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http://math.stackexchange.com/questions/147451/for-n-an-odd-positive-square-free-integer-there-exists-an-odd-prime-p-wit?answertab=active | For $n$ an odd, positive, square-free integer, there exists an odd prime $p$ with $\left( \frac{n}{p} \right) = -1$
I'd like to prove that for $n$ an odd, positive, square-free integer, there exists an odd prime $p$ with $\left( \frac{n}{p} \right) = -1$
I'm drawing a complete blank here. Any help would be appreciated!
Thanks
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I would try using quadratic reciprocity and the infinitude of primes in arithmetic progressions with step coprime to the first element. Something like insisting that $p\equiv 1\pmod4$ and $p\equiv a_i\mod {p_i}$ for all prime factors $a_i$ of $n$, and suitably specified residues $a_i$. Reciprocity and $p\equiv 1\pmod4$ implies that $$\left(\frac {a_i} p\right)=\left(\frac p {a_i}\right).$$ – Jyrki Lahtonen May 20 '12 at 16:31
OOOOOPPSS! Substitute $p_i$ for $a_i$ on the last line. I would rewrite the comment, but need to catch a taxi to the airport in 5. See y'all! – Jyrki Lahtonen May 20 '12 at 16:46
Hint See this long interesting 1998/5/13 sci.math thread square in every $\mathbb Z/m$ implies square? – Bill Dubuque May 20 '12 at 16:54
@Bill, long, yes, but only 13 of the 61 messages in that thread were on-topic. The ones that were on-topic got the job done, so I second your recommendation. – Gerry Myerson May 21 '12 at 4:44
You can have a look at the proof of Theorem 5.2.3, p.57 in the book Ireland, Rosen: A Classical Introduction to Modern Number Theory, GTM 84. – Martin Sleziak May 21 '12 at 6:30
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1 Answer
To spell out some of what's in the comments:
Suppose that the Legendre symbol is 1 for all odd primes.
Then in particular it is 1 for all odd primes congruent to 1 modulo 4.
So by quadratic reciprocity, every prime congruent 1 mod 4 is a quadratic residue modulo n.
But given any b relatively prime to n, there is a prime congruent to b modulo n and congruent to 1 modulo 4 (using Dirichlet's Theorem on primes in arithmetic progressions, and the Chinese Remainder Theorem). The prime being 1 modulo 4 implies it's a quadratic residue modulo n, and the prime being b modulo n then says b is a quadratic residue modulo n, so we have just proved that every residue modulo n is a quadratic residue modulo n. But this is nonsense; it's easy to show that there are quadratic nonresidues modulo n.
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https://www.physicsforums.com/threads/impulse-and-rubber-balls.265966/ | # Impulse and rubber balls
1. Oct 21, 2008
### kallisti
1. The problem statement, all variables and given/known data
2 equal mass balls (one red one blue) are dropped from the same height and rebound off the floor. the red ball rebounds to a higher position. which ball is subjected to the greater magnitude of impulse during its collision with the floor?
a. it's impossible to tell since time intervals and forces are unknown.
b. both balls were subjected to the same3 magnitude impulse.
c. blue
d. red
2. Relevant equations
impulse = f*deltat
3. The attempt at a solution
i think the answer is b, but i'm not sure
2. Oct 21, 2008
### PhanthomJay
kallisti, welcome to PF! What's the equation that relates impulse to momentum?
3. Oct 21, 2008
### omgitsmonica
impulse equals change in momentum.
so are they the same, since momentum is conserved?
Have something to add?
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https://infoscience.epfl.ch/record/152319 | Infoscience
Journal article
# On the inapproximability of independent domination in 2P3-free perfect graphs
We consider the complexity of approximation for the INDEPENDENT DOMINATING SET problem in 2P(3)-free graphs, i.e., graphs that do not contain two disjoint copies of the chordless path on three vertices as all induced subgraph. We show that, if P not equal NP, the problem cannot be approximated for 2P(3)-freegraphs in polynomial time within a factor of n(1-epsilon) for any constant epsilon > 0, where n is the number of vertices ill the graph. Moreover, we show that the result holds even if the 2P(3)-free graph is restricted to being weakly chordal (and thereby perfect). (C) 2008 Elsevier B.V. All rights reserved. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8777146935462952, "perplexity": 583.3034255797519}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818686465.34/warc/CC-MAIN-20170920052220-20170920072220-00086.warc.gz"} |
http://cms.math.ca/cjm/msc/46?page=3 | location: Publications → journals
Search results
Search: MSC category 46 ( Functional analysis )
Expand all Collapse all Results 51 - 75 of 151
51. CJM 2010 (vol 62 pp. 961)
Aleman, Alexandru; Duren, Peter; Martín, María J.; Vukotić, Dragan
Multiplicative Isometries and Isometric Zero-Divisors For some Banach spaces of analytic functions in the unit disk (weighted Bergman spaces, Bloch space, Dirichlet-type spaces), the isometric pointwise multipliers are found to be unimodular constants. As a consequence, it is shown that none of those spaces have isometric zero-divisors. Isometric coefficient multipliers are also investigated. Keywords:Banach spaces of analytic functions, Hardy spaces, Bergman spaces, Bloch space, Dirichlet space, Dirichlet-type spaces, pointwise multipliers, coefficient multipliers, isometries, isometric zero-divisorsCategories:30H05, 46E15
52. CJM 2010 (vol 62 pp. 845)
Samei, Ebrahim; Spronk, Nico; Stokke, Ross
Biflatness and Pseudo-Amenability of Segal Algebras We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, $L^1(G)$, and the Fourier algebra, $A(G)$, of a locally compact group~$G$. Keywords:Segal algebra, pseudo-amenable Banach algebra, biflat Banach algebraCategories:43A20, 43A30, 46H25, 46H10, 46H20, 46L07
53. CJM 2010 (vol 62 pp. 827)
Ouyang, Caiheng; Xu, Quanhua
BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm. Keywords:BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spacesCategories:46E40, 42B25, 46B20
54. CJM 2010 (vol 62 pp. 889)
Xia, Jingbo
Singular Integral Operators and Essential Commutativity on the Sphere Let ${\mathcal T}$ be the $C^\ast$-algebra generated by the Toeplitz operators $\{T_\varphi : \varphi \in L^\infty (S,d\sigma )\}$ on the Hardy space $H^2(S)$ of the unit sphere in $\mathbf{C}^n$. It is well known that ${\mathcal T}$ is contained in the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$. We show that the essential commutant of $\{T_\varphi : \varphi \in \operatorname{VMO}\cap L^\infty (S,d\sigma )\}$ is strictly larger than ${\mathcal T}$. Categories:32A55, 46L05, 47L80
55. CJM 2010 (vol 62 pp. 595)
Martínez, J. F.; Moltó, A.; Orihuela, J.; Troyanski, S.
On Locally Uniformly Rotund Renormings in C(K) Spaces A characterization of the Banach spaces of type $C(K)$ that admit an equivalent locally uniformly rotund norm is obtained, and a method to apply it to concrete spaces is developed. As an application the existence of such renorming is deduced when $K$ is a Namioka--Phelps compact or for some particular class of Rosenthal compacta, results which were originally proved with \emph{ad hoc} methods. Categories:46B03, 46B20
56. CJM 2009 (vol 62 pp. 305)
Hua, He; Yunbai, Dong; Xianzhou, Guo
Approximation and Similarity Classification of Stably Finitely Strongly Irreducible Decomposable Operators Let $\mathcal H$ be a complex separable Hilbert space and ${\mathcal L}({\mathcal H})$ denote the collection of bounded linear operators on ${\mathcal H}$. In this paper, we show that for any operator $A\in{\mathcal L}({\mathcal H})$, there exists a stably finitely (SI) decomposable operator $A_\epsilon$, such that $\|A-A_{\epsilon}\|<\epsilon$ and ${\mathcal{\mathcal A}'(A_{\epsilon})}/\operatorname{rad} {{\mathcal A}'(A_{\epsilon})}$ is commutative, where $\operatorname{rad}{{\mathcal A}'(A_{\epsilon})}$ is the Jacobson radical of ${{\mathcal A}'(A_{\epsilon})}$. Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of Cowen-Douglas operators given by C. L. Jiang. Keywords:$K_{0}$-group, strongly irreducible decomposition, CowenâDouglas operators, commutant algebra, similarity classificationCategories:47A05, 47A55, 46H20
57. CJM 2009 (vol 62 pp. 646)
Rupp, R.; Sasane, A.
Reducibility in AR(K), CR(K), and A(K) Let $K$ denote a compact real symmetric subset of $\mathbb{C}$ and let $A_{\mathbb R}(K)$ denote the real Banach algebra of all real symmetric continuous functions on $K$ that are analytic in the interior $K^\circ$ of $K$, endowed with the supremum norm. We characterize all unimodular pairs $(f,g)$ in $A_{\mathbb R}(K)^2$ which are reducible. In addition, for an arbitrary compact $K$ in $\mathbb C$, we give a new proof (not relying on Banach algebra theory or elementary stable rank techniques) of the fact that the Bass stable rank of $A(K)$ is $1$. Finally, we also characterize all compact real symmetric sets $K$ such that $A_{\mathbb R}(K)$, respectively $C_{\mathbb R}(K)$, has Bass stable rank $1$. Keywords:real Banach algebras, Bass stable rank, topological stable rank, reducibilityCategories:46J15, 19B10, 30H05, 93D15
58. CJM 2009 (vol 62 pp. 242)
Azagra, Daniel; Fry, Robb
A Second Order Smooth Variational Principle on Riemannian Manifolds We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature. Keywords:smooth variational principle, Riemannian manifoldCategories:58E30, 49J52, 46T05, 47J30, 58B20
59. CJM 2009 (vol 61 pp. 1239)
Davidson, Kenneth R.; Yang, Dilian
Periodicity in Rank 2 Graph Algebras Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of $\mathrm{C}^*(\mathbb{F}^+_{\theta})$. The periodic $\mathrm{C}^*$-algebras are characterized, and it is shown that $\mathrm{C}^*(\mathbb{F}^+_{\theta}) \simeq \mathrm{C}(\mathbb{T})\otimes\mathfrak{A}$ where $\mathfrak{A}$ is a simple $\mathrm{C}^*$-algebra. Keywords:higher rank graph, aperiodicity condition, simple $\mathrm{C}^*$-algebra, expectationCategories:47L55, 47L30, 47L75, 46L05
60. CJM 2009 (vol 61 pp. 1262)
Dong, Z.
On the Local Lifting Properties of Operator Spaces In this paper, we mainly study operator spaces which have the locally lifting property (LLP). The dual of any ternary ring of operators is shown to satisfy the strongly local reflexivity, and this is used to prove that strongly local reflexivity holds also for operator spaces which have the LLP. Several homological characterizations of the LLP and weak expectation property are given. We also prove that for any operator space $V$, $V^{**}$ has the LLP if and only if $V$ has the LLP and $V^{*}$ is exact. Keywords:operator space, locally lifting property, strongly locally reflexiveCategory:46L07
61. CJM 2009 (vol 61 pp. 503)
Baranov, Anton; Woracek, Harald
Subspaces of de~Branges Spaces Generated by Majorants For a given de~Branges space $\mc H(E)$ we investigate de~Branges subspaces defined in terms of majorants on the real axis. If $\omega$ is a nonnegative function on $\mathbb R$, we consider the subspace $\mc R_\omega(E)=\clos_{\mc H(E)} \big\{F\in\mc H(E): \text{ there exists } C>0: |E^{-1} F|\leq C\omega \mbox{ on }{\mathbb R}\big\} .$ We show that $\mc R_\omega(E)$ is a de~Branges subspace and describe all subspaces of this form. Moreover, we give a criterion for the existence of positive minimal majorants. Keywords:de~Branges subspace, majorant, Beurling-Malliavin TheoremCategories:46E20, 30D15, 46E22
62. CJM 2009 (vol 61 pp. 282)
Bouya, Brahim
Closed Ideals in Some Algebras of Analytic Functions We obtain a complete description of closed ideals of the algebra $\cD\cap \cL$, $0<\alpha\leq\frac{1}{2}$, where $\cD$ is the Dirichlet space and $\cL$ is the algebra of analytic functions satisfying the Lipschitz condition of order $\alpha$. Categories:46E20, 30H05, 47A15
63. CJM 2009 (vol 61 pp. 241)
Azamov, N. A.; Carey, A. L.; Dodds, P. G.; Sukochev, F. A.
Operator Integrals, Spectral Shift, and Spectral Flow We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general \vNa s. For semifinite \vNa s we give applications to the Fr\'echet differentiation of operator functions that sharpen existing results, and establish the Birman--Solomyak representation of the spectral shift function of M.\,G.\,Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow. Categories:47A56, 47B49, 47A55, 46L51
64. CJM 2009 (vol 61 pp. 124)
Dijkstra, Jan J.; Mill, Jan van
Characterizing Complete Erd\H os Space The space now known as {\em complete Erd\H os space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the closed subspace of the Hilbert space $\ell^2$ consisting of all vectors such that every coordinate is in the convergent sequence $\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G. Oversteegen we present simple and useful topological characterizations of $\cerdos$. As an application we determine the class of factors of $\cerdos$. In another application we determine precisely which of the spaces that can be constructed in the Banach spaces $\ell^p$ according to the Erd\H os method' are homeomorphic to $\cerdos$. A novel application states that if $I$ is a Polishable $F_\sigma$-ideal on $\omega$, then $I$ with the Polish topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$, $\Z\times2^\omega$, or $\cerdos$. This last result answers a question that was asked by Stevo Todor{\v{c}}evi{\'c}. Keywords:Complete Erd\H os space, Lelek fan, almost zero-dimensional, nowhere zero-dimensional, Polishable ideals, submeasures on $\omega$, $\R$-trees, line-free groups in Banach spacesCategories:28C10, 46B20, 54F65
65. CJM 2009 (vol 61 pp. 50)
Chen, Huaihui; Gauthier, Paul
Composition operators on $\mu$-Bloch spaces Given a positive continuous function $\mu$ on the interval $0 Categories:47B33, 32A70, 46E15 66. CJM 2008 (vol 60 pp. 1010) Galé, José E.; Miana, Pedro J. $H^\infty$Functional Calculus and Mikhlin-Type Multiplier Conditions Let$T$be a sectorial operator. It is known that the existence of a bounded (suitably scaled)$H^\infty$calculus for$T$, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra$\Lambda_{\infty,1}^\alpha(\R^+)$. Such an algebra includes functions defined by Mikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for$T$. In this paper, we use fractional derivation to analyse in detail the relationship between$\Lambda_{\infty,1}^\alpha$and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence. Keywords:functional calculus, fractional calculus, Mikhlin multipliers, analytic semigroups, unbounded operators, quasimultipliersCategories:47A60, 47D03, 46J15, 26A33, 47L60, 47B48, 43A22 67. CJM 2008 (vol 60 pp. 1108) Lopez-Abad, J.; Manoussakis, A. A Classification of Tsirelson Type Spaces We give a complete classification of mixed Tsirelson spaces$T[(\mathcal F_i,\theta_i)_{i=1}^{r}]$for finitely many pairs of given compact and hereditary families$\mathcal F_i$of finite sets of integers and$0<\theta_i<1$in terms of the Cantor--Bendixson indices of the families$\mathcal F_i$, and$\theta_i$($1\le i\le r$). We prove that there are unique countable ordinal$\alpha$and$0<\theta<1$such that every block sequence of$T[(\mathcal F_i,\theta_i)_{i=1}^{r}]$has a subsequence equivalent to a subsequence of the natural basis of the$T(\mathcal S_{\omega^\alpha},\theta)$. Finally, we give a complete criterion of comparison in between two of these mixed Tsirelson spaces. Categories:46B20, 05D10 68. CJM 2008 (vol 60 pp. 975) Boca, Florin P. An AF Algebra Associated with the Farey Tessellation We associate with the Farey tessellation of the upper half-plane an AF algebra$\AA$encoding the `cutting sequences'' that define vertical geodesics. The Effros--Shen AF algebras arise as quotients of$\AA$. Using the path algebra model for AF algebras we construct, for each$\tau \in \big(0,\frac{1}{4}\big]$, projections$(E_n)$in$\AA$such that$E_n E_{n\pm 1}E_n \leq \tau E_n$. Categories:46L05, 11A55, 11B57, 46L55, 37E05, 82B20 69. CJM 2008 (vol 60 pp. 703) Toms, Andrew S.; Winter, Wilhelm $\mathcal{Z}$-Stable ASH Algebras The Jiang--Su algebra$\mathcal{Z}$has come to prominence in the classification program for nuclear$C^*$-algebras of late, due primarily to the fact that Elliott's classification conjecture in its strongest form predicts that all simple, separable, and nuclear$C^*$-algebras with unperforated$\mathrm{K}$-theory will absorb$\mathcal{Z}$tensorially, i.e., will be$\mathcal{Z}$-stable. There exist counterexamples which suggest that the conjecture will only hold for simple, nuclear, separable and$\mathcal{Z}$-stable$C^*$-algebras. We prove that virtually all classes of nuclear$C^*$-algebras for which the Elliott conjecture has been confirmed so far consist of$\mathcal{Z}$-stable$C^*$-algebras. This follows in large part from the following result, also proved herein: separable and approximately divisible$C^*$-algebras are$\mathcal{Z}$-stable. Keywords:nuclear$C^*$-algebras, K-theory, classificationCategories:46L85, 46L35 70. CJM 2008 (vol 60 pp. 520) Chen, Chang-Pao; Huang, Hao-Wei; Shen, Chun-Yen Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences Let$A=(a_{j,k})_{j,k \ge 1}$be a non-negative matrix. In this paper, we characterize those$A$for which$\|A\|_{E, F}$are determined by their actions on decreasing sequences, where$E$and$F$are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces:$\ell_p$,$d(w,p)$, and$\ell_p(w)$. The results established here generalize ones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour. Keywords:norms of matrices, normed Riesz spaces, weighted mean matrices, Nörlund mean matrices, summability matrices, matrices with row decreasingCategories:15A60, 40G05, 47A30, 47B37, 46B42 71. CJM 2008 (vol 60 pp. 189) Lin, Huaxin Furstenberg Transformations and Approximate Conjugacy Let$\alpha$and$\beta$be two Furstenberg transformations on$2$-torus associated with irrational numbers$\theta_1,\theta_2,$integers$d_1, d_2$and Lipschitz functions$f_1$and$f_2$. It is shown that$\alpha$and$\beta$are approximately conjugate in a measure theoretical sense if (and only if)$\overline{\theta_1\pm \theta_2}=0$in$\R/\Z.$Closely related to the classification of simple amenable \CAs, it is shown that$\af$and$\bt$are approximately$K$-conjugate if (and only if)$\overline{\theta_1\pm \theta_2}=0$in$\R/\Z$and$|d_1|=|d_2|.$This is also shown to be equivalent to the condition that the associated crossed product \CAs are isomorphic. Keywords:Furstenberg transformations, approximate conjugacyCategories:37A55, 46L35 72. CJM 2007 (vol 59 pp. 1135) Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari Sobolev Extensions of Hölder Continuous and Characteristic Functions on Metric Spaces We study when characteristic and H\"older continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and H\"older continuous functions into globally defined Sobolev functions. Keywords:characteristic function, Newtonian function, metric space, resolutivity, Hölder continuous, Perron solution,$p$-harmonic, Sobolev extension, Whitney coveringCategories:46E35, 31C45 73. CJM 2007 (vol 59 pp. 966) Forrest, Brian E.; Runde, Volker; Spronk, Nico Operator Amenability of the Fourier Algebra in the$\cb$-Multiplier Norm Let$G$be a locally compact group, and let$A_{\cb}(G)$denote the closure of$A(G)$, the Fourier algebra of$G$, in the space of completely bounded multipliers of$A(G)$. If$G$is a weakly amenable, discrete group such that$\cstar(G)$is residually finite-dimensional, we show that$A_{\cb}(G)$is operator amenable. In particular,$A_{\cb}(\free_2)$is operator amenable even though$\free_2$, the free group in two generators, is not an amenable group. Moreover, we show that if$G$is a discrete group such that$A_{\cb}(G)$is operator amenable, a closed ideal of$A(G)$is weakly completely complemented in$A(G)$if and only if it has an approximate identity bounded in the$\cb$-multiplier norm. Keywords:$\cb$-multiplier norm, Fourier algebra, operator amenability, weak amenabilityCategories:43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25 74. CJM 2007 (vol 59 pp. 1029) Kalton, N. J.; Koldobsky, A.; Yaskin, V.; Yaskina, M. The Geometry of$L_0$Suppose that we have the unit Euclidean ball in$\R^n$and construct new bodies using three operations --- linear transformations, closure in the radial metric, and multiplicative summation defined by$\|x\|_{K+_0L} = \sqrt{\|x\|_K\|x\|_L}.$We prove that in dimension$3$this procedure gives all origin-symmetric convex bodies, while this is no longer true in dimensions$4$and higher. We introduce the concept of embedding of a normed space in$L_0$that naturally extends the corresponding properties of$L_p$-spaces with$p\ne0$, and show that the procedure described above gives exactly the unit balls of subspaces of$L_0$in every dimension. We provide Fourier analytic and geometric characterizations of spaces embedding in$L_0$, and prove several facts confirming the place of$L_0$in the scale of$L_p\$-spaces. Categories:52A20, 52A21, 46B20
75. CJM 2007 (vol 59 pp. 897)
Bruneau, Laurent
The Ground State Problem for a Quantum Hamiltonian Model Describing Friction In this paper, we consider the quantum version of a Hamiltonian model describing friction. This model consists of a particle which interacts with a bosonic reservoir representing a homogeneous medium through which the particle moves. We show that if the particle is confined, then the Hamiltonian admits a ground state if and only if a suitable infrared condition is satisfied. The latter is violated in the case of linear friction, but satisfied when the friction force is proportional to a higher power of the particle speed. Categories:81Q10, 46N50
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http://www.gradesaver.com/enriques-journey/q-and-a/what-happened-at-the-end-of-the-attempt-in-chapter-3-238236 | # What happened at the end of the attempt in Chapter 3?
What happened at the end of the attempt in Chapter 3? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8656612634658813, "perplexity": 444.761256984329}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812259.18/warc/CC-MAIN-20180218192636-20180218212636-00479.warc.gz"} |
http://mathoverflow.net/questions/68825/quantum-field-theory-in-solovay-land/68996 | # Quantum field theory in Solovay-land
Constructing quantum field theories is a well-known problem. In Euclidean space, you want to define a certain measure on the space of distributions on R^n. The trickiness is that the detailed properties of the distributions that you get is sensitive to the dimension of the theory and the precise form of the action.
In classical mathematics, measures are hard to define, because one has to worry about somebody well-ordering your space of distributions, or finding a Hamel basis for it, or some other AC idiocy. I want to sidestep these issues, because they are stupid, they are annoying, and they are irrelevant.
Physicists know how to define these measures algorithmically in many cases, so that there is a computer program which will generate a random distribution with the right probability to be a pick from the measure (were it well defined for mathematicians). I find it galling that there is a construction which can be carried out on a computer, which will asymptotically converge to a uniquely defined random object, which then defines a random-picking notion of measure which is good enough to compute any correlation function or any other property of the measure, but which is not sufficient by itself to define a measure within the field of mathematics, only because of infantile Axiom of Choice absurdities.
So is the following physics construction mathematically rigorous?
Question: Given a randomized algorithm P which with certainty generates a distribution $\rho$, does P define a measure on any space of distributions which includes all possible outputs with certain probability?
This is a no-brainer in the Solovay universe, where every subset S of the unit interval [0,1] has a well defined Lebesgue measure. Given a randomized computation in Solovay-land which will produce an element of some arbitrary set U with certainty, there is the associated map from the infinite sequence of random bits, which can be thought of as a random element of [0,1], into U, and one can then define the measure of any subset S of U to be the Lebesgue measure of the inverse image of S under this map. Any randomized algorithm which converges to a unique element of U defines a measure on U.
Question: Is it trivial to de-Solovay this construction? Is there is a standard way of converting an arbitrary convergent random computation into a measure, that doesn't involve a detour into logic or forcing?
The same procedure should work for any random algorithm, or for any map, random or not.
EDIT: (in response to Andreas Blass) The question is how to translate the theorems one can prove when every subset of U gets an induced measure into the same theorems in standard set theory. You get stuck precisely in showing that the set of measurable subsets of U is sufficiently rich (even though we know from Solovay's construction that they might as well be assumed to be everything!)
The most boring standard example is the free scalar fields in a periodic box with all side length L. To generate a random field configuration, you pick every Fourier mode $\phi(k_1,...k_n)$ as a Gaussian with inverse variance $k^2/L^d$, then take the Fourier transform to define a distribution on the box. This defines a distribution, since the convolution with any smooth test function gives a sum in Fourier space which is convergent with certain probability. So in Solovay land, we are free to conclude that it defines a measure on the space of all distributions dual to smooth test functions.
But the random free field is constructed in recent papers of Sheffield and coworkers by a much more laborious route, using the exact same idea, but with a serious detour into functional analysis to show that the measure exists (see for instance theorem 2.3 in http://arxiv.org/PS_cache/math/pdf/0312/0312099v3.pdf). This kind of thing drives me up the wall, because in a Solovay universe, there is nothing to do--- the maps defined are automatically measurable. I want to know if there is a meta-theorem which guarantees that Sheffield stuff had to come out right without any work, just by knowing that the Solovay world is consistent.
In other words, is the construction: pick a random Gaussian free field by choosing each Fourier component as a random gaussian of appropriate width and fourier transforming considered a rigorous construction of measure without any further rigamarole?
EDIT IN RESPONSE TO COMMENTS: I realize that I did not specify what is required from a measure to define a quantum field theory, but this is well known in mathematical physics, and also explicitly spelled out in Sheffield's paper. I realize now that it was never clearly stated in the question I asked (and I apologize to Andreas Blass and others who made thoughtful comments below).
For a measure to define a quantum field theory (or a statistical field theory), you have to be able to compute reasonably arbitrary correlation functions over the space of random distributions. These correlation functions are averages of certain real valued functions on a randomly chosen distribution--- not necessarily polynomials, but for the usual examples, they always are. By "reasonably arbitrary" I actually mean "any real valued function except for some specially constructed axiom of choice nonsense counterexample". I don't know what these distribtions look like a-priory, so honestly, I don't know how to say anything at all about them. You only know what distributions you get out after you define the measure, generate some samples, and seeing what properties they have.
But in Solovay-land (a universe where every subset S of [0,1] is forced to have Lebesgue measure equal to the probability that a randomly chosen real number happens to be an element of S) you don't have to know anything. The moment you have a randomized algorithm that converges to an element of some set of distributions U, you can immediately define a measure, and the expectation value of any real valued function on U is equal to the integral of this function over U against that measure. This works for any function and any distribution space, without any topology or Borel Sets, without knowing anything at all, because there are no measurability issues--- all the subsets of [0,1] are measurable. Then once you have the measure, you can prove that the distributions are continuous functions, or have this or that singularity structure, or whatever, just by studying different correlation functions. For Sheffield, the goal was to show that the level sets of the distributions are well defined and given by a particular SLE in 2d, but whatever. I am not hung up on 2d, or SLE.
If one were to suggest that this is the proper way to do field theory, and by "one" I mean "me", then one would get laughed out of town. So one must make sure that there isn't some simple way to de-Solovay such a construction for a general picking algorithm. This is my question.
EDIT (in response to a comment by Qiaochu Yuan): In my view, operator algebras are not a good substitute for measure theory for defining general Euclidean quantum fields. For Euclidean fields, statistical fields really, you are interested any question one can ask about typical picks from a statistical distribution, for example "What is the SLE structure of the level sets in 2d"(Sheffield's problem), "What is the structure of the discontinuity set"? "Which nonlinear functions of a given smeared-by-a-test-function-field are certainly bounded?" etc, etc. The answer to all these questions (probably even just the solution to all the moment problems) contains all the interesting information in the measure, so if you have some non-measure substitute, you should be able to reconstruct the measure from it, and vice-versa. Why hide the measure? The only reason would be to prevent someone from bring up set-theoretic AC constructions.
For the quantities which can be computed by a stochastic computation, it is traditional to ignore all issues of measurability. This is completely justified in a Solovay universe where there are no issues of measurability. I think that any reluctance to use the language of measure theory is due solely to the old paradoxes.
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If you really want to sidestep all the set-theoretic issues, why are you using measure theory as a conceptual framework at all? – Qiaochu Yuan Jun 26 '11 at 2:14
One can effectively replace a measure space by a suitable algebra of random variables on it (see for example en.wikipedia.org/wiki/Abelian_von_Neumann_algebra and terrytao.wordpress.com/2010/02/10/245a-notes-5-free-probability), and it is possible that a suitable generalization of this construction may produce a "generalized measure theory" suitable for QFT. I have no idea if it's expected that this works, but my point is the assumption that measure theory is a reasonable framework for QFT seems to be the assumption you should be, but aren't, challenging. – Qiaochu Yuan Jun 26 '11 at 2:39
It's unclear to me why you want a measure defined on all subsets of $U$. As Andreas explained, your randomized algorithm does define a probability measure on some $\sigma$-algebra of subsets of $U$. Is there any reason to believe this $\sigma$-algebra is insufficient? – François G. Dorais Jun 26 '11 at 20:16
Sufficient for what? Please add some substance to your questions... – François G. Dorais Jun 27 '11 at 4:42
Dear Ron Maimon: if you cannot refrain from using language like "dinky" or "infantile" or "this answer is too trivial", then it might be best if you found another place to ask your questions. – S. Carnahan Jun 27 '11 at 15:31
I don't know anything about the space of all distributions dual to smooth test functions, but do know a fair bit about computable measure theory (from a certain perspective).
First, you mention that you have a computable algorithm which generates a probability distribution. I believe you are saying that you have a computable algorithm from $[0,1]$ (or technically the space of infinite binary sequences) to some set $U$ where $U$ is the space of distributions of some type.
Say your map is $f$. How are you describing the element $f(x) \in U$? In computable analysis, there is a standard way to talk about these things. We can describe each element of $U$ with an infinite code (although each element has more than one code). Then $f$ works as follows: It reads the bits of $x$; from those bits, it starts to write out the code for the $f(x)$. The more bits of $x$ known, the more bits of the code for $f(x)$ known.
(Note, not every space has such a nice encoding. If the space isn't separable, there isn't a good way to describe each object while still preserving the important properties, namely the topology. Is say, in your example above, the space of distributions that are dual to smooth test functions, is it a separable space--maybe in a weak topology? Does the encoding you use for elements of $U$ generate the same topology?)
The important property of such a computable map is that it must be continuous (in the topology generated by the encoding, but these usually coincide with the topology of the space). Since $f$ is continuous, we know we can induce a Borel measure on $U$ as follows. If $S$ is an open set then $f^{-1}(S)$ is open and $\mu(f^{-1}(S))$ is known. Similarly, with any Borel sets, hence you have a Borel measure.
Borel measures are sufficient for most applications I can think of (you can integrate continuous functions and from them, define and integrate the L^p functions), but once again, I don't know anything about your applications.
Also, if the function $f$ doesn't always converge to a point in $U$, but only does so almost everywhere, the function $f$ is not continuous, but it is still fairly nice and I believe stuff can be said about the measure, although I need to think about it.
Update: If $f$ converges with probability one, then the set of input points that $f$ converges on is a measure one $G_{\delta}$ set, in particular it is Borel. The function remains continuous on that domain (in the restricted topology). Hence there is still an induced Borel measure on the target space. (Take a Borel set; map it back. It is Borel on the restricted domain, and hence Borel on [0,1]).
Update: Also, I am assuming that your algorithm directly computes the output from the input. I will give an example what I mean. Say one want to compute a real number. To compute it directly, I should be able to ask the algorithm to give me that number within $n$ decimal places with an error bound of $1/10^n$. An indirect algorithm works as follows: The computer just gives me a sequence of approximations that converge to the number. The computer may say $0,0,0,...$ so I think it converges to 0, but at some point it starts to change to $1,1,1,...$. I can never be sure if my approximation is close to the final answer. Even if your algorithm is of the indirect type, it doesn't matter for your applications. It will still generate a Borel map, albeit a more complex one than continuous, and hence it will generate a Borel measure on the target space. (The almost everywhere concerns are similar; they also go up in complexity, but are still Borel.) Without knowing more about your application it is difficult for me to say much specific to your case.
Am I correct in my understanding of your construction, especially the computable side of it? For example, is this the way you describe the computable map from $[0,1]$ to $U$?
On a more general note, much of measure theory has been developed in a set theoretic framework. This isn't very helpful with computable concerns. But using various other definitions of measures, one is able to once again talk about measure theory with an eye to what can and cannot be computed.
I hope this helps, and that I didn't just trivialize your question.
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Yes on everything. The only topology I was thinking about is what you call the topology generated by the encoding. The only sticking point left is the you mention about almost everywhere convergence. – Ron Maimon Jun 28 '11 at 5:37
I made some edits to the post addressing the almost everywhere convergence. (They are marked Update.) – Jason Rute Jun 28 '11 at 18:36
Thank you for explaining this constructive measure theory business. Although your answer is self-contained, I was wondering if you can add a literature pointer, just for my own edification. For the specific questions: I didn't think about topology on the space of distributions, because Solovay guarantees that the measure will be defined on all subsets without worrying about topology. But, as you pointed out, there is the implicit topology in the statement that the random-picking algorithm converges. This allows you to easily make a countable dense set in the support. – Ron Maimon Jul 1 '11 at 1:49
Ron, there are two books: Computability in Analysis and Physics by Pour-El and Richards and Computable Analysis by Klaus Weihrauch. The first has less material, but might be easier to read. The second is heavy on notation. The first also has a section on Fourier transforms which may be of interest to you. – Jason Rute Jul 4 '11 at 15:21
The question is not very clear, but the paragraph following it suggests that you might mean the following. Suppose we have an operation $P$ that takes as input an infinite sequence of binary digits (or, almost equivalently, a number in $[0,1]$) and always produces an output in some set $U$ (of distributions). Does this induce a measure defined on all subsets of $U$? In general (with or without Solovay, and regardless of what the elements of $U$ are), such a $P$ induces a measure on some subsets of $U$, namely those whose inverse-image under $P$ is Lebesgue measurable. In Solovay's model where all sets of reals are Lebesgue measurable, the induced measure is thus defined on all subsets of $U$. In a universe where not all sets of reals are Lebesgue measurable, the natural induced measure on $U$ will not, in general, be defined on all subsets of $U$. For example, $P$ might be the identity map of $[0,1]$. Or, if you insist on $U$ being a set of distributions, $P$ could send each $x\in[0,1]$ to the Dirac delta-distribution concentrated at $x$.
Instead of asking whether the natural induced measure on $U$ is defined on all subsets, one could ask (and maybe you meant to ask) whether there is a reasonable extension of this measure to all subsets of$U$. In that case, I'd like to know what would count as reasonable.
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The question is as follows: you can define the random free field (for example) by picking all Fourier components as Gaussian random with width $k^2/L^2$. This is a complete definition in Solovay universe. Is it a complete definition in the standard universe? – Ron Maimon Jun 26 '11 at 14:26
It seems very unlikely to me that you can extend the measure in the presence of choice to arbitrary subsets, precisely because the measure has certain translation properties which should allow a Vitali set (although I didn't do an explicit construction). – Ron Maimon Jun 26 '11 at 14:34
In answer to your first question, I think that what you propose is a complete definition in the sense that nothing more would need to be said to specify which measure you mean. The measure it defined would not, however, have as its domain the collection of all subsets of $U$. – Andreas Blass Jun 26 '11 at 15:20
In answer to your second question: Your mention of translation invariance looks like at least the beginning of saying "what would count as reasonable". There are no translation-invariant extensions of Lebesgue measure to all sets (in the standard universe where AC holds), but there might be extensions that are not translation-invariant. This is one reason why I would want to know what sort of extensions you're willing to consider. – Andreas Blass Jun 26 '11 at 15:22
The construction of theorem 2.3 in the Sheffield paper is what I want to avoid. This is a theorem of Gross which is cited, which constructs a probability measure which has the properties of the random picking measure. This theorem is trivial in Solovay-land, because the definition in the statement of the theorem automatically constructs the measure, but it is obviously considered nontrivial by Sheffield et. al. Is there a way to transfer the trivial proof to the usual universe, and avoid this Gross thing. – Ron Maimon Jun 26 '11 at 18:36
I don't know anything about the Solovay land, but I can say a little about the random functions and this may be related to what you're going for.
People have for a while been considering random functions which are generated in this way in the context of nonlinear dispersive equations. Probably the most interesting examples are the Gibbs measures associated to infinite dimensional Hamiltonian systems like nonlinear Schrödinger or wave equations. See for example these slides of Jim Colliander:
http://blog.math.toronto.edu/colliand/files/2010/08/2010_08_Colliander_Istanbul_Final.pdf
which has an outline of the technicalities to determine on which Banach space your measure will be supported. More can be found from this lecture of Gigliola Staffilani:
http://www-math.mit.edu/~gigliola/Milan-lecture3-4.pdf
Now, if you want to put a measure on an infinite dimensional space (like the space of distributions), its support will necessarily be extremely thin even if you do get a dense subset. So for instance if you start with some $f \in L^2$ and randomize its Fourier coefficients to make a random function $f^\omega$, you get an increased integrability $f^\omega \in L^p$ for all $p < \infty$ almost surely. There are other measures you can use besides Gaussian measures where the same phenomenon will occur (random $\pm 1$'s will also do the trick by the well-known Khintchine's inequality).
Using Gibbs measures (or just ad hoc randomizations like randomized Fourier coefficients), people have been able to establish almost surely globally defined flows for random data which is "supercritical" when measured in a Sobolev space. The Gibbs measures just come with the nice feature of being invariant under the flow provided you can construct the flow. In physical space, the random data looks much better than a typical element in the Banach space. For this reason, you can show solutions to nonlinear evolution equations exist almost surely and even establish some kind of almost sure well-posedness when you deterministically would not have such a result. But there is a limit to what can be achieved with this freedom. In particular, if your variances not only fail to decay but even grow with the frequency, then it can be very difficult or simply impossible to construct a solution to the equation. The example you gave of variance like $k^2$, for instance, is likely to be too large at frequency infinity -- i.e. too irregular -- for any sensible solution to a familiar nonlinear evolution equation to exist, even though, of course, it will make sense as a measure on the space of distributions and have support in some negative Sobolev space you can explicitly compute. Since Fourier multipliers applied to the random Fourier series will also be random Fourier series, you will also be able to solve linear PDE with the random data with no difficulty, and these solutions will also possess higher integrability than you would ever sensibly ask for. But if the equation you have in mind is the cubic nonlinear Schrödinger and you're insistent about very large data, what you're asking for could likely be hopeless for reasons much more serious than this axiom of choice stuff. You have to pay attention to the space because even cubing -- let alone solving the equation -- may be impossible.
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I inverted the variance by accident--- I meant inverse variance is k^2. This is the typical free bosonic quantum field variance, and it is the same as a Boltzmann distribution for an elastic sheet. The inverse k^2 variance is still irregular at high frequencies at high dimensions, it is only continuous (Brownian) in 1d, and somewhat regular in dimension 2. – Ron Maimon Jun 28 '11 at 12:50
I am aware of the issues with nonlinear functions of quantum fields. Those need to be dealt with by using the appropriate renormalization at the computational stage. I just wanted to make sure that given a solution to the computational problem (admittedly the more difficult one), that there are no further AC type difficulties in defining the theory. – Ron Maimon Jun 28 '11 at 12:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9438447952270508, "perplexity": 234.6269352252344}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657125113.78/warc/CC-MAIN-20140914011205-00052-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
https://ru.scribd.com/document/74126176/Simple-Linear-Regression-Analysis | Вы находитесь на странице: 1из 7
# Simple Linear Regression Analysis
I. Correlation Analysis
Goal: measure the strength and direction of a linear association between two variables. Basic concepts Scatter diagram plot of individual pairs of observations on a two-dimensional graph; used to visualize the possible underlying linear relationship. Example: Consider the following hypothetical data: x y 1 2 3.5 4.5 2 2 4 3.5 3 3 7 6.5 3 8 3 4 6 7.9 4 5 6 7 9.4 9.3 6 6 7 7 11 10.5 12.4 11.5 7 10 8 15 8 8 11 13.7
## The scatter plot is as follows:
16 14 12
10
8 6 4 y
2 0
0 2 4 x 6 8 10
Linear correlation coefficient () a measure of the strength of the linear relationship existing between two variables, say X and Y, that is independent of their respective scales of measurements. Some characteristics of : It can only assume values between -1 and 1. The sign describes the direction of the linear relationship between X and Y: If is positive, the line slopes upward to the right, i.e., as X increases, the value of Y also increases.
If is negative, the line slopes downward to the right, and so as X increases, the value of Y decreases. If =0, then there is NO LINEAR RELATIONSHIP between X and Y. If is -1 or 1, there is perfect linear relationship between X and Y and all the points (x,y) fall on a straight line. A that is close to 1 or -1 indicates a strong linear relationship. A strong linear relationship does not necessarily imply that X causes Y or Y causes X. it is possible that a third variable may have caused the change in both X and Y, producing the observed relationship. The Pearson product moment correlation coefficient between X and Y, denoted by r, is defined as: =1 =1 =1 = 2 =1 2 =1 2 2 =1 =1 Example: Compute for r of the hypothetical data given above. Solution: x 1 2 2 2 3 3 3 3 4 4 5 6 6 6 7 7 7 8 8 8 95 y xy 3.5 3.5 4.5 9 4 8 3.5 7 7 21 6.5 19.5 8 24 6 18 7.9 31.6 7 28 9.4 47 9.3 55.8 11 66 10.5 63 12.4 86.8 11.5 80.5 10 70 15 120 11 88 13.7 109.6 171.7 956.3 x^2 y^2 1 12.25 4 20.25 4 16 4 12.25 9 49 9 42.25 9 64 9 36 16 62.41 16 49 25 88.36 36 86.49 36 121 36 110.25 49 153.76 49 132.25 49 100 64 225 64 121 64 187.69 553 1689.21
Sum
## We obtain the following values: n=20
=1 = =1 =95 =1 =171.7
956.3
2 =1 = 2 =1 =
553 1689.21
Substituting these values to the formula, we have: = 20 956.3 95(171.7) 20 553 952 (20 1689.21 (171.72 ) = 0.9511
Tests of Hypotheses for Null Hypothesis Ho =o Alternative Hypothesis Ha <o >o o = Test Statistic Critical Region (i.e., Reject Ho if) < ( = 2) > ( = 2) > /2 ( = 2)
( ) 2 1 2
Example: Consider the hypothetical data given above. Suppose that the linear correlation coefficient between X and Y in the past is 0.9. Determine if the correlation has significantly increased compared to the past. a. Ho: =0.90 b. =0.05 c. = d. = vs Ha: >0.90
## ( ) 2 1 2 (0.9511 0.9) 18 10.95112
= 0.7019
e. Decision rule: Reject Ho if > = 2 = .05 18 = 1.734 f. Since t = 0.7019 is not greater than .05 18 = 1.734, we do not reject Ho. At 0.05 level of significance, there is a sufficient evidence to conclude that the correlation coefficient between X and Y is 0.9. NOTE: Even if two variables are highly correlated, it is not a sufficient proof of causation. One variable may cause the other or vice versa, or a third factor is involved, or a rare event may have occurred.
II.
## Simple Linear Regression Analysis
Goal: To evaluate the relative impact of a predictor on a particular outcome. The simple linear regression model is given by the equation = + 1 +
Where
- the value of the response variable for the ith element - the value of the explanatory variable for the ith element - regression coefficient that gives Y- intercept of the regression line. 1 - regression coefficient that gives the slope of the line - random error for the ith element, where are independent, normally distributed with mean 0 and variance 2 for i=1, 2, , n n number of elements
Remark: The model tells us that two or more observations having the same value for X will not necessarily have the same value for Y. However, the different values of Y for a given value of X, say x i, will be generated by a normal distribution whose mean is + 1 , that is, = + 1 . This is known as the regression equation where the parameters and 1 are interpreted as follows: is the value of the mean of Y when X=0 1 is the amount of change in the mean of Y for every unit increase in the value of X. The random error It may be thought of as a representation of the effect of other factors, that is, apart from X, not explicitly stated in the model but do affect the response variable to some extent. Sources of random error: o Other response variables not explicitly stated in the model o Inherent and inevitable variation present in the response variable o Measurement errors Satisfies the following: o The error terms are independent from one another; o The error terms are normally distributed; o The error terms all have a mean of 0; and o The error terms have constant variance, 2 .
Typical steps in doing a simple linear regression analysis: 1. Obtain the equation that best fits the data. 2. Evaluate the equation to determine the strength of the relationship for prediction and estimation. 3. Determine if the assumptions on the error terms are satisfied. 4. If the model fits the data adequately, use the equation for prediction and for describing the nature of the relationship between the variables.
Obtaining the equation: Method of Least Squares The best-fitting line is selected as the one that minimizes the sum of squares of the deviations of the observed value of Y from its expected value. That is we want to estimate and 1 such that =1 2 is smallest, where = = + 1 Based on this criterion, the following formulas for b o , the estimate for , and b1 , the estimate for 1 , are obtained: =
=1 =1 =1 2 =1 2 =1
= 1 Thus, the estimated regression equation is given by = + 1 Remarks: The estimated regression equation is appropriate only for the relevant range of X, i.e., for the values of X used in developing the regression model. If X=0 is not included in the range of the sample data, the will not have a meaningful interpretation. Example: Consider the given hypothetical example where we fit a linear model of the form = + 1 + Using the method of least squares, the following values are needed to estimate and 1 :
n=20
=1 = =1 =95
=1 =171.7
956.3
2 =1 =
553
## We get the values of bo and b1 as: 1 = 20 956.3 95 (171.7) = 1.383 20(553) 95 2
= 8.585 1.383 4.75 = 2.016 Hence, the prediction equation is given by: = 2.016 + 1.383 Interpretation: For every 1 unit increase in X, the mean of Y is estimated to increase by 1.383. Note that bo =2.016 has no meaningful interpretation since X=0 is not within the range of values used in the estimation.
Mean Square Error The common variance of and Y, denoted by 2 , is given by: =
2
=1
2 2
where SSE stands for sum of squares due to error and MSE stands for mean square error. The MSE is the variance of the data, Y, about the estimated regression line, .
Determining the strength of relationship between X and Y A (1-)100% Confidence Interval for 1 is (1 = 2 1 , 1 + = 2 1 )
2 2
Where 1 =
2 =1 2 =1
## A (1-)100% Confidence Interval for o is ( = 2 , + = 2 )
2 2 ( 2 ) =1 2 2 =1 =1
Where =
Test of Hypothesis concerning 1 Null Hypothesis Ho 1 =0 Alternative Hypothesis Ha 1 <0 1 >0 1 0 Test Statistic 1 1 Critical Region (i.e., Reject Ho if) < ( = 2) > ( = 2) > /2 ( = 2)
Coefficient of Determination (R2 ) The proportion of the variability in the observed values of the response variable that can be explained by the explanatory variable through their linear relationship. The realized value of the coefficient of determination, r 2 , will be between 0 and 1. If a model has perfect predictability, then R2 =1; but if a model has no perfect predictive capability, then R2 =0. Interpretation: R2 *(100%) of the variability in the response variable, Y, can be explained by the explanatory variable, X, through the simple linear regression model.
Residual (di) The difference between the observed value and predicted value of the response variable. That is, = . If indeed the variances of the error terms are constant, then the plot of the residuals versus X should tend to form a horizontal band, i.e., spread of the residuals should not increase or decrease with values of the independent variable. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9084799885749817, "perplexity": 334.32464059117825}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107889173.38/warc/CC-MAIN-20201025125131-20201025155131-00160.warc.gz"} |
https://crypto.ethz.ch/publications/JoMaRi18.html | Information Security and Cryptography Research Group
Information-Theoretic Secret-Key Agreement: The Asymptotically Tight Relation Between the Secret-Key Rate and the Channel Quality Ratio
Daniel Jost , Ueli Maurer, and João L. Ribeiro
Theory of Cryptography — TCC 2018, LNCS, Springer International Publishing, vol. 11239, pp. 345–369, Nov 2018.
Information-theoretically secure secret-key agreement between two parties Alice and Bob is a well-studied problem that is provably impossible in a plain model with public (authenticated) communication, but is known to be possible in a model where the parties also have access to some correlated randomness. One particular type of such correlated randomness is the so-called satellite setting, where a source of uniform random bits (e.g., sent by a satellite) is received by the parties and the adversary Eve over inherently noisy channels. The antenna size determines the error probability, and the antenna is the adversary's limiting resource much as computing power is the limiting resource in traditional complexity-based security. The natural assumption about the adversary is that her antenna is at most $Q$ times larger than both Alice's and Bob's antenna, where, to be realistic, $Q$ can be very large.
The goal of this paper is to characterize the secret-key rate per transmitted bit in terms of $Q$. Traditional results in this so-called satellite setting are phrased in terms of the error probabilities $\epsilon_A$, $\epsilon_B$, and $\epsilon_E$, of the binary symmetric channels through which the parties receive the bits and, quite surprisingly, the secret-key rate has been shown to be strictly positive unless Eve's channel is perfect ($\epsilon_E=0$) or either Alice's or Bob's channel output is independent of the transmitted bit (i.e., $\epsilon_A=0.5$ or $\epsilon_B=0.5$). However, the best proven lower bound, if interpreted in terms of the channel quality ratio $Q$, is only exponentially small in $Q$. The main result of this paper is that the secret-key rate decreases asymptotically only like $1/Q^2$ if the per-bit signal energy, affecting the quality of all channels, is treated as a system parameter that can be optimized. Moreover, this bound is tight if Alice and Bob have the same antenna sizes.
Motivated by considering a fixed sending signal power, in which case the per-bit energy is inversely proportional to the bit-rate, we also propose a definition of the secret-key rate per second (rather than per transmitted bit) and prove that it decreases asymptotically only like $1/Q$.
BibTeX Citation
@inproceedings{JoMaRi18,
author = {Daniel Jost and Ueli Maurer and João L. Ribeiro},
title = {Information-Theoretic Secret-Key Agreement: The Asymptotically Tight Relation Between the Secret-Key Rate and the Channel Quality Ratio},
editor = {Beimel, Amos and Dziembowski, Stefan},
booktitle = {Theory of Cryptography --- TCC 2018},
pages = {345--369},
series = {LNCS},
volume = {11239},
year = {2018},
month = {11},
publisher = {Springer International Publishing},
} | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8992100954055786, "perplexity": 1029.1731146864095}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499888.62/warc/CC-MAIN-20230131154832-20230131184832-00276.warc.gz"} |
http://www.conservapedia.com/Cauchy_sequence | # Cauchy sequence
x2 − 5x + 6 = 0 x = ? This article/section deals with mathematical concepts appropriate for a student in mid to late high school.
The reader should be familiar with the material in the Limit (mathematics) page.
A Cauchy sequence (pronounced CO-she) is an infinite sequence that converges in a particular way. This type of convergence has a far-reaching significance in mathematics. Cauchy sequences are named after the French mathematician Augustin Cauchy (1789-1857).
There is an extremely profound aspect of convergent sequences. A sequence of numbers in some set might converge to a number not in that set. The famous example of this is that a sequence of rationals might converge, but not to a rational number. For example, the sequence
1.4
1.41
1.414
1.4142
1.41421
consists only of rational numbers, but it converges to $\sqrt{2}\,$, which is not a rational number. (See real number for an outline of the proof of this.)
The sequence given above was created by a computer, and it could be argued that we haven't really exhibited the sequence. But we can put such a sequence on a firm theoretical footing by using the Newton-Raphson iteration. This would give us
$A_0 = 1\,$
$A_{n+1} = \frac{1}{2}(A_n + 2/A_n)\,$
so that
$A_1= 3/2 = 1.5\,$
$A_2= 17/12 = 1.4166666...\,$
...
These aren't the same as the sequence given previously, but they are all rational numbers, and they converge to $\sqrt{2}\,$.
So if we lived in a world in which we knew about rational numbers but had never heard of the real numbers (the ancient Greeks sort of had this problem) we wouldn't know what to do about this. Recall that, for a sequence (an) to converge to a number A, that is
$\lim_{n\to \infty}a_n = A\,$
we would need to use the definition of a limit—we would need a number A such that, for every ε > 0, there is an integer M such that, whenever $n > M, |a_n-A| < \varepsilon\,$.
There is no such rational number A.
But there is clearly a sense in which $(a_n)\,$ converge. The definition of Cauchy convergence is this:
A sequence $(a_n)\,$ converges in the sense of Cauchy (or is a Cauchy sequence) if, for every ε > 0, there is an integer M such that any two sequence elements that are both beyond M are within ε of each other.
Whenever n > M and m > M, $|a_n-a_m| < \varepsilon\,$.
Note that there is no reference to the mysterious number A—the convergence is defined purely in terms of the sequence elements being close to each other. The example sequence given above can be shown to be a Cauchy sequence.
## Construction of the Real Numbers
What we did above effectively defined $\sqrt{2}\,$ in terms of the rationals, by saying
"The square root of 2 is whatever the Cauchy sequence given above converges to."
even though that isn't a "number" according to our limited (rationals-only) understanding of what a number is.
The real numbers can be defined this way, by saying that a real number is defined to be a Cauchy sequence of rational numbers.
There are many details that we won't work out here; among them are:
• There are different Cauchy sequences that converge to the same thing; we gave two sequences above that converged to $\sqrt{2}\,$. So a real number is actually an "equivalence class" of Cauchy sequences, under a carefully defined equivalence. This is a bit tricky.
• We have to show how to add, subtract, multiply, and divide Cauchy sequences. This is a bit tricky.
• We have to give the Cauchy sequences corresponding to rational numbers. This is easy—5/12 becomes (5/12, 5/12, 5/12, ...).
Once we have done that, the payoff is enormous. We have defined an extension to the rationals that is metrically complete—that extension of the rationals is the real numbers. Metrically complete means that every Cauchy sequence made from the set converges to an element which is itself in the set. The reals are the metric completion of the rationals.
The use of Cauchy sequences is one of the two famous ways of defining the real numbers, that is, completing the rationals. The other method is Dedekind cuts | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 13, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9728090763092041, "perplexity": 199.66104602508236}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115860608.29/warc/CC-MAIN-20150124161100-00022-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://www.vosesoftware.com/riskwiki/NormalapproximationtotheBetadistribution.php | # Normal approximation to the Beta distribution
The Beta distribution is difficult to calculate, involving a Beta function in its denominator, so an approximation is often welcome. A Taylor series expansion of the Beta distribution probability density function shows that the Beta(a1, a2) distribution can be approximated by the Normal distribution when a1 and a2 are sufficiently large. More specifically, the conditions are:
and
A pretty good rule of thumb is that a1 and a2 are both equal to 10 or more, but they can be as low as 6 if a1 » a2. In such cases, an approximation using the Normal distribution works well where we use the mean and standard deviations from the exact Beta distribution:
Beta(a1, a2)» Normal
Examples of a Normal approximation to a Beta distribution | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9966410994529724, "perplexity": 232.1240903552116}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986676227.57/warc/CC-MAIN-20191017200101-20191017223601-00097.warc.gz"} |
https://en.wikipedia.org/wiki/Morlet_wavelet | # Morlet wavelet
Real-valued Morlet wavelet
Complex-valued Morlet wavelet
In mathematics, the Morlet wavelet (or Gabor wavelet)[1] is a wavelet composed of a complex exponential (carrier) multiplied by a Gaussian window (envelope). This wavelet is closely related to human perception, both hearing[2] and vision.[3]
## History
In 1946, physicist Dennis Gabor, applying ideas from quantum physics, introduced the use of Gaussian-windowed sinusoids for time-frequency decomposition, which he referred to as atoms, and which provide the best trade-off between spatial and frequency resolution.[1] These are used in the Gabor transform, a type of short-time Fourier transform.[2] In 1984, Jean Morlet introduced Gabor's work to the seismology community and, with Goupillaud and Grossmann, modified it to keep the same wavelet shape over equal octave intervals, resulting in the first formalization of the continuous wavelet transform.[4] (See also Wavelet history)
## Definition
The wavelet is defined as a constant ${\displaystyle \kappa _{\sigma }}$ subtracted from a plane wave and then localised by a Gaussian window:[5]
${\displaystyle \Psi _{\sigma }(t)=c_{\sigma }\pi ^{-{\frac {1}{4}}}e^{-{\frac {1}{2}}t^{2}}(e^{i\sigma t}-\kappa _{\sigma })}$
where ${\displaystyle \kappa _{\sigma }=e^{-{\frac {1}{2}}\sigma ^{2}}}$ is defined by the admissibility criterion, and the normalisation constant ${\displaystyle c_{\sigma }}$ is:
${\displaystyle c_{\sigma }=\left(1+e^{-\sigma ^{2}}-2e^{-{\frac {3}{4}}\sigma ^{2}}\right)^{-{\frac {1}{2}}}}$
The Fourier transform of the Morlet wavelet is:
${\displaystyle {\hat {\Psi }}_{\sigma }(\omega )=c_{\sigma }\pi ^{-{\frac {1}{4}}}\left(e^{-{\frac {1}{2}}(\sigma -\omega )^{2}}-\kappa _{\sigma }e^{-{\frac {1}{2}}\omega ^{2}}\right)}$
The "central frequency" ${\displaystyle \omega _{\Psi }}$ is the position of the global maximum of ${\displaystyle {\hat {\Psi }}_{\sigma }(\omega )}$ which, in this case, is given by the solution of the equation:
${\displaystyle (\omega _{\Psi }-\sigma )^{2}-1=(\omega _{\Psi }^{2}-1)e^{-\sigma \omega _{\Psi }}}$
The parameter ${\displaystyle \sigma }$ in the Morlet wavelet allows trade between time and frequency resolutions. Conventionally, the restriction ${\displaystyle \sigma >5}$ is used to avoid problems with the Morlet wavelet at low ${\displaystyle \sigma }$ (high temporal resolution).
For signals containing only slowly varying frequency and amplitude modulations (audio, for example) it is not necessary to use small values of ${\displaystyle \sigma }$. In this case, ${\displaystyle \kappa _{\sigma }}$ becomes very small (e.g. ${\displaystyle \sigma >5\quad \Rightarrow \quad \kappa _{\sigma }<10^{-5}\,}$) and is, therefore, often neglected. Under the restriction ${\displaystyle \sigma >5}$, the frequency of the Morlet wavelet is conventionally taken to be ${\displaystyle \omega _{\Psi }\simeq \sigma }$.
The wavelet exists as a complex version or a purely real-valued version. Some distinguish between the "real Morlet" vs the "complex Morlet".[6] Others consider the complex version to be the "Gabor wavelet", while the real-valued version is the "Morlet wavelet".[7][8]
## Matlab function
[PSI,X] = morlet(LB,UB,N) returns values of the Morlet wavelet on an N point regular grid in the interval [LB,UB].
Output arguments are the wavelet function PSI computed on the grid X, and the grid X.
This wavelet has [-4 4] as effective support. Although [-4 4] is the correct theoretical effective support, a wider effective support, [-8 8], is used in the computation to provide more accurate results.[9]
## Use in medicine
The Morlet wavelet transform method presented offers an intuitive bridge between frequency and time information which can clarify interpretation of complex head trauma spectra obtained with Fourier transform. The Morlet wavelet transform, however, is not intended as a replacement for the Fourier transform, but rather a supplement that allows qualitative access to time related changes and takes advantage of the multiple dimensions available in a free induction decay analysis.[10]
## Use in music
The Morlet wavelet transform method is applied to music transcription. It produces very accurate result that is not possible before using Fourier transform. It is able to capture short bursts of repeating and alternate music note. Each note has a clear start and end time in Morlet transform.
## Application in the electrocardiogram
The application of the Morlet wavelet analysis in the electrocardiogram (ECG) is mainly to discriminate the abnormal heartbeat behavior. Since the variation of the abnormal heartbeat is a non-stationary signal, then this signal is suitable for wavelet-based analysis.
## References
1. ^ a b A Real-Time Gabor Primal Sketch for Visual Attention "The Gabor kernel satisfies the admissibility condition for wavelets, thus being suited for multi-resolution analysis. Apart from a scale factor, it is also known as the Morlet Wavelet."
2. ^ a b Time-Frequency Dictionaries, Mallat
3. ^ J. G. Daugman. Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters. Journal of the Optical Society of America A, 2(7):1160–1169, July 1985.
4. ^ http://rocksolidimages.com/pdf/gabor.pdf
5. ^ John Ashmead (2012). "Morlet Wavelets in Quantum Mechanics". Quanta. 1 (1): 58–70. doi:10.12743/quanta.v1i1.5.
6. ^ Matlab Wavelet Families - "Morlet Wavelet: morl" and "Complex Morlet Wavelets: cmor"
7. ^ Mathematica documentation: GaborWavelet
8. ^ Mathematica documentation: MorletWavelet
9. ^ "morlet". www.mathworks.com.
10. ^ http://cds.ismrm.org/ismrm-2001/PDF3/0822.pdf
• P. Goupillaud, A. Grossman, and J. Morlet. Cycle-Octave and Related Transforms in Seismic Signal Analysis. Geoexploration, 23:85-102, 1984
• N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torrésani. Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies. IEEE Trans. Inf. Th., 38:644-664, 1992 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 17, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8061084151268005, "perplexity": 2709.251632074903}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982905736.38/warc/CC-MAIN-20160823200825-00288-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/400073/why-gravitational-waves-are-not-part-of-thermal-phenomena | # Why gravitational waves are not part of thermal phenomena?
Electromagnetic waves are part of thermal phenomena in the form of thermal radiations.
But why gravitational waves do not show up as a thermal phenomenon, for example, why gravitational waves do not (directly) contribute to the making of temperature?
And if there are gravitons, what gravitons have to be to avoid being part of thermal phenomena?
• What makes you think they aren't? – Nathaniel Apr 15 '18 at 23:08
• I can't find any known thermal phenomena that can be attributed to gravitational wave. And if gravitational waves are thermal, they should dominate the background temperature in the universe but that is not the case. – southwind Apr 16 '18 at 1:00
• Gravitational waves are very, very weak in comparison to other contributions under any but the most extreme circumstances. Detecting gravitational waves is only barely in our grasp now, but we've known about the much more obvious forms of energy for a long time. Size matters, at least when you're trying to measure it. :-) – StephenG Apr 16 '18 at 2:35
• Then the gravitational wave being so weak is in itself a problem. How can a very strong and long range gravitational force having such a weak gravitational wave? Where exactly the does gravitational force come from? – southwind Apr 16 '18 at 2:47 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8969536423683167, "perplexity": 460.7712946223949}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257939.82/warc/CC-MAIN-20190525084658-20190525110658-00023.warc.gz"} |
http://math.stackexchange.com/questions/361140/fail-of-optional-sampling-theorem?answertab=votes | # Fail of optional sampling theorem
Could anyone help me see why the optional sampling theorem ($E(M_{\tau}\mid\mathcal{F}_{\sigma})=M_{\sigma}$ a.s.) fails for certain stopping times $\sigma\leq\tau$ for the not uniformly integrable martingale $$e^{aW_{t}-a^2t/2}$$ where $a$ is not 0 and $W$ is a Brownian Motion.
Why does $E(M_{\tau}\mid\mathcal{F}_{\sigma})=M_{\sigma}$ not hold for this martingale?
Really having trouble with this one so help is very much appreciated!
-
have you proved the martingale inequality question you asked yesterday? if not, i am not dishing out hints again when I think you are not actually following them. – Lost1 Apr 14 '13 at 10:16
Yes, I have as a matter of fact. I needed a bit of a push into the right direction and it took a while but I finally did manage. So thank you for your help on that question I do appreciate it as I had tried so many approaches but non seemed to work. For this question I can only say I've come quite far by proving its not integrable and that it satisfies the martingale property but on this last part again I need a bit of a push. However I realize this is non of your concern, so there is no need to help me out if you'd rather not. – user70267 Apr 14 '13 at 10:36
okay, sorry. good. I am glad. too often i find people don't actually try the question themselves when I/others only gives hints rather than complete solutions. I would just not waste time replying to them in the future. as for this question, have you understood what i wrote? – Lost1 Apr 14 '13 at 10:42
Well i'd again take $\sigma=0$ so on the righthandside one would get the value 1 in the equation. And then for $\tau$ i'd like a stopping time that tends to infinity so that the almnost sure limit of the martingale sets in and the left side becomes 0. But I'm a bit confused as there is an extra $t$ in the martingale besides from the brownian motion one. For a brownian motion It's rather easy. – user70267 Apr 14 '13 at 11:09
I had tried so many approaches but non seemed to work... For your questions to come, you might want to include in them a description of these "so many approaches" you tried. So far you are remarkably silent on these. – Did Apr 14 '13 at 11:24
Optional sampling theorem only works for bounded stopping times in general. For any arbitary $\tau$ and $\sigma$, we can actually define bounded stopping time by $\tau\wedge t$ and $\sigma\wedge t$. Now. If we apply optional sampling theorem to $\tau\wedge t$ and $\sigma\wedge t$, we will get
$\mathbb{E}(M_{\tau\wedge t}|\mathcal{F}_{\sigma\wedge t})=M_{\sigma\wedge t}$
assuming $\tau\geq\sigma$ almost surely.
In order to arrive at your expression. what we do is to take $t\rightarrow\infty$ on both handside. In order to do that on the left, we need uniform integrability.
Okay so, are you aware if you take a Brownian motion, and take $\tau$ to be the first time it hits 1 and $\sigma = 0$, you will get a contradiction? because 1 side would be 1, the other would be 0.
The exponential martingle you wrote down converges to 0 almost surely. can you try something similiar?
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https://www.physicsforums.com/threads/calculating-the-flux-of-f.27311/ | # Calculating the Flux of F
1. May 24, 2004
### vas85
Calculating the Flux of F (Vector Calculus)
Let S be the surface of the solid which is enclosed by the cone
\begin{align*} z = \sqrt{x^2+y^2} \end{align}
and the sphere $$x^2+y^2+(z-1)^2 = 1$$, and which lies above the
cone and below the sphere.
Let \begin{align*} \mathbf{F} = xz\mathbf{i}+yz\mathbf{j}-2\mathbf{k} \end{align}
Calculate the flux of \begin{align*} \mathbf{F} \end{align} , outwards through S
I am currently trying to work on the problem dont know were to approach it from exactly if any1 can help Great Appreciation :)
Last edited: May 24, 2004
2. May 24, 2004
### vas85
I am thinking whether they want me to evaluate it as half an orange with a cone missing. any help is greatly appreciated
3. May 24, 2004
### arildno
The region is the solid "ice-cream" formed.
(Letting x,y be fixed, the z-value will have its lowest value on the cone, and its highest value at a point on the spherical shell)
You should in all probability use the divergence theorem (Gauss' theorem) to evaluate this integral.
4. May 24, 2004
### arildno
To help you on a bit, here's a derivation of the planes of intersection between the cone and the sphere:
$$z=\sqrt{1-(z-1)^{2}}\rightarrow{z}^{2}=2z-z^{2}\rightarrow{z}=0,1$$
Hence, the portion of the sphere directly above the cone is a hemisphere!
5. May 24, 2004
### vas85
thanks Arildno
6. May 24, 2004
### vas85
umm Arildno do i have to do it in TWO parts? like asin Triple Integral over V1 UNION Triple Integral over V2???
becuase of the fact that its a ICECREAM
7. May 24, 2004
### arildno
Not at all!
Let z lie between the values:
$$\sqrt{x^{2}+y^{2}}\leq{z}\leq{1}+\sqrt{1-x^{2}-y^{2}}, (x^{2}+y^{2}\leq{1})$$
8. May 24, 2004
### vas85
ummmm Arildno, dunno mayb i'm not a quick learner..
the DIV F i got to be 2z
umm you have told me the bounds of integration for Z but for X and Y? like can you draw down the triple integral i need to evaluate, Appreciate it
9. May 24, 2004
### arildno
Well, I wrote the bounds down:
$$x^{2}+y^{2}\leq{1}$$
Intgrating 2z between limits is easy (z^{2} evaluated on the given limits),
while the disk in the x-y plane is most easily evaluated by polar coordinates.
Hence, you get to evaluate the double integral:
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https://www.intechopen.com/chapters/58522 | Open access peer-reviewed chapter
# Collective Mode Interactions in Lorentzian Space Plasma
Written By
Submitted: April 21st, 2017Reviewed: October 23rd, 2017Published: December 20th, 2017
DOI: 10.5772/intechopen.71847
From the Edited Volume
## Kinetic Theory
Edited by George Z. Kyzas and Athanasios C. Mitropoulos
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## Abstract
Plasmas exhibit a vast variety of waves and oscillations in which moving charged particle produce fields which ultimately give rise to particle motion. These wave-particle effects are used in the acceleration heating methods of plasma particles, and in wave generation as well. Plasmas are often manipulated with EM waves, e.g., Alfvén waves are long-wavelength modes (drift-waves) where fluid theory is most reliable, while for short wavelength modes (e.g., Kinetic Alfvén waves), collisionless effects becomes important. In this chapter, the properties of kinetic Alfvén waves are aimed to study by employing two potential theory by taking particle streaming and Weibel instability with temperature anisotropy in a Lorentzian plasma.
### Keywords
• KAWs
• Lorentzian distribution
• streaming and temperature anisotropy
• dusty plasma
## 1. Introduction
This chapter addresses one of the intriguing topics of Astrophysics—the existence of kinetic Alfvén wave (KAW) and the important consequences for astrophysical and space science to explore and investigate the new avenues. Due to the fact that KAWs have non-zero electric field Ewhich is parallel to background magnetic field and possess anisotropic polarized and spatial structures which contribute to particle energization. It is an interesting mechanism that KAWs can accelerate the field-aligned charged particles and has been applied in the dissipation of solar wind turbulence, the acceleration, and heating of charged particles in both the filed-aligned and perpendicular directions and is anticipated to play a vital role in the particle energization in laboratory, space and astrophysical plasma. The progress reported here would have immense impact and hence a small step in particular direction.
The solar wind plasma is hot and weakly collisional, existing in a state far from thermal equilibrium [1] as observed in situ in the solar wind through its nonthermal characteristics of velocity distribution function (VDF). The electron VDFs measured at 1 AU have been used as boundary condition to determine the VDFs at different altitudes. It has been confirmed that for several solar radii, the suprathermal population of particles is present in the corona [2]. For low collision rates in such plasmas, the particles can develop temperature anisotropy and the VDFs become slanted and build up high energy tails and heat fluxes along the magnetic field direction especially in fast winds and energetic interplanetary shocks. Various processes in a collisionless solar wind plasma lead to the development of particle temperature anisotropy to generate plasma instabilities which are often kinetic in nature. The free energy sources associated with the deviation from the thermodynamical equilibrium distribution function could also excite plasma waves [3, 4, 5, 6, 7, 8, 9, 10, 11].
In general, the study of plasma waves and micro-instabilities in the solar wind shows that proton VDFs are prone to anisotropic instability and originate to be stable or marginally stable. Marsch [12] has discussed four significant electrostatic and electromagnetic wave modes and free energy sources to make them unstable. For example, the electrostatic ion acoustic wave may be destabilized by the ion beams and electrons and electron heat flux, [13] the electromagnetic ion Alfvén-cyclotron wave needs proton beam and temperature anisotropy, magnetosonic wave requires proton beam and ion differential streaming and whistler-mode and lower-hybrid wave [14] unstable solutions. Among several electromagnetic instabilities, the kinetic Alfvén wave instability is the most important one.
The satellite missions in space and astrophysical plasmas have confirmed the presence of non-Maxwellian high energy and velocity tails in the particle distribution function and found in the magnetosphere of Saturn, Mercury, Uranus and Earth [2, 15, 16, 17]. The non-Maxwellian distribution of charged particles has been observed to give a better fit to the thermal and superthermal part by employing kappa distribution, since it fits both thermal and suprathermal parts in the energy velocity spectra.
The subject area of this chapter involves the basic research of space plasma physics and in particular, focuses the investigations of electrostatic and electromagnetic waves in a multi-component dusty (complex) Maxwellian and non-Maxwellian plasmas. In the last few years, various power-law distribution functions (in velocity space), i.e., kappa and rqhave been used to investigate collective phenomena and associated instabilities, such as dust-acoustic waves, kinetic Alfvén waves, Weibel instabilities, dust charging processes (in linear and nonlinear regimes) in space and astrophysical situations for better fitting the observational data in comparison to Maxwell distribution. These distributions have relevance to space plasmas containing solar wind, interstellar medium, ionosphere, magnetosphere, auroral zones, mesosphere, lower thermosphere, etc.
When the intense radiations interact with plasmas, it ends up with many applications like instabilities, inertial confinement fusion [18], and pulsar emissions [19]. These instabilities further generate turbulent electromagnetic fields in plasma regimes. We can characterize instabilities as electrostatic as well as electromagnetic according to the conditions provided by nature [20]. In this chapter, we shall also discuss electromagnetic instability called Weibel instability in a Lorentzian plasma. The free energy source available for Weibel instability is temperature anisotropy and can be developed in magnetically confined and magnetic free plasma environment as well. First time Weibel [21] came up with the calculations of impulsive growing transverse waves with anisotropic velocity distribution function in 1958. This instability developed when the electrons in the fluctuating magnetic field generates momentum flux, this flux sequentially effects velocity <v>(and ultimately current density <J>) as to increase the fluctuating field [22]. The property of Weibel instability is that it is different from normal resonant wave-particle instabilities because it depends on effects in bulk plasma without any resonant particle contributions [23]. The particle distribution functions in kinetic model adequately describe a physical phenomenon in terms of time and phase space configurations providing more information to investigate plasma waves, instabilities, plasma equilibrium, Landau damping phenomena, etc. In this chapter, we shall review the kinetic/Inertial Alfvén waves and instabilities, the effects of dust grain charging as well as field aligned/cross field currents, streaming velocity and the non-Maxwellian power-law distribution and its effect on various electromagnetic modes. We intend to show that the presence of dust grains introduces a new cutoff frequency Ωdlhwhich is associated with the motion of mobile charged particles. Moreover, an interesting feature is to show that the employed model inhibits the temperature anisotropy and supports the velocity anisotropy. Further, we shall calculate the linear dispersion relation for Weibel instability in Lorentzian plasma B0=0B00by using linearized, nonrelativistic Vlasov equation. We shall solve Zκαby assuming α<1or α>1for κ=3,5,7.
## 2. Model and methodology
In long-wavelength modes the fluid theory is most reliable, while for short wavelength modes (like KAWs), collisionless effects are important, for example, Landau damping due to finite ion Larmor radius explains observed damping rate and in dusty plasmas and charge fluctuations. Kinetic Alfvén waves (KAWs) are small scale dispersive Alfvén waves (AWs) which plays a significant role in particle acceleration and plasma heating. A coupling mechanism between small-scale KAWs and large-scale AWs in the presence of superthermal particles has been discussed which in turns giving rise to the excitation of KAWS in a solar/stellar wind plasma have been studied in the past. In this chapter, we intend to show the relationship between the growth rates of excited anisotropic KAWs and perpendicular wavelength by taking charge fluctuation and Landau damping variations into account. Moreover, when the perpendicular component of the wavelength, when comparable to the ion gyroradius, a magnetic field aligned electric field plays a significant role in the plasma acceleration/heating. Utilizing a two potential theory along with kinetic description, the properties of kinetic Alfvén waves are aimed to investigate different modes in low beta plasmas by incorporating the streaming effects. We present overview of electromagnetic KAW streaming instability in a collisionless dusty magnetoplasma, whose constituents are the electrons, ions and negatively charged dust particles. The interaction between monochromatic electron/ion beam with plasma is also discussed under various conditions. Further, to calculate the linear dispersion relation for Weibel instability in unmagnetized Lorentzian plasma, we shall employ linearized, nonrelativistic Vlasov equation.
### 2.1. Two potential theory
In a low beta plasma, β<1,the electric field can be described by two potential theory or fields expressing the electromagnetic perturbations with shear perturbations only in the magnetic field. We may neglect the electromagnetic wave compression along the direction of magnetic field (B1z=0), which leads to the coupling of Alfvén-acoustic mode. Thus, we adopt a two potential theory which represents both the transverse and parallel components of the electric field as E=φand E=ψ/z,with φψ[23]. We shall also consider the charge on the dust grain which may fluctuate according to the plasma conditions. At equilibrium, the charge neutrality imposes the condition ne0ni0+Zd0nd0=0,where ne0ni0is the electron (ion) number density and Zd0is the equilibrium dust charging state.
The linearized Poisson and Maxwell equations in terms of parallel and perpendicular operators can be expressed as
2φ+z2ψ=1ϵ0ne1+Zd0nd1+nd0Zd1ni1,E1
and
z2φψ=μ0tJe1z+Ji1z+Jd1z,E2
where ϵ0μ0is the permittivity (permeability) of the free space and Jj1zrepresents the field aligned current density for jthspecies (j=efor electrons, ifor ions and d for dust grains). In obtaining Eq. (2), we have ignored the factor zEz2E.The main idea is to decouple the compressional Alfvén mode under the assumption ×Ez=B/tz=0,i.e., to highlight bending of line of force and minimize any change in field strength due to wave compression. Moreover, ×Ezengross only Eand the least restrictive assumption for ×Ezto vanish is E=φin which the perpendicular electric field Eis electrostatic, leaving an incompressible mode. When φ=ψ,the twist of the magnetic field lines vanishes, therefore, the incompressible shear modes have .u=0=B1z,and E=φ.
t+vfj1+qjmjE+v×B.vfj0=0,E3
fj1=qjkzψmjωkzvzfj0vz,E4
where fj0is the equilibrium distribution function. The dynamics of cold and magnetized dust is governed by set of fluid equations, i.e.,
tvd=ZdemdE+Vd×ωcdE5
and
tnd+divndVd=0E6
### 2.2. Number density and current density perturbations
Here, we may define the number density and current density as
nj1=n0fj1dv,j=e,i,dJj1=qjn0vfj1dv,E7
## 3. Dispersion and damping of kinetic Alfvén waves (KAW)
Kinetic Alfvén waves (KAWs) are small scale dispersive Alfvén waves (AWs) which plays a significant role in particle acceleration, turbulence, wave particle interaction and plasma heating. Kinetic processes prevail in the regimes where plasma is dilute, multi-component, and non-uniform. A coupling mechanism between small-scale KAWs and large-scale AWs with superthermal plasma species which in turns gives rise to the excitation of KAWS in a solar/stellar wind plasma has proved dispersive Alfvén waves responsible for the solar wind turbulence especially when the turbulence cascade of these electromagnetic waves transfer from larger to smaller scale as compared to proton gyro radius. Moreover, from spacecraft observations in ionospheric plasma, it is evident that Alfvénic Poynting flux is responsible to transfer the energy for particle acceleration. All the energized auroral particles accelerate in ionosphere, initiate Joule heating phenomenon and stream out into the magnetosphere [25, 26, 27, 28].
There are number of studies to show the relationship between the growth rates of excited anisotropic KAWs and perpendicular wavelength by taking charge fluctuation and Landau damping variations into account. Moreover, the perpendicular component of wavelength, when comparable to ion gyroradius, a magnetic field aligned electric field plays a significant role in plasma acceleration/heating.
One of the important features in astrophysical plasma is the transportation of electromagnetic energy through the wave interaction with thermal plasma ions [29, 30, 31]. The KAW plays a vital role to transfer the wave energy through Landau damping (when thermal electrons travel along the magnetic field lines), which is regarded as collisionless damping of low-frequency waves and during this process the particles gain kinetic energy from the wave. This process can only happen when the distribution function has a negative slope which results in the heating of plasmas or acceleration of electrons along the magnetic field direction [24]. Recent studies also suggest the impact of non-Maxwellian distribution functions on the dynamics of solar wind and auroral plasma [32]. This study shows that the plateau formation in the parallel electron distribution functions minimize the Landau damping rate significantly.
In this chapter, the properties of kinetic Alfvén waves would be discussed by employing two potential theory, Maxwell equations and Vlasov model to study different plasma modes and by taking streaming of charged particles along and across the field direction in a Maxwellian and Lorentzian plasma.
### 3.1. Kinetic Alfvén waves in Maxwellian plasma
The propagation of kinetic Alfvén waves in a dusty plasma with finite Larmor radius effects will be discussed using a fluid-kinetic formulation by taking charge variations of dust particles. The coupling of Alfvén-acoustic mode results in the formations of kinetic Alfvén wave which would be discussed in forth coming subsections. In a magnetized plasma, we shall consider the electrons are thermal and strongly magnetized obeying an equilibrium Maxwellian distribution, while ions are hot and magnetized so that finite Larmor radius can be taken into account. For ions, we may employ Vlasov equation by utilizing guiding center approach to obtain the perturbed distribution function for an electromagnetic wave when the electric field and the wave vector klie in the xzplane and, B0=00B0,k=k0kz,
fi1=ni0eTilnkzvzψ+nΩciφωnΩcikzvzexpinlθJnkvΩciJlkvΩcifi0,E8
where Jnis the Bessel function of first kind, having order nand fi0is the equilibrium distribution function. On using Eq. (7), we obtain the modified number and current densities for hot and magnetized ions and thermal electrons, i.e.,
ni1=eni0kzmivti2nkzvtiψ1+ξinZξin+nΩciφZξinInϑieϑi,E9
Ji1z=ni0e2Tikzn1+ξinZξinkzvtiξinψ+nΩciφInϑieϑi,E10
and
ne1=ene0Teψ(1+ξeZξe,E11
Je1=e2ne0ψmevteξeZ'ξe,E12
where Inis the modified Bessel function with argument ϑi,e=kvti,e2/2Ωci,eand Zξinis the usual dispersion function for a Maxwellian plasma with ξin=ωnΩci/kzvtiand Z'is the derivative of Zwith respect to its argument.
The dust component is considered to be cold and unmagnetized such that ω<<ωcj,kzvte>>ωand kzvti<<ω,therefore, we use hydrodynamical model with momentum balance equation and continuity equation For cold and unmagnetized dust and thus we obtain
nd1=nd0Zd0emdω2k2φ+kz2ψ,E13
and
Jd1z=nd0Qd02mdωkzψ.E14
To find the relation between φand ψ,the expressions of ne1,ni1and nd1are used into Eq. (1) and Ji1z,Je1z,and Jd1zinto Eq. (2) to obtain the following coupled equations:
+=0,+=0,E15
The coefficients in Eq. (15) are given by
A=kϜiω2ω134ϑi,B=λDe2kz2ωpd2/ω2,C=c2kz2k2,D=1kz2ω2λDe2kzε,E16
where Ϝi=ωpi2/ωci2,ω=ω2Ωdlhi2, ε=ωpd2+c2k2and Ωdlhi2=ωpd2/Ϝi. The solution of homogeneous Eq. (15) in the form of a biquadratic equation, i.e.,
pω4+Qω2+R=0E17
where,
p=2ωpe2kzvte2,Q=2ωpd2kzvte2Ωdlhi2kzϵkzvAi2λDe21+34ϑi,R=Ωdlhi2kzϵ+kz3vAi2ωpd21+34ϑiE18
where VAi=cωci/ωpiis the Alfvén velocity of ions. The solution of biquadratic equation in the form of kinetic Alfvén wave is as follows,
ω2=Ωdlhi2+kz2VAi21+34+TΛiϵc2k2ϑiE19
where, T=TeTi.and Λi=ni0/ne0This shows the dispersion relation of kinetic Alfvén waves in the presence of mobile dust that are the extension of shear Alfvén waves in the range of small perpendicular wavelength. The first term on the R. H. S appears due to dust dynamics, i.e., a new cut off frequency due to the hybrid dynamics of cold dust and magnetized ions which provides a limit to the propagation of electromagnetic wave. In a dustless plasma, i.e., ωpd=0,we obtain usual dispersion relation in electron-ion plasma. Expressing ωin terms of real and imaginary part, ω=ωr+,with ωr>>γ,we either obtain growth or damping of KAW satisfying the condition, ω/kz=vAvzthrough wave particle interactions [33, 34]. In a dusty plasma with dust charge fluctuation effect, the main mechanism of wave damping is associated with dust charge fluctuation effects as compared to Landau damping [34]. It is a well-known fact that if the particle thermal velocity exceeds the Alfvén velocity, then the particles interact with Alfvén wave as the result of wave particle interaction/resonance, the linear Landau damping prevails. In a dusty plasma, the massive dust grains move slowly as compared to Alfvén velocity, therefore they may interact with Alfvén wave through linear Landau damping (which is negligible in case of dust species) or charge fluctuation effects.
### 3.2. Lorentzian distribution function
A number of processes in a space based plasma lead to the development of particle anisotropy through streaming or temperature and are responsible for plasma instabilities in collision-free plasma which are frequently kinetic in nature and their persistent features have been confirmed by many spacecraft measurements, e.g., the electron energy spectra and the near-earth environment observations have witnessed the presence of superthermal populations. It is a well-known fact that the equilibrium Maxwell-Boltzmann distributions are associated with the Boltzmann collision term, but on the large scale Fokker-Plank model is not appropriate due to strong interaction and correlation in a collisionless plasma. The kinetic foundations of generalized Lorentzian statistical mechanics has been remarkably established by [35] with the generalization of Boltzmann collision term that is not based on binary collisions. The long range correlation between particles vindicates that power law distributions posses a particular thermodynamical equilibrium state. The mathematical form on isotropic Lorentzian distribution function is given by
fj0κ=Aκ1+1κvtjκ2vz2+v2κ1;κ>3/2,E20
where Aκ=ni01πκvtjκ232Γκ+1Γκ1/2
Due to the stated fact, the deviation from the Maxwellian equilibrium distribution function could also excite plasma waves by using free energy sources. Such distributions are frequently observed in solar and terrestrial environments and can be represented by anisotropy in temperature and velocity, i.e., [36]
fvz,v=Aκ1+1κvzv0vtjκz22+vv02vtjκ2κ1,E21
where vtjκ2=2κ3κvtj, vtj=kBT/mjis the thermal speed of jthplasma component, the number densities are represented by nand anisotropic temperatures components are represented as moment of second order
Tz=mnckBfκvz2d3v,T=mnckBfκv2d3v,E22
In the limit κ, the bi-Lorentzian function is reduced to bi-Maxwellian, fκvfMv.
### 3.3. Lorentzian current and number density perturbations
Many space and astrophysical plasmas have been found to have generalized Lorentzian particle distribution functions. It is of some interest to observe the impact of the high energy tail on the current and number densities of plasma species. By using Eqs. (4), (7) and (20), we get the modified expressions of number and current densities based on kappa distribution function, i.e.,
nj1=±2nj0mjvtjκ2κ+ξj0Zκξj0,E23
and
Jj1z=2e2ψnj0mjvtjκκξj0+ξj02Zξj0,E24
where Zκξj0=1π1/2Γκ+1κ3/2Γκ1/2+xdxxξj01+x2/κκ+1,is the plasma dispersion function and κ=2κ1/2κ..
### 3.4. KAW and instability in Lorentzian plasma
In a low βplasma, the kinetic Alfvén wave instability driven by field aligned currents has dependence on plasma βand streaming velocity of current carrying species which can be responsible for particle energization. In this subsection, we extend the above scenario of electromagnetic kinetic Alfvén wave by introducing the streaming of Lorentzian ions along an external magnetic field B0ẑwith constant ion drift velocity V0B0, strongly magnetized and hot electrons to be Maxwellian and cold unmagnetized dust. The plasma beta βeis assumed to be very small. The electric field and the wave vector klie in the xzplane, i.e., B0=00B0ẑ,V0=00V0ẑ,k=kx̂0kzẑ.We again solve the Vlasov equation For hot and magnetized electrons [33] to get the number and current density of electrons as obtained in the previous section. Making use of Eqs. (4), (7) and (21), we get the perturbed number density of Lorentzian type streaming ions,
ni1=2eni0ψmivtiκ2κ+ηZκη,E25
The longitudinal components of current density perturbation [7, 19, 37] is given by
Ji1z=2e2ni0ψmivtiκκη+η2Zκη,E26
where η=ωkzV0/kzvtiκ.
By incorporating the values of ni1and Ji1in Eqs. (1), (2), and using (15), the dispersion relation of KAW streaming instability in a Lorentzian dusty plasma is obtained as
1+2ωpe2kz2vte2δ1+2ωpi2kz2vtiκ2δ2+ωpd2kz2c2+K1+Kωpd2ω2=0,E27
where K=k2/kz2and
δ1=nϵnΩcek2c2kzvteZξenIneϑe+1ωvteξenc2kz+Ωcek2c21+ξenZξenIneϑeδ2=12vtiκωηc2kz1ωpd2ω2κ+ηZκη.E28
A visible modification can be noticed by the effect of superthermality via the kappa-modified plasma dispersion function and the appearance of dust lower hybrid frequency due to dust effects on the dispersion characteristics. Numerous standard wave modes can originate from the above dispersion equation by applying particular limits, i.e.,
(i) kB0V0:For n=0, ϑe1, a dispersion relation two stream instability (TSI) in unmagnetized plasma is obtained [37], i.e.,
1+2ωpe2kz2vte21+ξenZξen+2ωpi2kz2vtiκ2κ+ηZκηωpd2ω2=0.E29
In the limit κ,our results approach to its classical Maxwellian counterpart in a dustless plasma environment [38].
(ii) kB0,V0=0,ΩciωΩce:In a dustless plasma, we get whistler-like mode whose frequency is below the electron cyclotron frequency, i.e.,
ω=kzvph,
where vph=c2kzΩce/ωpe2is the phase velocity of whistler waves which is obviously not susceptible to the Lorentzian index κ. Again, in the limiting case ωΩce,ϑe1,and expanding plasma dispersion function Eq. (15) depicts the coupling of electromagnetic and electrostatic mode, i.e., shear Alfvén-acoustic mode due to thermal kinetic effects due to which shear Alfvén wave builds a longitudinal component, e.g.,
ω2Ωdlhe2κωpi2kzvtiκ2ω2ωkzV0kzϵkzVAe2λDiκ2ω2kz2ωpd2λDiκ2=0,E30
where Ωdlhe2=ωpd2/Ϝe, Ϝe=ωpe2/Ωce2, λDiκ=Ti4πni0e21cκand cκ=2κ1/2κ3.In the limit ω2/kz2VA20,V0=0,k2ρe21we get the dispersion relation of Lorentzian dust-acoustic waves, ω2=kz2CD2/cκ,where CD2=Zd0Ti/mdand VAe=B0/4πne0me1/2is the electron Alfvén speed with electron mass density. For a low beta plasma, the coupling between dust-acoustic and shear Alfvén wave becomes weak and two modes would decouple. In the limit κ,we approach to a Maxwellian DAW [40]. It is worthy to mention here that due to the contribution of Lorentzian particles, the KAW instability suppresses. As a matter of fact, the coupling mechanism enhances the unstable regions as the wave exchanges the energy, and we can deduce that in case of generalized Lorentzian plasma, the coupling between two modes becomes weak to some extent. Moreover for non-zero streaming velocity of ions, the unstable regions tend to grow. After simplifying Eq. (30), we get the mixed shear Alfvén-acoustic mode, i.e.,
ω2=Ωdlhe2+kz2VAe21+ϵΛi1ρe2c2cκ,E31
where ρe2=Ti/meΩce2,and VAe=cΩce/ωpe.In the limit βi1and for k2ρe21,the two modes decouple and we get,
ω2=ϵzΩdlhe2ωpd2,E32
where ϵz=ωpd2+kz2c2and Ωdlhe2=ωpd2/Ϝeis the dust lower hybrid frequency which arises due to the hybrid motion of magnetized electrons and unmagnetized dust grains and is referred as a cutoff frequency which gives rise to a limit for the propagation of electromagnetic waves in the presence of dust grains. For graphical representation, we have chosen parameters typical to space dusty environment, for example, we consider ni0=10104cm3,nd0=10102cm3,Zd0=10104,md=105108mi. For computational convenience, we introduce dimensionless parameters which are as follows: ω=Ωcω˜,kz=Ωck˜z/VA,V0=VAV˜0.It has been observed that the growth rates of KAW instability are significantly affected by the presence of superthermal population, i.e., instability suppresses due to energetic particles possessed by kappa distribution when compared to its Maxwellian counterpart as shown in Figure 1. Similarly, the effect of streaming velocity, dust number density and charge on the growth rates is depicted in Figures 24 respectively. The free energy is associated with the drift motion of ions along the field direction which is responsible for the excitation of KAW. In a streaming plasma the velocity of ions is directly coupled to dust-acoustic waves and through this coupling the maximum growth rate is obtained when the wave exchanges energy through the streaming of ions. Moreover, the presence of dust particles has a noticeable effect on the wave dynamics through dust charge Zdand number density nd, i.e., it modifies the wave propagation and excitation. We can observe that Zdand ndenhances the growth rates of KAW due to the reason that when dust concentration in plasma is introduced they attach the plasma electrons toward them and the electron loss rate increases which in particular enhances the drift velocity to facilitate the unstable wave structure.
### 3.5. Dust kinetic Alfvén waves (DKAWs)
DKAWs arise when the dispersion relation of ordinary Alfvén waves is modified by the finite Larmor radius effect of dust. This process is dominated by the collective dynamics of magnetized dust particles. We have investigated shear Alfvén waves and their coupling with dust-acoustic wave by considering magnetized dust and Lorentzian electrons and ions.
The perturbed current and number densities of cold and magnetized dust are obtained by using Eqs. (5) and (6)
nd1=nd0Qd0md1ω2ωcd2k2φ+kz2ψ/ω2E33
The parallel component of perturbed dust current density turns out to be from Eq. (14) Jd1z=ωpd2/ωϵ0kz2ψand the dispersion relation of kinetic Alfvén wave in the presence of magnetized dust is given by
ω2=kz2VDA21+ϵρd2c2cκE34
In the limit κ,we obtain classical results in a Maxwellian plasma.
#### 3.5.1. Lorentzian-type charging currents
The charging equation containing Lorentzian electron and ion currents is
Qd1t=Ie1κ+Ii1κE35
where the electron and ion currents are calculated using a surface integral through the dust grain surface of radius rdhaving potential φdare given as
Ie1κ=τ1ξe0κ+ξe0Zξe0meθe+2eφdZκξe0,E36
and
Ii1κ=τ2ξi0κ+ξi0Zξi0miθi2eφdZκξi0,E37
where τ1=ad2ψ/4miθiλDiκ2κ,τ2=ad2ψ/4miθiλDiκ2κand λe.iκ=1cκTe,i4πne,i0e212is the Debye wavelength in superthermal plasma which is much smaller than found for a Maxwellian plasma and has been shown by [39, 41] and adis the radius of dust grain.
The Lorentzian charging currents are derived by using Vlasov-kinetic model whose fluid version by Rubab and Murtaza [41] and in the limit κ,our results matched with Das et al., [32]. Now, by putting the value of perturbed dust grain charge, Qd1=±iωΩψ,in Eq. (1), the dispersion equation of DKAW becomes,
which clearly shows that charge fluctuation effects are insensitive to the form of the distribution function.
#### 3.5.2. Modified dust-acoustic wave
In the limit ω2/kz2VA20,the Eq. (16) after simplification turns out to be
ω2=cκ1kz2CD2cκ1+k2ρd2+±nd0rd2kz,E39
where ρd2=CD2/ωcd2,CD=Teff/md12and Teff=nd0Zd2Teni0+Tine0/ni0neo.Eq. (39) is the dispersion relation of dust-acoustic wave in a magnetized plasma whose Maxwellian version without dust charge fluctuation effects is given by Mahmood and Saleem [42]. It could be seen that the component of dust velocity in the direction of magnetic field Vdz, which finally turns out to be dust gyroradius, is responsible for the coupling of Lorentzian type DKAW with DAW. When the dust-acoustic wave frequency is very large compared to the dust gyroradius, then the dust is considered to be unmagnetized. In an unmagnetized plasma B0=0with Td=0,we get the dispersion relation of Lorentzian dust-acoustic wave (without dust charge fluctuation effects) which is exactly equal as discussed by [40]. The effect of Lorentzian index when growth rates are plotted as function of parallel and perpendicular wave number are depicted through graphical representation in Figure 5 and Figure 6 and shows that Maxwellian distribution functions are supportive to enhance the wave frequency.
#### 3.5.3. DKAW: Perpendicular streaming
We consider an electromagnetic dust kinetic Alfvén wave streaming instability in a collisionless electron-ion dusty magnetoplasma. The motion of DKAW is followed by considering thermal and magnetized Lorentzian electrons to be Maxwellian and Lorentzian ions drifting across the external magnetic field B0ẑwith a constant drift velocity V0x̂, i.e., V0B0.The dust is considered to be cold and magnetized ωωcdand the charge on the dust grain surface is taken to be constant. The wave vector associated with the electromagnetic wave lies within xzplane
B0=00B0,V0=V000,k=ksinθ0kcosθ
where
kz=kcosθ,k=ksinθ
The distribution function of Lorentzian ions where ions are streaming perpendicular to the field direction is given as,
fi0κ=Aκ1+1κvtiκ2vz2+vV02κ1;κ>3/2,E40
ni1=2ni0mivtiκ2κ+ηZη.E41
where η=ωkV0kvtiκ.As there are no ions along the field direction due to perpendicular streaming, therefore we may neglect the ions current density Ji1z=0.In the limit κ,our results reduce to Maxwellian distribution.
The dispersion relation with the aid of Eq. (15) is obtained by using Eqs. (23), (24), (33) and (41) in Eqs. (1) and (2), i.e.,
1+2ωpi2kz2vtiκ2κ'+η'Zκη'χ+2ωpe2kz2vteκ2κ+ξe0Zκξe0+ωvteκc2k3ξκ+ξe0Zκξe0+k2kz2χ1+ϜDωpd2ω2=0,E42
which is the general dispersion relation of kinetic Alfvén waves in the presence of perpendicular streaming ions and cold and magnetized dust. In the above equation, ϜD=ωpd2/ωcd2is responsible for the magnetized dust part.
For parallel propagation and in the limit ωpd2/c2k21,ϜD1,we get dispersion relation of two stream instability (TSI) in an unmagnetized dusty plasma. In a dust free plasma ωpd2=0,we get the classical well know relation of TSI, while in the absence of streaming ions, i.e., V0=0,we obtain the dispersion relation of dust kinetic Alfvén waves
where Ωdlhd2=ωpi/ϜD,Λd=nd0zd0/ne0and ρd=CD/ωcd.In the limit k2ρd21,we obtain modified shear Alfvén wave associated with the hybrid dynamics of the ions and magnetized dust through Ωdlhd2which provides a cut-off for the EM wave propagation, i.e.,
ω2=Ωdlhd21+λi2kz2E44
where λi=c/ωpi.
The dispersion relation for the DKAW instability is found to be dependent on the spectral index κwhich means Lorentzian plasma is able to support a number of unstable branches. Lorentzian index is found to be more effective in large wave length limit as compared to small wavelength where the tail of unstable region remains independent of κ. When a large number of dust grains are introduced, it will enhance the loss rate of electrons by attachment on a dust grain surface which reduces the wave activity. At the same time the electron loss rate increases the drift velocity which in turns helps to excite the DKAWs and a further increase will help to stabilize the system. Due to particular choice of equations which involves parallel current density, the ions electromagnetic response cant not take part which limits the existence of ions Weibel instability.
By using the same parameters as above, we have plotted the growth rates as the function of propagation vector for different values of kappa. We have seen that the cross-field streaming of superthermal ions inhibit the growth rate of instability as shown in Figure 7. Similarly, βdis found to support the unstable structure and the instability increases with the value of βdas shown in Figure 8.
## 4. Weibel instability in a Lorentzian plasma
The Weibel plasma instability has so many applications in astrophysical [43], and in laboratory plasmas as well [44]. The generation of magnetic field can be explained in the domain of gamma ray burst, galactic cosmic rays and supernovae [45, 48]. For the case of unmagnetized plasma, the Weibel instability [20] has been widely discussed in relativistic and nonrelativistic regimes. In 1989, Yoon [46, 47] generalized his work by using relativistic bi-Maxwellian plasma. Later, Schaerfer [48] have discussed this instability in relativistic regimes of plasma with arbitrary distributions and presented comparison with his previous works which was based on bi-Gaussian distribution functions. The Weibel instability was investigated by Califano [49, 50] with temperature anisotropy, produced by two counterstreaming electron populations. Davidson probed the multi species Weibel instability for the charged beam and intense ions in plasma [51].
In our work, we have derived the analytical expressions and compared the results numerically for the real and imaginary parts of the dielectric constant with the Maxwellian and kappa κdistributions under two conditions i.e., α=ωΘkz1and 1.
By using kinetic model, the linear dispersion relation for Weibel instability in unmagnetized plasma has been derived after solving the linearized, nonrelativistic Vlasov equation as below [52],
ω2c2k2ωpe2+πωpe2kmm3dvzωkvz0v3dv×f0κvz=0,E45
where f0is the distribution function and here we will discuss the different velocity distributions, i.e., Maxwellian distribution and κdistribution functions.
To calculate Weibel instability in a Lorentzian plasma, we use Eq. (21), for zero streaming velocity of particle, i.e., V0=0and using f0vzin Eq. (45), and performing perpendicular integration, we are left with parallel integral which is called modified plasma dispersion function for kappa distribution.
ω2c2k2ωpe21TTz+ωpe2πTTzΓκκ12Γκ12α1+x2κκxαdx=0,E46
where x=Θz1vz, α=ωΘzkzand Θz,=2κ3Tz,κm.
Applying same procedure as above and again using Plemelj’s formula,
1+x2κκxαdx=P1+x2κκxαdx+1+x2κκ,E47
the integration of principal part yields
P1+x2κκxαdx=πκ1/2Γκ12Γκ1+κΓκ32Γκ12.E48
The dispersion relation will be solved under two following conditions
For α>1:
ω4c2k2+ωpe2ω2ωpe2TTzvtz2kz2=0,E49
which shows the real part of Weibel instability is insensitive to the value of Lorentzian index and the imaginary part 1+α2κκ0.
For α<1:
The dispersion relation takes the form
ω2c2k2ωpe21TTz+ωpe2πTTzΓκκ12Γκ12α1+x2κκxαdx=0.E50
Now, we define a new plasma dispersion function, i.e.,
Zκα=1πΓκκ12Γκ121+x2κκxαdx,E51
the corresponding dispersion relation can be expressed as
ω2c2k2ωpe21TTz+ωpe2TTzαZκα=0.E52
We can Solve Zκαby taking κan integer and assuming α<1.So for κ=3,5,7we get the following three Zfunctions respectively.
Z3α=α1.660.370α2.+ι1.5391.539α2+..Z5α=α1.80.48α2.+ι1.6351.635α2+Z7α=α1.980.59α2..+ι1.7321.736α2+E53
So the three dispersion relations for the above three corresponding Z-functions are.
For κ=3,we get
c2k2+ωpe2TTzωpe21+1.66α2iωpe2TT1.539α=0γ=Imω=0.649kzvTzωpe2TzTc2k2+ωpe21TTzE54
Similarly, for κ=5and 7we obtain the followings
γ=Imω=0.7324kzvTzωpe2TzTc2k2+ωpe21TTzE55
and
γ=Imω=0.8152kzvTzωpe2TzTc2k2+ωpe21TTzE56
Using the Vlasov model, we have derived new dispersion relations based on κdistribution function in an unmagnetized plasma. The analytical expressions for the dielectric constant have been obtained under two conditions i.e., α1and α1, which finally give real and imaginary parts respectively. The real part if found to be insensitive to the value of Lorentzian index while imaginary part shows strong dependence on κ.A graphical representation has also been added for the comparison of non-Maxwellian distributions with that of the Maxwellian. The imaginary parts of the dispersion relation obtained above have been plotted for different values of κshowing the variation of the normalized frequencies, i.e., Imωωpeagainst ckωpe.Figure exhibits the comparison of the result of kappa distribution with that of the Maxwellian. For small κ,the growth rate also reduces but on other hand on increasing the κvalue, the growth rate enhances and finally approaches the Maxwellian results which is shown in Figure 9.
## 5. Collisional Weibel instability with non-zero magnetic field
The dispersion relation of Weibel instability for transverse waves propagating parallel to magnetic field is obtained as
ω2c2k2ωpe21Θ2Θz2+ωpe2kΘzω1Θ2Θz2ω±ΩZκα=0E57
where Θz,=2κ3κTz,mtaking the limit α>1,we obtain
ω2c2k2ωpe2ωω±Ω+ωpe21T2Tz2k2vTz2ω±Ω2=0E58
We notice that the final expression becomes independent of the spectral index κ.
However, for αsmall, the dispersion function Zκαis obtained by choosing specific values of κ.
For κ=5,7we get
Z5α=α1.860.560α2.+ι1.731.79α2+..Z7α=α2.050.734α2..+ι1.91.92α2+E59
The imaginary ωtherefore becomes
ω=i0.79kvTzωpe2TzTc2k2ωpe21TTz1TzTΩE60
and
ω=i0.88kvTzωpe2TzTc2k2ωpe21TTz1TzTΩE61
for κ=5and 7respectively.
Considering
ω=ωr+iωi+νe
ω=i0.79kvTzωpe2TzTc2k2ωpe21TTz1TzTΩνeE62
and
ω=i0.79kvTzωpe2TzTc2k2ωpe21TTz1TzTΩνeE63
where
νe=1pdpdt=1p0fκvmevzdvE64
It is obvious from the above relation that collision frequency for particles obeying kappa distribution differs from that of Maxwellian distribution and is dependent on the value of specie of choice j=e,i. It is seen that collision frequency increases with j=e,iand is less for kappa distributed particles than that of the Maxwellian particles. It is therefore justified to use appropriate collision frequency for such Kappa distributed particles.
## 6. Conclusion
In this chapter, we have described the electromagnetic waves and instabilities in a generalized Lorentzian plasma including particle streaming and finite and anisotropic thermal spread. It allows to grasp the practical understanding of a complex collisionless system in terms of spectra, bulk relative motion and instabilities. In particular, we have focused on kinetic Alfvén waves and instabilities in a dusty and Lorentzian plasma and several types of modes have been identified under various conditions. We have reviewed the kinetic waves and Weibel instabilities in a non-Maxwellian space and astrophysical plasmas by incorporating some basic concepts of dusty environments. We have found that dispersion characteristics involving kinetic Alfvén waves become significantly modified by superthermality effects and dust plasma parameters. The coupling of magnetized dust to the waves due to cyclotron resonance is shown to play a vital role on the wave dynamics. Moreover, the dust grain charging yield some additional plasma currents, which depends on the streaming velocity, Lorentzian index and plasma beta. The Lorentzian index is found to either enhance or quench the electromagnetic instabilities. The dust component is found to play an essential role in wave dynamics, i.e., introducing dust lower hybrid frequency when mobile dust particles are included in the plasma. We have seen that the temperature anisotropy in the distribution function has no effect on the wave characteristics, i.e., the employed model inhibits the temperature anisotropy, but supports the velocity anisotropy. Moreover, a brief analysis on Weibel instabilities in a non-Maxwellian plasma in is also presented.
Kinetic Alfvén turbulence are always present in the streaming solar wind near 1 AU and in situ measurements have confirmed the presence of non-Maxwellian proton distribution function. The present investigations show that the Lorentzian charged particle distributions in space lead to a essentially new physical situation as compared to the plasma with equilibrium distribution functions. Our results of the present analysis opens a new window of investigation to study various streaming and anisotropic modes in different plasma scenarios when Lorentzian distribution function is employed.
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https://math.stackexchange.com/questions/3197668/computing-partial-derivatives-of-fa-b-int-01axb-frac11x22 | # Computing partial derivatives of $f(a,b)= \int_{0}^{1}(ax+b+\frac{1}{1+x^2})^{2}dx$ using chain rule.
Let $$f(a,b)= \int_{0}^{1}(ax+b+\frac{1}{1+x^2})^{2}dx$$. I want to compute $$\frac{\partial{f(a,b)}}{\partial{a}}$$ and $$\frac{\partial{f}(a,b)}{\partial{b}}$$. I was told in the text that $$\frac{\partial{f}(a,b)}{\partial{a}}=2. \int_{0}^{1}(ax+b+\frac{1}{1+x^{2}})dx x$$ and $$\frac{\partial{f}(a,b)}{\partial{b}}=2. \int_{0}^{1}(ax+b+\frac{1}{1+x^{2}})dx$$ because the Chain rule but I cannot justify this. I was thinking this holds because if $$f(a,b)=F(H(a))$$ where $$F(t)=\int_{0}^{1}(t+b+\frac{1}{1+x^2})^{2}dx$$ and $$H(a)=ax$$, then $$\frac{\partial{f(a,b)}}{\partial{a}}=F'(H(a))H'(a)$$ but there are counterexamples where $$f(a,b) \neq F(H(a)$$. Can anyone help me fill the gaps about how these partial derivatives were obtained?? Thanks!
It is because the derivative with respect to $$a$$ of $$(ax+b+\frac{1}{1+x^2}$$) is $$x$$. Best keep the $$x$$ inside the integrand before the $$dx$$.
$$\frac{\partial f(a,b)}{\partial a} = \frac{\partial}{\partial a}\int_0^1 (ax+b+\frac{1}{1+x^2})^2 dx$$
$$= \int_0^1 \frac{\partial}{\partial a}(ax+b+\frac{1}{1+x^2})^2dx = \int_0^1 2(ax+b+\frac{1}{1+x^2})(ax+b+\frac{1}{1+x^2})' dx = \int_0^1 2(ax+b+\frac{1}{1+x^2})x dx$$
• $2$ is the derivative of $x^2$, and our expression is $g(a)^2$ so the derivative wrt $a$ of this is $2g(a)g'(a)$ – George Dewhirst Apr 22 at 23:44
• How is explicitily $g(a)$ expressed $g(a)=(ax+b+\frac{1}{1+x^2})$ or $g(a)= \int_{0}^{1} (ax+b+\frac{1}{1+x^2})$? – Cos Apr 22 at 23:48
• The first one. We are differentiating $\int_0^1 g(a,x)^2 dx$, we are allowed to take the partial derivative inside of the integral. – George Dewhirst Apr 22 at 23:49
• So $g(a,x)^{2}=(ax+b+\frac{1}{1+x^2})$?? and partial derivative of $f(a,b)$ as I defined is the same of diferentiating $\int_{0}^{1}g(a,x)^{2}$??? And differentiating this last one is $\int_{0}^{1} \frac{\partial{g(a,x)^{2}}}{\partial {a}}$ ???? – Cos Apr 23 at 0:00 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 17, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9891906380653381, "perplexity": 199.84127871485123}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256147.15/warc/CC-MAIN-20190520202108-20190520224108-00304.warc.gz"} |
http://wikinotes.ca/MATH_236/lecture-notes/winter-2013/thursday-february-7 | # Thursday, February 7, 2013 Polynomials, introduction to eigenvalues
## 1Polynomials¶
### 1.1Proposition 4.10: Complex conjugate roots¶
Suppose $p$ is a polynomial with real coefficents. If $\alpha \in \mathbb C$ is a root of $p$, so is $\overline \lambda$ (the complex conjugate).
Proof: write $p$ as $p(x) = a_0 + a_1x + \ldots + a_mx^m$, where $a_i \in \mathbb R$. Then, $\lambda \in \mathbb C$ being a root of $p$ means that $p(\lambda) = a_0 + a_1 \lambda + \ldots + a_m\lambda^m = 0$. Let's take the conjugate of both sides of that:
$$\overline{p(\lambda)} = \overline 0 = 0 \quad \therefore \overline{(a_0 + a_1\lambda + \ldots + a_m\lambda^m)} = 0$$
Now, we know that $\overline{a_i} = a_i$ for each $a_i$ since they're all real. Also, by the properties of the conjugate operator, we know that $\overline{u+v} = \overline u + \overline v$ and $\overline{u \cdot v} = \overline u \cdot \overline v$. Thus, we have that
$$a_0 + a_1\overline \lambda + \ldots + a_m\overline \lambda^m = 0 = p(\overline \lambda)$$
which tells us that $\overline \lambda$ is a root of $p$. $\blacksquare$
Let $\alpha, \beta \in \mathbb R$. If we can factor $x^2 + \alpha x + \beta$ as $(x-\lambda_1)(x-\lambda_2)$ where $\lambda_1, \lambda_2 \in \mathbb R$, then $\alpha^2 \geq 4\beta$ holds true. (This is really just an application of the quadratic formula to a less general form.)
Proof: ($\rightarrow$) Complete the square. $x^2 + \alpha x + \beta = (x+\alpha/2)^2 + (\beta - \alpha/4)^2$. Suppose that $\alpha^2 < 4\beta$. Then, $\beta - \alpha^2/4 > 0$, and so $(x + \alpha/2)^2$ will always be positive. Thus the function will always be above the $x$-axis, and so there are no roots. (This was a proof using the contrapositive.)
($\leftarrow$) Assume that $\alpha^2 \geq 4\beta$. Then
$$x^2 + \alpha x + \beta = (x+\alpha/2)^2 - \underbrace{(\alpha/4-\beta)}_{= c^2}$$
(We can set the latter term to some arbitrary constant $c^2$ for some $c \in \mathbb R$, since it's a positive term.) Then, $x^2 + \alpha x + \beta = (x + \alpha/2)^2 - c^2 = (x + \alpha/2 - c)(x + \alpha/2+c)$ and so $\lambda_1 = \alpha/2 -c$, $\lambda_2 = \alpha/2+c$. $\blacksquare$
### 1.3Theorem 4.14: Unique factorization¶
If $o \in P(\mathbb R)$ is a non-constant polynomial, then $p$ has a unique factorisation (ignoring differences in order) of the form
$$p(x) = c(x-\lambda_1)\ldots(x-\lambda_m)(x^2+\alpha_1x+\beta_1) \ldots (x^2+\alpha_nx+\beta_nx)$$
where the $\lambda$s are in $\mathbb C$, $c \in \mathbb R$, and $\alpha_i^2 < 4\beta_i$.
Thus, we can always factor any polynomial into its irreducible linear/quadratic factors.
Also, if we know that $(x-\lambda)$ for some complex (non-real) $\lambda$, then we also know that $(x-\overline \lambda)$ is a factor. So $(x-\lambda)(x -\overline \lambda)$ is a factor, and this happens to be a quadratic with real coefficients. So this will show up as a quadratic factor.
Proof: in the textbook. QED $\blacksquare$ $\checkmark$ $\dagger$ $\heartsuit$ $\sharp$ $\Im$ $\leadsto$
## 2Eigenvalues and invariant subspaces¶
Let $V$ be a finite-dimensional, non-trivial vector space. Recall that $\mathcal L(V) = \mathcal L(V, V)$ (the set of linear operators - that is, linear maps from a vector space to itself).
### 2.1Invariant subspaces¶
Let $T \in \mathcal L(V)$, and let $U$ be a subspace of $V$. We say that $U$ is invariant under $T$ if for every $u \in U$, $Tu \in U$.
#### 2.1.1Examples¶
$T: P_7(\mathbb R) \to P_7(\mathbb R), p(x) \mapsto \frac{dp(x)}{dx}$ (that is, it maps a polynomial to its derivative). Let $U = P_5(\mathbb R)$. Thus $U$ is invariant under $T$, obviously. (Any subspace would be ...) Although if the linear operator were integration and not differentiation, no subspace would be invariant under it.
For any $T \in \mathcal L(V)$, are its nullspace and range invariant? Yes, obviously - if $u \in \text{null}(T)$, then $Tu = 0$ and so $T(Tu) = T(0) = 0$. Similarly for the range: if $v \in \text{range}(T)$, then $Tv$ is also in the range (since $v \in V$, the domain).
### 2.2Eigenvalues¶
$\lambda \in \mathbb F$ is an eigenvalue of $T \in \mathcal L(V)$ if there exists a non-zero vector $v \in V$ such that $Tu = \lambda v$.
If $T$ has an eigenvalue, then $T$ has a one-dimensional invariant subspace, and vice versa. Proof: let $u \neq 0$, and let $U = \{au \mid a \in \mathbb F \}$, which is a one-dimensional subspace of $V$. Assume that $Tu = \lambda u$ for some $\lambda$. Then $U$ is invariant, by definition. Conversely, if $u \in U$ and $U$ is invariant, then we have $Tu \in U$. But then $Tu$ is just $au$ for some $a$. Like, $a=\lambda$. Thus, eigenvalue. $\heartsuit$
Notice that the equation $Tu = \lambda u$ is equivalent to $(T - \lambda I)u = 0$ (where $I$ is the identity operator, not matrix). Thus $\lambda$ is an eigenvalue of $T$ if and only if $T-\lambda I$ is not injective (otherwise its nullspace would consist only of the zero vector). Correspondingly, $T-\lambda I$ cannot be surjective or invertible either. Linear algebra is pretty cool.
(I got the last paragraph from the textbook. I vaguely remember something along those lines being written on the board, but at that point I had already put away my notebook and to bring it back again just to write down a few lines felt defeatist. If you have notes for that please let me know.) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9884136319160461, "perplexity": 138.41001397214987}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203493.88/warc/CC-MAIN-20190324210143-20190324232143-00525.warc.gz"} |
http://kmj.knu.ac.kr/journal/view.html?uid=2312&vmd=Full | Kyungpook Mathematical Journal 2018; 58(4): 781-788
Finslerian Hypersurface and Generalized β–Conformal Change of Finsler Metric
Shiv Kumar Tiwari*, and Anamika Rai
Department of Mathematics, K. S. Saket Post Graduate College, Ayodhya, Faizabad224 123, India
e-mail : [email protected] and [email protected]
*Corresponding Author.
Received: March 11, 2015; Accepted: February 13, 2018; Published online: December 23, 2018.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In the present paper, we have studied the Finslerian hypersurfaces and generalized β–conformal change of Finsler metric. The relations between the Finslerian hypersurface and the other which is Finslerian hypersurface given by generalized β–conformal change have been obtained. We have also proved that generalized β–conformal change makes three types of hypersurfaces invariant under certain conditions.
Keywords: generalized β-conformal change, generalized β-change, β-change, conformal change, Finslerian hypersurfaces, hyperplane of fi rst, second and third kinds.
1. Introduction
Let (Mn, L) be an n–dimensional Finsler space on a differentiable manifold Mn equipped with the fundamental function L(x, y). In 1984, Shibata [12] introduced the transformation of Finsler metric:
$L¯(x,y)=f(L,β),$
where β = bi(x) yi, bi(x) are components of a covariant vector in (Mn, L) and f is positively homogeneous function of degree one in L and β. This change of metric is called a β–change. In 2013, Prasad, B. N. and Kumari, Bindu [10] have considered the β–change of Finsler metric. In the year 2014 [13], we studied generalized β–change defining as
$L(x,y)→L¯(x,y)=f(L,β1),β2),…,βm)),$
where f is any positively homogeneous function of degree one in L, β1), β2), …, βm), where β1), β2), …, βm) are linearly independent one-form.
The conformal theory of Finsler spaces has been initiated by M. S. Knebelman [7] in 1929 and has been investigated in detail by many authors [1, 2, 3, 6] etc. The conformal change is defined as
$L(x,y)→eσ(x)L(x,y),$
where σ(x) is a function of position only and known as conformal factor.
We also studied the generalized β–conformal change of Finsler metric by taking
$L¯=f(eσ(x)L(x,y),β1),β2),…,βm)),$
where f is any positively homogeneous function of degree one in eσL, β1), β2), …, βm).
On the other hand, in 1985, M. Matsumoto investigated the theory of Finslerian hypersurface [8]. He has defined three types of hypersurfaces that were called a hyperplane of the first, second and third kinds.
In the year 2009, B. N. Prasad and Gauri Shanker [11] studied the Finslerian hypersurfaces and β–change of Finsler metric and obtained different results in his paper. In the present paper, using the field of linear frame [5, 4, 9], we shall consider Finslerian hypersurfaces given by a generalized β–conformal change of a Finsler metric. Our purpose is to give some relations between the original Finslerian hypersurface and the other which is Finslerian hypersurface given by generalized β–conformal change. We have also obtained that a generalized β–conformal change makes three types of hypersurfaces invariant under certain conditions.
2. Finslerian Hypersurfaces
Let Mn be an n–dimensional manifold and Fn = (Mn, L) be an n–dimensional Finsler space equipped with the fundamental function L(x, y) on Mn. The metric tensor gij(x, y) and Cartan’s C–tensor Cijk(x, y) are given by
$gij=12∂2L2∂yi∂yj, Cijk=12∂gij∂yk,$
respectively and we introduce the Cartan’s connection $CΓ=(Fjki,Nji,Cjki)$ in Fn.
A hypersurface Mn−1 of the underlying smooth manifold Mn may be parametrically represented by the equation xi = xi(uα), where uα are Gaussian coordinates on Mn−1 and Greek indices vary from 1 to n − 1. Here, we shall assume that the matrix consisting of the projection factors $Bαi=∂xi∂uα$ is of rank n − 1. The following notations are also employed:
$Bαβi=∂2xi∂uα∂uβ, B0βi=vαBαβi.$
If the supporting element yi at a point (uα) of Mn−1 is assumed to be tangential to Mn−1, we may then write $yi=Bαi(u)vα$, i.e. vα is thought of as the supporting element of Mn−1 at the point (uα). Since the function (u, v) = L{x(u), y(u, v)} gives rise to a Finsler metric of Mn−1, we get a (n − 1)–dimensional Finsler space Fn−1 = {Mn−1, (u, v)}.
At each point (uα) of Fn−1, the unit normal vector Ni(u, v) is defined by
$gijBαiNj=0, gijNiNj=1.$
If $Biα$, Ni is the inverse matrix of ($Bαi$, Ni), we have
$BαiBiβ=δαβ, BαiNi=0, NiNi=1 and BαiBjα+NiNj=δji.$
Making use of the inverse matrix (gαβ) of (gαβ), we get
$Biα=gαβgijBβj, Ni=gijNj.$
For the induced Cartan’s connection $ICΓ=(Fβγα,Nαβ,Cβγα)$ on Fn−1, the second fundamental h–tensor Hαβ and the normal curvature Hα are respectively given by [9]
$Hαβ=Ni(Bαβi+FjkiBαjBβk)+MαHβ,Hα=Ni(B0βi+NjiBβj),$
where
$Mα=CijkBαiNjNk.$
Contracting Hαβ by vα, we immediately get H0β = Hαβvα = Hβ. Furthermore the second fundamental v–tensor Mαβ is given by [8]
$Mαβ=CijkBαiBβiNk.$
3. Finsler Space with Generalized β–Conformal Change
Let (Mn, L) be a Finsler space Fn, where Mn is an n–dimensional differentiable manifold equipped with a fundamental function L. A change in fundamental metric L, defined by equation (1.4), is called generalized β–conformal change, where σ(x) is conformal factor and function of position only and β1), β2), …, βm) all are linearly independent one-form and defined as $βr)=bir)yi$.
Homogeneity of f gives
$eσLf0+frβr)=f,$
where the subscripts ‘0’ and ‘r’ denote the partial derivative with respect to L and βr) respectively. The letters r, s, t, r′ and s′ vary from 1 to m throughout the paper. Summation convention is applied for the indices r, s, t, r′ and s′. If we write n = (Mn, ), then the Finsler space n is said to be obtained from Fn by generalized β–conformal change. The quantities corresponding to n are denoted by putting bar on those quantities.
To find the relation between fundamental quantities of (Mn, L) and (Mn, ), we use the following results:
$∂˙i βr)=bir), ∂˙i L=li, ∂˙j li=L-1hij,$
where ∂̇i stands for $∂∂yi$ and hij are components of angular metric tensor of (Mn, L) given by
$hij=gij-li lj=L ∂˙i ∂˙j L.$
Differentiating (3.1) with respect to L and βs) respectively, we get
$eσL f00+f0rβr)=0$
and
$eσL f0s+frsβr)=0.$
The successive differentiation of (1.4) with respect to yi and yj give
$l¯i=eσf0li+frbir),$$h¯ij=eσf f0Lhij+e2σf f00lilj+eσf f0r(bjr)li+bir)lj)+f frsbir)bjs).$
Using equations (3.3) and (3.4) in equation (3.6), we have
$h¯ij=eσf f0Lhij+f frs (bir)-βr)Lli) (bjs)-βs)Llj).$
If we put $mir)=bir)-βr)Lli$, equation (3.7) may be written as
$h¯ij=eσf f0Lhij+f frsmir)mjs).$
From equations (3.5) and (3.8), we get the following relation between metric tensors of (Mn, L) and (Mn, )
$g¯ij=eσf f0Lgij+eσ (eσf02-f f0L) lilj+f frsmir)mjs)+eσf0fr(bir)lj+bjr)li)+frfsbir)bjs).$
Now,
$(a)∂˙imjr)=-1L(mir)lj+βr)Lhij),(b)∂˙if=eσf0li+frbir),(c)∂˙ifrs=eσfrs0li+frstbit).$
Differentiating equation (3.8) with respect to yk and using equations (3.2), (3.3), (3.4), (3.5), (3.9) and (3.10), we get
$C¯ijk=p0Cijk+p1(hijmkr)+hjkmir)+hkimjr))+p2mir)mjs)mkt),$
where
$p0=eσf f0LCijk, p1=eσ2L(f0fr+f f0r),p2=12(frsft+fstfr+ftrfs+f frst).$
4. Hypersurfaces Given by a Generalized β–Conformal Change
Consider a Finslerian hypersurface Fn−1 = {Mn−1, (u, v)} of the Fn and another Finslerian hypersurface n−1 = {Mn−1, (u, v)} of the n given by generalized β–conformal change. Let Ni be the unit vector at each point of Fn−1 and ($Biα$, Ni) be the inverse matrix of ($Biα$, Ni). The function $Biα$ may be considered as components of (n − 1) linearly independent tangent vectors of Fn−1 and they are invariant under generalized β–conformal change. Thus, we shall show that a unit normal vector i(u, v) of n−1 is uniquely determined by
$g¯ijBαiN¯j=0, g¯ijN¯iN¯j=1.$
Contracting (3.9) by NiNj and paying attention to (2.1) and the fact that liNi = 0, we have
$g¯ijNiNj=p0+p(bir)bjs)NiNj),$
where p = ffrs + frfs. Therefore, we obtain
$g¯ij(±Nip0+p(bir)bjs)NiNj)) (±Njp0+p(bir)bjs)NiNj))=1.$
Hence, we can put
$N¯i=Nip0+p(bir)bjs)NiNj),$
where we have chosen the positive sign in order to fix an orientation.
Using equations (3.9), (4.3) and from first condition of (4.1), we have
$Bαi(2p1Lli+pbir)).bjs)Njp0+p(bir)bjs)NiNj)=0.$
If $Bαi(2p1Lli+pbir)=0$, then contracting it by vα and using $yi=Bαivα$, we get L = 0 or βr) = 0 which is a contradiction with the assumption that L > 0. Hence $bjs)Nj=0$. Therefore equation (4.3) is written as
$N¯i=Nip0.$
Summarizing the above, we obtain
### Proposition 4.1
For a field of linear frame ($B1i,B2i,…,Bn-1i$, Ni) of Fn there exists a linear frame ($B1i,B2i,…Bn-1i,N¯i=Nip0$) of F̄n such that (4.1) is satisfied along F̄n−1and then$bir)$is tangential to both of the hypersurfaces Fn−1and F̄n−1.
The quantities $B¯iα$ are uniquely defined along n−1 by
$B¯iα=g¯αβg¯ijBβj$
where αβ is the inverse matrix of αβ. Let ($B¯iα$, i) be the inverse matrix of ($Bαi$, i), then we have
$BαiB¯iβ=δαβ, BαiN¯i=0, N¯iN¯i=1.$
Furthermore $BαiB¯jα+N¯iN¯j=δji$. We also get i = ijj which in view of (3.5), (3.9) and (4.5) gives
$N¯i=p0 Ni.$
We denote the Cartan’s connection of Fn and n by ($Fjki,Nji,Cjki$) and ($F¯jki,N¯ji,C¯jki$) respectively and put $Djki=F¯jki-Fjki$ which will be called difference tensor. We choose the vector field br)i in Fn such that
$Djki=Ajkbr)i+Bjkli+δjiDk+δkiDj,$
where Ajk and Bjk are components of a symmetric covariant tensor of second order and Di are components of a covariant vector. Since Nibr)i = 0, Nili = 0 and $δjiNiBαj=0$, from (4.7), we get
$NiDjkiBαjBβk=0 and NiD0kiBβk=0.$
Therefore, from (2.3) and (4.6), we get
$H¯α=p0 Hα.$
If each path of a hypersurface Fn−1 with respect to the induced connection also a path of the enveloping space Fn, then Fn−1 is called a hyperplane of the first kind. A hyperplane of the first kind is characterized by Hα = 0 [8]. Hence from (4.9), we have
### Theorem 4.1
If$bir)(x)$be a vector field in Fn satisfying (4.7), then a hypersurface Fn−1is a hyperplane of the first kind if and only if the hypersurface F̄n−1is a hyperplane of the first kind.
Next contracting (3.11) by $BαiN¯jN¯k$ and paying attention to (4.5), $mir)Ni=0$, hjkNjNk = 1 and $hijBαiNj=0$, we get
$M¯α=Mα+p1p0mir)Bαi.$
From (2.3), (4.6), (4.8), we have
$H¯αβ=p0 Hαβ.$
If each h–path of a hypersurface Fn−1 with respect to the induced connection is also h–path of the enveloping space Fn, then Fn−1 is called a hyperplane of the second kind. A hyperplane of the second kind is characterized by Hαβ = 0 [8]. Since Hαβ = 0 implies that Hα = 0 from (4.9) and (4.10), we have the following:
### Theorem 4.2
If$bir)(x)$be a vector field in Fn satisfying (4.7), then a hypersurface Fn−1is a hyperplane of the second kind if and only if the hypersurface F̄n−1is a hyperplane of the second kind.
Finally contracting (3.11) by $BαiBβjN¯k$ and paying attention to (4.5), we have
$M¯αβ=p0 Mαβ.$
If the unit normal vector of Fn−1 is parallel along each curve of Fn−1, then Fn−1 is called a hyperplane of third kind. A hyperplane of the third kind is characterized by Hαβ = 0, Mαβ = 0 [8]. From (4.10) and (4.11), we have:
### Theorem 4.3
If$bir)(x)$be a vector field in Fn satisfying (4.7), then a hypersurface Fn−1is a hyperplane of the third kind if and only if the hypersurface F̄n−1is a hyperplane of the third kind.
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11. Prasad, BN, and Shanker, Gauri (2009). Finslerian hypersurfaces and β–change of Finsler metric. Acta Cienc Indica Math. 35, 1055-1061.
12. Shibata, C (1984). On invariant tensors of β–change of Finsler metrics. J Math Kyoto Univ. 24, 163-188.
13. Tiwari, SK, and Rai, Anamika (2014). The generalized β-change of Finsler metric. International J Contemp Math Sci. 9, 695-702. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 73, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9814154505729675, "perplexity": 4395.697673688424}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986684854.67/warc/CC-MAIN-20191018204336-20191018231836-00468.warc.gz"} |
http://www.planetmath.org/homomorphicimageofgroup | # homomorphic image of group
The homomorphic image of a group is a group. More detailed, if $f$ is a homomorphism from the group $(G,\,\ast)$ to the groupoid$(\Gamma,\,\star)$, then the groupoid $(f(G),\,\star)$ also is a group. Especially, the isomorphic image of a group is a group.
Proof. Let $\alpha,\,\beta,\,\gamma$ be arbitrary elements of the image $f(G)$ and $a,\,b,\,c$ some elements of $G$ such that $f(a)=\alpha,\,f(b)=\beta,\,f(c)=\gamma$. Then
$\alpha\star\beta\;=\;f(a)\star f(b)\;=\;f(a\ast b)\;\in\;f(G),$
whence $f(G)$ is closed under$\star$”, and we, in fact, can speak of a groupoid $(f(G),\,\star)$.
Secondly, we can calculate
$\displaystyle(\alpha\star\beta)\star\gamma$ $\displaystyle\;=\;(f(a)\star f(b))\star f(c)$ $\displaystyle\;=\;f(a\ast b)\star f(c)$ $\displaystyle\;=\;f((a\ast b)\ast c)$ $\displaystyle\;=\;f(a\ast(b\ast c))$ $\displaystyle\;=\;f(a)\star f(b\ast c)$ $\displaystyle\;=\;f(a)\star(f(b)\star f(c))$ $\displaystyle\;=\;\alpha\star(\beta\star\gamma),$
whence the associativity is in in the groupoid $(f(G),\,\star)$.
Let $e$ be the identity element of $(G,\,\ast)$ and $f(e)=\varepsilon$. Then
$\varepsilon\star\alpha\;=\;f(e)\star f(a)\;=\;f(e\ast a)\;=\;f(a)\;=\;\alpha,$
$\alpha\star\varepsilon\;=\;f(a)\star f(e)\;=\;f(a\ast e)\;=\;f(a)\;=\;\alpha,$
and therefore $\varepsilon$ is an identity element in $f(G)$.
If $f(a^{-1})=\alpha^{\prime}$, then
$\alpha\star\alpha^{\prime}\;=\;f(a)\star f(a^{-1})\;=\;f(a\ast a^{-1})\;=\;f(e% )\;=\;\varepsilon,$
$\alpha^{\prime}\star\alpha\;=\;f(a^{-1})\star f(a)\;=\;f(a^{-1}\ast a)\;=\;f(e% )\;=\;\varepsilon.$
Thus any element $\alpha$ of $f(G)$ has in $f(G)$ an inverse.
Accordingly, $(f(G),\,\star)$ is a group.
Remark 1. If $(G,\,\ast)$ is Abelian, the same is true for $(f(G),\,\star)$.
Remark 2. Analogically, one may prove that the homomorphic image of a ring is a ring.
Example. If we define the mapping $f$ from the group $(\mathbb{Z},\,+)$ to the groupoid $(\mathbb{Z}_{9},\,\cdot)$ by
$f(n)\;:=\;\langle 4\rangle^{n},$
then $f$ is homomorphism:
$f(m\!+\!n)\;=\;\langle 4\rangle^{m+n}\;=\;\langle 4\rangle^{m}\langle 4\rangle% ^{n}\;=\;f(m)f(n).$
The image $f(\mathbb{Z})$ consists of powers of the residue class (http://planetmath.org/Congruences) $\langle 4\rangle$, which are
$\langle 4\rangle,\;\;\langle 16\rangle=\langle 7\rangle,\;\;\langle 64\rangle=% \langle 1\rangle.$
These apparently form the cyclic group of order 3.
Title homomorphic image of group HomomorphicImageOfGroup 2013-03-22 18:56:27 2013-03-22 18:56:27 pahio (2872) pahio (2872) 14 pahio (2872) Theorem msc 20A05 msc 08A05 GroupHomomorphism CorrespondenceBetweenNormalSubgroupsAndHomomorphicImages | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 47, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9781537652015686, "perplexity": 1623.1890089210722}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257645604.18/warc/CC-MAIN-20180318091225-20180318111225-00085.warc.gz"} |
https://www.acallard.net/talks/the-aperiodic-domino-problem | # The aperiodic Domino problem
The classical Domino problem asks whether there exists a tiling in which none of the forbidden patterns given as input appear. In this talk, we consider the aperiodic version of the Domino problem: given as input a family of forbidden patterns, does it allow an aperiodic tiling? The input may correspond to a subshift of finite type, a sofic subshift or an effective subshift.
The Aperiodic Domino is undecidable, because it reduces to the (classical) Domino problem. In this talk, we study the computational complexity of the Aperiodic Domino problem: $$\Pi_1^0$$-complete for $$\mathbb{Z}^2$$ subshifts (by a result from A. Grandjean, B. Hellouin and P. Vanier), it becomes $$\Sigma_1^1$$-complete (ie. much harder, namely analytic) in higher dimension: $$d \geq 4$$ in the finite type case, $$d \geq 3$$ for sofic and effective subshifts.
These results are surprising for two reasons: first, the Aperiodic Domino separates 2- and 3-dimensional subshifts, wheareas most subshift properties are the same in dimension 2 and higher; second, this gap unexpectedly large.
For more details, you can read the associated article! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8674742579460144, "perplexity": 1183.8600551942056}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572870.85/warc/CC-MAIN-20220817062258-20220817092258-00180.warc.gz"} |
http://blog.cupcakephysics.com/classical%20mechanics/2015/07/26/newtons-cupcake.html | ## Newton's Cupcake
### Newton’s Cupcake
Isaac Newton published a famous thought experiment in A Treatise of the System of the World. He imagined what would happen if one were to shoot a cannonball from the top of a very high mountain (so high up that air resistance could be ignored). If the canonball were launched with a relatively small velocity, it would strike the ground at some distance from the mountain. However, if the velocity of the canonball were much larger, it might “miss” the ground entirely and go into orbit around the planet. This thought experiment was the key to linking the force of gravity to orbital motion.
It is easy to determine the speed at which circular orbital motion will occur. Let $M$ be the mass of the planet, $m$ be the mass of the cannonball, and $r$ be the radius of the orbit. We know that the force of gravity provides the centripetal force for a body in uniform circular motion around the planet.
After learning about the HTML5 canvas tag in a workshop at the AAPT Summer Meeting, I decided to make a little simulation in order to visualize this thought experiment. The planet below has the same radius as the Earth; you can adjust the mass of the planet, the initial velocity of the projectile, and the angle of launch of the projectile. Sadly, we are out of cannonballs, but we do have a brave cupcake volunteer.
Initial Cupcake Speed 7500 m/s Planet Mass 6.0e24 kg Angle of Fire 0 degrees | {"extraction_info": {"found_math": true, "script_math_tex": 3, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 3, "x-ck12": 0, "texerror": 0, "math_score": 0.968575119972229, "perplexity": 357.5811699796491}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806832.87/warc/CC-MAIN-20171123123458-20171123143458-00074.warc.gz"} |
https://www.math.tu-berlin.de/fachgebiete_ag_diskalg/fachgebiet_algorithmische_algebra/v_menue/members/prof_dr_peter_buergisser/parameter/de/font2/maxhilfe/mobil/?tx_sibibtex_pi1%5Bcontentelement%5D=tt_content%3A721853&tx_sibibtex_pi1%5BshowUid%5D=1351093&cHash=3234ee38a804b85b5cd65005554c9e46 | Fachgebiet Algorithmische AlgebraProf. Dr. Peter Bürgisser
# Leitung
## Prof. Dr. Peter Bürgisser
Anschrift
Technische Universität Berlin
Institut für Mathematik
Sekretariat MA 3-2
Straße des 17. Juni 136
10623 Berlin
Büro
Raum MA 317 (3. OG)
Institut für Mathematik
## Kontakt
Sekretariat
Beate Nießen
Raum MA 318
Tel.: +49 (0)30 314 - 25771
eMail
[email protected]
Telefon
+49 (0)30 314 - 75902
Faxgerät
+49 (0)30 314 - 25839
Sprechstunde
Während der Vorlesungszeit: Do, 15-16 Uhr.
Während der vorlesungsfreien Zeit: nach Vereinbarung.
## Publikationen
Zitatschlüssel BCL-Coverage-Processes-On-Spheres-And-Condition-Numbers-For-Linear-Programming Peter Bürgisser and Felipe Cucker and Martin Lotz 570-604 2010 0091-1798 10.1214/09-AOP489 The Annals of Probability 38 2 This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let $p(n,m,a)$ be the probability that n spherical caps of angular radius $a$ in $S^m$ do not cover the whole sphere $S^m$. We give an exact formula for $p(n,m,a)$ in the case $a\in [\frac\pi2,\pi]$ and an upper bound for $p(n,m,a)$ in the case $a\in [0,\frac\pi2]$, which tends to $p(n,m,\frac\pi2)$ when $a\to\frac\pi2$. In the case $a\in [0,\frac\pi2]$ this yields upper bounds for the expected number of spherical caps of radius $a$ that are needed to cover $S^m$. Secondly, we study the condition number $CC(A)$ of the linear programming feasibility problem $\exists x\in\mathbb R^m+1\, Axłe 0,\, x\ne 0$, where $A\in\mathbb R^n× (m+1)$ is randomly chosen according to the standard normal distribution. We exactly determine the distribution of $CC(A)$ conditioned to $A$ being feasible and provide an upper bound on the distribution function in the infeasible case. Using these results, we show that $\mathbf E(łn CC(A))łe 2łn(m+1) + 3.31$ for all $n>m$, the sharpest bound for this expectancy as of today. Both agendas are related through a result which translates between coverage and condition. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9863136410713196, "perplexity": 761.3203402888057}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875143805.13/warc/CC-MAIN-20200218180919-20200218210919-00557.warc.gz"} |
http://matecolima.blogspot.com.au/ | ## sábado, 28 de febrero de 2015
### Roots of complex numbers
Consider $z=a+ib$ a nonzero complex number. The number $z$ can be written in polar form as
$z=r(\cos \theta +i \sin \theta)$
where $r=\sqrt{a^2+b^2}$ and $\theta$ is the angle, in radians, from the positive $x$-axis to the ray connecting the origin to the point $z$. Furthermore, $z$ can be written in exponential form as
$z=re^{i\theta}$
because $e^{i\theta}=\cos \theta +i \sin \theta$.
The $n$th roots of a nonzero complex number $z$ is given by the following expression
$z=\sqrt[n]{r}\;\mbox{exp}\left[i\left(\frac{\theta}{n}+\frac{2k\pi}{n}\right)\right]$
where $k=0,\pm 1,\pm 2, \ldots , n-1$.
In the following applet you can see a geometrical representation of the $n$th roots for a family of complex numbers. Change the values of the real and imaginary parts of $z$ and the $n$th root. Some examples: i) Re(z)=1/2, Im(z)=1.72, n=3; ii) Re(z)=sqrt(2), Im(z)=pi, n=7.
External link to GeoGebra applet: http://tube.geogebra.org/student/m298919
## sábado, 25 de octubre de 2014
### Relative velocity: Boat problems
Problem 1.
A river flows due East at a speed of 1.3 metres per second. A girl in a rowing boat, who can row at 0.4 metres per second in still water, starts from a point on the South bank and steers due North. The boat is also blown by a wind with speed 0.6 metres per second from a direction of N20ºE.
Figure 1: The red arrows represent the velocities of the boat (b), wind (w) and flow (r).
1. Find the resultant velocity of the boat and its magnitude.
2. If the river has a constant width of 10 metres, how long does it take the girl to cross the river, and how far upstream or downstream has she then travelled?
Problem 2.
A river flows due West at a speed of 2.5 metres per second and has a constant width of 1 km. You want to cross the river from point A (South) to a point B (North) directly opposite with a motor boat that can manage to a speed of 5 metres per second.
1. If you head out pointing your boat at an angle of 90 degrees to the bank. How long does it take to get from point A to point B?
2. After crossing the river you realised that it took longer than expected. In what direction should you point you motor boat in order to reduce the time to cross the river? How long will it take you to get from point A to point B? Is it a better time?
Applet GeoGebra
The following applet shows a representation of the problem 2, considering that the boat starts from a point A. It also shows the velocities (vectors) and their magnitudes (speeds) of the boat and current.
1. Move the sliders to change the magnitude and direction of vectors.
2. Click the 'Start' button to activate the motion of the boat.
3. Click the 'Reset' button to put back the boat to its original position.
4. You can also change the width of the river. Chose a number between 5 and 1000.
5. All velocities can be considered either as metres per second or km per second.
Open this applet in an external window: Relative Velocity: Boat Problem
## viernes, 26 de setiembre de 2014
### Geometrical approach to speed and acceleration (Complete)
Available on iTunes for iPad: Geometrical approach to speed and acceleration | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8137168288230896, "perplexity": 662.492878206474}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936463660.11/warc/CC-MAIN-20150226074103-00107-ip-10-28-5-156.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/questions/14721/how-can-one-localize-the-massless-fermions-in-dirac-materials?answertab=oldest | # How can one localize the massless fermions in Dirac materials?
I noticed that finite electric potential cannot localize the low energy excitations in a graphene sheet. Is it possible to localize the massless fermions in the surface band of topological insulators with a magnetic field?
I found a paper dealing with a similar problem: http://apl.aip.org/resource/1/applab/v98/i16/p162101_s1
-
What do you mean "localize"? Do you mean make bound states? – Ron Maimon Sep 16 '11 at 19:02
Yes. That is what I mean. – Z.Sun Sep 16 '11 at 19:50 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9447199106216431, "perplexity": 916.2871403773154}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375098059.60/warc/CC-MAIN-20150627031818-00134-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://knowridge.com/2016/07/see-solar-corona-3d/ | # See the solar corona in 3D
Solar corona is an aura of plasma surrounding the sun. It extends millions of kilometers into space and is most easily seen during a total solar eclipse.
People usually observe solar corona using coronagraph. This equipment can artificially block the disk of the sun and hence image the regions around it.
Nevertheless, the images of solar corona are usually 2D projections of the 3D emitting structure. This is because every instrument observes it from a single angle of vision or two the most.
Detailed knowledge of the 3D distribution of solar corona is very important to advance its modeling.
Fortunately, the problem has been solved due to the development of differential emission measure tomography (DEMT).
In a paper newly published in Advances in Space Research, the author reviewed all the related work in the field.
DEMT was first introduced 10 years ago. It refers to a technique using time series of extreme ultraviolet imaging (EUV) to determine the 3D distribution of corona.
The technique has two steps. First, the time series of full-sun EUV images is inversed to find the 3D distribution of the EUV emissivity in each filter band of the telescope. Emissivity is the ability of a surface to radiate energy.
Second, the emissivity found for all bands in a given coronal location is used as a constraint to infer the coronal local-DEM. All moments of the local DEM are taken and form the global maps of the solar corona.
DEMT has been used in observing coronal structures and validating coronal models. It also helps extrapolate the coronal photospheric magnetic field.
In the future, this technique will be used in comparative studies of the solar minima and research of the coronal radiative losses.
Citation: Vásquez AM (2016). Seeing the solar corona in three dimensions. Advances in Space Research, 57: 1286-1293. doi:10.1016/j.asr.2015.05.047
Figure legend: This Knowridge.com image is for illustrative purposes only. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8195949792861938, "perplexity": 1267.214701260734}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221209884.38/warc/CC-MAIN-20180815043905-20180815063905-00299.warc.gz"} |
http://mathonline.wikidot.com/connected-and-disconnected-metric-spaces | Connected And Disconnected Metric Spaces
# Connected and Disconnected Metric Spaces
Definition: A metric space $(M, d)$ is said to be Disconnected if there exists nonempty open sets $A$ and $B$ such that $A \cap B = \emptyset$ and $M = A \cup B$. If $M$ is not disconnected then we say that $M$ Connected. Furthermore, if $S \subseteq M$ then $S$ is said to be disconnected/connected if the metric subspace $(S, d)$ is disconnected/connected.
Intuitively, a set is disconnected if it can be separated into two pieces while a set is connected if it’s an entire piece.
For example, consider the metric space $(\mathbb{R}, d)$ where $d$ is the Euclidean metric on $\mathbb{R}$. Let $S = (a, b) \subset \mathbb{R}$, i.e., $S$ is an open interval in $\mathbb{R}$. We claim that $S$ is connected.
Suppose not. Then there exists nonempty open subsets $A$ and $B$ such that $A \cap B = \emptyset$ and $(a, b) = A \cup B$. Furthermore, $A$ and $B$ must be open intervals themselves, say $A = (c, d)$ and $B = (e, f)$. We must have that $A \cup B = (c, d) \cup (e, f)$. So $c = a$ or $e = a$ and furthermore, $d = b$ or $f = b$.
If $c = a$ then this implies that $f = b$ (since if $d = b$ then $A = (a, b)$ which implies that $B = \emptyset$). So if $A \cup B = (c, d) \cup (e, f) = (a, d) \cup (e, b)$ we must have that [[$a < d, e < b$. If $d = e$ then $A \cup B = (a, d) \cup (d, b)$ and so $d \not \in (a, b)$ so $A \cup B \neq = (a, b)$. If $d < e$ then $A \cup B = (a, d) \cup (e, b)$ and $(d, e) \not \in (a, b)$ so $A \cup B \neq (a, b)$. If $d > e$ then $A \cap B = (e, d) \neq \emptyset$. Either way we see that $(a, b) \neq A \cup B$.
We can use the same logic for the other cases which will completely show that $(a, b)$ is connected. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9959908723831177, "perplexity": 45.78742400862186}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891813608.70/warc/CC-MAIN-20180221103712-20180221123712-00759.warc.gz"} |
http://mathhelpforum.com/calculus/72335-logs-real-variables.html | # Thread: Logs and real variables.
1. ## Logs and real variables.
Hello, Iam hoping to get some help with a question I have for an assignment ...
Real variables x and y are related by the equation
"ln(2+y) = 5ln(3 - x) - 2 sqrt x .....(sorry, I haven't as yet got the hang on LaTeX)
Determine the range of values of x and y for which the expressions on each side of this equation are defined."
I haven't really been able to make an attempt at a solution. I think I have to take the exp of each side, but I am not sure excatly what way I go about this, so if someone could give me some advice, that would be fantastic.
Thanks in advance.
Sean
2. Since powers of $e$ are always positive, the natural logarithm function $\ln\, x$ is only defined for positive values of $x$. Similarly, since squares are always nonnegative, the function $\sqrt{x}$ is only defined for nonnegative real numbers.
For $\ln\,(2 + y)$ to be defined, $2 + y$ must therefore be positive.
For $5\,\ln\,(3 - x) - 2\sqrt{x}$ to be defined, both terms must have a value and thus be defined individually. Therefore, $3 - x$ must be positive, and $x$ must be nonnegative.
To view the LaTeX source code for any formula on this forum, you can click on it.
3. Thanks
I have it now. I thought it was more complicated than it was.
Cheers
Sean | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8849186897277832, "perplexity": 213.4817157089023}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320264.42/warc/CC-MAIN-20170624152159-20170624172159-00006.warc.gz"} |
http://www.r-bloggers.com/the-first-version-of-my-inference-from-iterative-simulation-using-parallel-sequences-paper/ | # The first version of my “inference from iterative simulation using parallel sequences” paper!
May 9, 2012
By
(This article was first published on Statistical Modeling, Causal Inference, and Social Science » R, and kindly contributed to R-bloggers)
From August 1990. It was in the form of a note sent to all the people in the statistics group of Bell Labs, where I’d worked that summer.
To all:
Here’s the abstract of the work I’ve done this summer. It’s stored in the file,
/fs5/gelman/abstract.bell, and copies of the Figures 1-3 are on Trevor’s desk.
Any comments are of course appreciated; I’m at [email protected]
On the Routine Use of Markov Chains for Simulation
Andrew Gelman and Donald Rubin, 6 August 1990
corrected version: 8 August 1990
1. Simulation
In probability and statistics we can often specify multivariate distributions
many of whose properties we do not fully understand–perhaps, as in the
Ising model of statistical physics, we can write the joint density function, up
to a multiplicative constant that cannot be expressed in closed form.
For an example in statistics, consider the Normal random
effects model in the analysis of variance, which can be
easily placed in a Bayesian framework with a conjugate prior distribution.
All the conditional densities of the resulting posterior distribution
are simple, but marginal densities can only be written in integral form and
can only be calculated approximately. (For details, see Kinderman and Snell
(1980) or Pickard (1987) for the Ising model, and Lindley and Smith (1972)
for the Bayesian random effects model.)
In such cases, we may not even be able to compute marginal moments of the
difficult distribution, let alone more complicated and interesting summaries
that would help us understand a probability model or posterior inference.
When direct methods such as analytic or numerical integration of “nuisance”
parameters are not computationally feasible, we might try Monte Carlo simulation;
in the simplest form, we draw a finite set of independent random samples from our
distribution, and then calculate desired distributional summaries as functions of
the sampled points. The Monte Carlo method is quite general and powerful; it is
easy to calculate arbitrary quantities of interest
such as the expected long-distance correlation in the Ising model or a posterior
95% confidence region for the largest block effect in a random effects model.
Any aspect of the distribution can be approximated to any desired accuracy if
the number of independently sampled points is large enough.
Simulation also has the advantage of flexiblility: once a sample is drawn, it
can be used to learn about any number of different distributional summaries.
2. Markov chain methods
Drawing independent random samples is a wonderful tool that is unfortunately not
available for every distribution; in particular, the Ising model and random
effects posterior distributions mentioned above do not permit direct
simulation. Fortunately, a form of indirect simulation method exists for
any multivariate distribution if we can calculate its joint density
(up to a multiplicative constant) or if we can sample from all its univariate
conditional densities. The first of these methods was introduced by
Metropolis et al. (1953) in the Journal of Chemical Physics. Our work focuses
on a similar and slightly simpler method called the Gibbs sampler by Geman and
Geman (1984) in an article for the IEEE.
Let F(x) be our distribution; the Metropolis algorithm takes a starting (vector)
point x0 and constructs a series x1, x2, . . ., that is a sample from an
ergodic Markov chain whose stationary distribution is F(x). Computer
simulation of the series requires calculation of the density f(x) (up to a
constant). These samples xj are not independent; however, the stationary
distribution of the Markov chain is
correct, so if we take a long enough series, the set of values {x1, . . ., xn}
takes the place of the distribution just as an
independent random sample does (although of course an independent sample
carries more information than a Markov chain sample of the
same length).
The Gibbs sampler is a similar algorithm, which produces a Markov chain that
converges to the desired distribution, this time requiring draws from all the
univariate conditional densities at each iteration.
3. Have we converged yet?
Markov chain simulation methods are attractive for many problems because they
enable us to flexibly summarize intractable multivariate distributions by making
full use of the mathematical structure we do know, using a tool we think we
understand–Monte Carlo simulation. Unfortunately, using a sample of a Markov
chain to estimate a distribution raises an immediate question: how long a series
is needed? After one or two steps, we are almost certainly still too close to
the starting point to hope for unbiased summaries. Asymptotically, the chain is
stationary, and all is OK (with some loss of efficiency compared to independent
samples, as mentioned above).
To obtain a feeling for the practical difficulties, we ran the Gibbs sampler for
2000 steps to simulate a case of the Ising model. To give the minimum of details:
x is a vector of binary variables defined on a 100 by 100 lattice; each step of
the Gibbs sampler took on the order of 10,000 computations; and we summarize
each iterate xj by the sample correlation r on the lattice–a function r(x) that
lies between -1 and 1. Theoretical calculations (Pickard, 1987) show that
under our model–the Ising model with beta = 0.5–the marginal distribution
of r is approximately Gaussian with mean around 0.85 or 0.9 and standard
deviation around 0.01. We’d like to know whether the set {r(x1), . . .,
r(x2000)} from the simulated Markov chain can serve as a substitute for the
marginal distribution of r.
Figure 1 shows the values of r(xj), for j=1 to 2000. (r(x0) = 0, and the first
few values are cut off to improve resolution on the graph.) The Markov chain
seems to have “converged to stationarity” after the thousand or so steps required
to shake off its initial state. How do we know it has converged, though? Figure
2 zooms in of the first 500 steps of the series, whose apparent convergence we
know to be illusory. For comparison we ran the Gibbs
sampler again for 2000 steps, but this time starting at a point x0 for which
r(x0) = 1; Figure 3 displays the series r(xj), which again seems to have
converged nicely. To destroy all illusions about convergence, hold
Figures 1 and 3 up to the light. The two Markov chains have “converged” to
different distributions! We are, of course, still observing transient
behavior.
Interestingly, the means of the series in Figures 1 and 3 differ, but
the variances are roughly equal. We’re not sure why, but it seems to be a
general feature in these Markov chain simulations that the variance converges
before the mean.
All simulations and plots were done using the New S Language:
A Programming Environment for Data Analysis and Graphics.
4. The answer: parallel Markov chains
To restate the general problem: we wish to summarize an intractable
distribution F(x) by running the Gibbs sampler (or a similar method such as the
Metropolis algorithm) until the distribution of the set of Markov chain
iterates is close to F. As shown in the previous section, convergence seems
impossible to monitor from a single finite realization of the Markov chain;
consequently, we follow the implicit suggestion of Figures 1 and 3 and track
several parallel sample paths.
Consider m independent runs of the Gibbs sampler, each of length n, starting
from m different initial states x10, . . ., xm0:
series 1: x11, . . ., x1n
. . .
series m: xm1, . . ., xmn.
Again, we focus attention on a univariate summary, say r(x); we want to
use the observed simulations rij to determine whether the series of r’s are
close to convergence after n steps.
To understand our method, consider the set of series as m blocks, each with
n observations, in the one-way analysis of variance layout (that is, ignore the
time ordering in the series). We will work with the total sum of squares
(with (mn-1) degrees of freedom) and the “within” sum of squares (with
(m-1)(n-1) degrees of freedom).
First assume for simplicity that the starting points of the simulated series
are themselves independent random samples from F(x). (Of course, if this
condition could be obtained in practice,
a Markov chain simulation method would not be
needed.) With independent starting points, all values of any series
are independent of all the values of any other series, and the unconditional
variance of any point rij is just the marginal variance var r under the
distribution F. We can then estimate var r, given the “data matrix” (rij),
by [total SS - (within SS)/m] / ((m-1)n). (Algebraic derivations appear
in the longer version of this article.) Given the assumption of initial
independence, this “between” estimate of variance
(not the same as the usual “between” estimate in ANOVA) is unbiased for finite
series of any length.
In contrast, the estimated variance within the series, (within SS) / ((m-1)(n-1)),
has expectation var r only in the limit as n -> infinity.
For finite series, the expected within mean square increases with n, assuming,
as is likely, that the random variables r(x1), r(x2), . . ., from the Markov
chain are positively correlated. The discrepancy between the two estimates
of var r suggests a test: declare the Markov chain to have converged when
the within mean square is close to the variance estimate between series, with
confidence intervals derived from classical ANOVA theory. Because of the
dependence within blocks, the degrees of freedom of the between and within
estimates are less than (m-1)n and m(n-1), respectively. We can
approximately correct for this information loss (once again, details will be
provided in the longer article).
Once we are close enough to convergence to be satisfied, the variance estimates
and degrees of freedom corrections alluded to above allow us to estimate the
marginal summaries E r, var r, and Normal-theory confidence intervals for our
Monte Carlo approximations. We can run the series longer if more precision
is desired, and can repeat the process to study the marginal distributions of
other parameters (without, of course, having to simulate any new series of x’s).
In practice, the starting points of the parallel series can never be sampled
independently with distribution F(x); the simulated series are thus no longer
stationary for any finite n, formally invalidating the above analysis. We
currently have two strategies designed to make the independence assumption
approximately true. First, we try to pick starting values that are far apart and,
if anything, more dispersed than independent random samples. The m parallel series
should then start far apart and grow closer as they approach stationarity, as in
Figures 1 and 3; since the variance between series declines with n, the
comparison-of-variances test should be conservative. Second, we reduce the
effect of the starting values by crudely throwing away the the first half
of each simulated series until approximate convergence has been reached.
Once again, Figures 1 and 3 illustrate how a few early steps
of the Markov chain can greatly distort estimates of means and variances
within series. We hope that the conservative strategies of starting with
dispersed points and throwing away early simulations will yield confidence
regions that are wider than those obtained by the ideal method, but that
still have good coverage properties.
The idea of comparing parallel simulations is not new; for
example, Fosdick (1959) applied the Metropolis algorithm to the Ising model
by simulating four series independently, from each of two different starting
points. Approximate convergence was declared when the two groups of series
became indistinguishable on the scale of a prechosen error bound.
5. Some references
Ehrman, J. R., Fosdick, L. D., and Handscomb, D. C. (1960).
Computation of order parameters in an Ising lattice by the Monte Carlo method.
{\em Journal of Mathematical Physics} {\bf 1} 547–558.
Fosdick, L. D. (1959). Calculation of order parameters in a binary
alloy by the Monte Carlo method. {\em Physical Review} {\bf 116}, 565–573.
Geman, S., and Geman, D. (1984). Stochastic relaxation, Gibbs
distributions, and the Bayesian restoration of images. {\em IEEE Transactions
on Pattern Analysis and Machine Intelligence} {\bf 6}, 721–741.
Hammersley, J. M., and Handscomb, D. C. (1964), chapter 9. {\em Monte Carlo
Methods}. London: Chapman and Hall.
Kinderman, R., and Snell, J. L. (1980). {\em Markov Random Fields and
their Applications}. Providence, R.I.: American Mathematical Society.
Lindley, D. V., and Smith, A. F. M. (1972). Bayes estimates for the linear
model. {\em Journal of the Royal Statistical Society B} {\bf 34}, 1–41.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and
Teller, E. (1953). Equation of state calculations by fast computing machines.
{\em Journal of Chemical Physics} {\bf 21}, 1087–1092.
Pickard, D. K. (1987). Inference for discrete Markov fields: the
simplest nontrivial case. {\em Journal of the American Statistical Association}
{\bf 82} 90–96.
Ripley, B. D. (1981). {\em Spatial Statistics}, p. 16ff. New York: Wiley.
Tanner, M. A., and Wong, W. H. (1987). The calculation of posterior
distributions by data augmentation. {\em Journal of the American Statistical
Association} {\bf 82}, 528–550.
I wrote the article but properly listed Rubin as coauthor, as the idea came about after many long phone conversations. I encountered the idea of between-within comparison in the 1959 paper by Fosdick (see above citations); I can’t remember how I found that paper but it must have been from a literature search, going backward from more recent sources. Anyway, when I brought up this idea, Rubin picked up on it right away, as it was close to methods he had developed for inference from multiple imputations. Once we had that connection, the idea was there. And I’d credit Rubin’s influence for my goal of estimating a potential scale reduction factor—that is, a numerical measure of lack of mixing—rather than formulating the problem as a hypothesis test.
The published article appeared over two years later in the journal Statistical Science, in a much expanded version.
In some ways, I prefer this short paper to the full version. I like the snappy style, and I like the clarity about what we believe and what we don’t know. I regret not submitting some version of the above article to a journal immediately, right then in Aug 1990. On the other hand, editors and reviewers for statistics journals can be very stuffy, and an article such as above with a concept but no theoretical derivations probably would’ve been shot down over and over and over. Maybe it just took two years to put in enough blah blah blah to make it publishable.
The above is more like a blog post than a journal article. It contains the key idea with no messing around.
P.S. You’ll notice above that I wrote, “Any comments are of course appreciated.” And you probably won’t be surprised to hear that I got no comments. It took me a long time to realize that most people don’t want to comment on things. When we were getting close to finishing the first edition of Bayesian Data Analysis back in 1994, I printed out copies and gave them to lots of prominent statisticians I knew, but very few gave any comments at all. It’s not about me; people just don’t like to read and make comments. We get some comments on the blog, but when you consider the number of comments and the number of readers, you’ll realize that most people don’t comment here either.
To leave a comment for the author, please follow the link and comment on his blog: Statistical Modeling, Causal Inference, and Social Science » R.
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If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook... | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8957765102386475, "perplexity": 1558.4922204052973}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510268660.14/warc/CC-MAIN-20140728011748-00374-ip-10-146-231-18.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/96810-algebra-problems-fun-40-a.html | # Thread: Algebra, Problems For Fun (40)
1. ## Algebra, Problems For Fun (40)
This is the last problem of this series of problems that I've been posting for a while now:
Let $F$ be a field and $R=\left \{\begin{pmatrix}x & 0 & y \\ 0 & x & z \\ 0 & 0 & x \end{pmatrix}: \ \ x,y,z \in F \right \}.$ It's easy to see that $R$ is a commutative ring with identity element. Find all maximal ideals of $R.$
Hint:
Spoiler:
$R$ has only one maximal ideal.
2. Clearly $S=\left\{\begin{pmatrix}x & 0 & 0\\
0 & x & 0\\
0 & 0 & x
\end{pmatrix}:x\in F\right\}$
is an ideal of $R$ and the quotient ring $R/S$ is a commutative ring with identity $I_3+S.$
Let’s let $A(x,y,z)$ denote the element $\begin{pmatrix}x & 0 & y\\
0 & x & z\\
0 & 0 & x
\end{pmatrix}+S$
in $R/S.$
Then if $x\ne0,$ $A(x,y,z)\cdot A(x^{-1},-yx^{-2},-zx^{-2})=I_3+S.$ Hence $R/S$ is a field and so $S$ is a maximal ideal.
3. Originally Posted by TheAbstractionist
Clearly $S=\left\{\begin{pmatrix}x & 0 & 0\\
0 & x & 0\\
0 & 0 & x
\end{pmatrix}:x\in F\right\}$
is an ideal of $R$ and the quotient ring $R/S$ is a commutative ring with identity $I_3+S.$
Let’s let $A(x,y,z)$ denote the element $\begin{pmatrix}x & 0 & y\\
0 & x & z\\
0 & 0 & x
\end{pmatrix}+S$
in $R/S.$
Then if $x\ne0,$ $A(x,y,z)\cdot A(x^{-1},-yx^{-2},-zx^{-2})=I_3+S.$ Hence $R/S$ is a field and so $S$ is a maximal ideal.
$S$ contains the identity element of $R.$ so if it was an ideal of $R,$ it would be equal to $R.$ but it's not! so $S$ is not an ideal of $R.$
you can also see that by checking that $S$ is not closed under multiplication by elements of $R.$
4. Originally Posted by NonCommAlg
This is the last problem of this series of problems that I've been posting for a while now:
Let $F$ be a field and $R=\left \{\begin{pmatrix}x & 0 & y \\ 0 & x & z \\ 0 & 0 & x \end{pmatrix}: \ \ x,y,z \in F \right \}.$ It's easy to see that $R$ is a commutative ring with identity element. Find all maximal ideals of $R.$
Hint:
Spoiler:
$R$ has only one maximal ideal.
(I've used $(x,y,z) = \begin{pmatrix}x & 0 & y \\ 0 & x & z \\ 0 & 0 & x \end{pmatrix}$, with the equivalent multiplication, because it's quicker to type...)
The subring with $x=0$ is clearly an ideal: $(x,y,z).(a,b,c) = (xa, xb+ya, xc+za)$ $\Rightarrow (0,y,z).(a,b,c) = (0, ya, za)=(a,b,c).(0,y,z)$.
However, this ideal is also maximal. This is because if $I$ is an ideal containing an element $(x,y,z)$, $x \neq 0$, then $[(x,y,z)(x^{-1},0,0)](1, -yx^{-1}, -zx^{-1}) = id$ and so $I=R$.
Subsequently this ideal contains all other proper ideals, as if it didn't then there would be an ideal with $x \neq 0$, a contradiction. Thus it is the only maximal ideal. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 50, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.94996178150177, "perplexity": 118.05528786990259}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917121644.94/warc/CC-MAIN-20170423031201-00638-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://paperity.org/search/?q=authors%3A%22Leszek+Gasi%C5%84ski%22 | # Search: authors:"Leszek Gasiński"
10 papers found.
Use AND, OR, NOT, +word, -word, "long phrase", (parentheses) to fine-tune your search.
#### Asymmetric (p, 2)-equations with double resonance
We consider a nonlinear Dirichlet elliptic problem driven by the sum of a p-Laplacian and a Laplacian [a (p, 2)-equation] and with a reaction term, which is superlinear in the positive direction (without satisfying the Ambrosetti–Rabinowitz condition) and sublinear resonant in the negative direction. Resonance can also occur asymptotically at zero. So, we have a double resonance ...
#### Positive Solutions for the Neumann p-Laplacian with Superdiffusive Reaction
We consider a generalized logistic equation driven by the Neumann p-Laplacian and with a reaction that exhibits a superdiffusive kind of behavior. Using variational methods based on the critical point theory, together with truncation and comparison techniques, we show that there exists a critical value $\lambda _*>0$ of the parameter, such that if $\lambda >\lambda _*$, the ...
#### Nonlinear, Nonhomogeneous Periodic Problems with no Growth Control on the Reaction
We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator, which includes as a particular case the scalar p-Laplacian. We assume that the reaction is a Carathéodory function which admits time-dependent zeros of constant sign. No growth control near ±∞ is imposed on the reaction. Using variational methods coupled with suitable truncation and ...
#### Existence and uniqueness of positive solutions for the Neumann p-Laplacian
We consider a nonlinear Neumann problem driven by the p-Laplacian and with a Carathéodory reaction which satisfies only a unilateral growth restriction. Using the principal eigenvalue of an eigenvalue problem involving the Neumann p-Laplacian plus an indefinite potential, we produce necessary and sufficient conditions for the existence and uniqueness of positive smooth solutions.
#### Multiplicity of positive solutions for eigenvalue problems of ( p , 2 ) -equations
Abstract We consider a nonlinear parametric equation driven by the sum of a p-Laplacian (p > 2 ) and a Laplacian (a (p,2)-equation) with a Carathéodory reaction, which is strictly ( p − 2 ) -sublinear near +∞. Using variational methods coupled with truncation and comparison techniques, we prove a bifurcation-type theorem for the nonlinear eigenvalue problem. So, we show that there ...
#### Multiple Solutions for Nonlinear Coercive Problems with a Nonhomogeneous Differential Operator and a Nonsmooth Potential
We consider a nonlinear elliptic problem driven by a nonlinear nonhomogeneous differential operator and a nonsmooth potential. We prove two multiplicity theorems for problems with coercive energy functional. In both theorems we produce three nontrivial smooth solutions. In the second multiplicity theorem, we provide precise sign information for all three solutions (the first ...
#### Neumann problems resonant at zero and infinity
We consider a semilinear Neumann problem with a reaction which is resonant at both zero and ±∞. Using a combination of methods from critical point theory, together with truncation techniques, the use of upper–lower solutions and of the Morse theory (critical groups), we show that the problem has at least five nontrivial smooth solutions, four of which have constant sign (two ...
#### Nonlinear Elliptic Equations with Singular Terms and Combined Nonlinearities
We consider nonlinear elliptic Dirichlet problems with a singular term, a concave (i.e., (p − 1)-sublinear) term and a Carathéodory perturbation. We study the cases where the Carathéodory perturbation is (p − 1)-linear and (p − 1)-superlinear near +∞. Using variational techniques based on the critical point theory together with truncation arguments and the method of upper and lower ...
#### A pair of positive solutions for the Dirichlet p(z)-Laplacian with concave and convex nonlinearities
We consider a nonlinear parametric Dirichlet problem driven by the anisotropic p-Laplacian with the combined effects of “concave” and “convex” terms. The “superlinear” nonlinearity need not satisfy the Ambrosetti-Rabinowitz condition. Using variational methods based on the critical point theory and the Ekeland variational principle, we show that for small values of the parameter, ...
#### Anisotropic nonlinear Neumann problems
We consider nonlinear Neumann problems driven by the p(z)-Laplacian differential operator and with a p-superlinear reaction which does not satisfy the usual in such cases Ambrosetti–Rabinowitz condition. Combining variational methods with Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of which have constant sign (one positive, the other ... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9420229196548462, "perplexity": 843.9146350957552}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084888878.44/warc/CC-MAIN-20180120023744-20180120043744-00419.warc.gz"} |
http://acscihotseat.org/index.php?qa=537&qa_1=existence-neutral-probabilities-implies-implied-arbitrage&show=550 | # Showing that the existence of risk neutral probabilities implies and is implied by no arbitrage
+1 vote
75 views
"Part of the fundamental theorem of asset pricing says that a market model is arbitrage free iff risk neutral probabilities exist: prove this for the one period binomial model"
the way I went about this was to use the condition d<$$e^r$$<u for no arbitrage and that probabilities must lie in the interval [0,1], this worked except that I didn't get strict inequalities on the arbitrage condition
commented May 23, 2017 by (4,010 points)
Hi Anonymous.
A tutor has looked at your question, but was not sure of the necessity regarding strict bounds. I have asked Alex and will post an Answer once I have one for you. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8719014525413513, "perplexity": 647.552863028749}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376832330.93/warc/CC-MAIN-20181219130756-20181219152756-00260.warc.gz"} |
https://astronomy.stackexchange.com/questions/7739/is-it-accurate-to-say-that-we-have-a-gravitational-attraction-towards-all-object | # Is it accurate to say that we have a gravitational attraction towards all objects in the known universe?
Its probably small, but is there a theoretical gravitational attraction between all objects in the universe? Light can move pretty far, so does that mean gravity can as well, and is the gravity blocked by the same way light is blocked by an object?
We can show this easily using Newtonian gravity. Newton's law of universal gravitation is formulated $$F=G\frac{m_1m_2}{r_{12}^2}$$ where $m_1$ and $m_2$ are the masses of the two objects - we'll say $m_1$ is the mass of a human, $m_h$, and $m_2$ is the mass of Earth, $m_E$. A human on Earth's surface is roughly $6,371 \text { km}$ from it's center, or $6,371,000 \text { meters}$. Mars, on the other hand, is, at its closest, $0.3814 \text { AU}$ from Earth - or roughly $57,210,000 \text { km}$, or $57,210,000,000 \text { meters}$. Care to do the math? Well, what with this distance being squared ($r^2$), you can tell at a glance that the force from Mars is many orders of magnitude less than the force from Earth. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9619044065475464, "perplexity": 194.82046351859572}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439739073.12/warc/CC-MAIN-20200813191256-20200813221256-00160.warc.gz"} |
http://math.stackexchange.com/questions/257377/are-my-derivatives-worked-correctly | Are my derivatives worked correctly?
I have 3 derivatives that I need to find:
1: $y = 3sin(x) - ln(x) + 2x^\frac{1}{2}$
$3cos(x) - \frac{1}{x} + (\frac{1}{2}) (2x^\frac{-1}{2})$
2: $y = 3e^{2x} * ln(2x + 1)$
$3e^{2x} (\frac{2}{2x+1}) + 6e^{2x} (ln(2x+1))$
3: $y = \frac{sin(2x)}{3x^2 + 5}$
$\frac{(3x^2 + 5)(2sin(2x)) - sin(2x)(6x)}{(3x^2 + 5)^2}$
I'm relatively positive I did these correctly, but please let me know if I haven't.
EDIT
3: $\frac{(3x^2 + 5)(2cos(2x)) - sin(2x)(6x)}{(3x^2 + 5)^2}$
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In 3: $(\sin{t})'=\cos{t}$ – M. Strochyk Dec 12 '12 at 20:02
Seems I overlooked that, thanks! Glad someone felt the need to downvote the question when I'm genuinely asking just to be checked so I know I'm ready for the exam. Aside from that the community has been quite nice about my questions. – StrugglingWithMath Dec 12 '12 at 20:06
When differentiating $\sin(2x)$ in 3., you use the chain rule and differentiate $\sin$ first. Also, you would (might) probably lose a point for not making obvious and easy simplifications in your first two answers. – David Mitra Dec 12 '12 at 20:07
@DavidMitra The professor has been quite nice about the simplifications. While I could simplify it, he just wants to make sure we know the required steps :) – StrugglingWithMath Dec 12 '12 at 20:09
Note:
In $(3)$: $\cos(2x)$ appears nowhere in your solution. That should tell you something about your solution to $(3)$.
You got it right in $(1)$, so I'm assuming you know that the derivative of $\sin(2x) = 2 \cos(2x)$.
Spend some more time on $(3)$, and see what you arrive at.
I'm assuming you have written intermediate steps, and simply posted your solutions. Be sure that if this is homework, that you hand in a more detailed, step-by-step solution. You can simplify a little bit, too, e.g., in $(1)$.
-
Indeed, I need to fix that! Thanks! – StrugglingWithMath Dec 12 '12 at 20:05
Yeah I have it written down in steps on my paper with each rule that I followed, I just didn't want to type up all the latex for it :) Most of my issues with calculus come from doing it too quick and not checking my answer. I'll have to work on that. – StrugglingWithMath Dec 12 '12 at 20:13
I figured you had shown your work, and it is perfectly understandable why you wouldn't want to type it all out! – amWhy Dec 12 '12 at 20:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8904330730438232, "perplexity": 706.9408551655897}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207929832.32/warc/CC-MAIN-20150521113209-00018-ip-10-180-206-219.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/91388/winbugs-plots-in-latex | # Winbugs plots in latex
I conducted a bayesian analysis and obtained some plots in winbugs. Does any one how to put the history graphs in WinBUGS in Latex? I could not save the plots as pdf.
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Maybe virtually print the plots with software like CutePDF and by \includegraphics or pdfpage include the graph in your Latex file? – La Raison Jan 11 '13 at 21:36
@Günal save/export the figure data in ascii .txt and plot in any software you like to get a PDF or EPS. May be combination of R and Sweave. I suggest post your question at Cross Validated.SX – texenthusiast Jan 11 '13 at 22:57
As it stands, this seems to be about WinBUGS (for which we could do with a link) rather than about TeX. It's borderline for on-topic. It would be useful if you could specify if th key part is to get the data out of WinBUGS or to reproduce the same type of graphic using LaTeX. – Joseph Wright Jan 12 '13 at 8:07 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8028228878974915, "perplexity": 2404.70259642346}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394678694108/warc/CC-MAIN-20140313024454-00098-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://www.jiskha.com/questions/1781029/A-meter-stick-is-positioned-on-a-horizontal-table-top-such-that-the-100cm-end-extends | # Physics
A meter stick is positioned on a horizontal table top such that the 100cm end extends over the edge of the table. A 200 g mass is produced on the 85 cm mark of the meter stick. When the 70 cm mark of the meter stick is on the edge of the table, the meter stick and 200 g mass system is perfectly balanced.
a. What is the mass of the meter stick?
b. What is the magnitude of the normal force (from the table) acting on the meter stick?
How do I solve these two parts?
1. 👍 0
2. 👎 0
3. 👁 102
1. the center of mass of the stick is at 50 cm , 20 cm from the balance point
a. 15 cm * 200 g = 20 cm * m
b. the normal force equals the weight of the stick and the 200 g mass
... n = g (m + 200 g)
1. 👍 0
2. 👎 0
posted by R_scott
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https://ccssmathanswers.com/multiplication-of-a-fraction-by-fraction/ | # Multiplication of a Fraction by Fraction – Definition, Examples | How do you Multiply a Fraction by a Fraction?
Have you stuck at some point while multiplying one fraction with another fraction and need help? Don’t bother as we are with you in this and compiled an article covering the fraction definition, how to multiply a fraction by another fraction, how to multiply two mixed fractions. Refer to the solved examples for multiplying fraction by fraction and try to solve related problems on your own.
## Fraction – Definition
Fractions are the numerical values or things that are divided into parts, then each part will be a fraction of a number. A fraction is denoted as $$\frac { a }{ b }$$, where a is the numerator and b is the denominator.
### Multiplication of a Fraction by Fraction
Multiplication of fraction by fraction starts with the multiplication of the given numerators followed by multiplication of the denominators. Then, the resultant fraction can be simplified further and reduced to its lowest terms if needed. Multiplication of fractions is not the same as addition or subtraction of fractions, where the denominator should be the same. Here any two fractions without the same denominator can also be multiplied.
### How to Multiply Two Fractions?
Follow the simple steps listed below to multiply a fraction by fraction. They are in the below fashion
Step 1: Simplify the fractions into their lowest terms.
Step 2: Multiply both the numerators of the given fractions to get a new numerator.
Step 3: Multiply both the denominators of the given fractions to get a new denominator.
Simplify the resulting fraction if needed.
Let a/b is one fraction and the other fraction is c/d. Multiplication of these fractions are
Multiplying fractions formula:
$$\frac { a }{ b }$$ * $$\frac { c }{ d }$$ = $$\frac { a * b }{ c * d }$$
### Multiplication of Fraction by another Fraction Examples
Example 1:
$$\frac { 1 }{ 3 }$$ is the first fraction. $$\frac { 2 }{ 3 }$$ is another fraction. What is the multiplication of both fractions?
Solution:
It is given that $$\frac { 1 }{ 3 }$$ is one fraction and $$\frac { 2 }{ 3 }$$ is another fraction.
When we multiply both these fractions, we multiply both the numerators and both the denominators.
Product of two fractions = (Product of numerators)/(Product of denominators).
That is (1*2) is the numerator. (3*3) is the denominator.
The final result is the product of numerators and denominators.
Therefore the final answer is $$\frac { 2 }{ 9 }$$ .
Example 2:
$$\frac { 12 }{ 5 }$$ is the first fraction. $$\frac { 23 }{ 9 }$$ is another fraction. What is the multiplication of both fractions?
Solution:
It is given that $$\frac { 12 }{ 5 }$$ is one fraction and $$\frac { 23 }{ 9 }$$ is another fraction.
When we multiply both these fractions, we multiply both the numerators and both the denominators.
Product of two fractions = (Product of the numerators)/(Product of the denominators).
That is (12*23) is the numerator. (5*9) is the denominator.
The final result is the product of numerators and denominators. The value is ($$\frac { 276 }{45 }$$ .
Therefore the final answer is $$\frac { 92 }{15 }$$ ,
Example 3:
$$\frac { 16 }{ 30 }$$ is the first fraction. $$\frac { 21 }{ 50 }$$ is another fraction. What is the multiplication of both fractions?
Solution:
It is given that $$\frac { 16 }{ 30 }$$ is one fraction and $$\frac { 21}{ 50 }$$ is another fraction.
When we multiply both these fractions, we multiply both the numerators and both the denominators.
Product of two fractions = (Product of numerators)/(Product of denominators).
That is (16 * 21) is the numerator. (30*50) is the denominator.
The final result is the product of numerators and denominators.
Therefore the final answer is $$\frac { 336 }{ 1500 }$$ .
### Multiplication of Mixed Fractions
A fraction that is represented by its quotient and remainder is a mixed fraction. So, it is a combination of a whole number and a proper fraction. Multiplication of mixed fractions will be difficult to change each number into an improper fraction.
### How to Multiply Mixed Fractions?
Go through the below-listed steps to multiply two mixed fractions. They are in the below fashion
Step 1: Convert given mixed fractions into improper fractions.
Step 2: Simplify the fractions into their lowest terms for easy calculations.
Step 3: Multiply both the numerators of the given fractions to get a new numerator.
Step 4: Multiply both the denominators of the given fractions to get a new denominator.
Simplify the resultant fraction if needed.
### Multiplication of Mixed Fraction by another Mixed Fraction Examples
Example 1:
Multiply mixed fractions 2$$\frac { 3 }{ 5 }$$ and 6 $$\frac { 7 }{ 8 }$$. What is the multiplication of both fractions?
Solution:
It is given that 2$$\frac { 3 }{ 5 }$$ is one fraction and the other fraction is 6$$\frac { 7 }{ 8 }$$
As we know mixed fractions cannot be multiplied. Simplify the given mixed fractions into improper fractions.
Now the given mixed fractions are $$\frac { 13 }{ 5 }$$ and the other fraction is $$\frac { 55 }{ 8 }$$.
When we multiply both these fractions, we multiply both the numerators and both the denominators.
Product of two fractions = (Product of numerators)/(Product of denominators).
That is (13 * 55) is the numerator. (5*8) is the denominator.
The final result is the product of numerators and denominators. The value is $$\frac { 715 }{ 40 }$$
Therefore the final answer is $$\frac { 715 }{ 40 }$$
Example 2:
Multiply mixed fractions 2$$\frac { 1}{ 8 }$$ and 6 $$\frac { 4 }{ 9 }$$. What is the multiplication of both fractions?
Solution:
It is given that 2$$\frac { 1 }{ 8 }$$ is one fraction and the other fraction is 6$$\frac { 4 }{ 9 }$$
As we know mixed fractions cannot be multiplied. Simplify the given mixed fractions into improper fractions.
Now the given mixed fractions are $$\frac { 17 }{ 8 }$$ and the other fraction is $$\frac { 58 }{ 9 }$$.
When we multiply both these fractions, we multiply both the numerators and both the denominators.
Product of two fractions = (Product of numerators)/(Product of denominators).
That is (17 * 58) is the numerator. (8*9) is the denominator.
The final result is the product of numerators and denominators. The value is $$\frac { 986 }{ 72 }$$
Therefore the final answer is $$\frac { 986 }{ 72 }$$.
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http://www.talkstats.com/threads/uncertain-about-degrees-of-freedom-in-value-with-estimated-standard-error.73932/ | # Uncertain about degrees of freedom in value with estimated standard error
#### DannyR
##### New Member
I am estimating an evolutionary metric for various genes. Each estimate has an associated standard error. Typically, I'll have a set of genes for which I want to compare its mean to the mean of the rest of the genes. The size of the sets are typically in the 10 to 100 range. What I have been doing is using the estimated standard error of each value within the set to weight each value, and using the weighted Welch's t-test in R to compare the weighted means.
However, if I want to compare the value of a single gene to the rest I'm not quite sure what the best way to do that is. I could just ask if the single value is significantly different than the mean of the other set, but I would like to incorporate the estimated standard error of the single value to make the test more conservative. Can I use this standard error (with N=1) in conjunction with the weighted standard error of the set to calculate the Welch's t statistic and degrees of freedom estimate? N=1 here seems wrong to me because this N should refer to the sampling used to derive the standard error estimate. This standard error is calculated by the curvature method from an optimization algorithm so there isn't really an N associated with it. Would it be fair (conservative) to just use N=1 here to calculate the t statistic and degrees of freedom? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8394567370414734, "perplexity": 250.91002459192862}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987769323.92/warc/CC-MAIN-20191021093533-20191021121033-00537.warc.gz"} |
https://openstax.org/books/university-physics-volume-2/pages/6-1-electric-flux | University Physics Volume 2
6.1Electric Flux
Learning Objectives
By the end of this section, you will be able to:
• Define the concept of flux
• Describe electric flux
• Calculate electric flux for a given situation
The concept of flux describes how much of something goes through a given area. More formally, it is the dot product of a vector field (in this chapter, the electric field) with an area. You may conceptualize the flux of an electric field as a measure of the number of electric field lines passing through an area (Figure 6.3). The larger the area, the more field lines go through it and, hence, the greater the flux; similarly, the stronger the electric field is (represented by a greater density of lines), the greater the flux. On the other hand, if the area rotated so that the plane is aligned with the field lines, none will pass through and there will be no flux.
Figure 6.3 The flux of an electric field through the shaded area captures information about the “number” of electric field lines passing through the area. The numerical value of the electric flux depends on the magnitudes of the electric field and the area, as well as the relative orientation of the area with respect to the direction of the electric field.
A macroscopic analogy that might help you imagine this is to put a hula hoop in a flowing river. As you change the angle of the hoop relative to the direction of the current, more or less of the flow will go through the hoop. Similarly, the amount of flow through the hoop depends on the strength of the current and the size of the hoop. Again, flux is a general concept; we can also use it to describe the amount of sunlight hitting a solar panel or the amount of energy a telescope receives from a distant star, for example.
To quantify this idea, Figure 6.4(a) shows a planar surface $S1S1$ of area $A1A1$ that is perpendicular to the uniform electric field $E→=Ey^.E→=Ey^.$ If N field lines pass through $S1S1$, then we know from the definition of electric field lines (Electric Charges and Fields) that $N/A1∝E,N/A1∝E,$ or $N∝EA1.N∝EA1.$
The quantity $EA1EA1$ is the electric flux through $S1S1$. We represent the electric flux through an open surface like $S1S1$ by the symbol $ΦΦ$. Electric flux is a scalar quantity and has an SI unit of newton-meters squared per coulomb ($N·m2/CN·m2/C$). Notice that $N∝EA1N∝EA1$ may also be written as $N∝ΦN∝Φ$, demonstrating that electric flux is a measure of the number of field lines crossing a surface.
Figure 6.4 (a) A planar surface $S1S1$ of area $A1A1$ is perpendicular to the electric field $Ej^Ej^$. N field lines cross surface $S1S1$. (b) A surface $S2S2$ of area $A2A2$ whose projection onto the xz-plane is $S1S1$.The same number of field lines cross each surface.
Now consider a planar surface that is not perpendicular to the field. How would we represent the electric flux? Figure 6.4(b) shows a surface $S2S2$ of area $A2A2$ that is inclined at an angle $θθ$ to the xz-plane and whose projection in that plane is $S1S1$ (area $A1A1$). The areas are related by $A2cosθ=A1.A2cosθ=A1.$ Because the same number of field lines crosses both $S1S1$ and $S2S2$, the fluxes through both surfaces must be the same. The flux through $S2S2$ is therefore $Φ=EA1=EA2cosθ.Φ=EA1=EA2cosθ.$ Designating $n^2n^2$ as a unit vector normal to $S2S2$ (see Figure 6.4(b)), we obtain
$Φ=E→·n^2A2.Φ=E→·n^2A2.$
Interactive
Check out this video to observe what happens to the flux as the area changes in size and angle, or the electric field changes in strength.
Area Vector
For discussing the flux of a vector field, it is helpful to introduce an area vector $A→.A→.$ This allows us to write the last equation in a more compact form. What should the magnitude of the area vector be? What should the direction of the area vector be? What are the implications of how you answer the previous question?
The area vector of a flat surface of area A has the following magnitude and direction:
• Magnitude is equal to area (A)
• Direction is along the normal to the surface ($n^n^$); that is, perpendicular to the surface.
Since the normal to a flat surface can point in either direction from the surface, the direction of the area vector of an open surface needs to be chosen, as shown in Figure 6.5.
Figure 6.5 The direction of the area vector of an open surface needs to be chosen; it could be either of the two cases displayed here. The area vector of a part of a closed surface is defined to point from the inside of the closed space to the outside. This rule gives a unique direction.
Since $n^n^$ is a unit normal to a surface, it has two possible directions at every point on that surface (Figure 6.6(a)). For an open surface, we can use either direction, as long as we are consistent over the entire surface. Part (c) of the figure shows several cases.
Figure 6.6 (a) Two potential normal vectors arise at every point on a surface. (b) The outward normal is used to calculate the flux through a closed surface. (c) Only $S3S3$ has been given a consistent set of normal vectors that allows us to define the flux through the surface.
However, if a surface is closed, then the surface encloses a volume. In that case, the direction of the normal vector at any point on the surface points from the inside to the outside. On a closed surface such as that of Figure 6.6(b), $n^n^$ is chosen to be the outward normal at every point, to be consistent with the sign convention for electric charge.
Electric Flux
Now that we have defined the area vector of a surface, we can define the electric flux of a uniform electric field through a flat area as the scalar product of the electric field and the area vector, as defined in Products of Vectors:
$Φ=E→·A→(uniformE→,flat surface).Φ=E→·A→(uniformE→,flat surface).$
6.1
Figure 6.7 shows the electric field of an oppositely charged, parallel-plate system and an imaginary box between the plates. The electric field between the plates is uniform and points from the positive plate toward the negative plate. A calculation of the flux of this field through various faces of the box shows that the net flux through the box is zero. Why does the flux cancel out here?
Figure 6.7 Electric flux through a cube, placed between two charged plates. Electric flux through the bottom face (ABCD) is negative, because $E→E→$ is in the opposite direction to the normal to the surface. The electric flux through the top face (FGHK) is positive, because the electric field and the normal are in the same direction. The electric flux through the other faces is zero, since the electric field is perpendicular to the normal vectors of those faces. The net electric flux through the cube is the sum of fluxes through the six faces. Here, the net flux through the cube is equal to zero. The magnitude of the flux through rectangle BCKF is equal to the magnitudes of the flux through both the top and bottom faces.
The reason is that the sources of the electric field are outside the box. Therefore, if any electric field line enters the volume of the box, it must also exit somewhere on the surface because there is no charge inside for the lines to land on. Therefore, quite generally, electric flux through a closed surface is zero if there are no sources of electric field, whether positive or negative charges, inside the enclosed volume. In general, when field lines leave (or “flow out of”) a closed surface, $ΦΦ$ is positive; when they enter (or “flow into”) the surface, $ΦΦ$ is negative.
Any smooth, non-flat surface can be replaced by a collection of tiny, approximately flat surfaces, as shown in Figure 6.8. If we divide a surface S into small patches, then we notice that, as the patches become smaller, they can be approximated by flat surfaces. This is similar to the way we treat the surface of Earth as locally flat, even though we know that globally, it is approximately spherical.
Figure 6.8 A surface is divided into patches to find the flux.
To keep track of the patches, we can number them from 1 through N . Now, we define the area vector for each patch as the area of the patch pointed in the direction of the normal. Let us denote the area vector for the ith patch by $δA→i.δA→i.$ (We have used the symbol $δδ$ to remind us that the area is of an arbitrarily small patch.) With sufficiently small patches, we may approximate the electric field over any given patch as uniform. Let us denote the average electric field at the location of the ith patch by $E→i.E→i.$
$E→i=average electric field over theith patch.E→i=average electric field over theith patch.$
Therefore, we can write the electric flux $ΦiΦi$ through the area of the ith patch as
$Φi=E→i·δA→i(ith patch).Φi=E→i·δA→i(ith patch).$
The flux through each of the individual patches can be constructed in this manner and then added to give us an estimate of the net flux through the entire surface S, which we denote simply as $ΦΦ$.
$Φ=∑i=1NΦi=∑i=1NE→i·δA→i(Npatch estimate).Φ=∑i=1NΦi=∑i=1NE→i·δA→i(Npatch estimate).$
This estimate of the flux gets better as we decrease the size of the patches. However, when you use smaller patches, you need more of them to cover the same surface. In the limit of infinitesimally small patches, they may be considered to have area dA and unit normal $n^n^$. Since the elements are infinitesimal, they may be assumed to be planar, and $E→iE→i$ may be taken as constant over any element. Then the flux $dΦdΦ$ through an area dA is given by $dΦ=E→·n^dA.dΦ=E→·n^dA.$ It is positive when the angle between $E→iE→i$ and $n^n^$ is less than $90°90°$ and negative when the angle is greater than $90°90°$. The net flux is the sum of the infinitesimal flux elements over the entire surface. With infinitesimally small patches, you need infinitely many patches, and the limit of the sum becomes a surface integral. With $∫S∫S$ representing the integral over S,
$Φ=∫SE→·n^dA=∫SE→·dA→(open surface).Φ=∫SE→·n^dA=∫SE→·dA→(open surface).$
6.2
In practical terms, surface integrals are computed by taking the antiderivatives of both dimensions defining the area, with the edges of the surface in question being the bounds of the integral.
To distinguish between the flux through an open surface like that of Figure 6.4 and the flux through a closed surface (one that completely bounds some volume), we represent flux through a closed surface by
$Φ=∮SE→·n^dA=∮SE→·dA→(closed surface)Φ=∮SE→·n^dA=∮SE→·dA→(closed surface)$
6.3
where the circle through the integral symbol simply means that the surface is closed, and we are integrating over the entire thing. If you only integrate over a portion of a closed surface, that means you are treating a subset of it as an open surface.
Example 6.1
Flux of a Uniform Electric Field
A constant electric field of magnitude $E0E0$ points in the direction of the positive z-axis (Figure 6.9). What is the electric flux through a rectangle with sides a and b in the (a) xy-plane and in the (b) xz-plane?
Figure 6.9 Calculating the flux of $E0E0$ through a rectangular surface.
Strategy
Apply the definition of flux: $Φ=E→·A→(uniformE→)Φ=E→·A→(uniformE→)$, where the definition of dot product is crucial.
Solution
1. In this case, $Φ=E→0·A→=E0A=E0ab.Φ=E→0·A→=E0A=E0ab.$
2. Here, the direction of the area vector is either along the positive y-axis or toward the negative y-axis. Therefore, the scalar product of the electric field with the area vector is zero, giving zero flux.
Significance
The relative directions of the electric field and area can cause the flux through the area to be zero.
Example 6.2
Flux of a Uniform Electric Field through a Closed Surface
A constant electric field of magnitude $E0E0$ points in the direction of the positive z-axis (Figure 6.10). What is the net electric flux through a cube?
Figure 6.10 Calculating the flux of $E0E0$ through a closed cubic surface.
Strategy
Apply the definition of flux: $Φ=E→·A→(uniformE→)Φ=E→·A→(uniformE→)$, noting that a closed surface eliminates the ambiguity in the direction of the area vector.
Solution
Through the top face of the cube, $Φ=E→0·A→=E0A.Φ=E→0·A→=E0A.$
Through the bottom face of the cube, $Φ=E→0·A→=−E0A,Φ=E→0·A→=−E0A,$ because the area vector here points downward.
Along the other four sides, the direction of the area vector is perpendicular to the direction of the electric field. Therefore, the scalar product of the electric field with the area vector is zero, giving zero flux.
The net flux is $Φnet=E0A−E0A+0+0+0+0=0Φnet=E0A−E0A+0+0+0+0=0$.
Significance
The net flux of a uniform electric field through a closed surface is zero.
Example 6.3
Electric Flux through a Plane, Integral Method
A uniform electric field $E→E→$ of magnitude 10 N/C is directed parallel to the yz-plane at $30°30°$ above the xy-plane, as shown in Figure 6.11. What is the electric flux through the plane surface of area $6.0m26.0m2$ located in the xz-plane? Assume that $n^n^$ points in the positive y-direction.
Figure 6.11 The electric field produces a net electric flux through the surface S.
Strategy
Apply $Φ=∫SE→·n^dAΦ=∫SE→·n^dA$, where the direction and magnitude of the electric field are constant.
Solution
The angle between the uniform electric field $E→E→$ and the unit normal $n^n^$ to the planar surface is $30°30°$. Since both the direction and magnitude are constant, E comes outside the integral. All that is left is a surface integral over dA, which is A. Therefore, using the open-surface equation, we find that the electric flux through the surface is
$Φ=∫SE→·n^dA=EAcosθ=(10N/C)(6.0m2)(cos30°)=52N·m2/C.Φ=∫SE→·n^dA=EAcosθ=(10N/C)(6.0m2)(cos30°)=52N·m2/C.$
Significance
Again, the relative directions of the field and the area matter, and the general equation with the integral will simplify to the simple dot product of area and electric field.
What angle should there be between the electric field and the surface shown in Figure 6.11 in the previous example so that no electric flux passes through the surface?
Example 6.4
Inhomogeneous Electric Field
What is the total flux of the electric field $E→=cy2k^E→=cy2k^$ through the rectangular surface shown in Figure 6.12?
Figure 6.12 Since the electric field is not constant over the surface, an integration is necessary to determine the flux.
Strategy
Apply $Φ=∫SE→·n^dAΦ=∫SE→·n^dA$. We assume that the unit normal $n^n^$ to the given surface points in the positive z-direction, so $n^=k^.n^=k^.$ Since the electric field is not uniform over the surface, it is necessary to divide the surface into infinitesimal strips along which $E→E→$ is essentially constant. As shown in Figure 6.12, these strips are parallel to the x-axis, and each strip has an area $dA=bdy.dA=bdy.$
Solution
From the open surface integral, we find that the net flux through the rectangular surface is
$Φ=∫SE→·n^dA=∫0a(cy2k^)·k^(bdy)=cb∫0ay2dy=13a3bc.Φ=∫SE→·n^dA=∫0a(cy2k^)·k^(bdy)=cb∫0ay2dy=13a3bc.$
Significance
For a non-constant electric field, the integral method is required.
If the electric field in Example 6.4 is $E→=mxk^,E→=mxk^,$ what is the flux through the rectangular area? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 88, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9798263311386108, "perplexity": 162.45803409931673}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107869933.16/warc/CC-MAIN-20201020050920-20201020080920-00324.warc.gz"} |
https://infocom.spbstu.ru/en/article/2011.18.14/ | Applying the adaptive importance sampling method to the trilateration problem
Authors:
Abstract:
The paper are devoted the adaptive method devised by the authors as applied to the trilateration problem in 2D-space. The method are compared with the important sampling method, that is usually used for analogous problems solving. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9562115669250488, "perplexity": 621.9696000766614}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232261326.78/warc/CC-MAIN-20190527045622-20190527071622-00334.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/114760-independent-random-variables.html | 1. ## independent random variables
Suppose that $A, B, C,$ are independent random variables, each being uniformly distributed over $(0,1).$
(a) What is the joint cumulative distribution function of $A, B, C?$
(b) What is the probability that all the roots of the equation $Ax^2 + BX + C = 0$ are real?
2. Hello,
Let x,y,z in $\mathbb{R}$
The cdf of (A,B,C) is $P(A\leq x,B\leq y,C\leq z)=P(\{A\leq x\}\cap \{B\leq y\} \cap \{C\leq z\})$
But since A,B,C are independent, the probability of the intersection is the product of the probabilities
So the cdf is $P(A\leq x)P(B\leq y)P(C\leq z)$
And since you certainly know the cdf of a uniform distribution, you'll be able to answer the question...
You have to find $P(B^2-4AC\geq 0)=\int_0^1\int_0^1 \int_{2\sqrt{ac}}^1 db ~da ~dc$
Try to understand that... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9488144516944885, "perplexity": 209.22393854802525}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738661327.59/warc/CC-MAIN-20160924173741-00295-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/log-substitution-problem.167575/ | # Log substitution problem
1. Apr 25, 2007
### brandon1
1. The problem statement, all variables and given/known data
(Idk how to put in the equation to make sense, therefore it is at the link below)
2. Relevant equations
3. The attempt at a solution
Here is all I have done. Something just isn't right...there should be 3 answers (in the back of the book) because there is a cube involved...
http://i123.photobucket.com/albums/o318/trashfile_bucket/Trash/2007-04-25-2009-26_edited.jpg
2. Apr 25, 2007
### mjsd
note:
$$(\log x)^3 \neq \log (x^3)$$ ie. 2nd step is wrong
hint: a make substitution: y = log x and solve for y first then x.
3. Apr 25, 2007
### Flux = Rad
The first thing you've done is to cube both sides. That's ok but it should give you
$$( log(x) )^3 = log(x)$$
Since the whole log(x) is cubed, you can't move the 3 down (that's only if the x was cubed).
But what you can do is take all the terms over to one side and then you just have to solve a cubic (which will give you 3 solutions). You may want to make it easier to see by introducing a new variable, $$u = log(x)$$ for example.
4. Apr 25, 2007
### brandon1
Good to go!
5. Apr 27, 2007
### sutupidmath
and the solutiions should probbably be, couse i just glanced at it,:
x_1=1
x_2=y,(if the base of the logarithm is y, couse i could not see it clear)
x_3=1/y
Last edited: Apr 27, 2007
Similar Discussions: Log substitution problem | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8144622445106506, "perplexity": 1576.4042802034035}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917121752.57/warc/CC-MAIN-20170423031201-00318-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://en.wikipedia.org/wiki/Schmidt-Samoa_cryptosystem | # Schmidt-Samoa cryptosystem
The Schmidt-Samoa cryptosystem is an asymmetric cryptographic technique, whose security, like Rabin depends on the difficulty of integer factorization. Unlike Rabin this algorithm does not produce an ambiguity in the decryption at a cost of encryption speed.
## Key generation
• Choose two large distinct primes p and q and compute ${\displaystyle N=p^{2}q}$
• Compute ${\displaystyle d=N^{-1}\mod {\text{lcm}}(p-1,q-1)}$
Now N is the public key and d is the private key.
## Encryption
To encrypt a message m we compute the ciphertext as ${\displaystyle c=m^{N}\mod N.}$
## Decryption
To decrypt a ciphertext c we compute the plaintext as ${\displaystyle m=c^{d}\mod pq,}$ which like for Rabin and RSA can be computed with the Chinese remainder theorem.
Example:
• ${\displaystyle p=7,q=11,N=p^{2}q=539,d=N^{-1}\mod {\text{lcm}}(p-1,q-1)=29}$
• ${\displaystyle m=32,c=m^{N}\mod N=373}$
Now to verify:
• ${\displaystyle m=c^{d}\mod pq=373^{29}\mod pq=373^{29}\mod 77=32}$
## Security
The algorithm, like Rabin, is based on the difficulty of factoring the modulus N, which is a distinct advantage over RSA. That is, it can be shown that if there exists an algorithm that can decrypt arbitrary messages, then this algorithm can be used to factor N.
## Efficiency
The algorithm processes decryption as fast as Rabin and RSA, however it has much slower encryption since the sender must compute a full exponentiation.
Since encryption uses a fixed known exponent an addition chain may be used to optimize the encryption process. The cost of producing an optimal addition chain can be amortized over the life of the public key, that is, it need only be computed once and cached. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 7, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8775198459625244, "perplexity": 1225.3654580504465}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203021.14/warc/CC-MAIN-20190323201804-20190323223804-00191.warc.gz"} |
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Rank Reduction and Volume Minimization Approach to State-Space Subspace System Identification
2006 (engelsk)Inngår i: Signal Processing, ISSN 0165-1684, E-ISSN 1872-7557, Vol. 86, nr 11, s. 3275-3285Artikkel i tidsskrift (Fagfellevurdert) Published
##### Abstract [en]
In this paper we consider the reduced rank regression problem
solved by maximum-likelihood-inspired state-space subspace system identification algorithms. We conclude that the determinant criterion is, due to potential rank-deficiencies, not general enough to handle all problem instances. The main part of the paper analyzes the structure of the reduced rank minimization problem and identifies signal properties in terms of geometrical concepts. A more general minimization criterion is considered, rank reduction followed by volume minimization. A numerically sound algorithm for minimizing this criterion is presented and validated on both simulated and experimental data.
##### sted, utgiver, år, opplag, sider
Elsevier, 2006. Vol. 86, nr 11, s. 3275-3285
##### Emneord [en]
Reduced rank regression, System identification, General algorithm, Determinant minimization criterion, Rank reduction, Volume minimization
##### Identifikatorer
OAI: oai:DiVA.org:liu-13191DiVA, id: diva2:18007
Tilgjengelig fra: 2008-04-29 Laget: 2008-04-29 Sist oppdatert: 2017-12-13
##### Inngår i avhandling
1. Algorithms in data mining using matrix and tensor methods
Åpne denne publikasjonen i ny fane eller vindu >>Algorithms in data mining using matrix and tensor methods
2008 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
##### Abstract [en]
In many fields of science, engineering, and economics large amounts of data are stored and there is a need to analyze these data in order to extract information for various purposes. Data mining is a general concept involving different tools for performing this kind of analysis. The development of mathematical models and efficient algorithms is of key importance. In this thesis we discuss algorithms for the reduced rank regression problem and algorithms for the computation of the best multilinear rank approximation of tensors.
The first two papers deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is
$\min_{\rank(X) = k} \det (B - X A)(B - X A)\tp,$
where $A$ and $B$ are given matrices and we want to find $X$ under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on $A$ and $B$ so that $(B - X A)(B - X A)\tp$ is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing a singular objective matrix.
Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third paper concerns with classification of handwritten digits in the context of tensors or multidimensional data arrays. Tensor and multilinear algebra is an area that attracts more and more attention because of the multidimensional structure of the collected data in various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98--99 \% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The resulting algorithm achieves 5\% in classification error with fairly low amount of computations.
The remaining two papers discuss computational methods for the best multilinear
rank approximation problem
$\min_{\cB} \| \cA - \cB\|$
where $\cA$ is a given tensor and we seek the best low multilinear rank approximation tensor $\cB$. This is a generalization of the best low rank matrix approximation problem. It is well known that for matrices the solution is given by truncating the singular values in the singular value decomposition (SVD) of the matrix. But for tensors in general the truncated HOSVD does not give an optimal approximation. For example, a third order tensor $\cB \in \RR^{I \x J \x K}$ with rank$(\cB) = (r_1,r_2,r_3)$ can be written as the product
$\cB = \tml{X,Y,Z}{\cC}, \qquad b_{ijk}=\sum_{\lambda,\mu,\nu} x_{i\lambda} y_{j\mu} z_{k\nu} c_{\lambda\mu\nu},$
where $\cC \in \RR^{r_1 \x r_2 \x r_3}$ and $X \in \RR^{I \times r_1}$, $Y \in \RR^{J \times r_2}$, and $Z \in \RR^{K \times r_3}$ are matrices of full column rank. Since it is no restriction to assume that $X$, $Y$, and $Z$ have orthonormal columns and due to these constraints, the approximation problem can be considered as a nonlinear optimization problem defined on a product of Grassmann manifolds. We introduce novel techniques for multilinear algebraic manipulations enabling means for theoretical analysis and algorithmic implementation. These techniques are used to solve the approximation problem using Newton and Quasi-Newton methods specifically adapted to operate on products of Grassmann manifolds. The presented algorithms are suited for small, large and sparse problems and, when applied on difficult problems, they clearly outperform alternating least squares methods, which are standard in the field.
##### sted, utgiver, år, opplag, sider
Matematiska institutionen, 2008. s. 29
##### Serie
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1178
##### Emneord
Volume, Minimization criterion, Determinant, Rank deficient matrix, Reduced rank regression, System identification, Rank reduction, Volume minimization, General algorithm, Handwritten digit classification, Tensors, Higher order singular value decomposition, Tensor approximation, Least squares, Tucker model, Multilinear algebra, Notation, Contraction, Tensor matricization, Newton's method, Grassmann manifolds, Product manifolds, Quasi-Newton algorithms, BFGS and L-BFGS, Symmetric tensor approximation, Local/intrinsic coordinates, Global/embedded coordinates;
##### Identifikatorer
urn:nbn:se:liu:diva-11597 (URN)978-91-7393-907-2 (ISBN)
##### Disputas
2008-05-27, Glashuset, B-huset, ing. 25, Campus Valla, Linköpings universitet, Linköping, 10:15 (engelsk)
##### Veileder
Tilgjengelig fra: 2008-04-29 Laget: 2008-04-29 Sist oppdatert: 2013-10-11
2. Algorithms in data mining: reduced rank regression and classification by tensor methods
Åpne denne publikasjonen i ny fane eller vindu >>Algorithms in data mining: reduced rank regression and classification by tensor methods
2005 (engelsk)Licentiatavhandling, med artikler (Annet vitenskapelig)
##### Abstract [en]
In many fields of science, engineering, and economics large amounts of data are stored and there is a need to analyze these data in order to extract information for various purposes. Data mining is a general concept involving different tools for performing this kind of analysis. The development of mathematical models and efficient algorithms is of key importance. In this thesis, which consists of three appended manuscripts, we discuss algorithms for reduced rank regression and for classification in the context of tensor theory.
The first two manuscripts deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is
where A and B are given matrices and we want to find X under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on A and B so that (B - XA)(B - XA)T is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing singularity on the objective matrix.
Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third appended manuscript concerns with classification of hand written digits in the context of tensors or multidimensional data arrays. Tensor theory is also an area that attracts more and more attention because of the multidimensional structure of the collected data in a various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98%- 99% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The amount of computations is fairly low and the performance reasonably good, 5% in error rate.
##### Serie
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1214
##### Identifikatorer
urn:nbn:se:liu:diva-30272 (URN)15789 (Lokal ID)91-85457-81-7 (ISBN)15789 (Arkivnummer)15789 (OAI)
Tilgjengelig fra: 2009-10-09 Laget: 2009-10-09 Sist oppdatert: 2013-11-06
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• rtf | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9165941476821899, "perplexity": 1686.7265023387424}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986658566.9/warc/CC-MAIN-20191015104838-20191015132338-00071.warc.gz"} |
http://www.computer.org/csdl/trans/tc/1998/02/t0264-abs.html | Subscribe
Issue No.02 - February (1998 vol.47)
pp: 264-269
ABSTRACT
<p><b>Abstract</b>—Recently, an unconventional clock distribution scheme, called Branch-and-Combine (<it>BaC</it>), has been proposed. The scheme is the first to guarantee constant skew upper bound irrespective of the clocked network's size. In <it>BaC</it> clocking, a set of interconnected nodes perform simple processing on clock signals such that the path from the source to any node is automatically and adaptively selected such that it is the shortest delay path. The graph underlying a <it>BaC</it> network is constrained by the requirement that each pair of adjacent nodes is in a cycle of length ≤<it>k</it>, where <it>k</it> is the feature cycle length. The graph representing such a network is called a <it>BaC</it>(<it>k</it>) graph. The feature cycle length (<it>k</it>) is an important parameter upon which skew bound and node function depend.</p><p>In this paper, we study the complexity of the general problem of designing a minimum cost <it>BaC</it> network for clocking a data processing network of arbitrary topology such that a certain feature cycle length is satisfied. We define two versions of the problem, differing in the way we are allowed to place edges in the graph representing the <it>BaC</it> network. We show that, in both cases, the general optimization problem is NP-hard. We also provide efficient heuristic algorithms for both versions of the optimization problem. When <it>k</it> = 2, the two versions of the optimization problem become the same and can be solved in polynomial time. For <it>k</it> = 3, the complexity is still unknown.</p>
INDEX TERMS
Clock network, optimal design, skew bound, computational complexity, branch-and-combine network, graph orientation.
CITATION
Ahmed El-Amawy, Priyalal Kulasinghe, "On the Complexity of Designing Optimal Branch-and-Combine Clock Networks", IEEE Transactions on Computers, vol.47, no. 2, pp. 264-269, February 1998, doi:10.1109/12.664212 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8871527314186096, "perplexity": 991.0940887415934}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398447860.26/warc/CC-MAIN-20151124205407-00082-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://www.researchgate.net/researcher/49415532_R_Lastowiecki | # R. Lastowiecki
University of Wroclaw, Vrotslav, Lower Silesian Voivodeship, Poland
Are you R. Lastowiecki?
## Publications (9)8.89 Total impact
• Source
##### Article: Quark matter in high-mass neutron stars?
R. Lastowiecki · D. Blaschke · T. Fischer · T. Klahn ·
[Hide abstract]
ABSTRACT: The recent measurements of the masses of the pulsars PSR J1614-2230 and PSR J0348-0432 provide independent proof for the existence of neutron stars with masses in range of 2 $M_\odot$. This fact has significant implications for the physics of high density matter and it challenges the hypothesis that the cores of NS can be composed of deconfined quark matter. In this contribution we study a description of quark matter based on the Nambu--Jona-Lasinio effective model and construct the equation of state for matter in beta equilibrium. This equation of state together with the hadronic Dirac-Brueckner-Hartree-Fock equation of state is used here to describe neutron star and hybrid star configurations. We show that compact stars masses of 2 $M_\odot$ are compatible with the possible existence of deconfined quark matter in their core.
Physics of Particles and Nuclei 03/2015; 46(5). DOI:10.1134/S1063779615050159 · 0.62 Impact Factor
• Source
##### Article: Finite-size effects at the hadron-quark transition and heavy hybrid stars
[Hide abstract]
ABSTRACT: We study the role of finite-size effects at the hadron-quark phase transition in a new hybrid equation of state constructed from an ab-initio Br\"uckner-Hartree-Fock equation of state with the realistic Bonn-B potential for the hadronic phase and a covariant non-local Nambu--Jona-Lasinio model for the quark phase. We construct static hybrid star sequences and find that our model can support stable hybrid stars with an onset of quark matter below $2 M_\odot$ and a maximum mass above $2.17 M_\odot$ in agreement with recent observations. If the finite-size effects are taken into account the core is composed of pure quark matter. Provided that the quark vector channel interaction is small, and the finite size effects are taken into account, quark matter appears at densities 2-3 times the nuclear saturation density. In that case the proton fraction in the hadronic phase remains below the value required by the onset of the direct URCA process, so that the early onset of quark matter shall affect on the rapid cooling of the star.
Physical Review C 03/2014; 89(6). DOI:10.1103/PhysRevC.89.065803 · 3.73 Impact Factor
• Source
##### Article: Crossover transition to quark matter in heavy hybrid stars
D. E. Alvarez-Castillo · S. Benic · D. Blaschke · R. Lastowiecki ·
[Hide abstract]
ABSTRACT: We study the possibility that the transition from hadron matter to quark matter at vanishing temperatures proceeds via crossover, similar to the crossover behavior found with lattice QCD studies at high temperatures. The purpose is to examine astrophysical consequences of this postulate by constructing hybrid star sequences fulfilling current experimental data.
Acta Physica Polonica B, Proceedings Supplement 11/2013; 7(1). DOI:10.5506/APhysPolBSupp.7.203
• Source
##### Article: Nonlocal PNJL models and heavy hybrid stars
[Hide abstract]
ABSTRACT: Nonlocal PNJL models allow for a detailed description of chiral quark dynamics with running quark masses and wave function renormalization in accordance with lattice QCD (LQCD) in vacuum. Their generalization to finite temperature T and chemical potential \mu{} allows to reproduce the \mu-dependence of the pseudocritical temperature from LQCD when a nonvanishing vector meson coupling is adjusted. This restricts the region for the critical endpoint in the QCD phase diagram and stiffens the quark matter equation of state (EoS). It is demonstrated that the construction of a hybrid EoS for compact star applications within a two-phase approach employing the nonlocal PNJL EoS and an advanced hadronic EoS leads to the masquerade problem. A density dependence of the vector meson coupling is suggested as a possible solution which can be adjusted in a suitable way to describe hybrid stars with a maximum mass in excess of 2 M_sun with a possible early onset of quark deconfinement even in the cores of typical (M ~ 1.4 M_sun) neutron stars.
• Source
##### Article: Neutron star matter in a modified PNJL model
R. Lastowiecki · D. Blaschke · J. Berdermann ·
[Hide abstract]
ABSTRACT: We discuss a three-flavor Nambu-Jona-Lasinio model for the quark matter equation of state with scalar diquark interaction, isoscalar vector interaction and Kobayashi-Maskawa-'t Hooft interaction. We adopt a phenomenological scheme to include possible effects of a change in the gluon pressure at finite baryon density by including a parametric dependence of the Polyakov-loop potential on the chemical potential. We discuss the results for the mass-radius relationships for hybrid neutron stars constructed on the basis of our model EoS in the context of the constraint from the recently measured mass of (1.97 {+-} 0.04) M{sub Circled-Dot-Operator} for the pulsar PSR J1614-2230.
Physics of Atomic Nuclei 07/2012; 75(7). DOI:10.1134/S106377881207006X · 0.51 Impact Factor
• Source
##### Article: Strangeness in the cores of neutron stars
R. Lastowiecki · D. Blaschke · H. Grigorian · S. Typel ·
[Hide abstract]
ABSTRACT: The measurement of the mass 1.97 +/- 0.04 M_sun for PSR J1614-2230 provides a new constraint on the equation of state and composition of matter at high densities. In this contribution we investigate the possibility that the dense cores of neutron stars could contain strange quarks either in a confined state (hyperonic matter) or in a deconfined one (strange quark matter) while fulfilling a set of constraints including the new maximum mass constraint. We account for the possible appearance of hyperons within an extended version of the density-dependent relativistic mean-field model, including the phi-meson interaction channel. Deconfined quark matter is described by the color superconducting three-flavor NJL model.
Acta Physica Polonica B, Proceedings Supplement 12/2011; 5(2). DOI:10.5506/APhysPolBSupp.5.535
• Source
##### Article: Compact Stars, Heavy Ion Collisions, and Possible Lessons For QCD at Finite Densities
Thomas Klahn · David Blaschke · Rafal Lastowiecki ·
[Hide abstract]
ABSTRACT: Large neutron star masses as the recently measured $1.97\pm0.04$ M$_\odot$ for PSR J1614-2230 provide a valuable lower limit on the stiffness of the equation of state of dense, nuclear and quark matter. Complementary, the analysis of the elliptic flow in heavy ion collisions suggests an upper limit on the EoS stiffness. We illustrate how this dichotomy permits to constrain parameters of effective EoS models which otherwise could not be derived unambiguously from first principles.
Acta Physica Polonica B, Proceedings Supplement 11/2011; 5(3). DOI:10.5506/APhysPolBSupp.5.757
• Source
##### Article: Hybrid Neutron Stars Based on a Modified PNJL Model
David Blaschke · Jens Berdermann · Rafal Lastowiecki ·
[Hide abstract]
ABSTRACT: We discuss a three-flavor Nambu--Jona-Lasinio (NJL) type quantum field theoretical approach to the quark matter equation of state (EoS) with scalar diquark condensate, isoscalar vector mean field and Kobayashi-Maskawa-'t Hooft (KMT) determinant interaction. While often the diquark and vector meson couplings are considered as free parameters, we will fix them here to their values according to the Fierz transformation of a one-gluon exchange interaction. In order to estimate the effect of a possible change in the vacuum pressure of the gluon sector at finite baryon density we exploit a recent modification of the Polyakov-loop NJL (mPNJL) model which introduces a parametric density dependence of the Polyakov-loop potential also at T=0, thus being relevant for compact star physics. We use a Dirac-Brueckner-Hartree-Fock (DBHF) EoS for the hadronic matter phase and discuss results for mass-radius relationships following from a solution of the TOV equations for such a hybrid EoS in the context of observational constraints from selected objects.
Progress of Theoretical Physics Supplement 09/2010; 186. DOI:10.1143/PTPS.186.81 · 1.25 Impact Factor
• Source
##### Article: How strange are compact star interiors ?
D. Blaschke · T. Klahn · R. Lastowiecki · F. Sandin ·
[Hide abstract]
ABSTRACT: We discuss a Nambu--Jona-Lasinio (NJL) type quantum field theoretical approach to the quark matter equation of state with color superconductivity and construct hybrid star models on this basis. It has recently been demonstrated that with increasing baryon density, the different quark flavors may occur sequentially, starting with down-quarks only, before the second light quark flavor and at highest densities also the strange quark flavor appears. We find that color superconducting phases are favorable over non-superconducting ones which entails consequences for thermodynamic and transport properties of hybrid star matter. In particular, for NJL-type models no strange quark matter phases can occur in compact star interiors due to mechanical instability against gravitational collapse, unless a sufficiently strong flavor mixing as provided by the Kobayashi-Maskawa-'t Hooft determinant interaction is present in the model. We discuss observational data on mass-radius relationships of compact stars which can put constraints on the properties of dense matter equation of state. Comment: 7 pages, 2 figures, to appear in the Proceedings of the International Conference SQM2009, Buzios, Rio de Janeiro, Brazil, Sep.27-Oct.2, 2009
Journal of Physics G Nuclear and Particle Physics 02/2010; 37(9). DOI:10.1088/0954-3899/37/9/094063 · 2.78 Impact Factor
#### Publication Stats
74 Citations 8.89 Total Impact Points
#### Institutions
• ###### University of Wroclaw
Vrotslav, Lower Silesian Voivodeship, Poland | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.961510181427002, "perplexity": 3273.4521972441366}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398448389.58/warc/CC-MAIN-20151124205408-00122-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://infoscience.epfl.ch/record/178543 | Infoscience
Journal article
# The Cosmic Linear Anisotropy Solving System (CLASS). Part II: Approximation schemes
Boltzmann codes are used extensively by several groups for constraining cosmological parameters with Cosmic Microwave Background and Large Scale Structure data. This activity is computationally expensive, since a typical project requires from 104 to 105 Boltzmann code executions. The newly released code CLASS (Cosmic Linear Anisotropy Solving System) incorporates improved approximation schemes leading to a simultaneous gain in speed and precision. We describe here the three approximations used by CLASS for basic Lambda CDM models, namely: a baryon-photon tight-coupling approximation which can be set to first order, second order or to a compromise between the two; an ultra-relativistic fluid approximation which had not been implemented in public distributions before; and finally a radiation streaming approximation taking reionisation into account. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8793913125991821, "perplexity": 2194.9369248907683}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171936.32/warc/CC-MAIN-20170219104611-00207-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://learnzillion.com/resources/73035-understand-the-probability-of-chance-events-7-sp-c-5 | # Understand the probability of chance events (7.SP.C.5)
Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9349071979522705, "perplexity": 879.4364809914699}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864343.37/warc/CC-MAIN-20180622030142-20180622050142-00183.warc.gz"} |
http://physics.stackexchange.com/questions/44379/what-is-the-physical-interpretation-of-the-density-matrix-in-a-double-continuous?answertab=active | # What is the physical interpretation of the density matrix in a double continuous basis $|\alpha\rangle$, $|\beta\rangle$?
(a) Any textbook gives the interpretation of the density matrix in a single continuous basis $|\alpha\rangle$:
• The diagonal elements $\rho(\alpha, \alpha) = \langle \alpha |\hat{\rho}| \alpha \rangle$ give the populations.
• The off-diagonal elements $\rho(\alpha, \alpha') = \langle \alpha |\hat{\rho}| \alpha' \rangle$ give the coherences.
(b) But what is the physical interpretation (if any) of the density matrix $\rho(\alpha, \beta) = \langle \alpha |\hat{\rho}| \beta \rangle$ for a double continuous basis $|\alpha\rangle$, $|\beta\rangle$?
I know that when the double basis are position and momentum then $\rho(p, x)$ is interpreted as a pseudo-probability. I may confess that I have never completely understood the concept of pseudo-probability [*], but I would like to know if this physical interpretation as pseudo-probability can be extended to arbitrary continuous basis $|\alpha\rangle$, $|\beta\rangle$ for non-commuting operators $\hat{\alpha}$, $\hat{\beta}$ and as probability for commuting ones.
[*] Specially because $\rho(p, x)$ is bounded and cannot be 'spike'.
EDIT: To avoid further misunderstandings I am adding some background. Quantum averages can be obtained in a continuous basis $| \alpha \rangle$ as
$$\langle A \rangle = \int \mathrm{d} \alpha \; \langle \alpha | \hat{\rho} \hat{A} | \alpha \rangle$$
(a) Introducing closure in the same basis $| \alpha \rangle$
$$\langle A \rangle = \int \mathrm{d} \alpha \int \mathrm{d} \alpha' \; \langle \alpha | \hat{\rho} | \alpha' \rangle \langle \alpha' | \hat{A} | \alpha \rangle = \int \mathrm{d} \alpha \int \mathrm{d} \alpha' \; \rho(\alpha,\alpha') A(\alpha',\alpha)$$
with the usual physical interpretation for the density matrix $\rho(\alpha,\alpha')$ as discussed above.
(b) Introducing closure in a second basis $| \beta \rangle$, we obtain the alternative representation
$$\langle A \rangle = \int \mathrm{d} \alpha \int \mathrm{d} \beta \; \langle \alpha | \hat{\rho} | \beta \rangle \langle \beta | \hat{A} | \alpha \rangle = \int \mathrm{d} \alpha \int \mathrm{d} \beta \; \rho(\alpha,\beta) A(\beta,\alpha)$$
When the two basis are momentum $| p \rangle$ and position $| x \rangle$ the density $\rho(p,x)$ is the well-known Wigner function whose physical interpretation is that of a pseudo-probability. My question is about the physical interpretation of $\rho(\alpha,\beta)$ in two arbitrary basis $| \alpha \rangle$, $| \beta \rangle$.
-
Probabilities have a physical meaning only in a context where measureemnt is possible. Between states of a pointer basis in a measurement context, the matrix elements of a density matrix have the standard probabilistic meaning.
In any other basis, they are just mathematical expressions intermediate to other calculations of interest. (I wouldn't give a penny for attempts to interpret these in terms of nonphysical pseudo-probabilities.)
-
Thank you! The part about measurements seems very related to my remark about the spikeness of the Wigner function, but my aim was not to discuss here the concept of pseudo-probability but to know if it can be extended beyond momentum-position basis. Moreover your answer seems to avoid the cases when the operators commute. My belief is that those would be true probabilities instead of pseudo. Am I mistaken? – juanrga Nov 19 '12 at 19:06 @juanrga: Yes, but you called them $x$ and $p$, which don't commute. - You had asked about a pseudo-probability interpretation. This is somewhat meaningful in the Wigner case because one can take the calssical limit where phase space probabilities result. But for general bases, one has no such limiting interpretation. – Arnold Neumaier Nov 19 '12 at 19:10 Well I referred to "arbitrary continuous basis $\alpha$ and $\beta$" and said I was interested in both cases when the operators commute and when do not. I asked about a pseudo-probability interpretation for non-commuting operators and about the interpretation "as probability for commuting ones". I used $x$ and $p$ only as illustrative example. I do not follow your argument about the classical limit (Wigner function does not reduce to classical phase space probabilities), but thank you. – juanrga Nov 19 '12 at 20:34
How is $\langle \alpha | \hat \rho | \beta \rangle$ different from $\langle \alpha | \hat \rho | \alpha^\prime \rangle$? Both representations are basis-independent, that is, you can choose any basis of your choice (position, momentum, you-name-it).
If your question is referring to the fact that it is sometimes useful to use two indices rather than one to enumerate the states (such as spin and momentum), realising that you can easily combine them into one index (which is then possibly multi-dimensional) and that you can similarly split any single basis index in multiple indices should resolve your problem.
-
$|\alpha \rangle$ and $|\alpha' \rangle$ are two elements of the same basis. $|\alpha \rangle$ and $|\beta \rangle$ are two elements of two different basis. – juanrga Nov 17 '12 at 12:04 Isn’t the Dirac representation meant to be basis-independent? Not to mention that having two different bases in the (matrix multiplication) expression doesn’t make much sense either to me. – Claudius Nov 17 '12 at 13:53 (i) Basis-independence does not mean that the physical interpretation remains unchanged in any representation and I am precisely asking about the physical interpretation of the density matrix $\rho(\alpha, \beta)$. (ii) It must not make much sense for you, but as stated in the original question $\rho(p, x)$ is interpreted as a pseudo-probability in the textbooks. – juanrga Nov 17 '12 at 17:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9668314456939697, "perplexity": 351.9599191469446}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368704234586/warc/CC-MAIN-20130516113714-00097-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://www.physicsforums.com/showpost.php?p=3241641&postcount=1 | View Single Post
P: 352 Please explain Euler's theorem. I don't get how he got this formula and how it can be used instead of trigonometry. Thanks | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.942287266254425, "perplexity": 324.9783158474986}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510267824.47/warc/CC-MAIN-20140728011747-00363-ip-10-146-231-18.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/205266-newtons-cooling-law-i-need-help-problem-thank-you.html | # Thread: newtons cooling law. I need help with this problem. Thank you.
1. ## newtons cooling law. I need help with this problem. Thank you.
A thermometer reading 70°F is placed in an oven preheated to
a constant temperature. Through a glass window in the oven
door, an observer records that the thermometer read 110°F after
.5 minute and 145°F after 1 minute. How hot is the oven?
using the following:
dT/dt = k(T- Tm)
2. ## Re: newtons cooling law. I need help with this problem. Thank you.
Hey ALAIN971.
This is a separable DE which means you move the T's to one side and the t's (and the differentials) to the other. Can you separate these variables and their differentials and integrate both sides?
3. ## Re: newtons cooling law. I need help with this problem. Thank you.
I have done this up to this step.
T(t)=Tm + Ce^(kt)
i get 2 equations with 2 variables but i am stuck.
145= Tm + Ce^(kt)
110= Tm + Ce^(0.5kt)
C and k are constants.
4. ## Re: newtons cooling law. I need help with this problem. Thank you.
sorry i meant :
I have done this up to this step.
T(t)=Tm + Ce^(kt)
i get 2 equations with 2 variables but i am stuck.
145= Tm + Ce^(k)
110= Tm + Ce^(0.5k)
C and k are constants.
5. ## Re: newtons cooling law. I need help with this problem. Thank you.
What about the information at t = 0? Can you use this information to get another piece of information?
6. ## Re: newtons cooling law. I need help with this problem. Thank you.
at t=o temperature is 70
7. ## Re: newtons cooling law. I need help with this problem. Thank you.
So now you have three equations for Tm, C and k.
70 = Tm + C.
145= Tm + Ce^(k)
110= Tm + Ce^(0.5k)
C = 70 - Tm
k = ln(145 - Tm) - ln(C)
k = 2*ln(110 - Tm) - 2*ln(C)
ln(145 - Tm) = 2*ln(110 - Tm) - ln(C)
145 - Tm = 70 + C
110 - Tm = 40 + C which means
ln(70 + C) = 2*ln(40 + C) - ln(C) which means
ln([70 + C]C) = ln([40 + C]^2) taking exponentials we get:
C[70+C] = [40+C]^2. and how you have a quadratic equation for C.
8. ## Re: newtons cooling law. I need help with this problem. Thank you.
ok how do i proceed then.
i found C=-160
pit it back in 145-Tm=70+C
and i found Tm=235 Farenheights
9. ## Re: newtons cooling law. I need help with this problem. Thank you.
I made a mistake, 145 - Tm = 75 + C.
You should be checking what I do not just following blindly. I am only trying to supplement the thinking process: not do the whole thing.
10. ## Re: newtons cooling law. I need help with this problem. Thank you.
thank you for your help. i returned back to school this fall and calculus is far away for me i have not done any for years.
i dont know if i did it correctly but i get 230 degrees now. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9160214066505432, "perplexity": 2132.8405261928538}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948609934.85/warc/CC-MAIN-20171218063927-20171218085927-00690.warc.gz"} |
https://www.physicsforums.com/threads/a-sub-thread-to-yomammas-infinity-issue.90549/ | A sub-thread to Yomammas Infinity issue
1. Sep 24, 2005
zanazzi78
Im relativly new to Physics but have heard of a technic used called renomalisation, the cancelling of infinties. If infinity is not a number how does this work?
(my appologies to the mods if ive posted this in the wrong place)
2. Sep 24, 2005
Hurkyl
Staff Emeritus
This exercise is something I've actually stumbled through at work -- it is fairly elementary, and I understand is a related sort of thing.
I wanted to get a Taylor series for the cumulative standard normal distribution. Recall that it's given by:
$$f(z) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^z e^{-t^2 / 2} \, dt$$
So, I replaced the exponential with its Taylor series, giving:
$$f(z) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^z \sum_{n = 0}^{\infty} \frac{1}{n!} \left( -\frac{t^2}{2} \right)^n \, dt$$
And then optimistically pulled the sum out of the integral:
$$f(z) = \frac{1}{\sqrt{2\pi}} \sum_{n=0}^{\infty} \frac{(-1)^n}{n!2^n} \int_{-\infty}^{z} t^{2n} \, dt$$
From which you "get":
$$f(z) = \frac{1}{\sqrt{2\pi}} \sum_{n=0}^{\infty} \frac{(-1)^n}{n!2^n} \left( \frac{z^{2n+1}}{2n+1} + \infty \right)$$
At which point I had the idea that I knew f(0) = 1/2, and since all of the ±∞ "are" constants, I could just collect them together into the correct constant and get the correct formula (assuming I haven't made any silly mistakes here):
$$f(z) = \frac{1}{2} + \frac{1}{\sqrt{2\pi}} \sum_{n=0}^{\infty} \frac{(-1)^n}{n!2^n} \frac{z^{2n+1}}{2n+1}$$
Of course, once I did all of this, I realized that I could do the thing properly by observing that:
$$\int_{-\infty}^z g(t) \, dt = \int_{-\infty}^0 g(t) \, dt + \int_0^z g(t) \, dt$$
3. Sep 24, 2005
Staff Emeritus
And the renormalization part of Hurkyl's derivation was formally collecting all those infinities into a normalizing constant. This is the way some people think of renormalization, and of course then they think it's just "sweeping the infinity under the rug". Actual renormalization in field theory is a little more sophisticated and is NOT sweeping under the rug.
First of all, renormalization concerns the infinities that come into a theory at very short scales; the electron interacts with itself and the theory adds up all the ways such interactions can happen and gets infinity. In quantum mechanics very small spaces are associated with very big energies, because of the uncertainty principle (your position is very sharply limited so your momentum can reach very high limits and the energy with it). So the physicists say, my theory is just not complete enough to handle these very high energies; I will put a cutoff in my theory that prevents it from wrongly trying to process them. This use of a cutoff is called regularization. So then they have a theory ( a bunch of integrals) with a cutoff in the form of an unspecified constant in them. And they work out the theory - simplify and transform the integrals to get the numbers they want to calculate - and the constant flows through all this work and will make it impossible to get the actual numbers. So at the last stage they remove the constant by combining it into a normalization constant. This is the process called renormalization. Note that it wasn't infinity that they moved, it was a finite but unknown constant that they got from putting a cutoff into the theory. And the process is philosophically justified by the admission that the theory was incomplete and unable to handle energies higher than some (very high) limit.
Last edited: Sep 24, 2005
4. Sep 24, 2005
Hurkyl
Staff Emeritus
And, incidentally, note that what I did would have been fully justified if I replaced the lower limit -∞ with some negative constant H.
Then, instead of infinities, I have finite numbers which add up to a finite value, which I could simply roll up into a constant.
(And my example is particularly nice, because the limit as H→-∞ worksout nicely)
5. Sep 24, 2005
JamesU
OMG....
6. Sep 24, 2005
zanazzi78
ditto :surprised
Thanx for the info and example (Hurkyl) but ...
could you clarify, i don`t understand the term normalization constant.
Last edited: Sep 24, 2005
7. Sep 24, 2005
EnumaElish
Yeah, is it similar to Hurkyl's having realized that he has a "boundary condition" f(0) = 1/2 and solving for the integration constant using that condition?
8. Sep 24, 2005
Staff Emeritus
Right Enuma. In a theory you typically have to set a scale, your integrals or whatever don't come out in any particular scale and you "normalize" them by multiplying them by some number that is chosen to make things come out right. This is the normalizing constant. Once you choose it you can't change it, but what you can do if you want to gent rid of that unknown cutoff constant (call it K) is to ASSUME that your good normalization constant N is composed of a factor M times K: N = MK. So when you normalize your theory, which is after you have done all the computations, you just say "...and N includes the cutoff."
9. Sep 24, 2005
Mk
You were sorry you ever clicked on that link. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8937251567840576, "perplexity": 765.3495187573436}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917119225.38/warc/CC-MAIN-20170423031159-00555-ip-10-145-167-34.ec2.internal.warc.gz"} |
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Jul 28, 2016In addition, one inch diameter pieces of pipe are clumsier than straws and take more room in the drawer. Have you ever tried to drink through one of those cheap coffee stirrers? Large diameter is easier up to a point. Edit It's also harder to slurp up a 12 ounce tumbler of milk when it takes 8 ounces just to fill the straw.How do we calculate inch dia of pipe? - QuoraWhat formula can be used to calculate the diameter of a How do we calculate inch dia of pipeQuora See more resultsCosting of Pipelines - 1 PIPING GUIDEApr 25, 2009The following example will illustrate the concept of the inch-meter and the inch-dia Q.1 Elbow 2" size, 20 nos., are to be fitted in a pipeline. Find out the inch dia? Ans Inch dia size x no. of weld joints x no. of elbows = 2 x 2 x 20 = 80 inch dia. Q. 2 Find out the inch meter for 20 m pipeline of 2" size. Ans Inch meter = Pipe size in How do we calculate inch dia of pipeQuoraHow do you calculate inch dia of pipe? - AnswersThe meaning of dia inch is the number of weld joints multiplied by the dia of a given pipe size, since the weld joints is on the 6" pipe, the calculations should be 1 weld x 6" = 6 dia inch.
How do you calculate inch dia of pipe? - Answers
The meaning of dia inch is the number of weld joints multiplied by the dia of a given pipe size, since the weld joints is on the 6" pipe, the calculations should be 1 weld x 6" = 6 dia inch.How do you calculate inch diameter for reinforcement pad How do we calculate inch dia of pipeQuoraHow do you calculate inch diameter for reinforcement pad in pipe? Free e-mail watchdog. Tweet. Answer this question. How do you calculate inch diameter for reinforcement pad in pipe? Answer for question Your name Answers. recent questions recent answers. Yes my new car; 6*10^5 is how many times as large as 3*10^3;How to Calculate tolerance values for shaft or a hole How do we calculate inch dia of pipeQuoraLets see how we can calculate Tolerances and the Fundamental deviations for the Hole or a Shaft. Procedure to Calculate tolerance values for shaft or a hole 1. Calculating Fundamental Deviation a) Calculation of Fundamental Deviation for Shafts. The upper deviation and the lower deviation for the shafts are represented as es, ei
How to Calculate tolerance values for shaft or a hole How do we calculate inch dia of pipeQuora
Lets see how we can calculate Tolerances and the Fundamental deviations for the Hole or a Shaft. Procedure to Calculate tolerance values for shaft or a hole 1. Calculating Fundamental Deviation a) Calculation of Fundamental Deviation for Shafts. The upper deviation and the lower deviation for the shafts are represented as es, eiHow to accurately measure the Diameter of a BallJun 02, 2019As an example, if you measure a yoga ball and its circumference measures 67.71 inches, the diameter of the yoga ball which youre measuring will be 21.65 inches. How to measure the diameter of a circle If youre not looking to measure the diameter of a three dimensional ball and are looking to calculate the diameter of a two dimensional How do we calculate inch dia of pipeQuoraHow to calculate water in pipe - QuoraMay 14, 2018We can calculate the volume or mass of water in pipe. before calculation we would have to know about pipe dimension and rate at which water is coming out from them pipe or velocity at exit of the pipe. Rate= density ×area×velocity of water Put all How do we calculate inch dia of pipeQuora
How to calculate wheel radius x-engineer
In order to calculate the diameter of the wheel, we need to know rim diameter (taken from size marking) tire sidewall height (calculated from aspect ratio) From (1) we can calculate the tire side wall height H [mm] $H = \frac{AR \cdot W}{100} \tag{2}$ The wheel diameter d w [mm] is the sum between the rim diameter and twice the height of How do we calculate inch dia of pipeQuoraHow to calculate wheel radius x-engineerIn order to calculate the diameter of the wheel, we need to know rim diameter (taken from size marking) tire sidewall height (calculated from aspect ratio) From (1) we can calculate the tire side wall height H [mm] $H = \frac{AR \cdot W}{100} \tag{2}$ The wheel diameter d w [mm] is the sum between the rim diameter and twice the height of How do we calculate inch dia of pipeQuoraLength & Diameter to Volume CalculatorCircle area to diameter; User Guide. This tool will calculate the volume of a cylindrical shaped object from the dimensions of length and diameter. No conversion needed, since length, diameter and volume units can be selected independently, so this calculator
People also askHow do you calculate pipeline size?How do you calculate pipeline size?Ans Inch dia size x no. of weld joints x no. of elbows. = 2 x 2 x 20 = 80 inch dia. Q. 2 Find out the inch meter for 20 m pipeline of 2" size. Ans Inch meter = Pipe size in inches x length in m.How do we calculate inch dia of pipe? - QuoraPipe Volume Calculator
We will calculate the volume of a 6-meter length pipe, with an inner diameter equal to 15 centimeters. The pipe is used to transport water. Let's put these data into the calculator to find the volume of water in the pipe, as well as its mass. First, enter the pipe's diameter inner diameter = 15 cm. Then, type in its length length = 6 m.Pipe Volume CalculatorWe will calculate the volume of a 6-meter length pipe, with an inner diameter equal to 15 centimeters. The pipe is used to transport water. Let's put these data into the calculator to find the volume of water in the pipe, as well as its mass. First, enter the pipe's diameter inner diameter = 15 cm. Then, type in its length length = 6 m.
Pulley Calculator. RPM, Belt Length, Speed, Animated
Enter any 3 known values to calculate the 4th If you know any 3 values (Pulley sizes or RPM) and need to calculate the 4th, enter the 3 known values and hit Calculate to find the missing value. For example, if your small pulley is 80mm diameter, and spins at 1000 RPM, and you need to find the second pulley size to spin it at 400 RPM, Enter Pulley1 80, Pulley 1 RPM 1000, Pulley 2 RPM 400, and How do we calculate inch dia of pipeQuoraPulley Calculator. RPM, Belt Length, Speed, Animated Enter any 3 known values to calculate the 4th If you know any 3 values (Pulley sizes or RPM) and need to calculate the 4th, enter the 3 known values and hit Calculate to find the missing value. For example, if your small pulley is 80mm diameter, and spins at 1000 RPM, and you need to find the second pulley size to spin it at 400 RPM, Enter Pulley1 80, Pulley 1 RPM 1000, Pulley 2 RPM 400, and How do we calculate inch dia of pipeQuoraTANK VOLUME CALCULATOR [How to Calculate Tank We can do this because our calculator is able to do the conversions for you, making it far easier for you! For each measurement there are multiple options that are available to use. For example, length can be calculated in terms of feet (ft), inches (in), yards (yd), meters (m) or centimeters (cm).
TREE VALUE CALCULATOR How Much Is Your Tree Worth?
The easiest way to approach this is to calculate an average height and diameter of one tree. This can be done by calculating the height and diameter of 10 trees and dividing the total by 10. This will give you the average height and diameter of one tree growing in the woodlot. Next you will need to do a tree count and multiply that number by How do we calculate inch dia of pipeQuoraTank Volume Calculator - Inch CalculatorThe volume or capacity of a tank can be found in a few easy steps. Of course, the calculator above is the easiest way to calculate tank volume, but follow along to learn how to calculate it yourself. Step One Measure the Tank. The first step is to measure the key dimensions of the tank. For round tanks find the diameter and length or height.What is pipe invert elevation? - QuoraJun 18, 2017There are different inverts when referencing pipe, not the generic pipe invert; here are the most common 1. The lowest point of the inside diameter is the storage invert. 2. The bottom of the outside diameter is the chamfer invert. 3. Nomi How do we calculate inch dia of pipeQuora
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Mail Us 24/7 For Customer Support At [email protected] | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.918656051158905, "perplexity": 2191.2684984787757}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323587623.1/warc/CC-MAIN-20211025030510-20211025060510-00306.warc.gz"} |
https://www.all-dictionary.com/sentences-with-the-word-insert | # Sentence Examples with the word insert
The ear has no external leaf whatever; and into the hole itself you can hardly insert a quill, so wondrously minute is it.
As he was studying it out, Starbuck took a long cutting-spade pole, and with his knife slightly split the end, to insert the letter there, and in that way, hand it to the boat, without its coming any closer to the ship.
The Egyptologist who has long lived in the realm of conjecture is too prone to consider any feries of guesses good enough to serve as a translation, and forgets to insert the notes of interrogation which would warn workers in other fields from implicit trust.
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So little was the collection considered as a literary work with a definite text that every one assumed a right to abridge or enlarge, to insert ideas of his own, or fresh scriptural quotations; nor were the scribes and translators by any means scrupulous about the names of natural objects, and even the passages from Holy Writ.
He therefore saw that it was a mistake to insert a potential-affected detector such as a coherer in between the base of the antenna and the earth because it was then subject to very small variations of potential between its ends.
The flexor digitorum sublimis muscle arises fleshy from the long elastic band which extends from the inner humeral condyle along the ventral surface of the ulna to the ulnar carpal bone, over which the tendon runs to insert itself on the radial anterior side of the first phalanx of the second digit.
It was followed by the Vein Shimbun (Pictorial Newspaper), the first to insert illus.
There is no difficulty in observing the temperature of the surface of the sea on board ship, the only precautions required being to draw the water in a bucket which has not been heated in the sun in summer or exposed to frost in winter, to draw it well forward of any discharge pipes of the steamer, to place it in the shade on deck, insert the thermometer immediately and make the reading without delay.
De Inclinationibus had for its object to insert a straight line of a given length, tending towards a given point, between two given (straight or circular) lines.
Assuming at the outset an opposition between the two, self and matter of knowledge, he is driven by the exigencies of the problem of reconciliation to insert term after term as means of bringing them together, but never succeeds in attaining a junction which is more than mechanical. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8069260716438293, "perplexity": 2003.4359277269882}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886110485.9/warc/CC-MAIN-20170822065702-20170822085702-00633.warc.gz"} |
https://www.coursehero.com/file/p74gokr/Example-3-illustrates-that-direct-proofs-of-even-rather-simple-limits-can-get/ | Example 3 illustrates that direct proofs of even rather simple limits can get
Example 3 illustrates that direct proofs of even
This preview shows page 9 - 12 out of 85 pages.
Example 3 illustrates that direct proofs of even rather simple limits can get complicated. With the limit theorems of §9 we would just write lim 4 n 3 + 3 n n 3 6 lim 4 + 3 n 2 1 6 n 3 lim 4 + 3 · lim( 1 n 2 ) lim 1 6 · lim( 1 n 3 ) 4 . Example 4 Show that the sequence a n ( 1) n does not converge. Discussion . We will assume that lim( 1) n a and obtain a con- tradiction. No matter what a is, either 1 or 1 will have distance at least 1 from a . Thus the inequality | ( 1) n a | < 1 will not hold for all large n . Formal Proof Assume that lim( 1) n a for some a R . Letting 1 in the definition of the limit, we see that there exists N such that n > N implies | ( 1) n a | < 1 . By considering both an even and an odd n > N , we see that | 1 a | < 1 and | − 1 a | < 1 .
2. Sequences 40 Now by the Triangle Inequality 3.7 2 | 1 ( 1) | | 1 a + a ( 1) | ≤ | 1 a |+| a ( 1) | < 1 + 1 2 . This absurdity shows that our assumption that lim( 1) n a must be wrong, so the sequence ( 1) n does not converge. Example 5 Let ( s n ) be a sequence of nonnegative real numbers and suppose that s lim s n . Note that s 0; see Exercise 8.9(a). Prove that lim s n s . Discussion . We must consider > 0 and show that there exists N such that n > N implies | s n s | < . This time we cannot expect to obtain N explicitly in terms of be- cause of the general nature of the problem. But we can hope to show such N exists. The trick here is to violate our training in algebra and “irrationalize the denominator”: s n s ( s n s )( s n + s ) s n + s s n s s n + s . Since s n s we will be able to make the numerator small [for large n ]. Unfortunately, if s 0 the denominator will also be small. So we consider two cases. If s > 0, the denominator is bounded below by s and our trick will work: | s n s | ≤ | s n s | s , so we will select N so that | s n s | < s for n > N . Note that N exists, since we can apply the definition of limit to s just as well as to . For s 0, it can be shown directly that lim s n 0 implies lim s n 0; the trick of “irrationalizing the denominator” is not needed in this case. Formal Proof Case I: s > 0. Let > 0. Since lim s n s , there exists N such that n > N implies | s n s | < s .
§8. A Discussion about Proofs 41 Now n > N implies | s n s | | s n s | s n + s | s n s | s < s s . Case II: s 0. This case is left to Exercise 8.3. Example 6 Let ( s n ) be a convergent sequence of real numbers such that s n 0 for all n N and lim s n s 0. Prove that inf {| s n | : n N } > 0. Discussion . The idea is that “most” of the terms s n are close to s and hence not close to 0. More explicitly, “most” of the terms s n are within 1 2 | s | of s , hence most s n satisfy | s n | ≥ 1 2 | s | . This seems clear from Figure 8.1, but a formal proof will use the triangle inequality.
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• Spring '14
• Limits, Limit of a sequence, Sn, lim sn | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9714728593826294, "perplexity": 436.383534245159}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400274441.60/warc/CC-MAIN-20200927085848-20200927115848-00698.warc.gz"} |
https://scicomp.stackexchange.com/questions/40130/solve-ivp-from-scipy-does-not-integrate-the-whole-range-of-tspan | # solve_ivp from scipy does not integrate the whole range of tspan
I'm trying to use solve_ivp from scipy in Python to solve an IVP. I specified the tspan argument of solve_ivp to be (0,10), as shown below. However, for some reason, the solutions I get always stop around t=2.5.
from scipy.integrate import solve_ivp
import numpy as np
import matplotlib.pyplot as plt
import scipy.optimize as optim
def dudt(t, u):
return u*(1-u/12)-4*np.heaviside(-(t-5), 1)
ic = [2,4,6,8,10,12,14,16,18,20]
sol = solve_ivp(dudt, (0, 10), ic, t_eval=np.linspace(0, 10, 10000))
for solution in sol.y:
y = [y for y in solution if y >= 0]
t = sol.t[:len(y)]
plt.plot(t, y)
You can examine the sol object to see why the integration failed. It provides the message 'Required step size is less than spacing between numbers.' This usually indicates an implementation error in the right-hand side function or a singularity in the ODE.
Your ODE is simple enough to find the exact solution. We can consider the scalar case because each component is completely independent. For the initial condition $$u(0)=2$$, the exact solution is $$u(t) = 6-2 \sqrt{3} \tan \left(\frac{t}{2 \sqrt{3}}+\tan ^{-1}\left(\frac{2}{\sqrt{3}}\right)\right).$$ The tangent function has a singularity at $$\frac{\pi}{2}$$ which occurs when $$t \approx 2.47$$. solve_ivp cannot handle singularities (nor should it be expected to).
You will have to decide if your ODE is correct and whether it should have this singularity. Even without it, some care will be needed to handle the discontinuity at $$t=5$$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 5, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8398239016532898, "perplexity": 919.3354510676575}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662510097.3/warc/CC-MAIN-20220516073101-20220516103101-00548.warc.gz"} |
http://math.stackexchange.com/questions/88870/prove-int-limits-01-fracta-1et-dt-converges-for-0a-t1 | # Prove: $\int \limits_{0}^{1}\frac{t^{a-1}}{e^t} dt$ converges for $0<a,t<1$
How do I prove that $$\int_{0}^{1}\frac{t^{a-1}}{e^t} dt$$ converges for $0<a,t<1$ I tried to bound it with couple of other integrals, for example: $$\int_{0}^{1}\frac{1}{te^t} dt$$ but it diverges.
My idea is that $t^{a-1}$ is bigger than 1 and that $$\int_{0}^{1}\frac{1}{e^t} dt$$ does converge..
Thanks for the help.
-
How about bounding with $t^{a-1}e^{-t} \le t^{a-1}$ ? Isn't that sufficient ? – Joel Cohen Dec 6 '11 at 12:52
You basically answered the question already: $a > 0$ means that $a-1 > -1$. You should be able to convince yourself that $\int \limits_0^1 t^p\,dt$ converges when $p > -1$. As you already noted, the $e^{-t}$ term doesn't cause any trouble. (Here's a better hint: can you bound $e^{-t}$ by some constant $C$ on the interval, so that $\int \limits_ 0^1 t^{a-1}e^{-t}\,dt \leq C\int_0^1t^{a-1}\,dt$ ? )
-
Later note: My first guess, below, as to what was desired, appears to be incorrect. See the further "later note" below.
If you mean "as $a\to\infty$" (the only guess I've got), then observe that $$0 < \int_0^1 \frac{t^{a-1}}{e^t}\;dt < \int_0^1 t^{a-1}\;dt$$ and let $a\to\infty$.
If you didn't mean as $a\to\infty$, then you'd better re-write your question to make it clear what you meant.
Later note: One of your other question suggests something about what you meant. Your phrase "for $0 < a,t<1$" is confusing. $a$ is a parameter that remains fixed as $t$ goes from $0$ to $1$. The variable $t$ is already "bound" by the expression $\int_0^1\cdots\cdots dt$, and anything that binds the variable $a$, such as "for $0 Let's try again: $$0<\int_0^1\frac{t^{a-1}}{e^t}\;dt < \int_0^1 t^{a-1}\;dt$$ and this is finite if$-1<a-1<0\$.
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.955694317817688, "perplexity": 281.8639962491634}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246636104.0/warc/CC-MAIN-20150417045716-00244-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.ee.iitb.ac.in/~trivedi/LatexHelp/latexsample.htm | Sample Latex Files:
I am distributing sample Latex files for the Report, Slides and Paper in one column and two column format. It will help you in your documentation in Latex.
• Latex files for Report writing:
For writing reports, we can make a latex file can be considered as main file which takes input from other files. The format of this Latex file is given here (Download). It takes inputs from various files like for an Abstract, it will take input from file abstract.tex. For chapters it will take input from different Latex files meant for that chapter only. For an example, the command \input{Algorithm.tex} will take input from file Algorithm.tex. For taking input from files, there is a command included in main Latex file. This command may be either \input{File_Name.tex} or \include{File_Name}. For an examplem, my Latex main file takes abstract.tex, Introduction.tex, Parallel.tex, Implementation_Details.tex, Performance.tex, Algorithm.tex, FutureWork.tex, acknow.tex and ref.tex. These Latex files need some other files like figures (fig1.ps, fig2.ps, fig3.ps, fig4.ps, fig5.ps, fig6.ps, fig7.ps, arch.ps, lgrain.ps and iitlogo1.ps) and other files (iitblogo.mf and iitblogo.tfm). The main Latex file (here in this case is Report.tex) can be compiled by giving following command:
latex Main_file_name.tex
and it generates a file named as Main_file_name.dvi. By using this, we can produce postscript file by giving the following command on the command prompt:
dvips -o Main_file_name.ps Main_file_name.dvi
For my files, I issued the command latex Report.tex. It generated a file named as Report.dvi. Using this file, a postscript file Report.ps was produced by using following command:
dvips -o Report.ps Report.dvi
Thus, I generated the postscript file of my report using the above given procedure. For an example purpose, I am giving postscript file of my report (Download).
NOTE: Please compile latex file two times so that the list of figures, list of tables and table of content can be appeared in main postscript file.
For beginners: Copy all the Latex files (having an extension tex) and files having extensions tfm and mf in a directory. Now, change the text of the Latex files at all the places you want. Now compile it using the above defined procedure and generate postscript file. You can view this postscript file using Ghostview in the following way:
gv File_Name.ps
e.g. gv Report.ps
You can also view the file having an extension dvi by using xdvi shown below:
xdvi File_Name.dvi
e.g. xdvi Report.dvi
• Latex file for Slides Preparation
For preparing slides for the presentation, one can use and modify the sample template file (Download). During compilation of this template Latex file, it is required to have these files also (fig1.ps, fig2.ps, fig3.ps, fig4.ps, fig5.ps, fig6.ps, fig7.ps, arch.ps, lgrain.ps and iitlogo1.ps). The final postscript file look like this one (Download). Compilation method is same as that described in previous section. For generating colour slides in postscript format, a sample template file (Download) is given. One can study this and change it according to one's need. The files needed during compilation of this colour template file are iitlogo1.ps, Multidec.ps, original.ps, part.ps, part1.ps and part2.ps. You can compile the template file with the files needed and then view the postscript file using GhostView. You are also suggested to do some experiments with the template file to understand the usage of commands.
• Latex file for writing article in one column format
For writing an article in one column format, one can use and modify the sample template file (Download). It requires Multidecomp.ps and fig6.ps during compilation. The title page of the article is also given (Download). The postscripts of the article (Download) and title page (Download) are also given. One can study the template file of the article and know about how to prepare an article in one column format.
• Latex file for writing article in two column format
For writing an article in two column format, one can use and modify the sample template file (Download). It requires Multidecomp.ps and fig6.ps during compilation. The postscripts of the article (Download) is also given. One can study the template file of the article and know about how to prepare an article in two column format. It may be asked when writing an article in two column format that the two columns on the last page should be of equal length. How to do this? One can do this by modifying the last Latex file fed to the main Latex file or the last section in an article. Here in this template file, the last section used is References. Thus, I made changes in the section References. I used Bibtex to generate bibliography. Thus, I did changes in file name having an extension bbl. Here it is hpcPaper2Col.bbl. The modified file is hpcPaper2ColMod.bbl. The changes introduced to the hpcPaper2Col.bbl file is putting \vspace*{6.5in} and \pagebreak at the appropriate place. Here it is noted that the size field (6.5in) in \vspace*{6.5in} command can be changed according to the need. You can adjust size field parameter in the way you want. If there is only one Latex file having all the documentation then these commands can be put in between the text of the last section at an appropriate place.
In two column format, page wide figures and page wide tables can also be incorporated using the \begin{figure*} and \begin{table*} invironment. A detailed help is given in the section Online Help. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8832079768180847, "perplexity": 3115.6180575233543}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917120101.11/warc/CC-MAIN-20170423031200-00608-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://codedump.io/share/VtOcfBxGv6Jb/1/how-to-handle-the-bug-in-maple-latex-command | Bravo Young - 7 months ago 31
LaTeX Question
# How to handle the bug in Maple latex command?
In Windows 7, I need to use Maple to export the Tex code into a text file.
In Command-line Maple, I type
latex(LambertW(x), "C:/Users/Bravo/Desktop/out.txt");
to do this, but the result is:
{\rm W} \left(x\right)
That is not right, why is this ? Is there any method to solve this problem ?
Yeah, that is a bug in Maple. You can try latex(subs(LambertW=lambertW,erf=Erf,arctanh=Artanh,LambertW(x))); | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9910986423492432, "perplexity": 1847.7414142996959}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917118713.1/warc/CC-MAIN-20170423031158-00555-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://mathhelpforum.com/algebra/206959-inequality-word-problem.html | 1. ## Inequality word problem
Given that a<b<c<d<e and p=abcde, a, b, c, d and e are positive integers. When just one of these integers is increased by 1 and multiplied by the other four, the new product is n. The difference (n - p) is greatest when which of the five integers is increased by 1?
How can I prove that it is a?
2. ## Re: Inequality word problem
Suppose that the numbers are r,s,t,u,v, where one of these is a second is b and so on, p = rstuv.
Increase r by 1 and you have (r+1)stuv = rstuv + stuv, so n - p = (r+1)stuv - rstuv = stuv.
n - p will take its greatest value when stuv takes its greatest value and that will be when s,t,u and v are the four greatest numbers (and r is the smallest).
3. ## Re: Inequality word problem
So how did we use the given inequality? I still don't see it how to prove it (I followed everything except the very end) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8090367317199707, "perplexity": 860.9230116144631}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982297973.29/warc/CC-MAIN-20160823195817-00183-ip-10-153-172-175.ec2.internal.warc.gz"} |
http://nrich.maths.org/4987/index?nomenu=1 | $AB$ is a diameter of a circle of radius 1 cm. Two circular arcs of equal radius are drawn with centres $A$ and $B$. These arcs meet on the circle, as shown. What is the shaded area? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9685213565826416, "perplexity": 87.20894582543707}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471983019893.83/warc/CC-MAIN-20160823201019-00287-ip-10-153-172-175.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/123859-proof-linear-independence-print.html | # Proof of linear independence
• January 14th 2010, 09:00 PM
Mollier
Proof of linear independence
problem:
(a)
Prove that the four vectors
$x=(1,0,0),\;y=(0,1,0),\;z=(0,0,1)\;u=(1,1,1)$ in $\mathbb{C}^3$ form a linearly dependent set, but any three of them are linearly independent.
(b)
$x(t)=1,\;y(t)=t,\;z(t)=t^2,\;u(t)=1+t+t^2$, prove that $x,y,z,u$ are linearly dependent, but any three of them are linearly independent.
attempt:
(a) To prove that the four vectors are linearly dependent, I guess I could set up four linear equations and solve them.
But I'm thinking it should be enough to show that:
$x+y+z-u=0$.
The thing I do not understand is the "any three of them are linearly independent" part.
How do I go about proving that without checking all combinations?
(b) $x+y+z-u=0$. Again, I don't know how to prove the "any three of them" part without checking all combinations.
Thanks!
• January 14th 2010, 10:04 PM
Black
For both (a) and (b), you're pretty much spot on for proving linear dependence.
For (a), x, y, and z being independent is clear. Therefore, we need to show that u and any two of x,y, and z are linearly independent. Let $x=\mathbf{e}_1, y=\mathbf{e}_2, z=\mathbf{e}_3$ and let $i,j \in \{1,2,3\}$, where $i \not= j$. If we have (for $a,b,c \in \mathbb{C}$)
$a\mathbf{e}_i+b\mathbf{e}_j+cu=0$ ,
then no matter what i and j are, we will always end up with the following system of equations:
$a+c=0$
$b+c=0$
$c=0$,
which implies $a=b=c=0$.
Working out part (b) is very similar.
• January 14th 2010, 11:37 PM
Mollier
Quote:
Originally Posted by Black
For (a), x, y, and z being independent is clear. Therefore, we need to show that u and any two of x,y, and z are linearly independent.
Can I use the statement that "x,y,z being independent is clear" in a proof?
As for (b), is this ok:
Let $x=t^0,\;y=t^1,\;z=t^2,\;i,j=\{0,1,2\},\;i\neq j$
If for $a,b,c \in \mathbb{C}$:
$at^i+bt^j+cu=0$
then for all i and j we get a=b=c=0.
Thanks.
• January 15th 2010, 03:59 AM
Black
Quote:
Originally Posted by Mollier
Can I use the statement that "x,y,z being independent is clear" in a proof?
I guess it depends on the teacher that's grading your work. You should prove that they are independent (to be on the safe side), but it's very straightforward.
Quote:
Originally Posted by Mollier
As for (b), is this ok:
Let $x=t^0,\;y=t^1,\;z=t^2,\;i,j=\{0,1,2\},\;i\neq j$
If for $a,b,c \in \mathbb{C}$:
$at^i+bt^j+cu=0$
then for all i and j we get a=b=c=0.
Thanks.
Yep, pretty much. No matter what i and j are, when you group the like terms together and compare coefficients, you'll end up with the same system of equations as part (a).
• January 15th 2010, 04:40 AM
Raoh
Quote:
Originally Posted by Mollier
problem:
(a)
Prove that the four vectors
$x=(1,0,0),\;y=(0,1,0),\;z=(0,0,1)\;u=(1,1,1)$ in $\mathbb{C}^3$ form a linearly dependent set, but any three of them are linearly independent.
(b)
$x(t)=1,\;y(t)=t,\;z(t)=t^2,\;u(t)=1+t+t^2$, prove that $x,y,z,u$ are linearly dependent, but any three of them are linearly independent.
attempt:
(a) To prove that the four vectors are linearly dependent, I guess I could set up four linear equations and solve them.
But I'm thinking it should be enough to show that:
$x+y+z-u=0$.
The thing I do not understand is the "any three of them are linearly independent" part.
How do I go about proving that without checking all combinations?
(b) $x+y+z-u=0$. Again, I don't know how to prove the "any three of them" part without checking all combinations.
Thanks!
hi
$u\in Span(z,y,x)$
and $(z,y,x)$ is a simple basis of $\mathbb{C}^3$ which means it's L.I.
• January 15th 2010, 04:47 AM
Raoh
Quote:
Originally Posted by Mollier
Can I use the statement that "x,y,z being independent is clear" in a proof?
As for (b), is this ok:
Let $x=t^0,\;y=t^1,\;z=t^2,\;i,j=\{0,1,2\},\;i\neq j$
If for $a,b,c \in \mathbb{C}$:
$at^i+bt^j+cu=0$
then for all i and j we get a=b=c=0.
Thanks.
hi
put,
$\lambda _0+\lambda _1t+\lambda _2t^2=0$
For $t=0$ ,you get $\lambda _0$.
Differentiate with respect to $t$,
$\lambda _1+2\lambda _2t=0.$
put $t=0$ and get $\lambda _1=0.$
....
...and you get $\lambda _2=0.$
• January 15th 2010, 04:57 AM
Mollier
Great stuff guys, thank you very much!
Edit-
Roah: just saw you last post, sweet! :) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 41, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8803802728652954, "perplexity": 580.3024095772374}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454702039825.90/warc/CC-MAIN-20160205195359-00223-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://it.cornell.edu/managed-servers/transfer-files-using-putty | # Transfer Files Using PuTTY
How to install the PuTTy Secure Copy client and use it to transfer files
PuTTY is the CIT-recommended application for secure file transfer using SCP between Windows clients and Windows or Unix servers. Its secure copy utility is called PuTTy Secure Copy Protocol (PSCP).
PSCP and PuTTY are available from PuTTY.org.
## Install PuTTY SCP (PSCP)
PSCP is a tool for transferring files securely between computers using an SSH connection. To use this utility, you should be comfortable working in the Windows Command Prompt.
1. Download the PSCP utility from PuTTy.org by clicking the file name link and saving it to your computer. (If you also want to use the PuTTY shell program, you can download and save putty.exe to your computer as well.)
2. The PuTTY SCP (PSCP) client does not require installation in Windows, but runs directly from a Command Prompt window. Move the client program file to a convenient location in your Programs folders and make a note of the location.
3. To open a Command Prompt window, from the Start menu, click Run.
In Windows 10, open the Start menu and type `cmd`. Click the Command Prompt search result item that appears.
1. A Command Prompt window will open. To be sure the utility launches correctly from any directory in the Command Prompt window, set up an environment path so your system knows where to look for it. You'll use the pscp.exe location that you made note of in Step 2. For example, if you've saved the pscp.exe file to the folder "C:\Program Files\PuTTy\", set up a path by entering `set PATH="%PATH%;%ProgramFiles%\putty"` at the prompt in the Command Prompt window.
2. Entering the path in this way only lasts for the duration of the current session (that is, while you have the Command Prompt window open). To set up an environment variable path permanently, open the System control panel in Windows and click Advanced system settings, then click Environment Variables. In the Environment Variables window, select Path from the list of User variables, then click Edit. (If no Path variable is listed, click New.)
3. In the Edit User Variable window, click New. Type or paste the directory path for the PSCP utility you noted in Step 2 (for example, `C:/Program Files/putty`) into the empty highlighted new line item.
4. Click OK to save the new entry, then click OK again to close the Environment Variables window. The PSCP program file location is set up in your system and will not need to be entered each time you open a Command Prompt window.
Many users will not have sufficient administrative privileges to add or edit the Path environment variable permanently in their Windows System settings. In this case, contact the IT Service Desk for assistance in setting up PSCP, PuTTy, and Windows environment path variables.
## Transfer files using PSCP
1. Open the Command Prompt window, and if necessary set up your path variable as shown above in Step 4.
2. To copy the local file c:\documents\info.txt as user username to the server server.example.com with destination directory /tmp/foo, type at the prompt:
`pscp c:\documents\info.txt [email protected]:/tmp/foo/info.txt`
Review the complete documentation for PSCP and PuTTY on the PuTTY.org site. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8738336563110352, "perplexity": 3778.0371548257754}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676592387.80/warc/CC-MAIN-20180721051500-20180721071500-00189.warc.gz"} |
https://encyclopediaofmath.org/wiki/Large_deviations | # Large deviations
Jump to: navigation, search
Copyright notice
This article large deviation principle (=Large deviations) was adapted from an original article by francis michel comets, which appeared in StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies. The original article ([http://statprob.com/encyclopedia/.html StatProb Source], Local Files: pdf | tex) is copyrighted by the author(s), the article has been donated to Encyclopedia of Mathematics, and its further issues are under Creative Commons Attribution Share-Alike License'. All pages from StatProb are contained in the Category StatProb.
Large deviations
Francis Comets
Université Paris Diderot, France
http://www.proba.jussieu.fr/~comets/
Large deviations is concerned with the study of rare events and of small probabilities. Let $X_i, 1 \leq i \leq n,$ be independent identically distributed (i.i.d.) real random variables with expectation $m$, and $\bar X_n=(X_1+\ldots X_n)/n$ their empirical mean. The law of large numbers shows that, for any Borel $A \subset {\mathbb R}$ not containing $m$ in its closure, $P(\bar X_n \in A) \to 0$ as $n \to \infty$, but does not tell us how fast the probability vanishes. Large deviations state it is exponential in $n$, and give us the rate of decay. Cramér's theorem states that, \begin{equation*} P(\bar X_n \in A) = \exp -n\big(\inf\{I(x); x \in A\} + o(1)\big) \end{equation*} as $n \to \infty$, for all interval $A$. The rate function $I$ can be computed as the Legendre conjugate of the logarithmic moment generating function of $X$, \begin{equation*} I(x)=\sup\{ \lambda x - \ln E \exp( \lambda X_1); \lambda \in {\mathbb R}\}, \end{equation*} and is called the Cramér transform of the common law of the $X_i$'s. The natural assumption is the finiteness of the moment generating function in a neighborhood of the origin, i.e., the property of exponential tails. The function $I: {\mathbb R} \to [0, +\infty]$ is convex with $I(m)=0$.
$\bullet$ In the Gaussian case $X_i \sim {\cal N}(m, \sigma^2)$, we find $I(x)=(x-m)^2/(2\sigma^2)$;
$\bullet$ In the Bernoulli case $P(X_i=1)=p=1-P(X_i=0)$, we find the entropy function $I(x)=x \ln(x/p)+(1-x) \ln(1-x)/(1-p)$ for $x\in [0,1]$, and $I(x)=+\infty$ otherwise.
To emphasize the importance of rare events, let us mention a consequence, the Erdös-Rényi law: consider an infinite sequence $X_i, i \geq 1$, of Bernoulli i.i.d. variables with parameter $p$, and define $R_n$ the length of the longest consecutive run, contained within the first $n$ tosses, in which the fraction of 1's is at least $a$ ($a >p$). Erdös and Rényi proved that, almost surely as $n \to \infty$, $$R_n/ \ln n \longrightarrow I(a)^{-1} ,$$ with the function $I$ from the Bernoulli case above. Though it may look paradoxical, large deviations are at the core of this event of full probability. This result is the basis of bioinformatics applications like sequence matching, and of statistical tests for sequence randomness.
The theory does not only apply to independent variables, but allows for many variations, including weakly dependent variables in a general state space, Markov or Gaussian processes, large deviations from ergodic theorems, non-asymptotic bounds, asymptotic expansions (Edgeworth expansions), \ldots.
Here is the formal definition. Given a Polish space (i.e., a separable complete metric space) $\mathcal X$, let $\{ \mathbb{P}_n\}$ be a sequence of Borel probability measures on $\mathcal X$, let ${a_n}$ be a positive sequence tending to infinity, and finally let $I:{\mathcal X} \to [0,+\infty]$ be a lower continuous functional on $X$ which level sets$\{x: I(x)\leq a\}$ are compact for all $a <\infty$. We say that the sequence $\{ \mathbb{P}_n\}$ satisfies a large deviation principle with speed ${a_n}$ and rate $I$, if for each measurable set $E \subset X$ $$-\inf_{x \in E^\circ} I(x) \le \varliminf_n a_n^{-1} \ln\mathbb{P}_n(E) \le \varlimsup_n a_n^{-1} \ln\mathbb{P}_n(E) \le -\inf_{x \in \bar{E}} I(x)$$ where $\bar{E}$ and $E^\circ$ denote respectively the closure and interior of $E.$ The rate function can be obtained as \begin{equation*} I(x)= - \lim_{\delta \searrow 0} \lim_{n \to \infty} a_n^{-1} \ln\mathbb{P}_n( B(x, \delta)), \end{equation*} with $B(x, \delta)$ the ball of center $x$ and radius $\delta$. Large deviation theory allows for an abstract version of Laplace method for estimating integrals: Varadhan's lemma states that, for any continuous function $F: X \to {\mathbb R}$ with $$\lim_{M \to \infty} \limsup_{n \to \infty} a_n^{-1} \ln \int_{F(x)\geq M} e^{a_nF(x)} dP_n(x) = - \infty$$ (a bounded $F$ is fine), we have $$\lim_{n \to \infty} a_n^{-1} \ln \int_X e^{a_nF(x)} dP_n(x) = \sup_x \{F(x)-I(x)\},$$ and the sequence of probability measures $e^{a_nF} dP_n/ \int e^{a_nF} dP_n$ concentrates on the set of maximizers.
Sanov's theorem and sampling with replacement: let $\mu$ be a probability measure on a set $\Sigma$ that we assume finite for simplicity, with $\mu(y)>0$ for all $y \in \Sigma$. Let $Y_i, i \geq 1$, an i.i.d. sequence with law $\mu$, and $N_n$ the score vector of the $n$-sample, $$N_n (y) = \sum_{i=1}^n '''1'''_{y}(Y_i).$$ By the law of large numbers, $N_n/n \to \mu$ a.s. From the multinomial distribution, one can check that, for all $\nu$ such that $n \nu$ is a possible score vector for the $n$-sample, $$(n+1)^{-|\Sigma|} e^{-n H(\nu | \mu)} \leq P(n^{-1} N_n = \nu ) \leq e^{-n H(\nu | \mu)} ,$$ where $H(\nu | \mu)=\sum_{y \in \Sigma} \nu(y) \ln \frac{\nu(y)}{\mu(y)}$ is the relative entropy of $\nu$ with respect to $\mu$. The large deviations theorem holds for the empirical distribution of a general $n$-sample, with speed $n$ and rate $I(\nu)=H(\nu | \mu)$ given by the natural generalization of the above formula. This result, due to Sanov, has many consequences in information theory and statistical mechanics [DZ, dH], and for exponential families in statistics. Applications in statistics also include point estimation (by giving the exponential rate of convergence of $M$-estimators) and for hypothesis testing (Bahadur efficiency) [K], and concentration inequalities [DZ].
Consider now a Markov chain $(Y_n, n \geq 0)$. For simplicity we assume that it is irreducible with a finite state space $\Sigma$. We denote by $Q=(Q(i,j); i,j \in \Sigma)$ the transition matrix, and for any $V: \Sigma \to {\mathbb R}$, $Q_V(i,j)=Q(i,j)e^{V(j)}$. By Perron-Frobenius theorem, $n^{-1} \ln \sum_j Q_V^n(i,j) \to \ln \lambda_V(Q)$ as $n \to \infty$, with $\lambda_V(Q)$ the principal eigenvalue of the positive matrix $Q_V$. By the ergodic theorem, $N_n/n$ converges to the (unique) invariant law for $Q$. The law of the empirical distribution $N_n/n$ satisfes a large deviation principle with speed $n$ and rate $I_Q$ given by $$I_Q(\nu)= \sup_V \{ \sum_j V(j) \nu(j) - \ln \lambda_V(Q)\}$$ for any law $\nu$ on $\Sigma$.
Consider next a Markov process $(Y_t, t \in {\mathbb R}^+)$. We assume it is irreducible on the finite state space $\Sigma$, and denote by $a(i,j)$ the transition rate from $i$ to $j$ ($a(i,j)\geq 0$ for $i \neq j$, $\sum_j a(i,j)=0$). Then, similarly to the time-discrete case, the law of the empirical distribution $(t^{-1} \int_0^t {\mathbf 1}_j(Y_s)ds; j \in \Sigma)$ satisfes a large deviation principle with speed $t$ and rate $I_a$ given by $$I_a(\nu)= \sup_V \{ \sum_j V(j) \nu(j) - \lambda_V(a)\},$$ with $\lambda_V(a)$ the principal eigenvalue of the matrix $(a(i,j)+\delta(i,j)V(j))_{i,j}$. Now, assume in addition that the process is reversible with respect to a probability measure $\pi$, i.e. $\pi(i) a(i,j)= \pi(j) a(j,i)$ for all $i,j$. Then, $\pi$ is the invariant measure for the process and $\pi(i)>0$ for all $i \in \Sigma$. Using the variational formula for eigenvalues of symmetric operators, Donsker and Varadhan found that the rate function takes a simple form in the reversible case, $$I_a(\nu)= \frac{1}{2} \sum_{i,j} \pi(i) a(i,j) \left( \sqrt{ \frac{\nu(i)}{\pi(i)}} -\sqrt{ \frac{\nu(j)}{\pi(j)}} \right)^2 ,$$ that is the value of the Dirichlet form of the reversible process on the square root of the density of $\nu$ with respect to the invariant measure.
The Freidlin-Wentzell theory deals with diffusion processes with small noise, $$d X^{\epsilon}_t = b(X^{\epsilon}_t) dt + \sqrt{\epsilon}\; \sigma(X^{\epsilon}_t) dB_t\; , \qquad X^{\epsilon}_0=y.$$ The coefficients $b, \sigma$ are uniformly lipshitz functions, and $B$ is a standard Brownian motion. The sequence $X^{\epsilon}$ can be viewed as $\epsilon \searrow 0$ as a small random perturbation of the ordinary differential equation (ODE) $$d x_t = b(x_t ) dt\;,\qquad x_0=y.$$ Indeed, $X^{\epsilon} \to x$ in the supremum norm on bounded time-intervals. Freidlin and Wentzell have shown that, on a finite time interval $[0,T]$, the sequence $X^{\epsilon}$ with values in the path space obeys the LDP with speed $\epsilon^{-1}$ and rate function $$I_{0,T}(\phi)= \frac{1}{2}\int_0^T \sigma(\phi(t))^{-2} \Big(\dot{\phi(t)}-b(\phi(t))\Big)^2 dt$$ if $\phi$ is absolutely continuous with square-integrable derivative and $\phi(0)=y$; $I(\phi)= \infty$ otherwise. (To fit in the above formal definition, take a sequence $\epsilon= \epsilon_n \searrow 0$, and for $\mathbb{P}_n$ the law of $X^{\epsilon_n}$.) Note that $I(\phi)=0$ if and only if $\phi$ is a solution of the above ODE. To follow closely any other path $\phi$ during a finite time $T$ is an event of probability $\exp \{-\epsilon^{-1} I_{0,T}(\phi)\}$ at leading order, and therefore is very rare for small $\epsilon$.
A simple case is $\sigma=1$ and $b(x)=- V'(x)$ with a smooth $V$; we view $V(x)$ as the height of $x$, and a key role is played by the local minima of $V$. With an overwhelming probability as $\epsilon \searrow 0$, the picture will be as follows in a generic situation. The process $X^\epsilon$ will stay close to the solution of the ODE starting from $y$, and will eventually come near the local minimum (say $z_0$) which attracts $y$, and stay around for times of order $\exp o(\epsilon^{-1})$. But, by ergodicity, it will leave the neighborhood of $z_0$ at some time, and, even more, it will visit all points: it is then important how these large deviations occur. Let $D$ be domain of attraction of $z_0$, $h$ be its depth (i.e., the height difference between $z_0$ and the lowest point on the boundary of $D$, that we call the lowest pass). Up to times of order $\exp (\epsilon^{-1} h)$, the process remains in the part of $D$ of relative height smaller than $h$, and will occupy this region with density proportional to $\exp -(\epsilon^{-1} V(\cdot))$. At some random time $\tau$ of order $\exp (\epsilon^{-1} h)$, it will leave $D$ through the lowest path, and fall down towards a new local minimum, following roughly a path of the ODE. The piece of path just before leaving $D$ is the time-reversed of an ODE path. The ratio of $\tau$ to its expected value converges to an exponential law.
The { Freidlin-Wentzell theory} has applications in physics (metastability phenomena, analysis of rare events) and engineering (tracking loops, statistical analysis of signals, stabilization of systems and algorithms) [FV, A, DZ, OV].
Acknowledgement: This article is based on an article from Lovric, Miodrag (2011), International Encyclopedia of Statistical Science. Heidelberg: Springer Science$+$Business Media, LLC
#### References
[A] Azencott, R. Grandes déviations et applications. (French) Eighth Saint Flour Probability Summer School---1978, 1--176, Lecture Notes in Math. 774, Springer, Berlin, 1980 [DZ] Dembo, Amir; Zeitouni, Ofer: Large deviations techniques and applications. Springer, New York, 1998. [DS] Deuschel, J.-D., Stroock, D. Large deviations. Academic Press, Inc., Boston, MA, 1989 [FK] Feng, J., Kurtz, T. Large deviations for stochastic processes. American Mathematical Society, Providence, RI, 2006 [dH] den Hollander, Frank: Large deviations. American Mathematical Society, Providence, RI, 2000. [FV] Freidlin, M. I.; Wentzell, A. D.: Random perturbations of dynamical systems. Springer-Verlag, New York, 1998. [K] Kester, A.: Some large deviation results in statistics. CWI Tract, 18. Centrum voor Wiskunde en Informatica, Amsterdam, 1985. [KL] Kipnis, C., Landim, C.: Scaling limits of interacting particle systems. Springer-Verlag, Berlin, 1999 [OV] Olivieri, Enzo; Vares, Maria Eulália: Large deviations and metastability. Cambridge University Press, 2005. [V] Varadhan, S. R. S.: Large deviations. Ann. Probab. 36 (2008), 397--419
How to Cite This Entry:
Large deviations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Large_deviations&oldid=38480 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9873657822608948, "perplexity": 271.4588941088171}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107880878.30/warc/CC-MAIN-20201023073305-20201023103305-00491.warc.gz"} |
https://forum.gwb.com/topic/1210-3-questions-on-act2/ | Geochemist's Workbench Support Forum
# 3 questions on ACT2
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Hi, I have 3 basic questions about ACT2:
(1) If I want to plot a ACT-ACT diagram on Cu, is that true that for the diagram species I can only input
total activity instead of molality? I see that it only gives me the choice of activity or log activity.
(2) If I input the total activity of 10^-3 for Cu, is the 10^-3 including both the dissolved and solid phases?
(3) How is the boundary between aqueous species and solid phases defined in ACT2? Garrels and
Christ, defined the boundary as the point where the “sum of the activities of the ions in equilibrium
with the solid exceeds some chosen value”. They chose 10^-6 as a default value on the basis that if it is less
than this value, the solid will tend to behave as an immobile constituent in the environment.
My question is whether the boundary is defined following Garrels and Christ (boundary at total dissolved Cu to be 10^-6)
or to be the line where mineral precipitation begins?
Thanks,
Peng
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Hi, I have 3 basic questions about ACT2:
Hi Peng:
1) Indeed, since this is an activity diagram, all species concentrations must be input as activities.
2) If you specify an activity for Cu++, you are constraining the non-ideal concentration (or activity) for the dissolved Cu++ species.
3) Any boundary in Act2 is simply the mass action equation between the species/minerals on either side of the boundary written in terms of the axis species. For example, on a Quartz solubility diagram with log SiO2 activity as the Y-axis and -log H+ activity (pH) as the X-axis, the boundary between dissolved silica and Quartz is expressed by the reaction:
Quartz = SiO2(aq)
Which has the mass action equation:
K = aSiO2(aq)
From the thermodynamic database, the log K value at 25C for this reaction is -3.9993. Substituting for K and aSiO2, and switching to log scale, we come up with the final equation as it's plotted in Act2:
Y = -3.999
Note that since the reaction is only written in terms of the Y-axis (ie. not in terms of H+ activity), there is a Y-intercept, but no line slope- that is, the line plots as a horizontal line on the activity diagram.
I hope this helps.
Regards,
Tom Meuzelaar
RockWare, Inc.
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Hi Tom,
If I set Cu activity in the system as 1e-6 and use "thermo" database to plot,
how are the boundaries of CuOH+ vs Tenorite and CuOH+ vs Cuprite defined?
Does the program assume the CuOH+ to be 1e-6 M when calculating the boundaries?
Or, it will speciate Cu++ and CuOH+ at certain pH value and used the real activity
of CuOH+ (say 1.5e-8 M) to calculate the boundaries?
Peng
Hi Peng:
1) Indeed, since this is an activity diagram, all species concentrations must be input as activities.
2) If you specify an activity for Cu++, you are constraining the non-ideal concentration (or activity) for the dissolved Cu++ species.
3) Any boundary in Act2 is simply the mass action equation between the species/minerals on either side of the boundary written in terms of the axis species. For example, on a Quartz solubility diagram with log SiO2 activity as the Y-axis and -log H+ activity (pH) as the X-axis, the boundary between dissolved silica and Quartz is expressed by the reaction:
Quartz = SiO2(aq)
Which has the mass action equation:
K = aSiO2(aq)
From the thermodynamic database, the log K value at 25C for this reaction is -3.9993. Substituting for K and aSiO2, and switching to log scale, we come up with the final equation as it's plotted in Act2:
Y = -3.999
Note that since the reaction is only written in terms of the Y-axis (ie. not in terms of H+ activity), there is a Y-intercept, but no line slope- that is, the line plots as a horizontal line on the activity diagram.
I hope this helps.
Regards,
Tom Meuzelaar
RockWare, Inc.
##### Share on other sites
Hi Tom,
If I set Cu activity in the system as 1e-6 and use "thermo" database to plot,
how are the boundaries of CuOH+ vs Tenorite and CuOH+ vs Cuprite defined?
Does the program assume the CuOH+ to be 1e-6 M when calculating the boundaries?
Or, it will speciate Cu++ and CuOH+ at certain pH value and used the real activity
of CuOH+ (say 1.5e-8 M) to calculate the boundaries?
Peng
Hi Peng:
You can look at the calculations behind an entire activity diagram by choosing the Run - View - .\Act2_output.txt menu option. I think this will answer all of your questions.
Regards,
Tom
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× You cannot paste images directly. Upload or insert images from URL. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8154908418655396, "perplexity": 3737.5163555779654}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499470.19/warc/CC-MAIN-20230128023233-20230128053233-00513.warc.gz"} |
http://math.stackexchange.com/questions/192806/how-to-prove-that-this-function-is-continuous-at-zero | # How to prove that this function is continuous at zero?
Assume that $g : [0, \infty) \rightarrow \mathbb R$ is continuous and $\phi :\mathbb R \rightarrow \mathbb R$ is continuous with compact support with $0\leq \phi(x) \leq 1$, $\phi(x)=1$ for $x \in [0,1]$ and $\phi(x)=0$ for $x\geq 2$.
I wish to prove that $$\lim_{x \rightarrow 0^-} \sum_{n=1}^\infty \frac{1}{2^n} \phi(-nx) g(-nx)=g(0).$$
I try in the following way. Let $f(x)=\sum_{n=1}^\infty \frac{1}{2^n} \phi(-nx) g(-nx)$ for $x \leq 0$.
For $\varepsilon>0$ there exists $n_0 \in \mathbb N$ such that $\sum_{n\geq n_0} \frac{1}{2^n} \leq \frac{\varepsilon}{2|g(0)|}$. For $x<0$ there exists $m(x) \in \mathbb N$, $m(x)> n_0$ such that $\phi(-nx)=0$ for $n>m(x)$.
Then $$|f(x)-f(0)|\leq \sum_{n=1}^\infty \frac{1}{2^n} |\phi(-nx)g(-nx)-\phi(0)g(0|= \sum_{n=1}^{m(x)} +\sum_{n\geq m(x)}$$ (because $\phi(0)=1$, $\sum_{n=1}^\infty \frac{1}{2^n}=1$). The second term is majorized by $\frac{\varepsilon}{2}$ , but I don't know what to do with the first one because $m(x)$ depend on $x$.
-
Fyi, $g(-nx)$ is not defined when $n\in\mathbb{N}$ and $x>0$ since $g:[0,\infty)\to\mathbb{R}$. Perhaps you need to change something? – J. Loreaux Sep 8 '12 at 15:56
I think there is a problem with signs. All $-n x$ should read $n x$, all$x\le0$ should read $x\ge0$, $x<0$ should read $x>0$. – Hagen von Eitzen Sep 8 '12 at 15:58
I want to show left side continuity of $f$ at zer0. – L.T Sep 8 '12 at 16:02
Let $h(x)=\phi(x)g(x)$. Then $h\colon[0,\infty)\to\mathbb R$ is continuous and bounded by some $M$ and $h(x)=0$ for $x\ge2$. Given $\epsilon>0$, find $\delta$ such that $x<\delta$ implies $|h(x)-h(0)|<\frac\epsilon3$.
Then for $m\in \mathbb N$ $$\sum_{n=1}^\infty \frac1{2^n} h(nx)-h(0)=\sum_{n=1}^{m} \frac1{2^n} (h(nx)-h(0))+\sum_{n=m+1}^\infty \frac1{2^n} h(nx)-\frac1{2^m}h(0)$$ If $m<\frac\delta x$, then $$\left |\sum_{n=1}^{m} \frac1{2^n} (h(nx)-h(0))\right |<\sum_{n=1}^\infty \frac 1{2^n}\frac\epsilon3=\frac\epsilon3.$$ For the middle part, $$\left|\sum_{n=m+1}^\infty \frac1{2^n} h(nx)\right|<\frac M{2^m}$$ and finally $\left|\frac1{2^m}h(0)\right|\le \frac M{2^m}$. If $m>\log_2(\frac {3M}\epsilon)$, we find that $$\left|\sum_{n=1}^\infty \frac1{2^n} h(nx)-h(0)\right|<\epsilon$$ for all $x$ with $x<\min\{\delta,\frac Mm\}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9846696853637695, "perplexity": 104.8534327545626}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246658376.88/warc/CC-MAIN-20150417045738-00280-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://mathematica.stackexchange.com/questions/114188/is-it-the-correct-syntax-with-transformeddistribution | Is it the correct syntax with TransformedDistribution?
I define a function as f[x_] := ...;which is in fact the PDF of a random variable $X$. The PDF if given by $f_X(x)$.
Let $Y$ is a random variable, which is the square of the random variable $X$, i.e., $Y=X^2$.
The PDF of $Y$ is expressed as $f_Y(y)$.
Let $I$ is a function of $Y$, which we express in MMA as I = ... (not as I[y_]:=)
I need to evaluate
$\exp\left(-\mathbb{E}[I]\right)$
How do I do it?
• Please provide a complete minimal working example. "..." as a function body is useless to readers in general. – ciao May 3 '16 at 6:37
I think you almost did right, but you need to give the distribution rather than the PDF of a distribution when evaluating the expectation.
For example:
f[x_] := Sqrt[2]/Pi/(1 + x^4)
distX = ProbabilityDistribution[f[x], {x, -Infinity, Infinity}];
distY = TransformedDistribution[X^2, X \[Distributed] distX];
distI = TransformedDistribution[2 Y + 1, Y \[Distributed] distY];
Exp[-Expectation[II, II \[Distributed] distI]]
(* 1/E^3 *) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.978965163230896, "perplexity": 1014.1442916484965}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487629632.54/warc/CC-MAIN-20210617072023-20210617102023-00591.warc.gz"} |
https://math.stackexchange.com/questions/3053610/riccati-ode-solution-path | # Riccati ODE Solution Path
In the Ricatti Ordinary Differential Equation introduction in this video, the form of the general solution is stated as $$y = y_1 + u$$ and the Ricatti ODE becomes a Bernoulli ODE, which eventually becomes linear, while in this video, the stated general solution is $$y = y_1 + \frac{1}{v}$$, and the Ricatti ODE becomes a linear ODE directly, with no intermediate Bernoulli step. Is the difference in the assumed general solution form the reason for the presence or absence of the intermediate Bernoulli step? If not, why is $$y = y_1 + \frac{1}{v}$$ sometimes preferred, and what conditions determine whether the Ricatti ODE will go through Bernoulli form as an intermediate step, or will become linear immediately?
The answer to "Is the difference in the assumed general solution form the reason for the presence or absence of the intermediate Bernoulli step?" is yes. Why? The Riccati equation always has the form: $$y' = A(x) + B(x)y + C(x)y^2$$ We will write it like this: $$y' = A + By + Cy^2$$
And (as the video says) if you use the sustitution $$y = y_P(x) + u$$ where $$y_P(x)$$ is a particular solution (and known) of the Ricatti equation, you have: $$y'_P(x) + u' = A + B[y_P(x)+u] + C[y_P(x) + u]^2$$ $$y'_P(x) + u' = A + \left[By_P(x) + Bu\right] + \left[Cy_P(x)^2 + 2Cy_P(x)u + Cu^2\right]$$
But $$y_P(x)$$ is solution: $$y'_P(x) = A + By_P(x) + Cy_P(x)^2$$ Then if you substitute $$y'_P(x)$$ as $$A + By_P(x) + Cy_P(x)^2$$ in the Ricatti equation: $$A + By_P(x) + Cy_P^2 + u' = A + By_P + Bu + Cy_P^2 + 2Cy_Pu + Cu^2$$ $$u' = Bu + 2Cy_Pu + Cu^2$$ This is a Bernoulli equation, so that we have a linear equation, we must use the substitution $$\displaystyle v=u^{1-2}=\frac{1}{u}$$. If you can notice: $$v=\frac{1}{u}=\frac{1}{y-y_P(x)}$$ $$v=\frac{1}{y-y_P(x)}$$ $$y-y_P(x)=\frac{1}{v}$$ $$y=y_P+\frac{1}{v}$$ Then if you had started with the substitution $$\displaystyle y=y_P+\frac{1}{v}$$ you would have arrived at the linear equation to have missed the Bernoulli step, but it is exactly the same. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 21, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.986708402633667, "perplexity": 105.91330249809604}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670512.94/warc/CC-MAIN-20191120060344-20191120084344-00160.warc.gz"} |
https://www.physicsforums.com/threads/superposition-and-coherence-in-quantum-physics.376461/ | # Superposition and Coherence in quantum physics
1. Feb 8, 2010
### neelakash
I want a clarification in the idea of superposition principle.Perhaps,I should rather say that the distinction of superposition principle in QM and in classical physics.
Few weeks back,I was spending time with Gottfried's book. He explains the novelty of superposition principle with the help of a two state system.He does this as for one particle systems, the superposition principle has some classical analogue,he comments.You can take a look at the experiment described in his text; it is available at google book review (Kurt Gottfried,p14).It took some time for me to digest the experiment,and I have written something in my blog:http://www.gradqm.blogspot.com/
But my present question is how superposition principle for one particle system [QM] (say, double slit experiment with electrons) different from the superposition principle in classical optics.In classical optics,the superposition occurs between two waves (generated from the different parts of the wavefront of the same primary wave) and interference effect is produced by the variation in the cross term(real of course).
In double slit experiment in QM(with photon or electron or whatever),we can reach a situation where a single state is exhibiting interference.It is interpreted as the actual state is a linear combination of two base states:
$$\psi\ =\ a\ |1>\ +\ b\ |2>$$
And interference occurs between the two base states beyond the slit.Here, the probability amplitude $$\ <1|2>$$ is complex.It looks conceptually similar to the classical optics to me.Am I missing something?Or is it that the wavefunction is complex and that is making all the difference?
While going through the same,I also found that the idea of coherence is modified in QM.In particular,Gottfried comments that coherence has a richer meaning in multi-particle system,which he did not explain.I found some rigorous treatment in Ballentine's book (many body theory).But what I want is some simpler way to look at it.How to conceive the meaning of coherence in double slit experiment and how it is different from its meaning in classical optics.And what possible modification is needed when we consider multi-particle system.Can anyone shed some light?
2. Feb 8, 2010
The answer that comes to my mind goes along the lines that you cannot decohere a photon easily. If you send a particle through a slit and you manage to hit it with a photon along the way you can upset its phase in a random manner, destroying the coherence pattern. The second answer is "Aharonov Bohm effect"
3. Feb 8, 2010
### neelakash
Can you be a little elaborate?
The superposition looks the same to me in single particle interference whether in classical physics or in QM.My question is for single particle interference,are they the same? It is often said that superposition principle in QM is different than in classical optics.
What did you mean by the "2nd question"?
4. Feb 9, 2010
The second answer is also for the first question.
Particles don't interfere classically, waves do. The superposition principle is a fact about systems described by linear equations. If you want to get an intuitive feeling for quantum mechanical interference maybe you should read the third part of the Feynman lectures. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8478756546974182, "perplexity": 764.5798557650187}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267864139.22/warc/CC-MAIN-20180621094633-20180621114633-00633.warc.gz"} |
https://chem.libretexts.org/Courses/University_of_California_Davis/UCD_Chem_110A%3A_Physical_Chemistry__I/UCD_Chem_110A%3A_Physical_Chemistry_I_(Larsen)/Worksheets/09B%3A_Multi-Electron_Wavefunctions_(Worksheet) | # 9B: Multi-Electron Wavefunctions (Worksheet)
Name: ______________________________
Section: _____________________________
Student ID#:__________________________
Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.
## ...and then there were Four
Solving the Schrödinger's equation for the hydrogen atom results in three quantum numbers: the principal quantum number ($$n$$), angular quantum number ($$l$$), and the magnetic quantum number ($$m_l$$). These quantum numbers describe the size, shape, and orientation of the spatial properties of the wavefunction. However, experimental evidence suggests that an atomic wavefunction can hold no more than two electrons.
To distinguish between the two electrons in an wavefunction, we introduce a fourth quantum number called the spin quantum number ($$m_s$$) because electrons behave as if they were spinning in either a clockwise or counterclockwise fashion. One of the electrons in an orbital is arbitrarily assigned an $$m_s = +1/2$$ (called $$\alpha$$), the other is assigned a $$m_s = -1/2$$ (called $$\beta$$). Hence, while it takes three quantum numbers to define an orbital, it takes four quantum numbers to describe a spin-orbital.
### Q1
For the electron configuration described below, identify the four quantum numbers for each electron. Do you need more information?
## Spin-Orbitals
The wavefunctions obtained by solving the hydrogen atom Schrödinger equation are associated with orbital angular motion and are often called spatial wavefunctions, to differentiate them from the spin wavefunctions. The complete wavefunction for an electron in a hydrogen atom must contain both the spatial and spin components. We refer to the complete one-electron orbital as a spin-orbital and a general form for this orbital is
$| \varphi _{n,l,m_l , m_s} \rangle = \psi _{2,1,0,+½} = | \psi _{n,l,m_l} (r, \theta , \psi ) \rangle | \sigma ^{m_s}_s \rangle \label {8.7.1}$
A spin-orbital for an electron in the $$2p_z$$ orbital with $$m_s = + \frac {1}{2}$$, for example, could be written as
$| \psi _{2pz_\alpha} \rangle = \psi _{2,1,0,-½} = | \psi _{2,1,0} (r, \theta, \psi) \rangle | \alpha \rangle \label{8.7.2}$
We can combine the 3-D coordinates ($$(r, \theta, \psi)$$) for electron 1 as $$r_1$$ and separate this "spin-orbital" into spatial and spin components:
• Spatial component: $| 1s(1) \rangle = | \psi_{1s}(r_1) \rangle$
• Spin component: $|\alpha (1) \rangle \, \text{for } m_s=+\frac{1}{2}$ and $|\beta (1) \rangle \, \text{for } m_s=-\frac{1}{2}$
With the following known orthogonormality integrals:
$\langle1s(1)|1s(1)\rangle=1 \label{ON1}$
$\langle1s(1)|1s(2)\rangle=0 \label{ON2}$
$\langle\alpha |\alpha\rangle=\langle\beta |\beta \rangle=1 \label{ON3}$
$\langle\alpha |\beta\rangle=0 \label{ON4}$
### Q2
In your own words, explain what Equations \ref{ON1} - \ref{ON4} mean. What do the numbers in Equations \ref{ON1} - \ref{ON2} mean?
## Spin-orbitals
Consider the two single-electron spin-orbitals:
$| 1s\alpha (1) \rangle = \psi_{1,0,0,+½} \label{Wave1}$
and
$|1s\beta (1) \rangle = \psi_{1,0,0,-½} \label{Wave2}$
and the two-electron spin-orbitals
$|1s\beta (1) \rangle |1s\beta (2) \rangle \label{Wave3}$
### Q3
On the blank configurations below, draw configuration for the three sping-orbitals described above.
## Wavefunctions can be Linear Combinations of Spin-orbitals
Equation \ref{Wave3} is one example of two-electron wavefunction. We can construct other two-electron wavefunctions as linear combinations of spin-orbitals:
$\psi_1=|1s \alpha (1) |1s \beta (2) \rangle +|1s \beta (1) \rangle |1s \alpha(2)\rangle \label{Wave4}$
$\psi_2=|1s \alpha (1) |1s \beta (2) \rangle - |1s \beta (1)\rangle |1s \alpha(2)\rangle \label{Wave5}$
### Q4
Is $$\psi_{100½}=1s\alpha (1)$$ an appropriate representation for an electron in an atom? Justify your answer.
### Q5
How can you combine $$\psi_{100½}=|1s\alpha (1) \rangle$$ and $$\psi_{100-½}=|1s\beta (1) \rangle$$ to make an appropriate representation for a single electron in a hydrogen atom?
## Symmetry in Multi-Electron Wavefunctions
Quantum mechanics allows us to predict the results of experiments. If we conduct an experiment with indistinguishable particles, a correct quantum description cannot allow anything which distinguishes between them. For example, if the wavefunctions of two particles overlap, and we detect a particle, which one is it? The answer to this is not only that we do not know, but that we cannot know. Quantum mechanics can only tell us the probability of finding a particle in a given region. The wavefunction must therefore describe both particles. The Schrödinger equation is then
$\hat{H} \psi(r_1, r_2) = E \psi(r_1, r_2)$
where the subscripts label each particle, and there are six coordinates, three for each particle and $$\psi(r_1, r_2)$$ is a wavefunction in six dimensions which contains the information we can measure: the probability of finding particles at $$r_1$$ and $$r_2$$, but is cannot tell us which particle is which.
Hence, multi-electron electron wavefunctions must be totally antisymmetric with respect to interchange of any two electrons.
$\psi(r_1,r_2) = - \psi(r_2,r_1)$
#### Q6
Are the three two-electron wavefunctions in Equation \ref{Wave3}-\ref{Wave5} antisymmetric with respect to interchange of any two electrons?
## Slater Determinants
To avoid getting a totally different function when we permute the electrons, we can make a linear combination of functions. A very simple way of taking a linear combination involves making a new function by simply adding or subtracting functions. A linear combination that describes an appropriately antisymmetrized multi-electron wavefunction for any desired orbital configuration is easy to construct for a two-electron system. However, interesting chemical systems usually contain more than two electrons. For these multi-electron systems a relatively simple scheme for constructing an antisymmetric wavefunction from a product of one-electron functions is to write the wavefunction in the form of a determinant. John Slater introduced this idea so the determinant is called a Slater determinant.
The Slater determitant can be generalized to as many electrons as needed:
$\psi(1,2,...,N)=\dfrac{1}{\sqrt{N!}}\begin{vmatrix} \mu_1(1) &\mu_2(1) &\cdots &\mu_N(1) \\\mu_1(2) &\mu_2(2) &\cdots &\mu_N(2) \\\vdots &\vdots &\ddots &\vdots \\\mu_N(N) &\mu_N(N) &\cdots &\mu_N(N) \end{vmatrix} \label{slater}$
where $$\mu_i(i)$$ is an orthonormal spin-orbital involving both spatial and spin components.
### Q7
What is the value of the following Slater determinant wavefunction?
$\psi(1,2)=\begin{vmatrix} 1s\alpha(1) &1s\beta(1) \\1s\alpha(2) &1s\beta(2) \end{vmatrix}=$
### Q8
What happens to the determinant when we interchange spin-orbitals (columns)?
$\psi(1,2)=\begin{vmatrix} 1s\beta(1) &1s\alpha(1) \\1s\beta(2) &1s\alpha(2) \end{vmatrix}=$
### Q9
What happens to the determinant when we interchange electrons (rows)?
$\psi(2,1)=\begin{vmatrix} 1s\alpha(2) &1s\beta(2) \\1s\alpha(1) &1s\beta(1) \end{vmatrix}=$
### Q10
Are $$\psi(1,2)$$ and $$\psi(2,1)$$ normalized? If not, what is the normalization constant?
### Q11
What happens to the Slater Determinant wavefunction if two of the spin-orbitals are identical?. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8808662295341492, "perplexity": 624.8120259079624}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670921.20/warc/CC-MAIN-20191121153204-20191121181204-00469.warc.gz"} |
http://rspa.royalsocietypublishing.org/content/early/2010/08/03/rspa.2010.0171.short?rss=1 | Auxetic behaviour from connected different-sized squares and rectangles
Joseph N. Grima , Elaine Manicaro , Daphne Attard
Abstract
Auxetic materials exhibit the unusual property of becoming fatter when uniaxially stretched and thinner when uniaxially compressed (i.e. they exhibit a negative Poisson ratio; NPR), a property that may result in various enhanced properties. The NPR is the result of the manner in which particular geometric features in the micro- or nanostructure of the materials deform when they are subjected to uniaxial loads. Here, we propose and discuss a new model made from different-sized rigid rectangles, which rotate relative to each other. This new model has the advantage over existing models that it can be used to describe the properties of very different systems ranging from silicates and zeolites to liquid-crystalline polymers. We show that such systems can exhibit scale-independent auxetic behaviour for stretching in particular directions, with Poisson’s ratios being dependent on the shape and relative size of different rectangles in the model and the angle between them.
1. Introduction
Unless a material has a zero Poisson ratio, it will change its thickness when it is uniaxially stretched or compressed. In fact, conventional materials having a positive Poisson ratio are observed to get thinner when stretched and fatter when compressed in a uniaxial direction. Although such behaviour is assumed to be the norm, the classical theory of elasticity suggests that Poisson’s ratio need not be positive. In fact, Poisson’s ratio for isotropic three-dimensional materials ranges from −1 to 0.5.
In recent years, there were significant developments in materials that exhibit a negative Poisson ratio (NPR), commonly known as auxetic (Evans et al. 1991). Such developments include work on cellular systems (Gibson et al. 1982; Gibson & Ashby 1997), foams (Lakes 1987; Evans et al. 1994; Scarpa et al. 2004aBezazi & Scarpa 2006; Grima et al. 2009a), polymers (Caddock & Evans 1989; Evans et al. 1991, 1995; Baughman & Galvao 1993; Grima & Evans 2000a,b; Alderson et al. 2001; He et al. 2005a,b; Ravirala et al. 2005), metals (Baughman et al. 1998), silicates (Keskar & Chelikowsky 1992; Yeganeh-Haeri et al. 1992; Kimizuka et al. 2000; Alderson & Evans 2001, 2002; Alderson et al. 2004, 2005) and zeolites (Grima 2000; Grima et al. 2000, 2005b, 2007c, 2009b; Sanchez-Valle et al. 2005; Williams et al. 2007).
Auxeticity can bring about various enhanced properties that include high-energy absorption properties (Bezazi & Scarpa 2006; Alderson & Alderson 2007), increased indentation resistance (Lakes & Elms 1993; Alderson 1999; Evans & Alderson 2000), enhanced sound and vibration absorption properties (Scarpa & Tomlinson 2000; Scarpa & Smith 2004; Scarpa et al. 2004a,b, 2005) and the ability to form dome-shaped structures (Lakes 1987; Evans et al. 1991; Alderson 1999; Evans & Alderson 2000) and to act as smart filters (Alderson et al. 1998a, 2001; Grima et al. 2000; Ravirala et al. 2005).
Research suggests that auxeticity can be explained in terms of geometric models, which describe how particular geometric features in the micro- or nanostructure of the materials deform when they are subjected to uniaxial loads. Auxeticity is also known to be a scale-independent property, which means that the same ‘geometry/deformation mechanism’ may be found to operate in systems ranging from macroscale to nano (atomic) scale. As a result of this, research in the field of auxetics often focuses on the analysis of mechanistic models, which result in auxetic behaviour, and in recent years, various geometry-based models that lead to auxeticity by deforming through particular deformation mechanisms have been proposed. These include models based on conventional honeycombs deforming through stretching (Evans et al. 1995; Masters & Evans 1996), re-entrant honeycombs deforming through hinging and/or flexure (Gibson et al. 1982; Evans et al. 1995; Masters & Evans 1996; Gibson & Ashby 1997), chiral honeycombs (Prall & Lakes 1997; Grima 2000; Spadoni et al. 2005; Grima et al. 2008a,b) and two-/three-dimensional rotating rigid/semi-rigid units (Grima et al. 1999, 2004, 2005a,b, 2006, 2007a, 2008a,b; Grima 2000; Grima & Evans 2000a; Ishibashi & Iwata 2000; Alderson & Evans 2001, 2002; Alderson et al. 2004, 2005; Attard & Grima 2008; Attard et al. 2009a,b). In addition, at smaller length scales, in particular, the micro-, meso- and nanoscale, other effects such as defects and disorder may become important and may affect auxeticity (Gaspar et al. 2003; Gaspar 2008; Horrigan et al. 2009), although in several cases, ordered macroscale models have been found to be very useful to account for observed auxeticity qualitatively and/or quantitatively.
Rotating polygons were first studied by Grima et al. (1999), who studied rotating square structures and used them to explain the NPR in various zeolites. This area of research was further developed through the study of auxeticity of rotating congruent rectangles (Grima et al. 2004, 2005a,b), equilateral triangles (Grima & Evans 2006), rhombi (Attard & Grima 2008; Grima et al. 2008a,b) and parallelograms (Williams et al. 2007; Grima et al. 2008a,b; Attard et al. 2009a). In the case of rotating squares, Poisson’s ratio was shown to be −1, irrespective of the direction of loading and dimensions of the squares. This work was of particular significance in view of the many naturally occurring crystalline materials, which have geometric features similar to the ‘rotating squares’ model, including the zeolite natrolite, which was recently confirmed to be auxetic (Sanchez-Valle et al. 2005; Grima et al. 2007c, 2009b; Williams et al. 2007). It was also shown that rectangles of the same shape and size can be connected through hinges at their vertices in two different ways: type I, in which the rectangles are connected in such a way that the empty spaces form rhombi, or type II, in which the empty spaces are parallelograms. Grima et al. (2004, 2005a,b) have shown that the type II rectangles are two-dimensionally isotropic with a Poisson ratio of −1, whereas the type I rectangles are anisotropic with a Poisson ratio being also dependent on the shape of the rectangles and the hinging angle between adjacent rectangles. Once again, such patterns and deformation mechanisms can be found in naturally occurring auxetics, where, for example, the type II rotating ‘rectangles model’ has been shown to be manifested in the naturally occurring silicate α-cristobalite.
Although this work had marked an important step forward as it highlighted the role of shape, size and connectivity of rectangles on the mechanical properties, in particular, Poisson’s ratios, it was limited due to the fact that all of the rectangles were required to be of equal size and thus could only be applied to very particular and highly idealized systems. In particular, one may consider a more general system that is made from two types of rectangles of dimensions a×b and c×d, which may be denoted by [a×b,c×d], as illustrated in figure 1, of which Grima’s type I and type II ‘rotating rectangles’ models are particular cases. (The type I systems may be achieved by letting c=b and d=a, i.e. structure [a×b,b×a], whereas the type II system is achieved by letting c=a and d=b, i.e. structure [a×b,a×b].)
Figure 1.
The system made from connected different-sized rectangles of dimensions a×b and c×d (denoted by [a×b,c×d]) discussed in this paper. Note that this system may be described in terms of two unit cells (UC1 and UC2) and in the derivation presented here, the system is aligned in the Ox1Ox2 plane in such a way that the side of the unit cell UC2 of length l2 is always parallel to the Ox2 direction. Grey dashed lines, UC1 and long dashed lines, UC2.
In view of all this, here we present a model that can predict the behaviour of a system made from rigid rectangles of dimensions a×b and c×d connected together at their vertices through simple hinges, as illustrated in figures 1 and 2 with the aim of predicting the extent of auxeticity of such systems. Special cases arising from such systems are also discussed.
Figure 2.
Illustration of the type of deformations that may be obtained from rigid rectangles connected together at their vertices through flexible hinges as a result of uniaxial loading in the Ox1 direction. Note that as the structure is stretched in the Ox1 direction, the rectangles rotate relative to each other and extend in the Ox2 direction, hence the NPR. This is also accompanied by a shearing of the structure. An animation of this figure is also provided (Anim-2.gif, electronic supplementary material).
2. Analytical model
A space-filling tessellation is formed by connecting each a×b rectangle to four c×d rectangles, as shown in figure 1. As illustrated in figure 2 (see animation in the electronic supplementary material), stretching of such systems in particular directions will result in a relative rotation of the rectangles. This may result in a more open structure (hence the NPR), if one assumes that the different-sized rectangles are perfectly rigid, but are connected to each other through simple hinges, which only permit relative rotation of the rectangles. Here, it should be noted that although this paper derives Poisson’s ratio in the setting of a uniaxial stretching/compression test, Poisson’s ratio, like any other elastic constant, will have an effect on any stress/strain state of the system.
Note that although the rectangles can be connected in two different ways such that the resulting empty spaces between the rectangles are either parallelograms of dimensions a×c and b×d or of a×d and b×c, these two cases are obviously identical as the length of the sides is chosen arbitrarily, with the result that one need not speak about type I rectangles or type II rectangles in such cases.
This tessellation illustrated in figure 1 can be described by either of the two unit cells, which have a different orientation in the global Ox1Ox2 coordinate system and are being highlighted as ‘UC1’ and ‘UC2’. Note that the UC2 unit cell is the smallest unit cell and contains only two rectangles, one of each type. In contrast, the unit cell UC1 contains four rectangles, i.e. two of each type of rectangles. Also note that these two unit cells are related such that the diagonal of the unit cell UC2 coincides with one of the sides of the unit cell UC1. In this study, the mechanical properties will be described using the unit cell UC2.
In our derivation, it shall be assumed that the structure is aligned in the Ox1Ox2 space in such a way that the unit cell side of length l2 is always parallel to the Ox2 direction, whereas the other unit cell side of length l1 can assume any direction. Under such assumptions, the projections of the parallelogramic unit cell UC2 in this orientation along the Oxi directions are given by 2.1 and 2.2 where X11 is the projection of the unit cell in the Ox1 direction, X22 is the projection of the unit cell in the Ox2 direction and α12 is the internal angle of the unit cell (α12=90−r+s, where r and s are the angles shown in figure 1). The lengths l1 and l2 and the angle α12 may be expressed in terms of geometric parameters a, b, c and d (the side lengths of the rectangles, which are assumed to be constants in this derivation) and the angle θ (the angle between the rectangles, which is a variable), and are given by 2.3 2.4and 2.5 Note that in the general case, the internal angle of the unit cell α12 is dependent on θ. This implies that, in general, the unit cell shears upon deformation, i.e. a change in θ as a result of an applied stress in the Ox1 or Ox2 direction will result in strains in the Ox1 and Ox2 directions as well as a shear strain. In two dimensions, the mechanical properties of such system can be described by a 3×3 compliance matrix S, which relates the applied stress σ to the resultant strain ε according to the following relationship: 2.6 where S is of the form (Daniel & Ishai 1994) 2.7 where ν12 and ν21 are Poisson’s ratios for loading in the Ox1 and Ox2 directions, respectively, E1 and E2 are Young’s moduli in the Ox1 and Ox2 directions, respectively, and G12 is the shear modulus in the Ox1Ox2 plane. η13, η31, η23 and η32 are the shear coupling coefficients defined as 2.8 where dγ is the shear strain and dεi are the normal strains in the Oxi directions (i=1,2).
In general, the unit cell may shear upon application of only normal strains in the Oxi directions (i=1,2), a property that results in non-zero shear coupling coefficients (cases in which the internal angle α12 of the unit cell is dependent on θ), such that, in general, one needs to derive six independent elastic constants (S is a symmetric matrix), e.g. Ei, νij, G12 and ηij, to fully characterize a system. However, this is not always the case, and there are certain configurations for which the internal angle of the unit cell is independent of θ, in which cases, the coupling coefficients have a null value. For example, there is no shearing of the unit cell in the type II rectangles [a×b,a×b], a system that has isotropic ν12=−1 (Grima et al. 2004, 2005a,b).
(a) Strains in the Oxi directions and on-axis Poisson ratios
Poisson’s ratio for loading in the Oxi direction is given by 2.9 where dεi is an infinitesimally small strain in the Oxi direction given by the ratio of the infinitesimally small change in the unit cell dimension dXii to the unit cell dimension Xii 2.10 Assuming that the only deformation mechanism is hinging, the geometry of the systems is dependent on the single variable θ and hence the strains can be re-written as 2.11
By substituting for l1, l2 and in equations (2.1) and (2.2) and differentiating with respect to θ, the strains along the Ox1 and Ox2 directions are found to be 2.12 and 2.13
Hence, Poisson’s ratio ν12 for loading in the Ox1 direction and Poisson’s ratio ν21 for loading in the Ox2 direction can be written as 2.14 where L is defined as 2.15
Note that Poisson’s ratio ν12(θ) has a removable discontinuity point (Thomson et al. 2001) at θ=90°, which can be removed by defining Poisson’s ratio at θ=90° equal to the limit L. This removable discontinuity corresponds to the physical situation in which the structure is locked: for θ between 0° and 90°, stretching in the Ox1 direction will not cause the structure to go past the point when θ=90°, as will be the case when θ is between 90° and 180° for compression, which highlights the fact that, as discussed elsewhere (Alderson et al. 1997; Smith et al. 1999), the expressions for Poisson’s ratios as a function of θ should not be treated as analogous to relationships of Poisson’s ratios with strain.
From equation (2.14), it is important to note that the on-axis Poisson ratios, ν12 and ν21 (for orienting the structure as described earlier) are always negative as a, b, c, d and (for 0<θ<180) are all positive so that the resulting Poisson ratio is negative.
(b) On-axis Young’s moduli
Young’s moduli of the structure can be derived through an energy-conservation approach. As we are assuming that the only deformation mechanism is hinging, the stiffness in the structure is solely due to the stiffness of the θ-hinges, which may be described through the stiffness constant Kh and defined as 2.16 where w is the work done at each hinge in changing the angle θ by dθ. As there are four θ-hinges in each unit cell, the total work done is 2.17
The strain energy due to an infinitesimally small strain dεi in the Oxi direction is given by 2.18 where Ei is Young’s modulus of the structure along the Oxi direction.
From the principle of conservation of energy, 2.19 where V is the volume of the unit cell given by 2.20 and z is the out-of-plane thickness of the rectangles.
By equating the two expressions for U, Young’s moduli are found to be 2.21 and 2.22
(c) On-axis shear strain
As already stated, in general, the unit cell shears upon deformation as the internal angle α12 is dependent on θ. This will result in a shear strain dγ, which can be defined in terms of dθ as follows (Grima 2000; Grima et al. 2007b): 2.23 where 2.24
Substituting for X11, X12, and in equation (2.23) and simplifying, the shear strain can be written as 2.25
(d) On-axis shear modulus
The shear modulus G12 can be derived using a similar approach used to derive Young’s modulus. The shear strain energy due to an infinitesimally small shear strain dγ is related to the total energy U stored in the system through the following equation: 2.26 such that the shear modulus may be written as 2.27
(e) On-axis shear coupling coefficients
The shear coupling coefficients ηij (defined by equation (2.8)) are given by 2.28
(f) Off-axis properties
The off-axis mechanical properties of the general rotating different-sized rectangles structure may be computed using standard axis-transformation techniques (Nye 1957). In particular, Poisson’s ratio for loading in any arbitrary direction in the xy plane (specifically at an angle +ξ to the x-axis) is given by 2.29 where 2.30
3. Discussion
The model presented earlier, which was validated using the Empirical modelling using dummy atoms (EMUDA) methodology, as described by Grima et al. (2005c), suggests that the systems presented here can exhibit a wide range of Poisson’s ratios and moduli, which can be fine-tuned to particular pre-desired values through careful choice of the geometric parameters a, b, c, d and θ. This is very significant as it provides us with a key to understand better the requirements for maximizing auxeticity for any particular system.
Let us now present some important observations that apply to the general model presented here (and hence also to all the special cases discussed in appendix A of the electronic supplementary material) by discussing in more detail the significance of equations (2.14) and (2.30) obtained in §2 for the on-axis and off-axis Poisson ratios of the generalized model. To facilitate the discussion, plots of Poisson’s ratios for typical systems are plotted in figure 3 (on-axis Poisson’s ratio ν12 versus angle θ between rectangles) and in figure 4 (off-axis Poisson’s ratio versus direction of loading ξ). In figure 3, a selection of representative ratios for a : b and c : d was made. These include the ratios 2 : 1, 1 : 2, 3 : 4, 4 : 3, 2 : 3 and 3 : 2. To fully exploit all possible combinations, the ratio a : b : c : d was considered, and while fixing the ratio for a : b, the ratio for c : d was varied. Note that if the ratio of a : b is interchanged with that for c : d (i.e. a : b : c : d= 2 : 1 : 3 : 4 and 3 : 4 : 2 : 1), this would result in the same structure, i.e. one would obtain the same Poisson ratio plots.
Figure 3.
Typical plots of the on-axis Poisson ratio ν12 (ordinate) against θ° (abscissa) for the general rotating different-sized rectangles structure [a×b,c×d].
Figure 4.
Plots of the off-axis Poisson ratio (ordinate) against the off-axis angle ξ° (abscissa).
Figure 3 clearly illustrates that Poisson’s ratio ν12 is always negative for loading on-axis, irrespective of the size and the degree of openness of the systems. However, the exact value of Poisson’s ratio will depend on the geometry of the system. It should also be noted that as ν21=(ν12)−1, ν21 is also always negative for all values of θ.
Moreover, as θ approaches 90°, the gradient of the curves for ν12 against θ (and also ν21 against θ) approaches zero, resulting in a ‘turning point’ at the point of the removable discontinuity (θ=90°), which corresponds to the point at which the rigid units cease to rotate. Referring to figure 5a, for any value of θ other than 90°, a stress applied along the Ox1 direction will result in a force component that is perpendicular to the side PQ, resulting in a moment that causes the rectangular units in the structure to rotate. This is not the case when θ=90° (figure 5b), at which point a stress applied along the Ox1 direction will result in a force parallel to the edge PQ, and hence there is no force component perpendicular to side PQ to create a moment. At this instant, the structure is ‘fully open’ and becomes locked (figure 5c), i.e. further tensile loading in the Ox1 direction will not result in any deformation if one assumes that the rectangles are perfectly rigid (i.e. no other mode of deformation can take place). Thus, loading in the Ox1 direction will not result in a change in the unit-cell dimensions, a property which in equations (2.12) and (2.13) is represented by the fact that there is a factor in the numerator of both expressions of dεi, which result in dεi=0 when θ=90°. The same applies when loading in the Ox2 direction. Moreover, it should be noted that it is still possible for the system to exist at angles θ in the range 90°<θ≤180°; however, the transition between the regions 0°θ<90° and 90°<θ≤180° requires forcing the rectangles to rotate past this barrier of θ=90° by, for example, rotating one of the rectangles or in some cases by shearing. Note that at θ=180°, the system is ‘fully closed’, i.e. the sides of the rectangles are touching such that unless overlapping of the rectangles is allowed, θ cannot be greater than 180°.
Figure 5.
(a) When θ<90°, applying a force along the x-axis will result in a component of force perpendicular to side PQ creating a moment. (b) When θ=90°, there is no component of the force perpendicular to side PQ such that the structure is locked and (c) the locked (fully open) structure.
Figure 3 also confirms that in the region 0°θ≤180°,
• — there is just one turning point at 90° in the relationship νij=νij(θ) and
• — the relationship is symmetric about the line θ=90°.
This means that unless νij=−1 for all θ (something that will occur in the special cases discussed subsequently and in appendix A of the electronic supplementary material), Poisson’s ratio will be dependent on the value of θ with maximum auxeticity either occurring at the points θ=0° and 180° or at θ=90°, depending on the geometry of the system. In fact, it may be shown that the nature of the turning point at 90° depends on whether ad3+cb3 is greater than bd(ab+cd) or smaller. In fact,
• — if ad3+cb3>bd(ab+cd), then the turning point for ν12 versus θ is a minimum turning point, which means that maximum auxeticity (i.e. a minimum point) for loading in the Ox1 direction occurs as θ=90° and
• — if ad3+cb3<bd(ab+cd), then the turning point for ν12 versus θ is a maximum turning point, which means that minimum auxeticity is obtained at θ=90° and that maximum auxeticity occurs at the boundaries, i.e. at θ=0° and at θ=180°.
It should also be noted that a maximum turning point for ν12 versus θ corresponds to a minimum turning point for ν21 versus θ since ν21=(ν12)−1. All this suggests that to maximize on-axis auxeticity for loading in the Ox1 direction, the structure must be such that the sides of the rectangles satisfy ad3+cb3<bd(ab+cd). In such cases, the on-axis Poisson ratio ν12 can assume values that are lower than −1. To maximize on-axis auxeticity for loading in the Ox2 direction, the structure must be such that the sides of the rectangles satisfy ad3+cb3>bd(ab+cd). In such cases, the greater the difference between the ratios a/b and c/d where bd, the greater the auxeticity.
If one looks at the dependence of Poisson’s ratio on the direction of loading, first of all one should note that for any particular structure, the plot showing variation of the off-axis Poisson ratio with ξ at any particular angle θ is the mirror image of the plot for the same structure when the angle is 180°θ. This is to be expected since the systems themselves are also mirror images of each other. Also the off-axis Poisson ratio is obviously periodic with a period of 180°. This is because loading at an angle +ξ is the same as loading at an angle of ξ−180°.
The plots of the off-axis Poisson ratio against ξ (figure 4) suggest that when a/b is almost equal to c/d, Poisson’s ratio is negative for loading at any off-axis angle, except as θ approaches 90°, in which case, there are both continuous and asymptotic transitions from negative to positive Poisson ratios and vice versa. Also interesting is the limiting case where a/b is equal to c/d (i.e. similar rectangles structure [(bc/db,c×d]), for which as discussed below and in appendix A of the electronic supplementary material, Poisson’s ratio is equal to −1 and such a structure is isotropic. As the difference between the values of a/b and c/d increases, continuous and asymptotic transitions start to be observed at smaller θ. The greater the difference between the ratios a/b and c/d, the smaller the θ, for which both positive Poisson ratios and NPRs are observed. The plots also suggest that for θ closer to 90°, the range of positive Poisson ratios increases.
Various special cases, which arise from particular combinations of size and aspect ratios of the rectangles in the generalized model, are presented in appendix A of the electronic supplementary material, in which we substitute into and simplify the equations derived in §2 to consider the special case of systems made from squares of the same size [a×a,a×a] (discussed earlier by Grima et al. 1999; Grima 2000; Grima & Evans 2000a); rectangles of the same size of type I [a×b,b×a] and type II [a×b,a×b] (discussed earlier by Grima et al. 2004, 2005a,b); systems composed of rotating different-sized squares [a×a,c×c] (figure 6a); systems composed of similar rectangles, where either a/b=c/d, i.e. structure [(bc/db,c×d] (figure 6d), or a/b=d/c, i.e. structure [(bd/cb,c×d] (figure 6c); systems made of rectangles of dimensions a×d and c×d[a×d,c×d] (figure 6e) and systems composed of rectangles and squares, i.e. structure [a×b,d×d] (figure 6b). Animations of these cases are available in the electronic supplementary material.
Figure 6.
Special cases of the different-sized rotating rectangles structure considered in appendix A of the electronic supplementary material: (a) rotating different-sized squares [a×a,c×c], (b) rotating squares and rectangles structure [a×b,d×d], (c) similar rectangles [bd/c×b,c×d], (d) similar rectangles [bc/d×b,c×d] and (e) rotating different-sized rectangles of dimensions a×d and c×d; [a×d,c×d] structure. Animations that illustrate how these systems deform when loading in the Ox1 direction are also supplied (see Anim-6a.gif, Anim-6b.gif, Anim-6c.gif, Anim-6d.gif and Anim-6e.gif, respectively, in the electronic supplementary material). Dotted line, UC1; dashed line, UC2.
In the case of the same-sized squares structure [a×a,a×a] and the type II rectangles [a×b,a×b], Poisson’s ratio is −1 for all values of θ (which means that the Poisson ratio of such a system is strain independent) and for all directions of loading (which suggest that such structures are isotropic in the plane of the structure). This is in accordance with previous work (Grima 2000; Grima & Evans 2000a; Grima et al. 2005a). For the type I [a×b,b×a] rectangles, the derived equations for the on-axis Poisson ratio complement those derived by Grima et al. (2004, 2005b), who derived the properties of this system in a different orientation. The two sets of expressions can be transformed to one another through standard axis-transformation techniques. Here, one should highlight the fact that Grima’s earlier derivation had shown that the sign of Poisson’s ratio in the directions corresponding to the lattice vector when using UC1 was dependent on the angle between the rectangles, something that is not found for the orientation used here where the on-axis Poisson ratios are always negative. This highlights the necessity to examine the anisotropy when looking for auxeticity.
As noted earlier and as discussed in appendix A of the electronic supplementary material, when the rectangles are similar in such a way that the ratio of the sides a/b is equal to c/d, (i.e. [(bc/db,c×d] structure), then such a structure is also two-dimensionally isotropic with a Poisson ratio equal to −1. Special cases of this structure are the type II rectangles and the same-sized and different-sized squares (the ratio a/b is equal to 1 in these latter cases). In fact, the mechanical properties of these structures are analogous to that of the ‘parent’ case, i.e. Poisson’s ratio is −1 and these structures are all isotropic.
Another interesting case is the structure [a×d,c×d]. In this case, the on-axis Poisson ratio is −1, but such a structure is anisotropic. Note that a special case of this structure is the type II rectangles (structure [a×b,a×b]), which is isotropic.
Let us now discuss some situations in which the model presented here may be used. An important observation that applies to the general model presented here (and hence also to all the special cases discussed in appendix A of the electronic supplementary material) is that all expressions derived for the mechanical properties are scale independent since if one lets b=r1a, c=r2a and d=r3a, then the expression can be re-written in terms of only ri and θ. This means that the deformation mechanism presented here can be implemented at different scales of structures, ranging from the nano (molecular) scale to the macroscale, i.e. the model can be used to predict the behaviour of systems and even design new systems that mimic its geometry and deformation on a smaller scale. On a nanolevel, the usefulness of these models has already been pointed out in other work (Ishibashi & Iwata 2000; Grima et al. 2005b, 2006, 2007a,c, 2009b; Williams et al. 2007), in which it has been shown that these models can explain, predict, as well as quantify, auxeticity in certain crystalline materials such as α-cristobalite (Grima et al. 2006) and natrolite (Grima et al. 2007c, 2009b; Williams et al. 2007) for loading in particular directions. This is possible because the relevant two-dimensional projections of these crystals have the same geometry as the rotating rectangles/squares model, and molecular-modelling simulations also suggest that these molecular systems deform with the same rotating mechanism (Grima et al. 2005b, 2006; Williams et al. 2007).
However, these earlier models based on rotating squares or rectangles were somewhat limited because they required all rectangles/squares to be of the same size. The model presented here has more degrees of freedom, allowing for variations in the size of adjacent units; therefore, one expects that this model may be applied to a wider range of materials. For example, it is interesting to note that at the extremity of having one set of rectangles with a very large aspect ratio relative to the other, then, as shown in figure 7, the resulting network would bear resemblance to the mechanisms proposed by He et al. (2005a,b) to explain auxeticity at the nanolevel in liquid-crystalline polymers. At this extremity, the model can also be considered as a network of interconnected nodules and fibrils, which could be of use to model the properties of auxetic microstructured polymers, which have long been studied through nodule–fibril models (Alderson & Evans 1995, 1997; Alderson et al. 1998b).
Figure 7.
Comparison of the model presented here (a) with that proposed by Griffin et al. (He et al. 2005a,b) to explain auxeticity in liquid-crystalline polymers (b).
Furthermore, the highly versatile model presented here can be of use to researchers who may use it as a ‘blueprint’ on which they can design and synthesize newly ‘designed’ auxetic materials. For example, a direct application of this work would be as a template for the design and manufacture of perforated sheets, which exhibit NPR in analogy to the recent work based on squares (Grima & Gatt 2010). On a microlevel, microstructures based on these models can be micromachined using techniques such as laser ablation and mechanical micromachining.
Another interesting application of such structures would be the possibility of using them as tunable filters and sieves. The pore size would be controlled by the amount of load applied, such that multi-purpose usage could be achieved. Another advantage of these filters over conventional filters is the fact that these can be easily unclogged by stretching the filter, thus increasing the lifetime of such filters.
Before we conclude, it is important to note some possible limitations of this model (and of other rotating rigid polygons mechanisms in general). Although it has been stated that such networks can be used to predict the mechanical properties of crystalline systems whose two-dimensional projections correspond to the geometry of the systems treated here, one must note that in a real material, it is very unlikely that the projected rectangles are completely rigid, and in addition to rotation, other deformation mechanisms such as stretching may occur concurrently so that the degree of auxeticity would depend on which deformation mechanism dominates. Furthermore, it should be appreciated that at the nanolevel, it is not correct to consider the nanostructure of a material as a purely mechanical system. Thus, at the nanolevel, it is not likely that the extremes of behaviour predicted by this highly idealized model will be manifested to the full. For example, it is not expected that any real nanostructured material will indeed exhibit an infinite shear modulus as predicted by the model of connected rectangles [a×b,a×b], although some features of the model would still be present, as, for example, observed in α-cristobalite. Besides, one must also keep in mind that the model presented here only offers a two-dimensional representation of three-dimensional systems. In reality, it is the three-dimensional units which are rotating with the result that projected rectangles in the plane of interest may change shape as a result of re-orientation of the three-dimensional units. Such effects are not captured in the model presented here.
4. Conclusion
In this work, the on- and off-axis mechanical properties of a structure made up of two different-sized rectangles have been derived. It was shown that some of the systems discussed, in particular, the similar rectangles structure where a/b is equal to c/d and the different-sized squares structure, exhibit isotropic and strain-independent Poisson ratios of −1, whereas some others have Poisson ratios that are dependent on the shape and relative size of the rectangles, the angle between the rectangles and also the direction of loading. It was also shown that since all systems exhibit negative on-axis Poisson ratios, all conformations based on rotating rectangles of this form are auxetic for loading in certain directions. We have also shown that this model could be used to elucidate the behaviour of a wide range of auxetics, ranging from liquid-crystalline polymers to silicates and zeolites.
Given the many benefits associated with having an NPR and the versatility of the proposed systems, it is hoped that this model will stimulate further work. This could, for example, lead to the manufacture of new man-made auxetics, which mimic the behaviour of the model structure proposed here. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9360311031341553, "perplexity": 1024.187988603877}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131299360.90/warc/CC-MAIN-20150323172139-00256-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://meridian.allenpress.com/jmar/article-abstract/14/1/79/80594/Product-Costing-and-Pricing-under-Long-Term?redirectedFrom=fulltext | We develop a model to analyze optimal product‐costing and pricing decisions in a dynamic information environment under long‐term‐capacity commitment. The arrival of new information about demand and cost parameters each period makes the problem complex. The optimal prices and capacity choices in our model cannot be decoupled as in Banker and Hughes' (1994) single‐period model.
The optimal prices are based on product costs that are adjusted each period to reflect changes in expected variable costs as well as utilization of fixed activity resources. The charge for each fixed resource is monotonically increasing in the expected demand for that resource in each state given the optimal capacity choice. The average optimal prices across periods and states are similar to Banker and Hughes' (1994) benchmark prices.
Finally, we investigate a two‐period version of the model to explore the optimality of carrying idle capacity. The optimal product‐cost charge for fixed capacity is strictly less in the first period than in the second period when the firm expects demand growth.
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https://www.lesswrong.com/posts/GYRBWJKTm42haE5FL/uml-final | # 23
(This is the fourteenth and final post in a sequence on Machine Learning based on this book. Click here for part I.)
The first part of this post will be devoted to multiclass prediction, which is the last topic I'll cover in the usual style. (It's on the technical side; requires familiarity with a bunch of concepts covered in earlier posts.) Then follows a less detailed treatment of the remaining material and, finally, some musings on the textbook.
# Multiclass Prediction
For this chapter, we're back in the realm of supervised learning.
Previously in this sequence, we've looked at binary classification (where ) and regression (where ). In multiclass prediction or multiclass categorization, we have but . For simplicity, we take the next simplest case, i.e., . (Going up from there is straight-forward.) More specifically, consider the problem of recognizing dogs and cats in pictures, which we can model by setting . For the purposes of this chapter, we ignore the difficult question of how to represent the instance space .
We take the following approach:
• train a function to compute a score for each label that indicates how well the instance fits that label
• classify each new domain point as the best-fitting label
In our case, this means we are interested in learning a function of the form . Then, if we have
this translates into something like " kinda looks like a dog, but a little bit more like a cat, and it does seem to be some animal." Each such function determines a predictor via the rule .
We will restrict ourselves to linear predictors. I.e., we assume that the feature space is represented as something like and our predictors are based on hyperplanes.
With the setting established, the question is how to learn .
## Approach 1: Reduction to Binary Classification
Suppose (unrealistically) that and our training data looks like this:
where
We could now train a separate scoring function for each label. Starting with cats, we would like a function that assigns each point a score based on how much it looks like a cat. For , the label corresponds to the label 1, whereas and both correspond to the label 0. In this setting, learning corresponds to finding a passable hyperplane. (Apply linear programming for Soft Support Vector Machines.)
Note that, while hyperplanes are used to classify points, they do, in fact, produce a numerical score based on how far on the respective side each point is.
Next, we train a function. Perhaps its hyperplane looks like so:
And, finally, a function in a similar fashion. Then, we define
And likewise with and . As before, we let .
Note that a point might be classified correctly, even if it is on the wrong side of the respective hyperplane. For example, consider this point:
thought this point was not a cat, but it thought it less intensely than thought that it was not a dog. Thus, depending on 's verdict, the that computes might still put out a cat label. Of course, the reverse is also true – a point might be classified as a cat by but also as a dog by , and if the dog classification is more confident, the will pick that as the label.
The reduction approach is also possible with predictors that merely put out a binary yes/no. However, numerical scores are preferable, precisely because of cases like the above. If we had used classifiers only, then for and we could do no better than to guess.
While the reduction approach can work, it won't have the best performance in practice, so we look for alternatives.
## Approach 2: Linear Programming
Using the fact that we are in the linear setting, we can rephrase our problem as follows: let , then we wish to find three vectors such that, for all training points , we have for all . In words: for each training point, the inner product with the vector measuring the similarity to the correct label needs to be larger than the inner product with the other vectors (in this case, the 2 others). This can be phrased as a linear program in much the same way as for Support Vector Machines (post VIII).
However, it only works if such vectors exist. In practice, this is only going to be the case if the dimension is extremely large.
## Approach 3: Surrogate Loss and Stochastic Gradient Descent
A general trick I first mentioned in post V is to apply Stochastic Gradient Descent (post VI) with respect to a surrogate loss function (i.e., a loss function that is convex and upper-bounds the first loss). In more detail (but still only sketched), this means we
• represent the class of hypotheses as a familiar set (easy for hyperplanes)
• for each labeled data point in our training sequence , construct the point-based loss function
• construct convex point-based loss functions such that
• compute the gradients and update our hypothesis step by step, i.e., set .
To do this, we have to change the framing somewhat. Rather than measuring the loss of our function (or of the predictor ), we would like to analyze the loss of a vector for some . However, we do not want to change our high-level approach: we still wish to evaluate all pairs and and separately.
This can be done by the following construction. Let be the number of coordinates that we use for each label, i.e., . We can "expand" this space to and consider it to hold three copies of each data point. Formally, we define the function by and and , where is the vector consisting of many zeros. Then we can represent all parts of our function as a single vector . If we take the inner product , it will only apply the part of that deals with the dog label since the other parts of are all 0s.
Thus, what was previously is now . The nontrivial information that was previously encoded in is now encoded in . Our predictor (which is fully determined by ) follows the rule . Thus, we still assign to each point the label for which it received the highest score. Note that makes use of all coordinates; there's no repetition.
Now we need to define loss functions. Since our goal is to apply Stochastic Gradient Descent, we focus on point-based loss functions that measure the performance of on the point only. (We write fat now since we specified that it's a vector.) One possible choice is the generalized 0-1 loss, that is 1 iff assigns the wrong label and 0 otherwise. However, this choice may not be optimal: if an image contains a cat, we probably prefer the predictor claiming it contains a dog to it claiming it doesn't contain any animal. Therefore, we might wish to define a function that measures how different two labels are. Then, we set . The function should by symmetric and non-negative, and it should be the case that . For our example, a reasonable choice could be
The loss function as defined above is non-convex: if and output different labels on , they will have different losses, which means that somewhere on the straight line between and , the loss makes a sudden jump (and sudden jumps make a function non-convex). Thus, our goal now is to construct a convex loss function such that, for each in the training sequence, upper-bounds .
To this end, consider the term
for any point . Recall that the predictor chooses its label precisely in such a way that the above inner product is maximized. It follows that the term above is non-negative. (It's 0 for ). Therefore, adding this term to our loss will upper-bond the original loss. We obtain
This term penalizes for (1) the difference between the predicted and the correct label; and (2) the difference between 's score for the predicted and the correct label. This would be a questionable improvement – we now punish twice for essentially the same mistake. But the term above is only an intermediate step. Now, instead of using in the term above, we can take the maximum over all possible labels ; clearly, this will not make the term any smaller. The resulting term will define our loss function. I.e.:
Note that, unlike before, the difference between the two inner products may be negative.
Let's think about what these two equations mean (referring two the two centered equations, let's call them E1 and E2). E1 (the former) punishes in terms of the correct label and the label which guessed: it looks both at how far its prediction is off and how wrong the label is. E2 does the same for , so it will always put out at least as high of an error. However, E2 also looks at all the other labels. To illustrate the effect of this, suppose that the correct label is , but guessed . The comparison of the correct label to the guessed label (corresponding to ) will yield an error of 0.2 (for the difference between both labels) plus the difference between 's score for both (which is inevitably positive since it guessed ). Let's say and , then the difference is another 0.3 and the total error comes out .
Now consider the term for . Suppose ; in that case, did think was a more likely label than , but only by 0.2. Therefore, . This is a negative term, which means it will be subtracted from the error. However, the loss function "expects" a difference of 1, because that's the difference between the labels and . Thus, the full term for comes out at , which is higher than for .
(Meanwhile, the trivial case of – i.e., comparing the correct label to itself – always yields an error of 0.)
Alas, the loss function ends up scolding the most not for the (slightly) wrong label it actually put out, but for the (very) wrong label it almost put out. This might be reasonable behavior, although it certainly requires that the numerical scores are meaningful. In any case, note that the primary point of formulating the point-based loss functions was to have it be convex, so even if the resulting function were less meaningful, this might be an acceptable price to pay. (To see that it is, in fact, convex, simply note that it is the maximum of a set of linear functions.)
With this loss function constructed, applying Stochastic Gradient Descent is easy. We start with some random weight vector . At step , we take and the labeled training point . Consider the term
The gradient of the maximum of a bunch of functions is the maximum of their gradients. Thus, if is the label for which the above is maximal, then
Therefore, our update rule is .
# Remaining Material
Now we get to the stuff that I read at least partially but decided to cover in less detail.
## Naive Bayes Classifier
Naive Bayes is a simple type of classifier that can be used for specific categorization tasks (supervised learning), both binary classification and multi-classification but not regression. The "naive" does not imply that Bayes is naive; instead, it is naive and uses Bayes' rule.
We usually frame our goal in terms of wanting to learn a function . However, we can alternatively ask to learn the conditional probability distribution . If we know this probability – i.e., the probability of observing a (point, label) pair once we see the point – it is easy to predict new points.
For the rest of this section, we'll write "probability" as rather than , and we'll write rather than .
Now. For any pair , we can express in terms of and and using Bayes' rule. This implies that knowing all the is sufficient to estimate any (the priors on and are generally not an issue). The difficulty is that is usually infinite and we only have access to some finite training sequence .
For this reason, Naive Bayes will only be applicable to certain types of problems. One of them, which will be our running example, is document classification in the bag of words model. Here, where is the number of words in our dictionary, and each document is represented as a vector . where bit indicates whether or not the -th word of our dictionary appears in [the document represented by ]. Thus, we throw away order and repetitions but keep everything else. Furthermore, is a set of possible topics, and the goal is to learn a predictor that sees a new document and assigns it a topic.
The crucial property of this example is that each coordinate of our vectors is very simple. On the other hand, the vectors themselves are not simple: we have (number of words in the dictionary), which means we have about parameters to learn (all ). The solution is the "naive" part: we assume that all feature coordinates are independent so that . This assumption is, of course, incorrect: the probabilities for and are certainly not independent. But an imperfect model can still make reasonable predictions.
Thus, naive Bayes works by estimating the parameters for any and . If consists of 100 possible topics, then we wish to estimate parameters, which may be feasible.
Given a labeled document collection (the training sequence ), estimating these parameters is now both simple and fast, which is the major strength of Naive Bayes. In fact, it is the fastest possible predictor in the sense that each training point only has to be checked once. Note that is the probability of encountering word in a document about topic . For example, it could be the probability of encountering the word "Deipnophobia" (fear of dinner parties) in a document about sports. To estimate this probability, we read through all documents about sport in our collection and keep track of how many of them contain this term (spoilers: it's zero). On that basis, we estimate that
Why the ? Well, as this example illustrates, the natural estimate will often be 0. However, a 0 as our estimate will mean that any document which contains this word has no chance to be classified as a sports document (see equations below), which seems overkill. Hence we "smooth" the estimate by the .
Small tangent: I think it's worth thinking about why this problem occurs. Why isn't the computed value our best guess? It's not because there's anything wrong with Bayesian updating – actually the opposite. While the output rule is Bayesian, the parameter estimation, as described above, is not. A Bayesian parameter estimation would have a prior on the value of , and this prior would be updated upon computing the above fraction. In particular, it would never go to 0. But, of course, the simple frequentist guess makes the model vastly more practical.
Now suppose we have all these parameters. Given an unknown document , we would like to output a document in the set . For each , we can write
where we got rid of the term because it won't change the ; it doesn't matter how likely it was to encounter our new document, and it will be the same no matter which topic we consider.
Now kicks in the naivety: we will choose a topic in the set
This is all stuff we can compute – we can estimate the prior based on our document collection, and the remaining probabilities are precisely our parameters.
In practice, one uses a slightly more sophisticated way to estimate parameters (taking into account repetitions). One also smoothes slightly differently (smoothing is the +1 step) and maps everything into log space so that the values don't get absurdly small, and we can add probabilities rather than multiplying them. Nonetheless, the important ideas are intact.
## Feature Selection
Every learning problem can be divided into two steps, namely
• (1) Feature Selection
• (2) Learning
This is true even in the case of unsupervised learning or online learning: before applying clustering algorithms onto a data set, one needs to choose a representation.
The respective chapter in the book does not describe how to select initial features, only how, given a (potentially large) set of features, to choose a subset . The approaches are fairly straight-forward:
• Measure the effectiveness of each feature by training a predictor based on that feature only; pick the best ones (local scoring)
• Start with an empty set of features, i.e., . In step , for each , test the performance of a predictor trained on . Choose to be the feature maximizing the performance, and set (greedy selection)
• Start with the full set of features, i.e., . In each step, drop one feature such that the loss in performance is minimal
It's worth pointing out that greedy selection can be highly ineffective since features and could be useless on their own but work well together. For an absurdly contrived example, suppose that , i.e., the target value is the sum of both features modulo some prime number . In that case, if and are uniformly distributed, and knowing either one of them is zero information about the target value. Nonetheless, both taken together determine the target value exactly.
(This example is derived from cryptography: splitting a number into additive parts modulo some fixed prime number is called secret sharing. The idea is that all shares are required to reconstruct the secret, but any subset of shares is useless. For example, if and all are uniformly distributed, then and are independently distributed (both uniform), and ditto with and . This remains true for an arbitrary number of shares. In general, Machine Learning and cryptography have an interesting relationship: the former is about learning information, the second about hiding information, i.e., making learning impossible. That's why cryptography can be a source of negative examples for Machine Learning algorithms.)
In this particular case, even the greedy algorithm wouldn't end up with features and since it has no motivation to pick up either of them to begin with – but the third approach would keep them both. On the other hand, if is useful on its own but only in combination with , then the local scoring approach would at best choose , while the greedy algorithm might end up with both. And, of course, it is also possible to construct examples where the third approach fails. Suppose we have 10 features that all heavily depend on each other. Together, they add more than one feature typically does but not ten times as much, so they're not worth keeping. Local scoring and greedy selection won't bother, but the third approach will be such with them: getting rid of one would worsen performance by too much.
## Regularization and Stability
This is a difficult chapter that I've read partially but skipped entirely in this sequence.
The idea of regularization is as follows: suppose we can represent our hypotheses as vectors , i.e., . Instead of minimizing the empirical loss , we minimize the regularized empirical loss defined as
where is a regularization function. A common choice is for some parameter . In that case, we wish to minimize a term of the form . This approach is called Regularized Loss Minimization. There are at least two reasons for doing this. One is that the presence of the regularization function makes the problem easier. The other is that, if the norm of a hypothesis is a measure for its complexity, choosing a predictor with a small norm might be desirable. In fact, you might recall the Structural Risk Minimization paradigm, where we defined a sequence of hypothesis classes such that . If we let , then both Structural Risk Minimization and Regularized Loss Minimization privilege predictors with smaller norm. The difference is merely that the tradeoff is continuous (rather than step-wise) in the case of Regularized Loss Minimization.
The problem
where is the squared loss can be solved with linear algebra – it comes down to inverting a matrix, where the term ensures that it is always invertible.
## Generative Models
Recall the general model for supervised learning: we have the domain set , the target set , and the probability distribution over that generates the labels. Our goal is to learn a predictor .
It took me reaching this chapter to realize that this goal is not entirely obvious. While the function is generally going to be the most interesting thing, it is also conceivable to be interested in the full distribution . In particular, even if the best predictor is known (that's the predictor given by ), one still doesn't know how likely each domain point is to appear. Differently put, the probability distribution over is information separate from the predictor; it's neither necessary nor sufficient for finding one.
The idea of generative models is to attempt to estimate the distribution directly. A way to motivate this is by Tegmark et. al.'s observation that the process that generates labeled points often has a vastly simpler description than the points themselves (chapter 2.3 of post X). This view suggests that estimating could actually be easier than learning the predictor . (Although not literally since learning implies learning the best predictor .)
The book has Naive Bayes as part of this chapter, even though it only involves estimating the , not the probability distribution over .
The book is structured in four parts. The first is about the fundamentals, which I covered in posts I-III. The second and third are about learning models, which I largely covered in posts IV-XIV. If there is a meaningful distinction between parts two and three, it's lost on me – they don't seem to be increasing in difficulty, and I found both easier than part one. However, the fourth part, which is titled advanced theory, is noticeably harder – too difficult for me to cover it at one post per week.
Here's a list of its chapters:
• Rademacher Complexities, which is an advanced theory studying the rate at which training sequences become representative (closely related to the concept of uniform convergence from post III).
• Covering numbers, a related concept, about bounding the complexity of sets
• Proofs for the sample complexity bounds in the quantitative version of the fundamental theorem of statistical learning (I've mentioned those a couple of times in this sequence)
• Some theory on the learnability of multiclass prediction problems, including another off-shoot of the VC-dimension.
• Compression Bounds, which is about establishing yet another criterion for the learnability of classes: if "a learning algorithm can express the output hypothesis using a small subset of the training set, then the error of the hypothesis on the rest of the examples estimates its true error."
• PAC-Bayes, which is another learning approach
# Closing Thoughts on the Book
I've mentioned before that, while Understanding Machine Learning is the best resource on the subject I've seen so far, it feels weaker than various other textbooks I've tried from Miri's list. Part one is pretty good – it actually starts from the beginning, it's actually rigorous, and it's reasonably well explained. Parts two and are mixed. Some chapters are good. Others are quite bad. Occasionally, I felt the familiar sense of annoyance-at-how-anyone-could-possibly-think-this-was-the-best-way-explain-this-concept that I get all the time reading academic papers very rarely with Miri-recommended textbooks. It feels a bit like the authors identified a problem (no good textbooks that offer a comprehensive theoretical treatment of the field), worked on a solution, but lost inspiration somewhere on the way and ended up producing several chapters that feel rather uninspired. The one on neural networks is a good example.
(Before bashing the book some more, I should note that the critically acclaimed Elements of Statistical Learning made me angry immediately when I tried to read it – point being, I appear to be unusually hard to please.)
The exercises are lacking throughout. I think the goal of exercises should be to help me internalize the concepts of the respective chapter, and they didn't really do that. It seems that the motivation for them was often "let's offload the proof of this lemma into the exercises to save space" or "here's another interesting thing we didn't get to cover, let's make it an exercise." (Note that there's nothing wrong with offloading the proof of a lemma to exercises per se, they can be perfectly fine, but they're not automatically fine.) Conversely, in Topology, the first exercises of each sub-chapter usually felt like "let's make sure you understood the definitions" and the later ones like "let's make sure you get some real practice with these concepts," which seems much better. Admittedly, it's probably more challenging to design exercises for Machine Learning than it is for a purely mathematical topic. Nonetheless, I think they could be improved dramatically.
Next up (but not anytime soon) are -calculus and Category theory. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9094365835189819, "perplexity": 498.24733992033464}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590348509264.96/warc/CC-MAIN-20200606000537-20200606030537-00247.warc.gz"} |
http://math.stackexchange.com/questions/59315/complex-number-trigo-1-tan3i-find-modulus-and-argument | # Complex number + trigo : $-1 + \tan(3)i$ , find modulus and argument
I have $-1 + \tan(3)i$ and must find its modulus and its argument. I tried to solve it by myself for hours, and then I looked at the answer, but I am still confused with a part of the solution.
Here is the provided solution: \begin{align} z &= -1 + \tan(3)i \\ &= -1 + \frac{\sin(3)}{\cos(3)}i \\ &= \frac1{\left|\cos(3)\right|} ( \cos(3) + i(-1)\sin(3)) \\ &= \frac1{\left|\cos(3)\right|} e^{-3i} \\ &= \frac1{\left|\cos(3)\right|} e^{(2\pi-3)i} \end{align}
I don't understand how we get to $$\frac1{\left|\cos(3)\right|}(\cos(3) + i(-1)\sin(3))$$ How did they get this modulus $1/|\cos(3)|$, and the $-1$ in the imaginary part? How did they reorder the previous expression to obtain this?
I also don't see why they developed the last equality. They put $2\pi-3$ instead of $-3$; OK, it is the same, but what was the aim of a such development?
Thanks!
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Given the solution, are you sure you don't have $-1 + i\tan(3)$ instead of $-1 + \tan(3)$? – Rahul Aug 23 '11 at 20:58
They notice that $\cos(3) < 0$, and use $\frac{\sin (3)}{\cos (3)} = \frac{\sin( 3) \operatorname{sgn}(\cos(3))}{ \vert \cos (3) \vert} = \frac{(-1) \sin( 3)}{ \vert \cos (3) \vert}$. – Sasha Aug 23 '11 at 21:00
@Rahul: you are right ! I mistyped – jlink Aug 23 '11 at 21:02
@Sasha : Thank you very much for your help ! I didn't pay attention to cos(3) < 0 – jlink Aug 23 '11 at 21:38
Let $z = -1 + \tan(3) \ i$. In the complex plane, this would be the point $(-1, \tan(3))$, which has length
$$|z| = \sqrt{(-1)^2 + \tan^2(3)} = \sqrt{1 + \frac{\sin^2(3)}{\cos^2(3)}} = \sqrt{\frac{1}{\cos^2(3)}} \sqrt{\cos^2(3) + \sin^2(3)} = \frac{1}{|\cos(3)|}$$
For the last equality, we used $\sin^2(x) + \cos^2(x) = 1$. Now we want to write
$$z = |z| \ e^{\omega i} = |z| \ (\cos(\omega) + i \sin(\omega))$$
for some $\omega$. It turns out this can be done easily by writing
$$z = \frac{1}{\cos(3)}(-\cos(3) + i \sin(3)) = \frac{1}{|\cos(3)|}(\cos(-3) + i \sin(-3)) = \frac{1}{|\cos(3)|} e^{-3i}$$
Since $-3 \notin [0, 2\pi)$ they decided to add $2 \pi$ to the angle, so that it is inside this interval.
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I misunderstood abs and module notation for |cos(3)| , I thought it was module... Thank you for this wonderful answer ! – jlink Aug 23 '11 at 21:27
@Jonathan: But the modulus and absolute value of a real number are the same thing... – Guess who it is. Aug 24 '11 at 3:45
Yes thank you... I started to mix everything.. Now I am ok with everything :) – jlink Aug 24 '11 at 8:32
Let me fill in some of the steps they have jumped over.
\begin{align} z &= -1 + i\frac{\sin 3}{\cos 3} \\ &= \frac 1 {\cos 3} (-\cos 3 + i \sin 3) \end{align} However, $1/\cos 3$ is a negative real number. To make that term positive, we negate both terms: \begin{align} z &= \frac 1 {-\cos 3} (\cos 3 - i \sin 3) \\ &= \frac 1 {\lvert \cos 3 \rvert} (\cos 3 - i \sin 3). \end{align}
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why do we want it to be positive ? – jlink Aug 23 '11 at 21:28
@Jon: Because moduli are intended to be positive. – Guess who it is. Aug 24 '11 at 2:50
oh yes i got it, because it it a length ! (and length can't be negative...) – jlink Aug 24 '11 at 8:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9998977184295654, "perplexity": 810.4814546222606}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375102712.76/warc/CC-MAIN-20150627031822-00156-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://brilliant.org/problems/a-simple-answer-to-a-complicated-x/ | # A simple answer to a complicated $$x$$
Algebra Level 3
$$x = \sqrt[3]{7+\sqrt{22}}+ \sqrt[3]{7- \sqrt{22}}$$. In the equation $$x^3 + bx + c = 0$$, what is $$-bc$$ if $$b$$ and $$c$$ are integers?
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9251590967178345, "perplexity": 482.8965965991099}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281332.92/warc/CC-MAIN-20170116095121-00560-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://www.hepdata.net/record/65336 | Forward-Backward Asymmetry of Drell-Yan Lepton Pairs in $pp$ Collisions at $\sqrt{s} = 7$ TeV
The collaboration
Phys.Lett.B 718 (2013) 752-772, 2013.
Abstract (data abstract)
A measurement of the forward-backward asymmetry (A[FB]) of Drell-Yan lepton pairs in pp collisions at sqrt(s) = 7 TeV is presented. The data sample, collected with the CMS detector, corresponds to an integrated luminosity of 5 inverse femtobarns. The asymmetry is measured as a function of dilepton mass and rapidity in the dielectron and dimuon channels. Combined results from the two channels are also presented. The AFB measurement in the dimuon channel and the combination of the two channels are the first such results obtained at a hadron collider. The measured asymmetries are consistent with the standard model predictions. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9965739846229553, "perplexity": 1687.6896008127696}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057227.73/warc/CC-MAIN-20210921191451-20210921221451-00099.warc.gz"} |
https://groupprops.subwiki.org/wiki/Formal_group_law | # Formal group law
This is a variation of group|Find other variations of group | Read a survey article on varying group
## Definition
### One-dimensional formal group law
Let $R$ be a commutative unital ring. A one-dimensional formal group law on $R$ is a formal power series $F$ in two variables, denoted $x$ and $y$, such that:
Condition no. Name Description of condition Interpretation
1 Associativity $\! F(x,F(y,z)) = F(F(x,y),z)$ as formal power series If $F$ is the binary operation denoting multiplication, then $F$ is associative.
2 Identity element $\! F(x,y) = x + y + xyG(x,y)$ for some power series $G$. Thus, $F(x,0) = x, F(0,y) = y$ The element $0$ is the identity element for multiplication.
3 Inverses There exists a power series $m(x)$ such that $m(0) = 0$ and $F(x,m(x)) = 0$. Every element has an inverse for multiplication.
Condition (3) is redundant, i.e., it can be deduced from (1) and (2).
A one-dimensional commutative formal group law is a one-dimensional formal group law $F$ such that $F(x,y) = F(y,x)$. Two important examples of commutative formal group laws, that make sense for any ring, are the additive formal group law and the multiplicative formal group law.
### Higher-dimensional formal group law
Let $R$ be a commutative unital ring. A $n$-dimensional formal group law is a collection of $n$ formal power series $F_i$ involving $2n$ variables $(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n)$ satisfying a bunch of conditions.
Before stating the conditions, we introduce some shorthand. Consider $x = (x_1,x_2,\dots,x_n)$ and $y = (y_1,y_2,\dots,y_n)$. Then, $F(x,y)$ is the $n$-tuple $(F_1(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n),F_2(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n),\dots,F_n(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n))$.
Condition no. Name Description of condition in shorthand Description of condition in longhand
1 Associativity $\! F(x,F(y,z)) = F(F(x,y),z)$ For each $i$ from $1$ to $n$, $F_i(x_1,x_2,\dots,x_n,F_1(y_1,y_2,\dots,y_n,z_1,z_2,\dots,z_n),F_2(y_1,y_2,\dots,y_n,z_1,z_2,\dots,z_n),\dots,F_n(y_1,y_2,\dots,y_n,z_1,z_2,\dots,z_n))$ equals $F_i(F_1(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n),F_2(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n),F_n(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n),z_1,z_2,\dots,z_n)$.
2 Identity element $\! F(x,y) = x + y +$ terms of higher degree, so $\! F(x,0) = F(0,x) = x$ For each $i$, $\! F_i(x_1,x_2,\dots,x_n,y_1,y_2,\dots,y_n) = x_i + y_i +$ terms of higher degree (each further term is a product that involves at least one $x_j$ and one $y_k$.
3 Inverse There exists $m$, a collection of $n$ formal power series in one variable, such that $F(x,m(x)) = 0$ formally. There exist $m_i, 1 \le i \le n$, all formal power series in one variable, such that $\! F(x_1,x_2,\dots,x_n,m_1(x_1,x_2,\dots,x_n),m_2(x_1,x_2,\dots,x_n),\dots,m_n(x_1,x_2,\dots,x_n)) = 0$.
Condition (3) is redundant, i.e., it can be deduced from (1) and (2).
A commutative formal group law is a formal group law $F$ such that $F(x,y) = F(y,x)$. Two important examples of commutative formal group laws, that make sense for any ring, are the additive formal group law and the multiplicative formal group law.
## Interpretation as group
### For power series rings
A one-dimensional formal group law over a commutative unital ring $R$ gives a group structure on the maximal ideal $\langle t \rangle$ in the ring $R[[t]]$ of formal power series in one variable over $R$.
A one-dimensional formal group law can also be interpreted to give a group structure over the image of the maximal ideal $\langle t \rangle$ in any quotient ring of $R[[t]]$; i.e., a ring of the form $R[[t]]/(t^n) \cong R[t]/(t^n)$.
A $n$-dimensional formal group law over a commutative unital ring $R$ gives a group structure on the set of $n$-tuples of formal power series in one variable over $R$.
### For arbitrary algebras over $R$
Further information: formal group law functor from commutative algebras to groups
More generally, for any commutative $R$-algebra $S$, if $N$ is the set of nilpotent elements of $S$, then any $n$-dimensional formal group law $F$ over $S$ gives a group structure on the set $N^n$ of $n$-tuples over $N$. The formal group law thus gives a functor from the category of commutative $R$-algebras to the category of groups.
A particular case of this is when $R$ is a local ring and $M$ is its unique maximal ideal. In this case, we get what is called a $R$-standard group.
## Examples
### Examples of one-dimensional formal group laws
Name of law Expression for law Crude explanation for associativity Additional properties
additive formal group law $x + y$ addition is associative in the base ring commutative formal group law
multiplicative formal group law $x + y + xy$ rewrite as $(x + 1)(y + 1) - 1$. In other words, if we translate by 1, this is just multiplication. Now use associativity of multiplication commutative formal group law | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 76, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8989102244377136, "perplexity": 155.8233277554335}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039550330.88/warc/CC-MAIN-20210421191857-20210421221857-00571.warc.gz"} |
http://mathhelpforum.com/math-software/119087-problem-nmaximize.html | # Math Help - problem with NMaximize
1. ## problem with NMaximize
Hi,
I'm trying to obtain the maximum value of the function and estimate the values of parameters in this function.
But when I ran the command NMaximize, I got this error:
NMaximize::nrnum: The function value 1.25645*10^6-370.708 I is not a real number at {r,x,\[Theta]} = {0.652468,0.266141,1.36563}. >>
Please see the code in the attachment | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9136846661567688, "perplexity": 1266.4628890709735}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398447773.21/warc/CC-MAIN-20151124205407-00142-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/224806-maximizing-volume-ellipsoid-given-surface-area.html | # Math Help - Maximizing volume of an ellipsoid with a given surface area.
1. ## Maximizing volume of an ellipsoid with a given surface area.
I have a problem for my vector calculus class where we have to build a storage tank that is the shape of a surface of revolution obtained by rotating
x^2 + a^2 y^2 = 1, x > 0, around the y-axis. Given a fixed surface area S the problem is to find the best value of a to maximize the volume.
I'm honestly so fried with abstract classes that I'm having trouble approaching this problem. Any help would be greatly appreciated.
2. ## Re: Maximizing volume of an ellipsoid with a given surface area.
Originally Posted by anoldoldman
I have a problem for my vector calculus class where we have to build a storage tank that is the shape of a surface of revolution obtained by rotating
x^2 + a^2 y^2 = 1, x > 0, around the y-axis. Given a fixed surface area S the problem is to find the best value of a to maximize the volume.
I'm honestly so fried with abstract classes that I'm having trouble approaching this problem. Any help would be greatly appreciated.
The basic idea is that you use your constraint, i.e. the surface area, to reduce the expression for the volume from 2 variables to 1 variable. Via the constraint you'll have an expression for y in terms of x or vice versa, whichever is more convenient, and you plug that into your volume formula.
That whole mess gives you the volume as the function of a single variable and you know how to maximize that. Just find the zero of the derivative.
Of course you have to come up with formulas for the surface area and volume of your surface of revolution but you can figure that out.
(in case you can't remember you make a surface by spinning a curve around the center 2pi radians. To make a volume you spin the area under a curve around the center 2pi radians.
In the first case the surface area is the length of the curve times 2pi. For the volume it's the area under the curve times 2pi.)
3. ## Re: Maximizing volume of an ellipsoid with a given surface area.
I'm trying to get the arc length of the curve but it is coming out to a horrible integral. I created the integral by solving for y and then running it through the standard arc length formula. Is there a better way of doing this that I am missing?
4. ## Re: Maximizing volume of an ellipsoid with a given surface area.
Originally Posted by anoldoldman
I'm trying to get the arc length of the curve but it is coming out to a horrible integral. I created the integral by solving for y and then running it through the standard arc length formula. Is there a better way of doing this that I am missing?
Looking into this I find that calculating the arc length of a 1/4 ellipse is a not trivial. Here is a reference.
Calculating the area of a 1/4 ellipse is at least easy. It's pi/(4a) based on how you've specified your ellipse. So the full area is just pi/a.
Your problem btw is equivalent to maximizing the area of the 1/4 ellipse subject to arc length S/2pi for the same quarter ellipse since the surface areas and volumes of your revolved object is just these times 2pi.
This is equivalent to maximizing the area of the ellipse for a constrained perimeter of (2S)/pi.
Here is a page on calculating the circumference of an ellipse.
5. ## Re: Maximizing volume of an ellipsoid with a given surface area.
I've been looking at this problem some more and it's poorly posed.
You want a function x^2 + a^2 * y^2 = 1 and you want to vary only "a" but doing so will inevitably change the arc length of the quarter ellipse and thus the surface area of your surface of revolution.
You need another scaling parameter so that you can select an ellipse eccentricity via "a" and then scale the ellipse so it's arc length remains constant.
Does this make sense? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8880937695503235, "perplexity": 248.07744106238633}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115864313.15/warc/CC-MAIN-20150124161104-00061-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/conservation-of-momentum-extremely-easy-but-confusing.267916/ | # Homework Help: Conservation of momentum (extremely easy but confusing)
1. Oct 29, 2008
### avenkat0
1. The problem statement, all variables and given/known data
If the mass of an object increases, its momentum
A. always increases
B. always decreases
C. sometimes increases, sometimes decreases
D. sometimes decreases, never increases
E. sometimes increases, never decreases
2. Relevant equations
MV=momentum
3. The attempt at a solution
I figured if one of the variables increased the whole would increases but it turned out to be wrong.
Then if the velocity changed with the mass momentum would remain the same. but that choice is nowhere to be found.
Thank you
2. Oct 29, 2008
### Hootenanny
Staff Emeritus
In my opinion, the question in itself is poorly worded as we have no information regarding the velocity. However, since we have no information on the velocity, we have no way of knowing if the momentum will increase or decrease so I would have to say C.
3. Oct 29, 2008
### avenkat0
Turned out to be wrong... :( that was my last shot. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.869606077671051, "perplexity": 1220.6331838850656}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794868248.78/warc/CC-MAIN-20180527111631-20180527131631-00562.warc.gz"} |
https://economics.stackexchange.com/questions/27823/walrasian-equilibrium-intuition-given-prices-and-some-initial-allocation | # Walrasian Equilibrium intuition given prices and some initial allocation
Suppose we have two agents who are each assigned some initial allocation of two different goods, where the prices of each good are given. Also, suppose the utility functions for each agent are weakly concave and strictly increasing.
If, after receiving their allocations, no trades occur between the two agents, can we infer that this initial allocation represents a competitive equilibrium (CE)?
My intuition says that it may not be a CE, because perhaps agent 1 has an optimal allocation, but maybe agent 2 wishes that he could trade with agent 1 to change his (agent 2's) allocation. Although this would not be a competitive equilibrium, this scenario would still be Pareto efficient. Is my analysis correct?
• Can you make your "analysis" a bit more rigorous, preferably more formal? E.g. "agent 1 has an optimal allocation", but what is an optimal allocation if utility functions are strictly increasing? – Giskard Apr 17 '19 at 14:57 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9198216199874878, "perplexity": 1024.3814124079659}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618039550330.88/warc/CC-MAIN-20210421191857-20210421221857-00367.warc.gz"} |
https://rd.springer.com/article/10.1140/epjc/s10052-014-3236-1 | # AIC, BIC, Bayesian evidence against the interacting dark energy model
• Marek Szydłowski
• Aleksandra Kurek
• Michał Kamionka
Open Access
Regular Article - Theoretical Physics
## Abstract
Recent astronomical observations have indicated that the Universe is in a phase of accelerated expansion. While there are many cosmological models which try to explain this phenomenon, we focus on the interacting $$\Lambda$$CDM model where an interaction between the dark energy and dark matter sectors takes place. This model is compared to its simpler alternative—the $$\Lambda$$CDM model. To choose between these models the likelihood ratio test was applied as well as the model comparison methods (employing Occam’s principle): the Akaike information criterion (AIC), the Bayesian information criterion (BIC) and the Bayesian evidence. Using the current astronomical data: type Ia supernova (Union2.1), $$h(z)$$, baryon acoustic oscillation, the Alcock–Paczynski test, and the cosmic microwave background data, we evaluated both models. The analyses based on the AIC indicated that there is less support for the interacting $$\Lambda$$CDM model when compared to the $$\Lambda$$CDM model, while those based on the BIC indicated that there is strong evidence against it in favor of the $$\Lambda$$CDM model. Given the weak or almost non-existing support for the interacting $$\Lambda$$CDM model and bearing in mind Occam’s razor we are inclined to reject this model.
### Keywords
Dark Matter Dark Energy Cosmic Microwave Background Baryon Acoustic Oscillation Dark Energy Component
## 1 Introduction
Recent observations of type Ia supernovae (SNIa) provide the main evidence that the current Universe is in an accelerating phase of expansion [1]. Cosmic microwave background (CMB) data indicate that the present Universe has also a negligible space curvature [2]. Therefore if we assume the Friedmann–Robertson–Walker (FRW) model in which the effects of nonhomogeneities are neglected, then the acceleration must be driven by a dark energy component $$X$$ (matter fluid violating the strong energy condition $$\rho _{{X}}+3p_{{X}}\ge 0)$$. This kind of energy represents roughly 70 % of the matter content of the current Universe. Because the nature as well as mechanism of the cosmological origin of the dark energy component are unknown some alternative theories try to eliminate the dark energy option by modifying the theory of gravity itself. The main prototype of this kind of models is a class of covariant brane models based on the Dvali–Gabadadze–Porrati (DGP) model [3] as generalized to cosmology by Deffayet [4]. The simplest explanation of a dark energy component is the cosmological constant with effective equation of state $$p=-\rho$$ but then the problem of its smallness appears and hence its relatively recent dominance. Although the $$\Lambda$$CDM model offers a possibility of explanation of the observational data it is only the effective theory which contains the enigmatic theoretical term—the cosmological constant $$\Lambda$$. Numerous other candidates for a dark energy description have also been proposed like the evolving scalar field [5], usually referred as quintessence, the phantom energy [6, 7], the Chaplygin gas [8] model, etc. Some authors believe that the dark energy problem belongs to the quantum gravity domain [9].
Recent Planck observations still favor the standard cosmological model [10], especially for the high multipoles. However, in this model there are some problems with understanding the values of the density parameters for both dark matter and dark energy. The question is why energies of vacuum energy and dark matter are of the same order for the current Universe. The very popular methodology to solve this problem is to treat the coefficient equation of state as a free parameter, i.e. the wCDM model, which should be estimated from the astronomical and astrophysical data. The observations from the CMB and baryon acoustic oscillation (BAO) data sets give $$w_x=-1.13^{+0.24}_{-0.23}$$ with 95 % confidence levels [10].
Alternative to this idea of the phantom dark energy mechanism of alleviating the coincidence problem is to consider the interaction between dark matter and dark energy; the interaction model. Many authors investigated observational constraints of the interaction model. Costa et al. [11] concluded that the interaction models become in agreement with the admissible observational data which can provide some argument toward consistency of the measured density parameters. Yang and Xu [12] constrained some interaction models under the choice of an ansatz for the transfer energy mechanism. From this investigation the joined geometrical tests show a stricter constraint on the interaction model if we include information from the large scale structure [$$f\sigma _{8}(z)$$ data] of the Universe. These authors have found the interaction rate in the $$3\sigma$$ region. This means that the recent cosmic observations favor it but with rather a small interaction between the both dark sectors. However, the measurement of the redshift-space distortion could rule out a large interaction rate in the $$1\sigma$$ region. Zhang and Liu [13] using the SNIa observations, $$H(z)$$ data (OHD), CMB, and secular Sandage–Loeb obtained the small value of the interacting parameter: $$\delta =-0.019\pm 0.01 (1 \sigma ), \pm 0.02 (2\sigma )$$.
In all interaction models the specific ansatz for a model of interaction is postulated. There are infinite many of such models with a different form of interaction and there is some kind of a theoretical bias or degeneracy, coming from the choice of the potential form in scalar field cosmology. Szydlowski [14] proposed the idea of the estimation of the interaction parameter without any ansatz for the model of the interaction.
These theoretical models are consistent with the observations; they are able to explain the phenomenon of the accelerated expansion of the Universe. But should we really prefer such models over the $$\Lambda$$CDM one? All observational constraints show that the $$\Lambda$$CDM model still shows a good fit to the observational data. But from these constraints the small value of the interaction is still admissible. To answer this question we should use some model comparison methods to confront the existing cosmological models having observations at hand. We choose the information and Bayesian criteria of the model selection which are based on Occam’s razor (principle), the well-known and effective instrument in science to obtain a definite answer of whether the interacting $$\Lambda$$CDM model can be rejected.
Let us assume that we have $$N$$ pairs of measurements $$(y_i,x_i)$$ and that we want to find the relation between the $$y$$ and $$x$$ variables. Suppose that we can postulate $$k$$ possible relations $$y\equiv f_i(x,\bar{\theta })$$, where $$\bar{\theta }$$ is the vector of the unknown model parameters and $$i=1,\dots ,k$$. With the assumption that our observations come with uncorrelated Gaussian errors with a mean $$\mu _i=0$$ and a standard deviation $$\sigma _i$$, the goodness of fit for the theoretical model is measured by the quantity $$\chi ^2$$ given by
\begin{aligned} \chi ^2=\sum _{i=1}^{N} \frac{(f_l(x_i,\bar{\theta }) - y_i)^2}{2\sigma _i^2}=-2\ln L, \end{aligned}
(1)
where $$L$$ is the likelihood function. For the particular family of models $$f_l$$ the best one to minimize the $$\chi ^2$$ quantity we denote $$f_l(x,\hat{\bar{\theta }})$$. The best model from our set of $$k$$ models $${f_1(x,\hat{\bar{\theta }}),\dots ,f_k(x,\hat{\bar{\theta }})}$$ could be the one with the smallest value of the quantity $$\chi ^2$$. But this method could give us misleading results. Generally speaking, for a more complex model the value of $$\chi ^2$$ is smaller, thus the most complex one will be the choice as the best from the set under consideration.
A clue is given by Occam’s principle known also as Occam’s razor: “If two models describe the observations equally well, choose the simplest one”. This principle has an aesthetic as well as empirical justification. Let us quote the simple example which illustrates this rule [15]. In Fig. 1 is observed a black box and a white one behind it. One can postulate two models: first, there is one box behind the black box, second, there are two boxes of identical height and color behind the black box. Both models explain our observations equally well. According to Occam’s principle we should accept the explanation which is simpler so that there is only one white box behind the black one. Is not it more probable that there is only one box than two boxes with the same height and color?
We could not use this principle directly because the situations when two models explain the observations equally well are rare. But in information theory as well as in the Bayesian theory there are methods for model comparison which include such a rule.
In information theory there are no true models. There is only reality which can be approximated by models, which depend on some number of parameters. The best one from the set under consideration should be the best approximation to the truth. The information lost when truth is approximated by the model under consideration is measured by the so called Kullback–Leibler (KL) information, so the best one should minimize this quantity. It is impossible to compute the KL information directly because it depends on the truth, which is unknown. Akaike [16] found an approximation to the KL quantity, which is called the Akaike information criterion (AIC), given by
\begin{aligned} \text {AIC}=-2\ln \mathcal {L} +2d, \end{aligned}
(2)
where $$\mathcal {L}$$ is the maximum of the likelihood function and $$d$$ is the number of model parameters. A model which is the best approximation to the truth from a set of models under consideration has the smallest value of the AIC quantity. It is convenient to evaluate the differences between the AIC quantities computed for the rest of the models from the set and the AIC for the best one. Those differences ($$\Delta _{\text {AIC}}$$) are easy to interpret and allow for a quick ‘strength of evidence’ for a considered model with respect to the best one. The models with $$0 \le \Delta _{\text {AIC}}\le 2$$ have substantial support (evidence), those where $$4<\Delta _{\text {AIC}}\le 7$$ have considerably less support, while models having $$\Delta _{\text {AIC}} > 10$$ have essentially no support with respect to the best model.
It is worth noting that the complexity of the model is interpreted here as the number of its free parameters that can be adjusted to fit the model to the observations. If models under consideration fit the data equally well according to the Akaike rule the best one is with the smallest number of model parameters (the simplest one in such an approach).
In the Bayesian framework the best model (from the model set under consideration) is that which has the largest value of probability in the light of the data (so-called posterior probability) [17]
\begin{aligned} P(M_{i}|D)=\frac{P(D|M_{i})P(M_{i})}{P(D)}, \end{aligned}
(3)
where $$P(M_{i})$$ is a prior probability for the model $$M_{i}$$, $$D$$ denotes the data, $$P(D)$$ is the normalization constant,
\begin{aligned} P(D)= \sum _{i=1}^{k} P(D|M_{i})P(M_{i}). \end{aligned}
(4)
$$P(D|M_{i})$$ is the marginal likelihood, also called the evidence,
\begin{aligned} P(D|M_{i})=\int P(D|\bar{\theta },M_{i})P(\bar{\theta }|M_{i}) \ \mathrm{d} \bar{\theta } \equiv E_{i}, \end{aligned}
(5)
where $$P(D|\bar{\theta },M_{i})$$ is the likelihood under model $$i$$, $$P(\bar{\theta }|M_{i})$$ is the prior probability for $${\bar{\theta }}$$ under model $$i$$.
Let us note that we can include Occam’s principle by assuming the greater prior probability for the simpler model, but this is not necessary and rarely used in practice. Usually one assumes that there is no evidence to favor one model over another which causes one to assign equal values of the prior for all models under consideration. It is convenient to evaluate the posterior ratio for models under consideration which in the case with a flat prior for the models is reduced to the evidence ratio, called the Bayes factor,
\begin{aligned} B_{ij} = \frac{P(D|M_i)}{P(D|M_j)}. \end{aligned}
(6)
The interpretation of twice the natural logarithm of the Bayes factor is as follows: $$0<2\ln B_{ij}\le 2$$ as weak evidence, $$2<2\ln B_{ij}\le 6$$ as positive evidence, $$6<2\ln B_{ij}\le 10$$ as strong evidence and $$2\ln B_{ij}> 10$$ as very strong evidence against model $$j$$ comparing to model $$i$$. This quantity is our Occam’s razor. Let us simplify the problem to illustrate how this principle works here [15, 18].
Assume that $$\bar{P}(\bar{\theta }|D,M)$$ is the non-normalized posterior probability for the vector $$\bar{\theta }$$ of the model parameters. In this notation $$E=\int \bar{P}(\bar{\theta }|D,M)d\bar{\theta }$$. Suppose that the posterior has a strong peak in the maximum: $$\bar{\theta }_{\text {MOD}}$$. It is reasonable to approximate the logarithm of the posterior by its Taylor expansion in the neighborhood of $$\bar{\theta }_{\text {MOD}}$$, so we finish with the expression
\begin{aligned} \bar{P}(\bar{\theta }|D,M)&= \bar{P}(\bar{\theta }_\text {MOD}|D,M) \nonumber \\&\quad \times \exp \left[ -(\bar{\theta }-\bar{\theta }_{\text {MOD}})^T C^{-1}(\bar{\theta }-\bar{\theta }_{\text {MOD}})\right] , \end{aligned}
(7)
where $$\left[ C^{-1} \right] _{ij} = -\left[ \frac{\partial ^2\ln \bar{P}(\bar{\theta }|D,M)}{\partial \theta _i\partial \theta _j}\right] _{\bar{\theta }=\bar{\theta }_\text {MOD}}$$. The posterior is approximated by the Gaussian distribution with the mean $$\bar{\theta }_\text {MOD}$$ and the covariance matrix $$C$$. The evidence then has the form
\begin{aligned} E&= \bar{P}(\bar{\theta }_{\text {MOD}}|D,M) \nonumber \\&\times \int \exp \left[ -(\bar{\theta }-\bar{\theta }_{\text {MOD}})^T C^{-1}(\bar{\theta }-\bar{\theta }_{\text {MOD}})\right] \ \mathrm{d} \bar{\theta }. \end{aligned}
(8)
Because the posterior has a strong peak near the maximum, the highest contribution to the integral comes from the neighborhood close to $$\bar{\theta }_\text {MOD}$$. The contribution from the other region of $$\bar{\theta }$$ can be ignored, so we can expand the limit of the integral to the whole of $$R^d$$. With this assumption one can obtain $$E=(2\pi )^{\frac{d}{2}}\sqrt{\det C}\bar{P}(\bar{\theta }_{\text {MOD}}|D,M)= (2\pi )^{\frac{d}{2}}\sqrt{\det C}P(D|\bar{\theta }_\text {MOD},M)P(\bar{\theta }_\text {MOD}|M)$$. Suppose that the likelihood function has a sharp peak in $$\hat{\bar{\theta }}$$ and the prior for $$\bar{\theta }$$ is nearly flat in the neighborhood of $$\hat{\bar{\theta }}$$. In this case $$\hat{\bar{\theta }}=\bar{\theta }_{\text {MOD}}$$ and the expression for the evidence takes the form $$E=\mathcal {L}(2\pi )^{\frac{d}{2}}\sqrt{\det \text {C}}P(\hat{\bar{\theta }}|M)$$. The quantity $$(2\pi )^{\frac{d}{2}}\sqrt{\det \text {C}}P(\hat{\bar{\theta }}|M)$$ is called the Occam factor (OF). When we consider the case with one model parameter with a flat prior, $$P(\theta |M)=\frac{1}{\Delta \theta }$$ the $$\mathrm{OF} =\frac{2\pi \sigma }{\Delta \theta }$$, this can be interpreted as the ratio of the volume occupied by the posterior to the volume occupied by the prior in the parameter space. The more parameter space wasted by the prior, the smaller value of the evidence. It is worth noting that the evidence does not penalize parameters which are unconstrained by the data [19].
As the evidence is hard to evaluate, an approximation to this quantity was proposed by Schwarz [20], the so-called Bayesian information criterion (BIC), and is given by
\begin{aligned} \text {BIC}=-2\ln \mathcal {L}+2d\ln N, \end{aligned}
(9)
where $$N$$ is the number of the data points. The best model from a set under consideration is the one which minimizes the BIC quantity. One can notice the similarity between the AIC and BIC quantities, though they come from different approaches to the model selection problem. The dissimilarity is seen in the so-called penalty term: $$ad$$, which penalizes more complex models (complexity is identified here as the number of free model parameters). One can evaluate the factor by which the additional parameter must improve the goodness of fit to be included in the model. This factor must be greater than $$a$$; equal to $$2$$ in the AIC case and equal to $$\ln N$$ in the BIC case. Notice that the latter depends on the number of data points.
It can be shown that there is the simple relation between the BIC and the Bayes factor,
\begin{aligned} 2 \ln B_{ij} = -(\text {BIC}_i - \text {BIC}_j). \end{aligned}
(10)
The quantity $$B_{ij}$$ is the Bayes factor for the hypothesis (model) $$i$$ against the hypothesis (model) $$j$$. We categorize this evidence against the model $$j$$ taking the following ranking. The evidence against the model $$j$$ is not worth than bare mentioning when twice the natural logarithm of the Bayes factor (or minus the difference between BICs) is $$0< 2\ln B_{ij} \le 2$$, is positive when $$2< 2\ln B_{ij} \le 6$$, is strong when $$6< 2\ln B_{ij}\le 10$$, and is very strong when $$2\ln B_{ij} > 10$$.
It should be pointed out that the model selection methods presented are widely used in the context of cosmological model comparisons [18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. We should keep in mind that the conclusions based on such quantities depend on the data at hand. Let us mention again the example with the black box. Suppose that we made a few steps toward this box so that we can see the difference between the height of the left and right side of the white box. Our conclusion changes now.
Let us quote the example taken from [30]. Assume that we want to compare the Newtonian and Einsteinian theories in the light of the data coming from a laboratory experiment where general relativistic effects are negligible. In this situation the Bayes factor between Newtonian and Einsteinian theories will be close to unity. But comparing the general relativistic and Newtonian explanations of the deflection of a light ray that just grazes the Sun’s surface gives the Bayes factor $$\sim 10^{10}$$ in favor of the first one (and even greater with more accurate data).
We share George Efstathiou’s opinion [41, 42, 43] that there is no sound theoretical basis for considering dynamical dark energy, whereas we are beginning to see an explanation for a small cosmological constant emerging from a more fundamental theory. In our opinion the $$\Lambda$$CDM model has the status of a satisfactory effective theory. Efstathiou argued why the cosmological constant should be given a higher weight as a candidate for a dark energy description than the dynamical dark energy. In this argumentation Occam’s principle is used to point out a more economical model explaining the observational data.
The main aim of this paper is to compare the simplest cosmological model—the $$\Lambda$$CDM model—with its generalization where the interaction between dark energy and matter sectors is allowed using the methods described above.
## 2 Interacting $$\Lambda$$CDM model
The interaction interpretation of the continuity condition (conservation condition) was investigated in the context of the coincidence problem since the paper Zimdahl [44], for recent developments in this area see Olivares et al. [45, 46]; see also Le Delliou et al. [47] for a discussion of recent observational constraints.
Let us consider two basic equations which determine the evolution of FRW cosmological models,
\begin{aligned} \frac{\ddot{a}}{a}&=-\frac{1}{6}(\rho +3p) ,\end{aligned}
(11)
\begin{aligned} \dot{\rho }&=-3H(\rho +p). \end{aligned}
(12)
Equation (11) is called the acceleration equation and Eq. (12) is the conservation (or adiabatic) condition. Equation (11) can be rewritten in a form analogous to the Newtonian equation of motion,
\begin{aligned} \ddot{a}=-\frac{\partial V}{\partial a}, \end{aligned}
(13)
where $$V=V(a)$$ is potential function of the scale factor $$a$$. To evaluate $$V(a)$$ from (13) via integration by parts it is useful to rewrite (12) in the new equivalent form
\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}(\rho a^3) + p \frac{\mathrm{d}}{\mathrm{d}t}(a^3)=0. \end{aligned}
(14)
From (11) we obtain
\begin{aligned} \frac{\partial V}{\partial a}=\frac{1}{12}(\rho +3p)\mathrm{d}(a^2). \end{aligned}
(15)
It is convenient to calculate the pressure $$p$$ from (14) and then substitute into (15). After simple calculations we obtain from (15)
\begin{aligned} \frac{\partial V}{\partial a} = - \frac{1}{6}\left[ a^2 \frac{\mathrm{d}\rho }{\mathrm{d}a}+\rho \mathrm{d}(a^2) \right] . \end{aligned}
(16)
Therefore
\begin{aligned} V(a)=-\frac{\rho a^2}{6}. \end{aligned}
(17)
In Eq. (17) $$\rho$$ means the effective energy density of the fluid filling the Universe.
We find a very simple interpretation of (11): the evolution of the Universe is equivalent to the motion of a particle of unit mass in the potential well parameterized by the scale factor. In the procedure of the reduction of the problem of the FRW evolution to the problem of the investigation of a dynamical system of a Newtonian type we only assume that the effective energy density satisfies the conservation condition. We do not assume the conservation condition for each energy component (or non-interacting matter sectors).
Equations (11) and (12) admit the first integral which is usually called the Friedmann first integral. This first integral has a simple interpretation in the particle-like description of the FRW cosmology, namely energy conservation. We have
\begin{aligned} \frac{\dot{a}^2}{2}+V(a)=E=-\frac{k}{2}, \end{aligned}
(18)
where $$k$$ is the curvature constant and $$V$$ is given by Eq. (17).
Let us consider the Universe filled with the two fluid components,
\begin{aligned} \rho =\rho _{{m}} + \rho _X, \quad p=0+w_X\rho _X, \end{aligned}
(19)
where $$\rho _{{m}}$$ means the energy density of the usual dust matter and $$\rho _X$$ denotes the energy density of dark energy satisfying the equation of state $$p_X=w_X\rho _X$$, where $$w_X=w_X(a)$$. Then Eq. (14) can be separated into the dark matter and dark energy sectors, which in general can interact
\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}(\rho _{{m}} a^3) + 0 \cdot \frac{\mathrm{d}}{\mathrm{d}t}(a^3)=\Gamma ,\end{aligned}
(20)
\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}(\rho _X a^3) + w_X(a)\rho _X \frac{\mathrm{d}}{\mathrm{d}t}(a^3)=-\Gamma . \end{aligned}
(21)
In our previous paper [48] it was assumed that
\begin{aligned} \Gamma =\alpha a^n \frac{\dot{a}}{a}, \end{aligned}
(22)
which enables us to integrate (20), which gives
\begin{aligned}&\rho _m = \frac{C}{a^3}+\frac{\alpha }{n}a^{n-3},\end{aligned}
(23)
\begin{aligned}&\frac{\mathrm{d} \rho _X}{\mathrm{d}a}+\frac{3}{a}(1+w_X(a))\rho _X=-\alpha a^{n-4}. \end{aligned}
(24)
The solution of the homogeneous equation (24) can be written in terms of the average $$\overline{w_X}(a)$$ as
\begin{aligned} \rho _X=\rho _{X,0}a^{-3(1+\overline{w_X}(a))}, \end{aligned}
(25)
where
\begin{aligned} \overline{w_X}(a)=\frac{\int w_X(a)\mathrm{d}(\ln a)}{\mathrm{d}(\ln a)}. \end{aligned}
(26)
The solution of the nonhomogeneous equation (24) is
\begin{aligned} \rho _X&= - \left[ \int _1^a a^{n-1+3\overline{w_X}(a)}\mathrm{d}a\right] a^{-3(1+\overline{w_X}(a))} \nonumber \\&\quad +\, \frac{C_X}{a^{3(1+\overline{w_X}(a))}}. \end{aligned}
(27)
Finally we obtain
\begin{aligned} \rho _{\text {eff}}&\equiv 3H^2 +3\frac{k}{a^2}=\rho _m+\rho _{X} \nonumber \\&= \frac{C_m}{a^3}+\frac{\alpha }{n}a^{n-3} +\frac{C_X}{a^{3(1+\overline{w_X}(a))}} \nonumber \\&- \left[ \int _1^a a^{n-1+3\overline{w_X}(a)}\mathrm{d}a\right] a^{-3(1+\overline{w_X}(a))}. \end{aligned}
(28)
The second and last terms originate from the interaction between the dark matter and dark energy sectors.
Let us consider the simplest case of $$\overline{w_X}(a)=$$ const$$~=w_X(a)$$. Then integration of (27) can be performed and we obtain
\begin{aligned} \rho _{\text {eff}}=\frac{C_m}{a^3}+\frac{C_X}{a^{3(1+w_X)}}+\frac{C_\text {int}}{a^{3-n}} \end{aligned}
(29)
where $$C_{\text {int}}=\frac{\alpha }{n}-\frac{\alpha }{n-3w_X}$$. In this case we obtain one additional term in $$\rho _{\text {eff}}$$ or in the Friedmann first integral scaling like $$a^{2-n}$$. It is convenient to rewrite the Friedmann first integral in a new form, using dimensionless density parameters. Then we obtain
\begin{aligned} \left( \frac{H}{H_0}\right) ^2&= \Omega _{{m},0}(1+z)^3+\Omega _{k,0}(1+z)^2 \nonumber \\&\quad +\, \Omega _{\text {int}}(1+z)^{3-n} + \Omega _{X,0}(1+z)^{3(1+w_X)}. \end{aligned}
(30)
Note that this additional power law term related to interaction can also be interpreted as the Cardassian or polytropic term [49, 50] (one can easily show that the assumed form of the interaction always generates a correction of type $$a^m, m=1-n$$, in the potential of the $$\Lambda$$CDM model and vice versa). Another interpretation of this term might originate from the Lambda decaying cosmology when the Lambda term is parametrized by the scale factor [51].
In the next section we draw a comparison between the above model with the assumption that $$\overline{w_X}(a)=\text {const}=-1$$ and the $$\Lambda$$CDM model.
## 3 Data
To estimate the parameters of the two models we used for our purposes the modified CosmoMC code [52, 53] with the implemented nested sampling algorithm Multinest [54, 55].
We used the observational data of 580 type Ia supernovae (the Union2.1 compilation [56]), 31 observational data points of the Hubble function from [57, 58, 59, 60, 61, 62, 63, 64, 65, 66] collected in [67], the measurements of BAO from the Sloan Digital Sky Survey (SDSS-III) combined with the 2dF Galaxy Redshift Survey [68, 69, 70, 71], the 6dF Galaxy Survey [72, 73], and the WiggleZ Dark Energy Survey [74, 75, 76]. We also used information coming from determinations of the Hubble function using the Alcock–Paczyński test [77, 78]. This test is very restrictive in the context of modified gravity models.
The likelihood function for the type Ia supernova data is defined by
\begin{aligned} L_{\text {SN}} \propto \exp \left[ \! -\! \sum _{i,j}\left( \mu _{i}^{\text {obs}} \!-\! \mu _{i}^{\text {th}}\right) C_{ij}^{-1} \left( \mu _{j}^{\text {obs}} \!-\! \mu _{j}^{\mathrm {th}}\right) \right] , \end{aligned}
(31)
where $$C_{ij}$$ is the covariance matrix with the systematic errors, $$\mu _{i}^{\text {obs}}=m_{i}-M$$ is the distance modulus, $$\mu _{i}^{\text {th}}=5\log _{10}D_{Li} + \mathcal {M}=5\log _{10}d_{Li} + 25$$, $$\mathcal {M}=-5\log _{10}H_{0}+25$$ and $$D_{Li}=H_{0}d_{Li}$$, where $$d_{Li}$$ is the luminosity distance which is given by $$d_{Li}=(1+z_{i})c\int _{0}^{z_{i}} \frac{dz'}{H(z')}$$ (with the assumption $$k=0$$).
For $$H(z)$$ the likelihood function is given by
\begin{aligned} L_{H_z} \propto \exp \left[ - \sum _i\frac{\left( H^{\text {th}}(z_i)-H^{\text {obs}}_i\right) ^2}{2 \sigma _i^2} \right] , \end{aligned}
(32)
where $$H^{\text {th}}(z_i)$$ denotes the theoretically estimated Hubble function, $$H^{\text {obs}}_i$$ is observational data.
The likelihood function for the BAO data is characterized by
\begin{aligned} L_{\text {BAO}} \propto \exp \left[ - \sum _{i,j}\left( d^{\text {th}}(z_i)-d^{\text {obs}}_i\right) C_{ij}^{-1} \left( d^{\text {th}}(z_j)-d^{\text {obs}}_j\right) \right] \end{aligned}
(33)
where $$C_{ij}$$ is the covariance matrix with the systematic errors, $$d^{\text {th}}(z_i)\equiv r_s(z_d) \left[ (1+z_i)^2 D_\mathrm{A}^2(z_i)\frac{cz_i}{H(z_i)} \right] ^{-\frac{1}{3}}$$, $$r_s(z_d)$$ is the sound horizon at the drag epoch, and $$D_\mathrm{A}$$ is the angular diameter distance.
The likelihood function for the information coming from the Alcock–Paczyński test is given by
\begin{aligned} L_{AP} \propto \exp \left[ - \sum _i\frac{\left( \mathrm{AP}^{\text {th}}(z_i)-\mathrm{AP}^{\text {obs}}_i\right) ^2}{2 \sigma _i^2} \right] \end{aligned}
(34)
where $$\mathrm{AP}^{\text {th}}(z_i)\equiv \frac{H(z_i)}{H_0 (1+z_i)}$$.
Finally, we used likelihood function for the CMB shift parameter $$R$$ [79], which is defined by
\begin{aligned} L_{\text {CMB}} \propto \exp \left[ -\frac{1}{2}\frac{(R^{\text {th}}-R^{\text {obs}})^2}{\sigma _{\mathcal {A}}^2} \right] \end{aligned}
(35)
where $$R^{\text {th}}=\frac{\sqrt{\Omega _{{m}} H_0}}{c}(1+z_{*})D_\mathcal {A}(z_{*})$$, $$D_\mathcal {A} (z_{*})$$ is the angular diameter distance to the last scattering surface, $$R^{\text {obs}}=1.7477$$ and $$\sigma _{\mathcal {A}}^{-2}=48976.33$$ [80].
The total likelihood function $$L_{\text {tot}}$$ is defined as
\begin{aligned} L_{\text {tot}}=L_{\text {SN}}L_{H_z}L_{\text {BAO}}L_{\text {CMB}}L_{\mathrm{AP}}. \end{aligned}
(36)
## 4 Results
### 4.1 The model parameter estimation
The results of the estimation of the parameters of the $$\Lambda$$CDM and the interacting $$\Lambda$$CDM models are presented in Table 1. Given the likelihood function (31), first, we estimated the models parameters using the Union2.1 data only. Next, the parameter estimations with the joint data of the Union2.1, $$h(z)$$, BAO, Alcock–Paczyński test [likelihood functions (31)–(34)] have been performed. Finally, we estimated the model parameters with the joint data enlarged with the CMB data [the total likelihood function (36)].
Table 1
The mean of marginalized posterior PDF with 68 % confidence level for the parameters of the models. In the brackets are shown parameter values of the joint posterior probabilities. Estimations were made using the Union2.1, $$h(z)$$, BAO, determinations of Hubble function using the Alcock–Paczyński test and the CMB $$R$$ data sets
Union2.1 data only
Union2.1, $$h(z)$$, BAO, AP data
Union2.1, $$h(z)$$, BAO, AP, CMB data
Interacting model
$$\Omega _{{m},0} \in \langle 0,1 \rangle$$
$$0.3126^{+0.0064}_{-0.0343} \,(0.2952)$$
$$0.2770^{+0.0119}_{-0.0130} \,(0.2690)$$
$$0.2847^{+0.0107}_{-0.0115} \,(0.2725)$$
$$\Omega _{\text {int},0} \in \langle -1,1 \rangle$$
$$-0.0232^{+0.1070}_{-0.1018} \,(-0.3492)$$
$$0.0109^{+0.0146}_{-0.0267} \,(0.0734)$$
$$-0.0139^{+0.0244}_{-0.0056} \,(-0.0152)$$
$$m \in \langle -10,10 \rangle$$
$$-0.2687^{+1.2726}_{-0.3223} \,(-0.0528)$$
$$0.5622^{+0.7790}_{-0.5499} \,(0.9911)$$
$$0.3205^{+0.7826}_{-0.6730} \,(3.7364)$$
$$h \in \langle 0.6,0.8 \rangle$$
$$0.7004^{+0.0996}_{-0.1004} \,(0.7912)$$
$$0.6949^{+0.0121}_{-0.0148} \,(0.6937)$$
$$0.6957^{+0.0120}_{-0.0147} \,(0.7093)$$
$$\Lambda$$CDM model
$$\Omega _{{m},0} \in \langle 0,1 \rangle$$
$$0.2956^{+0.0035}_{-0.0034} \,(0.2955)$$
$$0.2777^{+0.0070}_{-0.0073} \,(0.2791)$$
$$0.2912^{+0.0043}_{-0.0045} \,(0.2904)$$
$$h \in \langle 0.6,0.8 \rangle$$
$$0.7000^{+0.1000}_{-0.1000} \,(0.6053)$$
$$0.6932^{+0.0048}_{-0.0049} \,(0.6922)$$
$$0.6858^{+0.0041}_{-0.0043} \,(0.6849)$$
The value of the interaction parameter $$\Omega _{\text {int},0}$$ is very small for all data sets. Especially the result for the second data set [Union2.1, $$h(z)$$, BAO, AP data] indicates that the interaction is probably negligible. There is also no indication of the direction of the interaction if it is a physical effect. While for the Union2.1 data set only the interaction parameter $$\Omega _{\text {int},0}$$ is negative and a greater value of $$\Omega _{{m},0}$$ in the interacting $$\Lambda$$ CDM model implies the flow from the dark energy sector to the matter sector, and for the data set consisting of all data the opposite.
The uncertainty of the each estimated model parameter is presented twofold: as 68 % confidence levels in Table 1 and as the marginalized probability distributions in Figs. 2 and 3.
### 4.2 The likelihood ratio test
We begin our statistical analysis with the likelihood ratio test. In this test one of the models (null model) is nested in a second model (alternative model) by fixing one of the second model parameters. In our case the null model is the $$\Lambda$$CDM model, the alternative model is the interactive $$\Lambda$$CDM model, and the parameter in question is $$\Omega _{\text {int}}$$. We have
\begin{aligned} H_0 :\Omega _{\text {int}}&= 0, \\ H_1 :\Omega _{\text {int}}&\ne 0. \end{aligned}
The statistic is given by
\begin{aligned} \lambda = 2 \ln \left( \frac{L(H_1|D)}{L(H_0|D)} \right) = 2\left( \frac{\chi ^2_{\text {int}}}{2} - \frac{\chi ^2_{\Lambda \text {CDM}}}{2} \right) \end{aligned}
(37)
where $$L(H_1|D)$$ is the likelihood of the interacting $$\Lambda$$CDM model, $$L(H_0|D)$$ is the likelihood of the $$\Lambda$$CDM model obtained using three different sets of data. The statistic $$\lambda$$ has the $$\chi ^2$$ distribution with $$df=n_1-n_0=2$$ degrees of freedom where $$n_1$$ is the number of parameters of the alternative model, $$n_0$$ is the number of parameters of the null model. The results are presented in Table 2. In all three cases the $$p$$ values are greater than the significance level $$\alpha = 0.05$$, which is why the null hypothesis cannot be rejected. In other words we cannot reject the hypothesis that there is no interaction between the dark matter and dark energy sector.
Table 2
The results of the likelihood ratio test for the $$\Lambda$$CDM model (null model) and the $$\Lambda$$CDM interacting model (alternative model). The values of $$\chi ^2_{\text {int}}$$, $$\chi ^2_{\Lambda \text {CDM}}$$, test statistic $$\lambda$$, and the corresponding $$p$$ value ($$df=4-2=2$$). Estimations were made using the Union2.1, $$h(z)$$, BAO, determinations of the Hubble function using the Alcock–Paczyński test and the CMB $$R$$ data sets
Data sets
$$\chi ^2_{\text {int}}/2$$
$$\chi ^2_{\Lambda \text {CDM}}/2$$
$$\lambda$$
$$p$$ value
Union2.1
$$272.5377$$
$$272.5552$$
$$0.0350$$
$$0.9826$$
Union2.1, $$h(z)$$, BAO, AP
$$282.2215$$
$$282.2555$$
$$0.0680$$
$$0.9667$$
Union2.1, $$h(z)$$, BAO, AP, CMB
$$282.3073$$
$$282.4912$$
$$0.3678$$
$$0.8320$$
### 4.3 The model comparison using the AIC, BIC, and Bayes evidence
To obtain the values of the AIC and BIC quantities we perform the $$\chi ^2=-2\ln L$$ minimization procedure after marginalization over the $$H_0$$ parameter in the range $$\langle 60,80 \rangle$$. They are presented in Table 3.
Table 3
Values of the $$\chi ^2$$, AIC, $$\Delta$$AIC (with respect to the $$\Lambda$$CDM model), BIC, and Bayes factor. Estimations were made using the Union2.1, $$h(z)$$, BAO, determinations of Hubble function using the Alcock–Paczyński test and the CMB $$R$$ data sets
Data sets
$$\chi ^2/2$$
AIC
$$\Delta \text {AIC}_{\text {int},\Lambda \text {CDM}}$$
BIC
$$2\ln B_{\Lambda \text {CDM,int}}$$
Interacting model
Union2.1
$$272.5377$$
$$553.0754$$
$$3.9650$$
$$570.5275$$
$$12.6910$$
Union2.1, $$h(z)$$, BAO, AP
$$282.2215$$
$$572.4430$$
$$3.9320$$
$$590.1683$$
$$12.7947$$
Union2.1, $$h(z)$$, BAO, AP, CMB
$$282.3073$$
$$572.6146$$
$$3.6322$$
$$590.3464$$
$$12.4981$$
$$\Lambda$$CDM model
Union2.1
$$272.5552$$
$$549.1104$$
$$557.8365$$
Union2.1, $$h(z)$$, BAO, AP
$$282.2555$$
$$568.5110$$
$$577.3736$$
Union2.1, $$h(z)$$, BAO, AP, CMB
$$282.4912$$
$$568.9824$$
$$577.8483$$
Regardless the data set the differences of the AIC quantities are in the interval $$(3.4, 4)$$ and are a little outside the interval $$(4,7)$$, which indicates the considerably smaller support for the interacting $$\Lambda$$CDM model. It means that while the $$\Lambda$$CDM model should be preferred over the interacting $$\Lambda$$CDM model, the latter cannot be ruled out.
However, we can arrive at a decisive conclusion employing the Bayes factor. The difference of BIC quantities is greater than 10 and have values in the interval $$(12,13)$$ for all data sets. Thus, the Bayes factor indicates strong evidence against the interacting $$\Lambda$$CDM model compared to the $$\Lambda$$CDM model. Therefore we are strongly convinced we should reject the interaction between dark energy and dark matter sectors due to Occam’s principle.
## 5 Conclusion
We considered the cosmological model with dark energy represented by the cosmological constant and the model with interaction between dark matter and dark energy (the interacting $$\Lambda$$CDM model). These models were studied statistically using the available astronomical data and then compared using the tools taken from information as well as Bayesian theory. In both cases the model selection is based on Occam’s principle, which states that if two models describe the observations equally well we should choose the simpler one. According to the Akaike and Bayesian information criteria the model complexity is interpreted in terms of the number of free model parameters, while according to the Bayesian evidence a more complex model has a greater volume of the parameter space.
Anyone using the Bayesian methods in astronomy and cosmology should be aware of the ongoing debate not only about pros but also cons of this approach. Efstathiou provided a critique of the evidence ratio approach indicating difficulties in defining models and priors [81]. Jenkins and Peacock [82] called attention to too much noise in the data, which does not allow one to decide to accept or reject a model based solely on whether the evidence ratio reaches some threshold value. That is the reason that we also used the AIC based on information theory.
The observational constraints on the parameter values, which we have obtained, have confirmed previous results that if the interaction between dark energy and matter is a real effect it should be very small. Therefore it seems to be natural to ask whether cosmology with interaction between dark energy and matter is plausible.
At the beginning of our model selection analysis we performed the standard likelihood ratio test. This test conclusion was to fail to reject the null hypothesis that there is no interaction between matter and dark energy sectors with the significance level $$\alpha =0.05$$. It was the first clue against the interacting $$\Lambda$$CDM model. The $$\Delta$$AIC between both models was less conclusive. While the $$\Lambda$$CDM model was more supported, the interacting $$\Lambda$$CDM cannot be rejected. On the other hand the Bayes factor has given a decisive result; there was a very strong evidence against the interacting $$\Lambda$$CDM model compared to the $$\Lambda$$CDM model. Given the weak or almost non-existing support for the interacting $$\Lambda$$CDM model and bearing in mind Occam’s razor we are inclined to reject this model.
We have also the theoretical argument against the interacting $$\Lambda$$CDM model. If we consider the $$H^2$$ formula which is a base for estimation there is a degeneracy because one cannot distinguish the effects of interaction from the effect $$w(z)$$—the case of varying equation of state depending on time or redshift.
As was noted by Kunz [83] there is a dark degeneracy problem. It means that the effect of interaction cannot be distinguished from the effect of an additional non-interacting fluid with the constant equation of state $$w_{\text {int}}=n/3-2$$. Therefore if we consider a mixture of all three non-interacting fluids we obtain the coefficient equation of state for the dark energy and interacting fluid in the form
\begin{aligned} w_{\text {dark}}&= \frac{(p_X+ p_{\text {int}})}{C_X(1+z)^{3(1+w_X)} + C_{\text {int}}(1+z)^{3-n}} \nonumber \\&= \frac{w_{X}(1+z)^{3(1+w_{X})} + C_{\text {int}}(1+z)^{3-n}w_{\text {int}}}{C_X(1+z)^{3(1+w_X)} + C_{\text {int}}(1+z)^{3-n}}. \end{aligned}
(38)
## Notes
### Acknowledgments
M. Szydłowski has been supported by the National Science Centre (Narodowe Centrum Nauki) Grant 2013/09/B/ST2/03455. M. Kamionka has been supported by the National Science Centre (Narodowe Centrum Nauki) Grant PRELUDIUM 2012/05/N/ST9/03857. We thank the referee for carefully going through our manuscript.
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## Authors and Affiliations
• Marek Szydłowski
• 1
• 3 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9368085265159607, "perplexity": 1202.1899177633568}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814787.54/warc/CC-MAIN-20180223134825-20180223154825-00090.warc.gz"} |
http://comunidadwindows.org/standard-deviation/standard-error-plus-minus-mean.php | Home > Standard Deviation > Standard Error Plus Minus Mean
Standard Error Plus Minus Mean
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Generated Sun, 30 Oct 2016 11:59:42 GMT by s_fl369 (squid/3.5.20) We know that 95% of these intervals will include the population parameter. Dividing the difference by the standard deviation gives 2.62/0.87 = 3.01. Economic Evaluations6. http://comunidadwindows.org/standard-deviation/standard-error-of-estimate-standard-deviation-of-residuals.php
Minus-plus sign The minus-plus sign (∓) is generally used in conjunction with the "±" sign, in such expressions as "x ± y ∓ z", which can be interpreted as meaning "x Minus-plus sign The minus-plus sign (∓) is generally used in conjunction with the "±" sign, in such expressions as "x ± y ∓ z", which can be interpreted as meaning "x For this purpose, she has obtained a random sample of 72 printers and 48 farm workers and calculated the mean and standard deviations, as shown in table 1. rgreq-8138eaae5a19cd0cbbdaefb600f69f26 false ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.9/ Connection to 0.0.0.9 failed. https://en.wikipedia.org/wiki/Plus-minus_sign
Mean Plus Or Minus Standard Deviation Excel
As noted above, if random samples are drawn from a population, their means will vary from one to another. My answer applies primarily to the usage quoted in the problem statement, i.e. Anything outside the range is regarded as abnormal. Swinscow TDV, and Campbell MJ.
Systematic Reviews5. Similarly, the trigonometric identity sin ( A ± B ) = sin ( A ) cos ( B ) ± cos ( A ) sin ( This section considers how precise these estimates may be. Minus Plus Sign We do not know the variation in the population so we use the variation in the sample as an estimate of it.
In TeX 'plus-or-minus' and 'minus-or-plus' symbols are denoted \pm and \mp, respectively. In this example, n = 5. Otherwise, it's only approximate, but its finite-sample performance is generally better than when using a z-statistic.tl;dr: the purpose of a sample is to estimate some property of the population from which https://en.wikipedia.org/wiki/Plus-minus_sign up vote 4 down vote favorite I have represented standard deviation as "±SD" before in publications.
A better method would be to use a chi-squared test, which is to be discussed in a later module. Plus Minus Symbol Word more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed This indicates that the equation has two solutions, each of which may be obtained by replacing this equation by one of the two equations x = +1 or x = −1. Only in the particular situation where the student was not clear about the exact requirement of the teacher, he went for an optimum solution.
Plus Or Minus Standard Deviation Or Standard Error
For many biological variables, they define what is regarded as the normal (meaning standard or typical) range. By using this site, you agree to the Terms of Use and Privacy Policy. Mean Plus Or Minus Standard Deviation Excel On Unix-like systems, it can be entered by typing the sequence compose + -. Mean Plus Minus Standard Deviation Calculator Typing On Windows systems, it may be entered by means of Alt codes, by holding the ALT key while typing the numbers 0177 or 241 on the numeric keypad.
On the other hand, if there are two instances of the "±" sign in an expression, it is impossible to tell from notation alone whether the intended interpretation is as two http://comunidadwindows.org/standard-deviation/standard-error-or-standard-deviation-on-graphs.php Besides, there's no way that you are going back to him in order to ask what measurement did he actually ask for since there is a high chance that he gave The "±" symbol is used to signal that the population quantity we are attempting to estimate from our sample is in some sense likely to lie in the interval [x - The rarer minus-plus sign (∓) is not generally found in legacy encodings and does not have a named HTML entity but is available in Unicode with codepoint U+2213 and so can How To Calculate Mean Plus Or Minus Standard Deviation
However, the more common chess notation would be only + and –. [3] If a difference is made, the symbol + and − denote a larger advantage than ± and ∓. To take another example, the mean diastolic blood pressure of printers was found to be 88 mmHg and the standard deviation 4.5 mmHg. Basically, you found a way to report an optimum answer while specifying the upper and lower limit to be as accurate as possible.Please note, I am not saying that it is his comment is here Video 1: A video summarising confidence intervals. (This video footage is taken from an external site.
Solutions? Plus Minus Symbol Keyboard Shortcut BMJ Books 2009, Statistics at Square One, 10 th ed. In chess, the sign indicates a clear advantage for the white player; the complementary sign ∓ indicates the same advantage for the black player.[3] The sign is normally pronounced "plus or
Confidence intervals The means and their standard errors can be treated in a similar fashion.
In this sense, a confidence interval tells you the set of true population means that are "likely" to be consistent with your sample.Finally, if you replace sigma/sqrt(N) with a consistent estimator In it, you'll get: The week's top questions and answers Important community announcements Questions that need answers see an example newsletter By subscribing, you agree to the privacy policy and terms On Macintosh systems, it may be entered by pressing option shift = (on the non-numeric keypad). Mean Plus Or Minus 2 Standard Deviations One of the printers had a diastolic blood pressure of 100 mmHg.
The ± notation helped him be closer to accuracy inspite of not knowing the exact amount of error he was including. The mean plus or minus 1.96 times its standard deviation gives the following two figures: We can say therefore that only 1 in 20 (or 5%) of printers in the population Given a sample of disease free subjects, an alternative method of defining a normal range would be simply to define points that exclude 2.5% of subjects at the top end and weblink Example 1 A general practitioner has been investigating whether the diastolic blood pressure of men aged 20-44 differs between printers and farm workers.
For example, 230 ± 10% V refers to a voltage within 10% of either side of 230V (207V to 253V). What does that "0.4 +- 0.33 mg/dl" thing mean ?UpdateCancelAnswer Wiki5 Answers Erik Madsen, A dismal scientistWritten 200w ago · Upvoted by Justin Rising, PhD in statisticsAt a very basic level, Similarly, the trigonometric identity sin ( A ± B ) = sin ( A ) cos ( B ) ± cos ( A ) sin ( Another example is x 3 ± 1 = ( x ± 1 ) ( x 2 ∓ x + 1 ) {\displaystyle x^{3}\pm 1=(x\pm 1)(x^{2}\mp x+1)} which represents two equations.
In statistics The use of ⟨±⟩ for an approximation is most commonly encountered in presenting the numerical value of a quantity together with its tolerance or its statistical margin of error.[1] Smart fellow, since by covering the maximum measure, minimum measure, as well as the optimum measure, he didn't leave any scope for the teacher to call his result "wrong". :)1.2k Views This can be proven mathematically and is known as the "Central Limit Theorem". In contrast to the quadratic formula example, both of the equations described by this identity are simultaneously valid.
Privacy policy About Wikipedia Disclaimers Contact Wikipedia Developers Cookie statement Mobile view Mean Values and Error Estimates Contents: Mean Value Variance and Standard Deviation Standard Deviation of the Mean Stating the That is, the maximum diameter that could be measured is 12 cm (outer circle), while the minimum is 10 cm. This probability is usually used expressed as a fraction of 1 rather than of 100, and written as p Standard deviations thus set limits about which probability statements can be made. In chess The symbols ± and ∓ are used in chess notation to denote an advantage for white and black respectively.
Finding the Evidence3. In chess, the sign indicates a clear advantage for the white player; the complementary sign ∓ indicates the same advantage for the black player.[3] The sign is normally pronounced "plus or We can conclude that males are more likely to get appendicitis than females. Please try the request again.
Generated Sun, 30 Oct 2016 11:59:42 GMT by s_fl369 (squid/3.5.20) ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.10/ Connection Did the teacher ask for the external diameter or the internal diameter? What would you call "razor blade"? Hot Network Questions Pythagorean Triple Sequence Raise equation number position from new line Who calls for rolls?
But I like to have opinions on this. By using this site, you agree to the Terms of Use and Privacy Policy. These characters may also be produced as an underlined or overlined + symbol (+ or +), but beware of the formatting being stripped at a later date, changing the meaning. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8077653050422668, "perplexity": 1260.3440561441043}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891815918.89/warc/CC-MAIN-20180224172043-20180224192043-00107.warc.gz"} |
https://www.physicsforums.com/insights/renormalisation-needs-cutoff/ | # Why Renormalisation in Quantum Theory Needs a Cutoff
## Introduction
This is a follow on from my paper explaining renormalization. A question was raised – why exactly do we need a cut-off. There is a deep reason to do with dimensional analysis, and the power series expansion used in perturbation theory. Along the way, we will see renormalization in a more general setting, and exactly why logarithms, like in the previous paper, so often crop up.
## A More General Look At Renormalisation
Suppose we have a function G(x) that depends on some parameter λ ie G(x,λ). Then, so perturbation theory can be used, expand it in a power series about λ:
G(x) = G0 + G1(x)*λ + G2(x)*λ^2 + ……..
In perturbation theory, for theoretical convenience, it is usual to define a new function F(x) = (G(x) – G0)/G1 so:
F(x) = λ + F2(x) *λ^2 + ……..
It makes things like formal inversion etc easier. This seems a pretty innocuous thing to do, but from dimensional analysis, has a consequence that lies at the heart of where QFT infinities come from, and the need for a cut-off. To see this, suppose x has some kind of dimension such as, for example, momentum squared, and λ is dimensionless. A number of theories fall into this class; such as:
quantum electrodynamics where the fine structure constant is dimensionless, and only high energies are considered, so the electron mass is negligible;
the Weinberg-Salam model of electroweak interactions;
the meson theory, again at high energies so the mass is negligible, used as an example in my previous paper. K^2 has dimensions of momentum squared and the coupling constant is dimensionless.
Suppose λ is small, then F(x) = λ, F has the dimensions of λ, so is dimensionless. This is also seen by its definition where G(x) – G0 is divided by G1(x). But lets expand F2(x) in a power series about x so F2(x) = F20 + F21*x + F22*x^2 + ……. = F20 + F21*x + O(x^2). Suppose x is small, so O(x^2) can be neglected, then F2(x) has the dimensions of x, hence to the second order of λ, F(x) has the dimensions of x. Here we have a dimensional mismatch. This is the exact reason the equations blow up – in order for it to be dimensionless it cant depend on x. This can only happen if F2(x) is a constant or infinity. Either of course is death for our theory – but nature seemed to choose infinity – the reason for which will be examined later.
Now for the solution. The only way to avoid this is to divide x by some parameter, Λ, of the same units as x, so it becomes dimensionless.
The correct equation is:
F(x/Λ) = λ + F2(x/Λ) *λ^2 + F2(x/Λ) *λ^2 +………+ Fi(x/Λ) *λ^i + ……………
We see, due to dimensional analysis of the perturbation methods used, we have neglected a parameter in our theory, which can be interpreted as a cut-off. It is this oversight that has caused the trouble all along.
## Consequence Of The Introduction Of Λ
To second order we have F(x/Λ) = λ + F2(x/Λ) *λ^2. (1)
The issue is while we know there is a Λ, we do not know its value so, as per the example in my previous paper, we want a formula without it. Similar to what was done before, we define the renormalized coupling constant:
λr = F(u/Λ) = λ + F2(u/Λ) *λ^2. (2)
Here u is some arbitrarily chosen value of x that yields a value of λr that can be measured.
Subtracting (2) from (1), and noting that to second-order λr^2 = λ^2, we get:
F(x/Λ) = λr + (F2(x/Λ) – F2(u/Λ))*λr^2.
We want this to not depend on Λ, so F2(x/Λ) – F2(u/Λ) = f(x,u), where f(x,u) depends on x and u, but not Λ. Theories, where this works to eliminate Λ, are called renormalizable. Not all theories are renormalizable – but as we will see, if it is, this imposes restrictions on the equations.
Let g(x) = f(x,1) = F2(x/Λ) – F2(1/Λ) ⇒f(x,u) = g(x) – g(u). Let K(x) = F2(x/Λ) – g(x) ⇒ K(x) – K(u) = 0 ⇒ K(x) = K(u). But since x and u are independent, K can’t depend on x or u, so must only depend on Λ ie K = K(Λ). Thus:
F2(x/Λ) = g(x) + K(Λ).
We see the renormalization condition, which is basically we want to get rid of the unknown Λ, determines the form of F2(x/Λ), namely it is the sum of a function of x and a function of Λ. The reason renormalization works are when you subtract (2) from (1) the Λ dependant term cancels.
## Why You Get Logarithms
An interesting consequence of this is it must involve logarithms. That the meson/meson scattering formula in the previous paper contained them is no accident.
Taking the derivative wrt to x in F2(x/Λ) = g(x) + K(Λ) ⇒ F2′(x/Λ)/Λ = g'(x). Let x =1. F2′(1/Λ)/Λ = g'(1) which will be called -α, where its conventional to use a minus sign because that’s what tends to occur in equations, such as the C in the previous paper. Let 1/Λ = y ⇒ F2′(y) = -α/y whose solution is F2(y) = -α*log(y) + C.
Hence we have:
F2(x/Λ) = -α*log (x/Λ) + C = α*log (Λ/x) + C = α*log (Λ) – α*log (x) + C.
As promised we see that α occurs in α*log (Λ) like the meson/meson scattering equation; justifying the negative sign.
This has exactly the same form as the equation for meson/meson scattering in the previous paper. However it can be simplified further to eliminate C. This is done by subtracting C*λ^2 from F(x/Λ) to give F(x/Λ) – C*λ^2. Using this new F we have:
F2(x/Λ) = α*log (Λ/x) = α*log (Λ) – α*log (x).
## Why Did This Take So Long To Sort Out
We have seen the use of perturbation theory secretly requires another parameter to make sense. If you don’t include it, dimensional analysis shows you will get nonsense, with this nonsense manifesting in the infinities.
Even worse was an incorrect assumption about the coupling constant λ. Measurements showed it was much less than 1, so it looked reasonable to use in a perturbation expansion. But now we know there is a neglected parameter, Λ, in our equations, let’s look at what happens to λ when that is taken into account.
To second order:
λ = λr + a*λr^2
F(x/Λ) = λ + α*log(Λ/x)*λ^2 = λr + a*λr^2 + α*log(Λ/x)*λr^2 = λr + (α*log(Λ/x) + a)*λr^2
But λr = F(u/Λ) = λr + (α*log(Λ/u) + a)*λr^2 ⇒ α*log(Λ/u) + a = 0 ⇒ a = -α*log(Λ/u) = α*log(u/Λ). Hence:
λ = λr + a*λr^2 = λr + α*log(u/Λ)*λr^2.
We see the coupling constant depends on this new parameter. Now, making the reasonable interpretation of Λ as a cut-off let’s remove it by taking the limit at infinity similar to the previous paper. When this is done, we see to first order, the coupling constant λ = λr, so, in our first order calculations, no problem arose. But at second order it blows up to -∞. It’s also interesting to note the other reasonable choice to get rid of Λ, taking the limit to zero, also leads to it blowing up – this time to ∞.
In perturbation theory, you want what we perturb about to be much less than one. But for it to actually be infinite – that’s really, really bad, and no wonder you get nonsense infinite answers.
Measurements gave small values of the coupling constant, which from the above equation, means Λ isn’t too large, or small. This is what fooled people all those years.
## Conclusion
We have seen there is a secret parameter in our theories required by dimensional analysis. The inclusion of this parameter, and the renormalization condition, leads to them having a certain form. For theories with a dimensional parameter, and a dimensionless coupling constant, to second-order it is F(x/Λ) = λ + α*log(Λ/x)*λ^2.
It’s very interesting that dimensional considerations show why there is a parameter missing. When it’s not introduced, you get nonsense. If it’s included, then requiring our equations to be renormalizable, constrains its form.
I posted the following paper before:
A Hint Of Renormalisation
It extends these ideas a lot further by calculating higher-order terms and investigating the important renormalization group. The trouble is it has a few (relatively minor) errors and isn’t 100% clear what’s going on in some areas.
I hope to do some further papers giving the third and higher-order terms, plus the renormalization group.
31 replies
1. atyy says:
[QUOTE=”Jimster41, post: 5120363, member: 517770″]I was associating the “cutoff” with a boundary on the space of “possible paths”, limiting the integration to run only over the ones with some minimum probability, rather than an infinite number of them.[/QUOTE]
The Feynman path integral must always sum over all paths (or all energies). We can split the sum into “partial” sums over high, intermediate and low energies, and each partial sum must enter the final sum. But how can we sum over all paths when we don’t know the true theory of everything? We cannot, but we can guess. The cutoff represents our guess as to what the sum looks like after the partial sum over the high energies (strings or whatever) has been done.
2. Jimster41 says:
I had the Feynman “Path Integral Formulation” roughly in mind, and I had a connection between his “probability amplitudes” and Schrodinger’s wave equation.
I was associating the “cutoff” with a boundary on the space of “possible paths”, limiting the integration to run only over the ones with some minimum probability, rather than an infinite number of them.
And that at the heart of that approach was a dependence on the natural “diffusivity” of the wave equation.
3. bhobba says:
[QUOTE=”atyy, post: 5120343, member: 123698″]For example, in quantum electrodynamics, the “absolutely necessary” (nonsharp) cutoff is given by the Landau pole and lies above the Planck scale. However, reality may intervene way before that, and cause our theory to be false. In particular, we expect quantum gravity to render QED already false below the Planck scale.[/QUOTE]
There is no may in QED – long before then the electroweak theory takes over – but for the electroweak theory, which I believe has its own Landau pole, that looks a real issue.
Thanks
Bill
4. bhobba says:
[QUOTE=”Jimster41, post: 5120323, member: 517770″]Is “the cutoff” whatever it needs to be, dimensionally, and in value, depending on the specific problem being solved?[/QUOTE]
In Wilson’s approach its an actual value – but the cut-off used depends on the coupling constant chosen and the regime you are working with. This will be a lot clearer when I do my paper on renormalisation group flow.
[QUOTE=”Jimster41, post: 5120323, member: 517770″]Is it correct to say that the cut-off is really along the “probability” axis regardless of the dimension in question?[/QUOTE]
I don’t know what you mean by that.
Thanks
Bill
5. atyy says:
[QUOTE=”Jimster41, post: 5120323, member: 517770″]Is “the cutoff” whatever it needs to be, dimensionally, and in value, depending on the specific problem being solved?
Or, is there a canonical “maximal/minimal” cutoff. For some reason I was thinking it was thelinit given by the Planck constant.
Is it correct to say that the cutoff is really along the “probability” axis regardless of the dimension in question? That renormalization is drawing a boundary around the peak of the “probability” wave. Even though the wave keeps going, the probability of observation is always? (or “generally”)decreasing, so introducing a “cutoff” is practical, and is just about how infinitesimal a degree of uncertainty is tolerable.
If this is the case, is it true, do I understand correctly that “many body” case is more worrisome?[/QUOTE]
First, the cutoff is not sharp, since it just represents an energy above which new degrees of freedom must enter. Roughly, the “absolutely necessary” (nonsharp) cutoff is determined by our guess of the low energy theory.
For example, in quantum electrodynamics, the “absolutely necessary” (nonsharp) cutoff is given by the Landau pole and lies above the Planck scale. However, reality may intervene way before that, and cause our theory to be false. In particular, we expect quantum gravity to render QED already false below the Planck scale.
Of course, the Planck scale is again the “absolutely necessary” (nonsharp) cutoff for quantum general relativity. But if there are low energy stringy effects, then quantum general relativity will be false even at energies far below the Plack scale.
6. bhobba says:
[QUOTE=”nrqed, post: 5120287, member: 15416″]Ah, your question was about what people in the field prefer to use to regularize their integrals?[/QUOTE]
That’s it. But just for the heck of it I did the integral from the table – yuck.
Interestingly, in the high energy regime its log (Λ^2/K^2) + ∫ log (α – α^2) dα – where the integral is from 0 to 1 and I dropped the C. ∫ log (α – α^2) dα is a bit nasty as well – but its a constant whatever it is. This is the constant I got in my paper from solving the differential equation for F2.
Thanks
Bill
7. Jimster41 says:
Is “the cutoff” whatever it needs to be, dimensionally, and in value, depending on the specific problem being solved?
Or, is there a canonical “maximal/minimal” cutoff. For some reason I was thinking it was thelinit given by the Planck constant.
Is it correct to say that the cutoff is really along the “probability” axis regardless of the dimension in question? That renormalization is drawing a boundary around the peak of the “probability” wave. Even though the wave keeps going, the probability of observation is always? (or “generally”)decreasing, so introducing a “cutoff” is practical, and is just about how infinitesimal a degree of uncertainty is tolerable.
If this is the case, is it true, do I understand correctly that “many body” case is more worrisome?
8. nrqed says:
[QUOTE=”bhobba, post: 5120280, member: 366323″]Do – yes – its actually in a table of integrals – plug and chug.
Thanks
Bill[/QUOTE]
Ah, your question was about what people in the field prefer to use to regularize their integrals? Then yes, dimensional regularization is almost always used, in practice. In an abelian gauge theory like QED, one may use a Pauli-Villars regularization but in non abelian gauge theories, one uses dimensional regularization and then, by habit, people use dim reg everywhere (after introducing the gauge fixing terms, Fadeed-Popov ghosts, etc, in the lagrangian)
9. bhobba says:
[QUOTE=”nrqed, post: 5120257, member: 15416″]You can use the online Mathematica integrator, for example.[/QUOTE]
Do – yes – its actually in a table of integrals – plug and chug.
Thanks
Bill
10. nrqed says:
[QUOTE=”bhobba, post: 5120236, member: 366323″]Hi Guys
Just a quick question how one actually does integrals like that. Zee used a Pauli-Villiars approximation and even then you end up with
C* ∫ log (Λ^2/(m^2 – α(1-α)*K^2) dα where the integral is from 0 to 1
Do you do it in dimensional regularisation?
Thanks
Bill[/QUOTE]
Using dimensional regularization would mean not introducing the cutoff and the integral would be quite different.
But I don’t see any problem carrying out this integral explicitly. It can be done explicitly. You can use the online Mathematica integrator, for example.
11. bhobba says:
Hi Guys
Just a quick question how one actually does integrals like that. Zee used a Pauli-Villiars approximation and even then you end up with
C* ∫ log (Λ^2/(m^2 – α(1-α)*K^2) dα where the integral is from 0 to 1
Do you do it in dimensional regularisation?
Thanks
Bill
12. bhobba says:
[QUOTE=”nrqed, post: 5120213, member: 15416″] there is no justification for dropping all masses and then, as I said, the cutoff does not cancel out unless we take it to infinity.[/QUOTE]
Again good point :smile::smile::smile::smile::smile::smile:
Thanks
Bill
13. nrqed says:
[QUOTE=”bhobba, post: 5120121, member: 366323″]Good point.
But what do you think of the following argument – take the integral before:
∫ dk^4 1/((k^2 – m^2)((K – k)^2 – m^2)).
We divide the integral into two bits – the sum of a finite integral bit and two high energy bits that are of the form ∫ dk^4 1/k^4 because k swamps the other terms – one from – ∞ the other to +∞. Then by subtracting from the finite integral ∫ dk^4 1/k^4 over that finite region, the improper integrals become ∫ dk^4 1/k^4 for -∞ to ∞. This has the form Limit Λ → ∞ C*log (Λ^2). In this case it has the form for my original argument to be applicable. Of course approximations are used – but what Zee used was full of them as well.
This was like in your example where Λ was large.
Thanks
Bill[/QUOTE]
Hi Bill,
Well, even if one does something like this, one has to make some approximation in some of the regions. One makes his type of approximation only when making hand waving arguments. The actual calculations should be done without such approximations. Or else, one should make clear the approximations made to reach a conclusion. But if we are serious about using a QFT to calculate physical results, there is no justification for dropping all masses and then, as I said, the cutoff does not cancel out unless we take it to infinity.
14. bhobba says:
[QUOTE=”nrqed, post: 5120120, member: 15416″]Your demonstration worked only because you were taking the high energy limit but it does not follow in general, which was my point.[/QUOTE]
Good point.
But what do you think of the following argument – take the integral before:
∫ dk^4 1/((k^2 – m^2)((K – k)^2 – m^2)).
We divide the integral into two bits – the sum of a finite integral bit and two high energy bits that are of the form ∫ dk^4 1/k^4 because k swamps the other terms – one from – ∞ the other to +∞. Then by subtracting from the finite integral ∫ dk^4 1/k^4 over that finite region, the improper integrals become ∫ dk^4 1/k^4 for -∞ to ∞. This has the form Limit Λ → ∞ C*log (Λ^2). In this case it has the form for my original argument to be applicable. Of course approximations are used – but what Zee used was full of them as well.
This was like in your example where Λ was large.
Thanks
Bill
15. nrqed says:
[QUOTE=”bhobba, post: 5119967, member: 366323″]Hi Patrick.
I am sure you are correct in general – but I was not considering the general case – I was considering the high energy scale of a one parameter theory. The example you gave has two parameters K1 and K2.
An example is the scattering amplitude from the Φ^4 theory which is what I considered in the first paper. From Zee page 145 – its
i*λ + 1/(2*(2π)^4) ∫ dk^4 1/((k^2 – m^2)((K – k)^2 – m^2)).
For simplicity we go to the high energy regime so m can be neglected
i*λ + 1/(2*(2π)^4) ∫ dk^4 1/(k^2*(K – k)^2)
Zee claims its i*λ + i*C*log(Λ^2/K^2) which is exactly the form I came up with.
However I have to say there were a number of approximations made in deriving it (Zee did it on page 151 and 152 – band its rather tricky). I don’t know if that’s a factor.
Thanks
Bill[/QUOTE]
Hi Bill,
Yes, my point was about not considering the high energy regime. I realize that things simplify greatly when we can neglect all the masses (and my k’s can be masses).
But I did not get the feeling that you were implying that your conclusions were only valid at the condition of working in the high energy regime. Such a result is then of limited interest since of course the real calculation includes the correction due to finite masses or a real calculation is not necesseraly done at high energy.
You wrote in your first blog
“The cut-off is gone. Everything is expressed in terms of quantities we know. That we don’t know what cut-off to use now doesn’t matter.”
My point is that this conclusion does not follow in general for a quantum field theory, even if it is renormalizable. Your demonstration worked only because you were taking the high energy limit but it does not follow in general, which was my point.
Regards,
Patrick
16. atyy says:
In the Wilsonian view, the cutoff is not simply a reasonable interpretation. The cutoff is truly a cutoff (unless it turns out the theory is asymptotically free or safe).
We basically start with a cutoff, because we assume that the true high energy theory is strings or something unknown to us. However, we guess that we can do physics at low energy by assuming (for example) that special relativity, electrons, positrons and the electromagnetic field approximately exist, even though they may not really do so in the true theory of everything. Then we write down all theories consistent with this assumption, and see what predictions we can make for physics at low energy and finite resolution. Because we write down all theories, all possible terms automatically appear in our initial guess. Even if we don’t write down all possible terms, we will find that the flow to low energy automatically generates them. Since we started with all terms, at low enough energy, we will find the traditional “renormalizable” terms as well as “nonrenormalizable” terms that we traditionally don’t want, but the nonrenormalizable terms will be suppressed in powers of the cutoff.
If for some reason we made such a fantastic guess that is potentially the true theory of everything, we will find that we can remove the cutoff. Regardless, it is not necessary in order to do low energy physics with finite resolution.
17. bhobba says:
[QUOTE=”stevendaryl, post: 5119955, member: 372855″]I guess it’s obvious that if $F(x)$ is dimensionless, but $x$ is not, then $F(x)$ can be rewritten as a function of $frac{x}{Lambda}$, where $Lambda$ is a scale factor with the same dimensions as $x$. But I don’t see what that shows about the interpretation of $Lambda$ as a cutoff.[/QUOTE]
Ahhhh.
That I agree with. Its simply a reasonable interpretation.
Thanks
Bill
18. bhobba says:
[QUOTE=”nrqed, post: 5119733, member: 15416″]I was talking about a general calculation, so I am not assuming that the external momenta are much larger than other physical scales (like the masses of the particles in the loops). My point is that in general, if one does a one loop QFT calculation[/QUOTE]
Hi Patrick.
I am sure you are correct in general – but I was not considering the general case – I was considering the high energy scale of a one parameter theory. The example you gave has two parameters K1 and K2.
An example is the scattering amplitude from the Φ^4 theory which is what I considered in the first paper. From Zee page 145 – its
i*λ + 1/(2*(2π)^4) ∫ dk^4 1/((k^2 – m^2)((K – k)^2 – m^2)).
For simplicity we go to the high energy regime so m can be neglected
i*λ + 1/(2*(2π)^4) ∫ dk^4 1/(k^2*(K – k)^2)
Zee claims its i*λ + i*C*log(Λ^2/K^2) which is exactly the form I came up with.
However I have to say there were a number of approximations made in deriving it (Zee did it on page 151 and 152 – but its rather tricky). I don’t know if that’s a factor.
Thanks
Bill
19. stevendaryl says:
[QUOTE=”bhobba, post: 5119845, member: 366323″]I simply expanded F2 to make it clearer what’s going on. You can argue it has dimension F2(X) – whatever F2 is – if its squared it’s dimensions x^2, if its √ it has dimensions x^1/2 etc. If its constant then its dimensionless. It looks obvious to me from what dimensions means in dimensional analysis.[/QUOTE]
I guess it’s obvious that if $F(x)$ is dimensionless, but $x$ is not, then $F(x)$ can be rewritten as a function of $frac{x}{Lambda}$, where $Lambda$ is a scale factor with the same dimensions as $x$. But I don’t see what that shows about the interpretation of $Lambda$ as a cutoff.
20. bhobba says:
[QUOTE=”stevendaryl, post: 5119666, member: 372855″]I don’t understand why you say that if $F$ actually depended on x, it would not be dimensionless.[/QUOTE]
I simply expanded F2 to make it clearer what’s going on. You can argue it has dimension F2(X) – whatever F2 is – if its squared it’s dimensions x^2, if its √ it has dimensions x^1/2 etc. If its constant then its dimensionless. It looks obvious to me from what dimensions means in dimensional analysis.
Thanks
Bill
21. nrqed says:
[QUOTE=”stevendaryl, post: 5119666, member: 372855″]I might be just being stupid, but I don’t understand this point. You have a dimensionless function of x, $F(x)$. It can be written as a power series in x, as follows:
$F(x) = F_0 + x F_1 + x^2 F_2 + …$
If x is small, then we can approximate F by just the first two terms, so:
$F(x) = F_0 + x F_1$
I don’t understand why you say that if $F$ actually depended on x, it would not be dimensionless. What it seems to me is that $F$ is dimensionless, and so is $F_0$, but $F_1$ has the dimensions of $frac{1}{D}$, where $D$ is the dimensions of x.
I agree that if you want all the $F_i$ to be dimensionless, then you can’t have an expansion in $x$, you have to have an expansion in $frac{x}{Lambda}$ where $Lambda$ has the same dimensions as $x$. But saying that $F$ is dimensionless doesn’t imply that $F_1$ is dimensionless.[/QUOTE]
You are absolutely correct, stevendaryl. I was going to make the same remark. The higher Fs come from a Taylor expansion and the terms in a Taylor expansion (the terms that multiply the powers of the variable in which we expand) all have different dimensions since they come from derivatives with respect to a dimensional quantity! So there is no problem with the dimensions of any of the F’s. We cannot conclude anything one way or another from the existence of the Taylor expansion.
22. nrqed says:
[QUOTE=”bhobba, post: 5119589, member: 366323″]But for large Λ its the same. I have gone through the exact calculations for the messon/meeson scattering in my original paper, and it, even without taking a large number approximation you get $ln( (Lambda^2/k_2^2)$ . Are you sure you are talking about the large energy approximation I am using in the paper?
Thanks
Bill[/QUOTE]
Hi Bill,
I was talking about a general calculation, so I am not assuming that the external momenta are much larger than other physical scales (like the masses of the particles in the loops). My point is that in general, if one does a one loop QFT calculation, the Lambda do not cancel out unless we take the infinite limit.
We could discuss a more specific example if you want, you could just give me the Feynman rules you were using. Or we could just consider a vacuum polarization in QED or a vertex correction or even a Higgs loop. Of course, if you assume that all the masses of the particles are negligible compared to external momenta, things simplify greatly. But one should also be able to calculate quantities where this is not a valid approximation. And even if it is a good approximation, one should be able to go beyond that limit.
Regards,
Patrick
23. stevendaryl says:
[QUOTE=”bhobba, post: 5119611, member: 366323″]No. Its because F must be dimensionless – but the expansion says it isn’t. This is an inconsistency – to accommodate it, it must be infinity or a constant – if it actually depended on x it woul not be dimensionless.[/QUOTE]
I might be just being stupid, but I don’t understand this point. You have a dimensionless function of x, $F(x)$. It can be written as a power series in x, as follows:
$F(x) = F_0 + x F_1 + x^2 F_2 + …$
If x is small, then we can approximate F by just the first two terms, so:
$F(x) = F_0 + x F_1$
I don’t understand why you say that if $F$ actually depended on x, it would not be dimensionless. What it seems to me is that $F$ is dimensionless, and so is $F_0$, but $F_1$ has the dimensions of $frac{1}{D}$, where $D$ is the dimensions of x.
I agree that if you want all the $F_i$ to be dimensionless, then you can’t have an expansion in $x$, you have to have an expansion in $frac{x}{Lambda}$ where $Lambda$ has the same dimensions as $x$. But saying that $F$ is dimensionless doesn’t imply that $F_1$ is dimensionless.
24. Jimster41 says:
[QUOTE=”bhobba, post: 5119611, member: 366323″]Its a Taylor series expansion – in applied math you generally assume you can do that.
Not quite – because of the division it creates something dimensionless – its different to a rescaling which would simply be a change of units.
“Suppose λ is small, then F(x) = λ, F has the dimensions of λ, so is dimensionless”
No. Its because F must be dimensionless – but the expansion says it isn’t. This is an inconsistency – to accommodate it, it must be infinity or a constant – if it actually depended on x it woul not be dimensionless.
I am sorry – but you cant do that. Its modelling something – nothing you can do can change what its modelling.
Thanks
Bill[/QUOTE]
Thanks for the direction Bill.
I need to chew on this more, but I feel like I’m learning something.
I just wanted to clarify, I didn’t mean to imply a conversion from length to Btu’s had some specific, real, quality of meaning, I just meant that the computer can be told to “recast” some value. Like, “hey computer, I know I said 10degF + 20 deltaF = 30 degF, but I just totally changed my mind. It equals 30 “Ice cream cones”. If I tell it not to care, It will let me do things that are dimensionally nonsensical. At the end of the day, I am the one telling it what “modeling something” means. But no, of course I would be disappointed and confused to say the least, if the temperature outside changed 20 degrees and I somehow expected “ice-cream cones”.
25. bhobba says:
[QUOTE=”Jimster41, post: 5119508, member: 517770″]But regardless, do I understand correctly that this is saying that G(x,λ) can be decomposed into a linear combination of functions ${ G }_{ i }(x)$ multiplied by powers of λ (That just what the power series expansion technique)?[/QUOTE]
Its a Taylor series expansion – in applied math you generally assume you can do that.
[QUOTE=”Jimster41, post: 5119508, member: 517770″]Do I understand correctly that this just normalizes (scales) the “power series representation of G(x,λ)” to the difference between the first to constants of expansion of G(x,λ)?[/QUOTE]
Not quite – because of the division it creates something dimensionless – its different to a rescaling which would simply be a change of units.
“Suppose λ is small, then F(x) = λ, F has the dimensions of λ, so is dimensionless”
[QUOTE=”Jimster41, post: 5119508, member: 517770″]This is because powers of small numbers go to zero in the limit, correct?[/QUOTE]
No. Its because F must be dimensionless – but the expansion says it isn’t. This is an inconsistency – to accommodate it, it must be infinity or a constant – if it actually depended on x it woul not be dimensionless.
[QUOTE=”Jimster41, post: 5119508, member: 517770″]I can declare something “Dimensional” to be suddenly “Dimensionless”, change length into Btu’s or whatever). After all it’s just a computer, I can make it do whatever I want. But it seems telling to me that without instructions for how/when/where to do this, the computer can’t “automatically” do so .[/QUOTE]
I am sorry – but you cant do that. Its modelling something – nothing you can do can change what its modelling.
Thanks
Bill
26. bhobba says:
[QUOTE=”nrqed, post: 5119334, member: 15416″]Instead, one gets typically something of the form $ln( (Lambda^2+k_1^2)/k_2^2)$.[/QUOTE]
But for large Λ its the same. I have gone through the exact calculations for the messon/meeson scattering in my original paper, and it, even without taking a large number approximation you get $ln( (Lambda^2/k_2^2)$ . Are you sure you are talking about the large energy approximation I am using in the paper?
Thanks
Bill
27. Jimster41 says:
FYI I can’t see the Tex in Patrick’s reply in the original Insights post.
Just trying to get a handhold (I would like to understand this)
“Suppose we have a function G(x) that depends on some parameter λ ie G(x,λ). Then, so perturbation theory can be used, expand it in a power series about λ:
G(x) = G0 + G1(x)*λ + G2(x)*λ^2 + ……..”
Why isn’t this written:
G(x,λ) = G0 + G1(x)*λ + G2(x)*λ^2 + …….. ?
But regardless, do I understand correctly that this is saying that G(x,λ) can be decomposed into a linear combination of functions ${ G }_{ i }(x)$ multiplied by powers of λ (That just what the power series expansion technique)?
(somewhat aside) It’s been a long time since I learned about power series expansions. But they have always bugged me because of their dependence on “convergence at infinity”. I get that there are lot’s of key tools that use infinite limits. But it has been a regular thorn in my side. To be honest I always sort of associated the QM “infinities” problem with this… that you had to “sum over histories” but that there was effectively no limit to the terms in the sum. Only recently have I realized that the “energy level” is associated with the “cuttoff”.
“In perturbation theory, for theoretical convenience, it is usual to define a new function F(x) = (G(x) – G0)/G1 so:
F(x) = λ + F2(x) *λ^2 + ……..”
Do I understand correctly that this just normalizes (scales) the “power series representation of G(x,λ)” to the difference between the first to constants of expansion of G(x,λ)?
“Suppose λ is small, then F(x) = λ, F has the dimensions of λ, so is dimensionless”
…This is also seen by its definition where G(x) – G0 is divided by G1(x). But lets expand F2(x) in a power series about x so F2(x) = F20 + F21*x + F22*x^2 + ……. = F20 + F21*x + O(x^2). Suppose x is small, so O(x^2) can be neglected, then F2(x) has the dimensions of x, hence to second order of λ, F(x) has the dimensions of x. Here we have a dimensional mismatch. This is the exact reason the equations blow up – in order for it to be dimensionless it cant depend on x. This can only happen if F2(x) is a constant or infinity. Either of course is death for our theory – but nature seemed to choose infinity – the reason for which will be examined later.
This is because powers of small numbers go to zero in the limit, correct?
I guess I find this confusing because (at least in the software I use) I wouldn’t be able to get away with just assuming the “dimension” x of my expression therefore completely vanishes? The software won’t “automatically start to neglect the dimension-ality of a system just because the value of the Range in that dimension is ensie-weensie, or whatever. This has always seemed onto-logically correct to me. Nor will it automatically add dimension.
I can declare something “Dimensional” to be suddenly “Dimensionless”, change length into Btu’s or whatever). After all it’s just a computer, I can make it do whatever I want. But it seems telling to me that without instructions for how/when/where to do this, the computer can’t “automatically” do so .
I guess I have assumed this was for a pretty deep reason, that somehow logically there is simply not enough information in any scalar value alone (even zero) to determine it’s dimensionality (or lack thereof)?
I [I]think[/I] this is clicking. Now “x” is a number of apples in the world of apples “Λ” :
“Now for the solution. The only way to avoid this is to divide x by some parameter, Λ, of the same units as x, so it becomes dimensionless.
The correct equation is:
F(x/Λ) = λ + F2(x/Λ) *λ^2 + F2(x/Λ) *λ^2 +………+ Fi(x/Λ) *λ^i + ……………
We see, due to dimensional analysis of the perturbation methods used, we have neglected a parameter in our theory, which can be interpreted as a cut-off. It is this oversight that has caused the trouble all along.”
I’m interested to see where the log’s come from now…
But… must… eat…
28. atyy says:
[QUOTE=”nrqed, post: 5119343, member: 15416″]This is the “old” approach to renormalization (pre Ken Wilson, say). The modern point of view is that the cutoff should not be taken to infinity. But then one must treat the theory as and effective field theory and there is an infinite of terms to be included in the lagrangian. This is for another post. But my point here was to convey that the cutoff does not go away even in renormalizable theories if we don’t take the limit cutoff goes to infinity.[/QUOTE]
Including the (usually finite) cutoff, and the infinite number of terms is so important conceptually. I don’t know why even modern texts like Srednicki or Schwartz put it so late, and even then make it hard to extract the key concept (well, Schwartz is pretty good, actually). On the other hand, the statistical mechanics texts do this right away.
29. nrqed says:
I am sorry, I messed up again by unintentionally including my first post in my reply, making the whole thing a mess. And I don't know how to go back and edit a post or or to deleter a post, so here is my final version!Watch out… the presentation is a bit misleading for the following reason:In actual calculations, when integrating loop diagrams, one almost never get a pure log of the formln ( Lambda / k) where k is some energy scale (could be a mass). It is almost never like this. Instead, one gets typically something of the form ln((Lambda^2 + k^2)/(u^2)) where u is another energy scale.And one can even have cases (when there are scalar bosons loops, for example) where in addition to these terms, one can have terms of the form 1/(k^2+Lambda^2).So after renormalizing, the cutoff does NOT go away if we keep it at a finite value, even when we are dealing with logs! Instead, one generically get terms fo the formln[ (Lambda^2 + k^2) / (Lambda^2+u^2)] or1/(Lambda^2+k^2) – 1/(Lambda^2 + u^2)We see that the cutoff does not go away, even if the theory is renormalizable!BUT we see that if we take the limit Lambda goes to infinity, THEN the cutoff disappears. This is the reason for taking this limit!This is the "old" approach to renormalization (pre Ken Wilson, say). The modern point of view is that the cutoff should not be taken to infinity. But then one must treat the theory as and effective field theory and there is an infinite of terms to be included in the lagrangian. This is for another post. But my point here was to convey that the cutoff does not go away even in renormalizable theories if we don't take the limit cutoff goes to infinity.Cheers,Patrick
30. nrqed says:
:-( my equations did not show up so let me write them without using TeX.Watch out… the presentation is a bit misleading for the following reason:In actual calculations, when integrating loop diagrams, one almost never get a pure log of the formln ( Lambda / k) where k is some energy scale (could be a mass). It is almost never like this. Instead, one gets typically something of the form ln((Lambda^2 + k^2)/(u^2)) where u is another energy scale.And one can even have cases (when there are scalar bosons loops, for example) where in addition to these terms, one can have terms of the form 1/(k^2+Lambda^2).So after renormalizing, the cutoff does NOT go away if we keep it at a finite value, even when we are dealing with logs! Instead, one generically get terms fo the form ln[ (Lambda^2 + k^2) / (Lambda^2+u^2)] or 1/(Lambda^2+k^2) – 1/(Lambda^2 + u^2)We see that the cutoff does not go away, even if the theory is renormalizable!BUT we see that if we take the limit Lambda goes to infinity, THEN the cutoff disappears. This is the reason for taking this limit!This is the "old" approach to renormalization (pre Ken Wilson, say). The modern point of view is that the cutoff should not be taken to infinity. But then one must treat the theory as and effective field theory and there is an infinite of terms to be included in the lagrangian. This is for another post. But my point here was to convey that the cutoff does not go away even in renormalizable theories if we don't take the limit cutoff goes to infinity.Cheers,Patrick
31. nrqed says:
Watch out… the presentation is a bit misleading for the following reason:In actual calculations, when integrating loop diagrams, one almost never get a pure log of the form $\ln(\Lambda/k)$ where k is some energy scale. It is almost never like this. Instead, one gets typically something of the form $\ln( (\Lambda^2+k_1^2)/k_2^2)$. And one can even have cases (when there are scalar bosons loops, for example) where in addition to these terms, one can have terms of the form $1/(k^2 + \Lambda^2)$. So after renormalizing, the cutoff does NOT go away if we keep it at a finite value, even when we are dealing with logs! Instead, one generically get terms fo the form $\ln((\Lambda^2+k_1^2)/(\Lambda^2+k_2^2))$ or $1/(\Lambda^2+k_1) – 1/(\Lambda^2 + k_2^2)$.We see that the cutoff does not go away, even if the theory is renormalizable. BUT we see that if we take the limit $\Lambda \rightarrow \infty$, THEN the cutoff disappears. This is the reason for taking this limit! This is the "old" approach to renormalization (pre Ken Wilson, say). The modern point of view is that the cutoff should not be taken to infinity. But then one must treat the theory as and effective field theory and there is an infinite of terms to be included in the lagrangian. This is for another post. But my point here was to convey that the cutoff does not go away even in renormalizable theories if we don't take the limit cutoff goes to infinity.Cheers,Patrick | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9170446991920471, "perplexity": 876.596491101523}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710662.60/warc/CC-MAIN-20221128203656-20221128233656-00012.warc.gz"} |
https://worldwidescience.org/topicpages/r/relativistic+mean-field+approximation.html | Sample records for relativistic mean-field approximation
1. Time-dependent Relativistic Mean-field Theory and Random Phase Approximation
P.Ring; D.Vretenar; A.Wandelt; NguyenVanGiai; MAZhong-yu; CAOLi-gang
2001-01-01
The relativistic random phase approximation (RRPA) is derived from the time-dependent relativistic mean field (TD RMF) theory in the limit of small amplitude oscillations. In the no-sea approximation of the RMF theory, the RRPA configuration space includes not only the usual particle-hole ph-states, but also ah configurations, i.e. pairs formed from occupied states in the Fermi sea and empty negative-energy states in the Dirac sea. The contribution of the negative energy states to the RRPA matrices is examined in a schematic model, and the large effect of Dirac sea states on isoscalar strength distributions is illustrated for the giant monopole resonance in 116Sn. It is shown that
2. Beyond the relativistic mean-field approximation (III): collective Hamiltonian in five dimensions
Niksic, T; Vretenar, D; Prochniak, L; Meng, J; Ring, P
2008-01-01
The framework of relativistic energy density functionals is extended to include correlations related to restoration of broken symmetries and fluctuations of collective variables. A model is developed for the solution of the eigenvalue problem of a five-dimensional collective Hamiltonian for quadrupole vibrational and rotational degrees of freedom, with parameters determined by constrained self-consistent relativistic mean-field calculations for triaxial shapes. The model is tested in a series of illustrative calculations of potential energy surfaces and the resulting collective excitation spectra and transition probabilities of the chain of even-even gadolinium isotopes.
3. Relativistic mean field description of cluster radioactivity
Bhagwat, A.; Gambhir, Y. K.
2005-01-01
Comprehensive investigations of the observed cluster radioactivity are carried out. First, the relativistic mean field (RMF) theory is employed for the calculations of the ground-state properties of relevant nuclei. The calculations reproduce the experiment well. The calculated RMF point densities are folded with the density-dependent M3Y nucleon-nucleon interaction to obtain the cluster-daughter interaction potential. This, along with the calculated and experimental Q values, is used in the WKB approximation for estimating the half-lives of the parent nuclei against cluster decay. The calculations qualitatively agree with the experiment. Sensitive dependence of the half-lives on Q values is explicitly demonstrated.
4. Diabatic constrained relativistic mean field approach
L"u, H F; Meng, J
2005-01-01
A diabatic (configuration-fixed) constrained approach to calculate the potential energy surface (PES) of the nucleus is developed in the relativistic mean field model. The potential energy surfaces of $^{208}$Pb obtained from both adiabatic and diabatic constrained approaches are investigated and compared. The diabatic constrained approach enables one to decompose the segmented PES obtained in usual adiabatic approaches into separate parts uniquely characterized by different configurations, to define the single particle orbits at very deformed region by their quantum numbers, and to obtain several well defined deformed excited states which can hardly be expected from the adiabatic PES's.
5. General Relativistic Mean Field Theory for rotating nuclei
Madokoro, Hideki [Kyushu Univ., Fukuoka (Japan). Dept. of Physics; Matsuzaki, Masayuki
1998-03-01
The {sigma}-{omega} model Lagrangian is generalized to an accelerated frame by using the technique of general relativity which is known as tetrad formalism. We apply this model to the description of rotating nuclei within the mean field approximation, which we call General Relativistic Mean Field Theory (GRMFT) for rotating nuclei. The resulting equations of motion coincide with those of Munich group whose formulation was not based on the general relativistic transformation property of the spinor fields. Some numerical results are shown for the yrast states of the Mg isotopes and the superdeformed rotational bands in the A {approx} 60 mass region. (author)
6. Relativistic mean-field mass models
Peña-Arteaga, D.; Goriely, S.; Chamel, N.
2016-10-01
We present a new effort to develop viable mass models within the relativistic mean-field approach with density-dependent meson couplings, separable pairing and microscopic estimations for the translational and rotational correction energies. Two interactions, DD-MEB1 and DD-MEB2, are fitted to essentially all experimental masses, and also to charge radii and infinite nuclear matter properties as determined by microscopic models using realistic interactions. While DD-MEB1 includes the σ, ω and ρ meson fields, DD-MEB2 also considers the δ meson. Both mass models describe the 2353 experimental masses with a root mean square deviation of about 1.1 MeV and the 882 measured charge radii with a root mean square deviation of 0.029 fm. In addition, we show that the Pb isotopic shifts and moments of inertia are rather well reproduced, and the equation of state in pure neutron matter as well as symmetric nuclear matter are in relatively good agreement with existing realistic calculations. Both models predict a maximum neutron-star mass of more than 2.6 solar masses, and thus are able to accommodate the heaviest neutron stars observed so far. However, the new Lagrangians, like all previously determined RMF models, present the drawback of being characterized by a low effective mass, which leads to strong shell effects due to the strong coupling between the spin-orbit splitting and the effective mass. Complete mass tables have been generated and a comparison with other mass models is presented.
7. Relativistic mean-field mass models
Pena-Arteaga, D.; Goriely, S.; Chamel, N. [Universite Libre de Bruxelles, Institut d' Astronomie et d' Astrophysique, CP-226, Brussels (Belgium)
2016-10-15
We present a new effort to develop viable mass models within the relativistic mean-field approach with density-dependent meson couplings, separable pairing and microscopic estimations for the translational and rotational correction energies. Two interactions, DD-MEB1 and DD-MEB2, are fitted to essentially all experimental masses, and also to charge radii and infinite nuclear matter properties as determined by microscopic models using realistic interactions. While DD-MEB1 includes the σ, ω and ρ meson fields, DD-MEB2 also considers the δ meson. Both mass models describe the 2353 experimental masses with a root mean square deviation of about 1.1 MeV and the 882 measured charge radii with a root mean square deviation of 0.029 fm. In addition, we show that the Pb isotopic shifts and moments of inertia are rather well reproduced, and the equation of state in pure neutron matter as well as symmetric nuclear matter are in relatively good agreement with existing realistic calculations. Both models predict a maximum neutron-star mass of more than 2.6 solar masses, and thus are able to accommodate the heaviest neutron stars observed so far. However, the new Lagrangians, like all previously determined RMF models, present the drawback of being characterized by a low effective mass, which leads to strong shell effects due to the strong coupling between the spin-orbit splitting and the effective mass. Complete mass tables have been generated and a comparison with other mass models is presented. (orig.)
8. Cluster decay in very heavy nuclei in Relativistic Mean Field
2008-01-01
Exotic cluster decay of very heavy nuclei has been studied in the microscopic Super-Asymmetric Fission Model. Relativistic Mean Field model with the force FSU Gold has been employed to obtain the densities of the cluster and the daughter nuclei. The microscopic nuclear interaction DDM3Y1, which has an exponential density dependence, and the Coulomb interaction have been used in the double folding model to obtain the potential between the cluster and the daughter. Half life values have been calculated in the WKB approximation and the spectroscopic factors have been extracted. The latter values are seen to have a simple dependence of the mass of the cluster as has been observed earlier. Predictions have been made for some possible decays.
9. Spurious Shell Closures in the Relativistic Mean Field Model
Geng, L S; Toki, H; Long, W H; Shen, G
2006-01-01
Following a systematic theoretical study of the ground-state properties of over 7000 nuclei from the proton drip line to the neutron drip line in the relativistic mean field model [Prog. Theor. Phys. 113 (2005) 785], which is in fair agreement with existing experimental data, we observe a few spurious shell closures, i.e. proton shell closures at Z=58 and Z=92. These spurious shell closures are found to persist in all the effective forces of the relativistic mean field model, e.g. TMA, NL3, PKDD and DD-ME2.
10. Quantum Corrections on Relativistic Mean Field Theory for Nuclear Matter
ZHANG Qi-Ren; GAO Chun-Yuan
2011-01-01
We propose a quantization procedure for the nucleon-scalar meson system, in which an arbitrary mean scalar meson field Φ is introduced.The equivalence of this procedure with the usual one is proven for any given value of Φ.By use of this procedure, the scalar meson field in the Walecka's MFA and in Chin's RHA are quantized around the mean field.Its corrections on these theories are considered by perturbation up to the second order.The arbitrariness of Φ makes us free to fix it at any stage in the calculation.When we fix it in the way of Walecka's MFA, the quantum corrections are big, and the result does not converge.When we fix it in the way of Chin's RHA, the quantum correction is negligibly small, and the convergence is excellent.It shows that RHA covers the leading part of quantum field theory for nuclear systems and is an excellent zeroth order approximation for further quantum corrections, while the Walecka's MFA does not.We suggest to fix the parameter Φ at the end of the whole calculation by minimizing the total energy per-nucleon for the nuclear matter or the total energy for the finite nucleus, to make the quantized relativistic mean field theory (QRMFT) a variational method.
11. Cranked Relativistic Mean Field Description of Superdeformed Rotational Bands
Afanasjev, A. V.; Lalazissis, G. A.; Ring, P.
1997-01-01
The cranked relativistic mean field theory is applied for a detailed investigation of eight superdeformed rotational bands observed in $^{151}$Tb. It is shown that this theory is able to reproduce reasonably well not only the dynamic moments of inertia $J^{(2)}$ of the observed bands but also the alignment properties of the single-particle orbitals.
12. Cluster decay in very heavy nuclei in Relativistic Mean Field
2008-01-01
Exotic cluster decay of very heavy nuclei has been studied in the microscopic Super-Asymmetric Fission Model. Relativistic Mean Field model with the force FSU Gold has been employed to obtain the densities of the cluster and the daughter nuclei. The microscopic nuclear interaction DDM3Y1, which has an exponential density dependence, and the Coulomb interaction have been used in the double folding model to obtain the potential between the cluster and the daughter. Half life values have been ca...
13. Modified Mean Field approximation for the Ising Model
Di Bartolo, Cayetano
2009-01-01
We study a modified mean-field approximation for the Ising Model in arbitrary dimension. Instead of taking a "central" spin, or a small "drop" of fluctuating spins coupled to the effective field of their nearest neighbors as in the Mean-Field or the Bethe-Peierls-Weiss methods, we take an infinite chain of fluctuating spins coupled to the mean field of the rest of the lattice. This results in a significative improvement of the Mean-Field approximation with a small extra effort.
14. Relativistic Mean Field Study on Halo Structures of Mirror Nuclei
LIANG Yu-Jie; LI Yan-Song; LIU Zu-Hua; ZHOU Hong-Yu
2009-01-01
Halo structures of some light mirror nuclei are investigated with the relativistic mean field (RMF) theory.The calculations show that the dispersion of the valence proton is larger than that of the valence neutron in its mirror nucleus,the difference between the root-mean-square (rms) radius of the valence nucleon in each pair of mirror nuclei becomes smailer with the increase of the mass number A,and all the ratios of the rms radius of the valence nucleon to that of the matter in each pair o~ mirror nuclei decrease almost linearly with the increase of the mass number A.
15. A New Parameter Set for the Relativistic Mean Field Theory
Nerlo-Pomorska, B; Nerlo-Pomorska, Bozena; Sykut, Joanna
2004-01-01
Subtracting the Strutinsky shell corrections from the selfconsistent energies obtained within the Relativistic Mean Field Theory (RMFT) we have got estimates for the macroscopic part of the binding energies of 142 spherical even-even nuclei. By minimizing their root mean square deviations from the values obtained with the Lublin-Srasbourg Drop (LSD) model with respect to the nine RMFT parameters we have found the optimal set (NL4). The new parameters reproduce also the radii of these nuclei with an accuracy comparable with that obtained with the NL1 and NL3 sets.
16. Relativistic Consistent Angular-Momentum Projected Shell-Model:Relativistic Mean Field
LI Yan-Song; LONG Gui-Lu
2004-01-01
We develop a relativistic nuclear structure model, relativistic consistent angular-momentum projected shellmodel (RECAPS), which combines the relativistic mean-field theory with the angular-momentum projection method.In this new model, nuclear ground-state properties are first calculated consistently using relativistic mean-field (RMF)theory. Then angular momentum projection method is used to project out states with good angular momentum from a few important configurations. By diagonalizing the hamiltonian, the energy levels and wave functions are obtained.This model is a new attempt for the understanding of nuclear structure of normal nuclei and for the prediction of nuclear properties of nuclei far from stability. In this paper, we will describe the treatment of the relativistic mean field. A computer code, RECAPS-RMF, is developed. It solves the relativistic mean field with axial-symmetric deformation in the spherical harmonic oscillator basis. Comparisons between our calculations and existing relativistic mean-field calculations are made to test the model. These include the ground-state properties of spherical nuclei 16O and 208Pb,the deformed nucleus 20Ne. Good agreement is obtained.
17. COMPRESSIBILITY OF NUCLEI IN RELATIVISTIC MEAN FIELD-THEORY
BOERSMA, HF; MALFLIET, R; SCHOLTEN, O
1991-01-01
Using the relativistic Hartree approximation in the sigma-omega model we study the isoscalar giant monopole resonance. It is shown that the ISGMR of lighter nuclei has non-negligible anharmonic terms. The compressibility of nuclear matter is determined using a leptodermous expansion.
18. Merging Belief Propagation and the Mean Field Approximation
Riegler, Erwin; Kirkelund, Gunvor Elisabeth; Manchón, Carles Navarro
2010-01-01
We present a joint message passing approach that combines belief propagation and the mean field approximation. Our analysis is based on the region-based free energy approximation method proposed by Yedidia et al., which allows to use the same objective function (Kullback-Leibler divergence...
19. Relativistic Mean-Field Models and Nuclear Matter Constraints
Dutra, M; Carlson, B V; Delfino, A; Menezes, D P; Avancini, S S; Stone, J R; Providência, C; Typel, S
2013-01-01
This work presents a preliminary study of 147 relativistic mean-field (RMF) hadronic models used in the literature, regarding their behavior in the nuclear matter regime. We analyze here different kinds of such models, namely: (i) linear models, (ii) nonlinear \\sigma^3+\\sigma^4 models, (iii) \\sigma^3+\\sigma^4+\\omega^4 models, (iv) models containing mixing terms in the fields \\sigma and \\omega, (v) density dependent models, and (vi) point-coupling ones. In the finite range models, the attractive (repulsive) interaction is described in the Lagrangian density by the \\sigma (\\omega) field. The isospin dependence of the interaction is modeled by the \\rho meson field. We submit these sets of RMF models to eleven macroscopic (experimental and empirical) constraints, used in a recent study in which 240 Skyrme parametrizations were analyzed. Such constraints cover a wide range of properties related to symmetric nuclear matter (SNM), pure neutron matter (PNM), and both SNM and PNM.
20. Back-reaction beyond the mean field approximation
Kluger, Y.
1993-12-01
A method for solving an initial value problem of a closed system consisting of an electromagnetic mean field and its quantum fluctuations coupled to fermions is presented. By tailoring the large N{sub f} expansion method to the Schwinger-Keldysh closed time path (CTP) formulation of the quantum effective action, causality of the resulting equations of motion is ensured, and a systematic energy conserving and gauge invariant expansion about the electromagnetic mean field in powers of 1/N{sub f} is developed. The resulting equations may be used to study the quantum nonequilibrium effects of pair creation in strong electric fields and the scattering and transport processes of a relativistic e{sup +}e{sup {minus}} plasma. Using the Bjorken ansatz of boost invariance initial conditions in which the initial electric mean field depends on the proper time only, we show numerical results for the case in which the N{sub f} expansion is truncated in the lowest order, and compare them with those of a phenomenological transport equation.
1. A new approach to spinel ferrites through mean field approximation
Yazdani, A. [Tarbyat Modares University, Tehran P.C 14115-175 (Iran, Islamic Republic of)]. E-mail: [email protected]; Jalilian Nosrati, M.R. [Islamic Azad University Central Tehran Branch, Tehran P.C 14168-94351 (Iran, Islamic Republic of); Ghasemi, R. [Islamic Azad University Central Tehran Branch, Tehran P.C 14168-94351 (Iran, Islamic Republic of)
2006-09-15
The magnetic behavior and specification of spinel ferrites regarding exchange interactions is being studied. The strength of interactions has been examined through the cation substitution with application of mean field approximation of exchange interaction J{sub ij} . Two correlation and approximation parameters have been defined: correlation length R {sub c} in super-exchange and the magnetic effect of ion on the electron fluctuation J {sub 0}.
2. Relativistic mean-field models and nuclear matter constraints
Dutra, M.; Lourenco, O.; Carlson, B. V. [Departamento de Fisica, Instituto Tecnologico de Aeronautica-CTA, 12228-900, Sao Jose dos Campos, SP (Brazil); Delfino, A. [Instituto de Fisica, Universidade Federal Fluminense, 24210-150, Boa Viagem, Niteroi, RJ (Brazil); Menezes, D. P.; Avancini, S. S. [Departamento de Fisica, CFM, Universidade Federal de Santa Catarina, CP. 476, CEP 88.040-900, Florianopolis, SC (Brazil); Stone, J. R. [Oxford Physics, University of Oxford, OX1 3PU Oxford (United Kingdom) and Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996 (United States); Providencia, C. [Centro de Fisica Computacional, Department of Physics, University of Coimbra, P-3004-516 Coimbra (Portugal); Typel, S. [GSI Helmholtzzentrum fuer Schwerionenforschung GmbH, Theorie, Planckstrasse 1,D-64291 Darmstadt (Germany)
2013-05-06
This work presents a preliminary study of 147 relativistic mean-field (RMF) hadronic models used in the literature, regarding their behavior in the nuclear matter regime. We analyze here different kinds of such models, namely: (i) linear models, (ii) nonlinear {sigma}{sup 3}+{sigma}{sup 4} models, (iii) {sigma}{sup 3}+{sigma}{sup 4}+{omega}{sup 4} models, (iv) models containing mixing terms in the fields {sigma} and {omega}, (v) density dependent models, and (vi) point-coupling ones. In the finite range models, the attractive (repulsive) interaction is described in the Lagrangian density by the {sigma} ({omega}) field. The isospin dependence of the interaction is modeled by the {rho} meson field. We submit these sets of RMF models to eleven macroscopic (experimental and empirical) constraints, used in a recent study in which 240 Skyrme parametrizations were analyzed. Such constraints cover a wide range of properties related to symmetric nuclear matter (SNM), pure neutron matter (PNM), and both SNM and PNM.
3. Green's function relativistic mean field theory for Λ hypernuclei
Ren, S.-H.; Sun, T.-T.; Zhang, W.
2017-05-01
The relativistic mean field theory with the Green's function method is extended to study Λ hypernuclei. Taking the hypernucleus Ca61Λ as an example, the single-particle resonant states for Λ hyperons are investigated by analyzing the density of states, and the corresponding energies and widths are given. Different behaviors are observed for the resonant states, i.e., the distributions of the very narrow 1 f5 /2 and 1 f7 /2 states are very similar to bound states while those of the wide 1 g7 /2 and 1 g9 /2 states are like scattering states. Besides, the impurity effect of Λ hyperons on the single-neutron resonant states is investigated. For most of the resonant states, both the energies and widths decrease with adding more Λ hyperons due to the attractive Λ N interaction. Finally, the energy level structure of Λ hyperons in the Ca hypernucleus isotopes with mass number A =53 -73 are studied; obvious shell structure and small spin-orbit splitting are found for the single-Λ spectrum.
4. Hot and dense matter beyond relativistic mean field theory
Zhang, Xilin
2016-01-01
Properties of hot and dense matter are calculated in the framework of quantum hadro-dynamics by including contributions from two-loop (TL) diagrams arising from the exchange of iso-scalar and iso-vector mesons between nucleons. Our extension of mean-field theory (MFT) employs the same five density-independent coupling strengths which are calibrated using the empirical properties at the equilibrium density of iso-spin symmetric matter. Results of calculations from the MFT and TL approximations are compared for conditions of density, temperature, and proton fraction encountered in astrophysics applications involving compact objects. The TL results for the equation of state (EOS) of cold pure neutron matter at sub- and near-nuclear densities agree well with those of modern quantum Monte Carlo and effective field-theoretical approaches. Although the high-density EOS in the TL approximation for neutron-star matter is substantially softer than its MFT counterpart, it is able to support a $2M_\\odot$ neutron star req...
5. Benchmarking mean-field approximations to level densities
Alhassid, Y; Gilbreth, C N; Nakada, H
2015-01-01
We assess the accuracy of finite-temperature mean-field theory using as a standard the Hamiltonian and model space of the shell model Monte Carlo calculations. Two examples are considered: the nucleus $^{162}$Dy, representing a heavy deformed nucleus, and $^{148}$Sm, representing a nearby heavy spherical nucleus with strong pairing correlations. The errors inherent in the finite-temperature Hartree-Fock and Hartree-Fock-Bogoliubov approximations are analyzed by comparing the entropies of the grand canonical and canonical ensembles, as well as the level density at the neutron resonance threshold, with shell model Monte Carlo (SMMC) calculations, which are accurate up to well-controlled statistical errors. The main weak points in the mean-field treatments are seen to be: (i) the extraction of number-projected densities from the grand canonical ensembles, and (ii) the symmetry breaking by deformation or by the pairing condensate. In the absence of a pairing condensate, we confirm that the usual saddle-point appr...
6. Relativistic heavy ion collisions with realistic non-equilibrium mean fields
Fuchs, C; Wolter, H H
1996-01-01
We study the influence of non-equilibrium phase space effects on the dynamics of heavy ion reactions within the relativistic BUU approach. We use realistic Dirac-Brueckner-Hartree-Fock (DBHF) mean fields determined for two-Fermi-ellipsoid configurations, i.e. for colliding nuclear matter, in a local phase space configuration approximation (LCA). We compare to DBHF mean fields in the local density approximation (LDA) and to the non-linear Walecka model. The results are further compared to flow data of the reaction Au on Au at 400 MeV per nucleon measured by the FOPI collaboration. We find that the DBHF fields reproduce the experiment if the configuration dependence is taken into account. This has also implications on the determination of the equation of state from heavy ion collisions.
7. Description of Drip-Line Nuclei within Relativistic Mean-Field Plus BCS Approach
2004-01-01
Recently it has been demonstrated, considering Ni and Ca isotopes as prototypes, that the relativistic mean-field plus BCS (RMF+BCS) approach wherein the single particle continuum corresponding to the RMF is replaced by a set of discrete positive energy states for the calculation of pairing energy provides a good approximation to the full relativistic Hartree-Bogoliubov (RHB) description of the ground state properties of the drip-line neutron rich nuclei. The applicability of RMF+BCS is essentially due to the fact that the main contribution to the pairing correlations is provided by the low-lying resonant states. General validity of this approach is demonstrated by the detailed calculations for the ground state properties of the chains of isotopes of O, Ca, Ni, Zr, Sn and Pb nuclei. The TMA and NL-SH force parameter sets have been used for the effective mean-field Lagrangian. Comprehensive results for the two neutron separation energy, rms radii, single particle pairing gaps and pairing energies etc. are pres...
8. Ground State Properties of Ds Isotopes Within the Relativistic Mean Field Theory
张海飞; 张鸿飞; 李君清
2012-01-01
The ground state properties of Ds (Z=110) isotopes (N=151-195) are studied in the framework of the relativistic mean field (RMF) theory with the effective interaction NL-Z2.The pairing correlation is treated within the conventional BCS approximation.The calculated binding energies are consistent with the results from finite-range droplet model (FRDM) and Macroscopic-microscopic method (MMM).The quadrupole deformation,α-decay energy,α-decay half-live,charge radius,two-neutron separation energy and single-particle spectra are analyzed for Ds isotopes to find new characteristics of superheavy nuclei (SHN).Among the calculated results it is rather distinct that the isotopic shift appears evidently at neutron number N=184.
9. Cluster decay in very heavy nuclei in a relativistic mean field model
2008-02-01
Exotic cluster decay of very heavy nuclei was studied in the microscopic Super-Asymmetric Fission Model. The Relativistic Mean Field model with the force FSU Gold was employed to obtain the densities of the cluster and the daughter nuclei. The microscopic nuclear interaction DDM3Y1, which has an exponential density dependence, and the Coulomb interaction were used in the double folding model to obtain the potential between the cluster and the daughter. Half-life values were calculated in the WKB approximation and the spectroscopic factors were extracted. The latter values are seen to have a simple dependence of the mass of the cluster as has been observed earlier. Predictions were made for some possible decays.
10. Multiple chiral doublet candidate nucleus $^{105}$Rh in a relativistic mean-field approach
Li, Jian; Meng, J; 10.1103/PhysRevC.83.037301
2011-01-01
Following the reports of two pairs of chiral doublet bands observed in $^{105}$Rh, the adiabatic and configuration-fixed constrained triaxial relativistic mean-field (RMF) calculations are performed to investigate their triaxial deformations with the corresponding configuration and the possible multiple chiral doublet (M$\\chi$D) phenomenon. The existence of M$\\chi$D phenomenon in $^{105}$Rh is highly expected.
11. Shape Coexistence for 179Hg in Relativistic Mean-Field Theory
WANG Nan; MENG Jie; ZHAO En-Guang
2005-01-01
The potential energy surface of179 Hg is traced and the multi-shape coexistence phenomenon in that nucleus is studied within the relativistic mean-field theory with quadrupole moment constraint. The calculation results of binding energies and charge radii of mercury isotopes are in good agreement with the experimental data.
12. One-Proton Halo in 31Cl with Relativistic Mean-Field Theory
蔡翔舟; 沈文庆; 任中洲; 蒋维洲; 方德清; 张虎勇; 钟晨; 魏义彬; 郭威; 马余刚; 朱志远
2002-01-01
We investigate proton-rich isotopes s1,32Cl using the nonlinear relativistic mean-field model. It is shown that this model can reproduce the properties of these nuclei well. A long tail appears in the calculated proton density distribution of 31 Cl. The results of relativistic density-dependent Hartree theory show a similar trend of tail density distribution. It is strongly suggested that there is a proton halo in 31Cl and it is indicated that there may be a proton skin in 32 Cl. The relation between the proton halo in 31Cl and the new proton magic number is discussed.
13. Properties and structure of N=Z nuclei within relativistic mean field theory
GAO Yuan; DONG Jian-Min; ZHANG Hong-Fei; ZUO Wei; LI Jun-Qing
2009-01-01
The axially deformed relativistic mean field theory with the force NLSH has been performed in the blocked BCS approximation to investigate the properties and structure of N=Z nuclei from Z=20 to Z=48.Some ground state quantities such as binding energies, quadrupole deformations, one/two-nucleon separation energies, root-mean-square (rms) radii of charge and neutron, and shell gaps have been calculated.The results suggest that large deformations can be found in medium-heavy nuclei with N=Z=38-42.The charge and neutron rms radii increase rapidly beyond the magic number N=Z=28 until Z=42 with increasing nucleon number, which is similar to isotope shift, yet beyond Z=42, they decrease dramatically as the structure changes greatly from Z=42 to Z=43.The evolution of shell gaps with proton number Z can be clearly observed.Besides the appearance of possible new shell closures, some conventional shell closures have been found to disappear in some region.In addition, we found that the Coulomb interaction is not strong enough to breakdown the shell structure of protons in the current region.
14. Nuclear matter fourth-order symmetry energy in relativistic mean field models
Cai, Bao-Jun
2011-01-01
Within the nonlinear relativistic mean field model, we derive the analytical expression of the nuclear matter fourth-order symmetry energy $E_{4}(\\rho)$. Our results show that the value of $E_{4}(\\rho)$ at normal nuclear matter density $\\rho_{0}$ is generally less than 1 MeV, confirming the empirical parabolic approximation to the equation of state for asymmetric nuclear matter at $\\rho_{0}$. On the other hand, we find that the $E_{4}(\\rho)$ may become nonnegligible at high densities. Furthermore, the analytical form of the $E_{4}(\\rho)$ provides the possibility to study the higher-order effects on the isobaric incompressibility of asymmetric nuclear matter, i.e., $K_{\\mathrm{sat}}(\\delta)=K_{0}+K_{\\mathrm{{sat},2}}\\delta ^{2}+K_{\\mathrm{{sat},4}}\\delta ^{4}+\\mathcal{O}(\\delta ^{6})$ where $\\delta =(\\rho_{n}-\\rho_{p})/\\rho$ is the isospin asymmetry, and we find that the value of $K_{\\mathrm{{sat},4}}$ is generally comparable with that of the $K_{\\mathrm{{sat},2}}$. In addition, we study the effects of the $E... 15. The Accuracy of Mean-Field Approximation for Susceptible-Infected-Susceptible Epidemic Spreading Qu, Bo 2016-01-01 The epidemic spreading has been studied for years by applying the mean-field approach in both homogeneous case, where each node may get infected by an infected neighbor with the same rate, and heterogeneous case, where the infection rates between different pairs of nodes are different. Researchers have discussed whether the mean-field approaches could accurately describe the epidemic spreading for the homogeneous cases but not for the heterogeneous cases. In this paper, we explore under what conditions the mean-field approach could perform well when the infection rates are heterogeneous. In particular, we employ the Susceptible-Infected-Susceptible (SIS) model and compare the average fraction of infected nodes in the metastable state obtained by the continuous-time simulation and the mean-field approximation. We concentrate on an individual-based mean-field approximation called the N-intertwined Mean Field Approximation (NIMFA), which is an advanced approach considered the underlying network topology. Moreove... 16. Systematic nuclear structure studies using relativistic mean field theory in mass region A ˜ 130 Shukla, A.; Åberg, Sven; Bajpeyi, Awanish 2017-02-01 Nuclear structure studies for even-even nuclei in the mass region \\backsim 130, have been performed, with a special focus around N or Z = 64. On the onset of deformation and lying between two closed shell, these nuclei have attracted attention in a number of studies. A revisit to these experimentally accessible nuclei has been made via the relativistic mean field. The role of pairing and density depletion in the interior has been specially investigated. Qualitative analysis between two versions of relativistic mean field suggests that there is no significant difference between the two approaches. Moreover, the role of the filling {{{s}}}1/2 orbital in density depletion towards the centre has been found to be consistent with our earlier work on the subject Shukla and Åberg (2014 Phys. Rev. C 89 014329). 17. Proton rich nuclei at and beyond the proton drip line in the Relativistic Mean Field theory Geng, L S; Meng, J 2003-01-01 The Relativistic Mean Field theory is applied to the analysis of ground-state properties of deformed proton-rich odd-Z nuclei in the region$55\\le Z \\le 73$>. The model uses the TMA and NL3 effective interactions in the mean-field Lagrangian, and describes pairing correlations by the density-independent delta-function interaction. The model predicts the location of the proton drip line, the ground-state quadrupole deformation, one-proton separation energy at and beyond the proton drip line, the deformed single-particle orbital occupied by the odd valence proton and the corresponding spectroscopic factor. The results are in good agreement with the available experimental data except for some odd-odd nuclei in which the proton-neutron pairing may become important and are close to those of Relativistic Hartree-Bogoliubov model. 18. Description of$^{178}$Hf$^{m2}$in the constrained relativistic mean field theory Wei, Zhang; Shuang-Quan, Zhang 2009-01-01 The properties of the ground state of$^{178}$Hf and the isomeric state$^{178}$Hf$^{m2}$are studied within the adiabatic and diabatic constrained relativistic mean field (RMF) approaches. The RMF calculations reproduce well the binding energy and the deformation for the ground state of$^{178}$Hf. Using the ground state single-particle eigenvalues obtained in the present calculation, the lowest excitation configuration with$K^\\pi=16^+$is found to be$\
19. Restoration of rotational symmetry in deformed relativistic mean-field theory
YAO Jiang-Ming; MENG Jie; Pena Arteaga Daniel; Ring Peter
2009-01-01
We report on a very recently developed three-dimensional angular momentum projected relativistic mean-field theory with point-coupling interaction (3DAMP+RMF-PC). Using this approach the same effective nucleon-nucleon interaction is adopted to describe both the single-particle and collective motions in nuclei.Collective states with good quantum angular momentum are built projecting out the intrinsic deformed meanfield states. Results for 24Mg are shown as an illustrative application.
20. Finite Size Corrected Relativistic Mean-Field Model and QCD Critical End Point
2012-01-01
The effect of finite size of hadrons on the QCD phase diagram is analyzed using relativistic mean field model for the hadronic phase and the Bag model for the QGP phase. The corrections to the EOS for hadronic phase are incorporated in a thermodynamic consistent manner for Van der Waals like interaction. It is found that the effect of finite size of baryons is to shift CEP to higher chemical potential values.
1. Relativistic mean field study of the superdeformed rotational bands in the A {approx} 60 mass region
Madokoro, Hideki [Dept. of Physics, Kyushu Univ., Fukuoka (Japan); Matsuzaki, Masayuki
1999-03-01
The superdeformed rotational bands in {sup 62}Zn, which were recently discovered, are examined using Relativistic Mean Field model. The experimental dynamical moments of inertia and deformations are well reproduced, but the calculated bands which seem to correspond to the experimental data do not become yrast. This seems to be connected with the wrong position of the g{sup 9/2} single neutron orbit. (author)
2. Antimagnetic rotation in 108,110In with tilted axis cranking relativistic mean-field approach
Sun, Wu-Ji; Xu, Hai-Dan; Li, Jian; Liu, Yong-Hao; Ma, Ke-Yan; Yang, Dong; Lu, Jing-Bing; Ma, Ying-Jun
2016-08-01
Based on tilted axis cranking relativistic mean-field theory within point-coupling interaction PC-PK1, the rotational structure and the characteristic features of antimagnetic rotation for ΔI = 2 bands in 108,110In are studied. Tilted axis cranking relativistic mean-field calculations reproduce the experimental energy spectrum well and are in agreement with the experimental I ∼ ω plot, although the calculated spin overestimates the experimental values. In addition, the two-shears-like mechanism in candidate antimagnetic rotation bands is clearly illustrated and the contributions from two-shears-like orbits, neutron (gd) orbits above Z = 50 shell and Z = 50, N = 50 core are investigated microscopically. The predicted B(E2), dynamic moment of inertia ℑ(2), deformation parameters β and γ, and ℑ(2)/B(E2) ratios in tilted axis cranking relativistic mean-field calculations are discussed and the characteristic features of antimagnetic rotation for the bands before and after alignment are shown. Supported by National Natural Science Foundation of China (11205068, 11205069, 11405072, 11475072, 11547308) and China Postdoctoral Science Foundation (2012M520667)
3. Nuclear Matter in Relativistic Mean Field Theory with Isovector Scalar Meson
Kubis, S
1997-01-01
Relativistic mean field (RMF) theory of nuclear matter with the isovector scalar mean field corresponding to the delta-meson [a_0(980)] is studied. While the delta-meson mean field vanishes in symmetric nuclear matter, it can influence properties of asymmetric nuclear matter in neutron stars. The RMF contribution due to delta-field to the nuclear symmetry energy is negative. To fit the empirical value, E_s=30 MeV, a stronger rho-meson coupling is required than in the absence of the delta-field. The energy per particle of neutron matter is then larger at high densities than the one with no delta-field included. Also, the proton fraction of beta-stable matter increases. Splitting of proton and neutron effective masses due to the delta-field can affect transport properties of neutron star matter.
4. MODEL STUDY OF THE SIGN PROBLEM IN A MEAN-FIELD APPROXIMATION.
HIDAKA,Y.
2007-07-30
We study the sign problem of the fermion determinant at nonzero baryon chemical potential. For this purpose we apply a simple model derived from Quantum Chromodynamics, in the limit of large chemical potential and mass. For SU(2) color, there is no sign problem and the mean-field approximation is similar to data from the lattice. For SU(3) color the sign problem is unavoidable, even in a mean-field approximation. We apply a phase-reweighting method, combined with the mean-field approximation, to estimate thermodynamic quantities. We also investigate the meanfield free energy using a saddle-point approximation [1].
5. Nuclear matter EOS with light clusters within the mean-field approximation
Ferreira, Márcio
2013-01-01
The crust of a neutron star is essentially determined by the low-density region ($\\rho<\\rho_0\\approx0.15-0.16\\unit{fm}^{-3}$) of the equation of state. At the bottom of the inner crust, where the density is $\\rho\\lesssim0.1\\rho_0$, the formation of light clusters in nuclear matter will be energetically favorable at finite temperature. At very low densities and moderate temperatures, the few body correlations are expected to become important and light nuclei like deuterons, tritons, helions and $\\alpha$-particles will form. Due to Pauli blocking, these clusters will dissolve at higher densities $\\rho\\gtrsim 0.1\\rho_0$. The presence of these clusters influences the cooling process and quantities, such as the neutrino emissivity and gravitational waves emission. The dissolution density of these light clusters, treated as point-like particles, will be studied within the Relativistic Mean Field approximation. In particular, the dependence of the dissolution density on the clusters-meson couplings is studied.
6. Mean-field approximation for spacing distribution functions in classical systems
González, Diego Luis; Pimpinelli, Alberto; Einstein, T. L.
2012-01-01
We propose a mean-field method to calculate approximately the spacing distribution functions p(n)(s) in one-dimensional classical many-particle systems. We compare our method with two other commonly used methods, the independent interval approximation and the extended Wigner surmise. In our mean-field approach, p(n)(s) is calculated from a set of Langevin equations, which are decoupled by using a mean-field approximation. We find that in spite of its simplicity, the mean-field approximation provides good results in several systems. We offer many examples illustrating that the three previously mentioned methods give a reasonable description of the statistical behavior of the system. The physical interpretation of each method is also discussed.
7. Neutron Stars in Relativistic Mean Field Theory with Isovector Scalar Meson
Kubis, S; Stachniewicz, S
1998-01-01
We study the equation of state of beta-stable dense matter and models of neutron stars in the relativistic mean field theory with the isovector scalar mean field corresponding to the delta-meson [a_0(980)]. A range of values of the delta-meson coupling compatible with the Bonn potentials is explored. Parameters of the model in the isovector sector are constrained to fit the nuclear symmetry energy, E_s=30 MeV. We find that the quantity most sensitive to the delta-meson coupling is the proton fraction of neutron star matter. It increases significantly in the presence of the delta-field. The energy per baryon also increases but the effect is smaller. The equation of state becomes slightly stiffer and the maximum neutron star mass increases for stronger delta-meson coupling.
8. Three-dimensional angular momentum projection in relativistic mean-field theory
Yao, J M; Ring, P; Arteaga, D Pena
2009-01-01
Based on a relativistic mean-field theory with an effective point coupling between the nucleons, three-dimensional angular momentum projection is implemented for the first time to project out states with designed angular momentum from deformed intrinsic states generated by triaxial quadrupole constraints. The same effective parameter set PC-F1 of the effective interaction is used for deriving the mean field and the collective Hamiltonian. Pairing correlations are taken into account by the BCS method using both monopole forces and zero range d-forces with strength parameters adjusted to experimental even-odd mass differences. The method is applied successfully to the isotopes 24Mg, 30Mg, and 32Mg.
9. Shell evolution at N=20 in the constrained relativistic mean field approach
2008-01-01
The shell evolution at N = 20, a disappearing neutron magic number observed experimentally in very neutron-rich nuclides, is investigated in the constrained relativistic mean field (RMF) theory. The trend of the shell closure observed experimentally towards the neutron drip-line can be reproduced. The predicted two-neutron separation energies, neutron shell gap energies and deformation parameters of ground states are shown as well. These results are compared with the recent Hartree-Fock-Bogliubov (HFB-14) model and the available experimental data. The perspective towards a better understanding of the shell evolution is discussed.
10. Fission Barrier for 240Pu in the Quadrupole Constrained Relativistic Mean Field Approach
L(U) Hong-Feng; GENG Li-Sheng; MENG Jie
2006-01-01
@@ The fission barrier for 240Pu is investigated beyond the second saddle point in the potential energy surface by the constrained relativistic mean field method with the newly proposed parameter set PK1. The microscopic correction for the centre-of-mass motion is essential to provide the correct potential energy surface. The shell effects that stabilize the nuclei against the fission is also investigated by the Strutinsky method. The shapes for the ground state, fission isomer and saddle-points, etc, are studied in detail.
11. Delta isobars in relativistic mean-field models with $\\sigma$-scaled hadron masses and couplings
Kolomeitsev, E E; Voskresensky, D N
2016-01-01
We extend the relativistic mean-field models with hadron masses and meson-baryon coupling constants dependent on the scalar $\\sigma$ field, studied previously to incorporate $\\Delta(1232)$ baryons. Available empirical information is analyzed to put constraints on the couplings of $\\Delta$s with meson fields. Conditions for the appearance of $\\Delta$s are studied. We demonstrate that with inclusion of the $\\Delta$s our equations of state continue to fulfill majority of known empirical constraints including the pressure-density constraint from heavy-ion collisions, the constraint on the maximum mass of the neutron stars, the direct Urca and the gravitational-baryon mass ratio constraints.
12. Pasta phases in neutron star studied with extended relativistic mean field models
Gupta, Neha
2013-01-01
To explain several properties of finite nuclei, infinite matter, and neutron stars in a unified way within the relativistic mean field models, it is important to extend them either with higher order couplings or with density-dependent couplings. These extensions are known to have strong impact in the high-density regime. Here we explore their role on the equation of state at densities lower than the saturation density of finite nuclei which govern the phase transitions associated with pasta structures in the crust of neutron stars.
13. Ground state properties of La isotopes in reflection asymmetric relativistic mean field theory
2009-01-01
The ground state properties of La isotopes are investigated with the reflection asymmetric relativistic mean field(RAS-RMF) model.The calculation results of binding energies and the quadrupole moments are in good agreements with the experiment.The calculation results indicate the change of the quadrupole deformation with the nuclear mass number.The "kink" on the isotope shifts is observed at A = 139 where the neutron number is the magic number N = 82.It is also found that the octupole deformations may exist in the La isotopes with mass number A ～ 145-155.
14. Ground state properties of La isotopes in reflection asymmetric relativistic mean field theory
WANG Nan; GUO Lu
2009-01-01
The ground state properties of La isotopes are investigated with the reflection asymmetric relativistic mean field (RAS-RMF) model.The calculation results of binding energies and the quadrupole moments are in good agreements with the experiment.The calculation results indicate the change of the quadrupole deformation with the nuclear mass number.The "kink" on the isotope shifts is observed at A=139 where the neutron number is the magic number N=82.It is also found that the octupole deformations may exist in the La isotopes with mass number A～ 145-155.
15. Tidal deformability of neutron and hyperon star with relativistic mean field equations of state
Kumar, Bharat; Patra, S K
2016-01-01
We systematically study the tidal deformability for neutron and hyperon stars using relativistic mean field (RMF) equations of state (EOSs). The tidal effect plays an important role during the early part of the evolution of compact binaries. Although, the deformability associated with the EOSs has a small correction, it gives a clean gravitational wave signature in binary inspiral. These are characterized by various love numbers kl (l=2, 3, 4), that depend on the EOS of a star for a given mass and radius. The tidal effect of star could be efficiently measured through advanced LIGO detector from the final stages of inspiraling binary neutron star (BNS) merger.
16. Proton and neutron skins of light nuclei within the Relativistic Mean Field theory
Geng, L S; Ozawa, A; Meng, J
2004-01-01
The Relativistic Mean Field (RMF) theory is applied to the analysis of ground-state properties of Ne, Na, Cl and Ar isotopes. In particular, we study the recently established proton skin in Ar isotopes and neutron skin in Na isotopes as a function of the difference between the proton and the neutron separation energy. We take the TMA effective interaction in the RMF Lagrangian, and describe pairing correlation by the density-independent delta-function interaction. We calculate single neutron and proton separation energies, quadrupole deformations, nuclear matter radii, and differences in proton radii and neutron radii, and compare these results with the recent experimental data.
17. Tidal deformability of neutron and hyperon stars within relativistic mean field equations of state
Kumar, Bharat; Biswal, S. K.; Patra, S. K.
2017-01-01
We systematically study the tidal deformability for neutron and hyperon stars using relativistic mean field equations of state (EOSs). The tidal effect plays an important role during the early part of the evolution of compact binaries. Although, the deformability associated with the EOSs has a small correction, it gives a clean gravitational wave signature in binary inspiral. These are characterized by various Love numbers kl(l =2 ,3 ,4 ), that depend on the EOS of a star for a given mass and radius. The tidal effect of star could be efficiently measured through an advanced LIGO detector from the final stages of an inspiraling binary neutron star merger.
18. Shell-model-like Approach (SLAP) for the Nuclear Properties in Relativistic Mean field Theory
MENG Jie; GUO Jian-you; LIU Lang; ZHANG Shuang-quan
2006-01-01
A Shell-model-like approach suggested to treat the pairing correlations in relativistic mean field theory is introduced,in which the occupancies thus obtained have been iterated back into the densities.The formalism and numerical techniques are given in detail.As examples,the ground state properties and low-lying excited states for Ne isotopes are studied.The results thus obtained are compared with the data available.The binding energies,the odd-even staggering,as well as the tendency for the change of the shapes in Ne isotopes are correctly reproduced.
19. Accuracy criterion for the mean-field approximation in susceptible-infected-susceptible epidemics on networks
Van Mieghem, P.F.A.; Van de Bovenkamp, R.
2015-01-01
Mean-field approximations (MFAs) are frequently used in physics. When a process (such as an epidemic or a synchronization) on a network is approximated by MFA, a major hurdle is the determination of those graphs for which MFA is reasonably accurate. Here, we present an accuracy criterion for Markovi
20. Multidimensionally constrained relativistic mean-field study of triple-humped barriers in actinides
Zhao, Jie; Lu, Bing-Nan; Vretenar, Dario; Zhao, En-Guang; Zhou, Shan-Gui
2015-01-01
Background: Potential energy surfaces (PES's) of actinide nuclei are characterized by a two-humped barrier structure. At large deformations beyond the second barrier, the occurrence of a third barrier was predicted by macroscopic-microscopic model calculations in the 1970s, but contradictory results were later reported by a number of studies that used different methods. Purpose: Triple-humped barriers in actinide nuclei are investigated in the framework of covariant density functional theory (CDFT). Methods: Calculations are performed using the multidimensionally constrained relativistic mean field (MDC-RMF) model, with the nonlinear point-coupling functional PC-PK1 and the density-dependent meson exchange functional DD-ME2 in the particle-hole channel. Pairing correlations are treated in the BCS approximation with a separable pairing force of finite range. Results: Two-dimensional PES's of 226,228,230,232Th and 232,235,236,238U are mapped and the third minima on these surfaces are located. Then one-dimensional potential energy curves along the fission path are analyzed in detail and the energies of the second barrier, the third minimum, and the third barrier are determined. The functional DD-ME2 predicts the occurrence of a third barrier in all Th nuclei and 238U . The third minima in 230 ,232Th are very shallow, whereas those in 226 ,228Th and 238U are quite prominent. With the functional PC-PK1 a third barrier is found only in 226 ,228 ,230Th . Single-nucleon levels around the Fermi surface are analyzed in 226Th, and it is found that the formation of the third minimum is mainly due to the Z =90 proton energy gap at β20≈1.5 and β30≈0.7 . Conclusions: The possible occurrence of a third barrier on the PES's of actinide nuclei depends on the effective interaction used in multidimensional CDFT calculations. More pronounced minima are predicted by the DD-ME2 functional, as compared to the functional PC-PK1. The depth of the third well in Th isotopes decreases
1. Relativistic mean field theory with density dependent coupling constants for nuclear matter and finite nuclei with large charge asymmetry
Typel, S.; Wolter, H.H. [Sektion Physik, Univ. Muenchen, Garching (Germany)
1998-06-01
Nuclear matter and ground state properties for (proton and neutron) semi-closed shell nuclei are described in relativistic mean field theory with coupling constants which depend on the vector density. The parametrization of the density dependence for {sigma}-, {omega}- and {rho}-mesons is obtained by fitting to properties of nuclear matter and some finite nuclei. The equation of state for symmetric and asymmetric nuclear matter is discussed. Finite nuclei are described in Hartree approximation, including a charge and an improved center-of-mass correction. Pairing is considered in the BCS approximation. Special attention is directed to the predictions for properties at the neutron and proton driplines, e.g. for separation energies, spin-orbit splittings and density distributions. (orig.)
2. New parameterization of the effective field theory motivated relativistic mean field model
Kumar, Bharat; Singh, S. K.; Agrawal, B. K.; Patra, S. K.
2017-10-01
A new parameter set is generated for finite and infinite nuclear system within the effective field theory motivated relativistic mean field (ERMF) formalism. The isovector part of the ERMF model employed in the present study includes the coupling of nucleons to the δ and ρ mesons and the cross-coupling of ρ mesons to the σ and ω mesons. The results for the finite and infinite nuclear systems obtained using our parameter set are in harmony with the available experimental data. We find the maximum mass of the neutron star to be 2.03M⊙ and yet a relatively smaller radius at the canonical mass, 12.69 km, as required by the available data.
3. Hyperons in neutron star matter within relativistic mean-field models
Oertel, M; Gulminelli, F; Raduta, A R
2014-01-01
Since the discovery of neutron stars with masses around 2 solar masses the composition of matter in the central part of these massive stars has been intensively discussed. Within this paper we will (re)investigate the question of the appearance of hyperons. To that end we will perform an extensive parameter study within relativistic mean field models. We will show that it is possible to obtain high mass neutron stars (i) with a substantial amount of hyperons, (ii) radii of 12-13 km for the canonical mass of 1.4 solar masses, and (iii) a spinodal instability at the onset of hyperons. The results depend strongly on the interaction in the hyperon-hyperon channels, on which only very little information is available from terrestrial experiments up to now.
4. B-Spline Finite Elements and their Efficiency in Solving Relativistic Mean Field Equations
Pöschl, W
1997-01-01
A finite element method using B-splines is presented and compared with a conventional finite element method of Lagrangian type. The efficiency of both methods has been investigated at the example of a coupled non-linear system of Dirac eigenvalue equations and inhomogeneous Klein-Gordon equations which describe a nuclear system in the framework of relativistic mean field theory. Although, FEM has been applied with great success in nuclear RMF recently, a well known problem is the appearance of spurious solutions in the spectra of the Dirac equation. The question, whether B-splines lead to a reduction of spurious solutions is analyzed. Numerical expenses, precision and behavior of convergence are compared for both methods in view of their use in large scale computation on FEM grids with more dimensions. A B-spline version of the object oriented C++ code for spherical nuclei has been used for this investigation.
5. Description of 178 Hfm2 in the Constrained Relativistic Mean Field Theory
ZHANG Wei; PENG Jing; ZHANG Shuang-Quan
2009-01-01
Properties of the ground state of 178 Hf and the isomeric state 178Hfn2 are studied within the adiabatic and diabatic constrained relativistic mean field (RMF) approaches. The RMF calculations reproduce well the binding energy and the deformation for the ground state of 178Hf. Using the ground state single-particle eigenvalues obtained in the present calculation, the lowest excitation configuration with Kπ = 16+ is found to be v(7/2- [514])-1 (9/2+ [624])1 π(7/2+ [404])-1 (9/2-[514])1. Its excitation energy calculated by the RMF theory with time-odd fields taken into account is equal to 2.801 MeV, i.e., close to the 178 Hfm2 experimental excitation energy 2.446 MeV. The self-consistent procedure accounting for the time-odd component of the meson fields is the most important aspect of the present calculation.
6. Study of reaction and decay using densities from relativistic mean field theory
2012-01-01
Relativistic mean field calculations have been performed to obtain nuclear density pro- file. Microscopic interactions have been folded with the calculated densities of finite nuclei to obtain a semi-microscopic potential. Life time values for the emission of proton, alpha particles and complex clusters have been calculated in the WKB approach assum- ing a tunneling process through the potential barrier. Elastic scattering cross sections have been estimated for proton-nucleus scattering in light neutron rich nuclei. Low en- ergy proton reactions have been studied and their astrophysical implications have been discussed. The success of the semi-microscopic potentials obtained in the folding model with RMF densities in explaining nuclear decays and reactions has been emphasized.
7. Ground-State Properties of Z = 59 Nuclei in the Relativistic Mean-Field Theory
ZHOU Yong; MA Zhong-Yu; CHEN Bao-Qiu; LI Jun-Qing
2000-01-01
Ground-state properties of Pr isotopes are studied in a framework of the relativistic mean-field (RMF) theory using the recently proposed parameter set TM1. Bardeen-Cooper-Schrieffer (BCS) pproximation and blocking method is adopted to deal with pairing interaction and the odd nucleon, respectively. The pairing forces are taken to be isospin dependent. The domain of the validity of the BCS theory and the positions of neutron and proton drip lines are studied. It is shown that RMF theory has provided a good description of the binding energy,isotope shifts and deformation of nuclei over a large range of Pr isotopes, which are in good agreement with those obtained in the finite-range droplet model.
8. Investigation of A＋c- and Ab-Hypernuclei in Relativistic Mean-Field Model
TANYu-Hong; CAIChong-Hai; LILei; NINGPing-Zhi
2003-01-01
We investigate the properties of A+c- and Ab-hypernuclei within the framework of the relativistic mean-field model (RMF). It is found that no A+c bound states can exist if the A+c potential well depth |UA+c| in nuclear matter is less than 10 MeV. If |UA+c|is less than 20 MeV, A+c cannot bind to the heavier nuclei with atomic number larger than 100. We suggest it is preferable to search the A+c-hypernuclei from medium-heavy nuclear systems in experiment. Very small spin-orbit splitting for the A+c in hypernuclei is a/so observed, and for the Ab it is nearly zero.
9. Magnetic moments of 33Mg in the time-odd relativistic mean field approach
2009-01-01
The configuration-fixed deformation constrained relativistic mean field approach with time-odd component has been applied to investigate the ground state properties of 33Mg with effective interaction PK1.The ground state of 33Mg has been found to be prolate deformed,β2=0.23,with the odd neutron in 1/2[330] orbital and the energy -251.85 MeV which is close to the data -252.06 MeV.The magnetic moment -0.9134 μN is obtained with the effective electromagnetic current which well reproduces the data -0.7456 μN self-consistently without introducing any parameter.The energy splittings of time reversal conjugate states,the neutron current,the energy contribution from the nuclear magnetic potential,and the effect of core polarization are discussed in detail.
10. K--nucleus relativistic mean field potentials consistent with kaonic atoms
Friedman, E.; Gal, A.; Mareš, J.; Cieplý, A.
1999-08-01
K- atomic data are used to test several models of the K- nucleus interaction. The t(ρ)ρ optical potential, due to coupled channel models incorporating the Λ(1405) dynamics, fails to reproduce these data. A standard relativistic mean field (RMF) potential, disregarding the Λ(1405) dynamics at low densities, also fails. The only successful model is a hybrid of a theoretically motivated RMF approach in the nuclear interior and a completely phenomenological density dependent potential, which respects the low density theorem in the nuclear surface region. This best-fit K- optical potential is found to be strongly attractive, with a depth of 180+/-20 MeV at the nuclear interior, in agreement with previous phenomenological analyses.
11. Simplified method for including spatial correlations in mean-field approximations
Markham, Deborah C.; Simpson, Matthew J.; Baker, Ruth E.
2013-06-01
Biological systems involving proliferation, migration, and death are observed across all scales. For example, they govern cellular processes such as wound healing, as well as the population dynamics of groups of organisms. In this paper, we provide a simplified method for correcting mean-field approximations of volume-excluding birth-death-movement processes on a regular lattice. An initially uniform distribution of agents on the lattice may give rise to spatial heterogeneity, depending on the relative rates of proliferation, migration, and death. Many frameworks chosen to model these systems neglect spatial correlations, which can lead to inaccurate predictions of their behavior. For example, the logistic model is frequently chosen, which is the mean-field approximation in this case. This mean-field description can be corrected by including a system of ordinary differential equations for pairwise correlations between lattice site occupancies at various lattice distances. In this work we discuss difficulties with this method and provide a simplification in the form of a partial differential equation description for the evolution of pairwise spatial correlations over time. We test our simplified model against the more complex corrected mean-field model, finding excellent agreement. We show how our model successfully predicts system behavior in regions where the mean-field approximation shows large discrepancies. Additionally, we investigate regions of parameter space where migration is reduced relative to proliferation, which has not been examined in detail before and find our method is successful at correcting the deviations observed in the mean-field model in these parameter regimes.
12. Mean-field approximation for the potts model of a diluted magnet in the external field
Semkin, S. V.; Smagin, V. P.
2016-07-01
The Potts model of a diluted magnet with an arbitrary number of states placed in the external field has been considered. Phase transitions of this model have been studied in the mean-field approximation, the dependence of the critical temperature on the external field and the density of magnetic atoms has been found, and the magnetic susceptibility has been calculated. An improved mean-field technique has been proposed, which provides more accurate account of the effects associated with nonmagnetic dilution. The influence of dilution on the first-order phase transition curve and the magnetization jump at the phase transition has been studied by this technique.
13. Core-crust transition properties of neutron stars within systematically varied extended relativistic mean-field model
Sulaksono, A; Agrawal, B K
2014-01-01
The model dependence and the symmetry energy dependence of the core-crust transition properties for the neutron stars are studied using three different families of systematically varied extended relativistic mean field model. Several forces within each of the families are so considered that they yield wide variations in the values of the nuclear symmetry energy $a_{\\rm sym}$ and its slope parameter $L$ at the saturation density. The core-crust transition density is calculated using a method based on random-phase-approximation. The core-crust transition density is strongly correlated, in a model independent manner, with the symmetry energy slope parameter evaluated at the saturation density. The pressure at the transition point dose not show any meaningful correlations with the symmetry energy parameters at the saturation density. At best, pressure at the transition point is correlated with the symmetry energy parameters and their linear combination evaluated at the some sub-saturation density. Yet, such corre...
14. Multidimensionally-constrained relativistic mean-field study of spontaneous fission: coupling between shape and pairing degrees of freedom
Zhao, Jie; Niksic, Tamara; Vretenar, Dario; Zhou, Shan-Gui
2016-01-01
Studies of fission dynamics, based on nuclear energy density functionals, have shown that the coupling between shape and pairing degrees of freedom has a pronounced effect on the nonperturbative collective inertia and, therefore, on dynamic (least-action) spontaneous fission paths and half-lives. Collective potentials and nonperturbative cranking collective inertia tensors are calculated using the multidimensionally-constrained relativistic mean-field (MDC-RMF) model. Pairing correlations are treated in the BCS approximation using a separable pairing force of finite range. Pairing fluctuations are included as a collective variable using a constraint on particle-number dispersion. Fission paths are determined with the dynamic programming method by minimizing the action in multidimensional collective spaces. The dynamics of spontaneous fission of $^{264}$Fm and $^{250}$Fm are explored. Fission paths, action integrals and corresponding half-lives computed in the three-dimensional collective space of shape and pa...
15. Treating Coulomb exchange contributions in relativistic mean field calculations: why and how
Van Giai, Nguyen; Gu, Huai-Qiang; Long, Wenhui; Meng, Jie
2014-01-01
The energy density functional (EDF) method is very widely used in nuclear physics, and among the various existing functionals those based on the relativistic Hartree (RH) approximation are very popular because the exchange contributions (Fock terms) are numerically rather onerous to calculate. Although it is possible to somehow 'mock up' the effects of meson-induced exchange terms by adjusting the meson-nucleon couplings, the lack of Coulomb exchange contributions hampers the accuracy of predictions. In this note, we show that the Coulomb exchange effects can be easily included with a good accuracy in a perturbative approach. Therefore, it would be desirable for future relativistic EDF models to incorporate Coulomb exchange effects, at least to some order of perturbation.
16. Parity Violating Electron Scattering in the Relativistic Eikonal Approximation
DONG Tie-Kuang; REN Zhong-Zhou
2008-01-01
The parity violating electron scattering is investigated in the relativistic Eikonal approximation. The parity violating asymmetry parameters for many isotopes are calculated. In calculations the proton and neutron densities are obtained from the relativistic mean-field theory. We take Ni isotopes as examples to analyse the behaviour of the parity violating asymmetry parameters. The results show that the parity violating asymmetry parameter is sensitive to the difference between the proton and neutron densities. The amplitude of the parity violating asymmetry parameter increases with the distance between the minima of proton and neutron form factors. Our results are useful for future parity violating electron scattering experiments. By comparing our results with experimental data one can test the validity of the relativistic mean-field theory in calculating the neutron densities of nuclei.
17. Particle-number projection in the finite-temperature mean-field approximation
Fanto, P; Bertsch, G F
2016-01-01
Calculation of statistical properties of nuclei in a finite-temperature mean-field theory requires projection onto good particle number, since the theory is formulated in the grand canonical ensemble. This projection is usually carried out in a saddle-point approximation. Here we derive formulas for an exact particle-number projection of the finite-temperature mean-field solution. We consider both deformed nuclei, in which the pairing condensate is weak and the Hartree-Fock (HF) approximation is the appropriate mean-field theory, and nuclei with strong pairing condensates, in which the appropriate theory is the Hartree-Fock-Bogoliubov (HFB) approximation, a method that explicitly violates particle-number conservation. For the HFB approximation, we present a general projection formula for a condensate that is time-reversal invariant and a simpler formula for the Bardeen-Cooper-Schrieffer (BCS) limit, which is realized in nuclei with spherical condensates. We apply the method to three heavy nuclei: a typical de...
18. Merging Belief Propagation and the Mean Field Approximation: A Free Energy Approach
Riegler, Erwin; Kirkelund, Gunvor Elisabeth; Manchón, Carles Navarro
2013-01-01
We present a joint message passing approach that combines belief propagation and the mean field approximation. Our analysis is based on the region-based free energy approximation method proposed by Yedidia et al. We show that the message passing fixed-point equations obtained with this combination...... correspond to stationary points of a constrained region-based free energy approximation. Moreover, we present a convergent implementation of these message passing fixed-point equations provided that the underlying factor graph fulfills certain technical conditions. In addition, we show how to include hard...
19. Relativistic Mean Field Description of Nuclear Collective Rotation -The Superdeformed Rotational Bands in the A$\\sim$60 Mass Region-
1997-01-01
Relativistic Mean Field Theory is applied to the description of rotating nuclei. Since the previous formulation of Munich group was based on a special relativistic transformation property of the spinor fields, we reformulate in a fully covariant manner using tetrad formalism. The numerical calculations are performed for 3 zinc isotopes, including the newly discovered superdeformed band in $^{62}$Zn which is the first experimental observation in this mass region.
20. Relativistic Mean Field Description of Nuclear Collective Rotation - The Superdeformed Rotational Bands in the A ~ 60 Mass Region -
Relativistic Mean Field Theory is applied to the description of rotating nuclei. Since the previous formulation of Munich group was based on a special relativistic transformation property of the spinor fields, we reformulate in a fully covariant manner using tetrad formalism. The numerical calculations are performed for 3 zinc isotopes, including the newly discovered superdeformed band in $^{62}$Zn which is the first experimental observation in this mass region.
1. A Study of Multi-Λ Hypernuclei Within Spherical Relativistic Mean-Field Approach
Rather, Asloob A.; Ikram, M.; Usmani, A. A.; Kumar, B.; Patra, S. K.
2017-09-01
This research article is a follow up of an earlier work by M. Ikram et al., reported in Int. J. Mod. Phys. E 25, 1650103 (2016) where we searched for Λ magic numbers in experimentally confirmed doubly magic nucleonic cores in light to heavy mass region (i.e., 16 O-208 P b) by injecting Λ's into them. In the present manuscript, working within the state of the art relativistic mean field theory with the inclusion of ΛN and ΛΛ interaction in addition to nucleon-meson NL 3∗ effective force, we extend the search of lambda magic numbers in multi- Λ hypernuclei using the predicted doubly magic nucleonic cores 292120, 304120, 360132, 370132, 336138, 396138 of the elusive superheavy mass regime. In analogy to well established signatures of magicity in conventional nuclear theory, the prediction of hypernuclear magicities is made on the basis of one-, two- Λ separation energy (S Λ,S 2Λ) and two lambda shell gaps (δ 2Λ) in multi- Λ hypernuclei. The calculations suggest that the Λ numbers 92, 106, 126, 138, 184, 198, 240, and 258 might be the Λ shell closures after introducing the Λ's in the elusive superheavy nucleonic cores. The appearance of new lambda shell closures apart from the nucleonic ones predicted by various relativistic and non-relativistic theoretical investigations can be attributed to the relatively weak strength of the spin-orbit coupling in hypernuclei compared to normal nuclei. Further, the predictions made in multi- Λ hypernuclei under study resembles closely the magic numbers in conventional nuclear theory suggested by various relativistic and non-relativistic theoretical models. Moreover, in support of the Λ shell closure, the investigation of Λ pairing energy and effective Λ pairing gap has been made. We noticed a very close agreement of the predicted Λ shell closures with the survey made on the pretext of S Λ, S 2Λ, and δ 2Λ except for the appearance of magic numbers corresponding to Λ = 156 which manifest in Λ effective
2. Nucleon Finite Volume Effect and Nuclear Matter Properties in a Relativistic Mean-Field Theory
R. Costa; A.J. Santiago; H. Rodrigues; J. Sa Borges
2006-01-01
Effects of excluded volume of nucleons on nuclear matter are studied, and the nuclear properties that follow from different relativistic mean-field model parametrizations are compared. We show that, for all tested parametrizations,the resulting volume energy a1 and the symmetry energy J are around the acceptable values of 16 MeV and 30 MeV,and the density symmetry L is around 100 Me V. On the other hand, models that consider only linear terms lead to incompressibility K0 much higher than expected. For most parameter sets there exists a critical point (ρc,δc), where the minimum and the maximum of the equation of state are coincident and the incompressibility equals zero. This critical point depends on the excluded volume parameter r. If this parameter is larger than 0.5 fm, there is no critical point and the pure neutron matter is predicted to be bound. The maximum value for neutron star mass is 1.85M⊙, which is in agreement with the mass of the heaviest observed neutron star 4U0900-40 and corresponds to r = 0.72 fm. We also show that the light neutron star mass (1.2M⊙) is obtained for r (≌) 0.9 fm.
3. A Second Relativistic Mean Field and Virial Equation of State for Astrophysical Simulations
Shen, G; O'Connor, E
2011-01-01
We generate a second equation of state (EOS) of nuclear matter for a wide range of temperatures, densities, and proton fractions for use in supernovae, neutron star mergers, and black hole formation simulations. We employ full relativistic mean field (RMF) calculations for matter at intermediate density and high density, and the Virial expansion of a non-ideal gas for matter at low density. For this EOS we use the RMF effective interaction FSUGold, whereas our earlier EOS was based on the RMF effective interaction NL3. The FSUGold interaction has a lower pressure at high densities compared to the NL3 interaction. We calculate the resulting EOS at over 100,000 grid points in the temperature range $T$ = 0 to 80 MeV, the density range $n_B$ = 10$^{-8}$ to 1.6 fm$^{-3}$, and the proton fraction range $Y_p$ = 0 to 0.56. We then interpolate these data points using a suitable scheme to generate a thermodynamically consistent equation of state table on a finer grid. We discuss differences between this EOS, our NL3 ba...
4. Multi-dimensional constraint relativistic mean field model and applications in actinide and transfermium nuclei
Lu, Bing-Nan; Zhao, En-Guang; Zhou, Shan-Gui
2013-01-01
In this contribution we present some results of potential energy surfaces of actinide and transfermium nuclei from multi-dimensional constrained relativistic mean field (MDC-RMF) models. Recently we developed multi-dimensional constrained covariant density functional theories (MDC-CDFT) in which all shape degrees of freedom $\\beta_{\\lambda\\mu}$ with even $\\mu$ are allowed and the functional can be one of the following four forms: the meson exchange or point-coupling nucleon interactions combined with the non-linear or density-dependent couplings. In MDC-RMF models, the pairing correlations are treated with the BCS method. With MDC-RMF models, the potential energy surfaces of even-even actinide nuclei were investigated and the effect of triaxiality on the fission barriers in these nuclei was discussed. The non-axial reflection-asymmetric $\\beta_{32}$ shape in some transfermium nuclei with $N=150$, namely $^{246}$Cm, $^{248}$Cf, $^{250}$Fm, and $^{252}$No were also studied.
5. Mean-Field Approximation to the Hydrophobic Hydration in the Liquid-Vapor Interface of Water.
Abe, Kiharu; Sumi, Tomonari; Koga, Kenichiro
2016-03-03
A mean-field approximation to the solvation of nonpolar solutes in the liquid-vapor interface of aqueous solutions is proposed. It is first remarked with a numerical illustration that the solvation of a methane-like solute in bulk liquid water is accurately described by the mean-field theory of liquids, the main idea of which is that the probability (Pcav) of finding a cavity in the solvent that can accommodate the solute molecule and the attractive interaction energy (uatt) that the solute would feel if it is inserted in such a cavity are both functions of the solvent density alone. It is then assumed that the basic idea is still valid in the liquid-vapor interface, but Pcav and uatt are separately functions of different coarse-grained local densities, not functions of a common local density. Validity of the assumptions is confirmed for the solvation of the methane-like particle in the interface of model water at temperatures between 253 and 613 K. With the mean-field approximation extended to the inhomogeneous system the local solubility profiles across the interface at various temperatures are calculated from Pcav and uatt obtained at a single temperature. The predicted profiles are in excellent agreement with those obtained by the direct calculation of the excess chemical potential over an interfacial region where the solvent local density varies most rapidly.
6. The effect of inclusion of $\\Delta$ resonances in relativistic mean-field model with scaled hadron masses and coupling constants
Maslov, K A; Voskresensky, D N
2016-01-01
Knowledge of the equation of state of the baryon matter plays a decisive role in the description of neutron stars. With an increase of the baryon density the filling of Fermi seas of hyperons and $\\Delta$ isobars becomes possible. Their inclusion into standard relativistic mean-field models results in a strong softening of the equation of state and a lowering of the maximum neutron star mass below the measured values. We extend a relativistic mean-field model with scaled hadron masses and coupling constants developed in our previous works and take into account now not only hyperons but also the $\\Delta$ isobars. We analyze available empirical information to put constraints on coupling constants of $\\Delta$s to mesonic mean fields. We show that the resulting equation of state satisfies majority of presently known experimental constraints.
7. Quantum mean-field approximation for lattice quantum models: Truncating quantum correlations and retaining classical ones
Malpetti, Daniele; Roscilde, Tommaso
2017-02-01
The mean-field approximation is at the heart of our understanding of complex systems, despite its fundamental limitation of completely neglecting correlations between the elementary constituents. In a recent work [Phys. Rev. Lett. 117, 130401 (2016), 10.1103/PhysRevLett.117.130401], we have shown that in quantum many-body systems at finite temperature, two-point correlations can be formally separated into a thermal part and a quantum part and that quantum correlations are generically found to decay exponentially at finite temperature, with a characteristic, temperature-dependent quantum coherence length. The existence of these two different forms of correlation in quantum many-body systems suggests the possibility of formulating an approximation, which affects quantum correlations only, without preventing the correct description of classical fluctuations at all length scales. Focusing on lattice boson and quantum Ising models, we make use of the path-integral formulation of quantum statistical mechanics to introduce such an approximation, which we dub quantum mean-field (QMF) approach, and which can be readily generalized to a cluster form (cluster QMF or cQMF). The cQMF approximation reduces to cluster mean-field theory at T =0 , while at any finite temperature it produces a family of systematically improved, semi-classical approximations to the quantum statistical mechanics of the lattice theory at hand. Contrary to standard MF approximations, the correct nature of thermal critical phenomena is captured by any cluster size. In the two exemplary cases of the two-dimensional quantum Ising model and of two-dimensional quantum rotors, we study systematically the convergence of the cQMF approximation towards the exact result, and show that the convergence is typically linear or sublinear in the boundary-to-bulk ratio of the clusters as T →0 , while it becomes faster than linear as T grows. These results pave the way towards the development of semiclassical numerical
8. Mean-field approximation for the Sznajd model in complex networks
Araújo, Maycon S.; Vannucchi, Fabio S.; Timpanaro, André M.; Prado, Carmen P. C.
2015-02-01
This paper studies the Sznajd model for opinion formation in a population connected through a general network. A master equation describing the time evolution of opinions is presented and solved in a mean-field approximation. Although quite simple, this approximation allows us to capture the most important features regarding the steady states of the model. When spontaneous opinion changes are included, a discontinuous transition from consensus to polarization can be found as the rate of spontaneous change is increased. In this case we show that a hybrid mean-field approach including interactions between second nearest neighbors is necessary to estimate correctly the critical point of the transition. The analytical prediction of the critical point is also compared with numerical simulations in a wide variety of networks, in particular Barabási-Albert networks, finding reasonable agreement despite the strong approximations involved. The same hybrid approach that made it possible to deal with second-order neighbors could just as well be adapted to treat other problems such as epidemic spreading or predator-prey systems.
9. On the connections and differences among three mean-field approximations: a stringent test.
Yi, Shasha; Pan, Cong; Hu, Liming; Hu, Zhonghan
2017-07-19
This letter attempts to clarify the meaning of three closely related mean-field approximations: random phase approximation (RPA), local molecular field (LMF) approximation, and symmetry-preserving mean-field (SPMF) approximation, and their use of reliability and validity in the field of theory and simulation of liquids when the long-ranged component of the intermolecular interaction plays an important role in determining density fluctuations and correlations. The RPA in the framework of classical density functional theory (DFT) neglects the higher order correlations in the bulk and directly applies the long-ranged part of the potential to correct the pair direct correlation function of the short-ranged system while the LMF approach introduces a nonuniform mimic system under a reconstructed static external potential that accounts for the average effect arising from the long-ranged component of the interaction. Furthermore, the SPMF approximation takes the viewpoint of LMF but instead instantaneously averages the long-ranged component of the potential over the degrees of freedom in the direction with preserved symmetry. The formal connections and the particular differences of the viewpoint among the three approximations are explained and their performances in producing structural properties of liquids are stringently tested using an exactly solvable model. We demonstrate that the RPA treatment often yields uncontrolled poor results for pair distribution functions of the bulk system. On the other hand, the LMF theory produces quite reasonably structural correlations when the pair distribution in the bulk is converted to the singlet particle distribution in the nonuniform system. It turns out that the SPMF approach outperforms the other two at all densities and under extreme conditions where the long-ranged component significantly contributes to the structural correlations.
10. Thermodynamics of the one-dimensional parallel Kawasaki model: Exact solution and mean-field approximations
Pazzona, Federico G.; Demontis, Pierfranco; Suffritti, Giuseppe B.
2014-08-01
The adsorption isotherm for the recently proposed parallel Kawasaki (PK) lattice-gas model [Phys. Rev. E 88, 062144 (2013), 10.1103/PhysRevE.88.062144] is calculated exactly in one dimension. To do so, a third-order difference equation for the grand-canonical partition function is derived and solved analytically. In the present version of the PK model, the attraction and repulsion effects between two neighboring particles and between a particle and a neighboring empty site are ruled, respectively, by the dimensionless parameters ϕ and θ. We discuss the inflections induced in the isotherms by situations of high repulsion, the role played by finite lattice sizes in the emergence of substeps, and the adequacy of the two most widely used mean-field approximations in lattice gases, namely, the Bragg-Williams and the Bethe-Peierls approximations.
11. Continuum theory of critical phenomena in polymer solutions: Formalism and mean field approximation
Goldstein, Raymond E.; Cherayil, Binny J.
1989-06-01
A theoretical description of the critical point of a polymer solution is formulated directly from the Edwards continuum model of polymers with two- and three-body excluded-volume interactions. A Hubbard-Stratonovich transformation analogous to that used in recent work on the liquid-vapor critical point of simple fluids is used to recast the grand partition function of the polymer solution as a functional integral over continuous fields. The resulting Landau-Ginzburg-Wilson (LGW) Hamiltonian is of the form of a generalized nonsymmetric n=1 component vector model, with operators directly related to certain connected correlation functions of a reference system. The latter is taken to be an ensemble of Gaussian chains with three-body excluded-volume repulsions, and the operators are computed in three dimensions by means of a perturbation theory that is rapidly convergent for long chains. A mean field theory of the functional integral yields a description of the critical point in which the power-law variations of the critical polymer volume fraction φc, critical temperature Tc, and critical amplitudes on polymerization index N are essentially identical to those found in the Flory-Huggins theory. In particular, we find φc ˜N-1/2, Tθ-Tc˜N-1/2 with (Tθ the theta temperature), and that the composition difference between coexisting phases varies with reduced temperature t as N-1/4t1/2. The mean field theory of the interfacial tension σ between coexisting phases near the critical point, developed by considering the LGW Hamiltonian for a weakly inhomogeneous solution, yields σ˜N-1/4t3/2, with the correlation length diverging as ξ˜N1/4t-1/2 within the same approximation, consistent with the mean field limit of de Gennes' scaling form. Generalizations to polydisperse systems are discussed.
12. A systematic sequence of relativistic approximations.
Dyall, Kenneth G
2002-06-01
An approach to the development of a systematic sequence of relativistic approximations is reviewed. The approach depends on the atomically localized nature of relativistic effects, and is based on the normalized elimination of the small component in the matrix modified Dirac equation. Errors in the approximations are assessed relative to four-component Dirac-Hartree-Fock calculations or other reference points. Projection onto the positive energy states of the isolated atoms provides an approximation in which the energy-dependent parts of the matrices can be evaluated in separate atomic calculations and implemented in terms of two sets of contraction coefficients. The errors in this approximation are extremely small, of the order of 0.001 pm in bond lengths and tens of microhartrees in absolute energies. From this approximation it is possible to partition the atoms into relativistic and nonrelativistic groups and to treat the latter with the standard operators of nonrelativistic quantum mechanics. This partitioning is shared with the relativistic effective core potential approximation. For atoms in the second period, errors in the approximation are of the order of a few hundredths of a picometer in bond lengths and less than 1 kJ mol(-1) in dissociation energies; for atoms in the third period, errors are a few tenths of a picometer and a few kilojoule/mole, respectively. A third approximation for scalar relativistic effects replaces the relativistic two-electron integrals with the nonrelativistic integrals evaluated with the atomic Foldy-Wouthuysen coefficients as contraction coefficients. It is similar to the Douglas-Kroll-Hess approximation, and is accurate to about 0.1 pm and a few tenths of a kilojoule/mole. The integrals in all the approximations are no more complicated than the integrals in the full relativistic methods, and their derivatives are correspondingly easy to formulate and evaluate.
13. Mean field approximation for biased diffusion on Japanese inter-firm trading network.
Watanabe, Hayafumi
2014-01-01
By analysing the financial data of firms across Japan, a nonlinear power law with an exponent of 1.3 was observed between the number of business partners (i.e. the degree of the inter-firm trading network) and sales. In a previous study using numerical simulations, we found that this scaling can be explained by both the money-transport model, where a firm (i.e. customer) distributes money to its out-edges (suppliers) in proportion to the in-degree of destinations, and by the correlations among the Japanese inter-firm trading network. However, in this previous study, we could not specifically identify what types of structure properties (or correlations) of the network determine the 1.3 exponent. In the present study, we more clearly elucidate the relationship between this nonlinear scaling and the network structure by applying mean-field approximation of the diffusion in a complex network to this money-transport model. Using theoretical analysis, we obtained the mean-field solution of the model and found that, in the case of the Japanese firms, the scaling exponent of 1.3 can be determined from the power law of the average degree of the nearest neighbours of the network with an exponent of -0.7.
14. Study of the Alpha-Decay Chain for7753 194Rn with Relativistic Mean-Field Theory
SHENG Zong-Qiang; GUO Jian-You
2008-01-01
The structures of the nuclei on the alpha-decay chain of 194Rn are investigated in the deformed relativistic mean-field theory with the effective interaction TMA. We put an emphasis on the ground state properties of 194Rn. The calculated alpha-decay energies and lifetimes are both very close to the experimental data for 186pb and 190po. For 194 Rn, the deviations are a little large on both the alpha-decay energy and the lifetime. We also calculate the alpha-decay energies for the isotopes 192～208Rn. The tendency for the change of the alpha-decay energies with neutron number is correctly reproduced in the relativistic mean-field theory (RMF). In general, the RMF theory can give a good description of the alpha decay chain of 194Rn.
15. Forward approximation as a mean-field approximation for the Anderson and many-body localization transitions
Pietracaprina, Francesca; Ros, Valentina; Scardicchio, Antonello
2016-02-01
In this paper we analyze the predictions of the forward approximation in some models which exhibit an Anderson (single-body) or many-body localized phase. This approximation, which consists of summing over the amplitudes of only the shortest paths in the locator expansion, is known to overestimate the critical value of the disorder which determines the onset of the localized phase. Nevertheless, the results provided by the approximation become more and more accurate as the local coordination (dimensionality) of the graph, defined by the hopping matrix, is made larger. In this sense, the forward approximation can be regarded as a mean-field theory for the Anderson transition in infinite dimensions. The sum can be efficiently computed using transfer matrix techniques, and the results are compared with the most precise exact diagonalization results available. For the Anderson problem, we find a critical value of the disorder which is 0.9 % off the most precise available numerical value already in 5 spatial dimensions, while for the many-body localized phase of the Heisenberg model with random fields the critical disorder hc=4.0 ±0.3 is strikingly close to the most recent results obtained by exact diagonalization. In both cases we obtain a critical exponent ν =1 . In the Anderson case, the latter does not show dependence on the dimensionality, as it is common within mean-field approximations. We discuss the relevance of the correlations between the shortest paths for both the single- and many-body problems, and comment on the connections of our results with the problem of directed polymers in random medium.
16. Beyond-mean-field corrections within the second random-phase approximation
Grasso, M.; Gambacurta, D.; Engel, J.
2016-06-01
A subtraction procedure, introduced to overcome double-counting problems in beyond-mean-field theories, is used in the second random-phase approximation (SRPA). Doublecounting problems arise in the energy-density functional framework in all cases where effective interactions tailored at leading order are used for higher-order calculations, such as those done in the SRPA model. It was recently shown that this subtraction procedure also guarantees that the stability condition related to the Thouless theorem is verified in extended RPA models. We discuss applications of the subtraction procedure, introduced within the SRPA model, to the nucleus 16O. The application of the subtraction procedure leads to: (i) stable results that are weakly cutoff dependent; (ii) a considerable upwards correction of the SRPA spectra (which were systematically shifted downwards by several MeV with respect to RPA spectra, in all previous calculations). With this important implementation of the model, many applications may be foreseen to analyze the genuine impact of 2 particle-2 hole configurations (without any cutoff dependences and anomalous shifts) on the excitation spectra of medium-mass and heavy nuclei.
17. Masses, Deformations and Charge Radii--Nuclear Ground-State Properties in the Relativistic Mean Field Model
Geng, L S; Meng, J
2005-01-01
We perform a systematic study of the ground-state properties of all the nuclei from the proton drip line to the neutron drip line throughout the periodic table employing the relativistic mean field model. The TMA parameter set is used for the mean-field Lagrangian density, and a state-dependent BCS method is adopted to describe the pairing correlation. The ground-state properties of a total of 6969 nuclei with $Z,N\\ge 8$ and $Z\\le 100$ from the proton drip line to the neutron drip line, including the binding energies, the separation energies, the deformations, and the rms charge radii, are calculated and compared with existing experimental data and those of the FRDM and HFB-2 mass formulae. This study provides the first complete picture of the current status of the descriptions of nuclear ground-state properties in the relativistic mean field model. The deviations from existing experimental data indicate either that new degrees of freedom are needed, such as triaxial deformations, or that serious effort is ne...
18. Mean-field effects on flows in relativistic heavy-ion collisions
Isse, M.; Ohnishi, A. [Hokkaido Univ., Graduate School of Science, Sapporo, Hokkaido (Japan); Otuka, N. [Hokkaido Univ., Graduate School of Engineering, Sapporo, Hokkaido (Japan); Sahu, P.K. [Istituto Nazionale di Fisica Nucleare, Sezione di Catania (Italy); Nara, Y. [Brookhaven National Laboratory, RIKEN BNL Research Center, Upton, NY (United States)
2002-09-01
At RHIC experiments, started in 2000, the data obtained recently seem to exhibit QGP formation, but the conclusion is not drawn yet. Here, we pay out attention to the collective flows at hadronic freeze-out as an evidence of QGP formation. To discuss it, the mean-field effect on the flows is not negligible. It is dominant at SIS or AGS energy, and our conjecture is that it is negligible at SPS or RHIC energy. We formed a model to investigate our assumption, and some simulated results are shown. (author)
19. Building relativistic mean field models for finite nuclei and neutron stars
Chen, Wei-Chia; Piekarewicz, J.
2014-10-01
Background: Theoretical approaches based on density functional theory provide the only tractable method to incorporate the wide range of densities and isospin asymmetries required to describe finite nuclei, infinite nuclear matter, and neutron stars. Purpose: A relativistic energy density functional (EDF) is developed to address the complexity of such diverse nuclear systems. Moreover, a statistical perspective is adopted to describe the information content of various physical observables. Methods: We implement the model optimization by minimizing a suitably constructed χ2 objective function using various properties of finite nuclei and neutron stars. The minimization is then supplemented by a covariance analysis that includes both uncertainty estimates and correlation coefficients. Results: A new model, "FSUGold2," is created that can well reproduce the ground-state properties of finite nuclei, their monopole response, and that accounts for the maximum neutron-star mass observed up to date. In particular, the model predicts both a stiff symmetry energy and a soft equation of state for symmetric nuclear matter, suggesting a fairly large neutron-skin thickness in Pb208 and a moderate value of the nuclear incompressibility. Conclusions: We conclude that without any meaningful constraint on the isovector sector, relativistic EDFs will continue to predict significantly large neutron skins. However, the calibration scheme adopted here is flexible enough to create models with different assumptions on various observables. Such a scheme—properly supplemented by a covariance analysis—provides a powerful tool to identify the critical measurements required to place meaningful constraints on theoretical models.
20. Building relativistic mean field models for finite nuclei and neutron stars
Chen, Wei-Chia
2014-01-01
Background: Theoretical approaches based on density functional theory provide the only tractable method to incorporate the wide range of densities and isospin asymmetries required to describe finite nuclei, infinite nuclear matter, and neutron stars. Purpose: A relativistic energy density functional (EDF) is developed to address the complexity of such diverse nuclear systems. Moreover, a statistical perspective is adopted to describe the information content of various physical observables. Methods: We implement the model optimization by minimizing a suitably constructed chi-square objective function using various properties of finite nuclei and neutron stars. The minimization is then supplemented by a covariance analysis that includes both uncertainty estimates and correlation coefficients. Results: A new model, FSUGold2, is created that can well reproduce the ground-state properties of finite nuclei, their monopole response, and that accounts for the maximum neutron star mass observed up to date. In particul...
1. Nonuniversal behavior for aperiodic interactions within a mean-field approximation.
Faria, Maicon S; Branco, N S; Tragtenberg, M H R
2008-04-01
We study the spin-1/2 Ising model on a Bethe lattice in the mean-field limit, with the interaction constants following one of two deterministic aperiodic sequences, the Fibonacci or period-doubling one. New algorithms of sequence generation were implemented, which were fundamental in obtaining long sequences and, therefore, precise results. We calculate the exact critical temperature for both sequences, as well as the critical exponents beta, gamma, and delta . For the Fibonacci sequence, the exponents are classical, while for the period-doubling one they depend on the ratio between the two exchange constants. The usual relations between critical exponents are satisfied, within error bars, for the period-doubling sequence. Therefore, we show that mean-field-like procedures may lead to nonclassical critical exponents.
2. Beyond mean-field approximations for accurate and computationally efficient models of on-lattice chemical kinetics
Pineda, M.; Stamatakis, M.
2017-07-01
Modeling the kinetics of surface catalyzed reactions is essential for the design of reactors and chemical processes. The majority of microkinetic models employ mean-field approximations, which lead to an approximate description of catalytic kinetics by assuming spatially uncorrelated adsorbates. On the other hand, kinetic Monte Carlo (KMC) methods provide a discrete-space continuous-time stochastic formulation that enables an accurate treatment of spatial correlations in the adlayer, but at a significant computation cost. In this work, we use the so-called cluster mean-field approach to develop higher order approximations that systematically increase the accuracy of kinetic models by treating spatial correlations at a progressively higher level of detail. We further demonstrate our approach on a reduced model for NO oxidation incorporating first nearest-neighbor lateral interactions and construct a sequence of approximations of increasingly higher accuracy, which we compare with KMC and mean-field. The latter is found to perform rather poorly, overestimating the turnover frequency by several orders of magnitude for this system. On the other hand, our approximations, while more computationally intense than the traditional mean-field treatment, still achieve tremendous computational savings compared to KMC simulations, thereby opening the way for employing them in multiscale modeling frameworks.
3. Investigation of Exotic Structure of the Largely Deformed Nucleus 23Al in the Relativistic-Mean-Field Model
CHEN Jin-Gen; ZHOU Xing-Fei; WANG Kun; MA Guo-Liang; TIAN Wen-Dong; ZUO Jia-Xu; MA Chun-Wang; CHEN Jin-Hui; YAN Ting-Zhi; SHEN Wen-Qing; CAI Xiang-Zhou; WANG Ting-Tai; MA Yu-Gang; REN Zhong-Zhou; FANG De-Qing; ZHONG Chen; WEI Yi-Bin; GUO Wei
2004-01-01
@@ A candidate for proton halo nucleus 23Al is investigated based on the constrained calculations in the framework of the deformed relativistic mean field (RMF) model with the NL075 parameter set. It is shown by the constrained calculations that the ground state of 23Al has a large deformation that corresponds to the prolate shape. With that large deformation, the non-constrained RMF calculation predicts that there appears an inversion between the 2s1/2 [211] and 1d5/2 [202] shells. The valence proton of 23Al is weakly bound and occupies 2s1/2 [211] and 1d5/2 [202] with the weights of 56% and 29%, respectively. The calculated RMS radius for matter is in agreement with the experimental one. It is also predicted that the difference between the proton RMS radius and the neutron one is very large. This suggests that there exists a proton halo in 23Al.
4. Investigation of the Mg isotopes using the shell-model-like approach in relativistic mean field theory
Bai, Hong-Bo; Zhang, Zhen-Hua; Li, Xiao-Wei
2016-11-01
Ground state properties for Mg isotopes, including binding energies, one- and two-neutron separation energies, pairing energies, nuclear matter radii and quadrupole deformation parameters, are obtained from the self-consistent relativistic mean field (RMF) model with the pairing correlations treated by a shell-mode-like approach (SLAP), in which the particle-number is conserved and the blocking effects are treated exactly. The experimental data, including the binding energies and the one- and two-neutron separation energies, which are sensitive to the treatment of pairing correlations and block effects, are well reproduced by the RMF+SLAP calculations. Supported by NSFC (11465001,11275098, 11275248, 11505058,11165001) and Natural Science Foundation of Inner Mongolia of China (2016BS0102)
5. Superdeformed rotational bands in the A ˜ 140-150 mass region: A cranked relativistic mean field description
Afanasjev, A. V.; König, J.; Ring, P.
1996-02-01
The cranked relativistic mean field approach is applied for a systematic investigation of superdeformed rotational bands observed in the A ˜ 140-150 mass region. The present investigation covers yrast and in some cases also excited superdeformed bands of all nuclei of this mass region in which such bands have been observed so far. Using the parameter set NL1, which has been adjusted ten years ago to a few spherical nuclei, reasonable agreement with experimental data is obtained throughout the mass region under investigation. It is shown that the calculated properties of superdeformed rotational bands such as the dependence of the dynamic moment of inertia J(2) with respect to the rotational frequency and the absolute value of the charge quadrupole moment Q0 depends sensitively on the number of occupied high- N intruder orbitals. This is agreement both with previous investigations within the cranked Nilsson-Strutinsky and the cranked Woods-Saxon-Strutinsky approaches and with available experimental data.
6. Relativistic continuum random phase approximation in spherical nuclei
Daoutidis, Ioannis
2009-10-01
Covariant density functional theory is used to analyze the nuclear response in the external multipole fields. The investigations are based on modern functionals with zero range and density dependent coupling constants. After a self-consistent solution of the Relativistic Mean Field (RMF) equations for the nuclear ground states multipole giant resonances are studied within the Relativistic Random Phase Approximation (RRPA), the small amplitude limit of the time-dependent RMF. The coupling to the continuum is treated precisely by calculating the single particle Greens-function of the corresponding Dirac equation. In conventional methods based on a discretization of the continuum this was not possible. The residual interaction is derived from the same RMF Lagrangian. This guarantees current conservation and a precise decoupling of the Goldstone modes. For nuclei with open shells pairing correlations are taken into account in the framework of BCS theory and relativistic quasiparticle RPA. Continuum RPA (CRPA) presents a robust method connected with an astonishing reduction of the numerical effort as compared to conventional methods. Modes of various multipolarities and isospin are investigated, in particular also the newly discovered Pygmy modes in the vicinity of the neutron evaporation threshold. The results are compared with conventional discrete RPA calculations as well as with experimental data. We find that the full treatment of the continuum is essential for light nuclei and the study of resonances in the neighborhood of the threshold. (orig.)
7. Quantum Dynamics of Dark and Dark-Bright Solitons beyond the Mean-Field Approximation
Krönke, Sven; Schmelcher, Peter
2014-05-01
Dark solitons are well-known excitations in one-dimensional repulsively interacting Bose-Einstein condensates, which feature a characteristical phase-jump across a density dip and form stability in the course of their dynamics. While these objects are stable within the celebrated Gross-Pitaevskii mean-field theory, the situation changes dramatically in the full many-body description: The condensate being initially in a dark soliton state dynamically depletes and the density notch fills up with depleted atoms. We analyze this process in detail with a particular focus on two-body correlations and the fate of grey solitons (dark solitons with finite density in the notch) and thereby complement the existing results in the literature. Moreover, we extend these studies to mixtures of two repulsively interacting bosonic species with a dark-bright soliton (dark soliton in one component filled with localized atoms of the other component) as the initial state. All these many-body quantum dynamics simulations are carried out with the recently developed multi-layer multi-configuration time-dependent Hartree method for bosons (ML-MCTDHB).
8. Thermal vacancies in random alloys in the single-site mean-field approximation
Ruban, A. V.
2016-04-01
A formalism for the vacancy formation energies in random alloys within the single-site mean-filed approximation, where vacancy-vacancy interaction is neglected, is outlined. It is shown that the alloy configurational entropy can substantially reduce the concentration of vacancies at high temperatures. The energetics of vacancies in random Cu0.5Ni0.5 alloy is considered as a numerical example illustrating the developed formalism. It is shown that the effective formation energy increases with temperature, however, in this particular system it is still below the mean value of the vacancy formation energy, which would correspond to the vacancy formation energy in a homogeneous model of a random alloy, such as given by the coherent potential approximation.
9. Tractable approximations for probabilistic models: The adaptive Thouless-Anderson-Palmer mean field approach
Opper, Manfred; Winther, Ole
2001-01-01
We develop an advanced mean held method for approximating averages in probabilistic data models that is based on the Thouless-Anderson-Palmer (TAP) approach of disorder physics. In contrast to conventional TAP. where the knowledge of the distribution of couplings between the random variables is r...... is required. our method adapts to the concrete couplings. We demonstrate the validity of our approach, which is so far restricted to models with nonglassy behavior? by replica calculations for a wide class of models as well as by simulations for a real data set....
10. Thermal vacancies in random alloys in the single-site mean-field approximation
Ruban, Andrei V
2015-01-01
A formalism for the vacancy formation energies in random alloys is outlined within the single-site mean-filed approximation where vacancy-vacancy interaction is neglected. It is shown that alloy entropy (without vacancies) can substantially reduce the concentration of vacancies at high temperatures. The energetics of vacancies in random Cu_0.5Ni_0.5 alloy is considered as a numerical example illustrating the developed formalism. It is shown that the effective formation energy is increases with temperature, however, in this particular system it is still below the mean value of the vacancy formation energy due to a large dispersion of the local vacancy formation energies.
11. Superdeformed rotational bands in the A{proportional_to}140-150 mass region: a cranked relativistic mean field description
Afanasjev, A.V. [Technische Univ. Muenchen, Garching (Germany). Physik-Department]|[Latvian Acad. of Sci., Salaspils (Latvia). Dept. of Math. Phys.]|[Lund Inst. of Tech. (Sweden). Dept. of Mathematical Physics; Koenig, J. [Technische Univ. Muenchen, Garching (Germany). Physik-Department; Ring, P. [Technische Univ. Muenchen, Garching (Germany). Physik-Department
1996-10-14
The cranked relativistic mean field approach is applied for a systematic investigation of superdeformed rotational bands observed in the A {proportional_to}140-150 mass region. The present investigation covers yrast and in some cases also excited superdeformed bands of all nuclei of this mass region in which such bands have been observed so far. Using the parameter set NL1, which has been adjusted ten years ago to a few spherical nuclei, reasonable agreement with experimental data is obtained throughout the mass region under investigation. It is shown that the calculated properties of superdeformed rotational bands such as the dependence of the dynamic moment of inertia J{sup (2)} with respect to the rotational frequency and the absolute value of the charge quadrupole moment Q{sub 0} depends sensitively on the number of occupied high-N intruder orbitals. This is in agreement both with previous investigations within the cranked Nilsson-Strutinsky and the cranked Woods-Saxon-Strutinsky approaches and with available experimental data. (orig.).
12. Systematic Calculation on Ground State Properties of Even-even Superheavy Nuclei Using Relativistic Mean Field Theory
ZhangHongfei; ZuoWei; SoojaeRenIm; ZhouXiaohong; LiJunqing
2003-01-01
In recent years the discovery of Super Heavy Element (SHE) with atomic number Z=108～116 has opened up a new era of research in nuclear physics, however, the extreme difficulties to synthesize SHE greatly restrict the experimental studies on it, so that the theoretical studies are very important. The Relativistic Mean Field theory (RMF) is proved to be a simple and successful theory due to its great success in describing the bulk properties at the β-stable valley, as well as nuclei far from the β-stable line, and gives good predictions for nuclei far beyond the end of the known periodic table. In the framework of RMF we have calculated the properties on SHN such as the binding energy, the deformation, single and double neutron separation energy, and the a-decay half-life and so on for nuclei Z=108～114 and N=156～190. The axial deformations considered by using the expansion of harmonic oscillator basis. The Lagrangian wc have used is as the following form:
13. Isoscalar Giant Resonances of 120Sn in the Quasiparticle Relativistic Random Phase Approximation
CAO Li-Gang; MA Zhong-Yu
2004-01-01
@@ The quasiparticle relativistic random phase approximation (QRRPA) is formulated based on the relativistic mean field ground state in the response function formalism. The pairing correlations are taken into account in the Bardeen-Cooper-Schrieffer approximation with a constant pairing gap. The numerical calculations are performed in the case of various isoscalar giant resonances of nucleus 120Sn with parameter set NL3. The calculated results show that the QRRPA approach could satisfactorily reproduce the experimental data of the energies of low-lying states.
14. Magnetic properties of double perovskite SrCrReO: Mean field approximation and Monte Carlo simulation
El Rhazouani, O.; Benyoussef, A.; Naji, S.; El Kenz, A.
2014-03-01
The double perovskite (DP) Sr2CrReO6, with its high Curie temperature, is a good candidate for magneto-electric and magneto-optic applications. Thus, a theoretical study by Monte Carlo Simulation (MCS) and Mean Field Approximation (MFA) in the context of the Ising model is important for a better understanding of the magnetic behavior of this material. The critical behavior of the magnetization and the susceptibility of this system have been determined. The phase diagrams depending on the exchange couplings and the crystal fields have been given. The values of critical exponents are also reported.
15. A systematic study of even-even nuclei in the nuclear chart by the relativistic mean field theory
Sumiyoshi, K.; Hirata, D.; Tanihata, I.; Sugahara, Y.; Toki, H. [Institute of Physical and Chemical Research, Wako, Saitama (Japan)
1997-03-01
We study systematically the properties of nuclei in the whole mass range up to the drip lines by the relativistic mean field (RMF) theory with deformations as a microscopic framework to provide the data of nuclear structure in the nuclear chart. The RMF theory is a phenomenological many-body framework, in which the self-consistent equations for nucleons and mesons are solved with arbitrary deformation, and has a potential ability to provide all the essential information of nuclear structure such as masses, radii and deformations together with single particle states and wave functions from the effective lagrangian containing nuclear interaction. As a first step toward the whole project, we study the ground state properties of even-even nuclei ranging from Z=8 to Z=120 up to the proton and neutron drip lines in the RMF theory. We adopt the parameter set TMA, which has been determined by the experimental masses and charge radii in a wide mass range, for the effective lagrangian of the RMF theory. We take into account the axially symmetric deformation using the constrained method on the quadrupole moment. We provide the properties of all even-even nuclei with all the possible ground state deformations extracted from the deformation energy curves by the constrained calculations. By studying the calculated ground state properties systematically, we aim to explore the general trend of masses, radii and deformations in the whole region of the nuclear chart. We discuss the agreement with experimental data and the predictions such as magicness and triaxial deformations beyond the experimental frontier. (author)
16. Electronic and magnetic properties of TbNi4Si: Ab initio calculations, mean field approximation and Monte Carlo simulation
Bensadiq, A.; Zaari, H.; Benyoussef, A.; El Kenz, A.
2016-09-01
Using the density functional theory, the electronic structure; density of states, band structure and exchange couplings of Tb Ni4 Si compound have been investigated. Magnetic and magnetocaloric properties of this material have been studied using Monte Carlo Simulation (MCS) and Mean Field Approximation (MFA) within a three dimensional Ising model. We calculated the isothermal magnetic entropy change, adiabatic temperature change and relative cooling power (RCP) for different external magnetic field and temperature. The highest obtained isothermal magnetic entropy change is of -14.52 J kg-1 K-1 for a magnetic field of H=4 T. The adiabatic temperature reaches a maximum value equal to 3.7 K and the RCP maximum value is found to be 125.12 J kg-1 for a field magnetic of 14 T.
17. Isoscalar giant monopole resonance for drip-line and super heavy nuclei in the framework of a relativistic mean field formalism with scaling calculation
Biswal, S K
2014-01-01
We study the isoscalar giant monopole resonance for drip-lines and super heavy nuclei in the frame work of a relativistic mean field theory with scaling approach. The well known extended Thomas-Fermi approximation in the non-linear $\\sigma$-$\\omega$ model is used to estimate the giant monopole excitation energy for some selected light spherical nuclei starting from the region of proton to neutron drip-lines. The application is extended to super heavy region for Z=114 and 120, which are predicted by several models as the next proton magic number beyond Z=82. We compared the excitation energy obtained by four successful force parameters NL1, NL3, NL3$^*$ and FSUGold. The monopole energy decreases toward the proton and neutron drip-lines in an isotopic chain for lighter mass nuclei contrary to a monotonous decrease for super heavy isotopes. The maximum and minimum monopole excitation energies are obtained for nuclei with minimum and maximum isospin, respectively in an isotopic chain.
18. Relativistic mean field study of the properties of Z=117 nucleus and the decay chains of $^{293,294}$117 isotopes
Bhuyan, M.; Gupta, S. K. Patra Raj K.
2010-01-01
We have calculated the binding energy, root-mean-square radius and quadrupole deformation parameter for the recently synthesized superheavy element Z=117, using the axially deformed relativistic mean field (RMF) model. The calculation is extended to various isotopes of Z=117 element, strarting from A=286 till A=310. We predict almost spherical structures in the ground state for almost all the isotopes. A shape transition appears at about A=292 from prolate to a oblate shape structures of Z=11...
19. The phase transition in hot $\\Lambda$ hypernuclei within relativistic Thomas-Fermi approximation
Hu, Jinniu; Bao, Shishao; Shen, Hong
2016-01-01
A self-consistent description for hot $\\Lambda$ hypernuclei in hypothetical big boxes is developed within the relativistic Thomas-Fermi approximation in order to investigate directly the liquid-gas phase coexistence in strangeness finite nuclear systems. We use the relativistic mean-field model for nuclear interactions. The temperature dependence of $\\Lambda$ hyperon density, $\\Lambda$ hyperon radius, excitation energies, specific heat, and the binding energies of $\\Lambda$ hypernuclei from $^{16}_{\\Lambda}$O to $^{208}_{\\Lambda}$Pb in phase transition region are calculated by using the subtraction procedure in order to separate the hypernucleus from the surrounding baryon gas. The $\\Lambda$ central density is very sensitive to the temperature. The radii of $\\Lambda$ hyperon at high temperature become very large. In the relativistic Thomas-Fermi approximation with the subtraction procedure, the properties of hypernuclei are independent of the size of the box in which the calculation is performed. The level de...
20. Static quadrupolar susceptibility for a Blume–Emery–Griffiths model based on the mean-field approximation
Pawlak, A., E-mail: [email protected] [Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61–614 Poznań (Poland); Gülpınar, G. [Department of Physics, Dokuz Eylül University, 35160 İzmir (Turkey); Erdem, R. [Department of Physics, Akdeniz University, 07058 Antalya (Turkey); Ağartıoğlu, M. [Institute of Science, Dokuz Eylül University, 35160 İzmir (Turkey)
2015-12-01
1. Stellar electron-capture rates calculated with the finite-temperature relativistic random-phase approximation
Niu, YiFei; Vretenar, Dario; Meng, Jie
2011-01-01
We introduce a self-consistent microscopic theoretical framework for modelling the process of electron capture on nuclei in stellar environment, based on relativistic energy density functionals. The finite-temperature relativistic mean-field model is used to calculate the single-nucleon basis and the occupation factors in a target nucleus, and $J^{\\pi} = 0^{\\pm}$, $1^{\\pm}$, $2^{\\pm}$ charge-exchange transitions are described by the self-consistent finite-temperature relativistic random-phase approximation. Cross sections and rates are calculated for electron capture on 54,56Fe and 76,78Ge in stellar environment, and results compared with predictions of similar and complementary model calculations.
2. Relativistic mean field study of the properties of Z=117 nucleus and the decay chains of $^{293,294}$117 isotopes
Bhuyan, M
2010-01-01
We have calculated the binding energy, root-mean-square radius and quadrupole deformation parameter for the recently synthesized superheavy element Z=117, using the axially deformed relativistic mean field (RMF) model. The calculation is extended to various isotopes of Z=117 element, strarting from A=286 till A=310. We predict almost spherical structures in the ground state for almost all the isotopes. A shape transition appears at about A=292 from prolate to a oblate shape structures of Z=117 nucleus in our mean field approach. The most stable isotope (largest binding energy per nucleon) is found to be the $^{288}$117 nucleus. Also, the Q-value of $\\alpha$-decay $Q_\\alpha$ and the half-lives $T_{\\alpha}$ are calculated for the $\\alpha$-decay chains of $^{293}$117 and $^{294}$117, supporting the magic numbers at N=172 and/ or 184.
3. The Vlasov formalism for extended relativistic mean field models: the crust-core transition and the stellar matter equation of state
Pais, Helena
2016-01-01
The Vlasov formalism is extended to relativistic mean-field hadron models with non-linear terms up to fourth order and applied to the calculation of the crust-core transition density. The effect of the nonlinear $\\omega\\rho$ and $\\sigma\\rho$ coupling terms on the crust-core transition density and pressure, and on the macroscopic properties of some families of hadronic stars is investigated. For that purpose, six families of relativistic mean field models are considered. Within each family, the members differ in the symmetry energy behavior. For all the models, the dynamical spinodals are calculated, and the crust-core transition density and pressure, and the neutron star mass-radius relations are obtained. The effect on the star radius of the inclusion of a pasta calculation in the inner crust is discussed. The set of six models that best satisfy terrestrial and observational constraints predicts a radius of 13.6$\\pm$0.3 km and a crust thickness of $1.36\\pm 0.06$km for a 1.4 $M_\\odot$ star.
4. Bound-state energy of double magic number plus one nucleon nuclei with relativistic mean-field approach
M MOUSAVI; M R SHOJAEI
2017-02-01
In this work, we have obtained energy levels and charge radius for the $\\beta$-stability line nucleus, in relativistic shell model. In this model, we considered a close shell for each nucleus containing double magicnumber and a single nucleon energy level. Here we have taken $^{41}$Ca with a single neutron in the $^{40}$Ca core as an illustrative example. Then we have selected the Eckart plus Hulthen potentials for interaction between the coreand the single nucleon. By using parametric Nikiforov–Uvarov (PNU) method, we have calculated the energy values and wave function. Finally, we have calculated the charge radius for 17O, $^{41}$Ca, $^{49}$Ca and $^{57}$Ni. Our results are in agreement with experimental values and hence this model can be applied for similar nuclei.
5. A study of Monte Carlo methods for weak approximations of stochastic particle systems in the mean-field?
Haji Ali, Abdul Lateef
2016-01-08
I discuss using single level and multilevel Monte Carlo methods to compute quantities of interests of a stochastic particle system in the mean-field. In this context, the stochastic particles follow a coupled system of Ito stochastic differential equations (SDEs). Moreover, this stochastic particle system converges to a stochastic mean-field limit as the number of particles tends to infinity. I start by recalling the results of applying different versions of Multilevel Monte Carlo (MLMC) for particle systems, both with respect to time steps and the number of particles and using a partitioning estimator. Next, I expand on these results by proposing the use of our recent Multi-index Monte Carlo method to obtain improved convergence rates.
6. Separable approximation method for two-body relativistic scattering
Tandy, P.C.; Thaler, R.M.
1988-03-01
A method for defining a separable approximation to a given interaction within a two-body relativistic equation, such as the Bethe-Salpeter equation, is presented. The rank-N separable representation given here permits exact reproduction of the T matrix on the mass shell and half off the mass shell at N selected bound state and/or continuum values of the invariant mass. The method employed is a four-space generalization of the separable representation developed for Schroedinger interactions by Ernst, Shakin, and Thaler, supplemented by procedures for dealing with the relativistic spin structure in the case of Dirac particles.
7. Separable approximation method for two-body relativistic scattering
Tandy, P. C.; Thaler, R. M.
1988-03-01
A method for defining a separable approximation to a given interaction within a two-body relativistic equation, such as the Bethe-Salpeter equation, is presented. The rank-N separable representation given here permits exact reproduction of the T matrix on the mass shell and half off the mass shell at N selected bound state and/or continuum values of the invariant mass. The method employed is a four-space generalization of the separable representation developed for Schrödinger interactions by Ernst, Shakin, and Thaler, supplemented by procedures for dealing with the relativistic spin structure in the case of Dirac particles.
8. The Post-Newtonian Approximation for Relativistic Compact Binaries
Futamase Toshifumi
2007-03-01
Full Text Available We discuss various aspects of the post-Newtonian approximation in general relativity. After presenting the foundation based on the Newtonian limit, we show a method to derive post-Newtonian equations of motion for relativistic compact binaries based on a surface integral approach and the strong field point particle limit. As an application we derive third post-Newtonian equations of motion for relativistic compact binaries which respect the Lorentz invariance in the post-Newtonian perturbative sense, admit a conserved energy, and are free from any ambiguity.
9. A multiscale variational approach to the kinetics of viscous classical liquids: The coarse-grained mean field approximation
Sereda, Yuriy V.; Ortoleva, Peter J.
2014-04-01
A closed kinetic equation for the single-particle density of a viscous simple liquid is derived using a variational method for the Liouville equation and a coarse-grained mean-field (CGMF) ansatz. The CGMF ansatz is based on the notion that during the characteristic time of deformation a given particle interacts with many others so that it experiences an average interaction. A trial function for the N-particle probability density is constructed using a multiscale perturbation method and the CGMF ansatz is applied to it. The multiscale perturbation scheme is based on the ratio of the average nearest-neighbor atom distance to the total size of the assembly. A constraint on the initial condition is discovered which guarantees that the kinetic equation is mass-conserving and closed in the single-particle density. The kinetic equation has much of the character of the Vlasov equation except that true viscous, and not Landau, damping is accounted for. The theory captures condensation kinetics and takes much of the character of the Gross-Pitaevskii equation in the weak-gradient short-range force limit.
10. Relativistic equation of state at subnuclear densities in the Thomas-Fermi approximation
Zhang, Z W
2014-01-01
We study the non-uniform nuclear matter using the self-consistent Thomas--Fermi approximation with a relativistic mean-field model. The non-uniform matter is assumed to be composed of a lattice of heavy nuclei surrounded by dripped nucleons. At each temperature $T$, proton fraction $Y_p$, and baryon mass density $\\rho_B$, we determine the thermodynamically favored state by minimizing the free energy with respect to the radius of the Wigner--Seitz cell, while the nucleon distribution in the cell can be determined self-consistently in the Thomas--Fermi approximation. A detailed comparison is made between the present results and previous calculations in the Thomas--Fermi approximation with a parameterized nucleon distribution that has been adopted in the widely used Shen EOS.
11. Spins and parities of the odd-A P isotopes within a relativistic mean-field model and elastic magnetic electron-scattering theory
Wang, Zaijun; Ren, Zhongzhou; Dong, Tiekuang; Xu, Chang
2014-08-01
The ground-state spins and parities of the odd-A phosphorus isotopes 25-47P are studied with the relativistic mean-field (RMF) model and relativistic elastic magnetic electron-scattering theory (REMES). Results of the RMF model with the NL-SH, TM2, and NL3 parameters show that the 2s1/2 and 1d3/2 proton level inversion may occur for the neutron-rich isotopes 37-47P, and, consequently, the possible spin-parity values of 37-47P may be 3/2+, which, except for P47, differs from those given by the NUBASE2012 nuclear data table by Audi et al. Calculations of the elastic magnetic electron scattering of 37-47P with the single valence proton in the 2s1/2 and 1d3/2 state show that the form factors have significant differences. The results imply that elastic magnetic electron scattering can be a possible way to study the 2s1/2 and 1d3/2 level inversion and the spin-parity values of 37-47P. The results can also provide new tests as to what extent the RMF model, along with its various parameter sets, is valid for describing the nuclear structures. In addition, the contributions of the upper and lower components of the Dirac four-spinors to the form factors and the isotopic shifts of the magnetic form factors are discussed.
12. Multi-dimensional potential energy surfaces and non-axial octupole correlations in actinide and transfermium nuclei from relativistic mean field models
Lu, Bing-Nan; Zhao, En-Guang; Zhou, Shan-Gui
2013-01-01
We have developed multi-dimensional constrained covariant density functional theories (MDC-CDFT) for finite nuclei in which the shape degrees of freedom \\beta_{\\lambda\\mu} with even \\mu, e.g., \\beta_{20}, \\beta_{22}, \\beta_{30}, \\beta_{32}, \\beta_{40}, etc., can be described simultaneously. The functional can be one of the following four forms: the meson exchange or point-coupling nucleon interactions combined with the non-linear or density-dependent couplings. For the pp channel, either the BCS approach or the Bogoliubov transformation is implemented. The MDC-CDFTs with the BCS approach for the pairing (in the following labelled as MDC-RMF models with RMF standing for "relativistic mean field") have been applied to investigate multi-dimensional potential energy surfaces and the non-axial octupole $Y_{32}$-correlations in N=150 isotones. In this contribution we present briefly the formalism of MDC-RMF models and some results from these models. The potential energy surfaces with and without triaxial deformatio...
13. Alpha-decay properties of superheavy elements $Z=113-125$ in the relativistic mean-field theory with vector self-coupling of $\\omega$ meson
Sharma, M M; Münzenberg, G
2004-01-01
We have investigated properties of $\\alpha$-decay chains of recently produced superheavy elements Z=115 and Z=113 using the new Lagrangian model NL-SV1 with inclusion of the vector self-coupling of $\\omega$ meson in the framework of the relativistic mean-field theory. It is shown that the experimentally observed alpha-decay energies and half-lives are reproduced well by this Lagrangian model. Further calculations for the heavier elements with Z=117-125 show that these nuclei are superdeformed with a prolate shape in the ground state. A superdeformed shell-closure at Z=118 lends an additional binding and an extra stability to nuclei in this region. Consequently, it is predicted that the corresponding $Q_\\alpha$ values provide $\\alpha$-decay half-lives for heavier superheavy nuclei within the experimentally feasible conditions. The results are compared with those of macroscopic-microscopic approaches. A perspective of the difference in shell effects amongst various approaches is presented and its consequences o...
14. Electron correlation within the relativistic no-pair approximation
Almoukhalalati, Adel; Knecht, Stefan; Jensen, Hans Jørgen Aa.; Dyall, Kenneth G.; Saue, Trond
2016-08-01
This paper addresses the definition of correlation energy within 4-component relativistic atomic and molecular calculations. In the nonrelativistic domain the correlation energy is defined as the difference between the exact eigenvalue of the electronic Hamiltonian and the Hartree-Fock energy. In practice, what is reported is the basis set correlation energy, where the "exact" value is provided by a full Configuration Interaction (CI) calculation with some specified one-particle basis. The extension of this definition to the relativistic domain is not straightforward since the corresponding electronic Hamiltonian, the Dirac-Coulomb Hamiltonian, has no bound solutions. Present-day relativistic calculations are carried out within the no-pair approximation, where the Dirac-Coulomb Hamiltonian is embedded by projectors eliminating the troublesome negative-energy solutions. Hartree-Fock calculations are carried out with the implicit use of such projectors and only positive-energy orbitals are retained at the correlated level, meaning that the Hartree-Fock projectors are frozen at the correlated level. We argue that the projection operators should be optimized also at the correlated level and that this is possible by full Multiconfigurational Self-Consistent Field (MCSCF) calculations, that is, MCSCF calculations using a no-pair full CI expansion, but including orbital relaxation from the negative-energy orbitals. We show by variational perturbation theory that the MCSCF correlation energy is a pure MP2-like correlation expression, whereas the corresponding CI correlation energy contains an additional relaxation term. We explore numerically our theoretical analysis by carrying out variational and perturbative calculations on the two-electron rare gas atoms with specially tailored basis sets. In particular, we show that the correlation energy obtained by the suggested MCSCF procedure is smaller than the no-pair full CI correlation energy, in accordance with the underlying
15. Rotating Bose-Einstein condensates with a finite number of atoms confined in a ring potential: Spontaneous symmetry breaking beyond the mean-field approximation
Roussou, A.; Smyrnakis, J.; Magiropoulos, M.; Efremidis, Nikolaos K.; Kavoulakis, G. M.
2017-03-01
Motivated by recent experiments on Bose-Einstein condensed atoms which rotate in annular and/or toroidal traps, we study the effect of the finiteness of the atom number N on the states of lowest energy for a fixed expectation value of the angular momentum, under periodic boundary conditions. To attack this problem, we develop a general strategy, considering a linear superposition of the eigenstates of the many-body Hamiltonian, with amplitudes that we extract from the mean-field approximation. This many-body state breaks the symmetry of the Hamiltonian; it has the same energy to leading order in N as the mean-field state and the corresponding eigenstate of the Hamiltonian, however, it has a lower energy to subleading order in N and thus it is energetically favorable.
16. Extended quasiparticle approximation for relativistic electrons in plasmas
V.G.Morozov
2006-01-01
Full Text Available Starting with Dyson equations for the path-ordered Green's function, it is shown that the correlation functions for relativistic electrons (positrons in a weakly coupled non-equilibrium plasmas can be decomposed into sharply peaked quasiparticle parts and off-shell parts in a rather general form. To leading order in the electromagnetic coupling constant, this decomposition yields the extended quasiparticle approximation for the correlation functions, which can be used for the first principle calculation of the radiation scattering rates in QED plasmas.
17. Relativistic quasiparticle random phase approximation in deformed nuclei
Pena Arteaga, D.
2007-06-25
Covariant density functional theory is used to study the influence of electromagnetic radiation on deformed superfluid nuclei. The relativistic Hartree-Bogolyubov equations and the resulting diagonalization problem of the quasiparticle random phase approximation are solved for axially symmetric systems in a fully self-consistent way by a newly developed parallel code. Three different kinds of high precision energy functionals are investigated and special care is taken for the decoupling of the Goldstone modes. This allows the microscopic investigation of Pygmy and scissor resonances in electric and magnetic dipole fields. Excellent agreement with recent experiments is found and new types of modes are predicted for deformed systems with large neutron excess. (orig.)
18. Approximate Relativistic Solutions for One-Dimensional Cylindrical Coaxial Diode
曾正中; 刘国治; 邵浩
2002-01-01
Two approximate analytical relativistic solutions for one-dimensional, space-chargelimited cylindrical coaxial diode are derived and utilized to compose best-fitting approximate solutions. Comparison of the best-fitting solutions with the numerical one demonstrates an error of about 11% for cathode-inside arrangement and 12% in the cathode-outside case for ratios of larger to smaller electrode radius from 1.2 to 10 and a voltage above 0.5 MV up to 5 MV. With these solutions the diode lengths for critical self-magnetic bending and for the condition under which the parapotential model validates are calculated to be longer than 1 cm up to more than 100 cm depending on voltage, radial dimensions and electrode arrangement. The influence of ion flow from the anode on the relativistic electron-only solution is numerically computed, indicating an enhancement factor of total diode current of 1.85 to 4.19 related to voltage, radial dimension and electrode arrangement.
19. Relativistic Quasiparticle Random Phase Approximation with a Separable Pairing Force
TIAN Yuan; MA Zhong-Yu; Ring Peter
2009-01-01
In our previous work [Phys. Lett. (to be published), Chin. Phys. Lett. 23 (2006) 3226], we introduced a separable pairing force for relativistic Hartree-Bogoliubov calculations. This force was adjusted to reproduce the pairing properties of the Gogny force in nuclear matter. By using the well known techniques of Talmi and Moshinsky it can be expanded in a series of separable terms and converges quickly after a few terms. It was found that the pairing properties can be depicted on almost the same footing as the original pairing interaction, not only in nuclear matter, but also in finite nuclei. In this study, we construct a relativistic quasiparticle random phase approximation (RQRPA ) with this separable pairing interaction and calculate the excitation energies of the first excited 2+ .states and reduced B(E2; 0+ → 2+) transition rates for a chain of Sn isotopes in RQRPA. Compared with the results of the full Gogny force, we find that this simple separable pairing interaction can describe the pairing properties of the excited vibrational states as well as the original pairing interaction.
20. A simple approximation for the current-voltage characteristics of high-power, relativistic diodes
Ekdahl, Carl
2016-06-01
A simple approximation for the current-voltage characteristics of a relativistic electron diode is presented. The approximation is accurate from non-relativistic through relativistic electron energies. Although it is empirically developed, it has many of the fundamental properties of the exact diode solutions. The approximation is simple enough to be remembered and worked on almost any pocket calculator, so it has proven to be quite useful on the laboratory floor.
1. Dynamic mean field theory for lattice gas models of fluids confined in porous materials: higher order theory based on the Bethe-Peierls and path probability method approximations.
Edison, John R; Monson, Peter A
2014-07-14
Recently we have developed a dynamic mean field theory (DMFT) for lattice gas models of fluids in porous materials [P. A. Monson, J. Chem. Phys. 128(8), 084701 (2008)]. The theory can be used to describe the relaxation processes in the approach to equilibrium or metastable states for fluids in pores and is especially useful for studying system exhibiting adsorption/desorption hysteresis. In this paper we discuss the extension of the theory to higher order by means of the path probability method (PPM) of Kikuchi and co-workers. We show that this leads to a treatment of the dynamics that is consistent with thermodynamics coming from the Bethe-Peierls or Quasi-Chemical approximation for the equilibrium or metastable equilibrium states of the lattice model. We compare the results from the PPM with those from DMFT and from dynamic Monte Carlo simulations. We find that the predictions from PPM are qualitatively similar to those from DMFT but give somewhat improved quantitative accuracy, in part due to the superior treatment of the underlying thermodynamics. This comes at the cost of greater computational expense associated with the larger number of equations that must be solved.
2. Rheology and orientational distributions of rodlike particles with magnetic moment normal to the particle axis for semi-dense dispersions (analysis by means of mean field approximation).
Satoh, Akira; Sakuda, Yasuhiro
2007-04-15
We have considered a semi-dense dispersion composed of ferromagnetic rodlike particles with a magnetic moment normal to the particle axis to investigate the rheological properties and particle orientational distribution in a simple shear flow as well as an external magnetic field. We have adopted the mean field approximation to take into account magnetic particle-particle interactions. The basic equation of the orientational distribution function has been derived from the balance of the torques and solved numerically. The results obtained here are summarized as follows. For a very strong magnetic field, the magnetic moment of the rodlike particle is strongly restricted in the field direction, so that the particle points to directions normal to the flow direction (and also to the magnetic field direction). This characteristic of the particle orientational distribution is also valid for the case of a strong particle-particle interaction, as in the strong magnetic field case. To the contrary, for a weak interaction among particles, the particle orientational distribution is governed by a shear flow as well as an applied magnetic field. When the magnetic particle-particle interaction is strong under circumstances of an applied magnetic field, the magnetic moment has a tendency to incline to the magnetic field direction more strongly. This leads to the characteristic that the viscosity decreases with decreasing the distance between particles, and this tendency becomes more significant for a stronger particle-particle interaction. These characteristics concerning the viscosity are quite different from those for a semi-dense dispersion composed of rodlike particles with a magnetic moment along the particle direction.
3. Mean-field models and exotic nuclei
Bender, M.; Buervenich, T.; Maruhn, J.A.; Greiner, W. [Inst. fuer Theoretische Physik, Univ. Frankfurt (Germany); Rutz, K. [Inst. fuer Theoretische Physik, Univ. Frankfurt (Germany)]|[Gesellschaft fuer Schwerionenforschung mbH, Darmstadt (Germany); Reinhard, P.G. [Inst. fuer Theoretische Physik, Univ. Erlangen (Germany)
1998-06-01
We discuss two widely used nuclear mean-field models, the relativistic mean-field model and the (nonrelativistic) Skyrme-Hartree-Fock model, and their capability to describe exotic nuclei. Test cases are superheavy nuclei and neutron-rich Sn isotopes. New information in this regime helps to fix hitherto loosely determined aspects of the models. (orig.)
4. Electron correlation within the relativistic no-pair approximation
Almoukhalalati, Adel; Knecht, Stefan; Jensen, Hans Jørgen Aa
2016-01-01
This paper addresses the definition of correlation energy within 4-component relativistic atomic and molecular calculations. In the nonrelativistic domain the correlation energy is defined as the difference between the exact eigenvalue of the electronic Hamiltonian and the Hartree-Fock energy....... In practice, what is reported is the basis set correlation energy, where the "exact" value is provided by a full Configuration Interaction (CI) calculation with some specified one-particle basis. The extension of this definition to the relativistic domain is not straightforward since the corresponding......-like correlation expression, whereas the corresponding CI correlation energy contains an additional relaxation term. We explore numerically our theoretical analysis by carrying out variational and perturbative calculations on the two-electron rare gas atoms with specially tailored basis sets...
5. Normal modes of relativistic systems in post Newtonian approximation
Sobouti, Y
1998-01-01
We use the post Newtonian (pn) order of Liouville's equation (pnl) to study the normal modes of oscillation of a relativistic system. In addition to classical modes, we are able to isolate a new class of oscillations that arise from perturbations of the space-time metric. In the first pn order; a) their frequency is an order q smaller than the classical frequencies, where q is a pn expansion parameter; b) they are not damped, for there is no gravitational wave radiation in this order; c) they are not coupled with the classical modes in q order; d) in a spherically symmetric system, they are designated by a pair of angular momentum eigennumbers, (j,m), of a pair of phase space angular momentum operators (J^2,J_z). Hydrodynamical behavior of these new modes is also investigated; a) they do not disturb the equilibrium of the classical fluid; b) they generate macroscopic toroidal motions that in classical case would be neutral; c) they give rise to an oscillatory g_{0i} component of the metric tensor that otherwi...
6. Mean field games
Gomes, Diogo A.
2014-01-06
In this talk we will report on new results concerning the existence of smooth solutions for time dependent mean-field games. This new result is established through a combination of various tools including several a-priori estimates for time-dependent mean-field games combined with new techniques for the regularity of Hamilton-Jacobi equations.
7. Nonrelativistic mean-field description of the deformation of Λ hypernuclei
2009-01-01
The deformations of light Λ hypernuclei are studied in an extended nonrelativistic deformed Skyrme-Hartree-Fock approach with realistic modern nucleonic Skyrme forces,pairing correlations,and a microscopical lambda-nucleon interaction derived from Brueckner-Hartree-Fock calculations.Compared to the large effect of an additional Λ particle on nuclear deformation in the light soft nuclei within relativistic mean field method,this effect is much smaller in the nonrelativistic mean-field approximation.
8. Application of an effective gauge-invariant model to nuclear matter in the relativistic Hartree-Fock approximation
Bernardos, P. [Universidad de Cantabria, Departamento de Matematica Aplicada y Ciencias de la Computacion, 39005, Santander (Spain); Fomenko, V.N. [St Petersburg University for Railway Engineering, Department of Mathematics, 190031, St Petersburg (Russian Federation); Marcos, S.; Niembro, R. [Universidad de Cantabria, Departamento de Fisica Moderna, 39005, Santander (Spain); Lopez-Quelle, M. [Universidad de Cantabria, Departamento de Fisica Aplicada, 39005, Santander (Spain); Savushkin, L.N. [St Petersburg University for Telecommunications, Department of Physics, 191186, St Petersburg (Russian Federation)
2001-02-01
An effective nuclear model describing {omega}-, {rho}- and axial-mesons as gauge fields is applied to nuclear matter in the relativistic Hartree-Fock approximation. The isoscalar two-pion exchange is simulated by a scalar field s similar to that used in the conventional relativistic mean-field approach. Two more scalar fields are essential ingredients of the present treatment: the {sigma}-field, the chiral partner of the pion, and the {sigma}-field, the Higgs field for the {omega}-meson. Two versions of the model are used depending on whether the {sigma}-field is considered as a dynamical variable or 'frozen', by taking its mass as infinite. The model contains four free parameters in the first case and three in the second one which are fitted to the nuclear matter saturation conditions. The nucleon and meson effective masses, compressibility modulus and symmetry energy are calculated. The results prove the reliability of the Dirac-Hartree-Fock approach within the linear realization of the chiral symmetry. (author)
9. Mean Field ICA
Petersen, Kaare Brandt
2006-01-01
This thesis describes investigations and improvements of a technique for Independent Component Analysis (ICA), called "Mean Field ICA". The main focus of the thesis is the optimization part of the algorithm, the so-called "EM algorithm". Using different approaches it is demonstrated that the EM...... Gradient Recipe is applicable to a wide selection of models. Furthermore, the Mean Field ICA model is extended to incorporate ltering over time in a so-called "convolutive ICA" model. Finally, by using mixture of Gaussians as source priors, the generative and ltering approach to ICA is compared...
10. An introduction to relativistic magnetohydrodynamics I. The force-free approximation
2005-12-01
This lecture summarizes basic equations of relativistic magnetohydrodynamics (MHD). The aim of the lecture is to present important relations and approximations that have been often employed and found useful in the astrophysical context, namely, in situations when plasma motion is governed by magnetohydrodynamic and gravitational effects competing with each other near a black hole.
11. The limits of the mean field
Guerra, E.M. de [Inst. de Estructura de la Materia, Consejo Superior de Investigaciones Cientificas (Spain)
2001-07-01
In these talks, we review non relativistic selfconsistent mean field theories, their scope and limitations. We first discuss static and time dependent mean field approaches for particles and quasiparticles, together with applications. We then discuss extensions that go beyond the non-relativistic independent particle limit. On the one hand, we consider extensions concerned with restoration of symmetries and with the treatment of collective modes, particularly by means of quantized ATDHF. On the other hand, we consider extensions concerned with the relativistic dynamics of bound nucleons. We present data on nucleon momentum distributions that show the need for relativistic mean field approach and probe the limits of the mean field concept. Illustrative applications of various methods are presented stressing the role that selfconsistency plays in providing a unifying reliable framework to study all sorts of properties and phenomena. From global properties such as size, mass, lifetime,.., to detailed structure in excitation spectra (high spin, RPA modes,..), as well as charge, magnetization and velocity distributions. (orig.)
12. Nonasymptotic mean-field games
Tembine, Hamidou
2014-12-01
Mean-field games have been studied under the assumption of very large number of players. For such large systems, the basic idea consists to approximate large games by a stylized game model with a continuum of players. The approach has been shown to be useful in some applications. However, the stylized game model with continuum of decision-makers is rarely observed in practice and the approximation proposed in the asymptotic regime is meaningless for networked systems with few entities. In this paper we propose a mean-field framework that is suitable not only for large systems but also for a small world with few number of entities. The applicability of the proposed framework is illustrated through a dynamic auction with asymmetric valuation distributions.
13. Nonasymptotic mean-field games
Tembine, Hamidou
2014-12-01
Mean-field games have been studied under the assumption of very large number of players. For such large systems, the basic idea consists of approximating large games by a stylized game model with a continuum of players. The approach has been shown to be useful in some applications. However, the stylized game model with continuum of decision-makers is rarely observed in practice and the approximation proposed in the asymptotic regime is meaningless for networks with few entities. In this paper, we propose a mean-field framework that is suitable not only for large systems but also for a small world with few number of entities. The applicability of the proposed framework is illustrated through various examples including dynamic auction with asymmetric valuation distributions, and spiteful bidders.
14. Dynamical Mean-Field Theory
Vollhardt, D.; Byczuk, K.; Kollar, M.
2011-01-01
The dynamical mean-field theory (DMFT) is a widely applicable approximation scheme for the investigation of correlated quantum many-particle systems on a lattice, e.g., electrons in solids and cold atoms in optical lattices. In particular, the combination of the DMFT with conventional methods for the calculation of electronic band structures has led to a powerful numerical approach which allows one to explore the properties of correlated materials. In this introductory article we discuss the ...
15. Electric dipole response of neutron-rich Calcium isotopes in relativistic quasiparticle time blocking approximation
Egorova, Irina A
2016-01-01
New results for electric dipole strength in the chain of even-even Calcium isotopes with the mass numbers A = 40 - 54 are presented. Starting from the covariant Lagrangian of Quantum Hadrodynamics, spectra of collective vibrations (phonons) and phonon-nucleon coupling vertices for $J \\leq 6$ and normal parity were computed in a self-consistent relativistic quasiparticle random phase approximation (RQRPA). These vibrations coupled to Bogoliubov two-quasiparticle configurations (2q$\\otimes$phonon) form the model space for the calculations of the dipole response function in the relativistic quasiparticle time blocking approximation (RQTBA). The results for giant dipole resonance in the latter approach are compared to those obtained in RQRPA and to available data. Evolution of the dipole strength with neutron number is investigated for both high-frequency giant dipole resonance (GDR) and low-lying strength. Development of a pygmy resonant structure on the low-energy shoulder of GDR is traced and analyzed in terms...
16. Calculation of indirect nuclear spin-spin coupling constants within the regular approximation for relativistic effects.
Filatov, Michael; Cremer, Dieter
2004-06-22
A new method for calculating the indirect nuclear spin-spin coupling constant within the regular approximation to the exact relativistic Hamiltonian is presented. The method is completely analytic in the sense that it does not employ numeric integration for the evaluation of relativistic corrections to the molecular Hamiltonian. It can be applied at the level of conventional wave function theory or density functional theory. In the latter case, both pure and hybrid density functionals can be used for the calculation of the quasirelativistic spin-spin coupling constants. The new method is used in connection with the infinite-order regular approximation with modified metric (IORAmm) to calculate the spin-spin coupling constants for molecules containing heavy elements. The importance of including exact exchange into the density functional calculations is demonstrated.
17. Mean field theory for fermion-based U(2) anyons
McGraw, P
1996-01-01
The energy density is computed for a U(2) Chern-Simons theory coupled to a non-relativistic fermion field (a theory of non-Abelian anyons'') under the assumptions of uniform charge and matter density. When the matter field is a spinless fermion, we find that this energy is independent of the two Chern-Simons coupling constants and is minimized when the non-Abelian charge density is zero. This suggests that there is no spontaneous breaking of the SU(2) subgroup of the symmetry, at least in this mean-field approximation. For spin-1/2 fermions, we find self-consistent mean-field states with a small non-Abelian charge density, which vanishes as the theory of free fermions is approached.
18. Continuous time finite state mean field games
Gomes, Diogo A.
2013-04-23
In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. For this mean field problem we prove a trend to equilibrium theorem, that is convergence, in an appropriate limit, to stationary solutions. Then we study an N+1-player problem, which the mean field model attempts to approximate. Our main result is the convergence as N→∞ of the mean field model and an estimate of the rate of convergence. We end the paper with some further examples for potential mean field games. © 2013 Springer Science+Business Media New York.
19. PADÉ APPROXIMANTS FOR THE EQUATION OF STATE FOR RELATIVISTIC HYDRODYNAMICS BY KINETIC THEORY
Tsai, Shang-Hsi; Yang, Jaw-Yen, E-mail: [email protected] [Institute of Applied Mechanics, National Taiwan University, Taipei 10764, Taiwan (China)
2015-07-20
A two-point Padé approximant (TPPA) algorithm is developed for the equation of state (EOS) for relativistic hydrodynamic systems, which are described by the classical Maxwell–Boltzmann statistics and the semiclassical Fermi–Dirac statistics with complete degeneracy. The underlying rational function is determined by the ratios of the macroscopic state variables with various orders of accuracy taken at the extreme relativistic limits. The nonunique TPPAs are validated by Taub's inequality for the consistency of the kinetic theory and the special theory of relativity. The proposed TPPA is utilized in deriving the EOS of the dilute gas and in calculating the specific heat capacity, the adiabatic index function, and the isentropic sound speed of the ideal gas. Some general guidelines are provided for the application of an arbitrary accuracy requirement. The superiority of the proposed TPPA is manifested in manipulating the constituent polynomials of the approximants, which avoids the arithmetic complexity of struggling with the modified Bessel functions and the hyperbolic trigonometric functions arising from the relativistic kinetic theory.
20. Parity violation in quasielastic electron-nucleus scattering within the relativistic impulse approximation
González-Jiménez, R; Donnelly, T W
2015-01-01
We study parity violation in quasielastic (QE) electron-nucleus scattering using the relativistic impulse approximation. Different fully relativistic approaches have been considered to estimate the effects associated with the final-state interactions. We have computed the parity-violating quasielastic (PVQE) asymmetry and have analyzed its sensitivity to the different ingredients that enter in the description of the reaction mechanism: final-state interactions, nucleon off-shellness effects, current gauge ambiguities. Particular attention has been paid to the description of the weak neutral current form factors. The PVQE asymmetry is proven to be an excellent observable when the goal is to get precise information on the axial-vector sector of the weak neutral current. Specifically, from measurements of the asymmetry at backward scattering angles good knowledge of the radiative corrections entering in the isovector axial-vector sector can be gained. Finally, scaling properties shown by the interference $\\gamma... 1. {beta}-decay rates of r-process nuclei in the relativistic quasiparticle random phase approximation Niksic, T.; Marketin, T.; Vretenar, D. [Zagreb Univ. (Croatia). Faculty of Science, Physics Dept.; Paar, N. [Technische Univ. Darmstadt (Germany). Inst. fuer Kernphysik; Ring, P. [Technische Univ. Muenchen, Garching (Germany). Physik-Department 2004-12-08 The fully consistent relativistic proton-neutron quasiparticle random phase approximation (PN-RQRPA) is employed in the calculation of {beta}-decay half-lives of neutron-rich nuclei in the N{approx}50 and N{approx}82 regions. A new density-dependent effective interaction, with an enhanced value of the nucleon effective mass, is used in relativistic Hartree-Bogolyubov calculation of nuclear ground states and in the particle-hole channel of the PN-RQRPA. The finite range Gogny D1S interaction is employed in the T=1 pairing channel, and the model also includes a proton-neutron particle-particle interaction. The theoretical half-lives reproduce the experimental data for the Fe, Zn, Cd, and Te isotopic chains, but overestimate the lifetimes of Ni isotopes and predict a stable {sup 132}Sn. (orig.) 2. Relativistic gravitational collapse in comoving coordinates: The post-quasistatic approximation Herrera, L 2010-01-01 A general iterative method proposed some years ago for the description of relativistic collapse, is presented here in comoving coordinates. For doing that we redefine the basic concepts required for the implementation of the method for comoving coordinates. In particular the definition of the post-quasistatic approximation in comoving coordinates is given. We write the field equations, the boundary conditions and a set of ordinary differential equations (the surface equations) which play a fundamental role in the algorithm. As an illustration of the method, we show how to build up a model inspired in the well known Schwarzschild interior solution. Both, the adiabatic and non adiabatic, cases are considered. 3. Collective Multipole excitations of exotic nuclei in relativistic continuum random phase approximation Yang, Ding; Ma, Zhongyu 2013-01-01 Journal of Combinatorial Theory, Series B, 98(1):173-225, 2008n exotic nuclei are studied in the framework of a fully self-consistent relativistic continuum random phase approximation (RCRPA). In this method the contribution of the continuum spectrum to nuclear excitations is treated exactly by the single particle Green's function. Different from the cases in stable nuclei, there are strong low-energy excitations in neutron-rich nuclei and proton-rich nuclei. The neutron or proton excess pushes the centroid of the strength function to lower energies and increases the fragmentation of the strength distribution. The effect of treating the contribution of continuum exactly are also discussed. 4. Spin symmetry in the relativistic modified Rosen-Morse potential with the approximate centrifugal term Chen Wen-Li; Wei Gao-Feng 2011-01-01 By applying a Pekeris-type approximation to the centrifugal term, we study the spin symmetry of a Dirac nucleon subjected to scalar and vector modified Rosen-Morse potentials. A complicated energy equation and associated twocomponent spinors with arbitrary spin-orbit coupling quantum number k are presented. The positive-energy bound states are checked numerically in the case of spin symmetry. The relativistic modified Rosen-Morse potential cannot trap a Dirac nucleon in the limiting case α→ 0. 5. Influence of magnetic interactions between clusters on particle orientational characteristics and viscosity of a colloidal dispersion composed of ferromagnetic spherocylinder particles: analysis by means of mean field approximation for a simple shear flow. Satoh, Akira 2005-09-01 We have theoretically investigated the particle orientational distribution and viscosity of a dense colloidal dispersion composed of ferromagnetic spherocylinder particles under an applied magnetic field. The mean field approximation has been applied to take into account the magnetic interactions of the particle of interest with the other ones that belong to the neighboring clusters, besides those that belong to its own cluster. The basic equation of the orientational distribution function, which is an integrodifferential equation, has approximately been solved by Galerkin's method and the method of successive approximation. Some of the main results obtained here are summarized as follows. Even when the magnetic interaction between particles is of the order of the thermal energy, the effect of particle-particle interactions on the orientational distribution comes to appear more significant with increasing volumetric fraction of particles; the orientational distribution function exhibits a sharper peak in the direction nearer to the magnetic field one as the volumetric fraction increases. Such a significant inclination of the particle in the field direction induces the large increase in viscosity in the range of larger values of the volumetric fraction. The above-mentioned characteristics of the orientational distribution and viscosity come to appear more significantly when the influence of the applied magnetic field is not so strong compared with that of magnetic particle-particle interactions. 6. Critical rotation of general-relativistic polytropic models simulating neutron stars: a post-Newtonian hybrid approximative scheme Geroyannis, Vassilis S 2014-01-01 We develop a "hybrid approximative scheme" in the framework of the post-Newtonian approximation for computing general-relativistic polytropic models simulating neutron stars in critical rigid rotation. We treat the differential equations governing such a model as a "complex initial value problem", and we solve it by using the so-called "complex-plane strategy". We incorporate into the computations the complete solution for the relativistic effects, this issue representing a significant improvement with regard to the classical post-Newtonian approximation, as verified by extended comparisons of the numerical results. 7. General Relativistic Radiant Shock Waves in the Post-Quasistatic Approximation H, Jorge A Rueda [Centro de Fisica Fundamental, Universidad de Los Andes, Merida 5101, Venezuela Escuela de Fisica, Universidad Industrial de Santander, A.A. 678, Bucaramanga (Colombia); Nunez, L A [Centro de Fisica Fundamental, Universidad de Los Andes, Merida 5101, Venezuela Centro Nacional de Calculo Cientifico, Universidad de Los Andes, CeCalCULA, Corporacion Parque Tecnologico de Merida, Merida 5101, Venezuela (Venezuela) 2007-05-15 An evolution of radiant shock wave front is considered in the framework of a recently presented method to study self-gravitating relativistic spheres, whose rationale becomes intelligible and finds full justification within the context of a suitable definition of the post-quasistatic approximation. The spherical matter configuration is divided into two regions by the shock and each side of the interface having a different equation of state and anisotropic phase. In order to simulate dissipation effects due to the transfer of photons and/or neutrinos within the matter configuration, we introduce the flux factor, the variable Eddington factor and a closure relation between them. As we expected the strong of the shock increases the speed of the fluid to relativistic ones and for some critical values is larger than light speed. In addition, we find that energy conditions are very sensible to the anisotropy, specially the strong energy condition. As a special feature of the model, we find that the contribution of the matter and radiation to the radial pressure are the same order of magnitude as in the mant as in the core, moreover, in the core radiation pressure is larger than matter pressure. 8. Roles of Antinucleon Degrees of Freedom in the Relativistic Random Phase Approximation Kurasawa, Haruki 2015-01-01 Roles of antinucleon degrees of freedom in the relativistic random phase approximation(RPA) are investigated. The energy-weighted sum of the RPA transition strengths is expressed in terms of the double commutator between the excitation operator and the Hamiltonian, as in nonrelativistic models. The commutator, however, should not be calculated with a usual way in the local field theory, because, otherwise, the sum vanishes. The sum value obtained correctly from the commutator is infinite, owing to the Dirac sea. Most of the previous calculations takes into account only a part of the nucleon-antinucleon states, in order to avoid the divergence problems. As a result, RPA states with negative excitation energy appear, which make the sum value vanish. Moreover, disregarding the divergence changes the sign of nuclear interactions in the RPA equation which describes the coupling of the nucleon particle-hole states with the nucleon-antinucleon states. Indeed, excitation energies of the spurious state and giant monop... 9. Complete equation of state for neutron stars using the relativistic Hartree-Fock approximation Miyatsu, Tsuyoshi; Cheoun, Myung-Ki [Department of Physics, Soongsil University, Seoul 156-743 (Korea, Republic of); Yamamuro, Sachiko; Nakazato, Ken' ichiro [Department of Physics, Faculty of Science and Technology, Tokyo University of Science (TUS), Noda 278-8510 (Japan) 2014-05-02 We construct the equation of state in a wide-density range for neutron stars within relativistic Hartree-Fock approximation. The properties of uniform and nonuniform nuclear matter are studied consistently. The tensor couplings of vector mesons to baryons due to exchange contributions (Fock terms) are included, and the change of baryon internal structure in matter is also taken into account using the quark-meson coupling model. The Thomas-Fermi calculation is adopted to describe nonuniform matter, where the lattice of nuclei and the neutron drip out of nuclei are considered. Even if hyperons exist in the core of a neutron star, we obtain the maximum neutron-star mass of 1.95M{sub ⊙}, which is consistent with the recently observed massive pulsar, PSR J1614-2230. In addition, the strange vector (φ) meson also plays a important role in supporting a massive neutron star. 10. Relativistic Three-Quark Bound States in Separable Two-Quark Approximation Öttel, M; Alkofer, R 2002-01-01 Baryons as relativistic bound states in 3-quark correlations are described by an effective Bethe-Salpeter equation when irreducible 3-quark interactions are neglected and separable 2-quark correlations are assumed. We present an efficient numerical method to calculate the nucleon mass and its covariant wave function in this quantum field theoretic quark-diquark model with quark-exchange interaction. Expanding the components of the spinorial wave function in terms of Chebyshev polynomials, the four-dimensional integral equations are in a first step reduced to a coupled set of one-dimensional ones. This set of linear and homogeneous equations defines a generalised eigenvalue problem. Representing the eigenvector corresponding to the largest eigenvalue, the Chebyshev moments are then obtained by iteration. The nucleon mass is implicitly determined by the eigenvalue, and its covariant wave function is reconstructed from the moments within the Chebyshev approximation. 11. Lagrangian theory of structure formation in relativistic cosmology I: Lagrangian framework and definition of a non-perturbative approximation Buchert, Thomas 2012-01-01 In this first paper we present a Lagrangian framework for the description of structure formation in general relativity, restricting attention to irrotational dust matter. As an application we present a self-contained derivation of a general-relativistic analogue of Zel'dovich's approximation for the description of structure formation in cosmology, and compare it with previous suggestions in the literature. This approximation is then investigated: paraphrasing the derivation in the Newtonian framework we provide general-relativistic analogues of the basic system of equations for a single dynamical field variable and recall the first-order perturbation solution of these equations. We then define a general-relativistic analogue of Zel'dovich's approximation and investigate consequences by functionally evaluating relevant variables. We so obtain a possibly powerful model that, although constructed through extrapolation of a perturbative solution, can be used to address non-perturbatively, e.g. problems of structu... 12. Approximate Harten-Lax-van Leer Riemann solvers for relativistic magnetohydrodynamics Mignone, Andrea; Bodo, G.; Ugliano, M. 2012-11-01 We review a particular class of approximate Riemann solvers in the context of the equations of ideal relativistic magnetohydrodynamics. Commonly prefixed as Harten-Lax-van Leer (HLL), this family of solvers approaches the solution of the Riemann problem by providing suitable guesses to the outermots characteristic speeds, without any prior knowledge of the solution. By requiring consistency with the integral form of the conservation law, a simplified set of jump conditions with a reduced number of characteristic waves may be obtained. The degree of approximation crucially depends on the wave pattern used in prepresnting the Riemann fan arising from the initial discontinuity breakup. In the original HLL scheme, the solution is approximated by collapsing the full characteristic structure into a single average state enclosed by two outermost fast mangnetosonic speeds. On the other hand, HLLC and HLLD improves the accuracy of the solution by restoring the tangential and Alfvén modes therefore leading to a representation of the Riemann fan in terms of 3 and 5 waves, respectively. 13. Kinetic mean-field theories Karkheck, John; Stell, George 1981-08-01 A kinetic mean-field theory for the evolution of the one-particle distribution function is derived from maximizing the entropy. For a potential with a hard-sphere core plus tail, the resulting theory treats the hard-core part as in the revised Enskog theory. The tail, weighted by the hard-sphere pair distribution function, appears linearly in a mean-field term. The kinetic equation is accompanied by an entropy functional for which an H theorem was proven earlier. The revised Enskog theory is obtained by setting the potential tail to zero, the Vlasov equation is obtained by setting the hard-sphere diameter to zero, and an equation of the Enskog-Vlasov type is obtained by effecting the Kac limit on the potential tail. At equilibrium, the theory yields a radial distribution function that is given by the hard-sphere reference system and thus furnishes through the internal energy a thermodynamic description which is exact to first order in inverse temperature. A second natural route to thermodynamics (from the momentum flux which yields an approximate equation of state) gives somewhat different results; both routes coincide and become exact in the Kac limit. Our theory furnishes a conceptual basis for the association in the heuristically based modified Enskog theory (MET) of the contact value of the radial distribution function with the ''thermal pressure'' since this association follows from our theory (using either route to thermodynamics) and moreover becomes exact in the Kac limit. Our transport theory is readily extended to the general case of a soft repulsive core, e.g., as exhibited by the Lennard-Jones potential, via by-now-standard statistical-mechanical methods involving an effective hard-core potential, thus providing a self-contained statistical-mechanical basis for application to such potentials that is lacking in the standard versions of the MET. We obtain very good agreement with experiment for the thermal conductivity and shear viscosity of several 14. Weakly coupled mean-field game systems Gomes, Diogo A. 2016-07-14 Here, we prove the existence of solutions to first-order mean-field games (MFGs) arising in optimal switching. First, we use the penalization method to construct approximate solutions. Then, we prove uniform estimates for the penalized problem. Finally, by a limiting procedure, we obtain solutions to the MFG problem. © 2016 Elsevier Ltd 15. A regularized stationary mean-field game Yang, Xianjin 2016-04-19 In the thesis, we discuss the existence and numerical approximations of solutions of a regularized mean-field game with a low-order regularization. In the first part, we prove a priori estimates and use the continuation method to obtain the existence of a solution with a positive density. Finally, we introduce the monotone flow method and solve the system numerically. 16. A mean field approach to watershed hydrology Bartlett, Mark; Porporato, Amilcare 2016-04-01 Mean field theory (also known as self-consistent field theory) is commonly used in statistical physics when modeling the space-time behavior of complex systems. The mean field theory approximates a complex multi-component system by considering a lumped (or average) effect for all individual components acting on a single component. Thus, the many body problem is reduced to a one body problem. For watershed hydrology, a mean field theory reduces the numerous point component effects to more tractable watershed averages, resulting in a consistent method for linking the average watershed fluxes to the local fluxes at each point. We apply this approach to the spatial distribution of soil moisture, and as a result, the numerous local interactions related to lateral fluxes of soil water are parameterized in terms of the average soil moisture. The mean field approach provides a basis for unifying and extending common event-based models (e.g. Soil Conservation Service curve number (SCS-CN) method) with more modern semi-distributed models (e.g. Variable Infiltration Capacity (VIC) model, the Probability Distributed (PDM) model, and TOPMODEL). We obtain simple equations for the fractions of the different source areas of runoff, the spatial variability of runoff, and the average runoff value (i.e., the so-called runoff curve). The resulting space time distribution of soil moisture offers a concise description of the variability of watershed fluxes. 17. Non-linear axisymmetric pulsations of rotating relativistic stars in the conformal flatness approximation Dimmelmeier, H; Font, J A; Dimmelmeier, Harald; Stergioulas, Nikolaos; Font, Jose A. 2005-01-01 We study non-linear axisymmetric pulsations of rotating relativistic stars using a general relativistic hydrodynamics code under the assumption of a conformal flatness. We compare our results to previous simulations where the spacetime dynamics was neglected. The pulsations are studied along various sequences of both uniformly and differentially rotating relativistic polytropes with index N = 1. We identify several modes, including the lowest-order l = 0, 2, and 4 axisymmetric modes, as well as several axisymmetric inertial modes. Differential rotation significantly lowers mode frequencies, increasing prospects for detection by current gravitational wave interferometers. We observe an extended avoided crossing between the l = 0 and l = 4 first overtones, which is important for correctly identifying mode frequencies in case of detection. For uniformly rotating stars near the mass-shedding limit, we confirm the existence of the mass-shedding-induced damping of pulsations, though the effect is not as strong as i... 18. Mean Field Game for Marriage Bauso, Dario 2014-01-06 The myth of marriage has been and is still a fascinating historical societal phenomenon. Paradoxically, the empirical divorce rates are at an all-time high. This work describes a unique paradigm for preserving relationships and marital stability from mean-field game theory. We show that optimizing the long-term well-being via effort and society feeling state distribution will help in stabilizing relationships. 19. Quantum correlations in nuclear mean field theory through source terms Lee, S J 1996-01-01 Starting from full quantum field theory, various mean field approaches are derived systematically. With a full consideration of external source dependence, the stationary phase approximation of an action gives a nuclear mean field theory which includes quantum correlation effects (such as particle-hole or ladder diagram) in a simpler way than the Brueckner-Hartree-Fock approach. Implementing further approximation, the result can be reduced to Hartree-Fock or Hartree approximation. The role of the source dependence in a mean field theory is examined. 20. Mean-field Ensemble Kalman Filter Law, Kody 2015-01-07 A proof of convergence of the standard EnKF generalized to non-Gaussian state space models is provided. A density-based deterministic approximation of the mean-field limiting EnKF (MFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given a certain minimal order of convergence between the two, this extends to the deterministic filter approximation, which is therefore asymptotically superior to standard EnKF for d < 2 . The fidelity of approximation of the true distribution is also established using an extension of total variation metric to random measures. This is limited by a Gaussian bias term arising from non-linearity/non-Gaussianity of the model, which arises in both deterministic and standard EnKF. Numerical results support and extend the theory. 1. Relativistic RPA in axial symmetry Arteaga, D Pena; 10.1103/PhysRevC.77.034317 2009-01-01 Covariant density functional theory, in the framework of self-consistent Relativistic Mean Field (RMF) and Relativistic Random Phase approximation (RPA), is for the first time applied to axially deformed nuclei. The fully self-consistent RMF+RRPA equations are posed for the case of axial symmetry and non-linear energy functionals, and solved with the help of a new parallel code. Formal properties of RPA theory are studied and special care is taken in order to validate the proper decoupling of spurious modes and their influence on the physical response. Sample applications to the magnetic and electric dipole transitions in$^{20}$Ne are presented and analyzed. 2. Quantum corrections to the Relativistic mean-field theory Maydanyuk, Sergei P; Bakry, Ahmed 2016-01-01 In this paper, we compare the RMF theory and the model of deformed oscillator shells (DOS) in description of the quantum properties of the bound states of the spherically symmetric light nuclei. We obtain an explicit analytical relation between differential equations for the RMF theory and DOS model, which determine wave functions for nucleons. On such a basis we perform analysis of correspondence of quantum properties of nuclei. We find: (1) Potential$V_{RMF}$of the RMF theory for nucleons has the wave functions$f$and$g$with joint part$h$coincident exactly with the nucleon wave function of DOS model with potential$V_{\\rm shell}$. But, a difference between$V_{RMF}$and$V_{\\rm shell}$is essential for any nucleus. (2) The nucleon wave functions and densities obtained by the DOS and RMF theories are essentially different. The nucleon densities of the RMF theory contradict to knowledge about distribution of the proton and neutron densities inside the nuclei obtained from experimental data. This indica... 3. Pedestrian Flow in the Mean Field Limit Haji Ali, Abdul Lateef 2012-11-01 We study the mean-field limit of a particle-based system modeling the behavior of many indistinguishable pedestrians as their number increases. The base model is a modified version of Helbing\\'s social force model. In the mean-field limit, the time-dependent density of two-dimensional pedestrians satisfies a four-dimensional integro-differential Fokker-Planck equation. To approximate the solution of the Fokker-Planck equation we use a time-splitting approach and solve the diffusion part using a Crank-Nicholson method. The advection part is solved using a Lax-Wendroff-Leveque method or an upwind Backward Euler method depending on the advection speed. Moreover, we use multilevel Monte Carlo to estimate observables from the particle-based system. We discuss these numerical methods, and present numerical results showing the convergence of observables that were calculated using the particle-based model as the number of pedestrians increases to those calculated using the probability density function satisfying the Fokker-Planck equation. 4. Mean field interaction in biochemical reaction networks Tembine, Hamidou 2011-09-01 In this paper we establish a relationship between chemical dynamics and mean field game dynamics. We show that chemical reaction networks can be studied using noisy mean field limits. We provide deterministic, noisy and switching mean field limits and illustrate them with numerical examples. © 2011 IEEE. 5. Electron-deuteron scattering in a current-conserving description of relativistic bound states formalism and impulse approximation calculations Phillips, D R; Devine, N K 1998-01-01 The electromagnetic interactions of a relativistic two-body bound state are formulated in three dimensions using an equal-time (ET) formalism. This involves a systematic reduction of four-dimensional dynamics to a three-dimensional form by integrating out the time components of relative momenta. A conserved electromagnetic current is developed for the ET formalism. It is shown that consistent truncations of the electromagnetic current and the$NN$interaction kernel may be made, order-by-order in the coupling constants, such that appropriate Ward-Takahashi identities are satisfied. A meson-exchange model of the$NN$interaction is used to calculate deuteron vertex functions. Calculations of electromagnetic form factors for elastic scattering of electrons by deuterium are performed using an impulse-approximation current. Negative-energy components of the deuteron's vertex function and retardation effects in the meson-exchange interaction are found to have only minor effects on the deuteron form factors. 6. Two Photon Exchange in Impact Parameter Space in the Relativistic Eikonal Approximation for Elastic e - N Scattering Alhalholy, Tareq 2016-01-01 Using the relativistic Eikonal approximation, we study the one and two photon exchange amplitudes in elastic electron-nucleon scattering for the case of transversely polarized nucleons with unpolarized electrons beam. In our approach, we utilize the convolution theory of Fourier transforms and the transverse charge density in transverse momentum space to evaluate the one and two photon exchange Eikonal amplitudes. The results obtained for the$2\\gamma$amplitude in impact parameter space are compared to the corresponding 4D case. We show that while the one and two photon cross sections are azimuthally symmetric, the interference term between them is azimuthally asymmetric, which is an indication of an azimuthal single spin asymmetry for proton and neutron which can be attributed to the fact that the nucleon charge density is transversely (azimuthally) distorted in the transverse plane for transversely polarized nucleons. In addition, the calculations of the interference term for proton and neutron show agreem... 7. Mean Field Games for Stochastic Growth with Relative Utility Huang, Minyi, E-mail: [email protected] [Carleton University, School of Mathematics and Statistics (Canada); Nguyen, Son Luu, E-mail: [email protected] [University of Puerto Rico, Department of Mathematics (United States) 2016-12-15 This paper considers continuous time stochastic growth-consumption optimization in a mean field game setting. The individual capital stock evolution is determined by a Cobb–Douglas production function, consumption and stochastic depreciation. The individual utility functional combines an own utility and a relative utility with respect to the population. The use of the relative utility reflects human psychology, leading to a natural pattern of mean field interaction. The fixed point equation of the mean field game is derived with the aid of some ordinary differential equations. Due to the relative utility interaction, our performance analysis depends on some ratio based approximation error estimate. 8. Variational Worldline Approximation for the Relativistic Two-Body Bound State in a Scalar Model Barro-Bergfl"odt, K; Stingl, M 2006-01-01 We use the worldline representation of field theory together with a variational approximation to determine the lowest bound state in the scalar Wick-Cutkosky model where two equal-mass constituents interact via the exchange of mesons. Self-energy and vertex corrections are included approximately in a consistent way as well as crossed diagrams. Only vacuum-polarization effects of the heavy particles are neglected. In a path integral description of an appropriate current-current correlator an effective, retarded action is obtained by integrating out the meson field. As in the polaron problem we employ a quadratic trial action with variational functions to describe retardation and binding effects through multiple meson exchange.The variational equations for these functions are derived, discussed qualitatively and solved numerically. We compare our results with the ones from traditional approaches based on the Bethe-Salpeter equation and find an enhanced binding. For weak coupling this is worked out analytically ... 9. Deterministic Mean-Field Ensemble Kalman Filtering Law, Kody J. H. 2016-05-03 The proof of convergence of the standard ensemble Kalman filter (EnKF) from Le Gland, Monbet, and Tran [Large sample asymptotics for the ensemble Kalman filter, in The Oxford Handbook of Nonlinear Filtering, Oxford University Press, Oxford, UK, 2011, pp. 598--631] is extended to non-Gaussian state-space models. A density-based deterministic approximation of the mean-field limit EnKF (DMFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given a certain minimal order of convergence k between the two, this extends to the deterministic filter approximation, which is therefore asymptotically superior to standard EnKF for dimension d<2k. The fidelity of approximation of the true distribution is also established using an extension of the total variation metric to random measures. This is limited by a Gaussian bias term arising from nonlinearity/non-Gaussianity of the model, which arises in both deterministic and standard EnKF. Numerical results support and extend the theory. 10. Worldline Variational Approximation: A New Approach to the Relativistic Binding Problem Barro-Bergflodt, K; Stingl, M 2004-01-01 We determine the lowest bound-state pole of the density-density correlator in the scalar Wick-Cutkosky model where two equal-mass constituents interact via the exchange of mesons. This is done by employing the worldline representation of field theory together with a variational approximation as in Feynman's treatment of the polaron. Unlike traditional methods based on the Bethe-Salpeter equation, self-energy and vertex corrections are (approximately) included as are crossed diagrams. Only vacuum-polarization effects of the heavy particles are neglected. The well-known instability of the model due to self-energy effects leads to large qualitative and quantitative changes compared to traditional approaches which neglect them. We determine numerically the critical coupling constant above which no real solutions of the variational equations exist anymore and show that it is smaller than in the one-body case due to an induced instability. The width of the bound state above the critical coupling is estimated analyt... 11. The relativistic spherical δ -shell interaction in R3: Spectrum and approximation Mas, Albert; Pizzichillo, Fabio 2017-08-01 This note revolves on the free Dirac operator in R3 and its δ -shell interaction with electrostatic potentials supported on a sphere. On one hand, we characterize the eigenstates of those couplings by finding sharp constants and minimizers of some precise inequalities related to an uncertainty principle. On the other hand, we prove that the domains given by Dittrich et al. [J. Math. Phys. 30(12), 2875-2882 (1989)] and by Arrizabalaga et al. [J. Math. Pures Appl. 102(4), 617-639 (2014)] for the realization of an electrostatic spherical shell interaction coincide. Finally, we explore the spectral relation between the shell interaction and its approximation by short range potentials with shrinking support, improving previous results in the spherical case. 12. Risk-sensitive mean-field games Tembine, Hamidou 2014-04-01 In this paper, we study a class of risk-sensitive mean-field stochastic differential games. We show that under appropriate regularity conditions, the mean-field value of the stochastic differential game with exponentiated integral cost functional coincides with the value function satisfying a Hamilton -Jacobi- Bellman (HJB) equation with an additional quadratic term. We provide an explicit solution of the mean-field best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean-field risk-neutral problem is formulated and the corresponding mean-field equilibria are characterized in terms of backward-forward macroscopic McKean-Vlasov equations, Fokker-Planck-Kolmogorov equations, and HJB equations. We provide numerical examples on the mean field behavior to illustrate both linear and McKean-Vlasov dynamics. © 1963-2012 IEEE. 13. Fully General Relativistic Simulations of Core-Collapse Supernovae with An Approximate Neutrino Transport Kuroda, Takami; Takiwaki, Tomoya 2012-01-01 We present results from the first generation of multi-dimensional hydrodynamic core-collapse simulations in full general relativity (GR) that include an approximate treatment of neutrino transport. Using a M1 closure scheme with an analytic variable Eddington factor, we solve the energy-independent set of radiation energy and momentum based on the Thorne's momentum formalism. To simplify the source terms of the transport equations, a methodology of multiflavour neutrino leakage scheme is partly employed. Our newly developed code is designed to evolve the Einstein field equation together with the GR radiation hydrodynamic equations. We follow the dynamics starting from the onset of gravitational core-collapse of a 15$M_{\\odot}$star, through bounce, up to about 100 ms postbounce in this study to study how the spacial multi-dimensionality and GR would affect the dynamics in the early postbounce phase. Our 3D results support the anticipation in previous 1D results that the neutrino luminosity and average neutri... 14. Extended Deterministic Mean-Field Games Gomes, Diogo A. 2016-04-21 In this paper, we consider mean-field games where the interaction of each player with the mean field takes into account not only the states of the players but also their collective behavior. To do so, we develop a random variable framework that is particularly convenient for these problems. We prove an existence result for extended mean-field games and establish uniqueness conditions. In the last section, we consider the Master Equation and discuss properties of its solutions. 15. Mean field approaches for$\\Xi^-$hypernuclei and current experimental data Sun, T T; Sagawa, H; Schulze, H -J; Meng, J 2016-01-01 Motivated by the recently observed hypernucleus (Kiso event)$^{15}_{\\Xi}$C ($^{14}$N$+\\Xi^-$), we identify the state of this system theoretically within the framework of the relativistic-mean-field and Skyrme-Hartree-Fock models. The$\\Xi N$interactions are constructed to reproduce the two possibly observed$\\Xi^-$removal energies,$4.38\\pm 0.25$MeV or$1.11\\pm 0.25$MeV. The present result is preferable to be$^{14}{\\rm N}({\\rm g.s.})+\\Xi^-(1p)$, corresponding to the latter value. 16. Mean Field Approach to the Giant Wormhole Problem Gamba, A.; Kolokolov, I.; Martellini, M. We introduce a gaussian probability density for the space-time distribution of worm-holes, thus taking effectively into account wormhole interaction. Using a mean-field approximation for the free energy, we show that giant wormholes are probabilistically suppressed in a homogenous isotropic “large” universe. 17. Critical fluctuations for quantum mean-field models Fannes, M.; Kossakowski, A.; Verbeure, A. (Univ. Louvain (Belgium)) 1991-11-01 A Ginzburg-Landau-type approximation is proposed for the local Gibbs states for quantum mean-field models that leads to the exact thermodynamics. Using this approach, the spin fluctuations are computed for some spin-1/2 models. At the critical temperature, the distribution function showing abnormal fluctuations is found explicitly. 18. Model-checking mean-field models: algorithms & applications Kolesnichenko, Anna Victorovna 2014-01-01 Large systems of interacting objects are highly prevalent in today's world. In this thesis we primarily address such large systems in computer science. We model such large systems using mean-field approximation, which allows to compute the limiting behaviour of an infinite population of identical o 19. Optimized$\\delta$expansion for relativistic nuclear models Krein, G I; Peres-Menezes, D; Nielsen, M; Pinto, M B 1998-01-01 The optimized$\\delta$-expansion is a nonperturbative approach for field theoretic models which combines the techniques of perturbation theory and the variational principle. This technique is discussed in the$\\lambda \\phi^4$model and then implemented in the Walecka model for the equation of state of nuclear matter. The results obtained with the$\\delta$expansion are compared with those obtained with the traditional mean field, relativistic Hartree and Hartree-Fock approximations. 20. Mean-field games for marriage Bauso, Dario 2014-05-07 This article examines mean-field games for marriage. The results support the argument that optimizing the long-term well-being through effort and social feeling state distribution (mean-field) will help to stabilize marriage. However, if the cost of effort is very high, the couple fluctuates in a bad feeling state or the marriage breaks down. We then examine the influence of society on a couple using mean-field sentimental games. We show that, in mean-field equilibrium, the optimal effort is always higher than the one-shot optimal effort. We illustrate numerically the influence of the couple\\'s network on their feeling states and their well-being. © 2014 Bauso et al. 1. Degenerate second order mean field games systems Tonon, Daniela; Cardaliaguet, Pierre; Graber, Philip,; Poretta, Alessio 2014-01-01 Parallel session; International audience; We consider degenerate second order mean field games systems with a local coupling. The starting point is the idea that mean field games systems can be understood as an optimality condition for optimal control of PDEs. Developing this strategy for the degenerate second order case, we discuss the existence and uniqueness of a weak solution as well as its stability (vanishing viscosity limit). Speaker: Daniela TONON 2. Higher-order approximate solutions to the relativistic and Duffing-harmonic oscillators by modified He's homotopy methods Belendez, A [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Pascual, C [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Fernandez, E [Departamento de Optica, FarmacologIa y AnatomIa, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Neipp, C [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain); Belendez, T [Departamento de Fisica, Ingenieria de Sistemas y Teoria de la Senal, Universidad de Alicante, Apartado 99, E-03080 Alicante (Spain) 2008-02-15 A modified He's homotopy perturbation method is used to calculate higher-order analytical approximate solutions to the relativistic and Duffing-harmonic oscillators. The He's homotopy perturbation method is modified by truncating the infinite series corresponding to the first-order approximate solution before introducing this solution in the second-order linear differential equation, and so on. We find this modified homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. The approximate formulae obtained show excellent agreement with the exact solutions, and are valid for small as well as large amplitudes of oscillation, including the limiting cases of amplitude approaching zero and infinity. For the relativistic oscillator, only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate frequency of less than 1.6% for small and large values of oscillation amplitude, while this relative error is 0.65% for two iterations with two harmonics and as low as 0.18% when three harmonics are considered in the second approximation. For the Duffing-harmonic oscillator the relative error is as low as 0.078% when the second approximation is considered. Comparison of the result obtained using this method with those obtained by the harmonic balance methods reveals that the former is very effective and convenient. 3. Atomic approximation to the projection on electronic states in the Douglas-Kroll-Hess approach to the relativistic Kohn-Sham method. Matveev, Alexei V; Rösch, Notker 2008-06-28 We suggest an approximate relativistic model for economical all-electron calculations on molecular systems that exploits an atomic ansatz for the relativistic projection transformation. With such a choice, the projection transformation matrix is by definition both transferable and independent of the geometry. The formulation is flexible with regard to the level at which the projection transformation is approximated; we employ the free-particle Foldy-Wouthuysen and the second-order Douglas-Kroll-Hess variants. The (atomic) infinite-order decoupling scheme shows little effect on structural parameters in scalar-relativistic calculations; also, the use of a screened nuclear potential in the definition of the projection transformation shows hardly any effect in the context of the present work. Applications to structural and energetic parameters of various systems (diatomics AuH, AuCl, and Au(2), two structural isomers of Ir(4), and uranyl dication UO(2) (2+) solvated by 3-6 water ligands) show that the atomic approximation to the conventional second-order Douglas-Kroll-Hess projection (ADKH) transformation yields highly accurate results at substantial computational savings, in particular, when calculating energy derivatives of larger systems. The size-dependence of the intrinsic error of the ADKH method in extended systems of heavy elements is analyzed for the atomization energies of Pd(n) clusters (n 4. Pion mean fields and heavy baryons Yang, Ghil-Seok; Polyakov, Maxim V; Praszałowicz, Michał 2016-01-01 We show that the masses of the lowest-lying heavy baryons can be very well described in a pion mean-field approach. We consider a heavy baryon as a system consisting of the$N_c-1$light quarks that induce the pion mean field, and a heavy quark as a static color source under the influence of this mean field. In this approach we derive a number of \\textit{model-independent} relations and calculate the heavy baryon masses using those of the lowest-lying light baryons as input. The results are in remarkable agreement with the experimental data. In addition, the mass of the$\\Omega_b^*$baryon is predicted. 5. Mean Field Games with a Dominating Player Bensoussan, A., E-mail: [email protected] [The University of Texas at Dallas, International Center for Decision and Risk Analysis, Jindal School of Management (United States); Chau, M. H. M., E-mail: [email protected]; Yam, S. C. P., E-mail: [email protected] [The Chinese University of Hong Kong, Department of Statistics (Hong Kong, People’s Republic of China) (China) 2016-08-15 In this article, we consider mean field games between a dominating player and a group of representative agents, each of which acts similarly and also interacts with each other through a mean field term being substantially influenced by the dominating player. We first provide the general theory and discuss the necessary condition for the optimal controls and equilibrium condition by adopting adjoint equation approach. We then present a special case in the context of linear-quadratic framework, in which a necessary and sufficient condition can be asserted by stochastic maximum principle; we finally establish the sufficient condition that guarantees the unique existence of the equilibrium control. The proof of the convergence result of finite player game to mean field counterpart is provided in Appendix. 6. Mean field games for cognitive radio networks Tembine, Hamidou 2012-06-01 In this paper we study mobility effect and power saving in cognitive radio networks using mean field games. We consider two types of users: primary and secondary users. When active, each secondary transmitter-receiver uses carrier sensing and is subject to long-term energy constraint. We formulate the interaction between primary user and large number of secondary users as an hierarchical mean field game. In contrast to the classical large-scale approaches based on stochastic geometry, percolation theory and large random matrices, the proposed mean field framework allows one to describe the evolution of the density distribution and the associated performance metrics using coupled partial differential equations. We provide explicit formulas and algorithmic power management for both primary and secondary users. A complete characterization of the optimal distribution of energy and probability of success is given. 7. Dynamical mean-field theory from a quantum chemical perspective. Zgid, Dominika; Chan, Garnet Kin-Lic 2011-03-07 We investigate the dynamical mean-field theory (DMFT) from a quantum chemical perspective. Dynamical mean-field theory offers a formalism to extend quantum chemical methods for finite systems to infinite periodic problems within a local correlation approximation. In addition, quantum chemical techniques can be used to construct new ab initio Hamiltonians and impurity solvers for DMFT. Here, we explore some ways in which these things may be achieved. First, we present an informal overview of dynamical mean-field theory to connect to quantum chemical language. Next, we describe an implementation of dynamical mean-field theory where we start from an ab initio Hartree-Fock Hamiltonian that avoids double counting issues present in many applications of DMFT. We then explore the use of the configuration interaction hierarchy in DMFT as an approximate solver for the impurity problem. We also investigate some numerical issues of convergence within DMFT. Our studies are carried out in the context of the cubic hydrogen model, a simple but challenging test for correlation methods. Finally, we finish with some conclusions for future directions. 8. Robust mean field games for coupled Markov jump linear systems Moon, Jun; Başar, Tamer 2016-07-01 We consider robust stochastic large population games for coupled Markov jump linear systems (MJLSs). The N agents' individual MJLSs are governed by different infinitesimal generators, and are affected not only by the control input but also by an individual disturbance (or adversarial) input. The mean field term, representing the average behaviour of N agents, is included in the individual worst-case cost function to capture coupling effects among agents. To circumvent the computational complexity and analyse the worst-case effect of the disturbance, we use robust mean field game theory to design low-complexity robust decentralised controllers and to characterise the associated worst-case disturbance. We show that with the individual robust decentralised controller and the corresponding worst-case disturbance, which constitute a saddle-point solution to a generic stochastic differential game for MJLSs, the actual mean field behaviour can be approximated by a deterministic function which is a fixed-point solution to the constructed mean field system. We further show that the closed-loop system is uniformly stable independent of N, and an approximate optimality can be obtained in the sense of ε-Nash equilibrium, where ε can be taken to be arbitrarily close to zero as N becomes sufficiently large. A numerical example is included to illustrate the results. 9. Obstacle mean-field game problem Gomes, Diogo A. 2015-01-01 In this paper, we introduce and study a first-order mean-field game obstacle problem. We examine the case of local dependence on the measure under assumptions that include both the logarithmic case and power-like nonlinearities. Since the obstacle operator is not differentiable, the equations for first-order mean field game problems have to be discussed carefully. Hence, we begin by considering a penalized problem. We prove this problem admits a unique solution satisfying uniform bounds. These bounds serve to pass to the limit in the penalized problem and to characterize the limiting equations. Finally, we prove uniqueness of solutions. © European Mathematical Society 2015. 10. Mean-field magnetohydrodynamics and dynamo theory Krause, F 2013-01-01 Mean-Field Magnetohydrodynamics and Dynamo Theory provides a systematic introduction to mean-field magnetohydrodynamics and the dynamo theory, along with the results achieved. Topics covered include turbulence and large-scale structures; general properties of the turbulent electromotive force; homogeneity, isotropy, and mirror symmetry of turbulent fields; and turbulent electromotive force in the case of non-vanishing mean flow. The turbulent electromotive force in the case of rotational mean motion is also considered. This book is comprised of 17 chapters and opens with an overview of the gen 11. Approximate, non-relativistic scattering phase shifts, bound state energies, and wave function normalization factors for a screened Coulomb potential of the Hulthen type Buehring, W. 1983-03-01 Non-relativistic scattering phase shifts, bound state energies, and wave function normalization factors for a screened Coulomb potential of the Hulthen type are presented in the form of relatively simple analytic expressions. These formulae have been obtained by a suitable renormalization procedure applied to the quantities derived from an approximate Schroedinger equation which contains the exact Hulthen potential together with an approximate angular momentum term. When the screening exponent vanishes, our formulae reduce to the exact Coulomb expresions. The interrelation between our formulae and Pratt's analytic perturbation theory for screened Coulomb potentials' is discussed. 12. Analysis of factorization in (e,ep) reactions. A survey of the relativistic plane wave impulse approximation Caballero, J.A. [Univ. de Sevilla (Spain). Dept. de Fisica Atomica, Molecular y Nucl.]|[Instituto de Estructura de la Materia, Consejo Superior de Investigaciones Cientificas, Serrano 123, Madrid 28006 (Spain); Donnelly, T.W. [Centre for Theoretical Physics, Laboratory for Nuclear Science and Dept. of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307 (United States); Moya de Guerra, E. [Instituto de Estructura de la Materia, Consejo Superior de Investigaciones Cientificas, Serrano 123, Madrid 28006 (Spain); Udias, J.M. [Departamento de Fisica Atomica, Molecular y Nuclear, Universidad Complutense de Madrid, Avda. Complutense s/n, Madrid 28040 (Spain) 1998-03-23 The issue of factorization within the context of coincidence quasi-elastic electron scattering is revisited. Using a relativistic formalism for the entire reaction mechanism and restricting ourselves to the case of plane waves for the outgoing proton, we discuss the role of the negative-energy components of the bound nucleon wave function. (orig.). 30 refs. 13. Asymptotic Approximation of Solutions and Eigenvalues of a Boundary Problem for a Singular Perturbated Relativistic Analog of Schroedinger Equation Amirkhanov, I V; Zhidkova, I E; Vasilev, S A 2000-01-01 Asymptotics of eigenfunctions and eigenvalues has been obtained for a singular perturbated relativistic analog of Schrdinger equation. A singular convergence of asymptotic expansions of the boundary problems to degenerated problems is shown for a nonrelativistic Schrdinger equation. The expansions obtained are in a good agreement with a numeric experiment. 14. Propagation peculiarities of mean field massive gravity S. Deser 2015-10-01 Full Text Available Massive gravity (mGR describes a dynamical “metric” on a fiducial, background one. We investigate fluctuations of the dynamics about mGR solutions, that is about its “mean field theory”. Analyzing mean field massive gravity (m‾GR propagation characteristics is not only equivalent to studying those of the full non-linear theory, but also in direct correspondence with earlier analyses of charged higher spin systems, the oldest example being the charged, massive spin 3/2 Rarita–Schwinger (RS theory. The fiducial and mGR mean field background metrics in the m‾GR model correspond to the RS Minkowski metric and external EM field. The common implications in both systems are that hyperbolicity holds only in a weak background-mean-field limit, immediately ruling both theories out as fundamental theories; a situation in stark contrast with general relativity (GR which is at least a consistent classical theory. Moreover, even though both m‾GR and RS theories can still in principle be considered as predictive effective models in the weak regime, their lower helicities then exhibit superluminal behavior: lower helicity gravitons are superluminal as compared to photons propagating on either the fiducial or background metric. Thus our approach has uncovered a novel, dispersive, “crystal-like” phenomenon of differing helicities having differing propagation speeds. This applies both to m‾GR and mGR, and is a peculiar feature that is also problematic for consistent coupling to matter. 15. Thermal Effects in Dense Matter Beyond Mean Field Theory Constantinou, Constantinos; Prakash, Madappa 2016-01-01 The formalism of next-to-leading order Fermi Liquid Theory is employed to calculate the thermal properties of symmetric nuclear and pure neutron matter in a relativistic many-body theory beyond the mean field level which includes two-loop effects. For all thermal variables, the semi-analytical next-to-leading order corrections reproduce results of the exact numerical calculations for entropies per baryon up to 2. This corresponds to excellent agreement down to sub-nuclear densities for temperatures up to$20$MeV. In addition to providing physical insights, a rapid evaluation of the equation of state in the homogeneous phase of hot and dense matter is achieved through the use of the zero-temperature Landau effective mass function and its derivatives. 16. Dynamical mean field theory of optical third harmonic generation Jafari, S. A.; Tohyama, T.; Maekawa, S. 2006-01-01 We formulate the third harmonic generation (THG) within the dynamical mean field theory (DMFT) approximation of the Hubbard model. In the limit of large dimensions, where DMFT becomes exact, the vertex corrections to current vertices are identically zero, and hence the calculation of the THG spectrum reduces to a time-ordered convolution, followd by appropriate analytic continuuation. We present the typical THG spectrum of the Hubbard model obtained by this method. Within our DMFT calculation... 17. Mean-field dynamo action in renovating shearing flows. Kolekar, Sanved; Subramanian, Kandaswamy; Sridhar, S 2012-08-01 We study mean-field dynamo action in renovating flows with finite and nonzero correlation time (τ) in the presence of shear. Previous results obtained when shear was absent are generalized to the case with shear. The question of whether the mean magnetic field can grow in the presence of shear and nonhelical turbulence, as seen in numerical simulations, is examined. We show in a general manner that, if the motions are strictly nonhelical, then such mean-field dynamo action is not possible. This result is not limited to low (fluid or magnetic) Reynolds numbers nor does it use any closure approximation; it only assumes that the flow renovates itself after each time interval τ. Specifying to a particular form of the renovating flow with helicity, we recover the standard dispersion relation of the α(2)Ω dynamo, in the small τ or large wavelength limit. Thus mean fields grow even in the presence of rapidly growing fluctuations, surprisingly, in a manner predicted by the standard quasilinear closure, even though such a closure is not strictly justified. Our work also suggests the possibility of obtaining mean-field dynamo growth in the presence of helicity fluctuations, although having a coherent helicity will be more efficient. 18. A relativistic symmetry in nuclei Ginocchio, J N [MS B283, Theoretical Division, Los Alamos National Laboratory Los Alamos, New Mexico 87545 (Mexico) 2007-11-15 We review some of the empirical and theoretical evidence supporting pseudospin symmetry in nuclei as a relativistic symmetry. We review the case that the eigenfunctions of realistic relativistic nuclear mean fields approximately conserve pseudospin symmetry in nuclei. We discuss the implications of pseudospin symmetry for magnetic dipole transitions and Gamow-Teller transitions between states in pseudospin doublets. We explore a more fundamental rationale for pseudospin symmetry in terms of quantum chromodynamics (QCD), the basic theory of the strong interactions. We show that pseudospin symmetry in nuclei implies spin symmetry for an anti-nucleon in a nuclear environment. We also discuss the future and what role pseudospin symmetry may be expected to play in an effective field theory of nucleons. 19. Mean-field and non-mean-field behaviors in scale-free networks with random Boolean dynamics Castro e Silva, A [Departamento de Fisica, Universidade Federal de Ouro Preto, Campus Universitario, 35.400-000 Ouro Preto, Minas Gerais (Brazil); Kamphorst Leal da Silva, J, E-mail: [email protected], E-mail: [email protected] [Departamento de Fisica, Universidade Federal de Minas Gerais, Caixa Postal 702, 30.161-970, Belo Horizonte, Minas Gerais (Brazil) 2010-06-04 We study two types of simplified Boolean dynamics in scale-free networks, both with a synchronous update. Assigning only the Boolean functions AND and XOR to the nodes with probabilities 1 - p and p, respectively, we are able to analyze the density of 1's and the Hamming distance on the network by numerical simulations and by a mean-field approximation (annealed approximation). We show that the behavior is quite different if the node always enters in the dynamics as its own input (self-regulation) or not. The same conclusion holds for the Kauffman NK model. Moreover, the simulation results and the mean-field ones (i) agree well when there is no self-regulation and (ii) disagree for small p when self-regulation is present in the model. 20. Mean-Field and Non-Mean-Field Behaviors in Scale-free Networks with Random Boolean Dynamics Silva, A Castro e 2009-01-01 We study two types of simplified Boolean dynamics over scale-free networks, both with synchronous update. Assigning only Boolean functions AND and XOR to the nodes with probability$1-p$and$p$, respectively, we are able to analyze the density of 1's and the Hamming distance on the network by numerical simulations and by a mean-field approximation (annealed approximation). We show that the behavior is quite different if the node always enters in the dynamic as its own input (self-regulation) or not. The same conclusion holds for the Kauffman KN model. Moreover, the simulation results and the mean-field ones (i) agree well when there is no self-regulation, and (ii) disagree for small$pwhen self-regulation is present in the model. 1. Mean-field learning for satisfactory solutions Tembine, Hamidou 2013-12-01 One of the fundamental challenges in distributed interactive systems is to design efficient, accurate, and fair solutions. In such systems, a satisfactory solution is an innovative approach that aims to provide all players with a satisfactory payoff anytime and anywhere. In this paper we study fully distributed learning schemes for satisfactory solutions in games with continuous action space. Considering games where the payoff function depends only on own-action and an aggregate term, we show that the complexity of learning systems can be significantly reduced, leading to the so-called mean-field learning. We provide sufficient conditions for convergence to a satisfactory solution and we give explicit convergence time bounds. Then, several acceleration techniques are used in order to improve the convergence rate. We illustrate numerically the proposed mean-field learning schemes for quality-of-service management in communication networks. © 2013 IEEE. 2. Mean field methods for cortical network dynamics Hertz, J.; Lerchner, Alexander; Ahmadi, M. 2004-01-01 We review the use of mean field theory for describing the dynamics of dense, randomly connected cortical circuits. For a simple network of excitatory and inhibitory leaky integrate- and-fire neurons, we can show how the firing irregularity, as measured by the Fano factor, increases with the stren...... cortex. Finally, an extension of the model to describe an orientation hypercolumn provides understanding of how cortical interactions sharpen orientation tuning, in a way that is consistent with observed firing statistics... 3. Bosonic Dynamical Mean-Field Theory Snoek, Michiel; Hofstetter, Walter 2013-02-01 We derive the bosonic dynamical mean-field equations for bosonic atoms in optical lattices with arbitrary lattice geometry. The equations are presented as a systematic expansion in 1/z, z being the number of lattice neighbours. Hence the theory is applicable in sufficiently high-dimensional lattices. We apply the method to a two-component mixture, for which a rich phase diagram with spin order is revealed. 4. Mean field games systems of first order Cardaliaguet, Pierre; Graber, Philip Jameson 2014-01-01 International audience; We consider a system of mean field games with local coupling in the deterministic limit. Under general structure conditions on the Hamiltonian and coupling, we prove existence and uniqueness of the weak solution, characterizing this solution as the minimizer of some optimal control of Hamilton-Jacobi and continuity equations. We also prove that this solution converges in the long time average to the solution of the associated ergodic problem. 5. 'Phase diagram' of a mean field game Swiecicki, Igor; Ullmo, Denis 2015-01-01 Mean field games were introduced by J-M.Lasry and P-L. Lions in the mathematical community, and independently by M. Huang and co-workers in the engineering community, to deal with optimization problems when the number of agents becomes very large. In this article we study in detail a particular example called the 'seminar problem' introduced by O.Gu\\'eant, J-M Lasry, and P-L. Lions in 2010. This model contains the main ingredients of any mean field game but has the particular feature that all agent are coupled only through a simple random event (the seminar starting time) that they all contribute to form. In the mean field limit, this event becomes deterministic and its value can be fixed through a self consistent procedure. This allows for a rather thorough understanding of the solutions of the problem, through both exact results and a detailed analysis of various limiting regimes. For a sensible class of initial configurations, distinct behaviors can be associated to different domains in the parameter space... 6. Dynamical mean-field theory for quantum chemistry. Lin, Nan; Marianetti, C A; Millis, Andrew J; Reichman, David R 2011-03-04 The dynamical mean-field concept of approximating an unsolvable many-body problem in terms of the solution of an auxiliary quantum impurity problem, introduced to study bulk materials with a continuous energy spectrum, is here extended to molecules, i.e., finite systems with a discrete energy spectrum. The application to small clusters of hydrogen atoms yields ground state energies which are competitive with leading quantum chemical approaches at intermediate and large interatomic distances as well as good approximations to the excitation spectrum. 7. Time-odd mean fields in covariant density functional theory: Rotating systems Afanasjev, A V; 10.1103/PhysRev.82.034329 2010-01-01 Time-odd mean fields (nuclear magnetism) and their impact on physical observables in rotating nuclei are studied in the framework of covariant density functional theory (CDFT). It is shown that they have profound effect on the dynamic and kinematic moments of inertia. Particle number, configuration and rotational frequency dependences of their impact on the moments of inertia have been analysed in a systematic way. Nuclear magnetism can also considerably modify the band crossing features such as crossing frequencies and the properties of the kinematic and dynamic moments of inertia in the band crossing region. The impact of time-odd mean fields on the moments of inertia in the regions away from band crossing only weakly depends on the relativistic mean field parametrization, reflecting good localization of the properties of time-odd mean fields in CDFT. The moments of inertia of normal-deformed nuclei considerably deviate from the rigid body value. On the contrary, superdeformed and hyperdeformed nuclei have ... 8. Mean field methods for cortical network dynamics Hertz, J.; Lerchner, Alexander; Ahmadi, M. 2004-01-01 We review the use of mean field theory for describing the dynamics of dense, randomly connected cortical circuits. For a simple network of excitatory and inhibitory leaky integrate- and-fire neurons, we can show how the firing irregularity, as measured by the Fano factor, increases...... with the strength of the synapses in the network and with the value to which the membrane potential is reset after a spike. Generalizing the model to include conductance-based synapses gives insight into the connection between the firing statistics and the high- conductance state observed experimentally in visual... 9. Mean-field behavior of cluster dynamics Persky, N.; Ben-Av, R.; Kanter, I.; Domany, E. 1996-09-01 The dynamic behavior of cluster algorithms is analyzed in the classical mean-field limit. Rigorous analytical results below Tc establish that the dynamic exponent has the value zSW=1 for the Swendsen-Wang algorithm and zW=0 for the Wolff algorithm. An efficient Monte Carlo implementation is introduced, adapted for using these algorithms for fully connected graphs. Extensive simulations both above and below Tc demonstrate scaling and evaluate the finite-size scaling function by means of a rather impressive collapse of the data. 10. Resummed mean-field inference for strongly coupled data Jacquin, Hugo; Rançon, A. 2016-10-01 We present a resummed mean-field approximation for inferring the parameters of an Ising or a Potts model from empirical, noisy, one- and two-point correlation functions. Based on a resummation of a class of diagrams of the small correlation expansion of the log-likelihood, the method outperforms standard mean-field inference methods, even when they are regularized. The inference is stable with respect to sampling noise, contrarily to previous works based either on the small correlation expansion, on the Bethe free energy, or on the mean-field and Gaussian models. Because it is mostly analytic, its complexity is still very low, requiring an iterative algorithm to solve for N auxiliary variables, that resorts only to matrix inversions and multiplications. We test our algorithm on the Sherrington-Kirkpatrick model submitted to a random external field and large random couplings, and demonstrate that even without regularization, the inference is stable across the whole phase diagram. In addition, the calculation leads to a consistent estimation of the entropy of the data and allows us to sample form the inferred distribution to obtain artificial data that are consistent with the empirical distribution. 11. Mean field magnetization of gapped anisotropic multiplet Paixão, L. S.; Reis, M. S. 2014-06-01 Some materials have a large gap between the ground and first excited states. At temperatures smaller than the gap value, the thermodynamic properties of such materials are mainly ruled by the ground state. It is also common to find materials with magnetocrystalline anisotropy, which arises due to interatomic interactions. The present paper uses a classical approach to deal large angular momenta in such materials. Based on analytical expressions for the thermodynamics of paramagnetic gapped anisotropic multiplets, we use mean field theory to study the influence of the anisotropy upon the properties of interacting systems. We also use Landau theory to determine the influence of the anisotropy in first and second order phase transitions. It is found that the anisotropy increases the critical temperature, and enlarges the hysteresis of first order transitions. We present analytical expressions for the quantities analyzed. 12. Invisible dynamo in mean-field models Reshetnyak, M. Yu. 2016-07-01 The inverse problem in a spherical shell to find the two-dimensional spatial distributions of the α-effect and differential rotation in a mean-field dynamo model has been solved. The derived distributions lead to the generation of a magnetic field concentrated inside the convection zone. The magnetic field is shown to have no time to rise from the region of maximum generation located in the lower layers to the surface in the polarity reversal time due to magnetic diffusion. The ratio of the maximum magnetic energy in the convection zone to its value at the outer boundary reaches two orders of magnitude or more. This result is important in interpreting the observed stellar and planetary magnetic fields. The proposed method of solving the inverse nonlinear dynamo problem is easily adapted for a wide class of mathematical-physics problems. 13. Mean-field models for disordered crystals Cancès, Eric; Lewin, Mathieu 2012-01-01 In this article, we set up a functional setting for mean-field electronic structure models of Hartree-Fock or Kohn-Sham types for disordered crystals. The electrons are quantum particles and the nuclei are classical point-like articles whose positions and charges are random. We prove the existence of a minimizer of the energy per unit volume and the uniqueness of the ground state density of such disordered crystals, for the reduced Hartree-Fock model (rHF). We consider both (short-range) Yukawa and (long-range) Coulomb interactions. In the former case, we prove in addition that the rHF ground state density matrix satisfies a self-consistent equation, and that our model for disordered crystals is the thermodynamic limit of the supercell model. 14. Fictive impurity approach to dynamical mean field theory Fuhrmann, A. 2006-10-15 A new extension of the dynamical mean-field theory was investigated in the regime of large Coulomb repulsion. A number of physical quantities such as single-particle density of states, spin-spin correlation, internal energy and Neel temperature, were computed for a two-dimensional Hubbard model at half-filling. The numerical data were compared to our analytical results as well as to the results computed using the dynamical cluster approximation. In the second part of this work we consider a two-plane Hubbard model. The transport properties of the bilayer were investigated and the phase diagram was obtained. (orig.) 15. Mean Field Evolution of Fermions with Coulomb Interaction Porta, Marcello; Rademacher, Simone; Saffirio, Chiara; Schlein, Benjamin 2017-03-01 We study the many body Schrödinger evolution of weakly coupled fermions interacting through a Coulomb potential. We are interested in a joint mean field and semiclassical scaling, that emerges naturally for initially confined particles. For initial data describing approximate Slater determinants, we prove convergence of the many-body evolution towards Hartree-Fock dynamics. Our result holds under a condition on the solution of the Hartree-Fock equation, that we can only show in a very special situation (translation invariant data, whose Hartree-Fock evolution is trivial), but that we expect to hold more generally. 16. Time dependent mean-field games Gomes, Diogo A. 2014-01-06 We consider time dependent mean-field games (MFG) with a local power-like dependence on the measure and Hamiltonians satisfying both sub and superquadratic growth conditions. We establish existence of smooth solutions under a certain set of conditions depending both on the growth of the Hamiltonian as well as on the dimension. In the subquadratic case this is done by combining a Gagliardo-Nirenberg type of argument with a new class of polynomial estimates for solutions of the Fokker-Planck equation in terms of LrLp- norms of DpH. These techniques do not apply to the superquadratic case. In this setting we recur to a delicate argument that combines the non-linear adjoint method with polynomial estimates for solutions of the Fokker-Planck equation in terms of L1L1-norms of DpH. Concerning the subquadratic case, we substantially improve and extend the results previously obtained. Furthermore, to the best of our knowledge, the superquadratic case has not been addressed in the literature yet. In fact, it is likely that our estimates may also add to the current understanding of Hamilton-Jacobi equations with superquadratic Hamiltonians. 17. Nonequilibrium dynamical mean-field theory Eckstein, Martin 2009-12-21 The aim of this thesis is the investigation of strongly interacting quantum many-particle systems in nonequilibrium by means of the dynamical mean-field theory (DMFT). An efficient numerical implementation of the nonequilibrium DMFT equations within the Keldysh formalism is provided, as well a discussion of several approaches to solve effective single-site problem to which lattice models such as the Hubbard-model are mapped within DMFT. DMFT is then used to study the relaxation of the thermodynamic state after a sudden increase of the interaction parameter in two different models: the Hubbard model and the Falicov-Kimball model. In the latter case an exact solution can be given, which shows that the state does not even thermalize after infinite waiting times. For a slow change of the interaction, a transition to adiabatic behavior is found. The Hubbard model, on the other hand, shows a very sensitive dependence of the relaxation on the interaction, which may be called a dynamical phase transition. Rapid thermalization only occurs at the interaction parameter which corresponds to this transition. (orig.) 18. Time-odd mean fields in the rotating frame microscopic nature of nuclear magnetism Afanasiev, A V 2000-01-01 The microscopic role of nuclear magnetism in rotating frame is investigated for the first time in the framework of the cranked relativistic mean field theory. It is shown that nuclear magnetism modifies the expectation values of single-particle spin, orbital and total angular momenta along the rotational axis effectively creating additional angular momentum. This effect leads to the increase of kinematic and dynamic moments of inertia at given rotational frequency and has an impact on effective alignments. 19. Shell Effect of Superheavy Nuclei in Self-consistent Mean-Field Models RENZhong-Zhou; TAIFei; XUChang; CHENDing-Han; ZHANGHu-Yong; CAIXiang-Zhou; SHENWen-Qing 2004-01-01 We analyze in detail the numerical results of superheavy nuclei in deformed relativistic mean-field model and deformed Skyrme-Hartree-Fock model. The common points and differences of both models are systematically compared and discussed. Their consequences on the stability of superheavy nuclei are explored and explained. The theoreticalresults are compared with new data of superheavy nuclei from GSI and from Dubna and reasonable agreement is reached.Nuclear shell effect in superheavy region is analyzed and discussed. The spherical shell effect disappears in some cases due to the appearance of deformation or superdeformation in the ground states of nuclei, where valence nucleons occupysignificantly the intruder levels of nuclei. It is shown for the first time that the significant occupation of vaJence nucleons on the intruder states plays an important role for the ground state properties of superheavy nuclei. Nuclei are stable in the deformed or superdeformed configurations. We further point out that one cannot obtain the octupole deformation of even-even nuclei in the present relativistic mean-field model with the σ，ω and ρ mesons because there is no parityviolating interaction and the conservation of parity of even-even nuclei is a basic assumption of the present relativistic mean-field model. 20. RELATIVISTIC TRANSPORT-THEORY MALFLIET, R 1993-01-01 We discuss the present status of relativistic transport theory. Special emphasis is put on problems of topical interest: hadronic features, thermodynamical consistent approximations and spectral properties. 1. Control and Nash Games with Mean Field Effect Alain BENSOUSSAN; Jens FREHSE 2013-01-01 Mean field theory has raised a lot of interest in the recent years (see in particular the results of Lasry-Lions in 2006 and 2007,of Gueant-Lasry-Lions in 2011,of HuangCaines-Malham in 2007 and many others).There are a lot of applications.In general,the applications concern approximating an infinite number of players with common behavior by a representative agent.This agent has to solve a control problem perturbed by a field equation,representing in some way the behavior of the average infinite number of agents.This approach does not lead easily to the problems of Nash equilibrium for a finite number of players,perturbed by field equations,unless one considers averaging within different groups,which has not been done in the literature,and seems quite challenging.In this paper,the authors approach similar problems with a different motivation which makes sense for control and also for differential games.Thus the systems of nonlinear partial differential equations with mean field terms,which have not been addressed in the literature so far,are considered here. 2. Topological properties of the mean-field ϕ4 model Andronico, A.; Angelani, L.; Ruocco, G.; Zamponi, F. 2004-10-01 We study the thermodynamics and the properties of the stationary points (saddles and minima) of the potential energy for a ϕ4 mean-field model. We compare the critical energy vc [i.e., the potential energy v(T) evaluated at the phase transition temperature Tc ] with the energy vθ at which the saddle energy distribution show a discontinuity in its derivative. We find that, in this model, vc≫vθ , at variance to what has been found in different mean-field and short ranged systems, where the thermodynamic phase transitions take place at vc=vθ [Casetti, Pettini and Cohen, Phys. Rep. 337, 237 (2000)]. By direct calculation of the energy vs(T) of the “inherent saddles,” i.e., the saddles visited by the equilibrated system at temperature T , we find that vs(Tc)˜vθ . Thus, we argue that the thermodynamic phase transition is related to a change in the properties of the inherent saddles rather than to a change of the topology of the potential energy surface at T=Tc . Finally, we discuss the approximation involved in our analysis and the generality of our method. 3. Mean-field vs. Stochastic Models for Transcriptional Regulation Blossey, Ralf; Giuraniuc, Claudiu 2009-03-01 We introduce a minimal model description for the dynamics of transcriptional regulatory networks. It is studied within a mean-field approximation, i.e., by deterministic ode's representing the reaction kinetics, and by stochastic simulations employing the Gillespie algorithm. We elucidate the different results both approaches can deliver, depending on the network under study, and in particular depending on the level of detail retained in the respective description. Two examples are addressed in detail: the repressilator, a transcriptional clock based on a three-gene network realized experimentally in E. coli, and a bistable two-gene circuit under external driving, a transcriptional network motif recently proposed to play a role in cellular development. 4. Mean-field versus stochastic models for transcriptional regulation Blossey, R.; Giuraniuc, C. V. 2008-09-01 We introduce a minimal model description for the dynamics of transcriptional regulatory networks. It is studied within a mean-field approximation, i.e., by deterministic ODE’s representing the reaction kinetics, and by stochastic simulations employing the Gillespie algorithm. We elucidate the different results that both approaches can deliver, depending on the network under study, and in particular depending on the level of detail retained in the respective description. Two examples are addressed in detail: The repressilator, a transcriptional clock based on a three-gene network realized experimentally in E. coli, and a bistable two-gene circuit under external driving, a transcriptional network motif recently proposed to play a role in cellular development. 5. Two stochastic mean-field polycrystal plasticity methods Tonks, Michael [Los Alamos National Laboratory 2008-01-01 In this work, we develop two mean-field polycrystal plasticity models in which the L{sup c} are approximated stochastically. Through comprehensive CPFEM analyses of an idealized tantalum polycrystal, we verify that the L{sup c} tend to follow a normal distribution and surmise that this is due to the crystal interactions. We draw on these results to develop the STM and the stochastic no-constraints model (SNCM), which differ in the manner in which the crystal strain rates D{sup c} are prescribed. Calibration and validation of the models are performed using data from tantalum compression experiments. Both models predict the compression textures more accurately than the FCM, and the SNCM predicts them more accurately than the STM. The STM is extremely computationally efficient, only slightly more expensive than the FCM, while the SNCM is three times more computationally expensive than the STM. 6. Modeling distributed axonal delays in mean-field brain dynamics Roberts, J. A.; Robinson, P. A. 2008-11-01 The range of conduction delays between connected neuronal populations is often modeled as a single discrete delay, assumed to be an effective value averaging over all fiber velocities. This paper shows the effects of distributed delays on signal propagation. A distribution acts as a linear filter, imposing an upper frequency cutoff that is inversely proportional to the delay width. Distributed thalamocortical and corticothalamic delays are incorporated into a physiologically based mean-field model of the cortex and thalamus to illustrate their effects on the electroencephalogram (EEG). The power spectrum is acutely sensitive to the width of the thalamocortical delay distribution, and more so than the corticothalamic distribution, because all input signals must travel along the thalamocortical pathway. This imposes a cutoff frequency above which the spectrum is overly damped. The positions of spectral peaks in the resting EEG depend primarily on the distribution mean, with only weak dependences on distribution width. Increasing distribution width increases the stability of fixed point solutions. A single discrete delay successfully approximates a distribution for frequencies below a cutoff that is inversely proportional to the delay width, provided that other model parameters are moderately adjusted. A pair of discrete delays together having the same mean, variance, and skewness as the distribution approximates the distribution over the same frequency range without needing parameter adjustment. Delay distributions with large fractional widths are well approximated by low-order differential equations. 7. Some Aspects of Nuclear Structure in Relativistic Approach MAZhong-Yu; RONGJian; CAOLi-Gang; CHENBao-Qiu; LIULing 2004-01-01 The nucleon effective interaction in the nuclear medium is investigated in the framework of the DiracBrueckner-Hartree-Fock (DBHF) approach. A new decomposition of the Dirac structure of nucleon self-energy in the DBHF is adopted for asymmetric nuclear matter. The properties of finite nuclei are investigated with the nucleon effective interaction. The agreement with the experimental data is satisfactory. The relativistic microscopic optical potential in asymmetric nuclear matter is investigated in the DBHF approach. The proton scattering from nuclei is calculated and compared with the experimental data. A proper treatment of the resonant continuum for exotic nuclei is studied. The width effect of the resonant continuum on the pairing correlation is discussed. The quasiparticle relativistic random phase approximation based on the relativistic mean-field ground state in the response function formalism is also addressed. 8. Relativistic models for quasielastic electron and neutrino-nucleus scattering Meucci Andrea 2012-12-01 Full Text Available Relativistic models developed within the framework of the impulse approximation for quasielastic (QE electron scattering and successfully tested in comparison with electron-scattering data have been extended to neutrino-nucleus scattering. Different descriptions of final-state interactions (FSI in the inclusive scattering are compared. In the relativistic Green’s function (RGF model FSI are described consistently with the exclusive scattering using a complex optical potential. In the relativistic mean field (RMF model FSI are described by the same RMF potential which gives the bound states. The results of the models are compared for electron and neutrino scattering and, for neutrino scattering, with the recently measured charged-current QE (CCQE MiniBooNE cross sections. 9. Dual mean field search for large scale linear and quadratic knapsack problems Banda, Juan; Velasco, Jonás; Berrones, Arturo 2017-07-01 An implementation of mean field annealing to deal with large scale linear and non linear binary optimization problems is given. Mean field annealing is based on the analogy between combinatorial optimization and interacting physical systems at thermal equilibrium. Specifically, a mean field approximation of the Boltzmann distribution given by a Lagrangian that encompass the objective function and the constraints is calculated. The original discrete task is in this way transformed into a continuous variational problem. In our version of mean field annealing, no temperature parameter is used, but a good starting point in the dual space is given by a ;thermodynamic limit; argument. The method is tested in linear and quadratic knapsack problems with sizes that are considerably larger than those used in previous studies of mean field annealing. Dual mean field annealing is capable to find high quality solutions in running times that are orders of magnitude shorter than state of the art algorithms. Moreover, as may be expected for a mean field theory, the solutions tend to be more accurate as the number of variables grow. 10. Relativistic diffusion. Haba, Z 2009-02-01 We discuss relativistic diffusion in proper time in the approach of Schay (Ph.D. thesis, Princeton University, Princeton, NJ, 1961) and Dudley [Ark. Mat. 6, 241 (1965)]. We derive (Langevin) stochastic differential equations in various coordinates. We show that in some coordinates the stochastic differential equations become linear. We obtain momentum probability distribution in an explicit form. We discuss a relativistic particle diffusing in an external electromagnetic field. We solve the Langevin equations in the case of parallel electric and magnetic fields. We derive a kinetic equation for the evolution of the probability distribution. We discuss drag terms leading to an equilibrium distribution. The relativistic analog of the Ornstein-Uhlenbeck process is not unique. We show that if the drag comes from a diffusion approximation to the master equation then its form is strongly restricted. The drag leading to the Tsallis equilibrium distribution satisfies this restriction whereas the one of the Jüttner distribution does not. We show that any function of the relativistic energy can be the equilibrium distribution for a particle in a static electric field. A preliminary study of the time evolution with friction is presented. It is shown that the problem is equivalent to quantum mechanics of a particle moving on a hyperboloid with a potential determined by the drag. A relation to diffusions appearing in heavy ion collisions is briefly discussed. 11. Nonequilibrium Dynamical Mean-Field Theory for Bosonic Lattice Models 2015-01-01 We develop the nonequilibrium extension of bosonic dynamical mean-field theory and a Nambu real-time strong-coupling perturbative impurity solver. In contrast to Gutzwiller mean-field theory and strong-coupling perturbative approaches, nonequilibrium bosonic dynamical mean-field theory captures not only dynamical transitions but also damping and thermalization effects at finite temperature. We apply the formalism to quenches in the Bose-Hubbard model, starting from both the normal and the Bos... 12. Verbalization of Mean Field Utterances in German Instructions Tayupova O. I. 2013-01-01 Full Text Available The article investigates ways of actualization of mean field utterances used in modern German instructions considering the type of the text. The author determines and analyzes similarities and differences in linguistic means used in mean field utterances in the context of such text subtypes as instructions to household appliances, cosmetic products directions and prescribing information for pharmaceutical drugs use. 13. Nuclear mean field from chiral pion-nucleon dynamics Kaiser, N; Weise, W 2002-01-01 Using the two-loop approximation of chiral perturbation theory, we calculate the momentum- and density-dependent single-particle potential of nucleons in isospin-symmetric nuclear matter. The contributions from one- and two-pion exchange diagrams give rise to a potential depth for a nucleon at rest of U(0,k sub f sub 0)=-53.2 MeV at saturation density. The momentum dependence of the real part of the single-particle potential U(p,k sub f sub 0) is nonmonotonic and can be translated into a mean effective nucleon mass of M*bar approx =0.8M. The imaginary part of the single-particle potential W(p,k sub f) is generated to that order entirely by iterated one-pion exchange. The resulting half width of a nucleon hole-state at the bottom of the Fermi sea comes out as W(0,k sub f sub 0)=29.7 MeV. The basic theorems of Hugenholtz-Van-Hove and Luttinger are satisfied in our perturbative two-loop calculation of the nuclear mean field. 14. On Social Optima of Non-Cooperative Mean Field Games Li, Sen; Zhang, Wei; Zhao, Lin; Lian, Jianming; Kalsi, Karanjit 2016-12-12 This paper studies the social optima in noncooperative mean-field games for a large population of agents with heterogeneous stochastic dynamic systems. Each agent seeks to maximize an individual utility functional, and utility functionals of different agents are coupled through a mean field term that depends on the mean of the population states/controls. The paper has the following contributions. First, we derive a set of control strategies for the agents that possess *-Nash equilibrium property, and converge to the mean-field Nash equilibrium as the population size goes to infinity. Second, we study the social optimal in the mean field game. We derive the conditions, termed the socially optimal conditions, under which the *-Nash equilibrium of the mean field game maximizes the social welfare. Third, a primal-dual algorithm is proposed to compute the *-Nash equilibrium of the mean field game. Since the *-Nash equilibrium of the mean field game is socially optimal, we can compute the equilibrium by solving the social welfare maximization problem, which can be addressed by a decentralized primal-dual algorithm. Numerical simulations are presented to demonstrate the effectiveness of the proposed approach. 15. Mean field dynamics of networks of delay-coupled noisy excitable units Franović, Igor, E-mail: [email protected] [Scientific Computing Laboratory, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade (Serbia); Todorović, Kristina; Burić, Nikola [Department of Physics and Mathematics, Faculty of Pharmacy, University of Belgrade, Vojvode Stepe 450, Belgrade (Serbia); Vasović, Nebojša [Department of Applied Mathematics, Faculty of Mining and Geology, University of Belgrade, PO Box 162, Belgrade (Serbia) 2016-06-08 We use the mean-field approach to analyze the collective dynamics in macroscopic networks of stochastic Fitzhugh-Nagumo units with delayed couplings. The conditions for validity of the two main approximations behind the model, called the Gaussian approximation and the Quasi-independence approximation, are examined. It is shown that the dynamics of the mean-field model may indicate in a self-consistent fashion the parameter domains where the Quasi-independence approximation fails. Apart from a network of globally coupled units, we also consider the paradigmatic setup of two interacting assemblies to demonstrate how our framework may be extended to hierarchical and modular networks. In both cases, the mean-field model can be used to qualitatively analyze the stability of the system, as well as the scenarios for the onset and the suppression of the collective mode. In quantitative terms, the mean-field model is capable of predicting the average oscillation frequency corresponding to the global variables of the exact system. 16. Mean-field Approach to the Derivation of Baryon Superpotential from Matrix Model Suzuki, H 2003-01-01 We discuss how to obtain the superpotential of the baryons and mesons for SU(N) gauge theories with N flavour matter fields from matrix integral. We apply the mean-field approximation for the matrix integral. Assuming the planar limit of the self-consistency equation, we show that the result almost agrees with the field theoretical result. 17. Analytically Solvable Mean-Field Potential for Stable and Exotic Nuclei Stoitsov, M. V.; S. S. Dimitrova(INRNE, Sofia); Pittel, S.; Van Isacker, P.(GANIL, CEA/DSM–CNRS/IN2P3, Bd Henri Becquerel, BP 55027, F-14076 Caen Cedex 5, France); Frank, A 1997-01-01 Slater determinants built from the single-particle wave functions of the analytically solvable Ginocchio potential are used to approximate the self-consistent Hartree-Fock solutions for the ground states of nuclei. The results indicate that the Ginocchio potential provides a good parametrization of the nuclear mean field for a wide range of nuclei, including those at the limit of particle stability. 18. Mean field dynamics of networks of delay-coupled noisy excitable units Franović, Igor; Todorović, Kristina; Vasović, Nebojša; Burić, Nikola 2016-06-01 We use the mean-field approach to analyze the collective dynamics in macroscopic networks of stochastic Fitzhugh-Nagumo units with delayed couplings. The conditions for validity of the two main approximations behind the model, called the Gaussian approximation and the Quasi-independence approximation, are examined. It is shown that the dynamics of the mean-field model may indicate in a self-consistent fashion the parameter domains where the Quasi-independence approximation fails. Apart from a network of globally coupled units, we also consider the paradigmatic setup of two interacting assemblies to demonstrate how our framework may be extended to hierarchical and modular networks. In both cases, the mean-field model can be used to qualitatively analyze the stability of the system, as well as the scenarios for the onset and the suppression of the collective mode. In quantitative terms, the mean-field model is capable of predicting the average oscillation frequency corresponding to the global variables of the exact system. 19. Mean-field instabilities and cluster formation in nuclear reactions Colonna, M; Baran, V 2016-01-01 We review recent results on intermediate mass cluster production in heavy ion collisions at Fermi energy and in spallation reactions. Our studies are based on modern transport theories, employing effective interactions for the nuclear mean-field and incorporating two-body correlations and fluctuations. Namely we will consider the Stochastic Mean Field (SMF) approach and the recently developed Boltzmann-Langevin One Body (BLOB) model. We focus on cluster production emerging from the possible occurrence of low-density mean-field instabilities in heavy ion reactions. Within such a framework, the respective role of one and two-body effects, in the two models considered, will be carefully analysed. We will discuss, in particular, fragment production in central and semi-peripheral heavy ion collisions, which is the object of many recent experimental investigations. Moreover, in the context of spallation reactions, we will show how thermal expansion may trigger the development of mean-field instabilities, leading to... 20. Mean Field Games Models-A Brief Survey Gomes, Diogo A. 2013-11-20 The mean-field framework was developed to study systems with an infinite number of rational agents in competition, which arise naturally in many applications. The systematic study of these problems was started, in the mathematical community by Lasry and Lions, and independently around the same time in the engineering community by P. Caines, Minyi Huang, and Roland Malhamé. Since these seminal contributions, the research in mean-field games has grown exponentially, and in this paper we present a brief survey of mean-field models as well as recent results and techniques. In the first part of this paper, we study reduced mean-field games, that is, mean-field games, which are written as a system of a Hamilton-Jacobi equation and a transport or Fokker-Planck equation. We start by the derivation of the models and by describing some of the existence results available in the literature. Then we discuss the uniqueness of a solution and propose a definition of relaxed solution for mean-field games that allows to establish uniqueness under minimal regularity hypothesis. A special class of mean-field games that we discuss in some detail is equivalent to the Euler-Lagrange equation of suitable functionals. We present in detail various additional examples, including extensions to population dynamics models. This section ends with a brief overview of the random variables point of view as well as some applications to extended mean-field games models. These extended models arise in problems where the costs incurred by the agents depend not only on the distribution of the other agents, but also on their actions. The second part of the paper concerns mean-field games in master form. These mean-field games can be modeled as a partial differential equation in an infinite dimensional space. We discuss both deterministic models as well as problems where the agents are correlated. We end the paper with a mean-field model for price impact. © 2013 Springer Science+Business Media New York. 1. Streamlined mean field variational Bayes for longitudinal and multilevel data analysis. Lee, Cathy Yuen Yi; Wand, Matt P 2016-07-01 Streamlined mean field variational Bayes algorithms for efficient fitting and inference in large models for longitudinal and multilevel data analysis are obtained. The number of operations is linear in the number of groups at each level, which represents a two orders of magnitude improvement over the naïve approach. Storage requirements are also lessened considerably. We treat models for the Gaussian and binary response situations. Our algorithms allow the fastest ever approximate Bayesian analyses of arbitrarily large longitudinal and multilevel datasets, with little degradation in accuracy compared with Markov chain Monte Carlo. The modularity of mean field variational Bayes allows relatively simple extension to more complicated scenarios. 2. Multivariate Ornstein-Uhlenbeck processes with mean-field dependent coefficients: Application to postural sway Frank, T. D.; Daffertshofer, A.; Beek, P. J. 2001-01-01 We study the transient and stationary behavior of many-particle systems in terms of multivariate Ornstein-Uhlenbeck processes with friction and diffusion coefficients that depend nonlinearly on process mean fields. Mean-field approximations of this kind of system are derived in terms of Fokker-Planck equations. In such systems, multiple stationary solutions as well as bifurcations of stationary solutions may occur. In addition, strictly monotonically decreasing steady-state autocorrelation functions that decay faster than exponential functions are found, which are used to describe the erratic motion of the center of pressure during quiet standing. 3. Calculating charge-carrier mobilities in disordered semiconducting polymers: Mean field and beyond Cottaar, J.; Bobbert, P. A. 2006-09-01 We model charge transport in disordered semiconducting polymers by hopping of charges on a regular cubic lattice of sites. A large on-site Coulomb repulsion prohibits double occupancy of the sites. Disorder is introduced by taking random site energies from a Gaussian distribution. Recently, it was demonstrated that this model leads to a dependence of the charge-carrier mobilities on the density of charge carriers that is in agreement with experimental observations. The model is conveniently solved within a mean-field approximation, in which the correlation between the occupational probabilities of different sites is neglected. This approximation becomes exact in the limit of vanishing charge-carrier densities, but needs to be checked at high densities. We perform this check by dividing the lattice in pairs of neighboring sites and taking into account the correlation between the sites within each pair explicitly. This pair approximation is expected to account for the most important corrections to the mean-field approximation. We study the effects of varying temperature, charge-carrier density, and electric field. We demonstrate that in the parameter regime relevant for semiconducting polymers used in practical devices the corrections to the mobilities calculated from the mean-field approximation will not exceed a few percent, so that this approximation can be safely used. 4. Mean field theories and dual variation mathematical structures of the mesoscopic model Suzuki, Takashi 2015-01-01 Mean field approximation has been adopted to describe macroscopic phenomena from microscopic overviews. It is still in progress; fluid mechanics, gauge theory, plasma physics, quantum chemistry, mathematical oncology, non-equilibirum thermodynamics. spite of such a wide range of scientific areas that are concerned with the mean field theory, a unified study of its mathematical structure has not been discussed explicitly in the open literature. The benefit of this point of view on nonlinear problems should have significant impact on future research, as will be seen from the underlying features of self-assembly or bottom-up self-organization which is to be illustrated in a unified way. The aim of this book is to formulate the variational and hierarchical aspects of the equations that arise in the mean field theory from macroscopic profiles to microscopic principles, from dynamics to equilibrium, and from biological models to models that arise from chemistry and physics. 5. Nuclear matter properties in the relativistic mean field model with\\sigma-\\omega$coupling Chung, K C; Santiago, A J; Zhang, J W 2001-01-01 The possibility of extending the linear sigma-omega model by introducing a sigma-omega coupling phenomenologically is explored. It is shown that, in contrast to the usual Walecka model, not only the effective nucleon mass M* but also the effective sigma meson mass m*_sigma and the effective omega meson mass m*_omega are nucleon density dependent. When the model parameters are fitted to the nuclear saturation point (the nuclear radius constant r_0=1.14fm and volume energy a_1=16.0MeV) as well as to the effective nucleon mass M*=0.85M, the model yields m*_sigma=1.09m_sigma and m*_omega=0.90m_omega at the saturation point, and the nuclear incompressibility K_0=501MeV. The lowest value of K_0 given by this model by adjusting the model parameters is around 227MeV. 6. Deformed neutron stars due to strong magnetic field in terms of relativistic mean field theories Yanase, Kota; Yoshinaga, Naotaka 2014-09-01 Some observations suggest that magnetic field intensity of neutron stars that have particularly strong magnetic field, magnetars, reaches values up to 1014-15G. It is expected that there exists more strong magnetic field of several orders of magnitude in the interior of such stars. Neutron star matter is so affected by magnetic fields caused by intrinsic magnetic moments and electric charges of baryons that masses of neutron stars calculated by using Tolman-Oppenheimer-Volkoff equation is therefore modified. We calculate equation of state (EOS) in density-dependent magnetic field by using sigma-omega-rho model that can reproduce properties of stable nuclear matter in laboratory Furthermore we calculate modified masses of deformed neutron stars. 7. Mean-field diffusivities in passive scalar and magnetic transport in irrotational flows Rädler, Karl-Heinz; Del Sordo, Fabio; Rheinhardt, Matthias 2011-01-01 Certain aspects of the mean-field theory of turbulent passive scalar transport and of mean-field electrodynamics are considered with particular emphasis on aspects of compressible fluids. It is demonstrated that the total mean-field diffusivity for passive scalar transport in a compressible flow may well be smaller than the molecular diffusivity. This is in full analogy to an old finding regarding the magnetic mean-field diffusivity in an electrically conducting turbulently moving compressible fluid. These phenomena occur if the irrotational part of the motion dominates the vortical part, the P\\'eclet or magnetic Reynolds numbers are not too large and, in addition, the variation of the flow pattern is slow. For both the passive scalar and the magnetic case several further analytical results on mean-field diffusivities and related quantities found within the second-order correlation approximation are presented as well as numerical results obtained by the test-field method, which applies independently of this a... 8. Uncertainty quantification for mean field games in social interactions Dia, Ben Mansour 2016-01-09 We present an overview of mean field games formulation. A comparative analysis of the optimality for a stochastic McKean-Vlasov process with time-dependent probability is presented. Then we examine mean-field games for social interactions and we show that optimizing the long-term well-being through effort and social feeling state distribution (mean-field) will help to stabilize couple (marriage). However , if the cost of effort is very high, the couple fluctuates in a bad feeling state or the marriage breaks down. We then examine the influence of society on a couple using mean field sentimental games. We show that, in mean-field equilibrium, the optimal effort is always higher than the one-shot optimal effort. Finally we introduce the Wiener chaos expansion for the construction of solution of stochastic differential equations of Mckean-Vlasov type. The method is based on the Cameron-Martin version of the Wiener Chaos expansion and allow to quantify the uncertainty in the optimality system. 9. Regularity theory for mean-field game systems Gomes, Diogo A; Voskanyan, Vardan 2016-01-01 Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields. 10. Accretion Disks and Dynamos: Toward a Unified Mean Field Theory Blackman, Eric G 2012-01-01 Conversion of gravitational energy into radiation near stars and compact objects in accretion disks the origin of large scale magnetic fields in astrophysical rotators have long been distinct topics of active research in astrophysics. In semi-analytic work on both problems it has been useful to presume large scale symmetries, which necessarily results in mean field theories; magnetohydrodynamic turbulence makes the underlying systems locally asymmetric and highly nonlinear. Synergy between theory and simulations should aim for the development of practical, semi-analytic mean field models that capture the essential physics and can be used for observational modeling. Mean field dynamo (MFD) theory and alpha-viscosity accretion disc theory have exemplified such distinct pursuits. Both are presently incomplete, but 21st century MFD theory has nonlinear predictive power compared to 20th century MFD. in contrast, alpha-viscosity accretion theory is still in a 20th century state. In fact, insights from MFD theory ar... 11. Nonequilibrium Dynamical Mean-Field Theory for Bosonic Lattice Models Strand, Hugo U. R.; Eckstein, Martin; Werner, Philipp 2015-01-01 We develop the nonequilibrium extension of bosonic dynamical mean-field theory and a Nambu real-time strong-coupling perturbative impurity solver. In contrast to Gutzwiller mean-field theory and strong-coupling perturbative approaches, nonequilibrium bosonic dynamical mean-field theory captures not only dynamical transitions but also damping and thermalization effects at finite temperature. We apply the formalism to quenches in the Bose-Hubbard model, starting from both the normal and the Bose-condensed phases. Depending on the parameter regime, one observes qualitatively different dynamical properties, such as rapid thermalization, trapping in metastable superfluid or normal states, as well as long-lived or strongly damped amplitude oscillations. We summarize our results in nonequilibrium "phase diagrams" that map out the different dynamical regimes. 12. Mean field limit for bosons and propagation of Wigner measures Ammari, Z 2008-01-01 We consider the N-body Schr\\"{o}dinger dynamics of bosons in the mean field limit with a bounded pair-interaction potential. According to the previous work \\cite{AmNi}, the mean field limit is translated into a semiclassical problem with a small parameter$\\epsilon\\to 0$, after introducing an$\\epsilon$-dependent bosonic quantization. The limit is expressed as a push-forward by a nonlinear flow (e.g. Hartree) of the associated Wigner measures. These object and their basic properties were introduced in \\cite{AmNi} in the infinite dimensional setting. The additional result presented here states that the transport by the nonlinear flow holds for rather general class of quantum states in their mean field limit. 13. Regularity Theory for Mean-Field Game Systems Gomes, Diogo A. 2016-09-14 Beginning with a concise introduction to the theory of mean-field games (MFGs), this book presents the key elements of the regularity theory for MFGs. It then introduces a series of techniques for well-posedness in the context of mean-field problems, including stationary and time-dependent MFGs, subquadratic and superquadratic MFG formulations, and distinct classes of mean-field couplings. It also explores stationary and time-dependent MFGs through a series of a-priori estimates for solutions of the Hamilton-Jacobi and Fokker-Planck equation. It shows sophisticated a-priori systems derived using a range of analytical techniques, and builds on previous results to explain classical solutions. The final chapter discusses the potential applications, models and natural extensions of MFGs. As MFGs connect common problems in pure mathematics, engineering, economics and data management, this book is a valuable resource for researchers and graduate students in these fields. 14. A mean field theory of coded CDMA systems Yano, Toru [Graduate School of Science and Technology, Keio University, Hiyoshi, Kohoku-ku, Yokohama-shi, Kanagawa 223-8522 (Japan); Tanaka, Toshiyuki [Graduate School of Informatics, Kyoto University, Yoshida Hon-machi, Sakyo-ku, Kyoto-shi, Kyoto 606-8501 (Japan); Saad, David [Neural Computing Research Group, Aston University, Birmingham B4 7ET (United Kingdom)], E-mail: [email protected] 2008-08-15 We present a mean field theory of code-division multiple-access (CDMA) systems with error-control coding. On the basis of the relation between the free energy and mutual information, we obtain an analytical expression of the maximum spectral efficiency of the coded CDMA system, from which a mean-field description of the coded CDMA system is provided in terms of a bank of scalar Gaussian channels whose variances in general vary at different code symbol positions. Regular low-density parity-check (LDPC)-coded CDMA systems are also discussed as an example of the coded CDMA systems. 15. Large amplitude motion with a stochastic mean-field approach Yilmaz Bulent 2012-12-01 Full Text Available In the stochastic mean-field approach, an ensemble of initial conditions is considered to incorporate correlations beyond the mean-field. Then each starting point is propagated separately using the Time-Dependent Hartree-Fock equation of motion. This approach provides a rather simple tool to better describe fluctuations compared to the standard TDHF. Several illustrations are presented showing that this theory can be rather effective to treat the dynamics close to a quantum phase transition. Applications to fusion and transfer reactions demonstrate the great improvement in the description of mass dispersion. 16. An Adaptive Filtering Algorithm using Mean Field Annealing Techniques Persson, Per; Nordebo, Sven; Claesson, Ingvar 2002-01-01 We present a new approach to discrete adaptive filtering based on the mean field annealing algorithm. The main idea is to find the discrete filter vector that minimizes the matrix form of the Wiener-Hopf equations in a least-squares sense by a generalized mean field annealing algorithm. It is indicated by simulations that this approach, with complexity O(M^2) where M is the filter length, finds a solution comparable to the one obtained by the recursive least squares (RLS) algorithm but withou... 17. Socio-economic applications of finite state mean field games Gomes, Diogo A. 2014-10-06 In this paper, we present different applications of finite state mean field games to socio-economic sciences. Examples include paradigm shifts in the scientific community or consumer choice behaviour in the free market. The corresponding finite state mean field game models are hyperbolic systems of partial differential equations, for which we present and validate different numerical methods. We illustrate the behaviour of solutions with various numerical experiments,which show interesting phenomena such as shock formation. Hence, we conclude with an investigation of the shock structure in the case of two-state problems. 18. Suppression of oscillations in mean-field diffusion Neeraj Kumar Kamal; Pooja Rani Sharma; Manish Dev Shrimali 2015-02-01 We study the role of mean-field diffusive coupling on suppression of oscillations for systems of limit cycle oscillators. We show that this coupling scheme not only induces amplitude death (AD) but also oscillation death (OD) in coupled identical systems. The suppression of oscillations in the parameter space crucially depends on the value of mean-field diffusion parameter. It is also found that the transition from oscillatory solutions to OD in conjugate coupling case is different from the case when the coupling is through similar variable. We rationalize our study using linear stability analysis. 19. A General Quadrature Solution for Relativistic, Non-relativistic, and Weakly-Relativistic Rocket Equations Bruce, Adam L 2015-01-01 We show the traditional rocket problem, where the ejecta velocity is assumed constant, can be reduced to an integral quadrature of which the completely non-relativistic equation of Tsiolkovsky, as well as the fully relativistic equation derived by Ackeret, are limiting cases. By expanding this quadrature in series, it is shown explicitly how relativistic corrections to the mass ratio equation as the rocket transitions from the Newtonian to the relativistic regime can be represented as products of exponential functions of the rocket velocity, ejecta velocity, and the speed of light. We find that even low order correction products approximate the traditional relativistic equation to a high accuracy in flight regimes up to$0.5cwhile retaining a clear distinction between the non-relativistic base-case and relativistic corrections. We furthermore use the results developed to consider the case where the rocket is not moving relativistically but the ejecta stream is, and where the ejecta stream is massless. 20. Noisy mean field game model for malware propagation in opportunistic networks Tembine, Hamidou 2012-01-01 In this paper we present analytical mean field techniques that can be used to better understand the behavior of malware propagation in opportunistic large networks. We develop a modeling methodology based on stochastic mean field optimal control that is able to capture many aspects of the problem, especially the impact of the control and heterogeneity of the system on the spreading characteristics of malware. The stochastic large process characterizing the evolution of the total number of infected nodes is examined with a noisy mean field limit and compared to a deterministic one. The stochastic nature of the wireless environment make stochastic approaches more realistic for such types of networks. By introducing control strategies, we show that the fraction of infected nodes can be maintained below some threshold. In contrast to most of the existing results on mean field propagation models which focus on deterministic equations, we show that the mean field limit is stochastic if the second moment of the number of object transitions per time slot is unbounded with the size of the system. This allows us to compare one path of the fraction of infected nodes with the stochastic trajectory of its mean field limit. In order to take into account the heterogeneity of opportunistic networks, the analysis is extended to multiple types of nodes. Our numerical results show that the heterogeneity can help to stabilize the system. We verify the results through simulation showing how to obtain useful approximations in the case of very large systems. © 2012 ICST Institute for Computer Science, Social Informatics and Telecommunications Engineering. 1. Mean-field versus microconvection effects in nanofluid thermal conduction. Eapen, Jacob; Williams, Wesley C; Buongiorno, Jacopo; Hu, Lin-Wen; Yip, Sidney; Rusconi, Roberto; Piazza, Roberto 2007-08-31 Transient hot-wire data on thermal conductivity of suspensions of silica and perfluorinated particles show agreement with the mean-field theory of Maxwell but not with the recently postulated microconvection mechanism. The influence of interfacial thermal resistance, convective effects at microscales, and the possibility of thermal conductivity enhancements beyond the Maxwell limit are discussed. 2. Two numerical methods for mean-field games Gomes, Diogo A. 2016-01-09 Here, we consider numerical methods for stationary mean-field games (MFG) and investigate two classes of algorithms. The first one is a gradient flow method based on the variational characterization of certain MFG. The second one uses monotonicity properties of MFG. We illustrate our methods with various examples, including one-dimensional periodic MFG, congestion problems, and higher-dimensional models. 3. Condition monitoring with Mean field independent components analysis Pontoppidan, Niels Henrik; Sigurdsson, Sigurdur; Larsen, Jan 2005-01-01 We discuss condition monitoring based on mean field independent components analysis of acoustic emission energy signals. Within this framework it is possible to formulate a generative model that explains the sources, their mixing and also the noise statistics of the observed signals. By using... 4. Mean-Field Versus Microconvection Effects in Nanofluid Thermal Conduction Eapen, Jacob; Williams, Wesley C.; Buongiorno, Jacopo; Hu, Lin-Wen; Yip, Sidney; Rusconi, Roberto; Piazza, Roberto 2007-08-01 Transient hot-wire data on thermal conductivity of suspensions of silica and perfluorinated particles show agreement with the mean-field theory of Maxwell but not with the recently postulated microconvection mechanism. The influence of interfacial thermal resistance, convective effects at microscales, and the possibility of thermal conductivity enhancements beyond the Maxwell limit are discussed. 5. Photoassociation of Atomic BEC within Mean-Field Approximation:Exact Solutions CAI Wei; JING Hui; ZHAN Ming-Sheng; XU Jing-Jun 2007-01-01 We propose an exactly solvable method to study the coherent two-colour photoassociation of an atomic BoseEinstein condensate,by linearizing the bilinear atom-molecule coupling,which allows su to conveniently probe the quantum dynamics and statistics of the system.By preparing different initial states of the atomic condensate,we can observe very different quantum statistical properties of the system by exactly calculating the quadraturesqueezed and mode-correlated functions. 6. Adaptive and self-averaging Thouless-Anderson-Palmer mean-field theory for probabilistic modeling Opper, Manfred; Winther, Ole 2001-01-01 We develop a generalization of the Thouless-Anderson-Palmer (TAP) mean-field approach of disorder physics. which makes the method applicable to the computation of approximate averages in probabilistic models for real data. In contrast to the conventional TAP approach, where the knowledge of the d......We develop a generalization of the Thouless-Anderson-Palmer (TAP) mean-field approach of disorder physics. which makes the method applicable to the computation of approximate averages in probabilistic models for real data. In contrast to the conventional TAP approach, where the knowledge...... distributions in the thermodynamic limit. On the other hand, simulations on a real data model demonstrate that the method achieves more accurate predictions as compared to conventional TAP approaches.... 7. Inverse Magnetic Catalysis in Nambu--Jona-Lasinio Model beyond Mean Field Mao, Shijun 2016-01-01 We study inverse magnetic catalysis in the Nambu--Jona-Lasinio model beyond mean field approximation. The feed-down from mesons to quarks is embedded in an effective coupling constant at finite temperature and magnetic field. While the magnetic catalysis is still the dominant effect at low temperature, the meson dressed quark mass drops down with increasing magnetic field at high temperature due to the dimension reduction of the Goldstone mode in the Pauli-Villars regularization scheme. 8. Point-coupling models from mesonic hyper massive limit and mean-field approaches Lourenco, O.; Dutra, M., E-mail: [email protected] [Departamento de Fisica, Instituto Tecnologico da Aeronautica - CTA, Sao Jose dos Campos, SP (Brazil); Delfino, Antonio, E-mail: [email protected] [Instituto de Fisica, Universidade Federal Fluminense, Niteroi, RJ (Brazil); Amaral, R.L.P.G. [Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA (United States) 2012-08-15 t In this work, we show how nonlinear point coupling models, described by a Lagrangian density containing only terms up to fourth order in the fermion condensate ({Psi}-bar{Psi}), are derived from a modified meson exchange nonlinear Walecka model. We present two methods of derivation, namely the hyper massive meson limit within a functional integral approach and the mean-field approximation, in which equations of state at zero temperature of the nonlinear point-coupling models are directly obtained. (author) 9. Mean field and collisional dynamics of interacting fermion-boson systems the Jaynes-Cummings model Takano-Natti, E R 1996-01-01 A general time-dependent projection technique is applied to the study of the dynamics of quantum correlations in a system consisting of interacting fermionic and bosonic subsystems, described by the Jaynes-Cummings Hamiltonian. The amplitude modulation of the Rabi oscillations which occur for a strong, coherent initial bosonic field is obtained from the spin intrinsic depolarization resulting from collisional corrections to the mean-field approximation. 10. Nonlinear Effects in Quantum Dynamics of Atom Laser: Mean-Field Approach JING Hui 2002-01-01 Quantum dynamics and statistics of an atom laser with nonlinear binary interactions are investigated inthe framework of mean-field approximation. The linearized effective Hamiltonian of the system is accurately solvable.It is shown that, although the input radio frequency field is in an ordinary Glauber coherent state, the output matterwave will periodically exhibit quadrature squeezing effects purely originated from the nonlinear atom-atom collisions. 11. A mean field theory for the cold quark gluon plasma applied to stellar structure Fogaca, D. A.; Navarra, F. S.; Franzon, B. [Instituto de Fisica, Universidade de Sao Paulo Rua do Matao, Travessa R, 187, 05508-090 Sao Paulo, SP (Brazil); Horvath, J. E. [Instituto de Astronomia, Geofisica e Ciencias Atmosfericas, Universidade de Sao Paulo, Rua do Matao, 1226, 05508-090, Sao Paulo, SP (Brazil) 2013-03-25 An equation of state based on a mean-field approximation of QCD is used to describe the cold quark gluon plasma and also to study the structure of compact stars. We obtain stellar masses compatible with the pulsar PSR J1614-2230 that was determined to have a mass of (1.97 {+-} 0.04 M{sub Circled-Dot-Operator }), and the corresponding radius around 10-11 km. 12. A simplified BBGKY hierarchy for correlated fermionic systems from a Stochastic Mean-Field approach Lacroix, Denis; Ayik, Sakir; Yilmaz, Bulent 2015-01-01 The stochastic mean-field (SMF) approach allows to treat correlations beyond mean-field using a set of independent mean-field trajectories with appropriate choice of fluctuating initial conditions. We show here, that this approach is equivalent to a simplified version of the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy between one-, two-, ..., N-body degrees of freedom. In this simplified version, one-body degrees of freedom are coupled to fluctuations to all orders while retaining only specific terms of the general BBGKY hierarchy. The use of the simplified BBGKY is illustrated with the Lipkin-Meshkov-Glick (LMG) model. We show that a truncated version of this hierarchy can be useful, as an alternative to the SMF, especially in the weak coupling regime to get physical insight in the effect beyond mean-field. In particular, it leads to approximate analytical expressions for the quantum fluctuations both in the weak and strong coupling regime. In the strong coupling regime, it can only be used for sho... 13. A simplified BBGKY hierarchy for correlated fermions from a stochastic mean-field approach Lacroix, Denis; Tanimura, Yusuke [Universite Paris-Sud, Institut de Physique Nucleaire, IN2P3-CNRS, Orsay (France); Ayik, Sakir [Tennessee Technological University, Physics Department, Cookeville, TN (United States); Yilmaz, Bulent [Ankara University, Physics Department, Faculty of Sciences, Ankara (Turkey) 2016-04-15 The stochastic mean-field (SMF) approach allows to treat correlations beyond mean-field using a set of independent mean-field trajectories with appropriate choice of fluctuating initial conditions. We show here that this approach is equivalent to a simplified version of the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy between one-, two-,.., N -body degrees of freedom. In this simplified version, one-body degrees of freedom are coupled to fluctuations to all orders while retaining only specific terms of the general BBGKY hierarchy. The use of the simplified BBGKY is illustrated with the Lipkin-Meshkov-Glick (LMG) model. We show that a truncated version of this hierarchy can be useful, as an alternative to the SMF, especially in the weak coupling regime to get physical insight in the effect beyond mean-field. In particular, it leads to approximate analytical expressions for the quantum fluctuations both in the weak and strong coupling regime. In the strong coupling regime, it can only be used for short time evolution. In that case, it gives information on the evolution time-scale close to a saddle point associated to a quantum phase-transition. For long time evolution and strong coupling, we observed that the simplified BBGKY hierarchy cannot be truncated and only the full SMF with initial sampling leads to reasonable results. (orig.) 14. Chaotic time series prediction using mean-field theory for support vector machine Cui Wan-Zhao; Zhu Chang-Chun; Bao Wen-Xing; Liu Jun-Hua 2005-01-01 This paper presents a novel method for predicting chaotic time series which is based on the support vector machines approach, and it uses the mean-field theory for developing an easy and efficient learning procedure for the support vector machine. The proposed method approximates the distribution of the support vector machine parameters to a Gaussian process and uses the mean-field theory to estimate these parameters easily, and select the weights of the mixture of kernels used in the support vector machine estimation more accurately and faster than traditional quadratic programming-based algorithms. Finally, relationships between the embedding dimension and the predicting performance of this method are discussed, and the Mackey-Glass equation is applied to test this method. The stimulations show that the mean-field theory for support vector machine can predict chaotic time series accurately, and even if the embedding dimension is unknown, the predicted results are still satisfactory. This result implies that the mean-field theory for support vector machine is a good tool for studying chaotic time series. 15. Pseudospin symmetry in finite nuclei within the relativistic Hartree-Fock framework Lopez-Quelle, M [Departamento de Fisica Aplicada, Universidad de Cantabria, E-39005 Santander (Spain); Savushkin, L N [Department of Physics, St Petersburg University for Telecommunications, 191186 St Petersburg (Russian Federation); Marcos, S [Departamento de Fisica Moderna, Universidad de Cantabria, E-39005 Santander (Spain); Niembro, R [Departamento de Fisica Moderna, Universidad de Cantabria, E-39005 Santander (Spain) 2005-10-01 In the present work, we analyse the behaviour of the pseudospin symmetry (PSS) in heavy nuclei ({sup 208}Pb) in the framework of the relativistic Hartree-Fock approximation (RHFA). The quasidegeneracy of the pseudospin partners and the similarity of the small F components of their respective Dirac spinors have a somewhat lower degree of accuracy than in the relativistic mean field approximation (RMFA). Both properties improve when the number of nodes of the small component increases, as happens in the RMFA. The behaviour of the single-particle potentials appearing in the Dirac equation of the pseudospin partners is analysed. There is no dominance of the pseudocentrifugal barrier (PCB) compared to the pseudospin-orbit potential (PSOP). In the RHFA, the PSS is an approximately satisfied symmetry in nuclei and its dynamical character is reinforced with respect to the RMFA. 16. Hadron resonance gas and mean-field nuclear matter for baryon number fluctuations Fukushima, Kenji 2014-01-01 We give an estimate for the skewness and the kurtosis of the baryon number distribution in two representative models; i.e., models for a hadron resonance gas and relativistic mean-field nuclear matter. We emphasize formal similarity between these two descriptions. The hadron resonance gas leads to a deviation from the Skellam distribution if quantum statistical correlation is taken into account at high baryon density, but this effect is not strong enough to explain fluctuation data seen in the beam-energy scan at RHIC/STAR. In the calculation of mean-field nuclear matter the density correlation with the vector \\omega-field rather than the effective mass with the scalar \\sigma-field renders the kurtosis suppressed at higher baryon density so as to account for the observed behavior of the kurtosis. We finally discuss the difference between the baryon number and the proton number fluctuations from correlation effects in isospin space. Our numerical results suggest that such effects are only minor even in the cas... 17. Entanglement spectrum in cluster dynamical mean-field theory Udagawa, Masafumi; Motome, Yukitoshi 2015-01-01 We study the entanglement spectrum of the Hubbard model at half filling on a kagome lattice. The entanglement spectrum is defined by the set of eigenvalues of a reduced thermal density matrix, which is naturally obtained in the framework of the dynamical mean-field theory. Adopting the cluster dynamical mean-field theory combined with continuous-time auxiliary-field Monte Carlo method, we calculate the entanglement spectrum for a three-site triangular cluster in the kagome Hubbard model. We find that the results at the three-particle sector well capture the qualitative nature of the system. In particular, the eigenvalue of the reduced density matrix, corresponding to the chiral degrees of freedom, exhibits a characteristic temperature scale Tchiral, below which a metallic state with large quasiparticle mass is stabilized. The entanglement spectra at different particle number sectors also exhibit characteristic changes around Tchiral, implying the development of inter-triangular ferromagnetic correlations in the correlated metallic regime. 18. Mean-field theory of echo state networks Massar, Marc; Massar, Serge 2013-04-01 Dynamical systems driven by strong external signals are ubiquitous in nature and engineering. Here we study “echo state networks,” networks of a large number of randomly connected nodes, which represent a simple model of a neural network, and have important applications in machine learning. We develop a mean-field theory of echo state networks. The dynamics of the network is captured by the evolution law, similar to a logistic map, for a single collective variable. When the network is driven by many independent external signals, this collective variable reaches a steady state. But when the network is driven by a single external signal, the collective variable is non stationary but can be characterized by its time averaged distribution. The predictions of the mean-field theory, including the value of the largest Lyapunov exponent, are compared with the numerical integration of the equations of motion. 19. Analytic Beyond-Mean-Field BEC Wave Functions Dunn, Martin; Laing, W. Blake; Watson, Deborah K.; Loeser, John G. 2006-05-01 We present analytic N-body beyond-mean-field wave functions for Bose-Einstein condensates. This extends our previous beyond-mean-field energy calculations to the substantially more difficult problem of determining correlated N-body wave functions for a confined system. The tools used to achieve this have been carefully chosen to maximize the use of symmetry and minimize the dependence on numerical computation. We handle the huge number of interactions when N is large (˜N^2/2 two-body interactions) by bringing together three theoretical methods. These are dimensional perturbation theory, the FG method of Wilson et al, and the group theory of the symmetric group. The wave function is then used to derive the density profile of a condensate in a cylindrical trap.This method makes no assumptions regarding the form or strength of the interactions and is applicable to both small-N and large-N systems. 20. Characterizing the mean-field dynamo in turbulent accretion disks Gressel, Oliver 2015-01-01 The formation and evolution of a wide class of astrophysical objects is governed by turbulent, magnetized accretion disks. Understanding their secular dynamics is of primary importance. Apart from enabling mass accretion via the transport of angular momentum, the turbulence affects the long-term evolution of the embedded magnetic flux, which in turn regulates the efficiency of the transport. In this paper, we take a comprehensive next step towards an effective mean-field model for turbulent astrophysical disks by systematically studying the key properties of magnetorotational turbulence in vertically-stratified, isothermal shearing boxes. This allows us to infer emergent properties of the ensuing chaotic flow as a function of the shear parameter as well as the amount of net-vertical flux. Using the test-field method, we furthermore characterize the mean-field dynamo coefficients that describe the long-term evolution of large-scale fields. We simultaneously infer the vertical shape and the spectral scale depen... 1. Nuclear collective vibrations in extended mean-field theory Lacroix, D. [Lab. de Physique Corpusculaire/ ENSICAEN, 14 - Caen (France); Ayik, S. [Tennessee Technological Univ., Cookeville, TN (United States); Chomaz, Ph. [Grand Accelerateur National d' Ions Lourds (GANIL), 14 - Caen (France) 2003-07-01 The extended mean-field theory, which includes both the incoherent dissipation mechanism due to nucleon-nucleon collisions and the coherent dissipation mechanism due to coupling to low-lying surface vibrations, is briefly reviewed. Expressions of the strength functions for the collective excitations are presented in the small amplitude limit of this approach. This fully microscopic theory is applied by employing effective Skyrme forces to various giant resonance excitations at zero and finite temperature. The theory is able to describe the gross properties of giant resonance excitations, the fragmentation of the strength distributions as well as their fine structure. At finite temperature, the success and limitations of this extended mean-field description are discussed. (authors) 2. Schrödinger Approach to Mean Field Games Swiecicki, Igor; Gobron, Thierry; Ullmo, Denis 2016-03-01 Mean field games (MFG) provide a theoretical frame to model socioeconomic systems. In this Letter, we study a particular class of MFG that shows strong analogies with the nonlinear Schrödinger and Gross-Pitaevskii equations introduced in physics to describe a variety of physical phenomena. Using this bridge, many results and techniques developed along the years in the latter context can be transferred to the former, which provides both a new domain of application for the nonlinear Schrödinger equation and a new and fruitful approach in the study of mean field games. Utilizing this approach, we analyze in detail a population dynamics model in which the "players" are under a strong incentive to coordinate themselves. 3. Mean-field theory of a recurrent epidemiological model. Nagy, Viktor 2009-06-01 Our purpose is to provide a mean-field theory for the discrete time-step susceptible-infected-recovered-susceptible (SIRS) model on uncorrelated networks with arbitrary degree distributions. The effect of network structure, time delays, and infection rate on the stability of oscillating and fixed point solutions is examined through analysis of discrete time mean-field equations. Consideration of two scenarios for disease contagion demonstrates that the manner in which contagion is transmitted from an infected individual to a contacted susceptible individual is of primary importance. In particular, the manner of contagion transmission determines how the degree distribution affects model behavior. We find excellent agreement between our theoretical results and numerical simulations on networks with large average connectivity. 4. Communication patterns in mean field models for wireless sensor networks 2015-01-01 Wireless sensor networks are usually composed of a large number of nodes, and with the increasing processing power and power consumption efficiency they are expected to run more complex protocols in the future. These pose problems in the field of verification and performance evaluation of wireless networks. In this paper, we tailor the mean-field theory as a modeling technique to analyze their behavior. We apply this method to the slotted ALOHA protocol, and establish results on the long term... 5. Dynamical mean-field theory for flat-band ferromagnetism Nguyen, Hong-Son; Tran, Minh-Tien 2016-09-01 The magnetically ordered phase in the Hubbard model on the infinite-dimensional hyper-perovskite lattice is investigated within dynamical mean-field theory. It turns out for the infinite-dimensional hyper-perovskite lattice the self-consistent equations of dynamical mean-field theory are exactly solved, and this makes the Hubbard model exactly solvable. We find electron spins are aligned in the ferromagnetic or ferrimagnetic configuration at zero temperature and half filling of the edge-centered sites of the hyper-perovskite lattice. A ferromagnetic-ferrimagnetic phase transition driven by the energy level splitting is found and it occurs through a phase separation. The origin of ferromagnetism and ferrimagnetism arises from the band flatness and the virtual hybridization between macroscopically degenerate flat bands and dispersive ones. Based on the exact solution in the infinite-dimensional limit, a modified exact diagonalization as the impurity solver for dynamical mean-field theory on finite-dimensional perovskite lattices is also proposed and examined. 6. HBT Pion Interferometry with Phenomenological Mean Field Interaction Hattori, K. 2010-11-01 To extract information on hadron production dynamics in the ultrarelativistic heavy ion collision, the space-time structure of the hadron source has been measured using Hanbury Brown and Twiss interferometry. We study the distortion of the source images due to the effect of a final state interaction. We describe the interaction, taking place during penetrating through a cloud formed by evaporating particles, in terms of a one-body mean field potential localized in the vicinity of the source region. By adopting the semiclassical method, the modification of the propagation of an emitted particle is examined. In analogy to the optical model applied to nuclear reactions, our phenomenological model has an imaginary part of the potential, which describes the absorption in the cloud. In this work, we focus on the pion interferometry and mean field interaction obtained using a phenomenological pipi forward scattering amplitude in the elastic channels. The p-wave scattering wit h rho meson resonance leads to an attractive mean field interaction, and the presence of the absorptive part is mainly attributed to the formation of this resonance. We also incorporate a simple time dependence of the potential reflecting the dynamics of the evaporating source. Using the obtained potential, we examine how and to what extent the so-called HBT Gaussian radius is varied by the modification of the propagation. 7. Mean field study of a propagation-turnover lattice model for the dynamics of histone marking Yao, Fan; Li, FangTing; Li, TieJun 2017-02-01 We present a mean field study of a propagation-turnover lattice model, which was proposed by Hodges and Crabtree [Proc. Nat. Acad. Sci. 109, 13296 (2012)] for understanding how posttranslational histone marks modulate gene expression in mammalian cells. The kinetics of the lattice model consists of nucleation, propagation and turnover mechanisms, and exhibits second-order phase transition for the histone marking domain. We showed rigorously that the dynamics essentially depends on a non-dimensional parameter κ = k +/ k -, the ratio between the propagation and turnover rates, which has been observed in the simulations. We then studied the lowest order mean field approximation, and observed the phase transition with an analytically obtained critical parameter. The boundary layer analysis was utilized to investigate the structure of the decay profile of the mark density. We also studied the higher order mean field approximation to achieve sharper estimate of the critical transition parameter and more detailed features. The comparison between the simulation and theoretical results shows the validity of our theory. 8. Mean-field dynamics of a population of stochastic map neurons Franović, Igor; Maslennikov, Oleg V.; Bačić, Iva; Nekorkin, Vladimir I. 2017-07-01 We analyze the emergent regimes and the stimulus-response relationship of a population of noisy map neurons by means of a mean-field model, derived within the framework of cumulant approach complemented by the Gaussian closure hypothesis. It is demonstrated that the mean-field model can qualitatively account for stability and bifurcations of the exact system, capturing all the generic forms of collective behavior, including macroscopic excitability, subthreshold oscillations, periodic or chaotic spiking, and chaotic bursting dynamics. Apart from qualitative analogies, we find a substantial quantitative agreement between the exact and the approximate system, as reflected in matching of the parameter domains admitting the different dynamical regimes, as well as the characteristic properties of the associated time series. The effective model is further shown to reproduce with sufficient accuracy the phase response curves of the exact system and the assembly's response to external stimulation of finite amplitude and duration. 9. State-of-the-art of beyond mean field theories with nuclear density functionals Egido, J Luis 2016-01-01 We present an overview of beyond mean field theories (BMFT) based on the generator coordinate method (GCM) and the recovery of symmetries used in nuclear physics with effective forces. After a reminder of the Hartree-Fock-Bogoliubov (HFB) theory a discussion of the shortcomings of any mean field approximation (MFA) is presented. The recovery of the symmetries spontaneously broken in the HFB approach, in particular the angular momentum, is necessary, among others, to describe excited states and transitions. Particle number projection is needed to guarantee the right number of protons and neutrons. Furthermore a projection before the variation prevents the pairing collapse in the weak pairing regime. The lack of fluctuations around the average values of the MFA is a shortcoming of this approach. To build in correlations in BMFT one selects the relevant degrees of freedom: quadrupole, octupole and the pairing vibrations as well as the single particle ones. In the GCM the operators representing these degrees of f... 10. Spectral properties of the one-dimensional Hubbard model: cluster dynamical mean-field approaches Go, Ara; Jeon, Gun Sang 2011-03-01 We investigate static and dynamic properties of the one-dimensional Hubbard model using cluster extensions of the dynamical mean-field theory. It is shown that the two different extensions, the cellular dynamical mean-field theory and the dynamic cluster approximation, yield the ground-state properties which are qualitatively in good agreement with each other. We compare the results with the Bethe ansatz results to check the accuracy of the calculation with finite sizes of clusters. We also analyze the spectral properties of the model with the focus on the spin-charge separation and discuss the dependency on the cluster size in the two approaches. This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2010-0010937). 11. Oscillating epidemics in a dynamic network model: stochastic and mean-field analysis. Szabó-Solticzky, András; Berthouze, Luc; Kiss, Istvan Z; Simon, Péter L 2016-04-01 An adaptive network model using SIS epidemic propagation with link-type-dependent link activation and deletion is considered. Bifurcation analysis of the pairwise ODE approximation and the network-based stochastic simulation is carried out, showing that three typical behaviours may occur; namely, oscillations can be observed besides disease-free or endemic steady states. The oscillatory behaviour in the stochastic simulations is studied using Fourier analysis, as well as through analysing the exact master equations of the stochastic model. By going beyond simply comparing simulation results to mean-field models, our approach yields deeper insights into the observed phenomena and help better understand and map out the limitations of mean-field models. 12. Skymapping with OSSE via the Mean Field Annealing Pixon Technique Dixon, D D; Zych, A D; Cheng, L X; Johnson, W N; Kurfess, J D; Pina, R K; Pütter, R C; Purcell, W R; Wheaton, W A; Wheaton, Wm. A. 1997-01-01 We present progress toward using scanned OSSE observations for mapping and sky survey work. To this end, we have developed a technique for detecting pointlike sources of unknown number and location, given that they appear in a background which is relatively featureless or which can be modeled. The technique, based on the newly developed concept and mean field annealing, is described, with sample reconstructions of data from the OSSE Virgo Survey. The results demonstrate the capability of reconstructing source information without any a priori information about the number and/or location of pointlike sources in the field-of-view. 13. Small-world network spectra in mean-field theory. Grabow, Carsten; Grosskinsky, Stefan; Timme, Marc 2012-05-25 Collective dynamics on small-world networks emerge in a broad range of systems with their spectra characterizing fundamental asymptotic features. Here we derive analytic mean-field predictions for the spectra of small-world models that systematically interpolate between regular and random topologies by varying their randomness. These theoretical predictions agree well with the actual spectra (obtained by numerical diagonalization) for undirected and directed networks and from fully regular to strongly random topologies. These results may provide analytical insights to empirically found features of dynamics on small-world networks from various research fields, including biology, physics, engineering, and social science. 14. Time-Dependent Mean-Field Games with Logarithmic Nonlinearities Gomes, Diogo A. 2015-10-06 In this paper, we prove the existence of classical solutions for time-dependent mean-field games with a logarithmic nonlinearity and subquadratic Hamiltonians. Because the logarithm is unbounded from below, this nonlinearity poses substantial mathematical challenges that have not been addressed in the literature. Our result is proven by recurring to a delicate argument which combines Lipschitz regularity for the Hamilton-Jacobi equation with estimates for the nonlinearity in suitable Lebesgue spaces. Lipschitz estimates follow from an application of the nonlinear adjoint method. These are then combined with a priori bounds for solutions of the Fokker-Planck equation and a concavity argument for the nonlinearity. 15. A mean-field game economic growth model Gomes, Diogo A. 2016-08-05 Here, we examine a mean-field game (MFG) that models the economic growth of a population of non-cooperative, rational agents. In this MFG, agents are described by two state variables - the capital and consumer goods they own. Each agent seeks to maximize his/her utility by taking into account statistical data about the whole population. The individual actions drive the evolution of the players, and a market-clearing condition determines the relative price of capital and consumer goods. We study the existence and uniqueness of optimal strategies of the agents and develop numerical methods to compute these strategies and the equilibrium price. 16. Mean-field theory and self-consistent dynamo modeling Yoshizawa, Akira; Yokoi, Nobumitsu [Tokyo Univ. (Japan). Inst. of Industrial Science; Itoh, Sanae-I [Kyushu Univ., Fukuoka (Japan). Research Inst. for Applied Mechanics; Itoh, Kimitaka [National Inst. for Fusion Science, Toki, Gifu (Japan) 2001-12-01 Mean-field theory of dynamo is discussed with emphasis on the statistical formulation of turbulence effects on the magnetohydrodynamic equations and the construction of a self-consistent dynamo model. The dynamo mechanism is sought in the combination of the turbulent residual-helicity and cross-helicity effects. On the basis of this mechanism, discussions are made on the generation of planetary magnetic fields such as geomagnetic field and sunspots and on the occurrence of flow by magnetic fields in planetary and fusion phenomena. (author) 17. Asymptotics of Mean-Field O( N) Models Kirkpatrick, Kay; Nawaz, Tayyab 2016-12-01 We study mean-field classical N-vector models, for integers N≥2. We use the theory of large deviations and Stein's method to study the total spin and its typical behavior, specifically obtaining non-normal limit theorems at the critical temperatures and central limit theorems away from criticality. Important special cases of these models are the XY (N=2) model of superconductors, the Heisenberg (N=3) model [previously studied in Kirkpatrick and Meckes (J Stat Phys 152:54-92, 2013) but with a correction to the critical distribution here], and the Toy (N=4) model of the Higgs sector in particle physics. 18. Asymptotics of the mean-field Heisenberg model Kirkpatrick, Kay 2012-01-01 We consider the mean-field classical Heisenberg model and obtain detailed information about the magnetization by studying the model on a complete graph and sending the number of vertices to infinity. In particular, we obtain Cram\\er- and Sanov-type large deviations principles for the magnetization and the empirical spin distribution and demonstrate a second-order phase transition in the Gibbs measures. We also study the asymptotics of the magnetization throughout the phase transition using Stein's method, proving central limit theorems in the sub- and supercritical phases and a nonnormal limit theorem at the critical temperature. 19. A mechanical approach to mean field spin models Genovese, Giuseppe 2008-01-01 Inspired by the bridge pioneered by Guerra among statistical mechanics on lattice and analytical mechanics on 1+1 continuous Euclidean space-time, we built a self-consistent method to solve for the thermodynamics of mean-field models defined on lattice, whose order parameters self average. We show the whole procedure by analyzing in full details the simplest test case, namely the Curie-Weiss model. Further we report some applications also to models whose order parameters do not self-average, by using the Sherrington-Kirkpatrick spin glass as a guide. 20. Angular momentum projection for a Nilsson mean-field plus pairing model Wang, Yin; Pan, Feng; Launey, Kristina D.; Luo, Yan-An; Draayer, J. P. 2016-06-01 The angular momentum projection for the axially deformed Nilsson mean-field plus a modified standard pairing (MSP) or the nearest-level pairing (NLP) model is proposed. Both the exact projection, in which all intrinsic states are taken into consideration, and the approximate projection, in which only intrinsic states with K = 0 are taken in the projection, are considered. The analysis shows that the approximate projection with only K = 0 intrinsic states seems reasonable, of which the configuration subspace considered is greatly reduced. As simple examples for the model application, low-lying spectra and electromagnetic properties of 18O and 18Ne are described by using both the exact and approximate angular momentum projection of the MSP or the NLP, while those of 20Ne and 24Mg are described by using the approximate angular momentum projection of the MSP or NLP. 1. Mean field theory for U(n) dynamical groups Rosensteel, G, E-mail: [email protected] [Department of Physics, Tulane University, New Orleans, LA 70118 (United States) 2011-04-22 Algebraic mean field theory (AMFT) is a many-body physics modeling tool which firstly, is a generalization of Hartree-Fock mean field theory, and secondly, an application of the orbit method from Lie representation theory. The AMFT ansatz is that the physical system enjoys a dynamical group, which may be either a strong or a weak dynamical Lie group G. When G is a strong dynamical group, the quantum states are, by definition, vectors in one irreducible unitary representation (irrep) space, and AMFT is equivalent to the Kirillov orbit method for deducing properties of a representation from a direct geometrical analysis of the associated integral co-adjoint orbit. AMFT can be the only tractable method for analyzing some complex many-body systems when the dimension of the irrep space of the strong dynamical group is very large or infinite. When G is a weak dynamical group, the quantum states are not vectors in one irrep space, but AMFT applies if the densities of the states lie on one non-integral co-adjoint orbit. The computational simplicity of AMFT is the same for both strong and weak dynamical groups. This paper formulates AMFT explicitly for unitary Lie algebras, and applies the general method to the Lipkin-Meshkov-Glick su(2) model and the Elliott su(3) model. When the energy in the su(3) theory is a rotational scalar function, Marsden-Weinstein reduction simplifies AMFT dynamics to a two-dimensional phase space. 2. Simulated Tempering and Swapping on Mean-Field Models Bhatnagar, Nayantara; Randall, Dana 2016-08-01 Simulated and parallel tempering are families of Markov Chain Monte Carlo algorithms where a temperature parameter is varied during the simulation to overcome bottlenecks to convergence due to multimodality. In this work we introduce and analyze the convergence for a set of new tempering distributions which we call entropy dampening. For asymmetric exponential distributions and the mean field Ising model with an external field simulated tempering is known to converge slowly. We show that tempering with entropy dampening distributions mixes in polynomial time for these models. Examining slow mixing times of tempering more closely, we show that for the mean-field 3-state ferromagnetic Potts model, tempering converges slowly regardless of the temperature schedule chosen. On the other hand, tempering with entropy dampening distributions converges in polynomial time to stationarity. Finally we show that the slow mixing can be very expensive practically. In particular, the mixing time of simulated tempering is an exponential factor longer than the mixing time at the fixed temperature. 3. Non-local correlations within dynamical mean field theory Li, Gang 2009-03-15 The contributions from the non-local fluctuations to the dynamical mean field theory (DMFT) were studied using the recently proposed dual fermion approach. Straight forward cluster extensions of DMFT need the solution of a small cluster, where all the short-range correlations are fully taken into account. All the correlations beyond the cluster scope are treated in the mean-field level. In the dual fermion method, only a single impurity problem needs to be solved. Both the short and long-range correlations could be considered on equal footing in this method. The weak-coupling nature of the dual fermion ensures the validity of the finite order diagram expansion. The one and two particle Green's functions calculated from the dual fermion approach agree well with the Quantum Monte Carlo solutions, and the computation time is considerably less than with the latter method. The access of the long-range order allows us to investigate the collective behavior of the electron system, e.g. spin wave excitations. (orig.) 4. Nonlinear regimes in mean-field full-sphere dynamo Pipin, V V 2016-01-01 The mean-field dynamo model is employed to study the non-linear dynamo regimes in a fully convective star of mass 0.3M_{\\odot}$rotating with period of 10 days. The differential rotation law was estimated using the mean-field hydrodynamic and heat transport equations. For the intermediate parameter of the turbulent magnetic Reynolds number,$Pm_{T}=3$we found the oscillating dynamo regimes with period about 40Yr. The higher$Pm_{T}$results to longer dynamo periods. The meridional circulation has one cell per hemisphere. It is counter-clockwise in the Northen hemisphere. The amplitude of the flow at the surface around 1 m/s. Tne models with regards for meridional circulation show the anti-symmetric relative to equator magnetic field. If the large-scale flows is fixed we find that the dynamo transits from axisymmetric to non-axisymmetric regimes for the overcritical parameter of the$\\alpha$effect. The change of dynamo regime occurs because of the non-axisymmetric non-linear$\\alpha$-effect. The situation pe... 5. Kinetic and mean field description of Gibrat's law Toscani, Giuseppe 2016-01-01 We introduce and analyze a linear kinetic model that describes the evolution of the probability density of the number of firms in a society, in which the microscopic rate of change obeys to the so-called law of proportional effect proposed by Gibrat. Despite its apparent simplicity, the possible mean field limits of the kinetic model are varied. In some cases, the asymptotic limit can be described by a first-order partial differential equation. In other cases, the mean field equation is a linear diffusion with a non constant diffusion coefficient that models also the geometric Brownian motion and can be studied analytically. In this case, it is shown that the large-time behavior of the solution is represented, for a large class of initial data, by a lognormal distribution with constant mean value and variance increasing exponentially in time at a precise rate. The relationship between the kinetic and the diffusion models allow to introduce an easy-to- implement expression for computing the Fourier transform o... 6. The effectiveness of mean-field theory for avalanche distributions Lee, Edward; Raju, Archishman; Sethna, James We explore the mean-field theory of the pseudogap found in avalanche systems with long-range anisotropic interactions using analytical and numerical tools. The pseudogap in the density of low-stability states emerges from the competition between stabilizing interactions between spins in an avalanche and the destabilizing random movement towards the threshold caused by anisotropic couplings. Pazmandi et al. have shown that for the Sherrington-Kirkpatrick model, the pseudogap scales linearly and produces a distribution of avalanche sizes with exponent t=1 in contrast with that predicted from RFIM t=3/2. Lin et al. have argued that the scaling exponent ? of the pseudogap depends on the tail of the distribution of couplings and on non-universal values like the strain rate and the magnitude of the coupling strength. Yet others have argued that the relationship between the pseudogap scaling and the distribution of avalanche sizes is dependent on dynamical details. Despite the theoretical arguments, the class of RFIM mean-field models is surprisingly good at predicting the distribution of avalanche sizes in a variety of different magnetic systems. We investigate these differences with a combination of theory and simulation. 7. Kinetic and mean field description of Gibrat's law Toscani, Giuseppe 2016-11-01 I introduce and analyze a linear kinetic model that describes the evolution of the probability density of the number of firms in a society, in which the microscopic rate of change obeys to the so-called law of proportional effect proposed by Gibrat (1930, 1931). Despite its apparent simplicity, the possible mean field limits of the kinetic model are varied. In some cases, the asymptotic limit can be described by a first-order partial differential equation. In other cases, the mean field equation is a linear diffusion with a non constant diffusion coefficient that can be studied analytically, by virtue of a transformation of variables recently utilized in Iagar and Sánchez (2013) to study the heat equation in a nonhomogeneous medium with critical density. In this case, it is shown that the large-time behavior of the solution is represented, for a large class of initial data, by a lognormal distribution with constant mean value and variance increasing exponentially in time at a precise rate. 8. Relativistic astrophysics Demianski, Marek 2013-01-01 Relativistic Astrophysics brings together important astronomical discoveries and the significant achievements, as well as the difficulties in the field of relativistic astrophysics. This book is divided into 10 chapters that tackle some aspects of the field, including the gravitational field, stellar equilibrium, black holes, and cosmology. The opening chapters introduce the theories to delineate gravitational field and the elements of relativistic thermodynamics and hydrodynamics. The succeeding chapters deal with the gravitational fields in matter; stellar equilibrium and general relativity 9. RELATIVISTIC CALCULATIONS OF THE SUPERHEAVY NUCLEUS 114-298 BOERSMA, HF 1993-01-01 We investigate ground-state properties of the superheavy nucleus with N = 184 and Z = 114, (298)114, using conventional relativistic mean-field theory and density-dependent mean-field theory, which reproduces Dirac-Brueckner calculations in nuclear matter. Our calculations provide support for N = 18 10. Exact Analytical Solutions to the Two-Mode Mean-Field Model Describing Dynamics of a Split Bose-Einstein Condensate WU Ying; YANG Xiao-Xue 2002-01-01 We present the analytical solutions to the two-mode mean-field model for a split Bose Einstein condensate.These explicit solutions completely determine the system's dynamics under the two-mode mean-field approximation for all possible initial conditions. 11. Mean Field Limit of Interacting Filaments and Vector Valued Non-linear PDEs Bessaih, Hakima; Coghi, Michele; Flandoli, Franco 2017-03-01 Families of N interacting curves are considered, with long range, mean field type, interaction. They generalize models based on classical interacting point particles to models based on curves. In this new set-up, a mean field result is proven, as N→ ∞. The limit PDE is vector valued and, in the limit, each curve interacts with a mean field solution of the PDE. This target is reached by a careful formulation of curves and weak solutions of the PDE which makes use of 1-currents and their topologies. The main results are based on the analysis of a nonlinear Lagrangian-type flow equation. Most of the results are deterministic; as a by-product, when the initial conditions are given by families of independent random curves, we prove a propagation of chaos result. The results are local in time for general interaction kernel, global in time under some additional restriction. Our main motivation is the approximation of 3D-inviscid flow dynamics by the interacting dynamics of a large number of vortex filaments, as observed in certain turbulent fluids; in this respect, the present paper is restricted to smoothed interaction kernels, instead of the true Biot-Savart kernel. 12. Hydrodynamic mean-field solutions of 1D exclusion processes with spatially varying hopping rates Lakatos, Greg; O'Brien, John; Chou, Tom 2006-03-01 We analyse the open boundary partially asymmetric exclusion process with smoothly varying internal hopping rates in the infinite-size, mean-field limit. The mean-field equations for particle densities are written in terms of Ricatti equations with the steady-state current J as a parameter. These equations are solved both analytically and numerically. Upon imposing the boundary conditions set by the injection and extraction rates, the currents J are found self-consistently. We find a number of cases where analytic solutions can be found exactly or approximated. Results for J from asymptotic analyses for slowly varying hopping rates agree extremely well with those from extensive Monte Carlo simulations, suggesting that mean-field currents asymptotically approach the exact currents in the hydrodynamic limit, as the hopping rates vary slowly over the lattice. If the forward hopping rate is greater than or less than the backward hopping rate throughout the entire chain, the three standard steady-state phases are preserved. Our analysis reveals the sensitivity of the current to the relative phase between the forward and backward hopping rate functions. 13. Towards a nonequilibrium Green's function description of nuclear reactions: one-dimensional mean-field dynamics Rios, Arnau; Buchler, Mark; Danielewicz, Pawel 2010-01-01 Nonequilibrium Green's function methods allow for an intrinsically consistent description of the evolution of quantal many-body body systems, with inclusion of different types of correlations. In this paper, we focus on the practical developments needed to build a Green's function methodology for nuclear reactions. We start out by considering symmetric collisions of slabs in one dimension within the mean-field approximation. We concentrate on two issues of importance for actual reaction simulations. First, the preparation of the initial state within the same methodology as for the reaction dynamics is demonstrated by an adiabatic switching on of the mean-field interaction, which leads to the mean-field ground state. Second, the importance of the Green's function matrix-elements far away from the spatial diagonal is analyzed by a suitable suppression process that does not significantly affect the evolution of the elements close to the diagonal. The relative lack of importance of the far-away elements is tied t... 14. Hydrodynamic mean-field solutions of 1D exclusion processes with spatially varying hopping rates Lakatos, Greg; O' Brien, John; Chou, Tom [Department of Biomathematics and Institute for Pure and Applied Mathematics, UCLA, Los Angeles, CA 90095 (United States) 2006-03-10 We analyse the open boundary partially asymmetric exclusion process with smoothly varying internal hopping rates in the infinite-size, mean-field limit. The mean-field equations for particle densities are written in terms of Ricatti equations with the steady-state current J as a parameter. These equations are solved both analytically and numerically. Upon imposing the boundary conditions set by the injection and extraction rates, the currents J are found self-consistently. We find a number of cases where analytic solutions can be found exactly or approximated. Results for J from asymptotic analyses for slowly varying hopping rates agree extremely well with those from extensive Monte Carlo simulations, suggesting that mean-field currents asymptotically approach the exact currents in the hydrodynamic limit, as the hopping rates vary slowly over the lattice. If the forward hopping rate is greater than or less than the backward hopping rate throughout the entire chain, the three standard steady-state phases are preserved. Our analysis reveals the sensitivity of the current to the relative phase between the forward and backward hopping rate functions. 15. Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource Graber, P. Jameson, E-mail: [email protected] [Baylor University, Department of Mathematics (United States) 2016-12-15 We study a general linear quadratic mean field type control problem and connect it to mean field games of a similar type. The solution is given both in terms of a forward/backward system of stochastic differential equations and by a pair of Riccati equations. In certain cases, the solution to the mean field type control is also the equilibrium strategy for a class of mean field games. We use this fact to study an economic model of production of exhaustible resources. 16. How to do mean field theory in Feynman gauge and doing it for U(1) with corrections to fourth order Flyvbjerg, H. 1984-07-02 It is demonstrated how mean field theory with corrections from fluctuations may be applied to lattice gauge theories in covariant gauges. By fixing the gauge at tree level, the importance of fluctuations is decreased. This is understood as inclusion of terms of next-to-leading-order in d in the definition of the mean field tree approximation, d being the dimension of the lattice. The gauge group U(1) and Wilson's action are used as testing ground. Tree and one-loop results comparable to those previously obtained in axial gauge are obtained in for d=4. The next three correction terms to the free and plaquette energies are evaluated in Feynmann gauge. The truncated asympotic series thus obtained is compared to that of the ordinary weak coupling expansion. The mean field series gives, to those orders studied, a much better approximation. The location of phase transitions in 4d and 5d are predicted with 1% error bars. 17. Neural Population Dynamics Modeled by Mean-Field Graphs Kozma, Robert; Puljic, Marko 2011-09-01 In this work we apply random graph theory approach to describe neural population dynamics. There are important advantages of using random graph theory approach in addition to ordinary and partial differential equations. The mathematical theory of large-scale random graphs provides an efficient tool to describe transitions between high- and low-dimensional spaces. Recent advances in studying neural correlates of higher cognition indicate the significance of sudden changes in space-time neurodynamics, which can be efficiently described as phase transitions in the neuropil medium. Phase transitions are rigorously defined mathematically on random graph sequences and they can be naturally generalized to a class of percolation processes called neuropercolation. In this work we employ mean-field graphs with given vertex degree distribution and edge strength distribution. We demonstrate the emergence of collective oscillations in the style of brains. 18. Spectral Synthesis via Mean Field approach Independent Component Analysis Hu, Ning; Kong, Xu 2015-01-01 In this paper, we apply a new statistical analysis technique, Mean Field approach to Bayesian Independent Component Analysis (MF-ICA), on galaxy spectral analysis. This algorithm can compress the stellar spectral library into a few Independent Components (ICs), and galaxy spectrum can be reconstructed by these ICs. Comparing to other algorithms which decompose a galaxy spectrum into a combination of several simple stellar populations, MF-ICA approach offers a large improvement in the efficiency. To check the reliability of this spectral analysis method, three different methods are used: (1) parameter-recover for simulated galaxies, (2) comparison with parameters estimated by other methods, and (3) consistency test of parameters from the Sloan Digital Sky Survey galaxies. We find that our MF-ICA method not only can fit the observed galaxy spectra efficiently, but also can recover the physical parameters of galaxies accurately. We also apply our spectral analysis method to the DEEP2 spectroscopic data, and find... 19. Non-mean-field screening by multivalent counterions Loth, M S; Shklovskii, B I, E-mail: [email protected] [Department of Physics, University of Minnesota, Minneapolis, MN 55455 (United States) 2009-10-21 Screening of a strongly charged macroion by its multivalent counterions cannot be described in the framework of a mean-field Poisson-Boltzmann (PB) theory because multivalent counterions form a strongly correlated liquid (SCL) on the surface of the macroion. It was predicted that a distant counterion polarizes the SCL as if it were a metallic surface and creates an electrostatic image. The attractive potential energy of the image is the reason why the charge density of counterions decreases faster with distance from the charged surface than in PB theory. Using the Monte Carlo method to find the equilibrium distribution of counterions around the macroion, we confirm the existence of the image potential energy. It is also shown that, due to the negative screening length of the SCL, -2xi, the effective metallic surface is actually above the SCL by |xi|. 20. Explicit Solutions for One-Dimensional Mean-Field Games Prazeres, Mariana 2017-04-05 In this thesis, we consider stationary one-dimensional mean-field games (MFGs) with or without congestion. Our aim is to understand the qualitative features of these games through the analysis of explicit solutions. We are particularly interested in MFGs with a nonmonotonic behavior, which corresponds to situations where agents tend to aggregate. First, we derive the MFG equations from control theory. Then, we compute explicit solutions using the current formulation and examine their behavior. Finally, we represent the solutions and analyze the results. This thesis main contributions are the following: First, we develop the current method to solve MFG explicitly. Second, we analyze in detail non-monotonic MFGs and discover new phenomena: non-uniqueness, discontinuous solutions, empty regions and unhappiness traps. Finally, we address several regularization procedures and examine the stability of MFGs. 1. Mean field games with nonlinear mobilities in pedestrian dynamics Burger, Martin 2014-04-01 In this paper we present an optimal control approach modeling fast exit scenarios in pedestrian crowds. In particular we consider the case of a large human crowd trying to exit a room as fast as possible. The motion of every pedestrian is determined by minimizing a cost functional, which depends on his/her position, velocity, exit time and the overall density of people. This microscopic setup leads in the mean-field limit to a parabolic optimal control problem. We discuss the modeling of the macroscopic optimal control approach and show how the optimal conditions relate to the Hughes model for pedestrian flow. Furthermore we provide results on the existence and uniqueness of minimizers and illustrate the behavior of the model with various numerical results. 2. Nuclear Level Density: Shell Model vs Mean Field Sen'kov, Roman 2015-01-01 The knowledge of the nuclear level density is necessary for understanding various reactions including those in the stellar environment. Usually the combinatorics of Fermi-gas plus pairing is used for finding the level density. Recently a practical algorithm avoiding diagonalization of huge matrices was developed for calculating the density of many-body nuclear energy levels with certain quantum numbers for a full shell-model Hamiltonian. The underlying physics is that of quantum chaos and intrinsic thermalization in a closed system of interacting particles. We briefly explain this algorithm and, when possible, demonstrate the agreement of the results with those derived from exact diagonalization. The resulting level density is much smoother than that coming from the conventional mean-field combinatorics. We study the role of various components of residual interactions in the process of thermalization, stressing the influence of incoherent collision-like processes. The shell-model results for the traditionally... 3. Metabifurcation analysis of a mean field model of the cortex Frascoli, Federico; Bojak, Ingo; Liley, David T J 2010-01-01 Mean field models (MFMs) of cortical tissue incorporate salient features of neural masses to model activity at the population level. One of the common aspects of MFM descriptions is the presence of a high dimensional parameter space capturing neurobiological attributes relevant to brain dynamics. We study the physiological parameter space of a MFM of electrocortical activity and discover robust correlations between physiological attributes of the model cortex and its dynamical features. These correlations are revealed by the study of bifurcation plots, which show that the model responses to changes in inhibition belong to two families. After investigating and characterizing these, we discuss their essential differences in terms of four important aspects: power responses with respect to the modeled action of anesthetics, reaction to exogenous stimuli, distribution of model parameters and oscillatory repertoires when inhibition is enhanced. Furthermore, while the complexity of sustained periodic orbits differs ... 4. Double binding energy differences: Mean-field or pairing effect? Qi, Chong 2012-10-01 In this Letter we present a systematic analysis on the average interaction between the last protons and neutrons in atomic nuclei, which can be extracted from the double differences of nuclear binding energies. The empirical average proton-neutron interaction Vpn thus derived from experimental data can be described in a very simple form as the interplay of the nuclear mean field and the pairing interaction. It is found that the smooth behavior as well as the local fluctuations of the Vpn in even-even nuclei with N ≠ Z are dominated by the contribution from the proton-neutron monopole interactions. A strong additional contribution from the isoscalar monopole interaction and isovector proton-neutron pairing interaction is seen in the Vpn for even-even N = Z nuclei and for the adjacent odd-A nuclei with one neutron or proton being subtracted. 5. Mean-field games with logistic population dynamics Gomes, Diogo A. 2013-12-01 In its standard form, a mean-field game can be defined by coupled system of equations, a Hamilton-Jacobi equation for the value function of agents and a Fokker-Planck equation for the density of agents. Traditionally, the latter equation is adjoint to the linearization of the former. Since the Fokker-Planck equation models a population dynamic, we introduce natural features such as seeding and birth, and nonlinear death rates. In this paper we analyze a stationary meanfield game in one dimension, illustrating various techniques to obtain regularity of solutions in this class of systems. In particular we consider a logistic-type model for birth and death of the agents which is natural in problems where crowding affects the death rate of the agents. The introduction of these new terms requires a number of new ideas to obtain wellposedness. In a forthcoming publication we will address higher dimensional models. ©2013 IEEE. 6. Glauber Dynamics for the mean-field Potts Model Cuff, Paul; Louidor, Oren; Lubetzky, Eyal; Peres, Yuval; Sly, Allan 2012-01-01 We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with$q\\geq 3$states and show that it undergoes a critical slowdown at an inverse-temperature$\\beta_s(q)$strictly lower than the critical$\\beta_c(q)$for uniqueness of the thermodynamic limit. The dynamical critical$\\beta_s(q)$is the spinodal point marking the onset of metastability. We prove that when$\\beta\\beta_s(q)$the mixing time is exponentially large in$n$. Furthermore, as$\\beta \\uparrow \\beta_s$with$n$, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of$O(n^{-2/3})$around$\\beta_s$. These results form the first complete analysis of the critical slowdown of a dynamics with a first order phase transition. 7. Glauber Dynamics for the Mean-Field Potts Model Cuff, P.; Ding, J.; Louidor, O.; Lubetzky, E.; Peres, Y.; Sly, A. 2012-11-01 We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q≥3 states and show that it undergoes a critical slowdown at an inverse-temperature β s ( q) strictly lower than the critical β c ( q) for uniqueness of the thermodynamic limit. The dynamical critical β s ( q) is the spinodal point marking the onset of metastability. We prove that when β β s ( q) the mixing time is exponentially large in n. Furthermore, as β↑ β s with n, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of O( n -2/3) around β s . These results form the first complete analysis of mixing around the critical dynamical temperature—including the critical power law—for a model with a first order phase transition. 8. Quasi-isotropic cascade in MHD turbulence with mean field Grappin, Roland; Gürcan, Özgür 2012-01-01 We propose a phenomenological theory of incompressible magnetohydrodynamic turbulence in the presence of a strong large-scale magnetic field, which establishes a link between the known anisotropic models of strong and weak MHD turbulence We argue that the Iroshnikov-Kraichnan isotropic cascade develops naturally within the plane perpendicular to the mean field, while oblique-parallel cascades with weaker amplitudes can develop, triggered by the perpendicular cascade, with a reduced flux resulting from a quasi-resonance condition. The resulting energy spectrum$E(k_\\parallel,k_\\bot)$has the same slope in all directions. The ratio between the extents of the inertial range in the parallel and perpendicular directions is equal to$b_{rms}/B_0$. These properties match those found in recent 3D MHD simulations with isotropic forcing reported in [R. Grappin and W.-C. M\\"uller, Phys. Rev. E \\textbf{82}, 26406 (2010)]. 9. Mean field theory for U(n) dynamical groups Rosensteel, G. 2011-04-01 Algebraic mean field theory (AMFT) is a many-body physics modeling tool which firstly, is a generalization of Hartree-Fock mean field theory, and secondly, an application of the orbit method from Lie representation theory. The AMFT ansatz is that the physical system enjoys a dynamical group, which may be either a strong or a weak dynamical Lie group G. When G is a strong dynamical group, the quantum states are, by definition, vectors in one irreducible unitary representation (irrep) space, and AMFT is equivalent to the Kirillov orbit method for deducing properties of a representation from a direct geometrical analysis of the associated integral co-adjoint orbit. AMFT can be the only tractable method for analyzing some complex many-body systems when the dimension of the irrep space of the strong dynamical group is very large or infinite. When G is a weak dynamical group, the quantum states are not vectors in one irrep space, but AMFT applies if the densities of the states lie on one non-integral co-adjoint orbit. The computational simplicity of AMFT is the same for both strong and weak dynamical groups. This paper formulates AMFT explicitly for unitary Lie algebras, and applies the general method to the Lipkin-Meshkov-Glick {\\mathfrak s}{\\mathfrak u} (2) model and the Elliott {\\mathfrak s}{\\mathfrak u} (3) model. When the energy in the {\\mathfrak s}{\\mathfrak u} (3) theory is a rotational scalar function, Marsden-Weinstein reduction simplifies AMFT dynamics to a two-dimensional phase space. 10. Dynamical Mean-Field Theory and Its Applications to Real Materials Vollhardt, D.; Held, K.; Keller, G.; Bulla, R.; Pruschke, Th.; Nekrasov, I. A.; Anisimov, V. I. 2005-01-01 Dynamical mean-field theory (DMFT) is a non-perturbative technique for the investigation of correlated electron systems. Its combination with the local density approximation (LDA) has recently led to a material-specific computational scheme for the ab initio investigation of correlated electron materials. The set-up of this approach and its application to materials such as (Sr,Ca)VO3, V2O3, and Cerium is discussed. The calculated spectra are compared with the spectroscopically measured electronic excitation spectra. The surprising similarity between the spectra of the single-impurity Anderson model and of correlated bulk materials is also addressed. 11. Second-order corrections to mean-field evolution of weakly interacting Bosons, II Grillakis, M; Margetis, D 2010-01-01 We study the evolution of a N-body weakly interacting system of Bosons. Our work forms an extension of our previous paper I, in which we derived a second-order correction to a mean-field evolution law for coherent states in the presence of small interaction potential. Here, we remove the assumption of smallness of the interaction potential and prove global existence of solutions to the equation for the second-order correction. This implies an improved Fock-space estimate for our approximation of the N-body state. 12. How reliable is the mean-field nuclear matter description for supporting chiral effective lagrangians? Delfino, A; Frederico, T 1996-01-01 The link between non-linear chiral effective Lagrangians and the Walecka model description of bulk nuclear matter [1] is questioned. This fact is by itself due to the Mean Field Approximation (MFA) which in nuclear mater makes the picture of a nucleon-nucleon interaction based on scalar(vector) meson exchange, equivalent to the description of a nuclear matter based on attractive and repulsive contact interactions. We present a linear chiral model where this link between the Walecka model and an underlying to chiral symmetry realization still holds, due to MFA. 13. Mean-Field Semantics for a Process Calculus for Spatially-Explicit Ecological Models Mauricio Toro 2016-03-01 Full Text Available We define a mean-field semantics for S-PALPS, a process calculus for spatially-explicit, individual-based modeling of ecological systems. The new semantics of S-PALPS allows an interpretation of the average behavior of a system as a set of recurrence equations. Recurrence equations are a useful approximation when dealing with a large number of individuals, as it is the case in epidemiological studies. As a case study, we compute a set of recurrence equations capturing the dynamics of an individual-based model of the transmission of dengue in Bello (Antioquia, Colombia. 14. A Mean-Field Treatment in Studying Nuclear Matter Through a Thermodynamic Consistent Resummation Scheme 舒崧; 李家荣 2012-01-01 We used the Cornwall, Jackiw and Tomboulis （CJT） resummation scheme to study nuclear matter. In the CJT formalism the meson propagators are treated as the bare propagators and the the higher order loop corrections of the thermodynamic potential are evaluated at the Hartree approximation, while the vacuum fluctuations are ignored. Under these treatments in the CJT formalism we derived exact mean-field theory （MFT） results for the nuclear matter. The results are thermodynamically consistent, and our study indicates that the MFT result is the lowest order resummation result in the CJT resummation scheme. The relation between CJT formalism and MFT is clearly presented through the calculations. 15. Low Complexity Sparse Bayesian Learning for Channel Estimation Using Generalized Mean Field Pedersen, Niels Lovmand; Manchón, Carles Navarro; Fleury, Bernard Henri 2014-01-01 constrain the auxiliary function approximating the posterior probability density function of the unknown variables to factorize over disjoint groups of contiguous entries in the sparse vector - the size of these groups dictates the degree of complexity reduction. The original high-complexity algorithms......We derive low complexity versions of a wide range of algorithms for sparse Bayesian learning (SBL) in underdetermined linear systems. The proposed algorithms are obtained by applying the generalized mean field (GMF) inference framework to a generic SBL probabilistic model. In the GMF framework, we... 16. Variational mean-field algorithm for efficient inference in large systems of stochastic differential equations. Vrettas, Michail D; Opper, Manfred; Cornford, Dan 2015-01-01 This work introduces a Gaussian variational mean-field approximation for inference in dynamical systems which can be modeled by ordinary stochastic differential equations. This new approach allows one to express the variational free energy as a functional of the marginal moments of the approximating Gaussian process. A restriction of the moment equations to piecewise polynomial functions, over time, dramatically reduces the complexity of approximate inference for stochastic differential equation models and makes it comparable to that of discrete time hidden Markov models. The algorithm is demonstrated on state and parameter estimation for nonlinear problems with up to 1000 dimensional state vectors and compares the results empirically with various well-known inference methodologies. 17. Deterministic Methods for Filtering, part I: Mean-field Ensemble Kalman Filtering Law, Kody J H; Tempone, Raul 2014-01-01 This paper provides a proof of convergence of the standard EnKF generalized to non-Gaussian state space models, based on the indistinguishability property of the joint distribution on the ensemble. A density-based deterministic approximation of the mean-field EnKF (MFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given a certain minimal order of convergence k between the two, this extends to the deterministic filter approximation, which is therefore asymptotically superior to standard EnKF for d<2k. The fidelity of approximation of the true distribution is also established using an extension of total variation metric to random measures. This is limited by a Gaussian bias term arising from non-linearity/non-Gaussianity of the model, which arises in both deterministic and standard EnKF. Numerical results support and extend the theory. 18. Second order corrections to mean field evolution for weakly interacting Bosons in the case of 3-body interactions Chen, Xuwen 2010-01-01 In this paper, we consider the Hamiltonian evolution of N weakly interacting Bosons. Assuming triple collisions with singular potentials, its mean field approximation is given by a quintic Hartree equation. We construct a second order correction to the mean field approximation using a kernel k(t,x,y) and derive an evolution equation for k. We show the global existence for the resulting evolution equation for the correction and establish an apriori estimate comparing the approximation to the exact Hamiltonian evolution. Our error estimate is global and uniform in time. Comparing with the work in [20,11,12] where the error estimate grows in time, our approximation tracks the exact dynamics for all time with an error of the order O(1/$\\sqrt{N}$). 19. The quark mean field model with pion and gluon corrections Xing, Xueyong; Shen, Hong 2016-01-01 The properties of nuclear matter and finite nuclei are studied within the quark mean field (QMF) model by taking the effects of pion and gluon into account at the quark level. The nucleon is described as the combination of three constituent quarks confined by a harmonic oscillator potential. To satisfy the spirit of QCD theory, the contributions of pion and gluon on the nucleon structure are treated in second-order perturbation theory. For the nuclear many-body system, nucleons interact with each other by exchanging mesons between quarks. With different constituent quark mass,$m_q$, we determine three parameter sets about the coupling constants between mesons and quarks, named as QMF-NK1, QMF-NK2, and QMF-NK3 by fitting the ground-state properties of several closed-shell nuclei. It is found that all of the three parameter sets can give satisfactory description on properties of nuclear matter and finite nuclei, meanwhile they can also predict the larger neutron star mass around$2.3M_\\odot$without the hypero... 20. Quark mean field model with pion and gluon corrections Xing, Xueyong; Hu, Jinniu; Shen, Hong 2016-10-01 The properties of nuclear matter and finite nuclei are studied within the quark mean field (QMF) model by taking the effects of pions and gluons into account at the quark level. The nucleon is described as the combination of three constituent quarks confined by a harmonic oscillator potential. To satisfy the spirit of QCD theory, the contributions of pions and gluons on the nucleon structure are treated in second-order perturbation theory. In a nuclear many-body system, nucleons interact with each other by exchanging mesons between quarks. With different constituent quark mass, mq, we determine three parameter sets for the coupling constants between mesons and quarks, named QMF-NK1, QMF-NK2, and QMF-NK3, by fitting the ground-state properties of several closed-shell nuclei. It is found that all of the three parameter sets can give a satisfactory description of properties of nuclear matter and finite nuclei, moreover they also predict a larger neutron star mass around 2.3 M⊙ without hyperon degrees of freedom. 1. One-Dimensional Forward–Forward Mean-Field Games Gomes, Diogo A. 2016-11-01 While the general theory for the terminal-initial value problem for mean-field games (MFGs) has achieved a substantial progress, the corresponding forward–forward problem is still poorly understood—even in the one-dimensional setting. Here, we consider one-dimensional forward–forward MFGs, study the existence of solutions and their long-time convergence. First, we discuss the relation between these models and systems of conservation laws. In particular, we identify new conserved quantities and study some qualitative properties of these systems. Next, we introduce a class of wave-like equations that are equivalent to forward–forward MFGs, and we derive a novel formulation as a system of conservation laws. For first-order logarithmic forward–forward MFG, we establish the existence of a global solution. Then, we consider a class of explicit solutions and show the existence of shocks. Finally, we examine parabolic forward–forward MFGs and establish the long-time convergence of the solutions. 2. A Mean-Field Theory for Coarsening Faceted Surfaces Norris, Scott A 2009-01-01 A mean-field theory is developed for the scale-invariant length distributions observed during the coarsening of one-dimensional faceted surfaces. This theory closely follows the Lifshitz-Slyozov-Wagner theory of Ostwald ripening in two-phase systems [1-3], but the mechanism of coarsening in faceted surfaces requires the addition of convolution terms recalling the work of Smoluchowski [4] and Schumann [5] on coalescence. The model is solved by the exponential distribution, but agreement with experiment is limited by the assumption that neighboring facet lengths are uncorrelated. However, the method concisely describes the essential processes operating in the scaling state, illuminates a clear path for future refinement, and offers a framework for the investigation of faceted surfaces evolving under arbitrary dynamics. [1] I. Lifshitz, V. Slezov, Soviet Physics JETP 38 (1959) 331-339. [2] I. Lifshitz, V. Slyozov, J. Phys. Chem. Solids 19 (1961) 35-50. [3] C. Wagner, Elektrochemie 65 (1961) 581-591. [4] M. von S... 3. Spectral Synthesis via Mean Field approach to Independent Component Analysis Hu, Ning; Su, Shan-Shan; Kong, Xu 2016-03-01 We apply a new statistical analysis technique, the Mean Field approach to Independent Component Analysis (MF-ICA) in a Bayseian framework, to galaxy spectral analysis. This algorithm can compress a stellar spectral library into a few Independent Components (ICs), and the galaxy spectrum can be reconstructed by these ICs. Compared to other algorithms which decompose a galaxy spectrum into a combination of several simple stellar populations, the MF-ICA approach offers a large improvement in efficiency. To check the reliability of this spectral analysis method, three different methods are used: (1) parameter recovery for simulated galaxies, (2) comparison with parameters estimated by other methods, and (3) consistency test of parameters derived with galaxies from the Sloan Digital Sky Survey. We find that our MF-ICA method can not only fit the observed galaxy spectra efficiently, but can also accurately recover the physical parameters of galaxies. We also apply our spectral analysis method to the DEEP2 spectroscopic data, and find it can provide excellent fitting results for low signal-to-noise spectra. 4. Metastability for the Exclusion Process with Mean-Field Interaction Asselah, Amine; Giacomin, Giambattista 1998-12-01 We consider an exclusion particle system with long-range, mean-field-type interactions at temperature 1/β. The hydrodynamic limit of such a system is given by an integrodifferential equation with one conservation law on the circle C: it is the gradient flux of the Kac free energy functional F β. For β≤1, any constant function with value m ∈ [-1, +1] is the global minimizer of F β in the space \\{ u:int_C {u(x)} dx = m\\} . For β>1, F β restricted to \\{ u:int_C {u(x)} dx = m\\} may have several local minima: in particular, the constant solution may not be the absolute minimizer of F β. We therefore study the long-time behavior of the particle system when the initial condition is close to a homogeneous stable state, giving results on the time of exit from (suitable) subsets of its domain of attraction. We follow the Freidlin-Wentzell approach: first, we study in detail F β together with the time asymptotics of the solution of the hydrodynamic equation; then we study the probability of rare events for the particle system, i.e., large deviations from the hydrodynamic limit. 5. One-Dimensional Forward–Forward Mean-Field Games Gomes, Diogo A., E-mail: [email protected]; Nurbekyan, Levon; Sedjro, Marc [King Abdullah University of Science and Technology (KAUST), CEMSE Division (Saudi Arabia) 2016-12-15 While the general theory for the terminal-initial value problem for mean-field games (MFGs) has achieved a substantial progress, the corresponding forward–forward problem is still poorly understood—even in the one-dimensional setting. Here, we consider one-dimensional forward–forward MFGs, study the existence of solutions and their long-time convergence. First, we discuss the relation between these models and systems of conservation laws. In particular, we identify new conserved quantities and study some qualitative properties of these systems. Next, we introduce a class of wave-like equations that are equivalent to forward–forward MFGs, and we derive a novel formulation as a system of conservation laws. For first-order logarithmic forward–forward MFG, we establish the existence of a global solution. Then, we consider a class of explicit solutions and show the existence of shocks. Finally, we examine parabolic forward–forward MFGs and establish the long-time convergence of the solutions. 6. Mean-field study of$^{12}$C+$^{12}$C fusion Chien, Le Hoang; Khoa, Dao T 2016-01-01 The nuclear mean-field potential arising from the$^{12}$C+$^{12}$C interaction at the low energies relevant for the astrophysical carbon burning process has been constructed within the double-folding model, using the realistic nuclear ground-state density of the$^{12}$C nucleus and the effective M3Y nucleon-nucleon (NN) interaction constructed from the G-matrix of the Paris (free) NN potential. To explore the nuclear medium effect, both the original density independent M3Y-Paris interaction and its density dependent CDM3Y6 version have been used in the folding model calculation of the$^{12}$C+$^{12}$C potential. The folded potentials at the different energies were used in the optical model description of the elastic$^{12}$C+$^{12}$C scattering at the energies around and below the Coulomb barrier, as well as in the barrier penetration model to estimate the fusion cross section and astrophysical$S$factor of the$^{12}$C+$^{12}$C reactions at the low energies. The obtained results are in good agreement wit... 7. Individual based and mean-field modeling of direct aggregation Burger, Martin 2013-10-01 We introduce two models of biological aggregation, based on randomly moving particles with individual stochasticity depending on the perceived average population density in their neighborhood. In the firstorder model the location of each individual is subject to a density-dependent random walk, while in the second-order model the density-dependent random walk acts on the velocity variable, together with a density-dependent damping term. The main novelty of our models is that we do not assume any explicit aggregative force acting on the individuals; instead, aggregation is obtained exclusively by reducing the individual stochasticity in response to higher perceived density. We formally derive the corresponding mean-field limits, leading to nonlocal degenerate diffusions. Then, we carry out the mathematical analysis of the first-order model, in particular, we prove the existence of weak solutions and show that it allows for measure-valued steady states. We also perform linear stability analysis and identify conditions for pattern formation. Moreover, we discuss the role of the nonlocality for well-posedness of the first-order model. Finally, we present results of numerical simulations for both the first- and second-order model on the individual-based and continuum levels of description. 2012 Elsevier B.V. All rights reserved. 8. Real-space renormalized dynamical mean field theory Kubota, Dai; Sakai, Shiro; Imada, Masatoshi 2016-05-01 We propose real-space renormalized dynamical mean field theory (rr-DMFT) to deal with large clusters in the framework of a cluster extension of the DMFT. In the rr-DMFT, large clusters are decomposed into multiple smaller clusters through a real-space renormalization. In this work, the renormalization effect is taken into account only at the lowest order with respect to the intercluster coupling, which nonetheless reproduces exactly both the noninteracting and atomic limits. Our method allows us large cluster-size calculations which are intractable with the conventional cluster extensions of the DMFT with impurity solvers, such as the continuous-time quantum Monte Carlo and exact diagonalization methods. We benchmark the rr-DMFT for the two-dimensional Hubbard model on a square lattice at and away from half filling, where the spatial correlations play important roles. Our results on the spin structure factor indicate that the growth of the antiferromagnetic spin correlation is taken into account beyond the decomposed cluster size. We also show that the self-energy obtained from the large-cluster solver is reproduced by our method better than the solution obtained directly for the smaller cluster. When applied to the Mott metal-insulator transition, the rr-DMFT is able to reproduce the reduced critical value for the Coulomb interaction comparable to the large cluster result. 9. Combining Few-Body Cluster Structures with Many-Body Mean-Field Methods Hove, D.; Garrido, E.; Jensen, A. S.; Sarriguren, P.; Fynbo, H. O. U.; Fedorov, D. V.; Zinner, N. T. 2017-03-01 Nuclear cluster physics implicitly assumes a distinction between groups of degrees-of-freedom, that is the (frozen) intrinsic and (explicitly treated) relative cluster motion. We formulate a realistic and practical method to describe the coupled motion of these two sets of degrees-of-freedom. We derive a coupled set of differential equations for the system using the phenomenologically adjusted effective in-medium Skyrme type of nucleon-nucleon interaction. We select a two-nucleon plus core system where the mean-field approximation corresponding to the Skyrme interaction is used for the core. A hyperspherical adiabatic expansion of the Faddeev equations is used for the relative cluster motion. We shall specifically compare both the structure and the decay mechanism found from the traditional three-body calculations with the result using the new boundary condition provided by the full microscopic structure at small distance. The extended Hilbert space guaranties an improved wave function compared to both mean-field and three-body solutions. We shall investigate the structures and decay mechanism of ^{22}C (^{20}C+n+n). In conclusion, we have developed a method combining nuclear few- and many-body techniques without losing the descriptive power of each approximation at medium-to-large distances and small distances respectively. The coupled set of equations are solved self-consistently, and both structure and dynamic evolution are studied. 10. Mean field lattice model for adsorption isotherms in zeolite NaA Ayappa, K. G.; Kamala, C. R.; Abinandanan, T. A. 1999-05-01 Using a lattice model for adsorption in microporous materials, pure component adsorption isotherms are obtained within a mean field approximation for methane at 300 K and xenon at 300 and 360 K in zeolite NaA. It is argued that the increased repulsive adsorbate-adsorbate interactions at high coverages must play an important role in determining the adsorption behavior. Therefore, this feature is incorporated through a "coverage-dependent interaction" model, which introduces a free, adjustable parameter. Another important feature, the site volume reduction, has been treated in two ways: a van der Waal model and a 1D hard-rod theory [van Tassel et al., AIChE J. 40, 925 (1994)]; we have also generalized the latter to include all possible adsorbate overlap scenarios. In particular, the 1D hard-rod model, with our coverage-dependent interaction model, is shown to be in best quantitative agreement with the previous grand canonical Monte Carlo isotherms. The expressions for the isosteric heats of adsorption indicate that attractive and repulsive adsorbate-adsorbate interactions increase and decrease the heats of adsorption, respectively. It is concluded that within the mean field approximation, our simple model for repulsive interactions and the 1D hard-rod model for site volume reduction are able to capture most of the important features of adsorption in confined regions. 11. Relativistic Quantum Communication Hosler, Dominic 2013-01-01 In this Ph.D. thesis, I investigate the communication abilities of non-inertial observers and the precision to which they can measure parametrized states. I introduce relativistic quantum field theory with field quantisation, and the definition and transformations of mode functions in Minkowski, Schwarzschild and Rindler spaces. I introduce information theory by discussing the nature of information, defining the entropic information measures, and highlighting the differences between classical and quantum information. I review the field of relativistic quantum information. We investigate the communication abilities of an inertial observer to a relativistic observer hovering above a Schwarzschild black hole, using the Rindler approximation. We compare both classical communication and quantum entanglement generation of the state merging protocol, for both the single and dual rail encodings. We find that while classical communication remains finite right up to the horizon, the quantum entanglement generation tend... 12. Maximizing Influence in an Ising Network: A Mean-Field Optimal Solution Lynn, Christopher 2016-01-01 The problem of influence maximization in social networks has typically been studied in the context of contagion models and irreversible processes. In this paper, we consider an alternate model that treats individual opinions as spins in an Ising network at dynamic equilibrium. We formalize the Ising influence maximization (IIM) problem, which has a physical interpretation as the maximization of the magnetization given a budget of external magnetic field. Under the mean-field (MF) approximation, we develop a number of sufficient conditions for when the problem is convex and exactly solvable, and we provide a gradient ascent algorithm that efficiently achieves an$\\epsilon$-approximation to the optimal solution. We show that optimal strategies exhibit a phase transition from focusing influence on high-degree individuals at high interaction strengths to spreading influence among low-degree individuals at low interaction strengths. We also establish a number of novel results about the structure of steady-states i... 13. On the genesis of spike-wave oscillations in a mean-field model of human thalamic and corticothalamic dynamics Rodrigues, Serafim [Department of Mathematical Sciences, Loughborough University, Leicestershire, LE11 3TU (United Kingdom); Terry, John R. [Department of Mathematical Sciences, Loughborough University, Leicestershire, LE11 3TU (United Kingdom)]. E-mail: [email protected]; Breakspear, Michael [Black Dog Institute, Randwick, NSW 2031 (Australia); School of Psychiatry, UNSW, NSW 2030 (Australia) 2006-07-10 In this Letter, the genesis of spike-wave activity-a hallmark of many generalized epileptic seizures-is investigated in a reduced mean-field model of human neural activity. Drawing upon brain modelling and dynamical systems theory, we demonstrate that the thalamic circuitry of the system is crucial for the generation of these abnormal rhythms, observing that the combination of inhibition from reticular nuclei and excitation from the cortical signal, interplay to generate the spike-wave oscillation. The mechanism revealed provides an explanation of why approaches based on linear stability and Heaviside approximations to the activation function have failed to explain the phenomena of spike-wave behaviour in mean-field models. A mathematical understanding of this transition is a crucial step towards relating spiking network models and mean-field approaches to human brain modelling. 14. Relativistic impulse approach for proton elastic scattering with sup 5 sup 8 Ni and sup 1 sup 2 sup 0 Sn at E sub p =200, 300 and 400 MeV Kaki, K 2001-01-01 We calculate proton elastic scattering with sup 5 sup 8 Ni and sup 1 sup 2 sup 0 Sn at various intermediate energies with relativistic impulse approximation (RIA). We use the ground-state wave functions of the relativistic mean-field (RMF) calculation with the use of the TMA parameter set. We found good agreement with experimental data for the elastic scattering observables. In addition to the standard scalar and vector densities, we include also the tensor density. We study the Pauli effect and the vacuum polarization effect on the elastic scattering observables using the prescription of Horowitz and Serot at lower energy. 15. Conservation in two-particle self-consistent extensions of dynamical mean-field theory Krien, Friedrich; van Loon, Erik G. C. P.; Hafermann, Hartmut; Otsuki, Junya; Katsnelson, Mikhail I.; Lichtenstein, Alexander I. 2017-08-01 Extensions of dynamical mean-field theory (DMFT) make use of quantum impurity models as nonperturbative and exactly solvable reference systems which are essential to treat the strong electronic correlations. Through the introduction of retarded interactions on the impurity, these approximations can be made two-particle self-consistent. This is of interest for the Hubbard model because it allows to suppress the antiferromagnetic phase transition in two dimensions in accordance with the Mermin-Wagner theorem, and to include the effects of bosonic fluctuations. For a physically sound description of the latter, the approximation should be conserving. In this paper, we show that the mutual requirements of two-particle self-consistency and conservation lead to fundamental problems. For an approximation that is two-particle self-consistent in the charge and longitudinal spin channels, the double occupancy of the lattice and the impurity is no longer consistent when computed from single-particle properties. For the case of self-consistency in the charge and longitudinal as well as transversal spin channels, these requirements are even mutually exclusive so that no conserving approximation can exist. We illustrate these findings for a two-particle self-consistent and conserving DMFT approximation. 16. Generalized One-Dimensional Point Interaction in Relativistic and Non-relativistic Quantum Mechanics Shigehara, T; Mishima, T; Cheon, T; Cheon, Taksu 1999-01-01 We first give the solution for the local approximation of a four parameter family of generalized one-dimensional point interactions within the framework of non-relativistic model with three neighboring$\\delta$functions. We also discuss the problem within relativistic (Dirac) framework and give the solution for a three parameter family. It gives a physical interpretation for so-called high energy substantially differ between non-relativistic and relativistic cases. 17. Relativistic corrections to molecular dynamic dipole polarizabilities Kirpekar, Sheela; Oddershede, Jens; Jensen, Hans Jørgen Aagaard 1995-01-01 Using response function methods we report calculations of the dynamic isotropic polarizability of SnH4 and PbH4 and of the relativistic corrections to it in the random phase approximation and at the correlated multiconfigurational linear response level of approximation. All relativistic corrections... 18. Relativistic calculation of nuclear magnetic shielding tensor using the regular approximation to the normalized elimination of the small component. III. Introduction of gauge-including atomic orbitals and a finite-size nuclear model Hamaya, S.; Maeda, H.; Funaki, M.; Fukui, H. 2008-12-01 The relativistic calculation of nuclear magnetic shielding tensors in hydrogen halides is performed using the second-order regular approximation to the normalized elimination of the small component (SORA-NESC) method with the inclusion of the perturbation terms from the metric operator. This computational scheme is denoted as SORA-Met. The SORA-Met calculation yields anisotropies, Δσ =σ∥-σ⊥, for the halogen nuclei in hydrogen halides that are too small. In the NESC theory, the small component of the spinor is combined to the large component via the operator σ⃗ṡπ⃗U/2c, in which π⃗=p⃗+A⃗, U is a nonunitary transformation operator, and c ≅137.036 a.u. is the velocity of light. The operator U depends on the vector potential A⃗ (i.e., the magnetic perturbations in the system) with the leading order c-2 and the magnetic perturbation terms of U contribute to the Hamiltonian and metric operators of the system in the leading order c-4. It is shown that the small Δσ for halogen nuclei found in our previous studies is related to the neglect of the U(0,1) perturbation operator of U, which is independent of the external magnetic field and of the first order with respect to the nuclear magnetic dipole moment. Introduction of gauge-including atomic orbitals and a finite-size nuclear model is also discussed. 19. Coagulation kinetics beyond mean field theory using an optimised Poisson representation. Burnett, James; Ford, Ian J 2015-05-21 Binary particle coagulation can be modelled as the repeated random process of the combination of two particles to form a third. The kinetics may be represented by population rate equations based on a mean field assumption, according to which the rate of aggregation is taken to be proportional to the product of the mean populations of the two participants, but this can be a poor approximation when the mean populations are small. However, using the Poisson representation, it is possible to derive a set of rate equations that go beyond mean field theory, describing pseudo-populations that are continuous, noisy, and complex, but where averaging over the noise and initial conditions gives the mean of the physical population. Such an approach is explored for the simple case of a size-independent rate of coagulation between particles. Analytical results are compared with numerical computations and with results derived by other means. In the numerical work, we encounter instabilities that can be eliminated using a suitable "gauge" transformation of the problem [P. D. Drummond, Eur. Phys. J. B 38, 617 (2004)] which we show to be equivalent to the application of the Cameron-Martin-Girsanov formula describing a shift in a probability measure. The cost of such a procedure is to introduce additional statistical noise into the numerical results, but we identify an optimised gauge transformation where this difficulty is minimal for the main properties of interest. For more complicated systems, such an approach is likely to be computationally cheaper than Monte Carlo simulation. 20. Non-perturbative heterogeneous mean-field approach to epidemic spreading in complex networks Gomez, Sergio; Moreno, Yamir; Arenas, Alex 2011-01-01 Since roughly a decade ago, network science has focused among others on the problem of how the spreading of diseases depends on structural patterns. Here, we contribute to further advance our understanding of epidemic spreading processes by proposing a non-perturbative formulation of the heterogeneous mean field approach that has been commonly used in the physics literature to deal with this kind of spreading phenomena. The non-perturbative equations we propose have no assumption about the proximity of the system to the epidemic threshold, nor any linear approximation of the dynamics. In particular, we first develop a probabilistic description at the node level of the epidemic propagation for the so-called susceptible-infected-susceptible family of models, and after we derive the corresponding heterogeneous mean-field approach. We propose to use the full extension of the approach instead of pruning the expansion to first order, which leads to a non-perturbative formulation that can be solved by fixed point it... 1. Elementary proof of convergence to the mean-field model for the SIR process. Armbruster, Benjamin; Beck, Ekkehard 2016-12-21 The susceptible-infected-recovered (SIR) model has been used extensively to model disease spread and other processes. Despite the widespread usage of this ordinary differential equation (ODE) based model which represents the mean-field approximation of the underlying stochastic SIR process on contact networks, only few rigorous approaches exist and these use complex semigroup and martingale techniques to prove that the expected fraction of the susceptible and infected nodes of the stochastic SIR process on a complete graph converges as the number of nodes increases to the solution of the mean-field ODE model. Extending the elementary proof of convergence for the SIS process introduced by Armbruster and Beck (IMA J Appl Math, doi: 10.1093/imamat/hxw010 , 2016) to the SIR process, we show convergence using only a system of three ODEs, simple probabilistic inequalities, and basic ODE theory. Our approach can also be generalized to many other types of compartmental models (e.g., susceptible-infected-recovered-susceptible (SIRS)) which are linear ODEs with the addition of quadratic terms for the number of new infections similar to the SI term in the SIR model. 2. Quantum de Finetti theorems and mean-field theory from quantum phase space representations Trimborn, F.; Werner, R. F.; Witthaut, D. 2016-04-01 We introduce the number-conserving quantum phase space description as a versatile tool to address fundamental aspects of quantum many-body systems. Using phase space methods we prove two alternative versions of the quantum de Finetti theorem for finite-dimensional bosonic quantum systems, which states that a reduced density matrix of a many-body quantum state can be approximated by a convex combination of product states where the error is proportional to the inverse particle number. This theorem provides a formal justification for the mean-field description of many-body quantum systems, as it shows that quantum correlations can be neglected for the calculation of few-body observables when the particle number is large. Furthermore we discuss methods to derive the exact evolution equations for quantum phase space distribution functions as well as upper and lower bounds for the ground state energy. As an important example, we consider the Bose-Hubbard model and show that the mean-field dynamics is given by a classical phase space flow equivalent to the discrete Gross-Pitaevskii equation. 3. Effects of anisotropy of turbulent convection in mean-field solar dynamo models Pipin, V V 2013-01-01 We study how anisotropy of turbulent convection affects diffusion of large-scale magnetic fields and the dynamo process on the Sun. The effect of anisotropy is calculated in a mean-field magneto-hydrodynamics framework using the minimal$\\tau$-approximation. We examine two types of mean-field dynamo models: the well-known benchmark flux-transport model, and a distributed-dynamo model with the subsurface rotational shear layer. For both models we investigate effects of the double-cell meridional circulation, recently suggested by helioseismology. We introduce a parameter of anisotropy as a ratio of the radial and horizontal intensity of turbulent mixing, to characterize the anisotropy effects. It is found that the anisotropy of turbulent convection affects the distribution of magnetic fields inside the convection zone. The concentration of the magnetic flux near the bottom and top boundaries of the convection zone is greater when the anisotropy is stronger. It is shown that the critical dynamo number and the d... 4. Stable oscillations of a predator-prey probabilistic cellular automaton: a mean-field approach Tome, Tania; Carvalho, Kelly C de [Instituto de FIsica, Universidade de Sao Paulo, Caixa Postal 66318, 05315-970 Sao Paulo (Brazil) 2007-10-26 We analyze a probabilistic cellular automaton describing the dynamics of coexistence of a predator-prey system. The individuals of each species are localized over the sites of a lattice and the local stochastic updating rules are inspired by the processes of the Lotka-Volterra model. Two levels of mean-field approximations are set up. The simple approximation is equivalent to an extended patch model, a simple metapopulation model with patches colonized by prey, patches colonized by predators and empty patches. This approximation is capable of describing the limited available space for species occupancy. The pair approximation is moreover able to describe two types of coexistence of prey and predators: one where population densities are constant in time and another displaying self-sustained time oscillations of the population densities. The oscillations are associated with limit cycles and arise through a Hopf bifurcation. They are stable against changes in the initial conditions and, in this sense, they differ from the Lotka-Volterra cycles which depend on initial conditions. In this respect, the present model is biologically more realistic than the Lotka-Volterra model. 5. Resonances and reactions from mean-field dynamics Stevenson P. D. 2016-01-01 Full Text Available The time-dependent version of nuclear density functional theory, using functionals derived from Skyrme interactions, is able to approximately describe nuclear dynamics. We present time-dependent results of calculations of dipole resonances, concentrating on excitations of valence neutrons against a proton plus neutron core in the neutron-rich doubly-magic 132Sn nucleus, and results of collision dynamics, highlighting potential routes to ternary fusion, with the example of a collision of 48Ca+48Ca+208Pb resulting in a compound nucleus of element 120 stable against immediate fission. 6. Relativistic hydrodynamics Luciano, Rezzolla 2013-01-01 Relativistic hydrodynamics is a very successful theoretical framework to describe the dynamics of matter from scales as small as those of colliding elementary particles, up to the largest scales in the universe. This book provides an up-to-date, lively, and approachable introduction to the mathematical formalism, numerical techniques, and applications of relativistic hydrodynamics. The topic is typically covered either by very formal or by very phenomenological books, but is instead presented here in a form that will be appreciated both by students and researchers in the field. The topics covered in the book are the results of work carried out over the last 40 years, which can be found in rather technical research articles with dissimilar notations and styles. The book is not just a collection of scattered information, but a well-organized description of relativistic hydrodynamics, from the basic principles of statistical kinetic theory, down to the technical aspects of numerical methods devised for the solut... 7. Mean-field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo Schrinner, M; Schmitt, D; Rheinhardt, M; Christensen, U R 2006-01-01 A comparison is made between mean-field models and direct numerical simulations of rotating magnetoconvection and the geodynamo. The mean-field coefficients are calculated with the fluid velocity taken from the direct numerical simulations. The magnetic fields resulting from mean-field models are then compared with the mean magnetic field from the direct numerical simulations. 8. Diabatic Mean-Field Description of Rotational Bands in Terms of the Selfconsistent Collective Coordinate Method Shimizu, Y R; Shimizu, Yoshifumi R.; Matsuyanagi, Kenichi 2000-01-01 Diabatic description of rotational bands provides a clear-cut picture for understanding the back-bending phenomena, where the internal structure of the yrast band changes dramatically as a function of angular momentum. A microscopic framework to obtain the diabatic bands within the mean-field approximation is presented by making use of the selfconsistent collective coordinate method. Applying the framework, both the ground state rotational bands and the Stockholm bands are studied systematically for the rare-earth deformed nuclei. An overall agreement has been achieved between the calculated and observed rotational spectra. It is also shown that the inclusion of the double-stretched quadrupole-pairing interaction is crucial to obtain an overall agreement for the even-odd mass differences and the rotational spectra simultaneously. 9. Second-order corrections to mean field evolution for weakly interacting Bosons. I Grillakis, Manoussos G; Margetis, Dionisios 2009-01-01 Inspired by the works of Rodnianski and Schlein and Wu, we derive a new nonlinear Schr\\"odinger equation that describes a second-order correction to the usual tensor product (mean-field) approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation. We show that our new equation, if it has solutions with appropriate smoothness and decay properties, implies a new Fock space estimate. We also show that for an interaction potential$v(x)= \\epsilon \\chi(x) |x|^{-1}$, where$\\epsilon$is sufficiently small and$\\chi \\in C_0^{\\infty}$, our program can be easily implemented locally in time. We leave global in time issues, more singular potentials and sophisticated estimates for a subsequent part (part II) of this paper. 10. Mean-field state population study for iron-based superconductors Wang, Zhigang; Fu, Zhen-Guo; Zheng, Fa-Wei; Zhang, Ping 2017-02-01 The occupation number distribution in momentum space are theoretically studied within a two-orbital model, which can be unified describing the low-energy physics of the iron pnictides and iron chalcogenides. The mean-field approximation of Hubbard interaction is employed. By tuning the hopping parameters, the difference between the iron pnictides and iron chalcogenides in their occupation number distribution behavior can be clearly observed. The results show that when the pairing interaction tends to zero, the occupation number n (k) ≈ 0 at Γ point for iron chalcogenides while n (k) ≈ 2 at Γ point for iron pnictides. By increasing the strength of the pairing interaction to a large value, the change of n (k) at Γ point for iron chalcogenides (pnictides) is remarkable (unremarkable). In addition, we find that the effect of the nearest-neighbor coupling between the two layers, contained in the S4 model [Hu and Hao, (2012) [33 11. β-decay of magic nuclei: Beyond mean-field description Niu, Yifei, E-mail: [email protected] [Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621900 (China); INFN, Sezione di Milano, via Celoria 16, I-20133 Milano (Italy); Niu, Zhongming [School of Physics and Material Science, Anhui University, Hefei 230601 (China); Colò, Gianluca [Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, I-20133 Milano (Italy); INFN, Sezione di Milano, via Celoria 16, I-20133 Milano (Italy); Vigezzi, Enrico [INFN, Sezione di Milano, via Celoria 16, I-20133 Milano (Italy) 2015-10-15 Nuclear β-decay plays an important role not only in nuclear physics but also in astrophysics. The widely used self-consistent Random Phase Approximation (RPA) models tend to overestimate the half-lives of magic nuclei. To overcome this problem, we go beyond the mean-field description and include the effects of particle-vibration coupling (PVC) on top of the RPA model. The β-decay half-lives of {sup 34}Si, {sup 68}Ni, {sup 78}Ni, and {sup 132}Sn are studied within this approach in the case of the Skyrme interaction SkM*. It is found that the low-lying Gamow-Teller (GT) strength is shifted downwards with the inclusion of the PVC effect, and as a consequence, the half-lives are reduced due to the increase of the phase space available for β-decay, which leads to a good agreement between theoretical and experimental lifetimes. 12. Hall current effects in mean-field dynamo theory Lingam, Manasvi 2016-01-01 The role of the Hall term on large scale dynamo action is investigated by means of the First Order Smoothing Approximation. It is shown that the standard$\\alpha$coefficient is altered, and is zero when a specific double Beltrami state is attained, in contrast to the Alfv\\'enic state for MHD dynamos. The$\\beta$coefficient is no longer positive definite, and thereby enables dynamo action even if$\\alpha$-quenching were to operate. The similarities and differences with the (magnetic) shear-current effect are pointed out, and a mechanism that may be potentially responsible for$\\beta < 0$is advanced. The results are compared against previous studies, and their astrophysical relevance is also highlighted. 13. Slave-boson mean-field theory versus variational-wave-function approach for the periodic Anderson model Yang, Min-Fong; Sun, Shih-Jye; Hong, Tzay-Ming 1993-12-01 We show that a special kind of slave-boson mean-field approximation, which allows for the symmetry-broken states appropriate for a bipartite lattice, can give essentially the same results as those by the variational-wave-function approach proposed by Gula´csi, Strack, and Vollhardt [Phys. Rev. B 47, 8594 (1993)]. The advantages of our approach are briefly discussed. 14. Relativistic Kinematics Sahoo, Raghunath 2016-01-01 This lecture note covers Relativistic Kinematics, which is very useful for the beginners in the field of high-energy physics. A very practical approach has been taken, which answers "why and how" of the kinematics useful for students working in the related areas. 15. Mean-field dynamo in a turbulence with shear and kinetic helicity fluctuations. Kleeorin, Nathan; Rogachevskii, Igor 2008-03-01 We study the effects of kinetic helicity fluctuations in a turbulence with large-scale shear using two different approaches: the spectral tau approximation and the second-order correlation approximation (or first-order smoothing approximation). These two approaches demonstrate that homogeneous kinetic helicity fluctuations alone with zero mean value in a sheared homogeneous turbulence cannot cause a large-scale dynamo. A mean-field dynamo is possible when the kinetic helicity fluctuations are inhomogeneous, which causes a nonzero mean alpha effect in a sheared turbulence. On the other hand, the shear-current effect can generate a large-scale magnetic field even in a homogeneous nonhelical turbulence with large-scale shear. This effect was investigated previously for large hydrodynamic and magnetic Reynolds numbers. In this study we examine the threshold required for the shear-current dynamo versus Reynolds number. We demonstrate that there is no need for a developed inertial range in order to maintain the shear-current dynamo (e.g., the threshold in the Reynolds number is of the order of 1). 16. The standard mean-field treatment of inter-particle attraction in classical DFT is better than one might expect Archer, Andrew J.; Chacko, Blesson; Evans, Robert 2017-07-01 In classical density functional theory (DFT), the part of the Helmholtz free energy functional arising from attractive inter-particle interactions is often treated in a mean-field or van der Waals approximation. On the face of it, this is a somewhat crude treatment as the resulting functional generates the simple random phase approximation (RPA) for the bulk fluid pair direct correlation function. We explain why using standard mean-field DFT to describe inhomogeneous fluid structure and thermodynamics is more accurate than one might expect based on this observation. By considering the pair correlation function g(x) and structure factor S(k) of a one-dimensional model fluid, for which exact results are available, we show that the mean-field DFT, employed within the test-particle procedure, yields results much superior to those from the RPA closure of the bulk Ornstein-Zernike equation. We argue that one should not judge the quality of a DFT based solely on the approximation it generates for the bulk pair direct correlation function. 17. Global study of beyond-mean-field correlation energies in covariant energy density functional theory using a collective Hamiltonian method Lu, K Q; Li, Z P; Yao, J M; Meng, J 2015-01-01 We report the first global study of dynamic correlation energies (DCEs) associated with rotational motion and quadrupole shape vibrational motion in a covariant energy density functional (CEDF) for 575 even-even nuclei with proton numbers ranging from$Z=8$to$Z=108$by solving a five-dimensional collective Hamiltonian, the collective parameters of which are determined from triaxial relativistic mean-field plus BCS calculation using the PC-PK1 force. After taking into account these beyond mean-field DCEs, the root-mean-square (rms) deviation with respect to nuclear masses is reduced significantly down to 1.14 MeV, which is smaller than those of other successful CEDFs: NL3* (2.96 MeV), DD-ME2 (2.39 MeV), DD-ME$\\delta$(2.29 MeV) and DD-PC1 (2.01 MeV). Moreover, the rms deviation for two-nucleon separation energies is reduced by$\\sim34\\%$in comparison with cranking prescription. 18. Mean-field approach to collective excitations in deformed sd-shell nuclei using realistic interactions Erler, Bastian 2012-07-18 Realistic nucleon-nucleon interactions transformed via the Unitary Correlation Operator Method (UCOM) or the Similarity Renormalization Group (SRG) have proven to be a suitable starting point for the description of closed-shell nuclei via mean-field methods like Hartree-Fock (HF). This allows the treatment of a number of heavy nuclei with realistic nucleon-nucleon interactions, which would otherwise only be possible with phenomenological interactions. To include three-nucleon forces in an approximate way, the UCOM or SRG transformed interactions can be augmented by a three-body contact interaction, which is necessary to reproduce measured charge radii. However, many interesting nuclei, including those near the neutron drip line, are far away from closed shells. These nuclei are of great importance for modeling nucleosynthesis processes in the universe, but experiments can only be performed at a few research facilities. In this work, the Hartree Fock (HF) approach with realistic interactions is extended to light deformed nuclei. Pairing correlations are not taken into account. A crucial step in this process is to allow deformed ground states on the mean-field level, as only nuclei with at least one closed shell can be described with spherical HF ground states. To restore the rotational symmetry in the lab frame, exact angular-momentum projection (AMP) is implemented. Constrained HF calculations are used for an approximate variation after projection approach. The AMP-HF description of open-shell nuclei is on par with the pure HF description of closed-shell nuclei. Charge-radii and systematics of binding energies agree well with experiment. However, missing correlations, lead to an underestimated absolute value of the binding energy. Projection on higher angular momenta approximately reproduces the energy systematics of rotational bands. To describe collective excitations, the Random Phase Approximation (RPA) constitutes a well tested approach, which can also be 19. Effective nucleon mass, incompressibility, and third derivative of nuclear binding energy in the nonlinear relativistic mean field theory Kouno, H.; Kakuta, N.; Noda, N.; Koide, K.; Mitsumori, T.; Hasegawa, A.; Nakano, M. (Department of Physics, Saga University, Saga 840 (Japan)) 1995-04-01 We have studied the equations of state of nuclear matter using the nonlinear [sigma]-[omega] model. At the normal density, there is a strong correlation among the effective nucleon mass [ital M][sub 0][sup *], the incompressibility, [ital K] and the third derivative [ital K][prime] of binding energy. The results are compared with the empirical analysis of the giant isoscalar monopole resonances data. It is difficult to fit the data when [ital K][approx lt]200 MeV, using the model. It is also found that [ital K]=300[plus minus]50 MeV is favorable to account for the volume-symmetry properties of nuclear matter. 20. Alpha-decay chains of$^{288}_{173}115$and$^{287}_{172}115$in the Relativistic Mean Field theory Geng, L S; Meng, J 2003-01-01 The results of experiments designed to synthesize element 115 in the$^{243}$Am+$^{48}$Ca reaction are reported at Dubna in Russia \\cite{ogan.03}. With a beam dose of$4.3\\times 10^{18}$248-MeV$^{48}$Ca projectiles, three similar decay chains consisting of five consecutive$\\alpha$-decays are observed. At a higher bombarding energy of 253 MeV, with an equal$^{48}$Ca beam dose, a different decay chain of four consecutive$\\alpha$-decays are detected. The decay properties of these synthesized nuclei are consistent with consecutive$\\alpha$-decay originating from the parent isotopes of the new element 115,$^{288}115$and$^{287}115$, respectively. In the present work, the recently developed deformed RMF+BCS method with a density-independent delta-function interaction in the pairing channel is applied to the analysis of these newly synthesized superheavy nuclei$^{288}115$,$^{287}115$, and their$\\alpha$-decay daughter nuclei. The calculated$\\alpha$-decay energies and half-lives agree well with the experime... 1. Magnetic Field Line Random Walk in Isotropic Turbulence with Varying Mean Field Sonsrettee, W.; Subedi, P.; Ruffolo, D.; Matthaeus, W. H.; Snodin, A. P.; Wongpan, P.; Chuychai, P.; Rowlands, G.; Vyas, S. 2016-08-01 In astrophysical plasmas, the magnetic field line random walk (FLRW) plays an important role in guiding particle transport. The FLRW behavior is scaled by the Kubo number R=(b/{B}0)({{\\ell }}\\parallel /{{\\ell }}\\perp ) for rms magnetic fluctuation b, large-scale mean field {{\\boldsymbol{B}}}0, and coherence scales parallel ({{\\ell }}\\parallel ) and perpendicular ({{\\ell }}\\perp ) to {{\\boldsymbol{B}}}0. Here we use a nonperturbative analytic framework based on Corrsin’s hypothesis, together with direct computer simulations, to examine the R-scaling of the FLRW for varying B 0 with finite b and isotropic fluctuations with {{\\ell }}\\parallel /{{\\ell }}\\perp =1, instead of the well-studied route of varying {{\\ell }}\\parallel /{{\\ell }}\\perp for b \\ll {B}0. The FLRW for isotropic magnetic fluctuations is also of astrophysical interest regarding transport processes in the interstellar medium. With a mean field, fluctuations may have variance anisotropy, so we consider limiting cases of isotropic variance and transverse variance (with b z = 0). We obtain analytic theories, and closed-form solutions for extreme cases. Padé approximants are provided to interpolate all versions of theory and simulations to any B 0. We demonstrate that, for isotropic turbulence, Corrsin-based theories generally work well, and with increasing R there is a transition from quasilinear to Bohm diffusion. This holds even with b z = 0, when different routes to R\\to ∞ are mathematically equivalent; in contrast with previous studies, we find that a Corrsin-based theory with random ballistic decorrelation works well even up to R = 400, where the effects of trapping are barely perceptible in simulation results. 2. Active matter beyond mean-field: ring-kinetic theory for self-propelled particles. Chou, Yen-Liang; Ihle, Thomas 2015-02-01 Recently, Hanke et al. [Phys. Rev. E 88, 052309 (2013)] showed that mean-field kinetic theory fails to describe collective motion in soft active colloids and that correlations must not be neglected. Correlation effects are also expected to be essential in systems of biofilaments driven by molecular motors and in swarms of midges. To obtain correlations in an active matter system from first principles, we derive a ring-kinetic theory for Vicsek-style models of self-propelled agents from the exact N-particle evolution equation in phase space. The theory goes beyond mean-field and does not rely on Boltzmann's approximation of molecular chaos. It can handle precollisional correlations and cluster formation, which are both important to understand the phase transition to collective motion. We propose a diagrammatic technique to perform a small-density expansion of the collision operator and derive the first two equations of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. An algorithm is presented that numerically solves the evolution equation for the two-particle correlations on a lattice. Agent-based simulations are performed and informative quantities such as orientational and density correlation functions are compared with those obtained by ring-kinetic theory. Excellent quantitative agreement between simulations and theory is found at not-too-small noises and mean free paths. This shows that there are parameter ranges in Vicsek-like models where the correlated closure of the BBGKY hierarchy gives correct and nontrivial results. We calculate the dependence of the orientational correlations on distance in the disordered phase and find that it seems to be consistent with a power law with an exponent around -1.8, followed by an exponential decay. General limitations of the kinetic theory and its numerical solution are discussed. 3. Coagulation kinetics beyond mean field theory using an optimised Poisson representation Burnett, James [Department of Mathematics, UCL, Gower Street, London WC1E 6BT (United Kingdom); Ford, Ian J. [Department of Physics and Astronomy, UCL, Gower Street, London WC1E 6BT (United Kingdom) 2015-05-21 Binary particle coagulation can be modelled as the repeated random process of the combination of two particles to form a third. The kinetics may be represented by population rate equations based on a mean field assumption, according to which the rate of aggregation is taken to be proportional to the product of the mean populations of the two participants, but this can be a poor approximation when the mean populations are small. However, using the Poisson representation, it is possible to derive a set of rate equations that go beyond mean field theory, describing pseudo-populations that are continuous, noisy, and complex, but where averaging over the noise and initial conditions gives the mean of the physical population. Such an approach is explored for the simple case of a size-independent rate of coagulation between particles. Analytical results are compared with numerical computations and with results derived by other means. In the numerical work, we encounter instabilities that can be eliminated using a suitable “gauge” transformation of the problem [P. D. Drummond, Eur. Phys. J. B 38, 617 (2004)] which we show to be equivalent to the application of the Cameron-Martin-Girsanov formula describing a shift in a probability measure. The cost of such a procedure is to introduce additional statistical noise into the numerical results, but we identify an optimised gauge transformation where this difficulty is minimal for the main properties of interest. For more complicated systems, such an approach is likely to be computationally cheaper than Monte Carlo simulation. 4. Gravitation relativiste Hakim, Rémi 1994-01-01 Il existe à l'heure actuelle un certain nombre de théories relativistes de la gravitation compatibles avec l'expérience et l'observation. Toutefois, la relativité générale d'Einstein fut historiquement la première à fournir des résultats théoriques corrects en accord précis avec les faits. 5. Relativistic Astrophysics Jones, Bernard J. T.; Markovic, Dragoljub 1997-06-01 Preface; Prologue: Conference overview Bernard Carr; Part I. The Universe At Large and Very Large Redshifts: 2. The size and age of the Universe Gustav A. Tammann; 3. Active galaxies at large redshifts Malcolm S. Longair; 4. Observational cosmology with the cosmic microwave background George F. Smoot; 5. Future prospects in measuring the CMB power spectrum Philip M. Lubin; 6. Inflationary cosmology Michael S. Turner; 7. The signature of the Universe Bernard J. T. Jones; 8. Theory of large-scale structure Sergei F. Shandarin; 9. The origin of matter in the universe Lev A. Kofman; 10. New guises for cold-dark matter suspects Edward W. Kolb; Part II. Physics and Astrophysics Of Relativistic Compact Objects: 11. On the unification of gravitational and inertial forces Donald Lynden-Bell; 12. Internal structure of astrophysical black holes Werner Israel; 13. Black hole entropy: external facade and internal reality Valery Frolov; 14. Accretion disks around black holes Marek A. Abramowicz; 15. Black hole X-ray transients J. Craig Wheeler; 16. X-rays and gamma rays from active galactic nuclei Roland Svensson; 17. Gamma-ray bursts: a challenge to relativistic astrophysics Martin Rees; 18. Probing black holes and other exotic objects with gravitational waves Kip Thorne; Epilogue: the past and future of relativistic astrophysics Igor D. Novikov; I. D. Novikov's scientific papers and books. 6. Linear$\\Sigma$Model in the Gaussian Functional Approximation Nakamura, I 2001-01-01 We apply a self-consistent relativistic mean-field variational Gaussian functional'' (or Hartree) approximation to the linear$\\sigma$model with spontaneously and explicitly broken chiral O(4) symmetry. We set up the self-consistency, or gap'' and the Bethe-Salpeter equations. We check and confirm the chiral Ward-Takahashi identities, among them the Nambu-Goldstone theorem and the (partial) axial current conservation [CAC], both in and away from the chiral limit. With explicit chiral symmetry breaking we confirm the Dashen relation for the pion mass and partial CAC. We solve numerically the gap and Bethe-Salpeter equations, discuss the solutions' properties and the particle content of the theory. 7. Monte Carlo Mean Field Treatment of Microbunching Instability in the FERMI@Elettra First Bunch Compressor Bassi, G.; /Liverpool U. /Cockroft Inst.; Ellison, J.A.; Heinemann, K.; /New Mexico U.; Warnock, R.; /SLAC 2009-05-07 Bunch compressors, designed to increase the peak current, can lead to a microbunching instability with detrimental effects on the beam quality. This is a major concern for free electron lasers (FELs) where very bright electron beams are required, i.e. beams with low emittance and energy spread. In this paper, we apply our self-consistent, parallel solver to study the microbunching instability in the first bunch compressor system of FERMI{at}Elettra. Our basic model is a 2D Vlasov-Maxwell system. We treat the beam evolution through a bunch compressor using our Monte Carlo mean field approximation. We randomly generate N points from an initial phase space density. We then calculate the charge density using a smooth density estimation procedure, from statistics, based on Fourier series. The electric and magnetic fields are calculated from the smooth charge/current density using a novel field formula that avoids singularities by using the retarded time as a variable of integration. The points are then moved forward in small time steps using the beam frame equations of motion, with the fields frozen during a time step, and a new charge density is determined using our density estimation procedure. We try to choose N large enough so that the charge density is a good approximation to the density that would be obtained from solving the 2D Vlasov-Maxwell system exactly. We call this method the Monte Carlo Particle (MCP) method. 8. Brownian regime of finite-N corrections to particle motion in the XY Hamiltonian mean field model Ribeiro, Bruno V.; Amato, Marco A.; Elskens, Yves 2016-08-01 We study the dynamics of the N-particle system evolving in the XY Hamiltonian mean field (HMF) model for a repulsive potential, when no phase transition occurs. Starting from a homogeneous distribution, particles evolve in a mean field created by the interaction with all others. This interaction does not change the homogeneous state of the system, and particle motion is approximately ballistic with small corrections. For initial particle data approaching a waterbag, it is explicitly proved that corrections to the ballistic velocities are in the form of independent Brownian noises over a time scale diverging not slower than {N}2/5 as N\\to ∞ , which proves the propagation of molecular chaos. Molecular dynamics simulations of the XY-HMF model confirm our analytical findings. 9. Brownian regime of finite-N corrections to particle motion in the XY hamiltonian mean field model Ribeiro, Bruno V; Elskens, Yves 2016-01-01 We study the dynamics of the N-particle system evolving in the XY hamiltonian mean field (HMF) model for a repulsive potential, when no phase transition occurs. Starting from a homogeneous distribution, particles evolve in a mean field created by the interaction with all others. This interaction does not change the homogeneous state of the system, and particle motion is approximately ballistic with small corrections. For initial particle data approaching a waterbag, it is explicitly proved that corrections to the ballistic velocities are in the form of independent brownian noises over a time scale diverging not slower than$N^{2/5}$as$N \\to \\infty$, which proves the propagation of molecular chaos. Molecular dynamics simulations of the XY-HMF model confirm our analytical findings. 10. Exact mean-field theory of ionic solutions: non-Debye screening Varela, Luis M.; García, Manuel; Mosquera, Víctor 2003-07-01 The main aim of this report is to analyze the equilibrium properties of primitive model (PM) ionic solutions in the formally exact mean-field formalism. Previously, we review the main theoretical and numerical results reported throughout the last century for homogeneous (electrolytes) and inhomogeneous (electric double layer, edl) ionic systems, starting with the classical mean-field theory of electrolytes due to Debye and Hückel (DH). In this formalism, the effective potential is derived from the Poisson-Boltzmann (PB) equation and its asymptotic behavior analyzed in the classical Debye theory of screening. The thermodynamic properties of electrolyte solutions are briefly reviewed in the DH formalism. The main analytical and numerical extensions of DH formalism are revised, ranging from the earliest extensions that overcome the linearization of the PB equation to the more sophisticated integral equation techniques introduced after the late 1960s. Some Monte Carlo and molecular dynamic simulations are also reviewed. The potential distributions in an inhomogeneous ionic system are studied in the classical PB framework, presenting the classical Gouy-Chapman (GC) theory of the electric double layer (edl) in a brief manner. The mean-field theory is adequately contextualized using field theoretic (FT) results and it is proven that the classical PB theory is recovered at the Gaussian or one-loop level of the exact FT, and a systematic way to obtain the corrections to the DH theory is derived. Particularly, it is proven following Kholodenko and Beyerlein that corrections to DH theory effectively lead to a renormalization of charges and Debye screening length. The main analytical and numerical results for this non-Debye screening length are reviewed, ranging from asymptotic expansions, self-consistent theory, nonlinear DH results and hypernetted chain (HNC) calculations. Finally, we study the exact mean-field theory of ionic solutions, the so-called dressed-ion theory 11. Exact mean-field theory of ionic solutions: non-Debye screening Varela, L.M.; Garcia, Manuel; Mosquera, Victor 2003-07-01 The main aim of this report is to analyze the equilibrium properties of primitive model (PM) ionic solutions in the formally exact mean-field formalism. Previously, we review the main theoretical and numerical results reported throughout the last century for homogeneous (electrolytes) and inhomogeneous (electric double layer, edl) ionic systems, starting with the classical mean-field theory of electrolytes due to Debye and Hueckel (DH). In this formalism, the effective potential is derived from the Poisson-Boltzmann (PB) equation and its asymptotic behavior analyzed in the classical Debye theory of screening. The thermodynamic properties of electrolyte solutions are briefly reviewed in the DH formalism. The main analytical and numerical extensions of DH formalism are revised, ranging from the earliest extensions that overcome the linearization of the PB equation to the more sophisticated integral equation techniques introduced after the late 1960s. Some Monte Carlo and molecular dynamic simulations are also reviewed. The potential distributions in an inhomogeneous ionic system are studied in the classical PB framework, presenting the classical Gouy-Chapman (GC) theory of the electric double layer (edl) in a brief manner. The mean-field theory is adequately contextualized using field theoretic (FT) results and it is proven that the classical PB theory is recovered at the Gaussian or one-loop level of the exact FT, and a systematic way to obtain the corrections to the DH theory is derived. Particularly, it is proven following Kholodenko and Beyerlein that corrections to DH theory effectively lead to a renormalization of charges and Debye screening length. The main analytical and numerical results for this non-Debye screening length are reviewed, ranging from asymptotic expansions, self-consistent theory, nonlinear DH results and hypernetted chain (HNC) calculations. Finally, we study the exact mean-field theory of ionic solutions, the so-called dressed-ion theory 12. Investigation of Properties of Exotic Nuclei in Non-relativistic and Relativistic Models 2001-01-01 Properties of exotic nuclei are described by non-relativistic and relativistic models. The relativistic mean field theory predicts one proton halo in 26,27,28P and two proton halos in 27,28,29S, recently, one proton halo in 26,27,28P has been found experimentally in MSU lab. The relativistic Hartree-Fock theory has been used to investigate the contribution of Fock term and isovector mesons to the properties of exotic nuclei. It turns out that the influence of the Fock term and isovector mesons on the properties of neutron extremely rich nuclei is very different from that of near stable nuclei. Meanwhile, the deformed Hartree-Fock-Bogoliubov theory has been employed to describe the ground state properties of the isotopes for some light nuclei. 13. Amplitude pattern synthesis for conformal array antennas using mean-field neural networks Castaldi, G.; Gerini, G. 2001-01-01 In this paper, we deal with the synthesis problem of conformai array antennas using a mean-field neural network. We applied a discrete version of mean-field neural network proposed by Vidyasagar [1], This technique is used to find the global minimum of the objective function, which represents the sq 14. Rigorous mean-field dynamics of lattice bosons: quenches from the Mott insulator M. Snoek 2011-01-01 We provide a rigorous derivation of Gutzwiller mean-field dynamics for lattice bosons, showing that it is exact on fully connected lattices. We apply this formalism to quenches in the interaction parameter from the Mott insulator to the superfluid state. Although within mean-field the Mott insulator 15. On Mean-Field Theory of Quantum Phase Transition in Granular Superconductors Simkin, M V 1996-01-01 In previous work on quantum phase transition in granular superconductors, where mean-field theory was used, an assumption was made that the order parameter as a function of the mean field is a convex up function. Though this is not always the case in phase transitions, this assumption must be verified, what is done in this article. 16. Gibbs Properties of the Fuzzy Potts Model on Trees and in Mean Field Häggström, O.; Külske, C. 2004-01-01 We study Gibbs properties of the fuzzy Potts model in the mean field case (i.e. on a complete graph) and on trees. For the mean field case, a complete characterization of the set of temperatures for which non-Gibbsianness happens is given. The results for trees are somewhat less explicit, but we do 17. Mean-field theory of random-site q-state Potts models van Enter, Aernout; Hemmen, Jan Leonard van; Pospiech, C. 1988-01-01 A class of random-site mean-field Potts models is introduced and solved exactly. The bifurcation properties of the resulting mean-field equations are analysed in detail. Particular emphasis is put on the relation between the solutions and the underlying symmetries of the model. It turns out that, in 18. Superconductivity a new approach based on the Bethe-Salpeter equation in the mean-field approximation Malik, G P 2016-01-01 Given the Debye temperature of an elemental superconductor (SC) and its Tc, BCS theory enables one to predict the value of its gap 0 at T = 0, or vice versa. This monograph shows that non-elemental SCs can be similarly dealt with via the generalized BCS equations (GBCSEs) which, given any two parameters of the set {Tc, 10, 20 > 10}, enable one to predict the third. Also given herein are new equations for the critical magnetic field and critical current density of an elemental and a non-elemental SC — equations that are derived directly from those that govern pairing in them. The monograph includes topics that are usually not covered in any one text on superconductivity, e.g., BCS-BEC crossover physics, the long-standing puzzle posed by SrTiO3, and heavy-fermion superconductors — all of which are still imperfectly understood and therefore continue to avidly engage theoreticians. It suggests that addressing the Tcs, s and other properties (e.g., number densities of charge carriers) of high-Tc SCs via GBCSE... 19. Relativistic and non-relativistic geodesic equations Giambo' , R.; Mangiarotti, L.; Sardanashvily, G. [Camerino Univ., Camerino, MC (Italy). Dipt. di Matematica e Fisica 1999-07-01 It is shown that any dynamic equation on a configuration space of non-relativistic time-dependent mechanics is associated with connections on its tangent bundle. As a consequence, every non-relativistic dynamic equation can be seen as a geodesic equation with respect to a (non-linear) connection on this tangent bundle. Using this fact, the relationships between relativistic and non-relativistic equations of motion is studied. 20. Nuclear mean field and double-folding model of the nucleus-nucleus optical potential Khoa, Dao T; Loan, Doan Thi; Loc, Bui Minh 2016-01-01 Realistic density dependent CDM3Yn versions of the M3Y interaction have been used in an extended Hartree-Fock (HF) calculation of nuclear matter (NM), with the nucleon single-particle potential determined from the total NM energy based on the Hugenholtz-van Hove theorem that gives rise naturally to a rearrangement term (RT). Using the RT of the single-nucleon potential obtained exactly at different NM densities, the density- and energy dependence of the CDM3Yn interactions was modified to account properly for both the RT and observed energy dependence of the nucleon optical potential. Based on a local density approximation, the double-folding model of the nucleus-nucleus optical potential has been extended to take into account consistently the rearrangement effect and energy dependence of the nuclear mean-field potential, using the modified CDM3Yn interactions. The extended double-folding model was applied to study the elastic$^{12}$C+$^{12}$C and$^{16}$O+$^{12}$C scattering at the refractive energies, wher... 1. Renormalized parameters and perturbation theory in dynamical mean-field theory for the Hubbard model Hewson, A. C. 2016-11-01 We calculate the renormalized parameters for the quasiparticles and their interactions for the Hubbard model in the paramagnetic phase as deduced from the low-energy Fermi-liquid fixed point using the results of a numerical renormalization-group calculation (NRG) and dynamical mean-field theory (DMFT). Even in the low-density limit there is significant renormalization of the local quasiparticle interaction U ˜, in agreement with estimates based on the two-particle scattering theory of J. Kanamori [Prog. Theor. Phys. 30, 275 (1963), 10.1143/PTP.30.275]. On the approach to the Mott transition we find a finite ratio for U ˜/D ˜ , where 2 D ˜ is the renormalized bandwidth, which is independent of whether the transition is approached by increasing the on-site interaction U or on increasing the density to half filling. The leading ω2 term in the self-energy and the local dynamical spin and charge susceptibilities are calculated within the renormalized perturbation theory (RPT) and compared with the results calculated directly from the NRG-DMFT. We also suggest, more generally from the DMFT, how an approximate expression for the q ,ω spin susceptibility χ (q ,ω ) can be derived from repeated quasiparticle scattering with a local renormalized scattering vertex. 2. Mean-field and Monte Carlo studies of the magnetization-reversal transition in the Ising model Misra, Arkajyoti [Saha Institute of Nuclear Physics, Bidhannagar, Calcutta (India)]. E-mail: [email protected]; Chakrabarti, Bikas K. [Saha Institute of Nuclear Physics, Bidhannagar, Calcutta (India)]. E-mail: [email protected] 2000-06-16 Detailed mean-field and Monte Carlo studies of the dynamic magnetization-reversal transition in the Ising model in its ordered phase under a competing external magnetic field of finite duration have been presented here. An approximate analytical treatment of the mean-field equations of motion shows the existence of diverging length and time scales across this dynamic transition phase boundary. These are also supported by numerical solutions of the complete mean-field equations of motion and the Monte Carlo study of the system evolving under Glauber dynamics in both two and three dimensions. Classical nucleation theory predicts different mechanisms of domain growth in two regimes marked by the strength of the external field, and the nature of the Monte Carlo phase boundary can be comprehended satisfactorily using the theory. The order of the transition changes from a continuous to a discontinuous one as one crosses over from coalescence regime (stronger field) to a nucleation regime (weaker field). Finite-size scaling theory can be applied in the coalescence regime, where the best-fit estimates of the critical exponents are obtained for two and three dimensions. (author) 3. State-of-the-art of beyond mean field theories with nuclear density functionals Egido, J. Luis 2016-07-01 We present an overview of different beyond mean field theories (BMFTs) based on the generator coordinate method (GCM) and the recovery of symmetries used in many body nuclear physics with effective forces. In a first step a short reminder of the Hartree-Fock-Bogoliubov (HFB) theory is given. A general discussion of the shortcomings of any mean field approximation (MFA), stemming either from the lack of the elementary symmetries (like particle number and angular momentum) or the absence of fluctuations around the mean values, is presented. The recovery of the symmetries spontaneously broken in the HFB approach, in particular the angular momentum, is necessary, among others, to describe excited states and transitions. Particle number projection is also needed to guarantee the right number of protons and neutrons. Furthermore a projection before the variation prevents the pairing collapse in the weak pairing regime. A whole chapter is devoted to illustrate with examples the convenience of recovering symmetries and the differences between the projection before and after the variation. The lack of fluctuations around the average values of the MFA is a big shortcoming inherent to this approach. To build in correlations in BMFT one selects the relevant degrees of freedom of the atomic nucleus. In the low energy part of the spectrum these are the quadrupole, octupole and the pairing vibrations as well as the single particle degrees of freedom. In the GCM the operators representing these degrees of freedom are used as coordinates to generate, by the constrained (projected) HFB theory, a collective subspace. The highly correlated GCM wave function is finally written as a linear combination of a projected basis of this space. The variation of the coefficients of the linear combination leads to the Hill-Wheeler equation. The flexibility of the GCM Ansatz allows to describe a whole palette of physical situations by conveniently choosing the generator coordinates. We discuss the 4. Real-space, mean-field algorithm to numerically calculate long-range interactions Cadilhe, A.; Costa, B. V. 2016-02-01 Long-range interactions are known to be of difficult treatment in statistical mechanics models. There are some approaches that introduce a cutoff in the interactions or make use of reaction field approaches. However, those treatments suffer the illness of being of limited use, in particular close to phase transitions. The use of open boundary conditions allows the sum of the long-range interactions over the entire system to be done, however, this approach demands a sum over all degrees of freedom in the system, which makes a numerical treatment prohibitive. Techniques like the Ewald summation or fast multipole expansion account for the exact interactions but are still limited to a few thousands of particles. In this paper we introduce a novel mean-field approach to treat long-range interactions. The method is based in the division of the system in cells. In the inner cell, that contains the particle in sight, the 'local' interactions are computed exactly, the 'far' contributions are then computed as the average over the particles inside a given cell with the particle in sight for each of the remaining cells. Using this approach, the large and small cells limits are exact. At a fixed cell size, the method also becomes exact in the limit of large lattices. We have applied the procedure to the two-dimensional anisotropic dipolar Heisenberg model. A detailed comparison between our method, the exact calculation and the cutoff radius approximation were done. Our results show that the cutoff-cell approach outperforms any cutoff radius approach as it maintains the long-range memory present in these interactions, contrary to the cutoff radius approximation. Besides that, we calculated the critical temperature and the critical behavior of the specific heat of the anisotropic Heisenberg model using our method. The results are in excellent agreement with extensive Monte Carlo simulations using Ewald summation. 5. Spin-1 Blume-Capel model with longitudinal random crystal and transverse magnetic fields:A mean-field approach Erhan Albayrak 2013-01-01 The spin-1 Blume-Capel model with transverse Ω and longitudinal external magnetic fields h,in addition to a longitudinal random crystal field D,is studied in the mean-field approximation.It is assumed that the crystal field is either turned on with probability p or turned off with probability 1-p on the sites of a square lattice.Phase diagrams are then calculated on the reduced temperature crystal field planes for given values of γ =-Ω/J and p at zero h.Thus,the effect of changing γ and p are illustrated on the phase diagrams in great detail and interesting results are observed. 6. Relativistic semi-classical theory of atom ionization in ultra-intense laser 2001-01-01 A relativistic semi-classical theory (RSCT) of H-atom ionizationin ultra-intense laser (UIL) is proposed. A relativistic analytical expression for ionization probability of H-atom in its ground state is given. This expression, compared with non-relativistic expression, clearly shows the effects of the magnet vector in the laser, the non-dipole approximation and the relativistic mass-energy relation on the ionization processes. At the same time, we show that under some conditions the relativistic expression reduces to the non-relativistic expression of non-dipole approximation. At last, some possible applications of the relativistic theory are briefly stated. 7. Relativistic Pseudospin Symmetry as a Supersymmetric Pattern in Nuclei Leviatan, A 2004-01-01 Shell-model states involving several pseudospin doublets and intruder'' levels in nuclei, are combined into larger multiplets. The corresponding single-particle spectrum exhibits a supersymmetric pattern whose origin can be traced to the relativistic pseudospin symmetry of a nuclear mean-field Dirac Hamiltonian with scalar and vector potentials. 8. Special Relativistic Hydrodynamics with Gravitation Hwang, Jai-chan; Noh, Hyerim 2016-12-01 Special relativistic hydrodynamics with weak gravity has hitherto been unknown in the literature. Whether such an asymmetric combination is possible has been unclear. Here, the hydrodynamic equations with Poisson-type gravity, considering fully relativistic velocity and pressure under the weak gravity and the action-at-a-distance limit, are consistently derived from Einstein’s theory of general relativity. An analysis is made in the maximal slicing, where the Poisson’s equation becomes much simpler than our previous study in the zero-shear gauge. Also presented is the hydrodynamic equations in the first post-Newtonian approximation, now under the general hypersurface condition. Our formulation includes the anisotropic stress. 9. Special relativistic hydrodynamics with gravitation Hwang, Jai-chan 2016-01-01 The special relativistic hydrodynamics with weak gravity is hitherto unknown in the literature. Whether such an asymmetric combination is possible was unclear. Here, the hydrodynamic equations with Poisson-type gravity considering fully relativistic velocity and pressure under the weak gravity and the action-at-a-distance limit are consistently derived from Einstein's general relativity. Analysis is made in the maximal slicing where the Poisson's equation becomes much simpler than our previous study in the zero-shear gauge. Also presented is the hydrodynamic equations in the first post-Newtonian approximation, now under the {\\it general} hypersurface condition. Our formulation includes the anisotropic stress. 10. Relativistic magnetohydrodynamics Hernandez, Juan; Kovtun, Pavel 2017-05-01 We present the equations of relativistic hydrodynamics coupled to dynamical electromagnetic fields, including the effects of polarization, electric fields, and the derivative expansion. We enumerate the transport coefficients at leading order in derivatives, including electrical conductivities, viscosities, and thermodynamic coefficients. We find the constraints on transport coefficients due to the positivity of entropy production, and derive the corresponding Kubo formulas. For the neutral state in a magnetic field, small fluctuations include Alfvén waves, magnetosonic waves, and the dissipative modes. For the state with a non-zero dynamical charge density in a magnetic field, plasma oscillations gap out all propagating modes, except for Alfvén-like waves with a quadratic dispersion relation. We relate the transport coefficients in the "conventional" magnetohydrodynamics (formulated using Maxwell's equations in matter) to those in the "dual" version of magnetohydrodynamics (formulated using the conserved magnetic flux). 11. Relativistic Achilles Leardini, Fabrice 2013-01-01 This manuscript presents a problem on special relativity theory (SRT) which embodies an apparent paradox relying on the concept of simultaneity. The problem is represented in the framework of Greek epic poetry and structured in a didactic way. Owing to the characteristic properties of Lorenz transformations, three events which are simultaneous in a given inertial reference system, occur at different times in the other two reference frames. In contrast to the famous twin paradox, in the present case there are three, not two, different inertial observers. This feature provides a better framework to expose some of the main characteristics of SRT, in particular, the concept of velocity and the relativistic rule of addition of velocities. 12. From infinity to one: The reduction of some mean field games to a global control problem Guéant, Olivier 2011-01-01 This paper presents recent results from Mean Field Game theory underlying the introduction of common noise that imposes to incorporate the distribution of the agents as a state variable. Starting from the usual mean field games equations introduced by J.M. Lasry and P.L. Lions and adapting them to games on graphs, we introduce a partial differential equation, often referred to as the Master equation, from which the MFG equations can be deduced. Then, this Master equation can be reinterpreted using a global control problem inducing the same behaviors as in the non-cooperative initial mean field game. 13. Constant entropy hybrid stars: a first approximation to cooling evolution Mariani, M; Vucetich, H 2016-01-01 We study the possibility of a hadron-quark phase transition in the interior of neutron stars, taking into account different schematic evolutionary stages at finite temperature. We also discuss the strange quark matter stability in the quark matter phase. Furthermore, we analyze the astrophysical properties of hot and cold hybrid stars, considering the recent constraint on maximum mass given by the pulsars PSR J1614-2230 and PSR J1614-2230. We have developed a computational code to construct semi-analytical hybrid equations of state at fixed entropy per baryon to obtain different families of hybrid stars. An analytical approximation of the Field Correlator Method is used for the quark matter equation of state. For the hadronic ecuation of state, we use a table based on the relativistic mean field theory without hyperons. The phase transition is obtained imposing the Maxwell conditions, by assuming a high surface tension at the interface hadron-quark. The relativistic structure equations of hydrostatic equilibr... 14. Mean-field dynamos: the old concept and some recent developments Rädler, K -H 2014-01-01 This article reproduces the Karl Schwarzschild lecture 2013. Some of the basic ideas of electrodynamics and magnetohydrodynamics of mean fields in turbulently moving conducting fluids are explained. It is stressed that the connection of the mean electromotive force with the mean magnetic field and its first spatial derivatives is in general neither local nor instantaneous and that quite a few claims concerning pretended failures of the mean-field concept result from ignoring this aspect. In addition to the mean-field dynamo mechanisms of$\\alpha^2$and$\\alpha\\Omega$type several others are considered. Much progress in mean-field electrodynamics and magnetohydrodynamics results from the test-field method for calculating the coefficients that determine the connection of the mean electromotive force with the mean magnetic field. As an important example the memory effect in homogeneous isotropic turbulence is explained. In magnetohydrodynamic turbulence there is the possibility of a mean electromotive force t... 15. On the existence of classical solutions for stationary extended mean field games Gomes, Diogo A. 2014-04-01 In this paper we consider extended stationary mean-field games, that is mean-field games which depend on the velocity field of the players. We prove various a-priori estimates which generalize the results for quasi-variational mean-field games in Gomes et al. (2012). In addition we use adjoint method techniques to obtain higher regularity bounds. Then we establish the existence of smooth solutions under fairly general conditions by applying the continuity method. When applied to standard stationary mean-field games as in Lasry and Lions (2006), Gomes and Sanchez-Morgado (2011) or Gomes et al. (2012) this paper yields various new estimates and regularity properties not available previously. We discuss additionally several examples where the existence of classical solutions can be proved. © 2013 Elsevier Ltd. All rights reserved. 16. Dynamical Mean-Field Theory of Electronic Correlations in Models and Materials Vollhardt, Dieter 2010-11-01 The concept of electronic correlations plays an important role in modern condensed matter physics. It refers to interaction effects which cannot be explained within a static mean-field picture as provided by Hartree-Fock theory. Electronic correlations can have a very strong influence on the properties of materials. For example, they may turn a metal into an insulator (Mott-Hubbard metal-insulator transition). In these lecture notes I (i) introduce basic notions of the physics of correlated electronic systems, (ii) discuss the construction of mean-field theories by taking the limit of high lattice dimensions, (iii) explain the simplifications of the many-body perturbation theory in this limit which provide the basis for the formulation of a comprehensive mean-field theory for correlated fermions, the dynamical mean-field theory (DMFT), (v) derive the DMFT self-consistency equations, and (vi) apply the DMFT to investigate electronic correlations in models and materials. 17. On the mean-field theory of the Karlsruhe Dynamo Experiment K.-H. Rädler 2002-01-01 Full Text Available In the Forschungszentrum Karlsruhe an experiment has been constructed which demonstrates a homogeneous dynamo as is expected to exist in the Earth's interior. This experiment is discussed within the framework of mean-field dynamo theory. The main predictions of this theory are explained and compared with the experimental results. Key words. Dynamo, geodynamo, dynamo experiment, mean-field dynamo theory, a-effect 18. Can realistic interaction be useful for nuclear mean-field approaches? Nakada, H.; Sugiura, K. [Chiba University, Department of Physics, Graduate School of Science, Inage, Chiba (Japan); Inakura, T. [Chiba University, Department of Physics, Graduate School of Science, Inage, Chiba (Japan); Kyoto University, Yukawa Institute of Theoretical Physics, Sakyo, Kyoto (Japan); Niigata University, Department of Physics, Niigata (Japan); Margueron, J. [Universite de Lyon 1, CNRS/IN2P3, Institut de Physique Nucleaire de Lyon, Villeurbanne (France) 2016-07-15 Recent applications of the M3Y-type semi-realistic interaction to the nuclear mean-field approaches are presented: (i) Prediction of magic numbers and (ii) isotope shifts of nuclei with magic proton numbers. The results exemplify that the realistic interaction, which is derived from the bare 2N and 3N interaction, furnishes a new theoretical instrument for advancing nuclear mean-field approaches. (orig.) 19. Crowd-Averse Cyber-Physical Systems: The Paradigm of Robust Mean Field Games Bauso, D.; Tembine, H. 2015-01-01 For a networked controlled system we illustrate the paradigm of robust mean-field games. This is a modeling framework at the interface of differential game theory, mathematical physics, and H1-optimal control that tries to capture the mutual influence between a crowd and its individuals. First, we establish a mean-field system for such games including the effects of adversarial disturbances. Second, we identify the optimal response of the individuals for a given population behavior. Third, we... 20. A Stochastic Maximum Principle for Risk-Sensitive Mean-Field Type Control Djehiche, Boualem 2015-02-24 In this paper we study mean-field type control problems with risk-sensitive performance functionals. We establish a stochastic maximum principle (SMP) for optimal control of stochastic differential equations (SDEs) of mean-field type, in which the drift and the diffusion coefficients as well as the performance functional depend not only on the state and the control but also on the mean of the distribution of the state. Our result extends the risk-sensitive SMP (without mean-field coupling) of Lim and Zhou (2005), derived for feedback (or Markov) type optimal controls, to optimal control problems for non-Markovian dynamics which may be time-inconsistent in the sense that the Bellman optimality principle does not hold. In our approach to the risk-sensitive SMP, the smoothness assumption on the value-function imposed in Lim and Zhou (2005) needs not be satisfied. For a general action space a Peng\\'s type SMP is derived, specifying the necessary conditions for optimality. Two examples are carried out to illustrate the proposed risk-sensitive mean-field type SMP under linear stochastic dynamics with exponential quadratic cost function. Explicit solutions are given for both mean-field free and mean-field models. 1. Exact Solution Versus Gaussian Approximation for a Non-Ideal Bose Gas in One-Dimension Tommasini, P; Natti, P L 1997-01-01 We investigate ground-state and excitation spectrum of a system of non-relativistic bosons in one-dimension interacting through repulsive, two-body contact interactions in a self-consistent Gaussian mean-field approximation which consists in writing the variationally determined density operator as the most general Gaussian functional of the quantized field operators. There are mainly two advantages in working with one-dimension. First, the existence of an exact solution for the ground-state and excitation energies. Second, neither in the perturbative results nor in the Gaussian approximation itself we do not have to deal with the three-dimensional patologies of the contact interaction . So that this scheme provides a clear comparison between these three different results. PACS numbers : 05.30.-d, 05.30.Jp, 67.40.Db 2. Bosonic particle-correlated states: A nonperturbative treatment beyond mean field Jiang, Zhang; Tacla, Alexandre B.; Caves, Carlton M. 2017-08-01 Many useful properties of dilute Bose gases at ultralow temperature are predicted precisely by the (mean-field) product-state Ansatz, in which all particles are in the same quantum state. Yet, in situations where particle-particle correlations become important, the product Ansatz fails. To include correlations nonperturbatively, we consider a new set of states: the particle-correlated state of N =l ×n bosons is derived by symmetrizing the n -fold product of an l -particle quantum state. Quantum correlations of the l -particle state "spread out" to any subset of the N bosons by symmetrization. The particle-correlated states can be simulated efficiently for large N , because their parameter spaces, which depend on l , do not grow with n . Here we formulate and develop in great detail the pure-state case for l =2 , where the many-body state is constructed from a two-particle pure state. These paired wave functions, which we call pair-correlated states (PCS), were introduced by A. J. Leggett [Rev. Mod. Phys. 73, 307 (2001), 10.1103/RevModPhys.73.307] as a particle-number-conserving version of the Bogoliubov approximation. We present an iterative algorithm that solves for the reduced (marginal) density matrices (RDMs), i.e., the correlation functions, associated with PCS in time O (N ) . The RDMs can also be derived from the normalization factor of PCS, which is derived analytically in the large-N limit. To test the efficacy of PCS, we analyze the ground state of the two-site Bose-Hubbard model by minimizing the energy of the PCS state, both in its exact form and in its large-N approximate form, and comparing with the exact ground state. For N =1000 , the relative errors of the ground-state energy for both cases are within 10-5 over the entire parameter region from a single condensate to a Mott insulator. We present numerical results that suggest that PCS might be useful for describing the dynamics in the strongly interacting regime. 3. Variational and perturbative formulations of quantum mechanical/molecular mechanical free energy with mean-field embedding and its analytical gradients Yamamoto, Takeshi 2008-12-01 Conventional quantum chemical solvation theories are based on the mean-field embedding approximation. That is, the electronic wavefunction is calculated in the presence of the mean field of the environment. In this paper a direct quantum mechanical/molecular mechanical (QM/MM) analog of such a mean-field theory is formulated based on variational and perturbative frameworks. In the variational framework, an appropriate QM/MM free energy functional is defined and is minimized in terms of the trial wavefunction that best approximates the true QM wavefunction in a statistically averaged sense. Analytical free energy gradient is obtained, which takes the form of the gradient of effective QM energy calculated in the averaged MM potential. In the perturbative framework, the above variational procedure is shown to be equivalent to the first-order expansion of the QM energy (in the exact free energy expression) about the self-consistent reference field. This helps understand the relation between the variational procedure and the exact QM/MM free energy as well as existing QM/MM theories. Based on this, several ways are discussed for evaluating non-mean-field effects (i.e., statistical fluctuations of the QM wavefunction) that are neglected in the mean-field calculation. As an illustration, the method is applied to an SN2 Menshutkin reaction in water, NH3+CH3Cl→NH3CH3++Cl-, for which free energy profiles are obtained at the Hartree-Fock, MP2, B3LYP, and BHHLYP levels by integrating the free energy gradient. Non-mean-field effects are evaluated to be reaction in water. 4. Asymmetric Neutrino Reaction in Magnetized Proto-Neutron Stars in Fully Relativistic Approach Yasutake Nobutoshi 2012-02-01 Full Text Available We calculate asymmetric neutrino absorption and scattering cross sections on hot and dense magnetized neutron-star matter including hyperons in fully relativistic mean-field theory. The absorption/scattering cross sections are suppressed/enhanced incoherently in the direction of the magnetic field B = Bẑ. The asymmetry is 2–4% at the matter density ρ0 ≤ ρB ≤ 3ρ0 and temperature T ≤ 40MeV for B = 2 × 1017G. Then we solve the Boltzmann equation for the neutrino transport in 1D attenuation approximation, and get the result that the kick velocity becomes about 300 km/s for the proto-neutron star with 168 solar mass at T = 20MeV. 5. Finite-size and correlation-induced effects in Mean-field Dynamics Touboul, Jonathan 2010-01-01 The brain's activity is characterized by the interaction of a very large number of neurons that are strongly affected by noise. However, signals often arise at macroscopic scales integrating the effect of many neurons into a reliable pattern of activity. In order to study such large neuronal assemblies, one is often led to derive mean-field limits summarizing the effect of the interaction of a large number of neurons into an effective signal. Classical mean-field approaches consider the evolution of a deterministic variable, the mean activity, thus neglecting the stochastic nature of neural behavior. In this article, we build upon a recent approach that includes correlations and higher order moments in mean-field equations, and study how these stochastic effects influence the solutions of the mean-field equations, both in the limit of an infinite number of neurons and for large yet finite networks. We show that, though the solutions of the deterministic mean-field equation constitute uncorrelated solutions of... 6. Mean-field dynamos: The old concept and some recent developments. Karl Schwarzschild Award Lecture 2013 Rädler, K.-H. This article elucidates the basic ideas of electrodynamics and magnetohydrodynamics of mean fields in turbulently moving conducting fluids. It is stressed that the connection of the mean electromotive force with the mean magnetic field and its first spatial derivatives is in general neither local nor instantaneous and that quite a few claims concerning pretended failures of the mean-field concept result from ignoring this aspect. In addition to the mean-field dynamo mechanisms of α2 and α Ω type several others are considered. Much progress in mean-field electrodynamics and magnetohydrodynamics results from the test-field method for calculating the coefficients that determine the connection of the mean electromotive force with the mean magnetic field. As an important example the memory effect in homogeneous isotropic turbulence is explained. In magnetohydrodynamic turbulence there is the possibility of a mean electromotive force that is primarily independent of the mean magnetic field and labeled as Yoshizawa effect. Despite of many efforts there is so far no convincing comprehensive theory of α quenching, that is, the reduction of the α effect with growing mean magnetic field, and of the saturation of mean-field dynamos. Steps toward such a theory are explained. Finally, some remarks on laboratory experiments with dynamos are made. 7. Stationary Relativistic Jets Komissarov, S S; Lyutikov, M 2015-01-01 In this paper we describe a simple numerical approach which allows to study the structure of steady-state axisymmetric relativistic jets using one-dimensional time-dependent simulations. It is based on the fact that for narrow jets with v~c the steady-state equations of relativistic magnetohydrodynamics can be accurately approximated by the one-dimensional time-dependent equations after the substitution z=ct. Since only the time-dependent codes are now publicly available this is a valuable and efficient alternative to the development of a high-specialized code for the time-independent equations. The approach is also much cheaper and more robust compared to the relaxation method. We tested this technique against numerical and analytical solutions found in literature as well as solutions we obtained using the relaxation method and found it sufficiently accurate. In the process, we discovered the reason for the failure of the self-similar analytical model of the jet reconfinement in relatively flat atmospheres a... 8. Relativistic diffusive motion in random electromagnetic fields Haba, Z, E-mail: [email protected] [Institute of Theoretical Physics, University of Wroclaw, 50-204 Wroclaw, Plac Maxa Borna 9 (Poland) 2011-08-19 We show that the relativistic dynamics in a Gaussian random electromagnetic field can be approximated by the relativistic diffusion of Schay and Dudley. Lorentz invariant dynamics in the proper time leads to the diffusion in the proper time. The dynamics in the laboratory time gives the diffusive transport equation corresponding to the Juettner equilibrium at the inverse temperature {beta}{sup -1} = mc{sup 2}. The diffusion constant is expressed by the field strength correlation function (Kubo's formula). 9. Mean field theory for a balanced hypercolumn model of orientation selectivity in primary visual cortex Lerchner, A; Hertz, J; Ahmadi, M 2004-01-01 We present a complete mean field theory for a balanced state of a simple model of an orientation hypercolumn. The theory is complemented by a description of a numerical procedure for solving the mean-field equations quantitatively. With our treatment, we can determine self-consistently both the firing rates and the firing correlations, without being restricted to specific neuron models. Here, we solve the analytically derived mean-field equations numerically for integrate-and-fire neurons. Several known key properties of orientation selective cortical neurons emerge naturally from the description: Irregular firing with statistics close to -- but not restricted to -- Poisson statistics; an almost linear gain function (firing frequency as a function of stimulus contrast) of the neurons within the network; and a contrast-invariant tuning width of the neuronal firing. We find that the irregularity in firing depends sensitively on synaptic strengths. If Fano factors are bigger than 1, then they are so for all stim... 10. Exact mean field dynamics for epidemic-like processes on heterogeneous networks Lucas, Andrew 2012-01-01 We show that the mean field equations for the SIR epidemic can be exactly solved for a network with arbitrary degree distribution. Our exact solution consists of reducing the dynamics to a lone first order differential equation, which has a solution in terms of an integral over functions dependent on the degree distribution of the network, and reconstructing all mean field functions of interest from this integral. Irreversibility of the SIR epidemic is crucial for the solution. We also find exact solutions to the sexually transmitted disease SI epidemic on bipartite graphs, to a simplified rumor spreading model, and to a new model for recommendation spreading, via similar techniques. Numerical simulations of these processes on scale free networks demonstrate the qualitative validity of mean field theory in most regimes. 11. Mean-Field Backward Stochastic Evolution Equations in Hilbert Spaces and Optimal Control for BSPDEs Ruimin Xu 2014-01-01 Full Text Available We obtain the existence and uniqueness result of the mild solutions to mean-field backward stochastic evolution equations (BSEEs in Hilbert spaces under a weaker condition than the Lipschitz one. As an intermediate step, the existence and uniqueness result for the mild solutions of mean-field BSEEs under Lipschitz condition is also established. And then a maximum principle for optimal control problems governed by backward stochastic partial differential equations (BSPDEs of mean-field type is presented. In this control system, the control domain need not to be convex and the coefficients, both in the state equation and in the cost functional, depend on the law of the BSPDE as well as the state and the control. Finally, a linear-quadratic optimal control problem is given to explain our theoretical results. 12. Relativistic description of electron scattering on the deuteron Hummel, E 1994-01-01 Within a quasipotential framework a relativistic analysis is presented of the deuteron current. Assuming that the singularities from the nucleon propagators are important, a so-called equal time approximation of the current is constructed. This is applied to both elastic and inelastic electron scattering. As dynamical model the relativistic one boson exchange model is used. Reasonable agreement is found with a previous relativistic calculation of the elastic electromagnetic form factors of the deuteron. For the unpolarized inelastic electron scattering effects of final state interactions and relativistic corrections to the structure functions are considered in the impulse approximation. Two specific kinematic situations are studied as examples. 13. Mean-field theory of globally coupled integrate-and-fire neural oscillators with dynamic synapses Bressloff, P. C. 1999-08-01 We analyze the effects of synaptic depression or facilitation on the existence and stability of the splay or asynchronous state in a population of all-to-all, pulse-coupled neural oscillators. We use mean-field techniques to derive conditions for the local stability of the splay state and determine how stability depends on the degree of synaptic depression or facilitation. We also consider the effects of noise. Extensions of the mean-field results to finite networks are developed in terms of the nonlinear firing time map. 14. Another mean field treatment in the strong coupling limit of lattice QCD Ohnishi, Akira; Miura, Kohtaroh; Nakano, Takashi Z. 2011-01-01 We discuss the QCD phase diagram in the strong coupling limit of lattice QCD by using a new type of mean field coming from the next-to-leading order of the large dimensional expansion. The QCD phase diagram in the strong coupling limit recently obtained by using the monomer-dimer-polymer (MDP) algorithm has some differences in the phase boundary shape from that in the mean field results. As one of the origin to explain the difference, we consider another type of auxiliary field, which corresp... 15. Skyrme mean-field studies of nuclei far from the stability line Heenen, P H; Cwiok, S; Nazarewicz, W; Valor, A 1999-01-01 Two applications of mean-field calculations based on 3D coordinate-space techniques are presented. The first concerns the structure of odd-N superheavy elements that have been recently observed experimentally and shows the ability of the method to describe, in a self-consistent way, very heavy odd-mass nuclei. Our results are consistent with the experimental data. The second application concerns the introduction of correlations beyond a mean-field approach by means of projection techniques and configuration mixing. Results for Mg isotopes demonstrate that the restoration of rotational symmetry plays a crucial role in the description of 32Mg. 16. Dynamical mean field theory-based electronic structure calculations for correlated materials. Biermann, Silke 2014-01-01 We give an introduction to dynamical mean field approaches to correlated materials. Starting from the concept of electronic correlation, we explain why a theoretical description of correlations in spectroscopic properties needs to go beyond the single-particle picture of band theory.We discuss the main ideas of dynamical mean field theory and its use within realistic electronic structure calculations, illustrated by examples of transition metals, transition metal oxides, and rare-earth compounds. Finally, we summarise recent progress on the calculation of effective Hubbard interactions and the description of dynamical screening effects in solids. 17. Time-Dependent Mean-Field Games in the Subquadratic Case Gomes, Diogo A. 2014-10-14 In this paper we consider time-dependent mean-field games with subquadratic Hamiltonians and power-like local dependence on the measure. We establish existence of classical solutions under a certain set of conditions depending on both the growth of the Hamiltonian and the dimension. This is done by combining regularity estimates for the Hamilton-Jacobi equation based on the Gagliardo-Nirenberg interpolation inequality with polynomial estimates for the Fokker-Planck equation. This technique improves substantially the previous results on the regularity of time-dependent mean-field games. 18. On the convergence of finite state mean-field games through Γ-convergence Ferreira, Rita C. 2014-10-01 In this study, we consider the long-term convergence (trend toward an equilibrium) of finite state mean-field games using Γ-convergence. Our techniques are based on the observation that an important class of mean-field games can be viewed as the Euler-Lagrange equation of a suitable functional. Therefore, using a scaling argument, one can convert a long-term convergence problem into a Γ-convergence problem. Our results generalize previous results related to long-term convergence for finite state problems. © 2014 Elsevier Inc. 19. Non-mean-field effects in systems with long-range forces in competition. Bachelard, R; Staniscia, F 2012-11-01 We investigate the canonical equilibrium of systems with long-range forces in competition. These forces create a modulation in the interaction potential and modulated phases appear at the system scale. The structure of these phases differentiate this system from monotonic potentials, where only the mean-field and disordered phases exist. With increasing temperature, the system switches from one ordered phase to another through a first-order phase transition. Both mean-field and modulated phases may be stable, even at zero temperature, and the long-range nature of the interaction will lead to metastability characterized by extremely long time scales. 20. Obtaining Arbitrary Prescribed Mean Field Dynamics for Recurrently Coupled Networks of Type-I Spiking Neurons with Analytically Determined Weights Wilten eNicola 2016-02-01 Full Text Available A fundamental question in computational neuroscience is how to connect a network of spiking neurons to produce desired macroscopic or mean field dynamics. One possible approach is through the Neural Engineering Framework (NEF. The NEF approach requires quantities called decoders which are solved through an optimization problem requiring large matrix inversion. Here, we show how a decoder can be obtained analytically for type I and certain type II firing rates as a function of the heterogeneity of its associated neuron. These decoders generate approximants for functions that converge to the desired function in mean-squared error like 1/N, where N is the number of neurons in the network. We refer to these decoders as scale-invariant decoders due to their structure. These decoders generate weights for a network of neurons through the NEF formula for weights. These weights force the spiking network to have arbitrary and prescribed mean field dynamics. The weights generated with scale-invariant decoders all lie on low dimensional hypersurfaces asymptotically. We demonstrate the applicability of these scale-invariant decoders and weight surfaces by constructing networks of spiking theta neurons that replicate the dynamics of various well known dynamical systems such as the neural integrator, Van der Pol system and the Lorenz system. As these decoders are analytically determined and non-unique, the weights are also analytically determined and non-unique. We discuss the implications for measured weights of neuronal networks 1. Relativistic Fractal Cosmologies Ribeiro, Marcelo B 2009-01-01 This article reviews an approach for constructing a simple relativistic fractal cosmology whose main aim is to model the observed inhomogeneities of the distribution of galaxies by means of the Lemaitre-Tolman solution of Einstein's field equations for spherically symmetric dust in comoving coordinates. This model is based on earlier works developed by L. Pietronero and J.R. Wertz on Newtonian cosmology, whose main points are discussed. Observational relations in this spacetime are presented, together with a strategy for finding numerical solutions which approximate an averaged and smoothed out single fractal structure in the past light cone. Such fractal solutions are shown, with one of them being in agreement with some basic observational constraints, including the decay of the average density with the distance as a power law (the de Vaucouleurs' density power law) and the fractal dimension in the range 1 <= D <= 2. The spatially homogeneous Friedmann model is discussed as a special case of the Lemait... 2. Relativistic Fluid Dynamics Cattaneo, Carlo 2011-01-01 This title includes: Pham Mau Quam: Problemes mathematiques en hydrodynamique relativiste; A. Lichnerowicz: Ondes de choc, ondes infinitesimales et rayons en hydrodynamique et magnetohydrodynamique relativistes; A.H. Taub: Variational principles in general relativity; J. Ehlers: General relativistic kinetic theory of gases; K. Marathe: Abstract Minkowski spaces as fibre bundles; and, G. Boillat: Sur la propagation de la chaleur en relativite. 3. Test of Relativistic Eigenfunctions for Pseudospin Symmetry Ginocchio, Joseph N. 2001-10-01 Pseudospin symmetry has been shown to be a relativistic symmetry of the Dirac Hamiltonian [1] and the generators of this symmetry have been determined [2]. Although the measured energy splittings between pseudospin doublets are small, the eigenfunctions of the doublets have been examined only recently [3]. We show to what extent the pseudospin partners of realistic relativistic mean field eigenfunctions [4] are themselves eigenfunctions of the same Dirac Hamiltonian. 1) J. N. Ginocchio, Phys. Rev. Lett. 78, 436 (1997). 2) J. N. Ginocchio and A. Leviatan, Phys. Lett. B 425, 1 (1998). 3) J. N. Ginocchio and A. Leviatan, to be published in Phys. Rev. Lett. (2001). 4) J. N. Ginocchio and D. G. Madland, Phys. Rev. C 57, 1167 (1998). 4. A Two-Mode Mean-Field Optimal Switching Problem for the Full Balance Sheet Boualem Djehiche 2014-01-01 a two-mode optimal switching problem of mean-field type, which can be described by a system of Snell envelopes where the obstacles are interconnected and nonlinear. The main result of the paper is a proof of a continuous minimal solution to the system of Snell envelopes, as well as the full characterization of the optimal switching strategy. 5. Another mean field treatment in the strong coupling limit of lattice QCD Ohnishi, Akira; Nakano, Takashi Z 2010-01-01 We discuss the QCD phase diagram in the strong coupling limit of lattice QCD by using a new type of mean field coming from the next-to-leading order of the large dimensional expansion. The QCD phase diagram in the strong coupling limit recently obtained by using the monomer-dimer-polymer (MDP) algorithm has some differences in the phase boundary shape from that in the mean field results. As one of the origin to explain the difference, we consider another type of auxiliary field, which corresponds to the point-splitting mesonic composite. Fermion determinant with this mean field under the anti-periodic boundary condition gives rise to a term which interpolates the effective potentials in the previously proposed zero and finite temperature mean field treatments. While the shift of the transition temperature at zero chemical potential is in the desirable direction and the phase boundary shape is improved, we find that the effects are too large to be compatible with the MDP simulation results. 6. Mean Field Theory, Ginzburg Criterion, and Marginal Dimensionality of Phase-Transitions Als-Nielsen, Jens Aage; Birgenau, R. J. 1977-01-01 By applying a real space version of the Ginzburg criterion, the role of fluctuations and thence the self‐consistency of mean field theory are assessed in a simple fashion for a variety of phase transitions. It is shown that in using this approach the concept of ’’marginal dimensionality’’ emerges... 7. Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion Gomes, Diogo A. 2017-01-05 Here, we consider one-dimensional first-order stationary mean-field games with congestion. These games arise when crowds face difficulty moving in high-density regions. We look at both monotone decreasing and increasing interactions and construct explicit solutions using the current formulation. We observe new phenomena such as discontinuities, unhappiness traps and the non-existence of solutions. 8. Phase behaviour of colloids suspended in a near-critical solvent : A mean-field approach Edison, John R.; Belli, Simone; Evans, Robert; Van Roij, René; Dijkstra, Marjolein 2015-01-01 Colloids suspended in a binary solvent may, under suitable thermodynamic conditions, experience a wide variety of solvent-mediated interactions that can lead to colloidal phase transitions and aggregation phenomena. We present a simple mean-field theory, based on free-volume arguments, that describe 9. Going Beyond a Mean-field Model for the Learning Cortex: Second-Order Statistics Steyn-Ross, Moira L.; Steyn-Ross, D. A.; Sleigh, J. W. 2008-01-01 Mean-field models of the cortex have been used successfully to interpret the origin of features on the electroencephalogram under situations such as sleep, anesthesia, and seizures. In a mean-field scheme, dynamic changes in synaptic weights can be considered through fluctuation-based Hebbian learning rules. However, because such implementations deal with population-averaged properties, they are not well suited to memory and learning applications where individual synaptic weights can be important. We demonstrate that, through an extended system of equations, the mean-field models can be developed further to look at higher-order statistics, in particular, the distribution of synaptic weights within a cortical column. This allows us to make some general conclusions on memory through a mean-field scheme. Specifically, we expect large changes in the standard deviation of the distribution of synaptic weights when fluctuation in the mean soma potentials are large, such as during the transitions between the “up” and “down” states of slow-wave sleep. Moreover, a cortex that has low structure in its neuronal connections is most likely to decrease its standard deviation in the weights of excitatory to excitatory synapses, relative to the square of the mean, whereas a cortex with strongly patterned connections is most likely to increase this measure. This suggests that fluctuations are used to condense the coding of strong (presumably useful) memories into fewer, but dynamic, neuron connections, while at the same time removing weaker (less useful) memories. PMID:19669541 10. Metastates in Finite-type Mean-field Models : Visibility, Invisibility, and Random Restoration of Symmetry Iacobelli, Giulio; Kuelske, Christof We consider a general class of disordered mean-field models where both the spin variables and disorder variables eta take finitely many values. To investigate the size-dependence in the phase-transition regime we construct the metastate describing the probabilities to find a large system close to a 11. Mean-field description of the structure and tension of curved fluid interfaces Kuipers, Joris 2009-01-01 This thesis described the interfacial properties of curved fluid interfaces mainly employing mean-field models. Investigations of Tolman's length in simple systems and systems in contact with a hard wall are presented. Both the interfacial properties as well as the wetting behavior of phase-separate 12. Ground state correlations and mean field using the exp(S) method Heisenberg, J H; Heisenberg, Jochen H.; Mihaila, Bogdan 1999-01-01 This document gives a detailed account of the terms used in the computation of the ground state mean field and the ground state correlations. While the general approach to this description is given in a separate paper (nucl-th/9802029) we give here the explicite expressions used. 13. Automating the mean-field method for large dynamic gossip networks Bakhshi, Rena; Endrullis, Jörg; Endrullis, Stefan; Fokkink, Wan; Haverkort, Boudewijn 2010-01-01 We investigate an abstraction method, called mean- field method, for the performance evaluation of dynamic net- works with pairwise communication between nodes. It allows us to evaluate systems with very large numbers of nodes, that is, systems of a size where traditional performance evaluation meth 14. New a priori estimates for mean-field games with congestion Evangelista, David 2016-01-06 We present recent developments in crowd dynamics models (e.g. pedestrian flow problems). Our formulation is given by a mean-field game (MFG) with congestion. We start by reviewing earlier models and results. Next, we develop our model. We establish new a priori estimates that give partial regularity of the solutions. Finally, we discuss numerical results. 15. Mean field limit for bosons and infinite dimensional phase-space analysis Zied, Ammari 2007-01-01 This article proposes the construction of Wigner measures in the infinite dimensional bosonic quantum field theory, with applications to the derivation of the mean field dynamics. Once these asymptotic objects are well defined, it is shown how they can be used to make connections between different kinds of results or to prove new ones. 16. Mott-Hubbard and Anderson transitions in dynamical mean-field theory Byczuk, Krzysztof [Institute of Theoretical Physics, Warsaw University, ul. Hoza 69, PL-00-681 Warsaw (Poland)]. E-mail: [email protected]; Hofstetter, Walter [Condensed Matter Theory Group, Massachusetts Institute of Technology, Cambridge, MA 02139 (United States); Vollhardt, Dieter [Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute for Physics, University of Augsburg, D-86135 Augsburg (Germany) 2005-04-30 The Anderson-Hubbard Hamiltonian at half-filling is investigated within dynamical mean-field theory at zero temperature. The local density of states is calculated by taking the geometric and arithmetic mean, respectively. The non-magnetic ground state phase diagrams obtained within the different averaging schemes are compared. 17. Mean-field cosmological dynamos in Riemannian space with isotropic diffusion de Andrade, L Garcia 2009-01-01 Mean-field cosmological dynamos in Riemannian space with isotropic diffusion}} Previous attempts for building a cosmic dynamo including preheating in inflationary universes [Bassett et al Phys Rev (2001)] has not included mean field or turbulent dynamos. In this paper a mean field dynamo in cosmic scales on a Riemannian spatial cosmological section background, is set up. When magnetic fields and flow velocities are parallel propagated along the Riemannian space dynamo action is obtained. Turbulent diffusivity${\\beta}$is coupled with the Ricci magnetic curvature, as in Marklund and Clarkson [MNRAS (2005)], GR-MHD dynamo equation. Mean electric field possesses an extra term where Ricci tensor couples with magnetic vector potential in Ohm's law. In Goedel universe induces a mean field dynamo growth rate${\\gamma}=2{\\omega}^{2}{\\beta}$. In this frame kinetic helicity vanishes. In radiation era this yields${\\gamma}\\approx{2{\\beta}{\\times}10^{-12}s^{-1}}$. In non-comoving the magnetic field is expressed as$B\\ap...
18. Non-thermal quenched damage phenomena: The application of the mean-field approach for the three-dimensional case
Abaimov, Sergey G.; Akhatov, Iskander S.
2016-09-01
In this study, we apply the mean-field approach to the three-dimensional damage phenomena. The model approximates a solid as a polycrystalline material where grains are assumed isotropic. While the stiffness properties are considered homogeneous, the heterogeneous distribution of grains' strengths provides the quenched statistical variability generating non-thermal fluctuations in the ensemble. Studying the statistical properties of the fluctuations, we introduce the concept of susceptibility of damage. Its divergence in the vicinity of the point of material failure can be treated as a catastrophe predictor. In accordance with this criterion, we find that damage growth in reality is much faster than it could be expected from intuitive engineering considerations. Also, we consider avalanches of grain failures and find that due to the slowing down effect the characteristic time of the relaxation processes diverges in the vicinity of the point of material failure.
19. Diagrammatic Monte Carlo approach for diagrammatic extensions of dynamical mean-field theory -- convergence analysis of the dual fermion technique
Gukelberger, Jan; Hafermann, Hartmut
2016-01-01
The dual-fermion approach provides a formally exact prescription for calculating properties of a correlated electron system in terms of a diagrammatic expansion around dynamical mean-field theory (DMFT). It can address the full range of interactions, the lowest order theory is asymptotically exact in both the weak- and strong-coupling limits, and the technique naturally incorporates long-range correlations beyond the reach of current cluster extensions to DMFT. Most practical implementations, however, neglect higher-order interaction vertices beyond two-particle scattering in the dual effective action and further truncate the diagrammatic expansion in the two-particle scattering vertex to a leading-order or ladder-type approximation. In this work we compute the dual-fermion expansion for the Hubbard model including all diagram topologies with two-particle interactions to high orders by means of a stochastic diagrammatic Monte Carlo algorithm. We use benchmarking against numerically exact Diagrammatic Determin...
20. Finite-State Mean-Field Games, Crowd Motion Problems, and its Numerical Methods
2017-09-10
In this dissertation, we present two research projects, namely finite-state mean-field games and the Hughes model for the motion of crowds. In the first part, we describe finite-state mean-field games and some applications to socio-economic sciences. Examples include paradigm shifts in the scientific community and the consumer choice behavior in a free market. The corresponding finite-state mean-field game models are hyperbolic systems of partial differential equations, for which we propose and validate a new numerical method. Next, we consider the dual formulation to two-state mean-field games, and we discuss numerical methods for these problems. We then depict different computational experiments, exhibiting a variety of behaviors, including shock formation, lack of invertibility, and monotonicity loss. We conclude the first part of this dissertation with an investigation of the shock structure for two-state problems. In the second part, we consider a model for the movement of crowds proposed by R. Hughes in [56] and describe a numerical approach to solve it. This model comprises a Fokker-Planck equation coupled with an Eikonal equation with Dirichlet or Neumann data. We first establish a priori estimates for the solutions. Next, we consider radial solutions, and we identify a shock formation mechanism. Subsequently, we illustrate the existence of congestion, the breakdown of the model, and the trend to the equilibrium. We also propose a new numerical method for the solution of Fokker-Planck equations and then to systems of PDEs composed by a Fokker-Planck equation and a potential type equation. Finally, we illustrate the use of the numerical method both to the Hughes model and mean-field games. We also depict cases such as the evacuation of a room and the movement of persons around Kaaba (Saudi Arabia).
1. Relativistic radiative transfer in relativistic spherical flows
Fukue, Jun
2017-02-01
Relativistic radiative transfer in relativistic spherical flows is numerically examined under the fully special relativistic treatment. We first derive relativistic formal solutions for the relativistic radiative transfer equation in relativistic spherical flows. We then iteratively solve the relativistic radiative transfer equation, using an impact parameter method/tangent ray method, and obtain specific intensities in the inertial and comoving frames, as well as moment quantities, and the Eddington factor. We consider several cases; a scattering wind with a luminous central core, an isothermal wind without a core, a scattering accretion on to a luminous core, and an adiabatic accretion on to a dark core. In the typical wind case with a luminous core, the emergent intensity is enhanced at the center due to the Doppler boost, while it reduces at the outskirts due to the transverse Doppler effect. In contrast to the plane-parallel case, the behavior of the Eddington factor is rather complicated in each case, since the Eddington factor depends on the optical depth, the flow velocity, and other parameters.
2. Skyrme's interaction beyond the mean-field. The DGCM+GOA Hamiltonian of nuclear quadrupole motion
Kluepfel, Peter
2008-07-29
This work focuses on the microscopic description of nuclear collective quadrupole motion within the framework of the dynamic Generator-Coordinate-Method(DGCM)+Gaussian-Overlap-Approximation(GOA). Skyrme-type effective interactions are used as the fundamental many-particle interaction. Starting from a rotational invariant, polynomial and topologic consistent formulation of the GCM+GOA Hamiltonian an interpolation scheme for the collective masses and potential is developed. It allows to define the collective Hamiltonian of fully triaxial collective quadrupole dynamics from a purely axial symmetric configuration space. The substantial gain in performance allows the self-consistent evaluation of the dynamic quadrupole mass within the ATDHF-cranking model. This work presents the first large-scale analysis of quadrupole correlation energies and lowlying collective states within the DGCM+GOA model. Different Skyrme- and pairing interactions are compared from old standards up to more recent parameterizations. After checking the validity of several approximations to the DGCM+GOA model - both on the mean-field and the collective level - we proceed with a detailed investigation of correlation effects along the chains of semi-magic isotopes and isotones. This finally allows to define a set of observables which are hardly affected by collective correlations. Those observables were used for a refit of a Skyrme-type effective interaction which is expected to cure most of the problems of the recent parameterizations. Preparing further work, estimates for the correlated ground state energy are proposed which can be evaluated directly from the mean-field model. (orig.)
3. On the relativistic mass function and averaging in cosmology
Ostrowski, Jan J; Roukema, Boudewijn F
2016-01-01
The general relativistic description of cosmological structure formation is an important challenge from both the theoretical and the numerical point of views. In this paper we present a brief prescription for a general relativistic treatment of structure formation and a resulting mass function on galaxy cluster scales in a highly generic scenario. To obtain this we use an exact scalar averaging scheme together with the relativistic generalization of Zel'dovich's approximation (RZA) that serves as a closure condition for the averaged equations.
4. Isotropic wave turbulence with simplified kernels: Existence, uniqueness, and mean-field limit for a class of instantaneous coagulation-fragmentation processes
Merino-Aceituno, Sara
2016-12-01
The isotropic 4-wave kinetic equation is considered in its weak formulation using model (simplified) homogeneous kernels. Existence and uniqueness of solutions is proven in a particular setting where the kernels have a rate of growth at most linear. We also consider finite stochastic particle systems undergoing instantaneous coagulation-fragmentation phenomena and give conditions in which this system approximates the solution of the equation (mean-field limit).
5. General relativistic corrections and non-Gaussianity
Villa, Eleonora; Matarrese, Sabino
2014-01-01
General relativistic cosmology cannot be reduced to linear relativistic perturbations superposed on an isotropic and homogeneous (Friedmann-Robertson-Walker) background, even though such a simple scheme has been successfully applied to analyse a large variety of phenomena (such as Cosmic Microwave Background primary anisotropies, matter clustering on large scales, weak gravitational lensing, etc.). The general idea of going beyond this simple paradigm is what characterises most of the efforts made in recent years: the study of second and higher-order cosmological perturbations including all general relativistic contributions -- also in connection with primordial non-Gaussianities -- the idea of defining large-scale structure observables directly from a general relativistic perspective, the various attempts to go beyond the Newtonian approximation in the study of non-linear gravitational dynamics, by using e.g., Post-Newtonian treatments, are all examples of this general trend. Here we summarise some of these ...
6. Relativistic calculations of coalescing binary neutron stars
Joshua Faber; Phillippe Grandclément; Frederic Rasio
2004-10-01
We have designed and tested a new relativistic Lagrangian hydrodynamics code, which treats gravity in the conformally flat approximation to general relativity. We have tested the resulting code extensively, finding that it performs well for calculations of equilibrium single-star models, collapsing relativistic dust clouds, and quasi-circular orbits of equilibrium solutions. By adding a radiation reaction treatment, we compute the full evolution of a coalescing binary neutron star system. We find that the amount of mass ejected from the system, much less than a per cent, is greatly reduced by the inclusion of relativistic gravitation. The gravity wave energy spectrum shows a clear divergence away from the Newtonian point-mass form, consistent with the form derived from relativistic quasi-equilibrium fluid sequences.
7. Energy spectra in relativistic electron precipitation events.
Rosenberg, T. J.; Lanzerotti, L. J.; Bailey, D. K.; Pierson, J. D.
1972-01-01
Two events in August 1967, categorized as relativistic electron precipitation (REP) events by their effect on VHF transmissions propagated via the forward-scatter mode, have been examined with regard to the energy spectra of trapped and precipitated electrons. These two substorm-associated events August 11 and August 25 differ with respect to the relativistic, trapped electron population at synchronous altitude; in the August 25 event there was a nonadiabatic enhancement of relativistic (greater than 400 keV) electrons, while in the August 11 event no relativistic electrons were produced. In both events electron spectra deduced from bremsstrahlung measurements (made on a field line close to that of the satellite) had approximately the same e-folding energies as the trapped electron enhancements. However, the spectrum of electrons in the August 25 event was significantly harder than the spectrum in the event of August 11.
8. Density functional theory and dynamical mean-field theory. A way to model strongly correlated systems
Backes, Steffen
2017-04-15
The study of the electronic properties of correlated systems is a very diverse field and has lead to valuable insight into the physics of real materials. In these systems, the decisive factor that governs the physical properties is the ratio between the electronic kinetic energy, which promotes delocalization over the lattice, and the Coulomb interaction, which instead favours localized electronic states. Due to this competition, correlated electronic systems can show unique and interesting properties like the Metal-Insulator transition, diverse phase diagrams, strong temperature dependence and in general a high sensitivity to the environmental conditions. A theoretical description of these systems is not an easy task, since perturbative approaches that do not preserve the competition between the kinetic and interaction terms can only be applied in special limiting cases. One of the most famous approaches to obtain the electronic properties of a real material is the ab initio density functional theory (DFT) method. It allows one to obtain the ground state density of the system under investigation by mapping onto an effective non-interacting system that has to be found self-consistently. While being an exact theory, in practical implementations certain approximations have to be made to the exchange-correlation potential. The local density approximation (LDA), which approximates the exchange-correlation contribution to the total energy by that of a homogeneous electron gas with the corresponding density, has proven quite successful in many cases. Though, this approximation in general leads to an underestimation of electronic correlations and is not able to describe a metal-insulator transition due to electronic localization in the presence of strong Coulomb interaction. A different approach to the interacting electronic problem is the dynamical mean-field theory (DMFT), which is non-perturbative in the kinetic and interaction term but neglects all non
9. Large pseudocounts and L2-norm penalties are necessary for the mean-field inference of Ising and Potts models
Barton, J. P.; Cocco, S.; De Leonardis, E.; Monasson, R.
2014-07-01
The mean-field (MF) approximation offers a simple, fast way to infer direct interactions between elements in a network of correlated variables, a common, computationally challenging problem with practical applications in fields ranging from physics and biology to the social sciences. However, MF methods achieve their best performance with strong regularization, well beyond Bayesian expectations, an empirical fact that is poorly understood. In this work, we study the influence of pseudocount and L2-norm regularization schemes on the quality of inferred Ising or Potts interaction networks from correlation data within the MF approximation. We argue, based on the analysis of small systems, that the optimal value of the regularization strength remains finite even if the sampling noise tends to zero, in order to correct for systematic biases introduced by the MF approximation. Our claim is corroborated by extensive numerical studies of diverse model systems and by the analytical study of the m-component spin model for large but finite m. Additionally, we find that pseudocount regularization is robust against sampling noise and often outperforms L2-norm regularization, particularly when the underlying network of interactions is strongly heterogeneous. Much better performances are generally obtained for the Ising model than for the Potts model, for which only couplings incoming onto medium-frequency symbols are reliably inferred.
10. Large pseudocounts and L2-norm penalties are necessary for the mean-field inference of Ising and Potts models.
Barton, J P; Cocco, S; De Leonardis, E; Monasson, R
2014-07-01
The mean-field (MF) approximation offers a simple, fast way to infer direct interactions between elements in a network of correlated variables, a common, computationally challenging problem with practical applications in fields ranging from physics and biology to the social sciences. However, MF methods achieve their best performance with strong regularization, well beyond Bayesian expectations, an empirical fact that is poorly understood. In this work, we study the influence of pseudocount and L(2)-norm regularization schemes on the quality of inferred Ising or Potts interaction networks from correlation data within the MF approximation. We argue, based on the analysis of small systems, that the optimal value of the regularization strength remains finite even if the sampling noise tends to zero, in order to correct for systematic biases introduced by the MF approximation. Our claim is corroborated by extensive numerical studies of diverse model systems and by the analytical study of the m-component spin model for large but finite m. Additionally, we find that pseudocount regularization is robust against sampling noise and often outperforms L(2)-norm regularization, particularly when the underlying network of interactions is strongly heterogeneous. Much better performances are generally obtained for the Ising model than for the Potts model, for which only couplings incoming onto medium-frequency symbols are reliably inferred.
11. Relativistic Remnants of Non-Relativistic Electrons
Kashiwa, Taro
2015-01-01
Electrons obeying the Dirac equation are investigated under the non-relativistic $c \\mapsto \\infty$ limit. General solutions are given by derivatives of the relativistic invariant functions whose forms are different in the time- and the space-like region, yielding the delta function of $(ct)^2 - x^2$. This light-cone singularity does survive to show that the charge and the current density of electrons travel with the speed of light in spite of their massiveness.
12. Asynchronous stochastic approximation with differential inclusions
David S. Leslie
2012-01-01
Full Text Available The asymptotic pseudo-trajectory approach to stochastic approximation of Benaïm, Hofbauer and Sorin is extended for asynchronous stochastic approximations with a set-valued mean field. The asynchronicity of the process is incorporated into the mean field to produce convergence results which remain similar to those of an equivalent synchronous process. In addition, this allows many of the restrictive assumptions previously associated with asynchronous stochastic approximation to be removed. The framework is extended for a coupled asynchronous stochastic approximation process with set-valued mean fields. Two-timescales arguments are used here in a similar manner to the original work in this area by Borkar. The applicability of this approach is demonstrated through learning in a Markov decision process.
13. Beyond the mean field in the particle-vibration coupling scheme
Baldo, M; Colo', G; Rizzo, D; Sciacchitano, L
2015-01-01
The Energy Density Functional theory is one of the most used methods developed in nuclear structure. It is based on the assumption that the energy of the ground state is a functional only of the density profile. The method is extremely successful within the effective force approach, noticeably the Skyrme or Gogny forces, in reproducing the nuclear binding energies and other bulk properties along the whole mass table. Although the Density Functional is in this case represented formally as the Hartree-Fock mean field of an effective force, the corresponding single-particle states in general do not reproduce the phenomenology particularly well. To overcome this difficulty, a strategy has been developed where the effective force is adjusted to reproduce directly the single particle energies, trying to keep the ground state energy sufficiently well reproduced. An alternative route, that has been developed along several years, for solving this problem is to introduce the mean field fluctuations, as represented by t...
14. Mean field theory for Lyapunov exponents and KS entropy in Lorentz lattice gases
Ernst, M H; Nix, R; Jacobs, D; Ernst, M H; Dorfman, J R; Nix, R; Jacobs, D
1994-01-01
automata lattice gases are useful systems for systematically exploring the connections between non-equilibrium statistical mechanics and dynamical systems theory. Here the chaotic properties of a Lorentz lattice gas are studied analytically and by means of computer simulations. The escape-rates, Lyapunov exponents, and KS entropies are estimated for a one- dimensional example using a mean field theory. The results are compared with simulations for a range of densities and scattering parameters of the lattice gas. The computer results show a distribution of values for the dynamical quantities with average values that are in good agreement with the mean field theory and consistent with the escape-rate formalism for the coefficient of diffusion.
15. Mean-Field Limit and Phase Transitions for Nematic Liquid Crystals in the Continuum
Bachmann, Sven; Genoud, François
2017-08-01
We discuss thermotropic nematic liquid crystals in the mean-field regime. In the first part of this article, we rigorously carry out the mean-field limit of a system of N rod-like particles as N→ ∞, which yields an effective `one-body' free energy functional. In the second part, we focus on spatially homogeneous systems, for which we study the associated Euler-Lagrange equation, with a focus on phase transitions for general axisymmetric potentials. We prove that the system is isotropic at high temperature, while anisotropic distributions appear through a transcritical bifurcation as the temperature is lowered. Finally, as the temperature goes to zero we also prove, in the concrete case of the Maier-Saupe potential, that the system converges to perfect nematic order.
16. Mean-field analysis of phase transitions in the emergence of hierarchical society
Okubo, Tsuyoshi; Odagaki, Takashi
2007-09-01
Emergence of hierarchical society is analyzed by use of a simple agent-based model. We extend the mean-field model of Bonabeau [Physica A 217, 373 (1995)] to societies obeying complex diffusion rules where each individual selects a moving direction following their power rankings. We apply this mean-field analysis to the pacifist society model recently investigated by use of Monte Carlo simulation [Physica A 367, 435 (2006)]. We show analytically that the self-organization of hierarchies occurs in two steps as the individual density is increased and there are three phases: one egalitarian and two hierarchical states. We also highlight that the transition from the egalitarian phase to the first hierarchical phase is a continuous change in the order parameter and the second transition causes a discontinuous jump in the order parameter.
17. On the dynamics of mean-field equations for stochastic neural fields with delays
Touboul, Jonathan
2011-01-01
The cortex is composed of large-scale cell assemblies sharing the same individual properties and receiving the same input, in charge of certain functions, and subject to noise. Such assemblies are characterized by specific space locations and space-dependent delayed interactions. The mean-field equations for such systems were rigorously derived in a recent paper for general models, under mild assumptions on the network, using probabilistic methods. We summarize and investigate general implications of this result. We then address the dynamics of these stochastic neural field equations in the case of firing-rate neurons. This is a unique case where the very complex stochastic mean-field equations exactly reduce to a set of delayed differential or integro-differential equations on the two first moments of the solutions, this reduction being possible due to the Gaussian nature of the solutions. The obtained equations differ from more customary approaches in that it incorporates intrinsic noise levels nonlinearly ...
18. Analytical slave-spin mean-field approach to orbital selective Mott insulators
Komijani, Yashar; Kotliar, Gabriel
2017-09-01
We use the slave-spin mean-field approach to study particle-hole symmetric one- and two-band Hubbard models in the presence of Hund's coupling interaction. By analytical analysis of the Hamiltonian, we show that the locking of the two orbitals vs orbital selective Mott transition can be formulated within a Landau-Ginzburg framework. By applying the slave-spin mean field to impurity problems, we are able to make a correspondence between impurity and lattice. We also consider the stability of the orbital selective Mott phase to the hybridization between the orbitals and study the limitations of the slave-spin method for treating interorbital tunnelings in the case of multiorbital Bethe lattices with particle-hole symmetry.
19. Heterogeneous mean field for neural networks with short-term plasticity
di Volo, Matteo; Burioni, Raffaella; Casartelli, Mario; Livi, Roberto; Vezzani, Alessandro
2014-08-01
We report about the main dynamical features of a model of leaky integrate-and-fire excitatory neurons with short-term plasticity defined on random massive networks. We investigate the dynamics by use of a heterogeneous mean-field formulation of the model that is able to reproduce dynamical phases characterized by the presence of quasisynchronous events. This formulation allows one to solve also the inverse problem of reconstructing the in-degree distribution for different network topologies from the knowledge of the global activity field. We study the robustness of this inversion procedure by providing numerical evidence that the in-degree distribution can be recovered also in the presence of noise and disorder in the external currents. Finally, we discuss the validity of the heterogeneous mean-field approach for sparse networks with a sufficiently large average in-degree.
20. Macroscopic and large scale phenomena coarse graining, mean field limits and ergodicity
This book is the offspring of a summer school school “Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity”, which was held in 2012 at the University of Twente, the Netherlands. The focus lies on mathematically rigorous methods for multiscale problems of physical origins. Each of the four book chapters is based on a set of lectures delivered at the school, yet all authors have expanded and refined their contributions. Francois Golse delivers a chapter on the dynamics of large particle systems in the mean field limit and surveys the most significant tools and methods to establish such limits with mathematical rigor. Golse discusses in depth a variety of examples, including Vlasov--Poisson and Vlasov--Maxwell systems. Lucia Scardia focuses on the rigorous derivation of macroscopic models using $\\Gamma$-convergence, a more recent variational method, which has proved very powerful for problems in material science. Scardia illustrates this by various basic examples and a mor... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9027737379074097, "perplexity": 1324.745318611674}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257648177.88/warc/CC-MAIN-20180323024544-20180323044544-00520.warc.gz"} |
http://tex.stackexchange.com/questions/105721/aligning-parts-of-equations/105725 | # Aligning parts of equations
So this is a continuation of my previous question but I didn't know if I should edit that post and ask a new question or just post a new question entirely. Anyway, I have this bit of latex:
\begin{align*}
C_{in-nand3} & = \frac{5.493 * 1.21739}{3.3098} = 2.02 \rightarrow & \frac{2.02}{2.75} = .735 \\
C_{in-nand2} & = \frac{(2.02 + 12.97) * 1.21739}{3.3098} = 5.514 \rightarrow & \frac{5.514}{2.75} = 2.005 \\
C_{in-nand1} & = \frac{5.514 * 1.21739}{3.3098} = 2.0279 \rightarrow & \frac{2.0279}{2.75} = .7374
\end{align*}
And that gives me:
But what I really want is the answers to the first equation to be centered down a middle column. So for example, I want =2.02-> to be shifted to the right and aligned with all the following values such as =5.514-> down a column. I tried adding a & before the =2.02 but that gave me:
Which is not what I want, that is shifted waaay too much.
-
Have you tried using $$...$$? – cryptic0 Mar 28 '13 at 16:50
@cryptic0 do I have to wrap $$...$$ around every single line? Because I just posted 3 lines of equations, it's actually much more than that – Richard Mar 28 '13 at 17:08
perhaps this is closer to what you want?
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{alignat*}{5}
C_{in-nand3} & = \frac{5.493 * 1.21739}{3.3098} && = 2.02 \rightarrow{} & \frac{2.02}{2.75} & = .735 \\
C_{in-nand2} & = \frac{(2.02 + 12.97) * 1.21739}{3.3098} && = 5.514 \rightarrow{} & \frac{5.514}{2.75} & = 2.005 \\
C_{in-nand1} & = \frac{5.514 * 1.21739}{3.3098} && = 2.0279 \rightarrow{} & \frac{2.0279}{2.75} & = .7374
\end{alignat*}
\end{document}
edit: the question Aligning equations with text with alignat addresses a similar problem.
-
Okay this worked, thanks a lot ! – Richard Mar 28 '13 at 17:15
Maybe you could mention a link to the question Aligning equations with text with alignat and I'll delete my almost identical answer ? – Vincent Nivoliers Mar 28 '13 at 17:26
@VincentNivoliers -- done. thanks for the suggestion. – barbara beeton Mar 28 '13 at 19:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.89567631483078, "perplexity": 2463.599872778688}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223203841.5/warc/CC-MAIN-20140423032003-00537-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://www.loseyourmind.com/printersapprentice/help/Reference_Install_Fonts.aspx | Printer's Apprentice 8.1 Documentation
Contents
The Install Fonts dialog makes a font file available for use with other Windows applications.
Behind the Scenes
When a font is installed, the following steps take place behind the scenes.
1. If specified, the font file is copied to a target folder. This is usually %windir%\fonts.
2. The font is made available to Windows and other applications by executing the AddFontResource() API call.
3. An entry is added to the HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows NT\CurrentVersion\Fonts section in the Registry. This makes the font available the next time Windows is rebooted.
4. Other applications are notified of the font change by using a SendMessage(hwnd, WM_FONTCHANGE) call.
Font Destinations
The Install Fonts dialog has options that allow you select a destination folder before the fonts are installed. You might use this if you prefer to house fonts on a separate drive from the Windows OS.
When installing fonts from a font group, the destination options are not available. Fonts from a group are always copied to the %windir%\fonts folder.
Fonts to Install tab
This tab lists the fonts that will be installed. The list includes the source font file, the file size and the Registry entry that will be created. The Registry Entry column will change depending on the destination folder for the font.
The bottom of the tab displays a summary of the font install actions to be taken, depending on the Options selected.
Options tab - Copy font files to Windows Fonts folder
When this option is selected, the font file will be copied to the %windir%\fonts folder before it is installed. This is the recommended option to use.
Options tab - Copy font files to a different folder
When this option is selected, the font file will be copied to another folder before it is installed. The Registry Entry used will change to "My Font Name (TrueType) = c:\targetfolder\myfont.ttf."
Options tab - Leave fonts where they are
When this option is selected, the font file will not before it is installed. The Registry Entry used will change to "My Font Name (TrueType) = c:\sourcefolder\myfont.ttf."
Options tab - Create log file
Use this to create a log file detailing the results of the font install operation. This is helpful for debugging any problems that may arise.
Options tab - Show log file when finished
When checked, the log file will open in Notepad when the font install process is complete. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8607096076011658, "perplexity": 4546.988566222396}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141195198.31/warc/CC-MAIN-20201128070431-20201128100431-00319.warc.gz"} |
https://arizona.pure.elsevier.com/en/publications/effects-of-boundary-conditions-on-magnetization-switching-in-kine | # Effects of boundary conditions on magnetization switching in kinetic Ising models of nanoscale ferromagnets
Howard L. Richards, Miroslav Kolesik, Per Anker Lindgård, Per Arne Rikvold, M. A. Novotny
Research output: Contribution to journalArticle
36 Citations (Scopus)
### Abstract
Magnetization switching in highly anisotropic single-domain ferromagnets has been previously shown to be qualitatively described by the droplet theory of metastable decay and simulations of two-dimensional kinetic Ising systems with periodic boundary conditions. In this paper we consider the effects of boundary conditions on the switching phenomena, A rich range of behaviors is predicted by droplet theory: the specific mechanism by which switching occurs depends on the structure of the boundary, the particle size, the temperature, and the strength of the applied field. The theory predicts the existence of a peak in the switching field as a function of system size in both systems with periodic boundary conditions and in systems with boundaries. The size of the peak is strongly dependent on the boundary effects. It is generally reduced by open boundary conditions, and in some cases it disappears if the boundaries are too favorable towards nucleation. However, we also demonstrate conditions under which the peak remains discernible. This peak arises as a purely dynamic effect and is not related to the possible existence of multiple domains. We illustrate the predictions of droplet theory by Monte Carlo simulations of two-dimensional Ising systems with various system shapes and boundary conditions.
Original language English (US) 11521-11540 20 Physical Review B - Condensed Matter and Materials Physics 55 17 Published - May 1 1997 Yes
### Fingerprint
Ising model
Magnetization
Boundary conditions
boundary conditions
magnetization
Kinetics
kinetics
Nucleation
simulation
Particle size
nucleation
decay
predictions
### ASJC Scopus subject areas
• Condensed Matter Physics
### Cite this
Effects of boundary conditions on magnetization switching in kinetic Ising models of nanoscale ferromagnets. / Richards, Howard L.; Kolesik, Miroslav; Lindgård, Per Anker; Rikvold, Per Arne; Novotny, M. A.
In: Physical Review B - Condensed Matter and Materials Physics, Vol. 55, No. 17, 01.05.1997, p. 11521-11540.
Research output: Contribution to journalArticle
Richards, Howard L. ; Kolesik, Miroslav ; Lindgård, Per Anker ; Rikvold, Per Arne ; Novotny, M. A. / Effects of boundary conditions on magnetization switching in kinetic Ising models of nanoscale ferromagnets. In: Physical Review B - Condensed Matter and Materials Physics. 1997 ; Vol. 55, No. 17. pp. 11521-11540.
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title = "Effects of boundary conditions on magnetization switching in kinetic Ising models of nanoscale ferromagnets",
abstract = "Magnetization switching in highly anisotropic single-domain ferromagnets has been previously shown to be qualitatively described by the droplet theory of metastable decay and simulations of two-dimensional kinetic Ising systems with periodic boundary conditions. In this paper we consider the effects of boundary conditions on the switching phenomena, A rich range of behaviors is predicted by droplet theory: the specific mechanism by which switching occurs depends on the structure of the boundary, the particle size, the temperature, and the strength of the applied field. The theory predicts the existence of a peak in the switching field as a function of system size in both systems with periodic boundary conditions and in systems with boundaries. The size of the peak is strongly dependent on the boundary effects. It is generally reduced by open boundary conditions, and in some cases it disappears if the boundaries are too favorable towards nucleation. However, we also demonstrate conditions under which the peak remains discernible. This peak arises as a purely dynamic effect and is not related to the possible existence of multiple domains. We illustrate the predictions of droplet theory by Monte Carlo simulations of two-dimensional Ising systems with various system shapes and boundary conditions.",
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AU - Novotny, M. A.
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N2 - Magnetization switching in highly anisotropic single-domain ferromagnets has been previously shown to be qualitatively described by the droplet theory of metastable decay and simulations of two-dimensional kinetic Ising systems with periodic boundary conditions. In this paper we consider the effects of boundary conditions on the switching phenomena, A rich range of behaviors is predicted by droplet theory: the specific mechanism by which switching occurs depends on the structure of the boundary, the particle size, the temperature, and the strength of the applied field. The theory predicts the existence of a peak in the switching field as a function of system size in both systems with periodic boundary conditions and in systems with boundaries. The size of the peak is strongly dependent on the boundary effects. It is generally reduced by open boundary conditions, and in some cases it disappears if the boundaries are too favorable towards nucleation. However, we also demonstrate conditions under which the peak remains discernible. This peak arises as a purely dynamic effect and is not related to the possible existence of multiple domains. We illustrate the predictions of droplet theory by Monte Carlo simulations of two-dimensional Ising systems with various system shapes and boundary conditions.
AB - Magnetization switching in highly anisotropic single-domain ferromagnets has been previously shown to be qualitatively described by the droplet theory of metastable decay and simulations of two-dimensional kinetic Ising systems with periodic boundary conditions. In this paper we consider the effects of boundary conditions on the switching phenomena, A rich range of behaviors is predicted by droplet theory: the specific mechanism by which switching occurs depends on the structure of the boundary, the particle size, the temperature, and the strength of the applied field. The theory predicts the existence of a peak in the switching field as a function of system size in both systems with periodic boundary conditions and in systems with boundaries. The size of the peak is strongly dependent on the boundary effects. It is generally reduced by open boundary conditions, and in some cases it disappears if the boundaries are too favorable towards nucleation. However, we also demonstrate conditions under which the peak remains discernible. This peak arises as a purely dynamic effect and is not related to the possible existence of multiple domains. We illustrate the predictions of droplet theory by Monte Carlo simulations of two-dimensional Ising systems with various system shapes and boundary conditions.
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https://www.physicsforums.com/threads/nmr-theory-relaxation-and-precession.593376/ | # NMR Theory, Relaxation and Precession
1. ### collateral
4
Hello everyone, I'm having a really confusing time trying to get my head around these concepts, so I will try to explain what I can...
So, in proton NMR we have nuclei that can be in either spin-up or spin-down states. Nuclei align with an external magnetic field but precess. (This precession is more towards the z axis?). Applying a second, othogonal magnetic field at the precession frequency (Larmor), will cause the precession to go in the xy plane (Im not too sure about this either - sort of like a spinning top completely on its side if i can visualise it). When this happens, the nuclei can switch between states. Because of the small difference in states (Boltzmann), this can be seen via absorption at a particular RF.
So is spin-lattice relaxation when the precessing goes from xy plane (spinning top on it side) to the z plane (spinning top is now perpendicular to the surface)?
And where does spin-spin relaxation fit into this - from what I understand, its when spins of different nuclei don't correspond to each other. How is this different to T1 relaxation?
Thanks for any help!
2. ### marcusl
2,126
You are mixing up quantum and classical pictures of NMR. I suggest sticking with the classical picture (which has no concept of states) and stick with classical. The spins align with B0. An applied RF field B1 causes the spins to precess and increases their polar angle theta (angle away from the z axis) continuously as long as B1 is applied. The spins end up in the xy plane only if B1 is turned off when they reach theta=90°. As they continue to precess in the xy plane, a signal is induced in the pickup coil. If B1 is left on longer, the spins precess down to 180°, that is, the magnetic moment is opposite to B0.
Spin-lattice relaxation describes the transfer of energy from the spin system to the lattice. Following a 180° RF pulse, the spins are pointing along -z but they relax back to +z through this mechanism.
Spin-spin relaxation doesn't remove energy, but it changes the local magnetic field such that spins following a 90° become decoherent (neighboring spins no longer track each other). This causes the signal to disappear.
3. ### nez
10
Precession cone from xy plane to z axis is spin-spin relaxationT2(exchange of energy b/w nuclie spins) after 90o pulse applied to spin system.
This T2relaxation is insuffecient to detect practical useful decaying signal, but when 180o pulse is applied, in which magnetic moment will take initially -z axis direction, spin-lattice relaxationT1(exchange of energy with suroundings"atomic vibrations or molecular tumblings") takes place.
So, this is the practical technique(Multiple-Pulse FT):
We start with T1 process(180o pulse) then applyT2 process(90o pulse) after approperiate delay time(magnetic moment in +z direction) then the entensity of signal will be proportional to magnetic moment!
repeating this several times gives us our lovely spetrum by computer
4. ### collateral
4
Why is there is a signal induced when it's precessing in the transverse plane? I always thought this was because the spins with B0 all can go to spins against B0 when they absorb photons. Also, why do we use a 90° pulse as opposed to a continuous wave (because wouldn't a continuous wave saturate the higher energy state).
Why does precession at θ=90° allow the nuclei to from spin-up to spin-down?
I understand more, but I just wish to clear some stuff up and remove confusion which is frustating. Anyway thanks guys!
Last edited: Apr 5, 2012
5. ### DrDu
4,351
The spins cannot only bei either up or down but you can have any possible quantum mechanical superposition of these two states. The up and down states are solutions of the time independent Schroedinger equation for a spin in the magnetic field corresponding to two different energy eigenvalues. A superposition of the two will have a non-vanishing expectation value for the magentic moment having a component in the xy plane. The direction will precess with the Lamor frequency. Any rotating dipole will emit radiation which can be observed in the NMR and will drive it back to the z axis.
It is not so easy to understand this emission in the picture of photons getting absorbed or emitted. The main point is that the phase of the electromagnetic wave is due to the superposition of states with different photon number (just like the angle of the magnetic moment in the xy plane depends on the phase of the superposition of the up and down states). To get a classical picture you have to consider both superpositions of spin eigenstates and of photon number eigenstates.
6. ### collateral
4
OK, im a biochem, so I hope that explains my confusion. Looking at other threads, I realise that I was trying to find out why applying a B1 field orthogonal to B0 would cause the spins to change state. I realise that this is now in quantum territory, which I know nothing about - all I understand is that when the precession is in the xy plane, the chance that the spins can change states is increased compared to when the precession is near the z axis.
PS. what do you mean by non-vanishing expectation?
7. ### DrDu
4,351
I fear that is a term from quantum mechanics, too.
However if you are willing to forget for some time on what you know about spin up and down, NMR can be understood to a large part in purely classical terms:
http://en.wikipedia.org/wiki/Bloch_equations
8. ### DrDu
4,351
If you apply an additional field B1, the spins will rotate around the vectorial sum of B_0 and _1. How this appears in QM depends on what you measure. If you measure the projection of spin onto the z-axis, you will find that after application of B_1, a certain part of spins has flipped.
9. ### ZapperZ
30,166
Staff Emeritus
10. ### collateral
4
Thanks ZapperZ (and everyone else), I think I was mixing up quantum and classical too. The explanations helped as well.
I have another question now regarding T1...When we say that the states become saturated is it that all the spin states are in the higher energy state? Because if we lowered the temp to 0K, wouldn't nearly all of the spins be at the lower state?
Secondly, I read this online here
It has confused me: I thought T1 was dependent on z magnetisation so how does fluctuations in the xy plane affect it, and how is it not spontaneous?
I was also looking at some lecture notes and another thing which caught me was linewidth. The lecture says that line width is largely independent of field strength. But on another of the notes I was reading online, it mentioned that T2 and T1 have an effect on line width. Is this correct and if this is, doesn't field strength have an effect on relaxation?
Again thanks a lot guys, it has cleared up a lot but replaced it with more questions!
11. ### Mike H
491
Saturation refers to the situation when the populations of each state have been equalized. If you apply enough strong pulses continuously, the populations will equilibrate. If you read the discussion that was linked earlier by ZapperZ, you will note that ZapperZ mentions the situation at T = 0.
It may help to think about what T1 is - it tells you how long it takes for the system to be restored to its thermal equilibrium values. It has to interact with its surroundings to do that, as you just perturbed it with the RF pulses of an NMR experiment. The sample is in the presence of an incredibly strong static magnetic field that runs along the z-axis. Fluctuations along the z-axis are most likely going to be quite small when compared to the static magnetic field. Remember, for a 14.4 Tesla magnetic field, the proton Larmor frequency is 600 MHz - that means in one second, it precesses 600 million times about the static magnetic field. For something to be "spontaneous" - at least in a practical sense - would mean it occurs prior to a single precession.
I don't know what other notes you've been reading, but I can certainly say the following. In principle, at least for a simple enough system, one can show that the Lorentzian peak width at half-height is inversely related to T2. Of course, in the lab, things can be more complicated - there can be effects from imperfections in the static magnetic field, magnetic susceptibility of the sample, and I believe related instrumental issues whose details I am not remembering at the moment. Clearly, in certain cases, the two times can be basically the same magnitude (I'm thinking of quadrupolar nuclei), and relaxation becomes extremely efficient. Otherwise, you're going to have to point us to these other sets of notes you've been reading.
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https://brilliant.org/problems/fraction-meor-fraction-you/ | # fractions
Level pending
There is a tank filled with 1/3 water, 24 liters is poured into the tank and the water in the tank became 1/2. What is the total capacity of the tank?
× | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9566304087638855, "perplexity": 779.431517821317}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719542.42/warc/CC-MAIN-20161020183839-00549-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://emacs.stackexchange.com/questions/7157/how-to-debug-org-mode-export-to-beamer/7161 | # How to debug org-mode export to Beamer?
When using org-mode to create Beamer slides, sometimes the error shown is
"Pdf file ... wasn't produced"
or the previous PDF file generated is opened (without the changes made) when the command C-c C-e l o is issued to open the PDF directly.
However, the .tex file generated from export to Beamer when compiled separately does give the PDF.
Since there is no debug message given, I am not sure how to detect why the export to Beamer to open the slides directly is not working.
How can we detect the source of error in exporting org files ?
Thanks.
• Not sure if you made a typo, but the key sequence to create a beamer PDF and open it is C-x C-e l O and not C-x C-e l o, which is the sequence for creating a normal latex file and opening it. If that isn't the problem, do you get an error when you just do C-x C-e l P to generate the beamer PDF? If not, then at least you know the problem is likely with the view command or arguments and not with the PDF generation step – Tim X Feb 21 '15 at 9:10
• Sorry, should be C-c C-e l O etc. and not C-x C-e l O (I'm using the emacs SX package to browse the gorup and haven't worked out how to edit comments yet!) – Tim X Feb 21 '15 at 9:13
• @TimX There is no typo. C-c C-e with .org file shows all the options and difference between o and O is for latex and beamer option. The only difference between option o and p is whether the pdf is opened or not and therefore not the error or cause of error. Simply use the pen like symbol next to comment to edit. – Anusha Feb 21 '15 at 18:20
• I wasn't being clear enough. My point is that the command you show as causing the error is C-c C-e l o, but on my system, for a latex beamer presentation, it should be C-c C-e l O (upper case o). Finally, with Emacs SX package, I see no pen like symbol next to comment. – Tim X Feb 21 '15 at 22:32
As you mention, org-mode does not make use of the echo area to show export errors. Instead, it dumps all output that is produced during LaTeX export to a buffer called *Org PDF LaTeX Output*. If you want to get information about why the export process failed, you can switch to this buffer and search for error (or even warning).
### Example
Contents of .org file:
* Heading
\begin{algorithm}
...
\end{algorithm}
Contents of *Org PDF LaTeX Output* buffer (abbreviated to show relevant parts only):
This is pdfTeX, Version 3.14159265-2.6-1.40.15 ... (preloaded format=pdflatex) ...
LaTeX2e <2014/05/01>
...
LaTeX Warning: No \author given.
(.//beamer.toc)
! LaTeX Error: Environment algorithm undefined.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.32 \begin{algorithm}
! LaTeX Error: \begin{document} ended by \end{algorithm}.
See the LaTeX manual or LaTeX Companion for explanation.
Type H <return> for immediate help.
...
l.33 \end{algorithm}
...
In this case, the output tells us that the error was caused by the \begin{algorithm} declaration in line 32. Note that specifications of line numbers (l.32, l.33, etc.) refer to the .tex file, not to the original .org file.
• Thank you for the reply. But why does the .tex file generated compile properly ? It usually happens that the export to beamer is working properly and then after some more input, without changing the syntax and only plain text, or apparent reason, there is an error in org export to pdf but .tex file compiles. – Anusha Jan 8 '15 at 9:40
• @Anusha I can't really answer that because I've never had that happen. You could try checking the *Messages* buffer (C-h e) for complaints coming directly from org-mode when you run into this problem. If you want me (or anyone else who comes across your post) to be able to give you more detailed feedback, please add a minimal example to your question that allows us to reproduce the error (i.e., a version of the file that compiles cleanly and a portion of text that makes the export process fail when adding it to the file). – itsjeyd Jan 8 '15 at 10:06
• @Anusha You should also make sure that the problem isn't caused by your personal configuration. Try to reproduce the behavior with a minimal configuration that consists only of the code that is necessary to enable Beamer export (AFAIK, (require 'ox-beamer) is all you really need). Let me know if you need more help with that. – itsjeyd Jan 8 '15 at 10:22 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8396041393280029, "perplexity": 2482.250466581599}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655897707.23/warc/CC-MAIN-20200708211828-20200709001828-00274.warc.gz"} |
https://www.graphicmaths.com/gcse/geometry/tangent-radius/ | # Tangent and radius of a circle meet at 90°
Martin McBride
2020-08-16
A tangent is a line that just touches the circle at a single point on its circumference.
If we draw a radius that meets the circumference at the same point, the angle between the radius and the tangent will always be exactly 90°.
This theorem is covered in this video on circle theorems:
## Proof
You aren't required to learn this proof for GCSE, it is just here for information.
We want to prove that the angle between the radius AB and the tangent CD is a right angle.
The way we will do this is to take some other point P on the tangent and prove that the line AP cannot be perpendicular CD. If we prove that this cannot be true for any point P, then it follows that AB must be perpendicular to CD.
We will start by assuming that ∠APB is a right angle. We will then show that this leads to an impossibility and so cannot be true.
Consider the triangle formed by points A, B and P. Suppose the angle of the triangle at P (that is∠APB) was a right angle. This would mean that the line AB would be the hypotenuse of the triangle.
We know that the hypotenuse of a triangle is always longer than the other two sides, which means that AB must be longer than AP. But that cannot be true. The line AE is the same length as the line AB, because each one is a radius of the circle. AP is clearly longer than AE, therefore AB cannot be longer than AP. So, the angle ∠APB cannot be a right angle.
Since the line AP cannot be perpendicular to the tangent for any P, it follows that AB must be perpendicular to the tangent. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9259878993034363, "perplexity": 220.66890166711786}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662658761.95/warc/CC-MAIN-20220527142854-20220527172854-00103.warc.gz"} |
https://dorigo.wordpress.com/tag/sld/ | ## Latest global fits to SM observables: the situation in March 2009March 25, 2009
Posted by dorigo in news, physics, science.
Tags: , , , , , , , , , ,
A recent discussion in this blog between well-known theorists and phenomenologists, centered on the real meaning of the experimental measurements of top quark and W boson masses, Higgs boson cross-section limits, and other SM observables, convinces me that some clarification is needed.
The work has been done for us: there are groups that do exactly that, i.e. updating their global fits to express the internal consistency of all those measurements, and the implications for the search of the Higgs boson. So let me go through the most important graphs below, after mentioning that most of the material comes from the LEP electroweak working group web site.
First of all, what goes in the soup ? Many things, but most notably, the LEP I/SLD measurements at the Z pole, the top quark mass measurements by CDF and DZERO, and the W mass measurements by CDF, DZERO, and LEP II. Let us give a look at the mass measurements, which have recently been updated.
For the top mass, the situation is the one pictured in the graph shown below. As you can clearly see, the CDF and DZERO measurements have reached a combined precision of 0.75% on this quantity.
The world average is now at $M_t = 173.1 \pm 1.3 GeV$. I am amazed to see that the first estimate of the top mass, made by a handful of events published by CDF in 1994 (a set which did not even provide a conclusive “observation-level” significance at the time) was so dead-on: the measurement back then was $M_t=174 \pm 15 GeV$! (for comparison, the DZERO measurement of 1995, in their “observation” paper, was $M_t=199 \pm 30 GeV$).
As far as global fits are concerned, there is one additional point to make for the top quark: knowing the top mass any better than this has become, by now, useless. You can see it by comparing the constraints on $M_t$ coming from the indirect measurements and W mass measurements (shown by the blue bars at the bottom of the graph above) with the direct measurements at the Tevatron (shown with the green band). The green band is already too narrow: the width of the blue error bars compared to the narrow green band tells us that the SM does not care much where exactly the top mass is, by now.
Then, let us look at the W mass determinations. Note, the graph below shows the situation BEFORE the latest DZERO result;, obtained with 1/fb of data, and which finds $M_W = 80401 \pm 44 MeV$; its inclusion would not change much of the discussion below, but it is important to stress it.
Here the situation is different: a better measurement would still increase the precision of our comparisons with indirect information from electroweak measurements at the Z. This is apparent by observing that the blue bars have width still smaller than the world average of direct measurements (again in green). Narrow the green band, and you can still collect interesting information on its consistency with the blue points.
Finally, let us look at the global fit: the electroweak working group at LEP displays in the by now famous “blue band plot”, shown below for March 2009 conferences. It shows the constraints on the Higgs boson mass coming from all experimental inputs combined, assuming that the Standard Model holds.
I will not discuss this graph in details, since I have done it repeatedly in the past. I will just mention that the yellow regions have been excluded by direct searches of the Higgs boson at LEP II (on the left, the wide yellow area) and the Tevatron ( the narrow strip on the right). From the plot you should just gather that a light Higgs mass is preferred (the central value being 90 GeV, with +36 and -27 GeV one-sigma error bars). Also, a 95% confidence-level exclusion of masses above 163 GeV is implied by the variation of the global fit $\chi^2$ with Higgs mass.
I have started to be a bit bored by this plot, because it does not do the best job for me. For one thing, the LEP II limit and the Tevatron limit on the Higgs mass are treated as if they were equivalent in their strength, something which could not be possibly farther from the truth. The truth is, the LEP II limit is a very strong one -the probability that the Higgs has a mass below 112 GeV, say, is one in a billion or so-, while the limit obtained recently by the Tevatron is just an “indication”, because the excluded region (160 to 170 GeV) is not excluded strongly: there still is a one-in-twenty chance or so that the real Higgs boson mass indeed lies there.
Another thing I do not particularly like in the graph is that it attempts to pack too much information: variations of $\alpha$, inclusion of low-Q^2 data, etcetera. A much better graph to look at is the one produced by the GFitter group instead. It is shown below.
In this plot, the direct search results are introduced with their actual measured probability of exclusion as a function of Higgs mass, and not just in a digital manner, yes/no, as the yellow regions in the blue band plot. And in fact, you can see that the LEP II limit is a brick wall, while the Tevatron exclusion acts like a smooth increase in the global $\chi^2$ of the fit.
From the black curve in the graph you can get a lot of information. For instance, the most likely values, those that globally have a 1-sigma probability of being one day proven correct, are masses contained in the interval 114-132 GeV. At two-sigma, the Higgs mass is instead within the interval 114-152 GeV, and at three sigma, it extends into the Tevatron-excluded band a little, 114-163 GeV, with a second region allowed between 181 and 224 GeV.
In conclusion, I would like you to take away the following few points:
• Future indirect constraints on the Higgs boson mass will only come from increased precision measurements of the W boson mass, while the top quark has exhausted its discrimination power;
• Global SM fits show an overall very good consistency: there does not seem to be much tension between fits and experimental constraints;
• The Higgs boson is most likely in the 114-132 GeV range (1-sigma bounds from global fits). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 8, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.901347279548645, "perplexity": 733.4664632353577}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585186.33/warc/CC-MAIN-20211018000838-20211018030838-00048.warc.gz"} |
https://vulcanhammer.net/2018/08/14/jean-louis-briauds-pet-peeve-on-the-analysis-of-consolidation-settlement-results/?shared=email&msg=fail | Posted in Soil Mechanics
Jean-Louis Briaud’s “Pet Peeve” on the Analysis of Consolidation Settlement Results
In his recent, excellent article on the settlement (and subsidence) of the San Jacinto Monument east of Houston, Briaud (2018) takes an opportunity to vent a “pet peeve” of his relative to the way consolidation tests are reduced and consolidation properties reported:
A Chance to Share a Pet Peeve
The consolidation e versus log p’ curve is a stress-strain curve. Typically, stress-strain curves are plotted as stress on the vertical axis and strain on the horizontal axis. Both axes are on normal scales, not log scales. It’s my view that consolidation curves should be plotted in a similar fashion: effective vertical stresses on the vertical axis in arithmetic scale, and normal strain on the horizontal axis in arithmetic scale. When doing so, the steel ring confining the test specimen influences the the measurements and skews the stiffness data. Indeed the stress-strain curve, which usually has a downward curvature, has an upward curvature in such a plot. (p. 54)
Is this correct? And is he the only one who thinks this way? The two questions are neither the same nor linked. Although this problem will certainly not be solved in one blog post, it deserves some investigation.
Statement of the Problem
Let’s start with a text we use often here: Verruijt, A., and van Bars, S. (2007). Soil Mechanics. VSSD, Delft, the Netherlands. Early in the presentation on the subject, he presents the following plot:
As Jean-Louis would have us do, the strain (or negative strain, since we’re dealing with compression) is on the abscissa, and the dimensionless stress is on the ordinate. The difference between the two is that the stress is plotted logarithmically. But it’s a step. We’ll come back to that later.
Verruijt defines the relationship between the strain and stress ratio as follows:
$\epsilon = -\frac{1}{C}\ln\frac{\sigma}{\sigma_0}$
This relationship goes back to Terzaghi’s original tests and formulation of settlement and consolidation theory almost a century ago.
From a “conventional” standpoint there are two things wrong with this formulation. The first is that it is based on strain, not void ratio. The second is that it uses the natural logarithm rather than the common one. The last problem can be fixed by rewriting it as follows:
$\epsilon = -\frac{1}{C_{10}}\log\frac{\sigma}{\sigma_0}$
This formulation is essentially the same as is used in Hough’s Method for cohesionless soils, once the strains are converted to displacements by considering the thickness of the layer. So it is not as strange as it looks.
The first problem can be “fixed” by noting the following:
$\epsilon = \frac{e-e_0}{1+e_0}$
We can substitute this into the equation before it and, with judicious changes of the constants and other subsitutions, come up with the familiar, non-preconsolidated formula for consolidation settlement, or
$\Delta H = \frac{C_c H_0}{1+e_0}\log\frac{\sigma}{\sigma_0}$
When we reverse the axes, we then get the “classic” plot as follows:
But is there a problem with using strain? Verruijt explains the two conventions as follows:
In many countries, such as the Scandinavian countries and the USA, the results of a confined compression test are often described in a slightly different form, using the void ratio e to express the deformation, rather than the strain ε…It is of course unfortunate that different coefficients are being used to describe the same phenomenon. This can only be explained by the historical developments in different parts of the world. It is especially inconvenient that in both formulas the constant is denoted by the character C, but in one form it appears in the numerator, and in the other one in the denominator. A large value for $C_{10}$ corresponds to a small value for $C_c$. It can be expected that the compression index $C_c$ will prevail in the future, as this has been standardized by ISO, the International Organization.
As is often the case, the simplest way to help sort out this issue is with an example. Briaud (2018) actually has one, but we will use another.
Example of Settlement Plotting
An example we have used frequently in our teaching of Soil Mechanics is this one, from the Bearing Capacity and Settlement publication. It is a little more complex than the theory shown above because it involves a preconsolidated soil. The plot (with the simplifications for determination of $C_c$ and $C_r$ is shown below.
With this information in hand, we process the data as follows:
1. We convert the void ratio data to strains using the formula above.
2. We convert the stresses to dimensionless stresses by dividing them by the initial stress.
3. We “split” the data up into compression and decompression portions to allow us to develop separate trend lines for both.
First, the strain-dimensionless stress plot, using natural scales for both.
The result is similar to that in Briaud (2018). The compression portion best fits a second-order polynomial fit. (Not that we have thrown out the zero point to allow more fit options.) The decompression portion fits an exponential trend line best.
Below is the same plot with the stress scale now being logarithmic.
This is basically the original graph with the axes reversed. There is no effect using strain; we will discuss the advantages of doing so below.
Now let us look at the data from another angle: the tangent “modulus of elasticity,” defined of course by
$E = \frac{\Delta\sigma}{\Delta\epsilon}$
We consider natural scales for both modulus and strain. To obtain the slope, we used a “central difference” technique except at the ends.
It’s interesting to note that, except for the “kink” caused by preconsolidation, in compression the tangent modulus of elasticity increases somewhat linearly with strain, as it does with the decompression.
Discussion of the Results
There’s a great deal to consider here, and we’ll try to break it down as best as possible.
Use of Strain vs. Void Ratio
The graphs above show that there is no penalty in using strain instead of void ratio to plot the results. The advantage to doing so is both conceptual and pedagogical.
In the compression and settlement of soils, we traditionally conceive of it as a three-stage process: elastic settlement, primary consolidation settlement, and secondary consolidation settlement. Consolidation settlement is nothing more than the rearrangement of particles under load; the time it takes to do so is based in part on the permeability of the soil and its ability to expel pore water trapped in shrinking voids. Elastic settlement is due to the elastic modulus of the material, the strain induced in the material and the geometry of the system. This distinction, however, obscures the fact that we are dealing with one soil system and one settlement. Using strain for all types of settlement would both help unify the problem conceptually and ease the transition to numerical methods such as finite element analysis, where strain is used to estimate deflection. In the past we were able to use a disparate approach without difficulty, but that option is not as viable now as before.
The Natural Scale, Consolidation Settlement Stiffness, and the Ring
Both here and in Briaud (2018) the natural stress-strain curve experiences an upward curvature, which is obviously different from what we normally experience in theory of elasticity/plasticity. This comes into better focus if we consider the variation of the tangent modulus of elasticity, which (except for the aforementioned preconsolidation effect) linearly increases with stress. There are two possible explanations for this.
The first is to observe that, as soils compress in consolidation settlement, their particles come closer together, and thus more resistant to further packing.
The second, as suggested by Briaud (2018), is that the presence of the confining ring in the consolidation test augments the resistance of the particles to further compression. The issue of confinement is an interesting one because in other tests (unconfined compression tests, triaxial tests) confinement is either very flexible or non-existent. It should be observed that consolidation theory, as originally presented, is one-dimensional consolidation theory. For true one-dimensional consolidation, we assume a semi-infinite case where the infinite boundary “confines” the physical phenomena. The use of a confining ring assumes that the ring can replicate this type of confinement in the laboratory. Conditions in the field, with finite loads and variations in the surrounding soils, may not reflect this. While it would be difficult to replicate variations in confinement in the laboratory, these variations should be kept in mind by anyone using laboratory-generated consolidation data.
The “Modulus of Elasticity” for Consolidation Settlement
This may strike many geotechnical engineers (especially those in areas where void ratio is used to estimate consolidation settlement) as an odd concept, but if we consider the material strain vs. its deflection, it is a natural one. Varying moduli of elasticity are nothing new in geotechnical engineering; they have been discussed on this site in detail. The situation here is somewhat different for a wide variety of reasons, not the least of which is that here we are dealing with a tangent modulus while previously we looked at a secant one. Also, differing physical phenomena are at work; theory of elasticity implicitly assumes that particle rearrangement is at a minimum, while consolidation settlement (both primary and secondary) is all about particle rearrangement.
A more unified approach to settlement would probably reveal a process where the change in stress vs. the change in strain varies at differing points in the process along a stress path with multiple irreversibilities. Such an approach would require some significant conceptual changes in the way we look at settlement, but would hopefully result in more accurate results.
Conclusion
Consolidation settlement is a topic that has occupied geotechnical engineering for most of its modern history. While the theory is considered well established, changes in computational methodology will eventually force changes in the way the theory is applied. A good start of this process is to use strain (rather than void ratio) as the measure of the relative deflection of structures, and the example from Briaud (2018), along with the demonstration relative to natural scales, is an excellent start.
References
Briaud, J.-L. (2018) “The San Jacinto Monument.” Geostrata, July/August. Issue 4, Vol. 22, pp. 50-55.
2 thoughts on “Jean-Louis Briaud’s “Pet Peeve” on the Analysis of Consolidation Settlement Results”
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https://www.physicsforums.com/threads/structure-of-matter-in-quantum-field-theory.957085/ | # Structure of Matter in Quantum Field Theory
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This is a topic I've mentioned a few times before. Essentially the structure of matter in quantum gauge field theories is unclear to me. I have no clear question here, just some initial discussion points.
So at the first level, it seems a particle based view of quantum field theory is difficult to maintain. In brief on the Hilbert Spaces of interacting field theories there is no well-defined number operator ##N##. In some texts it's said that states in the interacting theory are a superposition of different particle numbers, but in reality the fact that an ##N## operator doesn't exist means there are no states with a well-defined particle number. You can see this explicitly in Glimm, Osterwalder and Schrader's work on ##\phi^{4}_{3}##, where the number operator for the interacting theory diverges as the cutoff is removed.
In addition to this we have Malament's theorem where he shows that relaitivistic quantum theories don't possess any states with the intuitive property of particles. In essence he shows that there are no local particle creation operators.
Then we have the usual picture of fields and particles being their excitations. Now if particle like states aren't elements of interacting field theory Hilbert spaces, I don't think one can say they are excitations of fields. What one can say is that most field theories can be shown to give rise to states that at late times can be proven to have a well defined number of "clicks" they cause in detectors, with the detectors being represented by specific local observables (details are to be found in the monographs of Araki and Haag, which we can go into, but I don't want to overload the initial post).
At this point we might turn to fields as the fundamental objects. However this remains tricky to me, let's look at gauge theories.
The problem with Gauge theories is that fields carrying the gauge charges cannot be local. If ##\Omega_{\Delta}## is a projector onto states confined to the region ##\Delta## of Minkowski spacetime and ##A## is an observable carrying the gauge charge, then it is always the case that ##\Omega_{\Delta}A \neq 0##. For this reason parameters like ##x## in the quark field ##\psi(x)## are formal, carried over from classical notation, but don't reflect actual localisation at a point.
If one wants ##\psi(x)## to be local, it can only be so as an operator on a Hilbert-Krein space, not on a Hilbert Space of physical states. If one does this there needs to be a condition selecting out the subset of the Hilbert-Krein space containing the physical states (BRST condition in non-rigorous language).
All the operators defined on the actual Hilbert space in gauge theories are necessarily loop like, such as Wilson operators.
All of this is to be found in Chapter 7 of Strocchi's "An introduction to Nonperturbative Foundations of Quantum Field Theory".
So in something like QED the electron and photon fields aren't well-defined local operators and thus actual physical states (which can be localised) can't be associated with them directly. In QCD it is even worse, where the quark and gluon fields, even ignoring this problem, would map one out of the physical Hilbert Space regardless as they carry color charge when all physical states are colorless.
Gauge symmetry itself is formulated in terms of these formal objects ##\psi(x), A_{\mu}(x)## carrying guage charges and most properties of such gauge symmetries (e.g. conserved charges) do not survive quantization. The only aspect of them that does are local Gauss laws (see Nakanishi "Covariant Operator Formalism of Gauge Theories of Quantum Gravity" and Strocchi's book).
So really QCD's physical content to me appears to be there are states which at early/late times make detectors click in a fashion that corresponds to the properties we give nucleons, i.e. late time states will act like particles with nucleon properties. There is a relation between the scattering angles and amplitudes of these nucleon like states that can be encoded in differential forms expressing local Gauss laws.
However the entire structure of quarks, gluons and to some degree even their local fields, seems like a crutch we use so that we have objects that give nice integrals and implement the Gauss law in a way that is easy to use (as Guage "charges") since it reduces it to group theoretic calculations. However the price is that these objects are a long way from the physical content of the theory.
This seems to render statements like "The proton is made of three quarks" into shorthand for "There is a state which at late times has localized properties x,y,z. The restriction the Gauss law imposes on its amplitudes can be modelled perturbatively by decomposing its creation operator into a product of three fields containing charges. Though the states corresponding to these fields are unphysical, lacking even positivity, we have conditions (BRST) to recover what we need for the late time particulate state"
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• #2
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All the operators defined on the actual Hilbert space in gauge theories are necessarily loop like, such as Wilson operators.
All of this is to be found in Chapter 7 of Strocchi's "An introduction to Nonperturbative Foundations of Quantum Field Theory".
Where does one find the first statement in this chapter? It seems to contradict the likely fact that in any gauge theory there should be local gauge invariant fields like the trace of the ''square'' of the field strength. This is the basis for the Clay Millennium problem on quantum gauge theory, which asks for a Wightman field theory for the vacuum sector, which should in particular give this field a rigorous definition. See, e.g., https://www.physicsoverflow.org/21846
In QED, the gauge group is abelian and more should hold: The field strength itself and the electron currents should be fields in a corresponding Wightman field theory for the vacuum sector.
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• #3
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There are certainly local field operators, however looking at QED and QCD (not pure Yang-Mills) how are these connected to the particle dynamics? How do you introduce local field theoretic couplings between the physical matter fields and the physical gauge fields?
A direct coupling between the field strengths and the electron field will either give chargeless electron fields or be nonlocal.
Certainly it can be done, but it would be more in terms of a nucleon Lagrangian in QCD's case having some sort of Gauss's law encoded in a differential form which is some function of the Nucleon fields.
A far cry from the fields that appear in the usual Lagragian. I doubt the physicality of the fields we typically use, not fields in general.
• #4
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Just to say I only have my phone at the moment, I'll give better references when I am back at my library.
• #5
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There are certainly local field operators, however looking at QED and QCD (not pure Yang-Mills) how are these connected to the particle dynamics? How do you introduce local field theoretic couplings between the physical matter fields and the physical gauge fields?
I believe that, just like particle numbers, whatever couplings are ''introduced'' by a Lagrangian ansatz, these are washed away by renormalization, hence are not rigorously meaningful notions. However the local fields that survive renormalization in a rigorous limit are of course correlated, unlike their free counterparts. This implies interaction.
In electrodynamics one does not need a notion of local charge but only those of a 4D charge current and an electromagnetic field, as these are the only fields that figure classically. Thus the vacuum sector of QED describes everything needed to recover macroscopic electrodynamics. Charged states cannot be found there, of course, but these can be taken to be an unphysical idealization. Rays of charged particles may be viewed as approximate notions that can probably be modelled by appropriate charge current distributions. I haven't seen this done but I don't see any obstacle in doing this.
I understand much less about QCD, but there the matter content should be describable too by local and gauge invariant quark currents. Again the vacuum sector should describe everything of true physical relevance. For example, current-current correlations are among the observables that reveal information testable by experiment. See, e.g., the book ''Quantum chromodynamics: an introduction to the theory of quarks and gluons'' by Yndurain.
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• #6
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I think possibly we are saying similar things, but I've rambled.
The currents exist (I'll have more to say on them later) and are local fields, but their decomposition or expression in terms of quark fields is not directly physically sensical, as the quark fields only exist on an enlarged Hilbert-Krein space.
This to me obscures the physical content of the standard description. Really we should say there are nucleon currents obeying Gauss laws. Quark fields are an unphysical expansion of the physical nuclei fields on a Hilbert-Krein space. So these aren't really quark currents and if we had full nonperturbative control of the theory quarks could be eliminated.
I'm saying something similar to what you say about virtual particles I guess. They're a concept useful for calculations, but not part of the theory's genuine physical content. Same with gluons.
• #7
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Quark fields are an unphysical expansion of the physical nuclei fields on a Hilbert-Krein space. So these aren't really quark currents
I agree that quark fields themselves are unphysical. But what is your argument that quark currents (the renormalized version of the corr. quadratics in the quark fields) cannot be physical? They should exists in the 6 quark flavors (or, ignoring spin, as a 6x6 matrix valued field), and hence should deserve to be called quark currents, even though the quarks themselves (as fields or particles) are virtual only.
• #8
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I think they are physical, but more so that they are really nuclei currents and the decomposition in terms of quark fields is an unphysical convenience. In a proper nonperturbative formulation they would probably be expressed as functions of nuclei fields.
Would this seem reasonable to you?
• #9
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I think they are physical, but more so that they are really nuclei currents and the decomposition in terms of quark fields is an unphysical convenience. In a proper nonperturbative formulation they would probably be expressed as functions of nuclei fields.
Would this seem reasonable to you?
I guess you mean baryon currents, since we are discussing QCD and not the standard model - nuclei are held together by the weak force.
But there are far too many baryons, and the flavor information is lost.
Thus I think more than just the asymptotic bound state structure, namely that of quark currents, is preserved in the local fields. Other uncharged fields would be expressed in terms of these and glueball fields, and charged fields would emerge as soliton-like fields in other sectors of the theory, not described by Wightman axioms.
• #10
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• #11
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Other uncharged fields would be expressed in terms of these and glueball fields
in an OPE-fashion
• #12
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Okay a nice reference is:
Taichiro Kugo, Izumi Ojima; Local Covariant Operator Formalism of Non-Abelian Gauge Theories and Quark Confinement Problem, Progress of Theoretical Physics Supplement, Volume 66, 1 February 1979, Pages 1–130
In particular in section 5.3 they discuss the physical content of QCD. Now their discussion has certain propositions they can't prove (if they could it would constitute a rigorous construction of Yang-Mills), but under these assumptions the Wightman axioms imply:
$$\mathcal{H}_{phys} = \overline{\mathcal{A}\left(\mathcal{O}\right)\Omega}$$
That is the physical Hilbert space can be constructed from the closure of local hadron fields operating on the vacuum. To be clear ##\mathcal{A}\left(\mathcal{O}\right)## is the local algebra of color singlet local fields, hence this is essentially a Reeh-Schlieder type theorem.
Could you clarify:
But there are far too many baryons, and the flavor information is lost.
Do you mean the flavor information becomes obscured or do you mean it is literally not present?
• #13
vanhees71
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I guess you mean baryon currents, since we are discussing QCD and not the standard model - nuclei are held together by the weak force.
But there are far too many baryons, and the flavor information is lost.
Thus I think more than just the asymptotic bound state structure, namely that of quark currents, is preserved in the local fields. Other uncharged fields would be expressed in terms of these and glueball fields, and charged fields would emerge as soliton-like fields in other sectors of the theory, not described by Wightman axioms.
I'm a bit puzzled by these statements... Of course the main force to keep "quarks and gluons" bound (even confined!) in hadrons, which are so far the only asymptotic states of QCD that can be measured, is the strong force. The only "ab-initio way" to understand hadrons from QCD are lattice-QCD calculations. These are Monte-Carlo evaluations of appropriate gauge-invariant correlation functions. One of the key achievements of this approach is a pretty accurate calculation of the hadronic mass spectrum of the empirically known as well as not-yet seen hadrons.
I've no clue what you mean by "the flavor information is lost". Of course to get the hadron spectrum the lattice-QCD calculations involve correlation functions with the right quantum numbers, including flavor, i.e., the corresponding valence-quark flavor.
Another approach to understand hadron phenomenology are of course effective low-energy models. The most important approach in the light-quark sector is chiral perturbation theory (and unitarized versions thereof). This uses the approximate chiral symmetry of QCD in the light-quark sector (flavor SU(2) for up and down or flavor SU(3) for up, down, and strange quarks), which is spontaneously broken in the vacuum and at low temperatures and or (net-baryon) densities. These models are governed by the symmetry properties of corresponding composite fields. Also these involve of course well-defined flavor degrees of freedom. For a nice introduction to this approach, see
https://arxiv.org/abs/nucl-th/9706075
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• #14
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nuclei are held together by the weak force.
I don't think so.
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• #15
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Where does one find the first statement in this chapter? It seems to contradict the likely fact that in any gauge theory there should be local gauge invariant fields like the trace of the ''square'' of the field strength. This is the basis for the Clay Millennium problem on quantum gauge theory, which asks for a Wightman field theory for the vacuum sector, which should in particular give this field a rigorous definition. See, e.g., https://www.physicsoverflow.org/21846
In QED, the gauge group is abelian and more should hold: The field strength itself and the electron currents should be fields in a corresponding Wightman field theory for the vacuum sector.
I just read this again and my phrasing was very bad. I meant to say the concept of local field is dubious for the quark and gluon fields and the only versions of them that are well-defined on the physical Hilbert space are loop-like, see Bert Schroer's paper here:
https://arxiv.org/abs/1601.04528
Hadrons however do have a local field theoretic description.
• #16
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To summarise, in QCD the real physical description involves a complicated non-Fock space on which we have only hadron operators.
However this leads into my next point. If quarks fields had been defined on the physical Hilbert Space, they would have formed a basis (in the operator theoretic sense) for all other local fields and thus some sense could be given to the notion that "protons are made of quarks, pions are made of quarks".
However since the only physical fields are hadrons what now is the picture of their composition? They seem fundamental. Perhaps there is a basic set of hadron fields that others can be expressed as functions of, but I don't know if there is a unique choice of such.
• #17
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since the only physical fields are hadrons
You didn't demonstrate this. To say that charged fields are not in the physical Hilbert space does not say anything about which local fields exist in the positive part of the Krein space. For example, glueballs should have local gauge invariant fields associated with them, namely the renormalized trace of the ''square'' of the field strength.
• #18
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I don't think so.
Oh, sorry. Of course, nuclei are held together by meson exchange, which is the strong force.
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• #19
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You didn't demonstrate this. To say that charged fields are not in the physical Hilbert space does not say anything about which local fields exist in the positive part of the Krein space. For example, glueballs should have local gauge invariant fields associated with them, namely the renormalized trace of the ''square'' of the field strength.
Sorry, hadron and glueballs then. More so the point is that what in the conventional way of writing the theory are composite fields are in fact fundamental and that this seems to me to cause a funny detail with the matter, i.e. hadron, fields in that without quarks they aren't all reducible to being "composed" of a unique common set of fields.
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• #20
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I'm a bit puzzled by these statements. [...]
I've no clue what you mean by "the flavor information is lost".
We are discussing here what can (probably) be said rigorously about QED and QCD in Minkowski space. Thus Euclidean lattice calculations do not help.
Do you mean the flavor information becomes obscured or do you mean it is literally not present?
What I was referring to is that the hadron spectrum reflects the flavor information only in a very rough way, probably without anything that can be used to recover the latter mathematically. The U(6) flavor symmetry is badly broken, and working with hadrons in place of quarks is like working with hydrogen bound states in place of the bare electrons, but worse since in the latter case one still has an exact symmetry. It seems that there are infinitely many hadrons in complicated states, and these would all have to be regarded on equal footing in a hadronic theory of QCD.
• #21
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We are discussing here what can (probably) be said rigorously about QED and QCD in Minkowski space. Thus Euclidean lattice calculations do not help.
What I was referring to is that the hadron spectrum reflects the flavor information only in a very rough way, probably without anything that can be used to recover the latter mathematically. The U(6) flavor symmetry is badly broken, and working with hadrons in place of quarks is like working with hydrogen bound states in place of the bare electrons, but worse since in the latter case one still has an exact symmetry. It seems that there are infinitely many hadrons in complicated states, and these would all have to be regarded on equal footing in a hadronic theory of QCD.
I agree, but is this not correct, i.e. since quark and gluon fields are an unphysical expansion of the physical (hadron and glueball) fields on a Hilbert-Krein space, then in a formulation of QCD that used no unphysical concepts we would simply have a hadron "zoo" with all hadrons on equal footing and thus is this not the actual picture QCD presents of the world?
• #22
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I agree, but is this not correct, i.e. since quark and gluon fields are an unphysical expansion of the physical (hadron and glueball) fields on a Hilbert-Krein space, then in a formulation of QCD that used no unphysical concepts we would simply have a hadron "zoo" with all hadrons on equal footing and thus is this not the actual picture QCD presents of the world?
That depends on which fields are local and gauge invariant. I believe that there may be local, gauge invariant current fields for all quark flavors, though I haven't checked yet whether this is likely to hold - too many other things to do now, at the beginning of our winter term. It is only at the level of asymptotic fields that we necessarily have the hadron zoo, with nuclear democracy between all bound states.
• #23
vanhees71
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We are discussing here what can (probably) be said rigorously about QED and QCD in Minkowski space. Thus Euclidean lattice calculations do not help.
What I was referring to is that the hadron spectrum reflects the flavor information only in a very rough way, probably without anything that can be used to recover the latter mathematically. The U(6) flavor symmetry is badly broken, and working with hadrons in place of quarks is like working with hydrogen bound states in place of the bare electrons, but worse since in the latter case one still has an exact symmetry. It seems that there are infinitely many hadrons in complicated states, and these would all have to be regarded on equal footing in a hadronic theory of QCD.
Well, this is the fundamental difference between physics and math. Physics works through observations first, and of course in the early 60ies one had no clue about quarks. This came only later when Gell-Mann discovered the possibility to order hadrons in terms of the approximate SU(3) flavor symmetry. Only somewhat later with the discovery of Bjorken Scaling at SLAC and Feynman's parton model came the notion that quarks are real constituents. The so far final description is QCD with the discovery of asymptotic freedom in the early 70ies and the "confinement conjecture". What's observable in QFTs are the asymptotic free states, and no colored "objects" are asymptotic free according to confinement. As I said, the only way to investigate this (not yet fully solved problem) of confinement is through lattice-gauge theory, which allows to study well certain aspects of it only, among them the hadron spectrum and in its "thermal" version the equation of state of strongly interacting matter. Of course, this is not mathematically rigorous, but physics is not about a mathematically rigorous formulation of QFT but of the application of unfortunately in some aspects not well-defined models based on QFT ideas. Euclidean lattice QCD may not help in finding a mathematically rigorous description of QFT (although, who knows in which way such a mathematically satisfiable version of non-Abelian gauge theories may be formulated in the future), but allows to answer many physically relevant questions, and it's very confirming for the phenomenologist that QCD doesn't only work in the perturbative regime of deep inelastic scattering but also in the low-energy non-perturbative regime of hadron physics, at least as far as it is accessible with lattice-QCD methods.
I still do not get what you are after concerning flavor. That's a well-defined quantum number, according to which all observable particles are described, including flavor in the sense of the electroweak sector, which is closely related with flavor in strong-interaction physics. Admittedly it's a bit hidden, and that's why it took a while to get the idea of quark (and today also neutrino) mixing right.
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• #24
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That depends on which fields are local and gauge invariant. I believe that there may be local, gauge invariant current fields for all quark flavors, though I haven't checked yet whether this is likely to hold - too many other things to do now, at the beginning of our winter term. It is only at the level of asymptotic fields that we necessarily have the hadron zoo, with nuclear democracy between all bound states.
No worries, I'll try to get current thinking on that. Assuming it were true let's say, would your view then be that QCD's fundamental objects are flavour current fields and glueball fields? Hadrons then being reducible (in some operator theoretic sense) to the flavour current fields.
• #25
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I still do not get what you are after concerning flavor. That's a well-defined quantum number, according to which all observable particles are described, including flavor in the sense of the electroweak sector, which is closely related with flavor in strong-interaction physics. Admittedly it's a bit hidden, and that's why it took a while to get the idea of quark (and today also neutrino) mixing right.
As I understand it, QCD has quarks of 6 flavors. What is the flavor of a proton or a neutron, or a kaon? You can describe it only as a tensor product of flavors.
But in a description of local hadron fields that DarMM is after, one would have an elementary field for each hadron, and hence for proton, neutron, kaon,.... How would you recover mathematically the flavor description from a description where these are elementary fields (without assuming anything about quarks)? I think it cannot be done.
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# Math
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A circle has a radius of 12.6 cm. What is the exact length of an arc formed by a central angle measuring 120°?
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$$\frac{\text{arc length}}{\text{circumference}}=\frac{\text{central angle measure}}{360^{\circ}} \\~\\ \frac{\text{arc length}}{2\pi * 12.6}=\frac{120^{\circ}}{360^{\circ}} \\~\\ \text{arc length}=\frac{120^{\circ}}{360^{\circ}}(2\pi *12.6)=\frac{42\pi}{5}\approx26.389 \text{ cm}$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.993495523929596, "perplexity": 697.4946777693922}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948589177.70/warc/CC-MAIN-20171216201436-20171216223436-00784.warc.gz"} |
http://math.stackexchange.com/questions/32843/eigen-decomposition-of-an-interesting-matrix-general-case?answertab=active | # eigen decomposition of an interesting matrix (general case)
Lets define:
$U=\left \{ u_j\right \} , 1 \leq j\leq N= b^{L},$ the set of all different sequences of length $L$ where each element of the sequence can be an integer in $\left \{ 0, 1, .., b-1 \right \}$.
$V=\left \{ v_i\right \} , 1 \leq i\leq M=\binom{L}{k}b^{k},$ the set of all different gaped sequences with $k$ known elements and $L-k$ gaps.
$A_{M*N}=[a_{i,j}]$ is a binary matrix defined as following:
$$a_{i,j} = \left\{\begin{matrix} 1 & \text{if } v_i \text{ matches } u_j \\ 0 & \text{otherwise } \end{matrix}\right.$$
now, the questions are:
i) What is the rank of the matrix $S_{M*M}=AA^{T}$?
ii) What are the eigenvectors and eigenvalues of $AA^{T}$?
Here is an example for $L=2, k=1, b=2$:
$$U = \left \{ 00,01,10,11\right \}$$ $$V = \left \{ 0.,1.,.0,.1\right \} ^*$$
$$A = \begin{bmatrix} 1 & 1 & 0 &0 \\ 0 & 0 & 1 &1 \\ 1 & 0 & 1 &0 \\ 0 & 1 & 0 &1 \end{bmatrix}$$
$$S = \begin{bmatrix} 2 & 0 & 1 &1 \\ 0 & 2 & 1 &1 \\ 1 & 1 & 2 &0 \\ 1 & 1 & 0 &2 \end{bmatrix}$$
For the special case $k=1$, it has been previously solved by joriki and the solution can be found here. For the special case of binary sequences $(b=2)$, the rank is given here by joriki, and a solution for the eigen vectors is given here by Siva.
$^{*}$ here dots denote gaps. a gap can take any value, and each gaped sequence with $k$ known elements and $(L−K)$ gaps in $V$, exactly matches to $b^{L−k}$ sequences in U, hence the sum of elements in each row of $A$ is $b^{L−k}$.
EDIT:
my guess is that $\text{rank}(AA^T)=\text{rank}(A)=\sum_{m=0}^k\left({L\atop m}\right)(b-1)^m\;\;$. and it has $\left({L\atop m}\right)(b-1)^m$ eigenvalues of $\binom{L-m}{k-m}* b^{L-k}$, and the corresponding eigenvectors can be constructed in a similar way as Siva showed for b=2.
EDIT2:
using Gram-Schmidt process I obtained the following orthogonal set from Siva's proposed set of eigenvectors. but I wonder if there is a simpler solution too. $$\Delta_{m}^{L,k}=\left[\begin{array}{c|c|cccc} & \mathrm{first\, bit\, Not\, picked} & & \mathrm{first\, bit\, picked}\\ & & a_{1}=1 & a_{1}=2 & \cdots & a_{1}=(b-1)\\ \hline 0\ldots & \Delta_{m}^{L-1,k-1} & \Delta_{m-1}^{L-1,k-1} & \frac{1}{2}\Delta_{m-1}^{L-1,k-1} & \cdots & \frac{1}{b-1}\Delta_{m-1}^{L-1,k-1}\\ \hline 1\ldots & \Delta_{m}^{L-1,k-1} & -\Delta_{m-1}^{L-1,k-1} & \frac{1}{2}\Delta_{m-1}^{L-1,k-1} & \cdots & \frac{1}{b-1}\Delta_{m-1}^{L-1,k-1}\\ 2\ldots & \Delta_{m}^{L-1,k-1} & 0 & -\Delta_{m-1}^{L-1,k-1} & \cdots & \frac{1}{b-1}\Delta_{m-1}^{L-1,k-1}\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots\\ (b-1)\ldots & \Delta_{m}^{L-1,k-1} & 0 & 0 & \cdots & -\Delta_{m-1}^{L-1,k-1}\\ \hline g\ldots & \Delta_{m}^{L-1,k} & 0 & 0 & \cdots & 0\end{array}\right]$$ $\Delta_{m}^{L,k}$ is the matrix containing orthogonal eigenvectors for L,k and m.
-
I think the guesses you gave above are all correct. The eigenvectors can be constructed similar to the $b=2$ case. For each $m=0,1,..,k$, pick $m$ bits from the total $L$ bits ($\binom{L}{m}$ ways to choose) and pick an $m$-bit vector, $a$, with elements from $\{1,2,..,b-1\}$ ($(b-1)^m$ ways to choose). Define a vector $x$ where $i^{th}$ element $x_i$ is $0$ if $v_i$ has a gap in any of the $m$ bits. And if $v_i$ has no gaps in these $m$ bits, consider the $m$-bit vector (say $b$) formed by these bits in $v_i$ and define $x_i$ as
$x_i = \begin{cases} 1, & \text{if } b \text{ has even number of non-zero values and if } b_j \neq 0 \implies b_j=a_j \forall j=0,1,..,m\\ -1, & \text{if } b \text{ has odd number of non-zero values and if } b_j \neq 0 \implies b_j=a_j \forall j=0,1,..,m\\ 0, & \text{otherwise} \end{cases}$
Each of the $\binom{L}{m}*(b-1)^m$ different vectors $x$ defined as above is an eigenvector of $AA^T$, with an eigenvalue of $\binom{L-m}{k-m}*b^{L-k}$. Proof is again similar to $b=2$ case.
Sum of all the eigenvalues obtained above is
$\sum_{m=0}^{k} \binom{L}{m} \binom{L-m}{k-m} * b^{L-k}*(b-1)^m = \binom{L}{k}*b^{L-k} \sum_{m=0}^{k} \binom{k}{m}*(b-1)^m = \binom{L}{k}*b^L$
This is equal to trace of $AA^T$. But $AA^T$ is symmetric positive semidefinite, hence has no negative eigenvalues. So all the remaining eigenvalues of $AA^T$ are $0$. So rank($A$) = $\sum_{m=0}^{k} \binom{L}{m}*(b-1)^m$.
-
Thanks Siva. Here is a couple of questions: – mghandi May 28 '11 at 14:31
For the case b=2, the eigenvectors are mutually orthogonal, but for the general b, the eigenvectros constructed by the above process are not necessary orthogonal, for example for b=3,L=2,k=1, we have both [1,-1,0,0,0,0] and [1,0,-1,0,0,0]. is there a way to construct an orthogonal set? – mghandi May 28 '11 at 14:38
Can you please give me some references where I can find more about this or similar stuffs. – mghandi May 28 '11 at 14:57 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9740775227546692, "perplexity": 271.1603875507054}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246657588.53/warc/CC-MAIN-20150417045737-00177-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=06-273 | Below is the ascii version of the abstract for 06-273. The html version should be ready soon.
Laurent NIEDERMAN
Prevalence of exponential stability among nearly-integrable Hamiltonian systems
(288K, PDF)
ABSTRACT. In the 70's, Nekhorochev proved that for an analytic nearly integrable Hamiltonian system, the action variables of the unperturbed Hamiltonian remain nearly constant over an exponentially long time with respect to the size of the perturbation, provided that the
unperturbed Hamiltonian satisfies some generic transversality condition known as {\it steepness}. Recently, Guzzo has given examples of exponentially stable integrable Hamiltonians which are non steep but satisfy a weak condition of transversality which involves only the affine subspaces spanned by integer vectors.
We generalize this notion for an arbitrary integrable Hamiltonian and prove the Nekhorochev's estimates in this setting. The point in this refinement lies in the fact that it allows to exhibit a {\it generic} class of real analytic integrable Hamiltonians which are exponentially stable with {\it fixed} exponents.
Genericity is proved in the sense of measure since we exhibit a prevalent set of integrable Hamiltonian which satisfy the latter property. This is obtained by an application of a quantitative Sard theorem given by Yomdin. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9861947894096375, "perplexity": 624.07127504423}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814827.46/warc/CC-MAIN-20180223174348-20180223194348-00224.warc.gz"} |
https://rd.springer.com/article/10.1186/s13662-015-0675-4 | , 2015:335
# Asymptotics of the number of eigenvalues of one-term second-order operator equations
Open Access
Research
## Abstract
We study one-term operator L acting in the space $$H_{1}=L_{2}([0,\infty );H)$$ generated by the operator-differential expression $$\mathcal{L}=-\frac{d}{dx} ( P(x)\frac{d}{dx})$$ and the boundary condition $$y(0)=0$$. We evaluate the asymptotic number of eigenvalues of the operator L under certain conditions.
## Keywords
operator equations eigenvalues Hilbert space
34K08
## 1 Introduction
### 1.1 Related work
The theory of operator-differential equations with unbounded operator-coefficients is a common tool for the study of infinite systems of ordinary differential equations, partial differential equations and integro-differential equations. The main task in this theory is to determine the behavior of the eigenvalues and eigenfunctions of the associated differential operator. The first significant investigation in this direction belongs to Kostyuchenko and Levitan [1]. They studied the asymptotic behavior of the spectrum of Sturm-Liouville operator with operator coefficient. Later, the subject of investigation has been developed by Gorbachuk [2], Gorbachuk and Gorbachuk [3, 4], Otelbayev [5], Solomyak [6], Maksudov et al. [7], Adiguzelov et al. [8] and Vladimirov [9].
In recent years, Maslov [10] has investigated the number of eigenvalues for a Gibbs ensemble of self-adjoint operators. Muminov [11] has studied the expression for the number of eigenvalues of a Friedrichs model. Also, Vladimirov [12] has calculated the eigenvalues of the Sturm-Liouville problem with a fractal indefinite weight.
### 1.2 Formulation of the problem
Let L denote the differential operator in the space $$H_{1}=L_{2}( [ 0,\infty ) ;H)$$ generated by the operator-differential expression
$$\mathcal{L}=-\frac{d}{dx} \biggl( P(x)\frac{d}{dx} \biggr)$$
(1)
with the boundary condition
$$y(0)=0,$$
(2)
where $$P(x)$$ ($$0\leq x<\infty$$) is a self-adjoint operator function in a Hilbert space H.
In this paper, we suppose that the operator L has a discrete spectrum. For instance, in [13], the authors present some conditions under which the operator L has a discrete spectrum.
The aim of the present paper is to study the asymptotic behavior of the eigenvalues of the operator L. The existing methods still are not capable to evaluate the number of eigenvalues of the operator L directly. The reason is as follows. It is impossible to apply Courant’s variational principle [14] directly because on a finite interval the operator, generated by the differential expression $$\mathcal{L}$$ and Neumann boundary conditions, has an infinite number of eigenvalues. (For example, 0 is an eigenvalue of infinite multiplicity.) In order to avoid this difficulty, we consider instead of the operator L its some relatively compact perturbation $$L_{\alpha}=L+P^{\alpha}$$, where $$P^{\alpha}$$ is the αth power of the operator $$P(x)$$. In this, we base on the study of Marcus and Matsaev [15], where under certain conditions the authors show that the main terms of the asymptotics of the eigenvalues of the operator L and the unbounded operator with a relatively compact perturbation $$L_{\alpha}$$ are the same.
## 2 Main results
Throughout the paper, we suppose that the operator-valued function $$P(x)$$ satisfies the following relative compactness (RC) conditions: There exist self-adjoint operators $$A\geq E$$ (here E denotes the identity operator) and $$B\geq E$$ with $$D ( P ( x ) ) \subset D ( A ) =D ( B )$$ and $$A^{-1}, B^{-1}\in{ \sigma}_{\infty}$$ (here $${\sigma}_{\infty}$$ denotes the set of compact operators in H); local integral functions $$q ( x ) \geq1$$, $$\varphi(x)\geq1$$ and constants $$0<\alpha\leq1/2$$, $$\beta>0$$ such that for any $$f\in D(P ( x ) )$$ the following inequalities are satisfied:
1. (a)
$$q ( x ) ( Af,f ) \leq ( P ( x ) f,f ) \leq\varphi(x) ( Bf,f )$$;
2. (b)
$$(B^{\alpha}f,f)\leq(A^{1-\beta}f,f)$$;
3. (c)
$$\lim_{N\rightarrow\infty}\int_{N}^{\infty }\frac{1}{q ( x ) }\int_{N}^{x}{\varphi^{\alpha} ( s ) \, ds\, dx=0.}$$
Below we present a range of lemmas, based on which we prove two main theorems. In Lemma 1, under certain conditions we prove that the operator $$P^{\alpha}$$ is compact with respect to the operator L. In Lemmas 2-6 we evaluate the asymptotics of the eigenvalues of $$L_{\alpha}=L+P^{\alpha}$$, which is the same as for the operator L.
First, we prove the following lemma.
### Lemma 1
The operator $$P^{\alpha} ( x )$$ in the space $$H_{1}$$ is compact relative to operator L under RC-conditions.
### Proof
Let us introduce the spaces $$L_{2}^{1} ( 0,N;P )$$ and $$L_{2}(0,N;P^{\alpha})$$ as a closure of H-valued smooth finite functions near $$x=0$$ and $$x=N$$ with metrics
$$\Vert y\Vert _{L_{2}^{1} ( 0,N;P ) }=\int_{0}^{N} \bigl(P(x)y^{\prime}, {y}^{\prime}\bigr)\,dx$$
and
$$\Vert y\Vert _{L_{2}(0,N;P^{\alpha})}=\int_{0}^{N} \bigl(P^{\alpha} ( x ) y,y\bigr)\,dx,$$
respectively.
We need to check the following two assertions to prove Lemma 1:
1. 1.
For any $$\varepsilon>0$$, there exists a natural number $$N(\varepsilon)$$ such that for any $$N\geq N ( \varepsilon )$$ and any $$y\in D ( L )$$ the following inequality holds:
$$\int_{0}^{N}\bigl(P^{\alpha} ( x ) y,y\bigr) \,dx\leq\int_{0}^{N}\bigl(P(x)y^{\prime}, {y}^{\prime}\bigr)\,dx.$$
2. 2.
Embedding operators from the space $$L_{2}^{1} ( 0,N;P )$$ to $$L_{2}(0,N;P^{\alpha})$$ are completely continuous.
To check assertion 1, we will use Lemma 1 from [13].
### Lemma
[13]
For every finite function y, defined on $$[0,\infty)$$ and taken from the domain $$D(\mathcal {L})$$, the following two inequalities hold:
\begin{aligned}& \int_{0}^{\infty} \biggl( \bigl\vert \bigl({y}, y^{\prime}\bigr)\bigr\vert \big/\int_{x}^{\infty} \frac{dt}{\gamma_{1}(t)} \biggr) \,dx \leq 2\int_{0}^{\infty} \bigl(P(x)y^{\prime}, y^{\prime}\bigr) \,dx, \\& \int_{0}^{\infty}({y}, y) \,dx \leq C\int _{0}^{\infty }\bigl({y}^{(n-1)}, {y}^{(n)}\bigr) x^{2n-1} \,dx. \end{aligned}
Let $$\gamma_{1}\leq\gamma_{2}\leq\gamma_{3}\leq\cdots\leq\gamma _{n}\leq\cdots$$ be the eigenvalues of the operator B. Then, using the above lemma from [13], under RC-conditions we obtain the following chain of inequalities:
\begin{aligned} \int_{N}^{\infty}\bigl(P^{\alpha} ( x ) y,y\bigr) \,dx \leq&\int_{N}^{\infty}\varphi^{\alpha} ( x ) \bigl(B^{\alpha}y,y\bigr)\,dx \\ =&\int _{N}^{\infty }\sum_{k=1}^{\infty} \gamma_{k}^{\alpha}\varphi^{\alpha} (x ) \bigl\vert y_{k}(x)\bigr\vert ^{2}\,dx \\ =&\sum_{k=1}^{\infty}\gamma_{k}^{\alpha} \int_{N}^{\infty }\varphi^{\alpha} ( x ) \bigl\vert y_{k}(x)\bigr\vert ^{2}\,dx \\ =&2\sum_{k=1}^{\infty}\gamma_{k}^{\alpha} \int_{N}^{\infty }\varphi^{\alpha} ( x ) \biggl\vert \int_{x}^{\infty }y_{k}^{\prime}(s)y_{k}(s) \, ds\biggr\vert \,dx \\ \leq&2\sum_{k=1}^{\infty}\gamma_{k}^{\alpha} \int_{N}^{\infty }\varphi^{\alpha} ( x ) \int _{x}^{\infty}\frac{\vert y_{k}^{\prime}{(s)y}_{k}(s)\vert \int_{s}^{\infty}{\frac {1}{q ( t ) }\,dt}}{\int_{s}^{\infty}{\frac{1}{q ( t ) }\,dt}}\,ds\,dx \\ \leq&2\sum_{k=1}^{\infty}\int _{N}^{\infty}\varphi^{\alpha }(x)\int _{x}^{\infty}{\frac{1}{q ( t ) }\, dt\, dx\int _{N}^{\infty }{q(s)\gamma_{k}^{\alpha} \bigl\vert y_{k}^{\prime}(s)\bigr\vert ^{2}}}\,ds \\ =&\int_{N}^{\infty}\frac{1}{q ( x ) }\int _{N}^{x}{\varphi ^{\alpha} ( s ) \,ds}\,dt\int _{N}^{\infty}q ( s ) \bigl( B^{\alpha}{y}^{\prime},{y}^{\prime} \bigr) \,ds. \end{aligned}
Since $$A\geq E$$ and $$1-\beta<1$$, by using condition (b) of relative compactness, we have $$(B^{\alpha}f,f)\leq(A^{1-\beta}f,f)\leq(Af,f)$$. Then from the above chain of inequalities we obtain
\begin{aligned} \int_{N}^{\infty}\bigl(P^{\alpha} ( x ) y,y\bigr) \,dx \leq& \int_{N}^{\infty}{\frac{1}{q ( x ) }\int _{N}^{x}{\varphi ^{\alpha} ( s ) \,ds}\,dt}\int _{N}^{\infty}q(s) \bigl(Ay^{\prime }, y^{\prime}\bigr)\,ds \\ \leq &\int_{N}^{\infty}{\frac{1}{q ( x ) }\int _{N}^{x}{\varphi ^{\alpha} ( s ) \,ds}\,dx}\int _{N}^{\infty}\bigl(P(x)y^{\prime }, y^{\prime} \bigr)\,dx. \end{aligned}
From these inequalities and part (b) of the RC-conditions we get assertion 1.
To establish assertion 2, we use Lemma 1 from [16].
### Lemma
[16]
If the operator function $$Q(x)$$ is Bochner integrable on the interval $$[0,N]$$ and its values are the essence of completely continuous operators in H, then the embedding operator from $$W_{2}^{1} ( 0,N )$$ to $$L_{2}(0,N;Q)$$ is completely continuous.
Since the operator $$\varphi^{\alpha}(x)A^{-\beta}$$ is completely continuous for all $$0\leq x<\infty$$, the above lemma from [16] implies that the embedding operator from $$L_{2}^{1} ( 0,N;E )$$ to $$L_{2}(0,N;\varphi^{\alpha}(x)A^{-\beta})$$ is completely continuous.
If function u is replaced by $$u=A^{\frac{1}{2}}y$$, we establish the continuity of the embedding operator from $$L_{2}^{1} ( 0,N;A )$$ to $$L_{2}(0,N;\varphi^{\alpha}(x)A^{1-\beta})$$.
From parts (a) and (b) of the RC-conditions we have
\begin{aligned}& \int_{0}^{N}\bigl(Ay^{\prime}, {y}^{\prime}\bigr)\,dx \leq \int_{0}^{N} \bigl(P(x)y^{\prime}, {y}^{\prime}\bigr)\,dx, \\& \int_{0}^{N}\bigl(P^{\alpha} ( x ) y,y\bigr) \,dx \leq \int_{0}^{N}\varphi ^{\alpha} ( x ) \bigl(B^{\alpha}y,y\bigr)\,dx\leq\int_{0}^{N} \varphi ^{\alpha} ( x ) \bigl(A^{1-\beta}y,y\bigr)\,dx. \end{aligned}
To finish the proof of Lemma 1, we use Theorem 17 from [17].
### Theorem
[17]
If a symmetric operator K and a positive operator $$K_{1}$$ are defined on $$D_{A}$$ and the inequality $$\vert (Kf, f)\vert \leq(K_{1}f, f)$$ holds for all $$f\in D_{A}$$, then the complete continuity of the operator $$K_{1}$$ with respect to A implies the complete continuity of the operator K with respect to A. Here $$D_{A}$$ denotes the domain of definition of the operator A.
From the theorem and the last integral inequalities it follows that the embedding operator from $$L_{2}^{1} ( 0,N;P )$$ to $$L_{2}(0,N;P^{\alpha})$$ is completely continuous, which proves Lemma 1. □
We now turn to the calculation of the asymptotics of the eigenvalues of operator $$L_{\alpha}$$ generated by the operator-differential expression
$$\mathcal{L}_{\alpha}(y)=-\bigl(P(x){y}^{\prime}\bigr)^{\prime}+P^{\alpha } ( x ) y$$
(3)
and the boundary condition
$$y(0)=0.$$
(4)
Let $$\gamma_{1} ( x ) \leq\gamma_{2} ( x ) \leq \cdots\leq \gamma_{n} ( x ) \leq\cdots$$ be the family of eigenvalues of the operator function $$P(x)$$. Suppose that the following conditions are satisfied:
1. (i)
$$\gamma_{1} ( x ) \geq C_{1}x^{5+\delta}$$ for large x, where $$C_{1}>0$$ and $$\delta>0$$.
2. (ii)
$$P(x_{1})\leq P(x_{2})$$ for $${x}_{1}< x_{2}$$.
3. (iii)
There exists a positive number $$m>0$$ such that $$\frac{1}{2+\delta }+m<\frac{1}{2}$$ and $$P^{-\alpha}(0)\in{\sigma}_{m}$$, where $${\sigma}_{m}= \{ K\in{\sigma}_{\infty} \mid \operatorname{tr} ( (K^{\ast}K)^{m/2} ) <\infty \}$$ and $$K^{\ast}$$ denotes the adjoint operator of K.
For our purpose we also need to consider the following operators:
1. 1.
Operator $$L_{\alpha}^{1}$$, acting in the space $$L_{2}([\lambda ^{\frac{1}{2+\delta}},\infty);H)$$, generated by expression (3) and the boundary condition
$${y}^{\prime}\bigl(\lambda^{\frac{1}{2+\delta}}\bigr)=0.$$
(5)
2. 2.
Operators $$L_{\alpha}^{\mathrm{I}}$$ and $$L_{\alpha}^{\mathrm{II}}$$, acting in the space $$L_{2}([\lambda^{\frac{1}{2+\delta}},\infty);H)$$, generated by expression (3) and the boundary conditions
\begin{aligned}& y(0) = y\bigl(\lambda^{\frac{1}{2+\delta}}\bigr)=0, \end{aligned}
(6)
\begin{aligned}& {y}^{\prime} ( 0 ) = {y}^{\prime} \bigl( \lambda^{\frac{1}{2+\delta}} \bigr) =0, \end{aligned}
(7)
respectively.
3. 3.
Operators $$L_{\alpha_{i}}^{\mathrm{I}}$$ and $$L_{\alpha_{i}}^{\mathrm{II}}$$ acting in the space $$L_{2}([x_{i-1},x_{i}];H)$$ and generated by expression (3) and the boundary conditions
\begin{aligned}& y(x_{i-1}) = y(x_{i})=0, \end{aligned}
(8)
\begin{aligned}& {y}^{\prime} ( x_{i-1} ) = {y}^{\prime} ( x_{i} ) =0, \end{aligned}
(9)
respectively.
### Lemma 2
If the function $$\gamma_{1} ( x )$$ satisfies (i), the intersection of the set (interval) $$(0,\lambda)$$ with the spectrum of the operator $$L_{\alpha}^{1}$$ is empty.
### Proof
After some algebra (one can find the details in [13], in the proof of Lemma 2), it can be shown that
$$\int_{0}^{N}(y, y)\,dx\leq N\int _{0}^{N}\frac{1}{\gamma_{1} ( x ) }\,dx\int _{0}^{N}\bigl(P(x)y^{\prime}, {y}^{\prime}\bigr)\,dx.$$
From this inequality and condition (i) we obtain
$$\int_{\lambda^{\frac{1}{2+\delta}}}^{\infty}\bigl(P(x)y^{\prime}, {y}^{\prime}\bigr)\,dx\geq C_{1}(4+\delta) \lambda^{1+\frac{1}{2+\delta}}\int_{\lambda^{\frac{1}{2+\delta}}}^{\infty} ( y,y ) \,dx> \lambda\int_{\lambda^{\frac{1}{2+\delta}}}^{\infty} ( y,y ) \,dx.$$
The last inequality holds for large λ. Lemma 2 is proved. □
Let λ be some positive number. Denote by $$N_{\alpha}(\lambda )$$, $$N_{\alpha}^{\mathrm{I}}(\lambda)$$ and $$N_{\alpha}^{\mathrm{II}}(\lambda)$$ the numbers of eigenvalues of operators $$L_{\alpha}$$, $$L_{\alpha}^{\mathrm{I}}$$ and $${L}_{\alpha }^{\mathrm{II}}$$, respectively, which are less than or equal to λ. Taking into account Courant’s variation principles [14], we find that
$$N_{\alpha}^{\mathrm{I}} ( \lambda ) \leq N_{\alpha} ( \lambda ) \leq N_{\alpha}^{\mathrm{II}} ( \lambda ) .$$
Let us split the interval $$[0,\lambda^{\frac{1}{2+\delta}}]$$ into subintervals of equal length ω. Let M be the number of the created subintervals, and $$0=x_{0}< x_{1}<\cdots<x_{M}=\lambda^{\frac{1}{2+\delta }}$$.
### Lemma 3
If the operator function $$P(x)$$ satisfies condition (ii) for any $$x_{1}< x_{2}$$, then for large λ the following inequality is valid:
$$n_{\alpha_{i}}^{\mathrm{II}}\leq\sum_{\gamma_{j}^{\alpha}(x_{i-1})\leq \lambda} \biggl\{ \frac{1}{\pi}\int_{x_{i-2}}^{x_{i-1}}\sqrt{ \frac {\lambda -\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}}\,dx+1 \biggr\} ,$$
(10)
where $$n_{\alpha_{i}}^{\mathrm{II}}$$ is the number of eigenvalues of $$L_{\alpha _{i}}^{\mathrm{II}}$$, which are less than or equal to λ, and $$i=1,2,\ldots,M$$.
### Proof
Since the operator function $$P(x)$$ satisfies condition (ii), we have $$P(x_{i-1})< P(x)$$ for all $$x\in(x_{i-1},x_{i})$$. Therefore, operator $$L_{\alpha_{i}}^{\mathrm{II}}$$ is not less than $$L_{\alpha_{i}}^{\ast\ast}$$, acting in the space $$L_{2}([x_{i-1},x_{i}];H)$$, generated by the expression
$$-\bigl(P(x_{i-1}){y}^{\prime}\bigr)^{\prime}+P^{\alpha} ( x_{i-1} ) y$$
and the boundary condition (9).
Let $$n_{\alpha_{i}}^{\ast\ast}$$ be the number of eigenvalues of the operator $$L_{\alpha_{i}}^{\ast\ast}$$, which are less than or equal to λ. Then
$$n_{\alpha_{i}}^{\mathrm{II}}< n_{\alpha_{i}}^{\ast\ast}.$$
(11)
Eigenvalues of the operator $$L_{\alpha_{i}}^{\ast\ast}$$ are of the form
$$\gamma_{j}(x_{i-1}) \biggl( {\frac{\pi k}{x_{i}-x_{i-1}}} \biggr) ^{2}+\gamma _{j}^{\alpha}(x_{i-1})\quad (k=0,1,2,\ldots, j=1,2,\ldots).$$
From here it directly implies that
$$n_{\alpha_{i}}^{\ast\ast}\leq\sum_{\substack{ j \\ \gamma _{j}^{\alpha }(x_{i-1})\leq\lambda}} \biggl\{ \frac{\omega}{\pi}\sqrt{\frac{\lambda -\gamma_{j}^{\alpha}(x_{i-1})}{\gamma_{j}(x_{i-1})}}+1 \biggr\} .$$
(12)
It follows from condition (ii) that all eigenvalues $$\gamma_{j} ( x )$$ are monotonically increasing. Therefore, on the interval $$(x_{i-2},x_{i-1})$$ we have
$$\gamma_{j } ( x ) \leq\gamma_{j } ( x_{i-1} ) .$$
Hence,
$$\omega\sqrt{\frac{\lambda-\gamma_{j}^{\alpha}(x_{i-1})}{\gamma _{j}(x_{i-1})}}\leq\int_{x_{i-2}}^{x_{i-1}} \sqrt{\frac{\lambda-\gamma _{j}^{\alpha}(x)}{\gamma_{j}(x)}}\,dx.$$
From inequalities (11) and (12) and from the last inequality we obtain
$$n_{\alpha_{i}}^{\mathrm{II}}\leq\sum_{\substack{ j \\ \gamma_{j}^{\alpha }(x_{i-1})\leq\lambda}} \biggl\{ {\frac{1}{\pi}}\int_{x_{i-2}}^{x_{i-1}}\sqrt{\frac{\lambda-\gamma_{j}^{\alpha}(x)}{\gamma _{j}(x)}}\,dx+1 \biggr\} .$$
Lemma 3 is proved. □
We denote by $$\psi_{j} ( \lambda )$$ ($$j=1,2,\ldots$$) the functions defined by the following equation:
$$\psi_{j} ( \lambda ) =\min \Bigl\{ \sup_{\gamma _{j}^{\alpha } ( x ) \leq\lambda}(x), \lambda^{\frac {1}{2+\delta}} \Bigr\} .$$
(13)
We prove the next lemma.
### Lemma 4
Under conditions of Lemma 3, the following inequality holds:
$$N_{\alpha}^{\mathrm{II}} ( \lambda ) \leq n_{\alpha_{1}}^{\mathrm{II}}+ \sum_{\substack{ j \\ \gamma_{j}^{\alpha}(x_{1})\leq\lambda}} \biggl\{ \int_{0}^{\psi_{j} ( \lambda ) }{ \frac{1}{\pi}\sqrt{\frac{\lambda-\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}}}\,dx+\frac{\psi _{j} ( \lambda ) }{\omega} \biggr\} .$$
### Proof
According to Courant’s variation principle, we have
$$N_{\alpha}^{\mathrm{II}} ( \lambda ) \leq\sum _{i=1}^{M}n_{\alpha _{i}}^{\mathrm{II}}.$$
By Lemma 3, from this inequality we obtain
$$N_{\alpha}^{\mathrm{II}} ( \lambda ) \leq n_{\alpha _{1}}^{\mathrm{II}}+ \sum_{i\geq2}\sum_{\substack{ j \\ \gamma _{j}^{\alpha }(x_{i-1})\leq\lambda}} \biggl\{ \int_{x_{i-2}}^{x_{i-1}}{\frac{1}{\pi} \sqrt {\frac{\lambda-\gamma_{j}^{\alpha}(x)}{\gamma _{j}(x)}}}\,dx+1 \biggr\} .$$
(14)
Furthermore, we have
\begin{aligned}& \sum_{i\geq2}\sum_{\substack{ j \\ \gamma_{j}^{\alpha }(x_{i-1})\leq\lambda}}\int _{x_{i-2}}^{x_{i-1}}\sqrt{\frac{\lambda -\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}}\,dx \\& \quad =\sum_{j}\sum_{i\geq2} \int_{x_{i-2}}^{x_{i-1}}\sqrt{ \frac{\lambda-\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}} \,dx= \sum_{\gamma_{j}^{\alpha}(x_{1})\leq\lambda}\int_{0}^{{{{x}_{j}^{0}}}} \sqrt{\frac{\lambda-\gamma_{j}^{\alpha}(x)}{\gamma _{j}(x)}}\,dx, \end{aligned}
where
$$x_{j}^{0}=\min \Bigl\{ \max_{\gamma_{j}^{\alpha}(x_{i-1})\leq \lambda} ( x_{i-1} ) , \lambda^{\frac{1}{2+\delta}} \Bigr\} .$$
(15)
On the other hand, $$x_{j}^{0}\leq\psi_{j} ( \lambda )$$, so
$$\sum_{i\geq2}\sum_{\substack{ j \\ \gamma_{j}^{\alpha }(x_{i-1})\leq\lambda}}\int _{x_{i-2}}^{x_{i-1}}{\sqrt{\frac{\lambda -\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}}\,dx< } \sum_{\substack{ j \\ \gamma _{j}^{\alpha}(x_{1})\leq\lambda}}\int_{0}^{\psi_{j} ( \lambda ) } \sqrt{\frac{\lambda-\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}}\,dx.$$
(16)
Taking into account (15), we estimate the sum
\begin{aligned} \sum_{i\geq2}\sum_{\substack{ j \\ \gamma_{j}^{\alpha }(x_{i-1})\leq\lambda}}1 =&\sum_{j}\sum_{\substack{ i\geq2 \\ \gamma_{j}^{\alpha}(x_{i-1})\leq\lambda}}1 \\ =& \frac{1}{\omega}\sum_{j}\sum _{\substack{ i\geq2 \\ \gamma _{j}^{\alpha}(x_{i-1})\leq\lambda}}(x_{i-1}-x_{i-2}) \\ =&\frac {1}{\omega } \sum_{\substack{ j \\ \gamma_{j}^{\alpha}(x_{1})\leq\lambda}} x_{j}^{0}\leq \frac{1}{\omega}\sum_{\substack{ j \\ \gamma_{j}^{\alpha }(x_{1})\leq\lambda}}\psi_{j} ( \lambda ) . \end{aligned}
From inequalities (14), (16) and the last inequality we obtain
$$N_{\alpha}^{\mathrm{II}} ( \lambda ) \leq n_{1}^{\mathrm{II}}+{ \sum_{\substack { j \\ \gamma_{j}^{\alpha}(x_{1})\leq\lambda}}} \biggl\{ \int_{0}^{\psi _{j} ( \lambda ) }{ \frac{1}{\pi}\sqrt{\frac{\lambda-\gamma _{j}^{\alpha}(x)}{\gamma_{j}(x)}}}\,dx+\frac{\psi_{j} ( \lambda ) }{\omega} \biggr\} .$$
Lemma 4 is proved. □
### Lemma 5
Under the conditions of Lemma 3 the following inequality holds:
$$n_{\alpha_{i}}^{\mathrm{I}}\geq\sum_{\gamma_{j}^{\alpha}(x_{i})\leq \lambda} \biggl\{ {\frac{1}{\pi}}\int_{x_{i}}^{\varphi_{i,j} ( \lambda ) }\sqrt {\frac{\lambda-\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}} \,dx-1 \biggr\} ,$$
where $$\varphi_{i,j}(\lambda)=\min \{ x_{i+1}, \psi_{j} ( \lambda ) \}$$ and $$i=1,2,\ldots,M$$.
### Proof
Since by our assumption the operator function $$P(x)$$ increases, then $$P(x)< P(x_{i})$$ on the interval $$(x_{i-1},x_{i})$$, which implies that the operator $$L_{\alpha_{i}}^{\mathrm{I}}$$ is not greater than the operator $$L_{\alpha_{i}}^{\ast}$$ acting in the space $$L_{2}([x_{i-1},x_{i}];H)$$ and generated by the expression
$$-\bigl(P(x_{i})y^{\prime}\bigr)^{\prime}+P^{\alpha} ( x_{i} ) y$$
and boundary condition (8). In this case
$$n_{\alpha_{i}}^{\mathrm{I}}>n_{\alpha_{i}}^{\ast},$$
(17)
where $$n_{\alpha_{i}}^{\ast}$$ is the number of eigenvalues of the operator $$L_{\alpha_{i}}^{\ast}$$, which are less than or equal to λ. Eigenvalues of the operator $$L_{\alpha_{i}}^{\ast}$$ are of the form
$$\gamma_{j}(x_{i}) \biggl(\frac{\pi k}{x_{i}-x_{i-1}} \biggr)^{2}+\gamma _{j}^{\alpha }(x_{i}), \quad \mbox{where }k=1,2,\ldots \mbox{ and }j=1,2,\ldots .$$
From the inequality
$$\gamma_{j}(x_{i}) \biggl( {\frac{\pi k}{x_{i}-x_{i-1}}} \biggr) ^{2}+\gamma _{j}^{\alpha}(x_{i})\leq \lambda,$$
it follows that
$$n_{\alpha_{i}}^{\ast}=\sum_{\gamma_{j}^{\alpha}(x_{i})\leq \lambda} \biggl\{ \frac{\omega}{\pi}\sqrt{\frac{\lambda-\gamma _{j}^{\alpha}(x)}{\gamma_{j}(x)}} \biggr\} \geq{\sum _{\substack{ j \\ \gamma_{j}^{\alpha} ( x_{i} ) \leq\lambda}}} \biggl\{ \frac{\omega}{\pi}\sqrt{ \frac{\lambda-\gamma_{j}^{\alpha}(x_{i})}{\gamma _{j}(x_{i})}}-1 \biggr\} ,$$
(18)
where $$\omega=x_{i}-x_{i-1}$$.
Since the function $$\gamma_{j} ( x )$$ monotonically increases, it is clear that when $$\gamma_{j}^{\alpha} ( x_{i+1} ) <\lambda$$, in other words, when $$x_{i+1}<\psi_{j} ( \lambda )$$,
$$\omega\sqrt{\frac{\lambda-\gamma_{j}^{\alpha}(x_{i})}{\gamma _{j}(x_{i})}}=\int_{x_{i}}^{x_{i+1}} \sqrt{\frac{\lambda-\gamma_{j}^{\alpha }(x_{i})}{\gamma_{j}(x_{i})}}\,dx\geq\int_{x_{i}}^{x_{i+1}} \sqrt{\frac{\lambda -\gamma _{j}^{\alpha}(x)}{\gamma_{j}(x)}}\,dx$$
and when $$x_{i}\leq\psi_{j} ( \lambda ) \leq x_{i+1}$$,
$$\omega\sqrt{\frac{\lambda-\gamma_{j}^{\alpha}(x_{i})}{\gamma _{j}(x_{i})}}\geq\int_{x_{i}}^{\psi_{j} ( \lambda ) } \sqrt{\frac{\lambda -\gamma_{j}^{\alpha}(x_{i})}{\gamma_{j}(x_{i})}}\,dx\geq\int_{x_{i}}^{\psi _{j} ( \lambda ) } \sqrt{\frac{\lambda-\gamma_{j}^{\alpha }(x)}{\gamma_{j}(x)}}\,dx.$$
Thus, we see that, for $$\gamma_{j}^{\alpha} ( x_{i+1} ) \leq \lambda$$,
$$\omega\sqrt{\frac{\lambda-\gamma_{j}^{\alpha}(x_{i})}{\gamma _{j}(x)}}\geq\int_{x_{i}}^{\varphi_{i,j} ( \lambda ) } \sqrt{\frac {\lambda -\gamma_{j}^{\alpha}(x_{i})}{\gamma_{j}(x)}}\,dx,$$
where $$\varphi_{i,j} ( \lambda ) =\min \{ x_{i+1}, \psi _{j} ( \lambda ) \}$$. From inequalities (17), (18) and from the last inequality we find
$$n_{\alpha_{i}}^{\mathrm{I}}\geq\sum_{\gamma_{j}^{\alpha} ( x_{i} ) \leq\lambda} \biggl\{ \frac{1}{\pi}\int_{x_{i}}^{\varphi _{i,j} ( \lambda ) }\sqrt {\frac{\lambda-\gamma_{j}^{\alpha } ( x ) }{\gamma_{j} ( x ) }}\,dx-1 \biggr\} .$$
(19)
Lemma 5 is proved. □
### Lemma 6
Let the operator function $$P(x)$$ satisfy conditions (ii) and (iii) for any $$x_{1}< x_{2}$$. Then the following inequality holds:
$$N_{\alpha}^{\mathrm{I}}(\lambda)\geq\sum_{j} \biggl\{ \frac{1}{\pi}\int_{\varphi_{0,j} ( \lambda ) }^{\psi_{j} ( \lambda ) } \sqrt{\frac{\lambda-\gamma_{j}^{\alpha} ( x ) }{\gamma _{j} ( x ) }}\,dx-\frac{\psi_{j} ( \lambda ) }{\omega } \biggr\} ,$$
(20)
where $$\varphi_{0,j} ( \lambda ) =\min \{ \omega, \psi _{j} ( \lambda ) \}$$.
### Proof
By Courant’s variation principles and Lemma 5, we have
\begin{aligned} N_{\alpha_{i}}^{\mathrm{I}} \geq&\sum_{i=1}^{M}n_{\alpha_{i}}^{\mathrm{I}} \geq \sum_{i\geq1}\sum_{\substack{ j \\ \gamma_{j}^{\alpha} ( x_{i} ) \leq\lambda}} \biggl\{ \frac{1}{\pi}\int_{x_{i}}^{\varphi _{i,j} ( \lambda ) }\sqrt {\frac{\lambda-\gamma_{j}^{\alpha }(x)}{\gamma_{j}(x)}}\,dx-1 \biggr\} \\ =&\sum_{j}\sum_{i\geq1} \biggl\{ \frac{1}{\pi}\int_{x_{i}}^{\varphi_{i,j} ( \lambda ) } \sqrt{\frac{\lambda -\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}}\,dx-1 \biggr\} . \end{aligned}
(21)
Let us estimate the first term on the right-hand side. Given that $$\varphi _{i,j} ( \lambda )$$ is of the form $$\varphi_{i,j} ( \lambda ) =\min \{ x_{i+1}, \psi_{j} ( \lambda ) \}$$, we get
\begin{aligned}& \sum_{j}\sum _{i\geq1}\frac{1}{\pi}\int_{x_{i}}^{\varphi _{i,j} ( \lambda ) } \sqrt{\frac{\lambda-\gamma_{j}^{\alpha }(x)}{\gamma_{j}(x)}}\,dx \\& \quad = \frac{1}{\pi}\sum_{j} \biggl\{ \int _{x_{1}}^{x_{2}}\sqrt{\frac{ \lambda-\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}}\,dx+ \int_{x_{2}}^{x_{3}}\sqrt{ \frac{\lambda-\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}}\,dx+\cdots+\int_{x_{i_{0}}}^{\psi_{j} ( \lambda ) } \sqrt{\frac{\lambda-\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}}\,dx \biggr\} \\& \quad = \sum_{\psi_{j} ( \lambda ) \geq x_{1}}\frac{1}{\pi }\int _{x_{1}}^{\psi_{j} ( \lambda ) }\sqrt{\frac{\lambda -\gamma _{j}^{\alpha}(x)}{\gamma_{j}(x)}}\,dx, \end{aligned}
(22)
where $$x_{i_{0}}$$ satisfies the condition $$x_{i_{0}}\leq\psi_{j} ( \lambda ) \leq x_{i_{0}+1}$$. Furthermore,
\begin{aligned}& \sum_{\psi_{j} ( \lambda ) \geq x_{1}}\frac{1}{\pi} \int _{x_{1}}^{\psi_{j} ( \lambda ) }\sqrt{\frac{\lambda -\gamma _{j}^{\alpha}(x)}{\gamma_{j}(x)}}\,dx \\& \quad = \frac{1}{\pi}\sum_{\psi_{j} ( \lambda ) \geq x_{1}} \biggl\{ \int _{0}^{\psi_{j} ( \lambda ) }\sqrt{\frac {\lambda -\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}}\,dx- \int_{0}^{x_{1}}\sqrt {\frac{\lambda-\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}} \,dx \biggr\} \\& \quad = \frac{1}{\pi}\sum_{j}\int _{0}^{\psi_{j} ( \lambda ) }\sqrt{ \frac{\lambda-\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}}\,dx-\frac {1}{\pi}\sum_{\psi_{j} ( \lambda ) \leq x_{1}}{ \int_{0}^{\psi _{j} ( \lambda ) }\sqrt{\frac{\lambda-\gamma_{j}^{\alpha }(x)}{\gamma_{j}(x)}} \,dx} \\& \qquad {} -\frac{1}{\pi}\sum_{\psi_{j} ( \lambda ) \geq x_{1}}\int _{0}^{x_{1}}\sqrt{\frac{\lambda-\gamma_{j}^{\alpha} ( x ) }{\gamma_{j} ( x ) }}\,dx \\& \quad = \frac{1}{\pi}\sum_{j} \biggl\{ \int _{0}^{\psi_{j} ( \lambda ) }\sqrt{\frac{\lambda-\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}} \,dx- \int_{0}^{\varphi_{0,j} ( \lambda ) }\sqrt{\frac{\lambda -\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}}\,dx \biggr\} \\& \quad = \frac{1}{\pi}\sum_{j}\int _{\varphi_{0,j} ( \lambda ) }^{\psi_{j} ( \lambda ) }\sqrt{\frac{\lambda-\gamma _{j}^{\alpha }(x)}{\gamma_{j}(x)}}\,dx. \end{aligned}
(23)
From inequalities (21) and (22) we find
\begin{aligned}& \sum_{j}\sum_{i\geq1} \frac{1}{\pi}\int_{x_{i}}^{\varphi _{i,j} ( \lambda ) }\sqrt{ \frac{\lambda-\gamma_{j}^{\alpha }(x)}{\gamma_{j}(x)}}\,dx \\& \quad =\frac{1}{\pi}\int_{\varphi_{0,j} ( \lambda ) }^{\psi_{j} ( \lambda ) } \biggl( {\sum_{j}\sqrt{ \frac{\lambda-\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}}}\,dx \biggr). \end{aligned}
(24)
For the second term on the right-hand side of (20), as before (in the proof of Lemma 4), we have
$$\sum_{i\geq2}\sum_{j}1 \leq\frac{1}{\omega}\sum_{j} \psi_{j} ( \lambda ) .$$
(25)
Finally, taking into account (21), (24) and (25), we obtain the desired inequality (20). Lemma 6 is proved. □
### Corollary
Under the conditions of Lemma 6, the following inequality holds:
$$N_{\alpha_{i}}^{\mathrm{I}}\geq\frac{1}{\pi}\sum _{j}{\int_{0}^{\psi _{j} ( \lambda ) }\sqrt {\frac{\lambda-\gamma_{j}^{\alpha } ( x ) }{\gamma_{j} ( x ) }}\,dx-\sqrt{\lambda}}\omega c_{1}- \frac{\psi_{1} ( \lambda ) }{\omega}l_{\lambda},$$
(26)
where $$c_{1}$$ is a constant and $$l_{\lambda}$$ is the number of eigenvalues of the operator $$P^{\alpha}(0)$$, which are less than or equal to λ, i.e.,
$$l_{\lambda}=\sum_{\gamma_{j}^{\alpha} ( 0 ) \leq \lambda }1.$$
### Proof
In fact, by Lemma 6
\begin{aligned} N_{\alpha}^{\mathrm{I}}(\lambda) \geq&\frac{1}{\pi}\sum _{j}\int_{0}^{\psi _{j} ( \lambda ) }\sqrt {\frac{\lambda-\gamma_{j}^{\alpha }(x)}{\gamma_{j}(x)}}\,dx \\ &{}-\frac{1}{\pi}\sum _{j}\int_{0}^{\varphi _{0,j}{ ( \lambda ) }}\sqrt {\frac{\lambda-\gamma_{j}^{\alpha} ( x ) }{\gamma_{j} ( x ) }}\,dx-\frac{1}{\omega}\sum _{j} \psi_{j} ( \lambda ) . \end{aligned}
(27)
Let us estimate the second term on the right-hand side of this inequality. Since all functions $$\gamma_{j} ( x )$$ ($$j=1,2,\ldots$$) monotonically increase on half axis $$[0,\infty)$$, we have
\begin{aligned}& \frac{1}{\pi}\sum_{j}\int _{0}^{\varphi_{0,j}{ ( \lambda )}}\sqrt{\frac{\lambda-\gamma_{j}^{\alpha} ( x ) }{\gamma _{j} ( x ) }}\,dx \\& \quad < \frac{1}{\pi}\sum_{j}\int_{0}^{\varphi _{0,j}{ ( \lambda ) }} \sqrt{\frac{\lambda}{\gamma_{j} ( x ) }}\,dx\leq \frac{\sqrt{\lambda}}{\pi}\sum_{\gamma_{j}^{\alpha} ( 0 ) \leq\lambda}\int _{0}^{\omega}\frac{1}{\sqrt{\gamma_{j} ( x ) }}\,dx \\& \quad \leq \frac{\omega\sqrt{\lambda}}{\pi}\sum_{\gamma_{j}^{\alpha} ( 0 ) \leq\lambda}\frac{1}{\sqrt{\gamma _{j} ( 0 ) }}\leq c_{1}\omega\sqrt{\lambda}. \end{aligned}
For the third term on the right-hand side of (26), we find
$$\frac{1}{\omega}\sum_{j}{\psi_{j} ( \lambda ) \leq \frac{\psi_{1} ( \lambda ) }{\omega}}\sum_{\gamma _{j}^{\alpha } ( 0 ) \leq\lambda}1= \frac{\psi_{1} ( \lambda ) }{\omega}l_{\lambda}.$$
(28)
From these inequalities, we obtain inequality (26), which proves the corollary. □
Assume that $$P^{-\alpha}(0)\in{\sigma}_{m}$$, where m is some positive number satisfying the condition $$\frac{1}{2+\delta}+m<\frac {1}{2}$$. Then we have
\begin{aligned} \mathrm{const} \geq&\sum_{\gamma_{j}^{-\alpha} ( 0 ) \geq \lambda ^{-1}}{\bigl( \gamma_{j}^{-\alpha} ( 0 ) \bigr)}^{m}\geq \sum _{\gamma_{j}^{-\alpha} ( 0 ) \geq\lambda^{-1}}{\lambda^{-m}} \\ \geq&\lambda^{-m}\sum_{\gamma_{j}^{-\alpha} ( 0 ) \geq\lambda^{-1}}1= \lambda^{-m}\sum_{\gamma_{j}^{\alpha } ( 0 ) \leq\lambda}1= \lambda^{-m}l_{\lambda}. \end{aligned}
Hence
$$l_{\lambda}\leq \mathrm{const}\cdot\lambda^{m}.$$
(29)
Now take the step ω as
$$\omega=\frac{\lambda^{\frac{1}{2+\delta}}}{ [ \lambda^{\frac{1}{ 2+\delta}+\theta} ] }, \quad \mbox{where }\frac{1}{2+\delta}+\theta +m< \frac{1}{2}.$$
(30)
Using this form of ω, we estimate the numbers $$N_{\alpha }^{\mathrm{I}}(\lambda)$$ and $$N_{\alpha}^{\mathrm{II}}(\lambda)$$.
From Lemma 4 it follows that
$$N_{\alpha}^{\mathrm{II}} ( \lambda ) \leq n_{\alpha_{1}}^{\mathrm{II}}+{ \sum_{\substack{ j \\ \gamma_{j}^{\alpha}(x_{1})\leq\lambda}}} \biggl\{ \int_{0}^{\psi_{j} ( \lambda ) }{ \frac{1}{\pi}\sqrt{\frac{\lambda-\gamma_{j}^{\alpha}(x)}{\gamma_{j}(x)}}}\,dx+\frac{\psi _{j} ( \lambda ) }{\omega} \biggr\} .$$
(31)
Using inequality (29), we estimate the number $$n_{\alpha _{1}}^{\mathrm{II}}$$,
\begin{aligned} n_{\alpha_{1}}^{\mathrm{II}} \leq&{\sum _{\substack{ j \\ \gamma_{j}^{\alpha } ( 0 ) \leq\lambda}}} \biggl( \frac{\omega}{\pi}\sqrt{ \frac{ \lambda-\gamma_{j}^{\alpha} ( 0 ) }{\gamma_{j} ( 0 ) }}+\frac{\psi_{j} ( \lambda ) }{\omega} \biggr) \\ \leq&\frac{\omega\sqrt{\lambda}}{\pi}{\sum_{\substack{ j \\ \gamma _{j}^{\alpha} ( 0 ) \leq\lambda}}} \frac{1}{\sqrt{\gamma _{j} ( 0 ) }}+{\sum_{\substack{ j \\ \gamma_{j}^{\alpha} ( 0 ) \leq\lambda}}1\leq}c_{2} \lambda^{\frac{1}{2}-\theta }+c_{3}\lambda^{m+\theta}, \end{aligned}
(32)
where $$c_{2}\geq\frac{1}{\pi}\operatorname{tr}(P^{-\frac{1}{2}} ( 0 ) )$$ and $$c_{3}\geq \operatorname{tr}(P^{-\alpha m} ( 0 ) )$$ (here, $$\operatorname{tr}(A)$$ denotes the trace of the operator A).
It follows from inequalities (31) and (32) and formula (13) that
\begin{aligned} N_{\alpha}^{\mathrm{II}} ( \lambda ) \leq&\sum _{j}{\frac {1}{\pi}\int_{0}^{\psi_{j} ( \lambda ) } \sqrt{\frac{\lambda}{\gamma _{j} ( x ) }}\,dx+\frac{\psi_{1} ( \lambda ) }{\omega }}l_{\lambda}+c_{2} \lambda^{\frac{1}{2}-\theta}+c_{3}\lambda^{m} \\ \leq&\sqrt{\lambda}\sum_{j}{\frac{1}{\pi} \int_{0}^{\psi _{j} ( \lambda ) }\frac{1}{\sqrt{\gamma_{j} ( x ) }} \,dx+{c_{1}\lambda}^{\frac{1}{2+\delta}+m+\theta}}+c_{2} \lambda^{\frac {1}{2}-\theta}+c_{3}\lambda^{m}. \end{aligned}
(33)
Using the corollary to Lemma 6, inequality (29) and formula (30), for $$N_{\alpha}^{\mathrm{I}} ( \lambda )$$, we obtain the following inequality:
$$N_{\alpha}^{\mathrm{I}} ( \lambda ) \geq\frac{1}{\pi}\sum _{j}{\int_{0}^{\psi_{j} ( \lambda ) } \sqrt{\frac{\lambda-\gamma _{j}^{\alpha}(x)}{\gamma_{j} ( x ) }}\,dx-}c_{4}\lambda^{\frac {1}{2}-\theta}-c_{3} \lambda^{\frac{1}{2+\delta}+m+\theta}.$$
(34)
Let us estimate the first term on the right-hand side:
\begin{aligned}& \frac{1}{\pi}\sum_{j}\int _{0}^{\psi_{j} ( \lambda ) }\sqrt{ \frac{\lambda-\gamma_{j}^{\alpha}(x)}{\gamma_{j} ( x ) }}\,dx \\& \quad = \frac{1}{\pi}\sum_{j}\int _{0}^{\psi_{j} ( \lambda ) }\sqrt{ \frac{{(\sqrt{\lambda})}^{2}-2\gamma_{j}^{\alpha} ( x ) \frac{\sqrt{\lambda}}{\sqrt{\lambda}}+\frac{\gamma_{j}^{2\alpha }(x)}{\lambda}-\frac{\gamma_{j}^{2\alpha}(x)}{\lambda}+\gamma _{j}^{\alpha}(x)}{\gamma_{j} ( x ) }}\,dx \\& \quad \geq \frac{1}{\pi}\sum_{j}\int _{0}^{\psi_{j} ( \lambda ) }\sqrt{\frac{{(\sqrt{\lambda})}^{2}-2\gamma_{j}^{\alpha} ( x ) \frac{\sqrt{\lambda}}{\sqrt{\lambda}}+\frac{\gamma _{j}^{2\alpha }(x)}{\lambda}}{\gamma_{j} ( x ) }}\,dx \\& \quad = \frac{1}{\pi}\sum_{j}\int _{0}^{\psi_{j} ( \lambda ) }\frac{\sqrt{\lambda}-\frac{\gamma_{j}^{\alpha}(x)}{\sqrt{\lambda }}}{\sqrt{\gamma_{j} ( x ) }}\,dx \geq \frac{1}{\pi}\sum_{j}{\sqrt{ \lambda}\int_{0}^{\psi _{j} ( \lambda ) }\frac{1}{\sqrt{\gamma_{j} ( x ) }} \,dx}-c_{6}\psi_{1} ( \lambda ) . \end{aligned}
(35)
From inequalities (34) and (35) we obtain
$$N_{\alpha}^{\mathrm{I}} ( \lambda ) \geq\frac{\sqrt{\lambda}}{\pi}\sum _{j}{\int_{0}^{\psi_{j} ( \lambda ) } \frac {1}{\sqrt{\gamma_{j} ( x ) }}\,dx-c_{1}^{\mathrm{I}}\lambda^{\frac{1}{2+\delta }}-}c_{2}^{\mathrm{I}} \lambda^{\frac{1}{2}-\theta}-c_{3}^{\mathrm{I}}\lambda^{\frac {1}{2+\delta }+m+\theta}.$$
(36)
By Lemma 4 and inequalities (33) and (36), we obtain
\begin{aligned}& \frac{1}{\pi}\sum_{j}{\int _{0}^{\psi_{j} ( \lambda ) }\frac{1}{\sqrt{\gamma_{j} ( x ) }} \,dx-c_{0}\lambda^{-\frac {1}{2}+\frac{1}{2+\delta}+m+\theta}} \\ & \quad \leq \frac{N_{\alpha}(\lambda)}{\sqrt{\lambda}}\leq\frac{1}{\pi}\sum _{j}\int_{0}^{\psi_{j} ( \lambda ) } \frac {1}{\sqrt{\gamma_{j} ( x ) }}\,dx+c_{1}\lambda^{-\frac{1}{2}+\frac{1}{2+\delta}+m+\theta}. \end{aligned}
Given that $$-\frac{1}{2}+\frac{1}{2+\delta}+m+\theta<0$$, we finally obtain the following relation for the number of eigenvalues of the operator $$L_{\alpha}$$:
$$\lim_{\lambda\longrightarrow\infty} {\frac{N_{\alpha }(\lambda)}{\sqrt{\lambda}}=} \frac{1}{\pi}\sum_{j}\int_{0}^{\infty } \frac{1}{\sqrt{\gamma_{j} ( x ) }}\,dx.$$
(37)
Thus we have proved the following theorem.
### Theorem 1
Let the operator L have discrete spectrum. Then under RC-conditions and (i)-(iii), the number $$N_{\alpha}(\lambda)$$ of eigenvalues of operator $$L_{\alpha}$$ satisfies relation (37).
The next theorem is Theorem 3.2 in [15] which has been proved by Marcus and Matsaev.
### Theorem
[15]
Let M be a normal operator with discrete spectrum; and all its eigenvalues, which lie in the corner $$\psi_{2\theta}= \{ \lambda: \vert \varphi \vert <2\theta \}$$ ($$0<\theta\leq\frac{\pi}{2}$$), are positive and their number is infinite. Also let B be an operator, which is compact with respect to M, and $$A=M+B$$. If $$\lim_{r\rightarrow\infty; \varepsilon\rightarrow0}\frac {N_{+}(r ( 1+\varepsilon ) ,M)}{N_{+}(r,M)}=1$$, then $$N ( r,\theta ,A ) \sim N_{+}(r,M)$$. Here, $$N_{+}(r,M)$$ denotes the number of positive eigenvalues of M, which are less than or equal to r.
Operator $$P^{\alpha}$$ is compact relative to operator $$L_{\alpha}$$ and all the conditions of the above theorem are satisfied for operators $$M=L_{\alpha }$$, $$B=-P^{\alpha}$$ and $$A=L_{\alpha}-P^{\alpha}$$. Then
$$N_{\lambda}(L_{\alpha})\sim N_{\lambda}(L).$$
By taking into account Lemma 1 and Theorem 3.2 from [15], we obtain the following main theorem.
### Theorem 2
Under the conditions of Theorem 1, the following relation is satisfied for the asymptotics of the eigenvalues of the operator L when $$\lambda\rightarrow\infty$$:
$$N_{\lambda}(L)=\frac{\sqrt{\lambda}}{\pi}\sum_{j} \biggl( \int_{0}^{\psi_{j} ( \lambda ) }\frac{1}{\sqrt{\gamma _{j} ( x ) }}\,dx+o(1) \biggr) .$$
## 3 Example
### Example
Consider the operator L generated by the differential expression
$$\mathcal{L}u={(-1)}^{k+1}\frac{\partial^{k+1}}{\partial x\, \partial y^{k}} \biggl( a(x,y) \frac{\partial^{k+1}u}{\partial y^{k}\, \partial x} \biggr)$$
(38)
and the boundary and initial conditions
\begin{aligned} &\frac{\partial^{j}u}{\partial y^{j}} ( x,\pm1 ) = 0, \quad i=0,1, \ldots,k-1; \\ &u ( 0,y ) = 0, \end{aligned}
(39)
where $$a(x,y)\geq1$$ for all $$0\leq x\leq\infty$$, $$-1\leq y\leq1$$.
This operator can be reduced to the operator generated by the operator-differential expression
$$\mathcal{L}u=-\frac{d}{dx} \biggl( P(x)\frac{d}{dx}u \biggr)$$
and the boundary condition
$$u ( 0 ) =0,$$
where
$$P(x)f= \biggl\{ {(-1)}^{k+1}\frac{d^{k}}{dy^{k}} \biggl( a(x,y) \frac {d^{k}f}{dy^{k}} \biggr) ; \frac{d^{i}f}{dy^{i}} ( \pm1 ) =0, i=0,1,\ldots,k-1 \biggr\} .$$
The operator function $$P(x)$$ acts in the space $$L_{2}(-1,1)$$ for all x.
Consider the following functions:
\begin{aligned}& a_{1} ( x ) = \min_{-1\leq y\leq1}a(x,y), \\& a_{2} ( x ) = \max_{-1\leq y\leq1}a(x,y). \end{aligned}
Assume that there exists such a number $$0<\alpha\leq\frac{1}{2}$$ for which the following condition is satisfied:
1. (1)
$$\lim_{N\rightarrow\infty}\int_{N}^{\infty}\frac {1}{a_{1} ( x ) }\int_{N}^{x}a_{2}^{\alpha} ( s ) \,ds\,dx=0$$.
Then the RC-conditions are satisfied, where operators A and B and functions $$q(x)$$ and $$\varphi(x)$$ are defined as follows:
\begin{aligned}& Af = Bf= \biggl\{ {(-1)}^{k}\frac{d^{2k}}{dy^{2k}}f; \frac {d^{i}f}{dy^{i}} ( \pm1 ) =0, i=0,1,\ldots,k-1 \biggr\} , \\& q ( x ) = a_{1} ( x ) , \qquad \varphi ( x ) =a_{2} ( x ) . \end{aligned}
Let the function $$a(x,y)$$ and the order k of differential operator $$P(x)$$ satisfy the following conditions:
1. (2)
$$a_{1} ( x ) \geq cx^{5+\delta}$$.
2. (3)
The function $$a ( x,y )$$ is monotonically increasing with respect to the variable x.
3. (4)
The order k of differential operator $$P(x)$$ is such that the condition $$\sum_{n=1}^{\infty}\frac{1}{n^{2m\alpha k}}<\infty$$ is satisfied for some number m, where $$\frac{1}{2+\delta}+m<\frac {1}{2}$$.
Then, for the asymptotics of the number of eigenvalues of the operator L, we have the following formula:
$$N_{\lambda}(L)=\frac{\sqrt{\lambda}}{\pi}\sum_{n=1}^{\infty } \biggl( \int_{0}^{\infty}\frac{1}{\sqrt{\alpha_{n} ( x ) }}\,dx+o(1) \biggr) ,$$
where $$\alpha_{n} ( x )$$ are eigenvalues of the operator $$P(x)$$.
## Notes
### Acknowledgements
I am grateful to the anonymous reviewers for their comments and suggestions, which have helped to improve the quality of this paper.
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Glazman, IM: Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators. Fizmatgiz, Moscow (1963) Google Scholar | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9994232654571533, "perplexity": 1420.8854773958699}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676590901.10/warc/CC-MAIN-20180719125339-20180719145339-00387.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/196297-distribution-fdd-random-process-xt-print.html | # Distribution (FDD) of random process Xt
• March 23rd 2012, 04:42 AM
Distribution (FDD) of random process Xt
Hello everyone!
The question I have is:
$\xi$ and $\eta$ are indpt standard normal random variables, we have a random process $X_t$ ,where:
$X_t=(\xi-\eta)\sqrt{t}, t\geq0$
I would like to find out the distribution (one dimension FDD) of $X_t$ ,for a fixed $t>0$
Here is what I did:
$F_t(x)=P((\xi-\eta)\sqrt{t}\leq x)$
$=\int P(\xi\leq\frac{x}{\sqrt{t}}+s|\eta=s)dF_\eta(s)$ by Total Probability Formula
$=\int P(\xi\leq\frac{x}{\sqrt{t}}+s)dF_\eta(s)$ by independence
$=\int F_\xi(\frac{x}{\sqrt{t}}+s)\cdot\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2} s^2} ds$
here is where I stuck...how should I deal with $F_\xi(\frac{x}{\sqrt{t}}+s)$, or is there other ways to this kind of things?
Could anyone help me? :)
• March 24th 2012, 04:44 AM
Re: Distribution (FDD) of random process Xt
I found a similar Q with answer, just post it on if it helps...
$\xi$ and $\eta$ are indpt standard normal random variables
$X_t=(\xi+\eta)t, t\geq0$
Find the n dimensional FDD.
if $0=t_1,
$F_{t_1,...,t_k}(x_1,...,x_k) = \left\{\begin{matrix}0,&\mbox{ if }x_1<0,\\\Phi(\frac{1}{\sqrt{2}}min\{\frac{x_2}{t_ 2},...,\frac{x_k}{t_k}\}),&\mbox{ if }x_1\geq 0,\end{matrix}\right$
if $0,
$F_{t_1,...,t_k}(x_1,...,x_k) = \Phi(\frac{1}{\sqrt{2}}min\{\frac{x_2}{t_2},..., \frac{x_k}{t_k}\})$
I understand where the $min$ comes from, take 2D case, like above:
$P(\xi\leq\frac{x_1}{t_1}-\eta\wedge\frac{x_2}{t_2}-\eta)$
again using TPF:
$\int P(\xi\leq\frac{x_1}{t_1}-s \wedge\frac{x_2}{t_2}-s|\eta = s)dF_\eta(s)$
but how does this becomes
$F_{t_1,t_2}(x_1,x_2) = \Phi(\frac{1}{\sqrt{2}}min\{\frac{x_1}{t_1},\frac{ x_2}{t_2}\})$???
I believe the previous should have the similar FDD, so could anyone explain this to me??
Thank you for your help :)
• March 28th 2012, 03:42 AM
Re: Distribution (FDD) of random process Xt
Looks like I am answering my own question again...
After a bit study, I found it is a very simple problem...maybe people won't type so much for a easy Q like this :P
$\xi$ and $\eta$ are indpt standard normal random variables,
therefore $(\xi - \eta) \sim N(1, (\sqrt{2})^2)$, so that $X_t \sim N(1, (\sqrt{2})^2)$
the 1D FDD is
$F_t(x)=P((\xi-\eta)\leq \frac{x}{\sqrt{t}})$
$=P(\frac{(\xi-\eta)}{\sqrt{2}}\leq \frac{1}{\sqrt{2}}\frac{x}{\sqrt{t}})$
$=P(z\leq \frac{1}{\sqrt{2}}\frac{x}{\sqrt{t}})$
$= \Phi(\frac{1}{\sqrt{2}}\frac{x}{\sqrt{t}}})$
this could be easily extend to nD FDD | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 30, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8470370769500732, "perplexity": 2808.655218799836}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645281115.59/warc/CC-MAIN-20150827031441-00347-ip-10-171-96-226.ec2.internal.warc.gz"} |
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