitter, but the accuracy is on the order of tens of femtoseconds [37]. Very recently, it has been shown that a buncher cavity powered by a laser-driven THz pulse may allow one to compress a keV beam without introducing such jitter [38]. However, it requires very intense THz source in order to apply this scheme to MeV electron beam. It has been demonstrated recently that a MeV electron beam can also interact with a THz pulse effectively through the inverse FEL mechanism for beam acceleration and manipulation, but this scheme requires a dedicated undulator and careful matching of the phase and group velocity of the THz pulse with the electron beam [39]. It should also be noted that few femtosecond relativistic electron beam with intrinsically small timing jitter has also been produced in a laser wakefield accelerator where the accelerating gradient is several orders of magnitude higher than that achieved in rf guns, but the beam quality and stability still need significant improvements in order to apply the beam for UED applications [40].
Here, we demonstrate a laser-driven THz streaking technique with which the arrival time of a 6 fs (rms) ultrashort electron beam was determined with 1.5 fs (rms) accuracy. With the newly developed rf deflector that provides about 2.5 fs (rms) temporal resolution, the rf buncher cavity is optimized to compress the relativistic electron beam from about 200 fs to well below 10 fs. With a narrow slit to both enhance the local THz field strength and reduce the electron beam intrinsic angular fluctuation, accurate measurement of the relativistic electron beam arrival time is achieved with a THz pulse with moderate field strength (∼100 kV/cm). Because the THz pulse used for time-stamping is tightly synchronized with the laser, the measured timing information can be directly used for machine optimizations and for correcting timing jitter in laser-pump electron-probe applications. Furthermore, a noninvasive method for correcting the timing jitter of a compressed beam through measurement of the compressed beam energy has been proposed and demonstrated. This non-invasive time-stamping method is easy to implement and can be applied to both keV and MeV UED 4 to significantly improve the temporal resolution to potentially sub-10 fs regime.
II. THZ STREAKING DEFLECTOGRAM
The set-up for THz streaking of a few-femtosecond relativistic electron beam is shown in The laser-driven THz streaking measurement combines the standard streak camera technique with the concept of laser streaking from attosecond metrology (see, e.g. [42,43]).
In this experiment, a 250 µm × 10 µm slit perforated on a 50 µm thick Al foil with laser machining is illuminated with a THz pulse with 30 degrees angle of incidence (Fig. 2). The electron beam gets a time-dependent angular kick from the THz pulse when it passes through the slit. For electrons that impinge the foil, due to multiple scattering, most of them are lost because their angles are larger than the acceptance of the vacuum pipe and a small fraction may arrive at the detector, forming a relatively uniform background. The measurement also resembles the streaking technique used to measure x-ray pulse width and timing jitter in FELs [8,11,44], except that here the electron beam angular distribution rather than energy distribution is changed by interaction with the THz pulse and the electron beam energy is several orders of magnitude higher than that of the photoelectrons. Very recently, such scheme has also been applied to keV energy electrons for characterizing the electron pulse width and arrival time [38].
The THz pulse is produced with a ∼2 mJ 800 nm femtosecond laser through optical rectification in LiNbO 3 crystal with the tilted-pulse-front-pumping scheme [41] where the pulse front of the laser is first tilted with a diffraction grating and then imaged onto the 5 LiNbO 3 with two cylindrical lenses (CL1 and CL2 in Fig. 1) for matching the laser phase velocity with the THz group velocity. The THz energy is measured to be about 0.5 µJ with a calibrated Golay cell detector. With a THz camera (IRXCAM-THz-384) the THz divergence at the exit of the crystal was quantified through measurement of the THz transverse size at various positions along the propagation direction. Then a focusing system that consists of 3 off-axis parabolic mirrors are used to guide and focus the THz to the slit. The THz transverse size at the slit is measured to be about 0.5 mm (rms 6 EOS [45] is shown in Fig. 3a with its corresponding spectrum shown in Fig. 3b. The THz field strength is calculated using the known thickness and electro-optic coefficients of ZnTe.
It should be pointed out that due to crystal defects and imperfect crystal orientation, the calculated peak value of about 90 kV/cm should be considered as the lower limit of the THz field strength. Alternatively, the upper limit of the THz peak electric field is estimated to be about 170 kV/cm using the measured THz energy, transverse beam size and waveform.
When this single-cycle THz pulse is focused onto the narrow slit with its polarization pointing along the slit's short axis, near-field enhancement occurs in the slit that increases the streaking strength (see, e.g. [46]). The simulated field (using CST Microwave Studio) in the center of the slit is shown in Fig. 3c where one can see that due to transmission Fig. 4b, which is in good agreement with the simulation result. It should be pointed out that the simulated deflection angle is slightly smaller than the experimental result, which is probably due to the fact that in the simulation the lower limit (90 kV/cm) of the peak electric field is used. of the streaking. The maximal streaking ramp (around t=4 ps region in Fig. 4b) is found to be 5.1 µrad/fs. The accuracy of the arrival time measurement is mainly affected by the fluctuation of the centroid divergence of the electron beam, resulting in temporal offset in the measurement. Benefiting from the narrow slit, the shot to shot fluctuation of the beam centroid divergence is found to be about 7.6 µrad, corresponding to an uncertainty of 1.5 fs in beam arrival time determination. It should be mentioned that in principle one may rotate the slit and THz polarization by 90 degrees to imprint energy modulation in the electron beam and determine the beam arrival time by monitoring the energy change of the electron beam imprinted by the THz pulse, similar to that used in [39]. However, the accuracy will be much lower because one can't benefit from the narrow slit (the slit does not reduce the beam energy fluctuation) and the buncher cavity increases the beam energy fluctuation (the buncher cavity does not increase the beam centroid divergence fluctuation).
electron beam and THz beam is varied and the measured streaking deflectogram is shown in
III. TIME-STAMPING OF AN ULTRASHORT RELATIVISTIC ELECTRON BEAM
It is worth mentioning that the electron beam temporal profile can also be retrieved from the broadening of the streaked beam angular distribution. However, with the beam intrinsic divergence being approximately 50 µrad, the temporal resolution in beam temporal profile measurement is estimated to be about 10 fs (rms), limited by the strength of the streaking field. Because the available c-band rf deflector can provide a much higher resolution, in our experiment the electron beam temporal profile is measured with the rf deflector. The rf deflector is an rf structure operating in TM11 mode which gives the beam a time-dependent angular kick (i.e. y ∝ t) after passing through at zero-crossing phase. The beam angular distribution is converted to spatial distribution after a drift section, and the vertical axis on the phosphor screen (P1 and P2 in Fig. 1) becomes the time axis (y ∝ t). (Fig. 5b) was measured at screen P2 downstream of the energy spectrometer. In this measurement, the electron beam is bent in horizontal direction in the energy spectrometer such that the horizontal axis on the phosphor screen P2 becomes the energy axis (x ∝ E).
Then the beam longitudinal phase space is mapped to the transverse distribution at screen P2. The absolute time is calibrated by scanning the rf phase and recording the vertical beam centroid motion on the screens (1 degree change in rf phase corresponds to about 0.5 ps change in time). The absolute energy is calibrated from the measured magnetic field of the energy spectrometer and the dispersion at the phosphor screen.
As can be seen in Fig. 5b, initially the beam longitudinal phase space has positive chirp (bunch head having higher energy than bunch tail) which is caused by space charge force that accounts for bunch lengthening. As the buncher voltage is gradually increased (V b = 0.84 MV), the energy chirp is reversed to negative, enabling bunch compression after a drift. has been stably compressed to about 6 fs (rms). In this measurement, the voltage of the rf deflector is about 1.8 MV and a 20 microns narrow slit is used to improve the temporal resolution of the beam temporal profile measurement to about 2.5 fs (rms), as limited by the intrinsic beam size with the rf deflector off. Because both the beam intrinsic divergence and high order effects in the rf deflector contribute to the measured vertical beam size with the deflector on [30], the estimated value of 6 fs (rms) should be considered as the upper limit of the bunch length.
Though the bunch length is compressed to a few fs, the timing jitter is likely to be at a consecutive measurement of beam arrival time with THz streaking is shown in Fig. 6a and the timing jitter at full compression collected over 500 shots is estimated to be about 140 fs (rms), as shown in Fig. 6b. Fortunately, such timing jitter can be corrected with femtosecond precision (as shown in Fig. 6a the jitter for most of the shots can be corrected with an accuracy better than 3 fs), which significantly improves the temporal resolution in laser-pump electron-probe applications.
IV. REAL-TIME NONINVASIVE TIME-STAMPING
The current set-up with a 10 micron slit is best suited for machine optimizations, because the narrow slit reduces the useful number of electrons in UED. While with a more intense THz source, a wider slit may be used, the dynamic range (about 600 fs in this experiment) may still hinder its applications to cases where the maximal arrival time difference is larger than half the period of the streaking field. Motivated by the fact that the timing jitter is primarily caused by energy variations after the buncher cavity, here we quantify their correlation and demonstrate that the timing jitter related to bunch compression may be corrected in a non-invasive way through measurement of the shot by shot beam energy fluctuation. In a separate experiment, the phase of the rf buncher was varied and the measured beam distribution on screen P2 is shown in Fig. 7a. In this measurement, the electron beam is streaked vertically by the THz pulse and is bent horizontally by the energy spectrometer. It should be noted that changing buncher phase is equivalent to scanning the delay time between the THz pulse and electron beam. This is the main reason that the energy-deflection map in Fig. 7a is quite similar to the time-deflection map in Fig. 4b.
Combining the time information from Fig. 4b and energy information from Fig. 7a, the correlation between the arrival time and centroid energy of the beam is shown in Fig. 7b where one can see that the beam timing jitter ∆t is indeed linearly correlated with the beam energy jitter ∆E/E, i.e. ∆t=R × ∆E/E with R determined to be -117 ps, in good agreement with the momentum compaction of the drift [50], i.e. R 56 = −L/cγ 2 ≈ − 120 ps with L ≈ 1.6 m being the distance from the buncher cavity to the THz slit and γ being the Lorentz factor of the electron beam. With the coefficient determined with THz streaking, the timing jitter may be corrected in a non-invasive way (e.g. using the un-diffracted beam) and the dynamical range of the jitter measurement is no longer limited by the wavelength of the THz pulse and thus even picosecond jitter may be corrected with femtosecond precision. For this method, the accuracy is limited by the uncertainty of beam energy at the entrance to the buncher cavity.
In our experiment, with the beam energy stability to be about 0.02% and the distance between the cathode to the buncher being 0.8 m (corresponding to a momentum compaction of about -60 ps), the accuracy is estimated to be about 12 fs. Note, for keV UED where the beam energy stability at the entrance to the buncher cavity is orders of magnitude higher, the accuracy of using beam energy jitter to correct timing jitter should be well below 10 fs.
14
V. CONCLUSIONS AND OUTLOOK
In conclusion, we have experimentally demonstrated a novel method for time-stamping of relativistic electron beams. A non-invasive, easy-to-implement method for correcting timing jitter with high accuracy through measurement of the un-diffracted electron beam centroid energy has also been proposed and demonstrated. Together with the available few-cycle optical lasers for exciting the dynamics, the demonstrated technique should allow one to advance UED towards sub-10 fs frontier. In the future, the rf buncher voltage may be increased to produce sub-femtosecond beam. With stronger streaking field (note, LiNbO 3
based THz pulse with electric field exceeding 1 MV/cm has been achieved [51]) the resolution of the demonstrated method may also be extended to well beyond sub-femtosecond, making attosecond electron diffraction metrologies capable of visualizing attosecond structural dynamics within reach.
VI. ACKNOWLEDGMENTS
The authors want to thank S. Li, Z. Tian and X. Su for help in THz source design. This
Fig. 1 .
1A ∼50 fs (FWHM) Ti:sapphire laser at 800 nm is first split into two pulses with a 50%-50% beam splitter (BS1). One pulse is frequency tripled to produce electron beam ina 1.5 cell S-band (2856 MHz) photocathode rf gun. The other pulse is further split into two parts with a 10%-90% beam splitter (BS2) with the main pulse (∼2 mJ) used to produce THz radiation through optical rectification in LiNbO 3 crystal [41] and the remaining part for in situ characterization of the THz pulse at the interaction region through EOS technique. The ∼ 3.4 MeV electron beam is compressed by a C-band (5712 MHz) rf buncher cavity and the electron beam arrival time is measured with THz streaking in a narrow slit. In this experiment, both the THz and electron beam are running at 50 Hz. The electron beam charge is measured to be about 30 fC with a Faraday cup.
FIG. 1 :
1THz-streaking of a relativistic electron beam experiment setup. The electron beam is produced in a photocathode rf gun by illuminating the cathode with a UV laser and longitudinally compressed with an rf buncher by imprinting a negative energy chirp (i.e. with the bunch head having lower energy than the bunch tail) in the beam phase space. A set of off-axis parabolic (OAP) mirrors allows tight focus of the THz pulse onto the slit. The electron beam experiences transverse Lorenz force when passing through the slit and the THz-induced angular deflection is converted into spatial shift at the screen P1 after a drift of 1.8 m. In general, the electron beam is streaked in vertical direction with its time information mapped into spatial distribution on screen P1. Alternatively, the streaked electron beam may be sent through an energy spectrometer for measuring the longitudinal phase space at screen P2. The slit for streaking, Zinc Telluride (ZnTe) crystal for EOS, and a transmission electron microscope (TEM) grid for synchronization, are mounted on a remote-controlled manipulator.
FIG. 3 :
3resonance, the field strength is increased by about a factor of 4 and the single-cycle THz pulse becomes multi-cycle resonating at the wavelength of approximately twice the length of the long axis of the slit, i.e. the cut off wavelength of a 250 µm × 10 µm rectangular waveguide. This enhancement also limits the effective THz-electron interaction within the region of the slit. The simulated electron beam centroid deflection for various time delay is found with the integral of the Lorentz force along the THz-electron interaction region, as shown inFig. 4a. Because the effective interaction region is much smaller than the wavelength of the oscillation field, the deflection (Fig. 4a) just closely follows the electric field (Fig. 3c) which dominates over the magnetic field in our interaction configuration. Effective interaction between the THz pulse and electron beam is achieved when electron FIG. 2: Schematic of the THz-electron interaction at the narrow slit.7and THz beam overlap both spatially and temporally in the slit. This is done with the help of the 800 nm laser used for EOS. First, the ZnTe crystal is put in the center of the interaction chamber and the EOS signal is maximized when the 800 nm laser is well overlapped with the THz pulse both in space and time. Then the ZnTe crystal is removed from the beam path and a TEM grid is inserted. The BS2 is replaced with a mirror so that the energy of 800 nm laser is sufficient to produce transient plasmas around the interaction point on the TEM grid. The time of the laser and THz pulse is varied with a delay stage until considerable perturbation to the electron beam transverse profile from the transient electromagnetic field associated with the transient plasma is observed (see, e.g.[47][48][49]). Note, the delay stage does not change the relative timing between the laser and THz pulse and with this technique temporal overlap between the electron beam and the THz pulse is achieved. The TEM grid is then removed from the beam path and the narrow slit is inserted. The position of the slit is varied until the 800 nm laser passes through the slit. Finally the electron beam is steered to pass through the narrow slit and both spatial and temporal overlap between the THz pulse and electron beam is then achieved. After this procedure, the delay between the Measured THz waveform (a) at the streaking interaction position with EOS, the corresponding THz spectrum (b) and the simulated THz electric field and magnetic field in the center of the slit (c).
This time-dependent angular streaking allows one to map the electron beam time information into spatial distribution on a downstream screen. Similar to THz streaking in FELs, the electron beam should overlap with the THz pulse near the zero-crossing of the deflectogram (i.e. the rate of angular change is approximately linear) for measurement of beam timing jitter and the dynamic range of the measurement is limited to half of the wavelength FIG. 4: Simulated beam centroid deflection as a function of time delay between the electron beam and THz pulse (a) and the measured streaking deflectogram as a function of time delay (in 67 fs steps) between the electron beam and THz pulse. The measured beam centroid deflection at each time delay is shown with white points. In the measurement a collimator (3 mm upstream of the slit) is used to reduce the beam size to about 20 micron (full width) at the slit such that the electron beam feels uniform streaking force and each time slice is integrated over 50 single-shot measurements.
Fig
. 5a shows measurements of 30 fC electron beam temporal profile for various voltages of the buncher cavity, corresponding to no compression (V b = 0), under compression (V b = 0.84 MV) and full compression (V b = 1.0 MV). The corresponding beam longitudinal phase space
FIG. 5 :
5With the buncher voltage set to V = 1.0 MV, the bunch tail exactly catches up with the bunch head, and the shortest bunch length was achieved. Under full compression condition, 100 consecutive measurements of the raw beam profile (with vertical axis converted into time) with rf deflector off and on are shown in Fig. 5c, where one can see that the beam Longitudinal compression of a relativistic electron beam to sub-10 fs. (a) Electron beam temporal profile for various buncher voltages measured with the rf deflector. (b) Corresponding beam longitudinal phase space (bunch head to the left). (c) 100 consecutive measurements of the beam profile with rf deflector off (the first 10 shots) and on (the rest 90 shots). The average beam profiles with the deflector off (black line) and on (white line) are also shown. The number of electrons in the bunch is about 2 × 10 5 .
much larger time scale. In this particular measurement, we ignore variations in the beam temporal profile and the arrival time of the electron beam under full compression condition is determined by recording the fluctuations of the beam centroid with THz streaking. 100
FIG. 6 :
6Time-stamping of a relativistic electron beam with THz streaking. (a) 100 consecutive measurement of beam arrival time with THz streaking using the single-valued streaking ramp (white curve). Scales on the right correspond to the accuracy in beam arrival time determination which depends on the relative time with respect to the zero-crossing of the streaking ramp. The arrival time of the shots within ±300 fs is determined with an accuracy higher than half the bunch length, i.e. 3 fs (rms). (b) Distribution of the electron beam arrival time collected over 500 shots.A Gaussian fit (magenta line) to the distribution within ±300 fs yields a timing jitter of about 140 fs (rms) between the electron beam and THz pulse. The distribution with red color has uncertainty larger than 3 fs and is not used in fitting.
FIG. 7 :
7Energy-time correlation map. (a) Correlation between electron beam energy and beam angular distribution measured by scanning the phase of the rf buncher cavity (voltage set to V b = 0.5 MV for minimizing beam energy spread). (b) Beam arrival time vs beam energy after converting the angular shift into time delay. To increase the accuracy, only the equivalent time delay for each zero-crossing points (position indicated by dashed line) of the streaking ramp is used. The data are linearly correlated and the coefficient is found to be in good agreement with the momentum compaction of the drift from the rf buncher to the THz streaking position.
work was supported by the Major State Basic Research Development Program of China (Grants No. 2015CB859700) and by the National Natural Science Foundation of China (Grants No. 11327902, 1150423211655002 and 11721091). One of the authors (DX) would like to thank the support of grant from the office of Science and Technology, Shanghai Municipal Government (No. 16DZ2260200). [email protected] [email protected]
). The THz waveform measured byEOS
Tripler
Photocathode
rf gun
BS1
50%
Ti:sapphire
NIR laser
UV laser
50%
90%
10%
BS2
Grating
LiNbO3
λ/2
Delay between
UV& NIR/THz
EOS Delay
ZnTe
Slit
TEM
Grid
rf buncher
Energy
Spectrometer
THz
OAP1
OAP2
OAP3
P1
P2
time
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|
[] |
[
"Understanding the Effect of the COVID-19 Pandemic on the Usage of School Buildings in GreeceUsing an IoT Data-Driven Analysis",
"Understanding the Effect of the COVID-19 Pandemic on the Usage of School Buildings in GreeceUsing an IoT Data-Driven Analysis"
] |
[
"Georgios Mylonas \nIndustrial Systems Institute Athena Research & Innovation Center Patras\nGreece\n",
"Dimitrios Amaxilatis [email protected] \nDpt. of Computer, Control & ManagementEngin\nSparkWorks ITC Ltd Derbyshire\nUnited Kingdom\n",
"Ioannis Chatzigiannakis \nSapienza University of Rome Rome\nItaly\n"
] |
[
"Industrial Systems Institute Athena Research & Innovation Center Patras\nGreece",
"Dpt. of Computer, Control & ManagementEngin\nSparkWorks ITC Ltd Derbyshire\nUnited Kingdom",
"Sapienza University of Rome Rome\nItaly"
] |
[] |
The COVID-19 pandemic has brought profound change in the daily lives of a large part of the global population during 2020 and 2021. Such changes were mirrored in aspects such as changes to the overall energy consumption, or long periods of sustained inactivity inside public buildings. At the same time, due to the large proliferation of IoT, sensors and smartphones in the past few years, we are able to monitor such changes to a certain degree over time. In this paper, we focus on the effect of the pandemic on school buildings and certain aspects in the operation of schools. Our study is based on data from a number of school buildings equipped with an IoT infrastructure. The buildings were situated in Greece,a country that faced an extended lockdown during both 2020 and 2021.Our results show that as regards power consumption there is room for energy efficiency improvements since there was significant power consumption during lockdowns, and that using other sensordata we can also infer interesting points regarding the buildings and activity during the lockdown.
|
10.1109/smartiot52359.2021.00068
|
[
"https://arxiv.org/pdf/2206.01330v1.pdf"
] | 238,750,416 |
2206.01330
|
4959b254636f5482487a9295785780a80d770091
|
Understanding the Effect of the COVID-19 Pandemic on the Usage of School Buildings in GreeceUsing an IoT Data-Driven Analysis
Georgios Mylonas
Industrial Systems Institute Athena Research & Innovation Center Patras
Greece
Dimitrios Amaxilatis [email protected]
Dpt. of Computer, Control & ManagementEngin
SparkWorks ITC Ltd Derbyshire
United Kingdom
Ioannis Chatzigiannakis
Sapienza University of Rome Rome
Italy
Understanding the Effect of the COVID-19 Pandemic on the Usage of School Buildings in GreeceUsing an IoT Data-Driven Analysis
10.1109/SmartIoT52359.2021.00068Preprint submitted to 2021Index Terms-resilient citiesIoTbuildingssmart citysus- tainability
The COVID-19 pandemic has brought profound change in the daily lives of a large part of the global population during 2020 and 2021. Such changes were mirrored in aspects such as changes to the overall energy consumption, or long periods of sustained inactivity inside public buildings. At the same time, due to the large proliferation of IoT, sensors and smartphones in the past few years, we are able to monitor such changes to a certain degree over time. In this paper, we focus on the effect of the pandemic on school buildings and certain aspects in the operation of schools. Our study is based on data from a number of school buildings equipped with an IoT infrastructure. The buildings were situated in Greece,a country that faced an extended lockdown during both 2020 and 2021.Our results show that as regards power consumption there is room for energy efficiency improvements since there was significant power consumption during lockdowns, and that using other sensordata we can also infer interesting points regarding the buildings and activity during the lockdown.
I. INTRODUCTION
The COVID-19 pandemic has proved to be one of the biggest challenges on a global scale since the Spanish flu of 1918 on many levels, including public health and the global economy. During 2020 and 2021, a very large part of the global population [1] experiencedsome form of lockdown or similar set of strict measures,at a scale believed to be unseen before. In this context, a large part of our daily activity has been affected to a variable degree, depending on the austerity of the lockdown and the spread of the pandemic. At the same time, this situation has provided an opportunity to the researchcommunity for monitoring directly the effect of these changes,or other phenomenathat have surfaced as side effects of the pandemic and whose realization would otherwise a coordinated effort in order to achieve. An example of such aspects are energy and water consumption inside public and private buildings during the pandemic,or the pandemic's effect on air quality inside urban areas, just to name a few.
This has coincided with the fact that in the last few years, before the pandemic, we have seen the adoption of IoT and smart city technologies at a rapid pace throughout the world.
The divide between the natural and the digital worlds has lessened considerably, and there are now multiple sensing endpoints in our offices, homes or even on our body, which continuously produce streams of sensor data. Privacy matters aside, this hasmadeit tangibly easier to monitor the effects of the pandemic on a range of aspects of our everyday activity and the overall impact of the pandemic on our communities. As examples,Google [2] and Apple [3] producereports for the changesin the mobility patterns of the usersof mobile devices, based on requests to these companies' mapping services and mobile device location history data.
At the same time as the pandemic, there is an increasing interest in raising awareness about climate change and energy efficiency, with the EU announcing the European Green Deal [4] and a number of other countries making similar pledges. In this context, there hasbeena lot of interest on how the pandemic affected energy and water consumption overall, as well as in more specific areas such as public buildings, offices and homes. Regarding overall energy consumption, initial results that have shown a considerable drop in the first months of the pandemic, havecontrasted with later results that showed a rebound in consumption [5].
In this setting of great interest about energy in general, the importance of the educational sector is self-evident, in terms of size and significance. Sustainable development and energy-saving behaviors are gradually becoming a part of educational programs. An example of recent activity to promote sustainability concepts in the educational sector through a structured curriculum is the United Nations' Climate Change Learning Partnership ( [6]). Simultaneously and in the context of the pandemic, as mentioned in [1], "schools, universities and colleges have closed either on a nationwide or local basis in 63 countries", a fact revealing a major disruption in the educational sector in terms of delivering classes, but also in terms of how schools function overall and consumeresources, such as water and electricity.
In this work, we study the effect of the COVID-19 pandemic on certain aspects of the operation of a number of school buildings in Greece.We utilize the existing infrastructure and data produced by work kickstarted by the Green Awarenessin Action [7] (GAIA) Horizon 2020 Project. GAIA produceda framework comprising IoT infrastructure in school buildings, applications, as well as educational content and action plans for increasing sustainability awarenessamong students, while also aiming for energy savings through behavioral changesin schools. Essentially, this infrastructure, which was deployed in most casesat least 1 year before the start of the pandemic, can provide us with data to help us draw comparisons regarding energy consumption, indoor comfort, indoor noise and air quality levels, among other.
In terms of research questions, and based on the data that we had available, we compiled the following list:
How did the lockdown during 2020 and 2021 affect the power consumption inside this set of school buildings? Can use the GAIA dataset to detect potential behavior changes during these periods? Are there any other useful findings or correlations that we can infer using data like humidity, temperature and luminosity measurements? Our results indicate that there is a considerable amount of power consumption inside the school buildings involved in our study, in the order of 20-40% of the normal power consumption during weekdays in most cases. As regards indoor noise level measurements, they can be used to track activity inside schools and correlate well with power consumption, but data at this point are inconclusive with respect to detecting behavior changes. With respect to other kind of measurementstypically used in IoT deployments inside buildings, like temperature,humidity and luminosity, they can be usedto infer someinteresting points regarding the buildings themselvesand activity during the lockdown.
II. RELATED WORK
As mentioned in the previous section, although the pandemic is still ongoing, there is a lot of activity from the researchcommunity and, in relation to this work, specifically as regards energy consumption. An overview of the changes in energy consumption during 2020 at a global level was presentedin [8]. Although there were somesteepdrops during the early part of the pandemic, energy consumption picked up pace during the second half of the year. At a global scale, a 6.4% overall reduction compared to 2019 was recorded, with areas like the EU and the US registering 7.7.% and 12.9% drops respectively. Similar results about the first lockdown period until April 2020 are also reported in [9], with a 17% reduction, with consumption returning to pre-COVID levels after mid-June 2020.
Regarding studies focusing on more specific examples of buildings, [10] presented a study on the energy use of several types of municipal buildings in Florianopolis, Brazil, during a period of almost 3.5 months of lockdown, discovering that almost half of the energy consumption in buildings like administrative buildings or elementary schools are not directly related to the presenceof people inside them. Their findings to a certain degree agree with our own, as reported in the following sections. [11] focused on monitoring energy and hot water consumption patterns in a social housing building in Canada. The authors report that during the first, more strict, months of the lockdown overall consumption changed slightly, with the most notable change being the changein the time of the day as regards demand, noting that consumption moved to work hours instead of evening. A report [5] by the International Energy Agency (IEA) agreeswith thesefindings, providing further details into the timing of the bounce of energy consumption back to 2019 levels in several countries
In this work, we focus on the effect of the pandemic on energy consumption specifically in school buildings, as well asthermal comfort and indoor noise levels. We focus on school buildings only in Greece, having identical or similar periods of lockdown and normal operation. Such aspectsreflect either directly or indirectly the effect of the pandemicand lockdown on the behavior of school staff and students,aswell asprovide insights to the overall operation of the schools.
III. THE GAIA PROJECT-INFRASTRUCTURE AND HARDWARE DESCRIPTION
The work presentedhere was conducted in the context of the GAIA Project [12]. Its main objective consisted in increasing awarenessand promoting responsible behavior towards energy efficiency in educational communities. Sensorsand data processingtechnologies were usedto realize educational activities on top of data collected from an IoT infrastructure. Sensors were installed in school buildings to help studentsmonitor the energy consumption of their building, or specific rooms, and become aware of the impact of environmental parameters and their behaviour as regards energy consumption. Overall, 25 schools (from primary to high school level) located in Greece, Italy and Sweden were involved.
This IoT infrastructure currently comprises over 1200 IoT monitoring endpoints and utilizes several hardware and software technologies [13], [14]. In each building, a set of sensors are deployed to monitor the following parameters: active power, energy, internal environmental parameters(e.g., luminosity, noise), external weather and pollution parameters. The actual deployment at each site is customized and adaptedto the specific characteristics of the building (e.g., size, number of students,orientation, building plan). Measurements are continuously acquired by the GAIA IoT Platform which implements data processing services, with sensor data streams processed and aggregates extracted in real time.
IV. METHODS
In this work, we focus on studying the effects of the pandemic on the school buildings in Greece for which we had available data both for the pandemic, as well as previous school years. This allows us to study such phenomenawithin a set of public schools that had similar changes in their operational environment, since they are located in the same country. Some differences in their operation were due to the different educational level of the schools, since we had data available from primary to high school level. The overall timeline of the pandemic, lockdown period and the respective effect on schools in Greece can be seen in Fig. 2. After someinitial caseswere reported, all schools were closed on March 10, 2020 and a strict lockdown began on March 23. Schools startedto reopengradually on May 11, until the end of the school year in June. After the summer break, schools reopened in September 14 and closed again during November, in 2 stages. On January 2021, schools reopened in 2 stages, while in some areasof the country schools were closed and reverted to remote class mode, since these areas were designated as "red" (i.e., with a high number of cases). On May 10, 2021, all schools in the country reopened, with any new closures decided for each class and school separately upon the detection of COVID-19 cases,basedon a number of criteria related to the number of cases. In terms of the question of building diversity and whether the school buildings we use in this study are representative of the schools in Greece,we should note that this set of buildings comprises schools in several parts of the country, including both urban and rural areas, as well as schools from mainland and island areas. They also include primary. junior high and high schools, covering a wide range of cases.
As mentioned in the previous sections, in terms of things that we will be investigating, thesedatasetsfirst of all present an opportunity to study the baseline of schools in Greece with regards to power consumption. Although some conclusions regarding this aspect can be drawn by data from regular periods, the lockdown periods present an interesting scenario, since in several casesalthough lessons were entirely remotely, the buildings in many caseswere open to use from the educational staff, especially in the looser periods of the lockdown. There is also the additional dimension of studying potential differences in energy consumption patterns before, during and after lockdown periods, if any. Indoor noise levels before, during and after lockdown periods present another interesting scenario, especially in terms of correlating them to activity inside schools and seeing whether there are tangible differences that could be traced to behavior change patterns.
In addition, variations in humidity, temperature and luminosity, can be used to indirectly check for behavior changes, aswell asfor studying the effect of the lockdown and COVID-19 measures on indoor comfort levels inside the schools. Specifically for comfort levels, we will utilize the Predicted Mean Vote [15] (PMV) indicator to study this aspect.
The sensors in the utilized infrastructure were sampled at least every 30 seconds, with each batch of samples from all the available sensors reported back in near real time. The collected data were then aggregatedand stored in our backend, either as raw data or as aggregatedhistorical data in multiple intervals (5 minutes, 1 hour, 1 day). The analysis presented in the next section is mainly based on an extracted dataset containing the 1 hour aggregateddatafrom all school buildings and for a period from 1/1/2019 until late May 2021. To clean and process the data in the extracted dataset we use Project Jupyter [16] in Python, together with tools like Pandas [17] and Matplotlib [18].
V. RESULTS -DISCUSSION
A. Power Consumption Comparison
Regarding electric power consumption in our set of school buildings, asmentioned above we utilized measurementstaken every 30 seconds,which were then processedto produce daily sum values. The measurementsin the majority of these school buildings represent either total building consumption, or a significant part of the building, e.g., including all but onefloor of the building. In other words, we have a very representative dataset of the energy consumption in these schools, although we do not include other energy sources,such asgas.Moreover, for the examples included in our figures, which cover the period from March to May, useof gas/oil is minimal, or close to zero, in Greece in such buildings, since they are mainly used for heating.
In Fig. 3, we include some characteristic examples of the consumption of school buildings of various levels. The data displayed show averagepower consumption over the duration of a day, between March 23 and May 5, for 3 consecutive years. In the first 2 examples, at first glance, we see a more or less expected picture; consumption during 2020 and 2021 is significantly lesser than 2019, with the one in 2021 a bit higher compared to 2020, due to schools being open at least for some days. However, we can also spot other interesting findings, with the consumption during off-class hours being higher in 2021 than 2020, and2019, in the first example, while in the second, 2020 displays a lower consumption overall.
Moving on to the rest of the examples, we can see a somewhat different picture. Some schools present a higher consumption during off-hours in 2020, in somecasesprobably due to the use of external lighting for security reasonsduring night time, such as in the case of school H1. In the last example, we see a school that is quite optimized for offhours with minimal consumption, and the consumption in 2021 being close to the one of 2019, indicating quite high levels of activity.
Furthermore, when looking at the averagepicture of power consumption during normal periods and during the lockdown in Fig. 4, in our set of schools we see that the power consumptionduring night time in all 3 yearsis mostly similar, and is close to 40% of the typical power consumption during class time. We also seethat during the lockdown of 2020 when all schools were closed down uniformly power consumption during daytime is smaller than the night time, but is again close to 30% of the normal daytime power consumption. Thesetwo findings could be explained by the use of additional lighting for security reasons during night time, and the non-optimal use of infrastructure inside the school building.
B. Noise levels comparison
Moving on to noise levels, several different types of hardware are utilized in the school infrastructure, due to the fact that the deployment roll-out spanned across a number of years. Such hardware ranges from very basic types of audio sensorsto purpose-built noise level sensors.In this sense,it is difficult to draw comparisons acrossthe whole set of schools, however it should be sufficient for comparing measurements in a relative manner, spanning across the 3 last years for more general observations. Fig. 5 and Fig. 6 provide an overview of the noise levels in several schools between January 2019 and May 2020 and during the 2020-21 respectively. We can observe that noise levels align almost perfectly with the activity in schools,break periods and the lockdown. In this sense, we can use the noise levels to further characterize the activity inside school buildings, and get a better senseof the overall activity levels in the schools, together with the power consumption data. Another interesting finding is that here are examples of noise levels being both similar and lower after the lockdown period. Lower noise levels could also be attributed to certain classes being shut down due to COVID-19 cases. In Fig. 6 such differences are more pronounced, with some schools being completely shut down, while others had somerangeof activity. We can also some bigger noise level spikes in 2 schools after reopening, while in the other noise spikes are similar or a bit smaller than the pre-lockdown levels. Therefore, at this point, results seem to be on a school-by-school basis.
C. Indoor comfort levels and PMV
PMV indicates comfort for a group of subjects given a particular combination of air temperature, mean radiant temperature, relative humidity, air speed, metabolic rate, and clothing. The PMV implementation we use for this analysis uses an eleven-point scale from cold ( 5) to hot (+5) to represent the thermal conditions inside school buildings with more detail in contrast to other implementations with that use the [-3,3] range. A PMV equal to 0 represents thermal neutrality, and the comfort zoneis defined by the combinations of the 6 parametersfor which PMV is between 1 and +1.
Observing the data presented in Fig. 7, we can observe how during the first lockdown period in 2020, the conditions in all schools deteriorated, as expected, as the schools were completely closed and heating was switched off completely. As this period wasduring the endof the winter, the PMV value decreasedfrom an average around -1 to a value lower than -3 in most cases. After that negative peak, conditions gradually improved as the weather conditions were better over time.
For the second lockdown, starting in November 2020, the PMV value started again from a value around -1 to a slightly lower value. This time around the conditions did not deteriorate so fast, as conditions in Greece were still good, with not so low environmental temperatures. Once the primary schools re-opened in January 2021 (middle of winter for Greece), the schools were ordered to keep their windows open at all times to help ventilation. This lead to significant variations in the conditions inside the school buildings as heating was used to keep the classrooms hot enough for the students while the windows were open for more time and heat losses were expected. This behavior is more or less observed mostly in primary schools. Secondary schools have more consistent PMV valuesfor the duration of the lockdown, as the buildings were closed for the whole period.
VI. CONCLUSIONS
In this work, we focused on studying the effect of the COVID-19 pandemic on a diverse set of schools in Greece. This was made possible via an IoT infrastructure installed in previous years for the purposesof sustainability awarenessand energy savings, which provided us with datasets containing power consumption, noise and air quality levels, temperature, humidity and luminosity data. From our results, it is obvious that the way school buildings operate could be further optimized in terms of energy efficiency. During the period of the lockdown, buildings consumeda significant amount of energy close to, or exceeding, 20-40% of the energy consumed on average in normal periods. Furthermore, regarding noise levels they can be used to infer the levels of activity inside the schools, together with power consumption, but results are inconclusive at this point in our study as regards behavior change. Moreover, indoor conditions, as regards comfort, have shifted considerable during the lockdown. Regarding our future work, we intend to delve deeper into the available datasets and further study the impact of the pandemic on school operation, and especially as regards potential behavior changes in students and educators.
Fig. 1 :
1Examples of nodesand installations in schools involved in the study.
Fig. 2 :
2Timeline of the pandemic in Greece and its effect on schools' operation from March 2020 to May 2021.
Fig. 3 :
3Average Hourly Power Consumption for schools between 23/3 and 5/5 for 2019, 2020 and 2021.
Fig. 4 :
4Average Hourly Power Consumption for primary and secondary schools before, during and after the 2020 42-day lockdown.
Fig. 5 :
5Noise levels during 2019 and 2020 lockdown on 5 schools of all grades.
ACKNOWLEDGMENT This work has been supported by the European Union's research project "European Extreme Performing Big Data Stacks" (E2Data), funded by the European Commission (EC) under the Horizon 2020 framework and contract number 780245,andthe "Green AwarenessIn Action" (GAIA) project, funded by the European Commission and the Executive Agency for Small and Medium-sized Enterprises (EASME) under the Horizon 2020 framework and contract number 696029. This document reflects only the authors' views and the EC and EASME are not responsible for any use that may be made of the information it contains.
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Along the design process of interactive system many intermediate artefacts (such as user interface prototypes, task models describing user work and activities, dialog models specifying system behavior, interaction models describing user interactions …) are created, tested, revised and improved until the development team produces a validated version of the full-fledged system. Indeed, to build interactive systems there is a need to use multiple artefacts/models (as they provide a complementary view). However, relevant information for describing the design solution and/or supporting design decisions (such as rational about the design, decisions made, recommendations, etc.) is not explicitly capturable in the models/artefacts, hence the need for annotations. Multi-artefacts approaches usually argue that a given information should only be present in one artefact to avoid duplication and increase maintainability of the artefacts. Nonetheless, annotations created on one artefact are usually relevant to other artefacts/models. So that, there is a need for tools and techniques to coordinate annotations across artefacts/models which is the contribution of the present work. In this paper, we propose a model-based approach that was conceived to handle annotations in a systematic way along the development process of interactive systems. As part of the solution, we propose an annotation model built upon the W3C's Web Annotation Data Model. The feasibility of the approach is demonstrated by means of a tool suite featuring a plugin, which has been deployed and tested over the multi-artefacts. The overall approach is illustrated on the design of an interactive cockpit application performing two design iterations. The contribution brings two main benefits for interactive systems engineering: i) it presents a generic pattern for integrating information in multiple usually heterogenous artefacts throughout the design process of interactive systems; and ii) it highlights the need for tools helping to rationalize and to document the various artefacts and the related decisions made during interactive systems design.CCS CONCEPTS • Human-centered computing • Human computer interaction (HCI)Additional Keywords and Phrases: Development process of interactive systems, prototyping, model-based approach, annotations.ACM Reference Format:Authors. 2022. Engineering Annotations: a generic framework for gluing design artefacts of interactive systems. ACM, New York, NY, USA, 12 pages. https://doi.org/XX.XXXX/XXXXXXX.XXXXXXX.
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10.48550/arxiv.2205.01333
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"https://arxiv.org/pdf/2205.01333v1.pdf"
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2205.01333
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21a4ef6e373496937c1b169f86b4c37c5088a98b
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Engineering Annotations: A Generic Framework For Gluing Design Artefacts in Models of Interactive Systems
Marco Winckler
Université Côte d'Azur * Philippe Palanque
Universite Paul Sabatier -Toulouse III
France
Eric Barboni
Jean-Luc Hack
SofteamFrance
Olivier Nicolas
Universite Paul Sabatier -Toulouse III
France
France Laurent Softeam
Universite Paul Sabatier -Toulouse III
France
Gonçalves
Universite Paul Sabatier -Toulouse III
France
France Softeam
Universite Paul Sabatier -Toulouse III
France
Engineering Annotations: A Generic Framework For Gluing Design Artefacts in Models of Interactive Systems
Along the design process of interactive system many intermediate artefacts (such as user interface prototypes, task models describing user work and activities, dialog models specifying system behavior, interaction models describing user interactions …) are created, tested, revised and improved until the development team produces a validated version of the full-fledged system. Indeed, to build interactive systems there is a need to use multiple artefacts/models (as they provide a complementary view). However, relevant information for describing the design solution and/or supporting design decisions (such as rational about the design, decisions made, recommendations, etc.) is not explicitly capturable in the models/artefacts, hence the need for annotations. Multi-artefacts approaches usually argue that a given information should only be present in one artefact to avoid duplication and increase maintainability of the artefacts. Nonetheless, annotations created on one artefact are usually relevant to other artefacts/models. So that, there is a need for tools and techniques to coordinate annotations across artefacts/models which is the contribution of the present work. In this paper, we propose a model-based approach that was conceived to handle annotations in a systematic way along the development process of interactive systems. As part of the solution, we propose an annotation model built upon the W3C's Web Annotation Data Model. The feasibility of the approach is demonstrated by means of a tool suite featuring a plugin, which has been deployed and tested over the multi-artefacts. The overall approach is illustrated on the design of an interactive cockpit application performing two design iterations. The contribution brings two main benefits for interactive systems engineering: i) it presents a generic pattern for integrating information in multiple usually heterogenous artefacts throughout the design process of interactive systems; and ii) it highlights the need for tools helping to rationalize and to document the various artefacts and the related decisions made during interactive systems design.CCS CONCEPTS • Human-centered computing • Human computer interaction (HCI)Additional Keywords and Phrases: Development process of interactive systems, prototyping, model-based approach, annotations.ACM Reference Format:Authors. 2022. Engineering Annotations: a generic framework for gluing design artefacts of interactive systems. ACM, New York, NY, USA, 12 pages. https://doi.org/XX.XXXX/XXXXXXX.XXXXXXX.
INTRODUCTION
Design is a problem-solving process whose main objective is to find a way to implement requirements, respecting constraints, and ensure good quality. According to the ISO standard 9241-210 [1], the design process of an interactive * Authors' addresses: Marco Winckler, [email protected], Université Côte d'Azur, SPARKS/wimmics team, CNRS/INRIA I3S, France; Philippe Palanque, [email protected], Universite Paul Sabatier -Toulouse III, IRIT, France; [email protected], Universite Paul Sabatier -Toulouse III, IRIT, France; Jean-Luc Hack, [email protected], Softeam, France; Olivier Nicolas, Softeam, [email protected], France; Laurent Gonçalves, [email protected], Softeam, France. systems is iterative: needs and requirements are analyzed, design solutions are identified, created, tested and improved until the development team produces a deployable version of the full-fledged interactive system that meets users' goals and needs. This process produces two types of results: a specification of the design solution to be implemented (the interactive system) and a set of design decisions that drive the evolution of the design along the iteration cycles.
We claim that the information explicitly described by models/artefacts are not enough to describe all the information required to produce interactive systems, hence the need of annotations. To illustrate our point, consider the case of user interface prototypes that are the most common type of artefact used to describe design solutions for interactive systems. In all the possible forms (i.e. low-fidelity, hi-fidelity, executable, etc.) prototypes feature a concrete (yet partial) representation of an interactive system and they can be used to explore many design alternatives before implementing the final product [2]. In early phases of the development process, drawings and wire frames are useful and desirable [3] to support ideation of the product but as the process advances, they are replaced by interactive specifications and by executable prototypes [4]. Prototypes are useful and necessary but they are not sufficient to fully describe an interactive system. On one hand, prototypes are not self-explicative, which is illustrated by the fact that annotations are widely used to explain, for instance, the use of icons in a design [5]. On the other hand, prototypes cannot directly inform aspects of the interactive system such as the goals they support (and the ones they don't) and how the system will compute the information provided by users. One might be tempted to assume that lack of expressiveness of prototypes would be solved by using multiple artefacts/models (such as task models, dialog models and interaction technique models) to provide complementary views. Nonetheless, handling multiple artefacts implies that related artefacts should be updated together during the iterations in order to keep a consistent and integrated view of the interactive system [12]. For example, adding a new button to a low fidelity prototype should affect the tasks performed by users and the dialog model describing the inner system behavior.
Therefore, artefacts must co-evolve along the development process [10].
Whilst decisions made by the development team occur iteratively throughout the process and will result in evolutions of multiple artefacts, the ISO standard 9241-210 is very silent about how to record and process such design decisions. Empirical observation [14] [15] have shown that development teams often make extensive use of annotations. Annotations are flexible, they can assume many forms (such as text, sketching, etc.) and be attached to any type of artefacts, making them a suitable mechanism to provide complementary information throughout the development process. The study performed by Gutierrez et al. [15] pointed out that annotations are used by members of development teams to: i) record the results of discussion including decisions and upcoming tasks, communicate and inform other team members of the work done, ii) gather internal and external feedback on artefacts stored in the workspace, iii) conduct usability evaluations by documenting information and by recording conversation between design teams and UX experts, and justify design choices, and document the design choices by describing them retrospectively. While previous work [15] argued for the use of annotations for user interface prototypes, this paper extends the use of annotations made over various artefacts (including prototypes) by providing a generic engineering solution to connect them across multiple artefacts. More precisely this work presents a design and engineering solution that allows using annotations to achieve the following goals to:
i) Enrich artefacts used to specify an interactive system with any kind of complementary information that would be used to support decision making along the development process;
ii) Cross-reference and cross-check design decisions to multiple artefacts; iii) Follow the evolution and ensure the consistency of design decisions along the development of interactive systems.
To achieve these goals, we put annotations as first-class information throughout the design process. We propose a model-based approach offering a unified view of annotations throughout the interactive system iterative design process. The meta-model extends the W3C's Web Annotation Data Model by revising concepts (such as authors, versioning and annotation types) to better support interactive systems designers making it possible to track changes made in artefacts and their information. Our approach is generic to any type of artefact but the present work focuses on three types of artefacts: task models, dialog models and prototypes.
Hereafter we start by revising the concept of annotations (section 2) and we present an overview of existing studies about the use of annotations over prototypes. Section 3 review existing annotation models proposes an extension to the W3C's Web Annotation Data Model [16] to support the annotations on multiple artefacts. Section 4 introduces an approach and a tool support (called ARMADILLO) for connecting multiple artefacts using annotations according to annotations model presented in section 3. In order illustrate the feasibility of our approach, we illustrate the use of the tools on a case study on the domain of aircraft cockpits. The last section concludes the paper and introduces perspectives and future work.
RELATED WORK
In this section we present an overview of the literature concerning the structure and the use of annotation for the design of interactive systems, including tool support for prototyping activities.
Anatomy of annotations
The first studies about annotations started with the identification of common practices by university students on their paper textbooks [17][18] [19]. Many of the elements of paper-based annotations were then transposed to electronic documents. Therefore, the common definition of annotation such that provided by Bringay et al. [19] below, refers to documents: "An Looking to more robust definition of annotations for digital documents, we have found annotations describe as "user made statements", consisting in a body (i.e. text note or graphical content), a link (or anchor) to a target including a location within the document as well as other metadata [22]. In the Open Annotation Data model and the Web Annotation Data model proposed by Sanderson et al. [20], [21] annotations are considered as a set of resources in which the body is related to one or several targets (document annotated) as illustrated by Figure 1. All those definitions acknowledge the distinction between an element being in relation to another: the annotation body and the annotated document, which are linked somehow by an anchor. Agosti and N. Ferro [13] stress the importance of the linking mechanism to formally describe annotations. This linking can be made in several ways, depending on the support of each part of the annotation. For example, when annotating books, notes can be written in the margins or in postits. Marshall [18] identified 3 different mechanisms of association on textbooks including: arrows to connect the document to its annotation, marks such as bracket and brace, and the proximity of the annotation and the target by writing in the margin or the interline. However, as for digital annotations, the artefact annotated and the annotation can be located in the same or in two different files. For that, we talking about electronic documents we add to this list of linking mechanisms the usage of reference, which allows a nonintrusive way to annotate a document. Moreover, the linking between a body and a target created by an anchor might convey particular meaning for the annotations, as suggest by S. Bringay et al. [19]:
• The placement of an annotation can be explicit (when the target is clearly visible in document), or tacit anchor (when placed in the document but not connected to a particular element in the document). • One-target versus multi-target anchor, with respect to the many possible targets for an annotation;
• Conventional and not conventional anchor, with respect to the existence (or not) of an agreement for interpreting annotations (red marks means might refer to a convent for important topics).
As we shall see, these three elements (body, target and anchor) are core concepts not only for paper-based or electronic documents but they are essential to understand how annotations applies to the many artefacts used to build interactive system as well. In [36], Li et al. have defined a classification of annotation approach for Computer-aided Design. This classification identifies the following categories of attributes that complete the specification of annotation: targeted media, audience, rendering system, usage and function, representation, and storage location. This classification of annotations brings another complementary view of annotations.
Function, meaning, and uses of annotations of digital artefacts
Based on the review of the literature, we summarize hereafter three main functions played by annotations: to enrich a document, to support communication and to support an intention/activity carried out by the author of the annotation. Whilst most of the literature in the matter refers to text documents, we suggest that the following classification is relevant for the development of interactive systems as it can help to add semantics to the widget used for creating annotations over artefacts.
A Mean to Enrich Documents
When adding an annotation, the document is augmented somehow. Based on the analysis of the writing contents of the body of an annotation and the relationship created to the target, Zacklad [24] suggests there are three types of annotation:
• An attentional-annotation draws the attention of future readers of the document by emphasizing and pointing out some part of the document. For example, highlighted text, underlined text, symbols.
• An associative-annotation connects an existing element to another one. This annotation can be represented with arrows or references that are not necessarily located in the same document.
• A contributive-annotation is a new information created in reaction or in response of a segment of the annotated document. This annotation either complete this document or discuss it and it requires a link with its initial document by using an associative-annotation. The contribution is either added to the document or in the next edition of the document (which is a similar concept of the elaboration annotations proposed by Lortal et al. [26]).
To this list, we add descriptive-annotation such as semantic annotation as defined in the W3C [21]; which consists in adding metadata to make it easier to process by computer (e.g. indexing, researching) and make them interoperable between different systems. For example, a document is extended with structured information such as the author, the title, the creation date. G. Lortal in [26] qualify those annotations as "Computational level" annotation.
A Mean to Support Communication
Annotations play an important role for the communication between diverse actors (author, reader, reviewer, coordinator, etc.). In an iterative design process, the interaction between actors will ultimately make artefacts and annotations themselves to evolve over time. Bringay et al. in [20] defines collaborative annotations as a way to help actors to communicate in a collaborative work to accomplish three main goals: i) Editorial help: annotations can be used as a guide for the creation of a document by indicating instruction or constraints. They can be used as a set of guidelines for the creation of the document or in a revision of the document after a review; ii) Argumentation: annotations can also be used to discuss and argue between collaborators about the document; and iii) Planning: annotations can be used to coordinate the project, plan tasks to do and manage the people working in the project. Three attributes are related to this type of annotation: the task to carry out, the time to complete the task and the person or group of people in charge of the task [27].
A Mean to Convey the Large Variety of Authors' Intentions
Another classification of annotations refers to the authors' intention and/or activity carried out by the author of the annotation. In this respect, the classifications proposed by Naghsh et al [24] and by Agosti and Ferro [10] are worthy of mention. Naghsh et al [27] identified 6 different usages of annotations that match with the categories defined above:
• Clarifying and explaining the design.
• Verifying and request a verification from other designers or users.
• Exploring by asking questions to obtain more details on end users' needs.
• Altering or requesting an alteration proposed by the end users.
• Confirming and give feedback on a design.
• Understanding by asking questions to the designers.
Agosti and Ferro [13] encompasses three goals:
• Comprehension and study. The intention here is to understand and analyze the document. It can relate to attentionalannotations in which the annotator highlights parts of the document he found interesting or ask questions to help his comprehension. Those annotations do not add information on the document. • Interpretation and elucidation. This usage refers to the annotations made to add information, explain a document according to the annotator understanding in order to make it easier to understand and then discuss about it. It could be an analysis or an argumentation for instance.
• Cooperation and revision. Annotations can also be used for sharing ideas and opinions about a text. This can be done through evaluation of the document, feedbacks on it or tasks planning for example.
As we shall see, goals described for annotations might be present in design activities on interactive systems.
Use of annotations for the design of interactive systems
During the design process of an interactive system, the development team gather information about users and produce artefacts to represent the design solutions. The fully-fledge system is the result of iterations of cycle of design an evaluation of the artefact produced. Design decisions are made along the development process and influence the next step of specification of the system. As illustrated Figure 2, annotations are indeed commonplace and might have multiple uses during the development of user interface prototypes. Given the associative nature of annotations, it is quite natural to consider annotation as a possible design solution to the problem of tracing design decisions to artefacts. The annotations presented in Figure 2.c, for example, should be interpreted as a design decision.
Gutierrez et al. [28] conducted a study to investigate how annotations could affect the development of interactive systems in a UCD process. For the purpose of that study, they developed an independent tool called Helaba, which allow the design teams to store and organize artefacts on a common workspace and to connect their decisions to them. The ultimate goal was to support the traceability of design along the design process. Annotations are materialized by Heleba by the means of "Decision cards", "Notes" and "Conversation thread" that can be attached as references to artefacts in order to add content or discuss about an artefact or a specific part of it. They observed that participants stored a curated selection of artefacts that were representative of the process in the shared workspace during the different activities of the UCD and that the annotations provided in her tool were used by the participants of the study to "build a narrative of their design process, especially in relation to how artefacts linked to each other". According to [28], annotations were used to:
• Record the results of discussion including the outcome of those discussion, decisions and upcoming tasks.
• Communicate and inform other team members of the work done.
• Gather internal and external feedback on artefacts stored in the workspace.
• Conduct usability evaluations by documenting information and by recording design team's comments.
• Remind and to justify choices that was made during the process "in the late stages of the project".
• Help to document the design choices by describing them retrospectively.
Overall, this study showed the usage of a shared workspace of annotations that were used along the design process (in particular for User analysis, Task analysis, Lo-Fi prototype and Hi-Fi prototype). Nonetheless, Heleba works as a repository of artefacts and annotations that are not directly connected to the tools used to build the artefacts.
It is interesting to notice that annotations must be considered a special case of artefact required to follow the development process of interactive systems. One particular aspect of the annotations in a UCD process is that annotations can be related to certain versions of the artefact but not each of its version (e.g. an annotation indicating to fix an error). Moreover, annotations (especially contributive or organizational annotations) can influence the evolution of other artefacts but also affect by the evolution of the artefacts. Thus, while annotation have their own lifecycle, this lifecycle can be affected by the related artefact's lifecycle as well. a) Explaining the design. b) Record results of usability evaluation. c) Document design decisions.
Tools for supporting annotations of prototypes and other artefacts
In this section we present a summary of an analysis of 80 tools for annotating prototypes. Using the keyword "annotations tools", "prototyping tools", and "wireframing tools", we have retrieved publications from conferences (including CHI, UIST, DIS, EICS, INTERACT and IAnnotate) and then we extended the search to find other tools indexed by google.com. These tools can be generally classified into two categories: a) generic annotation tools that are loosely coupled with prototypes and artefacts used for the design of interactive systems, and b) prototyping tools that embed annotations as native functions for designing interactive systems.
SnapUp, and Atomic).
A full comparative analysis of these tools is to be found in [29]. Hereafter, we provide a summary of the most striking findings that we consider relevant for understanding the contribution presented in this paper. During our review, we observed that annotation support in the tools were limited for our studies. Indeed, we did not find a satisfying tool providing support for our needs on evolving artefacts and annotations management in a UCD process.
The main objective of the annotating tools we reviewed is to provide a medium that can be used to communicate over a document. Most of the annotating tools only allow to annotate web pages. As for prototyping tools, annotations features were either promoted for helping collaboration within the design team or to gather feedback from external users. The overall result of this study is that while annotations are supported in many tools, their implementations are limited and not always suited in the context of the iterative process of a UCD approach.
Hereafter we list the main findings of this review:
• Annotations can only be related to one artefact at a time. Other artefacts can only be loosely referenced by citing them or enclosing a copy. Thus, creating a unidirectional relationship between those artefacts. Only 6 tools allowed to set several targets on one annotation.
• The targeting of annotation is either made by selecting text, defining areas, positioning the annotation, positioning a marker or citing the related fragment of artefact.
• Temporal evolution of artefacts is rarely acknowledged and it is managed by only 9 tools.
• Fewer tools manage to keep a consistent link between the annotation and its artefact either by extracting a snapshot of the targeted fragment or by implementing a limited recognition system. • Lifecycle of annotations has been considered in only 19 tools through a status system allowing users to manage manually annotations. The other tools forces user to dispose manually the annotations once they have been processed which imply a loss in the traceability of the management of annotations.
• Textual annotations are the most adopted form of annotation followed by graphical representation of either shapes, icons or markers. Only three tools enable to enclose files in annotations (JustInMind , MockFlow and Notism).
• There is no integrated support for documenting a custom semantic. Thus, those semantics are either implicit inside the design team or defined in an external document. This semantic can be used for managing the annotations or give them a weight. When available, semantic is limited to the use of predefined tags (e.g. Amaya , iRise , Concept.Ly , Quilt [30], Cacoo, Neonion [31]).
• As for the management of annotations, 29 tools feature a list of the annotations. Following this list, we noted that only 17 tools supported a filtering option and 15 tools supported a search feature on the annotation. A navigation feature between annotations and their targets is not systematic in every tool we reviewed.
• For the collaboration, 55 tools supported a synchronous collaboration over artefacts. In 27 tools, it is possible to restrict users' right on the artefact or in the annotations. In 42 tools, author of an annotation can be identified.
Pitfalls of existing annotations approaches
This study about annotations raises some interesting questions about the role of annotations for engineering interactive systems. It is worthy of notice that annotation is a concept easy to understand and common place in many professional scenarios where a piece of information cannot be represented as being part or the artefact. Empirical studies demonstrate the important role of annotations in communication between teams and support for decision making processes along the development process of interactive system; and yet quite often annotations are still considered as a second-class piece of information that is not systematically treated (or even mentioned) by most development approaches.
Annotations can have different formats and be used for different purposes with multiple artefacts. Nonetheless, existing tools only support a few types of annotation formats (mainly text and drawings) that lack of semantics to support decision making process.
Annotation tools are often very specialized for a single type of artefact. Moreover, there is a strong binding between the annotation tool and the editor. Whilst such strong binding make sense for an interaction point of view, it is not possible to reuse annotations outside the editor which a major drawback for tracking annotation created on multiple artefacts.
Development processes using multiple artefacts are quite common and yet, they fail to provide a uniform view for annotations that could make sense multiple artefacts at a time. It is interesting to notice that previous work [51] pointed similar problems when dealing with multiple UML models; however, we could not find in the literature a solution for unified view on annotations.
Conversely to other approach, the solutions we propose were designed to support systematic use annotations of multiple artefacts along the development process. First of all, our solution allows to connect multiple artefacts (as illustrated in section 5). We rely on standards (described in section 3) which allows the export/import of annotations into diverse tools. We differ from other approaches where the annotation mechanisms are a functionality of the editor of artefacts (strong binding between artefacts and annotations), by proposing a distributed architecture (section 4) where annotations created using plugins are stored independently into a central storage, thus keeping annotations available for reuse by other tools (including the creation of dedicated tools for creating an overview of all annotations in different artefacts of a project). The architecture of plugin is meant to extend editors of artefacts. Last but not least, as part of our work we have proposed (see section 4) a set of annotations formats that go beyond text or drawings (including markers, voting mechanisms, and testing scenarios) that features the semantics for the use of annotations.
ANNOTATION MODEL
Currently, the Web Annotation Data Model proposed by the W3C [16] is the standard model for describing annotations. The W3C annotation model provides a common format for describing interoperability of annotations through the web. In this section, we propose an extension of the Web Annotation Data Model [16] to cope with the idiosyncrasies of annotations of interactive system design and development.
Rationale for extending the Web Annotation Model
The Web Annotation model [16] was initially designed to support the annotation of Web documents and URIs (Universal Resources Identifiers). As such, the Web annotation model is quite generic for annotation of image and text documents.
That model is quite flexible and meant to be extended according to the particular cases of use [21]. By extending the Web Annotation Model, we aim at reaching the following idiosyncrasies of interactive system design and development:
•
Annotations concern diverse design artefacts: during the context of the design of an interactive system, a large variety of artefacts is produced. Annotations should work with all sort of artefacts required for the design and development of interactive systems. In the present work, we are particularly interested in artefacts including prototypes, tasks models, system models, specification documents and any other document produced or gathered during the design process. Each artefact is likely to receive annotations either as a support for active reading, for communicating, for reviewing, for planning or for editing.
•
Independence with respect to the targets: annotations are often considered part of the artefact and/or strongly tied to them. Such as a strong binding prevents reuse and analysis of annotations outside the environment where they were created. Some level of independence from the target might allow the export of annotations, so that they can be used as a first-class information to reason about comments and decisions made during the development process. • Connecting multiple targets: an annotation might concern different artefacts. For example, a typo in a form field not only affect the user interface prototype but also the data model. Enabling the connection to multiple targets helps to reduce redundancy of annotations referring to the same problem and also foster the connection between artefacts that are implicitly related. • Support the evolution of artefacts: artefacts are produced through iterations and continuously evolve along the development process. The development team might need to trace of the outcomes of annotations attached to evolving versions of artefacts.
•
Consider the life cycle of annotations: along the development process, annotations might be created, revised by the creator and/or other members of the development team, and disposed when no longer needed. For that, annotation must be part considered as living components that can change the status according to a specific life cycle that might be independent of the evolution of targets (the design artefacts).
•
Enabling many forms of annotations: annotations might be created using diverse forms (such as text, brushing/highlight markers, drawings, icons, …). They can superimpose contents available in other sources, for example adding a layer information about usability problems found on design artefacts [11]. Complex annotations forms might include voting mechanisms as a mean to consolidate opinions around design options [14]. Each form of annotation might be more or less suitable to a particular type artefact. Nonetheless, each form of annotation must have a specialized selector that allow users to express an intention with respect to artefacts. Choosing a particular form of annotation (ex. text or drawing) might also imply a particular meaning and require a particular processing. • Make explicit the intentions: annotations can be adapted to many usages and functions like communication, planning, contribution in the edition of the artefact and so on. We take the benefice of the possibility of extending the classes of the model to implement different type of annotations (i.e. attentional, associative, and contributive annotations) as suggested by Zacklad [24]. The association of intentions to annotations is a means to make orient decisions along the development process. It is interesting to notice that the intentions for creating annotations might be different according the people involved in the development process. For example, end-users, clients (or product owners), as well other members of the development team might have a different perspective for the usage of the system [46]. So, knowing the profile of the creator of annotations is important for understanding the intentions. • Add semantics to the annotations: The integration of semantic annotations will help organizing information, building a structure for the information gathered and describing the interactive system being developed in its entirety instead of only considering localized annotations in artefacts as existing prototyping do tools do. The semantic annotations describing the relationship between artefacts, annotations for planning according to Bringay [19] as well as annotations for documenting the design process that are described by Gutierrez et al [15]. Our extended model aims to support both semantic annotations and non-semantic annotation on artefacts in the context of the design process of an interactive system. It is compatible with the attribute categories defined by Li et al. in [36].
These specificities of the design process of an interactive system indicates a particular use of annotation that goes beyond the usual annotation of documents. Indeed, we need to consider the evolution of the artefacts, the lifecycle of the annotations, and the evolution of the information on the interactive system being designed (which might include the successive versions of the artefact as well as the input provided by every actors of the design process).
Macroscopic view of the annotations model
Our The class artefact was added to the model to allow adapting an annotation to the diverse types of artifacts that might be used along the development of interactive systems. Is considered an artefact any file (or document) that is produced (or gathered) during the design process. It is composed by a set of versions that correspond to a versioned content of the artefact. Each version is also characterized by its metadata which contains various information like the creation date and the lifecycle status of this version that indicate if the artefact is being written, waiting for review, reviewed finished, being updated or archived. The content of the artefact refers to the type of information that is contained in the artefact. Each type of content reflects a different aspect of the interactive system that is being designed and each follow a different syntax. For instance, a task model can be described using a task model notation such as HAMSTERS [41].
In our model, an annotation considered is a particular type of artefact. This enable the independence of annotations compared to their evolving targets, it enables annotating different artefacts with one annotation and also enable annotating other annotations. Moreover, our model extends the elements body, type and target as defined in the Web Annotation Data Model. As for body, our model includes classes allowing adapting the visual aspect of the annotation. We consider a body might have a basic representation (such as text, drawing and images) or a structured representation composed of (at least) a label and an interactive element. Examples of annotations featuring structured body include: a vote (which contains a counter allowing every reader to cast a vote agree/disagree), a scenario (which features a list of text elements constrained by a grammar to express a sequence of tasks performed by users), an external file (which in addition to a visible label, has a link allowing to open the file), and markers (associating a glyph/icon next to the label). As for the type, this was extended to describe the function of the annotation as suggested by Zacklad [24] (i.e. attentional, organizational, contributive and associative annotations).
Like the Web Annotation Data model, a target is materialized by a selector which is used to specify the relevant parts of the artefact that is being annotated. A selector must be specialized to cope with the inner structure of the document being annotated. These extensions describe how the model can be adapted (using an adapter design pattern) to cope with the specificities of every development environment for editing the diverse artefacts used for building interactive systems. So far, we have extended this class to copy with HAMSTERS task models [41], PETSHOP dialog models [45] and PANDA prototype models [14]. We illustrate here how specialized selector are created in the model. Further details about these models are presented in the case study.
We also extend the entity creator defined in the W3C which was originally meant to identify the people or group of people that create the annotation. Indeed, we refine the entity creator to include a variety of roles involved in the development process of interactive systems. In our model, these roles are expressed by the means of metadata whose terms must be adapted according to the project. A basic list of roles having access to the annotations includes the client (or product owner), the end-users, the designer, and the developers (this should encompass the role of participants involved).
Metadata is used to assess the different annotations produced on the artefacts. Annotations can be created by any user and each annotation can express a variety of information on any domain (ex. Use cases, requirements, constraints, data, identification of problems, personal feedback, and personal opinion). The role of the creator of the annotation can help to determine both the expertise and legitimacy of the creator in the information given in his annotations. Other parameters can quantify (popularity of an opinion within an identified group) or be informally assessed such as the relevance and trustworthiness of information (a creator can emit hypothesis or facts based on unreliable sources).
TOOL SUPPORT: ARMADILLO AND PLUGINS
In order to demonstrate the feasibility of our approach we have developed a tool suite called ARMADILLO, which stands for "Annotating by Referencing Models, Artefacts, Documents to Identify Logically Linked Objects". ARMADILLO implements the extended annotation model described in section 3. The ARMADILLO tool suite was specifically designed to help the development team to accomplish two main tasks:
i) annotate artefacts used to specify interactive systems; and, ii) allow cross-referencing of annotations and multiple artefacts.
To support these tasks, ARMADILLO encompass two main components: a Project repository and a plugin. The project repository is used to manage all the annotations created on a project and deal with the cross-referencing of annotations created over diverse artefacts. The plugin allows to create annotations using the editor of artefacts used in the project. All these components of ARMADILLO were built as a Java application over the NetBeans Platform framework to facilitate the integration and distribution of the tools Figure 5 : Overview of ARMADILLO architecture and its components project repository and plugin.
ARMADILLO: project repository
The component project repository is the central element in the architecture of ARMADILLO. It was conceived as a central storage of files that contains the specification of the interactive system (i.e. the models as we follow a model-based approach) and all the annotations created over these artefacts. Annotations are stored as independent files in a central annotation repository. The central part of the Figure 5 shows the ARMADILLO repository which contains the list of models describing the artefacts (using generic names such as Model A1, Model A2, Model B1 and Model B2), the list of individual annotation files (one per artefact and per version of the artefact created, respectively Annotation A1, Annotation A2, and Annotation B2) and an index file (regrouping all files in a project). Each model corresponds to an artefact used to specific interactive systems (such as task models, user interface prototypes, dialog models, etc.).
Annotations are stored in ARMADILLO as XML files. Each annotation file contains references for the artefact it annotates, so that it is possible to navigate from a particular annotation to the corresponding the artefact. This feature is illustrated by Figure 6 where we can see an annotation file featuring a reference (see the arrow showing the place) to "pandaannotation", which is known in index files records as the artefact editor for PANDA models. With that annotation then, it is possible to know how to launch the corresponding editor (PANDA in this example) to see the annotations in the context where artefacts were built. Other fields in the annotation field (not displayed here) let us to know which project the annotation belong. This basic file-oriented system architecture allows to create annotations at any time along the development process. Moreover, it is possible to connect an annotation to any specification available in the development environment as far as it refers to a file available in the project repository; this is an important aspect allowing the scalability of the approach. When an annotation is created with ARMADILLO, it is recorded in the index file. ARMADILLO uses the index file to parse individual annotations files and generate a graph depicting annotations in a project. Figure 7 illustrates a particular case for an annotation in ARMADILLO that connects multiple targets (see Figure 7.c).
ARMADILLO: plugins
When annotations are created directly over the repository, the level of granularity of the target is a file that describes the artefacts. For a fine selection of elements (ex. zone, objects, etc.) it is necessary to have access to the editor tool used to create artefacts. For that, in addition to the project repository, we deliver a plugin allowing to create, edit and assign annotations to targets. So far, the ARMADILLO plugin is only available for the NetBeans Platform framework, which means that editor tools build with that same framework. Once connected to the editor, the ARMADILLO plugin provides access to a palette of annotations that includes: basic types of annotation (such as textual_annotation used for creating text annotations, drawing_annotation for hand free sketching) and structured annotations (such as scenario for describing a list of steps to follow with the artefact, annotation marks that allow to pinpoint elements that might require user attention, and references allowing to complete the description with an external file). Figure 8 shows the palette of annotations supported and made easily available through the ARMADILLO plugin. Each plugin is a particular instance of the class selector in the model shown Figure 4. Figure 9 illustrates the instantiation of the ARMADILLO plugin in the environment of the editor PANDA. The creation of annotations is then made by simple drag & drop interaction from the palette (Figure 9.a) to the main editor area (Figure 9.b), placed next to the artefacts ( Figure 9.c) where annotations (featuring yellow marks, Figure 9.d) can be customized and connected to specific targets.
So far we have developed ARMADILLO plugins for three editors: PANDA (for creating user interface prototypes), HAMSTERS (for describing task models), and PETSHOP (for describing dialog models). These three editors are delivered as part of the framework called CIRCUS [40]; they can be used alone or in combination. All these tools were built upon the Netbeans Platform framework. Hereafter we present a brief description of each tool used for the case study in section 5.
PANDA is a prototyping tool which allow to create medium-fidelity and interactive prototypes by specifying both the dialog and the presentation of an interactive system [14]. This tool support focuses on the modeling of prototypes to formally represent the interactive system as shown by Figure 12. PANDA allows to model the presentation (i.e. screens) and part of the dialog (i.e. navigation between screens) of the interactive system using automata. PANDA prototypes are partial representation of interactive system, thus, they convey a representative view of the current goal of the design process. This goal can be refined or altered during the course of the design process. We suggest that tracing the evolution of this artefact and the reasons of the evolutions would be feasible by structuring the different versions produced within the project workspace of the prototype and by using annotations to connect those evolutions of the prototypes with the other artefacts of the project.
HAMSTERS (Humancentered Assessment and Modelling to Support Task Engineering for Resilient Systems) [41]
is a tool-supported task modelling notation for representing human activities in a hierarchical and structured way. Task models [6] are useful artefacts supporting the analysis of user tasks and the description of the logical activities that have to be carried out in order to reach the user's goals. HAMSTERS notation is inspired by existing notations, especially CTT [48], but it has largely extended including pre-conditions associated to task executions, data flow across task models, more detailed interactive tasks… HAMSTERS models can be edited and simulated in an environment which also provides a dedicated API for observing editing and simulation events making it possible to connect task models to system models (such as ICO models). Figure 11 shows an illustration of task models used in the case study.
PETSHOP is a modeling environment that supports the edition, execution and verification of ICO models [40]. The ICO formalism is a formal description technique dedicated to the specification of the dialog part of interactive systems.
Dialog models [7] play a major role on design of interactive system by capturing the dynamic aspects of the user interaction with the system providing a specification of the relationship between presentation units (e.g. transitions between windows) as well as widgets (e.g. activate/deactivate buttons). As for interaction techniques models, they enable to describe precisely the events chain (i.e. fusion/fission of events) both at input and output levels [8], thus mapping events to actions according to predefined constraints enabling/disabling actions at runtime which is not covered by other type of artefacts [9]. ICO uses concepts borrowed from the object-oriented approach (dynamic instantiation, classification, encapsulation, inheritance, client/server relationship) to describe the structural or static aspects of systems, and uses high-level Petri nets to describe their dynamic or behavioral aspects. The ICO notation has evolved since to deal with the modelling of multimodal issues in interactive-system (e.g. event-based communication, temporal modelling and structuring mechanism based on transducers in order to deal with low level and higher lever events) and to address news challenges raised by the various application domains it has been applied to, for example VR systems [37] and cockpits in aircrafts [40]. Figure 13 is an illustration of ICO models used in the case study. ICO models can be executed along the application, making this a powerful tool for analysis of the inner system behavior of interactive systems.
CASE STUDY
This section illustrates the use of the tool ARMADILLO in the context of a real-life case study. Only a partial view of the project using the annotation is given hereafter. The evolution of the artefacts following the production of annotations is not presented here.
The application selected for a case study is a Weather radar (WXR) currently deployed in many cockpits of commercial aircrafts, as illustrated at Figure 10. It provides support to pilots' activities by increasing their awareness of meteorological phenomena during the flight journey, allowing them to determine if they may have to request a trajectory change, in order to avoid storms or precipitations for example. Figure 10.a) presents a screenshot of the weather radar control panel, used to operate the weather radar application. This panel provides two functionalities to the crew. The first one is dedicated to the mode selection of weather radar and provides information about status of the radar, in order to ensure that the weather radar can be set up correctly. The operation of changing from one mode to another can be performed in the upper part of the panel (named MODE SELECTION). The second functionality, available in the lower part of the window, is dedicated to the adjustment of the weather radar orientation (Tilt angle). This can be done in an automatic way or manually (Auto/manual buttons). Additionally, a stabilization function aims to keep the radar beam stable even in case of turbulences. Figure 10.b shows a screenshot of the weather radar display produced by the weather radar aircraft system.
Spots in the middle of the images show the current position, importance and size of the clouds (importance is capture by color coding). a) b) Figure 10 : Screenshots of the case study application Weather radar (WXR). At left (a) the weather radar control pane with the interactive radio button controls. At right (b) the display produced by the weather radar aircraft system.
Various Artefacts of WXR Project
Hereafter we present the various artefacts used during the project of the Weather Radar application (WXR). At this point, the artefacts do feature annotations. Each artefact was selected to provide a unique and complimentary view of the application:
• Task models using HAMSTERS [41] describe the different tasks of the pilot [10] as well as their goals while using the Weather Radar Application (as shown by Figure 11);
• System models using the ICO notation [42] formally describe the system behavior (see Figure 13);
• A medium-fidelity prototype of the user interface designed with PANDA [14] (as shown in Figure 12).
• A High-fidelity prototype of the panel implemented in Java (as shown in Figure 9).
The task model presented at Figure 11 describes the crew activities performed in order to check weather conditions. At the higher level of the tree, there is an iterative activity (circular arrow symbol) to "detect weather targets" that is interrupted (operator [>) by a cognitive task "mental model of current weather map is built". Other human tasks include perception (task "Perceive image") and motor (task "Turn knob"). Connection between crew's activities and cockpit functions is made through interactive tasks (as input "Turn knob" and output "Rendering of radar information"). The time required for performing the latter heavily depends on the radar type. Such behavioral aspects of systems can be modeled using ICO notation and PETSHOP tool as detailed in section below. The task "Manage WXR" is a subtask of a more global activity of flying crew understanding weather conditions ahead of the aircraft. This activity would include analyzing the image produced by the weather radar (shown in Figure 9 b). This task model corresponds to the manipulation of the user interface presented in Figure 9 a. From these models we can see that the tasks to be performed in order to check weather conditions in a given direction are rather complex. The time required to perform them depends on 3 elements: the operator's performance in terms of motor movements, perception and cognitive processing. Figure 11 : Task model using the notation HAMSTER describing the WXR project. Hierarchical view of user tasks supported by the application. Different icons represent different types of tasks. Figure 12 shows the user interface prototype that was designed to provide to the user to: • Switch between the five available modes (upper part of the figure) using radio buttons (the five modes being WXON to activate the weather radar detection, OFF to switch it off, TST to trigger a hardware checkup, STDBY to switch it on for test only and WXA to focus detection on alerts).
•
Select the tilt angle control mode (lower part of the figure) amongst three modes (fully automatic, manual with automatic stabilization and manual selection of the tilt angle). Figure 12 : Prototype created with PANDA for describing the user interface of the WXR project. The screenshot corresponds to an early version of the prototype that evolved along the development process.
The ICO model presented in Figure 13 describes to configure the weather radar using the mode and angle. In Figure 13 presents the behavior description of this part of the interactive cockpit using the ICO model into two parts:
• The Petri net in the upper part handles events received from the 5 radio buttons. The current selection (an integer value from 1 to 5) is carried by the token stored in MODE_SELECTION place and corresponds to one the possible radio buttons (OFF, STDBY, TST, WXON, WXA). The token is modified by the transitions (new_ms = 3 for instance) using variables on the incoming and outgoing arcs as formal parameters of the transitions. Each time the mode value is changed, the equipment part (represented by the variable wxr within the token) is set up accordingly.
•
The Petri net in the lower part handles events from the four buttons and the text field (modify tilt angle). Interacting with these buttons changes the state of the application. In the current state, this part of the application is in the state fully automatic (a token is in AUTO place). To reach the state where the text field is available for the angle modification, it is necessary to bring the token to the place STABILIZATION_OFF by successively fire the two transitions switchManual_T1 and switchStabOff_T1 (by using the two buttons MANUAL and OFF represented in Figure 12, making transition change_Angle_T1 available. The selected angle must belong to the correct range (-15 to 15), controlled by the three transitions angleIsLow, angleIsCorrect and angleIsHigh. When checked, the wxr equipment tilt angle is modified, represented by the method called wxr.setTiltangle. Figure 12 presents the UI of the WXR produced with PANDA, which was designed to evaluate the organization of widgets and information to be displayed in the panel. This prototype is composed of one window called "WXR" and features several labels and buttons. The upper part of the prototype lists the different mode that can be selected: OFF, STDBY, TST, WXON, and WXA. The lower part is used to change the stabilization and the tilt of the radar as well as display the angle of the radar. It is important to note that, with respect to the application in Figure 9.a some user interface elements are not identified yet e.g. the radio button group on the upper part of the window.
Annotating individual artefacts
Once installed, the ARMADILLO plugin features the same on all hosting editors. A palette for annotations (as shown at Figure 8) is embedded into the editor (as shown by Figure 9) and from there the user can annotate individual artefacts. The steps for creating an annotation are illustrated by Figure 14 using the tool PANDA. First of all, we should notice the ARMADILLO plugin is installed in the editor (Figure 14.a, at the right-side) proposing a palette with functionalities for creating the annotations. Annotations are created via a drag & drop from the palette to the main edition area (Figure 14.b).
From there, the user can select the annotation and type text describing it.
We shall see in the edition area the user interface prototype of the WXR application (Figure 14.c) and, next to it a series of five annotations (Figure 14.d), identifiable as yellow boxes vertically aligned immediately at the right of the WRX window. These five annotations describe each mode can be relevant in the other representations in which the abbreviation of the modes is used; they are, namely, "OFF = Switch OFF", "STDY = Switch for test only", "TST = trigger for hardware checkup", "WxON = activate radar detection", and "WXA = Focus detection on alert". These annotations were created as Explanations, meaning that they are intended to explain the different modes of the WXR panel on the prototype. An additional annotation (Figure 14.e) named "TODO: Ergonomic inspection reference" is also present and it is attached a reference to a PDF file of the Ergonomic Criteria for the Evaluation of Human-Computer Interfaces by Christian Bastien and Scapin [43]; that reference is proposed as a recall for that user interface still requires an ergonomic inspection. The annotations created with the ARMADILLO plugin are automatically saved in the ARMADILLO repository. Figure 14 : Annotations on the medium-fidelity prototype for the weather radar application.
Importing annotations from multiple artefacts
The annotation created for a specific artefact can be reused to annotated other artefacts in the same project. Hereafter we illustrate how annotations created using the PANDA editor (describing the user interface prototype) can be associated with ICO models (describing the system behavior). We present here how these annotations can be connected together and why connecting them is relevant. To start, we should select all the annotations created with PANDA and then make them to point to the file describing the ICO system model. This is made by selecting the annotation and informing the ICO system model as a target. The next step is to launch the tool PETSHOP and open the ICO model set as target. While opening the ICO model, the ARMADILLO plugin looks for annotation in the repository that have as target the ICO model. When annotations are found, they are displayed in the edition area of PETSHOP as shown at Figure 15.
As we shall see, the annotations in PETSHOP are initially displayed at the same coordinates as they used to be in PANDA. This initial position is meant to facilitate the visual identification of annotation freshly imported. However, different representations of the interactive system might have their own layout and the annotations are unlikely to be located at the same place for each artefact. Thus, it is necessary to move the annotation to an appropriate location. Figure 16 shows the annotations that have been moved in the PETSHOP environment to establish a relationship with the corresponding areas of the ICO model. Since we cannot use the selection of fragment for the annotations, we rely on the location of the annotation to associate them with the relevant fragment of the model. The location and the size of one annotation are properties that are specific to one target while the other properties like the content are shared. Thus, moving the annotations in this model will not affect their display on the low-fidelity prototype and vice-versa. Thus, the annotations created to explain the meaning of the radio button on the user interface prototype (for example OFF = Switch OFF") are now reused to explain which parts of the ICO model are responsible for describing the behavior of that button.
Overview of all annotations over multiple artefacts
The ARMADILLO plugin presented so far provides a local view of annotations created with a particular editor and annotations that were imported from multiple artefacts. In order to have a complete view of all annotations created for the project, possibly including multiple artefacts, we have created a tool called ARMADILLO viewer which is shown by Figure 17. The ARMADILLO viewer is available at the project management level of the CIRCUS environment. It is as a standalone tool that is able to parse the files describing annotations in the repository and recreate a graphic view that retraces all annotations to targets and artefacts. The Figure 17 illustrates such as an overview of all annotations created for the case study. As we shall see, there are five annotations (namely, , "OFF = Switch OFF", "STDY = Switch for test only", "TST = trigger for hardware checkup", "WxON = activate radar detection", and "WXA = Focus detection on alert") originally created using PANDA and then connected to ICO models using PETSHOP. Such as connections are indicated to a pointer to the ICO file "MPIA_WXR.xml" and another pointer to the PANDA prototyping file "WXR -V0.prstn".
Benefits of the tool from an end-user perspective
The contributions presented above were initiated by several research projects with different companies (mainly SoftTeam and Airbus). While previous sub-sections have presented the use of ARMADILLO and its underlying concepts to a small but real-life case study, this section highlights the genesis of this work and reports on feedback received from the various stakeholders and how this feedback was used to design and develop ARMADILLO.
Project context
We focus here on a set of research projects performed with different departments at Airbus. One was with interactive cockpits safety engineering group and the other one was with the Flight Warning System team. These research projects lasted over a decade and in total and resulted in publication co-authored with Airbus engineers and experts. As for the interactive cockpits formal models engineering the interested reader can refer to [52] or [53] for Flight Warning Systems modelling and [54] or [55] for formal description of interactive cockpits.
The aim of these projects was to develop a Model-Based System Engineering approach [59] dedicated to the modeling of complete set of interactive cockpits components, including graphical widgets (compatible with the ARINC 661 Specification standard [56]), interaction techniques and interactive applications (to be deployed in the new generation of interactive cockpits from Airbus manufacturer). During the recent years, the focus was on touch interactions design and modelling [57] and software architecture of interactive services connected to aircraft systems [58].
During these projects various notations and associated tools were designed and developed each of them focusing on specific behaviors (e.g. interactive system behavior [42] or operators' tasks descriptions [60]). More holistic descriptions of aircraft systems were using SysML-related modelling [58].
Identification of stakeholders needs
From the beginning of these projects it was that a quadruple problem had to be solved: -Designing notations that have an expressive power high enough to describe all the elements that had to be described;
-Designing notations that are semantically close enough to the behaviors to be modeled so that the models would remain meaningful and would encompass modeling primitives close to the information to be modelled;
-Designing connections between the notations so that information used in one model can be referred to in another model; -Support the editing of each kind of models with dedicated tools.
While multiple tools and notations were designed and built (presented in previous sections) e.g. PANDA for interactive applications prototypes, PetShop for interactive systems behavior and HAMSTERS for modeling operators' goals and tasks it was clear that some information would not be possible to integrate in the models. First, vernacular and informal requirements, that are produced in the early phases of the industrial project (such as early requirements for interactions to be used in aircraft cockpits) are too generic to appear in a behavioral model of a system or an operator. We use vernacular here instead of informal as engineers working for a long period of time in the same company develop a language that is exploiting a lot of acronyms and embeds the company culture. Second, requirements (due to their abstract nature) are imprecise and incomplete leaving a lot of space for interpretation. In that case, a given requirement considered in a given model might be differently integrated in another one. Third, these different views on the requirements requires discussions and argumentation in order to reach an agreement. Modeling tools usually don't integrate those aspects as argued in [61]. is DAL A [63] according to DO 178C classification [67]. In the field of critical systems, safety standards such as DO-178C
Two concrete examples
Informal requirements in different artefacts: Addressing different level of DAL in different artefacts
or IEC 61508 define Development Assurance Levels for software systems (or for functions of software systems). These levels are based on the analysis of consequences or effect of a malfunction. For instance, if a function failure has high consequences such as multiple fatalities, it is called catastrophic and certification authorities will require that the system manufacturer will provide a Development Assurance Level A (DO-178C standard for aeronautics [67]). If consequences are lower, the required level will decrease. Developing a system of a DAL A is extremely resource consuming and expensive and, as far as software is concerned, the use of formal description techniques is required [66]. In lower DALs, such expensive approaches are not required and for reaching levels such as DAL D rigorous software engineering approaches are sufficient
The right-hand side is less critical. As presented in [62] within the Airbus project we have proposed to engineer and model the barometer setting and to program the other part. This information was added to the ICO model that was only coping with the barometer. A question that arise was whether or not to make this information about the criticality salient or not on the user interface. In order to keep the user interface consistent with the other ones and due to the fact that the pilots are trained and know the importance of the barometer value it was decided not to change the appearance. This information was added to the user interface prototype and connected to the formal model to ensure completeness (the information is present in each artefact) and consistency (each artefact has processed the information, even though in that concrete case nothing was changed on the user interface).
Requirements addressed in different artefacts: The mutual exclusion of buttons
Stakeholders of the project expressed the need for describing information that was outside of the expressive power of the ICO notation [40] that was used for interactive widgets and interactive applications. The requirement was to ensure that two functions would always be triggered in a mutually exclusive way (e.g. engaging and disengaging the autopilot).
A procedural state-based notation like ICOs allows easily to model such a behavior as shown in Figure 19. In the current state (Disengaged) the only transition available is Engage. Triggering that transition would set a token in place Engaged making the transition Engage unavailable and the transition Disengage available. While formal analysis would demonstrate that these two transitions are mutually exclusive but it is not explicit in the model that this requirement is met. Annotations were immediately added to the ICO model to record that information in an explicit manner. In addition, it was claimed important by the stakeholders that this property should be reflected and ensured on the user interface using toggle buttons (instead of two different buttons one being enabled and the other one being disabled). That property was thus appearing also at the user interface prototype artefact that was connected to the ICO formal model.
Impact on the design and improvement of ARMADILLO
As presented in the two examples above the need to offer the possibility to engineers to use annotations was made salient from the very beginning of the project. Once the need for a tool support was settled, we start to work on the design starting with paper-based prototype and ultimate leading to the design of the tools previously presented. One of the challenges issues was where to place information (edition area, list of annotation created, controls). For that, we have investigated several design opnions. The process of creation of ARMADILLO was iterative and incremental. The development of our approach took almost four years. Along the project, we had regular meetings with the users, usually every two or three months. These meeting served as formative evaluation for the design of the tools. We collected feedback through informal interviews that follow a demonstration of a prototype. Interviews were not formalized but three questions we regularly made: what do you like with the prototype? what you do not like? do you have suggestions for improvements? Unfortunately, we did not have made transcripts of answers uttered by the participants. For the need of the project at that stage, a backlog of improvements to be made was considered enough. Nonetheless, the most important requests made by the users are reported hereafter as milestones for the development of the tool. Initially, the request was to have annotations for user interface prototypes. A series of prototypes was created until we found the best placement for annotations in the palette of PANDA (as shown in Figure 9). The second milestone was to create annotations formats that suits the user needs.
This was not a straightforward process. We start working with basic annotation types such as text and drawings, but through the meetings and discussions with engineers we have identified some complex annotations types (such as voting mechanisms, markers and scenarios) that trigger cycles of design and test for that particular annotations. The case of annotations for scenarios, in particular, was identified as a mean for defining non-regression. Indeed, when making changes in the prototypes, the engineers could break design solutions that worked.
So that, it was suggested to create annotations to indicate things that cannot be changed when prototypes evolve. Some studies investigating the use of annotations as testing scenarios are described in [64] [65]. At this point, once we had a suitable solution for annotating user interface prototypes, we generalize the solution using a plugin that could be reused to give to other tools the functions for annotating HAMSTERS models and the ICO models using PETSHOP. The last step was the creation of the ARMADILLO repository and the viewer allowing to build the overview of all annotations in a project.
Section 5.5. of this paper has highlighted the importance for the various stakeholders involved in various research projects to provide means of annotating the various interactive systems-related artefacts produced in the design, prototyping, specification and modeling of interactive systems. Beyond the simple addition of annotations these stakeholders highlighted the importance of structuring annotations, reusing them and connecting them to one or several artefacts. The ARMADILLO tools was iteratively designed and modified to support these needs and to make complex tasks easy to perform. As the tool was developed mainly with one company, the identified needs and tasks of stakeholders might not be complete with respect to other companies practices. By making the tool publicly available we hope that more needs will be identified and in that case, we will extend ARMADILLO to cover them.
CONCLUSIONS
This paper presented a tool-supported annotation model for the design and development of interactive systems. The paper builds on related work to demonstrate the importance of annotations the development of interactive systems and proposes a generic solution (a meta-model) to engineers the use of these annotations. As demonstrated by other authors, annotations play a critical role for communicating ideas, information and decisions along the development process. The meta-model proposed addresses the multi-form nature of annotations and may be further extended to cover new needs. The ARMADILLO tool supports that meta-model and provides additional features such as versioning of annotations. In order to support annotation sharing among multiple artefacts dedicated mechanisms have been presented. This supports the entire life-cycle of an annotation but support also the design process by having annotations as a first class information and not only (as this is usual the case) a side product to be handled as a side product.
While the W3C annotation data model [16] addresses web applications our extended version of it is specially targeting at interactive systems and supports the User Centered Design process of ISO 9241 part 210 iterative process. Beyond the model and the tool ARMADILLO that supports it; we have presented their integrability and integration in the existing tools in the CIRCUS platform [40] (which encompass the tools PANDA, HAMSTERS, and PETSHOP). This is done by a generic plugin that has been instantiated to every tool editor of CIRCUS platform and used to produce specific artefacts (e.g. task models or dialog models). While the code of our tools are not open source (industrial constraint), the ARMADILLO and the other editors in the CIRCUS platform are available and can be used by the community free of charge. So far the plugin has been deployed in a few editors of CIRCUS but the meta-model and the plugin concept are generic enough to be deployable in other tools. In order to support design and development activities, the plugin support the entire life cycle of annotations including their connection to the different artefacts produced by different stakeholders involved in the design process of interactive systems. On one hand, the choice of a plugin architecture is justified by some of the inner advantages such as consistent interaction with the annotation tool across multiple editors, reuse of the code, extensibility of annotations to other editors, etc. On the other hand, plugin are not a panacea and are tied to a specific development platform. For that we cannot claim that the plugin tools presented in this paper are an universal solution.
However, our approach illustrated by the means of the tools (including centralized repository for annotations, using standards for describing annotations with extensions for metadata and annotation types, dedicated tools for tracking the annotations distributes along multiple artefacts, etc.) can be used as an example to build other similar tools in other environments to solve some problems such as : how to systematically enrich and bind annotations to digital artefacts/models used to specify an interactive systems, thus supporting decision making along the development process; how to cross-reference and cross-check design decisions to multiple artefacts; how to follow the evolution and ensure the consistency of design decisions along the development of interactive systems.
Relationship with annotations in programming languages (R1). We can easily expand the section state-of-the-art to encompasses the uses of annotations in this context as we did the work as part of the industrial project. However, we decided not to include this in the current paper as the case study is only dealing with graphical notations. We propose to add a short paragraph to the state of the art explaining that our annotation model covers current practice in programming.
We have demonstrated the use of the annotation models and the ARMADILLO tool in a case study on the aerospace domain involving four different artefacts. The case study demonstrates that it is possible to collect several information that might be useful for supporting design decisions. This opens-up several perspectives for investigating the use of informal information collected and design decisions along the process. In future work, we will focus on traceability of the decisions and tools for supporting the visualization and rational design. Now that the concept is operational, we have the tools for a large case study including the collection of data describing the results of usability testing.
Figure 1 :
1Simplified model of annotations defined by the Web Annotation Data Model.
Figure 2 :
2Three examples of the use of annotations on prototypes.
Figure 3 :
3Number of tools supporting nine different features concerning annotations (i.e. multi-targeting, temporal evolution of artefacts, lifecycle of annotations, textual annotations, callout annotations, graphical annotations, semantic annotations, annotations management, collaborative word). in a comparative study of 80 annotations tools.
annotation model consider two types of classes of particular importance: the classes allowing to define the characteristics of annotation (which includes the type of annotation, the description of the intended audience for the annotation completed by the motivation and the purpose of the annotation stating the reasons of the creation of the annotation) and the classes describing annotation metadata (including information about date of creation and modification of an annotation and identity of its creator). We take the benefice of the possibility of extending the classes of the Web Annotation Model to implement different type of annotations. The core classes of our extended model are: annotation, target, creator and artefact. Notice that the first three classes are already present in the Web Annotation Model, while the class artefact is a new addition.
Figure 4
4provides a view at glance of our model, highlighting: i) classes defined in the W3C model (in orange), ii) new attributes that have been added to enrich metadata classes (in violet), iii) new classes proposed as an extension to the model (in blue), and iv) specialized classes that extend the use of the overall model for particular types of annotations used in our case study (in green).
Figure 4 :
4Overview of extensions made to the W3C's annotation model.
Figure 6 :
6Structure of an annotation file showing the references to with the plugin pandaannotation.
Figure 7 :
7Example of an annotations in ARMADILLO project repository and plugin.
Figure 8 :
8Palette of the ARMADILLO plugin featuring the diverse types of annotations supported by the tool.
Figure 9 :
9ARMADILLO plugin in context of use as integrated to the tool editor PANDA: (a) palette featuring diverse types of annotations, (b) edition area, (c) prototype artefact, (d) annotations (marks in yellow).
Figure 13 :
13The system model artefact of the WXR project using the ICO formalism within PETSHOP Tool.
Figure 15 :
15Annotating an ICO model using the editor PETSHOP: initial display of annotations imported from PANDA.
Figure 16 :
16Annotating an ICO model using the editor PETSHOP: repositioning annotations to target elements in the model.
These annotations are graphical representations of everything available in the ARMADILLO repository for the project used as case study. Such as annotations contains all the metadata describing the editor, authors and classpath allowing to open the corresponding editors so each annotation in the context of artefacts being annotated.
Figure 17 :
17Overall view of annotations over multiple artefacts using ARMADILLO.
Figure 18presents an envisioned application that could replace the current Flight Control Unit (FCU) panel below the windshield. That FCU panel allows multiple actions including entering parameters for the auto pilot and engaging and disengaging it. The user interface inFigure 18is composed of two parts. The upper part of the left panel of the EFIS (Electronic Flight Information System) is dedicated to the configuration of the barometer settings. The top right panel of the EFIS page enables the display of several navigation information (such as waypoints). The value of the barometer setting
Figure 18 :
18EFIS control panel (left panel is highly critical DAL A while right panel is less critical DAL C [63])
Figure 19 :
19Engaging and disengaging transitions which are mutually exclusive in an ICO model
annotation is a particular note linked to a target by an anchor. The target can be a collection of documents, a document, a segment of a document (paragraph, group of words, image, part of image, etc.), and another annotation. Each annotation has content, materialized by an inscription. It is the trace of the mental representation elaborated by the annotator about the target. The content of the annotation can be interpreted by another reader. The anchor links the annotation to the target (a line, a surrounded sentence, etc.)".
Among the 25 generic annotation tools, we have 18 academic tools which means tools described in scientific publications (i.e. Amaya, Quilt, sense.us, HyperImage 3, SparTag.us, D.note, LiquidText, List-it, Zydeco, Annotorious, ChronoViz, Domeo, Elias' prototype, GatherReader, Instant Annotation, Neonion, Dokieli, and Pundit Annotator) and 7 commercial tools (i.e. Annozilla (Annotea project), Diigo, Protonotes, Annotatorjs, Authorea, Hypothesis, Ponga). The complet references to these tools are available in the Annex I. Among the prototyping tools embeding functions for annotating prototypes, we have found 55 tools where 3 are academic tools (i.e. Silk, DEMAIS, ActiveStory Enhanced) for the other 52 have been distributed as commercial tools at some point (i.e. Rise, SoftAndGui, UxPin, AdobeXD, Microsoft visio, Smartdraw, Axure, GUI Design Studio, MockupScreens, JustinMind, Balsamiq, DesignerVista, InPreso Screens, MockingBird, PencilProject, Pidoco, ProtoShare, WireframeSketcher, Cacoo, Crank Storyboard Designer, Creately, FlairBuilder, ForeUI, Gliffy, Microsoft SketchFlow, iPlotz, BluePrint, FrameBox, HotGloo, LucidChart, Mockflow, Sketch, Antetype, Draw.io, Lumzy, MockupBuilder, Mockups.me, MockupTiger, PowerMockup, Proto.io, FluidUI, IndigoStudio, Moqups, Prototyping on Paper (Marvel), Alouka, Concept.Ly, InVision, NinjaMock, Notism, MockupPlus,
ANNEX I -List of annotations tool examined for the state of the art on annotations
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Prototyping tools and techniques. M Beaudouin-Lafon, W Mackay, The human-computer interaction handbook. Julie A. Jacko and Andrew SearsHillsdale, NJ, USAL. Erlbaum Associates IncBeaudouin-Lafon, M., Mackay, W. 2002. Prototyping tools and techniques. In The human-computer interaction handbook, Julie A. Jacko and Andrew Sears (Eds.). L. Erlbaum Associates Inc., Hillsdale, NJ, USA 1006-1031.
Prototyping for tiny fingers. Marc Rettig, 10.1145/175276.175288Communications of the ACM. 374Marc Rettig. 1994. Prototyping for tiny fingers. Communications of the ACM 37, 4 (April 1994), 21-27. DOI:https://doi.org/10.1145/175276.175288
Breaking the fidelity barrier: an examination of our current characterization of prototypes and an example of a mixed-fidelity success. Michael Mccurdy, Christopher Connors, Guy Pyrzak, Bob Kanefsky, Alonso Vera, 10.1145/1124772.1124959Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI '06). Paul Aoki, Ed Cutrell, Robin Jeffries, and Gary Olsonthe SIGCHI Conference on Human Factors in Computing Systems (CHI '06)Rebecca Grinter, Thomas Rodden; New York, NY, USAACMMichael McCurdy, Christopher Connors, Guy Pyrzak, Bob Kanefsky, and Alonso Vera. 2006. Breaking the fidelity barrier: an examination of our current characterization of prototypes and an example of a mixed-fidelity success. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems (CHI '06), Rebecca Grinter, Thomas Rodden, Paul Aoki, Ed Cutrell, Robin Jeffries, and Gary Olson (Eds.). ACM, New York, NY, USA, 1233-1242. DOI=http://dx.doi.org/10.1145/1124772.1124959.
The anatomy of prototypes: Prototypes as filters, prototypes as manifestations of design ideas. Youn-Kyung Lim, Erik Stolterman, Josh Tenenberg, 10.1145/1375761.1375762ACM Trans. Comput.-Hum. Interact. 157Youn-Kyung Lim, Erik Stolterman, and Josh Tenenberg. 2008. The anatomy of prototypes: Prototypes as filters, prototypes as manifestations of design ideas. ACM Trans. Comput.-Hum. Interact. 15, 2, Article 7 (July 2008), 27 pages. DOI:https://doi.org/10.1145/1375761.1375762
The Handbook of Task Analysis for Human-Computer Interaction. Diaper, D. & Stanton, N. A.Lawrence Erlbaum Associates650Diaper, D. & Stanton, N. A. (eds.) The Handbook of Task Analysis for Human-Computer Interaction. Lawrence Erlbaum Associates, 2004. 650 pages.
Cascading Dialog Modeling with UsiXML. International Workshop on the Design, Verification and Specification of Interactive Systems (DSVIS'. M Winckler, J Vanderdonckt, F Trindade, A Stanciulescu, LNCS. 5136SpringerWinckler, M., Vanderdonckt, J., Trindade, F., Stanciulescu, A. Cascading Dialog Modeling with UsiXML. International Workshop on the Design, Verification and Specification of Interactive Systems (DSVIS'2008). Kingston, Ontario, Canada, July 16-18 2008. Springer LNCS 5136. pp. 121-135.
Lexical and pragmatic considerations of input structures. William Buxton, 10.1145/988584.988586SIGGRAPH Comput. Graph. 17William Buxton. 1983. Lexical and pragmatic considerations of input structures. SIGGRAPH Comput. Graph. 17, 1 (January 1983), 31-37. DOI:https://doi.org/10.1145/988584.988586
10.1145/142394.142401A metamodel for the runtime architecture of an interactive system: the UIMS tool developers workshop. SIGCHI Bulletin. 24A metamodel for the runtime architecture of an interactive system: the UIMS tool developers workshop. SIGCHI Bulletin, 24, 1 (Jan. 1992), 32-37. DOI:https://doi.org/10.1145/142394.142401
A generic tool-supported framework for coupling task models and interactive applications. Célia Martinie, David Navarre, Philippe A Palanque, Camille Fayollas, EICS. 2015Célia Martinie, David Navarre, Philippe A. Palanque, Camille Fayollas. A generic tool-supported framework for coupling task models and interactive applications. EICS 2015: 244-253.
Enriching Task Models with Usability and User Experience Evaluation Data. Regina Bernhaupt, Philippe Palanque, Dimitri Drouet, Celia Martinie, Proceedings of the 7 th IFIP WG 13.2 International Working Conference, HCSE 2018. the 7 th IFIP WG 13.2 International Working Conference, HCSE 2018Sophia Antipolis, FranceSpringer11262Regina Bernhaupt, Philippe Palanque, Dimitri Drouet, Celia Martinie. Enriching Task Models with Usability and User Experience Evaluation Data. In: Proceedings of the 7 th IFIP WG 13.2 International Working Conference, HCSE 2018, Sophia Antipolis, France, September 3-5, 2018, Revised Selected Papers. Springer, LNCS, volume 11262, Pages 146-163.
Beyond modelling: an integrated environment supporting co-execution of tasks and systems models. Eric Barboni, Jean-François Ladry, David Navarre, Philippe A Palanque, Marco Winckler, EICS. 2010Eric Barboni, Jean-François Ladry, David Navarre, Philippe A. Palanque, Marco Winckler. Beyond modelling: an integrated environment supporting co-execution of tasks and systems models. EICS 2010: 165-174.
A historical and contemporary study on annotations to derive key features for systems design. Maristella Agosti, Giorgetta Bonfiglio-Dosio, Nicola Ferro, 10.1007/s00799-007-0010-0Int. J. Digit. Libr. 8Maristella Agosti, Giorgetta Bonfiglio-Dosio, and Nicola Ferro. 2007. A historical and contemporary study on annotations to derive key features for systems design. Int. J. Digit. Libr. 8, 1 (October 2007), 1-19. DOI=http://dx.doi.org/10.1007/s00799-007-0010-0
PANDA: prototyping using annotation and decision analysis. Jean-Luc Hak, Marco Winckler, David Navarre, EICS. 2016Jean-Luc Hak, Marco Winckler, David Navarre. PANDA: prototyping using annotation and decision analysis. EICS 2016: 171-176
Capturing Design Decision Rationale with Decision Cards. INTERACT (1). Marisela Gutierrez Lopez, Gustavo Rovelo, Mieke Haesen, Kris Luyten, Karin Coninx, Marisela Gutierrez Lopez, Gustavo Rovelo, Mieke Haesen, Kris Luyten, Karin Coninx. Capturing Design Decision Rationale with Decision Cards. INTERACT (1) 2017: 463-482
Web Annotation Data Model -W3. Web Annotation Data Model -W3. Available at: https://www.w3.org/TR/annotation-model
Annotation: from paper books to the digital library. C Catherine, Marshall, 131-140.DOI=10.1145/263690.263806Proceedings of the second ACM international conference on Digital libraries (DL '97). the second ACM international conference on Digital libraries (DL '97)New York, NY, USAACMCatherine C. Marshall. 1997. Annotation: from paper books to the digital library. In Proceedings of the second ACM international conference on Digital libraries (DL '97). ACM, New York, NY, USA, 131-140. DOI=10.1145/263690.263806 http://doi.acm.org/10.1145/263690.263806
Toward an ecology of hypertext annotation. C Catherine, Marshall, Proceedings of the ninth ACM conference on Hypertext and hypermedia : links, objects, time and space---structure in hypermedia systems: links, objects, time and space---structure in hypermedia systems (HYPERTEXT '98). the ninth ACM conference on Hypertext and hypermedia : links, objects, time and space---structure in hypermedia systems: links, objects, time and space---structure in hypermedia systems (HYPERTEXT '98)Catherine C. Marshall. 1998. Toward an ecology of hypertext annotation. In Proceedings of the ninth ACM conference on Hypertext and hypermedia : links, objects, time and space---structure in hypermedia systems: links, objects, time and space---structure in hypermedia systems (HYPERTEXT '98).
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Annotations: A new type of document in the Electronic Health Record. S Bringay, C Barry, J Charley, the 2nd International Conference on Document Research and Development in Sciences, arts and business: DOCAM. University of California, Berkeley, Etats-UnisPaper presented atBringay S., Barry C., Charley J., Annotations: A new type of document in the Electronic Health Record. Paper presented at the 2nd International Conference on Document Research and Development in Sciences, arts and business: DOCAM 2004, University of California, Berkeley, Etats-Unis, octobre 2004.
Open Annotation Data Model, Community Draft. Robert Sanderson, Paolo Ciccarese, Herbert Van de SompelRobert Sanderson, Paolo Ciccarese, Herbert Van de Sompel, 2013. Open Annotation Data Model, Community Draft. Available at: http://www.openannotation.org/spec/core/
Web Annotation Data Model. Robert Sanderson, Paolo Ciccarese, Benjamin YoungWorking DraftRobert Sanderson, Paolo Ciccarese, Benjamin Young, 2015. Web Annotation Data Model, Working Draft.
Annotea: an open RDF infrastructure for shared Web annotations. José Kahan, Marja-Ritta Koivunen, 10.1145/371920.372166Proceedings of the 10th international conference on World Wide Web (WWW '01). the 10th international conference on World Wide Web (WWW '01)New York, NY, USAACMJosé Kahan and Marja-Ritta Koivunen. 2001. Annotea: an open RDF infrastructure for shared Web annotations. In Proceedings of the 10th international conference on World Wide Web (WWW '01). ACM, New York, NY, USA, 623-632. DOI=http://dx.doi.org/10.1145/371920.372166
A formal model of annotations of digital content. M Agosti, N Ferro, http:/doi.acm.org/10.1145/1292591.1292594ACM Trans. Inform. Syst. 26Agosti M. and Ferro N. 2007. A formal model of annotations of digital content. ACM Trans. Inform. Syst. 26, 1, Article 3 (November 2007), 57 pages. DOI = 10.1145/1292591.1292594 http://doi.acm.org/10.1145/1292591.1292594
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|
[] |
[
"Molecular formations in ultracold mixtures of interacting and noninteracting atomic gases",
"Molecular formations in ultracold mixtures of interacting and noninteracting atomic gases"
] |
[
"T Nishimura ",
"A Matsumoto ",
"H Yabu ",
"\nDepartment of Physics\nDepartment of Physics\nTokyo Metropolitan University\n1-1 Minami-Ohsawa192-0397HachiojiTokyoJapan\n",
"\nRitsumeikan University\n525-8577KusatsuShigaJapan\n"
] |
[
"Department of Physics\nDepartment of Physics\nTokyo Metropolitan University\n1-1 Minami-Ohsawa192-0397HachiojiTokyoJapan",
"Ritsumeikan University\n525-8577KusatsuShigaJapan"
] |
[] |
Atom-molecule equilibrium for molecular formation processes is discussed for boson-fermion, fermion-fermion, and boson-boson mixtures of ultracold atomic gases in the framework of quasichemical equilibrium theory. After presentation of the general formulation, zero-temperature phase diagrams of the atom-molecule equilibrium states are calculated analytically; molecular, mixed, and dissociated phases are shown to appear for the change of the binding energy of the molecules. The temperature dependences of the atom or molecule densities are calculated numerically, and finitetemperature phase structures are obtained of the atom-molecule equilibrium in the mixtures. The transition temperatures of the atom or molecule Bose-Einstein condensations are also evaluated from these results. Quantum-statistical deviations of the law of mass action in atom-molecule equilibrium, which should be satisfied in mixtures of classical Maxwell-Boltzmann gases, are calculated, and the difference in the different types of quantum-statistical effects is clarified. Mean-field calculations with interparticle interactions (atom-atom, atom-molecule, and molecule-molecule) are formulated, where interaction effects are found to give the linear density-dependent term in the effective molecular binding energies. This method is applied to calculations of zero-temperature phase diagrams, where new phases with coexisting local-equilibrium states are shown to appear in the case of strongly repulsive interactions.
|
10.1103/physreva.77.063612
|
[
"https://arxiv.org/pdf/0710.5819v3.pdf"
] | 119,249,836 |
0710.5819
|
3c046202e5cd7e74ff187a36d63d057efb7bbc98
|
Molecular formations in ultracold mixtures of interacting and noninteracting atomic gases
6 Aug 2008
T Nishimura
A Matsumoto
H Yabu
Department of Physics
Department of Physics
Tokyo Metropolitan University
1-1 Minami-Ohsawa192-0397HachiojiTokyoJapan
Ritsumeikan University
525-8577KusatsuShigaJapan
Molecular formations in ultracold mixtures of interacting and noninteracting atomic gases
6 Aug 2008numbers: 0375Mn0530−d3115bt8260Hc * nimut@tmuacjp † yabu@seritsumeiacjp
Atom-molecule equilibrium for molecular formation processes is discussed for boson-fermion, fermion-fermion, and boson-boson mixtures of ultracold atomic gases in the framework of quasichemical equilibrium theory. After presentation of the general formulation, zero-temperature phase diagrams of the atom-molecule equilibrium states are calculated analytically; molecular, mixed, and dissociated phases are shown to appear for the change of the binding energy of the molecules. The temperature dependences of the atom or molecule densities are calculated numerically, and finitetemperature phase structures are obtained of the atom-molecule equilibrium in the mixtures. The transition temperatures of the atom or molecule Bose-Einstein condensations are also evaluated from these results. Quantum-statistical deviations of the law of mass action in atom-molecule equilibrium, which should be satisfied in mixtures of classical Maxwell-Boltzmann gases, are calculated, and the difference in the different types of quantum-statistical effects is clarified. Mean-field calculations with interparticle interactions (atom-atom, atom-molecule, and molecule-molecule) are formulated, where interaction effects are found to give the linear density-dependent term in the effective molecular binding energies. This method is applied to calculations of zero-temperature phase diagrams, where new phases with coexisting local-equilibrium states are shown to appear in the case of strongly repulsive interactions.
I. INTRODUCTION
The experimental success of the Bose-Einstein condensation (BEC) [1] of the trapped ultracold atomic gas has made much progress in the physics of quantum gases [2,3,4], which includes Fermi-degenerate systems and Bose-Fermi mixtures.
Recently, using the Feshbach-resonance method, molecular formations have been performed experimentally in ultracold atomic gases for two fermions [5,6,8] and two bosons [7,9];
Bose-Einstein condensations have been observed for the thus created molecules of two fermions [10,11] and two bosons [9]. In these experiments, the molecular binding energies can be tuned by continuous changes of the applied magnetic fields; especially, bound molecular states can be changed into resonances by shifting the binding energies above the atom-atom scattering thresholds.
One of the interesting applications of ultracold molecules is the observation of the BEC-BCS crossover in experiments with atomic Fermi gases: continuous crossover between the strong-coupling molecular BEC and weak-coupling BCS superconducting states [12,13,14,15,16,17]. In ultracold atomic-gas experiments, the crossover occurred by a change of the molecular binding energy through the Feshbach resonance method [18,19,20], and the experimental success has led to a lot of experimental and theoretical works on the crossover [21,22,23] and molecular BEC physics [24,25].
Molecules in optical lattice potentials are also an interesting problem. After the first observation of molecular formations using 87 Rb [26], many experiments have been performed on the long-lived state of the 87 Rb molecule [27] , 40 K difermion [28], and 87 Rb-40 K bosonfermion heteronuclear molecules [29]. Quantum degeneracy has also been observed in gases of 87 Rb molecules [30]. The phase structure of the lattice-trapped atomic gas, which is produced through interparticle correlations, has much interest in relation to the strongly correlated condensed-matter system described by the Hubbard model. Thus many experimental and theoretical works have been done: for example, on the superfluid-Mott insulator transition in bosonic [31,32,33,34,35], fermionic [36,37], and boson-fermion systems [38].
Other theoretical studies on ultracold molecules include coherent molecular solitons [39], coherent photoassociation [40], and coherent intercondensate exchange between atoms and molecules [41].
In this paper, we discuss atom-molecule equilibrium for boson-fermion, fermion-fermion, and boson-boson mixtures of ultracold atomic gases with quasichemical equilibrium theory, an extension of classical chemical equilibrium theory [42,43] for the quantum many-body problem, which was originally developed for the electron system in superconductors [12,13].
Our special interest is in applications of this method to boson-fermion and boson-boson mixtures, especially occurrences of atom or molecule BEC, as shown in some works for bosonfermion mixtures [44,45,46,47]. In addition, in contrast to the fermionic system, where many-body quantum calculations based on the microscopic model have been established, the singularities from the boson degrees of freedom sometimes cause problems in calculations of boson-fermion and boson-boson mixtures; quasichemical equilibrium theory can give definite solutions in such cases. As another interest of this approach, it should be pointed out that we can easily include interparticle interactions, especially atom-molecule and moleculemolecule ones, which are sometimes omitted in many-body calculations. The quasichemical approach gives the equilibrium structures in a less model-independent way. It turned out that the effects of these interactions should change the atom-molecule equilibrium structures drastically in the strong-coupling region.
In Sec. II, a general formulation of quasichemical equilibrium theory is presented for molecular formation or dissociation processes in noninteracting atomic-gas mixtures, and in Sec. III, the method is applied to boson-fermion, fermion-fermion, and boson-boson mixtures and the atom-molecule equilibrium structures are shown at zero and finite temperatures.
Special attention is paid to the condition on the occurrences of atom or molecule BEC; the shifts of the BEC transition temperatures are discussed from the molecular binding energy effects. In Sec. IV, the law of mass action, which is satisfied in chemical processes with classical Boltzmann statistics, is shown to deviate in ultracold molecular formation or dissociation processes by quantum-statistical effects (the law of quantum mass action). In Sec. V, we extend the quasichemical theory to include interparticle interactions for s-wave scattering processes (three kinds of atom-atom ones, two kinds of atom-molecule ones, and one kind of molecule-molecule one in combination) in molecular formation or dissociation processes.
In the mean-field approximation, the original six coupling constants of the interactions are shown to integrate into two parameters. It allows discussions of the interaction effects to be very clear. The formulations are applied to molecular formation or dissociation processes in interacting mixtures, and we discuss the change of the equilibrium structures at T = 0 through interaction effects; new phases with coexisting local-equilibrium states are shown to appear in the case of strongly repulsive interactions. Section VI is devoted to a summary and outlook.
We should comment that a relativistic extension is also possible of quasichemical theory and applications to diquark condensates in quark matter have been done in [48]. To develop a quasichemical equilibrium theory, we consider a molecular formation or dissociation process in the mixture:
A 1 + A 2 ←→ (A 1 A 2 ) = M,(1)
where M is a composite molecule with mass m M , which is bosonic in BB or FF mixtures and fermionic in BF mixtures.
The mass defect of the molecule is defined as ∆m M ≡ (m M − m 1 − m 2 ). The boundmolecule (∆m M < 0) is stable in both vacuum and gases, and has molecular binding energy ∆E = ∆m M c 2 , where c is the velocity of light in vacuum. In contrast, the resonance (∆m M > 0) is unstable at least in vacuum; however, resonance states can exist stably in gases, so that we consider both bound-molecule and resonance cases.
Here we take the quasiparticle picture wherein the system consists of atoms A 1 and A 2 and molecule M, which are quasiparticles [49]. In this picture, two-body interactions between "bare" atoms bring about two-body correlations in the mixture; their major effects are the creation of the composite molecule M with binding energy ∆E M and atoms and molecules, which are quasiparticle in the mixture, and interact through residual interactions, which are generally different from the original interaction between bare atoms. These quasiparticle interactions are generally regarded to be weak, and we neglect them in the first part of this paper. In Sec. IV, we introduce the quasiparticle interactions and discuss their effects on atom-molecule equilibrium within the mean-field approximation.
The equilibrium condition for the process (1) is given by
µ 1 + µ 2 − µ M = ∆E M ,(2)
where µ 1,2,M are chemical potentials of A 1 , A 2 , and M.
The molecular binding energy ∆E M in (2) is generally very small (∼ 10 −(5∼10) eV) in molecular formation or dissociation processes in ultracold atomic-gases, and it takes the same order of magnitude with the chemical potentials of atoms or molecules µ α at ultralow temperature (T = µK − nK). Thus, the term ∆E M cannot be omitted in (2).
For free uniform gases, the particle densities are given by the Bose and Fermi statistics:
n α = 1 (2π) 3 d 3 k e (εα−µα)/k B T − 1 + n (0) α ≡ 1 (λ T,α ) 3 B 3/2 (−µ α /k B T ) + n (0) α (for boson A α ),(3) n α = 1 (2π) 3 d 3 k e (εα−µα)/k B T + 1 ≡ 1 (λ T,α ) 3 F 3/2 (−µ α /k B T )(for fermion A α ),(4)
where k B is the Boltzmann constant and λ T,α is the thermal de Broglie wave length of particle A α at temperature T :
λ T,α = 2πh 2 m α k B T .(5)
The one-particle energy ε α in (3) and (4) is give by ε α =h 2 k 2 /(2m α ), where k is the wave-number vector of particle A α .
The B A and F A in (3) and (4) are the Bose and Fermi functions:
B A (ν) = 1 Γ(A) ∞ 0 x A−1 dx e x+ν − 1 ,(6)F A (ν) = 1 Γ(A) ∞ 0 x A−1 dx e x+ν + 1 ,(7)
where ν corresponds to the fugacity and Γ(A) is the gamma function. The Bose function B A can be written with the Appel function φ(z, s) as B A (ν) = φ(A, e ν ), and the Fermi function
is expressed with the Bose functions F A (ν) = B A (ν) − 2 1−A B A (2ν).
The Bose function B A in (6) converges in ν ≥ 0 (or µ α ≤ 0 in the chemical potential) and becomes singular at µ α = 0, with which a phase transition occurs to the Bose-Einstein condensate of the boson A α . In the BEC region, the chemical potential vanishes and the condensed density n (0) α in (3) takes a finite value (it vanishes in the normal region). In the process (1), the particle-number conservations for A 1,2 give the constraints
n 1 + n M = n 1,t , n 2 + n M = n 2,t ,(8)
where n 1,t and n 2,t are the total number densities of atoms A 1,2 , which consist of isolated atoms and constituents in the composite molecule M.
Solving Eqs. (2) and (8) with (3) and (4), we can determine the densities n 1,2,M in equilibrium at temperature T for the parameters m α , n α,t (α = 1, 2), and ∆E.
B. Scaled variables
We now introduce scaled dimensionless variables; they simplify the form of the above equations and greatly reduce the number of parameters.
The scaled massesm α are defined bym α = m α /m M , where m M = m 1 + m 2 + ∆m M . In the ultracold atomic-gas system, the mass defect (∆m M ∼ 10 −5 −10 −10 eV) is highly smaller than the atom or molecule masses (∼ GeV); thus, we use the approximation m M ∼ m 1 + m 2 (the conservation law of mass in chemical processes) throughout this paper.
We introduce n t ≡ n 1,t +n 2,t and E s ≡h 2 (n t ) 2/3 /m M as scaling quantities for the particlenumber densities and the energies, respectively: for example,ñ α = n α /n t (α = 1, 2, and Using the scaled quantities, the equilibrium conditions (2) and (8) becomẽ
M),T = k B T /E s , ∆Ẽ M = ∆E M /E s , etc.µ 1 +μ 2 −μ M = ∆Ẽ M ,(9)n 1 +ñ M =ñ 1,t ,ñ 2 +ñ M =ñ 2,t .(10)
The Bose and Fermi statistics formulas (3) and (4) becomẽ
n α = m αT 2π 3/2 B 3/2 (−μ α /T ) +ñ (0) α (for boson A α ),(11)
n α = m αT 2π
3/2 F 3/2 (−μ α /T ) (for fermion A α ).(12)
Now the problem of obtaining the atom-molecule equilibrium states is reduced to solving Eqs. (9) and (10) with (11) and (12) for temperatureT , molecular binding energy ∆Ẽ M , and mass and total density of A 1 (m 1 ,ñ 1,t ). The scaled atomic masses and the scaled total densities take the values of 0 ≤m α ,ñ α,t ≤ 1 and satisfỹ
m 1 +m 2 = 1,ñ 1,t +ñ 2,t = 1.(13)
These constraints play the role of reducing the number of independent parameters.
III. MOLECULAR FORMATIONS IN NONINTERACTING ATOMIC-GAS MIX-TURES
A. Bose-Einstein condensation and Fermi degeneracy Before presenting the calculational results, we discuss two interesting quantum effects:
BEC for bosons and Fermi degeneracy (FD) for fermions.
As discussed in the previous section, the phase transition to the BEC of the boson A α occurs whenμ α = 0 in (11); the transition temperature T C is
T C = 2π(ñ α ) 2/3 m α [ζ(3/2)] −2/3 ∼ 3.313 (ñ α ) 2/3 m α ,(14)
where ζ(3/2) is the Riemann zeta function. The condensed and thermal parts of the number density at T < T C are given by (11); with scaled quantities, they becomẽ
n (th) α = m αT 2π 3/2 ζ(3/2),ñ (0) α = 1 − T T C 3/2 ñ α .(15)
From these equations, we can find that, if bosons A α exist in the mixture atT = 0 (ñ α = 0), all A α should condense into the BEC (ñ (0) α =ñ α atT = 0). Different from bosons, fermions go into the FD state at very low temperatures; especially,
atT = 0, Eq. (12) becomesñ α = √ 2(m α ) 3/2 3π 2 (μ F,α ) 3/2 ,(16)
where µ F,α ≡μ α (T = 0) is the Fermi energy.
The transition into the FD state is not a phase transition, but a continuous one, so that, different from the BECT C in (14), no clear boundaries exist for FD. Instead, as an estimation of the occurrence of FD, we use the temperature atμ F,α = 0:
T F,α = (6π 2ñ F,α ) 2/3 2m F,α ∼ 7.569 (ñ α ) 2/3 m α .(17)
Because of the permitted ranges of the scaled densities 0 ≤ñ ≤ 1, we can findT C ∼T F from Eqs. (14) and (17) are obtained from the equilibrium condition (9) with Eqs. (10)- (12). Interparticle interaction effects are neglected, which will be discussed in a later section.
To understand the overall structure of the atom-molecule equilibrium, we consider the phase diagram at T = 0, which is obtained with analytical calculations as shown in Appendix B. Fig. 1 shows the T = 0 phase diagram of the BF mixture with the same atom masses In Fig. 1, we can read off the existence condition of the BEC of atom B; it is in the regions with the bracketed B because the bosons always condense into a BEC at T = 0.
We now turn to the atom-molecule equilibrium of the BF mixture at finite temperatures, which are obtained in numerical calculations using (4) and (9). Fig. 2 shows the temperature dependences of the scaled densitiesñ B andñ M in the BF mixture with massesm B =m F = 1/2 and the same total atomic-number densitiesñ B,t =ñ F,t = 1/2. The lines d, e, and f in give a free-energy reduction because of ∆Ẽ M < 0, but at low T , the molecules constitute the FD states, which have large kinetic energies coming from the occupied high-energy oneparticle states. In contrast, in dissociated states, in spite of the kinetic-energy contribution from the FD states of the fermions F , the bosons B can reduce the kinetic energy largely as they condense into the BEC at low-T . Thus, depending on the positive or negative value of ∆Ẽ M , the dissociated or molecular state becomes stable, and a mixed phase appears between them.
C. Molecular formations in the FF mixture
In the FF mixture (A 1 = F 1, A 2 = F 2), we consider atom-molecule equilibrium through the process F 1 + F 2 ↔ M = (F 1F 2) in the same way as in the BF mixture. In this section, we take the FF mixture with the same atomic massesm F 1 =m F 1 = 1/2. The topological structure of the equilibrium phases in Fig. 3 is the same as that in the BF mixture (Fig. 1); it consists of a "molecular phase" to the left of the border BAC, a "dissociated phase" to the right of the boundary BC, and a "mixed phase" between them.
The existence condition of BEC of the bosonic molecules M is also read off in Fig. 3 as the regions with the symbol M in brackets.
The equilibrium states at T = 0 can be expressed in the same way as in the BF mixture.
In Fig. 4, we show the temperature dependences ofñ We should comment on the relation of the present approach to the FF mixture with the BEC-BCS crossover theory [21,22,23]. In the crossover theory, two kinds of bare fermions become dressed quasifermions, and quasimolecule states (or Cooper-pair states) appear as physical degrees of freedoms. The strength change of the attraction between bare fermions gives the crossover between the BCS states (weak interaction) and the molecular BEC states (strong interaction). In the present quasiequilibrium approach, the fermions and the molecule should be considered as quasi-particles; the effects of bare-particle interactions are included as the existence of the molecule and its binding energy ∆Ẽ M . Really, the present approach is proved to give an approximated result on the strong interaction
µ B = k B T ln (λ 3 T n B )(18)
with λ T ∝ T −1/2 defined in (5). Eq. (18) shows that µ B is singular at (n B , T ) ∼ (0, 0) and
can take any negative value µ 0 if we take the limit through the pass
n B ∼ (λ T,B ) −3 e µ 0 /k B T .
Now let us go ahead to the atom-molecule equilibrium in the BB mixture through the process B1 + B2 ↔ M = (B1B2); then, the equilibrium condition (9) becomes
µ B1 +μ B2 −μ M = ∆Ẽ M .(19)
If all bosons condense into the BEC, the chemical potentials vanish, µ B1,B2,M = 0, at T = 0;
it gives a contradiction in (19) except for the case of ∆Ẽ M = 0 (no triple BEC theorem) [48].
IV. LAW OF QUANTUM MASS ACTION
In mixtures of classical Maxwell-Boltzmann gases, the molecular formation or dissociation processes through (1) satisfy the "law of mass action":
n 1 n 2 n M = K(T ),(20)
where n 1,2,M are the number densities of the particle A 1,2,M and K(T ) is an equilibrium constant, which depends on T , but not on n 1,t and n 2,t .
In the present calculations of ultracold molecular formation or dissociation processes, quantum-statistical effects play an important role, so that they give the density dependence of K(T ) in (20); we call it the "law of quantum mass action".
In order to make a comparison with the quantum cases, we derive an exact form of (20) in the case of ideal Maxwell-Boltzmann gases, where the density of the particle A α is given by
n (M B) α = λ −3 T,α e µα/k B T ,(21)
with the thermal de Broglie wave length defined in (5). Substituting Eq. (21) into the equilibrium condition (2), we obtain the classical law of mass action [43,49]: .
The ratio R generally depends onT andñ 1,t because of the constraint (13). To understand the deviations from the law of mass action, we expand the ratio R in the high-T region analytically:
R BB ∼ 1 + 1 2 3/2 −λ 3 T,1 n 1 − λ 3 T,2 n 2 + λ 3 T,M n M ,(24)R F F ∼ 1 + 1 2 3/2 +λ 3 T,1 n 1 + λ 3 T,2 n 2 + λ 3 T,M n M ,(25)R BF ∼ 1 + 1 2 3/2 −λ 3 T,1 n 1 + λ 3 T,2 n 2 − λ 3 T,M n M ,(26)
where R BB,F F,BF are for the BB,FF,BF mixtures. The leading-order terms R ∼ 1 in
Eqs. (24)- (26) correspond to the Maxwell-Boltzmann limit (law of mass action). Completely different contributions are found in the next-order terms (∝ T −3/2 ) for R BB and R F F in BB and FF mixtures, which are consistent with the results in Fig. 7(a) and Figs. 7(c) and 7(d)
at high T . The relatively small T dependence of the ratio R BF for the BF mixture [ Fig. 7
(b)]
at high T can be explained quantitatively from the cancellation of the boson and fermion contributions in the T 3/2 term in (26).
V. EFFECTS OF INTERPARTICLE INTERACTIONS ON EQUILIBRIUM
A. Interaction effects in the mean-field approximation
In the previous sections, we discussed atom-molecule equilibrium in noninteracting atomgas mixtures. If interactions exist between atom-atom, atom-molecule, and moleculemolecule (interacting) mixtures, they should modify the equilibration and sufficiently strong interactions could change the phase structures of the mixtures.
Theoretically, the effects of the interactions for the free energy can be divided into two kinds: mean-field and correlation ones. The mean-field effect can be evaluated, for example, using the Hartree-Fock approximation, and it can be represented as interaction terms (background energy) in the one-particle energies of atoms or molecules in the mixtures.
The correlation effects are defined as the contributions that cannot be introduced in the mean-field approximation. It should be noted that this division of interaction effects is a theoretical one and sometimes ambiguous. For example, in the BCS theory of superconductors, if we take the normal quasielectron theory of a normal Fermi-degenerate vacuum in the Hartree-Fock sense, then Cooper pairs and their condensations are created by correlation effects; however, if we take the Bogoliubov quasi-particles as dynamical degrees of freedom, the BCS states can be understood as a kind of mean-field theory. In this paper, we discuss interaction effects for atom-molecule equilibrium in the Hartree-Fock-like mean-field approximation.
For interparticle interactions, we take the ones coming through the two-body s-wave scattering processes, which give dominant contributions in ultracold atomic gases except the strongly interacting or spin-degenerate ones. They can be introduced through effective
interactions, for which we use the pseudopotentials[50]
V i,j = g i,j α,β δ 3 (r α − r ′ β ), (i, j = A1, A2, M)(27)
between the αth and βth particles of the i and j species, respectively. The coupling constants g i,j are determined from the s-wave scattering lengths a i,j between i and j species:
g i,j = 2πh 2 µ i,j a i,j ,(28)
where µ i,j ≡ m i m j /(m i + m j ) is the reduced mass.
Let us consider the contributions of the interactions in mixtures with equilibrium A1 + A2 ↔ M in the mean-field approximation.
The interaction effects are introduced into the free energy as the background energy:
F = E 0 + E int − i=A1,A2,M µ i n i ,(29)
where the E 0 is the kinetic energy, which exists also in noninteracting cases.
The contribution of the potential V i,j in (27) to the background energy E int is evaluated by E int i,j = V i,j . In the mean-field approximation, it is expressed by the number densities n i and n j . In the interaction between the same kinds of particles, it becomes
E BB = g BB 2 2n 2 B − (n (0) B ) 2 , E F F = 0,(30)
for the boson B and the fermion F . The n (0) B is the condensed density of B; when T ∼ 0, we can use the approximation that n To avoid unnecessary complexity in the formulation, we redefine g F F = 0 for the fermions.
The background energies coming from the interactions between different kinds of particles become E i,j = g i,j n i n j (i = j).
From the above considerations, the total background energy in the equilibrium through
A 1 + A 2 ↔ M can be approximated by E int = i=A1,A2,M g i,i 2 (n i ) 2 + g A1,A2 n A1 n A2 + g A1,M n A1 n M + g A2,M n A2 n M .(32)
Differentiating the free energy F in (29) with respect to the density n i , we obtain the single-particle energy
ǫ i = ǫ (0) i + j=A1,A2,M g i,j n j − µ i ,(33)
where ǫ (0) i is the kinetic energy of particle i:
ǫ (0) i = ∂E 0 ∂n i = (p i ) 2 2m i .(34)
With the effective chemical potential defined by
µ ′ i = µ i − j=A1,A2,M g i,j n j ,(35)
Eq. (33) becomes
ǫ i = ǫ (0) i − µ ′ i .(36)
It should be noticed that the ǫ i in (36) has the same form as that in the noninteracting case, so that the same Bose and Fermi statistic formulas (11) and (12) can be applied also in the interacting case using the effective chemical potential µ ′ i instead of µ ′ i :
n α = m αT 2π 3/2 B 3/2 (−μ ′ α /T ) (for boson A α ),(37)
n α = m αT 2π
3/2 F 3/2 (−μ ′ α /T ) (for fermion A α ).(38)
Using eq. (35), the equilibrium condition (9) for the interacting mixture becomes
µ ′ A1 + µ ′ A2 − µ ′ M = ∆E ′ M ,(39)
where the effective binding energy ∆E ′ M is given by the density of the M molecule
∆E ′ M = αn M + γ,(40)
where interaction effects are included in the two parameters α and γ:
α = i=A1,A2,M g i,i + 2(g A1,A2 − g A1,M − g A2,M ),(41)γ = ∆E + (g A1,M − g A1,A1 − g A1,A2 )n A1,t + (g A2,M − g A2,A2 − g A1,A2 )n A2,t .(42)
In the derivation of eqs. (40)-(42), we have used the constraints (13).
As has been done in the case of noninteracting mixtures in Sec. IIB, we introduce scaled variables for the coupling constants and the effective binding energies:
g i,j = g i,j n t E s ,(43)∆Ẽ ′ M = ∆Ẽ ′ M E s ,(44)
where n t = n 1,t + n 2,t and E s =h 2 (n t ) 2/3 /m M . For the effective binding energy ∆Ẽ ′ M , Eq. (40) becomes
∆Ẽ ′ M =αñ M +γ,(45)
where the scaled parametersα andγ are defined bỹ
α = i=A1,A2,Mg i,i + 2(g A1,A2 −g A1,M −g A2,M ),(46)γ = ∆Ẽ + (g A1,M −g A1,A1 −g A1,A2 )ñ A1,t + (g A2,M −g A2,A2 −g A1,A2 )ñ A2,t .(47)
Using the scaled variables, the atom-molecule equilibrium condition (39) becomes
µ 1 +μ 2 −μ M = ∆Ẽ ′ M ≡αñ M +γ,(48)
where Eq. (45) has been used.
The atom-molecule equilibrium states of the interacting mixtures can be obtained from eqs. (37)-(39) with the constraints (13). The interaction effects are included in the effective binding energy ∆E ′ M in (40) through the two parameters α and γ, which are determined from the coupling constants g i,j . Thus, we find that, in the mean-field approximation, one extra parameter is necessary in the equilibrium theory of interacting mixtures in comparison with that of noninteracting ones.
B. Phase structure changes by interaction effects
Before we give the numerical results for the atom-molecule equilibrium states of the interacting mixtures, we discuss qualitatively how the interactions shift and change the phase structures (PSs). We assume that all coupling constants are positive g i,j > 0 to avoid possible instabilities from the spatial fluctuations of densities [38].
Different from noninteracting cases where the equilibrium condition (9) has a unique solution, the existence of the density-dependent termαñ M in (48) results in two or more different solutions of (48), which correspond to the different equilibrium states (coexisting phases). These solutions include locally stable and unstable states, so that we have to examine the behavior of Eq. (48) and take out solutions corresponding to stable states.
Close examination shows that two critical points 0 >α c1 >α c2 exist in the parameterα for the BF, FF, and BB mixtures, and the T = 0 equilibrium structures can be classified into three regions with them as folloes, PS1 (α c1 <α), Eq. (48), has unique solutions for each value of the parametersα,γ, and Thus, interaction effects generally give complex phase structures in atom-molecule equilibrium with the phases with two locally stable states. The occurrence of these phases depends on the parametersα,γ, andñ 1,2,t , and the transitions of the states caused by the change of these parameters might be first order. In the phases with two locally stable states, the absolutely stable equilibrium state should be determined from a free-energy comparison of these states. However, the difference of the free energies of these states is generally small, so that effects that are not included in the present mean-field calculations (the correlation effect) may give comparative contributions, and, also, in real experiments, states which are not absolutely stable can occur through nonequilibrium and history effects. For these reasons, we should say that the stability of these phases is very subtle, and a study of the more detailed structures of these phases will not be done in this paper. for the BF mixture are given bỹ ∼ −28.7 wheñ m B =m F = 1/2, so that the phase diagrams in Fig. 8 are found to show the phase structure PS1; they have qualitatively similar structures as the diagram for the noninteracting mixture ( Fig. 1). Especially, the phase diagram forα = 0 (solid line in Fig. 8) is completely the same as that in Fig. 1 because of the vanishing density-dependent termαñ M in (48). We also find that the boundary between the dissociate and mixed phases, CB, in Fig. 8, which is given by (B17) in Appendix B, is independent of the values ofα; this is because the n M = 0 condition on this boundary eliminates theα dependence in (48). The boundaries between the mixed and molecular phases depend onα as shown in (B15) and (B18) in Appendix B.
α (BF ) c1 = − 2 4/3 3 3π 2 √ 2 2/3 1 + 1 2 1/4m 3/4 F ,(49)α (BF ) c2 = −2 1/3 3π 2 √ 2 2/3 1 + 1 m F .(50)
Theα-dependence of the mixed phase can be understand from the position of the end point A in Fig. 8
:γ = − 3π 2 2 √ 2 2/3 −α 2 ∼ −4.78 −α 2 ,(51)
which is given by (B20) in Appendix B. Eq. (51) shows that the area of the mixed phase increases whenα > 0 and decreases in the case ofα < 0. This behavior can be explained from (48); in the case ofα > 0, the molecule densityñ M has the effect of increasing ∆Ẽ ′ M , which makes molecular formation difficult, and the area of the mixed phase becomes large (theα < 0 case has the contrary effect).
The phase diagrams with the phase structures PS2 and PS3 are given in Fig. 9 (α = −25) and Fig. 10 (α = −30). In Fig. 9, the occurrence of two locally stable states gives the additional two phases with the boundaries (AB + DF ) and (DF + AB + EF ). The areas of these new phases in the PS2 structure are small; they are nothing more than substructures in the mixed phase. In the PS3 structure (Fig. 10), the left-shifted phase boundary AB crosses the boundary BC, and new kinds of phase can be produced between AB and BC: Fig. 10, for example, where dissociated and mixed states coexist. Fig. 10 also has a very complex substructure around the boundary BC. For extremely large values of α (Fig. 11,α = −70), the regions with substructures shrink to the small areas around the vertexes B and C in Fig. 11, and the whole phase structure becomes simple again; the phases where the dissociated and molecular states coexist appear in the central region, where the dissociated states include no molecule and the molecular states have as many molecules as possible.
region (b) in
D. Phase structures of interacting BB and FF mixtures
In this subsection, we briefly sketch the structures of the T = 0 phase diagrams of interacting BB and FF mixtures.
The T = 0 phase diagrams of interacting BB mixtures withm B1 =m B2 = 1/2 are shown in Fig. 12(a) (α = 20) and Fig. 12(b) (α = −20). In the case ofα > 0, the phase diagrams have the mixed phases [the region ABC in Fig. 12(a)], which do not exist in the noninteracting cases. The position of the end point A is given by (B25) in Appendix B:
γ = −α 2 ,(52)
which locates on the left side of the boundary BC in Fig. 12(a). We can understand that the phase structure inα > 0 is just PS1. Whenα < 0, Eq. (52) shows that point A moves across the boundary BC and locates to the right of it; the mixed phase disappears and new phases occur with coexisting locally stable equilibrium states [ Fig. 12(b)]. That means that the phase structure is PS3 forα < 0. As a result, we findα In the case of interacting FF mixtures, the critical valuesα
(F F ) c1 andα (F F ) c2 becomẽ α (F F ) c1 = − 2 3 m −3/4 F 1 +m −3/4 F 1 4/3 ,(53)α (F F ) c2 = −2 1/3 3π 2 √ 2 2/3 1 m F 1mF 2 ,(54)
which becomeα
(F F ) c1 ∼ −25.5 andα (F F ) c2
∼ −38.2 in the case ofm F 1 =m F 2 = 1/2.
In Fig. 13, we show the T = 0 phase diagrams for interacting FF mixtures with m F 1 =m F 2 = 1/2. The change of the phase structure is essentially similar to that in the interacting BF mixtures. In the PS1 structure [ Fig. 13(a)
VI. SUMMARY AND OUTLOOK
We have developed a quasichemical equilibrium theory for the molecular formation or dissociation processes in BF, FF, and BB mixtures of ultracold atomic gases and discussed atom-molecule equilibrium in these mixture.
The law of mass actions has also been examined for the mixtures; it is satisfied well at high T . We have shown that the quantum-statistical effects of the atoms and molecules, which become more effective in ultracold temperature, give deviations from the law (law of quantum mass action). The quantum-statistical effects are shown to give different deviations for bosons and fermions, and, in BF mixtures, both contributions have a tendency to cancel out at high T .
We have also discussed the effects of the interparticle interactions in the mixture within the mean-field approximation at T = 0 and evaluated the shifts of the T = 0 phase structures of atom-molecule equilibrium in the mixtures. Especially, in the case of large repulsive interactions between atoms and molecules, the phase structures have been shown to change qualitatively with the occurrence of coexisting local-equilibrium states. We have given the conditions for the coupling constants with which the phase-structure changes occur. The atom-molecule equilibrium in interacting mixtures can be calculated also at finite temperatures within the present framework, and the results are planned to be published in another paper.
The other correlation effects beyond the mean-field approximations should also be important-for example, in the BCS-BEC crossover problem. Combining the method of the Beth-Uhlenbeck approach [51,52] with the present quasichemical equilibrium theory, we should discuss the correlation effects and the crossover problem from a less modelindependent point of view. A study along these lines is now ongoing and will be presented in the near future.
The authors thank T. Suzuki and T. Takayama for many useful discussions.
APPENDIX A: ASYMPTOTIC BEHAVIORS OF BOSE OR FERMI FUNC-
TIONS
In this appendix, we derive some formulas of the asymptotic behaviors of the Bose and Fermi functions (6) and (7) at ν ∼ 0 and ν ∼ ±∞.
Before we discuss the asymptotic behavior, we prove the relation between B A and F A :
F A (ν) = B A (ν) − 2 1−A B A (2ν). (A1)
This formula is obtained by integrating both sides of the equation
x A−1 e x+ν + 1 = x A−1 e x+ν − 1 − 2x A−a e 2x+2ν − 1 . (A2)
Let us go to the asymptotic behavior of the functions B A (ν) and B A (ν) at ν ∼ ∞. Using the expansion of the integrand of B A ,
x A−1 e x+ν − 1 = ∞ k=1 x A−1 e −k(x+ν) ,(A3)
we obtain
B A (ν) = ∞ k=1 e −kν Γ(A) ∞ 0 x A−1 e −kx dx = ∞ k=1 e −kν k A ,(A4)
where we have used the formula of the gamma function:
Γ(A) = ∞ 0 z A−1 e −z dz.(A5)
Substituting (A4) into (A1), an expansion formula for F A (ν) can be obtained:
F A (ν) = ∞ k=1 (−1) k−1 e −kν k A .(A6)
Taking the first terms in Eqs. (A4) and (A6), we obtain the asymptotic behaviors of B A (ν) and F A (ν) at ν ∼ ∞:
B A (ν) ∼ F A (ν) ∼ e −ν .(A7)
We turn to the asymptotic behavior of B A (ν) around ν = 0. When 0 < α < 1, the point ν = 0 becomes an irregular singular point, so that only the asymptotic expansion is obtained for B A . For this purpose, we consider the Hankel-type complex integral
I = 1 Γ(A) (∞;−ν+) z A−1 dz e z+ν − 1 .(A8z A−1 e z+ν − 1 ∼ (−ν) A−1 z + ν . (A9)
The integration path does not pass the singular point z = −ν, so that we can evaluate the integral using the expansion (A2):
I = ∞ k=1 e −kν Γ(A) (∞;0+) z A−1 e −kz dz = ∞ k=1 e −kν k A (e 2πAi − 1),(A10)
where we have used the Hankel-integral representation of the gamma function.
On the other hand, we can deform the integration path and divide it into the path C −ν circulating around the singular point z = −ν and the Hankel-type path (∞; 0+) around the origin O:
I = 1 Γ(A) (∞;−ν+) z A−1 dz e z+ν − 1 = 1 Γ(A) C −ν + (∞;0+) z A−1 dz e z+ν − 1 . (A11)
The integral on C −ν is evaluated by the theorem of residue, and that on (∞; 0+) can be attributed to the real integral:
1 Γ(A) C −ν z A−1 dz e z+ν − 1 = 2πi Γ(A) (−ν) A−1 ,(A12)1 Γ(A) (∞;0+) z A−1 dz e z+ν − 1 = e 2πiA − 1 Γ(A) ∞ 0 x A−1 dx e x+ν − 1 = (e 2πiA − 1)B A (ν).(A13)
Combining Eqs. (A10), (A12), and (A13), we obtain the asymptotic expansion of B A (ν):
B A (ν) = π sin πA ν A−1 Γ(A) + ∞ k=1 e −kν k A .(A14)
Using the expansion of the series part in (A14),
∞ k=1 e −kν k A = ∞ k=1 1 k A ∞ n=0 (−kν) n n! = ∞ n=0 (−ν) n n! ∞ k=1 1 k A−n = ∞ n=0 (−1) n ζ(A − n) ν n n! ,(A15)
we obtain the power expansion formula by Opechowski [53,54]:
B A (ν) = π sin πA ν A−1 Γ(A) + ∞ n=0 (−1) n ζ(A − n) ν n n! . (A16)
Taking the leading term of the Opechowski formula, we obtain the asymptotic behavior of B A (ν) around ν = 0:
B A (ν) ∼ ζ(A) (A > 1), − ln ν (A = 1), π sin πA ν A−1 Γ(A) + ζ(A) (0 < A < 1). (A17)
Using Eq. (A1), the asymptotic formula of F A (ν) around ν = 0 becomes
F A (ν) ∼ (1 − 2 1−A )ζ(A).(A18)
It should be noted that the residue term in B A (ν) is canceled out in (A18); this is consistent with the fact that no such singular terms exist in F A (ν) originally.
The asymptotic formula of F A (ν) at ν ∼ −∞ is obtained by the Sommerfeld expansion
formula[54, 55] ∞ 0 φ ′ (u)du e u−α + 1 ∼ φ(α) + ∞ n=1 2F 2n (0)φ (2n) (α),(A19)
where φ(u) is a ∞-differentiable function and the coefficients F 2n (0), the Fermi function at ν = 0, are represented by the Bernoulli numbers B n :
2F 2n (0) = (1 − 2 1−2n )(2π) 2n (2n)! B n .(A20)
Using Eq. (A19), the asymptotic formula of F A (ν) at ν ∼ −∞ is obtained by Let us consider the mixed phase of the BF mixture of atom massesm B,F , which corresponds to the central region in the phase diagram (Fig. 1). The equilibrium condition (9) becomes 0 + 3π 2 √ 2
F A (ν) ∼ (−ν) A Γ(A + 1) .(A212/3 1 m F (ñ F ) 2/3 − 3π 2 √ 2 2/3 (ñ M ) 2/3 = ∆Ẽ M ,(B1)
where we have used µ B = 0 (BEC state) and the Fermi energy formula (16) for F and M withm M ∼ 1.
The boundaries AB, BC, and CA in Fig. 1 are obtained by settingñ B = 0,ñ M = 0, and n F = 0, respectively:
AB : ∆Ẽ M = 1 2m F [6π 2 (1 − 2ñ B,t )] 2/3 − 1 2 [6π 2ñ B,t ] 2/3 ,(B2)
BC :
∆Ẽ M = 1 2m F [6π 2ñ F,t ] 2/3 ,(B3)
CA :
∆Ẽ M = − 1 2 [6π 2ñ F,t ] 2/3 .(B4)
and the end points of the boundaries, A, B, and C, are obtained from (B4) withñ F,t = 1/2, (B3) withñ F,t = 1, and (B3) withñ F,t = 0:
A : ∆Ẽ M = − (3π 2 ) 2/3 2 ∼ −4.8,(B5)
B :
∆Ẽ M = 1 2m F (6π 2 ) 2/3 ∼ 7.6 m F ,(B6)
C :
∆Ẽ M = 0.(B7)
The boundaries and end points in Fig. 1 are obtained from the above formulas form F = 1/2.
Noninteracting FF mixture
In the case of the phase diagram of the FF mixture ( Fig. 3) with massesm F 1 andm F 2 , the equilibrium condition (9) in the mixed phase becomes
3π 2 √ 2 2/3 1 m F 1 (ñ F 1 ) 2/3 + 3π 2 √ 2 2/3 1 m F 2 (ñ F 2 ) 2/3 = ∆Ẽ M ,(B8)
where we have used µ M = 0 (BEC state) and the Fermi energy formula (16) for F 1 and F 2..
The boundaries AB, BC, and CA in Fig. 3 are obtained by settingñ F 2 = 0,ñ M = 0, andñ F 1 = 0, respectively:
AB : ∆Ẽ M = 1 2m F 1 [6π 2 (1 − 2ñ F 1,t )] 2/3 ,(B9)
BC :
∆Ẽ M = 1 2m F 1 [6π 2ñ F 1,t ] 2/3 + 1 2m F 2 [6π 2ñ F 2,t ] 2/3 ,(B10)
CA :
∆Ẽ M = 1 2m F 2 [6π 2 (2ñ F 1,t − 1)] 2/3 ,(B11)
and the points A, B and C are obtained from (B9) withñ F 1,t = 1/2, (B9) withñ F 1,t = 0, and (B11) withñ F 1,t = 1:
A : ∆Ẽ M = 0, (B12)
B : ∆Ẽ M = 1 2m F 1 (6π 2 ) 2/3 ∼ 7.6 m F 2 ,(B13)
C :
∆Ẽ M = 1 2m F 2 (6π 2 ) 2/3 . ∼ 7.6 m F 1 .(B14)
The boundaries and end points in Fig. 3 are obtained from the above formulas withm F 1 = m F 2 = 1/2.
Interacting BF mixture
The T = 0 phase diagram of the interacting BF mixture with massesm B,F is obtained by the equilibrium condition (48) for T = 0:
0 + 3π 2 √ 2 2/3 1 m F (ñ F ) 2/3 − 3π 2 √ 2 2/3 (ñ M ) 2/3 =αñ M +γ.(B15)
In the case ofα >α
CA :γ = − 3π 2 √ 2 2/3ñ 2/3 F,t −αñ F,t ,(B17)
for the phase boundaries (ñ B,t +ñ F,t = 1), and
A :γ = − 3π 2 2 √ 2 2/3 −α 2 ∼ −4.78 −α 2 ,(B19)B :γ = 1 m F 3π 2 √ 2 2/3 ∼ 7.6 m F ,(B20)
C :γ = 0.
The boundaries and end points in Fig. 7 are obtained from the above formulas withm F = m B = 1/2.
Interacting BB mixture
The T = 0 phase diagram of the interacting BB mixture with massesm B1,B2 is obtained by the equilibrium condition (48).
At T = 0, all chemical potentials appearing in (48), which are bosonic, can take two alternative possibilities:μ ′ k = 0 withñ k = 0 (BEC) orμ ′ k < 0 withñ k = 0 (k = B1, B2, M). In the mixed phase, the BEC conditionsμ ′ B1 =μ ′ B2 =μ ′ M = 0 giveαñ M +γ = 0; the boundaries, AB, BC, and CA, are obtained by substitutingñ B2 = 0,ñ M = 0, andñ B1 = 0, respectively:
AB :γ = −αñ B1,t ,(B22)
BC :γ = 0, (B23)
CA :γ = −α(1 −ñ B1,t ),(B24)
where we have used the constraintsñ B1 +ñ M =ñ B1,t andñ B2 +ñ M =ñ B2,t . The end points are obtained by It should be noted that the boundaries and the end points are independent of the atom massesm B1,B2 .
Interacting FF mixture
The T = 0 phase diagram of the interacting FF mixture with massesm B,F is obtained by the equilibrium condition (48). for T = 0:
3π 2 √ 2 2/3 1 m F 1 (ñ F 1 ) 2/3 + 3π 2 √ 2 2/3 1 m F 2 (ñ F 2 ) 2/3 =αñ M +γ.(B28)
In the case ofα >α
(F F ) c1
, the boundaries and end points of the phases can be obtained in the same manner as those in noninteracting cases.
The results are
AB :γ = 1 2m F 1 [6π 2 (1 − 2ñ F 1,t )] 2/3 −αñ F 1,t ,(B29)BC :γ = 1 2m F 1 [6π 2ñ F 1,t ] 2/3 + 1 2m F 2 [6π 2ñ F 2,t ] 2/3 ,(B30)CA :γ = 1 2m F 2 [6π 2 (2ñ F 1,t − 1)] 2/3 +α(ñ F 1.t − 1),(B31)
for the phase boundaries (ñ B,t +ñ F,t = 1), and
A :γ = −α 2 ,(B32)B :γ = 1 2m F 1 (6π 2 ) 2/3 ∼ 7.6 m F 2 ,(B33)C :γ = 1 2m F 2 (6π 2 ) 2/3 . ∼ 7.6 m F 1 .(B34)
The boundaries and end points in Fig. 11 are obtained from the above formulas withm F 1 = m F 2 = 1/2.
II. QUASICHEMICAL EQUILIBRIUM THEORY ON MOLECULAR FORMA-TION IN ATOMIC-GAS MIXTURES A. Molecular formation or dissociation process Let us take an atomic-gas mixture consisting of two atomic spices A 1 and A 2 with masses m 1 and m 2 ; in quantum statistics, A 1,2 are bosons or fermions, so that we have three kinds of combinations: boson-boson (BB), fermion-fermion (FF), and boson-fermion (BF).
E s can be interpreted as having the meaning of the Fermi energy for the fermionic matter of fermions with mass m M with density n t , but we use it simply as a scaling quantity with the dimension of energy.
as a rough estimation; quantum effects begin to appear at the same order of temperature in both noninteracting Bose and Fermi gases.
B. Molecular formations in the BF mixtureLet us consider the BF mixture (A 1 = B, A 2 = F ) with the molecular formation process B + F ↔ M = (BF ). The atom or molecule densitiesñ B,F,M in atom-molecule equilibrium
mFIG. 1 :
1B =m F = 1/2 for the molecular binding energy (∆Ẽ M ) and a total density of B (ñ B,t ). Because of the constraints (13), ∆Ẽ M andñ B,t are unique parameters to determine the equilibrium state. The bracketed letters in the regions or lines in Fig. 1 represent the species which exist in equilibrium at T = 0; for example, (B, F, M) in the central region means that atoms B and F and molecule M coexist there, etc. The T = 0 phase diagram of BF mixtures with the same atom masses in the ∆Ẽ M -ñ B,t plane. The bracketed letters show what kinds of particles exist in equilibrium at T = 0. From Fig. 1, we find that the BF mixture at T = 0 shows the structure that, in the area left of the border BAC (∆Ẽ M < ∼ 0), the states with molecules as many as possible become stable (molecular states) and, in the area right of the boundary BC, all molecules dissociate into atoms B and F (dissociated states). Between these two areas, mixed states of atoms and molecules become stable in the sense of equilibrium.
Fig. 2 FIG. 2 :
22are for several values of the molecular binding energies: ∆Ẽ M = 0, −4.78, and 9.57, respectively. The BEC line (14) and theμ F = 0 line, Eq. (17), are also drawn in Fig. 2 as lines a and b, respectively; they show the boundaries where quantum-(Color online) Temperature dependences of the number densitiesñ B of the bosonic atom B [(a), left] andñ M of the molecule M [(b), right] in the BF mixture with the same atomic masses and the same total atomic densities. The lines d (dashed line), e (double-dot-dashed line) and f (triple-dot-dashed line) are for mixtures with ∆Ẽ M = 0, 9.57 (resonance state), and −4.78 (bound state), respectively. The BEC T C line of B (line a) and theμ F = 0 lines for F (line b) and M (line c) are also drawn.
Fig. 2
2shows that, at the high-temperature limit,ñ M converge to 0 (complete dissociation of molecules into atoms). It can be explained by the energy-entropy balance in the free energy. The atom-molecule mixed states have a reduction in the energy part of the free energy; in contrast, the dissociated states have a reduction of the entropy part because the state density of B + F is larger than that of molecules. With increasing temperature, the entropy contribution becomes large and the reaction process swings to the dissociation of molecules. This mechanism can also be applied to BB or FF mixtures because the binding energy and quantum-statistical properties are less effective at high temperature.The equilibrium states have a large dependence on the binding energy ∆Ẽ M around and below the BEC T C andμ F = 0 lines. At T = 0, they converge to states belonging to the phases on the horizontalñ F,t = 1/2 line in Fig. 1: the molecular, mixed and dissociated phases divided by the points, ∆Ẽ M = ∆ 1 ≡ −(3π 2 ) 2/3 /2 ≈ −4.78 (point A) and ∆ 2 ≡ (3π 2 ) 2/3 ≈ 9.57 (point D). The temperature dependence in Fig. 2 is also classified with the values of ∆Ẽ M : (i) ∆Ẽ M ≤ ∆ 1 (lines f ). With decreasing T , theñ B becomes small and goes below the BEC T C line (no BEC of atom B for all temperatures). (ii) ∆ 1 < ∆Ẽ M < ∆ 2 (line d). With decreasing T ,ñ B (and alsoñ F ) andñ M run into the BEC andμ ≥ 0 regions, and converge into finite values. (iii) ∆ 2 ≤ ∆Ẽ M (line e). With decreasing T ,ñ M takes the maximum value on the right of theμ = 0 line (line c) and decreases toñ M = 0 along the line from below. The atom-molecule equilibrium in the BF mixtures can be explained by competition between the quantum-statistical effects and the binding energy of M; the molecule states
Fig. 3
3shows the T = 0 phase diagram of the FF mixture. The notation for each phase is the same as that of the BF mixture (seeFig. 1). Analytical expressions of the phaseboundary lines and points inFig. 3are given in Appendix B.
FIG. 3 :
3F 1 andñ M in the FF mixture with The T = 0 phase diagram of the FF mixture with the same atom masses in the ∆Ẽ M -ñ F 1,t plane. The bracketed letters show what kinds of particles exist in equilibrium at T = 0. online) Temperature dependences of the number densitiesñ F 1 of the atom F 1 [(a), left] andñ M of the molecule M [(b), right] in the FF mixture with the same atomic masses and the same total atomic densities. The lines c (dashed line) and d (double-dot-dashed line) are for mixtures with ∆Ẽ M = 0 and 19.1 (resonance state), respectively. Theμ F = 0 lines and the BEC T C lines are drawn for F 1 and F 2 (line a) and M (line b). n F 1,t =ñ F 2,t = 1/2 (the states on the lineñ F 1,t = 1/2 in Fig. 3) and ∆Ẽ M = 0 and 19.1. In the FF mixture, the T = 0 phases are classified with the value of ∆Ẽ M : (i) ∆Ẽ M < 0 (molecule phase), (ii) 0 < ∆Ẽ M < ∆ 3 (mixed phase), and (iii) ∆ 3 < ∆Ẽ M (dissociated phase), where the border points with ∆Ẽ M = 0 and ∆ 3 ≡ 2(3π 2 ) 2/3 ≈ 19.1 correspond to point A and the crossing point between line BC andñ F 1,t = 1/2 inFig. 3. The two lines c and d inFig. 4are for the border between two phases.
(
BEC) side (∆Ẽ M < ∼ 0)[21]. In the weak interaction (BCS) side, the BECT C in the present approach vanishes at ∆Ẽ M = ∆ 3 as shown inFig. 3and disagrees with the BCS T C . This is because the molecule states are treated as structureless bosons and no statistical correlations are introduced. It can be said that the present approach should give a good approximation in the strong interaction side and in the high-T region.D. Molecular formations in the BB mixture Before discussing the results, we should give some explanations of the singular property of the boson chemical potential at the double limit of n B = 0 and T = 0 in the no BEC region. Around the point of (n B , T ) ∼ (0, 0), ν = −µ B /k B T becomes large, so that the Bose function B 3/2 in the density formula (6) can be approximated by an asymptotic formula [Eq. (A9) in Appendix A]: then the chemical potential becomes
Fig. 5 .FIG. 5 :
55The negativeness of the chemical potentials determines the solutions of (19) at T = 0: (i)∆Ẽ M < 0 (molecular phase):μ M = 0 (the BEC of M) andμ α = ∆Ẽ M withñ α = 0 (α = B1 or B2). (ii) ∆Ẽ M > 0 (dissociated phase):μ B1 =μ B2 = 0 (the BECs of B1 and B2) and μ M = −∆Ẽ M withñ M = 0. The results are summarized in the T = 0 phase diagram in We should find that no triple BEC states exist except for the states on the boundary line of ∆Ẽ M = 0. Accordingly no mixed phase exists in the BB mixture. The T = 0 phase diagram of BB mixtures in the ∆Ẽ M -ñ B1,t plane. The bracketed letters show what kinds of particles exist in equilibrium at T = 0.
online) Temperature dependences of the number densitiesñ B1 of the atom B1 [(a), left] andñ M of the molecule M [(b), right] in the BB mixture with the same atomic masses and the same total atomic densities. The lines c (short-dashed line), d (dotted line), e (dot-dashed line) and f (double-dot-dashed line) are for mixtures with ∆Ẽ M = +0, −0, +1, and −1, respectively, where ∆Ẽ M = ±0 ≡ lim ǫ→+0 ±ǫ. The BEC T C lines of B1 and B2 (line a) and M (line b) are also drawn.
Fig. 6
6shows the temperature dependences ofñ B1 andñ M in the BB mixture withm B1 = m B2 = 1/2 andñ B1,t =ñ B2,t = 1/2 (the states on the lineñ B1,t = 1/2 inFig. 5) for ∆Ẽ M = ±0, ±1. The above-mentioned exclusive behaviors at T = 0 for BEC have influences also at T = 0 as shown in Fig. 6: in ∆Ẽ M < 0, only the molecules become the BEC (lines e); otherwise, only atomic BECs can occur (lines f ). The mixture with ∆Ẽ M = 0 has a singularity at point C; if we take the different kinds of limit ∆Ẽ M = ±0, BEC of atoms or molecules occurs, respectively.The exclusiveness between atom and molecule BECs in BB mixtures has been observed in a Cs experiment by the Innsbruck group[9]; a sudden interchange occurs between the atom and molecule BECs when the resonance becomes the bound molecule using the Feshbach resonance method. This observation shows that the exclusiveness is incomplete and some mixture of atom and molecule BECs exists on the bound molecule side; it might be explained by an interparticle interaction effect (see Sec. V) or nonequilibrium effect.
FIG. 7 :
7(Color online) The ratio R defined by (23) for the FF [(a), top left], BF [(b), top right], and BB [(c) and (d), bottom left and right] mixtures with the same atom masses. The mass defects are ∆m M = −1 for the FF and BF mixtures and ∆m M = −1 (c), +1 (d) for the BB mixtures. The lines are for temperatures: log 10T = ∞ (Maxwell-Boltzmann limit, solid line), 0.4 (long-dashed line), 0 (short-dashed line), and −0.4 (dotted line), respectively. In Fig. 7, we show theñ 1,t dependences of log R for several temperatures (logT = −0.4, 0, 0.4, ∞) in the case of FF, BF, and BB mixtures withm 1 =m 2 = 1/2. The mass defects are ∆m M = −1 for the FF and BF mixtures and ∆m M = +1 [Fig. 7(c)] and −1 [Fig. (7(d)] for the BB mixtures. In the BB mixture, the behaviors of R are found to be very different for molecules and resonances.
B
∼ n B . The vanishing background energy for the fermion F originates in forbidden s-wave scatterings by Pauli blocking effects at T ∼ 0.
t , and the T = 0 phase diagrams become similar in structure with those for the noninteracting mixtures, including the dissociate, mixed, and molecule phases. (As shown later, the BB mixture is somewhat exceptional. ) PS2 (α c2 <α <α c1 ), Eq. (48), has two stable (one unstable) solutions in the mixed phase.PS3 (α <α c2 ) has new coexisting phases appearing where both the mixed and molecular states become locally stable.
From
Eq. (46), the negative contributions toα can be given by large values ofg A1,M andg A2,M ; thus, atom-molecule interactions are found to be interesting and important in transitions to unstable phases. online) The T = 0 phase diagram of interacting BF mixtures with the same atom masses in theγ-ñ B,t plane forα = −20, 0, 20. The bracketed letters in the regions or lines show what kinds of particles exist in equilibrium at T = 0. C. Phase structures of the interacting BF mixture
Fig. 8 FIG
8shows the T = 0 phase diagram for interacting BF mixtures withm B =m F . 9: (Color online) The T = 0 phase diagram of interacting BF mixtures with the same atom masses in theγ-ñ B,t plane forα = −25. The bracketed letters in the regions or lines show what kinds of particles exist in equilibrium at T = 0.
FIG. 10: (Color online) The T = 0 phase diagram of interacting BF mixtures with the same atom masses in theγ-ñ B,t plane forα = −30 [(a), top left]. (b) and (c) (top right and bottom left) show the detailed structures of (a). (d) (bottom left) is the legend that explains the phases (i)-(n) in (a)-(c). 11: (Color online) The T = 0 phase diagram of interacting BF mixtures with the same atom masses in theγ-ñ B,t plane forα = −70, where the phase (i) and (m) are coexisting phases of (B, F ), (F, M ) and (B, F ), (B, M ) as explained in the legend in Fig. 10(d). FIG. 12: (Color online) The T = 0 phase diagram of interacting BB mixtures with the same atom masses in theγ-ñ B1,t plane forα = 20 [(a), left] andα = −20 [(b), right], where the phases (i) and (m) are coexisting phases of (B1, B2), (B2, M ) and (B1, B2), (B1, M ).
exist in the interacting BB mixtures.
], the phase structures are obtained by deformation from the noninteracting ones. The substructures with the coexisting states appear in the mixed phase in the PS2 structure [Fig. 13(b)]. The end point A inFig. 13(a) and 13(b) crosses the boundary BC between the mixed and dissociated phases and new phases appear in the PS3 structure with the coexisting equilibrium states,FIG. 13: (Color online) The T = 0 phase diagram of interacting FF mixtures with the same atom masses in theγ-ñ F 1,t plane forα = 20, 0, −20 [(a), top left], −30 [(b), top right], −40 [(c), bottom left], and −70 [(d), bottom right], where the phases (i), (m), and (n) are coexisting phases of (F 1, F 2), (F 2, M ); (F 1, F 2), (F 1, M ); and (F 1, F 2), (F 1, F 2, M ). which have complex substructures [Fig. 13(c)]; the structure becomes again simple for large negative values ofα [Fig. 13(d)].
) ν FIG. 14 :
ν14Hankel-type integral path in the complex-number plane. The integration path (∞; −ν+) and the cut line on the positive real axis are shown in Fig. 14. The phase branches of z A−1 are fixed at x A−1 or e 2πAi x A−1 on the upper or lower parts of the cut line. The integrand function has a pole at z = −ν:
)
APPENDIX B: DERIVATIONS OF T = 0 PHASE DIAGRAMS IN BF AND FF MIXTURES 1. Noninteracting BF mixture
,
the boundaries and end points of the phases can be obtained in the same manner as those in noninteracting cases.
e ∆G/k B T ≡ K(T ),(22)where ∆G is called the standard chemical affinity, and it becomes ∆G = ∆E M (the binding energy of the molecule A M ).To show the deviations from the law of quantum action by quantum effects, we evaluate the ratio of n 1 n 2 /n M calculated in the previous section (including the quantum effects) to
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|
[] |
[
"Identification of Ne lines in H-deficient (pre-) white dwarfs: a new tool to constrain the temperature of the hottest stars ⋆ ⋆⋆",
"Identification of Ne lines in H-deficient (pre-) white dwarfs: a new tool to constrain the temperature of the hottest stars ⋆ ⋆⋆"
] |
[
"K Werner \nKepler Center for Astro and Particle Physics\nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 172076TübingenGermany\n",
"T Rauch \nKepler Center for Astro and Particle Physics\nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 172076TübingenGermany\n",
"J W Kruk \nDepartment of Physics and Astronomy\nJohns Hopkins University\n21218BaltimoreMDUSA\n"
] |
[
"Kepler Center for Astro and Particle Physics\nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 172076TübingenGermany",
"Kepler Center for Astro and Particle Physics\nInstitut für Astronomie und Astrophysik\nUniversität Tübingen\nSand 172076TübingenGermany",
"Department of Physics and Astronomy\nJohns Hopkins University\n21218BaltimoreMDUSA"
] |
[] |
For the first time, we have identified Ne absorption lines in far-UV spectra of the hottest known (T eff > ∼ 150 000 K) hydrogendeficient (pre-) white dwarfs of spectral type PG1159. They are of photospheric origin and can be matched by synthetic non-LTE line profiles. We also show that a number of UV and optical emission lines in these stars can be explained as being photospheric Ne features and not, as hitherto suspected, as ultrahigh ionised O lines created along shock-zones in the stellar wind. Consequently, we argue that the long-standing identification of the same emission lines in hot [WR]-type central stars as being due to ultrahigh-ionised species (O -, C -) must be revised. These lines can be entirely attributed to thermally excited species (Ne -, N , O ). Photospheric Ne lines are also identified in the hottest known He-rich white dwarf (KPD 0005+5106), some of which were also attributed to O previously. This is a surprise because it must be concluded that KPD 0005+5106 is much hotter (T eff ≈200 000 K) than hitherto assumed (T eff ≈120 000 K). This is confirmed by a re-assessment of the He line spectrum. We speculate that the temperature is high enough to explain the mysterious, hard X-ray emission (1 keV) as being of photospheric origin.
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10.1051/0004-6361:20078152
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[
"https://arxiv.org/pdf/0709.0097v1.pdf"
] | 15,435,080 |
0709.0097
|
65e949ee7a388dfa1831d52d4b16b8c3af161446
|
Identification of Ne lines in H-deficient (pre-) white dwarfs: a new tool to constrain the temperature of the hottest stars ⋆ ⋆⋆
2 Sep 2007 February 1, 2008
K Werner
Kepler Center for Astro and Particle Physics
Institut für Astronomie und Astrophysik
Universität Tübingen
Sand 172076TübingenGermany
T Rauch
Kepler Center for Astro and Particle Physics
Institut für Astronomie und Astrophysik
Universität Tübingen
Sand 172076TübingenGermany
J W Kruk
Department of Physics and Astronomy
Johns Hopkins University
21218BaltimoreMDUSA
Identification of Ne lines in H-deficient (pre-) white dwarfs: a new tool to constrain the temperature of the hottest stars ⋆ ⋆⋆
2 Sep 2007 February 1, 2008Received; acceptedAstronomy & Astrophysics manuscript no. aaStars: abundances -Stars: atmospheres -Stars: evolution -Stars: AGB and post-AGB -White dwarfs
For the first time, we have identified Ne absorption lines in far-UV spectra of the hottest known (T eff > ∼ 150 000 K) hydrogendeficient (pre-) white dwarfs of spectral type PG1159. They are of photospheric origin and can be matched by synthetic non-LTE line profiles. We also show that a number of UV and optical emission lines in these stars can be explained as being photospheric Ne features and not, as hitherto suspected, as ultrahigh ionised O lines created along shock-zones in the stellar wind. Consequently, we argue that the long-standing identification of the same emission lines in hot [WR]-type central stars as being due to ultrahigh-ionised species (O -, C -) must be revised. These lines can be entirely attributed to thermally excited species (Ne -, N , O ). Photospheric Ne lines are also identified in the hottest known He-rich white dwarf (KPD 0005+5106), some of which were also attributed to O previously. This is a surprise because it must be concluded that KPD 0005+5106 is much hotter (T eff ≈200 000 K) than hitherto assumed (T eff ≈120 000 K). This is confirmed by a re-assessment of the He line spectrum. We speculate that the temperature is high enough to explain the mysterious, hard X-ray emission (1 keV) as being of photospheric origin.
Introduction
Observations of extremely hot post-AGB stars (T eff > ∼ 100 000 K) with the Far Ultraviolet Spectroscopic Explorer (FUSE) have led to the discovery of many unidentified spectral lines. Their identification is difficult, mostly because of the lack of accurate atomic data for highly ionized elements. On the other hand, their identification is rewarding because they are often the only features by which particular species are accessible for abundance determinations. Such species can be used to test stellar evolutionary models. For example, several lines of fluorine (F -), neon (Ne ), silicon (Si ) and argon (Ar ) were identified for the first time (Jahn et al. 2007;Werner et al. 2004aWerner et al. , 2005Werner et al. , 2007. In particular, the Ne λ 973 Å line occasionally displays a prominent P Cygni profile, which is an important tool for stellar wind analyses (Herald et al. 2005). In this paper, we report the identification of a number of Ne lines in the FUSE spectra of the hottest known hydrogen-deficient (pre-) white dwarfs. We show that the mere occurrence of such highly ionized neon lines puts a strict lower limit on the effective temperature (≈ 150 000 K).
Send offprint requests to: K. Werner e-mail: [email protected] ⋆ Based on observations made with the NASA-CNES-CSA Far Ultraviolet Spectroscopic Explorer. FUSE is operated for NASA by the Johns Hopkins University under NASA contract NAS5-32985. ⋆⋆ Some of the data presented in this paper were obtained from the Multimission Archive at the Space Telescope Science Institute (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NAG5-7584 and by other grants and contracts.
Ne is a one-valence electron, lithium-like ion with a relatively simple energy-term structure, however, a detailed compilation of accurate level energies and line wavelengths, which is a prerequisite of any quantitative work, became available only recently (Kramida & Buchet-Poulizac 2006). Our model atmosphere and spectrum synthesis calculations predict the presence of Ne lines in other wavelength regions, too. As a surprising result, we find that all previous identifications of ultra-high ionization (i.e., non-thermally excited) oxygen lines (O ) in UV and optical spectra of PG1159 stars, and the hottest known DO white dwarf, are wrong. We find that, instead, these lines are due to photospheric Ne . The earlier identification of these lines as O was motivated by the supposed occurrence of the same lines in early-type Wolf-Rayet central stars (i.e., spectral type [WCE]) and it was argued that they are formed in shocked wind regions. The results presented here provide evidence that also in [WCE] stars these, as well as other lines that were assigned to ultrahigh-ionized C and O, do probably stem from thermally excited neon.
We present our observations and line identifications in Sect. 2 and describe our modeling in Sect. 3. The results from line profile fits to individual objects are presented in Sect. 4. Implications for [WCE] stars are discussed in Sect. 5 and we conclude with Sect. 6.
Observations and Ne line identifications
FUSE observations and data reduction for most of our program stars were described in our previous work (Werner et al. 2004a(Werner et al. , 2004b Several lines of the n = 5 → 6 transition are detected, as labeled in detail at the uppermost spectrum. Overplotted are computed profiles with T eff and log g, as indicated. Middle and right panels: Optical spectral regions where we identified the Ne n = 8 → 9 and n = 9 → 10 transitions. All observed spectra are shifted such that the photospheric lines appear at their rest wavelengths.
spectrograph. Descriptions of the FUSE instrument, and channel alignment and wavelength calibration issues, are given by Moos et al. (2000) and Sahnow et al. (2000). The FUSE spectra cover the wavelength range from the Lyman edge up to 1187 Å with a spectral resolution of 0.05 Å. For spectra that are too faint to permit co-alignment of individual exposures, the spectral resolution may be degraded to 0.1 Å. The S/N ratio in the LiF2a spectra in the vicinity of the Ne lines being studied here ranges from a low of 17:1 per 0.05 Å resolution element for Longmore 4 to 70:1 per resolution element for RX J2117.1+3412; it was greater than 45:1 for the other objects. The S/N ratio in the LiF1b spectra at these wavelengths was typically ∼30% lower. UV spectra of KPD 0005+5106 and RX J2117.1+3412, taken with the FOS and GHRS spectro-graphs aboard the Hubble Space Telescope, were retrieved from the MAST archive. High-resolution optical spectra of K1-16, RX J2117.1+3412, KPD 0005+5106, and H1504+65 were obtained at the Keck observatory and the HIRES spectrograph. For details on data reduction see Zuckerman & Reid (1998). Spectra of NGC 246 and Longmore 4 were obtained with the ESO Very Large Telescope and the UVES spectrograph, in the framework of the SPY project (Napiwotzki et al. 2003). The starting point of our analysis was the identification of two strong and broad absorption features of Ne at λ = 1162 − 1165 Å in the FUSE spectra of five PG1159 stars and the DO white dwarf KPD 0005+5106 (left panel of Fig. 1). They are blends of several n = 5 → 6 lines with high angular quantum number l. Ne lines are not detected in the FUSE spec- tra of any other PG1159 star, indicating that the minimum T eff for exhibiting these lines is around 150 000 K (this limit will be assessed more closely below). In addition to these lines, several weaker Ne lines are detectable in the FUSE spectra of some objects, particularly in RX J2117.1+3412 (panels a)-c) in Fig. 2). These are low-l n = 5 → 6 lines and an isolated n = 6 → 8 line. The strongest, high-l, n = 6 → 7 transitions are located at λ 1932 Å. We found only one archival spectrum covering this wavelength position, namely an HST/FOS observation of KPD 0005+5106. An emission feature is barely detectable at this location (panel d) of Fig. 2), formerly attributed to O n = 6 → 7 (Sion & Downes 1992). We note that the line positions for O and Ne between levels with these high principal quantum numbers become indistinguishable. Both ions have the same core charge but their valence electrons are single 1s and 2s electrons, respectively. Based on our model computations described below, we attribute the previously discovered emission features at λλ 2977, 4340, and 6068 Å (Figs. 2 and 1) to the n = 7 → 8, n = 8 → 9, and n = 9 → 10 transitions of Ne , respectively. They, too, were previously thought to stem from O Werner et al. 1996). The λ 6068 Å emission line was also detected in the very hot PG1159 stars Longmore 3 and HE 1429−1209 (Werner et al. 1994(Werner et al. , 2004a. Table 2 summarizes all Ne lines that were detected in any of the examined objects. The NIST 1 wavelengths coincide with the photospheric rest wavelengths, with the exception of the 5p → 6d, 5 f → 6d, and 6p → 8d transitions. Their observed wavelengths are smaller by 0.16, 0.09, and 0.07 Å, respectively. This is corrected for in panels b) and c) of Fig. 2 Table 2.
and in
We note that we have not detected any Ne lines in the hottest (T eff ≈100 000 K) hydrogen-rich central stars or DA white dwarfs from which FUSE spectra are available. The reason is that their temperature is not sufficiently high (all have T eff <150 000 K). The same holds for all hot DO white dwarfs besides KPD 0005+5106. The possible presence of Ne lines in the FUSE spectra of [WCE] stars is discussed below (Sect. 5).
Model atmospheres and neon line formation calculations
We use a grid of line-blanketed non-LTE model atmospheres, which is described in detail in Werner et al. (2004a). In essence, the models include the main photospheric constituents, namely, 4 K. Werner et al.: Ne in the hottest hydrogen-deficient (pre-) white dwarfs He, C, O, and Ne. The neon model atom consists of the ionization stages Ne -. Of particular importance for the investigations presented in this paper is the Ne ion. It consists of 5 NLTE levels and 6 lines, which is sufficient for the atmospheric model structure computation, but considerable extensions were required in order to be able to compute profiles of the identified highly excited lines. Since we aimed at the computation of lines involving the n = 10 levels (the n = 9 → 10 emission at 6068 Å), we included all 77 levels up to n = 12 with all 511 line transitions between them. In addition, levels up to n = 14 are included as LTE levels. With this extended Ne model ion, we performed line-formation iterations for neon, i.e., keeping the atmospheric structure fixed. For the final spectrum synthesis, fine-structure splitting of levels and lines was considered, if possible. Energies for fine-structure splitting are not available for all levels. In particular, they are only partly available for levels involved in the n = 5 → 6 absorption lines at λ 1162 − 1165 Å. As a consequence, the detailed structure of the computed line profile cores within these absorption troughs cannot exactly match the observations and, indeed, a close inspection of Fig. 1 reveals this problem. Level energies were taken from NIST (Kramida & Buchet-Poulizac 2006). Oscillator strengths were taken from the Opacity (Seaton et al. 1994) and IRON (Hummer et al. 1993) Projects databases (TIPTOPbase 2 ), which are, however, not complete for all transitions of our model atom. For lines involving levels with n = 11 and 12, we extrapolated oscillator strengths from transitions to lower quantum numbers. For a lithium-like ion this appears reasonably accurate; the estimates are probably within a 10% error. Photoionization cross-sections are taken from the Opacity Project database when available or, otherwise, computed in hydrogen-like approximation. Electron collisional rates were calculated with usual approximation formulae. The Ne model atoms that were used for this analysis have been developed in the framework of the German Astrophysical Virtual Observatory (GAVO 3 ) project and are provided within the Tübingen Model-Atom Database TMAD 4 . The treatment of Stark broadening of Ne lines poses a severe problem. We employ an approximate formula to compute Stark widths, which we use routinely to compute profiles of lines between highly excited levels of lithium-like ions (C , O ; Werner et al. 1991), and which accounts for linear Stark broadening. While this formula gave reasonable results for the other Li-isoelectronic ions in our past work, we found that the n = 5 → 6 absorption lines in the FUSE spectral range came out much too narrow (by about a factor of three) compared to all of the observations. Interestingly, a similar notorious problem is encountered in the interpretation of Ne line widths of laboratory spectra. Measurements of the 3s → 3p transition at 2821 Å (Glenzer et al. 1992) display line profiles that are roughly a factor of two wider than predicted by theory, i.e., the Z −2 scaling of the line width, as predicted by the impact theory, significantly deviates from observations. Even today, this discrepancy is not resolved, despite of intense efforts with both improved experiments and quantum-mechanical calculations (Hegazy et al. 2003;Griem & Ralchenko 2006). Our finding from stellar spectra indirectly suggests that the line widths measured in laboratory are in fact correct, and that instead, the line-broadening theory is still inaccurate. It is necessary to note, however, that we observe different lines of Ne . Fig. 3. The Ne lines are strongly sensitive to T eff and log g. Generally, they become stronger with increasing T eff and decreasing log g. Top panel: T eff -sensitivity in DO models with log g =6.5. The lines disappear if T eff is below 160 000 K.
Middle panel: T eff -sensitivity in PG1159 models with lower gravity (log g =5.7). In this case, the lines are detectable down to T eff =140 000 K. Bottom panel: log g sensitivity in PG1159 models at T eff =160 000 K. The gravity must be sufficiently low, otherwise the Ne lines are not detectable. The numbers in the lower right corner of the panels give the model abundances of He/C/O/Ne in % mass fraction.
To cope with this problem we formally reduced the ionic core charge of Ne in our line broadening formula from Z = 8 to Z = 2. This yields widths of the n = 5 → 6 lines that coincide with the observations. Of course, this procedure is rather unsatisfactory and would prohibit an accurate abundance determination from these lines. In addition, a close inspection of the computed profiles for the optical lines (i.e., the higher-n transitions, Fig. 1) reveals that they become too broad with our crude Z-reduction. Fortunately, the neon abundance of all PG1159 stars discussed here is known to be of the order 2% (Werner et al. 2004a). The focus of our paper is on the extreme temperature sensitivity of the Ne lines and line broadening is a subordinate effect only.
Results
We now compare our computed Ne line profiles to the observations (Fig. 1). It is instructive to look at Fig. 3, in which we demonstrate that the strengths of the λλ 1162 − 1165 Å absorption lines strongly depend on T eff and log g. The minimum T eff for a line detection is a strong function of gravity, ranging from about 160 000 K at log g =6.5 to 140 000 K at log g =5.7. The mere detection of these Ne lines thus provides a strict lower limit to T eff . Generally, the overall characteristics of the Ne lines are reproduced by the models. We find that the UV lines are in absorption, while the optical lines are in emission.
Let us discuss in some detail each of the six objects in which we detected the UV Ne lines, particularly those Fig. 4. The He lines in KPD 0005+5106 are compared with two different models. Left panel: T eff =200 000 K, log g =6.5. Right panel: T eff =120 000 K, log g =7.0. The hotter model yields a much better fit. In particular, only this model matches the emission cores in the λλ 6560 and 4686 Å lines. The emission line at λ 4340 Å is due to Ne and is only exhibited in the hotter model. cases in which contradictions to T eff determinations from previous work appear. This mainly affects the DO white dwarf KPD 0005+5106, whereas for the PG1159 stars the Ne line features can be matched with model parameters that are in good or acceptable agreement with previous results.
PG1159 stars
Models for individual objects were computed with element abundances as given in Werner et al. (2005). As already mentioned, the neon abundance is kept fixed at 2%. We started with values for T eff and log g also taken from this reference; typical uncertainties are 10% and 0.5 dex, respectively.
K1-16
The previously determined parameters are T eff =140 000 K and log g =6.4. Clearly, a model with these values shows no Ne lines at all. We find an acceptable fit only at T eff and log g values that are slightly beyond the error bars, namely T eff =160 000 K and log g =5.8.
RX J2117.1+3412
We find a good fit to the Ne lines with models close to the literature values (T eff =170 000 K, log g =6.0). At these parameter values, the Ne absorption lines are strongest. This explains why we can identify a number of additional weak Ne lines in the FUSE spectrum (Fig. 2).
NGC 246 A good fit to the Ne lines is obtained at the literature values (T eff =150 000 K, log g =5.7). Note that NGC 246 is a fast rotator and the computed profiles were broadened with 70 km/s (Rauch & Werner 1997).
Longmore 4 The strong Ne lines in the FUSE spectrum suggest that this star is as hot as RX J2117.1+3412. We find a good fit with a T eff =170 000 K, log g =6.0 model. Therefore, Longmore 4 is significantly hotter than previously thought (T eff =120 000 K, log g =5.5). This result is, however, not surprising because we have already found independent hints that the temperature is at least about 150 000 K (Werner et al. 2004a). Note that Longmore 4 exhibited much more prominent optical Ne lines during its observed "outburst" and change of spectral type from PG1159 to [WCE] .
H1504+65 The Ne lines in the FUSE spectrum are shallow and broad. This is a consequence of the high gravity (8.0). The profiles confirm the extremely high T eff (200 000 K).
The DO white dwarf KPD 0005+5106
An analysis of optical and HST UV spectra of KPD 0005+5106 gave the result T eff =120 000 K and log g =7 (Werner et al. 1994). Although being the hottest known DO white dwarf, the identification of Ne lines is a big surprise. Figure 3 shows that at this relatively high gravity the temperature must be around 180 000 K. We find that models with T eff =180 000-200 000 K and log g =6.5-7.0 give the best fit to the Ne lines. Interestingly, this is not in contradiction with the He lines. We even find that a model with T eff =200 000 K and log g =6.5 gives better He line fits than the cooler 120 000 K model. This is shown in Fig. 4. To be more specific, the hot model is able to match the height of the central emissions reversals in the λλ 4686 and 6560 Å line cores. At the same time, the λ 1640 Å line is matched very well. We conclude that KPD 0005+5106 is significantly hotter than previously thought.
Unlike for the PG1159 stars, the neon abundance in KPD 0005+5106 is not known from previous analyses. The usual diagnostic lines are Ne λλ 973 and 3644 Å. The FUSE spectrum of KPD 0005+5106 is contaminated by interstellar H 2 absorption, making the detection of Ne λ 973 Å impossible. In a medium-resolution optical spectrum published by Werner et al. (1994), no line feature is detected at λ 3644 Å. We have verified that this is compatible with all our models presented here in the T eff =120 000-200 000 K range.
We assumed that the Ne abundance is determined by radiative levitation. The hottest model presented by Chayer et al. (1995) has T eff =100 000 K with log g =7.5 and predicts log(Ne/He) = −4 by number. It is difficult to extrapolate this to a significantly higher T eff but the tendency is that the Ne abundance increases with T eff . A further increase can be expected because of the smaller gravity. Although uncertain, it is not unreasonable that we set Ne=1% (by mass), that is, log(Ne/He) = −2.7 by number. We stress that the high T eff derived from the Ne lines is hardly affected by this assumption. In addition, the He line profiles are essentially independent of the neon abundance.
Implications for ultrahigh-ionisation emission line identifications in [WCE] stars
Since the supposed identification of several optical emission lines in the hot [WC] star Sand 3 as being due to ultrahigh ionised (i.e. non-thermally excited) C and O (C and O -) (Barlow et al. 1980), a number of authors claimed the identification of these features in other early-type [WC] stars. In the light of our results, we propose that most, and probably all, of these features stem from thermally excited ionisation stages of neon (Ne -), carbon and oxygen. Table 3 summarizes our proposed identifications. We give qualitative arguments for this, but detailed NLTE modeling with expanding model atmospheres will be necessary for a quantitative confirmation. A broad emission feature in Sand 3 and other [WCE]s at λ 3893 Å was attributed to the n = 7 → 8 transition of Table 3. List of lines for which we propose photospheric identifications as opposed to previously thought identifications as non-photospheric ultrahigh-ionisation features in PG1159 stars, [WCE] stars, and in the DO KPD 0005+5106. The O and N emission lines were identified before but thought to be blended by the ultrahigh ionisation features.
Wavelength / Å Old ultrahigh- New photospheric ionisation identification identification 1932 O n = 7 → 8 Ne n = 7 → 8 2977 O n = 6 → 7 Ne n = 6 → 7 3893 O n = 7 → 8 Ne 3p 3 P o → 3d 3 D plus Ne n = 7 → 8 4340 O n = 8 → 9 Ne n = 8 → 9 4500 C n = 8 → 10 O n = 8 → 10 4555 O n = 9 → 11 Ne n = 9 → 11 4945 C n = 6 → 7 N n = 6 → 7 5290 C n = 7 → 8 O n = 7 → 8 5665 O n = 8 → 9 Ne n = 8 → 9 6068
O n = 9 → 10 Ne n = 9 → 10 O . We propose that it originates from a prominent Ne multiplet recently identified in the hottest PG1159 central stars (Werner et al. 2004a), which has its strongest component located at λ 3892 Å. In addition, the Ne n = 7 → 8 transition contributes. The emission at λ 4340 Å is probably from Ne , not O . Emission lines at λλ 4555 Å and 5665 Å were assigned to the O transitions n = 9 → 11 and n = 8 → 9, respectively. We suggest that they are lines between high Rydberg states of Ne , with the same principal quantum numbers (level energies are given in NIST and Lapierre & Knystautas 1999). This is supported by our discovery of two high-l n = 8 → 9 emission lines of Ne at λ 5665 Å in some of our program stars (Fig. 5). Finally, the strong emission feature at λ 4945 Å was assigned to C n = 7 → 6. We think that it is the respective N transition, as in the case of KPD 0005+5106. Nitrogen is definitely present in Sand 3, because the 3s → 3p emission doublet at λλ 4602, 4620 Å is present, as well as a prominent P Cygni profile of the N V resonance line. Thus, the entire optical emission line spectrum of Sand 3 and other [WCE]s can be explained without invoking ultrahigh-ionisation features. Based on observations with the International Ultraviolet Explorer (IUE), several papers appeared in the literature claiming the presence of ultrahigh ionisation lines in the UV spectra of [WCE] stars. In Sand 3, as well as in NGC 5315 and NGC 6905, Feibelman (1996a, b) discovered the λλ 1932 Å and 2977 Å emission features that we have discussed above, and he also assigned them to O . As in KPD 0005+5106 and RX J2117.1+3412 (Fig. 2), they probably are from Ne .
Feibelman (1996b) even claimed the existence of an "O sequence" of planetary-nebulae nuclei, mainly based on the supposed O - identifications in PG1159 and [WCE] stars. Obviously, that idea must now be discarded. This is further corroborated by the fact that we detected the Ne lines at λ 1162 − 1165 Å in a [WCE] star too, namely in NGC 2371 (Fig. 6). The line strengths suggest T eff ≈ 150 000 K, which is, considering error limits, in good agreement with the result of a detailed analysis of FUSE and IUE spectra (135 000 Fig. 5. First identification of two high-l n = 8 → 9 emission lines of Ne (!). A strong and broad emission feature is seen in [WCE] stars at this location. We propose that it is due to these Ne transitions and not due to ultrahigh ionized O . The labels denote the theoretical line positions using energy levels that are not precisely known.
Summary and conclusions
We have identified Ne absorption lines in FUSE spectra of PG1159 stars and a DO white dwarf. Line profile fits confirm that the PG1159 stars are the very hottest members of their spectral class (T eff ≥150 000 K). We have shown that two well-known emission lines in the optical spectra of these stars are also from Ne and not, as previously thought, from O . While Ne is thermally excited in the hot photospheres, the existence of O would require temperatures of the order of one million K, thus, an unknown process (e.g., shock zones in the wind) was invoked to explain these emission lines. This is no longer necessary. We argue that these, and probably all, emission lines in earlytype [WC] central stars that are usually assigned to ultrahighionisation stages (O -, C ) originate from Ne - and other thermally excited species.
The discovery of Ne lines in the central star Longmore 4 suggests that its T eff is distinctively higher than previously thought (170 000 K instead of 120 000 K). To a smaller extent, the same tendency is seen for the PG1159 star K1-16 and the [WCE] star NGC 2371. This calls for a re-analysis of the complete spectra of these stars to see how the fit to other line features can be reconciled. The computation of model grids with varying parameters T eff , log g, and abundances of the most important elements (He, C, O, Ne) will be necessary. Since the stars display prominent P Cygni line profiles, it could be advantageous to use expanding atmosphere models with the available spectral data. As already mentioned, the neon abundance cannot be determined from the newly discovered Ne lines because of the uncertainties with the line broadening theory. Instead, the strong P Cygni profile of the Ne λ 973 Å line exhibited by these stars can be used as a sensitive tool (Bianchi & Herald 2007). Again, detailed parameter studies must be performed to find which neon abundance and which mass-loss rate yield good fits to this Ne line profile, while at the same time determining the T eff of the model that is high enough to produce detectable Ne lines.
A surprising result of our investigation is the identification of Ne lines in the hottest known DO white dwarf, KPD 0005+5106. We conclude that its temperature is close to T eff =200 000 K and, hence, that it is significantly hotter than hitherto thought (120 000 K). As in the case of the PG1159 stars, the optical Ne emission lines were previously assigned to ultrahigh-ionized O . The announcement of the discovery of a relatively soft X-ray emitting corona about KPD 0005+5106 seemed to support this assignment (Fleming et al. 1993). However, the analysis of a Chandra spectrum exhibiting flux in the range 20-80 Å proved that the soft X-rays stem from the photosphere of KPD 0005+5106 (Drake & Werner 2005). The deposition of the corona is now in accordance with the deposition of the ultrahigh-ionisation lines. A comprehensive re-analysis of all available data (Chandra, FUSE, HST, Keck) is required to tightly constrain the atmospheric parameters and metal abundances.
It remains to be seen how the extremely high effective temperature derived in our work relates to the observed hard X-ray emission at 1 keV (12 Å) from KPD 0005+5106 (O'Dwyer et al. 2003). It might be that this emission is also of photospheric origin. After all, the entire spectral properties of KPD 0005+5106 could be deciphered as thermal photospheric radiation. In any case, the star has turned out to be by far the hottest known DO white dwarf.
Fig. 2 .
2Details of other spectral regions of the PG1159-type central star RX J2117.1+3412 and the DO white dwarf KPD 0005+5106 displaying Ne lines. Panels a)-c) show n = 5 → 6 and n = 6 → 8 transitions (the blending C line in panel a) is not included in the model). Panel d) shows a barely detectable n = 6 → 7 emission feature in the HST/FOS spectrum of KPD 0005+5106, which was previously assigned to O . Panel e) displays HST/GHRS spectra with a Ne n = 7 → 8 emission line that was also thought to stem from O . The adjacent emission feature in KPD 0005+5106 is from nitrogen; it is not included in the model.
). Table 1 lists our program stars with references to results from previous analyses. The FUSE instrument consists of four coaligned telescopes, each with a prime-focus 2 K. Werner et al.: Ne in the hottest hydrogen-deficient (pre-) white dwarfsFig. 1. Left panel: Identification of Ne lines in the FUSE spectra of PG1159 stars and the DO white dwarf KPD 0005+5106.K1-16
160000/5.8
RX J2117.1+3412
170000/5.8
NGC 246
150000/5.7
Longmore 4
170000/6.0
H1504+65
200000/8.0
KPD0005+5106
200000/7.0
5d-6f
5f-6g
5g-6h
5g-6f 5f-6d
Ne VIII
0
1
2
3
4
5
6
1160 1161 1162 1163 1164 1165 1166 1167
relative flux
K1-16
160000/5.8
RX J2117.1+3412
170000/6.0
NGC 246
150000/5.7
Longmore 4
170000/6.0
H1504+65
200000/8.0
KPD0005+5106
200000/6.5
8k-9l etc.
(He II)
4335
4340
4345
wavelength / A
o
K1-16
160000/5.8
RX J2117.1+3412
170000/6.0
NGC 246
150000/5.7
Longmore 4
170000/6.0
H1504+65
200000/8.0
KPD0005+5106
200000/6.5
9l-10m etc.
6065
6070
Table 1 .
1Effectivetemperature and surface gravity of the
program stars as taken from the literature. As described in
the text, the discovery of Ne lines suggests that T eff for
KPD 0005+5106 and Longmore 4 must be significantly higher.
Object
Spectral
T eff
log g
Reference
Type
[K]
(cgs)
H1504+65
PG1159 200 000
8.0
A
RX J2117.1+3412 PG1159 170 000
6.0
B
NGC 246
PG1159 150 000
5.7
C
K1-16
PG1159 140 000
6.4
D
NGC 2371
[WCE]
135 000
5.5
E
Longmore 4
PG1159 120 000
5.5
F
KPD 0005+5106
DO
120 000
7.0
G
References: A: Werner et al. 2004b, B: Werner et al. 1996b, C: Rauch
& Werner 1997, D: Werner et al. 1992, E: Herald & Bianchi 2004, F:
Rauch & Werner 1997, G: Werner et al. 1994
Table 2 .
2Ne lines identified in our program stars.Wavelengths are computed from NIST level energies (except for
the 5p → 6d, 5 f → 6d, and 6p → 8d transitions, see text) and
given in vacuum or air for λ smaller or larger than 3000 Å, re-
spectively. The presence of the feature at λ 1932 Å is uncertain.
Wavelength / Å
Transition
1060.36, 1061.03
5s → 6p
1129.02, 1130.12, 1130.35 5p → 6d
1150.12, 1150.81, 1150.92 6p → 8d
1162.24, 1162.67
5d → 6 f
1164.54, 1164.75
5 f → 6g
1164.88
5g → 6h
1165.94, 1166.15, 1166.18 5 f → 6d
1170.04, 1170.29
6d → 8 f
(1932.04)
6h → 7i etc.
2976.75
7i → 8k etc.
4340.77
8k → 9l etc.
6068.63
9l → 10m etc.
2 http://vizier.u-strasbg.fr/topbase/ 3 http://www.g-vo.org 4 http://astro.uni-tuebingen.de/ ∼ rauch/TMAD/TMAD.htmlT eff =160 000 K
T eff =180 000 K
T eff =200 000 K
at log g=6.5
Ne VIII
99/0.3/0.06/1
0.5
1.0
T eff =130 000 K
T eff =140 000 K
T eff =150 000 K
at log g=5.7
Ne VIII
62/30/6/2
0.5
1.0
relative flux
log g=6.6
log g=6.0
log g=5.8
at T eff =160 000 K
Ne VIII
38/54/6/2
0.5
1.0
1160 1161 1162 1163 1164 1165 1166 1167
wavelength / A
o
K; Herald & Bianchi 2004). NGC 2371 was classified as [WC4] by Acker & Neiner (2003). These authors have based a new spectral classifi-K1-16
RX J2117.1+3412
NGC 246
KPD0005+5106
Ne VII 8i-9k, 8k-9l
0
1
5655
5660
5665
5670
5675
λ / A
o
relative flux
Fig. 6. Identification of Ne lines in the [WCE] central star NGC 2371 (top) and comparison to the PG1159-type central star Longmore 4 (bottom). cation system for the hottest [WC]s on the occurrence of O - lines in the optical spectrum. Our identification as Ne - lines means that the empirical classification criteria remain essentially correct because only the hottest [WC]s are able to show these lines.NGC 2371
150000/5.7
Longmore 4
170000/6.0
Ne VIII
0
1
1160 1161 1162 1163 1164 1165 1166 1167
wavelength / A
o
relative flux
http://physics.nist.gov/ PhysRefData/ASD/index.html
Acknowledgements. We are grateful to the referee, Luciana Bianchi, for a careful reading of the manuscript and for her constructive comments. We thank Uli Heber and Ralf Napiwotzki for putting their Keck and VLT spectra at our disposal. We thank A.E. Kramida
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|
[] |
[
"Critical points in the CP N −1 model",
"Critical points in the CP N −1 model"
] |
[
"Youness Diouane \nSISSA and INFN -Via Bonomea 265\n34136TriesteItaly\n\nICTP\nStrada Costiera 1134151TriesteItaly\n",
"Noel Lamsen \nSISSA and INFN -Via Bonomea 265\n34136TriesteItaly\n",
"Gesualdo Delfino \nSISSA and INFN -Via Bonomea 265\n34136TriesteItaly\n"
] |
[
"SISSA and INFN -Via Bonomea 265\n34136TriesteItaly",
"ICTP\nStrada Costiera 1134151TriesteItaly",
"SISSA and INFN -Via Bonomea 265\n34136TriesteItaly",
"SISSA and INFN -Via Bonomea 265\n34136TriesteItaly"
] |
[] |
We use scale invariant scattering theory to obtain the exact equations determining the renormalization group fixed points of the two-dimensional CP N −1 model, for N real. Also due to special degeneracies at N = 2 and 3, the space of solutions for N ≥ 2 reduces to that of the O(N 2 − 1) model, and accounts for a zero temperature critical point. For N < 2 the space of solutions becomes larger than that of the O(N 2 − 1) model, with the appearance of new branches of fixed points relevant for criticality in gases of intersecting loops.
|
10.1088/1742-5468/ac4983
|
[
"https://arxiv.org/pdf/2110.04014v1.pdf"
] | 246,774,709 |
2110.04014
|
671a1cc0ce12b7a88597db838be1204000777da7
|
Critical points in the CP N −1 model
8 Oct 2021
Youness Diouane
SISSA and INFN -Via Bonomea 265
34136TriesteItaly
ICTP
Strada Costiera 1134151TriesteItaly
Noel Lamsen
SISSA and INFN -Via Bonomea 265
34136TriesteItaly
Gesualdo Delfino
SISSA and INFN -Via Bonomea 265
34136TriesteItaly
Critical points in the CP N −1 model
8 Oct 2021
We use scale invariant scattering theory to obtain the exact equations determining the renormalization group fixed points of the two-dimensional CP N −1 model, for N real. Also due to special degeneracies at N = 2 and 3, the space of solutions for N ≥ 2 reduces to that of the O(N 2 − 1) model, and accounts for a zero temperature critical point. For N < 2 the space of solutions becomes larger than that of the O(N 2 − 1) model, with the appearance of new branches of fixed points relevant for criticality in gases of intersecting loops.
Introduction
Determining if and how an additional local symmetry affects the universality class of a statistical model is a relevant issue in the theory of critical phenomena. A basic example is provided by the RP N −1 model, in which N -component spin variables at each lattice site interact through an Hamiltonian invariant under global O(N ) rotations and local spin reversals. The local symmetry makes the difference with the usual O(N ) model and amounts to the head-tail symmetry characteristic of liquid crystals [1]. In three dimensions, the weak first order transition observed in numerical simulations of the ferromagnetic model [2] is consistent with the mean field scenario [1]. On the other hand, in the two-dimensional case -the one we focus on -fluctuations are stronger and minimize the reliability of mean field predictions (see e.g. [3]), as illustrated by the phase transition of the three-state Potts model, which becomes continuous on planar lattices [4]. For the RP N −1 model, the absence of spontaneous breaking of continuous symmetry in two dimensions [5] generically suggests that criticality is limited to zero temperature, and numerical studies for T → 0 show a fast growth of the correlation length which makes particularly hard to reach the asymptotic limit and draw conclusions about universality classes [6,7,8,9,10,11,12]. On the other hand, the possibility of finite temperature topological transitions similar to the Berezinskii-Kosterlitz-Thouless (BKT) one [13] -which should definitely occur for RP 1 ∼ O(2) -and mediated by "disclination" defects [14,15] has also been debated in numerical studies [16,17,18,19,20,21,22,23,24]. While two-dimensional criticality has allowed for an impressive amount of exact solutions thanks to lattice integrability [25,26] and conformal field theory [27,28], models with local symmetries traditionally remained outside the range of application of these methods. Recently, however, we showed in [29,30] how the renormalization group fixed points of the RP N −1 model can be accessed in an exact way in the scale invariant scattering framework [31], which implements in the basis of particle excitations the infinite-dimensional conformal symmetry characteristic of critical points in two dimensions and has provided in the last years new results for pure and disordered systems [32,33,34,35,36,37,38] (see [39] for a review). We found that only O(N (N + 1)/2 − 1) fixed points exist for 1 N ≥ 3 and account for zero temperature criticality, while a line of fixed points yielding a BKT transition exists only for N = 2. Our framework automatically yields results for N real, corresponding to the known fact that lattice models such as O(N ) admit loop gas formulations [3,26] in which N plays the role of loop fugacity and does not need to be an integer. For RP N −1 we found new branches of fixed points emerging below N = 2.24421.. [30].
In this paper we consider the basic lattice model with a continuous -U (1) -local symmetry, namely the CP N −1 model realized in terms of complex N -component spin vectors at lattice sites. In two dimensions, this model has been studied in the high energy context (since [40,41,42]) for the similarities -in particular asymptotic freedom -which it shares with quantum chromodynamics, in statistical mechanics in relation with loop gases [43], and in condensed matter in relation with quantum antiferromagnets (see e.g. [44]). The remarks that we made above for RP N −1 concerning the continuous nature of the symmetry, zero temperature criticality, the possibility of topological transitions, and the absence of previous exact results, apply to CP N −1 as well. We use scale invariant scattering to determine the exact fixed point equations for CP N −1 symmetry, and find that the only solutions for N ≥ 2 are of O(N 2 − 1) type, also due to a special degeneracy emerging for N = 2, 3. This is consistent, in particular, with the known correspondence CP 1 ∼ O(3). Our results again extend to real values of N , allowing us to see that quasi-long-range order and a BKT transition occur only for N = ± √ 3, where O(N 2 − 1) = O (2). Also here the space of solutions enlarges and new branches of fixed points appear below a threshold value of the symmetry parameter, which in this case turns out to be N = 2.
The paper is organized as follows. We recall the generalities of scale invariant scattering in section 2 and illustrate its application to the O(M ) model in section 3. Section 4 is devoted to the derivation of the fixed point equations for the CP N −1 model, whose space of solutions is analyzed in section 5. The results for the RP N −1 model are briefly recalled in section 6 for comparison, while the last section contains some final remarks. Two appendices complete the paper.
Generalities of scale invariant scattering
We begin our discussion by briefly recalling the generalities of scale invariant scattering [31], referring the reader to [39] for a review. The method relies on the fact that the continuum limit of a critical statistical system in two dimensions is described by a Euclidean field theory, which is the continuation to imaginary time of a quantum field theory defined in one space and one time dimension and exhibiting conformal invariance. In the quantum theory massless particles describe the excitations above the ground state (vacuum) and correspond to the fluctuation modes of the statistical system. Since in two dimensions conformal symmetry possesses infinitely many generators [28], the scattering processes of the particles are subject to an infinite number of conservation laws, which force the final state to be kinematically identical to the initial one (completely elastic scattering). Moreover, scale invariance implies that the scattering amplitude of a two-particle process is a constant, namely does not depend on the center of mass energy, which is the only relativistic invariant and is dimensionful.
These features are specific to two-dimensional criticality and substantially simplify the unitarity and crossing equations [45] that generally apply to relativistic scattering 2 . Let us denote by µ = 1, 2, . . . k the particle species, by S the scattering operator, and by S ρσ µν = ρσ|S|µν the scattering amplitude for a process with particles µ and ν in the initial state and particles ρ and σ in the final state (figure 1). Then the unitarity and crossing equations take the form [31] λ,τ S λτ µν S ρσ
λτ * = δ µρ δ νσ ,(1)S ρσ µν = S ρν µσ * ,(2)
respectively 3 . Invariance under charge conjugation, time reversal and spatial inversion provides the relations S ρσ µν = Sρσ µν = S σρ νµ = S µν ρσ .
Fixed points of the O(M) model
Before turning to the CP N −1 model it will be useful to briefly recall how scale invariant scattering applies to the O(M ) model [31,36], which is defined on the lattice by the Hamiltonian The O(M ) tensorial structure involved in the scattering of a particle a with a particle b is preserved once the scattering matrix is written as S cd ab = S 1 δ ab δ cd + S 2 δ ac δ bd + S 3 δ ad δ bc ,
H O(M ) = −J i,j s i · s j ,(4)Solution M ρ 1 ρ 2 cos φ I ± (−∞, ∞) 0 ±1 - II ± [−2, 2] 1 0 ± 1 2 √ 2 − M III ± 2 [0, 1] ± 1 − ρ 2 1 0
with the amplitudes S 1 , S 2 and S 3 accounting for annihilation, transmission and reflection, respectively (figure 2). The crossing equations (2) then yield
S 1 = S * 3 ≡ ρ 1 e iφ ,(6)S 2 = S * 2 ≡ ρ 2 ,(7)
and lead to the parametrization of the amplitudes in terms of ρ 2 and φ real, and ρ 1 ≥ 0. It follows that the unitarity equations (1) can be written in the form
ρ 2 1 + ρ 2 2 = 1 ,(8)ρ 1 ρ 2 cos φ = 0 ,(9)
M ρ 2 1 + 2ρ 1 ρ 2 cos φ + 2ρ 2 1 cos 2φ = 0 . Table 1 contains the solutions of equations (8)-(10) [31,36] (also shown in figure 3), which yield the renormalization group fixed points with O(M ) symmetry. While a detailed discussion of the solutions is given in [36], here we recall some basic features relevant for the remainder of the paper. The solutions II ± are characterized by nonintersecting particle trajectories (namely [3,26]. A particularly relevant feature of the loop formulation is that it implements on the lattice the continuation to noninteger values of M that we see realized by equations (8)-(10) directly in the continuum. The statistical properties of self-avoiding walks correspond to the limit M → 0 [47]. The correspondence between nonintersecting loop paths and nonintersecting particle trajectories was originally observed in [48] for the off-critical case. The fact that solutions III ± , defined only for M = 2, possess ρ 1 as a free parameter immediately identifies them with the line of fixed points at the origin of the BKT transition in the XY model [3,13]. III + and III − meet at ρ 1 = 1, the BKT transition point where the field driving the transition is marginal [31,36]; it is instead irrelevant along III + , so that this is the BKT phase in which correlations decay algebraically in the XY model (quasi-long-range order).
Finally, solutions I ± are purely transmissive (S 1 = S 3 = 0) and correspond to noninteracting bosons for S 2 = 1 and noninteracting fermions for S 2 = −1. I + describes zero temperature criticality in the nonlinear sigma model with reduced Hamiltonian
H SM = 1 T d 2 x (∇s) 2 , s 2 = 1 ,(11)
where s(x) replaces in the continuum the lattice variable s i . The sigma model with M > 2 describes the continuum limit of the O(M ) model in this range of M and is characterized by exponentially diverging correlation length and vanishing interaction for T → 0 (asymptotic freedom) [3,49]. The solution I − yields a realization of the symmetry in terms of M free fermions and is not relevant 4 for the critical behavior of the vector model (4) for generic M . The case M = 1, however, allows some observations that will be useful in the subsequent sections. The symmetry O(1) = Z 2 is that of the Ising model, which in two dimensions has a critical point described by a free neutral fermion [28]. The corresponding amplitude S 11 11 = −1 is of course realized by I − in the purely transmissive form S 11 11 = S 2 . On the other hand, it is also realized by II − in the form S 11 11 = S 1 + S 3 , as required by the fact that also the Ising partition function has a "geometrical" representation in terms of self-avoiding loops. This illustrates that a specific critical point may allow different diagrammatic realizations at the scattering level. Clearly, this is due to the fact that at N = 1 there is a single particle species, and transmission, reflection and annihilation are not physically distinguishable 5 . At the same time, some geometrical observables in the Ising model need to be computed in the limit N → 1 [50,51], and in this case solution II − provides the right analytic continuation.
Fixed point equations of the CP N −1 model
The CP N −1 lattice model is defined by the Hamiltonian
H CP N−1 = −J i,j |s i · s * j | 2 ,(12)
where s j is a N -component complex vector at site j satisfying s j · s * j = 1. The Hamiltonian (12) is invariant under global U (N ) transformations (s j → U s j , U ∈ U (N )) and site-dependent U (1) transformations (s j → e iα j s j , α j ∈ R). These symmetries are represented through the tensorial order parameter variable
Q ab i = s a i (s b i ) * − 1 N δ ab .(13)
The presence of an invariant linear in the order parameter components is excluded by the constraint s j · s * j = 1, which in turn makes Q ab i traceless. The implementation of scale invariant scattering for the two-dimensional CP N −1 model at criticality proceeds through steps analogous to those seen in the previous section for the vector model. We first of all observe that in the continuum limit the order parameter field is now the Hermitian tensor Q ab (x), which creates particles that we label by µ = ab, with a and b taking values from 1 to N . A state containing a particle ab transforms under the U (N ) symmetry as
|ab −→ |a ′ b ′ = a,b U a ′ ,a U * b ′ ,b |ab ,(14)
4 Notice that, due to the quadratic nature of the unitarity equations (1), solutions differing for a change of sign of all amplitudes are always simultaneously present. 5 In a relativistic scattering process only the initial and final states are observable [45]. Figure 4: Amplitudes entering (17). Time runs upwards so that a scattering amplitude S ef,gh ab,cd = ef, gh|S|ab, cd with particles ab and cd in the initial state and particles ef and gh in the final state transforms into
S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S 10 S 11S e ′ f ′ ,g ′ h ′ a ′ b ′ ,c ′ d ′ = a,b,c,d e,f,g,h U a ′ ,a U * b ′ ,b U c ′ ,c U * d ′ ,d U * e ′ ,e U f ′ ,f U * g ′ ,g U h ′ ,h S ef,gh ab,cd .(15)
Taking also into account the relations (3), which can now be written as S ef,gh ab,cd = S f e,hg ba,dc = S gh,ef cd,ab = S ab,cd ef,gh ,
U (N )-invariance corresponds to S ef,gh ab,cd = S 1 δ a,d δ b,c δ e,h δ f,g + S 2 δ a,e δ b,f δ c,g δ d,h + S 3 δ a,g δ b,h δ c,e δ d,f + S 4 (δ a,d δ b,f δ c,g δ e,h + δ b,c δ a,e δ d,h δ f,g ) + S 5 (δ b,c δ a,g δ d,f δ e,h + δ a,d δ b,h δ c,e δ f,g ) + S 6 (δ a,e δ b,h δ d,f δ c,g + δ b,f δ a,g δ c,e δ d,h ) + S 7 (δ a,b δ e,f δ c,g δ d,h + δ c,d δ g,h δ a,e δ b,f ) + S 8 (δ c,d δ e,f δ a,g δ b,h + δ a,b δ g,h δ c,e δ d,f ) + S 9 δ e,f (δ a,d δ b,h δ c,g + δ b,c δ a,g δ d,h ) + δ c,d (δ b,f δ a,g δ e,h + δ a,e δ b,h δ f,g ) δ a,b (δ d,f δ c,g δ e,h + δ c,e δ d,h δ f,g ) + δ g,h (δ a,d δ b,f δ c,e + δ b,c δ a,e δ d,f ) + S 10 δ a,b δ c,d δ e,f δ g,h + S 11 (δ a,b δ c,d δ e,h δ f,g + δ e,f δ g,h δ a,d δ b,c ) ,(16)
with amplitudes S 1 , . . . , S 11 depicted in figure 4. In this figure each incoming or outgoing particle has two terminals corresponding to its two indices, and a line connecting two indices corresponds to a Kronecker delta identifying them. Crossing symmetry (2) translates into
S ef,gh ab,cd = S ef,dc ab,hg * .(18)
The crossing equations for the amplitudes S i≤3 preserve the form (6) and (7), and we keep for these amplitudes the same parametrization in terms of ρ 1 , ρ 2 and φ. The crossing relations and the corresponding parametrizations for the remaining amplitudes are
S 4 = S * 6 ≡ ρ 4 e iθ ,(19)S 5 = S * 5 ≡ ρ 5 ,(20)S 7 = S * 7 ≡ ρ 7 ,(21)S 8 = S * 11 ≡ ρ 8 e iψ ,(22)S 9 = S * 9 ≡ ρ 9 ,(23)S 10 = S * 10 ≡ ρ 10 ,(24)
with ρ 5 , ρ 7 , ρ 9 , ρ 10 , θ and ψ real, and ρ 4 and ρ 8 nonnegative.
The unitarity condition (1) can be written as
N i,j=1 N k,l=1 S ij,kl ab,cd S ef,gh ij,kl * = δ a,e δ b,f δ c,g δ d,h ,(25)
and gives rise to the 11 independent equations
1 = ρ 2 1 + ρ 2 2 + 2ρ 2 4 ,(26)0 = 2ρ 1 ρ 2 cos φ + 2ρ 2 4 ,(27)0 = N 2 ρ 2 1 + 2ρ 2 1 cos 2φ + 2ρ 1 ρ 2 cos φ + 4N ρ 1 ρ 4 cos(θ − φ) + 4N ρ 1 ρ 5 cos φ + 2ρ 2 4 + 4ρ 4 ρ 5 cos θ + 2ρ 2 5 + 2N ρ 1 ρ 8 cos(ψ + φ) + 8ρ 1 ρ 9 cos φ + 4ρ 5 ρ 8 cos ψ + 8ρ 4 ρ 8 cos θ cos ψ + 8N ρ 8 ρ 9 cos ψ + N 2 ρ 2 8 + 8ρ 2 9 ,(28)0 = 2ρ 1 ρ 5 cos φ + 2ρ 2 ρ 4 cos θ + N ρ 2 4 + N ρ 2 5 + 8ρ 4 ρ 9 cos θ + 4ρ 5 ρ 9 + 2N ρ 2 9 ,(29)
0 = 2ρ 1 ρ 4 cos(θ + φ) + 2ρ 2 ρ 5 + 2N ρ 4 ρ 5 cos θ + 8ρ 4 ρ 9 cos θ + 4ρ 5 ρ 9 + 2N ρ 2 9 ,
0 = 2ρ 1 ρ 4 cos(θ − φ) + 2ρ 2 ρ 4 cos θ,(30)
0 = 2ρ 1 ρ 8 cos(ψ + φ) + 2ρ 2 ρ 7 + 4ρ 9 (ρ 4 cos θ + ρ 7 + ρ 8 cos ψ) + N (ρ 2 7 + ρ 2 8 + 2ρ 2 9 ),
0 = 2ρ 1 ρ 7 cos φ + 2ρ 2 ρ 8 cos ψ + 4ρ 9 (ρ 4 cos θ + ρ 7 + ρ 8 cos ψ) + 2N (ρ 7 ρ 8 cos ψ + ρ 2 9 ),
0 = 2ρ 1 ρ 9 cos φ + 2ρ 2 ρ 9 + ρ 2 4 e −2iθ + ρ 4 ρ 5 e −iθ + 2ρ 4 ρ 7 cos θ + 2ρ 4 ρ 8 e iψ cos θ + N ρ 4 ρ 9 e −iθ + ρ 5 ρ 7 + ρ 5 ρ 8 e iψ + N ρ 5 ρ 9 + N ρ 7 ρ 9 + N ρ 8 ρ 9 e iψ + 4ρ 2 9 ,
0 = 2ρ 1 ρ 10 cos φ + 2ρ 2 ρ 10 + 4ρ 4 ρ 8 cos(θ − ψ) + 8ρ 7 ρ 8 cos ψ + 4N ρ 7 ρ 10 + 6N ρ 8 ρ 10 cos ψ + 2ρ 2 7 + 4ρ 2 8 cos 2ψ + N 2 + 2 ρ 2 8 + 8N ρ 8 ρ 9 cos ψ + 8ρ 2 9 + 8ρ 9 ρ 10 + N 2 ρ 2 10 ,
0 = 2ρ 1 ρ 4 e −i(θ+φ) + 4e −iθ ρ 4 ρ 9 + 2e −iθ ρ 4 ρ 10 + 2e iθ ρ 4 ρ 10 + N 2 ρ 8 ρ 10 e iψ + N 2 ρ 1 ρ 8 e −i(ψ+φ) + 2N ρ 4 ρ 8 e −i(θ+ψ) + 2N ρ 2 8 e 2iψ + 2N ρ 5 ρ 8 e −iψ + 2N ρ 7 ρ 8 e iψ + 4N ρ 1 ρ 9 e −iφ + N ρ 1 ρ 10 e −iφ + N ρ 2 8 + 4N ρ 9 ρ 10 + ρ 2 ρ 8 e −iψ + ρ 2 ρ 8 e iψ + 4ρ 8 ρ 9 e −iψ + 8ρ 8 ρ 9 e iψ + ρ 1 ρ 8 e i(φ−ψ) + 3ρ 1 ρ 8 e i(ψ−φ) + 2ρ 1 ρ 7 e −iφ + 4ρ 5 ρ 9 + 4ρ 7 ρ 9 + 2ρ 5 ρ 10 .
The choices of the indices yielding these equations are given in table 2, where the notation ab implies a = b; we checked that no new constraints arise from different choices. (1), (25) to obtain the unitarity equations (26)- (36).
We still need to take into account that the field Q ab (x) that creates the particles is traceless. We do this requiring that the trace mode
T = N a=1 aa(37)
does not interact with the generic particle cd and can be discarded. This corresponds to
S|T cd = S 0 |T cd , S 0 = ±1,(38)
where the sign factor S 0 takes into account that the trace mode can decouple as a boson or a fermion. The last equation translates into a S ef,gh aa,cd = S 0 δ ef δ cg δ dh and yields the relations
S 0 = ρ 2 + N ρ 7 + 2ρ 9 ,(39)0 = ρ 1 e iφ + N ρ 8 e −iψ + 2ρ 9 ,(40)0 = 2ρ 4 cos θ + ρ 5 + N ρ 9 ,(41)0 = ρ 7 + 2ρ 8 cos ψ + N ρ 10 .(42)
These can be used to express S i≥7 in terms of S i≤6 through
ρ 7 = 1 N S 0 − ρ 2 + 2 N 2ρ 4 cos θ + ρ 5 ,(43)ρ 8 cos ψ = 1 N −ρ 1 cos φ + 2 N 2ρ 4 cos θ + ρ 5 ,(44)ρ 8 sin ψ = 1 N ρ 1 sin φ,(45)ρ 9 = − 1 N 2ρ 4 cos θ + ρ 5 ,(46)ρ 10 = 1 N 2 2ρ 1 cos φ + ρ 2 − S 0 − 6 N 2ρ 4 cos θ + ρ 5 .(47)Solutions N ρ 1 ρ 2 cos φ ρ 4 ρ 5 cos θ A1 ± R 0 ±1 − 0 0 − A2 ± [− √ 3, √ 3] 1 0 ± 1 2 √ 3 − N 2 0 0 − A3 ± ± √ 3 1 − ρ 2 2 [−1, 1] 0 0 0 − B ± 3 1 2 ± 1 2 ∓1 1 2 ρ 2 ±11 = ρ 2 1 + ρ 2 2 + 2ρ 2 4 ,(48)0 = 2ρ 1 ρ 2 cos φ + 2ρ 2 4 ,(49)0 = (N 2 − 1)ρ 2 1 + 2ρ 2 1 cos 2φ + 2ρ 1 ρ 2 cos φ + 4 N − 1 N ρ 1 (ρ 4 cos(θ − φ) + ρ 5 cos φ) − 4 N ρ 1 ρ 4 cos(θ + φ) + 8 N 2 ρ 2 4 cos 2θ + 2 1 + 4 N 2 ρ 4 (ρ 4 + 2ρ 5 cos θ) + 2 1 + 2 N 2 ρ 2 5 ,(50)0 = 2ρ 1 ρ 5 cos φ + 2ρ 2 ρ 4 cos θ − 4 N ρ 2 4 cos 2θ + N − 4 N ρ 2 4 − 8 N ρ 4 ρ 5 cos θ + N − 2 N ρ 2 5 ,(51)0 = 2ρ 1 ρ 4 cos(θ + φ) + 2ρ 2 ρ 5 − 4 N ρ 2 4 cos 2θ − 4 N ρ 2 4 + 2 N − 4 N ρ 4 ρ 5 cos θ − 2 N ρ 2 5 ,(52)0 = 2ρ 1 ρ 4 cos(θ − φ) + 2ρ 2 ρ 4 cos θ .(53)
The solutions of these equations, which we discuss in the next section, correspond to the renormalization group fixed points with CP N −1 symmetry in two dimensions. Notice that, since we derived the equations relying only on the symmetries of the Hamiltonian (12), the space of solutions contains both the fixed points of the ferromagnetic case (J > 0) and those of the antiferromagnetic case (J < 0). This point is explicitly illustrated in [33,39] for the case of the q-state Potts model.
Solutions
The Since continuous symmetries do not break spontaneously in two dimensions [5], the Hamiltonian (12) is expected to possess only a zero temperature fixed point for N ≥ 2. For N > 3 we only have solution A1, which corresponds to an O(N 2 − 1) fixed point 7 . For N = 3 the situation is apparently complicated by the existence of solution B. However, while solutions A1 and B clearly differ at the level of the amplitudes S 1 , . . . , S 11 , it can be checked that they yield the same scattering matrix (17). Hence, through the same mechanism we illustrated in section 3 for the Ising model, the solutions A1 and B of the CP 2 model correspond to the same O(8) fixed point 8 . This is possible because for N < 4 the particle indices do not take enough different values to make physically distinguishable all the terms entering the decomposition (17).
Having clarified what happens for N > 2, let us now consider N = 2. Figures 5 and 6 show that N = 2 is the value at which several pairs of solutions existing for N < 2 meet and terminate. The list of solutions at N = 2 is given in table 4 in appendix A. Such a proliferation is at first sight problematic, since we already argued that for N ≥ 2 the Hamiltonian (12) should possess only a zero temperature critical point. This is also fully consistent with the fact that CP 1 corresponds to the Riemann sphere, and then to O(3). We can then suspect that, by the same mechanism observed for solution B at N = 3, the solutions of table 4 reconstruct the same scattering matrix (17) than solution A1, and we checked that this is indeed the case. More specifically, solutions C3, C4, C7, C8, D3 and D4 correspond to A1 + , while C1, C2, C5, C6, D1 and D2 correspond to A1 − .
|ab, ba − 1 N |aa, bb ,(54)
which scatters into itself, i.e. satisfies S|ψ = λ|ψ , with an amplitude λ which is a phase by unitarity and is given by
λ = (N 2 − 1)S 1 + S 2 + S 3 + 2 N − 1 N (S 4 + S 5 ) − 2 N S 6 .(55)
Such a phase is related to the conformal dimension ∆ η of the chiral field that creates the particles as [31,39]
λ = e −2πi∆η .(56)
The values of ∆ η obtained through (55) and (56) We also see that the numerical solutions at N < 2 correspond to values of ∆ η -and then to fixed points -different from the O(N 2 − 1) ones (A1, A2, A3). It appears from figure 5 that the numerical solutions have nonvanishing ρ 2 and ρ 4 . Hence, they correspond to intersecting particle trajectories (see figure 4) and should describe criticality in gases of intersecting loops. Actually, the relevance of RP N −1 and CP N −1 models for gases of intersecting loops was discussed in [43]. Here we are finding the corresponding CP N −1 fixed points and showing that they exist up to N = 2.
We then see that the fixed points of the CP N −1 model coincide with those of the O(N 2 − 1) model only for N ≥ 2, where -at least for N integer -the notion of continuous symmetry holds and does not allow for long range order. O(N 2 − 1) fixed points are then obtained as a consequence of the fact that for N ≥ 2 there are only solutions with ρ 4 = ρ 5 = 0 (or equivalent to them at N = 2, 3). When moving away from criticality, on the other hand, ρ 4 and ρ 5 are expected 11 to develop nonvanishing values, thus producing deviations 12 from the off-critical O(N 2 − 1) behavior that vanish as T → 0. These conclusions parallel those we reached for the RP N −1 model in [29,30], whose basic findings we recall in the next section.
are descendants with dimension ∆ + n, n = 1, 2, . . . . In addition, the duplication of solutions pointed out in footnote 4 causes ∆η to go into itself under shifts by half-integers. 10 See [36] for details about ∆η in the O(M ) model. 11 Not for N = 2, given that CP 1 ∼ O(3). 12 In particular, contrary to the O(N 2 − 1) model [53], the CP N−1 model is not expected to be exactly solvable away from criticality [54,55].
Parallels with the RP N −1 model
We briefly point out similarities and differences between the above results for the CP N −1 model and those obtained for the RP N −1 model in refs. [29,30], to which we refer the reader for the detailed derivation. The RP N −1 model, defined by the lattice Hamiltonian
H RP N−1 = −J i,j (s i · s j ) 2 ,(57)
differs from CP N −1 for the fact that the spin variable s i is real, so that the model is invariant under global O(N ) transformations and local spin inversions. The scale invariant scattering description proceeds through steps analogous to those of the present paper, starting from an order parameter that is now a traceless symmetric tensor. This allows a larger number of contractions between pairs of particle indices, but there are still 11 amplitudes S 1 , . . . , S 11 parametrized as in (6) We can again use (56) to determine the conformal dimension ∆ η , taking into account that (55) is now replaced by in terms of the RP N −1 amplitudes S i given in [29,30]. The result for the different solutions is shown in figure 9. Notice that N = 2.24421.. is the threshold value below which solutions that are not (or not equivalent to) O(M ) solutions appear, a threshold that in CP N −1 occurs at N = 2. While in the previous section the correspondence CP 1 ∼ O(3) allowed us to anticipate that all the "threshold solutions" should be equivalent to A1, a similar argument is absent at the RP N −1 threshold, and indeed figure 9 illustrates that the solutions at N = 2.24421.. are not related to A1.
(N −1)(N +2) 2 S 1 + S 2 + S 3 + 2 (N −1)(N +2) N (S 4 + S 5 ) + 2 N −2 N S 6 ,(58)
Conclusion
In this paper we used scale invariant scattering theory to determine the exact fixed point equations of the two-dimensional CP N −1 model for real values of N . We found that only solutions of O(N 2 − 1) type exist for N ≥ 2, and account for a zero temperature critical point. Additional solutions existing at N = 2, 3 actually correspond to alternative scattering realizations of the same O(N 2 − 1) fixed points. We also found that a topological transition of BKT type only exists for N = ± has a finite temperature BKT transition, the space of fixed point solutions becomes larger than that of the O(N (N + 1)/2 − 1) model for N ≤ 2.24421.. [29,30].
It is also worth stressing that scale invariant scattering theory only exploits conformal invariance of critical points and the internal symmetry of the Hamiltonian, so that the space of solutions of the fixed point equations includes the critical points for both the ferromagnetic and the antiferromagnetic cases, a circumstance illustrated in more detail in [33,39] for the q-state Potts model.
We finally point out that our results for the fixed points of the renormalization group, characterized by diverging correlation length and scale invariance, add nothing to the debate [56,44,57] about the possibility of a first order transition for N large in CP N −1 and RP N −1 models, for which a first order transition at N = ∞ was deduced in [58,59] and shown to be absent in numerical simulations performed up to N = 40 [17].
A Analytic solutions
We list in this appendix the solutions of the fixed point equations (48)-(53) that we determined analytically. With respect to table 3, we also use the equations (43)- (47) to express the amplitudes S i≥7 .
• Solution A1a ± is defined for N ∈ R and reads ρ 2 = S 0 , ρ 1 = ρ 4 = ρ 5 = ρ 8 = ρ 7 = ρ 9 = ρ 10 = 0 .
• Solution A1b ± is defined for N ∈ R and reads ρ 2 = −S 0 , ρ 1 = ρ 4 = ρ 5 = ρ 8 = ρ 9 = 0 ,
ρ 7 = 2S 0 N , ρ 10 = − ρ 7 N ,(60)
• Solution A2 ± is defined for N ∈ [− √ 3, √ 3] and reads 13
ρ 1 = 1 , ρ 2 = ρ 4 = ρ 5 = ρ 9 = 0 , cos φ = (±) 1 2 3 − N 2 , sin φ = (±) 1 2 1 + N 2 , ρ 7 = S 0 N , ρ 8 = 1 |N | , cos ψ = −sgn(N ) cos φ, sin ψ = sgn(N ) sin φ , ρ 10 = 2 cos φ N 2 − S 0 N 2 .(61)
• Solution A3 ± is defined for N = ± √ 3 and reads
ρ 1 = 1 − ρ 2 2 , ρ 2 ∈ [−1, 1] , ρ 4 = ρ 5 = ρ 9 = 0 , φ = (±) π 2 , ψ = ±φ, ρ 8 = 1 |N | 1 − ρ 2 2 , ρ 7 = S 0 − ρ 2 N , ρ 10 = − ρ 7 N .
(62) 13 Signs enclosed in parenthesis are both allowed. • Solution B ± is defined for N = 3 and reads
ρ 1 = ρ 4 = ρ 8 = 1 2 , ρ 2 = ρ 5 = ρ 9 = ± 1 2 , φ = π 2 ± π 2 = θ + π = ψ + π, ρ 7 = ρ 2 + S 0 3 , ρ 10 = − ρ 7 3 ∓ 1 3 .(63)
In the next appendix we show that solutions (59) and (60) differ only for the way the trace mode decouples (as a free fermion or a free boson); this is why they both appear in table 3 as solution A1. Table 4 gives the solutions at N = 2.
B Mapping of nonmixing solutions
Equation (46) shows that the solutions with ρ 4 = ρ 5 = 0 also have ρ 9 = 0, and then S 4 = S 5 = S 6 = S 9 = 0. Figure 4 shows that the vanishing of these amplitudes eliminates the mixing of indices coming from different particles, and for this reason we refer to this type of solutions as "nonmixing". We now show how, through a change of basis, these nonmixing solutions can all be expressed as those of a system consisting of an O(N 2 − 1) vector and a scalar that are decoupled. The amplitudes for such a system, in which the scalar and the vector in general interact [37], are shown in figure 10 and take the form Figure 10: Scattering processes for a vector particle multiplet (continuous lines) and a scalar particle (dashed lines).
S ′ 1 = S ′ * 3 ≡ ρ ′ 1 e iφ ′ ,(64)S ′ 2 = S ′ * 2 ≡ ρ ′ 2 ,(65)S ′ 4 = S ′ * 6 ≡ ρ ′ 4 e iθ ′ ,(66)S ′ 5 = S ′ * 5 ≡ ρ ′ 5 ,(67)S ′ 7 = S ′ * 7 ≡ ρ ′ 7 .(68)S ′ 1 S ′ 2 S ′ 3 S ′ 4 S ′ 5 S ′ 6 S ′ 7
The change of basis that we perform in the CP N −1 model is
|Φ µ = |Φ 0 = 1 √ N N a=1
|aa ,
with Φ µ |Φ ν = δ µν , and the trace mode Φ 0 being the scalar of the vector-scalar system. The scattering matrix for the non-mixing case of the CP N −1 model can now be expressed as
S ρ,σ µ,ν = S ′ 1 δ µ,ν δ ρ,σ + S ′ 2 δ ρ µ δ σ ν + S ′ 3 δ σ µ δ ρ ν δ 0 µδ 0 νδ ρ 0δ σ 0 + S ′ 4 (δ µ,ν δ ρ 0 δ σ 0δ 0 µδ 0 ν + δ 0 µ δ 0 ν δ ρ,σδρ 0δ σ 0 ) + S ′ 5 δ 0 µ δ 0 ν δ ρ 0 δ σ 0 + S ′ 6 (δ σ µ δ 0 ν δ ρ 0δ 0 µδ σ 0 + δ 0 µ δ σ 0 δ ρ νδ 0 νδ ρ 0 ) + S ′ 7 (δ ρ µ δ 0 ν δ σ 0δ 0 µδ ρ 0 + δ 0 µ δ ρ 0 δ σ νδ 0 νδ σ 0 ) ,(70)
whereδ ν µ ≡ 1 − δ ν µ , and
S ′ 1 = Φ ν Φ ν |S|Φ µ Φ µ = S 1 ,(71)S ′ 2 = Φ µ Φ ν |S|Φ µ Φ ν = S 2 ,(72)S ′ 3 = Φ ν Φ µ |S|Φ µ Φ ν = S 3 ,(73)S ′ 4 = Φ 0 Φ 0 |S|Φ µ Φ µ = Φ ν Φ ν |S|Φ 0 Φ 0 = S 1 + N S 11 (74) S ′ 5 = Φ 0 Φ 0 |S|Φ 0 Φ 0 = S 1 + S 2 + S 3 + 2N (S 7 + S 8 ) + N 2 S 10 + 2N S 11 ,(75)S ′ 6 = Φ µ Φ 0 |S|Φ 0 Φ µ = Φ 0 Φ ν |S|Φ ν Φ 0 = S 3 + N S 8 ,(76)S ′ 7 = Φ 0 Φ µ |S|Φ 0 Φ µ = Φ ν Φ 0 |S|Φ ν Φ 0 = S 2 + N S 7 .(77)
Using the trace decoupling equations (43)-(47) the relations (71)-(77) reduce to
S ′ 1 = S 1 , S ′ 2 = S 2 , S ′ 3 = S 3 , S ′ 4 = S ′ 6 = 0 , S ′ 5 = S ′ 7 = S 0 ,(78)
which exhibit the decoupling between the vector and the scalar (recall that S 0 = ±1). Table 5 gives the explicit form of the CP N −1 nonmixing solutions in terms of the vector-scalar amplitudes. One sees, in particular, that solutions A1a ± and A1b ∓ only differ for the nature of the decoupled scalar (fermionic or bosonic). Table 5: Nonmixing solutions of the CP N −1 model in terms of the amplitudes of the vector-scalar system. Signs in parenthesis are both allowed, and S 0 = ±1.
Solution N 2 − 1 ρ ′ 1 ρ ′ 2 cos φ ′ ρ ′ 4 ρ ′ 5 ρ ′ 7 A1a ± R 0 S 0 − 0 S 0 S 0 A1b ± R 0 −S 0 − 0 S 0 S 0 A2 ± [−2, 2] 1 0 (±) 1 2 √ 3 − N 2 0 S 0 S 0 A3 ± 2 1 − ρ 2 2 [−1, 1] 0 0 S 0 S 0
Figure 1 :
1Pictorial representations of the scattering amplitude S ρσ µν (left) and of the product of amplitudes entering the unitarity equations (1) (right).
Figure 2 :
2Scattering amplitudes appearing in (5); time runs upwards.
1 :
1Solutions of equations (8)-(10), corresponding to the renormalization group fixed points with O(M ) symmetry. where s i is a real M -component unit vector at site i and the sum is taken over nearest neighbors. As usual, averages over configurations are performed with the Boltzmann weight e −H O(M ) /T , where T is the temperature. The order parameter variable s i corresponds in the scattering description to a vector multiplet of self-conjugated particles a = 1, 2, . . . , M .
Figure 3 :
3Solutions of the O(M ) fixed point equations (8)-(10). The two branches of II correspond to the critical lines for the dilute and dense phases of nonintersecting loops, III accounts for the BKT transition of the XY model, and the upper branch of I corresponds to the zero temperature critical point of the model for M > 2.
S 2
2= 0, see figure 2), are defined in the range M ∈ [−2, 2], and meet at M = 2. They are then identified as the critical lines of the dilute and dense regimes of the loop gas model, whose mapping on the partition function of the O(M ) model is well known
3 :
3Analytic solutions of the CP N −1 fixed point equations (48)-(53). When substituting (43)-(47) in (26)-(36), the imaginary parts of (34) and (36) vanish, while their real parts as well as (32), (33), (35) become linear combinations of the first six equations. This reduces the unitarity equations (26)-(36) to six independent equations given by
solutions of the equations (48)-(53) that we determined analytically are listed in appendix A and summarized in table 3. The remaining solutions, which we dermined numerically for N > 0, are shown infigure 5together with the analytical ones. The figure shows values of N up to 2, since it turns out that only the solutions A1 and B exists beyond this value. Another visualization of the solutions is given infigure 6.We start the discussion of the solutions observing that when ρ 4 = ρ 5 = 0 equations (48)-(53) reduce to the equations (8)-(10) of the O(M = N 2 − 1) model 6 . As a consequence, the CP N −1 6 N 2 − 1 is the number of independent real components of the order parameter variable(13).
Figure 5 :
5Solutions of the CP N −1 fixed point equations (48)-(53). model contains in particular the fixed points of the O(N 2 − 1) model. This immediately allows to identify the solutions A1, A2 and A3 of table 3 as corresponding to the O(M = N 2 − 1) solutions I, II and III, respectively, of table 1. The fact that N 2 − 1 = 2 when N = ± √ 3 explains the domain of definition of solutions A2 and A3.
Figure 6 :
6Solutions of the CP N −1 fixed point equations (48)-(53) in the parameter subspace (ρ 1 cos φ, ρ 2 ).We then see that for N ≥ 2 all solutions of the CP N −1 fixed point equations (48)-(53) correspond to O(N 2 −1) fixed points, and we already know that this is also the case for solutions A2 and A3. We finally need to consider the solutions that we determined numerically, which extend up to N = 2, where they meet in pairs (see figures 5 and 6). Since the meeting points at N = 2 are O(3) fixed points, and the O(M ) model does not possess branches of fixed points terminating at M = 3, we can anticipate that the CP N −1 branches terminating at N = 2 correspond to new universality classes. We illustrate this fact considering the U (N )
Figure 7 :
7for the different solutions of the fixed point equations (48)-(53) are shown in figure 7. Equation (56) defines ∆ η modulo integers 9 , and we plot the most relevant (in the renormalization group sense) interval ∆ η ∈ (0, 1). The values 0 The conformal dimension ∆ η for the different solutions of the CP N −1 fixed point equations (48)-(53). and 1/2 correspond to the O(N 2 −1) sigma model (solution A1 + ) and to the fermionic realization (solution A1 − ), respectively 10 . The figure clearly exhibits the collapse on the O(N 2 − 1) solution A1 of the additional solutions existing at N = 2, 3.
Figure 8 :
8Solutions of the fixed points equations for RP N −1 symmetry [29, 30]. The dashed vertical line indicates the value N = 2.24421.. .
, ( 7 )
7,(19)-(24). When ρ 4 = ρ 5 = 0, the fixed point equations reduce to those of the O(M N ) model, with M N = N (N + 1)/2 − 1. As a consequence, there are solutions A1, A2 and A3 that correspond to the solutions I, II and III, respectively, of table 1 with M = M N . A1 is the only solution for N > 2.24421.. . More precisely, at N = 3 there is an isolated solution B3, but we have now checked that it is equivalent to A1 by the same mechanism discussed in the previous section for solution B in CP 2 . At N = 2, solution A3 goes along with two additional solutions, B1 and B2, which also possess a free parameter and provide alternative realizations of the BKT phase in the RP 1 ∼ O(2) model. Finally, we show in figure 8 how for N < 2.24421.. there is a rich pattern of solutions that we determined numerically.
Figure 9 :
9∆ η for the different RP N −1 solutions.
√ 3 ,
3where O(N 2 − 1) = O(2). Several branches of fixed points that are not of O(N 2 − 1) type appear below N = 2 and are expected to describe criticality in loop gases. These solutions are characterized by amplitudes with nonvanishing transmission and correspond to gases of intersecting loops, examples of which were considered in [43] and related to the RP N −1 and CP N −1 models. It is interesting to observe how these new fixed points only emerge for N < 2, where the model only makes sense within the loop gas continuation to real values of N . For integer N ≥ 2, the continuous symmetry does not allow ordered phases in two dimensions, and the fixed point equations have solutions only in the subspace with ρ 4 = ρ 5 = 0, where they coincide with those of the O(N 2 − 1) model. On the other hand, for N > 2, these scattering parameters are expected to develop nonzero values, so that the O(N 2 − 1) behavior only arises asymptotically in the zero temperature limit. An exception is provided by N = 2, since CP 1 ∼ O(3). We recalled for comparison how in the RP N −1 model, where RP 1 ∼ O(2)
= kk , k = 1, . . . , N − 1,
Table
Table 2 :
2External indices used in
Table
Table 4 :
4Solutions of equations (48)-(53) at N = 2; we omit A1.
See section 6 below for a question mark that we had left at N = 3.
See[46] for a review on the off-critical regime in two dimensions.3 We denote byμ the antiparticle of µ.
When discussing the Hamiltonian (12) we refer to the bosonic realization A1+ of the symmetry. The fermionic realization A1− is not relevant for that Hamiltonian.8 In three dimensions, where the symmetry can break spontaneously, a finite temperature critical point in the O(8) universality class has been observed in numerical simulations of the antiferromagnetic CP 2 model[52].
This corresponds to the fact that in conformal field theory, given a primary field with dimension ∆, there
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|
[] |
[
"Mean-field dynamics of tumor growth and control using low-impact chemoprevention",
"Mean-field dynamics of tumor growth and control using low-impact chemoprevention"
] |
[
"Andrei R Akhmetzhanov \nInstitute of Evolutionary Sciences of Montpellier\nUMR 5554\nUniversity of Montpellier II\nPlace Eugéne BataillonCC065, 34095, Cedex 5MontpellierFrance\n\nDept. of Biology\nTheoretical Biology Lab\nMcMaster University\nL8S4K1HamiltonOntarioCanada\n",
"Michael E Hochberg \nInstitute of Evolutionary Sciences of Montpellier\nUMR 5554\nUniversity of Montpellier II\nPlace Eugéne BataillonCC065, 34095, Cedex 5MontpellierFrance\n\nSanta Fe Institute\n87501Santa FeNMUSA\n\nWissenschaftskolleg zu Berlin\nWallotst. 1914193BerlinGermany\n\nKavli Institute for Theoretical Physics\nUniversity of California\n93106-4030Santa BarbaraCAUSA\n"
] |
[
"Institute of Evolutionary Sciences of Montpellier\nUMR 5554\nUniversity of Montpellier II\nPlace Eugéne BataillonCC065, 34095, Cedex 5MontpellierFrance",
"Dept. of Biology\nTheoretical Biology Lab\nMcMaster University\nL8S4K1HamiltonOntarioCanada",
"Institute of Evolutionary Sciences of Montpellier\nUMR 5554\nUniversity of Montpellier II\nPlace Eugéne BataillonCC065, 34095, Cedex 5MontpellierFrance",
"Santa Fe Institute\n87501Santa FeNMUSA",
"Wissenschaftskolleg zu Berlin\nWallotst. 1914193BerlinGermany",
"Kavli Institute for Theoretical Physics\nUniversity of California\n93106-4030Santa BarbaraCAUSA"
] |
[] |
Cancer poses danger because of its unregulated growth, development of resistant subclones, and metastatic spread to vital organs. Although the major transitions in cancer development are increasingly well understood, we lack quantitative theory for how chemoprevention is predicted to affect survival. We employ master equations and probability generating functions, the latter well known in statistical physics, to derive the dynamics of tumor growth as a mean-field approximation. We also study numerically the associated stochastic birth-death process. Our findings predict exponential tumor growth when a cancer is in its early stages of development and hyper-exponential growth thereafter. Numerical simulations are in general agreement with our analytical approach. We evaluate how constant, low impact treatments affect both neoplastic growth and the frequency of chemoresistant clones. We show that therapeutic outcomes are highly predictable for treatments starting either sufficiently early or late in terms of initial tumor size and the initial number of chemoresistant cells, whereas stochastic dynamics dominate therapies starting at intermediate neoplasm sizes, with high outcome sensitivity both in terms of tumor control and the emergence of resistant subclones. The outcome of chemoprevention can be understood in terms of both minimal physiological impacts resulting in long-term control and either preventing or slowing the emergence of resistant subclones. We argue that our model and results can also be applied to the management of early, clinically detected cancers after tumor excision.Author summaryOne of the principal risks of aggressive chemotherapy is the selection of cells that are at the origin of relapse and are refractory to subsequent treatments. Alternative approaches based on management models usually target clinically detected tumours or residual cancers. These therapies carry the risk of not being able to control fast growing subclones or resistant lineages. We develop a mean-field approach to evaluate low impact chemoprevention. Effective management slows or prevents evolution though the incorporation of fitness-enhancing driver mutations, and the emergence of chemoresistance. Dynamics are highly predictable for sufficiently small or large initial tumour sizes, and increasingly stochastic for intermediate-sized tumours. Based on empirical parameter estimates, we predict that the optimal daily levels of reduction in tumour growth for sufficiently small neoplasms are between 0.1% and 0.2%. This corresponds to reducing the net growth rate of the existing tumour to below zero (0.1% growth reduction), but not so much so as to select for subsequent driver mutations (0.2% growth reduction). Satisficing based on chemoprevention offers an alternative approach for people at high risks of life-threatening cancers.
| null |
[
"https://arxiv.org/pdf/1408.6052v1.pdf"
] | 18,198,477 |
1408.6052
|
023d304bf777a333af2de6c67328adde2adc1138
|
Mean-field dynamics of tumor growth and control using low-impact chemoprevention
26 Aug 2014
Andrei R Akhmetzhanov
Institute of Evolutionary Sciences of Montpellier
UMR 5554
University of Montpellier II
Place Eugéne BataillonCC065, 34095, Cedex 5MontpellierFrance
Dept. of Biology
Theoretical Biology Lab
McMaster University
L8S4K1HamiltonOntarioCanada
Michael E Hochberg
Institute of Evolutionary Sciences of Montpellier
UMR 5554
University of Montpellier II
Place Eugéne BataillonCC065, 34095, Cedex 5MontpellierFrance
Santa Fe Institute
87501Santa FeNMUSA
Wissenschaftskolleg zu Berlin
Wallotst. 1914193BerlinGermany
Kavli Institute for Theoretical Physics
University of California
93106-4030Santa BarbaraCAUSA
Mean-field dynamics of tumor growth and control using low-impact chemoprevention
26 Aug 20141 2
Cancer poses danger because of its unregulated growth, development of resistant subclones, and metastatic spread to vital organs. Although the major transitions in cancer development are increasingly well understood, we lack quantitative theory for how chemoprevention is predicted to affect survival. We employ master equations and probability generating functions, the latter well known in statistical physics, to derive the dynamics of tumor growth as a mean-field approximation. We also study numerically the associated stochastic birth-death process. Our findings predict exponential tumor growth when a cancer is in its early stages of development and hyper-exponential growth thereafter. Numerical simulations are in general agreement with our analytical approach. We evaluate how constant, low impact treatments affect both neoplastic growth and the frequency of chemoresistant clones. We show that therapeutic outcomes are highly predictable for treatments starting either sufficiently early or late in terms of initial tumor size and the initial number of chemoresistant cells, whereas stochastic dynamics dominate therapies starting at intermediate neoplasm sizes, with high outcome sensitivity both in terms of tumor control and the emergence of resistant subclones. The outcome of chemoprevention can be understood in terms of both minimal physiological impacts resulting in long-term control and either preventing or slowing the emergence of resistant subclones. We argue that our model and results can also be applied to the management of early, clinically detected cancers after tumor excision.Author summaryOne of the principal risks of aggressive chemotherapy is the selection of cells that are at the origin of relapse and are refractory to subsequent treatments. Alternative approaches based on management models usually target clinically detected tumours or residual cancers. These therapies carry the risk of not being able to control fast growing subclones or resistant lineages. We develop a mean-field approach to evaluate low impact chemoprevention. Effective management slows or prevents evolution though the incorporation of fitness-enhancing driver mutations, and the emergence of chemoresistance. Dynamics are highly predictable for sufficiently small or large initial tumour sizes, and increasingly stochastic for intermediate-sized tumours. Based on empirical parameter estimates, we predict that the optimal daily levels of reduction in tumour growth for sufficiently small neoplasms are between 0.1% and 0.2%. This corresponds to reducing the net growth rate of the existing tumour to below zero (0.1% growth reduction), but not so much so as to select for subsequent driver mutations (0.2% growth reduction). Satisficing based on chemoprevention offers an alternative approach for people at high risks of life-threatening cancers.
Introduction
Mathematical models play an important role in describing and analyzing the complex process of carcinogenesis. Natural selection for increases in tumor cell population growth rate can be represented as the net effect of increased fission rates and/or decreased apoptosis (e.g., [1]). Relatively rare driver mutations confer such a net growth advantage, whereas numerically dominant passenger mutations with initially neutral or mildly deleterious effects [2][3][4] can only initially grow in frequency due to genetic hitchhiking. Amongst the many passengers in a growing tumor, some can contribute to cell chemoresistance, and a sufficiently large tumor will contain different clones that, taken as a group, can resist most, if not all, possible chemotherapies (see [5] for resistance to imatinib). Chemotherapeutic remission followed by relapse suggests that these resistant cells are often at low frequencies prior to therapy, either due to genetic drift or costs associated with resistance. Resistant phenotypes subsequently increase in frequency during chemotherapy, and through competitive release, they may incorporate one or more additional drivers, resulting in accelerated growth compared to the original tumor [6].
Previous mathematical studies have considered alternatives to attempting to minimize or eradicate clinically diagnosed cancers with maximum tolerated doses (MTD) of chemotherapeutic drugs. This body of work indicates that MTD is particularly prone to select for chemoresistance (e.g., [7][8][9]), and empirical studies support this basic prediction [10]. Numerous alternatives to the goal of cancer minimization/eradication have been investigated (e.g., [7,[11][12][13][14]). For example, Komarova and Wodarz [11] considered how the use of one or multiple drugs could prevent the emergence or curb the growth of chemoresistance. They showed that the evolutionary rate and associated emergence of a diversity of chemoresistant lineages is a major determinant in the success or failure of multiple drugs versus a single one. Foo and Michor [7] evaluated how different dosing schedules of a single drug could be used to slow the emergence of resistance given toxicity constraints. One of their main conclusions is that drugs slowing the generation of chemoresistant mutants and subsequent evolution are more likely to be successful than those only increasing cell death rates.
These and other computational approaches have yet to consider the use of chemoprevention to reduce cancer-associated morbidity and mortality. Prevention, more generally, encompasses life-style changes, interventions or therapies in the absence of detectable invasive carcinoma (e.g., [15][16][17][18]). In depth consideration of preventive measures and their likely impact on individual risk and epidemiological trends is important given the virtual certitude that all people have pre-cancerous lesions, some of which may transform into invasive carcinoma [19,20], and concerns as to whether technological advances will continue to make significant headway in treating clinically detected cancers [21,22].
Here we model how chemoprevention affects tumor progression and the emergence of chemoresistant lineages. Previous study has considered the effects of deterministic and stochastic processes on tumor growth and the acquisition of chemoresistance [2,11,23]. We consider both processes through exact solutions and numerical simulations of master equations, using the mean field approach. A mean field approach assumes a large initial number of cells [24] and averages any effects of stochasticity, so that an intermediate state of the system is described by a set of ordinary differential equations (i.e., master equations; [25]). Solutions to these are complex even in the absence of the explicit consideration of both drivers and passengers [26]. Our approach [27,28] follows the dynamics of the relative frequencies of subclones, composed of identical cells, instead of the fate of individual cells. We derive the dynamics for the expected total number of cells within a tumor at any given time.
We show that the expected mean tumor size can be substantially different from the median, since the former is highly influenced by outliers due to tumors of extremely large size. We then consider constant chemopreventive treatments, starting at a given tumor size and number of chemoresistant mutations. We find that treatment outcome can be highly sensitive to initial conditions. Not surprisingly, initially small tumors are more likely to be controlled than larger tumors employing low dose therapies, whereas large tumors follow deterministic growth and are both difficult to control in overall size and in the emergence of resistance. In contrast, there is a range of intermediate size tumors, where stochastic dynamics become significant, and clinical outcome is highly sensitive to the commencement time (i.e., initial tumor size) of treatment regimes. We study a low intensity, constant treatment regime that starts at time t = 0. First, we study meanfield dynamics by considering the distribution H t (x) of tumor sizes x at time moments t, and examine effects on the mean n(t) = H t (x) .
Results
Parameter
Using the master equations, we derive an analytical expression for the dynamics n(t). Namely, we use (8)-(9) and (13)-(14) (see Methods section) to obtain the dynamics of the expected tumor size
n(t) = n(0) (1 − κ) 1 + v 2 e (σ−c)t − 1 σ − c + κe (σ−c)t exp (s − σ)t + N ln 1 + u 2N e st − 1 s ,(1)
and the frequency of resistant cells within a tumor n res (t)
n(t) = (1 − κ) v 2 e (σ−c)t −1 σ−c + κe (σ−c)t (1 − κ)(1 + v 2 e (σ−c)t −1 σ−c ) + κe (σ−c)t .(2)
Here, time t is normalized so that any event occurs at rate equal to unity, or t = τ /T . In the following, we use the variable t as shorthand for t = τ /T . Fig. 1 (A and a) shows the excellent correspondence between numerical experiments and analytical results for σ on the order of s.
Equation (1) is simplified for two limiting cases. In the early stages of tumor growth, the value n(t) changes according to a hyper-exponential law:
n(t) ≈ n(0) (1 − κ) 1 + v 2 e (σ−c)t − 1 σ − c + κe (σ−c)t exp (s − σ)t + u 2 e st − 1 s .
while at later stages the most aggressive subclone persists, being sensitive if σ < c (n(t) ∝ e s(N +1)t ) and resistant otherwise (n(t) ∝ e s(N +1)t−c ). A more detailed study of the distribution H t (x) reveals that the mean n(t) diverges importantly from median behavior in the majority of cases, since the former is strongly influenced by outliers. Fig. 1(B) and Suppl. Fig. S1 illustrate examples where the mean trajectory deviates from the median, and exceeds the 95% confidence interval at approximately 22 years into the simulation. Treat. int., % 0.0 0.2 0.4 0.6 1.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Table 1 for other parameter values.
To see how alternative formulations affect the results, we investigate H x (t), which is the distribution of times t when a tumor reaches a given threshold size x. We assume x = M = 10 9 cells (i.e., the lower boundary for clinical detection of a tumor -approximately 1cm 3 in volume). Note that the results below can be generalized for other values of M .
Based on extensive numerical experiments, we find that the means of the distributions H t (x) and H x (t) (as well as other characteristics such as the mode and the median) are the same only in case of no treatment (σ = 0). The reason for these apparent discrepancies is that a tumor has two distinct subpopulations (sensitive and resistant), meaning that the distributions H t (x) and H x (t) are bimodal. Trivially, this does not occur for σ = 0, since the resistant part is negligible (at a mutation-selection balance).
We perform three sets of numerical experiments to study how variation in any one of the following parameters-the selective advantage s, the cost of resistance c, or the initial number of cells n(0)-influences the properties of the distribution H M (t).
Variation in the selective advantage s, with c = 0.1% and n(0) = 10 6 cells being fixed, leads to Fig. 2. We see that tumor growth is mainly driven by its non-resistant part for relatively low impact treatments σ < 2s. The tumor changes from being mainly non-resistant to resistant at σ ≈ 2s, which is reflected by the emergence of an inflection point in the trajectory of the median (indicated by C in Fig. 2). Notice that the detection times are also most variable at σ ≈ 2s. The median changes smoothly at high treatment levels (σ > 2s), tending to a horizontal asymptote. This is explained by the fact that the sensitive part is heavily suppressed at high treatment levels, meaning that the dynamics are strongly influenced by an actual time point when the resistance mutation occurs.
The inflection point at σ ≈ 2s is due to the accumulation of additional drivers within tumors and associated increases the likelihood that the tumor eventually resists treatment if no resistant cells were initially present. Since the initial population consists of 10 6 cells, in the absence of treatment, a new cell with one additional driver and associated fitness (2s − σ) will appear very rapidly. Such a tumor can only be suppressed only if we apply the treatment with σ > 2s. This is supported by additional numerical experiments, where we vary the maximal number of allowed driver mutations N , see Fig. 4(A). We see that the inflection point σ ≈ 2s disappears when N = 0. Similarly, we may expect inflection points around σ = 3s, 4s and so on, which is shown in Suppl. Fig. S2 or in Suppl. Video S3 and where the resistant mutation is knocked out. In contrast, when the resistant mutation is present, an appearance of many inflection points is blurred by higher growth of the resistant part of a tumor: only two humps are noticed in Fig. 2 for the largest value of cost of resistance c = 0.4% and one hump for smaller values of c. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Year of detection C q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 40 60 80 D q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Table 1. Note that the detection time in A is log-transformed.
Counterintuitively, if the cost of resistance is low to moderate then early-detected tumors are more likely to be resistant under constant treatments than those detected at later times (B, C and D in Fig. 2). While a resistance mutation emerges, the tumor grows faster under selection (despite the very low impact of therapy on sensitive cancer cells) and is therefore more likely to be detected at earlier times. By the time of detection, non-resistant tumors usually accumulate up to 4 additional drivers on average, while resistant tumors have fewer. For larger values of c, an additional non-regularity emerges (segment BCD in Fig. 3), appearing at σ ≈ 3s and is associated with tumors having a majority of cells with 3 total drivers. This region is also characterized by a different transition to complete resistance (compare Suppl. Video S1 and S2 for relatively low and high costs of resistance, respectively). For example, at point B tumors with a majority of non-resistance have less variable detection times than tumors with a majority of resistant cells (A and corresponding panel B in Fig. 3). Treatment levels along the segment BCD result in tumors that are more likely to be resistant as one goes from the center to the tails of the q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 80 110
Detection year C q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 20 50 80 110 D q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0% 50% 100% Resistance Figure 3. Higher costs of resistance lead to slower, but more variable growth. The median and 95% confidence intervals for detection times when the cost of resistance is varied. The selective advantage is fixed at 0.4%. Other conditions as for Fig. 2.
distribution H M (t). This differs qualitatively from the previous case of low cost of resistance, where the tumors are less resistant in a tail of the distribution. The initial cell number n(0) affects both the median and distribution of detection times Fig. 4(B). For large initial tumors, growth is deterministic and exponential. As the initial size is decreased (n(0) = 10 6 to 10 5 ), stochastic effects are increasingly manifested by the appearance of an inflection point in the trajectory of the median, as well as increased variability in detection times.
Whereas M = 10 9 cells are an approximate clinical detection threshold for many solid tumors, ap-proximatelyM = 10 11 cells could be attained for certain otherwise undetectable cancers, and are only discovered in late stage, metastatic disease [30]. Using numerical experiments, we find that the difference in times when a tumor reaches M andM respectively, is 4.7 ± 0.2 years (mean±s.d.).
A tumor is likely to be eradicated under a range of constant treatments if it has n(0) = 10 5 or fewer initial cells; a tumor is virtually certain to persist for n(0) = 10 7 cells or greater, as it is shown in Suppl. Fig. S3. In other words, our model indicates that tumors that are c. 1% of clinically detectable size will typically be impossible to eradicate.
The above analysis assumes zero initial resistance within a tumor. Given mutation rates assumed here we can expect that many tumors with one million cells will already contain resistant cells. We extend our study to other values of initial resistance level, denoted by κ. As shown in Fig. 4(C) larger values of κ create a transition from stochastic to deterministic tumor growth. As expected, larger κ results in worse control outcomes, with a threshold for treatment failure -tumors can only be eradicated for σ > 2s (see the inset Fig. 4(C)).
Finally, we briefly consider how competition affects the results presented above. We assume for simplicity that sensitive cells inhibit the growth of resistant cells (i.e., chemoresistance has both a constant cost and an additional cost in proportion to sensitive cell number). Such assumption in its simple form leads to more variability in tumor growth and, as expected, delays in cancer detection Fig. 4(D) and a positive correlation with treatment success, see the inset Fig. 4(D), which can be understood as the proportion of numerical experiments in which the tumor stays undetected. Detection year A q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Max add.drvs q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Init popul n(0) Treatment intensity, %
Detection year C q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Pre−resist., % Treatment intensity, % D q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Compet. param. Table 1.
Discussion
Maximum tolerated dose chemotherapies present numerous challenges, ostensibly the major one being the selection of resistant phenotypes, which are possible precursors for relapse [31]. Over the past decade, several alternative approaches have been proposed, where the objective is to manage rather than eradicate tumors (e.g., [8,11,13,14,32,33]. Tumor management attempts to limit cancer growth, metastasis, and reduce the probability of obtaining resistance mutations through micro-environmental modification or through competition with non-resistant cancer cell populations or with healthy cells. These approaches usually involve clinically diagnosed cancers: either inoperable tumors or residual cancers after tumor excision. In the former situation tumors are typically large enough in size to contain numerous resistance mutations. In many, if not most, cases these neoplasms will have metastasized, meaning greater variability both in terms of phenotypes and hence potential resistance to chemotherapies, and in penetrance of therapeutic molecules to targeted tumor cells [34,35]. The latter situation involves smaller, residual cancer cell populations, but composed of high frequencies of resistant variants or dormant cells [34]. Both scenarios are likely to involve cancer cell populations with large numbers of accumulated driver mutations, which ostensibly contribute to the speed of relapse. Thus, management of clinically detected tumors need not only limit the proliferation and spread of refractory subpopulations ( Fig. 2-4), but should also aim to control the growth of multi-driver clones (Suppl. Fig. S2). We mathematically investigated an alternative strategy, chemoprevention through satisficing, where a satisfactory objective is defined from the outset for patients at a high risk of contracting a life-threatening cancer. Such objectives can be complex, involving minimal side effects, defining acceptable risks of developing a lethal cancer at a later time, and realistic maximum expected frequencies of chemoresistant lineages. Neoplasms in our model system could correspond to pre-cancerous states of dyplasia, carcinoma in situ, or to invasive carcinoma, but the relative frequency of these different stages for tumors of the initial sizes modeled here are unknown; nor is it known how chemopreventive therapies affect cell populations in these different states. Several authors have previously argued for how constant or intermittent low toxicity therapies either before or after tumor discovery could be an alternative to maximum tolerated dose chemotherapies [18,36], but to our knowledge no study has actually quantified the modalities (treatment start time, dose) for such approaches using empirically derived parameter estimates [2,12,37].
Our model indicates that daily reductions in population growth of s < σ < 2s, corresponding to 0.1%-0.2%, is sufficient in most cases to control neoplasm expansion for tumors less than about 1 million cells, and harboring no resistance mutations and only one driver mutation at the start of therapy. We find that such nascent tumors can be managed for tens of years without growing to life-threatening levels, and that the duration of successful prevention is sensitive to both initial tumor size and treatment intensity. Specifically, tumors growing beyond approximately the reciprocal of the driver mutation rate (c 10 6 cells) are exponentially increasingly likely to produce a faster growing subclone with a new driver. Such a subclone is even more likely, once it reaches ∼1/u cells, to produce a new subclone with an additional driver, and so on. The result over sufficiently long periods is a hyper-exponential increase in tumor size. Given that mutation rates for chemoresistance are thought to be on the order of 10 −6 per cell division [11], this means that tumor size is also a sensitive predictor of the likelihood of chemoresistance, and thus the potential for chemopreventive management to slow the progression of a potentially lethal cancer.
Indeed, one of our central results is the sensitivity of tumor growth to size at the commencement of therapy ( Fig. 4(C)). Deterministic equations provide an accurate description of such growth for sufficiently small ( 10 5 ) or large (>10 6 ) initial tumor sizes. We found that sufficiently small tumors were controlled and sometimes eradicated if therapeutic reductions in population growth exceeded 2s. Larger tumors, though affected by therapy, were impossible to control or to eradicate, because of the presence of resistance mutations. This effect was mitigated to some extent by cell-cell competition (Fig. 4(D)), but the process of competition as modeled here was not sufficient to permit tumor control (but see [33]). Moreover, we found that tumors approximately between 10 5 and 10 6 cells had more variable, stochastic outcomes, meaning that a given preventive therapeutic regimen may or may not be successful due to the chance emergence of driver mutations and local extinctions. This result emphasizes not only the sensitivity of tumor control to initial size (i.e., time at which therapy commences), but also the accurate assessment of changes in cancer risk for different therapeutic alternatives. Although not investigated in the context of cancer therapies, the results of Bozic and colleagues [2] indicate that achieving a tumor size where driver mutations become probable, distinguishes patients harboring small tumors after 25 years from those developing life-threatening tumors over this same period (see Fig. 1 in [2]). In their simulations like ours, the time to emergence of the second driver mutation is a good predictor of future tumor growth.
Our theory also highlights two potentially contrasting objectives of cancer prevention: managing tumor size vs managing resistance mutations. We show the sensitivity of these two outcomes to treatment levels, especially near the threshold 2s. Treatments at or beyond 2s effectively offset or reduce sensitive cell growth, leaving a large subpopulation of resistant cells that are released from competition and can rapidly obtain additional driver mutations. Treatments just below 2s also reduce tumor growth, but maintain high frequencies of sensitive cells, which potentially compete with resistance cells, thereby reducing overall resistance cell numbers. Adopting the latter strategy could make a difference to longterm outcomes, especially in cases where the constant therapy is discontinued, or ulterior attempts are made at high dose chemotherapy.
Some empirical studies support the role of certain molecules in chemoprevention [17]. For example, Silva and colleagues [38] parameterized computational models to show how low doses of verapamil and 2-deoxyglucose could be administered adaptively to promote longer tumor progression times. These drugs are thought to increase the costs of resistance and the competitive impacts of sensitive on resistant cancer cell subpopulations. However, some of the most promising results have come from studies employing nonsteroidal anti-inflammatory drugs (NSAIDs), including experiments [39], investigations of their molecular effects [40,41], and their use [42]. For example, Ibrahim and coworkers [39] studied the action of NSAIDs and specifically sodium bicarbonate in reducing prostate tumors in male TRAMP mice (i.e. an animal model of transgenic adenocarcinoma of the mouse prostate). They showed that mice commencing the treatment at 4 weeks of age had significantly smaller tumor masses, and that more survived to the end of the experiment than either the controls or those mice commencing the treatment at an older age. Kostadinov and colleagues [40] showed how NSAID use in a sample of people with Barrett's esophagus is associated with reductions in somatic genomic abnormalities and their growth to detectable levels. It is noteworthy that it is not known to what extent reductions in cancer progression under NSAIDs is due to either cytotoxic or cytostatic effects, or both. Although we do not explicitly model cytotoxic or cytostatic impacts, therapies curbing net growth rates, but maintaining them at or above zero, could be interpreted as resulting from the action of either cytotoxic and/or cytostatic processes. In contrast, therapies reducing net growth rates below zero necessarily have a cytostatic component. Our model, or modifications of it to explicitly include cytotoxic and cytostatic effects, could be used in future research to make predictions about optimal dose and start times to achieve acceptable levels of tumor control, or the probability of a given tumor size by a given age. Lorz and coworkers [9] recently modeled the employment of cytotoxic and cytostatic therapies alone or in combination and showed how combination strategies could be designed to be superior in terms of tumor eradication and managing resistance than either agent used alone.
Decisions whether or not to employ chemopreventive therapies carry with them the risk of a poorer outcome than would have been the case had another available strategy, or no treatment at all, been adopted [43]. This issue is relevant to all preventive approaches, where alterations in life-style, removal or treatment of pre-cancerous lesions, or medications may result in unwanted side effects or potentially induce new invasive neoplasms (e.g., [44]). Chemopreventive management prior to clinical detection would be most appropriate for individuals with genetic predispositions, familial histories, elevated levels of specific biomarkers, or risk-associated behaviors or life-styles [16][17][18]45,46]. Importantly, our approach presupposes that the danger a nascent, growing tumor presents is proportional to its size and (implicitly, all else being equal) a persons age. Due caution is necessary in applying our results, since studies have argued that metastatic potential rather than tumor size may be a better predictor of future survival [47][48][49].
We have modeled preventive approaches to managing risks of future lethal cancers. However, our model also could be applied to scenarios where an invasive carcinoma is discovered early in progression. In such cases, tumor clones are likely to harbor greater numbers of driver mutations and show higher levels of genomic instability and standing genetic heterogeneity than the earlier stages targeted by chemoprevention. Higher aptitude for growth and mutation (adaptation) in clinical tumors could mean that outcomes are less sensitive to cancer cell numbers as we found in the prevention scenario. We suggest that the frequency distribution of driver mutations and the distribution of resistant subclones within these lineages could instruct decisions of the time course of treatment levels, with the aims of satisfactory tumor growth, metastasis, and resistance control. Although residual cancer cell populations from excised tumors and associated micro-metastases are often difficult to assess with accuracy [50], our results suggest that if order of magnitude estimates are possible, than low dose, constant approaches could be optimized, and according to our model, will always be superior to aggressive chemotherapies even if resistance mutations are likely to be present.
Models and methods
Conceptual framework
Let each cell in a population be described by two characteristics. The first is its resistance status, which is either "not resistant" (j = 0) or "resistant" (j = 1). The probability of a resistant mutant emerging during cell division is assumed constant (v). The second property is the number of accumulated driver mutations in a given cell line (maximum N ). The mutation rate at any locus resulting in the addition of a driver is u, and we assume no back mutation. Thus, the genome of a cell in our model is composed of N potential driver loci and one chemoresistance locus.
We initially assume that at each time step cells either divide or die, but do not compete for space or limiting resources. The fitness function f ij is the difference between the birth and death rates of a cell and is defined by the number of accumulated drivers (i = 0, 1, . . . , N ) and resistance status (j = 0, 1). A chemosensitive cancerous cell with a single driver has selective advantage s. Any additional drivers add s to fitness, while resistance is associated with a constant cost c. Exposure to a single chemotherapy treatment affects only non-resistant cells (j = 0), incurring a loss σ ∈ [0, 1] to their fitness. We assume that all parameters c, s and σ are arbitrarily small ( 1).
Fitness is
f ij = s(i + 1) − σ(1 − j) − cj .(3)
The assumption of driver additivity is a special case of multiplicative fitness, and both are approximately equivalent for very small s.
Numerical simulations
To simulate tumor growth, we adopt a discrete time branching process for the cell-division process [2,51], which is usually referred to as a discrete time Galton-Watson process [52]. For each numerical experiment we initiate a tumor of a given size with cells of a type i = 0, and proportion of resistant cells within a tumor κ (0 ≤ κ ≤ 1). Table 1 presents baseline parameter values employed in this study. At the beginning of each time step, the number of cells is n ij (t). The number of cells at next step (t+1) is then sampled by a multinomial distribution. If we let B ij be the number of births in the population, D ij the number of deaths, M and (i, j + 1) respectively, then the multinomial distribution is
P [(B ij , D ij , M (u) ij , M (v) ij ) = (k 1 , k 2 , k 3 , k 4 )] = n ij (t)! k 1 !k 2 !k 3 !k 4 ! (b ij (1 − u i − v j )) k1 d k2 ij (b ij u i ) k3 (b ij v j ) k4 , where u i = u(1 − i/N ) and v j = v(1 − j) (i = 0, . . . , N , j = 0, 1).
The number of cells of type (i, j) at time step t + 1 is now given by
n ij (t + 1) = n ij (t) + B ij − D ij + M (u) i−1,j + M (v) i,j−1 ,
where we assume M i,−1 = 0. We conducted numerical experiments of the above model, each with the same initial states, but each using a unique set of randomly generated numbers of a branching process.
Code
All calculations were made using programs, written in C, and the free, open-source statistical package R [53]. The color palette for figures was adopted from [54]. Code for all calculations, and for producing all of the figures, is available at [55] and can be used freely for non-commercial purposes.
Mean-field dynamics
We use the mean-field approach, see e.g. [24], which approximates the behavior of a system consisting of many cells, so that the effects of stochasticity are averaged and an intermediate state is described by a set of ordinary differential equations.
Master equations We write master equations to track the probability P ij (t) that a randomly chosen cell from the population of tumor cells will be of type (i, j) at time t.
The temporal dynamics of probabilities P ij (t), i = 0, 1, . . . , N , where N is the maximal number of additionally acquired drivers and j = 0, 1, are described by
dP ij (t) dt = P ij + uP (u) ij + vP (v) ij .
Here, the right-hand side is a superposition of probabilistic in-and out-flows from different mutational states to the current one (i, j). The function P ij describes the growth of subclone (i, j) and is proportional to the probability P ij (t), multiplied by the difference between f ij and the average fitness over the whole populationf = i,j f ij P ij (t). P ij , a driver is added from class (i − 1, j) to (i, j) in proportion to the probability P i−1,j (t), the probability of cell birth b i−1,j , and the probability of a zero locus being chosen from N total loci consisting of N −(i−1) other zero loci. A similar approach is used to define the outflow term for the probability from class (i, j) to (i + 1, j). The second term P (v) ij is the probability of mutating to therapeutic resistance ((i, j = 0) to (i, j = 1)), and is proportional to P i0 (t) and birth rate b i0 . Finally, all terms are summed, taking into account the initial conditions: P 00 (0) = 1 − κ, P 01 (0) = κ and P ij (0) = 0 for any other i or j.
The above elements lead to the following system of ordinary differential equations (ODEs):
dP ij (t) dt = (f ij −f )P ij (t) + u 1 − i − 1 N 1 + f i−1,j 2 P i−1,j (t) − 1 − i N 1 + f ij 2 P ij (t) − v(1 − 2j) 1 + f i0 2 P i0 (t) ,(4)
where some probabilities P ij could, theoretically, take on negative values, e.g. P −1,j (t), when i = 0, in which case, they are set to zero. A simple transformation
p ij (0) = P ij (0), p ij (t) = P ij (t) exp t 0f (r) dr ,
allows omitting the termf from equation (4) and to linearize the latter with respect to the new "transformed" probabilities p ij (t). This gives
dp ij (t) dt = f ij p ij (t) + u 1 − i − 1 N 1 + f i−1,j 2 p i−1,j (t) − 1 − i N 1 + f ij 2 p ij (t) + v 1 + f ij 2 (jp i,j−1 (t) + (j − 1)p ij (t)) ,(5)
where, for convenience, we write (jp i,j−1 (t) + (j − 1)p ij (t)) instead of (1 − 2j)p i0 (t).
Probability generating function approach With the master equations (5), we apply the probability generating function (p.g.f.) method [25,56] to transform the system of (2N + 1) ordinary differential equations to a Hamilton-Jacobi (HJ) equation, that is, a first order partial differential equation. We define the p.g.f. as the polynomial over all modified probabilities p ij of the form
G(ξ, η, t) = N i=0 1 j=0 ξ i η j p ij (t) ,(6)
where ξ and η are variables that can be viewed as the momentum of an auxiliary Hamiltonian system governing the leading-order stochastic dynamics of the system [57]. Notice that the function G(ξ, η, t) is linear with respect to η. Suppose that the function G(ξ, η, t) is defined, one can then obtain all characteristics of the stochastic process such as the average tumor size n(t) and the average frequency n res (t)/n(t) of resistant cells within a tumor. The former quantity is dn(t) dt = n(t)f (t) .
Using the normalization condition for the probability: i,j P ij (t) = 1, we obtain
G(ξ = 1, η = 1, t) = exp t 0f (r) dr ,
and then
n(t) = n(0) exp t 0f (r) dr = n(0)G(ξ = 1, η = 1, t) ,(7)
where the initial tumor size n(0) is sufficiently large. The latter quantity is written as follows
n res (t) n(t) = N i=0 P i1 (t) = N i=0 p i1 (t) exp − t 0f (r) dr = ∂G/∂η G(ξ, η, t) ξ=1,η=1 .(8)
Initial conditions yield p 00 (0) = 1 − κ, p 01 (0) = κ and p ij (0) = 0 for any other i or j, so that G(ξ, η, t = 0) = 1 − κ + κη.
To obtain the HJ equation related to the p.g.f. G(ξ, η, t), we multiply (5) on ξ i η j and sum up all equations for i = 0, 1, . . . , N and j = 0, 1. After some algebra, we obtain
∂G ∂t = s ξ ∂ ∂ξ + 1 − σ 1 − η ∂ ∂η − cη ∂ ∂η + u(ξ − 1) 2 1 − ξ N ∂ ∂ξ + v(η − 1) 2 1 − η ∂ ∂η G ,(9)
where only terms of order greater than or equal to u, v are retained, meaning that terms composed of the products s, c and u, v are omitted. Equation (9) is solved by the method of characteristics such that the HJ equation is transformed into a system of ordinary differential equations (i.e., the system of characteristics, see e.g.
[58]).
Constant treatment
We study the case for constant σ. Notice that this includes the case of no treatment (σ = 0).
First, we find the characteristics for the variables ξ and η. Namely, we write using (9):
dξ(t) dt = −sξ(t) + uξ(ξ − 1) 2N , dη(t) dt = (c − σ)η + vη(η − 1) 2 ,(10)
which gives
ξ(t) = s + u/(2N ) Ae (s+u/(2N ))t + u/(2N ) , η(t) = σ − c + v/2 Be (σ−c+v/2)t + v/2 ,(11)
where A and B are integration constants associated with initial values of ξ(0) and η(0) as
ξ(0) = s + u/(2N ) A + u/(2N ) , η(0) = σ − c + v/2 B + v/2 .(12)
The p.g.f. G(ξ, η, t) changes along the characteristic (11)- (12) according to the following ODE
dG(t) dt = s − σ + u(ξ(t) − 1) 2 + v(η(t) − 1) 2 G(t) ,
which is straightforward to integrate. Indeed, if we use (10), this yields: d ln G = (s(N + 1) − c)dt + N d ln ξ + d ln η. Then, with initial condition G(ξ(0), η(0), 0) = (1 − κ) + κη(0), κ is a level of resistance within a tumor (κ ∈ [0, 1]), (11) and (12), we finally obtain the solution to (9) of the following form G(ξ, η, t) = G(ξ(0), η(0), 0) exp (s − σ − (u + v)/2)t + N ln 1 + ξu 2N e (s+u/(2N ))t − 1 s + u/(2N ) + ln 1 + ηv 2
e (σ−c+v/2)t − 1 σ − c + v/2 .
Taking into account u, v s, c and assuming v σ − c, we can simplify (13) further and write its approximate form
G(ξ, η, t) ≈ 1 − κ + κηe (σ−c)t 1 + ηv 2 e (σ−c)t −1 σ−c exp (s − σ)t + N ln 1 + ξu 2N e st − 1 s + ln 1 + ηv 2 e (σ−c)t − 1 σ − c ,
which can be further simplified and written in the form
G(ξ, η, t) ≈ (1 − κ)(1 + ηv 2 e (σ−c)t − 1 σ − c ) + κηe (σ−c)t exp (s − σ)t + N ln 1 + ξu 2N e st − 1 s .(13)
As expected (13) is linear with respect to η.
The dynamics for the frequency of resistant cells within a tumor (8) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Log(Tumor size) C qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0% 50% 100% Resistance Figure S1. Tradeoff between growth and resistance under different treatment regimes. (A) Analytically-derived times for a tumor to reach 10 9 cells (see equation (1)). (B and C) Sample distributions for corresponding points B and C, shown in plot A. The bottom panel shows the mean number of additionally accumulated drivers for all detected tumors over intervals of 3 months. Light red points indicate tumors with a majority of resistant cells, while light blue points are for tumors dominated by non-resistant cells. The color-code indicates the level of resistance in detected tumors over 3 month intervals (see the colorbar on the right for details). Parameters otherwise as in Table 1. Treatment intensity, %
Detection year q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Mut rate to resist Figure S2. In the absence of resistance, higher treatment selects for faster growing subclones. The median (thick blue) and 95% confidence intervals (shaded blue area) for the distributions of detection times, when a resistant mutation is knocked out. (no initial level of resistance of a tumor (κ = 0) and zero mutation rate to acquire the resistance v = 0). For comparizon, the case of non-zero mutation rate v for the same initial conditions is shown besides in orange. The point color-code indicates the average number of additionally accumulated drivers within detected tumors. Probability of extinction, % Figure S3. Sufficiently small tumors can be driven to extinction by low dose therapies. The median (thick blue) and 95% confidence intervals (shaded area) for the distributions of extinction times. Red line indicates the probability of extinction, depending on initial cell number. Treatment level is 0.1%, and no pre-resistance κ = 0.0. Parameters otherwise as in Table 1.
Figure 1 .
1Mean field dynamics concord with numerical simulations. (A) Effect of treatment level and observation time on mean tumor size. (Inset) Mean frequency of resistant cells within tumors corresponding to three of the cases in A. Lines are analytically computed mean-field trajectories, while dots are numerical simulations (see Methods section for details). (B) Dynamics of mean and median tumor size, and percentiles around the mean (shaded areas), assuming a fixed constant treatment of 0.6%. Treatments start at t = 0, the maximal number of additionally accumulated drivers is 3. See
Figure 2 .
2Treatment level affects both detection time and frequency of resistance. (A) The median and 95% confidence intervals (shaded or hatched areas) of detection times for 0.4% and 0.8% selective advantages (see legend). (B, C and D) Three samples of the distribution of detection times for corresponding points B, C and D, shown in A. Dashed black line is the mean and the dotted line is the median. Bottom panel shows the mean number of additionally accumulated drivers within tumors over periods of 3 months. Light red points correspond to tumors with a majority of resistant cells; light blue points are for tumors with a majority of non-resistant cells. Color-code indicates the level of resistance in detected tumors over 3 month intervals. No pre-resistance is assumed. Other parameter values are as in
Figure 4 .
4Sensitivity analysis of several key parameters. The median (think line) and 95% confidence intervals (shaded areas with dashed boundaries) for the distribution of detection times. Parameter values are as inTable 1, except the one being varied: (A) maximal number of additionally accumulated drivers; (B) initial cell number n(0); (C) level of initial partial resistance of a tumor; (D) competitive parameter value α. The competition is implemented by introducing an exponential factor in the fitness calculation: f i0 = s(i + 1) − σ and f i1 = s(i + 1)e −αS − c, where S = i n i0 is the number of non-resistant cells and α characterizes the strength of competition. The insets show failure with respect to the change in the treatment level. The color-code for points indicates the average level of resistance within tumors, analogous toFig. 2. The insets show the percentage of cases in simulations, leading to detection of a tumor rather than its extinction or keeping its size below the detection threshold. For simplicity of representation, only the median is indicated in B, C and D for the baseline case, which is shown in green in A and with all parameter values, indicated in
the number of mutations from class (i, j) to classes (i + 1, j)
represent the probabilistic flows of mutations. For P (u)
Table 1. Canonical parameter values used in this study.Variable
Value
Reference
Time step (cell cycle length)
T
4 days
[2]
Selective advantage
s
0.4%
[2]
Cost of resistance
c
0.1%
Mutation rate to acquire an additional driver
u
3.4 × 10 −5
[2]
Mutation rate to acquire resistance
v
10 −6
[11]
Maximal number of additional drivers
N
5
Initial cell population
n(0)
10 6
Pre-resistance level
κ
0.01%
[29]
Number of replicate numerical simulations
-
10 6
(excl. the ones with extinction)
is then given by e (σ−c)t − 1 σ − c + κe (σ−c)t exp (s − σ)t + N ln 1 + ξu 2N 54. Color Brewer 2: color advice for cartography. URL http://colorbrewer2.org. 55. Code scripts used for numerical simulations. URL http://tiny.cc/tumor_mean-field_scripts. 56. Assaf M (2010) Theory of Large Fluctuations in Stochastic Populations. PhD thesis, Hebrew University of Jerusalem. URL http://guava.physics.uiuc.edu/~assaf/thesis.pdf. 57. Elgart V, Kamenev A (2004) Rare event statistics in reaction-diffusion systems. Phys Rev E Stat Nonlin Soft Matter Phys 70: 041106. 58. Melikyan AA (1998) Generalized characteristics of first order PDEs. Boston, MA: Birkhauser.∂G
∂η
≈ (1 − κ)
v
2
e st − 1
s
.
(14)
AcknowledgementThe authors are grateful to Athena Aktipis, Sylvain Gandon, Urszula Hibner, Patrice Lassus and Carlo Maley for discussions and helpful remarks. ARA thanks all members of Evolutionary Community Ecology group (University of Montpellier 2), especially Marie Vasse, Sarah Calba and Isabelle Gounand, for their support. This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET:www.sharcnet.ca) and Compute/Calcul Canada.
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"A tensorial description of the Turaev cobracket on genus 0 compact surfaces * Nariya Kawazumi",
"A tensorial description of the Turaev cobracket on genus 0 compact surfaces * Nariya Kawazumi"
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We give a tensorial description of the Turaev cobracket on any genus 0 compact surface by the standard group-like expansion, where the Bernoulli numbers appear.IntroductionThe free homotopy set of free loops on an oriented surface S,π =π(S) = [S 1 , S] = π 1 (S)/(conjugate), has rich structures. In the classical theory of Riemann surfaces, the algebraic intersection number of two free loops plays an central role. As a noncommutative generalization of the intersection number, the Goldman bracket [4] of two free loops appears in the Weil-Petersson symplectic geometry [23], the Poisson structure on the moduli space of flat bundles [4] and the Skein algebra of links in the 3-manifold S × [0, 1] [22]. In the case where S is a compact surface with connected boundary, Kuno and the author [8] gave a tensorial description of the Goldman bracket, and described Dehn twists on the surface S in terms of the Goldman Lie algebra. These results are generalized to any compact surfaces with non-empty boundary in [16] [9] [11].On the other hand, the Turaev cobracket δ is related to Turaev's earlier work[21], and was introduced by Turaev [22] in connection with the Skein algebra. It is a dual notion of the Goldman bracket, and measures the self-intersection of a single free loop. But little is known about the Turaev cobracket. As was discovered by Kuno and the author [10], the Turaev cobracket gives a geometric constraint of the images of the (higher) Johnson homomorphisms. In order to deduce some results from this fact, we need a tensorial description of the Turaev cobracket. In [10] and [17], the lowest degree term of the description was computed. When the preprint of this paper[7]was uploaded at the arXiv (June 10, 2015), there was no other full results on the tensorial description.In this paper we will give the tensorial description of the Turaev cobracket for any genus 0 compact surface with respect to the standard group-like expansion θ std . Unfortunately the expansion θ std does not reflect the topology of the surface enough, so that we cannot deduce topological consequences from our result.
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1506.03174
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0211460aeeb16559160e159ddc81df0fa14af351
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A tensorial description of the Turaev cobracket on genus 0 compact surfaces * Nariya Kawazumi
20 Jun 2016
A tensorial description of the Turaev cobracket on genus 0 compact surfaces * Nariya Kawazumi
20 Jun 2016The description is stated in Theorem1.2, where the Bernoulli numbers appear. In this paper, following the convention in [16], we agree that the function s(z) and the *AMS subject classifications: Primary 57N05; Secondary 20F3432G15 Keywords: Turaev cobracketBernoulli numbers
We give a tensorial description of the Turaev cobracket on any genus 0 compact surface by the standard group-like expansion, where the Bernoulli numbers appear.IntroductionThe free homotopy set of free loops on an oriented surface S,π =π(S) = [S 1 , S] = π 1 (S)/(conjugate), has rich structures. In the classical theory of Riemann surfaces, the algebraic intersection number of two free loops plays an central role. As a noncommutative generalization of the intersection number, the Goldman bracket [4] of two free loops appears in the Weil-Petersson symplectic geometry [23], the Poisson structure on the moduli space of flat bundles [4] and the Skein algebra of links in the 3-manifold S × [0, 1] [22]. In the case where S is a compact surface with connected boundary, Kuno and the author [8] gave a tensorial description of the Goldman bracket, and described Dehn twists on the surface S in terms of the Goldman Lie algebra. These results are generalized to any compact surfaces with non-empty boundary in [16] [9] [11].On the other hand, the Turaev cobracket δ is related to Turaev's earlier work[21], and was introduced by Turaev [22] in connection with the Skein algebra. It is a dual notion of the Goldman bracket, and measures the self-intersection of a single free loop. But little is known about the Turaev cobracket. As was discovered by Kuno and the author [10], the Turaev cobracket gives a geometric constraint of the images of the (higher) Johnson homomorphisms. In order to deduce some results from this fact, we need a tensorial description of the Turaev cobracket. In [10] and [17], the lowest degree term of the description was computed. When the preprint of this paper[7]was uploaded at the arXiv (June 10, 2015), there was no other full results on the tensorial description.In this paper we will give the tensorial description of the Turaev cobracket for any genus 0 compact surface with respect to the standard group-like expansion θ std . Unfortunately the expansion θ std does not reflect the topology of the surface enough, so that we cannot deduce topological consequences from our result.
Bernoulli numbers B 2m are defined by
s(z) = 1 e −z − 1 + 1 z = − 1 2 − ∞ m=1 B 2m (2m)! z 2m−1 = − 1 2 − 1 12 z + 1 720 z 3 − 1 30240 z 5 + · · · .
The appearance of the Bernoulli numbers comes from the tensorial description of the homotopy intersection form by Massuyeau-Turaev [16] (Theorem 2.3), and a formula for the coaction operation µ by Fukuhara-Kawazumi-Kuno [3] (Theorem 2.1). The Kashiwara-Vergne problem in the formulation by Alekseev-Torossian [2] looks for a group-like expansion of the fundamental group of a pair of pants which is compatible with all the boundary components and satisfies some equation involved with the Bernoulli numbers and the divergence cocycle. As the author announced in [6], a regular homotopy version of the Turaev cobracket on genus 0 compact surfaces includes the divergence cocycle. Hence the result in this paper seems to suggest the following conjecture.
Conjecture 0.1. The tensorial description of the Turaev cobracket with respect to any solution to the Kashiwara-Vergne problem is of simple expression. In particular, the description might be formal, namely, might equal its lowest degree term.
It is our working hypothesis for studying the higher Johnson homomorphisms that there is a symplectic expansion for a compact surface with connected boundary whose description of the Turaev cobracket equals the lowest degree term, i.e., Schedler's cobracket [20]. In fact, Kuno [13] already found such an expansion for the surface of genus 1 with connected boundary up to degree 10 by a computer calculation. If Conjecture 0.1 would be true, our hypothesis should be a positive genus analogue of the Kashiwara-Vergne problem.
After the preprint of this paper was uploaded, Alekseev, Kuno, Naef and the author [1] obtained a formal description of the Turaev cobracket by regarding solutions of the Kashiwara-Vergne problem as special expansions for genus 0 compact surfaces. This means that Conjecture 0.1 is true. Independently from our results, Massuyeau [15] obtained a formal description of the Turaev cobracket for genus 0 compact surfaces by the Kontsevich integral. Theorem 2.3 in this paper is a modification of a theorem of Massuyeau and Turaev [16]. It says that the value of a group-like expansion at the boundary loop of a surface with connected boundary completely determines the tensorial description of the homotopy intersection form by the expansion. As is showed by Naef [18], this fact can be generalized in the light of a non-commutative Poisson geometry, which is one of the foundations of the work [1].
The author thanks Anton Alekseev, Yusuke Kuno, Florian Naef and Shunsuke Tsuji for helpful discussions. The first draft of this paper was written during my stay at IRMA, Strasbourg, on the occasion of the JSPS-CNRS joint project on Teichmüller spaces and surface mapping class groups. He would like to express his gratitude to IRMA for kind hospitality. He is partially supported by the Grant-in-Aid for Scientific Research (S) (No.24224002), (B) (No.24340010) and (B) (No.15H03617) from the Japan Society for Promotion of Sciences.
Statement of the Result
Let S be a compact connected oriented surface with non-empty boundary. It is classified by its genus and the number of its boundary components, so that we may denote the surface S by the symbol Σ g,n+1 for some g, n ≥ 0. Here the genus of S is g, and the number of the boundary components is n + 1. The fundamental group of the surface S is free of rank 2g+n. In general, for a free group π of finite rank, we have the notion of group-like expansion. See [14]. In order to recall the definition of a grouplike expansion, we need to prepare some tensor algebra. Let H be the first rational homology group of π, i.e., H := (π/[π, π])⊗ Z Q. We denote [γ] := (γ mod [π, π])⊗1 ∈ H for any γ ∈ π. The completed tensor algebra T = T (H) := ∞ m=0 H ⊗m is endowed with the topology by the decreasing filtration T ≥p := m≥p H ⊗m , p ≥ 1, and has the strucuture of a complete Hopf algebra with an augmentation ε : T → Q, a coproduct ∆ : T → T ⊗ T and an antipode ι : T → T . They are defined to be the unique continuous algebra (anti)-homomorphisms satisfying ε(X) = 0, ∆(X) = X ⊗1+ 1 ⊗X and ι(X) = −X for any X ∈ H, respectively. The group ring Qπ is also a Hopf algebra. The augmentation ε : Qπ → Q, the coproduct Qπ → Qπ ⊗ Qπ and the antipode ι : Qπ → Qπ are the unique algebra (anti)-homomorphisms satisfying ε(γ) = 1, ∆(γ) = γ ⊗ γ and ι(γ) = γ −1 for any γ ∈ π, respectively. The completion of Qπ with respect to the augmentation ideal Iπ := Ker ε, Qπ := lim ← −p→∞ Qπ/(Iπ) p , is a complete Hopf algebra in a natural way. Definition 1.1 (See [16]). The map θ : π → T is a group-like expansion if the following three conditions hold:
1. The map θ is multiplicative, i.e., we have θ(γ 1 γ 2 ) = θ(γ 1 )θ(γ 2 ) for any γ 1 and γ 2 ∈ π.
2. For any γ ∈ π, θ(γ) ≡ 1 + [γ] (mod T ≥2 ).
3. For any γ ∈ π, θ(γ) ∈ T is group-like, i.e., ∆θ(γ) = θ(γ) ⊗θ(γ) ∈ T ⊗ T .
The linear extension of any group-like expansion induces an isomorphism of complete Hopf algebras θ : Qπ
∼ = −→ T ,
a γ γ → a γ θ(γ). The group-like expansion we study in this paper is defined as follows. Let S be the genus 0 compact surface Σ 0,n+1 for some n ≥ 0. Number the boundary components as ∂S = n k=0 ∂ k S, and choose a basepoint * ∈ ∂ 0 S. The standard generators γ k ∈ π 1 (S, * ), 1 ≤ k ≤ n, are given such that each γ k is a simple loop going around the k-th boundary ∂ k S in the positive direction, and the product γ 1 γ 2 · · · γ n ∈ π 1 (S, * ) is homotopic to a simple loop around the 0-th boundary ∂ 0 S in the negative direction. Here we read the product γ 1 γ 2 · · · γ n as a loop going along first γ 1 , next γ 2 , and finally γ n . Here we remark that ǫ( · γ k (0), · γ k (1)) = +1. The fundamental group π 1 (S, * ) is a free group of rank n with free generators γ k , 1 ≤ k ≤ n. We denote by x k := [γ k ] ∈ H = H 1 (S; Q), 1 ≤ k ≤ n, the homology class of γ k . Equivalently x k is the homology class of the k-th boundary ∂ k S, so that we define
x 0 := [∂ 0 S] = −[γ 1 γ 2 · · · γ n ] = − n k=1 x k ∈ H = H 1 (S; Q)
. Then we can consider the exponential
e x k = exp(x k ) = ∞ m=0 1 m! x k m ∈ T = T (H 1 (S; Q)).
We define the standard grouplike expansion θ std : π = π 1 (S, * ) → T = T (H 1 (S; Q)) as the unique group-expansion satisfying θ std (γ k ) = e x k , 1 ≤ ∀k ≤ n. Here we require these conditions only for k ≥ 1, not for k = 0. The reason why one can compute the tensorial description of the Turaev cobracket with respect to the expansion θ std is that we can apply Theorem 2.1 to x k = θ std (log(γ k )).
Let δ : Zπ ′ → Zπ ′ ⊗ Zπ ′ be the Turaev cobracket [22]. Here Zπ ′ := Zπ/Z1 is the quotient of the Z-free module over the setπ, Zπ, by the linear span of the constant loop 1 ∈π. We denote by | | ′ : Zπ 1 (S, p) → Zπ → Zπ/Z1 = Zπ ′ the quotient map for any p ∈ S. The definition of the Turaev cobracket will be stated in §2. The Goldman bracket and the Turaev cobracket make Zπ ′ a Lie bialgebra in the sense of Drinfel'd [22], so that we call it the Goldman-Turaev Lie bialgebra of the surface S. The bialgebra has a completion with respect to the augmentation ideal Iπ,
Qπ := lim ← −p→∞ Qπ ′ /|(Iπ) p | ′ . We have a natural continuous extension | | ′ : Qπ → Qπ.
The Goldman bracket and the Turaev cobracket extend continuously to Qπ [9] [10].
In particular, the Turaev cobracket is a continuous map δ : Qπ → Qπ ⊗ Qπ.
On the tensor algebra side, we denote by N ( T ) the quotient of T by the closure
of Q1 + [ T , T ], where [ T , T ] is the Q-linear subspace of T generated by the set {uv − vu; u, v ∈ T }. The vector space N ( T ) is naturally isomorphic to the space of cyclic invariants ∞ m=1 (H ⊗m ) Z/m ,
where the cyclic group Z/m acts on the space H ⊗m by cyclic permutation. We denote by | | ′ : T → N ( T ) the quotient map. Any group-like expansion θ induces a topological isomorphism θ : Qπ [9]. Thus we have the tensorial description δ θ of the Turaev cobracket with respect to θ defined by the diagram
∼ = → N ( T )Qπ δ − −−− → Qπ ⊗ Qπ θ θ ⊗θ N ( T ) δ θ − −−− → N ( T ) ⊗N ( T ).
Now we can formulate our result.
Theorem 1.2. Let δ std = δ θ std be the tensorial description of the Turaev cobracket with respect to the standard group-like expansion θ std for the surface S = Σ 0,n+1 . Then, for any m ≥ 1 and any k 1 , k 2 , . . . , k m ∈ {1, 2, . . . , n}, we have
δ std (x k 1 x k 2 · · ·x km ) = alt(| | ′ ⊗ | | ′ ) 1≤i<j≤m K k i k j (x k j+1 · · · x km x k 1 · · · x k i−1 ⊗x k i+1 · · · x k j−1 ) − 1 2 m i=1 x k 1 · · · x k i−1 x k i+1 · · · x km ⊗x k i + m i=1 ∞ q=1 B 2q (2q)! 2q−1 p=0 (−1) p 2q p x k 1 · · · x k i−1 x k i p x k i+1 · · · x km ⊗x k i 2q−p .
Here, for 1 ≤ k, l ≤ n, we denote
K k,l := (1 ⊗ι)∆ ǫ kl x k x l − δ kl x k 2 e −x k − 1 ∈ T ⊗ T ,
where δ kl is the Kronecker delta, ǫ kl is defined by
ǫ kl := 1, if k > l, 0, if k ≤ l,
and alt :
N ( T ) ⊗N ( T ) → N ( T ) ⊗N ( T ), u ⊗v → u ⊗v − v ⊗u,
is the alternating operator.
Preliminaries
Let S be a compact connected oriented surface with non-empty boundary. Choose a basepoint * ∈ ∂S, and denote π := π 1 (S, * ). We begin by recalling the coaction µ : Zπ → Zπ ⊗ Zπ ′ , which is introduced in [10] inspired by a construction of Turaev [21]. The alternating part of µ is just the Turaev cobracket δ, but µ is of multiplicative nature as stated below. Choose another point * + ∈ ∂S near * in the positive direction. For any γ ∈ π we regard it as a path from * to * + , and choose a representative of γ in general position. By abuse of notation, we also denote by γ the representative. Then the curve γ is an immersion, and its singularities are at worst transverse double points. For each double point p of γ we have a unique pair
0 < t p 1 < t p 2 < 1 of parameters such that γ(t p 1 ) = γ(t p 2 ) = p. Then µ(γ) ∈ Zπ ⊗ Zπ ′ is defined by µ(γ) := − p ε( · γ(t p 1 ), · γ(t p 2 ))(γ 0t p 1 γ t p 2 1 ) ⊗ |γ t p 1 t p 2 | ′ ,
where the sum runs over the set of self-intersection points of γ, ε( · γ(t p 1 ), · γ(t p 2 )) ∈ {±1} is the local intersection number with respect to the orientation of S, and γ s 1 s 2 is the restriction of γ to the interval [s 1 , s 2 ] ⊂ [0, 1] for any 0 ≤ s 1 < s 2 ≤ 1. The operation µ is well-defined, i.e., independent of the choice of a representative [10]. The Turaev cobracket δ : Zπ ′ → Zπ ′ ⊗ Zπ ′ [22] can be defined to be the alternating part of µ
δ • | | ′ = alt • (1 ⊗ | | ′ ) • µ : Zπ → Zπ ′ ⊗ Zπ ′ .(1)
Here alt : Zπ ′ ⊗ Zπ ′ → Zπ ′ ⊗ Zπ ′ is the alternating operator as above. The map µ extends continuously to the map µ : Qπ → Qπ ⊗ Qπ. For example, the extension µ is computed as follows.
µ(log γ) = 1 2 1 ⊗| log γ| ′ + ∞ m=1 B 2m (2m)! 2m−1 p=0 2m p (−1) p (log γ) p ⊗|(log γ) 2m−p | ′ .
We can define the tensorial description of the map µ θ : T → T ⊗N ( T ) with respect to any group-like expansion θ of the fundamental group π 1 (S, * ). Theorem 1.2 follows immediately from the following.
Theorem 2.2. Let δ std = δ θ std be the tensorial description of the Turaev cobracket with respect to the standard group-like expansion θ std for the surface S = Σ 0,n+1 . Then, for any m ≥ 1 and any k 1 , k 2 , . . . , k m ∈ {1, 2, . . . , n}, we have
µ std (x k 1 x k 2 · · · x km ) = (1 ⊗ | | ′ ) 1≤i<j≤m (x k 1 · · · x k i−1 ⊗1)K k i k j (x k j+1 · · · x km ⊗x k i+1 · · · x k j−1 ) − 1 2 m i=1 x k 1 · · · x k i−1 x k i+1 · · · x km ⊗x k i + m i=1 ∞ q=1 B 2q (2q)! 2q−1 p=0 (−1) p 2q p x k 1 · · · x k i−1 x k i p x k i+1 · · · x km ⊗x k i 2q−p .
Here it should be remarked |x k 1 · · · x k i−1 x k j+1 · · · x km | ′ = |x k j+1 · · · x km x k 1 · · · x k i−1 | ′ ∈ N ( T ). The rest of this paper is devoted to the proof of Theorem 2.2.
Our proof consists of Theorem 3.1, Theorem 2.1, and (a slight modification of) the tensorial description of the homotopy intersection form by Massuyeau-Turaev [16], which we will explain later in short. Let S be a (general) connected compact oriented surface with non-empty boundary. Choose basepoints * and * + in ∂S as above. Then, using a short path along the boundary from * to * + , we identify the fundamental groups π = π 1 (S, * ) and π 1 (S, * + ) with the homotopy set of path from * to * + and that from * + to * . Then the homotopy intersection form η : Zπ 1 (S, * )⊗Zπ 1 (S, * + ) → Zπ, introduced by Papakyriakopoulos [19] and Turaev [21] independently, is defined as follows. For γ 1 ∈ π 1 (S, * ) and γ 2 ∈ π 1 (S, * + ) we choose their representatives in general position. Then η(γ 1 , γ 2 ) ∈ Zπ is defined by
η(γ 1 , γ 2 ) := p∈γ∩δ ε p (γ 1 , γ 2 )(γ 1 ) * p (γ 2 ) p * + ,
where ε p (γ 1 , γ 2 ) ∈ {±1} is the local intersection number of γ 1 and γ 2 at the intersection point p, (γ 1 ) * p the segment of γ 1 from * to p, and (γ 2 ) p * + that of γ 2 from p to * + . We define a map κ : Zπ ⊗ Zπ → Zπ ⊗ Zπ by
κ(γ 1 , γ 2 ) := −(1 ⊗ γ 2 ) ((1 ⊗ ι)∆η(γ 1 , γ 2 )) (1 ⊗ γ 1 ) for γ 1 , γ 2 ∈ π. In other words, if we denote ∆u = u ′ ⊗ u ′′ and ∆v = v ′ ⊗ v ′′ for u, v ∈ Qπ, we define κ(u, v) = − (1 ⊗ v ′′ ) (1 ⊗ ι)∆η(u ′ , v ′ ) (1 ⊗ u ′′ ).(2)
Then we have a product formula µ(γ 1 γ 2 ) = µ(γ 1 )(γ 2 ⊗ 1) + (γ 1 ⊗ 1)µ(γ 2 ) + (1 ⊗ | | ′ )κ(γ 1 , γ 2 ).
More generally, we have
µ(u 1 u 2 · · · u m ) = m i=1 ((u 1 · · · u i−1 ) ⊗ 1)µ(u i )((u i+1 · · · u m ) ⊗ 1) + i<j ((u 1 · · · u i−1 ) ⊗ 1)(1 ⊗ | | ′ ) (κ(u i , u j )(u j+1 · · · u m ⊗ u i+1 · · · u j−1 )) ( * )
for any m ≥ 1 and any u 1 , u 2 , . . . , u m ∈ Zπ [10] (Corollary 4.3.4). Massuyeau and Turaev [16] gave explicitly the tensorial description of the homotopy intersection form η with respect to any symplectic expansion [14] in the case S = Σ g,1 , g ≥ 1, i.e., the boundary ∂S is connected. In this case, we denote by ⋆ ∈ ∂S a basepoint on the boundary, and by ζ ∈ π 1 (S, ⋆) the simple loop along the boundary in the negative orientation. The algebraic intersection number H ⊗ H → Q, X ⊗ Y → X · Y , is a non-degenerate pairing on H. The symplectic form
ω := g i=1 A i B i − B i A i ∈ H ⊗2 ⊂ T is independent of the choice of a symplectic basis {A i , B i } g i=1 ⊂ H = H 1 (Σ g,1 ; Q)
. Throughout this paper we omit the symbol ⊗ when it indicates the product in T . We have θ(ζ) ≡ 1 + ω (mod T ≥3 ) for any group-like expansion θ. Massuyeau [14] introduced the notion of a symplectic expansion: A group-like expansion θ :
π → T is symplectic if θ(ζ) = exp(ω)(= ∞ m=0 1 m! ω m ) ∈ T ,
i.e., log θ(ζ) = ω ∈ T . Symplectic expansions (in rational coefficients) exist [14] [12]. See also [5] for symplectic expansions in real coefficients. While their result deals only with symplectic expansions, but it is not hard to generalize it to any group-like expansion.
In order to give the tensorial description, Massuyeau and Turaev [16] introduced a continuous operation
• : T ≥1 × T ≥1 → T by (X 1 · · · X l−1 X l ) • (Y 1 Y 2 · · · Y m ) := (X l · Y 1 )X 1 · · · X l−1 Y 2 · · · Y m
for any l, m ≥ 1 and any X i , Y j ∈ H = H 1 (Σ g,1 ; Q). Minus the sympletic form is the unit for the operation • , i.e., (−ω)
• u = u • (−ω) = u for any u ∈ T ≥1 . The restriction of • to T ≥2 is associative, and • ( T ≥l × T ≥m ) ⊂ T ≥(l+m−2) . Hence, for any Z ∈ (−ω) + T ≥3 , there exists a unique Z −1 ∈ (−ω) + T ≥3 such that Z • Z −1 = Z −1 • Z = −ω.
Theorem 2.3 (Massuyeau-Turaev [16]). Let θ : π 1 (Σ g,1 , ⋆) → T be a group-like expansion. We denote Ω = Ω θ := log θ(ζ) ∈ ω + T ≥3 . Then the tensorial description of the homotopy intersection form η : Qπ × Qπ → Qπ with respect to the expansion θ, ρ θ , is given by
ρ θ (a, b) = (a − ε(a)) • ((−Ω) −1 + ωs(Ω)ω) • (b − ε(b))
for any a, b ∈ T .
Proof. We modify the proof of Theorem 10.4 in Massuyeau-Turaev [16]. The tensorial description ρ θ is characterized by the condition
∀X ∈ H, ρ θ (X, e −Ω ) = X.(3)
Since s(z)z − 1 = z(e −z − 1) −1 , we have ρ θ (X, e −Ω ) = ρ θ (X, Ω) e −Ω − 1 Ω = ρ θ (X, Ω)(s(Ω)Ω − 1) −1 .
Hence the condition (3) is equivalent to
∀X ∈ H, ρ θ (X, Ω) = Xs(Ω)Ω − X.
Now the map (a, b) ∈ T × T → (a − ε(a))s(Ω)(b − ε(b)) ∈ T is a Fox pairing in the sense of Massuyeau-Turaev [16]. Hence, if we introduce a unique Fox pairing ρ Ω : T × T → T characterized by the condition
∀X ∈ H, ρ Ω (X, Ω) = −X,(5)then we have ρ θ (a, b) = ρ Ω (a, b) + (a − ε(a))s(Ω)(b − ε(b))
for any a and b ∈ T . Let {A i , B i } g i=1 ⊂ H be a symplectic basis. The tensor
R Ω := g i,j=1 ( − B i ρ Ω (A i , A j )B j + B i ρ Ω (A i , B j )A j + A i ρ Ω (B i , A j )B j − A i ρ Ω (B i , B j )A j ) ∈ T ≥2 satisfies ρ Ω (a, b) = (a−ε(a)) • R Ω • (b−ε(b))
for any a and b ∈ T . Then the condition (5) is equivalent to R Ω • Ω = ω. This means R Ω = (−Ω) −1 . Therefore we have
ρ θ (a, b) = (a − ε(a)) • R Ω • (b − ε(b)) + (a − ε(a))s(Ω)(b − ε(b)) = (a − ε(a)) • ((−Ω) −1 + ωs(Ω)ω) • (b − ε(b)).
This proves the theorem.
Proof of the Result
Now we begin the proof of Theorem 2.2, from which Theorem 1.2 follows immediately by (1). Let S be the genus 0 compact surface Σ 0,n+1 for some n ≥ 0. We consider the standard group-like expansion θ std : π = π 1 (S, * ) → T = T (H 1 (Σ 0,n+1 ; Q)). Choose one point * k ∈ ∂ k S for each component ∂ k S and let ξ k ∈ π 1 (S, * k ) be the simple positive boundary loop for 1 ≤ k ≤ n. We can choose a simple path χ k from * ∈ ∂ 0 S to * k such that χ k ξ k χ k −1 = γ k ∈ π 1 (S, * ). We glue n copies of the surface Σ 1,1 to the surface S = Σ 0,n+1 along the boundary ∂ k S, 1 ≤ k ≤ n, such that the basepoints ⋆ and * k are identified with each other. This gluing yields a surfaceŜ ∼ = Σ n,1 . Let {α k , β k } be a symplectic generator of the fundamental group of the k-th copy of Σ 1,1 with basepoint ⋆. Then the set {χ k α k χ k −1 , χ k β k χ k −1 } n k=1 is a symplectic generator of the fundamental group π 1 (Ŝ, * ). If we denote
A k := [χ k α k χ k −1 ] and B k := [χ k β k χ k −1 ] ∈ H 1 (Ŝ; Q), then the set {A k , B k } g k=1
is a symplectic basis of the homology group H 1 (Ŝ; Q). The map ı :
T = T (H 1 (S; Q)) → T (H 1 (Ŝ; Q)) defined by ı(x k ) := A k B k − B k A k is an injective algebra homomorphism. See [9] §6.2.
Let θ k : π 1 (Σ 1,1 , ⋆) → T (H 1 (Σ 1,1 ; Q)) be a symplectic expansion for the k-th copy of Σ 1,1 . We identify the target with the completed tensor algebra T (QA k ⊕ QB k ) ⊂ T (H 1 (Ŝ; Q)), and define a group-like expansionθ :
π 1 (Ŝ, * ) → T (H 1 (Ŝ; Q)) byθ(χ k α k χ k −1 ) := θ k (α k ) andθ(χ k β k χ k −1 ) := θ k (β k ). Then the diagram π 1 (S, * ) θ std − −−− → T (H 1 (S; Q)) i * ı π 1 (Ŝ, * )θ − −−− → T (H 1 (Ŝ; Q))
commutes, where i : (S, * ) ֒→ (Ŝ, * ) is the inclusion. We haveθ(ζ) = n k=1 exp(A k B k − B k A k ) = ı( n k=1 exp(x k )). Here we denote by u * v the Baker-Campbell-Hausdorff series of u and v ∈ T ≥1 = T (H 1 (S; Q)) ≥1
u * v := log((exp u)(exp v)) = u + v + 1 2 [u, v] + 1 12 [u, [u, v]] + 1 12 [v, [v, u]] + · · · ,
and consider the element Ξ := x 1 * x 2 * · · · * x n ∈ T ≥1 . Then we obtain logθ(ζ) = ı(Ξ) ∈ T (H 1 (Ŝ; Q)), and, from the Massuyeau-Turaev theorem 2.3,
ρθ(a, b) = (a − ε(a)) • ((−ı(Ξ)) −1 + ωs(ı(Ξ))ω) • (b − ε(b))(6)
for any a, b ∈ T (H 1 (Ŝ; Q)). By the injective homomorphism ı, the Massuyeau-Turaev operation
• on T (H 1 (Ŝ; Q))
induces a continuous operation on T ≥1 = T (H 1 (Σ 0,n+1 ; Q)) ≥1 ,
• : T ≥1 × T ≥1 −→ T ≥1 ,
given by
x i 1 · · · x i l−1 x i l • x j 1 x j 2 · · · x jm = −δ i l j 1 x i 1 · · · x i l−1 x j 1 x j 2 · · · x jm
for l, m ≥ 1 and 1 ≤ i 1 , . . . , i l , j 1 , . . . , j m ≤ n. In fact, we have (
A k B k −B k A k ) • (A l B l − B l A l ) = −δ kl (A k B k − B k A k ) for 1 ≤ k, l ≤ n.
The operation • on T ≥1 is associative with unit x 0 = − n k=1 x k . Thus we can take the inverse element Z −1 of any Z ∈ x 0 + T ≥2 with respect to the operation
• , Z −1 • Z = Z • Z −1 = x 0 .
Consider the inverse element −Ξ −1 of −Ξ = −x 1 * x 2 * · · · * x n with respect to the operation • .
Theorem 3.1.
−Ξ −1 + x 0 s(Ξ)x 0 = x 0 − k>l x k x l + n k=1 s(x k )x k 2 = − k>l x k x l + n k=1 x k 2 e −x k − 1 .
Proof. We denote the left-hand side by
Y := −Ξ −1 + x 0 s(Ξ)x 0 = ∞ m=1 Y (m) , Y (m) ∈ H ⊗m . Since Ξ ≡ −x 0 + 1 2 k<l [x k , x l ] (mod T ≥3 ), we have Y (1) = x 0 and Y (2) = 1 2 k<l [x k , x l ] − 1 2 x 0 2 = 1 2 k<l (x k x l − x l x k ) − 1 2 k<l (x k x l + x l x k ) − 1 2 n k=1 x k 2 = − k>l x k x l − 1 2 n k=1 x k 2 .
To compute the higher degree term Y (m) for each m ≥ 3, we introduce a topological algebra automorphism Q of T defined by
Q(x k ) = −x n−k , 1 ≤ k ≤ n,
inspired by Kuno's work [13]. See also [12]
On the other hand, we have
Y −1 = −1 + e −Ξ = −1 + e −xn · · · e −x 2 e −x 1 .(8)In fact, Ξ = Ξ • Y • Y −1 = −Ξ • Ξ −1 • Y −1 + Ξs(Ξ)Y −1 = −Y −1 + Ξs(Ξ)Y −1 = Ξ e −Ξ − 1 Y −1 .
Since the algebra T has no zero divisor, we obtain (8).
Let W (resp. I) be the closed linear subspace in T ≥1 generated by the set {x k 1 x k 2 · · · x km ; k 1 ≥ k 2 ≥ · · · ≥ k m } (resp. {x k 1 x k 2 · · · x km ; ♯{k 1 , k 2 , . . . , k m } ≥ 2}). The subspace W (resp. I) is a subalgebra (resp. a two-sided ideal) of T ≥1 with respect to the multiplication • . Since
Y = x 0 + ∞ m=1 m times (x 0 − Y −1 ) • (x 0 − Y −1 ) • · · · • (x 0 − Y −1 )
. (8)
and x 0 − Y −1 ∈ W fromY −1 • (x 0 + n k=1 x k 2 s(x k )) = Y −1 • ( n k=1 x k x k e −x k − 1 ) ≡( n k=1 e −x k − 1) • ( n k=1 x k x k e −x k − 1 ) = − n k=1 (e −x k − 1) x k e −x k − 1 = x 0 .
Hence we have Y ≡ x 0 + n k=1 x k 2 s(x k ) (mod I), as was to be shown.
As a corollary, we conclude
ρ θ std (a, b) = (a − ε(a)) • (− k>l x k x l + n k=1 x k 2 e −x k − 1 ) • (b − ε(b))(9)
for any a, b ∈ T = T (H 1 (S; Q)). In particular, by (2), we have
κ std (x k , x l ) = −((1 ⊗ι)∆ ǫ kl x k x l − δ kl x k 2 e −x k − 1 = −K k,l ∈ T ⊗ T ,(10)
where κ std is the tensorial description of κ with respect to the standard exponential expansion θ std . Recall x k = log θ std (γ k ). Consequently, substituting (10) and Theorem 2.1 to the product formula (*), we obtain Theorem 2.2. This completes the proof.
Theorem 2. 1
1([3]). If γ ∈ π 1 (S, * ) is represented by a simple loop with ε(
Example 5 . 3 .
53It is clear to see Q(Ξ) = −Ξ and Qx 0 = −x 0 . Here we haveQ(u • v) = −(Qu) • (Qv)for any u and v ∈ T ≥1 . In fact, we compute (Qx k )• (Qx l ) = (−x n−k ) • (−x n−l ) = −δ kl x n−k = Q(δ kl x k ) = −Q(x k • x l )for any 1 ≤ k, l ≤ n. In particular, for anyZ ∈ x 0 + T ≥2 , we have x 0 = −Qx 0 = −Q(Z • Z −1 ) = (QZ) • (QZ −1 ), and so Q(Z −1 ) = (QZ) −1 . Moreover we have s(−z) = −1 − s(z). ThereforeQY = −(QΞ) −1 + x 0 s(QΞ)x 0 = Ξ −1 − x 0 2 − x 0 s(Ξ)x 0 = −Y − x 0 2 .
, we have Y ∈ W . It is clear that the direct sumdecomposition W = (W ∩ I) ⊕ n k=1 x k Q[[x k ]] holds, and so W ∩ Ker(Q + 1) ⊂ n k=1 x k Q[[x k ]], while we have Q(Y − Y (2) ) = −(Y − Y (2) ) from (7). Hence we have Y − Y (2) ∈ n k=1 x k Q[[x k ]]. This implies that it suffices to show the theorem modulo the ideal I. From (8) we have
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|
[] |
[
"Phenomenology of transverse spin: past, present and future",
"Phenomenology of transverse spin: past, present and future"
] |
[
"Mariaelena Boglione \nDipartimento di Fisica Teorica\nUniversità di Torino\nINFN\nSezione di Torino\nVia P. Giuria 1I-10125TorinoItaly\n",
"Alexei Prokudin \nDivision of Science\nPenn State Berks\n19610ReadingPAUSA\n"
] |
[
"Dipartimento di Fisica Teorica\nUniversità di Torino\nINFN\nSezione di Torino\nVia P. Giuria 1I-10125TorinoItaly",
"Division of Science\nPenn State Berks\n19610ReadingPAUSA"
] |
[] |
We summarize the most significant aspects in the study of transverse spin phenomena over the last few decades, focusing on Semi-Inclusive Deep Inelastic Scattering processes and hadronic production in e + e − annihilations. The phenomenology of transverse momentum dependent distribution and fragmentation functions will be reviewed in an in-depth analysis of the most recent developments and of the future perspectives.PACS. 13.88.+e Polarization in interactions and scattering -13.60.-r Inclusive production with identified hadrons -13.85.Ni Photon and charged-lepton interactions with hadrons
|
10.1140/epja/i2016-16154-6
|
[
"https://arxiv.org/pdf/1511.06924v2.pdf"
] | 54,989,137 |
1511.06924
|
5a2d21965bce45b4eb321f2f8b9d397cea09a514
|
Phenomenology of transverse spin: past, present and future
2 Dec 2015
Mariaelena Boglione
Dipartimento di Fisica Teorica
Università di Torino
INFN
Sezione di Torino
Via P. Giuria 1I-10125TorinoItaly
Alexei Prokudin
Division of Science
Penn State Berks
19610ReadingPAUSA
Phenomenology of transverse spin: past, present and future
2 Dec 2015Received: date / Revised version: dateEPJ manuscript No. (will be inserted by the editor)
We summarize the most significant aspects in the study of transverse spin phenomena over the last few decades, focusing on Semi-Inclusive Deep Inelastic Scattering processes and hadronic production in e + e − annihilations. The phenomenology of transverse momentum dependent distribution and fragmentation functions will be reviewed in an in-depth analysis of the most recent developments and of the future perspectives.PACS. 13.88.+e Polarization in interactions and scattering -13.60.-r Inclusive production with identified hadrons -13.85.Ni Photon and charged-lepton interactions with hadrons
Past
The importance of the transverse motion of partons bound inside the nucleon and the corresponding azimuthal effects were first pointed out in the 70's by Feynman, Field and Fox [1,2], who realized that the origin of transverse momentum in Drell-Yan processes could be either due to non zero intrinsic momentum of partons confined in the nucleon (non-perturbative effect) or to the recoil of gluons radiated off active quarks (perturbative effect). Their papers are the precursors of the Generalized Parton Model (GPM), which is a straightforward generalization of the parton model by inclusion of the transverse quark motion.
Azimuthal asymmetries in unpolarized reactions, such as Drell-Yan production and Semi-Inclusive Deep Inelastic Scattering (SIDIS), play an important role in testing the perturbative and non-perturbative aspects of strong interactions, as it was recognized in the early work by Georgi and Politzer [3], Mendez [4], and Kane, Pumplin, and Repko [5]. It was Robert Cahn [6,7] who first pointed out that cos φ asymmetries in SIDIS can easily be generated by intrinsic quark motion: the associated azimuthal modulation is called the "Cahn effect".
The related QCD evolution of the cross-sections was studied in the 80's, in the pioneering work by Collins-Soper-Sterman (CSS) [8,9]. It was realized that both nonperturbative and perturbative parts should be combined in order to achieve a reliable theoretical description of the corresponding experimental measurements. Yet, it took several decades to develop the appropriate QCD formalism [10] to describe transverse momentum dependent distribution and fragmentation functions (collectively called TMDs).
Simultaneously, an idea of multi-parton quantum mechanical correlations, or the Efremov-Teryaev-Qiu-Ster-man matrix elements [11,12,13,14], was born. These correlations are suppressed in the leading term contribution to the unpolarized cross-sections, but can be dominant in spin asymmetries; they are important in the so-called "twist-3" approach to factorization. It was later realized that TMD and twist-3 approaches are intimately related [15].
In the 90's two very important correlations of transverse motion and spin were proposed by Sivers [16,17] and Collins [18]. In order to describe the large (left-right) single spin asymmetries (SSAs) in pion production off hadronhadron scattering, Sivers suggested that they could originate, at leading twist, from the intrinsic motion of quarks in the colliding hadrons generating an inner asymmetry of unpolarised quarks in a transversely polarized hadron, the so-called Sivers effect. He proposed a new Transverse Momentum Dependent (TMD) distribution function, now commonly called the "Sivers function", which represents the number density of unpolarized partons inside a transversely polarized nucleon. This mechanism was criticized at first as it seemed to violate time-reversal invariance [18], however Brodsky, Hwang and Schmidt proved by an explicit calculation that initial-state interactions in Drell-Yan processes [19] and final-state interactions in SIDIS [20], arising from gluon exchange between the struck quark and the nucleon remnants, can generate a non-zero Sivers asymmetry. The situation was further clarified by Collins [21] who pointed out that, taking correctly into account the gauge links in the TMD distributions, time-reversal invariance does not imply a vanishing Sivers function, but rather a sign difference between the Sivers distribution measured in SIDIS and the same distribution measured in DY. This sign difference is one of the main goals of the next generation of DY measurements, soon to start at the COMPASS-II experiment at CERN [22], at RHIC (BNL) [23] and at Fermilab [24].
arXiv:1511.06924v2 [hep-ph] 2 Dec 2015
In a different approach, Collins proposed a mechanism based on a spin asymmetry in the fragmentation of transversely polarized quarks into a spinless hadron [18], which involved a transverse-momentum dependent (TMD) fragmentation function, called the "Collins function", which generates a typical azimuthal correlation, later denoted as the "Collins effect".
At the same time, and over the following years, the Torino-Cagliari group of Anselmino et al. proposed the first, pioneering phenomenological studies of asymmetries in hadron-hadron scattering [25,26,27,28]. In principle many different azimuthal correlations can contribute to the large single spin asymmetries measured in inclusive hadro-production from proton-proton scattering [29,30]: at first it was believed that the Sivers asymmetry would be largely dominant compared to the Collins effect [31], but later it turned out that this was not necessarily the case. Unfortunately, only one azimuthal angle is observed in the reaction, and this information is not sufficient to allow for the separation of the two effects. The situation might be clarified by a combined data analysis of the Sivers and Collins effects in polarized proton-proton scattering and in SIDIS, under the assumption that factorization holds also for hadronic processes, as proposed in Ref. [32]. A phenomenological overview and the experimental state-ofthe-art of polarized proton-proton scattering processes is reviewed in the contribution of E. Aschenauer, U. D'Alesio and F. Murgia to this Special Issue.
The idea of correlations and the corresponding transverse momentum dependent functions (TMDs) describing the nucleon structure came to its full fruition in 1995, when Kotzinian first [33] and later Mulders and Tangerman [34,35] developed a full theoretical description of Drell-Yan and Semi Inclusive Deep Inelastic Scattering cross sections in terms of TMDs. The three well known collinear distribution functions unfold, at leading order in 1/Q, into eight independent TMDs: the Sivers function is among them, together with the unpolarized and the helicity distribution functions and two manifestations of the transversity function, h 1 and h ⊥ 1T (the so called "pretzelosity"), related to the density number of transversely polarized partons inside a transversely polarized nucleon. In addition, we find the Boer-Mulders function, h ⊥ 1 , related to the density number of transversely polarized partons inside an unpolarized nucleon, and two "mixed" functions (later denominated "warm gear" functions) describing the distribution of transversely polarized partons inside longitudinally polarized nucleons, and vice-versa. The picture is simpler for the fragmentation TMDs where, considering only spinless hadrons, only two functions appear: the unpolarized and the Collins TMD FFs.
The phenomenological extraction of the Sivers and Boer-Mulders distribution functions, of transversity and the Collins function and of pretzelosity will be addressed in Sect. 2, together with a brief overview on the most recent extractions of unpolarized TMD PDFs and FFs.
It was only at the beginning of the 21 st century, when the new-generation dedicated SIDIS measurements were performed by the HERMES [36] and COMPASS [37] Col-laborations, that the framework of TMDs could reliably be experimentally tested for the first time. In particular, the first data collected by the HERMES Collaboration using a transversely polarized proton target, showed clear evidence of a non zero transverse SSAs. One of the main advantages of SIDIS is that the Collins and Sivers effects, as well as the other TMD effects, can easily be separated by appropriately weighting the SIDIS cross section: this generates different azimuthal asymmetries, which can be studied one by one. Contrary to what happens in hadroproduction, where all TMD effects occur and mix together in the same observable, in SIDIS each of them can be separated and extracted analyzing the same experimental cross section.
Much progress was achieved in the understanding of the 3D nucleon structure by successive data takings, followed by more and more refined analyses of SIDIS measurements [38,39]. The front end of 3D studies is presently being reached with the new multidimensional analyses and phenomenological studies of SIDIS multiplicities [40,41,42,43], azimuthal modulations [44,39,45] and new, pioneering multidimensional measurements of the Sivers and Collins single spin asymmetries [46].
Correlations between the spin of partons and the hadronic transverse momentum, can also be detected by measuring the azimuthal asymmetries generated in e + e − annihilations, when two final hadrons are produced in two (almost) opposite jets. In the process e + e − →qq the transverse polarizations of theqq pair are correlated, thus the Collins effect is expected to cause correlated azimuthal modulations of the hadrons into which the q and theq fragment. In 2006 the Belle Collaboration provided highprecision measurements [47] of such modulations which allowed, shortly after, the first combined extraction of the Collins function and of the transversity distribution [48,49], which was refined over the years with the successive re-analyses of the Belle data [50,51] and with the addition of higher statistics measurements of the BaBar Collaboration [52], in the works of several groups [53,54,55,56]. A similar procedure for the extraction of the transversity distribution, which combines SIDIS and e + e − data replacing the Collins functions with di-hadron fragmentation functions, has been adopted in Refs. [57,58,59].
From a more formal point of view, TMDs have recently received a renewed burst of interest concerning their Q 2 dependence: the Collins-Soper-Sterman (CCS) resummation scheme, originally devised to describe the Drell-Yan (DY) cross section over its full q T range, was revisited by Collins in his book [10] and by Rogers and Aybat in Ref. [60], and extended to the fully non-collinear case: evolution equations were then formulated for unpolarized TMD distribution and fragmentation functions. Further studies involving the TMD evolution of the Sivers and Collins functions where performed in the following years by several groups, see for example Refs. [61,62,63,54,55]. For a complete review of TMD factorization and evolution properties, and an exhaustive list of references, we refer the reader to the contribution of T.C. Rogers in this Special Issue.
Present
In this Section we will present some of the most recent phenomenological extractions of TMD distribution and fragmentation functions. As anticipated in Sect. 1, we will focus on the Sivers and the Collins functions, which are at present the most well known from a variety of different experimental measurements, followed by transversity (which at present can only be extracted from SIDIS data, in association with a chirally odd fragmentation function), and the Boer-Mulders and pretzelosity functions. First of all, however, it is important to start with the extraction of the unpolarized TMDs, which one has to rely on for the computation of (the denominator of) any azimuthal spin asymmetry.
Unpolarized TMD distribution and fragmentation functions
The fundamental role of TMDs is already evident in unpolarized cross sections, simply by looking at the transverse momentum distribution of the final hadron or, at order 1/Q, at the azimuthal dependence of the hadron around the proton direction, see Fig. 1. In Ref. [64] a first investigation of SIDIS unpolarised cross sections was performed, mainly based on the EMC Collaboration experimental data [65,66], gathered from SIDIS experiments at different energies and different targets. This analysis was updated last year [43], by the inclusion of the newest, multidimensional data on the SIDIS multiplicities measured by the HERMES [40] and COMPASS [41] Collaborations.
Let's consider the unpolarized SIDIS process l( ) + N (P ) → l ( ) + h(P h ) + X(P X ) , in the γ * -N center-ofmass frame, with the virtual photon moving in the positive z direction, as in Fig. 1. We denote by P T the transverse momentum of the produced hadron. The azimuthal angle of this hadron φ h is referred to the lepton scattering plane formed by l and l . The unpolarized differential cross section of SIDIS is then
d 5 σ dx B dydz h dP 2 T dφ h = σ 0 sy 2 F U U + + 2(2 − y) √ 1 − y (1 + (1 − y) 2 F cos φ U U cos φ h + 2(1 − y) (1 + (1 − y) 2 F cos(2φ h ) U U cos(2φ h ) ,(1)where σ 0 = 2πα 2 em Q 2 1+(1−y) 2 y
, and one uses the following standard variables
x B = Q 2 2 P · q , y = P · q P · l , z h = P · P h P · q ,(2)
where α is the fine structure constant, while Q 2 = −q 2 = −(l − l ) 2 is the virtuality of the exchanged photon. The structure functions F U U , F cos φ U U , F cos 2φ are associated to the cos φ h and cos 2φ h modulations, respectively, which will be discussed in Sect. 2.4. If k is the momentum of the quark inside the proton, and k ⊥ its transverse component with respect to the γ * N axis, in the kinematical region where P T ∼ k ⊥ Q, the transverse-momentum-dependent (TMD) factorization is known to hold. In this case the structure functions can be expressed in terms of TMD distribution and fragmentation functions, which depend on the light-cone momentum fractions x x B and z z h .
U U depend on x B , z h ,Q
Introducing the transverse momentum p ⊥ of the final hadron with respect to the direction of the fragmenting quark, to order O(k ⊥ /Q), one has
P T = zk ⊥ + p ⊥ .(3)
In the TMD factorization scheme the structure function F U U is given by
F U U = x q e 2 q d 2 k ⊥ d 2 p ⊥ δ (2) zk ⊥ + p ⊥ − P T × ×f q/p (x, k 2 ⊥ ) D h/q (z, p 2 ⊥ ) = x q e 2 q d 2 k ⊥ f q/p (x, k 2 ⊥ )D h/q (z, (P T − zk ⊥ ) 2 ),(4)
where f q/p (x, k 2 ⊥ ) and D h/q (z, p 2 ⊥ ) are the unpolarized TMD distribution and fragmentation function, respectively, for the flavor q (the sum is intended to be both over quarks and antiquarks). At this stage, the Q 2 dependence of all functions is omitted for simplicity.
In most phenomenological models, the x(z) and k ⊥ (p ⊥ ) dependences are factorized and the k ⊥ and p ⊥ dependences are assumed to be Gaussian, with one free parameter which fixes the Gaussian width, (8), with the parameters of Eq. (12), are compared with HER-MES measurements for π + SIDIS production off a proton target [40]. The shaded uncertainty bands correspond to a 5% variation of the total χ 2 . Plot from Ref. [43].
f q/p (x, k ⊥ ) = f q/p (x) e −k 2 ⊥ / k 2 ⊥ π k 2 ⊥(5)D h/q (z, p ⊥ ) = D h/q (z) e −p 2 ⊥ / p 2 ⊥ π p 2 ⊥ ·(6)
The integrated PDFs, f q/p (x) and D h/q (z), can be taken from the available fits of the world data. In general, the widths of the Gaussians could depend on x or z and might be different for different distributions. Ref. [43] assumes flavour independence and one obtains
F U U = x q e 2 q f q/p (x B ) D h/q (z h ) e −P 2 T / P 2 T π P 2 T(7)
where
P 2 T = p 2 ⊥ + z 2 h k 2 ⊥ .(8)
The constant Gaussian parameterization, supported by a number of experimental evidences [67] as well as by dedicated lattice simulations [68], has the advantage that the intrinsic transverse momentum dependence of the cross section can be integrated out analytically. The differential hadron multiplicity (according to the HERMES [40] definition) is
M h n (x B , Q 2 , z h , P T ) ≡ 1 d 2 σ DIS (x B , Q 2 ) dx B dQ 2 d 4 σ(x B ,Q 2 ,z h , P T ) dx B dQ 2 dz h dP T ·(9)
where the index n denotes the kind of target.
The Deep Inelastic Scattering (DIS) cross section has the usual leading order collinear expression,
d 2 σ DIS (x B , Q 2 ) dx B dQ 2 = y σ 0 q e 2 q f q/p (x B ) ·(10)
Then, multiplicities are simply given by (8), with the parameters of Eq. (12), are compared with HER-MES measurements for π − SIDIS production off a proton target [40]. The shaded uncertainty bands correspond to a 5% variation of the total χ 2 . Plot from Ref. [43].
d 2 n h (x B , Q 2 , z h , P T ) dz h dP 2 T = 1 2P T M h n (x B , Q 2 , z h , P T ) = π q e 2 q f q/p (x B ) D h/q (z h ) q e 2 q f q/p (x B ) e −P 2 T / P 2 T π P 2 T ,(11)
with P 2 T given in Eq. (8). Notice that, by integrating the above equation over P T , with its magnitude ranging from zero to infinity, one recovers the ratio of the usual leading order cross sections in terms of collinear PDFs and FFs. Its agreement with experimental data has been discussed, for instance, in Refs. [40] and [42].
In Fig. 2 we show, as an example, the comparison between the HERMES measurements of the multiplicities for π + SIDIS production off a proton target [40] and those obtained in Ref. [43] by best fitting the HERMES multidimensional data using the expressions of Eqs. (11) and (8). Notice that this fit, which is performed over a sample of about 500 experimental points, relies on two free parameters only: the two Gaussian widths of the k ⊥ and p ⊥ distributions of the unpolarized PDF and FF TMDs. The normalization is not fixed by adding extra-parameters, as it was done in other analyses like, for instance, Ref. [69]. This simple TMD Gaussian parameterization, with constant and flavour independent widths, delivers a very satisfactory description of the HERMES data points over large ranges of x, z, P T and Q 2 : the extracted reference values, corresponding to a total χ 2 dof = of 1.69, are k 2 ⊥ = 0.57 ± 0.08 GeV 2 p 2 ⊥ = 0.12 ± 0.01 GeV 2 .
These values are obtained by selecting 497 data points corresponding to the following requirements: Q 2 > 1.69 GeV 2 , 0.2 < P T < 0.9 GeV and z < 0.6. By relaxing the cuts on z in such a way to include one more bin, z < 0.7, which increases the number of fitted data points to 576, the quality of the fits deteriorates considerably, giving χ 2 dof = 2.62, and the extracted Gaussian widths recover values closer to those obtained in previous analyses, like [64] k 2 ⊥ = 0.46 ± 0.09 GeV 2 and p 2 ⊥ = 0.13 ± 0.01 GeV 2 . HERMES multiplicities do not show any significant sensitivity to additional free parameters: the fits do not Distribution of the values of the ratios P 2 ⊥,unf / P 2 ⊥,fav vs. P 2 ⊥,uK / P 2 ⊥,fav . The white squared box indicates the center of the 68% confidence interval for each ratio. The shaded area represents the two-dimensional 68% confidence region around the white box. The dashed lines correspond to the ratios being unity; their crossing point corresponds to the result with no flavor dependence. For all points, P 2 ⊥,fav < P 2 ⊥,unfav ∼ P 2 ⊥,uK . Plot from Ref. [42].
improve by introducing a z-dependence in the Gaussian widths of the TMD-FFs or by allowing a flavour dependence in the Gaussian widths of the TMD-PDFs. We only find a slight improvement in χ 2 by using different (constant) Gaussian widths in the TMD-FFs; the disfavoured fragmentation functions show a preference for a width slightly wider than that of the favoured fragmentation functions. These results are in agreement with a similar study, performed by Signori et al. in Ref. [42], in which more elaborate input parameterizations were used to model the PDF and FF TMDs ( k 2 ⊥ ( p 2 ⊥ ) were assigned a particular x(z) and flavour dependence). However, on the basis of a study performed by fitting 200 replicas of the original data points, the authors claim the evidence of a much stronger flavour dependence of the Gaussian p ⊥ distributions in the fragmentation functions, see Fig. 4.
It is important to observe that in the SIDIS multiplicities, the two free parameters k 2 ⊥ and p 2 ⊥ are strongly (anti)correlated, as they appear in the combination P 2 (7) and (11). Consequently, they can only be uniquely determined by fitting simultaneously two or more different observables. An attempt in this direction has been made by V. Barone et al. in Ref. [45], as we will discuss in Sect. 2.4.
T = z 2 h k 2 ⊥ + p 2 ⊥ , see Eq.
As anticipated above, the COMPASS collaboration has also provided their measurements of SIDIS multiplicities, in multidimensional bins of definite Q 2 and x B values, each for several values of z h and P T , with much higher statistics compared to the HERMES experiment. Fitting COMPASS data, however, turns out to be more difficult: while the Gaussian shape of the P T dependence is qualitatively well reproduced, there are some unresolved issues with their relative overall normalisation, possibly related to a mistreatment of radiative corrections. The COMPASS fit of Ref. [43], performed by applying an "ad hoc", ydependent correction of the bin normalization, returns a p 2 ⊥ TMD-FF Gaussian width slightly larger than that extracted from the HERMES multiplicities, while it delivers similar k 2 ⊥ values. Notice that this analysis has been performed on the 2004 run data, when the COM-PASS detector was not yet completely set up and no RICH was installed for final hadron separation. Future analyses of more recent COMPASS data with hadron identification and a proper treatment of the radiative corrections should help to clarify the situation.
The study of the Q 2 dependence of SIDIS multiplicities deserves a dedicated discussion.
In the analysis of Ref. [42] no scale dependence was taken into account, while in Ref. [43], with the phenomenological parameterization of Eqs. (5) and (6), the only dependence on Q 2 was included in the collinear part of the TMD, i.e. in the collinear PDF or FF factor. The width of the Gaussian, which gives the k ⊥ (p ⊥ ) dependence of the TMDs, did not include any scale dependence. However Anselmino et al. tried, in Ref. [43] an alternative parameterizations, to allow for a Q 2 and/or x-dependence of the Gaussian widths. As the SIDIS cross section is not sensitive to the individual contributions of k 2 ⊥ and p 2 ⊥ , but only to their linear combination, P 2 T , see Eqs. (7) and (8), a simplified form can be considered:
P 2 T = g 1 + z 2 [g 1 + g 2 ln(Q 2 /Q 2 0 ) + g 3 ln(10 e x)] . (13)
For the HERMES data they did not find any significant x or Q 2 dependence in the transverse momentum spectra, confirming the good agreement of the measured multiplicities with the most simple version of the Gaussian model. For the COMPASS data, instead, some improvement in the quality of the fit can actually be obtained. However, due to the unresolved normalization issues discussed above, it is difficult to give any clear interpretation of this sensitivity and to draw, at this stage, any definite conclusion. Indeed, it is quite possible that the span in Q 2 of the available SIDIS data is not yet large enough to perform a safe analysis of TMD evolution based only on these data. Another important issue is that, always considering the SIDIS data set, the values of P T , while being safely low, are sometimes close to Q and corrections to the TMD factorisation scheme might be still relevant.
As a matter of fact, in order to describe the SIDIS cross section over a wide region of P T (or, more appropriately, of q T = P T /z) soft gluon resummation has to be performed. This can be done, in the impact parameter b T space, using for instance the Collins-Soper-Sterman (CSS) formalism or the improved TMD framework of ref. [10]. However, its successful implementation is affected by a number of practical difficulties: the strong influence of the kinematical details of the SIDIS process, the possible dependence on the parameters used to model the non-perturbative content of the SIDIS cross section, the complications introduced by having to perform phenomenological studies in the b T space, where the direct connection to the conjugate q T space is lost. Then, a matching prescription has to be applied to achieve a reliable description of the SIDIS process over the full q T range, going smoothly from the region of applicability of resummation, or equivalently of the TMD description, to the region of applicability of perturbative QCD.
A very thorough study of the issues related to matching the perturbative and non-perturbative contributions in SIDIS processes was performed in Ref. [70]. To take care of the non-perturbative content, in Ref. [70] the socalled b * prescription was adopted in order to cure the problem of the Landau pole in the perturbative expansion, complementing it with the introduction of a properly defined non-perturbative function. Studying the dependence of the cross section on this non-perturbative contribution and on the details of the b * prescription, i.e. on b max , it was found that some kinematical configurations, similar to those of COMPASS or HERMES experiments for example, are completely dominated by these features. As a consequence, no matching can be achieved exploiting the usual "Y-term prescription", based on a smooth switch from the dσ N LO cross section, calculated perturbatively to NLO, to the next to leading logarithm (NLL) resummed cross section W N LL through the so called Yterm, defined as Y = dσ N LO − dσ ASY , see Fig. 5. Notice that, at large q T , dσ ASY becomes negative and therefore unphysical (we show the absolute value of the asymptotic NLO cross section in Fig. 5 as a dashed, green line). Consequently, the Y term can become much larger than the N LO cross section in that region. This is because the Y term, being calculated in perturbative QCD, does not include any non-perturbative content.
As the mismatch between W N LL and dσ ASY at q T ∼ Q is mainly due to the non-perturbative content of the cross section, which turns out to be non-negligible, one could experiment different and more elaborate matching prescriptions, which take into account the non-perturbative contributions to the total cross section. One could require, for instance, that in a region of sizable q T
dσ total = W N LL − W F XO + dσ N LO ,(14)
where W F XO is the NLL resummed cross section approximated at first order in α s , with a first order expansion of the Sudakov exponential. However, as it was shown in Ref. [70], this method still presents several difficulties and remains largely unsatisfactory. In order to find the origin of these difficulties, Boglione et al. [70] studied in detail the b T behavior of the perturbative Sudakov factor and found that in a COMPASS-like kinematical configuration the perturbative Sudakov exponential is larger than one, i.e. unphysical, over most of the b T range. Therefore any resummation scheme would be inadequate in this case, and hardly applicable. Indeed, being the non-perturbative details of such importance to the description of the cross sections, a critical re-examination of the definition and implementation of the Y -term is needed.
We conclude that, at this stage, it is of crucial importance to have experimental data available in order to test all the mechanisms developed in the resummation of soft gluon emissions and study the non-perturbative aspects of the nucleon. It is essential to have (and analyze) data from HERA( √ s = 300 GeV), Electron-Ion Collider ( √ s = 20 -100 GeV), COMPASS ( √ s = 17 GeV), HERMES ( √ s = 7 GeV), and Jefferson Lab 12 ( √ s = 5 GeV). In particular, it will be very important to study experimental data on q T distributions that span from the region of low q T Q up to the region of q T ∼ Q.
Sivers Function
Among all TMDs the Sivers function, which describes the number density of unpolarized quarks inside a transversely polarized proton, has so far received the widest attention, from both phenomenological and experimental points of view.
The Sivers function f ⊥ 1T is related to initial and final state interactions and could not exist without the contribution of the orbital angular momentum of partons to the spin of the nucleon, to which it can be related, in a model dependent way, through the so-called "lensing function" [71]. As such it encodes the correlation between the partonic intrinsic motion and the transverse spin of the nucleon, and it generates a dipole deformation in momentum space: Fig. 6, taken from the EIC White Paper [72], shows the density distribution of unpolarized up and down quarks in a transversely polarized nucleon. For an overview of studies on the parton orbital angular momentum we refer the reader to the contribution of Liu and Lorcé in this Topical Issue.
Over the years, the Sivers function has been extracted from SIDIS data by several groups, with consistent results [73,74,75,76,77,78]. However, until very recently, all phenomenological fits had been performed by using a simplified version of the TMD factorization scheme, in which the QCD scale dependence of the TMDs -which was unknown -was either neglected or limited to the collinear part of the unpolarized PDFs. While this might not be a serious numerical problem when considering only experimental data which cover limited ranges of low Q 2 values, it is not correct in principle, and taking into account the appropriate Q 2 evolution might be numerically relevant for Ref. [72] predictions at higher Q 2 values, like future electron-ion or electron-nucleon colliders (EIC/ENC) and Drell-Yan experiments.
Recently the issue of the QCD evolution of unpolarized TMDs and of the Sivers function has been studied in a series of papers [10,60,61,79,80] where a TMD factorization framework has been worked out for the treatment of SIDIS data and the extraction of polarized TMDs. The main difficulty, here, is due to the fact that the TMD formalism originally developed to describe the Q 2 evolution of the unpolarized TMDs cannot be directly applied to the spin dependent distribution functions, like the Sivers function [16], for which the collinear limit corresponds to twist-3 Qui-Sterman function T F . Compared to the unpolarized TMD evolution scheme, the extra aid of a phenomenological input function is required: this input function embeds the missing information on the evolved function, that, in the case of the Sivers function, is both of perturbative and non-perturbative nature.
The TMD Sivers distribution can be extracted by fitting the HERMES and COMPASS SIDIS data on the azimuthal moment A
sin(φ h −φ S ) U T
. The relevant part of the SIDIS cross-section for Sivers asymmetry reads:
d 5 σ(S ⊥ ) dx B dydz h d 2 P T = σ 0 (x B , y, Q 2 ) F U U + sin(φ h − φ s ) F sin(φ h −φs) U T + ... ,(15)
where S T is transverse polarization, and φ h , φ S are the azimuthal angles of the produced hadron and the polarization vector. The spin structure function F
sin(φ h −φ S ) U T
is a convolution of the Sivers function f ⊥ 1T with the unpolarized FF D h/q . The ellipsis in Eq. (15) denotes contributions from other spin structure functions. The experimentally measured Sivers asymmetry is then Fig. 7. Comparison between HERMES [82] and preliminary COMPASS data [83] for the z and PT dependence of the Sivers asymmetry. The solid line is the fit from Ref. [81]. The dashed curve is the result of evolving to the COMPASS scale using the TMD-evolution scheme of Ref. [61]. Plot from Ref. [62].
A sin(φ h −φ S ) U T ≡ 2 sin(φ h − φ S ) ∼ f ⊥ 1T ⊗ D h/q f q/p ⊗ D h/q(16)
A first application of the new TMD evolution equations of Ref. [61] to some limited samples of the HERMES and COMPASS data [62] was proposed by Aybat et al. in Ref. [62]. There, it was explicitly shown that the evolution of an existing fit of the Sivers SIDIS asymmetry [81] from the average value Q 2 = 2.4 GeV 2 (HERMES [82]) to the average value of Q 2 = 3.8 GeV 2 (COMPASS [83]), proved to be reasonably compatible with the TMD evolution equations of Ref. [61]. Their results are shown in Fig. 7.
Shortly afterwards Anselmino, Boglione and Melis [63] performed a complete best fit of the SIDIS Sivers asymmetries taking into account the different Q 2 values of each data point and the Q 2 dependence of the TMDs and compared their results with a similar analysis performed without the TMD evolution. By following Ref. [61], and denoting by F either the unpolarized parton distribution, the unpolarized fragmentation function, or the first derivative, with respect to the parton impact parameter b T , of the Sivers function, the QCD evolution of the TMDs in the coordinate space can be written as
F (x, b T ; Q) = F (x, b T ; Q 0 ) × R(Q, Q 0 , b T ) exp −g K (b T ) ln Q Q 0 ,(17)
with
R(Q, Q 0 , b T ) ≡ exp ln Q Q 0 µ b Q0 dµ µ γ K (µ ) + Q Q0 dµ µ γ F µ, Q 2 µ 2(18)
and the anomalous dimensions γ F and γ K given by
γ F (µ; Q 2 µ 2 ) = α s (µ) C F π 3 2 − ln Q 2 µ 2 γ K (µ) = α s (µ) 2 C F π ·(19)
The Q 2 evolution is driven by the functions g K (b T ) and R(Q, Q 0 , b T ). While the latter, Eq. (18), can be easily evaluated, numerically or even analytically, the former, is essentially unknown and will need to be taken from independent experimental inputs. The explicit expression of the TMDs in the momentum space, with the QCD Q 2 dependence, can be obtained by Fourier-transforming Eq. (17), obtaining [61]:
f q/p (x, k ⊥ ; Q) = 1 2π ∞ 0 db T b T J 0 (k ⊥ b T ) f q/p (x, b T ; Q)(20)D h/q (z, p ⊥ ; Q) = 1 2π ∞ 0 db T b T J 0 (k T b T ) D h/q (z, b T ; Q) (21) f ⊥f 1T (x, k ⊥ ; Q) = −1 2πk ⊥ ∞ 0 db T b T J 1 (k ⊥ b T ) f ⊥q 1T (x, b T ; Q),(22)
where J 0 and J 1 are Bessel functions. f ⊥q 1T is the Sivers distribution defined, for unpolarized partons inside a transversely polarized proton, as:
f q/p ↑ (x, k ⊥ , S; Q) = = f q/p (x, k ⊥ ; Q) − f ⊥q 1T (x, k ⊥ ; Q) ij k i ⊥ S j M p = (23) = f q/p (x, k ⊥ ; Q) + 1 2 ∆ N f q/p ↑ (x, k ⊥ ; Q) ij k i ⊥ S j k ⊥ ·(24)
The unknown functions inside Eq. (17), g K (b T ) and F (x, b T ; Q 0 ), are then parameterized as
g K (b T ) = 1 2 g 2 b 2 T(25)f q/p (x, b T ; Q 0 ) = f q/p (x, Q 0 ) exp −α 2 b 2 T ,(26)
where g 2 is a parameter which should be extracted from experimental data, while the value of α 2 is fixed by requiring the desired behavior of the distribution function in the transverse momentum space at the initial scale Q 0 : taking α 2 = k 2 ⊥ /4 one recovers
f q/p (x, k ⊥ ; Q 0 ) = f q/p (x, Q 0 ) 1 π k 2 ⊥ e −k 2 ⊥ / k 2 ⊥ ,(27)
in agreement with Eq. (5). Similar relations hold for the TMD FFs, with an additional z 2 factor. Analogously, we parameterize the Sivers function at the initial scale Q 0 as
f ⊥ 1T (x, b T ; Q 0 ) = −2 γ 2 f ⊥ 1T (x; Q 0 ) b T e −γ 2 b 2 T ,(28)
which, when Fourier-transformed according to Eq. (22), yields:
f ⊥ 1T (x, k ⊥ ; Q 0 ) = f ⊥ 1T (x; Q 0 ) 1 4 π γ 2 e −k 2 ⊥ /4γ 2 .(29)
Eq. (29) agrees with the usual parameterization of the Sivers function [78,81,84], at the initial scale Q 0 , taking:
4 γ 2 ≡ k 2 ⊥ S = M 2 1 k 2 ⊥ M 2 1 + k 2 ⊥ (30) f ⊥ 1T (x; Q 0 ) = − M p 2M 1 √ 2e ∆ N f q/p ↑ (x, Q 0 ) k 2 ⊥ S k 2 ⊥ ,(31)
where M 1 is a mass parameter, M p the proton mass and
∆ N f q/p ↑ (x, Q 0 )
is the x-dependent term of the Sivers function, evaluated at the initial scale Q 0 and written as [78,81,84]:
∆ N f q/p ↑ (x, Q 0 ) = 2 N q (x) f q/p (x, Q 0 ) ,(32)
where N q (x) is a function of x, properly parameterized. The final evolution equations of the unpolarized TMD PDFs and TMD FFs, in the configuration space, are then
f q/p (x, b T ; Q) = f q/p (x, Q 0 ) R(Q, Q 0 , b T ) × exp −b 2 T α 2 + g 2 2 ln Q Q 0 (33) D h/q (z, b T ; Q) = 1 z 2 D h/q (z, Q 0 ) R(Q, Q 0 , b T ) × exp −b 2 T β 2 + g 2 2 ln Q Q 0 ,(34)
with α 2 = k 2 ⊥ /4, β 2 = p 2 ⊥ /(4z 2 ), g 2 given in Eq. (25) and R(Q, Q 0 , b T ) in Eq. (18). The evolution of the Sivers function is obtained through its first derivative, inserting Eq. (28) into Eq. (17):
f ⊥ 1T (x, b T ; Q) = −2 γ 2 f ⊥ 1T (x; Q 0 ) R(Q, Q 0 , b T ) × b T exp −b 2 T γ 2 + g 2 2 ln Q Q 0(35)
with γ 2 and f ⊥ 1T (x; Q 0 ) given in Eqs. (30)-(32). Eqs. (33)- (35) show that the Q 2 evolution is controlled by the logarithmic Q dependence of the b T Gaussian width, together with the factor R(Q, Q 0 , b T ): for increasing values of Q 2 , they are responsible for the typical broadening effect already observed in Refs. [60] and [61].
It is important to stress that although the structure of Eq. (33) is general and holds over the whole range of b T values, the input function F (x, b T , Q 0 ) is only designed to work in the large-b T region, corresponding to low k ⊥ Sivers asymmetries applying TMD evolution (red, solid lines) are compared with the analogous results found by using DGLAP evolution equations (blue, dashed lines). The experimental data are from HERMES [82] and COMPASS [83] Collaborations.
values. Therefore, this formalism is perfectly suitable for phenomenological applications in the kinematical region we are interested in, but the parameterization of the input function should be revised in the case one wishes to apply it to a wider range of transverse momenta, like higher Q 2 processes where perturbative corrections become important, as discussed in Sect.
2.1
The results obtained in Ref. [63] are shown in Fig. 8. They showed that the recently proposed Q 2 TMD evolution scheme can already be observed in the available SIDIS data on the Sivers asymmetry.
A definite statement resulting from this analysis is that the best fit of all SIDIS data on the Sivers asymmetry using TMD-evolution, when compared with the same analysis performed with the simplified DGLAP-evolution, exhibits a smaller value of the total χ 2 . Not only, but when analyzing the partial contributions to the total χ 2 value of the single subsets of data, one realizes that such a smaller value mostly originates from the large Q 2 COM-PASS data, which are greatly affected by the TMD evo- lution. This is indeed an indication in favor of the TMD evolution.
Later, an analogous phenomenological analysis, extended to Drell-Yan as well as SIDIS processes, was performed by Sun and Yuan [85], using an alternative, approximated form of the Sudakov form factor as proposed in Ref. [86]. Their study of TMD evolution effects in DY processes showed that extracting the free parameters which regulate the variation of the k ⊥ shape of the Sivers function by fitting solely SIDIS experimental data, could induce a strong dilution of the DY asymmetries. As usual, special care should be used when blindly applying parameter values extracted from a process to a different one. In this case, for example, it turns out that Sivers SIDIS asymmetries are very little sensitive to the g 2 parameter, which fixes the Gaussian width of the g K function, see Eq. (25), while the analogous asymmetries in DY are strongly affected by small variation of the same parameter. We conclude that global analyses, which include experimental data from as many different process as possible, represent the only reliable strategies to reach the full picture of hadronic structure, including TMD evolution.
More recently, Echevarria et al. [87] have extracted the Sivers function using a CSS evolution scheme, but relating the first moment of the Sivers function to the twist-three Qiu-Sterman quark-gluon correlation function, T qF (x, x, µ) [88]. The knowledge of T qF (x, x, µ), i.e. the "collinear counterpart" of the Sivers function will be very important for the description of SSAs in pp scattering. The T qF (x, x, µ) twist-three function, as extracted in Ref. [87], is presented in Fig. 9.
It is interesting to point out, here, that the Sivers function measured in SIDIS should be directly related to the twist-three Qiu-Sterman quark-gluon correlation function, T qF (x, x, µ). It was noted, however, that the T F extracted from SIDIS would give a single spin asymmetry A N , in proton-proton scattering, with opposite sign with respect to that observed in experiments [89]. This observation is referred to as the "sign puzzle". The attempts to solve this puzzle by considering the fact that kinematical regions of pp and SIDIS experiments are different, or by allowing the Sivers function to change sign, as a function of transverse momentum, did not result in a satisfactory solution of the problem. The more complete twist-3 phenomenology suggests [90] that fragmentation functions may play a more important role and generate the asymmetries in pp.
Future Drell-Yan experiments at COMPASS, RHIC and
Fermilab are going to reveal both the sign and the evolution of the Sivers function with respect to SIDIS measurements. Dedicated studies of TMD phenomenology in DY processes [91,92,93] will then become of crucial importance. Notice that the GPM model predicts the same sign of Sivers function in DY and SIDIS, while analyses including gauge links and TMD factorizations [21,94] suggests that the sign will change in DY with respect to SIDIS.
The Gluon Sivers function will be important at EIC: dedicated studies can be found for example in Ref. [95].
Collins Function and Transversity
The transversity distribution h 1 is the only source of information on the tensor charge of the nucleon and the Collins FF H ⊥ 1 decodes the fundamental correlation between the transverse spin of a fragmenting quark and the transverse momentum of the final produced hadron.
The Collins fragmentation function can be studied in SIDIS experiments, where it appears convoluted with the transversity distribution, and where, being dependent on the hadronic intrinsic transverse momentum, it induces a typical azimuthal modulation, the Collins asymmetry. It can also induce azimuthal angular correlations between hadrons produced in opposite jets in e + e − annihilations: here two of such functions, corresponding to the two final hadrons, appear convoluted. Consequently, a simultaneous analysis of SIDIS and e + e − data allows the combined extraction of the transversity distribution and the Collins fragmentation functions [48,49,53]. Notice that this is made possible by the universality of fragmentation functions, soft factors, and parton densities between e + e − annihilation, semi-inclusive deep-inelastic scattering and the Drell-Yan process, which was proven in Refs. [96,97].
Recently, new data on the e + e − → h 1 h 2 X process have been published by the BaBar Collaboration, focusing on their z and p ⊥ dependence [52]. It is the first direct measurement of the transverse momentum dependence of an asymmetry, in e + e − processes, related to TMD functions. Moreover, the newest results from BESIII [98], at much lower Q 2 values with respect to Belle and BaBar data, allow to explore the sensitivity of these azimuthal correlations on Q 2 dependent effects. A review of the experimental measurements involving the TMD fragmentation functions can be found in the contribution of Garzia and Giordano in this Topical Issue.
As mentioned in Sect. 1, work along these lines has been and is being done by several groups [53,54,55,56]. Here we will briefly report on the main achievements in the phenomenological extraction of the Collins and transversity functions and on their TMD evolution properties.
Collins asymmetries in SIDIS are generated by the convolution of the transversity function ∆ T q or h 1 and the Collins TMD FF ∆ N D h/q ↑ or H ⊥ 1 . The Torino and Amsterdam group notations for the Collins function, are related by [99] The relevant contributions to the SIDIS cross-sections are
∆ N D h/q ↑ (z, p ⊥ ) = (2 p ⊥ /z m h ) H ⊥q 1 (z, p ⊥ ) .(36)d 5 σ(S ⊥ ) dx B dydz h d 2 P h⊥ = σ 0 (x B , y, Q 2 ) F U U + sin(φ h + φ s ) 2(1 − y) 1 + (1 − y) 2 F sin(φ h +φs) U T + ... .(37)
The polarized structure function F
d 5 σ e + e − →h1h2+X dz h1 dz h2 d 2 P h⊥ d cos θ = N c πα 2 em 2Q 2 1 + cos 2 θ Z h1h2 uu + sin 2 θ cos(2φ 0 )Z h1h2
collins (38) where θ is the polar angle between the hadron h 2 and the beam of e + e − , φ 0 is defined as the azimuthal angle of hadron h 1 relative to that of hadron h 2 , i.e. of the plane containing hadrons h 1 and h 2 relative to the plane containing hadron h 2 and the lepton pair (see Fig. 10), and P h⊥ is the transverse momentum of hadron h 1 in this frame. The polarized structure function Z h1h2 collins contains the convolution of two Collins functions, H ⊥ 1 ⊗ H ⊥ 1 . Two methods have been adopted in the experimental analysis of the Belle and BaBar data [50,52]:
the "thrust-axis method" where the jet thrust axis, in the e + e − c.m. frame, fixes theẑ direction and the e + e − → qq scattering defines the xz plane; ϕ 1 and ϕ 2 are the azimuthal angles of the two hadrons around the thrust axis, while θ is the angle between the lepton direction and the thrust axis the "hadronic-plane method", in which one of the produced hadrons (h 2 in our case) identifies theẑ direction and the xz plane is determined by the lepton and the h 2 directions; the other relevant plane is determined byẑ and the direction of the other observed hadron, h 1 , at an angle φ 1 with respect to the xz plane. Here θ 2 is the angle between h 2 and the e + e − direction.
In this paper we will only discuss results obtained in the latter. In this reference frame, the elementary process e + e − → qq does not occur in the xz plane, and thus the helicity scattering amplitudes involve an azimuthal phase ϕ 2 . Ratios of unlike/like and unlike/charged are built in order to avoid false asymmetries:
R U 0 R L(C) 0 = 1 + cos(2φ 0 ) A U L(C) 0 ,(39)
which can then be directly compared to the experimental measurements. All details and definitions can be found in Ref. [56], which we will follow here. For the unpolarised parton distribution and fragmentation functions the factorized forms of Eqs. (5) and (6) are assumed. For the transversity distribution, ∆ T q(x, k ⊥ ), and the Collins FF, ∆ N D h/q ↑ (z, p ⊥ ), similar factorized shapes [48] are adopted:
∆ T q(x, k ⊥ ; Q 2 ) = ∆ T q(x, Q 2 ) e −k 2 ⊥ / k 2 ⊥ T π k 2 ⊥ T ,(40)∆ N D h/q ↑ (z, p ⊥ ; Q 2 ) =∆ N D h/q ↑ (z, Q 2 ) h(p ⊥ ) e −p 2 ⊥ / p 2 ⊥ π p 2 ⊥ ,(41)
where ∆ T q(x) is the integrated transversity distribution and∆ N D h/q ↑ (z) is the z-dependent part of the Collins function. In order to easily implement the proper positivity bounds, these functions are written, at the initial scale Q 2 0 , as [48]
∆ T q(x, Q 2 0 ) = N T q (x, Q 2 0 ) 1 2 [f q/p (x, Q 2 0 ) + ∆q(x, Q 2 0 )](42)∆ N D h/q ↑ (z, Q 2 0 ) = 2 N C q (z, Q 2 0 ) D h/q (z, Q 2 0 ) .(43)
They are then evolved up to the proper value of Q 2 . In Ref. [56], for ∆ T q(x, Q 2 ) we employ a transversity DGLAP kernel and the evolution is performed by an appropriately modified Hoppet code [102]; for the Collins function, Anselmino et al. assumed that the only scale dependence is contained in D(z, Q 2 ), which is evolved with an unpolarised DGLAP kernel, while N C q does not evolve with Q 2 . This is equivalent to assuming that the ratiõ ∆ N D(z, Q 2 )/D(z, Q 2 ) is constant in Q 2 . The function h(p ⊥ ), defined as [48]
h(p ⊥ ) = √ 2e p ⊥ M C e −p 2 ⊥ /M 2 C ,(44)
allows for a possible modification of the p ⊥ Gaussian width of the Collins function with respect to the unpolarised FF; for the TMD transversity distribution, instead, we assume the same Gaussian width as for the unpolarised TMD, k 2 ⊥ T = k 2 ⊥ . In Ref. [56] a simplified model which implies no Q 2 dependence in the p ⊥ distribution is used. We will compare the results obtained using this approximation with those presented in Ref. [103] using a NLL TMD evolution scheme for the Collins function.
N T q (x) is parameterized as as functions of P1T in e + e − → h1 h2 X processes, as measured by the BaBar Collaboration [52], are compared to the curves obtained from a GPM model in Ref. [56] (left panel) and using NLL TMD evolution in Ref. [103] (rightt panel). The shaded areas correspond to the statistical uncertainty on the model parameters.
N T q (x) = N T q x α (1 − x) β (α + β) α+β α α β β (q = u v , d v )(45)
where −1 ≤ N T q ≤ +1, α and β are free parameters of the fit. Thus, the transversity distributions depend on a total of 4 parameters (N T uv , N T dv , α, β). The Collins function, is distinguished in favoured and disfavoured contributions, parameterised as asymmetries as functions of P 1T (p t0 in the notation used by the BaBar Collaboration). Fig. 11 shows our best fit of the BaBar A U L 0 and A U C 0 asymmetries as functions of P 1T . These data offer the first direct insight of the dependence of the Collins function on the parton intrinsic transverse momentum: in fact, global fits now deliver a more precise determination of the Gaussian width of the Collins function (through the M C parameter), which in previous fits was affected by a very large uncertainty. Fig. 11 shows the best fit of the BaBar A U L 0 and A U C 0 asymmetries as functions of P 1T , as obtained in Ref. [56]. All details on the analysis and the values of the extracted parameters can be found there.
N C fav (z) = N C fav z γ (1 − z) δ (γ + δ) γ+δ γ γ δ δ , N C dis (z) = N C dis . (46) where −1 ≤ N C fav/dis ≤ +1,
As shown in the left panel of Fig. 12, the u and d quark transversity functions extracted in Ref. [56] are compatible with the previous extractions [48,49,53], and with those obtained by a similar procedure, but involving the di-hadron fragmentation functions instead of the Collins function [57,58,59]. While the u valence transversity distribution has a clear trend, the d valence transversity still shows large uncertainties. Instead, the newly extracted Collins functions look different from those obtained in our previous analyses: this is mainly due to the fact that a different parameterisation for the disfavoured Collins function was exploited. This study indicates that the actual shape of the disfavored Collins function is still largely unconstrained by data. About the p ⊥ dependence of the Collins function, we have already mentioned that its Gaussian width can now be determined with remarkable precision. However, this extraction is still subject to a number of initial assumptions: a Gaussian shape for the TMDs, a complete separation between transverse and longitudinal degrees of freedom, a Gaussian width of the unpolarised TMD-FFs fixed solely by SIDIS data. Hopefully, higher statistics and higher precision multidimensional data, for asymmetries and unpolarised multiplicities, will help clarifying the picture. The first extraction of the transversity distribution and Collins fragmentation functions with TMD evolution was performed in Ref. [103]. It was demonstrated that the TMD evolution can describe the experimental data and constrain the nucleon tensor charge with improved theoretical accuracy. To achieve that, the most recent developments from both theory and phenomenology sides [10,104,105,106,79,107,108,87,109,69] were used, and the TMD evolution at NLL order within the Collins-Soper-Sterman (CSS) [8,9] formalism was applied to the data.
Applying the TMD evolution, F U U and F U T can be written as [8,9,110,106]
F U U = 1 z 2 h db b 2π J 0 P h⊥ b z h e −SPT(Q,b * )−S (SIDIS) NP (Q,b) × C q←i ⊗ f i 1 (x B , µ b )Ĉ (SIDIS) j←q ⊗D h/j (z h , µ b ),(47)F U T = − 1 2z 3 h db b 2 2π J 1 P h⊥ b z h e −SPT(Q,b * )−S (SIDIS) NP coll (Q,b) × δC q←i ⊗ h i 1 (x B , µ b ) δĈ (SIDIS) j←q ⊗Ĥ (3) h/j (z h , µ b ),(48)
where b is the Fourier conjugate variable to the measured final hadron momentum P h⊥ , J 1 is the Bessel function, µ b = c 0 /b * with c 0 1.12, and the symbol ⊗ represents the usual convolution in momentum fractions. The sum over quark flavors q weighted with quark charge, q e 2 q , Fig. 13. Transversity distribution for up and down quarks comparison of extraction in Ref. [103] and [53]. The band corresponds to the uncertainty of the extraction. and the sum over i, j = q,q, g, are implicit in all formulas for the structure functions. C,Ĉ and δC, δĈ are the coefficient functions for the unpolarized distribution and fragmentation functions, and for transversity and Collins FF, that can be calculated perturbatively.
The usual b * -prescription was used in Ref. [103] and non perturbative factors were introduced S (SIDIS) NP and S (SIDIS) NP coll that contain information on the initial conditions of evolution. The Collins fragmentation function [18] enters as a p ⊥ moment [104],
H (3) h/q (z h ) = d 2 p ⊥ |p 2 ⊥ | M h H ⊥ 1 h/q (z h , p ⊥ ) ,(49)
where H ⊥ 1 h/q (z h , p ⊥ ) is the quark Collins function defined in [104], and differs by a factor (−1/z h ) from the so-called "Trento convention" [99],
H ⊥ 1 h/j (z h , p ⊥ ) = − 1 z h H ⊥ 1 h/j (z h , p ⊥ )| Trento ,(50)
with p ⊥ the transverse component of the hadron with respect to the fragmenting quark momentum. Three important ingredients have to be included to achieve the NLL formalism for the above structure functions and asymmetries. First of all, the perturbative Sudakov form factor [111],
S PT (Q, b * ) = Q 2 µ 2 b dµ 2 µ 2 A ln Q 2 µ 2 + B ,(51)
with perturbative coefficients A (1,2) ∼ α The global fit of SIDIS and e + e − was performed and resulted in the total χ 2 /n d.o.f = 0.88, equally good for SIDIS and e + e − data.A plot showing the results obtained in Ref. [103] is presented in Fig. 11 (left panel), where they are compared with the results obtained in Ref. [56]. It is (bottom panel) asymmetries measured by the BESIII collaboration [98] at Q 2 = 13 GeV 2 . Plot from Ref. [98] very interesting to notice the strong similarity between the two curves obtained with and without evolution. As the asymmetries measured by BaBar and Belle are actually double ratios, this similarity might be an indication of possible cancellations of strong evolution effects between numerators and denominators. Fig. 13 shows the comparison of the results from [103] and [53]. The right panel of Fig. 13 shows the predictions for future measurements at an EIC.
The BESIII Collaboration has recently measured the cos 2φ 0 asymmetries observed by BaBar and Belle, but at the lower energy √ s = Q = 3.65 GeV [98], see Fig. 14. Their low Q 2 values, as compared with Belle and BaBar experiments, might help in assessing the importance of TMD evolution effects. It is therefore important to check how a model in which the Q 2 dependence of the TMD Gaussian width is not included [56] can describe these new sets of measurements, and compare these results with the description obtained by using a TMD evolution scheme [103]. In Fig. 15 the solid, black circles represent the A U C 0 and A U L 0 asymmetries measured by the BESIII Collaboration at Q 2 = 13 GeV 2 , in bins of (z 1 , z 2 ), while the solid blue circles (with their relative bands) correspond to the predictions obtained by using the results of Ref. [56]. These asymmetries are well reproduced at small z 1 and z 2 , where we expect our model to work, while they are underestimated at very large values of either z 1 or z 2 , or both. Notice that the values of z 1 , z 2 in the last bins are very large for an experiment with √ s = 3.65 GeV: such data points might be affected by exclusive production contributions, and other effects. Fig. 14 shows the predictions for the BESIII asymmetries obtained in Ref. [103], evolving the (right panel) asymmetries measured by the BESIII collaboration [98] at Q 2 = 13 GeV 2 , in bins of (z1, z2), while the solid blue circles (with their relative bands) correspond to the predictions obtained by using the Collins functions from our alternative fit. Collins function with a TMD equations. As in the previous case, there is a striking similarity with the predictions obtained in Ref. [56] with no TMD evolution (which gives almost identical asymmetries for different Q 2 .
The transversity distribution and the Collins FF, as extracted in Ref. [103], are shown in Fig. 16 as function of k ⊥ and p ⊥ at three different Q 2 scales. The typical broadening dilution of the curves as Q 2 increases is clearly visible. Note that Ref. [103] obtained quite slow a TMD evolution in the low Q 2 range by re-extracting the appropriate non perturbative kernel of TMD evolution for the data.
At this stage, it is quite difficult to draw any clear-cut conclusion: despite the sizeable difference in Q 2 among the different sets of e + e − data differences among the measured BESIII and BaBar-Belle asymmetries are mild and can be explained by the different kinematical configurations and cuts. Predictions obtained with and without TMD evolution are both in qualitatively good agreement with the present BESIII measurements, indicating that the data themselves do not show strong sensitivity to the Q 2 dependence in the transverse momentum distribution.
Effects of TMD evolution in e + e − annihilation into hadrons were recently studied in Ref [55].
Boer-Mulders function
The Boer-Mulder function [113], ∆f q ↑ /p or h ⊥ 1 in the Torino or Amsterdam notation respectively, measures the transverse polarization asymmetry of quarks inside an unpolarized nucleon. It can be extracted by analyzing the cos φ and cos 2φ azimuthal modulations that appear in the unpolarized SIDIS cross section, see Eq. (1). The structure function associated with the cos φ modulation turns out to be of order 1/Q.Neglecting the dynamical twist-3 contributions (the so-called "tilde" TMD functions, which arise from quark-gluon correlations), F cos φ U U can be written as the sum of two terms
F cos φ U U = F cos φ U U Cahn + F cos φ U U BM ,(52)
with (h ≡ P T /|P T |)
F cos φ U U Cahn = −2 q e 2 q x d 2 k ⊥ (k ⊥ · h) Q f q (x, k ⊥ )D q (z, p ⊥ ),(53)F cos φ U U BM = q e 2 q x d 2 k ⊥ k ⊥ Q P T − z(k ⊥ · h) k ⊥ ⊗ ∆f q ↑ /p (x, k ⊥ )∆D h/q ↑ (z, p ⊥ ).(54)
Eq. (53) is the Cahn term, which accounts for the noncollinear kinematics of quarks in the elementary subprocess q → q . Eq. (54) is the Boer-Mulders contribution, arising from the correlation between the transverse spin and the transverse momentum of quarks inside the unpolarized proton. In this term the Boer-Mulders distribution function ∆f q ↑ /p couples to the Collins fragmentation function ∆D h/q ↑ . The relations between these functions, as defined in the present paper, and the corresponding quantities in the Amsterdam notation is
∆f q ↑ /p (x, k ⊥ ) = − k ⊥ M p h ⊥ 1 (x, k ⊥ ),(55)∆D h/q ↑ (z, p ⊥ ) = 2 p ⊥ zM h H ⊥ 1 (z, p ⊥ ),(56)
where M p and M h are the masses of the proton and of the final hadron, respectively. The Boer-Mulders effect is also responsible of the cos 2φ modulation of the cross section, giving a leading-twist contribution (that is, unsuppressed in Q), which has the form
F cos 2φ U U BM = − q e 2 q x d 2 k ⊥ P T (k ⊥ · h) + z h k 2 ⊥ − 2(k ⊥ · h) 2 2 k ⊥ p ⊥ ×∆f q ↑ /p (x, k ⊥ )∆D h/q ↑ (z, p ⊥ ). (57)
The cos φ and cos 2φ asymmetries are given, in terms of the structure functions, by
A cos φ = 2(2 − y) √ 1 − y [1 + (1 − y) 2 ] F cos φ U U F U U ,(58)A cos 2φ = 2(1 − y) [1 + (1 − y) 2 ] F cos 2φ U U F U U .(59)
Up to order 1/Q, cos φ receives contributions from the Cahn and the Boer-Mulders effect, while cos 2φ is proportional to the sole Boer-Mulders effect:
A cos φ = A cos φ Cahn + A cos φ BM A cos 2φ = A cos 2φ BM
A few years ago, these azimuthal asymmetries in unpolarized SIDIS were measured by the COMPASS and HER-MES Collaborations for positive and negative hadrons, and presented as one-dimensional projections, with all variables (x B , z h , Q 2 , P T ) but one integrated over [114,115,116]. The one-dimensional data on the cos 2φ asymmetry were analyzed in Ref. [117], where it was shown that the larger asymmetry for π − (h − ) production, compared to π + (h + ), was an indication of the existence of a non-zero Boer-Mulders effect, in agreement with the earlier predictions of Ref. [118]. Moreover, the analysis of Ref. [117] revealed that both up and down-quark Boer-Mulders functions are negative, see Fig. 17, consistently with various theoretical expectations (impact-parameter approach [119], lattice results [120], large-N c predictions [121] and model calculations [122,123,124]. It was also pointed out that measurements at different values of Q 2 were essential, in order to disentangle higher-twist contributions from the twist-two Boer-Mulder term.
The HERMES and COMPASS Collaborations have recently provided multidimensional data in bins of x B , z h , Q 2 and P T for the multiplicities [40,41] and for the azimuthal asymmetries [39,44]. A study of the SIDIS azimuthal moments cos φ and cos 2φ was presented by Barone et al. in Ref. [45], in order to understand the role of the Cahn effect and to extract the Boer-Mulders function, which was parameterized as follows Refs. [117,125].
∆f q ↑ /p (x, k ⊥ ) = ∆f q ↑ /p (x) √ 2e k ⊥ M BM e −k 2 ⊥ / k 2 ⊥ BM π k 2 ⊥ ,(60)with ∆f q ↑ /p (x) = N q f q/p (x),(61)
and
k 2 ⊥ BM = k 2 ⊥ M 2 BM k 2 ⊥ + M 2 BM ·(62)
N q and M BM are free parameters to be determined by the fit. For the favored and the disfavored components of the Collins function, the parameters are fixed to the values obtained in a recent fit of the Collins asymmetries in SIDIS and e + e − annihilation [53], as described in Sect. 2.3. F U U and the Cahn contribution to cos φ involve only the unpolarized TMD distribution and fragmentation functions f q/p (x, k ⊥ ) and D h/q (z, p ⊥ ). These functions have been recently extracted in Ref. [43], as described Sect. 2.1. There it was observed that, since the multiplicities are sensitive only to the combination P 2 T = z 2 h k 2 ⊥ + p 2 ⊥ , Eq. (7), they cannot distinguish k 2 ⊥ from p 2 ⊥ . Instead, the azimuthal asymmetries involve k 2 ⊥ and p 2 ⊥ separately, and are sensitive to a z h -dependent p 2 ⊥ . Therefore, in principle, by fitting simultaneously the multiplicities and the cos φ and cos 2φ asymmetries one should be able to extract the separate values of k 2 ⊥ and p 2 ⊥ . Unfortunately, the analysis of Ref. [45] shows that, due to the huge contribution of the Cahn effect, the recent COMPASS and HERMES multidimensional data can only be reproduced by a very small value of k 2 ⊥ , namely 0.03-0.04 GeV 2 . This means that most of the transverse momentum of the outgoing hadron is due to the fragmentation, which must be described by a function with a zdependent width. This result, mainly driven by cos φ , could be modified by the presence of further twist-3 terms, which might not be negligible due to the relevance of the small-Q 2 region in the present measurements.
A somehow disappointing output of this fits is the indeterminacy on the extraction of the Boer-Mulders function, which seems to play a minor role in the asymmetries. This is seen in particular from cos 2φ , which is entirely deter- mined by the Boer-Mulders contribution but appears to be, within large errors, compatible with zero. On the other hand, the integrated cos 2φ data [44] show a non vanishing asymmetry, especially when plotted against z. The asymmetry is positive for π + and negative for π − , as expected from the Boer-Mulders effect [118]. Also the integrated data on cos φ show a different asymmetry for π + and π − (negative in the first case, positive in the other): this indicates a flavor dependence which can only be achieved with a non-zero Boer-Mulders effect since, within a flavor-independent Gaussian model with factorized x and k ⊥ dependences, the Cahn effect is flavor blind and can only generate identical contributions for positively or negatively charged pions. However, the sign of the u and d Boer-Mulders functions required for a successful description of cos 2φ appears to be incompatible with those required to generate the appropriate difference between π + and π − in the cos φ azimuthal moment. Unfortunately, not even a more refined model with flavor dependent Gaussian widths can help, given the precision of the current experimental data.
One should not forget about the existence of other higher-twist effects that could combine with the Boer-Mulders term and alter the simple picture considered here. In order to disentangle these contributions, it might be useful to integrate the asymmetry data on restricted kinematical ranges,as suggested in Ref. [126], so as to avoid the low-Q 2 region and meet the requirements of TMD factorization. Analyzing properly integrated data could help to clarify the origin of azimuthal asymmetries and possibly to get more information on the Boer-Mulders function.
The Boer-Mulders functions also generate the cos(2φ h ) asymmetry in Drell-Yan processes: this asymmetry is proportional to the convolution of the Boer-Mulders functions for quark and for anti-quark h ⊥ 1 ⊗h ⊥ 1 . In Ref. [125] the anti-quark Boer-Mulders distributions were extracted using the E866/NuSea measure ments of pp and pD unpolarized DY [127,128]. Possible effects of TMD evolution were also studied in Ref. [125], by varying the width of the functions. (solid red and dashed blue curves in Fig. 17).
Future developments will involve studies of the Boer-Mulders functions including TMD evolution effects.
Pretzelosity
The pretzelosity distribution function h ⊥ 1T [129] describes transversely polarized quarks inside a transversely polarized nucleon.
The part of the SIDIS cross section we are interested in reads [130,131,84]: (63) where the spin structure function F sin(3φ h −φ S ) U T contains the convolution of pretzelosity h ⊥ 1T and the Collins FF H ⊥ 1 .
d 5 σ(S ⊥ ) dx B dydz h d 2 P h⊥ = σ 0 (x B , y, Q 2 ) F U U + sin(3φ h − φ s ) 2(1 − y) 1 + (1 − y) 2 F sin(3φ h −φs) U T + ... ,
Pretzelosity is the only TMD distribution that gives a quadrupole modulation of the parton densities in the momentum space, as shown in Fig. 21.
The measured asymmetry in SIDIS contains the convolution of pretzelosity h ⊥ 1T and the Collins FF H ⊥ 1 : The plot is from Ref. [129] Notice that the knowledge of the Collins FF is needed for the extraction of pretzelosity. h ⊥ 1T was extracted in Ref. [129]: the results are shown in Fig. 21. Notice that the current knowledge of pretzelosity is very poor due to the suppression of this asymmetry by kinematical factors. Future data from Jefferson Lab will be crucial for the phenomenology of h ⊥ 1T . In a vast class of models with spherically symmetric nucleon wave function in the rest frame, the pretzelosity distribution is related to the orbital angular momentum of quarks by the following relation
A sin(3φ h −φ S ) U T ≡ 2 sin(3φ h − φ S ) ∼ h ⊥ 1T ⊗ H ⊥ 1 f 1 ⊗ D 1 .(64)L a z = − dx d 2 k ⊥ k 2 ⊥ 2M 2 h ⊥a 1T (x, k 2 ⊥ ) = − dx h ⊥(1)a 1T (x) .
(65) Even though the relation of Eq. 65 is indeed model dependent, it is interesting to explore it to gain more information on this effect.
Future
In the last few decades it was realized that a simple collinear picture of the nucleon, with partons that move along the direction of motion of the nucleon itself, and encode parton dynamics into the parton distribution and fragmentation functions, is not sufficient to explain all phenomena associated with the nucleon's structure. The explanation of large Spin Asymmetries, early observed in hadronic reactions and later in SIDIS and in e + e − annihilation processes, requires taking into account the transverse motion of partons with respect to the parent nucleon motion. This leads to the exploration of the three dimensional structure of the nucleon, which brings our knowledge of nuclear structure to a new and deeper level.
Correlations between spin and partonic intrinsic transverse momentum are encoded in the TMDs, transverse momentum dependent structure functions which play a fundamental role in unraveling the non-perturbative aspects of the hadronic structure of matter.
Having reviewed the state-of-the-art of TMD phenomenology, we give a brief summary of the forthcoming events which are presently foreseen in this field.
With HERMES data analyses being officially closed, and the COMPASS experiment entering phase 2, with the DY program, the flow of novel SIDIS data will rely on the last analyses and re-analysis of COMPASS data now on tape (2010-2012 data takings) and on the upgrade of the Jefferson Lab experiments from 6 to 12 GeV.
The Jefferson Lab 12 GeV program is going to explore the region of relatively high-x dominated by valence quarks. The description of the data will require a very good understanding of the non perturbative effects and of the kinematical corrections, such as phase space limitations and target mass corrections. Clearly, phenomenological studies of the non-perturbative TMD functions will be very important for the description of Jefferson Lab new data.
RHIC [23], COMPASS [22] and Fermilab [24] will provide data on polarized Drell-Yan and one will be able to incorporate these data in global analyses and investigate issues like the change of sign of the Sivers function [21], the flavour dependence of TMDs and eventual flavour asymmetries in the light quark sea. In particular, data on proton-proton scattering asymmetries from RHIC will be very important for TMD and twist-3 phenomenology [23,90]. The "sign" puzzle [89] will most probably be solved in future.
Future Electron Ion Collider will explore the region dominated by sea quarks and gluons and the data will provide a unique opportunity to study both sea quark and gluon TMDs and to study the evolution of asymmetries and TMDs [72]. For a detailed report on the future of TMDs (and GPDs) we refer the reader to the contribution of R. Ent to this Topical issue.
Finally, the Large Hadron Collider is going to provide an unprecedented amount of data relevant to three dimensional nucleon structure studies. Both gluon TMDs and quark TMDs will be important for LHC studies.
Combined studies from all facilities will result in the ultimate understanding of the mechanisms and the origin of spin asymmetries and will lead to a more profound knowledge of the origin of spin and the 3D nucleon structure.
Fig. 2 .
2The multiplicities M π + p obtained from Eqs.(11) and
Fig. 3 .
3The multiplicities M π − p obtained from Eqs.(11) and
Fig. 4. Distribution of the values of the ratios P 2 ⊥,unf / P 2 ⊥,fav vs. P 2 ⊥,uK / P 2 ⊥,fav . The white squared box indicates the center of the 68% confidence interval for each ratio. The shaded area represents the two-dimensional 68% confidence region around the white box. The dashed lines correspond to the ratios being unity; their crossing point corresponds to the result with no flavor dependence. For all points, P 2 ⊥,fav < P 2 ⊥,unfav ∼ P 2 ⊥,uK . Plot from Ref. [42].
Fig. 5 .
5dσ N LO , dσ ASY , W N LL and the sum W N LL +Y corresponding to the SIDIS kinematical configuration of the COM-PASS experiment. Plot from Ref.[70].
Fig. 6 .
6The 3D density, in the transverse-momentum plane, of unpolarized up and down quarks inside a transversely polarized proton, described by the Sivers function. Here the model of[73] is used and the longitudinal momentum fraction is fixed to x = 0.1. The color code indicates the probability of finding the up or down quarks: the deep red (blue) indicates large negative (positive) values for the Sivers function. Plot from
Fig. 8 .
8The results obtained in Ref.[63] from the fit of the SIDIS A sin (φ h −φ S ) U T
Fig. 9 .
9The TqF (x, x, µ) twist-three function as extracted in Ref.[87].
Fig. 10 .
10Kinematical configuration and conventions for e + e − processes.
Fig. 11 .
11The experimental data on the azimuthal correlations A U C 0
0 (
0γ and δ are free parameters of the fit.A best fit of the data on A sin(φ h +φ S ) U T (HERMES and COMPASS) and of the data on A U L,C Belle and BaBar) is then performed. It turns out to be a fit of excellent quality, with a total χ 2 d.o.f. = 0.84, equally good for SIDIS and e + e − data.Let's focus on the new BaBar measurements of A
Fig. 12 .
12Comparison of the best fit results obtained by Anselmino et al. in Ref. [56] (red, solid lines) for the valence u and d quark transversity distributions (left panel) and for the lowest p ⊥ moment of the favoured and disfavoured Collins functions (right panel), at Q 2 = 2.4 GeV 2 , with those from their previous analysis [53] (blue, dashed lines).
112,111]. Then, the scale evolutions of the quark transversity distribution and of the Collins fragmentation functions up to the scale of µ b .
Fig. 14 .
14Predictions using results of Ref. [54, 103] and comparison to A U C 0 (upper panel) and A U L 0
Fig. 15 .
15The solid, black circles represent the A
Fig. 16 .
16Transversity distribution h1 (left panel) and Collins fragmentation function H ⊥ 1 (right panel) at three different scales Q 2 = 2.4, 10, 1000 GeV 2 (solid, dotted, and dashed lines). The plots are from Ref. [103]
Fig. 17 .
17First moment of the Boer-Mulders distribution for up and down-quarks (left panel) and for anti-up and anti-down quarks (right panel) at Q 2 = 2.4 GeV 2 . These plots are from
Fig. 20 .Fig. 21 .
2021Tomographic slice of the pretzelosity distribution at x = 0.1 for up and down quarks. The plot is from Ref. [129] First moment of the pretzelosity distribution for up (left panel) and down (right panel) quarks at Q 2 = 2.4 GeV 2 .
2 , and P 2 T . F U U is the unpolarized structure functionφ h
P h
ℓ
ℓ'
γ*
Lepton plane
Hadron production plane
P T
P
z z z
x
Fig. 1. Kinematical configuration and conventions for SIDIS
processes.
which survives upon integration over φ h , over which we
are going to concentrate now, while F cos φ h
U U
and F cos 2φ h
U U
The Collins FFs generate azimuthal asymmetries in e + e − , where TMD factorization is appropriate, and read[100,101] sin(φ h +φs)
U T
contains the
convolution of transversity with the Collins function, h 1 ⊗
H ⊥
1 .
Fig. 18. Best fit curves for cos φ obtained by fitting COM-PASS multiplicities, cos φ and cos 2φ data. The Cahn effect in cos 2φ has been set to zero. Plot from Ref.[45].Fig. 19. Best fit curves for cos 2φ obtained by fitting COM-PASS multiplicities, cos φ and cos 2φ data. The Cahn effect in cos 2φ has been set to zero. Plot from Ref.[45].<2cos(φ)> h +
<2cos(φ)> h -
-0.3
-0.2
-0.1
0.0
0.1
<Q
2
> = 1.74 GeV 2
<Q
2
> = 2.74 GeV 2
<Q
2
> = 6.61 GeV 2
-0.3
-0.2
-0.1
0.0
0.1
<Q
2
> = 1.77 GeV 2
<Q
2
> = 2.79 GeV 2
<Q
2
> = 6.71 GeV 2
-0.3
-0.2
-0.1
0.0
0.1
<Q
2
> = 1.78 GeV 2
<Q
2
> = 2.83 GeV 2
<Q
2
> = 6.84 GeV 2
-0.3
-0.2
-0.1
0.0
0.1
<Q
2
> = 1.80 GeV 2
<Q
2
> = 2.85 GeV 2
<Q
2
> = 6.96 GeV 2
-0.3
-0.2
-0.1
0.0
0.1
0.3
0.5
0.7
<Q
2
> = 1.82 GeV 2
0.3
0.5
0.7
<Q
2
> = 2.92 GeV 2
0.3
0.5
0.7
<Q
2
> = 7.06 GeV 2
<x B >
<z h >
P T (GeV)
0.61
0.46
0.36
0.28
0.22
0.02
0.03
0.06
<2cos(2φ)> h +
<2cos(2φ)> h -
-0.1
0.0
0.1
0.2
0.3
<Q
2
> = 1.74 GeV 2
<Q
2
> = 2.74 GeV 2
<Q
2
> = 6.61 GeV 2
-0.1
0.0
0.1
0.2
0.3
<Q
2
> = 1.77 GeV 2
<Q
2
> = 2.79 GeV 2
<Q
2
> = 6.71 GeV 2
-0.1
0.0
0.1
0.2
0.3
<Q
2
> = 1.78 GeV 2
<Q
2
> = 2.83 GeV 2
<Q
2
> = 6.84 GeV 2
-0.1
0.0
0.1
0.2
0.3
<Q
2
> = 1.80 GeV 2
<Q
2
> = 2.85 GeV 2
<Q
2
> = 6.96 GeV 2
-0.1
0.0
0.1
0.2
0.3
0.3
0.5
0.7
<Q
2
> = 1.82 GeV 2
0.3
0.5
0.7
<Q
2
> = 2.92 GeV 2
0.3
0.5
0.7
<Q
2
> = 7.06 GeV 2
<x B >
<z h >
P T (GeV)
0.61
0.46
0.36
0.28
0.22
0.02
0.03
0.06
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|
[] |
[
"State-selective influence of the Breit interaction on the angular distribution of emitted photons following dielectronic recombination",
"State-selective influence of the Breit interaction on the angular distribution of emitted photons following dielectronic recombination"
] |
[
"Pedro Amaro \nPhysikalisches Institut\nUniversität Heidelberg\nD-69120HeidelbergGermany\n",
"Chintan Shah \nPhysikalisches Institut\nUniversität Heidelberg\nD-69120HeidelbergGermany\n\nMax-Planck-Institut für Kernphysik\nD-69117HeidelbergGermany\n",
"Rene Steinbrügge \nMax-Planck-Institut für Kernphysik\nD-69117HeidelbergGermany\n",
"Christian Beilmann \nPhysikalisches Institut\nUniversität Heidelberg\nD-69120HeidelbergGermany\n\nMax-Planck-Institut für Kernphysik\nD-69117HeidelbergGermany\n",
"Sven Bernitt \nMax-Planck-Institut für Kernphysik\nD-69117HeidelbergGermany\n",
"José R Crespo López-Urrutia \nMax-Planck-Institut für Kernphysik\nD-69117HeidelbergGermany\n",
"Stanislav Tashenov \nPhysikalisches Institut\nUniversität Heidelberg\nD-69120HeidelbergGermany\n"
] |
[
"Physikalisches Institut\nUniversität Heidelberg\nD-69120HeidelbergGermany",
"Physikalisches Institut\nUniversität Heidelberg\nD-69120HeidelbergGermany",
"Max-Planck-Institut für Kernphysik\nD-69117HeidelbergGermany",
"Max-Planck-Institut für Kernphysik\nD-69117HeidelbergGermany",
"Physikalisches Institut\nUniversität Heidelberg\nD-69120HeidelbergGermany",
"Max-Planck-Institut für Kernphysik\nD-69117HeidelbergGermany",
"Max-Planck-Institut für Kernphysik\nD-69117HeidelbergGermany",
"Max-Planck-Institut für Kernphysik\nD-69117HeidelbergGermany",
"Physikalisches Institut\nUniversität Heidelberg\nD-69120HeidelbergGermany"
] |
[] |
We report a measurement of KLL dielectronic recombination in charge states from Kr +34 through Kr +28 , in order to investigate the contribution of Breit interaction for a wide range of resonant states. Highly charged Kr ions were produced in an electron beam ion trap, while the electron-ion collision energy was scanned over a range of dielectronic recombination resonances. The subsequent Kα x rays were recorded both along and perpendicular to the electron beam axis, which allowed the observation of the influence of Breit interaction on the angular distribution of the x rays. Experimental results are in good agreement with distorted-wave calculations. We demonstrate, both theoretically and experimentally, that there is a strong state-selective influence of the Breit interaction that can be traced back to the angular and radial properties of the wavefunctions in the dielectronic capture.
|
10.1103/physreva.95.022712
|
[
"https://arxiv.org/pdf/1701.09044v1.pdf"
] | 119,358,737 |
1701.09044
|
60f70822c4046e464ce6ecc4f85e53a074c4538a
|
State-selective influence of the Breit interaction on the angular distribution of emitted photons following dielectronic recombination
Pedro Amaro
Physikalisches Institut
Universität Heidelberg
D-69120HeidelbergGermany
Chintan Shah
Physikalisches Institut
Universität Heidelberg
D-69120HeidelbergGermany
Max-Planck-Institut für Kernphysik
D-69117HeidelbergGermany
Rene Steinbrügge
Max-Planck-Institut für Kernphysik
D-69117HeidelbergGermany
Christian Beilmann
Physikalisches Institut
Universität Heidelberg
D-69120HeidelbergGermany
Max-Planck-Institut für Kernphysik
D-69117HeidelbergGermany
Sven Bernitt
Max-Planck-Institut für Kernphysik
D-69117HeidelbergGermany
José R Crespo López-Urrutia
Max-Planck-Institut für Kernphysik
D-69117HeidelbergGermany
Stanislav Tashenov
Physikalisches Institut
Universität Heidelberg
D-69120HeidelbergGermany
State-selective influence of the Breit interaction on the angular distribution of emitted photons following dielectronic recombination
(Dated: Received: October 26, 2018)numbers: 3230Rj3480Lx5220Fs
We report a measurement of KLL dielectronic recombination in charge states from Kr +34 through Kr +28 , in order to investigate the contribution of Breit interaction for a wide range of resonant states. Highly charged Kr ions were produced in an electron beam ion trap, while the electron-ion collision energy was scanned over a range of dielectronic recombination resonances. The subsequent Kα x rays were recorded both along and perpendicular to the electron beam axis, which allowed the observation of the influence of Breit interaction on the angular distribution of the x rays. Experimental results are in good agreement with distorted-wave calculations. We demonstrate, both theoretically and experimentally, that there is a strong state-selective influence of the Breit interaction that can be traced back to the angular and radial properties of the wavefunctions in the dielectronic capture.
I. INTRODUCTION
Highly charged ions are ideal atomic systems for investigating the relativistic details of dynamical atomic processes, such as electron-ion collisions. At the strong electromagnetic fields of these atomic systems, both incident and bound electrons can reach sizable fractions of the light speed, and consequently, retardation and magnetic terms of the electron-electron interaction can significantly change the collision dynamics. These relativistic terms are included in the Breit interaction [1,2], which corresponds to the next quantum electrodynamic term after Coulomb interaction [3,4]. The contribution of the Breit interaction to the electronic structure has been extensively calculated, using various methods [1,2,5,6], showing excellent agreement with experimental results [7,8]. Its typical contribution to the binding energy of a few percent [9,10] can be treated perturbatively. On the other hand, the electron-ion collision dynamics is much more dependent on the electron-electron interaction, which hampers perturbative approaches to the Breit contribution. Hence, recent experiments have been performed in several electron-ion atomic processes, namely, resonant transfer and excitation [11], Coulomb excitation [12] and electron impact excitation [13] in order to investigate the relativistic details of the respective atomic collisions.
Dielectronic recombination (DR) provides an even more sensitive probe of the Breit interaction. In this process, a free electron is captured by an ion with simultaneous excitation of a bound electron, which only occurs due to the electron-electron interaction. Deexcitation of the resonant state by photon emission completes the DR process. Recent investigations highlighted a pronounced role of the Breit interaction along the (usually dominant) Coulomb interaction, not only on the resonant strengths [14], but also on the angular distribution and polarization properties of the photon emission [15,16]. All of these investigations focused on one particular resonant state, [1s2s 2 2p 1/2 ] 1 , of initial Li-like ion. Apart from this unique case, only few resonant DR states of initial H-like ions were reported also having a dominant Breit influence [17,18].
Besides this fundamental interest on the relativistic details of electron-electron interaction, investigations of the DR are also mandatory for modeling hot astrophysical and laboratory plasmas, as DR often dominates the recombination rates due to its high resonance strength [19][20][21]. Resonances can thus be used for temperature and density plasma diagnostics [22][23][24][25]. Simultaneously, information about the angular distribution and polarization of the DR photons emitted by anisotropic plasmas could be used to infer the directionality of the plasma electrons. Moreover, an anisotropic emission may change the intensity of observed lines and can thus affect temperature diagnostics [26][27][28][29][30][31]. Besides DR, trielectronic and quadruelectric recombination was recently demonstrated to play an essential role in charge state distribution of hot plasmas [32].
Recently, the polarization of DR Kα x rays from highly charged Kr and Xe were measured with a novel Compton polarimeter [16,33]. The degree of linear polarization was determined for a few DR resonances and showed agreement with theoretical predictions of a measurable influence of the Breit interaction (BI). In this work, we complement those previous studies with a systematic investigation of the influence of BI on the angular distri- An electron beam is accelerated toward the drift tubes and compressed by a magnetic field. Highly charged Kr ions are then produced in the trap by successive electron impact ionization. The electron beam energy is scanned over the DR resonances and the subsequent photon emission is observed by two x-ray detectors located along (0 • ) and perpendicular to the electron beam axis (90 • ).
bution of emitted photons following DR for a wide range of DR resonances. For this purpose, as in earlier works, charge states of He-like ( Kr +34 ) to O-like (Kr +28 ) were produced in an electron beam ion trap (EBIT), and the electron beam energy was scanned over the KLL resonances. We measured the angular distribution of the subsequent Kα x rays through simultaneous observation in directions along and perpendicular to the electron beam axis. The alignment parameter of the resonant state was extracted from the acquired spectra. These results were compared with extensive distorted-wave calculations based on the Flexible Atomic code (FAC). Here, we focus only on the state-selective influence of the Breit interaction on the angular distribution, i.e., why some resonances display such strong BI influence, while others only manifest a negligible one. A previous study was performed for only the two states [1s2s 2 2p 1/2 ] 1 and [1s2s2p 2 1/2 ] 1 in Li-like ions [34].
II. THEORY
We give here a brief description of the theoretical framework used for describing the DR in few-electron ions, as this process has already been extensively discussed in the literature [15,[35][36][37]. Due to the collision of unidirectional electrons with the ions in the present setup, the formed resonant states are aligned along the beam axis, i.e., the magnetic sublevel population of a resonant state |α d J d M d is non-statistically distributed. Alignment parameters A d k are often used to describe the photon angular distribution of non-statistically distributed states. For the DC process, A d k has the following form:
A d k = 2J d + 1 × J d M d =−J d (−1) J d −M d J d M d , J d − M d |k 0 σ M d . (1)
Here,σ M d are normalized magnetic sublevel cross sections of the DC process (
σ M d = σ M d / M d σ M d ).
The parameters A d k have only nonzero values if k is even and k < J d . Therefore, only resonant states with J d > 1/2 can be aligned. Aligned states often emits anisotropic radiation, and for the DR process, the angular distribution is given by [38]
dS dΩ d→f = S 4π 1 + k=2,4... α k d→f A d k P k (cos θ) ,(2)
with S being the DR resonance strength. The polar angle θ is defined with respect to the electron beam axis and P k denotes the Legendre polynomial. Information concerning the photon emission between the intermediate resonant and final states, such as the multipole contributions, is included in the intrinsic photon coefficients α k d→f [38]. All resonant states listed in Table I contains an allowed and dominant electric dipole channel. Therefore, all terms α k d→f with k > 2 can be neglected (α 2 d→f ≡ α d→f , see Ref. [16]). In this investigation, we do not consider a possible alignment of the initial state that might arise from the collision processes leading to the production of the ion. For the present charge states, only the initial (ground) state with J i = 3/2 of N-like ions can be aligned.
The evaluation of the DC cross sections in Eq. (1) was performed with the FAC code, which treats resonant electron capture in a distorted-wave framework [39]. In this approach, the free electron is expanded in partial waves (εl j ) (e.g. [15,17,35]) and the DC cross sections are traced back to the matrix elements
α d J d M d |V | α i J i M i (εl j ) .
Here, the matrix element addresses the formation of a resonant state |α d J d by resonant capture of a free electron by an initial ion with state |α i J i , through the electron-electron interaction V . The influence of the Breit interaction can thus be investigated by performing calculations either with the full operator
V = V C +V B , or with Coulomb interaction only V = V C .
The allowed partial waves (εl j ) are restricted by angular and parity selection rules included in the matrix elements. Moreover, since V is a two-body operator only matrix elements with two active pairs of orbitals participating in the DC process are nonzero [6].
The DC process can be further reduced to a combination of Slater integrals between the radial components of two active pair-electrons. A Slater integral is given as
Λ k x−v,y−w = R x−v (r 2 )dr 2 R y−w (r 1 ) r k > r k+1 < dr 1 = R x−v (r 2 )v k (y, w, r 2 )dr 2 ,(3)
where R i−j (r) can be either a density overlap, R C i−j (r) = r 2 ρ = P i (r)P j v(r) + Q i (r)Q j (r) resulted from the Coulomb interaction, or can be overlaps resulting from the Breit interaction, which mixes the large (P ) and small (Q) components of the radial wavefunctions, i.e., [6,10]. The pairs x−v and y − w are the active orbitals that are changed during the DC process. r > = max(r 2 , r 1 ) and r < = min(r 2 , r 1 ). Here, m is a positive integer that depends on the angular decomposition of the matrix elements. As will be seen in Sec. V, the observed state-selective influence of Breit interaction in DR angular distribution can be traced back to the type of active orbitals and to the radial Coulomb and Breit overlaps between them.
R B i−j (r) = P i (r)Q j (r)±Q i (r)P j (r)
III. EXPERIMENT
The experiment was performed at the Max-Planck Institut for Nuclear Physics, where highly charged ions of Kr were generated at the FLASH EBIT [40,41]. In this experiment, an electron beam is emitted by an electron gun and accelerated towards a set of drift tubes with an applied high-voltage. This monoenergetic electron beam is simultaneously compressed by a magnetic field of 6 T to a diameter of ≈50 µm (calculated according to Ref. [42]). An electrostatic axial trap of 50 mm length is formed by biasing the central drift tube with a slight positive voltage, relative to the two surrounding drift tubes. At the trap, injected atoms are then multi-ionized by the compressed electron beam through electron impact. A scheme of this experimental setup is shown in Fig. 1.
The electron beam energy was continuously scanned over the KLL DR resonances by controlling the drift tube bias voltage according to a triangular wave function from 6.7 kV to 7.8 kV with a rate of 1.8 eV/s. In order to limit the energy spread of the electron-ion interaction, the electron beam current was set to a moderate value of 70 mA. The cathode of the electron gun had a negative bias of -2 kV.
The trap settings were optimized for both highest concentration of the highly charged ions and energy beam resolution. The second criterion is accomplished by further reducing the energy spread with the evaporative cooling technique [43] through the use of a shallow axial trap. However, shallow traps contains less ions than deeper ones, specially for He-like and Li-like ions. Therefore, we performed two types of measurements: (a) a deeper trap with higher concentration of high charge states (like He-like ions) and a lower resolution of 19 eV (full width at half maximum, FWHM); (b) A shallow trap with concentration of high charge states and a better resolution of 12 eV. The respective values of the trap voltage offset for settings (a) and (b) are 100 V and 130 V.
High-purity germanium detectors were mounted along and perpendicular to the electron beam axis, as shown in Fig. 1. The ratio between solid angles obtained from their location and active area is Ω 90 o /Ω 0 o ≈ 9.4. The photon energy resolution (FWMH) at 13 keV of the detectors at 90 • and 0 • are 180 eV and 200 eV, respectively.
IV. DATA ANALYSIS
A typical contour plot with the electron beam energy and photon energy as parameters is shown in Fig. 2. The electron beam energy was scanned through the KLL DR resonances in He-like to O-like Kr, which are visible as bright spots. Each resonance is identified by the initial charge state of Kr before DR. The background near the resonances is due to radiative recombination (RR) into the L shell (n = 2), while the lower background (at photon energies of ∼11 keV) is due to radiative recombination in other heavier elements at the trap, like barium or tungsten.
We obtained the intensity of DR x-rays by selecting the events of interest of both RR and DR contained in the region of Fig. 2 inside the black solid lines. These events were added up for a given value of the electron beam energy. The obtained intensity of DR x-rays is displayed in Fig. 3. Here, the background due to RR was fitted with a linear function and removed. The left (a) and right (b) sides of the plot correspond to a deep and shallow trap, respectively, as described in Sec. III.
The selected energy region of Fig. 3 contains a high density of resonant states that were identified with the help of theoretical calculations. Table I displays the theoretical energies for the resonant states with respective identification, initial ion charge state and recombination process. If two or more resonant states have the energy separated by less than half of the energy resolution, we considered a single Gaussian function, for fitting the peak formed by these states. For now on, we refer to any of the obtained Gaussian functions as a resonance. The total number of resonances is 33 as shown in Fig. 3. The beam energy resolution of the settings (a) and (b) (see Table I by not showing the experimental energy.
A linear calibration based on two theoretical resonances was employed for the electron beam energy. Measurement (a) in Fig. 3 For each resonance we compare in Table I [44] Kr and, more recently, for the case of H-like, He-like and Li-like ions [45]. We have a good agreement in all values of resonant energies with differences of less than 4 eV.
Since the detectors at 0 • and 90 • have different solid angles of detection, we calibrated both spectra using an isotropic resonance. We choose resonance 33 that is the most intense resonance among the isotropic ones. The obtained solid angle correction factor due to the different solid angles was C Ω = 9.36 ± 0.01.
We quantified the emission anisotropy as the ratio between the DR resonance amplitudes observed along and perpendicular directions to the electron beam axis, corrected with the solid angle factor, i.e., R = C Ω I(0 • )/I(90 • ). Table I list values of ratios R for each resonance along with the respective charge state of the initial ion and the recombination process that populated the resonant state.
The obtained values of the ratios for the resonance 4 is shown in Fig. 4. The error bar in each measurement contains the combined uncertainty (1σ) of the statistical uncertainty of fitting the amplitudes for both I(0 • ), I(90 • ) and C Ω .
According to the non-laminar optical model developed by Herrmann [42], the motion of an electron in an EBIT is described by an helical path collinear with the beam direction. Thus, the relative electron-ion collision is not aligned with the electron beam direction but deviates by a pitch angle γ that is given by [46] tan γ =
√ E ⊥ E beam − E ⊥ ,(4)
where E ⊥ is the transverse electron energy. For a cathode temperature of 1300 K and a cathode radius of 1.5 mm, the transverse electron energy after compression of the electron beam is E ⊥ ≈0.1 keV, which corresponds to a pitch angle of γ ≈ 6 • [46] and a deviation of 0.02 in R.
The final uncertainty of each R in Table I is the combined uncertainty of the statistical error (see Fig. 4) and this systematic uncertainty. Besides the (usual) DR process, we also observe resonant states populated by higher-order resonant recombination, such as trielectronic (TR) and quadruelectronic (QR) recombination processes [44,47] as identified by some resonances in Table I. In the present work we restrict to their energy identification. Investigations of influence of angular distribution and polarization properties to hot plasma model have been published elsewhere [32].
V. RESULTS AND DISCUSSION
The obtained values R =0.99±0.07 and R =1.01±0.06 for the isotropic resonances 1 and 3 listed in Table I are TABLE I. Measured values of ratios R = CΩI(0 • )/I(90 • ) for KLL resonances of highly charged Kr. The first column labels resonances in Fig. 3, which can consist of a single resonant state or an ensemble of unresolved ones. The resonant states are given in jj-coupling notation, Column P identifies the resonant process and the charge state refers to the initial ion. Experimental and theoretical resonant energies are identified by ER exp and ER theo , respectively. Theoretical values were obtained with FAC. From all resonances listed in Table I, we observe that the majority of them have R < 1, i.e, the photons are mostly emitted in a direction perpendicular to the elec-tron beam axis. According to Eq. (2) this is equivalent to the product α 2 d→f A d 2 being negative. Due to radiative and collision deexitation, the ions are mostly populated in the ground state, which for He-like, Be-like, and Clike ions have J i = 0, while for Li-like and B-like ions have J i = 1/2. Therefore, for He-like, Be-like, and Clike ions, the intermediate magnetic sublevel states are limited to M d = ±1/2. This leads to an orientation of J d being mostly perpendicular to the electron trajectory, and thus to a negative value of A d 2 . For Li-like and Blike ions, a stronger photon emission being perpendicular to the electron beam axis and α 2 d→f > 0 indicates that magnetic sublevels M d = 0 are more populated. Some observed resonances have R > 1. This is due to α df < 0 for photon transitions with J d = J f . Such resonances include 22, 25, 26, 29 and 30. Therefore, the rule-of-thumb is that resonant magnetic sublevels with M d = 0 and M d = ±1/2 are mostly populated. This can be expected, since the orbital angular momentum of the free electron, which is perpendicular to its trajectory, is transferred in the collision. This results in an orientation of J d that is mostly perpendicular to the electron beam axis.
Resonance 4 is an exception to this general observation by having R > 1 and α df > 0, indicating that magnetic sublevels with M d = ±1 are predominantly populated. This is a consequence of the sublevel M d = 0 being weakly populated due to LS selection rules of the term 3 P 1 [15].
For the investigation of the BI influence we restricted ourselves to a set of resonances with one well-separated resonant state. Moreover, resonances of initial N-like ions were also not considered here due to a possible alignment of the initial state (Sec. II). The list of selected resonant states sorted by J d is given in Table II together with experimental and theoretical values of A d 2 . The experimental values of A d 2 were extracted from the ratios R, listed in Table I, according to the following expression
A d 2 = R − 1 T R 2 + 1 ,(5)
which can be obtained from Eq. (2). Here, the term T stands for the average of α d→f weighted by the radiative decay rates, i.e., Initial He-like, Be-like and C-like ions with J i = 0 have the DC process restricted to one partial wave, due to selection rules, which reduces the alignment parameter to a geometrical factor [17]. Exact values of A d 2 are −1 and −4 1/14 ≈ −1.069 for J d = 3/2 and J d = 5/2, respectively. On the other hand, Li-like and B-like ions include two partial waves, and the interference between them makes the A d 2 dependent of the details of the electronelectron interaction [17,50]. As can be observed in Table II, A d 2 of resonant states 4, 19 and 20 depends on the contribution of BI. The resonant state 10 also contains a dependency that is smaller than the uncertainty shown in Table II. We observed that without the contribution of Breit interaction, the difference between experimental and theoretical values amounts 3.5σ. On the other hand, BI influence is too small to be observed in resonances 10, 19 and 20, which agree with predictions.
T = f W df r α d→f / f W df
As depicted in Fig, 5, even for an heavy element like U, the influence of BI can be regarded as a correction for the resonance 19. A pronounced influence of BI was predict for the resonant state 4 [15] and experimentally demonstrated for heavy elements [48,49]. For investigation of the state-selective influence of BI, we calculated the radial overlaps in Eq. (3) between the active electrons participating in the DC process. The respective radial overlaps were calculated with FAC for the resonant states 4, 10, 19 and 20, as well as for Kr and U. The result is displayed in Fig. 6 for the cases of Breit and Coulomb radial overlaps. The integer k of Eq. (3) for these overlaps is the minimal value allowed by selection rules. For Coulomb overlaps it is k = 1 in all cases, with the except of the case 4 (with k = 0), whereas for Breit overlap it is k = 0 for all cases.
It can be noticed for the resonance 4 that the Breit overlap is similar in magnitude to the Coulomb overlap by factors of 5 and 1, for Kr and U ions, respectively. On the other hand, for all other resonances the Breit overlap is regarded as a small correction to the Coulomb overlap. In the case of resonance 4, all active orbitals 1s, 2s and 2p 1/2 have their relativistic small components and large components mostly located in the same radial region (≈ a 0 /Z with a 0 being the Bohr radius). This is not the case for the other resonant states, where the small and large components of the active orbitals 1s, 2p 1/2 and 2p 3/2 are located in different radial regions, the last one mostly centered at ≈ 4a 0 /Z. This delocalization of the radial large and small components reduces the Breit overlap and makes it much smaller than the Coulomb overlap.
Therefore, the relative importance of the Breit interaction in the alignment of doubly excited states is mostly related to the radial localizations of the large and small components of the relativistic wavefunctions.
VI. SUMMARY
In this work we performed a systematic investigation of the angular distribution of the emitted photons produced by KLL DR of highly charged ion, from He-like to O-like Kr. The radiation was recorded along and perpendicular to the electron beam axis. Experimental alignment parameters were extracted from the data. Among the extensive set of observed resonant states, only one manifests an observable dependency of Breit interaction on the alignment parameter. The lack of dependency of BI by the other resonances can be traced back to the radial overlaps between the active electrons of the DC process.
FIG
. 1. (Color online) Scheme of the experimental setup.
Fig. 3 )
3was obtained from resonances 3 and 24, each one consisted of a single well-separated resonant state. Some weak resonances, such as resonances 5, 8 or 11 have low resonant strengths, but contribute to the fit by perturbing the profiles of nearby, more intense resonances. The positions and widths of these resonances were fixed to theoretical energy values and to the previously obtained experimental width, having only the amplitudes as freefit parameters. These resonances are identified in
online) Intensity of x rays in function of x-ray energy and electron beam energy for the photon detector mounted at 90 • to the electron beam direction. The solid lines delimitates the region of interest for the summation of events. The dashed lines delimitates the constant background of radiative recombination (RR) into n = 2. Each bright spot is one DR resonance identified by the charge state of the initial ion. and 24 (not shown), corresponding to energies of 8899 eV and 9427 eV, respectively. Similarly, case (b) in Fig. 3 was calibrated with resonances 17 (9238 eV) and 24.
the theoretical and experimental energy. The theoretical values were obtained from the FAC code, which follows a relativistic configuration interaction formalism ([39] and references therein). Other theoretical values based in MCDF and experimental values of resonant energies are available for DR resonances of C-like, N-like and O-like
FIG. 3 .
3(Color online) Intensity of DR x-rays. The black solid line corresponds to the observed perpendicular DR intensity. The red solid line shows the DR intensity observed along the electron beam axis times 9.4. Each resonance is identified by one or more resonant states of a given charge state inTable I. The left side (a) of the plot corresponds to a measurement performed with a deep trap while the right side (b) corresponds to a shallow trap.
FIG. 4 .
4(Color online) Ratios of resonance 4 for several independent measurements. The black solid line corresponds to the weighted average and the blue dashed line stands for plus or minus σ.
online) Alignment parameter A d 2 for the intermediate states [1s2s 2 2p 1/2 ]1 (resonance 4) and [1s2s 2 (2p 1/2 ) 2 2p 3/2 ]1 (resonance 19). Theoretical values were evaluated with (C+BI) and without Breit interaction (C), according to Eq. (1). The present experimental results are represented by the dark circles, while values provided by Refs.[48,49] for heavier elements are represented by the purple diamonds.
) with the DC cross sections obtained from FAC, with and without inclusion of Breit interaction in the calculations.Table II also lists the allowed partial waves that describe the free electron for the DR process of each resonant state, alongside the radiative decay rate W df r (calculated with FAC) to the final states necessary for calculating T . By comparing the results of A d 2 , calculated with the DC cross sections of FAC, with all respective values in Ref. [15], we noticed maximum differences of 5 % along all the isoelectronic sequence. Experimental results presented in Table II agree with the theoretical predictions within 1.5σ for all resonances.r .
Theoretical values of A d
2 were calculated according to
Eq. (1
TABLE II .
IIValues the experimental alignment parameter A exp label and total angular momentum J d . Allowed partial waves of the free electron are displayed in the third column. Total radiative decay widths W df r calculated with FAC are listed in the fifth column. The term T in the sixth column denotes the average value of α df weighted by W df r . Theoretical values of A theo2
for a selection of resonant states. Intermediate states are
identified by a 2
are displayed for the cases of including BI and
only Coulomb interaction (C).
Label J d
Partial wave Final state
W df
r (s −1 ) T
A
exp
2
A theo
2
(C) A theo
2
(C+BI)
4
1
εp 1/2 , εp 3/2
[1s 2 2s 2 ]0
3.591(14)
0.682 0.62 ±0.02
0.704
0.640
[1s 2 (2p 2
1/2 )0]0
1.235(13)
[1s 2 (2p 1/2 2p 3/2 )1]1
3.452(12)
[1s 2 (2p 1/2 2p 3/2 )2]2
8.865(12)
19
1
εs 1/2 , εd 3/2
[1s 2 2s 2 2p 2
1/2 ]0
1.138(15)
0.453 -0.66±0.04
-0.694
-0.681
[1s 2 2s 2 (2p 1/2 2p 3/2 )1]1
1.142(14)
[1s 2 2s 2 (2p 1/2 2p 3/2 )2]2
5.145(14)
[1s 2 2s 2 (2p 2
3/2 )2]2
2.771(12)
15
3/2 εd 3/2
[1s 2 2s 2 2p 1/2 ] 1/2
1.163(15)
0.318 -0.91±0.05
-1.000
-1.000
[1s 2 2s 2 2p 1/2 ] 3/2
2.861(14)
[1s 2 2p 3
3/2 ] 3/2
3.599(12)
[1s 2 2p 1/2 (2p 2
3/2 )2] 3/2
2.978(12)
[1s 2 2p 1/2 (2p 2
3/2 )2] 5/2
8.662(12)
10
2
εd 3/2 , εd 5/2
[1s 2 (2s2p 1/2 )1]1
1.192(15)
0.270 -0.5 ±0.1
-0.604
-0.604
[1s 2 (2s2p 3/2 )2]2
1.646(14)
[1s 2 (2s2p 3/2 )1]1
9.222(13)
20
2
εd 3/2 , εd 5/2
[1s 2 2s 2 (2p 1/2 2p 3/2 )1]1
5.114(14)
0.179 -0.84 ±0.09 -0.929
-0.939
[1s 2 2s 2 (2p 1/2 2p 3/2 )2]2
7.714(13)
[1s 2 2s 2 (2p 2
3/2 )2]2
1.272(14)
6
5/2 εd 5/2
[1s 2 2p 3/2 ] 3/2
4.076(14)
0.374 -0.9 ±0.2
-1.069
-1.069
14
5/2 εd 5/2
[1s 2 2p 3
3/2 ] 3/2
3.737(14)
0.354 -0.95±0.08
-1.069
-1.069
[1s 2 2p 1/2 (2p 2
3/2 )2] 5/2
3.012(12)
17
5/2 εd 5/2
[1s 2 2p 3
3/2 ] 3/2
5.128(14)
0.357 -0.99±0.05
-1.069
-1.069
[1s 2 2p 1/2 (2p 2
3/2 )2] 5/2
1.122(13)
24
5/2 εd 5/2
[1s 2 2s 2 2p 2
1/2 2p 3/2 ] 3/2
4.667(14)
0.210 -0.94 ±0.08 -1.069
-1.069
[1s 2 2s 2 2p 1/2 (2p 2
3/2 )2] 3/2 3.190(14)
[1s 2 2s 2 2p 1/2 (2p 2
3/2 )2] 5/2 2.023(14)
Coulomb overlap for Kr × 10 Breit overlap for Kr × 50 Coulomb overlap for U Breit overlap for U Coulomb overlap for Kr × 10 Breit overlap for Kr × 50 Coulomb overlap for U Breit overlap for U1/2
2p
1
((1s2s)
[
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Color online) Coulomb and Breit overlaps of active orbitals for resonances 4, 10, 19 and 20, as well as for Kr and U ions. The overlaps (3) are R2p 1/2 −εp 1/2 (r)v(1s, 2s, r), R 2p 3/2 −εd 3/2 (r)v(1s, 2p 1/2 , r). A Gumberidze, D B Thorn, C J Fontes, B Najjari, H L Zhang, A Surzhykov, A Voitkiv, S Fritzsche, D Banaś, H Beyer, W Chen, R D Dubois, S Geyer, R E Grisenti, S Hagmann, M Hegewald, S Hess, C Kozhuharov, R Märtin, I Orban, N Petridis, R Reuschl, A Simon, U Spillmann, M Trassinelli, S Trotsenko, G Weber, D F A Winters, N Winters, 1s2s [ FIG. 6.. R2p 3/2 −εs(r)v(1s, 2p 1/2 , r) and R 2p 3/2 −εd 3/2 (r)v(1s, 2p 3/2 , r) for resonances 4, 10, 19 and 20, respectively. Atomic units are usedA. Gumberidze, D. B. Thorn, C. J. Fontes, B. Najjari, H. L. Zhang, A. Surzhykov, A. Voitkiv, S. Fritzsche, D. Banaś, H. Beyer, W. Chen, R. D. DuBois, S. Geyer, R. E. Grisenti, S. Hagmann, M. Hegewald, S. Hess, C. Kozhuharov, R. Märtin, I. Orban, N. Petridis, R. Reuschl, A. Simon, U. Spillmann, M. Trassinelli, S. Trotsenko, G. Weber, D. F. A. Winters, N. Winters, (1s2s [ FIG. 6. (Color online) Coulomb and Breit overlaps of active orbitals for resonances 4, 10, 19 and 20, as well as for Kr and U ions. The overlaps (3) are R2p 1/2 −εp 1/2 (r)v(1s, 2s, r), R 2p 3/2 −εd 3/2 (r)v(1s, 2p 1/2 , r), R2p 3/2 −εs(r)v(1s, 2p 1/2 , r) and R 2p 3/2 −εd 3/2 (r)v(1s, 2p 3/2 , r) for resonances 4, 10, 19 and 20, respectively. Atomic units are used.
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[
"On the number of lambda terms with prescribed size of their De Bruijn representation *",
"On the number of lambda terms with prescribed size of their De Bruijn representation *"
] |
[
"Bernhard Gittenberger ",
"Zbigniew Gołębiewski "
] |
[] |
[] |
John Tromp introduced the so-called 'binary lambda calculus' as a way to encode lambda terms in terms of binary words. Later, Grygiel and Lescanne conjectured that the number of binary lambda terms with m free indices and of size n (encoded as binary words of length n)is o n −3/2 τ −n for τ ≈ 1.963448 . . .. We generalize the proposed notion of size and show that for several classes of lambda terms, including binary lambda terms with m free indices, the number of terms of size n is Θ n −3/2 ρ −n with some class dependent constant ρ, which in particular disproves the above mentioned conjecture. A way to obtain lower and upper bounds for the constant near the leading term is presented and numerical results for a few previously introduced classes of lambda terms are given.
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10.4230/lipics.stacs.2016.40
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[
"https://arxiv.org/pdf/1509.06139v1.pdf"
] | 14,524,993 |
1509.06139
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cf2e38cf330a2f4a1369956951a6262aa9dbd2ff
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On the number of lambda terms with prescribed size of their De Bruijn representation *
September 24, 2015
Bernhard Gittenberger
Zbigniew Gołębiewski
On the number of lambda terms with prescribed size of their De Bruijn representation *
September 24, 2015
John Tromp introduced the so-called 'binary lambda calculus' as a way to encode lambda terms in terms of binary words. Later, Grygiel and Lescanne conjectured that the number of binary lambda terms with m free indices and of size n (encoded as binary words of length n)is o n −3/2 τ −n for τ ≈ 1.963448 . . .. We generalize the proposed notion of size and show that for several classes of lambda terms, including binary lambda terms with m free indices, the number of terms of size n is Θ n −3/2 ρ −n with some class dependent constant ρ, which in particular disproves the above mentioned conjecture. A way to obtain lower and upper bounds for the constant near the leading term is presented and numerical results for a few previously introduced classes of lambda terms are given.
Introduction
The objects of our interest are lambda terms which are a basic object of lambda calculus. A lambda term is a formal expression which is described by the grammar M ::= x | λx.M | (M N ) where x is a variable, the operation (M N ) is called application, and using the quantifier λ is called abstraction. In a term of the form λx.M each occurrence of x in M is called a bound variable. We say that a variable x is free in a term M if it is not in the scope of any abstraction. A term with no free variables is called closed, otherwise open. Two terms are considered equivalent if they are identical up to renaming of the variables, i.e., more formally speaking, they can be transformed into each other by α-conversion.
In this paper we are interested in counting lambda terms whose size corresponds to their De Bruijn representation (i.e. nameless expressions in the sense of [3]). Definition 1. A De Bruijn representation is a word described by the following specification:
M ::= n | λM | M N
where n is a positive integer, called a De Bruijn index. Each occurrence of a De Bruijn index is called a variable and each λ an abstraction. A variable n of a De Bruijn representation w is bound if the prefix of w which has this variable as its last symbol contains at least n times the symbol λ, otherwise it is free. The abstraction which binds a variable n is the nth λ before the variable when parsing the De Bruijn representation from that variable n backwards to the first symbol.
For the purpose of the analysis we will use the notation consistent with the one used in [1]. This means that the variable n will be represented as a sequence of n symbols, namely as a string of n − 1 so-called 'successors' S and a so-called 'zero' 0 at the end. Obviously, there is a one to one correspondence between equivalence classes of lambda terms (as described in the first paragraph) and De Bruijn representations. For instance, the De Bruijn representation of the lambda-term λx.λy.xy (which is e.g. equivalent to λa.λb.ab or λy.λx.yx) is λλ21; using the notation with successors this becomes λλ((S0)0).
In this paper we are interested in counting lambda terms of given size where we use a general notion of size which covers several previously studied models from the literature. We count the building blocks of lambda terms, zeros, successors, abstractions and applications, with size a, b, c and d, respectively. Formally,
|0| = a, |Sn| = |n| + b, |λM | = |M | + c, |M N | = |M | + |N | + d.
Thus we have for the example given above |λλ((S0)0)| = 2a + b + 2c + d. Assigning sizes for the symbols like above covers several previously introduced notions of size:
• so called 'natural counting' (introduced in [1]) where a = b = c = d = 1, • so called 'less natural counting' (introduced in [1]) where a = 0, b = c = 1, d = 2.
• binary lambda calculus (introduced in [7
]) where b = 1, a = c = d = 2,
Assumption 1. Throughout the paper we will make the following assumptions about the constants a, b, c, d:
1. a, b, c, d are nonnegative integers, 2. a + d ≥ 1, 3. b, c ≥ 1, 4. gcd(b, c, a + d) = 1.
If the zeros and the applications both had size 0 (i.e. a + d = 0), then we would have infinitely many terms of the given size, because one can insert arbitrary many applications and zeros into a term without increasing its size. If the successors or the abstractions had size 0 (i.e. b or c equals to 0), then we would again have infinitely many terms of given size, because one can insert arbitrarily long strings of successors or abstractions into a term without increasing its size. The last assumption is more technical in its nature. It ensures that the generating function associated with the sequence of the number of lambda-terms will have exactly one singularity on the circle of convergence.
Notations. We introduce some notations which will be frequently used throughout the paper: If p is a polynomial, then RootOf {p} will denote the smallest positive root of p. Moreover, we will write [z n ]f (z) for the nth coefficient of the power series expansion of f (z) at z = 0 and f (z) ≺ g(z)
(or f (z) g(z)) to denote that [z n ] f (z) < [z n ] g(z) (or [z n ] f (z) ≤ [z n ] g(z)) for all integers n.
Plan of the paper. The primary aim of this paper is the asymptotic enumeration of closed lambda terms of given size with the size tending to infinity. In the next section we define several classes of lambda terms as well as the generating function associated with them, present our main results and prove several auxiliary results which will be important in the sequel. We derive the asymptotic equivalent of the number of closed terms of given size up to a constant factor. This is established by construction of upper and lower bounds for the coefficients of the generating functions. These constructions are done in Sections 3 and 4. To get fairly accurate numerical bounds we present a method for improving the previously obtained bounds in Section 5. Finally, Section 6 is devoted to the derivation of very accurate results for classes of lambda terms which have been previously studied in the literature.
Main results
In order to count lambda terms of a given size we set up a formal equation which is then translated into a functional equation for generating functions. For this we will utilise the symbolic method developped in [5].
Let us introduce the following atomic classes: the class of zeros Z, the class of successors S, the class of abstractions U and the class of applications A. Then the class L ∞ of lambda terms can be described as follows:
L ∞ = Seq(S) × Z + U × L ∞ + A × L 2 ∞ (1)
The number of lambda terms of size n, denoted by L ∞,n , is |{t ∈ L ∞ : |t| = n}|. Let L ∞ (z) = n≥0 L ∞,n z n be the generating function associated with L ∞ . Then specification (1) gives rise to a functional equation for the generating function L ∞ (z):
L ∞ (z) = z a ∞ j=0 z bj + z c L ∞ (z) + z d L ∞ (z) 2 .(2)
Solving (2) we get
L ∞ (z) = 1 − z c − (1 − z c ) 2 − 4z a+d 1−z b 2z d ,
which defines an analytic function in a neighbourhood of z = 0.
Proposition 1. Let ρ = RootOf (1 − z b )(1 − z c ) 2 − 4z a+d . Then L ∞ (z) = a ∞ + b ∞ 1 − z ρ 1 2 + O 1 − z ρ ,(3)
for some constants a ∞ > 0, b ∞ < 0 that depend on a, b, c, d.
Proof 2 we can observe that all three terms are negative for 0 < z < 1. Since 0 < ρ < 1, the function L ∞ (z) has an algebraic singularity of type 1 2 which means that its Newton-Puiseux expansion is of the form (3). Since L ∞ (z) is a power series with positive coefficients, we know that a ∞ = L ∞ (ρ) > 0 and b ∞ < 0.
. Let f (z) = (1 − z b )(1 − z c ) 2 − 4z a+d . Then ρ is the solution of f (z) = 0. If we compute derivative f (z) = −4(a + b)z a+b−1 − 2cz c−1 (1 − z b )(1 − z c ) − bz b−1 (1 − z c )Corollary 1. The coefficients of L ∞ (z) satisfy [z n ]L ∞ (z) ∼ Cρ −n n −3/2 , as n → ∞, where C = −b ∞ /(2 √ π).
Let us define the class of m-open lambda terms, denoted L m , as L m = {t ∈ L ∞ : a prefix of at most m abstractions λ is needed to close the term} .
The number of m-open lambda terms of size n is denoted by L m,n and the generating function associated with the class by L m (z) = n≥0 L m,n z n . Similarly to L ∞ , the class L m can be specified, and this specification yields the functional equation
L m (z) = z a m−1 j=0 z bj + z c L m+1 (z) + z d L m (z) 2(4)
Note that L 0 (z) is the generating function of the set L 0 of closed lambda terms.
Let K m = L ∞ \ L m and K m (z) = L ∞ (z) − L m (z)
. Then using (2) and (4) we obtain
K m (z) = z a ∞ j=m z bj + z c K m+1 (z) + z d K m (z)L ∞ (z) + z d K m (z)L m (z).(5)
which implies
K m (z) = z a+bm (1 − z b )(1 − z d (L ∞ (z) + L m (z))) + z c 1 − z d (L ∞ (z) + L m (z)) K m+1 (z).(6)
Note that K m (z) as well as L m (z) define analytic functions in a neighbourhood of z = 0. Let us state the main theorem of the paper:
Theorem 1. Let ρ = RootOf (1 − z b )(1 − z c ) 2 − 4zCn − 3 2 ρ −n ≥ 1 and lim sup n→∞ [z n ] L m (z) Cn − 3 2 ρ −n ≤ 1,(7)
Remark 1. In case of given a, b, c, d and m we can compute numerically such constants C and C. This will be done for some of the models mentioned in the introduction.
Before proving this theorem we will present the key ideas needed for our proof. We introduce the class L
(h) m,n = t ∈ L (h) m : |t| = n and L (h) m (z) = n≥0 L (h) m,n z n . Then L (h) m (z) satisfies the functional equation L (h) m (z) = z a m−1 j=0 z bj + z c L (h) m+1 (z) + z d L (h) m (z) 2 if m < h, z a h−1 j=0 z bj + z c L (h) h (z) + z d L (h) h (z) 2 if m ≥ h.(8)
Notice that for m ≥ h we have a quadratic equation for L (h)
m (z) = L (h) h (z) that has the solution L (h) h (z) = 1 − z c − (1 − z c ) 2 − 4z a+d 1−z bh 1−z b 2z d .
For m < h we have a relation between L
L (h) m (z) = 1 − r m (z) + 2z c r m+1 (z) + 2z c · · · r h−1 (z) + 2z c r h (z) 2z d (9) where r j (z) = 1 − 4z a+d 1−z jb 1−z b − 2z c if m ≤ j < h − 1, 1 − 4z a+d 1−z (h−1)b 1−z b − 2z c + 2z 2c if j = h − 1, (1 − z c ) 2 − 4z a+d 1−z bh 1−z b if j = h.
One can check that the dominant singularity ρ
m (z) = a (h) m +b (h) m 1 − z ρ (h) 1 2 +O 1 − z ρ (h) as z → ρ (h) m for some constants a (h) m , b (h)
m depending on m and h). Notice that for all m,
k ≥ 0 we have ρ (h) m = ρ (h) k
and that is why we can drop the lower index and write just
ρ (h) instead of ρ (h) m . Moreover, ρ (h) > ρ as well as lim h→∞ ρ (h) = ρ.
Let us begin with computing the radii of convergence of the functions K m (z) and L m (z). For the case of binary lambda calculus Lemmas 2 and 3 were already proven in [6]. To extend those results to our more general setting, we will use different techniques.
Lemma 1. For all m ≥ 0 the radius of convergence of K m (z) equals ρ (the radius of convergence of L ∞ (z)).
Proof. Inspecting (6) reveals that the key part is
1 1−z d (L∞(z)+Li(z))
. This is the generating function of a sequence of combinatorial structures associated with the generating function z d (L ∞ (z)+L i (z)). One can check that we are not in the supercritical sequence schema case (i.e. a singularity of considered fraction does not come from the root of its denominator, see [5, pp. 293
]) because 1 − ρ d (L ∞ (ρ) + L i (ρ)) > 0. This follows from ρ d (L ∞ (ρ) + L i (ρ)) ≤ 2ρ d L ∞ (ρ) = 1 − ρ c < 1.
The first inequality holds because L ∞ (ρ) ≥ L i (ρ) for all i ≥ 0 and the second one because ρ > 0. Moreover, the radius of convergence of L i (z) is larger than or equal to the radius of convergence of L ∞ (z) because L i ⊆ L ∞ . Therefore, for all m ≥ 0 the radius of convergence of K m (z) equals ρ, the radius of convergence of L ∞ (z).
Lemma 2.
All the functions L m (z), m ≥ 0, have the same radius of convergence.
Proof. Let ρ m denote the radius of convergence of the function L m (z). From the definition of the function L m (z) it is known that for all m ≥ 0 and for all n we have [z n ] L m (z) ≤ [z n ] L m+1 (z) and therefore ρ m ≥ ρ m+1 . Moreover, from (4) we know
L m+1 (z) = −z a−c m−1 j=0 z bj + z −c L m (z) − z d−c L m (z) 2 .
Notice that ρ m ≤ 1 because ρ m ≤ ρ (h) < 1 for h ≥ m. Then due to the fact that z −c L m (z) − z d−c L m (z) 2 has radius of convergence bigger or equal ρ m , we have ρ m ≤ ρ m+1 . In the next sections we will present how to obtain an upper and a lower bound for [z n ] L m (z). The idea is to construct auxiliary functions satisfying certain inequalities and to use them to construct further ones until we have the desired bound. The procedure follows the flowchart depicted in Fig. 1.
L (h,H) m (z) L m (z) 4 K (h,H) m (z) K m (z) 5 L m (z) L m (z) 3 2 L (h) m (z) L m (z) 1 K (h) m (z) K m (z) L m (z) L m (z)K (h) m (z) = z a ∞ j=m z bj + z c K (h) m+1 (z) + z d K (h) m (z)L ∞ (z) + z d K (h) m (z)L (h) m (z).(10)
In fact, what we did is that we replaced in the application operation every m-open lambda term (corresponding to the subterm z d K m (z)L m (z) of (5)) by an m-open lambda term where each string of successors has bounded length (corresponding to z d K m (z)L (h) m (z)). Solving (10) we get
K (h) m (z) = z a−cm 1 − z b ∞ j=m z j(b+c) j i=m 1 1 − z d L ∞ (z) + L (h) i (z) =: S m,∞ (z).(11)
Lemma 4. Let ρ, a ∞ , b ∞ be as in Proposition
1 andc i = 1/(1 − ρ d (a ∞ + L (h) i (ρ))) andd i = b ∞ ρ d /(1 − ρ d (a ∞ + L (h) i (ρ))) 2 . Then K (h) m (z) admits the expansion K (h) m (z) = c (h) m + d (h) m 1 − z ρ 1 2 + O 1 − z ρ , as z → ρ,(12)
where
c (h) m = S m,h−1 (ρ) + R c (h) m if m < h, ρ a+bm (1−ρ b ) 1−ρ b+c −ρ d a∞+L (h) h (ρ) else, d (h) m = ρ a−cm 1−ρ b h−1 j=m ρ j(b+c) j i=md ĩ ci j k=mc k + R d (h) m if m < h, b∞ρ a+bm+d (1−ρ b ) 1−ρ b+c −ρ d a∞+L (h) h (ρ) 2 else, with R c (h) m = ρ a+bh+c(h−m) (1 − ρ b ) 1 − ρ b+c − ρ d a ∞ + L (h) h (ρ) h−1 i=mc i , R d (h) m = b ∞ ρ a+bh+c(h−m)+d (1 − ρ b ) 1 − ρ b+c − ρ d a ∞ + L (h) h (ρ) h−1 i=mc i × h−1 i=mc i + 1 1 − ρ b+c − ρ d a ∞ + L (h) h (ρ) .
Proof. Let us recall that for all m ≥ h we have L
(h) m (z) = L (h)
h (z). Therefore we can split the infinite sum S m,∞ (z) in (11) into the finite one S m,h−1 (z) and the rest S h,∞ (z).
Case I: m < h. First, consider the finite sum S m,h−1 (z). As in the proof of Lemma 1 we identify the key term, show that we are not in the supercritical case, and expand by means of Proposition 1. Eventually, this yields
j i=m 1 1 − z d (L ∞ (z) + L (h) i (z)) =c m,j +d m,j 1 − z ρ 1 2 + O 1 − z ρ wherec m,j =S h,∞ (z) = h−1 i=m 1 1 − z d L ∞ (z) + L (h) i (z) · z a+bh+c(h−m) (1 − z b ) 1 − z b+c − z d L ∞ (z) + L (h) h (z)
. We already know how to handle the product part of this expression, so let us consider the fraction
z a+bh+c(h−m) (1−z b ) 1−z b+c −z d L∞(z)+L (h) h (z)
. Similarly to before, we have to check that the singularity of this function does not come from the root of the denominator but from L ∞ (z) (it cannot come from L (h) h because it has a bigger radius of convergence than L ∞ (z)). So, we have to show the inequality
1 − ρ b+c − ρ d L ∞ (ρ) + L (h) h (ρ) > 0. But from L ∞ (ρ) ≥ L (h) h (ρ) and 0 < ρ b+c < ρ c < 1 we obtain ρ d (L ∞ (ρ) + L (h) h (ρ)) ≤ 2ρ d L ∞ (ρ) = 1 − ρ c < 1 − ρ b+c
and hence the desired inequality indeed holds. Now, similarly to the previous case we use the Newton-Puiseux expansion of L ∞ (z) at ρ to derive an expansion of the infinite part of the sum in (11) and get the asserted result.
Case II: m ≥ h. This case is easier, because the finite part of the sum in (11) does not exist and the other part can be evaluated to a closed form which can be treated as above.
Using the transfer lemmas of [4] (applied to K
Cn − 3 2 ρ −n ≤ 1 where C = b ∞ − d (h) m Γ − 1 2 . 4 Lower bound for [z n ] L m (z)
The idea behind obtaining a lower bound for [z n ] L m (z) is similar to the one used for the upper bound. First we will find an upper bound for [z n ] K m (z) using the function
L (h,H) m (z) = n≥0 L (h,H) m,n z n = L ∞ (z) − K (h) m (z) if m < H, L ∞ (z) else.(13)K (h,H) m (z) = z a ∞ j=m z bj + z c K (h,H) m+1 (z) + z d K (h,H) m (z)L ∞ (z) + z d K (h,H) m (z)L (h,H) m (z).
Solving this equation and using (13) we get
K (h,H) m (z) = z a−cm 1 − z b H−1 j=m z j(b+c) j i=m 1 1 − z d L ∞ (z) + L (h,H) m (z) = z a−cm 1 − z b H−1 j=m z j(b+c) j i=m 1 1 − z d 2L ∞ (z) − K (h) i (z) + z a+bH+c(H−m) (1 − z d ) (1 − z b+c − 2z d L ∞ (z)) H−1 i=m 1 1 − z d 2L ∞ (z) − K (h) i (z) .(14)
Lemma 5. Let ρ be the radius of convergence of the function L ∞ (z). Then the generating function K (h,H) m (z) admits the following expansion
K (h,H) m (z) = c (h,H) m + d (h,H) m 1 − z ρ 1 2 + O 1 − z ρ ,(15)
where
c (h,H) m = ρ a−cm 1 − ρ b H−1 j=m ρ j(b+c) j i=m 1 1 − ρ d 2a ∞ − c (h) i + R c (h,H) m , d (h,H) m = ρ a−cm 1 − ρ b H−1 j=m ρ j(b+c) j i=m ρ d 2b ∞ − d (h) i 1 − ρ d 2a ∞ − c (h) i j i=m 1 1 − ρ d 2a ∞ − c (h) i + R d (h,H) m , a ∞ , b ∞ and c (h) i , d (h) i
come from the expansion of L ∞ (z) and K (h) i (z), respectively, at ρ (see Proposition 1 and the proof of Lemma 4) and
R c (h,H) m = ρ a+bH+c(H−m) (1 − ρ d ) (1 − ρ b+c − 2ρ d a ∞ ) H−1 i=m 1 1 − ρ d 2a ∞ − c (h) i , R d (h,H) m = ρ a+bH+c(H−m)+d (1 − ρ d ) (1 − ρ b+c − 2ρ d a ∞ ) H−1 i=m 1 1 − ρ d 2a ∞ − c (h) i × 2b ∞ 1 − ρ b+c − 2ρ d a ∞ + H−1 i=m 2b ∞ − d (h) i 1 − ρ d 2a ∞ − c (h) i .
As in the previous section, we apply the the transfer lemmas of [4] to K
Cn − 3 2 ρ −n ≥ 1 where C = b ∞ − d (h,H) m Γ − 1 2 .
Improvement of the bounds
The bounding functions L m (z) = L ∞ (z) − K These two function admit a representation in terms of nested radicals which is similar to (9):
L (h) m,M (z) = 1 2z d · 1− 1−4z a+d 1−z mb 1−z b −2z c +2z c ··· 1−4z a+d 1−z (M −1)b 1−z b −2z c +2z c L∞(z)−K (h) M (z) , L (h,H) m,M (z) = 1 2z d · 1− 1−4z a+d 1−z mb 1−z b −2z c +2z c ··· 1−4z a+d 1−z (M −1)b 1−z b −2z c +2z c L∞(z)−K (h,H) M (z) .
So, in L (z)
C (nat) n − 3 2 ρ −n ≤ 1(16)
where ρ = RootOf{−1 + 3x + x 2 + x 3 } ≈ 0.295598... and C (nat) , C (nat) are computable constants
with numerical values C (nat) ≈ 0.00404525 . . . and C (nat) ≈ 0.18086721 . . ..
Proof.
For 'natural counting' we have a = b = c = d = 1. From Theorem 1 we know that the radius of convergence of L (nat) 0
(z) equals ρ = RootOf −1 + 3x + x 2 + x 3 ≈ 0.295598 . . .. We can also easily get L ∞ (z) ∼ a ∞ + b ∞ 1 − z/ρ with a ∞ = 1 − ρ 2ρ ≈ 1.19149 . . . and b ∞ = 1 ρ − 1 1 + ρ + ρ 2 − ρ 3 2ρ ≈ 2.15093 . . . ,
and from the transfer lemmas of [4] we obtain Since we are most interested in the enumeration of closed lambda terms, we examine the multiplicative constants in the leading term of the asymptotical lower and upper bound for [z n ] L 0 (z). Table 1). As expected, the bigger h and H are, the more accurate is the bound we get (in this case for [z n ] K 0 (z)). Taking the values of d for the lower and the upper bound. where ρ = RootOf{−1 + x + 2x 2 − 2x 3 + 3x 4 + x 5 } ≈ 0.509308 . . . and C (bin) , C In order to proof this lemma it is enough to recall that in case of binary lambda calculus the size defining constants are b = 1 and a = c = d = 2. Then we used the functions L 13,13 0,13 (z), L 13,13 0,13 (z) to obtain the numerical constants stated in Lemma 8.
[z n ] L ∞ (z) ∼ b ∞ n −3/2 ρ −n /Γ(−1/2) (0.606767 . . .)· n − 3 2 (3.38298 . . .) n ,
Binary lambda calculus
a+d . Then there exist positive constants C and C (depending on a, b, c, d and m) such that the number of m-open lambda terms of size n satisfies lim inf n→∞ [z n ] L m (z)
lambda terms in L m where the length of each string of successors is bounded by a constant integer h. As before, set L
z) in terms of nested radicands (cf.[2]) after all. Indeed, for m < h we have
z) comes from the real root of r h (z) which is closest to the origin and that it is of type 1 2 (i.e. L (h)
Lemma 3 .
3For all m ≥ 0 the radius of convergence of L m (z) equals ρ. Proof. Take L (m) m , defined in (8). Recall that ρ (m) , ρ m and ρ denote the radii of convergence of L (m) m (z), L m (z) and L ∞ (z), respectively. Notice that for all m, n ≥ 0 we have [z n ] L (m) m (z) ≤ [z n ] L m (z) ≤ [z n ] L ∞ (z). Thus we also have ρ (m) ≥ ρ m ≥ ρ. Since ρ (m) is the smallest positive solution of the equation (1 − z) c − 4z a+d 1−z bm 1−z b = 0, we have lim m→∞ ρ (m) = ρ, therefore Lemma 2 implies ρ m = ρ.
3
Upper bound for [z n ] L m (z) Notice that for all integers h and m we have L (h) m ⊂ L m . Moreover, for all m, h ≥ 0 there exists n (h) m such that [z n ] L (h) m (z) = [z n ] L m (z) if n < n (h) m and [z n ] L (h) m (z) < [z n ] L m (z) else. We will use those properties of L (h) m (z) in order to derive a lower bound for the asymptotics of [z n ] K m (z).
Figure 1 :
1The diagram illustrates the idea how we obtain the upper and the lower bound (in terms of the coefficients) for the function L m (z). Starting point is denoted by a blue node, the finish nodes are red.Note that (5) corresponds to an equation of the form K m = F (K m , K m+1 , L ∞ , L m ). Now, define the new set K
h
neither depend on m nor on j and z a−cm 1−z b has poles only on the unit circle, for all m ≥ 0 we have S m,h−1 (z) =c m +d m 1 let us look on the infinite part of the sum in (11). We will refer to its contributions to the first and second coefficient of the Newton-Puiseux expansion (12) as the remainders R c (h) m and R d (h) m , respectively. Since for all m ≥ h we have L (z), the sum can be rewritten as
m
(z)) and [z n ] K(h) m (z) ≤ [z n ] K m (z), we get lim inf n→∞ ([z n ] K m (z)) · Γ(−1/2)n 3/2 ρ n /d (h) m ≥ 1.
Corollary 2 .
2The number of m-open lambda terms of size n satisfies lim sup n→∞ [z n ] L m (z)
.
Notice that for all m, h, H, n ≥ 0 we have [z n ] L m (z) ≤ [z n ] L From the construction of F and the above properties of L
) ≥ [z n ] K m (z) to arrive at the following result: Corollary 3. The number of m-open lambda terms of size n satisfies lim inf n→∞ [z n ] L m (z)
) and L m (z) = L ∞ (z) − K(h) m (z) derived in the previous sections can be used in a straightforward way to compute numerical values for C and C, given concrete values for a, b, c, d. If we choose h and H big enough, then this will give us proper bounds. But in practice they still leave a large gap, at least for values h and H that allow us to perform the computation within a few hours on a standard PC. For instance, in case of the natural counting for h = H = 15 we get C (nat) ≈ 0.00404525 . . . and C (nat) ≈ 0.18086721 . . ..We show a simple way to improve C and C. Let us introduce the functions L
M (z) and L (h,H)m,M (z) we are using exact expressions for L m (z) up to some constant M and then replace L M (z) by a function that is its upper and lower bound, respectively. Numerical results for some previously studied notions of size (see Sections 6.1 and 6.2) reveal a significant improvement in closing the gap between the constants C, C obtained by utilizing the functions L
From the formulas in Lemmas 4 and 5 we have computed the values for c different constants h and H (see
Figure 2 :
2Plot lower and upper bound for h = H = 15 and 10 ≤ n ≤ 150.
h = H = 15, Corollaries 3 and 2 yield C (nat) ≈ 0.00404525 . . . and C (nat) ≈ 0.18086721 . . .. Notice that using the values of d (h,H) 0 to compute C (nat) gives non-trivial values only for h > 7 (for 1 ≤ h ≤ 7 we get negative numbers because in this case we have d (h,H) 0 > |b ∞ |).
Figure 2
2illustrates the bounds we obtained and the exact values of the coefficients [z n ] L 10 ≤ n ≤ 150. Applying the approach discussed in Section 5 for h = H = M = 13 we get the following improvement. ρ = RootOf{−1 + 3x + x 2 + x 3 } ≈ 0.295598 . . . and C (nat) , C (nat) are computable constants with numerical values C (nat) ≈ 0.07790995266 . . . and C (nat) ≈ 0.07790998229 . . ..
Figure 3
3illustrates how the improvement discussed in Section 5 allows to reduce the gap between the constants C (nat) , C (nat)
Lemma 8 .CFigure 3 :
83The (bin) n − 3 2 ρ −n ≥ 1 and lim sup n→∞ [z n ] L Plot of the numerical values for the constants C (nat) , C (nat) for 8 ≤ h = H = M ≤ 13.
numerical values C (bin) ≈ 0.01252417 . . . and C (bin) ≈ 0.01254593 . . ..
as n → ∞.Table 1: Numbers rounded up to 7 digits.h, H
c
(h)
0
d
(h)
0
c
(h,H)
0
d
(h,H)
0
1
0.855448 −1.153959 1.086200 −3.803686
2
0.898032 −1.313246 0.979519 −2.581823
3
0.917305 −1.397536 0.958215 −2.324953
4
0.927248 −1.444672 0.950295 −2.236290
5
0.932849 −1.472308 0.946185 −2.192353
6
0.936128 −1.488826 0.943824 −2.167379
7
0.938055 −1.498647 0.942443 −2.152790
8
0.939174 −1.504385 0.941643 −2.144335
9
0.939813 −1.507673 0.941187 −2.139511
10
0.940172 −1.509525 0.940931 −2.136799
11
0.940372 −1.510556 0.940788 −2.135291
12
0.940482 −1.511125 0.940710 −2.134460
13
0.940543 −1.511438 0.940667 −2.134004
14
0.940576 −1.511608 0.940643 −2.133755
15
0.940594 −1.511701 0.940630 −2.133619
AppendixBelow we present the proof of Lemma 5.Proof. Throughout the proof we will make use of Equation(14). Let us first focus on the finite sum:. The key part of itis the generating function of a sequence of structures enumerated by the denominatorLike in case of Equation (11), one can check that we are not in the supercritical sequence schema case becauseSecond inequality holds becauseThe last inequality holds because we have ρ > 0 for all a, b, c, d that satisfies Assumption 1. Therefore, for all i from the Newton-Puiseux expansion of L ∞ (z) and K (h) i (z) at singularity ρ (see Proposition 1 and Lemma 4) we have :In case of the remainder:we proceed similarly to the proof of Lemma 4. We already know how to handle the product part of this expression, so let us consider the fraction:. Similarly to before, we have to check that the singularity of this function does not come from the root of the denominator but from the function L ∞ (z). So the following inequality has to hold:Second inequality holds because we have 0 < ρ b+c < ρ c for all a, b, c, d that satisfies Assumption 1. Now, similarly to the previous case using the Newton-Puiseux expansion of L ∞ (z) at ρ we can derive expansion of this part of Equation(14).
A natural counting of lambda terms. Maciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, Marek Zaionc, abs/1506.02367CoRRMaciej Bendkowski, Katarzyna Grygiel, Pierre Lescanne, and Marek Zaionc. A natural counting of lambda terms. CoRR, abs/1506.02367, 2015.
Lambda-terms of bounded unary height. Olivier Bodini, Danièle Gardy, Bernhard Gittenberger, Proceedings of the Eighth Workshop on Analytic Algorithmics and Combinatorics. Philippe Flajolet and Daniel Panariothe Eighth Workshop on Analytic Algorithmics and CombinatoricsSan Francisco, California, USASIAMOlivier Bodini, Danièle Gardy, and Bernhard Gittenberger. Lambda-terms of bounded unary height. In Philippe Flajolet and Daniel Panario, editors, Proceedings of the Eighth Workshop on Analytic Algorithmics and Combinatorics, ANALCO 2011, San Francisco, California, USA, January 22, 2011, pages 23-32. SIAM, 2011.
Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem. N G De Bruijn, Nederl. Akad. Wetensch. Proc. Ser. A 75=Indag. Math. 34N. G. de Bruijn. Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser theorem. Nederl. Akad. Wetensch. Proc. Ser. A 75=Indag. Math., 34:381-392, 1972.
Singularity analysis of generating functions. Philippe Flajolet, Andrew M Odlyzko, SIAM J. Discrete Math. 32Philippe Flajolet and Andrew M. Odlyzko. Singularity analysis of generating functions. SIAM J. Discrete Math., 3(2):216-240, 1990.
Analytic Combinatorics. Philippe Flajolet, Robert Sedgewick, Cambridge University PressNew York, NY, USA1 editionPhilippe Flajolet and Robert Sedgewick. Analytic Combinatorics. Cambridge University Press, New York, NY, USA, 1 edition, 2009.
Counting terms in the binary lambda calculus. Katarzyna Grygiel, Pierre Lescanne, DMTCS. 25th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms. Katarzyna Grygiel and Pierre Lescanne. Counting terms in the binary lambda calculus. In DMTCS. 25th International Conference on Probabilistic, Combinatorial and Asymptotic Meth- ods for the Analysis of Algorithms. Discrete Mathematics & Theoretical Computer Science, Jun 2014.
Binary lambda calculus and combinatory logic. John Tromp, Kolmogorov Complexity and Applications, number 06051 in Dagstuhl Seminar Proceedings, Dagstuhl, Germany, 2006. Internationales Begegnungsund Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl. Marcus Hutter, Wolfgang Merkle, and Paul M.B. VitanyiGermanyJohn Tromp. Binary lambda calculus and combinatory logic. In Marcus Hutter, Wolfgang Merkle, and Paul M.B. Vitanyi, editors, Kolmogorov Complexity and Applications, number 06051 in Dagstuhl Seminar Proceedings, Dagstuhl, Germany, 2006. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany.
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[] |
[
"Bouchaud-Mézard model on a random network",
"Bouchaud-Mézard model on a random network"
] |
[
"Takashi Ichinomiya \nDepartment of Biomedical Informatics\nGifu University Graduate School of Medicine\nYanagido 1-1501-1194GifuJapan\n\nPRESTO\nJapan Science and Technology Agency\n4-1-8 Honcho Kawaguchi332-0012SaitamaJapan\n"
] |
[
"Department of Biomedical Informatics\nGifu University Graduate School of Medicine\nYanagido 1-1501-1194GifuJapan",
"PRESTO\nJapan Science and Technology Agency\n4-1-8 Honcho Kawaguchi332-0012SaitamaJapan"
] |
[] |
We studied the Bouchaud-Mézard(BM) model, which was introduced to explain Pareto's law in a real economy, on a random network. Using "adiabatic and independent" assumptions, we analytically obtained the stationary probability distribution function of wealth. The results shows that wealth-condensation, indicated by the divergence of the variance of wealth, occurs at a larger J than that obtained by the mean-field theory, where J represents the strength of interaction between agents. We compared our results with numerical simulation results and found that they were in good agreement. * Electronic address: [email protected]
|
10.1103/physreve.86.036111
|
[
"https://arxiv.org/pdf/1209.2467v1.pdf"
] | 19,957,363 |
1209.2467
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987b6b560e4eafd395d9610e00a0d58b8e48c92e
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Bouchaud-Mézard model on a random network
12 Sep 2012 (Dated: May 2, 2014)
Takashi Ichinomiya
Department of Biomedical Informatics
Gifu University Graduate School of Medicine
Yanagido 1-1501-1194GifuJapan
PRESTO
Japan Science and Technology Agency
4-1-8 Honcho Kawaguchi332-0012SaitamaJapan
Bouchaud-Mézard model on a random network
12 Sep 2012 (Dated: May 2, 2014)
We studied the Bouchaud-Mézard(BM) model, which was introduced to explain Pareto's law in a real economy, on a random network. Using "adiabatic and independent" assumptions, we analytically obtained the stationary probability distribution function of wealth. The results shows that wealth-condensation, indicated by the divergence of the variance of wealth, occurs at a larger J than that obtained by the mean-field theory, where J represents the strength of interaction between agents. We compared our results with numerical simulation results and found that they were in good agreement. * Electronic address: [email protected]
I. INTRODUCTION
Researchers in the field of complex networks agree that a change in the network topology induces a critical change in dynamics. Pastor-Satorras and Vespignani first showed the absence of an epidemic threshold in a scale-free network [1], following which many researchers have focused on the dynamics on complex networks, such as synchronization [2,3], pattern formation [4], and other phenomena.
In this study, we focus on the Bouchaud-Mézard(BM) model on a complex network [5].
It is known that the wealth distribution in a real economy exhibits a power-law behavior, called Pareto's law [6]. With a view to this power law, Bouchaud and Mézard proposed a model, given by the following Stratonovich stochastic differential equation.
dx i = J N N j=1 (x j − x i )dt + √ 2σx i • dW i ,(1)
where x i , N, J, and σ 2 represent the wealth of the i-th agent, number of agents, coupling between agents, and variation of noise, respectively. In this model, the evolution of the wealth is determined by two processes: exchange of wealth and a random multiplicative process, respectively described by the first and second term in the right-hand side of Eq.(1).
Bouchaud and Mézard analyzed this model using the mean-field theory and calculated the probability density function(PDF) of wealth. They showed that the stationary distribution of normalized wealth x i / x , where · · · represents the average over all agents, exhibits the power-law behavior. They also found that "wealth-condensation," which is indicated by the divergence of the variance of x i / x , occurs at J ≤ J c = σ 2 . The divergence of the variance implies that wealth condenses to a few rich agents. On the other hand, if J > J c , the variance remains finite, and the wealth of many agents is close to the average.
In the original BM model, all agents are coupled with each other. However, in a real economy, agents can exchange their wealth with a limited number of agents. Therefore, it is natural to extend the BM model on a complex network in which the number of neighbors is limited. Some studies have already dealt with this subject. In their original study on the BM-model [5], Bouchaud and Mézard carried out numerical simulations on a regular random graph to estimate the exponent of the power-law behavior. They reported that the exponent obtained from numerical simulation becomes smaller than that obtained from mean-field theory. Some studies have also reported on the simulation of this model on a Barabási-Albert (BA) network and a Watts-Strogatz(WS) network [7,8]. In these studies, the authors discussed numerical simulation results, however, none of them proposed a quantitative theory that could explain these results.
This study aims to develop a quantitative theory of the BM model on a random network. The key assumption of our theory is "adiabatic and independent" assumptions on the stationary distribution function, which is explained in a later section. By using these assumptions and the central limit theorem, we analytically derive the equations that determine the stationary distribution function in the non-wealth-condensate phase. We compared our analytic results with those of the numerical simulation, and we found that our theory showed better agreement than did the mean-field theory.
The remainder of this paper is organized as follows. In the next section, we define the model we investigate in this paper. Then, we describe our theory and its results, first for the case of a regular random network and then for a random network with arbitrary degree distribution. In Sec. IV, we describe a comparison of our analysis and the numerical simulation. In the last section, we summarize our results and discuss the problem to be solved.
II. MODEL
The original BM model is expressed using the Stratonovich stochastic differential equation; however, we use the equivalent Ito stochastic equation for mathematical convenience.
We consider the BM model on a complex network described by the following Ito stochastic differential equations
dx i = [J N j=1 a ij (x j − x i )]dt + √ 2σx i dW i ,(2)
where x i , J, N, and σ 2 are the same as Eq.(1), and a ij represents the adjacent matrix. On the network model, we consider a random network in which the degree distribution is given by Q(k).
III. THEORY
In this section, we consider the stationary PDF of the normalized wealth of Eq.(2).
Eq. (2) is invariant under the change of scale x ′ i = αx i for any positive constant α, and we can assume x = 1 without loss of generality. We derive the analytic form of the stationary PDF of x i / x , the normalized wealth at node i. First, we explain our method on the regular random graph, in which all nodes have the same degree k. Then, we extend the analysis to the general random network model, whose degree distribution is given by Q(k).
A. Case of a regular random network
This subsection focuses on the analysis of the system when each node has the same degree k, in other words, Q(k) is a delta function. We assume that ρ i (x, t), the PDF of wealth at
node i, is independent of i, ρ i (x, t) = ρ(x, t).
First, we review the mean-field treatment of the BM model. We consider the distribution of wealth at node i. By using the mean-field theory, we approximate 1 k j a ij x j = x = 1 in Eq. (2). Under this approximation, ρ(x, t) satisfies the following Fokker-Planck equation,
∂ρ ∂t = − ∂ ∂x (Jk(1 − x) − σ 2 x)ρ + σ 2 ∂ ∂x x ∂ ∂x (ρx)(3)
and we find ρ M F eq (x), the stationary PDF obtained from this equation, as
ρ M F eq (x) = C exp(−α/x)x −2−α ,(4)
where α = Jk/σ 2 and C = α 1+α /Γ(1 + α). In this case, wealth-condensation, defined as the divergence of x 2 , occurs at Jk ≤ σ 2 .
To proceed beyond the mean-field approximation, we make what we call "adiabatic and independent" assumptions. We define the "local" fieldx i = 1 k j a ij x j . Because a ij is not 0 only if nodes i and j are connected,x i represents the local average of the wealth around node i.
Ifx i is fixed, the PDF of x at node i is given by solving the "local" Fokker-Planck equation
∂ρ ∂t = − ∂ ∂x Jk(x i − x)ρ − σ 2 xρ + σ 2 ∂ ∂x x ∂ ∂x (ρx) ,(5)
and the conditional PDF of x under the local fieldx is given by
ρ eq (x|x) = C(x) exp(−αx/x)x −2−α ,(6)
where C(x) = (αx) 1+α /Γ(1 + α).
Here we make the "adiabatic and independent" assumptions. First, we assume the static PDF ρ eq (x) can be approximated by
ρ eq (x) = dxP (x)ρ i (x|x),(7)
where P (x) is the PDF ofx. Ifx changes much slower than x, this condition is satisfied, and therefore we call it the "adiabatic" assumption. Under this assumption, the problem to calculate ρ eq (x) is reduced to the one to calculate P (x).
The second assumption is needed to calculate P (x). We assume that the random variables
x j , where j runs in the neighborhood of node i, are independent. This assumption enables us to use the central limit theorem to obtain P (x). We consider the case in which the variance of x is finite, and that the average and variance of x are 1 and s 2 , respectively. Using the central limit theorem, we can approximate
P (x) ∼ √ k √ 2πs exp − k(x − 1) 2 2s 2 .(8)
Inserting Eq. (8) into Eq. (7), we obtain
ρ eq (x) ∼ ∞ 0 dx √ k √ 2πs exp − k(x − 1) 2 2s 2 ρ eq (x|x).(9)
The final step is to check that x = 1 and to calculate s 2 . Using
∞ 0 dxρ eq (x|x)x =x and ∞ 0 dxρ eq (x|x)x 2 = α α−1x 2 for α > 1, we obtain xρ eq (x) = ∞ 0 dxP (x)x ∼ ∞ −∞x P (x)dx = 1,(10)
and
x 2 ρ eq = ∞ 0 dx α α − 1x 2 P (x) ∼ α α − 1 1 + s 2 k .(11)
Here we change the lower limit of integration from 0 to −∞, assuming k/s 2 to be large.
From Eq.(10), we conclude that x = 1 , and that there is no inconsistency. From Eqs. (10) and (11), the variation of x is given by 1 α−1 ( αs 2 k + 1). Therefore we find the self-consistency condition for s 2 as
[(k − 1)α − k]s 2 = k.(12)
Eqs.(6), (9), and (12) are the set of equations that determine ρ eq (x).
We should make several comments on these results. First, this theory gives a distribution that does not obey the power-law behavior. Eq. (9) shows that the PDF ρ eq (x) is written with the integral of P (x)ρ eq (x|x) overx. Because P (x) is Gaussian, ρ eq (x) does not exhibit the simple power-law behavior.
The next important suggestion of this theory concerns the wealth-condensation transition.
In our theory, s 2 diverges if (k − 1)α − k = 0, implying that the wealth-condensation occurs at this point. Using α = Jk/σ 2 , this leads to the conclusion that wealth condensation occurs at J c = σ 2 /(k − 1). On the other hand, the mean-field theory gives the divergence of the variance at J c = σ 2 /k. Therefore the difference between our theory and the mean-field theory can be tested by estimating J c from the simulation.
Finally, we note that our theory can be applied only for the non-wealth-condensation phase. We need the central limit theorem to obtain P (x), which is only applicable for the case in which the variance of x is finite. We discuss this point in greater detail in the final section.
B. Case of a general random network
In this subsection, we extend the developed method for a regular random graph to a general random network.
As in the case of a regular random network, we start from the mean-field theory. In this theory, we define ρ k (x, t) as the PDF of x on the node whose degree is k. Because this network is heterogeneous, the mean of x may depend on k. Therefore, we need to definē
x k = dxxρ k (x, t)
, the average of x on nodes with degree k, to perform the mean-field calculation. The mean-field Fokker-Planck equation is constructed in the same manner as in the case of the SIS model or Kuramoto oscillator [1,3], and we obtain
∂ρ k (x, t) ∂t = − ∂ ∂x J k ′ kk ′ Q(k ′ ) k (x k ′ − x)ρ k (x, t) − σ 2 xρ k (x, t) + σ 2 ∂ ∂x x ∂ ∂x (xρ k (x, t))(13)
We can simplify this equation by introducing the "weighted" average of x asx = k kQ(k) k x k , which leads to
∂ρ k ∂t = − ∂ ∂x [Jk(x − x)ρ k − σ 2 xρ k ] + σ 2 ∂ ∂x x ∂ ∂x (xρ k ) ,(14)
where we abbreviated ρ k (x, t) as ρ k . As in the case of a regular random graph, we obtain ρ M F eq,k (x), the stationary PDF by the mean-field theory, as
ρ M F eq,k (x) = C k (x) exp(−α kx /x)x −2−α k ,(15)where α k = Jk/σ 2 and C k (x) = (α kx ) 1+α k /Γ(1 + α k ).
Finally,x is determined to satisfy the condition x = 1. From dxxρ M F eq,k (x) =x, we obtainx = 1. Now we follow the same procedure as that described in the previous subsection to calculate the PDF more accurately. We consider the PDF of the wealth on node i, whose degree is k. If the average of sums of x in the neighborhood of node i is given byx, we find the conditional PDF of x on node i as
ρ k (x|x) = C k (x) exp(−α kx /x)x −2−α k ,(16)
Using the adiabatic assumption explained in the previous subsection, we assume
ρ k (x) = dxP k (x)ρ k (x|x).(17)
Next, we approximate P k (x) by Gaussian using the independent assumption. One difference between the regular random graph and the random graph with arbitrary degree distribution lies in that we need to assume Lindeberg's condition in this case. Suppose that there are independent variables y 1 , · · · y m , whose mean and variance are µ 1 , µ 2 , · · · µ m and s 2 1 , s 2 2 , · · · s 2 m respectively. Then, the PDF of 1 m (y 1 + y 2 + · · · y m ) converges to the Gaussian, whose mean and variance are 1 m (µ 1 + µ 2 + · · · + µ m ) and 1 m 2 (σ 2 1 + σ 2 2 + · · · σ 2 m ), for m → ∞, if Lindeberg's condition is satisfied [10]. Although m is finite in our case, we can approximate
P k (x) ∼ 1 √ 2πS k exp − (x − 1) 2 2S 2 k ,(18)
where S 2 k is a parameter that must be determined by the self-consistency condition. The final step is to obtain self-consistent equations for the mean and variance of x k .
Using dxxρ k (x) = 1 and dxx 2 ρ k (x) = α k α k −1 (1 + S 2 k ) obtained from Eqs.(16), (17) and (18), we respectively obtain µ k and s 2 k , the average and variance of x on nodes with degree k, as µ k = 1 and
s 2 k = α k S 2 k + 1 α k − 1 .(19)
k , and we obtain
S k = 1 k 2 k ′ kk ′ Q(k ′ ) k s k ′ 2 = u k ,(20)
where u represents the weighted average of the variance,
u = k kQ(k)s 2 k / k .(21)
From Eqs. (19) and (20), we obtain
s 2 k = α k u/k + 1 α k − 1 .(22)
Inserting this equation into Eq.(21), we obtain
u = k Q(k) k α k u + k α k − 1 ,(23)
which leads to our final result, The most important difference between the mean-field theory and ours lies in that there is no effect of Q(k) in the mean-field theory. In the mean-field theory, the PDF on the node with degree k given by Eq.(15) does not depend on Q(k), which causes a curious behavior. Suppose that we decrease J from a large value to 0. At J = σ 2 /k, the node with degree k goes into the wealth-condensate phase, whereas nodes with a larger degree do not condensate. Therefore, the mean-field theory predicts the coexistence of condensated and non-condensated nodes. In this theory, low-degree nodes condensate at large J, whereas high-degree nodes do not condensate until J becomes sufficiently small.
1 − k Q(k)α k k (α k − 1) u = k Q(k)k k (α k − 1)(24)
On the other hand, all PDFs with different degrees are connected through u in our theory.
In this theory, u diverges when k Q(k)α k k (α k −1) = 1, which implies the divergence of all s k from Eq.(22). In our theory, wealth-condensation occurs on all nodes simultaneously.
Finally, we comment on the behavior of the BM model on a scale-free network. Researchers in the field of complex networks may consider that a singular behavior occurs in a scale-free network upon the wealth-condensation transition, such as the divergence or disappearance of the transition point J c . Unfortunately,this is not the case. The main difference between our case and other models that show singularity, such as the SIS-model or Kuramoto transition, lies in that we impose the self-consistency condition on the variance of x, and not on its average. As shown in Eq.(20), we divide the weighted average of the variance s 2 k by k to calculate S k . This eliminates the singular behavior that we often observe in the dynamics on a scale-free network.
IV. SIMULATION
In this section, we test the analytic results obtained in the previous section by comparing them with numerical simulations.
Because we have presented two analytic results, one for a regular random network and the other for a heterogeneous random network, we carried out the simulations for both networks.
For the former, all nodes had the same degree, whereas for the latter, half of the nodes had degree 10 and the other half had degree 20.
We use these two models because it is easy to calculate the PDF in these models. However, the readers might find these models too artificial. To test our theory in a more realistic model, we show the result of simulations on BA-network [11]. Here we note that the direct calculation of wealth distribution is rather hard because we must calculate integral in Eq. (17) for many values of k. In this paper, we investigate s 2 k , the variation of x at a node with degree k to test our theory instead, because it is much easier to compute.
In the following simulation, we carried out numerical integration by the Euler-Maruyama algorithm to obtain the distribution of normalized wealth x/ x .
A. Case of a regular random network
In this subsection, we show the simulation result for a regular random network and compare it with our theory.
First we calculate the PDF of the normalized wealth from 10 simulation trials on networks that include 5000 nodes. In Fig.1, we show the obtained PDF for the network with degree k = 10, σ 2 = 1 and coupling J = 0.3, 0.4, and 0.5. For all J, the distribution coincides well with our theory, indicated by the solid line. The PDF given by the mean-field theory, On the other hand, our theory gives a slightly larger PDF than the simulation, although, it shows better agreement.
Here we investigate the temporal and spatial correlation of x to test the "adiabatic and independent" assumptions used to derive Eqs. (7) and (9). In the right-hand side of Fig.2, we plot the typical trajectory of x(t) andx(t). Because the model we study is described by stochastic differential equations, it is difficult to show that the change inx i is "slower" than that in x i , however, it seems that x i changes more quickly thanx i . Concerning the "independent" assumption, we show the scatter plot between x andx in the left-hand side of Fig.2. Although there exists a tendency for x to increases asx increases, it is not strong.
The correlation between x andx calculated by the numerical simulation is 0.238. Therefore we conclude that the "adiabatic" "independent" assumptions are fairly good. In Fig. 3, we show the PDF at J = 0.5, σ 2 = 1 and k = 4, 10, and 20. It is clear that the discrepancy between our theory and the numerical simulation increases as k decreases. This is not surprising, because the error caused by the application of the central limit theorem increases at small k. However, our theory appears to be better than the mean-field theory, even in the worst case. For example, when k = 4, the probability density obtained by the mean-field theory is smaller than 10 −5 at x = 0.1, whereas the numerical simulation and our theory shows that it is O(10 −1 ). On the other hand, the difference from the result of numerical simulation at large x is indistinguishable between our theory and mean-field theory.
Finally, we check the wealth-condensation transitions. Though our theory does not exhibit the exact power-law behavior, it suggests that variance of normalized wealth diverges at J c = σ 2 /(k − 1). Therefore, we can expect power law behavior ρ(x) ∝ x −3 for large x at this point. On the other hand, the mean-field approximation gives J c = σ 2 /k, and we can test our theory by estimating J c by checking the exponential tail. In Fig.4 we plot the exponent γ, ρ(x) ∝ x −γ at x ≫ 1, estimated from the fitting of the PDF obtained through 100 simulation trials on a regular random graph N = 1000, k = 4. It is clear that wealthcondensation occurs at J ∼ 0.35, which is closer to the prediction of our theory J c = 0.33, than that of the mean-field theory , J c = 0.25.
B. Case of a heterogeneous network
In this section, we present the simulation result for a heterogeneous network. As explained in Sec.III, the PDF ρ eq,k (x) does not depend on Q(k) in the mean-field approximation. In this approximation, the PDF of the wealth for nodes whose degree is k equals that for the regular random graph. On the other hand, our theory predicts that the PDF changes if Q(k) differs. To test our theory for heterogeneous network, we make the network that half of the node have degree k 1 = 10, and other half have degree k 2 = 20.
The PDF of the normalized wealth obtained by the simulation is shown in Fig.5 for J = 0.3 and σ 2 = 1. In this figure, we also plot the PDF obtained by our theory for a heterogeneous network and a regular random graph with degree k 1 or k 2 and by the meanfield theory and that by our theory for regular random graph with degree k 1 or k 2 . For the PDF on nodes with degree 10, the PDF obtained by our theory, indicated by the solid line, is suppressed at small x compared with the case of a regular random network, indicated by the dashed line. The PDF obtained by the simulation shows better agreement with our theory for a heterogeneous network. For the PDF on node with degree 20, the difference among the three theories is small; however, we can conclude that our theory can estimate the PDF very well. it is slightly difficult to show the PDF obtained from our theory. Instead of the PDF, we plot the s 2 k , the variation of x on nodes whose degree is k, obtained by simulation and that
V. SUMMARY
In this paper, we derived the PDF of the Bouchaud-Mézard model on a random network.
Using adiabatic and independent assumptions, we derived the equations for the PDF, Eqs. for a general random graph. It is difficult to solve these equations analytically; however, doing so provides considerable information about the wealth-distribution. In particular, we can analytically obtain the wealth-condensation point J c for a regular random graph.
These analytic results are compared with the numerical simulations in Sec. IV, and good agreement is found with our theory. Below, we discuss the problem to be solved.
First, we note that the approach we have used in this paper cannot be applied to a wealthcondensate phase. We use the central limit theorem to derive the equation for a stationary PDF; however, this approach is not applicable when the variance of ρ(x) diverges. To treat the wealth-condensate phase, we need a generalized central limit theorem that is applicable when the variance diverges. It is known that the PDF of the sum of independent identical random variables converges to a stable distribution function when the variance diverges [10].
It would be possible to make the theory for the wealth-condensate phase on a regular random network using this theorem; however, further work will be required toward this end. For a general random network, the generalized central limit theorem unfortunately remains still insufficient. To treat such a network , we require a generalized central limit theorem for non-identical independent random variables, which has not been yet established.
Second, one might be interested in better approximating the the PDF. As shown in Fig. 3, the PDF obtained by our method still has a large discrepancy with the numerical simulation, especially for nodes with small degree. To reduce these errors, we need to use the improved central limit theorem, which can deal with the rate of convergence. The application of Chebyshev's rate-of-convergence theorem [12] would lead to a better approximation.
Finally, we should comment on the "adiabatic and independent" assumptions. We have no proof to verify these assumptions; however, the coincidence with the simulations suggests that these assumptions work very well. If so, the theory developed in this study will be applicable to other dynamical systems. The essential part of our theory is to impose the selfconsistency condition on the average and variance of the dynamical variables on each node.
In the case of the BM model, the self-consistency condition for the average is automatically satisfied, and we need only to calculate the variance. This procedure is very general, and it will be applicable to analyze other dynamical behaviors such as synchronization or diffusion.
Eqs.(16), (17), (18), (20), and (24) are the set of equations that determine the stationary PDF.
FIG. 1 :FIG. 2 :
12Log-log plot of PDF of normalized wealth obtained by simulation when k = 10, σ 2 = 1, and J = 0.3, 0.4, and 0.5. The solid and dashed lines indicate the result obtained by our theory and that obtained by the mean-field theory,respectively. Temporal and spatial behavior of x. Left: time series of x andx at one node. Right: scatter plot of x andx. Both plots are obtained from the simulation at N = 5000, k = 10 and J = 0.3.indicated by the dashed line, strongly underestimates the probability density at small x/ x .
FIG. 3 :
3Log-log plot of PDF obtained by simulation when J = 0.5, σ 2 = 1 and k = 4, 10, and 20.The solid and dashed lines indicate the result obtained by our theory and that obtained by the mean-field theory, respectively .
FIG. 4 :
4Exponent obtained by simulation for k = 4, σ 2 = 1. The dotted line indicates the wealth-condensation transition.
C. Case of BA-networkTo test our theory in a more realistic network model, we show the simulation result on the BA-network model. The PDF obtained from simulations on a BA-model with minimum degree 4, J = 0.3 and σ 2 = 1 is given in the left-hand side ofFig.6. As we have noticed,
FIG. 5 :FIG. 6 :
56PDF obtained from numerical simulation on a heterogeneous network. The solid, dashed, and dotted lines indicate the PDF obtained by our theory for a heterogeneous network and a regular random graph, and by the mean-field theory, respectively Result of simulation for BA-network with minimum degree 4 and J = 0.3, σ 2 = 1.0. Left: PDF. Right: s 2 k , the variation of x on nodes with degree k. Thick line indicates the theoretical result obtained from Eq.(22).
obtained by Eq. (22), in the right-hand side of Fig. 6 . The theoretical prediction, indicated by the thick line, coincide well with the result of numerical simulation.
( 6 )
6, (9), and (12) for a regular random network, and Eqs.(16), (17), (18), (20), and (24)
AcknowledgmentsThe author thanks H. Nakao, S. Morita, and T. Aoki for fruitful discussions.
This work is financially supported by PRESTO. Japan Science and Technology Agency. This work is financially supported by PRESTO, Japan Science and Technology Agency.
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|
[] |
[
"Leaky exciton condensates in transition metal dichalcogenide moiré bilayers",
"Leaky exciton condensates in transition metal dichalcogenide moiré bilayers"
] |
[
"Benjamin Remez \nT.C.M. Group\nCavendish Laboratory\nUniversity of Cambridge\nJJ Thomson AvenueCB3 0HECambridgeUnited Kingdom\n",
"Nigel R Cooper \nT.C.M. Group\nCavendish Laboratory\nUniversity of Cambridge\nJJ Thomson AvenueCB3 0HECambridgeUnited Kingdom\n\nDepartment of Physics and Astronomy\nUniversity of Florence\nVia G. Sansone 150019Sesto FiorentinoItaly\n",
"D Kara ",
"C M Pursar ",
"A R ",
"-P Montblanch ",
"T F Heinz ",
"O Karni ",
"E Barré ",
"A Camacho-Guardian ",
"D Bennett "
] |
[
"T.C.M. Group\nCavendish Laboratory\nUniversity of Cambridge\nJJ Thomson AvenueCB3 0HECambridgeUnited Kingdom",
"T.C.M. Group\nCavendish Laboratory\nUniversity of Cambridge\nJJ Thomson AvenueCB3 0HECambridgeUnited Kingdom",
"Department of Physics and Astronomy\nUniversity of Florence\nVia G. Sansone 150019Sesto FiorentinoItaly"
] |
[] |
We show that the "dark condensates" that arise when excitons form a Bose-Einstein condensate in a material with an indirect bandgap are not completely dark to optical emission. Rather, such states are "leaky condensates" in which optical emission is facilitated by many-body interactions. We analyze the properties of these leaky condensates in the context of twisted bilayers of transition metal dichalcogenides, which host strongly interacting excitons and an indirect bandgap. We show that this interaction-driven "leaky" emission dominates photoluminescence at low temperatures, with distinctive qualitative features. Finally, we propose that in these materials, unique intervalley physics can lead to crystal symmetry-breaking excitonic ordering, with implications for optical processes. arXiv:2110.07628v2 [cond-mat.mes-hall]
|
10.1103/physrevresearch.4.l022042
|
[
"https://arxiv.org/pdf/2110.07628v2.pdf"
] | 239,009,543 |
2110.07628
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cd11ca0d39ee15aa2e1afc9b4d440bbfc5f7f15a
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Leaky exciton condensates in transition metal dichalcogenide moiré bilayers
Benjamin Remez
T.C.M. Group
Cavendish Laboratory
University of Cambridge
JJ Thomson AvenueCB3 0HECambridgeUnited Kingdom
Nigel R Cooper
T.C.M. Group
Cavendish Laboratory
University of Cambridge
JJ Thomson AvenueCB3 0HECambridgeUnited Kingdom
Department of Physics and Astronomy
University of Florence
Via G. Sansone 150019Sesto FiorentinoItaly
D Kara
C M Pursar
A R
-P Montblanch
T F Heinz
O Karni
E Barré
A Camacho-Guardian
D Bennett
Leaky exciton condensates in transition metal dichalcogenide moiré bilayers
(Dated: March 24, 2022)for fruit-ful discussions. The support of the Cambridge International Trust, of EPSRC Grant Nos. EP/P009565/1, EP/P034616/1 and of a Simons Investigator Award are gratefully acknowl-edged.
We show that the "dark condensates" that arise when excitons form a Bose-Einstein condensate in a material with an indirect bandgap are not completely dark to optical emission. Rather, such states are "leaky condensates" in which optical emission is facilitated by many-body interactions. We analyze the properties of these leaky condensates in the context of twisted bilayers of transition metal dichalcogenides, which host strongly interacting excitons and an indirect bandgap. We show that this interaction-driven "leaky" emission dominates photoluminescence at low temperatures, with distinctive qualitative features. Finally, we propose that in these materials, unique intervalley physics can lead to crystal symmetry-breaking excitonic ordering, with implications for optical processes. arXiv:2110.07628v2 [cond-mat.mes-hall]
Excitons, bound electron-hole (e-h) pairs, give rise to a plethora of quantum-coherent phenomena in solids, including light-matter hybridization [1,2], long-range order [3], phase coherence [4], and Bose-Einstein condensates (BECs) [5]. Novel atomically-thin transition metal dichalcogenide (TMD) structures [6], featuring tightly-bound excitons with long lifetime [7][8][9][10][11][12] and valley pseudospin with contrasting optical selection rules [13], have spearheaded a new generation of excitonic devices. In parallel, the maturing field of twistronics [14] predicts phenomena such as flat [15] and topological excitonic bands with chiral edge modes [16]. This versatility is promising for realizing quantum emitters [17,18], simulators [19], and exciton BECs [20,21], and manybody exciton physics is being explored in electrostatically gated, optically-inert excitonic insulators [22][23][24][25][26][27][28] and cavity exciton-polaritons [29][30][31][32][33].
In twisted TMD heterobilayers, interlayer excitons [34] formed by electrons and holes in opposite layers lie at low energies [35], and provide a compelling platform for pumped exciton condensates. Firstly, the spatial separation of electrons and holes leads to long exciton lifetimes. The interlayer twist then rotates electron bands in momentum space [36], resulting in an indirect bandgap [37] and lifetimes longer still [7,38]. Secondly, the misaligned layers form a large-scale moiré superlattice, with a spatially-modulated bandgap [17,[39][40][41][42][43] that traps excitons in localized orbitals [44][45][46][47][48]. This, and the excitons' interlayer electric dipole, place them in the stronglyinteracting regime [17]. These BECs thus merge strong correlations, quasi-equilibrium dynamics, opto-, twist-and valleytronics. Clearly, new approaches are called for.
In this Letter we show that the intersection of strong interactions and indirect gap leads to striking optical properties in these moiré BECs. The indirect bandgap suggests that the exciton ground state forms a so-called "dark condensate" that cannot emit light directly [49]. However, as we will show, no condensate is completely dark if interactions are considered. In this strongly-interacting system such effects are dominant, driving emission from the BEC even at vanishing temperature, which we describe as a "leaky condensate". These "leaks" give rise to distinctive qualitative features in the opti- Model.-We consider a tight-binding lattice model for the interlayer excitons on the TMD moiré superlattice, as proposed in previous works [17,40,[47][48][49]. The superlattice inherits the triangular symmetry of the underlying monolayers, and has three high-symmetry locations labeled A, B, and C, as seen in Fig. 1a [50]. Sites A and B are local energy minima hosting bound states while C is a higher local energy maximum. A and B are generally not degenerate, and their energetic ordering may depend on the choice of monolayer compounds and whether they are stacked near 0 or 180 degrees [17,40]. However, for simplicity we consider only the lowestenergy locale and our results hold whether it is A or B. Additionally, the triangular symmetry of each monolayer lends it a hexagonal Brillouin zone, with gapped valleys at the two inequivalent ±K-point corners. e-h pairs are pumped optically via vertical interband transitions, and so excitons can be photogenerated in either valley, labelled by τ = ±1 [13].
We thus consider the Bose-Hubbard [51] Hamiltonian
H = k,τ (E 0 + k )χ † kτχkτ + R,τ,σ U τσ 2χ † Rτχ † RσχRσχRτ +V LMI (1)
whereχ † Rτ is the bosonic creation operator of the lowest Wannier state in supercell R (at either locale A or B) and valley τ. χ † kτ , defined shortly, creates plane-wave states with dispersion k . We use k = − t[4 cos(k x a/2) cos( √ 3k y a/2)+2 cos(k x a)− 6], corresponding to nearest-neighbor hopping with amplitude t > 0 and moiré period a [50], and plotted in Fig. 1b. E 0 is the exciton formation energy (bandgap minus binding energy). U τσ > 0 are valley-dependent [52] on-site repulsion strengths; due to moiré localization a strongly-interacting system with t/U τσ < 0.1 is predicted [17,49], which we will treat accordingly.V LMI is the light-matter interaction, addressed below.
A specific gauge is fixed in Eq. (1): Exciton momentum eigenstates superpose e-h pairs with fixed momentum transfer p e − p h = p. Due to the indirect gap, the lowest-energy transition occurs at p = −τK 0, where K = K h − K e is the momentum mismatch between the τ = +1 valley extrema of the two layers [17,36,37], coinciding with the moiré Brillouin zone (MBZ) corner. It is convenient to define the shifted wave vector k = p + τK, so real-and k-space states are related bŷ
χ kτ = 1 √ N R e −i(k−τK)·Rχ Rτ(2)
with N the number of supercells. Indeed, the hopping amplitudes Rτ|Ĥ|R τ are complex. Their phases are fixed by the momentum mismatch [17], which guarantees [50] that in this unique gauge k is valley-independent and minimal at k = 0. This momentum mismatch implies that the excitonic ground states cannot recombine radiatively: The nearlyvertical photon dispersion gives rise to the well-known optical light cone (LC) of states with |p| E 0 / c that can recombine, with c the speed of light in the surrounding medium. The light cone of each valley is thereby centered at k = τK, i.e. the MBZ corners, see Fig. 1b. Crucially, the k = 0 states lie outside both cones, and thus they are momentum-dark.
Under a broad set of conditions, the Hamiltonian (1) has a many-body ground state that is Bose-condensed, with a large phase-coherent occupation of the single-particle ground states at k = 0. In direct-gap systems, where these states are bright, this would lead to a pronounced phase-coherent emission similar to superradiance [5]. Yet the twist-induced momentum mismatch renders the BEC a "dark condensate" from which direct optical emission is forbidden by translation symmetry.
Breaking translation symmetry can enable emission from momentum-dark excitons. This has been realized externally in TMDs, with electronic charge-order [53,54] and incommensurate substrates [55]. Indeed, translation symmetry breaking is a hallmark of indirect-gap exciton coherence [22,56,57]. The interplay of two interacting valleys in Hamiltonian (1) can cause the excitons to break translation symmetry and form exciton density waves, which we explore in the Supplementary Material [50]. We find that while these density waves have precisely the required geometry, three-fold rotation symmetry prevents direct emission and the condensate remains dark.
Leaky condensates.-While its coherent component cannot emit directly, the many-body condensed ground state is not completely dark. Rather, exciton-exciton interactions induce an incoherent component that can radiate. Emission is driven by excitonic collective modes [58], which have attracted recent attention as insightful probes of excitonic manybody states [57,[59][60][61][62][63][64]. Here, the collective modes supply the momentum necessary for the excitons to recombine. Collective modes exist in normal phases, and so Bose-coherence is not prerequisite for this mechanism. However, below we demonstrate that in twisted bilayers this is the dominant emission channel at low temperatures. Thus, many-body interaction effects will determine the optical character of the lowtemperature condensed phases, which we call "leaky condensates".
The essential physics of our mechanism is captured by a single-valley model. This may be realized experimentally by pumping a valley-contrasting circularly-polarized intralayer exciton resonance, followed by rapid interlayer charge transfer [65]. Excitons exhibit long valley depolarization times [11,48,65,66] thanks to the large e-h vertical separation [34], suggesting the exciton population can be treated as valleypolarized over radiative timescales. Projecting onto valley τ = +1 and suppressing τ henceforth,Ĥ reduces tô
H = k (E 0 + k )χ † kχk + U 2 Rχ † Rχ † RχRχR +V LMI . (3)
Eq.
(2) notwithstanding, all excitons now carry the same momentum mismatch so it is gauge-eliminable, and Eq. (3) realizes the usual Bose-Hubbard Hamiltonian. We model the light-matter interaction bŷ
V LMI = p ,p ⊥ ,σ ω pâ † pσâpσ + (g pσâ † pσχK+p + H.c.),(4)
whereâ † pσ creates a photon with momentum p with the indicated in-and out-of-plane components and polarization σ. We have incorporated the exciton momentum mismatch, and made the rotating-wave approximation [67]. The coupling constants g pσ are obtained from electronic interband transition dipole matrix elements, and depend on the exciton pairing wavefunction [38] etc. The recombination rate Γ k of each k mode inside the light cone can be computed [38] using Fermi's Golden Rule [68]. This defines a natural timescale, the lifetime of a localized exciton in a single moiré site,
τ −1 loc = 1 N k∈LC Γ k .
The relative size of the light cone compared to the total MBZ = 1 N k∈LC ∼ (E 0 a/ c) 2 gives the fraction of the localized wavefunction that is contained in the light cone, and provides the estimate τ loc ∼ ( c/E 0 a) 2 /Γ K ∼ 10 ns [17]. Note τ loc is not the exciton mean radiative lifetime, which is associated with a thermal averaging over Γ k [38].
Let us assume that exciton recombination is sufficiently slow to maintain quasi-equilibrium, with an associated chemical potential µ and grand-canonical potentialΞ =Ĥ − µN.
We will study the ground state of the excitonic sector ofΞ and treatV LMI as a weak perturbation that generates photons which probe it. This picture is made consistent by shifting the photon energies to ( ω p − µ) [50].
Weak interactions: Bogoliubov theory.-Though the excitons we consider are strongly-interacting [17,49], we first present our emission mechanism in the familiar weaklyinteracting Bogoliubov theory to build our intuition.
A BEC with all excitons at k = 0 will be depleted by interactions that eject pairs of counter-propagating excitons from the condensate. If one lands within the light cone, it may recombine. Bogoliubov theory lets us neatly resum these virtual processes and find the depleted ground state.
Consider a state with total filling ν and condensate filling ν c (ν). Eq. (3) leads to the standard [69] mean-field (MF) Bogoliubov-de-Gennes (BdG) HamiltonianΞ MF = k Ω kb † kbk with the familiar dispersion Ω 2 k = k ( k + 2ν c U), µ = E 0 + ν c U, and the Bogoliubov modeŝ
b k = cosh(θ k )χ k + sinh(θ k )χ † −k , sinh θ k = Ω k − k 2 √ Ω k k .(5)
The ground state of the theory is the BdG vacuum. Yet the mixing of particle creation and annihilation in Eq. (5) implies that it nevertheless contains some excitons, most notably inside the light cone.
Consider recombination in terms of the collective modes. Transcribed into BdG modes, Eq. (4) contains terms such as −g pσ sinh(θ K+p )â † pσb † −K−p , which represent a spontaneous emission of a photon and a BdG mode, the latter assuring momentum conservation in analogy to phonon-assisted exciton recombination [5]. Therefore, interactions enable an otherwise-dark exciton condensate to "leak" photons with
ω p = µ − Ω −K−p = E 0 + ν c U − Ω −K−p < E 0 .(6)
As expected, some energy is lost to the Bogoliubov mode. We compute the total emission rate Γ with Fermi's Golden Rule via transitions between the BdG vacuum and single quasiparticle states. The redshift in Eq. (6) is negligible compared to E 0 , so this is equivalent to counting the number of excitons present within the light cone,
Γ ≈ k∈LC Γ k n k ≈ Nτ −1 loc n K ≈ Nτ −1 loc 1 18 ν c t/U 2 .(7)
Here we approximated n k = χ † kχk ≈ n K = sinh 2 θ K inside the light cone, assuming it is much smaller than the MBZ, and expanded around small densities. The total filling is found by integrating n k [69], and in two dimensions
ν c = ν[1 − O(U/t)].
Thus, unlike spontaneous decay, Γ is quadratic in density.
The generality of this construction suggests that exciton condensates in any indirect-gap system are leaky. Moreover, emission is accomplished without additional degrees of freedom. This contrasts with external optical probing [70] and other mechanisms that involve phonons [71] or carrier exchange in larger exciton complexes [72].
Interactions deplete excitons into excited bands as well, and so leaky condensates can also occur in systems that are dark due to a spin-forbidden transition, etc. Previous studies have shown that, for sufficiently strong interactions (or, equivalently, above a threshold density), a dark condensate can transition into a so-called "gray condensate" [73][74][75][76]. Our finding of leaky emissions is distinct from these previous works: (i) Emission from the leaky condensate grows continuously with increasing density, with no threshold value. (ii) In indirect-gap materials, the bright and (momentum-)dark states are smoothly connected along the same Bloch band, unlike the usual scenario where they are spin-split and form separate bands. This prevents fragmentation into a gray condensate. (iii) The gray condensate emission is coherent whereas the leaky condensate emission is not, due to its entanglement with the generated collective modes.
Condensate depletion by interactions has recently gained significant experimental attention [77][78][79][80][81]. Relatedly, BdG modes are used to renormalize phonon-assisted photoluminescence line shapes, e.g. in Cu 2 O [5,82,83]. However, the role of collective modes in enabling recombination of indirect-gap excitons, to the best of our knowledge, has not been pointed out so far. Additionally, these descriptions focus on weaklyinteracting excitons.
Strong interactions: hard-core bosons.-We now explore how leaky condensates manifest under strong interactions. Consider the U → ∞ limit corresponding to hard-core bosons, which with the transformation (S = 1 2 henceforth implied)
S − R = e −iK·Rχ R ,Ŝ + R = e iK·Rχ † R ,Ŝ z R =χ † RχR − S ,(8)
map to a spin-1 2 XX ferromagnet in a transverse field. The grand-canonical potential becomeŝ
Ξ = −t R,R (Ŝ + R Ŝ − R +Ŝ + RŜ − R ) + (E 0 − µ) R (Ŝ z R + S ). (9)
The recombination rate remains
Γτ loc /N ≈ n K = Ŝ + KŜ − K whereŜ − K = 1 √ N R e −iK·RŜ −
R . This limit readily manifests ground state emission: With t = 0, Eq. (9) factorizes into independent sites each with MF solution √ ν |⇑ + √ 1 − ν |⇓ , yielding [50] n K (ν) = ν 2 . As nonzero hopping allows repelling excitons to separate, leading to anticorrelations, this is an upper bound. Additionally, an emergent particle-vacancy duality [50]Ŝ ± R →Ŝ ∓ R connects the ground states of Eq. (9) with fillings ν and 1 − ν, providing the identity
n K (ν) − n K (1 − ν) = 2ν − 1. Thus, n K (ν) > max(2ν − 1, 0)
. These bounds already demonstrate the interaction-driven nonlinearity of Γ(ν).
A quantitative treatment of emission is again found in terms of spontaneously excited collective modes, now taking the form of spin waves. We perform a Holstein-Primakoff (HP) 1/S expansion [84] similar to Bernardet et al. [85] with details provided in the Supplemental Material [50]. The qualitative features of Bogoliubov theory are reproduced: we obtain a quadraticΞ = Ω kb † kbk , now with dispersion [86] within the Holstein-Primakoff expansion [50] (solid black), compared with exact diagonalization on a 6×3 lattice (filled circles) and a least-squares fit derived from two-body states (dashed blue). The solid and dashed gray lines are upper and lower bounds explained in the text. Inset: The crossover temperature T between interaction-and thermally-dominated emission. The solid and dashed thin lines mark the Berezinskii-Kosterlitz-Thouless superfluid transition and quantum degeneracy crossover, estimated at kT/t = √ 3πν and 4 √ 3πν, respectively [50]. Interactions dominate the cold regimes.
Ω 2 k = • • • • • • • • • • • • • • • • • • 0.k [(2ν MF − 1) 2 k + 24ν MF (1 − ν MF )t] and µ = E 0 + 6(2ν MF − 1)
, where the filling ν MF is the mean-field order parameter [cf. ν c in the previous section]. The exciton occupation in the spinwave vacuum, or equivalently at temperature T = 0, is now
n k = [ν MF cosh θ k + (1 − ν MF ) sinh θ k ] 2 .(10)
θ k and the emission spectra are still given by Eqs. (5) and (6). We plot the T = 0 emission rate Γ ∝ n K in Fig. 2 [86]. Comparing it against small-scale exact diagonalization, we find very good agreement across a wide range of fillings. In the dilute limit
Γ(ν 1) ≈ 4 9 Nτ −1 loc ν 2 MF .(11)
The apparent quadratic dependence in Eq. (11) does not imply an absence of correlations, which cause ν MF (ν) and ν to differ. Exciton correlations can be inferred from two-body states [76], from which we deduce [50] the asymptotic form Γ ∼ (ν/ log ν) 2 , revealing the expected suppression. Numerics confirm [50] that ν MF ∼ (ν/ log ν) for ν 1, indicating correlations are successfully captured. Our HP theory also allows us to treat nonzero temperatures. Thermal excitations enhance emission and unlock a second channel whereby a collective mode is absorbed instead of emitted, leading to two emission lines. In the inset of Fig. 2 we plot the crossover temperature T at which depletion (due to interactions) and thermal excitations contribute equally [50] to the total emission rate, and below which interactions dominate. This crossover occurs above the Berezinskii-Kosterlitz-Thouless (BKT) transition [87,88], indicating that leaky emission will be the dominant emission channel characterizing the superfluid phase and much of the quantum-degenerate regime. Remarkably, in this strongly-interacting system, at small filling "leaks" dominate emission even in the hot gas phase that can be treated semi-classically. Unlike the Stokes and anti-Stokes lines in phonon-assisted emission [5], the two BdG processes have unequal matrix elements with different density dependences. The anti-Stokes-like line dominates above T , and the net annihilation of BdG modes may evaporatively cool the BEC, similarly to a mechanism recently suggested [89]. Finally, the strength of interactions can be inferred from the ratio of emission line intensities [50].
The leaky condensate picture thus predicts a distinctive property of moiré exciton emission: A dominant redshifted emission line with quadratic density dependence below T , compared to a dominant blueshifted line with linear density dependence above T . We do not expect a qualitative change at the BKT transition.
Experimental Consequences.-We assess the parameters under which a leaky condensate may be observed. Comparing Eq. (7) with (11) suggests that this mechanism saturates once U 12t, which should hold across a wide range of twist angles [17]. For a ≈ 10 nm (twist ≈ 2 • ) and corresponding t ≈ 0.2 meV [17], and at a demonstrated [90] photoexcited exciton densities of n ≈ 10 11 cm −2 (ν ≈ 0.1), T ≈ 5 K. Larger twist angles or intercalated hBN spacers would increase t and thus T [17,49], and mitigate inhomogeneity effects. Furthermore, electrostatic gating might modify the moiré symmetry [17] or its elastic reconstruction [91,92], allowing insitu tunability. Above T the two emission lines are split by ∼ 20t ∼ 4 meV and should be resolvable.
Leaky emission will also manifest in a quadratic loss ∂ t n = −γn 2 , with n the exciton number density. With Eq. (11) and τ loc ∼ 10 ns at a ∼ 10 nm [17] we estimate a rate constant γ ∼ a 2 /τ loc ≈ 10 −4 cm 2 /s. Exciton density is controlled with pumping fluence, and tracked with transient absorption [93,94], emission blueshift [cf. Eq. (6)] [70,74,90], or time-resolved photoluminescence (PL) [90]. These reveal such quadratic dependence, which is attributed to Auger recombination. In TMDs this process is associated with a highenergy conduction band that supports bound excitons at energies close to 2E 0 , making Auger recombination nearly resonant [95,96]. Yet crucially, for nonzero detuning Auger recombination freezes out at T → 0 [97] and leaky emission will dominate. Comparisons at nonzero temperature are not straightforward; estimates are available mostly for roomtemperature monolayers, suggesting an intrinsic Auger constant γ ∼ 10 −3 cm 2 /s [95,98,99]. However, moving to cryogenic temperatures or to bilayers could each reduce γ by orders of magnitude [97,100]. Thus, the dominance of leaky recombination over Auger could feasibly extend to T and above. This unique regime of emission linear in continuous fluence yet nonlinear in instantaneous density might be important in interpreting PL experiments. In settings demonstrating population lifetimes approaching microseconds [9][10][11][12], the short "leaky lifetime" τ leaks ∼ (γn) −1 ∼ 10 ns induced at high fluences could also play a significant role in the complex population dynamics.
Conclusions.-We have shown that strong interactions challenge the picture of dark condensates in moiré bilayers, where they dominate optical processes. This prompts further material-specific modelling to compare these effects to other recombination and loss mechanisms. Additionally, while here we mostly considered a single-valley model, the physics of two-valley moiré condensates is very rich. As we remark above, excitonic density waves put such condensates on the verge of optical activation which is prevented only by vestigial rotational symmetry [50]. Therefore, direct emission with long-range phase coherence may be achieved in these systems by external fields, strain, layer separation, and pressure, with possible sensing applications. We leave this and other novel intervalley phenomena to future work.
PRELIMINARIES
In this Section we briefly outline the basic parameters of the moiré lattice used throughout the main text.
Geometry.-We define the lattice primitive vectors
a 1 = ax, a 2 = 1 2 ax + √ 3 2 aŷ,(S1)
where a is the moiré period. The moiré lattice has symmetry group p3m1, with 3 high-symmetry points A, B, and C. These locales are found at relative displacements
d A = 0, d B = d, d C = 2d, d = (a 1 + a 2 )/3.(S2)
within each unit cell. The reciprocal basis vectors are
b 1 = 2π a (x − 1 √ 3ŷ ), b 2 = 2π a 2 √ 3ŷ .(S3)
The moiré Brillouin zone has two inequivalent corners which should be labeled consistently. We follow Refs. [17,40] which find the A and B sites host s-wave bound states with angular momenta A = τ and B = −τ, respectively. The C locale is a potential maximum and so does not host any states. The three-fold C 3 rotation symmetry of the bilayer (see below) dictates that the hopping amplitudes from an A site to its three nearest-neighbor B sites must be complex and transform like a d +2 wave (mod 3),
t(C 2 3 d) = e − 2πi 3 τ t(C 3 d) = e − 4πi 3 τ t(d),(S4)
where t(v) = R + v, τ|Ĥ|R, τ . On the other hand, the (intravalley) hopping amplitude phases are fixed by momentum mismatch [17] t
(v) = − |t(v)| e −iτK·v ,(S5)
where (−τK) is the exciton momentum mismatch, |t(v)| has three-fold symmetry, and we separated the usual minus sign. K coincides with one of the moiré Brillouin zone corners, and Eqs. (S4) and (S5) are reconciled by picking
K = 2 3 b 1 + 1 3 b 2 = 4π 3ax . (S6)
Kinetic energy.-The valley-dependent complex phases (S5) generally lead to valley-dependent dispersions. However, the gauge choice (2) uniquely gives a valley-independent dispersion,
k = k, τ|Ĥ|k, τ = 1 N R,v e −i(k−τK)v R + v, τ|Ĥ|R, τ = − v |t(v)| e −ik·v .(S7)
We see this gauge also conveniently fixes the minimum of k to k = 0. In this work we consider only the lowest energy locale, and hopping to the same locale in adjacent supercells. We thus set t(C n 6 a 1 ) = t and otherwise zero. In the main text and here throughout we absorb 0 into E 0 so that the band minimum is at 0 and K = 9t.
EXCITON DENSITY WAVES & SELECTION RULES
In the main text we explore the single-valley model of Eq. (3), and direct emission from the BEC is forbidden by translation symmetry since the k = 0 excitons are momentumdark. However, if translation symmetry is broken, recombination of these excitons will be permitted. Such brightening of large-momentum excitons has been demonstrated in TMD mono-and bilayers via doped electronic charge-ordered states or interfacing with an incommensurate substrate [53][54][55]. Motivated by these studies, we point out that the interplay between the two valleys in the full Hamiltonian of Eq. (1) gives rise to excitonic density order which reduces translation symmetry. However, emission from the condensate requires a fine-tuned momentum transfer between k = 0 and K. To determine whether the BEC becomes bright thanks to this density wave, or if it remains dark, we analyze the symmetries of the fullĤ, and those that might be spontaneously broken by a two-valley condensate.
Translation symmetry.-Translation symmetry breaking is a paradigmatic feature of indirect-gap exciton coherence [22,56,57]. Here, a BEC comprised of excitons in both τ = ±1 valley minima, having different momenta p = ∓K, is no longer an eigenstate of total momentumP, showing translation symmetry is lost and momentum is not conserved. This is understood as an interference between the e ∓iK·R components, leading to a spatially-moduled exciton density wave.
However, translation symmetry is not broken completely, and instead reduced to that of the larger density wave periodicity. This implies a zone-folding scheme rather than mixing between any two arbitrary momenta, and the density wave might not necessarily couple the BEC and the light cone. Crucially, in our case the necessary momentum-folding is achieved not because the two valleys have opposite mismatch p = −τK that sum to zero, but because their mismatch difference equals the BEC-light cone offset K, since ∆p = −2K = +K modulo moiré reciprocal lattice vectors. Therefore, a potential brightening of the BEC is also a special feature of the moiré geometry and valley configuration.
Additionally, we note that appearance of the density wave also necessitates interactions, as integrability imposes a significantly stronger symmetry than the conservation laws we consider here. Therefore, like the leaky emission, the density wave is understood as a feature of the strong-interactions regime of interlayer excitons. It is thus instructive to demonstrate explicitly the origin of the density wave. Eq. (1) contains, among others, cross-valley interaction terms like Rχ † Rτχ † RτχRτχRτ with the notationτ = −τ. The mean-field-decoupled Hamiltonian is obtained from the different waysĤ can be contracted. Recalling Eq. (2), upon condensation of the k = 0 states, χ Rτ ∝ e −iτK·R . Thus, in the presence of a two-valley condensate, the contraction
H MF U ττ R χ † τχτ e 2iK·Rχ † RτχRτ + H.c. (S8)
emerges. These ±2K Fourier components in the MF effective Hamiltonian demonstrate the translation-symmetry-breaking density wave. Thanks to the moiré geometry, 2K = 3K − K = 2b 1 + b 2 − K = −K modulo the moiré Brillouin zone. Thus this term represents Umklapp scattering of the excitons by ±K. Consequently all the high-symmetry momenta states at p = 0, ±K are coupled together, showing the light cones are folded onto the condensate. However, as we now explain, other symmetries ofĤ will prevent emission even in this translation-symmetry-broken BEC.
Intervalley symmetry.-The long exciton valley depolarization time [11,48,65,66] enabled by the large e-h separation [34] motivates an effective U(1) symmetry of separate particle conservation in each valley at short times, and Eq. (1) does not contain valley-flipping terms. Such symmetry would prohibit genuine momentum folding, as it implies that
K =P + K(N + −N − ),(S9)
with N τ the population in each valley, is a constant of motion with [K,Ĥ] = 0. UnlikeP,K remains conserved by the density wave (i.e. commutes withĤ MF ) since both valley minima are its k = 0 eigenstates. Thus modes of differing k cannot mix and momentum remains unfolded. We then consider a valley interconversion perturbation
∆Ĥ IV = k,τ J IV kτχ † k+τK,τχk+τK,τ(S10)
with intervalley tunneling coefficients J IV kτ . This is the exchange interaction that leads to longitudinal-transverse splitting of exciton modes in TMD monolayers [101], which should also exist for interlayer excitons, albeit with reduced size due to their spatially indirect character [48]. Using J IV kτ given by nearest-neighbor hopping commensurate with the crystal symmetries, we confirm by an explicit mean-field theory calculation that the two-valley BEC indeed leads to a translation-symmetry-broken effective Hamiltonian. However, despite being folded over each other, the condensate and light cones remain decoupled. Rotation symmetry.-This robust darkness arises from rotation symmetry. Since both the condensate and light cones lie at high-symmetry k points, they can be simultaneous eigenstates of both linear and angular momentum, and thus the combined action of both translation and rotation symmetry will set optical selection rules.
host (k, τ = +1) (k, τ = −1) locale op. −K 0 +K −K 0 +K C 3A +1 +1 +1 −1 −1 −1 AĈ 3B 0 −1 +1 −1 +1 0 C 3C −1 0 +1 −1 0 +1 C 3A 0 +1 −1 +1 −1 0 BĈ 3B −1 −1 −1 +1 +1 +1 C 3C +1 0 −1 +1 0 −1
The three-fold C 3 rotational symmetry of the monolayers is lifted to the moiré superlattice as well. Like the monolayers, the bilayer possesses multiple three-fold symmetries, herê C 3s around the three locales s = A, B, and C. Excitons inherit the optical selection rules dictated by the local stacking configuration, determining the angular momenta Aτ = +τ and Bτ = −τ for excitons localized in A and B locales, respectively [17,40], and showing they are circularly polarized.
Momentum states, however, will transform differently. The rotation operations around the three moiré locales are related by [cf. Fig. 1
]Ĉ 3B =T a 1Ĉ 3A ,Ĉ 3C =T a 1Ĉ 3B ,(S11)
withT a 1 = exp(−iP · a 1 ) an elementary lattice translation. It follows that if a k state is an angular momentum eigenstate with angular momenta numbers (A,B) kτ with respect to symme-triesĈ 3A,3B , they must be related by
e − 2πi 3 (B) kτχ † C 3 kτ =Ĉ 3Bχ † kτĈ † 3B =T a 1 e − 2πi 3 (A) kτχ † C 3 kτT † a 1 = e − 2πi 3 (A) kτ −i(k−τK)·a 1χ † C 3 kτ .(S12)
Conversely, k can be a rotational eigenstate only if C 3 k = k up to reciprocal lattice vectors. This holds only at the highsymmetry k points, that is the zone center and corners. Writing k = κK, κ = 0, ±1, we thereby find
(B) kτ − (A) kτ = a 1 · (k − τK) 2π/3 mod 3 = −(κ − τ) mod 3.
(S13) The same identity relates (B) kτ and (C) kτ . Using these relations, we list (s) kτ of all high-symmetry momenta states formed by the lowest-energy orbitals in the A and B moiré sites in Table I. We find that for the condensed k = 0 modes, (s) 0τ depend on the rotation center. This indicates rotation symmetry also prevents their recombination, as they are symmetrydistinguishable from photons, which have uniform angular momentum with respect to all three centers.
Yet like translation symmetry, the BEC invalidates rotation centers s for which (s) 0τ (s) 0τ , potentially enabling emission. We find thatĈ 3A,3B are broken, whileĈ 3C survives with (C) 0τ = (C) 0τ = 0. While only one now characterizes k = 0 modes, this vestigial symmetry prevents emission: To conserve energy, high-symmetry excitons must emit a photon perpendicular to the bilayer, i.e. with = ±1, and therefore (C) 0τ = 0 dictates that the condensate remains dark. This selection rule is protected by the time-reversal duality between valleys and the high symmetry of the momentum mismatch K.
A similar situation where momentum-folded states remain optically dark was observed in Smolensky et al., see Ref. [54]. In that experiment, translation symmetry is broken by the charge-ordered Wigner crystallization of doped electrons, with wavenumber k w . Since |k w | < |K|, six degenerate, lowsymmetry exciton states at p = C n 6 k w are folded on top of each other and the light cone. The six states then hybridize and redistribute into states of well-defined angular momentum. Of those, only two have = ±1 and are allowed to couple to the light cone and become bright, while four states have different and thus remain dark. In our case, the commensurability of the exciton density wave leads to only two independent states p = ±K being folded, which are themselves = 0 eigenvalues, and therefore we do not observe the same richness here.
In total, we find that in these twisted bilayers, two-valley condensates lie on the verge of optical activation, which is driven by valley interconversion, and stopped by a vestigial rotation symmetry. Therefore, bright BECs could be achieved by applying rotation symmetry breaking fields and strain, and further enhanced by out-of-plane pressure which increases valley-depolarizing e-h exchange interactions. We leave this, and other interesting intervalley phenomena, to future work.
Finally, we remark that recombination of k 0 excitons via momentum folding remains possible due to their low rotation symmetry. However, this presents a secondary incoherent emission channel that is dominated by the leaky emission we consider in the main text. We demonstrate this below.
INTERVALLEY EXCITONS
In our Letter we study the properties of excitons formed of electrons and holes in the same K valley of the constituent monolayers. These are known as intravalley, or K-K, excitons. Thus, the bandgap is rendered indirect by the interlayer twist. This holds, for example, in MoS 2 /WSe 2 [46,102,103], and our treatment applies directly.
For some combinations of materials and stacking configurations, hybridization between the two layers can lead to an indirect bandgap even at zero twist, subtended by different k points such as Γ-K, Γ-Λ, and K-Λ [104]. Thus, momentumdark intervalley excitons composed of electrons and holes in different valleys could be lower in energy, and would be those to condense. Their intervalley momentum mismatch is too large to be compensated by center-of-mass motion; thus, while strong dipole-dipole interactions may play a part in facilitating recombination, our theory does not apply to them.
However, our theory may nevertheless still apply depending on the relevant timescales: since excitons are generated optically, intravalley excitons are formed first. The extremely long valley depolarization times observed in experiment, up to hundreds of ns [11,48], would suggest the conversion between intra-and intervalley excitons is slow, and momentum-direct excitons will be persistent. For example, intravalley excitons still lead to pronounced photoluminescence in MoSe 2 /WSe 2 heterobilayers [44,105,106]. Therefore, it is feasible that over experimentally-relevant timescales comparable to our τ loc , we can consider a population of photogenerated intravalley excitons, and our theory remains applicable.
THE ROTATING FRAME
To study emission from condensed states, we must consider populated, i.e. excited states of Hamiltonian (3). Nevertheless, it is convenient to move to a frame in which the condensate is the ground state, so that various observables are given by ground state expectation values. Such a transition is implemented by the unitary operator e iµNt withN the particle counting operator. However, this would lead to a timedependent light-matter interactionV LMI (t) in the new frame. This is avoided by havingN count photons as well. The Hamiltonian is then transformed intô
Ξ =Ĥ − µN,N = kχ † kχ † k + p,σâ † pσâpσ ,(S14)
andΞ is time-independent since [Ĥ,N] = 0 thanks to the rotating-wave approximation in Eq. (4).Ξ then appears as a standard excitonic grand-canonical potential with chemical potential µ, perturbatively coupled to photons with energies ( ω p − µ). As this frame allows us to focus on the low-energy physics of the condensate, this shift sets the emission spectrum around the bilayer optical bandgap µ ≈ E 0 .
HARD-CORE BOSONS AND SPIN WAVES
In this Section we briefly outline the analysis of the hardcore boson and corresponding spin model studied in the main text.
Particle-hole symmetry.-The hard-core constraint gives rise to the notion of boson vacancies. A particle-hole-like symmetryPŜ ± RP =Ŝ ∓ R connects the ground state of Eq. (9) with filling ν to that with filling 2S − ν (here S = 1 2 throughout). Therefore, one can show
n k (ν) − n k (2S − ν) = 2 Ŝ z = 2ν − 2S .(S15)
This relation places a strong restriction on the resulting emission rate. For a finite system size N, n k (0) = n k 0 (1/N) = 0, leading to the leading-order dependences at low and high filling
n k 0 (ν) = 0 + O(ν 2 ), ν → 0 2ν − 2S + O((2S − ν) 2 ), ν → 2S . (S16)
These two asymptotes can be satisfied simultaneously only by a nonlinear dependence, thus establishing the role of interactions.
Mean-field theory.-The first refinement to Eq. (S16) is found in classical mean-field theory. We write the mean-field wavefunction
|Ψ MF = R cos ϑ 2 |⇑ + sin ϑ 2 |⇓ R (S17)
which corresponds a uniform polarization in the XZ plane at polar angle ϑ, and is the mean-field ground state for µ = E 0 + z cos ϑ, where z = 6 is the lattice coordination number. At mean-field level the filling is ν MF = S z + S = S (cos ϑ + 1). A straightforward computation yields
n k 0 (ν MF ) = Ŝ + kŜ − k = cos 4 ϑ 2 = ν 2 MF ,(S18)
which is consistent with Eq. (S16). The factorized wavefunction (S17) is totally uncorrelated. However, due to their strong repulsion, particles should be anticorrelated, leading to fewer scatterings and thus weaker emission. Therefore, this result is an upper bound on the emission rate. In addition, its independence of k motivates us to seek the next order correction. Spin-wave theory.-Our treatment closely follows that of Bernardet et al [85] and references therein with suitable adaptations to the present lattice geometry. We first rotate the spins in the XZ plane such that the mean-field state points in the negative z direction, and then perform a Holstein-Primakoff (HP) 1/S expansion [84]. We substitute for the spin fieldŝ
S x R = cos ϑ 2 2S −ĥ † RĥRĥR + sin ϑ 2 (S −ĥ † RĥR ) + H.c., S z R = − sin ϑ 2 2S −ĥ † RĥRĥR + cos ϑ 2 (S −ĥ † RĥR ) + H.c., S y R = 1 2i 2S −ĥ † RĥRĥR + H.c.(S19)
whereĥ R are the HP bosonic annihilation operators. Linearizing results in a quadratic Hamiltonian that is diagonalized by the mode expansionΞ = k Ω kb † kbk , with dispersion Ω 2 k = k ( k cos 2 ϑ + zt sin 2 ϑ) where k is the dispersion in the main text. In terms of these collective modes, the real exciton annihilation operator iŝ
χ k 0 = cos 2 ϑ 2 cosh θ k + sin 2 ϑ 2 sinh θ k b † −k − sin 2 ϑ 2 cosh θ k + cos 2 ϑ 2 sinh θ k b k + . . .(S20)
The BdG mixing angles are again given by the rhs of Eq. (5), writ explicitly
sinh 2 θ k = 1 2 1 + cos 2 ϑ k + zt sin 2 ϑ 2Ω k − 1 . (S21)
The ellipses in Eq. (S20) represent sub-leading terms that are higher powers ofb,b † . The HP picture naturally shows that higher order processes that leave behind more than one collective mode will appear in the theory. Yet these contributions to n K cannot be evaluated consistently without computing higher-order corrections to the ground state as well, so are dropped.
Under the same approximation of a nearly-uniform occupancy inside the light cone, we find
Γ(ν) = Nτ −1 loc × cos 2 ϑ 2 cosh θ K − sin 2 ϑ 2 sinh |θ K | 2 = Nτ −1 loc × cos ϑ 2 + 2 + cos 2ϑ √ 30 + 6 cos 2ϑ .(S22)
Note that θ k < 0 if k > zt, and here we have explicitly used that it is negative at K. This shows suppression of emission compared to the mean-field result. Γ may be expressed explicitly by radicals, and in the dilute limit
Γ(ν 1) ≈ Nτ −1 loc zt K 2 cos 4 ϑ 2 = 4 9 Nτ −1 loc ν 2 MF .(S23)
Truncation of the HP expansion generically breaks Hamiltonian symmetries [107], and here the particle-hole relation (S15) is violated. However, trigonometric identities show that Γ(ν) upholds (S15) at the level of the mean-field density ν MF . Except for very small fillings (see below), the numerical correction between ν and ν MF is merely quantitative, and we neglect it when evaluating Γ(ν) in the main text. This recovers the particle-hole symmetry of Γ(ν), e.g. in Fig. 2.
Nonzero temperature.-Thermal excitations will also lead to a population of zone-corner excitons, and are the primary channel considered for exciton loss, e.g. Ref. [38]. In terms of spin waves, this corresponds to processes wherein thermal collective modes are absorbed, rather than emitted, to sink excess momentum.
Thus, at T > 0 two emission lines emerge around E 0 , ω = µ ∓ Ω K = E 0 + zt(cos ϑ ∓ r 2 cos 2 ϑ + r sin 2 ϑ), (S24)
where r = K /zt (=3/2) and the signs correspond to processes that emit and absorb a spin wave, respectively. Interactions lead to a blueshift with increasing density,
ω emit ≈ E 0 − 5 2 zt + 3ztν, ω absorb ≈ E 0 + 1 2 zt + ztν. (S25)
The two emission peaks are split by 3zt ∼ 20t. For a representative t ∼ 0.2 meV [17], this separation is ∼ 4meV, and might be resolved spectroscopically.
The matrix element for the two processes are given by the parentheses in Eq. (S20). Unlike the usual Stokes and anti-Stokes lines in phonon-assisted emission [5], they are not equal. Rather, at high temperature the ratio of intensities reproduces the leaky condensate scaling Fig. 2) and with (dashed black line) spin-wave density corrections between ν MF and ν. Focusing on smaller filling allows us to exactly diagonalize a larger 6 × 6 system, which we fit with the functional form (S34) (dashed blue line). Note the spin-wave corrections give the correct asymptotic form for ν → 0. The remaining discrepancy could be due to finite-size effects.
correction to excellent agreement, with ν 0 ∼ 5.0. The close agreement between the two fitted values shows that the spinwave theory correctly captures the correlations at small filling. We then plot the emission rate in terms of the full density in Fig. S1b, showing good agreement between the spin-wave calculation and exact diagonalization data.
EMISSION AT SMALL k
Above we show that the coupling between k = 0 condensate modes and the density wave is forbidden byĈ 3 symmetry. Nonzero k modes have lower symmetry, and therefore may couple to the density wave to recombine. This is another interaction-driven incoherent emission channel that is nonlinear in exciton density, and therefore acts as an additional leaky emission channel, further reinforcing the dominance of interactions at low temperatures. Here we show its contribution is subleading compared to the primary channel we explore in the main text.
While this effect requires a two-valley model, results of our hard-core theory are still useful for estimating its magnitude. So far, leaky emission was facilitated by spontaneous creation of zone-corner Bogoliubov modes, which mix the creation and annihilation of real excitons. The momentum folding induced by the density wave allows zone-center Bogoliubov mode creationb † −k to mix an annihilation of a bright zone corner excitonχ k+K as well. Therefore, emission via this channel is given byΓ = k sinh 2 θ k χ k+Kb−k Γ k+K , where the first two factors are the squared amplitude for spontaneous production of the BdG mode, and its bright fraction, respectively. (The latter is unity for zone-corner emission.) The summation runs over the folded light cone |k| ≤ E 0 / c.
We now estimate all factors for small k. From Eq. (S21), sinh 2 θ ≈ Ω k /2 k ∼ ν/(ka) 2 . As we show in our symmetry analysis, a genuine momentum folding density wave necessitates intervalley conversion. Thus, χ k+Kb−k ∼ |νU J IV k+K / 2 K | 2 which for strong interactions saturates to |νJ IV k+K / K | 2 . Finally, theĈ 3 forbidden transition at k = 0 [compare |k = 0, τ = +1 and |k = −K, τ = −1 in Table I] manifests in the suppressed |J IV k+K | ∼ J 0 × ak around the zone center, where J 0 is the typical amplitude for intervalley conversion.
Collecting contributions, in total we find Γ ∼ N (S35) Here Γ is the leaky emission rate computed in the main text. λ ∼ c/E 0 ∼ 1 µm is the optical wavelength. Furthermore, that intervalley conversion is driven by e-h exchange interactions that also determine recombination allows us to estimate J 0 ≈ D 2 /(4π a 3 ) ∼ 0.01 meV, where D is the dipole transition matrix element, about 0.5 eÅ [40]. We thus have a/λ ≈ 10 −2 and J 0 / K ∼ 10 −2 -10 −1 , showing this emission channel is substantially weaker than that of the main text.
We remark that vertically polarized excitons (e.g. Ref. [108]) are notĈ 3 -forbidden from participating in the momentum folding, and in that case |J IV k+K | ∼ J 0 . However, the inner product between this polarization and the outgoing photon polarization reintroduces a k 2 scaling, leading to the same result as above. Moreover, while such exciton polarization is observed in monolayers, it might not readily realize in bilayers [109,110].
a) The moiré triangular superlattice formed by a twist of 3 • . The diamond outlines the moiré unit cell, with the three high-symmetry rotation centers A, B, and C highlighted in blue, green, and red. (b) The nearest-neighbor-hopping exciton dispersion. The blue and orange regions represent the condensate and the optical light cone (of τ = +1 excitons, not to scale) respectively. cal emission of TMD moiré excitons at low temperatures.
FIG. 2 :
2Emission rate Γ(ν) at T = 0 versus filling
Acknowledgments.-We thank M. Atatüre, D. Kara, C. M. Pursar, A. R.-P. Montblanch, T. F. Heinz, O. Karni, E. Barré, A. Camacho-Guardian and D. Bennett for fruitful discussions. The support of the Cambridge International Trust, of EPSRC Grant Nos. EP/P009565/1, EP/P034616/1 and of a Simons Investigator Award are gratefully acknowledged.
FIG
. S1: Left: Mapping between ν and ν MF . The dots are calculated numerically, and the solid line is a fit to ν MF ∝ −ν/ log(ν/ν 0 ), giving excellent agreement with ν 0 = 5.0. The dashed line marks ν MF = ν for comparison. Right: Asymptotics of emission rate at low filling. The emission rate Γ(ν) is plotted without (solid line, same as in
Supplemental Material: Leaky exciton condensates in transition metal dichalcogenide moiré bilayers Benjamin Remez 1 , Nigel R. Cooper 1,2 1 T.C.M. Group, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom 2 Department of Physics and Astronomy, University of Florence, Via G. Sansone 1, 50019 Sesto Fiorentino, Italy
TABLE I :
IAngular momentum quantum numbers (s) kτ of momentum states relative to the three rotation centers. The condensate remains dark since all k = 0 states have identicalĈ 3C eigenvalues of 0.
3πνt.Fig. 2shows that at all densities T > T BKT , demonstrating that interactions dominate the emission in the superfluid phase and much of the Bosedegenerate regime.Numerical results.-We compare our HP expansion against exact diagonalization by implementing the Hamiltonian (9) numerically. We set µ = E 0 for convenience, and compute numerically the ground state ofΞ in each of the fixed density sectors ν = 2/N, 3/N . . . 1 separately. We then evaluate Ŝ + KŜ − K in this state to obtain n K (ν) of each sector. Commensurability of K with periodic boundary conditions limits the triangular crystal dimensions to be N = 3n × 3m with n, m integers.Fig. 2shows the results for a 6 × 3 lattice.Asymptotic correlations.-We would like to estimate the emission rate for very low filling. In this regime, correlation lengths become large, yet n-particle correlations are suppressed by factors of ν n . Therefore, 2-particle correlations are dominant[76]. We thus consider a fixed number of N s = 2 hard-core bosons, while taking the system size N → ∞ to scan n K (ν) for ν = 2 N → 0. The order of thermodynamic limits is important, and we expect different results for n K (ν) if N s or ν are held fixed while taking N → ∞, and we therefore use this to extract scaling laws without precise numerical factors.The two-body ground state with energy E GS is generally writtenwhere ψ ∆ is a real and symmetric normalized two-body pairing wavefunction, with ψ 0 = 0 due to the hard-core constraint. The zone-corner occupation is thenSubstituting this state into the Hamiltonian (9) gives a discretized Schrodinger equation for ψ, solved byHere ε = E GS /2 − 0 is the mean energy of each particle relative to the non-interacting single-particle band minimum, determined implicitly by the hard-core conditionSo far this solution is exact. We proceed to evaluate n K asymptotically bywhere A is a proportionality constant, and N 0 and ν 0 are some positive constants associated with an infrared cutoff of the integral's logarithmic divergence at the origin, corresponding to omitting k = 0 from the initial sum. While this relation was derived for a two-particle problem, it provides a remarkably good description of n K (ν) for more particles, as seen inFig. 2. Demanding a smooth transition at the ν = 1 2 particle-hole symmetry point with dn K /dν = 1 eliminates A, and we obtain the formwith appropriate symmetrization for ν > 1 2 , and a single fitting parameter ν 0 (to which we ascribe no physical meaning). Fitting numerical results with Eq. (S34), we obtain ν 0 ∼ 5.5.Our spin-wave result Eq. (11) is consistent with Eq. (S34) only if ν MF ∼ −ν/ log(ν/ν 0 ) with a similar ν 0 . Following Bernardet et al[85], we numerically evaluate the density from the defining thermodynamic relation ν = − 1 N ∂ ∂µ Ξ . ν then differs from ν MF by the contribution of the spin-wave zeropoint motion to the ground state expectation value Ξ . We plot ν MF versus ν inFig. S1a, and fit the expected logarithmic
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[
"Quantum hoop conjecture and a natural cutoff for vacuum energy of a scalar field",
"Quantum hoop conjecture and a natural cutoff for vacuum energy of a scalar field"
] |
[
"Rongjia Yang \nCollege of Physical Science and Technology\nHebei University\n071002BaodingChina\n\nHebei Key Lab of Optic-Electronic Information and Materials\nHebei University\n071002BaodingChina\n\nInstitute of Theoretical Physics\nState Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina\n"
] |
[
"College of Physical Science and Technology\nHebei University\n071002BaodingChina",
"Hebei Key Lab of Optic-Electronic Information and Materials\nHebei University\n071002BaodingChina",
"Institute of Theoretical Physics\nState Key Laboratory of Theoretical Physics\nChinese Academy of Sciences\n100190BeijingChina"
] |
[] |
We propose here a quantum hoop conjecture which states: the de Broglie wavelength of a quantum system cannot be arbitrarily small, it must be larger than the characterized Schwarzschild radius of the quantum system. Based on this conjecture, we find an upper bound for the wave number (or the momentum) of a particle, which offers a natural cutoff for the vacuum energy of a scalar field.
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10.1016/j.rinp.2016.02.005
|
[
"https://arxiv.org/pdf/1512.02421v2.pdf"
] | 62,823,157 |
1512.02421
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0215fc7ac270f7f79f74d6b5d5bc65496d64ceda
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Quantum hoop conjecture and a natural cutoff for vacuum energy of a scalar field
11 Mar 2016
Rongjia Yang
College of Physical Science and Technology
Hebei University
071002BaodingChina
Hebei Key Lab of Optic-Electronic Information and Materials
Hebei University
071002BaodingChina
Institute of Theoretical Physics
State Key Laboratory of Theoretical Physics
Chinese Academy of Sciences
100190BeijingChina
Quantum hoop conjecture and a natural cutoff for vacuum energy of a scalar field
11 Mar 2016numbers: 0370+k0470-s9536+x
We propose here a quantum hoop conjecture which states: the de Broglie wavelength of a quantum system cannot be arbitrarily small, it must be larger than the characterized Schwarzschild radius of the quantum system. Based on this conjecture, we find an upper bound for the wave number (or the momentum) of a particle, which offers a natural cutoff for the vacuum energy of a scalar field.
I. INTRODUCTION
In the past years, a lot of independent cosmological observations, such as supernova (SN) Ia at high redshift [1,2], the cosmic microwave background (CMB) anisotropy, [3,4] and large-scale structure [5], have confirmed that the Universe is undergoing an accelerated expansion. In the framework of general relativity, an unknown energy component, usually called dark energy, has to be introduced to explain this phenomenon. The simplest and most theoretically appealing scenario of dark energy is the vacuum energy which is about ρ ovac ∼ (10 −3 eV) 4 = 10 −8 ergs/cm 3 matched from observational data. However, this model is confronted with a very difficult problem-cosmological constant problem [6][7][8][9][10] (may suffer from age problem as well [11]). To briefly illustrate this issue, we consider, for example, the vacuum energy density of a scalar field. It is well known that the total vacuum energy density of a scalar field with mass m is quartically divergent in the ultraviolet (UV)
ρ tvac = 0|ρ tvac |0 = dk k 2 4π 2 c 2 k 2 c 2 + m 2 c 4 / 2 .(1)
A usually used regularisation for this divergence is to artificially take a UV cutoff. But if we take different UV cutoffs, such as electroweak scale, grand unification scale, or Planck scale, we can get different values of vacuum energy density. Furthermore the differences between these values are huge, see for example, taking electroweak scale, we get ρ tvac ∼ (10 11 eV) 4 = 10 48 ergs/cm 3 ; taking Planck scale, we have ρ tvac ∼ (10 27 eV) 4 = 10 112 ergs/cm 3 . The ratio of theoretical to observational value of the vacuum energy ranges from 10 56 to 10 120 . This is the well known cosmological constant problem [6][7][8][9][10]. Which scale we should take is still an open problem. Can we find a UV cutoff from fundamental laws of physics? This is the major issue we will consider in this letter.
Here, combining with quantum and black hole physics, we find an upper bound for the wave number of a quantum particle, which gives a natural cutoff for the vacuum energy of a scalar field.
The rest of the paper is organized as follows. In next section, we will present the upper limit of the wave number from quantum and black hole physics and consider a cutoff of the vacuum energy of scalar field. Finally, we will briefly summarize and discuss our results in section III.
II. UPPER BOUND FOR WAVE NUMBER AND A NATURAL CUTOFF FOR VACUUM ENERGY
For a quantum particle with mass m, the de Broglie relation reads E = ω, p = k. According to the mass-energy relation in special relativity, the total energy of a particle is E 2 = p 2 c 2 + m 2 c 4 . Combining the de Broglie relation and the mass-energy relation, then we have
E 2 = 2 k 2 c 2 + m 2 c 4 .(2)
This equation indicates that E −→ ∞ for k −→ ∞. A natural question rises: is this result reasonable? In other words, because ω = k 2 c 2 + m 2 c 4 / 2 , the question can also be stated as: can a particle oscillate arbitrarily fast (or, can the de Broglie wavelength of a particle be arbitrarily small)? If we take into account the effect of gravitation, the answer may be not. Think of black hole physics, a system with total energy E has an effective mass E/c 2 , so it will be characterized with a Schwarzschild radius which is given by
r c = 2G c 3 2 k 2 + m 2 c 2 .(3)
The hoop conjecture in black hole physics states: if matter is enclosed in sufficiently small region, then the system should collapse to a black hole [12,13]. Similar assumptions were also suggested in [14][15][16]: for example, it argued that the energy of a system of size L must have an upper bound not to collapse into a black hole [14]. Here we generalize the hoop conjecture to the quantum case: the de Broglie wavelength of a quantum system can not be arbitrarily small, it should be larger than the characterized Schwarzschild radius of the quantum system. This can be called quantum hoop conjecture. This quantum hoop conjecture can get supports from earlier works in literatures. Possible connection between gravitation and the fundamental length was discussed in [17]. From quantum mechanics and classical general relativity, it was shown in [18,19] that any primitive probe or target used in an experiment must be larger than the Planck length, which implies a device independent limit on possible position measurements. Researches from string theory, black hole physics, and quantum gravity also predict that there exists a minimum measurable length scale which is approximately equivalent to the Planck length l p [20][21][22][23][24]. Based on these researches, we can conclude that the de Broglie wavelength of any quantum system must not be less than the minimum length scale. This conclusion is consistent with the quantum hoop conjecture proposed here: the de Broglie wavelength of a quantum system should be larger than its characterized Schwarzschild radius. In [25], a quantum hoop conjecture was also suggested by constructing the horizon wave-function for quantum mechanical states representing two highly boosted non-interacting particles, which is different from the conjecture we proposed here.
The quantum hoop conjecture suggested here provides: λ > r c , which gives an upper bound for the wave number
k = 2π λ < 2π r c = πc 3 G 2 k 2 + m 2 c 2 − 1 2 < πc 3 Gk .(4)
It is easy to get
k < √ πl −1 p ,(5)
where l p = G /c 3 is the Planck length. This bound only holds in the observer's reference frame. Bound (5) also gives an upper limit for the momentum of the particle: p < √ π l −1 p . Obviously, the wave number of a massive particle is less than that of a massless particle.
As an application, we apply the bound for the wave number (5) to the vacuum energy of a scalar field. For a quantum particle of a scalar field, there are three freedoms for oscillation: k = k 2 x + k 2 y + k 2 z . So we have k < 2 √ 3π rc < √ 3πl −1 p which offers a natural cutoff for the vacuum energy of a scalar field (1)
ρ tvac = 0|ρ tvac |0 = kmax 0 dk k 2 4π 2 c 2 k 2 c 2 + m 2 c 4 / 2 .(6)
For k ≫ m, integration (6) is approximatively equivalent to 3 /(16cl 4 p ) which closes to the value obtained by taking the Planck scale cutoff. Also based on black hole physics, a cutoff for vacuum energy of a scalar field was found in [26].
III. CONCLUSIONS AND DISCUSSIONS
In this letter, we suggested a quantum hoop conjecture: the de Broglie wavelength of a quantum system can not be arbitrarily small, it must be larger than the characterized Schwarzschild radius of the quantum system. This conjecture gives an upper bound for the wave number or the momentum of the quantum system. For application, we found a natural cutoff for the vacuum energy of a scalar field.
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Quantum hoop conjecture: Black hole formation by particle collisions. R Casadio, O Micu, F Scardigli, 10.1016/j.physletb.2014.03.037arXiv:1311.5698Phys. Lett. 732R. Casadio, O. Micu, F. Scardigli, Quantum hoop conjecture: Black hole formation by particle collisions, Phys. Lett. B732 (2014) 105-109. arXiv:1311.5698, doi:10.1016/j.physletb.2014.03.037.
The Zero point energy and gravitation, Braz. H Culetu, arXiv:hep-th/0410133J. Phys. 36H. Culetu, The Zero point energy and gravitation, Braz. J. Phys. 36 (2006) 562. arXiv:hep-th/0410133.
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[
"Generalized susceptibility of quasi-one dimensional system with periodic potential: model for the organic superconductor (TMTSF) 2 ClO 4",
"Generalized susceptibility of quasi-one dimensional system with periodic potential: model for the organic superconductor (TMTSF) 2 ClO 4"
] |
[
"Yasumasa Hasegawa \nDepartment of Material Science\nGraduate School of Material Science\nUniversity of Hyogo\n678-1297HyogoJapan\n",
"Keita Kishigi \nFaculty of Education\nKumamoto University\nKurokami 2-40-1860-8555KumamotoJapan\n"
] |
[
"Department of Material Science\nGraduate School of Material Science\nUniversity of Hyogo\n678-1297HyogoJapan",
"Faculty of Education\nKumamoto University\nKurokami 2-40-1860-8555KumamotoJapan"
] |
[] |
The nesting vector and the magnetic susceptibility of the quasi-one-dimensional system having imperfectly nested Fermi surface are studied analytically and numerically. The magnetic susceptibility has the plateau-like maximum in "sweptback " region in the momentum space, which is surrounded by Q = (2kF , π) + qi (kF is the Fermi wave number, i = 1, 3, 4, and q1, q3 and q4 are given in this paper). The best nesting vector, at which the susceptibility χ0(Q) has the absolute maximum at T = 0, is obtained near but not at the inflection point, Q = (2kF , π) + q4. The effect of the periodic potential V on the susceptibility is studied, which is important for the successive transitions of the field-induced spin density wave in (TMTSF)2ClO4. We obtain that the sweptback region (surrounded by q2, q3 and q4 when V > 0) becomes small as V increases and it shrinks togives the degree of imperfect nesting of the Fermi surface, i.e. the second harmonics of the warping in the Fermi surface. The occurrence of the sign reversal of the Hall coefficient in the field-induced spin density wave states is discussed to be possible only whenwhere t4 is the amplitude of the fourth harmonics of the warping in the Fermi surface. This gives the novel limitation for the magnitude of V .
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10.1103/physrevb.78.045117
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"https://arxiv.org/pdf/0805.2445v1.pdf"
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0805.2445
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c43d2615eacaec7aef127b488ae3e12cfb378257
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Generalized susceptibility of quasi-one dimensional system with periodic potential: model for the organic superconductor (TMTSF) 2 ClO 4
16 May 2008 (Dated: May 16, 2008)
Yasumasa Hasegawa
Department of Material Science
Graduate School of Material Science
University of Hyogo
678-1297HyogoJapan
Keita Kishigi
Faculty of Education
Kumamoto University
Kurokami 2-40-1860-8555KumamotoJapan
Generalized susceptibility of quasi-one dimensional system with periodic potential: model for the organic superconductor (TMTSF) 2 ClO 4
16 May 2008 (Dated: May 16, 2008)numbers: 7530Fv7830Jw7110Pm
The nesting vector and the magnetic susceptibility of the quasi-one-dimensional system having imperfectly nested Fermi surface are studied analytically and numerically. The magnetic susceptibility has the plateau-like maximum in "sweptback " region in the momentum space, which is surrounded by Q = (2kF , π) + qi (kF is the Fermi wave number, i = 1, 3, 4, and q1, q3 and q4 are given in this paper). The best nesting vector, at which the susceptibility χ0(Q) has the absolute maximum at T = 0, is obtained near but not at the inflection point, Q = (2kF , π) + q4. The effect of the periodic potential V on the susceptibility is studied, which is important for the successive transitions of the field-induced spin density wave in (TMTSF)2ClO4. We obtain that the sweptback region (surrounded by q2, q3 and q4 when V > 0) becomes small as V increases and it shrinks togives the degree of imperfect nesting of the Fermi surface, i.e. the second harmonics of the warping in the Fermi surface. The occurrence of the sign reversal of the Hall coefficient in the field-induced spin density wave states is discussed to be possible only whenwhere t4 is the amplitude of the fourth harmonics of the warping in the Fermi surface. This gives the novel limitation for the magnitude of V .
I. INTRODUCTION
Various interesting properties, such as field-induced spin density wave (FISDW), quantum Hall effect and superconductivity, have been observed in the quasi-onedimensional organic conductors, (TMTSF) 2 X, where X is PF 6 , ClO 4 etc. 1 The successive transitions between different FISDW phases occur as the magnetic field is increased. The FISDW has been understood as a consequences of the reduction of the dimensionality due to the magnetic field and the quantization of the nesting vector 2, 3,4,5,6,7,8,9,10,11 . The FISDW phases are characterized by the integer N , by which the wave number of FISDW is given as Q x = 2k F + N G, where k F is the Fermi wave number, G = beB/ , b is the lattice constant (we take b = 1 in this paper), e is the electron charge, B is the magnetic field and = h/2π (h is the Planck constant). We take = 1 hereafter in this paper. The Hall conductivity is quantized as σ xy = 2N e 2 /h with the quantum number N of the nesting vector 12,13,14 . The quantization of the x component of the nesting vector, Q x , can be seen as the sharp peaks in the susceptibility for the non-interacting system, χ 0 (Q), at Q x = 2k F +N G in the magnetic field.
The peaks of χ 0 (Q) in the magnetic field can be understood to some extent by the peaks of χ 0 (Q) in the absence of the magnetic field. If the nesting of the Fermi surface is perfect, χ 0 (Q) in the absence of the magnetic field diverges at the nesting vector as temperature becomes zero. In that case the successive transitions of FISDW does not happen. If the nesting of the Fermi surface is not perfect, the best nesting vector at B = 0, which gives the maximum of χ 0 (Q), is located in the reciprocal space at
Q = Q 0 + q,(1)
where
Q 0 = (2k F , π),(2)
and q = 0. If q x > 0, the quantum number N of FISDW is positive. If q x < 0 at the best nesting vector, however, the the negative N is possible in some region of the magnetic field 15 . Although (TMTSF) 2 PF 6 is well understood by the quasi-one-dimensional model, (TMTSF) 2 ClO 4 is a little more complicated. Below T AO ≈ 24 K the anion ClO 4 , which has no inversion symmetry, orders alternatively in y direction, resulting the periodic potential V in the electron system. Actually, the magnetic field and temperature phase diagram in (TMTSF) 2 PF 6 16,17,18,19,20 is different from that in (TMTSF) 2 ClO 4 21,22,23 . The origin of the different phase diagrams in (TMTSF) 2 PF 6 and (TMTSF) 2 ClO 4 is caused by the periodic potential, V . The magnitude of V is first estimated to be the order of 24,25 The suppression of the N = 0 FISDW state 24 and even-N FISDW states 25 has been shown by the perturbation in V . On the other hand, the magnitude of V has been estimated to be V = 0.83t b from the angle dependence of the magnetoresistance by Yoshino et al. 26 . By treating V not in perturbation, a lot of interesting features, such as existence of several nesting vectors 27,28,29,30 and the phase diagram of the FISDW states 31,32 , has been obtained. Recently, Yoshino et al. 33 has estimated the value to be V = 0.028t a (V = 0.34t b with their estimation t a = 12t b ). Lebed et al. 34,35 have estimated the value as V = 0.2t b . The novel estimation of V is given in this paper from the existence of the sign reversal of the Hall effect.
T AO = 24K, i.e. V ≪ t b .
In this paper we study the nesting vector and the susceptibility in the quasi-one dimensional system with imperfectly nested Fermi surface in the absence of the magnetic field. The analytic expression of the susceptibility and the nearly flat region in the reciprocal space are given analytically for the first time in the simple model with V = 0. The effect of V on the nesting vector and the susceptibility are studied in detail numerically.
II. MODEL
We neglect the small dispersion in k z direction and study the tight binding model in the square lattice with anisotropic transfer integral elements t a ≫ t b . We take the lattice constant to be 1. In the real system the crystal is triclinic and we have to consider the multipletransverse-transfer integrals 36 but most of the essential features are obtained by studying the simple model in the square lattice 1 . The energy dispersion can be linearized with respect to k x and we take account of the higher harmonic terms for k y as
ǫ(k) = v F (|k x | − k F ) + t ⊥ (k y ),(3)
where
t ⊥ (k y ) = − 2t b cos k y − 2t ′ b cos(2k y ) − 2t 3 cos(3k y ) − 2t 4 cos(4k y ).(4)
and we study the case t b , t ′ b , t 3 and t 4 to be positive. The terms proportional to t 3 and t 4 are thought to be essential 15,32 to understand the negative N phase 37,38 of FISDW in some region of the magnetic field. The Fermi surface consists of two "Fermi lines" near k x ≈ ±k F , as shown in Fig. 1. The Fermi surface is almost nested, i.e. when we translate the left part of the Fermi line with the vector Q ≈ Q 0 , it overlaps with the right part of the Fermi line, but the overlap is not perfect due to the t ′ b and t 4 terms.
The Brillouin zone is divided into halves in the k y direction by the periodic potential. The Hamiltonian is written as a 2 × 2 matrix with the anion potential V as
H = ǫ(k) V V ǫ(k + Q A ) ,(5)
where Q A = (0, π). The energy E(k) is given by
E(k) = 1 2 ǫ(k) + ǫ(k + Q A ) ± (ǫ(k) − ǫ(k + Q A )) 2 + 4V 2 ,(6)
and the Fermi surface consists of four lines as shown in Fig. 2. It is known 27 that the susceptibility χ 0 (Q) has max-
imum near Q ≈ Q 0 if V 1.5t b when t ′ b = 0.1t b (i.e. V 15t ′ b ), while the absolute maximum of χ 0 (Q) is located near Q ≈ (2k F ± 2V /v F , π/2) if V 1.5t b .
The peak of χ 0 (Q) near Q ≈ Q 0 is caused by the nesting between the outer Fermi surface and the inner Fermi surface (k In this paper we examine in detail the nesting properties of the quasi-one dimensional systems without and with the periodic potential (V 0.5t b ). Thus we focus on the nesting condition for only Q ≈ Q 0 .
III. NESTING OF THE FERMI SURFACE FOR
V = 0
In this section we study the nesting properties of the quasi-one dimensional system described by Eq. (3). The (qy)) and one maximum (q max x (qy)) of qx as a function of Ky for each 0 < |qy| < q4y, while only one minimum and one maximum of qx for |qy| > q4y as shown by dotted vertical lines.
Fermi surface consists of two curves (see Fig. 1). The right and left part of the Fermi surface are given as a function of k y ,
k (R) x (k y ) = k F − 1 v F t ⊥ (k y ),(7)k (L) x (k y ) = −k F + 1 v F t ⊥ (k y ).(8)
We translate the left part of the Fermi surface with the nesting vector, Q = Q 0 + q. The translated curve is given by (qy) (thick blue lines in q1x < qx < q4x and dashed lines in qx > q4x) and q max x (qy) (thick green lines in q3x < qx < q4x and dashed green lines in qx > q4x). We take t ′ b /t b = 0.1, t3 = t4 = 0 (upper figure) and t3/t b = 0.02, t4/t b = 0.002 (lower figure). In the sweptback region enclosed by q1, q3 and q4, χ0(Q) has large values.
k (L ′ ) x (k y ) = k F + q x + 1 v F t ⊥ (k y + q y + π).(9)
The difference of the right part of the Fermi surface and the translated left part of the Fermi surface is given by
k (L ′ ) x (k y ) − k (R) x (k y ) = q x + 1 v F (t ⊥ (k y ) + t ⊥ (k y + q y + π)) .(10)If t ′ b = t 4 = 0, the nesting of the Fermi surface is perfect with q x = q y = 0, i.e. k (L ′ ) x (k y ) − k (R)
x (k y ) = 0 for all values of k y . If t ′ b = 0 or t 4 = 0, the nesting of the Fermi surface is not perfect. In this case the Fermi surface intersect with the translated one with the nesting vector Q 0 + q, if q x and q y satisfy
q x = −1 v F [t ⊥ (k y ) + t ⊥ (k y + q y + π)] = 4 v F t b sin(K y ) sin( q y 2 ) + t ′ b cos(2K y ) cos(q y ) + t 3 sin(3K y ) sin( 3q y 2 ) + t 4 cos(4K y ) cos(2q y ) ,(11)
for some value of K y , where
K y = k y + q y 2 .(12)
Eq. (11) is the condition for the nesting vector (Q = Q 0 + q) to realize the intersection of the translated left part of the Fermi surface with the right part of the Fermi surface at k y . In Fig. 3 we plot q x vs K y for q y = 0. We define two vectors, q 1 and q 3 , as q 1y = q 3y = 0 and q 1x and q 3x being the minimum and the maximum of q x as a function of K y at q y = 0, respectively. When t 4 ≤ t ′ b /4 (in this paper we study only in this case), the maximum of q x as a function of K y for q y = 0 is given at K y = 0 and ±π, and the minimum of q x as a function of K y for q y = 0 is given at K y = ±π/2, as shown in Fig. 3;
q 1 = 4 v F (−t ′ b + t 4 ) , 0 ,(13)q 3 = 4 v F (t ′ b + t 4 ) , 0 .(14)
We define q 2 = q 1 for V = 0 and we will define q 2 for V = 0 in section V. We plot q x vs. K y (Eq. (11)) for some values of q y in Fig. 4. As seen in Fig. 4, q x as a function of K y has two minimums at K y = ±π/2 (q min(±) x (q y )) and one maximum at 0 ≤ K y ≤ π/2 (q max x (q y )), if 0 < |q y | < q 4y (q 4 will be given later). There are one minimum at K y = −π/2 and one maximum at K y = π/2 if |q y | > q 4y . We obtain q min(+) x (q y ) and q min(+) x (q y ) as
q min(+) x (q y ) = 4 v F −t ′ b cos q y + t b sin |q y | 2 − t 3 sin 3|q y | 2 + t 4 cos 2q y ,(15)q min(−) x (q y ) = 4 v F −t ′ b cos q y − t b sin |q y | 2 + t 3 sin 3|q y | 2 + t 4 cos 2q y .(16)
If t 3 and t 4 are finite, we have to solve the fourth-degree equation to obtain the expression of q max x (q y ), but it is easy to obtain q max x (q y ) numerically. We define q 4 = (q 4x , q 4y ) by the equation
q min(+) x (q 4y ) = q max x (q 4y ) = q 4x .(17)
If t 3 = t 4 = 0, the simple expressions of q max x (q y ) and q 4 are obtained as
q max x (q y ) = 4 v F (t ′ b cos q y + t 2 b sin 2 qy 2 8t ′ b cos q y ),(18)q 4x = 1 v F 24t ′ b 1 + 128 t ′ b t b 2 + 1 ,(19)
and
q 4y = ±2 sin −1 8 t ′ b t b 1 + 128 t ′ b t b 2 + 1 .(20)
Note that q max x (q y ) has the physical meaning only if |q y | ≤ q 4y , since the analytical form Eq. (18) obtained in the case of t 3 = t 4 = 0 and the numerically obtained values at |q y | > q 4y corresponds to the local maximum of q x as a function of sin(K y /2) at | sin(K y /2)| > 1. We plot q max x (q y ), q min(+) x (q y ) and q i (i = 1, 3, and 4) in Fig. 5. There are large overlap between the Fermi line and the translated one, if q is in the "sweptback " region with the apexes q 1 and q 4 enclosed by the thick lines in Fig. 5.
IV. SUSCEPTIBILITY IN THE Q1D SYSTEM WITH V = 0
The susceptibility
χ 0 (Q) = k f (E k+Q ) − f (E k ) E k − E k+Q ,(21)
where f (E k ) is the Fermi distribution function, is calculated at T = 0 as
χ 0 (Q) = π −π dk y 2π k (R) x (ky) k (L) x (ky) dk x 2π 2 ǫ(k − Q) − ǫ(k) = 1 π π −π dk y 2π 0 k (L) x (ky) dk x v F Q x + t ⊥ (k y − Q y ) − t ⊥ (k y ) + k (R) x (ky) 0 dk x v F (−2k x + Q x ) + t ⊥ (k y − Q y ) − t ⊥ (k y ) = 1 2πv F π −π dk y 2π v F k F − t ⊥ (k y ) v F Q x + t ⊥ (k y − Q y ) − t ⊥ (k y ) − 1 2 log v F (Q x − 2k F ) + t ⊥ (k y − Q y ) + t ⊥ (k y ) v F Q x + t ⊥ (k y − Q y ) − t ⊥ (k y )(22)
The susceptibility is finite at T = 0 and has the singularity (kinks) as a function of Q. The singularity of χ 0 (Q) comes from the integration of the logarithmic term in eq. (22). For Q y = π (i.e. q y = 0) and t 3 = t 4 = 0, the singular part of χ 0 (Q 0 + Q) is calculated as
χ 0,sing = 1 πv F π −π dk y 2π − 1 2 log v F q x − 4t ′ b cos 2k y 2k F v F = − 1 2πvF log qxvF + √ (qxvF ) 2 −(4t ′ b ) 2 4kF vF if |q x v F | > 4t ′ b − 1 2πvF log t ′ b kF vF if |q x v F | < 4t ′ b .(23)
It is obtained from Eq. (23), that χ 0 (q) has a plateau as a function of q x when t 3 = t 4 = 0 and q y = 0. If t 3 , t 4 and q y are not zero, we have to integrate Eq. (22) numerically. In Fig. 6 we plot χ 0 (Q) for several t 3 and t 4 and q y as a function of q x . It can be seen that if t 3 = t 4 = 0, nearly flat peak at q min(+) x (q y ) < q x < q max
x first increases as q y increases, and have the absolute maximum before q y reaches q 4y (= 0.2065π and v F q 4x /t b = 0.956 when t ′ b /t b = 0.1) as shown in the top figure in Fig. 6. If t 3 > 0, the peaks for q y = 0 are suppressed as shown in the middle figure in Fig. 6. If t 4 > 0, the degeneracy of χ 0 (Q 0 + Q) at q 1 and q 3 is lifted and the absolute maximum of χ 0 (Q 0 + q) is obtained at q 1 for the sufficiently large values of t 3 and t 4 , as seen in the bottom figure in Fig. 6.
As seen in Fig. 6, χ 0 (Q 0 +q) has plateau-like maximum in the region q min(+) x (q y ) < q x < q max x (q y ). The absolute maximum of χ 0 (Q 0 +q) occurs at q close to q 4 but not at q = q 4 , as seen in Figs. 7 and 8, where we plot χ 0 (Q 0 +q) as a function of q x or q y on the curves of q x = q min(+) x (q y ) and q x = q max x (q y ), respectively. The three-dimensional plot of χ 0 (Q 0 + q) is shown in Fig. 9. When t 3 = t 4 = 0 and q y = q 4y , eq. (11) becomes
q x = 1 k F 4t ′ b 6 − 16 sin 4 (K y − π 2 ) 1 + 128 t ′ b t b 2 + 1 .(24)
Therefore, q x as a function of K y has a maximum at K y = π/2 as q x ∝ 6−16(K y −π/2) 4 when q y = q 4y . With the vector Q = Q 0 + q 4 the nesting of the Fermi surface is better than other q's, which will make the expectation of the large χ 0 (Q 0 + q). However, the region of q y where χ 0 (Q 0 + q) is mainly contributed, is larger at q x q 4x and |q y | q 4y than at q = q 4 . This is the reason why the absolute maximum of χ 0 (Q 0 + q) is not located at the inflection point (q = q 4 ).
V. NESTING OF THE FERMI SURFACE FOR
V = 0
In this section we study the effects of periodic potential V on the nesting of the Fermi surface and the susceptibility. When V = 0, there are two pairs of the Fermi lines in k x − k y plane (see Fig. 2), which are given by k x as a function of k y , i.e., k L± x (k y ) and k R± x (k y ) for the left and the right parts of the Fermi lines, respectively. The and
k (L±) x (k y ) = −k F − 1 v F −t ⊥ (k y ) − t ⊥ (k y + π) ± [t ⊥ (k y ) − t ⊥ (k y + π)] 2 + 4V 2 ,(25)k (R±) x (k y ) = k F + 1 v F −t ⊥ (k y ) − t ⊥ (k y + π) ± [t ⊥ (k y ) − t ⊥ (k y + π)] 2 + 4V 2 .(26)
The condition for the Fermi surface intersect by the translation of the left part (Eq. (11) for V = 0) is written Note that the absolute maximum is not realized at (q4x, q4y). When q y = 0, we obtain Eq. (27) for (+, −) and (−, +) to be the same as that for V = 0 (Eq. (11)),
We take t ′ b /t b = 0.1, t3 = t4 = 0.q (±±) x = 1 2v F −t ⊥ (k y ) − t ⊥ (k y + π) − t ⊥ (k y + q y ) − t ⊥ (k y + q y + π) ± [t ⊥ (k y ) − t ⊥ (k y + π)] 2 + 4V 2 ± [t ⊥ (k y + q y ) − t ⊥ (k y + q y + π)] 2 + 4V 2 .(27)q (+−) x = q (−+) x = 1 v F [−t ⊥ (k y ) − t ⊥ (k y + π)].(28)
The condition for the intersect of (+, +) is obtained as (29) and the condition for the intersect of (−−) is obtained as
q (++) x = 1 v F [−t ⊥ (k y ) − t ⊥ (k y + π)] + 1 v F (t ⊥ (k y ) − t ⊥ (k y + π)) 2 + 4V 2 ,q (−−) x = 1 v F [−t ⊥ (k y ) − t ⊥ (k y + π)] − 1 v F (t ⊥ (k y ) − t ⊥ (k y + π)) 2 + 4V 2 . (30)
We define q 0x , q 1x , q 2x , and q 3x as the maximum (at K y = ±π/2) and the maximum of q (+−) x (at K y = 0 and π) as a function of K y when q y = 0 (q 0y = q 1y = q 2y = q 3y = 0), respectively (see Fig. 10), i.e.,
q 0 = 1 v F (−4t ′ b + 4t 4 − 2V ), 0 ,(31)q 1 = 1 v F (−4t ′ b + 4t 4 ), 0 ,(32)q 2 = 1 v F (−4t ′ b + 4t 4 + 2V ), 0 ,(33)q 3 = 1 v F (4t ′ b + 4t 4 ), 0 .(34)
When q y is given, the maximums and minimums of q (q y ) at V = 0, respectively (cf. Fig. 5). On the other hand q min(+) x (q y ) has no partner at V = 0, since the filled squares in Figs. 10, 11, 12 and 13 become not the minimum but just the crossing points due to the folding in K y as V becomes zero. We define q 4 as the crossing points of q min(++) x (q y ) and q max x (q y ), which is the extension of that in V = 0.
We plot χ 0 (Q 0 +q) as a function of q x for several values of q y in Fig. 18 (t 3 /t b = 0.02, t 4 /t b = 0.002) for V /t b = 0, 0.2 (V /t ′ b = 2) and 0.4 (V /t ′ b = 4). When 0 < V < 4t ′ b , q 1x < q 2x < q 3x . In this case χ 0 (Q 0 + q) has a plateau-like maximum in the "sweptback " region enclosed by q 2 , q 4 , and q 3 , as shown in Figs. 14 and 15. This region shrinks to the point q 3 when V /t ′ b = 4 as shown in Fig. 16. The absolute maximum of χ 0 (Q 0 + q) occurs near q 4 if t 3 = t 4 = 0. The effects of t 3 and t 4 on χ 0 (Q 0 + q) are the same as these at V = 0; A finite t 3 suppresses χ 0 (Q 0 + q) at q y = 0 and t 4 lifts the degeneracy at q 2x ≤ q x ≤ q 4x . If V > 4t ′ b , we obtain q 1x < q 3x < q 2x and there are no region where χ 0 (Q) has a plateau-like maximum as shown in Figs. 16, 17 and 19. In that case the effects of t 3 and t 4 are small. In Fig. 22, q 2 and q which gives the maximum of χ 0 (Q 0 +q) (i.e., the best nesting vector) are shown for some values of V /t b in the case of t 3 = t 4 = 0. The best nesting vector moves to q 3 as V /t b approaches to 0.4.
The negative Hall constant in some region of the magnetic field 37,38 has been explained by the t 3 and t 4 terms 15 . When V = 0, the terms with t 3 /t b = 0.02 and t 4 /t b = 0.002 make the absolute maximum of χ 0 (Q) in the zero magnetic field to be at q 1 (best nesting vec- tor), while the best nesting vector is located near q 4 if t 3 = t 4 = 0, as shown in Fig. 6. The negative Hall constant is possible, since q 1x < 0. If V /t ′ b > 0 and t 3 and t 4 are the same as above, the best nesting vector is q 2 (see the lower figures in Fig. 18 and the middle figure in Fig. 21). As far as V < 2t ′ b − 2t 4 , the negative Hall constant is possible since q 2x < 0. If V > 2t ′ b − 2t 4 , however, the best nesting vector q 2 has the positive x component, as seen in the lower figures in Fig. 18 and Fig. 19. Therefore, the negative Hall constant is difficult to be stabilized when V > 2t ′ b − 2t 4 . Recently, the authors 32 have numerically obtained the phase diagram for the quantum Hall effect as a function of the magnetic field and periodic potential V . We have shown that the negative Hall constant (N = −2) appears only in the region 0.03
V /t b 0.2 (0.3 V /t ′ b
2) for the parameters t ′ b /t b = 0.1, t 3 /t b = 0.02 and t 4 /t b = 0.002 (the upper figure of Fig. 12 in Ref. [32]). That result can be understood by the fact that for V > 2t ′ b − 2t 4 the best nesting vector has the positive x component. The existence of the negative Hall constant for V /t ′ b 0.3 is understood by the effect of V that will make χ 0 (Q 0 + q) at q ≈ q 4 to be smaller. Experimentally, a negative Hall effect is observed when the system is cooled slowly (less than 0.03K/s) and the external magnetic field region for the negative Hall effect becomes larger as the cooling rate becomes slower (the slowest cooling rate is 0.00009 K/s). 37 It is expected that the magnitude of the periodic potential V becomes larger at the slower colling rate. Therefore, we can conclude from the existence of the negative Hall effect in (TMTSF) 2 ClO 4 that V < 2t ′ b − 2t 4 . The value of V estimated from the magnetic-field-angle dependence of the conductivity 33,34,35 is close to the border of this condition.
VI. SUMMARY AND DISCUSSIONS
We have studied the nesting vector and χ 0 (Q) in the quasi-one dimensional systems having the imperfectly nested Fermi surface (the imperfectness is measured by t ′ b ). We have obtained the plateau-like maximum of χ 0 (Q) when Q is in the sweptback region with the apexes q 1 and q 4 . The absolute maximum of χ 0 (Q) is obtained near but not at Q = Q 0 + q 4 if t 3 = t 4 = 0. When the periodic potential V is finite but not as large as 4t ′ b (which is thought to be the case in (TMTSF) 2 ClO 4 ), the "sweptback" region (with apexes q 2 and q 4 ) becomes smaller as V increases and shrinks to q 3 when V = 4t ′ b . The best nesting vector moves to Q ≈ Q 0 + q 3 . The absolute maximum of χ 0 (Q) is located at Q = Q 0 + q 3 when V > 4t ′ b . The negative Hall coefficient observed in the field-induced spin density wave states in some region of the magnetic field is shown to be possible only when V < 2t ′ b − 2t 4 , in which case the vectors q's giving the plateau-like maximum of χ 0 (Q 0 + q) ("sweptback " region) can have the negative x component, (q 2x < 0). Therefore, we conclude that V should be smaller than 2t ′ b − 2t 4 in (TMTSF) 2 ClO 4 , where the sign reversal of the Hall effect has been observed.
Recently, a lot of interest is attracted by the quasione-dimensional conductor (Per) 2 M (mnt) 2 (where Per = perylene, mnt = maleonitriledithiolate and M = Au and Pt) 41,42,43,44,45 . The charge density wave (CDW) state is realized in (Per) 2 M (mnt) 2 , and the successive transitions of the field-induced CDW has been observed in high magnetic field 41 in contrast to the field-induced SDW in (TMTSF) 2 ClO 4 . This material has a similar band structure as (TMTSF) 2 ClO 4 , but the origin of the pairs of the quasi-one-dimensional Fermi surface in (Per) 2 M (mnt) 2 is different from that in (TMTSF) 2 ClO 4 . The origin of the four pairs of the quasi-one-dimensional Fermi surface in (Per) 2 M (mnt) 2 is the existence of four perylene molecules in the unit cell in the perpendicular plane to the conduction axis 45 , while the origin of the two pairs of the quasi-one-dimensional Fermi surface in (TMTSF) 2 ClO 4 is the periodic potential caused by the anion ordering. It will be interesting to study the similarity and the difference between two materials, since the spin susceptibility χ 0 (Q) and the charge susceptibility χ c (Q) for the non-interacting system have the same Q dependence caused by the nesting properties of the Fermi surface, except for the effects of the Zeemen splitting of the Fermi surface, which play important role only for CDW.
FIG. 1 :FIG. 2 :
12Fermi surface for V = 0. Fermi surface for V = 0.
))
, i.e., the red and blue arrows inFig. 2, while the peaks of χ 0 (Q) near Q ≈ (2k F ± 2V /v F , π/2) are caused by the nesting the outer Fermi surfaces (k31,39,40 . The maximum value of χ 0 (Q) near Q ≈ Q 0 depends weakly on V if V 0.4t b , and it decreases as V increases if V 0.4t b . Sengupta and Dupuis 28 and Zanchi and Bjelis 29 obtained the similar results.
FIG. 3 :
3qx vs Ky (Eq. 11) for qy = 0 and qy = π.
FIG. 4 :
4qx vs Ky (Eq. (11)) for some values of qy. There are two minimums (q min(±) x
FIG. 5: q min(−) x (qy) (dashed lines in qx < q1x), q min(+) x
nesting vectors are characterized into four types according to the pairs of the left and right parts of the Fermi lines, i.e. (+, −), (−, +), (+, +), and (−, −) as shown in Fig. 2. The left and right parts of the Fermi lines are given by
FIG. 6 :
6'/t b =0.1, t 3 /t b =0.02, t 4 /t b =0.002 q y /π=0,0.02,0.04,...χ0(Q) at T = 0 (eq. 22) as a function of qx. We take t ′ b /t b = 0 and, t3 = t4 = 0 (the upper figure), t3/t b = 0.02, t4 = 0 (middle figure), and t3/t b = 0.02, t4/t b = 0.002 (lower figure). As obtained by Zanchi and Montambaux 15 , t3 reduces the peak height near q4 and t4 lifts the degeneracy at q1 and q3.
FIG. 7 :
7χ0(Q) at T = 0 (eq. 22) as a function of qx for (qx, qy) on the curves (q min(+) x (qy), qy) (filled green diamonds) and (q max x (qy), qy) (open blue circles) in Fig. 5. For qx > q4x we use Eq. (18), although curves (q max x , qy) terminate at q = q4.
FIG. 8 :
8χ0(Q) at T = 0 (eq. 22) as a function of qy for (qx, qy) on the curves (q min(+) x (qy), qy) (filled green diamonds) and (q max x (qy), qy) (open blue circles) in Fig. 5. We take t ′ b /t b = 0.1, t3 = t4 = 0. as the four equations (++, +−, −+, and −−),
FIG. 9 :
93D plot of χ0(Q) at T = 0 (eq. 22) as a function of qx and qy. We take t ′ b /t b = 0.1, t3 = t4 = 0.
FIG. 11 :FIG. 12 :
1112FIG. 10: qx vs. Ky (Eq. 27) for qy = 0 and V /t b = 0.1. of q (−−) x(at K y = ±π/2), the minimum of q qx as a function of Ky for qy/π qx as a function of Ky for qy/π = 0.3 K y = ±π/2), the minimum of q (++) x
FIGFIG. 14 :
14as a function of K y as shown by the filled green circles and the filled squares in Figs. 10, 11, 12 and 13. We define q max x (q y ) and q min(+) x (q y ) by the maximums (filled green circles) and minimums (filled squares) of q (+−) x for each q y , respectively. We also define q min(−) x(q y ) by the value of q (−−) x at K y = ±π/2 (open black squares) and q min(++) x (q y ) by the value of . 13: qx as a function of Ky for qy/π = 0.0 and V /t b = 0The same as Fig. 5 for V /t b = 0.1. q1 is given as the minimum of q (+−) x as a function of Ky for each qy. q2 is given as q (++) x at Ky = π/2 for each qy. q3 is given as the maximum of q (+−) x as a function of Ky for each qy. q0 is given as q (−−) xat Ky = π/2 for each qy.
y ) in the plane of q x and q y for V /t b = 0
(V /t b = 0.2) and Fig. 19 (V /t b = 0.4) for the parameters t ′ b /t b = 0.1 and some values of t 3 and t 4 . The contour plots of the χ 0 (Q 0 + q) in the k x − k y plane are shown in Fig. 20 (t 3 = t 4 = 0) and Fig. 21
FIG. 16 :
16The same as Fig. 14 for V /t b = 0.4.
FIG. 17 :
17The same as Fig. 14 for V /t b = 0.5.
FIG. 18 :FIG. 20 :FIG. 21 :FIG. 22 :
18202122'/t b =0.1, t 3 /t b =0.02, t 4 /t b =0.002 q y /π=0,0.02,0.04,...χ0(Q) at T = 0 as a function of qx. The parameters are the same as in Fig. 6 but V /t b = 0.2. '/t b =0.1, t 3 /t b =0.02, t 4 /t b =0.002 q y /π=0,0.02,0.04,...The contour plot of χ0(Q0 + q). The filled circles show the location of the maximum (best nesting vector). The diamonds, the open circles, the triangles, and the squares are q1, q2, q3, and q4, respectively. We take t ′ b /t b = 0.1, t3 = t4 = 0. The same as Fig. 20 with t ′ b /t b = 0.1, t3/t b = 0.02 and t4/t b = 0Open diamond, open triangle and open squares are q1, q3 and q4 for V = 0, respectively. Open circles and closed circles are q2 and the locations of the maximum of χ0(Q) (best nesting vector), respectively, for V /t b = 0, 0.1, 0.2, 0.3 and 0.4. We take t ′ b /t b = 0.1, t3 = t4 = 0.
For a review, see T. Ishiguro, K. Yamaji, and G. Saito, Organic Superconductors, 2nd ed., (Springer-Verlag, Berlin,
AcknowledgmentsThis work is partly supported by a Grant-in-Aid for the Promotion of Science and Scientific Research on Priority Areas (Grant No. 18028021) from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
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[
"Hierarchical hydropathic evolution of influenza glycoproteins (N2, H3, A/H3N2) under relentless vaccination pressure",
"Hierarchical hydropathic evolution of influenza glycoproteins (N2, H3, A/H3N2) under relentless vaccination pressure"
] |
[
"J C Phillips [email protected] \nDept. of Physics and Astronomy\nRutgers University\n08854PiscatawayN. J\n"
] |
[
"Dept. of Physics and Astronomy\nRutgers University\n08854PiscatawayN. J"
] |
[] |
Hemagglutinin (HA) and neuraminidase (NA) are highly variable envelope glycoproteins. Here hydropathic analysis, previously applied to quantify common flu (H1N1) evolution (1934-), is applied to the evolution of less common but more virulent (avian derived) H3N2 (1968-), beginning with N2. Whereas N1 exhibited opposing migration and vaccination pressures, the dominant N2 trend is due to vaccination, with only secondary migration interactions. Separation and evaluation of these effects is made possible by the use of two distinct hydropathic scales representing first-order and second-order thermodynamic interactions. The evolutions of H1 and H3 are more complex, with larger competing migration and vaccination effects. The linkages of H3 and N2 evolutionary trends are examined on two modular length scales, medium (glycosidic) and large (corresponding to sialic acid interactions). The hierarchical hydropathic results complement and greatly extend advanced phylogenetic results obtained from similarity studies. They exhibit simple quantitative trends that can be 2 transferred to engineer oncolytic properties of other viral proteins to treat recalcitrant cancers.
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"https://arxiv.org/pdf/1303.4383v1.pdf"
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1303.4383
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f14109a36967de1d934250d40df2f340b93a89bb
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Hierarchical hydropathic evolution of influenza glycoproteins (N2, H3, A/H3N2) under relentless vaccination pressure
J C Phillips [email protected]
Dept. of Physics and Astronomy
Rutgers University
08854PiscatawayN. J
Hierarchical hydropathic evolution of influenza glycoproteins (N2, H3, A/H3N2) under relentless vaccination pressure
1
Hemagglutinin (HA) and neuraminidase (NA) are highly variable envelope glycoproteins. Here hydropathic analysis, previously applied to quantify common flu (H1N1) evolution (1934-), is applied to the evolution of less common but more virulent (avian derived) H3N2 (1968-), beginning with N2. Whereas N1 exhibited opposing migration and vaccination pressures, the dominant N2 trend is due to vaccination, with only secondary migration interactions. Separation and evaluation of these effects is made possible by the use of two distinct hydropathic scales representing first-order and second-order thermodynamic interactions. The evolutions of H1 and H3 are more complex, with larger competing migration and vaccination effects. The linkages of H3 and N2 evolutionary trends are examined on two modular length scales, medium (glycosidic) and large (corresponding to sialic acid interactions). The hierarchical hydropathic results complement and greatly extend advanced phylogenetic results obtained from similarity studies. They exhibit simple quantitative trends that can be 2 transferred to engineer oncolytic properties of other viral proteins to treat recalcitrant cancers.
Introduction
Earlier papers on the panoramic smoothing evolution of H1N1 glycoproteins showed that several dominant factors could be quantified by hydropathic analysis of the interactions between amino acid (aa) sequences and the water film that envelopes or packages them [1][2][3]. Prior to 1950, this evolution exhibited only small fluctuations due to direct antibody evasion. However, after the introduction of widespread vaccinations by the US Army in 1944, evolution after 1950 was dominated by vaccination evasion, which progressively smoothes the water-protein interface, and opposing migration pressures, which compact the protein and regressively increase the roughness of the water-protein interface. These opposing interactions accumulate and lead to Darwinian punctuations. Here we extend the analysis to H3N2, in order to develop further insights into smoothing mechanisms, with a view to engineering viral oncolytics.
There are several extensive studies of the genetic evolution of H3N2 based on counting individual aa mutations, separated according to whether they are synonomous or not (described as positive in BLAST) [4]. Similarities correlate geographically and temporally with local clustering effects measured indirectly (antigenically with antibody titres) [5]. On a panoramic scale these clustering effects are small compared to the fundamental genetic hydropathic trends of H1N1 [1][2][3] and H3N2 (discussed here) There are at least two much larger fundamental hydropathic effects. The first depends on the length scales of native glycosidic or sialic acid interactions, while the second depends on whether the mutations are thermodynamically nearly first order (treated by the KD scale) or 3 second order (MZ scale). Here we analyze these two effects quantitatively, and show that they display a number of mechanisms that have previously been described for H3N2 by schematic models [6,7]. These models exhibit canalization, convergence, modular and punctuation effects that have larger implications for RNA transcriptive mechanisms [8,9]. In contrast to these general qualitative models, the present results include all these effects quantitatively, and do not rely on adjustable parameters. Hydropathic analysis can be used to engineer similar viruses with enhanced oncolytic properties.
Methods
The methods used here are almost the same as were used in [1] for NA, but with a few important changes. The evolution of NA was monitored by calculating the roughnesses of R KD (W max ) and R MZ (W max ) with W max = 17 and 47 of the water packaging film with two scales, KD and MZ. The two length scales W max = 17 and 47 correspond respectively to membrane thickness and glycosidic spacing. These N1 roughnesses (especially with the MZ scale) showed well-defined plateaus connected by punctuated decreases (vaccination program successes) or increases (migration), culminating in vaccination-driven N1 convergence to a nearly common strain after the 2007 swine flu program. Here punctuation of N2 is not visible on our larger scale (although it might be visible in an elaborate statistical analysis that combines BLAST similarities [4,5] with our hierarchies). Instead we see canalization with the first-order KD scale, and migration fluctuations with the second-order MZ scale. These differences between H1N1 and H3N2 can be explained by the different origins of H1N1 (primarily human) and H3N2 (frequent intermixing with avian) strains [6]. 4 To obtain an unbiased survey with fewer strains, data were obtained from three sites, Memphis, Hong Kong, and the Netherlands, with significantly different climates. These have led to large geographically dependent and sparsely measured antigenic differences, as measured on ferret post-infection inhibition titres [6]. These antigenic differences [5] Table I, normalized to their average values.
Results (N2)
The largest effects on roughness occur for a large change in ψ(i) that occurs near a peak or valley in ψ(i,W). An example is shown in Fig. 5 for N336Y of Mem and HK 1972.
At first one could suppose that this MZ-KD collapse is accidental, but as it occurs for the entire panorama 1968-and for three different sites with different climates, this is most unlikely.
A more plausible explanation is that the KD and MZ scales do in fact describe first-order and second-order thermodynamic differences, as implied by their origins [1][2][3]. Common flu H1N1 has evolved over perhaps 500 or more years, and has approached a critical point, first through antibody evasion, and more recently through vaccination pressures (since 1945). The N1 5 panorama exhibits large (but still second-order) migration perturbations in its decreasing roughness. By contrast, here N2 roughness decreases steadily (1968-) using the KD scale. The N2 subtype has emerged by reassorting human and avian strains [6], and an initially rapid roughness descent would be better described by the first-order KD scale than by the second-order MZ scale that worked well for N1 and H1N1. The internal consistency of this picture provides support for distinguishing hydropathicity scales in thermodynamic terms.
The collapse of broadened MZ panoramic evolution curves onto channeled KD curves enables us to examine the latter in Fig. 4 for distinctive features, which are readily obvious. Widespread vaccination for H3N2 began in 1968 with a Hong Kong strain [6], which was replaced by an English strain in 1972. Substantial decreases in KD roughness appear for all three sites between 1972 and 1975, with the effects being much larger for W = 47 (Fig. 4) than for W = 17 (Fig. 3).
From 1975 to present, R KD (17) decreases nearly linearly by more than 20%, echoing a similar decrease for N1 of H1N1. Similarly R KD (47) decreases, but much more rapidly and steeply by more than 35% up to 1995, when it abruptly levels off. This larger decrease and leveling after 1995 suggests that vaccination pressure on H3N2 affects the glycosidic length scale W = 47 well before it affects the membrane length scale W = 17. This larger effect on W = 47 is not a surprising result, as we already saw [2] that H1 and N1 are strongly coupled, and that the dominant interactions in H1 occur on the even larger length scale of W =111, appropriate to sialic acid interactions. These longest range interactions are optimized first, while the weaker medium-range glycosidic interactions are apparently still being optimized at the W = 17 level in 2009, but could saturate soon (5-10 years). 6 The overall picture for N1 in H1N1 and N2 in H3N2 is that both glycoproteins have evolved under vaccination pressure to nearly minimal roughness. As discussed before, the panoramic evolution of N1 roughness has occurred in tandem with infection severity, as reflected most recently in the numbers of strains sequenced during the successive swine flu outbreaks (New
Results (H3)
The evolution of H1 roughness is qualitatively different from N1 evolution, but the two are linked [2,4]. The simple reductions in roughness of N1 and N2 are replaced by large block shifts (called proteinquakes) in the hydropathic profiles of H1 at the times when N1 is punctuated. H1 roughness decreases with time. With large fluctuations for H3N2, we find that for the HA1 chain H3 R(W) simply oscillates, with two large peaks in 1976 and around 2000 for W = 111 ( Fig. 6(a)). We can resolve the broad 2000 peak by reducing W to 47, as shown in Fig. 6(b), which shows that the peak occurs in 1999, and is sharper with the MZ scale, with an abrupt increase between 1995 and 1999.
We can identify the cause of these oscillations by studying the MZ 47 hydroprofiles of H3.
For H1 we already saw in Fig. 5 of [2] evidence for a proteinquake in 1976 associated with the sialic acid binding site. Here H3 has been channeled, which leads us to suspect that H3 could be hydrorigid, and exhibit much larger proteinquakes. How did vaccination pressure return Memphis 1999 R(47and 111) to the background level by 2008 (Fig. 6)? The 2008 high-resolution W = 47 hydroprofile is shown in Fig. 8. There is a large hydrophobic contraction between 130 and 230, which corresponds very well to the sialic acid binding region of H1, previously identified in Fig. 2 of [2], and similarly labeled here.
Note that this region was previously identified in H1 by connecting it to structural data, but here the H3 identification is based solely on similarity of H3 proteinquakes to H1
proteinquakes.
There is a striking difference between the Memphis 1999 R(47and 111) profiles (Figs. 7,8 and 9), corresponding to glycosidic and sialic acid interactions, respectively. The deep double R(47) hydrofault from 140 to 170 in 1999 narrows and shifts to 170 -190 in 1999. It would be worthwhile to study linear epitopes on H3 by peptide scanning using libraries of overlapping peptides against convalescent sera from H3N2 patients similar to those done for H1N1 [12] and H5N1 [13].
What are the individual amino acid mutations responsible for this hydrophobic 1999-2008
shift? There are four hydrophilic to hydroneutral mutations, K166I, K171N, K189Q, S202G, and one hydrophilic to hydrophobic, S209F. The shift is cut off at by W237R, hydrophobic to hydrophilic, which reduces the largest hydrophobic peak near 260. [14].
Given the overall consistency of the hydropathic approach, it appears that short-range ionic effects may be only a weaker secondary reaction to dominant long-range hydropathic elastic interactions, especially in determining sialic acid interactions.
Discussion
One could ask which glycoprotein, H3 or N2, plays a larger role in evolution of virulence of There is an interesting glass network analogy for the hydrophobic proteinquake of H3 between 1999 and 2008. Hydrophobic interfaces often dimerize [15], and one can suppose that a necessary plastic condition for such dimerization is that the molecular network interfaces involved are both nearly rigid and yet stress-free. Such networks have been widely observed in glass alloys composed of nearly isovolume chalcogenides, which are good inorganic analogues of protein amino acids (also nearly isovolume). A striking feature of these elastically optimized inorganic alloys is that their glass transitions are nearly reversible [16], a necessary condition for protein functionality, here involving dimerization.
Emergence of new strains can be detected by similarity tools using dynamical clustering [17].
If such a new strain should show a large uptick in N2 R(47), then its hydropathic clustering properties should be studied with care, as it might become dangerous.
Conclusions
Comparison of the present H3N2 results with those for H1N1 shows simpler smoothing for N2 here than for N1 in H1N1, confirming the widely held view of close H-N linking [4]. More interesting is the connection between canalization [7] and larger proteinquakes in H3. These are again dominated by sialic acid interactions, which lead to larger and more dramatic H3 proteinquakes than those previously identified for H1. The combined analysis of roughening evolution and hydroprofiles is most informative. It does not require new structural data, and it is extremely accurate in portraying the evolution of competing elastic and charge interactions.
may be reflected in the large scatters seen in the R MZ (W max ) with W max = 17 and 47 data (MZ scale) shown in Figs. 1 and 2. Unexpectedly this scatter disappears in R KD (W max ) with W max = 17 and 47, as shown in Figs. 3 and 4 (KD scale). One of the largest scatters occurs between R MZ (17 or 47) for Mem and HK 1972. Inspection of the corresponding hydroprofiles shows that this difference is caused by a single philic -phobic mutation, N336Y. Another large R MZ (17 or 47) difference occurs between Mem and Neth 1999, which is traced to two mutations, L370S (phobic-philic) and S414G (philic-neutral). The collapse of R MZ (17,47) onto R KD (17,47) occurs because of differences between the MZ and KD values [10] for ψ(Y) in the 1972 example, and ψ(S) in the 1999 example. For the reader's convenience, the MZ and KD scales are shown in
York 2003 ,
2003Berlin 2005, Houston 2007). Current strains are more convergent, as expected from SOC, and are historically the mildest. So long as widespread vaccination programs are in place, this stabile mildness should persist in relatively uncrowded areas like America and Europe.
Fig. 7 shows the HA1 chain profiles (W = 47) between the 1976 and 1999 peaks: there are very large H3 proteinquakes. The most 7 striking shift occurs between 1985 and 1999, as a deep hydrophilic valley or hydrofault appears between 140 and 160 near the center of the HA1 chain. The 1985-1999, 140-160 hydrofault occurs because of multiple softening mutations [I137N, G140S, Y153S, and VN(160,161)NK]). The strengthening of the largest hydrophobic peak near 260 is caused by a single mutation, S263C.
hydrophobic compaction. Most of the sequences very similar to Memphis 1999 are earlier, and the nearest later ones are 2002 New York. Apart from hydrophilic to hydroneutral mutations, another way of suppressing charge interactions is to replace charged NK(160,161) in 1999 Memphis by neutral DK(160,161) in 2002 New York, where the two charges neutralize each other. If one restricts one's attention to epitopic subsets defined by Euclidean contacts at active sites and counts only total numbers of charged amino acids, including those in neutralized pairs, replacing NK by DK increases the number of charged mutations
H3N2. At present the answer would appear to be N2. The roughening of H3 in 1999 Memphis because of the appearance of the 140-170 R(47) hydrofault does not seem to have alarmed virologists, as there were few H3 sequences in Memphis in the period 1995-2004, and most of these appeared in a single year (2003), probably incidentally. By 1995 N2 had reached its low level of roughness, and this may explain H3N2 stable and low virulence after 1995. By 1999 the rotating WHO recommended H3N2 vaccine composition was Moscow/10 (AFM72208), very similar to Hong Kong 1999 and similar to New York 1999-1997. The hydroprofile of this strain also resembles that of Memphis 2003.
12 Fig. 1 .
121Evolution of R MZ (17) at three sites with different climates and crowding conditions.
Fig. 2 .
2Evolution of R MZ (47) at three sites with different climates and crowding conditions. The agreement is slightly improved.
Fig. 3 .
3Evolution of R KD (17) at three sites with different climates and crowding conditions. The use of the first-order KD scale, instead of the second-order MZ scale, improves concordance.
Fig. 4 .
4New evolutionary features (see text) are brought out here.
Fig. 5 .
5The largest effects on roughness occur for a large change in ψ(i) that occurs near a peak or valley in ψ(I,W). An example is N336Y of Mem and HK 1972.
Fig. 6 .
6(a) Evolution of R KD (111) and R MZ (111) for HA1 chain of H3 with W = 111, (b) Same at higher resolution with W = 47. Note how the 2001 doublet of (a) has been resolved.
Fig. 7 .Fig. 9 .
79Evolution of chain HA1 profiles of H3 from 1976 (ABD16740) to 1985 (ABD61777) to 1999 (AAZ43405) exhibits a deep double hydrofault from 140 to 170 in 1999. Compared to R(47) in Figs. 7 and 8, here for R(111) the deep valley has narrowed and shifted to 170-190.
Table I. Hydropathicities according to the KD and MZ scales. To facilitate comparisons, the values have been normalized to their respective averages (<>). Note that Glycine is close to average on both scales, while Thyrosine (Y) is treated quite differently.MZ 99
MZ 85
MZ 08
H3
W=111
Memphis
19
KD/<>
MZ/<>
A
1.29
1.01
C
1.38
1.58
D
0.62
0.56
E
0.62
0.61
F
1.42
1.40
G
1.01
1.00
H
0.66
0.98
I
1.63
1.43
K
0.57
0.44
L
1.54
1.27
M
1.30
1.42
N
0.62
0.73
P
0.86
0.78
Q
0.62
0.68
R
0.49
0.50
S
0.96
0.64
T
0.97
0.87
V
1.59
1.53
W
0.95
1.12
Y
0.90
1.43
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|
[] |
[
"Mobility Based SIR Model For Pandemics -With Case Study Of COVID-19",
"Mobility Based SIR Model For Pandemics -With Case Study Of COVID-19"
] |
[
"Rahul Goel [email protected] \nInstitute of Computer Science\nInstitute of Computer Science\nUniversity of Tartu\nEstonia\n",
"Rajesh Sharma [email protected] \nUniversity of Tartu\nEstonia\n"
] |
[
"Institute of Computer Science\nInstitute of Computer Science\nUniversity of Tartu\nEstonia",
"University of Tartu\nEstonia"
] |
[] |
In the last decade, humanity has faced many different pandemics such as SARS, H1N1, and presently novel coronavirus . On one side, scientists are focusing on vaccinations, and on the other side, there is a need to propose models that can help us in understanding the spread of these pandemics as it can help governmental and other concerned agencies to be well prepared, especially from pandemics, which spreads faster like COVID-19. The main reason for some epidemic turning into pandemics is the connectivity among different regions of the world, which makes it easier to affect a wider geographical area, often worldwide. In addition, the population distribution and social coherence in the different regions of the world is non-uniform. Thus, once the epidemic enters a region, then the local population distribution plays an important role. Inspired by these ideas, we proposed a mobility-based SIR model for epidemics, which especially takes into account pandemic situations. To the best of our knowledge, this model is first of its kind, which takes into account the population distribution and connectivity of different geographic locations across the globe. In addition to presenting the mathematical proof of our model, we have performed extensive simulations using synthetic data to demonstrate our model's generalizability. To demonstrate the wider scope of our model, we used our model to forecast the COVID-19 cases for Estonia.
|
10.1109/asonam49781.2020.9381457
|
[
"https://arxiv.org/pdf/2004.13015v1.pdf"
] | 216,562,220 |
2004.13015
|
1fb864c8c0904e7d26a68844735180f2b108196d
|
Mobility Based SIR Model For Pandemics -With Case Study Of COVID-19
Rahul Goel [email protected]
Institute of Computer Science
Institute of Computer Science
University of Tartu
Estonia
Rajesh Sharma [email protected]
University of Tartu
Estonia
Mobility Based SIR Model For Pandemics -With Case Study Of COVID-19
Index Terms-Epidemic Based ModelingSIRPandemicsEpidemicsCOVID-19
In the last decade, humanity has faced many different pandemics such as SARS, H1N1, and presently novel coronavirus . On one side, scientists are focusing on vaccinations, and on the other side, there is a need to propose models that can help us in understanding the spread of these pandemics as it can help governmental and other concerned agencies to be well prepared, especially from pandemics, which spreads faster like COVID-19. The main reason for some epidemic turning into pandemics is the connectivity among different regions of the world, which makes it easier to affect a wider geographical area, often worldwide. In addition, the population distribution and social coherence in the different regions of the world is non-uniform. Thus, once the epidemic enters a region, then the local population distribution plays an important role. Inspired by these ideas, we proposed a mobility-based SIR model for epidemics, which especially takes into account pandemic situations. To the best of our knowledge, this model is first of its kind, which takes into account the population distribution and connectivity of different geographic locations across the globe. In addition to presenting the mathematical proof of our model, we have performed extensive simulations using synthetic data to demonstrate our model's generalizability. To demonstrate the wider scope of our model, we used our model to forecast the COVID-19 cases for Estonia.
I. INTRODUCTION
In this modern age, pandemics are not a rare phenomenon. As in the last decade, we have seen several pandemics such as H1N1, SARS, EBOLA, and presently in 2020 humanity is facing its biggest crisis due to COVID-19. The severity of these pandemics can be understood by the death toll claimed by them. According to WHO, the pandemic H1N1/09 virus resulted in 18,036 deaths [1]. On the other hand, the CDC estimate between 151,700 to 575,400 deaths due to the pandemic H1N1/09 virus [2]. Currently, the coronavirus (COVID- 19) pandemic, which started in December 2019 from Wuhan, China has infected 2,404,249 individuals and claimed 165,229 (as of 20 th April 2020) deaths worldwide [3] [4]. Pandemics are different from epidemics in terms of their geographic spread. An epidemic affects many people at the same time. It spreads from person to person and remains local to a specific region. In comparison, when an epidemic engulfs an entire country, continent, or the whole world, it is termed as pandemic.
In the past, various models have been proposed for understanding the epidemic spreads. These models can be broadly classified into two categories, that is agent-based modeling [5] [6] [7] and compartmental models [8] [9] [10]. The agent-based modeling is used for simulating the actions and interactions of autonomous agents as a whole [11]. These agents can be both individual or collective entities such as organizations or groups. In contrast, differential equations are used in compartmental models, where the population is divided into different compartments such as suspected (S), infected (I), and recovered (R) [8]. Several other variants of these models have also been proposed such as SI [12], SIS [13], SIR [8], SIRS [14], etc.
Compartmental models are often being criticized by the agent-based model researchers because they struggle to capture the connectivity between different regions of the globe, and different real-world population characteristic, such as worldwide population distribution [15] [16]. In this study, we proposed a mobility-based model, an extension to the classical SIR based epidemic model, which considers the realworld population distribution across different regions of the world. Most importantly, the model also takes into account the connectivity factor among various regions of the world, which is the key cause in accelerating the process of transforming epidemics into pandemics. We model the regions in a 2dimensional lattice, where each cell represents the mobility parameter (or direct connectivity) from one region to another. Along with presenting the mathematical proof of our model, we have performed extensive simulations on synthetic data and forecast the COVID-19 cases in Estonia 1 by inferring the interaction among individuals through call data records between Estonian counties to demonstrate the model's ability to generalize on different types of data.
The proposed model is composed of (local) transmission rate of the infection β and to cover the mobility aspect, we introduce parameters: 1) 'α' which is a social connectivity parameter that signifies how well individuals are socially linked with each others, and 2) 'c (i,j) ' that represents individuals mobility from some region j to another region i. Thus, the infection can transfer within the region with the transmission rate β and can also be introduced from other regions through global transmission rate which depends upon α, c (i,j) , I j (fraction of infected at region j) and β. With the help of Figure 1, we illustrate our proposed model for better understanding. We applied our model on synthetic network as well as on a real network of Estonia considering the population density and the connectivity among counties, which is created using call data records (CDR) to investigate the following questions:
• How social connectivity parameter 'α' affects the fraction of individuals in different compartments (susceptible, infected and recovered) during a pandemic? We address this question by carefully examining the effect of α while keeping all the other parameters constant (Section IV-B). • What are the outcomes of restricting mobility from the top-X percentile of strongly connected regions? We explore the outcomes of mobility restriction with the model and found that restricting the mobility of top-10 percentile of connected regions can reduce the number of infected individuals between 18% to 27% (Section IV-B). • What is the relationship between social connectivity parameter 'α' and mobility restriction (of top-X percentile) of strongly connected regions? To address this question, we performed numerical simulation on the proposed mean-field equations (Section IV-B, Figure 4). • How efficiently this model can perform in real scenarios?
We answer this question by projecting the expected COVID-19 cases in Estonia using the model and compared the results with the real cases (Section IV-B3). The limitation of classical compartmental epidemiological models is that they do not take into account the importance of reducing social connectivity (or isolation) and the significance of mobility restriction during the spreading of a pandemic such as COVID-19. This limitation is overcome in the proposed model. We found that the reproduction number R 0 for a pandemic depends upon the social connectivity and mobility parameter. We also discovered that during a pandemic, restricting mobility reduces the fraction of individuals in an infected compartment and reducing the social connectivity (or isolation) delays the peak and also reduces the number of infected individuals from the pandemic. We believe that this model can help to adopt a balanced strategy to address a pandemic crisis.
The rest of the paper is organized as follows. Next, we discuss related works with respect to epidemic modeling. We then describe the model preliminaries and derivations in Section III. Section IV presents the evaluation results of our model and we conclude with a discussion of future directions in Section V.
II. RELATED WORK
In this section, we discuss relevant literature with respect to epidemic modeling which involves two different lines of work. First involving agent-based modeling and the second using compartmental based modeling. In the agent-based modeling, authors model epidemics by simulating the actions and interactions of autonomous agents (both individual or collective entities such as organizations or groups) with a view of assessing their effects on the system as a whole [11] by using transportation systems such as road networks [16], airways [15] etc. These models have been used for understanding various epidemics such as smallpox [17], influenza [18], cholera [19], and very recently about COVID-19 [15].
In contrast to agent-based modeling, a differential equation based compartmental models have also been used for understanding epidemics, which is the basis of this work. This line of literature is mainly based on the classical SIR model proposed by Kermack and McKendrick [8] followed by [20] [21]. In [20], the authors considered the host population as a dynamic variable rather than constant, as conventionally assumed, which provides a broader understanding of the population behavior during infectious disease. In their work in [21], authors discuss the idea of the basic reproductive rate, threshold about host densities, and modes of transmission.
Different variations of SIR model have also been proposed to capture various real-world scenarios. For example, introducing a delay in the model to capture the incubation period during the spreading [22]- [25] or the introduction of interventions such as antiviral drugs [26]. In a different work to represent non-linear nature of epidemic spread, a SIR rumor spreading model was proposed in which tie strengths were dependent on nodes' degree [27]. Apart from SIR based models, there exist different flavors of compartmental models, which represent different scenarios such as SIS [13], where individuals do not recover and can become susceptible again. This model has also been studied using varying types of underlying topologies [28].
A set of works has also focused on exhibiting the epidemic spreading by using varying types of underlying network structures. For example, authors in [29], [30] and [31] used a scale-free network and in [32] a small-world evolving networks for evaluating their epidemiological framework. In their work in [33], authors combine a discrete, stochastic SEIR (E stands for exposed) model with a three-scale community network model to demonstrate that the different regional trends may be explained by different community mixing rates. A detailed study with respect to various epidemic models on varying topologies has been done in [34].
In another line of work, authors proposed models to understand epidemics based on the speed of growth. For example, in [35], authors applied their generalized-growth model to characterize the ascending phase of an outbreak on 20 different epidemics. Their findings revealed that sub-exponential growth is a common phenomenon, especially for pathogens that are not airborne. In another work [36], researchers explain the rapid spread of H1N1 in 2009 around the world by using a flexible Bayesian, space-time, Susceptible-Infected-Recovered (SIR) modeling approach. [37] developed a simulation model of a pandemic (H1N1) 2009 outbreak in a structured population using demographic data from a medium-sized city in Ontario and epidemiologic influenza pandemic data. In comparison to previous works, the proposed model introduces mobility and social connectivity parameters, the key characteristics for turning epidemics into pandemics.
III. MODEL PRELIMINARIES AND DERIVATIONS
In this section, we first explain the classical SIR model and then discuss its limitations with respect to the absence of mobility and social connectivity parameters. Next, we describe our proposed model to understand the spreading of an infection during a pandemic.
In 1926, Kermack and McKendrick [8] proposed the clas-sical SIR model as follows:
ds(t) dt = −βs(t)i(t) (1) di(t) dt = βs(t)i(t) − µi(t) (2) dr(t) dt = µi(t)(3)
where, s(t), i(t), r(t) is the fraction of susceptible, infected and recovered population at time t. However, the classical SIR epidemic model does not consider the heterogeneity and topology of the real-world network. To overcome this limitation, we introduce the mobility and social connectivity parameters in our proposed model. Let, 'l' represents the total number of locations and 'c' denotes the connection (or individuals' mobility) between locations. The propagation of infection at each location is explained as: Each healthy individual can get the infection either from an infected individual located in the same location (local transmission) or from an individual visiting from other connected locations (global transmission). The local transmission rate of infection is represented by β and the recovery rate as µ and, β and µ ∈ [0,1]. In the next section, we discuss the local transmission of infection and then the global transmission is discussed in detail in the Section III-B.
B. Global Transmission
Let, j (j ⊂ l) represents a set of locations, which are connected to location i. Therefore, j N j is the maximum possible number of individuals connected to location i, from all the locations j. The parameter c i,j reflects the mobility of individuals from locations j to location i. Global transmission depends upon this mobility parameter of individuals from one location to another. Similar to local transmission, I j is the number of individuals in infected compartment in all the locations j. Hence, total mobility of infected individuals from all the other connected locations to location i is j c i,j Ij Nj . Considering the above description, the chances of transmission of infection from all the connected locations to location i is j c i,j Ij Nj β. This transmission further depends upon the social connectivity (α) of individuals at location i. Therefore, the proportion of healthy individuals at location i which can get infected from infected individuals from location j is α j ci,j I j N j β Ni+ j ci,j . Thus, the mean-field equations for the dynamics of the pandemic, based on the above discussed interactions:
dS i (t) dt = − βS i (t)I i (t) N i (t) − αS i (t) j c i,j Ij (t) Nj (t) β N i (t) + j c i,j (4) dI i (t) dt = βS i (t)I i (t) N i (t) + αS i (t) j c i,j Ij (t) Nj (t) β N i (t) + j c i,j − µI i (t) N i (t) (5) dR i (t) dt = µI i (t) N i (t)(6)
Where, Eq. 4 describes the rate of change of susceptible individuals at location i, and Eq. 5 refers to rate of change of infected individuals, and Eq. 6 explains the rate of change of recovered individuals at location i. Please refer Table I for notations and their meaning.
C. Dynamical Behaviour Of The Model
Eq. (4-6) represents nonlinear dynamical system of pandemic spreading, where at any time t,
S i (t) + I i (t) + R i (t) = N i (t)(7)
In order to solve mean-field Eq. (4-6), following assumptions are made (Please note that these assumptions are not considered during our experiments): 1) Initially, the population at all locations is equal to N(t) at time t. 2) Individuals in infected compartments are equal to I(t) at all locations at time t and j I j = |j|.I j = kI j , where, k is the number of locations connected to location i, that is, k = |j|.
3) The mobility of individuals from one location to another
location is a fraction of total population N . Let, the sum of fraction of population mobility from |k| locations is n. Then, the total individuals mobility from set of locations j to i is n * N . Therefore, j c i,j = nN . By considering the above assumptions, Eq. 4 and 6 can be written as
dS i (t) dt = − βS i (t)I(t) N (t) − αS i (t)nN (t)k I(t) N (t) β N (t) + nN (t) (8) dR i (t) dt = µI(t) N (t)(9)
From Eq. 8 and 9
dS i (t) dR i (t) = − βS i (t) µ − αS i (t)nkβ µ(1 + n) (10) = − βS i (t) µ 1 + αnk 1 + n (11) = − βS i (t) µ 1 + (1 + αk)n 1 + n(12)
For simplicity, Eq. 12 can be written as:
dS(t) dR(t) = − βS(t) µ 1 + (1 + αk)n 1 + n(13)
Eq. 13 can be rewritten as
S = S 0 e − β µ R[ 1+(1+αk)n 1+n ] (14) dR dt = µ(N − R − S 0 e − β µ R[ 1+(1+αk)n 1+n ] )(15)
Solving the Eq. 15, we get
t = 1 µ R 0 dR N − R − S 0 e − β µ R[ 1+(1+αk)n 1+n ](16)
As pandemic arrives at steady state when t −→ ∞ hence dR dt = 0 and R ∞ = constant
N − R ∞ = S 0 e − β µ R∞[ 1+(1+αk)n 1+n ](17)
Let initial conditions are R(0) = 0, I(0) = I and S(0) = N − I ≈ N . Therefore, Eq. 17 can be written as
R ∞ = N − N e − β µ R∞[ 1+(1+αk)n 1+n ](18)
Normalizing the Eq. 18
r ∞ = 1 − 1e −R0r∞(19)
Therefore, the reproduction number R 0 is
R 0 = β µ 1 + (1 + αk)n 1 + n(20)
In case there is no social connectivity to other locations (α = 0 or k = 0 or n = 0) then the mobility SIR model will become the standard SIR model and the reproduction number is R 0 = β µ . Therefore, the reproduction number is directly proportional to social connectivity parameter α, number of connected locations k and depends upon individuals' mobility during a pandemic.
IV. EVALUATION
In this section, we first explain our experimental setup and next, we discuss the results of our simulation conducted using the proposed model on synthetic networks. In addition, we also applied our model for predicting the real-time Estonian COVID-19 cases.
A. Experimental Setup
For the analysis, we created an aggregated flow matrix of individuals per day from Origin to Destination (OD), which follows random distribution. Furthermore, three different techniques are considered for selecting the seed infection location:
1) Pandemics origin from a random location: In this, a random location is selected as seed infection location and a small fraction of individuals were infected at that location. 2) Pandemics origin from a weakly connected location:
Here, seed location is selected strategically, which is weakly connected to other locations. That implies least mobility of individuals from this location to other locations. 3) Pandemics origin from a strongly connected location: In this also, seed location is selected strategically, which is strongly connected to other locations. This signifies that, highest mobility of individuals from this location to other locations. Our simulation is oriented towards addressing the following questions:
• How social connectivity parameter 'α' affects the fraction of individuals in different compartments (susceptible, infected and recovered) during a pandemic? • What are the outcomes of restricting the mobility (for top-X percentile) of strongly connected locations? • What is the relationship between social connectivity parameter 'α' and the mobility restriction (top-X percentile of strongly connected locations? • How efficiently this model can perform in real scenarios?
We answer this question by projecting the expected COVID-19 cases in Estonia.
B. Results
We perform various simulation experiments to explain the proposed model on OD network by using previously discussed techniques for selecting the seed infection location. It is to be noted that, if α = 0, then the model will behave as a standard SIR model. Also, if the mobility is reduced to 100 percentile (that is no mobility allowed) from strongly connected locations, then also model will act as a standard SIR model. the α decreases, and it also takes longer to reach its peak. This indicates that there is a positive impact of lock-down in controlling a pandemic. The effect of restricting the mobility from the top-X percentile of highly connected locations with other locations is shown in Fig. 3. Fig. 3a to 3d displays the pandemic dynamics with different percentile of mobility restrictions of highly connected locations starting with 0% to 30% (keeping α = 0.5). We observe that in case of pandemic, restricting the mobility from the top-10 percentile of highly connected locations can reduce the number of individuals who can get infected to 27%. Therefore, quarantine plays a vital role during pandemics.
In order to understand the relationship between α and mobility restriction from strongly connected locations, we performed the numerical simulation of the proposed mean-field equations (see Figure 4). We can infer that social connectivity parameter 'α' and mobility both plays an important role during pandemics. Therefore, it is advisable to follow a dual strategy approach during a pandemic outbreak as controlling mobility reduces the fraction of infected individuals and α delays the peak. Furthermore, we analysed the number of days required to reach the point where highest fraction of individuals get infected (see Figure 5). This indicates that mobility restrictions and minimal social contact will postpone the pandemic's peak and will give sufficient time for the preparations especially for the health sector.
2) Pandemics Origin From a Weakly and Strongly Connected Locations: Fig. 6 displays the influence of the social communication parameter 'α' while keeping the other parameters constant for both weakly and strongly connected locations. Fig. 6a to 6l shows the pandemic dynamics with different values of α starting with α = 1 to α = 0.1.
It can be noted that when a pandemic originates from a weakly connected location, it takes longer to reach its peak compared to when it starts from a strongly connected location. This shows that location of origin also plays an important role during pandemic. Similar to random location, reducing mobility from the highly connected locations by 10 percentile can reduce the number of infected individuals between 18% to 27% for weakly and strongly connected locations.
3) Case Study Of Estonia:
To demonstrate the usability of the model, we applied it on a real-time data of Estonia's to fit COVID-19 cases. Fig. 7 shows the actual number of cases and the cases forecast by the model using different values for α and mobility percentile. For example, when alpha = 0.95, this indicates that social connectivity of individuals are reduced by 5% and also top-5% of strongly connected locations are restricted from mobility. Similarly, α = 0.7, implies that social connectivity of individuals are reduced by 30% and also the top-30% of strongly connected locations have introduced restricted mobility.
For simulation, we created the OD matrix between counties of Estonia using call data records [38]. Furthermore, these call interactions are converted into population mobility between counties using Estonian population data [39]. For the local transmission of the virus (within the county), we consider the reproduction number R 0 = 2.5 [40]. Cases reported until 11 th March, 2020 are considered as initial condition for the model. The reason behind selecting 11 th March, 2020 as initial condition is that, till this date no local transmission of the virus was reported 2 . Till the day of initial condition, the Estonian Health Board confirmed 13 cases in Harju and two cases in Tartumaa and Saaremaa each 3 . During the simulation, the number of cases in all other counties are initialized to zero. The infection rate β and recovery rate µ are adjusted according to the value of R 0 for COVID-19. By 10 th April 2020, reported cases in Estonia and forecast cases using the model are shown in Fig. 7. It can be noticed that the model predicted much higher cases of COVID-19 if no restrictions are introduced (α = 1). However, as the restrictions were introduced by the Government 4 the number of cases got damped (Actual). Thus, the applicability of this model is to forecast a range of predicted number of cases which can help the governmental and health agencies to understand the impact and introduce proportional interventions to restrict the spread of the epidemic.
V. CONCLUSION
Classical compartmental epidemic models are unable to describe the spreading pattern of pandemics such as COVID-19 as they do not take into account the effect of social connectivity and mobility in spreading of the virus. Our proposed mobility based SIR model shows the significance of social connectivity and mobility during pandemics by taking into consideration the local and the global transmission rate of the infection. We have simulated the proposed model considering three different origins of the infection, namely random location, weakly connected location and strongly connected location. Our simulation shows that limiting the social connectivity reduces and delays the peak of the infected compartment.
Our analysis also shows that restricting the mobility from the top-10 percentile of connected locations can reduce the number of infected individuals between 18% to 27%. From the mathematical proof for our proposed model, we obtained that the reproduction number R 0 directly depends upon social connectivity of individuals, number of connected locations and individuals mobility between locations which is in line with our simulations' results. This indicates that introducing isolation and quarantine is effective in fighting a pandemic crisis. Using the proposed model, we also simulated the real world scenario by considering the COVID-19 cases in Estonia. Simulation reveals that the mobility based SIR model can be helpful to forecast the expected number of cases after some proportion of isolation and quarantine is introduced in the society.
We plan to include various future directions for this work such as by simulating the model using additional dynamic networks. Another direction would be to use additional mobility data such as transportation network for better understanding the pandemic behavior. Importantly, we plan to introduce infection delay and recovery delay simultaneously in our future studies.
Fig. 2 :Fig. 3 :
23Pandemic Origin From Random Location: Effect of Social Connectivity Parameter 'α' Pandemic Origin From Random Location: Effect of Quarantine Strongly Connected Locations A. Local Transmission Let, N i be the population at location i, where i ∈ l and the total population is divided into three compartments. The compartments for location i at time t are as follows: 1) S i (t): the number of individuals susceptible or not yet infected. This compartment is referred as susceptible compartment. 2) I i (t): the number of infected individuals which can further spread the disease to the individuals present in the susceptible compartment. This compartment is referred to as infected compartment. 3) R i (t): the number of individuals who have been recovered from infected compartment. This compartment is referred as recovered compartment. Our assumptions regarding the transmission of an individual from one compartment to another compartment are as follows:1) A healthy individual after becoming infected moves from susceptible to the infected compartment. 2) An individual can recovered spontaneously at any time with the recovery rate µ. The recovery of an individual is independent of healthy and infected compartment individuals. 3) Once the individual gets recovered, it will become immune to the disease and thus, will not transmit the infection to individuals in the susceptible compartment. 4) In addition, this model ignores the demography that is birth or death of individuals. Therefore, the population remains constant.
1 )Fig. 4 :Fig. 5 :
145Pandemics Origin From a Random Location: Fig. 2 displays the influence of the social connectivity parameter 'α' while retaining the other parameters constant. Fig. 2a to 2f shows the pandemic dynamics with different values of α starting with α = 1 to α = 0.1. We observe that the peak of the infected compartment decreases significantly, as Pandemic Origin From Random Location: Numerical simulation of relationship between α and quarantine For different combinations of α and quarantine percentile, number of days required to reach peak of infected compartment.
Fig. 6 :
6Pandemic Origin From Weak and Strongly Location: Effect of Social Connectivity Parameter 'α'
TABLE I :
IParameters description Number of susceptible individual at location i at time t I i (t) Number of infected individual at location i at time t R i (t) Number of recovered individual at location i at time t N i (t)Notations
Meaning
l
Number of locations
c
Connection between locations
S i (t)
Population at location at time t i
α
Social connectivity parameter
β
Infection rate
µ
Recovery rate
c i,j
Individuals mobility from location j to i
Fig. 7: COVID-19 Cases In Estonia10-03
11-03
12-03
13-03
14-03
15-03
16-03
17-03
18-03
19-03
20-03
21-03
22-03
23-03
24-03
25-03
26-03
27-03
28-03
29-03
30-03
31-03
01-04
02-04
03-04
04-04
05-04
06-04
07-04
08-04
09-04
10-04
Days
0
1000
2000
3000
4000
5000
6000
Cases
COVID-19 Pandemic Estonia
Actual
Alpha1
Alpha 0.95
Alpha 0.9
Alpha 0.8
Alpha 0.7
https://www.err.ee/1063204/terviseamet-eestis-on-kinnitatud-27koroonajuhtu-ja-kohalik-levik 3 https://www.terviseamet.ee/et/uuskoroonaviirus 4 https://www.valitsus.ee/en/news
ACKNOWLEDGMENT This research was funded by ERDF via the IT AcademyResearch Programme and SoBigData++.
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|
[] |
[
"Simulating Authenticated Broadcast in Networks of Bounded Degree",
"Simulating Authenticated Broadcast in Networks of Bounded Degree"
] |
[
"Shaolin Yu ",
"Jihong Zhu [email protected] ",
"Cn ",
"Jiali Yang [email protected] ",
"Yue Ma ",
"\nTsinghua University\nBeijingChina\n",
"\nTsinghua University\nBeijingChina\n",
"\nTsinghua University\nBeijingChina\n",
"\nTsinghua University\nBeijingChina\n"
] |
[
"Tsinghua University\nBeijingChina",
"Tsinghua University\nBeijingChina",
"Tsinghua University\nBeijingChina",
"Tsinghua University\nBeijingChina"
] |
[] |
The authenticated broadcast is simulated in the bounded-degree networks to provide efficient broadcast primitives for building efficient higher-layer Byzantine protocols. A general abstraction of the relay-based broadcast system is introduced, in which the properties of the relay-based broadcast primitives are generalized. With this, fault-tolerant propagation is proposed as a building block of the broadcast primitives. Meanwhile, complementary systems are proposed in complementing fault-tolerant propagation and localized communication. Analysis shows that efficient fault-tolerant propagation can be built with sufficient initiation areas. Meanwhile, by integrating fault-tolerant propagation and localized communication, efficient broadcast primitives can be built in bounded-degree networks.
|
10.1109/icpads53394.2021.00098
|
[
"https://arxiv.org/pdf/2203.03314v1.pdf"
] | 247,292,188 |
2203.03314
|
9fab552a98a77a4d7202bcca9b23d163bcdc65f5
|
Simulating Authenticated Broadcast in Networks of Bounded Degree
Shaolin Yu
Jihong Zhu [email protected]
Cn
Jiali Yang [email protected]
Yue Ma
Tsinghua University
BeijingChina
Tsinghua University
BeijingChina
Tsinghua University
BeijingChina
Tsinghua University
BeijingChina
Simulating Authenticated Broadcast in Networks of Bounded Degree
Index Terms-authenticated broadcastbounded-degree net- workssecure communicationfault-tolerant propagationcom- plementary systems
The authenticated broadcast is simulated in the bounded-degree networks to provide efficient broadcast primitives for building efficient higher-layer Byzantine protocols. A general abstraction of the relay-based broadcast system is introduced, in which the properties of the relay-based broadcast primitives are generalized. With this, fault-tolerant propagation is proposed as a building block of the broadcast primitives. Meanwhile, complementary systems are proposed in complementing fault-tolerant propagation and localized communication. Analysis shows that efficient fault-tolerant propagation can be built with sufficient initiation areas. Meanwhile, by integrating fault-tolerant propagation and localized communication, efficient broadcast primitives can be built in bounded-degree networks.
a sufficient number of correct nodes in several synchronous communication rounds. However, in building higher-layer Byzantine protocols such as Byzantine agreement (BA) with secure communication as the core primitive, the overall message complexity, computational complexity, and required communication rounds are still high.
In this paper, we explore how to simulate authenticated broadcasts in bounded-degree networks effectively. Firstly, we provide a simple system abstraction of a rich family of relay-based broadcast systems (broadcast systems for short), which also includes the original one provided in [1] upon fully-connected networks. With this abstraction, we identify the general properties of the broadcast systems. Then, to extend the original broadcast system, we investigate the almost everywhere (a.e.) broadcast problem upon bounded-degree networks. To derive efficient a.e. broadcast solutions, we explore the a.e. propagation problem upon strong enough expanders [7]. For efficient sublinear-degree broadcast solutions, we investigate the so-called complementary system which relatively complements the merits of efficient a.e. propagation and localized communication protocols [8,9,10]. With the proposed complementary system, more efficient broadcast systems can be built by integrating localized communication protocols and a.e. propagation. By extending the classical relay-based broadcast system to bounded-degree networks, various Byzantine protocols [1,11,12,13] can be further built upon bounded-degree networks in a simple way.
The rest of this paper is constructed as follows. The related work and the system model are respectively given in Section II and Section III. In Section IV, the general broadcast problem is proposed in arbitrarily connected networks. Then, efficient broadcast solutions are explored in Section V. Lastly, we conclude the paper in Section VI.
II. RELATED WORK
In the literature, [1] provides the first broadcast primitive that simulates the authenticated broadcast in fully connected reliable peer-to-peer networks. This primitive facilitates building much easier-understood BA algorithms like [11] in comparing with some early explorations like [14]. In [15], the broadcast primitive is extended to the bounded-delay model in constructing self-stabilizing BA and other higher-layer realtime protocols [12,13]. However, in real-world communica-tion networks, the reliability of the communication channels and high network connectivity can hardly be both provided. For example, bus-based networks can simply simulate fully connected networks, but the reliability of the shared communication channels is often low. Switch-based networks with traffic shaping can provide independence of communication channels, but it is still hard to support high connectivity in realworld applications. As a result, most large networks are also networks with some bounded node degrees. In these networks, no broadcast primitive can be applied yet.
As an alternative, secure communication is proposed as a core primitive in building higher-layer Byzantine protocols in bounded-degree networks. In [8], almost everywhere Byzantine protocols (or saying incomplete Byzantine protocols [16] in a broader meaning) are first intuitively provided upon some constant-degree networks. However, the original a.e. Byzantine solution [8] tolerates only O(n/ log n) faults even when n is much small. In [9], the a.e. Byzantine solutions can tolerate a linear number of faults with a linear number of poor nodes upon some constant-degree networks. However, some faulttolerant operation with very high computational complexity is demanded. In [10], the computational complexity and the number of the poor nodes are both asymptotically reduced by taking a multi-layer transmission scheme and allowing the node-degree to be polylogarithmic. However, the communication network constructed corresponding to the specially designed multi-layer transmission scheme is rather complex and lacks simplicity. Also, in running the transmission scheme, each node needs not only to transmit the passing messages but run some sub-layer fault-tolerant protocols [17] for the passing messages, which still generates considerable computation, time, and message complexities. In [18], it is shown that there exist more efficient transmission schemes and communication networks with allowing polylogarithmic-degrees, but the construction of such networks is not explicit nor deterministic yet. Also, in considering the overall efficiency, all these a.e. Byzantine solutions aim only at secure communication between the so-called privileged nodes. In constructing upperlayer Byzantine protocols like BA, the time needed to execute the low-layer communication protocol is often a factor of the overall execution time. In this sense, the overall complexity of the secure-communication-based deterministic BA is at least polynomial. In breaking these barriers, only probabilistic solutions are further investigated [19,20].
III. THE SYSTEM MODEL
The synchronous system S consists of n nodes, denoted as V (let V being represented as {1, 2, . . . , n} for convenience), in which up to f = αn nodes can fail arbitrarily (we assume 0 α < 1 and ignore all the trivial rounding problems). All nodes other than the faulty ones are correct. The bidirectional connections between the n nodes are represented as the edgeset E of the undirected graph G = (V, E).
During each basic synchronous round (round for short), each correct node i ∈ V can send one or more messages to all its neighbor nodes (denoted as N i and we assume i ∈ N i for convenience), receive all the valid messages sent from N i during the same round and complete all needed process according to the provided algorithms before the beginning of the next round. In any round, a faulty node i can send arbitrarily inconsistent valid messages, invalid messages, or nothing to any subset of N i . For simplicity and without loss of generality, here we always assume that in each round, each correct node i would distribute a valid message m contains a value v ∈ V that can be correctly extracted in every correct node in N i during the same round. In the basic system settings, we set V = {0, 1}. In this case, when a node sends no message to a correct node in a round, we assume the corresponding value would be extracted as 0. And whenever a correct node i extracts a value v = 0 from any node (might be faulty), i would set v as 1. Thus, when a correct node sends a message m, we can always assume m contains the value 1. We can see this would simplify the basic discussions.
For all the cases, we assume the faulty nodes being under the complete control of a malicious strong adversary who knows everything of the system. Namely, this strong adversary knows the network topology G, the algorithms provided for the correct nodes in S, the initial state of S and all events generated during every execution of S even before these events being generated. With this, the adversary can arbitrarily select any subset T ⊂ V from V with |T | f at the first round and send arbitrary messages from T during each round in every execution of S.
We say S is a broadcast system upon G if and only if (iff) S can simulate the authenticated broadcast [1] upon G. In measuring the Byzantine resilience, S is an α-resilient broadcast system iff the desired authenticated broadcast can be simulated in all correct nodes in the presence of αn Byzantine nodes. For bounded-degree networks, S is an (α, µ)-resilient incomplete broadcast system with 1 < µ < α −1 iff desired authenticated broadcast can be simulated in at least (1−µα)n correct nodes. For convenience, an α-resilient broadcast system is also the (α, µ)-resilient broadcast system with µ = 1. It should be noted that although we allow µ > 1 in the incomplete systems, all the systems discussed in this paper are deterministic, i.e., we consider only the solutions for the worst cases.
With these, the problem is to establish the broadcast system upon the bounded-degree network G.
IV. THE BROADCAST PROBLEM
In this section, we first extend the broadcast problem under a general system structure, with which the broadcast systems upon arbitrarily connected networks can be further explored.
A. A general system structure
In the broadcast system S, each node i ∈ V can be viewed as a local system D (i) running on the discrete-time k ∈ Z. In the context where only one execution of the broadcast system is considered, the discrete-time k can be directly viewed as the round numbers in N = {0} ∪ Z + , i.e., we can interchange the word time and round in this context. With the discrete time k, a signal is defined as a function s : Z → V which gives a unique value s(k) ∈ V for each k ∈ Z. With this, the input signal of D (i) corresponds to the extracted message values u i (k) that come from the broadcaster (also referred to as the General). The output signal of D (i) corresponds to the yielded decision values y i (k), as is shown in Fig. 1a. We say D (i) is correct iff i ∈ U . To support the desired properties of the broadcast system, a correct D (i) also generates a signal x i during each execution. By definition, this signal is intended to indicate other local systems about the current local state x i (k) of node i in round k. In the provided broadcast system upon K n [1], the signal x i and y i are all monotonically increasing. For simplicity, in the general broadcast systems, we still assume the signal x i and y i of the correct D (i) are monotonically increasing.
To process the signals, a correct D (i) is composed of several basic blocks, as is shown in Fig. 1b. Generally, the block D
(i) u transfers the raw General input u i (k) into a valid valuē u i (k) ∈ V in node i at round k. Thenū i (k) is added to the temporally computed local state x i (k) = D (i) x (ˆ x (i) (k − 1)) in current round k to update the current local state of node i as x i (k) = x i (k) +ū i (k). The vectorˆ x (i) (k − 1)
is the previous round estimation of system state in node i. The current local state x i (k) in each node i is exchanged with that of the neighbor nodes N i in block E (i) to collect an estimation of current system state in each node i asˆ x (i) (k). As the network G can be arbitrarily connected, the block E (i) can only output the states of the neighbours of node i in G. Then, the block D (i)
y transfers thisˆ x (i) (k) into the decision value y i (k) = D (i) y (ˆ x (i) (k)). In the correct D (i) , D (i) u , D (i) x and D (i)
y are all stateless and can only output values in V. Meanwhile, a faulty (i.e., not correct) local system D (i ) can generate not only arbitrary local state x i (k) but also make x i (k) being inconsistently measured as x (i) i (k) in the correct nodes i ∈ N i . In this sense, there might be no actually unique system state in any round in considering the multifaced faulty local systems. But equivalently, we can always assume that a unique system state x(k) = x 1 (k), . . . , x n (k) is first generated by all n correct local systems and is then interfered by some noises
υ (i) (k) = υ (i) 1 (k), . . . , υ (i) n (k) with υ (i) j (k) ∈ F = {0, 1, −1} before it entering E (i) at round k.
Similarly, the decision vector y(k) = y 1 (k), . . . , y n (k) can also be assumed being yielded by n correct local systems. In other words, a faulty node i still has the chance to behave as a correct node whenever it likes. For simplicity, we assumē u ≡ u and the signals u, y are extended to negative time (when k < 0) with value 0. In this manner, the general broadcast system upon G is represented as the system D whose structure is shown in Fig. 2.
B. Basic equations
With this, we can represent D (i) for every i ∈ V as
x i (k) = D x (ˆ x (i) (k − 1)) + D u (u i (k)) (1) x (i) (k) = E (i) ( x(k) + υ (i) (k)) (2) y i (k) = D y (ˆ x (i) (k))(3)
whereˆ x (i) (k) is the estimated system state in node i at round k and the functions D x , D y and D u are uniformly defined in all nodes (as only uniform solutions are considered). Here, the operator + is a function from
F × F to V = {0, 1}, with a + b = 0 iff a = −b.
In every execution of the broadcast system, as there can be multi-faced faulty local systems, the noise vectors υ (i) (k) in (5) for different i 1 , i 2 ∈ V can be different with the same k. In such an execution, the noises υ (1) (k), . . . , υ (n) (k) in round k can be represented as an n × n noise matrix F (k). The ith column vector of F (k) represents a state noise vector υ (i) (k) measured in node i. And if node j is correct, the jth row vector in F (k) equals to 0. Otherwise, if node j is faulty, the jth row vector in F (k) can have arbitrary values in F. These arbitrarily valued rows in F (k) are called Byzantine rows. As there can be up to f faulty nodes in round k, up to f Byzantine rows can be scattered on F (k) in all possible combinations. Here, a noise matrix F with up to f Byzantine rows is referred to as an f -Byzantine matrix F [f ] . The set of all f -Byzantine matrices are denoted as Υ [f ] . By this definition,
we have F [f0] ∈ Υ [f ] when f 0 f .
With this, the broadcast system D can be represented as
x(k) = D x (ˆ x(k − 1)) + D u ( u(k)) (4) x(k) = E ( 1 T ⊗ x(k) + F (k)) (5) y(k) = D y (ˆ x(k)) (6) whereˆ x(k) = [ˆ x (1) (k), . . . ,ˆ x (n) (k)], F (k) ∈ Υ [f ] , E = I n +
A (here we slightly abuse the edge-set E as also a matrix for convenience when it is not confusing), A is the adjacency matrix of G, I r is the r × r identity matrix with r 1, ⊗ is the Kronecker product, and the mask operator always computes a matrix with elements m i,j = p i,j q i,j for the samesized matrices, where x i,j denotes the element in the ith row and jth column of a matrix. As a special case, the broadcast system upon K n can be viewed as D with E = J n,n , where J r,s is the r × s all-ones matrix. As the system is distributed, D u , D x , and D y can use only information in each column of the function-input in computing the corresponding value of each element in the function-output vectors.
Denoting the set of all possible executions of D as Λ D , if all noise matrices in an execution
χ ∈ Λ D are in Υ [f ] , χ is referred to as an f -Byzantine execution, denoted as χ ∈ Λ [f ] D . For different χ 1 , χ 2 ∈ Λ [f ]
D , the output signals in D can be different as the input signals and the noises can all be different.
C. Required properties
In the broadcast system upon complete graph K n , if the correct General initiates a broadcast at time k 0 , it would always generate a valid input signal in every
D (i) as u i (k) ≡ δ[k − k 0 ]
where δ is the discrete Dirac function (i.e., with δ[0] = 1 and δ[k] = 0 for all k = 0). In this case, with the correctness property, every correct local system
D (i) should yield the decision signal y i (k) ≡ H[k − k 0 ], where H is the corresponding discrete Heaviside step function (i.e., H[k] ≡ k k =0 δ[k ])
. For convenience we also denote s +k0 as the signal s +k0 (k) ≡ s(k − k 0 ). With the unforgeability property, if the correct General does not initiate a broadcast before k 0 , every correct D (i) cannot yield a decision signal
y i (k) ≡ H[k − k ] with any k < k 0 . Thus, for a correct General, it requires (∃u : ∀i ∈ U : u i = u) → (∀i ∈ U : y i = u)(7)
where s is the integral of the signal s. This is referred to as the 1-Heaviside integral (1-Heaviside for short) property of D. Otherwise, if the General is faulty, with the relay property, in every χ ∈ Λ
[f ] D it requires ∃k 0 : ∀i, j ∈ U : |y i − y j | δ +k(8)
holds, where |s| is naturally defined as the absolute-value signal of the signal s (i.e., |s|(k) ≡ |s(k)| holds, and all other operators on the signals are naturally defined similarly). In other words, the y i signals yielded in all correct D (i) are allowed up to one δ apart (viewed as the δ-distance of the signals). This is referred to as the 1-Dirac differential (1-Dirac for short) property of D (Note that the Dirac differential property here should not be confused with that of the δdifferential consensus in [17] where δ describes a property of the initial values of the consensus). In the broadcast system upon the arbitrarily connected G, there are differences. Firstly, if the correct General initiates a broadcast at k 0 , not all correct local systems can be input with δ +k0 . Instead, only the local systems in some I 0 ⊆ V can be initiated by the correct General. In this situation, we always have u(k 0 ) |I 0 |, where r is the 0-norm of a vector r (i.e., the number of nonzero elements in r). More specifically, defining b a,A = 1 if a ∈ A and b a,A = 0 otherwise, we have u i = b i,I0 δ +k0 when the General is correct and u i = c i b i,I0 δ +k0 with c i ∈ V being arbitrarily valued when the General is not correct. Here, I 0 is called an initiation set upon G. The set of all initiation sets upon G is denoted as I G .
Secondly, in this case, with the extended correctness and unforgeability property, for every χ ∈ Λ
[αn] D it requires
∀I 0 ∈ I G : (∃u : ∀i ∈ V : u i = b i,I0 u) → (∃P ⊆ U : |P | (1 − µα)n ∧ ∀i ∈ P : ∃0 k 1 < k H : y i = u +k1 )(9)
with a bounded k H and sufficiently small µ. This is referred to as the k H -Heaviside property of an (α, µ)-resilient D.
Thirdly, with the extended relay property, for every χ ∈ Λ (10) with a bounded k δ . And this is referred to as the k δ -Dirac property of an (α, µ)-resilient D.
[αn] D it requires ∃P ⊆ U, k 1 · · · k m < k 1 + k δ : |P | (1 − µα)n ∧ ∀i, j ∈ P : |y i − y j | m r=1 δ +kr
With this, D is an (α, µ)-resilient (k H , k δ ) broadcast system upon G iff the (α, µ)-resilient D satisfies the k H -Heaviside and k δ -Dirac properties with all initiation sets in I G .
V. THE BROADCAST SYSTEMS UPON G One significant problem with the complete network is that the node-degrees are linear to n and thus are not bounded. Practically, with the increase of n, it is crucial to maintain the required node-degrees of the network within some affordable scale. In this section, we investigate the d-regular networks with f = αn, where α > 0 should be independent of n, and d should be sublinear to f . We can see that these requirements exclude some natural solutions, such as the ones allowing d = Ω(f ). However, this requirement also naturally comes from the real world. Firstly, with the increasing numbers of unreliable components in distributed systems, the allowed numbers of faulty components should be increased accordingly. As real-world common networking products are always with a restricted number of communication channels, d should remain affordable despite increasing system scales.
A. The fault-tolerant propagation upon G Under the system structure of D, when a correct node i ∈ V 0 for some V 0 ∈ I G is input with the Dirac signal δ, the local state x i should be set as the Heaviside signal according to the monotonic assumption. This can be viewed as the node i being excited by the input δ. From this point on, the execution of D upon G can be intuitively viewed as the propagation of the excitation signals in some excitable media [21] with the topology G. In satisfying the k H -Heaviside property required in (9), the excitation signals of an arbitrary initiation set should be propagated to at least (1 − µα) area of the whole excitable media within at most k H discrete-time in the presence of an arbitrarily distributed α area being arbitrarily faulty. Besides, the propagation should be prevented in such a (1 − µα) area when there is no initial excitation in this area. And to satisfy the k δ -Dirac property required in (10), whenever the output signal y i of an npc node i is triggered at time k 0 (i.e., y i ≡ H +k0 ), the output signals of the correct nodes in at least (1 − µα) area of G should also be triggered before k 0 +k δ , with which all the npc nodes would be triggered. Here, the natural idea is first to design some desired propagation protocol, denoted as P, upon which the desired k H -Heaviside and k δ -Dirac properties can be built after that.
Firstly, to conveniently observe the desired propagation, we can always rearrange the order of the n nodes at any time k to make the state vector x(k) of D being in the form 1, . . . , 1, 0, . . . , 0 , i.e., the excited nodes always with smaller index than the unexcited ones. With this, the n 2 elements in A (the adjacency matrix of G) can be rearranged accordingly. As x is monotonic, we can make the matrix E = I n +A being constant with respect to the time k during any execution of D. So with (5), we havê
x(k) = E X(k) + E F (k)(11)
where the n × n matrix X(k) is in the form [J n,m 0] T with m = x(k) . So X(k) can be viewed as a mask matrix for E that always propagates 1 from the left side to the right side.
To satisfy the k δ -Dirac property, together with (4) and (6),
when x(k 0 ) min{ x | D y ( x + υ) = 1}, we require x(k + 1) = D x (E X(k) + E F (k)) min{ x(k) + 1, (1 − µα)n}(12)
holds for all k k 0 in all executions of D. Further, in uniform solutions where the nodes are all equally weighted in all correct nodes, this condition can be simplified as
∃k 0 : D y (E · (X(k 0 ) + F (k 0 ))) = 0 → ∀k k 0 : D x (E · X(k) + E · F (k)) min{ x(k) + 1, (1 − µα)n}(13)
where · is the common matrix multiplication operator and D x , D y are all Heaviside functions. We can see that this is just the natural extension of the relay strategy (proposed in [1]) for satisfying the k δ -Dirac property upon G.
To satisfy the k H -Heaviside property, we still require that when no correct node is initially excited in some node-set P ⊆ U with |P | (1 − µα)n, then no node in P would be excited during the propagation. It should be noted that even under such simplification, the problem of satisfiability of (13) and the k H -Heaviside property upon the generally connected network G is still nontrivial.
B. A sufficient condition for a.e. -incomplete propagation
For large-scale systems, [9] shows that by explicitly constructing a d-regular Ramanujan network G (i.e., with λ = max{|λ i |} 2 √ d − 1 where {λ i } are the eigenvalues of the adjacency matrix of G in the (−d, d) interval [7]) [22] with a sufficiently large d, O(n) Byzantine faults can be tolerated in reaching a.e. -BA among the n connected ddegree nodes. Concretely, following [7], for any two primes p ≡ q ≡ 1 mod 4 with Legendre symbol ( p q ) = −1, we can construct a (p + 1)-regular bipartite Ramanujan network with n = q(q 2 − 1) nodes. And when ( p q ) = 1, a nonbipartite Ramanujan network with n = q(q 2 − 1)/2 nodes can also be constructed. Upon this, the basic fault-tolerant strategies provided in [9] can be employed in the d-regular non-bipartite networks for any d p + 1. Here for simplicity, we assume d = p + 1 and first aim for providing an a.e.
-broadcast system upon d-regular non-bipartite Ramanujan networks (also referred to as Ramanujan networks).
Firstly, it is known that the non-bipartite Ramanujan networks have the following basic property [22]. Lemma 1 ([22]): If G = (V, E) is a connected non-bipartite Ramanujan network, then for every S ⊆ V with |S| = θn
|e(S) − θ 2 dn/2| √ d − 1θ(1 − θ)n(14)
holds, where e(S) = |E∩(S×S)| is the number of the internal edges of the subgraph of G induced by S. Proof: As G is a connected non-bipartite Ramanujan network [7], d is a simple eigenvalue of A (the adjacency matrix of G) and the absolute-values of all the other n − 1 eigenvalues of A are no more than 2 √ d − 1. So the conclusion holds with Lemma 2.3 of [22]. Now to provide the desired system, in ease of the propagation of the excitation signals in any V 0 ∈ I G , |V 0 | should be as large as possible, and the propagation condition (such as the threshold function D x = H +m ) should be as loose as possible. However, in d-regular networks, it is impractical to require |V 0 | being larger than d + 1 in the absence of the underlying communication protocol. Meanwhile, in considering Byzantine faults, it is also nonsense to require m 1. In this situation, let m = βd+1 in the threshold function D x = H +m , where β ∈ (0, 1) is called the propagation coefficient. Namely, a correct node i ∈ U would be excited at time k iff i receives at least βd excitation signals from the 1-neighbors of i (defined as N i \ i) at some k 0 k. Similarly, let D y = H +(β2d+1) where β 2 ∈ (0, 1) is called the triggering coefficient. Now, we show that by taking a sufficiently large d, the propagation can reach almost everywhere of the excitable media when the initial excited area is relatively small. Meanwhile, when almost everywhere of the excitable media is not excited, it remains to be unexcited.
Firstly, the P function introduced in [9] can be generalized to construct the smallest node-set Z with satisfying T ⊆ Z and {i ∈ V | |N i ∩ Z| β 0 d} ⊆ Z for every T , where β 0 ∈ (0, 1) is called the immunity coefficient. Here we assume G = (V, E) is an n-vertex d-regular non-bipartite Ramanujan network and denote such constructed Z as Z(T, β 0 ) and the set of the npc nodes as P (T, β 0 ) = V \ (Z(T, β 0 ) ∪ T ) just following [9]. Then Lemma 1 of [9] can be generalized with β 0 as follows.
Lemma 2: For any α, β 0 ∈ (0, 1), if
β 0 − 2αβ 0 √ d − 1/d(15)
then there exists µ < 2β 0 /α, such that ∀T ⊂ V : |T | αn → |P (T, β 0 )| > n − µ|T |. Proof: Let |Z(T, β 0 ) ∪ T | = µ 0 |T |. Then for every µ ∈ (1, µ 0 ), as the subgraph of G induced by any S ⊆ Z(T, β 0 )∪T with |S| = µ|T | has at least (µ−1)|T |β 0 d internal edges, with Lemma 1, |β 0 (µ − 1)/µ − αµ/2| < √ d − 1/d holds. Denote g(x) = β 0 (x − 1)/x − αx/2 and suppose µ 0
2β 0 /α. As g( 2β 0 /α) = β 0 − √ 2αβ 0 , β 0 − √ 2αβ 0 < √ d − 1/d holds for µ = 2β 0 /α. A contradiction.
It is somewhat weird to see µ being taken as 2β 0 /α, which apparently says that a smaller β 0 promises a smaller µ. This is because that such µ is taken as the peak of the g function defined in Lemma 2. Actually, the nontrivial lowerbound of β 0 is restricted by (15), where β 0 is required to be sufficiently large to make the edges between the nodes in Z(T, β 0 ) ∪ T being sufficiently dense. Also, note that in Lemma 2, an implicit condition is 2β 0 /α > 1. This can also be deduced from the condition (15) required in Lemma 2. Now, to satisfy (15), as lim d→+∞ √ d − 1/d = 0, i.e., for
every 0 > 0 there exists d > 0 making √ d − 1/d 0 , a sufficiently large d would do iff β 0 > √ 2αβ 0 , for which β 0 > 2α should be satisfied.
Now we show that the -incomplete propagation can be accomplished upon this Ramanujan network G if the initiation set is affordable, providing that β is sufficiently small.
Lemma 3: For any α, β, β 0 , β 2 , θ 0 ∈ (0, 1), if
β + 3(β 0 − 2β 0 α) < θ 0(16)
and the inequality (15) hold and ∃P 1 ⊆ P (T, β 0 ) :
|P 1 | θ 0 n ∧ ∀i ∈ P 1 : x i (k 0 ) = 1, then ∀i ∈ P (T, β 0 ) : x i (k) = 1 holds for all k k 0 + |P (T, β 0 )| in D upon G.
Proof: Inspired by [9], now suppose that there exists S ⊆ P (T, β 0 ) with |S| = θn and ∀i ∈ S : x i = 0∧|N i ∩(P (T, β 0 )\ S)| < βd. Then there are less than βd|S| edges between S and P (T, β 0 )\S. As each node i ∈ S has more than (1−β 0 )d 1-neighbors in P (T, β 0 ), the subgraph of G induced by S has more than (
(1 − β 0 )d|S| − βd|S|)/2 = (1 − β − β 0 )d|S|/2 internal edges. With Lemma 1, |(1−β−β 0 )dθn/2−dθ 2 n/2| √ d − 1θ(1 − θ)n < √ d − 1θn holds. So 1 − β − β 0 − θ < 2 √ d − 1/d should be satisfied. Now with the existence of P 1 , we have |S| |P (T, β 0 )| − |P 1 | (1 − µα)n − θ 0 n) for µ < 2β 0 /α. So θ 1− √ 2β 0 α−θ 0 and thus (1−β −β 0 )− (1 − √ 2β 0 α − θ 0 ) = √ 2β 0 α + θ 0 − β − β 0 < 2 √ d − 1/d.
But this cannot hold together with (15) and (16). So the condition required in (13) is satisfied, and thus the conclusion holds.
With this, the desired properties of the broadcast system can be directly supported upon G with θ 0 n = (β 2 − β 0 )d + 1.
Lemma 4: For any α, β, β 0 , β 2 ∈ (0, 1), if
min{β, β 2 , 1 − β 2 } β 0(17)
and the inequalities (15) and (16) hold for θ 0 = ((β 2 − β 0 )d + 1)/n, the k H -Heaviside and k δ -Dirac properties are satisfied in D upon G with bounded k H and k δ . Proof: Firstly, if a node j ∈ P (T, β 0 ) initiates a broadcast (as a General) at k 0 , as j ∈ P (T, β 0 ) is a correct General, there is P j ⊆ P (T, β 0 ) ∩ N j satisfying |P j | (1 − β 0 )d + 1 and ∀i ∈ P j : x i (k 0 ) = 1. So with Lemma 3 we have ∀i ∈ P (T, β 0 ), k k 0 +k H : x i (k) = 1 for some k H |P (T, β 0 )|.
Next, if no node in P (T, β 0 ) initiates any broadcast before k 0 , as β β 0 and there are less than β 0 d 1-neighbors of any node i ∈ P (T, β 0 ) being out of P (T, β 0 ), no such i would be excited. So as β 2 β 0 , no node in P (T, β 0 ) would be triggered before k 0 .
Thirdly, if any node i ∈ P (T, β 0 ) is triggered at k 0 , there are at least (β 2 − β 0 )d + 1 nodes in N i ∩ P (T, β 0 ) are excited. So with Lemma 3, all nodes in P (T, β 0 ) would be excited since k 0 + k H . As every node i ∈ P (T, β 0 ) has more than (1 − β 0 )d 1-neighbors in P (T, β 0 ), with β 2 1 − β 0 every such i would be triggered no later than k 0 + k δ for some k δ k H + 1.
As we can make µ < 2β 0 /α, we would have lim n→∞ (µ−1)α/(1−α) lim n→∞ ( √ 2β 0 α−α)/(1−α) = 0 if α = n − 1 for some 1 > 0. So by definition this P protocol upon G is an a.e. -incomplete protocol, providing that β, β 0 and β 2 can be solved with (15), (16) and (17). Also, as the nodes need not know the network's actual topology, the propagation can run in dynamical networks, providing that the corresponding eigenvalues of the adjacency matrix of the continuously changing (and unknown) network are always sufficiently small.
Note, however, to satisfy (15), (16) and (17), there are implicit limitations. Firstly, as 1 − β 2 β 0 and β β 0 , we can set at most β 2 = 1 − β 0 and at least β = β 0 in making rooms for setting β 0 . Secondly, by taking β 2 = 1−β 0 , β = β 0 and θ 0 n = (β 2 − β 0 )d + 1 into (16) and then adding (15), we get (d+1)/(3n) > β 0 /3 > 2α/3 and thus d+1 > 2αn = 2f . This means that the pure-propagation-based broadcast system can at best be built upon linear-degree networks. In breaking this, the most trivial idea might be to enlarge the initial excitation area by directly adding extra edges to connect at least s = θ 0 n nodes for each node. However, by doing this, the degrees of the nodes would also be increased to at least θ 0 n = O(t), which is still linear to n. So we should make some further efforts to break this situation.
((1 − 2β 0 )d + 1)/(3n) > β 0 /3 + β 0 − √ 2β 0 α to
C. Complementing a.e. propagation with localized communication
For sublinear-degree solutions, we look again to the a.e. propagation upon the Ramanujan network G. The implicit linear-degree limitation mainly comes from θ 0 being set as ((β 2 − β 0 )d + 1)/n, where the excitation of a very small area ((1−2β 0 )d) is required to be propagated to almost everywhere of G. From Lemma 3 we also see that if the initial excitation area can be somehow larger than O(d), the condition on β 0 and β could be much looser. So the initial excitation area is the bottleneck of the fault-tolerant propagation. Meanwhile, the advantage of fault-tolerant propagation is that, once the initial excitation area is sufficiently large, the cost of a.e. propagation is much lower than that of many other faulttolerant communication protocols (such as the secure communication [9,10]). In a word, fault-tolerant propagation has the advantage of propagating to distant nodes when the propagated area is large. While on the other side, many fault-tolerant communication protocols (including secure communication, Byzantine agreement, and so on) have the advantage of providing efficient fault-tolerance when the communication range is small. So it is interesting to complement the advantages of distant-area propagation and nearby-region communication relatively. Here, similar to the complementary filters used in the frequency domain, we call such a relatively complemented D as a complementary system.
To construct a complementary system, we show that if θ 0 can be sufficiently large, a.e. propagation can be reached in logarithmic time upon sublinear-degree networks.
Lemma 5: For every d-regular connected non-bipartite Ra-
manujan network G, if √ d > 4/(θ 0 + 6α − 4 √ 2α)(18)
and
θ 0 > β + 3β 0 1 − − 3 − 1 − 2αβ 0(19)
hold for some constant ∈ (0, 1), then there exists D upon G such that for all P 1 ⊆ P (T, β 0 ) with |P 1 | θ 0 n, if ∀i ∈ P 1 : x i (k 0 ) = 1, then ∀i ∈ P (T, β 0 ) : x i (k) = 1 holds for all k k 0 + k δ with some k δ = O(log n). Proof: With Lemma 3, we need only to show β 0 , β can be solved with (18). Concretely, to satisfy (15) and (16)
with β = β 0 , as β 0 + 3(β 0 − √ 2β 0 α) < 4β 0 − 6α, β 0 − √ 2αβ 0 > β 0 − √ 2α and √ d − 1/d < 1/ √ d, we need only to show 4(1/ √ d + √ 2α) 4β 0 < θ 0 + 6α. So 4(1/ √ d + √ 2α) < θ 0 + 6α would suffice.
For the worst-case propagation time, as the subgraph of G induced by P (T, β 0 ) is an expander, by extending the proof of Lemma 3 with (19), with which we first suppose (and then get the similar contradiction) that there is only S ⊂ S satisfying |S | (1 − )|S| and ∀i ∈ S : |N i ∩ (P (T, β 0 ) \ S)| < βd, we have k δ = O(log n).
With Lemma 5, if only θ 0 > 4 √ 2α − 6α, there would exist a constant d satisfying (18) for the k δ -round a.e. propagation. Furthermore, it is easy to extend Lemma 5 to all d-regular strong enough expander G with the second largest eigenvalue of the adjacency matrix of G being λ = O(d 1/2 ) (see [9], and other results for the explicitly constructed Ramanujan networks can also be extended similarly). With this, the remaining problem is to construct some s-localized communication protocol C upon G to support the desired θ 0 with s n. Namely, with the s-localized C protocol running for some s-sized vertex-set of G, the end-to-end communication between the s nodes in each such vertex-set would be localized. Besides, it would be better if all the related communication paths in the localized communication protocol can also be localized in O(log s). Further, it would be even better if all the end-toend communication between the s nodes can be accomplished between the same s nodes. Moreover, it would be optimal if all the related communication paths are with length O(1).
In realizing the s-localized communication protocols, there can be different strategies. Firstly, we can directly employ some incomplete secure communication protocol as C. With this, each node i ∈ P (T, β 0 ) is expected to communicate with and only with up to s nodes in V (denoted as S i , i ∈ S i ). For efficiency, these s nodes can be selected in the c-neighborhood of i (with c = O(log n) in worst cases). The c-neighborhood of
i is defined as N (c) i = ∪ r c L (r) i , where L (r) i = {j | d G
i,j = r} is the set of all r-neighbors of i, with d G i,j being the length of the shortest path between i and j in G. Alternatively, we can also try to construct easier localized communication protocols other than secure communication. As is limited here, we only discuss how to complement the a.e. propagation with the general s-localized communication protocol C.
Firstly, for a.e. propagation, with Lemma 5, it is desired that Ω(αn) npc nodes should be initially excited. Here we show that this can be satisfied by initiating the broadcast with the slocalized C protocol. For this, we show that for every T ∈ V f , there can always be Ω(αn) npc nodes in some c-neighborhood of every npc node in G. = O(log n) for all i, j ∈ P (T, β 0 ). As the constant β 0 < 1/2, we have a = Ω(d). As i ∈ P (T, β 0 ), we always have |N Proof: Firstly, with Lemma 6, there exists a sufficiently large c = O(log n) such that for all T ∈ V f , if i ∈ P (T, β 0 ), then |N (c) i ∩ P (T, β 0 )| = Ω(αn) holds. So for every T ∈ V f , if an npc-General broadcasts at k 0 , at least Ω(αn) npc nodes would be excited before k 0 + O(log n). So with Lemma 5 we have ∀i ∈ P (T, β 0 ) : x i (k) = 1 holds for all k k 0 + O(log n). So by setting D y = H +u with a sufficiently large u = µαn + 4 √ 2αn and selecting S i ⊂ V with s = u + µαn for each i ∈ V , we have ∀i ∈ P (T, β 0 ) : y i (k) = 1 holds for all k k 0 + O(log n). And as no npc node would be excited if no npc-General broadcasts in the underlying P protocol, the Heaviside property is satisfied. For the Dirac property, if y i (k 0 ) = 1 holds for any i ∈ P (T, β 0 ), we have at least 4 √ 2αn npc nodes being excited no later than k 0 in this case. So again with Lemma 5 we have ∀i ∈ P (T, β 0 ) : x i (k) = 1 for all k k 0 + O(log n). And again with s u + µαn, ∀i ∈ P (T, β 0 ) : y i (k) = 1 holds for all k k 0 + O(log n).
VI. CONCLUSION
In this paper, we have investigated the broadcast problem upon bounded-degree networks with a simple but nontrivial system model. In providing the relay-based broadcast systems upon bounded-degree networks, the a.e. propagation and the complementary systems are proposed upon strong enough expanders. In building a.e. propagation upon the expanders, a general analysis of the fault-tolerant propagation is presented, and the related parameters are analysed. In providing efficient broadcast systems, complementary systems are constructed by relatively complementing a.e. propagation and localized communication. It is shown that by integrating a.e. propagation and localized communication protocols, more efficient broadcast systems can be built upon sublinear-degree networks than with only incomplete communication protocols. This approach can go further to show to what extent the complexity of the Byzantine protocols and the node-degree of the networks can be lowered. With the result of this paper, this mainly depends on the efficiency of the localized communication protocols.
Fig. 1: A Local System.
Fig. 2 :
2The Generalized Broadcast System upon G.
Lemma 6 :
6If G = (V, E) is a connected non-bipartite Ramanujan graph and (15) holds, then there exists c = O(log n) such that for every T ∈ V f , |N (c) i ∩ P (T, β 0 )| = Ω(αn) holds for all i ∈ P (T, β 0 ). Proof: As G = (V, E) is a connected non-bipartite Ramanujan graph and (15) holds, for every T ∈ V f , with Lemma 2 we have |P (T, β 0 )| (1 − µα)n > (1 − √ 2β 0 α)n. With the proof of Lemma 2 of [9], the subgraph of G induced by P (T, β 0 ), denoted as G(T, β 0 ), is a (vertex) expander graph with an a = Ω((1/2 − β 0 )d) expansion coefficient.
∩
P (T, β 0 )| < n/2. So we have |N (c) i ∩ P (T, β 0 )| ((a + 1) c+1 − 1)/a = Ω(d c ). So for every i ∈ P (T, β 0 ) there is c = O(log n) such that |N (c) i ∩ P (T, β 0 )| = Ω(αn).Now we show that there exists a.e. broadcast systems upon sublinear-degree networks by complementing the s-localized communication protocol and a.e. propagation. Theorem 1: If there is an s-localized communication protocol C upon the d-regular G with s u + µαn and the premise of Lemma 6 holds, then a.e. broadcast system D exists upon some d -regular G with d = d + O(1) with k δ = O(log n) and k H = O(log n).
Simulating authenticated broadcasts to derive simple fault-tolerant algorithms. T K Srikanth, S Toueg, Distributed Computing. 22T. K. Srikanth and S. Toueg, "Simulating authenticated broadcasts to derive simple fault-tolerant algorithms," Distributed Computing, vol. 2, no. 2, pp. 80-94, 1987.
The byzantine generals problem. L Lamport, R Shostak, M Pease, Acm Transactions on Programming Languages and Systems. 43L. Lamport, R. Shostak, and M. Pease, "The byzantine generals problem," Acm Transactions on Programming Languages and Systems, vol. 4, no. 3, pp. 382-401, 1982.
Authenticated algorithms for byzantine agreement. D Dolev, H R Strong, Siam Journal on Computing. 124D. Dolev and H. R. Strong, "Authenticated algorithms for byzantine agreement," Siam Journal on Computing, vol. 12, no. 4, pp. 656-666, 1983.
Elections in the presence of faults. M Merritt, Proceedings of the Third Annual ACM Symposium on Principles of Distributed Computing, ser. PODC '84. the Third Annual ACM Symposium on Principles of Distributed Computing, ser. PODC '84New York, NY, USAAssociation for Computing MachineryM. Merritt, "Elections in the presence of faults," in Proceedings of the Third Annual ACM Symposium on Principles of Distributed Computing, ser. PODC '84. New York, NY, USA: Association for Computing Ma- chinery, 1984, p. 134-142.
A framework for agreement. K J Perry, Lecture Notes in Computer Science. 312K. J. Perry, "A framework for agreement," Lecture Notes in Computer Science, vol. 312, pp. 57-75, 1988.
The byzantine generals strike again. D Dolev, Journal of Algorithms. 31D. Dolev, "The byzantine generals strike again," Journal of Algorithms, vol. 3, no. 1, pp. 14-30, 1982.
Ramanujan graphs. A Lubotzky, R Phillips, P Sarnak, Combinatorica. 83A. Lubotzky, R. Phillips, and P. Sarnak, "Ramanujan graphs," Combinatorica, vol. 8, no. 3, pp. 261-277, 1988.
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Improved fault tolerance and secure computation on sparse networks. N Chandran, J Garay, R Ostrovsky, Automata, Languages and Programming, S. Abramsky, C. Gavoille, C. Kirchner, F. Meyer auf der Heide, and P. G. SpirakisSpringerBerlin, Heidelberg; Berlin HeidelbergN. Chandran, J. Garay, and R. Ostrovsky, "Improved fault tolerance and secure computation on sparse networks," in Automata, Languages and Programming, S. Abramsky, C. Gavoille, C. Kirchner, F. Meyer auf der Heide, and P. G. Spirakis, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010, pp. 249-260.
Fast distributed agreement. S Toueg, K J Perry, T K Srikanth, Siam Journal on Computing. 163S. Toueg, K. J. Perry, and T. K. Srikanth, "Fast distributed agreement," Siam Journal on Computing, vol. 16, no. 3, pp. 445-457, 1987.
Self-stabilizing pulse synchronization inspired by biological pacemaker networks. A Daliot, D Dolev, H Parnas, Self-Stabilizing Systems, Proceedings. 2704A. Daliot, D. Dolev, and H. Parnas, "Self-stabilizing pulse synchronization inspired by biological pacemaker networks," Self-Stabilizing Systems, Proceedings, vol. 2704, pp. 32-48, 2003.
Byzantine self-stabilizing pulse in a bounded-delay model. D Dolev, E N Hoch, Stabilization, Safety, and Security of Distributed Systems, Proceedings. 4838234D. Dolev and E. N. Hoch, "Byzantine self-stabilizing pulse in a bounded-delay model," Stabilization, Safety, and Security of Distributed Systems, Proceedings, vol. 4838, pp. 234-+, 2007.
An efficient algorithm for byzantine agreement without authentication. D Dolev, M J Fischer, R Fowler, N A Lynch, H R Strong, Information and Control. 523D. Dolev, M. J. Fischer, R. Fowler, N. A. Lynch, and H. R. Strong, "An efficient algorithm for byzantine agree- ment without authentication," Information and Control, vol. 52, no. 3, pp. 257-274, 1982.
Self-stabilizing byzantine agreement. A Daliot, D Dolev, Proceedings of the Twenty-Fifth Annual ACM Symposium on Principles of Distributed Computing, ser. PODC '06. the Twenty-Fifth Annual ACM Symposium on Principles of Distributed Computing, ser. PODC '06New York, NY, USAAssociation for Computing MachineryA. Daliot and D. Dolev, "Self-stabilizing byzantine agreement," in Proceedings of the Twenty-Fifth Annual ACM Symposium on Principles of Distributed Comput- ing, ser. PODC '06. New York, NY, USA: Association for Computing Machinery, 2006, p. 143-152.
Asymptotically optimal distributed consensus. P Berman, J A Garay, Automata, Languages and Programming. Ausiello, M. Dezani-Ciancaglini, and S. R. Della RoccaBerlin, Heidelberg; Berlin HeidelbergSpringerP. Berman and J. A. Garay, "Asymptotically optimal distributed consensus," in Automata, Languages and Pro- gramming, G. Ausiello, M. Dezani-Ciancaglini, and S. R. Della Rocca, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 1989, pp. 80-94.
Efficient player-optimal protocols for strong and differential consensus. M Fitzi, J A Garay, Proceedings of the Twenty-Second Annual Symposium on Principles of Distributed Computing, ser. PODC '03. the Twenty-Second Annual Symposium on Principles of Distributed Computing, ser. PODC '03New York, NY, USAAssociation for Computing MachineryM. Fitzi and J. A. Garay, "Efficient player-optimal pro- tocols for strong and differential consensus," in Proceed- ings of the Twenty-Second Annual Symposium on Prin- ciples of Distributed Computing, ser. PODC '03. New York, NY, USA: Association for Computing Machinery, 2003, p. 211-220.
Efficient constructions for almost-everywhere secure computation. S Jayanti, S Raghuraman, N Vyas, Advances in Cryptology -EUROCRYPT 2020. A. Canteaut and Y. IshaiSpringer International PublishingS. Jayanti, S. Raghuraman, and N. Vyas, "Efficient con- structions for almost-everywhere secure computation," in Advances in Cryptology -EUROCRYPT 2020, A. Can- teaut and Y. Ishai, Eds. Cham: Springer International Publishing, 2020, pp. 159-183.
Agreement in the presence of faults, on networks of bounded degree. M Ben-Or, D Ron, Information Processing Letters. 576M. Ben-Or and D. Ron, "Agreement in the presence of faults, on networks of bounded degree," Information Processing Letters, vol. 57, no. 6, pp. 329-334, 1996.
Breaking the o(n2) bit barrier: Scalable byzantine agreement with an adaptive adversary. V King, J Saia, J. ACM. 584V. King and J. Saia, "Breaking the o(n2) bit barrier: Scal- able byzantine agreement with an adaptive adversary," J. ACM, vol. 58, no. 4, Jul. 2011.
Nonlocal control of pulse propagation in excitable media. C Bachmair, E Schöll, The European Physical Journal B. 8711276C. Bachmair and E. Schöll, "Nonlocal control of pulse propagation in excitable media," The European Physical Journal B, vol. 87, no. 11, p. 276, 2014.
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"Generation of Optical Coherent State Superpositions by Number-Resolved Photon Subtraction from Squeezed Vacuum"
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"Thomas Gerrits \nNational Institute of Standards and Technology\n80305BoulderCOUSA\n\nNational Institute of Standards and Technology\n80305BoulderCOUSA\n",
"Scott Glancy \nNational Institute of Standards and Technology\n80305BoulderCOUSA\n\nNational Institute of Standards and Technology\n80305BoulderCOUSA\n",
"Tracy S Clement \nNational Institute of Standards and Technology\n80305BoulderCOUSA\n\nNational Institute of Standards and Technology\n80305BoulderCOUSA\n",
"Brice Calkins \nNational Institute of Standards and Technology\n80305BoulderCOUSA\n\nNational Institute of Standards and Technology\n80305BoulderCOUSA\n",
"Adriana E Lita \nNational Institute of Standards and Technology\n80305BoulderCOUSA\n\nNational Institute of Standards and Technology\n80305BoulderCOUSA\n",
"Aaron J Miller \nAlbion College\n49224AlbionMIUSA\n\nNational Institute of Standards and Technology and Joint Quantum Institute\nUniv. of Maryland\n20899, 20742Gaithersburg, College ParkMD, MDUSA, USA\n",
"Alan L Migdall \nNational Institute of Standards and Technology\n20899GaithersburgMDUSA\n\nJoint Quantum Institute\nUniv. of Maryland\n20742College ParkMDUSA\n\nAlbion College, Albion\n49224MIUSA\n",
"Sae Woo Nam \nNational Institute of Standards and Technology\n80305BoulderCOUSA\n\nNational Institute of Standards and Technology\n80305BoulderCOUSA\n",
"Richard P Mirin \nNational Institute of Standards and Technology\n80305BoulderCOUSA\n\nNational Institute of Standards and Technology\n80305BoulderCOUSA\n",
"Emanuel Knill \nNational Institute of Standards and Technology\n80305BoulderCOUSA\n\nNational Institute of Standards and Technology\n80305BoulderCOUSA\n"
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"National Institute of Standards and Technology\n80305BoulderCOUSA",
"National Institute of Standards and Technology\n80305BoulderCOUSA",
"National Institute of Standards and Technology\n80305BoulderCOUSA",
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"National Institute of Standards and Technology\n20899GaithersburgMDUSA",
"Joint Quantum Institute\nUniv. of Maryland\n20742College ParkMDUSA",
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We have created heralded coherent state superpositions (CSS), by subtracting up to three photons from a pulse of squeezed vacuum light. To produce such CSSs at a sufficient rate, we used our highefficiency photon-number-resolving transition edge sensor to detect the subtracted photons. This is the first experiment enabled by and utilizing the full photon-number-resolving capabilities of this detector. The CSS produced by three-photon subtraction had a mean photon number of 2.75 +0.06 −0.24 and a fidelity of 0.59 +0.04 −0.14 with an ideal CSS. This confirms that subtracting more photons results in higher-amplitude CSSs.A coherent state of the electromagnetic field is often considered the most classical-like pure state, but a superposition of two coherent states with opposite phases has interesting quantum features. For example, coherent state superpositions (CSS) can be exploited for performing quantum information tasks and high precision measurements. CSSs are also of fundamental interest: When they contain many photons they are superpositions of macroscopically distinguishable states often called "Schrödinger cat states". Schrödinger's Gedanken experiment of 1935 described a cat apparently held in a superposition of alive and dead states [1], but many researchers now use "Schrödinger cat" to refer to a quantum state that is a superposition of two highly distinguishable classical states such as a CSS of high amplitude or mean number of photons [2]. CSSs have been prepared in traveling optical modes with a mean of up to 2.0 optical photons by heralding [3][4][5][6][7]. With sufficiently high quality and well characterized CSSs, one can in principle quantum compute using simple linear optical components and homodyne measurements[8]. Less ambitiously, they can serve as flying qubits for quantum communication.In addition to potentially simple processing, advantages of CSSs in traveling optical modes include fast linear manipulations, transport over large distances, robustness if loss is controlled, and simple conversion to entangled optical states, all at room temperature.The CSSs that we discuss here are superpositions of two coherent states | ± α of a single mode of light, where +α and −α are the states' complex mode amplitudes. Our experiments aim to prepare two special instances of these CSSs: the odd and even CSSs defined as the superpositions |−α ± |α (unnormalized). These are distinguished by having only even (+) or odd (−) numbers of photons. For |α| 1, the states' mean number of photons, n , is approximately |α| 2 . Two quality mea-sures for experimental CSSs are the fidelity of the created state with the nearest ideal CSS and the magnitude of the amplitude of this ideal CSS. There are two reasons to aim for large amplitude CSSs. The first is that to be useful for superresolution metrology, the probability p 0 = 1 − exp(−2|α| 2 ) with which the superposed coherent states can be distinguished must be close to one. To achieve p 0 > 0.99 requires |α| > 1.52. The second is that a minimum size estimated as |α| > 1.2 is required for fault tolerant quantum computing[9]. Because operation close to the lower bound is unrealistic due to excessive resource requirements, we are motivated to produce bigger CSSs. Similarly, high fidelity is required to avoid excessive overheads for eliminating unwanted errors due to deviations from an ideal CSS. The highest fidelity CSS achieved so far has |α| = 1.1 and a fidelity F = 0.76 [7], while the largest has an effective size of |α| = 1.4 and fidelity F = 0.60 [7]. We have created CSSs with amplitudes and fidelities of |α| = 1.76 +0.02 −0.19 and F = 0.59 +0.04 −0.14 , and |α| = 1.32 +0.01 −0.02 and F = 0.522 +0.004 −0.010 . Unlike the experiment reported in [7], our CSSs are generated in pulsed rather than continuous-wave mode. Pulsed operation is required for many applications to avoid the effects of light in neighboring modes in subsequent manipulations and measurements of the states.To create the CSSs, we used the photon subtraction scheme depicted inFig. 1. A squeezed vacuum state is prepared and sent through a weakly reflecting beam splitter. Reflected photons that are detected herald an approximate CSS in the transmitted beam. Because higher amplitude and fidelity CSSs can be created by heralding on detecting multiple photons at once [10, 11], we used a photon-number-resolving transition edge sensor (TES)[12,13]for subtracting two or three photons. The TES used in our experiment has an efficiency of 85 ± 2 % and can resolve up to 10 photons. This enabled obtain-arXiv:1004.2727v4 [quant-ph] 1 Feb 2011 2FIG. 1.(color online) Scheme for optical coherent state superposition (CSS) creation. An upconverted laser pulse enters an optical parametric amplifier (OPA) to create a squeezed vacuum state in section (A). After spectral filtering, this state is sent to a weakly reflecting beamsplitter R in (B). Reflected photons that are detected herald a CSS emerging from R into (C). Its quadratures are measured by homodyne detection in (C).ing higher amplitude CSS at practical rates. We also subtracted one and two photons using avalanche photodiodes (APDs) for comparison.For the experiments, we used a cavity-dumped 861.8 nm laser with transform-limited pulses of 140 fs (typical), pulse energies of 40 nJ and a repetition frequency of 548 kHz. A fraction of each pulse with > 10 9 photons was used as the local oscillator (LO) in the homodyne detector. The rest pumped a temperature-tuned 150 µm KNbO 3 crystal to generate a second-harmonic pump pulse (efficiency 25 %) for the optical parametric amplifier (OPA) shown inFig. 1. The OPA consists of a temperature-tuned 200 µm long KNbO 3 crystal. We determined that the squeezed vacuum state generated can be modeled as a pure squeezed state with minimum quadrature variance V 0 = −6.8 dB subjected to a loss of γ s = 0.36. We define the squeezing purity as η s = 1 − γ s . We used a variable beam splitter (R inFig. 1) made with a half-wave-plate and a polarizing beamsplitter and configured to send from 2.5 % (one-photon subtraction) to 20 % (three-photon subtraction) of the light to the photon subtraction arm. Photons in this arm were spectrally filtered by a fiber Bragg grating with a bandwidth of 1.5 nm in a polarization-based circulator before being coupled to the photon detector/counter. The other arm of the variable beam splitter delivers the heralded CSS to a conventional homodyne detector for measuring the quadrature at the phase of the LO. The CSS temporal shape is significantly different from that of the original pump due to the large mismatch in group velocity in our KNbO 3 crystals. To compensate, we expanded the temporal width of the LO by up to a factor of 2 with a pulse-shaping setup[14].
|
10.1103/physreva.82.031802
|
[
"https://arxiv.org/pdf/1004.2727v4.pdf"
] | 40,767,881 |
1004.2727
|
b5e5157026ce24b79f290bdc0f3f3c21be8524a6
|
Generation of Optical Coherent State Superpositions by Number-Resolved Photon Subtraction from Squeezed Vacuum
Thomas Gerrits
National Institute of Standards and Technology
80305BoulderCOUSA
National Institute of Standards and Technology
80305BoulderCOUSA
Scott Glancy
National Institute of Standards and Technology
80305BoulderCOUSA
National Institute of Standards and Technology
80305BoulderCOUSA
Tracy S Clement
National Institute of Standards and Technology
80305BoulderCOUSA
National Institute of Standards and Technology
80305BoulderCOUSA
Brice Calkins
National Institute of Standards and Technology
80305BoulderCOUSA
National Institute of Standards and Technology
80305BoulderCOUSA
Adriana E Lita
National Institute of Standards and Technology
80305BoulderCOUSA
National Institute of Standards and Technology
80305BoulderCOUSA
Aaron J Miller
Albion College
49224AlbionMIUSA
National Institute of Standards and Technology and Joint Quantum Institute
Univ. of Maryland
20899, 20742Gaithersburg, College ParkMD, MDUSA, USA
Alan L Migdall
National Institute of Standards and Technology
20899GaithersburgMDUSA
Joint Quantum Institute
Univ. of Maryland
20742College ParkMDUSA
Albion College, Albion
49224MIUSA
Sae Woo Nam
National Institute of Standards and Technology
80305BoulderCOUSA
National Institute of Standards and Technology
80305BoulderCOUSA
Richard P Mirin
National Institute of Standards and Technology
80305BoulderCOUSA
National Institute of Standards and Technology
80305BoulderCOUSA
Emanuel Knill
National Institute of Standards and Technology
80305BoulderCOUSA
National Institute of Standards and Technology
80305BoulderCOUSA
Generation of Optical Coherent State Superpositions by Number-Resolved Photon Subtraction from Squeezed Vacuum
(Dated: February 2, 2011)numbers: 4250Dv4250Xa0365Ta0365Wj
We have created heralded coherent state superpositions (CSS), by subtracting up to three photons from a pulse of squeezed vacuum light. To produce such CSSs at a sufficient rate, we used our highefficiency photon-number-resolving transition edge sensor to detect the subtracted photons. This is the first experiment enabled by and utilizing the full photon-number-resolving capabilities of this detector. The CSS produced by three-photon subtraction had a mean photon number of 2.75 +0.06 −0.24 and a fidelity of 0.59 +0.04 −0.14 with an ideal CSS. This confirms that subtracting more photons results in higher-amplitude CSSs.A coherent state of the electromagnetic field is often considered the most classical-like pure state, but a superposition of two coherent states with opposite phases has interesting quantum features. For example, coherent state superpositions (CSS) can be exploited for performing quantum information tasks and high precision measurements. CSSs are also of fundamental interest: When they contain many photons they are superpositions of macroscopically distinguishable states often called "Schrödinger cat states". Schrödinger's Gedanken experiment of 1935 described a cat apparently held in a superposition of alive and dead states [1], but many researchers now use "Schrödinger cat" to refer to a quantum state that is a superposition of two highly distinguishable classical states such as a CSS of high amplitude or mean number of photons [2]. CSSs have been prepared in traveling optical modes with a mean of up to 2.0 optical photons by heralding [3][4][5][6][7]. With sufficiently high quality and well characterized CSSs, one can in principle quantum compute using simple linear optical components and homodyne measurements[8]. Less ambitiously, they can serve as flying qubits for quantum communication.In addition to potentially simple processing, advantages of CSSs in traveling optical modes include fast linear manipulations, transport over large distances, robustness if loss is controlled, and simple conversion to entangled optical states, all at room temperature.The CSSs that we discuss here are superpositions of two coherent states | ± α of a single mode of light, where +α and −α are the states' complex mode amplitudes. Our experiments aim to prepare two special instances of these CSSs: the odd and even CSSs defined as the superpositions |−α ± |α (unnormalized). These are distinguished by having only even (+) or odd (−) numbers of photons. For |α| 1, the states' mean number of photons, n , is approximately |α| 2 . Two quality mea-sures for experimental CSSs are the fidelity of the created state with the nearest ideal CSS and the magnitude of the amplitude of this ideal CSS. There are two reasons to aim for large amplitude CSSs. The first is that to be useful for superresolution metrology, the probability p 0 = 1 − exp(−2|α| 2 ) with which the superposed coherent states can be distinguished must be close to one. To achieve p 0 > 0.99 requires |α| > 1.52. The second is that a minimum size estimated as |α| > 1.2 is required for fault tolerant quantum computing[9]. Because operation close to the lower bound is unrealistic due to excessive resource requirements, we are motivated to produce bigger CSSs. Similarly, high fidelity is required to avoid excessive overheads for eliminating unwanted errors due to deviations from an ideal CSS. The highest fidelity CSS achieved so far has |α| = 1.1 and a fidelity F = 0.76 [7], while the largest has an effective size of |α| = 1.4 and fidelity F = 0.60 [7]. We have created CSSs with amplitudes and fidelities of |α| = 1.76 +0.02 −0.19 and F = 0.59 +0.04 −0.14 , and |α| = 1.32 +0.01 −0.02 and F = 0.522 +0.004 −0.010 . Unlike the experiment reported in [7], our CSSs are generated in pulsed rather than continuous-wave mode. Pulsed operation is required for many applications to avoid the effects of light in neighboring modes in subsequent manipulations and measurements of the states.To create the CSSs, we used the photon subtraction scheme depicted inFig. 1. A squeezed vacuum state is prepared and sent through a weakly reflecting beam splitter. Reflected photons that are detected herald an approximate CSS in the transmitted beam. Because higher amplitude and fidelity CSSs can be created by heralding on detecting multiple photons at once [10, 11], we used a photon-number-resolving transition edge sensor (TES)[12,13]for subtracting two or three photons. The TES used in our experiment has an efficiency of 85 ± 2 % and can resolve up to 10 photons. This enabled obtain-arXiv:1004.2727v4 [quant-ph] 1 Feb 2011 2FIG. 1.(color online) Scheme for optical coherent state superposition (CSS) creation. An upconverted laser pulse enters an optical parametric amplifier (OPA) to create a squeezed vacuum state in section (A). After spectral filtering, this state is sent to a weakly reflecting beamsplitter R in (B). Reflected photons that are detected herald a CSS emerging from R into (C). Its quadratures are measured by homodyne detection in (C).ing higher amplitude CSS at practical rates. We also subtracted one and two photons using avalanche photodiodes (APDs) for comparison.For the experiments, we used a cavity-dumped 861.8 nm laser with transform-limited pulses of 140 fs (typical), pulse energies of 40 nJ and a repetition frequency of 548 kHz. A fraction of each pulse with > 10 9 photons was used as the local oscillator (LO) in the homodyne detector. The rest pumped a temperature-tuned 150 µm KNbO 3 crystal to generate a second-harmonic pump pulse (efficiency 25 %) for the optical parametric amplifier (OPA) shown inFig. 1. The OPA consists of a temperature-tuned 200 µm long KNbO 3 crystal. We determined that the squeezed vacuum state generated can be modeled as a pure squeezed state with minimum quadrature variance V 0 = −6.8 dB subjected to a loss of γ s = 0.36. We define the squeezing purity as η s = 1 − γ s . We used a variable beam splitter (R inFig. 1) made with a half-wave-plate and a polarizing beamsplitter and configured to send from 2.5 % (one-photon subtraction) to 20 % (three-photon subtraction) of the light to the photon subtraction arm. Photons in this arm were spectrally filtered by a fiber Bragg grating with a bandwidth of 1.5 nm in a polarization-based circulator before being coupled to the photon detector/counter. The other arm of the variable beam splitter delivers the heralded CSS to a conventional homodyne detector for measuring the quadrature at the phase of the LO. The CSS temporal shape is significantly different from that of the original pump due to the large mismatch in group velocity in our KNbO 3 crystals. To compensate, we expanded the temporal width of the LO by up to a factor of 2 with a pulse-shaping setup[14].
We have created heralded coherent state superpositions (CSS), by subtracting up to three photons from a pulse of squeezed vacuum light. To produce such CSSs at a sufficient rate, we used our highefficiency photon-number-resolving transition edge sensor to detect the subtracted photons. This is the first experiment enabled by and utilizing the full photon-number-resolving capabilities of this detector. The CSS produced by three-photon subtraction had a mean photon number of 2.75 +0.06 −0.24 and a fidelity of 0.59 +0.04 −0.14 with an ideal CSS. This confirms that subtracting more photons results in higher-amplitude CSSs. A coherent state of the electromagnetic field is often considered the most classical-like pure state, but a superposition of two coherent states with opposite phases has interesting quantum features. For example, coherent state superpositions (CSS) can be exploited for performing quantum information tasks and high precision measurements. CSSs are also of fundamental interest: When they contain many photons they are superpositions of macroscopically distinguishable states often called "Schrödinger cat states". Schrödinger's Gedanken experiment of 1935 described a cat apparently held in a superposition of alive and dead states [1], but many researchers now use "Schrödinger cat" to refer to a quantum state that is a superposition of two highly distinguishable classical states such as a CSS of high amplitude or mean number of photons [2]. CSSs have been prepared in traveling optical modes with a mean of up to 2.0 optical photons by heralding [3][4][5][6][7]. With sufficiently high quality and well characterized CSSs, one can in principle quantum compute using simple linear optical components and homodyne measurements [8]. Less ambitiously, they can serve as flying qubits for quantum communication.
In addition to potentially simple processing, advantages of CSSs in traveling optical modes include fast linear manipulations, transport over large distances, robustness if loss is controlled, and simple conversion to entangled optical states, all at room temperature.
The CSSs that we discuss here are superpositions of two coherent states | ± α of a single mode of light, where +α and −α are the states' complex mode amplitudes. Our experiments aim to prepare two special instances of these CSSs: the odd and even CSSs defined as the superpositions |−α ± |α (unnormalized). These are distinguished by having only even (+) or odd (−) numbers of photons. For |α| 1, the states' mean number of photons, n , is approximately |α| 2 . Two quality mea-sures for experimental CSSs are the fidelity of the created state with the nearest ideal CSS and the magnitude of the amplitude of this ideal CSS. There are two reasons to aim for large amplitude CSSs. The first is that to be useful for superresolution metrology, the probability p 0 = 1 − exp(−2|α| 2 ) with which the superposed coherent states can be distinguished must be close to one. To achieve p 0 > 0.99 requires |α| > 1.52. The second is that a minimum size estimated as |α| > 1.2 is required for fault tolerant quantum computing [9]. Because operation close to the lower bound is unrealistic due to excessive resource requirements, we are motivated to produce bigger CSSs. Similarly, high fidelity is required to avoid excessive overheads for eliminating unwanted errors due to deviations from an ideal CSS. The highest fidelity CSS achieved so far has |α| = 1.1 and a fidelity F = 0.76 [7], while the largest has an effective size of |α| = 1.4 and fidelity F = 0.60 [7]. We have created CSSs with amplitudes and fidelities of |α| = 1.76 +0.02 −0.19 and F = 0.59 +0.04 −0.14 , and |α| = 1.32 +0.01 −0.02 and F = 0.522 +0.004 −0.010 . Unlike the experiment reported in [7], our CSSs are generated in pulsed rather than continuous-wave mode. Pulsed operation is required for many applications to avoid the effects of light in neighboring modes in subsequent manipulations and measurements of the states.
To create the CSSs, we used the photon subtraction scheme depicted in Fig. 1. A squeezed vacuum state is prepared and sent through a weakly reflecting beam splitter. Reflected photons that are detected herald an approximate CSS in the transmitted beam. Because higher amplitude and fidelity CSSs can be created by heralding on detecting multiple photons at once [10,11], we used a photon-number-resolving transition edge sensor (TES) [12,13] for subtracting two or three photons. The TES used in our experiment has an efficiency of 85 ± 2 % and can resolve up to 10 photons. This enabled obtain- creation. An upconverted laser pulse enters an optical parametric amplifier (OPA) to create a squeezed vacuum state in section (A). After spectral filtering, this state is sent to a weakly reflecting beamsplitter R in (B). Reflected photons that are detected herald a CSS emerging from R into (C). Its quadratures are measured by homodyne detection in (C).
ing higher amplitude CSS at practical rates. We also subtracted one and two photons using avalanche photodiodes (APDs) for comparison.
For the experiments, we used a cavity-dumped 861.8 nm laser with transform-limited pulses of 140 fs (typical), pulse energies of 40 nJ and a repetition frequency of 548 kHz. A fraction of each pulse with > 10 9 photons was used as the local oscillator (LO) in the homodyne detector. The rest pumped a temperature-tuned 150 µm KNbO 3 crystal to generate a second-harmonic pump pulse (efficiency 25 %) for the optical parametric amplifier (OPA) shown in Fig. 1. The OPA consists of a temperature-tuned 200 µm long KNbO 3 crystal. We determined that the squeezed vacuum state generated can be modeled as a pure squeezed state with minimum quadrature variance V 0 = −6.8 dB subjected to a loss of γ s = 0.36. We define the squeezing purity as η s = 1 − γ s . We used a variable beam splitter (R in Fig. 1) made with a half-wave-plate and a polarizing beamsplitter and configured to send from 2.5 % (one-photon subtraction) to 20 % (three-photon subtraction) of the light to the photon subtraction arm. Photons in this arm were spectrally filtered by a fiber Bragg grating with a bandwidth of 1.5 nm in a polarization-based circulator before being coupled to the photon detector/counter. The other arm of the variable beam splitter delivers the heralded CSS to a conventional homodyne detector for measuring the quadrature at the phase of the LO. The CSS temporal shape is significantly different from that of the original pump due to the large mismatch in group velocity in our KNbO 3 crystals. To compensate, we expanded the temporal width of the LO by up to a factor of 2 with a pulse-shaping setup [14]. The phase of the LO was adjusted by a piezo-mounted mirror displaced at a frequency of 2.75 Hz with a saw-tooth profile to obtain a complete phase space measurement of the CSS. Further technical details are in [15].
We reconstructed the states produced by photon subtraction immediately after the subtracting beam splitter by maximum likelihood quantum state estimation as discussed in Ref. [16]. For this purpose, we considered the homodyne measurement setup including all of its losses such as those associated with the initial beamsplitter and imperfect spatial mode matching to the LO as a monolithic lossy quadrature measurement. This requires knowing the loss γ h , which we experimentally determined to be γ h = 15 ± 2 %. The uncertainty in γ h propagates to an uncertainty in the reported CSS parameters. In particular, the fidelities differ by up to ±0.02 if the boundary values for γ h are used. However, the main uncertainty in our state reconstructions is due to finite sample statistics. We estimated this statistical uncertainty by parametric-bootstrap resampling [17]. We report inferred values such as fidelities in the form F U −F −(F −L) , where F is the fidelity of the maximum likelihood estimate from the experiment's data, U is the 84 th percentile of the fidelities of the states estimated from resampled data sets, and L is the 16 th percentile. We obtained 100 resampled data sets for one-and two-photon subtraction and 1000 for three-photon subtraction. There is a significant bias toward more mixed states in the resampling procedure and the amount of bias increases with the purity of the state from which samples are generated. We did not correct for this bias in our reconstruction of the states, but note that it suggests that the true fidelities are above the reported ones.
The reconstructed states have well-defined average photon numbers, n . The reported amplitudes are those of the nearest even or odd CSS, which is found by maximizing the fidelity with respect to the reconstructed state. The reported fidelities are these maximized ones. Table I summarizes our results. Fig. 2 shows the reconstructed Wigner function for a one-photon-subtracted state heralded by an APD. The quantum character of this state can be identified by its negativity near the origin of the Wigner function, whose minimum has a value of W min = −0.041 +0.009 −0.001 . The state's fidelity is F = 0.522 +0.004 −0.010 with respect to an odd CSS with |α| = 1.32 +0.01 −0.02 . This fidelity is higher than the maximum fidelity of F = 0.487 that any coherent state can have with the |α| = 1.32 odd CSS. (Note that this is also the highest fidelity that any mixture of coherent states can have. The maximum fidelity of a coherent state with a CSS depends on the CSS's |α| and whether the CSS is even or odd. As |α| increases, this fidelity approaches 0.5 from above for even CSSs but from below for odd CSSs.) The amplitude of the CSS is notably larger than the |α| = 0.88, F = 0.70, W min = −0.13 state described in Ref. [3]. The lower fidelity is primarily due to a lower squeezing purity η s in our experiment.
We obtained even CSSs by two photon subtraction. We performed two experiments, the first used a TES, the second used two APDs at the two outputs of a 50/50 beamsplitter. For the APDs, coincidence heralds the presence of two photons in the subtraction arm. The reconstructed states are shown in Fig. 3. The TES measurement yielded a smaller CSS (|α TES | = 1.16 +0.04 −0.04 ) than the APD measurement (|α APD | = 1.30 +0.04 −0.02 ). The fidelities are F TES = 0.531 +0.017 −0.018 and F APD = 0.523 +0.022 −0.014 . For comparison, the maximum fidelity of coherent states with an |α| = 1.16 (|α| = 1.30) even CSS is 0.552 (0.522). Earlier studies [7] showed the continuous wave generation of even CSSs with |α| = 1.41 and F = 0.60.
The fidelity of the heralded CSSs is affected not only by low squeezing purity, but also by unwanted photons not matching the LO mode but still visible to the detectors. In addition to stray light (which can in principle be controlled) such photons come from temporally similar modes that are also squeezed in the OPA. When squeezed light is produced by down-conversion of a pulsed pump laser, multiple spatial-temporal modes may be squeezed, and none of these modes is guaranteed to match the mode of the LO [18]. These other modes have similar spectra to the LO mode and therefore cannot be conventionally filtered. Detections due to photons in these modes degrade the fidelity of the CSSs. We quantify the effect of unwanted photons with the "modal purity" ξ n of n photon subtraction -the probability that, when the subtraction detector registers n photons, these n photons were from the mode matching the LO. To estimate the modal purities, we used a single-mode photon subtraction model to fit our data [15]. From this we determined ξ 2,TES = 0.62 and ξ 2,APD = 0.85, compared to ξ 1 = 0.91 for the onephoton subtraction experiment. The reason for the lower modal purity of the TES experiment is its greater sensitivity to stray photons from the LO. With the APDs, we can gate the detections to reject slightly delayed LO photons arising from downstream reflections. The TES is slower, so such gating is not possible. The main advantages of the TES are the greater efficiency and the ability to directly count photons. In the two-photon subtraction experiments, this higher efficiency resulted in improving the rate at which CSSs were heralded by a factor of three.
Three-photon subtraction events are extremely rare in our experiment. Nevertheless, using the TES we were able to detect 1087 three photon events over a period of approximately 60 hours. With three multiplexed APDs we would have collected only about 120 events. Fig. 4 shows the odd CSS. To increase the three photon event rate, we increased the reflectivity of the photon subtraction beam splitter to 20 %, sacrificing the fidelity of the CSS. The reconstructed state shows a negative min- −0.24 . The state has fidelity F = 0.59 +0.04 −0.14 with an ideal CSS of |α| = 1.76 +0.02 −0. 19 . The estimated modal purity in this experiment is ξ 3 = 0.84. Thus, we observed the predicted increase in CSS amplitude for three-photon subtraction, but the increase in fidelity is not statistically significant.
In conclusion, we have measured heralded optical CSSs created by subtracting up to three photons from a squeezed vacuum state, using APDs for one-and two-photon subtraction and a TES for two and three. It was only by taking advantage of the high efficiency and the direct photon counting capability of the TES that we were able to successfully subtract three photons with a sufficiently high rate of CSS production. The CSSs produced were analyzed by homodyne measurement and maximum-likelihood state estimation. The quality of the CSSs can be improved by reducing the losses experienced by the squeezed vacuum state before reaching the photon-subtraction beam splitter. For multi-photon subtraction, however it is crucial to reduce the presence of unfilterable photons in unwanted modes. A promising route that addresses both problems is to tailor the squeezing source to create squeezed light only in a single mode matched to the LO. This route is being pursued in the photon-pair generation community [19][20][21]. Based on our findings, we propose that the combination of pure vacuum squeezing and high efficiency detectors with photon-number-resolving capabilities can yield high rate, amplitude and fidelity CSSs to support quantum information processing and metrology beyond the quantum limit.
Added note: Recently, the authors became aware of a similar measurement that made use of photon-numberresolving transition edge sensors [22]. This work was supported by the NIST Innovations in Measurement Science Program. T.G. thanks P. Grangier and A. Ourjoumtsev for discussions. This is a contribution of NIST, an agency of the U.S. government, not subject to copyright.
This supplementary material gives more technical experimental details and deeper insight into the analysis than was given in the main paper. First, we give a detailed description of the experimental setup and our efforts to improve the fidelity of our squeezed vacuum and coherent state superpositions (CSSs). Then we present an analytical model of subtraction of up to three photons from a squeezed vacuum. Last we give a detailed analysis of experimental parameters and results. Figure A1 shows our experimental setup. We use a cavity-dumped femtosecond laser with transform-limited pulses of typically 140 fs duration and a repetition frequency of 548 kHz. The center wavelength of the cavity dumper output is λ 0 = 861.8 nm. Typical pulse energies are 40 nJ at the output port of the cavity dumper. We spatially filter the laser beam by sending it through a 30 µm diameter pinhole (PH). At the 90/10 beam splitter (BS) we split the laser beam into two parts. The weaker part of the beam is the strong local oscillator (LO) for the homodyne detection (>10 9 photons/pulse). The stronger part pumps a 150 µm thick KNbO 3 crystal (SHG) to generate the secondharmonic pump photons with a conversion efficiency of 25%. To eliminate any fundamental photons from the laser itself, we spectrally filter (SF) the second-harmonic pump. Then, the pump is focused into a 200 µm thick down-converting KNbO 3 crystal (OPA). Both crystals are temperature-tuned with stability better than 0.05°C for optimum phasematching. We optimized both crystal temperatures to achieve purest squeezing (T OPA = 28.3°C; T SHG = 27.5°C). During the course of one measurement we observe a decrease in the squeezing from the OPA, which is probably caused by a photorefractive effect present in our crystals [1]. This decrease typically happens within the first hour of the measurement and remains stable afterwards if we keep the pump focused at the same position. Our optics in the homodyne detection arm after the down-converter eliminate most (>99.9%) of the pump, and its contribution to the homodyne signal is negligible. Therefore, no further spectral filtering is required after the OPA. The squeezed vacuum is sent to our photon-subtraction beam-splitting components. The beam splitter consists of a half-wave-plate and a polarizing beam splitter cube (HWP 1 /PBS 1 ). This arrangement is helpful, as we can adjust the beam splitter's transmissivity, rather than having a fixed splitting ratio. The s-polarized waves experience less than 1% loss under reflection and the p-polarized waves undergo a 5% loss due to transmission through the polarizing beam splitter. The reflected photons are sent to the homodyne detection arm, and hence the loss in this arm is minimized. The subtracted photons are directed to our spectral filter setup. It consists of a fiber Bragg grating (FBG) and a circulator. The FBG has a bandwidth of ∆λ FWMH = 1.5 nm. The circulator is a combination of a free-space polarizing beam splitter and quarter-waveplate (PBS 2 /QWP). This setup allows for good filtering of the subtracted photons. However, two fiber couplers are in the path of the subtracted photons. This decreases the overall detection efficiency of the subtracted photons due to the limited fiber coupling efficiency. The subtracted photons that are within the FBG bandwidth reach the photon-number-resolving detector with a probability of about 20%. We used two different photon detectors: avalanche photo diodes (APDs) for the one and two photon subtraction, or one transition-edge sensor (TES) for the two and three photon subtraction experiments. Upon detection of at least one photon at the photon detector, we know that we have prepared an approximation of a CSS. Homodyne measurements are recorded regardless of whether a photon is subtracted or not. When no photons are subtracted, a noisy squeezed state is created, which we use to calibrate the phase of the LO. When a photon subtraction event occurs, the coincident homodyne measurement is tagged.
I. Experimental Setup
The temporal width of the strong LO is controlled by a pulse-shaping setup [2] so that we can compensate for the large mismatch in group velocity in our KNbO 3 crystals, which is ~1.2 ps/mm [3]. This group velocity mismatch causes the temporal width of the squeezing pulse to be about double that of the LO. Using two gratings (G), two lenses (L) and a spatial filter (PDF), we can tune the temporal width of the LO from 140 fs to about 300 fs. The spatial filter is a polka dot filter (PDF) with Gaussian transmission profile to minimize possible chirp imposed by the pulse-shaping setup. We lithographically made these polka dot filters on chromium masks. In order to minimize and randomize the interference of the LO, which was sent through the PDF, we randomly distributed 4 µm × 4 µm squares along the z-axis of the filter. The squares' density changed according to a Gaussian envelope transmission profile along the x-axis of the filter. The filter design is shown in inset (a) of figure A1. Different LO temporal widths are achieved by different PDF designs. The highest fidelity CSS results were obtained for an LO width of 230 fs. However, this is about 40 fs shorter than the width that results in the most pure squeezed states, where the squeezing is measured by direct homodyne detection. The phase of the LO is adjusted by a piezo-mounted mirror (φ). We continuously displace the mirror with a frequency of 2.75 Hz. The mirror's displacement is a saw-tooth profile with amplitude of about 2λ 0 . This allows a complete phase space measurement of our CSS. The LO and prepared CSS are combined at PBS 3 . A half-wave-plate (HWP 2 ) and a polarizing beam splitter (PBS 4 ) constitute the 50/50 beam splitter necessary for the homodyne detection setup. Adjustment of HWP 2 allows for very accurate balancing of the homodyne detection system. Both photodiodes are high-speed pin-Si photodiodes with high detection efficiency. Out of a set of 10 same-wafer photodiodes, we chose the two photodiodes that have the best matching temporal response. The electrical homodyne circuit is shown in inset (b) of figure A1. It consists of a charge integration stage and a 10× amplification stage. The charge integration includes one field effect transistor (FET) bridged across the integrating capacitor. The FET circuit induces a random offset charge. We correct for this charge by subtracting the voltage before from the voltage after each laser pulse. An intensity plot of a quadrature probability distribution for a one photon subtraction experiment is shown in inset (c). The quadrature data consist of 324,000 heralded events. These data are then processed, and maximum likelihood estimation gives the density matrix of the measured state.
The overall efficiency of the homodyne detection is η h = 0.853 ± 0.028. We estimate this by separately measuring four efficiencies η o , η d , η w , and η e and computing η h = η o η d η w η e . The four efficiencies are defined as follows: 1) The efficiency of all optical elements after the photon subtraction beam splitter to the face of the homodyne detector's photodiodes is η o = 0.94 ± 0.005. 2) The mean efficiency of the photodiodes is η d = 0.976 ± 0.022, where the difference in efficiency between the two photodiodes is less than 0.5%. 3) The efficiency of the mode matching between the squeezed mode and the LO is η w = 0.95 ± 0.005. η w was determined by interfering a probe beam with the LO.
However, the probe beam does not travel through the up-conversion crystal and its temporal width has not been altered. Therefore, for this measurement we did not alter the LO's temporal width. The LO is simply sent through the pulse shaper without a polka dot filter in place. (We are unable to measure the mismatch between the temporal width of the CSS and the LO, so it is not included in η h .) The error bar for η w accounts only for the statistics of the measurement, not for likely systematic biases. The effect of such biases on the inferred states is well below the states' statistical error as discussed in the paper. 4) η e is the efficiency that is formally equivalent to the electronic background noise. The electrical background noise of the homodyne detectors and electronics is e = V e /(V e + V v ) = 0.021 ± 0.001, where V e is the variance of voltages measured when no light enters the photodiodes and V e + V v is the variance observed when only the local oscillator is present (V e + V v includes both electronic noise and shot noise of the LO). This is formally equivalent to an efficiency η e = (1-e) = 0.979 ± 0.001 [4]. Our calibration of the overall efficiency of the homodyne detection, η h , does not include any excess noise from the down-conversion process itself, nor does it include the reflectivity of the beam splitter used for photon subtraction.
We model the squeezed vacuum generated in the experiment as a pure squeezed vacuum state with squeezed quadrature variance V 0 , which has passed through a medium with transmissivity η s . This state is then measured with the homodyne system whose efficiency is η h ; hence the measured efficiency is η m = η h η s . (When measuring the squeezing directly, we set the photon subtraction beam splitter reflectivity to 0.) We can calculate η m and V 0 by use of [5]:
( ) ( ) 0 1 1 V V m p − = − η ,(A1)and p q p q m V V V V − − − − = 2 ) 1 )( 1 ( η ,(A2)
where V p is the observed squeezed quadrature's variance, and V q is the observed antisqueezed quadrature's variance. When we measured the squeezing by homodyne detection directly, we observed V q = 3.129 (+5.0 dB) and V p = 0.565 (-2.5 dB). After correcting for the overall homodyne detection efficiency η h , we found η s = 0.64 and an inferred squeezing variance of V 0 = 0.205 (-6.8 dB), based on equations A1 and A2.
II. Photon subtraction from squeezed vacuum (model calculations)
In the following, we present an analytical model of the photon subtracted squeezed states. The model is similar to the model published by Ourjoumtsev et al. [1]. However, it considers up to three photons subtracted from the squeezed vacuum and does not assume that the efficiency of the photon subtraction detector is small. A2. Schematic of the model for calculation of the two-mode Wigner function. A squeezed state is sent into a beam splitter (BS 1 ), with transmissivity η s , to model the loss of the input noisy squeezed state. BS 2 models the photon subtraction at a beam splitter with reflectivity R. BS 3 represents the loss in the photon subtraction mode given by the subtraction arm efficiency (µ). The two triangles represent ideal photon counters, one of which detects k photons in mode 3, and the other detects n -k photons in orthogonal modes (all of which we label as mode 6). BS 4 models the known optical efficiency in the homodyne subtraction arm η h , which includes the optical loss, photodiode efficiency, wavefront overlap and electrical background.
FIG.
Unlike the model in [6], we do not consider multimode interference effects -we treat photons in modes not matched to the local oscillator as if they are equivalent to dark counts in the subtraction detector. The APDs are unable to distinguish the n photon subtraction events from n + 1 or higher numbers, so in this experiment and our model we accept a heralding of an n photon subtraction event when the detector registers n or more photons. Although the TES detectors can distinguish n from n + 1 photons we use the same model to analyze the TES experiments. Because the frequency of the n + 1 events is so much smaller than the n photon events, their contribution is negligible. Figure A2 shows a schematic for the model. Beginning with the pure squeezed state |S〉 (with squeezed variance V 0 ) in mode 1 and vacuum in modes 2, 3, 4, and 5 we calculate the state of this system as it evolves through the beam splitters, trace out the lost modes, and project mode 3 onto a k photon Fock state. We will account for the effect of dark counts in the subtraction detectors and photons in other modes entering the subtraction detector by including a second virtual photon counter, which registers n -k photons, so that the total number of photons detected is n. (A4)
We apply beam splitters 1 through 4 to this system and then perform the partial trace over modes 2, 4, and 5, leaving the state of modes 1 and 3:
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] hη µ η ρ η µ η ρ R R − − = . (A5)
Next we project mode 3 onto the k photon Fock state
( ) , ) ( 3 3 , 1 3 1 k P k k k ρ ρ = (A6)
where P(k)=Tr[ 3 〈k|ρ 1,3 |k〉 3 ] is the probability that ρ 1,3 contains k photons in mode 3.
Suppose mode 6 contains x photons with probability q(x). The probability that the two detectors together register n photons is In fact τ 1 (m) does not depend on all of the details of the distribution q(x), and we simplify these expressions using modal coupling parameters ξ n , for all n such that n is an integer, and 0 ≤ n ≤ m. We define the ξ n as where ρ id =Tr 3 (ρ 13 ). We call ξ m the "modal purity" because it is the probability that the m or more observed photons were actually subtracted from the mode matched to the local oscillator.
( ) ( ) ( ) ( ) ( ) ≤ ≤ − − =
In practice we calculate these states and their associated probabilities by use of their Wigner functions, according to the methods described in [7]. Eventually we obtain W(q, p), the Wigner function for τ 1 (m), which depends on V 0 , η s , R, µ, η h , the ξ n 's, and m.
We then perform a least squares fit of these Wigner functions to the Wigner functions reconstructed by maximum likelihood from the homodyne measurements to obtain estimates for the parameters V 0 , η s and the ξ n 's. From separate measurements we know R, µ and η h , and fix these parameters in the fitting routine. A package to calculate the Wigner functions based on the above model can be obtained from the authors upon request.
III. Experimental findings
We have used the above model to fit our experimental data and obtain the experimental parameters. Figure A3 shows the Wigner functions obtained by maximum likelihood reconstruction from the homodyne data. We used a maximum number of 20 photons for the reconstruction algorithm. Figure A3 also shows Wigner functions obtained by fitting our model to the maximum likelihood reconstructions. Table I lists our experimental findings. The beam splitter reflectivities were adjusted from 2.5 % to 20 % in the different photon-number-subtraction experiments. Higher fidelities are predicted for lower reflectivities, but we increased the reflectivity to increase the frequency of higher photon number subtraction events. For the fitting routine, the overall detection efficiency µ was set to 0.17 for the TES and to 0.08 for the APD measurements. Inferring other experimental parameters by fitting the model described above to the one photon subtraction measurements, we find η s = 0.72, V 0 = 0.229 (-6.4 dB), V q = 3.423 (+5.3 dB), V p = 0.445 (-3.5 dB) and ξ 1 = 0.91. The model finds a better fit using a higher squeezing purity than is directly measured with homodyne detection. This may be attributed to using our single-mode model to describe our multi-mode states. A thorough investigation based on the multi-mode model in [6] may clarify this discrepancy. The fits to the data reveal a constant squeezing purity of about η s = 0.72 for all one and two photon subtraction experiments. When we fit the model to the three photon subtraction data, we find that a squeezing purity of η s = 1 provides the best fit. However, we know from other measurements that η s < 0.75. The failure of the model in the three photon subtraction experiment may be caused by numerical difficulty obtaining the correct fit and/or multimode effects [6]. The modal purity ξ m of the subtracted photons is at least 0.84, except for the two photon TES experiment, where spurious LO photons scattered into the TES contribute to 38% false heralds, as determined from the modal purity. The scattered LO photons arrive 5 ns after the true signal photons; the delay is determined by the beam paths. Therefore, gating the APD with a window smaller than 5 ns suppresses the spurious LO contribution. This gating is possible only because of the APD's small jitter (≈400 ps). The TES (jitter ≈ 100 ns) does not allow for such accurate gating. Therefore the modal purity of the subtracted photons is lower in the TES case. Note that the lower modal purity could be improved. In our case the spurious photons originate from a reflection off one output port of the polarizing beam splitter that combines the CSS and the LO (PBS 3 in figure A1). For example, we could use a slightly wedged output port surface which would ensure the reflection into a spatial mode that is orthogonal to the subtraction arm's spatial mode.
The reason for the higher modal purity in the three photon subtraction experiment is that after increasing the subtracting beam splitter's reflectivity, the rate of subtracting three "good" photons was increased while the rate of detecting scattered LO photons was decreased. When reporting the fidelity of the states produced in our experiment, we maximize the fidelity over all ideal CSSs, obtaining the amplitude α of the highest fidelity CSS. The mean photon number 〈n〉 α of that CSS is calculated via ( ) ( )
FIG. 1 .
1(color online) Scheme for optical coherent state superposition (CSS)
FIG. 2 .
2(color online) Maximum likelihood estimate of an odd CSS generated by one-photon subtraction from a squeezed vacuum. The graph shows the unitless Wigner function value W (q, p) as a function of the unitless quadratures of the electromagnetic field.
FIG. 3 .
3(color online) Wigner functions of the maximum likelihood estimates of even CSS created by two-photon subtraction and heralded with (a) one transition edge sensor and (b) two multiplexed APDs.
FIG. 4 .
4(color online) Maximum likelihood estimate of an odd CSS after the subtraction of three photons from a squeezed vacuum. The reconstructed state has a fidelity of F = 0.592 +0.036 −0.142 with a CSS of amplitude |α| = 1.76 +0.02 −0.19. Inset: Wigner function of an ideal odd CSS with |α| = 1.76 imum of its Wigner function W min = −0.116 +0.073 −0.019 and a mean photon number of 2.75 +0.06
FIG. A1 .
A1Experimental Setup. The experiment setup consists of six main sections, labeled A-F. A: secondharmonic conversion of fundamental beam; B: squeezed vacuum generation; C: photon subtraction; D: spectral filtering of subtracted photons and detection thereof; E: temporal pulse-shaping of local oscillator; F: homodyne detection. The inset (a) shows the polka dot filter used in the pulse-shaping setup. Inset (b) shows the charge-integration circuit for homodyne detection, and inset (c) shows quadrature data for a full scan of the phase of a CSS created by one photon subtraction. PH: pinhole; BS: beam splitter; SHG: second-harmonic generation; SF: spectral filter; OPA: down-conversion crystal; HWP: half-wave-plate; PBS: polarizing beam splitter; QWP: quarter-wave-plate; FBG: fiber Bragg grating; #n: photon-numberresolving detector; PD: photodiode; HDC: homodyne detection circuit; φ: Piezo stage; G: grating; L: lens; PDF: polka dot filter
a heralding of an m photon subtracted state whenever m or more photons are detected, so we consider the statistical mixture of all σ 1 (n) for n≥m:
the modal coupling parameters can be interpreted as probabilities. Using the modal coupling parameters we can rewrite τ 1 (
for an odd CSS.
PACS numbers: 42.50.Dv, 42.50.Xa, 03.65.Ta, 03.65.Wj
TABLE I .
IResults for the one-, two-and three-photon subtraction experiments. Wmin and n are the minimum value and the mean photon number of the reconstructed state, respectively. F is the fidelity of the reconstructed state compared to a theoretical CSS with amplitude |α|W min
n
F
|α|
One-photon experiment:
APD
−0.041 +0.009
−0.001 1.96 +0.05
−0.04 0.522 +0.004
−0.010 1.32 +0.01
−0.02
Ref. [3] −0.13 ± 0.01
0.70
0.89
Two-photon experiments:
APDs −0.018 +0.002
−0.002 2.34 +0.06
−0.05 0.523 +0.022
−0.014 1.30 +0.04
−0.02
TES
−0.010 +0.001
−0.001 1.89 +0.05
−0.06 0.531 +0.017
−0.018 1.16 +0.04
−0.04
Ref. [7]
0.60
1.4
Three-photon experiment:
TES
−0.116 +0.073
−0.019 2.75 +0.06
−0.24
0.59 +0.04
−0.14
1.76 +0.02
−0.19
Table IExperimental findings. V 0 , η s and ξ n were obtained from a least squares fit of the above model. α, 〈n〉 α and F were obtained by comparing the maximum likelihood state estimate with a theoretical CSS that gave highest fidelity. 〈n〉 ρ is the average photon number in the reconstructed state. W min is the minimum of the reconstructed Wigner function. The photon subtraction beam splitter reflectivity R was determined by a separate measurement.datapoints
324,000
25,000
39,000
1087
integration time
~3 hours
~24 hours
~120 hours
~60 hours
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|
[] |
[
"Monolithic or hierarchical star formation? A new statistical analysis",
"Monolithic or hierarchical star formation? A new statistical analysis"
] |
[
"Marios Kampakoglou ",
"Roberto Trotta ",
"Joseph Silk "
] |
[] |
[
"Mon. Not. R. Astron. Soc"
] |
We consider an analytic model of cosmic star formation which incorporates supernova feedback, gas accretion and enriched outflows, reproducing the history of cosmic star formation, metallicity, supernovae type II rates and the fraction of baryons allocated to structures. We present a new statistical treatment of the available observational data on the star formation rate and metallicity that accounts for the presence of possible systematics. We then employ a Bayesian Markov Chain Monte Carlo method to compare the predictions of our model with observations and derive constraints on the 7 free parameters of the model. We find that the dust correction scheme one chooses to adopt for the star formation data is critical in determining which scenario is favoured between a hierarchical star formation model, where star formation is prolonged by accretion, infall and merging, and a monolithic scenario, where star formation is rapid and efficient. We distinguish between these modes by defining a characteristic minimum mass, M 10 11 M ⊙ , in our fiducial model, for early type galaxies where star formation occurs efficiently. Our results indicate that the hierarchical star formation model can achieve better agreement with the data, but that this requires a high efficiency of supernova-driven outflows. In a monolithic model, our analysis points to the need for a mechanism that drives metal-poor winds, perhaps in the form of supermassive black hole-induced outflows. Furthermore, the relative absence of star formation beyond z ∼ 5 in the monolithic scenario requires an alternative mechanism to dwarf galaxies for reionizing the universe at z ∼ 11, as required by observations of the microwave background. While the monolithic scenario is less favoured in terms of its quality-of-fit, it cannot yet be excluded.
|
10.1111/j.1365-2966.2007.12747.x
|
[
"https://arxiv.org/pdf/0709.1104v2.pdf"
] | 14,649,746 |
0709.1104
|
8bae44a0f96dcb9ee9530c7f2dfaa2efc98c67db
|
Monolithic or hierarchical star formation? A new statistical analysis
1 February 2008
Marios Kampakoglou
Roberto Trotta
Joseph Silk
Monolithic or hierarchical star formation? A new statistical analysis
Mon. Not. R. Astron. Soc
00000001 February 2008(MN L A T E X style file v2.2)Cosmology: galaxy formation history-Galaxies: star formation
We consider an analytic model of cosmic star formation which incorporates supernova feedback, gas accretion and enriched outflows, reproducing the history of cosmic star formation, metallicity, supernovae type II rates and the fraction of baryons allocated to structures. We present a new statistical treatment of the available observational data on the star formation rate and metallicity that accounts for the presence of possible systematics. We then employ a Bayesian Markov Chain Monte Carlo method to compare the predictions of our model with observations and derive constraints on the 7 free parameters of the model. We find that the dust correction scheme one chooses to adopt for the star formation data is critical in determining which scenario is favoured between a hierarchical star formation model, where star formation is prolonged by accretion, infall and merging, and a monolithic scenario, where star formation is rapid and efficient. We distinguish between these modes by defining a characteristic minimum mass, M 10 11 M ⊙ , in our fiducial model, for early type galaxies where star formation occurs efficiently. Our results indicate that the hierarchical star formation model can achieve better agreement with the data, but that this requires a high efficiency of supernova-driven outflows. In a monolithic model, our analysis points to the need for a mechanism that drives metal-poor winds, perhaps in the form of supermassive black hole-induced outflows. Furthermore, the relative absence of star formation beyond z ∼ 5 in the monolithic scenario requires an alternative mechanism to dwarf galaxies for reionizing the universe at z ∼ 11, as required by observations of the microwave background. While the monolithic scenario is less favoured in terms of its quality-of-fit, it cannot yet be excluded.
INTRODUCTION
Massive galactic spheroids form either by hierarchical buildup or monolithically. The former scenario is favoured by the theory of, and evidence for, cold dark matter (CDM), the latter by some observations, most notably the evidence for down-sizing in stellar mass and on chemical evolution timescales from the [α/Fe] enhancement at high spheroid masses (Worthey et al. (1992)). The conventional view of hierarchical build-up fits in with semi-analytical galaxy formation simulations provided that suitable feedback models are prescribed (Croton et al (2006), Bower et al. (2006)). Gas cooling drives star formation below a critical halo mass and AGN quenching of star formation occurs at higher masses where the cooling is inefficient. This model reproduces the shape of the galaxy luminosity function at the high mass Not all studies agree on the role of major mergers. Not all luminous starbursts are triggered by such mergers. Indeed, in the case of disk galaxy formation, their role is likely to be small. There is evidence that major mergers are not exclusively responsible for the dominant star formation episodes in starbursts forming stars at up to 200 M⊙/yr where thick disks are seen (e.g. Hammer et al. (2005)). Such high star formation rates, also found in disks at z ∼ 2 by Forster-Schreiber et al. (2006), may favour an interpretation in terms of bulge formation since the disks are already present. The observed major merger rate is small out to z ∼ 1.2 despite the strong increase in the comoving star formation rate (Lotz et al. (2006)). Of course, ULIRGs, with star formation rates of > ∼ 500M⊙/yr, are examples of ma-jor merger-induced star formation. However, for the bulk of star formation in the early universe, there is little evidence to suggest that major mergers play a significant role. This motivates a hierarchical picture of prolonged minor mergers to build up mass. One can ask more generally whether minor mergers or gas cloud accretion are responsible for the gas supply. With regard to minor mergers, the answer seems to be negative, because the time-scale for supplying the gas would be many dynamical time-scales. Yet there are two persuasive arguments for a short star formation time-scale in massive galaxies. SED modelling suggests down-sizing, the most massive galaxies forming first and with star formation time-scales of order a dynamical time. The [α/Fe] enhancement with increasing spheroid mass independently reinforces this conclusion, since most of the star formation must have occurred before dilution of the star-forming gas by supernovae type Ia ejecta occurred. This result is not, or at least not yet, found in semi-analytical modelling of massive galaxy formation. It provides a strong argument for monolithic formation of massive galaxies, although one cannot of course exclude the possibility that hierarchical formation models will reproduce a similar result once more complex star formation rules are introduced. Indeed, simple spherically symmetric accretion models have been proposed (cf. Birnboim et al. (2007)) that allow the gas to accumulate in an isothermal halo atmosphere prior to an accretion-triggered burst of star formation. In essence, this approach introduces a monolothic formation model in combination with hierarchical gas accumulation. Unfortunately there is little evidence today for such gas-rich halos, suggesting that this process, if important, would only have been influential at very early epochs.
To save hierarchical models, one needs an efficient way of converting gas into stars. The typical gas-to-star efolding time in nearby well-studied sites of star formation, namely spiral galaxies, is several Gyr. By contrast, at a redshift of 2 − 3, the star formation time-scale is as short as 0.2 Gyr from both spectrophotometric SED filling (Maraston et al. (2006)) and [α/Fe] analyses (Thomas et al. (2005)). The AGN phenomenon is invoked for stopping star formation by quenching the gas supply. This may keep massive galaxies red, but may not be enough to produce massive galaxies sufficiently early. One resolution may be to invoke positive feedback from AGN-driven outflows that overpressure protogalactic clouds and trigger star formation on a rapid time-scale (Silk (2005)). Whether or not this particular solution turns out to resolve the dilemma (if indeed the problem persists in the perspective of improved star formation modelling) is not the issue we focus on in this paper. Rather we revisit the case for hierarchical versus monolithic galaxy formation in terms of accounting for the data on star formation rate and chemical evolution.
In this paper we apply a novel statistical approach that rigorously treats the key parameters in semi-analytical galaxy formation theory, with emphasis on reproducing the cosmic star formation and chemical evolution histories. Numerical modelling via semi-analytical techniques of a large box of the universe takes up so much computer time and memory that it is impossible to test the robustness of the results. Here we focus on probing the key parameter space by means of analytical techniques combined with appropriate binning of the full data sets. We find that the characteristic minimum mass for the building blocks of massive galaxies plays a central role. Specifically, the dust correction scheme one chooses to adopt for the cosmic star formation history data is one of the most critical factors in determining the balance of evidence in support of a hierarchical star formation model as opposed to a monolithic scenario, where star formation happens predominantly in massive spheroids. Our results indicate that the hierarchical star formation model can achieve better agreement with the data, but that this requires a high efficiency of supernova-driven outflows. In a monolithic model, our analysis points to the need for a mechanism that drives metal-poor winds, perhaps in the form of supermassive black hole-induced outflows. While the monolithic scenario is less favoured in terms of its quality-of-fit, it cannot yet be excluded.
This paper is organized as follows: our star formation model is introduced in section 2, and the dependence of the observable quantities on the free parameters of the model explored in section 3. We then describe the data employed and our statistical procedure in section 4. Our results are presented in section 5 and our conclusions discussed in section 6. Appendix A gives details of our binning procedure for star formation rate and metallicity data which accounts for undetected systematics.
STAR FORMATION MODEL
In this section we present a physical model of the cosmic star formation incorporating supernova feedback, gas accretion and enriched outflows. Our model builds upon the model described in Daigne et al. (2005), and more specifically their "Model 0". The main difference at the model level is our choice of using a different chemical evolution model. In fact, we adopt the instantaneous recycling approximation whereas in Daigne et al. (2005) the metallicity is computed under the delayed enrichment approximation. There are several differences in terms of the statistical treatment of the data and the fitting procedure, that in this work are significantly more sophisticated, as explained in section 4 and Appendix A.
Governing equations
The description of baryons in the Universe and the processes that define the evolution of the baryonic mass are of fundamental importance for our model. Following Daigne et al. (2005), we employ three baryon reservoirs in the model, encompassing the interstellar medium (gas), the mass in stars and the intergalactic medium (IGM). We denote by Mgas the mass of gas, by M⋆ the mass in stars and by Mstruct = Mgas + M⋆ the total mass in collapsed structures. The IGM and the structures exchange mass through accretion and outflow, while the interaction between stars and gas is governed by star formation and ejection of enriched gas. In the instantaneous recycling approximation adopted here, the accretion rate of the mass in stars is simply equal to the star formation rate (SFR), Ψ(t), i.e.
dM⋆ dt = Ψ(t).(1)
We then have the following set of differential equations governing the evolution of the mass in the three reservoirs:
dMIGM dt = − dMstruct dt (2) dMstruct dt = a b (t) − o(t) (3) dMgas dt = dMstruct dt − dM⋆ dt (4)
In the above equations, a b (t) is the rate of baryon accretion while o(t) is the rate of baryon outflow. The latter includes two terms, accounting for winds powered by stellar explosions and supernova ejecta. We neglect supernova ejecta since this effect was found to be subdominant (Daigne et al. (2004)). The relation between physical time t and redshift z is given by
dt dz = 9.78h −1 Gyr (1 + z) p ΩΛ + Ωm(1 + z) 3 ,(5)
where we have assumed a flat Universe. In the following, we will use Eq. (5) with parameters fixed to the values of the ΛCDM concordance model, i.e. a matter density parameter Ωm = 0.27 and a cosmological constant energy density ΩΛ = 0.73 (both in units of the critical energy density of the Universe), and we will take for the Hubble constant H0 = 100h km/sec/Mpc = 71.
Accretion
We adopt the hierarchical scenario of structure formation, where small structures are formed first. At redshift z, the comoving density of dark matter halos in the mass range [M, M +dM ] is fps(M, z)dM , normalized in such a way that
Z ∞ 0 dM M fps(M, z) = ρDM,(6)
where ρDM is the comoving dark matter density. The distribution function of halos fps(M, z) is computed using the method described in Jenkins et al. (2001) using code provided by A. Jenkins. It follows the standard theory (Press & Schechter (1974)) including the modification of Sheth & Tormen (1999). We assume a primordial power spectrum of fluctuation with a power law index nS = 1 and the fitting formula to the exact transfer function for non-baryonic cold dark matter given by Bond &Efstathiou (1999). For the rms amplitude we adopt a value σ8 = 0.9 for mass density fluctuations in a sphere of radius 8h −1 Mpc. Using the above expressions for the distribution function of dark matter halos, we can calculate the fraction of baryons at redshift z that is allocated to structures, by assuming that the baryon density is proportional to the dark matter density, with a proportionality factor given by the ratio of visible to dark matter density -in other words, we assume that light traces matter with no bias. The fraction of baryons in star-forming structures at redshift z is then given by
f bar (z) = R ∞ M min dM M fps(M, z) R ∞ 0 dM M fps(M, z) ,(7)
where Mmin is a free parameter controlling the minimum mass (in units of solar masses) of the collapsed structures where star formation can occur.
The accretion rate is then given by (Daigne et al. (2005)):
a b (t) = Ω b 3H 2 0 8πG " dt dz « −1 | df bar dz | (8)
Given a value of Mmin (that we adopt as a free parameter, see below), we fix the redshift at which star formation begins, zinit by the requirement that f bar (zinit) = 0.01. In other words, the first stars form in collapsed haloes of mass larger than Mmin when the fraction of baryons allocated to such structures is more than 1%. We adopt a fixed baryonic density parameter of Ω b = 0.044 (from the posterior mean of WMAP 3-years data combined with all other datasets, Spergel et al. (2006)).
Outflow
The adopted stellar initial mass function (IMF) is of the form
Φ(m) = B " m M ⊙ « −(1+x) , for m l < m < mu(9)
where the normalization constant B is fixed by the requirement that
Z mu m l mΦ(m)dm = M⋆(10)
For the limits of integration we fix m l = 0.1M ⊙ and mu = 100M ⊙ (Pagel (1997)). Therefore the only parameter needed to define the IMF is its power-law index, x.
The quantity x is used as a free parameter in this model. We model the outflow powered by stellar explosions as follows:
o(t) = 2ǫ v 2 esc (z) Z 100M ⊙ m 0 dmΦ(m)Ψ(t−τs(m))E kin (m) (11) where m0 = max(8M⊙, m d (t))(12)
and Φ(m) is the IMF defined above, τs(m) is the lifetime of a star of mass m and m d (t) is the mass of stars that die at age t. Furthermore, E kin (m) is the kinetic energy released by the explosion of a star of mass m, that we take to be a fixed constant independent of mass, E kin (m) = 10 51 ergs (a mass-dependence could easily be taken into account). The free parameter ǫ controls the fraction of the kinetic energy of supernovae that is available to power the winds, and v 2 esc (z) is the mean square of the escape velocity of structures at redshift z.
In order to compute the stellar lifetime τs(m) we assume it to be equal to the time that a star of mass m spends on the main sequence. So the age of a star of mass m is given by
τs(m) = (m/M⊙) −2.5 t⊙(13)
where t⊙ is the total time that a star of mass M = M⊙ will spend on the main sequence and we adopt a value t⊙ = 9 Gyr. To compute m d (t) in Eq. (12) we solve Eq. (13) for m, thereby obtaining the mass of stars m d (t) that die at age t.
The escape velocity is obtained by assuming virialized halos and averaging over the distribution function, thus ob-
taining 1 : v 2 esc (z) = R ∞ M min dM M fps(M, z)(2GM/R(M )) R ∞ M min dM M fps(M, z)(14)
where R(M ) is the radius of a dark matter halo of mass M given by the following expression:
R(M ) = " 3M 178ρc(Ωm(1 + z) 3 + ΩΛ)4π « 1/3 ,(15)
where ρc is the critical density of the universe today.The factor 178 is the overdensity (relative to the critical density) at virialization for an Einstein-de Sitter model (Coles & Lucchin (1995)).
Star formation and supernova rate
Following Daigne et al. (2005), we adopt an exponentially decreasing SFR:
Ψ(t) = νMstruct(t) exp(−(t − tinit)/τ ),(16)
where tinit is the time corresponding to the redshift zinit when star formation starts (as defined above), τ is a characteristic time scale that we take as a free parameter and ν is a normalization parameter (with dimensions of inverse time). The supernova rate (SNR) is strongly linked to the star formation rate because of the short lifetime of massive progenitors with M > 8M⊙. We can therefore assume that core collapse supernovae (SNe) are strongly correlated with instantaneous SFR, and the supernovae type II rate Ψsn(t) is given by
Ψsn(t) = Z mu 8M ⊙ Φ(m)Ψ(t − τs(m))dm.(17)
Chemical evolution model
Chemical evolution is included in the model using the instantaneous recycling approximation. i.e. we assume that all processes involving stellar evolution, nucleosynthesis and recycling take place instantaneously on the timescale of galactic evolution. The equation of galactic chemical evolution is (Pagel (1997))
Σg dZ dt = qΨ(t) + (ZF − Z(t))a b (t) − (η − 1)Z(t)o(t),(18)
where Σg is the density of the gas (in units of M⊙/Mpc 3 ), η is a multiple of the nucleosynthetic yield that parameterises the metallicity Z of the SN ejecta (Dalcanton (2006)) (also called "the load factor", adopted here as a free parameter) and q is the yield. We fix the value of the yield to q = 0.02 and assume that the mass accreted to the disk has zero metallicity, i.e. we fix ZF = 0). Furthermore, we normalise the metallicities to the solar value for which we adopt Z⊙ = 0.02. The chemical evolution of the ISM, described by Eq. (18) contains three terms. The first one represents the chemical enrichment due to the evolution of stars. The second term represents the dilution of metallicity (if ZF < Z(t)) or the chemical enrichment (if ZF > Z(t)) of the ISM due to accreted material. The last term describes the dilution of metallicity (if η > 1) or the chemical enrichment (if η < 1) of the ISM due to galactic winds powered by stellar explosions.
In recent theoretical work, what has been dubbed "the missing metals problem" has received considerable attention (see Prochaska et al. (2003) for an extensive discussion of this problem), namely the fact that the mean metallicity is ∼ 10 times lower than the value expected from the inferred star formation history. This problem may indicate a serious flaw in our understanding of the interplay between star formation and metal enrichment. Therefore, we have introduced in our chemical evolution model an extra parameter f dil accounting phenomenologically for these effects. This allows the metallicity predictions of Eq. (18) to be adjusted to match observational data. Thus we rescale the metallicity values given by solving Eq. (18) by a factor f dil , i.e.
Z(t) = Z(t)/f dil .(19)
Summary of model parameters
To summarize, our model is characterized by a set of 7 free parameters, that we denote by θ:
θ = (log Mmin, ǫ, x, ν, τ, η, f dil )(20)
The free parameters of the model (and the ones that we have chosen to fix) are summarized in Table 1, where we also give the prior ranges for our statistical analysis, i.e. the ranges within which their values are allowed to vary (see 4.2 for more details).
INFLUENCE OF MODEL PARAMETERS ON OBSERVABLE QUANTITIES
In this section, we discuss the impact of each of the 7 free parameters in our model (given in Eq. (20) and in the upper part of Table 1) on the physical observables introduced above, namely the SFR, SN type II rate, metallicity and baryonic fraction in structures. We also present a physical interpretation of the observed behaviour of these quantities. As a fiducial model we fix the parameter values to the following values: log Mmin = 8, ǫ = 0.1, x = 1.7, τ = 3, ν = 1.4, η = 10 and f dil = 2. We then proceed to vary one of the parameters at a time to get a feeling for the physical impact of each of them. Figure 1 shows the model dependence on the minimum mass of collapsed dark matter halos, log Mmin. Smaller values of this parameter describe a scenario where star formation is hierarchical and follows the growth of structures, while higher values of log Mmin correspond to star formation occurring in massive spheroids. Correspondingly, for small log Mmin star formation begins earlier, as apparent from the top panel of Figure 1. At small redshift, a smaller log Mmin leads to reduced SFR, since the relatively strong winds (ǫ = 0.1 in this example) drive the gas out of the system for shallower potentials. For large log Mmin, the buildup of metals is delayed in time but the metallicity can reach larger values, since supernova-powered winds are less important in massive systems (middle panel). As it is clear from Eq. (7), the percentage of baryons allocated to dark matter halos f bar increases for decreasing log Mmin (bottom panel). Due to the short lifetime of massive progenitors, the SN rate is essentially identical to the SFR, and we therefore do not display it.
In Figure 2 we show the model sensitivity to the parameter ǫ, defining the percentage of supernovae energy that goes to the ISM. This parameter essentially describes the strength of galactic winds. The physical interpretation of high values of ǫ is that strong winds driven by feedback energy are maintained in dark matter halos. For increasing value of ǫ, galactic winds become stronger and the star formation rate is reduced since less gas is available to make stars (top panel). This effect is more important for the shallower gravitational potential of the low mass halos, i.e. for smaller log Mmin (in this example, log Mmin = 8). As already remarked above, higher values of ǫ corresponds to winds sweeping out metals from the ISM and thus to lower metallicity (bottom panel). Again, this effect is most important for the low mass halos where their gravitational potential is relatively shallow.
The sensitivity to the parameter x, giving the slope of the initial mass function, is shown in Figure 3. We can see a strong influence of x on the supernovae type II rates (middle panel), a consequence of Eq. (17). Decreasing the value of x (i.e., making the IMF shallower) corresponds to a larger number of more massive stars, and hence the supernovae type II rate increases. Taking into account that in our model each supernova gives a constant percentage of its energy to the ISM, small values of x result in stronger galactic winds and so in smaller star formation rates (top panel) and a less enriched ISM, hence smaller metallicity (bottom panel). For the extreme case that x = 1 (very flat IMF) the supernovae type II rate is very large at high redshift causing very strong winds that reduce the SFR quickly. This causes the spike in Figure 3 (middle panel). Figure 4 shows the model sensitivity to the parameter ν, connected with star formation through the proportionality factor that defines the efficiency of star formation, see Eq. (16). Increasing the value of ν the star formation becomes more efficient and the interstellar medium becomes highly enriched in metals by evolving stars. On the contrary, smaller values of ν lead to a less efficient star formation.
The influence of the parameter τ , defining the characteristic timescale of star formation, is displayed in Figure 5. Decreasing the value of τ leads to the star formation activity ending sooner and to an ISM which is therefore poorer in metals. Larger values of τ result in an enriched ISM since galaxies are active, in terms of star formation, for a longer period.
The influence of the parameter η, controlling the metallicity of the ejecta, is displayed in Figure 6. A larger value of η leads to a decrease in metallicity of the system, since the metallicity of the winds is increased by a factor of η wrt the mean metallicity, see Eq. (18). Finally, we do not display the impact of the dilution factor f dil , since its value merely rescales the metallicity by a multiplicative factor, see Eq. (19).
We now turn to discuss the data employed and the details of our statistical treatment and fitting procedure.
DATA AND STATISTICAL ANALYSIS
Combining different types of observations to maximise their constraining power on multi-dimensional parameter spaces has become a common approach in cosmology. Following an approach similar in spirit, in this work we perform a simultaneous analysis of star formation history, SN rates, metallicity and baryonic fraction data in order to find tight constraints on the parameters of our model, Eq. (20). One of the aim of this paper is to provide the first complete statistical analysis of existing metallicity, SFR, SNII rate and local collapse baryon fraction data in a realistic model. We first describe the data employed in section 4.1, then we outline the Bayesian fitting procedure that vastly improves on usual fixed-grid scans in section 4.2.
Observational constraints
The usual compilations of measurements for the SFR and metallicity observations (such as the ones used e.g. in Daigne et al. (2005) are unsuitable for a robust statistical analysis, because of the large systematic differences among measurements at about the same redshift performed over a range of different systems. In fact, when using such a "raw" data compilation the statistical fit is usually dominated by only a few data points with very small errorbars, while the large majority of observations carry almost no statistical weight. This is clearly less then satisfactory. To cure this effect, it becomes important to bin the observations in such a way as to account for possible systematic uncertainties among different measurements at the same redshift. This problem is addressed here for the first time by employing a Bayesian procedure that accounts for possible systematic differences between measurements, based on the treatment given in (Press (1996)). The details of the method are given in Appendix A.
We apply the binning procedure described in the Appendix to the data points for the metallicity in the interstellar medium (ISM) given by Prochaska et al. (2003). By using Eq. (A3) we place the 125 measurements in 8 bins, ranging in redshift from z = 0.85 to z = 4.45. The bin distribution and spacing has been chosen to obtain a rea- sonable large number of points in each bin, while simultaneously having a sufficiently small redshift spacing between bins. The measurements of [M/H] number density relative to solar metallicity obtained after the statistical rebinning, are summarized in Table 2.
For the case of cosmic SFR data, our statistical rebinning is modified in order to take into account the redshift uncertainty in the raw data. Details are given in Appendix A2. We take the compilation of "raw" data out to z ∼ 5 from Hopkins (2004), excluding only one measurement corresponding to the cosmic star formation at z = 0.005±0.005, reported by Condon (1989). Instead, we replace this point by more recent measurement at the same redshift as reported by the same author (Condon et al. (2002)). Both these measurements use as cosmic star formation estimator counts at 1.4 GHz. From the raw data we derive binned values in 12 redshift bins, with centers ranging fron z = 0.035 to z = 5.12 by using Eq. (A6). The resulting bins with their errors are summarized in Table 3.
Furthermore, the SFR predictions of our model are corrected to account for dust absorption. There are large uncertainties associated with dust absorption correction, this is why at low redshift (i.e., for bins with z b 3) we employ both a "normal dust correction" of 1.0 mag and a "large dust correction" of 1.8 mag. These two choices are made in view of the fact that they seem to bracket the expected values valid over a broad range of systems (Schiminovich at al. (2005)). For bins at a higher redshift (z b > 3) we adopt a fixed dust correction of 0.4 mag, following Schiminovich at al. (2005). We shall see in the next section that the dust absorption correction scheme one adopts has a crucial impact on the resulting physical scenario.
The present-day fraction of baryons in structures, as estimated by Fukugita & Peebles (2004) is taken to be f bar (z = 0) = 0.61±0.11.
The data for the core collapse supernovae are taken from the Great Observatories Origins Deep Survey (GOODS, Dahlen et al. (2004)). The GOODS core collapse supernovae rates have been placed in two bins at z = 0.3±0.2 and z = 0.7±0.2. For the local rate (at z = 0) we adopt the value from Cappellaro et al. (1999). We convert the local rate from supernovae units as described in Figure 6. Sensitivity of the metallicity to the parameter η. Curves are for η = 5, 10, 15, 20, from thin to thick. Table 3. SFR density data after our statistical binning of the the "raw" SFR data compilation (see Appendix A2 for details). No dust correction has been applied to this values. Dahlen et al. (2004). The 3 above mentioned data points are summarized in Table 4.
Redshift SFR density z b log(ρ⋆)[M ⊙ yr −1 Mpc −3 ] 0
Bayesian Markov Chain Monte Carlo analysis
After the statistical rebinning of the data described above, the likelihood function P (d|θ) is the sum of four independent terms, describing the observations of the SFR, the metallic- ity, the SN rate and the baryonic fraction, respectively:
P (d|θ) = LSFR + Lmet + LSN + L b .(22)
We model each of the above four terms as a product of the data points for each observable, taken to be independent and with Gaussian noise
χ 2 obs = −2 ln L obs = N obs X i=1 (yi − di) 2 σ 2 i(23)
where "obs" stand for SFR, metallicity, SN or baryon fraction in structures, and the means di and standard deviations σi of the data points are given in Tables 2-4 and Eq. (21). The normalization constant does not matter, as we are only interested in the relative posterior probability density, as we now discuss. From the likelihood function of Eq. (23) we obtain the posterior probability for the parameters of interest, P (θ|d), via Bayes' theorem,
P (θ|d) = P (d|θ)P (θ) P (d) ,(24)
where P (θ) is the prior probability distribution ("prior" for short) and P (d) is a normalization constant that does not depend on the parameters and can therefore be neglected in the following (see Trotta (2007a); Trotta (2007b) for more details on Bayesian parameter inference and model comparison). We adopt flat (i.e., top-hat) priors on our set of parameters θ given in Eq. (20) in the ranges given in Table 1, which means that the posterior probability distribution function (pdf) is simply proportional to the likelihood. In order to explore efficiently our our 7-dimensional parameter space, we employ a Markov Chain Monte Carlo (MCMC) procedure, with some of the routines adapted from the publicly available cosmomc package 2 . The great advantages of MCMC methods are that the computational time scales approximately linearly with the number of dimensions of the parameter space, and that the marginalized posterior distribution for the parameters of interest and their correlations can be simply recovered by plotting histograms of the sample list. We follow the procedure outlined in de Austri et al. (2006), to which we refer for further details. Here we only briefly sketch the main points.
The aim of an MCMC is to produce a series of samples in parameter space (a Markov Chain) with the property that the density of points is proportional to the probability distribution (the target density) one is interested in mapping, in our case the posterior pdf of Eq. (24). There are several algorithms that can produce a chain with the required properties. Here we employ the Metropolis-Hastings algorithm (Metropolis et al. (1953);Hastings (1970)): the chain is started from a random point in parameter space, θ0, and a new point θ1 is proposed with an arbitrarily proposal density distribution q(θn, θn+1). The transition kernel T (θn, θn+1) gives the conditional probability for the chain to move from θn to θn+1, and it must satisfy the "detailed balance" condition P (θn+1|d)T (θn+1, θn) = P (θn|d)T (θn, θn+1)
so that the posterior P (θ|d) is the stationary distribution of 2 Available from cosmologist.info the chain. This is achieved by defining the transition kernel as T (θn, θn+1) ≡ q(θn, θn+1)α(θn, θn+1),
α(θn, θn+1) ≡ min 1, P (θn+1|d)q(θn+1, θn) P (θn|d)q(θn, θn+1) ff ,(26)
where α(θn, θn+1) gives the probability that the new point is accepted. Since P (θ|d) ∝ P (d|θ)P (θ) and for the usual case of a symmetric proposal density, q(θn, θn+1) = q(θn+1, θn), the new step is always accepted if it improves on the posterior, otherwise it is accepted with probability P (d|θn+1)P (θn+1)/P (d|θn)P (θn). The result is a sample list from the target distribution, from which all the statistical quantities of interest can readily be evaluated. Further details about MCMC methods can be found e.g. in MacKay (2003). Our Bayesian MCMC analysis allows us to not only to determine efficiently the best-fit value of the parameters, but also to explore correlations between the model parameters and estimate marginalized high probability regions, to which we now turn our attention.
RESULTS AND DISCUSSION
As mentioned above, we investigate two different dust correction schemes for SFR data at low (z < 3) redshift, one termed "normal dust correction" and the other "high dust correction". This is expected to roughly bracket the range of possible corrections. The outcome of our analysis is strongly dependent on which dust correction one chooses to employ, with the normal dust correction implying hierarchical star formation, while the high dust correction favours the monolithic scenario.
Best fit models and parameter constraints
The values of the best-fit model parameters for both dust correction schemes are given in Table 5, and the corresponding SFR, SN rate, metallicity evolution and baryonic fraction in structures are shown in Figure 7. The 1-dimensional posterior probability distributions (with all other parameters marginalized, i.e., integrated over) are plotted in Figure 8. We first discuss the case with the normal dust correction applied. In order to fit the (dust-corrected) SFR at both high and small redshifts, the model requires a small minimal mass (log Mmin ∼ 6) and strong winds (ǫ ∼ 0.3). Although the value of the supernova energy transfer parameter is quite large, it is not too far away from theoretical predictions, which give an upper limit of ǫ = 0.22 (Larson (1974)). An IMF power-law index x ∼ 1.8, slightly larger then the Scalo IMF, is also preferred, which translates in fewer available supernovae. This is linked to the high value of ǫ, since the energy transfer is so efficient that a large number of supernovae is not needed to get the appropriate feedback energy to reproduce the data sets. The metallicity load factor η can be connected with the IMF power-law index x and Dalcanton (2006) gives η values for a variety of IMFs. The value of η for the Scalo (1986) IMF (x = 1.7, close to our best fit value, x = 1.77) is η = 16.8 − 18.6, in reasonable agreement with our value, η = 8.74. This leaves metal-rich outflows as Table 5. In the top left panel, showing the SFR, the low redshift (z 3) data have been corrected for dust employing a normal dust correction (1.0 mag, lower data points) or a high dust correction (1.8 mag, upper data points). The high dust correction data have bee shifted slightly to the right for display purposes.
the only viable mechanism for producing the low effective yields observed in gas-rich galaxies. in agreement with suggestions presented in Dalcanton (2006). The dilution factor f dil is of order 2, which again is very reasonable, given the complex physics this parameter is supposed to summarize. The value of the chi-square of the best fit model in this case is 26.60 for 17 degrees of freedom, which suggests that our model captures the essential features of the data. Figure 7 shows the best fit models for normal dust correction (solid line) and high dust correction (dashed line). Both models provide an acceptable fit to the data, although in the normal dust correction case the low-redshift metallicity and the present-day baryon fraction in structures appear in better agreement with the data. For a redshift above z ≈ 5, the metallicity of the hierarchical model drops very sharply to 0 because of the very significant winds.
Turning now to the high dust correction case, we notice that the preferred values of the parameters in our model are very different from the previous case. Most importantly, a high dust correction at small redshift boosts the value of the SFR for z 3, and this pushes our model to very large values of log Mmin, of the order log Mmin ∼ 11 − 12. This implies that star formation occurs monolithically in heavy spheroids, as discussed in the introduction. We expect dry mergers to play a significant role in the build-up of massive log Mmin ∼ 13 ellipticals in agreement with observations showing that present-day spheroidal galaxies on average have undergone between 0.5 and 2 major dry mergers since z ∼ 0.7 (Bell et al. (2006)). Furthermore, we see from Figure 7 (dashed curves) that the onset of both the SFR and metals build-up is significantly delayed in this scenario, until about z ≈ 5. The supernovae energy transfer parameter ǫ becomes essentially irrelevant for such large values of the minimum mass, since the potential is deep enough to retain the ejected gas. The peak in the probability distribution for ǫ observed in Figure 8 is therefore mostly a consequence of a volume effect of our Bayesian MCMC scanning technique. The star formation timescale τ ∼ 3.5 Gyr is in good agreement with theoretical models for Milky Way size disk galaxies (with virial mass log Mvir close to our best fit value for log Mmin). The IMF index is tilted towards extreme values, thus reducing the SN rates but boosting the SFR (compare Figure 3). This in turns increases the metallicity, and a large dilution factor, f dil ∼ 20, is required to bring the predictions in line with observations. We notice that this agrees within a factor of two with the value already found in previous Table 5. Best-fit parameter values and marginalized 68% and 95% intervals for the normal (1.0 mag for z 3) and high (1.8 mag for z 3) dust corrections. For cases where only an upper or lower limit is found within our prior ranges, we give one-tail intervals. We also give the best-fit chi-square and the reduced chi-square, where the number of degrees of freedom (dof) is 17 (for 24 data points and 7 free parameters).
works on the metallicity of SN ejecta, which was of order 10. However, the extremely steep IMF that this model prefers (x ≈ 2) appears to exclude the possibility that stellar explosions are the main mechanism that drive galactic winds. This is reasonable, since supernova-driven gas flows cannot escape from massive galaxies' potential wells. A resolution to this wind dilemma could come from the hypothesis of supermassive black hole (SMBH) induced outflows (Silk (2005)). In fact, the very low value of the load factor (η = 0.02) is consistent with this scenario since the SMBH undergoes most of its growth in the gas rich phase and its outflow expels mostly unprocessed gas. Although our model does not include the physics of SMBHs, it is tempting to say that our best fit model suggests that SMBHs should play a key role in the evolution of massive spheroids.
In general, we observe that the high dust correction case seems to stretch our model parameters to extreme values, suggesting either a strong tension between datasets (mostly SFR and metallicity data) or a failure of the model to fully encapsulate all of the relevant physical processes. Even though with a reduced chi-square per dof of 2.0 this scenario is less favoured than the hierarchical star formation model discussed above, it appears that the monolothic formation model cannot be dismissed yet. It is interesting that our 7 parameter model is able to describe both cases, and that the SFR dust correction plays a major role in defining which scenario is preferred.
Correlations among parameters
We now turn to discuss the most relevant correlations among the model parameters in light of their physical interpretation Figure 9. Contours enclosing joint 2D 68% and 95% regions, with all other parameters marginalized, for both the "high dust correction" (dashed, monolithic scenario) and the "normal dust correction" case (solid, hierarchical star formation). and of their impact on the observables, as shown in section 3. Figure 9 shows a selection of 2D joint posterior probability distributions for log Mmin, ǫ, x, ν, τ , and f dil , thus giving complementary information to the 1D distributions plotted in Figure 8. The contours enclose joint 2D 68% and 95% regions, with all other parameters marginalized, for both the "high dust correction" (dashed) and the "normal dust correction" case (solid).
In the first panel on the left of Figure 9, showing the x − ǫ plane, we observe a positive correlation between the IMF power-law index and the SN type II energy efficiency factor. This is expected, since an IMF with a higher powerlaw index produces less SNe, each of which has to contribute more energy, leading to higher values for the parameter ǫ (compare Figures 2 and 3, panels showing the SFR and metallicity dependence). For the"high dust correction" case (dashed lines) this correlation is weaker, confirming our conclusion that SNe type II cannot drive the winds in the massive spheroids. The ǫ − log Mmin plane shows that for structures of smaller mass ("normal dust correction" case, solid lines) the parameter ǫ needs to be large, while for high mass structures (as preferred in the"high dust correction" case, dashed curves), ǫ is essentially unconstrained, indicating that SNe feedback is irrelevant for massive spheroids. The different physical processes taking place in small structures and massive spheroids can be further investigated by looking at the correlations in the f dil − ν plane. We expect to find a positive correlation among ν and the dilution factor f dil , as larger ν increases the SFR (compare Figure 4) thus leading to a more metal-rich ISM. To bring this back in line with the data, a larger dilution factor is needed. The above line of reasoning explains the strong positive correlation one observes for the high mass structures (dashed) where winds do not play a strong role and metals cannot escape from the structure. In contrast, metal-rich winds are dominant for smaller structures (solid curves), thus expelling most of the metals produced. This results in almost no correlation between f dil and ν, since the impact of ν on the SFR and metallicity predictions can be mimicked by a different combination of values for ǫ and log Mmin. Finally, in the right-most panel of Figure 9, we display the probability distribution in the τ −ν plane, which exhibits a strong negative correlation. Again, this is expected on the grounds that large values of the parameter ν increase the SFR (compare Figure 4) and a smaller time-scale is thus required in order to quench star formation fast enough (see Figure 5).
CONCLUSIONS
We have presented a well-motivated physical model of the cosmic star formation incorporating supernova feedback, gas accretion and enriched outflows. We computed the cosmic star formation history and the chemical evolution in the interstellar medium of forming galaxies as a function of redshift, and we presented for the first time a full statistical treatment of the observational data, which accounts for the possibility of systematic errors in the data sets.
We have employed four different observational datathe observed cosmic star formation rate up to z ∼ 5, the observed rate of type II supernova up to z ∼ 0.7, the present baryon fraction in structures and the evolution of the metal content in the ISM -to derive constraints on the free parameters of our model. After employing a Bayesian procedure to rebin the SFR and SN rate data, we found that the low redshift (z 3) SFR dust correction adopted has a critical impact on the scenario favoured by the data.
For what we have termed "normal dust correction", the hierarchical star formation model is preferred, where star formation occurs in small structures first and supernovae winds are important. While the winds load factor remains poorly constrained, we can conclude that larger values are preferred, in agreement with previous work (Dalcanton (2006)). Applying a larger dust correction at small redshifts, we found that the data on the contrary favour high values for the minimum mass of a dark halo of the collapsed structures (monolithic star formation scenario). This case requires a large dilution factor, a rather extreme IMF slope and a fairly small winds load factor, at the model parameters are pushed at the boundaries of the available range. We have suggested that this might be interpreted in terms of the presence of outflow from supermassive black holes, but this possibility will require further investigation. It is worth noticing that the monolithic star formation scenario has very little star formation beyond z ∼ 5. Observations of the E-mode polarization power spectrum of the cosmic microwave background however indicate that the Universe was re-ionized around z ∼ 11 (Spergel et al. (2006)). This means that in this scenario the reionization mechanism has to be found elsewhere than in massive UV-emitting stars. Several alternatives have been explored in the literature, for example reionization by decaying particles (Hansen & Haiman (2004)), or a high-redshift population of mini-quasars that can reionize the IGM up to 50% ionisation fraction (Dijkstra et al. (2004)).
For both models, the IMF slope is large. Unfortunately, this does not help in distinguishing one model from the other, since observations have so far not yielded convincing results concerning the form of the stellar IMF or its variations in space and time (Scalo (1998)). The most important difference among the IMFs is that the fraction of high-mass stars is larger for a shallower IMF. Since only high-mass stars emit significant amount of ultraviolet light, this results in a spectrum which is more shifted towards the ultraviolet for a typical galaxy with an e.g. Salpeter IMF (x = 1.35) as compared with a Scalo IMF (x = 1.7). In turn, this leads to a different reionization history, which can be in principle compared with the optical depth to reionization as inferred from cosmic microwave background polarization measurements.
While the monolithic scenario is less preferred in terms of quality of fit, it is clear that more work is required to be able to draw firm conclusions as to the viability of the two different models. Of particular importance remains the statistical treatment of the data, for which we have here presented a new procedure that we hope will prove useful for future work. Let us consider the measurement of a quantity y b in a top-hat bin b, 1 b B -in our case, this represents the metallicity value at the redshift of the bin, z b , and we assume we can neglect the redshift uncertainty of the measurements (this issue is addressed in the next section). Each measurement consists of a central value di and a statistical error σi, 1 i N b , for N b different measurements within bin b. If the i-th datum does not suffer from a systematic error (or where the systematic error, Si, is negligible compared with the quoted statistical error), the likelihood function is modelled as a Gaussian with the quoted standard deviation σi:
Pi,g(di|y b ) = 1 √ 2πσi exp " − 1 2 " di − y b σi « 2 # .(A1)
For the sake of brevity, let us denote such measurements as "good" measurements, as indicated by the subscript g.
If the datum suffers from an undetected systematic, i.e. the dominant error is Si ≫ σi, the likelihood is instead given by (neglecting the statistical error wrt the systematic one)
Pi,s(di|y b ) = 1 √ 2πSi exp " − 1 2 " di − y b Si « 2 # ,(A2)
where the subscript s denotes "systematics", or "spoiled" measurements, for brevity. Now of course we do not know which measurements suffer from systematic, but this can be determined statistically using the following procedure (adapted from Press (1996)). We denote by p the probability that each of the measurements i in bin b is a "good" one. Conversely, 1 − p is the probability that the datum suffers from systematics. Furthermore, we include a binary vector V = (V1, . . . , VN b ), whose elements Vi (1 i N b ) can either be 0 or 1, determining whether the datum i is a good one (for Vi = 1) or a spoiled one (for Vi = 0). We can then compute the posterior probability for the value of the observed quantity y b in bin b by multiplying the individual contributions of the measurements in the same bin and marginalizing over the unknown quantities p and V (see Eq. (16) in Press (1996)):
P (y b |d b ) ∝ Z dp N b Y i=1 [pPg,i + (1 − p)Ps,i] ,(A3)
where d b denotes the collection of measurements in bin d, i.e. d b = (d1, . . . , dN b ). For the prior probability on p we have assumed a flat prior distribution between 0 p 1 and the proportionality factor might be determined by requiring that the likelihood be normalized to unity, but this is not necessary in our application. The precise numerical value of the error associated with systematics, Si does not matter, as long as Si ≫ σi. In our case, we take Si to be unity on a log scale, corresponding to one order of magnitude uncertainty on the observable.
From the posterior distribution (A3), the central value of the bin b is obtained as the peak of the distribution, while the standard deviation is defined as the range enclosing 68.4% (1σ range) of the probability. These values are given in Table 2 for the metallicity data, and are then used for the likelihood function employed in the fit of the model. Of course one could as well employ the full probability distribution of Eq. (A3) as the likelihood function, but for simplicity we have summarized it as a Gaussian with mean and Figure A2. Raw SFR data and the binned values after the statistical treatment including redshift uncertainties. No dust correction has been applied to the data at this stage. standard deviation computed as described above. The collection of raw, unbinned metallicity data and the resulting bins form our statistical treatment are shown in Figure A1.
A2 Accounting for redshift uncertainty
When the observed quantity suffers from a substantial redshift uncertainty, as in the case of the SFR data, we need to take into account the redshift error in our binning procedure, as this introduces a further uncertainty as to which bin a given datum belongs to. The above procedure is then modified as follows.
The probability that an observations with central redshift zi and redshift uncertainty τi belongs to the b-th redshift bin (centered at redshift z b ) is modelled as a Gaussian, i.e.
P (zi|z
b ) = 1 √ 2πτi exp " − 1 2 " zi − z b τi « 2 # .(A4)
Given the uncertainty on the location of the measurements in redshift, it is now impossible to assign data points to top-hat bins. Instead, one needs to marginalize over all possible assignments of data points among redshift bins, with each point's contribution weighted by the conditional probability of Eq. (A4). For each bin, let us introduce a new binary variable, Z = (Zi, . . . , ZN ), whose elements indicate whether the i-th datum (1 i N ) belongs to the bin under consideration (Zi = 1) or not (Zi = 0). If we knew which datum belongs to which redshift bin, then we could assign an exact binary sequence to Z (this corresponds to the case considered in the previous section). Instead, we sum (marginalize) over all possibilities, writing for the posterior probability of the SFR value y b at redshift z b , given d, the full collection of data points at all redshifts P (y b |d) = X Z P (y b , Z|d) = X Z P (y b |Z, d)P (zi|z b , Zi = 1), (A5) where the conditional probability P (y b |Z, d) is given by Eq. (A3), given a specific assignment for Z. The sum over Z can be replaced by a product of binomial terms, so that
Figure 1 .
1Dependence of the SFR, metallicity and baryonic fraction in structures (panels from top to bottom) on the minimum mass of collapsed dark matter halos, log M min . The curves are for log M min = 6, 8, 10, 12, from thin to thick.
Figure 2 .
2Dependence of the SFR (top panel) and of the metallicity (bottom panel) on the model parameter ǫ, describing the strength of galactic winds. The curves are for ǫ = 0.05, 0.15, 0.25, 0.35, from thin to thick.
Figure 3 .
3Dependence of the SFR (top panel), SN rate (middle panel) and of the metallicity (bottom panel) on the model parameter x, controlling the slope of the IMF. Curves are for x = 1.0, 1.4, 1.7, 2.0, from thin to thick.
Figure 4 .
4Dependence of the SFR (top panel) and the metallicity (bottom panel) on the parameter ν, controlling the efficiency of star formation. The curves are for ν = 0.5, 2, 3.5, 5, from thin to thick.
Figure 5 .
5Sensitivity of the model to the parameter τ , giving the characteristic timescale of star formation. The curves are for τ = 1, 2, 3, 4, from thin to thick.
Figure 7 .
7Best-fitting models for the normal (solid line, hierarchical star formation, χ 2 = 26.60) and high (dashed, monolithic scenario, χ 2 = 33.3) dust corrections, with parameters as in
Figure 8 .
81-dimensional marginalised posterior probability distributions of the model parameters (normalized to their peak values). Solid histograms are for the normal dust correction case (hierarchical scenario), dotted for the high dust correction (monolithic model).
Figure A1 .
A1Raw metallicity data and the binned values after the statistical treatment.
Table 1. Upper part: free model parameters and priors used in the analysis. Top-hat (flat) priors have been adopted on the parameter ranges indicated. Lower part: model parameters that have been fixed.Quantity
Symbol
Defined
Prior range or value
Minimum mass of collapsed haloes (M min in M ⊙ )
log M min
Sec. 2.2
5 log M min 13
SN type II energy efficiency factor
ǫ
Eq. (11) 0.01 ǫ 0.45
IMF power-law index
x
Eq. (9)
3 x 2
SFR normalization parameter (Gyr −1 )
ν
Eq. (16) 0.01 ν 5
SFR timescale (Gyr)
τ
Eq. (16) 1 τ
5
Winds load factor
η
Eq. (18) 0 η 30
Metals dilution factor
f dil
Eq. (19) 1 f dil 30
Baryon density parameter
Ω b
0.044
Matter density parameter
Ωm
0.27
Cosmological constant density parameter
Ω Λ
0.70
Hubble constant (km/sec/Mpc)
H 0
71
Rms fluctuations amplitude
σ 8
0.9
Dark matter to baryons bias parameter
b
1.0
Minimum fraction of baryons when star formation begins
f bar (z init )
0.01
Kinetic energy from stellar explosions
E kin
10 51 ergs
Yield
q
0.02
Metallicity of accreted material
Z F
0
c 0000 RAS, MNRAS 000, 000-000
This is the escape velocity from R to infinity, not the escape velocity from the sites of star formation that are deeper in the potential well. This approximation does not affect the results for the massive spheroids since even the shallower potential at R is deep enough to prevent winds from being effective, see the discussion in section 5. For less massive systems we expect this approximation to result in slightly smaller values for the parameter ǫ than one would otherwise obtain.
ACKNOWLEDGMENTSThe authors would like to thank an anonymous referee for several interesting comments. RT is supported by the Royal Astronomical Society through the Sir Norman Lockyer Fellowship, and by St Anne's College, Oxford.APPENDIX A: BINNING OF DATA ACCOUNTING FOR UNDETECTED SYSTEMATICSA1 No redshift uncertaintyWe wish to define B bins in redshift space. Within each bin b, 1 b B, we have a collection of measurements (in our case, metallicity or SFR data), each with its own statistical accuracy and possibly an unspecified systematic error. That systematic differences above the quoted statistical errors dominate the raw data is apparent from a plot of the unbinned metallicity or SFR observations, that show a scatter of up to an order of magnitude for observations at about the same redshift. The origin of the systematic discrepancy can vary, from underestimated statistical errors in the observation to intrinsic dispersion in the observed systems to differences in the way the data are collected. In the presence of systematic errors, we cannot simply take the weighted average of the data within each bin. Instead, we model the presence of unknown systematics as follows.(A6) Notice that the product is here over all points in the dataset, not just over the ones in a bin, as in Eq. (A3).Since we include the full dataset for each bin, the resulting errors are in principle correlated across bins. However, the Gaussian term of Eq. (A4) ensures that only "nearby" points give a non-negligible contribution to the value of bin b. We therefore consider it acceptable to ignore the correlation among bins when using the mean and standard deviation of Eq. (A6) for the likelihood function for the SFR. The results from this procedure are tabulated inTable 3, and are plotted alongside with the raw, unbinned data inFigure A2.
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|
[] |
[
"Detection of a cyclotron line in SXP 15.3 during its 2017 outburst",
"Detection of a cyclotron line in SXP 15.3 during its 2017 outburst"
] |
[
"C Maitra \nMax-Planck-Institut für extraterrestrische Physik\nGiessenbachstraße 185748GarchingGermany\n",
"B Paul \nRaman Research Institute\nC.V. Raman Avenue560064Sadashivanagar, BangaloreIndia\n",
"F Haberl \nMax-Planck-Institut für extraterrestrische Physik\nGiessenbachstraße 185748GarchingGermany\n",
"G Vasilopoulos \nMax-Planck-Institut für extraterrestrische Physik\nGiessenbachstraße 185748GarchingGermany\n"
] |
[
"Max-Planck-Institut für extraterrestrische Physik\nGiessenbachstraße 185748GarchingGermany",
"Raman Research Institute\nC.V. Raman Avenue560064Sadashivanagar, BangaloreIndia",
"Max-Planck-Institut für extraterrestrische Physik\nGiessenbachstraße 185748GarchingGermany",
"Max-Planck-Institut für extraterrestrische Physik\nGiessenbachstraße 185748GarchingGermany"
] |
[
"MNRAS"
] |
We report the results of AstroSat and NuSTAR observations of the Be/X-ray binary pulsar SXP 15.3 in the Small Magellanic Cloud during its outburst in late 2017, when the source reached a luminosity level of ∼10 38 erg s −1 , close to the Eddington limit. The unprecedented broadband coverage of the source allowed us to perform timing and spectral analysis between 3 and 80 keV. The pulse profile exhibits a significant energy dependence, and morphs from a double peaked profile to a single broad pulse at energies > 15 keV. This can be explained by a spectral hardening during an intensity dip seen between the two peaks of the pulse profile. We detect a Cyclotron Resonance Scattering Feature (CRSF) at ∼5 keV in the X-ray spectrum, independent of the choice of the continuum model. This indicates a magnetic field strength of 6 × 10 11 G for the neutron star.
|
10.1093/mnrasl/sly141
|
[
"https://arxiv.org/pdf/1807.10696v1.pdf"
] | 119,431,190 |
1807.10696
|
27fa33cc8d315c5b520f90a4285c351962c09d76
|
Detection of a cyclotron line in SXP 15.3 during its 2017 outburst
2018
C Maitra
Max-Planck-Institut für extraterrestrische Physik
Giessenbachstraße 185748GarchingGermany
B Paul
Raman Research Institute
C.V. Raman Avenue560064Sadashivanagar, BangaloreIndia
F Haberl
Max-Planck-Institut für extraterrestrische Physik
Giessenbachstraße 185748GarchingGermany
G Vasilopoulos
Max-Planck-Institut für extraterrestrische Physik
Giessenbachstraße 185748GarchingGermany
Detection of a cyclotron line in SXP 15.3 during its 2017 outburst
MNRAS
0002018Accepted XXX. Received YYY; in original form ZZZPreprint 30 July 2018 Compiled using MNRAS L A T E X style file v3.0stars: neutron -pulsars: individual: SMC -galaxies: individual: SXP 153 - X-rays: binaries
We report the results of AstroSat and NuSTAR observations of the Be/X-ray binary pulsar SXP 15.3 in the Small Magellanic Cloud during its outburst in late 2017, when the source reached a luminosity level of ∼10 38 erg s −1 , close to the Eddington limit. The unprecedented broadband coverage of the source allowed us to perform timing and spectral analysis between 3 and 80 keV. The pulse profile exhibits a significant energy dependence, and morphs from a double peaked profile to a single broad pulse at energies > 15 keV. This can be explained by a spectral hardening during an intensity dip seen between the two peaks of the pulse profile. We detect a Cyclotron Resonance Scattering Feature (CRSF) at ∼5 keV in the X-ray spectrum, independent of the choice of the continuum model. This indicates a magnetic field strength of 6 × 10 11 G for the neutron star.
INTRODUCTION
SXP 15.3 (aka RX J0052.1-7319) is a transient X-ray binary pulsar located in the Small Magellanic Cloud (SMC), first detected using Einstein observations (Wang & Wu 1992). Later the source was also detected in the ROSAT-PSPC data as a hard and highly variable source and classified as a transient X-ray binary candidate (Kahabka & Pietsch 1996). Coherent pulsations with a period of 15.3 s were discovered in 1996 using ROSAT and CGRO observations with a pulse fraction of 27% at a luminosity (0.1-2 keV) of ∼10 37 erg s −1 (Lamb et al. 1999;Finger et al. 2001). Subsequently Kahabka (2000) investigated the ROSAT-HRI observations in 1995 and 1996 and found a large variation in the flux by a factor of ∼200, further ascertaining its transient nature. The optical counterpart to the source was identified as a likely Be star by Israel et al. (1999), which was later confirmed as an O9.5IIIe star (V = 14.6 mag, Covino et al. 2001). The source has not been reported in an outburst or a bright state ever since until July 25 2017, when the Swift SMC Survey (S-CUBED) detected a brightening of the source (Kennea et al. 2017). Pulsations at 15.253 s were detected, and the absorption corrected luminosity (0.5-10 keV) corresponded to 2.4 × 10 37 erg s −1 . The optical counterpart also exhibited a corresponding brightening. The source re-brightened again in November 2017, reaching a higher X-ray luminosity of 3.9 × 10 37 erg s −1 (Ducci et al. 2017). This triggered several Target of Opportunity observations (ToO).
In this letter, we present the broadband timing and spectral characteristics of SXP 15.3 for the first time, using AstroSat and NuSTAR observations performed during the recent outburst which E-mail: [email protected] started in October 2017. In Sect. 2 we describe the observations and data reduction. We present the results of a timing analysis (Sect. 3), a spectral analysis (Sect. 4) and pulse phase-resolved spectroscopy (Sect. 5). Discussions and conclusions are presented in Sect. 6.
OBSERVATIONS AND DATA REDUCTION
Following the report of an outburst in November 2017, SXP 15.3 was observed with NuSTAR (Harrison et al. 2013(Harrison et al. ) on 2017 for ∼70 ks as a ToO observation. A simultaneous Swift/XRT (Burrows et al. 2005) observation was also carried out for 3 ks (Obsid 00088639001). In addition, we triggered a ToO observation of the source with AstroSat (Agrawal 2006;Singh et al. 2014). The observation was performed on 2017-12-08 with an exposure of 60 ks. The simultaneous Swift and NuSTAR observation will be referred hereafter as Obs. 1 and the AstroSat observation as Obs. 2.
NuSTAR consists of two independent focal plane modules FPMA and FPMB. The data were processed from both the modules using the standard NuSTARDAS software (version 1.8.0 of HEASOFT v.6.22.1 and CALDB version 20171002) to extract barycentercorrected light curves, spectra, response matrices and effective area files. The source events were extracted using a circular region of radius 49 and background events were extracted using a circle of same radius, away from the source.
The Swift/XRT data were analysed following standard procedures described in the Swift data analysis guide 1 . The source and background events were extracted using circles of radii 45 . The task xrtpipeline was used to generate the Swift/XRT spectrum. The response file was generated by using the task XRTMKARF.
AstroSat consists of five independent instruments for performing simultaneous broadband observations. We analysed here data from the Soft X-ray Telescope (SXT) and the Large Area Xenon Proportional Counter (LAXPC). SXT is a focusing telescope consisting of a CCD camera working in the energy range of 0.3-8 keV (Singh et al. 2014). We used level2 data (reprocessed from the level2 pipeline version 1.4a) and merged the event files using sxtevtmergertool. We extracted the spectrum thereafter using XSELECT v2.4 2 . The source events were extracted using an annular region between 1 and 16 , and an appropriate on-axis ARF was used for the spectral analysis. Blank sky SXT observations are used to extract the background spectrum.
The LAXPC consists of three co-aligned identical X-ray proportional counters having an absolute time resolution of 10 µs in the energy range 3.0-80.0 keV (Antia et al. 2017;Agrawal et al. 2017). Data were obtained in the Event Analysis mode, level1 products of 2 http://www.tifr.res.in/~astrosat_sxt/page1_data_ analysis.php which were reprocessed with the LAXPC data analysis software 3 to produce light curves and spectral files. The events were filtered for the times corresponding to Earth occultation, passage of the South Atlantic anomaly (SAA) and for large angle offsets of the detectors pointing away from source. The LAXPC background was taken from the same observation in periods when the source was occulted by Earth. LAXPC20 had a different gain during this observation, as determined from the k-fluorescent pulse amplitudes of the double events. A gain factor was used to accommodate the same. All the three LAXPCs were used for the timing analysis. Due to a gas leakage in LAXPC30 leading to a loss of efficiency, LAXPC30 was excluded from the spectral analysis.
TIMING ANALYSIS
We extracted light curves at 100 ms time resolution for the timing analysis. We used the pulse folding and χ 2 -maximisation method to estimate the barycentric corrected pulse period of the pulsar. Pulsations were detected at 15.2563±0.0005 s in Obs. 1 and 15.2575±0.0009 s in Obs. 2 respectively. The errors correspond to 1σ confidence. In the case of the LAXPC detectors (AstroSat), background subtraction was performed by subtracting the background count rates in the energy bands concerned. Fig. 1 shows the background subtracted pulse profiles from AstroSat and NuS-TAR. Pulsations are detected up to 50 keV, and the pulse profiles from both the observations exhibit a double peaked profile which morphs to a single broad pulse at energies > 15 keV (Fig. 2). The pulse fraction was computed as the ratio of (I max − I min )/(I max + I min ). The pulsed fraction increases from 30% in the energy range of 3-5 keV to 60% in the energy range of 30-50 keV in Obs. 1, and from 20% in the energy range of 3-5 keV to 40% in the energy range of 30-50 keV in Obs. 2. The disappearance of the double-peaked nature of the pulse profile with energy motivated us to investigate the hardness ratio (HR) along the spin phase of the system in two energy bands, 3-15 keV and 15-50 keV (Fig. 2). The HR was defined as the ratio of intensity in the 15-50 keV band divided by the 3-15 keV band. The HR shows significant evolution with the spin phase with a spectral hardening seen at the dip phase (phase ∼0.5).
BROADBAND SPECTRAL ANALYSIS
Spectral analysis was performed using XSPEC v12.9.1. We grouped the spectra to achieve a minimum of 20 counts per spectral bin for the analysis. We investigated the broadband spectrum of SXP 15.3 using simultaneous Swift/XRT and NuSTAR data (Obs. 1) and SXT and LAXPC data (Obs. 2). The spectra were modelled with standard continuum models like a power-law with quasi exponential high energy cutoff having various functional forms (XSPEC models 'highecut', 'bknpow', 'fdcut' and 'newhcut'). Other continuum models are a combination of two power laws with different photon indices but a common cutoff energy value called the Negative and Positive power laws with Exponential model (XSPEC model 'NPEX' ), and a thermal Comptonization model (XSPEC model 'CompTT'). In order to account for the photoelectric absorption by the interstellar gas, two components were used. The first component was fixed to the Galactic value of 6×10 20 cm −2 (Dickey & Lockman 1990). The second component was left free to account for the absorption within the SMC. For the latter component, the metal abundances were fixed at 0.2 solar, as is typical in the SMC (Russell & Dopita 1992). The atomic cross-sections were adapted from Verner et al. (1996). The X-ray absorption was modelled using the XSPEC model 'tbabs'. Finally, to account for inter-calibration uncertainties of the instruments and small flux variations of the source during not fully simultaneous observing intervals (Obs. 1) we introduce normalisation factors between instruments.
We obtained the best fit to the continuum with the 'newhcut' model. This model is a modified version of 'highecut' which has a smoothed region around the cutoff energy. The smoothing function is a third order polynomial with continuous derivatives (Burderi et al. 2000). An iron fluorescence emission line at ∼6.4 keV was detected in the NuSTAR spectrum. Additionally, a narrow absorption feature was visible at ∼5 keV in the broadband spectra of Obs. 1 and Obs. 2, irrespective of the continuum model used (Fig. 3).
Addition of a Gaussian absorption feature (XSPEC model 'Gabs') improved the fit significantly and the reduced χ 2 after adding the absorption feature decreased to ∼1. An absorption feature detected in the energy spectrum of HMXB pulsars is reminiscent of a cyclotron resonance scattering feature (CRSF). A careful modelling of the broadband continuum spectrum is essential to detect and model shallow features such as the CRSFs (see for example Müller et al. 2013). Although in this case the addition of the line was required for all the tested continuum models, the line width was narrowest and best constrained with the 'newhcut' model (Fig. 3). The improvement in χ 2 after adding the CRSF to the 'newhcut' model was significant, with ∆χ 2 = 75.39 for 3 d.o.f. in the case of Obs. 1, and ∆χ 2 = 155 for 3 d.o.f. in the case of Obs. 2, respectively. Although, the CRSF was detected more prominently in Obs. 2, we avoided further interpretations of the CRSF and its parameters with this observation as the line lies at the edge of the energy bands for both the SXT and LAXPC detectors and needs to be treated with caution. However an independent detection of the line at the same energy and with an independent instrument gives us confidence on the obtained results.
PHASE-RESOLVED SPECTROSCOPY
The variation of the HR with spin phase (Fig. 2) indicates a dependence of the spectral parameters on the changing viewing angle of the neutron star. Motivated by this we performed pulse phaseresolved spectroscopy using the NuSTAR observation. We created good-time-interval files (gti) using the measured pulse period of NuSTAR to extract phase-resolved spectra into five equally spaced phase bins. As the Swift/XRT data lacked the required statistics for the phase-resolved analysis of SXP 15.3 only NuSTAR data were used for the purpose. The 'newhcut' continuum model was used for the spectral fits. The SMC N H , the iron line energy and width, and the CRSF width (σ c ) were frozen to the phase averaged value in each phase bin. Fig. 6 shows the variation of the spectral parameters with pulse phase. The spectrum is harder at the dip phase as compared to the peaks, i.e. Γ = 1.54 ± 0.05 at phase ∼0.3 to Γ = 1.40 ± 0.03 at phase ∼0.5. This is consistent with the results obtained from the investigation of the variation of the HR with the pulse phase. The CRSF centroid energy E c is variable with pulse phase with E c rising to ∼8 keV at the dip phase. E c varies by a factor of 1.5 ± 0.3 between the dip and the adjacent phase bin (phase ∼0.7). The CRSF is not detected at the off-pulse phase which might be due to insufficient statistics in that phase bin. In order to obtain an upper limit on the CRSF depth at this phase, we froze the CRSF energy and width to the phase averaged value and obtained τ c ∼ 0.4. The variation of CRSF parameters with phase is typically seen in many HMXB pulsars, with the pattern of the variations revealing important information on the beaming geometry and the magnetic field geometry of the HMXB pulsar (see Maitra 2017, for a comprehensive summary of phase-resolved analysis of CRSFs).
DISCUSSION AND CONCLUSIONS
In this letter we report the broadband X-ray timing and spectral properties of SXP 15.3 for the first time, and at the brightest state of the source detected till today. We also report the discovery of a CRSF at ∼5 keV. This makes it only the second Magellanic pulsar after SMC X-2 (Jaisawal & Naik 2016) with a cyclotron line detection, and hence a confirmed magnetic field strength of the neutron star. The spin period measurements with the NuSTAR and AstroSat observations are consistent within errors precluding the detection of any spin-up during the current outburst. The net spin-up rate between the CGRO and ROSAT observations separated by 123 days was −1.64 × 10 −9 s s −1 (Finger et al. 2001). The long term trend in the spin evolution as inferred from the spin period measurements between the CGRO and AstroSat observations however indicate a much reduced spin-up rate of −2.92 × 10 −11 s s −1 .
The magnetic field strength of the neutron star can be determined from the observed CRSF centroid energy E cyc (determined from Obs. 1), and is given as:
E cyc = 11.57 keV 1 + z × B 12 (1)
where B 12 is the field strength in units of 10 12 G; z ∼ 0.3 is the gravitational red shift in the scattering region for standard neutron star parameters. This implies a magnetic field strength of the neutron star of B = 6 × 10 11 G, assuming the line forming region lies close to the neutron star surface. The obtained field strength is consistent with the estimate obtained by Christodoulou et al. (2017) assuming that SXP 15.3 was in the propeller state at its lowest detected X-ray luminosity (L x ∼ 6.8 × 10 33 erg s −1 as detected from a Chandra observation).
The unabsorbed bolometric X-ray luminosity of SXP 15.3 during the observations indicate that the source was accreting near its Eddington limit of 2×10 38 erg s −1 for a typical neutron star mass of 1.4 M . In highly magnetised accretion powered pulsars, the location and geometry of the radiation emitting region are believed to be dependent on the mass accretion rate (Basko & Sunyaev 1976). At a luminosity of ∼10 38 erg s −1 SXP 15.3 is expected to be in the super-critically accreting or radiation-dominated regime. In the super-critical regime, a radiation-dominated shock is formed, after which the accreted matter settles to the neutron star surface in a magnetically confined accretion column. The radiation in this case predominantly escapes from the optically thin sides of the accretion column in a fan-beam like pattern. The critical-luminosity (L c ), which divides the two regimes of sub and super-critical accretion is a function of the surface magnetic field strength of the neutron star and can be approximated as (Becker et al. 2012 (2) M, R, and B surf are, the mass, radius, and surface magnetic field strength of the neutron star, w = 1 characterises the shape of the photon spectrum inside the column, and Λ is the mode of accretion. Λ = 0.1 approximates the case of disk accretion, and Λ = 1.0 is more appropriate for wind accretors. Assuming Λ = 0.1 and B surf = 6 × 10 11 G, results in L c = 9 × 10 36 erg s −1 for typical neutron star mass and radius values 4 . This ascertains that SXP 15.3 is accreting in the super-critical regime. The double-peaked pulse profile of SXP 15.3 observed in this work is in further support of the predominance of a fan-beam like emission. The disappearance of the double peak to a single broad peak at higher energies is most likely due to the intrinsic nature of the emission rather than being caused by a local absorbing matter phase locked to the neutron star. This is because we found no evidence of an additional absorption component in the spectral fit of SXP 15.3. A further indication of the fan-beam emission is obtained from the shape of the CRSF. A deep and narrow CRSF, as seen in SXP 15.3 is expected for viewing angles perpendicular to the magnetic field axis, a.k.a. fan-beam like emissions (Schwarm et al. 2017).
The luminosity of SXP 15.3 during Obs. 2 was ∼30% higher than in Obs. 1, with an indication of spectral softening with increasing luminosity. This behaviour is expected in the super-critical regime, and can be understood either due to a decrease in the plasma temperature with rising accretion column (Becker et al. 2012), or alternatively a lower fraction of the radiation reflected by the neutron star surface in the case of a taller accretion column at higher intensities (Postnov et al. 2015). The CRSF parameters show little variations with pulse phase. This may indicate no gradient of the properties of the line forming region across the viewing angles. Alternatively, this might also be due to the effect of gravitational light bending near the neutron star surface (Beloborodov 2002) which would smear out the pulsephase dependence, with a particular viewing angle having contributions from multiple emission regions. The only variable CRSF parameter is the centroid energy E c which is significantly higher at the dip phase. This suggests that the line forming region at this phase may offer a deep and a more direct view into the emission region along the magnetic axis which is consistent with fan-beam like emission. An indication of spectral hardening at the dip phase is further consistent with a direct view into the emission region along the magnetic axis (see for e.g. Pravdo et al. 1978).
In summary we present the broadband timing and spectral results of SXP 15.3 for the first time using AstroSat and NuSTAR ToO observations performed during the recent outburst in October 2017. We also report the discovery of a CRSF at ∼5 keV, establishing the magnetic field of the neutron star at 6 × 10 11 G. The CRSF centroid energy varies with pulse phase, with an increase in energy during an intensity dip. This is accompanied with a spectral hardening during the dip. The two signatures mentioned above and the double-peaked pulse profile of SXP 15.3 indicate a fan-beam like geometry dominating the emitting region as is expected for supercritically accreting sources.
Figure 1 .Figure 2 .
12Background subtracted pulse profiles of SXP 15.3, obtained from the FPMA detector of NuSTAR (3-79 keV, black) and LAXPC10 of As-troSat (3-80 keV, red). Background subtracted pulse profiles of SXP 15.3 obtained from LAXPC10 of AstroSat in the two energy bands of 3-15 keV and 15-50 keV and the HR variation with the pulse phase.
Figure 3 .
3The residuals of the spectral fits with Swift/XRT (in green), and FPMA ( in black) and FPMB (in red) detectors onboard NuSTAR (Obs. 1). The continuum models are mentioned in the panels. An absorption feature at ∼5 keV is not included in the fits.
Figure 4 .
4The upper panel shows the best-fit spectral model of SXP 15.3 using spectra from Swift/XRT (in green), and FPMA ( in black) and FPMB (in red) detectors onboard NuSTAR (Obs. 1). The second panel shows the residuals after the fit without taking into account the CRSF and the Fe line. The third panels shows the residuals after including all the model components.
Figure 5 .
5Same as inFig. 4using spectra from the SXT (in black), LAXPC10 (in red) and LAXPC20 (in green) onboard AstroSat (Obs. 2). The lower panel shows the residuals after including all the model components.
Figure 6 .
6Spectral parameters of SXP 15.3 obtained from the pulse phaseresolved spectroscopy from Obs. 1. The parameters are plotted with 90% confidence. The top panel shows the pulse profile obtained from the FPMA detector (3-79 keV).
Table 1
1summarises the best-fit broadband spectral parameters and Figs. 4 and 5 show the broadband spectra from Obs. 1 and Obs. 2, respectively. The continuum parameters are consistent between Obs. 1 and Obs. 2. with an indication of spectral softening in Obs. 2. The absorption corrected broadband luminosity (0.5-50 keV) is 9.1 × 10 37 ergs s −1 for Obs. 1 and 1.2 × 10 38 ergs s −1 for Obs. 2 respectively.
Table 1 .
1Best-fitting parameters (with 90% errors) obtained from the spectral fitting with the newhcut continuum model with an iron emission line and cyclotron absorption line. a : Equivalent hydrogen column density (in 10 22 atoms cm −2 ); b : Absorption corrected luminosity (0.5-50 keV) in 10 38 ergs s −1 , assuming a distance of 61 kpc.Parameter
Obs. 1
Obs. 2
NEWHCUT NEWHCUT×GABS NEWHCUT NEWHCUT×GABS
SMC N H
a
0.40±0.09
0.6±0.1
0.31±0.06
0.45±0.06
Photon index
1.40±0.01
1.48 +0.06
−0.03
1.40±0.02
1.40±0.02
E cut (keV)
16.4±0.6
16.9±0.8
18.8±2.0
20.9±2.3
E fold (keV)
26.8±1.4
28.5 +2.2
−1.7
37.1 +12.8
−10.4
33.4 +17.6
−12.8
Fe line energy (keV)
6.39±0.08
6.37±0.08
-
-
Fe line eq. width (eV)
89.7±18
91.4±18
-
-
Cycl. line energy (E c ) (keV)
-
5.7 +0.3
−0.6
-
5.2±0.2
Cycl. line width (σ c ) (keV)
-
1.7 +0.8
−0.5
-
0.67±0.17
Cycl. line strength (τ c )
-
0.4 +0.5
−0.2
-
0.6±0.1
Luminosity b
-
0.91±0.05
-
1.2±0.1
Reduced-χ 2 (d.o.f)
1.19 (450)
1.04 (447)
1.39 (582)
1.13 (579)
http://www.swift.ac.uk/analysis/xrt/ c 2018 The Authors arXiv:1807.10696v1 [astro-ph.HE] 27 Jul 2018 2 C. Maitra et al.
http://www.rri.res.in/~rripoc/
MNRAS 000, 1-5 (2018)
A more detailed treatment of critical-luminosity can be found inMushtukov et al. (2015). We however verified that our obtained L c is consistent between the two works for the estimated B 12 value.
ACKNOWLEDGEMENTSWe thank the reviewer for providing useful comments. The authors would like to thank the AstroSat operations team members for scheduling the ToO observation. The authors would also like to thank all the NuSTAR and Swift team members for ToO observations. CM acknowledges Gulab Dewangan for useful discussions and suggestions on the analysis of the SXT data.
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|
[] |
[
"Sequence-to-Action: End-to-End Semantic Graph Generation for Semantic Parsing",
"Sequence-to-Action: End-to-End Semantic Graph Generation for Semantic Parsing"
] |
[
"Bo Chen [email protected] \nScience Institute of Software\nState Key Laboratory of Computer\nChinese Academy of Sciences\nBeijingChina\n\nUniversity of Chinese Academy of Sciences\nBeijingChina\n",
"Le Sun [email protected] \nScience Institute of Software\nState Key Laboratory of Computer\nChinese Academy of Sciences\nBeijingChina\n",
"Xianpei Han [email protected] \nScience Institute of Software\nState Key Laboratory of Computer\nChinese Academy of Sciences\nBeijingChina\n"
] |
[
"Science Institute of Software\nState Key Laboratory of Computer\nChinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina",
"Science Institute of Software\nState Key Laboratory of Computer\nChinese Academy of Sciences\nBeijingChina",
"Science Institute of Software\nState Key Laboratory of Computer\nChinese Academy of Sciences\nBeijingChina"
] |
[
"Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Long Papers)"
] |
This paper proposes a neural semantic parsing approach -Sequence-to-Action, which models semantic parsing as an endto-end semantic graph generation process. Our method simultaneously leverages the advantages from two recent promising directions of semantic parsing. Firstly, our model uses a semantic graph to represent the meaning of a sentence, which has a tight-coupling with knowledge bases. Secondly, by leveraging the powerful representation learning and prediction ability of neural network models, we propose a RNN model which can effectively map sentences to action sequences for semantic graph generation. Experiments show that our method achieves state-of-the-art performance on OVERNIGHT dataset and gets competitive performance on GEO and ATIS datasets.
|
10.18653/v1/p18-1071
|
[
"https://www.aclweb.org/anthology/P18-1071.pdf"
] | 51,873,800 |
1809.00773
|
f2527a624729e81a76d2db971c5309851199ec06
|
Sequence-to-Action: End-to-End Semantic Graph Generation for Semantic Parsing
Association for Computational LinguisticsCopyright Association for Computational LinguisticsJuly 15 -20. 2018. 2018
Bo Chen [email protected]
Science Institute of Software
State Key Laboratory of Computer
Chinese Academy of Sciences
BeijingChina
University of Chinese Academy of Sciences
BeijingChina
Le Sun [email protected]
Science Institute of Software
State Key Laboratory of Computer
Chinese Academy of Sciences
BeijingChina
Xianpei Han [email protected]
Science Institute of Software
State Key Laboratory of Computer
Chinese Academy of Sciences
BeijingChina
Sequence-to-Action: End-to-End Semantic Graph Generation for Semantic Parsing
Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Long Papers)
the 56th Annual Meeting of the Association for Computational Linguistics (Long Papers)Melbourne, AustraliaAssociation for Computational LinguisticsJuly 15 -20. 2018. 2018
This paper proposes a neural semantic parsing approach -Sequence-to-Action, which models semantic parsing as an endto-end semantic graph generation process. Our method simultaneously leverages the advantages from two recent promising directions of semantic parsing. Firstly, our model uses a semantic graph to represent the meaning of a sentence, which has a tight-coupling with knowledge bases. Secondly, by leveraging the powerful representation learning and prediction ability of neural network models, we propose a RNN model which can effectively map sentences to action sequences for semantic graph generation. Experiments show that our method achieves state-of-the-art performance on OVERNIGHT dataset and gets competitive performance on GEO and ATIS datasets.
Introduction
Semantic parsing aims to map natural language sentences to logical forms (Zelle and Mooney, 1996;Zettlemoyer and Collins, 2005;Wong and Mooney, 2007;Lu et al., 2008;Kwiatkowski et al., 2013). For example, the sentence "Which states border Texas?" will be mapped to answer (A, (state (A), next to (A, stateid ( texas )))).
A semantic parser needs two functions, one for structure prediction and the other for semantic grounding. Traditional semantic parsers are usually based on compositional grammar, such as CCG Collins, 2005, 2007), DCS (Liang et al., 2011), etc. These parsers compose structure using manually designed grammars, use lexicons for semantic grounding, and exploit fea- tures for candidate logical forms ranking. Unfortunately, it is challenging to design grammars and learn accurate lexicons, especially in wideopen domains. Moreover, it is often hard to design effective features, and its learning process is not end-to-end. To resolve the above problems, two promising lines of work have been proposed: Semantic graph-based methods and Seq2Seq methods.
Semantic graph-based methods (Reddy et al., 2014(Reddy et al., , 2016Bast and Haussmann, 2015;Yih et al., 2015) represent the meaning of a sentence as a semantic graph (i.e., a sub-graph of a knowledge base, see example in Figure 1) and treat semantic parsing as a semantic graph matching/generation process. Compared with logical forms, semantic graphs have a tight-coupling with knowledge bases (Yih et al., 2015), and share many commonalities with syntactic structures (Reddy et al., 2014). Therefore both the structure and semantic constraints from knowledge bases can be easily exploited during parsing (Yih et al., 2015). The main challenge of semantic graph-based parsing is how to effectively construct the semantic graph of a sentence. Currently, semantic graphs are either constructed by matching with patterns (Bast and Haussmann, 2015), transforming from dependency tree (Reddy et al., 2014(Reddy et al., , 2016, or via a staged heuristic search algorithm (Yih et al., 2015). These methods are all based on manuallydesigned, heuristic construction processes, making them hard to handle open/complex situations.
In recent years, RNN models have achieved success in sequence-to-sequence problems due to its strong ability on both representation learning and prediction, e.g., in machine translation . A lot of Seq2Seq models have also been employed for semantic parsing (Xiao et al., 2016;Dong and Lapata, 2016;Jia and Liang, 2016), where a sentence is parsed by translating it to linearized logical form using RNN models. There is no need for high-quality lexicons, manually-built grammars, and hand-crafted features. These models are trained end-to-end, and can leverage attention mechanism Luong et al., 2015) to learn soft alignments between sentences and logical forms.
In this paper, we propose a new neural semantic parsing framework -Sequence-to-Action, which can simultaneously leverage the advantages of semantic graph representation and the strong prediction ability of Seq2Seq models. Specifically, we model semantic parsing as an end-to-end semantic graph generation process. For example in Figure 1, our model will parse the sentence "Which states border Texas" by generating a sequence of actions [add variable:A, add type:state, ...]. To achieve the above goal, we first design an action set which can encode the generation process of semantic graph (including node actions such as add variable, add entity, add type, edge actions such as add edge, and operation actions such as argmin, argmax, count, sum, etc.). And then we design a RNN model which can generate the action sequence for constructing the semantic graph of a sentence. Finally we further enhance parsing by incorporating both structure and semantic constraints during decoding.
Compared with the manually-designed, heuristic generation algorithms used in traditional semantic graph-based methods, our sequence-toaction method generates semantic graphs using a RNN model, which is learned end-to-end from training data. Such a learnable, end-to-end generation makes our approach more effective and can fit to different situations.
Compared with the previous Seq2Seq semantic parsing methods, our sequence-to-action model predicts a sequence of semantic graph generation actions, rather than linearized logical forms. We find that the action sequence encoding can better capture structure and semantic information, and is more compact. And the parsing can be enhanced by exploiting structure and semantic constraints. For example, in GEO dataset, the action add edge:next to must subject to the semantic constraint that its arguments must be of type state and state, and the structure constraint that the edge next to must connect two nodes to form a valid graph.
We evaluate our approach on three standard datasets: GEO (Zelle and Mooney, 1996), ATIS (He and Young, 2005) and OVERNIGHT (Wang et al., 2015b).
The results show that our method achieves state-of-the-art performance on OVERNIGHT dataset and gets competitive performance on GEO and ATIS datasets.
The main contributions of this paper are summarized as follows:
• We propose a new semantic parsing framework -Sequence-to-Action, which models semantic parsing as an end-to-end semantic graph generation process. This new framework can synthesize the advantages of semantic graph representation and the prediction ability of Seq2Seq models.
• We design a sequence-to-action model, including an action set encoding for semantic graph generation and a Seq2Seq RNN model for action sequence prediction. We further enhance the parsing by exploiting structure and semantic constraints during decoding. Experiments validate the effectiveness of our method.
2 Sequence-to-Action Model for End-to-End Semantic Graph Generation
Given a sentence X = x 1 , ..., x |X| , our sequenceto-action model generates a sequence of actions Y = y 1 , ..., y |Y | for constructing the correct semantic graph. Figure 2 shows an example. The conditional probability P (Y |X) used in our Figure 2: An example of a sentence paired with its semantic graph, together with the action sequence for semantic graph generation.
model is decomposed as follows:
P (Y |X) = |Y | t=1 P (y t |y <t , X)(1)
where y <t = y 1 , ..., y t−1 . To achieve the above goal, we need: 1) an action set which can encode semantic graph generation process; 2) an encoder which encodes natural language input X into a vector representation, and a decoder which generates y 1 , ..., y |Y | conditioned on the encoding vector. In following we describe them in detail.
Actions for Semantic Graph Generation
Generally, a semantic graph consists of nodes (including variables, entities, types) and edges (semantic relations), with some universal operations (e.g., argmax, argmin, count, sum, and not). To generate a semantic graph, we define six types of actions as follows:
Add Variable Node: This kind of actions denotes adding a variable node to semantic graph. In most cases a variable node is a return node (e.g., which, what), but can also be an intermediate variable node. We represent this kind of action as add variable:A, where A is the identifier of the variable node.
Add Entity Node: This kind of actions denotes adding an entity node (e.g., Texas, New York) and is represented as add entity node:texas. An entity node corresponds to an entity in knowledge bases.
Add Type Node: This kind of actions denotes adding a type node (e.g., state, city). We represent them as add type node:state.
Add Edge: This kind of actions denotes adding an edge between two nodes. An edge is a binary relation in knowledge bases. This kind of actions is represented as add edge:next to.
Operation Action: This kind of actions denotes adding an operation. An operation can be argmax, argmin, count, sum, not, et al. Because each operation has a scope, we define two actions for an operation, one is operation start action, represented as start operation:most, and the other is operation end action, represented as end operation:most. The subgraph within the start and end operation actions is its scope.
Argument Action: Some above actions need argument information. For example, which nodes the add edge:next to action should connect to. In this paper, we design argument actions for add type, add edge and operation actions, and the argument actions should be put directly after its main action.
For add type actions, we put an argument action to indicate which node this type node should constrain. The argument can be a variable node or an entity node. An argument action for a type node is represented as arg:A.
For add edge action, we use two argument actions: arg1 node and arg2 node, and they are represented as arg1 node:A and arg2 node:B.
We design argument actions for different operations.
For operation:sum, there are three arguments: arg-for, arg-in and arg-return. For operation:count, they are arg-for and arg-return. There are two arg-for arguments for operation:most.
We can see that each action encodes both structure and semantic information, which makes it easy to capture more information for parsing and can be tightly coupled with knowledge base. Furthermore, we find that action sequence encoding is more compact than linearized logical form (See Section 4.4 for more details).
Neural Sequence-to-Action Model
Based on the above action encoding mechanism, this section describes our encoder-decoder model for mapping sentence to action sequence. Specifically, similar to the RNN model in Jia and Liang (2016), this paper employs the attentionbased sequence-to-sequence RNN model. Figure 3 presents the overall structure. Encoder: The encoder converts the input sequence x 1 , ..., x m to a sequence of contextsensitive vectors b 1 , ..., b m using a bidirectional RNN . Firstly each word x i is mapped to its embedding vector, then these vectors are fed into a forward RNN and a backward RNN. The sequence of hidden states h 1 , ..., h m are generated by recurrently applying the recurrence:
h i = LST M (φ (x) (x i ), h i−1 ).(2)
The recurrence takes the form of LSTM (Hochreiter and Schmidhuber, 1997). Finally, for each input position i, we define its context-sensitive embedding as
b i = [h F i , h B i ].
Decoder: This paper uses the classical attentionbased decoder , which generates action sequence y 1 , ..., y n , one action at a time. At each time step j, it writes y j based on the current hidden state s j , then updates the hidden state to s j+1 based on s j and y j . The decoder is formally defined by the following equations:
s 1 = tanh(W (s) [h F m , h B 1 ]) (3) e ji = s T j W (a) b i (4) a ji = exp(e ji ) m i =1 exp(e ji )(5)c j = m i=1 a ji b i(6)P (y j = w|x, y 1:j−1 ) ∝ exp(U w [s j , c j ]) (7) s j+1 = LST M ([φ (y) (y j ), c j ], s j )(8)
where the normalized attention scores a ji defines the probability distribution over input words, indicating the attention probability on input word i at time j; e ji is un-normalized attention score. To incorporate constraints during decoding, an extra controller component is added and its details will be described in Section 3.3. Action Embedding. The above decoder needs the embedding of each action. As described above, each action has two parts, one for structure (e.g., add edge), and the other for semantic (e.g., next to).
As a result, actions may share the same structure or semantic part, e.g., add edge:next to and add edge:loc have the same structure part, and add node:A and arg node:A have the same semantic part. To make parameters more compact, we first embed the structure part and the semantic part independently, then concatenate them to get the final embedding. For instance, φ (y) (add edge:
next to ) = [ φ (y) strut ( add edge ), φ (y) sem ( next to )].
The action embeddings φ (y) are learned during training.
Constrained Semantic Parsing using Sequence-to-Action Model
In this section, we describe how to build a neural semantic parser using sequence-to-action model. We first describe the training and the inference of our model, and then introduce how to incorporate structure and semantic constraints during decoding.
Training
Parameter Estimation. The parameters of our model include RNN parameters W (s) , W (a) , U w , word embeddings φ (x) , and action embeddings φ (y) . We estimate these parameters from training data. Given a training example with a sentence X and its action sequence Y , we maximize the likelihood of the generated sequence of actions given X. The objective function is:
n i=1 log P (Y i |X i )(9)
Standard stochastic gradient descent algorithm is employed to update parameters. Logical Form to Action Sequence. Currently, most datasets of semantic parsing are labeled with logical forms. In order to train our model, we convert logical forms to action sequences using semantic graph as an intermediate representation (See Figure 4 for an overview). Concretely, we transform logical forms into semantic graphs using a depth-first-search algorithm from root, and then generate the action sequence using the same order. Specifically, entities, variables and types are nodes; relations are edges. Conversely we can convert action sequence to logical form similarly. Based on the above algorithm, action sequences can be transformed into logical forms in a deterministic way, and the same for logical forms to action sequences. Mechanisms for Handling Entities. Entities play an important role in semantic parsing (Yih et al., 2015). In Dong and Lapata (2016), entities are replaced with their types and unique IDs. In Jia and Liang (2016), entities are generated via attention-based copying mechanism helped with a lexicon. This paper implements both mechanisms and compares them in experiments.
Inference
Given a new sentence X, we predict action sequence by:
Y * = argmax Y P (Y |X)(10)
where Y represents action sequence, and P (Y |X) is computed using Formula (1). Beam search is used for best action sequence decoding. Semantic graph and logical form can be derived from Y * as described in above.
Incorporating Constraints in Decoding
For decoding, we generate action sequentially. It is obviously that the next action has a strong correlation with the partial semantic graph generated to current, and illegal actions can be filtered using structure and semantic constraints. Specifically, we incorporate constraints in decoding using a controller. This procedure has two steps: 1) the controller constructs partial semantic graph using the actions generated to current; 2) the controller checks whether a new generated action can meet Figure 5: A demonstration of illegal action filtering using constraints. The graph in color is the constructed semantic graph to current.
all structure/semantic constraints using the partial semantic graph.
Structure Constraints. The structure constraints ensure action sequence will form a connected acyclic graph. For example, there must be two argument nodes for an edge, and the two argument nodes should be different (The third candidate next action in Figure 5 violates this constraint). This kind of constraints are domain-independent. The controller encodes structure constraints as a set of rules.
Semantic Constraints. The semantic constraints ensure the constructed graph must follow the schema of knowledge bases. Specifically, we model two types of semantic constraints. One is selectional preference constraints where the argument types of a relation should follow knowledge base schemas. For example, in GEO dataset, relation next to's arg1 and arg2 should both be a state. The second is type conflict constraints, i.e., an entity/variable node's type must be consistent, i.e., a node cannot be both of type city and state. Semantic constraints are domain-specific and are automatically extracted from knowledge base schemas. The controller encodes semantic constraints as a set of rules.
Experiments
In this section, we assess the performance of our method and compare it with previous methods.
Datasets
We conduct experiments on three standard datasets: GEO, ATIS and OVERNIGHT. GEO contains natural language questions about US geography paired with corresponding Prolog database queries. Following Zettlemoyer and Collins (2005), we use the standard 600/280 instance splits for training/test. ATIS contains natural language questions of a flight database, with each question is annotated with a lambda calculus query. Following Zettlemoyer and Collins (2007), we use the standard 4473/448 instance splits for training/test. OVERNIGHT contains natural language paraphrases paired with logical forms across eight domains. We evaluate on the standard train/test splits as Wang et al. (2015b).
Experimental Settings
Following the experimental setup of Jia and Liang (2016): we use 200 hidden units and 100dimensional word vectors for sentence encoding. The dimensions of action embedding are tuned on validation datasets for each corpus. We initialize all parameters by uniformly sampling within the interval [-0.1, 0.1]. We train our model for a total of 30 epochs with an initial learning rate of 0.1, and halve the learning rate every 5 epochs after epoch 15. We replace word vectors for words occurring only once with an universal word vector. The beam size is set as 5. Our model is implemented in Theano (Bergstra et al., 2010), and the codes and settings are released on Github: https://github.com/dongpobeyond/Seq2Act. We evaluate different systems using the standard accuracy metric, and the accuracies on different datasets are obtained as same as Jia and Liang (2016).
Overall Results
We compare our method with state-of-the-art systems on all three datasets. Because all systems using the same training/test splits, we directly use the reported best performances from their original papers for fair comparison.
For our method, we train our model with three settings: the first one is the basic sequence-toaction model without constraints -Seq2Act; the second one adds structure constraints in decoding -Seq2Act (+C1); the third one is the full model which adds both structure and semantic GEO ATIS Previous Work Zettlemoyer and Collins (2005) constraints -Seq2Act (+C1+C2). Semantic constraints (C2) are stricter than structure constraints (C1). Therefore we set that C1 should be first met for C2 to be met. So in our experiments we add constraints incrementally. The overall results are shown in Table 1-2. From the overall results, we can see that:
1) By synthetizing the advantages of semantic graph representation and the prediction ability of Seq2Seq model, our method achieves stateof-the-art performance on OVERNIGHT dataset, and gets competitive performance on GEO and ATIS dataset. In fact, on GEO our full model (Seq2Act+C1+C2) also gets the best test accuracy of 88.9 if under the same settings, which only falls behind Liang et al. (2011)* which uses extra handcrafted lexicons and Jia and Liang (2016)* which uses extra augmented training data. On ATIS our full model gets the second best test accuracy of 85.5, which only falls behind Rabinovich et al. (2017) which uses a supervised attention strategy. On OVERNIGHT, our full model gets state-of-theart accuracy of 79.0, which even outperforms Jia and Liang (2016)* with extra augmented training data.
2) Compared with the linearized logical form representation used in previous Seq2Seq baselines, our action sequence encoding is more effective for semantic parsing. On all three datasets, (2016) OVERNGIHT, the Seq2Act model gets a test accuracy of 78.0, better than the best Seq2Seq baseline gets 77.5. We argue that this is because our action sequence encoding is more compact and can capture more information.
3) Structure constraints can enhance semantic parsing by ensuring the validity of graph using the generated action sequence. In all three datasets, Seq2Act (+C1) outperforms the basic Seq2Act model. This is because a part of illegal actions will be filtered during decoding. 4) By leveraging knowledge base schemas during decoding, semantic constraints are effective for semantic parsing. Compared to Seq2Act and Seq2Act (+C1), the Seq2Act (+C1+C2) gets the best performance on all three datasets. This is because semantic constraints can further filter semantic illegal actions using selectional preference and consistency between types.
Detailed Analysis
Effect of Entity Handling Mechanisms. This paper implements two entity handling mechanisms -Replacing (Dong and Lapata, 2016) which identifies entities and then replaces them with their types and IDs, and attention-based Copying (Jia and Liang, 2016). To compare the above two mechanisms, we train and test with our full model and the results are shown in Table 3. We can see that, Replacing mechanism outperforms Copying in all three datasets. This is because Replacing is done in preprocessing, while attention-based Copying is done during parsing and needs additional copy mechanism. Linearized Logical Form vs. Action Sequence. Table 4 shows the average length of linearized logical forms used in previous Seq2Seq models and the action sequences of our model on all three datasets. As we can see, action sequence encoding is more compact than linearized logical form encoding: action sequence is shorter on all three datasets, 35.5%, 9.2% and 28.5% reduction in length respectively. The main advantage of a shorter/compact encoding is that it will reduce the influence of long distance dependency problem.
Error Analysis
We perform error analysis on results and find there are mainly two types of errors. Unseen/Informal Sentence Structure. Some test sentences have unseen syntactic structures. For example, the first case in Table 5 has an unseen Gold Parse: answer(A, count (B, (const (C, stateid(iowa)), next to(C, B), state (B)), A)) Predicted Parse: answer (A, count(B, state(B), A))
Under-Mapping
Sentence: Please show me first class flights from indianapolis to memphis one way leaving before 10am Gold Parse:
(lambda x (and (flight x) (oneway x) (class type x first:cl) (< (departure time x) 1000:ti) (from x indianapolis:ci) (to x memphis:ci))) Predicted Parse: (lambda x (and (flight x) (oneway x) (< (departure time x) 1000:ti) (from x indianapolis:ci) (to x memphis:ci))) Table 5: Some examples for error analysis. Each example includes the sentence for parsing, with gold parse and predicted parse from our model. and informal structure, where entity word "Iowa" and relation word "borders" appear ahead of the question words "how many". For this problem, we can employ sentence rewriting or paraphrasing techniques (Chen et al., 2016;Dong et al., 2017) to transform unseen sentence structures into normal ones. Under-Mapping. As Dong and Lapata (2016) discussed, the attention model does not take the alignment history into consideration, makes some words are ignored during parsing. For example in the second case in Table 5, "first class" is ignored during the decoding process. This problem can be further solved using explicit word coverage models used in neural machine translation (Tu et al., 2016;Cohn et al., 2016)
Related Work
Semantic parsing has received significant attention for a long time (Kate and Mooney, 2006;Clarke et al., 2010;Krishnamurthy and Mitchell, 2012;Berant and Liang, 2014;Quirk et al., 2015;Artzi et al., 2015;. Traditional methods are mostly based on the principle of compositional semantics, which first trigger predicates using lexicons and then compose them using grammars. The prominent grammars include SCFG (Wong and Mooney, 2007;Li et al., 2015), CCG (Zettlemoyer and Collins, 2005;Kwiatkowski et al., 2011;Cai and Yates, 2013), DCS (Liang et al., 2011;Berant et al., 2013), etc. As discussed above, the main drawback of grammar-based methods is that they rely on high-quality lexicons, manually-built grammars, and hand-crafted features.
In recent years, one promising direction of semantic parsing is to use semantic graph as representation. Thus semantic parsing is modeled as a semantic graph generation process. Ge and Mooney (2009) build semantic graph by trans-forming syntactic tree. Bast and Haussmann (2015) identify the structure of a semantic query using three pre-defined patterns. Reddy et al. (2014Reddy et al. ( , 2016 use Freebase-based semantic graph representation, and convert sentences to semantic graphs using CCG or dependency tree. Yih et al. (2015) generate semantic graphs using a staged heuristic search algorithm. These methods are all based on manually-designed, heuristic generation process, which may suffer from syntactic parse errors (Ge and Mooney, 2009;Reddy et al., 2014Reddy et al., , 2016, structure mismatch (Chen et al., 2016), and are hard to deal with complex sentences (Yih et al., 2015).
One other direction is to employ neural Seq2Seq models, which models semantic parsing as an end-to-end, sentence to logical form machine translation problem. Dong and Lapata (2016), Jia and Liang (2016) and Xiao et al. (2016) transform word sequence to linearized logical forms. One main drawback of these methods is that it is hard to capture and exploit structure and semantic constraints using linearized logical forms. Dong and Lapata (2016) propose a Seq2Tree model to capture the hierarchical structure of logical forms.
It has been shown that structure and semantic constraints are effective for enhancing semantic parsing. Krishnamurthy et al. (2017) use type constraints to filter illegal tokens. Liang et al. (2017) adopt a Lisp interpreter with pre-defined functions to produce valid tokens. Iyyer et al. (2017) adopt type constraints to generate valid actions. Inspired by these approaches, we also incorporate both structure and semantic constraints in our neural sequence-to-action model.
Transition-based approaches are important in both dependency parsing (Nivre, 2008;Henderson et al., 2013) and AMR parsing (Wang et al., 2015a). In semantic parsing, our method has a tight-coupling with knowledge bases, and con-straints can be exploited for more accurate decoding. We believe this can also be used to enhance previous transition based methods and may also be used in other parsing tasks, e.g., AMR parsing.
Conclusions
This paper proposes Sequence-to-Action, a method which models semantic parsing as an end-to-end semantic graph generation process. By leveraging the advantages of semantic graph representation and exploiting the representation learning and prediction ability of Seq2Seq models, our method achieved significant performance improvements on three datasets. Furthermore, structure and semantic constraints can be easily incorporated in decoding to enhance semantic parsing.
For future work, to solve the problem of the lack of training data, we want to design weakly supervised learning algorithm using denotations (QA pairs) as supervision. Furthermore, we want to collect labeled data by designing an interactive UI for annotation assist like (Yih et al., 2016), which uses semantic graphs to annotate the meaning of sentences, since semantic graph is more natural and can be easily annotated without the need of expert knowledge.
Figure 1 :
1Overview of our method, with a demonstration example.
Figure 3 :
3Our attention-based Sequence-to-Action RNN model, with a controller for incorporating constraints.
Figure 4 :
4The procedure of converting between logical form and action sequence.
Soc. Blo. Bas. Res. Cal. Hou. Pub. Rec. Avg.Previous Work
Wang et al. (2015b)
48.2 41.9 46.3 75.9 74.4 54.0 59.0 70.8 58.8
Seq2Seq Models
Xiao et al. (2016)
80.0 55.6 80.5 80.1 75.0 61.9 75.8
-
72.7
Jia and Liang (2016)
81.4 58.1 85.2 76.2 78.0 71.4 76.4 79.6 75.8
Jia and Liang
Table 2 :
2Test accuracies on OVERNIGHT dataset, which includes eight domains: Social, Blocks, Bas-
ketball, Restaurants, Calendar, Housing, Publications, and Recipes.
our basic Seq2Act model gets better results than
all Seq2Seq baselines. On GEO, the Seq2Act
model achieve test accuracy of 87.5, better than
the best accuracy 87.1 of Seq2Seq baseline. On
ATIS, the Seq2Act model obtains a test accuracy
of 84.6, the same as the best Seq2Seq baseline. On
Table 3 :
3Test accuracies of Seq2Act (+C1+C2) on
GEO, ATIS, and OVERNIGHT of two entity han-
dling mechanisms.
Logical Form Action Sequence
GEO
28.2
18.2
ATIS
28.4
25.8
OVERNIGHT
46.6
33.3
Table 4 :
4Average length of logical forms and ac-
tion sequences on three datasets. On OVERNIGHT,
we average across all eight domains.
AcknowledgmentsThis research work is supported by the National Key Research and Development Program of China under Grant No.2017YFB1002104; and the National Natural Science Foundation of China under Grants no. 61572477 and 61772505. Moreover, we sincerely thank the reviewers for their valuable comments.
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|
[
"https://github.com/dongpobeyond/Seq2Act."
] |
[
"TRANSFER OF CHARACTERS FOR DISCRETE SERIES REPRESENTATIONS OF THE UNITARY GROUPS IN THE EQUAL RANK CASE VIA THE CAUCHY-HARISH-CHANDRA INTEGRAL",
"TRANSFER OF CHARACTERS FOR DISCRETE SERIES REPRESENTATIONS OF THE UNITARY GROUPS IN THE EQUAL RANK CASE VIA THE CAUCHY-HARISH-CHANDRA INTEGRAL"
] |
[
"Allan Merino "
] |
[] |
[] |
As conjectured by T. Przebinda, the transfer of characters in the Howe's correspondence should be obtained via the Cauchy-Harish-Chandra integral. In this paper, we prove that the conjecture holds for the dual pair (G = U(p, q), G ′ = U(r, s)), p + q = r + s, starting with a discrete series representation Π of U(p, q).2010 Mathematics Subject Classification. Primary: 22E45; Secondary: 22E46, 22E30.
| null |
[
"https://arxiv.org/pdf/2101.02063v1.pdf"
] | 230,770,105 |
2101.02063
|
158f5639b84f527a616c538c2c20df518bf1a4ef
|
TRANSFER OF CHARACTERS FOR DISCRETE SERIES REPRESENTATIONS OF THE UNITARY GROUPS IN THE EQUAL RANK CASE VIA THE CAUCHY-HARISH-CHANDRA INTEGRAL
6 Jan 2021
Allan Merino
TRANSFER OF CHARACTERS FOR DISCRETE SERIES REPRESENTATIONS OF THE UNITARY GROUPS IN THE EQUAL RANK CASE VIA THE CAUCHY-HARISH-CHANDRA INTEGRAL
6 Jan 2021
As conjectured by T. Przebinda, the transfer of characters in the Howe's correspondence should be obtained via the Cauchy-Harish-Chandra integral. In this paper, we prove that the conjecture holds for the dual pair (G = U(p, q), G ′ = U(r, s)), p + q = r + s, starting with a discrete series representation Π of U(p, q).2010 Mathematics Subject Classification. Primary: 22E45; Secondary: 22E46, 22E30.
Introduction
Let W be a finite dimensional vector space over R endowed with a non-degenerate, skew-symmetric, bilinear form ·, · , Sp(W) be the corresponding group of isometries, Sp(W) be the metaplectic cover of Sp(W) (see [1,Definition 4.18]) and (ω, H ) be the corresponding Weil representation (see [1,Section 4.8]). For every irreducible reductive dual pair (G, G ′ ) in Sp(W), R. Howe proved (see [14,Theorem 1]) that there is a bijection between R( G, ω) and R( G ′ , ω) whose graph is R( G · G ′ , ω) (where R( G, ω) is defined in Section 2). More precisely, to every Π ∈ R( G, ω), we associate a representation finitely generated admissible representation Π ′ 1 of G ′ which has a unique irreducible quotient Π ′ such that Π ⊗ Π ′ ∈ R( G · G ′ , ω). We denote by θ : R( G, ω) ∋ Π → Π ′ = θ(Π) ∈ R( G ′ , ω) the corresponding bijection. As proved by Harish-Chandra (see [7,Section 5] or Section 4), all the representations Π of G (resp. Π ′ of G ′ ) appearing in the correspondence admit a character, i.e. a G-invariant distribution Θ Π on G (in the sense of Laurent Schwartz) given by a locally integrable function Θ Π on G which is analytic on G reg (the set of regular elements of G). The character Θ Π determines the representation Π. In particular, one way to understand the Howe correspondence, i.e. to make the map θ explicit, is to understand the transfer of characters. In his paper [23], T. Przebinda conjectured that the correspondence of characters should be obtained via the socalled Cauchy-Harish-Chandra integral that he introduced in [23]. We recall briefly the construction of this integral. Let T : Sp(W) → S * (W) be the embedding of the metaplectic group inside the space of tempered distributions on W as in [1,Definition 4.23] (see also Remark 2.2) and H 1 , . . . , H n be a maximal set of non-conjugate Cartan subgroups of G which are ι-invariant, where ι is a Cartan involution on G. Every Cartan subgroup H i can be decomposed as H i = T i A i , with T i maximal compact in H i . Let A ′ i and A ′′ i be the subgroups of Sp(W) defined by A ′ i = C Sp(W) (A i ) and A ′′ i = C Sp(W) (A ′ i ). One can easily check that (A ′ i , A ′′ i ) form a dual pair in Sp(W), which is not irreducible in general. For every function ϕ ∈ C ∞ c ( A ′ i ), we define Chc(ϕ) by
Chc(ϕ) = A ′′ i \W A ′′ i T(ϕ)(w)dw,
where dw is a measure on the manifold A ′′ i \ W A ′′ i defined in [23,Section 1]. As mentioned in [23, Section 2] (see also Section 2), Chc(Ψ) is well-defined and the corresponding map Chc : C ∞ c ( A ′ i ) → C is a distribution on A ′ i . For every regular elementh i ∈ H i reg , we denote by Chc˜h i the pull-back of Chc through the map G ′ ∋g ′ →h ig ′ ∈ A ′ i . Assume now that rk(G) ≤ rk(G ′ ). In [3] (see also Section 4), F. Bernon and T. Przebinda defined a map:
Chc * : D ′ ( G) G → D( G ′ ) G ′ ,
where D ′ ( G) G is the set of G-invariant distributions on G. More precisely, if Θ is a G-invariant distribution given by a locally integrable function Θ on G, then, for every ϕ ∈ C ∞ c ( G ′ ), we get:
Chc * (Θ)(ϕ) = n i=1 1 |W (H i )| H i reg Θ(h i )|det(1 − Ad(h −1 i )) g/h i |Chc˜h i (ϕ)dh i .
The conjecture can be stated as follows:
Conjecture 1.1. Let G 1 and G ′ 1 be the Zariski identity components of G and G ′ respectively. Let Π ∈ R( G, ω) satisfying Θ Π| G/ G 1 = 0 if G = O(V), where V is an even dimensional vector space over R or C. Then, up to a constant, Chc * (Θ Π ) = Θ Π ′ 1 on G ′ 1 . This result is well-known if the group G is compact and had been proved recently in [24] in the stable range. In this paper, we investigate the case (G, G ′ ) = (U(p, q), U(r, s)), p + q = r + s (p will always be assumed to be smaller or equal than q, in particular, the number of non-conjugate Cartan subgroups of G is p+1 (see Remark 2)) and Π ∈ R( G, ω) a discrete series representation of G. Let λ be the Harish-Chandra parameter of Π. In this case, using Li's result (see [17,Proposition 2.4] or Section 7), we get that Π ′ 1 = Π ′ and using [20], we know that Π ′ = θ(Π) is a discrete series representations of G ′ (with Harish-Chandra parameter λ ′ ), and the correspondence λ → λ ′ is known and explicit (see [20,Theorem 2.7]). In order to prove that, up to a constant, Chc * (Θ Π ) = Θ Π ′ 1 = Θ Π ′ , we use a parametrisation of discrete series characters provided by Harish-Chandra (see [8,Lemma 44]). More precisely, it follows from [3] and a result of Harish-Chandra (see [9,Theorem 2]) that the distribution Chc * (Θ Π ) is given by locally integrable function Θ ′ Π analytic on G ′ reg . Using [3, Theorem 2.2], we proved in Proposition 6.5 that the value of Θ ′ Π on H ′ reg , where H ′ is the compact Cartan subgroup of G ′ , is of the form:
∆(ȟ ′ )Θ ′ Π (p(ȟ ′ )) = C σ∈S r ×S s ε(σ)(σȟ ′ ) λ Π , (ȟ ′ ∈Ȟ ′reg ),
where C ∈ R,Ȟ ′ is a double cover of H ′ (see Section 3) chosen such that ρ ′ = 1 2 α>0 α is analytic integral,p is a map fromȞ ′ into H ′ (which is not an isomorphism of double covers in general), and λ Π is a linear form on ih ′ depending on Π which is conjugated to λ under S r+s . Moreover, using results of [2], we proved in Proposition 6.7 that
sup g ′ ∈ G ′ reg |D(g ′ )| 1 2 |Θ ′ Π (g ′ )| < ∞,
where D is the Weyl denominator defined in Notations 5.3. Finally, applying [3, Theorem 1.3] to our particular dual pair, it follows that zChc * (Θ Π ) = χ λ Π (z)Chc * (Θ Π ) for every z ∈ Z(U (g ′ C )), where χ λ Π is the character of Z(U (g ′ C )) obtained via the linear form λ Π as in Remark A.9 and then, using a result of Harish-Chandra (see [8,Lemma 44]) we get that Chc * (Θ Π ) is the character of a discrete series representations of G ′ with Harish-Chandra parameter λ Π . In Section 7, we prove, by using results of [22] (see also [17]), that T(Θ Π ) is a well-defined G · G ′ -invariant distribution on S * (W) and we get in Corollary 7.4 the following equality:
T(Θ Π ) = C Π⊗Π ′ T(Chc * (Θ Π )),
where C Π⊗Π ′ is a constant depending on Π and Π ′ . In particular, we can hope that the following diagram often commutes (up to a constant):
D ′ ( G) G Chc * / / T % % ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ D ′ ( G ′ ) G ′ T
y y s s s s s s s s s s S * (W) G· G ′ Moreover, according to Li's result (see [17] or Section 7), Π can be embedded in ω as a subrepresentation, and by projecting onto the ν ⊗ Π ′ -isotypic component (where ν is the lowest K-type of Π as in Theorem 5.4), we get (see Equation (13)) the following equality:
We say that a pair of subgroup (G, G ′ ) of Sp(W) is a dual pair if G is the centralizer of G ′ in Sp(W) and vice-versa. The dual pair is said to be reductive if both G and G ′ act reductively on W and irreducible if we cannot find an orthogonal decomposition of W = W 1 ⊕ W 2 where both W 1 and W 2 are G · G ′ -invariant. One can easily prove that the preimages G = π −1 (G) and G ′ = π −1 (G ′ ) in Sp(W) form a dual pair in Sp(W). Let (ω, H ) be the Weil representation of Sp(W) corresponding to a fixed unitary character of R and (ω ∞ , H ∞ ) be the corresponding smooth representation (see [1,Section 4.8]). For a subgroup H of Sp(W), we denote by R( H, ω) the set of conjugacy classes of irreducible admissible representations (Π, H Π ) of H which can be realized as a quotient of H ∞ by a closed ω ∞ ( H)-invariant subspace. As proved by R. Howe (see [14,Theorem 1]), for every reductive dual pair (G, G ′ ) of Sp(W), we have a one-to-one correspondence between R( G, ω) and R( G ′ , ω) whose graph is R( G · G ′ , ω). More precisely, if Π ∈ R( G, ω), we denote by N(Π) the intersection of all the closed G-invariant subspaces N such that Π ≈ H ∞ /N . Then, the space
H (Π) = H ∞ /N(Π) is a G · G ′ -module; more precisely, H (Π) = Π ⊗ Π ′ 1 , where Π ′ 1 is a G ′ -module, not irreducible in general, but Howe's duality theorem says that there exists a unique irreducible quotient Π ′ of Π ′ 1 with Π ′ ∈ R( G ′ , ω) and Π ⊗ Π ′ ∈ R( G · G ′ , ω).
We will denote by θ : R( G, ω) → R( G ′ , ω) the corresponding bijection.
Notation 2.1. We use here the notations of [1]. We denote by S * (W) the space of tempered distributions on W and by [1,Definition 4.23]). We denote by Sp c (W) the subset of Sp(W) given by g ∈ Sp(W), det(g − 1) 0 and by Sp c (W) its preimage in Sp(W).
T : Sp(W) → S * (W) the injection of Sp(W) into S * (W) (see
Remark 2.2. As explained in [1], for everyg ∈ Sp c (W), the distribution T(g) is defined by
T(g) = Θ(g)χ c(g) µ W ,
where Θ is the character of the Weil representation (ω, H ) defined in [1,Definition 4.23], χ c(g) : W → C is the function on W given by χ c(g) (w) = χ 1 4 (g + 1)(g − 1) −1 w, w with g = π(g) and µ W is the appropriately normalized Lebesgue measure on W. The map T can be extended to Sp(W) and to C ∞ c ( Sp(W)) by
T(ϕ) = Sp(W) ϕ(g)T(g)dg, (ϕ ∈ C ∞ c ( Sp(W))),
where dg is the Haar measure on Sp(W). As proved in [1, Section 4.8], for every ϕ ∈ C ∞ c ( Sp(W)), the distribution T(ϕ) on W is given by a Schwartz function on W still denoted by T(ϕ), i.e.
T(ϕ)(φ) = W T(ϕ)(w)φ(w)dµ W (w), (φ ∈ S(W)).
We now recall the construction of the Cauchy-Harish-Chandra integral introduced by T. Przebinda in [23]. We denote by H i , 1 ≤ i ≤ n be a maximal set of non-conjugate Cartan subgroups of G. As explained in [
φ ∈ C ∞ c (W) such that supp(φ) ⊆ W A ′′ i , W A ′′ i φ(w)dµ W (w) = A ′′ i \W A ′′ i A ′′ i φ(aw)dadw.
For every ϕ ∈ C ∞ c ( A ′ i ), we denote by Chc(ϕ) the following integral:
Chc(ϕ) = A ′′ i \W A ′′ i T(ϕ)(w)dw.
According to Remark 2.2, the previous integral is well-defined and as proved in [23,Lemma 2.9], the corresponding map Chc :
C ∞ c ( A ′ i ) → C defines a distribution on A ′
i . For everyh i ∈ H i , we denote by τ˜h i the map:
τ˜h i : G ′ ∋g ′ →hg ′ ∈ A ′ i and, forh i regular, by Chc˜h i = τ * h i (Chc), where τ * h i
is the pull-back of τ˜h i as defined in [13,Theorem 8.2.4]. In particular, for everyh i ∈ H i reg , Chc˜h i is a well-defined distribution on G ′ .
Explicit formulas of Chc for unitary groups
Let V = C p+q and V ′ = C r+s be two complex vector spaces endowed with non-degenerate bilinear forms (·, ·) and (·, ·) ′ respectively, with (·, ·) hermitian and (·, ·) ′ skew-hermitian, and let (p, q) (resp. (r, s)) be the signature of (·, ·) (resp. (·, ·) ′ ). We assume that p + q ≤ r + s. Let B V = {e 1 , . . . , e n }, n = p + q (resp. B V ′ = e ′ 1 , . . . , e ′ n ′ , n ′ = r + s) be a basis of V (resp. V ′ ) such that Mat((·, ·) , B V ) = Id p,q (resp. Mat((·, ·) ′ , B V ′ ) = iId r,s ). Let G and G ′ be the corresponding group of isometries, i.e. G = G(V, (·, ·)) ≈ g ∈ GL(n, C), g t Id p,q g = Id p,q , G ′ = G(V ′ , (·, ·) ′ ) ≈ g ∈ GL(n ′ , C), g t Id r,s g = Id r,s .
where ≈ is a Lie group isomorphism. Let H and H ′ be the diagonal compact Cartan subgroups of G and G ′ respectively. By looking at the action of H on the space V, we get a decomposition of V of the form:
V = V 1 ⊕ . . . ⊕ V n ,
where the spaces V a given by V a = Cie a are irreducible H-modules. We denote by J the element of h such that J = iId V and let J j = iE j, j . Similarly, we write
V ′ = V ′ 1 ⊕ . . . ⊕ V ′ n ′ , with V ′ b = Cie ′ b , J ′ the element of h ′ given by J ′ = iId V ′ and J ′ j = iE j, j . Let W = Hom C (V ′ , V)
endowed with the symplectic form ·, · given by:
w 1 , w 2 = tr C/R (w * 2 w 1 ), (w 1 , w 2 ∈ W),
where w * 2 is the element of Hom(V, V ′ ) satisfying:
w * 2 (v ′ ), v = v ′ , w 2 (v) ′ (v ∈ V, v ′ ∈ V ′ ).
The space W can be seen as a complex vector space by
(1) iw = J • w (w ∈ W).
We define a double cover GL C (W) of the complex group GL C (W) by:
GL C (W) = g = (g, ξ) ∈ GL C (W) × C × , ξ 2 = det(g) .
Because p + q ≤ r + s, we get a natural embedding of h C into h ′ C and we denote by Z ′ = G ′h the centralizer of h in G ′ . Let H ′ C be the complexification of H ′ in GL C (W). In particular, H ′ C is isomorphic to
h ′ C / n ′ j=1 2πx j J j , x j ∈ Z .
We denote byȞ ′ C the connected two-fold cover of H C isomorphic to (1) Let Ψ ′ := Ψ ′ (g ′ C , h ′ C ) be a set of positive roots corresponding to (g ′ C , h ′ C ), Ψ ′ (k ′ ) be a the set of compact roots in Ψ ′ , where k is the Lie algebra of K = U(r) × U(s), and Ψ ′ n be the set of non-compact roots of Ψ ′ , i.e. Ψ ′ n = Ψ ′ \ Ψ ′ (k). The reason why we are considering the double coverȞ ′ C of H ′ C is to make the form ρ ′ = 1 2 α∈Ψ ′ α analytic integral. For every analytic integral form γ on h ′ C , we will denote byȟ ′ →ȟ ′γ the corresponding character on H ′ C . (2) We know that, up to conjugation, the number of Cartan subgroups in U(r, s) is min(r, s) + 1. Those Cartan subgroups can be parametrized by some particular subsets of Ψ ′ n . Let Ψ ′st n be the set of strongly orthogonal roots in Ψ ′ (see [25,Section 2]). For every α ∈ Ψ ′st n , we denote by c(α) the element of G ′ C given by:
h ′ C / n ′ j=1 2πx j J ′ j , n ′ j=1 x j ∈ 2Z, x j ∈ Z .c(α) = exp π 4 (X −α − X α ) .
where X α (resp. X −α ) is in g ′ C,α (resp. g ′ C,−α ) and normalized as in [25,Equation 2.7]. For every subset S of Ψ ′st n , we denote by c(S) the following element of G ′ C defined by c(S) = α∈S c(α), and let h ′ (S) = g ′ ∩ Ad(c(S))(h ′ C ). We denote by H ′ (S) the analytic subgroup of G ′ whose Lie algebra is h ′ (S). Then, H ′ (S) is a Cartan subgroup of G ′ and one can prove that all the Cartan subgroups are of this form (up to conjugation). For every S ⊆ Ψ ′st n , we will denote by H ′ S the subgroup of H ′ C given by:
H ′ S = c(S) −1 H ′ (S)c(S)
. Assume that r ≤ s. Then, we define Ψ ′ = e i − e j , 1 ≤ i < j ≤ r + s , where e i is the linear form on h ′ C = C r+s given by e i (λ 1 , . . . , λ r+s ) = λ i . In this case, the set Ψ ′st n is equal to {e t − e r+t , 1 ≤ t ≤ r}. In particular, H ′ (∅) = H ′ and if S t = {e 1 − e r+1 , . . . , e t − e r+t }, we get:
(2) H ′ S t = h = diag(e iX 1 −X r+1 , . .
. , e iX t −X r+t , e iX t+1 , . . . , e iX r , e iX 1 +X r+1 , . . . , e iX t +X r+t , e iX r+t+1 , . . . , e iX r+s ), X j ∈ R .
Fix a subset S ∈ Ψ ′st n . We denote byȞ ′ S the preimage of H ′ S inȞ ′ C . For every ϕ ∈ C ∞ c ( G ′ ), we denote by H S ϕ the function ofȞ ′ S defined by:
H S ϕ(ȟ ′ ) = ε Ψ ′ S,R (ȟ ′ )ȟ ′ 1 2 α∈Ψ ′ α α∈Ψ ′ (1 −ȟ ′−α ) G ′ /H ′ (S) ϕ(g ′ c(S)p(ȟ ′ )c(S) −1 g ′−1 )dg ′ H ′ (S) (ȟ ′ ∈Ȟ ′ S ),
where Ψ ′ S,R is the subset of Ψ ′ consisting of real roots for H ′ S and ε Ψ ′ S,R is the function defined onȞ ′reg S by
ε Ψ ′ S,R (ȟ ′ ) = sign α∈Ψ ′ S,R (1 −ȟ ′−α ) ,
To simplify the notations, we denote by ∆ Ψ ′ (ȟ ′ ) the quantity
∆ Ψ ′ (ȟ ′ ) =ȟ ′ 1 2 α∈Ψ ′ α α∈Ψ ′ (1 −ȟ ′−α ) (ȟ ′ ∈Ȟ ′ S ). We define ∆ Φ ′ similarly, where Φ ′ = −Ψ ′ . Remark 3.2. (1) For everyȟ ′ ∈Ȟ ′reg S , ∆ Φ ′ (ȟ ′ )∆ Ψ ′ (ȟ ′ ) = α∈Ψ ′+ (1 −ȟ ′α )(1 −ȟ ′−α )
Note that if S = ∅, we get for every α ∈ Ψ ′ andȟ ′ ∈Ȟ ′ thatȟ ′α =ȟ ′−α . In particular,
∆ Φ ′ (ȟ ′ )∆ Ψ ′ (ȟ ′ ) = α∈Ψ ′ (1 −ȟ ′α )(1 −ȟ ′α ) = α∈Ψ ′ |1 −ȟ ′α | 2 = |det(Id − Ad(ȟ ′ )) g ′ /h ′ |.
Similarly, if S ∅, we get that for every α ∈ Ψ ′ andȟ ′ ∈Ȟ ′ , there exists β ∈ Φ ′ , independant onȟ ′ , such thatȟ ′α =ȟ ′β . In particular, we get
∆ Φ ′ (ȟ ′ )∆ Ψ ′ (ȟ ′ ) = α∈Ψ ′ |1 −ȟ ′α | 2 .
For everyȟ ′ ∈Ȟ ′ S , we denote by
|∆ G ′ (ȟ ′ )| 2 = ∆ Φ ′ (ȟ ′ )∆ Ψ ′ (ȟ ′ ).
(2) One can easily check that two Cartan subalgebras h ′ (S 1 ) and h ′ (S 2 ), with S 1 , S 2 ⊆ Ψ ′st n , are conjugate if and only if there exists an element of σ ∈ W sending S 1 ∪ (−S 1 ) onto S 2 ∪ (−S 2 ) (see [25,Proposition 2.16]).
The Weyl's integration formula can be written with the previous notations as follows Proposition 3.3 (Weyl's Integration Formula). For every ϕ ∈ C ∞ c ( G ′ ), we get:
(3) G ′ ϕ(g ′ )dg ′ = S∈Ψ ′st n m S Ȟ′ S ε Ψ ′ S,R (ȟ ′ )∆ Φ ′ (ȟ ′ )H S ϕ(ȟ ′ )dȟ ′ .
where m S are complex numbers. Here, the subsets S of Ψ ′st n are defined up to equivalence (see Remark 3.2).
Proof. See [3, Section 2, Page 3830].
Remark 3.4. In particular, if we fix S ⊆ Ψ ′st n and ϕ ∈ C ∞ c ( G ′ ) such that supp(ϕ) ⊆ G ′ · H ′ (S) reg , the previous formula can be written as follow:
G ′ ϕ(g ′ )dg ′ = m S Ȟ′ S ε Ψ ′ S,R (ȟ ′ )∆ Φ ′ (ȟ ′ )H S ϕ(ȟ ′ )dȟ ′ .
Let H C , G ′ C ⊆ GL C (W) the complexifications of H and G ′ . We denote by G ′i C the subgroup of G ′ C consisting of elements commuting with the element i introduced in Equation (1). As proved in [3, Section 2], the character Θ defined in [1,Definition 4.23] extends to a rational function on H C · G ′i C given by
Θ(hg ′ ) = (−1) u det 1 2 (hg ′ ) det(1 − hg ′ ) (h ∈ H C ,g ′ ∈ G ′i C )
, where 2u is the maximal dimension of a real subspace of W on which the symmetric form J·, · is negative definite. More precisely, according to [3, Proposition 2.1], we get:
Proposition 3.5. For everyȟ ∈Ȟ C andȟ ′ ∈Ȟ ′ C , we get: det k 2 (ȟ) W h ∆ Ψ (ȟ)Θ(p(ȟ)p(ȟ ′ ))∆ Φ (ȟ ′ ) = σ∈W (H ′ C ) (−1) u+α sign(σ) |W (Z ′ C , H ′ C )| det − k 2 (σ −1 (ȟ ′ )) W h ∆ Φ(Z ′ ) (σ −1 (ȟ ′ )) det(1 − p(h)p(h ′ )) σW h ,
where α ∈ {0, −1} depends only on the choice of the positive roots Ψ and Φ ′ , k ∈ {0, −1} is defined by
k = −1 if n ′ − n ∈ 2Z 0 otherwise
and W h is the set of elements of W commuting with h.
Remark 3.6. One can easily check that the space W h is given by
W h = n i=1 Hom(V ′ i , V i ).
For every S ⊆ Ψ ′st n , we denote by S the subset of {1, . . . , r + s} given by S = j, ∃α ∈ S such that α(J ′ j ) 0 . Let σ ∈ S n ′ and S ⊆ Ψ ′st n , we denote by Γ σ,S the subset of h ′ defined as
(4) Γ σ,S = Y ∈ h ′ , Y·, · σW h ∩ j S Hom(V ′ j ,V) > 0 ,
and let E σ,S = exp(iΓ σ,S ) the corresponding subset of H ′ C , where exp is a choice of exponential map exp : h ′ C → H ′ C obtained by choosing an element 1 in π −1 {1}.
Theorem 3.7. For everyȟ ∈Ȟ =Ȟ ∅ and ϕ ∈ C ( G ′ ), we get:
det k 2 (ȟ) W h ∆ Ψ (ȟ) G ′ Θ(p(ȟ)g ′ )ϕ(g ′ )dg ′ = σ∈W (H ′ C ) S⊆Ψ ′st n M S (σ) lim r∈E σ,S r→1 Ȟ′ S det − k 2 (σ −1 (ȟ ′ )) W h ∆ Φ ′ (Z ′ ) (σ −1 (ȟ ′ )) det(1 − p(h)rp(h ′ )) σW h ε Φ ′ S,R (ȟ ′ )H S (ϕ)(ȟ ′ )dȟ ′ , where M S (σ) = (−1) u ε(σ)m S |W (Z ′ C , H ′ C )| .
The theorem 3.7 tells us how to compute Chc˜h for an elementh in the compact Cartan H = H(∅). Using [2], it follows that the value of Chc on the other Cartan subgroups can be computed explicitely by knowing how to do it for the compact Cartan (we will assume, without loss of generality, that p ≤ q, in particular, the number of Cartan subgroups of G, up to conjugation, is p + 1).
Notation 3.8. For every i ∈ max(p, min(r, s)), we define the set S i
S i = {e 1 − e α+1 , . . . , e i − e α+i } if r ≤ s e 1 − e β+1 , . . . , e i − e β+i otherwise , where α = p if r ≤ p r otherwise and β = p if s ≤ p s otherwise .
For every i ∈ [|0, p|] and j ∈ [|0, min(r, s)|], we denote by H(S i ) and H ′ (S j ) the Cartan subgroups of G and G ′ respectively and let H(
S i ) = T(S i )A(S i ) and H ′ (S j ) = T ′ (S j )A ′ (S j ) be the decompositions of H(S i ) and H ′ (S j ) as in [26, Section 2.3.6]. In particular, H(S k ) = H ′ (S k ) for every k ∈ [|0, min(p, min(r, s))|].
To simplify, we assume that r ≤ s. We denote by V 0,i the subspace of V on which A(S i ) acts trivially and by V 1,i the orthogonal complement of V 0,i in V. Let V 1,i = X i ⊕ Y i be a complete polarization of V 1,i . We assume that we have a natural embedding of V 1,i into V ′ such that X i ⊕ Y i is a complete polarization with respect to (·, ·) ′ (i.e i ≤ r). Let U i be the orthogonal complement of V 1,i in V ′ ; in particular, we get a natural embedding:
GL(X i ) × G(U i ) ⊆ G ′ = U(r, s).
We denote by T 1 (S i ) the maximal subgroup of T(S i ) which acts trivially on V 0,i and let
T 2 (S i ) the subgroup of T(S i ) such that T(S i ) = T 1 (S i ) × T 2 (S i ) with T 2 (S i ) ⊆ G(V 0,i ). In particular,(5)H(S i ) = T 1 (S i ) × A(S i ) × T 2 (S i ).
Similarly, we get a decomposition of H ′ (S i ) os the form:
(6) H ′ (S i ) = T ′ 1 (S i ) × A ′ (S i ) × T ′ 2 (S i )
. Let η(S i ) and η ′ (S i ) be the nilpotent Lie subalgebras of u(p, q) and u(r, s) respectively given by
η(S i ) = Hom(X i , V 0,i ) ⊕ Hom(X i , Y i ) ∩ u(p, q), η ′ (S i ) = Hom(U i , X i ) ⊕ Hom(X i , Y i ) ∩ u(r, s).
We will denote by W 0,i the subspace of W defined by Hom(U i , V 0,i ) and by P(S i ) and P ′ (S i ) the parabolic subgroups of G and G ′ respectively whose Levi factors L(S i ) and L ′ (S i ) are given by
L(S i ) = GL(X i ) × G(V 0,i ) L ′ (S i ) = GL(X i ) × G(U i ),
and by N(S i ) := exp(η(S i )) and N ′ (S i ) := exp(η ′ (S i )) the unipotent radicals of P(S i ) and P ′ (S i ) respectively.
Remark 3.9. One can easily check that the forms on V 0,i and U i have signature (p − i, q − i) and (r − i, s − i)
respectively.
As proved in [2, Theorem 0.9], for everyh =t 1ãt2 ∈ H(S i ) reg (using the decomposition of H(S i ) given in Equation (5)) and ϕ ∈ C ∞ c ( G ′ ), we get:
(7) |det(Ad(h) − Id) η(S i ) |Chc˜h(ϕ) = Cd S i (h)ε(t 1ã ) GL(X i )/T 1 (S i )×A(S i ) G(U i ) ε(t 1ãỹ )Chc W 0,i (t 2ỹ )d ′ S i (gt 1ã g −1ỹ )ϕ K ′ N ′ (S i ) (gt 1ã g −1ỹ )dỹdg, where C is a constant defined in [2, Theorem 0.9], ε is the character defined in [2, Lemma 6.3], d S i : L(S i ) → R and d ′ S i : L ′ (S i ) → R are given by d S i (l) = |det(Ad(l) η(S i ) )| 1 2 , d ′ S i (l ′ ) = |det(Ad(l ′ ) η ′ (S i ) )| 1 2 , (l ∈ L(S i ),l ′ ∈ L ′ (S i )), and ϕ K ′ N ′ (S i )
is the Harish-Chandra transform of ϕ, i.e. the function on L ′ (S i ) defined by:
ϕ K ′ N ′ (S i ) (l ′ ) = N ′ (S i ) K ′ ϕ(kl ′ñk−1 )dkdñ, (l ′ ∈ L ′ (S i )).
One can easily check that (G(V 0,i ), G(U i )) is an irreducible dual pair in Sp(W 0,i ) of the same type of G, G ′ . Moreover, the elementt 2 is contained in the compact Cartan of G(V 0,i ). In particular, it follows from Theorem 3.7 that the integral
G(U i ) ε(t 1ãỹ )Chc W 0,i (t 2ỹ )d ′ S i (gt 1ã g −1ỹ )ϕ K ′ N ′ (S i ) (gt 1ã g −1ỹ )dỹ
can be seen as a finite sum of integrals, where the test function ϕ is replaced by
ε(ỹ)d ′ S i (gt 1ã g −1ỹ )ϕ K ′ N ′ (S i ) (gt 1ã g −1ỹ ),ỹ ∈ G(U i ). Notation 3.10. For every j ∈ {1, . . . , p} and k ∈ {0, . . . , p − j}, we denote by S j k = {e j+1 − e p+ j+1 , . . . , e j+k − e p+ j+k } the subset of Ψ st n (g(V 0, j ) C , t 2 (S j ) C ) and by H(S j k ) the corresponding Cartan subgroup of G(V 0, j ). By convention, H(S j 0 ) = T 2 (S j ) is the compact Cartan subgroup of G(V 0, j ). Assume that r ≤ s. For j ∈ {1, . . . , r}, and k ∈ {0, . . . , r − j}, we denote by S j k = {e j+1 − e r+ j+1 , . . . , e j+k − e r+ j+k } the subset of Ψ ′st n (g(U j ) C , t ′ 2 (S j ) C ) and by H ′ (S j k ) the corresponding Cartan subgroup of G(U j ). By convention, H ′ (S j 0 ) = T ′ 2 (S j ) is the compact Cartan subgroup of G(U j ). Remark 3.11. Let i ∈ {0, . . . , min(p, r)} and j ≤ i. We denote by X j , Y j , X i , Y i the subspaces of V as before. There exists subspaces X j,i and Y j,i such that X i = X j ⊕ X j,i and Y i = Y j ⊕ X j,i . In particular, U j = U i ⊕ X j,i ⊕ Y j,i . The Cartan subgroup H ′ (S i ) is included in Levi factor L ′ (S j ) of the parabolic P ′ (S j ) of G. Let H ′ (S j i− j )
be the Cartan subgroup of G(U j ) as in Notation 3.10. As in Equation (5), we have:
H ′ (S j i− j ) = T ′ 1 (S j i− j ) × A ′ (S j i− j ) × T ′ 2 (S j i− j ),
and then we get the following decomposition of H ′ (S i ):
(8) H ′ (S i ) = T ′ 1 (S i ) × A ′ (S i ) ⊆GL(X i ) × T ′ 2 (S i ) ⊆G(U i ) = T ′ 1 (S j ) × A ′ (S j ) ⊆GL(X j ) × T ′ 1 (S j i− j ) × A ′ (S j i− j ) ⊆GL( X j,i ) × T ′ 2 (S i ) ⊆G(U i ) ⊆GL( X j,i )×G(U i )⊆G(U j ) ⊆GL(X j )×G(U j )=L ′ (S j )
.
Finally, one can see easily that T ′ 2 (S i ) = T ′ 2 (S j i− j ) and then,
H ′ (S i ) = T ′ 1 (S j ) × A ′ (S j ) × H ′ (S j i− j ).
We finish this section with a technical lemma which will be useful in Section 6.
Lemma 3.12. For every f ∈ C ∞ c ( G ′ ) andh ′ ∈ H ′ (S i ), we get: GL(X j )/T ′ 1 (S j )×A ′ (S j ) G(U j )/H ′ (S j i− j ) f K ′ N ′ (S j ) (g 1 g 2h ′ g −1 2 g −1 1 )dg 2 dg 1 = D L ′ (S i ) (h) D L ′ (S j ) (h) GL(X i )/T ′ 1 (S i )×A ′ (S i ) G(U i )/T ′ 2 (S i ) f K ′ N ′ (S i ) (g 1 g 2h ′ g −1 2 g −1 1 )dg 2 dg 1 ,
where D L ′ (S j ) and D L ′ (S i ) are given by:
D L ′ (S j ) (h ′ ) = |det(Id − Ad(h ′ ) −1 ) l ′ (S j )/h ′ (S j ) | 1 2 , D L ′ (S i ) (h ′ ) = |det(Id − Ad(h ′ ) −1 ) l ′ (S i )/h ′ (S i ) | 1 2 .
Proof. As explained in [2, Appendix A], we have:
G ′ / H ′ (S i ) f (ghg −1 )dg = D L ′ (S i ) (h) D L ′ (S 0 ) (h) L ′ (S i )/H ′ (S i ) f K ′ N ′ (S i ) (l ′h l ′−1 )dl ′ = D L ′ (S i ) (h) D L ′ (S 0 ) (h) GL(X i )/T ′ 1 (S i )×A ′ (S i ) G(U i )/T 2 (S i ) f K ′ N ′ (S i ) (g 1 g 2h g −1 2 g −1 1 )dg 2 dg 1 where D L ′ (S 0 ) (h) = D G ′ (h ′ ) = |det(Ad(h) −1 − Id) g ′ /h ′ (S i ) | 1 2 . Similarly, using that H ′ (S i ) ⊆ P ′ (S j )
, we get:
G ′ / H ′ (S j ) f (ghg −1 )dg = D L ′ (S j ) (h) D L ′ (S 0 ) (h) L ′ (S j )/H ′ (S i ) f K ′ N ′ (S j ) (l ′h l ′−1 )dl ′ = D L ′ (S j ) (h) D L ′ (S 0 ) (h) GL(X j )/T ′ 1 (S j )×A ′ (S j ) G(U j )/H ′ (S j i− j ) f K ′ N ′ (S j ) (g 1 g 2h g −1 2 g −1 1 )dg 2 dg 1 ,
and the lemma follows.
Transfer of invariant eigendistributions
We start this section by recalling the notion of invariant eigendistributions. We keep the notations of Appendix A. Let G be a connected real reductive Lie group, D ′ (G) be the space of distributions of G, i.e. the continuous linear forms on C ∞ c (G) and D G G (G) the space of bi-invariant differential operators on G as in Notations A.5. For every f ∈ C ∞ c (G) and g ∈ G, we denote by f g the function of
C ∞ c (G) defined by f g (x) = f (gxg −1 ), x ∈ G. We say that T ∈ D ′ (G) is a G-invariant distribution if T ( f g ) = T ( f ) for every f ∈ C ∞ c (G) and g ∈ G.
Definition 4.1. A distribution T on G is an eigendistribution if there exists an algebra homomorphism χ T :
D G G (G) → C such that D(T ) = χ T (D)T for every D ∈ D G G (G).
As proved by Harish-Chandra (see [9,Theorem 2]), for every invariant eigendistribution T on G, there exists a locally integrable function f T on G which is analytic on G reg such that T = f T , i.e. for every ϕ ∈ C ∞ c (G),
T (ϕ) = G f T (g)ϕ(g)dg.
Remark 4.2.
(1) Using the isomorphism defined in Appendix A, Theorem A.6, an eigendistribution T is an invariant distribution such that there exists a character χ T of Z(U (g C )) such that zT = χ T (z)T for every z ∈ Z(U (g C )).
(2) Let (Π, H ) be a representation of G. Following [6], we say that the representation Π is permissible if Π(z) is a scalar multiple of the unit operator for every z ∈ Z(G) ∩ D, where D is the analytic subgroup of G corresponding to Z(k) (k being the Lie algebra of a maximal compact subgroup K of G). A permissible representation is said quasi-simple if there exists an homomorphism χ of Z(U (g C )) into C such that dΠ(z)(η) = χ(z)η for every z ∈ Z(U (g C )) and η in the Garding space Gar(Π, H ) (for the definition of Gar(Π, H ), see [6,Part II]). In particular, for such representations, Harish-Chandra proved that for every ϕ ∈ C ∞ c (G), the operator Π(ϕ) is a trace class operator (see [7,Section 5]) and the corresponding map
Θ Π : C ∞ c (G) ∋ ϕ → tr(Π(ϕ)
) ∈ C is a distribution in the sense of Laurent Schwartz (see [7,Section 5]); the map Θ Π is called the global character of Π. Using that Θ Π is an invariant eigendistribution, it follows that there exists a locally integrable function Θ Π on G, analytic on G reg , such that
Θ Π (ϕ) = G Θ Π (g)ϕ(g)dg, (ϕ ∈ C ∞ c (G)).
The function Θ Π is the character of Π. As proved in [18], every irreducible unitary representation is quasisimple, in particular, every discrete series representations (see Section 5) has a character, whose value on H reg is explicit (see Theorem 5.4).
Notation 4.3.
For every reductive group G, we denote by I (G) the space of orbital integrals on G as in [4,Section 3]. Roughly speaking, the set I (G) is a subset of C ∞ (G reg ) G satisfying 4 conditions (see [4,). This space is endowed with a natural topology (see [4,Section 3.3]). We denote by J G the map J G :
C ∞ c (G) → I (G)
given as follow: for every γ ∈ G reg , there exists a unique, up to conjugation, Cartan subgroups H(γ) of G such that γ ∈ H(γ), and for every ϕ ∈ C ∞ c (G), we define J G (ϕ)(γ) by:
J G (ϕ)(γ) = |det(Id − Ad(γ −1 )) g/h(γ) | 1 2 G/H(γ) ϕ(gγg −1 )dg.
As proved in [4, Theorem 3.2.1], the map:
J G : C ∞ c (G) → I (G)
is well-defined and surjective. We denote by I ′ (G) the set of continuous linear forms on I (G) and by J t G :
I ′ (G) → D ′ (G) the transpose of J G defined by J t G (T )(ϕ) = T (J G (ϕ)), (T ∈ I ′ (G), ϕ ∈ C ∞ c (G)).
As proved in [4, Theorem 3.2.1], J t G (T ) is a G-invariant distribution on G and the corresponding map:
J t G : I ′ (G) → D ′ (G) G is bijective.
Let (G, G ′ ) be an irreducible dual pair in Sp(W) such that rk(G) ≤ rk(G ′ ) and (I ( G), J G ), (I ( G ′ ), J G ′ ) be the corresponding space of orbital integrals on G and G ′ respectively. To simplify, we assume that both G and G ′ are connected). For every function ϕ ∈ C ∞ c ( G ′ ), we denote by Chc(ϕ) the G-invariant function on G reg given by:
Chc(ϕ)(h i ) = Chc˜h i (ϕ), (h i ∈ H i reg ).
In [3], the authors proved the following results:
Theorem 4.4. For every ϕ ∈ C ∞ c ( G ′ )
, Chc(ϕ) ∈ I ( G) and the corresponding map
Chc : C ∞ c ( G ′ ) → I ( G) is continuous. Moreover, if J G ′ (ϕ) = 0, we get that Chc(ϕ) = 0, i.e. the map Chc : C ∞ c ( G ′ ) → I ( G) factors through I ( G ′ )
and we get a transfer of orbital integrals:
Chc : I ( G ′ ) → I ( G).
By dualizing the previous map, we get Chc t :
I ′ ( G) → I ′ ( G ′ ) given by Chc t (τ)(φ) = τ(Chc(φ)) (τ ∈ I ′ ( G), φ ∈ I ( G ′ )).
By using the isomorphisms J t G and J t G ′ , we get a map Chc * :
D ′ ( G) G → D ′ ( G ′ ) G ′ given by Chc * = J t G ′ •Chc t •(J t G ) −1 .
We denote by Eigen( G) (resp. Eigen( G ′ )) the set of invariant eigendistributions on G (resp. G ′ ).
Theorem 4.5. The map Chc * : D ′ ( G) G → D ′ ( G ′ ) G ′ sends Eigen( G) G into Eigen( G ′ ) G ′ .
Remark 4.6. If Θ is a distribution on G given by a locally integrable function Θ on G ′ , we get for every ϕ ∈ C ∞ c ( G ′ ) the following equality:
Chc * (Θ)(ϕ) = n i=1 1 |W (H i )| H i reg Θ(h i )|det(1 − Ad(h −1 i )) g/h i |Chc(ϕ)(h i )dh i .
where H 1 , . . . , H n is a maximal set of non-conjugate Cartan subgroups of G.
We recall the following conjecture.
Conjecture 4.7. Let G 1 and G ′ 1 be the Zariski identity components of G and G ′ respectively.
Let Π ∈ R( G, ω) satisfying Θ Π| G/ G 1 = 0 if G = O(V), where V is an even dimensional vector space over R or C. Then, up to a constant, Chc * (Θ Π ) = Θ Π ′ 1 on G ′ 1 .
In few cases, the conjecture is well-known: if G is compact (see [23]) and if (G, G ′ ) is in the stable range (see [24]). In this paper, we are investigating the case rk(G) = rk(G ′ ), with Π a discrete series representation of G. We will focus our attention on the dual pair of unitary groups satisfying rk(G) = rk(G ′ ), using some results of A. Paul that we recall in the next section.
Discrete series representations and a result of A. Paul
Let G be a connected real reductive Lie group.
Definition 5.1. We say that an irreducible representation (Π, (H , ·, · )) is a discrete series representation if all the functions τ u,v , u, v ∈ H , are in L 2 (G), where
τ u,v : G ∋ g → g(u), v ∈ C.
Remark 5.2. One can prove that the condition given in the previous definition is equivalent to say that the representation (Π, H ) is equivalent with a direct summand of the right regular representation of G on L 2 (G). Moreover, as recalled in [15,Section 9.3], for such a representation (Π, H ), there exists a positive number d Π ( depending on the Haar measure dg on G), called the formal degree of Π, such that for every u 1 ,
u 2 , v 1 , v 2 ∈ H , G Π(g)u 1 , v 1 Π(g)u 2 , v 2 dg = u 1 , u 2 v 1 , v 2 d Π .
In his papers [8] and [10], Harish-Chandra gave a classification of the discrete series representations of G. First of all, he proved that G has discrete series if and only if G has a compact Cartan subgroup (see [10,Theorem 13]). Let K be a maximal compact subgroup of G and H a Cartan subgroup of K. He also proved that the set of discrete series is indexed by a lattice of ih * . We say few words about this now. Let Ψ = Ψ(g C , h C ) be the set of roots of g, Ψ(k) = Ψ(k C , h C ) be the set of compact roots of g, ρ = 1 2 α∈Ψ + α and ρ(k) = 1 2 α∈Φ + (k) α.
Notation 5.3. For every g ∈ G, we denote by D g the function on R given by D g (t) = det((t + 1)Id g − Ad(g)) (t ∈ R).
In particular, D g (t) = n i=0 t i D i (g), with n = dim(G). The D ′ i s are analytic on G and let l be the least integer such that D l 0. The integer l is the rank of g. We denote by D(g) the coefficient of t l in the previous polynomial and by G reg the set of g ∈ G such that D(g) 0.
Theorem 5.4. Let λ be an element of ih * such that λ + ρ is analytic integral. Then, there exists a discrete series representation (Π λ , H λ ) of G such that:
(1) The representation Π λ has infinitesimal character χ λ as in Remark A.9, (2) The linear form ν = λ + ρ − 2ρ(k) is the highest weight of the lowest K-type for Π λ| K and the multiplicity of the corresponding representation Π ν in Π λ| K is one.
The parameter λ is called the Harish-Chandra parameter of Π λ . Moreover, if we denote by Θ λ the distribution character of Π and by Θ λ the corresponding locally integrable function on G reg , we get that the restriction of Θ λ of Π to H reg is given by the following formula
Θ λ (exp(X)) = (−1) dim(G)−dim(K) 2 w∈W (k) ε(w) e (wλ)(X) α>0 (e α(X) 2 − e − α(X) 2 )
, (X ∈ h reg ).
Remark 5.5. As proved in [8], for every discrete series Π of G with Harish-Chandra parameter λ, we get:
sup g∈G reg |D(g)| 1 2 |Θ λ (g)| < ∞.
The previous properties of Θ λ characterize the discrete series characters inside the space of invariant distributions of G. More precisely, as proved in [8, Lemma 44], we have the following result.
Theorem 5.6. Let Θ λ be G-invariant distribution on G such that:
(1) zΘ λ = γ(λ)(z)Θ λ , z ∈ Z(U (g C )),(2)sup g∈G reg |D(g)| 1 2 |Θ λ (g)| < ∞, (3) Θ λ = 0 pointwise on H reg .
Then, Θ λ = 0.
The previous theorem will be central for us in Section 6 to prove the conjecture 4.7 for discrete series representations in the equal rank case. We now recall a key result of A. Paul for unitary groups. Let (G, G ′ ) = (U(p, q), U(r, s)) be a dual pair of unitary groups in Sp(2(p + q)(r + s), R). As explained in [19, Section 1.2], the double cover of U(p, q) is isomorphic to
(9) U(p, q) ≈ (g, ξ) ∈ U(p, q) × C * , ξ 2 = det(g) r−s .
In particular, all the genuine admissible representations of U(p, q) are the form Π ⊗ det
r−s 2 , where det r−s 2
is the genuine character of U(p, q) given by det r−s 2 (g, ξ) = ξ and Π is an admissible representation of U(p, q). From now on, we fix p and q and let r and s vary under the condition that p + q = r + s. In particular, under this condition, it follows from Equation (9) that the double cover U(p, q) stays the same when r and s vary.
In [19,Section 6], A. Paul proved the following theorem: Theorem 5.7. For every genuine irreducible admissible representation (Π, H Π ) of U(p, q), there exists a unique pair of integers (r, s) = (r Π , s Π ) such that p + q = r + s with θ r,s (Π) 0.
She also obtained more precise results for discrete series representations (see [19,Theorem 6.1] or [20, Theorem 2.7]).
Notation 5.8. We fix a basis {e 1 , . . . , e n } of h * . In particular, every linear form λ on h can be written as λ = n i=1 λ i e i or also as λ = (λ 1 , . . . , λ n ).
Theorem 5.9. Let Π be a discrete series representation of U(p, q), the corresponding representation θ r Π ,s Π (Π) is a discrete series representation of U(r Π , s Π ). More precisely, if the Harish-Chandra parameter of Π is of the form λ = λ a,b = (α 1 , . . . , α a , β 1 , . . . , β p−a , γ 1 , . . . , γ b , δ 1 , . . . , δ q−b ),
with α i , β j , γ k , δ l ∈ Z + 1 2 such that α 1 > . . . > α a > 0 > β 1 > . . . > β p−a and γ 1 > . . . > γ b > 0 > δ 1 > . . . > δ q−b , then (r Π , s Π ) = (a + q − b, b + p − a) and the corresponding Harish-Chandra parameter λ ′ = λ ′ a,b of θ r Π ,s Π (Π) is of the form: λ ′ a,b = (α 1 , . . . , α a , δ 1 , . . . , δ q−b , γ 1 , . . . , γ b , β 1 , . . . , β p−a ).
6. Proof of Conjecture 4.7 for discrete series representations in the equal rank case
In this section, we are interested in the dual pair (G, G ′ ) = (U(p, q), U(r, s)) such that p + q = r + s. Without loss of generality, we assume that p ≤ q. In particular, the number of Cartan subgroups of G, up to conjugation, is p + 1. We denote by n = p + q. Let (V = C p+q , (·, ·)) and (V ′ = C r+s , (·, ·) ′ ) be the hermitian and skew-hermitian spaces corresponding to G and G ′ respectively. In this case, the space W = Hom(V ′ , V) = M((r + s) × (p + q), C) and for every w ∈ W, there exists a unique element w * ∈ Hom(V, V ′ ) = M((p + q) × (r + s), C) such that:
w(v ′ ), v = v ′ , w * (v) ′ , (v ∈ V, v ′ ∈ V ′ ).
One can prove that w * = iId p,q w t Id r,s and the symplectic form ·, · on W w, w ′ = Re(tr(w ′ * w)) = −Im(tr(Id p,q w ′ t Id r,s w)) (w, w ′ ∈ W).
Let V i = V ′ i = Ce i .
The subspaces h and h ′ of g and g ′ respectively given by:
h = h ′ = {y = (iX 1 , . . . , iX n ), X i ∈ R}
are Cartan subalgebras. Moreover, we get:
W h = n i=1 Hom(V i , V ′ i ) = n i=1 iRE i,i .
Let Π be a discrete series of U(p, q), Θ Π be the corresponding element of D ′ ( G) G , Θ Π the corresponding locally integrable function on G such that Θ Π = T Θ Π and χ Π the infinitesimal character of Π. As recalled in Theorem 4.5, Chc * (Θ Π ) is an element of Eigen( G ′ ) G ′ . According to [9, Theorem 2], the distribution Θ ′ Π = Chc * (Θ Π ) is given by a locally integrable function Θ ′ Π on G ′ , analytic on G ′ reg .
∆ Ψ ′ (ȟ ′ )Θ ′ Π (p(ȟ ′ )) = C |W (H)| σ∈S r+s ε(σ)det 1 2 (σ(ȟ ′ )) W h ′ lim r→1 r∈E σ,∅ Ȟ Θ Π (p(ȟ))∆(ȟ)det 1 2 (ȟ) det(1 − p(ȟ)rp(ȟ ′ )) σW h dȟ (ȟ ′ ∈Ȟ ′reg ),
where H = H(∅) is the compact Cartan of G and C = χ Π ( −1)Θ( −1)(−1) u .
Proof. Let ϕ be a function in C ∞ c ( G ′ ). According to Remark 4.6, we get that
Θ ′ Π (ϕ) = p i=0 1 |W (H(S i ))| H(S i ) reg Θ Π (h i )|det(1 − Ad(h −1 i )) g/h i |Chc(ϕ)(h i )dh i .
where H(S i ) is a set of Cartan subgroups as in Remark 2, and let H = H(∅) the compact Cartan of G. Now, if we assume that supp(ϕ) ⊆ G ′ · H ′ , then
Θ ′ Π (ϕ) = 1 |W (H)| H reg Θ Π (h)|det(1 − Ad(h −1 )) g/h |Chc(ϕ)(h)dh
According to [3,Equation 8] and Theorem 3.7, we get:
Θ ′ Π (ϕ) = 1 |W (H)| H reg Θ Π (h)|det(1 − Ad(h −1 i )) g/h |Chc(ϕ)(h)dh = (−1) u C |W (H)| Ȟreg Θ Π (p(ȟ))|∆ G (ȟ)| 2 G ′ Θ(p(ȟ)g ′ )ϕ(g ′ )dg ′ dȟ = (−1) u+1 C |W (H)| Ȟreg Θ Π (p(ȟ))∆ Ψ (ȟ)det 1 2 (ȟ) det − 1 2 (ȟ)∆ Ψ (ȟ) G ′ Θ(p(ȟ)g ′ )ϕ(g ′ )dg ′ dȟ = − Cm 0 |W (H)| σ∈S r+s ε(σ) lim r→1 r∈E σ,∅ Ȟreg Θ Π (p(ȟ))∆ Ψ (ȟ)det 1 2 (ȟ) Ȟ′ det 1 2 (σ −1 (ȟ ′ )) W h det(1 − p(ȟ)rp(ȟ ′ )) σW h H ∅ (ϕ)(ȟ ′ )dȟ ′ dȟ
With such assumptions on the support of ϕ, we get using Equation (3)
Θ ′ Π (ϕ) = G ′ Θ ′ Π (g ′ )ϕ(g ′ )dg ′ = m 0 Ȟ′ ∆ Ψ ′ (ȟ ′ )H ∅ (Θ ′ Π ϕ)(ȟ ′ )dȟ ′ = −m 0 Ȟ′ Θ ′ Π (p(ȟ ′ ))∆ Ψ ′ (ȟ ′ )H ∅ (ϕ)(ȟ ′ )dȟ ′
By identifications, we get, up to a constant, that:
∆ Ψ ′ (ȟ ′ )Θ ′ Π (p(ȟ ′ )) = C |W (H)| σ∈S r+s ε(σ)det 1 2 (σ −1 (ȟ ′ )) W h lim r→1 r∈E σ,∅ Ȟreg Θ Π (p(ȟ))∆ Ψ (ȟ)det 1 2 (ȟ) det(1 − p(ȟ)rp(σ(ȟ ′ ))) W h dȟ
and the theorem follows.
We know that the set of roots for (g, h) is given by
±(e i − e j ), 1 ≤ i < j ≤ n .
Let K = U(p) × U(q) be a maximal compact subgroup of G. Let Ψ(k) = Ψ(k C , h C ) be a set of compact positive roots given by:
Ψ(k) = e i − e j , 1 ≤ i < j ≤ p ∪ e i − e j , p + 1 ≤ i < j ≤ n . The compact Weyl group W (k) = W (K, H) is S p × S q . Let λ = p+q i=1
λ i e i be the Harish-Chandra parameter of Π.
Using Theorem 5.4, the value of Θ Π on H reg is given by:
Θ Π (p(ȟ)) = (−1) α p,q β∈S p ×S q ε(β) (βȟ) λ α>0 (ȟ α 2 −ȟ − α 2 ) , (ȟ ′ ∈Ȟ reg ), with α p,q = dim(G)−dim(K) 2 = pq. Using that W h = n i=1
Hom(V ′ i , V i ), we get:
det(1 − p(h)rp(h ′ )) σW h = n i=1 1 − h i (rh ′ ) −1 σ(i) = (−1) n n i=1 (rh ′ ) −1 σ(i) n i=1 h i − (rh ′ ) σ(i) , and det 1 2 (σ −1 (ȟ ′ )) W h = n i=1 h ′− 1 2 σ(i) , det 1 2 (ȟ) W h = n i=1 h 1 2 i .
To simplify the notations, we will denote by ξ the element of h * C given by ξ = n i=1 1 2 e i . We recall a basic Cauchy integral formula. Lemma 6.3. Let k ∈ Z and a ∈ C * \ S 1 . Then,
1 2iπ S 1 z k z − a dz = a k if k ≥ 0 and |a| < 1 −a k if k < 0 and |a| > 1 0 otherwise
For everyȟ ′ ∈Ȟ ′reg , we get from Theorem 6.2:
∆ Ψ ′ (ȟ ′ )Θ ′ Π (p(ȟ ′ )) = C σ∈S r+s β∈S p ×S q ε(σ)ε(β) n i=1 h ′− 1 2 σ(i) n i=1 h ′ σ(i) lim r→1 r∈E σ,∅ Ȟ (βȟ) λ n i=1 h 1 2 i n i=1 (h i − (rh ′ ) σ(i) ) dȟ = C σ∈S r+s β∈S p ×S q ε(σ)ε(β) n i=1 h ′ 1 2 σ(i) lim r→1 r∈E σ,∅ Ȟ (βȟ) λ+ξ n i=1 (h i − (rh ′ ) σ(i) ) dȟ = 2 C σ∈S r+s β∈S p ×S q ε(σ)ε(β) n i=1 h ′ 1 2 σ(i) lim r→1 r∈E σ,∅ H n i=1 h λ i + 1 2 β −1 (i) n i=1 (h i − (rh ′ ) σ(i) ) dh = 2 C n i=1 h ′ 1 2 i (2iπ) n σ∈S r+s β∈S p ×S q ε(σ)ε(β) lim r→1 r∈E σ,∅ n i=1 S 1 z λ i − 1 2 z − (rh ′ ) σ(β −1 (i)) dz , where C = (−1) pq C W (H) .
Lemma 6.4. For every σ ∈ S r+s , the space E σ,∅ is given by
E σ,∅ = h ′ = (e −X 1 , . . . , e −X n ) ∈ H ′ C , X σ(i) > 0 if i ∈ {1, Proof. Let w = n i=1 w i E i,i ∈ W h , σ ∈ S r+s and y = (iX 1 , . . . , iX n ) ∈ h ′ , with X j ∈ R. Then, y(σ(w)), σ(w) = y n i=1 w i E i,σ(i) , n j=1 w j E j,σ( j) = − n i=1 w i y σ(i) E i,σ(i) , n j=1 w j E j,σ( j) j = n i=1 n j=1 Im(tr(w j Id p,q E σ( j), j Id r,s w i y σ(i) E i,σ(i) )) = n i=1 Im(tr(Id p,q w i E i,σ(i) Id r,s w i y σ(i) E σ(i),i )) = p i=1 Im(tr(|w i | 2 y σ(i) E i,σ(i) Id r,s E σ(i),i )) − n i=p+1 Im(tr(|w i | 2 y σ(i) E i,σ(i) Id r,s E σ(i),i )) = p i=1 σ(i)∈{1,...,r} |w i | 2 X σ(i) − p i=1 σ(i)∈{r+1,...,n} |w i | 2 X σ(i) − n i=p+1 σ(i)∈{1,...,r} |w i | 2 X σ(i) + n i=p+1 σ(i)∈{r+1,...,n} |w i | 2 X σ(i)
In particular, using Equation (4), we get:
Γ σ,∅ = y = (iX 1 , . . . , iX n ) ∈ h ′ , X σ(i) > 0 if i ∈ {1,
The result follows using that E σ,∅ = exp(iΓ σ,∅ ). , q), ω) be a discrete series representation of Harish-Chandra parameter λ a,b as in Theorem 5.9 and let (r, s) = (r Π , s Π ) the unique integers such that θ r,s (Π) 0. The value of Θ ′ Π on H ′ reg = H ′ (∅) reg is given by
Proposition 6.5. Let Π ∈ R( U(p∆ Ψ ′ (ȟ ′ )Θ ′ Π (p(ȟ ′ )) = 2(−1) n−a−b ε(τ a,b ) C σ∈S r ×S s ε(σ)(σȟ ′ ) τ a,b λ a,b ,
where τ a,b ∈ S r+s is defined by:
• If r ≤ p, τ a,b = (a + 1, p + b + 1)(a + 2, p + b + 2) . . . (r, p + q), • If p + 1 ≤ r ≤ p + b, τ a,b ∈ Stab S r+s ({1, .
. . , a} ∪ {r + 1, . . . , p + b}) and satisfies:
τ a,b (a + 1) = p + b + 1, . . . , τ a,b (r) = r + s, τ a,b (p + b + 1) = a + 1, . . . , τ a,b (p + q) = r.
• If r ≥ p + b + 1, τ a,b ∈ Stab S r+s ({1, . . . , a} ∪ {p + b + 1, . . . , r}) and satisfies τ a,b (a + 1) = r + 1, . . . , τ a,b (p + b) = r + s, τ a,b (r + 1) = a + 1, . . . , τ a,b (r + s) = p + b + 1. Proof. To simplify the notations, we will denote by R(σ, λ a,b , β), σ ∈ S p+q , β ∈ S p × S q , the following term:
R(σ, λ a,b , β) = lim r→1 r∈E σ,∅ n i=1 S 1 z λ i − 1 2 z − (rh ′ ) σ(β −1 (i)) dz
According to Lemmas 6.3 and 6.4, we get that R(σ, λ a,b , β) 0 if and only if
σ • β −1 ∈ {i 1 ,...,i q−b }⊆{1,...,r} { j 1 ,..., j p−a }⊆{r+1,...,r+s} S {i 1 ,...,i q−b } c {1,...,a} × S { j 1 ,..., j p−a } {a+1,...,p} × S { j 1 ,..., j p−a } c {p+1,...,p+b} × S {i 1 ,...,i q−b } {p+b+1,...,p+q} .
We first assume that r ≤ p. In this case, using that {a + 1, . . . , p} = {a + 1, . . . , r} ∪ {r + 1, . . . , p}, we get where σ 1 = (a + 1, p + b + 1)(a + 2, p + b + 2) . . .(r, p + q). For every β ∈ S p × S q , there exists exactly r!s! elements in σ ∈ S r+s such that σ • β −1 ∈ S r × S s • σ 1 .Then,
S r × S s = σ∈S r+s β∈S p ×S q ε(σβ) lim r→1 r∈E σ,∅ p+q i=1 S 1 z λ i − 1 2 z − (rh ′ ) σ(β −1 (i)) dz = (−1) p+q−a−b (2iπ) p+q p!q! τ∈S r ×S s •σ 1 ε(τ) p+q i=1 h ′λ i − 1 2 τ(i) = (−1) p+q−a−b (2iπ) p+q p!q! τ∈S r ×S s ε(τσ −1 1 ) p+q i=1 h ′λ i − 1 2 τ(σ −1 1 (i)) = (−1) p+q−a−b (2iπ) p+q p!q! τ∈S r ×S s ε(τσ −1 1 )(σ 1 τ −1ȟ′ ) λ a,b −ξ = (−1) p+q−a−b (2iπ) p+q p!q!ε(σ 1 ) τ∈S r ×S s ε(τ)(τȟ ′ ) σ 1 (λ a,b −ξ) , where ξ = n i=1 1 2 e i , i.e. ∆ Ψ ′ (ȟ ′ )Θ ′ Π (p(ȟ ′ )) = 2(−1) p+q−a−b ε(σ 1 ) C τ∈S r ×S s ε(τ)(τȟ ′ ) σ 1 (λ a,b ) .
Now assume that r > p. We distinguish two cases. If p + 1 ≤ r ≤ p + b, then, given by σ 2 (a + 1) = p + b + 1, . . . , σ 2 (r) = r + s, σ 2 (p + b + 1) = a + 1, . . . , σ 2 (p + q) = r.
S r × S s = {i
This element satisfy the previous conditions and we get: λ a,b ) . This element satisfies the previous conditions and we get:
∆ Ψ ′ (ȟ ′ )Θ ′ Π (p(ȟ ′ )) = 2(−1) p+q−a−b ε(σ 2 ) C τ∈S r ×S s ε(τ)(τȟ ′ ) σ 2 (Similarly, r ≥ p + b + 1, S r × S s = ∆ Ψ ′ (ȟ ′ )Θ ′ Π (p(ȟ ′ )) = 2(−1) p+q−a−b ε(σ 3 ) C τ∈S r ×S s ε(τ)(τȟ ′ ) σ 3 (λ a,b ) .
Proposition 6.7. For every Π ∈ R ( U(p, q), ω), we get
sup g ′ ∈ G ′ reg |D(g ′ )| 1 2 |Θ ′ Π (g ′ )| < ∞.
We first need to introduce some notations.
Notation 6.8. Let k ∈ [|1, min(r, s)|], we denote by η ′ S k = Ad(c(S k ) −1 )(η ′ (S k )) ⊆ η ′+ C , where η ′+ C = α∈Ψ ′+ g ′ Cα ,
where g ′ Cα is the eigenspace corresponding to α ∈ Ψ ′ . By keeping the notations of Section 3, we get that Ψ ′ can be decomposed as follow:
(10) Ψ ′ = Ψ ′ (gl(X k )) ∪ Ψ ′ (g(U k )) ∪ Ψ ′ (η ′ (S k )), (k ∈ [|1, min(p, min(r, s))|]).
Finally, we denote by W (g(U k )) the Weyl group corresponding to (g(U k ) C , t 2 (S k ) C ).
Lemma 6.9. For everyh ′ ∈ H ′ (S k ) reg , det(Id − Ad(h ′ )) η ′ (S k ) ∈ R * + .
Proof. To make things easier, we will consider S k = {e 1 − e r+s−k+1 , . . . , e k − e r+s }. As in Equation (2), we have:
H ′ S k = c(S k ) −1 H ′ (S k )c(S k ) = h ′ = (e iθ 1 −X 1 , . . . , e iθ k −X k , t 1 , . . . , t r+s−2k , e iθ 1 +X 1 , . . . , e iθ k +X k ), t i ∈ U(1), θ i , X i ∈ R .
We denote by h ′ 1 = c(S k ) −1 h ′ c(S k ). Obviously, we get:
det(Id − Ad(h ′ )) η ′ (S k ) = det(Id − Ad(h ′ )) η ′ (S k ) = det(Id − Ad(h ′ 1 )) η ′ S k = det(Id − Ad(ȟ ′ 1 )) η ′ S k . We know that det(Id − Ad(h 1 )) η ′ S k = α∈Ψ ′ (η ′ (S k )) (1 − h −α 1 ),
and that
Ψ ′ (η ′ (S k )) = {e i − e j , 1 ≤ i ≤ k, k + 1 ≤ j ≤ r + s − k} ∪ {e i − e j , k + 1 ≤ i ≤ r + s − k, r + s − k + 1 ≤ j ≤ r + s}.
Let α 1 = e i − e j , with 1 ≤ i ≤ k, k + 1 ≤ j ≤ r + s − k and let α 2 = e j − e r+s−i+1 . Then,
(1 − h α 1 1 )(1 − h α 2 1 ) = (1 − e iθ i −X i t −1 j−k )(1 − t j−k e −iθ i −X i ) = |1 − e iθ i −X i t −1 j−k | 2 ,
and the result follows.
Proof of Proposition 6.7. Without loss of generality, we can assume that r ≤ s. We distinguish two cases. We first start with p ≤ r. Note that in this case,
H S i = H ′ S i (resp. H(S i ) = H ′ (S i )) for every 0 ≤ i ≤ p, with S i = {e 1 − e r+1 , .
. . , e i − e r+i } as in Notation 3.8. In this case, we get, using [2,Corollary A.4], that for every ϕ ∈ C ∞ c ( G ′ ):
Θ ′ Π (ϕ) = G ′ Θ ′ Π (g ′ )ϕ(g ′ )dg ′ = r i=0 m i Ȟ′ S i ε S i ,R (ȟ ′ )∆ Φ ′ (ȟ ′ )H S i (Θ ′ Π ϕ)(ȟ ′ )dȟ ′ = r i=0 m i Ȟ′ S i ε S i ,R (ȟ ′ )∆ Φ ′ (ȟ ′ ) ε S i ,R (ȟ ′ )∆ Ψ ′ (ȟ ′ ) G ′ /H ′ (S i ) Θ ′ Π (c(S i )p(ȟ ′ )c(S i ) −1 )ϕ(g ′ c(S i )p(ȟ ′ )c(S i ) −1 g ′−1 )dg ′ dȟ ′ = r i=0 m i Ȟ S i Θ ′ Π (c(S i )p(ȟ ′ )c(S i ) −1 )|∆ G ′ (ȟ ′ )| 2 G ′ /H ′ (S i ) ϕ(g ′ c(S i )p(ȟ ′ )c(S i ) −1 g ′−1 )dg ′ dȟ ′ = m 0 Ȟ′ S 0 Θ ′ Π (p(ȟ ′ ))|∆ G ′ (ȟ ′ )| 2 G ′ /H ′ (S 0 ) ϕ(g ′p (ȟ ′ )g ′−1 )dg ′ dȟ ′ + p i=1 m i Ȟ′ S i Θ ′ Π (c(S i )p(ȟ ′ )c(S i ) −1 )|∆ G ′ (ȟ ′ )| 2 Λ(c(S i )p(ȟ ′ )c(S i ) −1 ) L ′ (S i )/H ′ (S i ) ϕ K ′ N ′ (S i ) (l ′ c(S i )p(ȟ ′ )c(S i ) −1 l ′−1 )dl ′ dȟ ′ (11) + r i=p+1 m i Ȟ′ S i Θ ′ Π (c(S i )p(ȟ ′ )c(S i ) −1 )|∆ G ′ (ȟ ′ )| 2 L ′ (Sp)/H ′ (S i ) ϕ K ′ N ′ (Sp) (g ′ c(S i )p(ȟ ′ )c(S i ) −1 g ′−1 )dg ′ dȟ ′ where |∆ G ′ (ȟ ′ )| 2 = ∆ Φ ′ (ȟ ′ )∆ Ψ ′ (ȟ ′ ) as in Remark 3.2, L ′ (S i ) is defined in Section 3 and Λ(c(S i )p(ȟ ′ )c(S i ) −1 ) is given by: Λ(c(S i )p(ȟ ′ )c(S i ) −1 ) = D L ′ (S i ) (c(S i )p(ȟ ′ )c(S i ) −1 ) D L ′ (S 0 ) (c(S i )p(ȟ ′ )c(S i ) −1 ) = |det(Id − Ad(c(S i )p(ȟ ′ )c(S i ) −1 ) −1 ) l ′ (S i )/h ′ (S i ) | 1 2 |det(Id − Ad(c(S i )p(ȟ ′ )c(S i ) −1 ) −1 ) g ′ /h ′ (S i ) | 1 2 .
Using that Θ ′ Π = Chc * (Θ Π ), we get:
Θ ′ Π (ϕ) = p j=0 H(S j ) Θ Π (h)|det(Id − Ad(h) −1 ) g/h(S j ) | 2 Chc˜h(ϕ)dh.
Using Equation (7), we get that:
Θ ′ Π (ϕ) = p j=0 T 1 (S j ) A(S j ) T 2 (S j ) Θ Π (t 1ãt2 )|det(Id − Ad(t 1ãt2 ) −1 ) g/h(S j ) | 2 Chct 1ãt2 (ϕ)dt 2 dãdt 1 = p j=0 C j T 1 (S j ) A(S j ) T 2 (S j ) Θ Π (t 1ãt2 )|det(Id − Ad(t 1ãt2 ) −1 ) g/h(S j ) | 2 ε(t 1ãỹ )d S j (t 1ãt2 ) |det(Id − Ad(t 1ãt2 ) −1 ) η(S j ) |(12)GL(X j )/T ′ 1 (S j )×A ′ (S j ) G(U j ) Chc W 0, j (t 2ỹ )ε(t 1ãỹ )d ′ S j (gt 1ã g −1ỹ )ϕ K ′ N ′ (S j ) (gt 1ã g −1ỹ )dỹdgdt 2 dãdt 1 Let i ∈ [|1, r|]
and ϕ ∈ C ∞ c ( G ′ ) such that supp(ϕ) ⊆ G ′ · H ′ (S i ). On one hand, using Equation (11), we get:
Θ ′ Π (ϕ) = m i Ȟ′ S i Θ ′ Π (c(S i )p(ȟ ′ )c(S i ) −1 )|∆ G ′ (ȟ ′ )| 2 Λ(c(S i )p(ȟ ′ )c(S i ) −1 ) L ′ (S i )/H ′ (S i ) ϕ K ′ N ′ (S i ) (l ′ c(S i )p(ȟ ′ )c(S i ) −1 l ′−1 )dl ′ dȟ ′ = m i Ť′ 1,S i Ǎ′ S i Ť′ 2,S i Θ ′ Π (c(S i )p(ť ′ 1ǎ ′ť′ 2 )c(S i ) −1 )|∆ G ′ (ť ′ 1ǎ ′ť′ 2 )| 2 Λ(c(S i )p(ť ′ 1ǎ ′ť′ 2 )c(S i ) −1 ) GL(X i )/T ′ 1 (S i )×A ′ (S i ) G(U i )/T ′ 2 (S i ) ϕ K ′ N ′ (S i ) (g 1 g 2 c(S i )p(ť ′ 1ǎ ′ť′ 2 )c(S i ) −1 g −1 2 g −1 1 )dg 2 dg 1 dť ′ 2 dǎ ′ dť ′ 1 .
In particular, for every j < i, we get from Equation (8):
Θ ′ Π (ϕ) = m i Ť′ 1,S j Ǎ′ S j Ť′ 1,S j i− j Ǎ′ S j i− j Ť′ 2,S i Θ ′ Π (c(S i )p(ť jǎ jȟ jb jťi )c(S i ) −1 )|∆ G ′ (ť jǎ jȟ jb jťi )| 2 Λ(c(S i )p(ť jǎ jȟ jb jťi )c(S i ) −1 ) GL(X i )/T ′ 1 (S i )×A ′ (S i ) G(U i )/T ′ 2 (S i ) ϕ K ′ N ′ (S i ) (g 1 g 2 c(S i )p(ť jǎ jȟ jb jťi )c(S i ) −1 g −1 2 g −1 1 )dg 2 dg 1 dť j dǎ j dȟ j db j dť i .
On the other hand, it follows from Equation (12) that
Θ ′ Π (ϕ) = min(i,p) j=0 C i T 1 (S j ) A(S j ) T 2 (S j ) Θ Π (t 1ãt2 )|det(Id − Ad(t 1ãt2 ) −1 ) g/h(S j ) | 2 ε(t 1ã )d S j (t 1ãt2 ) |det(Id − Ad(t 1ãt2 )) η(S j ) | GL(X j )/T ′ 1 (S j )×A ′ (S j ) G(U j ) Chc W 0, j (t 2ỹ )ε(t 1ãỹ )d ′ S j (gt 1ã g −1ỹ )ϕ K ′ N ′ (S j ) (gt 1ã g −1ỹ )dỹdgdt 2 dãdt 1 = i j=0 C j Ť 1,S j Ǎ S j Ť 2,S j Θ Π (c(S j )p(ť 1ǎť2 )c(S j ) −1 )|det(Id − Ad(c(S j )p(ť 1ǎť2 )c(S j ) −1 ) −1 ) g/h(S j ) | 2 ε(c(S j )p(ť 1ǎ )c(S j ) −1 )d S j (c(S j )p(ť 1ǎť2 )c(S j ) −1 ) |det(Id − Ad(c(S j )p(ť 1ǎť2 )c(S j ) −1 )) η(S j ) | GL(X j )/T ′ 1 (S j )×A ′ (S j ) G(U j ) Chc W 0, j (p(ť 2 )ỹ)ε(c(S j )p(ť 1ǎ )c(S j ) −1ỹ )d ′ S j (gc(S j )p(ť 1ǎ )c(S j ) −1 g −1ỹ )ϕ K ′ N ′ (S j ) (gc(S j )p(ť 1ǎ )c(S j ) −1 g −1ỹ )dỹdgdť 2 dǎdť 1 = min(i,p) j=0 C j Ť 1,S j Ǎ S j Ť 2,S j Θ Π (c(S j )p(ť 1ǎť2 )c(S j ) −1 )|det(Id − Ad(c(S j )p(ť 1ǎť2 )c(S j ) −1 ) −1 ) g/h(S j ) | 2 ε(c(S j )p(ť 1ǎ )c(S j ) −1 )d S j (c(S j )p(ť 1ǎť2 )c(S j ) −1 ) |det(Id − Ad(c(S j )p(ť 1ǎť2 )c(S j ) −1 )) η(S j ) | σ∈W (g(U j )) ε(σ) lim r→1 r∈E σ,S j i− j GL(X j )/T ′ 1 (S j )×A ′ (S j ) det(ť 2 ) W t 2,S j ∆ Ψ ′ (g(U j )) (ť 2 ) −1 Ȟ′ S j i− j det − 1 2 (σ −1 (ȟ ′ )) W t 2,S j det(1 − p(ȟ ′ )r p(ť 2 )) σW t 2,S j G(U j )/H ′ (S j i− j ) ∆ Ψ ′+ (g(U j )) (ȟ ′ )ε(c(S j )p(ť 1ǎ )c(S j ) −1 g 2 c(S j i− j )p(ȟ ′ )c(S j i− j ) −1 g −1 2 )d ′ S j (g 1 c(S j )p(ť 1ǎ )c(S j ) −1 g −1 1 g 2 c(S j i− j )p(ȟ ′ )c(S j i− j ) −1 g −1 2 ) ϕ K ′ N ′ (S j ) (g 1 c(S j )p(ť 1ǎ )c(S j ) −1 g −1 1 g 2 c(S j i− j )p(ȟ ′ )c(S j i− j ) −1 g −1 2 )dg 2 dȟ ′ dg 1 dť 2 dǎdť 1
Using Lemma 3.12 and the equality c(S j )c(S j i− j ) = c(S i ), we get:
Θ ′ Π (ϕ) = min(i,p) j=0 C j Ť 1,S j Ǎ S j Ť 2,S j Θ Π (c(S j )p(ť 1ǎť2 )c(S j ) −1 )|det(Id − Ad(c(S j )p(ť 1ǎť2 )c(S j ) −1 ) −1 ) g/h(S j ) | 2 ε(c(S j )p(ť 1ǎ )c(S j ) −1 )d S j (c(S j )p(ť 1ǎť2 )c(S j ) −1 ) |det(Id − Ad(c(S j )p(ť 1ǎť2 )c(S j ) −1 )) η(S j ) | σ∈W (g(U j )) ε(σ) lim r→1 r∈E σ,S j i− j GL(X i )/T ′ 1 (S i )×A ′ (S i ) det(ť 2 ) W t 2,S j ∆ Ψ ′ (g(U j )) (ť 2 ) −1 Ȟ′ S j i− j det − 1 2 (σ −1 (ȟ ′ )) W t 2,S j ∆ Ψ ′+ (g(U j )) (ȟ ′ ) det(1 − p(ȟ ′ )r p(ť 2 )) σW t 2,S j D L ′ (S i ) (c(S i )p(ť 1ǎȟ ′ )c(S i ) −1 ) D L ′ (S j ) (c(S i )p(ť 1ǎȟ ′ )c(S i ) −1 ) G(U i )/T ′ 2 (S i ) ε(c(S i )p(ť 1ǎȟ ′ )c(S i ) −1 )d ′ S j (g 1 g 2 c(S i )p(ť 1ǎȟ ′ )c(S i )g −1 2 g −1 1 )ϕ K ′ N ′ (S j ) (g 1 g 2 c(S i )p(ť 1ǎȟ ′ )c(S i ) −1 g −1 2 g −1 1 )dg 2 dȟ ′ dg 1 dť 2 dǎdť 1 .
As explained in Remark 3.11, for every 0 ≤ j ≤ i, we have the following decomposition H ′ (
S i ) = T ′ 1 (S j ) ×A ′ (S j ) × T ′ 1 (S j i− j ) × A ′ (S j i− j ) × T ′ 2 (S i ).
In particular, every element h ∈ H ′ (S i ) can be written as h = t j a j h j b j t i . We get similar results for H ′ S i . In particular, we get that the value of Θ ′ Π on H ′ (S i ) is given by:
Θ ′ Π (c(S i )p(ȟ ′ )c(S i ) −1 )|∆ G ′ (ȟ ′ )| 2 Λ(c(S i )p(ȟ)c(S i ) −1 ) = min(i,p) j=0 C j σ∈W (g(U j )) ε(σ) ∆ Ψ ′+ (g(U j )) (ȟ jb jťi )det − 1 2 (σ −1 (ȟ jb jťi )) W t 2,S j D L ′ (S i ) (c(S i )p(ť jǎ jȟ jb jťi )c(S i ) −1 ) D L ′ (S j ) (c(S i )p(ť jǎ jȟ jb jťi )c(S i ) −1 ) ε(c(S jp (ȟ jb jťi )c(S j ) −1 ) lim r→1 r∈E σ,S j i− j Ť 2,S j d S j (c(S j )p(ť jǎ jȟ )c(S j ) −1 )Θ Π (c(S j )p(ť jǎ jȟ )c(S j ) −1 )|det(Id − Ad(c(S j )p(ť jǎ jȟ )c(S j ) −1 ) −1 ) g/h(S j ) | 2 det(ȟ) 1 2 W t 2,S j ∆ Ψ ′ (g(U j )) (ȟ)det(Id − Ad(c(S j )p(ť jǎ jȟ )c(S j ) −1 )) η(S j ) det(1 − p(ȟ jb jťi )rp(ȟ)) σW t 2,S j dȟ .
Using Equation (10), we get
D L ′ (S i ) (c(S i )p(ť jǎ jȟ jb jťi )c(S i ) −1 ) = α∈Ψ ′+ (l ′ (S i )) |1 − (ť jǎ jȟ jb jťi ) α | = α∈Ψ ′+ (gl ′ (X i )) |1 − (ť jǎ jȟ jb j ) α | α∈Ψ ′+ (g(U i )) |1 − (ť i ) α | and D L ′ (S j ) (c(S i )p(ť jǎ jȟ jb jťi )c(S i ) −1 ) = α∈Ψ ′+ (l ′ (S j )) |1 − (ť jǎ jȟ jb jťi ) α | = α∈Ψ ′+ (gl ′ (X j )) |1 − (ť jǎ j ) α | α∈Ψ ′+ (g(U j )) |1 − (ȟ jb jťi ) α |, so in particular D L ′ (S 0 ) (c(S i )p(ť jǎ jȟ jb jťi )c(S i ) −1 ) D L ′ (S j ) (c(S i )p(ť jǎ jȟ jb jťi )c(S i ) −1 )|∆ G ′ (ť jǎ jȟ jb jťi )| 2 = 1 α∈Ψ ′+ |1 − (ť jǎ jȟ jb jťi ) α | α∈Ψ ′+ (gl ′ (X j )) |1 − (ť jǎ j ) α | α∈Ψ ′+ (g(U j )) |1 − (ȟ jb jťi ) α | .
Finally,
∆ Ψ ′ (ȟ ′ )Θ ′ Π (c(S i )p(ȟ ′ )c(S i ) −1 ) = min(i,p) j=0 C j σ∈W (g(U j )) ε(σ) ∆ Ψ ′+ (g(U j )) (ȟ jb jťi )det − 1 2 (σ −1 (ȟ jb jťi )) W t 2,S j ∆ Ψ ′ (ť jǎ jȟ jb jťi )ε(c(S jp (ȟ jb jťi )c(S j ) −1 ) α∈Ψ ′+ |1 − (ť jǎ jȟ jb jťi ) α | α∈Ψ ′+ (gl(X j )) |1 − (ť jǎ j ) α | α∈Ψ ′+ (g(U j )) |1 − (ȟ jb jťi ) α | lim r→1 r∈E σ,S j i− j Ť 2,S j Θ Π (c(S j )p(ť jǎ jȟ )c(S j ) −1 )∆ Ψ ′ (ť jǎ jȟ ) ∆ Φ ′ (ť jǎ jȟ )det(ȟ ′ ) 1 2 W t 2,S j ∆ Ψ ′ (g(U j )) (ȟ)det(Id − Ad(c(S j )p(ť jǎ jȟ )c(S j ) −1 )) η(S j ) det(1 − p(ȟ jb jťi )rp(ȟ)) σW t 2,S j dȟ
Again, we get from Equation (10) that
∆ Ψ ′ (ť jǎ jȟ ) = ∆ Ψ ′ (gl(X i ) (ť jǎ j )∆ Ψ ′ (g(U j )) (ȟ)∆ Ψ ′ (η ′ (S j )) (ť jǎ jȟ ) = (ť jǎ j ) −ρ ′ (gl(X i ))−ρ ′ (η ′ (S j ))ȟ−ρ ′ (η ′ (S j )) α∈Ψ ′+ (gl(X j )) (1 − (ť jǎ j ) α ) α∈Ψ ′+ (η ′ (S j )) (1 − (ť jǎ jȟ ) α )∆ Ψ ′ (g(U j )) (ȟ),
then,
∆ Ψ ′ (ȟ ′ )Θ ′ Π (c(S i )p(ȟ ′ )c(S i ) −1 ) = min(i,p) j=0 σ∈W (g(U j )) ε(σ)det − 1 2 (σ −1 (ȟ jb jťi )) W t 2,S j ε(c(S jp (ȟ jb jťi )c(S j ) −1 ) α∈Ψ ′+ 1 −ȟ ′α α∈Ψ ′+ |1 −ȟ ′α | α∈Ψ ′+ (gl(X i )) 1 − (ť jǎ j ) α α∈Ψ ′+ (gl(X i )) |1 − (ť jǎ j ) α | α∈Ψ ′+ (g(U j )) 1 − (ť jǎ jȟ jb jťi ) α α∈Ψ ′+ (g(U j )) |1 − (ť jǎ jȟ jb jťi ) α | lim r→1 r∈E σ,S j i− j Ť 2,S j Θ Π (c(S j )p(ť jǎ jȟ )c(S j ) −1 )∆ Ψ ′ (ť jǎ jȟ ) det(ȟ) 1 2 W t 2,S j det(1 − p(ȟ jb jťi )rp(ȟ)) σW t 2,Θ Π (c(S j )p(ť jǎ jȟ )c(S j ) −1 )∆ Ψ ′ (ť jǎ jȟ ) = w∈S p+q c(w)(ť jǎ jȟ ) wλ = w∈S p+q c(w)(ť jǎ j ) wλȟwλ ,
where c(w) are complex numbers. Using that
ε(c(S jp (ȟ jb jťi )c(S j ) −1 ) α∈Ψ ′+ 1 −ȟ ′α α∈Ψ ′+ |1 −ȟ ′α | α∈Ψ ′+ (gl(X i )) 1 − (ť jǎ j ) α α∈Ψ ′+ (gl(X i )) |1 − (ť jǎ j ) α | α∈Ψ ′+ (g(U j )) 1 − (ť jǎ jȟ jb jťi ) α α∈Ψ ′+ (g(U j )) |1 − (ť jǎ jȟ jb jťi ) α |
is of norm 1, it follows from Lemma 6.3 and Theorem 5.6 that
lim r→1 r∈E σ,S j i− j Ť 2,S j Θ Π (c(S j )p(ť jǎ jȟ )c(S j ) −1 )∆ Ψ ′ (ť jǎ jȟ ) det(ȟ) 1 2 W t 2,S j det(1 − p(ȟ jb jťi )rp(ȟ)) σW t 2,S j dȟ
is a finite sum of bounded exponentials, and then, for every i,
sup h ′ ∈ H ′ (S i ) |D(h ′ )| 1 2 |Θ ′ Π (h ′ )| < ∞.
The proof is similar if r ≤ p. Note that in this case,
H S i = H ′ S i (resp. H(S i ) = H ′ (S i )) for every 0 ≤ i ≤ r, with S i = {e 1 −e p+1 , .
. . , e i −e p+i }, and in particular, the Cauchy-Harish-Chandra integrals Chc˜h i ,h i ∈ H(S i ), r+1 ≤ i ≤ p, does not give any contribution to Θ ′ Π on the different Cartan subgroups H ′ (S k ), 0 ≤ k ≤ r. It concludes the proof of the proposition. Lemma 6.10. Let (G = U(p, q), G ′ = U(r, s)), p + q = r + s and Π ∈ R( U(p, q), ω) a discrete series representation with Harish-Chandra parameter λ a,b as in Theorem 5.9. For every z ∈ Z(U (gl(r + s, C))), we get:
zΘ ′ Π = χ λ ′ a,b (z)Θ ′ Π , where χ λ ′ a,b = λ ′ a,b (γ(z)) as in Appendix A, Remark A.9.
Proof. Obviously, in the equal rank case, U (g C ) = U (g ′ C ). It follows from [3, Theorem 1.4] that Chc * (zΘ Π ) = zChc(Θ Π ). Because λ a,b and λ ′ a,b are conjugated under S r+s , the result follows from Theorem A.10.
Corollary 6.11. For every discrete series representation Π of U(p, q), we get
Chc * (Θ Π ) = CΘ θ r,s (Π) .
with C ∈ C.
Proof. Using Theorem 5.6, it follows from Propositions 6.5 and 6.7 and Lemma 6.10 that, up to a scalar, Θ ′ Π = Chc * (Θ Π ) is either the character of a discrete series representations of U(r, s) with Harish-Chandra parameter τ a,b (λ a,b ) if (r, s) = (r Π , s Π ) or 0 if (r, s) (r Π , s Π ). The result follows from Theorem 5.4 because τ a,b (λ a,b ) and λ ′ a,b (as in Theorem 5.9) are conjugated under S r × S s .
We get the following proposition.
Proposition 7.3. Let (G, G ′ ) = (U(V), U(V ′ )) with dim(V) ≤ dim(V ′ )
and Π be a discrete series representation of G. The intertwining distribution is given by
f Π⊗Π ′ = T(Θ Π ), where T(Θ Π ) = G Θ Π (g)T(g)dg.
Proof. As explained in [22,Theorem 3.1], the previous Lemma follows if the following condition G |Ω(g)||Θ Π (g)|dg < ∞ is satisfied, where Ω is defined in Appendix B. Using Lemma B.1, it follows that there exists C Ω > 0 such that
G |Ω(g)||Θ Π (g)|dg ≤ C Ω G Ξ(g)|Θ Π (g)|dg.
Using the fact that every discrete series satisfies the strong inequality (see [26, Corollary 7.4. Assume that (G, G ′ ) = (U(p, q), U(r, s)), with p + q = r + s, and let Π be a discrete series representation of G. Then, there exists a constant C Π⊗Π ′ ∈ C such that T(Θ Π ) = C Π⊗Π ′ T(Chc * (Θ Π )).
Proof. The proof of this corollary follows from Corollary 6.11 and Proposition 7.3.
We finish this section with a remark concerning the global character Θ Π ′ , Π ′ = θ(Π) and Π ∈ R( G, ω) a discrete series representation. We proved in Corollary 6.11 that Chc * (Θ Π ) = Θ Π ′ if rk(G) = rk(G ′ ). But the global character Θ Π ′ can be obtained via Θ Π in a different way. As before, we assume that rk(G) ≤ rk(G ′ ). In particular, every discrete series representation Π ∈ R( G, ω) is a sub-representation of ω and let H (Π) be the Π-isotypic component of H . As explained in Proposition 7.3, T(Θ Π ) is well-defined. Moreover, using [1, Section 4.8], the operator ω(Θ Π ) is a well-defined operator of H and one can check that P Π := ω(d Π Θ Π ) is a projection operator onto H (Π). As a G × G ′ -modules, we get H (Π) = Π ⊗ Π ′ . Let λ be the Harish-Chandra parameter of Π and ν the lowest K-type of Π | K . In particular, according to Theorem 5.4, as a K × G ′ , we get:
H (Π) = ξ∈ K Π m ξ Π ξ ⊗ Π ′ = Π ν ⊗ Π ′ ⊕ ξ ν ξ∈ K Π m ξ Π ξ ⊗ Π ′ ,
where Π ξ is a K-module of highest weight ξ and K Π is the set of irreducible representations of K such that Hom K (H ξ , H ) {0}. We denote by H (Π)(ν) the ν-isotypic component of H (Π). We denote by P ν : H (Π) → H (Π)(ν) the corresponding projection operator and let P Π,ν = P ν • P Π . Clearly,
P Π,ν = Π(d ν Θ Π ν ) • ω(d Π Θ Π ) = ω(d ν Θ Π ν ) • ω(d Π Θ Π ).
In particular, for every ϕ ∈ C ∞ c ( G ′ ), we get:
Θ Π ′ (ϕ) = 1 d ν tr(Id H ν ⊗ Π ′ (ϕ)) = 1 d ν tr(P Π,ν • ω(ϕ)),
i.e.
Θ Π ′ (ϕ) = d Π tr K G G ′ Θ Π ν (k)Θ Π (g)ϕ(g ′ )ω(kgg ′ )dg ′ dgdk .
In particular, if rk(G) = rk(G ′ ), we get using Corollary 6.11 that:
(13) p i=0 1 |W (H i )| H i Θ Π (h i )|det(Id − Ad(h i ) −1 ) g/h i |Chc˜h i (ϕ)dh i = d Π tr K G G ′ Θ Π ν (k)Θ Π (g)ϕ(g ′ )ω(kgg ′ )dg ′ dgdk .
Appendix A. Some standard isomorphisms A.0.1. Universal envelopping algebra of g as differential operators on G. Let M be a real connected manifold of dimension n. We denote by C ∞ (M) the space of smooth functions and C ∞ c (M) the space of compactly supported function on C ∞ (M). We denote by X (M) the set of derivations of C ∞ (M), i.e.
X (M) = {X : C ∞ (M) → C ∞ (M), X( f g) = X( f )g + f X(g)} .
The space X (M) is the set of C ∞ -vectors fields of M. From now on, we assume that G = M is a connected Lie group. We denote by (L, L 2 (G, dg)) the left regular representation. Obviously, the space C ∞ c (G) is G-invariant. We define an action of G on D(G) by
(τ(g)D)( f ) = L g • D( f • L g −1 ) (g ∈ G, f ∈ C ∞ c (G), D ∈ D(G)). Definition A.4. We say that D ∈ D(G) is left-invariant if τ(g)D = D for all g ∈ G, i.e. L g • D( f ) = D( f • L g ).
We denote by D G (G) the set of left-invariant differential operators of G. Similarly, we say that D is right invariant if τ 1 (g)(D) = D for every g ∈ G, where
(τ 1 (g)D)( f ) = R g −1 • D( f • R g ) ( f ∈ C ∞ c (G).
The operator D is is said to be bi-invariant if τ(g)τ 1 (h)(D) = D for every g, h ∈ G, i.e.
R h −1 • L g • D(L g −1 • f • R h ) = D( f ) for every f ∈ C ∞ c (G).
Notation A.5. We denote by D G (G) the set of right-invariant differential operators and by D G G (G) the set of biinvariant differential operators.
We recall the following result.
Theorem A.6. The natural embedding g → D G (G) extends to an algebra isomorphism
U (g) → D G (G).
Moreover, its restriction to Z(U (g)) is isomorphic D G G (G). Proof. The proof of this result can be found in [12].
A.0.2. Harish-Chandra isomorphism. Let g be a complex reductive Lie algebra and h be a Cartan subalgebra of g. We denote by W = W (g, h) the corresponding Weyl group. We denote by η + the subalgebra of g given by η + = α∈Ψ + (g,h)
CX α , where g α = CX α . Similarly, we denote by N and P the following subspaces of U (g) given by:
N = α∈Φ + (g,h) Y α U (g) P = α∈Φ + (g,h) X α U (g).
where Y α is a basis of g −α .
Lemma A.7. We get the following decomposition:
(14) U (g) = U (h) ⊕ (P + N ).
We denote by p 1 : U (g) → U (h) the natural projection corresponding to Equation (14). We restrict this map to Z(U (g)). We denote by ζ 1 the map:
ζ 1 : h ∋ h → ζ 1 (h) = h − ρ(h).1 ∈ S(h), where ρ = 1 2 α∈Φ + (g,h) α ∈ h * .
Using the universal property, we can extend the map ζ 1 to S(h). We denote by γ the map:
γ = ζ 1 • p 1 : Z(U (g)) → S(h).
Theorem A.8. The map γ is an algebra homomorphism which is injective. Moreover, Im(γ) = S(h) W and then:
γ : Z(U (g)) → S(h) W .
is a bijection.
Remark A.9. Harish-Chandra's isomorphism classify all the possible infinitesimal character. Indeed, let λ : h → C be a linear map. Using he universal property of the symmetric algebra [5, Appendix C], the linear form λ can be extended to a linear map λ : S(h) → C and by using the map λ, we get a map χ λ : Z(U (g)) → C given by:
χ λ (z) = λ(γ(z)) (z ∈ Z(U (g))).
We recall the following theorem.
Theorem A. 10. Let g be a complex reductive Lie algebra and h a Cartan subalgebra of g. Then every homomorphism of Z(U (g)) into C sending 1 to 1 is of the form χ λ , λ ∈ h * . If λ and λ ′ ∈ h * , then χ λ = χ λ ′ if and only if λ and λ ′ are in the same W -orbit. In particular, Spec(Z(U (g))) ≈ h * /W .
The proof of this result can be found in [16].
Appendix B. A general lemma for unitary groups
Let U be a maximal compact subgroup of Sp(W). It is well-known that the restriction of ω to U is a direct sum of irreducible representations whose multiplicity is one. Moreover, the lowest U-type V ω is one-dimensional. Let x be a non-zero vector in V ω and let Ω be the function on Sp(W) given by Ω(g) = ω(g)x, x , (g ∈ Sp(W)).
We denote by ξ ω the (unitary) character of K such that ω(k)x = ξ ω (k)x,k ∈ K. One can check that for everỹ k 1 ,k 2 ∈ K andg ∈ G, Ω(k 1gk2 ) = ω(k 1gk2 )x, x = ω(gk 2 )x, ω(k −1 1 )x = ξ ω (k 2 k −1 1 ) ω(g)x, x = ξ ω (k 2 k −1 1 )Ω(g). In particular, the map G ∋g → |Ω(g)| ∈ C is K-bi-invariant. In particular, using the decomposition Sp(W) = K A K as in [26, Section 3.6.7], with A = Cl(A + ), A + = exp(a + ), a the maximal split Cartan subalgebra of sp(W) and a + = {H ∈ a, α(H) > 0, α ∈ Ψ + }, we get for everyg =k 1ãk2 ∈ K A K that |Ω(g)| = |Ω(ã)|. We denote by Ξ the K-bi-invariant function defined in [26,Section 4.5.3]. Proof. We start by determining the value of Ω for the dual pair (G, G ′ ) = (U(1, 1), U(1)). Let G = KAK be the decomposition of G as in [26,Section 3.6.7]. In this case,
A = ch(X) sh(X) sh(X) ch(X) , X ∈ R * + .
Let a(X) ∈ A c and b(X) ∈ a c such that a(X) = c(b(X)). One can easily check that
b(X) = 0 α(X) α(X) 0 ,
where α(X) = sh(X) ch(X) − 1 . Note that α(X) = 1 th( X 2 )
. Let B = {e 1 , e 2 } be a basis of V such that Mat B (·, ·) = Id 1,1 . Then, using that B R = {e 1 , e 2 , ie 1 , ie 2 } is a basis of the real vector space V R , it follows that:
det R (Id − b(X)) = det 1 −α(X) 0 0 −α(X) 1 0 0 0 0 1 −α(X) 0 0 −α(X) 1 = (1 − α(X) 2 ) 2 .
Similarly, using that
0 α(X) 0 0 α(X) 0 0 0 0 0 0 α(X) 0 0 α(X) 0 = det 0 0 −1 0 0 0 0 −1 1 0 0 0 0 1 0 0 − 0 0 0 α(X) 0 0 −α(X) 0 0 −α(X) 0 0 α(X) 0 0 0 = det 0 0 −1 −α(X) 0 0 α(X) −1 1 α(X) 0 0 −α(X) 1 0 0 = (1 + α(X) 2 ) 2
Using that th( X 2 ) 2 − 1 th( X 2 ) 2 − 1
= − 1 ch(X) ,
it follows from Remark B.2 that there exists C > 0 such that:
|Ω(c(b(X)))| = C 1 − α(X) 2 1 + α(X) 2 = C ch(X) .
In particular, for the dual pair (G, G ′ ) = (U(1, 1), U(n)), we get for every a(X) = c(b(X)) ∈ A c ⊆ G that:
|Ω(c(b(X)))| = C ch(X) n .
From [26,Theorem 4.5.3], we know thatã(X) −ρ ≤ Ξ(ã(X)), withã(X) =c(b(X)). In our case, we get that a(X) −ρ = e −X and using that for every n ≥ 1, ch(X) n ≥ ch(X) ≥ e X , it follows that:
|Ω(c(b(X)))| = C ch(X) n ≤ C ch(X) ≤ Ce −X ≤ CΞ(ã(X)).
In particular, using the K-bi-invariance of Ω and Ξ, we get that Ω(g) ≤ CΞ(g) for everyg ∈ G for (G, G ′ ) = (U(1, 1), U(n)). One can easily check that G ′ can be replaced by U(r, s) and the computations are similar.
We can now extend it to (G, G ′ ) = (U(p, p), U(n)). In this case,
A = D = D 1 (X) D 2 (X) D 2 (X) D 1 (X) , X ∈ R * + p
where for X = (X 1 , . . . , X p ), D 1 (X) = diag(ch(X 1 ), . . . , ch(X p )) and D 2 (X) = diag(sh(X 1 ), . . . , sh(X p )). One can easily check that there exists C > 0 such that
|Ω(c(b(X)))| = C p i=1 ch(X i ) n .
In this case, ρ = 2p i=1 2p − 2i + 1 2 e i , and from Equation (2), we get a(X) −ρ = diag(e −X 1 , . . . , e −X p , e X 1 , . . . , e X p ) −ρ = p k=1 e −2pX k
If n ≥ 2p, it follows that ch(X k ) n ≥ ch(X k ) 2p ≥ e 2pX k for every k ∈ [|1, p|] and then,
|Ω(c(b(X)))| = C p i=1 ch(X i ) n ≤ C p k=1
e −2pX k ≤ Ξ(ã(X)).
Again, U(n) can be replaced by U(r, s) as long as r + s ≥ 2p. Finally, it G = U(p, q), with p ≤ q, we get that:
A = D = D 3 (X) D 4 (X) D t 4 (X) D ⋄ 3 (X) , X ∈ R * + p
where D 3 (X) = diag(ch(X 1 ), . . . , ch(X p )), D 4 (X) = (diag(sh(X 1 ), . . . , sh(X p )), 0 p,q ), where 0 p,q is the zero matrix of Mat(p, q − p), and D ⋄ 3 (X) = diag(ch(X 1 ), . . . , ch(X p )) 0 p,q−p 0 q−p,p 0 q−p,q−p .
and one can check that the computations done for U(p, p) extends easily to U(p, q). The lemma follows.
Remark B.3. One can easily see that the condition dim C (V) ≤ dim C (V ′ ) of Lemma B.1 does not mean that similar estimates cannot be obtained in some cases if dim C (V) > dim C (V ′ ). Indeed, it follows from the proof of Lemma B.1 that the inequality (15) can be obtained for (G, G ′ ) = (U(1, 1), U(1)).
Notation 6. 1 .
1From now on, we fix an element −1 be an element of π −1 ({−1}). Let c : g c → G c the Cayley transform, where g c and G c are defined as in Section 2. As explained in[21, Lemma 3.5], there exists a unique smooth mapc : g c → G c such that π •c = c andc(0) = −1.
Theorem 6. 2 .
2The value of Θ ′ Π on the compact Cartan H ′ = H ′ (∅) is given by the following formula:
. . . , p} and σ(i) ∈ {1, . . . , r} X σ(i) < 0 if i ∈ {1, . . . , p} and σ(i) ∈ {r + 1, . . . , r + s} X σ(i) < 0 if i ∈ {p + 1, . . . , p + q} and σ(i) ∈ {1, . . . , r} X σ(i) > 0 if i ∈ {p + 1, . . . , p + q} and σ(i) ∈ {r + 1, . . . , r + s}
. . . , p} and σ(i) ∈ {1, . . . , r} X σ(i) < 0 if i ∈ {1, . . . , p} and σ(i) ∈ {r + 1, . . . , n} X σ(i) < 0 if i ∈ {p + 1, . . . , n} and σ(i) ∈ {1, . . . , r} X σ(i) > 0 if i ∈ {p + 1, . . . , n} and σ(i) ∈ {r + 1, . . . , n}
Notation 6. 6 .
6For every subset {i 1 , . . . , i k } of {1, . . . , p} (resp. {p + 1, . . . , p + q}, {1, . . . , r} or {r + 1, . . . , r + s}), we denote by {i 1 , . . . , i k } c the set {1, . . . , p} \ {i 1 , . . . , i k } (resp. {p + 1, . . . , p + q} \ {i 1 , . . . , i k }, {1, . . . , r} \ {i 1 , . . . , i k } or {r + 1, . . . , r + s} \ {i 1 , . . . , i k }). For two subsets {a 1 , . . . , a w } and {b 1 , . . . , b w } of {1, . . . , p + q}, we denote by S {b 1 ,...,b w } {a 1 ,...,a w } the groups of bijections between {a 1 , . . . , a w } and {b 1 , . . . , b w }. Similarly, for every β ∈ S p × S q , we denote by S (β) {b 1 ,...,b w } {a 1 ,...,a w } the groups of bijections between {β(a 1 ), . . . , β(a w )} and {b 1 , . . . , b w }. Obviously, S p × S q = {i 1 ,...,i a }⊆{1,...,p} { j 1 ,..., j b }⊆{p+1,...,p+q} S {i 1 ,...,i a } c {1,...,p−a} × S {i 1 ,...,i a } {p−a+1,...,p} × S { j 1 ,..., j b } c {p+1,...,p+q−b} × S { j 1 ,..., j b } {p+q−b+1,...,p+q}for every 1 ≤ t ≤ p.
{i 1 ,...,i q−b }⊆{1,...,r} { j 1 ,..., j p−a }⊆{r+1,...,r+s} S {i 1 ,...,i q−b } c {1,...,a} × S { j 1 ,..., j p−a } {a+1,...,p} × S { j 1 ,..., j p−a } c {p+1,...,p+b} × S {i 1 ,...,i q−b } {p+b+1,...,p+q} • σ 1 ,
{i 1 ,...,i q−b }⊆{1,...,r} { j 1 ,..., j p−a }⊆{r+1,...,r+s}S {i 1 ,...,i q−b } c {1,...,a} × S { j 1 ,..., j p−a } {a+1,...,p} × S { j 1 ,..., j p−a } c {p+1,...,p+b} × S {i 1 ,...,i q−b } {p+b+1,...,p+q} • η,for every η ∈ S r+s satisfying η{1, . . . , a} = {1, . . . , a}, η{p+b+1, . . . , r} = {p+b+1, . . . , r}, η{a+1, . . . , p+b} ⊆ {r+ 1, . . . , r+s} and η{r+1, . . . , r+s} ⊆ {a+1, . . . , p+b+1}. Let σ 3 be the element of Stab S r+s ({1, . . . , a} ∪ {p + b + 1, . . . , r}) given by σ 3 (a + 1) = r + 1, . . . , σ 3 (p + b) = r + s, σ 3 (r + 1) = a + 1, . . . , σ 3 (r + s) = p + b + 1.
Definition A.1. A continuous endomorphism D of C ∞ c (M) is called a differential. operator if whenever U is an open set in M and f a function of C ∞ c (M). vanishing on U, then D f vanishes on U. Proposition A.2. Let D be a differential operator on M.For each p ∈ M and each open connected neighbourhood U of p on which the local coordinates system Ψ : x → (x 1 , . . . , x n ) is valid, there exists a finite set of functions a α of class C ∞ such that for each f ∈ C ∞ c (M) with support contained in U, α 1 ,...,α n )a α (x)D α f • Ψ −1 (x) if x ∈ U 0 otherwiseProof. The proof of this result can be found in[11, Proposition 1].Notation A.3. We denote by D(M) the set of differential operators.
Lemma B. 1 .
1Let (G, G ′ ) = (U(V), U(V ′ )) ⊆ Sp((V ⊗ V ′ ) R ) be a dual pairs of unitary groups such that dim C (V) ≤ dim C (V ′ ).Then, there exists a constant C Ω > 0 such that:
( 15 )
15|Ω(g)| ≤ C Ω Ξ(g), (g ∈ G). Remark B.2. As explained in[22, Equation 6.4], for an irreducible reductive dual pair (G, G ′ ), there exist a constantC = C d,d ′ > 0 such that |Ω(c(X))| = C|det R (Id − X)| d ′ 2 |det(iId − JX)| − d ′ 2 , (X ∈ g c ),where d = dim D (V) and d ′ = dim D (V ′ ).
J
= Mat B R (·, ·)
26, Section 2.3.6], for every 1 ≤ i ≤ n, the Cartan subgroup H i can be decomposed as H i = T i A i , with T i maximal compact in H i (A i is called the split part of H i ). For 1 ≤ i ≤ n, we denote by A ′ i and A′′
i the subgroups of Sp(W)
given by A ′
i = C Sp(W) (A i ) and A
′′
i = C Sp(W) (A
′
i ). As recalled in [23, Section 1], there exists an open and dense
subset W A ′′
i
, which is A
′′
i -invariant and such that A
′′
i \ W A ′′
i
is a manifold, endowed with a measure dw such that
for every
1 ,...,i q−b }⊆{1,...,r} { j 1 ,..., j p−a }⊆{r+1,...,r+s} for every η ∈ S r+s satisfying η{1, . . . , a} = {1, . . . , a}, η{r+1, . . . , p+b} = {r+1, . . . , p+b}, η{a+1, . . . , r} ⊆ {p+b+ 1, . . . , r+s} and η{p+b+1, . . . , p+q} ⊆ {a+1, . . . , r}. Let σ 2 be the element of Stab S r+s ({1, . . . , a} ∪ {r + 1, . . . , p + b})S
{i 1 ,...,i q−b } c
{1,...,a}
× S
{ j 1 ,..., j p−a }
{a+1,...,p} × S
{ j 1 ,..., j p−a } c
{p+1,...,p+b} × S
{i 1 ,...,i q−b }
{p+b+1,...,p+q}
• η,
Section 5.1.1]), it follows from [26, Lemma 5.1.3] that G Ξ(g)|Θ Π (g)|dg < ∞, and the proposition follows.
Corollary 6.12. If (G, G ′ ) = (U(p, q), U(r, s)), p + q = r + s and Π ∈ R( G, ω) a discrete series representation of G. Then, the conjecture 4.7 holds.Proof. It follows from Theorem 7.2 because Π ′ = Π ′ 1 .7.A commutative diagram and a remark on the distribution Θ Π ′ We start this section by recalling a result of T. Przebinda (see[22]). Let (G, G ′ ) = (G(V, (·, ·) , G(V ′ , (·, ·) ′ ))) be an irreducible reductive dual pair in Sp(W). As proved in[14](see also Section 2), the representations appearing in the correspondence are realized as quotients of H ∞ , the set of smooths vectors of the metaplectic representation (ω, H ). Let Π ∈ R( G, ω), Π ′ the corresponding element of R( G ′ , ω) andIn particular,i.e. there exists a unique element, up to a constant,Remark 7.1. Let W = X⊕Y be a complete polarization of W. It is well-known that we can realize the representation ω on H = L 2 (X): this is the Schrodinger model. Moreover, the space of smooth vectors of ω is the Schwartz space S(X) of X. Using the isomorphisms K : S * (W) → S * (X × X) and Op : S * (X × X) → Hom(S(X), S * (X)) (see[1,Equations (143)and(146). The distribution f Π⊗Π ′ is called the intertwining distribution corresponding to Π ⊗ Π ′ .As explained in[17,Section 2], the situation turns out to be slightly easier when dim(V) ≤ dim(V ′ ) and (Π, H Π ) a discrete series representation of G. Under those hypothesis, the space H ∞ ⊗ H ∞ Π has a natural structure of G-modules. Using the scalar products on H and H Π , we get a natural inner product ·, · on H ∞ ⊗ H ∞ Π . We denote by ·, · Π the following form on H ∞ ⊗ H ∞ Π :One can easily prove that in this context, the previous integral converges absolutely. We denote by R(Π) the radical of the form ·, · Π and we still denote by ·, · Π the non-degenerate form we got on H(Π) = H ∞ ⊗ H ∞ Π /R(Π). The group G ′ acts naturally on H(Π) and we denote by θ 0 (Π) the corresponding G ′ -module. (1) There exists u 0 , v 0 ∈ H ∞ and x, y ∈ H ∞ Π such thatMoreover, we get:(2) The representation Π can be embedded in ω as an irreducible subrepresentation and θ 0 (Π) defines an irreducible unitary representation on the completion of H(Π) (completion with respect to ·, · Π ). (3) The map Π → θ 0 (Π * ) coincide with Howe's duality correspondence.
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A Cauchy Harish-Chandra integral, for a real reductive dual pair. Tomasz Przebinda, Invent. Math. 1412Tomasz Przebinda. A Cauchy Harish-Chandra integral, for a real reductive dual pair. Invent. Math., 141(2):299-363, 2000.
The character and the wave front set correspondence in the stable range. Tomasz Przebinda, J. Funct. Anal. 2745Tomasz Przebinda. The character and the wave front set correspondence in the stable range. J. Funct. Anal., 274(5):1284-1305, 2018.
On the characters of the discrete series. The Hermitian symmetric case. Wilfried Schmid, Invent. Math. 301Wilfried Schmid. On the characters of the discrete series. The Hermitian symmetric case. Invent. Math., 30(1):47-144, 1975.
Real reductive groups. I. Nolan R Wallach, Pure and Applied Mathematics. 132Academic Press, IncDepartment of Mathematics, National University ofRepublic of Singapore Email address: [email protected] R. Wallach. Real reductive groups. I, volume 132 of Pure and Applied Mathematics. Academic Press, Inc., Boston, MA, 1988. Department of Mathematics, National University of Singapore, Block S17, 10, Lower Kent Ridge Road, Singapore 119076, Republic of Singapore Email address: [email protected]
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[] |
[
"ON THE COEFFICIENTS IN AN ASYMPTOTIC EXPANSION OF (1 + 1/x) x",
"ON THE COEFFICIENTS IN AN ASYMPTOTIC EXPANSION OF (1 + 1/x) x"
] |
[
"T M Dunster ",
"Jessica M Perez "
] |
[] |
[] |
The function g(x) = (1+1/x) x has the well-known limit e as x → ∞. The coefficients c j in an asymptotic expansion for g(x) are considered. A simple recursion formula is derived, and then using Cauchy's integral formula the coefficients are approximated for large j. From this it is shown that |c j | → 1 as j → ∞.
|
10.2140/involve.2021.14.775
|
[
"https://arxiv.org/pdf/2105.03794v2.pdf"
] | 234,332,486 |
2105.03794
|
352323410f9e74fc8cfb29c494199f0049f740cd
|
ON THE COEFFICIENTS IN AN ASYMPTOTIC EXPANSION OF (1 + 1/x) x
10 Aug 2021
T M Dunster
Jessica M Perez
ON THE COEFFICIENTS IN AN ASYMPTOTIC EXPANSION OF (1 + 1/x) x
10 Aug 2021Series expansionsAsymptotic representations in the complex planeIntegrals of Cauchy type AMS subject classifications 41A5830E1530E20
The function g(x) = (1+1/x) x has the well-known limit e as x → ∞. The coefficients c j in an asymptotic expansion for g(x) are considered. A simple recursion formula is derived, and then using Cauchy's integral formula the coefficients are approximated for large j. From this it is shown that |c j | → 1 as j → ∞.
Introduction and main results. Chen and Choi
[3]
presented a method to estimate Euler's constant e, accurate to as many decimal places as desired. Their starting point was the well known limit lim x→∞ g(x) = e, where (1. 1) g(x) = 1 + 1 x
x .
Their method was based on an asymptotic expansion of g(x) for large values of x, namely
(1.2) g(x) ∼ ∞ j=0 c j x j .
We shall show that this asymptotic expansion converges for x > 1.
In [3] they proved that c j = ea j , where a j are rational numbers which alternate in sign, and are given explicitly by (1.3) a j = (−1) j 1 k 1 !k 2 ! · · · k j ! 1 2 k1 1 3 k2 · · · 1 j + 1 kj .
Here for each j the sum is taken over all possible combinations of nonnegative integers k 1 , k 2 , k 3 , · · · , k j that satisfy the relation j l=1 lk l = j. We remark that Ponomorenko [7] gave a much simpler proof of (1.3) using Faà di Bruno's formula generalizing the chain rule to higher derivatives.
The number of terms in the sum (1.3) is the partition function P (j) [4, Sect. 27.14(i)]. Hardy and Ramanujan [5] gave the asymptotic formula
(1.4) P (j) ∼ exp(π 2j/3) 4j √ 3
as j → ∞. It is therefore evident that the number of terms in the formula (1.3) grows exponentially in √ j, and so is only practicable for small or moderate values of j.
We derive a new way of computing the coefficients c j , by a simple recursion formula. We also provide a simple asymptotic approximation for the coefficients as j → ∞, and this shows that in absolute value they approach the value 1.
Our main results read as follows. In section 2 we prove (1.5), and in section 3 we prove (1.6).
c j+1 = 1 j + 1 j l=0 (−1) j−l+1 j − l + 1 j − l + 2 c l (j = 0, 1, 2, · · · ).
Moreover as j → ∞
(1.6) c j = (−1) j 1 + 1 j + O ln(j) j 2 . Remark 1. Since lim x→∞ g(x) = e it is clear from (1.2) that c 0 = e.
2. Proof of the recursion formula (1.5). Define z = 1/x and f (z) = g(1/z) so that (1.2) is written in the new form
(2.1) f (z) = (1 + z) 1/z = ∞ j=0 c j z j .
and in this we consider z complex. Moreover, on writing it as
(2.2) f (z) = exp z −1 ln(1 + z) ,
we note that it has a removable singularity at z = 0, and therefore can be considered analytic at z = 0 by assuming f (0) = lim z→0 f (z) = c 0 = e. We also see it has one finite singularity (a logarithmic branch point) at z = −1. Therefore the radius of convergence of the series (2.1) is 1, i.e. it converges for |z| < 1. Thus (1.2) converges for x > 1, as asserted. By taking the principal branch of the logarithm in (2.2) we have that f (z) is analytic on the cut plane C \ (−∞, −1]. Next, on taking the natural logarithm of both sides of (2.1), we get
(2.3) 1 z ln(1 + z) = ln ∞ j=0 c j z j .
Then expand the ln(1 + z) term by its Maclaurin expansion valid for |z| < 1 and we arrive at
(2.4) ∞ j=1 (−1) j z j j + 1 = ln ∞ j=0 c j z j .
Next differentiate both sides with respect to z to yield
(2.5) ∞ j=1 (−1) j jz j−1 j + 1 = ∞ j=1 jc j z j−1 ∞ j=0 c j z j−1 −1 .
By shifting indices of the series starting at j = 1 to start at j = 0, and taking the series ∞ j=0 c j z j to the left-hand side, we see this is equivalent to
(2.6) ∞ j=0 c j z j ∞ j=0 d j z j = ∞ j=0 (j + 1)c j+1 z j ,
where d j is given by
(2.7) d j = (−1) j+1 j + 1 j + 2 .
We now use the Cauchy product, which is the discrete convolution of two infinite series. It is given by the formula [2, Sect. 73]
(2.8) ∞ j=0 C j ∞ j=0 D j = ∞ j=0 j l=0 C l D j−l .
Applying this to the left-hand side of (2.6) we combine both power series to the following single power series
(2.9) ∞ j=0 c j z j ∞ j=0 d j z j = ∞ j=0 j l=0 c l d j−l z j .
Finally substitute this into the left-hand side of (2.6), and equate coefficients of z j , to obtain (2.10) j l=0 c l d j−l = (j + 1)c j+1 , and then using (2.7) this leads to (1.5).
3. Proof of the asymptotic approximation (1.6). We will use the famous Cauchy integral formula [4, Eq. 1.9.31] to obtain an integral representation for the coefficients c j . If C r is the positively orientated circle {z : |z| = r} for arbitrary r ∈ (0, 1) then from (2.1) and (2.2)
(3.1) c j = 1 2πi Cr (1 + z) 1/z z j+1 dz = 1 2πi Cr exp z −1 ln(1 + z) z j+1 dz.
In the second integral of (3.1) we rewrite the exponential term using the geometric series
(3.2) 1 z = − 1 1 − (1 + z) = −(1 + δ + δ 2 + δ 3 + · · · ),
where δ = 1 + z, assuming 0 < |δ| < 1. So from (2.2)
(3.3) f (z) = exp − ln(δ) 1 + ∞ j=1 δ j = 1 δ exp − ln(δ) ∞ j=1 δ j .
Note that δ j ln(δ) → 0 as δ → 0 for j = 1, 2, 3, · · · by L'Hopital's rule. So using the Maclaurin expansion of the exponential function along with (3.3) this function has the expansion
(3.4) f (z) = 1 δ 1 − v + v 2 2! − v 3 3! + · · · , for 0 < |δ| < 1, where v = ln(δ) ∞ j=1 δ j .
From this one deduces for small δ that
(3.5) v = −δ ln(δ) − δ 2 ln(δ) + O δ 3 ln(δ) , v 2 = δ 2 ln 2 (δ) + O δ 3 ln 2 (δ) , and (3.6) v j = O δ 3 ln 3 (δ) (j = 3, 4, 5, · · · ).
Recalling δ = 1 + z we consequently have from (3.4) -(3.6)
(3.7) f (z) = (1 + z) −1 − ln (1 + z) + R(z), where (3.8) R(z) = (1 + z) 1/z − (1 + z) −1 + ln(1 + z) = 1 2 ln (1 + z) 2 − ln (1 + z) (1 + z) + O ln (1 + z) 3 (1 + z) 2 ,
as z → −1. Now substitute (3.7) into (3.1) to get (3.9) c j = I 1,j + I 2,j + η j , where (3.10)
I 1,j = 1 2πi Cr 1 z j+1 (1 + z) dz, I 2,j = − 1 2πi Cr ln (1 + z) z j+1 dz, and (3.11) η j = 1 2πi Cr R(z) z j+1 dz.
The integrals in (3.10) can readily be evaluated by residue theory. For the first we have by the geometric series expansion (3.12)
I 1,j = Res z=0 1 z j+1 (z + 1) = Res z=0 ∞ s=0 (−1) s z s−j−1 = (−1) j .
Likewise for I 2,j one finds that (3.13)
I 2,j = −Res z=0 ln(1 + z) z j+1 = (−1) j j .
For the integral (3.11) we make a change of variable w = − (z + 1) to obtain the following (3.14) The contour C ′ r in the w plane is now the circle {w : |w + 1| = r} for 0 < r < 1, and is positively orientated. This lies in the left half plane and encircles w = −1. The integrand of (3.14) has a branch point at w = 0 and a pole at w = −1, and is analytic elsewhere in the w plane having a cut along the non-negative real axis. So we can deform the contour to a new one, called Γ ǫ,ρ , as seen in Figure 1.
η j = (−1) j+1 2πi C ′ r R(−1 − w) (1 + w) j+1 dw.
This contour consists of circles γ ǫ and γ ρ centered at w = 0, of radius ǫ and ρ (respectively), where 0 < ǫ < 1 < ρ, and horizontal line segments l 1 and l 2 with end points w = ǫ ± i0 and w = ρ ± i0 above and below the cut. Now from (3.8) we see that the integrand of (3.14) is O{w ln(w) 2 } as w → 0, and O{w −j−1 ln(w)} as w → ∞. Hence the contributions of γ ǫ and γ ρ vanish as ǫ → 0 and ρ → ∞ . Then the only contribution will be along l 1 ∪ l 2 , where now these lines extend from 0 to ∞. We are therefore left with
(3.15) η j = (−1) j+1 2πi l1∪l2 R(−1 − w) (1 + w) j+1 dw.
Next, the contributions of real terms in the integrand of (3.15) cancel. Hence, using ℑ{ln(z) 2 } = 2 arg(z) ln(|z|) (z ∈ C \ {0}), we have from (3.8) ℑ{R(−1 − w)} = O{w ln(w)} uniformly for w ∈ l 1 ∪ l 2 , and so from (3.15) for unbounded j
(3.16) η j = O ∞ 0 w ln(w) (1 + w) j+1 dw .
We then let 1 + w = e t , and consequently from (3.16) obtain
(3.17) η j = O ∞ 0 e −jt (e t − 1) ln(e t − 1) dt .
Now split the integral in (3.17) into two integrals, one from t = 0 to t = 1 and the other from t = 1 to t = ∞. For the first use e t − 1 = O(t) for 0 ≤ t ≤ 1, and for the second use ln(e t − 1) = O(t) for 1 ≤ t < ∞. Thus as j → ∞ we deduce that
(3.18) η j = O 1 0 e −jt t ln(t) dt + O ∞ 1 e −(j−1)t t dt = O ln(j) j 2 + O 1 je j = O ln(j) j 2 ,
where the third O term comes from [6, Chap. 9, Thm. 1.1], and the fourth O term came from integration by parts. Finally, from (3.9), ( 3.12), (3.13) and (3.18) we arrive at (1.6).
Addendum. Christian Berg kindly made us aware of his recent joint paper [1]. In this they study a more general function h α (z) = (1 + 1/z) αz where α > 0. They obtain a number of results for the coefficients in a Maclaurin series in z for e −α h α (−1/z), which are polynomials in α. In particular they use properties of the exponential Bell partition polynomials to obtain a recursion relation, which for α = 1 reduces to our formula (1.5). In comparison we only require elementary techniques to obtain (1.5).
Theorem 1. 1 .
1For x > 1 the expansion (1.2) converges, where the coefficients c j are given recursively by c 0 = e and (1.5)
Fig. 1 .
1Contour Γǫ,ρ
Acknowledgments. TMD acknowledges financial support from Ministerio de Ciencia e Innovación, Spain, projects MTM2015-67142-P (MINECO/FEDER, UE) and PGC2018-098279-B-I00 (MCIU/AEI/FEDER, UE).
A family of entire functions connecting the Bessel function J 1 and the Lambert W function. C Berg, E Massa, A P Peron, Constr. Approx. 53C. Berg, E. Massa, and A. P. Peron, A family of entire functions connecting the Bessel function J 1 and the Lambert W function, Constr. Approx., 53 (2021), pp. 121-154.
J W Brown, R V Churchill, Complex variables and applications. BostonMcGraw-Hill Higher Education9J. W. Brown and R. V. Churchill, Complex variables and applications, Boston: McGraw-Hill Higher Education, 9 ed., 2014.
An asymptotic formula for (1 + 1/x) x based on the partition function. C.-P Chen, J Choi, Amer. Math. Monthly. 44C.-P. Chen and J. Choi, An asymptotic formula for (1 + 1/x) x based on the partition function, Amer. Math. Monthly, 44 (2014), pp. 338-343.
. R F Schneider, C W Boisvert, Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClainSchneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.
Asymptotic formulae in combinatory analysis. G H Hardy, S Ramanujan, Proc. London Math. Soc. 17G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc., 17 (1918), pp. 75-115.
Asymptotics and special functions. F W J Olver ; Akp Classics, Peters Ltd, M A Wellesley, Academic PressNew YorkReprint of the 1974 originalF. W. J. Olver, Asymptotics and special functions, AKP Classics, A K Peters Ltd., Wellesley, MA, 1997. Reprint of the 1974 original [Academic Press, New York].
Asymptotic formula for (1 + 1/x) x , revisited. V Ponomarenko, Amer. Math. Monthly. 122587V. Ponomarenko, Asymptotic formula for (1 + 1/x) x , revisited, Amer. Math. Monthly, 122 (2015), p. 587.
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[] |
[
"The fate of interaction-driven topological insulators under disorder",
"The fate of interaction-driven topological insulators under disorder"
] |
[
"Jing Wang \nDepartment of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiP.R. China\n\nInstitute for Theoretical Solid State Physics, IFW Dresden\nHelmholtzstr. 2001069DresdenGermany\n\nDepartment of Physics\nTianjin University\n300072TianjianP.R. China\n",
"Carmine Ortix \nInstitute for Theoretical Solid State Physics, IFW Dresden\nHelmholtzstr. 2001069DresdenGermany\n\nInstitute for Theoretical Physics\nCenter for Extreme Matter and Emergent Phenomena\nUtrecht University\nPrincetonplein 53584 CCUtrechtNetherlands\n",
"Jeroen Van Den Brink \nInstitute for Theoretical Solid State Physics, IFW Dresden\nHelmholtzstr. 2001069DresdenGermany\n\nInstitute for Theoretical Physics\n01069Dresden, DresdenTUGermany\n",
"Dmitry V Efremov \nInstitute for Theoretical Solid State Physics, IFW Dresden\nHelmholtzstr. 2001069DresdenGermany\n"
] |
[
"Department of Modern Physics\nUniversity of Science and Technology of China\n230026HefeiAnhuiP.R. China",
"Institute for Theoretical Solid State Physics, IFW Dresden\nHelmholtzstr. 2001069DresdenGermany",
"Department of Physics\nTianjin University\n300072TianjianP.R. China",
"Institute for Theoretical Solid State Physics, IFW Dresden\nHelmholtzstr. 2001069DresdenGermany",
"Institute for Theoretical Physics\nCenter for Extreme Matter and Emergent Phenomena\nUtrecht University\nPrincetonplein 53584 CCUtrechtNetherlands",
"Institute for Theoretical Solid State Physics, IFW Dresden\nHelmholtzstr. 2001069DresdenGermany",
"Institute for Theoretical Physics\n01069Dresden, DresdenTUGermany",
"Institute for Theoretical Solid State Physics, IFW Dresden\nHelmholtzstr. 2001069DresdenGermany"
] |
[] |
We analyze the effect of disorder on the weak-coupling instabilities of quadratic band crossing point (QBCP) in two-dimensional Fermi systems, which, in the clean limit, display interactiondriven topological insulating phases. In the framework of a renormalization group procedure, which treats fermionic interactions and disorder on the same footing, we test all possible instabilities and identify the corresponding ordered phases in the presence of disorder for both single-valley and two-valley QBCP systems. We find that disorder generally suppresses the critical temperature at which the interaction-driven topologically non-trivial order sets in. Strong disorder can also cause a topological phase transition into a topologically trivial insulating state.
|
10.1103/physrevb.96.201104
|
[
"https://arxiv.org/pdf/1710.09632v1.pdf"
] | 119,096,639 |
1710.09632
|
735ee8043f5ad5b311041e7ed395fc3eaa8b0063
|
The fate of interaction-driven topological insulators under disorder
26 Oct 2017
Jing Wang
Department of Modern Physics
University of Science and Technology of China
230026HefeiAnhuiP.R. China
Institute for Theoretical Solid State Physics, IFW Dresden
Helmholtzstr. 2001069DresdenGermany
Department of Physics
Tianjin University
300072TianjianP.R. China
Carmine Ortix
Institute for Theoretical Solid State Physics, IFW Dresden
Helmholtzstr. 2001069DresdenGermany
Institute for Theoretical Physics
Center for Extreme Matter and Emergent Phenomena
Utrecht University
Princetonplein 53584 CCUtrechtNetherlands
Jeroen Van Den Brink
Institute for Theoretical Solid State Physics, IFW Dresden
Helmholtzstr. 2001069DresdenGermany
Institute for Theoretical Physics
01069Dresden, DresdenTUGermany
Dmitry V Efremov
Institute for Theoretical Solid State Physics, IFW Dresden
Helmholtzstr. 2001069DresdenGermany
The fate of interaction-driven topological insulators under disorder
26 Oct 2017(Dated: October 27, 2017)numbers: 7343Nq7155Jv7110-w1130Qc
We analyze the effect of disorder on the weak-coupling instabilities of quadratic band crossing point (QBCP) in two-dimensional Fermi systems, which, in the clean limit, display interactiondriven topological insulating phases. In the framework of a renormalization group procedure, which treats fermionic interactions and disorder on the same footing, we test all possible instabilities and identify the corresponding ordered phases in the presence of disorder for both single-valley and two-valley QBCP systems. We find that disorder generally suppresses the critical temperature at which the interaction-driven topologically non-trivial order sets in. Strong disorder can also cause a topological phase transition into a topologically trivial insulating state.
The study of topological phases of matter is one of the most active research areas in contemporary condensed matter physics. The explanation of the quantum Hall effect in terms of the topological properties of the Landau levels [1,2] in the 1980's triggered an intense research effort in the theoretical prediction [3][4][5] and the experimental discovery [6,7] of a plethora of different topologically non-trivial quantum phases. In two-dimensional (2D) insulating systems only two distinct topological non-trivial phases can be realized according to the well-established classification of topological insulators and superconductors [8,9] : (i) the quantum anomalous Hall state (QAH) [3] with a time-reversal symmetry-broken ground state and topologically protected chiral edge states and (ii) the time-reversal invariant quantum spin Hall (QSH) state [4,5], which possesses helical edge states with counterpropagating electrons of opposite spins.
In recent years, attention has gradually shifted from non-interacting topological states of matter towards interaction-driven topological phases: many-particle quantum ground-states in which chiral orbital currents or spin-orbit couplings are spontaneously generated by electronic correlations. These states of matter possess both conventional order, characterized by an order parameter and a broken symmetry, and protected edge states associated with a topological quantum number. Interactiondriven QAH and QSH phases were first conceived in the context of 2D honeycomb lattice Dirac fermions [10] assuming sufficiently strong electronic repulsions although more recent analytical and numerical works question the proposal for this particular model [11][12][13][14].
On the contrary, it has been proposed that 2D systems with a quadratic band crossing point (QBCP) are unstable to electronic correlation because of the finite density of states at the Fermi level leading to the possibility of (Color online) Schematic renormalization group flows of the coupling constant ratios (g * 0 , g * − , g * 2 )/g * + for the checkerboard lattice in the clean limit (red) and in the presence of random chemical potential(green and blue). In clean limit the QAH fixed point (g * 0 , g * − , g * 2 )/g * + = (0, −3.73, 7.46) (red point) corresponds to a QAH state. In the presence of the weak random chemical potential the coupling constants flow to the QAH-II fixed point (green) (g * 0 , g * − , g * 2 )/g * + = (0, −0.2, 6.5) corresponding again to the QAH state and in the presence of the strong random chemical potential to the NSN fixed point (green) (g * 0 , g * − , g * 2 )/g * + = (0, 0, −1.09) corresponding to the NSN (site) state. The dashed lines represent all virtual trajectories to change the fixed points with impurities. The relationship between fixed points QAH and QAH-II is provided in the inset figure of Fig. 2(a).
weak-coupling interaction-driven topological insulating phases [15][16][17]. And, indeed, QAH and QSH phases generated by electronic repulsions occur both in the checkerboard lattice model [15,18], and in two-valley QBCP models for bilayer graphene [19,20]. A question that naturally arises is whether and how these weak-coupling interaction-driven states are affected by the presence of disorder, which is well-known to induce prominent phe-nomena such as Anderson localization, metal-insulator transition and phase transitions between superconducting phases [21][22][23][24][25][26][27]. For 2D topological states of matter this question is of particular importance, as the global topological nature of the ground state should render such states in principle robust against the local effects of disorder [28].
In this work, we analyze the fate of the interactiondriven topological insulators in Fermi systems with a QBCP under the effect of three different types of disorders, which preserve time reversal symmetry [23]. Depending on their couplings with fermions we refer to these as random chemical potential, random mass, and random gauge potential [29]. These different sorts of disorders have been shown to give rise to distinct behaviors of fermionic systems [30][31][32][33][34][35][36][37][38][39][40][41][42][43]. In general, the effect of disorder is essential in 2D for itinerant systems since it may lead to localization. Therefore interactions and disorder must be treated on equal footing and we subsequently go beyond a mean-field analysis of disorder and employ the perturbative renormalization group (RG) technique [44][45][46]. The renormalization flow procedure starts at high energy, when the ground state is known, and ends with a leading instability, which is characterized by a corresponding fixed point (FP). The analysis of the interplay of the phases well below T c is out of scope of the present paper. The central result of our calculations is schematically illustrated in Fig. 1 for the random chemical potential. With disorder the fixed points evolve to new positions, which correspond to topological phase transitions to trivial insulating states in the strong disorder regime. Moreover, the analysis of the evolution of the FP shows that disorder generally suppresses the critical temperature at which the interaction-induced topological insulating states set in.
Checkerboard lattice -The low-energy theory of spin one-half fermions on a checkerboard lattice in the presence of disorder is described by the Hamiltonian H = H 0 + H int + H dis , where H 0 is the kinetic energy, which is invariant under the C 4v point group and time-reversal symmetry [15]. It reads:
H 0 = |k|<Λ σ=↑↓ ψ † kσ H 0 (k)ψ kσ ,(1)H 0 (k) = t I k 2 τ 0 + 2t x k x k y τ 1 + t z (k 2 x − k 2 y )τ 3 ,(2)
where Λ is the momentum cut-off, while ψ kσ has two components corresponding to the two sublattices of the checkerboard lattice and τ i are Pauli matrices. Without loss of generality, we will consider in the remainder the parameter set t I = 0 and t x = t z = t, which corresponds to a particle-hole symmetric QBCP [15,18,47] and the parameter t is rescaled by 1/2m (here and below we assume = 1). The interacting part of the Hamiltonian (Color online) Schematic phase diagram for the checkerboard lattice in the presence of random chemical potential and which in the clean limit give (a): (g * 0 , g * − , g * 2 ) = (0, −3.73, 7.46)g + and (b): (g * 0 , g * − , g * 2 ) = (0, 0, −1.09)g + . The other two cases are presented in the SM [50]. Insets: Evolution of the (a) QAH fixed points (g * 0 , g * − , g * 2 ) = (0, −3.73, 7.46)g+ and (b) QSH (g * 0 , g * − , g * 2 ) = (0, 0, −1.09) with increase of the strength of the bare random mass disorder potential for the checkerboard lattice. The designations QAH and QAH-II (or QSH, QSH-II and QSH-III) FPs show the regimes, where the FPS are stable with increase of impurity scattering rates. The inset QAH and QSH represent the clean limit case.
H int has the general form [15,18,[47][48][49]:
H int = 2π m 3 i=0 g i d 2 x σ=↑↓ ψ † σ (x)τ i ψ σ (x) 2 .(3)
As mentioned above, we will consider three types of disorder: 1) random chemical potential, 2) random gauge potential and 3) random mass. Its general representation adopted from Refs. 23, 29, and 43, is:
H dis = ν m d 2 xψ † (x)M ψ(x)A(x).(4)
Here M = τ 0 is the random chemical potential, M = τ 1 and M = τ 3 the random gauge potential (two components), and M = τ 2 the random mass disorders. The field A(x) represents a quenched, Gauss-white potential determined by
A(x) = 0, while A(x)A(x ′ ) = n 0 δ(x − x ′ ),
where n 0 is the impurity (defects) concentration. The impurity scattering rate we quantify by τ −1 = n 0 ν 2 m /t, which will be measured by Λ E = tΛ 2 (for more details see Supplementary materials (SM) [50]). In general, a complete analysis should contain all possible fermion bilinears including those appearing due to interaction effects. However, we focus here on the suppression of the topological phases by disorder. Therefore for the sake of simplicity we restrict ourself to the effect of the aforementioned types of disorder separately.
RG analysis and fixed points -Within the Wilsonian renormalization group theory [19,[44][45][46][51][52][53][54][55], we derive the flow equations by integrating out the fields in the momentum shell e −l Λ < k < Λ, with l > 0 the running scale, integrating over all frequencies at the same time. The disorder contributes both to the fermion selfenergy renormalization and vertices renormalization. As a result, we obtain a system of flow equations for the coupling constants g i , disorder potentials ν m and t in the form [15,18,47]:
dg i dl = jk A ijk g j g k + j B ij n 0 ν 2 m πt 2 g j , dν m dl = D 0 ν m + t i D i g i n 0 ν 2 m πt 2 ,(5)dt dl = −C n 0 ν 2 m πt 2 t,
where the coefficients A ijk , B ij , C, D 0 and D i are provided in the SM [50]. The fixed points (FPs) are subsequently determined from the numerical analysis of the flow equations Eq. (5). Solving the flow equations in the clean limit [18,47] leads to three fixed points (g * 0 , g * − , g * 2 ) = (0, −3.73, 7.46)g + , (g * 0 , g * − , g * 2 ) = (0, 3.73, 7.46)g + and (g * 0 , g * − , g * 2 ) = (0, 0, −1.09)g + , where g ± = (g 3 ± g 1 )/2. The first two fixed points correspond to the QAH order while the last one to QSH. Under the influence of the disorder the fixed points move in the space of the coupling constants. The evolution of the FP (0, −3.73, 7.46)g + and FP (g * 0 , g * − , g * 2 ) = (0, 0, −1.09)g + with increasing the strength of the bare random chemical potential are shown in the inset figures of Fig. 2. They gradually change with the increase of the bare values of disorder and saturate with an intermediate plateau, designated as QAH (QSH) and QAH-II (QSH-II,III) FPs in Fig. 2 (here and below for easy identification we label the fixed points by the corresponding ground states QAH and QSH). We found that other evolutions are also possible (for disorders of the random mass and random gauge potential types see [50]).
Susceptibilities and phase diagrams -To find the phases that are realized at the FPs, we next solve the flow equations for the order parameters ∆ i corresponding to the different long-range orders allowed by the symmetries of the corresponding crystal structures. All possible order parameters for the checkerboard and honeycomb lattices are listed in the SM [50]. In particular, a finite expectation value of the time-reversal symmetry-breaking order parameter ∆ QAH corresponds to a QAH phase with quantized Hall conductivity, as it can be shown, in the clean limit and at the mean-field level, by integrating the Berry curvature in the full BZ [15,50]. Similarly, the ground state with order parameter ∆ QSH signals the onset of the spin-rotation symmetry-breaking QSH phase, which is characterized by the spin Chern number (C ↑ −C ↓ )/2 [56].
Employing the relation δχ i (l) = − [18,47,57], where f is the free energy, we can obtain the corresponding susceptibilities approaching the FPs. One finds that near the RG scale l * c , where the couplings
∂ 2 δf ∂∆i(0)∂∆ * i (0)g i (l) di- verge, χ i (l) ∼ (l * c −l) −γ .
For instance, their behavior as a function of the RG flow around the QAH FP is depicted in Fig. 3 and others are provided in SM [50]. Therefore the ground state may be obtained as the state characterized by the susceptibility with the strongest divergence or by comparison of the corresponding critical indexes γ i [18,47]. We checked that both ways give the same results.
Using this procedure we determine the resulting ground state as a function of the ratio of the bare interaction strength g 3 /g 0 and the disorder strength. The phase diagram for the checkerboard lattice is shown in Fig. 4. Please note that the boarder lines are drawn schematically and are the matter of further investigations. In the clean limit the g 3 /g 0 < 0.26 corresponds to QSH state, while g 3 /g 0 > 0.26 to QAH state [18]. Considering the QAH state in the presence of the chemical potential disorder, one sees that it is changed at certain value of disorder by the spin nematic (NSN) site order. Further increase of disorder potential leads to the non-ordered state. In contrast, the QSH state is suppressed by disorder without changes to intermediate phases. For the -TABLE I. Stability of the phases against different types of disorder: chemical potential (C), random mass (M) and random gauge potential (G). Here τ −1 (l * c0 ) is taken at the energy of the first instability in the clean limit.
QAH QSH NSN CDW C τ −1 (l * ) ∼ T 0 c τ −1 (l * ) ∼ T 0 c τ −1 (l * ) ≫ T 0 c τ −1 (l * ) ≫ T 0 c M τ −1 (l * ) ∼ T 0 c τ −1 (l * ) ∼ T 0 c - - G τ −1 (l * ) ∼ T 0 c τ −1 (l * ) ∼ T 0 c -
random mass potential we found no intermediate phases.
To further understand the possible consequences of the fixed point evolution, we subsequently determine the phase diagram with an effective T -dependence, which is linked to the transformation T = T 0 e −l [52,55]. As critical temperature we use the value T c = T 0 e −l * c , the results are presented in Fig. 2. Considering the effective critical temperature as a function of the random chemical potential disorder, one notes a considerable change of the T c slope at the QAH → NSN transition in Fig. 2(a). Surprisingly, the slopes also considerably change with evolution of the fixed points within the same phase. This comes from the fast crossovers from one FP to another. An example of such a slope change is provided by the evolution between the QSH-II and QSH-III FPs in Fig. 2(b). The evolution of the FP is gradual and one does not see any characteristic features on T c [50]. The situation of random mass and random gauge potential is detailed in the SM [50]. Before summarizing the results we have to note that the scattering rate τ −1 (l) is strongly renormalized in 2D together with the interaction [50]. Therefore the experimentally relevant values of the impurity scattering rate are not the bare τ 0 but τ −1 (l * c ) at the characteristic energy of the instability. The comparison of values of the scattering rate τ −1 (l * c ) necessary to suppress the effective critical temperature twice with the effective critical temperature T 0 c in the clean limit is summarized in Table I. As one can see from the table, the critical impurity scattering rates for a complete suppression of the topological phases are of the order of the critical temperature in the clean limit. By considering that the latter corresponds to the dynamically generated gap, we can conclude that, in perfect analogy with non-interacting topological insulating states [28], the stability of interaction-driven topological insulators is relatively immune to non-magnetic impurities.
Bilayer graphene -We subsequently generalize our analysis to the honeycomb lattice model for bilayer graphene. At the two inequivalent K and K ′ points of the Brillouin zone, the low-energy bands touch parabolically [16,19,20,58,59], and realize a two-valley QBCP system with an effective Hamiltonian H = k ψ † kσ H 0 ψ kσ (the noninteracting Hamiltonian and other information are provided in SM [50]). The effective action [20,47] is similar to the one for the checkerboard lattice and provided explicitly in the SM [50]. Despite the effective theory for the checkerboard and the bilayer honeycomb lattice are similar, the latter exhibits many more instabilities [47] due to the additional valley degree of freedom. In order to fully understand the leading instabilities for the bilayer honeycomb lattice, we calculate the susceptibilities for all 18 possible orderings [3,19,20,35,47,[60][61][62][63][64][65][66][67][68][69]. The resulting disorder phase diagrams for the bilayer honeycomb lattice are shown in Fig. 5. The temperature-dependent phase diagrams at bare g 1 (l = 0) = g 2 (l = 0) = 0 are qualitatively similar to the ones of the checkerboard lattice, but with the NSN (site) and QSH phases in Fig. 2 replaced by a charge-density wave (CDW) phase [50].
We have analyzed the fate of the weak-coupling interaction-driven topological insulators phases realized in 2D Fermi systems with a QBCP, under the influence of disorder. By means of the RG approach and unbiasedly studying the fermion-interacting couplings and disorders, we build the coupled flow equations of the fermioninteracting couplings and disorder strength. We established that the different types of disorder generally suppress the critical temperature at which the interactiondriven topological states set in. In particular cases, strong disorder can even induce phase transition from a topological to a non-topologically ordered state. Disorder in interaction-driven topological systems thus gives rise to a distinct set of phenomena that can be looked for and studied experimentally. Moreover, the response to disorder might be used as an experimental signature that a material is actually in a, so-far unobserved, interactiondriven topologically insulating state of matter.
H = ij (−t ij )c † i c j + V c † i c † j c j c i ,(6)
where t ij is the hopping amplitude between sites i and j while V > 0 is the nearest-neighbor repulsion. Moreover, t ij = t, t ′ , t ′′ , respectively for nearest neighbors, and next-nearest neighbors connected or not by a diagonal bond. Since the checkerboard lattice has two sublattices A an B, it is useful to introduce a spinor Ψ † i = (c † iA , c † iB ). Then the free particle Hamiltonian reads:
H 0 = 3 l=0 kσ ǫ l (k)ψ † kσ τ l ψ kσ(7)
where ǫ 0 (k) = −(t ′ + t ′′ ) (cos k x + cos k y ), ǫ 1 (k) = 4t cos ( kx 2 ) cos ( ky 2 ), and ǫ 3 (k) = −(t ′ − t ′′ ) (cos k x − cos k y ). Low energy sector-The noninteracting Hamiltonian for the checkerboard lattice in the low energy sector can be obtained expanding the tight-binding model near the corner of the Brillouin zone, i.e. at the M = (π, π) point, and is given by [15]
H 0 = |k|<Λ σ=↑↓ ψ † kσ H 0 (k)ψ kσ ,(8)
where
H 0 (k) = t I k 2 I + 2t x k x k y τ 1 + t z (k 2 x − k 2 y )τ 3 .(9)
The parameters of the continuum Hamiltonian are related to the hopping amplitudes by t x = t/2, t I = (t ′ + t ′′ )/2, and t z = (t ′ − t ′′ )/2. The primary interacting part is written as [15,18,47],
H int = i 2π m g i d 2 x σ=↑↓ ψ † σ (x)τ i ψ σ (x) 2(10)
with τ i being the Pauli matrices, which is allowed by the symmetries [15,18,[47][48][49]. The eigenvalues of H 0 (k) are derived as [15,18]
E ± k = k 2 √ 2m λ ± cos 2 η cos 2 θ k + sin 2 η sin 2 θ k ,(11)where m = 1 √ 2(t 2 x +t 2 z ) , λ = tI √ t 2 x +t 2 z , cos η = tz √ t 2 x +t 2 z , and sin η = tx √ t 2 x +t 2 z [18].
Here ψ kσ has two components, which in the case of a checkerboard lattice correspond to sublattices A and B, above equation describes one upward and one downward dispersing band at |t I | < min(|t x |, |t z |) [15,18]. The Hamiltonian possesses two touching parabolically at k = 0 and is invariant under the C 4 point group and time-reversal symmetry [15,18]. We here stress that the disorder A(x) is a quenched, Gaussian white noise potential defined by the following correlation functions
A(x) = 0; A(x 1 )A(x 2 ) = n 0 δ 2 (x 1 − x 2 ),(12)
the dimensionless parameter n 0 represents the concentration of impurity. Without lost of generality and also in order to compare with the according results in Ref. [18], we here primarily concentrate on the case in the limit of particle-hole symmetry (λ = 0) and rotational invariance (η = π 4 ). After the Fourier transformations and involving above analysis, we finally obtain the effective action in the presence of disorder,
S eff = +∞ −∞ dω 2π d 2 k (2π) 2 σ=↑↓ ψ † σ (ω, k)[−iω + 2tk x k y τ 1 + t(k 2 x − k 2 y )τ 3 ]ψ σ (ω, k) + 2π m 3 i=0 g i +∞ −∞ dω 1 dω 2 dω 3 (2π) 3 × Λ d 2 k 1 d 2 k 2 d 2 k 3 (2π) 6 σ,σ ′ =↑↓ ψ † σ (ω 1 , k 1 )τ i ψ σ (ω 2 , k 2 )ψ † σ ′ (ω 3 , k 3 )τ i ψ σ ′ (ω 1 + ω 2 − ω 3 , k 1 + k 2 − k 3 ) +ν m +∞ −∞ dω 2π d 2 kd 2 k ′ (2π) 4 ψ † (k, ω)M ψ(k ′ , ω)A(k − k ′ ).(13)
with m = 1/(2t) and M = τ 0 being the random chemical potential, M = τ 1 and M = τ 3 the random gauge potential (two components), and M = τ 2 the random mass [29,43], respectively. Here the parameter ν m measures the strength of a single impurity and the corresponding impurity scattering rate can be expressed as τ −1 ∼ n 0 ν 2 m /t, which will be measured by Λ E = tΛ 2 with t rescaled by 2 /2m. The free propagators are represented in the Fig. 6.
Coupled flow equations and fixed points
By including the disorder corrections and considering the RG theory [46,52,53], we obtain the revised re-scaling transformation as given in the main text. In the presence of disorder, the fermions receive self-energy corrections from the fermion-disorder interaction as shown in Fig. 7. In addition, the one-loop corrections to the fermion interacting couplings and the fermion-disorder vertex in presence of different sorts of disorders as presented in Fig. 8 and Fig. 9. After calculating the one-loop corrections paralleling the steps in Refs. [43,46,52,53], we derive the coupled flow equations for all parameters. Denoting g + = g3+g1 2 and g − = g3−g1 2 [18,47], we obtain the reduced flow equations for all parameters, as listed in the following. In the presence of random chemical potential with M = τ 0 , the coupled flow equations are In the presence of random gauge potential with M = τ 1,3 whose flow equations are the same, the coupled flow equations look like: for both M = τ 1 and M = τ 3 ,
dt dl = − n 0 ν 2 m 4πt 2 t,(14)dg 0 dl = −4g + − 3n 0 ν 2 m 2πt 2 g 0 ,(15)dg + dl = −(g 0 − g + ) 2 − (g 2 − g + ) 2 − 6g 2 + − n 0 ν 2 m 2πt 2 g + ,(16)dg 2 dl = 4(g 0 g 2 − g 2 2 − g 2 − + g 2 + − 3g 2 g + ) − 2n 0 ν 2 m πt 2 g 2 ,(17)dg − dl = 2g − (g 0 − 3g 2 − 2g + ) − n 0 ν 2 m 2πt 2 g − ,(18)dν m dl = [n 0 ν 2 m − 8πt(g 0 + g 2 + 2g + )] 4πt 2 ν m .(19)dt dl = − n 0 ν 2 m 4πt 2 t,(20)dg 0 dl = −4g + g 0 + n 0 ν 2 m 2πt 2 g 0 ,(21)dg + dl = −(g 0 − g + ) 2 − (g 2 − g + ) 2 − 6g 2 + ,(22)dg 2 dl = 4(g 0 g 2 − g 2 2 − g 2 − + g 2 + − 3g 2 g + ) + n 0 ν 2 m 2πt 2 g 2 ,(23)dg − dl = 2g − (g 0 − 3g 2 − 2g + ) ,(24)dν m dl = − n 0 ν 2 m 4πt 2 ν m .(25)
Finally, the coupled flow equations in the presence of random mass with M = τ 2 read:
dt dl = − n 0 ν 2 m 4πt 2 t,(26)dg 0 dl = −4g + g 0 ,(27)dg + dl = −(g 0 − g + ) 2 − (g 2 − g + ) 2 − 6g 2 + − n 0 ν 2 m 2πt 2 g + ,(28)dg 2 dl = 4(g 0 g 2 − g 2 2 − g 2 − + g 2 + − 3g 2 g + ) + n 0 ν 2 m 2πt 2 g 2 ,(29)dg − dl = 2g − (g 0 − 3g 2 − 2g + ) − n 0 ν 2 m 2πt 2 g − ,(30)dν m dl = [8πt(g 0 − 2g + + g 2 ) − 3n 0 ν 2 m ] 4πt 2 ν m .(31)
Performing numerical calculations of above coupled flow equations, we get the fixed points. The trajectories towards to the fixed points in clean limit have already studied by Murray and Vafek [18]. For completeness, we provide the 10. Flows of g0/g+, g2/g+ and g−/g+ in clean limit at some representatively initial values. The (g0, g2, g−)/g+ finally towards three fixed points: (a) (g * 0 , g * − , g * 2 )/g+ = (0, −3.73, 7.46); (b) (g * 0 , g * − , g * 2 )/g+ = (0, 3.73, 7.46) and (c) (g * 0 , g * − , g * 2 )/g+ = (0, 0, −1.09)g + . Inset: the enlarged regime for the fixed point.
corresponding trajectories in Fig. 10. Next, we consider the fixed points in the presence of different types of disorder. The evolution of fixed points in the presence of random chemical potential, and random mass are presented in Fig. 12 and Fig. 13 respectively. In distinction to the other two types of disorders, we find that both the QAH fixed point (g * 0 , g * − , g * 2 ) = (0, −3.73, 7.46) and the QSH fixed point (g * 0 , g * − , g * 2 ) = (0, 0, −1.09) are robust against the random gauge potential and do not evolve with increase of the disorder. By these reasons we do not show them here.
Mean field order parameter
In order to investigate possible types of symmetry breaking, we collect both the charge and spin source terms into the action [18,47]:
S ∆ = dτ d 2 x 3 i=0 ∆ c i ψ † M c i ψ + ∆ s i · ψ † M s i ψ .(32)
FIG. 11. One-loop corrections to the fermion-source terms ∆ c,s .
The matrices M c,s define the various fermion bilinears in charge and spin channels [18]. One-loop corrections to the fermion-source terms ∆ c,s can be derived by computing the diagrams in Fig. 11. For the charge channel, the matrixes
M c 1 = τ 1 , M c 2 = τ 2 , and M c 3 = τ 3(33)
correspond to the nematic (bond), QAH, and nematic (site), respectively [18]. Besides, for the spin channel, the matrixes M s 0 = 1 s, M s 1 = τ 1 s, M s 2 = τ 2 s, and M s 3 = τ 3 s refer to the FM, NSN (bond), QSH, and NSN (site), respectively [18]. Since the susceptibilities of M c,s 0 are independent of the energy scale which are not primary instabilities, we will only pay significance to the instabilities for cases M c,s 1,2,3 in the following text.
Chern number-The topological invariant in the insulating phase with finite expectation value of M c 2 can be computed using a relation between the Chern number and the 2π Berry flux carried by a symmetry-protected quadratic band crossing point. The latter corresponds to the topological charge of thed vector vortex withd = 2k x k y , k 2
x − k 2 y . Using a gauge fixing procedure [56], one can indeed show that the Chern number acquired by breaking time-reversal symmetry can be related to the topological charge W as C = W/2 [56]. This equivalence allows to obtain the Chern number for a quadratic band crossing point even though in this model the momentum space cannot be one-point compactified to a unit sphere S 2 . And indeed, the Chern number computed in this way corresponds precisely to the Chern number obtained at the mean field level using the lattice formulation [15]. A similar analysis can be performed for the QSH phase using that in each spin channel carries an opposite Hall conductivity and hence Chern number.
Another approach of calculation of the Chern number is based on mapping of the continuous model onto a tight binding model. Then the Chern number is given by the integration over the Brillouin zone (BZ):
N Chern = 1 4π BZ dk x dk y n · ∂n ∂k x × ∂n ∂k y .(34)
For the introduced tight binding model Eq. (7) and the mean-field order parameter of the QAH state M c 2 = sin( kx 2 ) sin( ky 2 )τ 2 the vector n = d/d with
d = t 4 cos k x 2 cos k y 2 , ∆ QAH 2t (sin k x 2 sin k y 2 ), −(cos k x − cos k y ) .(35)
Calculating the integral Eq. (34) we find that the Chern number coincides with the discussed above continuous model N Chern = −sign(∆ QAH ).
Susceptibilities in the presence of disorder
After fulfilling the one-loop corrections to the source terms 11, we can get the flow equations of the source terns [18] (the matrixes M s 0 = 1 s, M s 1 = τ 1 s, M s 2 = τ 2 s, and M s 3 = τ 3 s refer to the FM, NSN (bond), QSH, and NSN (site), respectively) i) random chemical potential:
M = τ 0 d ln ∆ c 1 dl = 2 + 4m 16π g 0 − 12m 16π g 1 − 4m 16π g 2 − 4m 16π g 3 ,(36)d ln ∆ c 2 dl = 2 + 8m 16π g 0 − 8m 16π g 1 − 24m 16π g 2 − 8m 16π g 3 − n 0 v 2 m 2πt 2 ,(37)d ln ∆ c 3 dl = 2 + 4m 16π g 0 − 4m 16π g 1 − 4m 16π g 2 − 12m 16π g 3 ,(38)
and d ln ∆ s
1 dl = 2 + 4m 16π g 0 + 4m 16π g 1 − 4m 16π g 2 − 4m 16π g 3 ,(39)d ln ∆ s 2 dl = 2 + 8m 16π g 0 − 8m 16π g 1 + 8m 16π g 2 − 8m 16π g 3 − n 0 v 2 m 2πt 2 ,(40)d ln ∆ s 3 dl = 2 + 4m 16π g 0 − 4m 16π g 1 − 4m 16π g 2 + 4m 16π g 3 ;(41)dl = 2 + 4m 16π g 0 − 12m 16π g 1 − 4m 16π g 2 − 4m 16π g 3 ,(42)d ln ∆ c 2 dl = 2 + 8m 16π g 0 − 8m 16π g 1 − 24m 16π g 2 − 8m 16π g 3 + n 0 v 2 m 2πt 2 ,(43)d ln ∆ c 3 dl = 2 + 4m 16π g 0 − 4m 16π g 1 − 4m 16π g 2 − 12m 16π g 3 ,(44)
and d ln ∆ s
1 dl = 2 + 4m 16π g 0 + 4m 16π g 1 − 4m 16π g 2 − 4m 16π g 3 ,(45)d ln ∆ s 2 dl = 2 + 8m 16π g 0 − 8m 16π g 1 + 8m 16π g 2 − 8m 16π g 3 + n 0 v 2 m 2πt 2 ,(46)
d ln ∆ s (Color online) Schematic phase diagram for the checkerboard lattice in the presence of random mass potential and which in the clean limit give (a): (g * 0 , g * − , g * 2 ) = (0, −3.73, 7.46)g + and (b): (g * 0 , g * − , g * 2 ) = (0, 0, −1.09)g + . The parameter τ −1 ∼ n0ν 2 m /t designates the impurity scattering rate and ΛE = tΛ 2 with t rescaled by 2 /2m. The phases QAH (or QSH) and QAH-II (QSH-II) are the same phase but with different critical temperatures caused by distinct of FPs as depicted in Fig. 13.
3 dl = 2 + 4m 16π g 0 − 4m 16π g 1 − 4m 16π g 2 + 4m 16π g 3 ;(47)dl = 2 + 4m 16π g 0 − 12m 16π g 1 − 4m 16π g 2 − 4m 16π g 3 ,(48)d ln ∆ c 2 dl = 2 + 8m 16π g 0 − 8m 16π g 1 − 24m 16π g 2 − 8m 16π g 3 − n 0 v 2 m 2πt 2 ,(49)d ln ∆ c 3 dl = 2 + 4m 16π g 0 − 4m 16π g 1 − 4m 16π g 2 − 12m 16π g 3 ,(50)
and d ln ∆ s
1 dl = 2 + 4m 16π g 0 + 4m 16π g 1 − 4m 16π g 2 − 4m 16π g 3 ,(51)
d ln ∆ s 2 dl = 2 + 8m 16π g 0 − 8m 16π
g 1 + 8m 16π g 2 − 8m 16π g 3 − n 0 v 2 m 2πt 2 ,(52)
d ln ∆ s
3 dl = 2 + 4m 16π g 0 − 4m 16π g 1 − 4m 16π g 2 + 4m 16π g 3 .(53)
Some of the corresponding results are provided in Figs. 12,13,14,15, and 16 besides the figures presented in the main text.
FIG. 1 .
1FIG. 1. (Color online) Schematic renormalization group flows of the coupling constant ratios (g * 0 , g * − , g * 2 )/g * + for the checkerboard lattice in the clean limit (red) and in the presence of random chemical potential(green and blue). In clean limit the QAH fixed point (g * 0 , g * − , g * 2 )/g * + = (0, −3.73, 7.46) (red point) corresponds to a QAH state. In the presence of the weak random chemical potential the coupling constants flow to the QAH-II fixed point (green) (g * 0 , g * − , g * 2 )/g * + = (0, −0.2, 6.5) corresponding again to the QAH state and in the presence of the strong random chemical potential to the NSN fixed point (green) (g * 0 , g * − , g * 2 )/g * + = (0, 0, −1.09) corresponding to the NSN (site) state. The dashed lines represent all virtual trajectories to change the fixed points with impurities. The relationship between fixed points QAH and QAH-II is provided in the inset figure of Fig. 2(a).
FIG. 2. (Color online) Schematic phase diagram for the checkerboard lattice in the presence of random chemical potential and which in the clean limit give (a): (g * 0 , g * − , g * 2 ) = (0, −3.73, 7.46)g + and (b): (g * 0 , g * − , g * 2 ) = (0, 0, −1.09)g + . The other two cases are presented in the SM [50]. Insets: Evolution of the (a) QAH fixed points (g * 0 , g * − , g * 2 ) = (0, −3.73, 7.46)g+ and (b) QSH (g * 0 , g * − , g * 2 ) = (0, 0, −1.09) with increase of the strength of the bare random mass disorder potential for the checkerboard lattice. The designations QAH and QAH-II (or QSH, QSH-II and QSH-III) FPs show the regimes, where the FPS are stable with increase of impurity scattering rates. The inset QAH and QSH represent the clean limit case.
FIG. 3 .
3(Color online) Susceptibilities to particle-hole phases as functions of the RG flow parameter l by approaching the QAH FP (g * 0 , g * − , g * 2 )/g+ = (0, −3.73, 7.46) for checkerboard lattice.
FIG
. 4. (Color online) Schematic phase diagram as a function of disorder and interaction strength g3/g0 for the checkerboard lattice. (a): random chemical potential and (b): random mass. The change of the color from dark to light is deduced from the evolution of the FPs as shown in Fig. 2.
FIG
. 5. (Color online) Schematic phase diagrams for the honeycomb lattice for bare couplings g1(l = 0) = g2(l = 0) = 0 in the presence of (a): random chemical potential and (b): random mass. The change of the color from dark to light is deduced from the evolution of the FPs as shown in Fig. 2.
We acknowledge J. M. Murray and O. Vafek for useful correspondence and C. Fulga for helpful discussions. J.W. is supported by the National Natural Science Foundation of China under Grant 11504360, the China Postdoctoral Science Foundation under Grants 2015T80655 and 2014M560510, the Fundamental Research Funds for the Central Universities, and the Program of Study Abroad for Young Scholar sponsored by China Scholarship Council. C.O. acknowledges the financial support of the Future and Emerging Technologies Programme for Research of the European Commission under FET-Open grant number: 618083 (CNTQC), and the Deutsche Forschungsgemeinschaft under Grant No. OR 404/1-1. D.V.E. and J.v.d.B would like to acknowledge the financial support provided by the German Research Foundation (Deutsche Forschungsgemeinschaft) through the program DFG-Russia, BR4064/5-1. J.v.d.B is also supported by SFB 1143 of the Deutsche Forschungsgemeinschaft. D.V.E would also acknowledge the VW foundation for partial financial support. model -The minimal model on the checkerboard lattice is:
FIG. 6 .
6Free propagators for (i) fermion, (ii) fermionic interaction, (iii) disorder, and (iv) source field.
FIG. 7 .
7One-loop corrections to the fermion propagator. FIG. 8. One-loop corrections to the fermion interacting couplings due to fermionic interactions.
FIG. 9 .
9One-loop corrections to the fermion interacting couplings due to the fermionic interactions and disorder effects.
FIG. 10. Flows of g0/g+, g2/g+ and g−/g+ in clean limit at some representatively initial values. The (g0, g2, g−)/g+ finally towards three fixed points: (a) (g * 0 , g * − , g * 2 )/g+ = (0, −3.73, 7.46); (b) (g * 0 , g * − , g * 2 )/g+ = (0, 3.73, 7.46) and (c) (g * 0 , g * − , g * 2 )/g+ = (0, 0, −1.09)g + . Inset: the enlarged regime for the fixed point.
12. (Color online) Evolution of the fixed points (g * 0 , g * − , g * 2 ) for the checkerboard lattice in the presence of random chemical potential around (a): the QAH fixed point (0, −3.73, 7.46) and (b): the QSH fixed point (0, 0, −1.09) at some representatively initial values of disorder strength of the random chemical potential. FP represents the fixed points and The parameter τ −1 ∼ n0ν 2 m /t designates the impurity scattering rate and ΛE = tΛ 2 with t rescaled by 2 /2m.
. 13. (Color online) Evolution of the fixed points (g * 0 , g * − , g * 2 ) for the checkerboard lattice in the presence of random mass around (a): the QAH fixed point (0, −3.73, 7.46)g+ and (b): the QSH fixed point (0, 0, −1.09)g+ at some representatively initial values of disorder strength of the random chemical potential. FP represents the fixed points and The parameter τ −1 ∼ n0ν 2 m /t designates the impurity scattering rate and ΛE = tΛ 2 with t rescaled by 2 /2m. ii) Random gauge potential: M = τ 1 and M = τ 3 d ln ∆ c 1
14. (Color online) Evolution of the fixed points (g * 0 , g * − , g * 2 ) for the checkerboard lattice in the presence of random gauge potential around (a): the QAH fixed point (0, −3.73, 7.46)g+ and (b): the QSH fixed point (0, 0, −1.09)g+ at some representatively initial values of disorder strength of the random chemical potential. FP represents the fixed points and the parameter The parameter τ −1 ∼ n0ν 2 m /t designates the impurity scattering rate and ΛE = tΛ 2 with t rescaled by 2 /2m.
FIG. 15 .
15FIG. 15. (Color online) Schematic phase diagram for the checkerboard lattice in the presence of random mass potential and which in the clean limit give (a): (g * 0 , g * − , g * 2 ) = (0, −3.73, 7.46)g + and (b): (g * 0 , g * − , g * 2 ) = (0, 0, −1.09)g + . The parameter τ −1 ∼ n0ν 2 m /t designates the impurity scattering rate and ΛE = tΛ 2 with t rescaled by 2 /2m. The phases QAH (or QSH) and QAH-II (QSH-II) are the same phase but with different critical temperatures caused by distinct of FPs as depicted in Fig. 13.
FIG. 16. (Color online) Schematic phase diagram for the checkerboard lattice in the presence of random gauge potential and which in the clean limit give (a): (g * 0 , g * − , g * 2 ) = (0, −3.73, 7.46)g + and (b): (g * 0 , g * − , g * . 17. (Color online) Flows of all charge and spin susceptibilities for the bilayer honeycomb lattice. UP: in clean limit in the vicinity of the fixed point QAH (g * 0 , g * − , g * 2 ) = (0, −3.73, 7.46)g + (a) and QSH (g * 0 , g * − , g * 2 ) = (0, 0, −1.09)g + (b); DOWN: in the presence of random chemical potential in the vicinity of the QAH fixed point (g * 0 , g * − , g * 2 ) = (0, −3.73, 7.46)g + , (a): small bare v 0 m . The leading instability is QAH; (b): large bare v 0 m . The leading instability is CDW. Susceptibilities for other fixed points and disorders can be derived similarly and are not shown here. iii) Random mass: M = τ 2 d ln ∆ c 1
) = (0, 0, −1.09)g + . The parameter τ −1 ∼ n0ν 2 m /t designates the impurity scattering rate. The phases QAH (or QSH) and QAH-II (QSH-II) are the same phase but with different critical temperatures caused by distinct of FPs as depicted inFig. 14.
Results for the bilayer honeycomb latticeEffective theory for bilayer grapheneThe tight-binding Hamiltonian for electrons hopping on the bilayer honeycomb lattice with Bernal stacking can be described as[19,20],where t represents the hopping amplitudes connecting the next-nearest sites in a plan and n(r) = c † (r)c(r). We then transfer above Hamiltonian to momentum space and gain[19,20,58,59],and H 0 can be described as[19,20,58,59]which can straightforwardly be diagonalized to[19,20]Two of these four bands are parabolically touching at k = 0[20], which can also considered as a QBCP system and whose band structure is similar to the checkerboard's. The effective action of noninteracting terms in clean limit for the bilayer graphene would be given by[20,58,59]The Pauli matrices σ i act on the layer indices 1-2 and the τ matrices act on the valley indices K − K ′ . The effective mass is m = 2t ⊥ 9t 2 and ψ represents N 2 copies of the four component pseudospinor. N = 4 for spin 1/2. All marginal interactions areIn order to compare with the effective theory of the checkerboard lattice, we introduce the new "Pauli" matrices by defining σ ′ 0 ≡ τ 0 σ 0 , σ ′ 1 ≡ τ 3 σ 2 , σ ′ 2 ≡ τ 3 σ 3 and σ ′ 3 ≡ τ 0 σ 1 , which also have the same symmetries of the Pauli matrices as σ ′ µ σ ′ ν = 1 4 δ µν + iǫ µνλ σ ′ λ . We finally obtain the effective action in the presence of disorders after carrying out the fourier transformation,with M = σ ′ 0 being the random chemical potential, M = σ ′ 1 and M = σ ′ 3 being the random gauge potential (two components), and M = σ ′ 2 being the random mass.Fixed points and susceptibilities for the bilayer honeycomb lattice in the presence of disorderWe emphasize that there are 12 other possible orders besides the 6 orders in the checkerboard lattice for the honeycomb lattice as provided in Ref.[47], which are listed here for completeness:Charge channelSpin channel τ 0 ⊗ σ 0 : charge instability τ 0 ⊗ σ 0 s: FM τ 0 ⊗ σ 1 : nematic (bond)[20,60]τ 0 ⊗ σ 1 s: NSN (bond) τ 3 ⊗ σ 3 : QAH[3,61]τ 3 ⊗ σ 3 s: QSH[66][67][68][62]τ 3 ⊗ σ 0 s: Staggered spin current τ 0 ⊗ σ 3 : Layer-polarized[63,64]τ 0 ⊗ σ 3 s: Layer AF[19,35,69] τ 3 ⊗ σ 1 : Loop current II τ 3 ⊗ σ 1 s: Loop spin current II τ 1 ⊗ σ 1 : Kekulé[65]τ 1 ⊗ σ 1 s: Spin Kekulé τ 1 ⊗ σ 2 : Kekulé current τ 1 ⊗ σ 2 s: Spin Kekulé current τ 1 ⊗ σ 0 : CDW τ 1 ⊗ σ 0 s: SDW Additional, there is the third fixed point in the honeycomb lattice besides the two fixed points considered in the checkerboard lattice due to the nonzero initial values of g 1 and g 2 . For instance, the fixed point (g * 0 , g * − , g * 2 ) = (0, 3.73, 7.46)g +[18]. By considering all these facets and paralleling the similar steps employed in the checkerboard lattice, we provide the primary results of the susceptibilities for both clean limit and impurity case in the vicinity of the some representative fixed points as presented inFig. 17. We summarize all the information from the susceptibilities and plot the schematic phase diagram, namelyFig. 5in the main text.
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[
"Dependence of the liquid-vapor surface tension on the range of interaction: a test of the law of corresponding states",
"Dependence of the liquid-vapor surface tension on the range of interaction: a test of the law of corresponding states"
] |
[
"Patrick Grosfils ",
"James F Lutsko ",
"\nChimie Physique E.P. CP 165/62\nAv.F.D\nCenter for Nonlinear Phenomena and Complex Systems CP 231\nMicrogravity Research Center\nUniversité Libre de Bruxelles\nRoosevelt 501050BrusselsBelgium. †\n",
"\nAlso at Center for Nonlinear Phenomena and Complex Systems CP 231\nUniversité Libre de Bruxelles\nBlvd. du Triomphe1050BrusselsBelgium ‡\n",
"\nUniversité Libre de Bruxelles\nBlvd. du Triomphe1050BrusselsBelgium ‡ Electronic\n"
] |
[
"Chimie Physique E.P. CP 165/62\nAv.F.D\nCenter for Nonlinear Phenomena and Complex Systems CP 231\nMicrogravity Research Center\nUniversité Libre de Bruxelles\nRoosevelt 501050BrusselsBelgium. †",
"Also at Center for Nonlinear Phenomena and Complex Systems CP 231\nUniversité Libre de Bruxelles\nBlvd. du Triomphe1050BrusselsBelgium ‡",
"Université Libre de Bruxelles\nBlvd. du Triomphe1050BrusselsBelgium ‡ Electronic"
] |
[] |
The planar surface tension of coexisting liquid and vapor phases of a fluid of Lennard-Jones atoms is studied as a function of the range of the potential using both Monte Carlo simulations and Density Functional Theory. The interaction range is varied from r * c = 2.5 to r * c = 6 and the surface tension is determined for temperatures ranging from T * = 0.7 up to the critical temperature in each case. The results are shown to be consistent with previous studies. The simulation data are well-described by Guggenheim's law of corresponding states but the agreement of the theoretical results depends on the quality of the bulk equation of state.
|
10.1063/1.3072156
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[
"https://arxiv.org/pdf/0807.0107v1.pdf"
] | 36,773,596 |
0807.0107
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c943b8e2eed69f65a745ff43d68c93c400998ec4
|
Dependence of the liquid-vapor surface tension on the range of interaction: a test of the law of corresponding states
Jul 2008
Patrick Grosfils
James F Lutsko
Chimie Physique E.P. CP 165/62
Av.F.D
Center for Nonlinear Phenomena and Complex Systems CP 231
Microgravity Research Center
Université Libre de Bruxelles
Roosevelt 501050BrusselsBelgium. †
Also at Center for Nonlinear Phenomena and Complex Systems CP 231
Université Libre de Bruxelles
Blvd. du Triomphe1050BrusselsBelgium ‡
Université Libre de Bruxelles
Blvd. du Triomphe1050BrusselsBelgium ‡ Electronic
Dependence of the liquid-vapor surface tension on the range of interaction: a test of the law of corresponding states
Jul 2008(Dated: July 1, 2008)1PACS numbers: 6803Cd,6520De * Electronic address: pgrosfi@ulbacbe †
The planar surface tension of coexisting liquid and vapor phases of a fluid of Lennard-Jones atoms is studied as a function of the range of the potential using both Monte Carlo simulations and Density Functional Theory. The interaction range is varied from r * c = 2.5 to r * c = 6 and the surface tension is determined for temperatures ranging from T * = 0.7 up to the critical temperature in each case. The results are shown to be consistent with previous studies. The simulation data are well-described by Guggenheim's law of corresponding states but the agreement of the theoretical results depends on the quality of the bulk equation of state.
I. INTRODUCTION
One of the most fundamental properties of a fluid is the surface tension at the liquid-vapor interface. It would seem that such a fundamental property would be an ideal candidate for study via computer simulation. However, the determination of the surface tension from simulation turns out to be frought with difficulties so that even today there is still a substantial amount of effort directed towards the development of more reliable algorithms and the refinement of the reported values even for the paradigmatic case of a simple fluid modeled with the Lennard-Jones interaction [1]. One of the primary difficulties is that in all simulations, the potential is truncated at some finite range and it happens that the surface tension is very sensitive to the value of the cutoff. For that reason, an important part of the development of algorithms has focused on the calculation of the corrections needed to get the infinite-ranged limit from data obtained using a truncated potential (see, e.g., ref. [1,2]). This sensitivity is therefore a nuisance when the goal is to get the infinite range result, but in other ways it can made useful. In particular, one of the important reasons to determine the surface tension from simulation is that it provides a baseline against which theories of inhomogeneous liquids can be tested [3,4]. For this application, the sensitivity of the surface tension to the range of the potential can be used as a test of the generality of a theory which was probably motivated in the first place by its agreement with some existing simulation data. Furthermore, there has recently been a significant increase in interest in short-ranged potentials in their own right. This is due to the fact that certain complex fluids, in particular globular proteins, can, in a first approximation, be modeled as a simple fluid with a very short ranged interaction [5]. It is therefore interesting to study the properties of fluids with these kinds of interactions and to test that existing theories work in this new domain of interest. For these reasons, we present in this paper a systematic study of the dependence of the surface tension of a Lennard-Jones fluid as a function of the range of the potential.
In this paper, we describe the results of Monte Carlo (MC) simulations of a Lennard-Jones fluid with the potential truncated at several different points. We have chosen to truncate and shift the Lennard-Jones potential, v LJ (r), so that the potential used in this work is v(r; r c ) = v LJ (r) − v LJ (r c ) for r < r c and v(r) = 0 for r ≥ r c . If this shift is not performed, then there is an impulsive contribution to the pressure when atoms move across the r = r c boundary that would have to be taken into account [6]. We do not shift the force, i.e. we do not useṽ(r;
r c ) = v LJ (r) − v LJ (r c ) − (r − r c )v ′ LJ (r c )
with v ′ LJ (r) = dv LJ (r)/dr inside the cutoff, as is usually done in molecular dynamics simulations to avoid impulsive forces: our potential is truncated and shifted but the force is not shifted. This choice was made in order to allow for comparison with previous MC studies.
In the simulations a slab of liquid is bounded on both sides by vapor. The surface tension is determined using the method of Bennett [6,7] as there seems to be some evidence that this method is more robust than other commonly used techniques [8]. It is often the case that the quantity of interest is the surface tension for the infinite-ranged potential. Since simulations almost always make use of truncated potentials, various techniques have been developed to approximate the so-called long range corrections, i.e. the difference between quantities calculated with the truncated potential and the infinite-ranged quantities [2]. We do not include any such corrections here since our goal is actually to study the truncated potentials. Thus, each value of the cutoff defines a different potential with its own coexistence curve and thermodynamics.
In Section II, we present the simulation techniques used in our work. Section III contains a discussion of our results including a comparison to previous work. Since one of the motivations for this work is to provide a baseline for testing theories of the liquid state, we illustrate this by comparing our results to Density Functional Theory (DFT) calculations and by testing the law of corresponding states. We give our conclusions in the last Section.
which is truncated and shifted so that the potential simulated is
v(r) = v LJ (r) − v LJ (r c ) : r < r c 0 : r ≥ r c(2)
where r c is the cutoff radius. Each simulation starts from a rectangular box (L x = L y = L, L z = 4 L) filled with a dense disordered arrangement of particles (ρ * ≡ ρσ 3 = 0.8)
surrounded along the z-direction by two similar rectangular boxes containing particles in a low density state (ρ * ∼ 0.01) . The total simulation box has sides of length L x = L y = 9.15σ
and L z = 109.63σ. The liquid film located in the middle of the box has a thickness ∆z ≃ 27σ so that the two interfaces do not influence each others. The system is first equilibrated during Although several methods are available for the computation of the surface tension, the Bennett's approach has been chosen because of its accuracy [8]. In the Bennett's method the calculation of the surface tension follows from the definition
γ = ∂F ∂A N,V,T(3)
where F is the free energy and A is the area of the liquid-vapor interface. In its implementation the method requires that one performs two simulations: one for system 0 of interface area A 0 , and another for system 1 of interface area A 1 = A 0 + ∆A. In this work ∆A/A = 5 × 10 −4 . The free energy difference ∆F between the two systems is evaluated by the method of acceptation ratio which starts with the computation of ∆E 01 = E 01 − E 00 which is the difference between E 00 , the energy of a configuration of system 0, and E 01 , the energy of a new configuration obtained from the previous one by rescaling the positions of the particles [6,9] :
x ′ = x (A 1 /A 0 ) 1 2 , y ′ = y (A 1 /A 0 ) 1 2 , and z ′ = z (A 0 /A 1 ).
Similarly one computes ∆E 10 = E 10 − E 11 obtained from a configuration of system 1 following an inverse rescaling of the positions. ∆F is obtained by requiring that
n 0 f (∆E 01 − ∆F ) = n 1 f (∆E 10 + ∆F )(4)
where n 0 ( n 1 ) is a sum over the configurations of systems 0 (1), and f (x) = (1 + exp(βx)) −1 . Then, taking into account the fact that the system contains two flat interfaces, the value of the surface tension is given by γ = ∆F/(2∆A) . The open circles are our data, the filled circles are from Duque et al [11], the squares are from Mecke et al [1], the diamonds are from Potoff et al [12] and the triangles are from [10]). Note that the Mecke and Potoff data both include long-ranged corrections.
III. RESULTS
A. Comparison to previous results
Our results for the surface tension as a function of the cutoff are given in Table 1. Note that all quantities are reported in reduced units so that the reduced temperature is T * = T /ǫ, the reduced cutoffs are r * c = r c /σ and the reduced surface tension is γ * = γσ 2 /ǫ. In Fig. 1 we show our results for cutoffs of r * c = 2.5 and 6.0 compared to the MC data of Haye and Bruin [10] for the shorter cutoff and to the MD data of Duque et al [11] (who appear to shift the forces) and Potoff et al [12] and Mecke et al [1]. The latter two are shown even though they include long-ranged corrections. Our data are seen to be very consistent with the MC data obtained without long-ranged corrections and to lie slightly below the corrected data, as expected.
In Fig. 2, we compare our results to the predictions of a recently proposed Density Functional Theory model [4]. The DFT requires knowledge of the bulk equation of state and the figure shows results using two different inputs: the 33-parameter equation of state of Johnson, Zollweg and Gubbins (JZG) [13] and first order Barker-Henderson thermodynamic perturbation theory [14,15]. Both versions of the theory are in good qualitative agreement with the data, showing the decrease in surface tension as the range of the potential decreases.
It might be thought that use of an empirical equation of state should automatically give superior results to an approximation, like thermodynamic perturbation theory, but this is not necessarily the case since the equation of state is fitted to data for the infinite-ranged potential. The finite cutoff is accounted for using simple mean-field corrections [6,13] and these become increasing inaccurate as the cutoff becomes shorter and, for fixed cutoff, as the fluid density becomes higher. The latter condition means, in the present context, increasing inaccuracy as the temperature decreases. Both of these trends are confirmed by the figure.
The decrease in accuracy with decreasing cutoff can be seen in the fact that the critical point (corresponding to the temperature at which the surface tension extrapolates to zero) is less accurately estimated for the smaller cutoffs than for the larger cutoffs.
The perturbation theory, on the other hand, takes the cutoff into account more accurately and consistently so that no strong change in accuracy is expected as the cutoff decreases.
However, the theory itself is expected to be less accurate for higher densities so again a drop in accuracy with decreasing temperature would be expected and that is indeed seen in the figure. Furthermore, perturbation theory is in general going to be inaccurate near the critical point as it does not take into account renormalization effects which tend to lower the critical point. These effects are less pronounced for shorter-ranged potentials and indeed the perturbation theory seems more consistent with shorter-ranged potential.
C. Corresponding states
The principle of corresponding states is a generalization of the results of the van der Waals equation of state [16]. The idea is that the properties of simple liquids should be universal functions of the state variables, density and temperature, scaled to the critical [17,18]. In some applications [19], higher order terms are included but we did not feel that the accuracy of our data warranted use of anything but the lowest order function. The critical density was then estimated using the law of the rectilinear diameter, 1 2 (ρ l +ρ v ) = ρ c +B(T c −T ) [16]. There are again higher order corrections to this formula which can be calculated using RG methods, but for the reasons just given, we have not attempted to include them. The results of these fits are illustrated in Fig. 3 and summarized in Table 2. The largest errors in this procedure are in the determination of the critical density. Figure 4 shows the surface tensions, as determined from simulation and theory using the JZG equation of state, scaled to the critical density and temperature as a function of distance from the critical temperature. Despite the wide range of cutoffs and the mixture of data from simulations and theory, it is nevertheless seen that the data do in fact obey the law of corresponding states to a good approximation. However, the same scaling of the theoretical calculations using the equation of state from thermodynamic perturbation theory, shown in r * c = 3 appear to coincide, the data for the larger cutoffs does not. This appears to be due, at least in part, to the fact that the estimate of the critical density as a function of the cutoff calculated using the perturbation theory is not monotonic (see Table II) which is at odds with the quantities as determined from simulation which clearly are monotonic in the cutoff.
result, 1 2 (ρ l − ρ v ) = A(T c − T ) 0.325
IV. CONCLUSIONS
We have presented our determination of the liquid-vapor surface tension in a Lennard-Jones fluid as a function of the range of the potential. The data give a systematic picture of the variation of surface tension with the cutoff and are in agreement with previous studies.
It is hoped that this can serve as a useful benchmark for the development of theories of inhomogeneous liquids. Indeed, the results were compared here to calculations made using a recently developed DFT and the strengths and weaknesses of the theory are evident: while it gives a good semi-quantitative estimate of the surface tension for all cutoffs, errors on the order of 10% are present indicating that further improvement is possible.
We have also tested the law of corresponding states by showing our results from both simulation and theory scaled to the critical density and temperature. For the simulation data and the theoretical calculations based on an empirical equation of state, the law of corresponding states appears to be obeyed. However, the calculations based on the equation of state from first order perturbation theory do not appear to scale well at all. This failure appears to be due to poor behavior of the critical density as a function of the cutoff and indicates that care must be exercised before using the law of corresponding states to extrapolate calculations.
II. SIMULATION METHODSSimulationsare performed with a standard Metropolis Monte-Carlo algorithm (MC-NVT) for a system of N = 2000 particles of mass m at temperature T in a volume V = L x L y L z where L x , L y , and L z are the dimensions of the rectangular simulation cell. Periodic boundary conditions are used in all directions. Particles interact via the Lennard-Jones potential,
5×10 5
5Monte-Carlo cycles (one cycle = N updates) after which the positions of the particles are saved every 20 cycles during 5 × 10 5 cycles. This ensemble of 2.5 × 10 4 configurations is used to compute the density profile and the surface tension by the Bennett's method.
FIG. 1 :
1(Color online)The surface tension as a function of temperature for two different cutoffs.
online)Comparison of our simulation data to a DFT model[4]. The panel on the left shows the results of the theory using an empirical equation of state while the results on the right were obtained using thermodynamic perturbation theory.The lines are ordered from the smallest cutoff (lowest lines) to the largest cutoff (highest lines) and were calculated for r * c = 2.5, 3, 4, 6, ∞.The data is represented by circles(2.5), squares(3), diamonds(4), filled diamonds (4 -larger system) and triangles(6).point. In this section, we test this hypothesis by applying it to the surface tension. The first step is therefore to determine the critical temperatures and densities of the various truncated potentials. Since the theoretical calculations require as input an equation of state, the critical points are easily determined. To determine them from the simulations, we took five independent averages over 5000 configurations and fitted the density profiles in each case to a hyperbolic tangent and then from these extract the coexisting vapor and liquid densities at each temperature. The five values obtained at each temperature were averaged and the variance used as an estimate of the errors in the values. The critical temperature was then estimated by using the lowest order renormalization group (RG)
Figure 5 ,FIG
5does not give a single curve. While the data for the shorter cutoffs, r * c = 2.. 3: (Color online) The fit of the difference in liquid and vapor densities, as determined from simulation (symbols), to the RG functional form (lines). The data are shown as circles (R c = 6.0), open squares (R c = 4.0, larger cell), filled squares (R c = 4.0, smaller cell), diamonds (R c = 3.0) and triangles (R c = 2.5).
a function of distance from the critical temperature. The left panel includes the theoretical curves, shown as full line (R * c = ∞), dotted line (R * c = 8), dashed line (R * c = 6), dash-dot line (R * c = 4), dash-dot-dot line (R * c = 3), and line+circles (R * c = 2.5), and the simulation data, shown as circles (R * c = 6), filled squares (R * c = 4, 2000 atoms), open squares (R * c = 4, 8000 atoms), diamonds (R * c = 3) and triangles (R * c = 2.5). The right hand panel shows only the data from simulation as well as the estimated error. In both cases, the thin line is a best fit to all of the data (theory and simulation) of the form γ * = γ * 0 (1 − T /T c ) with γ * temperature as calculated using the perturbative equation of state displayed as a full line (R * c = ∞), dashed line (R * c = 6), dash-dot line (R * c = 4), dash-dot-dot line (R * c = 3), and line+circles (R * c = 2.5). FL, USA, 2001).
TABLE I :
ISurface tension determined from simulation as a function of temperature for different cutoffs.Temperature
r *
c = 2.5 a
r *
c = 3 a
r *
c = 4 a
r *
c = 4 b
r *
c = 6 b
0.70
0.584 (27)
0.770 (21)
0.964(46)
0.914(30)
1.070(13)
0.72
0.561 (26)
0.726 (25)
0.899(22)
1.034(7)
0.75
0.511 (20)
0.698 (28)
0.825(31)
0.959(8)
0.80
0.421 (19)
0.608 (16)
0.748(25)
0.736 (32)
0.847(13)
0.85
0.315 (13)
0.480 (12)
0.633(24)
0.90
0.228 (13)
0.384 (15)
0.542(16)
0.484(33)
0.638(8)
0.95
0.181 (11)
0.314 (12)
0.443(22)
1.0
0.106 (7)
0.234 (8)
0.348(11)
0.313 (59)
0.438(15)
1.05
0.156 (11)
0.258(23)
1.1
0.111 (16)
0.200 (12)
0.261(13)
1.15
0.074 (10)
0.108 (14)
1.2
0.054 (6)
0.067 (15)
0.063 (19)
0.105(5)
a using approximately 2000 atoms.
b using approximately 8000 atoms.
TABLE II :
IIThe critical points for the LJ potential truncated at different values. The theoretical values were determined using the empirical JZG equation of state[13] (JZG) and the first order Barker-Henderson perturbation theory (BH). a using approximately 2000 atoms. b using approximately 8000 atoms. c From ref.[20].R *
c
MC
Theory -JZG
Theory -BH
T *
c
ρ *
c
T *
c
ρ *
c
T *
c
ρ *
c
2.5
1.10 (1) a
0.31 (9) a
1.04
0.26
1.18
0.325
3.0
1.18 (1) a
0.31 (7) a
1.15
0.28
1.27
0.342
4.0
1.26 (1) a
0.31 (5) a
1.25
0.32
1.34
0.341
4.0
1.25 (2) b
0.30 (6) b
-
-
-
-
6.0
1.30 (2) b
0.32 (9) b
1.29
0.35
1.38
0.341
∞
1.31 c
0.317 c
1.311
0.351
1.40
0.312
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. J Pérez-Pellitero, P Ungerer, G Orkoulas, A D Mackie, J. Chem. Phys. 12554515J. Pérez-Pellitero, P. Ungerer, G. Orkoulas, and A. D. Mackie, J. Chem. Phys. 125, 054515 (2006).
|
[] |
[
"Galactic seismology: the evolving \"phase spiral\" after the Sagittarius dwarf impact",
"Galactic seismology: the evolving \"phase spiral\" after the Sagittarius dwarf impact"
] |
[
"Joss Bland-Hawthorn \nSydney Institute for Astronomy\nSchool of Physics\nThe University of Sydney\nA28, 2006NSWAustralia\n\nCenter of Excellence for All Sky Astrophysics in Three Dimensions (ASTRO-3D)\nAustralia\n",
"Thor Tepper-García \nSydney Institute for Astronomy\nSchool of Physics\nThe University of Sydney\nA28, 2006NSWAustralia\n\nCenter of Excellence for All Sky Astrophysics in Three Dimensions (ASTRO-3D)\nAustralia\n\nCentre for Integrated Sustainability Analysis\nSchool of Physics\nThe University of Sydney\n2006NSWAustralia\n"
] |
[
"Sydney Institute for Astronomy\nSchool of Physics\nThe University of Sydney\nA28, 2006NSWAustralia",
"Center of Excellence for All Sky Astrophysics in Three Dimensions (ASTRO-3D)\nAustralia",
"Sydney Institute for Astronomy\nSchool of Physics\nThe University of Sydney\nA28, 2006NSWAustralia",
"Center of Excellence for All Sky Astrophysics in Three Dimensions (ASTRO-3D)\nAustralia",
"Centre for Integrated Sustainability Analysis\nSchool of Physics\nThe University of Sydney\n2006NSWAustralia"
] |
[
"MNRAS"
] |
One of the most important discoveries to emerge from the ESA Gaia astrometric survey is the "phase spiral" pattern detected in the z − V z plane throughout the solar neighbourhood. In Galactic cylindrical coordinates (R, φ, z), individual stars have velocities (V R , V φ , V z ) and oscillation frequencies (Ω R , Ω φ , Ω z ) = (κ, Ω, ν). The phase spiral is most evident when each point of the z − V z phase plane is represented by either V R or V φ , averaged over the volume, indicating that the disturbance has locked the vertical oscillations with the in-plane epicyclic motions. We revisit Binney & Schönrich (2018) who use an analytic model of a disc-crossing satellite (2 × 10 10 M ) to explain the phase spiral and its coupled behaviour in V R and V φ . With the aid of the agama (initialization) and ramses (simulation) codes, we explore a high-resolution N-body model that is true to the analytic calculation. We find that the impulse triggers a superposition of two distinct bisymmetric (m = 2) modes − a density wave and a bending wave − that wrap up at different rates. Stars in the faster density wave wrap up with time T according to φ D (R, T ) = (Ω D (R) + Ω o ) T where φ D describes the spiral pattern and Ω D = Ω(R) − κ(R)/2. While the pattern speed Ω o is small, it is non-zero indicating that this is a dynamic density wave. The slower bending wave wraps up according to Ω B ≈ Ω D /2 producing a corrugated wave, a phenomenon now well established in the Galaxy. The bunching effect of the density wave triggers the "phase spiral" as it rolls up and down on the bending wave ("rollercoaster" model). The phase spiral emerges slowly about 380 Myr after impact, and symbiotically in V R and V φ . It appears to be a long-lived, disc-wide phenomenon that continues to evolve over the entire 1 Gyr simulation, becoming increasingly wound up with the passage of time.
|
10.1093/mnras/stab704
|
[
"https://arxiv.org/pdf/2009.02434v1.pdf"
] | 221,516,573 |
2009.02434
|
80c4e01d695ae91742606507c4c17a8a9b37ef79
|
Galactic seismology: the evolving "phase spiral" after the Sagittarius dwarf impact
2020
Joss Bland-Hawthorn
Sydney Institute for Astronomy
School of Physics
The University of Sydney
A28, 2006NSWAustralia
Center of Excellence for All Sky Astrophysics in Three Dimensions (ASTRO-3D)
Australia
Thor Tepper-García
Sydney Institute for Astronomy
School of Physics
The University of Sydney
A28, 2006NSWAustralia
Center of Excellence for All Sky Astrophysics in Three Dimensions (ASTRO-3D)
Australia
Centre for Integrated Sustainability Analysis
School of Physics
The University of Sydney
2006NSWAustralia
Galactic seismology: the evolving "phase spiral" after the Sagittarius dwarf impact
MNRAS
0002020Preprint 8 September 2020 Compiled using MNRAS L A T E X style file v3.0Surveys -the Galaxy -stars: dynamics -stars: kinematics -methods: N-body simulations -methods: analytic
One of the most important discoveries to emerge from the ESA Gaia astrometric survey is the "phase spiral" pattern detected in the z − V z plane throughout the solar neighbourhood. In Galactic cylindrical coordinates (R, φ, z), individual stars have velocities (V R , V φ , V z ) and oscillation frequencies (Ω R , Ω φ , Ω z ) = (κ, Ω, ν). The phase spiral is most evident when each point of the z − V z phase plane is represented by either V R or V φ , averaged over the volume, indicating that the disturbance has locked the vertical oscillations with the in-plane epicyclic motions. We revisit Binney & Schönrich (2018) who use an analytic model of a disc-crossing satellite (2 × 10 10 M ) to explain the phase spiral and its coupled behaviour in V R and V φ . With the aid of the agama (initialization) and ramses (simulation) codes, we explore a high-resolution N-body model that is true to the analytic calculation. We find that the impulse triggers a superposition of two distinct bisymmetric (m = 2) modes − a density wave and a bending wave − that wrap up at different rates. Stars in the faster density wave wrap up with time T according to φ D (R, T ) = (Ω D (R) + Ω o ) T where φ D describes the spiral pattern and Ω D = Ω(R) − κ(R)/2. While the pattern speed Ω o is small, it is non-zero indicating that this is a dynamic density wave. The slower bending wave wraps up according to Ω B ≈ Ω D /2 producing a corrugated wave, a phenomenon now well established in the Galaxy. The bunching effect of the density wave triggers the "phase spiral" as it rolls up and down on the bending wave ("rollercoaster" model). The phase spiral emerges slowly about 380 Myr after impact, and symbiotically in V R and V φ . It appears to be a long-lived, disc-wide phenomenon that continues to evolve over the entire 1 Gyr simulation, becoming increasingly wound up with the passage of time.
INTRODUCTION
In April 2018, the ESA Gaia astrometric mission (Perryman et al. 2001;Prusti et al. 2016;Soubiran et al. 2018) presented its second data release (DR2). These observations have revolutionized both stellar and Galactic studies (DR2: Gaia Collaboration et al. 2018). After only two years of observations, the remarkable precision of Gaia's measured stellar parameters has led to new discoveries and new fields of study. These data are being complemented inter alia by large spectroscopic surveys -RAVE (Steinmetz et al. 2006), APOGEE (Majewski et al. 2017), Gaia-ESO (Gilmore et al. 2012), LAMOST (Deng et al. 2012)
and GALAH (De Silva
Contact e-mail: [email protected] et al. 2015). In all respects, this is the golden age of Galactic archaeology (Freeman & Bland-Hawthorn 2002).
We highlight the remarkable discovery of an unexpected phase-space signal in the local stellar disc by the Gaia team (Antoja et al. 2018). In a volume element defined by (∆R, R ∆φ, ∆z) = (±0.1, ±0.1, ±1) kpc 3 centred on the Sun, Antoja et al. (2018) detect a coherent spiral pattern in the phase plane defined by z and V z . This phenomenon 1 is in-dicative of a system that is settling from a mildly disturbed state to a stationary configuration through the process of phase mixing (Lynden-Bell 1967).
Since its discovery two years ago, up to a dozen independent research papers seek to explain or shed light on the phase-spiral phenomenon. Explanations to date invoke three different mechanisms: (i) partial coherence imposed by the recent dissolution of several star clusters (Candlish et al. 2014;Michtchenko et al. 2019); (ii) vertical oscillations excited by the strong buckling of a stellar bar (Khoperskov et al. 2019); and (iii) vertical and in-plane oscillations induced by a massive disc-crossing perturber (Binney & Schönrich 2018;Darling & Widrow 2019a). The first two mechanisms are of internal origin; the third external mechanism has received the majority of attention to date, not least because the Sgr dwarf galaxy is observed to be undergoing disruption as it crosses the disc. This is the focus of our attention because we consider it to be the most likely interpretation for the phase-spiral phenomenon (Laporte et al. 2019;Bland-Hawthorn et al. 2019). The effect is now observed over too large a surface area (see below) to be a coherent phenomenon arising from dissolving star clusters. Moreover, the extremely tight alignment of the plane of the stellar bar with the inner disc plane (Bland-Hawthorn & Gerhard 2016) imposes strong constraints on the idea of a buckling bar.
Three independent observing campaigns confirm the phase-space pattern and study it in more detail. With the aid of the GALAH survey, Bland-Hawthorn et al. (2019, hereafter B19) dissected the phase spiral as a function of orbit actions, stellar ages and stellar abundances. They were able to detect the pattern over a larger radial range (7 < R < 9 kpc) and azimuthal angle (δφ ±15 • ). The LAMOST survey provides some evidence for a changing phase-spiral pattern over the radial range 6 < R < 12 kpc, as predicted in actionbased, analytic models (e.g. B19,Fig. 20) and observed in the highest contrast numerical simulation to date (Laporte et al. 2019, hereafter L19). Both Li & Shen (2020) and B19 present evidence that the phase spiral is mostly confined to cold orbits and not associated with the hotter, α-enhanced disc, further underscored by the 2-component disc model in B19. In these studies, the spiral pattern in phase space is found to be somewhat patchy and uneven. This is due in part to the well-documented kinematic (Widrow et al. 2012;Williams et al. 2013;Carlin et al. 2013) and density asymmetries (Yanny & Gardner 2013;Slater et al. 2014;Xu et al. 2015) when comparing the upper and lower sides of the Galactic plane. However, when the z − V z plane is encoded as the angular momentum about the z-axis, L z (action J φ ), Khanna et al. (2019) find that the phase-spiral pattern is smooth and varies systematically as a function of disc location. This suggests that the emergence of the phase spiral is somehow related to the angular momentum properties of the disc, in accord with the BS18 model.
Galactic seismology, 2 a subset of Galactic archaeology, 2019). The advantage of the more compact language becomes apparent when used as an adjective, e.g. phase-spiral evolution, phase-spiral dynamics, etc. 2 Widrow et al. (2012) coined the term 'galactoseismology' to describe the study of dynamical modes arising from disc perturba-is useful for exploring past interactions and structural properties of the Galaxy. Specifically, we seek a better understanding of how a differentially-rotating stellar disc responds to a powerful impulse and how this sets up the phase-spiral phenomenon. If a satisfactory model can be achieved, it must surely provide evidence that dark matter, which is dominant in the regions of interest, absorbs energy and momentum from moving massive bodies like ordinary matter. Then we would have conclusive evidence that dark matter comprises non-degenerate fermions rather than something altogether more exotic (q.v. Hui et al. 2017;Milgrom & Sanders 2008).
In what follows, we start by reviewing the Binney & Schönrich (2018, hereafter BS18) model (Sec. 2), which provides the impetus for our new work, before discussing the overall strategy (Sec. 3). The main goal is to design and carry out a numerical simulation that bridges between the simplicity of an analytic model (BS18) and the complexity of a realistic numerical simulation (e.g. L19). In Sec. 4, we discuss the adopted parameters of the Galactic model and the intruder. The Galactic model is set up carefully with specific initial conditions that guarantee long-term stability in an N-body simulation. In Sec. 5, with reference to web links to our simulations, we describe our analysis and the new results. In the final discussion (Sec. 6), we show that the new simulations are a major improvement over earlier work (L19, B19), although many questions remain.
BINNEY & SCHÖNRICH (2018) MODEL
BS18 develop a purely analytic (toy) model to investigate the response of the Galactic stellar disc to the perturbation induced by an external massive object. They employ actionbased modelling defined by distribution functions f (J) that are analytic functions of the action integrals J = (J R , J φ , J z ). With recourse to the Action-based GAlaxy Model Architecture (agama) package (Vasiliev 2019), they produce a highly self-consistent, multi-component model of the Galaxy. Their vertical density profile through the solar neighbourhood is an excellent match to the observed two-component disc profile (Gilmore & Reid 1983). This is important if the model is to match the vertical actions (J z ) and frequencies (Ω z ≡ ν) given the anharmonic nature of the vertical gravitational potential. They choose phase-space coordinates for a million stars in the solar neighbourhood and backwards integrate them to a time when the perturber first appears. The intruder is moving with a downward velocity of v P ≈ 300 km s −1 , has a mass of M P = 2 × 10 10 M and impacts the disc at R P = 18 kpc along the Sun's radius vector. They are then able to compute the acceleration imposed by the intruder on the particles relative to the unperturbed acceleration.
BS18 observe that the largest off-planar perturbation is expected at φ − φ o ∼ ±π/2 because these stars are either moving towards (φ − φ o < 0) or away (φ − φ o > 0) from the perturber as it transits the disc. (Here, φ o = 0 because the intruder transits along the Sun's radius vector.) Thus, stars approaching φ = φ o feel a net upward pull, and stars receding from φ = φ o feel a net downward pull. This action sets up a strong low-order bending mode across the disc. As tions, both internal and external. This term is used interchangeably with Galactic seismology when it relates to the Milky Way. the perturber transits the disc, stars near the crossing point experience a strong in-plane tug towards the perturber's trajectory, which binds the in-plane and vertical actions.
With the aid of Figs. 1-3, we illustrate a key insight from the BS18 model. The coordinate system of the z − V z plane is shown in Fig. 1 superposed on the surfaces of section for different stellar orbits confined to tori about the Galactic Centre. In Fig. 2, sloshing in the z − V z plane induced by the impact leads to a stellar overdensity that is offset with respect to the original stratification at a mean angle of θ z ≈ θ z,o (Fig. 2). Whereas before stars had constant values of J z that were independent of the conjugate angle θ z , the entire distribution now depends on θ z and the assigned J z associated with the new ellipse. Since for a disc potential, Ω z declines with √ J z (e.g. Candlish et al. 2014, BS18), a phase-spiral structure starts to develop. Crucially, BS18 note that the phase spiral is not apparent in stellar overdensity, although L19 detect a weak signal, in agreement with the original study (Antoja et al. 2018). Another mechanism is needed for the pattern to emerge so clearly in V φ and V R .
In Fig. 3, we illustrate how elliptic orbits from the inner and outer disc are able to enter the local volume. The vertical oscillation frequency Ω z declines smoothly with increasing radius because of the disc's exponential decline in the local surface density. Stars from the inner disc reach us at their aphelia, and these are characterised by lower V φ and higher Ω z ; the converse is true for stars coming in from the outer disc. The dislodged stars trace ellipses in the z−V z plane with a conjugate angle that grows with time t as θ z = Ω z t + θ z,o (Fig. 2). But Ω z is tied to V φ (and V R ) such that stars with lower V φ move ahead of stars with higher V φ . Thus, the phase spiral is defined by a change in V φ (and V R ) as we move clockwise around each ellipse in the z − V z plane.
MOTIVATION & STRATEGY
The motivation for our new work can be stated as follows. We seek to understand the phase-spiral phenomenon with the simplest possible model that is true to the BS18 toy model while incorporating the advantages of the N-body approach. First, our work demands a more rigorous treatment for setting up the initial conditions. We require the coldest possible synthetic disc that comes into equilibrium at the start of the simulation and does not develop significant substructure or change its properties over the course of many disc rotations. Darling & Widrow (2019b) demonstrate that the emergence of the phase spiral is a competition between phase mixing and the disc's self gravity. Consistent with our approach, a lower self gravity leads to a stronger signal (higher contrast with respect to the background population) although a higher self gravity appears to sustain the signature for longer.
Secondly, we require a better treatment of the impulse approximation by an N-body solver. In particular, the heavy point mass must be 'live' on its first and only approach, and during the transit, and thereafter it must not continue to disturb the disc. In our earlier work (B19), the disc response can be more erratic in the post-transit phase due to the sustained force applied by the intruder. While there is an in-built asymmetry to our approach, this is true of all Sgr-based orbit families because of dynamical friction and Figure 1. The natural coordinate system in the z − V z plane: the conjugate angle θ z is the angle from the vertical; the radial distance of a star from the centre of this plane is proportional to the action √ J z . These are overlaid onto the projected stellar density in the solar neighbourhood where the simulated distribution arises from the vertical dispersion σ z profile and the scale height of the disc (Sec. 4). The blue ellipses are surfaces of section for orbits with a fixed vertical action J z . The vertical disc frequency Ω z of the orbit determines the rate at which a star moves around the centre (z, V z ) = (0, 0). Figure 2. In the BS18 model, the impact causes the local stellar distribution to "slosh" in the z − V z plane with respect to the underlying large-scale potential. Each star now has a different vertical frequency Ω z compared to its position in the unperturbed distribution and starts to circulate out of equilibrium (phase mix) in the z − V z plane.
the perturber's mass loss along its orbit (Law et al. 2005;Nichols et al. 2014). The underlying mechanisms involved in the disc response to an extended, disrupting perturber are obfuscated by many overlapping processes (e.g. L19, B19).
In contrast, the BS18 toy model uses a theoretical description of the Galaxy that is analysed in terms of distribu- Figure 3. Stars from the inner and outer disc are able to reach the solar neighbourhood along elliptic orbits. We show an inner elliptic orbit (IEO) and an outer elliptic orbit (OEO); these precess about the Galactic Centre but remain confined to an annulus about the guiding radius (R g ) at their midpoint. In the BS18 model, the tangential velocity V φ of a star in the Local Volume is intimately associated with the vertical frequency Ω z of the star's orbit. Stars arriving from the inner disc typically have lower V φ but higher Ω z due to the higher disk surface density; the converse is true for stars arriving from the outer disc. Stars from the inner disc circulate faster in the z − V z plane ( Fig. 1) compared to stars from the outer disc. (The chosen colour scheme is consistent with BS18.) tion functions defined in action space, as discussed in Sec. 2. The latter model, while computationally efficient, lacks the realism of the actual system, most notably the complex recoil of both Sgr and the Milky Way to their mutual attraction and, importantly, the self-gravity and 'elasticity' of the dominant stellar disc. How the vertical and in-plane motions emerge and work together during the interaction are unclear (D'Onghia et al. 2016;Darling & Widrow 2019b). But by using a model that bridges the divide between analytic and N-body models, we seek to shed more light on this process.
If we are to compare the BS18 model meaningfully with an N-body simulation, the basic parameters of their set-up must be matched closely (Sec. 4.1). In our model, a point mass is used in place of an extended perturber to maintain the comparison and is placed on a hyperbolic orbit to minimize ongoing interactions with the disc. Aguilar & White (1985) note that in order to recreate the impulse approximation within an N-body simulation, the interval during which the mutual gravitational forces are significant is short compared to the crossing time within each system (see also Binney & Tremaine 2008, §8.2). The "impulsiveness" is further ensured (see Sec. A) by a perturber mass that is constant until the point of disc transit, and then declines exponentially in time such that the mass is negligible at the second disc crossing (Sec. 4.1). In the next section, we explore a new suite of models to simulate a clean impulse of a disc-crossing perturber.
In order to achieve the fidelity of the phase spiral in the BS18 models, these authors recommend that any N-body simulation use ∼ 10 8 disc particles, an order of magnitude larger than any matched, impulsive disc simulation to date. BS18 observe a synthetic phase spiral over an area in the z − V z plane that is comparable to Antoja et al. (2018), i.e. ±60 km s −1 and ±1 kpc, a fourfold improvement over L19 and B19. After following BS18's recommendation, we also achieve phase spiral patterns over the smaller region.
N-BODY SIMULATIONS
The model
We conduct a series of N-body simulations using the tried and tested ramses simulation package (version 3.0 last described by Teyssier 2002). The Galaxy is approximated by a three-component system consisting of a dark-matter (DM) host halo, a stellar bulge, and a cold stellar disc; a complete list of parameters is given in Tab. 1. The stellar disc is sampled with 5 × 10 7 particles, the largest impulsive model of this kind to date. The model Galaxy has properties that are consistent with their observed counterparts (q.v. Bland-Hawthorn & Gerhard 2016). Its total mass is M tot ≈ 1.45×10 12 M with a distribution that yields a nearly flat rotation curve out to R = 20 kpc; the tangential velocity remains bounded within 240 to 250 km s −1 (Fig. 4, top right). The density and integrated mass profiles of each component are also shown (top left), along with the surface density profile of the disc alone (bottom right).
The LAMOST and GALAH studies reveal that the phase spiral is mostly confined to stellar populations with the coldest orbits (B19, Li & Shen 2020). In order to enhance the contrast of the phase-spiral signature in the stellar component, we achieve one of the coldest, long-lived disc simulations to date (Sec. 4.1). The disc is stabilised by our choice of Toomre (1964)'s stability parameter
Q ≡ σ R κ 3.36 G Σ ,(1)
where κ and Σ are the epicyclic frequency and stellar surface density,that provides local axisymmetric stability if we ensure Q 1.3 throughout the disc. Our choice of Q and disc mass fraction (Sec. 4.3) ensures that no bar or spiral arms form in isolation over a time frame of 4 Gyr (Fujii et al. 2018, see Sec. 4.3 below). These complicating factors, that were not considered by BS18, are deferred to a follow-up paper. The stellar dispersion profiles σ R (R) and σ z (R) are presented in Fig. 4 (bottom left); these apply to the unperturbed Galaxy model in isolation, but they are identical to the corresponding properties of the interacting Galaxy at the start of the simulation.
In line with BS18, the Sagittarius dwarf galaxy (Sgr) is represented by a point-like object. The point mass is placed on a hyperbolic orbit that intersects the Galactic plane at ∼ 18 kpc, at which point Sgr is travelling at ∼ 330 km s −1 . These constraints are consistent with Sgr's trefoil orbit parameters at the last crossing (e.g. Tepper-García & Bland-Hawthorn 2018). The motivation for using a hyperbolic orbit is to limit the interaction between the Galactic disc and the point mass to a single impact (impulse approximation) and then trace the disc's evolution over a timespan of up to 1 Gyr. This extended time frame is in line with earlier work that shows the phase spiral may be long-lived (Darling & Widrow 2019b, B19). Table 1. Galaxy model parameters: In columns 1 and 2, we show the Galactic component and the intended functional forms; these are only approximate because they share the same gravitational potential. In column 3, the function variable is shown where (R, z) are cylindrical coordinates, and r is the spherical radius. The total mass, scale length and cut-off radius are indicated in columns 4, 5, 6 respectively. Column 7 is the number of collisionless particles used in the simulation. (Navarro et al. 1997;Hernquist 1990). The scale height of the stellar disc is z t ≈ 250 pc; the local instability parameter is everywhere Q 1.3 (Toomre 1964). . The sub-panel shows the relative change (in percent) in the total rotation curve between these time frames. The gray-shaded area indicates the rms value around the mean. (Bottom left) The radial σ R (solid) and vertical σ z (dot-dashed) velocity dispersion profiles of the cold, synthetic disc extracted from the N-body simulation. In comparison to the Galaxy (Bland-Hawthorn & Gerhard 2016), σ R ≈ 35 km s −1 (+) and σ z ≈ 25 km s −1 (×) at the solar circle (R = 8.2 kpc), as indicated. The sub-panel shows the mean relative change (in percent) in each of the velocity dispersions between these time frames (horizontal lines: ∆(σ R ) ≈ 13%; ∆(σ z ) ≈ 3%) and the corresponding rms value around the mean (gray-shaded area). (Bottom right) The surface density of the disc: note that the red line is identical to the red curve in the top left panel. The sub-panel displays the relative change in the surface density between these two time frames. The gray-shaded area indicates the rms value around the mean. The adopted orbital parameters given above imply that the perturber is moving on a bound orbit, and must therefore cross the disc more than once in the allotted time frame. To circumvent this problem, we impose on Sgr the following mass evolution:
M P (t) = M P (0) × 1 t < t • exp [−(t − t • )/τ s ] t t •
where M P (0) = 2 × 10 10 M and τ s = 30 Myr, roughly a disc crossing time. The first disc crossing happens at ∼ 100 Myr so we set t • = 150 Myr. Thus the intruder mass remains constant until 40−50 Myr after the first disc transit, and declines by a factor of ∼ 10 6 at the time of the second disc crossing (t ≈ 450 Myr). The perturber's post-transit impact on the disc is entirely negligible. We experimented with these parameters and found that the impulse approximation is robust to our assumptions (Sec. A).
Initial conditions and evolution
In this work, we pay particular attention to the initial conditions (ICs) that specify our N-body models at the outset. A recurrent problem with N-body simulations to date, with rare exceptions (Widrow & Dubinski 2005), is that the initially specified, multi-component system evolves to a new configuration before long-term stability is achieved; the simulator must then accept a model that is less than ideal and not what was specified. This has been a longstanding problem with numerical simulations that the agama code originally set out to resolve (Vasiliev 2019). We exploit agama to generate the positions and velocities for all components of our model (Table 1) in a selfconsistent fashion. This guarantees that the ICs are virtually stationary from the outset 3 .
The ICs of the perturber are calculated using a backwards integration scheme in the two-body approximation where the Galaxy is described by an extended rigid body (for details, see Tepper-García et al. 2020). This is subject to the condition that the point mass crosses the Galactic plane after 95 Myr at a Galactocentric distance of about 18 kpc along the negative x-axis, with its velocity vector perfectly perpendicular to the Galactic plane and a magnitude of 330 km s −1 . The initial position and velocity of the perturber with respect to the Galactic Centre thus obtained are r P ≈ (−11.1, 0, 28.6) and v P = (−145.4, 0, −220.2) respectively.
The compound (Galaxy + point mass) N-body ICs are evolved with the ramses code (Teyssier 2002), which incorporates adaptive mesh refinement (AMR). The system is placed into a cubic box spanning 600 kpc on a side. The AMR grid is maximally refined up to level 14, implying a limiting spatial resolution of 600 kpc/2 14 ≈ 36 pc. The total simulation time is 1 Gyr for the phase-spiral enactments. We also explored a more expensive simulation with 2 pc resolution, but there were no perceptible benefits for our phase-spiral study, so we did not proceed further. 4 Due to the presence of the simulation mesh, the force exerted by (and onto) any particle is necessarily softened, at least by a length equivalent to the side of cubic volume element (cell) equal to the limiting spatial resolution. We choose to soften the force exerted by the point-like perturber using a length equal to two such elements ( s ≈ 72 pc); for all other particles, s is half this value. In addition, the force between the perturber and all the other particles is calculated by a direct N-body solver, rather than the particle-mesh (PM) method used otherwise.
From its starting point above the Galactic plane, the perturber reaches the stellar disc travelling at about 350 km s −1 after 95 Myr from an initial galactocentric distance of about 40 kpc above the disc. We provide a link to the movie in the footnote. 5 It crosses the disc at R ≈ 18 kpc (i.e. (x, y) ≈ (−18, 0) kpc), and transits below the Galactic plane to a vertical distance of about z ≈ 35 kpc some 250 Myr after the disc crossing, then falls back towards the Galaxy − see Fig. 5. At t • = 150 Myr, i.e. shortly after it crosses the Galactic plane, we artificially decrease its mass over an e-folding timescale, τ s = 30 Myr (Sec. 4.1), such that by the time the intruder crosses the disc again about 450 Myr after the first transit, its mass is entirely negligible.
Model stability
We have studied the long-term stability of our models with a series of expensive simulations for an isolated galaxy run over 4 Gyr and at different resolutions. We made this investment because cold discs are susceptible to the emergence and evolution of spurious substructure (e.g. clumps, bars, spiral arms) as the disc evolves (D'Onghia et al. 2013;Fujii et al. 2018). To underscore the importance of this test, the reader is encouraged to compare two disc simulations with N = 10 6 and N = 10 8 particles (see footnote 5). By T ∼ 1 Gyr, the lowresolution simulation excites complex bending modes in the outer disc that only get worse as time marches on, whereas no such effect is apparent in the high-resolution simulation. In the high-resolution simulation, the surface density profiles are reasonably constant over the full timespan (< 5 percent, rms), which amounts to about 20 disc rotations at the solar radius (Fig. 4, bottom right). The other panels show that deviations from the original rotation profiles are less than 1 percent at all radii.
We also examined the stellar dispersion profiles in both R and z. The vertical dispersion profile σ z (R) is essentially unchanged at all radii (< 5 percent), which shows that there is no disc heating induced by the coarseness of the finite resolution at any radius. In separate tests, we find that we can avoid internal disc heating when the number of disc particles exceeds about 10 million particles. The radial dispersion profile σ R (R) shows a slight excess (13 percent on average) over 4 Gyr; we deem this to be acceptable given the goals of the current work. From our perspective, the parametric profiles are essentially constant, such that the ultrathin disc retains the imposed ICs over many rotation periods, providing confirmation of the power of agama in setting up robust models.
ANALYSIS AND RESULTS
Early stage evolution
In analysing the simulations, our first goal is to understand just how the cold disc experiences the impulsive force from the disc-crossing intruder. Our second goal is to determine how efficiently the perturber's impulse is able to trigger a response in the cold disc. This must depend both on the infall velocity ( v P ) and the perturber mass (M P ), with lower velocities and higher masses triggering a stronger response. Earlier simulations (Darling & Widrow 2019c) employ perfect coupling by inserting the bending modes by hand. But we seek to understand how the phase-spiral signal builds over time as a diagnostic for age-dating the phenomenon.
First, we investigate how well the BS18 picture is reproduced in our simulations. Movies of the disc-perturber interaction are available at our website; see footnote 5. We stress that, at each timestep, all results are shown in the reference frame of the simulated galaxy's centre of mass where the disc's angular momentum vector is aligned with the positive z−axis. The perturber moves parallel to the +z axis towards the plane.
In Fig. 6, the response of the disc is presented in the x − y plane at T = 67 Myr, T = 95 Myr (time of transit), T = 143 Myr and T = 190 Myr. Each time step shows three panels: (1) the projected stellar density; (2) the projected mean z-height z of the stars; (3) the projected mean zmotions V z of the stars. In the x − z projection, even before the disc crossing (T = 67 Myr), stars move in z towards the perturber along the impact trajectory. These are mostly stars that were moving away (φ − φ o > 0) from the impact point (φ o = 0) in the disc plane (cf. BS18, Fig. 1) after transit (T > 95 Myr).
Conversely, after the perturber has transited, stars are observed to pursue the perturber along its impact trajectory. These are mostly stars that were moving towards (φ−φ o < 0) the impact point in the disc plane at the time of transit. This picture is in good agreement with the BS18 model, although it misses an important ingredient, as explained below.
The early phase of the disc's evolution is broadly consistent with the BS18 picture summarised in Sec. 2, except they do not consider the influence of the shearing disc due to differential rotation. We believe this to be crucial to how the phase spiral emerges. At T = 143 − 190 Myr, the first signs of a density wave become evident in the projected density map. Moreover, the impulse sets up a strong, low-order (m = 1) bending mode across the disc seen initially as the sine-wave warp of the outer disc in the z and V z panels. Of particular interest is the transition from m = 1 in the outer disc to an m = 2 bending mode inside R ≈ 10 kpc that wraps up with the differential rotation. (We explore this behaviour with a model in a later section.)
The early stages set up a sequence of events that unfold in the remaining time steps for which we show representative images sampled at intervals of 95 Myr (Fig. 7). The perturber triggers both a density wave (left panels) and a bending wave (middle and right panels) that wrap up as time passes. The panels are encoded in the same way as Fig. 6 but now zoomed to show the inner disc (R < 11 kpc) more clearly. Note that the vertical motion V z is out of phase with the vertical displacement z . This is expected for a corrugated wave that oscillates up and down as it wraps up. Observe also that the z displacement increases with radius as expected (B19, Figs. 20 and 21).
The twelve circles shown in each of the panels of Fig. 7 are the sampled volumes in our later discussion of phasespiral evolution. Note three things: (i) our solar neighbourhood is "volume 1" (x = −8.2 kpc) and the intruder crossing point is at x = −18.0 kpc; (ii) at T = 95 Myr, these lie along the same radius vector at disc transit; (iii) at T ≈ 200 Myr, volume 1 has moved to the other side of the disc since the impact, consistent with the orbital period at the Sun's radius (T ≈ 210 Myr), while the crossing point (T C ≈ 460 Myr orbital period) has only moved by about 80 • bringing it in line with volume 10. After a full orbit of volume 1 (T = 305 Myr in our time frame), the crossing point now lies along the radius vector with volume 7.
Middle to late stage evolution
Kinematic density wave
For eighty years, we have known that galaxy interactions excite trailing spiral arms in disc galaxies (Holmberg 1941). Thereafter, Lindblad discovered that the force fields of disc galaxies preserve twofold symmetric (bisymmetric) structures against dissolution from the differential rotation (Lindblad 1956(Lindblad , 1960. In an isolated galaxy, stellar orbits are mostly elliptic and precess about the centre. An external perturber can organize the elliptic orbits into a coordinated precession (2-arm spiral pattern) that depends on the local angular frequency Ω(R) and radial frequency κ(R), subject to the shearing disc's lowest order resonances, Ω(R) ± κ(R)/2 (Kalnajs 1973). These features are observed in numerical simulations to be associated with density waves that wind up slowly, rather than material arms that wind up rapidly with the disc's differential rotation (Sundelius et al. 1987;Oh et al. 2008;Dobbs & Baba 2014).
The spiral pattern is the natural response of a cold, differentially rotating disc subjected to a strong impulsive force (Binney & Tremaine 2008, §6.2). For an m = 2 density wave with trailing arms (Ω > κ/2), as seen from the positive z axis, the spiral pattern φ D (R) (R in units of kpc) winds up Figure 7. Early to middle stage evolution of the disc seen from the z > 0 axis; the time sequence continues in the next figure. The first panel is the projected stellar density field in the x − y plane. The second and third panels are the average vertical distance z and the average vertical velocity V z respectively in the x − y plane. The first panel shows the m = 2 spiral density wave; the second and third panels reveal the m = 2 bending mode. The numbered circles are 4 kpc in diameter and spread along the Solar Circle (R = 8.2 kpc). These relate to our discussion of the phase spiral and show the rotation of the disc.
following φ D (R, T ) = Ω D (R) + Ω o 1 km s −1 kpc −1 T 978 Myr + φ D,o(2)
where
Ω D (R) = Ω(R) − 1 2 κ(R)(3)
and Ω o is included in the fitting process to account for any pattern speed due to figure rotation. (φ D,o is an arbitrary offset.) All frequencies are in units of km s −1 kpc −1 (equivalent to (978 Myr) −1 ), T is in units of Myr, and angles are in radians.
If Ω o = 0, this form describes a "kinematic" density wave because it depends only on the kinematics of elliptic orbit crowding at aphelia. In this instance, the spiral pattern is not a propagating wave mode, it has no group velocity, and the complicating effects of self-gravity and swing amplification are negligible. For a flat rotation curve, a kinematic density wave wraps up at about 30% of the angular rate of a material arm carried by the shearing disc (Ω D = Ω).
Interestingly, this simple form is not exactly what is observed in simulations of disc mergers and perturbed discs to date (Struck et al. 2011). For example, the "grand design" spiral in M51 is a common target for modellers. Numerical simulations of the 2-arm spiral pattern are found to wind up at a higher rate than indicated by Eq. 2, with a non-linear dependence on radius (Oh et al. 2008;Dobbs et al. 2010;Salo & Laurikainen 2000).
So what do we find? In the last section, we saw how the density wave developed in the projected stellar density map (Fig. 7). For the wave to be kinematic, it must simply wind up with no additional contribution from, say, figure rotation with an associated pattern speed. In Fig. 9, the model described by Eq. 2 is overlaid on the density map where both the arm (blue: φ D ) and counter arm (red: φ D + π) are shown. The parameters Ω(R) and κ(R) are measured from the isolated disc simulation and presented in Fig. 10 (left panel). After determining Ω = V φ /R, we calculate
κ 2 (R) = ∂ 2 Φ eff ∂R 2 z=0 = R dΩ 2 dR + 4Ω 2 z=0(4)
where Φ(R) eff is the effective gravitational potential taken from the simulation, and the subscript z = 0 indicates that all measurements are made in the plane. In the early stages, at the time of the disc transit, the spiral pattern is highly asymmetric, but the pattern clearly settles down after T = 190 Myr and starts to exhibit 2-fold symmetry. After T = 450 Myr, the Lindblad-Kalnajs pattern (Fig. 9) reveals itself and tracks the evolution of the spiral pattern for the remainder of the simulation. Once the functional form of the pattern is fixed, there are no free parameters for any of the other time steps shown. But this statement requires two qualifications: (i) The time variable T has an arbitrary offset T that determines the degree of winding when the spiral pattern first appears. An initial line of points is assumed to lie along y > 0 and x = 0; the points are then advanced in time to match the spiral pattern at some epoch (Binney & Tremaine 2008). For our model to match, we set T = 550 Myr. (ii) Intriguingly, we find that the spiral pattern does have some figure rotation. We measure this to be π/24 rad every 100 Myr once the density wave has settled down from the initial impulse. Using the conversion factor above, the rigid pattern speed is equivalent to Ω o = 1.3 km s −1 kpc −1 . While Ω o is very slow, it is non-zero and indicates that the spiral arms are propagating as a dynamical wave mode, rather than a kinematic anomaly (cf. Lin & Shu 1966). Myr after the impulse, the non-symmetric spiral pattern is evolving towards 2-fold symmetry. After t = 300 Myr, the bisymmetry is revealed and the correspondence is very good, although a vestigial non-bisymmetry exists at all time steps. The pattern is winding up as φ(R, t) ∝ (Ω(R) − 1 2 κ(R)) t. This is the natural response of a cold disc perturbed by a strong impulse. Figure 10. (Left) The angular frequency (Ω) and the planar Lindblad resonance (Ω − κ/2) as a function of radius R measured from the isolated disc simulation in the mid-plane (z = 0). (Right) The angular frequency and vertical Lindblad resonance (Ω − ν/2) measured from the same simulation in the mid-plane (z = 0). Note that Ω (solid curve) is identical in both panels.
In summary, the evolution of the x − y spiral has two components: (a) a wind-up rate given by the simple Lindblad-Kalnajs formula (eq. 3); (b) an additional slow figure rotation Ω o that rotates the fixed pattern in a prograde (anticlockwise) direction as one moves forward in time. The combined effect is much slower than the wind-up rate of material arms.
As far as we can establish, the most basic form for the spiral evolution described in Eq. 2 has not strictly been observed before in simulations, certainly not over the long time frames considered here. This may reflect the high level of self-consistency made possible by the agama initialisation of our cold disc simulations. Furthermore, it appears to be a travelling density wave − the 'Lindblad-Kalnajs' propagating wave mode.
Kinematic bending wave
In addition to in-plane spiral density waves, corrugated waves are now well established in the Galaxy (Sec. 1). How these work together, if at all, is entirely unclear. A bending wave is observed in our new simulation, and appears to wrap up like the density wave discussed in the last section. They appear to share the same common origin, but are these a superposition that propagate without interference from the other? Like spiral density waves, bending modes may be transient on a timescale of a few rotation periods at the Sun's radius (Hunter & Toomre 1969a;Sparke 1984;Binney et al. 1998).
The progress that has been made in understanding spiral density waves is restricted to 2D razor-thin discs. Just how this action extends to 3D discs is largely unknown (Fouvry et al. 2017). It is clear that the fundamentally in-plane mode must involve V z in addition to V φ because stars will be drawn down to regions of high density (Masset & Tagger 1997). These motions will remain conjectural until the theory of spiral structure has been extended from razor-thin discs, in which vertical motion is impossible, to discs of finite thickness, a very difficult proposition (Binney & Tremaine 2008).
The available formalism relating to the second kind of mode, the corrugation or bending wave, is even more primitive than the current theory of spiral structure because it involves neglecting epicyclic oscillations in addition to taking the disc to be razor thin (Hunter & Toomre 1969b). Hence we really have very little idea what a proper theory of corrugation waves would look like. It is plausible that the V φ motions are coupled with V z motions because warps (that can be considered as concentric rings) arise from torques exerted by one ring on another.
Both wave phenomena are defined by natural oscillation frequencies and are described by dispersion relations that look similar, although the dependence on the disc self gravity has the opposite sign (Binney & Tremaine 2008). The vertical frequency has a similar form to the epicyclic frequency in Eq. 4, viz.
ν 2 (R) = ∂ 2 Φ eff ∂z 2 z=0(5)
The vertical height of any bending mode with parameter m is described by (e.g. Binney & Tremaine 2008)
z B (R, φ, T ) = z o (R) cos[m(φ − Ω B T )] cos νT(6)
The kinematic bending modes are subject to winding up much like the density waves. The pattern speed of the bending wave is
Ω B = Ω(R) ± ν(R)/m(7)
We now investigate the bending mode in our simulation because of its role in the phase-spiral phenomenon discussed below. First, we measure ν(R) directly from the isolated disc simulation using Eq. 5; the results are presented in Fig. 10. Our expectation is that the winding-up of the m = 2 bending model will lag behind the m = 2 spiral density wave. In a spherical gravitational potential with average density ρ s , κ = ν; for a flattened disc potential with density ρ d , ν 2 /κ 2 ≈ (3/2) (ρ d /ρ s ). For example, along the solar circle, ν/κ ≈ 2 (Binney & Tremaine 2008). Thus we expect Ω−ν/2 < Ω−κ/2 and this is indeed what we find when comparing both panels in Fig. 10. Fig. 7 presents a comparison of the projected stellar density, the average vertical height z and the average vertical velocity V z in the x − y plane. The slow wrapping up of the bending wave through the oscillatory behaviour in z and V z is clearly evident in the last two columns.
We now model these patterns using Eq. 6 and the measured properties of the disc (Fig. 10). The spiral patterns in z and V z are out of phase by π/2 consistent with their oscillatory motion. We present our toy model for V z in Fig. 11 to be compared to the last column in Fig. 7. The early time step is included to illustrate the important transition that was discussed in Sec. 5.1. The m = 1 integral sign warp is seen early on, but a stronger m = 2 mode develops across the inner disc by T = 190 Myr and dominates the later evolution. After this time, the outer disc behaviour is complex, but the inner disc settles down to a well organized m = 2 bending mode that wraps up slowly.
Our model for the m = 2 bending wave is not as trivial as the form used for the spiral density wave. The equivalent form would be Ω B = Ω(R) − ν(R)/2 but this is at odds with what we find. In Fig. 10, note that Ω(R) − ν(R)/2 ≈ 0 in the range 4 R 6 kpc. But the simulated disc continually winds up for all time over this radial range. Other values of m > 2 wrap up too rapidly. In its place, we use a slightly modified form, i.e.
Ω B = max[ Ω(R) − ν(R)/2, Ω(R)/6 ](8)
such that the measured trend is used in the inner disc, but beyond R > 3 kpc, the angular frequency transitions to Ω B = Ω/6 to ensure the 'drop out' near R = 5 kpc is never reached (red curve in Fig. 10). Our choice of this minimum threshold can be defended. The transient density wave model is defined by Ω D ≈ Ω/3; the bending mode must be slower and therefore we adopt Ω B ≈ Ω D /2 after some experimentation. This approximation tracks the observed wrapping up of the bending mode quite well. Because of our assumption, we are not able to detect reliably any fixed pattern speed by analogy with the density wave, and therefore we do not fit for it. In Fig. 11, the model for the evolving spiral density wave is superimposed on the bending wave in each time step. There is clearly a complex interaction between both wave modes as they wrap up at different rates. We return to this behaviour below as it is central to our explanation for the phase spiral phenomenon. Figure 11. A comparison of the evolution of the bending wave (colour image) with the wrapping up of the kinematic density wave (spiral lines) for R 10 kpc. The colour image represents the mean vertical velocity field v z from −6 km s −1 (blue) to +6 km s −1 (red). The interplay of the two distinct wave modes wrapping up at different rates is the origin of the phase spiral, as we discuss. Both are model fits to the disc simulations. At the first time step (T = 190 Myr), we include a composite model that is to be compared to the inset. This shows how the m = 1 bending mode (R > 8 kpc) relates to the inner m = 2 bending mode (R < 8 kpc).
Phase spiral evolution
We now describe the origin and evolution of the phase spiral as observed in our impulse-driven simulation. In Fig. 12, the z − V z plane is presented for 12 spherical volumes (4 kpc diameter) spread along the solar circle (Fig. 7). The volumes are numbered 1 to 12 in an anti-clockwise direction, which serves to identify the column in Fig. 12. After averaging over the phase-space volume, we form maps of V φ and V R . The numbered volumes are much larger than the original discovery volume (100 × 100 × 1000 pc).
Each row is a time step separated by 47.5 Myr intervals for the entire duration of the simulation (951 Myr), i.e. 21 time steps where the reference frame is the Rotational Standard of Rest (RSR). The simulated solar neighbourhood (volume 1) is in column 1. Our focus on the solar circle is important for several reasons: (i) the local volume retains the clearest signal in the Gaia radial velocity data; (ii) to illustrate the phase spiral evolution, we remove the Galactic rotation that can only be carried out at one radius if we are to avoid complex resampling of the simulation.
The solar circle at the time of the impact is shown in Region A (Fig. 12). As already seen (Fig. 6), the early stage evolution reveals that the disc is highly disturbed up to at least T = 380 Myr. At T = 238 Myr, we observe a thick, red outer band (Region B) from a 'feather' (in the language of L19), i.e. a part of the disc that separates and lifts away from the disc during the settling process in the outer disc. A fixed amplitude, bending wave propagates along the filament; this leads to an outer ring or annulus in phase space as illustrated in Fig. 13.
The phase spiral starts to emerge in both V φ and V R simultaneously around T = 476 Myr. The phenomenon comes and goes at all radii from R 3 kpc to the outer disc. For the first time, we detect the phase spiral over the same extent in the z − V z plane and at the same intrinsic resolution as the discovery paper; earlier work (e.g. L19, B19) had intrinsically lower resolution and a z−range twice as large compared to observations. The inversion of the relative z and V z extent as a function of radius is precisely as predicted in B19 (Fig. 20). But here the focus is on evolution along the solar circle, which is sufficient for our purposes.
In Fig. 12, diagonal bands from upper right to lower left are apparent in both mosaics. These track along the same line as the two sequences of yellow (left) and black (right) boxes. The simulated solar neighbourhood (volume 1) lies along the same radius vector as the impact site at T = 95 Myr. But this site increasingly lags behind volume Figure 12. Along each row, the ith volume in Fig. 7 corresponds to the ith column. Each column corresponds to a fixed volume moving with the rotation: the solar neighbourhood is in column 1; its antipode is in column 7. Each row corresponds to a time step separated by 47.5 Myr as indicated. Moving to the right along a row corresponds to moving counter-clockwise by 30 • per column along the solar circle, and the figure wraps around from the last to the first column on the left. The colour coding indicates the mean value of V Φ (left) and V R (right); the ranges are (230, 250) km s −1 and (-10,+10) km s −1 respectively. Regions A-E are discussed in the paper. The yellow (left) and black (right) boxes indicate the volumes that align with the original impact site, which lags further behind the solar neighbourhood as time passes. Higher resolution images are available at the website (footnote 5).
1 (period T = 210 Myr) as the disc rotates, but realigns at a later time. The boxes indicate which volumes at specific timesteps are precisely aligned with the radius vector to the original impact site.
The period at the impactor radius (R = 18 kpc) is about T C = 460 Myr, such that volume 1 realigns with the impact site roughly every T ,C = (T T C )/(T C − T ) ≈ 380 Myr. Interestingly, this is precisely the time delay for a phase spiral to emerge along the solar circle. The continued interaction of the solar neighbourhood with the region of the disc around the impact site is a symptom of the wrapping up of the bending mode and the density wave.
In Fig. 12, we observe that the phase spiral first emerges at T = 476 Myr. In Region C at T = 523 Myr, we see the phase spiral clearly in the antipode region (cols. 7-9), something that is not seen diametrically opposite in the solar neighbourhood (cols. 1-3). The subsequent evolution tracks along the diagonal bands, and so is tied to the original impact site. Even though the density wave and bending mode appear bisymmetric by nature, the impulsive event is intrinsically one-sided. Fig. 14 shows how difficult it is to associate the phase spiral with the observed properties of the disc in the x − y plane at any given epoch. The numbered z − V z diagrams around the outside are taken from row D (Fig. 12). The x − y plane at T = 618 Myr looks highly bisymmetric in projected stellar density, and in z and V z . But the phase spiral becomes stronger in both V R and V φ as one moves around the clock (volumes 1 to 11).
There is an inherent asymmetry in the strength of the disturbance, as we saw in Sec. 5.1, in addition to the underlying m = 1 mode. The phase spiral in each volume, to borrow from R.P. Feynman, is a sum over history. The disc crossing occurred in the distant past but the transit region (albeit stretched by the disc's shear) lies immediately outside of volumes 8-9 at this time (cf. Fig. 12). The imprint of the one-sided disturbance is preserved for at least 500 Myr! At later times, with respect to the solar circle, the phase spiral develops both in the upstream (against rotation) and downstream direction (with rotation) due to the disc's shear. Once triggered, an increasing number of panels in Fig. 12 show the phase spiral with the passage of time, and its fidelity also improves as more wraps emerge. By the end of the simulation, the phase spiral has taken hold of the entire solar circle (Row E). In this respect, we can use the ubiquity of the phase spiral to age-date the impulsive event, and even to locate the earlier crossing point in the outer disc. These Figure 13. Examples of how the z − V z plane can be filled by three distinct bending wave modes: a fast wave with constant amplitude, a composite wave with mixed amplitudes, and a standing wave. To the right, the idealized z − V z phase plane is shown. The set of wave modes operating within a volume determine how the z − V z plane is filled. Figure 14. A density-weighted z map at time step T = 618 Myr. In the central 11 × 11 kpc panel, the colour coding shows z , the mean height (kpc) of the stars above and below the plane. The vertical undulations arising from the bending mode are clearly seen. Individual spiral arms roll up and down as the bending mode moves underneath them. The pink-coloured inner arms passing through volumes 7 and 12 curve inwards avoiding the next volume; note how the bending mode under both inner arms crosses over to the counter arm. The outer collages for V φ (z, V z ) and V R (z, V z ) are taken from Fig. 12; the ranges are (230, 250) km s −1 and (-10,+10) km s −1 respectively. Figure 15. A comparison of the z − V z plane presented in the discovery paper (Antoja et al. 2018, top) with volumes 5 and 6 in our last time frame, T = 951 Myr (middle & bottom); we have preserved the colour schemes from the earlier paper. Our volumes are much larger than the Gaia volume (see text) but we match the extent in the z − V z and the range in values for the first time; the fine structure of the phase spiral is evident in both also. The simulation produces fewer wraps in V φ (left) than in V R (right), which may reflect the need for more disc particles and/or longer timespans in future simulations.
are themes that we anticipate will be addressed in future papers on Galactic seismology.
DISCUSSION
The Gaia discovery of the phase spiral was entirely unanticipated (Antoja et al. 2018). As described in Sec. 4.1, the most natural interpretation is incomplete phase mixing after a major disturbance of the disc. But the remarkable signature raises important questions: What is the precise mechanism that explains it? Is the signature fragile or robust? How widespread is it and can we learn from it?
On the balance of evidence presented, Galactic seismology, a subset of Galactic archaeology, has a rich future. But our work raises more questions than answers. In the closing comments of the last section, we make the case for follow-up studies. Within the broad remit of Galactic seismology, an important goal is to age-date the origin of the phase spiral. Using two distinct mechanisms, B19 and Khoperskov et al. (2019) show that once the corrugation is excited, it can last at least 2 Gyr. But repeated disc crossings from a massive perturber may wipe out the phase coherence imposed by the previous crossing (L19, B19).
Over a timeline as long as 1 Gyr, the phase-spiral signal appears to be quite fragile, coming into view in one time step (e.g. Fig. 12) and then either evolving in situ, migrating in azimuth (co-moving RSR frame), or fading altogether in the next time step. Although no panel in Fig. 12 shows a close match in both V φ and V R simultaneously, we find features that are ubiquitous and reminiscent of the discovery paper (see Fig. 15). Note here that the simulated panels show weak evidence of a second phase spiral emerging out of sync with the dominant one. This can arise from having two spiral arms moving through the same volume, which becomes increasingly likely as the density wave becomes highly wrapped at late times. There is a case for repeating the simulations with more particles allowing for smaller sample volumes.
The simulated phase spiral encoded by V R is acceptable, but the phase spiral encoded by V φ shows up to one less wrap than observed. Khanna et al. (2019) found that the Gaia phase spiral is particularly prominent and well behaved in L z , but we were unable to see any improvement in our simulated signal when encoded in the same way. Given the central role of orbital angular momentum in generating the phase spiral (Sec. 2), it is not clear why this should be. While our simulations achieve the recommended particle count of N ∼ 10 8 (Binney & Schönrich 2018), our test simulations with smaller intrinsic spatial resolution (Sec. 4) did not alter the outcome significantly. But more particles would allow us to better match the Gaia RVS volume.
There is one important consequence of our "rollercoaster" model that has been glossed over. The interaction of density waves and bending modes implies that the younger stellar populations are those that are most likely to be caught up in the phase-spiral action. It is probable that older stars also contribute to the signal, but the feature is likely to be more enhanced for the younger stars. While there is existing evidence that this is true (B19, Li & Shen 2020), the dominant population of the phase spiral needs to be addressed more carefully (e.g. better stellar ages) in future studies (Sharma et al. 2016;Miglio et al. 2020).
This brings us full circle to the original prediction of the phase spiral (with two arms rather than the singular arm observed by Gaia) arising from dissolving star clusters (Candlish et al. 2014) that predated the Gaia era. Spiral arms in the Milky Way can be identified from star-forming regions (e.g. Lépine et al. 2001), as they can be in external galaxies, but this is no guarantee that young star clusters ( 100 Myr) are entirely restricted to the spiral arms. An inventory of almost 2000 star clusters confirmed by Gaia (Cantat-Gaudin et al. 2020, Fig. 8) shows that star clusters of all ages inhabit our neighbourhood within a few kiloparsecs, with the youngest star clusters confined to broad bands rather than ridge lines. Most clusters that form appear to dissolve within 100 Myr (Krumholz et al. 2019;Adamo et al. 2020) such that cluster dissolution can conceivably contribute to the innermost wraps of the phase spiral (cf. Li & Shen 2020). Cluster dissolution is unlikely account for the signature in toto, not least because the models to date predict a 2-arm phase spiral (Candlish et al. 2014), and that has not been observed so far.
Intriguingly, Michtchenko et al. (2019) identify parts of the local phase spiral as arising from several moving groups, including Pleaides, Hyades, Sirius (Ursa Major) and Coma Berenices. These are four of the closest clusters, all of which appear to be unbound. What is particularly striking is that their ages range from 100 to 600 Myr (Famaey et al. 2008) with the youth carrying the stronger signal, in agreement with B19. More work is needed to assess the detailed structure of the local corrugation (e.g. Friske & Schönrich 2019; Beane et al. 2019) and its relationship to the local spiral arms (e.g. Miyachi et al. 2019;Griv et al. 2020). How these relate to the local young clusters and star-forming regions also becomes important (Quillen et al. 2020;Cantat-Gaudin et al. 2020). It is likely that our cold-disc simulations, while very carefully constructed, are too simplistic. In the next phase, our simulations introduce a central bar, giant molecular clouds, and allow for internally-driven density waves, with the overarching goal of producing a more realistic framework for future study. This work will inform the nature of future studies, leading to better targetted surveys rather than all-sky surveys to test different non-equilibrium dynamical models.
To date, we have not considered the contribution of the cool gas. Another prediction from B19 (Sec. 8.2) is that the corrugation may be evident in the Galactic HI or the molecular hydrogen distribution, and may even explain Gould's belt and other local phenomena. Connecting this information with star clusters at different stages of their evolution may also provide important insights (e.g. Quillen et al. 2020).
In the spirit of Widrow et al. (2012), there is a case for extending galactoseismology to the study of nearby galaxies. Recent arguments have been made for undulations in the Galactic rotation curve (Martinez-Medina et al. 2019) being correlated with the spectacular R − V φ ridges seen by Gaia (Antoja et al. 2018;Khanna et al. 2019). With careful disc modelling and subtraction, it may be possible to identify the signatures of bending modes in substantial numbers of nearby disc galaxies (cf. Matthews & Uson 2008). In such instances, it will be important to establish whether the bending mode wraps up with the disc rotation, as we show here. Morphologically, the wrapping of the spiral density wave is more easily determined and thus it may even be possible to establish whether the interaction of these two wave modes is a regular phenomenon.
Figure 4 .
4(Top left) The density and integrated mass profiles of our three-component Galaxy model. The curves corresponding to the bulge's and disc's density distribution truncate because of the limited spatial extension of these components. The DM halo extends out to 300 kpc. Note that the density distribution of the halo and bulge are given in M kpc −3 (volume density), while the disc's density distribution is given in M kpc −2 (surface density).(Top right) The flat rotation curve (blue) of the synthetic Galaxy taken from the N-body simulation showing an acceptable resemblance to the Milky Way (Bland-Hawthorn & Gerhard 2016). The contributions to the rotation are also shown: bulge (green dot-dashed line), disc (red dotted line), dark halo (orange dashed line)
Figure 5 .
5Orbital history of a point mass perturber (Sgr) around the Galaxy in the N-body simulation. (Left) Total distance (black solid curve) and speed (red dot-dashed curve) relative to the Galactic Centre. (Right) Vertical distance z (blue solid curve) and vertical speed V z (gray dot-dashed curve) relative to the Galactic plane. The disc crossings are identified by the intersection of the blue curve and z = 0 (dashed horizontal line) and are clearly visible at T ≈ 100 Myr, and T ≈ 500 Myr (bottom). The black dots on top of the black curve (top) and of the blue curve (bottom) are indicative of Sgr's mass along its orbit. The size of the symbol is directly proportional to log 10 (mass) relative to its initial value.
Figure 6 .
6The projected stellar density distribution of the disc (left), the average height of the stars (middle), and the average vertical speed of the stars (right) at four time steps(T = 67, 95, 143, 190 Myr). The disc's rotation is counter-clockwise in this projection. The point mass crosses the disc at (x, y) ≈ (−18, 0) kpc (T = 95 Myr). The disturbance is already apparent at T = 67 Myr when the perturber is about 20 kpc from the impact point. Stars in the vicinity of the impact point are initially lifted towards the intruder. A bending wave is triggered by the impulse and it immediately begins to wrap up with the differential rotation.
Figure 8 .
8Middle stage to late stage evolution of the disc seen from the z > 0 axis (continues from previous figure).
Figure 9 .
9Middle and late stage evolution of the kinematic density wave in the N-body simulation. The rotating galaxy is viewed at each time step from an inertial (non-rotating) frame. The solar circle is indicated with the dot-dashed line. After the phase of the pattern is fixed at t = 571 Myr, there are no free parameters for the functional form of the spiral model in the other time steps. For 200
MNRAS 000, 1-18 (2020)
To encourage more extensive use, our agama setup files are available upon request. 4 Our ramses setup files are available upon request.
http://www.physics.usyd.edu.au/~tepper/proj_galseismo. html MNRAS 000, 1-18 (2020)
ACKNOWLEDGEMENTSJBH is supported by an ARC Australian Laureate Fellowship (FL140100278) and the ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO-3D) through project number CE170100013. JBH & TTG acknowledge the resources and services from the National Computational Infrastructure (NCI), which is supported by the Australian Government. We thank E. Vasiliev for his valuable assistance and continued conversation with the use and development of the AGAMA code. We are indebted to Romain Teyssier for assisting us with the idea of a time-dependent impulsive mass, both in terms of our additions to the code, and for allowing sink particles for the first time to operate with pure N-body simulations. We thank Shourya Khanna and Sanjib Sharma for providing the Gaia DR2 data necessary to recreate the discovery images fromAntoja et al. (2018). We are particularly grateful to Ken Freeman for many years of collegiality and for insightful and helpful comments on the manuscript. This work has made used of Matplotlib, a Python-based plotting package(Hunter 2007), and Pynbody(Pontzen et al. 2013).APPENDIX A: IMPULSE APPROXIMATIONOur set-up ensures that we are working entirely within the impulse approximation. InFig. A1, we show the force experienced by a star in the immediate vicinity of Sgr's crossing point (R ≈ 18 kpc) as a function of time before and after Sgr's disc transit (solid curve). The top x-axis displays Sgr's vertical distance to the disc plane along its orbit. The grayshaded area indicates the width of the stellar disc, here given by twice its scale height (2z t ≈ 500 pc). The interaction between stars at the crossing point is clearly impulsive. TheFigure A1. The force experienced by a star close to Sgr's crossing point (R ≈ 18 kpc) as a function of time (solid curve). This is taken from the N-body simulation data dumped to storage roughly every 10 Myr. Since the time step in the simulation is orders of magnitude smaller, the impulsive response to the intruder is vastly narrower in the simulation. This is illustrated by the red curve, which indicates the theoretical result. The small asymmetry imposed by time-dependent intruder mass is clear. Note that the curves have been normalised to a peak height of unity. The shaded area indicates the width of the stellar disc defined as twice its scale height (2z t ≈ 500 pc). apparent asymmetry in the impulse is the result of the reduced mass after transit (cf. Sec. 4.1). The impulsive force in the simulation is much narrower in practice, but appears broadened due to the coarse time resolution (∆t ≈ 10 Myr) adopted for storing the timesteps. To emphasize this point, we include in the figure the theoretical result (red curve). The latter corresponds to the force experienced by a body in the vicinity of a point mass, softened by s , i.e. ∼ r/(r 2 + 2 s ) 3/2 where r is the radial separation.
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|
[] |
[
"The Cauchy problem for Lie-minimal surfaces",
"The Cauchy problem for Lie-minimal surfaces"
] |
[
"Emilio Musso "
] |
[] |
[] |
In the present paper we study the Lie sphere geometry of Legendre surfaces by the method of moving frame and we prove an existence theorem for real-analytic Lie-minimal Legendre surfaces.In §1 we consider the Kepler manifold 1 K, that is the isotropic Grassmannian of the null-planes through 0 ∈ R (4,2) , acted upon transitively by the Lie sphere group G = SO(4, 2)/ ± I of contact transformations. We also consider the Lie quadric Q (i.e. the projectivization of the light cone L of R (4,2) ) and the Dupin manifold D. We apply moving frame to study Legendre surfaces in the Kepler manifold. Any Legendre surface M can be parameterized by a pair of mappings φ 0 , φ 1 : M → Q satisfying φ 0 , φ 1 = 0 and dφ 0 , φ 1 = 0. Most of the Lie-invariant properties of M are determined by the sheaf S of quadratic forms spanned by dφ 0 , dφ 0 and by dφ 1 , dφ 1 . If the stalks S m are 2-dimensional, for every m ∈ M then, the tautological bundlehas a natural splitting into the direct sum of two line sub-bundles Σ 0 (M ) and Σ 1 (M ). The maps σ 0 , σ 1 : M → Q induced by Σ 0 (M ) and by Σ 1 (M ) are the two curvature sphere mappings of the surface. If σ 0 and σ 1 are everywhere of maximal rank, then M is said to be non-degenerate. Geometrically, this condition means that the Euclidean projection of M can not be obtained as the envelope of a 1-parameter family of spheres. In §2 we indicate how to construct on any non-degenerate Legendre surface M ⊂ K a canonical lift A : M → G to the Lie sphere group (the normal frame field along M ). By the means of the normal frame field we recover the Blaschke co-frame (α 1 , α 2 ) of M and we introduce a complete set of local differential invariants (q 1 , q 2 , p 1 , p 2 , r 1 , r 2 ), the invariant functions of the surface. From the structural equations of the group G we deduce the compatibility conditions fulfilled by the normal co-frame and by the invariant functions. In §3 we analyze the Lie-invariant Gauss map and we write the Euler-Lagrange equations of the variational problem in terms of the invariant functions. Subsequently we set up a Pfaffian differential system (I, Ω) on P = G × R 6 with the defining property that its integral manifolds are the canonical lifts of Lie-minimal surfaces. In §4 we prove that the differential system (I, Ω) is in involution and that its general integral manifolds depend on six functions in one variable. In the last part of the paper we prove our main result :Theorem. Let Γ ⊂ K be a real-analytic Legendre curve and let U (Γ) → Γ be the corresponding tautological bundle. Let L ⊂ U (Γ) be a real-analytic line sub-bundle of U (Γ) and let h, w : Γ → R be two real-analytic functions. If Γ and L are suitably general, then there exist a real-analytic Lie-minimal surface M ⊂ K containing Γ such thatMoreover, M is unique in the sense that any other Legendre surface with these properties agrees with M on an open neighborhood of Γ.1We adopt the terminology introduced by J.M Souriau in[16]and by Guillelmin and Sternberg in [11]
| null |
[
"https://export.arxiv.org/pdf/math/0303073v1.pdf"
] | 118,173,522 |
math/0303073
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e40ad30a6499d6e6663d533efa7794c22b17eae8
|
The Cauchy problem for Lie-minimal surfaces
5 Mar 2003 March 29, 2022
Emilio Musso
The Cauchy problem for Lie-minimal surfaces
5 Mar 2003 March 29, 2022
In the present paper we study the Lie sphere geometry of Legendre surfaces by the method of moving frame and we prove an existence theorem for real-analytic Lie-minimal Legendre surfaces.In §1 we consider the Kepler manifold 1 K, that is the isotropic Grassmannian of the null-planes through 0 ∈ R (4,2) , acted upon transitively by the Lie sphere group G = SO(4, 2)/ ± I of contact transformations. We also consider the Lie quadric Q (i.e. the projectivization of the light cone L of R (4,2) ) and the Dupin manifold D. We apply moving frame to study Legendre surfaces in the Kepler manifold. Any Legendre surface M can be parameterized by a pair of mappings φ 0 , φ 1 : M → Q satisfying φ 0 , φ 1 = 0 and dφ 0 , φ 1 = 0. Most of the Lie-invariant properties of M are determined by the sheaf S of quadratic forms spanned by dφ 0 , dφ 0 and by dφ 1 , dφ 1 . If the stalks S m are 2-dimensional, for every m ∈ M then, the tautological bundlehas a natural splitting into the direct sum of two line sub-bundles Σ 0 (M ) and Σ 1 (M ). The maps σ 0 , σ 1 : M → Q induced by Σ 0 (M ) and by Σ 1 (M ) are the two curvature sphere mappings of the surface. If σ 0 and σ 1 are everywhere of maximal rank, then M is said to be non-degenerate. Geometrically, this condition means that the Euclidean projection of M can not be obtained as the envelope of a 1-parameter family of spheres. In §2 we indicate how to construct on any non-degenerate Legendre surface M ⊂ K a canonical lift A : M → G to the Lie sphere group (the normal frame field along M ). By the means of the normal frame field we recover the Blaschke co-frame (α 1 , α 2 ) of M and we introduce a complete set of local differential invariants (q 1 , q 2 , p 1 , p 2 , r 1 , r 2 ), the invariant functions of the surface. From the structural equations of the group G we deduce the compatibility conditions fulfilled by the normal co-frame and by the invariant functions. In §3 we analyze the Lie-invariant Gauss map and we write the Euler-Lagrange equations of the variational problem in terms of the invariant functions. Subsequently we set up a Pfaffian differential system (I, Ω) on P = G × R 6 with the defining property that its integral manifolds are the canonical lifts of Lie-minimal surfaces. In §4 we prove that the differential system (I, Ω) is in involution and that its general integral manifolds depend on six functions in one variable. In the last part of the paper we prove our main result :Theorem. Let Γ ⊂ K be a real-analytic Legendre curve and let U (Γ) → Γ be the corresponding tautological bundle. Let L ⊂ U (Γ) be a real-analytic line sub-bundle of U (Γ) and let h, w : Γ → R be two real-analytic functions. If Γ and L are suitably general, then there exist a real-analytic Lie-minimal surface M ⊂ K containing Γ such thatMoreover, M is unique in the sense that any other Legendre surface with these properties agrees with M on an open neighborhood of Γ.1We adopt the terminology introduced by J.M Souriau in[16]and by Guillelmin and Sternberg in [11]
Introduction
In his analysis [3] of Lie sphere geometry of surfaces W.Blaschke proposed to study the variational problem for the functional
B : M ⊂ R 3 → M ∂ 1 k 1 ∂ 2 k 2 (k 1 − k 2 ) 2 du 1 ∧ du 2 ,(1)
on immersed surfaces M ⊂ R 3 with no umbilical points, where k 1 and k 2 are the principal curvatures and where (u 1 , u 2 ) are curvature line coordinates. He also showed that the functional is invariant under Lie sphere transformations. Recently, E.Ferapontov [9] reconsidered this classical variational problem and showed that the critical points of (1) do admit a spectral deformation. This work was taken up by F.Burstall and U.Hertrich-Jeromin [4], who introduced a Lie-invariant Gauss map D : M → D with values in the Dupin manifold D, that is the symmetric space consisting of all 3-dimensional subspaces of signature (2,1) in R (4,2) . They showed that M ⊂ R 3 is a critical point of the functional (1) if and only if its Lie-invariant Gauss map is harmonic. This explains the origin of the spectral deformation discovered by Ferapontov and suggests the existence of a dressing action on the space of the critical points of the Blaschke functional (see [4]). In this paper, we study the Cauchy problem for Lie-minimal surfaces using the invariance by Lie sphere transformations from the outset. From this point of view the relevant objects of study are the Legendre lifts in the space of contact elements rather than the surfaces itself.
1 Legendre Surfaces 1.1 Lie sphere geometry.
Let us begin with some basic facts. Consider the vector space R (4,2) with the inner product of signature (4,2) defined by
V, V = −2v 0 v 5 − 2v 1 v 4 + (v 2 ) 2 + (v 3 ) 2 = g IJ v I v J(2)
where v 0 , ..., v 5 are the components of V with respect to the standard basis (ǫ 0 , ..., ǫ 5 ). We let G be the identity component of the pseudo-orthogonal group of (2) and we let g be its Lie algebra. For each A ∈ G we denote by A J = A · ǫ J the J-th column vector of A. Thus, (A 0 , ..., A 5 ) is a positive-oriented basis of R (4,2) such that A I , A J = g IJ , I, J = 0, ..., 5.
Expressing the exterior derivative dA J in terms of the basis (A 0 , ..., A 5 ) we obtain
dA J = ω I J A I , J = 0, ..., 5,(3)
where ω = (ω I J ) is the g-valued Maurer-Cartan form A −1 dA on the group G. Taking the exterior derivative in (3) yields the structure equations
dω = −ω ∧ ω.(4)
The Lie quadric is the space Q ⊂ RP 5 of the isotropic lines through 0 ∈ R (4,2) and the Lie sphere group is defined to be the group G = G/{±I} of all projective transformations of RP 5 which send Q into itself. Elements of G are equivalence classes of matrices A ∈ G. Given any such matrix A, its equivalence class in G is denoted by [A]. Thus,
[A] = [B] iff A = ±B.
Since the Maurer-Cartan form ω = A −1 dA is bi-invariant under the action of {±I}, then we can identify the Lie algebra of G with g and we may think of ω as being the Maurer-Cartan form of the Lie sphere group G.
Remark 1.1
The role of the Lie quadric is to represent the oriented spheres of R 3 (including point spheres, oriented planes and the "point at infinity"). Given a point p ∈ R 3 and a real number r we let σ(p, r) be the oriented sphere with center p and signed radius r. Similarly, for every p ∈ R 3 and every − → n ∈ S 2 , we let π(p, − → n ) be the oriented plane passing through p and orthogonal to the unit vector − → n . Then, the correspondence between the points of Q and oriented spheres is given by
σ(p, r) → 1, r+p 1 √ 2 , p 2 , p 3 , r−p 1 √ 2 , p·p−r 2 2 , π(p, − → n ) → 0, 1+n 1 2 , n 2 √ 2 , n 3 √ 2 , 1−n 1 2 , n·p √ 2 , ∞ → [ǫ 5 ](5)
Definition 1.2 The Kepler manifold K is defined to be the isotropic Grassmannian consisting of all null planes through the origin of R (4,2) . Remark 1.3 Two oriented spheres corresponding to [V ], [V ′ ] ∈ Q are in oriented contact if and only if V, V ′ = 0. Geometrically, this means that a null plane ℓ ∈ K represents a pencil of oriented spheres in oriented contact. Thus, we may think of K as the space of the parabolic pencils of oriented spheres . Another classical model of the Kepler manifold is "R 3 × S 2 with a 2-dimensional sphere S 2 ∞ at the infinity". The sphere S 2 ∞ is the set of the null planes through the isotropic line [ǫ 5 ]. The complement
K 0 of S 2 ∞ is identified with R 3 × S 2 by F : (p, − → n ) ∈ R 3 × S 2 → [F 0 (p) ∧ F 1 (p, − → n )] ∈ K,(6)
where
F 0 (p) = ǫ 0 + p 1 √ 2 ǫ 1 + p 2 ǫ 2 + p 3 ǫ 3 − p 1 √ 2 ǫ 4 + p·p 2 ǫ 5 , F 1 (p, − → n ) = 1+n 1 2 ǫ 1 + n 2 √ 2 ǫ 2 + n 3 √ 2 ǫ 3 + 1−n 1 2 ǫ 4 + n·p √ 2 ǫ 5 .(7)
Let G act on K in the usual way : given a null plane [V ∧V ′ ] spanned by a pair V, V ′ of isotropic vectors, and given
A ∈ G, then A · [V ∧ V ′ ] = [AV ∧ AV ′ ]. The projection map π K : [A] ∈ G → [A 0 ∧ A 1 ] ∈ K (8) makes G into a G 0 -principal fibre bundle over K, where G 0 = {[A] ∈ G : A I 0 = A I 1 = 0, I = 2, ..., 5}.
Remark 1.4 For later use we observe that the elements of G 0 can be written as
X(D, B, Y, b) = D DY * JB Z(D, Y, b) 0 B Y 0 0 (D * ) −1 , where D ∈ GL + (2, R), B ∈ SO(2), Y ∈ gl(2, R), b ∈ R and where Z(D, Y, b) = 1 2 DJ Y t Y + 0 −b b 0 , J = 0 1 1 0 , X * = J t XJ.
The forms {ω 2 0 , ω 3 0 , ω 2 1 , ω 3 1 , ω 4 0 } are linearly independent and span the semi-basic forms for the projection π K . In particular, ω 4 0 is well-defined up to a positive multiple on K and, from the structure equations, we get
dω 4 0 = ω 2 0 ∧ ω 2 1 + ω 3 0 ∧ ω 3 1 + (ω 0 0 + ω 1 1 ) ∧ ω 4 0 .
From this we infer that ω 4 0 defines a G-invariant contact structure on K.
We let D be the Dupin manifold , that is the manifold of all 3-dimensional linear subspaces of R (4,2) of signature (2,1). Then, G act transitively on D by
A · [V ∧ V ′ ∧ V ′′ ] = [AV ∧ AV ′ ∧ AV ′′ ], for every A ∈ G and every [V ∧ V ′ ∧ V ′′ ] ∈ D. The projection map π D : A ∈ G → [A 0 ∧ A 3 ∧ A 5 ] ∈ D
gives on G the structure of a principal fibre bundle with structure group
S((O(2, 1) × O(2, 1)) ∼ = {A ∈ G : A · [ǫ 0 ∧ ǫ 3 ∧ ǫ 5 ] = [ǫ 0 ∧ ǫ 3 ∧ ǫ 5 ]}.
Thus, D can be viewed as the pseudo-riemannian symmetric space
SO(4, 2)/S (O(2, 1) × O(2, 1)) .
The canonical pseudo-Riemannian metric g D on D induced by the fibering π D is represented by the tensorial quadratic form on G given by
ω 1 0 ω 0 1 + ω 0 4 ω 4 0 + ω 2 0 ω 0 2 + 2ω 1 3 ω 3 1 − 1 2 (ω 3 2 ) 2 .(9)
1.2 Legendre Surfaces Definition 1.5 An oriented, connected immersed surface M ⊂ K is said to be Legendrian if it is tangent to the contact distribution on K.
Locally, there exist two smooth mappings φ 0 , φ 1 : U ⊂ M → R (4,2) such that ℓ = [φ 0 (ℓ) ∧ φ 1 (ℓ)], for every ℓ ∈ U , and that
φ 0 2 = φ 1 2 = φ 0 , φ 1 = 0, dφ 0 , φ 1 = 0.
Then, dφ 0 , dφ 0 and dφ 1 , dφ 1 span a sheaf S of quadratic forms on M . Throughout the paper we shall assume that the fiber S m of S is two-dimensional, for every point m ∈ M . We consider
F 0 (M ) = {(ℓ, A) ∈ M × G | ℓ = [A 0 ∧ A 1 ]}
the pull back of the fiber bundle π K : G → K to the surface M . The local cross sections of F 0 (M ) are called local frame fields along M . They can be considered as smooth
maps A : U → G, where U is an open subset of M , such that ℓ = [A 0 (ℓ) ∧ A 1 (ℓ)]
, for every ℓ ∈ U . For every local frame field A : U → G we let α = (α I J ) be the pull-back of the Maurer-Cartan form. Any other local frame field A on U is given by A = A · X, where X = U → G 0 is a smooth map. Thus, the 1-forms α and α are related by the gauge transformation α = X −1 dX + X −1 αX.
A frame field A : U → G is of first order if, with respect to it
α 3 0 ∧ α 2 0 > 0, α 2 0 = α 3 1 = 0.(11)
From (10) it follows that first order frames exist on a neighborhood of any point of M . The totality of first order frames is a principal
G 1 -bundle F 1 (M ) → M where G 1 = X(D, B, Y, b) ∈ G 1 : B = ǫId 2×2 , D = r 0 0 s , ǫ = ±1, rs > 0 .
Notation. The elements of G 1 will be denoted by
Y ǫ (r, s, Y, b), where ǫ = ±1, Y ∈ gl(2, R), b, r, s ∈ R and rs > 0. We let σ 0 , σ 1 : M → Q be defined by σ 0 | U = [A 0 ] and by σ 1 | U = [A 1 ]
, for every first order frame A : U → G. We follow the classical terminology and we call σ 0 and σ 1 the curvature sphere maps of the surface. We remark that σ 0 and σ 1 define a splitting of the tautological vector bundle
U (M ) = {(ℓ, V ) ∈ M × R (4,2) : ℓ ∈ M, V ∈ ℓ} into the direct sum Σ 0 (M ) ⊕ Σ 1 (M ) of the line sub-bundles Σ a (M ) = {(ℓ, V ) ∈ M × R (4,2) : ℓ ∈ M, V ∈ σ a (ℓ)}, a = 0, 1.α 4 0 = α 2 0 = α 3 1 = α 3 2 = α 1 0 − α 2 1 = α 0 1 − α 3 0 = α 0 2 = α 1 3 = 0,(12)
and the independence condition
α 3 0 ∧ α 2 1 > 0.(13)
Proof. If A : U → G is a first order frame then, the linear differential forms α 1 = α 3 0 and α 2 = α 2 1 give a positive-oriented co-framing on U so that we may write
α = P 1 α 1 + P 2 α 2 ,
where P 1 , P 2 : U → g are smooth maps. The components of P a are denoted by P I Ja , where a = 1, 2 and where I, J = 0, ..., 5. If A, A : U → G are first order frames on U and if the corresponding transition function is of the form Y ǫ (r, s, Y, b) : U → G 1 , theñ
α 1 = ǫrα 1 ,α 2 = ǫsα 2 ,α 1 0 = r s −1 α 1 0 − Y 2 1 α 1 ,α 0 1 = s r −1 α 0 1 − Y 1 2 α 2 .(14)
This implies
P 1 01 = ǫ(s −1 P 1 01 − Y 2 1 ), P 1 02 = ǫrs −2 P 1 02 , P 0 12 = −ǫ(r −1 P 0 12 + Y 1 2 ), P 0 11 = ǫsr −2 P 0 11 .(15)
From (15) we see that for every point ℓ ∈ M there exist a first order frame field A : U → G defined on an open neighborhood U of ℓ with respect to which P 1 01 = P 0 12 = 0. Such first order frame fields are said to be of second order . In addition, any other second order frame field on U is of the form A = A · X, where X : U → G 2 is a smooth map and
G 2 = Y ǫ (r, s, Y, b) ∈ G 1 : Y = p 0 0 q , p, q ∈ R .
Notation. The elements of G 2 will be denoted by Y ǫ (r, s, p, q, b), where ǫ = ±1, p, q, r, s, b ∈ R and rs > 0.
Second order frame fields are the cross sections of a reduced sub-bundle F 2 (M ) of F 1 (M ) with structural group G 2 . Differentiating α 2 0 = α 3 1 = 0 and applying the structure equations and Cartan's Lemma, we have that α 3 2 = 0, for every second order frame field A. Taking the exterior derivative of α 3 2 = 0 and using again the structure equations and the Cartan's lemma we have α 0 2 ∧ α 1 − α 1 3 ∧ α 2 = 0. This implies P 0 22 = −P 1 31 and hence we may write
α 0 2 = P 0 21 α 1 + P 0 22 α 2 , α 1 3 = −P 0 22 α 1 + P 1 32 α 2 .(16)
If A and A are second order frame fields on U ⊂ M and if Y ǫ (r, s, p, q, b) : U → G 2 is the corresponding transition function, we then have
α 0 2 = ǫr −1 spα 0 1 + ǫr −1 α 0 2 − sbα 2 , α 1 3 = ǫs −1 rqα 1 0 + ǫs −1 α 1 3 + rbα 1 .(17)
From this we obtain
P 0 21 = r −2 P 0 21 + spP 0 11 , P 0 22 = 1 rs P 0 22 − ǫb, P 1 32 = s −2 P 1 32 + rqP 1 02 .(18)
Thus, for every point ℓ ∈ M there exist a second order frame field U → G defined on an open neighborhood of ℓ with respect to which
P 0 22 = P 1 31 = 0.(19)
Such frame fields are said to be of third order . Now, (18) implies that these frame fields are the local cross sections of a reduced sub-bundle
F 3 (M ) of F 2 (M ). The structure group of F 3 (φ) is G 3 = {Y ǫ (r, s, p, q, b) ∈ G 2 : b = 0}.
Since M is non-degenerate, then the functions P 1 02 and P 0 11 are nowhere vanishing. Thus, from (15) we infer that for every ℓ ∈ M there exist a third order frame field A : U → G defined on an open neighborhood U of ℓ such that
α 0 1 = α 1 , α 1 0 = α 2 .(20)
A third order frame field satisfying (20) is said to be of fourth order . If A is a fourth order frame field on U , then any other is given by A = AX, where X : U → G 4 and
G 4 = {Y ǫ (r, s, p, q, 0) ∈ G 3 : r = s = ǫ}.
The elements of G 4 are denoted by Y ǫ (p, q), where p, q ∈ R. From this we immediately see that the third order frame fields define a G 4 sub-bundle F 4 (M ) of F 3 (M ). Now, using (18) we see that for every ℓ ∈ M there exist a fourth order frame A : U → G in an open neighborhood U of ℓ satisfying α 0 2 = α 1 3 = 0. Such frame fields are of fifth order . Notice that fifth order frame satisfy (12) and (13) We then consider the linearly independent 1-forms α 1 = α 3 0 and α 2 = α 2 1 and we call (α 1 , α 2 ) the canonical co-frame of the Legendre surface.
α 1 = 1 k 1 −k 2 g 11 g 22 (∂ 1 k 1 ) 2 ∂ 2 k 2 ) 1 3 du 1 , α 2 = −1 k 1 −k 2 g 22 g 11 ∂ 1 k 1 (∂ 2 k 2 ) 2 1 3 du 2 .(21)
Taking the exterior derivatives of (12) and using the Maurer-Cartan equations it follows that there exist smooth functions q 1 , q 2 , p 1 , p 2 and r 1 , r 2 such that
α 0 0 = −2q 1 α 1 + q 2 α 2 , α 1 1 = −q 1 α 1 + 2q 2 α 2 , α 0 3 = r 1 α 1 + p 2 α 2 , α 1 2 = p 1 α 1 + r 2 α 2 , α 0 4 = −r 2 α 1 + r 1 α 2 ,(22)
We call q 1 , q 2 , p 1 , p 2 and r 1 , r 2 the invariant functions of the Legendre surface. Using once more the Maurer-Cartan equations we obtain
dα 1 = α 0 0 ∧ α 1 , dα 2 = α 1 1 ∧ α 2 , dα 1 2 = −α 1 1 ∧ α 1 2 , dα 0 3 = −α 0 0 ∧ α 0 3 ,(23)
and
dα 0 0 = (α 2 − α 0 3 ) ∧ α 1 , dα 1 1 = (α 1 − α 1 2 ) ∧ α 2 , dα 0 4 = −(α 0 0 + α 1 1 ) ∧ α 0 4 .(24)
We may rewrite these equations in terms of the invariant functions
dα 1 = −q 2 α 1 ∧ α 2 , dα 2 = −q 1 α 1 ∧ α 2 (25) − 2dq 1 ∧ α 1 + dq 2 ∧ α 2 = (p 2 − q 1 q 2 − 1)α 1 ∧ α 2 , − dq 1 ∧ α 1 + 2dq 2 ∧ α 2 = (−p 1 + q 1 q 2 + 1)α 1 ∧ α 2 ,(26) dr 1 ∧ α 1 + dp 2 ∧ α 2 = (2q 2 r 1 + 3q 1 p 2 )α 1 ∧ α 2 , dp 1 ∧ α 1 + dr 2 ∧ α 2 = (2q 1 r 2 + 3q 2 p 1 )α 1 ∧ α2, − dr 2 ∧ α 1 + dr 1 ∧ α 2 = 4(q 1 r 1 − q 2 r 2 )α 1 ∧ α 2 .dr 1 ∧ α 2 − 4q 1 r 1 α 1 ∧ α 2 = 0, dr 2 ∧ α 1 − 4q 2 r 2 α 1 ∧ α 2 = 0.(28)
Proof. Without loss of generality we assume that D is an embedding and we identify M with its image in D. We extend the normal frame field A to a local cross section A : U → G of π D : G → D defined on an open neighborhood U ⊂ D of M . If we set α = A −1 d A then
β 1 = α 1 0 , β 2 = α 0 1 , β 3 = α 0 4 , β 4 = α 4 0 , β 5 = α 2 0 β 6 = α 0 2 , β 7 = α 1 3 , β 8 = α 3 1 , β 9 = α 3 2 ,(29)
is a co-frame on U . We let B 1 , ..., B 9 be the local trivialization of T (D) dual to (β 1 , ..., β 9 ). Then, (22) implies that
X 1 = B 2 | M − r 2 B 3 | M , X 2 = B 1 | M + r 1 B 3 | M(30)
is the trivialization 2 of T (M ) dual to the canonical co-frame (α 1 , α 2 ) and that
B 3 = B 3 | M B 4 = B 4 | M + r 2 B 1 | M − r 1 B 2 | M , B 5 = B 5 | M , ...., B 9 = B 9 | M(31)
is a trivialization of the normal bundle N → M . Taking the exterior derivative of (29) and using the Maurer-Cartan equations we compute the covariant derivatives of the vector fields B 1 , ..., B 9 . We then have and
∇B 1 = ( α 1 1 − α 0 0 )B 1 + α 2 1 B 5 − α 0 3 B 7 , ∇B 2 = ( α 0 0 − α 1 1 )B 2 − α 1 2 B 6 + α 3 0 B 8 , ∇B 3 = ( α 0 0 + α 1 1 )B 3 − α 2 1 B 6 + α 3 0 B 7 ,(32) ∇B 4 = −( α 0 0 + α 1 1 )B 4 + α 1 2 B 5 − α 0 3 B 8 , ∇B 5 = α 1 2 B 1 + α 2 1 B 4 − α 0 0 B 5 + α 0 3 B 9 , ∇B 6 = − α 2 1 B 2 − α 1 2 B 3 + α 0 0 B 6 + α 3 0 B 9 , ∇B 7 = − α 3 0 B 1 + α 0 3 B 3 + α 1 1 B 7 − α 2 1 B 9 , ∇B 8 = α 0 3 B 2 − α 3 0 B 4 − α 1 1 B 8 − α 1 2 B 9 , ∇B 9 = α 3 0 B 5 + α 0 3 B 6 − α 1 2 B 8 − α 2 1 B 9 .(33)
Thus, from (30), (31), (32) and (33) we infer that the shape operator
S ∈ Γ(M, Hom(T M, Ω 1 (M ) ⊗ N )) of M ⊂ D is given by S(X 1 ) = − dr 2 + 2r 2 (−q 1 α 1 + 2q 2 α 2 ) B 3 + α 1 −p 1 B 6 − r 2 B 7 + B 8 , S(X 2 ) = dr 1 + 2r 1 (−2q 1 α 1 + q 2 α 2 ) B 3 + α 2 B 5 − r 1 B 6 − p 2 B 7 .(34)
In particular, we obtain the following formula for the mean curvature vector
H = 1 2 (S(X 1 )(X 2 ) + S(X 2 )(X 1 )) = (−dr 2 (X 2 ) − 4r 2 q 2 + dr 1 (X 1 ) − 4p 1 q 1 ) B 3 (35)
From this we deduce that D : (M, Φ) → (D, g D ) is harmonic if and only if − dr 2 ∧ α 1 + 4q 2 r 2 α 1 ∧ α 2 + dr 1 ∧ α 2 − 4r 1 q 1 α 1 ∧ α 2 = 0.
From (27) and (36) we get the required result. ▽
The differential system of Lie-minimal surfaces
Let P be the configuration space G × R 6 and let denote by (q 1 , q 2 , p 1 , p 2 , r 1 , r 2 ) the coordinates on R 6 . On P we consider the Pfaffian ideal I ⊂ Ω * (P ) generated (as a differential ideal) by the linear differential forms
η 1 = ω 4 0 , η 2 = ω 2 0 , η 3 = ω 3 1 , η 4 = ω 3 2 , η 5 = ω 1 0 − ω 2 , η 6 = ω 0 1 − ω 1 , η 7 = ω 0 2 , η 8 = ω 1 3 ,(37)η 9 = ω 0 0 + 2q 1 ω 1 − q 2 ω 2 , η 10 = ω 1 1 + q 1 ω 1 − 2q 2 ω 2 ,(38) η 11 = ω 0 3 − r 1 ω 1 − p 2 ω 2 , η 12 = ω 1 2 − p 1 ω 1 − r 2 ω 2 , η 13 = ω 0 4 + r 2 ω 1 − r 1 ω 2 ,(39)
and by the exterior differential 2-forms
Θ 1 = dr 1 ∧ α 2 − 4q 1 r 1 α 1 ∧ α 2 , Θ 2 = dr 2 ∧ α 1 − 4q 2 r 2 α 1 ∧ α 2 ,(40)
together with the independence condition Ω = ω 1 ∧ ω 2 , where ω 1 = ω 3 0 and ω 2 = ω 2 1 . Taking the exterior derivatives of (37),(38),(39) and using the structural equations of G we obtain the quadratic equations
dη 1 ≡ ..... ≡ dη 8 ≡ dη 13 ≡ 0,(41)
and
dη 9 ≡ 2π 1 ∧ ω 1 − π 2 ∧ ω 2 + (−1 + p 2 − q 1 q 2 )ω 1 ∧ ω 2 , dη 10 ≡ π 1 ∧ ω 1 − 2π 2 ∧ ω 2 + (1 − p 1 + q 1 q 2 )ω 1 ∧ ω 2 , dη 11 ≡ ζ 1 ∧ ω 1 + υ 2 ∧ ω 2 − (2r 1 q 2 + 3q 1 p 2 )ω 1 ∧ ω 2 , dη 12 ≡ υ 1 ∧ ω 1 + ζ 2 ∧ ω 2 − (3p 1 q 2 + 2r 2 q 1 )ω 1 ∧ ω 2 ,(42)
where ≡ denotes equality up to the algebraic ideal generated by η 1 , ...., η 13 , Θ 1 , Θ 2 and where π i = dq i , υ i = dp i , ζ i = dr i , i = 1, 2.
If we set
Ω 1 = 2π 1 ∧ ω 1 − π 2 ∧ ω 2 + (−1 + p 2 − q 1 q 2 )ω 1 ∧ ω 2 , Ω 2 = π 1 ∧ ω 1 − 2π 2 ∧ ω 2 + (1 − p 1 + q 1 q 2 )ω 1 ∧ ω 2 , Ω 3 = ζ 1 ∧ ω 1 + υ 2 ∧ ω 2 − (2r 1 q 2 + 3q 1 p 2 )ω 1 ∧ ω 2 , Ω 4 = υ 1 ∧ ω 1 + ζ 2 ∧ ω 2 − (3p 1 q 2 + 2r 2 q 1 )ω 1 ∧ ω 2 .(43)
then
{η 1 , ...., η 13 , Θ 1 , Θ 2 , Ω 1 , Ω 2 , Ω 3 , Ω 4 , dΘ 1 , dΘ 2 } (44)
is a set of algebraic generators of the differential ideal I. The integral manifolds of this system are two-dimensional submanifolds M ⊂ P such that η a = 0, Θ 1 = Θ 2 = Ω 1 = Ω 2 = Ω 3 = 0 = Ω 4 , Ω = 0.
Thus, the map
φ : ([A], q 1 , q 2 , p 1 , p 2 , r 1 , r 2 ) ∈ M → [A 0 ∧ A 1 ] ∈ K (45)
is a non-degenerate Legendre immersion. Since our arguments are local, we identify M with its image M = φ( M ) ⊂ K. Thus, M a Lie-minimal surface with normal frame field A :
([A], q 1 , q 2 , p 1 , p 2 , r 1 , r 2 ) ∈ M → [A] ∈ G.
Conversely, if M is a Lie-minimal surface with normal frame field A : M → G and with invariant functions q 1 , ..., r 2 , then the map ℓ ∈ M → (A(ℓ), q 1 (ℓ), q 2 (ℓ), p 1 (ℓ), p 2 (ℓ), r 1 (ℓ), r 2 (ℓ)) ∈ P, ∀ℓ ∈ M (46)
defines an integral manifold of the differential system (I, Ω). To summarize :
Proposition 2.6 Lie-minimal surfaces M ⊂ K may be regarded as being the integral submanifolds of the differential system (I, Ω) on P .
3 The Cauchy problem 3.1 Involutivity of the differential system
On P we consider the parallelization
∂ ∂ω i , ∂ ∂η a , ∂ ∂π i , ∂ ∂υ i , ∂ ∂ζ , i = 1, 2, a = 1, ..., 13
dual to the co-frame (ω i , η a , π i , υ i , ζ i ). We define
V 1 (Ω) = {(z, E 1 ) ∈ G 1 (T (P )) : (ω 1 ) 2 + (ω 2 ) 2 | E 1 = 0}, V 2 (Ω) = {(z, E 2 ) ∈ G 2 (T (P )) : Ω| E 2 = 0}(47)
and we set
V 1 = S 1 × R 16 , V 2 = R(13, 2) ⊕ R(2, 2) ⊕ R(2, 2) ⊕ R(2, 2) ∼ = R 38 , with coordinates z = (cos(θ), sin(θ), x a , y i , u i , v i ), Z = (X a j , Y i j , U i j , V i j ), a = 1, ..., 13, i, j = 1, 2.
Then, we identify V 1 (Ω) with P × V 1 and V 2 (Ω) with P × V 2 by the means of
(z, z) ∈ P × V 1 → (z, E 1 (z, z)) ∈ V 1 (Ω), (z, Z) ∈ P × V 2 → (z, E 2 (z, Z)) ∈ V 2 (Ω),(48)
where
E 1 (z, z) = cos(θ) ∂ ∂ω 1 | z + sin(θ) ∂ ∂ω 2 | z + x a ∂ ∂η a | z + y i ∂ ∂π i | z + u i ∂ ∂υ i | z + v i ∂ ∂ζ i | z and where E 2 (z, Z) = [T 1 (z, Z) ∧ T 2 (z, Z)] , T 1 (z, Z) = ∂ ∂ω 1 | z + X a 1 ∂ ∂η a | z + Y i 1 ∂ ∂π i | z + U i 1 ∂ ∂υ i | z + V i 1 ∂ ∂ζ i | z , T 2 (z, Z) = ∂ ∂ω 2 | z + X a 2 ∂ ∂η a | z + Y i 2 ∂ ∂π i | z + U i 2 ∂ ∂υ i | z + V i 2 ∂ ∂ζ i | z .
Thus, the space V 1 (I, Ω) consisting of the 1-dimensional integral elements can be identified with the submanifold of P ×V 1 defined by the linear equations x 1 = .... = x 13 = 0. Similarly, the space V 2 (I, Ω) consisting of all 2-dimensional integral elements is identified with the submanifold of P × V 2 defined by
X a i = 0, a = 1, ..., 13, i = 1, 2, 2Y 1 2 + Y 2 1 + (1 − p 2 + q 1 q 2 ) = Y 1 2 + 2Y 2 1 − (1 − p 1 + q 1 q 2 ) = 0, U 2 1 − V 1 2 − (2r 1 q 2 + 3q 1 p 2 ) = U 1 2 − V 2 1 + (3p 1 q 2 + 2r 2 q 1 ) = 0, V 1 1 − 4q 1 r 1 = V 2 2 − 4q 2 r 2 = 0.
From this we infer that all the integral elements of the system are K-ordinary. Let (z, E 1 ) be a 1-dimensional integral element such that
E 1 = a 1 ∂ ∂ω 1 | z + a 2 ∂ ∂ω 2 | z + y i ∂ ∂π i | z + u i ∂ ∂υ i | z + v i ∂ ∂ζ i | z .
Contracting Θ 1 , Θ 2 and Ω 1 , ..., Ω 4 with E 1 , we obtain the polar equations of the integral element E 1
η a = 0, a = 1, ..., 13, 2a 1 π 1 − a 2 π 2 − 2y 1 + a 2 (1 − p 2 + q 1 q 2 ) ω 1 + y 2 + a 1 (1 − p 2 + q 1 q 2 ) ω 2 = 0, a 1 π 1 − 2a 2 π 2 − y 1 − a 2 (1 − p 1 + q 1 q 2 ) ω 1 + 2y 2 − a 1 (1 − p 1 + q 1 q 2 ) ω 2 = 0, a 1 υ 1 + a 2 ζ 2 − u 1 + a 2 (3p 1 q 2 + 2r 2 q 1 ) ω 1 − v 2 − a 1 (3p 1 q 2 + 2r 2 q 1 ) ω 2 = 0, a 2 υ 2 + a 1 ζ 1 − v 1 + a 2 (2r 1 q 2 + 3q 1 p 2 ) ω 1 − u 2 − a 1 (2r 1 q 2 + 3q 1 p 2 ) ω 2 = 0, a 2 ζ 1 − 4a 2 q 1 r 1 ω 1 + (4a 1 q 1 r 1 + v 1 )ω 2 = 0, a 1 ζ 2 − (4a 2 q 2 r 2 + v 2 )ω 1 + 4a 1 q 2 r 2 ω 2 = 0.
(49) This shows that the polar space H(z, E 1 ) of a non-characteristic integral element 3 is two-dimensional. From the Cartan-Kaehler theorem we deduce Proposition 3.1 If Γ ⊂ P is a non-characteristic real-analytic integral curve of (I, Ω) then there exist a unique real-analytic integral manifold M ⊂ P such that Γ ⊂ M .
Legendre Curves.
Let Γ ⊂ K be a smooth Legendre curve. Locally, we have that ℓ = [V 0 (ℓ) ∧ V 1 (ℓ)], for every ℓ ∈ Γ, where V 0 , V 1 : Γ → R (4,2) are smooth maps such that
V 0 , = V 1 = V 0 , V 1 = 0, V 0 , dV 1 = 0.(50)
We say that Γ is linearly full in case 4
V 0 (ℓ) ∧ V 1 (ℓ) ∧ V ′ 0 (ℓ) ∧ V ′ 1 (ℓ) ∧ V ′′ 0 (ℓ) ∧ V ′′ 1 (ℓ) = 0 ∀ℓ ∈ N.(51)
We let U (Γ) → Γ be the tautological vector bundle of the curve, that is
U (Γ) = {(ℓ, V ) ∈ Γ × R (4,2) : V ∈ ℓ}.(52)
A cross section of U (Γ) is a smooth map V : Γ → R (4,2) such that V (ℓ) ∈ ℓ, for every ℓ ∈ Γ. Accordingly, a line sub-bundle L ⊂ U (Γ) can be viewed as a mapping σ L : Γ → Q such that σ L (ℓ) ⊂ ℓ, for each ℓ ∈ Γ. We say that
L ⊂ U (Γ) is fat if V (ℓ) ∧ V ′ (ℓ) ∧ .... ∧ V (v) (ℓ) = 0, ∀ℓ ∈ Γ,(53)
for every local trivialization V : U → R (4,2) of L. In this case the osculating space (4,2) has signature (2, 1), for every ℓ ∈ Γ. The map
δ L (ℓ) = [V (ℓ) ∧ V ′ (ℓ) ∧ V ′′ (ℓ)] ⊂ Rδ L : ℓ ∈ Γ → δ L (ℓ) ∈ D, ∀ℓ ∈ Γ.(54)
is called the directrix curve of L. If δ L is non-isotropic (i.e. δ * L (g D ) is nowhere vanishing) then L is said to be a polarization of the curve Γ.
ℓ = [R 0 (ℓ) ∧ R 1 (ℓ)], R 0 (ℓ) ∈ L| ℓ , ∀ℓ ∈ Γ(55)
and that
R −1 dR = k 0 1 0 k 1 k 3 0 −1 −k 0 k 2 0 0 −k 3 0 −1 0 0 k 2 0 1 0 0 0 0 k 1 0 0 −1 0 k 0 −1 0 0 0 1 1 −k 0 µ,(56)
where µ is a nowhere vanishing 1-form and where k 0 , k 1 , k 2 and k 3 are real-valued functions.
Proof. We consider the G 0 fiber bundle
R 0 (Γ, L) = {(ℓ, R) ∈ Γ × G : ℓ = [R 0 ∧ R 0 ], R 0 ∈ L| ℓ }.
The cross-sections of R 0 (Γ, L) are smooth maps R : U → G defined on an open subset U ⊂ Γ, such that
ℓ = [R 0 (ℓ) ∧ R 1 (ℓ)], R 0 (ℓ) ∈ L| ℓ , ∀ℓ ∈ U.
For each frame field R : U → G we let ρ be the g-valued 1-form R −1 dR. We say that
R : U → G is of first order if ρ 3 0 = 0, ρ 2 0 = ρ 3 1 = ρ 3 0 + ρ 2 1 = ρ 4 0 = 0.(57)
Since Γ is linearly full then, first order frames do exist near any point of Γ and they define a sub-bundle R 1 (Γ, L) of R 0 (Γ, L) with fiber
H 1 = {X ∈ G 0 : X = X(rǫI, ǫI, Y, b) : ǫ = ±1, r, b ∈ R, r = 0, Y ∈ gl(2, R)}.
If R and R are first order frames such that R = RX(rǫI, ǫI, Y, b) then
ρ 0 1 = ρ 0 1 + rY 2 1 ρ 3 0 , ρ 1 0 = ρ 1 0 − rY 1 2 ρ 3 0 , ρ 3 2 = ρ 3 2 + ǫr(Y 1 2 + Y 2 1 )ρ 3 0 .(58)
This shows that near any point of Γ there exist first order frames such that
ρ 1 0 + ρ 0 1 = ρ 3 2 = 0.(59)
Frame fields satisfying (59) are said to be of second order . From (58) it follows that the totality of second order frames defines a fiber bundle R 2 (Γ, L) with fiber
H 2 = {X = X(rǫI, ǫI, Y, b) ∈ H 1 : Y 1 2 = Y 2 1 = 0}.
Notice that the 1-form ρ 0 1 is independent on the choice of the second order frame and hence there exist µ ∈ Ω 1 (Γ) such that µ| U = ρ 0 1 . At this juncture it is convenient to recall that the pseudo-riemannian metric g D of D is represented by the tensorial quadratic form on G defined by
ω 1 0 ω 0 1 + ω 0 4 ω 4 0 + ω 2 0 ω 0 2 + 2ω 1 3 ω 3 1 − 1 2 (ω 3 2 ) 2 .
Thus, δ * L (g D ) = −µ 2 and hence µ is nowhere vanishing. From (58) it follows that, locally, there exist second order frames such that
ρ 3 0 = −ρ 2 1 = −ρ 1 0 = ρ 0 1 = µ.(60)
Frame fields satisfying (60) are of third order . The totality of third order frames originates a principal fiber bundle R 3 (Γ, L) with structural group
H 3 = {X = X(rǫI, ǫI, Y, b) ∈ H 2 : r = 1}.
If R and R are third order frame fields then
ρ 0 0 = ρ 0 0 − ǫY 2 2 ρ 3 0 , ρ 1 1 = ρ 1 1 + ǫY 1 1 ρ 3 0 .
Therefore, near any point of Γ there exist a third order frame field R such that
ρ 1 1 + ρ 0 0 = 0.(61)
Frame fields satisfying (61) define a reduced sub-bundle R 4 (Γ, L) with structure group
H 4 = {X = X(ǫI, ǫI, Y, b) ∈ H 3 : Y 1 1 = Y 2 2 }.
Consider two local cross sections R and R of R 4 (Γ, L), we then have
ρ 0 2 = ρ 0 2 + ǫ(Y 1 1 + b 2 )µ, ρ 1 3 = ρ 1 3 − ǫ(Y 1 1 − b 2 )µ.(62)
This implies that there exist fourth order frame fields with respect to which
ρ 0 2 = ρ 1 3 = 0.(63)
Fourth order frame fields satisfying (63) are said to be of fifth order . The totality of fifth order frames generates a reduced sub-bundle
The Cauchy problem
Theorem 3.5 Let (Γ, L) be a real-analytic polarized Legendre curve and let h, w : Γ → R be two real-analytic functions. Then, there exist a real-analytic Lie-minimal surface M ⊂ K containing Γ such that
Γ * (α 1 + α 2 ) = 0, L = Σ 0 (M )| Γ , h = −3(q 1 + q 2 )| Γ , w = 1 3 (p 1 − p 2 )| Γ .
This manifold is unique in the sense that any other Legendre surface with these properties agrees with M on an open neighborhood of Γ.
Proof. Let R : Γ → G be the Frenet frame field along (Γ, L). We set
X(h) = 1 0 0 −h/2 h/2 h 2 /8 0 1 h/2 0 h 2 /8 −h/2 0 0 1 0 h/2 0 0 0 0 1 0 h/2 0 0 0 0 1 0 0 0 0 0 0 1 .(64)
and we consider the frame field From (66) and (68) it follows that Γ is a 1-dimensional integral manifold of the differential system (I, Ω). We set dq j = q * j µ, dp j = p * j µ, dr j = r * j µ,
where q * j , p * j , r * j are real-analytic functions. From (66) we infer that
Γ * ∂ ∂µ = ∂ ∂ω 1 − ∂ ∂ω 2 + q * i ∂ ∂π i + p * i ∂ ∂υ i + r * i ∂ ∂ζ i .(71)
Thus, Γ is a non-characteristic K-regular integral curve of (I, Ω). Therefore, there exist a unique 2-dimensional integral manifold M ⊂ P such that Γ ⊂ M . We consider the Legendre immersion
φ : ([A], q, p, r) ∈ M → [A 0 ∧ A 1 ] ∈ K.(72)
Since our arguments are local in nature, we suppose that φ is one-to-one and we identify M with its image M = φ( M ). Then, the map
A : ([A], q, p, r) ∈ M → [A] ∈ G(73)
is the normal frame field along M . From this we deduce that M is a Lie-minimal surface. By construction, Γ is contained in M and R = A| Γ , α 1 | Γ = −α 2 | Γ = µ, q i = q i | Γ , p i = p i | Γ , r i = r i | Γ , i = 1, 2. (74)
In particular, the 1-form α 1 + α 2 vanishes identically along Γ. Combining (68) and (74) we deduce w = 1 3 (p 1 − p 2 )| Γ , f = −3(q 1 + q 2 )| Γ , σ L = σ 0 | Γ .
Let us recall that the curvature sphere mappings σ 0 and σ 1 are represented by [A 0 ] and by [A 1 ] respectively. On the other hand, L is spanned by the first row vector of the framing R so that σ L = [ R 0 ]. This implies
σ 1 | Γ = [A 0 ]| Γ = [ R 0 ] = σ L .
From this we infer that M satisfies the required properties. The uniqueness of M follows from the uniqueness of the real-analytic integral manifold M containing Γ. ▽
Definition 1. 6
6We say that M is non-degenerate if σ 0 and σ 1 are immersions of M into the Lie quadric. Theorem 1.7 Let M ⊂ K be a non-degenerate Legendre surface. Then there exist a unique lift A : M → G to the group G satisfying the Pfaffian equations
. Moreover, if A and A are fifth order frame fields on an open neighborhood U , then A = ǫA, where ǫ = ±1. This implies that the canonical lift A : M → G is defined by A| U = [A], for every fifth order frame field A : U → G. ▽ Definition 1.8 The map A : M → G is said to be the normal frame field along M .
Remark 1. 9
9If M is the contact lift of f : M → R 3 and if (u 1 , u 2 ) are curvature lines coordinates then α 1 and α 2 coincide with the Blaschke'
Gauss map and the Euler-Lagrange equations Definition 2.1 The Gauss map of a generic Legendre surface M ⊂ K is defined by D : ℓ ∈ M → [A 0 (ℓ) ∧ A 3 (ℓ) ∧ A 5 (ℓ)], ∀ℓ ∈ M, where A = [A] is the normal frame field along M .
Remark 2. 2
2If we consider on M the quadratic form Φ = α 1 α 2 , then(9) and(12) imply that the Gauss map D : (M, Φ) → (D, g D ) is an isometric immersion. Definition 2.3 A non-degenerate Legendre surface M ⊂ K is said to be Lie-minimal if it is a critical point of the functional It is known (cfr.[4]) that Lie-minimal surfaces are characterized by the harmonicity of the Gauss map.
Theorem 2. 5
5Let M ⊂ K be a non-degenerate Legendre surface. Then, M is Lieminimal if and only if
2
Here T (M ) is viewed as a sub-bundle of D * T (D)
Proposition 3. 2
2Let (Γ, L) be a polarized Legendre curve. Then, there exist a unique map R : Γ → G such that
R 5 (
5Γ, L) with fiber Z 2 = {±I} and henceforth there exist a map R : Γ → G such that R| U = [R], for every fifth order frame field R : U → G. From (57), (59),(60),(61) and (63) it follows that R satisfies the required properties. ▽ Definition 3.3 The lift R : N → G is said to be the Frenet frame field of (Γ, L). The 1-form µ is the Lie-invariant line element and the functions k 0 , k 1 , k 2 and k 3 are the generalized curvatures of (Γ, L).
Remark 3. 4
4This proposition shows that polarized Legendre curves are completely determined, up to the action of the Lie sphere group, by the curvatures k 0 , ..., k 3 .
i.e. an integral element such that a 1 a 2 = 0 4 We use the notation dV = V ′ dζ, where dζ is a nowhere vanishing 1-form on N .
We then haveWe define q i , p i , r i : Γ → R, i = 1, 2, byLet us now consider the embedding Γ = ( R, q, p, r) : Γ → P.
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Lie sphere geometry: with applications to submanifolds. T E Cecil, Springer-VerlagNew YorkT.E.Cecil, Lie sphere geometry: with applications to submanifolds, Springer-Verlag, New York (1992).
Web Geometry. S S Chern, Bull. Amer. Math. Soc. 6S.S.Chern, Web Geometry, Bull. Amer. Math. Soc., 6(1982), pp. 1-8.
Moving Frames. S S Chern, Elie Cartan et les Mathematiques d'aujourd'hui. S.S.Chern,Moving Frames, in Elie Cartan et les Mathematiques d'aujourd'hui, Asterisque, (1985), pp. 67-77.
Real hypersurfaces in complex manifolds. S S Chern, J Moser, Acta Math. 133S.S.Chern, J.Moser, Real hypersurfaces in complex manifolds, Acta Math., 133(1974), pp. 219-271.
Lie sphere geometry and integrable systems. E Ferapontov, Tohoku Math. J. 52E.Ferapontov, Lie sphere geometry and integrable systems, Tohoku Math. J., 52(2000), pp. 199-233.
On Lie's higher sphere geometry, L'Einsegnement mathém., t.XXV, fasc. J P Fillmore, J.P.Fillmore, On Lie's higher sphere geometry, L'Einsegnement mathém., t.XXV, fasc. 1-2(1978), pp. 77-114.
Variations on a theme by Kepler. V Guillelmin, S Sternberg, Colloquium Publications AMS42Providence, Rhode IslandV.Guillelmin, S.Sternberg,Variations on a theme by Kepler, Colloquium Publications AMS, Vol.42, Providence, Rhode Island 1990.
The Evaston Colloquium Evaston, Ill 1894. F Klein, Lectures on Mathematics. Mac Millan and CoF.Klein,Lectures on Mathematics, The Evaston Colloquium Evaston, Ill 1894, Mac Millan and Co.
Lie Sphere Geometry of Wave Fronts and Moving Frames. E Musso, preprintE.Musso, Lie Sphere Geometry of Wave Fronts and Moving Frames, preprint (2003).
Laguerre Geometry of Surfaces with Plane Lines of Curvature. E Musso, L Nicolodi, Abh.Math.Sem.Univ. Hamburg. 69E.Musso, L.Nicolodi, Laguerre Geometry of Surfaces with Plane Lines of Curvature, Abh.Math.Sem.Univ. Hamburg 69(1999), pp. 123-138.
R Penrose, W Rindler, Spinors and Space-Time. Cambridge University PressR.Penrose, W.Rindler,Spinors and Space-Time, Cambridge University Press, 1984.
. J M Souriau, Sur La Variete De Kepler, Atti Acc, Scienze di Torino. 117J.M.Souriau, Sur la variete de Kepler, Atti Acc. Scienze di Torino 117(1983), pp. 369-418.
|
[] |
[
"ABTREE: AN ALGORITHM FOR SUBGROUP-BASED TREATMENT ASSIGNMENT",
"ABTREE: AN ALGORITHM FOR SUBGROUP-BASED TREATMENT ASSIGNMENT"
] |
[
"Derek Feng es:[email protected] \nDepartment of Statistics\nYale Institute for Network Science Yale University\nNew HavenCT\n\nDepartment of Mathematics and Statistics Amherst College\nXIAOFEI WANG\nAmherstMA\n"
] |
[
"Department of Statistics\nYale Institute for Network Science Yale University\nNew HavenCT",
"Department of Mathematics and Statistics Amherst College\nXIAOFEI WANG\nAmherstMA"
] |
[] |
Given two possible treatments, there may exist subgroups who benefit greater from one treatment than the other. This problem is relevant to the field of marketing, where treatments may correspond to different ways of selling a product. It is similarly relevant to the field of public policy, where treatments may correspond to specific government programs. And finally, personalized medicine is a field wholly devoted to understanding which subgroups of individuals will benefit from particular medical treatments. We present a computationally fast tree-based method, ABtree, for treatment effect differentiation. Unlike other methods, ABtree specifically produces decision rules for optimal treatment assignment on a per-individual basis. The treatment choices are selected for maximizing the overall occurrence of a desired binary outcome, conditional on a set of covariates. In this poster, we present the methodology on tree growth and pruning, and show performance results when applied to simulated data as well as real data.
| null |
[
"https://arxiv.org/pdf/1605.04262v1.pdf"
] | 88,521,716 |
1605.04262
|
b39ec6153530ffdf7905bfbdd6c54e2af27252c7
|
ABTREE: AN ALGORITHM FOR SUBGROUP-BASED TREATMENT ASSIGNMENT
13 May 2016
Derek Feng es:[email protected]
Department of Statistics
Yale Institute for Network Science Yale University
New HavenCT
Department of Mathematics and Statistics Amherst College
XIAOFEI WANG
AmherstMA
ABTREE: AN ALGORITHM FOR SUBGROUP-BASED TREATMENT ASSIGNMENT
13 May 20161 2 ABTREE: AN ALGORITHM FOR SUBGROUP-BASED TREATMENT ASSIGNMENTand phrases CARTsubgroup analysisA/B testingsegmentationtreespersonalized medicinetreatment effect differentiationheterogeneous treatment effectstreatment efficacy
Given two possible treatments, there may exist subgroups who benefit greater from one treatment than the other. This problem is relevant to the field of marketing, where treatments may correspond to different ways of selling a product. It is similarly relevant to the field of public policy, where treatments may correspond to specific government programs. And finally, personalized medicine is a field wholly devoted to understanding which subgroups of individuals will benefit from particular medical treatments. We present a computationally fast tree-based method, ABtree, for treatment effect differentiation. Unlike other methods, ABtree specifically produces decision rules for optimal treatment assignment on a per-individual basis. The treatment choices are selected for maximizing the overall occurrence of a desired binary outcome, conditional on a set of covariates. In this poster, we present the methodology on tree growth and pruning, and show performance results when applied to simulated data as well as real data.
Introduction
Knowing one's customers is of great importance to small and large businesses alike. The field of market segmentation is devoted to the identification of subgroups of a target audience whose consumption behaviors -direct mail response rates, purchases, frequency of store visits, types of purchases, and so on -differ. In many cases, the goal is to predict a categorical or quantitative response variable for individual targets based on various predictors. Tree-based methods are particularly desirable for tackling such problems, because they gracefully handle the problem of variable selection and present outputs in an intuitive way. [14] advocated the use of Classification and Regression Trees (CART), first introduced in [2], for classifying potential customers into one of two classes. [5] argued that Chi-Square Automatic Interaction Detection (CHAID), first introduced in [7], performs equally well and is simpler to use. [12] showed that the closely related Multivariate Automatic Interaction Detection (MAID) can be used to segment customers when the response variable is quantitative and multivariate.
With the advent of the internet, the E-commerce boom presents additional opportunities, not only for understanding customers, but also for responding to customer preferences. On a website, it is possible to experiment with two alternate store-fronts, randomly displaying one or the other to incoming visitors, and directly measure success from each. The field of A/B testing addresses the question: when there are two options to choose from, A and B, which one produces the better outcome? Given n visits, each individual visit to a website generates data consisting of (Y i , X i , T i ), where X i is a multivariate vector of characteristics relating to the user, such as type of web browser, operating system, and time of visit, T i ∈ {A, B} is the version of the website displayed to the user, and Y i corresponds to some profit measure. Profit measures can be quantitative -such as visit duration or number of ad clicks -or categorical -such as whether the user made a purchase. The standard problem involves determining the single version T ∈ {A, B} that maximizes expected profit across all future users. Specifically,
T = argmax t∈{A,B} ∞ n+1 E(Y i |T i = t).
One drawback of this approach is that it inherently ignores individual characteristics X i . For this reason, it is worth reformulating A/B testing as a market segmentation problem; rather than choosing a single T to be applied to all future customers, differences in customer characteristics should lead to different preferences, and therefore a meaningful objective is to arrive at a series of decision rules for T i tailored to the i-th customer (with characteristic vector x i ):
(1)
T i = argmax ti∈{A,B} E(Y i |X i = x i , T i = t i ), i > n.
In this paper, we present a novel method, ABtree, that yields a decision tree for determining T i . ABtree assumes Y i ∈ {0, 1} and covariates X are quantitative, categorical, or both. We begin by reviewing relevant literature in related fields, followed by describing our methodology in detail, fully specifying our algorithm with the splitting and pruning procedures. We then evaluate the performance of our method on a diverse set of simulated A/B testing data. We also show the results of ABtree on a real-world employment assistance dataset. We finally conclude with a discussion of the extensibility of A/B testing to other problems.
Related Work
We find parallels in recent advances in personalized medicine, wherein a chief concern is determining subgroups of patients who would benefit from a particular medical treatment. In this setup, X are the patient characteristics, such as weight, blood pressure, and cholesterol, and T are the treatments received. The response variable Y in such endeavors is typically far more complex. For example, when dealing with patient outcomes, tree-based methods often need to consider survival models and censoring [11]. A few approaches assume quantitative Y [13,3,8]. We briefly review the approaches that consider categorical Y . The Virtual Twins (VT) method takes a two-step approach, first using random forests to predict probabilities of success for each individual assuming both treatments, and then using the differences in those probabilities as a response variable in CART [4]. Another method, Subgroup Identification based on Differential Effect Search (SIDES), yields regions in the covariate space that are likely to have different effect sizes via repeated hypothesis testing [10]. Both SIDES and VT target the exploration of possible subgroup structures but neither proposes the assignment of treatments.
The area of treatment effect heterogeneity has received considerable and continued attention from myriad sources [1,6,15]. In particular, [6] proposes an SVM-based method for modeling treatment effects while considering multi-way interactions between the treatment and covariates. We note that while treatment effect heterogeneity is a driving force in treatment selection, we are not interested in the magnitude of treatment effects per se, but rather in an optimal assignment of treatments for the utilitarian goal of profit maximization. None of these methods address this objective.
Methodology
Formally, suppose we had access to
n training examples {(Y i , X i , T i ) : i = 1, . . . , n}, where the profit Y i ∈ {0, 1} is the response variable, X i = (X i1 , . . . , X ip ) ∈ X is the associated p-dimensional covariate vector, and T i ∈ {A, B} indicates the treat- ment received.
An individualized treatment rule is a function π : X → {A, B} that, given a customer with X = x, yields a treatment π(x) for that customer. Our goal is to find the optimal choice of treatment rule π -that is, the π that maximizes the expected profit,
(2) π = arg max π E [Y |X = x, T = π(x)] .
This optimization is intractable though, so we instead adopt the heuristics of decision trees, and consider a data-driven approach to finding a good approximation to π . Hence, we seek aπ that is piecewise constant on recursive binary partitions of the covariate space, and is comparable to π in expected profit. That is,π estimates a constantπ(x) to all x ∈ S j . For conciseness, we defineπ(S j ) :=π(x) for x ∈ S j , which yields the following simplified expected profit for any individual with x ∈ S j ,
P (x,π(S j )) := E [Y |X = x, T =π(S j )]
.
Given the n training examples {(Y i , X i , T i ) : i = 1, .
. . , n}, an empirical counterpart to this expectation is
(3)P (S j ,π(S j )) := 1 | {i : x i ∈ S j , T i =π(S j )} | i:x i ∈Sj ,Ti=π(Sj ) y i ,
which is the empirical mean profit for data points where the covariate is in the subspace S j , and the treatment applied isπ(S j ). Thus, the optimal choice ofπ(S j ) to maximize (3) has a simple solution: pickπ(S j ) to be the treatment t ∈ {A, B} with the largerP (S j , t).
More generally, by aggregating across all subspaces, the optimal choice ofπ is
max π j |S j | ·P (S j , π(S j )) = j |S j | max P (S j , A),P (S j , B)(4)
It will be helpful later to define the summand in (4):
Q(S j ) := |S j | max P (S j , A),P (S j , B) .
The mechanism for determining the set of regions S is described in the next subsections. For the most part, we follow the conventions laid out in CART [2]:
we build an initial tree, prune it, and then select the best tree via a holdout set (alternatively, through cross-validation). Thus, to fully delineate our algorithm, it suffices to describe the splitting criteria at each node, the stopping conditions for the tree growth, and the pruning technique.
3.1. Splitting Criteria. A split is induced by a threshold τ on the k-th predictor of X i , which we denote X ik . If X ik is continuous, then the split corresponds to the binary question X ik ≤ τ . When X ik is categorical (V defining the set of all possible categories), the split corresponds to the binary question X ik ∈ U ⊆ V . While this poses computational issues when the search is made over the power set of V , there are ways to circumvent such computational barriers. The general technique is to order the categories according to the mean response within each category, and then proceed as if X ik were ordinal. Another approach is to simply consider splits that correspond to the binary question X ik = τ . We adopt the latter approach, as the implementational advantages outweigh the sometimes marginal gains in tree performance.
Suppose we are in the subspace S ⊆ X . Let us fix a choice of predictor k, and assume that X ik is continuous for now. Then, the goal is to pick a τ such that the subspaces induced by the split are maximal with respect to (4). In other words, we are solving the following optimization:
(5) max τ {Q(L τ,k ∩ S) + Q(R τ,k ∩ S)} , where L τ,k = {X i : X ik ≤ τ } , R τ,k = {X i : X ik > τ }. Note thatP (S, T ) involves
only two simple quantities,
| {i : x i ∈ S, T i = T } | =: n T S , i:x i ∈S,Ti=T y i =: y T S .
Thus, (5) can be rewritten as There is a minimum split parameter, which is the minimum number of observations needed from both treatment groups in a node in order for a split to be considered.
Relative to standard methods, our parameter effectively doubles the number of observations needed in a node. Similarly, our minimum bucket parameter -the minimum number of observations in a terminal node -applies simultaneously for both treatments. This also ensures that the quantities in (6) are well defined.
3.3.
Pruning. We follow the pruning technique proposed by [2], whereby a sequence of optimal subtrees is formed by iteratively removing the weakest link of the tree. In our case, this corresponds to the pair of leaf nodes having a common parent that produce the smallest increase in (5).
Having formed a sequence of trees, the optimal tree can be chosen either using a hold-out set, or by cross-validation. Given the nature of the applications for our algorithm, where data is by no means limited in quantity, we recommend using a hold-out set. Unfortunately, as this is an unsupervised learning task, there is no clear measure for the performance of a tree. Our proxy for the performance is the fraction of treatment assignments predicted by the tree that match the assignment in the hold-out set.
Simulations
In this section, we evaluate ABtree's treatment selection results to those of (1) a random treatment allocation and (2) the existing A/B test treatment allocation.
We measure performance using a mean profit score. In the next subsections, we Uniform(0, 1)
Logit(p) log p 1−p φ k (X, T ) Profit Y Bernoulli e p 1+e p
begin by describing the simulation data and the comparison procedure, with a detailed review of the methods under consideration. We then discuss our specific scoring method before showing the actual simulation results.
We implemented our simulations in R. An R package ABtree is under development.
Simulation Data.
We consider generative models for our simulated data.
Tables 1 and 2 summarize the parameters used. To mimic the conditions of A/B testing, we assign T to be either 0 or 1 with equal probability. The covariates are generated independently from T . φ k fully specifies the relationship between the probability of success p, and a set of covariates X and choice of treatment T . We can then simulate from Bernoulli(p) to determine the outcome Y .
4.2.
Procedure. For each φ k setting, we simulate 50 datasets. Each dataset has 5000 rows, five covariates, one treatment, and one response, the profit. We divide the dataset into three parts: 50% training set, 25% validation set, and 25% test set. For each method, we proceed in the following way on a given dataset:
(1) Model-based treatment assignment. Use the model to generate treatment choices T on the test set.
(2) Counterfactual simulation. SimulateỸ using φ(X, T ).
(3) Evaluation. Calculate mean profitP (Ỹ ).
Step 1 will vary in complexity depending on the method used. We discuss steps 2 and 3 in greater detail in the next subsection.
The four methods under consideration appear below.
(1) Random assignment. Treatments are assigned at random to all individuals regardless of X. The model is simply Bernoulli(0.5).
(2) A/B testing. A single treatment is assigned to all T i = T . Without loss of generality, we assume the control and treatment options are A and B
respectively. We run a one-sided hypothesis test on the combined training and validation set to determine if there is evidence at the significance level of 0.05 that the treatment yields a higher average profit. If so, we assign T = B; otherwise, we assign T = A. (4) ABtree. We use the same covariates as in the previous approach. We run ABtree on the training set and prune the resulting tree using the validation set. We then generate treatment choices T i for each individual in the test set.
Performance Evaluation.
For either choice of treatment T i assigned to observation (Y i , X i , T i ) under regime k in the test set, we can simulate a counterfactual profitỸ i via Bernoulli(logit −1 (φ k (X i , T i ))). Consistent with the objective of A/B testing, we score each method on basis of mean counterfactual profit, defined as:
P (Ỹ ) := 1 nỸ
Intuitively, a method that assigns treatment choices resulting in higher average profits is preferred. Therefore, we equateP (Ỹ ) with better performance. We note that such a performance score is only meaningful in simulation settings where treatment effects are pre-specified. In real data examples, counterfactual profits
k φ k (X, T ) 1 2T · sgn(X i1 − 0.2) + X i3 + X i4 2 2T · sgn(X i1 ) · sgn(X i2 − 0.3) + X i2 + 0.2X i3 + 0.5X i4 3 3T · sgn(X i1 ) + 2X i2 + X i3 + 0.5X i5 4 3T · sgn(X i1 ) + T · sgn(X i2 ) + X i3
cannot be calculated and therefore accurate comparisons against other methods are impossible. Figure 1. It is immediately clear from the figure that our algorithm comfortably beats both random assignment and the standard procedure in A/B testing. This means ABtree is successfully capturing the treatment effects found in the φ k . The performance is comparable across the different φ k , though φ 1 and φ 2 are noticably lower than φ 3 and φ 4 , owing in part to the increased signal of the treatment effect (3 vs 2 in the coefficient). It is worth pointing out that ABtree is able to handle the existence of external noise (as in φ 3 ). On the other hand, the A/B testing procedure fails to beat random assignment for all cases other than φ 1 .
Simulation Results. A boxplot of the results is shown in
An interesting observation is that the improvements due to pruning are very minor at best. Given the computational cost associated with pruning, it might be prudent in certain applications to forgo the pruning step in lieu of using a (considerably) larger sample size.
Application: National Supported Work Demonstration
Due to the proprietary nature of A/B testing data, we are unable to provide an example of ABtree applied to an actual A/B testing dataset. We instead examine a National Supported Work Demonstrated dataset [9], subsequently explored in [6]. The National Supported Work Demonstration (NSW) was a large-scale national and private program designed to provide work experience for disadvantaged vidual earnings increased during the term of the experiment. We refer interested readers to [9] for full details.
We ran ABtree to model whether individual earnings increased conditional on treatment group with covariates of subject-specific information recorded at the start of the study in 1975; these include quantitative variables of age, years of education, and log of income, as well as categorical variables of race, marital status, attainment of high school degree, and unemployment status. Figure 2 shows the results, starting with an initial split by age above and below 18. The tree suggests that individuals younger than or equal to 18 would not benefit from the NSW program. Among those older than 18 years of age, the sample is further divided by log of earnings in 1975. Most interestingly, despite the wealth of information passed to the algorithm, age is the most important covariate for deciding whether an individual will benefit from the program.
We note that we cannot compare our results with those in [6] whereas ours is aimed at assigning treatments to maximize the occurrence of a desired outcome.
Discussion
In this paper, we presented a novel method, ABtree, for identifying subgroups of individuals that respond more positively to one treatment versus another. ABtree was motivated by the A/B testing problem; in simulations, we showed that ABtree is superior to the existing A/B testing solution that completely ignores individual characteristics. For contextual purposes, we opted to focus on the case where our response variable is binary. However, our results extend naturally with minimal modifications to the continuous case.
In addition to its ability to take a decision-oriented approach on analyzing treatment effects, ABtree has many core strengths. For one, as a tree-based method, it has the advantage of yielding highly interpretable decision rules that are easy to understand. The degree of interpretability can be adjusted by increasing or decreasing the maximum tree depth. In cases where interpretability is not a chief consideration, ensemble learning approaches such as bagging and random forests can be directly applied with ABtree, which helps to reduce variance and produces smooth decision boundaries. Secondly, the proposed treatment assignment step is entirely apparent given the ABtree structure; differences in profit within the terminal leaves are exposed in a plot of the final, preferably pruned, tree. Finally, ABtree has no additional tuning parameters other than the standard tree algorithm parameters such as maximum depth size, minimum node size for considering a split, and minimum node size in a leaf, and runs remarkably fast.
In addition to use in A/B testing, we also showed that ABtree produces intuitive and credible results on a real-world dataset from the NSW Demonstration study.
In fact, there are numerous disciplines in which it is desirable to identify subgroups of individuals for whom a particular treatment is beneficial. In policymaking, for example, it is important to identify subgroups of individuals that will benefit most from a particular subsidy, so that the rules for qualifying for the subsidy can be used to optimize for efficacy. In personalized medicine, as another example, it is important to identify subgroups of individuals that will benefit from a particular drug; ABtree can assist with identifying such individuals.
Acknowledgments
We would like to thank Sahand N. Negahban for helpful suggestions.
( 3 )
3ABtree (no pruning). We run ABtree on the combined training and validation set. In fitting the model only the first four covariates {X j } 4 j=1 are used. For some φ k , the fifth X 5 is important in determining the treatment effect. We exclude X 5 from modeling to mimic real world examples where true drivers of treatment effects may not be measured. We then generate treatment choices T i for each individual in the test set.
Figure 1 .
1Boxplot of mean profit across the different methods and different response functions φ k workers in the hopes of improving employability. A randomized experiment was conducted in the mid-1970s, in that qualified individuals were randomly allocated to a treatment group, receiving the full benefits of the program, and a control group, receiving nothing. The outcome of interest is a binary indicator of whether indi-
Figure 2 .
2NSW segmentation results via ABtree. Treatment A is the control (no benefits) and treatment B is the provision of assistance under the program. The yellow boxes in the leaf nodes summarize the proportion of success and sample size under each treatment.
maxτ
max
y A
S∩L τ,k
n A
S∩L τ,k
,
y B
S∩L τ,k
n B
S∩L τ,k
, + max
y A
S∩R τ,k
n A
S∩R τ,k
,
y B
S∩R τ,k
n B
S∩R τ,k
,
Because the terms in (6) depend only on simple counts, this optimization can be
solved efficiently. Thus, we iterate over the p predictors and pick the one with
the largest value of (5). For categorical predictors, the only difference lies in the
definition of L τ,k , R τ,k .
3.2. Tree Growth. To ensure that the comparisons performed in (6) are fair ones,
and we have a good representation of both treatments in every node of the tree, we
adopt similar stopping conditions to those found in standard decision tree methods.
Table 1 .
1Simulation Setup OverviewVariable
Notation Distribution
Treatment T
Bernoulli(0.5)
Covariates {X j }
5
j=1
Table 2 .
2Response Function Summary
because their method targets estimating treatment effect heterogeneity on a per-individual basisage
log.income75
> 18
p_A: 0.34
n_A:135
p_B: 0.41
n_B:111
> 7.7
age
<= 7.7
age
> 19
age
> 20
p_A: 0.55
n_A:42
p_B: 0.85
n_B:27
> 30
race
<= 30
p_A: 0.89
n_A:18
p_B: 0.73
n_B:15
!= Black
p_A: 0.53
n_A:103
p_B: 0.7
n_B:74
== Black
p_A: 0.68
n_A:19
p_B: 0.55
n_B:11
<= 20
p_A: 0.62
n_A:29
p_B: 0.91
n_B:11
<= 19
p_A: 0.73
n_A:79
p_B: 0.65
n_B:48
<= 18
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|
[] |
[
"SHARP SOBOLEV TRACE INEQUALITIES FOR HIGHER ORDER DERIVATIVES",
"SHARP SOBOLEV TRACE INEQUALITIES FOR HIGHER ORDER DERIVATIVES"
] |
[
"Qiaohua Yang "
] |
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Motivated by a recent work of Ache and Chang concerning the sharp Sobolev trace inequality and Lebedev-Milin inequalities of order four on the Euclidean unit ball, we derive such inequalities on the Euclidean unit ball for higher order derivatives. By using, among other things, the scattering theory on hyperbolic spaces and the generalized Poisson kernel, we obtain the explicit formulas of extremal functions of such inequations. Moreover, we also derive the sharp trace Sobolev inequalities on half spaces for higher order derivatives. Finally, we compute the explicit formulas of adapted metric, introduced by Case and Chang, on the Euclidean unit ball, which is of independent interest.
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1901.03945
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SHARP SOBOLEV TRACE INEQUALITIES FOR HIGHER ORDER DERIVATIVES
13 Jan 2019
Qiaohua Yang
SHARP SOBOLEV TRACE INEQUALITIES FOR HIGHER ORDER DERIVATIVES
13 Jan 2019
Motivated by a recent work of Ache and Chang concerning the sharp Sobolev trace inequality and Lebedev-Milin inequalities of order four on the Euclidean unit ball, we derive such inequalities on the Euclidean unit ball for higher order derivatives. By using, among other things, the scattering theory on hyperbolic spaces and the generalized Poisson kernel, we obtain the explicit formulas of extremal functions of such inequations. Moreover, we also derive the sharp trace Sobolev inequalities on half spaces for higher order derivatives. Finally, we compute the explicit formulas of adapted metric, introduced by Case and Chang, on the Euclidean unit ball, which is of independent interest.
Introduction
It is well known that the Sobolev inequalities and sharp constants play an important role in problems in analysis and conformal geometry. An elementary example is the following classical Sobolev inequality on the standard sphere (S n , g S n ) for n ≥ 3:
1 ω n S n |f | 2n n−2 dσ n−2 n ≤ 4 n(n − 2)ω n S n | ∇f | 2 dσ + 1 ω n S n |f | 2 dσ, (1.1) where dσ is Lebesgue measure on S n , ω n = 2π n+1 2 Γ( n+1 2 ) is the volume of S n and ∇ is the sphere gradient on (S n , g S n ). Using the conformal invariance, one observes inequality (1.1) is equivalent to the sharp Sobolev inequality on R n given as follows
Γ( n+2 2 ) Γ( n−2 2 ) ω 2 n n R n |f | 2n n−2 dx n−2 n ≤ R n |∇f | 2 dx, (1.2)
where ∇f is the gradient of f with respect to the Euclidean metric. In a limiting case, namely n = 2, (1.1) takes the form of Moser-Onfori inequality ( [34,36])
log 1 4π S 2 e f dσ ≤ 1 16π S 2 | ∇f | 2 dσ + 1 4π S 2 |f | 2 dσ. (1.3)
Inequality (1.3) has been widely used in analysis and conformal geometry, in particular, in the problem of prescribing Gaussian curvature on the sphere (see [9,37,38,39]).
Another example is the Sobolev trace inequality. Denote by B n+1 the unit ball on Euclidean space R n+1 with S n as the boundary. The Sobolev trace inequality on B n+1 reads as follow (see [18]): for f ∈ C ∞ (S n ) and n ≥ 2,
n − 1 2 ω 1 n n S n |f | 2n n−1 dσ n−1 n ≤ B n+1 |∇v| 2 dx + n − 1 2 S n |f | 2 dσ, (1.4)
where v is a smooth extension of f to B n+1 . By the conformal invariance, one observes inequality (1.4) is also equivalent to the sharp Sobolev inequality on half space (see [18]):
Γ( n+1 2 ) Γ( n−1 2 ) ω 1 n n R n |U(x, 0)| 2n n−1 dx n−1 n ≤ R n+1 +
|∇U(x, y)| 2 dxdy, (1.5) where R n+1 + = R n × R + = {(x, y) : x ∈ R n , y > 0}. In a limiting case, namely n = 1, (1.4) becomes the classical Lebedev-Milin inequality ( [30])
log 1 π S 1 e f dσ ≤ 1 4π B 2 |∇v| 2 dx + 1 π S 1 |f | 2 dσ. (1.6)
Such Sobolev trace inequalities and Lebedev-Milin inequality have also been widely used in analysis and geometry, such as Yamabe problem on manifolds with boundary (see [19]), the Bieberbach conjecture ( [16]) and the compactness of isospectral planar domains (see [37,38,39]).
Notice that the operators involved in (1.1)-(1.6), either the Laplace operator or the conformal Laplace operator, are order two. Recently, the role played by these operators of order two has been extended to operators of higher order, such as Paneiz operator, poly-Laplacian and GJMS operator. In particular, Beckner [2] established the higher order Sobolev inequality on the standard sphere (S n , g S n ). We state it as follow Theorem 1.1 (Beckner). Let ∆ be the Laplace-Beltrame operator on the standard sphere (S n , g S n ) and define , c ∈ R, ζ ∈ B n+1 , ξ ∈ S n .
If γ = n 2 , then ln 1 ω n S n e f −f dσ ≤ 1 2n!ω n S n f P n f dσ, (1.9) where f = 1 ωn S n f dσ is the integral average of f on S n . Equality holds only for functions of the form −n ln |1 − ζ, ξ | + c, ζ ∈ B n+1 , ξ ∈ S n , c ∈ R.
By the conformal invariance, one observes inequality (1.8) is equivalent to the following sharp Sobolev inequality on R n ( [32], see also [15]):
c(λ 2 + |x − x 0 | 2 ) − n−2γ 2 , x ∈ R n ,
where c ∈ R, λ > 0 and x 0 is some point in R n .
Recently, Ache and Chang [1] established sharp trace Sobolev inequality of order four on B n+1 for n ≥ 3. As an aplication, they used this inequality to characterize the extremal metric of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on B 4 (see [11,12]). We state the results as follow. Theorem 1.3 (Ache and Chang). Given f ∈ C ∞ (S n ) with n > 3, suppose v is a smooth extension of f to the unit ball B n+1 which also satisfies the Neumann boundary condition
∂v ∂n S n = − n − 3 2 f. (1.11)
Then we have the inequality
2 Γ( n+3 2 ) Γ( n−3 2 ) ω 3 n n S n |f | 2n n−3 dσ n−3 n ≤ B n+1 |∆v| 2 dx + 2 S n | ∇f | 2 dσ + (n + 1)(n − 3) 2 S n |f | 2 dσ,(1.e 3(f −f ) dσ ≤ 3 16π 3 B 4 |∆v| 2 dx + 3 8π 2 S 3 | ∇f | 2 dσ. (1.14)
Moreover, equality holds if and only if v is a biharmonic extension of a function of the form − log |1 − z 0 , ξ | + c, where c is a constant, ξ ∈ S 3 , z 0 is some point in B 4 and v satisfies the boundary condition (1.13).
The proof of Theorem 1.3 and 1.4 relies on the use of scattering theory on hyperbolic space (B n+1 , g B ), where g B = 4 (1−|x| 2 ) 2 g 0 and g 0 = |dx| 2 is the Euclidean metric, and the right choice of distance function and adapted metric. We remark that the adapted metric, introduced by Case and Chang [7], is a 'natural' metric in the study of Sobolev inequalities (see [1,13,14]). The explicit formulas of adapted metric is computed by Ache and Chang [1] only in the case γ = 1 2 and γ = 3 2 . To the best of our knowledge, the explicit formulas of adapted metric is unknown for the rest of cases.
Very recently, Ngô, Nguyen and Pham [35] show that (1.12) is equivalent to the following sharp Sobolev trace inequality on half space via Möbius transform:
Theorem 1.5. Let U ∈ W 2,2 (R n+1 + ) be satisfied the Neumann boundary condition ∂ y U(x, 0) = 0, (1.15) Where W 2,2 (R n+1 + )
is the usually Sobolev space. Then we have the sharp trace inequality
2 Γ( n+3 2 ) Γ( n−3 2 ) ω 3 n n R n |U(x, 0)| 2n n−3 dx n−3 n ≤ R n+1 + |∆U(x, y)| 2 dxdy. (1.16)
Furthermore, equality in (1.10) holds if and only if U is a biharmonic extension of a function of the form c(1 + |x − x 0 | 2 ) −(n−3)/2 , where c is a constant, x ∈ R n , x 0 is some fixed point in R n and U fulfills the boundary condition (1.15).
In the same paper, Ngô, Nguyen and Pham [35], among other results, propose a slightly different approach to prove Sobolev trace inequality of order six, while Case and Luo [8] obtained, among other results, the same sharp Sobolev trace inequalities by deeply work of on the boundary operators. However, it seems that the argument in [35,8] would become increasingly delicate when the order of the operator is large. A clear question is "What is the situation for sharp Sobolev trace inequality of higher order?" Another question is " what is the explicit formula of extremal function of such inequations?" In this paper, we shall give the answer of both questions.
The main results of this paper are the following three theorems.
Theorem 1.6. Let n > 3 and m ≥ 1 with 2m + 1 < n. Given f ∈ C ∞ (S n ) , suppose v is a smooth extension of f to the unit ball B n+1 which also satisfies the Neumann boundary condition:
∆ k v| S n = (−1) k Γ(m + 1)Γ(m − k + 1 2 ) Γ(m + 1 2 )Γ(m − k + 1) P 2m+1 P 2m+1−2k f ; 0 ≤ k ≤ [ m 2 ]; ∂ ∂n ∆ k v| S n =(−1) k+1 n − 1 − 2m + 2k 2 · Γ(m + 1)Γ(m − k + 1 2 ) Γ(m + 1 2 )Γ(m − k + 1) P 2m+1 P 2m+1−2k f, 0 ≤ k ≤ [ m − 1 2 ].
(1.17)
Then we have the inequality
Γ(m + 1)Γ( 1 2 ) Γ(m + 1 2 ) Γ( n+2m+1 2 ) Γ( n−2m−1 2 ) ω 2m+1 n n S n |f | 2n n−2m−1 dσ n−2m−1 n ≤ B n+1 |∇ m+1 v| 2 dx + S n f T m f dσ, (1.18) where ∇ m+1 = ∆T m = n − 1 2 Γ(m + 1)Γ( 1 2 ) Γ(m + 1 2 ) P 2m+1 P 1 + m−1 2 k=1 (m − 2k) Γ(m + 1) 2 Γ(m + 1 2 ) 2 · Γ(k + 1 2 )Γ(m − k + 1 2 ) Γ(k + 1)Γ(m − k + 1) P 2 2m+1 P 2m+1−2k P 2k+1 ;
if m is a even integer, then
T m = n − 1 2 Γ(m + 1)Γ( 1 2 ) Γ(m + 1 2 ) P 2m+1 P 1 + n − 1 − m 2 Γ(m + 1)Γ( m+1 2 ) Γ(m + 1 2 )Γ( m 2 + 1) 2 P 2 2m+1 P 2 m+1 + m 2 −1 k=1 (m − 2k) Γ(m + 1) 2 Γ(m + 1 2 ) 2 Γ(k + 1 2 )Γ(m − k + 1 2 ) Γ(k + 1)Γ(m − k + 1) P 2 2m+1 P 2m+1−2k P 2k+1 .
Moreover, equality holds if and only if
v(x) = c S n (1 − |x| 2 ) 2m+1 |x − ξ| n+1+2m |1 − x 0 , ξ | 2m+1−n 4 dσ, (1.19)
where c is a constant and x 0 is some point in B n+1 . When f = 1, inequality (1.18) is attained by the function v(x) = m k=0
( n−1 2 −m) k (−m) k (−2m) k (2ρ) k k! ,
where (a) k is the rising Pochhammer symbol defined in Section 2.
Theorem 1.7. Let n ≥ 3 be an odd integer. Given f ∈ C ∞ (S n ), suppose v is a smooth extension of f to the unit ball B n+1 which also satisfies the Neumann boundary condition
∆ k v| S n = (−1) k Γ( n+1 2 )Γ( n 2 − k) Γ( n 2 )Γ( n+1 2 − k) P n P n−2k f ; 0 ≤ k ≤ [ n − 1 4 ]; ∂ ∂n ∆ k v| S n = (−1) k+1 k Γ( n+1 2 )Γ( n 2 − k) Γ( n 2 )Γ( n+1 2 − k) P n P n−2k f, 0 ≤ k ≤ [ n − 3 4 ].
(1.20)
Then we have the inequality
log 1 ω n S n e n(f −f ) dσ ≤ n 2 n+1 π n+1 2 Γ( n+1 2 ) B n+1 |∇ n+1 2 v| 2 dx + S n f Tn−1 2 f dσ ,(1.v(x) = π − n 2 Γ(n) 2 n Γ( n 2 ) S n (1 − |x| 2 ) n |x − ξ| 2n (− ln |1 − x 0 , ξ | + c)dσ (1.22)
where c is a constant and x 0 is some point in B n+1 . Remark 1.8. We remark that one can also replace the Neumann boundary condition (1.17)
or (1.20) by ∂ n v| S n = ∂ n V m | S n , ∂ 2 n v| S n = ∂ 2 n V m | S n , · · · .
When k is small, we can compute the value through (4.2) and (4.3). For example, we have
∂ n V m | S n = 2m + 1 − n 2 f, ∂ 2 n V m | S n = ∆f 2m − 1 + (n − 1 − 2m)[(m − 1)n − m(2m − 1)] 2(2m − 1) f.
However, the argument would become increasingly delicate when k is large.
Theorem 1.9. Let n > 2m + 1 ≥ 3 and U(x, y) ∈ W m+1,2 (R n+1 + ) be satisfied the Neumann boundary condition
∆ k U m (x, y)| y=0 = Γ(m + 1)Γ(m + 1 2 − k) Γ(m − k − 1)Γ(m + 1 2 ) ∆ k x f, 0 ≤ k ≤ [ m 2 ]; ∂ y ∆ k U m (x, y)| y=0 =0, 0 ≤ k ≤ [ m − 1 2 ].
(1.23)
Then we have the sharp trace inequality
Γ(m + 1)Γ( 1 2 ) Γ(m + 1 2 ) Γ( n+2m+1 2 ) Γ( n−2m−1 2 ) ω 2m+1 n n R n |U(x, 0)| 2n n−2m−1 dx n−2m−1 n ≤ R n+1 + |∇ m+1 U| 2 dxdy.
Furthermore, equality holds if and only
U(x, y) = c R n y 1+2m (|x − ξ| 2 + y 2 ) n+1 2 +m (λ 2 + |ξ − ξ 0 | 2 ) − n−2m−1 2 dξ, (1.24)
where λ > 0, c is a constant and ξ 0 is some fixed point in R n . Remark 1.10. We remark that, because of Lemma 5.4, one can also replace the Neumann boundary condition (1.23) by the following:
∂ 2k y U(x, y)| y=0 = Γ(k + 1 2 )Γ(m − k + 1 2 ) Γ( 1 2 )Γ(m + 1 2 ) ∆ k x f, 0 ≤ k ≤ [ m 4 ]; ∂ 2k+1 y U(x, y)| y=0 =0, 0 ≤ k ≤ [ m − 1 4 ].
(1.25)
Notice that the boundary condition (1.25) is different to that given by R. Yang ([41]).
This article is organized as follows: In Section 2, we briefly quote some of properties of special functions, such as hypergeometric function and Gegenbauer polynomials, and Funk-Hecke formula for spherical harmonics which will be used in the paper. In Section 3 we first review the connection between scattering theory and conformally invariant objects on their boundaries. Next we compute the explicit formulas of the solution of Poisson equation and the adapted metrics on the model case (B n+1 , S n , g B ).
In Section 4, we prove Theorem 1.6 and 1.7. The proof of Theorem 1.9 is given in Section 5.
Preliminaries
In this section, we quote some preliminary facts which will be needed in the sequel.
2.1. Hypergeometric function. We use the notation F (a, b; c; z) to denote
F (a, b; c; z) = ∞ k=0 (a) k (b) k (c) k z k k! , (2.1)
where c = 0, −1, · · · , −n, · · · and (a) k is the rising Pochhammer symbol defined by
(a) 0 = 0, (a) k = a(a + 1) · · · (a + k − 1), k ≥ 1.
If either a or b is a nonpositive integer, then the series terminates and the function reduces to a polynomial.
Here, we only list some of properties of hypergeometric function which will be used in the rest of paper. For more information about of these functions, we refer to [25], section 9.1 and [17], Chapter II.
• The hypergeometric function F (a, b; c; z) satisfies the hypergeometric differential equation
z(1 − z)F ′′ + (c − (a + b + 1)z)F ′ − abF = 0. (2.2) • If Re(c − a − b) > 0, then F (a, b; c; 1) exists and F (a, b; c; 1) = Γ(c)Γ(c − a − b) Γ(c − a)Γ(c − b) . (2.3)
• Transformation formulas (1):
F (a, b; c; z) = (1 − z) c−a−b F (c − a, c − b; c; z). (2.4) • Transformation formulas (2): if c − a − b is not an integer, then F (a, b; c; z) = Γ(c)Γ(c − a − b) Γ(c − a)Γ(c − b) F (a, b; a + b − c + 1; 1 − z)+ (1 − z) c−a−b Γ(c)Γ(a + b − c) Γ(a)Γ(b) F (c − a, c − b; c − a − b + 1; 1 − z).
(2.5)
• Differentiation formula:
d k dz k F (a, b; c; z) = (a) k (b) k (c) k F (a + k, b + k; c + k; z), k ≥ 1. (2.6)
2.2. Gegenbauer polynomials. We use the notation C α k (x) to denote the Gegenbauer polynomial of degree k which can be defined in terms of the generating function:
1 (1 − 2xt + t 2 ) α = ∞ k=0 C α k (x)t k . (2.7)
Here, we also list some of properties of Gegenbauer polynomial and refer to [40] and [25], section 8.93 for more information about this polynomial.
• Rodrigues formula:
C α k (x) = (−1) k 2 k k! Γ(α + 1 2 )Γ(k + 2α) Γ(2α)Γ(α + k + 1 2 ) (1 − x 2 ) −α+ 1 2 d k dx k (1 − x 2 ) k+α− 1 2 . (2.8) • Orthogonality and normalization: if k = m, then 1 −1 C α k (x)C α m (x)(1 − x 2 ) α− 1 2 dx = 0; (2.9) if k = m, then 1 −1 [C α k (x)] 2 (1 − x 2 ) α− 1 2 dx = π2 1−2α Γ(k + 2α) k!(k + α)[Γ(α)] 2 . (2.10) • Differentiation formulas: d m dx m C α k (x) = 2 m Γ(α+m) Γ(α) C α+m k−m (x), k − m ≥ 0; 0, k − m < 0. (2.11)
Finally, we recall an integral (see [25], page 407, 3.665)
π 0 sin 2µ−1 θ (1 − 2t cos θ + t 2 ) α dθ = Γ(µ)Γ( 1 2 ) Γ(µ + 1 2 ) F (α, α − µ + 1 2 ; µ + 1 2 ; t 2 ), Reµ > 0, |t| < 1.
Using the expansion (2.1) and (2.7), we have, for k ≥ 0,
π 0 C α 2k (cos θ) sin 2µ−1 θdx = Γ(µ)Γ( 1 2 ) Γ(µ + 1 2 ) (α) k (α − µ + 1 2 ) k (µ + 1 2 ) k 1 k! ; π 0 C α 2k+1 (cos θ) sin 2µ−1 θdx =0.
(2.12) 2.3. Funk-Hecke formula. It is known that L 2 (S n ) can be decomposed as follow
L 2 (S n ) = ∞ l=0 H l ,
where H l is the space of spherical harmonics of degree l (see [40]). For n ≥ 2, the Funk-Hecke formula reads as follow (see e.g. [2,24])
S n K( ξ, η )Y l (η)dσ(η) =λ l Y l , λ l =(4π) n−1 2 l!Γ( n−1 2 ) Γ(l + n − 1) 1 −1 K(t)C n−1 2 l (t)(1 − t 2 ) n−2 2 dt, (2.13) where K ∈ L 1 ((−1, 1), (1 − t 2 ) n−2 2 dt) and Y l ∈ H l . Moreover, if Y l ∈ H l , then − ∆Y l = l(n − 1 + l)Y l (2.14)
and thus
BY l = l + n − 1 2 Y l , P 2γ Y l = Γ(l + n 2 + γ) Γ(l + n 2 − γ) Y l , (2.15)
where B and P 2γ defined in (1.7).
adapted metrics
Firstly, we briefly review the definition of the fractional GJMS operator via scattering theory (see [28]). A triple (X n+1 , M n , g + ) is a Poincaré-Einstein manifold if (1) X n+1 is (diffeomorphic to) the interior of a compact manifold X n+1 with boundary ∂X = M n , (2) X n+1 is complete with Ric(g + ) = −ng + , and (3) there exists a nonnegative ρ ∈ C ∞ (X) such that ρ −1 (0) = M n , dρ = 0 along M, and the metric g := ρ 2 g + extends to a smooth metric on X n+1 .
A function ρ satisfying (3) above is called a defining function. It is obvious that the conformal class [h] := [g| T M ] on M is well-defined for a Poincaré-Einstein manifold because ρ is only determined up to multiplication by a positive smooth function on X.
Given a Poincaré-Einstein manifold (X n+1 , M n , g + ) and a representative [h] on the conformal boundary, there is a uniquely defining function ρ such that
g + = ρ −2 (dρ 2 + h ρ ) on M × (0, δ), where h ρ is a one-parameter family of metrics on M satisfying h 0 = h. Given f ∈ C ∞ (M)
. It has been shown (see [33,28]) that the Poisson equation
−∆ g + u − s(n − s)u = 0 (3.1)
has a unique solution of the form
u = F ρ n−s + Hρ s , F, H ∈ C ∞ (X), F | ρ=0 = f, (3.2)
where s ∈ C and s(n − s) do not belongs to the pure point spectrum of −∆ g + . The scattering operator on M is defined as S(s)f = H| M . If Re(s) > n 2 , then the scattering operator is a meromorphic family of pseudo-differential operators. Graham and Zworski [28] defined the fractional GJMS operator P 2γ (γ ∈ (0, n 2 ) \ N) as follow
P 2γ f := d γ S n 2 + γ f, d γ = 2 2γ Γ(γ) Γ(−γ) . (3.3)
Here we denote by N the set of all natural numbers and N 0 = N \ {0}. In term of P 2γ , the the fractional Q-curvature Q 2γ is defined by
Q 2γ := 2 n − 2γ P 2γ (1).
If γ ∈ N 0 , then P 2γ is nothing but the GJMS operator on M (see [26]). It has been also shown by Graham and Zworski [28] that the principal symbol of P 2γ is is exactly the principal symbol of the fractional Laplacian (−∆) γ and satisfy an important conformal covariance property: for a conformal change of metric h = e 2τ h, we have
P 2γ f = e − n+2γ 2 τ P 2γ e n−2γ 2 f , ∀f ∈ C ∞ (M). (3.4)
Next we recall the adapted metric, introduced by Case and Chang [7], on a conformally compact Poincaré-Einstein manifold (X n+1 , ∂X, g + ). This metric is introduced for any parameter s = n 2 + γ with γ ∈ (0, n 2 ) and s = n if n is odd. For such an s, we denote by ϑ s the solution of Poisson equation (3.1) with Dirichlet condition f ≡ 1. Notice that if the Yamabe constant of the boundary metric h is positive, then by a result of Lee (see [31], Theorem A), we have ϑ s > 0 so that one can take ρ s := (ϑ s ) 1 n−s as a defining function. The metric g s = ρ 2 s g + is called adapted metric. In the limiting case, namely s = n and n is an odd integer, the adapted metric, appeared in [21], is defined as g * = e 2τ , where
τ = − d ds ϑ s | s=n . (3.5)
We remark that τ satisfies −∆ g + τ = n. For more information about GJMS operator and adapted metric, we refer to [4,5,6,7,8,10,13,14,21,23,27,28,29,31,41] In the rest of this section, we shall consider the model case (B n+1 , S n , g B ), where
g B = 4
(1−|x| 2 ) 2 g 0 and g 0 = |dx| 2 is the Euclidean metric. The defining function is ρ = 1−|x| 2 2 . Firstly, we give the explicit formula of the solution of Poisson equation (3.1). The main result is the following theorem:
Theorem 3.1. Let γ ∈ (0, n 2 ), s = n 2 + γ and ρ = 1−|x| 2 2 .
The solution of the following Poisson equation on the hyperbolic space
(B n+1 , g B ) −∆ g B u − s(n − s)u = 0 in B n+1 , u = F ρ n−s + Hρ s , F | ∂B n+1 = f (ξ), . (3.6) is u(x) = π − n 2 Γ( n 2 + γ) Γ(γ) S n 1 − |x| 2 2|x − ξ| 2 s f (ξ)dσ. (3.7) Furthermore, if f has an expansion in spherical harmonics, f = ∞ l=0 Y l , where Y l is a spherical harmonic of degree l, then (here we set r = |x|) u(x) = ρ n−s ∞ l=0 ϕ l (r 2 )r l Y l , (3.8) where ϕ l (r) = Γ(γ + 1 2 ) Γ(2γ) Γ(l + n 2 + γ) Γ(l + n+1 2 ) F (l + n 2 − γ, 1 2 − γ, l + n + 1 2 ; r) (3.9) satisfying ϕ l (1) = 1. Proof. Set V (x) =π − n 2 Γ( n 2 + γ) Γ(γ) ρ 2γ S n 1 |x − ξ| 2 s f (ξ)dσ =π − n 2 Γ( n 2 + γ) Γ(γ) ρ 2γ S n f (ξ) (1 − 2x · ξ + |x| 2 ) n 2 +γ dσ. By Funk-Hecke formula (2.13), if f = ∞ l=0 Y l , then V (x) =π − n 2 Γ( n 2 + γ) Γ(γ) ρ 2γ ∞ l=0 λ l Y l , (3.10) where λ l =(4π) n−1 2 l!Γ( n−1 2 ) Γ(l + n − 1) 1 −1 1 (1 − 2rt + r 2 ) n 2 +γ C n−1 2 l (t)(1 − t 2 ) n−2 2 dt =(4π) n−1 2 l!Γ( n−1 2 ) Γ(l + n − 1) ∞ k=0 r k 1 −1 C n 2 +γ k (t)C n−1 2 l (t)(1 − t 2 ) n−2 2 dt.
(3.11)
Using the Rodrigues formula (2.8) and differentiation formula (2.11), we have
∞ k=0 r k 1 −1 C n 2 +γ k (t)C n−1 2 l (t)(1 − t 2 ) n−2 2 dt = (−1) l 2 l l! Γ( n 2 )Γ(l + n − 1) Γ(n − 1)Γ(l + n 2 ) ∞ k=0 r k 1 −1 C n 2 +γ k (t) d l dt l (1 − t 2 ) l+ n−2 2 dt = 1 2 l l! Γ( n 2 )Γ(l + n − 1) Γ(n − 1)Γ(l + n 2 ) ∞ k=l 2 l Γ( n 2 + γ + l) Γ( n 2 + γ) r k 1 −1 C n 2 +γ+l k−l (t)(1 − t 2 ) l+ n−2 2 dt = 1 l! Γ( n 2 )Γ(l + n − 1) Γ(n − 1)Γ(l + n 2 ) Γ( n 2 + γ + l) Γ( n 2 + γ) ∞ k=0 r l+k 1 −1 C n 2 +γ+l k (t)(1 − t 2 ) l+ n−2 2 dt.
(3.12)
Substituting t = cos θ, we have, by (2.12), l!Γ( n−1 2 ) Γ(l + n − 1)
∞ k=0 r l+k 1 −1 C n 2 +γ+l 0 (t)(1 − t 2 ) l+ n−2 2 dt = ∞ k=0 r l+k π 0 C n 2 +γ+l k (cos θ) sin 2l+n−1 θdθ = ∞ k=0 r l+2k π 0 C n 2 +γ+l 2k (cos θ) sin 2l+n−1 θdθ = Γ(l + n 2 )Γ( 1 2 ) Γ(l + n+1 2 ) ∞ k=0 r l+2k ( n 2 + γ + l) k (γ + 1 2 ) k (l + n+1 2 ) k 1 k! .1 l! Γ( n 2 )Γ(l + n − 1) Γ(n − 1)Γ(l + n 2 ) Γ( n 2 + γ + l) Γ( n 2 + γ) Γ(l + n 2 )Γ( 1 2 ) Γ(l + n+1 2 ) · ∞ k=0 r l+2k ( n 2 + γ + l) k (γ + 1 2 ) k (l + n+1 2 ) k 1 k! =(4π) n−1 2 Γ( n−1 2 )Γ( n 2 )Γ( 1 2 ) Γ(n − 1) Γ( n 2 + γ + l) Γ( n 2 + γ)Γ( n+1 2 + l) F (γ + 1 2 , n 2 + l + γ, l + n + 1 2 ; r 2 )r l =(4π) n−1 2 Γ( n−1 2 )Γ( n 2 )Γ( 1 2 ) Γ(n − 1) Γ( n 2 + γ + l) Γ( n 2 + γ)Γ( n+1 2 + l) · (1 − r 2 ) −2γ F (l + n 2 − γ, 1 2 − γ, l + n + 1 2 ; r 2 )r l =2 −2γ (4π) n−1 2 Γ( n−1 2 )Γ( n 2 )Γ( 1 2 ) Γ(n − 1) Γ(2γ) Γ(γ + 1 2 )Γ( n 2 + γ) ρ −2γ ϕ l (r 2 )r l .
(3.14)
Therefore,
V (x) =π − n 2 Γ( n 2 + γ) Γ(γ) ρ 2γ ∞ l=0 λ l Y l =2 n−1−2γ Γ(2γ) Γ(γ)Γ(γ + 1 2 ) Γ( n−1 2 )Γ( n 2 ) Γ(n − 1) ρ 2γ ρ −2γ ∞ l=0 ϕ l (r 2 )r l Y l = ∞ l=0 ϕ l (r 2 )r l Y l . (3.15)
To get the last equation above, we use the duplication formula
Γ(2z) = 2 2z−1 Γ(z)Γ(z + 1 2 ) Γ( 1 2 )
.
V (x)| r=1 = f (x).
By the uniqueness of the solution, to finish the proof, it is enough to show (3.17) or, equivalently,
−∆ g B ρ s−2γ V (x) − s(n − s)ρ s−2γ V (x) = 0,−∆ g B ρ n−s ϕ l (r 2 )r l Y l − s(n − s)ρ n−s ϕ l (r 2 )r l Y l = 0, ∀l ≥ 0. (3.18)
Recall the conformal laplacian on (B n+1 , g B ) is
L gB = −∆ g B + n − 1 4n Scal(g B ),
where Scal(g B ) ia scalar curvature on (B n+1 , g B ). Since (B n+1 , g B ) has the constant sectional curvature −1, we have Scal(g B ) = −n(n + 1) and thus
L g B = −∆ g B − n 2 − 1 4 . (3.19)
By the conformal covariant property of the conformal Laplacian for the change of metric, we have
L g B f = ρ n+3 2 (−∆) ρ − n−1 2 f , ∀f ∈ C ∞ (B n+1 ), (3.20)
where ∆ is the Laplacian on Euclidean space. We have, by (3.19) and (3.20),
∆ g B + n 2 − 1 4 ρ n−s ϕ l (r 2 )r l Y l = ρ n+3 2 ∆ ρ − n−1 2 ρ n−s ϕ l (r 2 )r l Y l =ρ n+3 2 ∆ ρ 1 2 −γ ϕ l (r 2 )r l Y l =ρ n+3 2 ∆ ρ 1 2 −γ ϕ l (r 2 ) r l Y l + 2 ∇ ρ 1 2 −γ ϕ l (r 2 ) , ∇r l Y l .
(3.21)
To get the last equation, we use the fact ∆r l Y l = 0 since Y l is the spherical harmonic of degree l. Substituting the polar coordinate formula ∆ = ∂ 2 ∂r 2 + n r ∂ ∂r + 1 r 2 ∆ into (3.21), we have
∆ g B + n 2 − 1 4 ρ n−s ϕ l (r 2 )r l Y l =ρ n+3 2 ρ 1 2 −γ ϕ l (r 2 ) ′′ + n + 2l r ρ 1 2 −γ ϕ l (r 2 ) ′ r l Y l . (3.22)
We compute
ρ 1 2 −γ ϕ l (r 2 ) ′ =2rρ 1 2 −γ ϕ ′ l (r 2 ) + (γ − 1 2 )rρ − 1 2 −γ ϕ l (r 2 ); ρ 1 2 −γ ϕ l (r 2 ) ′′ =4r 2 ρ 1 2 −γ ϕ ′′ l (r 2 ) + 2ρ − 1 2 −γ ρ − (1 − 2γ)r 2 ϕ ′ l (r 2 )+ (γ − 1 2 )ρ − 1 2 −γ + (γ 2 − 1 4 )r 2 ρ − 3 2 −γ ϕ l (r 2 ).
(3.23)
Substituting (3.23) into (3.22), we obtain
∆ g B + n 2 − 1 4 ρ n−s ϕ l (r 2 )r l Y l =ρ n 2 −γ 4r 2 ρ 2 ϕ ′′ l (r 2 ) + 2ρ l + n + 1 2 − l + n + 3 2 − 2γ r 2 ϕ ′ l (r 2 ) + (n + 2l + 1)(γ − 1 2 )ρ + (γ 2 − 1 4 )r 2 ϕ l (r 2 ) r l Y l .
(3.24) By (2.2), ϕ l (r 2 ) satisfies the hypergeometric differential equation
2r 2 ρϕ ′′ l (r 2 ) = − l + n + 1 2 − l + n + 3 2 − 2γ r 2 ϕ ′ l (r 2 )+ l + n 2 − γ 1 2 − γ ϕ l (r 2 ).
(3.25)
Substituting (3.25) into (3.24), we obtain
∆ g B + n 2 − 1 4 ρ n−s ϕ l (r 2 )r l Y l =ρ n 2 −γ (1 + 2γ)(γ − 1 2 )ρ + (γ 2 − 1 4 )r 2 ϕ l (r 2 )r l Y l =ρ n 2 −γ (γ 2 − 1 4 )ϕ l (r 2 )r l Y l .
This proves equality (3.18). The roof of Theorem 3.1 is thereby completed.
Before we compute the explicit formulas of adapted metric in term of ρ, we need the following Lemma: Lemma 3.2. Let u(x) be the solution of (3.6) and f = ∞ l=0 Y l be the expansion in spherical harmonics. If γ ∈ (0, n 2 ) \ 1 2 N, where 1 2 N = {0, 1 2 , 1, · · · , n 2 , · · · }, then
u(x) =ρ n−s ∞ l=0 F (l + n 2 − γ, 1 2 − γ, 1 − 2γ; 2ρ)r l Y l + ρ s Γ(−λ) 2 2γ Γ(γ) ∞ l=0 Γ(l + n 2 + γ) Γ(l + n 2 − γ) F ( 1 2 + γ, l + n 2 + γ, 1 + 2γ; 2ρ)r l Y l .
(3.26)
If γ = m + 1 2 with m = [γ] < n−1 2 , then u(x) = ρ n−1 2 −m ∞ l=0 m k=0 (l + n−1 2 − m) k (−m) k (−2m) k (2ρ) k k! r l Y l ,(3.
27)
Proof. If γ ∈ (0, n 2 ) \ 1 2 N, we have, by (2.4) and (2.5),
ϕ l (r 2 ) = Γ(γ + 1 2 ) Γ(2γ) Γ(l + n 2 + γ) Γ(l + n+1 2 ) F (l + n 2 − γ, 1 2 − γ, l + n + 1 2 ; r 2 ) = Γ(γ + 1 2 ) Γ(2γ) Γ(l + n 2 + γ) Γ(l + n+1 2 ) (1 − r 2 ) 2γ F ( 1 2 + γ, l + n 2 + γ, l + n + 1 2 ; r 2 ) = Γ(γ + 1 2 ) Γ(2γ) Γ(l + n 2 + γ) Γ(l + n+1 2 ) (1 − r 2 ) 2γ · Γ(l + n+1 2 )Γ(−2γ) Γ(l + n 2 − γ)Γ( 1 2 − γ) F ( 1 2 + γ, l + n 2 + γ, 1 + 2γ; 1 − r 2 ) + (1 − r 2 ) −2γ Γ(l + n+1 2 )Γ(2γ) Γ(l + n 2 + γ)Γ( 1 2 + γ) F (l + n 2 − γ, 1 2 − γ, 1 − 2γ; 1 − r 2 ) =2 2γ ρ 2γ Γ(γ + 1 2 )Γ(l + n 2 + γ)Γ(−2γ) Γ(2γ)Γ(l + n 2 − γ)Γ( 1 2 − γ) F ( 1 2 + γ, l + n 2 + γ, 1 + 2γ; 2ρ)+ F (l + n 2 − γ, 1 2 − γ, 1 − 2γ; 2ρ).
Using the duplication formula (3.16), we have
ϕ l (r 2 ) =ρ 2γ Γ(−λ) 2 2γ Γ(γ) Γ(l + n 2 + γ) Γ(l + n 2 − γ) F ( 1 2 + γ, l + n 2 + γ, 1 + 2γ; 2ρ) + F (l + n 2 − γ, 1 2 − γ, 1 − 2γ; 2ρ).
(3.28)
Substituting (3.28) into (3.8), we get (3.26).
If γ = m + 1 2 , then ϕ l (r) = Γ(m + 1) Γ(2m + 1)
Γ(l + m + n+1 2 ) Γ(l + n+1 2 ) F (l + n − 1 2 − m, −m; l + n + 1 2 ; r)
is a polynomial of degree m and thus
ϕ l (r) = m k=0 ϕ (k) l (1) k! (r − 1) k .Γ(l + m + n+1 2 ) Γ(l + n+1 2 ) (l + n−1 2 − m) k (−m) k (l + n+1 2 ) k · F (l + n − 1 2 − m + k, −m + k; l + n + 1 2 + k; 1) = Γ(2m + 1 − k) Γ(2m + 1) (l + n − 1 2 − m) k (−m) k =(−1) k (l + n−1 2 − m) k (−m) k (−2m) k .ϑ s (ρ) =ρ n 2 −γ F ( n 2 − γ, 1 2 − γ; 1 − 2γ; 2ρ)+ ρ n 2 +γ Γ(−λ) 2 2γ Γ(γ) Γ( n 2 + γ) Γ( n 2 − γ) F ( 1 2 + γ, n 2 + γ; 1 + 2γ; 2ρ).
(3.31)
If γ = m + 1 2 with m = [γ] < n−1 2 , then ϑ s (ρ) =ρ n−1 2 −m m k=0 ( n−1 2 − m) k (−m) k (−2m) k (2ρ) k k! . (3.32)
Now we can compute the explicit formulas of adapted metric g * on the model case (B n+1 , S n , g B ).
Proposition 3.4. Let γ ∈ (0, n 2 ) and s = n 2 + γ. On the model case (B n+1 , S n , g B ) we have
(1) if γ ∈ (0, n 2 ) \ 1 2 N, then g * = ψ 2 n−s γ |dx| 2 = ψ 4 n−2γ γ |dx| 2 , where ψ γ =F ( n 2 − γ, 1 2 − γ, 1 − 2γ; 2ρ) + ρ 2γ 1 d γ Γ( n 2 + γ) Γ( n 2 − γ) F ( 1 2 + γ, n 2 + γ, 1 + 2γ; 2ρ); (3.33) (2) if γ = m + 1 2 < n 2 with m = [γ] , then g * = ψ 4 n−2m−1 m+ 1 2 |dx| 2 , where ψ m+ 1 2 = m k=0 ( n−1 2 − m) k (−m) k (−2m) k (2ρ) k k! ; (3.34)
(3) if γ = n 2 and n is an odd integer, then
g * = exp 2 Γ( n+1 2 ) Γ(n) (n−1)/2 k=1 Γ(n − k) Γ( n+1 2 − k)k (2ρ) k |dx| 2 . (3.35)
Proof. By the definition of g * , we have g * = ϑ
τ = lim γ→ n 2 ρ n 2 −γ n 2 − γ F ( n 2 − γ, 1 2 − γ; 1 − 2γ; 2ρ) − 1 + lim γ→ n 2 ρ n 2 −γ − 1 n 2 − γ lim γ→ n 2 ρ n 2 +γ n 2 − γ Γ(−λ) 2 2γ Γ(γ) Γ( n 2 + γ) Γ( n 2 − γ) F ( 1 2 + γ, n 2 + γ; 1 + 2γ; 2ρ) = lim γ→ n 2 1 n 2 − γ F ( n 2 − γ, 1 2 − γ; 1 − 2γ; 2ρ) − 1 + ln ρ ρ n Γ(− n 2 )Γ(n) 2 n Γ( n 2 )
F ( n + 1 2 , n; 1 + n; 2ρ).
(3.37)
To get the last equation abouve, we use the fact ( n 2 − γ)Γ( n 2 − γ) = Γ( n 2 − γ + 1) → 1 as γ → n 2 . We compute
lim γ→ n 2 1 n 2 − γ F ( n 2 − γ, 1 2 − γ; 1 − 2γ; 2ρ) − 1 = lim γ→ n 2 1 n 2 − γ ∞ k=1 ( n 2 − γ) k ( 1 2 − γ) k (1 − 2γ) k (2ρ) k k! = (n−1)/2 k=1 (k − 1)!( 1−n 2 ) k (1 − n) k (2ρ) k k! + 1 2 ∞ k=n (k − 1)!( 1−n 2 ) n−1 2 (k − n+1 2 )! (1 − n) n−1 (k − n)! (2ρ) k k! = (n−1)/2 k=1 (k − 1)!( 1−n 2 ) k (1 − n) k (2ρ) k k! + (−1) n−1 2 2 n−1 Γ( n+1 2 ) Γ(n) ρ n ∞ k=0 (k + n−1 2 )! (k + n)k! (2ρ) k .
(3.38)
Using the duplication formula (3.16), we have
(−1) n−1 2 2 n−1 Γ( n+1 2 ) Γ(n) ρ n ∞ k=0 (k + n−1 2 )! (k + n)k! (2ρ) k =(−1) n−1 2 2 n−1 Γ( n+1 2 ) 2 Γ(n + 1) ρ n F ( n + 1 2 , n; 1 + n; 2ρ) =(−1) n−1 2 Γ( 1 2 )Γ( n+1 2 ) Γ( n 2 + 1)
ρ n F ( n + 1 2 , n; 1 + n; 2ρ). F ( n + 1 2 , n; 1 + n; 2ρ)
=ρ n Γ(− n 2 )Γ( n+1 2 ) 2Γ( 1 2 )
F ( n + 1 2 , n; 1 + n; 2ρ)
=(−1) n+1 2 Γ( 1 2 )Γ( n+1 2 ) Γ( n 2 + 1)
ρ n F ( n + 1 2 , n; 1 + n; 2ρ).
(3.40)
To get the last equation above, we use Γ( 1 2 ) = √ π and the Euler's reflection formula
Γ − n 2 Γ 1 + n 2 = π − sin n 2 π = (−1) n+1 2 π.
Substituting (3.38) into (3.37) and using (3.39)-(3.40), we get
τ = (n−1)/2 k=1 (k − 1)!( 1−n 2 ) k (1 − n) k (2ρ) k k! + ln ρ = Γ( n+1 2 ) Γ(n) (n−1)/2 k=1 Γ(n − k) Γ( n+1 2 − k)k (2ρ) k + ln ρ.
The desired result follows.
We remark that one can also find the metric g * in dimension 2m+1 by a "dimension continuity" in the spirit of the work of Branson (see [3])), by computing the limit
e 2τ ρ −2 = lim n→2m+1 (ψ m+ 1 2 ) 4 n−2m−1 .
4. Proof of Theorem 1.6 and 1.7
In this section, we let γ = m + 1 2 with m = [γ]. As in the proof of Theorem 3.1, we set
V m (x) =π − n 2 Γ( n 2 + γ) Γ(γ) ρ 2m+1 S n 1 |x − ξ| 2 n+1 2 +m f (ξ)dσ (4.1) such that ρ n−1 2 −m V m (x) is the solution of (3.6). Moreover, if f = ∞ l=0 Y l , then V m (x) = ∞ l=0 ϕ l (r 2 )r l Y l , (4.2) where ϕ l (r) = Γ(m + 1) Γ(2m + 1) Γ(l + n+1 2 + m) Γ(l + n+1 2 ) F (l + n − 1 2 − m, −m, l + n + 1 2 ; r).∆ k V m (x) =4 k ∞ l=0 Γ(m + 1)Γ(l + n+1 2 + m) Γ(l + n+1
2 )Γ(2m + 1)
(l + n − 1 2 − m) k (−m) k · F (l + n − 1 2 − m + k, −m + k, l + n + 1 2 , r 2 )r l Y l .
(4.4)
In particular, we have
∆ m+1 V m (x) = 0. (4.5)
because of (−m) m+1 = 0.
Proof. We shall prove (4.4) by induction. It is easy to see that (4.4) is valid for k = 0. Now suppose that equation (4.4) is valid for k ≥ 0. Then we have
∆ k+1 V m (x) =4 k ∞ l=0 Γ(m + 1)Γ(l + n+1 2 + m) Γ(l + n+1 2 )Γ(2m + 1) (l + n − 1 2 − m) k (−m) k · ∆ F (l + n − 1 2 − m + k, −m + k, l + n + 1 2 , r 2 )r l Y l .
(4.6)
We compute
∆ F (l + n − 1 2 − m + k, −m + k, l + n + 1 2 , r 2 )r l Y l = ∂ rr + n r ∂ r + 1 r 2 ∆ m−k j=0 (l + n−1 2 − m + k) j (−m + k) j (l + n+1 2 ) j 1 j! r 2j+l Y l = m−k j=0 (l + n−1 2 − m + k) j (−m + k) j (l + n+1 2 ) j 1 j! 4j n − 1 2 + j r 2j+l−2 Y l =4(l + n − 1 2 − m + k)(−m + k)· F (l + n − 1 2 − m + k + 1, −m + k + 1, l + n + 1 2 , r 2 )r l Y l .
(4.7)
Substituting (4.7) into (4.6), we have
∆ k+1 V m (x) =4 k+1 ∞ l=0 Γ(m + 1)Γ(l + n+1 2 + m) Γ(l + n+1 2 )Γ(2m + 1) (l + n − 1 2 − m) k+1 (−m) k+1 · F (l + n − 1 2 − m + k + 1, −m + k + 1, l + n + 1 2 , r 2 )r l Y l .
This proves the Lemma 4.1.
Lemma 4.2. There holds, for 0 ≤ k ≤ m,
∆ k V m | r=1 =(−1) k Γ(m + 1)Γ(m − k + 1 2 ) Γ(m + 1 2 )Γ(m − k + 1) P 2m+1 P 2m+1−2k f (4.8)
and for 0 ≤ k ≤ m − 1,
∂ r ∆ k V m | r=1 =(−1) k+1 n − 1 − 2m + 2k 2 Γ(m + 1)Γ(m − k + 1 2 ) Γ(m + 1 2 )Γ(m − k + 1) P 2m+1 P 2m+1−2k f. (4.9)
In the case k = m, we have
∂ r ∆ m V m | r=1 =(−1) m Γ(m + 1)Γ( 1 2 ) Γ(m + 1 2 ) P 2m+1 − n − 1 2 P 2m+1 P 1 f (4.10)
Proof. By Lemma 4.1 and (2.3), we have, for 0 ≤ k ≤ m,
∆ k V m | r=1 =4 k ∞ l=0 Γ(m + 1)Γ(l + n+1 2 + m) Γ(l + n+1 2 )Γ(2m + 1) (l + n − 1 2 − m) k (−m) k · Γ(l + n+1 2 )Γ(2m − 2k + 1) Γ(m + 1 − k)Γ(l + n+1 2 + m − k) Y l =4 k Γ(m + 1)Γ(2m − 2k + 1) Γ(2m + 1)Γ(m − k + 1) (−m) k · ∞ l=0 Γ(l + n+1 2 + m)Γ(l + n−1 2 − m + k) Γ(l + n−1 2 − m)Γ(l + n+1 2 + m − k) Y l =4 k Γ(m + 1)Γ(2m − 2k + 1) Γ(2m + 1)Γ(m − k + 1) (−m) k P 2m+1 P 2m+1−2k f. (4.11)
To get the last equation, we use (2.15). Moreover, by duplication formula (3.16), we have
4 k Γ(m + 1)Γ(2m − 2k + 1) Γ(2m + 1)Γ(m − k + 1) (−m) k =(−1) k Γ(m + 1)Γ(m − k + 1 2 ) Γ(m + 1 2 )Γ(m − k + 1)
.
(4.12)
Substituting (4.12) into (4.13), we get (4.8). Similarly, using (2.6) and (2.3), we have, for 0 ≤ k ≤ m − 1,
∂ r ∆ k V m | r=1 = 4 k ∞ l=0 Γ(m + 1)Γ(l + n+1 2 + m) Γ(l + n+1 2 )Γ(2m + 1) (l + n − 1 2 − m) k (−m) k · 2 (l + n−1 2 − m + k)(−m + k) l + n+1 2 Γ(l + n+1 2 + 1)Γ(2m − 2k) Γ(m + 1 − k)Γ(l + n+1 2 + m − k) + l Γ(l + n+1 2 )Γ(2m − 2k + 1) Γ(m + 1 − k)Γ(l + n+1 2 + m − k) Y l = − n − 1 2 − m + k ∆ k V m | r=1 .
(4.13)
These prove (4.9). Now we prove (4.10). By Lemma 4.1,
∆ m V m (x) =4 k Γ(m + 1) Γ(2m + 1) (−m) m ∞ l=0 Γ(l + n+1 2 + m) (l + n−1 2 )Γ(l + n−1 2 − m) r l Y l .
Therefore, by (2.15), we have
∂ r ∆ m V m | r=1 =4 m Γ(m + 1) Γ(2m + 1) (−m) m ∞ l=0 l (l + n−1 2 ) Γ(l + n+1 2 + m) Γ(l + n−1 2 − m) Y l =4 m Γ(m + 1) Γ(2m + 1) (−m) m ∞ l=0 1 − n−1 2 l + n−1 2 · Γ(l + n+1 2 + m) Γ(l + n−1 2 − m) Y l =(−1) m 4 m Γ(m + 1) 2 Γ(2m + 1) P 2m+1 − n − 1 2 P 2m+1 P 1 f =(−1) m Γ(m + 1)Γ( 1 2 ) Γ(m + 1 2 ) P 2m+1 − n − 1 2 P 2m+1 P 1 f.
(4.14)
To get the last equation, we use duplication formula (3.16). The desired result follows.
Proof of Theorem 1. 6 We firstly prove (1.18) when v = V m . By Lemma 4.2 and Green's formula (see e.g. [20], Appendix C), we have
0 = B n+1 V m ∆ m+1 V m dx = B n+1 ∆V m ∆ m V m dx + S n V m ∂ r ∆ m V m dσ − S n ∂ r V m ∆ m V m dσ = B n+1 ∆V m ∆ m V m dx + (−1) m Γ(m + 1)Γ( 1 2 ) Γ(m + 1 2 ) S n f P 2m+1 f dσ− (−1) m n − 1 2 Γ(m + 1)Γ( 1 2 ) Γ(m + 1 2 ) S n f P 2m+1 P 1 f dσ. (4.15)
If m is an odd integer, then by Lemma 4.2 and and Green's formula, we have If m is a even integer, then also by Lemma 4.2 and and Green's formula, we have
B n+1 ∆V m ∆ m V m dx = B n+1 |∆ m+1 2 V m | 2 dx + m−1 2 k=1 (m − 2k) Γ(m + 1) 2 Γ(m + 1 2 ) 2 · Γ(k + 1 2 )Γ(m − k + 1 2 ) Γ(k + 1)Γ(m − k + 1) S n f P 2 2m+1 P 2m+1−2k P 2k+1 f dσ.B n+1 ∆V m ∆ m V m dx = − B n+1 |∇∆ m 2 V m | 2 dx − m−2 2 k=1 (m − 2k) Γ(m + 1) 2 Γ(m + 1 2 ) 2 · Γ(k + 1 2 )Γ(m − k + 1 2 ) Γ(k + 1)Γ(m − k + 1) S n f P 2 2m+1 P 2m+1−2k P 2k+1 f dσ− n − 1 − m 2 Γ(m + 1)Γ( m+1 2 ) Γ(m + 1 2 )Γ( m 2 + 1) 2 S n f P 2 2m+1 P 2 m+1 f dσ.Γ(m + 1)Γ( 1 2 ) Γ(m + 1 2 ) S n f P 2m+1 f dσ = B n+1 |∇ m+1 V m | 2 dx + S n f T m f dσ, (4.18)
where T m is defined in Theorem (1.6). Therefore, by Theorem 1.1, we prove the Theorem 1.6 when v = V m and in this case the only extremal functions is given by (1.19). For general v with the Neumann boundary condition (1.17), we claim
B n+1 |∇ m+1 V m | 2 dx ≤ B n+1 |∇ m+1 v| 2 dx.
In fact, we have
0 ≤ B n+1 |∇ m+1 (v − U m )| 2 dx = B n+1 |∇ m+1 v| 2 dx − B n+1 |∇ m+1 U m | 2 dx − 2 B n+1 ∇ m+1 (v − U m ) · ∇ m+1 U m dx.
Since v and U m have the same Neumann boundary condition (1.17), we have These prove the claim.
B n+1 ∇ m+1 (v − U m ) · ∇ m+1 U m dx = (−1) m+1 B n+1 (v − U m )∆ m+1 U m dx = 0. Therefore, B n+1 |∇ m+1 v| 2 dx = B n+1 |∇ m+1 (v − U m )| 2 dx + B n+1 |∇ m+1 U m | 2 dx ≥ B n+1 |∇ m+1 U m | 2 dx.
Finally, we prove the uniqueness of the extremal functions. If v is any extremal function with the Neumann boundary condition (1.17), then by (4.19) it must satisfies ∇ m+1 (v − U m ) = 0 and thus ∆ m+1 v = ∆ m+1 U m = 0. By the uniqueness of the solution, we get v = V m (x). Thus, by Theorem 1.1, the only extremal function is that given by (1.19). The proof of Theorem 1.6 is thereby completed.
Proof of Theorem 1.7 With the same argument in the proof of Theorem 1.6, we need only consider the case v = V m (x) with m = n−1 2 . Using (4.15)-(4.18), we get
Γ( n+1 2 )Γ( 1 2 ) Γ( n 2 ) S n f P n f dσ = B n+1 |∇ n+1 2 V m | 2 dx + S n f Tn−1 2 f dσ. (4.20)
Therefore, by Theorem 1.1, we have . With the same argument in Theorem 1.6 and using Theorem 1.1, we have get the only only extremal function is that given by (1.22). The proof of Theorem 1.7 is thereby completed.
ln 1 ω n S n e n(f −f ) dσ ≤ n 2(n − 1)!ω n S n f P n f dσ ≤ n 2(n − 1)!ω n Γ( n 2 ) Γ( n+1 2 )Γ( 1 2 ) B n+1 |∇ n+1 2 V m | 2 dx + S n f Tn−1 2 f dσ = n 2 n+1 π n+1 2 Γ( n+1 2 ) B n+1 |∇ n+1 2 V m | 2 dx + S n f Tn−1 2 f dσ .
Proof of Theorem 1.9
Since the Möbius transform M : (R n+1
+ , g H ) → (B n+1 , g B ), where g H = |dx| 2 +|dy| 2 y 2 , defined by M(x, y) = 2x (1 + y) 2 + |x| 2 , 1 − |x| 2 − y 2 (1 + y) 2 + |x| 2 ,
is an isometry between the two models of hyperbolic space, by Theorem 3.1, we can solve the Poisson equation (3.1) on (R n+1 + , g H ):
Theorem 5.1. Let γ ∈ (0, n 2 ) and s = n 2 + γ. The solution of the Poisson equation on the hyperbolic space (R n+1
+ , g H ) −∆ g H u − s(n − s)u = 0 in R n+1 , u = F y n−s + Hy s , F | ∂R n+1 + = f (x),
.
(5.1) is u(x, y) = π − n 2 Γ( n 2 + γ) Γ(γ) R n y |x − ξ| 2 + y 2 s f (ξ)dξ, (5.2)
where f and its derivatives have fast decay at infinity (for example, in certain fractional Sobolev spaces). Now we let γ = m + 1 2 with m an integer. In this case, we can find out the relationship between the kernel in (5.2) and the Poisson kernel. In fact, we have the following: Therefore,
Lemma 5.2. There holds, for (x, y) ∈ R n+1 + and m ∈ N, m k=0 2 k k! Γ(2m − k + 1) Γ(m − k + 1) (−y) k d k dy k y (|x| 2 + y 2 ) n+1 2 =2 2m Γ(m + n+1 2 ) Γ( n+1 2 ) y 1+2m (|x| 2 + y 2 )(n + 1 + 2m) y 3+2m (|x| 2 + y 2 ) n+1 2 +m+1 = 1 + 2m − y d dy y 1+2m (|x| 2 + y 2 ) n+1 2 +m =2 −2m Γ( n+1 2 ) Γ(m + n+1 2 ) · 1 + 2m − y d dy m k=0 2 k k! Γ(2m − k + 1) Γ(m − k + 1) (−y) k d k dy k y (|x| 2 + y 2 ) n+1 2 . (5.5) Since 1 + 2m − y d dy m k=0 2 k k! Γ(2m − k + 1) Γ(m − k + 1) (−y) k d k dy k = m k=0 2 k k! Γ(2m − k + 1) Γ(m − k + 1) (1 + 2m − k)(−y) k d k dy k + m k=0 2 k k! Γ(2m − k + 1) Γ(m − k + 1) (−y) k+1 d k+1 dy k+1 = m k=0 2 k k! Γ(2m − k + 1) Γ(m − k + 1) (1 + 2m − k)(−y) k d k dy k + m+1 k=1 2 k−1 (k − 1)! Γ(2m − k + 2) Γ(m − k + 2) (−y) k d k dy k = 1 2 m+1 k=0 2 k k! Γ(2m − k + 1) Γ(m − k + 1) (−y) k d k dy k y (|x| 2 + y 2 ) n+1 2 ,(5.6)
We have, by (5.5) and (5.6),
m+1 k=0 2 k k! Γ(2m − k + 3) Γ(m − k + 2) (−y) k d k dy k y (|x| 2 + y 2 ) n+1 2 =2 2m+2 Γ(m + 1 + n+1 2 ) Γ( n+1 2 )
y 3+2m
(|x| 2 + y 2 ) n+1 2 +m+1 . These completes the proof of Lemma 5.2.
In the rest paper, we let ∆ x = n i=1 ∂ x i x i and ∆ = ∆ x + ∂ yy . Since the Poisson kernel e −y √ −∆x on R n+1 + is given by (see e.g. [40])
e −y √ −∆x = π − n 2 Γ( n+1 2 ) Γ( 1 2 ) y (|x| 2 + y 2 ) n+1 2 , (5.7)
we have, by Theorem 5.1 and Lemma 5.2, that the solution of (5.1) is
u(x, y) =y n−1 2 −m 2 −2m Γ( 1 2 ) Γ(m + 1 2 ) m k=0 2 k k! Γ(2m − k + 1) Γ(m − k + 1) (−y) k d k dy k e −y √ −∆x f =y n−1 2 −m 2 −2m Γ( 1 2 ) Γ(m + 1 2 ) m k=0 2 k k! Γ(2m − k + 1) Γ(m − k + 1) y k (−∆ x ) k 2 e −y √ −∆x f. (5.8) Lemma 5.3.
Let U m (x, y) = u(x, y)y − n−1 2 +m , where u(x, y) is given by (5.8). Then for 0 ≤ k ≤ m + 1, we have
∆ k U m (x, y) =(−1) k 2 2k−2m Γ(m + 1)Γ( 1 2 ) Γ(m − k + 1)Γ(m + 1 2 ) · m−k j=0 2 j j! Γ(2m − 2k − j + 1) Γ(m − k − j + 1) y j (−∆ x ) k+ j 2 e −y √ −∆x f. (5.9)
In particular, ∆ m+1 u(x, y) = 0 because of (−m) m+1 = 0. Moreover,
∆ k U m (x, y)| y=0 = Γ(m + 1)Γ(m + 1 2 − k) Γ(m − k − 1)Γ(m + 1 2 ) ∆ k x f, 0 ≤ k ≤ m; ∂ y ∆ k U m (x, y)| y=0 =0, 0 ≤ k ≤ m − 1; ∂ y ∆ m U m (x, y)| y=0 =(−1) m Γ(m + 1)Γ( 1 2 ) Γ(m + 1 2 ) (−∆ x ) m+ 1 2 f. (5.10)
Proof. We prove (5.9) by induction. Obviously, (5.9) is valid for k = 0. Suppose (5.9) is valid for k. Then
∆ k+1 U m (x, y) = (−1) k 2 2k−2m Γ(m + 1)Γ( 1 2 ) Γ(m − k + 1)Γ(m + 1 2 ) · m−k j=0 2 j j! Γ(2m − 2k − j + 1) Γ(m − k − j + 1) ∆ y j (−∆ x ) k+ j 2 e −y √ −∆x f =(−1) k 2 2k−2m Γ(m + 1)Γ( 1 2 ) Γ(m − k + 1)Γ(m + 1 2 ) m−k j=0 2 j j! Γ(2m − 2k − j + 1) Γ(m − k − j + 1) · j(j − 1)y j−2 (−∆ x ) j 2 − 2jy j (−∆ x ) 1+j 2 (−∆ x ) k e −y √ −∆x f =(−1) k 2 2k−2m Γ(m + 1)Γ( 1 2 ) Γ(m − k + 1)Γ(m + 1 2 ) m−k j=2 2 j (j − 2)! Γ(2m − 2k − j + 1) Γ(m − k − j + 1) (−∆ x ) j 2 − 2 m−k j=1 2 j (j − 1)! Γ(2m − 2k − j + 1) Γ(m − k − j + 1) (−∆ x ) 1+j 2 (−∆ x ) k e −y √ −∆x f. A simple calculation shows m−k j=2 2 j (j − 2)! Γ(2m − 2k − j + 1) Γ(m − k − j + 1) (−∆ x ) j 2 − 2 m−k j=1 2 j (j − 1)! Γ(2m − 2k − j + 1) Γ(m − k − j + 1) (−∆ x ) 1+j 2 = m−k−2 j=0 2 j+2 j! Γ(2m − 2k − j − 1) Γ(m − k − j − 1) (−∆ x ) 1+ j 2 − 2 m−k−1 j=0 2 j+4 j! Γ(2m − 2k − j) Γ(m − k − j) (−∆ x ) 1+ j 2 =4(k − m) m−k−1 j=0 2 j j! Γ(2m − 2k − j − 1) Γ(m − k − j) y j (−∆ x ) k+1+ j 2 e −y √ −∆x f.
and thus we have ∆ k x f.
∆ k+1 U m (x, y) =(−1) k+1 2 2k+2−2m Γ(m + 1)Γ( 1 2 ) Γ(m − k)Γ(m + 1 2 ) · m−k−1 j=0 2 j j! Γ(2m − 2k − j − 1) Γ(m − k − j) y j (−∆ x ) k+1+ j 2 e −
Similarly, ∂ y ∆ k U m (x, y)| y=0 = 0 when 0 ≤ k ≤ m − 1 and ∂ y ∆ m U m (x, y)| y=0 = (−1) m Γ(m + 1)Γ( 1 2 ) Γ(m + 1 2 ) (−∆ x ) m+ 1 2 f.
These complete the proof of Lemma 5.3
We can also compute the Neumann boundary condition ∂ k y U m | y=0 for 0 ≤ k ≤ 2m. In fact, we have the following: ∆ k x f (5.11) and for 0 ≤ k ≤ m − 1, ∂ 2k+1 y U m (x, y)| y=0 =0. (5.12) Proof. Firstly, we prove (5.11) by induction. It is easy to see (5.11) is valid for k = 0. Suppose that (5.11) is valid for k, i.e., lim y→0+ ∂ 2k y Γ(m + 1)
Γ(2m + 1) m j=0 2 j j! Γ(2m − j + 1) Γ(m − j + 1) y j (−∆ x ) j 2 e −y √ −∆x f = Γ(k + 1 2 )Γ(m − k + 1 2 ) Γ( 1 2 )Γ(m + 1 2 )
∆ k x f. ∆ k x f.
(5.14)
By (5.9), we have
∂ 2k+2 y U m (x, y) = ∂ 2k y [∆U m (x, y) − ∆ x U m (x, y)] =∂ 2k y − m m − 1 2 Γ(m) Γ(2m − 1) m−1 j=0 2 j j! Γ(2m − 2k − j + 1) Γ(m − k − j + 1) y j (−∆ x ) 1+ j 2 e −Γ(k + 1 2 )Γ(m − k − 1 2 ) Γ( 1 2 )Γ(m − 1 2 ) ∆ k+1 x f − Γ(k + 1 2 )Γ(m − k + 1 2 ) Γ( 1 2 )Γ(m + 1 2 ) ∆ k+1 x f = Γ(k + 1 + 1 2 )Γ(m − k − 1 2 ) Γ( 1 2 )Γ(m + 1 2 )
∆ k+1 x f.
These prove (5.11). The proof of (5.12) is completely analogous to that of (5.11) and we omit.
Proof of Theorem 1.9 With the same argument in the proof of Theorem 1.6, we need only consider the case u = U m (x, y). By Lemma 5.3 and Green's formula, we have
0 = R n+1 + U m ∆ m+1 U m dxdy = R n+1 + ∆U m ∆ m U m dxdy + R n U m ∂ y ∆ m U m dx − R n ∂ y U m ∆ m U m dx = R n+1 + ∆U m ∆ m U m dxdy + (−1) m Γ(m + 1)Γ( 1 2 ) Γ(m + 1 2 ) R n U m (x, 0)(−∆ x ) m+ 1 2 U m (x, 0)dx =(−1) m−1 R n+1 + |∇ m+1 U m | 2 dxdy + (−1) m Γ(m + 1)Γ( 1 2 ) Γ(m + 1 2 ) R n U m (x, 0)(−∆ x ) m+ 1 2 U m (x, 0)dx.
Therefore, by Theorem 1.2, we have R n+1
+ |∇ m+1 U m (x, y)| 2 dxdy = Γ(m + 1)Γ( 1 2 ) Γ(m + 1 2 ) R n U m (x, 0)(−∆ x ) m+ 1 2 U m (x, 0)dx ≥ Γ(m + 1)Γ( 1 2 ) Γ(m + 1 2 ) Γ( n+2m+1 2 ) Γ( n−2m−1 2 ) ω 2m+1 n n R n |U m (x, 0)| 2n n−2m−1 dx n−2m−1 n .
With the same argument in Theorem 1.6 and using Theorem 1.2, we have get the only only extremal function is that given by (1.24). These complete the proof of Theorem 1.9.
−∆) γ f | 2 dx. (1.10)Equality holds only for functions of the form
T m is an operator of order 2m defined as follow: if m is an odd integer, then
, we have ϕ l (1) = 1 and thus
3.30) into (3.29) and using (3.8), we get(3.27).
Corollary 3. 3 .
3Let ϑ s be the solution of (3.6) when f = 1. If γ ∈ (0, n 2 ) \ 1 2 N, then
. By Corollary 3.3, we get (1) and(2). Now we prove(3.35). By the definition of g * , we have g * = e 2τ ρ −2 |dx| 2 , where τ = − d ds ϑ s | s=n = − lim s→n ϑ s − ϑ n = 1. Therefore, substituting (3.31) into (3.36), we have
.
There holds, for 0 ≤ k ≤ m + 1,
the last equation, we use the fact ω n
Proof. We shall prove it by induction. It is easy to see (5.3) is valid for m = 0. Suppose (5.3) is valid for m ≥ 0.
U m (x, y)| y=0 =(−1) k 2 2k−2m Γ(
Lemma 5. 4 .
4There holds, for 0 ≤ k ≤ m, ∂ 2k y U m (x,
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[
"Microscopic studies on nuclear spin-isospin properties-a personal perspective on covariant density functional theory Microscopic studies on nuclear spin-isospin properties in CDFT",
"Microscopic studies on nuclear spin-isospin properties-a personal perspective on covariant density functional theory Microscopic studies on nuclear spin-isospin properties in CDFT"
] |
[
"Haozhao Liang [email protected] ",
"Haozhao Liang ",
"\nGraduate School of Science\nRIKEN Nishina Center\n351-0198WakoJapan\n",
"\nthe University of Tokyo\n113-0033TokyoJapan\n"
] |
[
"Graduate School of Science\nRIKEN Nishina Center\n351-0198WakoJapan",
"the University of Tokyo\n113-0033TokyoJapan"
] |
[
"The 26th International Nuclear Physics Conference 11-16 September"
] |
Spin and isospin are essential degrees of freedom in nuclear systems, and the relevant studies on their properties play important roles not only in nuclear physics but also in nuclear astrophysics, particle physics, and so on. In this presentation for the IUPAP Young Scientist Prize 2016, I would like to introduce the microscopic studies on nuclear spin-isospin properties in the framework of covariant density functional theory (DFT), by taking a few works that I have been joining in as examples. It is seen that the covariant scheme plays an important role in describing the spin properties in a consistent way, such as the spin-orbit splitting, the pseudospin symmetry, etc. Meanwhile, the Fock terms play important roles in describing the isospin properties fully self-consistently, such as the Gamow-Teller and spin-dipole resonances, the isospin-symmetrybreaking corrections to nuclear superallowed β transitions, etc. To connect the covariant DFT to more fundamental theories, an ab initio relativistic Brueckner-Hartree-Fock theory for finite nuclei is introduced, and a personal perspective for the coming decade is also illustrated.
|
10.22323/1.281.0361
|
[
"https://arxiv.org/pdf/1703.08347v1.pdf"
] | 55,902,298 |
1703.08347
|
e5380e9a358fad1ce624349f9f52d09e68b1e432
|
Microscopic studies on nuclear spin-isospin properties-a personal perspective on covariant density functional theory Microscopic studies on nuclear spin-isospin properties in CDFT
2016
Haozhao Liang [email protected]
Haozhao Liang
Graduate School of Science
RIKEN Nishina Center
351-0198WakoJapan
the University of Tokyo
113-0033TokyoJapan
Microscopic studies on nuclear spin-isospin properties-a personal perspective on covariant density functional theory Microscopic studies on nuclear spin-isospin properties in CDFT
The 26th International Nuclear Physics Conference 11-16 September
Adelaide, Australia * Speaker2016
Spin and isospin are essential degrees of freedom in nuclear systems, and the relevant studies on their properties play important roles not only in nuclear physics but also in nuclear astrophysics, particle physics, and so on. In this presentation for the IUPAP Young Scientist Prize 2016, I would like to introduce the microscopic studies on nuclear spin-isospin properties in the framework of covariant density functional theory (DFT), by taking a few works that I have been joining in as examples. It is seen that the covariant scheme plays an important role in describing the spin properties in a consistent way, such as the spin-orbit splitting, the pseudospin symmetry, etc. Meanwhile, the Fock terms play important roles in describing the isospin properties fully self-consistently, such as the Gamow-Teller and spin-dipole resonances, the isospin-symmetrybreaking corrections to nuclear superallowed β transitions, etc. To connect the covariant DFT to more fundamental theories, an ab initio relativistic Brueckner-Hartree-Fock theory for finite nuclei is introduced, and a personal perspective for the coming decade is also illustrated.
Introduction
Atomic nucleus is quite a unique quantum-mechanical many-body system, in which three fundamental interactions out of four-the strong, weak, and electromagnetic interactions-interplay each other in a large range of time and energy scales with the co-existence of single-particle and collective characteristics. It holds profound properties as a quantum system, a many-body system, a finite system, as well as an open system. One of the best examples to identify these features is the halo structure in neutron drip-line nuclei. For more than a half century, we learnt in textbooks that the size of a nucleus would be essentially proportional to its mass number because of the saturation property of nuclear force. However, the nature surprised us in 1985. It was found that the size of 11 Li is almost the same as that of 208 Pb due to its exotic neutron-halo structure [1]. From then on, our knowledge in nuclear physics is rapidly expanding together with the constructions and upgrades of the radioactive-ion-beam facilities all around the world.
To study such a complicated system, of most importance is to catch the essential degrees of freedom, for example, the spin and isospin degrees of freedom. On one hand, the spin degree of freedom is one of the cornerstones in nuclear physics. Only after the strong spin-orbit interaction was taken into account, the traditional magic numbers in stable nuclei were understood. Meanwhile, by examining the single-particle spectra, Hecht and Adler [2] and Arima, Harvey, and Shimizu [3] found the near degeneracy between two single-particle states with quantum numbers (n, l, j = l + 1/2) and (n − 1, l + 2, j = l + 3/2). They introduced the concept of pseudospin symmetry to describe such an approximate degeneracy. On the other hand, the isospin degree of freedom distinguishes protons and neutrons in nuclei. The physics becomes much richer once there exist two different but similar kinds of fermions in one system. Note that nuclei will never be self-bound if they are composed of only protons or neutrons. It comes the concept of symmetry energy in nuclear equation of state, and this isospin property becomes one of the frontiers in nuclear physics and astrophysics, since it is crucial to understand why there are two-solar-mass neutron stars in our Universe.
Instead of investigating the nuclear spin and isospin properties separately, experimentally, one of the best probes to study these two essential degrees of freedom together is the so-called nuclear spin-isospin excitations [4], such as the Gamow-Teller (GT) and spin-dipole (SD) resonances. These excitations correspond to the transitions from an initial state of the nucleus (N, Z) to the final states in its isobaric neighboring nuclei (N −1, Z +1) and (N +1, Z −1) in the isospin lowering and raising channels, respectively. They can take place spontaneously in nature, like the well-known β decays, or be induced by external fields in laboratory, like the charge-exchange reactions, e.g., (p, n), (n, p), ( 3 He, t), which have been intensive studied, e.g., in RIKEN, RCNP, MSU, GSI, TRI-UMF, CERN, etc. That is because these excitations are important to understand the questions like "What are the spin and isospin properties of nuclear force and nuclei?" "Where and how does the rapid neutron-capture process (r-process) happen?" "Does Cabibbo-Kobayashi-Maskawa (CKM) matrix satisfy the unitary condition?" etc. These are among the top questions in nuclear physics, nuclear astrophysics, and particle physics. One of our main tasks is to understand these questions from the theoretical side.
Covariant density functional theory
For these studies, one of our favorite research tools is the nuclear density functional theory (DFT), in particular, its covariant (or the so-called relativistic) version [5]. Its fundamental is the Kohn-Sham DFT and its scheme is the Yukawa meson-exchange picture. The starting point of covariant DFT is an effective Lagrangian density with a typical form as
L =ψ iγ µ ∂ µ − M − g σ σ − g ω γ µ ω µ − g ρ γ µ τ · ρ µ − f π m π γ 5 γ µ ∂ µ π · τ − eγ µ 1 − τ 3 2 A µ ψ + 1 2 ∂ µ σ ∂ µ σ − 1 2 m 2 σ σ 2 − 1 4 Ω µν Ω µν + 1 2 m 2 ω ω µ ω µ − 1 4 R µν · R µν + 1 2 m 2 ρ ρ µ · ρ µ + 1 2 ∂ µ π · ∂ µ π − 1 2 m 2 π π · π − 1 4 F µν F µν ,(2.1)
in which nucleons are described as Dirac spinors that interact each other via the exchanges of the σ , ω, ρ, and π-mesons and photons. Some details can be found in recent Reviews [6,7,8,9,10,11] and the references therein. One of the key reasons for choosing DFT is because it is applicable to almost the whole nuclear chart, and not only for the ground states but also for the excited states. In particular, for choosing the covariant version, a dream here is starting with an effective Lagrangian we are eventually able to connect the nuclear DFT to more fundamental theories, such as QCD at low energy. Even before that, from the practical point of view, by using the Dirac equation as the nucleons' equation of motion instead of the Schrödinger one, the spin degree of freedom and a large part of the three-body effect can be taken into account in a consistent way. Moreover, the Lorentz covariant symmetry holds a unified description of the time-even and time-odd components of the energy density functionals.
Therefore, a great effort has been devoted to the nuclear covariant DFT since the Walecka model proposed in the 1970s [5]. At the beginning of the year 2016, a kind of summary for the cutting-edges in this field was published in the latest volume of International Review of Nuclear Physics-Relativistic Density Functional for Nuclear Structure [12]. Some of the highlights have been covered by Professor S.G. Zhou in his plenary talk in this conference. Here I show a few works that I have been joining in as examples.
An example is for the pseudospin symmetry. It is found that although it is deeply hidden in the original Hamiltonian, the origin of pseudospin symmetry can be traced in its supersymmetric partner Hamiltonian. For the first time, the pseudospin-orbit splitting can be understood in an explicit and quantitative way [13,14,10].
Another example is for the single-particle resonances. In order to probe the single-particle resonances in the relativistic scheme, a method for solving the Dirac equation in the complex momentum space was developed recently. It has been found that this method is not only very effective for the narrow resonances, but also can be reliably applied to the broad resonances [15,16]. This is potentially crucial for understanding some exotic properties of halo nuclei.
One more example is for the nuclear rotation. By developing the two-dimensional tilted-axis cranking model with the covariant DFT, the shears mechanism in the nuclear magnetic rotation and the two-shears-like mechanism in the anti-magnetic rotation can be described and understood in a self-consistent and microscopic way [17,18,19].
In short, it is seen that the nuclear covariant DFT achieves a great success in describing and understanding various kinds of nuclear ground-state and excited-state properties. Nevertheless, for simplicity, in most of versions of covariant DFT, only the local Hartree terms are kept, and the involved non-local Fock terms are neglected. As will be shown below, these Hartree-only approaches show their limitations in some important features, like the properties in the spin-isospin channel and the tensor effects of nuclear interaction [20]. Back to my Ph.D. study, one of our key tasks was to develop a covariant DFT with both Hartree and Fock terms.
Nuclear spin-isospin excitations
The most typical nuclear spin-isospin excitation is the so-called Gamow-Teller resonances. Their typical response functions are composed of a low-energy peak (around the β -decay window in unstable nuclei) and a high-energy broad peak as a giant resonance. The absolute positions of these peaks tell us the isospin properties of the target nucleus, and the relative distance between these two peaks tells us its spin properties. As shown in the left panel of Fig. 1, the GT resonances in 208 Pb can be reproduced by the random-phase-approximation calculations based on the relativistic Hartree-Fock theory (denoted as RHF+RPA). With both Hartree and Fock terms, it was achieved for the first time in the relativistic scheme in a fully self-consistent way [21]. To discover the physics behind, we switched on and off individual residual interactions provided by each meson. It has been found that in the present self-consistent scheme the isoscalar mesons play the essential roles via the Fock terms. By using the Fierz transformation, one can also understand the reason why the previous calculations with only Hartree terms work with a free parameter g being around 0.6 [20].
The predictive power of this RHF+RPA approach can be further examined by more delicate spin-isospin excitations, e.g., the spin-dipole excitations, which have three different components instead of one. In the right panel of Fig. 1, the SD excitations in 16 O are shown as an example. Note that the three components could not be distinguished experimentally until an experiment with the high-quality polarized proton beam performed in RCNP in 2011 [23]. It is seen that this delicate excitation structure can be described by the fully self-consistent RHF+RPA approach in details [22]. This is not only important to understand the nuclear structure, but also important for some interdisciplinary studies, such as the β -decay have-lives of neutron-rich nuclei for the r-process [24] and the β + decays and electron captures of proton-rich nuclei [25]. For example, we pointed out a remarkable speeding up of r-matter flow, which leads to the enhanced r-process abundances of the elements with A ≥ 140 [24].
Unitarity test of CKM matrix
Another interesting application of RHF+RPA approach is for evaluating the isospin-symmetrybreaking corrections δ c to nuclear superallowed β transitions [26], which is shown to be crucial for the unitarity test of the CKM matrix.
The CKM matrix links between the quark eigenstates of week interaction and its eigenstates of mass. In the latest versions of Review of Particle Physics by Particle Data Group (PDG) [27], it is shown that nowadays the most precise unitarity test for the CKM matrix comes from the square sum of its first-row elements V 2 ud + V 2 us + V 2 ub , and the most precise determination of the leading element V ud comes from nuclear superallowed β transitions, through which nuclear physics involves. Moreover, in order to extract the V ud value from these superallowed β transitions, not only the experimental data but also several theoretical corrections are needed, including the isospinsymmetry-breaking corrections δ c . In finite nuclei, the isospin symmetry is broken mainly due to the Coulomb interaction.
By using the self-consistent RHF+RPA approach, we performed a systematic calculation on the isospin-symmetry-breaking corrections δ c for nine different superallowed β transitions [26]. It has been found that the δ c values are not sensitive to specific nuclear effective interactions but to the proper treatment of the Coulomb part. The Coulomb exchange term [28] plays an important role in this specific topic. The V ud value was then deduced by combining with the experimental data and theoretical radiative corrections at that time. In the left panel of Fig. 2, I show the values of V ud as a function of year. For over two decades, PDG takes the isospin-symmetry-breaking corrections δ c from the shell-model calculations. However, our RHF+RPA result [26] as well as the results obtained by the projected DFT by Satuła et al. [29,30] are substantially different from the shell-model results. This discrepancy has also been pointed out by PDG [27].
Combining the V us and V ub values shown in PDG2016, the corresponding square sums of the first-row elements are shown in the right panel of Fig. 2. It can be seen that the unitarity condition is satisfied within the error bar by using the δ c values from the shell-model calculations, while this conclusion is, however, much less clear if one uses the δ c values from the DFT calculations. But it is important to remark that in the DFT scheme the same nuclear effective interaction is used globally for all different superallowed β transitions.
At this moment, the error bars are still large in other independent methods for determining the V ud value. In particular, the central value obtained from the neutron-decay measurements has been shifted up substantially, comparing with the value given several years ago, which is because a shorter neutron lifetime was found. Therefore, it is still an open question, at least to me, whether (or how) the CKM matrix satisfies the unitarity condition. This gives us one more strong motivations for improving the nuclear density functional theories.
Relativistic Brueckner-Hartree-Fock theory for finite nuclei
One of the ongoing projects concerns the tensor effects of nuclear interaction. It is known that such tensor effects are crucial, in particular, for the properties of exotic nuclei, such as the new magic numbers. Within the scheme of covariant DFT, on one hand, some fingerprints of tensor interaction can been seen on the two-proton separation energies [31] or the shell evolutions [32,33]. However, on the other hand, the tensor interaction is totally unwelcome if it is treated as a free parameter to fit the nuclear masses [34]. In order to solve this puzzle, one may look carefully at the region on the nuclear chart where both ab initio methods and DFT work. In such cases, the ab initio calculations could serve as a strong guild line for the development of DFT.
Another strong motivation comes from the latest progress in the lattice QCD simulations. As mentioned above, to link the covariant DFT to more fundamental theories is one of the dreams, which may be indeed not very far away. For example, during the last five years, our colleagues in HAL QCD collaboration have improved the simulations from a small box to a quite large box, from a heavy pion mass to almost the physical pion mass [35]. Therefore, we should also be ready for such progress from our side.
In the relativistic scheme, we chose the relativistic Brueckner Hartree-Fock (RBHF) theory as a benchmark. In particular, we are developing the RBHF theory for finite nuclei. For the first time, the Bethe-Goldstone equation was solved self-consistently in the two-body frame and in a pure relativistic scheme [36]. Figure 3 shows the results of RBHF calculations for 16 O with Bonn A interaction. First of all, the non-relativistic BHF calculation without the three-body force is away from the data. For many years, different kinds of approximation were introduced, in order to meet the computational power at that time. Here shows the result from the effective density approximation (EDA), which is a kind of hybrid model between the non-relativistic and relativistic schemes. Meanwhile, from the early 1990s, different kinds of local density approximation (LDA) were introduced by different groups. Nevertheless, within the same framework by using the same interaction for the same nucleus, as shown with the triangles, the results are found to be very different from each other. By avoiding these approximations, the present result would sever as a solid benchmark for various EDA/LDA calculations. Furthermore, it is remarkable that the spin-orbit splitting is well reproduced from
R B H F B H F L D A : D D R H
B r o c k m a n n ( 1 9 9 2 ) F r i t z ( 1 9 9 3 , 1 9 9 4 ) B o e r s m a ( 1 9 9 4 ) S h i ( 1 9 9 5 ) F u c h s ( 1 9 9 5 ) S h e n ( 1 9 9 7 ) U l r y c h ( 1 9 9 7 ) H o f m a n n ( 2 0 0 1 ) M a ( 2 0 0 2 ) V a n D a l e n ( 2 0 1 1 ) the bare nucleon-nucleon interaction, as we expect that the spin degree of freedom can be taken into account in a consistent way by using the Dirac equation in the relativistic framework. The capability of RBHF calculations now extends to 48 Ca in this ongoing project [37].
L D A : D D R H F
Perspectives
In this special occasion, let me try to close this talk with a dream for another decade. One of our dreams is to develop a new-generation nuclear DFT oriented by quantum field theories. As illustrated in Fig. 4, we are considering in this scheme, (i) the energy density functional is derived from the effective action with Legendre transform, (ii) the non-perturbative nature of nuclear force is handle by the renormalization group with flow equations, and (iii) the theoretical uncertainties come with the idea of effective field theory with proper power counting (e.g., cf. Refs. [38,39,40,41,42,43]). Within such a scheme, indeed we are able to share lots of knowledge and techniques with various fields, such as (lattice) QCD, hadron physics, cold atom physics, condensed matter, quan-tum chemistry, and so on. Moreover, in the coming years, we will get a strong support from the developments of supercomputers. Last but not least, with the construction and upgrades of newgeneration experimental facilities all around the world, we will be able to benchmark and challenge each other.
Figure 1 :
1Strength distributions of the Gamow-Teller resonances in 208 Pb (left) and the spin-dipole excitations in 16 O (right) by the self-consistent RHF+RPA approach. Taken from Refs.[21,22].
Figure 2 :
2(Left) Values of V ud as a function of year. (Right) Square sums of V 2 ud + V 2 us + V 2 ub obtained by different methods.
Figure 3 :
3Energy per particle and charge radius (left) and single-particle spectra (right) of16 O by the RBHF theory with Bonn A interaction, comparing to the EDA (open circle), LDA (triangles), and non-relativistic (open square) calculations and the experimental data. Taken from Refs.[36,37].
EDF from effective action E HK [ρ] ~ Γ[ρ]/β (Legendre transform) theoretical uncertainties from EFT Γ (2) , Γ (3) , Γ (4) … (power counting) non-perturbative nature by renormalization group ∂ k Γ k [ρ] = Tr{…} (flow eq.)
Figure 4 :
4A dream for nuclear DFT oriented by quantum field theories.
. 3 2 . 4 2 . 5 2 . 6 2 . 7 2 . 8
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[
"Low-energy spin dynamics in the [YPc 2 ] 0 S = 1/2 antiferromagnetic chain",
"Low-energy spin dynamics in the [YPc 2 ] 0 S = 1/2 antiferromagnetic chain"
] |
[
"F Branzoli \nDepartment of Physics \"A. Volta\"\nUniversity of Pavia-CNISM\n27100PaviaItaly\n",
"P Carretta \nDepartment of Physics \"A. Volta\"\nUniversity of Pavia-CNISM\n27100PaviaItaly\n",
"M Filibian \nDepartment of Physics \"A. Volta\"\nUniversity of Pavia-CNISM\n27100PaviaItaly\n",
"S Klyatskaya \nInstitute of Nanotechnology\nKarlsrhue Institute of Technology (KIT)\n76344Eggenstein-LeopoldshafenGermany\n",
"M Ruben \nInstitute of Nanotechnology\nKarlsrhue Institute of Technology (KIT)\n76344Eggenstein-LeopoldshafenGermany\n"
] |
[
"Department of Physics \"A. Volta\"\nUniversity of Pavia-CNISM\n27100PaviaItaly",
"Department of Physics \"A. Volta\"\nUniversity of Pavia-CNISM\n27100PaviaItaly",
"Department of Physics \"A. Volta\"\nUniversity of Pavia-CNISM\n27100PaviaItaly",
"Institute of Nanotechnology\nKarlsrhue Institute of Technology (KIT)\n76344Eggenstein-LeopoldshafenGermany",
"Institute of Nanotechnology\nKarlsrhue Institute of Technology (KIT)\n76344Eggenstein-LeopoldshafenGermany"
] |
[] |
1 H nuclear magnetic resonance (NMR) measurements in [YPc2] 0 , an organic compound formed by radicals stacking along chains, are presented. The temperature dependence of the macroscopic susceptibility, of the NMR shift and of the spin-lattice relaxation rate 1/T1 indicate that the unpaired electron spins are not delocalized but rather form a S = 1/2 antiferromagnetic chain. The exchange couplings estimated from those measurements are all in quantitative agreement. The low-energy spin dynamics can be described in terms of diffusive processes and the temperature dependence of the corresponding diffusion constant suggests that a spin-gap around 1 K might be present in this compound.
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10.1103/physrevb.83.174419
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[
"https://arxiv.org/pdf/1011.1141v2.pdf"
] | 119,306,023 |
1011.1141
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4f96cef61c758a0a44e69738300997362384e253
|
Low-energy spin dynamics in the [YPc 2 ] 0 S = 1/2 antiferromagnetic chain
10 Jan 2011
F Branzoli
Department of Physics "A. Volta"
University of Pavia-CNISM
27100PaviaItaly
P Carretta
Department of Physics "A. Volta"
University of Pavia-CNISM
27100PaviaItaly
M Filibian
Department of Physics "A. Volta"
University of Pavia-CNISM
27100PaviaItaly
S Klyatskaya
Institute of Nanotechnology
Karlsrhue Institute of Technology (KIT)
76344Eggenstein-LeopoldshafenGermany
M Ruben
Institute of Nanotechnology
Karlsrhue Institute of Technology (KIT)
76344Eggenstein-LeopoldshafenGermany
Low-energy spin dynamics in the [YPc 2 ] 0 S = 1/2 antiferromagnetic chain
10 Jan 2011numbers: 7660Es7510Pq7540Gb7510Jm
1 H nuclear magnetic resonance (NMR) measurements in [YPc2] 0 , an organic compound formed by radicals stacking along chains, are presented. The temperature dependence of the macroscopic susceptibility, of the NMR shift and of the spin-lattice relaxation rate 1/T1 indicate that the unpaired electron spins are not delocalized but rather form a S = 1/2 antiferromagnetic chain. The exchange couplings estimated from those measurements are all in quantitative agreement. The low-energy spin dynamics can be described in terms of diffusive processes and the temperature dependence of the corresponding diffusion constant suggests that a spin-gap around 1 K might be present in this compound.
I. INTRODUCTION
Molecular solids have attracted much interest since decades owing to the possibility to easily tune their properties either through a chemical bottom up approach or by varying physical parameters as the external pressure. 1 One of the most versatile families of molecular solids is the one based on phthalocyanine (Pc= C 32 H 16 N 8 ) molecules. 2 In fact, the employment of these materials in different areas, ranging from the fabrication of organic light emitting diodes, to contrast agents or spintronics materials, has been envisaged. Pc-based compounds have attracted significant interest in the last decade after it has been suggested that high temperature superconductivity could be induced in these materials by alkali doping 3 and, more recently, when it has been recognized that neutral [LnPc 2 ] 0 molecules, with Ln a lanthanide ion, are molecular nanomagnets with extremely long coherence times at liquid nitrogen temperature. [4][5][6][7][8] Owing to the flat shape of Pc molecules, the structure of Pc-based materials is typically characterized by chains along which Pc molecules tend to stack. 9 Accordingly some of the Pc-based materials show many similarities to the Beechgaard salts. 10 Bis(phthalocyaninato) yttrium [YPc 2 ] 0 compound can be considered the parent compound of the aforementioned [LnPc 2 ] 0 molecular magnets. In fact, it is characterized by the absence of localized f electrons and the microscopic properties are mainly associated with the presence of an unpaired electron delocalized in the a 2 π orbital, due to the one-electron oxidation of the [YPc 2 ] − unit. 11 Thus, [YPc 2 ] 0 allows to investigate the spin dynamics associated only with this unpaired electron spin, independently from the one due to f electrons. One of the most suitable tools to address this aspect is nuclear magnetic resonance (NMR) technique. In this work we present an experimental study of the magnetic properties of [YPc 2 ] 0 compound by means of magnetization and nuclear magnetic resonance (NMR) measurements. The temperature dependence of the macroscopic susceptibility, of the NMR shift and of the spin-lattice relaxation rate 1/T 1 clearly show that this system is a prototype of a S = 1/2 antiferromagnetic chain, characterized by a diffusive low-frequency spin dynamics and, possibly, by the presence of a low-energy spin-gap.
II. EXPERIMENTAL RESULTS AND DISCUSSION
[YPc 2 ] 0 polycristalline samples were synthesized by using some modifications of the protocol published in Ref.
12. All reagents were purchased from Across or Aldrich and used without further purification. A mixture of 1,2dicyanobenzene (42 mmol), Y(acac) 3 ·4H 2 O (5 mmol), and 1,8-diazabicyclo [5,4,0]undec-7-ene (DBU) (21 mmol) in 50 mL of 1-pentanol was refluxed for 1.5 days. The solution was allowed to cool to room temperature. The precipitate was collected by filtration and washed with n-hexane and Et 2 O. The crude purple product was redissolved in 800 ml of CHCl 3 /MeOH (1/1) and undissolved PcH 2 was filtered off. Both forms, blue (anionic [YPc 2 ] − ) and green (neutral [YPc 2 ] 0 ), were obtained simultaneously, as revealed by electronic absorption spectra. In order to convert the unstabilized anionic form to the neutral one, the reaction mixture was presorbed on active (H 2 O-0%) basic alumina oxide. Purification was carried out by column chromatography on basic alumina oxide (deactivated with 4.6% H 2 O, level IV) with chloroform methanol mixture (10:1) as eluent. In general, the yield was 30-35%. According to microelemental analysis based on atomic spectroscopic methods (ICP) performed at Mikroanalytisches Labor Pascher, the powder sample contains molecules of [YPc 2 ] 0 , water and CH 2 Cl 2 in ratio 1:1:1/3. The molecules crystallized in the space group P2 1 2 1 2 1 (γ-phase), as reported in Ref. 13.
DC magnetization (M ) measurements have been performed by using an MPMS-XL7 Quantum Design superconducting quantum interference device (SQUID) magnetometer. The magnetization was found to depend linearly on the magnetic field intensity H, for H ≤ 5 kGauss, over all the explored temperature range and, accordingly, the macroscopic static uniform susceptibility can be written as χ S = M/H. The temperature dependence of χ S reveals the presence of antiferromagnetic correlations. In fact, χ S (T) can be nicely reproduced by a Curie-Weiss (CW) law
χ S (T ) = C T + Θ + χ 0 ,(1)where C = g 2 µ 2 B S(S +1)N A /(3k B )
is Curie constant (µ B the Bohr magneton, g the Landé factor, N A Avogadro's number and k B Boltzmann constant). Θ is the CW temperature and χ 0 a temperature independent term mainly due to diamagnetic and Van-Vleck corrections. By fitting the data, leaving all three parameters free, we found an antiferromagnetic CW temperature Θ = 5.37 ± 0.04 K (Fig. 1) and a Curie constant C = 0.342 ± 0.002 erg · K/G 2 , quite close to the value 0.375 erg·K/G 2 , expected for a S = 1/2 system. If we fixed C = 0.375 erg·K/G 2 the fit was still good and the CW temperature Θ = 6.18 ± 0.03 K. The temperature dependence of χ S shows that the unpaired electron spins are localized along the chains formed by [YPc 2 ] 0 molecules, although a certain overlap of the π orbitals of adjacent molecules must be present in order to justify the magnitude of the antiferromagnetic exchange coupling J e . In fact, although this system should present a narrow half-filled band, the sizeable Hubbard Coulomb repulsion U ∼ 1eV prevents the electron delocalization along the chain. 14 In this limit, J e = Θ = 4t 2 /U , with t the hopping integral among adjacent molecules. From the estimated value of Θ one would derive a band width formed by the overlap of a 2 orbitals in adjacent molecules W = 4t ∼ 0.05 eV≪ U , justifying the spin localization along the chain.
The 1 H NMR spectra were obtained in the 1.6-300 K temperature range for magnetic field intensities H = 9 T, 1 T and 0.3 T. The spectra were derived from the Fourier transform of half of the echo formed after a π/2 − τ − π pulse sequence, when the full NMR line could be irradi- ated or, otherwise, from the envelope of the echo amplitude upon varying the irradiation frequency. The lineshape was gaussian in all the investigated temperature range. For H = 9 T and H = 1 T a broadening of the spectrum can be observed at low temperature, which is more pronounced at higher field intensities (Fig. 2). On the other hand, for H = 0.3 T the linewidth ∆ν is practically temperature independent and the broadening is likely to be due just to nuclear dipole-dipole interaction. The increase of the linewidth with H suggests that the low temperature line broadening originates from some anisotropy in the hyperfine coupling, which for a powder gives rise to a linewidth proportional to the susceptibility. In fact, it is noticed that if we subtract the T -independent contribution at H = 0.3 Tesla from the raw data and divide the linewidth by the Larmor frequency ν 0 , the data at different fields overlap (inset to Fig.2). This result also indicates that there is not an additional internal field due to the onset of a long-range magnetic order down to 1.6 K.
¡ ¢ ¡ £ ¡ ¡ £ ¢ ¡ ¤ ¡ ¡ ¤ ¢ ¡ ¥ ¡ ¡ ¦ ¦ § ¦ ¨ ∆ν (kHz) T(K) H= 9 T, H= 1 T, H= 0.3 T ∆ν/ν 0 T(K) FIG
The NMR paramagnetic shift ∆K = (ν R −ν 0 )/ν 0 , with ν R the resonance frequency, shows a more pronounced temperature dependence (Fig. 3). As expected, it was found to increase upon cooling, according to
∆K = Aχ S 2µ B N A ,(2)
namely the temperature dependence of ∆K should be the same of the macroscopic spin susceptibility. In fact, also ∆K(T ) is found to obey a Curie-Weiss law with a Curie-Weiss temperature Θ = 7.4K ± 0.3 K, close to the one derived from SQUID magnetization measurements. The small difference between those two type of measure- ments could be associated with a tiny amount of impurities which might affect the macroscopic susceptibility. Accordingly, the measurement of the microscopic susceptibility with paramagnetic shift measurements is expected to provide a more reliable estimate of the static uniform spin susceptibility and of the Curie-Weiss temperature. By plotting ∆K as a function of χ S a linear trend is attained (Fig. 3) and from the slope it is possible to estimate the isotropic term of the hyperfine coupling tensor A = 180 ± 10 Gauss. The 1 H nuclear spin-lattice relaxation rate 1/T 1 was measured in the 1.6 -300 K temperature range and for different values of the external field. 1/T 1 was extracted from the recovery of nuclear magnetization after a saturation recovery pulse sequence. The recovery law was found to be a single exponential in all the explored temperature range (Fig. 4, inset). This result is an evidence that the unpaired electron is delocalized onto a π orbital within the molecule. In fact, since in the two phthalocyanine rings a large number of inequivalent proton sites is present, if the electron was on a more localized orbital a distribution of hyperfine couplings would be present and, accordingly, a stretched exponential recovery law should be observed. Moreover, the fact that hyperfine coupling seems quite isotropic indicates that it could originate from the contact interaction between the unpaired electron spin in the a 2 π orbital and the 1 H nuclei.
The temperature dependence of 1/T 1 at different magnetic fields is shown in Fig. (4). In general, for a relaxation process driven by electron spin fluctuations one can write where γ is the nuclear gyromagnetic ratio, |A q | 2 the form factor describing the hyperfine coupling with spin excitations at wave-vector q and S α,α (q, ω L ) (α = x, y, z) the component of the dynamical structure factor at the Larmor frequency. In the high temperature limit, namely when the thermal energy is much larger than the exchange energy (T ≫ Θ), the 1/T 1 of a spin S = 1/2 antiferromagnet becomes temperature and field independent and is given by 15
1 T 1 = γ 2 2N α,q (|A q | 2 S α,α (q, ω L )) ⊥ ,(3)1 T 1 = γ 2 2 (A 2 x + A 2 y ) S(S + 1) 3 √ 2π ω H ,(4)
where A x ≃ A y ≃ A are the components of the hyperfine coupling tensor which is basically isotropic, while ω H = (J e k B / ) 2zS(S + 1)/3 is the Heisenberg exchange frequency, with z = 2 the number of nearest neighbour spins along the chain. By taking the measured value of 1/T 1 ≃ 20 s −1 at high temperature, from Eq. (4) it is possible to estimate the exchange frequency ω H ≃ 9.2 · 10 11 rad/s, corresponding to an exchange coupling constant J e ≃ 7.0 K, in quite good agreement with the value which can be estimated from the NMR shift measurements. Upon decreasing the temperature, for 200 K≥ T ≥ 30 K, one observes a progressive slow increase of 1/T 1 (Fig. (4)). In particular, it is noticed that nuclear spin-lattice relaxation rate increases on decreasing temperature according to
1/T 1 ∝ ln 1/2 (T 0 /T ) .(5)
In fact, in At about 20 K a peak in the nuclear spin-lattice relaxation rate appears (Fig. 4), whose intensity decreases by increasing the external field intensity. Eventually, below T ≃ 5 K, the 1/T 1 is only weakly temperature dependent. The maximum in 1/T 1 , not associated with molecular motions, could be due to a form factor, which partially filters out the antiferromagnetic fluctuations as the system gets more and more correlated. The magnetic field dependence of 1/T 1 (Fig. 6) can originate from the diffusive nature of the spin correlation function, which in one dimension is characterized by longtime tails yielding to a divergence of the low-frequency spectral density J(ω). 17 In fact, in the presence of diffusive processes for the spin excitations 1/T 1 can be written in terms of the spectral density for the spin excitations according to the following equation 18 : pling, which hereafter shall be neglected since A 2 ≫ A 2 d . Then just the second term in square bracket can be considered. In Eq.6 χ 0 is the static uniform susceptibility per spin and ω e = ω 0 γ e /γ is the electron resonance frequency. This means that during the nuclear relaxation process a simultaneous flip of the electron and nuclear spins occur, involving an energy exchange (ω e ± ω 0 ), and 1/T 1 thus probes the spin excitations at a frequency close to ω e .
1 T 1 = γ 2 2 k B T χ 0 (gµ B ) 2 3 5 A 2 d J(ω 0 ) + A 2 + 7 5 A 2 d J(ω e ± ω 0 ) ,(6)
In a one dimensional system, the spectral density at ω e is characterized by a low-frequency divergence given by 19
J(ω e ) = 1 √ 2D ( ω c + ω 2 e + ω 2 c ω 2 e + ω 2 c ) 1/2 ,(7)
where ω c is a low-frequency cutoff accounting for the finite spin anisotropy and/or inter-chain coupling, while D is the spin diffusion rate. In Fig. (6), the 1/T 1 is plotted as a function of ν −1/2 0 . The observed linear trend further proves the one-dimensional nature of the antiferromagnetic correlations. Moreover, the absence of a low-frequency flattening in 1/T 1 plot indicates that spin diffusion occurs in the electronic frequency range ω c ≪ ω e ≪ D. Thus, from the slopes of the curves it is possible to deduce the spin diffusion coefficient at different temperatures (Fig. 7) considering A ≃ 180 Gauss and neglecting ω c ≪ ω e in Eqs (6-7). The estimated spin diffusion coefficient is of the order of the exchange frequency ω H and it is found to progressively decrease with temperature and to become nearly constant above 20 K. It is interesting to observe that D ∝ exp(∆/T ), namely the behaviour expected for one-dimensional antiferromagnets in the presence of a spin-gap ∆ between singlet and triplet excitations. 20 Here we find that ∆ = 1.2 ± 0.4 K suggesting that a small gap, either due to competing exchange interactions or to a dimerization might be present in [YPc 2 ] 0 . It is interesting to observe that, at low temperature, when the Zeeman energy ω e ≃ ∆ the breakdown of Eq.7 is noticed. In fact, in Fig. (6) one clearly notices that at T = 1.6 K the linear behaviour is no longer obeyed at high fields (i.e. low values for 2π/ω 0 ) and 1/T 1 ceases to decrease with increasing field. This could be due to the modifications in the spin correlations induced by the magnetic field for ω e ≃ ∆, possibly associated with the progressive closure of the spin gap.
In conclusion, from magnetization, 1 H NMR paramagnetic shift and T 1 measurements we have derived the magnitude of the antiferromagnetic exchange interaction in [YPc 2 ] 0 compound and found an overall good agreement. The low-energy spin excitations are of diffusive character and characteristic of one-dimensional antiferromagnets. From the temperature dependence of the spin diffusion rate derived from 1/T 1 vs. H measurements it was found that a spin-gap around 1 K might be present in this compound.
FIG. 1 :
1Temperature dependence of static uniform susceptibility χS for [YPc2] 0 complex, derived from SQUID magnetization measurements. The solid line shows the best fit of the data to Curie-Weiss law.
FIG. 3 :
3Temperature dependence of 1 H paramagnetic shift ∆K in [YPc2] 0 . The solid line shows the best fit of the data to Curie-Weiss law. In the inset ∆K is reported as a function of the macroscopic susceptibility. The solid line is the best fit according for a linear dependence of ∆K vs χS.
FIG. 4 :
41 H nuclear spin-lattice relaxation rate temperature dependence for [YPc2] 0 compound measured for different values of the applied field. In the inset the recovery of the nuclear magnetization as a function of the delay τ between the saturating and the echo readout sequences is shown at two different temperatures. The solid lines show the best fit for a single exponential recovery.
FIG. 5 :
5Fig. (5) one observes that (1/T 1 ) 2 is a linear function of 1/T , when reported in logarithmic scale. Remarkably, this logarithmic increase of 1/T 1 is expected in The 1/T1 squared is plotted as a function of T −1 , in logarithmic scale, for two values of the external field (H = 9 T, circles and H = 1 T, squares). The dashed lines represent the best fits to Eq. (5).
FIG. 6 :
6The 1 H spin-lattice relaxation time 1/T1 in [YPc2] 0 is plotted as a function of (ω0/2π) −1/2 for different selected temperatures. The solid lines show the best fit according to Eqs.6 and 7 in the text.
FIG. 7 :
7where A d is the anisotropic term of the hyperfine cou-Temperature dependence of the ratio D/ωH between the spin diffusion coefficient D and the exchange frequency ωH in [YPc2] 0 compound as derived from the slopes in the 1/T1 vs (ω0/2π) −1/2 plots inFig. 6. The solid line gives the best fit according to D ∝ exp(∆/T ) with ∆ = 1.2 ± 0.4 K.
. 2: Temperature dependence of 1 H full NMR linewidth at half intensity in [YPc2] 0 , at three different magnetic fields. In the inset the linewidth is normalized by the Larmor frequency after subtracting the constant linewidth due to nuclear dipoledipole interaction.
AcknowledgementsThe research activity in Pavia was supported by Fondazione Cariplo (Grant N. 2008-2229) research funds.
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[] |
[
"Using TDHF to study quasifission dynamics",
"Using TDHF to study quasifission dynamics"
] |
[
"A S Umar ",
"C Simenel ",
"\nPhysics and Astronomy\nDepartment of Nuclear Physics\nVanderbilt University\n37235NashvilleTNUSA\n",
"\nThe Australian National University\n2601CanberraACTAustralia\n"
] |
[
"Physics and Astronomy\nDepartment of Nuclear Physics\nVanderbilt University\n37235NashvilleTNUSA",
"The Australian National University\n2601CanberraACTAustralia"
] |
[] |
We show that the microscopic TDHF approach provides an important tool to shed some light on the nuclear dynamics leading to the formation of superheavy elements. In particular, we discuss studying quasifission dynamics and calculating ingredients for compound nucleus formation probability calculations.
|
10.1142/9789813229426_0023
|
[
"https://arxiv.org/pdf/1612.08917v1.pdf"
] | 118,877,582 |
1612.08917
|
6d7f3e18eef993705343e50b6d058b50bb7d6714
|
Using TDHF to study quasifission dynamics
November 23, 2018
A S Umar
C Simenel
Physics and Astronomy
Department of Nuclear Physics
Vanderbilt University
37235NashvilleTNUSA
The Australian National University
2601CanberraACTAustralia
Using TDHF to study quasifission dynamics
November 23, 201822:22 WSPC Proceedings -9in x 6in sanibel˙umar page 1 1TDHFSuperheavy nucleiQuasifission
We show that the microscopic TDHF approach provides an important tool to shed some light on the nuclear dynamics leading to the formation of superheavy elements. In particular, we discuss studying quasifission dynamics and calculating ingredients for compound nucleus formation probability calculations.
Introduction
The time-dependent Hartree-Fock (TDHF) theory has provided a possible means to study the diverse phenomena observed in low energy nuclear physics 1 . As a result of theoretical approximations (single Slater determinant), TDHF describes reaction channels in a common mean-field, usually corresponding to the dominant reaction channel. For instance, it describes above-barrier fusion and average transfer dynamics 2 . To obtain multiple reaction channels or widths of observables one must go beyond TDHF [3][4][5] . In connection with superheavy element formation, the theory predicts best the cross-section for a particular process which dominates the reaction mechanism. This is certainly the case for studying capture cross-sections and quasifission.
In recent years has it become numerically feasible to perform TDHF calculations on a 3D Cartesian grid without any symmetry restrictions and with much more accurate numerical methods 6,7 . During the past several years, a novel approach based on TDHF called the density constrained timedependent Hartree-Fock (DC-TDHF) method was developed to compute heavy-ion potentials 8,9 and excitation energies 10 directly from TDHF timeevolution. This method was applied to calculate capture cross sections for fusion reactions leading to superheavy element Z = 112 11 . Furtermore, within the last few years the TDHF approach has been utilized for studying the dynamics of quasifission [12][13][14][15][16][17][18] . The study of quasifission is showing a great promise to provide insight based on very favorable comparisons with experimental data.
Recent quasifission studies
One of the major questions that is asked by the experimental superheavy element community is why a 48 Ca beam is so crucial in forming such systems and whether one could produce new superheavy nuclei using projectiles different than 48 Ca and actinide targets. Our first work in this area focused on the quasifission studies for the 40,48 Ca+ 238 U system 12,13 , showing that for neutron-rich 48 Ca beams quasifission is substantially reduced. While, the above work is also in the time span of this proposal, below we discuss the more recent studies of the 48 Ca+ 249 Bk and 50 Ti+ 249 Bk systems 14 . The reaction 48 Ca + 249 Bk creates superheavy isotopes of element 117 with cross-sections of 2-3 picobarns. However, the 50 Ti + 249 Bk reaction so far has not produced any superheavy isotopes of element 119, with an upper cross-section limit of 50 fb 19 .
We have calculated the microscopic DC-TDHF nucleus-nucleus potential barriers for the 48 We have also used DCTDHF to calculate the excitation energy of each fragment directly from the TDHF density evolution. This gives us new information on the repartition of the excitation energy between the heavy and light fragments which is not available in standard TDHF calculations. In Fig. 1 (c)
Shape evolution and collective dynamics of quasifission
The proper characterization of fusion-fission and quasifission is one of the most important tasks in analyzing reactions leading to superheavy elements. Experimental analysis of fusion-fission and quasifission fragment angular distributions W (θ) is commonly expressed in terms of a two-component expression 22 ,
W (θ) = JCN J=0 F (F F ) J (θ, K 0 (F F )) + Jmax J=JCN F (QF ) J (θ, K 0 (QF )) .(1)
Here, J CN defines the boundary between fusion-fission and quasifission, assuming a sharp cutoff between the angular momentum distributions of each mechanism. The quantum number K is known to play an important role in fission. It is the projection of the total angular momentum onto the deformation axis. In the Transition State Model (TSM), the characteristics of the fission fragments are determined by the K distribution at scission. The argument K 0 entering Eq. (1) is the width of this distribution which is assumed to be Gaussian. It obeys K 2 0 = T ef f / 2 , where the effective moment of inertia, ef f , is computed from the moments of inertia for rotations around the axis parallel and perpendicular to the principal deformation axis 1 ef f = 1 − 1 ⊥ , and T is the nuclear temperature at the saddle point. The physical parameters of the fusion-fission part are relatively well known from the liquid-drop model 23 . In contrast, the quasifission process never reaches statistical equilibrium. In principle, it has to be treated dynamically, while Eq. (1) is based on a statistical approximation. In addition, the usual choice for the nuclear moment of inertia for the quasifission component, 0 / ef f = 1.5 24 , is somewhat arbitrary. Here, 0 is the moment of inertia of an equivalent spherical nucleus.
We have developed methods to extract the moment of inertia of the system (the main collective observable of interest for fission and quasifission) directly from TDHF time-evolution of collisions resulting in quasifission. The proper way to calculate the moment-of-inertia for such time-dependent densities (particularly for non-zero impact parameters) is to directly diagonalize the moment-of-inertia tensor represented by a 3 × 3 matrix with elements
ij (t)/m = d 3 r ρ(r, t)(r 2 δ ij − x i x j ) ,(2)
where ρ is the local number-density calculated from TDHF evolution, m is the nucleon mass, and x i=1,2,3 denote the Cartesian coordinates. Numerical diagonalization the matrix gives three eigenvalues. One eigenvalue corresponds to the moment-of-inertia for the nuclear system rotating about the principal axis. The other two eigenvalues define the moments of inertia for rotations about axes perpendicular to the principal axis. Using the time-dependent moment-of-inertia obtained from the TDHF collision one can calculate the so-called effective moment-of-inertia defined above. We have calculated the moment-of-inertia ratio for the 48 Ca + 249 Bk noncentral collisions at E c.m. = 218 MeV. At the point of final touching configuration the moment-of-inertia ratios are in the range 1.4-1.8, suggesting a relatively strong impact parameter dependence which should be accounted for in future extensions to the TSM.
Ca+ 249 Bk and 50 Ti+ 249 Bk systems for two extreme orientations of the 249 Bk nucleus (tip and side). For the 48 Ca+ 249 Bk system, the tip orientation of 249 Bk results in a significantly lower barrier, E B (tip)= 191.22 MeV, as compared to the side orientation, E B (side)= 204.36 MeV. This reaction has been studied at E c.m. = 204-218 MeV in Dubna 20 and E c.m. = 211-218 MeV with GSI-TASCA 21 . Thus, the highest experimental energy E c.m. = 218 MeV is above both barriers but the lowest experimental energy E c.m. = 204 MeV is slightly below the barrier for the side orientation of 249 Bk. Similarly, the corresponding potential barriers for the 50 Ti + 249 Bk system are E B (tip)= 211.2 MeV and E B (side)= 224.6 MeV. Experimentally, E c.m. = 233.2 MeV was used in the GSI-TASCA experiment 21 , well above both barriers.
Figure 1 (
1a) (left) shows the contact time as a function of center-ofmass energy for central collisions of 48 Ca with 249 Bk. For the tip orientation of the 249 Bk nucleus (dashed line) we observe contact times of order 10-12 zs which are essentially constant over a wide range of energies, E c.m. = 191-230 MeV. Only at energies below the potential barrier, E B (tip)= 192.2 MeV, do the contact times drop off
Fig. 1 .
1(a) Contact time, (b) mass and charge of the light fragment, and (c) excitation energy E * of the heavy and light fragments as a function of Ec.m. for central collisions of 48 Ca with 249 Bk (left) and 50 Ti with 249 Bk (right). Solid lines are for the side orientation of the deformed 249 Bk nucleus, and dashed lines are for the tip orientations.because these events correspond to inelastic scattering and few-nucleon transfer reactions. A dramatically different picture emerges for the side orientation of the 249 Bk nucleus (solid line): At energies above the barrier E B (side)= 205.4 MeV, the contact times rise very steeply with energy and reach values up to 22 zs at E c.m. = 210 MeV. For energies above this value, fusion is observed which we define by a large contact time exceeding 35 zs and a mononuclear shape without a neck.Figure 1(b) (left) shows the corresponding mass and charge of the light fragment. We observe that the mass and charge transfer to the light fragment are roughly proportional to the nuclear contact time. In particular, for the side orientation of 249 Bk, we find quasielastic collisions at energies below E c.m. = 204 MeV. Quasifission is limited to a small range of energies E c.m. = 209-211 MeV, whereas for energies above 211 MeV we find fusion. Naturally, non-central impact parameters can show quasifission in the range where we see fusion. The quasifission results are very different for the tip orientation of 249 Bk, ranging over a much wider energy domain E c.m. = 191-230 MeV with a lower maximum mass and charge transfer compared to the side orientation of 249 Bk.
we show the excitation energies of the heavy and light fragments which contain approximately 55-60 MeV and 30-45 MeV of excitation energy (side orientation) and 50 MeV and 30 MeV (tip orientation), respectively, for c.m. energies corresponding to quasifission. Right panel of Fig. 1 shows the corresponding results for central collisions of 50 Ti with 249 Bk. The contact times and the masses and charges of the light fragment show a similar behavior as a function of energy as compared to the 48 Ca + 249 Bk reaction. For the tip orientation, we find quasifission for E c.m. ≥ 214 MeV, with excitation energies of E * H = 57-69 MeV for the heavy fragment and E * L = 27-41 MeV for the light fragment, respectively. The mass and charge of the fragments indicate a strong influence of the shell effects in the 208 Pb region, as in reactions with 48 Ca. However, N = 50 does not seem to play a role here. For the side orientation, we find inelastic and multi-nucleon transfer reactions at energies E c.m. = 223-227 MeV. Quasifission is confined to an extremely narrow energy window around E c.m. = 227.4-227.7 MeV, with excitation energies of E . At energies E c.m. > 228 MeV, fusion sets in.
November 23, 2018 22:22 WSPC Proceedings -9in x 6in sanibel˙umar page 3
AcknowledgmentsThis work has been supported by the U.S. Department of Energy under grant No. DE-SC0013847 with Vanderbilt University and by the Australian Research Councils Future Fellowship (project number FT120100760) and Discovery Projects (project number DP160101254) funding schemes.
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[] |
[
"ON D ℓ -EXTENSIONS OF ODD PRIME DEGREE ℓ",
"ON D ℓ -EXTENSIONS OF ODD PRIME DEGREE ℓ"
] |
[
"Henri Cohen ",
"Frank Thorne "
] |
[] |
[] |
Generalizing the work of A. Morra and the authors, we give explicit formulas for the Dirichlet series generating function of D ℓ -extensions of odd prime degree ℓ with given quadratic resolvent. Over the course of our proof, we explain connections between our formulas and the Ankeny-Artin-Chowla conjecture, the Ohno-Nakagawa relation for binary cubic forms, and other topics.Date: September 30, 2016.
|
10.1112/plms.12339
|
[
"https://arxiv.org/pdf/1609.09153v2.pdf"
] | 119,172,468 |
1609.09153
|
8243d969ffc7e1fa733924583ef721b39f0d9e38
|
ON D ℓ -EXTENSIONS OF ODD PRIME DEGREE ℓ
28 Sep 2016
Henri Cohen
Frank Thorne
ON D ℓ -EXTENSIONS OF ODD PRIME DEGREE ℓ
28 Sep 2016
Generalizing the work of A. Morra and the authors, we give explicit formulas for the Dirichlet series generating function of D ℓ -extensions of odd prime degree ℓ with given quadratic resolvent. Over the course of our proof, we explain connections between our formulas and the Ankeny-Artin-Chowla conjecture, the Ohno-Nakagawa relation for binary cubic forms, and other topics.Date: September 30, 2016.
Introduction
The theory of cubic number fields is, in many respects, well understood. One reason for this is that the Delone-Faddeev [18] and Davenport-Heilbronn [17] correspondences parametrize cubic fields in terms of binary cubic forms, up to equivalence by an action of GL 2 (Z), and satisfying certain local conditions. Therefore questions about counting cubic fields can be reduced to questions about counting lattice points, and this idea has led to asymptotic density theorems as well as other interesting results.
In more recent work, Bhargava [5,6] obtained similar parametrization and counting results for S 4 -quartic and S 5 -quintic fields. However, generalizing this work to number fields of arbitrary degree ℓ seems difficult, if not impossible: the parametrizations of S 3 -cubic, S 4 -quartic, and S 5 -quintic fields are all by prehomogeneous vector spaces, and for higher degree fields there is no apparent prehomogeneous vector space for which one could hope to establish a parametrization theorem.
In [12] and [14], A. Morra and the authors contributed to the cubic theory by giving explicit formulas for the Dirichlet generating series of discriminants of cubic fields having given resolvent. where ω(p) is equal to 2 or −1 if p is totally split or inert in the unique cubic field of discriminant 321, determined by the polynomial x 3 − x 2 − 4x + 1, and ω(p) = 0 otherwise. Similar formulas hold when −107 is replaced by any other fundamental discriminant D; the formula has one main term, and one additional Euler product for each cubic field of discriminant −D/3, −3D, and −27D.
The proofs involve class field theory and Kummer theory; see also work of Bhargava and Shnidman [7] obtaining related results through a study of binary cubic forms.
Let L/k be an extension 1 of odd prime degree ℓ, let N = L be a Galois closure of L, and assume that Gal(N/k) ≃ D ℓ , the dihedral group with 2ℓ elements. We will refer to any such L as a D ℓ -extension of k, or a D ℓ -field when k = Q. Below we also refer to F ℓ -extensions with the analogous meaning.
There exists a unique quadratic subextension K/k of N/k, called the quadratic resolvent of L, with Gal(N/K) ≃ C ℓ , and a nontrivial theorem of J. Martinet involving the computation of higher ramification groups (see Propositions 10.1.25 and 10.1.28 of [8]) tells us that its conductor f (N/K) is of the form f (N/K) = f (L)Z K , where f (L) is an ideal of the base field k, and that the relative discriminant d(L/k) of L/k is given by the formula d(L/k) = d(K/k) (ℓ−1)/2 f (L) ℓ−1 .
We study the set F ℓ (K) of D ℓ -extensions of k whose quadratic resolvent field is isomorphic to K. (Here and in the sequel, extensions are always considered up to k-isomorphism.) More precisely, we want to compute as explicitly as possible the Dirichlet series 2
Φ ℓ (K, s) = 1 ℓ − 1 + L∈F ℓ (K) 1 N (f (L)) s ,
where N (f (L)) = N k/Q (f (L)) is the absolute norm of the ideal f (L).
Our most general result is Theorem 6.1, which we specialize to a more explicit version in the case k = Q as Theorem 7.3. This should be considered as the most important result of this paper. In Section 9 we prove that our formulas can always be brought into a form similar to (1.2). Two sample results are as follows:
(1.3) Φ 5 (Q(√ 51 + ω E (p) p s ,
where E is the field defined by x 5 + 5x 3 + 5x − 1 = 0 of discriminant 5 7 , and ω E (p) = −1, 4, or 0 according to whether p is inert, totally split, or other in E. We also have where the products are over suitable primes p (see Example 9.8), E is the field defined by x 5 +5x 3 +5x−3 = 0, and ω E (p) is as before.
In a companion paper, joint with Rubinstein-Salzedo [13], we investigate a curious twist to this story. Taking the n = 1 term of formula (1.2) (or, rather, its generalization to any D) yields the nontrivial identity This identity was previously conjectured by Ohno [33] and then proved by Nakagawa [31], as a consequence of an 'extra functional equation' for the Shintani zeta function associated to the lattice of binary cubic forms. Our generalization of (1.2) thus subsumes the Ohno-Nakagawa theorem (1.5).
Our proof there used the Ohno-Nakagawa theorem, but in [13] we further develop some of the techniques of this paper (in particular, of Section 8) to give another proof of (1.5) and give a generalization to any prime ℓ ≥ 3. For ℓ > 3 our work relates counts of D ℓ -fields (the right-hand side of (1.5)) to counts of F ℓ -fields ℓ (the left-hand side), where F ℓ is the Frobenius group of order ℓ(ℓ − 1), whose definition is recalled in Section 9. (Note that S 3 = D 3 = F 3 .) The result involves a technical (Galois theoretic) condition on the F ℓ -fields which is not automatically satisfied for ℓ > 3, and we defer to [13] for a complete statement of the results. It is however important to note that, as for the cubic case, even the case n = 1 of our Dirichlet series identities such as (1.4) gives interesting results: for instance, for any negative fundamental discriminant −D coprime to 5, we have
(1.6) N F 5 (−1) 0 5 3 D 2 + N F 5 (−1) 0 5 5 D 2 + N F 5 (−1) 0 5 7 D 2 = N D 5 (−D) 2 + N D 5 (−5D) 2 ,
and if instead D > 1, then we have
(1.7) N F 5 (−1) 2 5 3 D 2 + N F 5 (−1) 2 5 5 D 2 + N F 5 (−1) 2 5 7 D 2 = 5 N D 5 D 2 + N D 5 (5D) 2 + 2 .
In the above, N G (X) denotes the number of G-fields with discriminant exactly equal to X, and (−1) r specifies the number of pairs of complex embeddings. If we want an identity counting D 5 -fields of discriminant (±D) 2 or (±5D) 2 alone, then the left side of (1.6) and (1.7) becomes more complicated, and involves the Galois condition mentioned above. The relevance to the present paper is that it is precisely those F ℓ -fields counted by this identity that yield Euler products. We describe this in more detail in Section 9.
There is one further curiosity that emerges in our work: a connection to a well-known conjecture attributed to 3 Ankeny, Artin, and Chowla [1] which states that if ℓ ≡ 1 (mod 4) is prime and ǫ = (a + b √ ℓ)/2 is the fundamental unit of Q( √ ℓ), then ℓ ∤ b. As we will see, the truth or falsity of the conjecture will be reflected in our explicit formula for D ℓ -extensions having quadratic resolvent Q( √ ℓ). Note that the conjecture is known to be true for ℓ < 2 · 10 11 , but on heuristic grounds it should be false: if we assume independence of the divisibility by ℓ, the number of counterexamples for ℓ ≤ X should be around log(log(X))/2; in addition, numerous counterexamples can easily be found for "fake" quadratic fields, see e.g., [15,32].
Our work follows several other papers studying dihedral field extensions. Much of the theory (such as Martinet's theorem) is described in the first author's book [8]. Another reference is Jensen and Yui [23], who studied D ℓ -extensions from multiple points of view. They proved that if ℓ ≡ 1 (mod 4) is a regular prime, then no D ℓ -extension of Q has discriminant a power of ℓ; our proof uses similar ideas, and we will recover and strengthen their result. Jensen and Yui also studied the problem of constructing D ℓ -extensions, and gave several examples.
Another relevant work is the paper of Louboutin, Park, and Lefeuvre [28], who developed a general class field theory method to construct real D ℓ -extensions. These problems have also been addressed in the function field setting by Weir, Scheidler, and Howe [38].
Since some of the proofs are quite technical, we give a detailed overview of the contents of this paper. We begin in Section 2 with a characterization of the fields L ∈ F ℓ (K) using Galois and Kummer theory. These fields are in bijection with elements of K z := K(ζ ℓ ) modulo ℓth powers, satisfying certain restrictions which guarantee that the associated Kummer extensions of K z descend to degree ℓ extensions of k. Writing such an extension as K z ( ℓ √ α) with αZ Kz = 0≤i≤ℓ−2 a g i i q ℓ , we further characterize these fields in terms of conditions on the a i and an associated member u of a Selmer group associated to K z . These conditions are described in terms of the group ring F ℓ [G], where G = Gal(K z /k). Groups such as
K * z /K * z ℓ , Cl(K z )[ℓ]
, and the Selmer group are naturally F ℓ [G]-modules, and our conditions correspond to being annihilated by certain elements of F ℓ [G] (see Definition 2.2).
In Section 2 we also study the subfields of K z /k, with particular attention to a degree ℓ−1 extension K ′ /k called the mirror field of K; we will see that much of the arithmetic of prime splitting in various extensions can be conveniently expressed in terms of K ′ .
The reader who is willing to take our technical computations for granted is advised to look only at the necessary definitions in the intermediate sections and to skip directly to Section 6.
In Section 3, we give an expression for the 'conductor' f (L) in terms of the quantities a i and u defined in Section 2. The main result, Theorem 3.8, was proved by the first author, Diaz y Diaz, and Olivier in [11] in their study of cyclic extensions of degree ℓ, and we also prove a few additional related lemmas and propositions. Unfortunately the results of that section are rather complicated to state, and oblige us to introduce a fair amount of notation.
In Section 4 we begin to study the fundamental Dirichlet series using the results proved in Section 3. That section is mostly elementary and combinatorial (but messy), and in Section 5 we study the size of a certain Selmer group appearing in our formulas. That section is heavily algebraic and again appeals heavily to the results of [11].
In Section 6, we put everything together to obtain our most general formula (Theorem 6.1) for Φ ℓ (K, s), a generalization of the main theorem of [12].
In the remainder of the paper we further study this formula with the aim of making everything more explicit; for the most part we now specialize to the case k = Q. In Section 7 we compute various quantities appearing in Theorem 6.1 for k = Q, leading to Theorem 7.3, a more explicit specialized version of Theorem 6.1. This also allows us to obtain asymptotics for the number of D ℓ -extensions of Q, proved in Corollary 7.5.
The formula of Theorem 7.3 falls short of being explicit in one important aspect: it involves a sum (of Euler products) over the character group of a somewhat complicated group G b . So in Section 8 we further study its size. The main result is the Kummer pairing of Theorem 8.2, familiar from (for example) the proof of the Scholz reflection principle, and fairly simple to prove. One important input (Proposition 8.1) is a very nice relationship, due essentially to Kummer and Hecke, between the conductor of Kummer extensions of K z , and congruence properties of the ℓth roots used to generate them. This section culminates in an explicit formula for the size of G b .
Some of our work in Section 8 (including Theorem 8.2) is also critical in [13], and to avoid redundancy we only sketch a few results whose complete proofs are given there.
In Section 8 we also explore the connection to the conjecture of Ankeny, Artin, and Chowla mentioned above. The truth or falsity of this conjecture will then be reflected in our explicit formula (Proposition 9.2) for Φ ℓ (Q( √ ℓ), s), and in Corollary 9.4 we will give a proof of an observation of Lemmermeyer, that the existence of D ℓ -fields ramified only at ℓ is equivalent to the falsity of the Ankeny-Artin-Chowla conjecture.
In Section 9 we further study the characters of the group G b , and prove (in Theorem 9.1) that each such character corresponds to an F ℓ -extension E/k, such that the values of χ correspond to the splitting types of primes in E. This was done for ℓ = 3 and k = Q in [14], but in Theorem 9.1 we do not require k = Q.
It is here that the connection to the Ohno-Nakagawa theorem emerges; for ℓ = 3 and k = Q, we established in [14] (using Ohno-Nakagawa) that the set of characters of G b corresponds precisely to a suitable and easily described set of fields E. For ℓ > 3 we require the generalization of Ohno-Nakagawa established in [13], and so in Section 9 we say a bit more about the results of [13] and explain their relevance. We also prove an explicit formula valid for the 'special case' K = Q( √ ℓ).
Galois and Kummer Theory
2.1. Galois and Kummer theory, and the Group Ring. We will use the results of [11], but before stating them we need some notation. We denote as usual by ζ ℓ a primitive ℓth root of unity, we set
K z = K(ζ ℓ ), k z = k(ζ ℓ ), N z = N (ζ ℓ )
, and we denote by τ , τ 2 , and σ generators of k z /k, K/k, and N/K respectively, with τ ℓ−1 = τ 2 2 = σ ℓ = 1. The number ζ ℓ could belong to k, or to K, or generate a nontrivial extension of K of degree dividing ℓ − 1. These essentially correspond respectively to cases (3), (4), and (5) of [12] (cases (1) and (2) correspond to cyclic extensions of k of degree ℓ, which have been treated in [11]). Cases (3) and (4) are considerably simpler since we do not have to adjoin ζ ℓ to K to apply Kummer theory.
We are particularly interested in the case k = Q, in which case either [K z : K] = ℓ − 1, or [K z : K] = (ℓ − 1)/2, i.e., K ⊂ k z , which is equivalent to K = Q( √ ℓ * ) with ℓ * = (−1) (ℓ−1)/2 ℓ. To balance generality and simplicity, we assume that k is any number field for which [k z : k] = ℓ − 1. Then, as for k = Q there are two possible cases: either [K z : K] = ℓ − 1, which we call the general case, or K ⊂ k z = K z and [K z : K] = (ℓ − 1)/2, which we will call the special case. Note that if ℓ = 3 this means that ζ ℓ ∈ K, so we are in case (4), but there is no reason to treat this case separately. It should not be particularly difficult to extend our results to any base field k, as was done in [11]. We set the following notation:
• We let g be a primitive root modulo ℓ, and also denote by g its image in F * ℓ = (Z/ℓZ) * . • We let G = Gal(K z /k). Thus in the general case G ≃ (Z/2Z) × (Z/ℓZ) * , while in the special case G = Gal(k z /k) ≃ (Z/ℓZ) * . We denote by τ the unique element of Gal(k z /k) such that τ (ζ ℓ ) = ζ g ℓ , so that τ generates Gal(k z /k), and we again denote by τ its lift to K z or N z .
The composite extension N z = N K z is Galois over k, and σ and τ naturally lift to N z . In the general case, τ and σ commute; in the special case, τ 2 is a generator of Gal(K z /K) and τ 2 can be taken to be any odd power of τ , for instance τ itself, so that τ στ −1 = σ −1 .
This information is summarized in the two Hasse diagrams below, depicting the general and special cases respectively.
N z <τ > • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • <σ> N z <τ 2 > t t t t t t t t t t t t t t t t t t t t t t <σ> N <τ 2 > ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ <σ>≃C ℓ N <τ > ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ ✞ <σ>≃C ℓ K z ⊇ p z <τ > ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ <τ 2 > ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ K z = k z ⊇ p z <τ 2 > ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ <τ > ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ℓL ℓ ℓL ℓ K ⊇ p <τ 2 > ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ k z = k(ζ ℓ ) ⊇ p k <τ > ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ K ⊇ p <τ > ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ k ⊇ p k ⊇ p
In the above p, p, p k , p z indicate our typical notation (to be used later) for primes of k, K, k z , K z respectively.
Lemma 2.1. For a mod (ℓ − 1) and b mod 2, set
e a = 1 ℓ − 1 j mod (ℓ−1) g aj τ −j ∈ F ℓ [G]
and, in the general case, e 2,b = 1 2 j mod 2 (−1) bj τ −j 2 .
The e a form a complete set of orthogonal idempotents in F ℓ [G], as do the e 2,b in the general case, so in the general case any F ℓ [G]-module M has a canonical decomposition M = a mod (ℓ−1), b mod 2 e a e 2,b M , while in the special case we simply have M = a mod (ℓ−1) e a M .
Proof. Immediate and classical; see, e.g., Section 7.3 of [19].
We set the following definitions: (1) We define ι(τ 2 + 1) = e 2,1 = 1 2 (1 − τ 2 ), and for any a we define ι(τ − g a ) = e a , so that for instance ι(τ + g) = e (ℓ+1)/2 . (1) For any t ∈ T we have t • ι(t) = ι(t) • t = 0, where the action of t and ι(t) is on M .
To conclude, we must prove that each α ∈ (K * z /K * z ℓ )[T ] determines such an L ∈ F ℓ (K). Again write θ = ℓ √ α with σ(θ) = ζ ℓ θ and N z = K z (θ). Define an automorphism τ of N z , agreeing with τ on K z , by writing τ (θ) = ηθ εg (ε = ±1 as before), where η ℓ = τ (α)/α εg ∈ K ℓ z , so that η ∈ K z is well-defined up to an ℓth root of unity, and we make an arbitrary such choice.
Computations show that τ σ ε (θ) = στ (θ) and that τ ℓ−1 (θ) is θ times a root of unity. Each τ σ i is also a lift of τ .
In the general case, we check that there is a unique such lift, which we denote simply by τ , for which τ ℓ−1 (θ) = θ. Write τ 2 (θ) = η 2 /θ with η ℓ 2 = ατ 2 (α) where η 2 is in K z and indeed k z . We check that τ 2 2 (θ) = θ and τ 2 σ(θ) = σ −1 τ 2 , so that by rewriting τ 2 as τ 2 we see that N z /k is Galois with Galois group C ℓ−1 × D ℓ , as required. Here the choice of lift τ 2 is not uniquely determined: D ℓ has ℓ elements of order 2, corresponding to the ℓ conjugate subextensions L/k of degree ℓ.
In the special case, rewriting τ as τ we now have τ ℓ−1 = 1 regardless of the choice of lift: we have τ ℓ−1 (θ) = ζ i ℓ θ for some integer i, so that unless i ≡ 0 (mod ℓ), τ is of order ℓ(ℓ − 1). We already know that N z /k is Galois, as the τ r σ s are distinct automorphisms of N z /k for 0 ≤ τ < ℓ − 1, 0 ≤ σ < ℓ. We have already proved that Gal(N z /k) is nonabelian, and in particular noncyclic, hence i = 0. So τ ℓ−1 = 1 and Gal(N z /k) has the required presentation.
Recall from [8] the following definition:
Definition 2.5. We denote by V ℓ (K z ) the group of (ℓ-)virtual units of K z , in other words the group of u ∈ K * z such that uZ Kz = q ℓ for some ideal q of K z , or equivalently such that ℓ | v pz (u) for any prime ideal p z of K z . We define the (ℓ-)Selmer group S ℓ (K z ) of K z by S ℓ (K z ) = V ℓ (K z )/K * z ℓ .
The following lemma shows in particular that the Selmer group is finite. Lemma 2.6. We have a split exact sequence of F ℓ [G]-modules
1 −→ U (K z ) U (K z ) ℓ −→ S ℓ (K z ) −→ Cl(K z )[ℓ] −→ 1 ,
where the last nontrivial map sends u to the ideal class of q such that uZ Kz = q ℓ .
Proof. Exactness follows from the definitions, and the sequence splits because ℓ ∤ |G| (see for example [10,Lemma 3.1] for a proof).
t(α) is a virtual unit, its image t(α) is annihilated by ι(t) in the Selmer group. By Lemma 2.3 applied to M = S ℓ (K z ), we have t(α) = t(β) for some β ∈ S ℓ (K z ), giving the first result. For the second, we replace each of the modules M by M [t ′ ]: since t and t ′ commute, if α ∈ M is annhilated by t ′ , so is t(α).
Proposition 2.8.
(1) There exists a bijection between elements L ∈ F ℓ (K) and equivalence classes of ℓ-tuples (a 0 , . . . , a ℓ−2 , u) modulo the equivalence relation
(a 0 , . . . , a ℓ−2 , u) ∼ (a −i , . . . , a ℓ−2−i , u g i )
for all i (with the indices of the ideals a considered modulo ℓ − 1), where the a i and u are as follows: (a) The a i are coprime integral squarefree ideals of K z such that if we set a = 0≤i≤ℓ−2 a g i i then the ideal class of a belongs to Cl(K z ) ℓ , and a ∈ (I(K z )/I(K z ) ℓ )[T ], where as usual I(K z ) denotes the group of (nonzero) fractional ideals of K z . (b) u ∈ S ℓ (K z ) [T ], and in addition u = 1 when a i = Z Kz for all i.
(2) Given (a 0 , . . . , a ℓ−2 ), a, and u as in (a), the field L ∈ F ℓ (K) is determined as follows: There exist an ideal q 0 and an element α 0 ∈ K z such that aq ℓ
0 = α 0 Z Kz with α 0 ∈ (K * z /K * z ℓ )[T ].
Then L is any of the ℓ conjugate degree ℓ subextensions of N z = K z ( ℓ √ α 0 u), where u is an arbitrary lift of u.
Proof. Given L, associate N z = K z ( ℓ √ α) as in Proposition 2.4. We may write uniquely αZ Kz = 0≤i≤ℓ−2 a g i i q ℓ , where the a i are coprime integral squarefree ideals of K z , and they must satisfy the conditions of (a).
Each a which thus occurs satisfies aq ℓ = α 0 Z Kz for some α 0 with α 0 ∈ (K * z /K * z ℓ )[T ], and for each a we arbitrarily associate such an α 0 . Given aq ℓ = αZ Kz , u := α/α 0 is a virtual unit; writing u for its class in S ℓ (K z ), u is annhiliated by T because both α and α 0 are. This establishes the bijection, and we conclude by observing the following:
• The elements α and β give equivalent extensions if and only if β = α g i γ ℓ for some element γ and some i modulo ℓ − 1, and then if α 0 Z Kz = j a g j j q ℓ and α = α 0 u, we have on the one hand βZ Kz = j a g j j−i q ℓ 1 for some ideal q 1 , so the ideals a j are permuted cyclically, and on the other hand β = (α 0 u) g i γ ℓ = α g i 0 u g i γ ℓ , so u is changed into u g i , giving the equivalence described in (1). • The only fixed point of the transformation (a 0 , . . . , a ℓ−2 , u) → (a ℓ−2 , a 0 , . . . , a ℓ−3 , u g ) is obtained with all the a i equal and u = u g , but since the a i are pairwise coprime this means that they are all equal to Z Kz , and u = u g i for all i, and so u = 1.
Remark 2.9. Note that condition (a) implies that a ∈ (Cl(K z )/Cl(K z ) ℓ )[T ], and for any modulus m coprime to a also that a ∈ (Cl
m (K z )/Cl m (K z ) ℓ )[T ].
Lemma 2.10. Keep the above notation, and in particular recall that a = 0≤i≤ℓ−2 a g i i . The condition a ∈ (I(K z )/I(K z ) ℓ )[T ] is equivalent to the following:
(1) In the general case τ (a i ) = a i−1 (equivalently, a i = τ −i (a 0 )), and τ (ℓ−1)/2 (a 0 ) = τ 2 (a 0 ).
(2) In the special case τ (a i ) = a i+(ℓ−3)/2 , so that a 2i = τ −2i (a 0 ) and a 2i+1 = τ −2i (a 1 ), with the following conditions on (a 0 , a 1 ):
• If ℓ ≡ 1 (mod 4) then a 1 = τ (ℓ−3)/2 (a 0 ), or equivalently a 0 = τ (ℓ+1)/2 (a 1 ). • If ℓ ≡ 3 (mod 4) then τ (ℓ−1)/2 (a 0 ) = a 0 and τ (ℓ−1)/2 (a 1 ) = a 1 .
Proof. Since τ (a) = i τ (a i ) g i and the τ (a i ) are integral, squarefree and coprime ideals, this is the canonical decomposition of τ (a) (up to ℓth powers). On the other hand a g = i a g i i−1 . Assume first that we are in the general case. Since τ (a)/a g is an ℓth power, by uniqueness of the decomposition we deduce that τ (a i ) = a i−1 . A similar proof using that g (ℓ−1)/2 ≡ −1 (mod ℓ) shows that τ 2 (a i ) = a i+(ℓ−1)/2 , and putting everything together proves (1). Assume now that we are in the special case, so that τ (a)/a −g is an ℓth
power. Since −g ≡ g (ℓ+1)/2 (mod ℓ), the same reasoning shows that τ (a i ) = a i−(ℓ+1)/2 = a i+(ℓ−3)/2 , so in particular τ 2 (a i ) = a i−(ℓ+1) = a i−2 ,
and the other formulas follow immediately.
Corollary 2.11. Let p z be a prime ideal of K z dividing some a i , denote by p the ideal of K below p z , and in the general case denote by p k the ideal of k z below p z .
(1) In all cases p is totally split in the extension K z /K. In addition:
(2) In the general case p k is split in the quadratic extension K z /k z .
(3) In the special case with ℓ ≡ 1 (mod 4), if we denote by p the ideal of k below p, then p is totally split in the extension K z /k (equivalently p is split in the quadratic extension K/k).
Proof. Assume first that we are in the general case. Then τ acts transitively on the a i , all of which are squarefree and coprime, and so any p dividing a i must have ℓ − 1 nontrivial conjugates (including p itself), establishing (1). Similarly, τ 2 (a i ) = a i+(ℓ−1)/2 , and for the same reason the prime ideals of K z dividing the a i come from prime ideals p k of k z which split in K z /k z .
In the special case, if p splits as a product of h conjugate ideals in K z , the decomposition group D(p z /p) has cardinality ef = (ℓ − 1)/h hence is the subgroup of Gal
(K z /k) generated by τ h since [K z : k] = ℓ − 1. Since τ h (a i ) = a (ℓ−3)h/2+i and τ h fixes p z , it follows as before that (ℓ − 1) | (ℓ − 3)h/2. Now evidently (ℓ − 1, (ℓ − 3)/2) is equal to 1 if ℓ ≡ 1 (mod 4) and to 2 if ℓ ≡ 3 (mod 4). Thus when ℓ ≡ 1 (mod 4) we deduce as above that (ℓ − 1) | h hence that e = f = 1, so that p is totally split in K z /k. On the other hand if ℓ ≡ 3 (mod 4) we only have (ℓ − 1)/2 | h. If h = ℓ − 1 then p is again totally split.
On the other hand, if h = (ℓ − 1)/2 then ef = 2, so p is either inert or ramified in the quadratic extension K/k, so p is totally split in K z /K. This leads to the following definition: Definition 2.12. We define D (resp., D ℓ ) to be the set of all prime ideals p of k with p ∤ ℓ (resp., with p | ℓ) such that the prime ideals p z of K z above p satisfy the above conditions (in other words p totally split in K z /K, and in addition in the general case p k split in K z /k z , and in the special case with ℓ ≡ 1 (mod 4), p split in K/k).
Thus the above corollary says that the prime ideals p of k below prime ideals of K z dividing one of the a i belong to D ∪ D ℓ .
2.3. The Mirror Field. We now introduce the mirror field of K. When ℓ = 3 this notion is classical and well-known; the mirror field of Q( √ D) is Q( √ −3D) and the Scholz reflection principle establishes that the 3-ranks of their class groups differ by at most 1.
In the case ℓ > 3 this notion is less well known but does appear in the literature (see for instance the works of G. Gras [20] and [21]), and in particular Scholz's theorem can be generalized to this context, see for instance [25] for the case ℓ = 5.
Definition 2.13. In the general case, we define the mirror field K ′ of K (implicitly, with respect to the prime ℓ) to be the degree ℓ − 1 subextension of K z /k fixed by τ (ℓ−1)/2 τ 2 .
We do not define the mirror field for the special case, although we could say that it is k z = K z , so in this subsection we assume that we are in the general case.
Lemma 2.14. Write K = k( √ D) for some D ∈ k * \ k * 2 . (1) The extension K ′ /k is cyclic of degree ℓ − 1, and K ′ = k √ D(ζ ℓ − ζ −1 ℓ ) . (2) The field K ′ is a quadratic extension of k(ζ ℓ + ζ −1 ℓ ), more precisely K ′ = k(ζ ℓ + ζ −1 ℓ ) −D(4 − α 2 ) , where α = ζ ℓ + ζ −1 ℓ . Proof. Straightforward; for (2), note that −D(4 − α 2 ) = D(ζ ℓ − ζ −1 ℓ ) 2 .
The point of introducing the mirror field is the following result: Proposition 2.15. Assume that we are in the general case. As before, let p be a prime ideal of k, p z an ideal of K z above p, and p k and p the prime ideals below p z in k z and K respectively. The following are equivalent:
(1) The ideals p k and p are both totally split in K z /k z and K z /K respectively (in other words p ∈ D∪D ℓ ).
(2) The ideal p is totally split in K ′ /k. In particular (by Corollary 2.11), (1)-(2) are true if p z divides some a i . Moreover, these conditions imply that exactly one of the following is true:
(a) p is split in K/k and totally split in k z /k. (b) p is inert in K/k and split in k z /k as a product of (ℓ − 1)/2 prime ideals of degree 2.
(c) p is above ℓ, is ramified in K/k, and its absolute ramification index e(p/ℓ) is an odd multiple of (ℓ − 1)/2 (equivalently e(p/ℓ) is an odd multiple of ℓ − 1). (2): We see that any nontrivial elements of D(p z /p) must be of the form τ i τ 2 with i ≡ 0 (mod ℓ − 1), and squaring we have
Proof. (1) if and only if
τ 2i ∈ D(p z /p), so 2i ≡ 0 (mod ℓ − 1), so D(p z /p) ⊂ {1, τ (ℓ−1)/2 τ 2 }, yielding (2)
. The converse is proved similarly.
To prove the last statement, first recall from [10] the following result:
Lemma 2.16. Let K be any number field and K z = K(ζ ℓ ). The conductor of the extension K z /K is given by the formula
f(K z /K) = p|ℓ (ℓ−1)∤e(p/ℓ) p .
It follows in particular that if p ∤ ℓ, or if p | ℓ and (ℓ − 1) | e(p/ℓ) then p is unramified in k z /k, and therefore also (arguing via inertia groups) in K/k, since otherwise the ideal p k would be ramified in K z /k z . Thus, assuming (2), the only prime ideals p which can be ramified in K/k are with p | ℓ and (ℓ − 1) ∤ e(p/ℓ).
If p is split or inert in K/k, we check that f (p z |p) equals 1 or 2 respectively, showing (a) and (b). If p is ramified, then (3.1) implies that (ℓ − 1) | e(p/ℓ) = e(p/p)e(p/ℓ). Since (ℓ − 1) ∤ e(p/ℓ) we conclude that e(p/ℓ) = n(ℓ − 1)/2 with n odd.
The following corollaries are immediate: Corollary 2.17. Let p be a prime ideal of k below a prime ideal p z of K z dividing some a i . If p is ramified in the quadratic extension K/k then p is above ℓ.
Corollary 2.18. In both the general and special cases, assume that for any prime ideal p of k above ℓ the absolute ramification index e(p/ℓ) is not divisible by (ℓ − 1)/2. Then all the ideals a i defined above are coprime to ℓ.
Note that for ℓ = 3 this corollary is empty, but the conclusion of the corollary always holds when ℓ > 2[k : Q] + 1, and in particular when k = Q and ℓ ≥ 5.
Proposition 2.19. There exists an ideal a α of K such that 0≤i≤ℓ−2 a i = a α Z Kz . In addition:
(1) In the general (resp., special) case, a α is stable by τ and τ 2 (resp., by τ ).
(2) If either the assumption of Corollary 2.18 is satisfied (for instance when ℓ > 2[k : Q] + 1), or we are in the special case with ℓ ≡ 1 (mod 4),
then a α = a ′ α Z K for some ideal a ′ α of k. Proof. (1). In the general case, since τ (a i ) = a i−1 we have 0≤i≤ℓ−2 a i = a α Z Kz with a α = N Kz/K (a 0 ), and since τ 2 (a i ) = a i+(ℓ−1)/2 , a α is stable by τ 2 . In the special case, since τ 2 (a i ) = a i−2 we have 0≤i<(ℓ−1)/2 a 2i = N Kz/K (a 0 )Z Kz and 0≤i<(ℓ−1)/2 a 2i+1 = N Kz/K (a 1 )Z Kz , so that 0≤i<ℓ−1 a i = a α Z Kz with a α = N Kz/K (a 0 a 1 )
an ideal of K, and since τ (a i ) = a i+(ℓ−3)/2 , a α is stable by τ .
(2). In the special case with ℓ ≡ 1 (mod 4) then (ℓ − 3)/2 is odd, so since a 1 = τ (ℓ−3)/2 (a 0 ) it follows that τ (N Kz/K (a 0 )) = N Kz/K (a 1 ), so that 0≤i≤ℓ−2 a i = N Kz/k (a 0 )Z Kz = a ′ α Z Kz with a ′ α an ideal of the base field k. On the other hand, if the assumption of Corollary 2.18 is satisfied then a α is coprime to ℓ, hence by Corollary 2.17 it is not divisible by any prime ramified in K/k, and since it is stable by Gal(K/k) it comes from an ideal a ′ α of k.
Hecke Theory: Conductors
Our goal (see Theorem 3.8) is to give a usable expression for the "conductor" f (L) in terms of the fundamental quantities (a 0 , · · · , a ℓ−2 , u) given by Proposition 2.8, where we recall that the conductor of the
C ℓ -extension N/K is equal to f (N/K) = f (L)Z K and that d(L/k) = d(K/k) (ℓ−1)/2 f (L) ℓ−1 .
In this section we will denote by p a prime ideal of k over ℓ, by p a prime ideal of K above p, by p z a prime ideal of K z above p, and in the general case, by p k a prime ideal of k z below p z .
We first recall from [10] and [11] some results concerning the cyclotomic extensions k z /k and K z /K. Remark 3.1. By and large we stick to the notation of [11] except that the notation m(p) of [11] is the same as M (p) here, which corresponds to numbers A α , while our m(p) corresponds to numbers a α . Proposition 3.2. As above, let p be a prime of k over ℓ, and let e(p) and e(p) be the respective absolute ramification indices over ℓ. Then we have
(3.1) e(p z /p) = ℓ − 1 (ℓ − 1, e(p))
and
e(p z /ℓ) ℓ − 1 = e(p) (ℓ − 1, e(p))
.
Proof. Immediate from Theorem 2.1 of [11]. (1) If pZ K = p 2 in K/k we set p 1/2 = p, and if pZ Kz = p e(pz /p) z in K z /K, we set p z = p 1/e(pz /p) . (2) We say that an ideal p of k divides some Gal(K z /k)-invariant ideal b of K (resp., of K z ) when (pZ K ) 1/e(p/p) (resp., (pZ K ) 1/e(pz /p) ) does, or equivalently when p (resp., p 1/e(pz /p) ) does, where this last condition is independent of the choice of ideal p of K above p. (3) If e is an integer, write r(e) for the unique integer such that e ≡ r(e) (mod ℓ−1) and 1 ≤ r(e) ≤ ℓ−1.
(4) We write M (p) = ℓe(p z /ℓ) ℓ − 1 = ℓe(p) (ℓ − 1, e(p)) ∈ Z , m(p) = M (p) e(p z /p) = ℓe(p) ℓ − 1 . (5) Denote by D n the congruence x ℓ /α ≡ 1 (mod * p n z ) in K z . (6) Define quantities A α (p) and a α (p) as follows: • If D n is soluble for n = M (p), we set A α (p) = M (p) + 1 and a α (p) = m(p). • Otherwise, if n < M (p) is the largest exponent for which it is soluble, we set A α (p) = n and we define a α (p) = A α (p) − r(e(p))/(ℓ − 1, e(p)) e(p z /p) = A α (p) e(p z /p) − 1 ∈ Z . z n ), with y = (h(x)/γ) (−1) j g −i (resp., y = (h(x)/γ) (−1) i g −i )
, proving the lemma.
Proposition 3.6. (1) We have ℓ ∤ A α (p), and if A α (p) ≤ M (p) (equivalently, if A α (p) ≤ M (p) − 1) then A α (p) ≡ e(p) (ℓ − 1, e(p)) mod ℓ − 1 (ℓ − 1, e(p))
.
(2) We have a α (p) = m(p) if A α (p) = M (p) + 1, and otherwise 0 ≤ a α (p) ≤ ℓe(p) ℓ − 1 − ℓ − 1 + r(e(p)) ℓ − 1 < ℓe(p) ℓ − 1 − 1 = m(p) − 1 .
Proof.
(1) follows from Proposition 3.8 of [11], and (2) follows from the definitions and from (3.1).
Remark 3.7. As mentioned in [12], the congruence (1), or equivalently the integrality of a α (p) (when A α (p) < M (p)) comes from a subtle although very classical computation involving higher ramification groups; see Proposition 3.6 of [11] along with Chapter 4 of [36].
We can now quote the crucial result from [11] which gives the conductor of the extension N/K: Theorem 3.15] Assume that (a 0 , . . . , a ℓ−2 ) are as in Proposition 2.8, so that 0≤i≤ℓ−2 a i = a α Z Kz with a α an ideal of K stable by τ 2 (resp., by τ in the special case), and sometimes coming from k (see Proposition 2.19). Then the conductor of the associated field extension N/K is given as follows:
Theorem 3.8. [11,f (N/K) = ℓa α p|ℓ p ⌈e(p)/(ℓ−1)⌉ p|ℓ , p∤aα p ⌈aα(p)⌉ . Remark 3.9.
One can now draw additional conclusions about the a α (p). For example, suppose that p is a prime ideal k above ℓ with pZ K = p 2 , p ∤ a α and a α (p) < m(p). Then v p (f (N/K)/ℓ) ≡ 0 (mod 2), as f (N/K) = f (L)Z K for an ideal f (L) of k, and it follows from the theorem and Proposition 3.6 that
(3.2) a α (p) ≡ ⌈e(p)/(ℓ − 1)⌉ (mod 2).
Definition 3.10. Let a equal either m(p), or an integer with 0 ≤ a < m(p) − 1, and define
h(0, a, p) = 0 if (ℓ − 1) ∤ e(p) or a = m(p) , 1 if (ℓ − 1) | e(p) and a < m(p) ; h(1, a, p) = 1 if (ℓ − 1) ∤ e(p) , 2 if (ℓ − 1) | e(p) . Remark 3.11. Note that if ℓ > 2[k : Q] + 1, for instance when k = Q and ℓ ≥ 5, we have e(p) < ℓ − 1 so (ℓ − 1) ∤ e(p)
. Thus in this case we simply have h(ε, a, p) = ε, independently of a and p. We will also see in Remark 4.7 that a number of other formulas simplify.
Lemma 3.12. Let p be a prime ideal of K above ℓ and denote by C n the congruence x ℓ /α ≡ 1 (mod * p n ) in K z . Then a α (p) is equal to the unique value of a as in the previous definition such that C n is soluble for n = a + h(0, a, p) and not soluble for n = a + h(1, a, p), where this last condition is ignored if a + h(1, a, p) > m(p).
Proof. By Lemma 3.5 the solubility of D n is equivalent to that of C n/e(pz/p) . If a = a α (p) = m(p), then D n is soluble for n = ℓe(p z /ℓ)/(ℓ − 1), which is equivalent to C m(p) = C a as desired.
If a = a α (p) < m(p), we have A α (p) = ae(p z /p) + r(e(p))/(ℓ − 1, e(p)), and Proposition 3.6 (1) implies that the solubility of D n for n = A α (p) is equivalent to that of D n ′ when A α (p) − (ℓ − 1)/(ℓ − 1, e(p)) < n ′ ≤ A α (p). If (ℓ − 1) ∤ e(p) we have r(e(p)) < ℓ − 1 and choose n ′ = ae(p z /p), while if (ℓ − 1) | e(p) we choose n ′ = n = ae(p z /p) + 1. Thus the solubility of D Aα(p) and D n ′ is equivalent to that of C n ′′ , where
n ′′ = n ′ /e(p z /p) = a + h(0, a, p) by definition of h(0, a, p). (Recall that e(p z /p) = 1 when (ℓ − 1) | e(p).) Furthermore, since D n is not soluble for n = A α (p) + 1, we also have that D n ′ is not soluble, where n ′ = n if (ℓ − 1) | e(p) and n ′ = a(p z /p) + (ℓ − 1)/(ℓ − 1, e(p)) ≥ n ′ otherwise. The solubility of D n ′ is equivalent to that of C n ′′ where n ′′ = n ′ /e(p z /p) = a + h(1, a, p), as desired.
Finally, we conclude by checking that the conditions are mutually exclusive.
The Dirichlet Series
Since f (N/K) = f (L)Z K for some ideal f (L) of k, we have N K/Q (f (N/K)) = N k/Q (f (L)) 2 . To emphasize the fact that we are mainly interested in the norm from k/Q, we set the following definition (norms from extensions other than k/Q will always indicate the field extension explicitly): Recall that we set
Φ ℓ (K, s) = 1 ℓ − 1 + L∈F ℓ (K)
1 N (f (L)) s , and f (N/K) = f (L)Z K is given by Theorem 3.8. By Proposition 2.4, we have
(ℓ − 1)Φ ℓ (K, s) = α∈(K * z /K * z ℓ )[T ] 1 N (f (L)) s , where L = K z ( ℓ √ α) G (including α = 1 corresponding to L = K G z = k with f (L) = Z k
and N (f (L)) = 1), so by Proposition 2.8, we have
(ℓ − 1)Φ ℓ (K, s) = (a 0 ,...,a ℓ−2 )∈J u∈S ℓ (Kz)[T ] 1 N (f (L)) s ,
where J is the set of (ℓ − 1)uples of ideals satisfying condition (a) of Proposition 2.8, and f (L) is the conductor of the extension corresponding to (a 0 , . . . , a ℓ−2 , u). Thus, replacing f (L) by the formula given by Theorem 3.8, recalling that p|ℓ N (p) e(p) = ℓ [k:Q] , and writing e(p) = (⌈e(p)/(ℓ − 1)⌉ − 1)(ℓ − 1) + r(e(p)) , we obtain
(4.1) (ℓ − 1)Φ ℓ (K, s) = ℓ − ℓ ℓ−1 [k:Q]s p|ℓ N (p) − ℓ−1−r(e(p)) ℓ−1 s (a 0 ,...,a ℓ−2 )∈J S α (s) N (a α ) s , where S α (s) = u∈S ℓ (Kz)[T ] p|ℓ p∤aα N (p) ⌈aαu(p)⌉s ,
and where α is any element of K * z such that α ∈ (K * z /K * z ℓ )[T ] and q ℓ 0 0≤i≤ℓ−2 a g i i = αZ Kz for some ideal q 0 . Definition 4.2. For α ∈ K * z and an ideal b of K z , we introduce the function
f α (b) = |{u ∈ S ℓ (K z )[T ], x ℓ /(αu) ≡ 1 (mod * b) soluble in K z }| , with the convention that f α (b) = 0 if b ∤ (1 − ζ ℓ ) ℓ Z Kz .
Let p i for 1 ≤ i ≤ n = n(α) be the prime ideals of k above ℓ and not dividing a α , and for each i let a i be such that either
a i = m(p i ), or 0 ≤ a i ≤ m(p i ) − (ℓ−1)+r(e(p i )) ℓ−1 = ⌈m(p i )⌉ − 2 with a i ∈ Z,
where as usual p i is an ideal of K above p i , and let A be the set of such (a 1 , . . . , a n ). Noting that thanks to the convention of Definition 4.1 we have p i |p i N (p i ) = N (p i ) 1/e(p i /p i ) , we thus have
(4.3) S α (s) = (b 1 ,...,bn)∈B(α) f α 1≤i≤n p b i /e(p i /p i ) i 1≤i≤n N (p i ) ⌈b i ⌉s/e(p i /p i ) Q(p b i /e(p i /p i ) i , s) ,
where Q(p b/e(p/p) , s) is defined as follows. Let as usual p be an ideal of K above p and define q = N (p) 1/e(p/p) . (1) We let B be the set of formal products of the form
Then if b = m(p) or 0 ≤ b < m(p) with b ∈ Z: (1) If (ℓ − 1) ∤ e(p) we set Q(p b/e(p/p) , s) = 1 if b = 0 , 1 − 1/q s if 1 ≤ b ≤ ⌈m(p)⌉ − 2 , −1/q s if b = ⌈m(p)⌉ − 1 , 1 if b = m(p) . (2) If (ℓ − 1) | e(p) we set Q(p b/e(p/p) , s) = 0 if b = 0 , 1/q s if b = 1 , 1/q s − 1/q 2s if 2 ≤ b ≤ m(p) − 1 , 1 − 1/q 2s if b = m(p) .b = p i |ℓ p b i /e(p i /p i ) i , where the b i are such that 0 ≤ b i ≤ m(p i ) and b i ∈ Z ∪ {m(p i )}.
(2) We will consider any b ∈ B as an ideal of K, where by abuse of language we accept to have fractional powers of prime ideals of K, and we will set b z = bZ Kz , which is a true ideal of K z stable by τ , and also by τ 2 in the general case.
(3) If b ∈ B as above, we set ⌈N ⌉(b) = p i |b N (p i ) ⌈b i ⌉/e(p i /p i ) and P (b, s) = p i |b Q(p b i /e(p i /p i ) i , s) .
where Q(p b/e(p/p) , s) := Q(p b/e(p/p) , s) except in the case (ℓ − 1) | e(p) and b = 0, where we set Q(p b/e(p/p) , s) = 1.
We thus obtain
f α (b) N (a α ) s .
The case p ∤ b, (ℓ − 1) | e(p), and p ∤ a α is precisely that for which Q(p b/e(p/p) , s) = 0 and Q(p b/e(p/p) , s) = 1. By excluding this case we may substitute Q for Q with Q(p 0 , s) = 1. (1) For b as above we define
r e (b) = p|ℓZ K , p∤b (ℓ−1)|e(p)
p .
(2) We set d ℓ = p∈D ℓ p (see Definition 2.12).
Remark 4.7. Since e(p) = e(p/p)e(p) ≤ 2[k : Q], we note that if ℓ > 2[k : Q] + 1 then r e (b) is always trivial. This will in particular be the case for k = Q and ℓ ≥ 5, which we will study later, and if we specialized to this case now we would avoid some of the subsequent complications. In particular, in view of the next lemma, when r e (b) is trivial all the ideals a i and a α are coprime to ℓ.
Lemma 4.8. For each a α appearing in the inner sum of (4.4) we have
(4.5) (a α , ℓZ K ) = r e (b) = p∈D ℓ (p,b)=1 p|p p , so that r e (b) | d ℓ . Additionally, in the special case with ℓ ≡ 1 (mod 4) we have r e (b) = p∈D ℓ (p,b)=1 p.
Proof. If p ∤ b and (ℓ − 1) | e(p) then clearly p | a α . Conversely, let p | a α be above ℓ. Since (a α , b) = 1 we know that p ∤ b. If we had (ℓ − 1) ∤ e(p), Proposition 3.2 would imply that e(p z /p) > 1, contradicting Corollary 2.11. This proves the first equality of (4.5), and the second equality and formula in the special case follow similarly.
Thus we obtain (4.6)
(a 0 ,...,a ℓ−2 )∈J S α (s) N (a α ) s = b∈B r e (b)|d ℓ ⌈N ⌉(b) s P (b, s) (a 0 ,...,a ℓ−2 )∈J (aα,ℓZ K )=r e (b) f α (b) N (a α ) s .
To compute f α (b) we set the following definition: Definition 4.9. For any ideal b ∈ B, and for any subset T of F ℓ [G], we set
S bz (K z )[T ] = {u ∈ S ℓ (K z )[T ], x ℓ /u ≡ 1 (mod * b z ) soluble} ,
where u is any lift of u coprime to b z , and the congruence is in K z . Lemma 4.10. Let (a 0 , . . . , a ℓ−2 ) satisfy condition (a) of Proposition 2.8, suppose that α satisfies the condition described before Definition 4.2, and recall that we set a = i a g i i . We have
f α (b) = |S bz (K z )[T ]| if a ∈ Cl bz (K z ) ℓ , 0 otherwise.
Proof. The lemma and its proof are a direct generalization of Lemma 5.3 of [12], and we omit the details.
Computation of |S bz (K z )[T ]|
In this section we compute the size of the group S bz (K z )[T ] appearing in Lemma 4.10, as well as several related quantities.
Lemma 5.1. Set Z bz = (Z Kz /b z ) * . Then |S bz (K z )[T ]| = |(U (K z )/U (K z ) ℓ )[T ]||(Cl bz (K z )/Cl bz (K z ) ℓ )[T ]| |(Z bz /Z ℓ bz )[T ]| , and in particular |S ℓ (K z )[T ]| = |(U (K z )/U (K z ) ℓ )[T ]||(Cl(K z )/Cl(K z ) ℓ )[T ]| .
Proof. This is a minor variant of Corollary 2.13 of [11], proved in the same way.
The quantity |(U (K z )/U (K z ) ℓ )[T ]| is given by the following lemma. (1) For any number field M we have
rk ℓ (U (M )) = r 1 (M ) + r 2 (M ) − 1 if ζ ℓ / ∈ M , r 1 (M ) + r 2 (M ) if ζ ℓ ∈ M . (2) We have |(U (K z )/U (K z ) ℓ )[T ]| = ℓ RU (K) , where RU (K) := r 2 (K) − r 2 (k) in the general case, r 1 (k) + r 2 (k)
in the special case with ℓ ≡ 3 (mod 4) , r 2 (k)
in the special case with ℓ ≡ 1 (mod 4) .
(3) In particular, if k = Q we have RU (K) = r 2 (K) in all cases.
Proof.
(1) is Dirichlet's theorem, and (3) is a consequence of (2). To prove (2) in the general case, where T = {τ 2 + 1, τ − g}, we apply the exact sequence
(5.1) 1 −→ U (k z ) U (k z ) ℓ [τ − g] −→ U (K z ) U (K z ) ℓ [τ − g] −→ U (K z ) U (K z ) ℓ [τ 2 + 1, τ − g] −→ 1 ,
where the last nontrivial map sends ε to τ 2 (ε)/ε. Surjectivity follows from Lemma 2.3, and (τ 2 +1)(τ 2 −1) = 0 implies that the two nontrivial maps compose to zero. Finally, suppose ε ∈ U (K z ) satisfies τ 2 (ε) = εη ℓ for some η ∈ K z . Applying τ 2 to both sides we see that ητ 2 (η) = ζ a ℓ for some a, and replacing η with η 1 = ηζ b ℓ with a + 2b ≡ 0 (mod ℓ), we obtain η 1 τ 2 (η 1 ) = 1 and τ 2 (ε) = εη ℓ 1 . By Hilbert 90 there exists η 2 with η 1 = η 2 /τ 2 (η 2 ), so that ε 1 = εη ℓ 2 satisfies τ 2 (ε 1 ) = ε 1 , in other words ε 1 ∈ k z , proving exactness of (5.1). By a nontrivial theorem of Herbrand (see Theorem 2.3 of [11] (2) in the general case.
), we have |(U (K z )/U (K z ) ℓ )[τ −g]| = ℓ r 2 (K)+1 and |(U (k z )/U (k z ) ℓ )[τ − g]| = ℓ r 2 (k)+1 , establishing
In the special case, with T = {τ + g} = {τ − g (ℓ+1)/2 }, (2) follows directly from Herbrand's theorem applied to the extension k z /k = K z /k, for which τ generates the Galois group.
Note that for ℓ = 3 the same is true except that in the special case we have RU (K) = r 1 (k) + r 2 (k) − 1. This follows from the shape of [11, Theorem 2.3], or may be easily verified from [12,Lemma 5.4].
We now need to compute |(Z bz /Z ℓ bz )[T ]|. Proof. See Proposition 2.6 and Theorem 2.7 of [11], or Lemma 1.5.6 of [29].
Theorem 5.4. We have in the general case
(5.2) |(c z /b z )[τ − g j ]| = p|bz N K/Q (p) x j (p) , where (5.3) x j (p) = v p (b) − je(p) ℓ − 1 − ⌈e(p z /p)v p (b)/ℓ⌉(ℓ − 1, e(p)) − je(p) ℓ − 1 .
In the special case, (5.3) holds with p and K replaced throughout by p and k respectively. Finally, in the general case, then (5.3) is also true with respect to k z /k. In this case one must replace p, K, b z , and c z respectively by p, k, b k := c z ∩ k z , and
(5.4) c k := c z ∩ Z kz = p k ⊂kz p k |b k p ⌈vp k (b k )/ℓ⌉ k .
Proof. This is the result at the bottom of [11, p. 177], applied to K z /K, K z /k, and k z /k respectively. As in [11, Theorem 2.7] the result may be simplified if v p (b) is either an integer or equal to ℓe(p) ℓ−1 , and in particular always in the general case with respect to K z /K, but in other cases v p may be a half integer.
Finally, the equality in (5.4) is readily verified.
Recall from [11, Theorem 2.1] and (3.1) that e(p z /p) = ℓ−1 (ℓ−1,e(p)) and e(p k /p) = ℓ−1 (ℓ−1,e(p)) . In the special case, this theorem together with Lemma 5.3 gives the cardinality of (Z bz /Z ℓ bz )[T ] by choosing j = (ℓ + 1)/2. In the general case we require the following additional lemma: Lemma 5.5. Assume that we are in the general case and set c k = c z ∩ k z and b k = b z ∩ k z . We have
|(Z bz /Z ℓ bz )[T ]| = |(c z /b z )[τ − g]|/|(c k /b k )[τ − g]| ,
where the two terms on the right-hand side are given by Theorem 5.4.
Proof. We have an exact sequence of
F ℓ [G]-modules 1 −→ c z b z [τ 2 − 1][τ − g] −→ c z b z [τ − g] −→ c z b z [T ] −→ 1 ,
the last map sending x to x − τ 2 (x). It therefore suffices to argue that (
c z /b z )[τ 2 − 1] = (c z ∩ k z )/(b z ∩ k z ): if
x ∈ c z satisfies τ 2 (x) = x + y for some y ∈ b z , then applying τ 2 we see that τ 2 (y) = −y, hence τ 2 (x + y/2) = x + y/2. Moreover x + y/2 ≡ x (mod b z ), because 2 is invertible modulo ℓ hence modulo b.
Definition 5.6. We set G b = (Cl bz (K z )/Cl bz (K z ) ℓ )[T ].
We conclude with one additional lemma which will be needed in the next section.
Lemma 5.7. In the general case set u = ι(τ 2 + 1)ι(τ − g) and in the special case set u = ι(τ + g).
(1) The map I → u(I) induces a surjective map from Cl bz (K z )/Cl bz (K z ) ℓ to G b , of which a section is the natural inclusion from G b to Cl bz (K z )/Cl bz (K z ) ℓ . (2) Any character χ ∈ G b can be naturally extended to a character of Cl bz (K z )/Cl bz (K z ) ℓ by setting χ(I) = χ(u(I)), which we again denote by χ by abuse of notation. (3) Let as usual a = 0≤i≤ℓ−2 a g i i with the a i satifying condition (a) of Proposition 2.8. • In the general case and in the special case when ℓ ≡ 1 (mod 4), we have χ(a) = χ(a 0 ) −1 ; • In the special case when ℓ ≡ 3 (mod 4), we have χ(a) = χ(a 0 a g 1 ) (ℓ−1)/2 , where χ on the right-hand side is defined in (2).
Proof.
(1) and (2) are immediate from Lemma 2.3. For (3), assume that we are in the special case. Using Lemma 2.10 we have a 2i = τ −2i (a 0 ), a 2i+1 = τ −2i (a 1 ), and χ(τ 2 (I)) = χ(I) g 2 , so that
χ(a) = 0≤i<(ℓ−1)/2 χ(τ −2i (a 0 a g 1 )) g 2i = 0≤i<(ℓ−1)/2 χ(a 0 a g 1 ) = χ(a 0 a g 1 ) (ℓ−1)/2 .
If in addition ℓ ≡ 1 (mod 4) we have a 1 = τ (ℓ−3)/2 (a 0 ) and χ(τ (I)) = χ(I −g ), giving χ(a 1 ) = χ(a 0 ) (−g) (ℓ−3)/2 = χ(a 0 ) −g (ℓ−3)/2 and χ(a g 1 ) = χ(a 0 ), so χ(a 0 a g 1 ) (ℓ−1)/2 = χ(a 0 ) ℓ−1 = χ(a 0 ) −1 . The general case of (3) is proved similarly, with a i = τ −i (a 0 ).
Semi-Final Form of the Dirichlet Series
We can now put everything together, and obtain a complete analogue of the main theorem of [12]: Theorem 6.1. Recall that for any (true or formal) ideal b of K as above we set
G b = (Cl bz (K z )/Cl bz (K z ) ℓ )[T ]. We have Φ ℓ (K, s) = ℓ RU (K) (ℓ − 1)ℓ ℓ ℓ−1 [k:Q]s p|ℓ N (p) − (ℓ−1−r(e(p)) ℓ−1 s · · b∈B r e (b)|d ℓ ⌈N ⌉(b) N (r e (b)) s P (b, s) |(Z bz /Z ℓ bz )[T ]| χ∈ G b F (b, χ, s) , where F (b, χ, s) = p|r e (b) p∈D ′ ℓ (χ) (ℓ − 1) p|r e (b) p∈D ℓ \D ℓ ′ (χ) (−1) p∈D ′ (χ) 1 + ℓ − 1 N (p) s p∈D\D ′ (χ) 1 − 1 N (p) s ,
and D ′ (χ) (resp. D ′ ℓ (χ)) is the set of p ∈ D (resp. D ℓ ) such that χ(p z ) = 1, where p z is any prime ideal of K z above p.
Proof. We begin with the formula for Φ ℓ (K, s) given by (4.1) and (4.6). By Remark 2.9 we have a ∈ (Cl bz (K z )/Cl bz (K z ) ℓ )[T ] with a = 0≤i≤ℓ−2 a g i i . Thus a ∈ Cl bz (K z ) ℓ if and only if χ(a) = 1 for all characters χ ∈ G b . The number of such characters being equal to |G b |, by orthogonality of characters and Lemmas 4.10, 5.1, and 5.2 we obtain
Φ ℓ (K, s) = ℓ RU (K) (ℓ − 1)ℓ ℓ ℓ−1 [k:Q]s p|ℓ N (p) − ℓ−1−r(e(p)) ℓ−1 s b∈B r e (b)|d ℓ ⌈N ⌉(b) s P (b, s) |(Z bz /Z ℓ bz )[T ]| χ∈ G b H(b, χ, s) , with H(b, χ, s) = (a 0 ,··· ,a ℓ−2 )∈J ′ (aα,ℓZ K )=r e (b) χ(a) N (a α ) s ,
where a α was defined in Proposition 2.19, and J ′ is the set of (ℓ − 1)uples of coprime squarefree ideals of K z , satisfying condition (a) of Proposition 2.8, but now without the condition that the ideal class of a belongs to Cl(K z ) ℓ , so satisfying the condition of Lemma 2. 10.
Assume first that we are in the general case. By Lemma 2.10 we can replace the sum over J ′ by a sum over ideals a 0 of K z . The conditions and quantities linked to a 0 are then as follows:
(a) The ideal a 0 is a squarefree ideal of K z such that τ (ℓ−1)/2 (a 0 ) = τ 2 (a 0 ). (b) The ideals a 0 and τ i (a 0 ) are coprime for (ℓ − 1) ∤ i. (c) If p z is a prime ideal of K z dividing a 0 , p the prime ideal of K below p z , and p the prime ideal of k below p z then by Corollary 2.11 we have p ∈ D ∪ D ℓ . Conversely, if this is satisfied then the ideals a i = τ −i (a 0 ) must be pairwise coprime since otherwise a α would be divisible by some p 2 z which is impossible since p is unramified in K z /K. Thus if we denote temporarily by J ′′ the set of ideals a 0 of K z satisfying the first three conditions above, we have
H(b, χ, s) = a 0 ∈J ′′ (N Kz /K (a 0 ),ℓZ K )=r e (b) χ −1 (a 0 ) N (N Kz/K (a 0 )) s .
So that we can use multiplicativity, write a 0 = cd, where c is the ℓ-part of a 0 and d is the prime to ℓ part (recall that a 0 is squarefree). The condition (N Kz/K (a 0 ), ℓZ K ) = r e (b) is thus equivalent to N Kz/K (c) = r e (b).
Thus H(b, χ, s) = S c S d with S c = c∈J ′′ N Kz /K (c)=r e (b) χ −1 (c) N (N Kz/K (c)) s and S d = d∈J ′′ (N Kz /K (d),ℓZ K )=1 χ −1 (d) N (N Kz/K (d)) s .
Consider first the sum S d . By multiplicativity we have S d = p∈D S d,p with
S d,p = d|pZ Kz τ (ℓ−1)/2 (d)=τ 2 (d) χ −1 (d) N (N Kz/K (d)) s .
As p is not above ℓ, it is unramfied in K/k by Proposition 2.15 and we consider the remaining two cases:
(1) Assume that pZ K = p, i.e, that p is inert in K/k. Since p is totally split in K z /K we have pZ Kz = 0≤i≤ℓ−2 τ i (p z ) for some prime ideal p z of K z . Furthermore, since p z /p k (with our usual notation) is split we have τ 2 (p z ) = p z , and since p is stable by τ 2 , τ 2 (p z ) is again above p, so we deduce that τ 2 (p z ) = τ (ℓ−1)/2 (p z ). Since d is squarefree and coprime to its K z /K-conjugates, we see that d = Z Kz or d = τ i (p z ) for some i, with N (N Kz/K (d)) equal to 1 or N (p) respectively. In the latter case we have
(6.1) S d,p = 1 + 0≤i≤ℓ−2 χ(p z ) −g i N (p) s = 1 + 1≤j≤ℓ−1 χ(p z ) j N (p) s ,
so that S d,p = 1 + (ℓ − 1)/N (p) s if χ(p z ) = 1, and S d,p = 1 − 1/N (p) s otherwise. (2) If instead pZ K = pτ 2 (p) is split in K/k, then similarly either d = Z Kz or d = τ i (p z τ ℓ−1 2 τ 2 (p z )) for some i and p z . We have that χ(τ (ℓ−1)/2 (τ 2 (p z ))) = χ −1 (τ 2 (p z )) = χ(p z ), and hence obtain the same result as above. Consider now the sum S c . By multiplicativity, since b is stable by τ 2 , and applying Lemma 4.8 we have
S c = 1 N (r e (b)) s c∈J ′′ N Kz /K (c)=r e (b) χ −1 (c) = 1 N (r e (b)) s p∈D ℓ (p,b)=1 S c,p , with S c,p = c|pZ Kz τ (ℓ−1)/2 (c)=τ 2 (c) N Kz /K (c)= p|p p χ −1 (c) .
Our analysis is essentially the same as before, except p can now be ramified in K/k and the possibility c = Z Kz is now excluded. In all cases we obtain that S c,p = ℓ − 1 if χ(p z ) = 1 and −1 otherwise.
Putting everything together proves the theorem in the general case.
In the special case with ℓ ≡ 1 (mod 4) the proof is similar; condition (a) is absent and (d) becomes N Kz/k (a 0 ) = N (a α ). Imitating the inert case of the previous argument, we obtain the same results.
In the special case with ℓ ≡ 3 (mod 4), we replace the sum over J ′ by a sum over pairs (a 0 , a 1 ) of ideals of K z satisfying suitable conditions:
• In place of (a), a 0 and a 1 are fixed by τ (ℓ−1)/2 .
• In place of (b), the ideals a 0 , a 1 , τ 2i (a 0 ), and τ 2i (a 1 ) must all be coprime.
• In place of (d), we have N Kz/K (a 0 a 1 ) = N (a α ).
• In place of (e), we have χ(a) = χ(a 0 a g 1 ) (ℓ−1)/2 . We must again consider all splitting types in K/k, and the arguments are similar. If p is inert, we compute that
S d,p = 1 + 0≤i≤ ℓ−3 2 χ(p z ) g 2i ·(ℓ−1)/2 N (p) s + 0≤i≤ ℓ−3 2 χ(p z ) g 2i+1 ·(ℓ−1)/2 N (p) s ,
equal to the same expression as before. If p is split, recall that by Proposition 2.19 a α must be stable by τ ; the relevant computation is
χ(p z τ (ℓ−1)/2 (p z )) (ℓ−1)/2 = χ(p 1−g (ℓ−1)/2 z ) (ℓ−1)/2 = χ(p z ) −1 ,
and again we obtain the same results. For p ∈ D ℓ the argument is similar, once again considering all three cases and obtaining the same result.
As mentioned in Remarks 3.11, if ℓ > 2[k : Q] + 1, and in particular if k = Q and ℓ ≥ 5, we always have r e (b) = (1). The theorem simplifies and gives the following: Corollary 6.2. Keep the same notation, and assume that ℓ ≥ 2[k : Q] + 3, for instance that k = Q and ℓ ≥ 5. We have
Φ ℓ (K, s) = ℓ RU (K) (ℓ − 1)ℓ ℓ ℓ−1 [k:Q]s p|ℓ N (p) − ℓ−1−r(e(p)) ℓ−1 s b∈B ⌈N ⌉(b) s P (b, s) |(Z bz /Z ℓ bz )[T ]| χ∈ G b F (b, χ, s) , where F (b, χ, s) = p∈D ′ (χ) 1 + ℓ − 1 N (p) s p∈D\D ′ (χ) 1 − 1 N (p) s .
In the general case, we now prove that the group G b can be described in somewhat simpler terms, in terms of the mirror field K ′ of K. (See also Theorems 9.1 and Theorem 9.7 for a further characterization.) Proposition 6.3. There is a natural isomorphism
Cl b (K z ) Cl b (K z ) ℓ [T ] → Cl b ′ (K ′ ) Cl b ′ (K ′ ) ℓ [τ − g], where b ′ = b ∩ K ′ .
Moreover, using this isomorphism to regard a character χ of Cl b (Kz) Cl b (Kz) ℓ [T ] as a character χ of
Cl b ′ (K ′ ) Cl b ′ (K ′ ) ℓ [τ −g], the condition χ(p z ) = 1 defining D(χ) ∩ D ′ ℓ (χ) is equivalent to the condition χ(p K ′ ) = 1 for the unique prime p K ′ of K ′ below p z .
Proof. The first statement is also proved in [13,Proposition 3.6], so we will be brief. As τ (ℓ−1)/2 τ 2 acts trivially on G b , it can be checked that elements of G b can be represented by an ideal of the form aτ 2 τ (ℓ−1)/2 a, which is of the form a ′ Z Kz for some ideal a ′ of K ′ . We therefore obtain a well-defined injective map
Cl b (Kz) Cl b (Kz) ℓ [T ] → Cl b ′ (K ′ ) Cl b ′ (K ′ ) ℓ [τ − g]
, which may easily be shown to be surjective as well. The latter statement follows because the condition χ(p K ′ ) = 1 is equivalent to χ(p K ′ Z Kz ) = 1, which is easily seen to be equivalent to χ(p z ) = 1 for any splitting type of p z |p K ′ .
Specialization to k = Q
We now specialize all of the results of this paper to the case where the base field is k = Q, where we will obtain more explicit results. Henceforth we assume that K = Q( √ D) is a quadratic field with discriminant D.
By definition, B = {1, (ℓ), (ℓ) ℓ/(ℓ−1) } in the general case with ℓ ∤ D, and B = {1, (ℓ) 1/2 , (ℓ), (ℓ) ℓ/(ℓ−1) } in the special case or in the general case with ℓ | D. Equivalently we may write
(7.1) b z ∈ Z Kz , (1 − ζ ℓ ) (ℓ−1)/2 Z Kz , (1 − ζ ℓ ) ℓ−1 Z Kz = ℓZ Kz , (1 − ζ ℓ ) ℓ Z Kz ,
with the second entry removed in the former case. Throughout, we use the notation (−, −, −, −) to describe quantities depending on B, with asterisks denoting 'not applicable'.
Proposition 7.1. We have that |(Z bz /Z ℓ bz )[T ]| is equal to (1, * , ℓ, ℓ) or (1, 1, ℓ, ℓ) for ℓ ∤ D or ℓ|D respectively, unless ℓ = 3 in the special case, in which case |(Z bz /Z 3 bz )[T ]| = (1, 1, 1, 3). Proof. This follows from Theorem 5.4 and Lemma 5.5. In the general case we obtain |(Z bz /Z ℓ bz )[T ]| = (0, * , 2, 2) − (0, * , 1, 1) = (0, * , 1, 1) , |(Z bz /Z ℓ bz )[T ]| = (0, 1, 2, 2) − (0, 1, 1, 1) = (0, 0, 1, 1) , depending on whether ℓ ∤ D or ℓ|D respectively; in the special case we obtain
|(Z bz /Z ℓ bz )[T ]| = (0, 0, 1, 1) , |(Z bz /Z ℓ bz )
[T ]| = (0, 0, 0, 1) , depending on whether ℓ ≥ 5 or ℓ = 3 respectively.
Recall that the mirror field of K = Q( √ D) with respect to ℓ is the degree ℓ−1 field K ′ = Q( √ D(ζ ℓ −ζ −1 ℓ )). The following is immediate from the results of Section 2: Lemma 7.2. Let p be a prime different from ℓ.
• We have p ∈ D if and only if p ≡ D p (mod ℓ). • In the general case, this is equivalent to p splitting completely in K ′ /Q. • In the special case with ℓ ≡ 1 (mod 4), this is equivalent to p ≡ 1 (mod ℓ).
• In the special case with ℓ ≡ 3 (mod 4), this is equivalent to p ≡ ±1 (mod ℓ).
We come now to the analogue of Theorem 3.2 of [14]. The case ℓ = 3, which is slightly different, is treated in loc. cit.: Theorem 7.3. Assume that ℓ ≥ 5 and let K = Q( √ D). We have
Φ ℓ (K, s) = ℓ r 2 (D) ℓ − 1 b∈B A b (s) χ∈ G b F (b, χ, s) ,
where the A b (s) are given by the following table:
Condition on D A (1) (s) A ( √ (−1) (ℓ−1)/2 ℓ) (s) A (ℓ) (s) A (ℓ ℓ/(ℓ−1) ) (s) ℓ ∤ D ℓ −2s 0 −ℓ −2s−1 1/ℓ ℓ | D ℓ −3s/2 ℓ −s − ℓ −3s/2 −ℓ −s−1 1/ℓ F (b, χ, s) = p≡( D p ) (mod ℓ), p =ℓ 1 + ω χ (p) p s ,
where we set:
ω χ (p) = ℓ − 1 if χ(p z ) = 1 −1 if χ(p z ) = 1 ,
where as usual p z is any ideal of K z above p.
Proof. The computation is routine, given the following consequences of our previous results:
• We have k = Q so ℓ ℓ ℓ−1 [k:Q]s = ℓ ℓs/(ℓ−1) . • The factor p|ℓ N (p) ... is equal to ℓ −(ℓ−2)s/(ℓ−1)) if ℓ ∤ D and to ℓ −(ℓ−3)s/(2(ℓ−1)) if ℓ | D.L∈F ℓ (K) 1 f (L) s = φ D (s) + 1 (ℓ − 1)ℓ 1−r 2 (D) L ℓ (s) p≡( D p ) (mod ℓ), p =ℓ 1 + ℓ − 1 p s .
Proof. Same as in Proposition 7.5 of [12]: The main term is the contribution of the trivial characters, and φ D (s) is the contribution of the nontrivial characters: we first regard each χ ∈ G b as a character of
Cl b ′ (K ′ ) Cl b ′ (K ′ ) ℓ
by Proposition 6.3 and then by setting χ equal to 1 on the orthogonal complement of
Cl b ′ (K ′ ) Cl b ′ (K ′ ) ℓ [τ − g]
. By the previous lemma, the primes occurring in the product are precisely those for which p is totally split in K ′ . Therefore, for each set of nontrivial characters χ, χ 2 , . . . , χ ℓ−1 ∈ G b , the sum of products F (b, χ, s) may be written as g(s) + χ L(s, χ), where L(s, χ) is the (holomorphic) Hecke L-function associated to χ, and g(s) is a Dirichlet series supported on squarefull numbers, absolutely convergent and therefore holomorphic in ℜ(s) > 1/2. Therefore φ D (s) is holomorphic in ℜ(s) > 1/2 as well. We also note that the product of the main term may similarly be written as h(s) + L(s, ω 0 ), where ω 0 is the trivial Hecke character, and h(s) satisfies the same properties as g(s).
The ℓ = 3 case is slightly different due to the nontriviality of r e (b); see [12].
This brings us to our asymptotic formulas:
Corollary 7.5. Assume that ℓ ≥ 5 and denote by M ℓ (D; X) the number of L ∈ F ℓ (Q( √ D)) such that f (L) ≤ X. Set c 1 (ℓ) = 1/((ℓ − 1)ℓ 1−r 2 (D) ), c 2 (ℓ) = (ℓ 2 + ℓ − 1)/ℓ 2 when ℓ ∤ D or c 2 (ℓ) = 2 − 1/ℓ when ℓ | D.
(1) In the general case, or in the special case with ℓ ≡ 1 (mod 4), for any ε > 0 we have
M ℓ (D; X) = C ℓ (D)X + O D (X 1− 2 ℓ+3 +ε ) , with C ℓ (D) = c 1 (ℓ)c 2 (ℓ)Res s=1 p≡( D p ) (mod ℓ) 1 + ℓ − 1 p s ,
and in the special case the product is equivalently over p ≡ 1 (mod ℓ). (2) In the special case with ℓ ≡ 3 (mod 4), for any ε > 0 we have
M ℓ (D; X) = C ℓ (D)(X log(X) + C ′ ℓ (D)) + O D (X 1− 2 ℓ+3 +ε ) , with C ℓ (D) = c 1 (ℓ)c 2 (ℓ) lim s→1 + (s − 1) 2 p≡±1 (mod ℓ) 1 + ℓ − 1 p s ,
and C ′ ℓ (D) can also be given explicitly if desired. Proof. In the general case, using the same proof as in [12], we see that the result follows, with C ℓ (D) equal to the residue at s = 1 of Φ ℓ (K, s). Note that since we assume that ℓ ≥ 5, the condition e(p) = f (p) = 1 implies that p = ℓ, otherwise it must simply be added. Note the marked difference in the asymptotics when ℓ ≡ 1 (mod 4) and ℓ ≡ 3 (mod 4).
We briefly recall how to obtain the error term. By the proof of Corollary 7.4, it equals (up to an implied constant depending on D and ℓ) the error made in estimating partial sums of Hecke L-functions of degree ℓ − 1.
We do this in the standard way, subject to the limitation that we may not shift any contour to ℜ(s) ≤ 1/2. We have by Perron's formula, for each Hecke L-function ξ(s) = n a(n)n −s and any c > 1, and we shift the portion of the contour from c − iT to c + iT to ℜ(s) = σ for σ ∈ (1/2, 1) and T > 0 to be determined. By convexity we have |ξ(s)| ≪ T ℓ−1 2 (1−σ+ε) , and choosing c = 1 + ε, σ = 1/2 + ε our integral is ≪ T ℓ−1
2 ( 1 2 +2ε) X 1 2 +ε + X 1+2ε
T ; then choosing T = X 2 ℓ+3 we obtain an error term of X 1− 2 ℓ+3 +ε .
In a separate paper by the first author [9], one explains how to compute the constants C ℓ (D) to high accuracy (100 decimal digits, say) for reasonably small values of |D|. For example, we have C 3 (−3) = 0.0669077333013783712918416 · · · , C 3 (−4) = 0.1362190676241212841449867 · · · .
Writing N ± D ℓ (X) for the number of degree ℓ fields L with Galois group D ℓ , |Disc(L)| ≤ X, and whose quadratic resolvent is respectively real or imaginary, it is natural to ask whether we can obtain estimates for N ± D ℓ (X). A plausible guess is that for some C ℓ > 0 we have
(7.2) N − D ℓ (X) ∼ C ℓ X 2/(ℓ−1) and N + D ℓ (X) ∼ C ℓ ℓ X 2/(ℓ−1) .
By Davenport-Heilbronn this is known for ℓ = 3 with C 3 = 1/(4ζ (3)), but it is unclear how to recover this value of C 3 , even heuristically, from our work. As such we still seem to be far from a proof of (7.2).
Study of the Groups G b
In this section, where we continue to assume that k = Q and also assume that ℓ ≥ 5, we study the groups G b appearing in Theorem 7.3. In particular, each Euler product appearing in Theorem 7.3 corresponds to a character of G b , and so we want to study the size of this group.
We are indebted to Hendrik Lenstra for help in this section. This was not done in [12], but much of this was done in our paper [13] with Rubinstein-Salzedo on the Ohno-Nakagawa relation. Accordingly we give only a brief account of those results which are proved there.
We recall a few of the important notations used previously:
• K z is an abelian extension of Q containing the ℓth roots of unity, with G = Gal(K z /Q) = τ, τ 2 or τ in the general and special cases respectively. • As in Proposition 2.4, N z = K z ( ℓ √ α) is a cyclic extension, for which we wrote αZ Kz = q ℓ 0≤i≤ℓ−2 a g i i and (in Proposition 2.19) 0≤i≤ℓ−2 a i = a α Z Kz for an ideal a α of K. • We recall the possibilities for b (equivalently, b z ) from (7.1), and we continue to use the notation (−, −, −, −) for quantities depending on b. For any b as in (7.1) we define b * := (1 − ζ ℓ ) ℓ /b z . Proposition 8.1. With the notation above, we have f(N z /K z ) | b z if and only if α ∈ S b * (K z ).
Proof. This is very classical, and essentially due to Kummer and Hecke: for instance, by Theorem 3.7 of [11] we have
f(N z /K z ) = (1 − ζ ℓ ) ℓ a α / pz|ℓ, pz∤aα p Aα(pz )−1 z .
Thus, since a α is coprime to the product then f(N z /K z ) | (1 − ζ ℓ ) ℓ if and only if a α = Z K , i.e., if and only if α is a virtual unit. If this is the case, then f(N z /K z ) | b z if and only if the product is a multiple of (1 − ζ ℓ ) ℓ /b z = b * , and by the definition of A α and the congruence in Proposition 3.6, this is equivalent to the solubility of the congruence x ℓ /α ≡ 1 (mod * b * ), hence to α ∈ S b * (K z ).
Theorem 8.2. [13, Corollary 3.2] Writing C b := Cl bz (K z )/Cl bz (K z ) ℓ , so that G b = C b [T ]
, and µ ℓ for the group of ℓth roots of unity, there exists a perfect, G-equivariant pairing of F ℓ [G]-modules
C b × S b * (K z ) → µ ℓ .
Proof. This is the Kummer pairing: given a ∈ C b , let σ a denote its image under the Artin map; given α ∈ S b * (K z ), let α be any lift; then define the pairing by (a, α) → σ a ( ℓ √ α)/ ℓ √ α ∈ µ ℓ .
G b × S b * (K z )[T * ] → µ ℓ . In particular, we have |G b | = |S b * (K z )[T * ]| .
Proof. Recalling that τ (ζ ℓ ) = ζ g ℓ , for any j the preceding corollary yields a perfect pairing
C b [τ − g j ] × S b * (K z )[τ − g 1−j ] → µ ℓ .
We conclude by taking j = 1 and j = (ℓ + 1)/2 in the general and special cases respectively.
Proposition 8.4. In the special case, we have
Cl(Q z )/Cl(Q z ) ℓ [τ + 1] = {1} .
Proof. We first show that there exists an isomorphism
Cl(Q z )/Cl(Q z ) ℓ [τ + 1] ≃ Cl(K)/Cl(K) ℓ [τ + 1] .
By Lemma 2.3 (which also applies to t = τ + 1), the left side consists of those classes which may be represented by ideals of the form N Qz/K (a)/τ (N Qz /K (a)). We therefore obtain a well-defined, injective map to Cl(K)/Cl(K) ℓ [τ + 1]. Any ideal in the target space may be represented by an ideal of the form c/τ (c), which is equivalent to (c/τ (c)) (ℓ−1) 2 , and c (ℓ−1) 2 = N Qz/K (c 2(ℓ−1) Z Qz ), so that the map is surjective as well. Now it suffices to show that ℓ ∤ h(±ℓ), where h(D) denotes the class number of Q( √ D), and this follows from the classical and easy fact that h(±ℓ) < ℓ for all prime ℓ.
Remark 8.5. For ℓ ≡ 3 (mod 4) it is also possible to prove the proposition via the Herbrand-Ribet theorem and a congruence for Bernoulli numbers. Now suppose that ℓ ≡ 1 (mod 4). Then the Ankeny-Artin-Chowla conjecture (AAC) [1,30] states that if ǫ = (a + b √ ℓ)/2 is the fundamental unit of Q( √ ℓ), then ℓ ∤ b. We will use the statement of the conjecture directly, but we note that Ankeny and Chowla [2] and Kiselev [24] proved that it is equivalent to the condition ℓ ∤ B (ℓ−1)/2 , which is trivially true if ℓ is a regular prime, a result first proved by Mordell [30]. It has been verified for ℓ ≤ 2 · 10 11 by van der Poorten, te Riele, and Williams [35], but as mentioned in the introduction, on heuristic grounds it it probably false. Lemma 8.6. Suppose that the AAC conjecture is true for ℓ. Then the congruence x ℓ ≡ ε (mod (1− ζ) k Z Qz ) is solvable for k = (ℓ − 1)/2, and not for any larger value of k.
Proof. Write ε = (a+ b √ ℓ)/2 with a, b in Z. Note first that (1− ζ) (ℓ−1)/2 Z Qz = √ ℓZ Qz , and ε ≡ a/2 ≡ c ≡ c ℓ (mod √ ℓZ K ) with c ≡ a/2 (mod ℓ), so the congruence is indeed solvable with k = (ℓ − 1)/2. Assume that it is soluble for a strictly larger k, hence modulo
√ ℓ(1 − ζ)Z Qz . If x = 0≤i≤ℓ−2 a i ζ i with a i ∈ Z (or even in Z ℓ ), we have x ℓ ≡ 0≤i≤ℓ−2 a i (mod ℓ), so x ℓ ≡ m (mod ℓ) for some integer m. Thus, if x ℓ ≡ ε (mod √ ℓ(1 − ζ)Z Qz ) we have a + b √ ℓ ≡ 2m (mod √ ℓ(1 − ζ)Z Qz ).
In particular a ≡ 2m (mod √ ℓ), and since they are both integers we deduce that a ≡ 2m (mod ℓ), so our congruence gives b √ ℓ ≡ 0 (mod √ ℓ(1−ζ)Z Qz ), i.e., b ≡ 0 (mod (1 − ζ)Z Qz ). Again since b is an integer this implies that ℓ | b, contradicting AAC and proving (1).
Remark 8.7. If AAC is false for ℓ, then the congruence is soluble for all k: it may trivially be solved for k = 3(ℓ − 1)/2 with x ∈ Z, and then a Newton-Hensel iteration as in [8, Lemma 10.2.10] settles the matter.
We now return to the groups S b * (K z )[T * ].
Proposition 8.8.
(1) In the general case we have S b * (K z )[T * ] ≃ S b * ∩K (K). Proof.
(1) [13,Proposition 3.4]. We have an injection S b * ∩K (K) ֒−→ S b * (K z )[τ − 1] which we prove is surjective by Hilbert 90 and some elementary computations, yielding an isomorphism
S b * (K z )[τ −1, τ 2 +1] ≃ S b * ∩K (K)[τ 2 + 1]. Furthermore, we have S b * ∩K (K) = S b * ∩K (K)[τ 2 + 1] ⊕ S b * ∩K (K)[τ 2 − 1] ,
and we argue that S ℓ (K)[τ 2 −1] is trivial (and a fortiori all the S b * ∩K [τ −1]), again using Hilbert 90, finishing the proof of (1).
(2) and (3). Assume now that we are in the special case, so that K z = Q z = Q(ζ ℓ ). By Proposition 8.4 we have (
Cl(K z )/Cl(K z ) ℓ )[τ + 1] = {1}, so that by Lemma 2.6 we have S ℓ (K z )[T * ] ≃ (U (K z )/U (K z ) ℓ )[τ − g (ℓ−1)/2 ]
. By Theorem 2.3 of [11] we deduce that S ℓ (K z )[T * ] is trivial if ℓ ≡ 3 (mod 4), ℓ = 3, and when ℓ ≡ 1 (mod 4) that it is an F ℓ -vector space of dimension 1. If ε is a fundamental unit of K = Q( √ ℓ), then since τ acts on ε as Galois conjugation of K/Q, we have ετ (ε) = N K/Q (ε) = ±1, which is an ℓth power. It
follows that S ℓ (K z )[T * ] = {ε j , j ∈ F ℓ }.
The sizes of the ray Selmer groups are then established by Lemma 8.6 and Remark 8.7.
Remarks 8.9.
(1) The assumption that ℓ = 3 is required when applying Theorem 2.3 of [11], and indeed (2) of the proposition is false for ℓ = 3 (see Proposition 7.3 of [12]).
(2) In Corollary 9.4 we will apply our computations to conclude that AAC is equivalent to the nonexistence of D ℓ -fields ramified only at ℓ.
This proposition, in combination with Corollary 8.3, gives the size of |G b | in the special case, with possible exceptions ℓ ≡ 1 (mod 4) larger than 2 · 10 11 . In the general case we have the following: (1) We have a canonical isomorphism G b ≃ Hom(S b * ∩K (K), µ ℓ ).
(2) In particular
|G b | = ℓ r(b) with r(b) = 1 − r 2 (D) − z(b) + rk ℓ (Cl b * ∩K (K)) ,
with z(b) = (2, 1, 0, 0) respectively, the second case occurring only if ℓ|D. Proof. (1) is immediate. Lemma 2.6, and Proposition 2.12 of [11], the proofs of which adapt to K without change, yield
|S b 1 (K)||Z b 1 /Z ℓ b 1 | = ℓ 1−r 2 (D) |Cl b 1 (K)/Cl b 1 (K) ℓ | , where Z b 1 = (Z K /b 1 ) * and b 1 = b * ∩ K. This gives (2) with z(b) = dim F ℓ (Z b 1 /Z ℓ b 1 )
, and to finish we compute for b as in (7.1):
• If ℓ ∤ D, we have b * ∩ K = (ℓ 2 Z K , * , ℓZ K , Z K ). • If ℓ | D with ℓZ K = p 2 ℓ , we have b * ∩ K = (p 3 ℓ , p 2 ℓ , p ℓ , Z K ).
Note that (3) is a generalization of Proposition 7.7 of [12]. Since the triviality of G b for all b is equivalent to the vanishing of the "remainder term" φ D (s) of Corollary 7.4, we conclude that Φ ℓ (K, s) is given by a single Euler product in a wide class of examples:
Corollary 8.11. Assume that ℓ ≥ 5, D < 0, and that either we are in the special case (so that ℓ ≡ 3 (mod 4)), or that we are in the general case with ℓ ∤ h(D). Then we have
L∈F ℓ (K) 1 f (L) s = − 1 ℓ − 1 + 1 ℓ − 1 L ℓ (s) p≡( D p ) (mod ℓ), p =ℓ 1 + ℓ − 1 p s ,
where L ℓ (s) is as above.
Note that for ℓ = 3, which we have excluded here, the possible nontriviality of r e (b) forces us to also distinguish between D ≡ 3 and D ≡ 6 (mod 9).
Examples with ℓ = 5: 1 + 4 p s .
L∈F 5 (Q( √ −1)) 1 f (L) s = −
Transformation of the Main Theorem
We now prove, as we did in [14] for the case of ℓ = 3, that the characters of G b appearing in Theorem 6.1 can be given a simpler description, in terms of the splitting of primes in degree ℓ extensions of k. Our main result along these lines extends Theorem 4.1 of [14] and Proposition 3.7 of [13], and does not assume that k = Q, and thus is new even for ℓ = 3.
For the case k = Q we will further specialize the result and obtain an explicit formula, relying (in the general case) on the results of [13]. We will assume that we are in either the general case or in the special case with ℓ ≡ 1 (mod 4). Recall that in the special case with k = Q, ℓ ≡ 3 (mod 4), and ℓ > 3, G b is trivial and Corollary 8.11 already gives a simple description of Φ ℓ (K, s). For simplicity's sake we will omit the special case with k = Q, ℓ ≡ 3 (mod 4); as we will see below the group theory would work out a bit differently.
We first recall a bit of group theory, and introduce some notation. The Frobenius group F ℓ = C ℓ ⋊ C ℓ−1 is the non-abelian group of order ℓ(ℓ − 1) given by the presentation τ, σ : τ ℓ−1 = σ ℓ = 1, τ στ −1 = σ h , for any primitive root h (mod ℓ). As may be easily checked, C ℓ−1 is not normal in F ℓ , nor is any nontrivial subgroup of C ℓ−1 ; moreover, there are ℓ subgroups isomorphic to C ℓ−1 , generated by τ σ i for 0 ≤ i ≤ ℓ − 1, and all of these subgroups are conjugate.
We say that a degree ℓ field extension E/k is an F ℓ -extension if its Galois closure has Galois group F ℓ over k. Now, let K, K z , τ, τ 2 be defined as before. In the general case recall that K ′ was defined to be the mirror field of K, e.g., the subfield of K z fixed by τ (ℓ−1)/2 τ 2 ; in the special case write K ′ = K z = k z . We chose τ ∈ Gal(k z /k) and a primitive root g (mod ℓ) with τ (ζ ℓ ) = ζ g ℓ . In the general case τ lifts uniquely to an element of Gal(K z /K) and restricts to a unique element of Gal(K ′ /k), so in either case the choice of g (mod ℓ) uniquely determines τ ∈ Gal(K ′ /k).
Theorem 9.1. Assume, if ℓ ≡ 3 (mod 4), that we are in the general case. For each b ∈ B (as in Theorem 6.1), there exists a bijection between the following sets:
• Characters χ ∈ G b , up to the equivalence relation χ ∼ χ a for each a coprime to ℓ.
• Subgroups of index ℓ of G b .
• F ℓ -extensions E/k (up to isomorphism), whose Galois closure E ′ contains K ′ and whose conductor f(E ′ /K ′ ) divides b ′ = b ∩ K, and such that τ στ −1 = σ g for τ ∈ Gal(K ′ /k) as described above and any generator σ of Gal(E ′ /K ′ ).
Moreover, for each corresponding pair (χ, E) and each prime p ∈ D ∪ D ℓ , the following is true: we have p ∈ D ′ (χ) ∪ D ′ ℓ (χ) if and only if p is totally split in E; equivalently, p ∈ D ′ (χ) ∪ D ′ ℓ (χ) if and only if p is totally inert or totally ramified in E.
Recall (Definition 2.12) that D ∪ D ℓ was defined in terms of splitting conditions in K z /k, so that this theorem describes each Euler factor in Φ ℓ (K, s) in terms of splitting conditions in a fixed set of number fields.
Proof. The proof borrows heavily from those of Proposition 4.1 of [14] and Proposition 3.7 of [13].
The correspondence between the first two sets is immediate: G b is elementary ℓ-abelian, and characters correspond to their kernels. By Proposition 6.3, regard G b as
Cl b ′ (K ′ ) Cl b ′ (K ′ ) ℓ [τ ∓ g],
where the sign is − in the general case and + in the special case. If we set
G ′ b = Cl b ′ (K ′ )/Cl b ′ (K ′ ) ℓ ,G ′ b = G b ×G ′′ b , where G ′′ b
is the direct sum of all of the other eigenspaces for the actions of τ . Thus, subgroups of G b of index ℓ correspond to subgroups B of Cl b ′ (K ′ ) of index ℓ containing G ′′ b . By class field theory, there exists a unique abelian extension E ′ /K ′ , with Galois group C ℓ and conductor dividing b ′ , for which the Artin map induces an isomorphism Cl b ′ (K ′ )/B ≃ Gal(E ′ /K ′ ). The uniqueness forces E ′ to be Galois over k; here we use that b ′ , B, and Cl b ′ (K ′ ) are preserved by Gal(K ′ /k). For each fixed b, we obtain a different E ′ for each B.
Because the action of Gal(K ′ /k) on Cl b ′ (K ′ )/B χ matches its conjugation action on Gal(E ′ /K ′ ), we have
(9.1) Gal(E ′ /k) = τ, σ : τ ℓ−1 = σ ℓ = 1, τ στ −1 = σ ±g ≃ F ℓ ,
and we take E to be the fixed field of τ (or, alternatively, of any conjugate subgroup). Note that −g is not a primitive root if ℓ ≡ 3 (mod 4), so that in the special case with ℓ ≡ 3 (mod 4) the group (9.1) contains τ (ℓ−1)/2 in its center and is not isomorphic to F ℓ .
It must finally be proved that whether p ∈ D ′ (χ) or not is determined by its splitting in E. Proposition 2.15 or Corollary 2.11 implies that D ∪ D ℓ is precisely the set of primes p which split completely in K ′ /k, and by definition D ′ (χ) ∪ D ′ ℓ (χ) is the set of primes p ∈ D ∪ D ℓ for which one (equivalently, all) of the primes p K ′ of K ′ above p split completely in E ′ . If p K ′ splits completely in E ′ , then so does p, so p also splits completely in E/k. Conversely, if any p K ′ is completely ramified or inert in K z , then p must also do the same in each E, since ramification and inertial degrees are multiplicative and [E ′ : E] = ℓ − 1.
For ℓ = 3 and k = Q in the general case, in [14] we further applied a theorem of Nakagawa to give a precise description of all the extensions E/Q occurring in the statement of Theorem 9.1 in terms of their discriminants. Using this, we obtained the formula 9.1. Explicit computations for k = Q in the special case. For ℓ = 3, we have the following explicit formula (corresponding to pure cubic fields), which was previously proved in [12]. Proof. Immediate by inspecting the Dirichlet series of the proposition; the proposition also shows that for any ℓ not satisfying the conjecture, the field is unique and has discriminant ℓ The connection to the Ankeny-Artin-Chowla conjecture was previously observed by Lemmermeyer [27], who suggested that a proof of Corollary 9.4 may exist somewhere in the literature.
Before beginning the proof of Proposition 9.2 we establish the following: Lemma 9.6. We have
Disc(N z ) = ℓ (3ℓ 2 −2ℓ−3)/2 if AAC is true, ℓ ℓ(ℓ−2)
if AAC is false.
In addition, in the extension N z /Q z the prime ideal (1 − ζ)Z Qz is totally ramified if AAC is true and totally split otherwise.
Proof. The field N z is a Kummer extension of K z = Q z with defining equation x ℓ − ε = 0, so that
Disc(N z ) = ±N Qz/Q (d(N z /Q z ))Disc(Q z ) ℓ = ±ℓ ℓ(ℓ−2) N Qz/Q (f(N z /Q z )) ℓ−1 ,
where f(N z /Q z ) is the conductor. By [11,Theorem 3.7] applied to K = Q z and α = ε which is a unit, we have f(N z /Q z ) = (1 − ζ) ℓ+1−Aε , where A ε = ℓ + 1 if x ℓ ≡ ε (mod (1 − ζ) ℓ ) has a solution in Q z , and otherwise A ε is the maximal k such that x ℓ ≡ ε (mod (1 − ζ) k ) has a solution. By Lemma 8.6 we have A ε = (ℓ − 1)/2 (resp., A ε = ℓ + 1) if AAC is true (resp., false), hence f(N z /Q z ) = (1 − ζ) (ℓ+3)/2 Z Qz (resp., f(N z /Q z ) = Z Qz ), from which the formula follows (note that the sign of the discriminant is positive since Q z hence N z is totally complex). In addition, if AAC is false, so that C k is soluble for all k, then Hecke's Theorem [8, 10.2.9] (an extension of [11,Theorem 3.7]) implies that (1 − ζ)Z Qz is totally split, while if AAC is true then it is totally ramified.
Proof of Proposition 9.2. The result follows for an undetermined E by Theorem 9.1 and Proposition 8.8. To determine E, observe that Proposition 8.1 and the proof of Proposition 8.8 imply that N z = K z (ǫ 1/ℓ ), and that the considerations in the proof of Theorem 9.1 allow us to take E to be any of the (conjugate) degree ℓ subfields of N z , so that it suffices to exhibit one.
We take E = Q(ε 1/ℓ − ε −1/ℓ ) for any fixed choice of ε 1/ℓ , recalling that the fundamental unit has norm −1. Then the minimal polynomial of E is P (x) − Tr(ε) by construction, or more precisely by (9.3).
It remains only to argue that Disc(E) = ℓ (3ℓ−1)/2 if AAC is true, ℓ ℓ−2 if AAC is false.
We assume that AAC is true (if false, a similar proof applies). On the one hand we have Disc(N z ) = Disc(E) ℓ−1 N E/Q (d(N z /E)) , in other words taking valuations and using the proposition:
(ℓ − 1)v ℓ (Disc(E)) = (3ℓ 2 − 2ℓ − 3)/2 − v ℓ (N E/Q (d(N z /E))) = (ℓ − 1)(3ℓ − 1)/2 + ℓ − 2 − v ℓ (N E/Q (d(N z /E))) .
On the other hand, the extension N z /E is of degree ℓ − 1 hence tame, so v ℓ (N E/Q (d(N z /E))) ≤ ℓ − 2. Divisibility by ℓ − 1 thus implies the result, together with the additional result that N E/Q (d(N z /E)) = ℓ ℓ−2 = Disc(Q z ).
We conclude by establishing the statements made in the remarks. For (1), it is easily seen that our construction still produces a degree ℓ subfield E. (2) follows because ℓ is totally ramified in E.
To prove (3), we again apply Hecke's theorem 10.2.9 of [8]: p is totally split in E iff it is in N z /Q z , hence by Hecke iff x ℓ ≡ ε (mod p) is soluble in Q z . (Here p is any prime of Q z above p, which must have degree 1 since p ≡ 1 (mod ℓ) is totally split in Q z .) This is equivalent to ε (p−1)/ℓ ≡ 1 (mod p), which by Galois theory will then be true for all primes p above p since for any σ ∈ Gal(Q z /Q) we have either σ(ε) = ε or σ(ε) = −ε −1 , and (p − 1)/ℓ is even so the sign disappears. Hence this is equivalent to the condition ε (p−1)/ℓ ≡ 1 (mod p), as desired.
Finally, (4) follows from Eisenstein's reciprocity law.
9.2. Explicit computations for k = Q in the general case. Let k = Q. In Theorem 9.1 we saw that characters χ of G b (up to the equivalence χ ∼ χ a for (a, ℓ) = 1) correspond to degree ℓ fields E having certain properties. In our companion paper [13] with Rubinstein-Salzedo, we further proved the following:
Theorem 9.7. [13] Suppose that k = Q and K = Q( √ D) with D = 1, ±ℓ, so that we are in the general case, and as before let K ′ be the mirror field of K.
Then the fields E enumerated in Theorem 9.1 are precisely those F ℓ -fields E whose Galois closure contains K ′ , subject to the condition τ στ −1 = σ g described there, satisfying the following additional conditions:
• E is totally real if D < 0, and has ℓ−1 2 pairs of complex embeddings if D > 0. Moreover, if E is any F ℓ -field satisfying these last two properties, then its Galois closure automatically contains K ′ .
Recall that we have b ∈ B = {1, (ℓ) 1/2 , (ℓ), (ℓ) ℓ/(ℓ−1) }, with the possibility (ℓ) 1/2 occurring only if ℓ|D. The complete list of fields enumerated in Theorem 9.7 corresponds to b = (ℓ) ℓ/(ℓ−1) . A careful reading of the proof of Theorem 9.7 (in [13]), with k having the same meaning above as in [13,Section 4], shows that the remaining b correspond to the following possibilities for k in (9.4):
Condition on D b = 1 b = (ℓ) 1/2 b = (ℓ) b = (ℓ) ℓ/(ℓ−1) ℓ ∤ D k = 0 -k = 0, 2 k = 0, 2 ℓ | D and ℓ ≡ 1 (mod 4) k = 0 k = 0 k = 0, (ℓ + 3)/2 k = 0, (ℓ + 3)/2 ℓ | D and ℓ ≡ 3 (mod 4) k = 0 k = 0 k = 0, (ℓ + 5)/2 k = 0, (ℓ + 5)/2
One exception occurs for ℓ = 3: Only k = 0 corresponds to b = (ℓ) when ℓ | D; this is because the inequality (ℓ + 5)/2 ≤ ℓ − 1 is true for all ℓ ≡ 3 (mod 4) except for ℓ = 3. (Note also for ℓ = 3 that this result is equivalent to part of Proposition 4.1 in [14].) This is sufficient to obtain an explicit formula for Φ ℓ (K, s) for any K and ℓ, provided that the appropriate F ℓ -fields can be tabulated. We present two examples, which we also double-checked numerically using a program written in PARI/GP [34].
(1. 5 )
5N 3 (D * ) + N 3 (−27D) = N 3 (D) if D < 0, 3N 3 (D) + 1 if D > 0,for any fundamental discriminant D, where D * = −3D if 3 ∤ D and D * = −D/3 if 3 | D. (Note that there are no cubic fields of discriminant −3D if 3 | D.)
Definition 2. 2 .
2In the group ring F ℓ [G], we set T = {τ 2 + 1, τ − g} in the general case , {τ + g} in the special case .
( 2 )
2For any F ℓ [G] module M , we denote as usual by M [T ] the subgroup annihilated by all the elements of T . Lemma 2.3. Let M be an F ℓ [G]-module.
Definition 3. 3 .
3Suppose that p, p, and p z are as above, so that e(p z /p) | (ℓ − 1). Moreover, let α ∈ (K * z /K * z ℓ )[T ] be as in Proposition 2.4.
Definition 4. 1 .
1If a is an ideal of k, we set N (a) = N k/Q (a), while if a is an ideal of K, we set N (a) = N K/Q (a) 1/2 .In particular, for each ideal a of k we have N (a) = N (aZ K ).
1 ,...,an)∈A 1≤i≤n N (p i ) ⌈a i ⌉s/e(p i /p i ) u∈S ℓ (Kz)[T ] ∀i, aαu(p i )=a i 1 . By Lemma 3.12, we have a αu (p i ) ≥ a i if and only if u is counted by f α (p b i i ), where b i = a i + h(0, a, p i ), and we rewrite p b i i = p b i /e(p i /p i ) i . Let B(α) be the set of n-uples (b 1 , . . . , b n ) with 0 ≤ b i ≤ m(p i ), b i ∈ Z ∪ {m(p i )}. By inclusion-exclusion we obtain the following:
Remark 4. 4 .
4There are conditions on the a i , e.g.(3.2), such that the inner sum in (4.2) vanishes for impossible choices of the a i . One can use this to prove alternate versions of Lemma 4.3 that are nonobviously equivalent. In particular, if (ℓ − 1) | e(p) then one can restrict to b i ∈ 2Z ∪ {m(p i )} with suitably modified Q(p b/e(p/p) , s).
0 ,...,a ℓ−2 )∈J S α (s) N (a α ) s = b∈B ⌈N ⌉(b) s P (b, s) (a 0 ,...,a ℓ−2 )∈J (aα,b)=1 p∤b and (ℓ−1)|e(p)⇒p|aα
Lemma 5. 2 .
2Assume that ℓ > 3, the case ℓ = 3 being treated in [12, Lemma 5.4]. For any number field M , write rk ℓ (U (M )) := dim F ℓ (U (M )/U (M ) ℓ ), and denote by r 1 (M ) and r 2 (M ) the number of real and pairs of complex embeddings of M .
Lemma 5. 3 .
3Let b ∈ B satisfy b z | (1 − ζ ℓ ) ℓ , and define c z = pz⊂Kz pz|bz p ⌈vp z (bz)/ℓ⌉ z . We have |(Z bz /Z ℓ bz )[T ]| = |(c z /b z )[T ]| , the latter being considered as an additive group.
(d) We have N Kz/K (a 0 ) = a α .(e) By Lemma 5.7 we have χ(a) = χ −1 (a 0 ).
Multiplied by the first factor this gives ℓ −2s if ℓ ∤ D and ℓ −3s/2 if ℓ | D. • We have ℓ RU (K) = ℓ r 2 (D) by Lemma 5.2, with r 2 (D) := r 2 (Q( √ D)). • By Definitions 4.5 and 4.1, we have ⌈N ⌉(b) = (1, * , ℓ, ℓ 2 ) and ⌈N ⌉(b) = (1, ℓ 1/2 , ℓ, ℓ 3/2 ) for ℓ ∤ D and ℓ | D respectively. • As already mentioned, if k = Q and ℓ > 3 we have r e (b) = (1), so the terms and conditions involving r e (b) disappear (in other words we use Corollary 6.2). • By Lemma 7.2, we have p ∈ D if and only if p ≡ D p (mod ℓ) and p = ℓ, and D ℓ = ∅ when ℓ = 3 by what we have just said. • By Lemma 4.3 and Definition 4.5 of P (b, s), when ℓ ∤ D and ℓ | D respectively. we have P (b, s) = (1, * , −ℓ −s , 1), and P (b, s) = (1, 1 − ℓ −s/2 , −ℓ −s/2 , 1) for ℓ ∤ D respectively for the usual sequence of b. • The values of |(Z bz /Z ℓ bz )[T ]| are given in Proposition 7.1. Corollary 7.4. Assume that ℓ ≥ 5, and set L ℓ (s) = 1 + (ℓ − 1)/ℓ 2s if ℓ ∤ D and L ℓ (s) = 1 + (ℓ − 1)/ℓ s if ℓ | D. There exists a function φ D (s) = φ D,ℓ (s), holomorphic for ℜ(s) > 1/2, such that
Corollary 8. 3 .
3[13, Corollary 3.3 (in part)] In the general case, where T = {τ − g, τ 2 + 1}, define T * = {τ − 1, τ 2 + 1}, and in the special case, where T = {τ + g}, define T * = {τ + 1}. Then we have a perfect pairing
( 2 )
2In the special case with ℓ ≡ 3 (mod 4), we have S b * (K z )[T * ] = {1} for all b.(3) In the special case with ℓ ≡ 1 (mod 4), if the Ankeny-Artin-Chowla conjecture is true for ℓ, then we have |S b * (K z )[T * ]| = (1, 1, ℓ, ℓ) for b as in (7.1). If Ankeny-Artin-Chowla is false for ℓ, then we have instead |S b * (K z )[T * ]| = (ℓ, ℓ, ℓ, ℓ).
Corollary 8 .
810.[13, Corollary 3.5] Assume that we are in the general case.
( 3 )
3In particular still, if D < 0 and ℓ ∤ h(D) then G b is trivial for all b ∈ B.
L 3 (D) is the set of all cubic fields of discriminant −D/3, −3D, and −27D; ω E (p) is 2 or −1 depending on whether p is split or inert in E, as in Theorem 9.1; and M 1 (s) and M 2,E (s) are 3-adic factors (a sum of the appropriate A b (s)).
Corollary 9. 4 .
4Let ℓ ≡ 1 mod 4. Then there exist D ℓ -fields ramified only at ℓ if and only if the Ankeny-Artin-Chowla conjecture is false for ℓ.
. 5 .
5This corollary recovers and strengthens a result of Jensen and Yui [23, Theorem I.2.2], who proved that if ℓ ≡ 1 (mod 4) is regular, then there are no D ℓ -fields with discriminant a power of ℓ. (This can also be seen for ℓ ≡ 3 (mod 4) from Corollary 8.11.)
• |Disc(E)| has the form ℓ k+b D ℓ−1 2 , where k and b satisfy(9.4) k ∈ {0, 2}, b = ℓ − 2 if ℓ ∤ D , k ∈ {0, (ℓ + 3)/2} , b = ℓ−3 2 if ℓ | D and ℓ ≡ 1 (mod 4) , k ∈ {0, (ℓ + 5)/2} , b = ℓ−5 2if ℓ | D and ℓ ≡ 3 (mod 4) .
by Lemmas 2.1 and 2.3 we have the orthogonal decomposition
Ankeny, Artin, and Chowla did not conjecture this in [1], although they did explicitly ask if it is true. Mordell[30] attributed the conjecture to them in followup work, where he proved the conjecture for regular primes.
The object of the present paper is to generalize the theory developed in[12]and[14]to degree ℓ extensions having Galois group D ℓ , for any odd prime ℓ.AcknowledgementsWe would like to thank Michael Filaseta, David Harvey, Franz Lemmermeyer, Hendrik Lenstra, David Roberts, and John Voight for helpful comments and suggestions.This material is based upon work supported by the National Science Foundation under Grant No. DMS-1201330 and by the National Security Agency under a Young Investigator Grant.where E is the cubic field defined by x 3 − 3x − 1 = 0 (discriminant 34, Galois group C 3 ), andif p is totally split in E , 0 otherwise.In fact, since E is cyclic cubic, we never have ω E (p) = 0, and ω E (p) = 2 if and only if p ≡ ±1 (mod 9).For ℓ ≡ 3 (mod 4) and ℓ > 3, a generalization was proved in Corollary 8.11. For ℓ ≡ 1 (mod 4), the generalization is more complicated due to the nontriviality of G b . Define a polynomialHere T ℓ (x) is the Chebyshev polynomial of the first kind, satisfyingAssume that ℓ ≡ 1 (mod 4) satisfies the Ankeny-Artin-Chowla conjecture, and let ε be a fundamental unit of Q( √ ℓ). Then we havewhere E is the F ℓ -field defined by P (x) − Tr(ε) = 0 of discriminant ℓ (3ℓ−1)/2 , andotherwise.If ℓ ≡ 1 (mod 4) does not satisfy the Ankeny-Artin-Chowla conjecture, we have the same formula, but with Disc(E) = ℓ ℓ−2 and ω E (ℓ) = ℓ − 1.Before presenting the proof, we make some additional observations (to be proved after the proposition).Remarks 9.3.(1) In the equation for E we may replace Tr(ε) by Tr(±ε m ) for any odd m ∈ Z coprime to ℓ.(2) Assuming AAC, the last product may be written as(3) When p ≡ 1 (mod ℓ) then p is totally split in E if and only if ε (p−1)/ℓ ≡ 1 (mod p) , and otherwise p is inert in E. (4) If in addition Q z = Q(ζ ℓ ) has class number 1, then p is totally split in E if and only if p = N Qz/Q (π) for some π ≡ 1 (mod ℓ) in Q z .Example 9.8. Let K = Q( √ 13) and ℓ = 5. Then, we havewhere the products are over primes p ≡ 1,16,19,24,34,36,44,51,54,56,59, 61 (mod 65), E is the field defined by the polynomial x 5 + 5x 3 + 5x − 3, and 24131 = 59 · 409.Example 9.9. Let K = Q( √ −7 · 41) and ℓ = 7. Then, we have = 1 + 7 · 301 −s + 7 · 337 −s + 7 · 581 −s + 7 · 791 −s + · · · + 42 · 296897 −s + · · · , where the products are over primes p ≡ D p (mod ℓ) excluding p = ℓ, E is the field defined by the polynomial x 7 − 14x 5 + 56x 3 − 56x − 15, and 296897 = 337 · 881.
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"e Macroeconomic E ects of Corporate Tax Reforms",
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[
"Francesco Furno [email protected] \nDepartment of Economics\nNew York University\n\n",
"Jaroslav Borovička ",
"Jess Benhabib ",
"Simon Gilchrist ",
"Tim Christensen ",
"Gian Luca Clementi ",
"Francesco Daveri ",
"Francesco Giavazzi ",
"Stefano Rossi ",
"Tom Sargent ",
"Bàlint ",
"\nNYU\n\n"
] |
[
"Department of Economics\nNew York University\n",
"NYU\n"
] |
[] |
is paper extends a standard general equilibrium framework with a corporate tax code featuring two key elements: tax depreciation policy and the distinction between c-corporations and pass-through businesses. In the model, the stimulative e ect of a tax rate cut on c-corporations is smaller when tax depreciation policy is accelerated, and is further diluted in the aggregate by the presence of pass-through entities. Because of a highly accelerated tax depreciation policy and a large share of pass-through activity in 2017, the model predicts small stimulus, large payouts to shareholders, and a dramatic loss of corporate tax revenues following the Tax Cuts and Jobs Act (TCJA-17). ese predictions are consistent with novel micro-and macro-level evidence from professional forecasters and sectoral tax returns. At the same time, because of less-accelerated tax depreciation and a lower pass-through share in the early 1960s, the model predicts sizable stimulus in response to the Kennedy's corporate tax cutsalso supported by the data. e model-implied corporate tax multipliers for Trump's TCJA-17 and Kennedy's tax cuts are +0.6 and +2.5, respectively.JEL Codes: E06, H02, H06
|
10.2139/ssrn.4023666
|
[
"https://arxiv.org/pdf/2111.12799v1.pdf"
] | 244,709,261 |
2111.12799
|
9bb93aff6a11a6891f45706048a1043ac642b74c
|
e Macroeconomic E ects of Corporate Tax Reforms
November 24, 2021 24 Nov 2021
Francesco Furno [email protected]
Department of Economics
New York University
Jaroslav Borovička
Jess Benhabib
Simon Gilchrist
Tim Christensen
Gian Luca Clementi
Francesco Daveri
Francesco Giavazzi
Stefano Rossi
Tom Sargent
Bàlint
NYU
e Macroeconomic E ects of Corporate Tax Reforms
November 24, 2021 24 Nov 2021am grateful to my advisorsSzőke for their en-couragement. I also thank Mark Gertler, Federico Kochen, Virgiliu Midrigan, Diego Perez, and participants to the NYU Macro Lunch, the BFI's Macro-Financial Modeling Summer Session, and the St. Louis FED's Dissertation Workshop for their comments.Corporate TaxMacroeconomicsTax DepreciationPass-rough Businesses
is paper extends a standard general equilibrium framework with a corporate tax code featuring two key elements: tax depreciation policy and the distinction between c-corporations and pass-through businesses. In the model, the stimulative e ect of a tax rate cut on c-corporations is smaller when tax depreciation policy is accelerated, and is further diluted in the aggregate by the presence of pass-through entities. Because of a highly accelerated tax depreciation policy and a large share of pass-through activity in 2017, the model predicts small stimulus, large payouts to shareholders, and a dramatic loss of corporate tax revenues following the Tax Cuts and Jobs Act (TCJA-17). ese predictions are consistent with novel micro-and macro-level evidence from professional forecasters and sectoral tax returns. At the same time, because of less-accelerated tax depreciation and a lower pass-through share in the early 1960s, the model predicts sizable stimulus in response to the Kennedy's corporate tax cutsalso supported by the data. e model-implied corporate tax multipliers for Trump's TCJA-17 and Kennedy's tax cuts are +0.6 and +2.5, respectively.JEL Codes: E06, H02, H06
Introduction
is paper presents a model of the macroeconomic e ects of corporate tax reforms and uses it to analyze the Trump's Tax Cuts and Jobs Act of 2017 and the Kennedy's corporate tax cuts of the early 1960s. e theoretical framework consists of a standard macroeconomic environment augmented with two key elements: tax depreciation policy and the distinction between c-corporations and pass-through businesses. For each reform, novel empirical evidence is collected and used to validate the model's predictions. e rst key ingredient of the analysis is tax depreciation policy, which de nes the set of rules that businesses are required to follow to deduct investment from their tax base. As I document in Furno (2021), the vast majority of corporate tax codes around the world do permit businesses to fully recover the cost of investment from their tax base, but only over time according to a tax depreciation schedule. As a result, di erences in tax depreciation policies across space and time boil down to how fast investment can be deducted. When investment is allowed to be deducted over a short period of time, the tax depreciation schedule is said to be 'accelerated', and recent empirical contributions have documented the ability of accelerated tax depreciation policy to stimulate rms' investment. 1 Pass-through businesses are the second key ingredient. In the US, only c-corporations are subject to corporate income taxation. All other forms of organization (s-corporations, partnerships, and sole-proprietorships) are 'pass-through', in the sense that their earnings are not subject to rm-level taxation and are 'passed through' to their owners. I estimate that roughly 40% of economic activity took place in the pass-through sector in 2017, compared to 25% in the early 1960s. 2 In the model, corporate tax changes a ect the economy primarily through the investment decision of c-corporations, which is a ected not only by the tax rate but also by tax depreciation policy. Speci cally, the possibility to deduct investment from the tax base (partially) counteracts the distortion introduced by the tax rate: the faster investment is deducted from the tax base, the smaller the distortion to the rate of return on investment. As a result, when tax depreciation policy is very accelerated -like it was in 2017 -the rate of return on investment is almost una ected by corporate tax policy, and a reduction in the corporate tax rate is not particularly expansionary. However, irrespective of how much stimulus is provided to investment, a corporate tax cut always entails a transfer of resources from the government to c-corporations. When pre-reform tax depreciation policy is very accelerated, the tax-savings from a rate cut are not used for investment and are distributed to the shareholders. When pre-reform tax depreciation policy is not very accelerated, instead, the extra cash is used for investment. Moreover, since a change to the corporate tax rate a ects only c-corporations, the aggregate e ect is diluted by the presence of pass-through businesses. A er a rate reduction, pass-through entities are not only excluded from the tax cut, but they are also put at a competitive disadvantage. is happens because they compete with c-corporations in the production of (imperfectly) substitutable goods, which further ampli es the shi of economic activity from pass-through businesses to c-corporations and reduces the aggregate e ect even more. To test my theory, I collect empirical evidence on the TCJA-17. In particular, I compare pre-reform professional forecasts with actual outcomes for both macroeconomic aggregates and c-corporations' aggregates constructed from rm-level data. I also use publiclyavailable tax returns from the IRS to compare the response of c-corporations and passthrough businesses. When simulating the TCJA-17, my model predicts small stimulus, large payouts to shareholders, and a dramatic loss of corporate tax revenues -in line with the empirical evidence.
is is due to highly accelerated tax depreciation policy and a large share of pass-through businesses in 2017. At the same time, when used to analyze the corporate provisions of the Kennedy's tax cuts, the model predicts sizable stimulus to GDP and investment, and a small increase in payouts to shareholders -in line with time-series descriptive evidence. is is due to less-accelerated tax depreciation policy and to a smaller share of pass-through activity in the early 1960s. I then compute model-implied corporate tax multipliers for each tax reform and nd that, for every dollar of lost corporate tax revenues, the Kennedy's tax cuts stimulated output roughly four times more than the Tax Cuts and Jobs Act. A large part of this di erence can be a ributed to di erences in pre-reform tax depreciation policy. To be er understand how corporate taxes work in my model, I o er some analytic insights. I rst provide a formal mapping between my corporate tax and the familiar concept of a "capital tax", and show that corporate taxes can be thought of as capital taxes accompanied by investment subsidies. I then prove that the government can allow fullexpensing of investment and still collect tax revenues in the steady-state. Finally, I derive the analytic steady-state and use it to characterize the distortions introduced by US corporate tax policy over the last few decades. is last exercise shows that US policy-makers have by now removed most of the distortions introduced by corporate taxes, but this also implies that they are running out of ammunition: further reductions to corporate tax rates are unlikely to provide strong stimulus to the economy.
Empirical Evidence on the TCJA-17
is section presents evidence on the e ects of the Tax Cuts and Jobs Act of 2017. I rst examine aggregate and c-corporate variables by comparing their actual paths with prereform forecasts. I then look at the response of both c-corporations and pass-through businesses using tax returns. A distinctive feature of the analysis is that it explicitly distinguishes between c-corporations -that pay corporate taxes -and pass-through businesses -that do not. Usually, data collection and analysis is organized around the distinction between corporations (c-corporations and s-corporations) and non-corporations (partnerships and sole-proprietorships). When studying corporate tax reforms, however, this categorization is problematic because ccorporations and s-corporations are aggregated together, but the la er do not pay corporate income taxes. At the aggregate level, the evidence suggests small stimulus and a sharp reduction in corporate tax revenues. e response is larger at the c-corporate level, but the percentage increase in payouts to shareholders outweighs the percentage increase in investment, suggesting that a sizable portion of the tax-savings from the reform were distributed to shareholders. Finally, tax returns shows a shi of economic activity from pass-through businesses to c-corporations.
is section proceeds as follows. e various tax changes introduced by the TCJA-17 are summarized in subsection 2.1, while the empirical evidence is presented and discussed in subsection 2.2 and subsection 2.3.
e TCJA-17: Corporate Provisions
It is common for recent major US tax reforms to include provisions a ecting a variety of tax instruments, and the Tax Cuts and Jobs Act of 2017 is no exception. For example, the reform included changes to individual income taxation, to the estate tax exemption, to the individual mandate penalty, to international tax rules, and introduced a deduction for pass-through income. 3 Since corporate taxation is the main focus of this paper, I summarize the corporate provisions of the TCJA-17 in Table 1. e two main corporate provisions introduced by the TCJA-17 are a permanent cut to the statutory corporate tax rate from 35% to 21% and a temporary ve-year increase in bonus depreciation for assets with an estimated life less than 20 years -basically any xed capital asset except buildings. 4 ese two provisions constitute the focus of the theoretical analysis carried out later in the paper. Another important provision reduces the ability of businesses to deduct interest payments on debt from their tax base, while the remaining provisions are aimed at re-organizing the tax code in an overall revenue-neutral fashion. 5 e Tax Cuts and Jobs Act was signed into law on December 22 2017, and the vast majority of its provisions became e ective in January 2018.
Aggregate and C-Corporate Response
To assess the response of the US economy to the TCJA-17, I compare actual realizations of macroeconomic and c-corporate variables with pre-reform professional forecasts, and interpret the di erence as the estimated e ect of the reform. 6 One issue with this approach is that the TCJA-17 is not the only shock hi ing the US 4 Bonus depreciation will gradually phase out starting in 2023. 5 Bonus depreciation, together with the newly-introduced pass-through income deduction for the individual income tax, a ect pass-through businesses as well. Goodman et al. (2021) document almost no response of pass-through businesses to the pass-through income deduction, and pass-through businesses tend to be less capital-intensive then c-corporations. us, I abstract from these two pass-through provisions in my main theoretical analysis. 6 e idea behind this exercise is the same as in Kopp et al. (2019) whose focus is on aggregate business investment. economy in 2018. To assess the potential impact of unforeseen shocks, I compute historical forecast errors and use them to construct con dence intervals for the forecasts. ese intervals are directly informative about the errors made by forecasters in the past and, to the extent that these errors re ect unanticipated shocks, the intervals do as well. Another important concern is that pre-reform forecasts might incorporate expectations of an imminent reform, thereby biasing the estimated e ect. To assess the extent of anticipation e ects in pre-reform forecasts, I rst look at the probability of an imminent reform from be ing markets data. Panel (a) of Figure 1 reports the probability of a corporate tax cut from the election of former President Trump to the passage of the TCJA-17. A corporate tax cut was perceived as almost certain in the rst few months a er the election, arguably as a re ection of electoral campaign promises. However, as months went by without any legislative action, the perceived probability decreased to around 30% in the summer of 2017. It then picked up once the rst dra of the TCJA-17 reform bill was introduced into Congress in the fall of 2017, and increased quickly as the bill passed congressional vote and eventually became law in December 2017. Based on the probability from be ing markets, it appears that forecasts made in the summer of 2017 are the least likely to incorporate anticipation e ects. It is possible, however, that be ing market participants' beliefs di er systematically from those of professional forecasters. To mitigate this concern, I examine the dynamic evolution of professional forecasts from IBES in Panel (b) of Figure 1. e plot reports the evolution over time of forecasts of capital expenditure growth for 2018. e series exhibits a strong correlation with be ing market probabilities, which suggest a similar evolution of beliefs between be ing market participants and professional forecasters. While it is not possible to completely rule out anticipation e ects in pre-reform forecasts, it is important to realize that there was no detailed dra of the reform before the fall of 2017, and thus no clear indication of the magnitude and composition of a possible policy intervention. is consideration further mitigates concerns of anticipation e ects.
Results for Macroeconomic Aggregates
Forecasts for macroeconomic aggregates come from the Survey of Professional Forecasters (SPF) and are compared to their NIPA counterparts -except for corporate tax revenues where both actuals and forecasts come from the Update to the Budget and Economic Outlook produced by the Congressional Budget O ce (CBO). e results are reported in Figure 2. Notes: "Forecast" refers to the median forecast in the SPF, and the point forecast made by the CBO. "FE Distribution" adds to the median forecast the 90% central mass of the forecast error distribution, computed by ing a kernel density over the forecast errors over the period 2011-2018. Corporate tax revenues are adjusted by subtracting an estimate for tax collection on repatriated pro t. Repatriation is measured from the BEA Balance of Payment, and the tax revenue adjustment is computed using a 15% tax rate. All values are normalized to 100 in 2017.
e gure shows small stimulus -a couple of percentage points at best -to output, consumption, employment, and investment. Interestingly, the response of non-residential investment appears larger than that of investment, which is consistent with the idea that the macroeconomic response is driven by the investment decision of the productive sector. e loss of corporate tax revenues, instead, is dramatic -especially when they are adjusted to lter out the e ect of pro t repatriation by multinational companies. 7 Since the theoretical framework in this paper features a closed-economy and abstracts from crossborder operations, it is important to have an empirical counterpart that can be used to assess the predictive power of the theory.
Results for C-Corporations
Unfortunately, actuals and forecasts for the c-corporate sector are not readily available. e proposed solution is to aggregate rm-level data from IBES and Compustat and construct measures of economic activity in the c-corporate sector from the micro data. e IBES database contains professional forecasts for large c-corporations. Similarly, Compustat contains detailed information for a large sample of c-corporations. Notes: Perfectly-balanced panel of ≈ 800 rms accounting for ≈ 25% of non-residential investment and ≈ 15% of employment. Forecasts for employment and buyback component of payout are extrapolated using the 2-year growth rate. "FE Distribution" adds to the median forecast the 90% central mass of the forecast error distribution, computed by ing a kernel density over the forecast errors over the period 2011-2017.
Since Compustat contains information on a large number of c-corporations but not forecasts, my strategy is to rst compare actuals with pre-reform forecasts using the IBES dataset, and then compare actuals between IBES and Compustat to assess the representativeness of the former. Figure 3 reports the results for the IBES sample. e stimulus to output, investment and employment is larger for the c-corporate sector than for the aggregate economy. In particular, the response of investment in 2018 exceeds pre-reform forecasts by more than 10%, which is consistent with the idea that the investment decision of c-corporations plays a key role. It is also useful to compare the response of pre-tax income, measured by EBITDA, and a er-tax income, measured by net income. While pre-tax income in 2018 is in line with forecasts, a er-tax income exceeds forecasts because of the reduction in tax-liabilities due to the TCJA-17. Furthermore, the large response of payouts to shareholders -measured as the sum of dividends and share buybacks -suggests that a big share of those tax-savings were transferred to owners of c-corporations. Since the IBES sample is skewed towards large c-corporations, I then compare it to a larger sample from Compustat, and the results are reported in Figure 4. Notes: Perfectly-balanced panel of ≈ 800 rms accounting for ≈ 25% of non-residential investment and ≈ 15% of employment. Forecasts for employment and buyback component of payout are extrapolated using the 2-year growth rate. "FE Distribution" adds to the median forecast the 90% central mass of the forecast error distribution, computed by ing a kernel density over the forecast errors over the period 2011-2017. Compustat aggregate comprises a perfectly-balanced panel of ≈ 5000 rms accounting for ≈ 50% of non-residential investment and ≈ 30% of employment.
e response of c-corporations in the Compustat sample is similar to the IBES sample. One notable exception is a er-tax income, but this is probably due to accounting di erences between the two samples. Overall, the IBES sample covers 25% of aggregate business investment and 15% of aggregate employment, while the Compustat sample covers 50% and 30%, respectively. In the rest of the paper, I consider the IBES sample representative of the population of c-corporations.
C-Corporations vs Pass-rough Businesses
Businesses in the US can choose to operate under one of four major legal forms of organization: sole-proprietorship, partnership, s-corporation and c-corporation. ere are several di erences between them, but what ma ers for this paper is how each legal form is taxed. e rst three forms of organization are pass-through for tax purposes: the business is not taxed directly, but its income is passed through to the owners who are taxed at the individual income level. C-corporations, instead, are taxed directly with the corporate income tax. In 2017, approximately 40% of US economic activity was carried out by pass-through businesses and was not subject to corporate income taxation. Similarly, one fourth of economic activity in the corporate sector was not subject to corporate taxation. ese magnitudes highlight the importance of considering explicitly the pass-through sector when analyzing corporate tax reforms. Panel (b) of Figure 5 shows the evolution of the share of economic activity that is subject to corporate income taxation since the early 1960s. ere are two clear trends. e rst one is the steady increase in pass-through economic activity since the tax reforms of the 1980s. e second one is the rise of c-corporations in the two decades before. While the dynamic evolution of the pass-through sector re ects intriguing technological, legal and tax considerations, a satisfactory analysis of this phenomenon is beyond the scope of the analysis. What this paper emphasizes is that, at any point in time, the aggregate impact of a corporate tax reform depends on the share of economic activity taking place in the pass-through sector, and this shares has experienced large uctuations over the last decades.
e Shi of Activity from Pass-rough Businesses to C-Corporations
I turn to publicly-available business tax returns from the IRS to assess the response of c-corporations and pass-through businesses to the TCJA-17. e results are displayed in Figure 6. Notes: All values are computed from the publicly-available IRS SOI aggregated tax returns. "Output" is measured by "Business Receipts". "Investment", which is not available for sole-proprietorships, is measured by capital expenditure and is computed as "Depreciable Assets" in year minus year − 1 plus "Depreciation" in year . "Income Reported by Individuals" de ned as the sum of "Ordinary Dividends" and " ali ed Dividends" for c-corporations, and as the sum of "Business or Profession Net Income" and "Partnership and S-Corporation Net Income" for pass-through businesses.
e top row compares the response of output, investment and income reported by individuals for c-corporations and pass-through businesses, while the bo om row reports the share of c-corporate activity for each of these variables. Tax returns suggest an expansion of the c-corporate sector relative to the pass-through sector in response to the TCJA-17, and this is especially clear when one looks at the share of activity happening in the ccorporate sector. e decline in the years before the reform is consistent with the 'secular rise' of pass-through businesses, but the trend is reversed in 2018 a er the TCJA-17.
eoretical Framework
is section introduces the theoretical framework and documents its ability to explain the empirical evidence presented in section 2. To illustrate the main mechanism, I introduce a frictionless "baseline model", which is essentially a two-sector neoclassical growth model augmented with tax policy. Despite its simplicity, the model can rationalize the qualitative response of macroeconomic and c-corporate variables to the TCJA-17. I then enrich the baseline model to improve its quantitative t, and use this "extended model" to assess the relative importance of the TCJA-17's two main corporate tax provisions: the tax rate cut and bonus depreciation.
Baseline Model
e model economy is deterministic and populated by a productive sector, a representative household, and a government. e productive sector is further divided into a representative c-corporate sector and a representative pass-through sector. e former is subject to corporate income taxation and distributes its a er-tax cash-ows to its shareholders. e la er is not directly subject to taxation, and its cash-ows are 'passed-through' to its shareholders. In the rest of the paper, variables relating to the pass-through sector will be denoted with a tilde. e representative household solves the following optimization problem:
max +∞ ∑︁ =0ˆ1 − 1 − . .ˆ= ·˜1 − +˜+ Δ +1 + Δ˜+ 1˜= (1 − ) · +˜˜+ +˜˜ + Transfer +˜= 1, = ,˜=Λ + , ≡ · (ˆ+ ) (ˆ)
where is consumption of goods from c-corporations,˜is consumption of goods from pass-through businesses, andˆis a consumption bundle constructed using a Cobb-Douglas aggregator. e good produced by the c-corporate sector is the numeraire, and is the (relative) price of the good produced by pass-through businesses. e household supplies labor inelastically to each sector, and receives wages equal to and˜each period. She also invests in shares of each sector, that trade at prices and˜. 9 Ownership of the 9 In equilibrium, the supply of each type of shares will be xed and normalized to one. productive sector entitles the household to dividends from c-corporations, and passthrough income˜from pass-through businesses. Finally, the household pays individual income taxes and receives transfers from the government. For simplicity, I assume that there is a uniform individual income tax rate on labor income, dividends and passthrough income. 10 Finally, the household's intertemporal marginal rate of substitution Λ , + will be used by the productive sector when making intertemporal decisions.
To be er understand how corporate tax reforms a ect the economy, I impose as much symmetry as possible between c-corporations and pass-through businesses. Each sector accumulates its own capital stock through investment, hires labor competitively, and produces a nal good using a constant return-to-scale technology. However, only ccorporations pay corporate income taxes.
C-Corporations
max +∞ ∑︁ =0 Λ 0, . . = − = − − +1 = (1 − ) + = · 1− = · = − − Pass-rough Businesses max +∞ ∑︁ =0 Λ 0,. .˜== ·˜−˜˜− ·+ 1 = (1 −˜)˜+=˜˜· 1−C
orporate income taxes are computed by multiplying the corporate income tax base by the statutory corporate income tax rate . e corporate income tax base di ers from corporate cash-ows because investment is usually not treated as an expense, but is deducted according to a tax depreciation schedule. 11 As a result, a fraction of present and past investment is deducted from the tax base each period, and this represents the investment deduction allowed by the tax code.
In general, the investment deduction for a generic period is given by:
= +∞ ∑︁ =0 · −
where the policy parameters { } +∞ =0 represent the percentage of investment from periods ago that can be deducted from the tax base. Investment is eventually deducted from the tax base in full, so that the policy parameters sum up to one. To improve tractability and build intuition, I approximate the tax depreciation schedule using a declining-balance tax depreciation schedule, which permits the aggregation of all non-depreciated past investment into an auxiliary variable . 12 e investment deduction can then be rewri en as
= · ( + ) where +1 = (1 − ) · ( + )
e auxiliary variable represents the stock of past investment that has not been depreciated for tax purposes yet, and is now the only policy parameter summarizing the tax depreciation schedule. 13 In this way, the corporate tax code is fully summarized by the pair ( , ). To close the model, I introduce a government that collects tax revenues that can go into wasteful spending or into transfers to the representative household:
= + = · Transfer = (1 − ) ·
where are individual income tax revenues, are total tax revenues, and is wasteful spending. e parameter ∈ [0, 1] determines the share of tax revenues that go into wasteful spending. When = 0, all tax revenues are distributed back to the representative household.
12 Winberry (2021) adopts the same approximation. In Furno (2021), I show that the error due to this approximation is negligible in standard economic environments. 13 Notice that = · (1 − ) .
e Investment Decision and the Tax Bill of C-Corporations
In the baseline model, the investment decision of the c-corporate sector is driven by the following Euler Equation:
1 = Λ , +1 1 − +1 1 − ≈1 · (1 − ) + 1 − 1 − "Corporate Tax Wedge" · +1 where = +∞ ∑︁ =0 Λ , + · (1 − ) ·
PDV of tax depreciation schedule e distortion to the investment decision introduced by the corporate tax code shows up in the form of a wedge, that I label as the "corporate tax wedge".
is wedge is jointly determined by the statutory tax rate and the present discounted value of the tax depreciation schedule. is result mirrors Hall and Jorgenson (1967), and can be thought of as an extension to general equilibrium thereof. 14 A higher value of re ects a more accelerated tax depreciation policy, which in turn implies that both and the corporate tax wedge are closer to one. As a result, even when the statutory tax rate is high, the distortions to the investment decision can be small if tax depreciation policy is highly accelerated. e tax rate and tax depreciation policy also determine the tax bill of c-corporations:
= · − − +∞ ∑︁ =0 · (1 − ) · −
However, changes to the corporate tax code do not a ect the investment decision and the tax bill in the same way. It is possible -and this is key to understand the TCJA-17 -to conceive a corporate tax reform that leaves the corporate tax wedge almost unchanged, while producing a big change to the corporate tax bill.
Calibration to the US Economy before the TCJA-17
I calibrate the model to the US economy in 2017, just before the TCJA-17. Several parameters -such as the discount rate, the household's IES, economic depreciation and the labor 14 is happens because the baseline model is a neoclassical model. In general, when the economic environment is enriched with frictions, it is not possible to summarize the distortions to the investment decision in such a clear-cut way.
share -are standard. I calibrate labor supply and the exponents of the Cobb-Douglas consumption aggregator to match the relative size of the c-corporate and pass-through sectors. Mimic a debt-nanced tax cut e tax code is calibrated as follows. e corporate tax rate is set equal to the statutory corporate tax rate. e tax depreciation rate is set in such a way that it matches the present discounted value of a representative tax depreciation schedule computed using the same methodology proposed in Zwick and Mahon (2017). is present discounted value averages tax depreciation schedules for di erent types of capital assets, and includes the 50% bonus depreciation that was in place in 2017 -see subsection A.1 for the details. e individual income tax rate is set equal to the average e ective tax rate computed from publicly available individual income tax returns from the IRS. Finally, in order to mimic a debt-nanced tax cut, I assume that all tax revenues are transferred back to the representative household by se ing = 0. Table 3 shows that the calibrated model's deterministic steady-state is able to reproduce four important empirical moments: corporate pro ts, dividends, corporate tax revenues and individual income tax revenues as a share of GDP. ese moments are not explicitly targeted by the calibration, but the model can match them well because the way the variables are de ned in the model is a good approximation of what happens in practice. Matching these four untargeted moments ensures that the size of the corporate sector and of the government's tax collection in the model is representative of the US economy before the TCJA-17.
e TCJA-17: Model vs Data
e Tax Cuts and Jobs Act of 2017 is simulated by starting from the calibration in Table 2 and introducing an unanticipated permanent change to the following policy parameters:
• A permanent reduction in the corporate tax rate from 35% to 21%. • A permanent increase in the tax depreciation rate from 0.4823 to 0.8305. e change to the tax depreciation rate increases the present discounted value of the representative tax depreciation schedule in steady-state from ≈ 0.94 to ≈ 0.99. 15 While the TCJA-17 increased bonus depreciation only temporarily, US policy-makers have repeatedly extended expiring bonus depreciation over the last couple of decades. It is not unreasonable to believe that bonus depreciation will be extended upon expiration, which justi es the assumption of a permanent change. Importantly, since the increase in bonus depreciation only applies to new investment, I introduce auxiliary variables to distinguish between old and new investment for tax purposes -see subsection A.2 for details.
e results from the model are presented and compared to the empirical evidence in Figure 7. e rst column describes the response estimated in the data, and the second column the response from the model. e rst row focuses on macroeconomic aggregates, and the second on c-corporate ones. e model successfully reproduces the relative responses of aggregate and c-corporate variables estimated in the data. At the aggregate level, the model predicts a small response of output and investment, and a large fall in corporate tax revenues. Moreover, the response of investment is larger than that of output. At the c-corporate level, the model predicts an increase in payouts to shareholders larger than investment -in line with the data. Again, the response of investment is larger than that of output.
e intuition behind what happens can be broken down into two pieces. e rst piece clari es the response of c-corporations. Because of highly accelerated tax depreciation policy before the TCJA-17, the pre-reform corporate tax wedge was close to one (≈ 0.97 under the proposed calibration). As a result, the ability of the reform to further remove distortions was very limited in the rst place, and ended up providing li le stimulus to c-corporate investment. At the same time, the tax-savings due to the reform were large, and c-corporations found themselves with a sizable amount of additional cash. Given their limited desire to increase investment, they distributed a big share of this extra cash to their shareholders. e second piece of intuition helps understand the even smaller response at the aggregate level. On the one hand, given a large share of pass-through businesses, the corporate provisions in the TCJA-17 applied to only 60% of the productive sector (measured in terms of economic activity). On the other hand, the remaining 40% was not only not stimulated, but was in fact put at a competitive disadvantage relative to prior the reform, which produced a shi of economic activity from pass-through businesses to c-corporations. Overall, this resulted in further dilution of the aggregate stimulus.
Improving Fit: An Extended Model
e baseline model can reproduce the overall pa ern of macroeconomic and c-corporate responses, but is not able to o er a good quantitative t for the response of some of the variables. In particular, Figure 8 shows that the response of output and investment for ccorporations is smaller than in the data. is is partly due to the assumption of exogenous labor supply, which reduces the ability of c-corporations to respond to the stimulus by hiring more workers. To improve the quantitative t, I alter the baseline model in three ways. First, I endogenize labor supply and assume it is mobile across the two sectors. Second, I assume a more general CES consumption aggregator for the representative household. ird, I assume variable capital utilization. e additional parameters are calibrated in a standard way and the details can be found in subsection A.3. e "extended model" response is given by the green lines in Figure 8. Endogenous labor supply that can move across the two sectors facilitates re-allocation of economic activity across sectors. Similarly, a CES consumption bundle allows household's spending to shi towards the goods produced by c-corporations -which are now relatively cheaper. Finally, variable capital utilization ampli es the response of c-corporate output as it gives an additional margin of adjustment to the c-corporate sector.
3.4.2 Decomposing the TCJA-17: Tax Rate Cut vs Bonus Depreciation I use the "extended model" to perform a counterfactual assessment of the importance of each of the two main corporate provisions in the TCJA-17, and the results for c-corporate investment and corporate tax revenues are reported in Figure 9. First, the expansionary e ect of each provision on the investment of c-corporations is similar, as both are aimed at removing distortions to the investment decision. Second, the interaction between these two provisions is negative. A cut to the corporate tax rate is more expansionary when the present discounted value of the tax depreciation schedule is lower. Similarly, the e ect of bonus depreciation is larger when the tax rate is higher. By reducing the tax rate while accelerating the depreciation schedule, the two provisions partially o set each other. ird, the e ect of these two provisions on corporate tax revenues is similar on impact, but is di erent in the long-run. A reduction of the tax rate produces a permanent loss of corporate tax revenues. An acceleration of the tax depreciation schedule, instead, results in a transitory one.
Trump vs Kennedy
is section compares the recent Trump's TCJA-17 with the Kennedy's corporate tax cuts of the early 1960s through the lens of the theoretical framework proposed in the previous section.
e Kennedy's tax cuts were implemented between 1962 and 1965. e Revenue Act of 1962 introduced a 7% investment tax credit for businesses. In the same year, the IRS also issued a new set of more accelerated tax depreciation guidelines.
e Revenue Act of 1964 then reduced the top individual tax rate from 91% to 70%, reduced individual tax rates across brackets, created the standard deduction, and reduced the corporate tax rate from 52% to 48%. In this section, I focus on the three corporate income tax provisions and follow Romer and Romer (2010) in classifying them as debt-nanced. 16 1960 1961 1962 1963 1964 1965 1966 Year 80 While it is di cult to obtain estimates of the e ects of the Kennedy's tax cuts due to 16 For additional details on the Kennedy's tax cuts see Greenberg et al. (2016). data availability, the time-series of investment and payouts to shareholders reported in Figure 10 reveal an interesting pa ern. e increase in payouts to shareholders outweighs the increase in investment a er the recent TCJA-17, but not a er the Kennedy's tax cuts.
is can be observed both at the aggregate and at the c-corporations level. In fact, payouts does not appear to deviate much from the pre-reform trend, unlike investment which exhibits a clear acceleration.
e increase in capital formation is gigantic: aggregate business investment is 50% higher in 1966 than in 1963, and c-corporations' investment is 80% higher.
Model-Implied Corporate Tax Multipliers
I use the "extended model" to assess the e ects of each reform on GDP, aggregate investment and payouts to shareholders. By construction, the counterfactual experiment explains di erent macroeconomic outcomes through pre-existing di erences in the corporate tax code, in the size of the pass-through sector, and in the composition of the policy intervention.
e TCJA-17 is simulated in the same way as before. e Kennedy's corporate tax cuts are simulated as follow. I start from the calibration for 2017 and adjust the corporate tax rate, the tax depreciation rate, and the weights of the CES consumption aggregator to match corporate tax policy and the pass-through share in 1961. I then simulate the Kennedy's tax cuts as unanticipated permanent changes to the following policy parameters:
• A permanent reduction in the corporate tax rate from 52% to 48%. • A permanent increase in the tax depreciation rate from 0.10 to 0.1857.
As for the TCJA-17, the new tax depreciation rate applies only to new investment and further details can be found in subsection A.2. e results are reported in Figure 11. In response to the Kennedy's tax cuts, the model predicts a large increase in GDP and investment, and a small e ect on payouts to shareholders: the opposite of Trump's Tax Cuts and Jobs Act. Similarly, the corporate tax multiplier for Kennedy's tax cuts is around 2.5 for GDP, 1.85 for investment, and close to zero for payouts to shareholders. For the TCJA-17, the multiplier is around 0.6 for each variable. For every dollar of lost corporate tax revenues, the Kennedy's corporate tax cuts stimulated GDP four times more than the TCJA-17. e intuition behind these results is the following. In the early 1960s, the corporate tax rate was high and tax depreciation policy was not accelerated as it was mimicking economic depreciation. As a result, the corporate tax wedge was well below one (around 0.72) before the reform. e Kennedy's tax cuts increased the wedge signi cantly (to around 0.84), thus providing strong stimulus to the investment of c-corporations. Moreover, since around 75% of economic activity was taking place in the c-corporate sector, the aggregate e ect was less diluted than in 2017. To be er understand how each factor (i.e. tax rate, tax depreciation, pass-through share, policy intervention) contributed to the outcomes reported in Figure 11, I perform another counterfactual experiment. First, I control for di erences in the policy intervention by simulating the exact same reform in both 1961 and 2017: an unanticipated permanent reduction in the corporate tax rate by 10%. en, I start from the calibration for 2017 and simulate the reform a er changing one of the tax rate, tax depreciation rate and pass-through share at a time. So, for example, I take the calibration for 2017, set the tax depreciation rate equal to that in 1961, and simulate the reform. I repeat the same for the tax rate and the pass-through share. e results are reported in Figure 12. e exercise shows that di erences in tax depreciation policy between the early 1960s and 2017 account for most of the di erence in the macroeconomic response to the reform. Looking at long-run changes, di erences in pre-reform corporate tax rates and in the prereform share of pass-through businesses contribute similarly to the di erence between the two reforms. e interaction between these three factors, instead, can be assessed by looking at the di erence between the rst and the second vertical bar for each variable. For example, under the 1961 calibration, the long-run investment response is +14.24%, while the response under the 2017 calibration with each factor introduced at a time is only +8.39%, which implies an interaction e ect of +5.85%. For the corporate tax multiplier, interaction e ects appear to be smaller. Moreover, the multiplier is una ected by the size of the pass-through sector. is happens because a smaller pass-through sector implies larger aggregate stimulus a er a corporate tax cut, but also a larger loss of corporate tax revenues -since a larger share of the economy receives the tax cut. e multiplier takes into account both e ects, which almost perfectly o set each other in this speci c experiment.
GDP
Analytic Insights
is section o ers analytic results to build intuition about speci c aspects of the theory. To obtain these results, I start from the "baseline model" and assume away pass-through businesses and individual income taxes by se ing = = 1 and = 0. With these restrictions, I recover a neoclassical growth model featuring a corporate tax levied on the entire productive sector.
Since it is common in macroeconomics to think about corporate taxation as a form of capital taxation, I rst provide a formal mapping between capital taxes and corporate taxes. I show that corporate taxes are isomorphic to a capital tax accompanied by investment subsidies. I then prove that, in the steady-state, it is possible for the government to collect corporate tax revenues and remove distortions to investment at the same time. Finally, I derive the analytic steady-state of the model and use it to characterize corporate tax distortions over the last few decades in the US.
Corporate vs Capital Taxes
It is possible to relate the proposed corporate tax with the familiar concept of a "capital income tax", i.e. a tax imposed on the income produced by the productive factor "capital". Under a constant return-to-scale technology, it is possible to unambiguously de ne capital income using Euler eorem: (1 − ) − which clari es that a corporate tax in this framework is isomorphic to a capital tax accompanied by a set of subsidies to current and past investment. e size of the subsidies is dictated by the tax depreciation schedule. To x ideas, consider now the special case where = 0, i.e. investment is not deductible from the tax base: the investment deduction is zero, and the corporate tax becomes equivalent to a capital tax. Consider then the special case where = 1, i.e. investment is fully and immediately deductible from the tax base: the corporate tax is equivalent to capital tax accompanied by a subsidy to current investment at the same rate.
Corporate Tax Revenues
Under full-expensing of investment (i.e. = 1), the corporate tax wedge becomes one and the distortion to the Euler Equation for capital accumulation disappears. It is interesting to see whether the corporate tax can actually collect revenues in such a case. Notice that, in light of the equivalence pointed out in subsection 5.1, this is fundamentally the same question asked in Abel (2007). It is possible to prove that the corporate tax can collect tax revenues in the steady-state even with full-expensing of investment. Consider the corporate tax base in steady-state:
= · − Since = ,1 = 1 − + which implies that − = 1 − 1 ≡
where > 0 is the rate of time preferences. erefore > 0 and the corporate tax can collect revenues in a non-distortionary fashion.
Corporate Tax Distortions over Time
It is possible to solve analytically for the deterministic steady-state of the model and express it as a function of its 'undistorted' counterpart, i.e. the deterministic steady-state in the absence of corporate taxation (i.e. when = 0). Long-run output ( ) can be expressed as
= * · 1− where = 1 − 1 − and = (1 + ) +
Undistorted long-run output is given by * , and is the corporate tax wedge in steadystate. Notice that is the present discounted value of the tax depreciation schedule in steady-state. In this frictionless environment, distortions to production are summarized by the corporate tax wedge -properly adjusted for capital-intensity. Interestingly, corporate tax revenues and payouts to shareholders depend on the tax code in a more complicated way: = * · · 1− · (1 + · (1 − )) = * · (1 − ) · 1− · (1 + · (1 − )) and this duality further clari es that the corporate tax code can di erentially a ect incentives and cash-ows. For convenience, I then de ne the following measure of long-run distortions to output Distortion = 1 − * and represent it in the corporate tax policy space in Figure 13. e gure displays a contour map of 'isodistortions' for each combination of the corporate tax rate ( ) and the present discounted value of the tax depreciation schedule ( ). Red dots representing the corporate tax code in di erent years are superimposed to assess the evolution of corporate tax distortions over time.
e exercise reveals a steady elimination of distortions by US policy-makers over time, captured by the movement towards the south-east corner of the map. For example, output was roughly 16% lower than its undistorted counterpart before the Kennedy's tax cuts, but only 1.7% lower before the TCJA-17. is improvement have been achieved through several rounds of statutory tax rate cuts, changes to tax depreciation rules, and repeated use of bonus depreciation over the decades. 15.9% (1961) 8.12% (1980) 2.38% (2002) 1.7% (2017) 0.29% ( While the numbers reported in the gure should be taken with a grain of salt, they teach two important lessons. On the one hand, corporate tax policy has become less distortionary over time. On the other hand, policy-makers are now running short of ammunition. Given that the current level of distortions is almost zero, further reductions of the statutory corporate tax rate and/or acceleration of the tax depreciation schedule will produce li le stimulus to output and investment.
Long-Run Output Distortion
Conclusions
is paper has focused on tax depreciation policy and the distinction between c-corporations and pass-through businesses to understand the e ects of major corporate tax reforms in the US. However, these two elements are not speci c to the US and can be found in basically every corporate tax code around the world. is implies that the analysis presented can be easily replicated and extended to other countries. Moreover, while the proposed theoretical framework is intendedly stylized in order to make the transmission mechanism as robust and transparent as possible, it can be enriched along several dimensions. For example, two candidates are the introduction of sectoral heterogeneity and the analysis of corporate debt and of the interest-payment deduction. Preliminary results suggest that these two extensions do not alter the overall predictions of the model, but they do allow the theory to generate additional implications for di erent sectors or for corporate leverage.
is could be of interest on its own, or could be used to discipline the theory further by exploiting empirical evidence from the cross-section of rms or of industries. e theoretical framework has implications for time-series exercises as well. e fact that distortions to investment and tax-liabilities are di erentially a ected by a corporate tax reform, implies that neither the tax rate nor the tax-liabilities changes fully summarize corporate tax shocks. Arguably, both the corporate tax wedge and the change to tax-liabilities should be introduced in an empirical speci cation to properly estimate the causal e ect of a reform. Finally, the empirical analysis carried out in this paper highlights the need for further data collection by legal form of organization. While this is not necessary for many research questions, it becomes essential whenever business taxation needs to be taken into account, directly or indirectly.
H , R. E.
A er the tax cuts, they estimate a PDV of the depreciation schedule equal to 0.726 in 1965, which is accompanied by an investment tax credit equal to 0.0657. e investment tax credit can be introduced by simply increasing the PDV of the tax depreciation schedule, which becomes 0.7917. e associated tax depreciation rate is = 0.1857.
A.2 Simulating Bonus Depreciation on New Investment
To capture the fact that bonus depreciation applies to new investment -as opposed to past investment not depreciated yet -I introduce auxiliary variables. Let , and , be the tax depreciation rate and the stock of un-depreciated investment before the reform. Let , and , be the same variables a er the reform. Finally, let take value equal to one a er the reform and equal to zero before. I then rewrite the investment deduction as follows = , · (1 − ) · + , + , · · + , where , +1 = (1 − , ) · (1 − ) · + , , +1 = (1 − , ) · · + , is modeling strategy ensures that -a er the reform -past investment that has not been depreciated yet can still be depreciated using the old depreciation schedule, while new investment is depreciated using the new depreciation schedule.
A2
A.3 Extended Model Details e 'extended model' starts from the 'baseline model' and introduces: 1) endogeneous labor supply that is mobile across sectors; 2) a CES consumption aggregator; 3) variable capital utilization in the c-corporate sector. e representative household solves the following optimization problem: A3 e new parameters introduced are , 0 , 1 , 2 , and . I set = 4, which implies a Frisch elasticity of 0.25. e steady-state economic depreciation for c-corporations is given by 0 = 0.10 since I set 1 = 1 − (1 − 0 ) = 0.1638. e parameter 2 is set equal to 0.10 to target a steady-state elasticity of depreciation to utilization of approximately 0.60, which is basically the mid-point between the values in Basu and Kimball (1997) and King and Rebelo (1999). I set = 0.33 to target an elasticity of substitution between the goods produced by the two sectors of approximately 1.5%.
is implies some substitutability between the two varieties. Given , I use to calibrate the target the relative size of c-corporations. I set = 0.55 for the TCJA-17, and = 0.70 for the Kennedy's tax cuts.
Figure 1 :
1Perceptions of a Corporate Tax Reform before the TCJA-17 Source: IBES, PredictIt.
Figure 2 :
2Response of Macroeconomic Aggregates to the TCJA-17
Figure 3 :
3Response of C-Corporations Aggregates to the TCJA-17
Figure 4 :
4Response of C-Corporations Aggregates to the TCJA-17: IBES vs Compustat
Figure 5 :
5of the Pass-rough Sector Panel (a) of Figure 5 o ers a decomposition of US economic activity in 2017 by legal form of organization. e Size of the Pass-rough Sector in the US Economy Notes: Economic activity is measured by "Business Receipts" from publicly-available aggregated tax returns from IRS SOI. Data before 1980 have been manually collected from scanned version of SOI's Business Income Tax Return Reports and Corporation Income Tax Return Reports.
Figure 6 :
6e Shi of Economic Activity from Pass-rough Businesses to C-Corporations
Figure 7 :
7e TCJA-17: Model vs Data Notes: Empirical moments are computed as the di erence between the actual realizations and the pre-reform forecasts from section 2. e empirical response of corporate tax revenues is adjusted to eliminate the e ect of pro t repatriation.
Figure 8 :
8antitative Fit of the Baseline and Extended Model
Figure 9 :
9Decomposing the TCJA-17: Tax Rate Cut vs Bonus Depreciation
Figure 10 :
10Investment and Payouts for Kennedy's and Trump's ReformsNotes: Macro aggregates come from the NIPA, while data for c-corporations from Compustat. For the Kennedy's tax cuts, a perfectly-balanced sample of ≈ 760 c-corporations accounts for ≈ 35% of business investment. For the TCJA-17, a perfectly-balanced sample of ≈ 5000 c-corporations accounts for ≈ 50% of business investment.
Figure 11 :
11e TCJA-17 vs Kennedy's Corporate Tax CutsNotes: e long-run change is computed as the 20-year cumulative deviation from the steady-state, obtained by summing the level of each variable for 20 years a er the reform and dividing it by its counterpart in the absence of the reform. e corporate tax multiplier is computed as the cumulative change in the level of each variable and divided by the cumulative change in corporate tax revenues.
Figure 12 :
12Understanding the Di erence between the TCJA-17 and Kennedy's Reforms
Since the labor market is competitive, = . By Euler eorem, − = . Writing the investment deduction as a function of current and past investment, the corporate tax base becomes:
Figure 13 :
13Corporate Tax Distortions over Time Notes: Values for 1961 and 1980 are computed from Cummins et al. (1994). Values for 2002, 2017 and 2021 are computed from Zwick and Mahon (2017). e only two parameters used are = 0.94 and = 0.35.
Transfer = (1 − ) ·
Table 1 :
1Corporate Tax Provisions in the TCJA-17 Source: JCT Conference Report for H.R.1.Provision
Static Revenue Change ($bln)
2018-2020
Corporate Tax Rate from 35% to 21%
−357.1
Bonus Depreciation Allowance from 50% to 100%
−93.6
Interest-Deduction Cap
+45.8
Small Business Reform (e.g. Section 179)
−34.6
Additional Changes to Deductions
+35.9
Changes to Loss Treatment
+27.5
AMT Repeal
−20.3
Changes for Insurances, Banks and Fin Instruments
+16.7
Changes to Business Credits
+2.1
Changes Accounting Methods
+5.6
Table 2 :
2Calibration of the Baseline ModelParameter Value
Notes
0.94
Rate of time preferences
1
IES
=˜0.10
Physical depreciation rate
=˜0.35
Labor share (= 0.65)
0.575 C-Corps share of salaries and wages
0.575
C-Corps share of business receipts
0.35
Statutory Corporate Tax Rate
0.4823
Tax Depreciation Rate
0.135
Average e ective tax rate
0
Table 3 :
3Fit of Key Untargeted MomentsNotes: Model (SS) refers to the deterministic steady-state of the model. Data comes from NIPA and span the period 2012-2017. Corporate pro t and dividends in the NIPA refer to both c-corporations and s-corporations, thus slightly over-estimating the value for c-corporations alone.Moment Model (SS) Data
/
0.08
0.10
/
0.05
0.05
/
0.03
0.02
/
0.10
0.08
the tax base can be rewri en as erefore, corporate tax revenue collection is positive if − > 0. In steady-state, the Euler Equation for capital accumulation becomes=
· (
− )
For example, see Zwick and Mahon (2017), Ohrn (2018), and Ohrn (2019). From a theoretical perspective, the importance of tax depreciation policy is known at least since Hall and Jorgenson (1967). 2 Several recent contributions have documented some of the implications and issues arising from this passthrough status. For example, see Cooper et al. (2016), Clarke and Kopczuk (2017), Chen et al. (2018), Smith et al. (2019), Barro and Wheaton (2020), Kopczuk and Zwick (2020), Bhandari and McGra an (2021), Smith et al. (2021).
For real-time assessments of the Tax Cuts and Jobs Act see Tax Foundation (2017), Barro and Furman (2018), Mertens (2018), Gale et al. (2018), Kopp et al. (2019).
To construct repatriated pro t I follow Smolyansky et al. (2019) and use a 15% tax repatriation rate to perform the adjustment.
Owners of c-corporations are also taxed through the dividend tax once corporate income is distributed, and through the capital-gains tax if they realize a capital gain thanks thanks to a share price increase.
In practice, dividends are taxed at a preferential rate, there are numerous deductions and exemptions, and there are tax brackets. Since my main theoretical experiments will involve changing the corporate tax rate while leaving the individual income tax rate unchanged, a uniform individual income tax rate will preserve my main conclusions. 11 In reality, rms use a mix of capital assets to produce their nal goods, and each asset category is potentially subject to a di erent tax depreciation schedule. erefore, the capital stock in the model should be interpreted as a representative business capital, and the tax depreciation schedule as a representative tax depreciation schedule for business capital.
I allow for 90% bonus depreciation, instead of 100%, to take into account the fact that the TCJA-17 placed some restrictions on asset eligibility. See subsection A.1 for the details.
In their paper, the de nition of PDV of depreciation deductions scales the PDV of the tax depreciation schedule by the statutory tax rate. For instance, they report a value of 0.3366 for 1960, which becomes 0.647 a er dividing by a statutory tax rate equal to 0.52.
D.W. J (1967): "Tax policy and investment behavior, " e American Economic Review, 57, 391-414.K , R. G. S. T. R (1999): "Resuscitating real business cycles, " Handbook of macroeconomics, 1, 927-1007.K,W. E. Z (2020): "Business incomes at the top, " Journal of Economic Perspectives, 34, 27-51.Appendix A Modeling DetailsA.1 Calibration of the Tax Depreciation ScheduleTo calibrate the tax depreciation schedule I choose the policy parameter so that the present discounted value (PDV) of the tax depreciation schedule in the steady-state of the model matches an empirical counterpart from the existing literature. Given a discount rate , the PDV of the tax depreciation schedule in steady-state is given bye tax depreciation rate that produces a given PDV in steady-state is given byTCJA-17To calibrate in 2017, I build on Zwick and Mahon (2017). I start from their cross-sectoral average of the investment-weighted PDV of MACRS depreciation rules. ey estimate a PDV for this object of 0.879. I then add an existing 50% bonus depreciation and compute the new PDV as follows:0.50 + (1 − 0.50) × 0.879 = 0.939 e associated is equal to 0.4823. To calibrate the new value of a er the TCJA-17, I increase bonus depreciation from 50% to 90%. is implies a PDV increase from 0.9395 to 0.9879, and a new value of = 0.8305.Kennedy's Tax CutsTo calibrate tax depreciation policy before and a er the Kennedy's tax cuts I followCummins et al. (1994). ey estimate a PDV of the tax depreciation schedule for equipment of 0.647 in 1960. 1 is is almost the PDV under economic depreciation, so I set = = 0.10.
Macroeconomic e ects of the 2017 tax reform. A , R J J , Brookings papers on economic activity. Optimal capital income taxationA , A. B. (2007): "Optimal capital income taxation, " . B , R. J. J. F (2018): "Macroeconomic e ects of the 2017 tax reform, " Brook- ings papers on economic activity, 2018, 257-345.
Taxes, incorporation, and productivity. B , R J B , Tax Policy and the Economy. 34B , R. J. B. W (2020): "Taxes, incorporation, and productivity, " Tax Policy and the Economy, 34, 91-111.
Cyclical productivity with unobserved input variation. B , S M S K ; B, A E R , e arterly Journal of Economics. 136Sweat equity in US private businessB , S. M. S. K (1997): "Cyclical productivity with unobserved input varia- tion, " . B , A. E. R. M G (2021): "Sweat equity in US private business, " e arterly Journal of Economics, 136, 727-781.
Business income and business taxation in the United States since the 1950s. C , D , S Q , D , American Economic Journal: Macroeconomics. C , C. W. K10Tax Policy and the EconomyC , D., S. Q , D. S (2018): "Corporate income tax, legal form of orga- nization, and employment, " American Economic Journal: Macroeconomics, 10, 270-304. C , C. W. K (2017): "Business income and business taxation in the United States since the 1950s, " Tax Policy and the Economy, 31, 121-159.
Business in the United States: Who Owns It, and How Much Tax Do ey Pay?. C , M , J P , R P , J S , D Y , O Z , E , Tax Policy and the Economy. 30C , M., J. M C , J. P , R. P , J. S , D. Y , O. Z , E. Z (2016): "Business in the United States: Who Owns It, and How Much Tax Do ey Pay?" Tax Policy and the Economy, 30, 91-128.
A reconsideration of investment behavior using tax reforms as natural experiments. C , J G , K A H , R J , Brookings papers on economic activity. C , J. G., K. A. H , R. G. H , R. E. H , R. J. C (1994): "A reconsideration of investment behavior using tax reforms as natural experiments, " Brookings papers on economic activity, 1994, 1-74.
Tax Depreciation Schedules: Facts and Modeling. F , F , Tech. repF , F. (2021): "Tax Depreciation Schedules: Facts and Modeling, " Tech. rep.
A preliminary assessment of the Tax Cuts and Jobs Act of 2017. G , W , H G , A K , M J M , E , National Tax Journal. 71G , W., H. G , A. K , M. J. M , E. T (2018): "A preliminary assessment of the Tax Cuts and Jobs Act of 2017, " National Tax Journal, 71, 589-612.
How Do Business Owners Respond to a Tax Cut? Examining the 199A Deduction for Pass-through Firms. G , L , K L , B S , A , National Bureau of Economic Research. Tech. rep.G , L., K. L , B. S , A. W (2021): "How Do Business Owners Respond to a Tax Cut? Examining the 199A Deduction for Pass-through Firms, " Tech. rep., National Bureau of Economic Research.
Modeling the Economic E ects of Past Tax Bills. G , S , J O , S J , Tax Foundation, Fiscal Fact. 527G , S., J. O , S. J. E (2016): "Modeling the Economic E ects of Past Tax Bills, " Tax Foundation, Fiscal Fact, 527.
|
[] |
[
"Software Licensing in the Cloud Age Solving the Impact of Cloud Computing on Software Licensing Models",
"Software Licensing in the Cloud Age Solving the Impact of Cloud Computing on Software Licensing Models"
] |
[
"Malcolm Mcroberts [email protected] \nEnterprise Architecture Core Technology Center\nSan Francisco State University\nCAU.S.A\n\nHarris Corporation\nGCSD Melbourne\nFloridaUSA\n"
] |
[
"Enterprise Architecture Core Technology Center\nSan Francisco State University\nCAU.S.A",
"Harris Corporation\nGCSD Melbourne\nFloridaUSA"
] |
[
"JSCSE]"
] |
Cloud computing represents a major shift in information systems architecture, combining both new deployment models and new business models. Rapid provisioning, elastic scaling, and metered usage are essential characteristics of cloud services, and they require cloud resources with these same characteristics. When cloud services depend on commercial software, the licenses for that software become another resource to be managed by the cloud. This paper examines common licensing models, including open source, and how well they function in a cloud services model. It discusses creative, new, cloud-centric licensing models and how they allow providers to preserve and expand their revenue streams as their partners and customers transition to the cloud. The paper concludes by identifying the next steps to achieve standardized, "cloud-friendly" licensing models.
|
10.7321/jscse.v3.n3.60
|
[
"https://arxiv.org/pdf/1401.5346v1.pdf"
] | 17,354,194 |
1401.5346
|
4f401ab31ddbdbbe099063220ba6acbd151c1dce
|
Software Licensing in the Cloud Age Solving the Impact of Cloud Computing on Software Licensing Models
2013
Malcolm Mcroberts [email protected]
Enterprise Architecture Core Technology Center
San Francisco State University
CAU.S.A
Harris Corporation
GCSD Melbourne
FloridaUSA
Software Licensing in the Cloud Age Solving the Impact of Cloud Computing on Software Licensing Models
JSCSE]
33201310.7321/jscse.v3.n3.60395software licensingcloud computingopen sourcestylingelastic scalingintellectual propertycomplianceSaaS
Cloud computing represents a major shift in information systems architecture, combining both new deployment models and new business models. Rapid provisioning, elastic scaling, and metered usage are essential characteristics of cloud services, and they require cloud resources with these same characteristics. When cloud services depend on commercial software, the licenses for that software become another resource to be managed by the cloud. This paper examines common licensing models, including open source, and how well they function in a cloud services model. It discusses creative, new, cloud-centric licensing models and how they allow providers to preserve and expand their revenue streams as their partners and customers transition to the cloud. The paper concludes by identifying the next steps to achieve standardized, "cloud-friendly" licensing models.
I. INTRODUCTION
Software licensing is a major part of IT service delivery. For server software, license fees alone can easily be three or four times the cost of server hardware. In addition, understanding the licensing options offered by vendors and how license terms apply to your operation, and then negotiating the price are all complex and time-consuming tasks. The management of these licenses, including auditing compliance, is a driving factor in IT policy, processes, and architecture.
Cloud computing promises lower costs, improved utilization, increased flexibility and extreme scalability. However, these same features change the software licensing landscape in a disruptive way. Some of these challenges are new and some are magnified by deployment in the cloud.
Organizations that have embraced virtualized infrastructure on a large scale have already experienced the complex and diverse world of licensing on virtual machines. Although based on virtualization, cloud computing represents a more dynamic IT infrastructure and a shift in roles. While it is not possible to capture all the complexity and nuance of this evolving field in a short paper, we can identify key issues that require the attention of the industry as a whole.
A. Characteristics of Cloud Computing (Benefits)
According to the National Institute of Standards and Technology (NIST) [1], cloud computing is a model for enabling ubiquitous, convenient, on-demand network access to a shared pool of configurable computing resources (e.g., networks, servers, storage, applications, and services) that can be rapidly provisioned and released with minimal management effort or service provider interaction.
This translates to a number of benefits for large and small organizations alike, as well as for individuals. Some of the main drivers for cloud adoption are discussed below: 1) Over/under provisioning Traditional IT resources represent a significant investment and long lead times to acquire and make operational. This not only applies to hardware but also to software. Sizing these resources becomes a dilemma. As shown in Fig. 1, sizing for peak loads results in excess capacity and poor utilization during off-peak periods, while sizing for nominal loads, as shown in Fig.2, will not be able to service peak demands.
By provisioning resources dynamically from large shared pools, cloud services can add or remove capacity on demand and only charge for resources actually used. This makes it possible to automatically scale resources (within minutes) to match the actual demand, as shown in Fig. 3. This type of auto-scaling requires fully automated provisioning, including any licensed software required. 2) Agility Cloud computing can greatly enhance the ability of small companies and small projects within large companies to develop and deploy new applications quickly and cheaply. This agility comes from the ability of projects or individual developers to rapidly provision and control the resources needed to develop, test, integrate, and deploy their applications.
Control typically includes network configuration, virtual machine life cycle, and even root access to guest operating systems. Public or community clouds are ideal for small companies; larger companies may choose to provide cloud services internally.
Virtualized infrastructure is often equated with cloud computing by corporate IT leadership. Through the use of virtualization, mature IT organizations can often provide relatively rapid (hours/days) provisioning of development resources, whereas, in a cloud, users can provision their own resources in minutes via APIs and web portals. Typically, when IT is responsible for provisioning development resources, it retains control of the resources providing developers with the limited access, in contrast to the delegated control provided by a true cloud. Central management of these resources becomes problematic, and most organizations end up with large virtual machine junkyards.
The benefits of prototyping in the cloud are so great that IT organizations must either embrace this new approach or be bypassed by shadow IT in the public cloud (software licensing issues will be addressed in subsequent sections).
3) Budgeting for software costs A traditional IT system typically requires a large, upfront capital investment (CapEx) that is amortized over several years and does not provide value until the system is operating at capacity. Cloud computing provides the ability to avoid purchasing infrastructure and instead pay for the cloud service as an operational expense (OpEx). As a result, a business conserves capital much like it would by leasing its office space. In addition, cloud expenses grow in proportion to the capacity (load) of the system.
The downside of this is that the scale is not fixed up front, and cloud charges can grow more rapidly than expected, a problem if not tied to a proportionally growing revenue stream.
B. Cloud Threats to Commercial Software
Cloud computing represents a more dynamic and flexible approach to providing IT services. It involves new distributed architectures, new ownership and control models, and new acquisition and pricing mechanisms. The disruptive influence of cloud computing on software licensing as been discussed in [2] and [3].
Traditional licensing models for commercial software products often make it difficult or impossible to use these products in the cloud. Independent software vendors (ISVs) that fail to adopt more flexible licensing schemes risk losing customers to other vendors. The more significant risk for vendors is that customers will move to completely open source products and avoid the issue entirely, as the most successful Internet companies have already done.
The flip side is that organizations struggling to adopt cloud computing or even large-scale virtualization are finding that software licensing is holding back their efforts. Some of the problems encountered are discussed below: 1) On-demand provisioning Cloud computing allows users to rapidly provision servers via APIs or GUI tools. Prior to cloud computing, users had to request resources through their IT organization. This allowed IT to ensure that software was properly licensed, and to perform server-specific activation. In the cloud, software licenses must either be automatically provisioned and managed by the cloud, or subscribers must provide their own, often a manual process.
2) Elastic scaling Clouds provide support for systems that automatically scale up when demand increases. Many products support the notion of a network pool of licenses, which is normally based on purchasing a fixed capacity (see Fig.1 and Fig. 2). In cloud bursting, scaling may involve more than one cloud infrastructure provider, further complicating license management.
3) Short-term resource (license) allocation Clouds can support both long-running, static deployment as well as short-term, task-based workloads. Resources can be provisioned, used, and then released all in the span of a few minutes. Most licensing schemes are not designed to handle this type of usage pattern. 4) Shared resource pools (multi-tenancy), delegation The pricing of many corporate software agreements is based on the organizational affiliation of the software user. Public and community clouds service many tenants (subscribers) from large, shared pools of resources. A cloud provider can only supply licenses if the license allows delegation to their tenants.
5) Compliance, indemnification and shadow IT
Software license compliance, along with security compliance, is a key factor driving centrally managed IT. While some software products provide their own license enforcement mechanisms, it is ultimately the responsibility of the user's company to ensure license compliance and maintain auditable compliance records. License agreements typically include the right of the vendor or representative to conduct audits and levy fines. SIIA [4] and BSA [5] are leading organizations that promote the active policing of license compliance. Large fines and even jail time for executives can result from infringement.
While cloud computing promises to allow users to provision and control their own resources, releasing them from the constraints of their IT group, it may also bypass IT's role in managing license compliance (and cyber security).
Even shops with centrally managed virtual infrastructure find that if users can install software themselves, their corporate license footprint can rapidly outpace their license agreement. Getting internal users to release licenses they no longer need is an ongoing challenge, as is getting them to release virtual machines (VMs).
A bigger threat to compliance comes from shadow IT. Projects that perceive corporate IT as too restrictive, expensive, slow, etc., often resort to obtaining resources outside corporate governance. In the past, this involved building labs with small hardware purchases, and using unauthorized network connections. Now anyone can create-in minutes-shadow IT in the cloud which is not tracked by corporate IT. Since most companies have no sanctioned mechanism for procuring cloud services and most cloud providers have a bring-your-own-license (BYOL) policy, users and their companies may be put at risk. 6) License enforcement Many products provide their own license enforcement mechanism; however, most of these are problematic for use in the cloud. These mechanisms may depend on features difficult to provide in the cloud, such as hardware keys (dongles), physical server IDs, CPU class, and global user identity. Where schemes can be implemented on one cloud, they cannot generally span multiple clouds (hybrid model).
7) Adoption-led acquisition models Traditional software licensing has evolved alongside traditional acquisition models. In this model, trades are performed and products are selected and purchased early in the development of a system. Changing out products can result in substantial loss of investment and impact schedules, and is only considered in extreme cases.
Many organizations are now opting for an adoption-led approach to acquisitions [6] and expect to license on a trybefore-you-buy basis. For instance, a company might deploy several different web meeting tools and license one or more depending on usage levels after a trial period.
Time-limited evaluation licenses attempt to address this need, but still have a large-scale, traditional license as the end goal.
8) Open source
Software licensed under open source models as defined by the Open Source Initiative [7] is in widespread use throughout the industry and continues to grow. This includes large shares in areas such as operation systems, databases, application servers, and hypervisors. Many factors drive the adoption of open source, including the cost and complexity of commercial licenses, the proliferation of third-party companies offering training and support services, and freedom from vendor lock-in.
Market leaders in cloud computing and Internet-based services in general have constructed their offerings almost exclusively from a combination of their own software and open source offerings. Open source allows these companies to start small and grow fast.
The shift to cloud computing only serves to highlight the advantages of open source licensing. 9) Commodity pricing Pricing and terms for commercial software are typically negotiated between an ISV and the individual licensee, based on each party's negotiating leverage. The trivial reproduction costs for software allow vendor account representatives tremendous freedom to set terms, using special pricing to lock in large accounts and effectively block use of competing products while integrating their products into the customer's operations and encouraging product-specific investment in training, custom tools, etc.
Larger customers are able to negotiate better pricing, reduce costs, and increase their competitiveness. Favorable reseller agreements directly impact the cost of the reseller's own products. Long-term, enterprise-wide standardization on a product enables corporate IT to develop streamlined support processes and avoid disruption in service.
The natural view of software in the cloud is as just another resource, which is billed at a standard hourly rate like CPU or storage. This threatens to undermine the power base of both the vendor and customer and introduces a degree of separation between them.
C. Rights versus Revenue
Software development requires a substantial investment of both time and money. Copyright laws are designed to encourage investment by giving creators the legal right to profit from their creation. Once a product has been developed, the focus should be on maximizing revenue generation. Restrictive license agreements and rigid license enforcement mechanisms may do a good job of protecting a vendor's intellectual property rights, but they also may negatively impact sales. Dongles are among the mechanisms that have been rejected by software customers.
Open source software licenses are designed to allow the software to be used by anyone free of charge, but most open source software projects are sponsored by commercial firms.
These firms have successful business models that are not dependent on the enforcement of license agreements.
A license scheme, pricing model, and enforcement mechanism should be designed to maximize revenue, not exclusively to protect property rights.
II. CLOUD SERVICE MODELS AND ROLES
Cloud computing is a rapidly evolving space that does not fit cleanly into a simple model. A wide variety of physical deployments, network topologies, business models, and software architectures are possible, including those yet to be conceived. Much like mobile phone service requires a complex network of service provides and vendors, cloud services are also made up of many parts. For cloud computing to mature, it must achieve the same level of industry standardization and consumer simplicity as the mobile phone industry.
A. NIST Models
NIST has led the way in developing a standard taxonomy for cloud computing [1], defining essential characteristics, service models, and deployment models. The NIST service model consists of three layers: infrastructure, platform, and software. All three layers are involved in delivering value to customers, but all may not take the form of a service. Each service layer may involve licensed software, thereby incurring license charges.
1) Infrastructure as a Service (IaaS) IaaS consists of CPU, storage, networking, and operating systems and supporting services provisioned on demand and delivered via the web. IaaS typically uses a hypervisor to provide virtual machine instances.
2) Platform as a Service (PaaS) PaaS is the least understood model. The platform consists of frameworks and services that host or support application software, but not the user-facing applications themselves. Databases, application servers (middleware), and business service APIs are common part of a PaaS.
3) Software as a Service (SaaS) In this model, users run applications and consume content over the network (Internet or intranet), much as they would use software installed on their personal devices. Some of the largest consumer content providers host their services on public cloud infrastructure.
B. Roles
The service layers previously described can all be provided by a single organization, or services can be layered on top of services from different providers, creating a type of supply chain or service map. Each step in the chain is defined by a service agreement that may involve software licensing. This perspective differs from the NIST taxonomy, which is focused primarily on security and systems engineering. Roles include: 1) Subscriber A subscriber is the entity that contracts for cloud services. These services may be consumed by the subscriber's IT systems or be directly used by its staff, partners, or customers.
2) End user An end user is the entity who uses software or content delivered over the network. Cloud services ultimately derive value by delivering service to end users.
3) ISV An ISV is a firm that develops proprietary software with the intent to license the software at a profit. ISVs typically have a portfolio of related products and supporting services, such as training and integration.
4) SaaS provider
The SaaS provider delivers application and content to end users over the network.
5) PaaS provider
The PaaS provider hosts applications developed by others and provides tools and frameworks for use in building applications.
6) IaaS provider The IaaS provider delivers servers (usually virtual) with operating systems, storage, and network connectivity on demand over the Internet. 7) Identity provider An identity provider allows users to establish a networkwide identity and access resources from multiple service providers based on a single authentication. Identity is used to manage access to personal data, billing, and user-based software/content licensing.
C. Relationships
The aforementioned roles are common to most cloud services, but the relationships between them determine software ownership and licensing patterns.
NIST defines four deployment models: public, community, private, and hybrid. Rather than defining these strictly in terms of network topology, we define them in terms of the relationship between the service provider and the subscriber. In a public cloud, the service is vended as a commodity on the open market. Any customer (subject to legal boundaries) can subscribe for the service. Public cloud providers may license their own proprietary software as part of their service in a pay-as-you-go (PAYG) model. The provider may also act as an ISV, licensing the same software outside of their cloud. Public cloud providers can supply licensing for third-party software or require subscribers to BYOL. In any case, subscribers are ultimately responsible for their own license compliance.
In a community cloud, the subscribers come from a community based on existing relationships with the provider and other subscribers. Such an organization may subsidize the cloud service, negotiate communitywide licenses, or even sponsor software development. They may also operate private networks to provide secure communication channels within the community.
In a private cloud, the provider and subscriber belong to the same organization, and services are typically only provided in the organization's intranet. In most cases, subscribers are not charged for service, and often usage is not even metered. Private clouds are usually operated by corporate IT, who is also accountable for its company's license compliance.
D. Portablity -Hybrid Cloud
Hybrid clouds combine private and public cloud services. The hybrid cloud can handle workload spikes by augmenting private cloud resources with resources from the larger public cloud. Cloud industry standards bodies have been working on the problem of workload portability between clouds, and commercial tools and services are available to manage hybrid clouds. But workload portability does not guarantee license portability. In practice, license terms and management tools differ between private and public clouds.
III. TRADITIONAL SOFTWARE LICENSE MODELS
This section discusses the most common types of commercial software licenses and issues arising from applying these in a cloud-based environment. This topic has also been discussed in [14].
A. Host ID-Based Model
In this scheme, one or more server hardware components are queried to generate a unique host ID. The customer obtains a license key from the vendor that is tied to the host ID. This scheme does not work well for cloud-based systems for several reasons. Most clouds are based on virtualization, meaning that the hardware is abstracted (hidden) from the cloud tenants. Even if a host ID can be generated for the physical device, the virtual server may be migrated to a different hardware as a function of cloud management.
Clouds are also designed to support automated provisioning and elastic scaling, which are not possible if the software vendor must be contacted each time a host is provisioned or migrated.
B. Token-Based Model
In this scheme, a physical license key, such as a dongle or copy-protected CD, must be physically attached to the host running the software. The advantage for workstation users is that the key can be easily moved between machines. Since clouds and most virtualized infrastructures do not allow client physical access to servers, this kind of scheme is generally not usable.
C. Named User Model
In this scheme, a license is tied to a specific user. That user is licensed to use the software on any device and, in some cases, multiple devices at the same time. Determining if a user is licensed for the software requires that the user authenticate against an identity provider in the domain where the software will run. Traditionally, licenses are tied to a company's internal directory service, but cloud-based services may utilize Internet identity providers. Services that deliver music and videos typically use this type of license. Managing identity-based licenses across domains, such as in a hybrid cloud, may be difficult.
Named user licenses are often sold to a company as a pool from which they can then assign and transfer as needed.
D. Concurrent Users Model
In this scheme, the customer purchases a pool of licenses, which users check out as they would (concurrent) copies of a library book. The definition of concurrency can depend on the type of software involved. A compiler, a word processor, and an application server may have very different concepts of concurrent use.
License check-in/checkout is usually automatic, tied to a web session or the start and stop of an application. This is usually implemented via application hooks that call out to a license manager service. Unfortunately, most software provides its own license manager, making consolidated management difficult. This type of licensing typically works the same in the cloud as in a private network, provided that the license terms permit this. In multi-tenant cloud services, it may be desirable to subdivide the license pool to enforce quotas on tenants. This may not be supported by license manager software. Among traditional license schemes, this is probably the best fits for the cloud computing paradigm.
E. Hardware Capacity-Based Model
The amount of work (e.g., transactions) that a software system can perform in a given time is bounded by the hardware on which it runs. This type of license equates software value with computing power. The formulas used to calculate power for licensing purposes are somewhat arbitrary and vary from vendor to vendor. This type of license does not usually provide any automated enforcement.
Optimum use of this type of license requires that hardware be scaled to fit expected load and dedicated for use by a single software package. Dedicating physical hardware to a specific application is contrary to the principle of cloud computing.
Fortunately, many vendors provide formulas for virtual machines and specific cloud services. IBM's bring your own software and license (BYOSL) strategy assigns power units to each type of Amazon EC2 instance. While this type of license supports use in the cloud, the allocation of licenses is still usually static.
F. Site-Wide Model
Site licenses are negotiated based on the general size of the customer site. The premise is that the site corresponds to a local or campus area network and the software can be freely installed and used on this network. Cloud computing explicitly hides physical site and network boundaries making this type of license a bad fit for cloud computing in general. Even private clouds usually support users from any location as long as they are on the private network, making this equivalent to an enterprise-wide license.
G. Enterprisewide Model
In this model, an ISV licenses its software to an entire enterprise. This type of license typically covers installation and use of software by employees of the enterprise on equipment owned by the enterprise. Exceptions may include employee-owned devices, known as bring your own device (BYOD) [9], use by subcontractors and employee home use.
Enterprise licenses are not intended to be used by service providers to resell the software to their customers. Some vendors have recognized this need and have provided licenses specifically for service providers [10].
While cloud providers may offer pay as you go licensing, large customers that have favorable enterprise license terms may prefer to use their own license. Some cloud providers and ISVs have partnered to provide explicit support for BYOL [11] [12].
H. "Commercial" Open Source Model
Open source software [7] may be copied and used by anyone free of charge. In spite of this, there is a large market for firms providing open source software. These firms do not sell licenses to software that is already free, but they do provide services that enterprise customers find valuable including:
1) Packaging, certification, accreditation Vendors providing these services take snapshots of multiple open source baselines, integrate them, test them, and certify them. In some cases they may also obtain security accreditations needed for use in government systems.
2) Indemnification Complex software products often embed components from multiple sources. The vendor evaluates all the licenses that apply to the integrated product and "indemnifies" the customer from any legal liability from embedded products.
3) Security patches (subscriptions) The security posture of any IT systems depends on the timely identification and installation of critical security patches. Vendors provide subscription services to ensure that systems are up to date. 4) Support Some vendors provide Tier 3 technical support, training, and consulting service, just like a large ISV. In the case of open source software, there may be multiple companies competing to provide support for the same software.
Commercial open source is sold under terms similar to those for commercial software. Since the price of this software tends to increase with the level of use, it may become more cost effective for an organization to provide these services internally. This would generally apply to service providers, although subscribers may choose to purchase software in this way to use in the cloud.
I. Free Open Source Model
Rather than pay commercial providers for open source, companies can obtain the software in source or binary form directly from open source project sites. This type of licensing is ideal for cloud service providers as well as subscribers. There are no restrictions on who may use the software, number of users, location, or size of hardware. Adopters of the free open source model take on the added responsibilities of tracking and integrating new releases, training and maintaining knowledgeable support staff, and ensuring the quality and security of the software they are using. Community support forums take the place of Tier 3 product support, but can often be equally responsive and helpful.
Leading internet companies have embraced open source both as users and as contributors. In order to prosper, commercial ISVs must provide unique features and exceptional quality to justify their cost, while also adopting licensing models that scale up and down for use in the cloud.
J. Ownership -Copyright Holder Model
Leading cloud service providers typically use a mixture o open source software and software from in-house development projects. Companies that hold the software copyrights either through creation or acquisition can use the software in any way they choose. Also ISVs may choose to become service providers to give their customers the option to pay for their software in the form of a service.
IV. CLOUD-CENTRIC LICENSE MODELS While the issues described earlier still pervade the industry, this section addressed cloud-centric licensing models currently in use by leading cloud providers.
A. Amazon Web Services -DevPay
Amazon has always been and continues to be a pioneer in the cloud computing market space. This extends to Amazon's innovative billing service, DevPay [13]. DevPay allows software vendors of any size to deliver their software as a service within the Amazon cloud and use DevPay to manage access and billing for that service. Users of DevPayenabled software must have an Amazon account and are billed for their software use by Amazon, much as mobile providers bill for applications. Software charges are normally computed as a surcharge on Amazon resources used, including virtual machine instances, storage, and bandwidth. Users can view rates and track charges on Amazon's dashboard. Software providers may also restrict access to registered customers. This mechanism is enabled by adding a few simple web service calls into the application software. This is especially attractive for small developers who can generate revenue, while avoiding the complexity of licensing and billing.
Amazon uses this same mechanism to bill for some of their higher value, platform services, which are built on top of the same infrastructure resources.
B. IBM SmartCloud
IBM Smart Cloud is a service based on hourly charges for virtual machine instances. IBM has chosen to lease a variety of products from their own middleware portfolio on a pay-as-you-go model. Machine images with these products and their hourly rates are listed in the Smart Cloud catalog. In addition, to encourage development in the Smart Cloud, development use only (DOU) licenses are available for select products free of charge.
C. Microsoft Azure
The Microsoft Azure cloud provides a platform for deploying custom applications as well as a host of services based largely on Microsoft products that have been engineered to work in the Azure cloud. Most of Microsoft's software products are available as services on Azure.
D. iTunes -Apple ID
Apple is a pioneer in end-user computing devices ranging from Macs to iPhones. The appeal of these devices is largely driven by their integration with Apple's cloud service. Apple has also pioneered the idea of software and content that is not bound to a device, but rather to a user identity in the cloud. For Apple, this is the user's Apple ID, which provides access to content and software belonging to that user from any compatible device. This is similar to intranet, named user licenses, but is global in scope, since it is based on a global ID and an Internet-based cloud service.
V. STEPS TOWARD STANDARDIZING CLOUD-FRIENDLY LICENSING Cloud computing is revolutionizing the way organizations pay for and use their IT resources. This paper has shown that while cloud computing has the potential to simplify the licensing and use of software, it has, in fact, only added to the problem.
For commercial software vendors to successfully move into the cloud age, they must now work as a group with cloud providers to standardize licensing in the cloud. Standards-developing organizations (SDOs) should govern the activity. A successful solution must address legal and financial concerns, as well a technical aspects of software licensing in the cloud. To do this effectively, the solution should provide the following:
A. Consistency and Confidence
Customers need to be able to use licensed software with confidence that their use will comply with the license; that the costs are understood, fair, and predictable; and that licensing is consistent between products and between cloud providers. Customers should be able to use software without resorting to guesswork or lawyers.
B. Inter-Cloud Portablity
Cloud industry standards bodies have been working toward standards to allow data and workloads to move seamlessly between clouds, both public and private. Software licenses must migrate seamlessly as well.
C. Protection of Intellectual Property
Cloud license agreements and technical standards must provide ISVs with the mechanisms they require to track the use of their IP, audit license compliance, collect fees and prevent software piracy.
D. License Model Options
ISVs must be able to choose from a range of standardized license models, the one that best fits their product. These would include iTunes and DevPay style licenses.
E. Pricing Flexibility
Service providers and subscribers should have the option to negotiate software pricing with ISVs or agents, or to pay list. Service providers that provide licenses should give subscribers the option to use their own.
F. Ability of Service Providers to Perform as Agents
ISVs must be able to delegate license management (tracking and enforcement), accounting and billing to service providers, based on standardized mechanisms. These mechanisms should support pass through to other service providers.
G. Automation through Standard APIs
Service providers need standard APIs to automate license management and make license management an integral part of their resource management systems. ISVs require standard APIs to allow use of their licensed software to be transparently managed by any service provider. APIs are also needed to support pass through and migration between service providers.
Once the industry has defined what makes up a "cloudfriendly" software license, the final step is to provide endorsements, much like OSI does for open source licenses. This endorsement would allow providers and consumers alike to select and use products based on functionality and price, without concern for the subtleties of license terms.
Figure 1 .Figure 2 .Figure 3 .
123Over-Provisioned Resources for Peak Demand The International Journal of Soft Computing and Software Engineering [JSCSE], Vol. Resources Provisioned for Nominal Demand Cloud Resource Auto-Scaling
Special Publication 800-145. The NIST Definition of Cloud Computing. National Institute of Standards and Technology (NIST)National Institute of Standards and Technology (NIST), Special Publication 800-145, "The NIST Definition of Cloud Computing" URL: http://csrc.nist.gov/publications/nistpubs/800-146/sp800- 146.pdf
Leah Gabriel Nurik, Cloud Computing Disrupting Software Licensing, Pricing. Channel InsiderLeah Gabriel Nurik, "Cloud Computing Disrupting Software Licensing, Pricing", Channel Insider, 2010-10-22
. Adam Stone, ; To, Software Cloud, Licensing Proves A Stumbling, Block, NextGov. Adam Stone, "IN MOVING TO THE CLOUD, SOFTWARE LICENSING PROVES A STUMBLING BLOCK", NextGov, February 28, 2011 URL:http://www.nextgov.com/cloud-computing/2011/02/in-moving- to-the-cloud-software-licensing-proves-a-stumbling-block/48596/
. Software and Information Industry Association. Software and Information Industry Association URL: http://siia.net/
URL. Business Software Alliance, Software Piracy and the Law. Business Software Alliance, "Software Piracy and the Law" URL:http://www.bsa.org/country/Anti- Piracy/Know%20the%20Law.aspx
Software Licensing In The Cloud. Stuart Charlton, Stuart Charlton, "Software Licensing In The Cloud", CloudWorld 2009, Aug 13, 2009 URL: http://www.slideshare.net/slideshow/embed_code/1857667
The Open Source Definition. Open Source InitiativeOpen Source Initiative, "The Open Source Definition" URL: http://opensource.org/osd
Cloud-enabling your software licenses. Joaquin Gamboa, Marc Lindsey, ComputerWorld. Joaquin Gamboa and Marc Lindsey, "Cloud-enabling your software licenses", ComputerWorld, September 8, 2008 URL: http://www.computerworld.com/s/article/323517/Cloud_enabling_Yo ur_Software_Licenses
Will BYOL (Bring Your Own License) Cripple BYOD?. Peter Silva, 4PETER SILVA, "Will BYOL (Bring Your Own License) Cripple BYOD?", Virtualizaton Journal, JULY 4, 2012 URL: http://virtualization.sys-con.com/node/2305543
. Microsoft Service Provider License Agreement URL. Microsoft Service Provider License Agreement URL: http://www.microsoft.com/hosting/en/in/licensing/splabenefits.aspx
. Amazon Relational Database Service Pricing URL. Amazon Relational Database Service Pricing URL: http://aws.amazon.com/rds/pricing/
. Amazon Devpay, Url , Amazon DevPay URL: http://aws.amazon.com/devpay/
. Dave Roberts, Gigaom, Dave Roberts, GIGAOM, Jul 8, 2012 http://gigaom.com/cloud/dont- let-your-cloud-app-become-a-software-licensing-hostage/
|
[] |
[
"Assessing Inconspicuous Smartphone Authentication for Blind People",
"Assessing Inconspicuous Smartphone Authentication for Blind People"
] |
[
"Diogo Marques [email protected] \nUniversity of Lisbon\nUniversity of Lisbon\nUniversity of Lisbon\n\n",
"Luís Carriço \nUniversity of Lisbon\nUniversity of Lisbon\nUniversity of Lisbon\n\n",
"Tiago Guerreiro \nUniversity of Lisbon\nUniversity of Lisbon\nUniversity of Lisbon\n\n"
] |
[
"University of Lisbon\nUniversity of Lisbon\nUniversity of Lisbon\n",
"University of Lisbon\nUniversity of Lisbon\nUniversity of Lisbon\n",
"University of Lisbon\nUniversity of Lisbon\nUniversity of Lisbon\n"
] |
[] |
As people store more personal data in their smartphones, the consequences of having it stolen or lost become an increasing concern. A typical counter-measure to avoid this risk is to set up a secret code that has to be entered to unlock the device after a period of inactivity. However, for blind users, PINs and passwords are inadequate, since entry 1) consumes a non-trivial amount of time, e.g. using screen readers, 2) is susceptible to observation, where nearby people can see or hear the secret code, and 3) might collide with social norms, e.g. disrupting personal interactions.Tap-based authentication methods have been presented and allow unlocking to be performed in a short time and support naturally occurring inconspicuous behavior (e.g. concealing the device inside a jacket) by being usable with a single hand. This paper presents a study with blind users (N = 16) where an authentication method based on tap phrases is evaluated. Results showed the method to be usable and to support the desired inconspicuity.
| null |
[
"https://arxiv.org/pdf/1506.00930v1.pdf"
] | 9,178,870 |
1506.00930
|
fb2d1d0e5ba1cf63bbd3612c1eadf4a082ea394a
|
Assessing Inconspicuous Smartphone Authentication for Blind People
Diogo Marques [email protected]
University of Lisbon
University of Lisbon
University of Lisbon
Luís Carriço
University of Lisbon
University of Lisbon
University of Lisbon
Tiago Guerreiro
University of Lisbon
University of Lisbon
University of Lisbon
Assessing Inconspicuous Smartphone Authentication for Blind People
Author Keywords Tappinginconspicuous interactionsmartphonesusable securityblind users ACM Classification Keywords H52 [Information interfaces and presentation]: User Interfaces -Input devices and strategies
As people store more personal data in their smartphones, the consequences of having it stolen or lost become an increasing concern. A typical counter-measure to avoid this risk is to set up a secret code that has to be entered to unlock the device after a period of inactivity. However, for blind users, PINs and passwords are inadequate, since entry 1) consumes a non-trivial amount of time, e.g. using screen readers, 2) is susceptible to observation, where nearby people can see or hear the secret code, and 3) might collide with social norms, e.g. disrupting personal interactions.Tap-based authentication methods have been presented and allow unlocking to be performed in a short time and support naturally occurring inconspicuous behavior (e.g. concealing the device inside a jacket) by being usable with a single hand. This paper presents a study with blind users (N = 16) where an authentication method based on tap phrases is evaluated. Results showed the method to be usable and to support the desired inconspicuity.
INTRODUCTION
The growing capabilities and increasing adoption of smartphones is transferring privacy and security risks from our desks to our pockets. A typical way to mitigate these risks is to set up an authentication barrier that has to be overcome after a period of device inactivity. The unlocking procedure comes, however, with costs. Unlocking takes time, which must be multiplied by the number of occasions one pulls the device from the pocket. Additionally, authenticating creates an interruption that obstructs a primary objective, e.g. reading an email, thus potentially causing frustration. Finally, mobile authentication is particularly susceptible to observation attacks -a third party discerning one's secret code -since devices are often used outside controlled environments like home or office.
To blind users, authenticating with touch-based smartphones raises even more usability and security barriers. Let us consider using a PIN with a screen reader, such as iPhone's VoiceOver 1 . As the user passes its fingers through the screen, a voice reads out the key. When the right key is found, the user can split or double tap it to select. Azenkot et al. [2] found that even in a sample of blind users familiarized with this technology, unlocking took in excess of 7 seconds, which may be too cumbersome for large-scale adoption. Furthermore, it is susceptible to observation: the process of selecting each digit is visible to bystanders if a method to hide the screen is not used (e.g., iPhone's screen curtain). This can be particularly troublesome if we consider situations where blind people are less aware of their surroundings than sighted people. Aural eavesdropping is also a risk -it is conceivable that blind people could use headphones, but that would not only require time to put them on but also reduce awareness of surroundings. Finally, limitations imposed by social norms should also be considered -using visible authentication mechanisms can be embarrassing or raise fears of being perceived as distrustful of others.
In this paper, we contribute with a study on inconspicuous authentication, based on tap phrases, for blind people. Besides feasibility and performance, we report strategies mentioned by participants on how these methods could be used to mitigate security threats.
RELATED WORK
Tap-based authentication has been previously addressed in the literature. Bianchi et al. [2] propose using taps to insert PINs recurring to the number of touch events. PassChords [1] is specifically targeted to blind users. In this system, multi-finger touches, using a chord analogy, are used. These works do not allow unbounded tap pattern lengths, nor take into consideration the tap and break time spans, instead being focused on tap counts and location. The first severely reduces the tap phrases selection space. The second is prone to screen smudge detection and most notably limits the space for inconspicuous interaction strategies. For example, it does not afford operations with a single hand, rendering the most accepted concealing strategies impossible. Forcing authenticating in conspicuous ways draws attention to the use of assistive technology, potentially leading to feelings of self-consciousness [7].
In TapSongs [8] users can configure tapping patterns representing songs. However, it uses a parametric approach, requiring at least 5 examples for configuration. For gestures, Li [4] points out that, only very few examples should be needed for subsequent accurate recognition. For taps, the case is even more evident. Using many examples and parametric approaches is prone to create a build-up of noise that prevents keeping the success rates and/or performance low, especially in low-end devices. Moreover, it discourages the configuration of long and securer codes. Also, to our knowledge, TapSongs has not been implemented in smartphones or similar devices, turning the user studies less realistic.
Ghomi et al. [3] attempt to generalize rhythmic authentication into a more general input method based on rhythm, making use of both taps and breaks. In this work the types of words that can exist are bounded to 5 -three varieties of tap and two of break, varying in canonical duration. Again the tap phrase selection space is limited. In our previous work [5], we devised a similar approach but enabling sequences of unbounded length and requiring only one template to record them.
Tap-based authentication has been previously presented as a method that enables non-visual usage. Still, particularly methods that allow concealing and one-hand usage, they have not been evaluated with blind people. It is paramount to understand how blind people cope with these tapping methods as well as their acceptance and concealing strategies.
USER STUDY
To validate tap phrase unlocking as a feasible approach to address the threats posed by traditional authentication, an exploratory user study with 16 blind participants was conducted. We addressed both the experience of using tap patterns for unlocking and the affordance of this method to inconspicuous interaction. Additionally, we extracted the patterns chosen by the users to develop an efficient matching algorithm, which we describe in section 5.
Specifically, our objectives were to understand if, after a short learning period, users could: 1) Perform authentication with a single hand easily and in a reasonable amount of time; and 2) confabulate strategies for inconspicuous authentication.
Authentication method
The tap-based authentication method used in this study was the one presented in [5]. This method uses tapping sequences on a touch screen. The Hamming distance-based matching approach accommodates rich tapping patterns of unbounded length, taking into consideration both the times where the user is pressing and the ones in which she releases. Additionally, it tries to match every pattern candidate as the user interacts with the device, thus dispensing timeouts for successful authentication.
Materials
For the experiments, a single Samsung Galaxy mini smartphone, with Android 2.3, was used. The prototype application for tap unlocking was used. A training mode was available in which data was not recorded and optional sound output (emission of a tone while the screen was being touched) was available. On data-recording mode, there were facilities for configuring a tap pattern template and afterwards to attempt unlocking, both without audio feedback. A short vibration was emitted on successful unlock. Success was determined by two factors: 1) the input and template having the same number of taps and 2) the input having total time span no more or less than 20% that of the template. This crude algorithm, not suited for authentication, was used so that users could have a more realistic understanding of the system. Logs were generated for offline analysis.
Paper questionnaires were employed to gather demographic data, register responses to the Single Ease Question (SEQ), and record concealment strategies suggested by participants. The SEQ, proposed by Sauro and Dumas [6], is known to have good psychometric properties, and was administered immediately after the task ("Overall, this task was?", Very Easy=7, Very difficult=1).
Participants
The 16 participants were volunteers recruited in a local training institution for blind people. Two participants had some residual vision. Ages varied between 26 and 64 years old, the average being 47 (SD = 12). Twelve participants were male and 4 female. Although all participants had mobile phones, they reported having none or very little experience with touch-screen devices. Eleven reported being very familiar with using PINs in electronic devices, albeit in physical keyboards. Participants reported never or rarely using headphones paired with their mobile devices.
Procedure
Participants were initially introduced to the concepts and explained the tasks they were asked to perform. At this stage, they were given no mention that the tap unlocking was method being proposed by the researchers. They were handed a device to feel and get accustomed to while being administered a short demographic questionnaire.
In a first stage, a training session lasting approximately 5 minutes was conducted, in the following steps:
1. Users freely explored the touch-screen area with their fingers. When they touched any point in the screen, an audio tone was emitted. Participants were explained that the whole screen acted as a single button and explored the fact that tap phrases are composed of taps and breaks lasting in time.
2. Users were asked to imagine tap phrases that they could record and later use for unlocking. They experimented freely, with sound enabled, until they were confident that they had grasped the concept.
3. Users were introduced to the vibrotactile feedback emitted on authentication success (short) and failure (long).
4. With sound output now removed, users conducted a complete dry-run, first configuring a template of their choice, then attempting to unlock.
5.
Step 2 was repeated, this time participants using the device with a single hand.
After training, still using a single hand, participants were asked to again configure a template and then try to unlock. This time, the interaction was recorded in log files. Immediately after completing this task, users responded to the Single Ease Question.
In the second stage of this study, participants were reminded of the observation threat and asked to imagine strategies they could use to conceal the input from potential observers. To facilitate this process, participants engaged in role-playing two scenarios: a meeting and a commute in public transportation. To that end, they were given a smartphone so they could simulate authentication. A facilitator gravitated at times around the participant to make him aware of possible visual observation angles. In the end, participants were asked to summarize the viable strategies they had identified.
Measures
For the unlocking task, we acquired:
1. The time it took to complete authentication with a single hand, measured from the moment a facilitator clicked a start button and initiated the unlocking application to the moment an input was accepted as the correct secret code;
2. The number of input errors, and;
3. The SEQ rating, from 1 to 7.
For the elicitation part of this study, the strategies indicated by participants were recorded and occurrences counted. Since the alternatives mentioned were clear and not very numerous, no special categorization was performed.
Results
After training, all users were able to authenticate in the first trial, so there were no input errors to record. Figure 1 presents the recorded tap phrases along with the attempt to replicate it for all 16 users.
The mean task completion time was 4.32s, with standard deviation 2.1s. A Shapiro-Wilk test suggests that the data is normally distributed (S-W = .890, df = 16, p = .056). A one-sample t test was conducted at an alpha level of .05 to evaluate if tap unlocking is faster than the 7.52s mean for PIN with VoiceOver found by Azenkot et al. [1]. The test has shown to be significant (t = -6.062, df = 15, p = .000, Cohen's d = -1.52), suggesting that unlocking with taps is faster than the current mainstream method (Voice Over).
The median SEQ score was 6 (IQR = 3), where 7 means "very easy" and 1 "very difficult", indicating that participants perceived tap unlocking as being easy to perform.
In the second stage of our study, we elicited inconspicuous authentication strategies through role-playing. The usersuggested approaches are summarized in Table 1. Each user contributed, on average, 3 strategies (SD = 1). The top suggestions, with 9 occurrences, were performing authentication under the table or inside a pocket.
Discussion
The results for task completion time and perceived easiness of authenticating with tap phrases are encouraging. Even so, the relatively large standard deviation in task completion time deserves a closer look. From our observations, there are two possible explanations for this fact: 1) some users, lacking the confidence and experience using smartphones, operated the device with an unusual level of caution, thus taking more time and 2) there may be, in fact, an extended initial period where a blind user needs to situate himself before starting tapping with confidence.
Strategy
Occurrences
Under the table 9
Inside pocket 9
Occluded by clothes (e.g. jacket) 7
Cover with one hand 5
Lean device against body 3
Inside bag / purse 3
Using device upside down 3
Move to an isolated location 2
Under the seat 1 Postpone 1 The top suggestions for inconspicuous authentication strategies include many cases that are made possible, or at least easier, by the tap phrase method. This is true not only for blind users, but in any situations where the visual channel is not available. Our emphasis focus on affording single hand interaction required by some of top suggestions, e.g. authenticating inside the pocket or using the free hand to cover the interaction. The feasibility of actually using some of the selected strategies must, however, be further evaluated in realistic settings. For example, using a pocket may not be possible because hand movements can be constrained. That's not the case for most of the others.
CONCLUSIONS
Mobile devices are increasingly seen as extensions of ourselves, permeating our personal and social life. But they also create new threats to our well-being that can have devastating effects. The ability to self-protect is a universal requirement in human endeavors, from which the ubiquitous computing enterprise cannot escape if it is to be successful and inclusive. In this paper, we presented a novel approach for mobile authentication that attempts to foster inclusion of blind people in the global trend towards smartphone adoption.
We propose a method in which authentication is accomplished by recognizing rich tapping patterns that can be performed with a single hand, using a smartphone's entire screen as a single button. We presented results of a user study that shows that this method is usable and that it affords inconspicuous interactions, thus not only offering increased security but also enabling compliance with social norms.
Clear avenues for further research were opened, namely exploring longer-term usage and real-world feasibility of inconspicuous authentication scenarios, measuring resistance to eavesdropping, and expanding the matching of tap patterns to accommodate other interactions other than authentication.
Figure 1 .
1Tap Phrase Authentication for all 16 participants. First line shows the recorded template and the next one shows the attempt to replicate it by the same participant.
Table 1 .
1Concealing strategies suggested by study participants.
http://www.apple.com/accessibility
ACKNOWLEDGMENTSWe thank Fundação Raquel and Martin Sain and the participants of the study.
PassChords: secure multi-touch authentication for blind people. S Azenkot, K Rector, R Ladner, J Wobbrock, Proc. ASSETS '12. ASSETS '12ACMAzenkot, S., Rector, K., Ladner, R., Wobbrock. J. PassChords: secure multi-touch authentication for blind people. In Proc. ASSETS '12, ACM (2012).
Counting clicks and beeps: Exploring numerosity based haptic and audio PIN entry. A Bianchi, I Oakley, D S Kwon, Interact. Comput. 24Bianchi, A., Oakley, I., Kwon, D.S. Counting clicks and beeps: Exploring numerosity based haptic and audio PIN entry. Interact. Comput. 24, 5 (2012), 409-422.
Using rhythmic patterns as an input method. E Ghomi, G Faure, S Huot, O Chapuis, M Beaudouin-Lafon, Proc. CHI '12. CHI '12ACM PressGhomi, E., Faure, G., Huot, S., Chapuis, O., and Beaudouin-Lafon, M. Using rhythmic patterns as an input method. In Proc. CHI '12. ACM Press (2012)
Protractor: a fast and accurate gesture recognizer. Y Li, Proc. CHI '10. CHI '10ACM PressLi, Y. Protractor: a fast and accurate gesture recognizer. In Proc. CHI '10, ACM Press (2010), 2169- 2172.
Under the table: tap authentication for smartphones. D Marques, T Guerreiro, L Duarte, L Carriço, Proc. of BCS HCI '13. of BCS HCI '13Swinton, UKMarques, D., Guerreiro, T., Duarte, L., Carriço, L. (2013). Under the table: tap authentication for smartphones. In Proc. of BCS HCI '13, Swinton, UK.
Comparison of three onequestion, post-task usability questionnaires. J Sauro, J S Dumas, Proc. CHI '09. CHI '09ACM PressSauro, J., Dumas, J.S. Comparison of three one- question, post-task usability questionnaires. In Proc. CHI '09, ACM Press (2009).
In the shadow of misperception: assistive technology use and social interactions. K Shinohara, J O Wobbrock, Proc. CHI '12. CHI '12ACM PressShinohara, K. & Wobbrock, J.O., 2011. In the shadow of misperception: assistive technology use and social interactions. In Proc. CHI '12. ACM Press (2012).
TapSongs: tapping rhythm-based passwords on a single binary sensor. J O Wobbrock, Proc. UIST '09. UIST '09ACM PressWobbrock, J.O. TapSongs: tapping rhythm-based passwords on a single binary sensor. In Proc. UIST '09, ACM Press (2009)
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[
"Construction and application of exact solutions of the diffusive Lotka-Volterra system: a review and new results",
"Construction and application of exact solutions of the diffusive Lotka-Volterra system: a review and new results"
] |
[
"Roman Cherniha \nInstitute of Mathematics\nNational Academy of Sciences of Ukraine\n3, Tereshchenkivs'ka Street01004KyivUkraine\n",
"Vasyl ' Davydovych \nInstitute of Mathematics\nNational Academy of Sciences of Ukraine\n3, Tereshchenkivs'ka Street01004KyivUkraine\n"
] |
[
"Institute of Mathematics\nNational Academy of Sciences of Ukraine\n3, Tereshchenkivs'ka Street01004KyivUkraine",
"Institute of Mathematics\nNational Academy of Sciences of Ukraine\n3, Tereshchenkivs'ka Street01004KyivUkraine"
] |
[] |
This review summarizes all known results (up to this date) about methods of integration of the classical Lotka-Volterra systems with diffusion and presents a wide range of exact solutions, which are the most important from applicability point of view. It is the first attempt in this direction. Because the diffusive Lotka-Volterra systems are used for mathematical modeling enormous variety of processes in ecology, biology, medicine, physics and chemistry, the review should be interesting not only for specialists from Applied Mathematics but also those from other branches of Science. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems.
|
10.1016/j.cnsns.2022.106579
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[
"https://export.arxiv.org/pdf/2209.08863v1.pdf"
] | 248,881,690 |
2209.08863
|
dcf0ce9bc45599cfab5401df894cf35e62312272
|
Construction and application of exact solutions of the diffusive Lotka-Volterra system: a review and new results
19 Sep 2022
Roman Cherniha
Institute of Mathematics
National Academy of Sciences of Ukraine
3, Tereshchenkivs'ka Street01004KyivUkraine
Vasyl ' Davydovych
Institute of Mathematics
National Academy of Sciences of Ukraine
3, Tereshchenkivs'ka Street01004KyivUkraine
Construction and application of exact solutions of the diffusive Lotka-Volterra system: a review and new results
19 Sep 2022Dedicated to the memory of Wilhelm Fushchych (1936-1997)Diffusive Lotka-Volterra systempopulation dynamicsexact solutiontravel- ing frontLie and conditional symmetry
This review summarizes all known results (up to this date) about methods of integration of the classical Lotka-Volterra systems with diffusion and presents a wide range of exact solutions, which are the most important from applicability point of view. It is the first attempt in this direction. Because the diffusive Lotka-Volterra systems are used for mathematical modeling enormous variety of processes in ecology, biology, medicine, physics and chemistry, the review should be interesting not only for specialists from Applied Mathematics but also those from other branches of Science. The obtained exact solutions can also be used as test problems for estimating the accuracy of approximate analytical and numerical methods for solving relevant boundary value problems.
Introduction
About 100 years ago, Alfred Lotka [1] and Vito Volterra [2] independently developed a mathematical model, which nowadays serves as a mathematical background for population dynamics, chemical reactions, ecology, etc. The model is based on a system of ordinary differential equations (ODEs) involving quadratic nonlinearities (typically two equations). Following some earlier papers, in which linear ODEs were used for mathematical modeling of chemical reactions (in particular, see [3,4,5]), Lotka has shown that the densities in periodic reactions can be adequately described by a model involving ODEs with quadratic nonlinearities. In contrast to Lotka, Volterra, as a mathematician, was inspired by the information that the amount of predatory fish caught in Italy varied periodically and suggested a prey-predator model for the interaction of two populations of fishes.
The classical Lotka-Volterra system consists of two nonlinear ODEs of the form
du dt = u(a − bv), dv dt = v(−c + du),(1)
where the functions u(t) and v(t) represent the numbers of prey and predators at time t, respectively, a, b, c and d are positive parameters, the interpretation of which is presented below. Verbally, the Lotka and Volterra model can be formulated as follows:
[ Later it was shown that two ODEs with quadratic nonlinearities describe some other types of population interaction, e.g., competition and mutualism, hence nowadays the Lotka-Volterra system is usually presented in the form
du dt = u(a 1 + b 1 u + c 1 v), dv dt = v(a 2 + b 2 u + c 2 v).(2)
In particular, three common types of interaction between two populations (predator-prey interaction, competition and mutualism) can be modeled, depending on the signs of coefficients in (2).
In the case of interaction of m species (cells, chemicals, etc.), a natural generalization of (2) can be formulated. Moreover, their diffusion in space should also be taken into account. Thus, the diffusive m-component Lotka-Volterra system is obtained
u i t = d i ∆u i + u i a i + m j=1 b ij u j , i = 1, . . . , m,(3)
where u 1 (t, x), u 2 (t, x), . . . , u m (t, x) are unknown functions, d i ≥ 0, a i and b ij are arbitrary constants (i, j = 1, . . . , m), while x = (x 1 , x 2 , . . . , x n ), u i t = ∂u i ∂t and ∆ is the Laplace operator
∂ 2 ∂x 2 1 + · · ·+ ∂ 2 ∂x 2 n .
Nowadays the diffusive Lotka-Volterra (DLV) system is used as the basic model for a variety of processes in biology, chemistry, ecology, medicine, economics, etc. [6,7,8,9,10,11,12]. Typically, the functions u j (j = 1, . . . , m) are nonnegative and describe concentrations of species in populations, cells and drugs in tissue (tumour, bones, etc.), chemicals in a volume.
Obviously, this model in the case n = 1 and m = 2 reads as
u t = d 1 u xx + u(a 1 + b 1 u + c 1 v), v t = d 2 v xx + v(a 2 + b 2 u + c 2 v),(4)
where the lower subscripts t and x denote differentiation with respect to (w.r.t.) these variables, u = u(t, x) and v = v(t, x) are to-be-found functions, a i , b i and c i are given parameters (some of them can vanish and various types of interactions arise depending on their signs), d 1 and d 2 are diffusion coefficients. If the diffusivities d 1 = d 2 = 0 then (4) reduces to the classical Lotka-Volterra system. In the case n = 1 and m = 3, the DLV system takes the form
u t = d 1 u xx + u(a 1 + b 1 u + c 1 v + e 1 w), v t = d 2 v xx + v(a 2 + b 2 u + c 2 v + e 2 w), w t = d 3 w xx + w(a 3 + b 3 u + c 3 v + e 3 w).(5)
Here u(t, x), v(t, x) and w(t, x) are again unknown concentrations of three different populations (cells, chemicals) moving with diffusivities d 1 , d 2 and d 3 , respectively. The parameters a k , b k , c k , and e k define the type of interaction between the populations. It should be noted that the threecomponent models describe an essentially larger number of interactions than those involving two components. For example, two populations can be predators w.r.t. the third population (the predator-predator-prey model), on the other hand, the first population can be predators w.r.t. the other two which can compete for the same food (the predator-prey-competition model).
In contrast to the Lotka-Volterra system (2), the DLV system attracted attention of scholars much later. To the best of our knowledge, its rigorous study started in the 1970s [13,14,15,16]. At the present time, there are a lot of recent works devoted to qualitative and numerical analysis of the DLV system (4) and multi-component systems of the form (3) (see, e.g., [17,18] and works cited therein).
However, the number of papers devoted to the construction of exact solutions of the nonlinear system (4) is relatively small. Probably the first work, in which exact solutions of the DLV system were constructed in an explicit form, was written by Rodrigo and Mimura [19]. The authors implicitly used a so-called tanh-method [20,21] (there are many recent papers, in which this method was rediscovered without proper citations) for identifying some traveling waves. In [23], the well-known solution of the Fisher equation [26] was used for finding traveling waves of the DLV system. Exact solutions of the DLV system (4) in the form of traveling waves were also constructed in [22,23,24,25]. In the case of the three-component DLV system, some traveling waves were found in [27,28,29], while the existence of traveling wave solutions was examined in [17,18,30,31]. Exact solutions with more complicated structures were derived only in [32,33] and [34,35] for the two-and three-component DLV systems, respectively. In [36], a natural generalization of system (4) involving additional linear and/or quadratic terms was studied and its exact solutions were derived. It should also be mentioned that systems of nonlinear ODEs for constructing exact solutions of (4) in the very special case when a 1 = a 2 , b 1 = b 2 , c 1 = c 2 , d 1 = d 2 are presented in the handbook [37]. However, relevant exact solutions are not presented therein.
Hence, the problem of construction of exact solutions of DLV systems, especially those with a biological, chemical or physical interpretation, is a hot topic. Notably construction of exact solutions with more complicated structures (compared to traveling waves) requires more sophisticated methods and techniques.
At the present time, the most useful methods for construction of exact solutions for nonintegrable nonlinear partial differential equations (PDEs) are symmetry-based methods (see Section 2). These methods are based on the Lie method, which was created by a famous Norwegian mathematician Sophus Lie in the late 19th century. The Lie method (the notions 'the Lie symmetry analysis', 'the Lie group analysis' and 'the group-theoretical analysis' are used as well) still attracts attention of many investigators and new results are published on a regular basis (see the recent monographs [38,39] and papers cited therein). On the other hand, many nonlinear PDEs and systems of PDEs arising in real-world applications have a poor Lie symmetry. The Lie method is not productive for such type equations since in this case exact solutions can be easily obtained without using this cumbersome method. be easily obtained without using this cumbersome method. The DLV system (4) is a typical example of such type systems because one admits a nontrivial Lie symmetry only under essential restrictions on the coefficients a i , b i and c i (see Section 3). As a result, direct application of the Lie method leads only traveling wave solutions for (4) (see Section 4), otherwise some coefficients in (4) must vanish.
Within recent decades, new symmetry-based methods were developed in order to solve nonlinear PDEs arising in applications, but possessing poor Lie symmetry. The method of nonclassical symmetries proposed by Bluman and Cole in 1969 [40] is one of the best-known among them. It should be noted that we use the terminology 'Q-conditional symmetry' instead of 'nonclassical symmetry'. In our opinion, this terminology, proposed by Fushchych in the 1980s [41,42] when he and his collaborators proposed a generalization of nonclassical symmetries, more adequately reflects the essence of the method.
Although the method suggested in [40] is rather simple, its successful applications for solving nonlinear systems of PDEs were accomplished only in the 2000s. Moreover, the majority of the papers devoted to conditional symmetries of reaction-diffusion systems (the DLV system (4) is a typical example) were published within the recent decade [32,34,43,44,45,46,47]. It happened so late because application of the nonclassical method [40] to nonlinear systems of PDEs leads to very complicated nonlinear overdetermined systems of differential equations to-be-solved. In other words, one needs to solve a much more complicated PDE system (a so-called system of determining equations (DEs)) comparing with the initial system of PDEs. In order to make essential progress in solving systems of DEs, a new definitions of Q-conditional symmetries and new algorithm were proposed in [44]. The algorithm follows from the notion of a Q-conditional symmetry of the first type. In recent papers, we successfully applied this algorithm for constructing new exact solutions of the DLV system (4) and (5) (see Sections 5,6 and 7). In Sections 4, 6 and 7, we also discuss the most interesting exact solutions derived by other authors using other techniques.
Main definitions
Let us consider the DLV system (3). First of all, we note that the DLV system (3) with d i > 0 (i = 1, . . . , m) in the (1 + 1)-dimensional case can be rewritten as
λ i u i t = u i xx + u i a i + m j=1 b ij u j , i = 1, . . . , m,(6)
by introducing the notations
d i → 1 λ i , a i → a i λ i , b ij → b ij λ i .
In what follows we study the DLV system in the form (6) because the terms with the higher-order derivatives do not involve any coefficients. Obviously, each result obtained for system (6) is valid for the DLV system (3) after introduction of the inverse notations
λ i → 1 d i , a i → a i d i , b ij → b ij d i .
Plane wave solutions form the most common class of the exact solutions of two-dimensional PDEs (system of PDEs) because such solutions are important from an applicability point of view. In particular, traveling fronts, i.e. plane wave solutions, which are nonnegative, bounded and satisfy the zero Neumann conditions at infinity, are the most interesting solutions for a wide range of applications. Properties of such solutions in the case of scalar nonlinear reactiondiffusion (RD) equations were extensively studied during the recent decades using different mathematical techniques (see, e.g., monographs [39,48] and references cited therein).In the case of systems of RD equations, the progress is rather modest especially in searching for the plane wave solutions in explicit forms (the main references are listed in Introduction). The corresponding ansatz (a special kind of substitutions) for search for plane waves of a given m-component RD system (including the DLV system) has the form
u i = ϕ i (ω), i = 1, . . . , m, ω = x − µt,(7)
where ϕ i are to-be-determined functions and the parameter µ means the wave speed. Obviously, each system of (1+1)-dimensional RD equations with coefficients that do not depend explicitly on time t and space x, is reducible to a system of ODEs via ansatz (7). The system obtained does not depend on the new variable ω. There are many techniques for solving such systems of ODEs, however, their applicability depends essentially on the structure of the system in question. We will demonstrate this in Section 4 for the ODE systems corresponding to the DLV system (6). The symmetry-based methods allow us to construct ansätze with more complicated structures than (7). It turns out that each new ansatz obtained via a symmetry also reduces the given RD system to a system of ODEs although the relevant reduction can be highly nontrivial in contrast to the reduction via (7). In what follows we restrict ourselves by the classical Lie method and the method of Q-conditional symmetries [39,42]. Both methods allow us to construct ansätze of the form
u i = g i (t, x) + G ij (t, x)ϕ j ω(t, x) , i = 1, . . . , m,(8)
provided the m-component RD system in question admits a Lie and/or Q-conditional symmetry.
Here ϕ i are to-be-determined functions of the variable ω(t, x), while g i (t, x) and G ij (t, x) are the known functions and a summation is assumed from 1 to m over the repeated index j.
Obviously, formulae (7) follow from (8) as a very particular case. Thus, in order to derive new reductions of the DLV system (6), one needs to construct its Lie and Q-conditional symmetries enabling ansätze of the form(8) to be found. Any Lie and Q-conditional symmetry has the form of a linear first-order differential operator (infinitesimal operator)
Q = ξ 0 (t, x, u 1 , . . . , u m )∂ t + ξ 1 (t, x, u 1 , . . . , u m )∂ x + η 1 (t, x, u 1 , . . . , u m )∂ u 1 + . . . + η m (t, x, u 1 , . . . , u m )∂ u m(9)
with the correctly specified coefficients ξ 0 , ξ 1 , η 1 , . . . , η m . Hereinafter we use the notations ∂ z = ∂ ∂z , z = t, x, u i , ... . It is well-known that in order to find a Lie symmetry of system (6), one needs to consider the system as the manifold
M = {S 1 = 0, S 2 = 0, . . . , S m = 0} , where S i ≡ λ i u i t − u i xx − u i a i + m j=1 b ij u j , i = 1, 2, . . . , m,
in the prolonged space of the variables
t, x, u i , u i t , u i x , u i tt , u i tx , u i xx , i = 1, 2, . . . , m.
Definition 1
The infinitesimal operator (9) is a Lie symmetry of system (6) (in other words the latter is invariant under the transformations generated by (9)) if the following invariance criterion is satisfied:
Q 2 (S i ) M = 0, i = 1, 2, . . . , m.(10)
The operator Q 2 is the second-order prolongation of the operator Q and its coefficients are expressed via the functions ξ 0 , ξ 1 , η 1 , . . . , η m by the well-known formulae (see any texbook/monograph devoted to Lie symmetries of PDEs).
The main idea used for introducing the notion of the Q-conditional symmetry is to change the manifold M in formulae (10). It was noted in [44] that there are several different possibilities to modify the manifold M in the case of PDE systems. The first possibility is natural and follows directly from the seminal work [40] (see more details in [49]). (9) is called a Q-conditional symmetry (nonclassical symmetry) for DLV system (6) if the following invariance criterion is satisfied:
Definition 2 Operator
Q 2 (S i ) Mm = 0, i = 1, 2, . . . , m.(11)
Here the manifold has the form
M m = S i = 0, Q u i = 0, ∂ ∂t Q u i = 0, ∂ ∂x Q u i = 0, i = 1, . . . , m , where Q u i ≡ ξ 0 u i t + ξ 1 u i x − η i .
Another possibility is to consider a manifold M * , which is between M and M m , i.e.
M ⊃ M * ⊃ M m .
There are several possibilities and the simplest one is
M * = M j 1 = S 1 = 0, S 2 = 0, . . . , S m = 0, Q u j = 0, ∂ ∂t Q u j = 0, ∂ ∂x Q u j = 0 , where j (1 ≤ j ≤ m) is a fixed number.
Definition 3 Operator (9) is called Q-conditional symmetry of the first type for DLV system (6) if the following invariance criterion is satisfied:
Q 2 (S i ) M j 1 = 0, i = 1, 2, . . . , m.(12)
In the case of the DLV system (6) with m > 2, there are more possibilities to construct new manifolds M * (see [44,50] for details).
The algorithms for search Lie and Q-conditional symmetries of the DLV system (6) are based on the definitions presented above and the standard methods for solving overdetermined systems of PDEs and linear systems of differential equations.
Lie symmetries of the DLV systems
The prominent Norwegian mathematician Sophus Lie was the first to develop and apply the method for finding Lie symmetries of PDEs. Nowadays this method is well-known and can be found together with examples in many monographs and textbooks (the most recent are [38,39,49,50,51]). Here we present results of its application to the DLV systems skipping excessive details.
System (6) in the case m = 2, i.e. a two-component DLV system, has the form (up to the notations)
λ 1 u t = u xx + u(a 1 + b 1 u + c 1 v), λ 2 v t = v xx + v(a 2 + b 2 u + c 2 v).(13)
Here we examine system (13) assuming that both equations are nonlinear and not autonomous, i.e.
b 2 1 + c 2 1 = 0, b 2 2 + c 2 2 = 0, c 2 1 + b 2 2 = 0.(14)
The above restrictions are natural. In fact, assuming c 2 1 + b 2 2 = 0, one obtains two autonomous equations, which cannot describe any kind of interaction between species (cells, chemicals). Setting b 2 1 + c 2 1 = 0 (or b 2 2 + c 2 2 = 0), we arrive at a system involving the linear diffusion equation with a linear source/sink. Such a system is not interesting from both the mathematical and applicability point of view.
It is obvious that the DLV system (13) with arbitrary coefficients admits a two-dimensional Lie algebra generated by the operators
P t = ∂ t , P x = ∂ x .(15)
Obviously, the above operators generate the following invariance transformations of (13):
t * = t + t 0 , x * = x + x 0 ,
where t 0 and x 0 are arbitrary parameters. It turns out that there are several cases when this nonlinear system with correctly-specified coefficients is invariant w.r.t. a three-and higher-dimensional Lie algebra.
Theorem 1 [23] The DLV system (13) with restrictions (14) admits three-and higherdimensional Lie algebra if and only if its nonlinear terms and the corresponding symmetry operator(s) have structures listed in Table 1. If the DLV system (13) with other reaction terms is invariant w.r.t. a nontrivial Lie algebra, then it is reduced to one of the forms presented in Table 1
1 u(b 1 u + c 1 u) D = 2tP t + xP x − 2(u∂ u + v∂ v ) v(b 2 u + c 2 v) 2 b 1 u 2 D, v∂ v b 2 uv 3 u(a 1 + b 1 u) v∂ v b 2 uv 4 u(a 1 + b 1 u) λ 1 = λ 2 v∂ v , u∂ v , (a 1 + b 1 u)e a 1 t ∂ v v(a 1 + b 1 u) 5 b 1 u 2 λ 1 = λ 2 v∂ v , u∂ v , D, R = b 1 tu∂ u + ∂ v b 1 uv
It can be seen from Table 1 that the DLV systems possessing nontrivial Lie symmetry are semi-coupled (see , except for Case 1. From the applicability point of view, the DLV system (13) with a 1 = a 2 = 0 is the most important among others.
System (6) in the case m = 3, i.e., three-component DLV system, has the form (up to the notations)
λ 1 u t = u xx + u(a 1 + b 1 u + c 1 v + e 1 w), λ 2 v t = v xx + v(a 2 + b 2 u + c 2 v + e 2 w), λ 3 w t = w xx + w(a 3 + b 3 u + c 3 v + e 3 w).(16)
Note that we want to exclude the system containing an autonomous equation from the study, hence, hereinafter the restrictions
c 2 1 + e 2 1 = 0, b 2 2 + e 2 2 = 0, b 2 3 + c 2 3 = 0(17)
are assumed. Similarly to the two-component case, the above restrictions are natural from the applicability point of view. A complete description of Lie symmetries of the three-component DLV system (16) was derived in [34]. Obviously, the DLV system (16) with arbitrary coefficients a k , b k , c k , e k and λ k admits the Lie algebra with the basic operators (15). In order to find all possible extensions of the Lie algebra (15), it is necessary to apply the invariance criterion (10), to solve the DEs obtained and to identify all possible restrictions on the coefficients λ i , ..., e i leading to extensions of the Lie algebra (15). Because all the coefficients of the DLV system (16) are constants, this problem can be solved by standard calculations (see section 3.3 in [34] for details). Table 2. Any other DLV system admitting three-and higher-dimensional Lie algebra is reducible to one of those from Table 2 by a transformation from the set:
u → c 11 exp(c 10 t)u + c 12 v + c 13 w, v → c 21 exp(c 20 t)v + c 22 u + c 23 w, w → c 31 exp(c 30 t)w + c 32 u + c 33 v, t → c 40 t + c 41 , x → c 50 x + c 51 ,(18)
where c ij (i = 1, . . . , 5, j = 0, . . . , 3) are correctly-specified constants (some of them vanish) that are defined by the DLV system in question.
It can be seen from Table 2 that the DLV system (16) admits three-and higher-order Lie algebra provided at least three coefficients vanish. It is questionable that such systems can arise in real-world applications. On the other hand, some of them result from an approximation of relevant models, e.g., the DLV system with a 1 = a 2 = a 3 = 0 (see Case 1) assumes zero natural birth/death rate for interacting species. It means an assumption on the equality of natural death rate and birth rate for each species.
Traveling wave solutions of the DLV systems
In this section, we look for traveling wave solutions of the DLV systems (13) and (16). Because the DLV systems (13) and (16) with arbitrary coefficients admits only the trivial algebra (15), the plane wave ansatz (7) can be easily derived. In fact, if one takes a linear combination of the operators (15) Q = P t + µP x (µ is a wave speed) and constructs the invariance surface condition
Q(u i ) ≡ u i t + µu i x = 0 then ansatz (7) is immediately obtained.
Ansatz (7) with m = 2 and m = 3 reduces the DLV systems (13) and (16) to the nonlinear ODE systems
ϕ ′′ 1 + αλ 1 ϕ ′ 1 + ϕ 1 (a 1 + b 1 ϕ 1 + c 1 ϕ 2 ) = 0, ϕ ′′ 2 + αλ 2 ϕ ′ 2 + ϕ 2 (a 2 + b 2 ϕ 1 + c 2 ϕ 2 ) = 0(19)1 u(b 1 u + c 1 v + e 1 w) D = 2t∂ t + x∂ x − 2(u∂ u + v∂ v + w∂ w ) v(b 2 u + c 2 v + e 2 w) w(b 3 u + c 3 v + e 3 w) 2 u(c 1 v + e 1 w) u∂ u v(a 2 + c 2 v + w) w(a 3 + v + e 3 w) 3 u(c 1 v + e 1 w) u∂ u , D v(c 2 v + w) w(v + e 3 w) 4 u(a 1 + bu + v) λ 2 = λ 3 = 1 exp(−a 2 t)v∂ w , w∂ w v(a 2 + u + cv) w(u + cv) 5 u(bu + v) λ 2 = λ 3 = 1 v∂ w , w∂ w , D v(u + cv) w(u + cv) 6 u(a 1 + u + v) λ 1 = λ 2 = λ 3 = 1, exp(−a 1 t)u∂ w , w∂ w , exp(−a 2 t)v∂ w , v(a 2 + u + v) a 1 a 2 (a 1 − a 2 ) = 0 (a 2 (u + a 1 ) + a 1 v)∂ w w(u + v) 7 u(a + u + v) λ 1 = λ 2 = λ 3 = 1, exp(−at)u∂ w , w∂ w , v∂ w , (u + a + avt)∂ w v(u + v) a = 0 w(u + v) 8 u(bu + v) λ 1 = λ 2 = λ 3 = 1, w∂ w , ((b − 1)u + (1 − c)v) ∂ w , D v(u + cv) (b − 1) 2 + (c − 1) 2 = 0 w(bu + cv) and ϕ ′′ 1 + αλ 1 ϕ ′ 1 + ϕ 1 (a 1 + b 1 ϕ 1 + c 1 ϕ 2 + e 1 ϕ 3 ) = 0, ϕ ′′ 2 + αλ 2 ϕ ′ 2 + ϕ 2 (a 2 + b 2 ϕ 1 + c 2 ϕ 2 + e 2 ϕ 3 ) = 0, ϕ ′′ 3 + αλ 3 ϕ ′ 3 + ϕ 3 (a 3 + b 3 ϕ 1 + c 3 ϕ 2 + e 3 ϕ 3 ) = 0(20)
(hereinafter the upper sign ′ denotes the derivation d dω ), respectively. To the best of our knowledge, the ODE systems (19) and (20) with arbitrary coefficients are not integrable. As a result, the recently published handbooks devoted to nonlinear ODEs, e.g., [55], do not contain their general solutions. Their exact solutions (solutions in closed forms) can be derived only under additional restrictions on parameters. For long time, these systems were studied using only qualitative and numerical methods. The papers devoted to search for exact solutions, especially those leading to traveling waves, were published only within the recent two decades.
A majority of the papers [19,23,24,25] devoted to search for the traveling wave solutions of the DLV system (13) are focused on the case when the system describes competition between two populations (cells, chemicals). It means that the signs of the parameters are fixed. Thus, introducing the new notations b k → −b k , c k → −c k we rewrite the DLV system (13) in the form
λ 1 u t = u xx + u(a 1 − b 1 u − c 1 v), λ 2 v t = v xx + v(a 2 − b 2 u − c 2 v),(21)
where the coefficients a i , b i and c i are nonnegative. The reduced ODE system corresponding to the DLV system (21) takes the form
ϕ ′′ 1 + αλ 1 ϕ ′ 1 + ϕ 1 (a 1 − b 1 ϕ 1 − c 1 ϕ 2 ) = 0, ϕ ′′ 2 + αλ 2 ϕ ′ 2 + ϕ 2 (a 2 − b 2 ϕ 1 − c 2 ϕ 2 ) = 0.(22)
As was mentioned above, [19] is the first study, in which exact solutions of the twocomponent DLV system were constructed. In order to solve the ODE system (22), the nonlocal ansatz [19]
ϕ ′ 1 = m i=0 α i ϕ i 1 , ϕ ′ 2 = n i=0 β i ϕ i 1 , m, n > 0
was used. Actually, after substitution of the ansatz into (22), the authors studied the special cases m = 1, 2 and n = 1, 2, which naturally lead to solutions in the form of tanh-functions (or coth-function). So, the authors used the tanh-method, which was developed earlier for similar purposes [20,21]. Here we present the main exact solutions obtained in [19].
Traveling wave solutions of the DLV system (21) with the parameters
λ 1 = 1, λ 2 = λ, a 1 = 1, a 2 = a, b 1 = 1, b 2 = 2λ + 5a 3 − aλ 3 , c 1 = 1 3 , c 2 = 1 and λ 1 = 1, λ 2 = 1 + a(c − 6) 5 − ac , a 1 = 1, a 2 = a, b 1 = 1, b 2 = ac + 1 − a, c 1 = c, c 2 = 1
given by [19] u(t, x) = 1
2 1 + tanh √ a 2 √ 6 x − a−6 √ 6a t , v(t, x) = a 4 1 − tanh √ a 2 √ 6 x − a−6 √ 6a t 2 ,(23)
and
u(t, x) = 1 4 1 + tanh √ 1+ac 2 √ 6 x − ac−5 √ 6+6ac t 2 , v(t, x) = a 4 1 − tanh √ 1+ac 2 √ 6 x − ac−5 √ 6+6ac t 2 ,(24)
respectively. Solution (23) and (24) are typical traveling fronts, which are positive and bounded for arbitrary x and t ≥ 0.
In [23], exact solutions of the ODE system (22) were constructed using the following condition:
ϕ 2 = β 0 + β 1 ϕ 1 ,(25)
where β 0 and β 1 are to-be-determined constants. Substituting (25) into (22), one obtains an overdetermined system, which possesses nonconstant solutions only under the restriction λ 1 = λ 2 = λ. Without loss of generality one can set λ = 1. Thus, the second-order ODE
ϕ ′′ 1 + αϕ ′ 1 + ϕ 1 (a − bϕ 1 ) = 0(26)
is obtained. Here the constants a and b depend on an additional parameter β 0 as follows
a = a 1 = a 2 , β 0 = 0, a 1 − a 2 c 1 c 2 , β 0 = a 2 c 2 , b = c 1 b 2 −b 1 c 2 c 1 −c 2 , β 0 = 0, b 1 + c 1 β 1 , β 0 = a 2 c 2 ,(27)β 1 = b 1 −b 2 c 2 −c 1 , c 1 = c 2 , b 1 = b 2 , − a 2 b 1 a 1 c 1 , c 1 = c 2 , b 1 = b 2 .(28)
ODE (26) is known as the reduced equation of the famous Fisher equation [52]. In particular, ODE (26) has the exact solution [26]
ϕ 1 = a b 1 + c exp ± a 6 ω −2 ,(29)
where α = 5 √ a √ 6 and c is an arbitrary constant. Assuming c > 0, taking into account formulae (25) and (7) and fixing the upper sign in (29), one obtains the exact solution in the form of traveling front
u = a 4b 1 − tanh a 24 x − 5a 12 t 2 , v = β 0 + β 1 u.(30)
Here the parameters a, b, β 0 and β 1 are defined by (27) and (28).
We want to point out that the traveling wave solution (30) was much later rediscovered in [25] (see formulae (18) and (24) therein). Notably this solution has essentially different properties depending on the value of the parameter β 0 , therefore one simulates different types of interaction between population (see examples below).
Setting c < 0 we observe that the exact solution (29) generates the following solution of the DLV system (21):
u = a 4b 1 − coth a 24 x − 5a 12 t 2 , v = β 0 + β 1 u.
In contrast to (30), this solution blows up at all points (t, x) belonging to the plane
a 24 x − 5a 12 t = 0.
Probably, solutions of such type may describe an unusual interaction when both populations grow unboundedly. Interestingly, that [56] is devoted to a special case of the DLV system (21) with a 2 = c 2 = 0, i.e. a so-called Belousov-Zhabotinskii system. The exact solution constructed in [56] can be obtained from (30) (for details see [23]).
An important feature of traveling waves follows from their property to satisfy no-flux conditions at infinity. No-flux conditions at boundaries are typical requirements for a wide range of real-world processes. As an example, we use the exact solution (30) for solving the Neumann boundary value problem (BVP) for the DLV system (21).
Theorem 3 [23] Let us consider the Neumann BVP with the governing equations (21), the initial conditions
u = a 4b 1 − tanh a 24 x 2 ≡ u 0 (x), v = β 0 + β 1 u 0 (x)(31)
and the Neumann conditions at infinity
u x (t, −∞) = u x (t, +∞) = v x (t, −∞) = v x (t, +∞) = 0 (32)
in the domain Ω = {(t, x) ∈ (0, +∞) × (−∞, +∞)} Then its bounded solution has the form (30).
In formulae (31) and (30), the coefficients a, b, β 0 and β 1 are defined by (27) and (28).
Now we want to suggest an example of biological interpretation of this theorem. First of all, we observe that two essentially different cases occur, namely: β 0 = 0 and β 0 = 0. If β 0 = 0 then solution (30) has the asymptotical behavior
(u, v) → a 1 b 1 , 0 as t → ∞,(33)
provided the following condition is satisfied: (30) and (33)). In population dynamics, such asymptotical behavior predicts an uncompromising competition between two populations of species u and v. In other words, any increase in population u leads to a decrease in species v. As a result, the species v completely disappear.
A > max{B, C},(34)where A = a 1 a 2 , B = b 1 b 2 , C = c 1 c 2 (note that the condition A(B − 1) = B(C − 1) follows from
It turns out that the opposite condition
A < min{B, C}(35)
leads to the competition with the same character. In this case, the species v dominates, while the species u eventually dies out. If β 0 = 0 (in this case, the restriction a 1 = a 2 = a follows from (27)) then solution (30) possesses the property
(u, v) → a(C − 1) b 2 (C − B) , a(1 − B) c 2 (C − B) , t → ∞.(36)
The restriction β 1 = b 1 −b 2 c 2 −c 1 > 0 must also be satisfied (see (28)), which guarantees that solution (30) is nonnegative. Obviously, formula (36) implies either the relation
B > A = 1 > C(37)
or the relation
C > A = 1 > B.(38)
The exact solution (30) possessing property (36) describes the case of a 'soft' competition between two populations that predicts an arbitrarily long (in time) coexistence of the species u and v.
We emphasize that all the exact solutions derived in [19,22,23,24,25,32,33] are not applicable for the description of the prey-predator interaction. It turns out that the sign restrictions for the parameters a 1 , a 2 , c 1 and b 2 in (21) (see the corresponding signs in the classical system (1)) do not guarantee positivity of the traveling fronts derived in the papers cited above. Motivated by this fact, we were able to construct an absolutely new example of a traveling front for the DLV system describing the prey-predator interaction of two populations. In fact, using the tanh-method [20,21], the exact solution
u(t, x) = 3a 1 +a 2 2(3b 1 +b 2 ) 1 + tanh a 1 b 2 −a 2 b 1 8(3b 1 +b 2 ) (x − αt) , v(t, x) = a 1 b 2 −a 2 b 1 4c(3b 1 +b 2 ) 1 + tanh a 1 b 2 −a 2 b 1 8(3b 1 +b 2 ) (x − αt) 2 ,(39)
of the DLV system
u t = u xx + u(a 1 − b 1 u − cv), λv t = v xx + v(−a 2 + b 2 u − 3cv)(40)
was discovered. Here the restrictions
a i > 0, b i > 0, c > 0, λ = a 2 (5b 1 + b 2 ) − 2a 1 b 2 a 2 b 1 − 3a 1 (2b 1 + b 2 ) > 0, α = a 2 b 1 − 3a 1 (2b 1 + b 2 ) 2(3b 1 + b 2 )(a 1 b 2 − a 2 b 1 )
should hold. In the DLV system (40), all parameters are assumed to be positive. Thus, solution (39) of the DLV system (40) can describe the prey-predator interaction. Since a 1 b 2 − a 2 b 1 > 0 (otherwise the component v is negative), we immediately obtain the restriction α < 0. With a 1 b 2 −a 2 b 1 > 0 and α < 0, the following asymptotical behavior of the above solution is obtained:
(u, v) → 3a 1 + a 2 3b 1 + b 2 , a 1 b 2 − a 2 b 1 c(3b 1 + b 2 ) , t → ∞.
Such a behavior predicts an arbitrarily long (in time) coexistence of the preys u and the predators v.
In order to finish this part about traveling fronts of the two-component DLV systems, we would like to point out the following. Theorems on the existence of solutions of the Neumann problem for DLV systems describing competition of two species (cells, chemicals, etc.) have been known for a long time (see, e.g., [57] and works cited therein) and new publications with pure mathematical results are published on regular basis (see, e.g., [17,18]). In particular, it has been established that the coefficient relations (34), (35), (37) and (38) play a key role in behavior of any solutions of (21). However, those papers typically do not present such solutions in an explicit form. Theorem 3 and the above discussion show such solutions in the closed form. Moreover the traveling waves presented here satisfy no-flux conditions (the zero Neumann conditions) at infinity. Now we present some information about traveling fronts of the three-component DLV systems. In contrast to the two-component DLV systems, there are very few papers [27,28,29] devoted to the search for traveling wave solutions of the three-component DLV systems.
Probably traveling waves of the DLV system (16) were for the first time identified in [27]. Those solutions were constructed under essential parameter restrictions. In particular, assuming that λ 1 = λ 2 = λ 3 = 1 in (16), i.e. diffusivities of all populations are the same, the traveling wave solution has the form [27]
u(t, x) = 2 + α − a 4 1 − tanh x − αt 2 , v(t, x) = a 4 1 + tanh x − αt 2 , w(t, x) = (a − 2 − α) 1 − tanh x − αt , provided a 1 = a 2 = a 3 = a, b 1 = −1, b 2 = a−24 8−a+4α , b 3 = a−4−2α 8−a+4α , c 1 = 4α−a−16 a , c 2 = −1, c 3 = 2α−a−4 a , e 1 = a−4−2α 2+α−a , e 2 = a−4+2α 2+α−a , e 3 = −1,
where a and α are arbitrary parameters. Obviously, the inequalities α + 2 < a < 4(α + 2) should hold in order to guarantee positivity of the components u, v and w. The above solution can be treated as a generalization of the traveling waves (23) and (24). Note that in [27] the traveling wave solution for arbitrary diffusion coefficients was constructed. More interesting traveling wave solutions of the DLV system (16) were derived in [28]. In particular, by setting the parameters as follows
λ 1 = λ 2 = λ 3 = 1, a 1 = a 2 = a 3 = a, b 1 = −1, b 2 = 8+3a+e(24−3a) a(e−1) , b 3 = 2(a+8e−ae)
a(e−1) ,
c 1 = 8(1−3e) a(e
the traveling wave
u(t, x) = a 2 1 + tanh x − αt , v(t, x) = a 4 1 − tanh x − αt 2 , w(t, x) = 4 e−1 1 − tanh 2 x − αt(42)
were obtained. Here α = a−4+20e−ae 2(e−1)
. In order to guarantee positivity of the components u, v and w, the inequalities e > 1 and a > 0 should hold.
Restrictions (41) define the signs of the parameters of (16). Depending on the parameter signs (16) can describe different type of interactions of species. It may also happen that formulae (41) lead to a system, which is not applicable for modeling any interaction. Here we present an example when the exact solution can be useful. It can be noted that the following additional restrictions on parameters a and e: a > 24, e > 1, a(e − 1) > 8e + 8 3 lead to negative b i , c i , e i in the DLV system (16). Thus, we conclude that the system models competition between three populations and the traveling fronts (42) describe their densities in time and space. Interestingly, the competition predicts extinction of the population w while other two population will survive or die out depending on the sign of the velocity α. In Fig. 1, three surfaces are presented for the exact solution (42) for the correctly-specified parameters satisfying the above restrictions. As one concludes from Fig. 1, the solution describes the competition, which leads to the extinction of the populations u and w, while the population v dominates as t → ∞. Finally, we present new traveling waves that describe another type of interaction between three populations (cells, chemicals). Assuming that the u and v species compete for the same resources and w is a predator for the above two species, one arrives at the DLV system
λ 1 u t = u xx + u(a 1 − b 1 u − c 1 v − e 1 w), λ 2 v t = v xx + v(a 2 − b 2 u − c 2 v − e 2 w), λ 3 w t = w xx + w(−a 3 + b 3 u + c 3 v − e 3 w),(43)
where all parameters are positive. The competition-prey-predator model (43) differs essentially from those studied in [27,28,29] and can be thought as a generalization of the two-component model (40). Applying the tanh-method, we found the traveling wave solutions of (43) with correctly-specified parameters. Omitting awkward calculations, we present only a result. Thus, the competition-prey-predator model (43) with the parameters
λ 1 = 2(4+a 1 ) 16−a 3 , λ 2 = 2(4+a 2 ) 16−a 3 , λ 3 = 1, e 1 = 1, e 2 = 1, e 3 = 3,
has traveling wave solutions of the form
u(t, x) = (8−a 2 )c 3 +(24+a 3 )c 2 2(b 3 c 2 −b 2 c 3 ) 1 + tanh x + a 3 −16 4 t , v(t, x) = (a 2 −8)b 3 −(24+a 3 )b 2 2(b 3 c 2 −b 2 c 3 ) 1 + tanh x + a 3 −16 4 t , w(t, x) = 2 1 + tanh x + a 3 −16 4 t 2 ,
provided the restriction on the parameters a k , b k and c k (k = 1, 2, 3)
(24 + a 3 )(b 1 c 2 − b 2 c 1 ) = (8 − a 1 )(b 2 c 3 − b 3 c 2 ) + (8 − a 2 )(b 3 c 1 − b 1 c 3 )
holds.
Obviously, there is an infinity number of parameter sets, which satisfy the above restriction and guarantee that the components u, v and w are nonnegative. For instance, setting a 1 = 11, a 2 = 9, a 3 = 4, b 1 = 1 2 , b 2 = 1 6 , b 3 = 5, c 1 = 6, c 2 = 2, c 3 = 7, one obtains λ 1 = 5 2 and λ 2 = 13 6 and the exact solution
u(t, x) = 147 53 (1 + tanh (x − 3 t)) , v(t, x) = 1 53 (1 + tanh (x − 3 t)) , w(t, x) = 2 (1 + tanh (x − 3 t)) 2 .
Notably, this solution predicts that all species die out as t → ∞.
Conditional symmetries of the DLV systems
Now we turn to analysis of Q-conditional symmetries of the DLV systems. It will be demonstrated that application of such symmetries for solving the DLV systems is more efficient in comparison with the Lie symmetries. First of all we note that (6) is a system of evolution equations. It is well-known that the problem of constructing its Q-conditional symmetries for systems of evolution equations essentially depends on the function ξ 0 in (9) (see, e.g., [33]). Thus, one should consider two different cases :
1. ξ 0 = 0. 2. ξ 0 = 0, ξ 1 = 0.
In Case 1 , one may set ξ 0 = 1 without loss of generality applying the well-known property of Q-conditional symmetry operators (see, e.g., Section 1.2 in [50]). Moreover, in this case the differential consequences of equations Q(u i ) = 0 (that are presented in the manifold M m ) w.r.t. the independent variables t and x lead to second-order PDEs involving the derivatives u i tt and the mixed derivatives u i tx . However, u i tt and u i tx do not occur in the invariance conditions (11). Thus, the manifold M m can be rewritten as {S i = 0, Q(u i ) = 0, i = 1, . . . , m}, i.e. the first-order differential consequences can be omitted.
It is well-known that the task of constructing the Q-conditional symmetries in Case 2 (ξ 0 = 0) for scalar evolution equations is equivalent to solving the equation in question [53]. This statement can be extended on evolution systems of PDEs. In other words, it means that application of the invariance criteria (11) to operator (9) with ξ 0 = 0 after cumbersome calculations leads to a system of DEs, which is equivalent to (6). So, in the case of nonlinear and nonintegrable equations (systems), one can identify only some particular cases of the Qconditional symmetry operators of the form (9) with ξ 0 = 0. In [33], the two-component DLV system (4) was examined in order to find Q-conditional symmetries in Case 2 (the so-called no-go case) using Definition 3.
The system of DEs for finding Q-conditional symmetries of the DLV system (13) for the first time was derived in [32]. An algorithm based on Definition 2 was applied for this purpose. In particular, it was shown that the structure of any Q-conditional symmetry of (13) can be specified as follows
Q = ∂ t + ξ∂ x + q 1 v + r 1 u + p 1 ∂ u + q 2 u + r 2 v + p 2 ∂ v ,(44)
where the functions q k (t, x), r k (t, x) and p k (t, x) (k = 1, 2) should be found from the remaining equations of the system of DEs (see equations (30)- (45) in [32]). The system is very complicated and was not completely integrated, however important results were derived. In particular, the following existence theorem was proved.
Theorem 4 [32] In the case λ 1 = λ 2 , DLV system (13) admits only such Q-conditional operators of the form (44), which are equivalent to the Lie symmetry operators. In the case λ 1 = λ 2 , DLV system (13) In order to find symmetries in explicit forms, the case λ 1 = λ 2 was examined. As a result, the following theorem was proved.
Theorem 5 [32] If bc = 0, b 2 + c 2 = 0 then system (13) and the Q-conditional symmetries (up to the transformations u → bu, v → exp a 2 λ 2 t v, b = 0 and u → exp a 1 λ 1 t v, cv → u, c = 0) have the forms
λ 1 u t = u xx + u(a 1 + u), λ 2 v t = v xx + vu, Q = ∂ t + 2α 1 λ 1 −λ 2 ∂ x + (ϕ(t) exp(α 1 x)u + exp(α 1 x) (λ 2 ϕ ′ (t) + a 1 ϕ(t) − α 2 1 ϕ(t)) + α 2 v) ∂ v ,
where the function ϕ(t) = 0 is the general solution of the linear ODE λ 2 2 ϕ ′′ + λ 2 (a 1 − 2α 2 1 )ϕ ′ + α 2 1 (α 2 1 − a 1 )ϕ = 0. If bc = 0 and the additional restrictions
q 1 x = q 2 x = 0(45)
hold then exactly three cases (up to the transformations u → bu, v → cv and u → v, v → u) exist when system (13) is invariant w.r.t. Q-conditional symmetry operators. These cases are listed in Table 3 Table 3: Q-conditional symmetries of the DLV system (13) with λ 1 = λ 2 and b 1 = b 2 = b, c 1 = c 2 = c, bc = 0.
DLV systems Restrictions Operators
1 λ 1 u t = u xx + u(a 1 + u + v) a 1 = a 2 (λ 1 − λ 2 )∂ t − (a 1 v + a 2 u + a 1 a 2 )(∂ u − ∂ v ), a 1 a 2 = 0; λ 2 v t = v xx + v(a 2 + u + v) (λ 1 − λ 2 )∂ t + (a 1 − a 2 )u(∂ u − ∂ v ); (λ 1 − λ 2 )∂ t − (a 1 − a 2 )v(∂ u − ∂ v ) 2 λ 1 u t = u xx + u(a + u + v) (λ 1 − λ 2 )∂ t − a(v + u + a)(∂ u − ∂ v ), a = 0; λ 2 v t = v xx + v(a + u + v) (λ 1 − λ 2 )t∂ t − (λ 1 v + λ 2 u)(∂ u − ∂ v ) 3 λ 1 u t = u xx + u(aλ 1 + u + v) a = 0 (λ 1 − λ 2 )∂ t − a(λ 1 v + λ 2 u + aλ 1 λ 2 )(∂ u − ∂ v ); λ 2 v t = v xx + v(aλ 2 + u + v) ∂ t + au(∂ u − ∂ v ); ∂ t − av(∂ u − ∂ v ); ∂ t + aα(λ 1 v+λ 2 u+aλ 1 λ 2 ) (e −at −α(λ 1 −λ 2 )) (∂ u − ∂ v ), α = 0
Remark 1 In contrast to Definition 2, applying Definition 3 leads to a complete description (i.e. without additional restrictions) of Q-conditional symmetries of the first type (with ξ 0 = 0) of the DLV system (13) (see Theorem 2 [32]). Unfortunately all the Q-conditional symmetries of the first type, which have been derived, coincide with those listed in Table 3.
Now we turn to Case 2 , i.e. the no-go case (ξ 0 = 0, see operator (9)), which was was investigated in [33]. As mentioned above, the algorithm based on Definition 2 leads to an unsolvable system of DEs in this case. So, we used Definition 3 in order to find all possible Q-conditional symmetries of the first type. The main result can be formulated as follows.
Theorem 6 [33] The DLV system (13) with restrictions (14) is invariant under Q-conditional symmetry operator(s) of the first type
Q = ξ(t, x, u, v)∂ x + η 1 (t, x, u, v)∂ u + η 2 (t, x, u, v)∂ v , ξ = 0,(46)
if and only if the system and the relevant operator(s) are as specified in Table 4. Any other DLV system (13) admitting a Q-conditional symmetry of the first type and the corresponding operator(s) are reducible to those listed in Table 4 by an appropriate transformation from the set
t * = t + t 0 , x * = e γ 0 (x + x 0 ), u * = β 11 e γ 1 t u + β 12 v, v * = β 22 e γ 2 t v + β 21 u,
where t 0 , x 0 , β ij and γ j are correctly-specified constants. Table 4, the upper indices u and v mean that the relevant Q-conditional symmetry operators satisfy Definition 3 in the case of the manifold M 1 1 (u 1 = u) and M 2 1 (u 2 = v), respectively. Table 4, ω = a 2 +c 2 v λv e a 2 λ t , θ = t + λ c 2 v ; h 1 , h 2 and h 3 are arbitrary smooth functions of the relevant variables, while the function p(t, x, v) is the general solution of the linear ODE
Remark 2 In
Remark 3 In
p t = p xx − v(a 2 + c 2 v) λ p v + vp,
the functions F and G form the general solution of the system
F F v − F x + av + v 2 = 0, G x = F G v ,
the function r(t, x) is the general solution of the Burgers equation
r t = r xx + 2rr x , while g i (t, x) = α 0 exp κ 2 t λ i + α 1 sin(κ x) + α 2 cos(κ x), if λ 1 a 2 −λ 2 a 1 λ 1 −λ 2 > 0, α 0 exp − κ 2 t λ i + α 1 e κx + α 2 e −κx , if λ 1 a 2 −λ 2 a 1 λ 1 −λ 2 < 0, α 0 + α 1 x + α 2 λ i x 2 + 2α 2 t, if λ 1 a 2 = λ 2 a 1 ,(47)
where i = 1, 2, κ = λ 1 a 2 −λ 2 a 1 λ 1 −λ 2 , α 0 , α 1 and α 2 are arbitrary constants.
λ 1 u t = u xx + u(a 1 + u + v) λ 1 = λ 2 Q u 1 = ∂ x + g 1 x g 1 u (∂ u − ∂ v ), λ 2 v t = v xx + v(a 2 + λ 2 λ 1 u + λ 2 λ 1 v) Q v 1 = ∂ x + g 2 x g 2 v (∂ v − ∂ u ) 2 u t = u xx + u(a + u + 2v) Q v 2 = G(x, v) (∂ x + F (x, v)(∂ u − ∂ v )) λv t = v xx + v(a + v) 3 u t = u xx + uv a 2 c 2 = 0 Q u 3 = ∂ x + r(t, x) u ∂ u , λv t = v xx + v(a 2 + c 2 v) Q v 3 = h 1 (ω) − 2th 2 (ω) ∂ x + (h 2 (ω)x + h 3 (ω))u + p(t, x, v) ∂ u 4 u t = u xx + uv c 2 = 0 Q u 3 , Q v 4 = h 1 (θ) − 2th 2 (θ) ∂ x λv t = v xx + c 2 v 2 + (h 2 (θ)x + h 3 (θ))u + p(t, x, v) ∂ u 5 u t = u xx + uv a 2 = 0 Q u 3 , Q v 3 , Q v 5 = ∂ x + e a 2 t u∂ v v t = v xx + v a 2 + v 2 + α u − e −a 2 t 2 v 2 − a 2 e −a 2 t v ∂ u 6 u t = u xx + uv Q u 3 , Q v 4 , v t = v xx + 1 2 v 2 Q v 6 = (α 1 t + α 0 )∂ x + (α 1 t + α 0 )u∂ v + α 2 − α 1 2 x u − α 1 t+α 0 2 v 2 − α 1 v ∂ u 7 u t = u xx + uv a 2 = 0 α 2 1 + α 2 2 = 0, Q u 3 , Q v 3 , v t = v xx + v(a 2 + v) Q u 7 = ∂ x + − x 2t u + α 1 t + α 2 e −a 2 t t + α 1 a 2 t v ∂ u 8 u t = u xx + uv α 2 1 + α 2 2 = 0 Q u 3 , Q v 4 , v t = v xx + v 2 Q u 8 = ∂ x + − x 2t u + α 1 t + α 2 t + α 1 v ∂ u
It should be noted that all the systems arising in Table 4, except that in Case 1, are semicoupled (see the second equation is each system). We point out that the second equation in Cases 2, 3, 5 and 7 is nothing else but the famous Fisher equation. Obviously, Case 1 from Table 4 is the most interesting from applicability point of view. In fact, it will be demonstrated in Section 6 that the system from Case 1 models competition of two populations of species (cells, chemicals) and the relevant exact solutions will be constructed. Now we turn to the three-component DLV system. A complete description of the Q-conditional (nonclassical) symmetry for the three-component DLV system is still unknown from the same reason as for the two-component system. To the best of our knowledge, there is only a single study [34] devoted only to the search for conditional symmetries of the threecomponent DLV system. In that paper, all possible Q-conditional symmetries of the first type were derived in Case 1 .
Theorem 7 [34] The DLV system (16) is invariant under Q-conditional symmetry of the first type
Q = ξ 0 (t, x, u, v, w)∂ t + ξ 1 (t, x, u, v, w)∂ x + η 1 (t, x, u, v, w)∂ u + η 2 (t, x, u, v, w)∂ v + η 3 (t, x, u, v, w)∂ w , ξ 0 = 0,
if and only if it and the relevant operators are as specified in Table 5. Any other DLV system admitting a Q-conditional symmetry operator of the first type is reduced to one of those from Table 5 by a transformation from the set (18).
In Table 5, the following designations are introduced:
Q 2 i = Q 4 i with α = 0, i = 1, . . . , 6; Q 4 1 = ∂ t + a 1 − a 2 λ 1 − λ 2 u(∂ u − ∂ v ) + αu(∂ v − ∂ w ), Q 4 2 = ∂ t + a 1 − a 2 λ 1 − λ 2 v(∂ v − ∂ u ) + αv(∂ u − ∂ w ), Q 4 3 = ∂ t + a 1 − a 3 λ 1 − λ 3 u(∂ u − ∂ w ) + αu(∂ v − ∂ w ), Q 4 4 = ∂ t + a 1 − a 3 λ 1 − λ 3 w(∂ w − ∂ u ) + αw(∂ u − ∂ v ), Q 4 5 = ∂ t + a 2 − a 3 λ 2 − λ 3 v(∂ v − ∂ w ) + αv(∂ u − ∂ w ), Q 4 6 = ∂ t + a 2 − a 3 λ 2 − λ 3 w(∂ w − ∂ v ) + αw(∂ u − ∂ v ); Q 5 1 = ∂ t + α 1 ∂ x + exp (λ 1 − λ 3 ) 2 4 α 2 1 − a 1 t λ 3 + λ 1 − λ 3 2 α 1 x u∂ w ; Q 6 1 = ∂ t + α 1 ∂ x + exp (λ 2 − λ 3 ) 2 4 α 2 1 − a 2 t λ 3 + λ 2 − λ 3 2 α 1 x v∂ w , Q 6 2 = ∂ t + a 1 − a 2 λ 1 − λ 2 u(∂ u − ∂ v ) + β exp (λ 1 − λ 3 )a 2 − (λ 2 − λ 3 )a 1 λ 3 (λ 2 − λ 1 ) t u∂ w , Q 6 3 = ∂ t + a 1 − a 2 λ 1 − λ 2 v(∂ v − ∂ u ) + β exp (λ 2 − λ 3 )a 1 − (λ 1 − λ 3 )a 2 λ 3 (λ 1 − λ 2 ) t v∂ w , Q 6 4 = ∂ t + a 2 λ 1 − a 1 λ 2 λ 3 (λ 2 − λ 1 ) w∂ w + exp (λ 3 − λ 2 )a 1 − (λ 3 − λ 1 )a 2 λ 3 (λ 1 − λ 2 ) t w(∂ u − ∂ v );
Q 9 1 = Q 5 1 with λ 3 = 1, Q 9 2 = Q 6 4 with λ 2 = λ 3 = 1, Table 5: Q-conditional symmetries of the first type of the DLV system (16) Reaction terms Restrictions Q-conditional symmetry operators
Q 9 3 = ∂ t + a 1 − a 2 λ 1 − 1 u(∂ u − ∂ v ) + (ϕ 1 (t)u + ϕ 2 (t)v + β 1 ) ∂ w ,1 u(a 1 + bu + bv + ew) (b − 1) 2 + (e − e 3 ) 2 = 0, ∂ t + a 1 −a 2 λ 1 −λ 2 u(∂ u − ∂ v ), v(a 2 + bu + bv + ew) a 1 = a 2 ∂ t + a 1 −a 2 λ 1 −λ 2 v(∂ v − ∂ u ) w(a 3 + u + v + e 3 w) 2 u(a 1 + u + v + w) (a 1 − a 2 ) 2 + (a 1 − a 3 ) 2 = 0 Q 2 i , i = 1, . . . , 6 v(a 2 + u + v + w) w(a 3 + u + v + w) 3 u(a 1 + u + v + w) (λ 2 − λ 3 )a 1 − λ 2 a 3 Q 2 i , i = 1, . . . , 6, v(a 2 + u + v + w) +λ 3 a 2 = 0, ∂ t + β exp a 2 −a 3 λ 2 −λ 3 t u(∂ v − ∂ w ) w(a 3 + u + v + w) a 2 = a 3 , β = 0 4 u(a 1 + u + v + w) (λ 2 − λ 3 )a 1 − (λ 1 − λ 3 )a 2 Q 4 i , i = 1, . . . , 6 v(a 2 + u + v + w) +(λ 1 − λ 2 )a 3 = 0, w(a 3 + u + v + w) (a 1 − a 2 ) 2 + α 2 = 0 5 u(a 1 + bu + v) (b − 1) 2 + (c − 1) 2 = 0 Q 5 1 v(a 2 + u + cv) w(bu + v) 6 u(a 1 + u + v) Q 5 1 , Q 6 i , i = 1, . . . , 4 v(a 2 + u + v) w(u + v) 7 u(a 1 + bu + cv) λ 2 = λ 3 = 1, b = 1, c = 1, ∂ t + ((1 − b)u + (1 − c)v + a 2 (1 − c)) ∂ w v(a 2 + u + v) a 1 (1 − b) = a 2 b(1 − c) w(bu + v) 8 u(a + bu + cv) λ 2 = λ 3 = 1, b = 1, c = 1, ∂ t + (1 − c)∂ w v(a + u + v) b(2 − c) = 1 + ((1 − b)u + (1 − c)v) ϕ 4 (t)∂ w w(bu + v) 9 u(a 1 + u + v) λ 2 = λ 3 = 1 Q 9 i , i = 1, . . . , 5 v(a 2 + u + v) w(u + v)
The coefficients λ k > 0 (k = 1, 2, 3) are assumed to be different in cases 1-6.
Q 9 4 = ∂ t + (ϕ 3 (t)u + ϕ 2 (t)v + β 1 ) ∂ w , Q 9 5 = ∂ t + a 1 − a 2 λ 1 − 1 v(∂ v − ∂ u );
where the functions ϕ i (t) (i = 1, . . . , 4) are as follows:
ϕ 1 (t) = β 1 t + β 2 , if a 2 = 0, β 2 exp(−a 2 t) + β 1 a 2 , if a 2 = 0, ϕ 2 (t) = β 1 t, if a 2 = 0, β 1 a 2 , if a 2 = 0, ϕ 3 (t) = β 1 t + β 2 , if a 1 = 0, β 2 exp(−a 1 t) + β 1 a 1 , if a 1 = 0, ϕ 4 (t) = t + β, if a = 0, β exp(−at) + 1 a ,
if a = 0, while α and β (with and without subscripts 1 and 2) are arbitrary constants. Table 5 guarantee that the Qconditional symmetries from the fourth column are not equivalent to any Lie symmetry presented in Table 2.
Remark 4 The inequalities listed in the third column of
We conclude that the three-component DLV system, depending on the coefficient restrictions, admits a wider range of Q-conditional symmetries of the first type compared to those for the two-component DLV system. In particular, there are cases when DLV system (16) admits sets consisting of 5, 6 and even 7 different symmetries. All these symmetries can be successfully used for finding exact solutions.
From the applicability point of view, the systems arising in cases 1-4 of Table 5 are most promising because their nonzero coefficients do not affect the biological sense of these systems. In the Section 7, we examine these systems.
Concluding this section, we present a short statement about the m-component DLV system (9). In the case of the DLV system (9) with m > 3, the problem of constructing conditional symmetries is still open. Some particular results can be obtained by a simple generalization of the results obtained for three-component system. In particular, we proved that the mcomponent system [34] λ i u i t = u i xx + u i (a i + u 1 + · · · + u m ), i = 1, 2, . . . , m admits m(m − 1) operators of the form
Q ij = ∂ t + a i − a j λ i − λ j u i (∂ u i − ∂ u j ) , i = j = 1, 2, . . . , m,
provided (a i − a j )(λ i − λ j ) = 0. One may consider the above system and the operators as a generalization of those presented in Case 2 of Table 5 on the case of the m-component DLV systems.
Exact solutions of the two-component DLV system
This section is devoted to the construction of exact solutions with more complicated structures than the traveling wave solutions presented in Section 4. It should be pointed out that the traveling wave solutions cannot be applied for solving practical models, in particular based on the DLV systems, describing processes in bounded domains. In fact, any traveling front does not satisfy typical boundary conditions like no-flux conditions or/and constant densities at a bounded interval. It means that an ansatz of the form (8) should be used for search for exact solutions.
It is well-known that using Q-conditional symmetries a given two-dimensional PDE (system of PDEs) can be reduced to an ODE (system of ODEs) via the same algorithm as for classical Lie symmetries. It means that the ansatz corresponding to the given operator Q can be constructed provided the linear (quasi-linear) first-order PDEs
Q(u) = 0, Q(v) = 0(48)
are solved. Theorems 5 and 6 give several possibilities for finding exact solutions of the DLV system with correctly-specified coefficients.
Let us consider the DLV system from Case 1 of Table 3, namely :
λ 1 u t = u xx + u(a 1 + u + v), λ 2 v t = v xx + v(a 2 + u + v), a 1 = a 2 ,(49)
and its Q-conditional symmetry operator
Q = (λ 1 − λ 2 )∂ t − (a 1 v + a 2 u + a 1 a 2 )(∂ u − ∂ v ).(50)
In the case of operator (50), system (48) takes the form
(λ 1 − λ 2 )u t = −(a 1 v + a 2 u + a 1 a 2 ), (λ 1 − λ 2 )v t = a 1 v + a 2 u + a 1 a 2 .(51)
It follows immediately from (51) that u t = −v t , hence
u(t, x) = −v(t, x) + ϕ 1 (x).(52)
Substituting (52) into the second equation of (51), one obtains the linear equation
(λ 1 − λ 2 )v t = (a 1 − a 2 )v + a 2 ϕ 1 (x) + a 1 a 2 .
Since a 1 = a 2 this equation has the general solution
v = 1 a 1 − a 2 exp a 1 − a 2 λ 1 − λ 2 t ϕ 2 (x) − a 2 ϕ 1 (x) − a 1 a 2 ,
therefore the ansatz
u = 1 a 1 −a 2 − exp a 1 −a 2 λ 1 −λ 2 t ϕ 2 (x) + a 1 ϕ 1 (x) + a 1 a 2 , v = 1 a 1 −a 2 exp a 1 −a 2 λ 1 −λ 2 t ϕ 2 (x) − a 2 ϕ 1 (x) − a 1 a 2(53)
is obtained. Here ϕ 1 and ϕ 2 are functions to be found.
To obtain the reduced system, we substitute ansatz (53) into (49). This means that we simply calculate the derivatives u t , v t , u xx , v xx , and insert them into (49). Making relevant calculations, one arrives at the ODE system
ϕ ′′ 1 + ϕ 2 1 + (a 1 + a 2 )ϕ 1 + a 1 a 2 = 0, ϕ ′′ 2 + a 2 λ 1 −a 1 λ 2 λ 1 −λ 2 ϕ 2 + ϕ 1 ϕ 2 = 0(54)
to find the functions ϕ 1 and ϕ 2 .
Remark 5 Using the second and third operators listed in Case 1 of Table 3, one can obtain reduced systems in a similar way and look for exact solutions. However, we have checked that the ODE systems obtained simply follow from (54) by removing the terms a 1 ϕ 1 + a 1 a 2 .
In order to construct exact solutions, now we examine the ODE systems obtained above. To the best of our knowledge, the general solution of the nonlinear ODE system (54) is unknown, therefore we look for its particular solutions. Setting ϕ 1 = α = const, we conclude α 2 + (a 1 + a 2 )α + a 1 a 2 = 0 ⇒ α 1 = −a 1 , α 2 = −a 2 from the first equation of system (54). So, setting ϕ 1 = −a 1 (the case ϕ 1 = −a 2 leads to the solution with the same structure) and substituting into the second equation of system (54), we obtain the linear ODE:
ϕ ′′ 2 − βλ 1 ϕ 2 = 0,(55)
where β = a 1 −a 2 λ 1 −λ 2 = 0. Depending on the sign of the parameter β the linear ODE (55) possesses two families of general solutions. These solutions and ansatz (53) lead to the following exact solutions of the DLV system (49):
u = −a 1 + 1 a 2 −a 1 C 1 exp √ βλ 1 x + C 2 exp − √ βλ 1 x e βt , v = 1 a 1 −a 2 C 1 exp √ βλ 1 x + C 2 exp − √ βλ 1 x e βt ,
if β > 0, and
u = −a 1 + 1 a 2 −a 1 C 1 cos √ −βλ 1 x + C 2 sin √ −βλ 1 x e βt , v = 1 a 1 −a 2 C 1 cos √ −βλ 1 x + C 2 sin √ −βλ 1 x e βt ,(56)
if β < 0 (hereinafter C 1 and C 2 are arbitrary constants). Now we demonstrate that extra exact solutions of (54) can be derived provided some restrictions on λ 1 and λ 2 take place. Indeed, we note that the substitution
ϕ 1 = ϕ − a 1(57)
simplifies the first equation of (54) to the form
ϕ ′′ + ϕ 2 + (a 2 − a 1 )ϕ = 0.(58)
Of course, (58) can be reduced to the first-order ODE
dϕ dx 2 = − 2 3 ϕ 3 + (a 1 − a 2 )ϕ 2 + C
with the general solution involving the Weierstrass function [54]. Now we set C = 0 in order to avoid cumbersome formulae, therefore the general solution is
ϕ = 3(a 1 − a 2 ) 2 1 − tanh 2 √ a 1 − a 2 2 x ,(59)
if a 1 > a 2 , and
ϕ = 3(a 1 − a 2 ) 2 1 + tan 2 √ a 2 − a 1 2 x ,(60)
if a 1 < a 2 . Thus, we can apply formulae (59) and (60) to solve the second ODE of (54). In the case of solution (59), this ODE takes the form
ϕ ′′ 2 + ϕ 2 (a 1 − a 2 ) λ 1 − 3λ 2 2(λ 1 − λ 2 ) − 3 2 tanh 2 √ a 1 − a 2 2 x = 0.(61)
The general solution of (61) with some restrictions on λ 1 and λ 2 can be found [55]:
ϕ 2 = f 1 (x) C 1 + C 2 1 f 2 1 (x) dx ,(62)
if λ 1 = 9 5 λ 2 , and
ϕ 2 = f 2 (x) C 1 + C 2 1 f 2 2 (x) dx ,(63)if λ 1 = 4 3 λ 2 , where f 1 (x) = cosh 3 √ a 1 − a 2 2 x , f 2 (x) = sinh √ a 1 − a 2 2 x cosh 3 √ a 1 − a 2 2 x .
Thus, substituting the functions ϕ 1 (x) and ϕ 2 (x) given by formulae (57), (59) and (62) into ansatz (53), one easily obtain exact solutions of the DLV system (49) (see [32] for details).
Let us consider an example. Example 1. Using the substitution
u → −bu, v → −cv (b > 0, c > 0),
one reduces the DLV system (49) to the system
λ 1 u t = u xx + u(a 1 − bu − cv), λ 2 v t = v xx + v(a 2 − bu − cv),(64)
which describes competition of two species (here a 1 > 0, a 2 > 0 and λ 1 = λ 2 ). Simultaneously this substitution transforms solution (56) with C 1 = 0 to the form
u(t, x) = a 1 b + 1 (a 1 −a 2 )b C 2 sin √ −βλ 1 x e βt , v(t, x) = 1 (a 2 −a 1 )c C 2 sin √ −βλ 1 x e βt ,(65)
where the coefficient restrictions β ≡ a 1 −a 2 λ 1 −λ 2 < 0, a 1 > 0, a 2 > 0 are assumed. Having this solution, we formulate the following theorem about the classical solution of a nonlinear BVP involving constant Dirichlet conditions. Theorem 8 [32] The classical solution of the nonlinear BVP formed by the competition system (64), the initial profile
u(0, x) = a 1 b + 1 (a 1 −a 2 )b C 2 sin √ −βλ 1 x , v(0, x) = 1 (a 2 −a 1 )c C 2 sin √ −βλ 1 x
and the boundary conditions
x = 0 : u = a 1 b , v = 0, x = π √ −βλ 1 : u = a 1 b , v = 0 in the domain Ω = (t, x) ∈ (0, +∞) × 0, π √ −βλ 1
is given by formulae (65).
The solution (65) with β < 0 has the time asymptotic Using biological terminology, this solution simulates competition between two populations of species when species u eventually dominates while species v dies out. An example of this competition with correctly-specified parameters is shown in Fig. 2. Now let us consider the DLV system from Case 1 of Table 4, namely :
(u, v) → a 1 b , 0 , t → +∞.λ 1 u t = u xx + u(a 1 + u + v), λ 2 v t = v xx + v a 2 + λ 2 λ 1 u + λ 2 λ 1 v , λ 1 = λ 2 .(66)
It should be noted that the Q-conditional symmetry operators Q u 1 and Q v 1 of system (66) lead to the same exact solutions (up to discrete transformation u → v, v → u). Thus, we use only the operator Q u 1 . The corresponding ansatz can be constructed by solving the linear first-order PDE system
u x = g 1 x g 1 u, v x = − g 1 x g 1 u.(67)
Integrating system (67) for each form of the function g 1 from (47), we obtain the ansatz
u = ϕ(t) α 0 + α 1 exp − κ 2 λ 1 t sin(κx) + α 2 exp − κ 2 λ 1 t cos(κx) , v = ψ(t) − ϕ(t) α 0 + α 1 exp − κ 2 λ 1 t sin(κx) + α 2 exp − κ 2 λ 1 t cos(κx) ,(68)if λ 1 a 2 −λ 2 a 1 λ 1 −λ 2 > 0, the ansatz u = ϕ(t) α 0 + α 1 exp κ 2 λ 1 t + κx + α 2 exp κ 2 λ 1 t − κx , v = ψ(t) − ϕ(t) α 0 + α 1 exp κ 2 λ 1 t + κx + α 2 exp κ 2 λ 1 t − κx ,(69)
if λ 1 a 2 −λ 2 a 1 λ 1 −λ 2 < 0, and the ansatz
u = ϕ(t) (α 0 + α 1 x + α 2 x 2 + 2d 1 α 2 t) , v = ψ(t) − ϕ(t) (α 0 + α 1 x + α 2 x 2 + 2d 1 α 2 t) ,(70)
if a 2 = a 1 λ 2 λ 1 . Now three reductions of the given DLV system to ODE systems can be obtained. In fact, inserting the above ansätze into the DLV system (66), we arrive at the ODE system
λ 1 dϕ dt = ϕ (a 1 + ψ) , λ 2 dψ dt = a 2 + λ 2 λ 1 ψ ψ + α 0 a 1 λ 2 λ 1 − a 2 ϕ,(71)
in the cases of formulae (68) and (69), while the system
λ 1 dϕ dt = ϕ (a 1 + ψ) , λ 2 dψ dt = λ 2 λ 1 (a 1 + ψ) ψ − 2α 2 (λ 1 − λ 2 ) ϕ,(72)
is obtained in the case of (70). Here ϕ(t) and ψ(t) are to-be-determined functions. It was proved that each of the ODE systems (71) and (72) can be integrated by reducing to a single second-order ODE (see [33] for details). Here we present exact solutions of the DLV system (66) with a 1 a 2 = 0 and λ 1 a 2 −λ 2 a 1 λ 1 −λ 2 > 0, namely:
u(t, x) = a 1 exp a 1 λ 1 t C 1 −α 0 exp a 1 λ 1 t +C 2 λ 2 exp a 2 λ 2 t α 0 + α 1 exp − κ 2 λ 1 t sin(κx) + α 2 exp − κ 2 λ 1 t cos(κx) , v(t, x) = α 0 a 1 exp a 1 λ 1 t −C 2 a 2 λ 1 exp a 2 λ 2 t C 1 −α 0 exp a 1 λ 1 t +C 2 λ 2 exp a 2 λ 2 t − u(t, x).(73)
Here α i , C 1 and C 2 are arbitrary constants, which should be specified using additional conditions/requirements satisfied by the exact solution (73). Example 2. Using the transformation u → −bu, v → −cv and introducing the notation α 0 → −b α 0 , α 1 → −b α 1 , one reduces the DLV system (66) to the form
λ 1 u t = u xx + u(a 1 − b u − c v), λ 2 v t = v xx + v a 2 − λ 2 b λ 1 u − λ 2 c λ 1 v .(74)
The nonlinear system (74) with positive parameters a 1 , a 2 , b and c can be applied for modeling competition of two population of species. Solution (73) (we set α 2 = 0 just for simplicity) after the above transformation reads as follows
u(t, x) = a 1 exp a 1 λ 1 t C 1 +α 0 b exp a 1 λ 1 t +C 2 λ 2 exp a 2 λ 2 t α 0 + α 1 exp λ 2 a 1 −λ 1 a 2 λ 1 (λ 1 −λ 2 ) t sin λ 1 a 2 −λ 2 a 1 λ 1 −λ 2 x , v(t, x) = 1 c α 0 a 1 b exp a 1 λ 1 t +C 2 a 2 λ 1 exp a 2 λ 2 t C 1 +α 0 b exp a 1 λ 1 t +C 2 λ 2 exp a 2 λ 2 t − b c u(t, x).(75)
In order to provide a biological interpretation, we introduce the following requirements: the u and v components are bounded and nonnegative in a domain because they represent densities of species. Let us consider the domain Ω = {(t, x) ∈ (0, +∞) × (−∞, +∞)}. It can be shown that both components are bounded and nonnegative if the coefficient restrictions
α 0 > |α 1 | , C 2 > max − α 0 b + C 1 λ 2 , ba 1 |α 1 | a 2 λ 1
hold. We also note that the exact solution (75) possesses the asymptotical behavior
(u, v) → a 1 b , 0 , if a 1 λ 2 > a 2 λ 1 , (u, v) → 0, a 2 λ 1 cλ 2 , if a 1 λ 2 < a 2 λ 1 , as t → +∞.(76)
Now one realizes that a 1 b , 0 and 0, a 2 λ 1 cλ 2 are steady state points of the competition model (74) and the asymptotical behavior (76) is in agreement with the qualitative theory of this model (see, e.g., [18] and papers cited therein).
In real-world applications, competition usually occurs in bounded domains. Let us consider the domain Ω * = {(t, x) ∈ (0, +∞) × (A, B)} , −∞ < A < B < +∞. Typically, zero flux conditions are prescribed at the boundaries:
x = A : u x = 0, v x = 0, x = B : u x = 0, v x = 0.
The zero flux conditions reflect a natural assumption that the competing species cannot cross the boundaries (e.g., a wide river could be a natural obstacle). One easily checks that the exact solution (75) satisfies the boundary conditions provided
A = π κ 1 2 + m 1 , B = π κ 1 2 + m 2 , m 1 < m 2 .
Here m 1 and m 2 are arbitrary integer parameters and κ = λ 1 a 2 −λ 2 a 1 λ 1 −λ 2 . Thus, we conclude that the exact solution (75) with correctly-specified parameters simulates the competition of two population of species in the bounded domain. An example is presented in Fig. 3.
Exact solutions of the three-component DLV system
This section is a natural continuation of the previous one. The only difference is that here a three-component DLV system is studied instead of a two-component system. It should be pointed out that the three-component DLV system (16) admits a much wider set of Qconditional symmetries compared to the two-component analogue. One may apply each Qconditional symmetry arising in Table 5 in order to find exact solutions for the biologically motivated DLV system. One notes that the DLV systems arising in Cases 1-4 of Table 5 can be reduced to those modeling different types of interaction between three populations of species (cells, chemicals etc.). Here we examine in details only Case 4 because the corresponding symmetry operators have the most complicated structure (Cases 1-3 can be examined in a quite similar way) and present the results derived in [34]. Obviously, the system from Case 4 of is reducible by the substitution u → −bu, v → −cv, w → −ew to the system
λ 1 u t = u xx + u(a 1 − bu − cv − ew), λ 2 v t = v xx + v(a 2 − bu − cv − ew), λ 3 w t = w xx + w(a 3 − bu − cv − ew),(77)
where the parameters a k , b, c and e are positive constants. System (77) can be used, in particular, for modeling three competing species in the population dynamics. Let us consider an example in the case when ϕ 0 2 (x) and ϕ 0 3 (x) are constants. It can easily checked that the constant solution ϕ 2 = v 0 , ϕ 3 = a 2 − v 0 of the second and third equations of (80) with a 2 = a 3 , generates the following solution of the three-component competition system (77) with a 1 = a 2 = a 3 :
u = ϕ 1 (x) b e δt , v = v 0 c + 1 c α δ − 1 ϕ 1 (x)e δt , w = a 2 −v 0 e − αϕ 1 (x) eδ e δt ,(81)
where ϕ 1 (x) is a solution of the linear ODE
ϕ ′′ 1 − λ 2 δ ϕ 1 = 0.(82)
Interestingly, the exact solution (81) is not obtainable by any Lie symmetry because system (77) admits the Li algebra (15), so that only traveling wave solutions can be constructed.
We point out that the general solution of ODE (82) essentially depends on the sign of δ. In the case δ > 0, unbounded (in time) solutions (see formulae (81)) are obtained and it is unlikely that they can describe a realistic competition between three populations.
On the other hand, equation (82) with δ < 0 has the general solution
ϕ(x) = C 1 cos −δλ 2 x + C 2 sin −δλ 2 x ,(83)
where C 1 and C 2 are arbitrary constants. Setting, for example, C 1 = 0 and C 2 = 1 in (83) and substituting ϕ(x) into (81), we obtain the exact solution
u = 1 b sin √ −δλ 2 x e δt , v = v 0 c + 1 c α δ − 1 sin √ −δλ 2 x e δt , w = a 2 −v 0 e − α eδ sin √ −δλ 2 x e δt(84)
of system (77) with a 1 = a 2 = a 3 , δ = a 1 −a 2 λ 1 −λ 2 . Let us provide a biological interpretation of the exact solution (84). For these purposes, we assume that the competition between three populations occurs at the space interval I = 0, π √ −δλ 2 . Obviously, the components of the exact solution (84) satisfy the boundary conditions
x = 0 : u = 0, v = v 0 c , w = a 2 −v 0 e ; x = π √ −δλ 2 : u = 0, v = v 0 c , w = a 2 −v 0 e .
These conditions predict that the densities of the species u, v and w are constant values at the boundaries (it means that an artificial regulation of the population densities holds in a vicinity of the x = 0 and x = π √ −δλ 2 points). Moreover, this exact solution tends to the steady-state
point 0, v 0 c , a 2 −v 0 e if t → +∞.
It can be checked that all the components in (84) are bounded and nonnegative for an arbitrary given t > 0 and x ∈ I provided the additional restrictions
0 ≤ v 0 ≤ a 2 − α δ , if α ≤ δ, 1 − α δ ≤ v 0 ≤ a 2 − α δ , if δ < α ≤ 0, 1 − α δ ≤ v 0 ≤ a 2 ,
if α > 0 hold. Thus, the exact solution (84) describes the following scenarios of the competition between three species:
(i) species v and w eventually coexist while species u dies out provided
0 < v 0 < a 2 ;
(ii) species v eventually dominates while species u and w die out provided v 0 = a 2 ;
(iii) species w eventually dominates while species u and v die out provided v 0 = 0.
Examples of scenarios (i) and (ii) are presented in Fig. 4 and Fig. 5, respectively.
An essential progress in constructing exact solutions of the three-component DLV system was achieved in [35]. New exact solutions were discovered when system (16) involves equal diffusivities (i.e. λ 1 = λ 2 = λ 3 ), positive a i and negative b i , c i , and e i parameters (i.e. describes competition of three populations). In this case, the DLV system (16) is reducible to the form
u t = u xx + u(1 − u − c 1 v − e 1 w), v t = v xx + c 2 v(1 − b 2 u − v − e 2 w), w t = w xx + e 3 w(1 − b 3 u − c 3 v − w) (85) Assuming that linear terms 1 − u − c 1 v − e 1 w, 1 − b 2 u − v − e 2 w and 1 − b 3 u − c 3 v − w arising
in the RHS of the system are linearly dependent, the following family of exact solutions was derived where w 0 is an arbitrary constant, while f (y) is an arbitrary continuous function such that the integral in the RHS of (86) converges. Although this result is formulated in the form of a cumbersome theorem (see Theorem 2.1 in [35]), the main idea is very simple and was implicitly used earlier in [23]. In fact, according to the assumption, there exist constants A, B, and C such that
u(t, x) = c 1 −1 c 1 b 2 −1 + e 1 −c 1 e 2 c 1 b 2 −1 w 0 + 1 √ 4πt ∞ −∞ exp − (x−y) 2 4t f (y)dy , v(t, x) = b 2 −1 c 1 b 2 −1 + e 2 −b 2 e 1 c 1 b 2 −1 w 0 + 1 √ 4πt ∞ −∞ exp − (x−y) 2 4t f (y)dy , w(t, x) = w 0 + 1 √ 4πt ∞ −∞ exp − (x−y) 2 4t f (y)dy,(86)A(1 − u − c 1 v − e 1 w) + B(1 − b 2 u − v − e 2 w) + C(1 − b 3 u − c 3 v − w) = 0.
So, taking the linear combination of equations from (86), we exactly arrive at the linear diffusion equation
U t = U xx , U = Au + Bv + Cw(87)
with correctly-specified A, B and C. Obviously, the integral in the RHS of (86) is the wellknown solution of (87).
In particular case, solution (86) with f (y) = β sin(γy) (here β and γ are nonzero constants) takes the form [35]
It can be seen that the exact solution (88) is a generalization of solution (84) on the case when all the diffusivities are equal.
Conclusions
This work summarizes all known results (up to this date) about methods of integration of the classical Lotka-Volterra systems with diffusion and presents a wide range of exact solutions, which are the most important from applicability point of view. To the best of our knowledge, it is the first attempt in this direction. Because the DLV systems are used for mathematical modeling of an enormous variety of processes in ecology, biology, medicine, chemistry, etc. (see, e.g., well-known books [6,7,8,9,10,11,12]), we believe that it is an appropriate time for such kind of a review.
We would like to point out that exact solutions always play an important role for any nonlinear model describing real-world processes. At the present time, there is no general theory for integrating nonlinear PDEs (system of PDEs). Thus, construction of particular exact solutions for these equations is a highly nontrivial and important problem. Identifying exact solutions in a closed form that have a physical (chemical, medical, biological etc.) interpretation is of fundamental importance. Even exact solution with questionable applications can be important for proper examination of software packages devoted to numerical solving of systems of PDEs. The obtained exact solutions can also be used as test problems to estimate the accuracy of approximate analytical methods for solving of boundary value problems for PDEs.
In this review, the main attention was paid to symmetry-based methods for exact solving the classical Lotka-Volterra systems with diffusion. We briefly presented the relevant theory (Section 2) and application of the theory to find Lie symmetries of the two-and three-component LV systems (Section 3). Furthermore, we applied the simplest Lie symmetries for constructing plane wave solutions, especially traveling fronts, which are the most popular type of exact solutions in the case of nonlinear evolution equations (Section 4). We also presented the most interesting traveling waves derived by other authors, including those from the pioneering work [19]. It turns out that Lie symmetries have rather a limited efficiency if one looks for exact solutions of the DLV systems, therefore we derived wide families of conditional symmetries of the DLV systems under study (Section 5). Finally, the conditional symmetries obtained were used to construct exact solutions with more complicated structures than the traveling fronts.
Moreover, examples of applications of some exact solutions for solving real-world models based on the DLV systems are successfully demonstrated (Sections 6 and 7). We also presented an interesting family of exact solutions derived in [35] by an ad hoc technique, which seems to be not related with symmetry-based methods.
In conclusion, we would like to highlight some unsolved problems. In this review, a majority of exact solutions are related to the DLV systems describing the competition of two (three) populations of species (cells). However, there are other types of interaction between species, cells, chemicals etc. In particular, the nonlinear system (6), in which all the parameters a i and b ij are nonnegative, is a model describing mutualism or cooperation (see, e.g., [6,58]). Obviously, the solutions presented in this work are useful for interactions of such type as well. On the other hand, these solutions are not applicable for the third most common type of interaction between species (cells, chemicals, etc.) leading to prey-predator models. In the two-component prey-predator model, the parameters satisfy the following typical restrictions a 1 a 2 < 0, c 1 b 2 < 0, b 1 ≤ 0 (see the DLV system (13)). It can be seen that Tables 1, 3 and 4 do not contain such types of systems, therefore the relevant exact solutions cannot be found. Moreover, we have checked that the exact solutions derived in the following studies [19,23,24,25,27,28,32,34] cannot describe the prey-predator interaction either (at least there are not examples highlighting applicability for the interaction of such type). Thus, the problem of finding exact solutions in a closed form for the DLV system (13) modeling the interaction between preys and predators is still unsolved. Probably, traveling fronts of the form (39) are the first example of such exact solutions.
Another problem of construction of exact solutions for the DLV type systems arise when one examines such systems with time-delay in order to take into account, for example, the age of species in the population. Some examples are presented in the very recent paper [59].
The authors are grateful to Yurko Holovach (Lviv, NAS of Ukraine) who brought our at-tention to the very old papers by Julius Hirniak [3,4]. R.Ch. is grateful to late Wilhelm Fushchych who encouraged him to study nonlinear systems of reaction-diffusion PDEs using the Lie symmetry method. Professor Fushchych passed away 25 years ago and the authors would like to dedicate this review to his memory.
change of u for a small time interval] = [the Malthusian law (with the exponent a) of growth of u without predation ] − [the quadratic law of loss of u due to predation with the coefficient b], [change of v for a small time interval] = − [the Malthusian law (with the exponent −c) of loss of v without prey] + [the quadratic law of growth of v due to predation with the coefficient b].
by a substitution of the form u → c 11 exp(c 10 t)u + c 12 , v → c 21 + c 22 exp(c 20 t)v (here c ki (k = 1, 2, i = 0, 1, 2,) are correctly-specified parameters).
Theorem 2
2[34] The DLV system(16) with restrictions (17) admits a nontrivial Lie algebra of symmetries if and only if the system and the corresponding Lie symmetry operators have the forms listed in
− 1 )
1, c 2 = −1, c 3 = 8(1−3e) a(e−1) , e 1 = −e, e 2 = (a−24)
Figure 1 :
1Surfaces representing the components u (blue), v (red) and w (green) of solution(42) with a = 25, e = 2, α = 11 2 of the DLV system (16) with the parameters defined by formulae(41).
is Q-conditionally invariant under operator (44) if and only if b 1 = b 2 = b and c 1 = c 2 = c.
Figure 2 :
2Surfaces representing the u (blue) and v (red) components of solution (65) with C 2 = 1 3 , β = −1 of system (64) with the parameters a 1 = 3, a 2 = 4, b = 1 2 , c = 1 5 , λ 1 = 2, λ 2 = 1.
Figure 3 :
3Surfaces representing the u (blue) and v (red) components of solution (75) with C 1 = −2, C 2 = 5, α 0 = 2, α 1 = 1 of system (74) with the parameters a 1 = 3, a 2 = 2,
Figure 4 :
4Surfaces representing the u (blue), v (red) and w (green) components of solution (84) with α = −1, v 0 = 3 2 , δ = − 5 2 of system (77) with the parameters a 1 = 9 2 , a 2 = a 3 = 2,
Figure 5 :
5Surfaces representing the u (blue), v (red) and w (green) components of solution (84) with α = 3 2 , v 0 = 2, δ = − 5 2 of system (77) with the parameters a 1 = 9 2 , a 2 = a 3 = 2, b = 1 2 , c = 3 4 , e = 1 7 , λ 1 = 1, λ 2 = λ 3 = 2.
+ β sin(γx)e −γ 2 t , v(t, x) = b 2 −1 c 1 b 2 −1 + e 2 −b 2 e 1 c 1 b 2 −1 w 0 + β sin(γx)e −γ 2 t ,w(t, x) = w 0 + β sin(γx)e −γ 2 t .
Table 1 :
1Lie symmetries of the DLV system(13).Reaction terms Restriction Lie symmetries extending algebra (15)
Table 2 :
2Lie symmetries of the DLV system(16) Reaction terms
Restrictions
Lie symmetries extending algebra (15)
Table 4 :
4Q-conditional symmetries of the first type of the DLV system(13) DLV systems
Restrictions
Operators
1
Let as assume that the coefficients a k and λ k (k = 1, 2, 3) satisfy the restrictions presented in Case 4 ofTable 5. It means that the system admits the symmetry operators Q 4 i (i = 1, . . . , 6), which have the same structure. Substituting u → −bu, v → −cv, w → −ew into, e.g., the Q-conditional symmetry operator Q 4 1 we obtainSo, using the standard algorithm to reduce the given PDE system to an ODE system via the known operator (78), one can easily obtain the ansatzwhere ϕ 1 (x), ϕ 2 (x) and ϕ 3 (x) are to-be-determined functions. Substituting ansatz (79) into (77) and taking into account the restrictionCase 4 of Table 5), we arrive at the reduced system of ODEsThus, exact solutions of the three-component competition system (77) can be obtained by substitution of arbitrary solutions of system (80) into ansatz (79). System (80) is three-component system of nonlinear second-order ODEs. To the best of our knowledge, its general solution is unknown. Let us assume that the triplet (ϕ 0 1 (x), ϕ 0 2 (x), ϕ 0 3 (x)) is a particular solution of (80). Moreover, we assume that the functions ϕ 0 k are nonnegative and bounded on a space interval I. Having this, we observe that the exact solution (79) with ϕ k = ϕ 0 k (k = 1, 2, 3) tends to the steady-state solution 0,
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"Nuclear Physics A Collective Effects: the viewpoint of HEP MC codes",
"Nuclear Physics A Collective Effects: the viewpoint of HEP MC codes"
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"Torbjörn Sjöstrand \nDepartment of Astronomy and Theoretical Physics\nLund University\nSölvegatan 14ASE-223 62LundSweden\n"
] |
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"Department of Astronomy and Theoretical Physics\nLund University\nSölvegatan 14ASE-223 62LundSweden"
] |
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"Nuclear Physics A"
] |
Collective effects are observed in high-multiplicity pp events, similar to the signals traditionally attributed to the formation of a Quark Gluon Plasma in heavy ion collisions. In core-corona models it is assumed that a partial plasma formation is indeed possible also in pp, but here the focus is on several recent models that attempt to explain pp data without invoking plasma formation. These attempts are partly successful, but there is still not a unified framework.
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10.1016/j.nuclphysa.2018.11.010
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"https://arxiv.org/pdf/1808.03117v1.pdf"
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1808.03117
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da4f3a5ab4bc377622579fc69462201ad9ae7fc8
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Nuclear Physics A Collective Effects: the viewpoint of HEP MC codes
9 Aug 2018. 2018
Torbjörn Sjöstrand
Department of Astronomy and Theoretical Physics
Lund University
Sölvegatan 14ASE-223 62LundSweden
Nuclear Physics A Collective Effects: the viewpoint of HEP MC codes
Nuclear Physics A
009 Aug 2018. 2018www.elsevier.com/locate/procedia XXVIIth International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2018)collectivityflowsmall systemsjet universalityquark gluon plasmaevent generatorshadronization
Collective effects are observed in high-multiplicity pp events, similar to the signals traditionally attributed to the formation of a Quark Gluon Plasma in heavy ion collisions. In core-corona models it is assumed that a partial plasma formation is indeed possible also in pp, but here the focus is on several recent models that attempt to explain pp data without invoking plasma formation. These attempts are partly successful, but there is still not a unified framework.
Introduction
The relationship between the pp and the AA communities at the LHC is changing. This has been brought about by a set of unexpected observations, wherein high-multiplicity pp events seem to attach smoothly to the behaviour observed in pA and AA collisions, with respect to flavour composition [1] and flow [2,3,4,5,6,7].
For pp we have been used to a simple picture of hadronization, wherein the density of colour fields and hadrons has been assumed low enough that interactions between them can be neglected. Then e + e − and pp events should share many common traits, "jet universality". This picture is embedded in the greater context of event generators [8], wherein also all other aspects of pp events are simulated. These programs provide an overall description of event properties that in the past appeared reasonably successful, and that still today can describe the bulk of data distributions quite well.
For the AA community the key concept has instead been the Quark Gluon Plasma (QGP), how it is created, what properties it has, and how it reverts back to ordinary matter. Catchwords include deconfinement, hydrodynamics, flow, perfect liquid, and more, all unknown in traditional pp approaches. This split has been encouraged by standard QGP theory, where the belief has been that pp collisions cannot generate a sufficiently large volume sufficiently long for a QGP to form [9,10,11].
Thus both communities have acted to keep a barrier between pp and AA physics. But now it is time to open up the discussion and ask some tough questions. Is a QGP formed in high-multiplicity pp events? If not, what other mechanisms could one imagine as being at the origin of the observed pp behaviour? How can these be tested? Specifically, which are the ironclad signals of QGP formation?
Answering these questions will keep us busy in the years to come. In this talk some of the attempts already made will be discussed, within the context of the traditional event generators used to describe pp collisions. Other presentations at this conference provide the view from other vantage points.
pp physics and generators
Hadronization is traditionally assumed to be environment-independent. Since it is a nonperturbative process, free parameters are needed for incalculable quantities, but these can be determined e.g. from LEP e + e − data and then be applied unchanged for LHC pp collisions. The modelling of the partonic state that is to hadronize has to be different in the two processes, of course. In e + e − events only final-state radiation and hadronization needs to be considered, while the composite nature of the proton additionally leads to parton distribution functions, initial-state radiation, beam remnants, and multiparton interactions (MPIs) [12].
MPIs imply several subcollisions in an average pp event, typically with the outgoing scattered partons having p ⊥ scales of a few GeV and being colour-connected to the beam remnants. Thereby a number of colour confinement fields -strings [13] -are stretched essentially longitudinally between the two remnants. A rule of thumb is that a single string gives about one charged particle per unit of rapidity, i.e. a typical LHC value dn charged /dy ≈ 6 would correspond to the order of six strings being pulled out. It is also useful to note that the tail to higher multiplicities is driven by events with many MPIs, rather than e.g. events with a pair of high-p ⊥ jets.
In such a picture there are no collective effects of any importance. The observation of a rising p ⊥ (n charged ) at the SppS was addressed by the introduction of Colour Reconnection (CR), however [14]. Here the colour fields of the event can be redirected, relative to the naive picture of colour-separated MPIs, in such a way that the total string length is reduced. Several different CR scenarios have been proposed, and for each such the P(n charged ), p ⊥ (n charged ) and other data can be used to tune free parameters. CR can give some effects that are of a collective-flow character, by providing transverse boosts to reconnected string pieces, that e.g. gives heavier hadrons higher p ⊥ , but CR does not address many other issues.
Currently the most successful realistic approach to collective effects in pp is the core-corona one, as implemented in the EPOS event generator [15,16]. In it the MPIs give rise to (mainly) longitudinally stretched strings. As long as these strings are well separated they hadronize independently, a "corona", but in case of close-packing they are assumed to collectively give rise to a local QGP, a "core", which expands according to hydrodynamics and hadronizes according to a statistical model. An event can be a mixture of the two. In low-multiplicity events only the corona may exist, but with increasing multiplicity the core fraction increases. EPOS works not only for pp events, but extends the same formalism to pA and AA, where the core QGP component dominates. Obviously this formalism is very economical, in that it does not require the introduction of any new principles. A smooth transition between two extreme behaviours is obtained by changing the admixture, but the core and corona components in a given event are discontinuously separated. EPOS is primarily a model for soft physics, however, not for hard processes and the underlying events associated with them, and therefore is not suited for much of the pp studies at the LHC.
The physics of EPOS is already well known within the AA community, and therefore it will not be discussed any further here. Instead the attention will be turned to other models that have been proposed, specifically within the context of the event generators normally used for pp studies. These are especially interesting insofar as they do not assume the formation of a QGP, but instead introduce alternative physics mechanisms that could be at play. The flip side is that these models are not yet extended to pA and AA collisions.
The three main pp event generators are Pythia [17,18], Herwig [19,20] and Sherpa [21]. While sharing the same overall common structure, there are still many physics differences, and philosophy ones.
• Pythia has its roots back in the string fragmentation studies in the late 70ies. A string can stretch e.g. from a quark end via a number of (colour-ordered) intermediate gluons to an antiquark end, and fragments along its full length [22]. Studies of soft physics have always been central, like MPIs and CR [12]. The Fritiof model for AA collisions [23,24] was a separate offshoot, but recently the related Angantyr pA/AA model is a fully integrated part of Pythia [25]. Many other event generators are built on top of Pythia code, and so are most of the alternative scenarios to be discussed here.
• Herwig was begun in the mid-80ies, to study coherent parton shower evolution [26]. Hadronization is based on cluster fragmentation [27], wherein gluons are forced to break up into quark-antiquark pairs at the end of the shower, such that lower-mass clusters replace the long strings. Both MPIs and CR are modelled, but along somewhat different lines than in Pythia.
• Sherpa grew out of matrix-element generator activities in the late 90ies, and the focus of attention has been in the matching and merging of matrix elements and parton showers. Cluster fragmentation is default, but the program can be linked to Pythia for string fragmentation. The default MPI/CR machinery is inspired by the Pythia one, but the KMR model implemented in the SHRiMPS code will be made available as an alternative [28].
Flavour composition
A significant strangeness enhancement is observed in high-multiplicity pp events [1]. This is visible in K 0 S and Λ production, but in particular in the multistrange Ξ and Ω production. The proton fraction, on the other hand, remains fairly constant, so the effect does not appear to be related to baryon number or particle mass. ALICE shows that this phenomenon is not described by the standard string fragmentation framework, which has an essentially multiplicity-independent particle composition, while the core-corona model in EPOS has the right trends but overshoots, and the rope model in DIPSY/Pythia [29] provides a decent description. Let us study expectations further.
In the standard string model, the string tension κ is assumed to be a constant, κ ≈ 1 GeV/fm. When such a string is pulled out between two receding colour charges, it can break by the production of a quarkantiquark pair that screens the endpoint colour charges. Such a break can be viewed as a tunneling process, where the pair is created in one common point but then q and q each has to tunnel out a distance d = m ⊥q /κ to become on shell, where m ⊥q = m ⊥q is the transverse mass of the quark. This gives a relative probability
P ∝ exp − πm 2 ⊥q κ = exp − πp 2 ⊥q κ × exp − πm 2 q κ ,(1)
i.e. a common Gaussian p ⊥ spectrum for all quarks, and a suppression of the production of heavy quarks. Quark masses are ill-defined, so the strangeness suppression is viewed as a free parameter, of the order of 0.2 -0.25, while charm and bottom are so suppressed that their nonperturbative production can be neglected. Other aspects also influence the meson production, such as the relative rate of pseudoscalars and vectors. The real problem is baryon production. In the simplest approach a colour antitriplet diquark is viewed as equivalent to an antiquark, and produced by the same tunneling process as above. Different diquarks are again suppressed in relation to their squared masses, with some free parameters to represent the uncertainty in diquark mass patterns. Unfortunately the diquark model gives too strong a suppression of multistrange and spin-3/2 baryons. An extension is the popcorn model [30], wherein quark-antiquark pairs are created one at a time, and the suppression of rare hadrons is not as extreme, but still too large e.g. for Ω production. There are also problems e.g. with azimuthal correlations in baryon pairs [31], so it is clear we still lack some fundamental insight on baryon production, at least in the string context.
The transverse size of a string is of the order of the proton radius. Therefore, when two protons collide and several strings are formed by MPIs, it is almost unavoidable that these strings come to overlap in spacetime. This has been used as an argument for colour reconnection, but otherwise the possibility of collective effects has largely been neglected. In the past few years some explicit models have appeared, however.
• The rope model [32,33,29] assumes that several nearby strings can be intertwined into a rope, which represents the field drawn out by the combination of several colour charges. Consider the example of two parallel strings, for which 3 ⊗ 3 = 6 ⊕ 3, where the sextet has a Casimir colour factor C (6) 2 = 5 2 C (3) 2 . In the first break of such a rope the effective string tension is proportional to C (6) 2 − C (3) 2 , i.e. κ eff = 3 2 κ should be used in eq. (1). For a second break (in the same region) the string tension is back to the normal one. For multiple (almost) collinear strings one could expect some kind of random walk in colour space, allowing higher colour charges to be reached, and thereby also larger κ eff . In those string breaks the mass suppression of strangeness and baryon production would be reduced. The rope model describes the production of (multi)strange baryons fairly well, as already mentioned, but does predict a rise of the p/π ratio with increasing multiplicity, in conflict with data.
• Most simple CR models do not change the hadron composition, but a QCD-colour-factor-based CR model does [34]. Again consider the relation 3 ⊗ 3 = 6 ⊕ 3, but now for the 3 possibility that two parallel strings may fuse to produce a normal string, although with the colour flow in the opposite direction. Near either endpoint, where either two q or two q are located, the fused string needs to split into two that stretch to the two endpoint quarks, and the point of splitting is a so-called junction. It becomes associated with a baryon number, and an antijunction at the other end with an antibaryon. Since the number of reconnections increases faster than the number of individual strings, it means that the baryon fraction increases with multiplicity. Furthermore, since a junction baryon consists of the flavours produced at the three separate string breaks closest to the junction in each of the three string legs out from it, production of multistrange baryons is not suppressed by a large strange diquark mass in the tunneling expression. Qualitatively it therefore describes the ALICE trends of Ξ and Ω being more common at high multiplicities, but unfortunately some of the rise is also present for p and Λ.
• From ISR days (pp collisions up to √ s = 62 GeV) it has been known that hadron production p ⊥ can be given a thermodynamical-like description, e.g. in terms of a mass-dependent p ⊥ spectrum dσ
d 2 p ⊥ = N exp − m ⊥had T , m ⊥had = m 2 had + p 2 ⊥ ,(2)
where N and T are (approximately) common for all hadrons. An effectively exponential fall-off could arise also starting from the Gaussian one in eq. (1), assuming that the string tension is fluctuating along its length, also in the absence of other strings [35]. An option has been added to the Pythia string model based on an exponential suppression, but with local flavour and p ⊥ conservation [36]. Such an ansatz gives an overall decent description of the particle composition with only a few free parameters, but does overestimate the rate of multistrange baryons. A variable string tension or "temperature" is used in cases of close-packing of strings, with a continuous change as strings become squeezed into smaller transverse areas, with results similar to those of the (discrete-step) rope model.
In summary, a few different ways have been introduced whereby the string model can be made to display a rising trend of multistrange baryon production. All of them share the problem that this rise inherently is accompanied by a rise of the overall baryon production rate, in contradiction with ALICE data.
In Herwig the cluster model has been improved in two ways [37]. Firstly, if three quark-antiquark clusters are aligned in parallel, then the three quarks can reconnect to a baryon cluster, and the three antiquarks to an antibaryon one. Secondly, nonperturbative g→ss branchings have been introduced, in addition to the conventional g→uu and g→dd ones. Together these two changes gives significant improvements in a number of respects, such as the K/π and p/π p ⊥ spectra. The rate of Λ, Ξ and Ω production is significantly increased, even if still below data. The fraction of strange baryons increases with multiplicity, since the chance of baryon reconnections increases in events with many MPIs, but unfortunately then so does the fraction of protons, just like for the Pythia modifications.
Collectivity and flow
Colour Reconnection can induce some of the signals often attributed to collectivity. The rising p ⊥ (n charged ) is a prime example. Without the CR, the event multiplicity would rise approximately proportionally to the number of MPIs, but with CR each further MPI contributes successively fewer further hadrons. Hence the perturbative p ⊥ associated with the MPIs is be shared among fewer hadrons, giving a larger average.
Heavier hadrons also have harder p ⊥ spectra, with K/π and p/π p ⊥ yield ratios that increase rapidly from almost zero at low p ⊥ scales to maximal at around 2-3 GeV [38,39], in decent agreement with EPOS and Pythia. Three effects contribute: resonance decays, which tends to produce many π at small p ⊥ values, (mini)jet fragmentation, wherein heavier hadrons take a larger fraction of the parton momentum, and transverse string (or cluster) boosts, wherein a string piece moving with a fix transverse velocity will impart that velocity (on the average) to the hadrons produced from its fragmentation. The latter mechanism is enhanced significantly by CR [5]. The p/π fraction drops above the peak position, and this drop is underestimated e.g. in Pythia [5], suggesting that the transition to baryon production in jets is not so well modelled.
In a related characterization, the p ⊥ is rising significantly as a function of the hadron mass [6], which could be interpreted as a sign of collective flow. The same three mechanisms as above combine to produce a decent description in Pythia and Sherpa. The trend is somewhat underestimated, however, and this gap is difficult to close [36]. A model with an exponential m ⊥ spectrum, e.g., intrinsically does give a steeper p ⊥ (m had ) than the Gaussian p ⊥ default, and so should fit better. But most pions come from decays of heavier particles and thus pion changes are more related to the m ρ scale than to the m π one, thereby suppressing differences. Likely some reasonable amount of hadronic rescattering in the final state is needed to bring agreement between data and models. This is already standard in AA generators, using programs such as UrQMD [40] or SMASH [41]. As a first step, the space-time production process in string fragmentation has recently been mapped out [42], confirming that indeed hadrons are produced very closely packed.
One of the most spectacular signals of collective behaviour is the ridge effect [2,3,4], which is not predicted in conventional pp models. It is possible to obtain a realistic model [43] based on the concept of shoving [44], i.e. that strings that overlap in space-time repel each other and thereby build up a transverse velocity. Initially strings are assumed to have zero width, but as they are pulled out in the longitudinal direction they also grow towards full transverse size. Therefore shoving always start at the middle of the local longitudinal rest frame and spreads outwards. For a practical implementation the event is sliced up into one unit wide rapidity ranges, and the net amount of shove on each string spanning that range is calculated. The resulting transverse momentum kick -balanced so as to preserve total momentum -is represented by a single gluon located along the string at the relevant rapidity. The shoving effects on azimuthal correlations become more important at higher multiplicities, in good agreement with data.
Collective flow can be characterized by the commonly-used v n coefficient, which are non-vanishing not only in AA but also in pp collisions [4,7]. Part of this "flow" in pp comes from trivial sources, such as back-to-back (mini)jet pairs, but also colour reconnection and string shoving can contribute to the overall level of the signal e.g. in v 2 {2} [45]. Unfortunately this signal is reduced appreciably when a gap is added to the v 2 {2} extraction, and the enhancement from CR is almost gone, while shove still gives a small positive contribution. Further studies are under way.
In summary, we have bits and pieces of an understanding how collective flow can arise in pp also without a QGP, but not yet a complete view.
Summary and outlook
While a number of interesting and revealing studies have appeared in recent years, the multiplicity dependence of different pp event properties could be investigated further. Are there signs of jet quenching at high multiplicities? A pattern of gradual Υ(1s, 2s, 3s) suppression? A changing temperature of a soft prompt photon spectrum? Changing flavour correlations, e.g. between baryons and antibaryons? Is the flavour composition in jets more similar to e + e − events or to the underlying event? And so on.
Some of these issues can be studied in the context of existing models coming from the AA side, such as core-corona ones. A shortcoming of implementations such as the EPOS generator is that these are focused on soft physics, and therefore do not currently have the full capability to study the interplay between hard and soft aspects. This is where traditional pp event generators such as Pythia, Herwig and Sherpa may offer an advantage. But for too long it was assumed that the combination of multiparton interactions and colour reconnection would generate all the collective effects needed to offer a reasonable description in pp, although we always knew that strings or clusters by necessity would come to be closely packed during the hadronization process, and that this should have repercussions. This is what we now try to remedy. Effects can occur before the strings/clusters start to hadronize, exemplified by shoving or junction formation, during it, like ropes or a gradual change of the string tension, or after it, like hadronic rescattering. So far the formation of a QGP has not been invoked, however, so comparisons with EPOS will remain relevant where possible.
Quite apart from the close-packing issues, but probably exacerbated by them, current models for baryon production fail to provide a convincing description. The string model with a Gaussian suppression of quark and diquark masses, or for that matter an alternative with a Gaussian suppression of hadron masses, suppresses the heavier multistrange baryons too much, while an alternative exponential formulation provides too little suppression. Correlations also are poorly described.
While we do have problems, recent and ongoing studies show that all is not hopeless. Some of the ideas do seem to provide a better understanding of data, but more is needed. It is also necessary to combine all the new pieces into a consistent framework. This could be achieved by upgrading existing AA-style generators to provide more complete descriptions of all kinds of pp event, or by extending pp generators to also simulate AA collisions. A simple stacking of (soft and hard) pp events here offers a possible starting point [46], but the Angantyr model is somewhat more sophisticated, with further developments intended to provide realistic descriptions of all pp/pA/AA collision types.
A key objective for current and future studies should be to better understand which experimental features are ironclad signatures of the formation of a quark gluon plasma, and which could be explained by other effects. The alternative explanations would likely also be of a collective character, like the models presented here, but not require a phase transition to another state of matter.
In summary, a whole new field of study has opened up in the last few years, and (seemingly?) made the borders between pp, pA and AA events crumble. Further experimental input will be crucial to understand what is going on, with model building in the context of event generators offering the main route to providing a global view of all possible effects.
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"A Quantum Paradox of Choice and Purported Classical Analogues",
"A Quantum Paradox of Choice and Purported Classical Analogues"
] |
[
"Emily Adlam \nCentre for Quantum Information and Foundations\nDAMTP\nCentre for Mathematical Sciences\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeU.K\n",
"Adrian Kent \nCentre for Quantum Information and Foundations\nDAMTP\nCentre for Mathematical Sciences\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeU.K\n\nPerimeter Institute for Theoretical Physics\n31 Caroline Street NorthN2L 2Y5WaterlooONCanada\n"
] |
[
"Centre for Quantum Information and Foundations\nDAMTP\nCentre for Mathematical Sciences\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeU.K",
"Centre for Quantum Information and Foundations\nDAMTP\nCentre for Mathematical Sciences\nUniversity of Cambridge\nWilberforce RoadCB3 0WACambridgeU.K",
"Perimeter Institute for Theoretical Physics\n31 Caroline Street NorthN2L 2Y5WaterlooONCanada"
] |
[] |
We recently considered the task of summoning an unknown quantum state and proved necessary and sufficient conditions for Alice to be able to guarantee to complete the task when there may be several possible calls, of which she need only respond to one. We showed that these are strictly stronger conditions than those previously established by Hayden and May for the case where Alice knows there will only be one call. We introduced the concept of a quantum paradox of choice to summarize the implications of these results: Alice is given more options to complete our version of the task, yet one can easily construct examples where our version is impossible and the apparently simpler version considered by Hayden-May is possible.Finkelstein has argued that one can identify analogous classical paradoxes of choice in a relativistic setting. We examine Finkelstein's proposed classical tasks and explain why they seem to us disanalogous.
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"https://arxiv.org/pdf/1509.08094v1.pdf"
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1509.08094
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ac413373d7062b8d3e7bafcb2f597c9673cb6299
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A Quantum Paradox of Choice and Purported Classical Analogues
27 Sep 2015
Emily Adlam
Centre for Quantum Information and Foundations
DAMTP
Centre for Mathematical Sciences
University of Cambridge
Wilberforce RoadCB3 0WACambridgeU.K
Adrian Kent
Centre for Quantum Information and Foundations
DAMTP
Centre for Mathematical Sciences
University of Cambridge
Wilberforce RoadCB3 0WACambridgeU.K
Perimeter Institute for Theoretical Physics
31 Caroline Street NorthN2L 2Y5WaterlooONCanada
A Quantum Paradox of Choice and Purported Classical Analogues
27 Sep 2015(Dated: September 2015)
We recently considered the task of summoning an unknown quantum state and proved necessary and sufficient conditions for Alice to be able to guarantee to complete the task when there may be several possible calls, of which she need only respond to one. We showed that these are strictly stronger conditions than those previously established by Hayden and May for the case where Alice knows there will only be one call. We introduced the concept of a quantum paradox of choice to summarize the implications of these results: Alice is given more options to complete our version of the task, yet one can easily construct examples where our version is impossible and the apparently simpler version considered by Hayden-May is possible.Finkelstein has argued that one can identify analogous classical paradoxes of choice in a relativistic setting. We examine Finkelstein's proposed classical tasks and explain why they seem to us disanalogous.
We recently considered the task of summoning an unknown quantum state and proved necessary and sufficient conditions for Alice to be able to guarantee to complete the task when there may be several possible calls, of which she need only respond to one. We showed that these are strictly stronger conditions than those previously established by Hayden and May for the case where Alice knows there will only be one call. We introduced the concept of a quantum paradox of choice to summarize the implications of these results: Alice is given more options to complete our version of the task, yet one can easily construct examples where our version is impossible and the apparently simpler version considered by Hayden-May is possible.
Finkelstein has argued that one can identify analogous classical paradoxes of choice in a relativistic setting. We examine Finkelstein's proposed classical tasks and explain why they seem to us disanalogous.
INTRODUCTION
We introduced a recent paper [1] with the following metaphor: "A Holistic Magician (HM) repeatedly performs the following trick. He first asks you to give him an object that you are sure he cannot copy. After working behind a curtain, he presents you with N boxes and asks you to choose one. Opening your chosen box, he reveals the original object inside.
You initially imagine that he has arranged some concealed mechanism that somehow passes the object sequentially through the boxes, allowing him to stop the mechanism and keep the object in one box if you select it. However, you are then puzzled to notice that he is unable to make the trick work if you select more than one box, even though you allow him to choose which of your selections to open. This argues against your mechanical explanation, and indeed seems to make any simple explanation problematic. How can giving the magician more freedom make him unable to complete the task?"
A comment by Finkelstein [2] inspires another version:
"A Faux-magician (F) repeatedly performs the following trick. He first describes a signal that can easily be generated and copied: a light flash, for example. After working behind a curtain, he presents you with N boxes with buttons and lights and asks you to choose one box and press its button. When you do so, its light flashes, and none of the other lights also flash. He stresses that he requires both of these conditions for the trick to have succeeded.
You are not at all intrigued or puzzled when F tells you that the trick (as he defines it) will not work if you press several buttons. This is because there seems to be an obvious explanation: each button is a switch for the corresponding light. Pressing several buttons would then cause several lights to flash, meaning that the trick as defined would not work. You thus tell F to get a better act and book HM for your next party."
Notice that the two magicians' tricks have something in common. In both cases you give them more ways of completing the trick by pressing several buttons, and in both cases this appears to prevent them from completing the task. But in the first case, this seems somewhat surprising, while in the second, it seems very unsurprising. The second story replicates one feature of the first while neglecting others. As a result, we think, no one is likely to call the second story puzzling or paradoxical.
RECAP
In Ref. [1], we considered the task of summoning [5] an unknown quantum state. This task involves two agencies, Alice and Bob, who may have collaborating agents distributed throughout space-time. Bob secretly prepares a quantum state, and he (i.e. his local agent) hands it over to Alice (i.e. her local agent) at some point s in space-time. Alice and Bob have agreed on a number of call points c i and corresponding return points r i . Bob may request the state at any call point c i , and Alice is then supposed to return it at the corresponding r i . In the most interesting version of the task, r i > c i and r i > s for each i, where > denotes the causal relation between space-time points. We proved necessary and sufficient conditions for Alice to be able to guarantee to complete the task when there may be several possible calls, of which she need only respond to one. We showed that these are strictly stronger conditions than those previously established by Hayden and May [6] for the case where Alice knows there will only be one call. We introduced the concept of a quantum paradox of choice to summarize the implications of these results: Alice is given more options to complete our version of the task, yet one can easily construct examples where our version is impossible and the apparently simpler version considered by Hayden-May is possible.
As we noted in Ref. [1], the discussion in that paper follows the tradition of using parables and apparent paradoxes to refine our understanding of quantum theory [8][9][10][11][12][13][14]. The results of Refs. [1,6] rely on relativistic causality as well as quantum theory, and we suggested in Ref. [1] that the apparent tension between them may be the first intrinsically relativistic quantum paradox.
There is also a counter-tradition (e.g. [3,4] of criticizing these parables and paradoxes, usually on the grounds that they do not seem (to the critics) even superficially paradoxical, or that they are not intrinsically quantum theoretic, or both. This too can be interesting and valuable: it is certainly worth reflecting on exactly what any proposed example, such as ours, really teaches us about physical principles. As we understand it, Finkelstein [2] follows this latter tradition by suggesting that there are precise classical analogues of our quantum paradox of choice. "Here is a simple example, for which quantum restrictions are not needed: Say there are two laboratories called L and R; let D be the distance between them, and T = R/c the time required for a signal traveling at the speed of light to go either from L to R or from R to L (all times and distances as measured in the frame in which both L and R are at rest). There are two tasks which B might request of A:
Task 1 Send a signal from L to arrive at R at time T , but do not send any signal from R to L. If this task is requested, the request is submitted to A in laboratory L at time t = 0.
Task 2 Send a signal from R to arrive at L at time T , but do not send any signal from L to R. If this task is requested, the request is submitted to A in laboratory R at time t = 0. Clearly it would not be possible for both requests to be fulfilled (just as in the [Adlam-Kent] example where the no-cloning restriction prevents more than one request from being fulfilled). The situation appears paradoxical because B, when making both requests, would be satisfied if either one were fulfilled."
Now, as stated, this example does not work, without imposing further restrictions on A. In particular, it does not work in the framework we consider [5,7] in which she may coordinate a network of collaborating agents distributed wherever she chooses in space-time. If A is allowed this power, she may station an agent A c on a line between laboratories L and R, and send all signals via this agent, who may instantaneously relay them. If B makes both requests, then A c receives two signals, and can choose to relay the first (or, if she is at the midpoint, may choose to relay either one) and intercept the other. B thus receives precisely one valid signal, at one of the two laboratories, whether he makes one request or two.
REFINING FINKELSTEIN'S EXAMPLE
However, Finkelstein's underlying point is clear, and the example can be simply refined to make it work in our framework. Suppose that A has two agents, A 0 and A 1 , at spatially well separated sites distance D apart, and B has agents B 0 adjacent to A 0 and B 1 adjacent to A 1 , so that each B i is separated from A i by distance ǫ ≪ D. All of these agents are stationary in some mutually agreed intertial frame. Now define the following non-local task. At time t = 0 in the agreed frame, each B i will send the corresponding A i a classical bit, 0 or 1. A is guaranteed that at least one 1 will be sent. Her task is to return to the B i , by time t = 2ǫ, two classical bits, one 0 and one 1, ensuring that the 1 is sent to an agent B i who sent her a 1. Now, if A were also guaranteed that only one 1 will be sent, the task is trivial: each A i simply needs to return the bit they are sent -and this is the only way of satisfying the task. However, if it is possible that two 1's will be sent, then A has no way of ensuring that she completes the task. This is true although, if A receives two 1's, there are two possible valid ways of completing the task.
DISCUSSION
In the last example, A may have more valid ways of completing the task, if two 1's are sent, but nonetheless this possibility makes it impossible for her to guarantee completion of the task. But is this in any way paradoxical? Successfully completing the task involves returning to the B i appropriately anti-correlated bits at space-like separated points. If the B i promise to supply these bits -in other words, if they promise to give the A i the data that complete the task -then the A i can indeed complete the task. If they do not, then the A i can not. We understand the term paradox to imply a challenge to pre-existing intuitions, and we suspect few readers will feel such a challenge here.
Compare the summoning task in our original discussion [1]. There Alice needs to get a single quantum state to some requested point in space-time. Yet it turns out that this is strictly harder if she may be given several options for return points. That is, there are strictly fewer sets of request and return points for which it can be achieved. This seems to us, and at least to some others with whom we have discussed the problem, an interesting and initially surprising feature of relativistic quantum theory. As discussed in Ref. [1], it challenges our understanding of whether and how quantum states can be localized in space-time.
Of course, the results are explicable. The relevant theorems were proven in Refs. [6] and [1], and the paradox was resolved as well as presented in Ref. [1]. To summarize: it turns out that, in the version of the task where only one summons is allowed, the anti-correlated data given to Alice by the summonses constitute an exploitable resource. However, to understand how and why this resource is relevant seems at present to require an intuitive understanding both of the possibilities given by iterative uses of teleportation and quantum secret sharing for transmitting quantum information in space-time and of the constraints implied by iterative uses of the no-signalling principle [1,6].
It is perhaps worth emphasizing that, in our view, not every situation (classical or quantum) in which more options make a task harder deserves to be termed a paradox of choice. A single path from A to B is straightforward to navigate, however twisty it may be. Extending it into a maze generally makes things harder, even when there are several paths through the maze. But we would not call this a paradox: to deserve that term requires a challenge to the intuition, and we see none here.
That said, it should also be acknowledged that scientific paradoxes can only be characterized in terms of the limitations of human cognition and of pre-existing mental models, and these may perhaps be different for different readers. There may perhaps be readers whose pre-existing intuitions about summoning relativistic quantum information assured them that the precise results of Refs. [1,6] must be true: if so, we salute them. Although we think it a bit less likely, there may perhaps also be readers whose pre-existing intuitions about classical information in space-time told them that the refined Finkelstein example above should not work. For any such readers (but, we would say, only for them), Finkelstein's term "classical paradox of choice" would indeed be appropriate.
For us, the key point of Ref. [1] is that considering summoning quantum states with single and multiple calls reveals a new and surprising physical distinction between the two, encapsulated in the quantum paradox of choice described therein. This feature is intrinsic to relativistic quantum theory: nothing like it arises in summoning quantum states in non-relativistic quantum mechanics, nor in summoning classical states in relativistic classical mechanics.
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Die gegenwärtige Situation in der Quantenmechanik. E Schrödinger, 10.1007/BF01491891Naturwissenschaften. 23E. Schrödinger, "Die gegenwärtige Situation in der Quantenmechanik," Naturwissenschaften 23, 807-812 (1935).
Quantum Cheshire Cats. Y Aharonov, S Popescu, D Rohrlich, P Skrzypczyk, 10.1088/1367-2630/15/11/113015arXiv:1202.0631New Journal of Physics. 15113015quant-phY. Aharonov, S. Popescu, D. Rohrlich, and P. Skrzypczyk, "Quantum Cheshire Cats," New Journal of Physics 15, 113015 (2013), arXiv:1202.0631 [quant-ph].
Perspectives on the Quantum Zeno paradox. M Itano, Journal of Physics: Conference Series. 19612018M. Itano, "Perspectives on the Quantum Zeno paradox," Journal of Physics: Conference Series 196, 012018 (2009).
Another look at the quantum mechanical entropy of mixing. D Dieks, V Van Dijk, American Journal of Physics. 56D. Dieks and V. van Dijk, "Another look at the quantum mechanical entropy of mixing," American Journal of Physics 56 (1988).
On the implications of the Quantum-Pigeonhole effect. A Rae, T Forgan, arXiv:1412.1333ArXiv e-prints. quant-phA. Rae and T. Forgan, "On the implications of the Quantum-Pigeonhole effect," ArXiv e-prints (2014), arXiv:1412.1333 [quant-ph].
Specker's parable of the overprotective seer: A road to contextuality, nonlocality and complementarity. Y-C Liang, R Spekkens, H Wiseman, Physics Reports. 506Y-C. Liang, R. Spekkens, and H. Wiseman, "Specker's parable of the overprotective seer: A road to contextuality, nonlocality and complementarity," Physics Reports 506, 1-39 (2011) .
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[] |
[
"Stora's fine notion of divergent amplitudes *",
"Stora's fine notion of divergent amplitudes *"
] |
[
"Joseph C Várilly \nEscuela de Matemática\nUniversidad de Costa Rica\n11501San José, Costa Rica\n",
"José M Gracia-Bondía \nDepartamento de Física Teórica\nUniversidad de Zaragoza\n50009ZaragozaSpain\n\nDepartamento de Física\nUniversidad de Costa Rica\n11501San Pedro, Costa Rica\n",
"Raymond Stora "
] |
[
"Escuela de Matemática\nUniversidad de Costa Rica\n11501San José, Costa Rica",
"Departamento de Física Teórica\nUniversidad de Zaragoza\n50009ZaragozaSpain",
"Departamento de Física\nUniversidad de Costa Rica\n11501San Pedro, Costa Rica"
] |
[
"Nuclear Physics B"
] |
Stora and coworkers refined the notion of divergent quantum amplitude, somewhat upsetting the standard power-counting recipe. This unexpectedly clears the way to new prototypes for free and interacting field theories of bosons of any mass and spin. *
|
10.1016/j.nuclphysb.2016.05.028
|
[
"https://arxiv.org/pdf/1605.00237v2.pdf"
] | 119,264,497 |
1605.00237
|
813bb05e4b6016dc26b85b95341e0fad2826d377
|
Stora's fine notion of divergent amplitudes *
1 Jun 2016 26 May 2016
Joseph C Várilly
Escuela de Matemática
Universidad de Costa Rica
11501San José, Costa Rica
José M Gracia-Bondía
Departamento de Física Teórica
Universidad de Zaragoza
50009ZaragozaSpain
Departamento de Física
Universidad de Costa Rica
11501San Pedro, Costa Rica
Raymond Stora
Stora's fine notion of divergent amplitudes *
Nuclear Physics B
1 Jun 2016 26 May 2016, 1930-2015: in memoriam
Stora and coworkers refined the notion of divergent quantum amplitude, somewhat upsetting the standard power-counting recipe. This unexpectedly clears the way to new prototypes for free and interacting field theories of bosons of any mass and spin. *
Exordium
One of us (JMG-B) learned of a flaw in the standard notion of "superficially divergent amplitude" from the lips of Raymond Stora, quickly becoming aware of some of the vistas opened by his alternative notion during intense conversations at CERN in the winter of 2013. In fairness, the notion should be attributed as well to Nikolay M. Nikolov and Ivan Todorov, with whom Raymond was working at the time on paper [1], wherein the matter is expounded in convincing detail. We shall refer to the new notion of convergent Feynman amplitude as the NST renormalization prescription.
We begin by a review of causal Riesz distributions as introduced in [2]. This prelude smooths the way for the new notion of divergent graph, valid for physical quantum fields (what "physical" means will be declared in due course). This helps to open the door to the brave new world of string-local fields. Finally, in Section 6 we show that, although homogeneity of the amplitudes is lost, the concept in [1] makes perfect sense for massive theories.
Causal Riesz distributions and massless field amplitudes
Let us invoke in somewhat simplified form a meromorphic family of distributions on Minkowski space M 4 studied in [2]:
G(x; α) := e −iπα Γ(−α) 4 α+2 π 2 Γ(α + 2) (t 2 − r 2 − i0) α ≡ e −iπα Γ(−α) 4 α+2 π 2 Γ(α + 2) (x 2 − i0) α .(1)
The distribution (x 2 − i0) α is well defined for −2 < ℜα < 0; it can be extended analytically to non-integer ℜα < −2 by repeated applications of ; so (x 2 − i0) α can be regarded as meromorphic in α with (simple) poles at −2 − n for n = 0, 1, 2, . . .. These are cancelled in (1) by the poles of Γ(α + 2). The extension prescription of analytic renormalization, obtained by discarding the pole part in the Laurent expansion of (x 2 − i0) α+ε , is therefore straightforward whenever ℜα > −2, i.e., there is a homogeneous extension. The relation
G(x; α) = G(x; α − 1)
holds, just as for the ordinary Riesz distributions. This is clear from
(x 2 − i0) α = 4α(α + 1)(x 2 − i0) α−1 ,
valid on the chosen domain, and then analytically extended. Note that iG(x; −1) = D F 0 (x), the Feynman propagator for massless scalars; so G(x; −l) for integer l ≥ 2 is proportional to l−2 δ (x). This is confirmed by a direct calculation of the residues at α = −2, −3, . . . . The first aim of this paper is to investigate a generalization of all this for massless particles of higher (integer) helicity. The quantum Maxwell field can be built from the helicity ±1 massless unirreps of the Poincaré group, under the form:
F µν (x) := i ∑ r dµ(p) e i(px) p µ e ν r (p) − p ν e µ r (p) a † r (p) − e −i(px) p µ e ν r (p) * − p ν e µ r (p) * a r (p) ,(2)
for appropriate creation operators a † r (p) and polarization vectors e ν r (p). With g µν denoting the Minkowski metric with (+−−−) signature, routine computation establishes for the vacuum expectation value of the two-point time-ordered product [3]:
T F µν (x) F ρσ (x ′ ) := 0 | T F µν (x) F ρσ (x ′ ) | 0 = g µρ ∂ ν ∂ σ − g νρ ∂ µ ∂ σ − g µσ ∂ ν ∂ ρ + g νσ ∂ µ ∂ ρ D F 0 (x − x ′ ) =: f µν,ρσ (∂ ) D F 0 (x − x ′ ) (3)
valid outside the diagonal x = x ′ . On the face of it, this expression seems logarithmically divergent, since it homogeneously scales like x −4 ; the field itself scales like x −2 .
For brevity, let us write x 2 ≡ x 2 − i0 hereinafter. In the Epstein-Glaser program [4], to renormalize a distribution like T F µν (x) F ρσ (x ′ ) in position space is to find a suitable extension to the diagonal. "Suitable" means keeping the scaling behaviour of the original distribution as much as possible. It also means satisfying physically motivated and mathematically convenient requirements, in particular Lorentz covariance and other symmetries.
Using translation invariance, extension of a distribution f (x − x ′ ) to the diagonal is equivalent to extending f (x), defined for x = 0, to the origin in Minkowski space. Then the distribution x 2α ≡ (x 2 ) α extends homogeneously for α > −2; and for integer α ≤ −2, its extensions can be determined by the complex-analytic methods in [1] or the real-variable methods in [5], adopted in [6]. Thus for instance the extensions of x −4 are given by:
R 4 [x −4 ] = − 1 4 x −2 log x 2 ℓ 2 − iπ 2 δ (x),
with a length scale ℓ. This is log-homogeneous of bidegree (−4, 1) in the terminology of [6].
(The Euclidean version is R 4 [x −4 ] = − 1 4 ∆(x −2 log(x 2 /ℓ 2 ))+π 2 δ (x)
; the two cases differ only in the coefficient of δ (x), arising from the fundamental solutions of the Laplacian, ∆(
x −2 ) = −4π 2 δ (x) in R 4 ; and of the d'Alembertian, (x −2 ) = 4iπ 2 δ (x) in M 4 .)
For two-point functions which are polynomials in x −2 , these procedures go a long way. For the sunset graph in massless φ 4 4 , demanding Lorentz invariance, one can show [6, (2.19)] that
R 4 [x −6 ] = − 1 32 2 x −2 log x 2 ℓ 2 − 5iπ 2 16 δ (x),
whose second term incidentally differs from the one in [1, Eq. (5.29)] due to the precise usage of the multiplicativity property of [5]. One concludes that while unrenormalized two-point amplitudes are homogeneous functions for x = x ′ , they admit log-homogeneous extensions to the diagonal. The second index in the bidegree indicates the power of the logarithm, counting the number of successive extensions for distributions presenting subdivergences, in general: the sunset graph is quadratically divergent, but still primitive in this dispensation. The matter was treated in detail for many graphs of the massless φ 4 4 theory in [6], albeit in the Euclidean signature; happily, only minor modifications are needed for the Minkowskian version. There has been a crop of relatively recent papers dealing with this kind of problem [1,6,7], reaching similar conclusions.
Things appear to be more complicated when the unrenormalized amplitude has an angular dependence, as in our present case (3).
Since ∂ µ ∂ ρ (x −2 ) = −2(g µρ x 2 − 4x µ x ρ )x −6 , we compute (for x = 0): g µρ ∂ ν ∂ σ − g νρ ∂ µ ∂ σ − g µσ ∂ ν ∂ ρ + g νσ ∂ µ ∂ ρ [x −2 ] = −4 (g µρ g νσ − g νρ g µσ )x 2 − 2(g µρ x ν x σ − g νρ x µ x σ − g µσ x ν x ρ + g νσ x µ x ρ ) x −6 =: h µν,ρσ (x) x −6 ,(4)
where each h µν,ρσ (x) is a homogeneous quadratic polynomial.
In fact, each of these polynomials is harmonic in the Minkowskian sense. To see that, it is enough to apply (x 2 ) = 8 and (x µ x ν ) = 2g µν to the quadratic polynomial in (4), to get
h µν,ρσ (x) = −4(8 − 8)(g µρ g νσ − g νρ g µσ ) = 0.
Actually, these h µν,ρσ form a basis for the vector space of quadratic harmonic polynomials on M 4 . Due to (skew)symmetry under the exchanges µ ↔ ν and ρ ↔ σ , and symmetry under (µ, ν) ↔ (ρ, σ ) and (µ, ν) ↔ (σ , ρ), there are 9 linearly independent h µν,ρσ ; whereas the harmonic homogeneous polynomials of degree k on M 4 (or on R 4 , for that matter) form a space of dimension (k + 1) 2 [8, Sect. 9.3].
The NST renormalization prescription
The task then becomes to extend to the origin functions of the form x 2α H k (x), where H k is a homogeneous polynomial of degree k that is also (Minkowskian) harmonic. There are two reasons to hope that the "radial" extensions of [1,6] may prove equal to the task. The first is the off-origin calculation:
(x 2α H k (x)) = (x 2α )H k (x) + 2∂ µ (x 2α ) ∂ µ (H k (x)) + x 2α (H k (x)) = 4α(α + 1)x 2α−2 H k (x) + 4αx 2α−2 x µ ∂ µ (H k (x)) = 4α(α + k + 1)x 2α−2 H k (x),(5)
where we have used harmonicity: H k = 0, and homogeneity: x µ ∂ µ H k = kH k . These relations show that the family of x 2α H k (x) also act like the causal Riesz distributions (1); a suitable normalization is
G(x; α, k) := e −iπα Γ(−α) 4 α+2 π 2 Γ(α + k + 2) x 2α H k (x);
and from (5) we get at once:
G(x; α, k) = G(x; α − 1, k).(6)
The extension prescription of analytic renormalization now tells us that there is a homogeneous extension whenever α > −k − 2. In particular, the case of interest (4) has α = −3 and k = 2. Since −3 > −4, the naïve power-counting recipe is overridden: the time-ordered product (3) does extend homogeneously to the origin, the result being none other than:
T F µν (x) F ρσ (x ′ ) = i 4 π 2 f µν,ρσ (∂ ) 1 (x − x ′ ) 2 − i0 ,
as many a physicist, taking a cue from the commutation relations [9, Aufgabe 7.5], would have written at the outset. In other words, the apparent singularity was removable; according to the lore of renormalization of massless amplitudes, truly renormalization has not taken place.
The general criterion [1,Corl. 5.4] is: a two-point unrenormalized Feynman amplitude in Minkowski space of the form h k (x)/(x 2 ± i0) s for x = 0 has an homogeneous extension if and only if its "degree of harmonicity" k is greater than the "degree of divergence" 2s − k − 4.
Furthermore, in this case the homogeneous extension is unique if we impose Lorentz covariance. This needs to be properly understood. Once a homogeneous extension of x 2α H k (x) has been found, any other such extension can differ from it only by a distribution P(∂ ) δ (x) supported at the origin, where P(x) is a homogeneous polynomial of degree 2(−α) − k − 4, the superficial degree of divergence. In our example, this degree is 0, so P(x) would be a constant. However, H k (x) is not constant: indeed, it transforms under a representation of the Lorentz group on the space of harmonic homogeneous polynomials of degree k, and P(x) must transform likewise. The upshot is that P(x) must be at least divisible by such a harmonic homogeneous polynomial, so that deg P ≥ k. Thus, again the condition k > 2(−α) − k − 4 [in the example: 2 > 0] is enough to ensure that the Lorentz-covariant extension of x 2α H k (x) is unique. In fine: the off-diagonal function (3) extends to a Lorentz-covariant time-ordered product, without ambiguity. Equivalently, one can argue in the spirit of the on-shell extension of amplitudes by Bahns and Wrochna [10]: the decisive fact is that the differential equation (6) is extended to the origin, too.
The prescription for higher helicities . . .
Similarly to the above, there is a free quantum field R αβ ρτ (x), the linearized Riemann tensor, corresponding to helicity-2 particles and transforming as a rank 4 tensor, with the symmetry properties:
R αβ κτ (x) = −R β ακτ (x) = −R αβ τκ (x) = R κταβ (x).
One analogously finds for this:
T R αβ κτ (x) R ρσ λ γ (x ′ ) = ∑ ±G β τ,σ γ ∂ α ∂ κ ∂ ρ ∂ λ D F 0 (x − x ′ ) + 15 similar terms =: 16π 8 3 h αβ κτ,ρσ λ γ (x) D F 0 (x − x ′ ) 5 ;(7)
where G β τ,σ γ := 1 8 g β σ g τγ + g β γ g τσ − g β τ g σ γ and the "similar terms" are obtained by permuting the indices under exchange of (α, β , ρ, σ ) with (κ, τ, λ , γ), (τ, κ, λ , γ), (κ, τ, γ, λ ) and (τ, κ, γ, λ ) respectively; the signs are those that respect the aforementioned symmetries. 1 Therefore, h αβ κτ,ρσ λ γ (x) = ∑ ±G β τ,σ γ q ακρλ (x) is likewise a sum of 16 quartic harmonic polynomials, coming from ∂ α ∂ κ ∂ ρ ∂ λ [x −2 ] =: q ακρλ (x) x −10 by direct calculation, such as:
q ακρλ (x) := 48x α x κ x ρ x λ + g ακ g ρλ + g αλ g κρ + g αρ g κλ x 4 − 6 g ακ x ρ x λ + g αρ x κ x λ + g αλ x κ x ρ + g κρ x α x λ + g κλ x α x ρ + g ρλ x α x κ x 2 .
The harmonic property q ακρλ (x) = 0 is easily checked directly, using:
(x 4 ) = 24x 2 , (x ρ x λ x 2 ) = 2g ρλ x 2 + 16x ρ x λ , (x α x κ x ρ x λ ) = 2g ακ x ρ x λ + 5 similar terms.
Just as before, these h αβ κτ,ρσ λ γ constitute a basis of the 25-dimensional space of quartic homogeneous harmonic polynomials on M 4 . Indeed, taking into account the 20 independent components of R αβ ρτ (x) and the four mentioned symmetries of the cross-indexes, the number of independent h • -polynomials in this case is (20) 2 /2 4 = 25. Now, on the face of it there is a quadratic divergence here -the field scales like x −3 . However, since 4 > 10 − 4 − 4, by the same token as above, the finer NST criterion shows that the vacuum expectation value of the time-ordered 2-point function for the R-tensor field is a convergent amplitude.
How to generalize to higher integer helicities should be clear now: among the free pointlocal fields for helicity h there are two tensor fields with apparently optimal ultraviolet behaviour in relative terms, namely, they scale like x −h−1 : the field strength F µ 1 ν 1 ,...,µ h ν h of rank 2h, symmetric under exchange of any of the pairs (µ i , ν i ) ↔ (µ j , ν j ) and skewsymmetric under exchange inside the pairs; and its potential A µ 1 ,...,µ h of rank h, which is totally symmetric [12,27]. The quantum fields associated to the representation (h, 0) ⊕ (0, h) are "physical" in that their classical counterparts are measurable.
"Apparently" we say, because in fact T F µ 1 ν 1 ,...,µ h ν h F α 1 β 1 ,...,α h β h is a convergent amplitude, as we have seen for h = 1, 2. Whereas the 2-point function for the potentials carries a problematic existence, due to gauge freedom (or slavery) and the impossibility, starting with the photon, for A µ 1 ,...,µ h to live on Hilbert space.
. . . and its consequence: a gauge-free world?
By abandoning point-localization, it is feasible to construct A-fields for any boson particle that share in the good ultraviolet properties of the field strengths. This fact has been known for over ten years now [13,14], and has the potentiality to drastically change the game of perturbative quantum field theory.
The field strengths remain pointlike. To keep notations simple, here we just exhibit a (lightlike) string-local potential field for the photon:
A µ (x, l) := ∞ 0 dt F µν (x + tl) l ν ,
with l = (l 0 , l l l) a null vector. The definition depends only on the ray of l, which is a point of the celestial sphere S 2 , or the light front uniquely associated to it.
A comment is in order here. Previous formulations of string-local fields were based on modular localization theory, which naturally suggests the use of spacelike strings [15]. However, in interacting models this leads to almost intractable complications at third order of perturbation theory. For purely massive models, there is a huge advantage in employing null strings, since then the field is actually a well-behaved function on the l-variable, not just a distribution like in the spacelike case. In models containing massless particles, use of null strings generates a sui generis ultraviolet-infrared problem, which needs to be and can be dealt with by appropriate recipes. Note that all null directions are on the same footing: each one carries its own cyclic subspace, and these are shuffled around by the Lorentz transformations -see right below.
The operator-valued distribution A "lives" on the same Fock space as F, and its main properties are the following:
⋆ Transversality: l A(x, l) = 0.
⋆ Pointlike differential: ∂ µ A ν (x, l) − ∂ ν A µ (x, l) = F µν (x).
⋆ Covariance: let U denote the lifting (or "second quantization") of Wigner's unirrep of the Poincaré group on the one-particle states. Then
U (a, Λ)A µ (x, l)U † (a, Λ) = A ν (Λx + a, Λl) Λ ν µ = (Λ −1 ) µ ν A ν (Λx + a, Λl). ⋆ Locality: [A µ (x, l), A ν (x ′ , l ′ )]
= 0 when the strings x + tl and x ′ + t ′ l ′ are causally disjoint.
The very concept of gauge disappears, since this potential vector, with all the good properties, is uniquely defined. The formalism appears more exotic than the usual one, in that a new variable is invoked. "The choice of what kind of field describes an observed particle is really a matter of choice: try what type of field describes best the observed data" [16]. It is however more mundane, in that it allows us to remain in physical Hilbert spaces: the ghosts can depart, since there is need for them no longer.
Of course, the string "ought not to be seen", and the program becomes to demonstrate whether, and how, this simple criterion is enough to determine interaction vertices and govern perturbative renormalization of string-local models of so-called (Abelian and non-Abelian) gauge interactions [17] from the Lie algebra structure, down to every relevant detail [18,19]. This includes models with massive intermediate vector bosons -see the following section.
The above construction works in a parallel way for all the other integer-helicity cases, like linear gravity, which now are gauge-free, and seen to possess the same ultraviolet properties as scalar particles. 2 What we realize is that the construction of string-local fields [13,14] rests on the bedrock of a never-ambiguous time-ordered product of the field strengths.
Massive field amplitudes
With a suitable change of the polarization vielbeins e ν r , the very formula (2) describes a skewsymmetric quantum field for massive spin 1 particles [3]. In the massive case, Eq. (3) holds as well. A small miracle is involved here, since
F µν (x) = ∂ µ B ν (x) − ∂ ν B µ (x),
where B denotes the Proca field, and for it, outside the diagonal x = x ′ :
T B µ (x)B ν (x ′ ) = i(g µν + ∂ µ ∂ ν /m 2 ) D F (x − x ′ ),
with just D F denoting the massive scalar Feynman propagator. Thus one would expect fourthorder derivatives (a quadratic divergence) in T FF ′ . But they all cancel, so the 2-point time-ordered function off the diagonal x = x ′ looks exactly like the one in (3):
T F µν (x) F ρσ (x ′ ) = g µρ ∂ ν ∂ σ − g νρ ∂ µ ∂ σ − g µσ ∂ ν ∂ ρ + g νσ ∂ µ ∂ ρ D F (x − x ′ ),(8)
but with the massive propagator replacing the massless one. That still looks logarithmically divergent. However, since the ultraviolet properties in both cases are the same, most physicists would conclude without hesitation that the formula makes sense and extends T F µν (x) F ρσ (x ′ ) to the diagonal. We cite Todorov in this context: "Introducing . . . masses in the analysis of small distance behaviour seems to be just adding technical details to the general picture" [21].
The conclusion is correct, and can be substantiated in at least two rather different ways.
⋆ We recall the expansion of D F in the vicinity of m = 0:
D F (x) = D F 0 (x) + m 2 f 1 (m 2 x 2 ) log(−m 2 (x 2 − i0)) + f 2 (m 2 x 2 ) ,(9)
where f 1 , f 2 are analytic. In [22,Sect. 6], it is shown that the basic postulate of Epstein-Glaser renormalization, to wit, that the renormalized amplitudes scale like the unrenormalized ones, up to logarithmic corrections, can be strengthened, in that these corrections -albeit necessarily introducing a new mass scale -do not change the dependence on m in (9); so (8) extends to the diagonal without further ado.
⋆ A method in the spirit of the present paper is as follows [23]. 3 We can modify G(x; α) in (1) by extracting the finite part of Γ(−α)x 2α for α = 0, 1, 2, . . .. This is equivalent to renormalizing the convolution powers of the massless Feynman propagator; these are all primitives, which means that only the first power of the logarithm appears in:
F(x; α) := G(x; α) for α = 0, 1, . . .;
F(x; n) := e −iπn x 2n 4 n+2 π 2 n!(n + 1)! log
m 2 x 2 4 − ψ(n + 2) − ψ(n + 1) − iπ ,
for n = 0, 1, . . .; where ψ is the digamma function.
Note the choice m = 1/l here. Now F(x; α) = F(x; α − 1) holds without restriction [24], so in fact we may write
F(x; α) = −i −1−α D F 0 (x),
for all α ∈ C, and we have a perfect generalization of Riesz theory. Moreover, the series ∑ ∞ n=−1 m 2n+2 F(x, n) solves the massive Klein-Gordon equation with the convolution unit as source [25,26]:
∞ ∑ n=−1 m 2n+2 F(x, n) = −im K 1 (m √ −x 2 ) 4π 2 √ −x 2 = D F (x).
So let us define, for H k homogeneous harmonic of order k:
F(x; α, k) = G(x; α, k) for α = 0, 1, . . .; F(x; n, k) := H k (x) e −iπn x 2n 4 n+2 π 2 n!(n + k + 1)! log m 2 x 2 4 − ψ(n + 2) − ψ(n + 1) − iπ .
Finally, it is clear that the formula
T F µν (x) F ρσ (x ′ ) = f µν,ρσ (∂ ) D F (x − x ′ ),
valid for x = x ′ , extends to the diagonal without further renormalization being necessary. What about higher spins? Following [27], we compute the expected value of the timeordered product of the linearized Riemann tensor for massive gravitons, with a result identical to (7), except that instead of G β τ,σ γ as in Sect. 4, one finds 1 8 g β σ g τγ + g β γ g τσ − 2 3 g β τ g σ γ . 4 This difference between the massive and the massless cases is immaterial for harmonicity since, as we remarked earlier, the polynomials q ακρλ are already harmonic. Therefore T R αβ κτ (x) R ρσ λ γ (x ′ ) extends to the diagonal, without further ado. We conjecture that our conclusions extend to all the massive F µ 1 ν 1 ,...,µ h ν h -fields.
Conclusion
Two small miracles do not a big miracle make. Nevertheless, it is surprising and gratifying that, against appearances, for massive or massless particles of respectively integer spin or helicity j, the quantum fields associated to the representation ( j, 0) ⊕ (0, j) enjoy the same optimal UV properties. These are inherited by the string-local true tensor fields A µ 1 ,...,µ h (x, l) constructed from them. 5 of string-local fields and to Ivan Todorov for bringing reference [29] to our attention. The referee's comments and questions were very instrumental in improving the paper. This research was generously helped by the program "Research in Pairs" of the Mathematisches Forschungsinstitut Oberwolfach in November 2015. The project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 690575, and from the COST Association through the COST Action QSPACE MP1405. JCV acknowledges support from the Vicerrectoría de Investigación of the Universidad de Costa Rica.
The expression for G β τ,σ γ appears in the graviton propagator, see for instance[11, Eq. 1.77].
It appears tempting to redo some of the graviton-scattering calculations in[20], performed in the framework of unimodular gravity, using the A(x, l)-field companion of the linearized Riemann tensor.
This is actually the same paper as[2], but in the published version the pertinent section was withdrawn, because the referee could not make head or tail of it.
A similar expression with the 2 3 coefficient appears in the massive graviton propagator given in [28, Sect. 1.5].5 As we were readying this paper for publication, we were made aware of the article[29]. It also seeks to transfer results from massless to massive models, in a direction different from ours.
AcknowledgementsWe are most grateful to Jens Mund for many discussions on string-local fields, raising some of the issues discussed here, and for making Ref. 27 available to us. Thanks are due to Michael Dütsch for timely comments on the manuscript, to Carmelo P. Martín for comments translating into our footnote 2, to Bert Schroer for constant prodding on the subject
Renormalization of massless Feynman amplitudes in configuration space. N M Nikolov, R Stora, I Todorov, Rev. Math. Phys. 26143000265 pagesN. M. Nikolov, R. Stora and I. Todorov, "Renormalization of massless Feynman amplitudes in configu- ration space", Rev. Math. Phys. 26 (2014), 1430002 (65 pages).
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|
[] |
[
"Maximum Leaf Spanning Trees of Growing Sierpinski Networks Models",
"Maximum Leaf Spanning Trees of Growing Sierpinski Networks Models"
] |
[
"Bing Yao \nCollege of Mathematics and Statistics\nNorthwest Normal University\n730070LanzhouChina\n",
"Xia Liu \nCollege of Mathematics and Statistics\nNorthwest Normal University\n730070LanzhouChina\n",
"Jin Xu \nSchool of Electronics Engineering and Computer Science\nPeking University\n100871BeijingChina\n"
] |
[
"College of Mathematics and Statistics\nNorthwest Normal University\n730070LanzhouChina",
"College of Mathematics and Statistics\nNorthwest Normal University\n730070LanzhouChina",
"School of Electronics Engineering and Computer Science\nPeking University\n100871BeijingChina"
] |
[] |
The dynamical phenomena of complex networks are very difficult to predict from local information due to the rich microstructures and corresponding complex dynamics. On the other hands, it is a horrible job to compute some stochastic parameters of a large network having thousand and thousand nodes. We design several recursive algorithms for finding spanning trees having maximal leaves (MLS-trees) in investigation of topological structures of Sierpinski growing network models, and use MLS-trees to determine the kernels, dominating and balanced sets of the models. We propose a new stochastic method for the models, called the edge-cumulative distribution, and show that it obeys a power law distribution.
| null |
[
"https://arxiv.org/pdf/1601.01465v1.pdf"
] | 8,016,658 |
1601.01465
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24262087b9babe0817a0ded1edea6caf660fa824
|
Maximum Leaf Spanning Trees of Growing Sierpinski Networks Models
7 Jan 2016
Bing Yao
College of Mathematics and Statistics
Northwest Normal University
730070LanzhouChina
Xia Liu
College of Mathematics and Statistics
Northwest Normal University
730070LanzhouChina
Jin Xu
School of Electronics Engineering and Computer Science
Peking University
100871BeijingChina
Maximum Leaf Spanning Trees of Growing Sierpinski Networks Models
7 Jan 2016Spanning treesscale-freeSierpinskialgorithm PACS 8975Da0545Df0210Ox8975Fb
The dynamical phenomena of complex networks are very difficult to predict from local information due to the rich microstructures and corresponding complex dynamics. On the other hands, it is a horrible job to compute some stochastic parameters of a large network having thousand and thousand nodes. We design several recursive algorithms for finding spanning trees having maximal leaves (MLS-trees) in investigation of topological structures of Sierpinski growing network models, and use MLS-trees to determine the kernels, dominating and balanced sets of the models. We propose a new stochastic method for the models, called the edge-cumulative distribution, and show that it obeys a power law distribution.
Introduction
In understanding complex networks, one must know the global properties of networks as well as the local properties such as the degree distribution. Bollobás and Riordan Ref. [2] consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number m of earlier vertices, where each earlier vertex is chosen with probability proportional to its degree. This process was introduced by Barabási and Albert in Ref. [1], as a simple model of the growth of real-world graphs such as the world-wide web.
It is well known that the dynamical phenomena of networks such as traffic and information flow are very difficult to predict from local information due to the rich microstructures and corresponding complex dynamics (Ref. [5]). On the other hands, it is difficult to implement miscellaneous measurements on complex networks, such as the betweenness centrality (BC) b(•) = i =j b(i,•,j) b(i,j) , where • is a node or an edge, and b(i, •, j) is the number of shortest (i, j)-paths through • and b(i, j) is the number of shortest (i, j)-paths. Clearly, this is a horrible job to compute the betweenness centrality of a network having thousand and thousand nodes. But Kim et al. (Ref. [5]), after studying the properties of the spanning trees with maximum total edge betweenness centrality, point out that the scale-free spanning trees represent the communication kernels on networks, and the scale-free spanning trees show robust characteristics in the degree correlation and the betweenness centrality distribution. They defined the communication kernel of a network as the spanning tree with a set of edges maximizing the summation of their edge BC's on the original networks, and investigated the structural and dynamical properties of the spanning tree of complex networks and the role of shortcuts in the networks, and find that the spanning trees show scale-free behavior in the degree distributions.
As known, the Maximum Leaf Spanning Tree (MLS-tree) problem, which asks to find, for a given graph, a spanning tree with as many leaves as possible, is one of the classical NP-complete problems in Ref. [8]. Fernau et al. (Ref. [7]) investigated MLS-trees based on an exponential time viewpoint that is equivalent to the Connected Dominating Set problem (CDSP), and present a branching algorithm whose running time of O(1.8966 n ) has been analyzed using the Measure-and-Conquer technique as well as a lower bound of Ω(1.4422 n ) for the worst case running time of their algorithm. By means of MLS-trees of growing networks in Ref. [11], a stochastic network model can be defined as M t = (p(u, k, t), G(t), U G) for t ∈ [a, b], where U G is the underlying graph of M t that contains all nodes and links appeared in M t for t ∈ [a, b]; p(u, k, t) is the probability of a node u being connected with other k nodes in M t ′ (t ′ ∈ [a, t)); G(t) is the connected topological graph of M t . A node of M t is called an alltime-hub node if it is not a leaf of any MLS-tree of M t at t ∈ [a, b]. A kernel of M t is an induced graph over the set of alltime-hub nodes.
We show our algorithms to find MLS-trees for investigating topological structures of growing Sierpinski network models (GSN-models) introduced in Ref. [13]. All graphs mentioned here are simple, undirected and finite. A leaf is a node of degree one. For a graph G, we let L(G) stand for the set of its leaves, and D(G) be the diameter of G, that is,
D(G) = min i =j {d(i, j)}, where d(i, j)
is the geodesic distance from node i to node j. Notation |X| is the number of elements of a set X.
GSN-models
The graph O in Figure 1 shows the result of a fractal-operation that will be used in the following. For a given triangle ∆ABC with three nodes A, B, C in the plane, we embed another triangle ∆abc with three nodes a, b, c in the inner face of ∆ABC, and then join A with b and c to produce two edges Ab, Ac; join B with c and a to form two edges Bc, Ba; and join C with a and b to generate two edges Ca, Cb. The resulting graph O is called the base, and it has six inner triangles that bound six regions I, III ′ II, I ′ , III, II ′ in the clockwise direction. For simpler statement, we call A, B, C three major nodes of the base O, where A is called the left major node, B the top major node, and C the right major node, and we call a, b, c to be three submajor nodes. We restate the construction of the networks N (t) shown in Ref. [13] by adding a labelling function below.
Construction of GSN-models
Let N (0) be the initial network pictured in Figure 1, and let V (0) be the node set of N (0). We define a labelling f such that f (α) = 0 for each node α ∈ V (0). Do a fractal-operation to the inner face of N (0) by adding a new triangle produces the second GSN-model N (1), and label f (β) = 1 for every node β ∈ V (1) \ V (0). To form the third GSN-model N (2) from N (1), we do a fractaloperation to each inner triangle ∆uvw of N (1) without f (u) = f (v) = f (w), and label each node Figure 1: The left is for illustrating a fractal-operation, and the right is the initial network.
x ∈ V (2) \ V (1) as f (x) = 2. Go on in this way, every GSN-model N (t) can be obtained from the previous GSN-model N (t − 1) for t ≥ 2 by doing a fractal-operation to each inner triangle ∆xyz of
N (t − 1) without f (x) = f (y) = f (z), and label each node w ∈ V (t) \ V (t − 1) as f (w) = t.
Let n v (t) and n e (t) be the numbers of nodes and edges of the network N (t), respectively. Clearly, each N (t) has an outer face ∆ABC and an inner face ∆abc for t ≥ 1 (see Figure 2). Another way to generate N (t) from N (t − 1) is the generalized MLS-TREE algorithm introduced later.
Deterministic statistics of GSN-models
Four first GSN-models N (0), N (1), N (2) and N (3) are shown in Figures 1 and 2, respectively. Notation n d (t) stands for the number of nodes of degree d and let ∆(t) be the maximum degree in N (t). For t ≥ 2, the degree spectrum of N (t) is that each number n d (t) of nodes of degree d = 1+3 k is equal to 3 · 6 t−k for k = 1, 2, . . . , t − 1, respectively; and the number of nodes of maximum degree ∆(t) = 1 + 3 t is n ∆(t) (t) = 6 that are the major nodes A, B, C and the subnodes a, b, c. It is easy to show n v (t) in the formula (1) by the degree spectrum of N (t). Since N (t) is a maximal planar graph, so n e (t) = 3n v (t) − 6, n v (t) = (3 · 6 t + 12)/5, n e (t) = (9 · 6 t + 6)/5 (1) as well as there are n e (t) − n v (t) + 1 inner triangles in N (t) by the famous Euler's formula on planar graphs. By the degree spectrum of N (t) and by the linear preferential attachment rule (Ref. [1], [4]), we can get the probability of joining a new node u out of
N (t) to a node v d(t) of degree d(t) in N (t) as follows, d(t) = k, P (u → v d(t) ) = k(v d(t) ) w∈V (t) k(w) = 3 k + 1 2n e (t) ≈ 5 18 · 3 k 6 t ,(2)where k(x) is the degree of node x in N (t). Again we obtain 3 t−k · P (u → v d(t) ) = P (u → v ∆(t) )
for large integers t > 0, which means that the nodes of large degrees play the role like hubs connecting the whole network together. This phenomenon is known as "the rich get richer" paradigm.
The average (mean) degree k of N (t) is defined as k = 2n e (t) n v (t) = 1 n v (t) v∈V (t) k(v).(3)
The [10] pointed that a giant component exists in the network if and only if k 2 − 2 k > 0. We verify that N (t) holds k 2 − 2 k > 0 when t ≥ 2.
average-square (mean-square) degree k 2 of N (t) is determined by k 2 = 1 nv(t) v∈V (t) k 2 (v). Newman
The edge-cumulative distribution
Motivated from the cumulative degree distribution that is an important character of scale-free networks (Refs. [9], [13]), we propose a deterministic statistic for 2 < t i < t, named as the edgecumulative distribution P e-cum (k), as follows
P e-cum (k) = 1 n e (t) t i j=0 n e (j) = 1 9 · 6 t + 6 t i j=0 (9 · 6 j + 6) = 1 9 · 6 t + 6 15 + 6t i + 54 5 (6 t i − 1) ≈ 6 5 6 t i −t(4)
Plugging t i = t − ln k ln 3 into Eq. (4) leads to P e-cum (k) ∝ 6 5 k −1−ln 2/ ln 3 , which means that P e-cum (k) follows a power law form with the exponent γ k = 1 + ln 2 ln 3 .
(α k , β k )-GSN models
By the degree spectrum of a GSN-model N (t), for 4 ≤ d ≤ 3 k + 1, adding numbers of nodes of
degree d ≤ k together is S(≤ k) = d≤k n d (t) = k i=1 3·6 t−i = 3 5 6 t−k (6 k −1)
, and adding degrees of
nodes of degree d ≤ k is equal to D(≤ k) = d≤k d · n d (t) = k i=1 3 · 6 t−i (3 i + 1) = 3 5 6 t (6 − 1 6 k − 5 2 k ). Thereby, the sum S(≥ k + 1) of numbers of nodes of degree d with 3 k+1 + 1 ≤ d ≤ 3 t + 1 is S(≥ k + 1) = n v (t) − S(≤ k), and the sum D(≥ k + 1) of their degrees is equal to D(≥ k + 1) = 2n e (t) − D(≤ k). If the node-number proportion S(≥k+1) nv(t) = α k , so we have α k = 3 · 6 t−k + 12 3 · 6 t + 12 ∼ 6 k = 1 α k .(5)
we get k = − ln α k ln 6 . From the node-number proportion S(≤k)
nv(t) = 1 − α k , we solve k = − ln α k ln 6 too. We call N (t) an (α k , β k )-GSN-model, where β k = D(≥k+1) 2ne(t) = 1 − D(≤k) 2ne(t) , and furthermore S(≥ k + 1)D(≥ k + 1) = 2α k β k n v (t)n e (t)
. As a test, we take α k = 1 2 · 1 10 6 , so k ≈ 8.0974 and the node-degree proportions are
D(≤ k) 2n e (t) ≈ 1 − 1 6 1 6 k + 5 2 k ≈ 0.997,(6)
and β k ≈ 0.003. The parameters α k , β k show a description of a GSN-model as: The nodes having degrees≤ k show a powerful controlling almost edges of N (t) as k is smaller. Conversely, the nodes with degrees≥ k + 1 are incident to fewer edges of N (t), however, they connect the the network model together. For example, deleting the nodes of N (k) from N (t) with t > k ≥ 1 produces 6 k−1 fragments. Moreover, the deletion of nodes of V (t−1) from N (t) make the remainder consisting of 6 t−1 components (triangles) in which the total number of nodes is equal to 3 · 6 t−1 = 5n v (t − 1) − 12. Obviously, such a deletion destroys the network made by a GSN-model into pieces, since 3 · 6 t−1 /n v (t) = 5 6 .
MLS-trees of GSN-models
In this section we will determine the kernels of GSN-models by MLS-trees in the models, and furthermore we present several algorithms for finding MLS-trees. Figure 1 and six graphs shown in Figure 3):
(a1) rename T M k 1 (i) ∈ M st (i) as I-T M k 1 (i)
and set its left major node c ← A, its top major node A ← B, and its right major node B ← C;
(a2) delete two edges AB, BC of T M k 2 (i) ∈ M st (i), and name the remainder as III ′ -T M k 2 (i), and then set its left major node c ← A, its top major node B ← B, and its right major node a ← C;
(a3) rename T M k 3 (i) ∈ M st (i) as II-T M k 3 (i)
, and set its left major node a ← A, its top major node B ← B, and its right major node C ← C;
(a4) delete two edges AB, BC of T M k 4 (i) ∈ M st (i) and, name the remainder as I ′ -T M k 4 (i), and set its left major node a ← A, its top major node C ← B, and its right major node b ← C;
(a5) rename T M k 5 (i) ∈ M st (i) as III-T M k 5 (i), and set its left major node b ← A, its top major node C ← B, and its right major node A ← C;
(a6) delete its two edges AB, BC of T M k 6 (i) ∈ M st (i) and, name the remainder as II ′ -T M k 6 (i), and set its left major node b ← A, its top major node A ← B, and its right major node c ← C.
Identify the major nodes of the above six graphs I-T M k 1 (i), III ′ -T M k 2 (i), II-T M k 3 (i), I ′ -T M k 4 (i), III-T M k 5 (i) and II ′ -T M k 6 (i) having the same letters into one node, respectively. The resulting is just an MLS-tree T M (i + 1) of N (i + 1), and T M (i + 1) has three submajor nodes a, b, c and three major nodes A, B, C (Ref. Figure 4).
Let f (A) ← 0, f (B) ← 0, f (C) ← 0, f (a) ← 1, f (b) ← 1, f (c) ← 1; f (x) ← f (x) + 1 for x ∈ V (T M (i + 1)) \ {a, b, c, A, B, C}; M st (i + 1) ← M st (i + 1) ∪ {T M (i + 1)}, P i ← P i \ {Q}, go to 3.
return M st (t).
Theorem 1. Every MLS-tree of a GSN-model N (t) for t ≥ 3 has 19 · 6 t−2 leaves.
Proof. It is not hard to verify that each MLS-tree of N (t) has 19 · 6 k−2 leaves for k = 2, 3. So, we can use the induction on time steps t ≥ 3. In the generalized MLS-TREE algorithm, we can select an MLS-tree T M (3) having 19 · 6 3−2 leaves and take k i = k j for all permutations (k 1 , k 2 , . . . , k 6 ) at each time step to obtain an MLS-tree T M (t) having 19 · 6 t−2 leaves for t ≥ 4. Note that N (t) is the resulting of overlapping six models N i (t − 1) ( ∼ = N (t − 1)) for i = 1, 2, . . . , 6, according to the generalized MLS-TREE algorithm. If N (t) has a spanning tree G such that |L(G)| > 19 · 6 t−2 , so G induces six spanning trees G i of N i (t − 1) for i = 1, 2, . . . , 6 by the generalized MLS-TREE algorithm. We have one spanning tree G i with |L(G i )| > 19 · 6 t−2 , which contradicts with the induction hypothesis.
The number |L(T M (t))| of leaves of a MLS-tree
T M (t) of N (t) is 19 · 6 t−2 = 95 18 n v (t − 1) − 38 3
, and also greater than the number
n v (t) − n v (t − 1) of adding new nodes to N (t − 1) since 19 · 6 t−2 = 6 t−2 + n v (t) − n v (t − 1).
Balanced sets in GSN-models
In Ref. [12], a non-empty node subset X of node set
V (t) of a GSN-model N (t) is called a T M - balanced set if |X ∩ T M i (t)| = |X ∩ T M j (t)|(7)
holds for any pair of MLS-trees T M i (t) and T M j (t) of N (t). Obviously, V (t) holds Eq. (7) true, but it is trivial. Our goal is to find T M -balanced sets X = V (t), and such sets are called proper T M -balanced sets.
Theorem 2. For every GSN-model N (t) with t ≥ 3 + k and k ≥ 0, each node of N (k) is not a leaf of any MLS-tree of N (t).
Proof. It is not hard to see that V (0) is not a proper T M -balanced set of N (1) and N (2) (Ref. these two models N (1) and N (2) shown in Figure 2).
i ) = d(A, z (r) i ) = 1 for i = 1, 2, . . . , 3 r−1 with 2 ≤ r ≤ t. In N (t) and N (t − 1), there are two triangles ∆x (t) i y (t) i z (t) i and ∆x (t−1) j y (t−1) j z (t−1) j such that three nodes of ∆x (t) i y (t) i z (t)
i are only connected with nodes y (t−1) j , z (t−1) j and A. In other words, nodes y
(t−1) j , z (t−1) j
are not leaves of T M (t). We can make another
spanning tree H of N (t) by joining A with y (t) i , z (t) i , joining y (t−1) j with x (t)
i , and make z (t−1) j to be a leaf of the new spanning tree. Note that we can do the above work in the rest regions I and III. Eventually, we obtain a spanning tree H * of N (t) such that |L(H * )| ≥ 2 + L(T M (t)), which violates the definition of T M (t) having maximal leaves in N (t). By contradiction to show that each node of V (k) is not a leaf of any MLS-tree T M (t) with t ≥ 3 + k. Assume that N (t) has an MLS-tree T M (t) such that V (k) ∩ L(T M (t)) = ∅ for some k with 1 ≤ k ≤ t − 3. So, we have a leaf x (k) s ∈ V (k) ∩ L(T M (t)) and a triangle ∆x A subset S of node set V (t) of a GSN-model N (t) is called a dominating set if every node of V (t) is adjacent to a node of S or belongs to S. By Theorem 2 we can confirm the following results Although the nodes with degrees≥ k + 1 do not control more edges by Eq. (6), but they are a controlling center in N (t) with t ≥ 3 + k. By Theorem 3 we can find a maximal proper connected T M -balanced set X 1 of N (t), and then get a connected model I 1 induced over X 1 . Next, I 1 has a maximal proper connected T M -balanced set X 2 that induces a connected model I 2 . In this way, we obtain a sequence of models I 1 , I 2 , . . . I m such that I i+1 ⊂ I i , and X i+1 is a maximal kernel of I i for i = 1, 2, . . . , m − 1. Clearly, I i ⊂ N (t − 2i).
(r) i y (r) i z (r) i of N (r) holding distances d(x (k) s , x (r) i ) = 2 and d(x (k) s , y (r) i ) = d(x (k) s , z (r) i ) = 1 for i = 1, 2, . . . , 3 r−k−1 with k + 2 ≤ r ≤ t.
A dynamic algorithm for finding MLS-trees of GSN-models
There are some algorithms to find spanning trees of networks in Ref. [11]. We will apply the Bread-first Search algorithm (BFSA) introduced in Ref. [3] to make our Dynamic First-first BFSA algorithm (DFF-BFSA algorithm) by the motivation of the linear preferential attachment rule (Ref. [4]). Predecessors' children are searched before successors' children, according to a rule of "priority has priority". j,i to the spanning tree T (k) in order to form new spanning tree T (k + 1); m(k + 1) ← max{m(k) + l(x) + 1 :
x ∈ V (k)}, V (k + 1) \ V (k) ← m(k+1) j=m(k)+1 V (k+1) j ; V (k + 1) ← k+1 l=0 m(l) j=m(l−1)+1 V (l)
j , go to 3. 3. If k + 1 = t, T (t) ← T (k + 1), go to 4; otherwise go to 2. Proof. Suppose that there is a cycle C = x 1 x 2 · · · x m x 1 in the graph T obtained by the DFF-BFSA algorithm. Without loss of generality, the level values l(x 1 ) < l(x i ) for x i = x 1 in C. So there are three nodes u, v, w such that l(u) < l(w) and l(v) < l(w), which mean that w ∈ nei(u) \ nei(s) and w ∈ nei(v) \ N (k), but it is impossible since (nei(u) \ nei(s)) ∩ (N (v) \ N (k)) = ∅. Thereby, T contains no cycle (Ref. y ∈ nei(x (l) j,i , k + 1) \ {w ∈ V (t) : l(w) exists}). Notice that N (t) is connected, and at each time step the DFF-BFSA algorithm scans all neighbors of a node. So, T is connected and a spanning tree.
We select the first node u 0 = B in N (2), so l(u 0 = B) = 0 and degree k(u 0 ) is equal to maximum degree ∆(k) = 3 k + 1 of N (k) for 2 ≤ k ≤ t by the DFF-BFSA algorithm. Notice that l(A) = l(C) = 1, so |k(A) − k(C)| = 1 and max{k(A), k(C)} = k(u 0 ) − 1 = 3 t . Three nodes A, B, C of the spanning tree T control other 3 k+1 + 1 nodes of N (k). For t = 0, 1, 2, it is not hard to see diameters D(T ) = 2t. For t ≥ 3, by the DFF-BFSA algorithm, every path P (A, w) from node A to a leaf w has at most length (t − 1) if it does not pass through node B, and each path P (C, w ′ ) from node C to a leaf w ′ has at most length (t − 1) if it does not pass through node B. Thereby, the path from w to w ′ has length 2t, which means D(T ) = 2t.
Notice that the spanning trees have 19 · 6 t−2 leaves for t = 0, 1, 2, 3 (Ref. Figure 5). We can confirm that for t ≥ 4, every spanning tree of N (t) obtained by the DFF-BFSA algorithm has 19 · 6 t−2 leaves according to Theorem 2. The theorem is covered.
Conclusion
For determining the kernels of GSN-models we focus on MLS-trees of GSN-models, and show the structures of some MLS-trees by our algorithms. Clearly, all MLS-trees T M (i + 1) having the shortest diameter 2(i + 1) can be constructed by our generalized MLS-TREE algorithm over all MLS-trees having shortest diameter D(T M (i)) = 2i in M st (i). Although our DFF-BFSA algorithm can not find spanning trees having maximal leaves in any growing network model, however, we verify it for some growing network models, and find out some interesting spanning trees.
Suppose that a triangle ∆xbc and another triangle ∆bcy have a common edge bc in a maximal planar graph G whose faces are triangular. We remove the edge bc and then join x with y by an edge, the resulting is still a maximal planar graph, written as G ′ and say 'flipping the edge bc'. We call the procedure of obtaining G ′ from G a flip operation. Note that every GSN-model N (t) is a maximal planar graph, and "Any pair of maximal planar graphs on n vertices can be transformed into each other by at most 5.2n − 24.4 flip operations (Ref. [6] )." We propose a problem: For what value of a positive integer m, does rewiring m edges of a GSN-model N (t) by the flip operation produce a scale-free network model? For larger integers t > 0, we guess that a maximal planar graph H * obtained from a GSN-model N (t) by flipping some edges having ends in V (t) \ V (t − 3) is scale-free. The above problem leads to a problem of graph theory: Determine finite maximal planar graphs G 1 , G 2 , . . . such that G t−1 is a proper subgraph of G t and each G t obeys a power law distribution.
Figure 2 :
2Three GSN-models N (1), N (2) and N (3).
The notation T M (t) denotes an MLS-tree of N (t), and L(T M (t)) stands for the set of leaves of T M (t). It is easy to verify that |L(T M (0))| = 2, |L(T M (1))| = 4 and |L(T M (2))| = 19.
4. 1
1Construction of MLS-trees of GSN-models Let M st (t) be the set of MLS-trees of N (t) such that three major nodes A, B, C of every MLS-tree of M st (t) are not leaves, and every one of M st (t) has two edges AB, BC and has no the edge AC. Since an MLS-tree of N (t) can be constructed by different MLS-trees of N (t − 1), so we call them non-uniformly MLS-trees. Generalized MLS-TREE algorithm Input: A GSN-model N (t) for t ≥ 3, M st (2) = {T M i (2) : i = 1, 2, . . . , m 2 }, where each i is called a footscript, and m 2 is the number of elements of M st (2). Output: M st (t). 1. M st (2) ← M st (2), P 2 ← {(k 1 , k 2 , . . . , k 6 )}, where (k 1 , k 2 , . . . , k 6 ) are permutations over all footscripts of MLS-trees of M st (2); i ← 2.
2 .
2If i < t, M st (i + 1) ← ∅, go to 3; otherwise M st (t) ← M st (i), go to 5. 3. If P i = ∅, go to 4; otherwise P i+1 ← {(k 1 , k 2 , . . . , k 6 )}, where (k 1 , k 2 , . . . , k 6 ) is a permutation of footscripts of MLS-trees of M st (i); i ← i + 1 go to 4. 4. Take a permutation Q ∈ P i , do (Ref. the graph O of
Figure 3 :
3Six graphs for illustrating the generalized MLS-TREE algorithm.
Figure 4 :
4An MLS-tree H M (3) obtained by six graphs shown in Figure 3 and the generalized MLS-TREE algorithm.
Case 1 .
1k = 0 and t ≥ 3. For any MLS-tree T M (t) of N (t), we say V (0) ∩ L(T M (t)) = ∅ when t ≥ 3. If it is not so, without loss of generality, the left major node A ∈ V (0) ∩ L(T M (t)), namely, the node A is a leaf of a MLS-tree T M (t) of the GSN-model N (t). In the region II ′ , there are triangles ∆x
Case 2 .
2k ≥ 1. The constraint t ≥ 3 + k is necessary, because there exists an MLS-tree T M (3) of the GSN-model N (3) such that (V (1) \ V (0)) ∩ L(T M (3)) = ∅. Furthermore, based on T M 1 (3), we can use the generalized MLS-TREE algorithm to obtain an MLS-trees H(m) such that (V (m − 2) \ V (m − 3)) ∩ L(H(m)) = ∅ with m ≥ 3.
these two triangles form a subgraph O * like the graph O shown in Figure 1. In this case, the position of the node x (k) i is as the same as the node A in Case 1. In the region II ′ of O * , there are triangles ∆x
Based on the same proof shown in Case 1, we can get a contradiction with the definition of T M (t).
Theorem 3 .
3Every GSN-model N (t) with t ≥ k + 3 and k ≥ 0 holds: (i) every V (k) is a proper connected T M -balanced set and induces a connected kernel of N (t); (ii) V (k) ⊂ V (T M i (t)) ∩ V (T M j (t)) for any two MLS-trees T M i (t) and T M j (t) of N (t); and (iii) V (t) \ L(T M (t)) is a connected dominating set of N (t).
A GSN-model N (t) for t ≥ 0. Output: A spanning tree T (t). 1. For the GSN-model N (0), BFSA outputs a spanning tree T (0) with V (0function l such that l(x) = j for x ∈ V well by BFSA for j = 1, 2, . . . , m(0).2. Let nei(x, k) be the neighborhood of a node x of N (k) at time step k. At time step k + 1, here, m(−1) = −1). Implementing BFSA do: For every ordered set V m(l,j) } with l ≤ k, from i = 1 to i = m(l, j), scan y ∈ nei(x (l)j,i , k + 1) \ {w ∈ V (t) : l(w) exists}, b ← l(x (l)j,i ) + 1, l(y) ← m(k) + b, and add y to the ordered set V (k+1) m(k)+b as the last node, add node y and edge yx (l)
4. return T (t) with a level function l.
Theorem 4 .
4The DFF-BFSA algorithm can find an MLS-tree T of a GSN-model N (t) such that T has 19 · 6 t−2 leaves, diameter D(T ) = 2t and maximum degree ∆(T ) = ∆(t) with t ≥ 2.
Figure 5 :
5The DFF-BFSA algorithm produces four MLS-trees with level functions in N (k) for k = 0, 1, 2, 3.
MLS-TREES OF GSN-MODELS
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|
[] |
[
"CVD Formation of Graphene on SiC Surface in Argon Atmosphere",
"CVD Formation of Graphene on SiC Surface in Argon Atmosphere"
] |
[
"Ma Lgorzata Wierzbowska \nInstitute of Theoretical Physics\nFaculty of Physics\nUniversity of Warsaw\nul. Hoża 6900-681WarszawaPoland\n",
"Adam Dominiak \nInstitute of Heat Engineering\nFaculty of Power and Aeronautical Engineering\nWarsaw University of Technology\nul. Nowowiejska 21/2500-665WarszawaPoland\n",
"Kamil Tokar \nInstitute of Theoretical Physics\nFaculty of Physics\nUniversity of Warsaw\nul. Hoża 6900-681WarszawaPoland\n"
] |
[
"Institute of Theoretical Physics\nFaculty of Physics\nUniversity of Warsaw\nul. Hoża 6900-681WarszawaPoland",
"Institute of Heat Engineering\nFaculty of Power and Aeronautical Engineering\nWarsaw University of Technology\nul. Nowowiejska 21/2500-665WarszawaPoland",
"Institute of Theoretical Physics\nFaculty of Physics\nUniversity of Warsaw\nul. Hoża 6900-681WarszawaPoland"
] |
[] |
We investigate the microscopic processes leading to graphene growth by the chemical vapor deposition of propane in the argon atmosphere at the SiC surface. Experimentally, it is known that the presence of argon fastens the dehydrogenation processes at the surface, in high temperature of about 2000 K. We perform ab-initio calculations, at zero temperature, to check whether chemical reactions can explain this phenomenon. Density functional theory and supporting quantum chemistry methods qualitatively describe formation of the graphene wafers. We find that the 4H-SiC(0001) surface exibits large catalytic effect in the adsorption process of hydrocarbon molecules, this is also supported by preliminary molecular dynamics results. Existence of the ArH+ molecule, and an observation from the Raman spectra that the negative charge transfers into the SiC surface, would suggest that presence of argon atoms leads to a deprotonization on the surface, which is necessary to obtain pure carbon add-layer. But the zero-temperature description shows that the cold environment is insufficient to promote the argon-assisted surface cleaning.
|
10.1039/c3cp44378g
|
[
"https://arxiv.org/pdf/1306.2027v1.pdf"
] | 14,958,729 |
1306.2027
|
f024abc150eee4d248a5543e7bde9d034e0a5b03
|
CVD Formation of Graphene on SiC Surface in Argon Atmosphere
9 Jun 2013
Ma Lgorzata Wierzbowska
Institute of Theoretical Physics
Faculty of Physics
University of Warsaw
ul. Hoża 6900-681WarszawaPoland
Adam Dominiak
Institute of Heat Engineering
Faculty of Power and Aeronautical Engineering
Warsaw University of Technology
ul. Nowowiejska 21/2500-665WarszawaPoland
Kamil Tokar
Institute of Theoretical Physics
Faculty of Physics
University of Warsaw
ul. Hoża 6900-681WarszawaPoland
CVD Formation of Graphene on SiC Surface in Argon Atmosphere
9 Jun 2013
We investigate the microscopic processes leading to graphene growth by the chemical vapor deposition of propane in the argon atmosphere at the SiC surface. Experimentally, it is known that the presence of argon fastens the dehydrogenation processes at the surface, in high temperature of about 2000 K. We perform ab-initio calculations, at zero temperature, to check whether chemical reactions can explain this phenomenon. Density functional theory and supporting quantum chemistry methods qualitatively describe formation of the graphene wafers. We find that the 4H-SiC(0001) surface exibits large catalytic effect in the adsorption process of hydrocarbon molecules, this is also supported by preliminary molecular dynamics results. Existence of the ArH+ molecule, and an observation from the Raman spectra that the negative charge transfers into the SiC surface, would suggest that presence of argon atoms leads to a deprotonization on the surface, which is necessary to obtain pure carbon add-layer. But the zero-temperature description shows that the cold environment is insufficient to promote the argon-assisted surface cleaning.
I. INTRODUCTION
Recent progress in nanotechnology attracts much attention to graphene 1-3 . Due to its elastic and electronic properties, this material is a very good candidate for novel devices with extraordinary features [4][5][6][7][8] . Preparation of pure, good quality, and large graphene wafers is of main technological interest. For many years, the SiC surfaces have been used for the graphene sheets growth in the epitaxy process by the Si sublimation 9 . This method, however, introduces many defects and causes that graphene does not possess satisfactory electronic transport properties. Structure of the epitaxial graphene and its interactions with the SiC surface have been studied by Raman spectroscopy 10 .
A new method of the epitaxy, by the chemical vapor deposition 11,12 (CVD), is much less sensitive to the surface defects and enables to obtain high electron mobilities in the graphene layers (up to 1800 cm 2 /Vs) and the grown wafers are large of even 150 mm in diameter 13 . Additionally, the graphene multilayers may be oriented in many stacking sequences 14 . A difference between the graphene growth on SiC by the sublimation and the CVD process is pronounced 15 . Very recent analysis of the experimental parameters in the CVD growth of the graphene and graphite sheets has been reported 16 . The CVD method has been also applied on the silicon dioxide substrate (SiO 2 ) 17 , copper [18][19][20] , nickel 21 and iron 22 . It enables to transfer graphene onto arbitrary substrates 23 .
In the technology, the gas mixture of Ar and propane (C 3 H 8 ), in a role of carbon precursor, is used as an ingredient in the graphene epitaxy by the CVD process 13,24,25 . Propane is used in role of carbon precursor in graphene layer creation process. It is desirable to understand how these compounds participate in the formation of the carbon layers, and especially, what is the mechanism of the removal of the hydrogen atoms from the Si-terminated SiC surface. The substrate surface must be very clean in order to obtain a good quality graphene. A possible functionalisation of graphene with the adsorbed hydrogen is a different issue 26,27 .
In this work, the chemical reactions behind the CVD process are described, and mechanisms of the surface dehydrogenation are checked. These mechanisms are closely related with the noble gases tendency to form the diatomic molecules with protons or, in specific conditions, with the neutral hydrogen atom. The propane molecule, obviously, chemisorbs neither on the Si-nor the C-terminated 4H-or 6H-SiC surface (4H and 6H means the hexagonal crystal structure with the stacking period in the z-axis of 4 and 6, respectively). This is because C 3 H 8 is a molecule with all chemical bonds saturated. It will be shown that it is absolutely sufficient to remove whichever hydrogen atom from the propane molecule in order to adsorb such created specie at the SiC surface. Further dehydrogenation of the molecule makes the adsorption stronger. The problem that arises is what is the propane dehydrogenation initialization event, since there are only the saturated propane molecules present in the gas phase. Therefore, we start from investigations of the reactions with isolated propane in the gas phase, and later we model the following chemical reactions at the surface. The possible role of argon in deprotonization reactions will be discussed. If the deprotonization scenario was true, then it would explain the Raman measurements, whose show that the charge transfers from the adsorbates to the SiC surface 28 .
II. CALCULATION DETAILS
All calculations in this work were performed with the density functional theory (DFT) 29 , using the plane-wave package Quantum ESPRESSO 30 . In order to verify the correctness of the results obtained by the DFT tool, used in the further studies, the solutions for the specific reac-
III. RESULTS
A. Molecular reactions in the gas phase
Initially, we investigated a scenario with the C 3 H 8 −→ C 3 H 7 + H and C 3 H 8 −→ C 3 H 6 + H 2 reactions in vacuum. The reaction energies presented in Table I were obtained with the schemes: restricted (open shell) Hartree-Fock, R(O)HF, without and with the second order perturbation corrections for the dynamical correlations at a level of the Möller-Plesset, MP2, method 33 (both by GAMESS) and the DFT (by Quantum ESPRESSO). Additionally, the dissociation energies of H 2 were calculated to complete a description of the reactions energetics. Details of a set-up used in the calculations are given in the supporting information.
Independently on the approximation level, the removal of one hydrogen from propane needs a considerable amount of energy provided into the system (ca. 4 eV). In a case of the propene molecule (C 3 H 6 ), a part of the energetical cost has been consumed by a formation of the H 2 diatomic bond. Because of high Ar concentration in the gas mixture, it is quite plausible that argon atoms could assist in the above reactions leading to free the hydrogen atom or a proton. This statement would be supported by the results of the quantum chemistry work on the dissociation of the HeH + molecule, led by Wolniewicz 34 , where the separation of proton is an exothermic reaction with about 2.04 eV achieved. Thus, a possibility of argon binding with a proton in our sys-tem was calculated. The results are presented in Table I. The energy gained from ArH + formation is smaller than the energy amount needed to remove one of the hydrogen atoms from the propane molecule. However, the hydrogen ionization energy is still necessary to be taken into account. Some energy might be obtained from any of the kinetic processes, which occur at high temperatures, or from the catalytic reaction with the SiC surface. Indeed, our preliminary results obtained with the MD support that fact. At averaged simulation temperature of Nosé thermostat around 1500 K, C 3 H 8 releases one hydrogen with the kinetic energy around 5 eV and the remaining C 3 H 7 moiety with the kinetic energy of 1.5 eV hits the surface zone and binds at the Si-site.
Fragmentation of the propane molecules might be also caused by the electron transfer from the neutral propane into the positively charged noble gases (with unpaired electrons in the valence shell), as it has been demonstrated experimentally 35,36 . On the other hand, at high temperatures in the range of 1300-1700 K, a similar decomposition of propane could be obtained without noble gases. This process was studied with IR laser absorption kinetic spectroscopy and discussed without any role of argon 37 . However, in the aforementioned experiment, the gas mixture of C 3 H 8 and Ar (as a major compound) has been used.
To complete overview of the argon role in the investigated microscopic mechanisms, it is needed to consider a possibility of the dehydrogenation assisted by the formation of the neutral ArH molecule. Such process seems to be forbiden, since the noble gases have closed valence shell and are expected not to form molecules with other atoms. We have checked, using the DFT and the ROHF methods, that indeed the neutral system ArH does not bind. However, the Van der Waals complexes of Ar with propane have been studied 38 , and also the HeH + and ArH + charged molecules can be formed due to this type of interaction. Moreover, there are also known the diatomic molecules of NeH + , KrH + and XeH + with the corresponding dissociation energies 2.08, 4.35 and 4.32 eV, 39 respectively. Even more interesting are the molecules containing the noble gases and some other atoms, where one or more ingredients are in the excited state. It is known from the experiment that the molecule HArF occurs as stable 40 and existence of HArCl and HHeF molecules have been predicted theoretically 41,42 to be stable too. Recently, the next two new molecules FArCCH and FArSiF 3 have also been proposed 43 .
The crucial information for our investigations of argon role comes from the multiconfigurational calculations for a dissociation of the ArH * molecule in the excited state, performed by Vance and Gallup 44 . The main results of the work mentioned above are summarized in the supporting information. Focusing on those data, we suppose that it is impossible that argon could build a diatomic molecule with neutral hydrogen in our system. This is because the curve minima, in the dissociation channels of the excited argon, are shallow with 1-1.5 eV energy. This energy is much less than the hydrogen binding energy to the surface or hydrocarbon, and the argon excitations are about 11.5 and 11.7 eV. Such energy excitations of the system cannot be accessible on this scale without a strong laser beam.
B. Adsorption at the surface
Assuming that, in a high temperature process, one hydrogen is removed from propane, the C 3 H 7 system can be adsorbed at the surface. Two possibilities of creating such specie were defined: by 1) symmetric or 2) nonsymmetric removal of the hydrogen atom from the original hydrocarbon molecule. Since an adsorption at the 4H-SiC(0001) surface occurs for the both cases, the symmetric CH 3 -CH-CH 3 molecule and the nonsymmetric CH 2 -CH 2 -CH 3 molecule, further removals of the hydro-gen atoms were considered and the adsorption energies were calculated. Following this procedure, the adsorption of a series of the species C 3 H 8−n , with n=1,...,7, was calculated. Finally, the hydrogen-free system, C 3 , was adsorbed at the 4H-SiC(0001) surface. This type of hydrocarbon molecular residues might serve as precursors for the graphene layer or a graphitic buffer layer 45 . The studied adsorbent species build one, two or three valence bonds with the Si-terminated SiC surface. For any studied molecule, the bond order formed with the surface atoms is strongly dependent on the specie-surface geometry and on number of hydrogens. Some of the adsorbent created C-C bonds have a double bond character. The relaxed geometries of the adsorbed species are presented in Figure 1.
All calculated adsorption energies, except the C 3 H 8 molecule, are negative, which means binding state. The modeled surface was considered to be metallic due to a saturation of the surface with hydrogens 46,47 .
Adsorption energies were obtained from a formula valid for the neutral and charged systems:
E ads. = E slab+mol. − E slab − E mol. − N µ e ,(1)
where N is the number of additional electrons in the charged systems (N =0 only in the cases presented in Figure 2). In the adsorption of charged molecules the total energies E slab+mol. and E mol were calculated with additional electrons, and the energy E slab corresponds to the neutral surface. For the chemical potential of electrons, i.e. µ e , we assumed the Fermi level of the pure slab (without the adsorbent) obtained from the quadrature of the electronic density to the proper number of valence electrons in the system with the used pseudopotentials.
Modeling interactions in crystals, using the periodic supercells, introduces spurious interactions between periodic images especially in the case of charged cells with the compensating charge uniform background. In order to take account of these effects, we use the Makov and Payne method 48 implemented in the Quantum ESPRESSO code. All geometries of the systems, taking a part in the adsorption process, were optimized separatelly and non of the configurations was fixed.
The resulting values of the energies for the first three species: C 3 H 7 , C 3 H 6 and C 3 H 5 are depicted in Figure 2. Since it has been assumed, that the dehydrogenation could be assisted by the ArH + molecule formation, the calculations for charged systems were also performed. It follows, that negatively charged species bind weaker to the surface. The binding energy depends on the number of bonds, but also on the local surface strain induced by the adsorbed molecules. For example, the symmetric configuration of C 3 H 6 group binds much stronger than the nonsymmetric one, due to a match of the Siterminated SiC surface lattice with the molecular C-C-C chain. On the other hand, the C 3 H 7 nonsymmetric molecule binds much stronger than the symmetric one, because the CH 3 group in this specie is more distant from the surface when the terminal hydrogen is removed from propane.
There exists a proposal of the charge transfer scenario from the deprotonized site to the SiC surface states (which have extended delocalized character) assisted by formation of the ArH + molecule. The experimental data showed 28 , that the charge distribution near the SiC surface is enhanced after the graphene layer adsorption. Also the binding energy of ArH + , of order 4.15 eV, is slightly larger than the adsorption energy of the hydrogen atom at the Si-site of the 4H-SiC(0001) surface, which amounts to 3.92 eV (from the DFT results). On the other hand, the energy of removal of a proton from the surface is higher of the H ionization potential, about 13.6 eV, minus the work-function of the SiC surface, circa 3.87 eV. Thus, the dissociation energy of a proton amounts to around 13.65 eV. This fact indicates that the zero temperature scenario with the argon-assisted surface chemical reactions does not take place.
Further, the adsorption energies of the species with four or less hydrogens were compared with the adsorption energies of rich hydrogenated molecules. In this comparison, the hydrogens dissociated from a molecule were adsorbed at the surface Si-sites nearby the molecule (somewhere in the middle of the primitive cell used in the calculations). The adsorption sites were distant enough that the adsorbed species do not interact chemically. Although, in an indirect way the surface deformations around the adsorbed molecules affect the adsorption energies. Thus, the final reaction was not just a sum of two separate reactions with the surface. Such picture corresponds to the experimental situation much better than a very separate adsorptions scenario, with hydrogens in the infinite distance from the molecule. The results of calculations for the aforementioned processes are included in the supporting information, since the barriers were calculated via the reactant in vaccum, and they do not include TABLE II. Barriers (in eV) for the reactions below, which occur at the SiC surface, for the symmetric and nonsymmetric adsorbates. The reaction directions are denoted by arrows (→) and (←) and defined by the differences between the highest energy configuration on the way from the left-to the right-hand side of given reaction and the energy of the starting (for →) or the final (for ←) configuration, respectively, calculated within the NEB approach.
Reaction the catalytic role of the surface.
C. Energy barriers for the surface catalyzed dehydrogenations
Since the dehydrogenation processes which occur via the geometric configurations in vacuum show very high transition energies (see the supporting information), we calculated also the minimum-energy paths for chosen reactions which take place at the surface. In order to obtain the barriers for the reactions close to the surface, we applied the climbing-image nudged-elastic-band method (NEB), implemented in the Quantum ESPRESSO code 30 . The results for chosen reactions are presented in Table II. Barrier energies are collected in columns corresponding to the symmetric and nonsymmetric geometries and to forward and backward reaction directions. The difference between the highest energy on the reaction path and the energy of the starting (or the final) geometric configuration gives the barrier for the reaction forward → (or backward ←). The energy differences between the starting and the final configurations can be obtained from the differences (←) -(→). The barriers obtained on the minimum-energy path are not high. This implies, that the surface acts as a strong catalyzer in the dehydrogenation process of the hydrocarbon molecules.
The preliminary MD simulations of processes after the adsorption of C 3 H 7 show also cascade of dissociations. First, the released hydrogen from C 3 H 8 , or some other H from the atmosphere, collides with the remaining middle H of C 3 H 7 , dissociating it and effectively creating H 2 outgoing back to the atmosphere. In the following dynamical evolution (time scale of 90-280 fs), one H atom from the tail CH 3 -group of remaining at the surface C 3 H 6 specie is released, and immediately attracted to the surface Si-site neighbouring to the adsorption site of just deprotonized C 3 H 5 .
IV. CONCLUSIONS
Role of argon and the SiC surface as catalysts in the dehydrogenation processes has been investigated. We started with a removal of one hydrogen atom from the C 3 H 8 molecule and found it to be sufficient to initiate the adsorption reactions, which may continue with further dehydrogenation of molecules and more strong binding, up to the C 3 moiety at the 4H-SiC(0001) surface. Barriers for the dehydrogenation of molecules at the surface, with one of the reactants in vacuum and other at the surface, are very high; except the first dehydrogenation of propane (see supporting information). On the other hand, the barriers obtained on the minimum-energy paths for the hydrogen transfer from the adsorbed hydrocarbons onto the nearest Si-site at the SiC surface are rather low. We conclude, that the SiC surface should act as a strong catalyzer in graphene epitaxy by the chemical vapor deposition process.
For the first time, we studied the chemical character of the dehydrogenation of molecules at the SiC slab, and not just the mechanical removing of the H atoms by the floating gas. We check a microscopic mechanism for the dehydrogenation of the SiC surface, assisted by the binding reaction of a proton to argon forming the ArH + molecule. After this process, the electronic charge could remain on the surface 28 . The zero-temperature description, how-ever, indicates that all proposed chemical reactions cannot occur without additional processes caused by the high temperature kinetics or by a strong laser beam.
Preliminary MD simulations without Ar in the atmosphere above the surface, performed at high temperature of about 1500 K, confirm the scenario with a cascade of dehydrogenations of the adsorbed hydrocarbons, and the fact that some of the dissociated hydrogens remain at the surface.
V. ACKNOWLEDGEMENT
We would like to thank Jacek Majewski for many useful discussions. This work has been supported by the European Funds for Regional Development within the SICMAT Project (
FIG. 1 .
1Adsorption geometries of propane and all transition C3H8−n species, where n=1,2,...,8 (up to the "naked" carbons) at the Si-terminated SiC surface. The starting and final configurations, C3H8 and C3, are in the first row. The second and third rows present the symmetric and the nonsymmetric cases, respectively, for the descending number of hydrogen atoms from the left-to the right-hand side.
FIG. 2 .
2Adsorption energies of the first three neutral and charged species at the surface obtained from a removal of the hydrogens from propane.
Contract No. UDA-POIG.01.03.01-14-155/09) and by the European Union in the framework of European Social Fund through The Didactic Development Program of The Faculty of Power and Aeronautical Engineering of The Warsaw University of Technology. Calculations have been performed in the Interdisciplinary Centre of Mathematical and Computer Modeling (ICM) of the University of Warsaw within the grant G47-7 and in Polish Infrastructure of Informatic Support for Science in European Scientific Space (PL-Grid) within the projects nr POIG.02.03.00-00-028/08-00 and MRPO.01.02.00-12-479/02.
TABLE I .
IReaction energies (in eV), defined as the total energies of the products minus the total energies of the substrates, for the removal of hydrogen from propane. The parameter re indicates the bond lengths (inÅ). C3H7 is obtained from C3H8 by a dissociation of H from the middle C. And C3H6 is the propene molecule (hydrogens are dissociated from the middle and terminal C of propane).Reaction
R(O)HF MP2 DFT exp. 33
C3H8 −→ C3H7 + H
3.538 4.128 4.208
-
C3H7 −→ C3H6 + H
1.671 1.341 1.850
-
C3H8 −→ C3H6 + H2 1.662 1.450 1.563
-
H2 −→ H + H
3.547 4.018 4.466 4.75
re (H2)
0.730 0.738 0.753 0.741
ArH + −→ Ar + H +
2.825 3.048 4.151
-
re (ArH + )
1.310 1.328 1.339
-
tions were validated by the all-electron calculations with
the quantum chemistry package GAMESS 31 , which em-
ploys the localized basis sets and treats the Coulomb in-
teractions by means of the perturbative and/or the multi-
configuration methods. To get insight into mechanisms of
hydrocarbon dehydrogenation on the surface, some pre-
liminary molecular dynamics (MD) simulations at ther-
mostat temperatures ≈ 1500 K were perfomed with the
SIESTA code 32 .
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[] |
[
"Non-Standard Extensions of Gradient Elasticity: Fractional Non-Locality, Memory and Fractality",
"Non-Standard Extensions of Gradient Elasticity: Fractional Non-Locality, Memory and Fractality"
] |
[
"Vasily E Tarasov [email protected] ",
"Elias C Aifantis ",
"\nSkobeltsyn Institute of Nuclear Physics\nLaboratory of Mechanics and Materials\nLomonosov Moscow State University\n119991MoscowRussia\n",
"\nAristotle University of Thessaloniki\n54006ThessalonikiGreece\n"
] |
[
"Skobeltsyn Institute of Nuclear Physics\nLaboratory of Mechanics and Materials\nLomonosov Moscow State University\n119991MoscowRussia",
"Aristotle University of Thessaloniki\n54006ThessalonikiGreece"
] |
[] |
Derivatives and integrals of non-integer order may have a wide application in describing complex properties of materials including long-term memory, non-locality of power-law type and fractality. In this paper we consider extensions of elasticity theory that allow us to describe elasticity of materials with fractional non-locality, memory and fractality. The basis of our consideration is an extension of the usual variational principle for fractional non-locality and fractality. For materials with power-law non-locality described by Riesz derivatives of non-integer order, we suggest a fractional variational equation. Equations for fractal materials are derived by a generalization of the variational principle for fractal media. We demonstrate the suggested approaches to derive corresponding generalizations of the Euler-Bernoulli beam and the Timoshenko beam equations for the considered fractional non-local and fractal models. Various equations for materials with fractional non-locality, fractality and fractional acceleration are considered.
|
10.1016/j.cnsns.2014.10.002
|
[
"https://arxiv.org/pdf/1404.5241v3.pdf"
] | 55,442,198 |
1404.5241
|
94304041461233f3765beeab7fa4ddc1f5e64a93
|
Non-Standard Extensions of Gradient Elasticity: Fractional Non-Locality, Memory and Fractality
3 Oct 2014
Vasily E Tarasov [email protected]
Elias C Aifantis
Skobeltsyn Institute of Nuclear Physics
Laboratory of Mechanics and Materials
Lomonosov Moscow State University
119991MoscowRussia
Aristotle University of Thessaloniki
54006ThessalonikiGreece
Non-Standard Extensions of Gradient Elasticity: Fractional Non-Locality, Memory and Fractality
3 Oct 20141 Emeritus Professor of Engineering, Michigan Tech, Houghton, MI 49931, USA Distinguished Adjunct Professor of King Abdulaziz University, Jeddah, 21589, SA 14510Hj6220Dc8140Jj Keywords: fractional continuum mechanics, fractional gradient elasticity, materials with mem- ory, fractal meterials
Derivatives and integrals of non-integer order may have a wide application in describing complex properties of materials including long-term memory, non-locality of power-law type and fractality. In this paper we consider extensions of elasticity theory that allow us to describe elasticity of materials with fractional non-locality, memory and fractality. The basis of our consideration is an extension of the usual variational principle for fractional non-locality and fractality. For materials with power-law non-locality described by Riesz derivatives of non-integer order, we suggest a fractional variational equation. Equations for fractal materials are derived by a generalization of the variational principle for fractal media. We demonstrate the suggested approaches to derive corresponding generalizations of the Euler-Bernoulli beam and the Timoshenko beam equations for the considered fractional non-local and fractal models. Various equations for materials with fractional non-locality, fractality and fractional acceleration are considered.
Introduction
Derivatives and integrals of non-integer orders [1]- [5] have wide applications in mechanics and physics [6]- [15]. The tools of fractional derivatives and integrals allow us to investigate the behavior of materials and systems that are characterized by power-law non-locality, power-law long-term memory and fractal properties. As concluded from the above listed works, there are different definitions of fractional derivatives such as Riemann-Liouville, Riesz, Caputo, Grünwald-Letnikov, Marchaud, Weyl, Sonin-Letnikov and others. The specific choice of fractional derivatives for a particular application, it thus depends on the taste of the investigator and the nature of the material or system at hand. Many properties of standard differentiation and integration do not extend in the fractional case and fractional counterpart of popular models need to be rederived, each on individual basis.
Usually non-local continuum mechanics are treated with two approaches [16]: The gradient elasticity theory (weak non-locality) and the integral non-local theory (strong non-locality). The fractional calculus can, in fact, be used to formulate a generalization of non-local theory of elasticity in both forms: fractional gradient elasticity (weak power-law non-locality) and fractional integral elasticity (strong power-law non-locality). In this paper, we consider fractional generalizations of the gradient elasticity theory only. In particular, we suggest fractional generalizations of a rather popular robust GRADELA model proposed by Aifantis and co-workers [17]- [22] for the following cases:
(1) The elasticity of materials with power-law non-locality that can be described by derivatives of non-integer order. Both 1D and 3D models are discussed.
(2) The elasticity of materials with power-law memory that can be described by fractional time derivatives for the internal inertia or combined strain-acceleration fractional gradient terms.
(3) The elasticity of materials with fractal structure that can be described by fractional integrals in the framework of fractional continuum models.
The basis of our consideration is an extension of the usual variational principle for materials with fractional non-locality, memory and fractality. For 3D spatial fractional models we also use the apparatus of fractional vector calculus. An extension of the traditional calculus of variations for systems described by fractional derivatives was first proposed by Agrawal in [23] for the Riemann-Liouville derivatives. Then it has been extended for other type of fractional derivatives [24]- [29], and fractional integrals [30]. For materials with power-law non-locality and memory, we suggest a new fractional variational principle for Lagrangians with Riesz fractional derivatives. A possible generalization of gradient elasticity theory for fractal materials was alluded in [31]. In this paper we describe fractal materials by using the fractional continuum formalism suggested in [32,33] (see also [34]- [37]). To obtain governing equations for fractional integral continuum models of fractal materials, we employ a generalization of the holonomic variational principle suggested in [36,35]. In this connection, we note that extremum and variational principles for non-gradient but fractal elastic materials within a fractional continuum model framework have been considered in [37,38].
The Euler-Bernoulli beam theory may be viewed as a benchmark example of the classical linear theory of elasticity. It provides tools for calculating the load-carrying and deflection characteristics of beams subjected to lateral loads only. In order to illustrate the implications of the suggested fractional approaches in this paper, we use a variational principle to derive the corresponding generalizations of the static and dynamic Euler-Bernoulli beam model, as well as that of the Timoshenko beam model for the fractional non-local and fractal cases. Solutions to some of these equations for fractional non-local and fractal beams are considered. Next, we list some non-standard generalizations of constitutive relations for gradient elasticity models. First, we recall the linear elastic constitutive relations for isotopic and homogeneous bodies, i.e.,
σ ij = λε kk δ ij + 2µε ij ,(1)
where σ ij is the stress tensor, ε ij is the strain tensor, whereas λ and µ are the Lame coefficients.
In [17]- [19] it was suggested a generalization of the constitutive relations (1) by a gradient modification that contains the Laplacian ∆ in the form
σ ij = λε kk δ ij + 2µε ij − l 2 s ∆ λε kk δ ij + 2µε ij ,(2)
where l s is an internal length scale parameter [16]. To describe complex materials characterized by non-locality of power-law type, long-term memory, and fractality, we should further generalize Eq. (1) and its gradient counterpart given by Eq. (2). In this paper, we consider the following non-standard generalizations of the gradient stress-strain relation.
(1) The fractional gradient elasticity models with power-law non-locality σ ij = λε kk δ ij + 2µε ij − l 2 s (α) (− R ∆) α/2 λε kk δ ij + 2µε ij ,
where (− R ∆) α/2 is the fractional generalization of the Laplacian in the Riesz form, and
σ ij = λε kk δ ij + 2µε ij − l 2 s (α) C ∆ α W λε kk δ ij + 2µε ij ,(4)
where C ∆ α W is the fractional Laplacian in the Caputo form. (2) The combined fractional strain gradient-internal inertia model with power-law memory and non-locality
σ ij = λε kk δ ij + 2µε ij − l 2 s (α) (− R ∆) α/2 + l 2 d (β) ( R D β t ) 2 λε kk δ ij + 2µε ij ,(5)
where ( R D β t ) 2 is the square of the derivative of non-integer order β with respect to time t, which describes acceleration with power-law memory.
(3) The gradient elasticity models for fractal materials
σ ij = λε kk δ ij + 2µε ij − l 2 F (D, d) ∆ (D,d) λε kk δ ij + 2µε ij ,(6)
where ∆ (D,d) is the "fractal-Laplacian" that takes into account the power-law density of states of the fractal medium under consideration.
The plan of the paper is as follows: In Section 2, we consider one-dimensional (1D) fractional gradient elasticity models. A variational principle for these models is suggested. Fractional Euler-Bernoulli and Timoshenko beam equations are derived. Solutions for the fractional static and dynamic Euler-Bernoulli beam governing equations are proposed. Corresponding fractional beam models with combined strain-internal inertia gradient terms are also considered. Moreover, solutions of the relevant generalized equation and dispersion law for this model are derived. In Section 3, three-dimensional (3D) fractional gradient elasticity models are formulated and discussed. In particular, 3D problems with spherical symmetry based on the Riesz fractional derivative are considered. In addition, fractional 3D gradient elasticity models based on fractional vector calculus are suggested. The operator split method for solving the relevant fractional gradient elasticity equations is formulated. To illustrate the potential of this method, a simple fractional gradient model is considered as an application and an explicit solution is provided. In Section 4, some basic concepts for extending gradient elasticity models to fractal media are suggested. The equilibrium equations for fractal materials are first derived. A variational principle for obtaining gradient elasticity equations for fractal materials is then proposed. Finally, in Section 5, generalizations of the Euler-Bernoulli and Timoshenko beam equations for fractal materials and the corresponding equations for the combined strain-acceleration gradients fractal beam models are derived.
2 Fractional 1D gradient elasticity Fractional elasticity models are those for which non-locality of power-law type is described by using derivatives and integrals of non-integer order. We can derive such phenomenological fractional elasticity models by using a variational principle for a Lagrangian with fractional derivatives. A generalization of the traditional calculus of variations for systems described by Riemann-Liouville fractional derivatives has been suggested by Agrawal in [23]. Then, extensions of variational calculus for the Riemann-Liouville derivatives [24], the Caputo derivative [25]- [27], the Hadamard derivative [28], the Riesz derivative [29], as well as fractional integrals [30], have been derived.
If we use the fractional derivatives of Riemann-Liouville, Caputo, Liouville, Marchaud, then we should take into account the left-sided and the right-sided fractional derivatives in the Lagrangian. The correspondent fractional Euler-Lagrange equations contain the left-sided and the right-sided fractional derivatives also. In addition, the integration by parts, which is used in the derivation of the Euler-Lagrange equations from the variational principle, transforms the left-sided derivatives into the right-sided (see Eq. 2.64 of [1]). As a result, we obtain a mixture of left-sided and the right-sided derivatives in the equations of motion. Unfortunately, these Euler-Lagrange equations can be solved for a very narrow class of Lagrangians only.
In this paper, we suggest a fractional variational principle for systems that are described by Riesz fractional derivatives [1,2]. The suggested principle differs from the one proposed in [29]. We take advantage of the fact that the Riesz derivative does not involve two forms, i.e., left-sided and right-sided derivatives. In addition, integration by parts transforms the Riesz fractional derivative into itself. The corresponding fractional Euler-Lagrange equations can be solved for a wide class of Lagrangians that describe nonlocal materials by the methods described in [2]. Moreover, the Riesz fractional derivatives naturally arise in the elasticity theory based on lattice models [52]- [56]. As an example, we derive the fractional gradient generalization of the Euler-Bernoulli beam model and provide some general solutions of the corresponding equations for both static and dynamics configurations.
Fractional 1D gradient elasticity from variational principle
To generalize standard variational principles for fractional nonlocal models, we write all expressions in dimensionless coordinate variables. We can introduce the dimensionless variables
x k = x ′ k /l 0 , r = r ′ /l 0 ,
where l 0 is a characteristic scale. This allows us to have usual physical dimensions of measured quantities.
The equation for the fractional gradient elasticity can be derived as the Euler-Lagrange equation of the following action
S[w] = dt dx L(w, D 1 t w, R D α 1 x w, R D α 2 x w),(7)
where L(w,
D 1 t w, R D α 1 x w, R D α 2 x w)
is the Lagrangian defining the 1D fractional elasticity model, w = w(x, t) denotes the displacement field, and x is the dimensionless coordinate.
The variation of the action functional (7) with respect to w(x, t) and its derivatives is given by
δS[w] = dt dx δL = dt dx ∂L ∂w δw + ∂L ∂D 1 t w δ(D 1 t w)+ + ∂L ∂ R D α 1 x w δ( R D α 1 x w) + ∂L ∂ R D α 2 x w δ( R D α 2 x w) ,(8)
where, in the absence of non-holonomic constraints, the variation and fractional derivatives commute, i.e.
δ(D 1 t w) = D 1 t (δw), δ( R D α 1 x w) = R D α 1 x (δw), δ( R D α 2 x w) = R D α 2 x (δw).
In order to utilize the fractional variational principle, we should perform the operation of integration by parts. Unfortunately, integration by parts transforms left-sided derivatives into right-sided ones for the most commonly used types of fractional derivatives. For the Liouville fractional derivatives
( L D α ± f )(x) = (−1) n Γ(n − α) d n dx n ∞ 0 f (x ∓ z) z α+1−n dz,(9)
the integration by parts (see Eq. 5.17 in Section 5.1 of [1]) has the form
+∞ −∞ f (x) ( L D α + g)(x) dx = +∞ −∞ ( L D α − f )(x) g(x) dx.(10)
For the Marchaud fractional derivatives, which is defined by
( M D α ± f )(x) = α Γ(1 − α) ∞ 0 f (x) − f (x ∓ z) z α+1 dz,(11)
the integration by parts (see Eq. 6.27 in Corollary 2 of Theorem 6.2 of [1]) has the form
+∞ −∞ f (x) ( M D α + g)(x) dx = +∞ −∞ ( M D α − f )(x) g(x) dx.(12)
This relation is valid for
functions f (x) ∈ L s (R), f (x) ∈ L t (R), such that ( M D α + g)(x) ∈ L p (R) and ( M D α − f )(x) ∈ L r (R), where p > 1, r > 1, 1 p + 1 r = 1 + α, 1 s = 1 p − α, 1 t = 1 r − α.
We suggest the use of Riesz fractional derivatives. It is known (see Section 20.1 of [1]) that the connection of the Riesz fractional derivative to the Marchaud fractional derivatives has the form
( R D α x f )(x) = 1 2 cos(απ/2) ( M D α + f )(x) + ( M D α − f )(x) ,(13)
where α > 0, and α = 1, 2, 3, .... Here R D α x is the Riesz fractional derivative defined by the equation
( R D α x f )(x) = − α 2Γ(1 − α) cos(απ/2) ∞ 0 f (x + z) − 2f (x) + f (x − z) z α+1 dz,(14)
where x ∈ R. Note that the Riesz derivative for an integer α = 2n gives
( R D 2n x f )(x) = (−1) n D 2 x f (x),(15)
where n ∈ N, i.e.
R D 2 x = − D 2 x , R D 4 x = D 4 x , R D 6 x = − D 6 x .(16)
Using relations (13) and (12), we obtain the equation of the integration by parts for the Riesz fractional derivative (13) in the form
+∞ −∞ f (x) ( R D α x g)(x) dx = +∞ −∞ ( R D α x f )(x) g(x) dx.(17)
As a result, integration by parts in Eq. (17) does not change the type of derivative, and also does not change the sign in front of the integral. Using the integration by parts given by Eq. (17), we can rewrite the variation in Eq. (8) as
δS[w] = dt dx ∂L ∂w δw − D 1 t ∂L ∂D 1 t w + R D α 1 x ∂L ∂ R D α 1 x w + R D α 2 x ∂L ∂ R D α 2 x w δw.
(18) Then, the stationary action principle in the form of the holonomic variational equation
δS[w] = 0 yields the equation ∂L ∂w − D 1 t ∂L ∂D 1 t w + R D α 1 x ∂L ∂ R D α 1 x w + R D α 2 x ∂L ∂ R D α 2 x w = 0.(19)
This is the fractional Euler-Lagrange equation for the model described by the Lagrangian
L = L(w, D 1 t w, R D α 1 x w, R D α 2 x w).
In the next section, we use this equation to establish a fractional generalization of the Euler-Bernoulli beam model.
Fractional Euler-Bernoulli beam equation from variational principle
The Lagrangian of Euler-Bernoulli beams with gradient power-law non-locality has the form
L(w, D 1 t w, R D α 1 x w, R D α 2 x w) = 1 2 µ D 1 t w(x, t) 2 − 1 2 (E I) R D α 1 x w(x, t) 2 − − 1 2 (E I) l 2 s (α 2 ) R D α 2 x w(x, t) 2 + q(x, t)w(x, t).(20)
The curve w(x) = u y (x) describes the deflection of the beam in the y direction at some position x. As we have already noted, x and l 2 s (α 2 ) are dimensionless values. The first term represents the kinetic energy, where µ = ρ A is the mass per unit length; the second term describes the potential energy due to internal forces (when considered with a negative sign); and the third term is the potential energy due to the external load q(x). Note that in the Lagrangian of Eq. (20) the second term has a negative sign, since integration by parts in Eq. (17) does not change the sign in front of the integral, in contrast to the standard case.
For the usual case of α 1 = 2 and α 3 = 3, the Lagrangian given by Eq. (20) is
L(w, D 1 t w, D 2 x w, D 3 x w) = 1 2 µ D 1 t w(x, t) 2 − 1 2 (E I) D 2 x w(x, t) 2 + + 1 2 (E I) l 2 s D 3 x w(x, t) 2 + q(x, t)w(x, t).(21)
For the fractional case, the Lagrangian (20) leads to the expressions
∂L ∂w = q(x, t) ∂L ∂D 1 t w(x, t) = µ D 1 t w(x, t),(22)∂L ∂ R D α 1 x w(x, t) = −(E I) R D α 1 x w(x, t), ∂L ∂ R D α 2 x w(x, t) = −(E I) l 2 s (α 2 ) R D α 2 x w(x, t).(23)
Substitution of Eqs. (22) and (23) into the Euler-Lagrange equation (19) gives
µ D 2 t w + R D α 1 x (E I) ( R D α 1 x )w + R D α 2 x (E I) l 2 s (α 2 ) R D α 2 x w − q(x, t) = 0,(24)
which is the governing equation for the dynamics of a fractional non-local Euler-Bernoulli beam. When the beam is homogeneous, E and I are independent of x, and the fractional Euler-Bernoulli beam equation assumes the simpler form
µ D 2 t w + (E I) ( R D α 1 x ) 2 w + (E I) l 2 s (α 2 ) ( R D α 2 x ) 2 w − q(x, t) = 0.(25)
For a wide class of functions w(x) the properties of the fractional Riesz derivatives allows us to write Eq. (25) as
µ D 2 t w + (E I) R D 2α 1 x w + (E I) l 2 s (α 2 ) R D 2α 2 x w − q(x, t) = 0.(26)
In general, we should consider an effective source term q ef f (x) instead of q(x), where q ef f (x) contains the function q(x) and deviations from the semigroup property for the Riesz derivatives as described for the fractional gradient model with Caputo derivatives dealt with in [54]. For materials without non-locality and memory, we have α 1 = 2, α 2 = 3, and then Eq. (25) obtains the form
µ D 2 t w + E I D 4 x w − E I l 2 s D 6 x w − q(x, t) = 0.(27)
This is the gradient elasticity Euler-Bernoulli beam equation derived earlier in [16] for the case of integer-order derivatives and non-fractal media.
Solution of fractional static Euler-Bernoulli beam equation
For the static case (D 1 t w = 0 and q(x, t) = q(x)), equation (26) has the form
R D 2α 1 x w + l 2 s (α 2 ) R D 2α 2 x w = (E I) −1 q(x).(28)
Using Corollary 5.14 of [2], we can state that a particular solution of equation (28) is
w(x) = (E I) −1 +∞ −∞ G 2α 1 ,2α 2 (x − x ′ ) q(x ′ )dx ′ ,(29)
where G α 1 ,α 2 (x) is a Green's type function of the form
G 2α 1 ,2α 2 (x) = ∞ 0 cos(λ|x|) λ 2α 1 + l 2 s (α 2 )λ 2α 2 dλ.(30)
Here α 1 > 0, α 2 > 0 and l 2 s (α 2 ) = 0. For a point load of intensity q 0 , i.e. a load q(x) of the form [66] q
(x) = q 0 δ(x),(31)
where δ(x) denotes the Dirac delta-function, the displacement field w(x) has a simple form w(x) = (q 0 /E I) G 2α 1 ,2α 2 (x) given by expression
w(x) = 2 q 0 π E I ∞ 0 cos(λ|x|) λ 2α 1 + l 2 s (α 2 )λ 2α 2 dλ,(32)
where the definition given by Eq. (30) for G 2α 1 ,2α 2 (x) has been used. For the usual nonfractional case, the solution of the static Euler-Bernoulli beam equation with the external point-load is given by Eq. (32) with α 1 = 2 and α 3 = 3.
Solution of fractional dynamic Euler-Bernoulli beam equation
For a plane wave traveling in a fractional non-local material with frequency ω, the governing fractional equation is
− µ ω 2 w p (x) + (E I) ( R D α 1 x ) 2 w p (x) + (E I) l 2 s (α 2 ) ( R D α 2 x ) 2 w p (x) − q p (x) = 0,(33)
where w(x, t) = e −iω t w p (x), and we have also used the notation q(x, t) = e −iω t q p (x). For a wide class of functions w p (x), Eq. (33) can be expressed as
R D 2α 1 x w p (x) + l 2 s (α 2 ) R D 2α 2 x w p (x) − µ ω 2 E I w p (x) = (E I) −1 q p (x).(34)
Using Theorem 5.24 of [2], we can obtain a particular solution of Eq. (34) as
w p (x, ω) = (E I) −1 +∞ −∞ G 2α 1 ,2α 2 (x − x ′ , ω)q p (x ′ )dx ′ ,(35)
where G
(ω) 2α 1 ,2α 2 (x) is a Green's type function of the form
G 2α 1 ,2α 2 (x, ω) = 2 ∞ 0 cos(λ |x|) λ 2α 1 + l 2 s (α 2 )λ 2α 2 − µ ω 2 /(E I) dλ.(36)
Here α 1 > 0, α 2 > 0, l 2 s (α 2 ) = 0 and µ ω 2 = 0, µ = ρ A. For the point-load case (31), the solution given by Eq. (35) is reduced to
w p (x, ω) = 2q 0 E I ∞ 0 cos(λ |x|) λ 2α 1 + l 2 s (α 2 )λ 2α 2 − µ ω 2 /(E I) dλ.(37)
For the usual non-fractional case, the solution of dynamic Euler-Bernoulli beam equation with an external point-load is given by Eq. (37) with α 1 = 2 and α 3 = 3.
Fractional gradient Timoshenko beam equations
In the Timoshenko beam theory the displacement vector u(x, y, z, t) of the beam is assumed to be given by
u x (x, y, z, t) = −z ϕ(x, t) u y (x, y, z, t) = 0, u z (x, y, t) = w(x, t),(38)
where (x, y, z) are the coordinates of a point in the beam, (u x , u y , u z ) are the corresponding components of the displacement vector, ϕ = ϕ(x, t) is the angle of rotation of the normal to the mid-surface of the beam, and w = w(x, t) is the displacement of the mid-surface in the z-direction.
To obtain a fractional generalization of the relevant gradient beam equation we use a fractional variational principle and a generalization of the Timoshenko beam Lagrangian. The appropriate form of such Lagrangian with fractional gradient non-locality, is
L = 1 2 ρ I D 1 t ϕ(x, t) 2 + 1 2 ρ A D 1 t w(x, t) 2 − − 1 2 (kGA) R D α 1 x w(x, t) − ϕ(x, t) 2 − 1 2 (E I) R D β 1 x ϕ(x, t) 2 − − 1 2 (kGA) l 2 s R D α 2 x w(x, t) − R D β 1 x ϕ) 2 − 1 2 (E I) l 2 s R D β 2 x ϕ(x, t) 2 ,(39)
where (x, y, z) are dimensionless coordinates. Note again that we use dimensionless coordinates. such that the relevant quantities of fractional models have the same physical dimension as as corresponding one for non-fractional models. Then, in view of the expressions
∂L ∂D 1 t w = ρ A D 1 t w, ∂L ∂ R D α 1 x w = − k G A R D α 1 x w − ϕ ,(40)∂L ∂ϕ = k G A R D α 1 x w − ϕ ∂L ∂D 1 t ϕ = ρ I D 1 t ϕ,(41)∂L ∂ R D β 1 x ϕ = −E I R D β 1 x ϕ + l 2 s k G A R D α 2 x w − l 2 s k G A R D β 1 x ϕ,(42)∂L ∂ R D α 2 x w = −l 2 s k G A R D α 2 x w − R D β 1 x ϕ , ∂L ∂ R D β 2 x ϕ = − l 2 s E I R D β 2 x ϕ,(43)
the stationary action principle gives the following Euler-Lagrange equations
∂L ∂w − D 1 t ∂L ∂D 1 t w + R D α 1 x ∂L ∂ R D α 1 x w + R D α 2 x ∂L ∂ R D α 2 x w = 0,(44)∂L ∂ϕ − D 1 t ∂L ∂D 1 t ϕ + R D β 1 x ∂L ∂ R D β 1 x ϕ + R D β 2 x ∂L ∂ R D β 2 x ϕ = 0.(45)
Equations (44) and (45) (45) gives the following fractional gradient Timoshenko beam equations for the displacement w = w(x) and the rotation ϕ = ϕ(x),
ρ A D 2 t w = R D α 1 x − k G A R D α 1 x w − ϕ + R D α 2 x −l 2 s k G A R D α 2 x w − R D β 1 x ϕ , (46) ρ I D 2 t ϕ = k G A R D α 1 x w − ϕ + R D β 1 x −E I R D β 1 x ϕ + l 2 s k G A R D α 2 x w − l 2 s k G A R D β 1 x ϕ + + R D β 2 x − l 2 s E I R D β 2 x ϕ .(47)
For homogeneous materials, Eqs. (46) and (47) take the form
ρ A D 2 t w = − k G A R D α 1 x R D α 1 x w − ϕ − l 2 s k G A R D α 2 x R D α 2 x w − R D β 1 x ϕ ,(48)ρ I D 2 t ϕ = k G A R D α 1 x w − ϕ − E I R D β 1 x R D β 1 x ϕ + l 2 s k G A R D β 1 x R D α 2 x w− − l 2 s k G A R D β 1 x R D β 1 x ϕ − l 2 s E I R D β 2 x R D β 2 x ϕ.(49)
For a wide class of functions w(x, t) and ϕ(x, t), Eqs. (48) and (49) can be rewritten as
ρ A D 2 t w = − k G A R D 2α 1 x w − R D α 1 x ϕ − l 2 s k G A R D 2α 2 x w − R D α 2 +β 1 x ϕ ,(50)ρ I D 2 t ϕ = k G A R D α 1 x w − ϕ − E I R D 2β 1 x ϕ + l 2 s k G A R D α 2 +β 1 x w− − l 2 s k G A R D 2β 1 x ϕ − l 2 s E I R D 2β 2 x ϕ.(51)
If α 1 = β 1 = 1, and α 2 = β 2 = 0, Eqs. (50)-(51) reduce to the well-known Timoshenko beam equations. If α 1 = β 1 = 1, and α 2 = β 2 = 2, Eqs. (50)-(51) reduce to the form of the gradient generalization of the Timoshenko beam equations. In general, the Riesz fractional derivatives do not commute and
R D α x R D β x = R D α+β x .(52)
In this case, Eqs. (48)- (49) give Eqs. (50)-(51) with an additional term in the form of an effective source terms that contain the deviations from the semigroup property for the Riesz derivatives as it was described in [54].
Combined strain-acceleration fractional gradients beam model
Let us now consider internal inertia effects, i.e. effects of combined strain-acceleration gradients on fractional nonlocal beams. We start with the governing equation of a gradient elasticity Euler-Bernoulli beam equation with internal inertia or acceleration gradients [16], i.e.,
ρ A D 2 t w + E I D 4 x w − E I l 2 s D 6 x w + ρ I l 2 d D 2 t D 4 x w − q(x, t) = 0,(53)
where (ρ, A, E, I) have their usual meaning, (x, t) are dimensionless variables, and (l 2 s , l 2 d ) are scale parameters. The fractional generalization of Eq. (53) can be written in the form
− ρ A R D 2β t w + E I R D 2α 1 x w + E I l 2 s (α 2 ) R D 2α 2 x w − ρ I l 2 d (α 3 ) R D 2β t R D 2α 3 x w − q(x, t) = 0,(54)
where R D 2β t is the Riesz fractional derivative [2] with respect to time. Using Eq. (16), Eq. (54) with α 1 = α 3 = 2, β = 1 and α 2 = 3 gives Eq. (53).
Equation (54) can be obtained from the stationary action principle and the correspondent fractional Euler-Lagrange equation
∂L ∂w + R D β t ∂L ∂ R D β t w + R D α 3 x R D β t ∂L ∂ R D α 3 x R D β t w + R D α 1 x ∂L ∂ R D α 1 x w + R D α 2 x ∂L ∂ R D α 2 x w = 0, (55) where the Lagrangian L = 1 2 ρ A R D β t w(x, t) 2 − 1 2 E I R D α 1 x w(x, t) 2 − − 1 2 E I l 2 (α 2 ) R D α 2 x w(x, t) 2 + 1 2 ρ I l 2 d (α 3 ) R D α 3 x R D β t w(x, t) 2 + q(x, t)w(x, t),(56)
is used and Eq. (16) is also taken into account.
In the above we use the Riesz fractional derivatives (13) with respect to time to derive Eq.(54) from a variational principle, instead of the Caputo derivatives that are commonly used. Moreover, the Riesz fractional derivatives allow us to obtain a general harmonic solution of the combined strain-acceleration fractional gradient beam model, as we will see in the sequel. At the same time, an interpretation of Riesz fractional derivatives with respect to time can be more complicated in comparison with the left-sided Caputo derivative. In any case, the Riesz fractional time derivative describes a special form of power-law material memory (acceleration with memory) and deserves to be explored in its own right.
Solution for the combined strain-acceleration fractional gradients beam model
Let us consider the Fourier transform F of the displacement field by utilizing the properties of the Riesz fractional derivative (see Property 2.34 in [2]) with respect to time
F R D α t w(x, t) (k) = |k| α (F w(x, t))(x, ω),(57)
where w(x, t) belongs to the space C ∞ 0 (R 2 ) of infinitely differentiable functions on R 2 with a compact support. Then Eq. (54) takes the form
− ρ A |ω| 2βŵ + E I R D 2α 1 xŵ + E I l 2 s (α 2 ) R D 2α 2 xŵ − ρ I l 2 d (α 3 ) ω 2β R D 2α 3 xŵ −q(x, ω) = 0, (58) whereŵ(x, ω) = (F w(x, t))(x, ω) andq(x, ω) = (F q(x, t))(x, ω)
. By rewriting this equation in the form
E I l 2 s (α 2 ) R D 2α 2 xŵ + E I R D 2α 1 xŵ − ρ I l 2 d (α 3 ) ω 2β R D 2α 3 xŵ − ρ A ω 2βŵ =q(x, ω),(59)
we can solve it by using Theorem 5.24 of [2] with the coefficients
a 0 = ρ A ω 2β , a 1 = ρ I l 2 d (α 3 ) ω 2β , a 2 = E I, a 3 = E I l 2 s (α 2 ).(60)
Noting that the Fourier transform of the Riesz fractional derivative with respect to coordinates is defined by
(F ( R D 2αŵ (x, ω))(k, ω) = |k| 2α (Fŵ)(k, ω),(61)
and applying F to both sides of Eq. (59) by also using Eq. (61), we obtain
(Fŵ)(k, ω) = a 3 |k| 2α 2 + a 2 |k| 2α 1 − a 1 |k| 2α 3 − a 0 −1 (Fŵ)(k, ω).(62)
Next, we define the fractional analogue of Green's function [2] as
G α (x) = F −1 a 3 |k| 2α 2 + a 2 |k| 2α 1 − a 1 |k| 2α 3 − a 0 −1 (x) = = R a 3 |k| 2α 2 + a 2 |k| 2α 1 − a 1 |k| 2α 3 − a 0 −1 e +ikx dk,(63)
where α = (α 1 , α 2 , α 3 ) is a multi-index. Then, the following relation holds
R n e i(k,r) f (|k|) d n k = (2π) n/2 |r| (n−2)/2 ∞ 0 f (λ) λ n/2 J n/2−1 (λ|r|) dλ(64)
for any function f such that the integral in the right-hand side of Eq. (64) is convergent (see Lemma 25.1 of [1]). Here J ν is the Bessel function of the first kind and we can use for n = 1 the expression
J −1/2 (z) = 2 π z cos(z).(65)
Using Eq. (64), the Green's function given by Eq. (63) can be rewritten (see Theorem 5.22 of [2]) in the form
G α (r) = (2π) 1/2 |x| 1/2 ∞ 0 λ 1/2 J −1/2 (λ|x|) dλ a 3 λ 2α 2 + a 2 λ 2α 1 − a 1 λ 2α 3 − a 0 ,(66)
where we used n = 1. If α 2 > 1, and a k = 0, then Eq. (59) is solvable [2]. A particular solution of Eq. (59) can be represented as the convolution of the functions G(x) and q(x), i.e.,
w(x, ω) = R G α (x − x ′ ) q(x, ω) dx ′ ,(67)
where the Green's function G α (z) is defined by Eq. (66). For the case q(x, ω) = q 0 δ(x), equation (67) giveŝ
w(x, ω) = 2q 0 ∞ 0 cos(λ|x|) dλ a 3 λ 2α 2 + a 2 λ 2α 1 − a 1 λ 2α 3 − a 0 .(68)
Using Eq. (60), we can write Eq. (68) in the form
w(x, ω) = 2q 0 E I ∞ 0 cos(λ|x|) dλ l 2 s (α 2 ) λ 2α 2 + λ 2α 1 − (ρ/E) l 2 d (α 3 ) ω 2β λ 2α 3 − (ρ A)/(E I) ω 2β .(69)
For a fractional non-local material without memory (β = 1) and fractional acceleration gradient
(l 2 d (α 3 ) = 0), the corresponding solution iŝ w(x, ω) = 2q 0 E I |x| ∞ 0 cos(λ|x|) dλ l 2 s (α 2 ) λ 2α 2 + λ 2α 1 − (ρ A)/(E I) ω 2 .(70)
This is in fact, the solution given by Eq. (37) for the fractional gradient Euler-Bernoulli beam equation of motion for the point load case of Eq. (31).
Dispersion law and general harmonic solution of the combined strain-acceleration fractional gradient beam model
Let us now obtain a general harmonic solution of the combined strain-acceleration fractional gradient beam model defined by Eq. (54). Using Property 2.34 in [2], the Fourier transform of ( R D α x w)(x) is given by Eq. (57), where w(x, t) belongs to the space C ∞ 0 (R 2 ) of infinitely differentiable functions on R 2 with a compact support.
The Fourier transform F of the fractional differential equation (54) with q = 0 gives
− ρA |ω| 2β + E I |k| 2α 1 + E I l 2 s (α 2 ) |k| 2α 2 − ρ I l 2 d (α 3 ) |k| 2α 3 |ω| 2β = 0,(71)
implying that
ω 2β = E ρ I A |k| 2α 1 + l 2 s (α 2 ) |k| 2α 2 1 + (I/A) l 2 d (α 3 ) |k| 2α 3 .(72)
As a result, we obtain
ω = C 1/β e R 1/β |k| α 1 /β 2β 1 + l 2 s (α 2 ) |k| 2(α 2 −α 1 ) 1 + R 2 l 2 d (α 3 ) |k| 2α 3 ,(73)
where
R = I/A, C e = R/ρ.(74)
The parameter R is called the gyration radius.
In the absence of memory, i.e. β = 1, equation (73) yields
ω = C e R |k| α 1 1 + l 2 s (α 2 ) |k| 2(α 2 −α 1 ) 1 + R 2 l 2 d (α 3 ) |k| 2α 3 .(75)
If α 1 = α 3 = 2, β = 1 and α 2 = 3, equation (73) gives
ω = C e R k 2 1 + l 2 s k 2 1 + R 2 l 2 d k 4 ,(76)
which is precisely the dispersion relation obtained earlier (Eq. (50) of [16])) for the nonfractional combined strain-acceleration gradient beam model. Using Eq. (73), we can obtain the group velocity C g = ∂ω(k)/∂k for the combined strain-acceleration fractional gradient beam model as
C g C e = 1 2β C (1−β)/β e R 1/β 1 + l 2 s (α 2 ) |k| 2(α 2 −α 1 ) (1−2β)/(2β) (1 + R 2 l 2 d (α 3 ) |k| 2α 3 ) (1+2β)/(2β) · 2α 1 |k| 2α 1 −1 + 2α 2 l 2 s (α 2 ) |k| 2α 2 −1 + + 2(α 1 + α 3 ) R 2 l 2 d (α 3 ) |k| 2(α 1 +α 3 )−1 + 2(α 2 + α 3 ) R 2 l 2 s (α 2 ) l 2 d (α 3 ) |k| 2(α 2 +α 3 )−1 .(77)
If α 1 = α 3 = 2, β = 1 and α 2 = 3, equation (77) is reduced to
C g C e = R k 2 + 3l 2 s k 2 + R 2 l 2 s l 2 d |k| 6 (1 + l 2 s k 2 ) 1/2 (1 + R 2 l 2 d k 4 ) 3/2 .(78)
This is the well-known normalized form of the corresponding group velocity (see Eq. (51) of [16]) for non-fractional counterpart of the model.
Toward 3D fractional gradient elasticity
To develop a fractional gradient elasticity theory in three-dimensions (3D), the following approaches may be used:
(1) An approach based on the Riesz fractional derivatives and integrals for R 3 [1,2,50]. This approach is best suited for 3D problems with spherical symmetry. The Riesz fractional derivative can be considered as a non-integer power of the Laplacian. Such a simple 3D fractional gradient elasticity model based on the Riesz fractional derivatives has already been recently considered by the authors in [54], and it can be naturally derived from lattice models with long-range interactions [52,53,55].
(2) An approach based on fractional vector calculus. Currently, however fractional vector calculus is formulated for a Cartesian coordinate system only [13,60]. The transformation from Cartesian to cylindrical, spherical or other coordinates is prohibitively complicated for fractional derivatives. It is connected with the fact that the formula of fractional derivative of a composite function (see Eq. 2.209 in Section 2.7.3 of [59]) is very complex, i.e.,
a D α x f (g(x)) = (x − a) α Γ(1 − α) f (g(x))+ ∞ k=1 C α k k!(x − a) k−α Γ(k − α + 1) k m=1 (D m g f )(g(x)) k r=1 1 a r ! (D r x g)(x) r! ar ,(79)
These two approaches which allow us to construct 3D fractional nonlocal models of gradient elasticity are briefly discussed below.
Fractional gradient elasticity based on Riesz derivative
Three-dimensional fractional gradient elasticity models based on the Riesz fractional derivative are possible due to the fact that this fractional derivative is a generalization of the Laplacian in R n and, in fact, it can be considered as a non-integer power of the Laplacian. The corresponding 3D fractional gradient elasticity model is described by the following equation (for details see [54])
c α ((− R ∆) α/2 u)(r) + c β ((− R ∆) β/2 u)(r) = f (r) (α > β),(81)
where r ∈ R 3 and r = |r| are dimensionless variables, and (− R ∆) α/2 is the Riesz fractional Laplacian of order α [2]. The coefficients (c α , c β ) are phenomenological constants and the rest of the symbols have their usual meaning, with u denoting the radial component of the displacement. For α > 0 and suitable functions u(r), r ∈ R 3 , the Riesz fractional derivative can be defined [2] in terms of the inverse Fourier transform F −1 by
((− R ∆) α/2 u)(r) = F −1 |k| α (F u)(k) ,(82)
where k denotes the wave vector, α > 0 and x ∈ R n . The fractional Laplacian in the Riesz form is usually defined in terms of the hypersingular integral
((−∆) α/2 f )(x) = 1 d n (m, α) R n 1 |z| α+n (∆ m z f )(x) dz,(83)
where m > α > 0, and (∆ m z f )(x) is a finite difference of order m of a function f (x) with a vector step z ∈ R n centered at the point x ∈ R n :
(∆ m z f )(x) = m k=0 (−1) k m! k!(m − k)! f (x − kz), where the constant d n (m, α) is defined by d n (m, α) = π 1+n/2 A m (α) 2 α Γ(1 + α/2)Γ(n/2 + α/2) sin(πα/2) , with A m (α) = m j=0 (−1) j−1 m! j!(m − j)! j α .
The definition given by Eq. (83) for the fractional Laplacian of order α does not depend on the choice of m > α. Its Fourier transform F satisfies the relationship (F (−∆) α/2 f )(k) = |k| α (F f )(k), which is valid for the Lizorkin space [1] and the space C ∞ (R n ) of infinitely differentiable functions on R n with compact support. If α = 4 and β = 2, we have the well-known equation of gradient elasticity [16] for the non-fractional case, i.e.,
c 2 ∆u(r) − c 4 ∆ 2 u(r) + f (r) = 0,(84)
where
c 2 = E, c 4 = ± l 2 E.(85)
Equation (81) is a fractional partial differential equation with a particular solution (Section 5.5.1. of [2]) of the form
u(r) = R 3 G 3 α,β (r − r ′ ) f (r ′ ) d 3 r ′ ,(86)
where the Green's type function is given by the expression
G 3 α,β (r) = R 3 1 c α |k| α + c β |k| β e +i(k,r) d 3 k.(87)
Using Lemma 25.1 of [1], the kernel function in Eq. (87) can be represented by the equation
G 3 α,β (r) = 1 (2π) 3/2 |r| ∞ 0 λ 3/2 J 1/2 (λ|r|) c α λ α + c β λ β dλ,(88)
where J 1/2 (z) = 2/(πz) sin(z) denotes Bessel function of the first kind. If we consider the deformation of an infinite elastic continuum due to an external field f (r) applied to a very small region, then for distances |r| which are large in comparison with the size of the region (neighborhood) of load application, we can suppose that f (r) is applied at a point [66]:
f (r) = f 0 δ(r).(89)
Then, the displacement field u(r) has a simple form u(r) = f 0 G 3 α,β (r) given by
u(r) = 1 (2π) 3/2 |r| ∞ 0 λ 3/2 J 1/2 (λ|r|) c α λ α + c β λ β dλ.(90)
3.2 Fractional vector calculus and 3D models
Fractional vector calculus
Fractional vector calculus is a very important tool for describing processes in complex media and materials with non-local properties. It allows us to formulate a dynamical theory of materials with non-locality of power-law type in three dimensions. At present, however, several formulations of fractional vector calculus are either incorrect or inconsistent, leading to errors. It seems that it is possible to define a generalization of grad, div and curl operators by using
a fractional derivative D α x k instead of the usual derivative D 1 x k , where D α x k are fractional (Li- ouville, Riemann-Liouville, Caputo, etc.) derivatives of order α with respect to x k , k = 1, 2, 3.
In such an approach, there is considerable arbitrariness in the definition of vector operators. The main problem in fractional vector calculus, however, appears when we try to generalize not only differential vector operators, but also the related integral theorems [60]. In general, a robust framework of fractional vector calculus must include generalizations of the differential operators (gradient, divergence, curl), the integral operations (flux, circulation), and the theorems of Gauss, Stokes and Green.
The main problem in the formulation of fractional integral vector operations is connected with the complex form of the fractional analogue of the Newton-Leibniz formula a I 1 b a D 1
x f (x) = f (b) − f (a).
In fact, the non-commutativity of D n x and a I α x does not allows us to derive a convenient Riemann-Liouville fractional counterpart of the Newton-Leibniz formula. For fractional Riemann-Liouville integrals and derivatives, we have the relation
a I α b a D α x f (x) = f (b) − n j=1 (b − a) α−j Γ(α − j + 1) (D n−j x a I n−α x f )(a)(91)holding almost everywhere in [a, b], where D n−j x = d n−j /dx n−j are integer derivatives, and n − 1 < α < n. Here f (x) is a Lebesgue measurable function on [a, b] for which a I 1 b f (x) < ∞, and a I n−α b
f (x) has absolutely continuous derivatives up to order (n − 1) on [a, b]. This relation was proved in [1] (see Theorem 2.4 of Section 2.6). For 0 < α < 1, Eq. (91) gives
a I α b a D α x f (x) = f (b) − (b − a) α−1 Γ(α) a I 1−α b f (x).(92)
Obviously, that Eqs. (92) and (91) do not have the usual form of the Newton-Leibniz formula. A consistent formulation of fractional vector calculus has been realized in [60] by using fractional derivatives and fractional integrals of different types. For this purpose, the Riemann-Liouville integration and the Caputo differentiation are used. The main property is that the Caputo fractional derivative provides an operation that is inverse to the Riemann-Liouville fractional integration from the left. As a result, we can formulate a fractional analogue of the Newton-Leibniz formula in the usual form if the integral is of Riemann-Liouville type and the derivative is of the Caputo type. i.e.,
a I α b C a D α x f (x) = f (b) − f (a), (0 < α < 1),(93)
where C a D α x is the Caputo fractional derivative defined by the equation
C a D α x F (x) = a I n−α x D n x F (x) = 1 Γ(n − α) x a dx ′ D n x ′ F (x ′ ) (x − x ′ ) 1+α−n ,
where n − 1 < α < n, and a I α x is the Riemann-Liouville fractional integral
a I α x f (x) := 1 Γ(α) x a f (x ′ ) (x − x ′ ) 1−α dx ′ . Here f (x) is a real-valued function defined on a closed interval [a, b] such that f (x) ∈ AC 1 [a, b] or f (x) ∈ C 1 [a, b].
For details, the reader may consult [60], where the fractional differential operators are defined such that fractional generalizations of integral theorems (Green's, Stokes', Gauss') can be realized. Using this fractional vector calculus [60], fractional differential equations for the conservation of mass, momentum and energy can be obtained for a continuum with power-law non-locality. This allows us to formulate 3D fractional models of continuum mechanics for fluids and solids with non-local properties. In the next subsection, we show how the fractional vector calculus can be used to formulate a fractional generalization of gradient elasticity for the 3D case.
Fractional differential vector operators
To properly define fractional vector operations, we will first introduce the operators that correspond to fractional differentiation and fractional integration. The left-sided Riemann-Liouville fractional integral operator is defined as
a I α x [x ′ ] := 1 Γ(α) x a dx ′ (x − x ′ ) 1−α , (α > 0).(94)
To designate that the operator given by Eq. (94) acts on a real-valued function f (x) ∈ L 1 [a, b], we employ the notation a I α x [x ′ ]f (x ′ ). We define the left-sided Caputo fractional differential operator on [a, b] in the form
C a D α x [x ′ ] := 1 Γ(n − α) x a dx ′ (x − x ′ ) 1+α−n ∂ n ∂x ′ n , (n − 1 < α < n).(95)
The Caputo operator defined by Eq. (95) acts on real-valued functions f (
x) ∈ AC n [a, b] as C a D α x [x ′ ]f (x ′ )
. We note that the Caputo operator can be represented as
C a D α x [x ′ ] = a I n−α x [x ′ ]D n [x ′ ], (n − 1 < α < n).
Equation (93) can be rewritten in the form We define a fractional generalization of nabla operator by
a I α b [x] C a D α x [x ′ ]f (x ′ ) = f (b) − f (a), (0 < α < 1).(96)∇ α W = C D α W = e 1 C D α W [x] + e 2 C D α W [y] + e 3 C D α W [z], (n − 1 < α < n),(97)
where C D α W [x m ] denotes the Caputo fractional derivative with respect to coordinates x m . For the parallelepiped W := {a x b, c y d, g z h}, we have
C D α W [x] = C a D α b [x], C D α W [y] = C c D α d [y], C D α W [z] = C g D α h [z].
Let us now give the definitions of fractional gradient, divergence and curl operators in Cartesian coordinates [13,60]). We assume that f (x) and F(x) are real-valued functions with continuous derivatives up to order (n − 1) on W ⊂ R 3 , such that their (n − 1) derivatives are absolutely continuous, i.e., f, F ∈ AC n [W ].
(1) The fractional gradient is defined by
Grad α W f = C D α W f = e l C D α W [x l ]f (x, y, z) = = e 1 C D α W [x]f (x, y, z) + e 2 C D α W [y]f (x, y, z) + e 3 C D α W [z]f (x, y, z),(98)
where f = f (x, y, z) is a (n − 1) times continuously differentiable scalar field such that the derivative D n−1 x l f is absolutely continuous. (2) The fractional divergence is defined by the equation
Div α W F = C D α W , F = C D α W [x l ]F l (x, y, z) = = C D α W [x]F x (x, y, z) + C D α W [y]F y (x, y, z) + C D α W [z]F z (x, y, z),(99)
where F(x, y, z) is a (n−1) times continuously differentiable vector field such that the derivatives D n−1 x l F l are absolutely continuous.
(3) The fractional curl operator is defined by
Curl α W F = C D α W , F = e l ε lmk C D α W [x m ]F k = e 1 C D α W [y]F z − C D α W [z]F y + + e 2 C D α W [z]F x − C D α W [x]F z + e 3 C D α W [x]F y − C D α W [y]F x ,(100)
where F k = F k (x, y, z) ∈ AC n [W ], (k = 1, 2, 3).
(4) Using the notation introduced in Eq. (97), the operator ( C D α W ) 2 can be considered as the fractional Laplacian of the Caputo type:
C ∆ α W = C D α W , C D α W = ( C D α W ) 2 = 3 l=1 ( C D α W [x l ]) 2 .(101)
Note that in the general case we have the inequality
( C D α W [x l ]) 2 = C D 2α W [x l ].(102)
Let us now give the basic relations for the fractional differential vector operators (for details of proofs see [13,60]).
(i) For the scalar field f = f (x, y, z), we have
Div α W Grad α W f = C D α W [x l ] C D α W [x l ]f = 3 l=1 ( C D α W [x l ]) 2 f.(103)
Using then the notations introduced in Eqs. (97) and (101), we conclude
Div α W Grad α W = C D α W , C D α W = C ∆ α W .(104)
(ii) The second relation for the scalar field f = f (x, y, z) is
Curl α W Grad α W f = 0.(105)
(iii) For the vector field F = e m F m , it is easy to prove the relation
Div α W Curl α W F(x, y, z) = 0.(106)
(iv) The following identity also holds for the double curl operator
Curl α W Curl α W F = Grad α W Div α W F − ( C D α W ) 2 F.(107)
(v) The Leibniz rule for fractional differential vector operators [58] does not hold, i.e.,
Grad α W f g = Grad α W f g + Grad α W g f,(108)
Div
α W f F = Grad α W f, F + f Div α W F.(109)
We define the fractional differential vector operators such that the fractional vector integral operators (circulation, flux, and volume integral) exist as inverse operations. This allows us to establish the fractional analogues of Green's, Stokes' and Gauss' integral theorems [13,60]. It is also noted that the fractional differential operators are nonlocal by definition. The fractional gradient, divergence and curl operators depend on the region W . This property allows for the use of fractional vector calculus to describe complex materials with power-law non-locality in three dimensional space. Note that these continuum fractional vector operators can be connected with the fractional-order operators on lattices with long-range interactions [61].
Fractional 3D gradient elasticity model
The simplest form of the stress-strain relation of gradient elasticity theory can be written [16] as
σ ij = C ijkl ε kl ± l 2 s ∆ε kl ,(110)
where C ijkl is the matrix of elastic modulus, l s is a length scale parameter, σ ij is the stress, and ε kl is the strain tensor. For homogenous and isotropic materials we have
C ijkl = λ δ ij δ kl + 2µ δ ik δ jl ,(111)
where λ and µ are the usual Lame constants, and δ ij is the Kronecker delta.
The equation of motion based on Eq. (110) has the form
C ijkl D 1 x j D 1 x l ± l 2 s D 1 x j (D 1 xm D 1 xm )D 1 x l u k + f i = ρ D 2 t u i ,(112)
where f i are the components of the external force field, and u k are the components of the displacement vector field. For homogenous and isotropic materials, Eq. (112) can be written as
λ D 1 x i D 1 x k ± l 2 s D 1 x i D 1 x k (D 1 xm D 1 xm ) u k + 2µ (D 1 x l D 1 x l ) ± l 2 s (D 1 x l D 1 x l )(D 2 xm D 1 xm ) u i + f i = ρ D 2 t u i .(113)
Using now operations of the vector calculus operators, this equation can be rewritten in the following vector form
λ 1 ± l 2 s ∆ grad div u + 2µ ∆ ± l 2 s ∆ 2 u + f = ρ D 2 t u.(114)
A formal fractional generalization of Eq. (112) can be obtained in the form
C ijkl C D α j W [x j ] C D α l W [x l ] ± l 2 s (α) C D α j W [x j ]( C D αm W [x m ] C D αm W [x m ]) C D α l W [x l ] u k + f i = ρ D 2 t u i ,(115)
where α = (α 1 , α 2 , α 3 ) is a multi-index. For the isotropic case (α 1 = α 2 = α 3 = α), we have the fractional equation
C ijkl C D α W [x j ] C D α W [x l ] ± l 2 s (α) C D α W [x j ]( C D α W [x m ] C D α W [x m ]) C D α W [x l ] u k + f i = ρ D 2 t u i .
(116) Using the properties of the fractional differential vector operators, Eq. (116) for homogenous and isotropic materials can be rewritten in the following vector form
λ 1 ± l 2 s (α) C ∆ α W Grad α W Div α u + 2µ C ∆ α W ± l 2 s (α)( C ∆ α W ) 2 u + f = ρ D 2 t u.(117)
Note that, in general, the following inequality holds
( C ∆ α W ) 2 = C ∆ 2α W ,(118)since ( C D α x ) 2 = C D 2α
x . In general, the fractional equations of motion may contain expressions of the form
A α (x j ) C D α W [x j ] with a given function A α (x) instead of the fractional derivative C D α W [x j ].
The explicit form of the function A α (x) is deduced by the conservation law for non-local media by using the fractional vector calculus [13,60]. In this case, the resulting 3D gradient elasticity models are more complicated and the corresponding equations of motion are much more difficult to solve. To solve the governing equations of 3D fractional models we should also take into account an explicit form of the violation of the semigroup property for the Caputo derivative [54] that gives the relationship between the product C D α a+ C D β a+ and the derivative C D α+β a+ . Using Eq. (107) in the form
Grad α W Div α W u = Curl α W Curl α W u + C ∆ W u,(119)
we can rewrite Eq. (117) as
λ 1 ± l 2 s (α) C ∆ α W Curl α W Curl α W u + (λ + 2µ) C ∆ α W ± l 2 s (α)( C ∆ α W ) 2 u + f = ρ D 2 t u. (120)
If we futher assume that the displacement vector u is radial and function of r = |r| alone (u k = u k (|r|)), we have Curl α W u = 0, and, as a result, Eq. (120) has the form
(λ + 2µ) 1 ± l 2 s (α) C ∆ α W C ∆ α W u + f = ρ D 2 t u.(121)
This is the governing fractional gradient elasticity equation for homogenous and isotropic materials with spherical symmetry.
The square of fractional derivative is not equal to a dual-order derivative
In order to solve the governing equations of fractional gradient elasticity, we should give first the explicit form of the relationship between the square of the Caputo derivative ( C D α a+ ) 2 and the Caputo derivative C D 2α a+ . To obtain this relation we use Eq. 2.4.6 of [2], in the form
( C D α a+ f )(x) = ( RL D α a+ f )(x) − n−1 k=0 (D k f )(a) Γ(k − α + 1) (x − a) k−α ,(122)
and Eq. 2.1.16 of [2], in the form
I α a+ (x − a) β = Γ(β + 1) Γ(α + β) (x − a) β+α ,(123)
where α > 0 and β > −1. The condition β > −1 gives another restriction for α in the form α < 1. The relationship between the square of the Caputo derivative of order α and the Caputo derivative of order 2α takes then the form
( C D α a+ ) 2 f (x) = C D 2α a+ f (x) + f ′ (a) Γ(1 − 2α) (x − a) 1−2α , (0 < α ≤ 1),(124)
where α = 1/2. Using Eq. (124), we can represent the fractional Laplacian of Caputo type as
C ∆ α W = 3 k=1 C D 2α x i + 3 k=1 (D 1 x k f )(a) Γ(1 − 2α) (x k − a k ) 1−2α .(125)
Note that the relation given by Eq. (124) cannot be used for α > 1. As a result, additional difficulties for solving fractional gradient equations arise. To solve these problems, we can use a generalization of the Ru-Aifantis operator split method [19,22].
Operator split method for fractional gradient elasticity
In 1993, Ru and Aifantis [19] suggested an operator split method to solve static problems of gradient elasticity. Let us consider a generalization of this method to solve the fractional gradient elasticity problems. For the static case, Eq. (117) can be written in the form
1 ± l 2 s (α) C ∆ α W λ Grad α W Div α +2µ C ∆ α W u + f = 0.(126)
By introducing l 2 s (α) = 0 in Eq. (126), we obtain the fractional differential equation
L (α) u + f = 0,(127)
where we use the fractional operator
L (α) = λ Grad α W Div α +2µ C ∆ α W .(128)
For the gradient-dependent case l 2 s (α) = 0, Eq. (126) has the form
1 ± l 2 s (α) C ∆ α W L (α) u + f = 0. (129)
In general, it is necessary to solve the fractional partial differential equation of order 4α, which has a very complex form caused by the inequality ( C ∆ α W ) 2 = C ∆ 2α W for the fractional Laplacian of Caputo type. The following observation can reduce the complexity of this task and greatly facilitate the obtaining of solutions in certain cases. For the radial displacement case (Curl α W u = 0), the operators L (α) and C ∆ α W commute, i.e.,
L (α) C ∆ α W − C ∆ α W L (α) = 0.
Therefore, we can see from Eq. (129) that the vector field 1 ± l 2 s (α) C ∆ α W u satisfies the nongradient expression of Eq. (127) for the field u. Thus, if 1 ± l 2 s (α) C ∆ α W u can be identified with the non-gradient displacement field u c of fractional non-gradient elasticity theory given by Eq. (127), which can be solved, then the original fractional gradient elasticity theory given by Eq. (126) is reduced to the following fractional equation
1 ± l 2 s (α) C ∆ α W u g = u c ,(130)
where u c is a classical ("non-gradient") solution of the fractional equation
L (α) u c + f = 0.
Obviously, the solution of Eq. (130) can be more conveniently obtained. This establishes a connection between the "gradient" (g) and the non-gradient "classical" (c) fractional elasticity solutions.
For the non-radial case (Curl α W u = 0), the fractional gradient elasticity theory given by Eq. (116) takes the form L
(α) ik 1 ± l 2 s (α) C ∆ α W u k + f i = 0,(131)
where L (α)
ik = C ijkl C D α W [x j ] C D α W [x l ].(132)
Using the operator split approach, Eq. (131) can be solved as an uncoupled sequence of two sets of fractional equations, that is L
(α) ik u c k + f i = 0 (133) followed by 1 ± l 2 s (α) C ∆ α W u g k = u c k ,(134)
where two separate displacement fields are distinguished. Firstly, u c k obeys the non-gradient fractional elasticity as given by Eq. (133). Secondly, u g k are the same as u k in Eq. (131), but they are now appended with a superscript g to emphasize that they incorporate fractional gradient effects.
Solutions by fractional operator split method
Unfortunately, the applicability of fractional vector calculus to solve 3D fractional differential equations, such as Eq. (117), is very limited due to the weak development of this area of mathematics. Therefore, we demonstrate an application of the suggested generalization of the operator split method to obtain solutions of fractional gradient elasticity equation for a 1D case only. The 1D counterpart of Eq. (121) reads
(λ + 2µ) 1 ± l 2 s (α) C ∆ α x C ∆ α x u(x) + f (x) = 0.(135)
Using the operator split method in Eq. (135), we derive two uncoupled fractional equations
L (α) u c (x) + f (x) = 0,(136)and 1 ± l 2 s (α) C ∆ α x u g (x) = u c (x),(137)
where the notation L (α) = (λ + 2µ) C ∆ α x was used. Let us first consider the equation for the non-gradient case. Using (124), equation (136) can be represented as
(λ + 2µ) C D 2α a+ u(x) + (λ + 2µ) u ′ (a) Γ(1 − 2α) (x − a) 1−2α + f (x) = 0.(138)
We can rewrite this equation in the form
(λ + 2µ) C D 2α a+ u(x) + f ef f (x) = 0,(139)
where we have used the effective body force given by the expression
f ef f (x) = (λ + 2µ) u ′ (a) Γ(1 − 2α) (x − a) 1−2α + f (x).(140)
If
f ef f (x) ∈ C γ [a; b] with 0 ≤ γ < 1u(x) = u c (x) = n−1 k=0 u (k) (a) k! (x − a) k − 1 (λ + 2µ) Γ(2α) x a f ef f (z) (x − z) 1−2α ,(141)
where n − 1 < 2α < n. Next, we consider the corresponding equation for the gradient case. Equation (137) can be rewritten as C D 2α
a+ u g (x) ± l −2 s (α) u g (x) = ± u c (x).(142)
Using (124), equation (142) can be represented as
C D 2α a+ u g (x) + u ′ (a) Γ(1 − 2α) (x − a) 1−2α ± l −2 s (α) u g (x) = ± u c (x),(143)
where u c (x) is defined by Eq. (141). We rewrite this equation in the form
C ∆ 3α x u g (x) ± l −2 s (α) u g (x) = u c ef f (x),(144)u c ef f (x) = ±u c (x) − u ′ (a) Γ(1 − 2α) (x − a) 1−2α .(145)
If u c ef f (x) ∈ C γ [a; b] with 0 ≤ γ < 1 and γ ≤ 2α, then (see Theorem 4.3 of [2]) equation (144) has a unique solution u g (x) belonging to the space C 2α,n γ [a; b], where n − 1 < 2α < n, defined by the expression
u g (x) = n−1 k=0 u g (a) (x − a) k E 2α,k+1 [∓ l −2 s (α) (x − a) 2α ]+ + x a (x − z) 2α−1 E 2α,2α [± l −2 s (α) (x − z) 2α ] u c ef f (z) dz.(146)
The quantity E α,β (z) is the Mittag-Leffler function [2] defined by the relation
E α,β [z] = ∞ k=0 z k Γ(αk + β) , (α > 0, β ∈ R).(147)
Note also that E 1,1 [z] = e z .
Toward gradient elasticity of fractal materials
Fractals are measurable metric sets with non-integer Hausdorff dimension [44,45] that should be observed on all scales. Real fractal materials can be characterized by an asymptotic relation between the mass M(W ) and the volume V (W ) of regions W of the fractal medium. For example, for a homogeneous fractal medium, a ball of radius R ≫ R 0 contains the mass M D (W ) = M 0 (R/R 0 ) D , where the number D is called the mass dimension, and R 0 is a characteristic size related to the arrangement of the medium particles. The mass dimension D does not depend on the shape of the region W , or on the packing of particles (close packing, random packing or porous packing with uniform distribution of holes). As a result, we can define a fractal material as a medium with non-integer mass (or number of particles) dimension. Although, the non-integer dimension does not reflect completely the geometric and dynamic properties of a fractal medium, it nevertheless permits a number of important conclusions about its behavior.
Fractional continuum model for fractal materials
In general, a fractal material cannot be considered as a usual continuum, since there are places and areas that are not filled with particles. Nevertheless it can be described by special continuum models [32,33,13] based on the use of the integrals with non-integer order. The order of these integrals should be defined by the fractal mass dimension. The kernel of the fractional integral operator describes a density of permitted states (permitted places) in space. The fractional-order integrals can be considered as integrals over a non-integer dimensional space up to a numerical factor by using the well-known formulas of dimensional regularization [46].
Fractional integral continuum models of fractal media may have a wide range of applications [13] due to the relatively small numbers of parameters that define a fractal material of great complexity and rich structure. One of the advantages of such models is the ability to describe dynamics of fractal materials and media (for details see [13]).
To describe fractal materials by a fractional integral continuum model, we use two different notions: the density of states c n (D, r) and the distribution function ρ(r, t).
(1) The function c n (D, r) is a density of states in the n-dimensional Euclidean space R n . The density of states describes how closely packed permitted states of particles in the space R n . The expression c n (D, r) dV n represents the number of states (permitted places) between V n and V n + dV n . We note that the symmetry of the density of states c n (D, r) must be the defined by the symmetry properties of the fractal medium.
(2) The function ρ(r, t) is a distribution function in the n-dimensional Euclidean space R n . It describes the distribution of physical values (for example, mass, electric charge, number of particles, probability) on a set of possible (permitted) states in the space R n .
For example, the mass of a region dV n in fractal media is defined by the equation dM(r, t) = ρ(r, t)c n (D, r)dV n .
In general, we cannot consider the value ρ(r, t)c n (D, r) as a new distribution function or a particle number density, since the notions of density of states and of distribution function are different. We cannot reduce all properties of the system to a description of the distribution function. This fact is well-known in statistical and condensed matter physics, where the density of states is usually considered as a density of energy states or as a density of wave vector states [57] that describe how closely packed the allowed states in energy or wave-vector spaces. For fractal distributions of particles in a coordinate space R n , we must use a density of states in this space. The density of states c n (D, r) in R n is chosen such that dµ D (r, n) = c n (D, r)dV n describes the number of states in dV n . We use the notations
dV D = c 3 (D, r)dV 3 , dS d = c 2 (d, r)dS 2 , dl β = c 1 (β, r)dl 1
to describe densities of states in n-dimensional Euclidean spaces with n = 1, 2, 3.
Mass of fractal materials
The cornerstone of fractal media is the non-integer mass dimension. One of the best static experimental methods to determine the mass dimension D of fractal materials is the boxcounting method (see, for example [49] and references therein). It involves the selection of a box of size R and counting the mass inside to estimate D from corresponding power law relation M ∼ R D .
Let us now consider a region W of a fractal material in the Euclidean space R 3 , with its boundary denoted by ∂W . Suppose that the medium in the region W has a mass dimension D, and the medium on the boundary ∂W has a dimension d. In general, the dimension d is not equal to (D − 1) and it is not equal to 2. The mass of the region W in the fractal medium is denoted by M D (W ). The fractality means that the mass in any region W ⊂ R 3 increases slower than the 3D volume of this region, i.e., according to the power law M D (W ) ∼ R D , where R is the radius of the ball used to measure D.
A fractal material is called homogeneous if the power law M D (W ) ∼ R D does not depend on the translation of the region W . In other words, for any two regions W 1 and W 2 of the homogeneous fractal material with equal volumes V D (W 1 ) = V D (W 2 ), the corresponding masses are equal M D (W 1 ) = M D (W 2 ). A wide class of fractal media satisfies the homogeneous property. Many porous materials, polymers, colloid aggregates, and aerogels can be considered as homogeneous fractal materials. However, the fact that a material is porous or random does not necessarily imply that this material is fractal. To describe fractal materials by a fractal integral continuum model, the fractality and homogeneity properties are implemented as follows:
• Homogeneity: The local density of a homogeneous fractal material can be described by the constant density ρ(r) = ρ 0 = const. This property means that if ρ(r) = const and V (W 1 ) = V (W 2 ), then M D (W 1 ) = M D (W 2 ).
• Fractality: The mass of the ball region W of a fractal homogeneous material obeys a power law relation M ∼ R D , where 0 < D < 3, and R is the radius of the ball. If V n (W 1 ) = λ n V n (W 2 ) and ρ(r, t) = const, then fractality implies that M D (W 1 ) = λ D M D (W 2 ).
These two conditions cannot be satisfied if the mass of the medium is described by an integral of integer order. In this case the mass is expressed by the fractional-order integral equation
M D (W, t) = W ρ(r, t)dV D , dV D = c 3 (D, r)dV 3 ,(148)
where r is a dimensionless vector variable. As already noted, ρ(r, t) is a distribution function, and c 3 (D, r) is a density of states in the Euclidean space R 3 . The order of the integral in Eq.
(148) is defined by the fractal mass dimension of the material. The kernel of the fractional integral operator describes a density of permitted states c 3 (D, r) in space, and its symmetry is defined by the symmetry of the material structure. The particular form (Riesz, Riemann-Liouville, etc.) of the function c 3 (D, r) is defined by the properties of the fractal material at hand. Note that the final field equations that relate the various physical variables of the system have a form that is independent of the numerical factor in the function c 3 (D, r). However the dependence on r is important in these equations. In addition, we note that for D = 2, we have the fractal mass distribution in 3D Euclidean space R 3 . In general, this case is not equivalent to the distribution on a 2D surface.
Moment of inertia for fractal materials
A method for calculating the moment of inertia of fractal materials has been suggested in [34]. The moment of inertia has two forms, a scalar form I(t), which is used when the axis of rotation is known, and a more general tensor form that does not require knowing the axis of rotation. The scalar moment of inertia (often called simply the "moment of inertia") of a rigid body with density ρ ′ (r ′ , t) with respect to a given axis is defined by the volume integral
I ′ (t) = W ρ ′ (r ′ , t) r ′ 2 ⊥ dV ′ 3 ,(149)
where (r ′ ) 2 ⊥ is the square of the perpendicular distance from the axis of rotation, and dV ′ 3 = dx ′ 1 dx ′ 2 dx ′ 3 . If r ′ = x ′ k e k denotes the position vector from the origin to a point (x ′ k , k = 1, 2, 3, are components of r ′ ), then the tensor form of the moment of inertia is
I ′ kl (t) = W ρ ′ (r ′ , t) (r ′ ) 2 δ kl − x ′ k x ′ l dV ′ 3 ,(150)
where δ kl is the Kronecker delta. We note that the SI units of I ′ kl is kg · m 2 , i.e., [I ′ kl ] = kg · m 2 . To generalize Eqs. (149) and (150) for fractional media, we express these equations through dimensionless coordinates. We thus introduce the dimensionless variables x k = x ′ k /l 0 , r = r ′ /l 0 , where l 0 is a characteristic length scale, and write the density as ρ(r, t) = l 3 0 ρ ′ (r l 0 , t) so its SI units is m, i.e., [ρ] = kg. We then define the following moments of inertia I kl (t) = l −2 0 I ′ kl (t), I(t) = l −2 0 I ′ (t) to finaly obtain the relations
I(t) = W ρ(r, t) r 2 ⊥ dV 3 , I kl (t) = W ρ(r, t) r 2 δ kl − x k x l dV 3 ,(151)
where dV 3 = dx 1 dx 2 dx 3 for Cartesian coordinates, and the variables x k , k = 1, 2, 3 are now dimensionless. We note that the SI units of I kl is kg, i.e., [I kl ] = kg. This representation allows us to generalize Eq. (151) to fractal materials in the form
I (D) (t) = W ρ(r, t) r 2 ⊥ dV D , I (D) kl (t) = W ρ(r, t) (r 2 δ kl − x k x l ) dV D ,(152)
where dV D = c 3 (D, r)dV 3 with D denoting, as usual, the mass dimension of the fractal material.
Equilibrium equations for fractal materials
Let us now derive the equilibrium equations for a fractal material with mass dimension D.
Consider a finite region W in the fractal material, supporting a volume force and a surface force. Let the density of force f(r, t) be a function of the dimensionless vector r, and time t. The volume or mass force F M (W ), i.e. the force acting on a region W of a fractal medium with dimension D, is defined by
F M (W ) = W f(r, t) dV D .(153)
The surface force F S (W ), i.e. the force acting on the surface ∂W with dimension d, is defined by
F S (W ) = ∂W σ n (r, t) dA d ,(154)
where σ = σ(r, t) is the traction vector on a surface with unit normal n. As already mentioned, in general the dimension d is not equal to (D − 1) and it is not equal to 2. The resultant force that acts on the region W is then
F Σ (W ) = F M (W ) + F S (W ),(155)
and by substituting Eqs. (153) -(154) into Eq. (155), we obtain
F Σ (W ) = W f(r, t) dV D + ∂W σ n (r, t) dA d .(156)
In component form, this equation reads
W f k (r, t) dV D + ∂W σ n k (r, t) dA d = 0,(158)
where we use f = f k e k and σ n = σ n k e k . Using the normal vector n = n j e j , we can represent σ n k in the form σ n i = σ ij n j , where σ ij is the stress tensor. The differential form of equilibrium equations follows directly from Eq. (158). Using the generalization of the Gauss theorem for fractal media [33], the surface integral can be represented as
∂W σ n dA d = ∂W c 2 (d, r) σ n dA 2 = W ∂(c 2 (d, r) σ l ) ∂x l c −1 3 (D, r) dV D = W ∇ (D,d) l σ l dV D ,(159)
where a generalization of the nabla operator for fractal materials [60] was also used in the form
∇ (D,d) k B = c −1 3 (D, r) ∂(c 2 (d, r)B) ∂x k ,(160)
where B = B(r) is a function of the coordinates. This operator will be called "fractal-nabla" operator. We note that the operator given by Eq. (160) is not a fractional derivative [2] or an operator on a fractal set [69]. For example, if we use the density of states c 3 (D, r) and c 2 (d, r) in the form
c 3 (D, r) = 2 3−D Γ(3/2) Γ(D/2) |r| D−3 ,(161)c 2 (d, r) = 2 2−d Γ(d/2) |r| d−2 ,(162)
then the "fractal-nabla" operator is given by
∇ (D,d) k B = 2 D−d−1 Γ(D/2) Γ(3/2)Γ(d/2) |r| 3−D ∂ ∂x k |r| d−2 B .(163)
For non-fractal materials (D = 3 and d = 2), we have
∇ (3,2) k B = ∂B ∂x k .
We note that the rule of term-by-term differentiation for the operator ∇
(D,d) k is not satisfied, i.e. ∇ (D,d) k (BC) = B∇ (D,d) k (C) + C∇ (D,d) k (B).
The operator ∇ (D,d) k satisfies the following rule
∇ (D,d) k (BC) = B∇ (D,d) k (C) + c(D, d, r) C ∇ 1 k B,(164)
where c(D, d, r) = c −1 3 (D, r)c 2 (d, r). For example, the density of states given by Eqs. (161) and (162), can be expressed as
c(D, d, r) = Note that, in general, ∇ (D,d) k (1) = 0 since ∇ (D,d) k (1) = c(D, d, r) (d − 2)
x k r 2 .
Using now Eq. (159), Eq. (157) takes the form
W f + ∇ (D,d) l σ l dV D = 0,(165)
or in components form (with f = f k e k , and σ n l = σ kl e k ), we have
W f k + ∇ (D,d) l σ kl dV D = 0, (k = 1, 2, 3).(166)
This equation is satisfied for all regions W . As a result, we have
∇ (D,d) l σ kl + f k = 0, (k = 1, 2, 3).(167)
Using the usual notation, we have
c −1 3 (D, r) D 1 x l c 2 (d, r) σ kl + f k = 0, (k = 1, 2, 3).(168)
These are the differential equations of equilibrium for fractal materials. Let us derive next, the equilibrium equation for the moment of forces. The moment M M (W ) of the mass force (153), can be written as
M M (W ) = W [r, f] dV D .(169)
The moment M S (W ) of the surface force (154) is given by
M S (W ) = ∂W [r, σ n ] dA d .(170)
In Eqs. (169)
M Σ (W ) = W [r, f] dV D + ∂W [r, σ n ] dA d .(172)
The equilibrium condition for the region W surrounded by its surface ∂W of a fractal material leads to M Σ (W ) = 0, yielding the fractional integral equation
W [r, f] dV D + ∂W [r, σ n ] dA d = 0.(173)
In component form, this equation reads
W ǫ ijk x j f k dV D + ∂W ǫ ijk x j σ kl n l dA d ,(174)
where ǫ ijk is the Levi-Civita symbol. Using then the generalization of Gauss theorem for fractal materials given by Eq. (159), we obtain
∂W ǫ ijk x j σ kl n l dA d = ∂W ǫ ijk x j σ kl n l c 2 (d, r) dA 2 = = ∂W ǫ ijk D 1 l x j σ kl c 2 (d, r) dV 3 = ∂W ǫ ijk c −1 3 (D, r) D 1 l x j c 2 (d, r) σ kl dV D = = ∂W c(D, d, r)ǫ ilk σ kl dV D + ∂W ǫ ijk x j ∇ (D,d) l σ kl dV D = = ∂W c(D, d, r)ǫ ilk σ kl dV D − ∂W ǫ ijk x j f k dV D ,(175)
where equation (167) is also used. Substitution of Eq. (175) into Eq, (174) gives
∂W c(D, d, r)ǫ ilk σ kl dV D = 0.(176)
This equation is satisfied for all regions W . Therefore we have the condition
ǫ ijk σ kj = 0,(177)
or, equivalent,
σ ij = σ ji .(178)
This equilibrium equation for the moment of the force in fractal materials is the same as for the non-fractal case, and suggests that the stress tensor is symmetric.
Conservation laws for fractal materials
In the framework of fractional integral continuum model, the fractional conservation laws for fractal media have been derived in [33] (see also [39,13]). For future reference, the differential equations of the conservation laws are also summarized below:
(1) The conservation law for mass
d dt (D,d) ρ = −ρ ∇ (D,d) k u k .(179)
(2) The conservation law for momentum
ρ d dt (D,d) u k = f k + ∇ (D,d) l σ kl .(180)
(3) The conservation law for energy
ρ d dt (D,d) e = c(D, d, r) σ kl D 1 l u k + ∇ (D,d) k q k .(181)
It is noted that these equations are differential equations with derivatives of integer order (see Eq. (163)). It is also pointed out that the generalized total time derivative is defined by
d dt (D,d) = ∂ ∂t + c(D, d, r) u l D 1 l ,(182)
where r = |r|, x k , k = 1, 2, 3, are dimensionless variables, the operator D 1 l is defined as usual by D 1 l = ∂/∂x k , and c(D, d, r) = c −1 3 (D, r) c 2 (d, r). The above listed differential equations of balance for the density of mass, the density of momentum, and the density of internal energy make up a set of five equations, which are not closed. In addition to the fields ρ(r, t), u(r, t), e(r, t), equations (180) and (181) include the tensor of stress σ kl (r, t) = σ lk (r, t) and the vector of thermal flux q k (r, t). It is also remarked that the conservation laws for fractal media, which are suggested in [39] are different from the conservation laws given by Eqs. (179-181) derived in [33,13]. In [39] all equations contain the derivatives c −1
1 (α x i , x i ) D 1 x i only, where the density of states c −1 1 (α x i , x i ) can be considered as c −1 3 (D, r)c 2 (D − α x i , r − x i e i )
Constitutive relations for fractal materials
For the theory of non-fractal gradient elasticity of isotropic materials the constitutive relations [17]- [19] has the form
σ ij = λε kk δ ij + 2µε ij − l 2 ∆ λε kk δ ij + 2µε ij ,(183)
where σ ij and ε ij are the stress and strain tensors and l denotes an internal length. As usual, λ and µ are the Lame coefficients; and ∆ is the Laplace operator defined by the scalar product of the nabla operators
∆ = (∇, ∇) = 2 k (∇ k ) 2 .(184)
It is easy to see that the balance equations for fractal media considered herein contain in addition to the usual derivatives D 1 k the "fractal-nabla" operator ∇
(D,d) k of Eq. (160), ∇ (D,d) k . = c −1 3 (D, r)∇ k c 2 (d, r) .(185)
that takes into account the density of states of fractal media with non-integer mass dimensions. Therefore, we can assume that corresponding generalizations of constitutive relations can be obtained by the replacement of the usual nabla operator by the "fractal-nabla" operator. For example, a fractal generalization of the gradient elasticity model given by Eq. (183) can be represented by the constitutive relations in the form
σ ij = λε kk δ ij + 2µε ij − l 2 F ∆ (D,d) λε kk δ ij + 2µε ij ,(186)
where we use the "fractal-Laplacian" that is defined by
∆ (D,d) = ∇ (D,d) , ∇ (D,d) = 2 k ∇ (D,d) k 2 .(187)
For non-fractal materials, we have D = 3, d = 2 and ∆ (3,2) = ∆. More generally, we can assume that the constitutive relations for fractal materials are of the form
σ ij = λε kk δ ij + 2µε ij − l 2 S ∆ λε kk δ ij + 2µε ij − l 2 F ∆ (D,d) λε kk δ ij + 2µε ij ,(188)
where two types of Laplacians are taken into account.
In general, fractal materials cannot be defined as media distributed over a fractal set. Naturally, in real materials the fractal structure cannot be observed on all scales. Materials demonstrate fractality only in a range of scales R min < R < R max . If the sample material has a size R S greater than R max , or the region of scales [R min , R max ] is narrow, then the material is "semi-fractal" material. The parameter l 2 S in constitutive relation given by Eq. (188) is a measure of spatial non-fractality of the material, whereas the parameter l 2 F is a measure of spatial fractality of material for the fractal gradient elasticity theory considered herein. Which of the two models of Eq. (186) or Eq. (188) is more appropriate to describe a particular fractal material, must be determined experimentally.
Strain-displacement relation for fractal materials
In [39]- [41] it is postulateed that the strain ε ij for small deformations of fractal materials is given in terms of the displacement u k by the equation
ε ij = 1 2 c −1 1 (α x i , x i ) D 1 x i u j + c −1 1 (α x j , x j ) D 1 x j u i .(189)
The one-dimensional analogue of Eq. (189) has been considered in [40,41] in the form
ε(x) = c −1 1 (α, x) D 1 x u(x),(190)
where c 1 (α, x) is the density of states. As a basis for using this definition, reference is made the differential form of a linear element dl α = c −1 1 (α x , x)dx, which takes into account the 1D density of states. Another argument [40,41] to support this choise is a possibility to obtain the same 1D elastic wave equation from a variational principle, as the wave equation obtained from the balance equations. However, it is not quite clear the necessity to consider the density of states in the definition of the strain.
It thus seems that the definitions given by Eqs. (189) or (190) are not sufficiently rigorously justified. The inclusion of the density of states c 1 (α, x) into the strain-displacement relation looks like an artificial reception. The relation between the strain tensor ε ij and the displacement vector u k should be derived directly from the relevant distance changes (for example, see Section 1.1 of [66]), and this relation should not be postulated in definition. For example, in the 1D case, the strain-displacement relation for fractal materials should be derived from the equation
(dl ′ α ) 2 = (dl α ) 2 (1 + 2ε(x))(191)
that describes the deformation of a linear element dl α = c 1 (α x , x)dx of 1D fractal medium. From Eq. (191) it is apparent that the strain ε(x) does not contain the density of states c 1 (α, x). The relation between strain and displacement should define the deformation of a volume element dV D = c 3 (D, r) dV 3 of a fractal material through the condition
dV ′ D = dV D [1 + ε 11 (x) + ε 22 (x) + ε 33 (x)],(192)
which is the fractal analogue of Eq. (1.6) of [66], we see that ε ii (x) does not contain the density of states also.
Variational principle for fractal materials
Another way to derive the governing equations for fractional integral continuum models for fractal materials is the use of variational principles. A holonomic variational principle for fractal materials has been suggested in [35,36] in the framework of a fractional integral continuum model. Variational principles for fractal elasticity are also considered in [37,38]. The equation for fractal elasticity can be derived as the Euler-Lagrange equations from a holonomic functional. Let us consider a fractional integral continuum model for fractal materials in R 3 that is described by the action
S F [u] = dt R 3 dV D L(u i , u i,t , u i,k , u i,kl , u i,klm )(193)
with Lagrangian L(u i , u i,t , u i,k , u i,kl , u i,klm ), where u i = u i (r, t) is the displacement vector. To take into account the fractality of the material in coordinate space R 3 , we use
dV D = c 3 (D, r)dV 3 ,
where the function c 3 (D, r) describes the density of states in R 3 . Note that x, y, z and r are dimensionless variables. The variation of the action functional given by Eq. (193) is
δS F [u] = dt R 3 dV D δL = dt R 3 dV D ∂L ∂u i δu i + ∂L ∂u i,t δu i,t + + ∂L ∂u i,k δu i,k + ∂L ∂u i,kl δu i,kl + ∂L ∂u i,klm δu i,klm ) .(194)
If the fractal material is not subjected to non-holonomic constraints, then the variation and fractional derivatives commute,
δu i,t = D 1 t (δw), δu i,k = ∇ k (δw), δu i,kl = ∇ k ∇ l (δw), δu i,klm = ∇ k ∇ l ∇ m (δw).
Using integration by parts, we can express Eq. (194) in the form
δS F [u] = dt R 3 dV 3 c 3 (D, r) ∂L ∂u i δu i − D 1 t c 3 (D, r) ∂L ∂u i,t + − ∇ k c 3 (D, r) ∂L ∂u i,k + ∇ k ∇ l c 3 (D, r) ∂L ∂u i,kl − ∇ k ∇ l ∇ m c 3 (D, r) ∂L ∂u i,klm δu i . (195)
Then, the stationary action principle, in the form of the holonomic variational equation δS F [u] = 0, gives the Euler-Lagrange equations for the fractional integral continuum model of the fractal material considered in the form
∂L ∂u i δu i − D 1 t ∂L ∂u i,t − c −1 3 (D, r) ∇ k c 3 (D, r) ∂L ∂u i,k + + c −1 3 (D, r) ∇ k ∇ l c 3 (D, r) ∂L ∂u i,kl − c −1 3 (D, r) ∇ k ∇ l ∇ m c 3 (D, r) ∂L ∂u i,klm = 0.(196)
It follows that a mathematical model for a fractal material is entirely determined by the choice of the Lagrangian. We demonstrate an application of this approach by considering the example of the Euler-Bernoulli fractal beam in the next section.
Gradient elasticity model for fractal beam
In this section we derive a gradient elasticity model for fractal materials in the form of the Euler-Bernoulli beam equation of motion by using the holonomic variational principle for fractal media [35,36]. We will consider the gradient fractal beam by using the fractional integral continuum approach suggested in [33,32,13]. In this connection, it is noted that a non-gradient fractal beam has been considered in [40,41] in the framework of a fractional integral continuum model.
Variational equation for 1-dimesional model of fractal materials
Let us consider a 1D fractional continuum model for fractal materials described by the action
S F [w] = dt dl αx L(x, t, w, D 1 t w, D 2 x w, D 3 x w)(197)
with Lagrangian L(x, t, w, D 1 t w, D 2 x w, D 3 x w), where dl αx = c 1 (α x , x) dx and x is dimensionless. The function c 1 (α x , x) denotes the density of states along the x-axis. For the Euler-Bernoulli fractal beam model, the field w(x) = u y (x) is the curve that describes the deflection of the beam in the y direction at some position x.
The variation of the action functional given by Eq. (197) is
δS F [w] = dt dx c 1 (α x , x) δL = dt dx c 1 (α x , x) ∂L ∂w δw + ∂L ∂D 1 t w δ(D 1 t w)+ + ∂L ∂D 2 x w δ(D 2 x w) + ∂L ∂D 3 x w δ(D 3 x w) .(198)
If non-holonomic constraints are not involved, the variation and fractional derivatives commute, i.e. δ(D 1
t w) = D 1 t (δw), δ(D 2 x w) = D 2 x (δw), δ(D 3 x w) = D 3
x (δw). Using integration by parts, we express Eq. (198) in the form
δS F [w] = dt dx c 1 (α x , x) ∂L ∂w δw − D 1 t c 1 (α x , x) ∂L ∂D 1 t w + + D 2 x c 1 (α x , x) ∂L ∂D 2 x w − D 3 x c 1 (α x , x) ∂L ∂D 3 x w δw.(199)∂L ∂w − c 1 (α x , x) D 1 t ∂L ∂D 1 t w + D 2 x c 1 (α x , x) ∂L ∂D 2 x w − D 3 x c 1 (α x , x) ∂L ∂D 3 x w = 0. (200)
This equation describes the fractional continuum model of a fractal material distributed in R 1 with dimension α z .
Euler-Lagrange equation for the Euler-Bernoulli fractal beam
The Lagrangian for the Euler-Bernoulli fractal beams has the form
L(x, t, w, D 1 t w, D 2 x w, D 3 x w) = 1 2 µ D 1 t w(x, t) 2 + 1 2 (E I (d) ) D 2 x w(x, t) 2 − − 1 2 (E I (d) ) l 2 F (d) D 3 x w(x, t) 2 − q(x, t)w(x, t).(201)
The first term represents the kinetic energy, where µ = ρ A is the mass per unit length; the second one represents the potential energy due to an internal forces (when considered with a negative sign); and the third term represents the potential energy due to the external load q(x, t). Note that (x, y, z) are dimensionless variables, and l 2 F (d) is a dimensionless parameter. The Lagrangian looks similar the usual Lagrangian for an Euler-Bernoulli gradient elastic beam. A difference is in the presence of the moment of inertia I (d) of the fractal material only. In the Lagrangian we used the second moment of area (I (d) = I (d) z ) of the fractal beam's cross-section defined by
I (d) = I (d) z = A y 2 dA x (d),(202)
where we take into account the density of states c 2 (d, y, z) in the expression of a fractal surface differential element, i.e. dA x (d) = c 2 (d, y, z) dA x .
In [40,41] it has been suggested to use the derivatives c −1 1 (α x , x) D 1 x instead of the usual derivatives D 1
x for fractal materials. If we use the derivatives c −1 1 (α x , x) D 1 x instead of D 1 x for fractal materials according to [40,41], then the Lagrangian for Euler-Bernoulli fractal beams takes the following form
L(x, t, w, D 1 t w, D 2 x w, D 3 x w) = 1 2 µ D 1 t w(x, t) 2 + 1 2 (E I (d) ) (c −1 1 (α x , x) D 1 x ) 2 w(x, t) 2 − − 1 2 (E I (d) ) l 2 F (d) (c −1 1 (α x , x) D 1 x ) 3 w(x, t) 2 − q(x, t)w(x, t).(203)
Using the Lagrangian (201), the corresponding terms in the relevant Euler-Lagrange equation, i.e. Eq. (200), are
∂L ∂w = −q(x, t) ∂L ∂D 1 t w(x, t) = µ D 1 t w(x, t),(204)
∂L ∂D 2
x w(x, t)
= (E I (d) ) D 2 x w(x, t), ∂L ∂D 3
x w(x, t)
= (E I (d) ) l 2 F (d) D 3 x w(x, t).
Gradient Euler-Bernoulli static equation for fractal beam
The gradient Euler-Bernoulli fractal homogeneous beam equation for the static case (D 1 t w = 0 and q(x, t) = q(x)) is obtained from Eq. (208) as
D 2 x c 1 (α x , x) D 2 x w − l 2 F (d) D 3 x c 1 (α x , x) D 3 x w = c 1 (α x , x) E I (d) q(x).(227)
For a non-fractal beam (α x = 1), the static gradient Euler-Bernoulli beam equation takes the form
D 4 x w − l 2 s D 6 x w = 1 E I (2) q(x).(228)
It is noted that Eq. (227) for a fractal beam is analogous to the static case of Eq. (207) for a non-fractal beam (α x = 1 and c 1 (α x , x) = 1), which is non-homogeneous such that the product E I (2) ef f depends on x as well as c 1 (α x , x), i.e. E I (2) ef f ∼ x αx−1 (0 < α x < 1). This effective static equation for a gradient Euler-Bernoulli non-homogeneous beam is expressed by
D 2 x (E I (2) ef f ) (D 2 x )w − l 2 s D 3 x (E I (2) ef f ) D 3 x w = q ef f (x) (229)
with the effective external load q ef f (x) = c 1 (α x , x)q(x). For the homogeneous case (q(x) = 0), equation (227) can be written in the form
x D 4 x w(x) + (α x − 1) D 4 x w(x) − l −2 F (d) x D 2 x w(x) = C 5 x 2−α + C 6 x 3−α ,(230)
where we take into account the form of the density of states c 1 (α x , x) = x αx−1 /Γ(α x ) and x > 0.
Here C 5 and C 6 are constants defined by the boundary conditions for the initial problem given by Eq. (227), which is a differential equation of 6th order. The general solution of Eq. (230) has the form w(x) = C 1 + C 2 x + C 3 1 F 2 −1/2; 1/2, α x /2 − 1; l −2 F (d)
x 2 /4 + + C 4 l −1 F (d) x 2−αx/2 K αx/2−1 (l −1 F (d) x) + l α−x/2−2 F (d) x I(l −1 F (d) x, α x ) ,(231)
where C 1 , C 2 , C 3 and C 4 are constants defined by appropriate boundary conditions; 1 F 2 [a 1 ; b 1 , b 2 ; c] denotes the hypergeometric function; K a (x) denotes the modified (hyperbolic) Bessel function of the second kind; and I(x, α) is the integral of the Bessel function of the form I(x, α) = x 1−αx/2 K a/2−1 (x) dx.
We can also use the fundamental solution for ordinary differential equations (2.105) in Kamke's book [51] for the case b < 0 and 0 < a < 2 where b = −l α−x/2−2 F (d) and a = α x − 1.
where w 0 is a constant, and C n (α) = cos(k n L α ) + cosh(k n L α ) sin(k n L α ) + sinh(k n L α ) , k n = 1 Γ(α + 1)
This trigonometric equation is solved numerically. The corresponding natural frequencies of vibration are ω n = k 2 n (E I (d) )/ρ A. For a non-trivial value of the displacement, w 0 ia assumed to be arbitrary, and the magnitude of the displacement is taked as unknown for free vibrations. Usually, w 0 = 1 is used when plotting mode shapes.
Combined strain-acceleration gradients for fractal beam
Let us consider a 1D model for a fractal material that is described by the action
S[w] = dt dl αx L(x, t, w, D 1 t w, D 2 x w, D 3 x w, D 2 x D 1 t w),(253)
with the Lagrangian L(x, t, w, D 1 t w, D 2 x w, D 3
x w) = 1 2 ρ A D 1 t w(x, t) 2 + 1 2 E I (d) D 2 x w(x, t) 2 − − 1 2 E I (d) l 2 F (d) D 3 x w(x, t) 2 − q(x, t)w(x, t),(254)
where dl αx = dx c 1 (α x , x), takes into account combined strain-acceleration gradients [16]. The stationary action principle δS F [w] = 0, gives the Euler-Lagrange equation in the form
∂L ∂w − c 1 (α x , x) D 1 t ∂L ∂D 1 t w + D 2 x c 1 (α x , x) ∂L ∂D 2 x w − − D 3 x c 1 (α x , x) ∂L ∂D 3 x w − D 1 t D 2 x c 1 (α x , x) ∂L ∂D 2 x D 1 t w = 0.(255)
For a homogeneous fractal beam, we obtain
ρ A D 2 t w + E I (d) D 4 x,αx w − l 2 F (d) E I (d) D 6 x,αx w + l 2 f (d) ρ I (d) D 2 t D 4
x,αx w − q(x, t) = 0, (256)
where the notation (210) was used.
In we use the fractional continuum model [33,32,13] with some changes suggested in [39]- [41], we derive the Euler-Lagrange equation in the form of Eq. (256), where the derivatives D 2n
x,αx are replaced by ∂ 2n
x,αx = (c −1 (α x , x) D 1
x ) 2n such that ρ A D 2 t w + E I (d) ∂ 4
x,αx w − l 2 F (d) E I (d) ∂ 6
x,αx w + l 2 f (d) ρ I (d) D 2 t ∂ 4
x,αx w − q(x, t) = 0.
Conclusions
In this paper, we consider non-standard generalizations of the gradient elasticity theory [17]- [22] for complex materials with power-law non-locality, long-term memory and fractality. These nonstandard generalizations may be important in describing unusual properties of nanomaterials [67,68].
To obtain the governing equations for the new fractional generalizations of gradient elasticity theory for materials with power-law non-locality, we use a new fractional variational principle for Lagrangians with Riesz fractional derivatives. New generalizations can also be obtained through extensions of the traditional variational calculus for Lagrangians by using other types of fractional derivatives [23]- [28], as well as with Riesz derivatives in the form suggested in [29]. We also assume that new fractional integral elasticity models can be derived by using the variational principle suggested in [30], where the Lagrangian contains fractional integrals instead of fractional derivatives.
The fractional approach, which is suggested in this paper, allows us to obtain exact analytical solutions of the fractional differential equations for models of a wide class of material with fractional gradient non-locality. A characteristic feature of the behavior of a fractional nonlocal continuum is the appearence of spatial power-tails of non-integer order. The fractional gradient models, which are suggested in this paper to describe complex materials with fractional non-locality, can be characterized by a common or universal spatial behavior of elastic materials in analogy to the universal temporal behavior of low-loss dielectrics [62]- [65].
The proposed generalization of gradient elasticity theory for fractal materials is based on the fractional continuum models proposed in [32]- [35] (see also [13,37]). In particular, equations for gradient models of fractal materials are obtained by a fractional integral generalization of the variational principle suggested in [35,36] (see also [13]). In the framework of the fractional integral continuum model for fractal materials, modified variational principles considered in [37,38] can also be used.
We assume that new non-standard generalizations of the gradient elasticity models of fractal materials can be obtained by using the analysis on fractals [69,70], as well as by using the methods of the vector calculus for non-integer-dimensional spaces [47,48], and by also using a generalization of fractal lattice models [71]- [73].
The approach proposed in this paper is based on fractional integral continuum models and it may have a wide application because of the relatively small numbers of parameters that define fractal media of great complexity and rich structure. The fractional continuum model of fractal elastic materials can be used not only to calculate global values and stationary characteristics, but also to describe dynamical properties of fractal materials.
where extends over all combinations of non-negative integer values of a 1 , a 2 , . . . ,
In the notations a I α b [x] and a D α x [x ′ ], we idicate the variable of integration by the brackets [ ], and the lower indices show the limits of integration. Note that in Eq. (96) the variable of integration is x since the result of the integration with respect to x ′ in the operator C a D α x [x ′ ] depends on x only. These notations are more convenient than the ones usually used (see Eq.(93)), since it allows us to take into account the variables of integration and the domain of the operators.
and γ ≤ 2α, then (see Section 4.1.3 and Theorem 4.3 of [2]) equation (139) has a unique solution u c (x) belonging to the space C 2α,n γ [a; b], where n − 1 < 2α < n, defined by the expression
This fractional integral equation represents the resultant force acting on any region W of the fractal material. For D = 3 and d = 2, Eq. (156) gives the usual equation for the resultant force in a non-fractal continuum. The force equilibrium condition for the region W requires F Σ (W ) = 0. Therefore, we have the fractional integral equation of equilibriumW f(r, t) dV D + ∂W σ n (r, t) dA d = 0.
and (170), the brackets [ . , . ] denotes vector product of vector fields. The resultant moment M Σ (W ) is the sum M Σ (W ) = M M (W ) + M S (W ). (171) Substituting Eqs. (169) -(170) into Eq. (171), we obtain
. Equations (179-181) contain two types of derivatives: D 1 l and ∇ (D,d) k .
Eqs. (204) and (205) into Eq. (200) givesµ D 2 t w + c −1 1 (α x , x) D 2 x c 1 (α x , x) (E I (d) ) (D 2 x )w −5.3 Second moment of area for fractal beam
boundary conditions given by Eqs. (248)-(249), the solution (250) exist only if k n are defined by cosh(k n L) cos(k n L) + 1 = 0.
The stationary action principle implies the holonomic variational equation δS F [w] = 0. This equation gives the Euler-Lagrange equation in the form
D−d−1 Γ(D/2) Γ(3/2)Γ(d/2) |r| d+1−D .
AcknowledgmentThe support of ERC-13 and the Aristeia II projects through the General Secretariat of Research and Technology (GSRT)of Greece is gratefully acknowledged.which is the governing equation of motion for a fractal Euler-Bernoulli beam. For a non-fractal beam, we have α x = 1, c −1 1 (α x , x) = 1, and the standard gradient elasticity Euler-Bernoulli beam equation is recoveredwhere the beam can be non-homogeneous, and E and I may depend on x.If the fractal beam is homogeneous (see Section 4.2), then E and I (d) are independent of x, and the beam equation has a simpler formThis equation can be expressed aswhere we have used the notationIf α x = 1, then c 1 (α x , x) = 1 and D 2nx,αx = D 2nx . Using the Lagrangian given by Eq. (203), the corresponding Euler-Lagrange equation has the form of Eq. (208), where the derivatives D 2nx,αx are replaced byFor non-fractal materials, we have α x = 1 and Eqs. (209), (211) have the formThis is the gradient elasticity Euler-Bernoulli beam equation for media without fractional nonlocality, memory and fractality[16].In this section, we give an example of computation a second moment of the fractal beam's cross-section by the method suggested in[34]. Let us consider a homogeneous fractal beam with circular cross-section. The second momentof the fractal beam's cross-section iswhere d = d yz is the fractal dimension of the circular cross-section of the beam. In Eq. (213) we take into account the density of states c 2 (d, y, z) in the fractal material through the relation dA x (d) = c 2 (d, y, z) dA x , where (x, y, z) are dimensionless variable. Let us derive the polar moment of inertia I (d) p for the circular cross-section. By using the equalitiesz , we find the moment of inertia by using the relationshipThe equation for the polar moment of inertia Ip can be written in the formwhere dA 2 = dydz, (x = x 1 , y = x 2 , z = x 3 ) are dimensionless Cartesian coordinates, and ρ 0 is the constant surface mass density. The fractional generalization of Eq. (215) is given by expressionwhereSubstitution of Eq. (217) into Eq.(216)givesIn equation (216) we use the numerical factor c(d) such that the limits d → (2 − 0) give the usual integral formula (215). For d = 2, Eq. (216) gives Eq. (215). The parameter d = d yz denotes the fractal mass dimension of the circular cross-section of the beam. This parameter can easily be calculated from the experimental data by using the box counting method for the cross-section of the beam.Let us now consider the circular region A that is defined byIn polar coordinates (φ, r), we haveSubstitution of Eq. (220) into Eq.(218)givesThis equation defines the second moment of the fractal beam's cross-section. If d = 2, we obtain the well-known equation IThe mass of the homogeneous fractal beam iswhere dA d is defined by equation(217), and ρ 0 is the constant surface mass density. Using the polar coordinates (220), we obtain the following mass expressionSubstituting (223) into (221), we getwhere d is the fractal mass dimension of the beam's circular cross-section (1 < d 2). If d = 2, we derive the well-known relation I(2) p = (1/2)MR 2 . If we consider a fractal beam with mass and radius that are equal to the mass and radius of a beam with integer mass dimension, then these second moments are connected by the equationwhere I(2) p is the moment for the homogeneous beam with the integer cross-section mass dimension d = 2.Using the relation (214), we getThis is the second moment of a circular cross-section of the fractal beam with cross-section in the yz-plane and fractal dimension d = d yz , which should be determined by experiment.Gradient Timoshenko equations for fractal beamIn this section we consider a gradient generalization of the Timoshenko beam equations for a fractal beam, as suggested in[40,41]. In the Timoshenko beam theory without axial effects, the displacement vector u(x, y, z, t) of the beam is assumed to be given bywhere (x, y, z) are the coordinates of a point in the beam, (u x , u y , u z ) are the components of the displacement vector u , ϕ = ϕ(x, t) is the angle of rotation of the normal to the mid-surface of the beam, and w = w(x, t) is the displacement of the mid-surface in the z-direction.In[40,41]it is suggested to use the derivativesinstead of the usual derivatives D 1 x and D n x for fractal materials. If we use the derivatives given by Eq. (234) for fractal materials according to[40,41], then the gradient Timoshenko equation for a fractal beam can be derived from the force and moment balance equationswith the bending moment M given byand the shear force Q isThen, the gradient Timoshenko equations for a homogeneous fractal beam have the formThe gradient Timoshenko fractal beam Eqs. (238) and (239) can also be derived from an appropriate variational principle. The Lagrangian for a Timoshenko fractal beam with gradient non-locality has the formThen, the stationary action principle gives the equationsEquations(241)If α = 1, then Eqs. (243)-(244) reduce to the gradient Timoshenko equations for a beam made by a homogeneous non-fractal material.For the models based on[39]-[43], solutions of equations for fractal materials can be obtained from solutions of equations for non-fractal materials. Let w c (x, t) and ϕ c (x, t) be solutions of Eqs. (243)-(244) with α = 1 and x > 0, i.e., of the gradient Timoshenko equations for homogeneous non-fractal beams. Then, the solutions w F (x, t) and ϕ F (x, t) of equations (243)-(244) for a fractal beam with 0 < α < 1 can be represented in terms of w c and ϕ c as follows:As an example, we consider the equation for an Euler-Bernoulli homogeneous fractal beam in the absence of a transverse load (q(x) = 0),This equation can be solved using the Fourier decomposition of the displacement into the sum of harmonic vibrations of the form w(x, t) = Re[w(x) exp(−iωt)], where ω is the frequency of vibration. Then, for each value of frequency, we can solve the ordinary differential equationThe boundary conditions for a cantilevered fractal beam of length L fixed at x = 0 are w(0) = 0, (∂ 1 x,α w)(0) = 0,(∂ 2 x,α w)(L) = 0, (∂ 3 x,α w)(L) = 0.The solution for the Euler-Bernoulli homogeneous fractal beam is defined by w F,n (x) = w 0 cosh(k n x α ) − cos(k n x α ) + C n (α) [sin(k n x α ) − sinh(k n x α )] , x ∈ [0; L], (250) This is the usual combined strain-acceleration gradient beam model[16]. Note that Eq. (256) for a fractal beam is analogous to the equation for the usual combined strain-acceleration gradient non-fractal beam (α x = 1 and c −1 1 (α x , x) = 1), which is nonhomogeneous, such that the product E I(2)ef f depends on x, as well as on c 1 (α x , x); i.e. E I(2)ef f ∼ x αx−1 (0 < α x < 1). Equation (257) can be solved by the method suggested in Section 5.5 from the solutions of Eq. (258) for non-fractal materials.
If the beam is non-fractal, then D = 3, α x = 1, c 1 (α x , x) = 1, and Eqs. (256) and (257). If the beam is non-fractal, then D = 3, α x = 1, c 1 (α x , x) = 1, and Eqs. (256) and (257)
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[] |
[
"A Generalized Volume-Correlation Subspace Detector and its application in Multiuser Detection",
"A Generalized Volume-Correlation Subspace Detector and its application in Multiuser Detection"
] |
[
"Hailong Shi ",
"Hao Zhang ",
"Xiqin Wang "
] |
[] |
[] |
The problem of subspace signal detection, i.e., detecting whether a signal lies within a subspace arises in a wide range of applications in the signal processing community, including communication, radar, array signal processing, hyperspectral image, etc. In this paper, a novel volume-correlation subspace detector is proposed to detect subspace signals contaminated by noises and subspace interferences. It is shown that, the volume of parallelotope, which is spanned by the signal subspace's basis vectors together with multi-dimensional received signal vectors, can be calculated and used to detect subspace signals with interferences. It is also proved in this paper that, with the knowledge of the signal subspace and the interference subspace, and by accumulating multi-dimensional signal observations, the volume-correlation subspace detector will eliminate the influence of noise as well as the influence of interference asymptotically. And this is an advantage that traditional subspace detectors don't have. Besides, the application of this volume-correlation subspace detector to Active User Identification in multiuser detection systems with multipath channels is discussed in this paper. Active User Identification is a well known and essential preprocess stage in multiuser detection systems, it can reduce the system complexity and improve the overall decoding performance. It is shown that the proposed volume-correlation subspace detector has its interior advantage to deal with the interference of other users in the system with a multipath channel, and provides asymptotically ideal performance in detection of active users. Numerical simulations is also provided to validate our conclusion. DRAFT• Determine the rank of Y (m) , r := rank(Y (m) ), m ≥ 1 June 6, 2014 DRAFT
| null |
[
"https://arxiv.org/pdf/1406.1286v1.pdf"
] | 124,361,867 |
1406.1286
|
c06bb5a7f830c7c1d1997ed60cd52f8f89c83f2f
|
A Generalized Volume-Correlation Subspace Detector and its application in Multiuser Detection
5 Jun 2014
Hailong Shi
Hao Zhang
Xiqin Wang
A Generalized Volume-Correlation Subspace Detector and its application in Multiuser Detection
5 Jun 20141Index Terms Subspace Signal DetectionMatched Subspace DetectorGeneralized Energy DetectorVolume-Correlation Sub- space DetectorActive User IdentificationMultiuser Detection
The problem of subspace signal detection, i.e., detecting whether a signal lies within a subspace arises in a wide range of applications in the signal processing community, including communication, radar, array signal processing, hyperspectral image, etc. In this paper, a novel volume-correlation subspace detector is proposed to detect subspace signals contaminated by noises and subspace interferences. It is shown that, the volume of parallelotope, which is spanned by the signal subspace's basis vectors together with multi-dimensional received signal vectors, can be calculated and used to detect subspace signals with interferences. It is also proved in this paper that, with the knowledge of the signal subspace and the interference subspace, and by accumulating multi-dimensional signal observations, the volume-correlation subspace detector will eliminate the influence of noise as well as the influence of interference asymptotically. And this is an advantage that traditional subspace detectors don't have. Besides, the application of this volume-correlation subspace detector to Active User Identification in multiuser detection systems with multipath channels is discussed in this paper. Active User Identification is a well known and essential preprocess stage in multiuser detection systems, it can reduce the system complexity and improve the overall decoding performance. It is shown that the proposed volume-correlation subspace detector has its interior advantage to deal with the interference of other users in the system with a multipath channel, and provides asymptotically ideal performance in detection of active users. Numerical simulations is also provided to validate our conclusion. DRAFT• Determine the rank of Y (m) , r := rank(Y (m) ), m ≥ 1 June 6, 2014 DRAFT
I. INTRODUCTION
The problem of subspace signal detection, i.e., detecting whether a signal lies within a subspace arises in a wide range of applications in the signal processing community, such as communication [1] [2][3] [4], radar [5][6] [7][8] [9], array signal processing [10] [11], as well as anomaly detection in hyperspectral imagery [12]. The basic problem of subspace detection can be described as a binary hypothesis test, in which the observation signal vectors r ∈ R P are given, and the problem is to decide whether the signal components in these observations lie within a known linear subspace, it is formulated as H 0 : r = x I + w,
H 1 : r = x s + x I + w,(1)
here w is the white Gaussian noise component, i.e., w ∼ N (0, σ 2 I P ); and it is assumed that the to-be-detected signal, i.e., x s , obeys the linear subspace model, which means
x s = X s α, X s ∈ R P ×d , α ∈ R d ,(2)
where the column vectors of matrix X s form the basis of the to-be-detected signal subspace, and α is the corresponding subspace coefficient vector, and d < P is the dimension of the to-be-detected signal subspace.
For convenience, we call the to-be-detected signal subspace as the target signal subspace, and denote it by X s := span(X s ).
In the same way, we also assume that the interference signal, i.e., x I , obeys the linear subspace signal model,
which means
x I = X I β, X I ∈ R P ×p , β ∈ R p ,(3)
where the column vectors of matrix X I form the basis of the interference signal subspace, and β denotes the corresponding subspace coefficient vector, and p < P is the dimension of the interference signal subspace. In the same way, the interference signal subspace is denoted by X I := span(X I ).
Therefore, our goal is to decide the existence of the target subspace signal x s = X s α from the received signal r ∈ R P with noise and interference x I = X I β. Without loss of generality, it is assumed that the target signal subspace X s is linearly independent and disjoint with the interference signal subspace X I , i.e., X s X I = ∅;
besides, the noise level σ is assumed to be known, while the subspace coefficients α and β are unknown.
The subspace signal detection described here is a generalization of the conventional signal detection problem, in which the matched filter is used as the common approach, the reason is that subspace signal detection problem concerns about the detection of multi-dimensional subspace signal x s = X s α, while the conventional signal detection problem detects signals in the form of individual vectors; in other words, the conventional signal detection problem is a one-dimensional special case of the subspace signal detection. The reason why subspace signal is needed to be dealt with is twofold. First, because of the physical process of signal transmitting and acquisition, for example, the wireless channel's multi-path effect [13] [2], the scattering mechanisms of radar's target signals and clutters [6] [14], as well as the interior connection between image's neighbor pixels [12], etc, signals we concern are usually multi-dimensional, and needed to be modeled as linear subspaces. Second, due to the randomness in signal modulation and transmitting, it is difficult to obtain the subspace coefficient α in the receiver, and thus the vector form of the target signal x s is usually unknown. For those reasons, more general approaches are needed to use knowledge of signal subspace X s and detect the subspace signal x s = X s α.
There have been well-known approaches aiming at detection subspace signals, which is firstly proposed by L. Scharf et.al., it is named as the Matched Subspace Detector [15], and already has tremendous variations and applications [16] [17][18] [19]. The core technique of the Matched Subspace Detector is to use the principal of Generalized Likelihood Ratio Test, with subspace coefficients α and β replaced by their maximum likelihood estimation. The test statistics of problem (1) given in [15] is:
t(r) = 1 σ 2 (P ⊥ XI − P ⊥ XsI )r 2 2 = 1 σ 2 P P ⊥ X I Xs r 2 2 ,(4)
where P ⊥ XI is the orthogonal projection operator to subspace X ⊥ I , P ⊥ XsI is the orthogonal projection operator to subspace (X s X I ) ⊥ , and P P ⊥ X I Xs is the orthogonal projection operator to subspace P ⊥ XI X s . The testing result is obtained by comparing (4) with thresholds derived according to certain conditions (such as Constant False Alarm Rate). (4) indicates that the test statistics is the energy of the part of r projected orthogonally onto subspace P ⊥ XI X s , therefore the detector in (4) is also called the Generalized Energy Detector [15].
However, the Generalized Energy Detector cannot eliminate the influence of the interference subspace, because the target signal subspace X s may not be orthogonal to the interference signal subspace X I , which will cause a loss of energy when projecting the target signal, to be more specific, under H 1 hypothesis, the projection in (4) will yield P P ⊥ X I Xs r = P ⊥ XI X s α + P P ⊥ X I Xs w,
As depicted in figure 1, if the interference subspace X I is "more close to" the target subspace X s , or less orthogonal to X s , then the signal component P ⊥ XI X s α in (5) will be weaker compared to noise, therefore more difficult to detect.
The reason why the detection performance of Generalized Energy Detector is influenced by the interference signal subspace is that the projection in (4) is onto the subspace P ⊥ XI X s , rather than X s . As a result, the Generalized Energy Detector cannot eliminate the influence of interference, while in this paper, we are going to propose a novel subspace detection technique to asymptotically eliminate the influence of interference. Before we introduce this technique, firstly we begin with a simple demonstration, if we consider the hypothesis test (1) without noise, then under H 1 hypothesis, the noiseless received signal is
y := x s + x I = X s α + X I β,(6)
then we have y ∈ X s X I .
Similarly, under H 0 hypothesis, we have y ∈ X I .
If we can get multiple observations of the noiseless received signal y, which means the coefficients α(or β) vary with different observations (without loss of generality, α and β can be assumed to be independent random vectors), denote the multiple observations of the received signal by y (1) , y (2) , · · · , then we have
y (i) := x (i) s + x (i) I = X s α (i) + X I β (i) , i = 1, 2, · · ·(9)
where α (i) and β (i) are the target and interference subspace coefficients corresponding to the i'th observation. Then we can have for m ≥ 1, (1) , y (2) , · · · , y (m) ]) ⊂ X I , H 1 : span([y (1) , y (2) , · · · , y (m) ]) ⊂ X s X I .
H 0 : span([y
Therefore, (10) reminds the use of multi-dimensional information from the multiple observations of the received signal, i.e., the subspace spanned by the columns of the observation matrix Y (m) := [y (1) , y (2) , · · · , y (m) ], to explore the relation between the received signal and the target subspace X s . It is not difficult to obtain and process multiple observations of the received signal in common signal processing systems. For example, in the Hyperspectral Imagery systems [12][20] [21], the received image signal is itself two-dimensional, and the sensors of hyperspectral imaging collects a set of images from different spectral channel bands, thus in this system a three-dimensional hyperspectral data cube is acquired for processing and analysis. Besides, it is well-known that array receivers or sensors, as well as MIMO systems have been widely used in communication, radar and other signal processing systems [10][11] [9], and this also makes it possible to exploit the multi-dimensional information hidden in the obtained multiple observations to gain a better performance for the subspace signal detection. Unfortunately, as far as we know, all these applications mentioned above relies on the traditional matched subspace detector approach, and only concerns about a single observation vector r (or in noiseless case y). As is said that the traditional matched June 6, 2014 DRAFT subspace detector approach is not interference-resistant, so in this paper, we will make use of the multi-dimensional information from multi-dimensional observations, i.e. the linear subspace spanned by the observations as in (10) [30]. Briefly speaking, these applications model the multi-dimensional signal as a linear subspace, and deal with the problem of subspace-oriented optimization or estimation using knowledge of an abstract manifold (i.e., the Grassmann Manifold, which is a topological space where each point is a linear subspace); the rich topological structure (such as the Tangent space, Geodesic Distance) as well as various definitions of metrics on Grassmann manifold make it possible to exploit multi-dimensional information and derive better optimization and estimation result. In the following sections, we will utilize the tool of principal angles of subspaces, as well as volumes of matrices to study the subspace signal detection problem (1).
II. PRELIMINARY BACKGROUND
A. Principal Angles between Subspaces
The well-known principal angles [31] can be used to describe the relationship between different subspaces, it is defined as follows:
Definition 1: For two linear subspaces X 1 and X 2 , with dimensions dim(X 1 ) = d 1 , dim(X 2 ) = d 2 . Take m = min(d 1 , d 2 )
, then the principal angles 0 ≤ θ 1 ≤ · · · ≤ θ m ≤ π/2 between these two subspaces are defined recursively by
cos θ i = max ui∈X1,vi∈X2 u T i v i , subject to u i 2 = v i 2 = 1, u T i u j = 0, v T i v j = 0, (j = 1, · · · , i − 1, i = 2, · · · m).
The principal angles are important mathematical tools to describe relations of subspaces. In fact, in the theory of Grassmann manifold [32][33], the Geodesic distance, which is an important metric measure, is defined using the principal angles [32]; besides, according to [33], there are numerous kinds of distance measures that are defined based on the principal angles. In this paper, we will use another distance measure, i.e., the volume, which is closely related to the principal angles, to construct our subspace detector.
B. The Volume of a matrix and the principal angles between subspaces
The d-dimensional (d < P ) volume of a full-rank matrix X ∈ R P ×d is defined as [34] vol
d (X) := d i=1 σ i ,(11)
where σ 1 ≥ σ 2 ≥ · · · ≥ σ d > 0 are the singular values of matrix X. The volume of the matrix X is also referred to as the d-dimensional parallelotope spanned by the column vectors of X. For the matrix X is of full column June 6, 2014 DRAFT rank, the volume is equivalently [34] [31] vol d (X) = det(X T X).
Volume provides a measure of separation between two linear subspaces and is closely related to the principal angles between subspaces, according to [31], for two subspaces X 1 and X 2 , with dimensions dim(X 1 ) = d 1 , dim(X 2 ) = d 2 , and basis matrix X 1 and X 2 , we have
vol d1+d2 ([X 1 , X 2 ]) vol d1 (X 1 ) vol d2 (X 2 ) = min(d1,d2) j=1 sin θ j (X 1 , X 2 ),(13)
where
0 ≤ θ j (X 1 , X 2 ) ≤ 2π, 1 ≤ j ≤ min(d 1 , d 2 )
are the principal angles of subspace X 1 and X 2 .
It can be seen intuitively from (13) that, the expression vol d1+d2 ([X 1 , X 2 ])/(vol d1 (X 1 ) vol d2 (X 2 )) can be used to derive a distance measure of subspace X 1 and X 2 . Because when X 1 and X 2 intersect, i.e.,
X 1 X 2 = ∅, we have vol d1+d2 ([X 1 , X 2 ]) = 0; when X 1 is perpendicular to X 2 , then vol d1+d2 ([X 1 , X 2 ]) = vol d1 (X 1 ) vol d2 (X 2 ).
The right side of (13) is the product of sines of the principal angles between subspaces X 1 and X 2 . As we know that various metric measures of linear subspaces are defined based on principal angles, typically such as • Chordal Distance or Projection distance
d proj := min(d1,d2) j=1 sin 2 θ j (X 1 , X 2 ) 1/2 , • Binet-Cauchy Distance d BC := 1 − min(d1,d2) j=1 cos 2 θ j (X 1 , X 2 ) 1/2 , • Procrustes Distance d proc := 2 min(d1,d2) j=1 sin 2 θ j (X 1 , X 2 ) 2 1/2 .
additionally, more kinds of distance measures can be defined using functions satisfying specific conditions [33].
Therefore, although we cannot validate whether (13) satisfies the condition of metric functions, i.e., the triangle inequality, following the terminology used in [33], and for simplicity of calculation and convenience, without verifying the triangle inequality, we regard the product of principal sines induced by volume in (13) as a generalized distance measure for linear subspaces in the following analyzes, and use it as the "volume-correlation", which plays a key role in our proposed subspace detector.
III. THE VOLUME-CORRELATION SUBSPACE DETECTOR
In this section we will introduce the Volume-Correlation Subspace Detector, as is discussed in the previous section, multi-dimensional observations of the received signal will be used to perform subspace detection, these multiple observations of problem (1) is of the form of:
H 0 : r (i) = X I β (i) + w (i) ,H 1 : r (i) = X s α (i) + X I β (i) + w (i) , i = 1, 2, · · · ,(14)
where α (i) and β (i) are subspace coefficients, X s ∈ R P ×d , X I ∈ R P ×p , and w (i) is the white Gaussian noise. We need to detect the existence of target subspace signal X s = span(X s ) in the observations r (i) ∈ R P , i = 1, 2, · · · .
Before the introduction of the subspace detector, we start with another demonstrative observation, as is mentioned previously, in the noiseless situation, and under H 1 hypothesis, the multiple observations
y (i) = X s α (i) + X I β (i) , i = 1, 2, · · · satisfy
span(Y (m) ) = span([y (1) , y (2) , · · · , y (m) ]) ⊂ X s X I . and we have
m ≤ d + p, rank(Y (m) ) = m, span(Y (m) ) X s X I ,(15)m > d + p, rank(Y (m) ) = d + p, span(Y (m) ) = X s X I .(16)
Similarly, in the H 0 hypothesis, we have
m ≤ p, rank(Y (m) ) = m, span(Y (m) ) X I ,(17)m > p, rank(Y (m) ) = p, span(Y (m) ) = X I .(18)
Then according to the previous analysis, under H 1 hypothesis, when m > p + d, then the volume of matrix
[Y (m) , X s ] will satisfy vol m+d ([Y (m) , X s ]) = 0.(19)
Similarly, under H 0 hypothesis, when m > p, because X s and X I are linearly independent, thus the volume of (19) and (20) implies that, in noiseless situations, when the target signal subspace X s and the interference subspace X I are linearly independent, m ≥ d + p observations will guarantee the difference of the volume of [Y (m) , X s ] between H 1 and H 0 hypotheses, and this leads to our Volume-Correlation Subspace Detector.
[Y (m) , X s ] will satisfy vol m+d ([Y (m) , X s ]) = 0.(20)
A. The Volume-Correlation Subspace Detector
In the noiseless situation, the Volume-Correlation Subspace Detector is:
Detector 1'. (w (i) = 0, i.e., noiseless scenario) • Obtain Y (m) = [y (1) , · · · , y (m) ], where H 0 : y (i) = X I β (i) ,H 1 : y (i) = X s α (i) + X I β (i) , i = 1, 2, · · · , m.
• Calculate the volume-correlation:
t(Y (m) ) := vol d+r ([X s , Y (m) ])/(vol d (X s ) vol r (Y (m) )),
• Test:
1/t(Y (m) ) H1 ≷ H0 1/τ (X s , X I ), where τ (X s , X I ) := vol d+p ([X s , X I ])/(vol d (X s ) vol p (X I )).
When there is noise, i.e., w (i) ∼ N (0, σ 2 I P ), then the Volume-Correlation Subspace Detector is:
Correlator 1. (w (i) ∼ N (0, σ 2 I)) • Obtain R (m) = [r (1) , · · · , r (m) ], where H 0 : r (i) = X I β (i) + w (i) ,H 1 : r (i) = X s α (i) + X I β (i) + w (i) , i = 1, 2, · · · , m.
• Eigenvalue decomposition of the sampled covariance matrix:
C (m) = 1 m m i=1 r (i) (r (i) ) T = P j=1λ (m) jû (m) j (û (m) j ) T , whereλ (m) 1 ≥λ (m) 2 ≥ · · · ≥λ (m) P ≥ 0 are eigenvalues andû (m) 1 , · · · ,û (m) P are corresponding eigenvectors of C (m) ,
• Estimate the dimension of the signal subspace: r,
• Extract the singal subspace:
U (m) sig := [û (m) 1 , · · ·û (m)
r ] is the eigenvector corresponding to the r largest eigenvalues ofĈ (m) .
• Calculate the volume-correlation:
t(R (m) ) := vol d+r ([X s ,Û (m) sig ])/(vol d (X s ) vol r (Û (m)
sig )),
• Test:
1/t(R (m) ) H1 ≷ H0 c/τ (X s , X I ),
where τ (X s , X I ) := vol d+p ([X s , X I ])/(vol d (X s ) vol p (X I )), and c ≥ 1 is a constant. • Remark 1. The Volume-Correlation Subspace Detector is given in two forms, i.e., Detector 1' in noiseless situation, and Detector 1 in noised situation. The reason why we propose the noiseless detector are twofolds.
First, by assuming a noiseless situation, we can have a deep insight into the nature of the Volume-Correlation Subspace Detector. Second, as shown in Detctor 1, the method of eliminating noise is the well-known eigendecomposition subspace method, and the theory and analysis about the subspace method is already mature and can be used directly. Therefore, the Volume-Correlation Subspace Detector is given in two forms and will be analyzed in both the noised and noiseless situations in following sections, respectively.
• Remark 2. The estimation of the dimension of the signal subspace (composed of both the target signal and interference signal subspaces) both in noised and noiseless situations is needed. This is because the dimension is a key parameter needed to calculate the volume. For example, in the noiseless situation, and H 1 hypothesis, when m ≤ p + d, the matrix Y (m) is full-rank; but when m > p + d, Y (m) ∈ R P ×m is not full-rank, then the m-dimensional volume vol m (Y (m) ) will reduces to 0, as a result, the Detector 1' won't work any more.
Therefore, the dimension of the signal subspace r is needed for our volume calculation.
• Remark 3. The detection result is given by comparing the reciprocal of the volume-correlation result 1/t(Y (m) )(or 1/t(R (m) )) with threshold 1/τ (X s , X I )(or c/τ (X s , X I )). The reason for use of reciprocals is to enlarge the difference of 1/t(Y (m) )(or 1/t(R (m) )) between two hypotheses, which will make the comparison to be easier.
Besides, in noised situation, an arbitrary constant c is choosen for the fine tuning the threshold, and to enhance the detection performance. The choosing and analysis of the threshold will be discussed in the following sections. The most important issue which should be noted here is that from (13), it can be seen that τ (X s , X I ) is a value decided only by the relation between subspaces X s and X I .
• Remark 4. In the noised situation, the Detector 1 uses the well-known eigen-decomposition subspace method as a preprocess for the estimation of signal subspace. As in these classic subspace methods such as MUSIC or PCA, the dimension as well as the basis of the signal subspace is estimated through the eigenvalue decomposition of sampling covariance matrix. As is known that the estimations of dimension and the basis are generally two independent stages, and for the first stage, i.e., the estimation of the dimension of signal subspace, there are already a lot of famous approaches, such as Akaike-Information Criterion (AIC) [35], Minimum Description Length (MDL) [36], Bayesian Information Criterion (BIC) [37], Predictive Description Length (PDL) [38] and the most recent Entropy Estimation of Eigenvalues(EEE) [39]. All those approaches are quite mature and ensure high precision in the dimension estimation. Therefore, we only focus on the second stage, and assume the dimension of signal subspace is precisely estimated, which is also a general assumption in other analysis of the subspace methods. In the following sections, we will analysis the asymptotic error of our Volume-Correlation Detector caused by the signal subspace estimation approach.
• Remark 5. There are two significant factors in the proposed volume-correlation detector: first, the detector makes use of multi-dimensional observations of the received signal, i.e., r (1) , · · · , r (m) (or y (1) , · · · , y (m) ), which will bring much more information than a single observation; second, the test statistics is calculated using volume of the matrix composed of the multi-dimensional observation data R (m) and the basis X s of target signal subspace X s . As far as we know, volume is firstly used to construct a subspace detector, and it has advantages in exploiting the multi-dimensional information. Specially, for two 1-dimensional vectors
x 1 and x 2 , vol 2 ([x 1 , x 2 ]) = x 1 2 x 2 2 sin θ(x 1 , x 2 )
, this means volume is the amplitude of the exterior product, which is in correspondence with the inner product x 1 , x 2 = x 1 2 x 2 2 cos θ(x 1 , x 2 ) (also known as "correlation", and is used in matched filter detectors). Besides, as is mentioned in the previous section, volume can be used to derive a generalized distance measure for multi-dimensional subspaces. As a result, we regard the volume-based test statistics
t(Y (m) ) := vol d+r ([X s , Y (m) ])/(vol d (X s ) vol r (Y (m) )),(21)
and
t(R (m) ) := vol d+r ([X s ,Û (m) sig ])/(vol d (X s ) vol r (Û (m) sig ))(22)
as a generalized volume-correlation, meaning that (21) and (22) imply a kind of measure of correlation between subspaces span(X s ) and span(Y (m) ) (or span(Û (m) sig )). As we know that the exterior product is the product June 6, 2014 DRAFT defined in Grassmann Algebra, and Grassmann Algebra is a powerful mathematical tool in describing multidimensional geometry; therefore our volume-correlation subspace detector inherits the advantage of Grassmann Algebra, and is good at dealing with multi-dimensional information by using knowledge of multi-dimensional geometry.
• Remark 6. Consider the traditional Generalized Energy Detector in (4), denote the orthogonal basis matrix of subspace P ⊥ XI X s by Q P ⊥ X I Xs ∈ R P ×d , then the projection operation can be written in a matrix form
Q P ⊥ X I Xs Q T P ⊥ X I
Xs r, then we have
P P ⊥ X I Xs r 2 2 = det r T Q P ⊥ X I Xs Q T P ⊥ X I Xs r = vol 2 1 Q T P ⊥ X I Xs r .(23)
This implies that the Generalized Energy Detector can be regarded as a special case of this paper's Volume-Correlation Subspace Detector. The main difference is that, the Volume-Correlation Subspace Detector makes
use of multi-dimensional observation data R (m) = [r (1) , · · · , r (m) ]
, while the Generalized Energy Detector only uses one-dimensional observation r and calculates the one-dimensional volume in (23) according to the GLRT principle. Although currently we cannot give the optimized form of our volume detector under certain principle, the usage of multi-dimensional information makes our volume-correlation detector indisputably advantageous, and the study of the optimality will be left for future work.
In the next section, we will give the theoretical analysis of the advantage of the Volume-Correlation Subspace
Detector.
IV. ANALYSIS OF THE VOLUME-CORRELATION SUBSPACE DETECTOR: THE NOISELESS SITUATION
In this section, we will show theoretically that, in the noiseless situation, by using multi-dimensional observations of the received signal, our Volume-Correlation Subspace Detector will totally eliminate the influence of the interference signal. The main result is the following theorems:
Theorem 1: In noiseless situation, under H 1 hypothesis, the multiple observations of received signal are of the form of
y (i) = X s α (i) + X I β (i) , i = 1, 2, · · · where X s ∈ R P ×d , α (i) n
∈ R d denotes the basis and coefficient of the target signal subspace, and X I ∈
R P ×p , β (i)
n ∈ R p denotes the basis and coefficient of the interference signal subspace, and span(X s ) is linearly independent of span(X I ); then for the multi-dimensional observation data matrix
Y (m) = [y (1) , · · · y (m) ] ∈ R P ×m ,
denote r := rank(Y (m) ), the following volume-correlation in Detector 1'
t(Y (m) ) := vol d+r ([X s , Y (m) ])/(vol d (X s ) vol r (Y (m) )),(24)
will satisfy:
• When 1 ≤ m < p, t(Y (m) ) ≥ t(Y (m+1) ); (25) • When m ≥ p + 1, t(Y (m) ) = 0.(26)
Theorem 1 demonstrates the advantage of our volume-correlation subspace detector. It shows that in noiseless situation, and under H 1 hypothesis, as the number of observations m increases, the volume-correlation t(Y (m) ) will generally descend to 0.
Under H 0 hypothesis, here is another theorem:
Theorem 2:
In noiseless situation, under H 0 hypothesis, the multiple observations of received signal are of the form of
y (i) = X I β (i) , i = 1, 2, · · · where X I ∈ R P ×p , β (i)
n ∈ R p denotes the basis and coefficient of the interference signal subspace; then for the multi-dimensional observation data matrix
Y (m) = [y (1) , · · · y (m) ] ∈ R P ×m , denote r := rank(Y (m) ), the following volume-correlation in Detector 1' t(Y (m) ) := vol d+r ([X s , Y (m) ])/(vol d (X s ) vol r (Y (m) )),(27)
will satisfy:
• When 1 ≤ m < p, t(Y (m) ) ≥ t(Y (m+1) ),(28)
and
t(Y (m) ) ≥ τ (X s , X I ) = min{d,p} j=1 sin θ j (X s , X I ).(29)
• When m ≥ p,
t(Y (m) ) = min{d,p} j=1 sin θ j (X s , X I ).(30)
where sin θ j (X s , X I ) (1 ≤ j ≤ min{d, p}) are the principal angles between subspaces X s and X I .
Theorem 2 shows the property of volume-correlation result under H 0 hypothesis in noiseless situation. It can be seen that as the number of observations m increases, the volume-correlation t(Y (m) ) will generally descend, not to 0, but to a threshold τ (X s , X I ) = min{d,p} j=1 sin θ j (X s , X I ). As is mentioned that min{d,p} j=1 sin θ j (X s , X I ) is only related to subspaces X s and X I , therefore, as long as X s and X I are linearly independent, then min{d,p} j=1 sin θ j (X s , X I ) will be greater than 0, which means under H 0 hypothesis t(Y (m) ) will never descend to 0.
Theorem 1 and 2 demonstrates the advantage of the volume-correlation subspace detector: in noiseless situation, as is shown by (25), (26) and (28), (29), the volume correlation t(Y (m) ) will descend to 0 under H 1 hypothesis, and to threshold τ (X s ,
X I ) = min{d,p} j=1 under H 0 hypothesis. Therefore τ (X s , X I ) = min{d,p} j=1
is chosen as the threshold for hypothesis test, and the difference of volume-correlation under these two hypotheses will ensure the validation of test result. The simulated volume-correlation values under both hypotheses are plotted in figures 2 and 3. In the simulation, we choose P = 1024, and d = 10, p = 40, the target and interference signal subspaces are chosen arbitrarily (simulation shows that τ (X s , X I ) ≈ 0.82), and subspace coefficients α and β are randomly generated, it can be seen that when m ≥ p = 40, t(Y (m) ) descends to the value predicted by (26) and (29). The simulation implies that in noiseless situation, the influence of interference signal subspace on the volume-correlation t(Y (m) ) is totally eliminated when the number of observations m ≥ p = 40. In addition, if the τ (X s , X I ) is chosen as the threshold for hypothesis test, in the current simulation environment, it is shown that only m > 3 observations will ensure the accurate subspace detection result.
However, theorem 1 and 2 is not enough to explain the advantage of our Volume-Correlation Subspace Detector over the Generalized Energy Detector, because currently we have only discussed the noiseless situation. Actually, the projection P P ⊥ X I Xs in the Generalized Energy Detector
t(r) = 1 σ 2 P P ⊥ X I Xs r 2 2 ,
comes from the maximum likelihood estimation of α and β in the GLRT test. Hence, we need to explore the performance of the Volume-Correlation Subspace Detector in noised situation. As is seen in Detector 1, in the noised situation, the classical eigenvalue decomposition methods are used to estimate the signal subspace span(U sig ) = X s ⊕ X I (orX I ). And it has been well-known that the estimated signal subspace span(Û (m) sig ) will approximate the signal subspace span(U sig ), therefore we can intuitively imagine that, in the noised situation, the volume-correlation t(R (m) ) will also approximate its values in noiseless situation, i.e., 0 or τ (X s , X I ). Detailed analysis will be given in the next section.
V. ANALYSIS OF THE VOLUME-CORRELATION SUBSPACE DETECTOR: THE NOISED SITUATION
In this section, we will analysis the asymptotic performance of the volume-correlation subspace detector, and discuss its advantage in noise and interference elimination.
Before we introduce the main theoretical results, we would like to recall the classical eigen-decomposition subspace method used in Detector 1. As is known, the eigen-decomposition subspace method deals with subspace signals contaminated by noise:
r = Xα + w,
where X ∈ R P ×r , α ∈ R r are basis and coefficient of the signal subspace, and w ∼ N (0, σ 2 I P ) is the white Gaussian noise, then the auto-correlation matrix C of the received signal r can be written as
C = E{rr T } = XE{αα T }X T + σ 2 I P = C sig + C noise .(31)
The eigenvalues of the auto-correlation matrix are: Fig. 2: simulation of the volume-correlation t(Y (m) ) in noiseless situation and the eigenvectors corresponding to λ 1 , λ 2 , · · · , λ r are
λ 1 ≥ λ 2 ≥ · · · ≥ λ r ≥ λ r+1 = · · · = λ P = σ 2 ,(32)1 m t(Y (m) ) Plot of t(Y (m) ) with respect to m (d=10,p=40) t(Y (m) ), H 1 Hypothesis, noiseless t(Y (m) ), H 0 Hypothesis, noiseless τ (X s , X I )u 1 , u 2 , · · · , u r(33)
then the matrix U sig := [u 1 , u 2 , · · · , u r ] ∈ R P ×r satisfies span(U sig ) = span(X), so span(U sig ) is called the signal subspace, and u 1 , u 2 , · · · , u r are called the signal eigenvectors; while the subspace spanned by the remaining noise eigenvectors u r+1 , · · · u P is called the noise subspace.
In real systems, the auto-correlation matrix C (m) can only be estimated by the sampled covariance matrix
C (m) := 1 m m i=1 r (i) (r (i) ) T ,
the eigenvectors corresponding to the r largest eigenvalues ofĈ (m) iŝ
u (m) 1 ,û (m) 2 , · · · ,û (m) r
and is proved to be an asymptotic estimation of the real signal eigenvectors u 1 , u 2 , · · · , u r [40][41] [42], which means the estimated signal subspace span(Û
H 1 : r (i) = X s α (i) + X I β (i) + w (i) , i = 1, 2, · · · , m.
where X s ∈ R P ×d , α (i) n ∈ R d are the basis and coefficient of the target signal subspace, and X I ∈ R P ×p , β
(i) n ∈ R p
are the basis and coefficient of the interference signal subspace; and w (i) ∼ N (0, σ 2 I P ) is the white Gaussian noise.
Then the volume-correlation of Detector 1 will satisfy:
Asymptotically (for large m), for any 0 < ε < 1 and δ > 0, if
m ≥ 1 + ε ( √ δ + 1 − 1) 2 p+d i,j=1 j =i λ i λ j (λ i − λ j ) 2 + p+d i=1 (P − p − d) λ i σ 2 (σ 2 − λ i ) 2 ,(34)
then there exists a constant C > 0, such that
t 2 (R (m) ) ≤ s p (Q T XsI P ⊥ Xs Q XsI )δ d + O(δ d+1 ).(35)
holds with probability
P ≥ 1 − exp{− (p + d) · P · ε 2 C }.(36)
Here λ 1 , λ 2 , · · · , λ r ≥ σ 2 are the corresponding eigenvalues of the auto-currelation matrix C shown in (32), and Q XsI denotes the orthogonal basis matrix of the subspace X s ⊕ X I ; in addition, s k (A) is defined as the kth elementary symmetric function of singular values of matrix A ∈ R n×n :
s k (A) := 1≤i1≤···≤i k ≤n σ i1 · · · σ i k , 1 ≤ k ≤ n.(37)
Theorem 4: In noised situation, under H 0 hypothesis, the multi-dimensional observations is of the form of
H 0 : r (i) = X I β (i) + w (i) ,(38)
where X I ∈ R P ×p , β (i) n ∈ R p are the basis and coefficient of the interference signal subspace; and w (i) ∼ N (0, σ 2 I P ) is the white Gaussian noise. Then the volume-correlation of Detector 1 will satisfy:
Asymptotically (for large m), for any 0 < ε < 1 and δ > 0, if
m ≥ 1 + ε ( √ δ + 1 − 1) 2 p i,j=1 j =i λ i λ j (λ i − λ j ) 2 + p i=1 (P − p) λ i σ 2 (σ 2 − λ i ) 2 ,(39)
then there exists a constant C > 0, such that
|t 2 (R (m) ) − τ 2 (X s , X I )| ≤ s p−1 (Q T XI P ⊥ Xs Q XI )δ + O(δ 2 ),(40)
holds with probability
P ≥ 1 − exp{− p · P · ε 2 C }.(41)
Also, λ 1 , λ 2 , · · · , λ r ≥ σ 2 are the corresponding eigenvalues of the auto-currelation matrix C shown in (32), and (35) and (40), together with (34) and (39), it implies that for a large enough number of observations m, the volume-correlation subspace detector will converge to its ideal performance without noise asymptotically, which also means the asymptotic elimination of influence of noise and interference. And this is the main advantage of our Volume-Correlation Subspace Detector.
It is noted that the asymptotic results in (35) and (40) are of the probabilistic form, i.e., they hold with a probability (36) and (41). The reason why the result is probabilistic is for the randomness of noise, and the probabilistic result also means that the asymptotic approximation of the volume-correlation is in the statistical sense, i.e., for large number of observations m, (35) and (40) will hold statistically.
The condition about the bound of the number of observations m in (34) and (39) is a sufficient condition, it means that for a given parameter δ, when m satisfies (34) and (39), then the volume-correlation will sufficiently satisfy (35) and (40) with a high probability. As we know that sufficient conditions are usually more conservative than reality, in practise the value of volume-correlation would converge much more faster than what is predicted in the condition (34) and (39), and this will also be shown in the later numerical simulations as a complementary demonstration.
The bound on the number of observations in (34) and (39) is influenced by three main factors: the probability parameter ε, the error toleration parameter of volume-correlation δ, as well as the eigenvalues of the auto-correlation matrix of the received signal:
H 0 : λ 1 ≥ λ 2 ≥ · · · ≥ λ p ≥ λ p+1 = · · · = λ P = σ 2 ,(42)H 1 : λ 1 ≥ λ 2 ≥ · · · ≥ λ d+p ≥ λ d+p+1 = · · · = λ P = σ 2 .(43)
For smaller error toleration parameter δ, we can see from (34) and (39) that only with more number of observations m can the volume-correlation satisfy (35) and (40). And similar analysis holds for the probability parameter ε. As for the eigenvalues λ 1 , λ 2 , · · · , the most noticeable ones among them that may affect the detector's performance are those that equals to the noise power σ 2 . A larger value of σ 2 will make the right side of (34) and (39) to be larger, meaning that only with more observations can the detector reach an ideal performance. Besides, the other eigenvalues λ 1 , λ 2 , · · · , are determined by the basis X s (and X I ) as well as the statistical property of α(and β), the analysis of their influence on the detector is not clear, and will be left for future work.
The main results (35) and (40) τ (X s , X I ). Therefore, as a whole, the simulation validates the result in theorem 3 and 4.
As is mentioned in the previous section, a manual constant c > 1 is need to fine tuning the threshold of detector 1, the reason can be obvious as shown in figure 5: if 1/τ (X s , X I ) is directly used as the threshold for detection, the false alarm rate will be high, therefore, we introduce the constant c > 1 to slightly raise the threshold in order to get a favorable false alarm rate. Simulation shows that a value of c slightly greater than 1 (e.g., c = 1.1) is enough for a satisfactory detection performance.
As a whole, we can conclude that, as the increase and accumulation of observations, the Volume-Correlation Subspace Detector will asymptotically converge to its ideal performance without noise, and eliminate the influence of interference; the accumulation of observations is the price to pay for an ideal detection performance.
VI. APPLICATION OF VOLUME-CORRELATION SUBSPACE DETECTOR IN MULTI-USER DETECTION
In this section, we will demonstrate the application of our Volume-Correlation Subspace Detector to the process of Active User Identification in multiuser detection systems with multipath channels. And we will demonstrate that our Volume-Correlation Subspace Detector has its advantage in dealing with other users' subspace interference in the multipath channel.
A. Active User Identification in multipath channels
Consider a typical multiuser detection system with N users, and each user transmitting the Direct sequence CDMA signal (DS-CDMA) [43], then the base band signature waveform of the n'th user is: where c n = [c n,1 , c n,2 , · · · , c n,L ] T ∈ R L , n = 1, · · · , N,
x n (t) = A n L l=1 c n,l p(t − lT c ), t ∈ [0, T s ),(44)
is the n'th user's signature sequence with length L, typically the length L can be smaller than the number of users In wireless communication, the multipath effect of wirelss channels is a common physical phenomenon that affects the communication quality, the impulse response of the n'th user's multipath channel is generally formulated as a number of delayed impulses:
h n (t) = kn s=1 h n,s δ(t − T n,s ), 1 ≤ n ≤ N,(46)
where k n is the maximum number of paths for the n'th user, h n,s is the amplitude of the s'th path's channel response for the n'th user, and T n,s denotes the channel delay of the s'th path for the n'th user. The channel response h n (t) will gradually vary with time due to different reasons, e.g., the mobility of each user and channel fading. Here we focus on the scenario of slow fading, i.e., the channel response h n (t) remains roughly constant in one symbol duration T s , but will evolve through several symbol durations.
In order to eliminate the inter-symbol interference, the signature sequence for each user is usually designed to be a pseudo-random sequence (denoted by s n ) with a cyclic prefix. Thus, the cyclic prefix technique is also used in this paper, if we denote the discrete path delay by τ n,s ⌊T n,s /T c ⌋ ∈ Z + , and the maximum discrete delay by τ max n,s {τ n,s }, with the assumption 1 < τ < L, then the transmitted signature sequence for the n'th user is designed by appending the last τ element of the original signature sequence s n to the head of s n , i.e., c n,l = s n,P −τ +l , 1 ≤ l ≤ τ,
c n,l = s n,l−τ , τ + 1 ≤ l ≤ L(47)
where s n := [s n,1 , · · · s n,L ] is the original pseudo-random sequence assigned for the nth user. As a result, any length P = L − τ sub-sequence of c n will be a cyclic shift version of s n . Then the signature waveform of the nth user in (44) is not affected by ISI in the time interval t ∈ (τ T c , T ). For convenience, we denote the jth cyclic shift operator by T j , i.e.,
T j s n = [s n,P −j+1 , · · · s n,P , s n,1 , · · · s n,P −j ]. 1 ≤ j ≤ τ
The active users transmit signals by modulating their signature waveform (44) with symbols b n , for simplicity, we assume the modulation method is BPSK, i.e., b n ∈ {−1, 1}. The set of active users is denoted by I ⊂ {1, 2, · · · , N }, with#|I| ≤ N , then the received signal will be:
y(t) = n∈I b n k s=1 h n,s x n (t − T n,s ) + w(t), t ∈ [0, T s ),(49)
where h n,s and T n,s are the channel response amplitude and path delay of the sth path for the nth user, respectively, and w(t) is the white Gaussian noise. Before the demodulation and decoding of each user's symbol b n from the received signal y(t), a preprocess is usually essential to identify the set of active users, i.e., identify the set I. The reason for this preprocess is that generally there are far less active users than the total N users, identification of the active users will reduce the system complexity, and improve the overall decoding performance [44] [45]. Therefore in this section, the goal is to identify the set I from the received signal y(t). At the receiver, the chip rate sampling front-end in figure 6 will yield a discrete sequence of samples of the receive signal, i.e., y 1 , y 2 , · · · , y L , and
y l = Tc 0 p(t − lT c )y(t)dt, l = 1, · · · , L,(50)
as is shown in figure 7, in order to eliminate ISI, the last P = L − τ samples of y 1 , y 2 , · · · , y L , is taken as the received signal sample, then the received signal vector is
r = n∈I A n b n x n + n(51)
where r := [y τ +1 , · · · , y L ], and x n = S n a n , with S n = [s n , T 1 s n , · · · , T τ s n ] ∈ C P ×(τ +1) ,
a n = [0, · · · , h n,1 , 0, · · · , h n,k , · · · , 0] T ∈ C τ +1 .
The coefficient vector a n implies information of the nth user's channel state, i.e., a n,j = h n,s , j = τ n,s , a n,j = 0, j = τ n,s . As a whole, the received signal from each user becomes a linear subspace signal due to the effect of mutipath channel, i.e., x n = S n a n . Each user's signature subspace S n := span(S n ) is spanned by the cyclic shift version of the signature sequence s n , and the subspace coefficient a n is determined by each user's channel response. As is mentioned previously, the channel fading effect makes a n to be unknown at the receiver, therefore our goal here is to detect the target subspace signal x n = S n a n only with the knowledge of S n .
It is obvious that our Volume-Correlation Subspace Detector has its advantage in dealing with this Active User Identification problem. For the nth user, its signature subspace S n = span(S n ) will be the target signal subspace X s , with dimension d := τ + 1; and the other users' signature subspace m∈I m =n S m will become the interference signal subspace X I , with dimension p := (u−1)(τ +1). As the Volume-Correlation Subspace Detector will asymptotically eliminate noise and interference using multi-dimensional observations R (m) = [r (1) , · · · , r (m) ], these observations will be chosen from different received signal vectors in different channel coherence intervals, i.e.,
r (i) = n∈I A (i) n b (i) n S n a (i) n + n (i) , i = 1, 2, · · ·(55)
as is shown in figure 8, where the coherence interval T coh is the period that the channel response, i.e., the coefficient a n , remains invariant.
B. Numerical Simulations
In this section, the performance of our Volume-Correlation Subspace Detector in the application of Active User Identification is demonstrated by monte-carlo simulation. In the simulation, the maximum number of active users is assumed to be 5, i.e., #|I| ≤ 5; and the user signature sequence s n is generated randomly, with values taking ±1 and P = 1024; the number of propagation paths for each user is assumed to be 10, which means the subspace basis and coefficient in (52) and (53) satisfies S n ∈ R 1024×10 and a n ∈ R 10 ; the channel is assumed to obey the slow fading Rayleigh model, therefore each element in a n is generated randomly from the Rayleigh distribution; and the amplitude of each user's transmit signal is assumed to be equal. For simplicity, in the simulation we mainly test the performance of our detector on one specific user, and therefore we take one arbitrary user's signature subspace as the target subspace, and the other 4 active users' signature subspaces as the interference subspace. We simulate two scenarios when the target user is active or inactive, while in both scenarios the other 4 interference users are at different SNR levels. It can be seen that even in a low SNR scenario, such as SNR=2.1dB, by accumulating as many observations, such as m ≥ 50, our detector will still ensure an ideal detection probability.
C. Further Discussion & Future Work
1) Flexibility about the dimensions of subspaces: As is discussed in the previous sections, in the calculation of volume-correlation in (21) and (22), it is assumed that the target signal subspace X s and interference signal subspace X I are known at the receiver, including their dimensions dim(X s ) = d and dim(X I ) = p. However, in real systems, there is also possibility that the signal and interference subspaces cannot be fully understood at the receiver, for example, the only knowledge could be some "larger" subspaces that the real subspaces X s and X I lie within. In other words, the real target and signal subspaces, i.e., X s and X I satisfy
X s ⊂ span(X s ), X s ∈ R P ×d , dim(X s ) = d s ≤ d;
(56)
X I ⊂ span(X I ), X I ∈ R P ×p , dim(X I ) = p I ≤ p.(57)
The X s and X I here are the priori knowledge of the target and interference subspace, and they are used in (21) and (22) to calculate the volume-correlation and perform detection; while the subspaces X s and X I are the real target subspaces and interference subspace in the received signal.
The example can be found in the previously mentioned Multiuser Detection system, (51) indicates that each user's signature subspace is spanned by its signature sequence s n and its cyclic shift versions, and the subspace coefficient a n is determined by the multipath channel response h n,s as in (54). From (54), it is obvious that generally the channel can not have paths with all possible delays, which means some elements in a n will remains zero, and thus each user's real signature subspace in the received signal is contained in (but does not equal to) span(S n ), then we have for the nth user X s ⊂ S n = span(S n ), with dim(X s ) = k n ≤ dim(S n ) = τ + 1;
X I ⊂ ⊕ m∈I m =n S m = ⊕ m∈I m =n span(S m ), with dim(X I ) = m∈I m =n k m ≤ dim(⊕ m∈I m =n S m ) = (u − 1)(τ + 1),
where X s is the real subspace of user n, and X I is the real subspace of the other active users.
Surprisingly, the volume-correlation subspace detector can still be reliable. And the reason is also quite obvious, and can be proved by simply redo the proof procedures of Theorem 1 and 2. In summary, for this scenario, the volume-correlation for the nth user in (21) will become
t(Y (m) ) := vol τ +1+r ([S n , Y (m) ])/(vol τ +1 (S n ) vol r (Y (m) )),
It can be proved that when m ≤ dim(X I ) = m =n k m , the volume correlation t(Y (m) ) descends with the increase of m; and when m ≥ m =n k m , the volume-correlation descends to 0 under H 1 hypothesis, and also satisfies inequality (29) under H 0 hypothesis, except that the equality may be unreachable. Besides, similar results
to Theorem 3 and 4 can also be derived. This advantage of the volume-correlation subspace detector is demonstrated by simulation in figure 11. Fig. 11: Output of the volume-correlation without and with noise in Active User Identification, when each user's real subspace is "smaller" than the known signature subspace span(S n ), and assume k n = 5
In the simulation, the same settings are used as the previous simulations of active user identification, except that actually each user only have k n = 5 channel paths, and the volume-correlation in (21) and (22) is calculated using the priori-known signature subspace in (52) which assumes channel paths with all delays. The simulation clearly shows that in noiseless situation the volume-correlation t(Y (m) ) descends to 0 under H 1 hypothesis when m ≥ 20, but satisfies (29) except the equality under H 0 hypothesis. Similar asymptotic results can also be seen for noised situations.
The above analysis and simulation indicates that, the volume-correlation subspace detector allows for significant flexibility about the priori knowledge of the target and interference signal subspace, and this flexibility is also an interior advantage of our detector, and this will be left for our future work.
2) The Phase Transition Phenomenon of random matrices and Future works: When the noise level is significantly high, i.e., the SNR is extremely low, analyzes have shown that there will be significant errors in the step of estimation of signal subspaces using the eigendecomposition subspace methods. The reason is for the so-called Phase Transition Phenomenon about eigenvalues and eigenvectors of random matrices [46][47] [48]. Analysis has shown that as P → ∞, and P/m → γ ∈ (0, 1), then we have the following results:
(a)If λ i /σ 2 > 1 + √ γ, then λ (m) i a.s. → λ i (1 + γσ 2 λ i − σ 2 ), i = 1, · · · d, or d + p (58) | û (m) i , u i | a.s. → 1 − γσ 4 (λ i − σ 2 ) 2 / 1 + γσ 2 λ i − σ 2 , i = 1, · · · d, or d + p. (59) (b) If λ i /σ 2 ≤ 1 + √ γ, thenλ (m) i a.s. → σ 2 (1 + √ γ) 2 , i = 1, · · · d, or d + p (60) | û (m) i , u i | a.s. → 0, i = 1, · · · d, or d + p,(61)
where λ i and u i are eigenvalues and eigenvectors of the auto-correlation matrix of the received signal, andλ surely. On the other hand, if the noise variance σ 2 is significantly high, such that λ i /σ 2 is lower than 1 + √ γ, then the estimated signal subspace span(Û (m) s ) will not asymptotically converge to the real signal subspace span(U s ), therefore the volume-correlation will not converge to 0 under H 1 hypothesis, or converge to τ (X s , X I ) under H 0 hypothesis. As a whole, the signal subspace estimation method in Detector 1 has its shortage in low SNR scenarios, so the estimation of signal subspace in a low SNR scenario remains an important future work for our volume-correlation subspace detector.
VII. CONCLUSION
In this paper, we proposed a novel volume-correlation subspace detector, and use it to deal with the problem of subspace signal detection with noise and interference. Subspace signal detection is a common problem in the signal processing community, it arises in a wide range of applications such as communication, radar, array signal processing and hyperspectral imagery. Our proposed volume-correlation subspace detector detects subspace signals by calculating the volume of parallelotope spanned by the basis of target signal subspace together with the multidimensional observations of the received signal. It is proved that by using the multi-dimensional knowledge of the target and interference signal subspace, as well as multi-dimensional observations, our detector can asymptotically eliminate the influence of noise and interference. Besides, we also discussed the application of the volume-correlation subspace detector to the problem of Active User Identification in multipath channels commonly seen in the multiuser detection systems. Numerical simulations were given to demonstrate the advantage and performance of our volumecorrelation subspace detector. As a whole, our volume-correlation subspace detector has its interior advantage in dealing with multi-dimensional subspace detection, and can converge asymptotically to its ideal performance with the accumulation of observations. APPENDIX A APPENDIX A. PROOF OF THEOREM 1
Proof:
According to the system model, under H 1 hypothesis, if m ≤ d+p, then rank(Y (m) ) = m, the volume-correlation in noiseless situation is:
t(Y (m) ) = vol d+m ([X s , Y (m) ])/(vol d (X s ) · vol m (Y (m) )),
According to the analysis in [31], we have:
vol d+m ([X s , Y (m) ]) vol d (X s ) · vol m (Y (m) ) = min(d,m) j sin θ j (X s , span(Y (m) )),(62)
where 0 ≤ θ j (X s , span(Y (m) )) ≤ 2π, 1 ≤ j ≤ min(d, m) are the principal angles of subspaces X s and span(Y (m) ).
Firstly we are going to prove (25), as is known, for 1 ≤ m ≤ p, we have
t(Y (m) ) = min(d,m) j=1
sin θ j (X s , span(Y (m) )).
t(Y (m) ) is the product of the sines of principal angles between subspaces X s and span(Y (m) ), thus we can replace the matrices X s and Y (m) by their orthogonal basis matrices in the volume-correlation expression (62), then equivalently, we have
t(Y (m) ) = vol d+m ([Q Xs , Q Y (m) ]),(64)
where Q Xs ∈ R P ×d and Q Y (m) ∈ R P ×m are the orthogonal bases of subspaces X s and span(Y (m) ), i.e.,
span(Q Xs ) = X s = span(X s ), Q T Xs Q Xs = I,(65)span(Q Y (m) ) = span(Y (m) ), Q T Y (m) Q Y (m) = I.(66)
In the same way, for m + 1, we have
t(Y (m+1) ) = vol d+m+1 ([Q Xs , Q Y (m+1) ]).(67)
Q Y (m+1) can be related to Q Y (m) , indeed, Q Y (m+1) can be written as
Q Y (m+1) = [Q Y (m) , q y (m+1) ],(68)
where
q y (m+1) := P ⊥ Y (m) y (m+1) / P ⊥ Y (m) y (m+1) 2 ,(69)
and P ⊥ Y (m) denotes the orthogonal projection operator onto subspace span(Y (m) ) ⊥ , therefore q y (m+1) is the component of y (m+1) that is perpendicular to the subspace span(Y (m) ).
Using knowledge of the determinant of block matrices, we have:
t(Y (m+1) ) (70) = vol d+m+1 ([Q Xs , Q Y (m+1) ]) = det Q T Xs Q T Y (m+1) [Q Xs , Q Y (m+1) ] 1/2 = det Q T Xs Q Xs Q T Xs Q Y (m+1) Q T Y (m+1) Q Xs Q T Y (m+1) Q Y (m+1) 1/2 = det Q T Xs Q Xs Q T Xs Q Y (m) Q T Xs q y (m+1) Q T Y (m) Q Xs Q T Y (m) Q Y (m) Q T Y (m) q y (m+1) q T y (m+1) Q Xs q T y (m+1) Q Y (m) q T y (m+1) q y (m+1) 1/2 = det Q T Xs Q Xs Q T Xs Q Y (m) Q T Y (m) Q Xs Q T Y (m) Q Y (m) 1/2 · det q T y (m+1) q y (m+1) − q T y (m+1) [Q Xs , Q Y (m) ] Q T Xs Q Xs Q T Xs Q Y (m) Q T Y (m) Q Xs Q T Y (m) Q Y (m) −1 Q T Xs Q T Y (m) q y (m+1) 1/2 = det Q T Xs Q Xs Q T Xs Q Y (m) Q T Y (m) Q Xs Q T Y (m) Q Y (m) 1/2 · P ⊥ [Xs,Y (m) ] q y (m+1) 2 = t(Y (m) ) · P ⊥ [Xs,Y (m) ] q y (m+1) 2 ,(71)
where P ⊥ [Xs,Y (m) ] is the orthogonal projection operator onto the subspace span([X s , Y (m) ]) ⊥ . Because
P ⊥ [Xs,Y (m) ] q y (m+1) 2 ≤ q y (m+1) 2 = 1,(72)therefore t(Y (m) ) ≥ t(Y (m+1) )
holds, and then (25) is proved.
Next we prove (26), as is known, the ith observation of the received signal is
y (i) = X s α (i) + X I β (i) , i = 1, 2, · · · , m.
then from the subspace perspective, it is known that span(Y (m) ) = span([y (1) , · · · , y (m) ]) ⊂ span(X s ) span(X I ),
When m = p + 1, we have
y (p+1) = X s α (p+1) + X I β (p+1) ,(74)
it can be seen that the second term in (74) satisfies X I β (p+1) ∈ span(X I ), and all the previous observations, i.e.,
y (i) = X s α (i) + X I β (i) , i = 1, · · · p,(75)
will satisfy span(X I β (1) , · · · , X I β (p) ) = span(X I ).
Therefore, there must exist a sequence of coefficients b j , j = 1, · · · p that are not all zero, such that
X I β (p+1) = p j=1 b j X I β (j) ,(77)
then y (p+1) can be written as
y (p+1) = X I β (p+1) + X s α (p+1) = p j=1 b j X I β (j) + X s α (p+1) = p j=1 b j (X s α (j) + X I β (j) ) + X s α (p+1) − p j=1 b j X s α (j) = p j=1 b j y (j) + X s (α (p+1) − p j=1 b j α (j) ).(78)
So it is obvious that
y (p+1) ∈ span([Y (p) , X s ]),(79)
where Y (p) = [y (1) , · · · , y (p) ]. As a result,
t(Y (p+1) ) = vol p+d+1 ([X s , Y (p+1) ]) vol d (X s ) vol p+1 (Y (p+1) ) , = vol p+d+1 ([X s , Y (p) , y (p+1) ]) vol d (X s ) vol p+1 (Y (p+1) ) , = 0.(80)
The same result can be derived for all m > p + 1, therefore, (26) is proved.
APPENDIX B APPENDIX B. PROOF OF THEOREM 2
Proof:
For simplicity, we assume d ≤ p in the following proof. Firstly, similar to the proof of theorem 1, (28) is easy to prove.
Therefore, for 1 ≤ m ≤ p, we have t(Y (1) ) ≥ · · · ≥ t(Y (d) ) ≥ · · · t(Y (p) ),
and for d ≤ m ≤ p, we have t(Y (m) ) = d j=1
sin θ j (X s , span(Y (m) )), so our focus is the relation between d j=1 sin θ j (X s , span(Y (m) )) and m. Define the orthogonal bases matrices Q Xs and Q Y (m) which are the same as (65) and (66), then according to the definition of principal angles in [31], the cosines of the principal angles between subspaces satisfy
cos θ j (X s , span(Y (m) )) = σ j (Q T Xs Q Y (m) ), 1 ≤ j ≤ d,(81)
where σ j (Q T Xs Q Y (m) ) are the singular vales of the matrix Q T Xs Q Y (m) , and is assumed to satisfy σ 1 ≥ · · · ≥ σ d . Then we have for m ≤ p,
span(Y (m) ) ⊂ span(X I ),(82)
Denote the orthogonal basis matrix of X I by Q XI , which satisfies span(Q XI ) = X I = span(X I ),
then span(Q Y (m) ) ⊂ span(Q XI ), and there exists a column-orthogonal matrix U (m)
XI ∈ R p×m , with (U (m) XI ) T U (m) XI = I, such that Q Y (m) = Q XI U (m) XI , d ≤ m ≤ p.(84)
In addition, the subspace span(Q Y (m) ) must have an orthogonal complement in subspace X I , i.e., there exists another column-orthogonal matrix V (m)
XI ∈ R p×(p−m) , such that span(Q Y (m) ) ⊥ span(Q XI V (m) XI ), d ≤ m ≤ p.(85)
and
U XI = [U (m) XI , V (m) XI ] ∈ R p×p ,(86)
satisfies U T XI U XI = I p , in other words, we have
U (m) XI = U XI · I m O ,(87)which means U (m)
XI is the sub-matrix of U XI , and
Q Y (m) = Q XI U XI · I m O ,(88)
So
Q T Xs Q Y (m) = Q T Xs Q XI U XI · I m O .(89)
According to the relation between singular values and eigenvalues, we have
σ 2 j (Q T Xs Q Y (m) ) = λ j (Q T Y (m) Q Xs Q T Xs Q Y (m) ) = λ j ([I m , O]U T XI · Q T XI Q Xs Q T Xs Q XI · U XI I m O ), 1 ≤ j ≤ d,(90)
where
λ j (Q T Y (m) Q Xs Q T Xs Q Y (m) ) are eigenvalues of the matrix Q T Y (m) Q Xs Q T Xs Q Y (m) . Then Q T Y (m) Q Xs Q T Xs Q Y (m) is the principal sub-matrix of the matrix U T XI · Q T XI Q Xs Q T Xs Q XI · U XI .
According to the Interlacing Theorem in [49], we have Lemma 1: For a Hermitian matrix X ∈ R N ×N , with its m × m principal matrix X m , the eigenvalues of X and X m satisfies:
λ N −k+1 (X) ≤ λ m−k+1 (X m ) ≤ λ m−k+1 (X), k = 1, · · · m,(91)
where λ N (X) ≤ · · · ≤ λ 1 (X) and λ m (X m ) ≤ · · · ≤ λ 1 (X m ) are eigenvalues of X and X m , respectively.
It has been mentioned that from (90)
that Q T Y (m) Q Xs Q T Xs Q Y (m) is the principal sub-matrix of U T XI ·Q T XI Q Xs Q T Xs Q XI · U XI , besides, it is known that for orthogonal matrix U XI , λ j (U T XI · Q T XI Q Xs Q T Xs Q XI · U XI ) = λ j (Q T XI Q Xs Q T Xs Q XI ) = σ 2 j (Q T Xs Q XI ),(92)
therefore, combining (91) and (92), we have
λ p−k+1 (Q T XI Q Xs Q T Xs Q XI ) ≤ λ m−k+1 (Q T Y (m) Q Xs Q T Xs Q Y (m) ) ≤ λ m−k+1 (Q T XI Q Xs Q T Xs Q XI )(93)
holds for k = 1, · · · , m, with d ≤ m ≤ p, then the right inequality implies that
λ d (Q T Y (m) Q Xs Q T Xs Q Y (m) ) ≤ λ d (Q T XI Q Xs Q T Xs Q XI ) . . . . . . λ 1 (Q T Y (m) Q Xs Q T Xs Q Y (m) ) ≤ λ 1 (Q T XI Q Xs Q T Xs Q XI ),
then cos θ j (X s , span(Y (m) )) ≤ cos θ j (X s , X I ), 1 ≤ j ≤ d.
As a result,
d j=1 sin θ j (X s , span(Y (m) )) ≥ d j=1 sin θ j (X s , X I ).(95)
So (29) is proved. And it is obvious that when m ≥ p, Q Y (m) = Q XI U XI , then the equality in (29)
2 , · · · ,û (m) r calculated from eigenvalue decomposition of the sampled covariance matrixĈ (m) are estimations of the real signal eigenvectors u 1 , u 2 , · · · , u r , a well-known result is: Lemma 2: [41] Consider the matrix whose columns are the r largest eigenvectors estimated from ofĈ (m) , i.e.,
U (m) sig = [û (m) 1 ,û (m) 2 , · · · ,û (m) r ],
its asymptotic distribution (for large m) is jointly Gaussian with mean
U sig = [u 1 , u 2 , · · · , u r ]
and covariance Σ (m)
1 , · · · , Σ (m) r , where Σ (m) i := λ i m r j=1 j =i λ j (λ i − λ j ) 2 u i u T i + P j=r+1 σ 2 (σ 2 − λ i ) 2 u i u T i , i = 1, · · · , r(96)
and
E(û (m) i − u i )(û (m) k − u k ) T = Σ (m) i · δ i,k , i, k = 1, · · · , r(97)
where r denotes the dimension of the signal subspace, and λ 1 ≥ λ 2 ≥ · · · , λ r ≥ λ r+1 = · · · λ P = σ 2 are eigenvalues of the auto-correlation matrix in (32), with u 1 , · · · , u P the corresponding eigenvectors.
Another lemma derived from the well known concentration inequality is need in our proof:
Lemma 3: For the random matrix E = [e 1 , · · · , e r ] ∈ R P ×r ,(98)
where e i ∼ N (0, Σ i ), 1 ≤ i ≤ d, and E(e i e T k ) = Σ i · δ i,k , then for any 0 < ε < 1, there exists a constant C > 0 that depends on Σ i , such that
E 2 F ≤ (1 + ε) r i=1 Tr(Σ i ),(99)
holds with probability
P ≥ 1 − exp{− r · P · ε 2 C }.(100)
proof of Lemma 3:
From the definition of F-norm, we know that
E 2 F = r i=1 e i 2 2 ,(101)
and for any 1 ≤ i ≤ r, e i ∼ N (0, Σ i ), denote the eigenvalue decomposition of Σ i ∈ R P ×P by
Σ i = V i Λ i V T i ,(102)
where the diagonal matrix Λ i := diag(σ 2 i,1 , · · · , σ 2 i,P ) and σ 2 i,1 ≥ σ 2 i,2 ≥ · · · ≥ σ 2 i,P ≥ 0 are eigenvalues of Σ i . If we letẽ
i = V T i e i ,(103)thenẽ i ∼ N (0, Λ i ), ẽ i 2 2 = e i 2 2 .(104)
Denote the elements of vectorẽ i byẽ
i = [ẽ i,1 , · · · , · · ·ẽ i,P ] T ,(105)
then differentẽ i,j are independent and satisfỹ
e i,j ∼ N (0, σ 2 i,j ), 1 ≤ j ≤ P.(106)
Now we stack all these vectorsẽ i , 1 ≤ i ≤ d into a single vector, i.e., we let
e := [ẽ T 1 ,ẽ T 2 , · · · ,ẽ T P ] T ∈ R r·P ,(107)
then we haveẽ ∼ N (0, Λ), Λ = diag(Λ 1 , · · · , Λ r ) = diag(σ 2 1,1 , · · · , σ 2 1,P , · · · , σ 2 r,1 , · · · , σ 2 r,P ).
Therefore, (101) is equivalent to
E 2 F = r i=1 ẽ i 2 2 = ẽ 2 2 .(109)
As is known, the norm of a Gaussian random vector will concentrate around its expectation, which is referred to as the concentration of measure phenomenon [50]. It has been proved that the norm of an i.i.d. Gaussian random vector will concentrate around its own expectation (Chapter 4, [51]). The problem of the concentration of ẽ 2 2 here only has a slight difference with the analysis from [51], i.e., the elements ofẽ have different variances, therefore, the proof will be quite similar with the proof of Theorem 4.2 in [51], and in the following proof we only give the part that are different from Theorem 4.2 in [51].
Firstly, we have
E{ ẽ 2 2 } = r i=1 P j=1 σ 2 i,j = r i=1 Tr(Σ i ),(110)
Then follow the same approach as Theorem 4.2 in [51], according to the Markov's Inequality, for any parameter β > 0 and λ > 0, we have
P{ ẽ 2 2 ≥ β r i=1 Tr(Σ i )} = P{exp(λ ẽ 2 2 ) ≥ exp(λβ r i=1 Tr(Σ i ))} = r i=1 P{exp(λ ẽ i 2 2 ) ≥ exp(λβ Tr(Σ i ))} ≤ r i=1 E{exp(λ ẽ i 2 2 )} exp(λβ Tr(Σ i )) = r i=1 P j=1 [ E{exp(λẽ 2 i,j )} exp(λβσ 2 i,j ) ],(111)
according to (106), the moment generating function of the Gaussian random variableẽ i,j is:
E{exp(λẽ 2 i,j )} = 1 1 − 2λσ 2 i,j ,(112)
If we let
σ max := max i,j σ i,j , σ min := min i,j σ i,j ,(113)
then we have
P{ ẽ 2 2 ≥ β r i=1 Tr(Σ i )} ≤ exp(−2λβσ 2 min ) 1 − 2λσ 2 max r·P/2(114)
holds for any λ > 0. The next steps of proof is the same as Theorem 4.2, by replacing λ with its optimal value such that exp(−2λβσ 2 min )/1 − 2λσ 2 max r·P/2 is minimized, and substituting some complicated formulas involving σ max and σ min for a constant C, we can derive the result of this lemma (which is also the result of Corollary 4.1
in [51]):
P{ ẽ 2 2 ≥ (1 + ε) r i=1 Tr(Σ i )} ≤ exp(− r · P ε 2 C ),(115)
holds for any 0 ≤ ε ≤ 1, where C > 0 is a constant depending on σ max and σ min . Therefore, (99) and (100) is proved.
Next, the theory of matrix perturbation will be used to help our analysis about the influence of the estimation error ofÛ (m) sig on the volume-correlation. Another lemma is need: Lemma 4: (Corollary 2.7 in [52]) For the matrix A ∈ R n×n , and the perturbation matrix E ∈ R n×n , we have
• If A is full-rank, then | det(A + E) − det(A)| ≤ n i=1 s n−i (A) E i 2 ,(116)
• If rank(A) = k for some 1 ≤ k ≤ n − 1, then
| det(A + E)| ≤ E n−k 2 k i=0 s k−i (A) E i 2 .(117)
where s k (A) is the kth elementary symmetric function of matrix A defined in (37).
Proof of Theorem 3:
As theorem 3 describes the situation where m is sufficiently large, therefore we naturally assume m ≥ d + p, the volume-correlation is of the form of:
t(R (m) ) = vol d+p+d ([X s ,Û (m) sig ])/(vol d (X s ) vol r (Û (m) sig )),(118)
Denote the orthogonal basis matrix of X s by Q Xs , i.e., Q T Xs Q Xs = I d and span(Q Xs ) = X s , then we have t(R (m) ) = vol d+p+d ([Q Xs ,Û
where P ⊥ Xs is the orthogonal projection operator onto the subspace X s . According to Lemma 2, the matrix constructed from estimated signal eigenvectorsÛ (m) sig ∈ R P ×(p+d) can be regarded as the matrix constructed from real signal eigenvectors plus a perturbation matrix, i.e.,Û (m)
sig = U sig + E (m) ,(120)Σ (m) i = λ i m p+d j=1 j =i λ j (λ i − λ j ) 2 u i u T i + P j=p+d+1 σ 2 (σ 2 − λ i ) 2 u i u T i .(122)
Therefore (119) = det (P ⊥ Xs U sig + P ⊥ Xs E (m) ) T (P ⊥ Xs U sig + P ⊥ Xs E (m) )
1/2 ,(123)
for convenience, let Q := P ⊥ Xs U sig , W := P ⊥ Xs E (m) , then (123) becomes t 2 (R (m) ) = det (Q + W ) T (Q + W ) ,
now the result in Lemma 4 is ready for use, let A := Q T Q, E = Q T W + W T Q + W T W , then t 2 (R (m) ) = det(A + E),
Therefore A = U T sig P ⊥ Xs U sig , E = U T sig P ⊥ Xs E (m) + (E (m) ) T P ⊥ Xs U sig + (E (m) ) T P ⊥ Xs E (m) ,(125)
As is known that, under H 1 hypothesis, U sig ∈ R P ×(p+d) and span(U sig ) = X s ⊕ X I , because X s and X I are Xs E (m)
F + P ⊥ Xs E (m) 2 F .(127)
Then, according to Lemma 2 and Lemma 3, we have
P ⊥ Xs E (m) ∼ N (0, P ⊥ Xs Σ (m) i (P ⊥ Xs ) T ),(128)
and for any ε > 0
P ⊥ Xs E (m) 2 F ≤ (1 + ε) p+d i Tr(P ⊥ Xs Σ i (P ⊥ Xs ) T ),(129)
holds with probability P ≥ 1 − exp{− r · P · ε 2 C }.
Consider the right side of (129), we have λ i λ j (λ i − λ j ) 2 Tr(P ⊥ Xs u j u T j P ⊥T Xs ) + P j=p+d+1 λ i σ 2 (σ 2 − λ i ) 2 Tr(P ⊥ Xs u j u T j P ⊥T Xs )
,(130)
because Tr(P ⊥ Xs u j u T j P ⊥T Xs ) = Tr(u T j P ⊥ Xs u j ) ≤ 1,
so p+d i Tr(P ⊥ Xs Σ i P Xs ) ≤ 1 m p+d i=1 p+d j=1 j =i λ i λ j (λ i − λ j ) 2 + p+d i=1 (P − p − d) λ i σ 2 (σ 2 − λ i ) 2 .(131)
Combine (132) and (129), because what we need is a sufficient condition, then for any ε > 0 and 0 ≤ δ < 1, we
let 1 m p+d i=1 p+d j=1 j =i λ i λ j (λ i − λ j ) 2 + p+d i=1 (P − p − d) λ i σ 2 (σ 2 − λ i ) 2 ≤ ( √ δ + 1 − 1) 2 1 + ε ,(133)
then equivalently, for any ε > 0 and 0 ≤ δ < 1, if
m ≥ 1 + ε ( √ δ + 1 − 1) 2 p+d i=1 p+d j=1 j =i λ i λ j (λ i − λ j ) 2 + p+d i=1 (P − p − d) λ i σ 2 (σ 2 − λ i ) 2 ,(134)
then
P ⊥ Xs E (m) 2 F ≤ ( √ δ + 1 − 1) 2 ,(135)
holds with probability P ≥ 1 − exp{− r · P · ε 2 C }.
Then combining (135) with (126), we get
E 2 ≤ δ,(136)
thus we have
t 2 (R (m) ) ≤ s p (U T sig P ⊥ Xs U sig )δ d + O(δ d+1 ),(137)
holds with probability P ≥ 1 − 2 exp{− r·P ·ε 2 C }. According to the definition of elementary symmetric function of singular values in (37), s k (A) is unitary-invariant, and span(U (m) sig ) = X s ⊕ X I = span(Q XsI ), therefore we can write s p (U T sig P ⊥ Xs U sig ) = s p (Q T XsI P ⊥ Xs Q XsI ). Then (35) of Theorem 3 is proved.
Proof of Theorem 4:
The result (40) of Theorem 4 can be similarly derived, as is known that under H 0 hypothesis, U sig ∈ R P ×p , span(U sig ) = X I , similarly we can let
A = U T sig P ⊥ Xs U sig , E = U T sig P ⊥ Xs E (m) + (E (m) ) T P ⊥ Xs U sig + (E (m) ) T P ⊥ Xs E (m) ,
and A is full rank, therefore, according to(116)
| det(A + E) − det(A)| ≤ s n−1 (A) E 2 + O( E 2 2 ),(138)
and according to (119), we have det(A) = det(U T sig P ⊥ Xs U sig ) = det(Q T XI P ⊥ Xs Q XI ) = vol 2 d+p ([Q Xs , Q XI ]) = τ 2 (X s , X I ),
Then combined with (127), and according to Lemma 2 and 3, for any 0 ≤ δ < 1 and ε > 0, if
m ≥ 1 + ε ( √ δ + 1 − 1) 2 p i=1 p j=1 j =i λ i λ j (λ i − λ j ) 2 + p i=1 (P − p) λ i σ 2 (σ 2 − λ i ) 2 ,(140)
then
P ⊥ Xs E (m) 2 F ≤ ( √ δ + 1 − 1) 2 ,(141)
hold with probability P ≥ 1 − 2 exp{− r · P · ε 2 C } therefore |t 2 (R (m) ) − τ 2 (X s , X I )| ≤ s p−1 (U T sig P ⊥ Xs U sig )δ + O(δ 2 ),
where in the same way we can write s p−1 (U T sig P ⊥ Xs U sig ) = s p−1 (Q T XI P ⊥ Xs Q XI ). P d P f
Fig. 1 :
1Depiction of the Generalized Energy Detector (right), and the influence of interference signal subspace on target signal subspace in Generalized Energy Detector (left)
2Fig. 3 :
3, · · · ,û (m) r ]) is also an asymptotic estimation of the real signal subspace span(U sig ). Combining the above analysis, the main theorems of this section are as follows. simulation of the reciprocal of volume-correlation 1/t(Y (m) ) in noiseless situation Theorem 3: In noised situation, under H 1 hypothesis, the multi-dimensional observations is of the form of
Q
XI denotes the orthogonal basis matrix of subspace X I . In addition, s k (A) is defined as the kth elementary symmetric function of singular values of matrix A in (37). Theorem 3 and 4 describe the asymptotic performance of our Volume-Correlation Subspace Detector in noised situation. The main result is shown in
of theorem 4 and 3 are also influenced by two main factors: firstly the deviation parameter δ, it is related to m as is shown in(34) and(39) and analyzed previously. The other important parameters are the elementary symmetry functions of singular values s p (Q T XsI P ⊥ Xs Q XsI )(and s p−1 (Q T XI P ⊥ Xs Q XI )). They are the coefficient of the first term of δ in(35) and(40), and according to their definition in(37), their values are decided only by the relation between subspaces X s and X I , and are finite-valued, therefore, for a given number of observations m, the deviation of the volume-correlation value from its ideal value is decided mainly by the order of δ in(35)and(40).The numerical simulation of detector 1 is demonstrated in figure 4 and 5. In the simulation, we we also choose P = 1024, and d = 10, p = 40, the target and interference signal subspaces are chosen arbitrarily. The average values of 100 monte-carlo simulations of the volume-correlation t(R (m) ) with respect to different m are plotted in figure 4 and 5, and each values of these 100 monte-carlo simulations are demonstrated by scatter diagram in the small sub-figures. It can be seen from the figures that in the noised situation, as the number of observations m increases, the value of the volume-correlation descends statistically, but slower than the ideal volume-correlation without noise; in addition, the values of the volume-correlation statistically concentrate around their average values, and these average values converge to the ideal values of the volume-correlation without noise, i.e., as m increases, the average of t(R (m) ) converges to 0 under H 1 hypothesis, while under H 0 hypothesis, t(R (m) ) converges to
Fig. 4 :
4t(Y (m) ) and t(R (m) ) with respect to m (d=10,p=40) t(Y (m) ), H 1 Hypothesis, noiseless t(Y (m) ), H 0 Hypothesis, noiseless average of t(R (m) ), H 1 Hypothesis average of t(R (m) ), H 0 Hypothesis τ (X s , X I ) simulation of the volume-correlation t(R (m) ) in noised situation
Fig. 5 :
5simulation of the reciprocal of volume-correlation 1/t(R (m) ) in noised situation
N
. The p(t) is the chip waveform, which satisfies Tc 0 |p(t)| 2 dt = 1, Tc 0 p(t − lT c )p(t − kT c )dt = 0, l = k; and T c is the chip duration, T s denotes the symbol duration, T c and T s satisfy LT c = T s ; A n is the transmit amplitude of the n'th user.
Fig. 6 :
6Diagram of chip rate sampling front-end
Fig. 7 :
7Eliminate the Intersymbol interference using cyclic prefix
Fig. 8 :
8Multi-dimensional observations of the received signal from different channel coherence intervals The detection performance of the proposed volume-correlation subspace detector in this application of Active User Identification is demonstrated in the next section.
assumed active. The detection and false alarm probability are calculated from 1000 monte-carlo simulations for different numbers of observations m and Signal-to-Noise Ratio (SNR). The SNR here is the ratio of signal power to noise power for each user, i.e., SN R = 10 log( s n 2 2 /σ 2 ). The threshold parameter c is chosen to be 1.1, which ensures an ideal false alarm probability.
1 )
1Given the number of observations m, the performance of the Volume-Correlation Subspace Detector at different level of SNR: The monte-carlo simulation of the detection probability and false alarm probability when SNR ranges from 0 to 30dB with m = 5, 10, 40, 50 is demonstrated in figure 9.It can be seen infigure 9that, even when there are only several number of observations, such as m = 5 shown in figure 9, our volume-correlation subspace detector still have a good performance for high SNR; and as the number of observations increases, the performance of our volume-correlation subspace detector increases significantly.
PfPlot of Pd and Pf with respect to m (d=10,p=40,m=50) Pd Pf
Fig. 9 :
9The Mote-Carlo simulation of Detection probability (Pd) and the False Alarm probability (Pf) for different SNR with m = 5, 10, 40, 50 2) Given the SNR, the performance of the Volume-Correlation Subspace Detector for increase number of observations m: The monte-carlo simulation of the detection probability and false alarm probability when m ranges from 1 to 50 with SN R = 2.1, 4.3, 8.3, 12.2 is demonstrated in figure 10.
Figure 10
10clearly demonstrated the increase of detection probability with increase of number of observations m
Pf Plot of Pd and Pf with respect to m (d=10,p=40,SNR=8.Pf Plot of Pd and Pf with respect to m (d=10,p=40,SNR=12.2) Pd Pf
Fig. 10 :
10The Mote-Carlo simulation of Detection probability (Pd) and the False Alarm probability (Pf) with increase of m from 1 to 50, with SN R = 2.1, 4.3, 8.3, 12.2
t(Y (m) ) with respect to m (ds=5,pI =20) t(Y (m) ), H1 Hypothesis, noiseless t(Y (m) ), H0 Hypothesis, noiseless τ (Xs, XI ) t(R (m) ) with respect to m (ds=5,pI =20) average of t(R (m) ), H1 Hypothesis average of t(R (m) ), H0 Hypothesis scatter of t(R (m) ), H1 Hypothesis scatter of t(R (m) ), H0 Hypothesis τ (Xs, XI )
corresponding estimation of λ i and u i calculated from eigenvalue decomposition of the sampled covariance matrix. This phase transition phenomenon means that when λ i /σ 2 is lower than a threshold, the eigenvalues and eigenvectors of the noise subspace almost
where (since it is under H 1 hypothesis) span(U sig ) = X s ⊕ X I ,
P d and P f with respect to m (d=10,p=40,SNR=14.9)
, and propose a new interference-resistant detection approach to detect subspace signals.Applying the theory of linear subspace to the domain of signal processing is not a new idea, actually, it hasbeen commonly applied in image processing and computer vision [22][23], machine learning [24][25][26], as well
as communication and radar [27][28][29]
holds. Before the proof of Theorem 3 and Theorem 4, several lemmas are required as intermediate results. As is mentioned, the signal eigenvectorsûAPPENDIX C
APPENDIX C. PROOF OF THEOREM 3 AND THEOREM 4
Proof:
(m)
1 ,û
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|
[] |
[
"Analysis of BaBar data for three meson tau decay modes using the Tauola generator",
"Analysis of BaBar data for three meson tau decay modes using the Tauola generator"
] |
[
"Olga Shekhovtsova \nInstitute of Nuclear Physics PAN ul\n152 31-342Radzikowskiego, KrakowPoland\n\nKharkov Institute of Physics and Technology 61108\nAkademicheskaya,1KharkovUkraine\n"
] |
[
"Institute of Nuclear Physics PAN ul\n152 31-342Radzikowskiego, KrakowPoland",
"Kharkov Institute of Physics and Technology 61108\nAkademicheskaya,1KharkovUkraine"
] |
[] |
The hadronic current for the τ − → π − π + π − ν τ decay calculated in the framework of the Resonance Chiral Theory with an additional modification to include the σ meson is described. Implementation into the Monte Carlo generator Tauola and fitting strategy to get the model parameters using the one-dimensional distributions are discussed. The results of the fit to one-dimensional mass invariant spectrum of the BaBar data are presented. This paper is based on [1].IntroductionThe precise experimental data for tau lepton decays collected at B-factories (both Belle and BaBar) provide an opportunity to measure the Standard Model (SM) parameters, such as the strong coupling constant, the quark-mixing matrix, the strange quark mass etc, and to search for new physics beyond SM. The leptonic decay modes of the tau lepton allow to test the universality of the lepton couplings to the gauge bosons. The hadronic decays (in fact, the tau lepton is the only one that can decay into hadrons due to its high mass) give an information about the hadronization mechanism and resonance dynamics in the energy region where the methods of the perturbative QCD cannot be applied. Also hadronic flavour-violating and CP violating decays of tau lepton allow to search for new physics scenario. Hadronic tau lepton decays are also a tool in high-energy physics. At the LHC and future linear colliders a correct simulation of the hadronic decay modes, mainly two pion and three pion modes, is needed to measure the Higgs spin and its CP properties.The implementation of the appropriate information on the hadronization of the QCD currents represents a key task of the TAUOLA library[2,3]. TAUOLA is a Monte Carlo generator (MC) dedicated to generating tau decays and it is used in the analysis of experimental data both at Bfactories and LHC. It is important to include in the analyses the information from QCD itself and not ad-hoc models that may hide the important information contained in the data. On the other hand the agreement with experimental data is essential and verifies a theoretical model. Resonance Chiral Theory (RChT)[4,5]provides such a reliable framework as it has been shown in many previous publications[6][7][8][9]. A set of RChT currents for the main two meson and three meson, namely, π − π 0 , K − π 0 , K 0 π − , π − π − π + , π 0 π 0 π − , K − π − K + , K 0 π −K0 and K − π 0 K 0 , was installed in TAUOLA. That set covers more than 88% of total hadronic τ width. The implementation of the currents, the related technical tests as well as the necessary theoretical concepts are documented in [10]. a
|
10.1051/epjconf/20148000054
|
[
"https://www.epj-conferences.org/articles/epjconf/pdf/2014/17/epjconf_qcd2014_00054.pdf"
] | 118,698,554 |
1410.2428
|
e1dfaa3ec91a10f7e39e313ed288ec9606d62fc4
|
Analysis of BaBar data for three meson tau decay modes using the Tauola generator
Olga Shekhovtsova
Institute of Nuclear Physics PAN ul
152 31-342Radzikowskiego, KrakowPoland
Kharkov Institute of Physics and Technology 61108
Akademicheskaya,1KharkovUkraine
Analysis of BaBar data for three meson tau decay modes using the Tauola generator
10.1051/epjconf/20148000054
The hadronic current for the τ − → π − π + π − ν τ decay calculated in the framework of the Resonance Chiral Theory with an additional modification to include the σ meson is described. Implementation into the Monte Carlo generator Tauola and fitting strategy to get the model parameters using the one-dimensional distributions are discussed. The results of the fit to one-dimensional mass invariant spectrum of the BaBar data are presented. This paper is based on [1].IntroductionThe precise experimental data for tau lepton decays collected at B-factories (both Belle and BaBar) provide an opportunity to measure the Standard Model (SM) parameters, such as the strong coupling constant, the quark-mixing matrix, the strange quark mass etc, and to search for new physics beyond SM. The leptonic decay modes of the tau lepton allow to test the universality of the lepton couplings to the gauge bosons. The hadronic decays (in fact, the tau lepton is the only one that can decay into hadrons due to its high mass) give an information about the hadronization mechanism and resonance dynamics in the energy region where the methods of the perturbative QCD cannot be applied. Also hadronic flavour-violating and CP violating decays of tau lepton allow to search for new physics scenario. Hadronic tau lepton decays are also a tool in high-energy physics. At the LHC and future linear colliders a correct simulation of the hadronic decay modes, mainly two pion and three pion modes, is needed to measure the Higgs spin and its CP properties.The implementation of the appropriate information on the hadronization of the QCD currents represents a key task of the TAUOLA library[2,3]. TAUOLA is a Monte Carlo generator (MC) dedicated to generating tau decays and it is used in the analysis of experimental data both at Bfactories and LHC. It is important to include in the analyses the information from QCD itself and not ad-hoc models that may hide the important information contained in the data. On the other hand the agreement with experimental data is essential and verifies a theoretical model. Resonance Chiral Theory (RChT)[4,5]provides such a reliable framework as it has been shown in many previous publications[6][7][8][9]. A set of RChT currents for the main two meson and three meson, namely, π − π 0 , K − π 0 , K 0 π − , π − π − π + , π 0 π 0 π − , K − π − K + , K 0 π −K0 and K − π 0 K 0 , was installed in TAUOLA. That set covers more than 88% of total hadronic τ width. The implementation of the currents, the related technical tests as well as the necessary theoretical concepts are documented in [10]. a
Publication by BaBar collaboration of the one-dimensional distributions for the π − π − π + mode [11] allows us to compare the RChT predicted spectra and modify the corresponding hadronic current to describe the experimental data. The main change is related with the σ meson inclusion. The paper is organized as follows. In Section 2 the hadronic currents for τ − → π − π − π + ν τ is presented. Fit of the three mass-invariant distributions for that process to BaBar data is presented in Section 3 where also the numerical tests are discussed. Summary, Section 4, closses the paper.
2 Hadronic current for τ − → π − π − π + ν τ mode For the final state of three pions π − (p 1 ), π − (p 2 ), π + (p 3 ) the Lorentz invariance determines the decomposition of the hadronic current to be [10]
J μ = N T μ ν (p 2 − p 3 ) ν F 1 − (p 3 − p 1 ) ν F 2 + q μ F 4 − i 4π 2 F 2 c 5 μ . νρσ p ν 1 p ρ 2 p σ 3 F 5 ,(1)
where: T μν = g μν −q μ q ν /q 2 denotes the transverse projector, and q μ = (p 1 + p 2 + p 3 ) μ is the momentum of the hadronic system. The normalization factor is N = cosθ Cabibbo /F, where F is the pion decay constant in chiral limit. In the isospin symmetry limit, the F 5 form factor for the three pion mode is zero due to G-parity conservation [12] and thus we will neglect it. The hadronic form factor, F i , are model dependent functions. In general they depend on three independent invariant masses that are constructed from the three meson four-vectors. We chose q 2 = (p 1 + p 2 + p 3 ) 2 and two invariant masses s 1 = (p 2 + p 3 ) 2 , s 2 = (p 1 + p 3 ) 2 built from pairs of momenta (then s 3 = (p 1 + p 2 ) 2 = q 2 − s 1 − s 2 + 3m 2 π ). In the framework of RChT every hadronic form factor consists of three parts: a chiral contribution (direct decay, without production of any intermediate resonance), single-resonance and doubleresonance mediated processes. The exact form of the form factors within RChT are written in [10], Eqs. (4)- (10). We would like to stress that only vector and axial-vector resonance contributions to the hadronic form factors were included [6,10]. The first comparison of the RChT results for the π − π − π + mode with the BaBar data [11], did not demonstrate a satisfactory agreement for the two pion invariant mass distributions, indicating that the lack of the f 0 (600) (or σ) meson contribution to the hadronic form factors might be responsible for that discrepancy [13].
As the σ meson is predominantly a teraquark state, it cannot be included in the RChT formalism. In view of this we have decided to describe the σ meson by a s-wave Breit-Wigner resonance function. In fact, a similar parametriztaion was used by the CLEO collaboration in the analysis of the three pion decay modess of the tau lepton [14]. The σ meson inclusion affects the F 1 (Q 2 , s, t) and F 2 (Q 2 , s, t) form factors in the following way
F R 1 → F R 1 + √ 2F V G V 3F 2 α σ BW σ (s 1 )F σ (q 2 , s 1 ) + β σ BW σ (s 2 )F σ (q 2 , s 2 ) ,(2)F RR 1 → F RR 1 + 4F A G V 3F 2 q 2 q 2 − M 2 a 1 − iM a 1 Γ a 1 (q 2 ) γ σ BW σ (s 1 )F σ (q 2 , s 1 ) + δ σ BW σ (s 2 )F σ (q 2 , s 2 ) , where BW σ (x) = M 2 σ M 2 σ − x − iM σ Γ σ (x) , Γ σ (x) = Γ σ σ π (x) σ π (M 2 σ ) , F σ (q 2 , x) = exp −λ(q 2 , x, m 2 π )R 2 σ 8q 2 ,
and σ π (q 2 ) ≡ 1 − 4m 2 π /q 2 and λ(x, y, z) = (x − y − z) 2 − 4yz. Bose symmetry implies that the form factors F 1 and F 2 satisfy the relation F 2 (q 2 , s 2 , s 1 ) = F 1 (q 2 , s 1 , s 2 ). The vertex coupling constants α σ , β σ , γ δ and δ σ as well as the mass (M σ ) and width (Γ σ ) of the σ mesons are parameters fitted to the data. More details about the modification to the RChT three pion current are presented in [1].
EPJ Web of Conferences
00054-p.2 M ρ M ρ Γ ρ M a 1 M σ Γ σ F F V Min 0.
As a further application we present here also a result for the differential τ → π − π − π + ν τ width:
dΓ dq 2 ds 1 ds 3 = G 2 F |V ud | 2 128(2π) 5 M τ F 2 M 2 τ q 2 − 1 2 W S A + 1 3 1 + 2 q 2 M 2 τ W A ,(3)
where
W A = −(V μ 1 F 1 + V μ 2 F 2 + V μ 3 F 3 )(V 1μ F 1 + V 2μ F 2 + V 3μ F 3 ) , W S A = q 2 |F 4 | 2 .
3 Fit to τ − → π − π − π + ν τ data from BaBar. Numerical results and tests
The three one-dimensional distributions, namely dΓ/dq 2 , dΓ/ds 1 and dΓ/ds 3 , were fitted to the BaBar data [11]. The corresponding distributions are obtained from the three-dimensional spectrum, Eq.(3), by integration over two parameters. The partial width is normalized to one measured by BaBar Γ = (2.00 ± 0.03%) · 10 −13 GeV [15]. The fit results are presented in table 1 and figure 1 and correspond to χ 2 /nd f = 6658/401. In our previous paper [13], χ 2 was computed using the combined statistical and systematic uncertainties since only the total covariance matrix was publicly available. For the present results we obtain χ 2 /nd f = 910/401, when the total covariance matrix is used and the conditions enabling direct comparisons are fulfilled. Thus is eight times better than the previous result [13].
The statistical uncertainties were determined using the HESSE routine from minuit [16] under the assumption that the correlations between distributions and the correlations related to having two entries per event in the π − π + distribution can be neglected. The fit results with estimated systematical and statistical errors, the statistical correlation matrix and the correlation matrix for systematic uncertainties are collected in tables 3, 4 and 5 of [1], correspondingly. A strong correlation (correlation coefficients moduli bigger than 0.95) was found between four parameters of the model M a 1 , F π , F V and β ρ . The correlation between these parameters can be explained by the underlying dynamics: the dominant contribution to the hadronic currents originates from the exchange a 1 → (ρ; ρ )π and, as a consequence, strong correlations between F V , F A , F π and also M a 1 and β ρ could have been expected, as it is the case for all of them except for F A which shows slightly smaller correlations (more details can be found in section V.A in [1]). Also the parameters β σ and Γ ρ are correlated (the corresponding correlation coefficients are larger tham 0.85).
The following test has been done to check whether the obtained minimum is a global one and does not depend on the starting parameter values. We start from a random scan of 2.1 * 10 5 points 00054-p.3 Table 1. Numerical ranges of the RChT parameters used to fit the BaBar data and the result of fit to BaBar data for three pion mode [11] .
QCD@Work 2014 Figure 1. The τ − → π − π − π + ν τ decay invariant mass distribution of the three-pion system (left panel) and twopion pairs (central and right panels). The BaBar measurements [11] are represented by the data points, with the results from the RChT current as described in the text (blue line) and the old tune from CLEO from Refs. [17] (red-dashed line) overlaid. At the bottom of the figures ratio of new RChT prediction to the data is given. The parameters used in our new model are collected in Table 1. and select 1000 events with the best χ 2 , out of which 20 points with maximum distance between them are retained and then these points are used as a start point for the full fit. We find that more than an half converges to the minimum (table 1), others either fall with number of parameters at their limits or converge to local minimum with higher χ 2 . Therefore, we conclude that the obtained result is stable and does not depend on an initial value of the fitting parameters. As an additional cross check we calculated the partial width resulting from the phase space integration of the matrix element Γ τ − →π − π − π + ν τ = 1.9974 · 10 −13 GeV which agrees with the one measured by BaBar Γ τ − →π − π − π + ν τ = (2.00 ± 0.03%) · 10 −13 GeV [15].
Comparison between the RChT results (after fit) and the BaBar spectra, presented in figure 1, demonstrates the possibility of missing resonances in the model (2). For the π + π − mass invariant spectrum it could be f2(1270) and f0(1370), as reported by CLEO [14,18] and suggested by figure 1. The largest discrepancies between data and the fitted distribution, which are responsible for a significant part of the total χ 2 , are observed also in the 3π invariant mass distribution. The slope and shape of the disagreement in the 3π invariant mass spectrum, in particular around 1.5 GeV in Fig. 1, indicates the possibility of an interference between a 1 (1260) and its excited state a 1 (1640).
Conclusion
In this paper we discussed the hadronic current for the τ − → π − π + π − ν τ decay within Resonance Chiral Theory and the modification to the current to include the sigma meson. The choice of this channel was motivated by its relatively large branching ratio, availability of unfolded experimental distribution and already non-trivial dynamics of three-pion final state. In addition, this channel is important for Higgs spin-parity studies through the associated di-τ decays. As a result, we improved agreement with the data by a factor of about eight.
To get the numerical values of the RChT parameters we fitted the one dimentional mass invariant distributions to the published BaBar data. Also we have tested that the obtained results correspond to a global minimum and that the fitting procedure does not depend on the initial values of the model parameters.
EPJ Web of Conferences 00054-p. 4 We have found discrepancies in the high mass region of the π + π − and π + π − π − invariant mass indicating the possibility of missing resonances in our RChT approach. This is consistent with the observation of additional resonances, more specifically the f 2 (1270) and f 0 (1370) and a 1 (1640), by CLEO in [14,18]. Although we could add phenomenologically the contribution of these resonances to the amplitude, we prefer not to do it at the moment to keep a compromise between the number of parameters, the stability of the fit and the amount of experimental data. Certainly, that type of improvements will be done in future analysis of multi-dimensional distributions.
The work on a generalization of the fitting strategy to a case of an arbitrary three meson tau decay is in progress.
AcknowledgementsThis research was supported in part by Foundation of Polish Science grant POMOST/2013-7/12, that is co-financed from European Union, Regional Development Fund and from funds of Polish National Science Centre under decisions DEC-2011/03/B/ST2/00107.
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. QCD@Work. QCD@Work 2014
|
[] |
[
"Digital Landscape of COVID-19 Testing: Challenges and Opportunities",
"Digital Landscape of COVID-19 Testing: Challenges and Opportunities"
] |
[
"Darshan Gandhi \nPathCheck Foundation\n02139CambridgeUSA\n",
"Rohan Sukumaran \nPathCheck Foundation\n02139CambridgeUSA\n",
"Priyanshi Katiyar \nPathCheck Foundation\n02139CambridgeUSA\n",
"Alex Radunsky \nPathCheck Foundation\n02139CambridgeUSA\n\nInstitute for Technology and Global Health\n02139CambridgeUSA\n",
"Sunaina Anand \nPathCheck Foundation\n02139CambridgeUSA\n",
"Shailesh Advani \nPathCheck Foundation\n02139CambridgeUSA\n\nNational Human Genome Research Institute\nNational Institutes of Health\n20892BethesdaUSA\n",
"Jil Kothari \nPathCheck Foundation\n02139CambridgeUSA\n",
"Kasia Jakimowicz \nPathCheck Foundation\n02139CambridgeUSA\n\nMassachusetts Institute of Technology\n02139CambridgeUSA\n\nAsh Center for Democratic Governance and Innovation\nHarvard Kennedy School\n02138CambridgeUSA\n",
"Sheshank Shankar \nPathCheck Foundation\n02139CambridgeUSA\n",
"Sethuraman T V ",
"Krutika Misra \nPathCheck Foundation\n02139CambridgeUSA\n",
"Aishwarya Saxena \nPathCheck Foundation\n02139CambridgeUSA\n\nBerkeley School of Law\nUniversity of California\n94720BerkeleyUSA\n",
"Sanskruti Landage \nPathCheck Foundation\n02139CambridgeUSA\n",
"Richa Sonker \nPathCheck Foundation\n02139CambridgeUSA\n",
"Parth Patwa \nPathCheck Foundation\n02139CambridgeUSA\n",
"Aryan Mahindra \nPathCheck Foundation\n02139CambridgeUSA\n",
"Mikhail Dmitrienko \nPathCheck Foundation\n02139CambridgeUSA\n",
"Kanishka Vaish \nPathCheck Foundation\n02139CambridgeUSA\n",
"Ashley Mehra \nPathCheck Foundation\n02139CambridgeUSA\n",
"Srinidhi Murali \nPathCheck Foundation\n02139CambridgeUSA\n",
"Rohan Iyer \nPathCheck Foundation\n02139CambridgeUSA\n",
"Joseph Bae \nPathCheck Foundation\n02139CambridgeUSA\n\nRenaissance School of Medicine\nStony Brook University\n11794 Stony BrookUSA\n",
"Vivek Sharma \nPathCheck Foundation\n02139CambridgeUSA\n\nMIT Media Lab\n02139CambridgeUSA\n\nMassachusetts Institute of Technology\n02139CambridgeUSA\n\nHarvard Medical School\n02115BostonUSA\n\nMIT Media Lab\n02139CambridgeUSA\n",
"Abhishek Singh \nMIT Media Lab\n02139CambridgeUSA\n\nMassachusetts Institute of Technology\n02139CambridgeUSA\n\nMIT Media Lab\n02139CambridgeUSA\n",
"Rachel Barbar \nMIT Media Lab\n02139CambridgeUSA\n\nMassachusetts Institute of Technology\n02139CambridgeUSA\n\nMIT Media Lab\n02139CambridgeUSA\n",
"Ramesh Raskar \nPathCheck Foundation\n02139CambridgeUSA\n\nMIT Media Lab\n02139CambridgeUSA\n\nMassachusetts Institute of Technology\n02139CambridgeUSA\n\nMIT Media Lab\n02139CambridgeUSA\n"
] |
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"PathCheck Foundation\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"Institute for Technology and Global Health\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"National Human Genome Research Institute\nNational Institutes of Health\n20892BethesdaUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"Massachusetts Institute of Technology\n02139CambridgeUSA",
"Ash Center for Democratic Governance and Innovation\nHarvard Kennedy School\n02138CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"Berkeley School of Law\nUniversity of California\n94720BerkeleyUSA",
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"PathCheck Foundation\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"Renaissance School of Medicine\nStony Brook University\n11794 Stony BrookUSA",
"PathCheck Foundation\n02139CambridgeUSA",
"MIT Media Lab\n02139CambridgeUSA",
"Massachusetts Institute of Technology\n02139CambridgeUSA",
"Harvard Medical School\n02115BostonUSA",
"MIT Media Lab\n02139CambridgeUSA",
"MIT Media Lab\n02139CambridgeUSA",
"Massachusetts Institute of Technology\n02139CambridgeUSA",
"MIT Media Lab\n02139CambridgeUSA",
"MIT Media Lab\n02139CambridgeUSA",
"Massachusetts Institute of Technology\n02139CambridgeUSA",
"MIT Media Lab\n02139CambridgeUSA",
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"MIT Media Lab\n02139CambridgeUSA",
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[] |
The COVID-19 Pandemic has left a devastating trail all over the world, in terms of loss of lives, economic decline, travel restrictions, trade deficit, and collapsing economy including real-estate, job loss, loss of health benefits, the decline in quality of access to care and services and overall quality of life. Immunization from the anticipated vaccines will not be the stand-alone guideline that will help surpass the pandemic and return to normalcy. Four pillars of effective public health intervention include diagnostic testing for both asymptomatic and symptomatic individuals, contact tracing, quarantine of individuals with symptoms or who are exposed to COVID-19, and maintaining strict hygiene standards at the individual and community level. Digital technology, currently being used for COVID-19 testing include certain mobile apps, web dashboards, and online self-assessment tools. Herein, we look into various digital solutions adapted by communities across universities, businesses, and other organizations. We summarize the challenges experienced using these tools in terms of quality of information, privacy, and user-centric issues. Despite numerous digital solutions available and being developed, many vary in terms of information being shared in terms of both quality and quantity, which can be overwhelming to the users. Understanding the testing landscape through a digital lens will give a clear insight into the multiple challenges that we face including data privacy, cost, and miscommunication. It is the destiny of digitalization to navigate testing for COVID-19. Block-chain based systems can be used for privacy preservation and ensuring ownership of the data to remain with the user. Another solution involves having digital health passports with relevant and correct information. In this early draft, we summarize the challenges and propose possible solutions to address the same.
| null |
[
"https://arxiv.org/pdf/2012.01772v1.pdf"
] | 227,254,598 |
2012.01772
|
b943ffa25ad1c5e137ca184d9baa11cbc8710867
|
Digital Landscape of COVID-19 Testing: Challenges and Opportunities
3 Dec 2020
Darshan Gandhi
PathCheck Foundation
02139CambridgeUSA
Rohan Sukumaran
PathCheck Foundation
02139CambridgeUSA
Priyanshi Katiyar
PathCheck Foundation
02139CambridgeUSA
Alex Radunsky
PathCheck Foundation
02139CambridgeUSA
Institute for Technology and Global Health
02139CambridgeUSA
Sunaina Anand
PathCheck Foundation
02139CambridgeUSA
Shailesh Advani
PathCheck Foundation
02139CambridgeUSA
National Human Genome Research Institute
National Institutes of Health
20892BethesdaUSA
Jil Kothari
PathCheck Foundation
02139CambridgeUSA
Kasia Jakimowicz
PathCheck Foundation
02139CambridgeUSA
Massachusetts Institute of Technology
02139CambridgeUSA
Ash Center for Democratic Governance and Innovation
Harvard Kennedy School
02138CambridgeUSA
Sheshank Shankar
PathCheck Foundation
02139CambridgeUSA
Sethuraman T V
Krutika Misra
PathCheck Foundation
02139CambridgeUSA
Aishwarya Saxena
PathCheck Foundation
02139CambridgeUSA
Berkeley School of Law
University of California
94720BerkeleyUSA
Sanskruti Landage
PathCheck Foundation
02139CambridgeUSA
Richa Sonker
PathCheck Foundation
02139CambridgeUSA
Parth Patwa
PathCheck Foundation
02139CambridgeUSA
Aryan Mahindra
PathCheck Foundation
02139CambridgeUSA
Mikhail Dmitrienko
PathCheck Foundation
02139CambridgeUSA
Kanishka Vaish
PathCheck Foundation
02139CambridgeUSA
Ashley Mehra
PathCheck Foundation
02139CambridgeUSA
Srinidhi Murali
PathCheck Foundation
02139CambridgeUSA
Rohan Iyer
PathCheck Foundation
02139CambridgeUSA
Joseph Bae
PathCheck Foundation
02139CambridgeUSA
Renaissance School of Medicine
Stony Brook University
11794 Stony BrookUSA
Vivek Sharma
PathCheck Foundation
02139CambridgeUSA
MIT Media Lab
02139CambridgeUSA
Massachusetts Institute of Technology
02139CambridgeUSA
Harvard Medical School
02115BostonUSA
MIT Media Lab
02139CambridgeUSA
Abhishek Singh
MIT Media Lab
02139CambridgeUSA
Massachusetts Institute of Technology
02139CambridgeUSA
MIT Media Lab
02139CambridgeUSA
Rachel Barbar
MIT Media Lab
02139CambridgeUSA
Massachusetts Institute of Technology
02139CambridgeUSA
MIT Media Lab
02139CambridgeUSA
Ramesh Raskar
PathCheck Foundation
02139CambridgeUSA
MIT Media Lab
02139CambridgeUSA
Massachusetts Institute of Technology
02139CambridgeUSA
MIT Media Lab
02139CambridgeUSA
Digital Landscape of COVID-19 Testing: Challenges and Opportunities
3 Dec 2020Gandhi et al. RESEARCH * Correspondence: Full list of author information is available at the end of the article Gandhi et al. Page 2 of 28COVID-19VaccinesHealthcare information managementPrivacy
The COVID-19 Pandemic has left a devastating trail all over the world, in terms of loss of lives, economic decline, travel restrictions, trade deficit, and collapsing economy including real-estate, job loss, loss of health benefits, the decline in quality of access to care and services and overall quality of life. Immunization from the anticipated vaccines will not be the stand-alone guideline that will help surpass the pandemic and return to normalcy. Four pillars of effective public health intervention include diagnostic testing for both asymptomatic and symptomatic individuals, contact tracing, quarantine of individuals with symptoms or who are exposed to COVID-19, and maintaining strict hygiene standards at the individual and community level. Digital technology, currently being used for COVID-19 testing include certain mobile apps, web dashboards, and online self-assessment tools. Herein, we look into various digital solutions adapted by communities across universities, businesses, and other organizations. We summarize the challenges experienced using these tools in terms of quality of information, privacy, and user-centric issues. Despite numerous digital solutions available and being developed, many vary in terms of information being shared in terms of both quality and quantity, which can be overwhelming to the users. Understanding the testing landscape through a digital lens will give a clear insight into the multiple challenges that we face including data privacy, cost, and miscommunication. It is the destiny of digitalization to navigate testing for COVID-19. Block-chain based systems can be used for privacy preservation and ensuring ownership of the data to remain with the user. Another solution involves having digital health passports with relevant and correct information. In this early draft, we summarize the challenges and propose possible solutions to address the same.
Introduction and Related Work
COVID-19, a global pandemic has high transmission and mortality rate, especially among the elderly, healthcare workers and those with underlying comorbidities. With the vaccines still undergoing final approvals, the mainstay of pandemic management depends on key public health intervention including early detection, mitigation through isolation and contact tracing. Countries across the world are taking strict measures to manage the overwhelming and unprecedented impact of the pandemic Digital health technology, the mainstay of pandemic strategy can achieve superior outcomes that are difficult to achieve otherwise. Integrating these digital tools into public health interventions has revolutionized surveillance, testing, contact tracing, healthcare access and quarantine measures aimed at control and mitigating COVID-19 spread thereby aiming to reduce COVID-19 infection and deaths Big data and artificial intelligence (AI), used widely in healthcare, have also helped in tracking people and thereby reducing the spread of infection. Migration maps including mobile phones, mobile payment applications and social media are used to collect real-time data on the location of people. Machine learning and disease spread models were developed to predict the regional transmission of the virus that help guide policy and containment measures. Efficient use of big data has contributed to reduced spread and early detection, thereby saving thousands of lives. The need to track the spread of COVID-19 has fuelled the innovation of data containing dashboards that display variables such as disease burden including number of cases, deaths, recovered, and active patients.s. Some dashboards could plot trends according to demographic and geographical characteristics. The limitations of AI need to be addressed and requires training with COVID-19 datasets. The accuracy, validity and reliability of these digital tools are a concern. This viewpoint provides an outline for the application of digital technology in the landscape of pandemic management and response. Also, highlighting the challenges and success of various digital technologies used by countries to manage the pandemic.
Since December 2019, the COVID-19 pandemic has contributed to an exponential growth in the development of diagnostic technologies and interdisciplinary and intercountry scientific data sharing. A recent bibliometric study has shown that the availability of evidence in response to the COVID-19 pandemic has been much more efficient than other recent epidemic events, such as the 2015-16 Zika virus epidemic and the 2014-16 Ebola virus outbreak due to technology and science advancements and rapid sharing of information. More than 2,500 articles related to COVID-19 were published in the first 4 months of the pandemic, compared to only 88 articles related to both Zika and Ebola viruses in the same epidemiological period. [1] The pace at which viral genomic sequences have been available to the public during the COVID-19 pandemic also illustrates the rapid pace of data sharing over time. [2] As of 31st December 2019, 19 gene sequences of the SARS-CoV-2 virus were already obtainable through the GISAID database (gisaid.org), which now has over 40,000 viral genome sequences shared by laboratories around the globe. As a comparison, it took almost 3 years for the number of sequential viral genomes to reach 1,500 sequences during the Ebola virus outbreak. The rapid developments of SARS-CoV-2 genomic sequences contributed to both the rapid development of the gold standard molecular diagnostic tests for COVID-19 and to the development of simplified protocols for complete viral genome sequencing and analysis and labbased serology assays using recombinantly produced SARS-CoV-2 proteins.
In this early draft, we aim to understand the digital landscape of the current testing scenarios. We go to the granularity of understanding the User agency and control, understanding the digital workflow of testing, how various organisations (or governments) are using the digital solutions for testing. Furthermore we look into understanding the major challenges in the current workflow including but not limited to -privacy, security, miscommunication, trust and laws. We look at various case studies across countries to understand the effects of different digital solutions and leave an open thread for the solutions.
User Agency and Control of Data
One of the key challenges in the digital management of testing has been how the agencies acquire and control the data or the users for their use and the privacy policies they follow. Confidentiality has been one of the most important issues and concerns of the digital revolution and given rapid use of digital platforms on COVID-19 management. Increased privacy concerns are attributed to several health platforms requiring patients to share their personal protected health information, including but not limited to the patient's name, phone number, email, race, country of birth, SSN, residential address and even travel history. This is problematic because of the lack of policies and regulations about the use of personal data. [3] Further, due to the sensitive nature of the data, users face the security risk of being tracked, man-in-middle attack and data snooping by a third party.
Thus, it becomes very important to have the right policies set which inform the user about the need for their data and consent of use from the patient is a must.
1.2 How other epidemic tests are managed (Ebola, HIV, malaria, Swine flu, zika, etc) The Zika outbreak in 2016 was a global public health concern and digital technology played a vital role alongside traditional public health approaches. Mathematical modelling helped in understanding the Aedes spp. Vectors in terms of transmission, host-virus interaction, and effectiveness of potential interventions. Geographical information systems (GIS) including epidemiological maps were used to estimate magnitude, transmission and spread of the Zika outbreak. The category of big data included social media analytics and web-based surveillance particularly on the OpenZika project, Digital Participatory Surveillance (DPS), and ProMED. DPS is a form of digital surveillance where the public reports signs and symptoms through a structured electronic form and the data is summarized via epidemiological maps. Zika tracker, a smartphone-based diagnostic platform could be used anywhere in the world to report suspected or confirmed cases of Zika. WHO's Zika App is a similar app that provides real-time data to all users. Researchers in Brazil developed MinION that is a novel real-time genome-sequencing device. An insecticide wearable device that releases Metofluthrin (traditional repellant of Aedes aegypti) reduced the number of mosquitoes attracted. [4] During the Ebola outbreak, diagnosis was difficult especially in asymptomatic people. The gold-standard diagnostic test for Ebola virus, the real-time RT-PCR needed substantial infrastructure, operation and maintenance of complex equipment, and expertise in molecular techniques. Limitations of the diagnostic tests were addressed by digital solutions. The Democratic Republic of Congo (DRC) used the National Health Information System to create a dashboard called PATH for monitoring Ebola cases. The US Center for Disease Control and Prevention (CDC), World Health Organization (WHO) and other organizations worked together to develop digital maps and graphs. In 2019, the Ministry of Health of DRC launched a national agency for clinical information and health informatics (ANICiiS). During the 2014-2016 epidemic, mHERO, an SMS system was set-up to share information with health workers. UNICEF also launched the Rapidpro free source platform that provided SMS messaging in Sierra Leone where it was rare to possess a smart-phone, internet, or even phone credit. [5][6] [7] HIV/AIDS control strategies included digital solutions such as mapping and maintaining surveillance of high risk behaviors, individual level intervention that targets key populations (during a low-level epidemic), providing voluntary testing and counseling (VTC) and screening for sexually transmitted infections (STIs).
Malaria remains endemic in several countries despite multiple interventions. Containing malaria outbreaks remained a challenge because it was difficult to predict who would contract the disease. As a result, resources were not deployed in the correct region. This caused the outbreak to grow. A quick and reliable digital solution was required. The National Aeronautics and Space Administration (NASA) used satellite data to help forecast malaria outbreaks by identifying areas of prime breeding grounds for these mosquitoes that overlapped with human population. They use the information on precipitation, temperature, soil moisture, and vegetation around the world and feed it into a prediction model and tell where a puddle is most likely to form. Simply mapping where people reported malaria is not enough as the place of infection is usually different. They also track humans to see where most people are getting infected i.e hotspots. But for the purposes of predicting a malaria outbreak, the map did not tell a complete story. They could successfully make projections down to the household level, allowing for resources to go where they're needed. The population models could be used for tracking not only malaria, but Zika and Dengue as well. [8] The Digital Solutions for Malaria Elimination (DSME) Community of Practice (COP) helped in building malaria surveillance systems that made accurate data reporting easier and thus improved decision-making processes. [9] 2 Testing Workflows
User experience
The rapid spread of COVID-19 has been worrisome for many individuals across the globe with respect to the safety of security, workflows and measures being taken to overcome these issues. There is an increasing emphasis on providing great user experience and information to the citizens so they stay informed and aware of the recent developments, policies, precautions and measures that have been taken up to fight the virus. In this table, we highlight the similarities and differences in user engagement within public/government sites, private clinics, employee/campus organizations, and entering venues (such as flights). It is important to understand the different facilities which are provided by the different organizations and institutions and how the public and government sites can be modified more to support the facility of giving out reminders and covering the insurances for the patients. Also the entry pass system at the public/ government sites should be more digitised to avoid the issues of data privacy and spoofing. Adding on, it is most important to provide the scheduling facility across all the different organizations to avoid the issue of overcrowding and have a systematic approach in place. Even the test results should be supported with a detailed report covering points ranging from a detailed analysis of virus to the precautionary measures that need to be taken by the patient to avoid getting infected in future, in case they are affected a report needs to be shared to indicate the exact steps the patient should take in order to stay protected and at the same time not transmit the infection to others. Table It is also important to understand more about the user's journey and the steps taken by them in order to get tested. First via social media campaigns, televisions and youtube videos the end user is educated about the different COVID-19 testing centres and sites. Using this information the user tries to explore more about the different sites which may be public, private, educational sites and draws a comparative analysis between them. Once they are certain about the type of site/ organisation they would want to visit next they analyse the different centres and hospitals available for them to visit and this decision is taken based on several parameters such as the type of testing available, cost of testing, time to get the results and one of the crucial points would be the location of thee testing sites. Lastly, once tested the patient/user tends to receive more updates and reminders on taking the necessary dosage and also receive timely reminders via messages. Figure 3 illustrates the standardised process flow for COVID-19 testing. The participants should flow in one direction and each of the stations should be separated by at least 6 feet. Participants begin with a temperature and symptom check. This is also where participants are given a face covering if they don't already have one. Then, they fill out all the necessary paperwork (pens and clipboards are disinfected in between participants). Participants proceed to the interview stations where they are screened for symptoms, after which they are put in testing stations where they are given a swab to collect their specimen. The specimen is then put into a vial and dropped into a transport bag. Lastly, participants are then given the information needed to receive their results [11].
User engagement summary table
There are many software and data management systems in use as well. Data management systems are essential during COVID-19 case investigations and contact tracing. This data is stored in different ways based on the guidelines of the local jurisdiction or testing sites, but typically is stored in storage platforms like cloud. It is important for the data to be stored in a way that preserves anonymity and protection of Personal Health Information (PHI). In addition, Lab Information Management Systems (LIMS), a type of software present in most modern laboratories, can be used at COVID-19 testing sites. LIMS could improve productivity and efficiency of laboratories by increasing the speed of test processing, reducing costs by automating, and streamlining lab workflows.
Test sites should be in large open areas (preferably outside) to allow for effective social distancing. It is also important for test sites to have centers to isolate patients who test positive for COVID-19. In a test site, keeping staff members safe is extremely important. The CDC recommends gloves and face masks for all staff members present at a test site, and gowns and N95 (or equivalent) masks for anyone who will be within 6 feet of any person being tested. The person being tested should also wear a face mask. Aggressive sanitation is necessary, including changing gowns, respirators, and face masks if they are soiled. In addition, these should be replaced in case they are accidentally touched or if they come in close contact with a person being tested.
Dissemination of results
to the user (website, app, phone call, email, text) (Comcare, Salesforce) Testing forms a crucial point of intervention for mitigating the COVID-19 pandemic. The results of one's COVID-19 test along with other relevant information will assist one's healthcare provider to make informed decisions regarding patient's health including recommending treatment and mitigation strategies. Timely and proper communication of results will also help in limiting the spread of COVID-19 to one's family and the larger community. The mechanism of disseminating results varies greatly across countries and can be through web based secure file transmission, an SMS, phone call, email or within an embedded app portal. Digital or electronic-based reporting would reduce unnecessary travel and risk of exposure. The timeframe for tests to be made available depends upon the type of test and the area it was done. The process for Rapid antigen tests is faster than RT-PCR. Point of care testing in community settings like schools, college campuses, nursing homes and employer organisations remain crucial for public health mitigation strategies.Testing sites ,generally employ the following methods for results notification:
• [13]. In order to gain access to the campus, one must agree to conditions like daily health attestation, and regular COVID-19 testing. Testing frequency is contingent on how often the person is on campus. COVID-19 Pass will notify ( through the email and mobile push notification) about testing status early each morning. This will help to keep track of when individuals need testing. Test results are generally provided within 1-2 days which will be available in COVID-19 Pass App on MIT Atlas and not in HealthElife. Though, tests done through appointments will have their results visible in HealthElife. If an individual does not yet have full authorization to use the COVID-19 Pass app, they may still see results at covidpass.mit.edu using MIT Kerberos authentication. A physical document of the result can also be requested by call at MIT Medical COVID-19 hotline. If tested positive, MIT Medical will contact the individual directly and the contact tracing process will begin. If no communication is made within 48 hours, we can presume tests are negative. Amidst a global pandemic, the MIT COVID-19 Pass is an efficient and time saving digital solution. In the hassle and rush of getting work done in the morning, when someone may forget that they need to be tested, the COVID-19 Pass app will notify them. • Minnesota Department of Health: It is a public health department that provides community testing [14]. If the test comes negative, one will receive a text and/or email notifying that one can access results online. If there is no access to the internet, there will be a number on which calls can be made for results. In case it is positive, one will receive a phone call from the triage line. • Henry Ford Health System: The Henry Ford Health System has provided an online COVID-19 screening tool to assess risk [15]. Remote monitoring and virtual consultation in the form of E-visits and video visits has been made possible. The Henry Ford Health System has provided an online COVID-19 screening tool to assess risk. Remote monitoring and virtual consultation in the form of E-visits and video visits has been made available. • Carewell Urgent Care: Carewell [16] is a community-based health care system that provides both patients and employers COVID-19 testing services.For employers, results will be communicated through fax. For others, in the case of the PCR test, results are displayed on the Quest Diagnostics Website while antigen test results are posted on carewell urgent care website itself. The dissemination of results is a crucial step in the user experience since it plays an important role to make sure the user is aware of the precautionary measures to be taken, will help make more informed decisions and also keep them updated about the current testing workflows and practices.
Efficacy, ethical and privacy concerns
An immunity passport based on vaccine-induced immunity would eliminate some of the moral hazards and ethical concerns raised by experts: that some non-infected individuals would see it as an incentive to expose themselves to the virus in order to develop antibodies [ref] (needs fix) as a way of receiving immunity passports and practice herd immunity. There's also a concern people would be penalized for low-risky behavior resulting in a lack of exposure to the virus, which would make them ineligible for a card based on infection-related immunity. Yet, other efficacy, ethical and privacy concerns remain relevant to digital passports based on vaccine-induced immunity. Evidence suggests [17] that, despite being immune, an individual might still transmit disease and infect others. Though different places are planning for non-discriminatory and equal access vaccine distribution, there are multiple unknown challenges along these lines. This could hinder the effectiveness of digital immunity passports . Otherwise, according to ethicists, digital immunity passports might increase racial and social inequalities [18] already exacerbated by the pandemic, repeating the history of immunity passports during "yellow fever" epidemics at the beginning of the 20th century. These were used as a means of racial discrimination, and oppression. Digital immunity passports could also facilitate social exclusion [19] and discriminatory work environments, where "an official stamp of immunity to COVID-19, or personal willingness to risk the disease" would "... become a prerequisite for employment". [18] Finally, the immunity certificates' credentials could create privacy and security risks. And the centralized nature of a digital credential system, where a single institution would be responsible for data collection and credential management, could lead to privacy violations and abuse of power. Dakota Greuner from Harvard Edmond J. Safra Centre for Ethics' COVID-19 Rapid Response Impact Initiative argues [20] that for any digital health passports to be effective, it should possess three features:
1 Be privacy-preserving -Personal data should only be managed by the user, stored on the device, and shared only after explicit user consent. It also should be encrypted and secured by biometrics. Selective disclosure of identifiers should be incorporated in the design (for example, age but not birth date etc.) [20] 2 Be portable and widely adopted -It should be based on open standards, so it is broadly usable across systems and devices. 3 Be trustworthy -This could be achieved with a digital signature of authenticity secured by the use of cryptography, so that information on the date and the credential could not be changed or hacked.
Decentralized systems of credentials deployed at a large scale are being suggested as a "privacy-protecting model of immunity certificates." [20] Owing to the decentralised and trustworthy framework, block chains have come across as a solution to deliver critical information in a secure and privacy preserving way.
With the nature of health records and issues like record linkage, privacy and misinterpreted results (due to false positives or miscommunication) it becomes difficult do extract insights from them. This coupled with the lack of trust (as health information is very personal) makes it difficult to have data shared. This results in data silos and many missed opportunities of information sharing and understanding of health information. Blockchain based solutions can help to mitigate some of these issues as the data ownership is still will the respective person (or hospital) and it ensures security and reliability. A few of the impactful applications for COVID-19 response which are contact tracing, sharing and handling patients data employ the use of Blockchain.
Digital Health Verification
From Ireland to Chile, efforts are underway worldwide to introduce health immunization passports as a way of easing out social distancing rules and allowing for individuals to go back to work, travel or go out to public venues, such as restaurants and sports events [21] [22]. These digital immunization passports are meant to be a digital equivalent of paper certificates [20], indicating the holder is at low risk of transmitting the coronavirus, providing evidence of acquired immunity to the virus, by having recently tested negative to the virus, having antibodies or been vaccinated. Immunity passports have been compared to international certificates of vaccination, such as the "Carte Jaune" for yellow fever. However, there are significant differences between the two types of documents to be noted. Such credentials could take the form of a wristband / wearable device, a digital certificate via smartphone application [17] or a biometric ID card, among others. The majority of initiatives focus on digital credentials that could be carried on users' smartphones (for instance in the form of the QR code). Individuals with an immunity passport could be exempt from physical restrictions and could return to work, school, and daily life. However, immunity passports pose considerable risks including, by not limited to, scientific risk, practical risk, equitable risk and legal challenges. Considering all these risks , the users should be wearing masks and follow minimum safety standards.
The immunity passports rely on two concepts: infection-related immunity and vaccine-induced immunity. Both raise issues of "the degree of immunity induced and the duration of immunity" [17]. WHO is not recommending [23] an introduction of infection-related immunity passports based on "the lack of sufficient evidence that people with COVID-19 antibodies are immune to new COVID-19 infections." [24] Vaccine-induced immunity as a basis for immunity passports is more promising, with recent results in animal trials revealing a correlation between vaccines and protection. A uniform vaccination, experts say, could provide a more "predictable pattern and duration of immunity."
Examples of digital health passports
As early as April 2020, countries and companies have started to look into digital health passports based on potential infection-related immunity. They did so in an effort to speed up economic recovery and help people to come back to work, travel or participate in sports. These and others are expected to be expanded to certify vaccination once it is available.
The list of available solutions is growing day-by-day with companies rushing to conquer the safe-back-to-work market, including IBM Watson Digital Health Passport [25], VHealth Passport by VST Enterprises [26], WIShelter by WISeKey [27], COVI-Pass by Tento Health [28], and other solutions introduced by German IDNow and UK companies Onfido and Yoti.
1 Chile released certificates: Chile has started a roll out of smartphonebased "release certificates" instead of "immunity passports," currently valid for 3 months and shared in the form of the QR codes [29] issued to people that did not have any symptoms for 14 days after testing positive for COVID-19. based on blockchain technology for a decentralized and secure data management and authentication of an digital International COVID-19 vaccination certificate based on the vaccine-related immunity. The system, based on a web platform and an app, will be tested via a 12-week pilot in Estonia, and if successful, it might become a standard of the future when the vaccine is available on a large-scale and vaccine-immunity is proven to be effective. 5 COVID Credentials Initiative: The COVID-19 Credentials Initiative (CCI) [34] is an international community of more than 300 members from over 100 organizations hoping to deploy and/or help deploy privacy-preserving verification-based credential projects in order to reduce the spread of COVID-19 and strengthen our societies and economies. The CCI has launched the usage of digital identity in order to curb the spread of COVID-19. Their aim involves developing "immunity passports" and much more. In this, The CCI uses verifiable credentials. Inspired by the functional utility of a physical credential (e.g. the cards in one's wallet), a verifiable credential (VC) is a declared assertion that contains documented claims about a person or a particular organization. The niche of VCs that make them so pivotal is that they're digitally native and cryptographically secure, providing a great privacy-preserving alternative to other types of credentials, if used responsibly. Upon accepting a VC (e.g. a driver's license) from a trusted issuer (e.g. a government body), holders can generate proof with minimal information (e.g. over 18) to validate to a verifier (e.g. a liquor store ) that they posses their own VCs with specific information (e.g. age), qualifying them for particular types of access defined by the verifier. And there is no need for direct contact between the issuer and verifier throughout the process. 6 Travel health passports: Airlines and tourism organizations are also introducing digital health passports for travelers to enable safer cross-border travel. United and Cathay Pacific [35] Digital Health Verification is one of the most factors that needs to be addressed to ensure the ethical, political and social concerns. The health passports might not eliminate the threat completely, but is an essential tool in this pandemic. Furthermore, these health passports should be privacy preserving, trustworthy and widely accepted. It is also important to be mindful of the potential social and racial discrimination that might entail a health passport based solution.
Dashboards and Tools for Public Health
Dashboards and other web tools play a very important role in sharing the message about the magnitude, severity of the COVID-19 crisis and its impact on health.
There are numerous key testing players which are leading the way and helping significantly with testing and analysing the pandemic situation. Combined with the testing efforts, having a visual representation of the number of country-wise cases along with a wide array of filters to study socio-demographic and geographical variations can be of great use. Dashboards enable us to serve this very purpose. With the purpose of getting a grasp on the COVID-19 pandemic, numerous dashboards have been created at university level, regional, national and global level. While some of them also give resource-related information that includes availability of beds, testing kits, PPEs, masks, and ventilators, dashboards also give diagnostic test-related information including number of tests and positivity rate. Government: The World Health Organization (WHO) dashboard is one of the most broadly-accessed and predicated on visualization techniques of the COVID-19 pandemic. The general populace can navigate this dashboard easily by discovering numbers across the world, such as infected, dead, and recoveries. In addition, the WHO dashboard assists users in evaluating COVID-19 data in real-time as graphics change based on the ever-changing situation. It gives information regarding COVID-19 cases globally in the form of Choropleth and bubble maps. It gives information on new cases, confirmed cases, and number of deaths. The data table covers different regions, states, and areas which also deals with transmission classification. The group demonstrated of COVID-19 on the WHO dashboard provides data on the follow-up guidance for comprehending the pandemic. For example, users can understand the method used to determine the COVID-19 information and to update the dashboard in real-time. The Explorer section directs the public on trying to navigate through the dashboard and offering to help them to make sense of the COVID-19 statistics.
Dashboard tools for labs
The Centers For Disease Control And Prevention (CDC) COVID-19 data tracker gives information about the total number of positive cases, deaths, trends including demographics and population factor. It also compares the trend in different states and helps in forecasting future trends for the United States. Interestingly, it also includes community impact of the pandemic in terms of mobility of individuals and social impact such as news reports on school closures, localized outbreaks, state of emergency declarations, etc. The CDC dashboard persists to be one of the most frequented systems, as the COVID-19 pandemic continues with the community using dashboard updated information to comprehend the virus. Unlike others, the CDC supports a framework of all factors necessary for the Pandemic Assessment, including the dates of infection and the racial groups affected. This dashboard comprises a segment at the top in which users could see a preview of existing COVID-19 data prior to actually moving ahead to other areas.
University level: The only way to grasp and understand the magnitude of the COVID-19 pandemic for the masses is through visualizing the data pertaining to the crisis. People, in general, can be overwhelmed by huge numbers and may find the patterns contradicting. Visualization of the COVID-19 scenario makes it easier for everyone to analyze and understand the situation in a better way. Experts at all levels, right from university-level to government organizations have published COVID-19 dashboards at global, country, state, and regional levels showcasing an array of parameters right from a comparative analysis of the COVID- 19 [54] as of 12th November 2020. These dashboards provide an overview of the key metrics for their vicinity. The daily cases, hospital capacity among them, average positive cases, and many other parameters are included, The student/faculty and in the campus cases are also displayed at a daily, weekly, and monthly level along with the number of isolation cases. These parameters combined with the many other parameters can help anyone visiting the vicinity to analyze the scenario and make an informed decision accordingly.
National level: The government, administration and healthcare experts need to keep an eye on the overall COVID-19 situation in the country. A dashboard, at the national level remains the cornerstone to monitor long term impacts and trends, quantify the burden and assist in decision and policy making. The CDC COVID-19 Dashboard [55] is easily one of the most visited dashboards across the United States of America with the public using it to understand the virus better as the pandemic continues to accelerate. The CDC Dashboard stands out amongst other dashboards since it considers all the factors which are necessary for pandemic evaluation like racial groups affected and infection dates as well. The dashboard by Carnegie Mellon University [56], CovidCast is another national level dashboard that shows a Mapbox level overview which is based on survey data. They show numerous parameters like hospitalized cases, percentage of the population with symptoms, etc. This data doesn't include the statistics for individual tests but has a view of the percentage of the population(corresponding to the survey) who have been tested and are positive/negative, based on antigen tests. However, the point that needs to be taken into account here is that this dashboard is based on survey data and not validated by the government or any other facility. Similarly, governments worldwide have shared public dashboards for effective analysis of the situation. The Canadian government website [57] can be used to analyze the general trend of the COVID-19 situation and shares immense knowledge right awareness resources to symptoms and treatments. Radio Canada [58] is another dashboard that provides a nationwide view of Canada in terms of COVID-19 along with the testing figures. The Government of India's Ministry of Health and Family Welfare's website [59] also provides nationwide data including daily active cases, discharged cases, deaths along with other information. New Zealand's Ministry of Health [60] also displays data related to the confirmed cases, a brief analysis of the current scenario, location wise distribution of cases, the tests data and much more. Etalab [61] offers a consolidated view of the official data provided by Public Health France for understanding the COVID-19 pandemic better in France. Furthermore, numerous dashboards have been developed at the local and district level to monitor community trends.
Global level: Since the epidemic of COVID-19, the John Hopkins dashboard has demonstrated clear and real-time updated information on the coronavirus. This dashboard offers insight into the information from countries accumulated by John Hopkins University including a rational view of the transmission of the disease from each location. This COVID-19 varies from country to country as seen on the dashboard and thus it helps us to compare trends across countries. Instances, deaths, and recoveries are confirmed on the side panel of the John Hopkins dashboard. Automatic virus dashboard updates encourage people to identify the nature of the virus and the spread of COVID-19. The John Hopkins chart provides virus hotspots through red marks around the world on various continents.
The Washington University dashboard gives an outline of COVID-19 infections worldwide by using confirmed cases and remaining infections to assess explanatory variables. Identifying an outbreak like COVID-19 includes the definition of an inflection point and the advantage of the public in comprehending the metamorphosis of the disease outbreak. The seafoam green slash texture on the dashboard enables users to see the movement of the pandemic and to estimate peak and curve levels. This dashboard produces information from diverse sources, including CDC and WHO, in relaying of coronavirus relevant information to the public. The simplified user design of this panel enables users to create a split second about disease outbreak by trying to compare figures. Infection, mortality, and recovery numbers on the dashboard guide users to explore the extent of the disease outbreak and to get a clear understanding of the geographical environment. The Washington University dashboard uses computational resources and technology to enhance the precision of observed trends.
Worldometer [62], another powerful global dashboard goes the extra mile and shows the number of serious/critical cases, total cases per million population, deaths per million population. The dashboard also gives updated country-wise information on total tests conducted and tests per 1 million population. It gives visual representation of newly infected vs newly recovered, death rate vs recovery rate, growth factor of daily new cases and trends based on age and gender. Interestingly, it gives information on fatality rate by age, sex and comorbidity. The dashboard is very exhaustive since it provides various important graphs for all countries. The Worldometer and other dashboards immensely contribute in taking into account and understanding the depth of the COVID-19 pandemic in our area, city, region, country, and even our respective continent Microsoft's Bing team has launched a website to follow the status of coronavirus outbreaks worldwide. The system provided all up-to-date infection statistics for each country. As an interactive map, the tracker allows people to browse the nation to see a particular number of cases and related articles from different publishers. Data are reported to be grouped from sources including WHO, the US Centers for Disease Control and Prevention (CDC), and the European Center for Disease Prevention and Control (ECDC).
The Coronavirus dashboard is developed by Thebaselab, and benefits by bringing a near-real-time, broad perspective of coronavirus. The color red is a tad alarmist, however, the white background is pretty well balanced. Like the JHU dashboard, The Coronavirus developed by Thebaselab [63] summarises known case stats on every country which has been affected so far. Thebaselab also began publishing stories to show how the dashboard tends to work and how coronavirus compares with other major epidemics. BBC is trying to offer a good highlighter on how coronavirus has spread in the last few months. Visuals are static and the BBC also remains an important source of assessing burden of COVID-19 among the most remote and inaccessible areas globally including regions of Iran, South Korea and Italy, Comparable to the BBC, the Gray Lady's very own dashboard is doing what the New York Times does better: it gives the public an easy-to-understand education about what's going on. There are no neat looking graphics or interactive charts, but there is still a useful breaking down of how each major continent has been affected and how they are struggling to contain the virus.
Websites related to COVID-19 testing: Many organizations have come forward to establish an online platform to cater to COVID-19 -19 related help. While plenty of them are involved in contact tracing, there are many other websites focusing on other aspects of COVID-19 services.
These websites provide common people a user-friendly, mobile means to gather the information they are on the lookout for. For example, some websites serve the purpose of self-assessment of COVID-19 for users. Coronavirus Self Checker [64] by CDC on Microsoft Azure platform employs an interactive tool where individuals respond to COVID-19 related questions. Based on the responses, individuals are provided advice on seeking medical care. COVID-19 Screening tool [65] by Apple helps identify COVID-19 related signs and suggests the next steps of action.
Some websites help users find COVID-19 test centers in their locality. Castlight came up with COVID-19 Test Site Finder [66] to provide users with information on test centers in their geographical area of interest. The website also includes a selfassessment provision. A government website [67] offers details about testing sites in each state. Verily has eased the process of booking appointments for COVID-19 tests through its website Project Baseline. [68] Hence, the need for a dashboard which provides granular, structured and well formatted information without overloading the user (public, organisations, governments etc) is increasing exponentially. The dashboards can further help to understand the trends and patterns of the virus and how it has been affecting the various regions and places around the world, at the level of a university, state, nation or even global.
Approaches and Challenges
With the introduction of different solutions digital or otherwise, we are bound to come across multiple challenges that impact both the individual and community as a whole. There are several challenges, regulations, norms and solutions which govern aspects of healthcare data collection, dissemination, analysis and its use in designing interventions or making policies which impact communities. These challenges transcend the means (physical or digital), space (geographical locations) and stakeholders (individuals, companies, governments, etc) by having effects on people who share the data, governments, healthcare providers etc. We set out to look at the different approaches currently in motion for protecting healthcare data and look at the various challenges that arise. Some of the challenges are partially addressed by existing solutions, while a large number of them still need improvisation. We examine scenarios at different socio-political levels where these challenges have unravelled differently, followed by a brief overview of their unintended consequences.
Current Techniques
Privacy Policies
Healthcare data in the US is protected primarily by the provisions of Health Insurance Portability and Accountability Act, 1996 ( "HIPAA" ) which operates on a federal level. Several states and sectors have their own laws which operate in tandem with HIPAA. The strongest of them being the recent California Consumer Protection Act 2018 ( "CCPR").
[69] HIPAA regulations even allow for the Preemption of State Law, wherein if a state law sets more stringent standards of protecting data and its disclosure, then the state law prevails over HIPAA. [70] The Family Educational Rights and Privacy Act of 1974 ( "FERPA" ), The Americans with Disabilities Act of 1990 ( "ADA" ) are some examples of sector specific laws which also operate on a federal level.
The General Data Protection Regulation ( "GDPR" ) by virtue of its extraterritorial effect [71] is also relevant in this scenario since it is applicable to any entity that collects or controls the healthcare data of EU citizens in the US.
Data breach/rogue employee (security protocols)
The pie chart represents the type of breaches occurred in 2019 alone according to the information provided in [72] Improper disposal of PHI 1.2% Theft 10.6% Unauthorised Access 28.8%
Hacking IT/Incidents 59.4%
Types of Healthcare breaches in 2019 Figure 4 Reasons for IT breaches in 2019 according to [72] In the past 10 years there has been an upward trend in the number of breaches which have happened. The breaches in 2019 were approximately twice more than the average number of breaches in the previous 9 years. A major cause of these breaches is owing to unauthorized access of confidential information, information technology incidents like Anthem Inc and data theft incidents, thefts etc. A brief distribution of incidents of data breach based on the type of breach is shown in the above pie chart. This is in accordance with the reports from [72]. The rapid use of digital technology to manage COVID-19 data provides a unique platform for data driven loss of privacy information through multiple channels.
The unauthorized entries from employees could be due to negligence, errors, or malicious intent. Furthermore, often the loss of information happens through stolen physical records or misplaced patient records which contain a lot of PII ("Personally Identifiable Information") and PHI ("Protected Healthcare Information"). The fact that medical data is more life-long and cannot be changed exacerbates the issues associated, unlike in the case of any financial data breaches. This longevity makes it a long-term threat for the person whose data was compromised. Another fact to consider is that the highly interconnected nature of health care devices like -insulin pumps, pacemakers, scanning devices, etc could leave threats open to not just the leakage of information but also on the medical health of people relying on such medical aids. [72] A survey from Accenture [73] in Feb 2017 revealed that 26% of U.S. consumers or more than one in every four Americans are a victim of the healthcare data breaches. Furthermore, a large percentage of the people who took the survey (about 50% of breach victims) suffered from medical identity theft, with an average of $2,500 outof-pocket costs due to the healthcare data breaches. To make matters worse, about 50% of the respondents shared that they were not informed by the company/legal institution about the breach, i.e., they learned of the breach themselves after they'd been alerted to an error on their benefits explanation, credit card statement, or similar documents. [74] These are possible pitfalls that will be accompanied by mismanagement of the large scale data collection happening in light of the COVID-19 pandemic.
Case studies
To inform our analysis we have conducted a few case-studies on the management of COVID-19 tests around the world with a special focus on South Korea, China, India, Vietnam, and Germany. Each of these countries has adopted different approaches to handling the pandemic and the sections below cover the approaches and measures taken by each of them to tackle the virus.
• India: India remains one of the world's hotspots for COVID-19 with rapid rise in infections (more than 9 million) [75] and high recovery rates. It also provides an interesting dynamic of constitutional, intellectual and social challenges. For eg. India is very densely populated constituting several religions, and has an already volatile economy. Beyond this, India is also a big exporter of pharmaceutical and medical equipment. Further, speaking at a press briefing in Geneva, Switzerland on Monday, Mike Ryan, WHO Emergencies Programme Director, said that it was essential for India to introduce ramped up measures at a public health and societal level, to control and suppress the disease. "India is a hugely populous country", he said. "The future of this pandemic will be determined by what happens to densely-populated countries".
[76] India at the moment has no legislation that requires the protection of healthcare data and data protection in general. Within the Constitution however, the right to privacy has been found to be a fundamental right [77] under the Right to Life and Liberty [78], as declared by the Supreme Court. The Information Technology Act, 2000 read with the Information Technology Rules 2011 -providing context of procedures, sensitive personal data and information reasoned with security practices -have been found to be inept. Nevertheless, there are two draft legislation: Personal Data Protection Bill,2019 (pending in Parliament) and Digital Information Security in Health Care Act (put out by the Health Ministry and specifically deals with sharing of healthcare data). These new pieces of legislation hope to bridge the gap between privacy considerations and public health. On account of the pandemic, the Indian Government launched a contact tracing app "Aarogya Setu" to tackle COVID-19. The software application clearly mentions data sharing upfront in a conspicuous place when consent is gained during user registration in the Application. This helps informs the potential users about data sharing through this Application. The details of the personal information gathered, the method in which it is gathered, and by whom as well as the purpose for which it will be used is set out in its Privacy Policy [79]. Besides, the Government has initiated a National Digital Health Mission as a part of which unique digital IDs will be provided as a single source of health information for the patients, government, and healthcare providers through interoperable health records. The NHA (National Health Authority) will play a crucial role in the development of these policies and managing NDHM. However, the lack of a legal framework for data protection in the country has increased the risks related to digitization. Privacy issues with Aarogya setu: 1) App fails to clarify which ministry or department will be accessing data suggesting interdepartmental exchanges of personal information. This makes ensuring accountability challenging. 2) No legal framework to regulate the application forcing users to accept T&C and privacy policy given by the government. 3) Uses location data via GPS trails in addition to Bluetooth, deviating from "privacy-focused global standards" which are confined to Bluetooth-based technology like TraceTogether app(Singapore) and framework suggested by MIT as a tool to monitor movement of patients and contact tracing. 4) The unique digital identity in Aarogya Setu is a static number, which increases the probability of identity breaches. • Vietnam: Vietnam lacks a unified comprehensive data protection legislation.
Instead, it has a very fragmented mechanism for regulating data privacy and required levels of protection through a number of laws like Civil Code, Penal Code, Law on Cyber Information Security, Law on Information Technology, Law on Cybersecurity, and sector-specific laws ("Vietnamese Data Privacy Laws"). These legal instruments, as a general principle, are personal data of individuals. While the laws fail to define the scope of personal data and the definition also varies between sectors, the laws are interpreted to protect, at a minimum, information that would enable the identification of an individual. The primary legislation regulating data protection is Cybersecurity law. It is to be noted that unlike cybersecurity laws in other jurisdictions (inspired by the GDPR of the EU) the Cybersecurity Law of Vietnam shares similarities with China's Cybersecurity Law enacted in 2017. Such legal precedents focus on providing the government with the ability to control the flow of information, instead of enforcing data privacy rights for individual data subjects. [80] In addition, the Law on Information Technology requires that the entity collecting personal data must provide data subjects with information about the form, scope, place, purpose of collecting, processing, and using personal data. Further, the collection, storage, use, and transfer of such data must have the consent of such a person [81]. Vietnam's Law on Information Technology (IT Law) requires organizations and individuals that collect, process and use personal information of other people in the network environment to obtain the consent of those people unless otherwise provided by law.Severe or Emerging Contagious Diseases fall within the definition of "Class A Infectious Diseases" under the Law on Prevention and Control of Infectious Diseases (LPCID) and individuals/ organizations in Vietnam are required to take sanitation, disinfection and sterilization measures according to instructions of competent health agencies. As a general principle, Vietnamese Data Privacy Laws protect information pertaining to or belonging to individuals or (to a lesser degree, organizations) that can serve to personally identify individuals (i.e., personal data). While Vietnamese Data Privacy Laws do not consistently define what information constitutes personal data and the definition also varies between sectors, the laws are interpreted to protect, at a minimum, information that would enable the identification of an individual. Specifically, the Law on Information Technology requires that the entity collecting personal data must provide data subjects with information about the form, scope, place, purpose of collecting, processing, and using personal data. Further, the collection, storage, use, and transfer of such data must have the consent of such a person. In the backdrop of COVID-19 , the Vietnamese gov. launched "BlueZone" (Vietnam's Ministry of Information and Communications and Ministry of Health)
[82]. The following are some of the main privacy protection features of Blue Zones: Data security: The Bluezone app stores data locally and doesn't send any data to the system, No location data collection: The Bluezone app does not collect geo-location data, Anonymity: All Bluezone app users are anonymous to each other. Only competent and verified health authorities know those whose infected and who are suspected of infection due to close contact with COVID-19 cases, Transparency: The Bluezone Project codebase is completely open source. • South Korea: The pandemic response of South Korea has been touted as an example to be followed by other nations. Nevertheless, its excessive testing and tracing strategy has come under the radar of privacy-related scrutiny. The data privacy in South Korea is regulated by three major legislations: Private Information Protection Act (PIPA), the Act on the Use and Protection of Credit Information ('Credit Information Act'), and the Act on the Promotion of Information and Communications Network Utilization, these amendments were to which were made recently. The Personal Information Protection Act (PIPA) implemented in South Korea imposes strict compliance requirements on entities that collect any potential information that can be used for identifying a specific person. Citizens also have the right to be forgotten, among other data ownership right. While organizations in the private and public sectors are legally obliged to comply with the PIPA, government agencies that require personal data for public interest purposes can collect and use data without the need to obtain consent. [83] Although it is expected that organizations will use personal information on justifiable grounds by evaluating the 'reasonable relevance' of personal data and information that they intend to use and maintaining and preserving relevant records. Countries that sought to follow this model can face challenges by their own domestic data privacy laws especially those subjected to GDPR.The mobile application South Korea is using monitors and tracks the location of all visitors to the country, as well. Individuals, including tourists, wear a location-tracking bracelet which is referred to as "smart city tech." The smart city tech platform shares information between cities on various things including traffic and pollution to find vulnerabilities and congestion, which makes it easier to identify hot spots for COVID-19. [84] The PIPA required data to be deleted post its utilization for the purpose it is collected.But the government has admitted that it will be permanently storing data of patients from the previous epidemic. • Germany: The German "Datenschutzkonferenz" which is a collective body composed of independent federal and state data protection authorities notified guidelines regarding COVID-19 and data protection. According to guidelines, sharing personal information of individuals infected with COVID-19 or those suspected of it would be lawful provided revealing their identity is exceptionally crucial for protecting people they were in contact with. Also, in such cases, reference should be taken to article 6(1)9c) or (f) of GDPR.For other instances, health data must be kept confidential and only be used for the purpose it was collected in the first place.Consent of data subject must be taken. To keep track of the number of lab tests regarding SARS-COV-2 carried out per week in Germany and how many among them are positive and negative , the RKI has initated a country-wide laboratory query in Germany. However, the quantity of laboratories reporting data seems to vary week to week. And by subtracting each change (per week) from the cumulative total, we retrospectively work out the cumulative totals at the end of each week. Because labs are able to post-check the tests of previous calendar weeks in the test number query of RKI, the initial figures may be revised upwards to a slight extent in subsequent reports. These sources explicitly state that the figures refer to tests conducted and that this wouldn't eqaute to the number of people tested, because of multiple tests can be done on one person. [85] Germany launched an official coronavirus tracing app in June 2020 which is touted to be so secure that even government ministers can use it, though developers acknowledged few discrepancies. Germany like other EU nations have chosen against centrally storing data through its tracing app. [86] This step was taken in compliance with data privacy standards. [87] Smartphone apps have been claimed as a high-tech tools for tracking down potential COVID-19 infections. Experts state that finding new cases quickly is key to clamping down fresh clusters of cases, especially as countries progressively emerge from lockdowns and try to avoid a second wave of deaths and infections. But governments and legislative authorities in Europe have run into hurdles of legal and cultural restrictions and are trying to reconcile the necessity for effective contact tracing with the continent's strict data privacy and digital security standards. Germany -where a person's right to their own data even after death is rooted in the constitution -has proven to be a particular challenge. Early government suggestions to use cell tower information and GPS coordinates for the app prompted a swift backlash. • China: Several legislations are in shape in the Republic of China for personal data protection such as National security law of the people's republic of china, General provision of the civil law of the people's republic of china, and cybersecurity law which in some way similar to General Data Protection Regulations by European Union. In addition, China has also introduced a draft data protection legislation recently. The CAC (Cyberspace Administration of China) released a notice on the protection of personal information when using big data joint support and defense [88]. The prime objective of this notice was to set some ground rules for the use of personal information of the users to control the spread of COVID-19. Specifically, this notice addresses Limiting Entities: Multiple entities dealing with personal data also raises privacy issues. Only entities authorized by the National Health Commission, in accordance with cybersecurity law, should be allowed to collect personal information for public health purposes, Data collected should have a targeted approach: Personal information which is collected for the purpose of COVID-19 tracing should not be used for any other purpose, Data minimization: Limited amount of data should be collected to prevent any further breach of privacy or misuse of that data collected, and Cybersecurity requirements: A strict and effective implementation of cybersecurity law will help to prevent personal information from being misused. From the above case-studies it is more than evident that we need different strategies to curb this virus across the globe, to ensure the citizens of the respective country stay safe and protected. Also, we can see the different measures taken to address the concerns of data privacy and protection, risk profiling analysis, social and racial discrimiation and economic development. These measures could help inform the decisions of countries or regions in similar situations.
3.2 Data silos: Missed opportunity: Due to anonymization, not possible to conduct detailed analysis Arising from anonymization efforts in medical centers is the existence of data silos, a collection of data that is accessible by only one group or organization. As a solution for data privacy of individuals testing at medical centers, data silos may be effective, yet they bring about additional problems. Data silos prevent a possibility to conduct detailed analysis of COVID-19 testing data, which would be effective in multiple ways, including evaluating the efficacy of testing methods in different demographics. Such analysis is made impossible by data silos' insularity, preventing health officials from seeing the bigger picture. [89][90] A solution, as part of our wishlist, is to create a data lake instead; a centralized data repository optimized for analysis. Data silos are by nature decentralized, which makes it difficult to enforce a uniform protocol for data governance. A data lake fixes this additional problem due to its centralized nature. [91] Rather than achieving anonymization through insularity, a more intelligent approach can be taken by protecting data privacy with fine-grained, data-centric control. Furthermore, one of the primary motives of sharing the data is to learn representations and draw insights from it. Privacy preserving machine learning has been on the rise in recent times. [92] Strategies like federated learning -a machine learning setting in which a centralized model is brought to localized data for training rather than data being brought to the model -can be used to ensure no explicit sharing of raw data. Split learning may also be used for this purpose; split learning is a specialized neural network method in which a designated layer in a neural network, the cut layer, 'splits' the network between client and server. These privacy solutions, among others, support data lakes as the best of both worlds in efficacy and privacy.
Solutions that overcome privacy, mismanagement and miscommunication
Many governments around the world have suggested a scheme of health passports/immunity passports since a clear solution for COVID-19 management including vaccine distribution remains in its early stages and will take months before it reaches the common man globally. It is proposed that the identification of antibodies in COVID-19 virus-causing persons, the SARS-CoV-2 virus, may be used as the basis for health/immunity passports globally [17]. Owing to the transient nature in terms of presence of antibodies, looking at the presence/absence of it alone could lead to issues. It might be a better option to look into vaccine based immunisation as an indicator. Adding on, we believe that a random number index is the best way to connect a user to their data, rather than a name, phone number, or email address. Later, the user will use a random number to validate their test results. The research needs to be done to understand the participant's knowledge about various aspects such as the use of masks, the use of hand sanitizer, avoiding going to school, and college for work can prevent the spread of viruses. There are a lot of upcoming strategies and areas in which scientists and researchers are trying to explore and venture to track and avoid the exponential spread of the virus [93]. These methods help to improve performance, shorten the timeline for obtaining outcomes, reduce costs and reduce the complexity involved in carrying out these studies. Some of the prominent references to current and upcoming test activities are STOPCovid [94] an initiative by MIT researchers and scientists to provide test facilities at a cheaper cost to non-laboratory experts to perform experiments in their home or private premises. Next, leveraging AI for testing, where simpler forms could be used to detect COVID, such as cough [95], breadth, and saliva. The use of these facilities will not only make the process simpler and easier to carry out, but will also allow the authorities to expand the reach of the testing and in turn, help control the available capital and supplies, but will also extend the testing to the community level, including mass population testing.
A few of the most important points to be addressed while developing the solutions are as follows:
1 Consent of the individual: It should be a practice to always obtain consent of the person whose data is being collected. Also, it is the right of the individual to know about the reason and time period for the data to be utilised. 2 Avoid collecting unnecessary data: It is also suggested to avoid collecting irrelevant information while carrying out the tests such as getting information about their location, previous health history 3 Authorized use of data by health officials: The data collected should be used by the public health authorities on a need base purpose only and who are trained for the same. 4 Minimize sharing: One should ensure sharing of data is minimized and is strictly limited by drafting appropriate laws and policies. 5 Decentralization: The data should be stored in a decentralised fashion using appropriate safeguarding techniques such as encryption, decentralization or de-identification to avoid or prevent any attempts to hack the data. 6 Training of the staff : The staff of the organization should be trained to take the precautionary measures for containing the virus such as checking the temperature of the employees, check if they have worn masks and are maintaining a social distance, maintaining the hygiene of the food and utensils being used.
Conclusion
The explosive growth of diagnostic technology is a necessary component of the Strategic Emergency Preparedness Plan for Epidemic. Subsequently, the technological landscape of the development of the diagnosis of COVID-19 is rapidly coming up with new knowledge that is formed on a daily basis. Numerous platforms for Open and rapid sharing of data has contributed to this rapid diagnostic development. In this early draft we have introduced the digital landscape of the testing workflows, the user centric issues and the challenges and consequences associated with each of these. Further we have enumerated multiple major dashboards contributing to the surveillance of COVID-19 and case studies pertinent to the digital management of resources from different countries. We have mapped the landscape of current testing systems, challenges and potential solutions.
Figure 2 Figure 3
23User journey table 2.1.3 Test site logistics (LIMS systems, Lab Info Management Systems) (Quest, LabCorp, Verily, etc) CDC recommended COVID-19 Flow Screening Process [10]
2 Estonia digital immunity passport: In Estonia [30], a digital immunity passport developed by the "MTÜ Back to Work" non-profit formed by tech companies such as Transferwise, Guardtime or and Bolt. Similarly, to a Chilean solution, users share the information about their test results and medical certificates via a QR-code generated after digital authentication. 3 Health Passport Ireland: Ireland began a digital health passport pilot in August [31]. It is based on privacy-preserving technologies, with only an authorized medical administrator allowed to create Health Passport Ireland accounts and make updates of the test results. The app displays green status if the test result was negative, red if positive and amber if a new test is required [21]. The color-code certification system is somewhat similar to the apps that have been introduced in China [22] and the UAE's Alshosn app [32]. 4 COVID-19 Vaccination Certificate pilot: Most recently, WHO established an international partnership between Estonia on WHO Digital COVID-19 Vaccination Infrastructure [33]
United Airlines as well as numerous sporting facilities, includes Health Pass [38] as a free feature. It allows for real-time screening for possible symptoms via surveys on the app and temperature-checks at kiosks installed at the venues and generates a QR-code with a verified identity and health-check status. In July, the Tourism Data Driven Solutions (TDDS), a Spanish soft-are piloting a program called Com-
monPass [36], a privacy-by-design, globally interoperable platform developed
by The Commons Project in collaboration with The World Economic Forum
and other public and private partners. It allows individuals to access via app
their test results. In the future, they also will be able to access their vac-
cination records, available from trusted sources [37] and give consent to use
those records to validate their COVID-19 status. ClearMe, a biometric iden-
tification solution developed by Secure Identity LLC, currently used by Delta
and ware company, developed Hi+ Card [39], a mobile app supported by the UN
World Tourism Organization (UNWTO), that is meant to serve as an interna-
tional health identification card. The solution uses blockchain technology and
encryption and is General Data Protection Regulation (GDPR) compliant.
Some other leading names when it comes to testing are Verily, BioReference Laboratories, and Mayoclinic to name a few. Verily[46] has introduced Project Baseline which is designed to support people right from screening through testing at respective community-based testing sites and receipt of individual test results. As of September 2020, 6L people have been tested through the Baseline COVID-19 program. BioReference Laboratories [47] have played a key role in the crisis as well as provided support to various patients as well as government agencies. Mayoclinic[48] which is an American non-profit organization focused on patient care, research and education has also played a significant role in terms of testing. It is crucial to understand about the various individuals who are involved in the testing workflow and how their contribution is vital to the success of preventing the spread of the virus and carrying out tests effectively and optimally.The COVID-19 diagnostic testing is carried out to check if an individual is infected
with the SARS-CoV-2 (the virus that causes COVID-19), a detailed overview about
the clinical landscape of testing has been covered by Gandhi et. al [40]. Mainly two
tests were aproved by The US Food and Drug Administration (FDA) for detecting
of COVID-19, which is the PCR test and the Rapid Antigen test. All the test-
ing laboratories are CLIA certified. CLIA, The Clinical Laboratory Improvement
Amendments are United states federal regulations and regulatory standards that
are applicable to every clinical laboratory testing, except clinical trials and basic
research which are carried out on humans in the United States of America.[41] Test-
ing is of utmost importance in current times and some of the leading players when it
comes to testing include Quest Diagnostics [42] which handles approximately 20% of
the testing efforts across the USA with over 40 years of proven effectiveness and ex-
perience in handling infectious disease testing, Medpace Central Labs [43] who have
been committed to assisting in fighting against the COVID-19 pandemic, Icon [44]
who offer a wide array of services right from testing to maintaining a coronavirus
observatory which is a powerful dashboard which provides updates regarding the
vaccine trails, regulatory updates, diagnostics and much more related to COVID-
19 via Twitter, Syneos Health, [45] the only fully integrated biopharmaceutical
solutions organizations have been taking extensive measures in helping battle the
pandemic. 2.2.2 Dashboard tools and software for public health, govt, and CDC/WHO
Dashboards are the landing pages for interactive maps and visuals showing where
the virus has spread. It gives a breakdown of which nations, states, cities, and
neighboring areas have positive cases and could predict which regions are likely to
see new outbreaks. This helps in predicting future trends and taking action such as
putting area restrictions or a complete lockdown. Data dashboards provide latest
figures on the number of positive cases (mild, moderate and severe), recoveries,
deaths, infection rate, and cases per million of the population in that area.
cases in the previous day or month to the number of recovered patients and much more. At the University level, the dashboard visualized by Amherst College [49] has made much noise by being the only college to receive an A++ rating by We Rate COVID-19 Dashboards [50] which is a panel of academic and medical experts who rate dashboards created by universities and colleges across the country. The top 5 universities who have excelled at providing data of the vicinity via dashboards according to the ratings are Amherst College [49], Wagner College [51], Tulane University [52], Ohio State University [53], and George Mason University
AcknowledgementsWe are grateful to Riyanka Roy Choudhury, CodeX Fellow, Stanford University, Adam Berrey, CEO of PathCheck Foundation, Dr. Brooke Struck, Research Director at The Decision Lab, Canada, Vinay Gidwaney, Entrepreneur and Advisor, PathCheck Foundation, and Paola Heudebert, co-founder of Blockchain for Human Rights, Alison Tinker, Saswati Soumya, Sunny Manduva, Bhavya Pandey, and Aarathi Prasad for their assistance in discussions, support and guidance in writing of this paper.Competing interestsThe authors declare that they have no competing interests.
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|
[] |
[
"Random Distances Associated with Trapezoids",
"Random Distances Associated with Trapezoids"
] |
[
"Maryam Ahmadi \nUniversity of Victoria\nBCCanada\n",
"Jianping Pan \nUniversity of Victoria\nBCCanada\n"
] |
[
"University of Victoria\nBCCanada",
"University of Victoria\nBCCanada"
] |
[] |
The distributions of the random distances associated with hexagons, rhombuses and triangles have been derived and verified in the existing work. All of these geometric shapes are related to each other and have various applications in wireless communications, transportation, etc. Hexagons are widely used to model the cells in cellular networks, while trapezoids can be utilized to model the edge users in a cellular network with a hexagonal tessellation. In this report, the distributions of the random distances associated with unit trapezoids are derived, when two random points are within a unit trapezoid or in two neighbor unit trapezoids. The mathematical expressions are verified through simulation. Further, we present the polynomial fit for the PDFs, which can be used to simplify the computation.
| null |
[
"https://arxiv.org/pdf/1307.1444v1.pdf"
] | 119,622,871 |
1307.1444
|
e5541abd1a44900e361d81da688ceb346b295c2e
|
Random Distances Associated with Trapezoids
Maryam Ahmadi
University of Victoria
BCCanada
Jianping Pan
University of Victoria
BCCanada
Random Distances Associated with Trapezoids
1Index Terms Random distancesdistance distribution functionstrapezoids
The distributions of the random distances associated with hexagons, rhombuses and triangles have been derived and verified in the existing work. All of these geometric shapes are related to each other and have various applications in wireless communications, transportation, etc. Hexagons are widely used to model the cells in cellular networks, while trapezoids can be utilized to model the edge users in a cellular network with a hexagonal tessellation. In this report, the distributions of the random distances associated with unit trapezoids are derived, when two random points are within a unit trapezoid or in two neighbor unit trapezoids. The mathematical expressions are verified through simulation. Further, we present the polynomial fit for the PDFs, which can be used to simplify the computation.
I. THE PROBLEM Denote a "unit trapezoid" as an isosceles trapezoid where the legs have the same length and equal to the length of the short base, 1, and the base angles are equal to π 3 . The length of the long base is 2. The distributions of the random distances between two points located within a unit trapezoid and in two neighbor unit trapezoids are of our interest. Several cases are considered depending on the arrangement of the trapezoids. As illustrated in Fig. 1, points A and B constitute the case where the two random points are located within a unit trapezoid, while CD, EF, and GH represent the cases where the two random points are located in two neighbor unit trapezoids. For CD, two neighbor unit trapezoids form a "unit hexagon" with side length 1. Two neighbor trapezoids share a leg for EF. In the last case, GH, two trapezoids share the short base, forming a concave polygon.
We derive the probability density functions (PDFs) and cumulative distribution functions (CDFs) of the 4 cases in the following section.
II. DISTANCE DISTRIBUTIONS ASSOCIATED WITH UNIT TRAPEZOIDS
A. PDF and CDF of the Random Distances within a Unit Trapezoid (Case |AB|)
A unit trapezoid in fact consists of three adjacent equilateral triangles. Since the distributions of the random distances within and between triangles are known [1], the PDF of the distances between two uniformly at random located points within a unit trapezoid can be derived using 3 the probabilistic sum, as follows.
f D AB (d) = 2d √ d 2 − 3 − 32 27 d − 32π 27 √ 3 √ 3 ≤ d ≤ 2 0 otherwise .(1)
The corresponding CDF is
F D AB (d) = 0 d < 0 4
B. PDF and CDF of the Distances between Two Uniformly at Random Located Points in
Neighbor Unit Trapezoids (Case |CD|)
In this case, the two neighbor trapezoids form a hexagon. Since the distributions of the random distances within a unit hexagon have been derived and verified in [2], the PDF of the distances between two uniformly at random located nodes in two neighbor unit trapezoids can be derived
as f D CD (d) = 2d − 4 27 1 + 5π 3 √ 3 d 2 + 32 27 d 0 ≤ d ≤ √ 3 2 − 1 3 √ 3 16 9 d 2 + 8 sin −1 √ 3 2d + 4 27 π 3 √ 3 − 1 d 2 − 28 27 √ 4d 2 − 3 + 32 27 d + 4π 3 √ 3 √ 3 2 ≤ d ≤ 1 − 4 9 √ 3 8 3 d 2 + 10 sin −1 √ 3 2d + 4 27 5π 3 √ 3 + 1 d 2 − 16 9 √ 4d 2 − 3 + 32 27 d + 52π 27 √ 3 + 4 9 1 ≤ d ≤ √ 3 8 9 √ 3 1 3 d 2 + 8 sin −1 √ 3 d − 8 27 π 3 √ 3 + 2 d 2 + 8 3 √ d 2 − 3 + 32 27 d − 64π 27 √ 3 − 8 3 √ 3 ≤ d ≤ 2 0 otherwise . (3) 5
The corresponding CDF is
F D CD (d) = 0 d < 0 − 2 27 5π 3 √ 3 + 1 d 4 + 64 81 d 3 0 ≤ d ≤ √ 3 2 − 8 27 √ 3 (d 2 + 9) d 2 sin −1 √ 3 2d + 2 27 π 3 √ 3 − 1 d 4 + 64 81 d 3 + 4π 3 √ 3 d 2 − 58d 2 +15 81 √ 4d 2 − 3 √ 3 2 ≤ d ≤ 1 − 8 27 √ 3 (2d 2 + 15)d 2 sin −1 √ 3 2d + 2 27 5π 3 √ 3 + 1 d 4 + 64 81 d 3 + 52π 27 √ 3 + 4 9 d 2 − 100d 2 +24 81 √ 4d 2 − 3 + 0.0359 1 ≤ d ≤ √ 3 4 27 √ 3 (d 2 + 48)d 2 sin −1 √ 3 d − 4 27 π 3 √ 3 + 2 d 4 + 64 81 d 3 − 64π 27 √ 3 + 8 3 d 2 + 148d 2 +168 81 √ d 2 − 3 + 0.7026 √ 3 ≤ d ≤ 2 1 d > 2 .
(4)
C. PDF and CDF of the Distances between Two Uniformly at Random Located Points in
Neighbor Unit Trapezoids (Case |EF |)
Knowing the PDF of the random distances between rhombuses [3] and within a unit trapezoid, the PDF of the random distances between two uniformly distributed points, one in each of the neighbor unit trapezoids in this case, is
6 f D EF (d) = 2d − 4 27 2π 3 √ 3 + 1 d 2 + 16 27 d 0 ≤ d ≤ √ 3 2 − 8 9 √ 3 2 3 d 2 + 1 sin −1 √ 3 2d + 4 27 4π 3 √ 3 − 1 d 2 + 16 27 d + 4π 9 √ 3 − 4 9 √ 4d 2 − 3 √ 3 2 ≤ d ≤ 1 8 9 √ 3 1 3 d 2 + 1 sin −1 √ 3 2d − 4 27 2π 3 √ 3 + 1 d 2 + 10 27 √ 4d 2 − 3 − 4π 27 √ 3 − 2 9 1 ≤ d ≤ √ 3 8 9 √ 3 1 3 d 2 − 2 sin −1 √ 3 2d − 8 9 √ 3 1 3 d 2 + 2 sin −1 √ 3 d + 4 27 π 3 √ 3 + 2 d 2 − 8 9 √ d 2 − 3 − 14 27 √ 4d 2 − 3 + 32π 27 √ 3 + 10 9 √ 3 ≤ d ≤ 2 8 9 √ 3 1 3 d 2 − 2 sin −1 √ 3 2d + 16 9 √ 3 sin −1 √ 3 d − 4 27 π 3 √ 3 − 1 d 2 − 14 27 √ 4d 2 − 3 + 16 27 √ d 2 − 3 + 2 9 2 ≤ d ≤ √ 7 8 9 √ 3 − 1 3 d 2 + 4 sin −1 √ 3 d + 4 27 π 3 √ 3 − 1 d 2 + 8 9 √ d 2 − 3 − 16π 27 √ 3 − 8 9 √ 7 ≤ d ≤ 2 √ 3 0 otherwise . (5) 7
The corresponding CDF is
F D EF (d) = 0 d < 0 − 2 27 2π 3 √ 3 + 1 d 4 + 32 81 d 3 0 ≤ d ≤ √ 3 2 − 8 27 √ 3 (d 2 + 3) d 2 sin −1 √ 3 2d + 2 27 4π 3 √ 3 − 1 d 4 + 32 81 d 3 + 4π 9 √ 3 d 2 − 26d 2 +3 81 √ 4d 2 − 3 √ 3 2 ≤ d ≤ 1 4 27 √ 3 (d 2 + 6)d 2 sin −1 √ 3 2d − 2 27 2π 3 √ 3 + 1 d 4 − 4π 27 √ 3 + 2 9 d 2 + 42d 2 +9 162 √ 4d 2 − 3 − 0.0561 1 ≤ d ≤ √ 3 4 27 √ 3 d 2 (d 2 − 12) sin −1 √ 3 2d − (d 2 + 12) sin −1 √ 3 d + 2 27 π 3 √ 3 + 2 d 4 + 2 9 16π 3 √ 3 + 5 d 2 − 54d 2 +27 162 √ 4d 2 − 3 − 52d 2 +24 81 √ d 2 − 3 − 0.0561 √ 3 ≤ d ≤ 2 4 27 √ 3 (d 2 − 12)d 2 sin −1 √ 3 2d + 16 9 √ 3 d 2 sin −1 √ 3 d − 2 27 π 3 √ 3 − 1 d 4 + 2 9 d 2 − 54d 2 +27 162 √ 4d 2 − 3 + 32d 2 +48 81 √ d 2 − 3 − 0.3528 2 ≤ d ≤ √ 7 − 4 27 √ 3 (d 2 − 24)d 2 sin −1 √ 3 d + 2 27 π 3 √ 3 − 1 d 4 − 8 9 2π 3 √ 3 + 1 d 2 + 44d 2 +120 81 √ d 2 − 3 − 1.6677 √ 7 ≤ d ≤ 2 √ 3 1 d > 2 √ 3 . (6) 8
D. PDF and CDF of the Distances between Two Uniformly at Random Located Points in Neighbor Unit Trapezoids (Case |GH|)
The PDF of the random distances between two nodes uniformly deployed in two neighbor unit trapezoids (case GH), is
f D GH (d) = 2d 4 27 π 3 √ 3 − 1 d 2 + 16 27 d 0 ≤ d ≤ √ 3 2 8 9 √ 3 2 3 d 2 − 1 sin −1 √ 3 2d − 4 27 5π 3 √ 3 + 1 d 2 − 4 27 √ 4d 2 − 3 + 16 27 d + 4π 9 √ 3 √ 3 2 ≤ d ≤ 1 8 3 √ 3 1 9 d 2 − 1 sin −1 √ 3 2d − 4 9 π 3 √ 3 − 1 d 2 − 22 27 √ 4d 2 − 3 + 28π 27 √ 3 + 2 3 1 ≤ d ≤ √ 3 − 8 9 √ 3 1 3 d 2 − 2 sin −1 √ 3 2d − 8 9 √ 3 1 3 d 2 − 2 sin −1 √ 3 d + 8 27 π 3 √ 3 − 1 d 2 + 14 27 √ 4d 2 − 3 + 8 27 √ d 2 − 3 − 16π 27 √ 3 − 10 9 √ 3 ≤ d ≤ √ 7 0 otherwise . (7) 9
The corresponding CDF is
F D GH (d) = 0 d < 0 2 27 π 3 √ 3 − 1 d 4 + 32 81 d 3 0 ≤ d ≤ √ 3 2 8 27 √ 3 (d 2 − 3) d 2 sin −1 √ 3 2d − 2 27 5π 3 √ 3 + 1 d 4 + 32 81 d 3 + 4π 9 √ 3 d 2 − 2d 2 +3 27 √ 4d 2 − 3 √ 3 2 ≤ d ≤ 1 4 27 √ 3 (d 2 − 18)d 2 sin −1 √ 3 2d − 2 9 π 3 √ 3 − 1 d 4 + 2 3 14π 9 √ 3 + 1 d 2 − 86d 2 +39 162 √ 4d 2 − 3 + 0.0176 1 ≤ d ≤ √ 3 − 4 27 √ 3 (d 2 − 12)d 2 sin −1 √ 3 2d + sin −1 √ 3 d + 4 27 π 3 √ 3 − 1 d 4 − 2 9 8π 3 √ 3 + 5 d 2 + 54d 2 +27 162 √ 4d 2 − 3 + 12d 2 +72 81 √ d 2 − 3 − 0.3157 √ 3 ≤ d ≤ √ 7 1 d > √ 7 .(8)
The results presented above are for the "unit" trapezoids. However, the results can be easily extended for scaled trapezoids. Assume each side of the unit trapezoid is scaled by a non-negative parameter s, then,
F sD (d) = P (sD ≤ d) = P (D ≤ d s ) = F D ( d s ).(9)
Therefore,
f sD (d) = F D ( d s ) = 1 s f D ( d s ).(10)
III. VERIFICATION Figure 2 demonstrates the PDF of the random distances associated with trapezoids, including the random distances within a unit trapezoid and between neighbor unit trapezoids. Figure 3 shows the comparison between the simulation and analytical results of the CDFs.
The simulation is done in Matlab, generating 10, 000 pairs of random points. As the figure suggests, there is a close match between the mathematical and simulation results, which verifies the accuracy of our analytical results. Table I. In Fig. 4, the approximated polynomial functions are shown with dots over the original PDF curves. One can observe that the polynomial functions
Fig. 1 :
1Random Distances Associated with Trapezoids.
Figure 4
4shows the PDFs of the random distances within a unit trapezoid as well as between two neighbor unit trapezoids, along with the fitted polynomial functions. The polynomial coefficients for the PDFs of the random distances associated with trapezoids, along with their norm of residual, are presented in
report, the closed-form expressions for the random distances within a unit trapezoid and between two neighbor unit trapezoids are given. The analytical results are verified through simulation. In addition, the polynomial fits for the PDFs are obtained. These polynomials offer a good fit and can be used instead of the original PDFs to reduce the computational complexity. ACKNOWLEDGMENT This work is supported in part by the NSERC, CFI and BCKDF. The authors would like to thank Lei Zhang, Tianming Wei and Fei Tong for their help.
Fig. 4 :
4Polynomial Fits.
TABLE I :
ICoefficients of the Polynomial Fit and the Norm of Residuals (NormR) PDF Polynomial Coefficients NormR [−6.3419 72.4436 − 356.68 988.25 − 1686.9 f D AB (d) 1824.8 − 1239.4 504.30 − 109.17 9.6850 − 5.1833 0.1537 4.7245 0.0021] [10.55774 − 124.22 632.96 − 1829.6 3302.5 f D CD (d) .6286 − 16.1105 1.9174 × 10 3 − 1.4062 × 10 3 f D GH (d) 7.1067 × 10 3 − 2.6205 × 10 4 7.2760 × 10 4 − 1.5469 × 10 5 0.1622 2.5338 × 10 5 − 3.1899 × 10 5 3.0528 × 10 5 − 2.1736e × 10 5 1.1081 × 10 5 − 3.7549 × 10 4 6.9028 × 10 3 41.6579 −3.2855 × 10 2 69.7237 − 4.9233 0.2052 − 0.0016] demonstrate a good match with the original PDFs. The polynomial functions are accurate and can be used instead of the original PDFs to reduce the computational complexity.−3864.6 2953.0 − 1452.6 443.86 − 80.1282 9.6421
0.1692
−0.27620.0024]
[0.0196 − 0.3853 3.2716 − 15.6480 46.2235
f D EF (d)
−87.0513104.33e − 77.147832.4560 − 6.2059
0.2405
0.6906 − 0.0154]
[0
Y Zhuang, J Pan, arXiv:1207.1511Random Distances Associated with Equilateral Triangles. Y. Zhuang and J. Pan, "Random Distances Associated with Equilateral Triangles," arXiv:1207.1511, 2012.
Random Distances Associated with Hexagons. Y Zhuang, J Pan, aarXiv:1106.2200Y. Zhuang and J. Pan, "Random Distances Associated with Hexagons," aarXiv:1106.2200, 2011.
Y Zhuang, J Pan, arXiv:1106.1257Random Distances Associated with Rhombuses. Y. Zhuang and J. Pan, "Random Distances Associated with Rhombuses," arXiv:1106.1257, 2011.
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[
"ON COVERING AND QUASI-UNSPLIT FAMILIES OF RATIONAL CURVES",
"ON COVERING AND QUASI-UNSPLIT FAMILIES OF RATIONAL CURVES"
] |
[
"Laurent Bonavero ",
"Cinzia Casagrande ",
"Stéphane Druel "
] |
[] |
[] |
We study extremality properties of covering families of rational curves on projective varieties. Among others, we show that on a normal and Q-factorial projective variety X with dim(X) ≤ 4, every covering and quasi-unsplit family of rational curves generates a geometric extremal ray of the Mori cone NE(X) of classes of effective 1-cycles.Question. Let V be a covering and quasi-unsplit family of rational 1-cycles on X. Is R ≥0 [V ] a geometric extremal ray of NE(X)?Note that this question is natural, since any family of rational 1-cycles such that the general member generates a geometric extremal ray of NE(X) is quasiunsplit. The converse is not true if the family is not covering (just think of a smooth blow-down of a smooth projective variety to a non projective one).Let V be any covering family of rational 1-cycles on X. Then V defines settheoretically an equivalence relation on X: two points x, x ′ are V -equivalent if there exist v 1 , . . . , v m ∈ V such that some connected component of C v 1 ∪ · · · ∪ C vm contains x and x ′ , where C v ⊂ X is the curve corresponding to v ∈ V .Date: 1st April 2005.
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"https://arxiv.org/pdf/math/0504056v1.pdf"
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math/0504056
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f7c484fee9520dd728208c78893ba2340eb46a47
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ON COVERING AND QUASI-UNSPLIT FAMILIES OF RATIONAL CURVES
4 Apr 2005
Laurent Bonavero
Cinzia Casagrande
Stéphane Druel
ON COVERING AND QUASI-UNSPLIT FAMILIES OF RATIONAL CURVES
4 Apr 2005arXiv:math/0504056v1 [math.AG]
We study extremality properties of covering families of rational curves on projective varieties. Among others, we show that on a normal and Q-factorial projective variety X with dim(X) ≤ 4, every covering and quasi-unsplit family of rational curves generates a geometric extremal ray of the Mori cone NE(X) of classes of effective 1-cycles.Question. Let V be a covering and quasi-unsplit family of rational 1-cycles on X. Is R ≥0 [V ] a geometric extremal ray of NE(X)?Note that this question is natural, since any family of rational 1-cycles such that the general member generates a geometric extremal ray of NE(X) is quasiunsplit. The converse is not true if the family is not covering (just think of a smooth blow-down of a smooth projective variety to a non projective one).Let V be any covering family of rational 1-cycles on X. Then V defines settheoretically an equivalence relation on X: two points x, x ′ are V -equivalent if there exist v 1 , . . . , v m ∈ V such that some connected component of C v 1 ∪ · · · ∪ C vm contains x and x ′ , where C v ⊂ X is the curve corresponding to v ∈ V .Date: 1st April 2005.
Introduction
Let X be a normal and uniruled complex projective variety. Consider an irreducible and closed subset V of Chow(X) such that:
• any element of V is a cycle whose irreducible components are rational curves; • V is covering (which means that for any point x ∈ X, there exists an element of V passing through x).
We call such a V a covering family of rational 1-cycles on X. If moreover, all irreducible components of the cycles parametrized by V are numerically proportional, we call V a covering and quasi-unsplit family of rational 1-cycles on X (see [CO04,Definition 2.13]). For any covering family V of rational 1-cycles on X, we will denote by [V ] the numerical class in NE(X) of the general cycle of the family V and by R ≥0 [V ] the half-line generated by [V ].
A geometric extremal ray of the Mori cone NE(X) is a half-line R ⊆ NE(X) such that if γ 1 + γ 2 ∈ R for some γ 1 , γ 2 ∈ NE(X), then γ 1 , γ 2 ∈ R (see Section 2 for precise definitions and notation).
In this situation, after Campana's results (see Section 2), there exists an almost holomorphic map q : X Y , to a projective algebraic variety, whose general fibers are V -equivalence classes.
We first prove the following result, which involves the dimension of the general fiber of q.
Theorem 1. Let X be a normal and Q-factorial complex projective variety of dimension n. Let V be a covering and quasi-unsplit family of rational 1-cycles on X, and let f V be the dimension of a general V -equivalence class.
If f V ≥ n−3, then R ≥0 [V ] is a geometric extremal ray of the Mori cone NE(X).
We then immediately get the following.
Corollary 1. Let X be a normal and Q-factorial complex projective variety of dimension n ≤ 4. Let V be a covering and quasi-unsplit family of rational 1-cycles on X. Then R ≥0 [V ] is a geometric extremal ray of the Mori cone NE(X).
As previously recalled, one can associate a rational map q : X Y to any covering family of rational 1-cycles on X. We call a geometric quotient for V a morphism q ′ : X → Y ′ , onto a normal projective variety Y ′ , such that every fiber of q ′ is a V -equivalence class. If such a quotient exists, then it is clearly unique up to isomorphism. On the other hand, even if X is smooth, a geometric quotient for V does not necessarily exist (see example 1).
The study of the extremal contraction given by the previous result leads to the following.
Theorem 2. Let X be a normal and Q-factorial complex projective variety, having canonical singularities, of dimension n. Let V be a covering and quasi-unsplit family of rational 1-cycles on X, and let f V be the dimension of a general Vequivalence class.
If
f V ≥ n − 3, then the Mori contraction of R ≥0 [V ], cont [V ] : X → Y ′ , is the geometric quotient for V . If moreover f V ≥ n − 2, then cont [V ] is equidimensional.
We finally consider the toric case, where we can prove both extremality and existence of the geometric quotient for a quasi-unsplit family in any dimension.
Theorem 3. Let X be a toric and Q-factorial complex projective variety, and let V be a quasi-unsplit covering family of rational 1-cycles in X. Then R ≥0 [V ] is a geometric extremal ray of the Mori cone NE(X), and the Mori contraction of
R ≥0 [V ], cont [V ] : X → Y ′ , is the geometric quotient for V .
The following is an immediate application of Theorems 1 and 3.
Corollary 2. Let X ⊂ P N be a normal and Q-factorial variety, covered by lines. Assume either that X is toric, or that X has canonical singularities and dim X ≤ 4. Let V be an irreducible family of lines covering X.
Then there exists a morphism q ′ : X → Y ′ , onto a normal, Q-factorial, projective variety Y ′ with ρ Y ′ = ρ X − 1, 2 such that all lines of V are contracted by q ′ .
Set-up on families of rational 1-cycles
Let X be a normal, irreducible, n-dimensional complex projective variety. We denote by N 1 (X) R (respectively, N 1 (X) Q ) the vector space of 1-cycles in X with real (respectively, rational) coefficients, modulo numerical equivalence. In N 1 (X) R , let NE(X) be the closure of the cone generated by classes of effective 1-cycles in X.
Recall that the existence of a covering family V of rational 1-cycles on X is equivalent to X being uniruled [Kol96, Proposition IV.1.3].
For such family V , we have a diagram given by the incidence variety C associated to V :
C π F / / X V (1)
where π and F are proper and surjective. We set
C v := F (π −1 (v)) for any v ∈ V .
The relation of V -equivalence on X induced by such a family was introduced and studied in [Cam81]; we refer the reader to [Cam04], [Deb01,§5.4] or [Kol96, §IV.4] for more details. In particular, there exists a rational map q : X Y associated to V , whose main properties are recalled now. By [Deb01, Theorem 5.9], there exists a closed and irreducible subset of Chow(X) whose normalization Y satisfies the following properties: (a) let Z ⊂ Y × X be the restriction of the universal family,
Z p e / / X q~~~~Ỹ (2)
then e is birational and q = p • e −1 is almost holomorphic (which means that the indeterminacy locus of q does not dominate Y ); (b) a general fiber of q is a V -equivalence class, (c) a general fiber of q, hence of p, is irreducible.
As a consequence of the existence of this map q, a general V -equivalence class is a closed subset of X. We denote by f V its dimension, so that dim Y = n − f V . Moreover, it is well known that any V -equivalence class is a countable union of closed subsets of X.
Definition 1. We say that a subset Z of X is V -rationally connected if every connected component of Z is contained in some V -equivalence class.
Lemma 1. Let X be a normal projective variety and V be a covering family of rational 1-cycles on X. Consider the diagram (2) above. Then e(p −1 (y)) is V -rationally connected for any y ∈ Y .
Proof. Let R ⊂ X × X the graph of the equivalence relation defined by V : it is a countable union of closed subvarieties since V is proper. The fiber product Z × Y Z is irreducible and thus (e × e)(Z × Y Z) ⊂ R thanks to properties (a) and (b) above. Therefore, for any x ∈ e(p −1 (y)), the cycle e(p −1 (y)) is contained in the V -equivalence class of x.
The following well known remark will be of constant use (see [Kol96,Proposition IV.3.13.3], or [ACO04, Corollary 4.2]).
Remark 1. If Z ⊂ X is V -rationally connected, every curve contained in Z is numerically equivalent in X to a linear combination with rational coefficients of irreducible components of cycles in V . In particular, if V is quasi-unsplit, the numerical class of every curve contained in a V -rationally connected subset Z of X belongs to R ≥0 [V ].
Finally, we will need the following.
Lemma 2. Let X be a normal projective variety and V be a covering and quasiunsplit family of rational 1-cycles on X. Then there exists a covering and quasiunsplit family V ′ of rational 1-cycles on X such that:
• the general cycle of V ′ is reduced and irreducible; • for any v ′ ∈ V ′ there exists v ∈ V such that C v ′ ⊆ C v ; in particular R ≥0 [V ] = R ≥0 [V ′ ].
Proof. Let C be the incidence variety associated to V as in (1). It is well-known that every irreducible component of C dominates V , let C ′′ be an irreducible component of C which dominates X too. Let C ′ be the normalization of C ′′ and C ′ → V ′ be the Stein factorization of the composite map C ′ → C ′′ → V . Since C ′ → V ′ has connected fibers and C ′ is normal, the general fiber of
C ′ → V ′ is irreducible. Moreover, the image in X of every fiber of C ′ → V ′ is contained in a cycle of V .
Since V ′ is normal, there is a holomorphic map V ′ → Chow(X). Then after replacing V ′ by its image in Chow(X) and C ′ by its image in Chow(X) × X, we get the desired family.
Properties of the base locus and extremality
Let V be a covering family of rational 1-cycles on X, and recall the diagram (2) associated to V .
Let E ⊂ Z be the exceptional locus of e, and B := e(E) ⊂ X. Observe that since X is normal, dim B ≤ n − 2.
Proposition 1. Let X be a normal and Q-factorial projective variety, and V be a covering and quasi-unsplit family of rational 1-cycles on X. Consider the associated diagram as in (2). Then:
(i) e(p −1 (y)) is a V -equivalence class of dimension f V for every y ∈ Y \ p(E); (ii) B is the union of all V -equivalence classes of dimension bigger than f V . Proof. Set X 0 := X \ B and Y 0 := Y \ p(E) = q(X 0 ).= q −1 (H ∩ Y 0 ), which is a Weil divisor in X. Since X is Q-factorial, some multiple of H defines a line bundle L on X.
Let now N := h 0 (L), and let s 1 , . . . , s N be general global sections generating L. For each i = 1, . . . , N , let H i ∈ |L| be the divisor of zeros of s i and H i in X as defined above.
Let's show that
H 1 ∩ · · · ∩ H N = B. If x ∈ B, then q is defined in x and there is some i 0 ∈ {1, . . . , N } such that q(x) ∈ H i 0 , so x ∈ H i 0 . Conversely, let x ∈ B and fix i ∈ {1, . . . , N }. Then e −1 (x) has positive dimension; let C ⊂ Z be an irreducible curve such that e(C) = x. Then p(C) is a curve in Y , hence H i ∩ p(C) = ∅ and p −1 (H i ) ∩ C = ∅. Now observe that p −1 (H i ) does not contain any component of E, hence e(p −1 (H i )) is a divisor in X which coincides with H i over X \ B. Then H i = e(p −1 (H i )) and x ∈ H i . Let i ∈ {1, . . . , N }. Observe that H i · [V ] = 0, because [V ]
is quasi-unsplit and any irreducible component of general cycle of the family is contained in a fiber of q disjoint from H i . This implies that H i is closed with respect to V -equivalence.
In fact, let C be an irreducible component of a cycle of V such that C ∩ H i = ∅. Since V is quasi-unsplit, we have H i · C = 0, which implies C ⊆ H i . Now since B = H 1 ∩ · · · ∩ H N and all H i 's are closed with respect to V - equivalence, we see that B is a union of V -equivalence classes. Observe that if C ⊂ X \ B is an irreducible curve such that H · C = 0 for some H ∈ U , then q(C) is a point. In fact, if q(C) is a curve, there exists H 0 ∈ U such that H 0 intersects q(C 0 )
in a finite number of points. Then H 0 intersects C without containing it, a contradiction, because H and H 0 are numerically equivalent, so C · H > 0. Now fix y 0 ∈ Y 0 . We know by Lemma 1 that e(p −1 (y 0 )) is contained in a V -equivalence class F . Since B is closed with respect to V -equivalence, we have F ⊂ X 0 . Consider an irreducible component C of a cycle of V such that C ⊆ F . Since V is quasi-unsplit, we have H · C = 0, hence q(C) is a point by what we proved above. Therefore q(F ) = y 0 and F = e(p −1 (y 0 )), so we have (i).
For any x ∈ X, let Y x := p(e −1 (x)) be the family of cycles parametrized by Y and passing through x, and Locus(Y x ) := e(p −1 (Y x )). Observe that for any y ∈ Y x , the subset e(p −1 (y)) contains x and is V -rationally connected by Lemma
1. Hence Locus(Y x ) is V -rationally connected for any x ∈ X. Since Z ⊂ X × Y , we have dim Y x = dim e −1 (x). Thus dim Y x > 0 if and only if x ∈ B, by Zariski's main Theorem. If so, Locus(Y x ) has dimension at least f V + 1. Now let F be a V -equivalence class contained in B, and x ∈ F . Then Locus(Y x ) has dimension at least f V + 1 and is contained in F , hence dim F ≥ f V + 1.
Let us remark that in general, if V is not quasi-unsplit, B is not closed with respect to V -equivalence.
Example 1. In P 2 fix two points x, y and the line L = xy. Consider P 2 × P 2 with the projections π 1 , π 2 on the two factors, and fix three curves R x , R y , L ′ such that:
• R x is a line in P 2 × x and R y is a line in P 2 × y; • π 1 (R x ) ∩ π 1 (R y ) is a point z ∈ P 2 ; • L ′ := z × L is the unique line dominating L via π 2 and intersecting both R x and R y . Let σ : W → P 2 × P 2 be the blow-up of R x and R y . In W , the strict transform of L ′ is a smooth rational curve with normal bundle O P 1 (−1) ⊕3 . Let X be the variety obtained by "flipping" this curve. Then X is a smooth toric Fano 4-fold with ρ X = 4 (this is Z 2 in Batyrev's list, see [Bat99, Proposition 3.3.5]).
X / / _ _ _ q A A A A W π 2 •σ P 2
The strict transform of a general line in a fiber of π 2 gives a covering family V of rational curves on X. The birational map X P 2 × P 2 is an isomorphism over P 2 × (P 2 \ L); if U ⊂ X is the corresponding open subset, then U is closed with respect to V -equivalence and every fiber of q : U → P 2 \ L is a V -equivalence class isomorphic to P 2 . Thus f V = 2.
Let T x and T y be the images in X of the exceptional divisors of σ in W . These two divisors are V -rationally connected, and they can not be contained in B because dim B ≤ 2. Moreover, P := T x ∩ T y is the P 2 with normal bundle O P 2 (−1) ⊕2 obtained under the flip. The map q : X P 2 can not be defined over P , so P ∩ B = ∅. Therefore B can not be closed with respect to V -equivalence.
Observe that the numerical class of V lies in the interior of NE(X), hence the unique morphism, onto a projective variety, which contracts curves in V , is X → {pt}.
A key observation is the following.
Proposition 2. Let X be a normal and Q-factorial projective variety, and V a covering and quasi-unsplit family of rational 1-cycles on X.
If B is V -rationally connected, then R ≥0 [V ] is a geometric extremal ray of NE(X). Let's show that H is nef. Assume by contradiction that there exists an irreducible curve C with C · H < 0.
Proof. Let
Claim. C ⊆ B.
Actually, either C is contained in a fiber of q, hence it is numerically proportional to [V ] which contradicts C · H < 0. Or C ∩ X 0 =: C 0 is an open subset of C, dim q(C 0 ) = 1, hence there exists H 0 ∈ U such that H 0 intersects q(C 0 ) in a finite number of points. Then H 0 intersects C without containing it, a contradiction, because H and H 0 are numerically equivalent, so C · H > 0.
Since B is V -rationally connected, C must be numerically proportional to V , impossible.
Let's finally show that C · H = 0 if and only if C is numerically proportional to [V ]: actually, if C · H = 0, the previous arguments show that either C ⊂ B or C is contained in a fiber of q, both are V -rationally connected, hence C is numerically proportional to V .
Unfortunately, B is not V -rationally connected in general as shown by the following example.
Example 2 (see [Kac97] Example 11.1 and references therein). Fix a point p 0 in P 3 and let P 0 := {Π ∈ (P 3 ) * | p 0 ∈ Π} ≃ P 2 be the variety of 2-planes in P 3 containing p 0 . Consider the variety X ⊂ P 3 × P 0 defined as X := {(p, Π) ∈ P 3 × P 0 | p ∈ Π}. Then X is a smooth Fano 4-fold, with Picard number 2 and pseudo-index 2. The two elementary extremal contractions are given by the projections on the two factors.
The morphism X → P 0 is a fibration in P 2 : the fiber over a point is the plane corresponding to that point.
Consider the morphism X → P 3 . If p = p 0 , the fiber over p is the P 1 of planes containing p and p 0 . But the fiber F 0 over p 0 is naturally identified with P 0 , hence it is isomorphic to P 2 . We have N F 0 /X = Ω 1 P 2 (1) and (−K X ) |F = O F (2).
C π=p F =e / / X q ′ q~~~~Ṽ ψ / / P 3
Here V → P 3 is the blow-up of p 0 and C → X is the blow-up of F 0 . Observe that V is a family of extremal irreducible rational curves of anticanonical degree 2. If we consider X × P 1 with the same family of curves, we have dim Y = 4, f V = 1 and B = F 0 × P 1 which is not V -rationally connected.
We finally get the following result: if B has the smallest possible dimension, then it is V -rationally connected.
Lemma 3. Let X be a normal and Q-factorial projective variety, and V be a covering and quasi-unsplit family of rational 1-cycles on X.
If dim B = f V + 1, then every connected component of B is a V -equivalence class.
Proof. By Proposition 1, we know that B is the union of all V -equivalence classes whose dimension is f V + 1. Since each of these equivalence classes must contain an irreducible component of B, they are in a finite number, and each is contained in a connected component of B.
So if B 0 is a connected component of B, we have B 0 = F 1 ∪ · · · ∪ F r , where each F i is a V -equivalence class. We want to show that r = 1.
Assume by contradiction that r > 1. Observe that the F i 's are disjoint and B 0 is connected, hence at least one F i is not a closed subset of X, assume it is F 1 .
Then F 1 is a countable union of closed subsets. Considering the decomposition of B 0 as a union of irreducible components, we find an irreducible component T of B 0 such that
T = m∈N K m
where each K m is a non empty proper closed subset of T . Since T is an irreducible complex projective variety, this is impossible.
We then reformulate in a single result what we proved so far, and show that it implies Theorem 1.
Proposition 3. Let X be a normal and Q-factorial projective variety, and V a covering and quasi-unsplit family of rational 1-cycles on X. Then:
(i) either B = ∅ or dim(B) ≥ f V + 1, (ii) if B = ∅ or if dim(B) = f V + 1 then R ≥0 [V ]
is a geometric extremal ray of the Mori cone NE(X).
Proof of Theorem 1. Just notice that if f V ≥ n−3 and B is not empty, Proposition 3 (i) gives dim B ≥ f V + 1 ≥ n − 2, so dim B = n − 2 = f V + 1. Then Proposition 3 (ii) gives that R ≥0 [V ] is a geometric extremal ray of the Mori cone NE(X).
Existence of a geometric quotient
Let V be a covering and quasi-unsplit family of rational 1-cycles on X, and assume that there exists a geometric quotient q ′ : X → Y ′ for V . Observe that q ′ has the following property: for any irreducible curve C in X,
q ′ (C) is a point if and only if [C] is proportional to [V ].
Conversely, we show that a morphism with the property above is quite close to be a geometric quotient.
Proposition 4. Let X be a normal and Q-factorial projective variety, and V a covering and quasi-unsplit family of rational 1-cycles on X.
Assume that there exists a morphism with connected fibers q ′ : X → Y ′ , onto a complete and normal algebraic variety Y ′ , such that for any irreducible curve C in X, q ′ (C) is a point if and only if [C] is proportional to [V ].
Then there exists a birational morphism ψ : Y → Y ′ that fits into the commutative diagram:
Z p e / / X q~} } } } q ′ Y ψ / / Y ′(3)
Moreover, if B ′ := q ′ (B), we have (q ′ ) −1 (B ′ ) = B, and
B ′ = {y ∈ Y ′ | dim(q ′ ) −1 (y) > f V } = {y ∈ Y ′ | dim ψ −1 (y) > 0}.
In particular, every fiber of q ′ over Y ′ \ B ′ is a V -equivalence class.
Observe that in example 2, ψ is not an isomorphism.
Proof. Let's show first of all that (q ′ ) −1 (B ′ ) = B. If C ⊂ X is an irreducible curve contained in a fiber of q ′ , then either C ∩B = ∅, or C ⊆ B. In fact, assume that C ∩ B = ∅. Let H 0 , . . . , H N be as in the proof of Proposition 1. Then for any i = 0, . . . , N , we have C · H i = 0 and C ∩ H i = ∅,
hence C ⊆ H i so C ⊆ B.
Since q ′ has connected fibers, we see that for every fiber F of q ′ , either F ∩B = ∅, or F ⊆ B. This means that (q ′ ) −1 (q ′ (B)) = B.
The existence of ψ as in (3) follows easily from the normality of Y and the fact that q ′ contracts all curves in V , hence all V -equivalence classes. Observe that ψ is surjective with connected fibers.
Let's show that p contracts to a point any fiber of q ′ • e over Y ′ \ B ′ . Let F be a fiber of q ′ over Y ′ \ B ′ , then we have F ⊂ X \ B. Let C ⊂ F be an irreducible curve, and choose an irreducible curve C ′ ⊂ X \ B which is a component of a cycle of the family V . Since q ′ (C) is a point, there exists λ ∈ Q >0 such that C ≡ λC ′ in X.
Set X 0 := X \B. Notice that e is an isomorphism over X 0 , so X 0 can be viewed also as an open subset of Z; in the same way the curves C and C ′ can be viewed also as a curves in Z. Let's show that C ≡ λC ′ still holds in Z.
Let L ∈ Pic Z, and write L |X 0 = O X 0 (D), D a Cartier divisor in X 0 . Let D be the closure of D in X (meaning, if D = i a i V i , that D = i a i V i ) and let m ∈ Z >0 be such that mD is Cartier in X. Then set M := e * (O X (mD)) ∈ Pic Z.
By construction, M ⊗ L ⊗(−m) is trivial on X 0 , so we can write
L ⊗m = M ⊗ O Z (G), where G is a Cartier divisor in Z with Supp G ⊆ E.
Now observe that C · G = C ′ · G = 0, because both curves are disjoint from E, and that C · M = λC ′ · M by the projection formula. Then we have C · L = λC ′ · L, so C ≡ λC ′ in Z.
Then p must contract C to a point, because Y is projective. Since e −1 (F ) is connected, we have shown that p contracts e −1 (F ) to a point. Since Y and Y ′ are normal, this implies that ψ is an isomorphism over Y ′ \ B ′ .
Finally, let y ∈ B ′ and let F ′ = (q ′ ) −1 (y). Then F ′ ⊆ B, so e has positive dimensional fibers on F ′ , and dim e −1 (F ′ ) > dim F ′ ≥ f V . Since e −1 (F ′ ) = p −1 (ψ −1 (y)) and p has all fibers of dimension f V , we must have dim ψ −1 (y) > 0.
We can finally prove our results.
Theorem 4. Let X be a normal and Q-factorial complex projective variety of dimension n, having canonical singularities. Let V be a covering and quasi-unsplit family of rational 1-cycles on X.
If dim B ≤ f V + 1, then R ≥0 [V ] is a geometric extremal ray of the Mori cone NE(X) and the Mori contraction of R ≥0 [V ], cont [V ] : X → Y ′ , is the geometric quotient for V .
Proof. If B is empty, then the statement is clear. Assume that B is not empty. Then Proposition 3 and Lemma 3 yield that dim B = f V + 1, every connected component of B is a V -equivalence class, and R ≥0 [V ] is a geometric extremal ray of NE(X).
We have to show that −K X ·[V ] > 0. Let V ′ be the covering family of rational 1cycles on X given by Lemma 2, and consider a resolution of singularities f : X ′ → X. The family V ′ determines a covering family V ′′ of rational 1-cycles in X ′ . If C 0 ⊂ X is a general element of the family V ′ , then C ′ := f −1 (C 0 \ Sing(X)) is a general element of V ′′ , and C 0 = f * (C ′ ).
Since C 0 is reduced and irreducible, so is C ′ . Moreover V ′′ is covering, so C ′ is a free curve in X ′ , and it has positive anticanonical degree.
Let m ∈ Z >0 be such that mK X is Cartier. Since X has canonical singularities, we have
mK X ′ = f * (mK X ) + i a i E i ,
where E i are exceptional divisors of f and a i ∈ Z ≥0 . Then
−mK X · C 0 = −f * (mK X ) · C ′ = −mK X ′ · C ′ + i a i E i · C ′ > 0.
This gives −K X · [V ′ ] > 0 and thus −K X · [V ] > 0. Since X has canonical singularities, the cone theorem and the contraction theorem hold for X (see [Deb01,Theorems 7.38 and 7.39]). Moreover, the extremal ray R ≥0 [V ] lies in the K X -negative part of the Mori cone, hence it can be contracted.
Let cont [V ] : X → Y ′ be the extremal contraction; then Y ′ is a normal, projective variety, and it is Q factorial by [Deb01, Proposition 7.44].
Applying Proposition 4, we see that all fibers of cont [V ] over Y ′ \ cont [V ] (B) are V -equivalence classes. Since connected components of B are V -equivalence classes, they are exactly the fibers of cont [V ] over cont [V ] (B), and we have the statement.
Observe that Theorem 2 is a straightforward consequence of Theorem 4.
The toric case: proof of Theorem 3
In the case ρ X = 1, the statement is true for q ′ : X → {pt}. In fact, using Proposition 4, we see that the geometric quotient Y must be a point.
Assume that ρ X > 1. Recall the diagram:
C π F / / X V Recall also that if D ⊂ X is a prime invariant Weil divisor, there is a natural inclusion i D : N 1 (D) R ֒→ N 1 (X) R .
Step 1 For
v ∈ V D , set G v := F D (π −1 D (v)). Then G v ∩ D = ∅, G v is connected, and G v · D = 0 because V is quasi-unsplit. This implies G v ⊆ D, hence F D (C D ) ⊆ D. Moreover, since W dominates D, we have F D (C D ) = D.
Since V D is normal, there is a holomorphic map V D → Chow(D). Then after replacing V D by its image in Chow(D) and C D by its image in Chow(D) × X, we get the desired family.
Step 2: there exists an invariant prime Weil divisor having intersection zero with
[V ].
In fact, let q : X Y be the rational map associated to V . Since ρ X > 1, Y is not a point. Let D be a prime divisor in Y intersecting q(X 0 ) and set D ′ := q −1 (D). Since there are curves of the family V disjoint from D ′ , we have D ′ · [V ] = 0. Moreover, D ′ is linearly equivalent to i a i D i , where a i ∈ Q >0 and D i are invariant prime Weil divisors. Hence the statement.
Step 3: we prove the statement.
Let Σ X be the fan of X in N ∼ = Z n , and let G X be the set of primitive generators of one dimensional cones in Σ X . It is well known that G X is in bijection with the set of invariant prime divisors of X; for any x ∈ G X , we denote D x the associated divisor. Recall that for any class γ ∈ N 1 (X) Q , we have x∈G X (γ · D x )x = 0 in N ⊗ Z Q, and that the association γ → x∈G X (γ · D x )x gives a canonical identification of N 1 (X) Q with the Q-vector space of linear relations with rational coefficients among G X .
Let m 1 x 1 + · · · + m h x h = 0 be the relation corresponding to [V ], with x i ∈ G X and m i non zero rational numbers for all i. Since V is covering and quasi-unsplit, all m i 's must be positive. For y ∈ G X , we have D y · [V ] = 0 if and only if y is different from x 1 , . . . , x h . So by Step 2, we know that G X \ {x 1 , . . . , x h } is non empty.
The following two statements are equivalent (see [Rei83,Theorem 2.4] and [Cas03, Theorem 2.2]):
(a) there exists a Q-factorial, projective toric variety Y ′ , and a flat, equivariant morphism q ′ : X → Y ′ , such that for any curve C in X, q ′ (C) is a point if and only if [C] is proportional to [V ]; (b) for any τ ∈ Σ X such that x 1 , . . . , x h ∈ τ , we have τ + x 1 , . . . ,x i , . . . , x h ∈ Σ X for all i = 1, . . . , h.
(4)
Let's show (b) by induction on the dimension of X. Clearly, it is enough to check (4) for any maximal τ in Σ X not containing any x i . Since {x 1 , . . . , x h } G X , such a maximal τ will have positive dimension.
Let y ∈ G X ∩ τ . We have D y · [V ] = 0, so by Step 1 there exists a quasi-unsplit, covering family V Dy in D y such that i Dy [V Dy ] is proportional to [V ].
Set N := N/Z · y and for any z ∈ N , write z for its image in N . The fan Σ Dy of D y is given by the projections in N ⊗ Z Q of all cones of Σ X containing y. The relation corresponding to [V Dy ] is λm 1 x 1 + · · · + λm h x h = 0, for some λ ∈ Q >0 . By induction, we know that (b) holds for V Dy in D y . In particular, the projection τ of τ is in Σ Dy , so we have τ + x 1 , . . . ,x i , . . . , x h ∈ Σ Dy for all i = 1, . . . , h.
This yields (4).
Finally, since q ′ is flat, all fibers must be V -equivalence classes and B = ∅.
X 0 := X \ B and Y 0 := Y \ p(E) = q(X 0 ). Let L be a very ample line bundle on Y . Let U ⊂ |L| be the open subset of divisors H that are irreducible and such that H ∩ Y 0 = ∅. For any H in U , we define H := q −1 (H ∩ Y 0 ) as in the proof of Proposition 1. Recall that H · [V ] = 0.
: let D ⊂ X be a prime invariant Weil divisor such that D · [V ] = 0. Then there exists a covering and quasi-unsplit familyV D of rational 1-cycles in D such that R ≥0 (i D [V D ]) = R ≥0 [V ].Choosean irreducible component W of F −1 (D) which dominates D. Set V ′ D := π(W ), and let C ′ D be an irreducible component of π −1 (V ′ D ) containing W . Consider the normalization C D of C ′ D , and let π D : C D → V D be the Stein factorization of the composite map C D → C ′ D → V ′ D . Finally let F D : C D → X be the induced map.
Choose a very ample line bundle L on Y , and let U ⊂ |L| be the open subset of divisors H that are irreducible and such that H ∩ Y 0 = ∅. For any H in U , we define H :
We denote by ρZ the Picard number of an algebraic variety Z.
Generalized Mukai conjecture for special Fano varieties. Marco Andreatta, Elena Chierici, Gianluca Occhetta, Central European Journal of Mathematics. 22Marco Andreatta, Elena Chierici, and Gianluca Occhetta. Generalized Mukai conjecture for special Fano varieties. Central European Journal of Mathematics, 2(2):272-293, 2004.
On the classification of toric Fano 4-folds. V Victor, Batyrev, Journal of Mathematical Sciences. 94Victor V. Batyrev. On the classification of toric Fano 4-folds. Journal of Mathematical Sciences (New York), 94:1021-1050, 1999.
Coréduction algébrique d'un espace analytique faiblement Kählérien compact. Frédéric Campana, Inventiones Mathematicae. 63Frédéric Campana. Coréduction algébrique d'un espace analytique faiblement Kählérien compact. Inventiones Mathematicae, 63:187-223, 1981.
Frédéric Campana. Orbifolds, special varieties and classification theory: an appendix. Annales de l'Institut Fourier. 543Frédéric Campana. Orbifolds, special varieties and classification theory: an appendix. Annales de l'Institut Fourier, 54(3):631-665, 2004.
Contractible classes in toric varieties. Cinzia Casagrande, Mathematische Zeitschrift. 243Cinzia Casagrande. Contractible classes in toric varieties. Mathematische Zeitschrift, 243:99-126, 2003.
The cone of curves of Fano varieties of coindex four. Elena Chierici, Gianluca Occhetta, math.AG/0401429PreprintElena Chierici and Gianluca Occhetta. The cone of curves of Fano varieties of coindex four. Preprint math.AG/0401429, 2004.
Higher-Dimensional Algebraic Geometry. Olivier Debarre, Springer-VerlagUniversitextOlivier Debarre. Higher-Dimensional Algebraic Geometry. Universitext. Springer- Verlag, 2001.
Extremal contractions from 4-dimensional manifolds to 3-folds. Yasuyuki Kachi, Annali della Scuola Normale Superiore di Pisa. Classe di Scienze24Yasuyuki Kachi. Extremal contractions from 4-dimensional manifolds to 3-folds. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze (4), 24(1):63-131, 1997.
Rational Curves on Algebraic Varieties. János Kollár, Ergebnisse der Mathematik und ihrer Grenzgebiete. 32Springer-VerlagJános Kollár. Rational Curves on Algebraic Varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1996.
Decomposition of toric morphisms. Miles Reid, Geometry, number 36 in Progress in Mathematics. BirkhäuserIIArithmetic and GeometryMiles Reid. Decomposition of toric morphisms. In Arithmetic and Geometry, vol. II: Geometry, number 36 in Progress in Mathematics, pages 395-418. Birkhäuser, 1983.
L B , [email protected] S.D. e-mail: [email protected] Institut Fourier, UFR de Mathématiques. 38402 Saint Martin d'Hères, FRANCE5582Université de Grenoble 1L.B. e-mail: [email protected] S.D. e-mail: [email protected] Institut Fourier, UFR de Mathématiques, Université de Grenoble 1, UMR 5582, BP 74, 38402 Saint Martin d'Hères, FRANCE
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[] |
[
"Cosmological surveys with multi-object spectrographs",
"Cosmological surveys with multi-object spectrographs"
] |
[
"Matthew Colless \nResearch School of Astronomy and Astrophysics\nAustralian National University\nCanberraAustralia\n"
] |
[
"Research School of Astronomy and Astrophysics\nAustralian National University\nCanberraAustralia"
] |
[] |
Multi-object spectroscopy has been a key technique contributing to the current era of 'precision cosmology'. From the first exploratory surveys of the large-scale structure and evolution of the universe to the current generation of superbly detailed maps spanning a wide range of redshifts, multi-object spectroscopy has been a fundamentally important tool for mapping the rich structure of the cosmic web and extracting cosmological information of increasing variety and precision. This will continue to be true for the foreseeable future, as we seek to map the evolving geometry and structure of the universe over the full extent of cosmic history in order to obtain the most precise and comprehensive measurements of cosmological parameters. Here I briefly summarize the contributions that multi-object spectroscopy has made to cosmology so far, then review the major surveys and instruments currently in play and their prospects for pushing back the cosmological frontier. Finally, I examine some of the next generation of instruments and surveys to explore how the field will develop in coming years, with a particular focus on specialised multi-object spectrographs for cosmology and the capabilities of multi-object spectrographs on the new generation of extremely large telescopes.
|
10.1117/12.2231829
|
[
"https://arxiv.org/pdf/1608.04454v1.pdf"
] | 119,199,178 |
1608.04454
|
aee46af811092a9b7671bf1483d4482d8929afc3
|
Cosmological surveys with multi-object spectrographs
Matthew Colless
Research School of Astronomy and Astrophysics
Australian National University
CanberraAustralia
Cosmological surveys with multi-object spectrographs
cosmologylarge-scale structuremulti-object spectroscopyspectrographtelescope
Multi-object spectroscopy has been a key technique contributing to the current era of 'precision cosmology'. From the first exploratory surveys of the large-scale structure and evolution of the universe to the current generation of superbly detailed maps spanning a wide range of redshifts, multi-object spectroscopy has been a fundamentally important tool for mapping the rich structure of the cosmic web and extracting cosmological information of increasing variety and precision. This will continue to be true for the foreseeable future, as we seek to map the evolving geometry and structure of the universe over the full extent of cosmic history in order to obtain the most precise and comprehensive measurements of cosmological parameters. Here I briefly summarize the contributions that multi-object spectroscopy has made to cosmology so far, then review the major surveys and instruments currently in play and their prospects for pushing back the cosmological frontier. Finally, I examine some of the next generation of instruments and surveys to explore how the field will develop in coming years, with a particular focus on specialised multi-object spectrographs for cosmology and the capabilities of multi-object spectrographs on the new generation of extremely large telescopes.
INTRODUCTION
Multi-object spectroscopy (MOS) has proved to be a richly fertile technique for probing the large-scale structure of the universe, from which it has proved possible to measure a wide range of cosmological parameters with increasing precision. For cosmological purposes, the key aspects of MOS are multiplex (the number of sources that can be simultaneously observed) and field of view (the area of sky that can be accessed at one time), since the primary requirements for cosmological surveys are the size and volume of the sample. Usually (though not invariably), the quantity being determined for each source is simply the redshift-the redward shift of spectral features resulting from the expansion of the universe during the time between emission and observation. This is (again, usually) a relatively undemanding measurement to make, requiring neither high signal-to-noise in the data nor sophistication in the analysis. Hence cosmological surveys typically reduce galaxy spectra to their redshifts and focus all their efforts on covering as large a sample as possible over as large a volume as possible.
The two key milestones in the use of MOS surveys for cosmology were, first, the step up to surveys that covered a statistically-representative volume of the relatively local universe and, second, the further step up to surveys that covered statistically-representative volumes over cosmologically significant ranges in redshift. The first step ushered in the era of 'precision cosmology' for MOS surveys, while the second step made MOS surveys a key tool for probing the nature of dark energy.
In the following sections I briefly review the past, present, and future of cosmological MOS surveys. An overview of the key cosmology surveys is provided in Figure 1, which gives a timeline of significant surveys showing the increasing multiplex and survey size resulting from the ongoing development of multi-object spectrograph systems (especially fibre MOS), and a map of the large-scale structure in the universe based on the various redshift surveys carried out with Australian Astronomical Observatory (AAO) facilities over the last 20 years.
COSMOLOGY SURVEYS TIMELINE
WiggleZ, 2.6, 5.
THE CLASSICAL PERIOD
Due to the modest field of view and multiplex of the early MOS instruments on (mostly) 4-metre class telescopes, the initial 3applications of MOS surveys tended to be studies of relatively small volumes (often clusters of galaxies 1 ) or 'pencil-beam' surveys over a limited range of redshifts (usually looking at galaxy evolution, 2-4 even though z 1).
The pioneering surveys by Geller, Huchra and collaborators 5-7 -which were done without the use of MOS instruments-demonstrated that there was a rich structure in the large-scale distribution of galaxies. As the theory of cosmological density perturbations 8,9 was developed, it was recognized that this rich structure encoded key cosmological parameters such as the overall density of the universe, and moreover could be used to distinguish the relative contributions of ordinary 'baryonic' matter, non-relativistic 'cold' dark matter, and relativistic 'hot' dark matter. This potentiality motivated the first MOS surveys with the explicit goal of measuring cosmological cosmological parameters, in contrast to previous surveys that either sought to explore the structure of the large-scale galaxy distribution or the evolution of the galaxy population.
The most extensive of this first generation of cosmological MOS surveys were the Las Campanas Redshift Survey 10-14 (LCRS), and the IRAS Point Source Catalogue Redshift Survey [15][16][17][18][19] (PSCz). The latter was a sparse all-sky survey that did not use MOS, but the LCRS was an important forerunner to all subsequent cosmological MOS surveys. It surveyed more than 26,000 galaxies over an area of 700 deg 2 and reached a median depth of z ≈ 0.1. The LCRS took 6 years on the Las Campanas 2.5-metre du Pont telescope, which had a MOS system with a multiplex of 112 and a field of view 2.1 deg in diameter. It was an eye-opener for observational cosmologists, demonstrating the potential of MOS for probing large-scale structure and measuring the galaxy luminosity function, the correlation function and power spectrum (in 2D and 3D), the pairwise velocity distribution, and producing a catalogue of groups and clusters. However it also illuminated the size of the challenge-despite being the largest redshift survey up to that time, it was nonetheless still an order of magnitude smaller than a survey of a statistically-representative volume of the universe needed to be.
THE ENLIGHTENMENT
Two groups took on the challenge of constructing the first truly cosmological survey of the nearby universe.
One was an British and Australian team that built a revolutionary new MOS, the 2-degree Field multifibre spectrograph (2dF) for the 4-metre Anglo-Australian Telescope. 20 As its name implies, 2dF had a 2-degree diameter field of view (the largest on any on a 4-metre-class telescope), a multiplex of 400, and used a robotic fibre positioner that facilitated rapid automatic reconfiguration. 21,22 The 2dF fibre positioning system was considered technically challenging and risky at the time it was being constructed (the early 1990s), but it proved effective and reliable in practice. 23 2dF's wide field of view was achieved with a corrector lens incorporating an atmospheric dispersion compensator, which was an essential innovation in a system aiming to achieve wide spectral coverage with small aperture fibres. The robot positioner placed fibres sequentially at the rate of one every 6 s with a precision of 0.3 arcsec (corresponding to 20 µm). Figure 2a shows the 2dF topend ring, with the corrector, positioner and the two spectrographs. Because it took about an hour to reconfigure a complete set of fibres, 2dF had a double-buffering system with two field plates each having 400 fibres. While the fibres on one field plate were being reconfigured by the robot, the second field plate was being observed; at the end of an observation the two field plates were tumbled into the other position and the process repeated. The 2.1 arcsec (140 µm) diameter fibres fed a pair of dual-channel spectrographs that offered spectral resolving powers (R=λ/FWHM) between R=500 and R=2000 covering wavelength ranges of 440 nm and 110 nm respectively. The overall throughput of the entire 2dF system peaked at about 5% around 600 nm. The layout of the 2dF spectrographs is shown in Figure 2b. The 2dF MOS system was designed from the outset to enable a massive galaxy redshift survey to test the cosmological model and measure its key parameters. Over a period of 5 years from 1997 to 2002, the 2dF Galaxy Redshift Survey (2dFGRS) measured 221,000 redshifts over ∼1500 deg 2 with a median depth of z ∼ 0.11 (corresponding to a volume of ∼0.12 Gpc 3 ), making it the first cosmological redshift survey to capture a statistically-representative sample of the universe. 24,25 The main cosmological results from 2dFGRS related to the nature of the large-scale structure, the overall density of the universe, and the nature of its massive constituents. [26][27][28][29][30][31] The survey precisely determined the statistical properties of the large-scale structure of the galaxy distribution (via the galaxy power spectrum or, equivalently, the galaxy correlation function) over size scales from about 1 Mpc to about 300 Mpc. The properties of the galaxy distribution confirmed the generally accepted paradigm that the large-scale structure grows by gravitational instability in a way that is qualitatively and quantitatively consistent with the standard model of gravitational amplification of quantum fluctuations emerging from the Big Bang. From the power . Front view of the SDSS telescope depicting the location of the spectrographs. Here a fiber cartridge is shown retracted from the spectrographs. The twin spectrographs, each with a mass of 320 kg, mount to the back of the Cassegrain instrument rotator adjacent to the focal plane with sufficient separation between the spectrographs to allow routine installation and removal of the imaging camera and fiber cartridges.
Hartmann Doors
The Hartmann doors are a simple bi-fold design. Each door pivots about an upper and lower bushing pressed into the optical bench, and is driven by a 90 • pneumatic rotary actuator (Bimba Pneu-Turn model PT-006090-A1M 40 ) located on the top surface of the bench. The doors are controlled by the spectrograph micro-controller and a bank of programmable solenoid valves (Clippard model number EMC-08 41 ) located in the spectrograph electronics box. Solid-state magnetic sensors (Bimba model number HSC-02) mounted to the actuators report the state of the doors (i.e., open or closed). Small manual flow control valves (Bimba model number FCP-1) at the inputs to each actuator set the speed of rotation.
Shutter
A shutter is required to set the exposure time for wavelength calibrations, flat fields, and science exposures. The shutter does not need to be fast or very accurate since science exposures include calibration standards that are used to determine the zero-point calibration at the time of the exposure. The minimum 40 The spectrograph shutter is located just upstream of the dichroic on a dividing wall in the optical bench residing between the central optics and the slithead; see Figure 14. This location is ideal since the entire bandpass can be blocked by a single, relatively small shutter. The shutter is a black anodized aluminum sliding door driven by a double-acting pneumatic cylinder (Bimba model BRM-02x-DXP). Slots in the Delrin door frame guide the door, making a light-tight seal when the door is closed. An oval window in the frame, slightly oversized to the beam, serves as a light baffle. Like the Hartmann doors, the shutter is controlled by the spectrograph micro-controller and solenoid valves on the Clippard EMC-08 board. Solid-state magnetic sensors (Bimba model number HSC-02) mounted to the cylinder indicate the state of the door (i.e., open or closed). Small manual flow control valves (Bimba model number FCP-1) at the inputs to cylinder set the speed at which the door opens and closes.
The shutter is accurate to approximately 0.5 s. Latency in the door motion due to temperature changes is responsible for most of the inaccuracy. The T-shaped optical bench is an enclosed aluminum (6061-T6) weldment with precision-machined interfaces to locate all five opto-mechanical subassemblies: the fiber slithead, the collimator, the central optics, and a the red and blue channel cameras. One of the two spectrographs also supports the guide camera. Electronics control chassis mount to the external walls of the bench. Three kinematic mounts on the top of the bench interface to the back of the Cassegrain instrument rotator.
Central Optical Assembly
The beamsplitter and grisms are mounted in a single optomechanical structure, the central optics assembly. The assembly is kinematically mounted inside the optical bench on three gusseted posts integral to the weldment and located on the top surface of the optical bench, thus the central optics assembly is suspended from the top of the bench. The faces of the three posts are precisely machined establishing a planar reference that is square to the collimator mounting surface and the two camera mounting surfaces. Two locating sleeves centered about two of the three posts establish the in-plane location of the assembly. These machined references precisely locate the assembly relative to the slithead, collimator, and cameras. A large port in the sidewall of the optical bench provides access for machining the post surfaces and installation. Figure 16 shows the details of the central optics assembly. The dichroic and both grisms are each located, without adjustment, by six machined reference surfaces (Kapton tape covers each surface to avoid metal-to-glass contact). Spring plungers seat the elements against these surfaces accommodating differential contraction between the glass optics and aluminum structure. A three-point-contact Alloy-39 block bonded to the top of each grism spreads the vertical load applied by a single large plunger embedded in the top plate, and provides a convenient lift point for installing the grism. To minimize tolerance stack-up and improve placement accuracy, all but 1 of the 18 reference surfaces (6 for each of 3 optics) is machined into a single component, the base plate, which interfaces to the optical bench. The single remaining reference surface, which controls the tip of the beamsplitter, is located on the top plate. Black anodized surfaces and thin light baffles at the exit faces of the assembly serve to mitigate stray light. The entire assembly has a mass of 39 kg.
Camera Opto-mechanics
The opto-mechanical design of the SDSS cameras, shown in Figure 17, was derived largely from the Norris spectrograph camera design (Cohen et al. 1988); a logical consequence of the two cameras having very similar optical designs, and the same optical designer, Harland Epps. The lenses and lens-groups are mounted in athermal cells. External reference surfaces on each cell are machined true to the opto-mechanical reference surfaces that locate each lens, thus establishing lens concentricity from cell-to-cell and accurate placement of the lenses along the spectrum and redshift-space distortions, 2dFGRS obtained an estimate for the total density of all types of matter in the universe of Ω M = 0.230.02; the uncertainty of less than 10% on this figure was one of major steps towards precision cosmology from redshift surveys. Moreover, 2dFGRS was able to show that the fraction of the total matter density in baryons is 18%, consistent with a baryon density of Ω B = 0.04, as found from the cosmic microwave background anisotropies and Big Bang nucleosynthesis models. On the other hand, relativistic matter such as neutrinos makes up less than 13% of the overall matter density, implying an upper limit on the total mass of the three neutrino species of 0.7 eV.
Contemporaneously with the 2dFGRS, an even more ambitious project, the Sloan Digital Sky Survey (SDSS), was being carried forward by a largely US-based team. 32,33 Whereas 2dFGRS relied on photographic sky surveys for its input target catalogue, the SDSS project paired a CCD imaging survey with a MOS spectral survey. SDSS used a purpose-built 2.5-metre telescope with a 3 deg diameter field of view at Apache Point Observatory, and its MOS instrument 34 had 640 fibres, each 3 arcsec (180 µm) in diameter, that could be positioned over the 7 deg 2 field (see Figure 3a). Unlike 2dF's robotic system, the SDSS fibre system was a plug-plate design requiring manual positioning of the fibres. The twin SDSS spectrographs (see Figure 3b) were each fed by 320 fibres, and utilized a simple optical layout with reflective collimators, gratings, all-refractive cameras, and state-of-the-art CCD detectors to record the spectra from these fibres simultaneously in two channels over the wavelength range from 390 nm to 910 nm at a resolving power R ≈ 2000. The overall efficiency of the spectrographs peaked at about 17% in the blue channel and 22% in the red channel.
The original SDSS survey 35 (comprising SDSS-I and SDSS-II) ran for 9 years from 2000 to 2008. It imaged an area of 11,663 deg 2 and obtained galaxy spectra over 8032 deg 2 , measuring redshifts for 930,000 galaxies with a median redshift of z ≈ 0.1 (corresponding to ∼0.5 Gpc 3 ). The cosmological results from SDSS 36-39 covered similar ground to those of 2dFGRS, but ultimately achieved higher precision due to both the larger size of the SDSS sample and the better quality of the CCD imaging and photometry.
With the 2dFGRS and SDSS surveys, MOS spectroscopy had made observations of large-scale structure a tool for 'precision cosmology'-though that initial level of 'precision' seems rather imprecise by today's standards.
One of the key outcomes from these surveys was the detection of the baryon acoustic oscillation (BAO) signature in the galaxy distribution. 30,38,39 The BAO 'standard ruler' has been a key tool in subsequent cosmological surveys seeking to probe the nature of dark energy, using the evolution of the expansion rate and geometry of the universe as means of determining the dark energy equation of state.
THE MODERN ERA
Following the outstanding success of the 2dFGRS and SDSS-I/II surveys there has been a procession of follow-on cosmological MOS surveys addressing a range of different issues. Table 1 lists the main recent MOS surveys (both cosmological and other) and the telescope/instrument combinations with which they have been carried out. Notable amongst the various cosmological MOS surveys in this list are the subsequent SDSS surveys (including BOSS and eBOSS), the WiggleZ survey on the Anglo-Australian Telescope, and the 6dFGS survey on the UK Schmidt Telescope.
6dFGS
The 6dF Galaxy Survey (6dFGS) is a cosmological survey of the relatively local (z < 0.1) universe, 40-43 differing from others in combining a redshift survey with a peculiar velocity survey. The redshift survey 41,44 was relatively standard, except that it covered whole southern hemisphere apart more than 10 deg from the Galactic Plane (at total area of 17,000 deg) and was restricted to the brightest 125,000 galaxies in the local universe. The peculiar velocity survey [45][46][47] used Fundamental Plane distances for early-type galaxies, in combination with their redshifts, to measure the peculiar velocities (i.e. non-Hubble-flow motions) for about 8000 galaxies at distances cz < 16, 000 km s −1 . Peculiar velocity surveys, by adding direct measurements of the motions of galaxies due to the effective gravitational force of the surrounding matter, provide additional, complementary information to redshift surveys regarding both the mass distribution and the nature of gravity. However, available methods for measuring galaxies' distances independent of their redshifts are not precise (e.g. the Fundamental Plane provides distances with typical errors of about 20%), and as a result they can only measure peculiar velocities effectively at relatively small distances (i.e. relatively low redshifts).
The 6dFGS observations were carried out over 5 years from 2001 to 2006 using the 6-degree Field (6dF) MOS system on the UK Schmidt Telescope (UKST). 41, 48-51 The 6dF system had 150 science fibers, each 6.7 arcec (100 µm) in diameter, that could be positioned over the 5.7-degree field of the UKST (see Figure 4a & b). This made 6dF the ideal instrument for full-sky spectroscopic surveys of relatively sparse (<50 deg −2 ), bright (V<17) objects. In terms of the AΩ (telescope aperture × field of view) figure of merit, 6dF on the 1.2-metre UKST has about 75% of the survey power of 2dF on the 3.9-metre Anglo-Australian Telescope (AAT), but with the operational advantage that it was entirely given over to survey observations. Because of the curved focal plane of the UKST, 6dF used a 3-axis r-θ-z robot (see Figure 4c) with a curved radial arm to position the individual fibres, which were contained in an adapted version of the Schmidt photograph plate holder. The robot was located off the telescope, and the plate holders had to be manually mounted in the UKST. The relatively large 'buttons' that magnetically held the fibres on the field plate, combined with the small platescale of the UKST, meant that targets closer than 5.7 arcmin could not be observed simultaneously. The fibres fed a floor-mounted, fixed format spectrograph (see Figure 4d), which used reflection gratings up to October 2002 and thereafter volume phase holographic gratings. The limited size of the CCD detector meant that, while all 150 fibres fitted on the detector simultaneously, full spectral coverage from ∼400 nm to ∼800 nm required two exposures with different gratings. The peak system efficiency was about 11% at wavelengths near the gratings' blaze angles.
(a) (b) (c) (d)
The main cosmological results from the 6dFGS relate to local (i.e. z ≈ 0) measurements of key parameters, which are important because they require little or no reliance on an assumed cosmological model to interpret them. Thus the 6dFGS redshift survey yielded a direct and independent estimate of the local Hubble constant 52 (H 0 )based on the standard ruler provided by the baryonic acoustic oscillations (BAO)-that agrees well with the value obtained by the Planck satellite extrapolating from the cosmic microwave background (CMB) at z ≈ 1100 assuming the standard ΛCDM cosmology. The 6dFGS similarly provide low-redshfit measurements of the product of the normalisation of the matter power spectrum (σ 8 ) and the growth rate of large-scale structure 53 derived AAOmega is a dual-beam spectrograph. Science fibres are arranged into a pseudo-slit which feeds into a single collimator and then separates into the blue and red arms of the system via a dichroic beam splitter. There are two dichroics, one operates at 570nm and one at 670nm. Each arm of AAOmega uses one of a selection of Volume Phase Holographic (VPH) gratings. The system shutter is in front of the fibre pseudo-slit, so both cameras must use the same exposure time.
AAOmega can be configured to observe the entire optical spectrum over the wavelength range 370nm-900nm, with a small overlap between the red and blue arms around the dichroic wavelength (570 or 670nm). The grating set available allows a range of resolutions between R ⇠ 1, 000 and R ⇠ 10, 000. The fibre spectra are recorded onto the 2K⇥4K E2V CCDs with light dispersed along the 2K axis (not the 4K axis). Hence, at low resolution the entire accessible spectral range is recorded at once, but at higher resolutions the user must tune the wavelength range to that which best suits their requirements.
AAOmega uses Volume-Phase Holographic (VPH) transmission gratings. These have flexible blaze angles. Each grating has a specific design blaze angle which will give the absolute from the redshift-space distortions in the two-point galaxy correlation function. This low-redshift constraint was combined with the very high-redshift CMB constraint to confirm that the growth of structure over the history of the universe is consistent with the model for gravity provided by General Relativity. By contrast, the peculiar velocity measurements have no high-redshift counterparts, but are complementary to the redshift survey. After careful calibration of the Fundamental Plane, 45 the 6dFGS was able to measure peculiar velocities for nearly 8000 early-type galaxies and map the large-scale velocity field 47 in the southern hemisphere out to ∼16,000 km s −1 . This observed velocity field was used to determine the rms bulk motions 54 on scales of 50-70 h −1 Mpc and to make a direct comparison with the predicted velocity field derived from the redshift survey density field. 55 The 6dFGS peculiar velocities also allowed the first-ever direct measurement of the velocity power spectrum, 56 which provided another means of measuring the growth rate of structure and showing that it is scale-independent up to scales of at least 300 h −1 Mpc.
WiggleZ
After the conclusion of the 2dFGRS, the original 2dF spectrographs were replaced with the AAOmega doublebeam spectrograph. [57][58][59] AAOmega is a bench-mounted system an f/3.15 Schmidt collimator, a dichroic beamsplitter, volume phase holographic (VPH) gratings, and articulated f/1.3 Schmidt cameras (see Figure 5). It accommodates 392 fibers and covers the wavelength range 370 nm to 950 nm at spectral resolutions from R=1000 to R=7500. It is floor-mounted in a thermally isolated environment with a fiber cable running 38m to the AAT's prime focus. Despite the long fibre run, AAOmega achieves a throughput of approximately 20% in both the blue and red arms, a gain of more than a factor of 2 over the 2dF spectrographs. This improvement is due to a number of factors: the use of highly efficient VPH gratings and optical coatings, and to new higher-performance CCDs. AAOmega also improved on 2dF by a factor of 2 in resolution, while the spectral stability is an order of magnitude better. AAOmega commenced science observations at the AAT in early 2006 and is still in high demand today, being used with the (refurbished) 2dF fiber positioning system, the KOALA 60, 61 wide-field IFU feed, and the SAMI 62, 63 multi-IFU system. AAOmega has been used for a number of galaxy surveys, including the ongoing GAMA 64 and SAMI 63,65 surveys of galaxy properties and evolution, and the 2SLAQ, 66 Wigglez, 67 and ongoing OzDES 68 surveys which had cosmological goals. Of the cosmological surveys using AAOmega, WiggleZ, which focussed on the evolution of the geometry of the universe and the growth of structure at redshifts up to z ≈ 1, has had the greatest impact.
WiggleZ 67, 69 measured redshifts for 238,000 galaxies starforming galaxies with 0.2 z 1 (z median =0.6) in 7 regions covering approximately 1000 deg 2 of sky with a total volume of ∼1 Gpc 3 . The main results from the WiggleZ survey were: (a) using the BAO standard ruler to measure the geometry of the universe, by mapping the distance-redshift relation and measuring the cosmic expansion history by combining the Alcock-Paczynski
BOSS & eBOSS
The current state-of-the-art in precision cosmology MOS surveys is represented by the recently completed Baryon Oscillation Spectroscopic Survey 81-84 (BOSS, part of SDSS-III) and the ongoing extended BOSS 85 (eBOSS, part of SDSS-IV). These surveys were enabled by a 2009 upgrade to the original SDSS spectrographs 34 on the Apache Point Observatory 2.5-metre telescope, just as WiggleZ and other AAT surveys were enabled by AAOmega replacing the original 2dF spectrographs. As with AAOmega, the upgrade to the SDSS spectrographs involved volume phase holographic gratings and more modern CCD detectors, and improved the peak efficiency by nearly a factor of 2, while extending the spectral range to 360-1000 nm and increasing the multiplex from 640 to 1000 fibers (500 per spectrograph). To achieve the higher multiplex, retain the spectral sampling given the smaller CCD pixels, and match the source size for the more distant BOSS galaxies, the fiber diameter was reduced from 3 arcsec to 2 arcsec. The overall layout of the upgraded spectrographs is the same as the original SDSS spectrographs (see Figure 3) but with a revised central optics assembly, as illustrated in Figure 6. As with the SAMI multi-IFU feed for AAOmega, the new SDSS spectrographs have been provided with multi-IFU capability as part of the MaNGA survey 86 that is obtaining spatially resolved spectroscopy for ∼10,000 galaxies.
BOSS observations were performed from 2008 to 2014, starting with imaging in 2008 and spectroscopy from 2009, when the upgraded SDSS spectrographs became available. The final data release 84 (SDSS DR12) was made public in 2015, and contained spectra for 1.37 million unique galaxies and QSOs, made up of 862,735 galaxies from the LOWZ target catalogue (0.15 < z < 0.4), 343,160 galaxies from the CMASS target catalogue (0.4 < z < 0.8), and 158,917 QSOs with 2.15 < z < 3.5 used to study large-scale structure in the Lyman-alpha forest. The BOSS spectroscopy covers an effective area of 9376 deg 2 over two large contiguous regions in the north and south Galactic caps.
Some of the main cosmological results of the BOSS survey, and relevant preceding surveys, are summarized 85 in Figure 7. The lefthand panel shows predictions and observations of the relation between comoving distance and
Redshift Space Distortions and Modified Gravity
Dark energy is often invoked to explain current CMB, SNe, and BAO observations that imply an accelerating Universe. It is also possible to explain the accelerated expansion of the Universe by modifying gravity at large scales. The galaxy redshifts used in spectroscopic BAO measurements of the expansion history help di↵erentiate these two possible e↵ects through measurements of the growth of structure via RSD (Kaiser 1987).
RSD arise because the gravitational pull of matter overdensities causes velocity deviations from the smooth Hubble flow expansion of the Universe. These peculiar velocities are imprinted in galaxy redshift surveys in which recessional velocity is used as the line-of-sight coordinate for galaxy positions. Although the correlation function of galaxies is isotropic in real space, the peculiar velocities lead to an increase in the amplitude of radial clustering relative to transverse clustering when the correlation function is measured in redshift space. The resulting anisotropy in the clustering of galaxies is correlated with the speed at which structure grows; deviations from GR causing slower or faster growth give smaller or larger anisotropic distortions in the observed redshift-space clustering.
In general, the amplitude of clustering at a given redshift is parameterized by 8(z), the rms fluctuations in spheres of radius 8 h 1 Mpc. The degree of anisotropy due to RSD depends on the rate of change of the amplitude of clustering. This change is typically parameterized as a function of the logarithm of the expansion scale parameter f 8 = @ 8/@ ln a, where a = (1 + z) 1 is the dimensionless cosmic expansion factor. Because RSD measurements are sensitive to the product of the growth rate and the amplitude of matter fluctuations, a wide range in redshift coverage is essential to constrain the evolution in clustering amplitude and directly probe gravity. Cosmological measurements from small-scale clustering are dependent on the accuracy of the modelling on quasi-linear and non-linear scales. The development and evaluation of analytic, phenomenological, and halo occupation models for anisotropic clustering remains a focus with the BOSS galaxy samples (e.g. Chuang et al. 2013;Beutler et al. 2014b;Guo et al. 2015). A study of several models in configuration-space tested against mock galaxy catalogs indicates that the clustering signal can be well characterized on scales in the range 40 < s < 80h 1 Mpc (White et al. 2015). Certain models, such as those based on Lagrangian perturbation theory, are able to fit the mock clustering samples without significant bias on scales above 25-30 h 1 Mpc. Continued development of theoretical models that allow use of smaller scale data may tighten the current BOSS constraints still further. 88,89 and a compilation of current SNe Ia measurements. 90 The righthand panel of Figure 7 shows the predicted growth rate of structure (f σ 8 ) as a function of redshift, and compares the Planck 87 ΛCDM model to the measurements based on redshift-space distortions (RSD) from 6dFGS, 53 2dFGRS, 30, 91 SDSS LRGs, 92 WiggleZ, 78 VIPERS 93 and BOSS. 84,[94][95][96][97] Because the growth rate simultaneously tests both the cosmological model and the theory of gravity, the panel alsos shows the predictions for theories of gravity predicting growth going as f = Ω γ , with γ differing slightly from the General Relativity value of 0.55, demonstrating the sensitivity of these measurements to such alternatives.
BOSS Constraints on RSD and Modified Gravity
All existing measurements are consistent with the standard ΛCDM model at the 5-10% level; the next generation of MOS surveys aim to tighten these constraints to a level approaching 1%. The first of the new surveys is eBOSS, 85, 98 a 6-year survey (part of the SDSS-IV program) that started in 2014 and plans to complement and extend BOSS. It has the goal of measuring the distance-redshift relation from the BAO ruler with a precision of a few percent in each of four redshift bins over the range 0.6 < z < 2.2. It will use four different tracers to cover this range: 250,000 luminous red galaxies with median redshift z ≈ 0.7; 195,000 emission line galaxies with median redshift z ≈ 0.9; 500,000 QSOs over 0.9 < z < 2.2; and Lyman-α forest measurements using 120,000 QSOs at z > 2.1. As well as determining the evolution of the geometry of the universe in order to determine the equation of state of dark energy, eBOSS aims to make stronger tests of General Relativity on cosmological scales through redshift-space distortion measurements, look for evidence of non-Gaussianity in the primordial density field, and tighten the constraints on the sum of the masses of all neutrino species. The constraints that eBOSS is predicted to yield 85 on the comoving distance and the growth rate of structure as functions of redshift (assuming the Planck 87 ΛCDM model) are shown in Figure 7. ark Energy -future progress
Weinberg+ (2013) (a)
Constraints from MOS Surveys
THE FUTURE
There are a number of proposed new cosmological MOS surveys planned for the near future, plus a wider variety of surveys aiming to explore the high-redshift universe using the next-generation Extremely Large Telescopes. The general program for these surveys is to continue the push for ever-larger samples over ever-larger volumes covering a wider range of redshfits. The focus of such surveys is ever-more-precise constraints on cosmological parameters, with a particular emphases on testing the nature of dark energy and the theory of gravity. Although all observations to date of the cosmic expansion history and the growth of structure are consistent with a flat ΛCDM+GR model with Ω M ≈ 0.3 and Ω Λ ≈ 0.7, a number of plausible alternative models are also consistent with existing data. However BAO and RSD have to the potential, for sufficiently large surveys, to provide ∼10× stronger constraints on the equation of state of dark energy and the nature of the gravitational force. This opportunity is quantified by improvement in the relevant figures of merit, expressed as the inverse of the uncertainties on the w a and w p parameterisations of the DE equation of state and deviations ∆γ from the GR value of γ = 0.55. These improvements for BAO and RSD measurements are shown for possible future MOS surveys 99 in Figure 8a; another view 100 of the constraints, based on the actual and predicted fractional errors in the distance-redshift relation for existing and future MOS surveys, is provided in Figure 8b. The TAIPAN facility will initially be used for the Taipan and FunnelWeb surveys described above, which motivated the development of TAIPAN and raised the funding required to realise its construction. The two surveys will split the observing time on the facility equally between them based on their scientific requirements, Taipan using the dark (Moon down) time and FunnelWeb the bright (Moon up) time. The operating costs of the facility will likewise be shared equally between the two teams; details of these funding arrangements are provided below in §2. The Taipan and FunnelWeb surveys will fully utilise the upgraded TAIPAN facility until at least the end of 2019, and through 2023 if the extended surveys are carried out as envisaged. Towards the end of the Taipan and FunnelWeb surveys, the UKST and TAIPAN will be the subject of a call for proposals for future exploitation. Such exploitation will be on the basis that projects will pay the AAO to operate and maintain the facility. Australian-led projects will be given preference, and it will be a requirement for all projects that astronomers at any Australian research organisation may join the project team (provided they contribute a suitable level of effort). Any future surveys with the UKST+TAIPAN will of course realise the full factor of 2 gain in survey speed enabled by the proposed upgrade.
Low Redshift Surveys
Other key points regarding the need for, and use of, the proposed TAIPAN upgrade include:
• In terms of the availability of, and access to, similar research infrastructure, there is simply no other facility, nationally or internationally, that has the wide-field, high-multiplex capability that allows UKST+TAIPAN to perform massive spectroscopic surveys over the whole hemisphere on a reasonable timescale.
• Both the Taipan survey and the FunnelWeb survey are open collaborations that astronomers at any Australian research organisation may join, provided only that they are willing to commit a suitable level of effort to the project.
• The strongest immediate scientific drivers for the upgraded TAIPAN facility are encapsulated in the goals of the Taipan and FunnelWeb surveys. However the long-term national benefit of the upgraded facility flows from its capacity to support the broader goals of: (a) maximising return on investment in the ASKAP radio telescope and the SkyMapper optical telescope by providing spectra for many of the sources catalogued by ASKAP and SkyMapper, and (b) leveraging all of these Australian facilities to extract the maximum scientific impact from investment in access to international 8-metre class telescopes and the 25-metre next-generation Giant Magellan Telescope. and uses autonomous piezo-electric micro-robots ('starbugs') to move the optical fibres about in the curved focal plane of the UKST. The TAIPAN spectrograph is a fixed-format, two-channel spectrograph covering the range 370-850 nm at R=2000-2400 and delivering velocity resolution σ v ≈ 60 km s −1 with a total efficiency of ∼30%.
(a) (b) (c) (d)
Covering essentially no redshift range, the Taipan survey cannot examine the evolution of such quantities, but it can measure them in the present-day universe without extrapolations from high redshift that require assumptions about the cosmological model. For example, the Planck 87 CMB observations yield a measurement of H 0 = 67.3 ± 0.7 km s −1 Mpc −1 , but this assumes a ΛCDM model with particular parameters in order to transform measurements made at z ≈ 1100 to parameters at z = 0. The Taipan survey, by contrast, will obtain redshifts for ∼1 million galaxies at redshifts z < 0.2 and directly measure H 0.1 with 1% precision from the BAO standard ruler that is imprinted in the large-scale structure. Taipan thus will provide a bookend measurement of the present-day expansion rate with a precision matching that of Planck's measurement of the expansion rate shortly after the Big Bang and that of ongoing and future cosmological surveys, such as the eBOSS and the Dark Energy Survey (DES), at high and intermediate redshifts. A recent review of the prospects in this field 106 stated that: A measurement of the local value of H 0 to 1% precision and accuracy would provide key new insights into fundamental physics questions and lead to potentially revolutionary discoveries. While ongoing programs 107 to measure H 0 at low redshift using supernovae as standard candles aim for a similar level of precision to Taipan, they have different and (arguably) greater systematic errors with which they must contend.
Similarly, Taipan is expected to obtain substantially better measurements at z ≈ 0 of the growth rate of structure (f σ 8 ) and the velocity field scaling parameter (β), both of which should be determined to better than 5% precision. These low-redshift measurements are important because, again, they provide bookends to CMB measurements, but also because they depend on both the cosmological model and the theory of gravity, allowing tests of general relativity against alternative models. Combining Taipan's low-redshift constraints with higher-redshift MOS surveys and the CMB observations will significantly tighten the constraints on all important extensions to the standard cosmological model-in particular: the nature of dark energy and its evolution with time, the curvature of the universe as a test of inflationary models, the mass of neutrinos, and the total number of families of relativistic particles. 99
Higher Redshift Surveys
The primary focus of cosmological MOS surveys at higher redshifts is tracing the evolution of the geometry of the universe and the growth of large-scale structure over a period encompassing the epoch during which the universe changed from being matter-dominated to dark-energy-dominated (this occurs at z ≈ 0.8). For direct redshift measurements, the upper limit on this range is set by the depth achievable with 4-metre and 8-metre telescopes under the constraint that for each redshift bin the survey must cover at least ∼1 Gpc 3 in order to determine the BAO signal, and hence the co-moving distance at that redshift, with sufficient precision to usefully constrain the dark energy equation of state. The most ambitious such survey currently planned is DESI, 109,110 which aims to provide at least an order of magnitude improvement over BOSS/eBOSS, both in the comoving volume of the universe probed and in the total number of galaxies mapped. DESI is planned to start in 2018, with observations running for 5 years. It will be complementary to other planned new cosmological surveys such as those using the Large Synoptic Survey Telescope (LSST) and the EUCLID and WFIRST satellite missions.
The DESI instrument 111 will be a fiber-fed spectrograph using a robotic positioner system and will be capable of taking up to 5000 simultaneous spectra over the range 360-980 nm. The main components of the system, and one of the fiber positioners, are shown in Figure 10. The instrument will be installed at the prime focus of the 4-metre Mayall telescope at Kitt Peak and requires a new wide-field corrector and an atmospheric disperion compensator, which produce a 3.2-degree diameter field of view with an average scale of 14.1 arcsec/mm. The 5000 fibers can be positioned over 7.5 deg 2 of the available 8.0 deg 2 in the field of view. The fiber density in the focal plane is 667 deg −2 and the individual fibers have 107 µm (1.5 arcsec) cores. The positioners are arrayed hexagonally, with a 10.4 mm (147 arcsec) pitch between fibers. Each positioner has two rotational degrees of freedom allowing it to reach any point within a 6 mm (85 arcsec) radius. The focal plane is divided into ten pie-slice-shaped petals containing 500 fibres, each feeding one of the ten spectrographs. Volume phase holographic gratings provide spectral dispersion in the three spectrograph channels (360-593 nm, 566-772 nm and 747-980 nm) with resolutions greater than 2000, 3200, and 4100 respectively. The blue arm of the spectrographs use CCDs from Imaging Technologies Lab, while the red and near-infrared channels use LBNL CCDs; all will We define ⌦ r for radiation and ⌦ DE for dark energy analogously. The curvature term ⌦ k = k/H 2 0 is defined so that General Relativity requires
⌦ r + ⌦ m + ⌦ k + ⌦ DE = 1 (2.5)
for a Universe with spatial curvature k. The expansion rate of the Universe is given by
H(a) ⌘ȧ a = H 0 ⇥ ⌦ r a 4 + ⌦ m a 3 + ⌦ k a 2 + ⌦ DE F (a) ⇤ 1/2 . (2.6)
The contribution from radiation, ⌦ r is negligible today and inflation predicts that the curvature is zero. The Hubble constant today is H 0 = h ⇥ 100 km/s/Mpc⇡ 70 km/s/Mpc. We have three possible explanations for the accelerating expansion of the Universe: a cosmological constant, equivalent to static dark energy with w = 1; a dynamical dark energy with w(a) 6 = 1; or a failure of General Relativity. DESI is designed to address this fundamental question about the nature of the Universe. The challenge of distinguishing the cosmological constant solution from dark energy with w near 1 is displayed in Figure 2.1.
The Dark Energy Spectroscopic Instrument (DESI) [3] will provide precise spectroscopic redshifts of more than thirty million objects. From these will come three-dimensional maps of the be 4k×4k with 15 µm pixels. The system will have overall throughput around 35-40% over most of the spectral range, except at the bluest wavelengths.
The DESI instrument will be used to conduct a 5-year survey covering 14,000 deg 2 that will observe four classes of objects in order to map the matter distribution over the widest possible range of redshifts. 110 At moderate redshifts (up to z ≈ 1), DESI will observe luminous red galaxies (LRGs) in dark time and a magnitude-limited sample of up to 10 million galaxies with median redshift z ≈ 0.2 in bright time. To map the matter distribution at higher redshifts, DESI will use emission line galaxies (ELGs, specifically those with strong [O II] lines) to reach z ≈ 1.7. At the highest redshifts, DESI will use QSOs both as direct tracers of the matter distribution and, at 2.1 < z < 3.5, as probes of the Ly-forest, tracing the distribution of neutral hydrogen. In total, DESI aims to obtain more than 30 million galaxy and QSO redshifts in order to measure the BAO feature and the matter power spectrum, including redshift space distortions, over most of cosmic history.
If DES achieves these observational goals, it will produce more than 30 separate measurements of the expansion rate of the universe, each with precision better than 1%, over the redshift range z=0.2 to z=3.5. In terms of the Dark Energy Task Force figure of merit (FoM), which measures the combined precision on the dark energy equation of state today, w 0 , and its evolution with redshift, w a , DESI's galaxy BAO measurements are predicted to achieve a FoM of 133, which is more than 3× better than the FoM for all previous galaxy BAO measurements combined. If the Lyman-α forest measurements are also included, then DESI's predicted FoM increases to 169. Finally, including galaxy broadband power spectrum measurements for wavenumbers k < 0.1 h Mpc 1 , DESI's FoM rises still further to 332, and to 704 if it is possible to obtain reliable measurements for k < 0.2 h Mpc 1 .
In addition to providing vastly improved constraints on dark energy, DESI will also measure the sum of neutrino masses with an uncertainty of just 0.02 eV (again, if the power spectrum is measured reliably for k < 0.2 h Mpc 1 ). This should be sufficient to make the first direct detection of the sum of the neutrino masses at 3σ significance and to rule out the the inverted mass hierarchy with 99% confidence (if the hierarchy is normal and the masses are minimal). The survey will also place tighter constraints on alternative (non-GR) theories of gravity and on models for inflation by measuring the spectral index n s and its running with wavenumber, α s , and also the velocity fields of in the infall regions around clusters of galaxies. The cosmological constraints that DESI is predicted to achieve, and the gains relative to existing results, are summarized in Figure 11.
Other facilities on 4-metre and 8-metre telescopes that have the capability to carry out similar surveys include: WEAVE 112 on the William Herschel 4.2-metre telescope at La Palma Observatory (1000 fibers over a 3 deg 2 field from 2017); 4MOST 113 on ESO's 4.1-metre VISTA facility at Paranal (2400 fibers over a 4 deg 2 field from 2021); and the Prime Focus Spectrograph 114 (PFS) on the Subaru 8.2-metre telescope on Mauna Kea (2400 fibers over a 1.3 deg 2 field from 2019). However these facilities generally have much broader scientific goals than just cosmology surveys, and often are general-use facilities rather than dedicated survey facilities.
ELT Surveys
At higher redshifts, cosmological surveys such as BOSS and DESI use the Lyman-α forest in quasar spectra to map large-scale structure and place constraints on the geometry of the universe and the growth of large-scale structure just as galaxy redshift surveys do at lower redshifts. This approach will be even more powerful for the new generation of Extremely Large Telescopes (ELTs). With collecting areas 6-15 times greater than the largest existing telescopes, ELTs can use substantially fainter (and therefore much more common) background sources such as Lyman-break galaxies to illuminate the Lyman-α forest. This means ELTs will be able to probe the matter distribution on much shorter length scales than surveys using quasars and build much larger samples. The disadvantage of ELTs is that they have significantly smaller fields of view than the wide-field 4-metre telescopes (2-3 deg diameter) and 8-metre telescopes (up to 1.5 deg diameter).
The largest ELT field of view is the 20 arcmin diameter provided by the 25-metre Giant Magellan Telescope (GMT). This field will be fully exploited by the MANIFEST fiber system [103][104][105] (see also the paper by Lawrence et al. in these proceedings), which will use starbugs (as on the prototype TAIPAN system on the UKST) to position hundreds of small integral field units (effectively, image-slicers). MANIFEST will feed both the GMACS medium-resolution optical spectrograph and the G-CLEF high-resolution optical spectrograph (see paper by Jacoby et al. in these proceedings). The image-slicing capability means that MANIFEST more than doubles the normal slit-limited spectral resolution of GMACS, providing a much better match to the resolved velocity structure in the Lyman-α forest.
GMT+MANIFEST could be used to perform a large survey to construct a 3D map of the IGM, using Lyman break galaxies (LBGs) as background sources with which to probe the Lyman-α forest. The crucial wavelength range for such a survey is 0.36-0.56 µm, corresponding to a redshift range 2.0 < z < 3.5. The intrinsic size of the LBGs is approximately 2 kpc or 0.3 arcsec, so they are well matched to the fiber sampling of MANIFEST, with each starbug having 17 fibers each sampling 0.2 arcsec. With 4-hour exposures, GMT+MANIFEST can achieve sufficient S/N for targets at r ≈ 25, enabling a ∼1 deg 2 survey of LBG targets (with a surface density of ∼4000 deg −2 to r ≈ 24) to be achieved in about 20 nights. This would be complemented by a sparsely-sampled galaxy redshift survey of the same area, to a depth of r ≈ 26 (corresponding to a surface density of ∼30,000 deg −2 ), taking about 40 nights. Such a combined survey would provide constraints on the growth of small-scale structure at high redshifts that are not achievable with 8-metre class telescopes that have to use brighter (and sparser) QSOs as the background sources for the Lyman-α forest.
In the ELT era, the greatest advantage of 8-metre-class telescopes will be their relatively large fields of view. Thus, like 4-metre telescopes in the 8-metre era, 8-metre telescopes in the ELT era will have strong reasons to develop wide-field capabilities. Since LSST will dominate the 8-metre imaging niche, other 8-metre telescopes will likely seek to develop some variety of wide-field MOS capability as their cutting-edge instrumentation, as Subaru is already doing with PFS.
CONCLUSIONS
The contributions of multi-object spectroscopy to cosmology over the past two decades have been profound. Redshift surveys of large-scale structure are one of the pillars supporting modern precision cosmology, and have considerable potential to increase their precision and range of applicability. MOS instruments on 4-metre, 8metre, and the next generation of extremely large telescopes will continue to be powerful tools for determining the evolution of the universe and attacking the fundamental question of the nature of dark energy.
Figure 1 .
1(a) A schematic history of cosmological redshift surveys, showing the approximatw order and dates of key surveys. The labels give the survey name and the log of the multiplex and sample size (log M and log N resp.); the colour indicates whether the survey was non-MOS (orange), slit-MOS (purple) or fibre-MOS (blue). (b) The redshift maps for the main MOS surveys carried out with AAO facilities.
Figure 2 .
2(a) Schematic of the 2dF topend ring for the 3.9-metre Anglo-Australian Telescope (AAT), showing the location of the two spectrographs, and a cutaway view of the corrector and positioner system. (b) A section view of one of the 2dF spectrographs showing the main subsystems.
Figure 13
13Figure 13. Front view of the SDSS telescope depicting the location of the spectrographs. Here a fiber cartridge is shown retracted from the spectrographs. The twin spectrographs, each with a mass of 320 kg, mount to the back of the Cassegrain instrument rotator adjacent to the focal plane with sufficient separation between the spectrographs to allow routine installation and removal of the imaging camera and fiber cartridges.
Figure 14 .
14Section views of the SDSS spectrographs.
14 Figure 3 .
143(a) Schematic of the Apache Point Observatory 2.5-metre telescope showing the location of the spectrographs,34 with a fiber cartridge in the retracted position. The twin spectrographs mount to the back of the Cassegrain instrument rotator adjacent to the focal plane with sufficient separation to allow installation and removal of the imaging camera and fiber cartridges. (b) Two views of the SDSS spectrographs 34 and the main opto-mechanical subassemblies: the fiber slithead, the collimator, the central optics, and the red and blue channel cameras.
Figure 4 .
4Images of the 6dF instrument. (a) The UK Schmidt Telescope (UKST) at Siding Spring Observatory. (b) The 6dF field plate holder, showing optical fibres and buttons on the convex field plate. (c) The 6dF r-θ-z robot. (d) The 6dF spectrograph with its designer, Fred Watson.
Figure 1 . 1 :
11Physical layout of the AAOmega spectrograph.
Figure 5 .
5(a) A schematic of the AAOmega spectrograph showing the main components of the instrument. (b) A photo of the AAOmega spectrograph in operation on the AAT.
Figure 6 .
6The optical layout for the upgraded SDSS spectrographs used for the BOSS and eBOSS surveys. 34 test and distant supernovae;[70][71][72][73] (b) measuring the growth rate of large-scale structure using redshift space distortions of the two-and three-point galaxy correlation functions; 74-77 (c) jointly measuring the expansion and growth history out to z ≈ 1; 78 (d) constraining the sum of the neutrino masses; 79 (e) tracking the transition to large-scale cosmic homogeneity; 80 and (f) fitting the galaxy power spectrum to determine six key cosmological parameters (Ω b h 2 , Ω CDM , H 0 , τ , A s , n s ) and five supplementary parameters (n run , r, w, Ω k , m ν ), and showing that all are consistent with the standard ΛCDM model.69
6 Figure 1 .
61Projections for eBOSS LRG, ELG, and quasar distance measurements on a Hubble Diagram presented in comoving distance (⌘) versus redshift. Current BAO measurements from BOSS, SDSS (Xu et al. 2013; Ross et al. 2015), 6dF Galaxy Survey (6dFGS), and WiggleZ (Parkinson et al. 2012) are compared to SNe Ia measurements (Betoule et al. 2014) and Planck predictions (solid curve) obtained by marginalizing over the full likelihood function.
Reid et al. (2012) andSamushia et al. (2013) presented the first measurements and cosmological interpretation of RSD in the BOSS DR9 galaxy sample. With these results, they constrain the parameter combination f 8 = 0.43±0.07. Using the larger DR11 sample,Samushia et al. (2014) constrain the parameter combination f 8 = 0.447 ± 0.
Figure 2 .Figure 7 .
27Current RSD constraints on the growth as a function of redshift compared to the projected measurements from eBOSS. The current measurements include those discussed in Section 2.3.1 and those for 6dFGS (Beutler et al. 2012), the main SDSS sample (Howlett et al. 2015), 2dFGRS (Song & Percival 2009), the SDSS LRG sample (Oka et al. 2014), a recent result from the BOSS CMASS sample (Alam et al. 2015b), WiggleZ (Blake et al. 2012), and VIPERS (de la Torre et al. 2013). Various models of modified gravity are shown, each with the same background expansion, the same comoving BAO position, and amplitude of the power spectrum normalized to that of the CMB at high redshifts. The black curve shows the growth in a ⇤CDM universe, assuming the Planck best fit model parameters. The yellow curve shows = 0.5 where f = ⌦ M (Linder 2005). The purple curve shows = 0.6. Measurements of the distance-redshift relation and the growth rate of structure from MOS surveys. 85 (a) The comoving distance η(z) versus redshfit z relation predicted from the Planck ΛCDM model (shown as the solid black curve, with 1σ and 2σ uncertainties in gray); the existing measurements from BAO surveys (6dFGS, SDSS, WiggleZ and BOSS); the expected measurements from eBOSS; and a compilation of SNe Ia measurements. (b) The growth rate of structure f σ8 versus redshfit z relation predicted from the Planck ΛCDM model (shown as the solid black curve, with 1σ and 2σ uncertainties in gray); the existing measurements from previous redshift-space distortions from redshift surveys (6dFGS, 2dFGRS, SDSS, WiggleZ, VIPERS and BOSS); and the expected measurements from eBOSS. Also shown are curves corresponding to theories of gravity predicting growth going as f = Ω γ , with γ differing slightly from the General Relativity value of 0.55. redshift, which constrains the equation of state of the universe. The prediction comes from the Planck 87 ΛCDM model, while the observed points are from previous BAO survey measurements by 6dFGS, 52 WiggleZ 69 and SDSS/BOSS,
Figure 8 .
8Current and predicted constraints from cosmological MOS surveys. (a) Potential improvements in figures of merit for constraints on the dark energy equation of state and the theory of gravity for existing and potential future BAO and RSD redshift surveys, and other methods. 99 (b) The fractional error in constraints on the co-moving distance versus redshift relation, at various redshifts, from current and future MOS surveys. 100eBOSS is currently the state of the art as far as MOS cosmological surveys are concerned. However there are, of course, plans afoot for even more potent MOS surveys providing still greater discrimination between cosmological models and yet higher precision in determining cosmological parameters.
At low redshifts, the Taipan survey 101 on the newly-refurbished 1.2-metre UK Schmidt Telescope (UKST) aims to obtain high-precision constraints on the z ≈ 0 values of key parameters such as the Hubble constant (H 0 ) and the growth rate of structure. The survey will use the new TAIPAN 102 fibre positioner and spectrograph, which are currently under construction at the Australian Astronomical Observatory. The TAIPAN positioner is a prototype for the MANIFEST 103-105 fibre positioning system planned for the Giant Magellan Telescope (GMT), The panels in the figure above show the components of the UKST+TAIPAN system: (a) the location of the TAIPAN fibre positioner within the UKST (in cutaway view); (b) a close-up view of the fibre positioner system containing the Starbugs; (c) the TAIPAN spectrograph layout, showing the blue and red channels covering the whole optical wavelength range; and (d) an image of some Starbugs being tested during the manufacturing process at AAO.
Figure 9 .
9The components of the UKST+TAIPAN system: (a) the location of the TAIPAN fibre positioner within the UKST (in cutaway view); (b) a close-up rendering of the fibre positioner system containing the Starbugs; (c) the TAIPAN spectrograph layout, showing the blue and red channels covering the whole optical wavelength range; and (d) an image of some Starbugs being tested during the manufacturing process at AAO.
Figure 1 . 2 :Figure 10 .
1210DESI block diagram. The DESI instrument.111 (a) Block diagram showing the main components of the instrument on the KPNO Mayall 4-metre telescope: 1. wide-field corrector and atmospheric dispersion compensator producing a 3.2-degree field of view at the telescope prime focus; 2. focal plane assembly, with 5000 fibers in 10 'petal' segments covering the field of view; 3. fiber management system and fiber cable run; 4. ten spectrographs, each taking 500 fibers from one of the 'petals' and each with three (blue/green/red) arms. (b) one of the robotic actuators for positioning individual fibres.
Figure 2 . 15 :Figure 11 .
21511Improvement in the measurements of wp, w 0 = wa, ⌦k, P m⌫ the sum of the neutrino masses, ns the spectral index, ↵s the running of the spectral index, and N⌫,e↵ the number of neutrino-like (relativistic) species. Predicted cosmological constraints and figures of merit for the DESI survey. 110 (a) Measurements of the expansion rate, comparing DESI to the best current BAO and supernovae measurements. The inset illustrates the very high precision needed to distinguish models with constant w ranging from 0.97 to 1.03. (b) The improvements predicted for DESI in determining wp, w = wa, Ω k , mν , ns, αs, and Nν,e, the number of neutrino-like (relativistic) species..
Bimba Manufacturing Company, http://www.bimba.com 41 Clippard Instrument Laboratory Inc., http://www.clippard.com exposure time, 4 s, is set by the arc lamps used for wavelength calibration. Flat fields require 30 s exposures. Science exposures are 15 minutes.
Table 1 .
1A list of the telescope:instrument combinations used to perform various MOS surveys (cosmological and other).Telescope : Instrument
Survey(s)
AAT : AAΩ, HERMES, SAMI
WiggleZ, GAMA, GALAH, SAMI
UKST : 6dF, TAIPAN
RAVE, 6dFGS, Taipan
SDSS 2.5m : SDSS, BOSS, MaNGA
SDSS, BOSS, eBOSS, MaNGA
WHT : ISIS, WEAVE
WEAVE
GTC : OSIRIS, EMIR, MEGARA
VLT : VIMOS, KMOS, MUSE,
MOONS
VANDELS, VIPERS, LEGA-C,…
VISTA : 4MOST
4MOST, WAVES
LAMOST
LEGUE
Calar Alto : CALIFA
CALIFA
Magellan : PRIMUS
PRIMUS
Subaru : FMOS, PFS
FastSound
HET : VIRUS
HETDEX
GAIA
GAIA
KPNO 4m : DESI
DESI
Euclid
Euclid
LSST:MOS / MSE / WFIRST
…
RedshiftBillions of Years from Today Scale of the UniverseRelative to Today's ScaleFigure 2.1: The expansion history of the Universe for di↵erent models of dark energy, holding the present-day Hubble constant fixed. The inset shows the spacing between five models with constant w ranging from 0.97 to 1.03, showing the exquisite precision required to distinguish these. Overlaid are measurements of the distance-redshift relation, translated into errors on lookback time at each redshift. Measurements from current supernovae, binned in redshift, are shown in blue; current BAO measurements from BOSS DR9, WiggleZ, and 6dF are shown in red; projections for DESI are shown in black. DESI measurements have the ability to make very tight constraints on dark energy, although we caution that this figure shows variations in only one cosmological parameter. Full forecasts, such as those presented in § 2.4.3, must marginalize over other cosmological parameters such as ⌦ m and H 0 .2 SCIENCE MOTIVATION AND REQUIREMENTS
4
SNe (binned)
Current BAO
DESI (predicted)
Alternative Universes
for Constant
ACKNOWLEDGMENTSThis work supported in part by Australian Research Council grants LE140100052 & DP160102075. The author thanks his collaborators on the 2dFGRS, 6dFGS, WiggleZ, and Taipan science teams and on the 2dF, 6dF, OzPoz, TAIPAN and MANIFEST instrument teams for making the last 20 years both productive and pleasurable.
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[] |
[
"The earliest O-type eclipsing binary in the Small Magellanic Cloud, AzV 476: A comprehensive analysis reveals surprisingly low stellar masses",
"The earliest O-type eclipsing binary in the Small Magellanic Cloud, AzV 476: A comprehensive analysis reveals surprisingly low stellar masses"
] |
[
"D Pauli \nInstitut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24/2514476PotsdamGermany\n",
"L M Oskinova \nInstitut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24/2514476PotsdamGermany\n",
"W.-R Hamann \nInstitut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24/2514476PotsdamGermany\n",
"V Ramachandran \nInstitut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24/2514476PotsdamGermany\n\nZentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\nMönchhofstr. 12-1469120Heidelberg, Ger-many\n",
"H Todt \nInstitut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24/2514476PotsdamGermany\n",
"A A C Sander \nArmagh Observatory and Planetarium\nCollege Hill, Northern IrelandBT61 9DGArmaghUK\n\nZentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\nMönchhofstr. 12-1469120Heidelberg, Ger-many\n",
"T Shenar \nInstitute of Astronomy\nKU Leuven\nCelestijnenlaan 200D3001LeuvenBelgium\n",
"M Rickard \nInstitut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24/2514476PotsdamGermany\n\nDepartment of Physics and Astronomy\nUniversity College London\nGower StreetWC1E 6BTLondonUK\n",
"J Maíz Apellániz \nCentro de Astrobiología. CSIC-INTA. Campus ESAC\nCamino bajo del castillo s/n. E-28 692 Villanueva de la CañadaMadridSpain\n",
"R Prinja \nDepartment of Physics and Astronomy\nUniversity College London\nGower StreetWC1E 6BTLondonUK\n"
] |
[
"Institut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24/2514476PotsdamGermany",
"Institut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24/2514476PotsdamGermany",
"Institut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24/2514476PotsdamGermany",
"Institut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24/2514476PotsdamGermany",
"Zentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\nMönchhofstr. 12-1469120Heidelberg, Ger-many",
"Institut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24/2514476PotsdamGermany",
"Armagh Observatory and Planetarium\nCollege Hill, Northern IrelandBT61 9DGArmaghUK",
"Zentrum für Astronomie\nAstronomisches Rechen-Institut\nUniversität Heidelberg\nMönchhofstr. 12-1469120Heidelberg, Ger-many",
"Institute of Astronomy\nKU Leuven\nCelestijnenlaan 200D3001LeuvenBelgium",
"Institut für Physik und Astronomie\nUniversität Potsdam\nKarl-Liebknecht-Str. 24/2514476PotsdamGermany",
"Department of Physics and Astronomy\nUniversity College London\nGower StreetWC1E 6BTLondonUK",
"Centro de Astrobiología. CSIC-INTA. Campus ESAC\nCamino bajo del castillo s/n. E-28 692 Villanueva de la CañadaMadridSpain",
"Department of Physics and Astronomy\nUniversity College London\nGower StreetWC1E 6BTLondonUK"
] |
[] |
Context. Massive stars at low metallicity are among the main feedback agents in the early Universe and in present-day star forming galaxies. When in binaries, these stars are potential progenitors of gravitational-wave events. Knowledge of stellar masses is a prerequisite to understanding evolution and feedback of low-metallicity massive stars. Aims. Using abundant spectroscopic and photometric measurements of an outstandingly bright eclipsing binary, we compare its dynamic, spectroscopic, and evolutionary mass estimates and develop a binary evolution scenario. Methods. We comprehensively studied the eclipsing binary system, AzV 476, in the Small Magellanic Cloud (SMC). The light curve and radial velocities were analyzed to obtain the orbital parameters. The photometric and spectroscopic data in the UV and optical were analyzed using the Potsdam Wolf-Rayet (PoWR) model atmospheres. The obtained results are interpreted using detailed binaryevolution tracks including mass transfer. Results. AzV 476 consists of an O4 IV-III((f))p primary and an O9.5: Vn secondary. Both components have similar current masses (20 M and 18 M ) obtained consistently from both the orbital and spectroscopic analysis. The effective temperatures are 42 kK and 32 kK, respectively. The wind mass-loss rate of log(Ṁ/( M yr −1 )) = −6.2 of the primary is a factor of ten higher than a recent empirical prescription for single O stars in the SMC. Only close-binary evolution with mass transfer can reproduce the current stellar and orbital parameters, including orbital separation, eccentricity, and the rapid rotation of the secondary. The binary evolutionary model reveals that the primary has lost about half of its initial mass and is already core helium burning. Conclusions. Our comprehensive analysis of AzV 476 yields a consistent set of parameters and suggests previous case B mass transfer. The derived stellar masses agree within their uncertainties. The moderate masses of AzV 476 underline the scarcity of bright massive stars in the SMC. The core helium burning nature of the primary indicates that stripped stars might be hidden among OB-type populations. 1 the shorter wavelengths in the FUSE range are largely contaminated by interstellar features that yield no information about the stellar parameters 2 https://ullyses.stsci.edu/ Article number, page 2 of 23 D. Pauli et al.: The earliest O-type eclipsing binary in the Small Magellanic Cloud, AzV 476.
|
10.1051/0004-6361/202141738
|
[
"https://arxiv.org/pdf/2201.09148v1.pdf"
] | 246,240,803 |
2201.09148
|
c618e5b4479ac4f529a8c198f86fcd0be3861115
|
The earliest O-type eclipsing binary in the Small Magellanic Cloud, AzV 476: A comprehensive analysis reveals surprisingly low stellar masses
January 25, 2022
D Pauli
Institut für Physik und Astronomie
Universität Potsdam
Karl-Liebknecht-Str. 24/2514476PotsdamGermany
L M Oskinova
Institut für Physik und Astronomie
Universität Potsdam
Karl-Liebknecht-Str. 24/2514476PotsdamGermany
W.-R Hamann
Institut für Physik und Astronomie
Universität Potsdam
Karl-Liebknecht-Str. 24/2514476PotsdamGermany
V Ramachandran
Institut für Physik und Astronomie
Universität Potsdam
Karl-Liebknecht-Str. 24/2514476PotsdamGermany
Zentrum für Astronomie
Astronomisches Rechen-Institut
Universität Heidelberg
Mönchhofstr. 12-1469120Heidelberg, Ger-many
H Todt
Institut für Physik und Astronomie
Universität Potsdam
Karl-Liebknecht-Str. 24/2514476PotsdamGermany
A A C Sander
Armagh Observatory and Planetarium
College Hill, Northern IrelandBT61 9DGArmaghUK
Zentrum für Astronomie
Astronomisches Rechen-Institut
Universität Heidelberg
Mönchhofstr. 12-1469120Heidelberg, Ger-many
T Shenar
Institute of Astronomy
KU Leuven
Celestijnenlaan 200D3001LeuvenBelgium
M Rickard
Institut für Physik und Astronomie
Universität Potsdam
Karl-Liebknecht-Str. 24/2514476PotsdamGermany
Department of Physics and Astronomy
University College London
Gower StreetWC1E 6BTLondonUK
J Maíz Apellániz
Centro de Astrobiología. CSIC-INTA. Campus ESAC
Camino bajo del castillo s/n. E-28 692 Villanueva de la CañadaMadridSpain
R Prinja
Department of Physics and Astronomy
University College London
Gower StreetWC1E 6BTLondonUK
The earliest O-type eclipsing binary in the Small Magellanic Cloud, AzV 476: A comprehensive analysis reveals surprisingly low stellar masses
January 25, 2022Received ; AcceptedAstronomy & Astrophysics manuscript no. mainbinaries: eclipsing -binaries: close -binaries: spectroscopy -stars: early-type -stars: fundamental parameters -stars: individual: AzV 476
Context. Massive stars at low metallicity are among the main feedback agents in the early Universe and in present-day star forming galaxies. When in binaries, these stars are potential progenitors of gravitational-wave events. Knowledge of stellar masses is a prerequisite to understanding evolution and feedback of low-metallicity massive stars. Aims. Using abundant spectroscopic and photometric measurements of an outstandingly bright eclipsing binary, we compare its dynamic, spectroscopic, and evolutionary mass estimates and develop a binary evolution scenario. Methods. We comprehensively studied the eclipsing binary system, AzV 476, in the Small Magellanic Cloud (SMC). The light curve and radial velocities were analyzed to obtain the orbital parameters. The photometric and spectroscopic data in the UV and optical were analyzed using the Potsdam Wolf-Rayet (PoWR) model atmospheres. The obtained results are interpreted using detailed binaryevolution tracks including mass transfer. Results. AzV 476 consists of an O4 IV-III((f))p primary and an O9.5: Vn secondary. Both components have similar current masses (20 M and 18 M ) obtained consistently from both the orbital and spectroscopic analysis. The effective temperatures are 42 kK and 32 kK, respectively. The wind mass-loss rate of log(Ṁ/( M yr −1 )) = −6.2 of the primary is a factor of ten higher than a recent empirical prescription for single O stars in the SMC. Only close-binary evolution with mass transfer can reproduce the current stellar and orbital parameters, including orbital separation, eccentricity, and the rapid rotation of the secondary. The binary evolutionary model reveals that the primary has lost about half of its initial mass and is already core helium burning. Conclusions. Our comprehensive analysis of AzV 476 yields a consistent set of parameters and suggests previous case B mass transfer. The derived stellar masses agree within their uncertainties. The moderate masses of AzV 476 underline the scarcity of bright massive stars in the SMC. The core helium burning nature of the primary indicates that stripped stars might be hidden among OB-type populations. 1 the shorter wavelengths in the FUSE range are largely contaminated by interstellar features that yield no information about the stellar parameters 2 https://ullyses.stsci.edu/ Article number, page 2 of 23 D. Pauli et al.: The earliest O-type eclipsing binary in the Small Magellanic Cloud, AzV 476.
Introduction
The most important parameter defining the evolution of a star is its mass. An often reported problem in stellar astrophysics is the "mass discrepancy" problem, which refers to the inconsistent masses derived from spectroscopy, evolutionary tracks, and, for binaries, from orbital motions (Herrero et al. 1992;Weidner & Vink 2010;Markova & Puls 2015). The analysis of O-type stars located in the Galaxy and the Large Magellanic Cloud (LMC) performed by Weidner & Vink (2010) reveal good agreement between the aforementioned mass estimates, but the study of Markova et al. (2018) suggests a mass discrepancy. Mahy et al. (2020) investigate a sample of O-type binaries in the LMC and find good agreement between spectroscopic and dynamic mass estimates while the evolutionary masses are at odds. These latter authors suggest previous binary interactions as a possible solution for this discrepancy.
However, the mass discrepancy problem has not yet been studied at metallicities lower than 1/2 Z . Sufficiently low metallicity is offered by the nearby Small Magellanic Cloud (SMC) galaxy (Z SMC ≈ 1/7 Z ; Hunter et al. (2007); Trundle et al. (2007)). Precise stellar masses in the SMC allow us to tackle questions about stellar evolution and feedback at low metallicity.
Stars with spectral types around O2-4 are expected to be very massive with M * 50 M (Martins & Palacios 2021). However, the true masses of the early-type O stars are only poorly known, and therefore spectral types and masses might be falsely mapped. So far, only a couple of the SMC eclipsing binaries with early spectral types have been studied. Morrell (2003) investigated the O6V+O4/5III(f) system Hodge 53-47 (alias MOA J010321.3-720538) and found dynamic masses of ≈ 26 M and ≈ 16 M for the primary and secondary, respectively. The eclipsing binary, OGLE SMC-SC10 108086, studied by Abdul-Masih et al. (2021) contains even less massive stars with ≈ 17 M and ≈ 14 M . Only one eclipsing multiple stellar system in the SMC, HD 5980, appears to contain stars with masses 30 M . Koenigsberger et al. (2014) and Hillier et al. (2019) studied this system in detail and found an inner eclipsing binary consisting of an LBV and a Wolf-Rayet (WR) star, and a third O-type supergiant star with a potential fourth companion. The orbital masses of the LBV and the WR star are ≈ 61 M and ≈ 66 M , respectively. Koenigsberger et al. (2014) suggest that there was little or no mass transfer, and that the WR star has formed via quasi-chemically homogeneous evolution. These examples highlight the complexity of massive star evolution and the need for advanced studies on the mass discrepancy problem in the high-mass regime.
For single stars, mass estimates rely on spectroscopic diagnostics or comparison with evolutionary tracks. Currently, standard stellar evolutionary models predict that the most massive stars, either single or in binary systems, start their lives on the main sequence (MS) as early-type O stars. Single stars expand and evolve away from the MS. During this phase, stars undergo strong mass loss where they might lose their entire hydrogenrich envelope, revealing the helium core and becoming WR stars. In the case of close binaries, evolutionary models predict that the expanding star is likely to interact with its companion. In the majority of cases, the expanding star will transfer its envelope to the companion, and become a binary-stripped helium star, possibly also with a WR-type spectrum (Dionne & Robert 2006;Shenar et al. 2020;Götberg et al. 2020).
The SMC hosts a handful of WR stars that have relatively high masses ranging from 10 M to 60 M (Shenar et al. 2016). These stars are so massive that their hydrogen-burning progenitors must have been early O-type (or WNL/Of) stars. However, in their recent study of the SMC OB-type population, Ramachandran et al. (2019) reveal a strong deficiency of massive stars (> 30 M ) close to the MS in the upper part of the empiric Hertzsprung-Russell diagram (HRD) (see Holgado et al. 2020;Ramachandran et al. 2018, for studies of Galactic and the LMC O star populations). O-type stars are hot and luminous and are therefore easily detectable. Hence, a paucity of the earliest Otype stars in the SMC cannot be explained either by selection effects or stellar evolution scenarios (Schootemeijer et al. 2021). Thus, the deficiency of most massive O stars strongly questions the formation process of the WR stars in the SMC, as well as our basic understanding of stellar evolution in low-metallicity environments. In this context it is crucial to quantify the budget of massive stars ( 30 M ) in the SMC.
Poorly constrained processes that affect the lives of massive stars include radiatively driven winds, stellar envelope inflation, core overshooting, and rotationally induced internal mixing. In particular their dependence on mass and metallicity are not yet fully understood. These effects can drastically alter the evolution of a star. Studies of objects in low-metallicity environments are needed to constrain these metallicity-dependent effects and thus allow the establishment of reliable stellar evolutionary tracks.
To address these outstanding questions, we selected one of the earliest subtype O stars in the SMC that is an eclipsing binary, allowing us to estimate its mass by various methods. The subject of this paper, AzV 476, is located in the cluster NGC 456 in the SMC Wing. The cluster contains an active star-forming region and hosts young stellar objects (Muraoka et al. 2017).
AzV 476 is embedded in a H ii region. The system was identified as an eclipsing binary by the Optical Gravitational Lensing Experiment (OGLE), which monitors stellar variability in the SMC and LMC (Pawlak et al. 2016). The primary star was classified by Massey et al. (2005) as an O2-3 V star based on its optical spectral appearance and is therefore one of the earliest type stars in the entire SMC. Here we present the first consistent analysis of AzV 476, based on photometric and spectroscopic data, and in particular accounting for its binary nature. Newly obtained high-resolution UV and optical data give us the opportunity to perform such an analysis, yielding estimates on masses and stellar and wind parameters.
This paper is structured as follows. In Sect. 2 we describe the observations and the known stellar and orbital parameters used in the analysis below. The results of the orbital and spectral analysis as well as those from evolutionary modeling are presented in Sects. 3, 4, and 5, respectively. Their implications, similarities, and disagreements on the stellar masses and other stellar parameters are discussed in Sect. 6, and conclusions are given in Sect. 7.
Observations
Spectroscopy
Over the last decade, a handful of AzV 476 spectra covering the UV, optical (VIS), and infrared (IR) have been obtained, yielding multi-epoch data well suited for measuring radial velocities (RVs) and spectroscopic analysis. We used a spectroscopic dataset consisting of 12 spectra and Table 1 gives a brief overview of these, their covered wavelength ranges, their observed date, and their associated orbital phase. In the remainder of this paper, we refer to the individual observed spectra by their ID in Table 1.
The wavelength regime 950 -1150 Å is covered by an archival FUSE (Oegerle et al. 2000) observations 1 . The FUSE spectrum (ID 1) was taken with a total exposure time of 21 567 s and a resolving power of R ≈ 20 000. Incidentally, the FUSE observation was taken close to the primary eclipse. This FUSE spectrum has a known but unsolved calibration issue. Therefore, we only used the LiF1A, LiF2A, and SiC2A channels, which appear to be the least affected by this problem. The spectrum is rectified by division through the combined continuum flux of our models.
AzV 476 is part of the ULLYSES program 2 . It was observed with the HST/COS (Hirschauer et al. 2021) using the G130M (1178 -1472 Å) and G160M (1383 -1777 Å) mediumresolution gratings (ID 2), R ≈ 19 000. The two spectra were taken sequentially with exposure times of 330 s and 1100 s, respectively. In addition, the star was re-observed in the UV with the HST/STIS spectrograph (Branton et al. 2021) using the E140M echelle graiting (ID 3) as part of the HST program 15837 (PI Oskinova). The spectrograph covers a wavelength regime of 1140 -1735 Å. The exposure time was 2707 s and the final resolving power is R ≈ 45 800.
There is another spectrum in the ULLYSES program (ID 4) taken with the HST/STIS spectrograph using the E230M echelle gratings covering a wavelength range 1574 -2673 Å. The exposure time was 2820 s and the resolving power is R = 30 000. For the optical and near-IR range, we use the publicly available spectra taken with the X-SHOOTER spectrograph (Vernet et al. 2011) mounted on the ESO Very Large Telescope (VLT). The spectrum (ID 5) was taken as part of the ESO 106.211Z program, which is part of the XSHOOTU program (their Paper I; Vink and the XShootU Collaboration, in prep.). The X-SHOOTER spectrograph consists of three different spectroscopic arms, which are optimized for the wavelength ranges in the UBV (3000 -5550 Å), VIS (5300 -10 000 Å), and near-IR (10 000 -25 000 Å). The resolving powers are R ≈ 6 600, R ≈ 11 000, and R ≈ 8 000, respectively. The spectra were obtained with exposure times of 750 s, 820 s, and 300 s for the UBV, VIS, and NIR arm, respectively.
The remaining seven spectra (ID 6 -12) used for our analysis are taken with the UVES spectrograph (Dekker et al. 2000) mounted on the ESO VLT. Each spectrum was taken with the DIC1 setup covering the wavelength ranges of 3000 -4000 Å and 5000 -11 000 Å. The exposure times are about 2895 s for both spectrographs and have a resolving power of R ≈ 65 000 and R ≈ 75 000, respectively. The UVES spectra were rectified by hand.
Photometry
The UBI photometry is adopted from the catalog of the SMC stellar population (Bonanos et al. 2010). For the VR photometry, we use the magnitudes from the fourth United States Naval Observatory (USNO) CCD Astrograph Catalog (UCAC4) (Zacharias et al. 2013). For completeness, we compare the values of the V-and I-band magnitudes to those listed in the IVth OGLE Collection of Variable Stars (Pawlak et al. 2016) (V OGLE and I OGLE ). Unfortunately, the OGLE magnitudes are published without error margins. Nonetheless, we find that they are in agreement with the V-and I-band magnitude from Zacharias et al. (2013) and Bonanos et al. (2010). JHK photometry is from the 2MASS catalog (Cutri et al. 2003). Additionally, we use the recent EDR3 Gaia photometry (Gaia Collaboration et al. 2016, 2021. A total list of the used magnitudes is shown in Table 2. Massey et al. (2005) estimated the extinction towards AzV 476 to be E B−V = 0.28 mag based on averaging the color excesses in B−V and U − B based on the spectral type. From our spectral energy distribution (SED) fit we find better agreement when using a lower extinction of E B−V = 0.26 mag (see Sect. 4.1). The slight difference can be due to the use of different reddening laws.
For the light curve modeling, we use the OGLE I-band photometry from the IVth OGLE Collection of Variable Stars (Pawlak et al. 2016). The photometric data are taken over the period from May 2010 to January 2014. We do not use the OGLE V-band photometry because it contains too few data points and therefore cannot be used to resolve the eclipses.
AzV 476 was observed by the TESS space telescope in 2018 (sector 2) and 2020 (sectors 27 and 28). However, the relatively low spatial resolution of TESS (21 px −1 ) precludes accurate point source photometry in the crowded region around our target. Therefore, the TESS data are not used for light-curve modeling. However, we use the TESS data to improve the ephemeris (see Sect. 3.1.1).
Distance and location
Our target, AzV 476 is part of the NGC 456 cluster which is located in the SMC Wing. The distance to the SMC Wing was estimated by Cignoni et al. (2009) d ≈ 55 kpc corresponding to a distance modulus of DM = 18.7 mag. This is in agreement with Nidever et al. (2013) who use red clump stars to study the structure of the SMC, including the SMC Wing, and find that it follows a bimodal distribution with a near component at a distance of d ≈ 55 kpc and a far component at d ≈ 67 kpc. Tatton et al. (2021) confirm this distance, but they speculate that young structures, such as the NGC 456 cluster, do not trace substructures that are associated with the intermediate-age populations and might be located in front of them (see Sect. 6.1). In this work we adopt a distance of d = 55 kpc.
The observed spectra of AzV 476 are corrected for barycentric motion, which was calculated with the tool described in Wright & Eastman (2014). Additionally, we shifted the spectra by the RV of the NGC 456 complex in the SMC SMC = 152 km s −1 determined by fitting Gaussians to several interstellar medium (ISM) lines that we associate with the environment of AzV 476. The RV is not uniform throughout the different regions in the SMC and our finding is in agreement with the results of De Propris et al. (2010). (Pawlak et al. 2016) with an orbital period of P OGLE = 9.366 319 8 d and an epoch of the primary eclipse of T 0, OGLE = 2457002.7608 in Heliocentric Julian Date (HJD). Unfortunately, the orbital period and the epoch of the primary eclipse are given without error margins. The observations that constitute the OGLE I-band light curve were taken with a cadence of ≈ 2 d, and therefore the individual eclipses are not well resolved. In contrast, the TESS light curve has a much finer time coverage of ≈ 5 h, allowing better constraint of the orbital period and the epoch of the primary eclipse.
Analysis of the binary orbit
The date at which an eclipse occurs can be expressed as
T (n) = P · n + T 0 ,(1)
where T is the time of the primary eclipse at orbital cycle n. We fitted Gaussians to the primary eclipses in the TESS light curve; the corresponding orbital cycles and dates of all eclipses are listed in Table 3. Using this procedure, we obtain P = (9.366 65 ± 0.000 25) d and T 0 = 2457002.7968 ± 0.0052. Figure 1 shows a part of the phased light curve centered on the primary eclipse for the two different ephemerides, those published in the OGLE catalog and those we obtain in this work. As can be seen, our newly obtained ephemeris adequately describes the OGLE as well as the TESS light curves, i.e., the data set that covers > 10 yr of observations.
In order to convert a date t at which a spectrum was taken to a phase Φ, the following formula is used With the improved ephemerides and their small error margins, the uncertainties on the phases are negligible and are therefore not taken into account here. The OGLE I-band light curve is shown in the upper panel of Fig. 2. The conjunctions are at phases φ = 0.0 and φ = −0.34, implying that the orbit is eccentric.
φ(t) = t − T 0 P mod 1 if φ < 0.5 t − T 0 P mod 1 − 1 otherwise.(2)
RVs
To measure the RVs, we use a Markov chain Monte Carlo (MCMC) method combined with a least-square fitting method. In the MCMC method, the individual synthetic spectra of the primary and secondary (see Sect. 4.1) are shifted by different RVs. The combined synthetic spectrum is then compared to the observation. Using a least-square likelihood function we estimate the quality of the used RVs. The MCMC method quickly explores a large parameter space of different RVs until it converges toward the true solution. In the vicinity of the true solution, the MCMC method calculates the probabilities of different combinations of the RVs. This yields the final probability distribution around the true solution. A more detailed explanation is given in Appendix A. Because the final probability distribution obtained with the MCMC method is not necessarily a Gaussian, we quote the error as the 68% confidence interval. The PHOEBE code cannot handle asymmetric errors, and therefore we only use the larger margin of the probability distribution as it is the safer choice. The uncertainties are included in the eclipsing binary modeling as described in Sect. 3.1.3.
The primary dominates the emission and absorption lines. In order to avoid uncertainties due to the intrinsic variability of lines formed in the stellar wind, the RVs listed in Table 1 are the averaged values of the RVs obtained from selected individual lines. A more detailed list of the RVs obtained from the individual lines in the different optical spectra is given in Tables A.1 and A.2.
All the absorption lines that are associated with the secondary show a contribution from the primary. Furthermore, the depths of these absorption lines are at the level of the noise, which introduces additional uncertainties; these are reflected in the larger error margins. Two selected optical spectra obtained at different phases are depicted in Fig. 3 to demonstrate how the spectral lines associated with the different binary components shift.
Modeling with PHOEBE
The Physics of Eclipsing Binaries (PHOEBE) v.2.3 modeling software (Prša & Zwitter 2005;Prša et al. 2016;Horvat et al. 2018;Jones et al. 2020;Conroy et al. 2020) is employed to derive dynamic masses as well as to obtain measures of the stellar parameters independently from the spectroscopic model. The si- Fig. 3. X-SHOOTER spectrum (ID 5 in Table 1) displayed in blue and one of the UVES spectra (ID 10 in Table 1) in red. The spectra are convolved with a Gaussian with an FWHM = 0.4 Å to reduce the noise and to make the RV shifts visible. The region containing Hγ, He i λ4387, and He i λ4471 lines is shown. In the spectrum shown by the red line, the primary's spectrum is redshifted (see Hγ), while the secondary's spectrum (broadened He i lines) is blueshifted. We note that the primary also partially contributes to the He i lines. multaneous fitting of the RV and light curve is done with the emcee sampler (Foreman-Mackey et al. 2013).
To reduce the parameter space, we fix the orbital period and the epoch of the primary eclipse to the values we derive from the TESS light curve; see Sect. 3.1.1. As we are using only one passband, the temperatures of the components cannot be obtained reliably. Therefore, the temperatures of the primary and secondary are fixed to the results from the spectral analysis (see Sect. 4.1), T eff, 1 = 42 kK and T eff, 2 = 32 kK, respectively.
The PHOEBE code assumes synchronous stellar rotation. The actual fast rotation of the secondary with sin i = 425 km s −1 is therefore not consistently taken into account for modeling the secondary eclipse. Gravitational darkening is modeled by a power law with a coefficient of β grav = 1, as recommended for radiative envelopes in the PHOEBE documentation 3 . Given the period of ∼ 10 d, this renders gravitational darkening unimportant.
The atmospheres of both components are approximated by a blackbody. We compared the blackbody flux to the SED of our atmospheric model and find that it is a valid approximation for the I-band flux. We calculated the emergent flux distribution using our spectral models (see Sect. 4.1) and fitted different types of limb-darkening laws (e.g., Diaz-Cordoves & Gimenez 1992). We find that the limb-darkening law that best describes the primary and secondary is a quadratic approximation in the form of
I(µ) = I(1) [1 − a i (1 − µ) − b i (1 − µ) 2 ],(3)
where µ = cos θ is the cosine of the directional angle θ, and a i and b i are the limb-darkening coefficients of each stellar component i. For the primary, the best fit is achieved with coefficients a 1 = 0.2032 and b 1 = 0.0275, while for the secondary a 2 = 0.1668 and b 2 = 0.0802 are required. In a binary where both stellar components have large radii, on the order of 20% of their separation, and rather similar temperatures, the so-called "reflection effect" becomes important (Wilson 1990). This effect accounts for the irradiation of the surface by the other component (Prša 2011;Prša et al. 2016). As AzV 476 is a close binary with two hot O-stars, the reflection effect is modeled with two reflections. The effect of ellipsoidal Fig. 3, but now for the region around the He i λ3819 and H lines. Most of the lines, including He i λ3926, are redshifted and associated with the primary, while the He i λ3819 line which is associated with the secondary is blueshifted. variability due to tidal interaction, which induces periodic variations in the light curve, is important in close binary systems with orbital periods on the order of a few days (Mazeh 2008). However, as none of the binary components are close to filling their Roche lobe (see Table 5) and the mass ratio is not extreme (q orb = 0.89) this effect is expected to be negligible (Gomel et al. 2021).
Resulting binary parameters
The best fitting model light curve and RV fit obtained with the PHOEBE code are shown in Fig. 2. The corresponding orbital parameters are listed in Table 4 and the stellar parameters of both components in Table 5. The orbital solution yields similar masses for both components, while the light curve fit reveals different T eff and L. This indicates that there was a prior masstransfer phase that has stripped away most of the primary's envelope, such that now has similar mass to its companion. From the light-curve fit, the fundamental stellar parameters -stellar radius and the surface gravity-are determined, giving us the opportunity to cross-check the spectral analysis. Furthermore, the PHOEBE code calculates the light ratio of the binary components in the observed band outside conjunctions, which is compared to the light ratio obtained from the spectroscopic analysis in Sect. 4.
Spectral analysis
Method: Spectral modeling with PoWR
Synthetic spectra for both stellar components were calculated with the Potsdam Wolf-Rayet (PoWR) model atmosphere code (Gräfener et al. 2002;Hamann & Gräfener 2004). In the following, we briefly describe the code. For further details, we refer to Gräfener et al. (2002), Hamann & Gräfener (2003), Todt et al. (2015), .
The PoWR code models stellar atmospheres and winds permitting departures from local thermodynamic equilibrium (non-LTE). The code has been widely applied to hot stars at various metallicities (e.g., Hainich et al. 2014Hainich et al. , 2015Oskinova et al. 2011;Reindl et al. 2014;Shenar et al. 2015). The models are assumed to be spherically symmetric, stationary, and in radiative equilibrium. Equations of statistical equilibrium are solved in turn with the radiative transfer in the co-moving frame. Consistency is achieved iteratively using the "accelerated lambda op-
ω 0 ( • ) 19 +2 −1 q orb 0.89 +0.06 −0.06 γ 12 +3 −3 i ( • ) 77.9 +0.3 −0.3 a (R ) 63 +2 −2f 2 / f 1 (OGLE I-band) 0.31 +0.05 −0.05
erator" technique. This yields the population numbers within the photosphere and wind. The emergent spectrum in the observer's frame is calculated with the formal integral in which the Doppler velocity is split into the depth-dependent thermal velocity and a "microturbulence velocity" ζ(r). The microturbulence grows from its photospheric value ζ ph = 10 km s −1 linearly with the wind velocity up to 0.1 ∞ .
By comparison of the synthetic spectrum with the observations, it is possible to determine the main stellar parameters. In addition to the chemical composition, the main stellar parameters that specify the model atmosphere are the luminosity L, stellar temperature T * , surface gravity g * , wind mass-loss ratė M, and the wind terminal wind velocity ∞ . The stellar temperature is defined as the effective temperature referring to the stellar radius R * by the Stefan-Boltzmann equation L = 4πσR 2 * T 4 * . The stellar radius is defined at the Rosseland mean optical depth of τ = 20. The differences between T * and the effective temperature T eff (referring to the radius where the optical depth τ = 2/3) are negligibly small, as the winds of OB-type stars are optically thin.
In the subsonic regions of the stellar atmosphere, the velocity law is calculated such that the density, related by the equation of continuity, approaches the hydrostatic stratification. The hydrostatic equation consistently accounts for the radiation pressure . In the supersonic regions, it is assumed that D. Pauli et al.: The earliest O-type eclipsing binary in the Small Magellanic Cloud, AzV 476.
33.0 17.5 logṀ [ M yr −1 ] −6.1 +0.2 −0.2 −8.8 +0.5 −0.5 - - −6.1 +0.3 −0.3 −7.9 +0.3 −0.3 −6.4 ( f ) −6.46 ( f ) ∞ [km s −1 ] 2500 +200 −200 2500 - - - - - - sin i [km s −1 ] 140 (d) 425 (d) - - 150 +26 −28 410 +60 −40 96 575 X H (by mass) 0.73 0.73 - - 0.74 +0.0 −0.1 0.74 +0.0 −0.1 0.47 0.74 X C /10 −5 (by mass) 2 +2 −1 21 (e) - - 27 +2 −5 7 +5 −3
3 20
X N /10 −5 (by mass) 45 +5 −10 3 (e) - - 6 +14 −6 61 +40 −(r) = ∞ (1 − f ) 1 − r 0 r β 1 + f 1 − r 1 r β 2 .(4)
In this work, we assume β 1 = 0.8, a typical value for O-stars, β 2 = 4, and a contribution of f = 0.4 of the second β-term. r 0 and r 1 are close to the stellar radius R * and are determined such that the quasi-hydrostatic part and the wind are smoothly connected. This choice of the wind velocity law leads to better agreement between the Hα absorption line and the C iv P Cygni line profile than the classical β-law (Castor et al. 1975).
Inhomogeneities within the wind are accounted for as optically thin clumps ("microclumping") which are specified by the "clumping factor" D, which describes by how much the density within the clumps is enhanced compared to a homogeneous wind with the same mass-loss rate (Hamann & Koesterke 1998b). In our analysis, we use a depth-dependent clumping which starts at the sonic point and increases outward until a clumping factor of D = 20 is reached at a radius of R D = 7 R * for the primary and at a radius R D = 10 R * for the secondary. The smaller radius at which the clumping factor is reached in the primary's wind profile is needed to model the observed O v λ1371 line properly.
The PoWR model atmospheres used here account for detailed model atoms of H, He, C, N, O, Mg, Si, P, and S. The iron group elements Sc, Ti, V, Cr, Mn, Fe, Co, and Ni are combined to one generic element "G" with solar abundance ratios, and treated in a superlevel approach (Gräfener et al. 2002). The abundances of Si, Mg, and Fe are based on Hunter et al. (2007, their table 17) and Trundle et al. (2007, their table 9). For the remaining elements, we divide the solar abundances of Asplund et al. (2005, their table 1) by seven to match the previously mentioned metallicity of the SMC. This yields the following mass fractions: X H = 0.73, X Si = 1.3 × 10 −4 , X Mg = 9.9 × 10 −5 , X P = 8.32 × 10 −7 , X S = 4.42 × 10 −5 , and X G = 3.52 × 10 −4 . The CNO individual abundances of both stellar components are not fixed but derived from the analysis. The complement mass fraction to unity is X He .
We determined the color excess E B−V and the luminosity L of the binary components by fitting the composite SED to photometry (top panel in Fig. 9). Reddening is modeled as a combined effect of the Galactic foreground, for which we adopt the reddening law of Seaton (1979) with E B−V = 0.03 mag, and the reddening law of Bouchet et al. (1985) for the SMC.
We use the iacob-broad tool (Simón-Díaz & Herrero 2014) in combination with the high S/N X-SHOOTER spectrum (S/N ∼ 100) to determine the rotation rates of the primary and the secondary. The helium lines are potentially pressure broadened and thus are not optimally suited for rotation broadening measurements. Therefore, for the primary, we use the few metal lines that are visible in the optical spectrum, namely Article number, page 7 of 23 Table 1) compared to different synthetic spectra. The observed spectrum, corrected for the velocity of the SMC and the barycentric motion, is shown as a solid blue line. The red dotted line is our best-fitting model which consists of the primary with T eff, 1 = 42 kK and log g * , 1 = 3.7 and the secondary with T eff, 2 = 32 kK and log g * , 1 = 4.0. The gray solid line is again a combined synthetic spectrum of the primary and secondary, but this time the surface gravity of the primary has been reduced to log g * , 1 = 3.6 (and the temperature and mass-loss rate are slightly adjusted so that the spectrum matches the observations) while the parameters of the secondary are kept fixed. The line identification marks correspond to the wavelengths in the rest frame. Panel (a) shows the Hγ line, in which the red wing is dominated by the primary and the blue wing by the secondary. Panel (b) shows the He ii λ5412 absorption line, which is dominated by the primary and is sensitive to surface gravity. Panel (c) shows the region of He ii λ6528 and Hα. While the Hα wings are only barely affected by the change in log g * , 1 , the He ii λ6528 line is more sensitive to it. the N iv λ3478 and the O iv λ3403 absorption lines and the N iv λ4058 emission line. We determine a projected rotation rate of 1 sin i = 140 km s −1 for the primary. The secondary only contributes to the He i lines. Fitting the He i λ4387 and He i λ4471 absorption lines -in which the contribution of the primary is smallest-yields a rotation rate of 2 sin i = 425 km s −1 for the secondary. Table 1) covering the wavelength range of the LiF2 channel -the same as used by Penny & Gies (2009) to determine the projected rotational velocity. The observed spectrum, corrected for the velocity of the SMC, is shown as a solid blue line. The red dotted line is our best-fitting model with the primary's projected rotational velocity 1 sin i = 140 km s −1 . The gray solid line is another model where the primary's projected rotational velocity is reduced to 1 sin i = 140 km s −1 . The line identification marks correspond to the wavelengths in the rest frame.
Previously, Penny & Gies (2009) determined the projected rotational velocity of the primary to 1 sin i = 65 km s −1 us-ing a cross-correlation of the FUSE spectrum in the far-UV with a template spectrum. We calculate tailored spectral models with the respective projected rotational velocities and find that also the FUSE spectrum is better reproduced when using 1 sin i = 140 km s −1 (see Fig. 6). We assume that the template spectrum used in Penny & Gies (2009) might not have been perfectly calibrated for our target and that the binary nature leads to additional uncertainties.
Resulting spectroscopic parameters
4.2.1. Temperature and surface gravity of the primary
AzV 476 was previously classified as O2-3 V plus a somewhat later O-type companion. This implies that nitrogen lines are expected in absorption and emission in the primary spectrum, while He i absorption lines should be present in the secondary spectrum. Indeed, we find that the N iv λ4057 (hereafter N iv) emission line, the N iv absorption lines at λλ 3463 Å, 3478 Å, 3483 Å, and 3485 Å, and the He ii λ6528 absorption can be entirely assigned to the primary. We observe only marginal N iii λλ4634, 4640 (hereafter N iii) emission and N v λλ4603, 4619 (hereafter N v) absorption; see Fig. B.1. Thus, the optical spectrum shows nitrogen only as N iv, while N iii as well as N v are virtually absent. This restricts the temperature of the primary to a narrow range (Rivero González et al. 2012). We also find that the O iv multiplets around 3400 Å are purely associated with the primary. Because these lines are highly contaminated by ISM absorption lines of Ti ii λ3385 and Co i λ3414, they are only used as a crosscheck of the oxygen abundance applied in our spectral model. A more detailed description of this line complex is given in Appendix B.2.
The secondary does not contribute noticeably to these weak metal lines. However, the secondary strongly dominates the He i absorption lines at λλ 3819 Å, 4387 Å, and 4471 Å, indicating a lower effective temperature. The primary star also contributes to these lines, giving additional constraints on the temperature of the primary.
The surface gravity g * , 1 of the primary is determined by fitting the wings of the Balmer lines using those UVES spectra with the least wavy patterns and highest RV shifts as well as the X-SHOOTER spectrum, which has a higher S/N. Because both stars contribute to the Balmer lines, the He ii λ5412 absorption line that is mostly originating from the primary and also sensitive to changes in surface gravity is used. Unfortunately, this line is only contained in the X-SHOOTER spectrum. Surface gravity affects the density structure and therefore the ionization balance of, for example, nitrogen and helium. As the temperature is adjusted such that the nitrogen lines are reproduced, changes in the surface gravity will not only affect the wings of the He ii lines but also change the depth of the He ii absorption lines. This gives the opportunity to use the He ii λ6528 absorption line in the UVES spectra as second criterion for the surface gravity estimate in the primary.
We tested different temperature values in the range T eff, 1 = 39 -50 kK and surface gravities in the range log(g * , 1 /(cm s −2 )) = 3.5 -4.1 and found that a temperature of T eff, 1 = (42 ± 3) kK and a surface gravity of log(g * , 1 /(cm s −2 )) = 3.7 ± 0.2 are most suitable for reproducing the primary's spectrum. The given error margins take into account that the measurements of temperature and surface gravity are not independent.
Different regions of the X-SHOOTER spectrum that are used to determine the surface gravity are shown in Fig. 5. The accuracy in surface gravity highly depends on the calibration of the spectrum. One can see that Hγ and He ii λ5412 line fits show preference to a surface gravity for the primary of log(g * , 1 /(cm s −2 )) = 3.7, while Hα and He ii λ6528 indicate that log(g * , 1 /(cm s −2 )) = 3.6 might be more suitable. However, from fitting the UVES data with the highest RV shifts we find that a surface gravity for the primary of log(g * , 1 /(cm s −2 )) = 3.7 yields the best fit.
Temperature and surface gravity of the secondary
It is more difficult to determine the temperature of the secondary, as only a few He i lines are associated with the secondary and all these line have a contribution from the primary. However, from the spectra with the highest RV shifts (e.g., UVES spectrum with ID 10), we find that the secondary does not contribute to the He ii λ4200 line, and therefore we have an additional constraint that can be used as an upper limit on the temperature of the secondary. In addition to the limited number of lines that are associated with the secondary, the ionization balance that changes depending on the surface gravity introduces another uncertainty in our temperature estimation. Therefore, we first adjust g * , 2 of the secondary such that the wings of the Balmer lines are well reproduced while keeping the surface gravity and temperature of the primary fixed. As in the case of the primary, in order to distinguish the contributions of the binary components to the Balmer wings, we employ the UVES spectra in the phase with the largest RV shifts. This gives additional information and a surface gravity of log(g * , 2 /(cm s −2 )) = 4.0 can be determined. Finally, the temperature of the secondary is adjusted such that the He i absorption lines at λλ 3819 Å, 4387 Å, and 4471 Å match the observation. Following this procedure, the spectrum of the secondary is best reproduced with a temperature of T eff, 2 = 32 kK. This method is accurate to ∆T eff, 2 = ±4 kK in temperature and ∆ log(g * , 2 ) = ±0.2 for the surface gravity. The obtained values of the surface gravity of the primary and the secondary are in agreement with those obtained from the orbital analysis (see Sect. 3). Table 1) (b) STIS observation (ID 3 in Table 1).
Luminosity
To estimate the light ratio, the luminosities of both stars are adjusted such that the shapes of the optical He ii lines match the observations. This is an iterative process done simultaneously with the temperature and surface gravity estimate. The total luminosity of the system is calibrated such that the calculated visual magnitude of the synthetic flux matches the observed one. This results in luminosities of log(L 1 /L ) = 5.65 for the primary and log(L 2 /L ) = 4.75 for the secondary. This estimate is sensitive to synthetic spectra, light ratio, and distance modulus. Therefore, we estimate that our measurements of the luminosity of each stellar component are accurate to ∆ log(L/L ) = ±0.2.
Taking these uncertainties into account, the model flux ratio in the OGLE I-band yields f 2 / f 1 = 0.26 ± 0.1 which is in agreement with the flux ratio derived using PHOEBE f 2 / f 1 = 0.31 ± 0.05 (see Sect. 3).
Mass-loss rates
After determining the temperatures, surface gravities, and luminosities of both binary components, we proceed to measuring the properties of their stellar winds. The Hα line in the optical as well as the C iv resonance line in the UV are used as the main diagnostic tools for measuring the mass-loss rate of the primary. In addition, we also pay attention to the appearance of the optical He i λ4686 and the UV He ii λ1640 line; see Fig B.3. When calibrating the mass-loss rate of the primary, the best fit is archived with a mass-loss rate of log(Ṁ 1 /( M yr −1 )) = −6.1 ± 0.2 and a terminal wind velocity of ∞, 1 = 2500 km s −1 .
We have two spectra covering the C iv resonance line at different phases, the COS and the STIS spectrum (ID 2 and 3 in Table 1), depicted in Fig. 7. The C iv P Cygni line profile in both spectra does not show any direct indications for the contribution from the secondary such as a double emission peak or a step in the absorption trough. The C iv resonance line in the STIS spectrum shows slightly weaker emission, which is most likely due to the sensitivity of the different instruments or weak wind variability. We use the bottom part of the P Cygni profile of C iv resonance line to confirm the continuum contribution of the secondary in the UV and therefore the used light ratio (L 2 /L 1 ).
Apparently the secondary does not contribute to the wind lines. Its mass-loss rate can therefore only be limited to log(Ṁ 2 /( M yr −1 )) ≤ −8.8 ± 0.5. For higher mass-loss rates, we Fig. 8. Observed X-SHOOTER spectrum (ID 5 in Table 1) and synthetic spectra of selected regions that show nitrogen emission lines. The observed spectrum is shown as a solid blue line, while the dotted red line is our best-fitting model. The observed spectrum is corrected for the velocity of the SMC and the barycentric motion. The line identification marks correspond to the wavelengths in the rest frame. Panel (a) shows the area around the N iv λ4057. Panel (b) shows the region of N iii λλ4511 − 4523, N v λλ4604, 4620, and N iii λλ4634, 4641. The N iii and N v lines are not observable in the spectrum. Panel (c) shows the region of the N iv λλ7103 − 7129 complex.
find that the secondary would noticeably contribute to the C iv resonance line in the UV. Due to the lack of indications of the secondary's wind we adopt the terminal velocity of the primary, ∞, 2 = 2500 km s −1 .
CNO surface abundances
As already explained in Sect. 4.1, the abundances of hydrogen and the iron group elements are fixed, while the CNO abundances of each stellar component are adjusted during spectral modeling. The nitrogen abundance is adjusted to find good agreement with the optical N iv absorption lines at wavelengths 3463 Å, 3478 Å, and 3483 Å, and the N iv λ4057 and N iv λλ7103 − 7129 emission lines, as well as the UV N v λλ1238, 1242 resonance doublet. The best agreement is achieved with a nitrogen mass fraction of X N = (45 +5 −10 ) × 10 −5 . The nitrogen lines seen in the optical are displayed in Fig. 8.
A carbon abundance of X C = (2 +2 −1 ) × 10 −5 reproduces the optical C iv λλ5801, 5812 lines best while maintaining the fit of the C iv resonance doublet, at least for mass-loss rates that do not conflict with other wind features. For lower abundances, the latter is no longer saturated. The oxygen abundance of the primary is calibrated with the oxygen absorption lines in the UV and the optical O iv multiplets around 3400 Å, resulting in X O = (80 +10 −20 ) × 10 −5 . There are no strong CNO lines in the secondary spectrum, and therefore we adopt initial CNO abundances scaled to the metallicity of the SMC (Z SMC = 1/7 Z ). Alternatively, one could assume that the surface abundances in the secondary are close to those in the primary as the accreted material is polluting the surface of the secondary. We explored both assumptions but could not find diagnostic lines that would allow us to pin down the composition of the atmosphere and wind of the secondary. We are only able to fix upper limits for the CNO abundances such that these elements would not produce features that are inconsistent with the observations: X N 50×10 −5 , X C 21×10 −5 and X O 110 × 10 −5 .
Required X-ray flux
Similar to other O-type stars, AzV 476 shows a notoriously strong P Cygni profile in the O vi λλ1032, 1038 resonance doublet. However, wind models do not predict a sufficient population of this high ion without the inclusion of additional physi-cal processes. This phenomenon, termed "super-ionization", was described by Cassinelli & Olson (1979) and interpreted as evidence for the presence of an X-ray field in stellar atmospheres.
In order to model the O vi doublet in the observed spectrum of AzV 476, we add a hot plasma component. The X-rayemitting plasma has an adopted temperature of T X = 3 MK and is distributed throughout the wind outside a radius of R X = 1.1 R * . Its constant filling factor is a free parameter; the best reproduction of the observation is achieved for a model with an emergent X-ray luminosity of log L X = 31.4 erg s −1 .
Current X-ray telescopes are not sensitive enough to detect individual O stars in the SMC. Our final model predicts an X-ray flux at earth of 1.4 × 10 −17 erg s −1 cm −2 Å −1 integrated over the X-ray band (≈ 6 − 60 Å). The region on the sky where AzV 476 is located was observed by both XMM-Newton and Chandra Xray observatories. However, despite the detection of AzV 476 in the UV by the XMM-Newton optical monitor, the star is not detected in X-rays. Hence, we set the upper limit on its X-ray flux at the median flux of detected sources in the XMMSSC -XMM-Newton Serendipitous Source Catalog (4XMM-DR10 Version); this is 5 × 10 −15 erg s −1 cm −2 Å −1 , which is far from being sufficiently sensitive for our predicted source.
The O vi resonance doublet is not the only feature that is sensitive to X-rays. Other ions which might be populated by Auger ionization are N v and N iv. We find that the N v λλ1242, 1238 resonance doublet in our model is not significantly affected by the applied X-ray field, while the N iv λλ1718, 1721 doublet is very sensitive. However, in the region around the N iv λλ1718, 1721 doublet, the observations are very noisy and can be explained by the model with X-rays as well as without.
The composed model spectrum as well as the individual spectra are shown in Figs. 5 -10. A summary of the stellar parameters including the abundances is given in Table 6.
Spectroscopic stellar masses
The spectroscopic masses derived from our analysis are M spec, 1 = 29 +17 −11 M for the primary and M spec, 2 = 22 +13 −8 M for the secondary. The spectroscopic masses, although a factor of 1.5 higher than the orbital masses, agree with these latter within their respective uncertainties. The large error margins on the spectroscopic masses arise mainly from the large uncertainties on surface gravity and luminosity (e.g., Fig. 5 (4), and X-SHOOTER (5) spectra as listed in Table 1. The light blue squares are the photometric UBVRIJHK data as listed in Table 2. The model SED composed of both stellar components is shown as a dashed red line, while the individual SEDs of the primary and the secondary are shown as dotted black and green lines, respectively. Lower panels: Normalized FUSE (1), COS (2), and STIS (4) spectra. The number in the upper left corner corresponds to the ID given to a specific spectrum as listed in Table 1. The line styles are the same as in the top panel. The synthetic spectra are calculated with the model parameters compiled in Table 6 ("Spectroscopy" columns).
(2) Lyα (is) N V C III Si II (is) Si III O I (is) Si II (is) C II (is) O IV O V Si IV (is) O III N IV Si V Si II (is) C IV(
ratio is q spec = 0.7 which is smaller than the one obtained from the orbital analysis.
It appears that the spectroscopically derived masses and luminosities are shifted systematically to higher values compared to the results of the orbital analysis. We discuss this in more detail in Sect. 6.1.
Nevertheless, a mass ratio close to unity, while the two stars in a close binary have different spectral types, strongly suggests mass transfer in the past. From our binary evolutionary models (Sect. 5), we expect that mass transfer removed most of the hydrogen-rich envelope from the primary. We calculated model spectra with strongly depleted surface hydrogen (X H = 0.2 and 0.5). However, these models yield poorer spectral fits as the he-Article number, page 11 of 23 Fig. 10. Same as Fig. 9, but now the normalized optical X-SHOOTER spectra (ID 5 in Table 1) are shown.
lium lines become too deep. Still, moderate hydrogen depletion and helium enrichment cannot fully be excluded.
Revisiting the spectral classification
Spectral classification aided by stellar atmosphere modeling
AzV 476 was previously classified on the basis of the appearance of its optical spectrum. However, during our spectroscopic analysis, we found that most of the classification lines are blended by the companion. Hence, including the additional information from our spectroscopic models and the available UV data allows us to reconsider the spectral classification. We used the Marxist Ghost Buster (MGB) spectral classification code (Maíz Apellániz et al. 2012; Maíz Apellániz 2019). MGB compares observed spectra with the criteria of Walborn et al. (2000Walborn et al. ( , 2002Walborn et al. ( , 2014 and Sota et al. (2011Sota et al. ( , 2014 while allowing the presence of two components by iteratively adjusting the spectral types, light ratio, RVs, and rotational indices. This approach poses two problems. First, the low metallicity of the SMC leads to weaker nitrogen lines compared to stars with similar type in the MW and LMC. Second, only the X-SHOOTER spectrum is covering the whole wavelength range of 3900 -5100 Å which is needed for a spectral classification. In addition we consider those UVES spectra with the highest RV shifts in order to better separate the individual binary components. Therefore, we are forced to use the X-SHOOTER spectrum first and then consider the UVES spectra with the highest RV shifts that cover the range from 3300 to 4500Å.
The MGB code provides the spectral type O4 IV((f)) for the primary. The main reason for the slightly later subtype assigned to the primary is its contribution to the He i λ4471 line. The suffix "((f))" is assigned because of the barely detectable N iii λλ4634, 4641 lines (see Fig. B.1). Regarding the secondary, the absence of the He ii λ4200 line suggests the latest O subtype, O9.5: Vn. The suffix "n" refers to the high rotational broadening of the lines of the secondary.
The MGB classification code only considers the optical range of the spectrum. If one inspects the UV spectrum, which mostly originates from the primary, and compares it to the N v and C iv UV wind-profile templates with those presented in Walborn et al. (2000), one might prefer a luminosity class "III" for the primary. The difference in the optical spectral appearance is explainable by the high L/M ratio which leads to very strong mass loss and thus to exceptionally strong wind lines in the UV. In light of these circumstances and the unusual chemical composition, we add the suffix "p" for this peculiar object. In the end, we therefore classify the system as O4 IV-III((f))p + O9.5: Vn.
Following the work of Weidner & Vink (2010), an isolated star with spectral type O4 III is expected to have a stellar mass of about M exp, 1 = 49 +7 −6 M , which is more than twice as high as that which we derive using two different methods (Sects. 3 and 4.1). For the O9.5: V secondary, the mass expected from the spectral type would be M exp, 2 = 16 +7 −3 M , which agrees within uncertainties with the findings of the orbital and spectroscopic analysis.
Stellar evolution modeling
From orbital and spectroscopic analysis, we find that the mass ratio is close to unity, although the two binary components have distinctly different luminosities, effective temperatures, and rotation rates: the secondary is a fast rotator while the rotation of the primary is only moderate. These facts strongly indicate that the system has already undergone mass transfer.
Single-star models
As a first approach we investigate whether the stellar parameters of the individual components can be reproduced by singlestar evolutionary models. For this purpose, we employ the Bayesian statistic tool "The BONN Stellar Astrophysics Interface" (BONNSAI 4 ), (Schneider et al. 2014) in combination with the BONN-SMC tracks (Brott et al. 2011). To find a suitable model, we request the tool to match the current luminosity, effective temperature, rotation rate, and orbital mass of the two stars. Indeed, we find that the fast-rotating secondary can be partially explained by a single-star track. With respect to the primary, the BONNSAI tool cannot find any track that would explain its current low mass. Only when ignoring the mass constraint are we able to find a suitable track that reproduces the remaining stellar parameters. In that case, the predicted ages of the primary and secondary differ by a factor of two. This finding confirms our suggestion that the system has already undergone mass transfer. The best-fitting stellar parameters obtained with the BONNSAI tool -applicable only for single stars-are listed in Table 6.
Binary evolutionary models
The binary models are calculated with MESA v. 10398. To mimic the BONN-SMC tracks we adjusted MESA in a similar way to that described by Marchant (2016). We follow most of the physical assumptions from the BONN-SMC models (e.g., overshooting, thermohaline mixing, etc.) and adopt them from Marchant (2016) with two exceptions. In our models, we use a more efficient semi-convection with α sc = 10 and calculate the mass transfer according to the "Kolb" scheme (Kolb & Ritter 1990), as it allows us to include the eccentricity enhancement mechanism meaning that the evolution of the eccentricity can be modeled properly. We want to emphasize that our goal is to check whether binary evolutionary models can explain the stellar parameters of AzV 476 and especially its masses. Therefore, we only explore a narrow parameter space and compute a small set of models with initial primary masses in the range of M ini, 1 = (25 -38) M and secondary masses M ini, 2 = (10 -25) M , orbital periods in the range of P ini = 8 -25 d and initial eccentricities e ini = 0.0 -0.4 . The initial parameters are adjusted such that at some later stage the model binary has properties similar to AzV 476. We assume that the stellar components initially rotate with rot = 0.1 crit ≈ 65 km s −1 . These values are chosen arbitrarily but as we are dealing with a close binary, the tidal forces nevertheless lead to a tidal synchronization. Further important assumptions are as follows: Our models include the effect of inflation, which appears inside a stellar envelope when the local Eddington limit is reached 4 www.astro.uni-bonn.de/stars/bonnsai and exceeded, leading to a convective region and a density inversion (Sanyal et al. 2015). The stellar wind prescription is inspired by the work of Brott et al. (2011). The winds of hot H-rich stars are described according to the Vink et al. (2001) recipe. For stars with effective temperatures below the bi-stability jump, where mass-loss rates abruptly increase (see Vink et al. 2001), we use the maximuṁ M from either Vink et al. (2001) or Nieuwenhuijzen & de Jager (1990). WR mass-loss rates are according to Shenar et al. (2019) but are assumed to scale with metallicity asṀ ∝ Z 1.2 as recommended by Hainich et al. (2017). In the transition phase from a hot H-rich star (X H ≥ 0.7) to the WR stage (X H ≤ 0.4), the massloss rate is interpolated between the prescriptions of Vink et al. (2001) and Shenar et al. (2019).
Rotational mixing is modeled as a diffusive process including the effects of dynamical and secular shear instabilities, the Goldreich-Schubert-Fricke instability, and Eddington-Sweet circulations (Heger et al. 2000). In addition to the angular momentum transport by rotation, the transport via magnetic fields from the Tayler-Spruit dynamo (Spruit 2002) is also included.
Convection is described according to the Ledoux criterion and the mixing length theory (Böhm-Vitense 1958) with a mixing length parameter of α mlt = l/H P = 1.5. For hydrogen burning cores, a steep overshooting is used such that the convective core is extended by 0.335H P (Brott et al. 2011) where H P is the pres-sure scale height at the boundary of the convective core. Thermohaline mixing is included with an efficiency parameter of α th = 1 (Kippenhahn et al. 1980), as well as semiconvection with an efficiency parameter of α sc = 10 (Langer et al. 1983).
Mass transfer in a binary is modeled using the "Kolb" scheme (Kolb & Ritter 1990). This allows us to use the Soker eccentricity enhancement (Soker 2000), which assumes phasedependent mass loss and calculates the change in eccentricity due to the mass that is lost from the system, and the mass that is accreted by the companion. Tidal circularisation is taken into account throughout the entire evolution of the system.
The remaining mass collapses into a compact objectrepresented by a point mass-when helium is depleted in the stellar core. This simplification avoids numerical problems in the latest burning stages and the unknowns that come with a supernova explosion and a possible kick.
As the luminosities obtained from the spectroscopic and orbital analysis differ to some extent, we put additional focus on reproducing the orbital masses, which are our most reliable estimate. The best agreement with the empirically derived parameters is found for a system with initial masses M ini, 1 = 33 M and M ini, 2 = 17.5 M , initial orbital period P ini = 12.4 d, and initial eccentricity e ini = 0.14. The model closest to the current stellar parameters gives similar masses for the primary and secondary: M evol, 1 = 17.8 M and M evol, 2 = 18.2 M ,. The remaining stellar parameters of our favorite models are listed in Table 6.
The Hertzsprung-Russell diagram (HRD) displaying the best fitting stellar evolutionary tracks for both binary components is illustrated in Fig. 11. The binary evolution model is able to reproduce almost all empirically derived stellar parameters including the current orbital period (P model = 9.3 d) and the eccentricity of the system (e model = 0.25). However, the evolutionary models over-predict the rotation rate of the secondary and the surface abundances of the primary; see Table 6. We calculated PoWR atmosphere models with the best fitting parameters obtained with our MESA models. The synthetic spectra are shown in Fig. B.4.
A more fine-tuned model would likely solve some of these problems. For example, we did not include the initial rotation as a free parameter. This might solve the faster rotation of the primary; however, the secondary would still be expected to be rotating too fast as it spins up to criticality during mass transfer. According to our favorite evolutionary model, the primary in AzV 476 is currently evolving towards the helium zero age MS (ZAMS) and will possibly spend the rest of its life as a hot helium or WR-type star.
Discussion
Comparison of the orbital, spectroscopic, and evolutionary mass estimates
The orbital masses and the masses predicted by the binary evolutionary models agree well within their error margins and deviate only by 10%. On the other hand, the spectroscopic mass and luminosity of both components appear to be higher by a factor of 1.5 compared to the orbital solutions. The question therefore arises as to whether the discrepancy between the orbital and spectroscopic solutions is significant.
Orbital versus spectroscopic mass
The simplest way to resolve this discrepancy is to assume that the distance to the system is lower than the canonical SMC distance of (d = 55 kpc). As the luminosity depends on the assumed distance as L ∝ d −2 , the radius (R ∝ √ L ∝ d −1 ) and the spectroscopic mass also depend on distance (M spec ∝ R 2 ∝ L ∝ d −2 ). The SMC galaxy is extended, and the distances to its various structural parts are not well established. Tatton et al. (2021) suggest that young structures, such as the NGC 456 cluster, are not well represented by the intermediate age stellar populations, which are usually used for distance estimates. Instead, young clusters might be located in front of these latter. This is in agreement with the argumentation by Hammer et al. (2015), who showed that the interactions between the LMC and SMC could lead to multiple tidal and ram-pressure stripped structures, which spatially separate young star-forming regions from older stellar populations.
In order to bring the spectroscopic and orbital parameters of AzV 476 into accordance, the system needs to be shifted to a distance of 49 kpc. The resulting spectroscopic masses are then M shift, 1 ≈ 23 M and M shift, 2 ≈ 18 M . However, d = 49 kpc would imply that our target is located at the same distance as the LMC.
The LMC has higher metallcity (Z LMC = 1/2 Z ) than the SMC (Z SMC = 1/7 Z ), and therefore we speculate that stars formed in the interaction regions may have higher metallicity than the SMC. To test this, we increased the metallicity content of our spectroscopic models to the LMC metallicity while keeping the remaining stellar parameters unchanged. We find that the lines in the iron forest, especially in the far-UV, are too deep compared to the observations. Therefore, it cannot be confirmed that our target has LMC metallicity, but at the same time this does not provide additional constraints on its distance.
Another option we need to consider is the possibility of a third light contribution. AzV 476 is located in a very crowded region. Therefore, it is possible that the observed light is contaminated by a nearby object and hence the true luminosity (and mass) is lower. We inspected the HST acquisition image of our target and find no other nearby UV-bright stars. Alternatively our target could be a multiple system rather than a binary. If there were a third component, it would need to be of similar luminosity to the secondary in order to have an impact on our analysis. However, we do not see any indications of a third component in our spectra.
A third option is that the surface gravities determined from the spectral analysis are overestimated. The accuracy of the spectroscopic mass strongly depends on surface gravity, log g, which is mainly determined from fitting the pressure-broadened line wings. Such estimates suffer from limited S /N ratio in the spectra. In cases where the Balmer lines include contributions from both components, further uncertainties are induced. As explained in Sect. 4.2 and shown in Fig. 5, the He ii lines help to improve the estimates, but still do not allow us to pin point log g for each component separately. This leaves room for various combinations of log g able to reproduce the Balmer lines and is reflected in the large uncertainties of the spectroscopic masses.
Orbital versus evolutionary mass
The evolutionary modeling yields a slightly lower mass for the primary than the spectroscopic and orbital analyses. As can be seen in Table 6, the evolutionary mass is 2 M lower than the orbital one.
For a given initial mass, the main parameter defining how much mass is lost during mass transfer by the primary is the initial binary orbital period. As a rule of thumb, the longer the initial orbital period, the less mass is removed from the donor star (Marchant 2016). However, several effects (e.g., eccentricity) influence the response of the orbit to mass transfer. For instance, an eccentric orbit leads to phase-dependent mass transfer, which potentially increases the eccentricity and widens the orbit. Thus, the donor can expand more, and less mass is removed via Rochelobe overflow (Soker 2000;Vos et al. 2015).
The amount of mass lost during mass transfer is also reduced when the radial expansion of the star is minimized such that the star can stay underneath the Roche lobe. The radial expansion is affected by the mixing processes. For instance, a star with more efficient semi-convective mixing is expected to be more compact (e.g., Klencki et al. 2020;Gilkis et al. 2021). We computed a binary evolutionary model with a less efficient semi-convection of α sc = 1 and find that the post-mass-transfer (after detachment) mass of the primary is lowered by 1 M . This is only one example and there are a handful of other processes that impact the mixing efficiency, energy transport, and the expansion of the envelope.
One more process that affects the radial expansion is the mass-loss rate which is described by the adopted mass-loss recipe (see Sect. 5). Over-or underestimating the mass-loss rate influences stellar evolution before, during, and after mass transfer. Interestingly, we find approximate agreement between the empirically derived mass-loss rate of the primary and the prediction of the wind recipe (see Sect. 6.2.1).
The evolutionary mass of a star after mass transfer is sensitive to various assumptions. Therefore, a difference of 2 M between evolutionary and orbital mass appears to be within the range of the evolutionary model uncertainties. While a more fine-tuned binary evolutionary model is beyond the scope of this paper, our modeling confirms that the orbital masses of AzV 476 are well explained by a post-mass-transfer binary evolutionary model. Therefore, this unique system is a useful benchmark for improving the understanding of mass transfer and mixing processes in close massive binaries. We conclude that the orbital, spectroscopic, and evolutionary mass estimates agree within their uncertainties.
The empirical mass-loss rate
Stellar evolution depends on the mass-loss rate. Stellar evolution models rely on recipes that are only valid for a given evolutionary phase. In transition-phases one typically interpolates between corresponding mass-loss recipes (e.g., Sect. 5). In the following, we compare the mass-loss rates we empirically derive from spectroscopy with other estimates as well as with the various recipes.
In this work, the mass-loss rates are primarily derived from resonance lines in the UV and the Hα line while taking into account the morphology of the He ii λ1640 and He ii λ4686 lines. The winds of all hot stars are clumpy (e.g., Hamann et al. 2008) which enhances the emission lines fed by recombination cascades (e.g., Hamann & Koesterke 1998a) such as the Hα line. Resonance lines in the UV are mostly formed by line scattering, a process that scales linearly with density and is thus independent from microclumping. Considering both Hα and the resonance lines allows us to constrain the clumping factor D and the mass-loss rateṀ consistently.
The primary
The spectroscopically derived mass-loss rate of the primary is log(Ṁ/( M yr −1 )) = −6.1 ± 0.2. Massa et al. (2017) derive a mass-loss rate of log(Ṁ/( M yr −1 )) = −5.54 +0.12 −0.10 from the IR excess. However, the clumping effects were not included in their analysis. As the IR excess originates from free-free emission, it scales with √ D. Correcting the mass-loss rate with a clumping factor of D = 20, as assumed in our spectroscopic models, yields log(Ṁ/( M yr −1 )) = −6.19 +0.12 −0.10 , which is in agreement with the optical and UV analyses.
In the massive star evolutionary models, the terminal wind velocity is prescribed as ∞ / esc = 2.6 and the mass-loss rates of OB stars are prescribed according to the Vink et al. (2001) recipe. Using the stellar parameters obtained with spectroscopic analysis (see Table 6) and the prescribed ∞ = 1911 km s −1 , the "Vink's recipe" yields log(Ṁ/( M yr −1 )) = −6.1. Alternatively, applying the actual wind velocity measured from the UV spectra, ∞ = 2500 km s −1 , the theoretical prediction changes to log(Ṁ/( M yr −1 )) = −6.2, which is in agreement with our empirical result within the uncertainties.
However, the primary's current mass-loss rate adopted in our favored binary evolutionary model (log(Ṁ/( M yr −1 )) = −6.4) is lower. This is mainly caused by the interpolation between the mass-loss rate prescriptions for OB stars by Vink et al. (2001) and for WR stars by Shenar et al. (2019) which is employed when the surface hydrogen mass-fraction of the primary's evolutionary model already dropped below X H < 0.7. Additionally, the stellar parameters (e.g., mass and luminosity) that enter the different mass-loss recipes employed by the evolutionary code are slightly different than those obtained from our spectral analysis.
Based on dynamically consistent Monte Carlo wind models, Vink & Sander (2021) suggest an updated scaling of metallicity for their mass-loss rates. This scaling yields higher mass-loss rates of log(Ṁ/( M yr −1 )) = −5.7, or, using the measured ∞ = 2500 km s −1 , log(Ṁ/( M yr −1 )) = −5.9. Correcting in a simple way the former theoretical mass-loss rate for clumping results in log(Ṁ/( M yr −1 )) = −6.4, which is somewhat lower than our empirical measurement.
The spectroscopic measurements of mass-loss rates of Odwarfs in the SMC deliver useful empiric prescriptions (e.g., Bouret et al. 2003;Ramachandran et al. 2019). According to Ramachandran et al. (2019, see their figure 15), the massloss rate of the primary star in AzV 476 would be only log(Ṁ/( M yr −1 )) = −7.4, which is much lower than obtained from our analysis. We explain this discrepancy by the advanced evolutionary status: the primary has already a substantially reduced mass due to its close-binary evolution, leading to a high L/M ratio. This precludes the comparison with the recipes which use the mass-luminosity relations for single stars.
The secondary
There are no indications of wind lines of the secondary in the observed spectra. Therefore, we can only set an upper limit on the mass-loss rate log(Ṁ/( M yr −1 )) ≤ −8.8 ± 0.5. The Vink et al. (2001) recipe predicts a higher value of log(Ṁ/( M yr −1 )) = −8.2. Correcting for clumping with D = 20 yields log(Ṁ/( M yr −1 )) = −8.8 which is in agreement with the empirical result.
According to Vink & Sander (2021), the predicted massloss rate of the secondary is log(Ṁ/( M yr −1 )) = −7.8; correcting it for clumping yields log(Ṁ/( M yr −1 )) = −8.4, which is higher than the empirical upper limit. The empirical relation of Ramachandran et al. (2019, their figure 15,) suggests log(Ṁ/( M yr −1 )) = −8.6 ± 0.2, which is somewhat higher than our result but consistent with its error margins.
As can be seen in Table 6, the mass-loss rate used in the binary evolution calculation is higher than predicted by the recipes as well as empirically measured. This is because in the MESA models, the mass-loss rate of a star rotating close to critical is enhanced compared to slowly rotating stars (Paxton et al. 2013). The synthetic spectrum calculated with the parameters obtained by the binary evolutionary models displayed in Fig. B.4 shows strong P Cygni profiles in the spectrum of the secondary (e.g., Si iv λλ1393.8, 1402.8) that are not observed. We conclude that the mass loss of the secondary is not rotationally enhanced above the observed limit.
Comparison of the stellar parameters as predicted by MESA evolutionary tracks versus derived spectroscopically
In this work, AzV 476 is studied using two different approaches.
(1) The empiric approach is to model the observed light curves and spectra using the PHOEBE code and the stellar atmosphere model PoWR in order to derive orbital, stellar, and wind parameters.
(2) The evolutionary modeling approach is to use the MESA code to compute tracks reproducing the current stellar parameters of AzV 476 components (Fig. 11) simultaneously with its orbital parameters. Below we compare the outcomes of these two approaches.
In binary evolutionary models, during the mass-transfer phase, most of the hydrogen-rich envelope is stripped off such that products of the CNO burning process are exposed at the surface. The predicted surface abundances in our favorite model correspond to an intermediate stage between the initial CNO abundances and the CNO equilibrium at metallicity Z SMC = 1/7 Z . This means that most of the initial C, N, and O become N, and the CO abundances are depleted.
Indeed, the abundances derived spectroscopically are intermediate between the initial CNO abundances and the CNO equilibrium. The C and O abundances are reduced while the N abundance increased drastically. However, we find a factor of two difference between the predicted and observed N and O abundances (see Table 1). We note that the surface CNO abundances predicted by the MESA models are subject to various assumptions; for example, on mixing efficiencies or the mass removal by Roche-lobe overflow. Because of the scarcity of suitable photospheric absorption lines, the CNO abundances of the primary are partially derived from emission lines. Those lines are sensitive to many effects, such as the mass-loss rate, temperature, clumping, and turbulence, introducing additional uncertainties in the abundance measurement.
Furthermore, we find disagreement between the observed and predicted H mass-fraction. In evolutionary models, a significant amount of the H-rich envelope (i.e., with X H > 0.7) is removed during the mass-transfer phase, revealing the products of the established chemical gradient between the core and the envelope, leading to the depletion of surface hydrogen. However, from the spectroscopic analysis, we do not find such strong H depletion. As in the case of the CNO measurements, this indi-cates some deficiencies in the evolutionary models that are likely related to the mass-transfer phase and/or to the mixing.
For a better understanding, we are showing in Fig. 12 abundance profiles at two different stages. The left panel in Fig. 12 shows the abundances at core hydrogen depletion. At this stage, the star is hydrogen shell burning, which leads to the formation of an intermediate convection zone (ICZ, Langer (1987)), which ranges from the mass coordinates 21 -24 M . One can see that the H, He, O, and N abundances in this region are the same as the abundances that we observe; only the predicted C abundance is higher. However, the spectroscopically derived C abundance is the least accurate one, as the observed spectra have no photospheric C absorption lines and the C iv resonance line in the UV highly depends on the mass-loss rate. At this evolutionary stage, the envelope is not well mixed, which can be seen in the step-like structure between the different mixing regions.
The right panel in Fig. 12 shows the abundances at the onset of mass transfer. Mass transfer happens on the dynamical timescale, while the mixing and burning processes take place on longer timescales. Therefore, this plot can be used to decipher the surface composition of the final model if a specific amount of mass is removed from the star. Compared to the left panel, the stellar model has had more time to mix the CNO elements through the envelope. The ICZ is now largely extended and the CNO abundance ratios have changed; for instance N is now the most abundant element in this region. We would like to emphasize that this highly depends on the assumed mixing efficiencies, as a less efficient semi-convection (e.g., α sc = 1) would lead to less mixing and might preserve the step-like structure seen the left panel. Semi-convection does not only affect the mixing efficiency but also impacts the change of the stellar radius (Klencki et al. 2020), and therefore a star with more effective semi-convective mixing spends more time hydrogen shell burning before it fills the Roche lobe and initiates mass transfer. This additionally impacts the final chemical composition.
By now it should be evident that accurate prediction of the surface CNO abundances in binary evolutionary models is a nontrivial task and is beyond the scope of this paper. However, this underlines the importance of AzV 476, which can be used to get a better understanding of the different mixing efficiencies, as it reveals the abundances formed by burning stages that would normally be hidden from the observer, making them difficult to calibrate.
In order to understand how the predicted stellar parameters of the binary evolution models -including the surface abundances -compare to the observed spectrum, synthetic spectra of the primary and secondary are calculated using these parameters (see Table 6). The resulting spectrum is shown in Fig. B.4. When comparing the synthetic spectrum to the observed one, it is evident that the N emission lines are too strong and the helium absorption lines are too deep.
The abundances are not the only difference between the evolutionary model predictions and the observations. Evolutionary model predicts much higher rotation rates than observed. In the evolutionary model, the secondary has spun up to critical rotation, which is not observed, and also the rotation rate of the primary is somewhat slower than observed. Vanbeveren et al. (2018) and Shara et al. (2020) studied the rotation rates of O stars in WR+O star binaries at different metallicities and compared them to predictions from evolutionary models. This comparison revealed that the observed rotation rates of the O stars are lower than the predictions from the evolutionary models. These latter authors suggest that there is a process during the fast case A mass transfer that removes the angular momentum of the star; their favored explanation is an angular momentum loss induced by a magnetic field that only develops during the mass-transfer phase.
The disagreements on the chemical abundances and the rotation rates between evolutionary models and the empiric analysis suggest some unconsidered physics during the mass-transfer phase. A more fine-tuned model would likely solve some of these problems. For example, we did not include the initial rotation as a free parameter. This might solve the faster rotation of the primary; however, the secondary would still be expected to be rotating too fast as it spins up to criticality during mass transfer.
6.4. AzV 476 and its future evolution in the context of the most massive star population in the SMC In the future, the primary in AzV 476 will evolve bluewards on the HRD, but it will never be able to lose its entire H-rich envelope. As its luminosity is on the edge of the least luminous observed WR star of the SMC (Shenar et al. 2016(Shenar et al. , 2020, it is unclear as to whether or not it will be able to develop an optically thick wind and become a WR star. Our evolutionary model predicts that the primary will have a final mass of M final, 1 = 16 M at core helium exhaustion. The primary will probably collapse and form a BH (Heger et al. 2003). To our knowledge, no BH has yet been identified in the SMC.
After the primary collapse and the formation of a compact object, the system will consist of a MS star with a compact object. The MESA model predicts that the secondary will have T eff = 28 kK and log(L/L ) = 4.85, at the halfway point of its remaining MS lifetime. We calculated a spectroscopic model of the secondary in this evolutionary stage and find that the secondary would be classified as a B0 V star. Its critical rotation makes the system a likely progenitor of a high-mass X-ray binary (HMXB) with a Be donor star. After the stellar model evolves beyond the MS, it undergoes a mass-transfer phase, offering an additional opportunity to show up as a HMXB. After large fractions of the secondary's envelope are stripped off, it will remain as a helium star with some hydrogen left on the surface with a luminosity around log(L/L ) ≈ 4.9. Using the recent estimates from Sander & Vink (2020) for massive hydrogen-free stars, the L/M-ratio will be significantly too low to yield a sufficiently strong stellar wind for it to become optically thick at this metallicity (even when accounting for additional hydrogen and larger radii). Thus, our secondary star will most likely never appear as a WR star. With a final mass of M final, 2 = 7 M at core helium depletion, the secondary is expected to explode as a supernova leaving a NS. With a final orbital period of about ≈ 100 d, this system is not expected to merge within a Hubble time. The impact of a supernova kick is neglected in our binary models; nevertheless, in principle, it could change the final orbital period of the system drastically.
Summary and conclusions
In this work we study the earliest known O-type eclipsing binary in the SMC, AzV 476. We derived the masses of its companions using a spectroscopic analysis and by modeling its light-and RV curves. We compare the results with binary evolutionary models.
We conducted a quantitative analysis of the multi-phase spectra in the optical and UV obtained by the ESO VLT and the HST and supplemented by the FUSE spectroscopy in the far-UV. To this end we used the non-LTE stellar atmosphere model code PoWR. The spectra of both components were disentangled, allowing us to determine their masses, stellar, and wind parameters. Independently, we used the PHOEBE code to derive the stellar and the orbital parameters from the light curve in the Iband as well as RV curves. We employed the MESA code to compute detailed binary evolutionary models that are able to reproduce the observed stellar parameters in significant detail (see Table 6 and Fig. 11).
Our conclusions can be summarized as follows:
(1) The eclipsing binary AzV 476 harbors one of the most luminous O stars in the SMC. It consists of an O4 IV-III((f))p type primary and an O9.5: Vn type secondary. Single stars with similar spectral types typically have masses of M exp, 1 = 49M and M exp, 2 = 16M .
(2) By analyzing the light curve and the RV curve consistently, we derive orbital masses of M orb, 1 = 20M and M orb, 2 = 18M for the binary components. We find that the primary has less than half of the mass that is expected from its spectral type. The modest mass of one of the most luminous stars in the SMC highlights the conspicuous deficiency of very massive O-type stars in this metal-poor galaxy.
(3) The spectroscopic masses agree within the uncertainties with the more reliable orbital masses, and they confirm a mass ratio of close to unity. The observed spectra reveal that the surface N abundance of the primary is enhanced while C and O abundances are reduced. These two aspects suggest a prior masstransfer phase.
(4) The spectroscopic analysis uncovers that the mass-loss rate of the primary with log(Ṁ/( M yr −1 )) = −6.1 is ten times stronger compared to recent empirical prescriptions for single O stars in the SMC. Likely, this is due to its high L/M ratio. Our empirical mass-loss rate is a factor of two higher compared to the one used in the evolutionary models. While for our primary the result as such is in line with standard theoretical predictions, this underlines that the present treatment of wind mass loss in stellar evolution models needs to be improved to properly account for the products of binary evolution.
(5) The current moderate mass of the primary can only be explained by binary evolutionary models. According to our favorite binary model, the initial mass of the primary was M ini, 1 = 33 M , while the secondary formed with M ini, 2 = 18 M . The system is ∼ 6 Myr old. (6) The binary evolutionary model confirms that this binary has undergone case B mass transfer and is now in a detached phase. The binary evolutionary model reveals that the primary must be core helium burning and that the observed CNO abundances correspond to those of the hydrogen shell burning layers. (7) According to our binary evolution model, the primary will become a helium star (or maybe a WR star) with a portion of the hydrogen remaining in the envelope (X H < 50%) and finally collapse to a BH or NS. If the binary system stays bound after core-collapse, it might show up as a HMXB with a rapidly rotating Be-type donor. After core hydrogen depletion, the secondary is expected to expand and transfer mass onto the compact object, stripping off most of its hydrogen-rich envelope. The secondary will spend most of its late evolutionary phases in the blue part of the HRD as a helium star with a small fraction of hydrogen left in the envelope. However, due to its low L/M ratio, it will never be able to display a WR-like spectrum. Finally, the secondary will collapse, and there is a small chance it will form a binary of compact objects that could potentially merge within a Hubble time.
The three different methods that we used to derive the stellar masses of the binary components show that the different mass estimates agree within their respective uncertainties (e.g., Table 6 and Fig. 11). Nonetheless, the different methods yield somewhat different results. We point out several difficulties that come along with the different methods applied, for instance when measuring surface gravity (see Fig. 5). Finally, we conclude that the earliest type eclipsing binary in the SMC, AzV 476, provides a unique laboratory for studying massive binaries at low metallicity.
A&A proofs: manuscript no. main
Appendix A: RV determination
To determine the RV shift of a star during a specific phase we use a MCMC approach. In our code we use the synthetic spectra obtained from the spectral analysis and shift them until the shifted synthetic spectrum best matches the observed spectrum. In our procedure, we use the method described in the emcee documentation 5 with small adjustments such that it fulfills our needs.
The probability function, a measure of how well the synthetic spectrum fits the observed one, consists of the combination of a prior function (i.e., a function that limits the parameter space to reasonable numbers) and a likelihood function. Uniform distributions are used as prior functions to avoid biases towards specific RV shifts,
log p prior ( 1 , 2 , f ) =
if −50 km s −1 < 1 < 600 km s −1 0 and −600 km s −1 < 2 < 50 km s −1 and − ∞ < log f < 1.0 − ∞ otherwise.
(A.1)
For the likelihood function we use the least square function and assume that the variance is underestimated by some fractional amount f , log p likelihood = − 1 2 n (y n − y model, n ) 2 s 2 n + log 2πs 2 n , (A.2) where s 2 n = σ 2 n + f (y model, n ) 2 . (A.3)
Using the above-mentioned functions, the probability function can be expressed as log P = log p prior + log p likelihood .
(A.4)
As recommended in the emcee documentation 5 the starting values are initialized as tiny Gaussian balls around 1 = 300 km s −1 , 2 = −300 km s −1 and log f = 0.0. To ensure a good converged posterior distribution, 32 walkers (i.e., individual Makrov chains) are used and each one is iterated for 5000 steps. We cut off the first 1000 steps to ensure that the chains are "burned-in" such that the remaining distribution resembles the posterior distribution. The uncertainties are based on the 16th, 50th, and 84th percentiles of the samples, corresponding to approximately 1σ deviation. To make these values readable for the PHOEBE code, which requires symmetric error margins, only the larger uncertainty is used. For each of the components, we take the mean value of the RVs obtained from several absorption lines. The primary is causing most of the He ii lines seen in the composite spectrum. Only the He ii λ4025 line shows contributions from both components. The secondary contributes the majority of the observed He i absorption lines, but in many of them also the primary can be seen. We use the He ii λ4025 line to fit both components and to obtain their RVs simultaneously, such that this line can be used as a cross-check to see if these fits are in agreement with the other (Table 1). The observed spectrum is shown as a solid blue line, our best fitting model as a dotted red line, the unweighted synthetic spectrum of the primary as a dashed black line, and the unweighted spectrum of the secondary as a dotted green line. The observed spectrum is convolved with a Gaussian with a FWHM = 0.2 Å and is corrected for the velocity of the SMC and the barycentric motion. The line identification marks correspond to the wavelengths of the absorption lines in the rest frame.
lines that are purely from the primary, like He ii λ4200, or the lines that are associated with the secondary. This gives us confidence that the RVs of the secondary obtained with this method are trustworthy. The obtained RVs of the primary and secondary can be found in Table A.1 and A.2.
This fitting procedure turns out to work well even for line complexes with contributions from both stars. Figure A.1 shows one of our fits of the He i λ3819 region. The He i λ3819 absorption line is present in both spectra. The absorption line of the primary is shifted redwards and the rotationally broadened absorption line of the secondary is shifted bluewards. This complex is blended by the blueshifted He ii λ3813 absorption line of the primary.
This method requires sufficiently high S/N observations. The MCMC code struggles with some of the He i lines that are associated with the secondary. In most cases, the problems emerge when the depth of the absorption lines is of the same order as the noise (i.e., the UVES spectra have a S/N ∼ 25). Moreover, some of the UVES spectra show a wavy pattern from the echelle orders. In some cases, these waves overlap with absorption lines, making an accurate RV determination almost impossible. All fits are inspected by eye and the observations that show the aforementioned issues are discarded from our analysis. D. Pauli et al.: The earliest O-type eclipsing binary in the Small Magellanic Cloud, AzV 476. (a) The He i λ3819 is present in the primary and secondary spectrum. We note that the observed He i λ3819 line also contains the He ii λ3813 line from the primary. (b) The He ii λ4025 line has a strong contribution from the secondary. (c) The mean value is calculated as i=n i=1 RV i /n where RV i is the RV of line i and n is the total number of lines used. The errors of the mean values are calculated via Gaussian error propagation. As main diagnostic wind lines, the C iv resonance doublet in the UV and the Hα line in the optical are used. However, the He ii λ1640 in the UV and the He ii λ4868 in the optical are also sensitive to the stellar wind and are taken into account when adjusting the stellar parameters of our synthetic model. These lines are shown in Fig. B.3. Table 1) and synthetic spectra of the region around 3400 Å showing the O iv and N iv lines. The observed spectrum (blue) shows clear indications of ISM absorption lines. The black line is the synthetic spectrum without a contribution of ISM lines. The spectrum with the modeled ISM lines is shown as red dotted line. The interstellar absorption arise in the Galactic foreground and in the SMC.
Fig. 5 .
5Selected regions of the X-SHOOTER spectrum (ID 5 in
Fig. 6 .
6Part of the observed FUSE spectrum (ID 1 in
Fig. 7 .
7Observed (blue) and synthetic (red) C iv resonance doublet of our final model. The individual primary and secondary spectra are shown as dotted black and green lines, respectively. The sharp interstellar absorptions arise in the Galactic foreground and in the SMC, and are also modeled with their respective RV shifts. (a) COS observation (ID 2 in
Fig. 9 .
9Spectral fit for AzV 476. Top panel: SED. Flux-calibrated observations (blue lines) are the FUSE (1), COS (2), STIS
Fig. 11 .
11HRD showing the positions of each binary component according to the spectroscopic and orbital analysis, and the best fitting binary evolutionary tracks. The results of the spectroscopic analysis are shown as triangles and the orbital analysis as upside-down triangles. The positions of the primary and secondary are marked in green and red, respectively. The shaded areas indicate the respective error-ellipses. Evolutionary tracks (solid lines) of both binary components are according to the best fitting binary model calculated with MESA (see Sect. 5.2). The tracks are labeled by their initial masses. The black dots on the tracks correspond to equidistant time-steps of 0.1 Myr to emphasize the most probable observable phases. The light blue dots mark the best-fitting model.
Fig. 12 .
12Abundance profiles of the primary at different evolutionary stages as predicted by the evolutionary models. The legend at the top of each plot explains the colors indicating the dominating mixing processes. The H, He, C, N, and O abundances of the model are shown as solid lines while the observed surface CNO abundances are shown as dotted lines of the same respective color. The vertical black dotted line indicates the mass of the primary after mass transfer. Left panel: Primary has depleted hydrogen in the core and is at the terminal age MS. Right panel: Primary fills its Roche lobe and starts transferring mass.
Fig. A. 1 .
1Radial-velocity fit of the synthetic spectrum to the observed He ii λ3813 and He i λ3819 absorption lines of the UVES spectrum with ID 10
Fig
Same as Fig. 8, but now zoomed on optical N iii and N v lines. . B.2. Observed (X-SHOOTER spectrum; ID 5 in
Fig
. B.3. Observed (blue) and synthetic (red) He ii lines that are sensitive to the wind parameters. The sharp interstellar absorptions arise in the Galactic foreground and in the SMC, and are also modeled with their respective RV shifts. (a) He ii λ1640 line in the COS spectrum (ID
et al. arXiv:2201.09148v1 [astro-ph.SR] 22 Jan 2022 A&A proofs: manuscript no. mainArticle number, page 1 of 23
Table 1 .
1List of all spectra of AzV 476 used in this work and their associated orbital phase. RVs and the wavelengths ranges and lines used for their measurements are also listed. The listed RVs are already corrected for the barycentric motion and the velocity of the NGC 456 complex with SMC = 152 km s −1 .spectral ID Instrument
Wavelength
MJD (a)
Phase φ (b)
RV 1
RV 2
[Å]
[d]
[km s −1 ]
[km s −1 ]
1
FUSE
950 -1150 Å 52478.3849
0.0192
-
-
2
HST/COS
1178 -1777 Å 56185.2387 −0.2305
194 ± 5 (c)
-
3
HST/STIS
1140 -1735 Å 59022.5673 −0.3124
91 ± 5 (c)
−86 ± 12 (c)
4
HST/STIS
1574 -2673 Å 57321.6698
0.0968
−55 ± 5 (d)
64 ± 45 (d)
5
X-SHOOTER 3000 -10 000 Å 59163.2501 −0.2928
112 ± 14
−110 ± 10
6
UVES
3000 -11 000 Å 57703.2868 −0.1610
223 ± 9
−171 ± 15
7
UVES
3000 -11 000 Å 57748.0479 −0.3823
60 ± 16
−42 ± 17
8
UVES
3000 -11 000 Å 57749.0558 −0.2747
126 ± 8
−104 ± 29
9
UVES
3000 -11 000 Å 57749.0923 −0.2708
141 ± 16
−95 ± 37
10
UVES
3000 -11 000 Å 57750.1132 −0.1618
201 ± 13
−184 ± 34
11
UVES
3000 -11 000 Å 57750.1492 −0.1580
207 ± 9
−186 ± 43
12
UVES
3000 -11 000 Å 57751.0782 −0.0588
148 ± 12
−71 ± 37
(a) Mid-exposure in HJD -2400000.5 (b) Calculated with Eq. (2). (c) Obtained from a fit over the range
1360 -1405 Å. (d) Obtained from a fit over the range 2100 -2230 Å.
Table 2 .
2UBVRIJHK photometry of AzV 476.Band Apparent magnitude
[mag]
U
12.49 ± 0.04
B
13.54 ± 0.07
V
13.48 ± 0.01
V OGLE
13.49
R
13.72 ± 0.06
I
13.54 ± 0.20
I OGLE
13.56
J
13.70 ± 0.03
H
13.72 ± 0.04
K
13.86 ± 0.06
G
13.51 ± 0.06
G BP
13.46 ± 0.01
G RP
13.57 ± 0.01
3.1. Method: Eclipse light-curve and RV curve modeling 3.1.1. Light curve Our target, AzV 476, is listed in the IVth OGLE Collection of Variable Stars
Table 3 .
3Dates of the primary eclipses in the TESS light curve.orbital cycle n
MJD (a)
[d]
145
58360.4633 ± 0.0027
146
58369.8256 ± 0.0030
147
58379.1905 ± 0.0029
218
59044.2165 ± 0.0016
219
59053.5942 ± 0.0016
220
59062.9590 ± 0.0014
221
59072.8303 ± 0.0014
222
59081.6913 ± 0.0015
(a) MJD = HJD − 2400000.5. The TESS data are given in
TBJD = BJD − 2457000.0 and that we converted the BJD to
HJD to be comparable to the OGLE data.
Table 4 .
4Orbital parameters obtained from the RV and light curves by the PHOEBE codeParameter
Value
P (d)
9.36665 (fixed)
HJD 0
7002.7968 (fixed)
e
0.240 +0.002
−0.002
Table 5 .
5Stellarparameters obtained from the RV and light curves by
the PHOEBE code
Parameter
Primary
Secondary
T eff [kK]
42 (input) 32 (input)
K [km s −1 ]
161 +13
−13
181 +14
−14
M orb [M ]
20.2 +2.0
−2.0
18.0 +1.8
−1.8
R [R ]
10.7 +0.4
−0.4
7.0 +0.4
−0.4
R RL [R ]
24.5 +0.9
−0.9
23.3 +0.8
−0.8
log g [cm s −2 ]
3.69 +0.06
−0.06
4.00 +0.07
−0.07
log L [L ]
5.51 +0.13
−0.13
4.67 +0.22
−0.22
M bol [mag]
−8.94 +0.32
−0.32
−6.83 +0.56
−0.56
Table 6 .
6Summary of the stellar parameters of both stellar components obtained from the different methods.Spectroscopic analysis
Orbital analysis (a)
Single star evolution (b) Binary evolution (c)
star 1
star 2
star 1
star 2
star 1
star 2
star 1
star 2
T eff [kK]
42 +3
−3
32 +4
−4
42 (fix)
32 (fix)
40.1 +1.7
−1.5
32.5 +1.8
−1.8
42.0
31.6
log g [cm s −2 ]
3.7 +0.2
−0.1
4.0 +0.2
−0.2
3.69 +0.06
−0.06
4.00 +0.07
−0.07
3.80 +0.09
−0.08
4.03 +0.10
−0.11
3.56
4.01
log L [L ]
5.65 +0.2
−0.2
4.75 +0.2
−0.2
5.51 +0.13
−0.13
4.67 +0.22
−0.22
5.60 +0.18
−0.13
4.67 +0.11
−0.14
5.55
4.66
R [R ]
12.6 +1.0
−1.0
7.8 +2.0
−2.0
10.7 +0.4
−0.4
7.0 +0.4
−0.4
12.94 +2.2
−1.7
6.57 +0.8
−0.9
11.3
7.2
M [M ]
29 +17
−11
22 +13
−8
20.2 +2.0
−2.0
18.0 +1.8
−1.8
39.2 +8.8
−5.8
17.4 +1.4
−1.6
17.8
18.2
M ini [M ]
-
-
-
-
40.2 +9.1
−6.4
17.4 +1.4
−1.6
Results from the PHOEBE code, seeSect. 3.2. (b) Results from the BONNSAI tool, see Sect. 5.1. (c) Models calculated with MESA, see Sect. 5.2. No uncertainties are given, as we did no in-depth analysis.(d) Obtained with the iacob broad tool. (e) Initial CNO abundances scaled to SMC metallicity; for more details seeSect. 4.2. ( f ) According to the mass-loss recipe implemented in our evolutionary models.(g) Values taken from the spectroscopic model that is calculated with the parameters of the binary evolutionary models.30
91
7
X O /10 −5 (by mass)
80 +10
−20
110 (e)
-
-
91 +4
−14
70 +7
−37
35
110
log Q H [ s −1 ]
49.43
47.88
-
-
-
-
49.34 (g) 47.72 (g)
log Q He i [ s −1 ]
48.72
45.83
-
-
-
-
48.62 (g) 45.76 (g)
log Q He ii [ s −1 ]
40.80
41.38
-
-
-
-
43.63 (g) 35.75 (g)
age [Myr]
-
-
-
-
2.8 +0.5
−0.4
5.9 +1.5
−1.5
6.0
6.0
(a) the wind velocity field can be described by a so-called double
β-law as was first introduced by Hillier & Miller (1999):
A&A proofs: manuscript no. mainHe II
Hγ
0.7
0.8
0.9
1.0
1.1
4320
4340
4360
λ / A
o
Normalized flux
(a)
He II 7-4
5400
5410
5420
λ / A
o
(b)
He II
He II
Hα
0.7
0.8
0.9
1.0
1.1
6520
6540
6560
6580
6600
λ / A
o
Normalized flux
(c)
). The spectroscopic mass Article number, page 10 of 23 D. Pauli et al.: The earliest O-type eclipsing binary in the Small Magellanic Cloud, AzV 476.AV 476
O4 IV-III((f))p + O9.5: Vn
E B-V = 0.27 mag
-16
-14
-12
3.0
3.2
3.4
3.6
3.8
4.0
4.2
4.4
4.6
log λ/A
o
log f
λ [erg s -1
cm -2
A
o
-1
]
0.0
0.5
1.0
1.5
1000
1100
λ/A
o
Normalized flux
(1)
Lyγ (is)
Ne VII
C III
He II
He II
Lyβ (is)
O VI
F V
He II
P V
S V
1200
1300
1400
1500
1600
λ/A
o
A&A proofs: manuscript no. main0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
3400
3600
3800
4000
4200
4400
4600
4800
5000
5200
5400
λ / A
o
Normalized flux
(5)
O IV
N IV
H I
He I
He I
He II / Hε
He I
He II
He I
N IV
He II
Hδ
He I
He II
He II
Hγ
He I
N III
He II
N V
N III
C III
O II
He II
He I
He II
Hβ
He I
[O III] (neb)
He I
He I
N IV
He II 7-4
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
6000
6500
7000
λ / A
o
Normalized flux
(5)
C III
C IV
He I
He II
He II
He II
He II
Hα
He I
He II
He II
He II
He I
N IV
The binary components of AzV 476 are among the most luminous O stars in the entire SMC. Nevertheless, our analysis reveals that the mass of the O4 IV-III((f))p primary star is only M orb, 1 ≈ 20M . We estimate its initial mass to be M ini, 1 ≈ 33 M which is significantly lower than the most massive O stars in the Galaxy and the LMC (e.g.,Ramírez-Agudelo et al. 2017;Gruner et al. 2019). This system therefore further exacerbates the problem with the deficiency of very massive O stars in the SMC(Ramachandran et al. 2019;Schootemeijer et al. 2021).
Normalized flux v rad,1 : 201 km/s v rad,2 : -169 km/sHe II
He I
0.94
0.96
0.98
1.00
1.02
3810
3815
3820
3825
λ / A
o
Table A
A.1. RVs of the primary determined from different spectral lines. spectral ID He i λ3819 (a) He i λ3935 He ii λ4025 (b) N iv λ4058 He ii λ4200 mean value(c) [km s −1 ] [km s −1 ] [km s −1 ] [km s −1 ] [km s −1 ] [km s −1 ]5
114 ± 63
117 ± 22
120 ± 8
103 ± 7
105 ± 4
112 ± 14
6
-
229 ± 26
236 ± 8
217 ± 21
210 ± 4
223 ± 9
7
48 ± 63
36 ± 47
53 ± 8
46 ± 9
55 ± 8
47 ± 16
8
140 ± 13
115 ± 34
134 ± 11
117 ± 18
122 ± 4
126 ± 8
9
145 ± 59
-
153 ± 10
134 ± 22
131 ± 5
141 ± 16
10
201 ± 53
215 ± 25
209 ± 11
195 ± 18
183 ± 8
201 ± 13
11
223 ± 32
212 ± 26
214 ± 9
192 ± 13
196 ± 4
207 ± 9
12
162 ± 18
167 ± 32
131 ± 14
137 ± 45
144 ± 8
148 ± 12
Table A . 2 .
A2RVs of the secondary determined from different spectral lines. spectral ID He i λ3819 (a) He ii λ4025 (b) He i λ4143 He i λ4387 He i λ4471 mean value(c) Same footnotes as inTable A.1.[km s −1 ]
[km s −1 ]
[km s −1 ]
[km s −1 ]
[km s −1 ]
[km s −1 ]
5
−88 ± 29
−122 ± 31
-
−117 ± 20 −113 ± 18
−110 ± 13
6
-
−164 ± 22
-
−198 ± 52 −152 ± 24
−171 ± 21
7
−41 ± 83
−33 ± 23
-
−32 ± 35
−35 ± 31
−35 ± 25
8
−137 ± 89
−79 ± 41
−110 ± 55
-
−91 ± 24
−104 ± 29
9
−108 ± 112
-
−95 ± 88
−81 ± 37
−97 ± 24
−95 ± 37
10
−169 ± 52
−199 ± 43
-
-
-
−184 ± 34
11
−194 ± 34
−151 ± 26
−213 ± 121
-
-
−186 ± 43
12
−149 ± 42
−125 ± 79
−146 ± 68 −130 ± 66
132 ± 42
−132 ± 27
www.phoebe-project.org Article number, page 5 of 23 A&A proofs: manuscript no. main
Article number, page 14 of 23 D. Pauli et al.: The earliest O-type eclipsing binary in the Small Magellanic Cloud, AzV 476.
https://emcee.readthedocs.io
inTable 1). (b) He ii λ4868 line in the X-SHOOTER spectrum (ID 5 inTable 1). Article number, page 22 of 23 D. Pauli et al.: The earliest O-type eclipsing binary in the Small Magellanic Cloud, AzV 476.
Acknowledgements. We dedicate this publication to the late Dr. Rodolfo H. Barba, whose passing is a tragic loss to the astrophysics community. As the original referee of this paper, he provided an outstandingly detailed and substantial report, which was extremely helpful and led to significant improvements of this work. Dr. Barba also provided us with the extracted TESS light curve of AzV 476 which resulted in the update in the orbital period. The authors thank Dr. Alex Fullerton for useful discussion and advise related to the FUSE spectroscopy and Soetkin Janssens for very helpful advises regarding the PHOEBE code. We gratefully thank the staff of the FUSE, HST, ESO telescopes for making the useful spectroscopic data publicly available. The results presented in this paper are based on observations obtained with the NASA/ESA Hubble Space Telescope, retrieved from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute (STScI). STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555. Furthermore, its conclusions are based on observations collected at the European Southern Observatory (ESO) under the program 098.A-0049. The authors thank Andrea Mehner for preparing the OBs of the XSHOOTU project and the people on the management committee of XSHOOTU. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/ gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. DP acknowledges financial support by the Deutsches Zentrum für Luft und Raumfahrt (DLR) grant FKZ 50 OR 2005. AACS is Öpik Research Fellow at Armagh Observatory & Planetarium. AACS and VR acknowledge support by the Deutsche Forschungsgemeinschaft (DFG -German Research Foundation) in the form of an Emmy Noether Research Group (grant number SA4064/1-1, PI Sander). JMA acknowledges support from the Spanish Government Ministerio de Ciencia through grant PGC2018-095 049-B-C22.Appendix B: Additional figuresAppendix B.1: Additional optical N iii and N v linesFigure B.1 shows N iii and N v lines in our spectrum with the highest S/N, the X-SHOOTER spectrum (ID 5 inTable 1, S/N ∼ 100). These lines are very weak and therefore barely visible in the spectrum. N v λ4604 is marginally in absorption, while N iii λλ4634, 4641 appears in slight emission. Although nearly hidden in the noise, these weak features are reproduced by our final model (red-dotted).Appendix B.2: Optical O iv and N iv linesWalborn et al. (2004)suggest to use the optical O iv multiplets around 3400 Å and the N iv absorption lines at wavelengths 3463 Å, 3478 Å, 3483 Å and 3485 Å to derive the N and O abundances in hot massive stars. Luckily, the X-SHOOTER spectrum extends to such short wavelengths.Figure B.2 shows that the N iv lines of our final model are in agreement with the observation. However, for the O iv multiplet at wavelengths 3403 Å and 3414 Å the synthetic spectrum is in emission and not in absorption as observed. For the O iv λ3414 Å line, there is an ISM contribution from Co i that explains the observed absorption. Regarding the O iv 3403 Å line, we are not aware of any blending lines. However, this particular line is formed close to the onset of the wind and is sensitive to different details within the wind (e.g., temperature stratification, wind velocity law). As the other O iv multiplet λλ3381 -3389 Å can be well reproduced, we trust our determined O abundance, which is supported by other absorption lines in the UV. This plot demonstrates that the predicted surface abundances of the binary evolutionary models are unable to reproduce the observed spectrum. The nitrogen abundance in this model is too strong as it overestimates all optical N iv lines and also predicts unseen N v absorption. The reduced hydrogen abundance leads to an increased helium abundance. This enrichment would produce too deep He ii lines in the spectrum of the primary.
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"T Moraschini ",
"J G Raftery ",
"J J Wannenburg "
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It is proved that every finitely subdirectly irreducible De Morgan monoid A (with neutral element e) is either (i) a Sugihara chain in which e covers ¬e or (ii) the union of an interval subalgebra [¬a, a] and two chains of idempotents, (¬a] and [a), where a = (¬e) 2 . In the latter case, the variety generated by [¬a, a] has no nontrivial idempotent member, and A/[¬a) is a Sugihara chain in which ¬e = e. It is also proved that there are just four minimal varieties of De Morgan monoids. This theorem is then used to simplify the proof of a description (due to K.Świrydowicz) of the lower part of the subvariety lattice of relevant algebras. The results throw light on the models and the axiomatic extensions of fundamental relevance logics.which allows us to recover the whole of ⊢ R t from its theorems.
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10.1016/j.jpaa.2018.09.015
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1801.06650
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874fef45683a430beeb4d9283ead3a84aeadd54c
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20 Jan 2018
T Moraschini
J G Raftery
J J Wannenburg 20 Jan 2018arXiv:1801.06650v1 [math.LO] VARIETIES OF DE MORGAN MONOIDS: MINIMALITY AND IRREDUCIBLE ALGEBRAS2010 Mathematics Subject Classification Primary: 03B4706D9906F05 Secondary: 03G2506D3008B15 Key words and phrases De Morgan monoidSugihara monoidresiduated latticerele- vance logic
It is proved that every finitely subdirectly irreducible De Morgan monoid A (with neutral element e) is either (i) a Sugihara chain in which e covers ¬e or (ii) the union of an interval subalgebra [¬a, a] and two chains of idempotents, (¬a] and [a), where a = (¬e) 2 . In the latter case, the variety generated by [¬a, a] has no nontrivial idempotent member, and A/[¬a) is a Sugihara chain in which ¬e = e. It is also proved that there are just four minimal varieties of De Morgan monoids. This theorem is then used to simplify the proof of a description (due to K.Świrydowicz) of the lower part of the subvariety lattice of relevant algebras. The results throw light on the models and the axiomatic extensions of fundamental relevance logics.which allows us to recover the whole of ⊢ R t from its theorems.
Introduction
De Morgan monoids are commutative monoids with a residuated distributive lattice order and a compatible antitone involution ¬, where a a 2 for all elements a. They form a variety, DMM.
The explicit study of residuated lattices goes back to Ward and Dilworth [73] and has older antecedents (see the citations in [9,24,29]). Much of the interest in De Morgan monoids stems, however, from their connection with relevance logic, discovered by Dunn [15] and recounted briefly below in Section 4.1 (where further references are supplied). A key fact, for our purposes, is that the axiomatic extensions of Anderson and Belnap's logic R t and the varieties of De Morgan monoids form anti-isomorphic lattices, and the latter are susceptible to the methods of universal algebra.
Slaney [59,60] showed that the free 0-generated De Morgan monoid is finite, and that there are only seven non-isomorphic subdirectly irreducible 0-generated De Morgan monoids. No similarly comprehensive classification is available in the 1-generated case, however, where the algebras may already be infinite. In 1996, Urquhart [70, p. 263] observed that "[t]he algebraic theory of relevant logics is relatively unexplored, particularly by comparison with the field of algebraic modal logic." Acquiescing in a paper of 2001, Dunn and Restall [17,Sec. 3.5] wrote: "Not as much is known about the algebraic properties of De Morgan monoids as one would like." These remarks pre-date many recent papers on residuated lattices-see the bibliography of [24], for instance. But the latter have concentrated mainly on varieties incomparable with DMM (e.g., Heyting and MV-algebras), larger than DMM (e.g., full
Lambek algebras) or smaller (e.g., Sugihara monoids).
A De Morgan monoid A, with neutral element e, is said to be idempotent or anti-idempotent if it satisfies x 2 = x or x (¬e) 2 , respectively. The idempotent De Morgan monoids are the aforementioned Sugihara monoids, and their structure is very well understood. Anti-idempotence is equivalent to the demand that no nontrivial idempotent algebra belongs to the variety generated by A (Corollary 3.6), hence the terminology.
It is well known that a De Morgan monoid is finitely subdirectly irreducible iff the element e is join-prime. The first main result of this paper shows that any such De Morgan monoid A is either (i) a totally ordered Sugihara monoid in which e covers ¬e or (ii) the union of an interval subalgebra [¬a, a] and two chains of idempotent elements, (¬a] and [a), where a = (¬e) 2 . In the latter case, the anti-idempotent subalgebra is the e-class of a congruence θ such that A/θ is a totally ordered Sugihara monoid in which ¬e = e, and all other θ-classes are singletons. (See Theorem 5.17 and Remark 5.19.) Subalgebra structure aside, another measure of the complexity of a De Morgan monoid A is the height, within the subvariety lattice of DMM, of the variety generated by A. Accordingly, the present paper initiates an analysis of the lattice of varieties of De Morgan monoids. We prove that such a variety consists of Sugihara monoids iff it omits a certain pair of four-element algebras (Theorem 5.21). This implies the paper's second main result: DMM has just four minimal subvarieties, all of which are finitely generated (Theorem 6.1). The covers of these atoms will be investigated in a subsequent paper.
For philosophical reasons, relevance logic also emphasizes a system called R, which lacks the so-called Ackermann truth constant t (corresponding to the neutral element of a De Morgan monoid). The logic R is algebraized by the variety RA of relevant algebras.Świrydowicz [65] has described the bottom of the subvariety lattice of RA. We simplify the proof of his result (see Theorem 7.8), using the above identification of the minimal subvarieties of DMM.
These findings have implications for the extension lattices of both R and R t . For instance,Świrydowicz's theorem has been applied recently to show that no consistent axiomatic extension of R is structurally complete, except for classical propositional logic [54]. The situation for R t is very different and is the subject of ongoing investigation by the present authors.
Residuated Structures
Definition 2.1. An involutive (commutative) residuated lattice, or briefly, an IRL, is an algebra A = A; ·, ∧, ∨, ¬, e comprising a commutative monoid A; ·, e , a lattice A; ∧, ∨ and a function ¬ : A − → A, called an involution, such that A satisfies ¬¬x = x and (1) x · y z ⇐⇒ ¬z · y ¬x, cf. [24]. Here, denotes the lattice order (i.e., x y abbreviates x ∧ y = x) and ¬ binds more strongly than any other operation; we refer to · as fusion.
Setting y = e in (1), we see that ¬ is antitone. In fact, De Morgan's laws for ¬, ∧, ∨ hold, so ¬ is an anti-automorphism of A; ∧, ∨ . If we define
x → y : = ¬(x · ¬y) and f : = ¬e, then, as is well known, every IRL satisfies
x · y z ⇐⇒ y x → z (the law of residuation), (2) ¬x = x → f and x → y = ¬y → ¬x and x · y = ¬(x → ¬y).
Definition 2.2. A (commutative) residuated lattice-or an RL-is an algebra A = A; ·, →, ∧, ∨, e comprising a commutative monoid A; ·, e , a lattice A; ∧, ∨ and a binary operation →, called residuation, where A satisfies (2).
Thus, up to term equivalence, every IRL has a reduct that is an RL. Conversely, every RL can be embedded into (the RL-reduct of) an IRL; see [26] and the antecedents cited there. Every RL satisfies the following well known laws. Here and subsequently, x ↔ y abbreviates (x → y) ∧ (y → x).
x · (x → y) y and x (x → y) → y (4)
(x · y) → z = y → (x → z) = x → (y → z) (5) (x → y) · (y → z) x → z (6) x · (y ∨ z) = (x · y) ∨ (x · z)(7)
x y =⇒
x · z y · z and z → x z → y and y → z x → z (8)
x y ⇐⇒ e x → y (9)
x = y ⇐⇒ e x ↔ y (10) e x → x and e → x = x (11) e x ⇐⇒ x → x x.(12)
By (10), an RL A is nontrivial (i.e., |A| > 1) iff e is not its least element, iff e has a strict lower bound. A class of algebras is said to be nontrivial if it has a nontrivial member.
Another consequence of (10) is that a non-injective homomorphism h between RLs must satisfy h(c) = e for some c < e.
(Choose c = e ∧ (a ↔ b), where h(a) = h(b) but a = b.)
In an RL, we define x 0 : = e and x n+1 : = x n · x for n ∈ ω. Lemma 2.3. If a (possibly involutive) RL A has a least element ⊥, then ⊤ : = ⊥ → ⊥ is its greatest element and, for all a ∈ A,
a · ⊥ = ⊥ = ⊤ → ⊥ and ⊥ → a = ⊤ = a → ⊤ = ⊤ 2 .
In particular, {⊥, ⊤} is a subalgebra of the ·, →, ∧, ∨ (, ¬) reduct of A.
Proof. See [50,Prop. 5.1], for instance. (We infer ⊤ = ⊤ 2 from (8), as e ⊤. The lattice anti-automorphism ¬, if present, clearly switches ⊥ and ⊤.)
If we say that ⊥, ⊤ are extrema of an RL A, we mean that ⊥ a ⊤ for all a ∈ A. An RL with extrema is said to be bounded. In that case, its extrema need not be distinguished elements, so they are not always retained in subalgebras. The next lemma is a straightforward consequence of (2).
Lemma 2.4. The following conditions on a bounded RL A, with extrema ⊥, ⊤, are equivalent.
(i) ⊤ · a = ⊤ whenever ⊥ = a ∈ A. (ii) a → ⊥ = ⊥ whenever ⊥ = a ∈ A. (iii) ⊤ → b = ⊥ whenever ⊤ = b ∈ A.
Definition 2.5. Following Meyer [46], we say that an RL is rigorously compact if it is bounded and satisfies the equivalent conditions of Lemma 2.4. Lemma 2.6. Let A be an IRL, with a ∈ A. Then e a = a 2 iff a · ¬a = ¬a iff a = a → a.
Proof. The second and third conditions are equivalent, by the definition of → and involution properties. Also, a 2 a and a · ¬a ¬a are equivalent, by (1). From e a and (8) we infer ¬a = e · ¬a a · ¬a. Conversely, a → a a and (11) yield e a, and therefore a a 2 .
The class of all RLs and that of all IRLs are finitely axiomatized varieties. They are arithmetical (i.e., congruence distributive and congruence permutable) and have the congruence extension property (CEP). These facts can be found, for instance, in [24, Sections 2.2 and 3.6].
Square-Increasing IRLs
An RL is said to be square-increasing if it satisfies x x 2 . Every squareincreasing RL can be embedded into a square-increasing IRL [43]. Moreover, Slaney [63] has shown that if two square-increasing IRLs have the same RLreduct, then they are equal. The following laws obtain in all square-increasing IRLs (and not in all IRLs):
x ∧ y x · y (13) x, y e =⇒ x · y = x ∧ y (14) e x ∨ ¬x.
The lemma below generalizes another result of Slaney [59, T36, p. 491] (where only the case a = f was discussed, and A satisfied an extra postulate).
Lemma 3.1. Let A be a square-increasing IRL, with f a ∈ A. Then a 3 = a 2 .
In particular, f 3 = f 2 .
Proof. As f a, we have ¬a = a → f a → a, by (3) and (8), so (16) a → ¬a a → (a → a) = a 2 → a, by (8) and (5). By the square-increasing law, (16), (8) and (6),
a → ¬a (a → ¬a) 2 (a 2 → a) · (a → ¬a) a 2 → ¬a.
Thus, ¬(a 2 → ¬a) ¬(a → ¬a), i.e., a 2 · a a · a (see (3)), i.e., a 3 a 2 . The reverse inequality follows from the square-increasing law and (8).
The first assertion of the next theorem has unpublished antecedents in the work of relevance logicians. A corresponding result for 'relevant algebras' is reported in [66,Prop. 5], but the claim and proof below are simpler.
Theorem 3.2. Every finitely generated square-increasing IRL A is bounded. More informatively, let {a 1 , . . . , a n } be a finite set of generators for A, with
c = e ∨ f ∨ i ≤ n (a i ∨ ¬a i ), and b = c 2 .
Then ¬b a b for all a ∈ A.
Proof. By De Morgan's laws, every element of A has the form ϕ A (a 1 , . . . , a n ) for some term ϕ(x 1 , . . . , x n ) in the language ·, ∧, ¬, e. The proof of the present theorem is by induction on the complexity #ϕ of ϕ. We shall write x and a for the respective sequences x 1 , . . . , x n and a 1 , . . . , a n .
For the case #ϕ ≤ 1, note that e, a 1 , . . . , a n c b, by the squareincreasing law. Likewise, f, ¬a 1 , . . . , ¬a n c b, so by involution properties, ¬b e, a 1 , . . . , a n . Now suppose #ϕ > 1 and that ¬b ψ A (a) b for all terms ψ with #ψ < #ϕ. The desired result, viz.
¬b ϕ A (a) b,
follows from the induction hypothesis and basic properties of IRLs if ϕ has the form ¬ψ(x) or ψ 1 (x)∧ ψ 2 (x). We may therefore assume that ϕ is ψ 1 (x)·ψ 2 (x) for some less complex terms ψ 1 (x), ψ 2 (x).
By the induction hypothesis and (8), (¬b) 2 ϕ A (a) b 2 . As ¬b e, we have (¬b) 2 = ¬b, by (14). And since f c, Lemma 3.1 gives c 3 = c 2 , so b 2 = c 4 = c 2 = b. Therefore, ¬b ϕ A (a) b, as required.
In a square-increasing IRL, the smallest subalgebra B (generated by ∅) has top element (e ∨ f ) 2 = f 2 ∨ e (by Theorem 3.2 and (7)). This is a lower bound of f → f 2 (by (2) and Lemma 3.1), so f 2 ∨ e = f → f 2 . That the extrema of B can be expressed without using ∧, ∨ is implicit in [46, p. 309]. Note also that e ↔ f = f ∧ ¬(f 2 ) is the least element of B.
An element a of an [I]RL
A is said to be idempotent if a 2 = a. We say that A is idempotent if all of its elements are. In the next result, the key implication is (ii) ⇒ (iii). A logical analogue of (ii) ⇔ (iii) is stated without proof in [46, p. 309]. Theorem 3.3. In a square-increasing IRL A, the following are equivalent.
(i) f 2 = f . (ii) f e. (iii) A is idempotent.
Consequently, a square-increasing non-idempotent IRL has no idempotent subalgebra (and in particular, no trivial subalgebra).
Proof. In any IRL, (i) ⇒ (ii) instantiates (1) (as ¬f = e), and (iii) ⇒ (i) is trivial.
(ii) ⇒ (iii): Suppose f e, and let a ∈ A. It suffices to show that a 2 a, or equivalently (by (1)), that a · ¬a ¬a. Now, by the square-increasing law, (8), the associativity of fusion, (3) and (4),
a · ¬a a · (¬a) 2 = (a · (a → f )) · ¬a f · ¬a e · ¬a = ¬a.
In a partially ordered set, we denote by [a) the set of all upper bounds of an element a (including a itself), and by (a] the set of all lower bounds.
A deductive filter of a (possibly involutive) RL A is a lattice filter G of A; ∧, ∨ that is also a submonoid of A; ·, e . Thus, [e) is the smallest deductive filter of A. The lattice of deductive filters of A and the congruence lattice Con A of A are isomorphic. The isomorphism and its inverse are given by
G → ΩG : = { a, b ∈ A 2 : a → b, b → a ∈ G}; θ → {a ∈ A : a ∧ e, e ∈ θ}.
For a deductive filter G of A and a, b ∈ A, we often abbreviate A/ΩG as A/G, and a/ΩG as a/G, noting that
(17) a → b ∈ G iff a/G b/G in A/G.
In the square-increasing case, the deductive filters of A are just the lattice filters of A; ∧, ∨ that contain e, by (13). This yields the following lemma.
(i) if e b ∈ A, then [b) is a deductive filter of A, e.g., (ii) [¬(f 2 )) is a deductive filter of A.
Here, (ii) follows from (i), because e ¬(f 2 ) follows from f f 2 .
Theorem 3.5. Let G be a deductive filter of a square-increasing IRL A. Then
A/G is idempotent iff ¬(f 2 ) ∈ G. In particular, A/[¬(f 2 )) is idempotent. Proof. A/G is idempotent iff f /G e/G (by Theorem 3.3), iff f → e ∈ G (by (17)), iff ¬(f 2 ) ∈ G (as ¬(f 2 ) = ¬(f · ¬e) = f → e).
We say that a square-increasing IRL is anti-idempotent if it satisfies x f 2 (or equivalently, ¬(f 2 )
x). This terminology is justified by the corollary below.
Corollary 3.6. Let K be a variety of square-increasing IRLs. Then K has no nontrivial idempotent member iff it satisfies x f 2 .
Proof. (⇒): As K is homomorphically closed but lacks nontrivial idempotent members, Theorem 3.5 shows that the deductive filter [¬(f 2 )) of any A ∈ K coincides with A, i.e., K satisfies ¬(f 2 ) x.
(⇐): If A ∈ K is idempotent, then f 2 = f e = ¬f = ¬(f 2 ), by Theorem 3.3, so by assumption, A is trivial.
Recall that an algebra A is subdirectly irreducible (SI) if its identity relation id A = { a, a : a ∈ A} is completely meet-irreducible in its congruence lattice; see for instance [5,Thm. 3.23]. If id A is merely meet-irreducible in Con A, then A is said to be finitely subdirectly irreducible (FSI), whereas A is simple if |Con A| = 2. (Thus, trivial algebras are FSI, but are neither SI nor simple.) By Birkhoff's Subdirect Decomposition Theorem [5,Thm. 3.24], every algebra is isomorphic to a subdirect product of SI homomorphic images of itself. (Even a trivial algebra is a copy of the direct product of an empty family.) Also, every algebra embeds into an ultraproduct of finitely generated subalgebras of itself [12, Thm. V.2.14]. Consequently, every variety is generated-and thus determined-by its SI finitely generated members, so we need to understand these algebras in the present context. The following result is well known; see [25,Cor. 14] and [50,Thm. 2.4], for instance. Here and subsequently, an RL A is said to be distributive if its reduct A; ∧, ∨ is a distributive lattice. (i) A is FSI iff e is join-irreducible in A; ∧, ∨ . Therefore, subalgebras and ultraproducts of FSI [I]RLs are FSI. (ii) When A is distributive, it is FSI iff e is join-prime (i.e., whenever a, b ∈ A with e a ∨ b, then e a or e b). (iii) If there is a largest element strictly below e, then A is SI. The converse holds if A is square-increasing. (iv) If e has just one strict lower bound, then A is simple. The converse holds when A is square-increasing.
An [I]RL is said to be semilinear if it is isomorphic to a subdirect product of totally ordered algebras; it is integral if e is its greatest element, in which case it satisfies e = x → x = x → e. A Brouwerian algebra is an integral idempotent RL, i.e., an RL satisfying x · y = x ∧ y. Such an algebra is determined by its lattice reduct, and is distributive, by (7). The variety of relative Stone algebras comprises the semilinear Brouwerian algebras; it is generated by the Brouwerian algebra on the chain of non-negative integers.
De Morgan Monoids
Definition 4.1. A De Morgan monoid is a distributive square-increasing IRL. 1 The variety of De Morgan monoids shall be denoted by DMM.
The following lemma is well known and should be contrasted with the previous section's concluding remarks about involutionless algebras.
Lemma 4.2. A De Morgan monoid is integral iff it is a Boolean algebra (in which the operation ∧ is duplicated by fusion).
Proof. Sufficiency is clear. Conversely, by (15) and De Morgan's laws, the fusionless reduct of an integral De Morgan monoid is a complemented (bounded) distributive lattice, i.e., a Boolean algebra, and · is ∧, by (14).
An algebra is said to be n-generated (where n is a cardinal) if it has a generating subset with at most n elements. Thus, an IRL is 0-generated iff it has no proper subalgebra.
Infinite 1-generated De Morgan monoids exist. Indeed, the integer powers of 2, with the usual order and ordinary multiplication as fusion, can be extended to an algebra of this kind. The larger varieties of distributive and of square-increasing IRLs each have infinite 0-generated members as well [62], but Slaney proved that the free 0-generated De Morgan monoid has just 3088 elements [59]. His arguments show that, up to isomorphism, only eight 0generated De Morgan monoids are FSI; they are exhibited in [60]. As the seven nontrivial 0-generated FSI De Morgan monoids are finite, they are just the 0-generated SI De Morgan monoids.
A theorem of Urquhart [69] implies that the equational theory of DMM is undecidable, whereas results in [10,36,47] show that the respective varieties of distributive and of square-increasing IRLs are generated by their finite members, whence their equational theories are decidable. (In the squareincreasing case, the complexity of any decision procedure is known to be immense [71].)
Recall that a quasivariety is a class of similar algebras closed under isomorphic images, subalgebras, direct products and ultraproducts. Equivalently, it is the model class of some set of pure quasi-equations
(α 1 = β 1 & . . . & α n = β n ) =⇒ α = β
in an algebraic signature. Here n ∈ ω, i.e., quasi-equations have finite length and encompass equations. Although a quasivariety need not be homomorphically closed (i.e., it need not be a variety), it must contain a trivial algebra, viz. the direct product of its empty subfamily.
Relevance Logic and De Morgan Monoids.
For present purposes, a logic is a substitution-invariant finitary consequence relation ⊢ over sentential formulas in an algebraic signature, cf. [8,13,19,20]. The general connections between residuated structures and substructural logics are explained in [24]. In the case of De Morgan monoids, the connection is with the older family of relevance logics. The monographs and survey articles on this subject include [2,3,11,17,39,40,55,56,58]. The correspondence is as follows.
For each subquasivariety K of DMM, there is a logic ⊢ K with the same signature, defined thus: for any set Γ ∪ {α} of formulas, Γ ⊢ K α iff there exist n ∈ ω and γ 1 , . . . , γ n ∈ Γ such that every algebra in K satisfies
e γ 1 ∧ . . . ∧ γ n =⇒ e α.
The elements (also called the derivable rules) of ⊢ K are just the pairs Γ/α for which this is true. In particular, the theorems of ⊢ K (i.e., the formulas α for which ∅ ⊢ K α) are just the IRL terms that take values in [e) whenever their variables are interpreted in any member of K.
Because DMM satisfies (10), the logic ⊢ K is algebraizable in the sense of [8], with K as its unique equivalent quasivariety. The map K → ⊢ K is a lattice anti-isomorphism from the subquasivarieties of DMM to the extensions of the relevance logic R t of [2], carrying the subvarieties of DMM onto the axiomatic extensions. In particular, R t itself is algebraized by DMM.
The relationship between R t and DMM was essentially established by Dunn [15] (see his contributions to [2], as well as [48]). Strictly speaking, R t denotes a formal system of axioms and inference rules, not a consequence relation. Here, however, we routinely attribute to a formal system F the significant properties of its deducibility relation ⊢ F . 2 Although relevance logic has multiple interpretations (see for instance [57,64,67,68,70]), it was originally intended as a framework in which the socalled paradoxes of material implication could be avoided. These include the weakening axiom p → (q → p). The unprovability of this postulate in R t reflects the fact that De Morgan monoids need not be integral, and Lemma 4.2 says in effect that classical propositional logic is the extension of R t by the weakening axiom. Another relevance logic R, and its connection with De Morgan monoids, will be discussed in Section 7.
The Structure of De Morgan Monoids
In the relevance logic literature, a De Morgan monoid is said to be prime if it is FSI. The reason is Lemma 3.7(ii), but we continue to use 'FSI' here, as it makes sense for arbitrary algebras. The next result is easy and well known, but note that it draws on all the key properties of De Morgan monoids.
Theorem 5.1. Let A be a De Morgan monoid that is FSI, with a ∈ A. Then e a or a f . Thus, A = [e) ∪ (f ].
Proof. As A is square-increasing, e a ∨ ¬a, by (15). So, because A is distributive and FSI, e a or e ¬a, by Lemma 3.7(ii). In the latter case, a f , because ¬ is antitone. The following result about bounded De Morgan monoids was essentially proved by Meyer [46,Thm. 3], but his argument assumes that the elements ⊥, ⊤ are distinguished, or at least definable in terms of generators. To avoid that presupposition, we give a simpler and more direct proof.
Theorem 5.3. Let A be a bounded FSI De Morgan monoid. Then A is rigorously compact (see Definition 2.5).
Proof. Let ⊥ = a ∈ A, where ⊥, ⊤ are the extrema of A. It suffices to show that ⊤ · a = ⊤. As e · a ⊥, we have ⊤ · a f , by (1), so (18) e ⊤ · a, by Theorem 5.1. Recall that ⊤ 2 = ⊤, by Lemma 2.3. Therefore,
⊤ = ⊤ · e ⊤ 2 · a (by (18)) = ⊤ · a ⊤, whence ⊤ · a = ⊤.
Corollary 5.4. If a De Morgan monoid is FSI, then its finitely generated subalgebras are rigorously compact.
Proof. This follows from Lemma 3.7(i) and Theorems 3.2 and 5.3.
At this point, we need to recall a few concepts and results from universal algebra. The class operator symbols I, H, S, P, P S and P U stand, respectively, for closure under isomorphic and homomorphic images, subalgebras, direct and subdirect products, and ultraproducts. Also, V and Q denote varietal and quasivarietal closure, i.e., V = HSP and Q = ISPP U . We abbreviate V({A}) as V(A), etc.
A variety K is said to be finitely generated if K = V(A) for some finite algebra A (or equivalently, K = V(L) for some finite set L of finite algebras). Every finitely generated variety is locally finite, i.e., its finitely generated members are finite algebras [12, Thm. II. 10.16].
Recall that P U (L) ⊆ I(L) for any finite set L of finite similar algebras. Given a class L of algebras, let us denote by L FSI the class of all FSI members of L.
Jónsson's Theorem [33,35] asserts that, if L is contained in a congruence distributive variety, then V(L) FSI ⊆ HSP U (L). In particular, if L consists of finitely many finite similar algebras and V(L) is congruence distributive, then V(L) FSI ⊆ HS(L).
As RLs are congruence distributive, Jónsson's Theorem shows that, whenever L consists of totally ordered [I]RLs, then so does V(L) FSI , whence V(L) consists of semilinear algebras. Indeed, since total order is expressible by a universal positive first order sentence, it persists under the operators H, S and P U . The variety SM of Sugihara monoids is well understood, largely because of Dunn's contributions to [2]; see [16] also. It is locally finite, but not finitely generated. In fact, SM is the smallest variety containing the unique Sugihara monoid
S * = {a : 0 = a ∈ Z}; ·, ∧, ∨, −, 1
on the set of all nonzero integers such that the lattice order is the usual total order, the involution − is the usual additive inversion, and the term function of |x| : = x → x is the natural absolute value function. In this algebra,
a · b = the element of {a, b} with the greater absolute value, if |a| = |b|; a ∧ b if |a| = |b|,
and the residual operation → is given by
a → b = (−a) ∨ b if a b; (−a) ∧ b if a b.
Note that e = 1 and f = −1 in S * . The remark before Definition 5.5 yields:
Lemma 5.6. Every FSI Sugihara monoid is totally ordered. In particular, Sugihara monoids are semilinear.
Definition 5.7. An IRL A is said to be odd if f = e in A.
Theorem 5.8. Every odd De Morgan monoid is a Sugihara monoid.
Proof. By Theorem 3.3, every square-increasing odd IRL is idempotent.
In the Sugihara monoid S = Z; ·, ∧, ∨, −, 0 on the set of all integers, the operations are defined like those of S * , except that 0 takes over from 1 as the neutral element for ·. Both e and f are 0 in S, so S is odd. It follows from Theorem 5.8 and Dunn's results in [2,16] that the variety of all odd Sugihara monoids is the smallest quasivariety containing S, and that SM is the smallest quasivariety containing both S * and S.
For each positive integer n, let S 2n denote the subalgebra of S * with universe {−n, . . . , −1, 1, . . . , n} and, for n ∈ ω, let S 2n+1 be the subalgebra of S with universe {−n, . . . , −1, 0, 1, . . . , n}. Note that S 2 is a Boolean algebra. The results cited above yield: Theorem 5.9. Up to isomorphism, the algebras S n (1 < n ∈ ω) are precisely the finitely generated SI Sugihara monoids, whence the algebras S 2n+1 (0 < n ∈ ω) are just the finitely generated SI odd Sugihara monoids.
We cannot embed S (nor even S 2n+1 ) into S * , owing to the involution. Nevertheless, S is a homomorphic image of S * , and S 2n+1 is a homomorphic image of S 2n+2 , for all n ∈ ω. In each case, the kernel of the homomorphism identifies −1 with 1; it identifies no other pair of distinct elements. Also, S 2n−1 is a homomorphic image of S 2n+1 if n > 0; in this case the kernel collapses −1, 0, 1 to a point, while isolating all other elements. Thus, S 3 is a homomorphic image of S n for all n ≥ 3. In particular, every nontrivial variety of Sugihara monoids includes S 2 or S 3 .
Corollary 5.10. The lattice of varieties of odd Sugihara monoids is the following chain of order type ω + 1 :
V(S 1 ) V(S 3 ) V(S 5 ) . . . V(S 2n+1 ) . . . V(S).
Proof. See [2,Sec. 29.4] or [27,Fact 7.6].
Odd Sugihara monoids are categorically equivalent to relative Stone algebras [27,Thm. 5.8]. The equivalence sends an odd Sugihara monoid to the set of lower bounds of its neutral element e, redefining residuation as (x → y) ∧ e and restricting the other RL-operations, as well as all morphisms. An analogous but more complex result for arbitrary Sugihara monoids is proved in [28,Thm. 10.5 As the structure of Sugihara monoids is very transparent, we concentrate now on De Morgan monoids that are not idempotent.
Lemma 5.11. Let A be a non-idempotent FSI De Morgan monoid, and let a be an idempotent element of A. If a f , then a > e. In particular, f 2 > e.
Proof. Suppose a 2 = a f . As A is not idempotent, f 2 = f , by Theorem 3.3, so a = f . Therefore, a f , whence e a, by Theorem 5.1. As f a, we cannot have a = e, by Theorem 3.3, so e < a. The last claim follows because f 2 is an idempotent upper bound of f (by Lemma 3.1).
Theorem 5.12. Let G be a deductive filter of a non-idempotent FSI De Morgan monoid A, and suppose ¬(f 2 ) ∈ G. Then A/G is an odd Sugihara monoid.
Proof. By Theorems 3.5 and 3.3, A/G is idempotent and f /G e/G. By Lemma 5.11, f 2 > e, i.e., ¬(f 2 ) < f , whence f ∈ G, i.e., e → f ∈ G. Then e/G f /G (by (17)), so e/G = f /G, as required.
Lemma 5.13. Let A be a De Morgan monoid that is FSI, with f a, b ∈ A, where a and b are idempotent. Then a b or b a. Proof. If A is a Sugihara monoid, the result follows from Lemma 5.6. We may therefore assume that A is not idempotent, so e < a, b, by Lemma 5.11. Then a · ¬a = ¬a and b · ¬b = ¬b, by Lemma 2.6, so (a · ¬b) ∧ (b · ¬a) (a · ¬b) · (b · ¬a) (by (13)) = (a · ¬a) · (b · ¬b) = ¬a · ¬b (by the above) = ¬a ∧ ¬b (by (14), as ¬a, ¬b e).
Therefore, by De Morgan's laws, ). Then, since A is FSI, Lemma 3.7(ii) and (9) yield e a → b or e b → a, i.e., a b or b a.
¬(¬a ∧ ¬b) ¬((a · ¬b) ∧ (b · ¬a)) = ¬(a · ¬b) ∨ ¬(b · ¬a) = (a → b) ∨ (b → a) and e < a ∨ b = ¬(¬a ∧ ¬b), so e < (a → b) ∨ (b → a
The subalgebra of an algebra A generated by a subset X of A shall be denoted by Sg A X.
Lemma 5.14. Let A be a De Morgan monoid that is FSI, and let f a ∈ A, where a < f 2 . Then a is idempotent.
Proof. By Lemma 3.1, f 2 is idempotent, so assume that a = f 2 . From f f 2 and a f 2 , we infer a f . Then e a, by Theorem 5.1, so e, f ∈ [¬a, a] : = {b ∈ A : ¬a b a}. Therefore, ¬(a 2 ) x a 2 for all x ∈ Sg A {a}, by Theorem 3.2. By Corollary 5.4, Sg A {a} is rigorously compact. In particular, (19) a 2 · x = a 2 whenever ¬(a 2 ) < x ∈ Sg A {a}. As a a 2 and a f 2 , we have a 2 f 2 . But a 2 and f 2 are idempotent, by Lemma 3.1, so f 2 < a 2 , by Lemma 5.13. Thus, ¬(a 2 ) < ¬(f 2 ) ∈ Sg A {a}, so (20) a 2 = a 2 · ¬(f 2 ), by (19). As A/[¬(f 2 )) is idempotent (by Theorem 3.5), ¬(f 2 ) a 2 → a, i.e., a 2
· ¬(f 2 ) a, by (17) and (2). Then (20) gives a 2 a, and so a 2 = a. Theorem 5.15. Let A be a non-idempotent FSI De Morgan monoid, with f 2 a ∈ A. Then ¬a < a and the interval [¬a, a] constitutes a subalgebra of A. In particular, [¬(f 2 ), f 2 ] is the universe of a subalgebra of A.
Proof. In A, we have ¬(f 2 ) e, as noted after Lemma 3.4, while e < f 2 , by Lemma 5.11. Of course, ¬a ¬(f 2 ), so ¬a < a. Thus, [¬a, a] includes e, and it is obviously closed under ∧, ∨ and ¬. Closure under fusion follows from (8) and the square-increasing law, because a is idempotent (by Lemma 5.14). Proof. This follows from Lemma 5.6 when the algebra is idempotent. In the opposite case, the idempotent upper bounds of f are exactly the upper bounds of f 2 (by (8) and Lemma 5.14), and they are comparable with all upper bounds of f (by Lemmas 5.14 and 5.13). (14). Suppose, with a view to contradiction, that there exists a ∈ A such that a / ∈ (¬(f 2 )] ∪ [¬(f 2 ), f 2 ] ∪ [f 2 ). By Theorem 5.1, e < a or a < f . By involutional symmetry, we may assume that e < a. Then a is incomparable with
f 2 (as a / ∈ [¬(f 2 ), f 2 ] ∪ [f 2 )), so f 2 ∨ a > f 2 . Also, since f 2 , a e, we have f 2 · a f 2 ∨ a, by (8), so f 2 · a > f 2 . Because a > e, we have f · a f . If f · a ∈ [¬(f 2 ), f 2 ], then f 2 · a (f · a) 2 f 4 = f 2 (by Lemma 3.1),
a contradiction. So, by Theorem 5.16, f ·a is idempotent and f ·a > f 2 . Then f · a > e, f , and by Theorem 5.15, ¬(f · a) < f · a. This, with Theorem 3.2, shows that f · a is the greatest element of the algebra C : = Sg A {f · a}, and ¬(f · a) is the least element of C. Note that ¬(f · a) < ¬(f 2 ), as f 2 < f · a. Now C is rigorously compact, by Corollary 5.4, so ¬(f 2 ) · (f · a) = f · a > f 2 .
Thus, ¬(f 2 ) · (f · a) a, as f 2 a. Nevertheless, as ¬(f 2 ) · f = ¬(f 2 ), we have (¬(f 2 ) · f ) · a = ¬(f 2 ) · a a, because ¬(f 2 ) e. This contradicts the associativity of fusion in A. Therefore,
A = (¬(f 2 )] ∪ [¬(f 2 ), f 2 ] ∪ [f 2 ).
Recall from (14) that fusion and meet coincide on the lower bounds of e in any De Morgan monoid. For the algebras in Theorem 5.17, the behaviour of fusion is further constrained as follows.
Theorem 5.18. Let A be a non-idempotent FSI De Morgan monoid, and let f a, b ∈ A. Then
a · b = f 2 if a, b f 2 ; max {a, b} otherwise.
If, moreover, a < b and f 2 b, then a · ¬b = ¬b = b · ¬b and b · ¬a = b.
Proof. If a, b f 2 , then f 2 a · b f 4 = f 2 , by (8) and Lemma 3.1, so a ·b = f 2 . We may therefore assume (in respect of the first claim) that a f 2 or b f 2 . Then a and b are comparable, by Theorem 5. 16. By symmetry, we may assume that a b and hence that b f 2 , so e < f 2 < b = b 2 , by Theorems 5.15 and 5.16.
If a = b, then a · b = b 2 = b = max {a, b}, so we may assume that a = b. Thus, b > a f , and so ¬b < ¬a e < b.
As b is an idempotent upper bound of e, f, a, ¬a, ¬b, Theorem 3.2 shows that b is the greatest element of Sg A {a, b}, and ¬b is the least element.
By Corollary 5.4, Sg A {a, b} is rigorously compact. We shall therefore have a · b = b = max {a, b}, provided that ¬b = a. This is indeed the case, as we have seen that ¬a < b.
Finally, suppose a < b and f 2 b. Again, Theorems 5.15 and 5.16 show that ¬b, b are the (idempotent) extrema of the algebra Sg A {a, b}, whose nonextreme elements include ¬a, a, so the remaining claims also follow from the rigorous compactness of Sg A {a, b}.
Remark 5.19. The foregoing results imply that, for an FSI De Morgan monoid A, there are just two possibilities.
The first is that f < e, in which case, by Theorems 3.3 and 5.1 and Lemmas 3.7(iii) and 5.6, A is a totally ordered SI Sugihara monoid whose fusion resembles that of S * , because the latter operation is definable by universal first order sentences, and because A ∈ ISP U (S * ). (See the remarks preceding Definition 5.5 and recall that the absolute value function on S * is the term function of x → x.) The improvement here on A ∈ HSP U (S * ) is due to the assumption f < e. Indeed, a nontrivial congruence on any B ∈ SP U (S * ) must identify f with e, because e covers f in S * , and therefore in B.
The second possibility is that A is the 'rigorous extension' of its antiidempotent subalgebra (on [¬(f 2 ), f 2 ]) by an (idempotent) totally ordered odd Sugihara monoid. More precisely, in this case, if θ = Ω [¬(f 2 )), then A/θ is a totally ordered odd Sugihara monoid (and is therefore determined by its e, reduct), while [¬(f 2 ), f 2 ] is the congruence class e/θ and no two distinct non-elements of [¬(f 2 ), f 2 ] are identified by θ (an easy consequence of Theorem 5.18). Thus, when ¬(f 2 ) and f 2 are identified in ¬(f 2 )] ∪ [f 2 ), we obtain a copy of A/θ; . Both A/θ and the algebra on [¬(f 2 ), f 2 ] are FSI, by Lemma 3.7(i), and may be trivial. By the last assertion of Theorem 3.3, A/θ is not a retract of A, unless A is odd (i.e., θ = id A ). There is no further constraint on [¬(f 2 ), f 2 ], while the e, reduct of A/θ may be any chain with a self-inverting antitone bijection, having a fixed point. Then A is a union of successive rigorously compact two-point extensions of [¬(f 2 ), f 2 ] (as many as A/θ requires). This largely reduces the study of irreducible De Morgan monoids to the anti-idempotent case.
We depict below the two-element Boolean algebra 2 (= S 2 ), the threeelement Sugihara monoid S 3 , and two 0-generated four-element De Morgan monoids, C 4 and D 4 . In each case, the labeled Hasse diagram determines the structure, in view of Lemma 2.3, Theorem 5.3 and the definitions. That C 4 and D 4 are indeed De Morgan monoids was noted long ago in the relevance logic literature, e.g., [45,46]. All four algebras are simple, by Lemma 3.7(iv).
C 4 : s s ❅ ❅ s s ❅ ❅ f 2 e f ¬(f 2 ) D 4 :
The next theorem is implicit in the findings of Slaney [59,60] mentioned after Lemma 4.2, but it is easier here to give a self-contained proof. Proof. Because A is simple (hence nontrivial) and 0-generated, {e} is not a subuniverse of A, so e = f and e has just one strict lower bound in A (Lemma 3.7(iv)). Suppose A ∼ = 2. As every simple Boolean algebra is isomorphic to 2, Lemma 4.2 shows that A is not integral. Equivalently, f is not the least element of A, so f e. Then by Theorem 3.3, A is not idempotent and f < f 2 , hence ¬(f 2 ) < e, so ¬(f 2 ) is the least element of A, i.e., f 2 is the greatest element. Consequently, a · ¬(f 2 ) = ¬(f 2 ) for all a ∈ A, by Lemma 2.3, and a · f 2 = f 2 whenever ¬(f 2 ) = a ∈ A, by Theorem 5.3.
There are two possibilities for the order: e < f or e f . If e f , then e ∧ f < e, whence e ∧ f is the extremum ¬(f 2 ) and, by De Morgan's laws,
e ∨ f = f 2 . Otherwise, ¬(f 2 ) < e < f < f 2 . Either way, {¬(f 2 )
, e, f, f 2 } is the universe of a four-element subalgebra of A, having no proper subalgebra of its own, so
A = {¬(f 2 ), e, f, f 2 }, as A is 0-generated. Thus, A ∼ = C 4 if e < f , and A ∼ = D 4 if e f .
We remark that both V(C 4 ) and V(D 4 ) are categorically equivalent to the variety V(2) of all Boolean algebras. (Equivalently, C 4 and D 4 are primal algebras, as they generate arithmetical varieties and are finite, simple and lack proper subalgebras and nontrivial automorphisms; see [22,32,41].) Proof. Necessity is clear. Conversely, suppose C 4 , D 4 / ∈ K and let A ∈ K be SI. It suffices to show that A is a Sugihara monoid. Suppose not. Then, by Theorem 5.15 and Remark 5. 19, ¬(f 2 ) < f 2 and the subalgebra B of A on [¬(f 2 ), f 2 ] is nontrivial, whence the 0-generated subalgebra E of A is nontrivial. Recall that every nontrivial finitely generated algebra of finite type has a simple homomorphic image [34,Cor. 4.1.13]. Let G be a simple homomorphic image of E, so G ∈ K. By assumption, neither C 4 nor D 4 is isomorphic to G, but G is 0-generated, so 2 ∼ = G, by Theorem 5.20. Thus, 2 ∈ HS(B). Then 2 must inherit from B the anti-idempotent identity x f 2 . This is false, however, so A is a Sugihara monoid.
In what follows, some features of C 4 will be important. Proof. As B is 0-generated, h is surjective. Suppose h is not an isomorphism. By the remarks preceding Lemma 2.3, h(a) = e for some a ∈ A with a < e. By Theorem 5.1, a f , so h(a) h(f ), i.e., e f in B. As B is 0-generated but not trivial, it cannot satisfy e = f , so e < f in B. Then C 4 embeds into B, by Lemma 5.22(ii), so B ∼ = C 4 , again because B is 0-generated.
Minimality
A quasivariety is said to be minimal if it is nontrivial and has no nontrivial proper subquasivariety. If we say that a variety is minimal (without further qualification), we mean that it is nontrivial and has no nontrivial proper subvariety. When we mean instead that it is minimal as a quasivariety, we shall say so explicitly, thereby avoiding ambiguity. Proof. Each X ∈ {2, S 3 , C 4 , D 4 } is finite and simple, with no proper nontrivial subalgebra, so the nontrivial members of HS(X) are isomorphic to X. Thus, the SI members of V(X) belong to I(X), by Jónsson's Theorem, because DMM is a congruence distributive variety. As varieties are determined by their SI members, this shows that V(X) has no proper nontrivial subvariety, and that V(X) = V(Y ) for distinct X, Y ∈ {2, S 3 , C 4 , D 4 }. As V(2) and V(S 3 ) are the only minimal varieties of Sugihara monoids, Theorem 5.21 shows that they, together with V(C 4 ) and V(D 4 ), are the only minimal subvarieties of DMM.
s ❏ ❏ ❏ s ❜ ❜ ❜ ❜ s ✡ ✡ ✡ s ✧ ✧ ✧ ✧ s s trivial V(C 4 ) V(S 3 ) V(D 4 ) V(2)
DMM
Bergman and McKenzie [6] showed that every locally finite congruence modular minimal variety is also minimal as a quasivariety. Thus, by Theorem 6.1, V(2), V(S 3 ), V(C 4 ) and V(D 4 ) are minimal as quasivarieties. (In a subsequent paper, we shall show that DMM has just 68 minimal subquasivarieties.) With a view to axiomatizing the varieties in Theorem 6.1, consider the following (abbreviated) equations.
e (x → y) ∨ (y → x) (21) e (x → (y ∨ ¬y)) ∨ (y ∧ ¬y)
(22) e (f 2 → x) ∨ (x → e) ∨ ¬x (23) x ∧ (x → f ) (f → x) ∨ (x → e)(24)
x → e x ∨ (f 2 → ¬x) (25) It is shown in [30] that an [I]RL A is semilinear (i.e., a subdirect product of chains) iff it is distributive and satisfies (21). Proof. Let X ∈ {2, S 3 , C 4 , D 4 }. It can be verified mechanically that X satisfies the proposed axioms for V(X). Let A be an SI De Morgan monoid satisfying the same axioms, and let a be the largest element of A strictly below e, which exists by Lemma 3.7(iii). By involution properties, ¬a is the smallest element of A strictly above f . It suffices to show that A ∼ = X.
When X is 2, this follows from Lemma 4.2, as every SI Boolean algebra is isomorphic to 2.
If X is S 3 or C 4 , then A is totally ordered (because it is semilinear, by (21), and SI).
Suppose that X = S 3 . In A, since e = f , we have a < e < ¬a, and there is no other element in the interval [a, ¬a]. We claim, moreover, that ¬a has no strict upper bound in A. Suppose, on the contrary, that ¬a < b ∈ A. By (22) and since e is join-prime (Lemma 3.7(ii)), we have e b → (a ∨ ¬a) or e a ∧ ¬a. But a ∧ ¬a = a < e, so by (9), b a ∨ ¬a = ¬a, a contradiction. This vindicates the above claim. By involutional symmetry, a has no strict lower bound in A. As A is totally ordered, this shows that A = {a, e, ¬a}.
Then A ∼ = S 3 , in view of Lemma 2.3.
We may now assume that X is C 4 or D 4 , so A satisfies ¬(f 2 ) x f 2 and is therefore rigorously compact (Theorem 5.3) and not idempotent (Corollary 3.6), whence f < f 2 and f e in A (Theorem 3.3).
Suppose X = D 4 . By assumption, b ∧ ¬b = ¬(f 2 ) for any b ∈ A.
If e < f , then e = e ∧ f = ¬(f 2 ), i.e., e is the bottom element of A, forcing A to be trivial (see the remarks before Lemma 2.3). This contradiction shows that e and f are incomparable in A.
As a is the greatest strict lower bound of e, we now have a < f , by Corollary 5.2. Then a = e ∧ f = ¬(f 2 ) and, by involution properties, no element lies strictly between f and f 2 . Suppose b ∈ A, with ¬(f 2 ) < b < f . By (23),
e (f 2 → ¬b) ∨ (¬b → e) ∨ b.
Since A is rigorously compact and ¬b = f 2 , we have f 2 → ¬b = ¬(f 2 ). So, because e is join-prime, e ¬b → e or e b. The last disjunct is false, for otherwise e b < f . Therefore, ¬b e, i.e., f b, contrary to assumption. Thus, no element of A lies strictly between ¬(f 2 ) and f and, by involution properties, no element lies strictly between e and f 2 . It follows that A = {¬(f 2 ), e, f, f 2 }, in view of Theorem 5.1. In this case, A ∼ = D 4 .
Lastly, suppose X = C 4 . Note that C 4 embeds into A, by Lemma 5.22. As a < e, it follows from (25) that e a → e a ∨ (f 2 → ¬a), but e is join-prime and e a, so f 2 ¬a, whence a = ¬(f 2 ). Thus, no element of A lies strictly between ¬(f 2 ) and e, nor strictly between f and f 2 .
Suppose, with a view to contradiction, that b ∈ A \ {¬(f 2 ), e, f, f 2 }. By the previous paragraph and since A is totally ordered, e < b < f . Then e b → f , so by (24),
e b ∧ (b → f ) (f → b) ∨ (b → e).
Now join-primeness of e gives f b or b e, a contradiction, so A ∼ = C 4 . Theorem 6.1 says, in effect, that for each axiomatic consistent extension L of R t , there exists B ∈ {2, S 3 , C 4 , D 4 } such that the theorems of L all take values e on any evaluation of their variables in B. Postulates for the four maximal consistent axiomatic extensions of R t follow systematically from Theorem 6.2. For example, (21) becomes the axiom (p → q) ∨ (q → p), while (25) becomes (p → t) → (p ∨ (f 2 → ¬p)).
Relevant Algebras
The relevance logic literature is equivocal as to the precise definition of a De Morgan monoid. Our Definition 4.1 conforms with Dunn and Restall [17], Meyer and Routley [49,57], Slaney [59] and Urquhart [69], yet other papers by some of the same authors entertain a discrepancy. In all sources, the neutral element of a De Morgan monoid A is assumed to exist but, in [60,61,62] for instance, it is not distinguished, i.e., the symbol for e (and likewise f ) is absent from the signature of A. That locally innocuous convention has global effects: it would prevent DMM from being a variety, as it would cease to be closed under subalgebras, and the tight correspondence between axiomatic extensions of R t and subvarieties of DMM would disappear. 3 This may explain why we have found in the literature no analysis of the subvariety lattice of DMM (despite interest in the problem discernable in [45,46]), and in particular no statement of Theorem 6.1, identifying the only four maximal consistent axiomatic extensions of R t (although the algebras defining these extensions were well known to relevance logicians).
The practice of not distinguishing neutral elements stems from the formal system R of Anderson and Belnap [2], which differs from R t only in that it lacks the sentential constant t (corresponding to e) and its postulates. The omission of constants from R produces a desirable variable sharing principle for 'relevant' implication:
if ⊢ R α → β, then α and β have a common variable [4].
The corresponding claim for R t is false, e.g.,
⊢ R t t → (p → p) and ⊢ R t (p ∧ t) → (t ∨ q).
Definition 7.1. A relevant algebra is an algebra A; ·, ∧, ∨, ¬ such that A; · is a commutative semigroup, A; ∧, ∨ is a distributive lattice and ¬¬a = a a · a, a b iff ¬b ¬a, a · b c iff a · ¬c ¬b, a a · (¬(b · ¬b) ∧ ¬(c · ¬c)), for all a, b, c ∈ A. The class of all relevant algebras is denoted by RA.
The two defining postulates of RA that are not pure equations can be paraphrased easily as equations, so RA is a variety. It is congruence distributive (since its members have lattice reducts) and congruence permutable (see for instance [72,Prop. 8.3]).
The main motivation for RA is that it algebraizes the logic R. The algebraization process for R t and DMM carries over verbatim to R and RA, provided we use (12) as a formal device for eliminating all mention of e. Further work on relevant algebras can be found in [18,21,37,38,52,54,65,66].
Because RA is closed under subalgebras, its study accommodates the variable sharing principle of relevance logic, without sacrificing the benefits of accurate algebraization. For the algebraist, however, RA has some forbidding features. It lacks the congruence extension property (CEP), for instance, as does its class of finite members (see [14, p. 289]), whereas DMM has the CEP. Also, De Morgan monoids have much in common with abelian groups (residuals being a partial surrogate for multiplicative inverses), but relevant algebras are less intuitive, being semigroup-based, rather than monoid-based.
The following facts are therefore noteworthy. . We can infer (i) from (ii), as every algebra embeds into an ultraproduct of finitely generated subalgebras of itself (or see [31,Cor. 4.11]). Theorem 7.2(i) reflects the fact that the t-free fragment of ⊢ R t is just ⊢ R , so there is a smooth passage from either system to the other. In particular, the variable sharing principle holds for the t-free formulas of R t .
The e-free reduct A; ·, ∧, ∨, ¬ of a De Morgan monoid A shall be denoted by A − . Also, if K is a class of De Morgan monoids, then K − shall denote the class of e-free reducts of the members of K. In this case, on general grounds,
(26) V(K) − ⊆ V(K − ).
Indeed, every equation satisfied by K − is an e-free identity of K, and therefore of V(K), and therefore of V(K) − . Because a De Morgan monoid and its e-free reduct have the same congruence lattice, we also obtain:
Lemma 7.3. A De Morgan monoid A is SI [resp. FSI; simple] iff the same is true of A − .
Crucially, however, subalgebras of the e-free reduct of a De Morgan monoid need not contain e, and they need not be reducts of De Morgan monoids themselves, unless they are finitely generated. For instance, the free ℵ 0generated relevant algebra is such a subreduct, and it lacks a neutral element, because the variable sharing principle rules out theorems of R of the form α → (p → p) whenever p is a variable not occurring in the formula α.
Still, because of Theorem 7.2, it is often easiest to obtain a result about relevant algebras indirectly, via a more swiftly established property of De Morgan monoids. This is exemplified below in Corollary 7.4, and more strikingly in Theorem 7.8. (We extend our use of the terms 'bounded' and 'rigorously compact' to relevant algebras in the obvious way, noting that the existence of a neutral element is not needed in the proof of Lemma 2.4.) Corollary 7.4. ( [66]) Every finitely generated relevant algebra A is bounded.
Proof. By Theorem 7.2(ii), A is a reduct of a De Morgan monoid with the same finite generating set, so A is bounded, by Theorem 3.2.
In contrast with this argument, the only published proof of Corollary 7.4, viz. [66,Prop. 5], is quite complicated. (For one generator, the indicated bounds are built up using all six of the inequivalent implicational one-variable formulas of R, determined in [42].) The result is attributed in [66] to Meyer and to Dziobiak (independently).
Corollary 7.5. Every nontrivial relevant algebra A has a copy of 2 − as a subalgebra.
Proof. Let B be the subalgebra of A generated by an arbitrary pair of distinct elements of A. By Corollary 7.4, B has (distinct) extrema ⊥, ⊤. By Theorem 7.2(ii), B is the e-free reduct of a De Morgan monoid, so by Lemma 2.3, {⊥, ⊤} is the universe of a subalgebra of B, isomorphic to 2 − . Clearly, when a Boolean algebra A is thought of as an integral De Morgan monoid, it has the same term operations as its e-free reduct A − , because e is definable as x → x. Thus, the variety of Boolean algebras can be identified with V(2 − ) = Q(2 − ). This reconfirms, of course, that classical propositional logic is the largest consistent extension of R. With Theorem 7.2(ii), it also yields the following. Theorem 7.7. There is a join-preserving (hence isotone) surjection from the lattice of subvarieties of DMM to that of RA, defined by K → V(K − ).
Moreover, this map remains surjective when its domain is restricted to the varieties that contain 2, together with the trivial variety.
Proof. Preservation of joins follows from Jónsson's Theorem and the following two facts: (i) an ultraproduct of reducts of members of a class C is the (corresponding) reduct of an ultraproduct of members of C, and (ii) an ultraproduct of members of the join of two varieties belongs to one of the two varieties. To prove surjectivity, let L be a variety of relevant algebras, and L FG its class of finitely generated members. By Theorem 7.2(ii), each A ∈ L FG is the e- Whereas the above argument about joins would apply in any context where the indicated reduct class is a congruence distributive variety, the surjectivity of the (restricted) function in Theorem 7.7 is a special feature of relevant algebras, reliant on Theorem 7.2(ii). The restricted function is not injective, however. Indeed, Jónsson's Theorem shows that V(2, S 2n+1 ) V(S 2n , S 2n+1 ) for all integers n > 1, but these two varieties have the same image under the map K → V(K − ), because S − 2n embeds into S − 2n+1 (although S 2n does not embed into S 2n+1 ). 4 This failure of injectivity limits the usefulness of the above function when we analyse the subvariety lattice of RA. Nevertheless, we can already derivé Swirydowicz's description of the lower part of that lattice by a mathematically simpler argument, based on the situation for De Morgan monoids. In particular, we avoid use of the complex ternary relation semantics for R (see [57]), employed in [65].
Proof. Let X ∈ {S − 3 , C − 4 , D − 4 }, so X is simple (by Lemma 7.
3) and X has just one nontrivial proper subalgebra, which is isomorphic to 2 − . Then every SI member of V(X) is isomorphic to 2 − or to X, by Jónsson's Theorem (cf. the proof of Theorem 6.1). So, there are no subvarieties of RA strictly between V(2 − ) and V(X), and V(
X) = V(Y ) for X = Y ∈ {S − 3 , C − 4 , D − 4 }.
Conversely, let K be a subvariety of RA, not consisting entirely of Boolean algebras. We must show that V(X) ⊆ K for some
X ∈ {S − 3 , C − 4 , D − 4 }.
As V(2 − ) K are varieties, there exists a finitely generated SI algebra A ∈ K \ V(2 − ). Now A is the e-free reduct of some A + ∈ DMM, by Theorem 7.2(ii). Note that A + is SI (by Lemma 7.3) and finitely generated. At least one of 2, S 3 , C 4 and D 4 belongs to V(A + ), by Theorem 6.1. By (26),
V(A + ) − ⊆ V(A) ⊆ K,
so it suffices to show that one of S 3 , C 4 or D 4 belongs to V(A + ).
Suppose C 4 , D 4 / ∈ V(A + ). Then A + is a Sugihara monoid, by Theorem 5.21. As A + is SI, finitely generated, and not a Boolean algebra, it is isomorphic to S n for some n ≥ 3, by Theorem 5.9. Then S 3 ∈ H(A + ) ⊆ V(A + ) (by the remarks preceding Corollary 5.10), completing the proof.
V(C − 4 ) V(S − 3 ) V(D − 4 )
RA
This means that the logics algebraized by V(S − 3 ), V(C − 4 ) and V(D − 4 ) are exactly the maximal non-classical axiomatic extensions of R, as was observed in [65]. The proof of Theorem 7.8 in [65] relies on a lemma, which says that every bounded SI relevant algebra is rigorously compact [65,Lem. 8]. In [65], the proof of the lemma uses the ternary relation semantics for R. As the lemma is itself of some interest, we supply an algebraic justification of it here. The key to the argument is that the subalgebras of FSI relevant algebras are still FSI, but that fact is concealed by the failure of the CEP and the lack of an obvious analogue for Lemma 3.7(i) in RA. One way to circumvent these difficulties is to extend the concept of deductive filters to relevant algebras. Definition 7.9. A subset F of a relevant algebra A is called a deductive filter of A if F is a lattice filter of A; ∧, ∨ and |a| : = a → a ∈ F for all a ∈ A.
Clearly, the set of deductive filters of A is closed under arbitrary intersections and under unions of non-empty directed subfamilies, so it is both an algebraic closure system over A and the universe of an algebraic lattice DFil A, ordered by inclusion. We denote by DFg A X the smallest deductive filter of A containing X, whenever X ⊆ A. Thus, the compact elements of DFil A are just the finitely generated deductive filters of A, i.e., those of the form DFg A X for some finite X ⊆ A.
The deductive filters of a relevant algebra A are just the subsets that contain all A-instances of the axioms of R and are closed under the inference rules-modus ponens and adjunction-of R. (This is easily verified, using (12) and Theorem 7.2(i).) Therefore, by the theory of algebraization [8,Thm. 5.1], and since RA is a variety, we have DFil A ∼ = Con A, for all A ∈ RA. Proof. (i) It suffices, by Theorem 7.2(i) and (12), to show that e |a| ∧ |b| in any De Morgan monoid that contains A as a subreduct. And this follows from (11).
(ii) Let F = {c ∈ A : a ∧ |d| c for some d ∈ A}. Then a ∈ F , since a ∧ |a| a. We claim that F is a deductive filter of A. Clearly, F is upward closed. Suppose c, c ′ ∈ F , so there exist d, d ′ ∈ A such that a ∧ |d| c and a∧|d ′ | c ′ . Then c∧c ′ a∧|d|∧|d ′ | a∧||d|∧|d ′ ||, by (i), so c∧c ′ ∈ F . Also, for any d ∈ A, we have a ∧ |d| |d|, so |d| ∈ F . This vindicates the claim. It remains to show that F is the smallest deductive filter of A containing a. So, let G ∈ DFil A, with a ∈ G, and let c ∈ F . Choose d ∈ A with a ∧ |d| c. Since a, |d| ∈ G, we have a ∧ |d| ∈ G, whence c ∈ G, as required.
(iii) Certainly, a ∨ b ∈ DFg A {a} ∩ DFg A {b}, as a, b a ∨ b. Now suppose c ∈ DFg A {a} ∩ DFg A {b}. Choose d, d ′ ∈ A, with a ∧ |d| c and b ∧ |d ′ | c. Then c a ∧ |d| ∧ |d ′ |, b ∧ |d| ∧ |d ′ |, so c (a ∧ |d| ∧ |d ′ |) ∨ (b ∧ |d| ∧ |d ′ |) = (a ∨ b) ∧ (|d| ∧ |d ′ |) (by distributivity) (a ∨ b) ∧ ||d| ∧ |d ′ || (by (i)).
Thus, c ∈ DFg A {a ∨ b} and the result follows.
Corollary 7.11. The class of FSI relevant algebras is closed under subalgebras (and ultraproducts).
Proof. Let A ∈ RA. Clearly, DFg A {a 1 , . . . , a n } = DFg A {a 1 ∧ . . . ∧ a n } for all a 1 , . . . , a n ∈ A, so every finitely generated deductive filter of A is principal. Therefore, by Theorem 7.10(iii), the intersection of any two compact (i.e., finitely generated) elements of DFil A is compact. The same applies to the lattice Con A, as it is isomorphic to DFil A (and since lattice isomorphisms between complete lattices preserve compactness). Now the result follows from the well known theorem below. Theorem 7.12. ( [7]) In any congruence distributive variety K, the following conditions are equivalent.
(i) For any A ∈ K, the intersection of any two compact (i.e., finitely generated) congruences of A is compact. (ii) K FSI is closed under S and P U (i.e., it is a universal class).
Finally, as promised, a slight generalization of [65,Lem. 8] follows easily from Corollary 7.11: Theorem 7.13. Every bounded FSI relevant algebra A is rigorously compact.
Proof. Let ⊥, ⊤ be the extrema of A, and consider ⊥ = a ∈ A. We must show that ⊤ · a = ⊤. Observe that B : = Sg A {⊥, a, ⊤} is FSI, by Corollary 7.11. As B is finitely generated, it is a reduct of a (bounded) De Morgan monoid B + , by Theorem 7.2(ii), which is also FSI, by Lemma 7.3. Now B + is rigorously compact, by Theorem 5.3, so ⊤ · a = ⊤.
Corollary 7.14. Every finitely generated subalgebra of an FSI relevant algebra is rigorously compact.
Proof. Use Corollaries 7.4 and 7.11 and Theorem 7.13.
Lemma 3. 4 .
4In a square-increasing IRL A,
Lemma 3 . 7 .
37Let A be a (possibly involutive) RL.
Corollary 5 . 2 .
52Let A be a De Morgan monoid that is SI. Let c be the largest element of A strictly below e (which exists, by Lemma 3.7(iii)). Then c f .
Definition 5.5. A Sugihara monoid is an idempotent De Morgan monoid, i.e., an idempotent distributive IRL.
Theorem 5 . 16 .
516In any FSI De Morgan monoid, the filter [f ) is the union of the interval [f, f 2 ] and a chain whose least element is f 2 . The elements of this chain are just the idempotent upper bounds of f .
.
Any non-idempotent FSI De Morgan monoid is the union of the interval subuniverse [¬(f 2 ), f 2 ] and two chains of idempotents, (¬(f 2 )] and [f 2 ). Proof. Let A be a non-idempotent FSI De Morgan monoid. Theorem 5.15 shows that e, f ∈ [¬(f 2 ), f 2 ] and (with Lemma 2.3) that ¬(f 2 ) · f = ¬(f 2 ). Note that [f 2 ) and (¬(f 2 )] are both chains of idempotents, by Theorem 5.16, involution properties and
Theorem 5 . 20 .
520Let A be a simple 0-generated De Morgan monoid. Then A ∼ = 2 or A ∼ = C 4 or A ∼ = D 4 .
Theorem 5.21. A variety K of De Morgan monoids consists of Sugihara monoids iff it excludes C 4 and D 4 .
Lemma 5 . 22 .
522Let A be a nontrivial square-increasing IRL.
( i )
iIf e f and a f 2 for all a ∈ A, then e < f . (ii) If e < f in A, then C 4 can be embedded into A. Proof. (i) Suppose A satisfies e f and x f 2 . Then A is not idempotent, by Corollary 3.6, so f = e, by Theorem 3.3, i.e., e < f . (ii) Suppose e < f in A. Then f < f 2 , by Theorem 3.3, i.e., ¬(f 2 ) < e. Thus, {¬(f 2 ), e, f, f 2 } is closed under ∧, ∨ and ¬, and ¬(f 2 ) is idempotent, by (14). By Lemma 3.1, f 2 is an idempotent upper bound of e, so f 2 ·¬(f 2 ) = ¬(f 2 ), by Lemma 2.6. Closure of {¬(f 2 ), e, f, f 2 } under fusion follows from these observations and (8), so C 4 embeds into A. Theorem 5.23. (Slaney [60, Thm. 1]) Let h : A − → B be a homomorphism, where A is an FSI De Morgan monoid, and B is nontrivial and 0-generated. Then h is an isomorphism or B ∼ = C 4 .
Theorem 6 . 1 .
61The distinct classes V(2), V(S 3 ), V(C 4 ) and V(D 4 ) are precisely the minimal varieties of De Morgan monoids.
(i) V(2) is axiomatized by adding x e to the axioms of DMM; (ii) V(S 3 ) by adding e = f , (21) and (22); (iii) V(D 4 ) by adding x f 2 , x ∧ ¬x y and (23); (iv) V(C 4 ) by adding x f 2 , e f , (21), (24) and (25).
RA coincides with the class of all e-free subreducts of De Morgan monoids (i.e., all subalgebras of reducts A; ·, ∧, ∨, ¬ of De Morgan monoids A). (ii) If a relevant algebra is finitely generated, then it is the e-free reduct A; ·, ∧, ∨, ¬ of a De Morgan monoid A. In this case, the unique neutral element of A is the greatest lower bound of all a → a, where a ranges over any finite generating set for A; ·, ∧, ∨, ¬ . Here, (ii) is a specialization of [51, Thm. 5.3], but it algebraizes a much older logical result of Anderson and Belnap [2, p. 343], already implicit in the proof of [1, Lem. 2]
Corollary 7 . 6 .
76Boolean algebras constitute the smallest nontrivial (quasi) variety of relevant algebras.
free reduct of a unique De Morgan monoid A + . Let M = {A + : A ∈ L FG }. Then L FG = M − ⊆ V(M) − , while (26) shows that V(M) − ⊆ V(L FG ). Thus, L = V(V(M) − ), as varieties are determined by their finitely generated members. If L is nontrivial, then 2 − ∈ L FG , by Corollary 7.5, so 2 ∈ M.
Theorem 7.8. ([65, Thm. 12]) V(S − 3 ), V(C − 4 ) and V(D − 4 ) are exactly the covers of V(2 − ) in the subvariety lattice of RA.
Theorem 7 . 10 .
710Let A be a relevant algebra, with a, b ∈ A. Then
(i) ||a| ∧ |b|| |a| ∧ |b|; (ii) DFg A {a} = {c ∈ A : a ∧ |d| c for some d ∈ A}; (iii) DFg A {a} ∩ DFg A {b} = DFg A {a ∨ b}.
] and refined in[23, Thm. 2.24]. Every quasivariety of odd Sugihara monoids is a variety[27, Thm. 7.3]. (For a stronger result, see[50, Thm. 9.4].)
But see the first paragraph of Section 7.
The general theory of algebraization[8] applies only to consequence relations. This is in contrast with a tradition-prevalent in relevance logic and elsewhere-of identifying a 'logic' with its set of theorems alone, leaving its rules of derivation under-determined in the absence of further qualification. The same tradition privileges axiomatic extensions. No serious ambiguity ensues in the case of R t , thanks to the so-called enthymematic deduction theorem[44], viz.
The meanings of statements about 'n-generated De Morgan monoids' would also change. For instance,[60, Thm. 5] says that every FSI De Morgan monoid on one idempotent generator is finite, but this is false when e is distinguished, as the proof of [60, Thm. 6] makes clear.
The function sending a subvariety W of RA to the variety generated by the De Morgan monoids whose e-free reducts lie in W is an injective join-preserving right-inverse for the function in Theorem 7.7, but it is not surjective.
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The complexity of decision procedures in relevance logic II. A Urquhart, J. Symbolic Logic. 64A. Urquhart, The complexity of decision procedures in relevance logic II, J. Symbolic Logic 64 (1999), 1774-1802.
Rule separation and embedding theorems for logics without weakening. C J Van Alten, J G Raftery, Studia Logica. 76C.J. van Alten, J.G. Raftery, Rule separation and embedding theorems for logics with- out weakening, Studia Logica 76 (2004), 241-274.
. M Ward, R P Dilworth, Trans. Amer. Math. Soc. 45Residuated latticesM. Ward, R.P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc. 45 (1939), 335-354.
. Czech Republic. E-mail address: [email protected] Department of Mathematics and Applied Mathematics. 20Institute of Computer Science, Academy of Sciences of the Czech Republic ; University of PretoriaPrivateInstitute of Computer Science, Academy of Sciences of the Czech Republic, Pod Vodárenskou věží 2, 182 07 Prague 8, Czech Republic. E-mail address: [email protected] Department of Mathematics and Applied Mathematics, University of Preto- ria, Private Bag X20, Hatfield, Pretoria 0028, South Africa E-mail address: [email protected] [email protected]
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[
"MODELS OF TYPE THEORY BASED ON MOORE PATHS *",
"MODELS OF TYPE THEORY BASED ON MOORE PATHS *"
] |
[
"Ian Orton \nDepartment of Computer Science and Technology\nUniversity of Cambridge\nUK\n",
"Andrew M Pitts [email protected] \nDepartment of Computer Science and Technology\nUniversity of Cambridge\nUK\n"
] |
[
"Department of Computer Science and Technology\nUniversity of Cambridge\nUK",
"Department of Computer Science and Technology\nUniversity of Cambridge\nUK"
] |
[
"Logical Methods in Computer Science"
] |
This paper introduces a new family of models of intensional Martin-Löf type theory. We use constructive ordered algebra in toposes. Identity types in the models are given by a notion of Moore path. By considering a particular gros topos, we show that there is such a model that is non-truncated, i.e. contains non-trivial structure at all dimensions. In other words, in this model a type in a nested sequence of identity types can contain more than one element, no matter how great the degree of nesting. Although inspired by existing non-truncated models of type theory based on simplicial and cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types.
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10.23638/lmcs-15(1:2)2019
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[
"https://arxiv.org/pdf/1802.05629v3.pdf"
] | 57,721,110 |
1802.05629
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82fc58f5b9ff30e74b221058ad73ded41d902616
|
MODELS OF TYPE THEORY BASED ON MOORE PATHS *
2019
Ian Orton
Department of Computer Science and Technology
University of Cambridge
UK
Andrew M Pitts [email protected]
Department of Computer Science and Technology
University of Cambridge
UK
MODELS OF TYPE THEORY BASED ON MOORE PATHS *
Logical Methods in Computer Science
15124201910.23638/LMCS-15(1:2)2019Submitted Feb. 16, 2018
This paper introduces a new family of models of intensional Martin-Löf type theory. We use constructive ordered algebra in toposes. Identity types in the models are given by a notion of Moore path. By considering a particular gros topos, we show that there is such a model that is non-truncated, i.e. contains non-trivial structure at all dimensions. In other words, in this model a type in a nested sequence of identity types can contain more than one element, no matter how great the degree of nesting. Although inspired by existing non-truncated models of type theory based on simplicial and cubical sets, the notion of model presented here is notable for avoiding any form of Kan filling condition in the semantics of types.
Introduction
Homotopy Type Theory [Uni13] has re-invigorated the study of the intensional version of Martin-Löf type theory [Mar75]. On the one hand, the language of type theory helps to express synthetic constructions and arguments in homotopy theory and higher-dimensional category theory. On the other hand, the geometric and algebraic insights of those branches of mathematics shed new light on logical and type-theoretic notions. One might say that the familiar propositions-as-types analogy has been extended to propositions-as-types-as-spaces. In particular, under this analogy there is a path-oriented view of intensional (i.e. proofrelevant) equality: proofs of equality of two elements x, y of a type A correspond to elements of a Martin-Löf identity type Id A x y and behave analogously to paths between two points x, y in a space A. The complicated internal structure of intensional identity types relates to the homotopy classes of path spaces. To make the analogy precise and to exploit it, it helps to have a wide range of models of intensional type theory that embody this path-oriented view of equality in some way. This paper introduces a new family of such models, constructed from Moore paths [Moo55] in toposes.
Let R + = {r ∈ R | r ≥ 0} be the real half-line with the usual topology. Classically, a Moore path between points x and y in a topological space X is a pair p = (f, r) where r ∈ R + and f : R + −→ X is a continuous function with f 0 = x and f r = y for all r ≥ r.
We will write x ∼ y for the set of Moore paths from x to y, with X understood from the context. Clearly there is a Moore path from x to y iff there is a conventional path, that is, a continuous function f : [0, 1] −→ X with f 0 = x and f 1 = y. The advantage of Moore paths is that they admit degeneracy and composition operations that are unitary and associative up to equality; whereas for conventional paths these identities only hold up to homotopy. Specifically, one has the following Moore paths: idp x :≡ (λ t x, 0) ∈ x ∼ x and for all p = (f, r) ∈ x ∼ y and q = (g, s) ∈ y ∼ z q • p :
≡ (g • r f, r + s) ∈ x ∼ z where (g • r f ) t :≡ f t if t ≤ r g(t − r) if r ≤ t (t ∈ R + )
These definitions satisfy p • idp x = p = idp y • p and r • (q • p) = (r • q) • p. In Section 3 we abstract from R some simple order-algebraic [Bou81, chapter VI] structure sufficient for the above definitions to work in a constructive algebraic setting, rather than a classical topological one. Initially the structure of an ordered abelian group in some topos [Joh02,MM92] suffices and then we extend that to an ordered commutative ring to ensure the models satisfy function extensionality.
In Sections 4-6 we use this structure in toposes to develop a family of models of intensional Martin-Löf type theory with: identity types given by Moore paths, Σ-types, Π-types satisfying function extensionality, inductive types (we just consider disjoint unions and W -types) and Tarski-style universes. By considering a particular gros topos [GV72] in Section 7.2 we get a non-truncated instance of our model construction, in other words one where iterated identity types Id A , Id Id A , Id IdId A , . . . can be non-trivial to any depth of iteration.
The observation that the strictly associative and unitary nature of composition of Moore paths aids in the interpretation of Martin-Löf identity types is not new; see for example [vG12,Sections 5.1 and 5.5]. However, the fact that function extensionality can hold for identity based on Moore paths (Theorem 5.1) is new and quite surprising, given that such paths carry an intensional component, namely their "shape" (Definition 3.1). Another novelty of our approach concerns the fact that existing non-truncated models of type theory typically make use of some form of Kan filling condition [Kan58] to define a class of fibrant families of types with respect to which path types behave as identity types. One of the contributions of this paper is to show that one can avoid any form of Kan filling and still get a non-truncated model of intensional Martin-Löf type theory. Instead we use path composition and a simple notion of fibrant family phrased just in terms of the usual operation of transporting elements along equality proofs (Definition 4.1). As a consequence every type, regarded as a family over the terminal type, is fibrant in our setting. In particular, this means that intervals are first-class types in our models, something which is not true for existing path-oriented models, such as the classical simplicial [KL16] and constructive cubical sets [BCH14, CCHM18, ABC + 17] models; and constructing universes in our setting does not need proofs of their fibrancy.
Informal type theory. The new models of type theory we present are given in terms of categories with families (CwF) [Dyb96,Hof97]. Specifically, we start with an arbitrary Vol. 15:1 MODELS OF TYPE THEORY BASED ON MOORE PATHS 2:3 topos E, to which can be associated a CwF, for example as in [LW15,Awo16]. We write E(Γ) for the set of families indexed by an object Γ ∈ E and E(Γ A) for the set of elements of a family A ∈ E(Γ). One can make E(Γ) into a category whose morphisms between two families A, B ∈ E(Γ) are elements in E(Γ.A B), where Γ.A is the comprehension object associated with A. The category E(Γ) is equivalent to the slice category E/Γ, the equivalence being given on objects by sending families A ∈ E(Γ) to corresponding projection morphisms Γ.A −→ Γ. We then construct a new CwF by considering families in E equipped with certain extra structure (the transport-along-paths structure of Definition 4.1); the elements of a family in the new CwF are just those of the underlying family in E. One could describe this construction using the language of category theory. Instead, as in [OP18,LOPS18] we find it clearer to express the construction using an internal language for the CwF associated with E -a combination of higher-order predicate logic and extensional type theory, with an impredicative universe of propositions given by the subobject classifier in E; see [Mai05]. This use of internal language allows us to give an appealingly simple description of the type constructs in the new CwF. In the text we use this language informally (analogously to the way that [Uni13] develops Homotopy Type Theory). In particular the typing contexts of the judgements in the formal version, such as [x 0 : A 0 , x 1 : A 1 (x 0 ), x 2 : A 2 (x 0 , x 1 )], become part of the running text in phrases like "given x 0 : A 0 , x 1 : A 1 (x 0 ) and x 2 : A 2 (x 0 , x 1 ), then. . . "; and when we refer to a "function in the topos" (as opposed to one of its morphisms) we mean a term of function type in its internal language.
The arguments we give in Sections 3-6 are all constructively valid and in fact do not require the impredicative aspects of topos theory; indeed we have used Agda [Agd] (with uniqueness of identity proofs and postulates for quotient types) as a tool to develop and experiment with the material presented in those sections. The specific model presented in Section 7.2 uses topological spaces within classical mathematics.
Ordered Rings in a Topos
Let E be a topos with a natural number object [Joh02,MM92]. A total order on an object R ∈ E is given by a subobject R × R which is not only reflexive, transitive and antisymmetric, but also satisfies that the join of the subobject and its opposite • π 2 , π 1 is the whole of R × R; in other words the following formula of the internal language of E is satisfied:
(∀i, j : R) i j ∨ j i (2.1) As mentioned in the Introduction, in this paper we use such formulas of the internal language to express properties of E instead of giving the category-theoretic version of the property. Note that since E may not be Boolean, we do not necessarily have the trichotomy property (∀i, j : R) i < j ∨ i = j ∨ j < i for the associated strict order relation i < j :≡ ¬(j i). So when defining functions by cases using (2.1) we have to verify that the clauses for i j and for j i agree on the overlap, where i = j holds by antisymmetry. For example, the positive cone Total order
R + :≡ {i : R | 0 i}((∀i : R) i i (2.4) (∀i, j, k : R) i j ∧ j k ⇒ i k (2.5) (∀i, j : R) i j ∧ j i ⇒ i = j (2.6) (∀i, j : R) i j ∨ j i (2.7)
Abelian group
(∀i, j, k : R) i + (j + k) = (i + j) + k (2.8) (∀i : R) 0 + i = i (2.9) (∀i, j : R) i + j = j + i (2.10) (∀i : R) i + (−i) = 0 (2.11)
Addition is order preserving
(∀i, j, k : R) i j ⇒ k + i k + j (2.12) Multiplicative abelian monoid (∀i, j, k : R) i · (j · k) = (i · j) · k (2.13) (∀i : R) 1 · i = i (2.14) (∀i, j : R) i · j = j · i (2.15)
Multiplication distributes over addition
(∀i, j, k : R) i · (j + k) = (i · j) + (i · k) (2.16)
Positive elements are closed under multiplication (∀i, j : R) 0 i ∧ 0 j ⇒ 0 i · j (2.17) Figure 1: Ordered commutative ring axioms associated with R has a binary operation of minimum min : R + × R + −→ R + well-defined by the following properties
(∀i, j : R + ) i j ⇒ min(i, j) = i j i ⇒ min(i, j) = j (2.3)
In order to define Moore paths in E with respect to R we need it to have additive structure compatible with the total order; later, to get function extensionality we also need multiplicative structure and to construct universes we need a connectedness property for R.
Definition 2.1 (order-ringed topos). An order-ringed topos (E, R) is a topos E together with an ordered commutative ring object [Bou81, chapter VI] R in E. Thus R comes equipped with a subobject R × R and morphisms
0 : 1 −→ R + : R × R −→ R − : R −→ R 1 : 1 −→ R · : R × R −→ R
satisfying the axioms in Fig. 1.
Specific examples of order-ringed toposes (E, R) will be considered in Section 7. In the next section we develop properties of Moore paths in E based on R.
Moore Paths in a Topos
Fix an order-ringed topos (E, R). Recall from the Introduction that there is a CwF associated with E [LW15, Awo16] whose families at an object Γ ∈ E we denote by E(Γ). We continue to use an informal internal language based on extensional type theory to describe constructions and properties of the topos and its associated CwF. The development in this and the next section only makes use of the order and additive structure of the positive cone R + = {i : R | 0 i} of the ordered commutative ring object R. (We will need to use its multiplicative structure in Section 5.)
Definition 3.1 (Moore path objects). For each object Γ ∈ E, the family (x ∼ y | x, y : Γ) ∈ E(Γ × Γ) of Moore path objects is defined by:
x ∼ y :≡ {(f, i) : (R + Γ) × R + | f 0 = x ∧ (∀j i) f j = y} (3.1)
We have the following functions associated with Moore paths:
| | : (x ∼ y) R + (3.2) |(f, i)| = i at : (x ∼ y) R + Γ (3.3) (f, i) at j = f j Following [Bro09,
Section 2], we call |p| the shape of the path p : x ∼ y. Thus if p : x ∼ y, then p at 0 = x and p at |p| = y. From now on we will just write p at i as p i.
Note that morphisms in E respect Moore paths in the sense that for each γ : Γ −→ ∆ in E there is a function mapping paths in Γ to those in ∆:
γ ' : (x ∼ y) (γ x ∼ γ y) (3.4) γ ' (f, i) :≡ (γ • f, i)
Definition 3.2 (Degenerate paths). For each Γ ∈ E, the degenerate path at x : Γ is denoted idp x :
x ∼ x and is well-defined by the requirements:
|idp x| = 0 (3.5) (∀i : R + ) (idp x) i = x (3.6)
Definition 3.3 (Path composition). Given A ∈ E, if i : R + and f, g :
R + A satisfy f i = g 0, then there is a function g • i f : R + A satisfying (g • i f ) j = f j if j i g (j − i) if i j (
where, as usual, we write j + (-i) as j − i). This is well-defined because when i = j, then f j = f i = g 0 = g(j − i). In particular, given paths p = (f, i) : x ∼ y and q = (g, j) : y ∼ z,
then g • i f : R + A satisfies (∀k i + j) (g • i f ) k = z, because i + j k implies i = i + 0 i + j k and j = (i + j) − i k − i; and hence (g • i f ) k = g(k − i) = g j = z.
Therefore we get a well-defined path q • p : x ∼ z satisfying Lemma 3.4. For each Γ ∈ E, given x, y, z, w : Γ, p : x ∼ y, q : y ∼ z, r : z ∼ w and γ : Γ Γ , one has:
|q • p| = |p| + |q| (3.7) (∀i : R + ) i |p| ⇒ (q • p) i = p i |p| i ⇒ (q • p) i = q (i − |p|)(p • (idp x) = p = (idp y) • p (3.9) (r • q) • p = r • (q • p) (3.10) γ ' (idp x) = idp(γ x) (3.11) γ ' (q • p) = (γ ' q) • (γ ' p) (3.12)
Proof. One just has to check that these properties follow constructively from the axioms (2.4)-(2.12). For example (3.10) holds because
|(r•q)•p| = |p|+|r•q| = |p|+(|q|+|r|) = (|p|+|q|)+|r| = |q•p|+|r| = |r•(q•p)| and ((r • q) • p) i = p i if i |p| q (i − |p|) if |p| i and i − |p| |q| r ((i − |p|) − |q|) if |p| i and |q| i − |p| (r • (q • p)) i = p i if i |p| + |q| and i |p| q (i − |p|)
if i |p| + |q| and |p| i r (i − (|p| + |q|)) if |p| + |q| i which are equal, because one can use axioms (2.4)-(2.12) to show that
i |p| ⇔ i |p| + |q| ∧ i |p| |p| i ∧ i − |p| |q| ⇔ i |p| + |q| ∧ |p| i |p| i ∧ |q| i − |p| ⇔ |p| + |q| i (i − |p|) − |q| = i − (|p| + |q|).
As well as composing Moore paths one can reverse them. To define this operation it is convenient to use the operation of truncated subtraction · − : R + × R + −→ R + , which is well-defined by the following properties:
(∀i, j : R + ) i j ⇒ i · − j = 0 j i ⇒ i · − j = i − j (3.13)
Lemma 3.5 (path reversal). For each Γ ∈ E, given x, y : Γ and p : x ∼ y, there is a reversed path rev p : y ∼ x well-defined by the requirements
|rev p| = |p| (∀i : R + ) (rev p) i = p (|p| · − i)
and satisfying rev(idp x) = idp x (3.14) Proof. As for the previous lemma, this follows straightforwardly from the axioms (2.4)-(2.12) within constructive logic.
rev(q • p) = (rev p) • (rev q)(
Although the definition of rev p is standard, the above equational properties are not often mentioned in the literature. However, they are crucial for the construction of identity types in Section 4 to work. What usually gets a mention is the fact that up to homotopy rev p is a two-sided inverse for p with respect to the • operation. Paths (rev p) • p ∼ idp x and p • (rev p) ∼ idp y can be constructed using the path contraction operation given below; we do not bother to do that, because they are also a consequence of the path induction [Uni13, 1.12.1] property of identity types that follows from Theorem 4.10.
Definition 3.6 (bounded abstractions). The following binding syntax is very convenient for describing Moore paths. For each Γ ∈ E, if λi ϕ(i) describes a function in R + Γ, then for each j : R + using the min function (2.3) we get a path i j ϕ(i) : ϕ(0) ∼ ϕ(j) in Γ by defining:
i j ϕ(i) :≡ (λi ϕ(min(i, j)) , j) (3.18) (i is bound in the above expression). It is easy to see that this form of bounded abstraction has the following properties:
γ ' i j ϕ(i) = i j γ(ϕ(i)) (3.19) i 0 ϕ(i) = idp(ϕ(0)) (3.20) i |p| (p i) = p (3.21)
Lemma 3.7 (path contraction). Given Γ ∈ E, for any path p : x ∼ y in Γ and i : R + , there is a path p i : x ∼ p i satisfying
p 0 = idp x (3.22) (∀i |p|) p i = p (3.23)
Proof. Using the bounded abstraction notation and the min function (2.3), we define
p i :≡ j min(|p|, i) (p j) (3.24)
Since p (min(|p|, i)) = p i, this does give a path x ∼ p i; and it has the required properties by (3.20) and (3.21).
Remark 3.8. Using p i we get for each x : Γ that y:
Γ (x ∼ y) is path-contractible [Uni13,
Section 3.11] with centre (x, idp x), since for each y : Γ and p :
x ∼ y we have a path i |p| (p i, p i ) in (x, idp x) ∼ (y, p)
. This is part of the more general fact that Moore paths model identity types, which we show in the next section (see Theorem 4.10).
Transport along Paths
In this section we continue with the assumptions of the previous one: E is a topos (with associated CwF) containing an ordered commutative ring object R. We want objects of Moore paths with respect to R (Definition 3.1) to give a model of identity types, as well as other type formers. Recall that in Martin-Löf Type Theory elements of identity types Id Γ x y give rise to transport functions A x A y between members of a family of types (A x | x : Γ) over a type Γ; see for example [Uni13, Section 2.3]. Therefore, for each object Γ ∈ E, we should restrict attention to families A ∈ E(Γ) that come equipped at least with some sort of transport operation taking a path p : x ∼ y in Γ and an element a : A x to an element p * a : A y. This leads to the following definition.
Definition 4.1 (tap fibrations). Given an object Γ ∈ E, a transport-along-paths (tap) structure for a family
A ∈ E(Γ) is a (Γ × Γ)-indexed family of morphisms (( ) * : x ∼ y (A x A y) | x, y : Γ) satisfying for all x : Γ and a : A x (idp x) * a = a (4.1)
We write Fib(Γ) for the families over Γ equipped with a tap structure and call them fibrations. They are stable under re-indexing: given γ :
∆ −→ Γ in E and A ∈ Fib(Γ), then A γ :≡ (A(γ x) | x : ∆) ∈ E(∆)
has a tap structure via the congruence operation (3.4), taking the transport of a :
(A γ) x = A(γ x) along p : x ∼ y to be (γ ' p) * a : A(γ y)
. This re-indexing of tap structure respects composition and identities in E. So Fib inherits the structure of a CwF from that of E, with the set of elements of a fibration being the elements of the underlying family in E, that is Fib
(Γ A) = E(Γ A).
We show that the CwF Fib inherits some type structure from E. To do so involves definitions and calculations using the bounded abstraction formalism of Definition 3.6.
To describe Σ-and Π-types in Fib we have to lift paths in Γ to paths in comprehension
objects Γ.A = x:Γ A x of fibrations A ∈ Fib(Γ): Lemma 4.2 (path lifting). Given Γ ∈ E and A ∈ Fib(Γ), for each path p : x ∼ y in Γ and each a : A x, there is a path lift(p, a) : (x, a) ∼ (y, p * a) in Γ.A satisfying lift(idp x, a) = idp(x, a) (4.2)
and stable under re-indexing along morphisms ∆ −→ Γ in E.
Proof. We can use path contraction (Lemma 3.7) to express path lifting using the bounded abstraction notation from Definition 3.6:
lift(p, a) :≡ i |p| (p i, (p i ) * a) (4.3)
Since the path contraction operation satisfies p 0 = idp x and p |p| = p, this does indeed give a path in (x, a) ∼ (y, p * a) . The desired properties of lift follow from corresponding properties of bounded abstraction and path contraction (3.19)-(3.23). Specifically, we have:
lift(idp x, a) = i |idp x| (idp x i, ((idp x) i ) * a) = i |0| (x, ((idp x) i ) * a) = idp(x, ((idp x) 0 ) * a) = idp(x, (idp x) * a) = idp(x, a)
where x : Γ and a : A x. We also have:
lift A (γ ' p, a) = i |γ ' p| ((γ ' p) i, ((γ ' p) i ) * a) = i |p| (γ(p i), (γ ' p i ) * a) = i |p| (γ × id)(p i, (γ ' p i ) * a) = (γ × id) ' ( i |p| (p i, (γ ' p i ) * a)) = (γ × id) ' (lift Aγ (p, a))
where γ : ∆ −→ Γ, p : x ∼ y in ∆, a : A(γ x) and we write lift A for the lifting operation on A ∈ Fib(Γ) and lift Aγ for the lifting operation on Aγ ∈ Fib(∆). Proof. Given a path p : x ∼ y in Γ, we get functions Σ A B x Σ A B y and Π A B x Π A B y using path lifting and (in the second case) path reversal:
p * (a, b) :≡ (p * a, lift(p, a) * b) ∈ Σ A B y where (a, b) : Σ A B x (4.4) (p * f ) a :≡ (rev(lift(rev p, a))) * f ((rev p) * a) ∈ B(y, a) where f : Π A B x and a : A y (4.5)
To see why (4.5) has the correct type, consider an arbitrary p : This allows us to transport along this path to get
x ∼ y in Γ, f : Π A B x(rev(lift(rev p, a))) * f ((rev p) * a) ∈ B(y, a)
as required.
Note that these definitions satisfy the required property when transporting along the identity path. That is, given x : Γ and (a, b) : Σ A B x, then using property (4.2) we have:
(idp x) * (a, b) = ((idp x) * a, lift((idp x), a) * b) = ((idp x) * a, (idp(x, a)) * b) = (a, b)
Similarly, given x : Γ, f : Π A B x and a : A x then using (4.2) and the properties of path reversal given in Lemma 3.5 we have:
((idp x) * f ) a = (rev(lift(rev(idp x), a))) * f ((rev(idp x)) * a) = (rev(lift(idp x, a))) * f ((idp x) * a) = (rev(idp(x, a))) * f (a) = (idp(x, a)) * f (a) = f (a)
Finally, the stability of these definitions under re-indexing comes from the fact that rev and lift are both stable under re-indexing.
Theorem 4.4 (Empty, unit, Boolean and natural number types). Given Γ ∈ E, for each A ∈ E, the constant family (A | x : Γ) has a tap fibration structure given by p * a = a (for any a : A) and this is stable under re-indexing. Taking A to be the initial object ∅, the terminal object 1, the coproduct 1 + 1 and the natural number object of the topos, we have that the CwF Fib supports empty, unit, Boolean and natural number types [Hof97, Exercises E3.24-E3.26].
Theorem 4.5 (Sum types). Given Γ ∈ E and A, B ∈ Fib(Γ), the family A ⊕ B :≡ (A x + B x | x : Γ) of sum types has a tap fibration structure that is stable under re-indexing along morphisms ∆ −→ Γ in E. Hence the CwF Fib supports sum types. Proof. Given a path p : x ∼ y in Γ, we get a function (A⊕B) x −→ (A⊕B) y by case analysis on the elements of (A ⊕ B) x = A x + B x. Thus if inl and inr denote the constructors of the sum type A x + B x, we have
p * (inl a) = inl(p * a) where a : A x (4.6) p * (inr b) = inr(p * b) where b : B x (4.7)
and clearly this definition inherits property (4.1) from the tap fibration structure of A and B, and is stable under re-indexing.
Since we only consider toposes with a natural number object, the CwF associated with E can interpret types of well-founded trees (W -types) [Uni13, Section 5.3]; see [MP00, Propositions 3.6 and 3.8]. We write W x:A B x for the object of well-founded trees determined by a family B ∈ E(A), with constructor sup : y:
A (B y W x:A B x) W x:A B x.
Theorem 4.6 (W -types). Given A ∈ Fib(Γ) and B ∈ Fib(Γ.A), the family
W A B :≡ (W a:A x B(x, a) | x : Γ) ∈ E(Γ)
has a tap fibration structure that is stable under re-indexing along morphisms ∆ −→ Γ in E. Therefore the CwF Fib supports W -types.
Proof. Given a path p : x ∼ y in Γ, we get a function W A B x W A B y via the following well-founded recursion equation:
p * sup(a , f ) = sup (p * a , λb p * f ((rev(lift(p, a))) * b))
where a : A x and f : B(x, a) W A B x (4.8)
Note that p * a : A y. Therefore, assuming that the second argument to the sup constructor has type B(y, p * a) W A B y, then the overall type of the constructor will be W A B y as required. To see why the second component does have this type, consider an arbitrary b : B(y, p * a) and observe that rev(lift(p, a)) : (y, p * a) ∼ (x, a). Therefore we have (rev(lift(p, a))) * b : B(x, a) and hence f ((rev(lift(p, a))) * b) : W A B x. Finally, recursively transporting along p gives us p * f ((rev(lift(p, a))) * b) : W A B y as required.
Well-founded inductions using the properties of reversal and lifting given in Lemmas 3.5 and 4.2 suffice to show that this inherits the properties of a tap structure from those for A and B:
(idp x) * sup(a , f ) = sup ((idp x) * a , λb (idp x) * f ((rev(lift(idp x, a))) * b)) = sup ((idp x) * a , λb (idp x) * f ((idp(x, a)) * b)) = sup (a , λb f (b)) = sup(a , f )
The fact that this definition is stable under re-indexing follows immediately from the fact that rev and lift are both stable under re-indexing.
So far we have considered type structure that lifts from the CwF associated with E to the CwF Fib. Now we consider identity types, where the structure of interest in Fib is not the one inherited from E. Since E is a topos, it certainly has extensional identity types [Mar84], inhabitation of which coincides with judgemental equality, and those could be lifted to Fib. However, we wish to show that Moore path objects give the intensional version of identity types in Fib, the family of types (Id A x y | x, y : A) inductively generated by a single constructor Refl A : x:A Id A x x. We will use Hofmann's version of the structure in a CwF needed to model such types [Hof97]. To do so, let us fix some notation for a CwF C.
Re-indexing of a family A ∈ C(Γ) and an element α ∈ C(Γ A) along a morphism γ : ∆ −→ Γ will just be denoted by A γ ∈ C(∆) and α γ ∈ C(∆ A γ); and given an element β ∈ C(∆ A γ), then γ , β denotes the unique morphism ∆ −→ Γ.A whose composition with fst : Γ.A −→ Γ is γ and whose re-indexing of the generic element snd ∈ C(Γ.A A fst) is β:
fst • γ , β = γ snd γ , β = β.
Definition 4.7. Following Hofmann [Hof97, Definition 3.19], we say that a CwF C supports the interpretation of intensional identity types if for each object Γ ∈ C and each family A ∈ C(Γ) the following data is given and is stable under re-indexing along any γ : ∆ −→ Γ:
• a family Id A ∈ C(Γ.A.A fst), • a morphism Refl A : Γ.A −→ Γ.A.A fst. Id A such that fst • Refl A equals the diagonal morphism id , snd : Γ.A −→ Γ.A.A fst, • a function mapping each B ∈ C(Γ.A.A fst. Id A ) and β ∈ C(Γ.A B Refl A ) to an element J A B β ∈ C(Γ.A.A fst. Id A B) such that the re-indexing (J A B β) Refl A equals β.
Given an object Γ of the topos E and a family A ∈ E(Γ), we can use the family of Moore path objects ∼ (Definition 3.1) to define a family Id A ∈ E(Γ.A.A fst) as follows:
Id A :≡ (a 1 ∼ a 2 | ((x, a 1 ), a 2 ) : Γ.A.A fst) (4.9)
We will show that this together with suitable Refl and J operations give an instance of Definition 4.7 for the CwF Fib. In particular Id A has a tap fibration structure when A does. To see this we first need to analyse paths in Γ.A in terms of paths in Γ and in the fibres A x (for x : Γ). and stable under re-indexing along any γ : ∆ −→ Γ in E.
Proof. We use a reversed version of the path-contraction operation from Lemma 3.7: for each path q : x ∼ y and each i : R + , define q i : q i ∼ y by
q i :≡ rev (rev q) (|q| · −i) (4.11)
Given p : (x, a) ∼ (y, b), we get fst ' p : x ∼ y and hence for each i :
R + we have (fst ' p) i : (fst ' p) i ∼ y; and since snd(p i) : A(fst(p i)) = A((fst ' p) i), we get ((fst ' p) i ) * snd(p i) : A y. So we can define snd(p) = i |p| (((fst ' p) i ) * snd(p i)) (4.12)
to get a path (fst ' p) * a ∼ b in A y. Property (4.10) holds since, using (3.20), we have Stability under re-indexing follows from (3.19) and the fact that rev is stable under reindexing.
snd(idp(x, a)) = i |idp(x, a)| (((fst ' idp(x, a)) i ) * snd(idp(x, a) i)) = i 0 (((idp x) i ) * snd(x, a)) = idp (((idp x) 0 ) * a) = idp ((idp x) * a) =
Lemma 4.9. Given Γ ∈ E and a fibration A ∈ Fib(Γ), the family Id A ∈ E(Γ.A.A fst) defined in (4.9) has a tap fibration structure that is stable under re-indexing along morphisms ∆ −→ Γ in E.
Proof. First note that re-indexing the fibration A ∈ Fib(Γ) along fst : Γ.A −→ Γ we get A fst ∈ Fib(Γ.A). Given a path p : ((x, a 1 ),
a 2 ) ∼ ((y, b 1 ), b 2 ) in Γ.A.A fst, define p :≡ fst ' (fst ' p) : x ∼ y
We can apply Lemma 4.8 to the paths fst ' p : (x, a 1 ) ∼ (y, b 1 ) and fst • fst , snd ' p : (x, a 2 ) ∼ (y, b 2 ) in Γ.A to get paths in A y
p 1 :≡ snd(fst ' p) : p * a 1 ∼ b 1 p 2 :≡ snd( fst • fst , snd ' p) : p * a 2 ∼ b 2 (using the fact that fst • fst • fst , snd = fst • fst). Thus for each path q : a 1 ∼ a 2 in A x, we have A x p * − − → A y a 1 a 2 q p * a 1 p * a 2 b 1 b 2 (p * )'q p 2 rev(p 1 )
and can compose together the paths in A y to get p 2 • ((p * ) ' q) • rev(p 1 ) : b 1 ∼ b 2 . So altogether we get a function p * : Id A ((x, a 1 ), a 2 ) Id A ((y, b 1 ), b 2 ) defined by:
p * q = snd( fst • fst , snd ' p) • (((fst ' (fst ' p)) * ) ' q) • rev(snd(fst ' p)) (4.13)
The properties of Moore path reversal (Lemma 3.5) together with Lemma 4.8 suffice to show that this definition inherits the property of a tap fibration structure (4.1) from the one for A and that it is stable under re-indexing. For example, given x : Γ, a 1 , a 2 : A x and q : a 1 ∼ a 2 we have:
(idp((x, a 1 ), a 2 )) * q = snd( fst • fst , snd ' idp((x, a 1 ), a 2 ))
• (((fst ' (fst ' idp((x, a 1 ), a 2 ))) * ) ' q)
• rev(snd(fst ' idp((x, a 1 ), a 2 ))) = snd(idp(x, a 2 )) • (((idp x) * ) ' q) • rev(snd(idp(x, a 1 ))) = (idp a 2 ) • q • (idp a 1 ) = q as required. (x, a), a), idp a) (4.14) Note that (fst • Refl A )(x, a) = ((x, a), a) = id , snd (x, a), as required.
To define J we combine transport along paths with the fact that singleton types are contractible (Remark 3.8). More specifically, given B ∈ Fib(Γ.A.A fst. Id A ) and β ∈ E(Γ.A B Refl A ), for each path p : a 1 ∼ a 2 in A x using Lemma 3.7 we have the following path in Γ.A.A fst. Id A :
i |p| (((x, a 1 ), p i), p i ) : (((x, a 1 ), a 1 ), idp a 1 ) ∼ (((x, a 1 ), a 2 ), p) Since B has a tap fibration structure we can transport β(x, a 1 ) : B (((x, a 1 ), a 1 ), idp a 1 ) along this path to get an element of B(((x, a 1 ), a 2 ), p). So we can define J A B β ∈ E(Γ.A.A fst. Id A B) by:
J A B β (((x, a 1 ), a 2 ), p) :≡ ( i |p| (((x, a 1 ), p i), p i )) * β(x, a 1 ) (4.15)
J has the required computation property, because , a), a), idp a) by (4.14)
((J A B β) Refl A )(x, a) = J A B β (((x
= ( i 0 (((x, a), a) idp a)) * β(x, a) by (4.15) and (3.24) = idp (((x, a), a) idp a) * β(x, a) by (3.20)
= β(x, a) by (4.1)
Stability of Refl under re-indexing follows from the fact that the congruence operations γ ' preserve degenerate paths; and stability of J uses (3.19) and the fact that the path contraction operation ( ) i is preserved by the congruence operations γ ' .
Remark 4.11 (Associative tap fibrations). Definition 4.1 does not require the transport action (( ) * : x ∼ y (A x A y) | x, y : Γ) of a tap fibration A ∈ Fib(Γ) to be strictly associative. In other words, for all paths p : x ∼ y and q : y ∼ z and all a : A x we do not necessarily have (q • p) * a = q * (p * a) (4.16) As we have seen, that property is not needed to prove that fibrations give a model of type theory. Nevertheless, if one changes the definition by requiring (4.16), then the analogues of Theorems 4.3-4.6 and 4.10 do hold for this stronger notion of fibration, although we do not prove that here.
With Definition 4.1 as it stands, one has associativity up to homotopy, i.e. there is a path (q • p) * a ∼ q * (p * a). This is a consequence of Theorem 4.10, which allows one to use path induction [Uni13, Section 2.9]) to construct a path (q • p) * a ∼ q * (p * a) for any p from the case when p = idp x, where one has the degenerate path for (q • (idp x)) * a = q * a = q * ((idp x) * a).
Remark 4.12 (Weak fibrations). In the definition of tap fibration, if one replaces the equality (4.1) by a homotopy, (idp x) * a ∼ a, the resulting weak notion of tap fibration satisfies versions of Theorems 4.3-4.6, albeit with more complicated proofs. The same is true for Theorem 4.10 except that one gets only propositional identity types, where there is a path between (J A B β) Refl A and β, rather than an equality in E (cf. [CCHM18, Section 9.1] and [van18]).
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Vol. 15:1
Function Extensionality
Let (E, R) be an order-ringed topos (Definition 2.1). In this section we show that in the CwF Fib constructed from (E, R) as in the previous section, it is the case that functions behave extensionally with respect to the intensional identity types given by Moore paths. So far we have only used the order and additive structure of R (axioms (2.4)-(2.12) in Fig. 1). For function extensionality to hold, we need more than that. To see why, consider the constant function K 0 = λi 0 : R + R + and the identity function id : R + R + . 1 If equality means existence of a Moore path, then these functions are extensionally equal, because we have λi j i j : i:R+ (K 0 i ∼ id i). So if the principle of function extensionality is to hold with respect to Moore paths, then there will have to be a path p : K 0 ∼ id in R + R + . What is its shape |p|? Since p does not depend upon any assumptions, |p| would have to be a global element 1 −→ R + in the topos E. Axioms (2.4)-(2.12) only guarantee the existence of one such global element, namely 0. However, if |p| = 0 then K 0 = id, so (∀i : R) 0 = i and the model of type theory is degenerate. Therefore we need some extra assumptions about R if function extensionality is to hold without collapsing everything. This is where axioms (2.13)-(2.17) come into play. These may not be the minimal assumptions needed for function extensionality, but they are a well-known part of (constructive, ordered) Algebra [Bou81, chapter VI] that does the job, and this makes easier the task of finding specific models with good properties (which we address in Section 7.2).
Given an object Γ ∈ E and a family A ∈ E(Γ), if p : f ∼ g is a path between dependent functions f, g : x:Γ A x, then for each x : Γ we can apply the path congruence operation (3.4) to λf f x : ( x:Γ A x) A x and p to obtain a path (λf f x) ' p : f x ∼ g x in A x. This gives us the following function (which coincides with the canonical function obtained by path induction as in [Uni13, Section 2.9]):
happly : (f ∼ g)
x:Γ (f x ∼ g x) happly p x :≡ (λf f x) ' p (where f, g : x:Γ A x) (5.1)
Theorem 5.1 (function extensionality modulo ∼). Given an order-ringed topos (E, R), an object Γ ∈ E and a family A ∈ E(Γ), for all f, g :
x:Γ A x there is a function funext : ( x:Γ (f x ∼ g x)) (f ∼ g). Furthermore this function is quasi-inverse [Uni13, Definition 2.4.6] to happly, that is, that for all e : x:Γ (f x ∼ g x) and p : f ∼ g there are paths ε e : happly(funext e) ∼ e in x:Γ (f x ∼ g x) and η p : p ∼ funext(happly p) in f ∼ g.
Proof. If e : x:Γ (f x ∼ g x), then for all x : Γ we have e x (0 · |e x|) = e x 0 = f x and e x (1 · |e x|) = e x |e x| = g x. So there is a path funext e : f ∼ g in x:Γ A x given by:
funext e = i 1 λx e x (i · |e x|) (5.2)
Note that by (3.19) we have for each x : Γ
happly(funext e) x = (λf f x) ' i 1 λx e x (i · |e x|) = i 1 (e x (i · |e x|))
whereas e x = i |e x| (e x i) by (3.21). So to get a path from happly(funext e) x to e x we need to interpolate shapes from 1 to |e x| while at the same time interpolating the 1 Function extensionality for Fib only concerns functions between fibrant families; but as we noted in Section 4, in Fib all objects, and in particular R+, are fibrant as trivial families over the terminal object. argument of e x from i · |e x| to i · 1 = i. So for each j : R with 0 j 1, consider
u j,x :≡ (1 − j) + j · |e x| and v j,x :≡ (1 − j) · |e x| + j
Calculating with the ring axioms we find that u j,x · v j,x = |e x| + j · (1 − j) · (|e x| − 1) 2 . Since 0 j and 0 1 − j by assumption and since squares are always positive in an ordered ring [Bou81, VI.19], we have that |e x| u j,x · v j,x ; hence e x (u j,x · v j,x ) = g x. So for all j : R with 0 j 1 we have a path i u j,x (e x (i · v j,x )) : f x ∼ g x in A x. When j = 0 this path is happly(funext e) x (because u 0,x = 1 and v 0,x = |e x|); when j = 1 the path is e x (because u 1,x = |e x| and v 1,x = 1). Therefore we can define
ε e :≡ j 1 λx i u j,x (e x (i · v j,x )) (5.3)
to get the desired path happly(funext e) ∼ e in x:Γ (f x ∼ g x). Since for any p : f ∼ g it is the case that p = i |p| (p i) and funext(happly p) = i 1 (p (i · |p|)), a similar argument to the one for ε shows that
η p :≡ j 1 i (1 − j) · |p| + j (p (i · (1 − j + j · |p|))) (5.4)
gives a path p ∼ funext(happly p) in f ∼ g.
Universes
Let (E, R) be an order-ringed topos (Definition 2.1). In this section we show how to construct Tarski-style universes [Mar84,p. 88] in the CwF Fib constructed from (E, R) as in Section 4. To do so we assume that (the CwF associated with) E supports inductive-recursive definitions [Dyb00]. This will be the case if E is a Grothendieck topos, that is, a category of Set-valued sheaves on a site, if we assume Set is a model of a sufficiently strong set theory; see [DS99, section 6]. As well as the topos E, we also need to assume something about the ordered ring R, namely that it is a connected object in E; see the definition below. The examples of order-ringed toposes that we give in Section 7 have these properties. For simplicity, we confine attention to universes containing a type of Booleans and closed under taking dependent function and identity types (given by Moore paths); other typing constructs can be dealt with in the same way. Let the object U ∈ E and the family T ∈ E(U) be defined simultaneously by inductionrecursion in E so that U has inductive constructors bool : U pi : u:U (T u U) U eq : u:U T u T u U (6.1) and T satisfies the following recursion equations
T bool = 1 + 1 (6.2) (∀u : U)(∀f : T u U) T(pi u f ) = x:T u T(f x) (6.3) (∀u : U)(∀x, y : T u) T(eq u x y) = x ∼ y (6.4)
Here x ∼ y is the Moore path object (Definition 3.1) and 1 + 1 is the coproduct of the terminal object 1 ∈ E with itself-this gives an object of Booleans in E with elements true : 1 + 1 false : 1 + 1 (6.5)
Recall from the proof of Theorem 4.4 that any object of E, and in particular U, has a tap fibration structure when regarded as a family over 1. So the main task is to show that there is a tap fibration structure for the family T ∈ E(U). The way that we will construct this structure agrees with the tap fibration structure for Boolean, Π-and identity types (as 2:16
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Vol. 15:1 in Theorems 4.4, 4.3 and 4.10 respectively) when it is re-indexed along the coding functions bool, pi and id. In this way we get a Tarski-style universe T ∈ Fib(U) in Fib containing a type of Booleans and closed under taking dependent function and identity types.
Recalling the definition of a tap structure (Definition 4.1), given a Moore path p : u 0 ∼ u 1 in U we wish to construct a function p * : T u 0 T u 1 which is the identity when p is degenerate. To do so we recurse over the structure of u 0 : U, which is of the form bool, or pi v 0 f 0 (for some unique v 0 and f 0 ), or eq v 0 x 0 y 0 (for some unique v 0 , x 0 and y 0 ). In each case we would like the whole path p to remain within whichever constructor form u 0 has, so that previously constructed transport functions can be combined appropriately using the recipes in the proofs of Theorems 4.4, 4.3 and 4.10. This property of paths in U holds if R is connected in the following sense.
Definition 6.1 (Connected objects in a topos). For each object X of the topos E, consider the morphism K : 1 + 1 −→ (X 1 + 1) from the Booleans 1 + 1 into the exponential of 1 + 1 by X which is the transpose of the first projection π 1 : (1 + 1) × X −→ 1 + 1; thus K sends a Boolean b : 1 + 1 to the constant function with value b. We will say that X is connected if K is an isomorphism.
If (E, R) is an order-ringed topos for which R is connected, then given a predicate ϕ : R + Ω that is decidable ((∀i : R + ) ϕ i ∨ ¬(ϕ i)), consider the function f : R 1 + 1 well-defined by
f x = true if (x 0 ∧ ϕ 0) ∨ (0 x ∧ ϕ x) false if (x 0 ∧ ¬ϕ 0) ∨ (0 x ∧ ¬ϕ x) (x : R)
Since R is connected, either f = K true, or f = K false. If ϕ 0 holds, then f 0 = true and therefore we cannot have f = K false; so in this case we must have f = K true. Thus R being connected implies that the half-line R + satisfies (∀ϕ : R + Ω) ((∀i : R + ) ϕ i ∨ ¬(ϕ i)) ⇒ ϕ 0 ⇒ (∀i : R + ) ϕ i (6.6)
We use this property to construct the required tap fibration structure for T.
Theorem 6.2 (Universes). Given an order-ringed topos (E, R), assuming the topos E supports the inductive-recursive definition (6.1)-(6.4) and that R is connected, then T ∈ E(U) has a tap fibration structure that make it into a Tarski-style universe in Fib containing a Boolean type and closed under dependent function and identity types.
Proof. We can construct a transport function
(∀u 0 , u 1 : U)(∀p : u 0 ∼ u 1 )(∀a : T u 0 ) p * a : T u 1 (6.7)
and prove (∀u : U)(∀a : T u) (idp u) * a = a (6.8) simultaneously by recursion and induction on the size (number of nested constructors) of elements of U. Given (f, i) : f 0 ∼ f i and a : T(f 0) consider the structure of f 0:
Case f 0 = bool. Let ϕ : R + Ω be ϕ j :≡ (f j = bool). Since U is the disjoint union of the images of the constructors bool, pi and eq, we have that ϕ is decidable; and it holds when j = 0. Therefore by (6.6) we have f i = bool. So a : T(f 0) = T bool = T(f i) and in this case we can define (f, i) * a :≡ a. Case f 0 = pi v 0 g 0 , for some v 0 , g 0 . Letting ϕ : R + Ω be ϕ j :≡ (∃v : U, g : T v U) f j = pi v g once again this is a decidable predicate that holds at 0. Therefore by (6.6) and the fact that pi is injective, we have that there are functions v : R + U and g : j:
R+ T(v j) U with (∀j : R + ) f j = pi (v j)(g j) ∧ (v, i) : v 0 ∼ v i ∧ g 0 = g 0 ∧ (∀j i) g j = g i Note that the type of a is T(f 0) = x:T v 0 T(g 0 x) = x:T(v 0) T(g 0 x).
To transport this to an element of T(f i) = x:T(v i) T(g i x) we can use the construction (4.5) from the proof of Theorem 4.3. More explicitly, for each x : T(v i) and k : R + we have a path j i · − k (v(i · − j)) : v i ∼ v k and by recursion we can form
x k :≡ ( j i · − k (v(i · − j))) * x : T(v k)
Hence we get g : R + U and a 0 : T(g 0) by defining
g k :≡ g k (x k) a 0 :≡ a(x 0)
By the induction hypothesis x satisfies (∀k i) x k = x and hence (g, i) : g 0 ∼ g i x. So by recursion once again, we get (g, i) * a 0 : T(g i x); and by induction hypothesis, this is equal to a x in the case i = 0. Abstracting over x gives the required element (f, i) * a of
x:T(v i) T(g i x)
.
Case f 0 = eq v 0 x 0 y 0 , for some v 0 , x 0 , y 0 . The argument is similar to the previous case, but using the decidable predicate ϕ j :≡ (∃v : U, x : T u, y : T u) f j = eq v x y and the construction (4.13) from the proof of Lemma 4.9.
Remark 6.3 (Univalence). An element u of the universe U is a code for the corresponding type T u. By the above theorem, every Moore path p : u 0 ∼ u 1 in U induces a transport function p * : T u 0 T u 1 . Because Moore paths give identity types in Fib, these functions are equivalences [Uni13, Chapter 4], the inverse equivalence being given by transport along the reverse path rev p. Were T to satisfy Voevodsky's univalence axiom [Uni13, Section 2.10], every equivalence between T u 0 and T u 1 would have to be of the form p * for some path p : u 0 ∼ u 1 . This cannot be the case, because we have seen that connectedness of R implies that given p : u 0 ∼ u 1 , the elements u 0 and u 1 must have equal outermost constructor form; whereas it is quite possible for 1 + 1 to be equivalent (indeed, isomorphic) to, for example, a Π-type named by a code in U. So the universes constructed in this section do not satisfy the interesting form of extensionality embodied by the univalence axiom. We return to this point in the Conclusion.
Models
In the previous sections we have seen how any order-ringed topos (E, R) (Definition 2.1) gives rise to a model of Martin-Löf type theory with intensional identity types given by Moore paths on R. If R is trivial, that is, satisfies 0 = 1, then existence of a Moore path just coincides with extensional equality in the topos E. If the object R is non-trivial, but has decidable equality in E (for example, if it is the object of integers, or of rationals), then there is a path i 1 (if i = 0 then true else false) : true ∼ false in the object 1 + 1 of Booleans and so in this case the model of type theory we get is logically degenerate. Therefore, when searching for examples of order-ringed toposes, we should at least look for ones with R non-trivial and not decidable. We first give a simple example of such an order-ringed topos, where the underlying topos is a presheaf category. Then we give a more sophisticated example, using a sheaf topos and for which the associated model of type theory has identity types that are not necessarily truncated at any level; in other words, ones in which iterated identity types Id A , Id Id A , Id IdId A , . . . can be homotopically non-trivial for any level of iteration.
7.1. A presheaf model. Consider the category whose objects are ordered rings in the topos Set (sets and functions) and whose morphisms are ordered ring homomorphisms, that is, functions preserving the order and the ring operations 0, 1, +, ·. Let C be a small full subcategory of this category. For the purposes of this example it is not important which ordered rings C contains, so long as it contains the terminal ordered ring 1 (which has one element 0 = 1) and a non-trivial ordered ring (one for which 0 = 1); for definiteness let us assume that C contains the reals R with the usual order and ring operations.
7.1.1. The topos. We use the topos Set C of covariant presheaves on C. Thus the objects of Set C are functors C −→ Set and the morphisms are natural transformations between such functors. Let ∆ : Set −→ Set C denoted the functor assigning to each set S the constant presheaf ∆ S : C −→ Set, whose value at each X ∈ C. is ∆ S(X) = S. ∆ is the inverse image part of the unique geometric morphism from the topos Set C to Set; its direct image part, the right adjoint to ∆, is the global sections functor Set C (1, ) : Set C −→ Set. Below we will also need to use the fact that ∆ has a left adjoint π 0 : Set C −→ Set (7.1) π 0 (F ) :≡ F (1) (The adjunction π 0 ∆ follows from the fact that 1 is a terminal object in C.) 7.1.2. The ordered commutative ring object R. Let R : C −→ Set denote the forgetful functor sending each ordered ring in C to its underlying set. As an object of Set C , R has the structure of an ordered ring object:
• R × R is the sub-presheaf of R × R whose value each object X ∈ C is the subset (X) ⊆ R(X) × R(X) = X × X given by the order on X:
(X) :≡ {(x, y) ∈ X × X | x y} (7.2)
Note that these subsets do form a subpresheaf, because each morphism in C is in particular an order-preserving function. Furthermore, R × R is a total order (2.4)-(2.7) in the topos Set C , because disjunction in a presheaf topos is computed component-wise and each (X) is a total order on X.
• The ring structure on R is given component-wise by the ring structure on each X ∈ C.
For example, the addition morphism + : R × R −→ R has component at X ∈ C given by addition in X: + X (x, y) = x + y; these are the components of a morphism in Set C because they are natural in X, since each morphism θ in C satisfies θ(x + y) = θ x + θ y.
Similarly, this structure on R satisfies axioms (2.8)-(2.17) because each R(X) = X is an ordered ring. 7.1.3. R is connected. To apply the results of Section 6, we need to verify that R is a connected object in Set C (Definition 6.1). Since the object of Booleans 1+1 in the presheaf topos Set C is isomorphic to the constant functor ∆ 2 : C −→ Set, where 2 = {0, 1} is a two-element set, we have to show that K : ∆ 2 −→ (R ∆ 2) is an isomorphism in Set C . To see this, we just need to show that K X : ∆ 2(X) −→ (R ∆ 2)(X) is a bijection for each X ∈ C.
Letting Y : C op −→ Set C denote the Yoneda embedding and using the adjunction π 0 ∆ mentioned in Sec. 7.1.1, there are bijections ∆ 2(X) = 2 (definition of ∆) ∼ = Set(C(X, 1) × 1, 2) (since 1 is terminal in C)
= Set((YX × R)(1), 2) (definition of YX × R) = Set(π 0 (YX × R), 2) (definition of π 0 ) ∼ = Set C (YX × R, ∆ 2) (π 0 left adjoint to ∆) ∼ = Set C (YX, R ∆ 2) (universal property of exponential) ∼ = (R ∆ 2)(X) (Yoneda Lemma [Mac71, III.2])
and one can check that their composition is K X . So (Set C , R) is an order-ringed topos from which we can construct a model Fib of intensional Martin-Löf Type Theory with universes as in the previous sections. Since we assumed that C contains the non-trivial ordered ring R, it is not the case that R is trivial; that is, the sentence 0 = 1 is not satisfied by R (because it is not satisfied at component X = R). 2 More than this, the associated model of Type Theory built from (Set C , R) as in the previous sections is not logically trivial: there is no proof in it of Id Bool true false (so in particular, R cannot be a decidable object of Set C ). To see this, note that giving a proof of Id Bool true false in the CwF Fib associated with (Set C , R) is the same as giving an R-based Moore path from true : 1 −→ ∆ 2 to false : 1 −→ ∆ 2. There is no such path because we saw above that R ∆ 2 ∼ = ∆ 2 and hence every path in ∆ 2 is constant. 7.2. A gros topos model. Being logically non-trivial is rather a weak condition for models of type theory with intensional identity types. A more interesting one is that the model contains types A whose iterated identity types Id A , Id Id A , Id IdId A , . . . are all non-trivial (not isomorphic to the unit type). An interesting way of demonstrating that for the model associated with an order-ringed topos (E, R) is to show that the homotopy types of a rich collection of topological spaces (including all the n-dimensional spheres, say) are faithfully represented by the internal, R-based notion of homotopy on a corresponding collection of objects of the topos E. We give one such order-ringed topos in this section. 7.2.1. The topos. We use a topos of sheaves Sh(T, J ) [MM92, III] for the following site (T, J ). Let Haus denote the category of Hausdorff topological spaces and continuous functions. We take the small category T to be the least full subcategory of Haus containing the reals R and closed under the following operations: (a) If X, Y ∈ T, then T contains the product space X × Y .
℘ A :≡ {(f, i) : (R + A) × R + | (∀j i) f j = f i} (7.4) ∂ -, ∂ + : ℘ A −→ A ∂ -(f, i) :≡ f 0 ∂ + (f, i) :≡ f i
is the total object of the family of Moore path objects (3.1) with associated source and target morphisms. Note that since {(x, y) ∈ R × R | x ≤ y} is a closed subset of R × R, it is an object of T and the order relation on R in Sh(T, J ) is given by the representable
Y{(x, y) | x ≤ y} Y(R × R) ∼ = R × R.
It follows that the positive cone of R is also a representable sheaf: R + ∼ = Y(R + ). Recall that, as well as preserving finite limits, the Yoneda embedding preserves any exponentials that happen to exist. It follows that for each representable sheaf YX its total path object is representable by the Moore path space of X (7.3): ℘(YX) ∼ = Y(M X); and under this isomorphism the source and target morphisms ∂ -, ∂ + correspond to the representable morphisms induced by the usual source and target functions for M X. Since Y is full and faithful, it follows that for two continuous functions f, g : X −→ X in T, the morphisms Yf and Yg get identified in Ho(Sh(T, J )) iff f and g are homotopic in the classical sense. Thus Y induces a full and faithful embedding of the category Ho(T) of homotopy types of spaces in T into Ho(Sh(T, J )). Since T contains all the spheres S n , we deduce that identity types in this particular Fib are not necessarily truncated at any level of iteration.
Related Work
The classical topological notion of Moore path is a standard, if somewhat niche topic within homotopy theory. The Schedule Theorem of Dyer and Eilenberg [DE88] for globalising Hurewicz fibrations is a nice example of their usefulness. They have been used in connection 2:22
I. Orton and A. M. Pitts
Vol. 15:1 with higher-dimensional category theory by Kapranov and Voevodsky [KV91] and by Brown [Bro09]. Although our use of constructive algebra within toposes to make models of intensional type theory appears to be new, we are not the only ones to consider using some form of path with strictly unitary and associative composition to model identity types with a judgemental computation rule. Van den Berg and Garner [vG12] use topological Moore paths and a simplicial version of them to get instances of their notion of path object category for modelling identity types. The results of Sections 3 and 4 show that any ordered abelian group in a topos induces a path object category structure on that topos; and since the notion of fibration we use (Definition 4.1) is closely related to the one used in [vG12] (see Proposition 6.1.5 of that paper), one can get alternative, more abstract categorical proofs of Theorems 4.3 and 4.10 from the work of Van den Berg and Garner. However, the concrete calculations in the internal language that we give are quite simple by comparison; and this approach proves its worth in Section 5, whose results on obtaining function extensionality from the ordered ring structure are new.
The PhD thesis of North [Nor17] uses a category-theoretic abstraction of the notion of Moore paths, called Moore relation systems, as part of a complete analysis of when a weak factorization system gives a model (in terms of display map categories, rather than CwFs) of identity-, Σ-and Π-types. A Moore relation system is a piece of category theory comparable to our use of ordered abelian groups in categorical logic in Section 3; it would be interesting to see if it can be extended in the way we extended from groups to rings in Section 5 in order to validate function extensionality.
Spitters [Spi16, Section 3] uses a somewhat different formulation of Moore path in the cubical topos [CCHM18]. His notion is the reflexive-transitive closure of the usual path types given by the bounded interval. For better properties and to get a closer relationship with our version, one would like to quotient these cubical "Spitters-Moore" path objects up to degenerate paths; but the undecidability of degeneracy seems to stop one being able to do that while retaining the (uniform) Kan-fibrancy of such path objects. Here we can side-step such issues, since notably our models manage to avoid using a notion of Kan fibrancy at all.
Conclusion
We have shown that any connected ordered ring in a topos gives rise to a model of Martin-Löf's intensional Type Theory with universes in which proofs of identity are given by Moore paths. We gave an example to show that such models of type theory can contain highly non-trivial identity types that faithfully represent the homotopy types of a wide class of topological spaces and in particular are not truncated at any level of iteration.
It is an open question whether there is an order-ringed topos that gives rise to such a model of type theory containing a univalent [Uni13, Section 2.10] universe. (We saw in Section 6 that the universes constructed there are not univalent.) The known examples of non-truncated univalent universes, such as the classical simplicial sets model [KL16] and the various constructive cubical sets models [BCH14, CCHM18, ABC + 17] make use of a modified form of the Hofmann-Streicher [HS99] construction in presheaf categories. Streicher [Str05] points out that the basic Hofmann-Streicher universe construction works for sheaf toposes through a suitable use of sheafification. So there are Hofmann-Streicher universes in both of the example toposes from Section 7, one of which is a presheaf topos and one a sheaf topos. However, the analysis of [LOPS18,Section 5] of univalence mentioned above, one gets from the Hofmann-Streicher universe to a univalent universe classifying fibrations (of various kinds) by using the fact that in those models the path functor ℘( ) has a right adjoint. Unfortunately this is not the case for the models in Section 7, where it seems that the very property of the interval (half-line) that allows us to avoid all uses of Kan filling in favour of path composition when building our models of type theory, namely the total order (2.1), prevents the interval from being "tiny" and hence prevents ℘( ) from having a right adjoint.
.
Given Γ ∈ E, A ∈ Fib(Γ) and B ∈ Fib(Γ.A), the families Σ A B :≡ ( a:A x B(x, a) | x : Γ) and Π A B :≡ ( a:A x B(x, a) | x : Γ) in E(Γ) have tap fibration structures that are stable under re-indexing along morphisms ∆ −→ Γ in E. Hence the CwF Fib supports Σ-and Π-types [Hof97, Definitions 3.15 and 3.18].
and a : A y. We have (rev p) : y ∼ x, and hence (rev p) * a : A x and f ((rev p) * a) : B(x, (rev p) * a). Next, we have lift(rev p, a) : (y, a) ∼ (x, rev p * a) and therefore rev(lift(rev p, a)) : (x, rev p * a) ∼ (y, a)
Lemma 4. 8 .
8Given Γ ∈ E and A ∈ Fib(Γ), for each path p : (x, a) ∼ (y, b) in Γ.A, there is a path snd(p) : (fst ' p) * a ∼ b in A y satisfying snd(idp(x, a)) = idp a (4.10)
Theorem 4 . 10 (
410Identity types). The CwF Fib of Definition 4.1 supports the interpretation of intensional identity types (Definition 4.7), given by Moore path objects as in (4.9).
Vol. 15:1 MODELS OF TYPE THEORY BASED ON MOORE PATHS 2:11
Proof. In view of Lemma 4.9, it just remains to define the Refl and J operations as in Definition 4.7. Given Γ ∈ E and A ∈ Fib(Γ), we get Refl A : Γ.A −→ Γ.A.A fst. Id A byVol. 15:1
MODELS OF TYPE THEORY BASED ON MOORE PATHS
2:13
defining
Refl A (x, a) :≡ ((
Vol. 15:1 MODELS OF TYPE THEORY BASED ON MOORE PATHS 2:15
Vol. 15:1 MODELS OF TYPE THEORY BASED ON MOORE PATHS 2:19
shows that in the examples Vol. 15:1 MODELS OF TYPE THEORY BASED ON MOORE PATHS 2:23
On the other hand, since we assume C also contains the trivial ring 1, neither is it the case that the sentence ¬(0 = 1) is satisfied by R. Note that Set C is not a Boolean topos -it does not satisfy the Law of Excluded Middle.
This fact is the motivation for considering a site based on covers by closed subsets rather than the more familiar case of open covers used for Giraud's gros topos [GV72, IV, 2.5]. However, as Spitters has pointed out [private communication], in view of [Joh79, Theorem 8.1] we could have used the reals in Johnstone's topological topos instead of Sh(T, J ) in this section.
AcknowledgementWe are very grateful to Benno van den Berg and Bas Spitters for discussions about the material in this paper. Orton was supported by a PhD studentship from the UK EPSRC funded by grants EP/L504920/1 and EP/M506485/1.2:20I. Orton and A. M. PittsVol. 15:1 (b) If X ∈ T and C ⊆ X is a closed subset, then C (with the subspace topology) is in T.(c) If X, Y ∈ T and X is locally compact, then T contains the exponential space Y X (the set of continuous functions from X to Y endowed with the compact-open topology). Since the spaces in T are Hausdorff, equalizers of continuous functions give closed subspaces and hence by (a) and (b) we have that T is closed under taking finite limits in Haus. Hence T contains R + (as well as [0, 1] and all the spheres S n for any n ∈ N). Then by (c) we have that T is closed under taking Moore path spaces (with their usual topology):We define a coverage [Joh02, Definition A2.1.9] on T as follows. Following Dyer and Eilenberg[DE88]we say that a set S of closed subsets of a topological space X is a local cover if for each x ∈ X there is a finite subset {C 1 , . . . , C n } ⊆ S with x in the interior of C 1 ∪ · · · ∪ C n . Of course every finite cover of X by closed subsets is trivially a local cover.NoteGiven a functor F : T op −→ Set, if C ⊆ X is a closed subspace of a space in T, then for each x ∈ F X we will just write x| C for the element F i x ∈ F C, where the T-morphism i : C −→ X is the inclusion function. Recall that F is a sheaf with respect to J iff for all X ∈ T, S ∈ J (X) and all S-indexed families (The topos Sh(T, J ) is by definition the full subcategory of the functor category Set T op whose objects are sheaves. 7.2.2. The ordered commutative ring object R. Let Y : T −→ Set T op denote the Yoneda embedding for the small category T. Because elements of J (X) are local covers of X ∈ T, for each S ∈ J (X) if we have a family of continuous functions f C : C −→ X for each C ∈ S that agree where they overlap (f C | C∩D = f D | C∩D ), then the unique function f : X −→ X that agrees with each of them (f C = f | C ) is necessarily continuous. Hence each representable presheaf Y X = T( , X ) is a sheaf (in such a case one says that the coverage J is subcanonical ) and the Yoneda embedding gives a functor Y : T −→ Sh(T, J ).The axioms for partially ordered commutative rings are those inFig. 1except for (2.7). They make sense in any category with finite limits once we have an interpretation of the binary operation , the constants 0, 1 and the operations +, -, · ; and satisfaction of those axioms is preserved by functors that preserve finite limits. Since {(x, y) ∈ R × R | x ≤ y} is a closed subset of R × R and the usual ring operations on R are continuous, it follows that R is a partially ordered commutative ring object in the finitely complete category T. Then since Y preserves finite limits, the representable sheaf R :≡ YR is a partially ordered commutative ring object in the topos Sh(T, J ). In fact it is totally ordered, that is, satisfies (2.7). This is simply because we have a local cover {{(x, y) | x ≤ y}, {(x, y) | y ≤ x}} ∈ J (R × R). 3
. Models, Type, Based, Moore, 221MODELS OF TYPE THEORY BASED ON MOORE PATHS 2:21
First note that T contains the discrete spaces ∅, 1 and 2 = {0, 1}. Since ∅ is covered by the empty family of closed sets and 2 is covered by the two closed inclusions {0} → 2 ← {1}. To apply the results of Section 6, we need to verify that R is a connected object in Sh(T, J ) (Definition 6.1). for any sheaf F ∈ Sh(T, J ) we have that F (∅)2.3. R is connected. To apply the results of Section 6, we need to verify that R is a connected object in Sh(T, J ) (Definition 6.1). First note that T contains the discrete spaces ∅, 1 and 2 = {0, 1}. Since ∅ is covered by the empty family of closed sets and 2 is covered by the two closed inclusions {0} → 2 ← {1}, for any sheaf F ∈ Sh(T, J ) we have that F (∅)
)(1 + 1, F ) naturally in F . So the object of Booleans in Sh(T, J ) is representable by 2 ∈ T. Thus to see that R is connected, it suffices to show that K : Y2 −→ (R Y2) is an isomorphism in Sh(T, J ). Sh(T, J )(Y2, F ) ∼ = F (2) ∼ = F (1) × F (1) ∼ = Sh(T, J )(Y1 + Y1, F ) ∼ = Sh(T, JBut R = YR is also representable and Y preserves exponentials. It follows that K : Y2 −→ (R Y2is topologically connected; and hence so is K : Y2 −→ (R Y2) in Sh(T, J )Sh(T, J )(Y2, F ) ∼ = F (2) ∼ = F (1) × F (1) ∼ = Sh(T, J )(Y1 + Y1, F ) ∼ = Sh(T, J )(1 + 1, F ) naturally in F . So the object of Booleans in Sh(T, J ) is representable by 2 ∈ T. Thus to see that R is connected, it suffices to show that K : Y2 −→ (R Y2) is an isomorphism in Sh(T, J ). But R = YR is also representable and Y preserves exponentials. It follows that K : Y2 −→ (R Y2is topologically connected; and hence so is K : Y2 −→ (R Y2) in Sh(T, J ).
Let Ho(Sh(T, J )) denote the associated quotient category. By Theorem 5.1, two morphisms γ, δ : B −→ A are identified in Ho(Sh(T, J )) iff x:B (γ x ∼ δ x) has a global section. This is equivalent to requiring the existence of a morphism H : B −→ ℘ A in Sh(T, J ) satisfying ∂ -• H = γ and ∂ + • H = δ, where References. (t Sh, J ) ; C Angiuli, G Brunerie, T Coquand, K.-B Hou, ; R Harper, D R Licata, Homotopy types in Sh(T, J ). From the order-ringed topos. Consider the congruence on the category Sh(T, J ) that identifies morphisms γ, δ : B −→ A when there is a global section of Id B A γ δ. Cartesian cubical type theory2.4. Homotopy types in Sh(T, J ). From the order-ringed topos (Sh(T, J ), YR) we get a CwF Fib modelling Martin-Löf Type Theory with universes and with intensional identity types Id A given by Moore paths. Consider the congruence on the category Sh(T, J ) that identifies morphisms γ, δ : B −→ A when there is a global section of Id B A γ δ. Let Ho(Sh(T, J )) denote the associated quotient category. By Theorem 5.1, two morphisms γ, δ : B −→ A are identified in Ho(Sh(T, J )) iff x:B (γ x ∼ δ x) has a global section. This is equivalent to requiring the existence of a morphism H : B −→ ℘ A in Sh(T, J ) satisfying ∂ -• H = γ and ∂ + • H = δ, where References [ABC + 17] C. Angiuli, G. Brunerie, T. Coquand, K.-B. Hou (Favonia), R. Harper, and D. R. Li- cata. Cartesian cubical type theory. https://github.com/dlicata335/cart-cube/blob/master/ cart-cube.pdf, December 2017.
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This work is licensed under the Creative Commons Attribution License. B Van Den, R Berg, Garner, ACM Trans. Comput. Logic. 131Suite. or Eisenacher Strasse 2, 10777B. van den Berg and R. Garner. Topological and simplicial models of identity types. ACM Trans. Comput. Logic, 13(1):3:1-3:44, 2012. This work is licensed under the Creative Commons Attribution License. To view a copy of this license, visit https://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, 171 Second St, Suite 300, San Francisco, CA 94105, USA, or Eisenacher Strasse 2, 10777 Berlin, Germany
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"Kim Nygård \nDepartment of Chemistry and Molecular Biology\nUniversity of Gothenburg\nSE-412 96GothenburgSweden\n",
"Sten Sarman \nDepartment of Materials and Environmental Chemistry\nStockholm University\nSE-106 91StockholmSweden\n",
"Roland Kjellander \nDepartment of Chemistry and Molecular Biology\nUniversity of Gothenburg\nSE-412 96GothenburgSweden\n"
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"Department of Chemistry and Molecular Biology\nUniversity of Gothenburg\nSE-412 96GothenburgSweden",
"Department of Materials and Environmental Chemistry\nStockholm University\nSE-106 91StockholmSweden",
"Department of Chemistry and Molecular Biology\nUniversity of Gothenburg\nSE-412 96GothenburgSweden"
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Packing frustration for confined fluids, i.e., the incompability between the preferred packing of the fluid particles and the packing constraints imposed by the confining surfaces, is studied for a dense hard-sphere fluid confined between planar hard surfaces at short separations. The detailed mechanism for the frustration is investigated via an analysis of the anisotropic pair distributions of the confined fluid, as obtained from integral equation theory for inhomogeneous fluids at pair correlation level within the anisotropic Percus-Yevick approximation. By examining the mean forces that arise from interparticle collisions around the periphery of each particle in the slit, we calculate the principal components of the mean force for the density profile -each component being the sum of collisional forces on a particle's hemisphere facing either surface. The variations of these components with the slit width give rise to rather intricate changes in the layer structure between the surfaces, but, as shown in this paper, the basis of these variations can be easily understood qualitatively and often also semi-quantitatively. It is found that the ordering of the fluid is in essence governed locally by the packing constraints at each single solid-fluid interface. A simple superposition of forces due to the presence of each surface gives surprisingly good estimates of the density profiles, but there remain nontrivial confinement effects that cannot be explained by superposition, most notably the magnitude of the excess adsorption of particles in the slit relative to bulk.
|
10.1063/1.4894137
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1409.1448
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00d38650198806627e412e9d6053db34abdbd9ad
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Packing frustration in dense confined fluids
4 Sep 2014. 2014
Kim Nygård
Department of Chemistry and Molecular Biology
University of Gothenburg
SE-412 96GothenburgSweden
Sten Sarman
Department of Materials and Environmental Chemistry
Stockholm University
SE-106 91StockholmSweden
Roland Kjellander
Department of Chemistry and Molecular Biology
University of Gothenburg
SE-412 96GothenburgSweden
Packing frustration in dense confined fluids
J. Chem. Phys
141945014 Sep 2014. 201410.1063/1.4894137(Dated: 5 September 2014)
Packing frustration for confined fluids, i.e., the incompability between the preferred packing of the fluid particles and the packing constraints imposed by the confining surfaces, is studied for a dense hard-sphere fluid confined between planar hard surfaces at short separations. The detailed mechanism for the frustration is investigated via an analysis of the anisotropic pair distributions of the confined fluid, as obtained from integral equation theory for inhomogeneous fluids at pair correlation level within the anisotropic Percus-Yevick approximation. By examining the mean forces that arise from interparticle collisions around the periphery of each particle in the slit, we calculate the principal components of the mean force for the density profile -each component being the sum of collisional forces on a particle's hemisphere facing either surface. The variations of these components with the slit width give rise to rather intricate changes in the layer structure between the surfaces, but, as shown in this paper, the basis of these variations can be easily understood qualitatively and often also semi-quantitatively. It is found that the ordering of the fluid is in essence governed locally by the packing constraints at each single solid-fluid interface. A simple superposition of forces due to the presence of each surface gives surprisingly good estimates of the density profiles, but there remain nontrivial confinement effects that cannot be explained by superposition, most notably the magnitude of the excess adsorption of particles in the slit relative to bulk.
I. INTRODUCTION
Spatial confinement of condensed matter is known to induce a wealth of exotic crystalline structures. [1][2][3][4][5][6] In essence this can be attributed to a phenomenon coined packing frustration; an incompability between the preferred packing of particles -whether atoms, molecules, or colloidal particles -and the packing constraints imposed by the confining surfaces. As an illustrative example, we can consider the extensively studied system of hard spheres confined between planar hard surfaces at a close separation of about five particle diameters or less. This is a convenient system for studies on packing frustration, because its phase diagram is determined by entropy only. While the phase diagram of the bulk hard-sphere system is very simple, 7 the dense packing of hard-sphere particles in narrow slits has been found to induce more than twenty novel thermodynamically stable crystalline phases, including exotic ones such as buckled and prism-like crystalline structures. [2][3][4]6 In the case of spatially confined fluids, the effects of packing frustration are more elusive. Nevertheless, extensive studies on the fluid's equilibrium structure has brought into evidence this phenomenon; confinementinduced ordering of the fluid is suppressed when the short-range order preferred by the fluid's constituent para) [email protected] b) [email protected] c) [email protected] ticles is incompatible with the confining surface separation (see, e.g., Ref. 8 for illustrative examples). Packing frustration also influences other properties of the confined fluid, such as a strongly suppressed dynamics because of caging effects. [9][10][11][12] However, little is known to date about the underlying mechanisms of frustration in spatially confined fluids.
A stumbling block when elucidating the mechanisms of packing frustration in fluids is the hierarchy of distribution functions; 13 a mechanistic analysis of distribution functions requires higher-order distributions as input. While density profiles (i.e., singlet distributions) in inhomogeneous fluids are routinely determined today, either by theory, simulations, or experiments, structural studies are only seldom extended to the level of pair distributions. [14][15][16][17][18][19][20] The overwhelming majority of theoretical work in the literature has been done on the singlet level where pair correlations from the homogeneous bulk fluid are used in various ways as approximations for the inhomogeneous system. Moreover, even in the cases where the pair distributions for the inhomogeneous fluid have been explicitly determined, [14][15][16][17][18] the mechanistic analysis of ordering is hampered by the sheer amount of variables. A conceptually simple scheme for addressing ordering mechanisms in inhomogeneous fluids is therefore much needed.
In this work, we deal with the mechanisms of packing frustration in a dense hard-sphere fluid confined between planar hard surfaces by means of first-principles statistical mechanics at the pair distribution level. For this purpose we introduce principal components of the mean force acting on a particle, and study their behavior as a function of confining slit width. This provides a novel and conceptually simple scheme to analyze the mechanisms of ordering in inhomogeneous fluids. In contrast to the aforementioned multitude of exotic crystalline structures induced by packing frustration, we obtain compelling evidence that even for a dense hard-sphere fluid in narrow confinement, as studied here, the ordering is in essence governed by the packing constraints at a single solidfluid interface. Nonetheless, there are also some common features for the structures in the fluid and in the solid phases. Finally, we demonstrate how subtleties in the ordering may lead to important, nontrivial confinement effects.
The calculations in this work are done in integral equation theory for inhomogeneous fluids at pair correlation level, where the density profiles and anisotropic pair distributions are calculated self-consistently. The only approximation made is the closure relation used for the pair correlation function of the inhomogeneous fluid. We have adopted the Percus-Yevick closure, which is suitable for hard spheres. The resulting theory, called the Anisotropic Percus-Yevick (APY) approximation, has been shown to give accurate results for inhomogeneous hard sphere fluids in planar confinement. 15,21 In principle, pair distribution data for confined fluids could also be obtained from particle configurations obtained by computer simulation, e.g. Grand-Canonical Monte Carlo (GCMC) simulations. However, even with the computing power presently available, one would need impracticably long simulations in order to obtain a reasonable statistical accuracy for the entire pair distribution, which for the present case has three independent variables. For the confined hard sphere fluid, the pair distribution function has narrow sharp peaks (see Ref. 8 for typical examples), which are particularly difficult to obtain accurately. The alternative to use, for example, the Widom insertion method to calculate the pair distribution point-wise by simulation is very inefficient for dense systems. It should be noted, however, that in cases where direct comparison is feasible in practice, simulations and anisotropic integral equation theory are in excellent agreement in terms of pair distributions. 22 In these cases, for a corresponding amount of pair-distribution data of essentially equal accuracy, the integral-equation approach was found to be many thousands times more efficient in CPU time than the simulations. Finally we note that other highly accurate theoretical approaches, such as fundamental measure theory (see, e.g., Ref. 23 for a recent review), have not yet been extended to the level of pair distributions in numerical applications.
II. SYSTEM DESCRIPTION, THEORY AND COMPUTATIONS
Within the present study, we focus on a dense hardsphere fluid confined between two planar hard surfaces. The sphere diameter is denoted by σ, the surface separation by H, and the reduced slit width by L. The gray region depicts the excluded volume around the left particle, which in this figure is in contact with the bottom wall. The red arrow shows the collisional force exerted by the right particle on the left one. The force acts in the radial direction. etry, we refer to Fig. 1. The particle diameter is denoted by σ and the surface separation by H. The space available for particle centers is given by the reduced slit width, which is defined as L = H − σ. The z coordinate is perpendicular while the x and y coordinates are parallel to the confining surfaces. The system has planar symmetry and therefore the number density profile n(z) only depends on the z coordinate.
Except when explicitly stated otherwise, the confined fluid is kept in equilibrium with a bulk reservoir of number density n b = 0.75σ −3 . The average volume fraction of particles in the slit, φ av = (πσ 3 /6H) L 0 n(z)dz, then varies between about 0.33 and 0.37 depending on the surface separation in the interval L = 1.0σ − 4.0σ. 8 Due to the planar symmetry all pair functions depend on three variables only, e.g., the pair distribution function g(r 1 , r 2 ) = g(z 1 , z 2 , R 12 ), where R 12 = |R 12 | with R 12 = (x 2 − x 1 , y 2 − y 1 ) denotes a distance parallel to the surfaces. In graphical representations of such functions, we let the z axis go through the center of a particle at r 2 , i.e., we select r 2 = (0, 0, z 2 ). Then the function g(z 1 , z 2 , R 12 ) states the pair distribution function at position r 1 = (R 12 , z 1 ) = (x 1 , y 1 , z 1 ), given a particle at position (0, 0, z 2 ). Likewise, n(z 1 )g(z 1 , z 2 , R 12 ) gives the average density at r 1 around a particle located at (0, 0, z 2 ). We plot for clarity also negative values of R 12 , i.e., in the following plots of pair functions R 12 is to be interpreted as a coordinate along a straight line in the xy plane through the origin.
Throughout this study, we make use of integral equation theory for inhomogeneous fluids on the anisotropic pair correlation level. Following Refs. 15 and 21, we determine the density profiles n(z 1 ) and pair distribution functions g(z 1 , z 2 , R 12 ) of the confined hard-sphere fluid by solving two exact integral equations self-consistently: the Lovett-Mou-Buff-Wertheim equation,
d[lnn(z 1 ) + βv(z 1 )] dz 1 = c(z 1 , z 2 , R 12 ) dn(z 2 ) dz 2 dz 2 dR 12 ,(1)
and the inhomogeneous Ornstein-Zernike equation,
h(r 1 , r 2 ) = c(r 1 , r 2 ) + h(r 1 , r 3 )n(z 3 )c(r 3 , r 2 )dr 3 ,(2)
where h = g − 1 is the total and c the direct pair correlation function, while v denotes the hard particle-wall potential
v(z) = 0 if 0 ≤ z ≤ L, ∞ otherwise.(3)
As the sole approximation, we thereby make use of the Percus-Yevick closure for anisotropic pair correlations, c = g − y, where y(r 1 , r 2 ) denotes the cavity correlation function that satisfies g = y exp(−βu) and u is the hard particle-particle interaction potential,
u(r 1 , r 2 ) = 0 if |r 1 − r 2 | ≥ σ, ∞ if |r 1 − r 2 | < σ.(4)
This set of equations constitutes the APY theory. The confined fluid is kept in equilibrium with a bulk fluid reservoir of a given density by means of a special integration routine, in which the rate of change of the density profile for varying surface separation is given by the exact relation 15,21
∂n(z 1 ; L) ∂L = −βn(z 1 ; L) ∂v(z 1 ; L) ∂L + n(z 2 ; L)h(z 1 , z 2 , R 12 ; L) ∂v(z 2 ; L) ∂L dz 2 dR 12(5)
under the condition of constant chemical potential. Here we have explicitly shown the L dependence of the functions, which is implicit in the previous equations. For a concise review of the theory and details on the computations we refer to Ref. 8.
III. RESULTS AND DISCUSSION
A. Density profiles and pair densities
The theoretical approach adopted in this work has recently been shown 24 to be in quantitative agreement with experiments at the pair distribution level for a confined hard-sphere fluid in contact with a bulk fluid of the same density, n b = 0.75σ −3 , as used in the current work. Both the anisotropic structure factors from pair correlations and the density profiles agree very well with the experimental data. For higher densities, we compare in Fig. 2(a) our result with the density profile obtained from GCMC simulations by Mittal et al. 9 for an average volume fraction in the slit φ av = 0.40 and at L = 1.40σ. For this extreme particle density, which is virtually at phase separation to the crystalline phase at this surface separation, 4 there are quantitative differences, but our theoretical profile agree semi-quantitatively with the simulation data. For L = 1.0σ and 2.0σ and at the same φ av , the deviations between our profiles and the GCMC profiles by Mittal et al. are larger (not shown). In the rest of this paper we shall, however, treat cases with lower particle concentrations in the slit: φ av between about 0.33 and 0.37, which are less demanding theoretically. In Ref. 21 we showed, for a wide range of slit widths, that our theory is in very good agreement with GCMC simulations for the confined hard sphere fluid in equilibrium with a bulk density 0.68σ −3 , which is only slightly lower than what we consider in this work. Furthermore, our main concern in this paper are cases with surface separations about halfway between integer multiples of sphere diameters, as in Fig. 2(a).
Returning to the system in equilibrium with a bulk with density n b = 0.75σ −3 , we illustrate the concept of packing frustration in spatially confined fluids by presenting the number density profile n(z) for reduced slit widths of L = 1.05σ, 1.40σ, and 1.60σ in Fig. 2(b). The fluid in the narrowest slit exhibits strong ordering, as illustrated by well-developed particle layers close to each solid surface. Such ordering is observed for the hard-sphere fluid in narrow hard slits when the surface separation is close to an integer multiple of the particle diameter σ. In this specific case, the average volume fraction φ av = 0.35 is about 82% of the volume fraction for phase separation to the crystalline phase at this surface separation, 4 and the "areal" number density near each solid surface is L/2 0 n(z)dz ≈ 0.69σ −2 , about 77% of the freezing density for the two-dimensional hard-sphere fluid. 2,4 For slit widths intermediate between integer multiples of σ, the confined fluid develops into a relatively disordered fluid in the slit center, despite the confining slit being narrow enough to support ordering across the slit. In the particular case shown in Fig. 2(b), we observe shoulders in the density peaks close to each surface which evolve with increasing slit width into two small (or secondary) density peaks in the slit center. At slightly larger slit widths (to be investigated below), these two small peaks merge and form a fairly broad layer in the middle of the slit. For L ≈ 2.0σ there is strong ordering again; the layers at either wall are then very sharp and the midlayer is quite sharp (more profiles for L = 1.0σ − 4.0σ for the current case can be found in Ref. 8 and as a video in Ref. 25). Such a change of ordering at the intermediate separations is usually interpreted as a signature of packing frustration, and in this paper we will address its mechanisms.
How can we understand these observations? The starting point for our discussion will be the pair density n(r 1 )g(r 1 , r 2 ), i.e., the density at position r 1 given a particle at position r 2 . As will become evident below, the pair densities allow us to analyze the mechanisms leading to the detailed structure of the layers in confined, inhomogeneous fluids. Here, we shall in particular investigate the mechanisms of packing frustration in dense hard-sphere fluids under spatial confinement. Fig. 3 shows examples of contour plots of the pair density n(z 1 )g(z 1 , z 2 , R 12 ) for three reduced slit widths, L = 1.50σ, 1.65σ, and 1.80σ, when a particle (the "central" particle) is located on the z axis at coordinate r 2 = (0, 0, z 2 ). The density profiles for these three slit widths are shown in the right hand side of the plot. In Fig. 3(a) there is a shoulder in the profile on either side of the midplane, while in Fig. 3(b) two small, but distinct, peaks have formed near the slit center. In Fig. 3(c) these secondary peaks have merged into one peak in the middle. These changes in the density profile occur within a variation in surface separation of only 0.3σ. In the contour plots, the central particle's center is in all cases situated at a distance of 1.55σ (about three particle radii) from the bottom surface, z 2 = 1.05σ, marked by a filled circle in the profile. Particles that form the main layer in contact with the bottom surface can then touch the central particle; the latter is penetrating just the edge of this layer. Note that the position of the secondary maximum for the middle case, Fig. 3(b), is also located at z 1 = 1.05σ.
Thus, it can be understood that particles forming the small secondary peak in Fig. 3(b) are in contact with, but barely penetrating, the main layer of particles at the bottom surface. The particles of this secondary peak are at the same time strongly penetrating the main layer at the top surface. The same is, however, true for the particles around the same z coordinate (1.05σ) in Figs. 3(a) and 3(c), but with a markedly different outcome for the profile. Our task here is to understand the reason for such differences.
In the contour plots of the pair density n(z 1 )g(z 1 , z 2 , R 12 ) in Fig. 3 we see that in all three cases the particle density in the wedge-like section formed between the central particle and the upper wall is strongly enhanced, resulting in a local number density of up to 17, 20, and 24 σ −3 , respectively, in the three cases. This enhancement in ng relative to the density n at the same z coordinate is given by the pair distribution function g, which is about 4 -4.5 in the inner part of the wedge-like section for all these cases. In the region near the bottom surface, where the central particle is in contact with the main bottom layer, there is also an enhancement in density, but to much smaller extent than at the top. Note that the density distribution ng near the bottom is very similar in all three cases.
B. Mean force
To understand why the profiles differ so much in these three cases, we investigate the mean force F (z) that acts on a particle with its center at position z. The potential of mean force, w, is related to the density n by
n = n b exp(−βw), where β = (k B T ) −1 , k B is Boltz- mann's constant, and T the absolute temperature. This implies that F ≡ −∇w = k B T ∇ ln n.
Due to the planar symmetry, n depends on z only and the total force components in the x and y directions are zero. The mean force F is then directed parallel to the z axis and we have βF (z) = d ln n(z)/dz = n ′ (z)/n(z), where n ′ = dn/dz. Thus, an understanding of the behavior of the profile can be obtained from an analysis of F . The sign of F tells whether n is increasing or decreasing and extremal points of n correspond to points where F is zero.
For hard-sphere fluids, the forces exerted on a particle by the other particles in the system are simply due to collisions. Due to planar symmetry, the density distribution in the vicinity of a particle has rotational symmetry around the z axis through the particle center. The average force at each point is acting in the direction normal to the sphere surface and for a particle located at z 2 the average force from all collisions along the sphere periphery at coordinate z 1 is proportional to the contact density n(
z 1 )g cont (z 1 , z 2 ), where g cont (z 1 , z 2 ) ≡ g(z 1 , z 2 , R 12 )| R 2 12 =σ 2 −(z1−z2) 2
is the contact value of the pair distribution function at the particle surface. Note that the force in, for example, the x direction on one side of the periphery is cancelled by the force in the −x direction on the opposite side. Thus only the z component of the net force on the particle contributes as expected. Using the first Born-Green-Yvon equation one can show 15 that
βF (z 2 ) = 2πσ z2+σ z2−σ n(z 1 )g cont (z 1 , z 2 ) (z 2 − z 1 ) σ dz 1 (6) (the integral is over the range |z 2 − z 1 | ≤ σ where g cont is FIG. 3.
Contour plot of the pair density n(r1)g(r1, r2) ≡ n(z1)g(z1, z2, R12) at coordinate r1 = (R12, z1) around a particle in the slit between two hard surfaces, when the particle is located on the z axis at coordinate r2 = (0, 0, z2). One surface is 0.5σ above the top and one 0.5σ below the bottom of each subplot (cf. Fig. 1). The system is in equilibrium with a bulk fluid of density n b = 0.75σ −3 [same as in Fig. 2(b)]. The gray region is the excluded volume zone around the particle. Data are shown for different reduced slit widths: (a) L = 1.50σ, (b) 1.65σ, and (c) 1.80σ. The number density profile n(z1) for each case is also shown for clarity to the right. The particle position z2 (shown as filled circle in the profile plots) is in all cases positioned at a distance of 1.55σ from the bottom surface (at z coordinate 1.05σ). The arrows in the gray region depict z components of the collisional forces acting on the particle (corresponding to the z projection of the red arrow in Fig. 1). The arrows displayed at a certain z1 coordinate here represent the entire force acting on the sphere periphery in a dz interval around this coordinate. In subplot (a) the sum of all arrows (with signs) is > 0, in (b) = 0 and in (c) < 0. Let us now return to the intriguing formation of secondary density maxima for L ≈ 1.65σ. For this purpose, we present in Fig. 3 the z component of the contact forces acting on the particle. They are represented by the arrows along the sphere periphery. In these plots, there are two major contributions to the net force acting on the particle, namely the repulsive forces exerted by the particle layers close to each confining wall. For L = 1.65σ and the chosen position of the central particle in Fig. 3(b), z 2 = 1.05σ, these force contributions cancel each other: the sum of the arrows (with signs) is zero and hence dn/dz = 0 at this z coordinate, as shown to the right in the figure. It is the subtle interplay between these forces for neighboring z 2 values which leads to the secondary density maximum.
The situation is, however, markedly different for L = 1.50σ and 1.80σ. While the total force exerted by the particles in the main layer at the bottom surface is practically equal for all three cases, the magnitude of the force exerted by the particles in the main layer at the upper surface varies strongly with L. This variation is partly due to the different magnitude of the contact densities in the wedge-like region mentioned above and partly due to the change in angle between the normal vector to the sphere surface there and the z axis. Recall that the contact force acts along this normal vector, so the z component is dependent on this angle. For L = 1.50σ, Fig. 3(a), the z component of the contact force from the upper layer is smaller than for L = 1.65σ. The sum of the arrows is then positive, i.e. the total average force is directed towards the upper wall and hence dn/dz > 0 at this z coordinate. For L = 1.80σ, Fig. 3(c), this z component is larger compared to L = 1.65σ, thereby pushing the central particle towards the slit center. Hence dn/dz < 0 at this z coordinate.
C. Principal components of mean force
In order to gain more insight into the formation of the secondary maxima, we present in Fig. 4 the net force acting on a particle for all positions z 2 in the same three cases as discussed above, L = 1.50σ, 1.65σ, and 1.80σ. To facilitate the interpretation, the principal force contributions acting in positive (denoted F ↑ ) and negative (F ↓ ) directions are shown separately. The total net force is F = F ↑ − F ↓ , where F ↑ originates from collisions on the lower half of the sphere surface and F ↓ on the upper half (F ↑ and F ↓ correspond to the absolute values of the sums of arrows in respective hemisphere in Fig. 3). In Fig. 4 the red (solid) and black (dashed) curves are each other's mirror images with respect to the vertical dashed axis at z 2 = L/2, which shows the location of the slit center.
Since the variation in F ↑ (and F ↓ ) is very similar for all slit widths in Fig. 4, the following discussion will hold for all three cases. For z 2 = 0 we have F ↑ = 0, because no spheres can collide from below since the confining surface precludes them from being there (cf. Fig. 1). With increasing z 2 we observe a monotonically increasing F ↑ , which can be attributed both to the increasing area exposed to collisions on the lower half of the sphere surface and the decrease in angle of the sphere normal there relative to the z axis. With further increase in z 2 , we eventually observe a decrease in the exerted force induced by a decrease in contact density n(z 1 )g cont (z 1 , z 2 ). Around z 2 ≈ 1.0σ we observe a sudden onset of a rapid decrease for F ↑ . This is a consequence of a rapid decrease in contact density, that occurs when the particle at z 2 loses contact with the dense particle layer at the bottom wall. For even larger z 2 , where the particle is close to the top surface, collisions with particles in the slit center around the entire lower half of the sphere surface become important so that F ↑ increases again.
The three red curves are compared in the bottom panel, where F ↑ from the first two panels (L = 1.50σ and 1.65σ) are shown as dotted curves. We see that the curves are nearly identical apart from in a small region to the far right. The analogous statement is true, of course, for the black dashed curves. Thus, apart from small z 2 intervals to the extreme left and right, the behavior of F = F ↑ − F ↓ for the different slit widths can be understood in terms of a horizontal shift of the red and the black curves relative to each other. The formation of secondary density maxima can then be explained from the resulting balance of the force contributions. For L = 1.65σ and z 2 > L/2 [i.e., the right half of Fig. 4(b)], the two curves intersect at two points where the forces cancel each other and where dn/dz = 0. The intersection marked by the filled circle gives a local maximum of n(z) and the next one to the right gives a minimum. Together with the minimum at the slit center, z 2 = L/2, where the curves also cross each other, these features give rise to the secondary peak of the density profile as we have seen in Fig. 3(b). This subtle balance of forces, and hence the formation of secondary maxima, is only observed in a narrow range of slit widths, as evidenced by the force profiles for L = 1.50σ and 1.80σ. In the latter case, the intersection at z 2 = L/2 corresponds to a local maximum and the other one to a minimum. Together they give one peak in the middle as seen in Fig. 3(c). The formation of secondary maxima is for z 2 > L/2 accordingly a consequence of two phenomena: (i) the rapid decrease of F ↑ followed by the subsequent increase of F ↑ and (ii) the monotonic decrease of F ↓ in the same region. Together these effects lead to the force curves intersecting twice in the manner they do for L = 1.65σ. The rapid decrease of F ↑ is, as we have seen, due to the loss of contact of the particle with the well-developed bottom layer, while the monotonic decrease of F ↓ occurs when the particle approaches the top surface.
For comparison, we present in Fig. 5 the principal force components for a set of larger slit widths: L = 3.45σ, 3.60σ, and 3.75σ. There are no secondary maxima in this case. Instead we observe for L = 3.60σ a broad region in the center of the slit where F ↑ and F ↓ virtually cancel each other and where, as a consequence, dn/dz ≈ 0. Hence, this observation implies an essentially constant n in the slit center, as can be seen in the third full curve of Fig. 6, where density profiles are shown for various cases.
The course of events shown in Fig. 5 when we increase L from 3.45σ to 3.75σ implies the formation of a layer at the slit center. The crossing of the principal force curves in Fig. 5(a) at the slit center, z 2 = L/2, corresponds to a density minimum, while that in Fig. 5(c) corresponds to a density maximum. Note that for L ≈ 3.0σ there are four layers in the slit (two very sharp ones at the walls and two less sharp on either side of the slit center) and for L ≈ 4.0σ there are five layers. The fifth layer that forms in the middle for the intermediate separations arises via the broad flattening of the density profile in the middle, and signals the packing frustration in this case.
The data of Figs. 4 and 5 indicate a qualitative difference in n(z) for L ≈ 1.65σ and ≈ 3.60σ. In the transition from 2 → 3 particle layers, the third layer is formed via the occurrence of secondary layers close to each surface, which merge to form a central layer with increasing L. This contrasts the transitions from 4 → 5 particle layers just discussed, where the new particle layer forms directly in the slit center. The secondary peaks for L ≈ 1.65σ are also evident in qualitatively different anisotropic structure factors S(q) for L = 1.60σ and 3.50σ presented in our previous work, Ref. 8. S(q) for confined fluids is governed by an ensemble average of the anisotropic pair density correlations n(z 1 )h(z 1 , z 2 , R 12 ) (see Ref. 25 for more slit widths). In order to address these differences in n(z) with L, we will in the following analyze further the principal force component F ↑ .
D. Superposition approximation
In both Figs. 4 and 5, the principal force components F ↑ (and F ↓ ) for different L nearly coincide for most z values. In order to investigate this further, we plot F ↑ for a wider set of slit widths, L = 3.0σ − 4.0σ, in Fig. 7(a). Indeed, apart from rather small deviations at large z, all data fall on a master curve given by F ↑ for L = ∞, i.e., the force component for the single solid-fluid interface [the former curves are also shown separated in Fig. 7(b)].
Although not shown here, we have verified that this observation holds reasonably well for L ≥ 1.0σ, implying the same ordering mechanism irrespective of slit width.
In order to gain further insight into the ordering mechanism, we have determined density profiles obtained in a simple superposition approximation. [27][28][29][30] Within this approximation, the potential of mean force w in the slit is calculated as the sum of the corresponding potentials from two single hard surfaces, i.e. w(z) ≈ w ∞ (z) + w ∞ (L−z), where w ∞ denotes the potential of mean force for the fluid at a single solid-fluid interface in contact with a bulk fluid of density n b . This implies the superposition for the mean force:
F (z) ≈ F ∞ (z) − F ∞ (L − z).
where we have explicitly shown that the density profile for the slit, n(z) ≡ n(z; L), depends on L, and where superscript sp indicates "superposition" and n ∞ (z) is the density profile outside a single surface. In Fig. 6 we compare n(z) for reduced slit widths of L = 1.65σ, 2.60σ, and 3.60σ obtained via the full theory (solid lines) and the superposition approximation thus obtained (dotted lines). Note that there are density peaks at z ≈ 1.05σ for all three slit widths and that they approximately coincide with the location of a density peak for the single solid-fluid interface (also shown in Fig. 6). This implies that the density peak at z ≈ 1.05σ is strongly correlated with the bottom solid surface. Although the profiles obtained via the superposition approximation deviate quantitatively from those of the full theory, especially for narrow slit widths, the qualitative agreement implies that the main features of n(z) -the density peaks and shoulders of Fig. 6 -are rather uncomplicated confinement effects.
To substantiate this conclusion, we present in Fig. 7(b) the principal force components F ↑ for L = 3.0σ − 4.0σ, obtained both using the full theory and the superposition approximation. The agreement is equally good as for the density profile of the L = 3.60σ case in Fig. 6. A significant point is now that the superposition allows us to separate the contributions to F ↑ from each surface in a simple manner, that will provide insights into what happens during confinement. As shown in Appendix A, F ↑ can be decomposed in this approximation into two components: a major contribution from the lower surface, F L ↑ , and a correction due to the presence of the upper surface, ∆F U ↑ . The former is the same as the average force component for the single solid-fluid interface plotted in Fig. 7(a) (denoted as "master curve" above). We have
F sp ↑ (z 2 ; L) = F L ↑ (z 2 ) + ∆F U ↑ (z 2 ; L),(8)where ∆F U ↑ (z 2 ; L) = F U ↑ (z 2 ; L)−F b ↑ , see Eq. (A3).
Here, F U ↑ is the average force for the case of a single solid-fluid interface (U) and F b ↑ is the force that acts on one side of a hard sphere (i.e. on one half) in the bulk fluid. Note that in F sp ↑ it is only F U ↑ that depends on L. In Fig. 7(c) we show F L ↑ and ∆F U ↑ for the same surface separations as before. The L dependence of the latter is simply a parallel displacement along z. When F L ↑ and ∆F U ↑ are added we obtain the dotted curves in Fig. 7(b). Thus the differences between each black curve and the red curve in Fig. 7(a) is essentially contained in the contribution ∆F U ↑ from the upper surface (for smaller surface separations there will remain a minor difference as indicated by the small deviations for the superposition approximation in Fig. 6).
To see in more detail what this means, we have in Fig. 8 shown schematically how these force contributions act on a sphere. In the presence of only one solid-fluid interface (L), the total force in the direction away from the surface (upwards) is F ↑ = F L ↑ , i.e., the force on the bottom half of the sphere shown as red in the figure. Let us now place the second surface (U) some distance from the other, at the location indicated in the figure. The change in the upwards force due to this second surface is given by ∆F ↑ ≈ ∆F U ↑ in the superposition approximation. Note that the former force, F L ↑ , acts on the hemisphere that is facing the surface L, while the latter, ∆F U ↑ , is a force that acts on the hemisphere away from the corresponding surface U and in the direction towards this surface.
If the lower wall were not present when we place the upper wall at the indicated position, the initial state would be a homogeneous bulk fluid and the final state a single solid-fluid interface (U) in contact with the bulk. In this situation ∆F U ↑ equals the actual change in the average force on the red hemisphere. In Eq. (8) we have adopted this value as an approximation for the corresponding change when placing the upper wall in the presence of the lower one, i.e., when the initial state is an inhomogeneous fluid in contact with the lower surface and the final state is a fluid simultaneously affected by both surfaces. Since this approximation obviously is very good, it follows that the inhomogeneity due to one surface has only a small influence on the effects from the other surface throughout the entire slit.
We saw in section III C that the seemingly complicated changes in structure as the surface separation varies around half-integer σ values of L (i.e., [m + 0.5]σ with m = integer), can be mainly explained by a parallel displacement of upward and downwards force curves along the z direction. There was, however, some variation in these force curves near one of the surfaces (the upper surface for the upward forces and the lower surface for the downwards forces) that remained unexplained there. In the current section we have seen that this variation too can be mainly explained by a parallel displacement -in this case a displacement of the contributions to F ↑ (or F ↓ ) due to each surface as seen in Fig. 7(c).
To summarize our results in this section we make two important conclusions: First, by considering the mean force due to one surface (here the lower one) and by treating the influence from the other (upper) surface as a correction ∆F U ↑ according the the superposition approximation, one obtains nearly quantitative agreement with the full theory. Our approach of defining principal components of the mean force thereby provides a means to understand the contributions of each confining surface. Second, the principal force components obtained within the full theory and the superposition approximation are virtually in quantitative agreement for L ≥ 3.0σ. For narrower slit widths (down to L = 1.0σ), quantitative discrepancies become more pronounced. These quantitative differences, which will be discussed in the next subsection, are nontrivial confinement effects. Nevertheless, the semi-quantitative agreement in the whole range of slit widths, down to L = 1.0σ, further strengthens the notion that ordering of confined hard-sphere fluids can, to a good approximation, be explained as a single-wall phenomenon. In essence, the fluid conforms locally with only one of the confining surfaces at a time. In some local regions it will thereby conform to one surface and in other regions to the other surface -regions that are continuously changing (recall that the distributions we calculate are time averages of the various possible structures). We emphasize that this reasoning holds for all slit widths, irrespective of whether L is close to an integer or a half-integer multiple of the particle diameter (cf. Fig. 7). In other words, from a mechanistic point of view there is little difference between ordering in frustrated and more ordered confined hard-sphere fluids. In the latter case, the local ordering near one surface essentially agrees with the local ordering at the other one, whereby for the density profiles there appear only small mutual effects of the ordering from both surfaces beyond what is given by superposition.
An interesting similarity between the structures observed in the fluid and solid phases should be mentioned. In some of the exotic crystalline structures observed under confinement -most notably the prism-like structures 3,4,6 -the particles locally conform with one of the solid surfaces. This is reminiscent of the situation in the fluid phase discussed above, although in the latter case the structures are less ordered and constantly changing locally. In particular, the adaptive prism phase 2P A found in Ref. 6 would yield an average density profile with secondary peaks on either side of the midplane, similar of those shown in Fig. 3(b) but much sharper.
The fact that the superposition approximation works surprisingly well for these rather large densities and gives a large part of the effects of confinement, means that it is simple to obtain good estimates of the density profiles for a confined fluid given an accurate density profile for a single solid-fluid interface. To obtain the latter is, however, computationally nontrivial and requires fairly advanced theories. Furthermore, as we shall see below, not all important properties of the confined fluid can be explained by superposition.
E. Nontrivial confinement effects
We have shown that the density profile n(z) of confined hard-sphere fluids is, to a large extent, determined by packing constraints at a single solid-fluid interface. In this respect, the ordering is a trivial confinement effect. However, subtle deviations in n(z) do remain in the superposition approximation, and these may lead to important, nontrivial confinement effects. The two most prominent nontrivial effects of confinement in, for example, Fig. 6 are the slit width dependence of the contact density at the walls, n cont , and the total number of par- ticles per unit area in the slit N = L 0 n(z)dz, which is a fundamental quantity for many properties of the confined fluid. In the following, we will discuss these two and related quantities in more detail.
In Fig. 9(a), we present the excess adsorption Γ(L) = L 0 [n(z) − n b ]dz of particles in the slit as a function of reduced slit width L, determined via both the full theory and the superposition approximation. The discrepancy between the two theoretical schemes is striking; since Γ is an integrated quantity, minor systematic deviations in n(z) accumulate to a large effect in the total number of particles. The superposition approximation gives, for example, in the interval L = 1.0σ to 2.0σ an estimate of N that is wrong by a factor that varies between 1.36 and 0.84. We note that, e.g., dynamic quantities such as diffusion coefficients 31 and relaxation times 32 in simple confined fluids have been found to scale with particle packing, as quantified by the excess entropy. Consequently, a systematic error in the packing of particles (especially for very narrow confinement), as evidenced by systematic quantitative differences in the number density n(z; L) and an ensuing large discrepancy in Γ(L) between the full theory and the superposition approximation, will have a substantial impact on many properties of the confined fluid obtained theoretically. Fig. 9(b) shows the contact density n cont = n(0) as a function of L, again obtained both via the full theory and the superposition approximation. This is an important quantity, because it yields the pressure between the walls, P in = k B T n(0), according to the contact theorem. Consequently, n cont is related to the net pressure acting on the confining surfaces, Π(L) = P in (L) − P b with P b denoting the bulk pressure, and hence to the extensively studied oscillatory surface forces. 33,34 While the superposition approximation explains reasonably well the magnitude of n cont , there is a nontrivial systematic phase shift with respect to L of about 0.1σ. This effect has been observed by one of us (S.S.) already earlier, 30 and in the following we will provide a mechanistic explanation of the phenomenon. A similar phase shift can also be seen in Γ(L), Fig. 9(a).
In the superposition approximation, Eq. (7) yields the contact density for the wall at z = 0 as
n sp cont (L) = n sp (0; L) = n ∞ (0)n ∞ (L) n b .(9)
Thus, the contact density for a reduced slit width L is in this approximation proportional to the density at z = L outside a single surface. To analyze the L dependence further we will need the following equation that is equivalent to Eq. (1),
d[ln n(z 1 ) + βv(z 1 )] dz 1 = − β n(z 2 )h(z 1 , z 2 , R 12 ) dv(z 2 ) dz 2 dz 2 dR 12 .(10)
[The two equations can be transformed into each other by the Ornstein-Zernike equation (2).] For a single hard wall-fluid interface located at z = 0, Eq. (10) yields
dn ∞ (z 1 ) dz 1 = n ∞ (z 1 )n ∞ (0) h ∞ (z 1 , 0, R 12 )dR 12 ,(11)
where h ∞ is the total pair correlation function for the fluid outside the single surface. By inserting z 1 = L, this equation together with Eq. (9) imply that
dn sp cont (L) dL = n sp cont (L)n ∞ (0) h ∞ (z 1 , 0, R 12 )dR 12 z1=L .(12)
Apart from the factors in front of the integral, we see that the main difference is that in the superposition approximation the total pair correlation function for a single wall is evaluated at coordinate z 1 = L outside the wall, while for the exact case the correlation function for the fluid in the slit is evaluated at the opposite surface (also at z 1 = L). The oscillatory behavior of the contact density as a function of L implies that its derivative changes sign with the same periodicity. Since the prefactors are positive, the phase shift for n sp cont relative to n cont must originate from the integrals.
In Fig. 10 we have plotted the total pair correlation function h(z 1 , 0, R 12 ) in the slit when the central particle is in contact with the lower surface (i.e., at coordinate 0) for the cases L = 1.25σ, 2.50σ, and 3.75σ together with the corresponding function for a single hard wall-fluid interface. The first impression is a striking similarity of these plots, despite that there is an upper surface present in the first three cases. There are only small differences in the entire slit compared to the single surface case for the corresponding z 1 values. When looking closely, one can, however, see some systematic differences in the h function induced by the presence of the upper surface. Most importantly, we will investigate h for z 1 = L, which occurs in the integral in Eq. (13), and compare this with the values at the same z 1 coordinates for the single surface case, occurring in Eq. (12). These z 1 values are marked with red arrows in the left hand side of Figs. 10(a)-(c) and with red lines in Fig. 10(d). Fig. 11 shows R 12 × h(z 1 , 0, R 12 ) with z 1 = L for the cases in Figs. 10(a)-(c) and these curves are compared to R 12 × h ∞ (z 1 , 0, R 12 ) for the same z 1 coordinates (shown as blue dotted lines in the figure). The factor R 12 is included so the areas under the curves in Fig. 11 are proportional to the values of the integrals of Eqs. (12) and (13); this factor originates from the area differential dR 12 = 2πR 12 dR 12 . The L values in Figs. 10 and 11 are selected such that we cover cases where dn cont (L)/dL and dn sp cont (L)/dL in Fig. 9(b) are negative (L = 1.25σ) and positive (L = 3.75σ). There is also one case (L = 2.50σ) with dn sp cont (L)/dL ≈ 0. These signs can be verified by inspection of the areas under the curves in Fig. 11 (the contributions around R 12 = 0 are most important for the sign; there are substantial cancellations in the tail region due to the oscillations).
We can see in the figure that the full curves and the blue dotted ones do not agree, which means that the values of the integrals and hence of dn cont (L)/dL are different, as expected. If we instead plot the values of R 12 × h ∞ (z 1 , 0, R 12 ) for z 1 = L + 0.1σ (red dotted lines in the figure) we obtain better agreement. Thus the presence of the upper surface makes h(z 1 , 0, R 12 ) "compressed" in the z direction by about 0.1σ compared to h ∞ (z 1 , 0, R 12 ). This compression gives rise to the phase shift observed in Fig. 9. There are also some other small differences between h and h ∞ and, in addition, there are different prefactors in Eqs. (12) and (13). This gives the remaining differences in n cont (L) and n sp cont (L) seen in Fig. 9(b).
The nontrivial confinement effects are accordingly due to rather delicate changes in the pair distribution func- tion g(z 1 , z 2 , R 12 ) due to the presence of a second solid surface. The packing of particles in the slit around each individual particle is described by the pair density n(z 1 )g(z 1 , z 2 , R 12 ) and the changes in ng can be large, even for small variations in g, in regions where the density profile n is large. Conversely, since there are large variations in the density profiles with surface separation, the packing is strongly altered even when the change in g is small.
IV. SUMMARY AND CONCLUSIONS
The self-consistent calculation of density profiles and anisotropic pair distribution functions, as provided by integral equation theories at the pair correlation level (like the APY theory used in this paper), gives efficient tools for the investigation of the structure of inhomogeneous fluids in confinement. This is exemplified in this paper by a detailed examination of the mechanism behind the packing frustration for a dense hard-sphere fluid confined between planar hard walls at short separations.
When the width of the slit between the walls is close to an integer multiple of sphere diameters, the layer structure is optimal and the number density profile n(z) between the walls has sharp peaks. For slit widths near half-integer multiples of sphere diameters ([m + 0.5]σ with m = integer), the layer structure is much weaker and the packing frustration is large. The density profile shows considerable intricacy when the slit width is varied around these latter values. For example, when the reduced slit width L is increased from 1.0σ, there appear secondary density peaks close to the main peaks at each wall. These secondary peaks merge into a single peak at the slit center when L approaches 2.0σ. The mechanism behind these and other structural changes have been investigated in this paper, using the tools provided by the anisotropic pair distribution function theory.
The number density profile is determined by the mean force F (z) on the particles in the slit via the relationship d ln n(z)/dz = βF (z). For the hard-sphere fluid the mean force, which acts on a particle located at z, originates from collisions by other particles at the surface of the former. The average collisional force on the sphere periphery is proportional to the contact density there, which varies around the surface since the fluid is inhomogeneous. The sum of the average collisional forces constitutes the mean force F and since we have access to the pair distribution, and thereby the contact density at the sphere surface, we can investigate the origin of any variations in F and thereby in n. Of particular interest here are the variations when the slit width is changed.
By introducing the two principal components F ↑ and F ↓ of F , each of which is the sum of the average collisional forces on the particle hemisphere facing one of the walls, we extract sufficient information from the pair distributions to obtain a lucid description of the causes for the structural changes due to varying degree of packing frustration. We show that most features of the structural changes, including the appearance and merging of the secondary peaks mentioned above, can be explained by a simple parallel displacement of the F ↑ and F ↓ curves when the slit width is varied around half-integer σ values. The underlying reasons for this simple behavior is revealed via a detailed investigation of the pair distribution, that gives information about how the contact densities around the sphere periphery varies for different positions z of a particle in the slit.
It is found that the components F ↑ and F ↓ , and thereby the ordering of the fluid, are essentially governed by the packing conditions at each single solid-fluid interface. The fluid in the slit thereby conforms locally with only one of the confining surfaces at a time. In some local regions it will conform to one surface and in other regions to the other surface -regions that are constantly changing (the calculated distributions are averages of the various possible structures). This picture holds for all slit widths, irrespective of whether L is close to an integer or a half-integer multiple of the particle diameter.
As a consequence of these local packing conditions, the force components F ↑ and F ↓ , and thereby the total mean force F = F ↑ − F ↓ acting on a particle in the slit, can to a surprisingly good approximation be written as a superposition of contributions due to the presence of each individual solid-fluid interface at the walls. When the slit width is varied, this superposition can be expressed in terms of a parallel displacement of force curves due to either surface.
There are, however, some important properties of the inhomogeneous fluid that cannot be described by a simple superposition, but are instead determined by nontrivial confinement effects. In this paper, we exemplify such quantities by the number of particles per unit area in the slit N , the excess adsorption Γ, the contact density of the fluid at the wall surfaces n(0), and the net interaction pressure between the walls Π. In the superposition approximation, N and Γ disagree to a large extent compared to the accurate values, while n(0) and Π are mainly off by a phase shift in their oscillations. The analysis show that these nontrivial confinement effects are due to rather delicate changes in the anisotropic pair distribution function g(z 1 , z 2 , R 12 ) when the wall separation is changed.
In the total F sp ↑ there is a further contribution. From Eq. (7) we see that the total βF sp is equal to the derivative of ln n sp (z; L) = ln n ∞ (z) + ln n ∞ (L − z) − ln n b . While the last term gives zero for βF sp , i.e., the mean force in bulk is zero, this is not the case for βF sp ↑ . The mean force on one half of the sphere surface in bulk, F b ↑ , is non-zero; it is only the sum of the forces on both halves that are zero. Thus we have
F sp ↑ (z 2 ; L) = F L ↑ (z 2 ) + F U ↑ (z 2 ; L) − F b ↑ (A3) with βF b ↑ = πσ 2 n b g cont b , where g cont b
is the contact value for the pair distribution in bulk. When L → ∞, the presence of the last term makes F sp ↑ go to the single surface force F L ↑ , as it should in this limit.
FIG. 1 .
1For a schematic representation of the confinement geom-Schematic of hard spheres between planar hard walls.
FIG. 2 .
2Number density profiles n(z) for the hard-sphere fluid confined between hard planar surfaces. (a) Data for the average volume fraction φav = 0.40 of particles in the slit of width H = 2.4σ (reduced slit width L = 1.4σ), which is virtually at phase separation to the crystalline state for this surface separation. The solid line depicts theoretical data within the Anisotropic Percus-Yevick (APY) approximation, while the crosses show data from the Grand-Canonical Monte Carlo simulation of Ref. 9. (b) Theoretical data from APY approximation for a confined fluid in equilibrium with a bulk reservoir of number density n b = 0.75σ −3 . The reduced slit widths are L = 1.05σ (dotted line), 1.40σ (solid line), and 1.60σ (dashed line). The average volume fraction φav is here 0.35, 0.34, and 0.33, respectively.
defined). The role of the factor (z 2 − z 1 )/σ is to project out the z component of the contact force. (This line of reasoning is readily extended to systems exhibiting soft interaction potentials, such as Lennard-Jones fluids or electrolytes; in such cases, however, one also needs to include the interactions with the walls and all other particles in the system, see e.g. Refs.16 and 26.)
FIG. 4 .
4Net forces acting on a particle for the systems inFig. 3. The force contributions acting in positive (F ↑ , full curve) and negative (F ↓ , dashed curve) z directions are presented separately as functions of particle position z2 across the confining slit. Data are shown for reduced slit widths L = 1.50σ, 1.65σ, and 1.80σ. The values of the forces for a particle at the z2 coordinates inFig. 3are shown by filled circles. The dashed vertical line denotes the slit center, while the solid vertical line on the right-hand side indicates the upper limit for possible z2 coordinates of the particle in the slit. For comparison of all three cases, F ↑ is also shown for L = 1.50σ (blue dots) and 1.65σ (black dots) in the bottom panel.
FIG. 5 .
5As Fig. 4, but for reduced slit widths L = 3.45σ, 3.60σ, and 3.75σ.
FIG. 6 .
6Number density profiles n(z) for the confined hardsphere fluid. The reduced slit widths are L = 1.65σ (offset vertically by 3.0σ −3 ), 2.60σ (offset by 2.0σ −3 ), and 3.60σ (offset by 1.0σ −3 ). The systems are otherwise the same as inFig. 2(b). The solid and dotted lines depict results based on the full APY theory and the superposition approximation, respectively. The density profile at a single solid-fluid interface (L = ∞) is also shown for comparison.
Since the density profile is given by n(z) = n b exp[−βw(z)] the superposition approximation implies n(z; L) ≈ n sp (z; L) = n ∞ (z)n ∞ (L − z) n b ,
FIG. 7 .
7Principal mean force component F ↑ for the hardsphere fluid between planar hard surfaces. The reduced separations are L = 3.00σ, 3.25σ, 3.50σ, 3.75σ, and 4.00σ. The systems are otherwise the same as inFig. 2(b). (a) F ↑ for the confined fluids (black lines) and for a single solid-fluid interface (L = ∞, red line). (b) F ↑ for the confined fluids (each offset vertically by m = 0...4 units for clarity), obtained via the full APY theory [solid lines, same as in (a)] and the superposition approximation (dotted lines). (c) Force components of the superposition approximation, F L ↑ and ∆F U ↑ , for different reduced slit widths (the latter curves are vertically offset by m for clarity). F L ↑ is the same as the red curve in (a).
FIG. 8 .
8A sketch illustrating the force contributions F L ↑ and ∆F U ↑ in the superposition approximation. Each arrow represents a force that acts on the entire red half of the sphere (the location of the arrow has no significance in this sketch). The lower wall is shown in black and the upper wall is to be placed on the location indicated by the striped rectangle. The dashed line that connects each arrow to the respective surface indicates from which wall the influence originates.
FIG. 9 .
9(a) Excess adsorption Γ and (b) contact density ncont = n(0) of hard spheres between two hard planar surfaces as functions of reduced surface separation. The system is in equilibrium with a bulk fluid of density n b = 0.75σ −3 [same as inFig. 2(b)]. Data are shown for both the full APY theory (solid lines) and the superposition approximation (dashed lines).
For the exact case, the corresponding equation can be obtained from Eq. (5), which yields dn cont (L) dL = [n cont (L)] 2 h(z 1 , 0, R 12 )dR 12 z1=L .
FIG. 10 .
10Contour plot of the total pair correlation function h(z1, 0, R12) at coordinate (R12, z1) around a particle located on the z axis at coordinate 0, i.e., in contact with the bottom surface. Data are shown for different reduced slit widths: (a) L = 1.25σ, (b) 2.50σ, (c) 3.75σ, and (d) the single solid-fluid interface (L = ∞). The systems are otherwise the same as in Fig. 2(b). A small interval around h = 0 is shown as gray in the contour scale and the black areas denote the core region where h = −1. The red horizontal lines on the left hand side in (d) show the z coordinate for spheres in contact with the top surface in subplots (a)−(c), i.e., at coordinate z1 = L (cf. the red arrow in each of these subplots).
FIG. 11 .
11R12 × h(L, 0, R12) as function of R12 for the systems inFig. 10(a)-(c) with reduced surface separations L = 1.25σ, L = 2.50σ, and L = 3.75σ. The data are obtained via full APY theory (solid line), superposition approximation (blue dotted line), and shifted superposition approximation (L → L + 0.1σ, red dotted line). In the latter two cases, R12 × h∞(z1, 0, R12) is plotted for the appropriate z1 values (see text). Note the different scales on the y axis in the subplots. The curves go to zero at R12 = 0 because of the factor R12.
ACKNOWLEDGMENTSWe thank Tom Truskett for providing the simulation data inFig. 2Appendix A: Force subdivision in superposition approximationFor a hard sphere fluid in the slit between two hard walls, the force on, for example, the lower hemisphere of a hard sphere, F ↑ , can in the superposition approximation be divided into contributions due to either wall surface. The contribution F L ↑ from the lower surface is given by [cf. Eq.(6)]is the contact value of the pair distribution for the fluid outside a single surface. Likewise, the contribution F U ↑ from the upper surface is given by
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[
"ON THE EXISTENCE OF HOMOGENEOUS GEODESICS IN HOMOGENEOUS KROPINA SPACES",
"ON THE EXISTENCE OF HOMOGENEOUS GEODESICS IN HOMOGENEOUS KROPINA SPACES"
] |
[
"M Hosseini ",
"H R Salimi Moghaddam "
] |
[] |
[] |
Recently, it is shown that each regular homogeneous Finsler space M admits at least one homogeneous geodesic through any point o ∈ M . The purpose of this article is to study the existence of homogeneous geodesics on singular homogeneous (α, β)-spaces, specially, homogeneous Kropina spaces. We show that any homogeneous Kropina space admits at least one homogeneous geodesic through any point. It is shown that, under some conditions, the same result is true for any (α, β)-homogeneous space. Also, in the case of homogeneous Kropina space of Douglas type, a necessary and sufficient condition for a vector to be a geodesic vector is given. Finally, as an example, homogeneous geodesics of 3-dimensional nonunimodular real Lie groups equipped with a left invariant Randers metric of Douglas type are investigated.
|
10.1007/s41980-019-00269-5
|
[
"https://arxiv.org/pdf/1710.02407v3.pdf"
] | 119,331,044 |
1710.02407
|
5bd14f0bcad49803838420f8a1c1fa9a8b50467c
|
ON THE EXISTENCE OF HOMOGENEOUS GEODESICS IN HOMOGENEOUS KROPINA SPACES
15 Jan 2019
M Hosseini
H R Salimi Moghaddam
ON THE EXISTENCE OF HOMOGENEOUS GEODESICS IN HOMOGENEOUS KROPINA SPACES
15 Jan 2019
Recently, it is shown that each regular homogeneous Finsler space M admits at least one homogeneous geodesic through any point o ∈ M . The purpose of this article is to study the existence of homogeneous geodesics on singular homogeneous (α, β)-spaces, specially, homogeneous Kropina spaces. We show that any homogeneous Kropina space admits at least one homogeneous geodesic through any point. It is shown that, under some conditions, the same result is true for any (α, β)-homogeneous space. Also, in the case of homogeneous Kropina space of Douglas type, a necessary and sufficient condition for a vector to be a geodesic vector is given. Finally, as an example, homogeneous geodesics of 3-dimensional nonunimodular real Lie groups equipped with a left invariant Randers metric of Douglas type are investigated.
Introduction
γ(t) = φ(t, γ(0)), t ∈ R.
The problem of the existence of homogeneous geodesics on homogeneous manifolds is an interesting and relatively old problem in differential geometry (see [12] and the references therein).
In [15], in the case of Lie groups equipped with left invariant metrics, Kajzer showed that there exists at least one homogeneous geodesic passing the identity element. Szenthe, in [29], proved that if the Lie group is also compact, connected, semi-simple and of rank greater than one, then there are infinitely many homogeneous geodesics through the identity. One year later, in 2001 (see [30]), he generalized Kajzer's result for left invariant Lagrangian on compact connected Lie groups. Also in the case of compact connected semi-simple Lie groups of rank greater than one, he proved that there are infinitely many homogeneous geodesics for left invariant Lagrangian. In [17], Kowalski and Szenthe extended Kajzer's result to homogeneous Riemannian manifolds. They also proved that a homogeneous Riemannian manifold M = G/H with semi-simple group G admits m = dim(M ) mutually orthogonal homogeneous geodesics. On the other hand, homogeneous geodesics have found many applications in mechanics. They are usually called relative equilibria or stationary geodesic in physics literature (for more details see [2,6,7,20,24,25,31]). For example in the cases of Lagrangian and Hamiltonian systems, Tóth studied trajectories which are orbits of a one-parameter symmetry group G. He obtained criteria under which an orbit of a one-parameter subgroup of a symmetry group is a solution of the Euler-Lagrange or Hamiltonian equations (see [31]). As another example, see [20], where Lacomba has used homogeneous geodesics in the framework of Smale's mechanical systems with symmetry. Recently, homogeneous geodesics on homogeneous Finsler spaces have been studied by many researchers. In the case of homogeneous Finsler space, a criterion for a nonzero vector to be a geodesic vector, was given by Latifi in [21]. Yan and Deng [33] investigated the existence of homogeneous geodesics on Randers spaces. They proved that every homogeneous Randers space admits at least a homogeneous geodesic through any point. In 2017, Yan extended this result to any homogeneous Finsler space of odd dimension (see [32]
Preliminaries
This section contains some preliminaries about Finsler spaces. As Chern mentioned in the title of his article [4], "Finsler geometry is just Riemannian geometry without the quadratic restriction". In fact, in 1854, Riemann has introduced a metric as a positive (when the vector y is not equal to zero) function on the tangent bundle which is homogeneous of degree one in y. Finsler worked on this general case in his doctoral thesis (see [13]). But Finsler's name established in differential geometry with the book [3] written by Cartan. Consider a differentiable manifold M and denote its tangent bundle by T M . A Finsler metric on M is a smooth function F :
T M \ {0} −→ R >0 such that i) F (x, λy) = λF (x, y), for all λ > 0,
ii) The hessian matrix (g ij ) = 1 2 ∂ 2 F 2 ∂y i ∂y j be positive definite for all (x, y) ∈ T M \{0}. An important family of Finsler metrics is the class of (α, β)-metrics which arises from Riemannian metrics and 1-forms (see [22]). More precisely, (α, β)-metrics can be expressed in the form F = αφ( β α ), where φ : (−b 0 , b 0 ) −→ R + is a C ∞ function and α(x, y) = a ij y i y j and β(x, y) = b i y i , and a and β are a Riemannian metric and a 1-form respectively, as follows
a = a ij dx i ⊗ dx j , (2.1) β = b i dx i . (2.2)
It can be shown that, F is a regular Finsler metric if φ satisfying
(2.3) φ(s) − sφ ′ (s) + (b 2 − s 2 )φ ′′ (s) > 0, |s| ≤ b < b 0 , and β x α = a ij (x)b i (x)b j (x) < b 0 for any x ∈ M ,
where (a ij (x)) is the inverse matrix of (a ij (x)) (for more details see [5]).
In the above definition, if φ does not satisfy the condition 2.3 or φ(0) is not defined, then the (α, β)-metric is called a singular finsler metric. For instance, if φ(s) = 1 + s or φ(s) = 1 s , then we obtain two important classes of Finsler metrics, called Randers metrics F = α + β (which are regular) and Kropina metrics F = α 2 β (which are singular), respectively. Easily we can see the Riemannian metric a induces an inner product on any cotangent space T * x M . This inner product induces a linear isomorphism between T * x M and T x M (see [10]). Using this isomorphism, the 1-form β corresponds to a vector fieldX on M such that (2.4) a(y,X(x)) = β(x, y).
Now we can rewrite the Randers and Kropina metrics as follows
(2.5) F (x, y) = a (y, y) + a X (x) , y , (2.6) F (x, y) = a (y, y) a X (x) , y .
Recently, Kropina metrics are known as a singular solution of the Zermelo's navigation problem on some Riemannian manifold (M, h) under the influence of a vector field W with W h = 1 . The pair (h, W ) is called the navigation data of Kropina metric. In fact, there is one to one correspondence between Kropina metric F and the navigation data (h, W ) (see [36]). In the present paper, we focus on a class of Kropina metrics which are obtained from the initial Zermelo's navigation problem in terms of a Riemannian metric h and a unit vector field W . It is well known that any homogeneous Riemannian manifold is a reductive homogeneous space (see [16,17]). The same proposition is true for any homogeneous Finsler space (see [21]). Now we study the relation between the isometry group of Kropina metrics and the isometry group the Riemannian metric h. As in the Riemannian homogeneous spaces the definition of homogeneous geodesic can be extended to homogeneous Finsler spaces as follows:
Definition 2.3. Let (M = G/H, F ) be a homogeneous Finsler space. A non-zero vector X ∈ g is said to be a geodesic vector if the curve γ(t) = exp tX.o is a geodesic of (M = G/H, F ).
Consider a Riemannian homogeneous space (M = G/H, a ) with a reductive decomposition g = m + h. In [18], Kowalski and Vanhecke proved that X ∈ g \ {0} is a geodesic vector if and only if
(2.7) a([X, Y ] m , X m ) = 0, ∀Y ∈ m,
where the subscript m indicates the projection into the subspace m. This proposition is generalized to homogeneous Finsler spaces by Latifi as follows (see [21]):
Theorem 2.4. A nonzero X ∈ g is a geodesic vector if and only if (2.8) g Xm ([X, Y ] m , X m ) = 0, ∀Y ∈ g,
where g denotes the fundamental tensor of F on m.
As a consequence he proved the following corollary.
Corollary 2.5. A nonzero X ∈ g is a geodesic vector if and only if
(2.9) g Xm ([X, Y ] m , X m ) = 0, ∀Y ∈ m.
Notation 2.6. For simplicity, from now on we use the notation , for the Riemannian metric a.
Homogeneous geodesic in homogeneous Kropina spaces
As already mentioned in the introduction, Kowalsky and Szenthe showed that any homogeneous Riemannian space admits at least one homogeneous geodesic on each origin point (see [17]). Yan and S. Deng generalized this result to homogeneous Randers spaces in [33]. In this section we show that the same result is true for homogeneous Kropina spaces. Firstly, we give a necessary and sufficient condition for a nonzero vector in a homogeneous Kropina spaces to be a geodesic vector.
[Y, Z] m , 2 F (Y m ) Y m − X = 0 (3.1)
holds for every Z ∈ m.
Proof. A nonzero vector Y ∈ g is a geodesic vector if and only if
g Ym ([Y, Z] m , Y m ) = 0 ∀Z ∈ m.
Using the formula (2.5) in [26] we have
g Ym ([Y, Z] m , Y m ) = 1 X, Y m 4 {(2 Y m , [Y, Z] m Y m , X − [Y, Z] m , X Y m , Y m ) (2 Y m , Y m Y m , X − Y m , X Y m , Y m ) + Y m , Y m Y m , X (2 [Y, Z] m , Y m Y m , X + 2 Y m , Y m [Y, Z] m , X − 2 Y m , X , Y m , [Y, Z] m ) − 2 Y m , Y m [Y, Z] m , X (2 Y m , Y m Y m , X − Y m , X Y m , Y m )} = Y m , Y m X, Y m 3 [Y, Z] m , 2 Y m , X Y m − Y m , Y m X = F 3 (Y m ) Y m , Y m [Y, Z] m , 2 F (Y m ) Y m − X . For Y m = 0 the inequality F 3 (Ym) Ym,Ym 2 = 0 implies that for any Z ∈ m, g Ym ([Y, Z] m , Y m ) = 0 if and only if [Y, Z] m , 2 F (Y m ) Y m − X = 0.
A direct consequence of the above proposition is the following corollary. Proof. According to the theorem 3.
[Y, Z] m , 2 F (Y m ) Y m − X = [Y, Z] m , 2 F (Y m ) Y m .
The above equation completes the proof.
In [33], it is shown that on any homogeneous Randers space (G/H, F ), there is at least one homogeneous geodesic issuing from each origin point. In a similar way we show that the same result is true for any homogeneous Kropina space. Proof. Let G be a connected Lie group of isometries acting transitively on M , and H be the isotropy group of a point o ∈ M . Suppose that K and radK denote the Killing form and it's null space, respectively. We recall that the null space radK is a solvable ideal of g, so it is included in the radical of g denoted by r (see [14]). Let m = h ⊥ . Then, since K is nondegenerate on h, g = h + m is a reductive decomposition and radK ⊆ m (see [17]). Assume that F is defined by a G-invariant Riemannian metric , and a vector fieldX such that X =X(H). Also, we denote the corresponding scalar product on m and the Ad(H)-invariant vector on m by , and X, respectively. There are two cases:
Case 1 (radK = m): If radK = m then there is a solvable Lie group L which acts transitively on M . Using the same argument as in the case (A) of the proof of proposition 3 in [17] shows that there exists a reductive decomposition g = h + m such that the m-projection [g, g] m is a proper subspace of m. Let Y belongs to [g, g] ⊥ m , with respect to , , such that Y, Y = 1. Thus, W = 1 2 ( X, X Y + X) is a geodesic
vector. An easy computation shows that F (W ) = 1 and so for any Z ∈ m we have
[W, Z] m , 2 F (W m ) W m − X = [W, Z] m , X, X Y + X − X = 0.
Therefore there exists one homogeneous geodesic through o. Case 2 (radK m): If radK is a proper subspace of m, the same argument as in the case (2) of the proof of theorem 2.9 in [33] shows that m has an eigenspace decomposition
m = V 0 +V 1 +...+V s with respect to a K-symmetric endomorphism θ : m −→ m defined by K(X, Y ) = θ(X), Y such that V 0 = radK. Let {f 1 , f 2 , ..., f r } be a , -orthonormal basis of V = V 1 + V 2 + ... + V s such that θ(f i ) = λ i f i for i = 1, 2, ..., r. Suppose that X = X 0 + Σ r i=1 x i f i and Y = Y 0 + Σ r i=1 y i f i , where X 0 , Y 0 ∈ V 0 and x i , y i ∈ R.
By proposition 3.1, Y is a geodesic vector of (M, F ) if and only if the following equation equals zero,
[Y, Z] m , 2 F (Y m ) Y m − X = [Y, Z] m , 2 F (Y m ) (Y 0 + Σ r i=1 y i f i ) − X 0 − Σ r i=1 x i f i = [Y, Z] m , 2 F (Y m ) Y 0 − X 0 + [Y, Z] m , 2 F (Y m ) Σ r i=1 y i f i − Σ r i=1 x i f i = [Y, Z] m , 2 F (Y m ) Y 0 − X 0 + [Y, Z] m , 2 F (Y m ) Σ r i=1 y i θ(f i ) λ i − Σ r i=1 x i θ(f i ) λ i = [Y, Z] m , 2 F (Y m ) Y 0 − X 0 + K [Y, Z] m , 2 F (Y m ) Σ r i=1 y i f i λ i − Σ r i=1 x i f i λ i = [Y, Z] m , 2 F (Y m ) Y 0 − X 0 + K [Y, Z], 2 F (Y m ) Σ r i=1 y i f i λ i − Σ r i=1 x i f i λ i = [Y, Z] m , 2 F (Y m ) Y 0 − X 0 + K Z, 2 F (Y m ) Σ r i=1 y i f i λ i − Σ r i=1 x i f i λ i , Y = [Y, Z] m , 2 F (Y m ) Y 0 − X 0 + K Z, 2 F (Y m ) Σ r i=1 y i f i λ i − Σ r i=1 x i f i λ i , Σ r i=1 y i f i .
The above equation equals zero if the following equations have a solution of the form (Y 0 , y 1 , y 2 , · · · , y r , t):
(3.2) F (Y ) = 2, Y 0 = X 0 , y i − x i λ i = ty i .
In the case X = X 0 , we easily obtain the following solution:
Y 0 = X 0 , y 1 = X 0 , X 0 , t = 1 λ 1 , y i = 0, i = 2, ..., r.
In the case X = X 0 , without loss of generality, we can assume that |λ 1 | ≥ |λ 2 | ≥ · · · ≥ |λ r | > 0.
Let Y 0 = X 0 , y i (t) = x i 1 − tλ i , i = 1, 2, · · · , r, Y (t) = X 0 + r i=1 y i (t)f i .
We can easily see that M (t) = F (Y (t)) − 2 is a continues function on ( −1 |λ 1 | , 1 |λ 1 | ) and M (0) < 0 and lim t→ 1 λ 1 M (t) = +∞. On the other hand for any t ∈ ( −1 |λ 1 | , 1 |λ 1 | ) we have F (Y (t)) > 0. So the intermediate value theorem states that there exists t = t 0 such that M (t 0 ) = 0 and F (Y (t 0 )) > 0. We observe that Y 0 = X 0 , y i = y i (t 0 ) and t = t 0 is a solution and it completes the proof.
Homogeneous Geodesics in Homogeneous (α, β)-Spaces
In this section we investigate homogeneous geodesics on homogeneous Finsler spaces equipped with (α, β)-metrics and obtain interesting results. Proof. X is a geodesic vector of the Riemannian manifold (M, , ). So, by theorem 2.2 of [28], X is a geodesic vector of (M, F ). Hence at least one homogeneous geodesic through o exists.
The next proposition develops the main result of previous section to (α, β)-metrics with some conditions. We use some ideas from [17] in the proof. Proof. The main idea is similar to that used in the proof of proposition 3 of [17]. Assume that G/H is an arbitrary representation of M , where G is a transitive Lie group of isometries. If we put o = H then there are two cases:
(1) There exists i ≥ 0 such that T e π(g (i) ) = T o M and T e π(g (i+1) ) T o M , (2) For every i ≥ 0, T e π(g (i) ) = T o M .
In the case (1), by attention to the part (A) of the proof of proposition 3 in [17], there exists a geodesic vector Y = 0 for the Riemannian manifold (M, , ). Now, theorem 2.3 of [28] shows that Y is a geodesic vector of (M, F ). Thus, there exists at least one homogeneous geodesic through o. In the case (2), by attention to the part (B) of the proof of proposition 3 in [17], the Lie group G is semi-simple.
Next, we discuss the existence of homogeneous geodesic through o on (M = G/H, F ) where G is a semi-simple group of isometries and F is an invariant (α, β)-metric induced by an invariant Riemannian metric , . For this, we consider the Killing form of g, denoted by K. We observe that K is a nondegenerate form on g and moreover, its restriction to h is nondegenerate. If g = m + h is the reductive decomposition such that m is the orthogonal complement of h with respect to K, then K is nondegenerate on m, too. Let , be the corresponding scalar product on m induced by the Riemannian metric , . Now we define an isomorphism θ of m onto m as follows:
(4.1) θ(X), Y = K(X, Y ), X, Y ∈ m.
It is easy to show that the corresponding matrix of θ and the corresponding matrix of K are equal with respect to a , -orthonormal basis of m. So m admits an , -orthonormal basis (f 1 , · · · , f m ) of eigenvectors (for more details see [17] ). Proof. Likewise the proof of theorem 1 in [17], we can show that each eigenvector f i is a geodesic vector of (M, , ). So according to the theorem 2.3 in [28], it is a geodesic vector of (M, F ) and this completes the proof. Then the nonzero elements of ρ are geodesic vectors of (M, F ).
Proof. According to the theorem 2 in [17], all nonzero element of ρ are geodesic vectors with respect to , . So, using the theorem 2.3 in [28], show that these elements are geodesic vectors of (M, F ).
(4.2) b(X, [Z, Y ] m ) + b([Z, X] m , Y ) = 0 X, Y, Z ∈ m,
where b is the induced inner product on m by a and the subscript m means the corresponding projection.
In the literature, there are two different definitions of naturally reductive Finsler manifolds (see [9] and [21]). One of them which is defined by Deng and Hou in [9], is as follows Proof. By theorem 2 in [17], all element belonging to m are geodesic vectors of (M, , ). Hence with the assumptions and notations of theorem 2.3 in [28], these are geodesic vectors of (M, F ) and so by theorem 3.1 in [11] (M, F ) is naturally reductive.
In the end we discuss concerning homogeneous geodesic vectors on 3-dimensional nonunimodular Lie groups equipped with invariant Randers metric of Douglas type. Suppose that G is a 3-dimensional non-unimodular Lie group with Lie algebra g equipped with a left invariant Riemannian metric , . Then, there exists an orthonormal basis {e 1 , e 2 , e 3 } of g such that the bracket is expressed as where α, β, γ, δ are real numbers such that α + δ = 0 and αγ + βδ = 0. This basis also diagonalizes the Ricci form (for more details see [23]).
Denote D = (β + γ) 2 − 4αδ such that all Ricci eigenvalues are distinct. Then, up to a reparametrization, the space (G, , ) admits just one or just two or just three homogeneous geodesic through a point if D < 0 or D = 0 or D > 0, respectively (see [19], proposition 3.2).
Here we generalize this result as follow Proposition 4.8. Let (G, F ) be a 3-dimensional non-unimodular Lie group with a Randers metric of Douglas type induced by a left invariant Reimannian metric , and a left invariant vector fieldX. If D is as above then, up to a reparametrization, (G, F ) admits just one or just two or just three homogeneous geodesics through a point for D < 0 or D = 0 or D > 0, respectively. For the case D = 0 they are mutually orthogonal and for D > 0, they are linearly independent but never mutually orthogonal.
Proof. Using the corollary 2.7 in [33], U is a geodesic vector of (G, F ) if and only if it is a geodesic vector of (G, , ). So we can use the same argument as in the proof of proposition 3.2 of [19] and prove this result.
A geodesic γ : R −→ M of a Finslerian manifold (M, F ) is called a homogeneous geodesic if there exists a one-parameter group of isometries φ : R × M −→ M such that
Homogeneous Finsler spaces are defined similar to the Riemannian case. A homogeneous Finsler space is a Finslerian manifold (M, F ) on which its isometry group I(M, F ) acts transitively on M . So, a connected homogeneous Finsler space M can be considered of the form M = G/H where G is a connected Lie group of isometries of M , acting transitively on M , and H is the isotropy subgroup of a point in M . A homogeneous space M = G/H is said to be reductive if there exists an Ad(H)-invariant decomposition g = m + h, where g and h denote the Lie algebras of G and H, respectively and + is the direct sum of subspaces.
Proposition 3. 1 .
1Suppose that (G/H, F ) is a homogeneous Finsler space and F is a Kropina metric arising from an invariant Riemannian metric , and an invariant vector fieldX such that X =X(H). Then, a nonzero vector Y ∈ g is a geodesic vector if and only if
Corollary 3. 2 .
2Consider the assumption of the previous proposition. Then the vector X is a geodesic vector of (G/H, , ) if and only if it is a geodesic vector of (G/H, F ).
Corollary 3. 3 .
3Suppose that (G/H, F ) is a homogeneous Kropina space as proposition 3.1 such that F is of Douglas type. Then a nonzero vector Y ∈ g is a geodesic vector of (G/H, F ) if and only if it is a geodesic vector of (G/H, , ).
Proposition 3. 4 .
4Suppose that (M, F ) is a homogeneous manifold with a Kropina metric F . Then there is at least one homogeneous geodesic issuing from each origin point.
Proposition 4 . 1 .
41Suppose that (M = G/H, F ) is a homogeneous Finslerian manifold and F is an invariant (α, β)-metric defined by an invariant Riemannian metric , and an invariant vector fieldX on M such that X =X(H) is a geodesic vector with respect to the Riemannian metric , . Then (M, F ) admits a homogeneous geodesic through any point.
Proposition 4 . 2 .
42Let (M = G/H, F ) be a homogeneous Finslerian manifold. Suppose that F is an invariant (α, β)-metric, arisen from an invariant Riemannian metric , and an invariant vector fieldX on M . If for any Y ∈ g \ {0} and Z ∈ m the equality X, [Y, Z] m = 0 holds and moreover ϕ ′′ (r m ) ≤ 0, where r m = X,Ym √ Ym,Ym , then either at least one homogeneous geodesic through any o ∈ M exists or M = G/H such that G is a semi-simple group.
Proposition 4. 3 .
3Suppose that (M = G/H, F ) is a homogeneous Finslerian manifold, dimM = m and G is a semi-simple group of isometries. Let F be a (α, β)-metric which is defined by an invariant Riemannian metric , and an invariant vector fieldX on M . If for any Y ∈ g \ {0} and Z ∈ m the equality X, [Y, Z] m = 0 holds and moreover ϕ ′′ (r m ) ≤ 0, where r m = X,Ym √ Ym,Ym, then m mutually orthogonal homogeneous geodesics issuing from any origin exist.
Proposition 4. 4 .
4Let (M, F ) be as in the previous proposition and ρ ⊆ g satisfies the following conditions a) Ad(H)(ρ) ⊆ ρ, b) ρ is irreducible with respect to the restriction of adjoint representation to H, c) K is nondegenerate on ρ, and ρ is K-orthogonal to h.
Definition 4. 5 .
5A homogeneous Riemannian manifold G/H with an invariant Riemannian metric a is called a naturally reductive homogeneous space if there exists a reductive decomposition g = m + h such that
Definition 4 . 6 .
46Let (M = G/H, F ) be a homogeneous Finsler space with an invariant Finsler metric. We call (M = G/H, F ) is a naturally reductive homogeneous space if there exists an invariant Riemannian metric a which (M = G/H, a ) is naturally reductive and the connections of F and a coincide.
Corollary 4. 7 .
7Suppose that (M = G/H, F ) is a homogeneous Finslerian manifold such that G is a semi-simple group of isometries and F is a (α, β)-metric arisen from an invariant Riemannian metric , and an invariant vector fieldX on M . Let for any Y ∈ g \ {0} and Z ∈ m the equality X, [Y, Z] m = 0 holds and moreover ϕ ′′ (r m ) ≤ 0, where r m = X,Ym √ Ym,Ym . If (M, , ) is an isotropy irreducible homogeneous manifold, then (M, F ) is naturally reductive.
[e 1
1, e 2 ] = αe 2 + βe 3 , [e 1 , e 3 ] = γe 2 + δe 3 , [e 2 , e 3 ] = 0,
specially, homogeneous Kropina spaces. It is shown that, there exists at least one homogeneous geodesic through any point of an arbitrary homogeneous Kropina space. We prove that, under some conditions, the same result is true for any (α, β)homogeneous space. In the case of homogeneous Kropina space of Douglas type, we show that a nonzero vector is a geodesic vector of the Kropina metric if and only if it is a geodesic vector of it's base Riemannian metric. Homogeneous geodesics of 3-dimensional non-unimodular real Lie groups equipped with left invariant Randers metrics of Douglas type are investigated at the end of the paper.).
Recently, in [34], Yan and Huang proved that any regular homogeneous Finsler spaces admits
at least one homogeneous geodesic through each point. They have used Legendre transforma-
tion, which is a bijection for regular Finsler metrics, to find nonzero geodesic vectors.
The family of (α, β)-metrics is an interesting class of Finsler metrics which have many applica-
tions in physics. In this paper we study the problem of existence of homogeneous geodesics on
singular homogeneous (α, β)-spaces,
the Randers metricF which arises from the Riemannian metric h and the vector field X on M . We claim that F andF have the same isometry group. It suffices to show that φ ∈ I(M, F ) if and only if φ ∈ I(M,F ). Let φ ∈ I(M, F ), then according to lemma 4 of[35], φ is an isometry of h which preserves W and so X is φ-invariant. It follows from proposition 3.2 of[27], φ belongs to the isometry group of (M,F ). Conversely, if φ ∈ I(M,F ), then using proposition 7.1 of[8] and proposition 3.2 of[27], φ ∈ I(M, h) and W is φ-invariant. Again, with lemma 4 of[35], φ ∈ I(M, F ). This finally implies by Proposition 7.1 of[8] that I(M, F ) is a closed subgroup of the isometry group of the Riemannian manifold (M, h).Remark 2.2. According to the previous proposition, I(M, F ) = I(M,F ). This implies that every such a homogeneous Kropina space is reductive.Proposition 2.1. Suppose that (M, F ) is a Kropina space which arises from a navigation
data (h, W ). Then the isometry group of (M, F ) is a closed subgroup of the isometry group of
Riemannian manifold (M, h).
Proof. If (h, W ) is the navigation data of F , then W h = 1. We put X =
1
2
W and consider
2 of [1], F is of Douglas type if and only if X is orthogonal to [m, m] m . Therefore we have
Acknowledgment.We are grateful to the office of Graduate Studies of the University of Isfahan for their support.
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X Zhang, Y-B Shen, -Iran. E-mail address: hoseini [email protected] Department of Mathematics, Faculty of Sciences. Isfahan31Department of Mathematics, Faculty of Sciences, University of Isfahan, Isfahan ; University of IsfahanOn Einstein-Kropina metricsX. Zhang and Y-B Shen, On Einstein-Kropina metrics, Differential Geom. Appl., 31(2013) 80-92. Department of Mathematics, Faculty of Sciences, University of Isfahan, Isfahan, 81746-73441- Iran. E-mail address: hoseini [email protected] Department of Mathematics, Faculty of Sciences, University of Isfahan, Isfahan, 81746-73441- Iran. E-mail address: [email protected] and [email protected]
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[
"ON THE LAST QUESTION OF STEFAN BANACH",
"ON THE LAST QUESTION OF STEFAN BANACH"
] |
[
"Pasha Zusmanovich "
] |
[] |
[] |
We discuss the last question of Banach, posed by him shortly before his death, about extension of a ternary operation to superposition of a binary one. We try to put things into the context of Polish mathematics of that time.
|
10.1016/j.exmath.2015.12.005
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[
"https://arxiv.org/pdf/1408.2982v6.pdf"
] | 118,911,582 |
1408.2982
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9e132dc29404194f78eb82f3f9ebd2d8a791512b
|
ON THE LAST QUESTION OF STEFAN BANACH
14 Oct 2014
Pasha Zusmanovich
ON THE LAST QUESTION OF STEFAN BANACH
14 Oct 2014arXiv:1408.2982v2 [math.HO]
We discuss the last question of Banach, posed by him shortly before his death, about extension of a ternary operation to superposition of a binary one. We try to put things into the context of Polish mathematics of that time.
Introduction
At the end of 1944, shortly before his death at August 1945, Stefan Banach regularly met with Andrzej Alexiewicz, then a fresh PhD from the (underground at that time) Lvov University. During these meetings, they discussed a lot of mathematics, and Alexiewicz kept a diary whose mathematical part, including questions posed by Banach, is available in [A]. The last entry in this diary, dated December 29, 1944, reads: "There exists a nontrivial example of ternary multiplication, which is not generated by binary multiplication (Banach). Can any finite set with ternary commutative multiplication be extended so that ternary multiplication is generated by a binary multiplication?"
In what follows, the claim and the question from this passage will be referred as "Banach's claim" and "Banach's question". This seems to be the last question of Banach which appeared in the "literature" (broadly interpreted). It is interesting that among all other questions posed by Banach -in his papers, in the Scottish Book [Sc], in other diary entries of [A], in the problem sections of Fundamenta Mathematicae and Colloquium Mathematicum (the latter being entered posthumously) -this is the only one which does not belong to the field of analysis. (The only possible exception is Question 47 from [Sc] about permutations of infinite matrices, which, incidentally, also deals with (im)possibility of building a certain class of maps from "simpler" ones). On the other hand, that Banach was interested in such sort of questions -belonging, somewhat vaguely, to a crossroad of universal algebra, discrete mathematics, logic † , and combinatorics -is, perhaps, not accidental at all, as such a crossroad, along the functional analysis, the main Banach's occupation, was another "Polish speciality" at that time (and long thereafter).
It is the purpose of this note to discuss (and answer) the possible interpretations of this question. For the historical context, in particular, for the last years of Banach in Lvov during two Russian and one German occupations, for the general atmosphere of mathematical Lvov, and for a unique interaction of mathematics and logic in inter-war Poland, we refer the reader to [Ba3], to introductory chapters of [Sc], and to [Mu], [Wo1], and [Wo2], respectively.
A 1955 paper by Loś, Hilbert's 13th problem, and functional completeness
There is at least another mentioning of a variant of Banach's question in the literature, namely by Jerzy Loś, a student at the Lvov University during 1937-1939, in [ L2, §16]. There, also with reference to Alexiewicz, Loś writes:
Date: last revised September 30, 2014. arXiv:1408.2982. † According to Hugo Steinhaus (cf. [St]), Banach "did not relish any logic research although he understood it perfectly". On the other hand, in [Mu,p. 40] several instances of Banach's involvement into contemporary logical activity are given. Anyhow, "logic" is a vast field, and, as we try to argue below, there are certain connections between Banach's question and some logical investigations cultivated in Poland (especially Warsaw) between the two world wars.
"During the last war Banach showed that not every ternary semigroup is reducible and he put forward the problem whether every ternary semigroup may be extended to a reducible one". Here by a ternary semigroup one means a set X with a ternary map
(1) f : X × X × X → X,
subject to a ternary variant of associativity:
f (f (x, y, z), u, v) = f (x, f (y, z, u), v) = f (x, y, f (z, u, v))
for any x, y, z, u, v ∈ X, and by a reducible ternary semigroup one means a ternary semigroup (X, f ) with multiplication given by
(2) f (x, y, z) = (x * y) * z,
where (X, * ) is an ordinary (i.e., binary) semigroup structure on the same underlying set X. Loś answers the question in affirmative as a consequence of his general results about extensibility of firstorder logical models. Later an alternative and more constructive proof was given in [MS,Theorems 1 and 2]. The latter paper contains also examples of ternary (in fact, n-ary for any positive integer n) semigroups not representable in the form (2), thus validating this variant of Banach's claim. However, this variant of Banach's question is more narrow in scope than those presented in [A]. Can the latter question be interpreted in a different way? In the absence of additional qualifications, the most general reading is the following: "multiplication" on a set X means an arbitrary ternary map (1), without associativity, or any other, for that matter, constraint (by abuse of terminology, sometimes we will call a ternary map the pair (X, f ) rather than just f ). But what does "generation" mean?
Virtually all Banach's works are devoted to (proper) analysis, with a few exceptions of set theory (including the famous Banach-Tarski paradox), computable analysis (a constructive approach to analysis, developed in a joint unpublished work with Stanis law Mazur), and the only short paper [Ba1], which touches, seemingly, a similar question: in this paper Banach gives a shorter proof of an earlier result of Wac law Sierpiński [Si2] to the effect that any countable number of unary maps on an infinite set can be generated, with respect to superposition, by just two maps. This statement is, essentially, about 2-generation of a countable transformation semigroup, and was apparently rediscovered in the literature several times afterwards. It admits various generalizations to other semigroups and groups, some of them, especially in topological setting (where "generation" is understood up to the closure), were pursued by Sierpiński and two Banach's students, Stanis law Ulam (PhD from the Lvov Polytechnic, 1933), and Józef Schreier (PhD from the Lvov University, 1934; cf. [GP] for an interesting discussion, and the bestseller [U2, p. 82] for Ulam's account how his joint work with Schreier secured him a place in the Harvard Society of Fellows at 1936).
Yet the statement about unary maps proved by Sierpiński and Banach admits generalizations in another direction -to the multiary maps. (Note the drastic difference between superposition of unary maps, which reduces to a mere composition and hence is associative, and the general case of superposition of multiary maps for which associativity is even not well defined). This may suggest that "generation" in Banach's question can be interpreted as an arbitrary superposition of the maps. In this general setting, however, Banach's claim becomes false. Indeed, any map of arbitrary (finite) arity on any set -in fact, any countable set of such maps -can be generated (with respect to superposition) by one binary operation -a multiary analog of Sierpiński's result cited above. This was proved several times in the literature: for the first time, in [We1] for the case of finite sets (the required binary operation is a multiary generalization of the Sheffer stroke). In the case of infinite sets, a different, and much shorter † proof was presented, again, by Loś [ L1] (a similar, and yet simpler, proof was rediscovered more than half century later in [G]). These proofs, in its turn, are based on † A quote from [E]: "Now it frequently happens in problems of this sort that the infinite dimensional case is easier to settle than the finite dimensional analogues. This moved Ulam and me to paraphrase a well known maxim of the American armed forces in WWII: 'The difficult we do immediately, the impossible takes a little longer', viz: 'The infinite we do immediately, the finite takes a little longer' ".
an another result of Sierpiński [Si3] to the effect that any map is generated by (possibly, several) binary maps.
The latter paper of Sierpiński appeared around the same time as Banach's question under discussion: the same 1945 issue of Fundamenta Mathematicae in which it was published, the first one after the 6-year break occurred during WWII, contains an announcement about Banach's death. The same paper contains also another elementary, but interesting for us result: for any binary bijection g : X × X → X on a (necessary infinite) set X, and for any ternary map (1) on X, there is another binary map h : X × X → X such that
(3) f (x, y, z) = g(h(x, y), z)
for any x, y, z ∈ X (the statement readily generalizes to n-ary maps f ). The result was, however, not new at that time: it appeared as the solution to the problem 119, imaginatively entitled "Are there actually functions of 3 variables?", of Part 2 in the first 1925 edition of the famous book [PS].
Sierpiński's interest in this topic stems, evidently, from Hilbert's 13th problem -see, for example, his earlier paper [Si1] where the statement (3) is proved for the case where X is the set of real numbers (albeit without using axiom of choice which is necessary in the general case), with reference to the (in)famously erroneous paper by Bieberbach about Hilbert's 13th problem. Ulam, along with Mark Kac (PhD from the Lvov University, 1937, with Banach as a member of examining committee), also an active participant of the Lvov mathematical scene until the end of 1930s † (cf. [Sc], [U2] and [Fe]), has at least a cursory interest in Hilbert's 13th problem as well. In the collection of Ulam's problems (dedicated to Schreier's memory) [U1, Chapter IV, Problem 2 and Chapter VI, Problem 5], which he positions as a sort of successor to the Scottish Book, as well in the joint Kac's and Ulam's popular book [KU,p. 163], after noting a remarkable result of Kolmogorov and Arnold (all continuous functions of any number of real variables can be represented as superposition of continuous functions of at most 2 variables), they ask about various extensions of the problem, for example: whether a bijective continuous (smooth, analytic, etc.) function on an n-dimensional real space can be represented as a superposition of functions from the respective class (continuous, smooth, analytic, etc.) in a smaller number of variables? In [U1], Ulam acknowledges Banach, among others, for "the pleasure of past collaboration", but it is unclear whether this interest in the circle of questions related to Hilbert's 13th problem goes back to the Lvov years, and if yes, whether it has something to do with Banach's question.
The questions whether that or another set of maps generates all maps within a given class, framed in terms of functional completeness of various multivalued propositional calculi ("multivalued logics", or "logistics", as it was called then) and the corresponding truth tables, was also popular among Polish logicians at that time. In particular, Jan Lukasiewicz was concerned about functional completeness of his famous 3-valued logic -that is, "implication", the binary map L on the 3-element set {0, 1 2 , 1} given by "multiplication table" 0 1 2 1 0 1 1 1 1 2 1 2 1 1 1 0 1 2 1 together with "negation", the unary map N defined by 0 → 1, 1 2 → 1 2 , 1 → 0, and similar systems. (While Lukasiewicz put emphasis on the philosophical significance of manyvalued logics -cf., e.g., [Mu,] -most of the questions related to them, including question of functional completeness, naturally have a pure formal, i.e. mathematical, character). His student Jerzy S lupecki proved in his PhD thesis (cf. [S l]) functional incompleteness of the set { L, N} (what † Among the two, Ulam was more actively engaged in the Lvov mathematical life, and not only because he was a few years Kac's senior, but by a more prosaic reason. Speaks Kac (cf. [Fe]): "I was less of a habitue of the Scottish Café... I was financially somewhat less affluent than Stan -I was ... independently poor. And it did cost a little to visit in the Café".
amounts to the fact that these two maps do not generate all possible multiary maps on the set of 3 elements), and, in the positive direction, established functional completeness of the set { L, N, T }, where T is the unary map sending all 3 elements to 1 2 . (It is interesting to note that the same 1936-1937 issue of Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie in which S lupecki's paper was published, contains the paper [We2] of Donald L. Webb, a fresh PhD from Caltech and the author of the already mentioned [We1], in which he extends the results of the latter paper. This, among other things, suggests that developments in this branch of logical calculi on both sides of Atlantic did not proceed in isolation at that time † ). Earlier, another Lukasiewicz student, Mordchaj Wajsberg has reported at the Lukasiewicz-Tarski seminar ‡ some related results -see a brief description at the very end ("Anmerkung") of [Wa] (cf. also [Su, §ix]). At the same paper, more binary maps generating all maps of arbitrary arity on a finite set are presented without proof. Some of the Wajsberg's results apparently go back as early as 1927, i.e. almost a decade before Webb.
More logical calculi, both complete and incomplete, were developed around this time and thereafter, first of all, by Emil Leon Post (born in Poland, otherwise not related to that country § ), as well as by Lukasiewicz, S lupecki, Boles law Sobociński (another PhD student of Lukasiewicz), Eustachẏ Zyliński (professor of the Lvov University at 1919-1941), Zygmunt Zawirski, and others -cf., e.g., [BB,] for a survey. Post's purely formal treatment of many-valued logics ("a combinatorial scheme" in the words of [Mo,p. 3]) is, perhaps, closer in spirit to mathematical questions considered here than the philosophical attitude of Lukasiewicz.
Another feature of the work of Lukasiewicz's school and Polish logicians in general, was the constant quest for minimal, as far as possible, systems of axioms for that or another logical calculus. Minimality was understood both in terms of the number of axioms and their length (cf., e.g., [Wo1, and [Su, §v]) ¶ . Banach's claim and question resonate well with this line of thought, as they can be phrased as follows: if a certain 3-term operation in an n-valued logic is not generated by a 2-term one (alas, as we have seen, this is wrong), the next best thing to ask is to extend the 3-term operation so that it will be generated that way.
Of course, many further results about Hilbert's 13th problem, from one side, and functional completeness and incompleteness of various logical calculi, from the other, were obtained afterwards (cf., e.g., the survey [V] for the former, and the book [L] for the latter), but their discussion will bring us far away from our main topic. † The personal contacts started, probably, with the visit of Willard Van Orman Quine (then at Harvard) to Warsaw at 1932. Ernest Nagel (then at Columbia) made public (cf. [Nag]) his interesting impressions of visiting Poland in 1935, but, written from the philosopher's rather than logician's standpoint ("the logical researches both at Warsaw and Lwów are extraordinary specialized and technical"), these impressions, probably, contributed little to interchange of logical and mathematical ideas between the countries. Ulam was moving back and forth between US (Princeton and then Harvard) and Lvov in 1935-1939, but his interests, at least at that time, were outside logic. Webb's thesis advisor, Eric Temple Bell, an enthusiastic, albeit not always precise, writer of popular mathematical books, praised Lukasiewicz's 3-valued logic as one of the four major contributions "on the nature of truth" during the last 6000 years (cf. [Be,), so it is, perhaps, not accidental that Webb, a young Californian, published in a relatively obscure Warsaw journal.
‡ "Professor Lukasiewicz's seminar at Warsaw was crowded with competent young men, incomparably better equipped in logic than students of like age in America, who were expected to write as seminar exercises papers which elsewhere would be thought important enough for publication", reports Nagel.
Clones of analytic and ordered maps, counting superpositions, examples to Banach's claim
The "generation" in the results above is understood in the sense of the theory of clones, i.e. when forming superposition of maps, the repeated variables (and, hence, superpositions of arbitrary length) are allowed. Thus, Banach's claim can not be interpreted in terms of the clone of all maps on the underlying set X, as all the clone, including its ternary fragment, is generated by its binary fragment. It should be noted that if we assume that X possess some additional -topological, analytic, order, etc. -structure, and consider not the clone of all maps, but the clone of maps on X preserving this structure -then the statement above about generation of all the maps by binary ones is no longer true. Some sporadic examples of various degree of sophistication:
(i) Not every real analytic function in 3 variables can be represented as a superposition of analytic functions in two variables (cf. [V, §3]) -a statement made already by Hilbert in his original formulation of the 13th problem, whose proof utilizes some counting similar to elementary counting in the case of X finite, and also in the case of polynomials over a finite field in Example 2, see below. (ii) Superpositions of smooth functions in 2 variables satisfy various differential equations, not satisfied by all smooth functions in 3 variables (cf. solution of Problem 119a, Part 2 of [PS] for the relevant calculations). (iii) Algebraic functions in sufficiently high number of variables can not be represented as superposition of algebraic functions in sufficiently low (in particular, 2) number of variables -a suite of deep results due to V.I. Arnold and his followers, obtained by interpreting cohomology classes of a suitable braid group as obstructions to such representation (cf. [Nap] for references). (iv) There is an 8-element poset whose clone of monotone maps cannot be generated not only by its binary fragment, but by any finite set of maps (due to G. Tardos, cf. [L, §11.5]). However, as Banach is apparently interested in the case of X finite, and does not impose on X any additional structure, such interpretation seems to be unlikely.
If, however, we will understand the superposition in a more "operadic-like", "multilinear" fashion, where each variable occurs only once, the only possible ways to generate the ternary map (1) by a binary one * : X × X → X, are:
(4L) f (x, y, z) = (x * y) * z,
i.e. the same as in the version of the question from the Loś paper [ L2] discussed above, and
(4R) f (x, y, z) = x * (y * z).
This is also in line with Sierpiński's result (3). Of course, any map representable in the form (4L) gives rise, via permutation of arguments, to a map representable in the form (4R), and vice versa. Indeed, the equality (4R) can be rewritten as
(5) f (13) (x, y, z) = (x * (12) y) * (12) z, where f σ (x 1 , . . . , x n ) = f (x σ(1) , . . . , x σ(n) ),
for a permutation σ belonging to S n , the symmetric group in n variables. Banach is concerned with the case of "commutative" maps, which, by all accounts, are n-ary maps f which are stable under any σ ∈ S n : f = f σ (usually, such maps are called symmetric). As it follows from (5), a commutative map f is representable in the form (4L) if and only if it is representable in the form (4R). In this sense, Banach's claim becomes true. In the case of a finite set X, this is obvious from an elementary counting: the number of binary maps is |X| |X| 2 , so the number of ternary maps of the form (4L) and (4R) is less than 2|X| |X| 2 (less, because these maps may coincide for different * 's), while the number of all ternary maps is |X| |X| 3 .
As an aside note, it seems to be an interesting question to estimate more exactly the number of different maps of the form (4L) and (4R) on an n-element set. A computer count produces the following table, where the second column contains the number of all binary maps on an n-element set, T L (n) denotes the number of ternary maps of the form (4L), T LR (n) denotes the number of ternary maps both of the form (4L) and (4R), and T comm (n) denotes the number of commutative ternary maps of the form (4L) (what coincides with the number of such maps both of the form (4L) and (4R)) † (so, for n > 1 we have the obvious inequalities T comm (n) < T L (n) < T LR (n) < 2T L (n)): n n n 2 T L (n) T LR (n) T comm (n) We do not know the general formulas for T L (n), T LR (n), and T comm (n). It is easy to manufacture examples of a ternary map f which cannot be generated, in the sense of (4L), by any binary operation * , thus explicitly confirming Banach's claim (in examples below, the ternary maps are commutative, but they are not representable as superposition, in the sense of (4L), of any binary map, commutative or not).
Example 1. Let X be a set containing more than 2 elements, and a, b ∈ X be two distinct elements of X. Define a ternary map f on X by f (x, y, z) = a, if all x, y, z are distinct from a b, if at least one of x, y, z coincides with a.
Suppose (4L) holds for some binary map * on X. If for any two elements x, y ∈ X, both distinct from a, x * y = a, then for any 3 elements x, y, z ∈ X, each distinct from a, we have a = f (x, y, z) = (x * y) * z = a, a contradiction. Hence there are u, v ∈ X, both distinct from a, such that u * v = a. Then a * a = (u * v) * a = f (u, v, a) = b, and then for any x ∈ X, b * x = (a * a) * x = f (a, a, x) = b, and, finally,
a = f (b, u, v) = (b * u) * v = b * v = b, a contradiction.
Example 2. Let X be a set of q = p n elements, where p is a prime, and f a ternary map on X. Endow X with the structure of the finite field GF(q). The question is whether there exists or not a binary map g on GF(q) such that (6) f (x, y, z) = g(g(x, y), z)
for any x, y, z ∈ GF(q). Since, due to Lagrange interpolation formula, each k-ary function on GF(q) can be represented as a polynomial in k variables with coefficients in GF(q), and the degree < q in each variable (cf., e.g., [LN,, and, moreover, two such polynomial maps are equal if and only if their coefficients are equal, the condition (6) can be rewritten as a system of quadratic equations in polynomial coefficients of g. Suitably choosing f , one can obtain a system not having solutions in GF(q) (in fact, "most" of the f 's will do, as the system consists of q 3 equations in q 2 unknowns). For example, defining a ternary map f : GF(2) × GF(2) × GF(2) → GF(2) by f (x, y, z) = xy + xz + yz, and writing g(x, y) = a + bx + cy + dxy for some a, b, c, d ∈ GF(2), we arrive at the system a + ab = 0, b 2 = 0, bc = 0, c + ad = 0, bd = 1, cd = 1, d 2 = 0 which, evidently, does not have solutions.
Of course, nothing in these examples if specific to 3 variables, and they can be easily extended to n-ary maps for arbitrary n, and, moreover, to non-representability in the form (x * y) • z for two binary maps * and •, and similar n-ary expressions.
Answer to Banach's question
The following answers the question of Banach, interpreted as above -i.e. about extensions of arbitrary ternary maps to those having the form (4L) -in affirmative. (We deal with arbitrary, not necessary commutative, maps). The elementary idea behind the answer is based on various, related, constructions of envelopes of Lie and other triple systems (cf., e.g., [Fi] and references therein), and goes back to the pioneering paper [J] of Nathan Jacobson (born in Warsaw, but otherwise not related to Poland), published only a few years later than the question was posed.
We will say that a ternary map f : X × X × X → X on a set X admits a binary extension (Y, * ), where Y is a set, and * : Y × Y → Y is a binary map on it, if X ⊆ Y , (X * X) * X ⊆ X, and the restriction to X × X × X of a ternary map Y × Y × Y → Y defined by (x, y, z) → (x * y) * z for x, y, z ∈ Y , coincides with f . Theorem 1. Any ternary map (X, f ) admits a binary extension (Y, * ). Moreover, if X is finite, then Y can be chosen to be finite.
Proof. Without loss of generality, we may assume that X contains a "neutral element" e with the property f (e, x, y) = f (x, e, y) = f (x, y, e) = e
for any x, y ∈ X. Indeed, we can always extend X and f by adjoining element with such property. Define Y as the Cartesian product X × M, where M is the set consisting of maps X → X of the form m x,y : z → f (x, y, z)
for all x, y ∈ X. Note that for any x ∈ X, m e,x = m x,e = m e,e is the map sending everything to e. Define a binary map * :
Y × Y → Y by (x, g) * (y, h) = (g(y)
, m x,y ) for x, y ∈ X, g, h ∈ M. Identify X with a subset of Y via the bijection x ↔ (x, m e,e ). For any x, y, z ∈ X we have The statement about finiteness of Y is obvious.
Varying the construction employed in this proof, one may obtain similar statements in classes of maps satisfying various conditions. For example, while a binary extension constructed above is, generally, not associative, even if the initial ternary map is, its slight modification allows to provide an alternative proof of a positive answer to Banach's question in the narrower -"associative"version, given in the papers [ L2] and [MS] as mentioned at the beginning of §1. Our proof differs from both of them and, as we hope, is shorter and simpler.
Theorem 2 ( Loś, Monk-Sioson). Any commutative ternary semigroup admits a binary extension which is a commutative semigroup.
Proof. Let (X, f ) be a commutative ternary semigroup. Let M be the set of maps m x,y for all x, y ∈ X, as in the proof of Theorem 1, and M the subsemigroup of all maps X → X generated by M with respect to composition. Note that commutativity and associativity of f imply m x,y • m u,v = m u,v • m x,y for any x, y, u, v ∈ X, so M is a commutative semigroup.
Let K be an arbitrary ring (one may take, for example, K = GF(2) to construct, within this framework, an extension as minimal as possible), and Y a free K-module generated by X and M . Define a binary map * on the free generators of Y as follows:
x * y = m x,y x * g = g * x = g(x) g * h = g • h for x, y ∈ X, g, h ∈ M (• denotes the composition of maps), and extend * on the whole Y by linearity. As f is commutative, m x,y = m y,x , and hence * is commutative.
Obviously, X ⊂ Y . For any x, y, z ∈ X, we have (x * y) * z = m x,y (z) = f (x, y, z), so (Y, * ) is a binary extension of (X, f ). It remains to check the associativity of * . For 3 terms which all belong to X, the associativity of * follows from the commutativity of f . Similarly, for 3 terms which all belong to M , the associativity of * follows from the associativity of •. If two terms, say, g and h, belong to M , and one, say x, belongs to X, the associativity of * follows from the commutativity of • in M :
(g * h) * x = (g • h) * x = g(h(x)) = g * (h(x)) = g * (h * x), (g * x) * h = g(x) * h = h(g(x)) = g(h(x)) = g * (h(x)) = g * (x * h).
In the remaining cases, where one term, g = m u,v (u, v ∈ X), belongs to M , and two terms, x and y, belong to X, assuming additionally z ∈ X, and utilizing commutativity and associativity of f , we have:
((x * y) * g)(z) = (m x,y * g)(z) = (m x,y • g)(z) = f (x, y, g(z)) = f (x, y, f (u, v, z)) = f (x, f (u, v, y), z) = f (x, g(y), z) = m x,g(y) (z) = (x * g(y))(z) = (x * (y * g))(z) and ((x * g) * y)(z) = (g(x) * y)(z) = m g(x),y (z) = f (g(x), y, z) = f (f (u, v, x), y, z) = f (x, f (u, v, y), z) as above = (x * (g * y))(z), what completes the proof.
One can deal similarly with not necessary commutative ternary semigroups (one needs then to consider, instead of m x,y , both "left" and "right" multiplications), with ternary groups (thus recovering a part of Post's results [Pos]), etc.
Some of the statements of this section may be suitably extended, via a straightforward iterative procedure, to the maps f of arbitrary arity, but we will not venture into this.
§ A quote from[Wo1]: "When Tarski met Emil Post for the first time(in 1939 or 1940) he told him: 'You are the only logician who achieved something important in propositional calculus without having anything to do with Poland'. Post answered: 'Oh, no, I was born in Bia lystok' ". ¶ This fascination with minimal systems of logical axioms was not shared by everyone in Poland. An anti-utopian novel "Nienasycenie" by Stanis law Ignacy Witkiewicz (a close friend of Leon Chwistek, professor of logic at the Lvov University, as well as of Alfred Tarski), written in 1927 and depicting conquest of Poland by enemy forces and establishment there of a totalitarian regime by the end of XX century, features a grotesque figure of logician Afanasol Benz (a Jew, stresses Witkiewicz) who invented a single axiom that nobody but him could understand, and from which all mathematics follows by a mere formal combination of symbols (cf.[Wi, p. 93]).
(x, m e,e ) * (y, m e,e ) = (e, m x,y ) and, consequently, (x, m e,e ) * (y, m e,e ) * (z, m e,e ) = (f (x, y, z), m e,e ), as desired.
† A simple Perl program which computes these numbers for small values of n, is available as http://justpasha.org/math/binary-ternary.pl
AcknowledgementsThanks are due to Nadezda Bazunova, Martin Goldstern, Kateryna Pavlyk, and Jan Woleński for useful remarks and pointers to the literature. The financial support of FAPESP (grant 13/12050-2) and Estonian Science Foundation (grant ETF9038) is gratefully acknowledged.
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M Wajsberg, Metalogische Beiträge, Ossolineum, Wroc law. ed. S.J. Surma43English translation in [Pol. reprinted in Logical WorksM. Wajsberg, Metalogische Beiträge, Wiadomości Matematyczne 43 (1937), 131-168; English translation in [Pol], 285-318; reprinted in Logical Works (ed. S.J. Surma), Ossolineum, Wroc law, 1977, 172-200.
Generation of any n-valued logic by one binary operation. D L Webb, Proc. Nat. Acad. Sci. USA. 21D.L. Webb, Generation of any n-valued logic by one binary operation, Proc. Nat. Acad. Sci. USA 21 (1935), 252-254.
The algebra of n-valued logic. Compt. Rend. Séances Soc. Sci. Lett. Varsovie, Cl. III. 29, The algebra of n-valued logic, Compt. Rend. Séances Soc. Sci. Lett. Varsovie, Cl. III 29 (1936-1937), 153-168.
S I Witkiewicz, Dom Nienasycenie, Akcyjna Ksiażki Polskiej Spó Lka, English translation: Insatiability. WarszawaUniv. Illinois PressS.I. Witkiewicz, Nienasycenie, Dom Ksiażki Polskiej Spó lka Akcyjna, Warszawa, 1930 (in Polish); English translation: Insatiability, Univ. Illinois Press, 1977.
Mathematical logic in Poland 1900-1939: people, circles, institutions, ideas. J Woleński, History of Logic and Logical Philosophy. 5Jagiellonian Univ. PressModern LogicJ. Woleński, Mathematical logic in Poland 1900-1939: people, circles, institutions, ideas, Modern Logic 5 (1995), 363-405; reprinted in Essays in the History of Logic and Logical Philosophy, Jagiellonian Univ. Press, Kraków, 1999.
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[] |
[
"On the classification of consistent boundary conditions for f (R)-Gravity",
"On the classification of consistent boundary conditions for f (R)-Gravity"
] |
[
"H Khodabakhshi \nDepartment of Physics\nUniversity of Tehran\nTehranIran\n",
"F Shojai \nDepartment of Physics\nUniversity of Tehran\nTehranIran\n\nSchool of Physics\nFoundations of Physics Group\nInstitute for Research in Fundamental Sciences (IPM)\nTehranIran\n",
"A Shirzad \nDepartment of Physics\nIsfahan University of Technology\nIsfahanIran\n\nSchool of Particles and Accelerators\nInstitute for Research in Fundamental Sciences (IPM)\nTehranIran\n"
] |
[
"Department of Physics\nUniversity of Tehran\nTehranIran",
"Department of Physics\nUniversity of Tehran\nTehranIran",
"School of Physics\nFoundations of Physics Group\nInstitute for Research in Fundamental Sciences (IPM)\nTehranIran",
"Department of Physics\nIsfahan University of Technology\nIsfahanIran",
"School of Particles and Accelerators\nInstitute for Research in Fundamental Sciences (IPM)\nTehranIran"
] |
[] |
Using a completely covariant approach, we discuss the role of boundary conditions (BCs) and the corresponding Gibbons-Hawking-York (GHY) terms in f (R)-gravity in arbitrary dimensions. We show that f (R)-gravity, as a higher derivative theory, is not described by a degenerate Lagrangian, in its original form. Hence, without introducing additional variables, one can not obtain consistent BCs, even by adding the GHY terms (except for f (R) = R).However, following the Ostrogradsky approach, we can introduce a scalar field in the framework of Brans-Dicke formalism to the system to have consistent BCs by considering appropriate GHY terms. In addition to the Dirichlet BC, the GHY terms for both Neumann and two types of mixed BCs are derived. We show the remarkable result that the f (R)-gravity is itself compatible with one type of mixed BCs, in D dimension, i.e. it doesn't require any GHY term. For each BC, we rewrite the GHY term in terms of Arnowit-Deser-Misner (ADM) variables.
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10.1140/epjc/s10052-018-6494-5
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[
"https://arxiv.org/pdf/1803.04306v3.pdf"
] | 118,918,098 |
1803.04306
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8d84d66951a0cea5ccbea80674a7d4f4a78043e8
|
On the classification of consistent boundary conditions for f (R)-Gravity
2 Oct 2018
H Khodabakhshi
Department of Physics
University of Tehran
TehranIran
F Shojai
Department of Physics
University of Tehran
TehranIran
School of Physics
Foundations of Physics Group
Institute for Research in Fundamental Sciences (IPM)
TehranIran
A Shirzad
Department of Physics
Isfahan University of Technology
IsfahanIran
School of Particles and Accelerators
Institute for Research in Fundamental Sciences (IPM)
TehranIran
On the classification of consistent boundary conditions for f (R)-Gravity
2 Oct 2018
Using a completely covariant approach, we discuss the role of boundary conditions (BCs) and the corresponding Gibbons-Hawking-York (GHY) terms in f (R)-gravity in arbitrary dimensions. We show that f (R)-gravity, as a higher derivative theory, is not described by a degenerate Lagrangian, in its original form. Hence, without introducing additional variables, one can not obtain consistent BCs, even by adding the GHY terms (except for f (R) = R).However, following the Ostrogradsky approach, we can introduce a scalar field in the framework of Brans-Dicke formalism to the system to have consistent BCs by considering appropriate GHY terms. In addition to the Dirichlet BC, the GHY terms for both Neumann and two types of mixed BCs are derived. We show the remarkable result that the f (R)-gravity is itself compatible with one type of mixed BCs, in D dimension, i.e. it doesn't require any GHY term. For each BC, we rewrite the GHY term in terms of Arnowit-Deser-Misner (ADM) variables.
Introduction
Since the theory of general relativity (GR) is a classical field theory of gravitation, the choice of BCs is of great importance. The role of surface integrals in GR has been investigated first in Dewitt and Dirac's papers [1,2] and then was covered deeply in the works of York, Regge and Teitelboim [3,4].
In Ref. [4], the authors indicated the crucial role of the surface integral in order to have a well-defined functional space of the gravitational field. Three years later, trying to quantize GR in path integral formalism, Gibbons and Hawking [5] showed that, a boundary term should be added to the Einstein-Hilbert (EH) action, in order to have a well-defined variational principle for an open manifold with Dirichlet BC, i.e. δg ab | Boundary = 0. Such terms, added to the EH action, or the action of generalized theories of gravity [6,7,8,9,10,11], are called GHY terms.
The Lagrangian of GR as well as f (R)-gravity contains second derivative of metric. Variational principle for these so called "Jerky mechanics" [12] is not well defined. In such actions, it is needed to apply Dirichlet and Neumann BCs simultaneously which may lead to destroy the Poisson structure of the system in a canonical treatment. However, care is needed to define momentum and go to a well defined phase space via an ordinary Legendre transformation [13,14].
GR is described by a degenerate Lagrangian, i.e. can be written as the sum of a quadratic part in the first derivatives of metric and a total derivative term. There are two approaches to deal with GR. The first one is the well-known ADM formalism which uses the Gauss-Codazzi equation to get rid of the second derivative terms of the Lagrangian [5,6,7,15]. The second one, which is more covariant, uses the holographic relation to manifest the quadratic Lagrangian by subtracting a suitable boundary term which can be removed by adding a GHY term [16,17].
For modified gravity models such as f (R)-gravity one needs to use the so-called Ostrogradsky approach by introducing enough number of fields to the theory such that the whole Lagrangian of the system includes at most the first derivatives of the fields. In this way one is able to go through a canonical approach and at the same time introduce consistent BCs. For f (R)-gravity without considering additional fields, one needs to consider the extrinsic curvature variation δK ij , as well as δg ij to vanish on the boundary, which is inconsistent since extrinsic curvature K ij , includes derivatives of the metric. However, by adding the famous GHY term −2 ∂M d D−1 yǫ √ hf ′ (R)δK (h and K are the trace of induced metric and the extrinsic curvature respectively, ǫ = ±1 depending on the timelike or spacelike nature of the boundary ∂M and f ′ (R) = ∂f ∂R ) to the action, the BCs reduce to vanish δR on the boundary simultaneously with the Dirichlet BC. But calculating δR (see appendix B) shows that the main problem is not resolved since R is not an independent field and its variation includes again variations of the derivatives of the metric.
In this paper we try in section 2 to investigate the more covariant ap-proach for f (R)-gravity. For this reason, we use the equivalent scalar-tensor formulation of f (R)-gravity and then using a suitable conformal transformation, we go to Einstein frame [18,19,20]. After imposing the holographic relation for Einsteinian curvature of space-time and changing back to the original variables [21], we obtain somehow a holographic-like relation for f (R)-gravity in which the bulk term is not quadratic. This shows that the f (R)-gravity by itself is a non-degenerate Lagrangian and the ordinary approach is not suitable for it. Then in section 3, we try to change f (R) Lagrangian into a degenerate one by Ostrogradsky approach. To do so, we write the f (R)-gravity in the Jordan frame of the Brans-Dicke action [18,19,20]. Then by using the holographic relation for the curvature of space-time, we find that the action of the theory is degenerate, though there is not a well defined holographic relation. Hence, by adding appropriate GHY terms, Dirichlet or other BCs can be achieved. Writing the boundary terms of the action in terms of fields and momentum fields, in a foliation independent approach, enables us to introduce the consistent GHY term for Dirichlet, Neumann and two types of mixed BCs in arbitrary dimensions. For one type of mixed BC, the GHY term vanishes. This may be interpreted that the f (R)-gravity is more consistent with this mixed type of BC in D dimension.
In this paper the Latin indices are used to show the space-time coordinates and the Greek ones are used to denote the space coordinates. The calculations are done in arbitrary dimensions of space-time and the signature of metric is (-,+,+,+).
2
Non-degeneracy of f (R)-Gravity
Similar to GR, f (R) Lagrangian includes second derivatives of the metric. The variational principle for this type of actions is not primarily well defined due to requirement of applying simultaneously Dirichlet and Neumann BCs. There is also no room in this type of Lagrangian to define the momentum and establish Hamiltonian structure via Legendre transformation. In dealing with these Lagrangians, there are two possibilities: 1) The Lagrangian is degenerate i.e. it can be written as a quadratic Lagrangian which is equivalent to the original one plus a total derivative term which can be removed by adding a GHY term and imposing Dirichlet, Neumann or mixed BCs.
2) The Lagrangian is non-degenerate in which Ostrogradsky method or other equivalent methods should be used [13,14]. In GR, Lagrangian is a degenerate Lagrangian, i.e. using the holographic relation, it can be written as
√ −gR = √ −gL quad (∂, ∂g) + 1 D/2 − 1 ∂ i −g ab ∂( √ −gL quad ) ∂(∂ i g ab ) ,(1)
where
L quad = 1 4 M abcdef ∂ a g bc ∂ d g ef M abcdef ≡ g ad g bc g ef − g be g cf + 2g af g be g cd − g bc g ed(2)
As can be seen, it can be written as a quadratic Lagrangian plus a total derivative term [16,17]. Unlike the GR Lagrangian, f (R)-gravity given by
S = d D x √ −gf (R),(3)
seems to be non-degenerate. In order to obtain the holographic relation in f (R)-gravity, first we try to write it in the GR form which we know how to work with it. To do so, we write the f (R) action using scalar-tensor theory as follows
S = d D x √ −g(φR − V (φ)),(4)in which φ = f ′ (R), V (φ) = R(φ)φ − f (R(φ)
) and we have assumed that f ′′ (R) = 0. This is, in fact, the action of Brans-Dicke theory in the Jordan frame with parameter ω = 0 [19,20,23,24]. As is well-known, using the conformal transformation [20]:
g ab = φ 2/(D−2) g ab dφ = 2(D − 1) (D − 2) dφ φ(5)
the action (4) changes to Einstein gravity minimally coupled to a scalar field. Thus, in the so-called Einstein frame, the separation of the Lagrangian into bulk and surface terms can be written as in Eq. (1). Then the obtained holographic relation can be restored into the Jordan frame by the inverse of transformation (5). To the end of this section, all quantities in the Einstein frame are denoted by ∼. Noting
R = φ −2/(D−2) R − 2(D − 1) (D − 2) φ φ + D − 1 D − 2 ∇ c φ∇ c φ φ 2(6)
and
√ −g = φ D/(D−2) √ −g, we can find S =S + 2(D − 1) D − 2 M d D x √ −g φ,(7)whereS = M d D x −g R − 1 2∇ aφ∇ aφ − U (φ) . (8) in which U (φ(φ)) = V (φ) φ D/(D−2)
. Now we can separate the action of f (R)gravity into a quadratic bulk term and a surface term. To do this, let us recall that
−gR = −gg ab Γ i jaΓ j ib −Γ i abΓ j ij + ∂ c [ −gṼ c ],(9)
whereṼ c =g ikΓc ik −g ckΓm mk [16]. Hence, the bulk term of (8) in the Einstein frame readsL
bulk =g ab Γ i jaΓ j ib −Γ i abΓ j ij − 1/2∇ aφ∇ aφ − U (φ).(10)
The second term of (9) is denoted asL Sur and leads to a surface term. Transforming back to the Jordan frame via Eq. (5), we obtain:
−gg ab Γ i jaΓ j ib −Γ i abΓ j ij = φ √ −gg ab Γ i ja Γ j ib − Γ i ab Γ j ij + φ √ −g Γ i ij ∂ i ln φ − g ab Γ i ab ∂ i ln φ + D − 1 D − 2 φ √ −g∂ i ln φ∂ i ln φ(11)
and
−gṼ c = φ √ −g g ik Γ c ik − g ck Γ m km − 2 D − 1 D − 2 φ √ −g∂ c ln φ.(12)
The above relations finally yield
√ −gL = √ −gL bulk + L sur ,(13)
where
L bulk = φg ab Γ i ja Γ j ib − Γ i ab Γ j ij + φ Γ i ij ∂ i ln φ − g ab Γ i ab ∂ i ln φ − V (φ)(14)
and [21,25]. Now let us find the holographic relation for f (R)-gravity similar to Eq.
L sur = ∂ c (φ √ −gV c ) (15) in which V c = g ik Γ c ik − g ck Γ i ik
(1) for GR. It is clear that the holographic relation is satisfied in the Einstein frame due to the minimal coupling of the scalar field to gravity. In order to write the holographic relation in the Jordan frame, let us start from Eqs. (14) and (15). A simple calculation shows
g ke ∂( √ −gL bulk ) ∂(∂ c g ke ) = − D − 2 2 φ √ −gV c + (D − 1)φ √ −g∂ c ln φ(16)
and
L sur = ∂ c (φ √ −gV c ) = − 2 D − 2 ∂ c g ke ∂( √ −gL bulk ) ∂(∂ c g ke ) − (D − 1) √ −g∂ c φ ,(17)
Inserting (16) and (17) into (13) and using φ = f ′ (R), finally one obtains
√ −gL = √ −gL bulk − 2 D − 2 ∂ c g ab ∂( √ −gL bulk ) ∂(∂ c g ab ) − (D − 1) √ −g∂ c f ′ (R) (18) where L bulk = f ′ (R)g ab Γ i ja Γ j ib − Γ i ab Γ j ij + f ′ (R) Γ j ij ∂ i ln f ′ (R) − g ab Γ i ab ∂ i ln f ′ (R) − (Rf ′ (R) − f (R)),(19)
Considering relation (18), we see that the surface part of the Lagrangian is not determined completely by its bulk part. Therefore, we called it "holographic-like" relation. This is in contrast to EH Lagrangian, or more generally Lanczos-Lovelock Lagrangians [21]. Furthermore, the bulk Lagrangian in f (R)-gravity is not necessarily a quadratic Lagrangian and contains an arbitrary function of the second order derivatives of metric. Hence, the f (R) Lagrangian is not a degenerate Lagrangian.
BC we have 1 δ M d D x √ −gf (R) = M d D x √ −gL ab δg ab + ∂M d D−1 y √ h − f ′ Π ij √ h + ǫ∇ a f ′ (h a j n i − n a h ij ) δh ij + ∂M d D−1 yǫf ′ δ(2K √ h)(20)
where
L ab ≡ − 1 2 f g ab + f ′ R ab − ∇ a ∇ b f ′ + g ab f ′ = 0 (21) is the equation of motion. Π ij = ǫ √ h(K ij − K h ij )
is the momentum conjugate to the h ij in GR and n i is the normal vector of the boundary. As can be seen, to obtain the equations of motion, imposing the Dirichlet BC which leads to δh ij | Boundary = 0, we can get rid of the first surface integral in the above equation. To remove the second surface integral, there are two possibilities: 1) substituting δK ij | Boundary = 0, 2) adding the usual GHY boundary term to the action as follows
S t = S + S GHY = M d D x √ −gf (R) − 2 ∂M d D−1 yǫ √ hf ′ K .(22)
The first choice implies simultaneously vanishing of the metric and its derivatives on the boundary which is inconsistent. To investigate the second choice let us vary the above action
δS t = M d D x √ −gL ab δg ab + ∂M d D−1 yǫ √ h f ′ Π ij √ h + ∇ a f ′ (h a j n i − n a h ij ) δh ij − 4 ∂M d D−1 y √ hK f ′ n i δn i − 2 ∂M d D−1 yǫ √ hK f ′′ δR.(23)
Hence, to get the equations of motion, we need to impose δR| Boundary = 0, in addition to δh ij | Boundary = δn i | Boundary = 0 which is Dirichlet BC because of g ab = h ab + ǫn a n b . It should be noted that although the normal to the boundary has a unit norm, this doesn't imply that the third term of (23) is zero. A simple calculation shows that
δn b = 1 2 ǫn b n i n j δg ij + h b i n j δg ij , thus n b δn b = 1 2 n i n j δg ij .
In appendix B we have shown that δR is a combination of variations δh ij , δK ij , δn i , ∇ i δK and δ(∇ a ∇ i n a ). Now we can ask if δR| Boundary = 0 is compatible with the Dirichlet BC?
To answer this question we need to define, in a consistent way, the momenta conjugate to the field variables in order to distinguish the Dirichlet and Neumann BCs where the momentum fields vanish on the boundary. However, this can be done only for degenerate theories, where the bulk term contains at most the first order derivatives of the fields. Noting Ostrogradsky approach, we should change the f (R) Lagrangian into a degenerate one as much as possible. To do so, using scalar-tensor formulation, by introducing an scalar field φ, we write f (R) as Lagrangian (4) which is not far form GR that is degenerate. Now substituting the holographic relation (1) in action (4), we have
S = M d D x √ −g (φL quad − V (φ))+ 1 D/2 − 1 M d D xφ∂ i −g ab ∂( √ −gL quad ) ∂(∂ i g ab )(24)
The first integral contains only the metric, the field φ and the first order derivatives of the metric. Integrating by parts, this is also the case for the second integral, and thus the above Lagrangian is degenerate. To see this, let us rewrite Eq. (24) as follows
S = M d D x √ −g (φL quad − V (φ)) + 1 D/2 − 1 M d D x∂ i φg ab M iab − 1 D/2 − 1 t d D−1 yφg ab P ab .(25)
where
M iab ≡ ∂( √ −gL quad ) ∂(∂ i g ab ) = √ −g 2 M iabpqr ∂ p g qr
and M iabpqr is defined in Eq. (2). Note that P ab ≡ ∂( √ −gL quad )/∂(∂ 0 g ab ) is
the canonical momentum of g ab in GR. Hereafter we have also assumed that ∂M contains two spacelike (D − 1)-dimensional surfaces at t = constant and one timelike surface on which the integral vanishes at large spatial distances. Now we are able to define the canonical momenta of φ and g ab as follows
P ab ≡ δS δ(∂ 0 g ab ) = φP ab + (D − 1) √ −g D − 2 (g i0 g ab − 2g ib g 0a )∂ i φ(26)andP φ ≡ δS δ(∂ 0 φ) = 1 D/2 − 1 P(27)
where P = g ab P ab and we have used the following relation
∂ i φg ab M iab0qr = (D − 1){g i0 g qr − 2g ir g 0q }∂ i φ.(28)
Considering the action (25), one can see that, regardless of the surface integral which is a GHY term, the Lagrangian contains fields and their first order derivatives. Therefore, we can be sure that the variational principle for this action is compatible with the Dirichlet BC. Before investigating in details the compatibility of the model, let us show explicitly the structure of the added GHY term in the ADM formalism. Consider the following relation
g ab P ab = √ −g 4 g ab g de,f M ab0def + M def ab0 = √ −g D − 2 2 g de,f g de g 0f − g df g 0e = D − 2 2 1 √ −g ∂ a (gg 0a ) = D − 2 2 √ −g −1 N 2 ∂ a (N 2 g 0a ) − g 0a ∂ a ln h = D − 2 2 √ −g −2K n 0 + ∂ α N α N 2(29)
where n a = N −1 (1, −N α ) and the lapse and shift functions are denoted by N and N α . In the last equality we have used the following two identities
2K n o = 2 N 2 √ h ∂ a (N 2 √ hg 0a ) = 2 N 2 ∂ a (N 2 g 0a ) + 2g 0a ∂ a ln √ h (30) 1 N 2 ∂ a (N 2 g 0a ) = ∂ α N α N 2(31)
Substituting the expression (29) into (25) gives
S = M d D x √ −g (φL quad − V (φ)) + 1 D/2 − 1 M d D x∂ i φg ab M iab − 2 t d D−1 y √ hφK + t d D−1 y √ hφ ∂ α N α N(32)
where the first surface integral in (32), is the same as GHY term of Refs. [6,7]. However, the second surface term is often lost in the literatures. We will come back to this point in the next subsection. Now let us consider the variations of the action (25). First, we rewrite it in terms of the momenta given in Eqs. (26) and (27). By adding and subtracting the following surface integral
2(D − 1) (D − 2) 2 t d D−1 y √ −gg ab (g i0 g ab −2g ib g 0a )∂ i φ = 2(D − 1) D − 2 t d D−1 y √ −g∂ 0 φ(33)
to the action (25), we get
S = M d D x √ −g (φL quad − V (φ)) + 2 D − 2 M d D x∂ i φg ab M iab − 2 D − 2 t d D−1 yg abP ab + 2(D − 1) D − 2 t d D−1 y √ −g∂ 0 φ(34)
Varying this action with respect to φ and g ab and using Eq. (28), after a little algebra, we obtain
δS = δ φ S + δ g S(35)
where
δ φ S = M d D x √ −g (L quad − ∂ φ V (φ)) + 1 D/2 − 1 ∂ i −g ab ∂( √ −gL quad ) ∂(∂ i g ab ) δφ
(36) and
δ g S = M d D xL ab δg ab + D − 4 D − 2 t d D−1 yP ab δg ab + D − 1 D − 2 t d D−1 y √ −g∂ 0 φg ab δg ab − 2 D − 2 t d D−1 yg ab δP ab + 2 D − 2 t d D−1 yg ab P ab δφ + 2(D − 1) D − 2 t d D−1 y √ −gδ(∂ 0 φ)(37)
in which
L ab = φ ∂( √ −gL quad ) ∂g ab − ∂ i φM iab − 1 2 √ −gg ab V (φ) + 2 D − 2 ∂ i φM iab + 2 D − 2 ∂ i φg kl H iabkl − 1 D − 2 ∂ p ( √ −g∂ i φg qr M iqrpab )(38)
and H iabkl ≡ ∂M iab /∂g kl . Substituting (36) and (37) in (35) gives
δS = M d D x √ −g (L quad − ∂ φ V (φ)) + 1 D/2 − 1 ∂ i −g ab ∂( √ −gL quad ) ∂(∂ i g ab ) δφ + M d D xL ab δg ab + D − 4 D − 2 t d D−1 yP ab δg ab − 2 D − 2 t d D−1 yg ab δP ab + D − 1 D − 2 t d D−1 y √ −g∂ 0 φg ab δg ab + t d D−1 yP φ δφ + 2(D − 1) D − 2 t d D−1 y √ −gδ(∂ 0 φ)(39)
As expected, without the GHY term, the undesirable BCs: δg ab | Boundary = δP ab | Boundary = δφ| Boundary = δ(∂ 0 φ)| Boundary = 0 should be assigned. In order to find the appropriate GHY term, let us discuss three different types of BCs leading to a consistent stationary action principle for f (R)-gravity.
Dirichlet BC
Considering the surface terms in Eq. (39), in order to impose the Dirichlet BC δg ab | Boundary = δφ| Boundary = 0, we need to modify the action (34) by adding the following GHY term
S D = S + S GHY D = S + 2 D − 2 t d D−1 yg abP ab − 2(D − 1) D − 2 t d D−1 y √ −g∂ 0 φ
(40) To see that the above action is compatible with the Dirichlet BC, let us vary it as follows
δS D = M d D x √ −g (L quad − ∂ φ V (φ)) + 1 D/2 − 1 ∂ i −g ab ∂( √ −gL quad ) ∂(∂ i g ab ) δφ + M d D xL ab δg ab + t d D−1 yP ab δg ab + t d D−1 yP φ δφ (41)
which gives the equation of motion subjected the Dirichlet BC. Note that φ = f ′ (R) gives δφ| Boundary = f ′′ (R)δR| Boundary = 0. Now we can surely say that δR| Boundary = 0 is compatible with the Dirichlet BC and is in fact part of it. This is a clear covariant verification of the result pointed in Ref. [6] in the framework of the ADM foliation. To be more concrete, we can determine the GHY term S GHY D in terms of ADM variables. Using (29) and substituting φ = f ′ (R), we have
S GHY D = −2 t d D−1 y √ hf ′ (R)K + t d D−1 y √ hf ′ (R) ∂ α N α N (42)
which are the same terms present in Eq. (32). Since in ADM formalism, the Dirichlet BC means δh ab | Boundary = δN µ | Boundary = δN | Boundary = 0, the last term of the above equation can be neglected and the first term suffices. However, note that this is correct only for the Dirichlet BC.
To complete our discussion, we can set φ = 1 and V (φ) = 0 in Eq. (39) to find the following result for the case of GR
δS (EH) = M d D xL ab δg ab + D − 4 D − 2 t d D−1 yP ab δg ab − 2 D − 2 t d D−1 yg ab δP ab (43) whereL ab = ∂( √ −gL quad ) ∂g ab − ∂ i M iab
Imposing the Dirichlet BC: δg ab | Boundary = 0, the action should be modified by the following GHY term to get the equations of motion,
S D(EH) = S (EH) + S GHY D(EH) = S (EH) + 2 D − 2 t d D−1 yg ab P ab(44)
Moreover, using Eq. (29), we can rewrite S GHY D(EH) in the familiar form
S GHY D(EH) = −2 t d D−1 y √ hK + t d D−1 y √ h ∂ α N α N (45)
where for the Dirichlet BC, the second term can be neglected [26].
Neumann BC
In order to obtain the GHY term related to the Neumann BC: δP ab | Boundary = δP φ | Boundary = 0, let us write (39) in a different form. From (26) and (27), we find that
P ab δg ab = −g ab δP ab + D − 2 2P φ δφ + D − 2 2 φδP φ + (D − 1)δ( √ −g∂ 0 φ) (46)
Inserting this into (39) gives
δS = M d D x √ −g (L quad − ∂ φ V (φ)) + 1 D/2 − 1 ∂ i −g ab ∂( √ −gL quad ) ∂(∂ i g ab ) δφ + M d D xL ab δg ab − t d D−1 yg ab δP ab + D − 4 2 t d D−1 yφδP φ + D − 2 2 t d D−1 yP φ δφ + D − 1 2 t d D−1 y √ −g∂ 0 φg ab δg ab + (D − 1) t d D−1 y √ −gδ(∂ 0 φ)(47)
This shows that the action (34) is consistent with the Neumann BC if we propose the following GHY term
S N = S + S GHY N = S − D − 2 2 t d D−1 yP φ φ − (D − 1) t d D−1 y √ −g∂ 0 φ (48)
Variation of (48) yields
δS N = M d D x √ −g (L quad − ∂ φ V (φ)) + 1 D/2 − 1 ∂ i −g ab ∂( √ −gL quad ) ∂(∂ i g ab ) δφ + M d D xL ab δg ab − t d D−1 yg ab δP ab − t d D−1 yφδP φ(49)
which gives the equations of motion using Neumann BC. Using (29) and inserting φ = f ′ (R), we can write the GHY term in (48) in the ADM formalism as
S GHY N = (D − 2) t d D−1 y √ hf ′ (R)K − D − 2 2 t d D−1 y √ hf ′ (R) ∂ α N α N − (D − 1) t d D−1 y √ hN ∂ 0 f ′ (R)(50)
It should be noted that unlike the case of Dirichlet BC, the second term in the above action can not be neglected unless for the coordinate system in which N α = 0. It is worth to compare (50) with the GHY term (42) for Dirichlet BC. It is easily seen that
S GHY N = − D − 2 2 S GHY D − (D − 1) t d D−1 y √ hN ∂ 0 f ′ (R)(51)
Mixed BC
There are two types of mixed BCs for f (R)-gravity: δP ab | Boundary = δφ| Boundary = 0 or δP φ | Boundary = δg ab | Boundary = 0. We begin with the first one. Using the variation of f (R)-gravity action, (39) or (47), the first type mixed BC would be consistent if we have added the following GHY term to the action
S MI = S + S GHY MI = S − D − 4 2 t d D−1 yφP φ − (D − 1) t d D−1 y √ −g∂ 0 φ (52)
Varying the above action gives
δS MI = M d D x √ −g (L quad − ∂ φ V (φ)) + 1 D/2 − 1 ∂ i −g ab ∂( √ −gL quad ) ∂(∂ i g ab ) δφ + M d D xL ab δg ab − t d D−1 yg ab δP ab + t d D−1 yP φ δφ(53)
As can be seen, the mixed BC: δP ab | Boundary = δφ| Boundary = 0 yields consistently the equations of motion. Now using Eq. (29) and φ = f ′ (R), we can write the GHY term of Eq. (52) in terms of ADM variables as
S GHY MI = (D − 4) t d D−1 y √ hf ′ (R)K − D − 4 2 t d D−1 y √ hf ′ (R) ∂ α N α N − (D − 1) t d D−1 y √ hN ∂ 0 f ′ (R)(54)
It is also worth noting here to find the relation between Dirichlet and the above mixed GHY boundary terms in f (R)-gravity. Comparing the above result with that obtained in Eq. (42), we see that
S GHY MI = − D − 4 2 S GHY D − (D − 1) t d D−1 y √ hN ∂ 0 f ′ (R)(55)
For GR, i.e. φ = 1 and V (φ) = 0, it is interesting that the newly defined action (52) is consistent with the Neumann BC
S N(EH) = S (EH) + S GHY N(EH) = S (EH) − D − 4 D − 2 t d D−1 yP(56)
This shows that the pure Neumann BC may be used for GR in arbitrary dimensions. Moreover in four dimensions, for GR with Neumann BC, there is no need to any GHY term in order to have a consistent theory. This point, explained here covariantly is also shown recently in [15] in ADM approach.
To clarify more, using Eq. (29), it is easy to see that the above GHY term with respect to the ADM variables takes the form
S GHY N(EH) = (D − 4) t d D−1 y √ hK − D − 4 2 t d D−1 y √ h ∂ α N α N .(57)
where again the second term in the above action or in (54), can be ignored only for the special choice of coordinate system mentioned in the previous subsection. Also, it can be easily seen that, in GR, in contrast to the Dirichlet case, the required GHY term, compatible with the Neumann BC, depends on the dimension of space-time and for D = 4, the coefficient of GHY term vanishes, as expected. Another interesting feature is the relation between Dirichlet and Neumann GHY term in GR. Comparing Eqs. (45) and (57), one finds
S GHY N(EH) = − D − 4 2 S GHY D(EH)(58)
Now let us look at the second type of mixed BC: δP φ | Boundary = δg ab | Boundary = 0. In order to discuss the consistency of f (R)-gravity with this BC, first we use (46) to substitute for g ab δP ab in (39). This leads to
δS MII = M d D x √ −g (L quad − ∂ φ V (φ)) + 1 D/2 − 1 ∂ i −g ab ∂( √ −gL quad ) ∂(∂ i g ab ) δφ + M d D xL ab δg ab + t d D−1 yP ab δg ab − t d D−1 yφδP φ(59)
Clearly by applying the BC: δP α | Boundary = δg ab | Boundary = 0, we can get the equations of motion without adding any GHY term to the above expression.
This means that f (R)-gravity with the above type of mixed BC is selfconsistent with no need to any GHY term in D dimension. To clarify this point better, let us return to the relation (20) in which the boundary terms are written in the ADM formalism. One can write these terms in term of the momenta conjugate to φ, h ij , N and N α . These are derived in details in appendix C and are as follows
Π N =Π N α = 0 Π φ = −2ǫ √ hK Π ij = ǫ √ h φ(K ij − K h ij ) + h ij N (φ − N α ∂ α φ)(60)
Substituting this into (20) and inserting φ = f ′ (R), we obtain
δ M d D x √ −gf (R) = M d D x √ −gL ab δg ab − ∂M d D−1 y √ hΠ ij δh ij − ∂M d D−1 yf ′ δΠ φ − ∂M d D−1 yǫ √ h∇ i φδn i(61)
where h ij δn i = −n i δh ij is used. It can be seen that the above surface terms, which are written in the ADM formalism, are completely in agreement with what we have derived by the covariant approach in (59). Regarding the above relation and by applying the mixed BC: δΠ φ | Boundary = δn i | Boundary = δh ij | Boundary = 0, we can get the equations of motion in D dimension without any GHY term.
conclusion
In this paper it is shown that unlike GR, the Lagrangian of f (R)-gravity does not follow a holographic relation which is the feature of the Lanczos-Lovelock Lagrangian. Moreover, the Lagrangian of f (R)-gravity can not be expressed as the sum of quadratic and total derivative terms. So f (R) Lagrangian is not degenerate. Following the Ostrogradsky approach, since f (R)-gravity is a theory with higher order derivatives of metric, it carries a single additional degree of freedom, which is the scalar field of equivalent Brans-Dicke action. Introducing this field, leads to a degenerate Lagrangian which is used to develop the problem of BC and the corresponding GHY terms in f (R)-gravity [6,7].
Here we have followed a foliation independent approach to find the GHY boundary terms in f (R)-gravity, required to make the BC variation problem well-defined. We have shown that in addition to the Dirichlet BC, the Neumann BC and two types of the mixed BCs can be introduced for the f (R)-gravity. The remarkable point which is one of the main results of this paper is about the mixed BCs. We have shown that one of the mixed BC: δP ab | Boundary = δφ| Boundary = 0 is reduced to the Neumann BC in the case of GR. This BC together the other mixed BC: δP φ | Boundary = δg ab | Boundary = 0 are self-consistent BCs, i.e. these do not need to any GHY term to be consistent with the theory, the first one for GR and the second one for f (R)-gravity, both in D dimension.
where D i is the spatial-covariant derivative defined on ∂M, U i ≡ n j h i
B Variation of the scalar curvature
It is instructive to find what does the condition δR | Boundary = 0 mean. To answer this question, first let's remind the Gauss-Codazzi equation:
R = (D−1) R − ǫ{K mn K mn − K 2 − 2∇ i (K n i + a i )} = (D−1) R + K 2 − K mn K mn − 2(∇ i K )n i − 2n a ∇ i ∇ a n i (B.1)
where (D−1) R is the scalar curvature of the (D − 1)-dimensional subspace and in the first line a i = n a ∇ a n i is the acceleration of the normal vector field. Then taking variation of (B.1) with ǫ = −1 gives the Palatini identity as follows
δR = δ (D−1) R + 2δK mn (K h mn − K mn ) − 2KK mn δh mn − 2∇ i (δK )n i − 2(∇ i K + ∇ a ∇ i ∇ a )δn i − 2n i δ(∇ a ∇ i n a ), (B.2)
where the variation of spatial scalar curvature reads
δ (D−1) R = (D−1) R ij δh ij + D a D d (−δh ad + h ad h ik δh ik ). (B.3)
As is obvious from (B.2) and (B.3), δR is a combination of δh mn , δn i , δK mn , ∇ i (δK ), δ(∇ a ∇ i n a ) and spatial-covariant derivatives of δh mn .
C The conjugate momenta in f (R)-gravity
To find the conjugate momenta, we write the f (R) action in the Brans-Dicke form and then using the Holographic relation as in (24), make it degenerate. To do this, substituting the Gauss-Codazzi equation in D dimension, (B.1), into (4), we find that
S = M d D xN √ h φ( (D−1) R − ǫ{K ij K ij − K 2 }) − V (φ) + 2 M d D x √ −gǫ∇ i (K n i + a i )φ (C.4)
By-part integration on the last term gives
S = M d D xN √ h φ( (D−1) R − ǫ{K ij K ij − K 2 }) − V (φ) −2 M d D xN √ hǫK n i ∇ i φ − 2 M d D x √ −gǫa i ∇ i φ +2 ∂M d D−1 y √ hǫn i (K n i + a i )φ = M d D xN √ h φ( (D−1) R − ǫ{K ij K ij − K 2 }) − V (φ) −2 M d D xN √ hǫK Dφ − 2 M d D x √ −gǫa i ∇ i φ + 2 ∂M d D−1 y √ hK φ (C.5)
where n i a i = 0, n i n i = ǫ and Dφ ≡ n i ∇ i φ have been used. Using following calculation
a i = n m ∇ m n i = −n m ∇ m (N ∇ i t) = 1 N n i n m ∇ m N + N n m ∇ i ( −1 N n m ) = 1 N (∇ i N + n i n m ∇ m N ) = 1 N h m i ∇ m N = 1 N D i N (C.6)
and n i = ( 1 N , −N α N ), we have
N Dφ = N n 0 ∂ 0 φ + N n α ∂ α φ =φ − N α ∂ α φ (C.7)
Substituting (C.6) and (C.7) into (21), we obtain
S = M d D x √ h{N φ( (D−1) R − ǫ{K ij K ij − K 2 }) − 2ǫK (φ − N α ∂ α φ) − 2ǫh ab D a N D b φ − N V (φ)} + 2 ∂M d D−1 y √ hK φ (C.8)
Now we can define the momenta conjugate to h αβ , N , N α and φ as
Π N =Π N α = 0 Π φ = −2ǫ √ hK Π ij = ǫ √ h φ(K ij − K h ij ) + h ij N (φ − N α ∂ α φ) (C.9) Ostrogradsky approach to f (R)-GravityAs it was mentioned in the previous section, f (R) Lagrangian is not degenerate. Varying the action (3) and integrating by part, without implying any
For a detailed calculations see Appendix A.
AcknowledgementsF. Shojai is grateful to the University of Tehran for supporting this work under a grant provided by the university research council.A Variation of f (R)-gravity action without BCThe variation of the action of f (R)-gravity givesThe first integral includes some terms of the equations of motion. Using the contracted form of Palatini equationand integrating by part in the second term of (A.1), we would haveNow we want to write the surface integral of (A.4) in ADM foliation of space-time. The first term gives) k δg jk and for the first equality see[16]. The third term of (A.5) is zero assuming the manifold is compact in D-1 dimension. The last term can be written asThen we have (A.5) aswhere h a j n j = 0, δn j = 1 2 ǫn j n k n e δg ke + n k n j ℓ δg kℓ and also δg ij = δh ij + ǫn i δn j +ǫn j δn i have been used. Eventually we can write the surface integrals of (A.4) as(A.9) Substituting (A.9) and (A.7) in (A.4), yields
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[
"ADAPTIVE LEJA SPARSE GRID CONSTRUCTIONS FOR STOCHASTIC COLLOCATION AND HIGH-DIMENSIONAL APPROXIMATION",
"ADAPTIVE LEJA SPARSE GRID CONSTRUCTIONS FOR STOCHASTIC COLLOCATION AND HIGH-DIMENSIONAL APPROXIMATION"
] |
[
"Akil Narayan ",
"John D Jakeman "
] |
[] |
[] |
We propose an adaptive sparse grid stochastic collocation approach based upon Leja interpolation sequences for approximation of parameterized functions with high-dimensional parameters. Leja sequences are arbitrarily granular (any number of nodes may be added to a current sequence, producing a new sequence) and thus are a good choice for the univariate composite rule used to construct adaptive sparse grids in high dimensions. When undertaking stochastic collocation one is often interested in constructing weighted approximation where the weights are determined by the probability densities of the random variables. This paper establishes that a certain weighted formulation of one-dimensional Leja sequences produces a sequence of nodes whose empirical distribution converges to the corresponding limiting distribution of the Gauss quadrature nodes associated with the weight function. This property is true even for unbounded domains. We apply the Leja-sparse grid approach to several high-dimensional and problems and demonstrate that Leja sequences are often superior to more standard sparse grid constructions (e.g. Clenshaw-Curtis), at least for interpolatory metrics.
|
10.1137/140966368
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[
"https://export.arxiv.org/pdf/1404.5663v2.pdf"
] | 37,065,188 |
1404.5663
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6453e1cd18e98b13845647dbe114f12f15488ba1
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ADAPTIVE LEJA SPARSE GRID CONSTRUCTIONS FOR STOCHASTIC COLLOCATION AND HIGH-DIMENSIONAL APPROXIMATION
Akil Narayan
John D Jakeman
ADAPTIVE LEJA SPARSE GRID CONSTRUCTIONS FOR STOCHASTIC COLLOCATION AND HIGH-DIMENSIONAL APPROXIMATION
We propose an adaptive sparse grid stochastic collocation approach based upon Leja interpolation sequences for approximation of parameterized functions with high-dimensional parameters. Leja sequences are arbitrarily granular (any number of nodes may be added to a current sequence, producing a new sequence) and thus are a good choice for the univariate composite rule used to construct adaptive sparse grids in high dimensions. When undertaking stochastic collocation one is often interested in constructing weighted approximation where the weights are determined by the probability densities of the random variables. This paper establishes that a certain weighted formulation of one-dimensional Leja sequences produces a sequence of nodes whose empirical distribution converges to the corresponding limiting distribution of the Gauss quadrature nodes associated with the weight function. This property is true even for unbounded domains. We apply the Leja-sparse grid approach to several high-dimensional and problems and demonstrate that Leja sequences are often superior to more standard sparse grid constructions (e.g. Clenshaw-Curtis), at least for interpolatory metrics.
Introduction
Stochastic collocation (SC) has become a standard tool for non-intrusively quantifying uncertainty in simulation models that are subject to a degree of uncertainty or randomness. Sources of uncertainty can include physical stochastic processes, parametric uncertainty or model form uncertainty [54,40,39]. If the sources of uncertainty can be parameterized by a set of random variables, the approaches used for uncertainty quantification (UQ) frequently reduce to computational methods that describe the behavior of a model with respect to those random variables.
In this paper we will focus on the use of stochastic collocation methods that utilize sparse grid interpolants [46,56,22,9] to approximate the dependence of model output on unknown model parameters. Sparse grid stochastic collocation involves constructing an ensemble of random variables realizations, solving a deterministic physical system for each corresponding realization on the sparse grid, and using the resulting model output to build an approximation of the model response to the uncertain parameters. Once constructed, the sparse grid can be evaluated inexpensively to predict the variability of the physical model with respect to the random parameters.
Obtaining the ensemble of model solutions is usually the most expensive part of the collocation procedure since the model under consideration is often very complicated (e.g., complex geometry, multiscale features, stiff time-stepping, etc.). This dominance of model execution time on the computational expense of UQ motivates the need to build approximations of the model response that require as few model runs as possible. Sparse grids attempt to minimize the number or parameter realizations by generating ensembles that are geometrically sparse in the state space of the random variable. Most efficient sparse grid constructions build up the ensemble adaptively by concentrating model evaluations in dimensions of the paramter space where the sparse grid approximation is poor [23,26,19].
One of the crucial considerations when building a sparse grid is the choice of ensemble for the random variable. The sparse grid is constructed via the union of judicious tensorizations of onedimensional ensemble grids, and so the identification of these univariate grids is of paramount importance. Common univariate choices are Clenshaw-Curtis or Chebyshev nodes and Gauss-quadraturetype nodes.
The desired characteristics in choosing a composite univariate rule for input into the sparse grid include:
• efficiency -high interpolation and/or quadrature accuracy with low cardinality sets • robustness -consistent and increasing accuracy when the grid is refined • monotonicity -fine-level grids are supersets of coarse-level grids • granularity -the number of nodes needed for refinement of a grid is as small as possible Efficiency and robustness are desired when using approximation grids in any context. The monotonicity property is motivated mainly by the high cost of solving deterministic physical simulations and the sparse grid algorithm: if monotonicity holds, then the tensorized sparse grid construction has many fewer total nodes. Granularity becomes important when several levels of refinement are necessary: it is much better to have the ability to add a small number nodes for each refinement step than to be required to, e.g., double the number of nodes.
In this paper we propose use of univariate Leja sequences for use in the sparse grid algorithm for interpolatory high-order and high-dimensional approximation. Leja rules are a sequence of interpolation/quadrature grids in one dimension that are strongly monotonic and granular: coarse grids are always strict subsets of fine grids, and refinement proceeds by adding a single node at a time. Leja sequences are very accurate as we show later, but they are not as efficient as Gauss-type rules in some cases (e.g., quadrature). Therefore, we argue that Leja sequences are a grid choice that serves as a good compromise of the above desired characteristics, in contrast to, e.g. a Gauss quadrature grid that is strong in efficiency and granularity, but very weak in monotonicity.
For many weight functions of interest (i.e. the random variable probability density function), we show that (our definition of) weighted Leja sequences produce a nodal sequence whose asymptotic distribution coincides with the asympototic distribution of Gauss quadrature nodes associated to the polynomial family orthogonal under the weight function. This result is known for the uniform-weight case; to our knowledge the weighted versions are novel. Additionally, we show that a contracted version of weighted Leja sequences are asymptotically weighted Fekete, meaning that their Vandermonde determinant grows comparably to the largest possible value (Fekete). That these results hold for unbounded domains is significant as it suggests that Leja sequences will be accurate for nested interpolatory approximation when the random variable state space is infinite.
In Section 2 we setup the problem and introduce notation and terminology. In Section 3 we introduce Leja sequences and formally present the above-mentioned properties. Section 4 develops the methodology for adaptive sparse grids. Finally, Section 5 shows that the Leja sparse grid algorithm produces results comparable to well-established sparse grid approximation methods, and in many cases is superior. The proof of our main result in Section 3 concerning the distribution of Leja sequence nodes is relatively involved, employing results from weighted potential theory; for this reason we leave this until the end in Section 7, serving somewhat as an appendix.
The first half of this paper (Section 3) proves certain results about one-dimensional weighted Leja sequences and compares them to other one-dimensional rules. The second half (Sections 4 and 5) uses these weighted Leja sequences for building adaptive multivariate sparse grids.
Setup
A model problem in the UQ community that serves as a motivating example is a parameterized elliptic equation where the parameters Z are random variables:
− d dx a(x, Z) du dx (x, Z) = f (x, Z), (x, Z) ∈ (0, 1) × I z(1)
This model describes the steady-state temperature distribution u in a one-dimensional domain where the domain has diffusivity coefficient a and experiences an external heat source defined by f . Here x is a spatial variable taking values in a one-dimensional interval domain. The variable Z is a random vector with density function ω(Z) on domain I z corresponding to a probability measure P on a complete probability space. The diffusion coefficient a is a model parameter that varies spatially, but is also influenced by uncertainty. Uncertainty in these kinds of models may arise from, e.g., imprecise knowledge of material parameters or external forcing. Under the assumption that the equation is well-posed almost surely, the solution u(x, Z) is random, and essentially depends on (1 + dim Z) variables. One goal in the UQ community is efficient and accurate prediction of u(x, Z) or some quantities of interest that depend on u (e.g. the temperature variance as a function of space x).
One popular technique is the generalized Polynomial Chaos (gPC) approach: we assume that the variation of u with respect to the random parameter Z can be described accurately by a finite-degree polynomial:
u(x, Z) u N (x, Z) = N n=1 u n (x)φ n (Z),
where φ n are polynomials that satisfy an orthogonality condition
E[φ n (Z)φ m (Z)] = Iz φ n (z)φ m (z)ω(z)dz = δ n,m ,
and the u n are coefficient functions that depend only on the spatial variable x. The determination of the functions u n is the challenge, and one straightforward procedure is to use a probabilistic sampling strategy to compute these coefficients: let z m for 1 ≤ m ≤ M be given samples of the variable Z. For each z m , equation (1) is a deterministic differential equation, and any computational simulation or experimental setup may be used to obtain u(x, z m ). (Frequently, this solution is a finite-dimensional quantity rather than a function of a continuum variable x, but this distinction does not affect the main focus of our discussion.) Once these realizations of u(x, Z) are collected, then we attempt to find u n (x) such that
N n=1 φ n (z m ) u n (x) ≈ u(x, z m ), m = 1, . . . , M.
If one considers u as a scalar, then this is a linear algebra problem, seeking a vector u that solves:
A u ≈ u.
This problem may be solved by defining ≈ in any convenient fashion: interpolation, least-squares regression, quadrature, or compressive sampling. See, e.g. [52,53,14,55]. Usually the particular choice made is dependent on the relationship between M and N (determined by the computational cost of computing each solution realization) and dependent on some a priori understanding of the accuracy for the choice. One major difficulty with this approach is the selection of z m when dim Z is large. While the spatial variable x is usually restricted to have dimension less than or equal to 3, it is not uncommon to have 100 or more parameters as the components of Z. High-dimensional approximation has been a persistent bottleneck in modern scientific computing. Methods that work very well for a small number of dimensions are rendered ineffective or impossible to implement in a large number of dimensions, owing to the curse of dimensionality: exponential dependence of functional complexity with respect to the parametric dimension (when the functional smoothness is fixed). Tensor product constructions and space-filling designs adopt this complexity with respect to dimension.
Attempting to circumvent the computationally onerous space-filling property is the main reason to consider alternatives such as sparse grids. The sparse grid construction still employs tensor product grids, but it does so in a way that attempts to control the cardinality of the mesh and delays the curse of dimensionality. Sparse grids are formed by (unions of) tensorized one-dimensional grids, and therefore an educated selection of the composite one-dimensional rules is of great importance.
In this paper we consider high-order interpolatory approximation using a sparse grid, and we employ weighted Leja grids as the one-dimensional composite rules. Weighted Leja grids are nested grids (they are a sequence), and we prove that the nodes distribute identically to the one-dimensional Gauss quadrature rules. Thus, Weighted Leja sequences distribute nodes in a way that emulates the Gauss quadrature rule, and have the advantage of being nested.
Gauss-Kronrod [29,11,20] and Gauss-Patterson [41] rules are likewise nested interpolatory schemes, but their computation is usually restricted to special weight functions because computation of the nested rules is relatively difficult. In contrast, weighted Leja sequences are simple and very easy to compute (exactly and approximately) even for exotic weight functions.
Our approach with weighted Leja sequences considers polynomial approximation on unbounded domains, but there are alternative high-order approaches. As described in [8], there are three popular approaches to high-order approximation on unbounded domains: (1) expansions on infinite domains using polynomial or non-polynomial complete basis sets [47,37] (2) domain trunction, where the full domain I z is replaced by a compact subset [7,6], and (3) mapping techniques [51,25,24] where standard methods on compact intervals are "transplanted" onto an infinite interval via a domain mapping. Each of these methods can perform accurate approximation on unbounded domains depending on the application. Our approach is most closely related to (1), but in principle one may use Leja sequences for any of the above methods. However, this application is outside the scope of this paper.
For the remainder of this paper, we replace the uppercase variable Z (traditionally denoting a random quantity) with its lowercase counterpart z: stochasticity does not affect our approach so in principle we may treat the random parameter as a deterministic parameter z with corresponding weight function ω.
Weighted Leja points
In this section we present and establish important properties of univariate Leja sequences. Consider approximation in the scalar variable z over the domain I in the presence of a weight function w. In the context of this paper, z represents one component of the vector-valued parameter z, I is the one-dimensional restriction of I z to the appropriate dimension, and w(z) is the marginal density computed from the joint density ω(z).
A Leja sequence (unweighted) is classically defined [16,30] as a sequence of points z n ∈ I = [−1, 1] ⊂ R for n = 1, 2, . . . , such that
z N +1 = argmax z∈[−1,1] N n=0 |z − z n | ,(2)
where the starting point of the sequence z 0 is arbitrarily chosen in [−1, 1]. We note that it is only sensible to define the above Leja sequences on bounded domains. We list below the properties of one-dimensional (unweighted) Leja sequences:
• Leja sequences are not unique. The initial point z 0 may be arbitrarily chosen and the objective function being maximized need not have a unique maximizer. • The Leja construction provides an interpolation sequence. Therefore if {z 1 , . . . , z 7 } are a Leja sequence, then we require only one more point z 8 to construct a higher-order interpolant. This addresses the granularity criterion for grids, and will be useful in minimizing the number of function evaluations necessary for approximation in high dimensions. • Maximizing the objective function (2) is equivalent to a greedy determinant maximization (e.g., [15]). With z 1 , . . . , z N −1 specified, let V N (z) be the N ×N interpolatory Vandermondelike matrix for the space Π N −1 at the collocation points z 1 , . . . , z N −1 , z. (The choice of basis for Π N −1 does not affect maximization.) Then (2) is equivalent to 1] |det V N (z)| Thus one can view Leja sequences as a greedy D-optimal experimental design [17]. • The Lebesgue constant for interpolation on a Leja sequence grows subexponentially [48].
z N +1 = argmax z∈[−1,
• Leja sequences are asymptotically Fekete [4]. (This is implied by the subexponentially growing Lebesgue constant.) Fekete points are those whose Vandermonde determinant is as large as possible. (These are known to be Gauss-Lobatto nodes in one dimension [18].) The asymptotically Fekete property essentially means that the Vandermonde determinant of a Leja sequence grows on par with that of true Fekete nodes.
• Any Leja sequence asymptotically distributes according to the Chebyshev (arcsine) measure.
(This is implied by the asymptotically Fekete property [4].) • In practice, optimization of (2) over a discrete candidate set can be accomplished in computationally efficient ways [1] and with standard numerical linear algebra routines: E.g., the row permutation information from a row-pivoted LU matrix decomposition gives the Leja sequence order, e.g. [5]. The Leja sequences introduced above are quite useful for unweighted polynomial interpolation. However, in the UQ context we are usually interested in interpolation involving a weight function (here, the marginal density of the random variables). Therefore, we are also interested in weighted Leja sequences. Let w(z) be a continuous and positive weight function on a univariate domain I, and let v(z) = w(z)
be its square root. If I is unbounded, we assume that polynomials are dense in the space of continuous functions, measured in the v-weighted supremum norm. For example, if v(z) ∝ exp(−|z| α ), then we require α ≥ 1, e.g. [31]. This density assumption is necessary in our context: we cannot hope to form an accurate polynomial approximation without polynomial density.
A weighted Leja sequence can be constructed via the optimization
z N +1 = argmax z∈I w(z) N n=0 |z − z n | = argmax z∈I v(z) N n=0 |z − z n | . (4a)
In the case where multiple choices of z maximize the objective, for concreteness we choose the one with smallest magnitude, i.e., The sequence of points z n produced by the above iteration is the central study of this paper, and we refer to this sequence as a weighted Leja sequence, or a w-weighted Leja sequence. This formulation still leaves a benign ambiguity when multiple maximizers differ only in sign. In this paper, Leja sequence optimization is one-dimensional, so in all that follows we optimize via (4) exactly (up to machine accuracy).
z N +1 = argmin z |z| | z ∈ f −1 (W ) , W = max z∈I f (z) max z∈I v(z) N n=0 |z − z n | . (4b)
In general there is no standard choice of how to incorporate the weight function into a Leja objective; we have chosen v = √ w, but alternatives have been proposed [15,45]. Our choice above is motivated by the fact that under this formulation the sequence of points produced has the same asymptotic distribution as w-Gauss quadrature nodes. We illustrate this property now: In Figure 1 we show that the distribution of a univariate 50-point weighted Leja sequence seems to converge to the distribution of the Gauss points associated with the family of polynomial orthogonal under the weight w. Therefore, the objective (4) produces points that are 'approximately' Gauss nodes, with the additional benefit of being nested. This distribution property alone does not guarantee that Leja sequences are useful, but we provide several examples in this paper that suggest that Leja points have utility. We emphasize that unlike the unweighted case (2), weighted Leja sequences are constructible on unbounded domains given our assumptions. In Figure 2 we show a graphical illustration of the iterative Leja procedure that produces a Leja sequence.
3.1. Limits of weighted Leja sequences. A significant concern for the unbounded case with the weighted Leja sequences we have introduced above is that they 'do the right thing', i.e. that they produce a set of nodes that is desirable from the approximation theory point of view. 1 The purpose of this subsection is to show that the weighted Leja sequences we have proposed (4) produce points whose asympototic distribution is identical to the Gauss quadrature nodes associated with the corresponding w-orthogonal polynomial family. To be precise, given a classical weight function w, it is known that there is another weight function v such that the empirical distribution of the w-Gauss quadrature nodes converges to the v-weighted potential equilibrium measure; roughly speaking, v ∼ √ w, modulo multiplicative polynomial factors. We show that for this same class of weight functions, the empirical measure of the Leja sequence converges to the same equilibrium 1 Alternative propositions for weighted Leja points [45] construct points on a compact subset of the domain. These points are asymptotically optimal if we are interested in approximation with a basis v n pn.But because we are interested in approximation on an unbounded domain with just vpn, we require samples to be produced on the entire domain.
measure. On unbounded domains with exponential weights, v is the square root of w. On bounded domains with Jacobi-type weights, v is the uniform weight. A comprehensive discussion of potential theory and equilibrium measures can be found in [45]; here and later in Section 7, which contains proofs, we give a brief account of this topic. Let v be a weight function on I that we will precisely relate to w later; define Q − log v. If v is admissible 2 on I, then there is a unique probability measure µ v that minimizes a weighted logarithmic energy. This measure µ v is the logarithmic potential equilibrium measure of the domain I in the presence of the external field (i.e. weight) Q. (See Section 7 for a more detailed discussion.) When v is the uniform measure on a compact interval I, then µ v is the arcsine, or Chebyshev measure.
For each N ∈ N, let ξ n,N with n = 1, . . . , N denote the N zeros of the degree-N polynomial orthogonal under w(z). I.e., ξ n,N are the N -point w-Gauss quadrature nodes. Let z n denote any sequence of weighted Leja nodes given by (4). The ξ n,N are a triangular array (n ≤ N ) while the z n are a sequence. We introduce a contraction factor k n defined in the following theorem; this contraction factor is used to define the empirical (counting) measure for an N -point Gauss (ξ n,N ) and Leja (z n ) grid, respectively:
ν G N = 1 N N n=1 δ (k N ξ n,N ) , ν L N = 1 N N n=1 δ (k N zn) ,
where δ z is the Dirac distribution centered at z. Our main result in this section shows that for most classical univariate weight functions of interest, ν G N and ν L N limit to the same measure.
Theorem 3.1. Let w(z) be a weight function on I.
(1) (Generalized Hermite) Let w(z) = z 2µ exp(−|z| α ) for any α ≥ 1, µ > − 1 2 on I = R. Define k n = n −1/α and v = exp − 1 2 |z| α . (2) (Laguerre) Let w(z) = z s exp(−|z|) for any s > −1 on I = [0, ∞). Define k n = n −1 and v(z) = exp − 1 2 z . (3) (Jacobi) Let w(z) = (1 − z) α (1 + z) β for any α, β > −1 on I = [−1, 1]. Define k n ≡ 1 and v(z) ≡ 1.
In all of the above cases, we have
lim n→∞ ν G n = µ v = lim n→∞ ν L n ,(5)
where equality holds in the weak sense.
Remark 3.1. The "Hermite" result from the Theorem above that lim n ν L n → µ v also holds for 0 < α < 1. However, since polynomials are not dense for this weight function [31], it is unclear if one should use polynomial approximation in this case.
The above theorem states that the Leja sequence produced by (4) produces a nested mesh whose samples distribute precisely like w-Gaussian quadrature nodes. We emphasize that while this property is promising, it does not guarantee a good approximation: for example, one can generate a grid according to the acrsine measure on I = [−1, 1] whose Lebesgue constant does not grow subexponentially [4]. However, an unweighted Leja sequence is known to have subexponentially growing Lebesgue constant (which is 'good enough' in a sense for approximating very smooth functions). In practice, unweighted Leja sequence have logarithmically-growing Lebesgue constant. However, to our knowledge it is presently unknown if weighted Leja sequences have subexponentially growing weighted Lebesgue constant.
2 v must be admissible in the sense of potential theory: (1) it is a non-negative upper semicontinuous function (2)
I ∩ v −1 ((0, ∞])
is nonpolar in the sense of potential theory (a polar is 'negligible' and has Lebesgue measure 0), and Table 2. Explicit formulae for special cases of Hermite/exponential-type asymptotic distributions [50]. See Table 1. The distribution function F (t) for α = 2 is plotted in the right-hand pane of Figure 1. The densities f (t) for these tabulated values of α are plotted in Figure 3.
(3) if I is unbounded, |z| v(z) → 0 as |z| → ∞ Class Domain Parameters Weight Contraction k n Equilibrium weight Hermite I = R α ∈ [1, ∞), µ ∈ − 1 2 , ∞ w(z) = z 2µ exp(−|z| α ) k n = n −1/α v(z) = exp − 1 2 |z| α Laguerre I = [0, ∞) s ∈ (−1, ∞) w(z) = z s exp(−z) k n = n −1 v(z) = exp − 1 2 z Jacobi I = [−1, 1] α, β ∈ (−1, ∞) w(z) = (1 − z) α (1 + z) β k n ≡ 1 v ≡ 1 Class supp(µ v ) = [a, b] Distribution F (t) = µ v [ (−∞, t] ] Density f (t) = dµ v dt Hermite −a = b = 2 α−1 B α 2 , α 2 1/α See Table 2 f (t) = α πb α b |t| u α−1 √ u 2 −t 2 du Laguerre a = 0, b = 4 F (t) = 2 π arcsin √ t 2 + 1 2π t(4 − t) f (t) = 1 2π 4−t t Jacobi a = −1, b = 1 F (t) = 1 2 + 1 π arcsin t f (t) = 1 π 1 √ 1−t 2Distribution F (t) = µ v [ (−∞, t] ] Density f (t) = dµ v dt t ∈ [−b(α), b(α)] t ∈ [−b(α), b(α)] α = 1 b = π 1 2 + 1 π arcsin t b + tf (t) 1 π 2 log b+ √ b 2 −t 2 |t| α = 2 b = √ 2 1 2 + 1 π arcsin t b + t 2 f (t) 1 π √ b 2 − t 2 w(z) = exp(−|z| α ) α = 3 b = π 2 1/3 1 2 + 1 π arcsin t b + t 3 f (t) 3 π 2 b √ b 2 − t 2 + t 2 log b+ √ b 2 −t 2 |t| b(α) = 2 α−1 B α 2 , α 2 1/α α = 4 b = 4 3 1/4 1 2 + 1 π arcsin t b + t 4 f (t) 1 π √ b 2 − t 2 2t 2 + b 2 α = 5 b = 3π 8 1/5 1 2 + 1 π arcsin t b + t 5 f (t) 5 3π 2 b 3t 2 + 2b 2 √ b 2 − t 2 + 3t 4 log b+ √ b 2 −t 2 |t| α = 6 b = 16 15 1/6 1 2 + 1 π arcsin t b + t 6 f (t) 3 8π √ b 2 − t 2 8t 4 + 4b 2 t 2 + 3b 4
The portion of (5) that relates the Gauss quadrature node distribution (zero distribution of orthogonal polynomials) to the measure µ v is well-known: [34,38,44,50,33,45]. That the unweighted Leja node distribution converges to the arcsine measure is likewise well-known. Our novel contribution to result (5) is for the limit for the weighted Leja formulation (4).
We give a summary of the weights w, contraction factors k n , and some details about the asymptotic measures µ v in Table 1. The formulas for the exponential weight w(z) = exp(−|z| α ) are not explicit for general α, so we explicitly compute and collect the density and distribution expressions for a selection of values for α in Table 2. Finally, the densities associated to µ v are plotted for these specials cases in Figure 3.
We note that by affine scaling, the limits of weighted Leja sequences in Tables 1 and 2 for other types of weights are readily derivable. I.e., suppose we have a new weight W defined on a new parameter Z that results from affine scaling of a canonical w, z pair from Theorem 3.1:
Z = T (z) Az + B, W (Z) = w T −1 (Z) ,
for some A, B ∈ R. Then the new limiting density g(s) and distribution G(s) for a W -weighted
Leja sequence on the Z domain T (I) can be expressed in terms of the weighted Leja asymptotics in Tables 1 and 2: Table 1 for a summary and Table 2 for explicit formulas.
g(s) = 1 A f T −1 (s) , G(s) = F T −1 (s) 0 1 2 3 4 0 1 2 3 4 t Density function f (t) −3 −2 −1 0 1 2 3 0 0.2 0.4 0.6 t Density function f (t) α = 1 α = 2 α = 3 α = 4 α = 5 α = 6
for s ∈ T (I). The contraction factor for Z will be the same as it was for z.
We prove Theorem 3.1 by leveraging a significant result from potential theory: nodal sets that are 'asymptotically weighted Fekete' distribute according to the weighted potential equilibium measure [45,49]. We prove that contracted versions of weighted Leja sequences are asymptotically weighted Fekete and essentially obtain (our novel contribution to) Theorem 3.1 as a corollary.
Let V (ξ 0 , . . . , ξ N ) denote the modulus determinant of the polynomial Vandermonde matrix W on any array of points ξ 0 , . . . , ξ N . W has entries (W ) j,k = ξ k−1 j for j, k = 0, . . . N . Given a weight function v and a positive integer N define the maximum attainable value for the following weighted determinant:
δ (N ) v = max (ξ0,...,ξ N )∈I N +1 V (ξ 0 , . . . , ξ N ) N j=0 v N (ξ j ) 2/(N 2 +N )
For any fixed N , a cardinality-N point set that achieves the maximum weighted determinant under the bracket is called a weighted Fekete set. It is known that the behavior of this maximum determinant value has a finite limit, the v-weighted transfinite diameter of I:
δ v = lim N →∞ δ (N ) v .
Any set of points whose asymptotic determinant limits to the transfinite diameter is called a set of asymptotically weighted Fekete points. In the cases of Theorem 3.1, the k N -contracted w-weighted Leja points we proposed in (4) are asymptotically v-weighted Fekete.
Theorem 3.2.
In all the cases of Theorem 3.1, the Leja sequence z n defined by (4) produces a set of points whose k N -contraction is asymptotically weighted Fekete:
lim N →∞ V (z 0,N , . . . , z n,N ) N n=0 v N (z n,N ) 2/(N 2 +N ) = δ v , z n,N = k N z n (6)
where v is the weight function corresponding to the limit measure in Theorem 3.1.
Theorem 3.2 is a stronger result than Theorem 3.1 (see Lemma 7.1) and is the result that we spend the most effort proving. Once this is established, it is well-known that asymptotically weighted Fekete points distribute according to the weighted equilibrium measure µ v . See Section 7 for details and the proof.
3.2.
Quadrature with Leja sequences. The construction of Leja points is motivated mainly by interpolation; however quadrature/cubature in a multidimensional sparse grid framework is very desirable. To this end, one may simply explicitly integrate an interpolant on Leja points to construct a quadrature rule.
Consider a weighted Leja sequence z n constructed using (4). We need only integrate the interpolant constructed from data on the z n . Let {p n }, n = 0, . . . , N − 1 denote family of polynomials orthonormal under w(z). There are two observations we need: (i) if we assume that w is a probability density function, then p 0 ≡ 1, and (ii) even if w is known only empirically and does not have a representation in terms of classical functions, there are simple and accurate methods to construct the p n in one dimension [28,20,36]. Given data f n we wish to interpolate at the sites z n , so we seek the coefficients c n solving the linear problem
Vc = f , V n,m = p m−1 (z n ). Since c = V −1 f , and Ω N n=1 c n p n−1 (z)ω(z)dz = c 1 Ω p 0 (z)ω(z)dz = c 1 ,
then we immediately conclude that the first row of the matrix V −1 gives us quadrature weights w n defining the Leja polynomial quadrature rule
Q 1 N −1 f N n=1 w n f (z n ).
The superscript '1' indicates that this quadrature rule applies to one-dimensional functions, and the subscript N − 1 refers to the 'level' of the quadrature rule; both of these indicators unnecessary at the moment, but are meaningful in coming sections. Naturally we wish to understand whether the Leja quadrature rules are useful in one dimension before proceeding to use them in higher-dimensional situations. We first verify that the quadrature rules are stable. The relative condition number of the quadrature rule Q 1 N −1 is given by the ∞-norm of the 1 × N matrix W with entries W 1,n = w n . Thus, the condition number of the quadrature rule is
κ 1 N −1 = N n=1 |w n | N n=1 w n = N n=1 |w n |,
where the last equality holds under the assumption that the p n are orthonormal with respect to a probability density function w. The metric κ 1 indicates the presence of negative weights, which make the computation susceptible to catastrophic cancellation. The left-hand pane of Figure 4 We perform one-dimensional global interpolation and refinement for these functions using the (uniform) Leja grids, nested Clenshaw-Curtis grids, and Legendre-Gauss-Patterson grids. Figure 5 shows results in the discrete maximum norm and the quadrature error. If f is the interpolatory approximation, then on a 10 4 -sized Clenshaw-Curtis grid x n with weights v n , these metrics are defined as
n v n f 1 (x n ) − f 1 (x n ) 2 (Discrete 2 error) max f 1 (x n ) − f 1 (x n ) (Discrete maximum error) n v n f 1 (x n ) − f 1 (x n ) (Quadrature error)
The left-hand pane of Figure 5 shows that the Leja grid is no less accurate than any of them in the maximum norm (being as accurate as the Clenshaw-Curtis grid). A discrete 2 error metric behaves similarly. The Leja sequence performs noticeably worse than the other two for the quadrature metric. We sacrifice quadrature accuracy in order to gain some dexterity in high-dimensional refinement: with the Clenshaw-Curtis or Legendre-Gauss-Patterson grid every refinement doubles the size of the (univariate) rule, whereas with the Leja procedure we can stop refinement at any size we choose. That the Gauss-Patterson grid performs so poorly for the maximum norm approximation can be explained by the fact that Gauss-Patterson nodes are constructed only to obtain a high degree of polynomial integration, not for interpolatory approximation. Gauss-Patterson grids that are formed for nested quadrature are not necessarily good for interpolation. This can be seem by comparing the left-and right-hand panels in Figure 5. The relatively good behavior of a Leja grid is not useful unless we have a good error metric from the hierarchical surplus. Now consider function f 2 , and the results in Figure 6, where we show the behavior of the surplus versus all the error metrics. The left-hand pane shows that the maximum Leja surplus is an excellent indicator of error in all three norms. The right-hand pane shows similar results for the Clenshaw-Curtis grid, but two observations are apparent: first, if using the surplus as a refinement technique, the surplus when N = 257 does not accurately reflect the actual error at the next level N = 513. This is true even if one were to employ a type of Richardson extrapolation to estimate the error. Second, the surplus indicator is a very conservative estimate of the error. In such a case, it is likely that we will refine more than is necessary in order to obtain an approximation. Both of these observations do not hold for the Leja surplus, which is sharper estimate of the error, and may be refined with arbitrary size.
Barycentric Interpolation.
A numerically robust method for computing polynomial interpolants is furnished by the Barycentric interpolation formula (see [3] for an accessible introduction). This method is computationally efficient with respect to computing and evaluating the interpolant, and is stable so long as the interpolation problem itself is stable.
On a set of nodes z 1 , . . . , z N , the degree-(N − 1) polynomial interpolant of a function f (z) with data f n = f (z n ) is given by
p(z) = N n=1 f n n (z) = N n=1 bnfn z−zn N n=1 bn z−zn ,(7)
where n (z) are the cardinal Lagrange interpolation basis and b n are the Barycentric weights, both defined as
n (z) = m =n z − z m z n − z m , b n = m =n (z n − z m ) −1 .
The Barycentric formula (7) allows evaluation of the interpolant p(z) in only O(N ) operations once the weights b n are precomputed. In addition, the symmetry of this formula allows one to renormalize all the weights b n by the same constant without affecting the interpolant. It is known that when the interpolation nodes z n correspond to a well-conditioned interpolation operator, then the weights b n all have comparable magnitude, leading to a well-conditioned numerical procedure. With Leja sequences, we are essentially interested in v-weighted polynomial interpolation. (Recall (3).) This means that while we want to produce a polynomial interpolant f , we do so by interpolating the function vf using v-weighted polynomials. With this in mind, it is straightforward to show that the v-weighted analogue of (7) is the following polynomial formula:
p(z) = 1 v(z) N n=1 v n f n v (z) = N n=1 b v n vnfn v−vn N n=1 b v n vn z−zn ,(8)
where
v n v(z n ), v are v-weighted Lagrange polynomials, and b v n are v-weighted Barycentric weights: v n (z) = v(z) v n n (z), b v n = b n v n .
For a sequence of Leja nodes generated according to (4), we use the Barycentric weights b v n given above to perform interpolation. We observe in practice that with this normalization that the weights b v n are all of comparable magnitude, just as we expect them to be for a well-conditioned interpolation problem.
Note that we do not necessarily avoid any troublesome numerical computations in the reformulated case (8); we have merely recast the problem into one that appears numerically well-conditioned. The actual process of interpolating f by an unweighted polynomial on an unbounded domain will still be mathematically ill-conditioned.
Sparse grids with univariate Leja rules
The approximation of a quantity depending on a finite number of Euclidean-like parameters is difficult when the parametric dimension d is large. It is well-known that an approximation to an r-times differentiable function converges with a rate of O(N −r/d ) [2]. Let f : Γ z → R for Γ z ⊂ R d be a function that we wish to approximate and Γ z be the domain of the possibly highdimensional parameter z upon which the function depends. Spatial and temporal variables are modeled separately. For simplicity we assume that Γ z is an isotropic tensor-product domain, so that the one-dimensional restricted variables z k with z = (z 1 , . . . , z d ) all take values on the same restricted one-dimensional space.
When f is approximated via a sampling procedure, the curse of dimensionality is readily apparent:
let Ξ 1 ⊂ R be an n-point nodal set in one dimension with associated quadrature weights w k . A tensorization of this quadrature rule over d dimensions yields the nodal set
Ξ d = d j=1 Ξ 1 , N = |Ξ d | = n d .(9)
The growth of the size of the tensorized set Ξ d usually makes it infeasible for usage in highdimensional approximation methods. (This is true both in cases when d is not small and fixed and n is increased, or when n is fixed and d is increased.) There are alternatives to tensor constructions, but approximation with any space-filling design requires O(n d ) samples, where n is the number of samples 'per dimension'.
An alternative to space-filling designs is the popular sparse grid, so named because of its geometrically dispersed distribution in Γ z . Like tensor constructions, sparse grids tensorize univariate nodal arrays, but sparse grids also attempt to delay the impact of the curse of dimensionality by taking only certain combinations of tensor products. We delay introduction of sparse grids until Section 4; for now we concentrate on motivating our choice of univariate rule: Leja sequences.
4.1.
Common univariate rules. The sparse grid construction requires specification of a univariate grid Ξ 1 l , for l = 0, 1, . . .. Several choices for these univariate grids work well, and among the most popular are a Clenshaw-Curtis (CC) grid, or grids associated with Gauss quadrature rules [22]. For concreteness, we consider the one-dimensional finite interval [−1, 1]. Then, for example, we may choose CC and we have the following grid for any level l:
Ξ l = {z l,n } N l n=1 , z l,N l +1−n = cos (n − 1)π N l − 1 ,
for n = 1, . . . , N l , and N l = 2 l + 1. This particular choice for Ξ 1 l is popular for two reasons: (i) Ξ l ⊂ Ξ l+1 allowing for hierarchical approximation and adaptive refinement with as few model evaluations as possible, and (ii) the sequence Ξ l is known to be both an excellent interpolatory and quadrature grid. Of course, one apparent concern is that |Ξ l+1 | − |Ξ l | = N l+1 − N l = 2 l , which grows exponentially with the level l. This means that each stage of refinement requires addition of a large number of points. In general, a large number of nodes is not necessarily adverse so long as the resulting grids have some optimality regarding, e.g., maximum degree of polynomial integration [42].
An alternative univariate rule that is competitive for quadrature purposes is the Gauss-Patterson grid [41] wherein one constructs a grid that is a subset of a given Gauss quadrature grid, and satisfies some polynomial integration optimality conditions. However, it is not always possible to construct such grids depending on (i) the cardinality of the subset X l and (ii) the weight function ω. Even when such construction is possible, construction of the Gauss-Patterson grid requires implementation of a nontrivial algorithm, and it is frequently easier to precompute and store the grids, making the method inflexible with respect to the choice of density ω.
There are several locally adaptive strategies for sparse grids that are also successful in combating the curse of dimensionality [27,32]. One method that has enjoyed recent success is the locally adaptive, high order, generalized sparse grid construction [27]. In this setup, one uses a highorder Lagrange polynomial basis as the univariate building block for a local high-order polynomial approximation; because the approximation is local, targeted adaptive strategies that utilize the grid hierarchical surpluses are naturally applicable and effective. However, the adaptation is usually (locally) uniform and it is well-known that high-order polynomial approximation on a uniform grid raises computational challenges.
We propose use of (weighted) Leja sequences as univariate building blocks for an adaptive Smolyak sparse grid constructions. Leja sequences can easily add an arbitrary number of samples at each stage, and have good interpolatory and quadrature properties, making them excellent ingredients for the Smolyak algorithm. Leja sequences have been used a sparse grid building blocks before: [13,12], but we believe this is the first investigation into adaptive hierarchical approximations for high-dimensional approximation.
Having discussed univariate Leja sequences at length in Section 3, we may now construct standard sparse grids using Leja sequences as building blocks. Let z = (z 1 , . . . , z d ) ∈ I z ⊆ R d be a random variable with probability density function ω(z) : I z → R, and assume that the components of X are mutually independent so that I z is a tensor-product domain
I z = d j=1 I zj .
We let ω i : I zj → R denote the marginal PDF of z i , so that ω(z) = d j=1 ω j (z j ). Sparse grids [9] approximate f via a weighted linear combination of basis functions (10) I
n [f ] := f n = n k=1 v k Ψ k (z)
The approximation is constructed on a set of anisotropic grids Ξ on the domain I z where = (l 1 , . . . , l d ) ∈ N d is a multi-index denoting the level of refinement of the grid in each dimension. These rectangular grids are Cartesian product of nested one-dimensional grid points Ξ l = {ξ l,i : i < 0 ≤ i ≤ m l } Ξ = Ξ l1 × · · · Ξ l d The number of points m l of a one-dimensional grid of a given level is dependent on the growth rate of the quadrature rule chosen.
The multivariate basis functions Ψ k are a tensor product of one dimensional basis functions. Adopting the multi-index notation used above we have
(11) Ψ ,i (z) = d n=1 ψ n ,in (z n )
where i determines the location of a given grid point. There is a one-to-one relationship between Ψ k in (10) and Ψ ,i and each Ψ ,i is uniquely associated with a grid point z ,i = (ξ 1 ,i1 , . . . , ξ d ,i d ) ∈ Ξ . Many different one-dimensional basis functions ψ n ,in (ξ n ) can be used. In the following we employ one-dimensional Lagrange polynomials for the functions ψ n,in . The multi-dimensional basis (11) spans the discrete space V ⊂ L 2 (I z )
V = span {Ψ ,i : i ∈ K } K = {i ∈ N d 0 : i k = 0, . . . , m l k , k = 1, .
. . , d} These discrete spaces can be further decomposed into hierarchical difference spaces
W = V \ V d n=0 V −en
The subspaces W consists of all basis functions Ψ ,i ∈ V which are not included in any of the spaces V k smaller than V , i.e. with k < with < the lexicographic partial ordering on multi-indices. These hierarchical difference spaces can be used to decompose the input space such that
V = k≤ W and L 2 (I z ) = ∞ k1=0 · · · ∞ k d =0 W k = k∈R d W k
For numerical purposes we must truncate the number of difference spaces used to construct V . Traditional isotropic sparse grids can be obtained by all hierarchical subspaces W with and index set that satisfy
(12) L = { : | | 1 ≤ l}
Given a truncation, such as the a priori one above or one which has been determined adaptively, f can be approximated by
f n = ∈L f , f = i∈I v ,i Ψ ,i (z) (13) where I = {i : Ψ ,i ∈ W }.
Here we note that the v ,j are the coefficient values of the hierarchical product basis, also known as the hierarchical surplus. The surpluses are simply the difference between the function value and the sparse grid approximation at a point, not already in the sparse grid. That is
v ,j = f (z ,i ) − f n (z ,i ), L ∩ = ∅
The particular choice of sparse grid in this paper is one constructed with univariate hierarchical Leja points: the sequence of points that we use for each dimension to evaluate the surplus and construct the interpolant is a univariate Leja sequence. In this way, the point sets z ,i are nested, and the number of points to add at each level can be as small or large as we wish. (I.e. the Leja choice allows a great deal of granularity for refinement.)
Dimension adaptivity.
The dimensional adaptivity of our algorithm in this section is based on the idea presented in [22]. We begin with a low-level isotropic sparse grid approximation with a set of subspaces representing the current approximation L and the set of active subspaces A that indicate the levels for potential refinement. Often L = W 0 and A = {W e k , k = 1 . . . , d}. We then choose W ∈ A with the largest error indicator γ and refine that subspace. Here we define the error indicator γ as
γ = Γz (f ) 2 dω(z) − Γz (f )dω(z) 2 , η = ∈A γ(14)
The indicator γ measures the contribution of the subspace to the variance of f n and the global indicator η measures the contribution of all active subspaces to the variance of f n . These indicators are calculated by transforming the Lagrange interpolant on each hierarchical subspace into a Polynomial Chaos Expansion that is orthogonal to the (possibly mixed) distribution weight ω(z). The cost of this transformation is linear in terms of the number of subspace points [10]. The chosen subspace for refinement with index is refined by adding all indices W k with k = +e n , n = 1, . . . , d that satisfy the following admissibility criterion (15) − e k ∈ L for 1 ≤ k ≤ d, l k > 1
The active set A is then rebuilt by adding each subspace corresponding to the indices from (15). This process continues until a computational budget limiting the number of model samples (grid points) is reached or a global error indicator drops below a predefined threshold. Pseudo-code for the dimension adaptive algorithm is shown in Algorithm 1.
The INDICATOR and TERMINATE routines in Algorithm 1 control which subspaces are added to the sparse grid via the use of a subspace error and global error metric. The indicators respectively provide estimates of the contribution of a subspace to reducing the error in the interpolant, and the error in the entire interpolant. A := A ∪ J % Add the forward neighbors to the active index set 10: end while
Numerical examples
We consider several multidimensional examples below that compare the Smolyak-Leja algorithm with a more standard Clenshaw-Curtis-Smolyak algorithm. Effectively, we see that the Leja construction is competitive (usually superior) to Clenshaw-Curtis when an interpolation metric is used. However, they appear suboptimal when a quadrature metric is evaluated. This is not surprising as Leja sequences are constructed with the goal of interpolation and not necessarily for quadrature.
Throughout these examples we compute discrete 2 errors ε 2 using 100,000 random samples taken in a Monte-Carlo fashion from the distribution of the input variable z. We also report absolute errors in the sparse grid mean ε µ and variance ε σ 2 where the exact moments are computed using a highresolution sparse grid that was refined so that its 2 error was in the order of machine precision. The error metric ε 2 is simply the discrete 2 error (RMSE). 5.1. Random oscillator. This section investigates the relative performance of the sparse grids when approximating the output from a model of linear oscillator subject to external forcing with six unknown parameters. That is,
(16) d 2 x dt 2 (t, z) + γ dx dt + kx = f cos(ωt),
subject to the initial conditions
(17) x(0) = x 0 ,ẋ(0) = x 1 ,
where we assume the damping coefficient γ, spring constant k, forcing amplitude f and frequency ω, and the initial conditions x 0 and x 1 are all uncertain. We solve (16) analytically to allow us to avoid consideration of discretization errors in our investigation. For this choice of random parameters any parameter realization in I z will produce an underdamped harmonic oscillator. Figure 7 compares the 2 accuracy in the sparse grid interpolants obtained using Clenshaw-Curtis nodes and Leja nodes. Although both univariate rules have similar interpolation properties in one-dimension, the one-at-a-time nestedness of the Leja rule produces, in this higher-dimensional setting, an approximation that is significantly more accurate than the approximation based upon the Clenshaw-Curtis quadrature rule. Figure 8 compares the accuracy of sparse grids based upon the univariate Clenshaw-Curtis and Leja nodes. Again the Leja interpolation sequence produces a more accurate interpolant for a given number of function evaluations, but suffers when evaluating quadrature quantities such as the mean shown in the left-hand pane. In this case as with many others, the Clenshaw-Curtis quadrature rule produces a more accurate estimate of the mean of the function. This statement is consistent with the one-dimensional results shown in Figure 5. Furthermore assume that the random diffusivity satisfies
(21) a(x, z) =ā + σ a d k=1 λ k φ k (x)z k where {λ k } d k=1 and {φ k (x)} d k=1
are, respectively, the eigenvalues and eigenfunctions of the covariance kernel
C a (x 1 , x 2 ) = exp − (x 1 − x 2 ) 2 l 2 c
The variability of the diffusivity field (21) is controlled by σ a and the correlation length l c which determines the decay of the eigenvalues λ k . Here we wish to approximate the solution u(1/3, z) when d = 40, σ a = 0.021 and l c = 1/14 and z k ∈ [−1, 1], k = 1, . . . , 40 to be independent and uniformly distributed random variables. We solve (21) non-intrusively: for each node z ,ı on our sparse grid, we use a finite-element discretization in x to compute the solution. The comparison between Leja and Clenshaw-Curtis Smolyak construction is shown in Figure 9. In this case, the Leja construction only performs marginally better than the CC approach for interpolation, and exhibits a now-familiar difficulty with quadrature. We explain this difference in the following way: for this equation, we certainly have dimensional anisotropy because the eigenvalues λ k decay. However, if we plot the Figure 10. Resistor network parameter indices l for the subspaces W that contribute significantly to the solution, we will see an ellipsoid shape in index space. Thus, extra refinement performed by CC in certain directions is not wasted because these degrees of freedom can properly resolve mixed terms in parameter space. In this example, the granularity offered by Leja sequences is not needed or utilized.
V 0 R 1 R 2 R P R P +1 R P +2 R 2P −1 R 2P V
5.4.
Resistor network. Consider the electrical resistor network shown in Figure 10 [43]. The network is comprised of d = 2P resistances of uncertain ohmage and the network is driven by a voltage source providing a known potential V 0 . We are interested in determining how the voltage V shown in Figure 10 depends on the d = 2P resistances, which we take as random parameters that are independent and identically distributed Gaussian random variables with mean µ = 1 and standard deviation σ = 0.005. Note for this value of σ, the probability that we encounter negative resistances is extremely small, and so apart from the obvious modeling error of possible negative resistances, no numerical difficulties are introduced. I.e. none of the sparse grid points or random samples used to generate the error resulted in a negative resistance. In this example we set d = 40 and set the reference potential V 0 = 1. In this case our comparison is not with CC, but with a nested Genz-Keister rule, which is one of the the standard ways to to peform nested interpolation and quadrature under a Gaussian weight [21]. Here our Leja rule is generated on an infinite domain with univariate weight function w(z j ) = exp(−(z j − µ) 2 /(2σ 2 )). In Figure 11 we see significant interpolatory improvement with the Leja rule, and even the quadrature results are competitive in this example.
Summary
We have used Leja interpolatory grids as one-dimensional composite rules for adaptive Smolyak sparse grid construction. In one dimension, Leja rules are excellent interpolation grids and are a sequence, allowing one to generated nested rules with arbitrary granularity. We have shown that for several classical one-dimensional weight functions of interest, a corresponding weighted Leja rule produces a sequence whose empirical distribution asymptotically coincides with the limiting distribution for the Gauss quadrature nodes of the same family.
Using Leja rules to build up sparse grids in multiple dimensions grants the user a greater dexterity in adaptive refinement compared to more standard composite rules such as Clenshaw-Curtis. We have shown via several examples that the Leja rule can outperform standard high-order Smolyak constructions in interpolatory metrics, but are suboptimal when considering quadrature metrics. Design of Leja-like rules that are effective for approximating integrals will be the subject of future investigation.
7. Proof of Theorem 3.2 7.1. Weighted potential theory. In this section we give the proof of Theorems 3.1 and 3.2. Indeed, if we show Theorem 3.2, then well-established results imply Theorem 3.1. We recall our notation: w is a given weight function associated to z. v is the square root of w. v is related to w through Theorem 3.1, and is essentially the square root of w, ignoring polynomial factors.
The proof of Theorem 3.2 relies on results from weighted potential theory. Potential theory is frequently explored in the complex plane C, but we will restrict ourselves to subsets of C lying on the real axis. An excellent exposition of univariate weighted potential theory with comprehensive historical references is given in [45].
Let domain I ⊆ R with non-negative weight function v be given. On rather mild assumptions on v and I then there exists a unique probability measure denoted µ v under which a weighted logarithmic energy for Q = − log v is minimized:
µ v = argmin µ : µ(I)=1 I I log 1 |η − ξ| v(η) v(ξ) dµ(η)dµ(ξ)(22)
Q(ξ)dµ(ξ)
A common physical analogy of the above is to find the minimum-energy electrostatic charge distribution (measure) in a region I when an external electrostatic field Q is applied. The measure µ v is the weighted equilibrium measure of I in the presence of the field Q. Even if I is unbounded, µ v always has compact support. A discrete version of the above optimization problem is furnished by the concept of Fekete points. An array of points which maximizes a weighted Vandermonde matrix determinant is called a set of Fekete points; this weighted determinant is a discrete, unnormalized Monte-Carlo-like estimate of the negative exponential of the integral in (22). A set of points ξ 0 , . . . , ξ n is an array of v-weighted
|z j − z k | v(z j ) v(z k ),(23)
where V (z 0 , . . . , z n ) is modulus determinant of the (n + 1) × (n + 1) Vandermonde matrix W with entries W r,s = z s r for 0 ≤ r, s ≤ n. Fekete sets are not necessarily unique, and are notoriously difficult to compute exactly. However, Fekete points are excellent interpolation/approximation nodal sets. The n → ∞ limiting behavior of the weighted Vandermonde determinant for Fekete nodes is described by the weighted transfinite diameter δ v (= weighted logarithmic capacity [35]). Let V n denote the maximum weighted determinant from (23) achieved by Fekete points. Then
lim n→∞ (V n ) 1/mn = δ v ,(24)
where m n 1 + 2 + · · · + n = n(n+1)
2
. Any triangular array of nodes whose determinant behaves like (24) is called asymptotically weighted Fekete, alluding to the fact that the determinant is not exactly maximum, but is asymptotically comparable to Fekete points.
The connection between Fekete nodes and the equilibrium measure is established by the following result: if a triangular array of nodes {ξ j,n } j≤n is asymptotically weighted Fekete, then its empirical measure distributionally converges to the weighted equilibrium measure: where δ z is the Dirac mass centered at z.
Here (and elsewhere when discussing convergence of measures) equality is in the weak- * sense. Our strategy is to first show Theorem 3.2, that a contracted version of the w-weighted Leja points from (4) are v-weighted asymptotically Fekete. This will allow us to immediately use Lemma 7.1 to conclude Theorem 3.1.
To proceed, we will need the notion of Chebyshev constants. Given a potential-theoretic admissible weight v on I, the weighted Chebyshev constant of order n is
τ (n) v = inf v n (z)p n (z) I | p n (z) = z n + q n−1 (z) ∀ q n−1 ∈ P n−1 ,
where · I is the sup-norm on the domain I, and P n is the space of polynomials of degree n or less.
The sequence of constants τ
(n) v 1/n
is a decreasing sequence with a limit:
inf n τ (n) v 1/n = lim n→∞ τ (n) v 1/n τ v
This limit is called the weighted Chebyshev constant [35], and coincides with the transfinite diameter δ v in the unweighted case, but is distinct in general weighted scenarios. Note that by definition, v n (z)(z n + q n−1 (z)) I ≥ (τ v ) n .
for any q n−1 ∈ P n−1 . In general, the relation between the Chebyshev constant τ v and the transfinite diameter δ v is given by
τ v = δ v exp S Q(z) dµ v (z) , S = supp µ v .(26)
For the Jacobi, Hermite, and Laguerre cases mentioned in Theorem 3.1, our goal is to prove the result (6). Our contraction proofs are different for the cases of bounded I versus unbounded I. We first consider the bounded Jacobi case. 7.2. Jacobi case. We prove that the Leja maximization scheme (4) produces nodes whose limiting distribution is the unweighted equilibrium measure, or the arcsine measure. For the unweighted case we have v ≡ 1, the contraction is k n ≡ 1 (and so is omitted), and the transfinite diameter δ and the Chebyshev constant τ are identical:
δ = τ(27)
We begin by considering the proof assuming α, β ≥ 0. We have v(z) = f (α/2,β/2) (z) = (1 − z) α/2 (1 + z) β/2 for α, β ≥ 0 over I = [−1, 1]. We will need the following constants:
C 1 = C 1 (α, β) f (α/2,β/2) ∞ < ∞ C 2 = C 2 (α, β) f ( α/2 , β/2 ) (z) f (α/2,β/2) (z) ∞ < ∞
With these constants, we have
1 ≥ v(z) C 1 = f (α/2,β/2) (z) C 1 ≥ f ( α/2 , β/2 ) (z) C 1 C 2 ,
We now note that α/2 and β/2 are integers, and for shorthand we write γ = α/2 + β/2 . Then the right-hand side of the above equation is a monic polynomial (modulo sign) of degree γ. Therefore, we have
V (z 0 , . . . , z n ) = n j=1 j−1 k=0 |z j − z k | ≥ 1 C n 1 n j=1 v(z j ) j−1 k=0 |z j − z k | = 1 C n 1 n j=1 v(z) j−1 k=0 |z − z k | I ≥ 1 C n 1 C n 2 n j=1 f ( α/2 , β/2 ) (z) j−1 k=0 |z − z k | I ≥ 1 C n 1 C n 2 n j=1 τ j+γ = τ mn τ nγ C n 1 C n 2
Thus, we have
τ mn τ γ C 1 C 2 n ≤ V (z 0 , . . . , z n ) ≤ max x0,...xn∈Ξ V (x 0 , . . . , x n )
We raise all the above to the 1/m n power, which yields We have thus proven that Jacobi-weighted Leja sequences for α, β ≥ 0 are v-asymptotically Fekete, i.e., we have proven (6).
which in turn implies that for our particular choice of family of weight functions:
v n (z j,n ) = v n−1 (z j,n−1 ) = · · · = v (z j,1 ) = v (z j )
We wish to prove that the array z j,n is v-weighted asymptotically Fekete. I.e., we wish to prove lim n→∞ |det V (z 0,n , . . . , z n,n )| n j=0 v n (z j,n )
1/mn = δ v ,(30)
We first consider the case with the parameter µ > 0. Using the explicit Vandermonde determinant formula (23) yields the following formula that we wish to prove:
lim n→∞ v n (z 0,n ) n j=1 v n (z j,n ) j−1 k=0 |z j,n − z k,n | 1/mn = δ v .(31)
As with the Jacobi case, showing that the limit is ≤ δ v is straightforward from the definition of δ v , so we concentrate on the inequality ≥. We have v n (z 0,n ) n j=1 v n (z j,n ) j−1 k=0 |z j,n − z k,n | (29) = v(z 0 )
n j=1 n −j/α v(z j ) j−1 k=0 |z j − z k | (a) .(32)
For the term (a), we note that z j = 0 for j ≥ 1 since µ > 0. Therefore, we may write this term as
(a) = |z j | −µ (aa) v(z j ) j−1 k=0 |z j − z k | (ab)
.
To compute lower bounds for terms (aa) and (ab) we will need the following notation: for the v-weighted equilibrium measure, we have S = supp µ v = [−c, c], S n = −n 1/α c, n 1/α c .
The constant c is 2 1/α b(α), with b(α) given in the "Hermite" case of Table 1, or in Table 2.
To bound (aa), we know that z j was computed from the optimization (4). This allows us to derive an upper bound for the magnitude of these weighted Leja points. Proof. We make use of the following result [34] that compactifies the set on which the supremum norm of a weighted polynomial "lives" for our exponential weights: v(z) (z n + q n−1 ) R = v(z) (z n + q n−1 ) Sn , ∀ q n−1 ∈ P n−1 (35a) v n (z) (z n + q n−1 ) R = v n (z) (z n + q n−1 ) S , ∀ q n−1 ∈ P n−1 (35b) where S and S n are given by (33). Note that z j is chosen as in (4) with
v(z j ) j−1 k=0 |z j − z k | = v(z) j−1 k=0 |z − z k | = z µ v(z) j−1 k=0 |z − z k |
If µ is an integer, then the argument under the norm is a v-weighted monic polynomial of degree µ + j. Thus, the extremum of the argument is achieved on the set S µ+j , which implies that the smallest-magnitude maximizer as stipulated in (4b) satisfies z j ∈ S µ+j . The result (34) follows.
If µ is not an integer, then consider the function
f (z) = |z| M v(z) j−1 k=0 |z − z k |
We know that z j is a maxmizer of |z| µ−M f (z). Let z * be the smallest-magnitude maximizer of f (z).
Suppose |z j | > |z * |; since µ − M < 0 we have
|z j | µ−M f (z j ) < |z * | µ−M f (z j ) < |z * | µ−M f (z * ),
which is a contradiction since z j maximizes the norm of z µ−M f (z). Therefore, |z j | ≤ |z * |. But z * maximizes f , which is a v-weighted polynomial of degree M + j. Therefore, by (35a) we have z * ∈ S M +j . This in turn implies z j ∈ S M +j , and again (34) follows.
We likewise have a bound for the (ab) term:
Lemma 7.3. For every j, we have the following bound for term (ab):
v(z j )
j−1 k=0 |z j − z k | ≥ j (j+µ)/α c M −µ (τ v ) j+M ,(36)
where c is the constant from (33).
Proof. Since, by (4), z j maximizes the left-hand side of (36), we are concerned with bounding v(z)
j−1 k=0 |z − z k | = z µ v(z) j−1 k=0 |z − z k | .
We contract z ← j −1/α z to obtain
z µ v(z) j−1 k=0 |z − z k | = j (j+µ)/α z µ v j (z) j−1 k=0 |z − z k,j | .(37a)
We also have the following properties: = v(z)
j−1 k=0 |z − z k | (37a) = j (j+µ)/α z µ v j (z) j−1 k=0 |z − z k,j | ≥ j (j+µ)/α z µ v j (z) j−1 k=0 |z − z k,j | S (37b) ≥ j (j+µ)/α 1 c M −µ z M v j (z) j−1 k=0 |z − z k,j | S (37c) ≥ j (j+µ)/α 1 c M −µ z M v j+M (z) j−1 k=0 |z − z k,j | S (35b) = j (j+µ)/α 1 c M −µ z M v j+M (z) j−1 k=0 |z − z k,j | R (25) ≥ j (j+µ)/α 1 c M −µ (τ v ) j+M .
With these two lemmas obtained, we can bound term (a) from (32)
≥ τ v exp − 1 2α = δ v ,
and so we have proven (31), that the weighted Leja points are asymptotically weighted Fekete. For the case µ < 0, we may repeat arguments for the Jacobi α, β < 0 case: µ < 0 implies that node z 1 = 0 is chosen (assuming z 0 = 0), which then reverts the w-weighted Leja objective (4) to one where µ > 0. 7.4. Laguerre case. The Laguerre result for weighted Leja sequences defined by (4) can easily be obtained by following the argument in Section 7.3, so we omit the details. We only mention that for weights of the form v(z) = exp(−|z|) on [0, ∞), we can directly obtain from (26):
τ v = √ eδ v .
I.e., (28) holds with α = 1 [33]. With this, the remainder of the proof follows in precisely the same fashion as the Hermite case.
Figure 1 .
1Theoretical asymptotic distribution of Gauss quadrature nodes for the family of polynomials orthogonal under weight ω, and empirical distributions for the first 50 associated Leja points (denoted Z 50 in both cases). Left: uniform weight w ≡ 1 with limiting (arcsine) distribution. Right: Gaussian density weight w(z) = exp(−z 2 ) with a 50-point Leja sequence, contracted to the origin by a factor of √ 50, along with the contracted limiting distribution of the Gauss quadrature nodes. (See tables 1 and 2)
Figure 2 .
2Sequential addition of nodes (top to bottom) via iteration on the Leja objective (4). Left: A Leja sequence on [−1, 1] with v = w ≡ 1 (i.e., unweighted). Right: A weighted Leja sequence on R with v 2 = w(z) = exp(−z 2 ).
Figure 3 .
3Plots of asymptotic densities for contracted weighted Leja (and Gauss) points. Left: Limiting density for the Laguerre weight w(z) = exp(−z). Right: Limiting density for the exponential weight w(z) = exp(−|z| α ) for various α. See
graphs κ 1 N
1−1 for three choices of w: the uniform density on [−1, 1], an oscillatory weight w on the same domain, and finally the Gaussian density function w ∝ exp(−z 2 ) on R. We see that the quadrature rules are all relatively well-conditioned. The accuracy of the rules are considered in the following examples:
Figure 4 .
4Left: Absolute condition number for Leja interpolatory quadrature rules for various weight functions ω: (top) ω ∝ 1 the uniform density, (middle) w(z) ∝ 1 2 I 0 (1 + z) + J 0 (50 + 50z), where J 0 (I 0 ) is the (modified) Bessel function of the first kind, (bottom) w(z) ∝ exp(−z 2 ) the Gaussian density. Right: plot of the second weight function, w(z) ∝ 1 2 I 0 (1 + z) + J 0 (50 + 50z).
Figure 5 .Figure 6 .
56Left: Maximum errors for the one-dimensional Runge function f 1 with Leja, Clenshaw-Curtis, and Gauss-Patterson nested refinements. Right: Maximum hierarchical surplus at each level. Left: Errors and the hierarchical surplus at each level for a one-dimensional Leja refinement of oscillatory function f 2 . Right: Errors and surplus using a Clenshaw-Curtis refinement.
Algorithm 1 J
1INTERPOLATE[f (z),L,A,τ ,n]→ f n 1: For a given L the points in the sparse grid are Ξ := ∈L Ξ . 2: The number of sparse grid points are N = #Ξ 3: while NOT TERMINATE[A,N ,τ ,n] do 4: W := argmax W ∈A γ % Determine the subspace with the highest priority 5: A := A \ W % Remove W from the active := REFINE[W ,L] % Find all admissible forward neighbors of W 8: γ := INDICATOR[W ] ∀ W ∈ J % Calculate the priority of the neighbors 9:
Let us choose our quantity of interest to be the position x(T ) of the osciallator at T = 20 seconds and let z = (γ, k, f, ω, x 0 , x 1 ) where γ ∈ [0.08, 0.12], k ∈ [0.03, 0.04], f ∈ [0.08, 0.12], ω ∈ [0.8, 1.2], x 0 ∈ [0.45, 0.55], x 1 ∈ [−0.05, 0.05].
Figure 7 .
7Convergence of RMSE in the sparse grid approximation of the oscillator position with respect to the number of model evaluations.5.2. Borehole model. For the next numerical demonstration, consider the following model of water flow through unkonwn parameters are uniform random variables z = (z 1 , . . . , z 8 ) with the following bounds: z 1 := r w ∈ [0.05, 0.15] (meters) denotes the radius of borehole, z 2 := r ∈ [100, 50000] (meters) the radius of influence, z 3 := T u ∈ [63070, 115600] (meters 2 /years) the transmissivity of upper aquifer, z 4 := H u ∈ [990, 1110] (meters) the potentiometric head of upper aquifer, z 5 := T l ∈ [63.1, 116] (meters 2 /years) the transmissivity of lower aquifer, z 6 := H l ∈ [700, 820] (meters) the potentiometric head of lower aquifer, z 7 := L ∈ [1120, 1680] (meters) the length of borehole, and z 8 := K w ∈ [9855, 12045] (meters/year) the hydraulic conductivity of borehole.
Figure 8 .
8Convergence of the mean and RMSE in the sparse grid approximation of the borehole model with respect to the number of model evaluations.5.3. Heterogeneous diffusion equation.In this section, we consider the heterogeneous diffusion equation in one-spatial dimension subject to uncertainty in the diffusivity coefficient. For d
Figure 9 .
9Convergence of variance and RMSE in the sparse grid approximation of the solution to the diffusion equation with respect to the number of model evaluations
Figure 11 .
11Convergence of the Mean and RMSE in the sparse grid approximation of V with respect to the number of model evaluations.
Fekete points if it satisfies {ξ 0 , . . . ξ n } = argmax z0,...,zn V (z 0 , . . . , z n ) n j=0 v n (z j ) = argmax z0,...,zn 0≤j<k≤n
Lemma 7.1 ([49,45]). Suppose ξ j,n is a triangular array of points satisfying lim n→∞ V (ξ 0,n , . . . , ξ n,n ) ξj,n = µ v ,
0 , . . . , z n )] 1/mn ≥ τ = δ lim n→∞ [V (z 0 , . . . , z n )] 1/mn ≤δ
Lemma 7 . 2 .
72Let M = µ . For each j, we have the following bound for term (aa):|z j | −µ ≥ (M + j) −µ/α c −µ(34)
j) µ/α c M (τ v ) j+MWe can therefore bound the entire weighted determinant from (32):
Table 1 .
1Asymptotic distributions and densities to which contracted weighted Leja sequences (and Gauss quadrature nodes) converge. Given a weighted Leja sequence zn, the formulae to which the quantities in this table correspond is given by Theorem 3.1.
For −1 < α, β < 0, we proceed without loss under the assumption that both α and β are negative. In this case then ξ 0 = ±1 is arbitrarily chosen so thatBut since v is infinite at the endpoints ±1, then the maximum is achieved at one of these points, say z 1 = +1. Then the Leja iteration continues:Now since z 1 = +1, then the last term is identical to (1 − z), and we may absorb this term into the weight, giving a more explicit formula:But since α/2 + 1 > 0, then the maximum (infinity) is now achieved at z = −1, which becomes z 2 . Now we choose z 3 and again combine the terms |z − z 1 | and |z − z 2 | into the weight function:Proceeding in this way, future Leja points are chosen according toIn other words, we choose z n as a Leja optimization with a new weight function f (α/2+1,β/2+1) , whose parameters are α/2 + 1 > 0 and β/2 + 1 > 0. This effectively reduces the problem to the case where α, β > 0. Then as before, this Leja sequence is asymptotically (unweighted) Fekete and so has empirical measure that converges to the arcsine measure µ.7.3.Hermite case. In this section, I = R (and for shorthand write · I = · ) and w(z) = z 2µ exp(−2|z| α ). (Compared to Theorem 3.1, in this section we have redefined z ← 2 1/α z to make the computations cleaner.) The contraction factor is k n = n −1/α . As usual, define v(z) = w(z). The limit weight v from Theorem 3.1 is v(z) = exp(−|z| α ). For this v, it is known that the weighted Chebyshev constant is related to the weighted transfinite diameter by the relationSee e.g.,[34,35]. We choose a weighted Leja sequence of points z n according to(4). The negative log-weight of v is Q = |z| α , and it is a homogeneous function with homogeneity exponent α:Then the following are easily verified: for any n > 0:Given the Leja sequence z n from (4), then for each n we define progressively contracted versions of the grid: z j,n = n −1/α z j , j, n = 1, . . .We will only need z j,n for j ≤ n: we view z j,n as a triangular array of points with j ≤ n. We note that one can easily transform between one contracted set of nodes and another:z j,n = n − 1 n 1/α z j,n−1 ,
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|
[] |
[
"Fractal-like Distributions over the Rational Numbers in High-throughput Biological and Clinical Data",
"Fractal-like Distributions over the Rational Numbers in High-throughput Biological and Clinical Data"
] |
[
"Vladimir Trifonov \nDepartment of Biomedical Informatics\nCenter for Computational Biology and Bioinformatics\nColumbia University\n10032New YorkNYUSA\n",
"Laura Pasqualucci \nInstitute for Cancer Genetics\nDepartment of Pathology and Cell Biology\nHerbert Irving Comprehensive Cancer Center\nColumbia University\n10032New YorkNYUSA\n",
"Riccardo Dalla-Favera \nInstitute for Cancer Genetics\nDepartment of Pathology and Cell Biology\nDepartment of Genetics and Development\nHerbert Irving Comprehensive Cancer Center\nColumbia University\n10032New YorkNYUSA\n",
"Raul Rabadan \nDepartment of Biomedical Informatics\nCenter for Computational Biology and Bioinformatics\nColumbia University\n10032New YorkNYUSA\n"
] |
[
"Department of Biomedical Informatics\nCenter for Computational Biology and Bioinformatics\nColumbia University\n10032New YorkNYUSA",
"Institute for Cancer Genetics\nDepartment of Pathology and Cell Biology\nHerbert Irving Comprehensive Cancer Center\nColumbia University\n10032New YorkNYUSA",
"Institute for Cancer Genetics\nDepartment of Pathology and Cell Biology\nDepartment of Genetics and Development\nHerbert Irving Comprehensive Cancer Center\nColumbia University\n10032New YorkNYUSA",
"Department of Biomedical Informatics\nCenter for Computational Biology and Bioinformatics\nColumbia University\n10032New YorkNYUSA"
] |
[] |
Recent developments in extracting and processing biological and clinical data are allowing quantitative approaches to studying living systems. High-throughput sequencing (HTS), expression profiles, proteomics, and electronic health records (EHR) are some examples of such technologies. Extracting meaningful information from those technologies requires careful analysis of the large volumes of data they produce. In this note, we present a set of fractal-like distributions that commonly appear in the analysis of such data. The first set of examples are drawn from a HTS experiment. Here, the distributions appear as part of the evaluation of the error rate of the sequencing and the identification of tumorogenic genomic alterations. The other examples are obtained from risk factor evaluation and analysis of relative disease prevalence and co-mordbidity as these appear in EHR. The distributions are also relevant to identification of subclonal populations in tumors and the study of quasi-species and intrahost diversity of viral populations. SUBJECT AREAS: BIOINFORMATICS COMPUTATIONAL BIOLOGY STATISTICS MATHEMATICS
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10.1038/srep00191
| null | 8,985,050 |
1010.4328
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7c3c168647ca05d6bfe3ab7ac5f0e1fdcb707162
|
Fractal-like Distributions over the Rational Numbers in High-throughput Biological and Clinical Data
Published 13 December 2011
Vladimir Trifonov
Department of Biomedical Informatics
Center for Computational Biology and Bioinformatics
Columbia University
10032New YorkNYUSA
Laura Pasqualucci
Institute for Cancer Genetics
Department of Pathology and Cell Biology
Herbert Irving Comprehensive Cancer Center
Columbia University
10032New YorkNYUSA
Riccardo Dalla-Favera
Institute for Cancer Genetics
Department of Pathology and Cell Biology
Department of Genetics and Development
Herbert Irving Comprehensive Cancer Center
Columbia University
10032New YorkNYUSA
Raul Rabadan
Department of Biomedical Informatics
Center for Computational Biology and Bioinformatics
Columbia University
10032New YorkNYUSA
Fractal-like Distributions over the Rational Numbers in High-throughput Biological and Clinical Data
Published 13 December 201110.1038/srep00191Received 8 August 2011 Accepted 31 October 2011Correspondence and requests for materials should be addressed to V.T. (vladot@c2b2. columbia.edu)
Recent developments in extracting and processing biological and clinical data are allowing quantitative approaches to studying living systems. High-throughput sequencing (HTS), expression profiles, proteomics, and electronic health records (EHR) are some examples of such technologies. Extracting meaningful information from those technologies requires careful analysis of the large volumes of data they produce. In this note, we present a set of fractal-like distributions that commonly appear in the analysis of such data. The first set of examples are drawn from a HTS experiment. Here, the distributions appear as part of the evaluation of the error rate of the sequencing and the identification of tumorogenic genomic alterations. The other examples are obtained from risk factor evaluation and analysis of relative disease prevalence and co-mordbidity as these appear in EHR. The distributions are also relevant to identification of subclonal populations in tumors and the study of quasi-species and intrahost diversity of viral populations. SUBJECT AREAS: BIOINFORMATICS COMPUTATIONAL BIOLOGY STATISTICS MATHEMATICS
T he large volumes of data obtained by recent technological developments, such as next-generation sequencing and expression profiles, are providing novel and complementary ways to studying biological systems. In order to extract meaningful, statistically significant information from such data, mathematical methods are being developed, implemented, and tested in various contexts. For example, it is believed that most tumors are due to somatic mutations that lead to an uncontrolled cell growth. Next-generation sequencing technologies produce hundreds of gigabases of genetic data, providing a way to identify genes responsible for the tumorigenic process by comparing the genome of the tumor and the normal tissue [1][2][3][4][5][6][7] .
In this note, we point out some interesting properties of the ratios of natural numbers obtained in a biological/ clinical setting. The ratios of interest can be seen as sampled from a distribution over the rational numbers in the unit interval. Consider pairs of positive integers, n and m, sampled from a distribution with probability f(n, m). The ratio q 5 n/(n 1 m) of one of these numbers by the sum of the two is a rational number in the unit interval. In this way the distribution f(n, m) gives rise to a distribution g(q) supported on the rational numbers in the unit interval. A case of particular interest is when the two integers are drawn independently from the same distribution h(n). As we are going to see, in this case and for h being certain common distributions, such as exponential and power-law, it is possible to have a closed-form expression for g. We will also see that the resulting distributions over the rational numbers possess certain self-similarity properties. Namely, the overall shape of those distributions is similar to Thomae's function (Figure 1, top left). Although irrelevant to our discussion we would like to point out that, similar to Thomae's function, the distributions which we study are rather interesting analytically, because, viewed as functions over the reals, they are continuous on the irrational numbers but not on the rationals.
We will illustrate the appearance of such distributions in real life data with two examples: 1) a next-generation sequencing experiment aimed at identifing genomic variations in cancers and 2) diagnosis data collected at the New York Presbyterian Hospital in several consecutive years. Although the presence of irregular shapes and spikes in empirically occuring distributions of ratios of natural numbers was reported before as a statistical artifact 8 , the authors of this previous work failed to acknowledge the interesting mathematical structure of the underlying distributions. In this work we propose the study of those naturally occurring distributions of rational numbers as an interesting mathematical topic with important clinical and biological applications.
Results
First example: identifying genomic alterations with next-generation sequencing. Our first example comes from a next-generation sequencing experiment of a diffuse large B-cell lymphoma (DLBCL) sample 6,7 . DLBCL is the most common B-cell non-Hodgkin lymphoma in adults, accounting for <40% of all new lymphoma diagnoses. Tumor DNA was extracted from a nodal tumor of a 63 year old female patient. The coding part the genome (the exome) was enriched using Roche NimbleGen Sequence Capture and the enriched product was sequenced using Roche 454 sequencing. The data produced from the experiment were 2 ? 10 6 reads (sequences of DNA) of average length 250 nucleotides. The reads were aligned to the hg18/NCBI36.1 reference human genome. This resulted in a coverage of about 10x of the human exome and the alignment was used to identify genomic variants distinguishing normal and tumor cells. Figure 1 (top right) shows a diagram of the alignment algorithm and the fractal-like distributions obtained from the sequencing experiment (bottom). Figure 2 (top, blue) shows the depth (5number of reads covering a particular position) distribution (coverage) after alignment of the reads. The figure also shows a negative binomial least-square fit of the data. If the reads were obtained from the genome independently and at random, one would expect the coverage to follow a Poisson distribution. As it is, even though restricted to a small part of the genome the coverage might be Poisson, overall, because of the way the sample was processed before sequencing, the means of the Poisson processes in different parts of the genome will vary. The result will be an overdispersion of the depth distribution and a better fit by the negative binomial, known to be a mixture of Poisson distributions with Gamma-distributed means.
Each of the 46 chromosomes of the human genome has two strands and, with the exception of the sex chromosomes X and Y, the human genome is diploid, i.e. each chromosome has a homologous copy. Since the reference genome is given as entirely haploid, the information about which copy of the genome a sample read originates from is not recovered by the alignment. Nonetheless, assuming that a read can originate from each copy of the genome with equal probability and given the coverage of the reference, one can obtain a theoretical coverage of a fixed copy of the genome. Thus the fraction of positions on a fixed copy of the genome covered with k reads is
p k ð Þ~X ? t~k q t ð Þ t k 2 {t ,
where q(t) is the fraction of positions with coverage t, as given in Figure 2 (top, blue). After a simple algebraic simplification it can be shown that, if q is Poiss(l), then p is Poiss(l/2). Furthermore, since the negative binomial is a mixture of Poissons with Gamma-distributed means, we can obtain that if q is NegBin(r, s), then p is NegBin(r, (s/2)/ (12s/2)). Figure 2 (top, green) shows the theoretical coverage of a fixed copy of the human genome obtained from these considerations. Similar reasoning leads us to a predicted coverage of a fixed strand of the human genome shown in Figure 2 (top, black). Although the alignment to the reference does not provide exact information about the origin of a read in the sample, we can still test the prediction about the coverage of a fixed copy of the cancer genome in the following way: take sufficiently many heterozygous positions, i.e. positions at which the two copies of the genome differ, and then consider the number of reads covering such a position and containing one of the variants at that position and the number of Since the information about the strand from which a sample read originates is also lost in the sequencing, here we used the orientation of a read when aligned to the reference as a surrogate for its strand. As can be seen, the predictions closely follow the data, confirming our intuition that the reads come from the four strands of the genome independently.
Our main observation is concerned with the heterozygous positions we used to obtain the data for Figure 2 (bottom). This time we consider the distribution of the ratios of the number of reads covering one of the variants at a particular position in the cancer genome to the total number of reads covering this position and the ratio of the number of reads covering one of the strands to the total number of reads covering the variant. The resulting distributions of ratios are given in Figure 1 (bottom, blue). There are two apparent features of the distributions which drew us to studying them: first, their fractallike self-similar structure, and second, the spikes they contain. We consider the topic of the self-similarity of the distributions in the Methods section and quantify it by computing the fractal dimension of related functions. From a biological point of view the spikes are interesting because at first sight one might decide that they show overrepresentation of certain ratios. For example, for the distribution of variant depth over the total depth, the spike at 0.5 is expected, since we are looking at heterozygous positions, but the spikes at 0.33 and 0.66 are harder to explain biologically since they would mean the significant presence of variants with ploidity other than 2. While such phenomena can occur in cancers because they can present genome aberrations known as copy number alterations, the scale at which the phenomenon is represented here is unusual. We will see that the spikes are due to the discreteness of the data and could actually be explained by a simple stochastic model. Hence regarding the biological conclusions one can draw from next-generation sequencing experiments, the message of our note is that when dealing with biological data the stochastic effects due to the discreteness of the data can be big and attention should be used when drawing conclusions lest one confuse such effects with real biological phenomena. A similar conclusion was drawn in 8 . In this note we further study the mathematical properties of the resulting distributions.
To formalize the situation we first define the convolution over the rational numbers of two functions defined over the natural numbers. Let
Q u~Q \ 0,1 ½ ~a azb : a [ N, b [ N, azbw0, a,b ð Þ~1 & '
be the set of rational numbers in the unit interval. For any two functions f,g : N?R define their convolution c f,g : Q u ?R to be
c f,g a azb ~X ? m~0 X ? n~0 f m ð Þg n ð Þd a azb { m mzn X ? t~1 f ta ð Þg tb ð Þ:
In Figure 1 (bottom left, red) we have also plotted the convolution c p,p of the negative-binomially distributed predicted coverage p of the two copies of the cancer genome as given in Figure 3 (bottom left, green). In Figure 1 (bottom right, red) we have done the same for the coverage of a fixed strand. As can be seen, the convolutions follow closely the empirical distributions of ratios. This observation is consistent with the null-hypothesis of reads originating from the four strands of the human genome independently and covering a particular position on the genome with a negative-binomial distribution. No further assumption seems to be necessary to explain the irregular shapes of the ratio distributions.
We would like to finish the exposition in this section by noting that the observed structures are not particular to the Roche 454 sequencing technology and can be observed in sequencing experiments performed with other sequencing platforms, e.g. Illumina's Solexa and Life Technologies' SOLiD.
Second example: electronic clinical data. The development and implementation of electronic clinical records has made available large amounts of longitudinal clinical data. The primary application of electronic clinical data is to improve the quality of health care provided to the individual patients 9 . Although using this data for uncovering large scale correlations and trends comes secondary to this, the impact such data mining will have on the public health is indisputable 10 . Some specific areas which will be influenced by such analyses are the creation of alert systems for emerging infectious diseases, identification of populations at risk, and measuring the efficacy and efficiency of public health measures. A recent example of this is provided by the 2009 H1N1 influenza pandemic. The first wave of the new influenza strain infected a considerable part of the world population at the end of spring 2009 and the beginning of the summer 2010 11,12 . Evaluating the impact of the new pandemic strain on the public health involved analyzing large clinical datasets [13][14][15] .
The New York Presbyterian Hospital has an electronic repository with the longitudinal clinical records of more than 2 million patients. An example of the large scale analysis enabled by this data is the identification of populations that are at higher risk of morbidity/ mortality from the new pandemic influenza virus versus seasonal influenza, for instance, people with asthma, children, pregnant women, etc 15 . The approach we took for this analysis was to compare the number of people with a given condition who were affected by seasonal or pandemic influenza at different time points. Towards this goal, for every two diseases identified by their ICD9 codes, we can obtain from the electronic health records the number of people who have been affected by both diseases. Although this might differ from the established terminology, for the purpose of this note we will call this number the co-morbidity of the two diseases. In this way for a fixed disease we can obtain its co-morbidity with all other possible diseases. If we do this for two diseases, which in our analysis we take to be seasonal and pandemic influenza, we can then compare the sets of co-morbidities and look for conditions enriched with respect to one of the diseases but not the other. Figure 3 (top left) shows the distribution of co-morbidites with seasonal and pandemic influenza. As can be seen, these distributions are long-tailed and can be modeled with power-law distributions. The results of the power-law fits are also shown in Figure 3 (top left).
For a particular health condition, an important measure of the risk of being infected by seasonal versus pandemic influenza for people who have had this condition is the ratio of the number people who have had both that condition and seasonal influenza, i.e. the comorbitity with seasonal flu, to the total number of people who have had the condition, i.e. the sum of the co-morbidities with seasonal and pandemic flu. We have plotted the distribution of these ratios in Figure 3 (top right, blue). As can be seen, its shape has the self-similar structure of interest to us. From the discussion so far one might be tempted to model this distribution as the convolution of the powerlaw distributions modeling the two sets of co-morbidities. The result of this attempt is shown in Figure 3 (top right, green). The graph shows that in this case the convolution is not a good model because the empirical ratios are shifted to the left, wheres the convolution is not. In Figure 3 (bottom) we have plotted the pairs of co-morbidities for all conditions. The Spearman correlation coefficient for the two sets is 0.83 and linear regression shows that the co-morbidities for pandemic influenza are 1.3 times the corresponding co-morbidities for the seasonal influenza. Hence one might suppose that the discrepancy is due to the fact that the pairs of co-morbidities are not independent -the convolution defined above assumes that the two distributions are independent.
To avoid this obstacle we reconsidered our model for the distribution of co-morbidities and asked the following question: what is the source of the long-tail of this distribution? Our stipulation is that 1) for a fixed pair of diseases the co-morbidity is Poisson distributed, if you observe it at different time points; 2) the means of these Poissons vary from pair to pair of diseases; and 3) the distribution of these means is long-tailed. The first two stipulations are trivial if one accepts the simplifying assumption that for every disease (or pairs of diseases) there is a fixed probability that a particular person will get afflicted with this disease at a particular moment. The third stipulation is supported by our experience with the electronic health records and is akin to the informal observation that there is no universal scale at which diseases happen in the human population. We use that the mixture of Poissons with power-law distributed means has a power-law distributed tail (see the Methods section) to model the long-tail distribution of the two sets of co-morbidities. In Figure 3 (top left, black) we have plotted the result of a mixture of Poissons with power-law distributed means.
Next we claim that the observed distribution of ratios is a mixture of convolutions of pairs of Poissons where the mixing is with the same power-law distribution used for the distribution of co-morbidities. More precisely, let's say that the co-morbidity of a fixed condition with seasonal influenza is Poisson with mean l s and its co-morbidity with the pandemic strain is Poisson with mean l p . From our observation on the dependance between the two sets of co-morbidities, we can say that l p 5 cl s for some c. Hence the risk ratio of this condition with the two kinds of influenza will be distributed according to the convolution of the two Poissons, which we denote with R ls,lp . Since the mean of R ls,lp is l s /(l s 1 l p ) 5 1/(1 1 c) (see the Methods section), for c ? 1 this mean will be shifted away from 1/2 depending on c. Our model of the distribution for pairs of co-morbidites is a power-law mixture of distributions choosing the two co-morbidities independently according to two Poissons, i.e. f n,m ð Þ~ð
? 1 g a l ð ÞP l n ð ÞP cl m ð Þdl,
where g a (l) / l 2a . Note that although f(n, m) is not a product distribution, i.e. its marginals are not independent, it is a mixture of such distributions. Finally, the distribution of risk ratios is given by
R a azb ~X ? m~0 X ? n~0 f m,n ð Þd a azb { m mzn ð ?
1 g a l ð ÞR l,cl a azb : Figure 3 (top right, green) shows the result of these considerations. We observe a good fit between the empirical distribution to the right of 1/2 and the new model and the predicted overall shift of the model to the left. The apparent discrepancy between the empirical and the mixture model for ratios less than 1/2 can be attributed to the discrepancy at low co-morbidities between the mixture and empirical co-morbidity distributions observed in Figure 3 (top left). Since the goal of this note is to give examples of and draw attention to the interesting self-similar distributions appearing in empirical data, rather than to explore one particular example in detail, we leave the further analysis of the distribution of co-morbidities and the risk ratios derived from them to a future work.
Closed form for the convolution. As a step towards understanding the mathematical properties of functions over the rational numbers in the unit interval obtained as the convolution of functions over the natural numbers, we attempted to obtain a closed form, i.e. in terms of known functions, for some of them. Ideally, given the considerations above, it would be interesting to obtain a closed form for the convolution of two negative binomials or two Poissons. Although we were not able to obtain a closed form in those cases, in the Methods section we present a general method for computing arbitrary moments of the convolution when moment generating functions are available. The most general class of distributions for which we were able to obtain a closed form is power-laws with geometric cut-off. Note that the power-law and the geometric distributions belong to this class, and it is known that the negative binomial is a sum of geometric distributions.
Let g be the probability mass function of a variable distributed according to a power-law with geometric cut-off with parameters a, b $ 0 such that b . 0 or a . 1 , i.e. Uniform. Although this example does not present a distribution appearing naturally in the discussion above, we believe it is fundamental enough to mention here. Furthermore, as discussed in the Methods section, this example is related to Thomae's function, because a certain infinite analogue of it has the same fractal dimension.
For a natural number L let f L be the probability mass function which is uniform on the set {1, 2, …, L}, i.e.
f L k ð Þ~1 =L, k [ 1,2, . . . ,L f g 0, o=w:
& Then c fL,fL a azb ~1
L 2 t L max a,b ð Þ s Thomae's function. f T a azb ~1
azb :
This function, supported on the rational numbers in the unit interval, is not a distribution. It is a classic example of a function which is constant almost everywhere and yet discontinuous on a dense set. It can be beautifully interpreted as the view from the corner of Euclid's orchard -an imaginary orchard which contains a tree at every point with integer coordinates. Although it probably is not the convolution of functions over the natural numbers, the fact that versions of it appeared in our empirical data was a pleasant surprise to us and one of the main motivations for this study. In the Methods www.nature.com/scientificreports SCIENTIFIC REPORTS | 1 : 191 | DOI: 10.1038/srep00191 section we will show that the graph of this function has a fractal dimension 3/2.
Discussion
We have presented a set of self-similar distributions supported on the rational numbers in the unit interval. These functions appear pervasively in the analysis of large datasets when models for the distribution of ratios of natural numbers are required. The examples presented in this manuscript are drawn from next-generation sequencing data obtained as part of a study on the identification of somatic mutations, on one hand, and understanding disease comorbidity as it is reflected in electronic clinical data, on the other. One can envisage further applications in clinical and biological settings in which the estimation of a frequency or ratio is necessary. Such examples are provided by the detection of subclonal populations in tumor samples, e.g. as part of a study on resistance to chemotherapy; the study of quasi-species and intrahost viral populations, e.g. in HIV and influenza; and studies of drug effectiveness, populations at risk in a pandemic, and other topics in clinical research approachable through the analysis of risk ratios. We hope that our presentation will stimulate further study of the functions presented here and provide a bridge between interesting theoretical work and important clinical applications.
Methods
Fractal dimensions. The distributions we considered in this note exhibit a self-similar fractal structure. We are interested in calculating the fractal dimension of those structures. More precisely, given a function f : Q u ?R, define G(f ) to be the set of line segments in the plane from (q, 0) to (q, f(q)) for q [ Q u . The fractal dimension of the set G(f ) is defined as
dim G f ð Þ~lim e?0 log N e ð Þ log 1=e
where N(e) is the number of squares of size e needed to cover G(f ). If f is such that P q[Qu f q ð Þv?, e.g. f is a probability distribution, then dim G(f ) 5 1. Hence, our attention will focus on the fractal dimension of more general non-normalizable functions defined on the rational numbers in the unit interval.
For a given a $ 0, let f a : Q u ?R f a a= azb ð Þ ð Þ ab ð Þ {a :
From the discussion on the closed form for the convolution follows that for a . 1, f a is normalizble, and hence, in this case, dim G(f a ) 5 1. Also trivially dim G(f 0 ) 5 2. It will be interesting to obtain dim G(f a ) for a g (0, 1]. The following calculations from 16 should be helpful in obtaining this dimension. Let f T : Q u ?R be Thomae's function f T (a/(a 1 b)) 5 1/(a 1 b). We will show that dim G(f T ) 5 3/2. Since max{a, b} 5 H (a 1 b), one can think of Thomae's function as the infinite analogue of the convolution of the uniform distribution on {1,…, L} extended to L 5 '.
Let F n be the n-th Farey sequence, i.e. F n~x0~0 vx 2 v Á Á Á v f x mn~1 g is the sequence of all rational numbers x i~ai = a i zb i ð Þã i =c i [ Q u , such that a i and c i # n, sorted in increasing order. Let A i ð Þ n be the area of the trapezoid between the x-axis and the line segment with points (x i21 , f T (x i21 )) and (x i , f T (x i )). Then
2A i ð Þ n~f T x i{1 ð Þzf T x i ð Þ ð Þ x i {x i{1 ð Þ c i{1 zc i c 2 i{1 c 2 i , where we use that x i 2x i-1 5 1/c i21 c i . Let A n~P mn i~1 A i ð Þ
n be the area under the piece-wise linear curve with points from F n . We will calculate A n 2A n-1 for n $ 3. Consider two consecutive members a i21 /c i21 and a i /c i of F n21 , which have an element y j 5 (a i21 1 a i )/(c i21 1 c i ) of F n inserted between them. Then c i21 1 c i 5 n and
2 A i ð Þ n{1 {A j ð Þ n {A jz1 ð Þ n
1=c i{1 c i n:
For every n . a . 0 if d 5 (a, n) there exist unique 0 , n9 , n and 0 # a9 , a such that d 5 (a9, n9), n9a2a9n 5 d 2 , a9 , n9, and a0 5 a2a9 # n2n9 5 n0. If a=n [ Q u { 0,1 f g, then (a, n) 5 1 and we have that a9/n9, a0/n0 g F n21 are consecutive and a/n g F n is inserted between them. Hence 1=c .
Since A 2 5 1 and lim kR' A k 5 0 we obtain that
A k~1 { X k n~2 G n n 2~X
? n~kz1 G n n 2 :
Since S bjn bG b 5 H n , where H n is the n-th harmonic number, from Möbius inversion follows that We are ready to obtain an asymptotic expression for A k . Namely Let e k 5 min i {x i 2x i21 } 5 1/k(k21), where the minimum is over the elements of F k . We need N k~H A k e 2 k À Á~H k 3 ln k À Á squares of size e k to cover the set G(f T ). Hence dim G(f T ) 5 3/2. Let F 0 k~y 0~0 vy 2 v Á Á Á vy m k~1 f g be the sequence of rational numbers x~a= azb ð Þ[ Q u , such that a, b # k, sorted in increasing order. Using similar arguments as above we can show that the length L a,k of the curve with points (y i , f a (y i )) satisfies
L a,k~X k a,b ð Þ~1 a,b~1 ab ð Þ {a < k 2 1{a ð Þ {k 1{a À Á log k f 2 ð Þ 1{a ð Þ :
Let A a,k be the area under the curve with points (y i , f a (y i )). Furhermore, let d k 5 min i {y i 2y i21 } 5 H(k 22 ) and N a,k be the number of squares of size d k necessary to cover G(f a ). Since N a,k~H A a,k d k
2 À Á V d k L a,k d k 2 À Á we obtain that for a g [0, 1] dim G f a ð Þ §2{a
We believe that this lower bound is an equality.
Moments of the convolution. In this section we derive an expression for the moments of the convolution of distributions on the natural numbers in terms of their moment generating functions. Using this expression we show that the mean of the convolution of any distribution with itself is 1/2. In the specific case of a convolution of two Poissons with means l and m we show that the mean is l/(l 1 m) and the variance is where Ein x ð Þ~ð
Figure 1 |
1Thomae's function, a self-similar function over the rational numbers in the unit interval (top left). The human genome is diploid with two strands per chromosome. The reads covering a position of the genome can originate from each of the four strands (top right). For every position, the ratio between the number of reads from one of the strands to the total number of reads from the chromosome and the ratio between the number of reads from the chromosome to the total number of reads covering the position are rational numbers. The distribution of each of these ratios follows a self-similar distribution (bottom). www.nature.com/scientificreports SCIENTIFIC REPORTS | 1 : 191 | DOI: 10.1038/srep00191 reads containing the other variant. Those two depth distributions should be close to the predicted distribution of the coverage of a fixed copy of the genome. Figure 2 (bottom left, blue and red) shows the result of these considerations. Here we took only the positions of exonic single nucleotide polymorphisms documented in the NCBI's dbSNP database, which are covered sufficiently well in the experiment (total of <3000 heterozygous positions). Figure 2 (bottom left, green) contains the predicted coverage of the two copies of the human genome as obtained earlier. Furthermore, Figure 2 (bottom right) shows similar plots for a fixed strand of the genome.
Figure 2 |
2Coverage in the cancer sequencing experiment (top). Coverage of the two copies of the cancer genome (bottom left). Coverage of the two strands of a fixed copy of the cancer genome (bottom right).
Figure 3 |
3Comparing the co-morbidity of various conditions with the 2009 H1N1 pandemic versus seasonal influenza. www.nature.com/scientificreports SCIENTIFIC REPORTS | 1 : 191 | DOI: 10.1038/srep00191
Power-law. Take b 5 0 and a .
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AcknowledgmentsWe would like to thank Ben Greenbaum, Arnold Levine, Bud Mishra, and Hossein Khiabanian for helpful discussions and comments, and Oliver Elliott for help in the preparation of the manuscript. The work of Trifonov and Rabadan is supported by the Northeast Biodefence Center (U54-AI057158) and the National Library of MedicineAuthor contributionsV.T. and R.R. analyzed the HTS and EHR data, and wrote the main manuscript text. L.P. and R.D.-F. designed the HTS study. L.P. conducted experiments, analyzed data and supervised the HTS study. All authors read and approved the manuscript.Additional informationCompeting financial interests: The authors declare no competing financial interests.
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PHF6 mutations in T-cell acute lymphoblastic leukemia. P Vlierberghe, Nature Genetics. 42Vlierberghe, P. et al. PHF6 mutations in T-cell acute lymphoblastic leukemia. Nature Genetics 42, 338-342 (2010).
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Statistical artifacts in the ratio of discrete quantities. R Johnston, S Schroder, A Mallawaaratchy, The Amer. Statistician. 49Johnston, R., Schroder, S. & Mallawaaratchy, A. Statistical artifacts in the ratio of discrete quantities The Amer. Statistician 49, 285-291 (1995).
Accelerating the use of electronic health records in physician practices. S Shea, G Hripcsak, N. Engl. J. Med. 362Shea, S. & Hripcsak, G. Accelerating the use of electronic health records in physician practices. N. Engl. J. Med. 362, 192-195 (2010).
Discovering disease associations by integrating electronic clinical data and medical literature. A Holmes, A Hawson, F Liu, C Friedman, H Khiabanian, R Rabadan, PLoS ONE. 6Holmes, A., Hawson, A., Liu, F., Friedman, C., Khiabanian, H. & Rabadan, R. Discovering disease associations by integrating electronic clinical data and medical literature. PLoS ONE 6 (2011).
Pandemic potential of a strain of influenza A (H1N1): early findings. C Fraser, Science. 324Fraser, C. et al. Pandemic potential of a strain of influenza A (H1N1): early findings. Science 324, 1557-1561 (2009).
Critical care services and 2009 H1N1 influenza in Australia and New Zealand. ANZIC Influenza Investigators. 361ANZIC Influenza Investigators. Critical care services and 2009 H1N1 influenza in Australia and New Zealand. N. Engl. J. Med. 361, 1925-1534 (2009).
H1N1 2009 influenza virus infection during pregnancy in the USA. D Jamieson, Lancet. 374Jamieson, D. et al. H1N1 2009 influenza virus infection during pregnancy in the USA. Lancet 374, 451-458 (2009).
Comparative epidemiology of pandemic and seasonal influenza A in households. B Cowling, N. Engl. J. Med. 362Cowling, B. et al. Comparative epidemiology of pandemic and seasonal influenza A in households. N. Engl. J. Med. 362, 2175-2184 (2010).
Signs of the 2009 influenza pandemic in the New York-Presbyterian Hospital electronic health records. H Khiabanian, A Holmes, B Kelly, M Gururaj, G Hripcsak, R Rabadan, PLoS ONE. 5Khiabanian, H., Holmes, A., Kelly, B., Gururaj, M., Hripcsak, G. & Rabadan, R. Signs of the 2009 influenza pandemic in the New York-Presbyterian Hospital electronic health records. PLoS ONE 5 (2010).
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[] |
[
"Inverse Design of Invisibility Cloaks using the Optical Theorem",
"Inverse Design of Invisibility Cloaks using the Optical Theorem"
] |
[
"Brian Slovick *[email protected] \nAdvanced Technology and Systems Division\nSRI International\n333 Ravenswood Avenue94025Menlo ParkCAUSA\n",
"Josh Hellhake \nAdvanced Technology and Systems Division\nSRI International\n333 Ravenswood Avenue94025Menlo ParkCAUSA\n"
] |
[
"Advanced Technology and Systems Division\nSRI International\n333 Ravenswood Avenue94025Menlo ParkCAUSA",
"Advanced Technology and Systems Division\nSRI International\n333 Ravenswood Avenue94025Menlo ParkCAUSA"
] |
[] |
We develop and apply an optimization method to design invisibility cloaks. Our method is based on minimizing the forward scattering amplitude of the cloaked object, which by the optical theorem, is equivalent to the total cross section. The use of the optical theorem circumvents the need to evaluate and integrate the scattering amplitude over angle at each iteration, and thus provides a simpler, more computationally efficient objective function for optimizing structures. We implement the approach using gradient descent optimization and present several gradient-permittivity cloaks that reduce scatter by metallic targets of different size and shape.
|
10.1364/prj.450937
|
[
"https://arxiv.org/pdf/2103.03979v1.pdf"
] | 232,148,019 |
2103.03979
|
29e18f355c2403b221199b69e05c7eb5722f6629
|
Inverse Design of Invisibility Cloaks using the Optical Theorem
Brian Slovick *[email protected]
Advanced Technology and Systems Division
SRI International
333 Ravenswood Avenue94025Menlo ParkCAUSA
Josh Hellhake
Advanced Technology and Systems Division
SRI International
333 Ravenswood Avenue94025Menlo ParkCAUSA
Inverse Design of Invisibility Cloaks using the Optical Theorem
We develop and apply an optimization method to design invisibility cloaks. Our method is based on minimizing the forward scattering amplitude of the cloaked object, which by the optical theorem, is equivalent to the total cross section. The use of the optical theorem circumvents the need to evaluate and integrate the scattering amplitude over angle at each iteration, and thus provides a simpler, more computationally efficient objective function for optimizing structures. We implement the approach using gradient descent optimization and present several gradient-permittivity cloaks that reduce scatter by metallic targets of different size and shape.
Introduction
The development of invisibility cloaks is a longstanding goal in electromagnetics [1][2][3][4][5][6][7]. An object that is invisible to electromagnetic waves would have widespread application in both military and commercial systems. At radio frequencies, cloaked objects have zero radar cross section [8,9], while in the optical band, displays with low scattering and reflection provide an improved viewing experience [10,11]. It has also been proposed that cloaked sensors may provide enhanced sensitivity owing to a reduced interaction with the environment [12][13][14][15][16].
More recently, the development of metamaterials and metasurfaces has expanded the design space to include materials with negative refractive index and plasmonic properties [17]. Materials with negative index can be used with transformation optics to design electromagnetic cloaks [18][19][20], and several implementations have been demonstrated [21][22][23]. Transformation optics has the considerable advantage of imposing no limitations on the size and shape of the object. However, the designs often require complex anisotropic materials with negative refractive index, which inevitably suffer from large absorption loss and dispersion [24].
Scattering cancellation is an alternative method for achieving invisibility [25]. Of course, antireflection coatings are used througout industry to reduce reflection by planar surfaces [10,11]. For nonplanar structures, such as spheres and cylinders, multilayered designs involving dielectric [26,27] and plasmonic materials [28,29] have been proposed. However, these designs are either limited to simple shapes or suffer from large absorption losses [24]. Patterned metasurfaces, or mantle cloaks, have been proposed as an alternative to thicker, multilayer designs [30,31]. However, their application is limited to subwavelength objects. Therefore, a more generalized design approach applicable to objects of any size and shape, that does not involve plasmonics or materials with negative index, would be a significant advance.
In recent years, inverse methods have become the predominant approach in photonic design [32][33][34][35][36][37][38][39]. Broadly described, inverse design is the optimization of an objective function with respect to the structure or material properties. Inverse design has been used to optimize photonic circuits [32][33][34] and nanophotonic resonant structures [35][36][37][38][39]. The key to developing an effective inverse design algorithm lies in the definition of the objective function. For instance, to design highly scattering structures, a suitable objective function is the determinant of the wave operator defining the poles of the scattering matrix [37]. On the other hand, nonscattering structures can be designed by minimizing the scattering cross section [40,41]. However, the need to calculate the cross section from the scattered field for all angles at each step in the optimization poses a considerable computational challenge, and limits the scale and type of objects that can be optimized.
In this Letter, we develop a more efficient optimization method for designing invisibility cloaks. Our inverse design approach is based on minimizing the forward scattering amplitude, which by the optical theorem is equal to the total cross section [42,43]. The use of the optical theorem circumvents the need to integrate the scattered fields over angle by evaluating the cross section from the forward scattering amplitude alone, and thus provides a simpler, more computationally efficient algorithm. Our method is completely general and makes no assumptions about the size, shape, or composition of the object, though large objects still present a computational challenge. For demonstration, we apply it to design gradient-permittivity cloaks to minimize scatter from metallic targets of different size and shape. We show that orders of magnitude reductions in cross section are achievable with subwavelength coatings. The ease and effectiveness of our approach enables the optimization of large-scale nonscattering structures and invisibility cloaks.
Approach
Consider an object in free space with relative permittivity (r). In the scalar approximation of electromagnetics, the field scattered by the object is obtained by solving the Helmholtz equation with radiation boundary conditions. The solution for the scattered field is [1][2][3]
(r) = ∫ (r − r ) (r ) (r ) 3 ,(1)
where (r − r ) is the Green's function given by
(r − r ) = |r−r | |r − r | ,(2)
where (r) = 2 [ (r) − 1]/(4 ) and (r) is the total field given by the sum of the scattered field and the incident field (r), which we assume is a plane wave of the form k ·r . A similar expression can be obtained for the full electromagnetic vector field [37,44], but here we only consider the scalar form. In the far field approximation, the Green's function reduces to
(r − r ) − k·r ,(3)
where = |r| and k = r/ is the scattered wavevector. The scattered far field is then
(r) = (k, k ),(4)
where (k, k ) is the scattering amplitude given by [1][2][3] (k, k ) =
∫ − k·r (r ) (r ) 3 .(5)
At this stage, the scattering cross section is normally evaluated by integrating the square modulus of the scattering amplitude over solid angle. This expression can be used as the objective function to design non-scattering structures [40,41]. However, the need to evaluate and integrate the scattering amplitude over angle leads to a significant computational expense. A much simpler method is to apply the optical theorem [42,43], which states that the total cross section is proportional to the imaginary part of the scattering amplitude evaluated in the forward (k = k ) direction as
= 4 Im[ (k , k )].(6)
The optical theorem implies that the scattering amplitude evaluated in just one direction can be used as the objective for a minimization function. However, to evaluate the scattering amplitude and cross section using Eqs. (5) and (6), we need an expression for the total field. This can be accomplished by discretizing space as r( ) = ℎ, where ℎ is the grid size. In discrete form, Eq.
(1) can be written as [44,45]
( ℎ) = ∑︁ ℎ 3 ( ℎ − ℎ) ( ℎ) ( ℎ),(7)
or in matrix-vector form
E = E,(8)
where is a diagonal matrix and ℎ 3 has been absorbed into the definition of . This matrix equation can be solved to obtain the total field as [37,44,45]
E = ( − ) −1 E .(9)
Substituting this expression into the discrete form of Eq. (5), we obtain
= 4 Im E * ( − ) −1 E .(10)
Equation (10) forms the objective function of our inverse design optimization algorithm. For a particular incident direction, we optimize nonscattering structures by minimizing with respect to the permittivity values using a nonlinear least squares algorithm. Note, that in the continuum limit ℎ → 0, Eq. (10) is an exact expression for the scattering amplitude, accounting for all multiple scattering and resonance effects. The most computationally intensive calculation in Eq. (10) is the matrix inversion, and the desire is to minimize the number of matrix inversions required to minimize the scattering cross-section. Conventional gradient descent algorithms would need to evaluate the derivative of the objective with respect to (r), which would involve a numerical evaluation of the Jacobian and an additional matrix inversion. Fortunately, since the objective function has an analytical form, we can obtain a closed-form expression for the Jacobian as
( − ) −1 E = + ( − ) −1 ( − ) −1 E .(11)
This expression provides a rapid determination of the Jacobian matrix at each step in the gradient descent, and more importantly, only requires the same matrix inversion already performed to calculate the cross-section. This removes extraneous matrix inversions at each step in the gradient descent and greatly speeds up the nonlinear least squares algorithm. We tested this methodology using Matlab on a standard PC, and the memory limitations provided a practical limitation on the size matrix that could be inverted, thus limiting the size of the scattering object compared to the wavelength. To relax this size limitation, we tested the methodology using a 2D scattering object. Using a 2D structure required rederiving Eq. (1) with an integral over two dimensions and the Green's function for the 2D Helmholtz operator [3,45] 2 (r − r ) = 4
(1)
0 ( |r − r |),(12)
where (1) 0 ( ) is a Hankel function of the first kind. While this form of the Green's function is used in the transfer matrix, the far field form of Eq. (4) used in the scattered field equation is
2 (r − r ) √︂ 8 − k·r(13)
Repeating the optical theorem derivation for the 2D case results in the same expression for the scattering cross-section as Eq. (10), with the volume elements replaced by area elements [42,43].
Results
To test this methodology for designing nonscattering objects, we considered various cases of an optimized cloak applied to a metal rod. The model was parameterized, with dimensions of the metal rod, the cloak, and the grid element dimensions all defined relative to a wavelength . The permittivity of the metal rod was chosen to be = −120 + 4 , based on the measured value for copper at 1.5 m [46]. However, we quickly found that neither the resulting cloak design or the scattering cross-section reduction were dependent on the copper permittivity for values ranging from microwave to infrared frequencies. Figure 1 shows the model setup, which in this case included the solid copper cylinder with 0.5 radius, a 0.5 thick cloak, and grid size of 0.0625 , resulting in 812 grid elements across the rod and cloak. The plane wave was incident in the +x direction with a polarization parallel to the length of the rod. The top row of plots (Fig. 1a) shows the permittivity distribution with no cloak as well as the scattered and total field magnitudes inside and outside the rod calculated using the discrete integral form of Eqs. (8) and (9). As expected, we see that the rod strongly scatters the incident wave creating interference patterns throughout the region around the rod.
The second row (Fig. 1b) shows results of a non-linear least squares algorithm computed using the lsqnonlin function in Matlab. As non-linear least squares algorithms require an initial seed value for the optimization, we defined the initial shell permittivity to be 1 throughout the cloak. The result was a cloak permittivity distribution of all real values ranging from -1 to +3, roughly centered around the initial seed value of +1, and with an expected mirror symmetry which was not imposed in the Matlab algorithm. This design shows extremely small scattered field magnitudes around the cloak, resulting in a total field magnitude of near unity. Calculating the scattering cross section with and without the cloak using Eq. (10) shows a reduction factor of ∼1900. This strong reduction was confirmed by calculating the scattering amplitude, shown in the final column of Fig. 1b. These results confirm the effectiveness of this methodology and were generated in less than 1 minute of computational time.
One limitation of the lsqnonlin function in Matlab is that it does not allow for bounds on the permittivity values when solving underdetermined problems such as Eq. (10). As a result, we ended up with permittivity values in the cloak that were real and less than 1, which is unrealistic. One way around this was to rerun the optimization with an initial seed permittivity of 4 throughout the cloak. The results of this run are shown in the bottom row (Fig. 1c). The resulting permittivity values generally trended higher, though in some cases still dropped below 1, and the permittivity distribution was more complex as compared to the simpler gradient for the prior seed permittivity. The resulting scatter reduction was still very good, with a reduction factor of ∼500, but clearly shows more structure in the near field scattering.
Having generated these promising results for a relatively thick cloak, we then performed similar optimizations with half the cloak thickness, 0.25 . With the smaller structure, the grid size was also reduced to 0.0417 . The overall results in scattering reduction shown in Fig. 2 were still very large, with reduction factors of ∼340 and ∼150 for the ℎ =1 and ℎ =4 seeds, respectively. However, the thinner cloak increased the range of permittivity values, resulting in permittivity values further below unity than in the previous design.
Lastly, we performed an additional cloak design for a metal rod with an elliptical cross-section in order to show applicability to general shapes aside from an ideal cylinder. In this case, the metal rod was an ellipse with a 1.2 major axis length and 0.4 minor axis length, with the major axis parallel to the incident field. The cloak was 0.2 thick around the elliptical rod, and the grid size was 0.025 . The optimization results in this case, shown in Figs. 3b and c, gave cross-section reduction factors of ∼30 for both the ℎ =1 and ℎ =4 seeds. While not as high as the previous cases, these may be sufficient reductions in cross-section for many applications. Interestingly, for the ℎ =1 seed, the permittivity distribution looks qualitatively similar to the permittivity distribution for the cylinder in Fig. 2b, but distorted around the elliptical shape and with different extrema in the range of permittivity values. However, the scattered fields look quite different within the cloak.
Overall, these cases all show that the optical theorem methodology for designing invisibility cloaks is highly effective. In all cases examined here, the optimization of the cloak resulted in cross-sections a small fraction of the initial metal rod, and the optimization was completed in minutes or less using a standard PC. Areas for future work include using a different or modified non-linear least squares algorithm which can employ bounds on the permittivity values, exploring the dependence of optimization seed values, and exploring the dependence of increasing grid density.
Summary
We developed and applied an inverse optimization method to design invisibility cloaks. The method is based on minimizing an objective function equal to the forward scattering amplitude of the cloaked object, which by the optical theorem, is equivalent to the total cross section. The use of the optical theorem greatly simplifies the optimization since only the forward scattering amplitude must be calculated at each iteration, in contrast to the traditional approach based on integrating the amplitude over angle. Using a nonlinear least squares gradient descent, we applied the method to design several gradient-permittivity cloaks to reduce scattering from metallic circular and elliptical cylinders. Our generalized approach provides a simple and effective design tool and enables the optimization of large-scale nonscattering structures and invisibility cloaks.
Funding
Defense Advanced Research Projects Agency (DARPA) (HR001118C0015).
Fig. 1 .
1Relative permittivity, scattered fields, total fields, and scattering amplitude for a 0.5 radius copper rod with (a) no cloak, (b) optimized 0.5 -thick cloak for an initial seed of ℎ = 1, and (c) optimized 0.5 -thick cloak for an initial seed of ℎ = 4.
Fig. 2 .
2Relative permittivity, scattered fields, total fields, and scattering amplitude for a 0.5 radius copper rod with (a) no cloak, (b) optimized 0.25 -thick cloak for an initial seed of ℎ = 1, and (c) optimized 0.25 -thick cloak for an initial seed of ℎ = 4.
Fig. 3 .
3Relative permittivity, scattered fields, total fields, and scattering amplitude for an elliptical copper rod with 1.2 and 0.4 major and minor axes, respectively, with (a) no cloak, (b) optimized 0.2 -thick cloak for an initial seed of ℎ = 1, and (c) optimized 0.2 -thick cloak for an initial seed of ℎ = 4.
DisclosuresThe authors declare no conflicts of interest.
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|
[] |
[
"UTCNN: a Deep Learning Model of Stance Classification on Social Media Text",
"UTCNN: a Deep Learning Model of Stance Classification on Social Media Text"
] |
[
"Wei-Fan Chen \nInstitute of Information Science\nInstitute of Information Science\nAcademia Sinica\nTaipeiTaiwan\n",
"Lun-Wei Ku [email protected] \nAcademia Sinica\nTaipeiTaiwan\n"
] |
[
"Institute of Information Science\nInstitute of Information Science\nAcademia Sinica\nTaipeiTaiwan",
"Academia Sinica\nTaipeiTaiwan"
] |
[
"Proceedings of COLING 2016, the 26th International Conference on Computational Linguistics: Technical Papers"
] |
Most neural network models for document classification on social media focus on text information to the neglect of other information on these platforms. In this paper, we classify post stance on social media channels and develop UTCNN, a neural network model that incorporates user tastes, topic tastes, and user comments on posts. UTCNN not only works on social media texts, but also analyzes texts in forums and message boards. Experiments performed on Chinese Facebook data and English online debate forum data show that UTCNN achieves a 0.755 macroaverage f-score for supportive, neutral, and unsupportive stance classes on Facebook data, which is significantly better than models in which either user, topic, or comment information is withheld. This model design greatly mitigates the lack of data for the minor class without the use of oversampling. In addition, UTCNN yields a 0.842 accuracy on English online debate forum data, which also significantly outperforms results from previous work as well as other deep learning models, showing that UTCNN performs well regardless of language or platform. This work is licenced under a Creative Commons Attribution 4.0 International License. License details: http:// creativecommons.org/licenses/by/4.0/
| null |
[
"https://www.aclweb.org/anthology/C16-1154.pdf"
] | 11,479,759 |
1611.03599
|
ba730bc2d057949902831abecafffe8510e57cb7
|
UTCNN: a Deep Learning Model of Stance Classification on Social Media Text
December 11-17 2016
Wei-Fan Chen
Institute of Information Science
Institute of Information Science
Academia Sinica
TaipeiTaiwan
Lun-Wei Ku [email protected]
Academia Sinica
TaipeiTaiwan
UTCNN: a Deep Learning Model of Stance Classification on Social Media Text
Proceedings of COLING 2016, the 26th International Conference on Computational Linguistics: Technical Papers
COLING 2016, the 26th International Conference on Computational Linguistics: Technical PapersOsaka, JapanDecember 11-17 2016
Most neural network models for document classification on social media focus on text information to the neglect of other information on these platforms. In this paper, we classify post stance on social media channels and develop UTCNN, a neural network model that incorporates user tastes, topic tastes, and user comments on posts. UTCNN not only works on social media texts, but also analyzes texts in forums and message boards. Experiments performed on Chinese Facebook data and English online debate forum data show that UTCNN achieves a 0.755 macroaverage f-score for supportive, neutral, and unsupportive stance classes on Facebook data, which is significantly better than models in which either user, topic, or comment information is withheld. This model design greatly mitigates the lack of data for the minor class without the use of oversampling. In addition, UTCNN yields a 0.842 accuracy on English online debate forum data, which also significantly outperforms results from previous work as well as other deep learning models, showing that UTCNN performs well regardless of language or platform. This work is licenced under a Creative Commons Attribution 4.0 International License. License details: http:// creativecommons.org/licenses/by/4.0/
Introduction
Deep neural networks have been widely used in text classification and have achieved promising results (Lai et al., 2015;Ren et al., 2016;Huang et al., 2016). Most focus on content information and use models such as convolutional neural networks (CNN) (Kim, 2014) or recursive neural networks (Socher et al., 2013). However, for user-generated posts on social media like Facebook or Twitter, there is more information that should not be ignored. On social media platforms, a user can act either as the author of a post or as a reader who expresses his or her comments about the post.
In this paper, we classify posts taking into account post authorship, likes, topics, and comments. In particular, users and their "likes" hold strong potential for text mining. For example, given a set of posts that are related to a specific topic, a user's likes and dislikes provide clues for stance labeling. From a user point of view, users with positive attitudes toward the issue leave positive comments on the posts with praise or even just the post's content; from a post point of view, positive posts attract users who hold positive stances. We also investigate the influence of topics: different topics are associated with different stance labeling tendencies and word usage. For example we discuss women's rights and unwanted babies on the topic of abortion, but we criticize medicine usage or crime when on the topic of marijuana (Hasan and Ng, 2014). Even for posts on a specific topic like nuclear power, a variety of arguments are raised: green energy, radiation, air pollution, and so on. As for comments, we treat them as additional text information. The arguments in the comments and the commenters (the users who leave the comments) provide hints on the post's content and further facilitate stance classification.
In this paper, we propose the user-topic-comment neural network (UTCNN), a deep learning model that utilizes user, topic, and comment information. We attempt to learn user and topic representations which encode user interactions and topic influences to further enhance text classification, and we also incorporate comment information. We evaluate this model on a post stance classification task on forumstyle social media platforms. The contributions of this paper are as follows: 1. We propose UTCNN, a neural network for text in modern social media channels as well as legacy social media, forums, and message boards -anywhere that reveals users, their tastes, as well as their replies to posts. 2. When classifying social media post stances, we leverage users, including authors and likers. User embeddings can be generated even for users who have never posted anything. 3. We incorporate a topic model to automatically assign topics to each post in a single topic dataset. 4. We show that overall, the proposed method achieves the highest performance in all instances, and that all of the information extracted, whether users, topics, or comments, still has its contributions.
2 Related Work 2.1 Extra-Linguistic Features for Stance Classification
In this paper we aim to use text as well as other features to see how they complement each other in a deep learning model. In the stance classification domain, previous work has showed that text features are limited, suggesting that adding extra-linguistic constraints could improve performance (Bansal et al., 2008;Hasan and Ng, 2013a;Walker et al., 2012). For example, Hasan and Ng as well as Thomas et al. require that posts written by the same author have the same stance (Hasan and Ng, 2013b; Thomas et al., 2006). The addition of this constraint yields accuracy improvements of 1-7% for some models and datasets. Hasan and Ng later added user-interaction constraints and ideology constraints (Hasan and Ng, 2013a): the former models the relationship among posts in a sequence of replies and the latter models inter-topic relationships, e.g., users who oppose abortion could be conservative and thus are likely to oppose gay rights.
For work focusing on online forum text, since posts are linked through user replies, sequential labeling methods have been used to model relationships between posts. For example, Hasan and Ng use hidden Markov models (HMMs) to model dependent relationships to the preceding post (Hasan and Ng, 2013b); Burfoot et al. use iterative classification to repeatedly generate new estimates based on the current state of knowledge (Burfoot et al., 2011);Sridhar et al. use probabilistic soft logic (PSL) to model reply links via collaborative filtering (Sridhar et al., 2015). In the Facebook dataset we study, we use comments instead of reply links. However, as the ultimate goal in this paper is predicting not comment stance but post stance, we treat comments as extra information for use in predicting post stance.
Deep Learning on Extra-Linguistic Features
In recent years neural network models have been applied to document sentiment classification (Socher et al., 2012;Socher et al., 2013;Kalchbrenner et al., 2014;Johnson and Zhang, 2015;Huang et al., 2016). Text features can be used in deep networks to capture text semantics or sentiment. For example, Dong et al. use an adaptive layer in a recursive neural network for target-dependent Twitter sentiment analysis, where targets are topics such as windows 7 or taylor swift (Dong et al., 2014a;Dong et al., 2014b); recursive neural tensor networks (RNTNs) utilize sentence parse trees to capture sentence-level sentiment for movie reviews (Socher et al., 2013); Le and Mikolov predict sentiment by using paragraph vectors to model each paragraph as a continuous representation (Le and Mikolov, 2014). They show that performance can thus be improved by more delicate text models.
Others have suggested using extra-linguistic features to improve the deep learning model. The userword composition vector model (UWCVM) (Tang et al., 2015b) is inspired by the possibility that the strength of sentiment words is user-specific; to capture this they add user embeddings in their model. In UPNN, a later extension, they further add a product-word composition as product embeddings, arguing that products can also show different tendencies of being rated or reviewed (Tang et al., 2015a). Their addition of user information yielded 2-10% improvements in accuracy as compared to the abovementioned RNTN and paragraph vector methods. We also seek to inject user information into the neural network model. In comparison to the research of Tang et al. on sentiment classification for product reviews, the difference is two-fold. First, we take into account multiple users (one author and potentially many likers) for one post, whereas only one user (the reviewer) is involved in a review. Second, we add comment information to provide more features for post stance classification. None of these two factors have been considered previously in a deep learning model for text stance classification. Therefore, we
. x n . x n U k U k T j T j .
..
x' 1 x' n X' w Figure 1: Document composition in a convolutional neural network with three convolutional filters and user-and topic-dependent semantic transformations. Respectively, x w is the word embedding of word w, x w is the word embedding of word w after transformation, U k and T j are user and topic matrix embeddings for user k and topic j.
propose UTCNN, which generates and utilizes user embeddings for all users -even for those who have not authored any posts -and incorporates comments to further improve performance.
Method
In this section, we first describe CNN-based document composition, which captures user-and topicdependent document-level semantic representation from word representations. Then we show how to add comment information to construct the user-topic-comment neural network (UTCNN).
User-and Topic-dependent Document Composition
As shown in Figure 1, we use a general CNN (Kim, 2014) and two semantic transformations for document composition 1 . We are given a document with an engaged user k, a topic j, and its composite n words, each word w of which is associated with a word embedding x w ∈ R d where d is the vector dimension. For each word embedding x w , we apply two dot operations as shown in Equation 1:
x w = [U k · x w ; T j · x w ](1)
where U k ∈ R du×d models the user reading preference for certain semantics, and T j ∈ R dt×d models the topic semantics; d u and d t are the dimensions of transformed user and topic embeddings respectively. We use U k to model semantically what each user prefers to read and/or write, and use T j to model the semantics of each topic. The dot operation of U k and x w transforms the global representation x w to a user-dependent representation. Likewise, the dot operation of T j and x w transforms x w to a topicdependent representation.
After the two dot operations on x w , we have user-dependent and topic-dependent word vectors U k · x w and T j · x w , which are concatenated to form a user-and topic-dependent word vector x w . Then the transformed word embeddings X w = [x 1 ; x 2 ; ...; x n ] are used as the CNN input. Here we apply three convolutional layers on the concatenated transformed word embeddings Figure 2: The UTCNN model. Assuming one post author, l likers and p topics, xd w is the word embedding of word w in the document; xc w is the word embedding of word w in the comments; U k and u k are the moderator matrix and vector embedding for moderator k; T j and t j are the topic matrix and vector embedding for topic j; R i and r i are the commenter matrix and vector embedding for commenter i. For simplicity we do not explicitly plot the topic vector embedding part for comments, but it does include a maximum pooling layer as with documents.
x c = [x m ; x m+1 ; ...; x m+l cf −1 ] ∈ R d·l cf : h cf = f W cf · x c + b cf (2) Document Composition u k t j
where m is the index of words; f is a non-linear activation function (we use tanh 2 ); W cf ∈ R len×d·l cf is the convolutional filter with input length d · l cf and output length len, where l cf is the window size of the convolutional operation; and h cf and b cf are the output and bias of the convolution layer cf , respectively. In our experiments, the three window sizes l cf in the three convolution layers are one, two, and three, encoding unigram, bigram, and trigram semantics accordingly.
After the convolutional layer, we add a maximum pooling layer among convolutional outputs to obtain the unigram, bigram, and trigram n-gram representations. This is succeeded by an average pooling layer for an element-wise average of the three maximized convolution outputs.
UTCNN Model Description
Figure 2 illustrates the UTCNN model. As more than one user may interact with a given post, we first add a maximum pooling layer after the user matrix embedding layer and user vector embedding layer to form a moderator matrix embedding U k and a moderator vector embedding u k for moderator k respectively, where U k is used for the semantic transformation in the document composition process, as mentioned in the previous section. The term moderator here is to denote the pseudo user who provides the overall semantic/sentiment of all the engaged users for one document. The embedding u k models the moderator stance preference, that is, the pattern of the revealed user stance: whether a user is willing to show his preference, whether a user likes to show impartiality with neutral statements and reasonable arguments, or just wants to show strong support for one stance. Ideally, the latent user stance is modeled by u k for each user. Likewise, for topic information, a maximum pooling layer is added after the topic matrix embedding layer and topic vector embedding layer to form a joint topic matrix embedding T j and a joint topic vector embedding t j for topic j respectively, where T j models the semantic transformation of topic j as in users and t j models the topic stance tendency. The latent topic stance is also modeled by t j for each topic.
As for comments, we view them as short documents with authors only but without likers nor their own comments 3 . Therefore we apply document composition on comments although here users are commenters (users who comment). It is noticed that the word embeddings x w for the same word in the posts and comments are the same, but after being transformed to x w in the document composition process shown in Figure 1, they might become different because of their different engaged users. The output comment representation together with the commenter vector embedding r i and topic vector embedding t j are concatenated and a maximum pooling layer is added to select the most important feature for comments. Instead of requiring that the comment stance agree with the post, UTCNN simply extracts the most important features of the comment contents; they could be helpful, whether they show obvious agreement or disagreement. Therefore when combining comment information here, the maximum pooling layer is more appropriate than other pooling or merging layers. Indeed, we believe this is one reason for UTCNN's performance gains. Finally, the pooled comment representation, together with user vector embedding u k , topic vector embedding t j , and document representation are fed to a fully connected network, and softmax is applied to yield the final stance label prediction for the post.
Experiment
We start with the experimental dataset and then describe the training process as well as the implementation of the baselines. We also implement several variations to reveal the effects of features: authors, likers, comment, and commenters. In the results section we compare our model with related work.
Dataset
We tested the proposed UTCNN on two different datasets: FBFans and CreateDebate. FBFans is a privately-owned 4 , single-topic, Chinese, unbalanced, social media dataset, and CreateDebate is a public, multiple-topic, English, balanced, forum dataset. Results using these two datasets show the applicability and superiority for different topics, languages, data distributions, and platforms.
The FBFans dataset contains data from anti-nuclear-power Chinese Facebook fan groups from September 2013 to August 2014, including posts and their author and liker IDs. There are a total of 2,496 authors, 505,137 likers, 33,686 commenters, and 505,412 unique users. Two annotators were asked to take into account only the post content to label the stance of the posts in the whole dataset as supportive, neutral, or unsupportive (hereafter denoted as Sup, Neu, and Uns). Sup/Uns posts were those in support of or against anti-reconstruction; Neu posts were those evincing a neutral standpoint on the topic, or were irrelevant. Raw agreement between annotators is 0.91, indicating high agreement. Specifically, Cohen's Kappa for Neu and not Neu labeling is 0.58 (moderate), and for Sup or Uns labeling is 0.84 (almost perfect). Posts with inconsistent labels were filtered out, and the development and testing sets were randomly selected from what was left. Posts in the development and testing sets involved at least one user who appeared in the training set. The number of posts for each stance is shown on the left-hand side of Table 1. About twenty percent of the posts were labeled with a stance, and the number of supportive (Sup) posts was much larger than that of the unsupportive (Uns) ones: this is thus highly skewed data, which complicates stance classification. On average, 161.1 users were involved in one post. The maximum was 23,297 and the minimum was one (the author). For comments, on average there were 3 comments per post. The maximum was 1,092 and the minimum was zero.
To test whether the assumption of this paper -posts attract users who hold the same stance to like them -is reliable, we examine the likes from authors of different stances. Posts in FBFans dataset are used for this analysis. We calculate the like statistics of each distinct author from these 32,595 posts. As the numbers of authors in the Sup, Neu and Uns stances are largely imbalanced, these numbers are normalized by the number of users of each stance. Table 4 shows the results. Posts with stances (i.e., not neutral) attract users of the same stance. Neutral posts also attract both supportive and neutral users, like what we observe in supportive posts, but just the neutral posts can attract even more neutral likers. These results do suggest that users prefer posts of the same stance, or at least posts of no obvious stance which might cause annoyance when reading, and hence support the user modeling in our approach.
The CreateDebate dataset was collected from an English online debate forum 5 discussing four topics: abortion (ABO), gay rights (GAY), Obama (OBA), and marijuana (MAR). The posts are annotated as for (F) and against (A). Replies to posts in this dataset are also labeled with stance and hence use the same data format as posts. The labeling results are shown in the right-hand side of Table 1. We observe that the dataset is more balanced than the FBFans dataset. In addition, there are 977 unique users in the dataset. To compare with Hasan and Ng's work, we conducted five-fold cross-validation and present the annotation results as the average number of all folds (Hasan and Ng, 2013b; Hasan and Ng, 2014).
The FBFans dataset has more integrated functions than the CreateDebate dataset; thus our model can utilize all linguistic and extra-linguistic features. For the CreateDebate dataset, on the other hand, the like and comment features are not available (as there is a stance label for each reply, replies are evaluated as posts as other previous work) but we still implemented our model using the content, author, and topic information.
Settings
In the UTCNN training process, cross-entropy was used as the loss function and AdaGrad as the optimizer. For FBFans dataset, we learned the 50-dimensional word embeddings on the whole dataset using GloVe 6 (Pennington et al., 2014) to capture the word semantics; for CreateDebate dataset we used the publicly available English 50-dimensional word embeddings, pre-trained also using GloVe. These word embeddings were fixed in the training process. The learning rate was set to 0.03. All user and topic embeddings were randomly initialized in the range of [-0.1 0.1]. Matrix embeddings for users and topics were sized at 250 (5 × 50); vector embeddings for users and topics were set to length 10.
We applied the LDA topic model (Blei et al., 2003) on the FBFans dataset to determine the latent topics with which to build topic embeddings, as there is only one general known topic: nuclear power plants. We learned 100 latent topics and assigned the top three topics for each post. For the CreateDebate dataset, which itself constitutes four topics, the topic labels for posts were used directly without additionally applying LDA.
For the FBFans data we report class-based f-scores as well as the macro-average f-score (F SNU 1 ) shown in equation 3. where P SN U and R SN U are the average precision and recall of the three class. We adopted the macroaverage f-score as the evaluation metric for the overall performance because (1) the experimental dataset is severely imbalanced, which is common for contentious issues; and (2) for stance classification, content in minor-class posts is usually more important for further applications. For the CreateDebate dataset, accuracy was adopted as the evaluation metric to compare the results with related work (Hasan and Ng, 2013a; Hasan and Ng, 2013b; Sridhar et al., 2015).
F SN U 1 = 2 · P SN U · R SN U P SN U + R SN U(
Baselines
We pit our model against the following baselines: 1) SVM with unigram, bigram, and trigram features, which is a standard yet rather strong classifier for text features; 2) SVM with average word embedding, where a document is represented as a continuous representation by averaging the embeddings of the composite words; 3) SVM with average transformed word embeddings (the x w in equation 1), where a document is represented as a continuous representation by averaging the transformed embeddings of the composite words; 4) two mature deep learning models on text classification, CNN (Kim, 2014) and Recurrent Convolutional Neural Networks (RCNN) (Lai et al., 2015), where the hyperparameters are based on their work; 5) the above SVM and deep learning models with comment information; 6) UTCNN without user information, representing a pure-text CNN model where we use the same user matrix and user embeddings U k and u k for each user; 7) UTCNN without the LDA model, representing how UTCNN works with a single-topic dataset; 8) UTCNN without comments, in which the model predicts the stance label given only user and topic information. All these models were trained on the training set, and parameters as well as the SVM kernel selections (linear or RBF) were fine-tuned on the development set. Also, we adopt oversampling on SVMs, CNN and RCNN because the FBFans dataset is highly imbalanced.
Results on FBFans Dataset
In Table 3 we show the results of UTCNN and the baselines on the FBFans dataset. Here Majority yields good performance on Neu since FBFans is highly biased to the neutral class. The SVM models perform well on Sup and Neu but perform poorly for Uns, showing that content information in itself is insufficient to predict stance labels, especially for the minor class. With the transformed word embedding feature, SVM can achieve comparable performance as SVM with n-gram feature. However, the much fewer feature dimension of the transformed word embedding makes SVM with word embeddings a more efficient choice for modeling the large scale social media dataset. For the CNN and RCNN models, they perform slightly better than most of the SVM models but still, the content information is insufficient to achieve a good performance on the Uns posts. As to adding comment information to these models, since the commenters do not always hold the same stance as the author, simply adding comments and post contents together merely adds noise to the model. Among all UTCNN variations, we find that user information is most important, followed by topic and comment information. UTCNN without user information shows results similar to SVMs -it does well for Sup and Neu but detects no Uns. Its best f-scores on both Sup and Neu among all methods show that with enough training data, content-based models can perform well; at the same time, the lack of user information results in too few clues for minor-class posts to either predict their stance directly or link them to other users and posts for improved performance. The 17.5% improvement when adding user information suggests that user information is especially useful when the dataset is highly imbalanced. All models that consider user information predict the minority class successfully. UCTNN without topic information works well but achieves lower performance than the full UTCNN model. The 4.9% performance gain brought by LDA shows that although it is satisfactory for single topic datasets, adding that latent topics still benefits performance: even when we are discussing the same topic, we use different arguments and supporting evidence. Lastly, we get 4.8% improvement when adding comment information and it achieves comparable performance to UTCNN without topic information, which shows that comments also benefit performance. For platforms where user IDs are pixelated or otherwise hidden, adding comments to a text model still improves performance. In its integration of user, content, and comment information, the full UTCNN produces the highest f-scores on all Sup, Neu, and Uns stances among models that predict the Uns class, and the highest macro-average f-score overall. This shows its ability to balance a biased dataset and supports our claim that UTCNN successfully bridges content and user, topic, and comment information for stance classification on social media text. Another merit of UTCNN is that it does not require a balanced training data. This is supported by its outperforming other models though no oversampling technique is applied to the UTCNN related experiments as shown in this paper. Thus we can conclude that the user information provides strong clues and it is still rich even in the minority class.
We also investigate the semantic difference when a user acts as an author/liker or a commenter. We evaluated a variation in which all embeddings from the same user were forced to be identical (this is the UTCNN shared user embedding setting in Table 3). This setting yielded only a 2.5% improvement over the model without comments, which is not statistically significant. However, when separating authors/likers and commenters embeddings (i.e., the UTCNN full model), we achieved much greater improvements (4.8%). We attribute this result to the tendency of users to use different wording for different roles (for instance author vs commenter). This is observed when the user, acting as an author, attempts to support her argument against nuclear power by using improvements in solar power; when acting as a commenter, though, she interacts with post contents by criticizing past politicians who supported nuclear power or by arguing that the proposed evacuation plan in case of a nuclear accident is ridiculous. Based on this finding, in the final UTCNN setting we train two user matrix embeddings for one user: one for the author/liker role and the other for the commenter role. Table 4 shows the results of UTCNN, baselines as we implemented on the FBFans datset and related work on the CreateDebate dataset. We do not adopt oversampling on these models because the Cre-ateDebate dataset is almost balanced. In previous work, integer linear programming (ILP) or linearchain conditional random fields (CRFs) were proposed to integrate text features, author, ideology, and user-interaction constraints, where text features are unigram, bigram, and POS-dependencies; the author constraint tends to require that posts from the same author for the same topic hold the same stance; the ideology constraint aims to capture inferences between topics for the same author; the user-interaction constraint models relationships among posts via user interactions such as replies (Hasan and Ng, 2013a; Hasan and Ng, 2013b). The SVM with n-gram or average word embedding feature performs just similar to the majority. However, with the transformed word embedding, it achieves superior results. It shows that the learned user and topic embeddings really capture the user and topic semantics. This finding is not so obvious in the FBFans dataset and it might be due to the unfavorable data skewness for SVM. As for CNN and RCNN, they perform slightly better than most SVMs as we found in Table 3 for FBFans.
Results on CreateDebate Dataset
Compared to the ILP (Hasan and Ng, 2013a) and CRF (Hasan and Ng, 2013b) methods, the UTCNN user embeddings encode author and user-interaction constraints, where the ideology constraint is modeled by the topic embeddings and text features are modeled by the CNN. The significant improvement achieved by UTCNN suggests the latent representations are more effective than overt model constraints.
The PSL model (Sridhar et al., 2015) jointly labels both author and post stance using probabilistic soft logic (PSL) (Bach et al., 2015) by considering text features and reply links between authors and posts as in Hasan and Ng's work. Table 4 reports the result of their best AD setting, which represents the full joint stance/disagreement collective model on posts and is hence more relevant to UTCNN. In contrast to their model, the UTCNN user embeddings represent relationships between authors, but UTCNN models do not utilize link information between posts. Though the PSL model has the advantage of being able to jointly label the stances of authors and posts, its performance on posts is lower than the that for the ILP or CRF models. UTCNN significantly outperforms these models on posts and has the potential to predict user stances through the generated user embeddings.
For the CreateDebate dataset, we also evaluated performance when not using topic embeddings or user embeddings; as replies in this dataset are viewed as posts, the setting without comment embeddings is not available. Table 4 shows the same findings as Table 3: the 21% improvement in accuracy demonstrates that user information is the most vital. This finding also supports the results in the related work: user constraints are useful and can yield 11.2% improvement in accuracy (Hasan and Ng, 2013a). Further considering topic information yields 3.4% improvement, suggesting that knowing the subject of debates provides useful information. In sum, Table 3 together with Table 4 show that UTCNN achieves promising performance regardless of topic, language, data distribution, and platform.
Conclusion
We have proposed UTCNN, a neural network model that incorporates user, topic, content and comment information for stance classification on social media texts. UTCNN learns user embeddings for all users with minimum active degree, i.e., one post or one like. Topic information obtained from the topic model or the pre-defined labels further improves the UTCNN model. In addition, comment information provides additional clues for stance classification. We have shown that UTCNN achieves promising and balanced results. In the future we plan to explore the effectiveness of the UTCNN user embeddings for author stance classification.
Table 1: Annotation results of FBFans and CreateDebate dataset.Dataset
FBFans
CreateDebate
Type
Sup
Neu
Uns All
ABO
GAY
OBA
MAR
F
A
F
A
F
A
F
A
Training
7,097 19,412 245 26,754 770.4 622.4 700.8 400.0 420.8 367.2 355.2 145.6
Development 155 2,785 11 2,951
-
-
-
-
-
-
-
-
Testing
252 2,619 19 2,890 192.6 155.6 175.2 100.0 105.2 91.8 88.8 36.4
All
7,504 24,816 275 32,595 963.0 778.0 876.0 500.0 526.0 459.0 444.0 182.0
Author
Post
Sup
Neu
Uns
Sup
58.5% 51.3% 29.4%
Neu
33.9% 43.5 % 9.3%
Uns
7.6%
5.2%
61.3%
Table 2 :
2Distribution of like behavior.
Table 3 :
3Performance of post stance classification on the FBFans dataset. *UTCNN (full) results are statistically significant (p-value < 0.005) with respect to all other methods except for UTCNN shared user embedding.
Table 4: Accuracies of post stance classification on CreateDebate dataset. *UTCNN results were statistically significant (p-value < 0.001) with respect to other UTCNN settings.Method
Features
Topics
AVG
Text User ABO GAY OBA MAR
Majority
.549 .634 .539
.695
.604
SVM -UniBiTrigram
√
.592 .569 .565
.673
.600
SVM -AvgWordVec
√
.559 .637 .548
.708
.613
SVM -AvgWordVec (transformed)
√
√
.859 .830 .800
.741
.808
CNN (Kim, 2014)
√
.553 .636 .557
.709
.614
RCNN (Lai et al., 2015)
√
.553 .637 .534
.709
.608
ILP (Hasan and Ng, 2013a)
√
.614 .626 .581
.669
.623
ILP (Hasan and Ng, 2013a)
√
√
.749 .709 .727
.754
.735
CRF (Hasan and Ng, 2013b)
√
√
.747 .699 .711
.754
.728
PSL (Sridhar et al., 2015)
√
√
.668 .727 .635
.690
.680
UTCNN without topic
√
√
.824 .851 .743
.814
.808
UTCNN without user
√
.617 .627 .599
.685
.632
UTCNN (full)
√
√
.878 .850 .857
.782 .842*
Here by saying document, we mean the user-generated content in a post or a comment.
Some papers suggest using ReLU as the activation function in deep CNNs with many layers. Nevertheless, we use tanh as the activation function, as our model is moderately deep and empirically we found the impact to be limited.3 Recently Facebook released a function allowing likes and comments on comments, but it was not available during the time we collected data. However, UTCNN works on this richer data, as comments are treated as posts under this framework.
Currently not released due to copyright and privacy issues.
http://www.createdebate.com/ 6 http://nlp.stanford.edu/projects/glove/
AcknowledgementsResearch of this paper was partially supported by Ministry of Science and Technology, Taiwan, under the contract MOST 104-2221-E-001-024-MY2.
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User modeling with neural network for review rating prediction. Duyu Tang, Bing Qin, Ting Liu, Yuekui Yang, Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence. the Twenty-Fourth International Joint Conference on Artificial IntelligenceDuyu Tang, Bing Qin, Ting Liu, and Yuekui Yang. 2015b. User modeling with neural network for review rating prediction. In Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, pages 1340-1346.
Get out the vote: Determining support or opposition from congressional floor-debate transcripts. Matt Thomas, Bo Pang, Lillian Lee, Proceedings of the 2006 Conference on Empirical Methods in Natural Language Processing. the 2006 Conference on Empirical Methods in Natural Language ProcessingAssociation for Computational LinguisticsMatt Thomas, Bo Pang, and Lillian Lee. 2006. Get out the vote: Determining support or opposition from con- gressional floor-debate transcripts. In Proceedings of the 2006 Conference on Empirical Methods in Natural Language Processing, pages 327-335. Association for Computational Linguistics.
Stance classification using dialogic properties of persuasion. A Marilyn, Pranav Walker, Robert Anand, Ricky Abbott, Grant, Proceedings of the 2012 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies. the 2012 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language TechnologiesAssociation for Computational LinguisticsMarilyn A Walker, Pranav Anand, Robert Abbott, and Ricky Grant. 2012. Stance classification using dialogic properties of persuasion. In Proceedings of the 2012 Conference of the North American Chapter of the Associ- ation for Computational Linguistics: Human Language Technologies, pages 592-596. Association for Compu- tational Linguistics.
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