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For any natural number \(k\geqq 3\) and any integer \(\lambda \), let c\( _{d}(k,\lambda )\) be the least integer such that the necessary conditions 2\( \lambda \)(\(\lambda -1)\equiv 0\pmod{(k-1)}\) and \(\lambda v(v-1)\equiv 0 \pmod{[k/2]}\) for the existence of a DB\((k,\lambda;v)\) are also sufficient, for all \(v \geqq c_{d}(k,\lambda ).\) In the paper, the estimate of c\(_{d}(k,\lambda )\) when \(k\) is a multiple of 4 and \(\lambda \) is arbitrary is given. Combining this result with that given by the authors in a previous paper [Discrete Math. 222, 27--40 (2000; Zbl 0967.05011)], one obtains an estimate of the number \(c_{d}(k,\lambda )\), for all \(k\geqq 3\) and all integers \( \lambda\). A directed balanced incomplete block design \(\text{DB}(v,k,\lambda)\) is a pair \((X,A)\), where \(X\) is a \(v\)-set and \(A\) is a collection of transitively ordered \(k\)-tuples of \(X\) (called blocks) such that every ordered pair of \(X\) appears in exactly \(\lambda\) blocks of \(A\). If \(k\) and \(\lambda\) are integers, \(k\geq 3\), \(\lambda\geq 1\), let \(c_d(k,\lambda)\) denote the smallest integer such that the necessary conditions \(2\lambda(v-1)\equiv 0\pmod{k-1}\) and \(\lambda v(v-1)\equiv 0\pmod{{k\choose 2}}\) for the existence of a \(\text{DB}(v,k,\lambda)\) are also sufficient for all \(v\geq c_k(k,\lambda)\). The authors provide an estimate for \(c_k(k,\lambda)\) when \(k\not\equiv 0\pmod 4\).

For any natural number \(k\geqq 3\) and any integer \(\lambda \), let c\( _{d}(k,\lambda )\) be the least integer such that the necessary conditions 2\( \lambda \)(\(\lambda -1)\equiv 0\pmod{(k-1)}\) and \(\lambda v(v-1)\equiv 0 \pmod{[k/2]}\) for the existence of a DB\((k,\lambda;v)\) are also sufficient, for all \(v \geqq c_{d}(k,\lambda ).\) In the paper, the estimate of c\(_{d}(k,\lambda )\) when \(k\) is a multiple of 4 and \(\lambda \) is arbitrary is given. Combining this result with that given by the authors in a previous paper [Discrete Math. 222, 27--40 (2000; Zbl 0967.05011)], one obtains an estimate of the number \(c_{d}(k,\lambda )\), for all \(k\geqq 3\) and all integers \( \lambda\). We propose a Multiple Neural Networks system for dynamic environments, where one or more neural nets could no longer be able to properly operate, due to partial changes in some of the characteristics of the individuals. We assume that each expert network has a reliability factor that can be dynamically re-evaluated on the ground of the global recognition operated by the overall group. Since the net's degree of reliability is defined as the probability that the net is giving the desired output, in case of conflicts between the outputs of the various nets the re-evaluation of their degrees of reliability can be simply performed on the basis of the Bayes Rule. The new vector of reliability will be used for making the final choice, by applying two algorithms, the Inclusion based and the Weighted one over all the maximally consistent subsets of the global outcome.

Let \(M\) be the Riemannian symmetric space \(\mathrm{SL}(2,{\mathbb H})/\mathrm{Sp}(3)\). The author studies the convolution operator \(T_{a,b}\) on \(M\) associated with the radial multipliers \(m_{a,b}\), defined by \[ m_{a,b}(\lambda)=(\|\lambda\|^{2}+\|\rho\|^{2})^{-b/2} e^{i(\|\lambda\|^{2}+\|\rho\|^{2})^{a/2}},\quad \Re b\geq 0,\quad a>0. \] The main results of the paper are conditions for the operator \(T_{a,b}\) to be bounded on \(L^{p}(M)\). These results are an extension of results of \textit{S. Giulini} and \textit{S. Meda} [J. Reine Angew. Math. 409, 93--105 (1990; Zbl 0696.43007)] for noncompact Riemannian symmetric spaces of rank one. Let G/K be a rank one symmetric space of the noncompact type. We consider the convolution operator \(T_{\alpha,\beta}\) associated to the radial multiplier \[ m_{\alpha,\beta}(\lambda)=(\lambda^ 2+\rho^ 2)^{- \beta /2}\exp (i(\lambda^ 2+\rho^ 2)^{\alpha /2}),\quad Re \beta \geq 0,\quad \alpha >0. \] The main result we prove is the following (I) If \(\alpha >1,\) \(T_{\alpha,\beta}\) is bounded on \(L^ p(G/K)\) if and only if \(p=2.\) (II) If \(\alpha =1,\) \(T_{\alpha,\beta}\) is bounded on \(L^ p(G/K)\) if \(| 1/p-1/2| \leq Re \beta /(n-1).\) (III) If \(\alpha <1,\) \(T_{\alpha,\beta}\) is bounded on \(L^ p(G/K)\) if \(| 1/p-1/2| <Re \beta /\alpha n.\) Claim (I) is a consequence of the properties of the spherical Fourier transform on symmetric spaces of the noncompact type. On the other side (II) and (III) depend on the fact that if \(\alpha\leq 1\) the ``part at infinity'' of \(T_{\alpha,\beta}\) behaves nicely, while the ``local part'' is essentially Euclidean.

Let \(M\) be the Riemannian symmetric space \(\mathrm{SL}(2,{\mathbb H})/\mathrm{Sp}(3)\). The author studies the convolution operator \(T_{a,b}\) on \(M\) associated with the radial multipliers \(m_{a,b}\), defined by \[ m_{a,b}(\lambda)=(\|\lambda\|^{2}+\|\rho\|^{2})^{-b/2} e^{i(\|\lambda\|^{2}+\|\rho\|^{2})^{a/2}},\quad \Re b\geq 0,\quad a>0. \] The main results of the paper are conditions for the operator \(T_{a,b}\) to be bounded on \(L^{p}(M)\). These results are an extension of results of \textit{S. Giulini} and \textit{S. Meda} [J. Reine Angew. Math. 409, 93--105 (1990; Zbl 0696.43007)] for noncompact Riemannian symmetric spaces of rank one. Let \(S\) denote a subset of \(\mathbb{R}\) and \(f: S\to S\), \(F: S\to \mathbb{R}\) and \(A: S\to \mathbb{R}\) be given functions. The results of this paper are concerned with the solution \(x(t)\) of the functional equation \(x(t)+ A(t)\cdot x(f(t))= F(t)\). Using regular summability methods \(T\), the author obtains some necessary and some sufficient conditions for the \(T\)- sum \(x(t)\) of the series \(\sum^\infty_{i= 0}(- 1)^i F(f^i(t))\) to be a solution of the above-mentioned equation under specific conditions on \(F(t)\).

Here we have the exercises and problems from Chapter 2, Integration, of the same author's text on differentiation and integration theory reviewed above (cf. Zbl 0913.00006), along with their solutions. The organization is the same as in the volume on exercises from Chapter 1, and the comments in that review (cf. Zbl 0913.00006) apply here as well. The book is divided into two large chapters. Chapter 1, of 115 pages, does the basics of differential calculus in Banach spaces, plus an introduction to differential geometry. The four sections are headed: Differentiable mapping, Derivatives of higher order, Implicit function theorems, and Varieties (manifolds). Chapter 2, pages 116-462, on integration, is divided into 11 sections: Measure theory, Lebesgue integral, Vector integration, Radon measure, Products of measure spaces, \(L^p\)-spaces, Absolutely continuous functions, Stokes' theorem, Fourier series, Fourier transforms, and Fredholm integral equations. The book is in basic ``definition, theorem, proof, remark'' format, with exercises and problems of varying levels of difficulty scattered liberally throughout. There is a short bibliography, a four-page index of notations and a short general alphabetical index (which one could wish were more complete). Although there is neither a preface nor remarks of any kind about intended use, the content and style of treatment make clear that this is a textbook for a course which might follow a standard introductory undergraduate real analysis course. There are two small separate volumes with solutions to the exercises and problems. (See the following reviews [Zbl 0913.00008] and [Zbl 0913.00009]).

Here we have the exercises and problems from Chapter 2, Integration, of the same author's text on differentiation and integration theory reviewed above (cf. Zbl 0913.00006), along with their solutions. The organization is the same as in the volume on exercises from Chapter 1, and the comments in that review (cf. Zbl 0913.00006) apply here as well. The investigation on the identification of outliers in linear regression models can be extended to those for circular regression case. In this paper, we propose a new numerical statistic called mean circular error to identify possible outliers in circular regression models by using a row deletion approach. Through intensive simulation studies, the cut-off points of the statistic are obtained and its power of performance investigated. It is found that the performance improves as the concentration parameter of circular residuals becomes larger or the sample size becomes smaller. As an illustration, the statistic is applied to a wind direction data set.

The authors consider in this well-written article conic-line (\(\mathcal{CL}\)) arrangements such that their singularities are nodes, tacnodes and ordinary triple points. All these three types of singularities are quasi-homogeneuous singularities. The first result is a Hirzebruch-type inequality for such a type of \(\mathcal{CL}\) arrangements. It says that if \(\mathcal{CL}\) is an arrangement in \(\mathbb{P}^2_{\mathbb{C}}\) of \(d\) lines and \(k\) smooth conics such that \(2k+d\geq 12\) with \(n_2\) nodes, \(t\) tacnodes, and \(n_3\) ordinary triple points then \(20k+n_2+\frac34n_3\geq d+4t\). This result is proved using Langer's variation on the Miyaoka-Yau inequality which involves the local orbifold Euler numbers \(e_{orb}\) of singular points. The second result provides bounds on the number of tacnodes and ordinary triple points using the properties of spectra of singularities as in the seminal paper of \textit{A. N. Varchenko} [Sov. Math., Dokl. 27, 735--739 (1983; Zbl 0537.14003); translation from Dokl. Akad. Nauk SSSR 270, 1294--1297 (1983)]. It says that if \(\mathcal{CL}\) is an arrangement in \(\mathbb{P}^2_{\mathbb{C}}\) of \(d\) lines and \(k\) smooth conics with \(n_2\) nodes, \(t\) tacnodes, and \(n_3\) ordinary triple points and \(m=d+2k=3m'+\epsilon, \epsilon\in \{1,2,3\}\) then \(t+n_3\leq \binom{m-1}{2}+k-\frac{m'(5m'-3)}{2}\) and \(n_3\leq (m'+1)(2m'+1)\). The third result gives a combinatorial constraint which says that if \(C=\{f=0\}\subset \mathbb{P}^2_{\mathbb{C}}\) is a reduced curve of degree \(m\) having only nodes, tacnodes and ordinary triple points as singularities, then \(mdr(f)\geq \frac 23m-2\) and in particular if \(C\) is free then \(m\leq 9\). Here \(mdr(f)\) is the minimal degree of a relation among the partial derivatives. Using this combinatorial constraint for a free reduced curve having such singularities, the authors completely classify free \(\mathcal{CL}\) arrangements having nodes, tacnodes and ordinary triple points as singularities. In particular they prove that such a free conic-line arrangement is determined up to a projective equivalence by the numerical/combinatorial data \(\{n_2,n_3,t\}\) where \(n_2\) is the number of nodes, \(n_3\) is the number of ordinary triple points and \(t\) is the number of tacnodes. Consequentially they prove that the Numerical Terao's Conjecture holds for conic-line arrangements with nodes, tacnodes and ordinary triple points. The conjecture says that if \(\mathcal{CL}_1,\mathcal{CL}_2\) are two such conic-line arrangements, \(\mathcal{CL}_1\) is free and both \(\mathcal{CL}_1,\mathcal{CL}_2\) have the same combinatorics \((m;n_2,t,n_3)\) with \(m\) being the degree of the arrangements then \(\mathcal{CL}_2\) is also free. Let \(Y\subset {\mathbb{C}}P^ n\) be an algebraic hypersurface of degree d, having only nondegenerate singular points. - Problem: What is a maximal number \(N_ n(d)\) of nondegenerate singular points which can be on a hypersurface of a degree d? - An answer is known only if \(n=1,2:\) if \(n=1\) then \(N_ 1(d)=[d/2],\) if \(n=2\) then \(N_ 2(d)=d(d-1)/2.\) The first nontrivial case is \(n=3\). In 1906, \textit{A. B. Basset} ''The maximum number of double points on a surface'', Nature 73, 246 (1906) proved \(N_ 3(d)\leq(d(d-1)^ 2-5-\sqrt{d(d-1)(3d-14)+25})/2.\) In this inequality the estimating number has as asymptotic \(d^ 3/2\), when \(d\to \infty\). Basset's estimation was improved and generalized in following works, but in all cases the estimating number has as asymptotic \(d^ n/2\), when \(d\to \infty\). In this article it is given an estimation \(N_ n(d)\leq A_ n(d)\) with a new asymptotic of \(A_ n(d)\). Namely, \(A_ n(d)=a_ nd^ n+(lower\quad \deg rees\quad of\quad d)\), and \(a_ 3=23/48, a_ n\sim \sqrt{6/\pi n},\) if \(n\to \infty\). The inequality \(N_ n(d)\leq A_ n(d)\) was conjectured by V. I. Arnol'd. The proofs are based on a theory of mixed Hodge structures in vanishing cohomologies. The estimation is a consequence of a general theorem that a spectrum of a quasihomogeneous critical point of a function is more dense than a sum of spectra of critical points, appearing on the same level in any lower deformation of quasihomogeneous critical point. Further results see in ''Semicontinuity of the singularity spectrum'' [Preprint 23, Math. Inst., Univ. Leiden, 1-9 (1983)] by \textit{J. Steenbrink}.

The authors consider in this well-written article conic-line (\(\mathcal{CL}\)) arrangements such that their singularities are nodes, tacnodes and ordinary triple points. All these three types of singularities are quasi-homogeneuous singularities. The first result is a Hirzebruch-type inequality for such a type of \(\mathcal{CL}\) arrangements. It says that if \(\mathcal{CL}\) is an arrangement in \(\mathbb{P}^2_{\mathbb{C}}\) of \(d\) lines and \(k\) smooth conics such that \(2k+d\geq 12\) with \(n_2\) nodes, \(t\) tacnodes, and \(n_3\) ordinary triple points then \(20k+n_2+\frac34n_3\geq d+4t\). This result is proved using Langer's variation on the Miyaoka-Yau inequality which involves the local orbifold Euler numbers \(e_{orb}\) of singular points. The second result provides bounds on the number of tacnodes and ordinary triple points using the properties of spectra of singularities as in the seminal paper of \textit{A. N. Varchenko} [Sov. Math., Dokl. 27, 735--739 (1983; Zbl 0537.14003); translation from Dokl. Akad. Nauk SSSR 270, 1294--1297 (1983)]. It says that if \(\mathcal{CL}\) is an arrangement in \(\mathbb{P}^2_{\mathbb{C}}\) of \(d\) lines and \(k\) smooth conics with \(n_2\) nodes, \(t\) tacnodes, and \(n_3\) ordinary triple points and \(m=d+2k=3m'+\epsilon, \epsilon\in \{1,2,3\}\) then \(t+n_3\leq \binom{m-1}{2}+k-\frac{m'(5m'-3)}{2}\) and \(n_3\leq (m'+1)(2m'+1)\). The third result gives a combinatorial constraint which says that if \(C=\{f=0\}\subset \mathbb{P}^2_{\mathbb{C}}\) is a reduced curve of degree \(m\) having only nodes, tacnodes and ordinary triple points as singularities, then \(mdr(f)\geq \frac 23m-2\) and in particular if \(C\) is free then \(m\leq 9\). Here \(mdr(f)\) is the minimal degree of a relation among the partial derivatives. Using this combinatorial constraint for a free reduced curve having such singularities, the authors completely classify free \(\mathcal{CL}\) arrangements having nodes, tacnodes and ordinary triple points as singularities. In particular they prove that such a free conic-line arrangement is determined up to a projective equivalence by the numerical/combinatorial data \(\{n_2,n_3,t\}\) where \(n_2\) is the number of nodes, \(n_3\) is the number of ordinary triple points and \(t\) is the number of tacnodes. Consequentially they prove that the Numerical Terao's Conjecture holds for conic-line arrangements with nodes, tacnodes and ordinary triple points. The conjecture says that if \(\mathcal{CL}_1,\mathcal{CL}_2\) are two such conic-line arrangements, \(\mathcal{CL}_1\) is free and both \(\mathcal{CL}_1,\mathcal{CL}_2\) have the same combinatorics \((m;n_2,t,n_3)\) with \(m\) being the degree of the arrangements then \(\mathcal{CL}_2\) is also free. The 3-versus-2 Keepaway soccer task represents a widely used benchmark appropriate for evaluating approaches to reinforcement learning, multi-agent systems, and evolutionary robotics. To date most research on this task has been described in terms of developments to reinforcement learning with function approximation or frameworks for neuro-evolution. This work performs an initial study using a recently proposed algorithm for evolving teams of programs hierarchically using two phases of evolution: one to build a library of candidate meta policies and a second to learn how to deploy the library consistently. Particular attention is paid to diversity maintenance, where this has been demonstrated as a critical component in neuro-evolutionary approaches. A new formulation is proposed for fitness sharing appropriate to the Keepaway task. The resulting policies are observed to benefit from the use of diversity and perform significantly better than previously reported. Moreover, champion individuals evolved and selected under one field size generalize to multiple field sizes without any additional training.

For a distribution \(u\in {\mathcal D}'(\mathbb{R}^n)\) and \(y\in \mathbb{R}^n\) denote by \(u\langle y\rangle\) the shifted by \(y\) distribution: \[ (u\langle y\rangle,\varphi):= (u,\varphi(\cdot-y)),\quad \varphi\in{\mathcal D}(\mathbb{R}^n). \] The author defines the integral of \(u\langle y\rangle\) with respect to \(y\) by \[ \Biggl(\int_M u\langle y\rangle d\mu_y,\varphi\Biggr):= \Biggl(u,\int_M \varphi(\cdot-y)d\mu_y\Biggr),\quad \varphi\in{\mathcal D}(\mathbb{R}^n) \] and ``differential and integral calculus'' for it is presented. Further, the integral of a distribution over a line in \(\mathbb{R}^n\) is considered; finally, Gauß' integral formula for distributions is proved. In this work the integral of \(u\langle y\rangle\) is a distribution while in previous works of \textit{G. Bruhn} and \textit{G. Budzick} [Der schwache Gaußsche Integralsatz, Preprint 1154, TH Darmstadt (1989); Integralsätze für Schwartzsche Distributionen, Diss., TH Darmstadt (1993; review above)] the integral of a distribution was a real number. In the present work, Gauß' integral formula holds for any distribution but not in arbitrary domains while in the above-mentioned works the formula holds in any Borel set but not for arbitrary distributions. The aim of this thesis is to prove weak versions of some remarkable integral formulas -- which hold for a rather wide class of distributions and domains. The main ideas are based on the work ``Der schwache Gaußsche Integralsatz'' [TH Darmstadt -- Preprint 1154 (1989)] by \textit{G. Bruhn} and \textit{G. Budzick} where a weak form of Gauß' integral formula is shown. The definitions of the above work are extended to more general cases and a generalization of Gauß' integral theorem is given. Further, trace distributions are defined, they are applied -- together with the weak form of Gauß' integral theorem -- to partial differential equations. Finally, generalizations of the theorem of Gauß-Stokes-Cartan and of Stokes' theorem are proved.

For a distribution \(u\in {\mathcal D}'(\mathbb{R}^n)\) and \(y\in \mathbb{R}^n\) denote by \(u\langle y\rangle\) the shifted by \(y\) distribution: \[ (u\langle y\rangle,\varphi):= (u,\varphi(\cdot-y)),\quad \varphi\in{\mathcal D}(\mathbb{R}^n). \] The author defines the integral of \(u\langle y\rangle\) with respect to \(y\) by \[ \Biggl(\int_M u\langle y\rangle d\mu_y,\varphi\Biggr):= \Biggl(u,\int_M \varphi(\cdot-y)d\mu_y\Biggr),\quad \varphi\in{\mathcal D}(\mathbb{R}^n) \] and ``differential and integral calculus'' for it is presented. Further, the integral of a distribution over a line in \(\mathbb{R}^n\) is considered; finally, Gauß' integral formula for distributions is proved. In this work the integral of \(u\langle y\rangle\) is a distribution while in previous works of \textit{G. Bruhn} and \textit{G. Budzick} [Der schwache Gaußsche Integralsatz, Preprint 1154, TH Darmstadt (1989); Integralsätze für Schwartzsche Distributionen, Diss., TH Darmstadt (1993; review above)] the integral of a distribution was a real number. In the present work, Gauß' integral formula holds for any distribution but not in arbitrary domains while in the above-mentioned works the formula holds in any Borel set but not for arbitrary distributions. No review copy delivered.

Author's abstract: In a previous paper, \textit{Y. Kannai} and \textit{J. Rosenmüller} [J. Math. Econ. 46, No. 2, 148--162 (2010; Zbl 1200.91307)] defined a strategic game where a finite number of players, endowed with money and a commodity, by and sell the commodity from each other by issuing strategic bids. Each player's utility function is the sum of his utility for commodity and either his utility for money or a bankruptcy penalty. A central bank issues loans and may print money to balance its books. Existence of a pure-strategy Nash equilibrium was proved in [loc. cit.], provided the game is played over one trading period only and the players' utility functions for money satisfy concavity properties. In this paper, we prove existence of a subgame-perfect pure-strategy equilibrium when the above game is played in series over three trading periods. We consider \(m\)-replica players with laziness in computing their best responses tending to zero as \(m\rightarrow \infty\) and with utility functions satisfying certain properties. We describe a financial market as a noncooperative game in strategic form. Agents may borrow or deposit money at a central bank and use the cash available to them in order to purchase a commodity for immediate consumption. They derive positive utility from consumption and from having cash reserves at the end of the day, whereas being bankrupt entails negative utility. The bank fixes interest rates. The existence of Nash equilibria (both mixed and pure) of the ensuing game is proved under various assumptions. In particular, no agent is bankrupt at equilibrium. Asymptotic behavior of replica markets is discussed, and it is shown that given appropriate assumptions, the difference between a strategic player and a price taker is negligible in a large economy.

Author's abstract: In a previous paper, \textit{Y. Kannai} and \textit{J. Rosenmüller} [J. Math. Econ. 46, No. 2, 148--162 (2010; Zbl 1200.91307)] defined a strategic game where a finite number of players, endowed with money and a commodity, by and sell the commodity from each other by issuing strategic bids. Each player's utility function is the sum of his utility for commodity and either his utility for money or a bankruptcy penalty. A central bank issues loans and may print money to balance its books. Existence of a pure-strategy Nash equilibrium was proved in [loc. cit.], provided the game is played over one trading period only and the players' utility functions for money satisfy concavity properties. In this paper, we prove existence of a subgame-perfect pure-strategy equilibrium when the above game is played in series over three trading periods. We consider \(m\)-replica players with laziness in computing their best responses tending to zero as \(m\rightarrow \infty\) and with utility functions satisfying certain properties. This paper deals with a nonsmooth semi-infinite multiobjective optimization problem (SIMOP, in brief) and discusses its duality. We focus on Mond-Weir type semi-infinite multiobjective dual problem of the SIMOP. And weak/strong/ converse duality results are obtained by imposing Clarke \(F\)-convexity hypotheses on some combinations of objective functions and constraint functions. Some of our results are new and generalize the conclusions in some former literatures.

The author gives a description of continuous paths in the invariant subspace lattice of the unilateral shift operator of arbitrary finite multiplicity. The main result of the paper reads as follows. {Theorem.} Let \(H^{2}({\mathbb C}^{n})\) be the Hardy space for some finite \(n\) and let \(\{ p_{t}\}_{t\in[0,1]}\) be a continuous in the uniform operator topology family of orthogonal projections on \(H^{2}({\mathbb C}^{n})\) such that \(p_{t}H^{2}({\mathbb C}^{n})\), \(t\in[0,1]\), are invariant subspaces of the unilateral shift operator of multiplicity \(n\). Then there exists an integer \(m\leq n\) and a family of inner operator-valued functions \( G_{t}\in H^{\infty}({\mathbb T},B({\mathbb C}^{m},{\mathbb C}^{n}))\), \(t\in[0,1]\), such that \(p_{t}H^{2}({\mathbb C}^{n})=G_{t}H^{2}({\mathbb C}^{m})\) and \(\{ G_{t}\}_{t\in[0,1]} \) is \(\sup\)-norm continuous. The paper generalizes the result of \textit{R.-W. Yang} [Integral Equations Oper. Theory 28, No. 2, 238--244 (1997; Zbl 0903.47002)] which was obtained for the unilateral shift of multiplicity one. Beurling's well known theorem connects the study of invariant subspaces to that of inner functions over the unit disc. In this paper, we will further explore this connection and, as a corollary of the result, show a one to one correspondence between the components of the invariant subspace lattice and the components of the space of inner functions.

The author gives a description of continuous paths in the invariant subspace lattice of the unilateral shift operator of arbitrary finite multiplicity. The main result of the paper reads as follows. {Theorem.} Let \(H^{2}({\mathbb C}^{n})\) be the Hardy space for some finite \(n\) and let \(\{ p_{t}\}_{t\in[0,1]}\) be a continuous in the uniform operator topology family of orthogonal projections on \(H^{2}({\mathbb C}^{n})\) such that \(p_{t}H^{2}({\mathbb C}^{n})\), \(t\in[0,1]\), are invariant subspaces of the unilateral shift operator of multiplicity \(n\). Then there exists an integer \(m\leq n\) and a family of inner operator-valued functions \( G_{t}\in H^{\infty}({\mathbb T},B({\mathbb C}^{m},{\mathbb C}^{n}))\), \(t\in[0,1]\), such that \(p_{t}H^{2}({\mathbb C}^{n})=G_{t}H^{2}({\mathbb C}^{m})\) and \(\{ G_{t}\}_{t\in[0,1]} \) is \(\sup\)-norm continuous. The paper generalizes the result of \textit{R.-W. Yang} [Integral Equations Oper. Theory 28, No. 2, 238--244 (1997; Zbl 0903.47002)] which was obtained for the unilateral shift of multiplicity one. The authors have undertaken for many years to evaluate explicitly matrix elements of unitary representations of Lie groups. Here the matrix elements of unitary irreducible representations of \(SO(n+1)\), and \(SO_ 0(n+1,1)\) in an SO(n) basis are explicitly calculated.

The authors extend Gosper's algorithm for indefinite hypergeometric summation [see e.g. (1998; Zbl 0909.33001)] for the case when the terms are simultaneously hypergeometric and multibasic hypergeometric. Furthermore they present algorithms to detect the polynomial as well as hypergeometric solutions of recurrence equations in the mixed case. Finally they generalize the concept of greatest factorial factorization to the mixed hypergeometric case. Note that a variant of the multibasic Gosper algorithm appeared also in [\textit{H. Böing} and \textit{W. Koepf}, J. Symb. Comput. 28, 777-799 (1999)]. The book deals with recently developed algorithmic techniques for hypergeometric summation. Such algorithms depend heavily upon use of a suitable computer algebra system, and the author decided to choose Maple. Some of the algorithms in the book are already included in Release V.4, and further packages are downloadable from the publisher. The three first chapters contain background material on the functions \(\Gamma\), \(_pF_q\), \(_r \phi_s\), and their special cases; some simple and useful algorithms are presented. Next comes an introductory consideration of holonomic recurrence equations for sums (a development of Sister Celine's idea), again set up as an algorithm. The main part of the book deals with the algorithms by Gosper (indefinite summation), Wilf-Zeilberger (hypergeometric identities), Zeilberger (holonomic recurrence relations), and Petkovšek (polynomial and hypergeometric term solutions of holonomic recurrence equations). Each of these is discussed in great detail, with Maple implementations, and examples. The corresponding \(q\)-cases are briefly considered. In the final chapters, the author proceeds from the discrete to the continuous case, from recurrence equations to differential equations. A number of differential and integral analogues are given, including the algorithm by Almkvist and Zeilberger finding a holonomic differential equation for the integral \(I(x)= \int^b_a F(x,t)dt\). Each chapter is accompanied by quite a few exercises. While most of them exemplify the algorithms, there are also some that deal more directly with implementations in Maple. Many results (in the exercises as well as in the main text) are of course classical ones whose proofs are well known. It is interesting to see that they are new derivable also by the new approaches. These do not always imply easier proofs than the classical ones, but clearly there is an enormous potential for considering cases that were hitherto beyond reach because of complexity. The book is recommended as a comprehensive and reasonably accessible treatment of the subject.

The authors extend Gosper's algorithm for indefinite hypergeometric summation [see e.g. (1998; Zbl 0909.33001)] for the case when the terms are simultaneously hypergeometric and multibasic hypergeometric. Furthermore they present algorithms to detect the polynomial as well as hypergeometric solutions of recurrence equations in the mixed case. Finally they generalize the concept of greatest factorial factorization to the mixed hypergeometric case. Note that a variant of the multibasic Gosper algorithm appeared also in [\textit{H. Böing} and \textit{W. Koepf}, J. Symb. Comput. 28, 777-799 (1999)]. A box approximation scheme is proposed using the lumped mass numerical integration technique on piecewise linear finite element spaces for a coupled nonlinear thermistor system. Existence and uniqueness of the approximation solution are proved via a fixed point theorem, and an \(H^1\)-error analysis is presented.

As the title suggests, the paper under review studies the \(h\)-triangles of sequentially \((S_r)\) simplicial complexes via algebraic shifting. The notion of a sequentially (\( S_r\)) simplicial complex, introduced by \textit{H. Haghighi} et al. [Proc. Am. Math. Soc. 139, No. 6, 1993--2005 (2011; Zbl 1220.13014)], gives a generalization of two properties for simplicial complexes: being sequentially Cohen-Macaulay and satisfying Serre's condition (\( S_r\)). Let \(\Delta\) be a \((d - 1)\)-dimensional \((S_r)\) simplicial complex with \(\Gamma\) (\(\Delta\)) as its algebraic shifting. When \(i\leq r-1\), the \(i\)th pure skeleton \(\Delta^{[i]}\) of \(\Delta\) is Cohen-Macaulay. This fact enables the study of the \(f\)-vectors of the sequential layers of \(\Delta\) and \(\Gamma(\Delta)\), which leads to the main result (Theorem 3.2), namely, algebraic shifting preserves the \(h\)-triangles: \( h_{i,j}(\Delta)= h_{i,j}(\Gamma(\Delta))\) for every \( i \) and \( j \) with \(0 \leq j \leq i \leq r - 1\). Several corollaries, including the study of iterated Betti numbers, are deduced from this main result. Let \(R=k[x_1,\dots,x_n]\) denote the polynomial ring in \(n\) variables over the field \(k\). Let \(\Delta\) denote a simplicial complex of \(n\) vertices. The authors use the dimension filtration as introduced by the reviewer [in Commutative algebra and algebraic geometry. Proceedings of the Ferrara meeting in honor of Mario Fiorentini on the occasion of his retirement, Ferrara, Italy. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 206, 245--264 (1999; Zbl 0942.13015)] in order to generalize the notion of sequentially Cohen-Macaulay as introduced by \textit{R. P. Stanley}, [Combinatorics and commutative algebra. 2nd ed. Progress in Mathematics 41. Basel: Birkhäuser (2005; Zbl 1157.13302)], (also denoted by Cohen-Macaulay filtered (see the reviewer's paper [loc. cit.])) in order to define sequentially \(S_r\) simplicial complexes. These are those complexes for which the Stanley-Reisner ring \(k[\Delta]\) has a dimension filtration such that each quotient satisfies Serre's condition \(S_r\). Then they use this notation in order to prove the following results: (1) \(\Delta\) is sequentially \(S_r\) if and only if its pure \(i\)-skeleton is \(S_r\) for all \(i\). (2) The Alexander dual of \(\Delta\) is sequentially \(S_r\) if and only if the minimal free resolution of \(k[\Delta]\) over \(R\) is componentwise linear in the first \(r\) steps. (3) \(\Delta\) is sequentially \(S_2\) if and only if the \(i\)-skeleton is connected for all \(i\geq 1\) and any link to a vertex is sequentially \(S_2\). Then these results are applied to graphs. Among various other results for a bipartite graph the following is shown to be equivalent: (i) \(G\) is vertex decomposable. (ii) \(G\) is shellable. (iii) \(G\) is sequentially Cohen-Macaulay. (iv) \(G\) is sequentially \(S_2\). With their results the authors extend various results known to the Cohen-Macaulay situation to the more general circumstances of the \(S_r\) condition.

As the title suggests, the paper under review studies the \(h\)-triangles of sequentially \((S_r)\) simplicial complexes via algebraic shifting. The notion of a sequentially (\( S_r\)) simplicial complex, introduced by \textit{H. Haghighi} et al. [Proc. Am. Math. Soc. 139, No. 6, 1993--2005 (2011; Zbl 1220.13014)], gives a generalization of two properties for simplicial complexes: being sequentially Cohen-Macaulay and satisfying Serre's condition (\( S_r\)). Let \(\Delta\) be a \((d - 1)\)-dimensional \((S_r)\) simplicial complex with \(\Gamma\) (\(\Delta\)) as its algebraic shifting. When \(i\leq r-1\), the \(i\)th pure skeleton \(\Delta^{[i]}\) of \(\Delta\) is Cohen-Macaulay. This fact enables the study of the \(f\)-vectors of the sequential layers of \(\Delta\) and \(\Gamma(\Delta)\), which leads to the main result (Theorem 3.2), namely, algebraic shifting preserves the \(h\)-triangles: \( h_{i,j}(\Delta)= h_{i,j}(\Gamma(\Delta))\) for every \( i \) and \( j \) with \(0 \leq j \leq i \leq r - 1\). Several corollaries, including the study of iterated Betti numbers, are deduced from this main result. The numerical simulation of the Vlasov equation using a phase space grid is studied. In contrast to the Particle-In-Cell (PIC) methods, a semi-Lagrangian method is proposed to discretize the Vlasov equation in two-dimensional phase space. Since this method requires a huge computational effort, one has to carry out the simulations on parallel machines. Therefore a method using patches which decompose the phase domain is used, each patch devoted to a processor, and Hermite boundary conditions allow for the reconstruction as a good approximation of the global solution. The numerical results demonstrate the good efficiency of the two-dimensional case and its good scalability with up to 64 processors.

The authors prove curvature estimates for complete isometric immersions \(f : M^m \rightarrow N^{n+ l} = P^n \times Q^l\) satisfying \(n+2l < 2m\) and which are cylindrically bounded, meaning that \(f(M) \subset B_P(R) \times Q\) where \(B_P(R)\) is an appropriately chosen geodesic ball of radius \(R\) in \(P\). The main tools used in the proof are Otsuki's Lemma, the Omori-Yau maximum principle, and the Hessian comparison theorem. This work generalizes a result by \textit{L. J. Alías} et al. [Trans. Am. Math. Soc. 364, No. 7, 3513--3528 (2012; Zbl 1277.53047)]. An isometric immersion \(\varphi\) of a Riemannian manifold \(M^m\) into the product \(N^{n-l}\times\mathbb R^l\) is said to be cylindrically bounded if there exists a geodesic ball \(B_N(r)\subset N\) centered at \(p\in N\) with radius \(r>0\), such that \(\varphi(M)\subset B_N(r)\times\mathbb R^l\). The authors extend the Jorge-Koutrofiotis theorem and give sharp sectional curvature estimates for complete immersed cylindrically bounded \(m\)-submanifolds \(\varphi:M^m\to N^{n-l}\times\mathbb R^l,\;n+l\leq 2m-1\), provided that either \(\varphi\) is proper with certain growth of the norm of the second fundamental form, or the scalar curvature of \(M\) has strong quadratic decay, moreover, the restriction on the codimension cannot be relaxed. In the case where \(M^m\) is compact, the radius of the smallest ball of \(N\) containing \(\pi_N(\varphi(M))\) is expressed in terms of the sectional curvatures of \(M\) and \(N\). For hypersurfaces, the growth rate of the norm of the second fundamental form is improved. The results are obtained via an application of a version of the Omori-Yau maximum principle for the Hessian of a Riemannian manifold due to \textit{S. Pigola} et al. [Mem. Am. Math. Soc. 822, 99 p. (2005; Zbl 1075.58017)].

The authors prove curvature estimates for complete isometric immersions \(f : M^m \rightarrow N^{n+ l} = P^n \times Q^l\) satisfying \(n+2l < 2m\) and which are cylindrically bounded, meaning that \(f(M) \subset B_P(R) \times Q\) where \(B_P(R)\) is an appropriately chosen geodesic ball of radius \(R\) in \(P\). The main tools used in the proof are Otsuki's Lemma, the Omori-Yau maximum principle, and the Hessian comparison theorem. This work generalizes a result by \textit{L. J. Alías} et al. [Trans. Am. Math. Soc. 364, No. 7, 3513--3528 (2012; Zbl 1277.53047)]. There is given a necessary and sufficient condition for the existence of an alternating Hamiltonian cycle in a complete bipartite graph whose edges are coloured with two colours. Some interesting consequences from it are consider too.

Let \(RG\) be the group ring of an Abelian group \(G\) over a commutative ring \(R\) with identity of prime characteristic \(p\) and let \(S(RG)\) be the \(p\)-component of the group of normalized units of \(RG\). Denote by \(\omega\) the first infinite ordinal. The author tries to prove the following result. Theorem. Suppose \(G\) is an Abelian group and \(R\) is a commutative ring with 1 of prime characteristic \(p\) so that \(R^{p^\omega}\) has nilpotent elements. Then \(S(RG)\) is quasi-complete if and only if \(G\) is a bounded \(p\)-group. For the establishment of this theorem the author uses at three places the way of the proofs of his paper [Tamkang J. Math. 34, No. 1, 87-92 (2003; Zbl 1035.16026)]. However, according to my review [Zbl 1035.16026], the last article contains the following essential inaccuracies and incompletenesses in the proofs of the results: The author ``makes a pseudo-proof of the Main Lemma, i.e., he does not manage to prove it. In proving the Main Lemma he assumes the equality \(\beta=\varepsilon\), i.e., what he must prove. Furthermore, the author establishes in the proof of the lemma some equalities which he does not use. The conclusions in the end of ``Proof'', ``nice property'', are generally groundless \dots. In this way the Main Lemma remains unproved. It is directly and indirectly used in the proofs of Theorems 1-4. Therefore, these theorems are not proved either''. In this way we cannot regard as proved the main result of this paper, stated in the Theorem. Let \(RG\) be the group ring of an Abelian group \(G\) over a commutative ring \(R\) of prime characteristic \(p\) and let \(S(RG)\) be the \(p\)-component of the group of the normalized units of \(RG\). The author tries to give necessary and sufficient conditions for \(S(RG)\) to be quasi-closed under some restrictions on \(G\) and \(R\) but he makes a pseudo-proof of the Main Lemma, i.e., he does not manage to prove it. In proving the Main Lemma he assumes the equality \(\beta=\varepsilon\), i.e., what he must prove. Furthermore, the author establishes in the proof of the lemma some equalities which he does not use. The conclusions in the end of ``Proof'', ``nice property'', are generally groundless, since the indicated expression is not connected with the ordinal \(\beta\): it is connected with \(\alpha\). In this way the Main Lemma remains unproved. It is directly and indirectly used in the proofs of Theorems 1-4. Therefore, these theorems are not proved either. The author discusses on page 90 a result of Nachev's and at the same time he is citing two of his own papers. This is incorrect. The ambigious concept of \(p\)-reduced Abelian group is used without a definition in the paper.

Let \(RG\) be the group ring of an Abelian group \(G\) over a commutative ring \(R\) with identity of prime characteristic \(p\) and let \(S(RG)\) be the \(p\)-component of the group of normalized units of \(RG\). Denote by \(\omega\) the first infinite ordinal. The author tries to prove the following result. Theorem. Suppose \(G\) is an Abelian group and \(R\) is a commutative ring with 1 of prime characteristic \(p\) so that \(R^{p^\omega}\) has nilpotent elements. Then \(S(RG)\) is quasi-complete if and only if \(G\) is a bounded \(p\)-group. For the establishment of this theorem the author uses at three places the way of the proofs of his paper [Tamkang J. Math. 34, No. 1, 87-92 (2003; Zbl 1035.16026)]. However, according to my review [Zbl 1035.16026], the last article contains the following essential inaccuracies and incompletenesses in the proofs of the results: The author ``makes a pseudo-proof of the Main Lemma, i.e., he does not manage to prove it. In proving the Main Lemma he assumes the equality \(\beta=\varepsilon\), i.e., what he must prove. Furthermore, the author establishes in the proof of the lemma some equalities which he does not use. The conclusions in the end of ``Proof'', ``nice property'', are generally groundless \dots. In this way the Main Lemma remains unproved. It is directly and indirectly used in the proofs of Theorems 1-4. Therefore, these theorems are not proved either''. In this way we cannot regard as proved the main result of this paper, stated in the Theorem. A general class of quantum improved stellar models with interiors composed of non-interacting (dust) particles is obtained and analyzed in a framework compatible with asymptotic safety. First, the effective exterior, based on the quantum Einstein gravity approach to asymptotic safety is presented and, second, its effective compatible dust interiors are deduced. The resulting stellar models appear to be devoid of shell-focusing singularities.

Put \(I=[0,1]\), denote by \(\Omega\) the set of all irrational numbers in I, let \(a_ k(\omega)\) \((k=1,2,...)\) be the partial quotients of \(\omega\) in its continued fraction expansion. Let \({\mathcal B}_ I\) be the set of all Borel subsets of I. Put \(i^{(k)}=(i_ 1,i_ 2,...,i_ k)\in {\mathbb{N}}\times {\mathbb{N}}\times...\times {\mathbb{N}}=N^{(k)}\), \(i_ r^{(k)}=(i_ r,i_{r+1},...,i_{r+k-1})(r=1,2,...)\). The symbols \(a^{(k)}(\omega)\), \(a_ r^{(k)}(\omega)\) have the analogous meaning, e.g., \(a_ r^{(k)}(\omega)=(a_ r(\omega)\), \(a_{r+1}(\omega),...,a_{r+k-1}(\omega))\). Let \(\gamma\) be the Gauss' probability measure on \({\mathcal B}_ I\), \(\gamma (A)=\int_{A}\frac{dx}{(1+x)\ell og 2}\) for \(A\in {\mathcal B}_ I\). If \(\mu\) is a probability measure on \({\mathcal B}_ I\), then \(\psi\)-mixing coefficients \(\psi_{\mu}(n)\) of \((a_ n)^{\infty}_{n=1}\) are defined as follows: \[ \psi_{\mu}(n)=\sup | \frac{\mu (a^{(k)}=i^{(k)},a^{(j)}_{k+n}=i^{(j)}_{k+n})}{\mu (a^{(k)}=i^{(k)})\cdot \mu (a^{(j)}_{k+n}=i^{(j)}_{k+n})}- 1| \quad, \] the supremum being taken over all \(i^{(k)}\in N^{(k)}\), \(i^{(j)}_{k+n}\in N^{(j)}\) for which the denominator is different from zero. Here \(\mu (a^{(k)}=i^{(k)})\) denotes the \(\mu\)- measure of all \(\omega\in \Omega\) for which \(a^{(k)}(\omega)=i^{(k)}\) a.s. 0. The paper is closely related to an earlier paper of \textit{W. Philipp} [Monatsh. Math. 105, No.3, 195-206 (1988; Zbl 0638.60039)], where the estimation \[ (1)\quad \psi_{\gamma}(n)\leq (0.8)^ n\quad (n\in {\mathbb{N}}) \] is proved. Using a treatment based on some ideas of P. Lévy the author improves (1) proving the estimation \[ \psi_{\gamma}(n)\leq (3.5-2\sqrt{2})^{n-1}\cdot \log 2\quad (n\in {\mathbb{N}}). \] We prove that the partial quotients \(a_ j\) of the regular continued fraction expansion cannot satisfy a strong law of large numbers of any reasonably growing norming sequence, and that the \(a_ j\) belong to the domain of normal attraction to a stable law with characteristic exponent 1. We also show that the \(a_ j\) satisfy a central limit theorem if a few of the largest ones are trimmed.

Put \(I=[0,1]\), denote by \(\Omega\) the set of all irrational numbers in I, let \(a_ k(\omega)\) \((k=1,2,...)\) be the partial quotients of \(\omega\) in its continued fraction expansion. Let \({\mathcal B}_ I\) be the set of all Borel subsets of I. Put \(i^{(k)}=(i_ 1,i_ 2,...,i_ k)\in {\mathbb{N}}\times {\mathbb{N}}\times...\times {\mathbb{N}}=N^{(k)}\), \(i_ r^{(k)}=(i_ r,i_{r+1},...,i_{r+k-1})(r=1,2,...)\). The symbols \(a^{(k)}(\omega)\), \(a_ r^{(k)}(\omega)\) have the analogous meaning, e.g., \(a_ r^{(k)}(\omega)=(a_ r(\omega)\), \(a_{r+1}(\omega),...,a_{r+k-1}(\omega))\). Let \(\gamma\) be the Gauss' probability measure on \({\mathcal B}_ I\), \(\gamma (A)=\int_{A}\frac{dx}{(1+x)\ell og 2}\) for \(A\in {\mathcal B}_ I\). If \(\mu\) is a probability measure on \({\mathcal B}_ I\), then \(\psi\)-mixing coefficients \(\psi_{\mu}(n)\) of \((a_ n)^{\infty}_{n=1}\) are defined as follows: \[ \psi_{\mu}(n)=\sup | \frac{\mu (a^{(k)}=i^{(k)},a^{(j)}_{k+n}=i^{(j)}_{k+n})}{\mu (a^{(k)}=i^{(k)})\cdot \mu (a^{(j)}_{k+n}=i^{(j)}_{k+n})}- 1| \quad, \] the supremum being taken over all \(i^{(k)}\in N^{(k)}\), \(i^{(j)}_{k+n}\in N^{(j)}\) for which the denominator is different from zero. Here \(\mu (a^{(k)}=i^{(k)})\) denotes the \(\mu\)- measure of all \(\omega\in \Omega\) for which \(a^{(k)}(\omega)=i^{(k)}\) a.s. 0. The paper is closely related to an earlier paper of \textit{W. Philipp} [Monatsh. Math. 105, No.3, 195-206 (1988; Zbl 0638.60039)], where the estimation \[ (1)\quad \psi_{\gamma}(n)\leq (0.8)^ n\quad (n\in {\mathbb{N}}) \] is proved. Using a treatment based on some ideas of P. Lévy the author improves (1) proving the estimation \[ \psi_{\gamma}(n)\leq (3.5-2\sqrt{2})^{n-1}\cdot \log 2\quad (n\in {\mathbb{N}}). \] Thermal and mechanical stability of solid circular plate which is made of saturated and unsaturated porous material with piezoelectric actuators is investigated. The edge of the plate is clamped, and the plate is assumed to be geometrically perfect. The effect of porosity on mechanical and thermal buckling loads is investigated in this study. The results show that the effect of porosity on thermal and mechanical stability is different and depends on some parameters. Effects of fluid thermal expansion and piezoelectric thermal expansion on critical thermal buckling load are investigated for a porous plate. The effects of porous, piezoelectric layers thicknesses, and applied electrical field on thermal and mechanical buckling load of the plate are investigated, too. Mechanical and thermal equilibria are derived based on classic plate theory. Thermal and mechanical critical loads are obtained for all the cases, and finally, there is a comparison between the result of this paper and previous literature.

Let \(G(k)\) denote the smallest positive integer \(s\) such that every sufficiently large natural number is a sum of at most \(s\) \(k\)-th powers of natural numbers. Recently, \textit{T. D. Wooley} obtained [Ann. Math. (2) 135, 131-164 (1992; Zbl 0754.11026)] that \[ G(12)\leq 79,\quad G(13)\leq 87\quad \text{and} \quad G(19)\leq 138. \] In the present paper the author refines some arguments of Wooley (loc. cit.) and is able to reduce the above bounds by 1, namely, 78, 86 and 137, respectively. This paper is a substantial improvement on Vaughan's ``New Iterative Method'' in Waring's problem which gave rise to remarkable improvements in this circle of ideas already, but the methods explained here essentially halve the number of variables required to solve Waring's problem. To be more precise, let \(G(k)\) be the smallest number \(s\) such that all large natural numbers are the sum of \(s\) \(k\)-th powers of natural numbers. A classical estimate of Vinogradov dating from the 1960's states that \(G(k)\leq 2k\log k+o(k\log k)\). Since then this has been improved only in the lower order terms, but in this paper it is shown that \[ G(k)\leq k\log k+k\log\log k+O(1). \] The method becomes effective already when \(k\geq 6\), in particular the new bound \(G(6)\leq 27\) is proved along the way. The new idea is explained in detail in \(\S2\) of the paper under review. Very roughly speaking, Vaughan's new iterative method depends on a procedure which preserves homogeneity in the more classical \(p\)-adic iterative method of Davenport. The price one has to pay is that only ``first differences'' of the relevant exponential sums can be taken effectively, in a sense. But now the author manages to handle higher differences nearly as effective as the first. This is where the improvement comes from. Crucial reading for the ``additive'' number theorist.

Let \(G(k)\) denote the smallest positive integer \(s\) such that every sufficiently large natural number is a sum of at most \(s\) \(k\)-th powers of natural numbers. Recently, \textit{T. D. Wooley} obtained [Ann. Math. (2) 135, 131-164 (1992; Zbl 0754.11026)] that \[ G(12)\leq 79,\quad G(13)\leq 87\quad \text{and} \quad G(19)\leq 138. \] In the present paper the author refines some arguments of Wooley (loc. cit.) and is able to reduce the above bounds by 1, namely, 78, 86 and 137, respectively. Let \(H\) be the class of functions regular in the unit disc. For two functions \(f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\), \(g(z)=\sum_{n=0}^{\infty}b_{n}z^{n}\) of the class \(H\) and for all complex \textit{m} defined the convolution \((f\star_{m} g)(z)=a_{0}b_{0}+m\sum_{n=1}^{\infty}a_{n}b_{n}z^{n}\). In this paper the author gave a solution to a problem of finding all \textit{m}\(\in{\mathcal C}\), for which convolution \(f\star_{m} g\) belongs to \(P_{ab}(X,Y)\) if \(f\in P_{a}(A,B)\) and \(g\in P_{b}(C,D),\) where \(P_{a}(A,B)\) defined as \(P_{a}(A,B)=\{f\in H; f(z)\prec\frac{a+Az}{1+Bz}\}\) for \(a\), \(A\), \(B\); \(aB\neq A\), \(|B|\leq 1\). The author managed to give new generalisations of results obtained by J. and Z. Stankiewicz and also by London.

This paper is concerned with the Hermite-type approximation of the linear elliptic equation \(-\text{div}(a\nabla u)=f\) on a bounded Lipschitz domain \(D\), where \(a\) has the so-called lognormal form \(a=\exp(b)\) with \(b\) a random function defined as \(b(y)= \sum_{j\geq 1}y_j\psi_j\) where \(y=(y_j)\in \mathbb{R}^N\) are i.i.d. standard scalar Gaussian variables and \(\psi_j\) are assumed to be in \(L^{\infty}(D)\). For the case where \(a\) has the so-called affine form see [\textit{M. Bachmayr} et al., ESAIM, Math. Model. Numer. Anal. 51, No. 1, 321--339 (2017; Zbl 1365.41003)]. This paper is concerned with the approximation of the elliptic parametric PDE of the form \(-\text{div}(a\nabla u)=f\) on a bounded Lipschitz domain \(D\), where \(a\) has the affine form \(a=a(y)= \bar{a} + \sum_{j\geq 1}y_j\psi_j\) with \(y=(y_j)\in U=[-1,1]^N\). The functions \(\bar{a}\) and \(\psi_j\) are assumed to be in \(L^{\infty}(D)\). Moreover, it is assumed that \(\sum_{j\geq 1}|\psi_j|\leq \bar{a} - r\) for some \(r>0\). Polynomial approximations of the solution map \(t\mapsto u(y)\in V:=H_0^1(D)\) are studied. It is known that, for any \(0<p<1\), the \(\ell^p\) summability of the \((\|\psi_j\|)\) implies the \(\ell^p\) summability of the \(V\)-norms of Taylor or Legendre coefficients (see [\textit{A. Cohen} et al., Anal. Appl., Singap. 9, No. 1, 11--47 (2011; Zbl 1219.35379)]). The authors considerably improve this result by providing sufficient conditions of \(\ell^p\) summability of the coefficient \(V\)-norm sequences and give a refined analysis which takes into account the amount of overlap between the supports of the \(\psi_j\). For instance, in the case of disjoint supports, it is shown that for all \(0<p<2\), the \(\ell^p\) summability of the coefficient \(V\)-norm sequences follows from the assumption that \((\|\psi_j\|)\) is \(\ell^q\) summable for \(q=2p/(2-p)\).

This paper is concerned with the Hermite-type approximation of the linear elliptic equation \(-\text{div}(a\nabla u)=f\) on a bounded Lipschitz domain \(D\), where \(a\) has the so-called lognormal form \(a=\exp(b)\) with \(b\) a random function defined as \(b(y)= \sum_{j\geq 1}y_j\psi_j\) where \(y=(y_j)\in \mathbb{R}^N\) are i.i.d. standard scalar Gaussian variables and \(\psi_j\) are assumed to be in \(L^{\infty}(D)\). For the case where \(a\) has the so-called affine form see [\textit{M. Bachmayr} et al., ESAIM, Math. Model. Numer. Anal. 51, No. 1, 321--339 (2017; Zbl 1365.41003)]. We study the admission control problem in general networks. Communication requests arrive over time, and the online algorithm accepts or rejects each request while maintaining the capacity limitations of the network. The admission control problem has been usually analyzed as a benefit problem, where the goal is to devise an online algorithm that accepts the maximum number of requests possible. The problem with this objective function is that even algorithms with optimal competitive ratios may reject almost all of the requests, when it would have been possible to reject only a few. This could be inappropriate for settings in which rejections are intended to be rare events. In this article, we consider preemptive online algorithms whose goal is to minimize the number of rejected requests. Each request arrives together with the path it should be routed on. We show an \(O(\log^{2} (mc))\)-competitive randomized algorithm for the weighted case, where \(m\) is the number of edges in the graph and \(c\) is the maximum edge capacity. For the unweighted case, we give an \(O(\log m \log c)\)-competitive randomized algorithm. This settles an open question of \textit{A. Blum} et al. [Internet Math. 1, No. 2, 165--176 (2004; Zbl 1077.94529)]. We note that allowing preemption and handling requests with given paths are essential for avoiding trivial lower bounds.{ }The admission control problem is a generalization of the online set cover with repetitions problem, whose input is a family of \(m\) subsets of a ground set of \(n\) elements. Elements of the ground set are given to the online algorithm one by one, possibly requesting each element a multiple number of times. (If each element arrives at most once, this corresponds to the online set cover problem.) The algorithm must cover each element by different subsets, according to the number of times it has been requested.{ }We give an \(O(\log m \log n)\)-competitive randomized algorithm for the online set cover with repetitions problem.{ }This matches a recent lower bound of \({\Omega}(\log m \log n)\) given by \textit{S. Korman} [On the use of randomization in the online set cover problem. Rechovot: Weizmann Institute of Science (M.S. Thesis) (2005), \url{http://www.wisdom.weizmann.ac.il/~feige/TechnicalReports/SimonKormanThesis.ps}] (based on [\textit{U. Feige}, J. ACM 45, No. 4, 634--652 (1998; Zbl 1065.68573)]) for the competitive ratio of any randomized polynomial time algorithm, under the \(\text{BPP} \neq \text{NP}\) assumption. Given any constant \(\epsilon > 0\), an \(O(\log m \log n)\)-competitive deterministic bicriteria algorithm is shown that covers each element by at least \((1 - \epsilon)k\) sets, where \(k\) is the number of times the element is covered by the optimal solution.

A \(C_n\)-move is a local move for oriented links. A \(C_1\)-move is a crossing change and a \(C_2\)-move is equivalent to a delta move. It is known that two knots have the same finite type invariants of order less than \(n\) if and only if they are related by a finite sequence of \(C_n\)-moves. The main results of this paper are the following, each provided with some examples. [Theorem 2.7] For two links \(L\) and \(M\) that are related by a \(C_n\)-move, the author calculates the difference of the HOMFLY polynomials \(P(L; t, z) - P(M; t, z)\) in terms of the HOMFLY polynomials obtained from the link \(L\) by changing or smoothing some crossings, which is a generalization of an earlier result by \textit{T. Kanenobu} and \textit{R. Nikkuni} [Topology Appl. 146--147, 91--104 (2005; Zbl 1080.57010)]. [Theorem 3.11] For two knots \(L\) and \(M\) that are related by a \(C_n\)-move, the author calculates the difference of the \(n\)th derivatives of the first HOMFLY coefficient polynomials \(P^{(n)}_0(L; t, z) - P^{(n)}_0(M; t, z)\) to be \(\pm 8\) if \(n = 2\) and \(0\) or \(\pm n!\cdot 2^n\) if \(n \geq 3\), where for a knot \(K\), \(P^{(n)}_{2i}(K; t, z)\) denotes the \(n\)th derivative of the \((2i)\)th coefficient of the HOMFLY polynomial \(P(K; t, z)\) in the expression \(P(K; t, z) = \sum_{i \geq 0} P_{2i}(K; t)z^{2i},\) which is a finite type invariant of order \(n\). In Section 4, the author obtains some results on the Conway polynomials and the constant terms of the Q polynomials. The delta move is an unknotting operation which substitutes a triangular entanglement by its rotation, introduced in [\textit{S. V. Matveev}, Mat. Zametki 42, 268--278 (1987; Zbl 0634.57006), resp., Math. Notes 42, 651--656 (1987; Zbl 0649.57010)], [\textit{H. Murakami} and \textit{Y. Nakanishi}, Math. Ann. 284, 75--89 (1989; Zbl 0646.57005)]. In this paper the authors give the relationship between the HOMFLY polynomials for the delta skein quadruple: namely the polynomials for the original oriented link \(L\), for the link \(M\) obtained by performing the delta move on \(L\), and smoothings \(L_0\) and \(M_0\), in the case that the arcs involved in the delta move belong to the same component. Using the relation, the authors also give the relationship between other finite type invariants for the quadruple. They further obtain a formula for the zero coefficient polynomials for the twist knots.

A \(C_n\)-move is a local move for oriented links. A \(C_1\)-move is a crossing change and a \(C_2\)-move is equivalent to a delta move. It is known that two knots have the same finite type invariants of order less than \(n\) if and only if they are related by a finite sequence of \(C_n\)-moves. The main results of this paper are the following, each provided with some examples. [Theorem 2.7] For two links \(L\) and \(M\) that are related by a \(C_n\)-move, the author calculates the difference of the HOMFLY polynomials \(P(L; t, z) - P(M; t, z)\) in terms of the HOMFLY polynomials obtained from the link \(L\) by changing or smoothing some crossings, which is a generalization of an earlier result by \textit{T. Kanenobu} and \textit{R. Nikkuni} [Topology Appl. 146--147, 91--104 (2005; Zbl 1080.57010)]. [Theorem 3.11] For two knots \(L\) and \(M\) that are related by a \(C_n\)-move, the author calculates the difference of the \(n\)th derivatives of the first HOMFLY coefficient polynomials \(P^{(n)}_0(L; t, z) - P^{(n)}_0(M; t, z)\) to be \(\pm 8\) if \(n = 2\) and \(0\) or \(\pm n!\cdot 2^n\) if \(n \geq 3\), where for a knot \(K\), \(P^{(n)}_{2i}(K; t, z)\) denotes the \(n\)th derivative of the \((2i)\)th coefficient of the HOMFLY polynomial \(P(K; t, z)\) in the expression \(P(K; t, z) = \sum_{i \geq 0} P_{2i}(K; t)z^{2i},\) which is a finite type invariant of order \(n\). In Section 4, the author obtains some results on the Conway polynomials and the constant terms of the Q polynomials. In this paper, we consider low rank matrix recovery with impulsive noise. We first study the difference of nuclear norm and Frobenius norm model and present a stable recovery result based on the matrix restricted isometry property. Then we find the truncated difference of nuclear norm and Frobenius norm model can also stably recover low rank matrices with impulsive noise.

Using a concept of a fuzzy feebly open set the authors define a fuzzy feebly continuous mapping and study its properties. For related results see also \textit{A. S. Mashhour, M. H. Ghanim} and \textit{M. A. Fath Alla} [Bull. Calcutta Math. Soc. 78, 57-69 (1986; Zbl 0604.54008)] and references therein. Note that in the terminology of Mashhour et al. a fuzzy feebly continuous mapping is called fuzzy \(\alpha\)-continuous. The study of weaker forms of fuzzy continuity was initiated by \textit{K. K. Azad} [J. Math. Anal. Appl. 82, 14-32 (1981; Zbl 0511.54006)]. The authors continue the study. They introduce a concept of a fuzzy \(\beta\)- continuous, a fuzzy \(\beta\)-open and a fuzzy \(\beta\)-closed mapping and give some of their properties.

Using a concept of a fuzzy feebly open set the authors define a fuzzy feebly continuous mapping and study its properties. For related results see also \textit{A. S. Mashhour, M. H. Ghanim} and \textit{M. A. Fath Alla} [Bull. Calcutta Math. Soc. 78, 57-69 (1986; Zbl 0604.54008)] and references therein. Note that in the terminology of Mashhour et al. a fuzzy feebly continuous mapping is called fuzzy \(\alpha\)-continuous. Compression of information in a visual analysis is one of the key problems. Orthogonal transformations in their turn present one of the methods of information compression. Here on a set of the orthogonal transformations a problem of choosing an optimal representative from the wavelet transformations is considered. A technology based on Haar's wavelet is described.

The author continues his study of belief change in the context of branching-time models, begun in the paper [``Axiomatic characterization of the AGM theory of belief revision in a temporal logic'', Artif. Intell. 171, No. 2--3, 144--160 (2007; Zbl 1168.03317)]. There, the AGM postulates for belief revision were represented in a language containing three modal operators: a belief operator, one representing the exact amount of information available to an agent and, crucially, a `next-time' operator in branching time. The language thus differs from those of the dynamic doxastic logics that have also been used to represent belief change in a modal setting. The paper under review studies the conditions under which a partial belief revision function induced by a state-time pair in a frame may be extended to a belief revision function defined over the entire frame and satisfying the AGM postulates. Final sections also explore the formulation of well-known principles of iterated belief change using the temporally modalized language. Since belief revision deals with the interaction of belief and information over time, branching-time temporal logic seems a natural setting for a theory of belief change. We propose two extensions of a modal logic that, besides the next-time temporal operator, contains a belief operator and an information operator. The first logic is shown to provide an axiomatic characterization of the first six postulates of the AGM theory of belief revision, while the second, stronger, logic provides an axiomatic characterization of the full set of AGM postulates.

The author continues his study of belief change in the context of branching-time models, begun in the paper [``Axiomatic characterization of the AGM theory of belief revision in a temporal logic'', Artif. Intell. 171, No. 2--3, 144--160 (2007; Zbl 1168.03317)]. There, the AGM postulates for belief revision were represented in a language containing three modal operators: a belief operator, one representing the exact amount of information available to an agent and, crucially, a `next-time' operator in branching time. The language thus differs from those of the dynamic doxastic logics that have also been used to represent belief change in a modal setting. The paper under review studies the conditions under which a partial belief revision function induced by a state-time pair in a frame may be extended to a belief revision function defined over the entire frame and satisfying the AGM postulates. Final sections also explore the formulation of well-known principles of iterated belief change using the temporally modalized language. Cryptography is the mathematics of secret codes. Symmetric-key block ciphers are the most fundamental elements in many cryptographic systems. They provide confidentiality which is used in a large variety of applications such as protection of the secrecy of login passwords, e-mail messages, video transmissions and many other applications. This paper includes both general concepts and details of specific cipher algorithms. It consists of the newest fundamental principles for designing and evaluating in block cipher algorithms.

The paper deals with the representation of the solutions of abstract Volterra equations in general Banach spaces. To be more precise, consider three Banach spaces \(X_i(\Omega_i, \Sigma_i,\mu_i; {\mathcal{Y}}_i)\), \(i=0,1,2\), where \(\mu_i\) is either a Borel measure, or the completion of a finite measure. It is assumed that the spaces used have finite measures and they are ideal in the sense that, given a measurable function \(y: \Omega_i\to {\mathcal{Y}}_i\), if for a certain \(x\in X_i(\Omega_i, \Sigma_i,\mu_i; {\mathcal{Y}}_i)\) it satisfies \(\| y(s)\| \leq\| x(s)\| \) for almost all \(s\), then we have \(y\in X_i(\Omega_i, \Sigma_i,\mu_i; {\mathcal{Y}}_i)\). Assume that \(X_0\) is the direct product \(X_1\otimes{\mathcal{Y}}_0\) and consider a linear operator \({\mathcal{L}}:X_0\to X_2\). In this paper, the author provides information about the integral representation of the solution of the operator equation \({\mathcal{L}}x=f\), when the operator \(\mathcal{L}\) is a so-called Volterra operator, a notion which is borrowed from previous work of the author and from the joint work of \textit{M.\,E.\thinspace Drakhlin, A.\,Ponosov} and \textit{E.\,Stepanov} [Proc.\ Edinb.\ Math.\ Soc., II.\ Ser.\ 45, No.\,2, 467--490 (2002; Zbl 1030.47045)]. (The notions of chain of measurable sets and of memory of an operator on a chain are also used.) An example of a delay differential equation illustrates the results. In the last section of the paper, some kernel properties of the integral representation of the solutions are given. Two classes of nonlinear operators are introduced. Both classes include local operators (in particular, Nemytskii operators) and are closed with respect to composition. Basic properties of these operators are studied.

The paper deals with the representation of the solutions of abstract Volterra equations in general Banach spaces. To be more precise, consider three Banach spaces \(X_i(\Omega_i, \Sigma_i,\mu_i; {\mathcal{Y}}_i)\), \(i=0,1,2\), where \(\mu_i\) is either a Borel measure, or the completion of a finite measure. It is assumed that the spaces used have finite measures and they are ideal in the sense that, given a measurable function \(y: \Omega_i\to {\mathcal{Y}}_i\), if for a certain \(x\in X_i(\Omega_i, \Sigma_i,\mu_i; {\mathcal{Y}}_i)\) it satisfies \(\| y(s)\| \leq\| x(s)\| \) for almost all \(s\), then we have \(y\in X_i(\Omega_i, \Sigma_i,\mu_i; {\mathcal{Y}}_i)\). Assume that \(X_0\) is the direct product \(X_1\otimes{\mathcal{Y}}_0\) and consider a linear operator \({\mathcal{L}}:X_0\to X_2\). In this paper, the author provides information about the integral representation of the solution of the operator equation \({\mathcal{L}}x=f\), when the operator \(\mathcal{L}\) is a so-called Volterra operator, a notion which is borrowed from previous work of the author and from the joint work of \textit{M.\,E.\thinspace Drakhlin, A.\,Ponosov} and \textit{E.\,Stepanov} [Proc.\ Edinb.\ Math.\ Soc., II.\ Ser.\ 45, No.\,2, 467--490 (2002; Zbl 1030.47045)]. (The notions of chain of measurable sets and of memory of an operator on a chain are also used.) An example of a delay differential equation illustrates the results. In the last section of the paper, some kernel properties of the integral representation of the solutions are given. [For the entire collection see Zbl 0556.00003.] Factorization theorems for the cardinal dimension k and for the large inductive dimension are proved. It is shown that the transfinite small inductive dimension is not defined for a large class of compact spaces.

The book is devoted to study those aspects of fractal geometry in \(\mathbb R^n\), which are connected to Fourier analysis, function spaces, and pseudodifferential operators. In the earlier book [\textit{D. E. Edmunds} and \textit{H. Triebel}, ``Function spaces, entropy numbers, differential operators'' (1996; Zbl 0865.46020)] the authors successfully applied estimates of entropy numbers of compact embeddings between function spaces to the spectral theory of degenerate pseudodifferential operators on bounded domains and on \(\mathbb R^n\). A good part of the book under review is based on similar techniques, but this time in the context of fractals. The exposition departs from some basic material on fractals with special emphasis on the \(d\)-sets. One of the central aims of the book is to introduce and study function spaces on \(d\)-sets. Let \(\Gamma\) be a \(d\)-set. The \(L^p(\Gamma)\) spaces are relatively easy to define since the measure on \(\Gamma\) is more or less uniquely determined, but their structure and relations to other function spaces are very complicated. This topic together with the introduction and study of the \(B^s_{p,q}(\Gamma)\) spaces is treated in detail in Chapter 4, and it needs a lot of a deep preliminary material, which is contained in Chapters 2 and 3. These chapters include entropy numbers on weighted \(\ell_p\) spaces with a dyadic block structure, and a new approach to the atomic decomposition of the spaces \(B_{p,q}^s\) and \(F_{p,q}^s\) on \(\mathbb R^n\), consisting of further atomizing of the atoms, which results in subatomic (or quarkonial) decomposition. A thorough study of asymptotic behaviour of entropy numbers of embedding between these function spaces is carried out next. It is worth mentioning that there is virtually no literature on this topic and hence the most of the presented material is published here the first time. The final Chapter 5 deals with spectra of pseudodifferential operators with fractal coefficients. On suitable function spaces, these operators are compact, and estimates for distribution of their eigenvalues and counting function can be obtained. Particular attention is paid to the \(n\)-dimensional drums with a compact fractal layer. This book, based on the results of the authors and their co-workers, deals with the following topics: 1) entropy numbers in quasi-Banach spaces, 2) distribution of eigenvalues of elliptic differential and pseudodifferential operators and 3) function spaces on \(R^n\) and in domains. The relationship between them is the main scope of the book. Two scales of function spaces are involved, the Besov-type spaces \(B^s_{pq}\) and the Triebel-Lizorkin type spaces \(F^s_{pq}\). Chapter 2 of the book deals with these function spaces. The reader can find here a condensed self-contained exposition of the basics of the theory together with some recent developments, concerning, e.g. embeddings, limiting cases, spaces in domains and others. Logarithmic Sobolev spaces are to be specially mentioned. Chapter 3 and 4 deal with entropy and approximation numbers of compact embeddings. One of the main typical example is as follows. Let \(B^{s_1}_{p_1q_1}(\Omega)\to B^{s_2}_{p_2q_2}(\Omega)\) be a compact embedding which is guaranteed by the conditions: \(s_1-s_2-n ({1\over p_1}-{1\over p_2})_+>0\), \(0< q_1\leq\infty\), \(0<q_2\leq\infty\), and let \(e_k\) be the entropy numbers of this embedding. Then \[ c_1k^{-(s_1-s_2)\setminus n}\leq e_k\leq c_1k^{-(s_1-s_2)\setminus n}. \] This goes back to M. S. Birman and M. Z. Solomiak, who first developed results of such a type, e.g. for the embedding \(W^s_p(\Omega)\to L_q(\Omega)\). In this connection the authors observe that they work purely within the frameworks of Fourier-analytic techniques. Chapter 5 contains applications of the results for the entropy and approximative numbers of compact embeddings to the problem of distribution of eigenvalues of degenerate elliptic differential and pseudodifferential operators. This is based on the results which go back to B. Carl's estimate \(|\mu_k|\leq \sqrt{2}e_k\), \(\mu_k\) and \(e_k\) being respectively the eigenvalues and the entropy numbers of a compact operator (in a quasi-Banach space). Presented in a clear, self contained and well organized manner, the book is easy to read, although it deals with rather complicated considerations. The contents are the following: Ch. 1. The abstract background. 1.1. Introduction. 1.2. Spectral theory in quasi-Banach spaces. 1.3. Entropy numbers and approximation numbers. Ch. 2. Function spaces. 2.1. Introduction. 2.2. The spaces \(B^s_{pq}\) and \(F^s_{pq}\) on \(R^n\). 2.3. Special properties. 2.4. Hölder inequalities. 2.5. The spaces \(B^s_{pq}\) and \(F^s_{pq}\) on domains. 2.6. The spaces \(L_p(\log L)_a\) and the logarithmic Sobolev spaces. 2.7. Limiting embeddings. Ch. 3. Entropy and approximation numbers of embeddings. 3.1. Introduction. 3.2. The embedding of \(\ell^m_p\) in \(\ell^m_q\). 3.3. Embeddings between function spaces. 3.4. Limiting embeddings in spaces of Orlicz type. 3.5. Embeddings in non-smooth domains. Ch. 4. Weighted function spaces and entropy numbers. 4.1. Introduction. 4.2. Weighted spaces. 4.3. Entropy numbers. Ch. 5. Elliptic operators. 5.1. Introduction. 5.2. Elliptic operators in domains: non-limiting cases. 5.3. Elliptic operators in domains: limiting cases. 5.4. Elliptic operators in \(R^n\).

The book is devoted to study those aspects of fractal geometry in \(\mathbb R^n\), which are connected to Fourier analysis, function spaces, and pseudodifferential operators. In the earlier book [\textit{D. E. Edmunds} and \textit{H. Triebel}, ``Function spaces, entropy numbers, differential operators'' (1996; Zbl 0865.46020)] the authors successfully applied estimates of entropy numbers of compact embeddings between function spaces to the spectral theory of degenerate pseudodifferential operators on bounded domains and on \(\mathbb R^n\). A good part of the book under review is based on similar techniques, but this time in the context of fractals. The exposition departs from some basic material on fractals with special emphasis on the \(d\)-sets. One of the central aims of the book is to introduce and study function spaces on \(d\)-sets. Let \(\Gamma\) be a \(d\)-set. The \(L^p(\Gamma)\) spaces are relatively easy to define since the measure on \(\Gamma\) is more or less uniquely determined, but their structure and relations to other function spaces are very complicated. This topic together with the introduction and study of the \(B^s_{p,q}(\Gamma)\) spaces is treated in detail in Chapter 4, and it needs a lot of a deep preliminary material, which is contained in Chapters 2 and 3. These chapters include entropy numbers on weighted \(\ell_p\) spaces with a dyadic block structure, and a new approach to the atomic decomposition of the spaces \(B_{p,q}^s\) and \(F_{p,q}^s\) on \(\mathbb R^n\), consisting of further atomizing of the atoms, which results in subatomic (or quarkonial) decomposition. A thorough study of asymptotic behaviour of entropy numbers of embedding between these function spaces is carried out next. It is worth mentioning that there is virtually no literature on this topic and hence the most of the presented material is published here the first time. The final Chapter 5 deals with spectra of pseudodifferential operators with fractal coefficients. On suitable function spaces, these operators are compact, and estimates for distribution of their eigenvalues and counting function can be obtained. Particular attention is paid to the \(n\)-dimensional drums with a compact fractal layer. We consider, on compact Riemann surfaces, singular extremal metrics whose Gauss curvatures have nonzero umbilical Hessians, which are usually called HCMU metrics. The singular sets of these HCMU metrics consist of conical and cusp singularities, both of which are finitely many. We show that these metrics exist with the prescribed singularities if and only if so do certain meromorphic 1-forms on the Riemann surfaces, which only have simple poles with real residues and whose real parts are exact outside their poles.

A left \(R\)-module \(M\) is called max-injective if for any maximal left ideal \(\mathfrak m\), any homomorphism \(f\colon\mathfrak m\to M\) can be extended to \(g\colon R\to M\) [\textit{M.-Y. Wang} and \textit{G. Zhao}, Acta Math. Sin., Engl. Ser. 21, No. 6, 1451-1458 (2005; Zbl 1108.16003)]. This is clearly equivalent to \(\text{Ext}_R^1(R/\mathfrak m,M)=0\) for any maximal left ideal \(\mathfrak m\). Similarly, a right \(R\)-module \(M\) is called max-flat if \(\text{Tor}_1^R(M,R/\mathfrak m)=0\) for any left maximal ideal of \(R\) [\textit{M.-Y. Wang}, Frobenius structure in algebra. Science Press, Beijing (2005)]. In the paper under review, a ring is defined to be left max-coherent if every maximal left ideal is finitely presented. Let \(\mathfrak{MI}\) denote the class of all max-injective \(R\)-modules and \(\mathfrak{MF}\) the class of all max-flat \(R\)-modules. A number of properties of left max-coherent rings and the classes \(\mathfrak{MI}\) and \(\mathfrak{MF}\) are proved and it is shown that if \(R\) is a left max-coherent ring, then every right \(R\)-module (resp. left \(R\)-module) has an \(\mathfrak{MF}\)-preenvelope (resp. \(\mathfrak{MI}\)-precover). For a left max-coherent ring, \(R\) being left max-injective is also shown to be equivalent to any left \(R\)-module having an epic \(\mathfrak{MI}\)-cover and to any right \(R\)-module having a monic \(\mathfrak{MF}\)-preenvelope. The main purpose of the next section is to consider the kernel of \(\mathfrak{MI}\)-(pre)covers and the cokernel of \(\mathfrak{MF}\)-(pre)envelopes. For this purpose, a left \(R\)-module \(M\) is called MI-injective if \(\text{Ext}_R^1(N,M)=0\) for any max-injective left \(R\)-module \(N\). A right \(R\)-module \(F\) is called MI-flat if \(\text{Tor}_1^R(F,N)=0\) for any max-injective left \(R\)-module \(N\). It is shown that \(R\) is a QF-ring if and only if every left \(R\)-module is MI-injective. For \(R\) a left max-coherent ring, the following results are proved: (1) If \(C\) is the cokernel of an \(\mathfrak{MF}\)-preenvelope \(f\colon M\to F\) of a right \(R\)-module \(M\) with \(F\) flat, then \(C\) is MI-flat. (2) If \(L\) is a finitely presented MI-flat right \(R\)-module, then \(L\) is the cokernel of an \(\mathfrak{MF}\)-preenvelope \(g\colon K\to P\) with \(P\) flat. In the final section \(\mathfrak{MI}\) resolutions are constructed and the \(\mathfrak{MI}\)-dimensions of modules and rings are investigated. It is shown that if every maximal left ideal of a ring \(R\) is finitely generated and every left \(R\)-module has an \(\mathfrak{MI}\)-cover, with the unique mapping property, then \(R\) is left max-coherent and gl. right \(\mathfrak{MI}\text{-}\dim_RM\leq 2\). A right \(R\)-module \(E\) is defined to be maximally injective (max-injective) if for any maximal right ideal \(m\) of \(R\), each right \(R\)-homomorphism \(f\colon m\to E\) can be extended to an \(R\)-homomorphism \(f'\colon R\to E\). A ring \(R\) is said to be right max-injective if \(R_R\) is max-injective. A right \(R\)-module \(E\) (\(R\) any ring) is shown to be right max-injective if and only if \(\text{Ext}_R^1(R/m,E)=0\) for every maximal right ideal \(m\leq R_R\). If a ring \(R\) is right max-injective, its Jacobson radical is contained in its right singular ideal. An example proves that a ring \(R\) may be max-injective, without being self-injective. Faith's conjecture [see \textit{C. Faith} and \textit{D. V. Huynh}, J. Algebra Appl. 1, No. 1, 75-105 (2002; Zbl 1034.16005)] states that every left (or right) perfect, right self-injective ring is QF. The authors show that Faith's conjecture is equivalent to the following statement: Any left perfect, right max-injective ring \(R\) is QF. They then prove that Faith's conjecture is true for the case where the ring \(R\) also satisfies the left *-condition, i.e. \(l(I)/l(m)\) is finitely generated for any pair \(I\leq m\) of right ideals such that \(m\) is maximal. Left perfect right max-injective rings are studied further. Every injective right \(R\)-module \(E\) over a left perfect right max-injective ring is shown to be the injective hull of a projective submodule. This result may be considered as an approximation to Faith's conjecture.

A left \(R\)-module \(M\) is called max-injective if for any maximal left ideal \(\mathfrak m\), any homomorphism \(f\colon\mathfrak m\to M\) can be extended to \(g\colon R\to M\) [\textit{M.-Y. Wang} and \textit{G. Zhao}, Acta Math. Sin., Engl. Ser. 21, No. 6, 1451-1458 (2005; Zbl 1108.16003)]. This is clearly equivalent to \(\text{Ext}_R^1(R/\mathfrak m,M)=0\) for any maximal left ideal \(\mathfrak m\). Similarly, a right \(R\)-module \(M\) is called max-flat if \(\text{Tor}_1^R(M,R/\mathfrak m)=0\) for any left maximal ideal of \(R\) [\textit{M.-Y. Wang}, Frobenius structure in algebra. Science Press, Beijing (2005)]. In the paper under review, a ring is defined to be left max-coherent if every maximal left ideal is finitely presented. Let \(\mathfrak{MI}\) denote the class of all max-injective \(R\)-modules and \(\mathfrak{MF}\) the class of all max-flat \(R\)-modules. A number of properties of left max-coherent rings and the classes \(\mathfrak{MI}\) and \(\mathfrak{MF}\) are proved and it is shown that if \(R\) is a left max-coherent ring, then every right \(R\)-module (resp. left \(R\)-module) has an \(\mathfrak{MF}\)-preenvelope (resp. \(\mathfrak{MI}\)-precover). For a left max-coherent ring, \(R\) being left max-injective is also shown to be equivalent to any left \(R\)-module having an epic \(\mathfrak{MI}\)-cover and to any right \(R\)-module having a monic \(\mathfrak{MF}\)-preenvelope. The main purpose of the next section is to consider the kernel of \(\mathfrak{MI}\)-(pre)covers and the cokernel of \(\mathfrak{MF}\)-(pre)envelopes. For this purpose, a left \(R\)-module \(M\) is called MI-injective if \(\text{Ext}_R^1(N,M)=0\) for any max-injective left \(R\)-module \(N\). A right \(R\)-module \(F\) is called MI-flat if \(\text{Tor}_1^R(F,N)=0\) for any max-injective left \(R\)-module \(N\). It is shown that \(R\) is a QF-ring if and only if every left \(R\)-module is MI-injective. For \(R\) a left max-coherent ring, the following results are proved: (1) If \(C\) is the cokernel of an \(\mathfrak{MF}\)-preenvelope \(f\colon M\to F\) of a right \(R\)-module \(M\) with \(F\) flat, then \(C\) is MI-flat. (2) If \(L\) is a finitely presented MI-flat right \(R\)-module, then \(L\) is the cokernel of an \(\mathfrak{MF}\)-preenvelope \(g\colon K\to P\) with \(P\) flat. In the final section \(\mathfrak{MI}\) resolutions are constructed and the \(\mathfrak{MI}\)-dimensions of modules and rings are investigated. It is shown that if every maximal left ideal of a ring \(R\) is finitely generated and every left \(R\)-module has an \(\mathfrak{MI}\)-cover, with the unique mapping property, then \(R\) is left max-coherent and gl. right \(\mathfrak{MI}\text{-}\dim_RM\leq 2\). The framework of budget-feasible mechanism design studies procurement auctions where the auctioneer (buyer) aims to maximize his valuation function subject to a hard budget constraint. We study the problem of designing truthful mechanisms that have good approximation guarantees and never pay the participating agents (sellers) more than the budget. We focus on the case of general (non-monotone) submodular valuation functions and derive the first truthful, budget-feasible, and \(O(1)\)-approximation mechanisms that run in polynomial time in the value query model, for both offline and online auctions. Prior to our work, the only \(O(1)\)-approximation mechanism known for non-monotone submodular objectives required an exponential number of value queries. At the heart of our approach lies a novel greedy algorithm for non-monotone submodular maximization under a knapsack constraint. Our algorithm builds two candidate solutions simultaneously (to achieve a good approximation), yet ensures that agents cannot jump from one solution to the other (to implicitly enforce truthfulness). The fact that in our mechanism the agents are not ordered according to their marginal value per cost allows us to appropriately adapt these ideas to the online setting as well. To further illustrate the applicability of our approach, we also consider the case where additional feasibility constraints are present, for example, at most \(k\) agents can be selected. We obtain \(O(p)\)-approximation mechanisms for both monotone and non-monotone submodular objectives, when the feasible solutions are independent sets of a \(p\)-system. With the exception of additive valuation functions, no mechanisms were known for this setting prior to our work. Finally, we provide lower bounds suggesting that, when one cares about nontrivial approximation guarantees in polynomial time, our results are, asymptotically, the best possible.

This paper further develops work of the author [Indag. Math., New Ser. 13, No. 1, 103--124 (2002; Zbl 1016.37003)] on particular orbits to give a more general understanding of a natural question about the linear action of \(SL_2(\mathbb R)\) on \(\mathbb R^2\). For a square \(\Omega\), an absolute error term is found for the number \(N(k,x)\) of matrices of norm not exceeding \(k\) which send \(x\) into \(\Omega\) in terms of the Diophantine properties of the angle (ratio of coordinates) of the vector \(x\). The method converts the question into a Diophantine counting problem. Methods from ergodic theory are exploited to give the asymptotic behavior as \(k\to\infty\). The distribution of orbits under the natural linear action of \(\mathrm{SL}(2,\mathbb Z\) on the plane is studied. An application/example is the evaluation of the asymptotic behaviour of the set \(\{A\in \mathrm{SL}(2,\mathbb Z)\mid Ax\in [0,r]^2, \|A\|\leq t\}\) for large \(r,t>0\), where \(x\) is an explicit vector with incommensurable entries. More general results in this direction have been obtained by \textit{F. Ledrappier} [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 1, 61--64 (1999; Zbl 0928.22012)] and others using the ergodic theory of Lie group actions. In this paper more restricted results are reached using number theory methods, giving sharper and more explicit error terms.

This paper further develops work of the author [Indag. Math., New Ser. 13, No. 1, 103--124 (2002; Zbl 1016.37003)] on particular orbits to give a more general understanding of a natural question about the linear action of \(SL_2(\mathbb R)\) on \(\mathbb R^2\). For a square \(\Omega\), an absolute error term is found for the number \(N(k,x)\) of matrices of norm not exceeding \(k\) which send \(x\) into \(\Omega\) in terms of the Diophantine properties of the angle (ratio of coordinates) of the vector \(x\). The method converts the question into a Diophantine counting problem. Methods from ergodic theory are exploited to give the asymptotic behavior as \(k\to\infty\). Let \(G\) be topological group. The classifying space \(BG\) plays an important role in applications to bundle theory and cohomology of groups. In this paper a filtration of \(BG\) is introduced by using the descending central series of free groups. The authors construct simplicial spaces \(B(q,G)\) that provide a filtration of \(BG\). In fact the construction gives a principal \(G\)-bundle which fits into a commutative diagram. The basic properties of \(B(q,G)\) are discussed, including the homotopy type and cohomology when \(G\) is either finite or a compact connected Lie group. For a finite group the construction gives rise to a covering space with monodromy related to a delicate result in group theory equivalent to the odd-order theorem of Feit-Thompson (which states that every finite group of odd order is solvable) (Proposition 7.2). A formula is given which provides information on the cardinality of the set of commuting elements in a finite group (Corollary 5.4). It is shown that \(B(q,G)\) is a natural colimit which is weakly equivalent to a more tractable homotopy colimit (Theorem 4.3). The basic properties of \(B(q,G)\) for connected Lie groups are analyzed in Section 6, including a calculation of the rational cohomology of \(B(2,G)\) when \(G\) is compact (Theorem 6.1). In a special case, the space \(B(2,G)\) is described for a particular class of groups namely the transitively commutative groups (Proposition 8.4).

[For part I see the authors, Rev. Math. Phys. 20, No.~1, 71--115 (2008; Zbl 1145.81053).] Devices exhibiting the integer quantum Hall effect can be modeled by one-particle Schrödinger operators describing the planar motion of an electron in a perpendicular, constant magnetic field, and under the influence of an electrostatic potential. The electron motion is confined to bounded or unbounded subsets of the plane by confining potential barriers. The edges of the confining potential barriers create edge currents. This is the second of two papers in which we review recent progress and prove explicit lower bounds on the edge currents associated with one- and two-edge geometries. In this paper, we study various unbounded and bounded, two-edge geometries with soft and hard confining potentials. These two-edge geometries describe the electron confined to unbounded regions in the plane, such as a strip, or to bounded regions, such as a finite length cylinder. We prove that the edge currents are stable under various perturbations, provided they are suitably small relative to the magnetic field strength, including perturbations by random potentials. The existence of, and the estimates on, the edge currents are independent of the spectral type of the operator. Devices exhibiting the integer quantum Hall effect can be modeled by one-electron Schrödinger operators describing the planar motion of an electron in a perpendicular, constant magnetic field, and under the influence of an electrostatic potential. The electron motion is confined to unbounded subsets of the plane by confining potential barriers. The edges of the confining potential barrier create edge currents. In this, the first of two papers, we prove explicit lower bounds on the edge currents associated with one-edge, unbounded geometries formed by various confining potentials. This work extends some known results that we review. The edge currents are carried by states with energy localized between any two Landau levels. These one-edge geometries describe the electron confined to certain unbounded regions in the plane obtained by deforming half-plane regions. We prove that the currents are stable under various potential perturbations, provided the perturbations are suitably small relative to the magnetic field strength, including perturbations by random potentials. For these cases of one-edge geometries, the existence of, and the estimates on, the edge currents imply that the corresponding Hamiltonian has intervals of absolutely continuous spectrum. In the second paper of this series, we consider the edge currents associated with two-edge geometries describing bounded, cylinder-like regions, and unbounded, strip-like, regions.

[For part I see the authors, Rev. Math. Phys. 20, No.~1, 71--115 (2008; Zbl 1145.81053).] Devices exhibiting the integer quantum Hall effect can be modeled by one-particle Schrödinger operators describing the planar motion of an electron in a perpendicular, constant magnetic field, and under the influence of an electrostatic potential. The electron motion is confined to bounded or unbounded subsets of the plane by confining potential barriers. The edges of the confining potential barriers create edge currents. This is the second of two papers in which we review recent progress and prove explicit lower bounds on the edge currents associated with one- and two-edge geometries. In this paper, we study various unbounded and bounded, two-edge geometries with soft and hard confining potentials. These two-edge geometries describe the electron confined to unbounded regions in the plane, such as a strip, or to bounded regions, such as a finite length cylinder. We prove that the edge currents are stable under various perturbations, provided they are suitably small relative to the magnetic field strength, including perturbations by random potentials. The existence of, and the estimates on, the edge currents are independent of the spectral type of the operator. In the case of small gravity, the shape of a drop on a flat horizontal plane is obtained by a formal expansion with respect to gravity. The leading term is the drop with the given volume in the absence of gravity, i. e., a spherical cup. A comparison with experimental data is presented.

Let \({f: M\to M}\) be a partially hyperbolic diffeomorphism (i.e., there exist certain continuous \(Df\)-invariant splittings \({TM = E^s\oplus E^c\oplus E^u}\)). The diffeomorphism \(f\) is called dynamically coherent if there exist \(f\)-invariant foliations \({\mathcal{W}^{cs}}\) and \({\mathcal{W}^{cu}}\) tangent to \({E^s\oplus E^c}\) and \(E^c\oplus E^u,\) respectively. The main result of the article states that if \(M\) is a Seifert 3-manifold with hyperbolic base and the induced action of \(f\) in the base has a pseudo-Anosov component, then \(f\) is not dynamically coherent. This result provides a new proof of the non-dynamical coherence of the examples constructed in [\textit{C. Bonatti} et. al., Geom. Topol. 24, No. 4, 1751--1790 (2020; Zbl 1470.37049)] and extends it to any partially hyperbolic diffeomorphism in the same isotopy class. Let \(M\) be a closed \(3\)-manifold admitting an Anosov flow. The two typical settings are geodesic flows on closed hyperbolic surfaces and Anosov flows on 3-manifolds which admit transverse tori. The paper develops a general method for constructing partially hyperbolic diffeomorphisms in many mapping classes of \(M\), by composing the time-one maps of Anosov flows with (arbitrarily many) Dehn twists. A criterion for partial hyperbolicity is given through the concept of \(h\)-transversality. In the case of the geodesic flows on \(T^1S\), the unit tangent bundle of a closed hyperbolic surface \(S\), the authors construct Theorem 1.1 partially hyperbolic diffeomorphisms whose mapping classes form a subgroup isomorphic to the mapping class group \(\mathcal{M}(S)\), and it is proved Corollary 3.7 that the set of all mapping classes realized by partially hyperbolic diffeomorphisms does not form a subgroup of the mapping class group \(\mathcal{M}(T^1S)\). In the case of Anosov flows on a 3-manifold \(M\), every mapping class in the subgroup of \(\mathcal{M}(M)\) generated by Dehn twists along the transverse tori, contains a partially hyperbolic representative (Theorem 1.3). In the example given in Corollary 1.4, a partially hyperbolic representative can be produced in virtually all mapping classes of \(M\). The anomalous dynamics of some examples of the above-constructed partially hyperbolic diffeomorphisms are investigated. For \(\varphi\in \mathcal{M}(S)\) pseudo-Anosov, examples of {dynamically incoherent} partially hyperbolic diffeomorphism \(f: T^1S\to T^1S\) are given (Theorem 1.2), disproving a conjecture by \textit{F. Rodriguez Hertz} et al. [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 33, No. 4, 1023--1032 (2016; Zbl 1380.37067)]. The dynamics of \(\varphi\) and \(f\) at infinity are used in the proof of the dynamical incoherence. The incompleteness of a center-unstable leaf (Corollary 5.12) and the minimality of the strong foliations (Proposition 5.14) of (a perturbation of) \(f\) are also studied. In summary, the paper is a significant work toward the classification of partially hyperbolic diffeomorphisms on 3-manifolds. For Part II, see [the authors, Invent. Math. 206, No. 3, 801--836 (2016; Zbl 1370.37069)].

Let \({f: M\to M}\) be a partially hyperbolic diffeomorphism (i.e., there exist certain continuous \(Df\)-invariant splittings \({TM = E^s\oplus E^c\oplus E^u}\)). The diffeomorphism \(f\) is called dynamically coherent if there exist \(f\)-invariant foliations \({\mathcal{W}^{cs}}\) and \({\mathcal{W}^{cu}}\) tangent to \({E^s\oplus E^c}\) and \(E^c\oplus E^u,\) respectively. The main result of the article states that if \(M\) is a Seifert 3-manifold with hyperbolic base and the induced action of \(f\) in the base has a pseudo-Anosov component, then \(f\) is not dynamically coherent. This result provides a new proof of the non-dynamical coherence of the examples constructed in [\textit{C. Bonatti} et. al., Geom. Topol. 24, No. 4, 1751--1790 (2020; Zbl 1470.37049)] and extends it to any partially hyperbolic diffeomorphism in the same isotopy class. A sequence \(\{X_j, j\geq 1\}\) of r.v.'s is called pairwise positively quadrant dependent (PQD) if for any real \(r_i\), \(r_j\), with \(i\neq j\), \[ P[X_i> r_i, X_j>r_j]\geq P[X_i> r_i] P[X_j> r_j], \] and it is said to be linearly positively quadrant dependent (LPQD), if for any disjoint sets \(A, B\subset \{1,2,\dots\}\) and positive numbers \(a_i\), \(b_j\), \(\sum_{i\in A}a_iX_i\) and \(\sum_{j\in B} b_jX_j\) are PQD. The concepts of PQD and LPQD can be easily extended to 2-parameter arrays of r.v.'s. Let \(\{X_{(j_1,j_2)}\}\) be a stationary 2-parameter array of LPQD r.v.'s such that \(EX_{(j_1,j_2)}= 0\), \(EX^2_{(j_1,j_2)}< \infty\) and let \[ S_{(j_1,j_2)}= \sum^{j_1}_{i= 1} \sum^{j_2}_{k= 1} X_{(i,k)},\quad S^*_{(m,n)}= \max\{S_{(j_1, j_2)}: 1\leq j_1\leq m, 1\leq j_2\leq n\}. \] The following maximal inequality is proved: for \(0\leq \lambda_1< \lambda_2< \infty\), \[ P[S^*_{(m, n)}\geq \lambda_2]\leq (3^{3/2}/2) (\lambda_2- \lambda_1)^{3/2} (ES^2_{(m, n)})^{3/4} \{P[S_{(m, n)}\geq \lambda_1]\}^{1/4}, \] and similarly when \(S_{(i, j)}\) is replaced by \(|S_{(i, j)}|\) on both sides. Moreover, let \(W_n(t_1, t_2)= n^{-1} \sum^{[nt_1]}_{j_1= 1} \sum^{[nt_2]}_{j_2= 1} X_{(j_1, j_2)}\), and let \(W(t_1, t_2)\) be the 2-parameter Wiener process, i.e. a mean zero Gaussian process with \[ \text{Cov}(W(t_1, t_2), W(s_1, s_2))= \sigma^2 \prod^2_{i= 1}\min(t_i, s_i),\quad (t_1,t_2)\in [0,1]^2. \] By an application of the above maximal inequality it is shown that \(W_n(t_1, t_2)\) converges weakly to \(W(t_1, t_2)\) as \(n\to\infty\).

Let H denote the regular functions in \(U=\{z:\) \(| z| <1\}\), and let \(A_ 0=\{f\in H:\) \(f(0)=0\), \(f'(0)=1\}\), \(A_ 1=\{f\in H:\) \(f(0)=1\), \(f'(0)\neq 0\}\). For \(a\in [0,1)\) let \(s_ a(z)=z(1-z)^{2(1- a)}=z+...,\) \(z\in U\), and let \(k\in A_ 1\) be a convex function with Re k(z)\(>0\). The author introduces the function classes \[ P_ a(k)=\{f\in A_ 0:\quad (f*s_ a)'\prec k\},\quad Q_ a(k)=\{f\in A_ 0:\quad zf'\in P_ a(k)\},\quad R_ a(k)=\{f\in A_ 0:\quad (f*s_ a)/z\prec k\}, \] where * denotes convolution and \(\prec\) denotes subordination. Several relationships among the classes are derived. For example, it is shown that \(Q_ a(k)\subset P_ a(k)\subset R_ a(k)\), which improves results of \textit{O. P. Ahuja} and \textit{S. Owa} [Math. Jap. 33, No.4, 485- 499 (1988; Zbl 0648.30013)]. Let \(A_ 1\) be the class of all functions f holomorphic in the open unit disc \(\Delta\) of the form \(f(z)=z+\sum^{\infty}_{2}a_ nz\) n. Let \(s_{\alpha}(z)=z/((1-z)^{\alpha (1-\alpha)})\). \(P_{\alpha}(\beta,\gamma)\) is the class of all \(f\in A\), satisfying \[ (f*S_{\alpha})'(z) \prec \frac{1+(2\beta -1)\gamma z}{1+\gamma z} \] for some \(\alpha\in [0,1)\), \(\beta\in [0,1)\) and \(\gamma\in (0,1]\). The authors define two new classes of functions: \[ Q_{\alpha}(\beta,\gamma)=\{f\in A_ 1: zf'(z)\in P_{\alpha}(\beta,\gamma)], \] \[ R_{\alpha}(\beta,\gamma)= \{f\in A_ 1: z^{-1}(f*S_{\alpha})(z) \prec \frac{1+(2\beta -1)\gamma z}{1+\gamma z}\}. \] First they prove that \(Q_{\alpha}(\beta,\gamma)\subset P_{\alpha}(\beta,\gamma)\subset R_{\alpha}(\beta,\gamma)\). They mainly determine the coefficient inequalities for functions belonging to these classes.

Let H denote the regular functions in \(U=\{z:\) \(| z| <1\}\), and let \(A_ 0=\{f\in H:\) \(f(0)=0\), \(f'(0)=1\}\), \(A_ 1=\{f\in H:\) \(f(0)=1\), \(f'(0)\neq 0\}\). For \(a\in [0,1)\) let \(s_ a(z)=z(1-z)^{2(1- a)}=z+...,\) \(z\in U\), and let \(k\in A_ 1\) be a convex function with Re k(z)\(>0\). The author introduces the function classes \[ P_ a(k)=\{f\in A_ 0:\quad (f*s_ a)'\prec k\},\quad Q_ a(k)=\{f\in A_ 0:\quad zf'\in P_ a(k)\},\quad R_ a(k)=\{f\in A_ 0:\quad (f*s_ a)/z\prec k\}, \] where * denotes convolution and \(\prec\) denotes subordination. Several relationships among the classes are derived. For example, it is shown that \(Q_ a(k)\subset P_ a(k)\subset R_ a(k)\), which improves results of \textit{O. P. Ahuja} and \textit{S. Owa} [Math. Jap. 33, No.4, 485- 499 (1988; Zbl 0648.30013)]. A closed-form solution using the actual distribution of the fiber aspect ratio is proposed for predicting the stiffness of aligned short fiber composite. The present model is the simplified form of Takao and Taya's model and the extended version of Taya and Chou's model, where Eshelby's equivalent inclusion method modified for finite fiber volume fraction is employed. The validity of using average fiber aspect ratio for predicting the composite stiffness is justified in terms of the scatter of fiber aspect ratio, fiber volume fraction, and constituents` Young's modulus ratio, comparing with the results by the present model. The guideline for selection of either the actual distribution or the average fiber aspect ratio is presented for the better prediction of the composite stiffness.

There is a certain kinship between the fine structure of canonical randomizations of Cantor sets on the one hand, and the range of subordinators on the other hand. The author constructs a subordinator whose range has the same exact Hausdorff gauge and satisfies the same integral test for the packing gauge as the random re-orderings of the Cantor set introduced by \textit{J. Hawkes} [Q. J. Math., Oxf. II. Ser. 35, 165-172 (1984; Zbl 0552.60005)]. Let \(\psi =\{a_ i\}\) be a sequence of positive real numbers with \(a_ 1\geq a_ 2\geq..\).. A compact set \(K\subset [0,1]\) is said to belong to \(\psi\) if its complement is a union of open intervals \(J_ i\) of lengths \(a_ i\), \(i=1,2,..\). \textit{A. S. Besicovitch} and \textit{S. J. Taylor} [J. Lond. Math. Soc. 29, 449-459 (1954; Zbl 0056.278)] have shown that the Hausdorff dimension of K satisfies \(0\leq \dim K\leq \alpha (\psi)\leq \beta (\psi)\), where \(\alpha\) and \(\beta\) are certain indices defined in terms of \(\psi\). The author, having proved before that \(\beta(\psi)\) equals the upper entropy dimension of K, now shows that \(\alpha(\psi)\) is equal to the lower entropy dimension. A second result states that, if the choice of the end-points of the intervals \(J_ i\) is randomised in a certain fashion, the resulting random set \(K(\omega)\) satisfies \(\dim K(\omega)=\alpha(\psi)\) almost surely.

There is a certain kinship between the fine structure of canonical randomizations of Cantor sets on the one hand, and the range of subordinators on the other hand. The author constructs a subordinator whose range has the same exact Hausdorff gauge and satisfies the same integral test for the packing gauge as the random re-orderings of the Cantor set introduced by \textit{J. Hawkes} [Q. J. Math., Oxf. II. Ser. 35, 165-172 (1984; Zbl 0552.60005)]. The discrete dynamics for competing populations of Lotka-Volterra type modelled as \[ \begin{aligned} N_1 (t+1) &= N_1 (t) \exp [r_1 (1- N_1- b_{12} N_2)],\\ N_2 (t+1) &= N_2 (t) \exp [r_2(1- N_2- b_{21} N_1)], \end{aligned} \] is considered. In the case of non- persistence the attractive behavior of the model has been discussed. Especially, there are two attractive sets when \(b_{ij}>1\), and the attractive behaviors are more complicated than those of the corresponding continuous model. The attracted regions are given. We prove that the model is also persistent in the degenerate case of \(b_{ij}=1\). In the persistence case of \(b_{ij}<1\), the existence and uniqueness for two- period points of the model are studied at \(r_1= r_2\). The condition for multi-pairs of two-period points is indicated and their influence on population dynamical behavior is shown.

Let \({\mathfrak g}\) be a finite-dimensional real Lie algebra. The purpose of the paper under review is to construct a certain explicit star product \(\star_\alpha^\Gamma\) on the Lie-Poisson space~\({\mathfrak g}^*\) and to investigate some of its basic properties. The method of construction extends the one developed by M.~Kontsevich. One proves that the product \(\star_\alpha^\Gamma\) is equivalent to the Kontsevich star product and also to the star product of \textit{M. A.~Rieffel} [Am. J. Math. 112, No.~4, 657--686 (1990; Zbl 0713.58054)]. Some of the properties of \(\star_\alpha^\Gamma\) pointed out in the present paper are that it is an analytic, strict, relative, closed, real star product which is nonsymmetric. One also proves that \(\star_\alpha^\Gamma\) is a deformation quantization by partial embeddings of the algebra of test functions \(C_c^\infty({\mathfrak g})\). In the case when \({\mathfrak g}\) is a nilpotent Lie algebra, the product~\(\star_\alpha^\Gamma\) is a strict deformation quantization on the Schwartz space~\({\mathcal S}({\mathfrak g})\). Si L est une algèbre de Lie de dimension finie, réelle, et \(L^*\) son dual, ce dermier admet une structure naturelle d'algèbre de Poisson. On remplace le crochet [X,Y] dans L (qui devient \(L_ h)\) par \(\hslash [X,Y]\), \(G_ h\) est le groupe simplement connexe associé à ce dernier crochet. On transporte par la transformation de Fourier la structure de Poisson de \(L^*\) à une structure de Poisson sur l'algèbre de convolution de L. Le produit de convolution obtenu de \(G_ h\) donne une quantification par déformation. Le cas ou L est nilpotent est étudié, ainsi que l'invariance de ces déformations par l'action coadjointe de G. On retrouve dans l'articles les démarches bien connues des spécialistes de la quantification par déformation.

Let \({\mathfrak g}\) be a finite-dimensional real Lie algebra. The purpose of the paper under review is to construct a certain explicit star product \(\star_\alpha^\Gamma\) on the Lie-Poisson space~\({\mathfrak g}^*\) and to investigate some of its basic properties. The method of construction extends the one developed by M.~Kontsevich. One proves that the product \(\star_\alpha^\Gamma\) is equivalent to the Kontsevich star product and also to the star product of \textit{M. A.~Rieffel} [Am. J. Math. 112, No.~4, 657--686 (1990; Zbl 0713.58054)]. Some of the properties of \(\star_\alpha^\Gamma\) pointed out in the present paper are that it is an analytic, strict, relative, closed, real star product which is nonsymmetric. One also proves that \(\star_\alpha^\Gamma\) is a deformation quantization by partial embeddings of the algebra of test functions \(C_c^\infty({\mathfrak g})\). In the case when \({\mathfrak g}\) is a nilpotent Lie algebra, the product~\(\star_\alpha^\Gamma\) is a strict deformation quantization on the Schwartz space~\({\mathcal S}({\mathfrak g})\). In this paper we investigate how to efficiently apply Approximate-Karush-Kuhn-Tucker proximity measures as stopping criteria for optimization algorithms that do not generate approximations to Lagrange multipliers. We prove that the KKT error measurement tends to zero when approaching a solution and we develop a simple model to compute the KKT error measure requiring only the solution of a non-negative linear least squares problem. Our numerical experiments on a Genetic Algorithm show the efficiency of the strategy.

As said in the preface of Volume 1, the dividing line between the two volumes could be drawn between what can be done before and after involving the holomorphic theory of JB*-triples and the structure theory of non-commutative JB\(^*\)-algebras. Volume 1 answered the question on what can be said of a unital \(B^*\)-algebra when the associativity of the product is removed from the axioms, by proving the following two results: Theorem 3.2.5. Any normed unital complex algebra \(A\) endowed with a conjugate-linear vector space involution * such that \(1^*= 1\) and \(\|a^*a\|=\|a\|^2\) is alternative and * in an algebra involution on \(A\). Theorem 3.3.11. Any norm-unital complete normed algebra \(A\) satisfying \(A = H(A) + iH(A)\), where \(H(A)\) denotes the set of all elements \(h \in A\) such that \(f(h) \in \mathbb R\) for every continuous linear functional \(f\) of \(A\) such that \(f(h) = 1 = \|f\|\), is a noncommutative JB\(^*\)-algebra. Different unit-free versions of Theorem 3.2.5 are also proved in Volume 1 (see Theorem 3.5.68). The main goal of the first chapter (Chapter 5) of Volume 2 is to prove what can be seen as a unit-free version of Theorem 3.3.11, namely that noncommutative JB\(^*\)-algebras are precisely those complete normed complex algebras having an approximate identity bounded by one, and whose open unit ball is a homogeneous domain. Some ingredients in the long proof of this result were already established in Volume 1. Among the new ingredients proved in this chapter we can find (1) Kaup's holomorphic characterization of JB\(^*\)-triples as those complex Banach spaces whose open unit ball is a homogeneous domain (Theorem 5.6.68), and (2) the Barton-Horn-Timoney basic theory of JBW\(^*\)-triples establishing the separate \(w^*\)-continuity of the triple product of a JBW\(^*\)-triple (Theorem 5.7.20) and the uniqueness of the predual (Theorem 5.7.38). The proofs given by the authors of these (and others) results are not always the original ones, although sometimes (as in the case of result (1)) the latter underlie the former. On the other hand, the proof of result (2) is new and, contrarily to what happens in the original one, avoids any Banach space result on uniqueness of preduals. One of the deepest results in the theory of JB-algebras is the fact (proved in the celebrated book of Hanche-Olsen and Stormer as a consequence of the representation theory of JB-algebras) asserting that the closed subalgebra generated by two elements of a JB-algebra is a JC-algebra. This result allowed the authors to develop a basic theory on non-commutative JB\(^*\)-algebras (including Theorem 3.1.11) without any implicit or explicit additional reference to representation theory. In Chapter 6, the authors conclude the basic theory of non-commutative JB\(^*\)-algebras, developing in depth the representation theory of these algebras, and, in particular, that of alternative \(C^*\)-algebras. Roughly speaking, this theory reduces the study of non-commutative JB\(^*\)-algebras (respectively, alternative \(C^*\)-algebras) to the knowledge of the so-called non-commutative JBW\(^*\)-factors (respectively, alternative \(W^*\)-factors). In particular, the study of alternative \(C^*\)-algebras reduces to that of (associative) \(C^*\)-algebras and the alternative \(C^*\)-algebra of complex octonions. Applying representation theory, and following a Zelmanovian approach, prime JB\(^*\)-algebras are described in Theorem 6.1.57, and prime noncommutative JB\(^*\)-algebras in Theorem 6.2.27. Chapter 7 deals with the analytic treatment of Zelmanov's prime theorems for Jordan systems, thus continuing the approach initiated in Section 6.1 for prime JB\(^*\)-algebras. The classical representation theory of JB\(^*\)-triples is revisited, and is applied (together with Zelmanovian techniques) to obtain the description of all prime JB\(^*\)-triples (see Theorem 7.1.41). Other applications of Zelmanov's prime theorems to the study of normed Jordan algebras and triples are fully surveyed in Section 7.2. The concluding chapter of the book (Chapter 8) develops some parcels of the theory of nonassociative normed algebras, not previously included in the book, and begins by revisiting one of the favorite topics of the authors in this field, namely the theory of \(H^*\)-algebras (see Section 8.1). Section 8.2 is devoted to show how, as in the case of \(H^*\)-algebras with zero annihilator, the study of certain normed (possibly non-star) algebras can be reduced to the knowledge of their minimal closed ideals (see Theorems 8.2.17 and 8.2.44). Other attractive topics, like the automatic continuity Theorem 8.3.9, the description of Banach Jordan algebras all elements of which have finite spectra (Theorem 8.3.21), and the non-associative study of the joint spectral radius of a bounded set in a normed algebra and of the so-called topologically nilpotent normed algebras (Section 8.4), conclude the book. Finally, let me express a personal commentary: This book is the fruit of the work of a whole life, under the light and the magic of that city Washington Irving fell in love with. When I was still a student, I had the privilege of attending a lecture on \(C^*\)-algebras given in the University of Málaga by an enthusiastic young mathematician from the University of Granada. The name of this mathematician was Angel Rodríguez Palacios. Forty years later, I have the honour of reviewing his impressive book with Miguel Cabrera on non-associative normed algebras. I dare say that this book is the first detailed treatise whose primary goal is the study of normed non-associative algebras. The two authors are recognized experts in this topic, which combines functional analysis, geometry of Banach spaces and what could be called structure theory of algebras which are nearly associative. The ``leitmotiv'' of the book is to answer the following question. What can be said of a unital \(B^*\)-algebra when the associativity of the product is removed from the axioms? To be more precise we should recall the two important characterizations of \(C^*\)-algebras. In what follows, by an algebra we will mean a real or complex vector space \(A\) endowed with a bilinear operation, its product, which is not necessarily associative. Sometimes we will use the term non-associative algebra instead to stress that the associativity of the product is not assumed. While by a \(C^*\)-algebra we understand any closed \(^*\)-subalgebra of the algebra of bounded linear operators on a complex Hilbert space, by a \(B^*\)-algebra we mean the abstract Gelfand-Naimark characterization of such operator algebras, i.e., a \(B^*\)-algebra is a complex Banach algebra endowed with a conjugate-linear algebra involution \(^*\) such that \(\|a^* a\| = \|a\|^2\). In his celebrated paper [J. Lond. Math. Soc., II. Ser. 22, 318--332 (1980; Zbl 0483.46050)], the second author proved that every unital non-associative \(B^*\)-algebra is actually alternative, namely, the subalgebra generated by two elements is associative. In Theorem 3.2.5 of the book, the authors give a statement and a proof slightly different from the original of this Theorem. Let \(A\) be a normed unital complex algebra endowed with a conjugate-linear vector space involution \(^*\) such that \(1^* = 1\) and \(\|a^* a\| = \|a\|^2\) for every \(a\in A\). Then \(A\) is alternative and \(^*\) is an algebra involution on \(A\). It is worth noticing (Proposition 2.6.8) that the algebra of complex octonions, endowed with a suitable norm and a suitable involution, becomes an alternative \(B^*\)-algebra which is not associative. The other important characterization of unital \(C^*\)-algebras is the well-known Vidav-Palmer theorem. Let \(A\) be a unital associative \(B^*\)-algebra. For an element \(h\in A\), the following conditions are equivalent: (i) \(h^* = h\), (ii) \(\| \exp(i\alpha h)\| = 1\) for every \(\alpha \in \mathbb R\), and (iii) \(f(h) \in \mathbb R\) for every continuous linear functional \(f\) of \(A\) such that \(f(h) = 1 = \|f \|\). While condition (ii) requires (at least) that the algebra is power-associative, condition (iii) makes sense in any norm-unital non-associative algebra. In such an algebra \(A\), an element \(h \in A\) is said to be Hermitian if it satisfies condition (iii). Denote by \(H(A)\) the set of Hermitian elements of \(A\). By a \(V\)-algebra we mean a norm-unital normed complex algebra \(A\) satisfying \(A = H(A) + iH(A)\). In fact, \(A = H(A) \oplus iH(A)\), so there is a unique conjugate linear vector space involution \(^*\) on \(A\) such that \(H(A) = \{a \in A : a^* = a\}\). Using this terminology, the Vidav-Palmer theorem (Theorem 2.3.32) can be stated as follows: Let \(A\) be a complete associative \(V\)-algebra. Then \(A\), endowed with its natural involution and its own norm, becomes a \(C^*\)-algebra. An immediate consequence of this result is the following Corollary 2.3.63: Every complete alternative \(V\)-algebra is an alternative \(B^*\)-algebra. Let us return to a non-associative \(V\)-algebra \(A\), with its natural involution denoted by \(^*\). It is proved in Theorem 2.3.8 that \(^*\) is in fact an algebra involution. Since nonassociative \(B^*\)-algebras are alternative, we can wonder whether there exists a similar result for \(V\)-algebras. The answer is given in Theorem 2.4.11: Any \(V\)-algebra \(A\) is a non-commutative Jordan algebra, i.e., its product satisfies the flexible law \((xy)x = x(yx)\) for all \(x, y \in A\), and its symmetrized product, \(x \circ y := \frac12 (xy + yx)\), satisfies the Jordan identity \((x^2 \circ y) \circ x = x^2 \circ (y \circ x)\) for all \(x, y \in A\). It is not difficult to check that every alternative algebra is a non-commutative Jordan algebra. The positive answer to the question on algebraic properties of \(V\)-algebras encourages us to another: Do \(V\)-algebras satisfy a Gelfand-Naimark-like axiom? The answer to this second question is also yes and can be expressed in the following terms (Theorem 3.3.11): Let \(A\) be a complete \(V\)-algebra. Then \(A\), endowed with its natural involution and its own norm, becomes a non-commutative \(JB^*\)-algebra, i.e., \(\|U_a (a^* )\| = \|a\|^3\) for every \(a \in A\), where \(U_a = L_a (L_a +R_a )-L_{a^2}\) with \(L_a\) and \(R_a\) denoting the left and right multiplication by \(a\), respectively. It is not difficult to check that, if \(A\) is an alternative algebra, then \(U_a = L_a R_a = R_a L_a\). Thus for an alternative algebra the \(B^*\) and the \(JB^*\) notions are equivalent. This review (understandably) does not cover the whole subject of the book and doubtlessly does not do justice to the richness of its contents, but I hope that it can catch the attention of people interested in functional analysis and/or in non-associative algebras.

As said in the preface of Volume 1, the dividing line between the two volumes could be drawn between what can be done before and after involving the holomorphic theory of JB*-triples and the structure theory of non-commutative JB\(^*\)-algebras. Volume 1 answered the question on what can be said of a unital \(B^*\)-algebra when the associativity of the product is removed from the axioms, by proving the following two results: Theorem 3.2.5. Any normed unital complex algebra \(A\) endowed with a conjugate-linear vector space involution * such that \(1^*= 1\) and \(\|a^*a\|=\|a\|^2\) is alternative and * in an algebra involution on \(A\). Theorem 3.3.11. Any norm-unital complete normed algebra \(A\) satisfying \(A = H(A) + iH(A)\), where \(H(A)\) denotes the set of all elements \(h \in A\) such that \(f(h) \in \mathbb R\) for every continuous linear functional \(f\) of \(A\) such that \(f(h) = 1 = \|f\|\), is a noncommutative JB\(^*\)-algebra. Different unit-free versions of Theorem 3.2.5 are also proved in Volume 1 (see Theorem 3.5.68). The main goal of the first chapter (Chapter 5) of Volume 2 is to prove what can be seen as a unit-free version of Theorem 3.3.11, namely that noncommutative JB\(^*\)-algebras are precisely those complete normed complex algebras having an approximate identity bounded by one, and whose open unit ball is a homogeneous domain. Some ingredients in the long proof of this result were already established in Volume 1. Among the new ingredients proved in this chapter we can find (1) Kaup's holomorphic characterization of JB\(^*\)-triples as those complex Banach spaces whose open unit ball is a homogeneous domain (Theorem 5.6.68), and (2) the Barton-Horn-Timoney basic theory of JBW\(^*\)-triples establishing the separate \(w^*\)-continuity of the triple product of a JBW\(^*\)-triple (Theorem 5.7.20) and the uniqueness of the predual (Theorem 5.7.38). The proofs given by the authors of these (and others) results are not always the original ones, although sometimes (as in the case of result (1)) the latter underlie the former. On the other hand, the proof of result (2) is new and, contrarily to what happens in the original one, avoids any Banach space result on uniqueness of preduals. One of the deepest results in the theory of JB-algebras is the fact (proved in the celebrated book of Hanche-Olsen and Stormer as a consequence of the representation theory of JB-algebras) asserting that the closed subalgebra generated by two elements of a JB-algebra is a JC-algebra. This result allowed the authors to develop a basic theory on non-commutative JB\(^*\)-algebras (including Theorem 3.1.11) without any implicit or explicit additional reference to representation theory. In Chapter 6, the authors conclude the basic theory of non-commutative JB\(^*\)-algebras, developing in depth the representation theory of these algebras, and, in particular, that of alternative \(C^*\)-algebras. Roughly speaking, this theory reduces the study of non-commutative JB\(^*\)-algebras (respectively, alternative \(C^*\)-algebras) to the knowledge of the so-called non-commutative JBW\(^*\)-factors (respectively, alternative \(W^*\)-factors). In particular, the study of alternative \(C^*\)-algebras reduces to that of (associative) \(C^*\)-algebras and the alternative \(C^*\)-algebra of complex octonions. Applying representation theory, and following a Zelmanovian approach, prime JB\(^*\)-algebras are described in Theorem 6.1.57, and prime noncommutative JB\(^*\)-algebras in Theorem 6.2.27. Chapter 7 deals with the analytic treatment of Zelmanov's prime theorems for Jordan systems, thus continuing the approach initiated in Section 6.1 for prime JB\(^*\)-algebras. The classical representation theory of JB\(^*\)-triples is revisited, and is applied (together with Zelmanovian techniques) to obtain the description of all prime JB\(^*\)-triples (see Theorem 7.1.41). Other applications of Zelmanov's prime theorems to the study of normed Jordan algebras and triples are fully surveyed in Section 7.2. The concluding chapter of the book (Chapter 8) develops some parcels of the theory of nonassociative normed algebras, not previously included in the book, and begins by revisiting one of the favorite topics of the authors in this field, namely the theory of \(H^*\)-algebras (see Section 8.1). Section 8.2 is devoted to show how, as in the case of \(H^*\)-algebras with zero annihilator, the study of certain normed (possibly non-star) algebras can be reduced to the knowledge of their minimal closed ideals (see Theorems 8.2.17 and 8.2.44). Other attractive topics, like the automatic continuity Theorem 8.3.9, the description of Banach Jordan algebras all elements of which have finite spectra (Theorem 8.3.21), and the non-associative study of the joint spectral radius of a bounded set in a normed algebra and of the so-called topologically nilpotent normed algebras (Section 8.4), conclude the book. Finally, let me express a personal commentary: This book is the fruit of the work of a whole life, under the light and the magic of that city Washington Irving fell in love with. Translation from Probl. Mat. Anal. 8, 26-35 (Russian) (1981; Zbl 0484.35061).

This article is motivated by Roberts' well-known example of a derivation of the polynomial ring in 7 variables over a characteristic zero field \(k\) having a kernel which is not of finite type [\textit{P. Roberts}, J. Algebra 132, 461-473 (1990; Zbl 0716.13013)]. This example has the form \((x^3\partial_s + y^3\partial_t + z^3\partial_u + (xzy)^2\partial_v)\) on the ring \(R[s,t,u,v]\), where \(R=k[x,y,z]\). This is \textit{elementary} in the sense that \(Ds,Dt,Du,Dv\in R\). A natural question is whether an elementary derivation of \(R[s,t,u]\) can have a kernel not of finite type, where \(R\) is a UFD. The author shows that the answer is negative for a large class of rings \(R\), including the case \(R\) is a polynomial ring over \(k\). As a corollary, he completes the proof that any elementary derivation of \(k^{[6]}\) (the polynomial ring in 6 variables) is finitely generated; the other cases were considered by \textit{A. van den Essen} and \textit{T. Janssen} [``Kernels of elementary derivations'', Univ. Nijmegen Tech. Report 9548 (1995)]. Let F be a field of characteristic zero, let R denote the polynomial ring in \(n\quad variables\), and let K be a subfield of the field of quotients of R. Hilbert's 14-th problem [see \textit{D. Hilbert}, C. R. 2me Congr. Internat. Math., Paris (1900), 58-114 (1902)] asked whether or not the ring \(K\cap R\) is a finitely generated algebra over F. In Proc. Int. Congr. Math. 1958, 459-462 (1960; Zbl 0127.263), \textit{M. Nagata} gave a counterexample, which is a ring of invariants of a linear group acting on a polynomial ring. - In his previous paper [see Proc. Am. Math. Soc. 94, 589-592 (1985; Zbl 0589.13008)] the author used this example in order to construct a prime ideal P in a regular ring R such that the symbolic blow-up \(S(P)=\oplus_{n\geq 0}P^{(n)} \), where \(P^{(n)}\) denotes the n-th symbolic power of P, is not an R-algebra of finite type. P is formally not a prime ideal. In the present paper the author constructs a prime ideal P in a complete local ring R such that S(P) is not an R-algebra of finite type. Moreover, it yields another counterexample of Hilbert's 14-th problem. The example is motivated by \textit{M. Hochster}'s monomial conjecture [see Conf. Board Math. Sci., Reg. Conf. Series Math. No.24 (1975; Zbl 0302.13003)], and its equations are explicitly given. The proof of the infinite generation follows by an ingenious construction of generators of the algebra.

This article is motivated by Roberts' well-known example of a derivation of the polynomial ring in 7 variables over a characteristic zero field \(k\) having a kernel which is not of finite type [\textit{P. Roberts}, J. Algebra 132, 461-473 (1990; Zbl 0716.13013)]. This example has the form \((x^3\partial_s + y^3\partial_t + z^3\partial_u + (xzy)^2\partial_v)\) on the ring \(R[s,t,u,v]\), where \(R=k[x,y,z]\). This is \textit{elementary} in the sense that \(Ds,Dt,Du,Dv\in R\). A natural question is whether an elementary derivation of \(R[s,t,u]\) can have a kernel not of finite type, where \(R\) is a UFD. The author shows that the answer is negative for a large class of rings \(R\), including the case \(R\) is a polynomial ring over \(k\). As a corollary, he completes the proof that any elementary derivation of \(k^{[6]}\) (the polynomial ring in 6 variables) is finitely generated; the other cases were considered by \textit{A. van den Essen} and \textit{T. Janssen} [``Kernels of elementary derivations'', Univ. Nijmegen Tech. Report 9548 (1995)]. The usual way to compute a low-rank approximant of a matrix \(H\) is to take its singular value decomposition (SVD) and truncate it by setting the small singular values equal to 0. However, the SVD is computationally expensive. This paper describes a much simpler generalized Schur-type algorithm to compute similar low-rank approximants. For a given matrix \(H\) which has \(d\) singular values larger than \(\varepsilon\), we find all rank \(d\) approximants \(\widehat {H}\) such that \(H- \widehat {H}\) has 2-norm less than \(\varepsilon\). The set of approximants includes the truncated SVD approximation. The advantages of the Schur algorithm are that it has a much lower computational complexity (similar to a QR factorization), and directly produces a description of the column space of the approximants. This column space can be updated and downdated in an on-line scheme, amenable to implementation on a parallel array of processors.

A Stieltjes string is a massless elastic thread bearing point masses. Small transversal vibrations of such a string are described by functions of time expressing the transversal displacement of the masses. These functions satisfy a linear system of differential equations. Assuming the harmonic character of oscillation, we come to a spectral problem. The paper deals with the case where the string forms a figure-of-eight graph consisting of two loops joined at the vertex. The continuity and the force balance conditions are imposed at the vertex. It is shown that the eigenvalues of the above problem are interlaced with elements of the spectra of the Dirichlet problems on separate parts of the graph, as well as the eigenvalues of the periodic problems. Note that spectral problems for Sturm-Liouville equations on the above graph were studied by \textit{A. M. Gomilko} and \textit{V. N. Pivovarchik} [Ukr. Mat. Zh. 60, No.~9, 1168--1188 (2008); translation in Ukr. Math. J. 60, No.~9, 1360--1385 (2008; Zbl 1240.34040)]. We study the inverse problem for the Sturm-Liouville equation on a graph that consists of two quasi-one-dimensional loops of the same length having a common vertex. As spectral data, we consider the set of eigenvalues of the entire system together with the sets of eigenvalues of two Dirichlet problems for the Sturm-Liouville equations with the condition of total reflection at the vertex of the graph. We obtain conditions for three sequences of real numbers that enable one to reconstruct a pair of real potentials from \(L_2\) corresponding to each loop. We give an algorithm for the construction of the entire set of potentials corresponding to this triple of spectra.

A Stieltjes string is a massless elastic thread bearing point masses. Small transversal vibrations of such a string are described by functions of time expressing the transversal displacement of the masses. These functions satisfy a linear system of differential equations. Assuming the harmonic character of oscillation, we come to a spectral problem. The paper deals with the case where the string forms a figure-of-eight graph consisting of two loops joined at the vertex. The continuity and the force balance conditions are imposed at the vertex. It is shown that the eigenvalues of the above problem are interlaced with elements of the spectra of the Dirichlet problems on separate parts of the graph, as well as the eigenvalues of the periodic problems. Note that spectral problems for Sturm-Liouville equations on the above graph were studied by \textit{A. M. Gomilko} and \textit{V. N. Pivovarchik} [Ukr. Mat. Zh. 60, No.~9, 1168--1188 (2008); translation in Ukr. Math. J. 60, No.~9, 1360--1385 (2008; Zbl 1240.34040)]. By a previous result of the author, any weakly compact operator on \(C=C[0,1)\) satisfies Daugavet's equation \(\| I-T\| =1+\| T\|.\) Consider the ideal \(J_ 0=\{T: C\to C|\) T factors through \(c_ 0\}\). It is shown that any \(T\in J_ 0\) satisfies Daugavet's equation as well. \(J_ 0\) is not contained in the weakly compact operators nor does the converse hold. Moreover, if J is any ideal in L(C) containing a non weakly compact operator, then \(J_ 0\subset J\).

Let \(k\) be an algebraically closed field, and \(\mathcal C\) a \(k\)-linear abelian category. By definition, \(\mathcal C\) is hereditary if it has homological dimension at most \(1\) (i.e., \(\text{Ext}^i(A,B)=0\) for \(A,B\in \mathcal C\) and \(i>1\)). A Serre functor for \(\mathcal C\) is an autoequivalence \(F:D^b({\mathcal C})\to D^b({\mathcal C})\) of derived categories, such that there are natural isomorphisms \(\text{Hom}(A,B)\simeq\text{Hom}(B, F(A))^*\) for \(A,B\in\mathcal C\), and \(\mathcal C\) is said to satisfy Serre duality if it has a Serre functor. The authors prove that \(\mathcal C\) has a Serre functor if and only if \(D^b({\mathcal C})\) has Auslander-Reiten triangles; moreover, if \(\mathcal C\) is hereditary, then \(\mathcal C\) has a Serre functor if and only if \(\mathcal C\) has almost split sequences and there is a one-one correspondence between the indecomposable projective objects \(P\) and the indecomposable injective objects \(I\), such that \(P/\text{rad} P\simeq\text{soc} I\). The main result of the paper is the classification of connected noetherian Ext-finite hereditary abelian categories satisfying Serre duality. A list of four categories is provided, and it is shown that any category satisfying the above conditions is one in the list. Such categories appear to be of interest both for non-commutative algebraic geometry and for the representation theory of finite-dimensional algebras. In the final part of the paper, applications of the classification theorem are given to saturated categories [introduced by \textit{A. I. Bondal} and \textit{M. M. Kapranov}, Izv. Akad. Nauk. SSSR, Ser. Mat. 53, No. 6, 1183-1205 (1989; Zbl 0703.14011)], and to graded rings. Let A be a triangulated category endowed with a filtration \(W_ 0A\subset W_ 1A\subset\dots\subset A\), where \(W_ iA\) is a thick subcategory of A such that the inclusion \(W_ iA\subset A\) admits an adjoint functor. For example, let A be the derived category \(D^ b_{coh}(P^ n)\) of all coherent sheaves on the projective space \(P^ n\). By a result of Beilinson, A is generated by the sheaves \({\mathcal O}(j),\quad j=0,1,\dots,n.\) Let \(W_ iA\) be the subcategory of A generated by \({\mathcal O}(j),\quad j=0,1,\dots,i.\) The aim of this paper is to study the data \((A,W_ 0A\subset W_ 1A\subset\dots\subset A).\) The philosophy is that many properties of the quotient categories \(W_ iA/W_{i-1}A\) are inherited by the category A itself. This is useful if one wants to study the representability properties of the cohomological functors \(A\to Vect\). The latter properties are also related to some duality questions. Another application of the above philosophy concerns the construction of a t-structure on A. This amounts to give on each quotient category \(W_ iA/W_{i-1}A\) a t-structure (via the theory of perverse sheaves).

Let \(k\) be an algebraically closed field, and \(\mathcal C\) a \(k\)-linear abelian category. By definition, \(\mathcal C\) is hereditary if it has homological dimension at most \(1\) (i.e., \(\text{Ext}^i(A,B)=0\) for \(A,B\in \mathcal C\) and \(i>1\)). A Serre functor for \(\mathcal C\) is an autoequivalence \(F:D^b({\mathcal C})\to D^b({\mathcal C})\) of derived categories, such that there are natural isomorphisms \(\text{Hom}(A,B)\simeq\text{Hom}(B, F(A))^*\) for \(A,B\in\mathcal C\), and \(\mathcal C\) is said to satisfy Serre duality if it has a Serre functor. The authors prove that \(\mathcal C\) has a Serre functor if and only if \(D^b({\mathcal C})\) has Auslander-Reiten triangles; moreover, if \(\mathcal C\) is hereditary, then \(\mathcal C\) has a Serre functor if and only if \(\mathcal C\) has almost split sequences and there is a one-one correspondence between the indecomposable projective objects \(P\) and the indecomposable injective objects \(I\), such that \(P/\text{rad} P\simeq\text{soc} I\). The main result of the paper is the classification of connected noetherian Ext-finite hereditary abelian categories satisfying Serre duality. A list of four categories is provided, and it is shown that any category satisfying the above conditions is one in the list. Such categories appear to be of interest both for non-commutative algebraic geometry and for the representation theory of finite-dimensional algebras. In the final part of the paper, applications of the classification theorem are given to saturated categories [introduced by \textit{A. I. Bondal} and \textit{M. M. Kapranov}, Izv. Akad. Nauk. SSSR, Ser. Mat. 53, No. 6, 1183-1205 (1989; Zbl 0703.14011)], and to graded rings. \(f(x)\) sei in \(\langle a,b\rangle \) stetig, nicht identisch Null, und es sei \(0\leqq f(x) \leqq M\). Dann gilt \[ 0<\Bigg (\int _a^b f(x)\,dx\,\Bigg )^2-\Bigg (\int _a^b f(x)\cos x\,dx\,\Bigg )^2\Bigg (\int _a^b f(x)\sin x\,dx\,\Bigg )^2\leqq M^2\frac {(b-a)^4}{12}. \]

Bisequential spaces were introduced by \textit{A.\,V.\,Arhangel'skii} [Trans. Mosc. Math. Soc. 55, 207--219 (1994; Zbl 0842.54004)]. Some typical results from the reviewed paper: (2.1) Weakly bisequential spaces coincide with weakly bi-quotient images of metrizable spaces. (2.4) There are two compact weakly bisequential spaces the product of which is not Fréchet-Urysohn. (2.6) A Fréchet-Urysohn weakly quasi-first countable space is weakly bisequential. (2.7) A space is weakly bisequential if it is weakly quasi-first countable and \(\alpha _4\). The well-known classical theorem of Čech which states that first countable spaces \(X\) and \(Y\) are homeomorphic iff their Stone-Čech compactifications \(\beta X\) and \(\beta Y\) are homeomorphic is extended to the class of bisequential spaces and to the class of scattered Fréchet-Urysohn spaces. A new class of weakly bisequential spaces, wider than the class of bisequential spaces, is introduced and studied. In particular, weakly bisequential Baire spaces are considered. It is shown that in such spaces the density coincides with the \(\pi\)-weight and their cardinality is \(\leq 2^{c(X)}\). The author shows that a topological group \(G\) is metrizable in every of the following cases: (i) \(G\) is bisequential; (ii) \(G\) is weakly bisequential with the Baire property; (iii) \(G\) has a compactification of countable tightness. Some criteria for metrizability of subspaces of topological groups in terms of frequency spectrum are obtained.

Bisequential spaces were introduced by \textit{A.\,V.\,Arhangel'skii} [Trans. Mosc. Math. Soc. 55, 207--219 (1994; Zbl 0842.54004)]. Some typical results from the reviewed paper: (2.1) Weakly bisequential spaces coincide with weakly bi-quotient images of metrizable spaces. (2.4) There are two compact weakly bisequential spaces the product of which is not Fréchet-Urysohn. (2.6) A Fréchet-Urysohn weakly quasi-first countable space is weakly bisequential. (2.7) A space is weakly bisequential if it is weakly quasi-first countable and \(\alpha _4\). This paper deals with the fourth order nonlinear neutral differential equation with quasi-derivatives of the form \[ L_4z(t)+q(t)G(y(t-\alpha))-h(t)H(y(t-\beta))=0\tag{\(\ast\)} \] and its corresponding non-homogeneous equation \[ L_4z(t)+q(t)G(y(t-\alpha))-h(t)H(y(t-\beta))=f(t)\tag{\(\ast\ast\)}, \] where \(z(t):=y(t)+p(t)y(t-\tau),\) \(L_0z(t):=z(t),\) \(L_1z(t):=r_1(t)\frac{d}{dt}L_0z(t),\) \(L_2z(t):=r_2(t)\frac{d}{dt}L_1z(t),\) \(L_3z(t):=r_3(t)\frac{d}{dt}L_2z(t)\), \(L_4z(t):=\frac{d}{dt}L_3z(t)\). Also, \(f, p\in C([0,+\infty),\mathbb{R})\), \(q, r_1, r_2, r_3\in C([0,+\infty),(0,+\infty)),\) \(h\in C([0,+\infty),[0,+\infty))\) and \(G, H\in C(\mathbb{R},\mathbb{R}).\) The authors provide sufficient conditions, among which \(\int_0^{\infty}r_j^{-1}(t)dt<\infty,\) for all \(j=1, 2, 3\), to guarantee that every unbounded solution \(y\) of (\(\ast\)) either oscillates, or \(\liminf_{t\to+\infty}y(t)=0,\) or \(\limsup_{t\to+\infty}y(t)=0,\) or \(\lim_{t\to+\infty}y(t)=\pm\infty.\) Conditions for the non-homogeneous equation \((**)\) are also provided for the solutions to have the same behavior as above. Some examples are given to illustrate the results.

In this article, we consider a class of state-dependent delay differential equations which models the dynamics of the number of adult trees in forests. We prove the boundedness and the dissipativity of the solutions for a single species model and an \(n\)-species model. For Part I, see [the authors, J. Evol. Equ. 18, No. 4, 1853--1888 (2018; Zbl 1416.35278)]. In this article, we investigate the semiflow properties of a class of state-dependent delay differential equations which is motivated by some models describing the dynamics of the number of adult trees in forests. We investigate the existence and uniqueness of a semiflow in the space of Lipschitz and \(C^1\) weighted functions. We obtain a blow-up result when the time approaches the maximal time of existence. We conclude the paper with an application of a spatially structured forest model.

In this article, we consider a class of state-dependent delay differential equations which models the dynamics of the number of adult trees in forests. We prove the boundedness and the dissipativity of the solutions for a single species model and an \(n\)-species model. For Part I, see [the authors, J. Evol. Equ. 18, No. 4, 1853--1888 (2018; Zbl 1416.35278)]. The AdS/CFT correspondence is central to efforts to reconcile gravity and quantum mechanics, a fundamental goal of physics. It posits a duality between a gravitational theory in Anti de Sitter (AdS) space and a quantum mechanical conformal field theory (CFT), embodied in a map known as the AdS/CFT dictionary mapping states to states and operators to operators. This dictionary map is not well understood and has only been computed on special, structured instances. In this work we introduce cryptographic ideas to the study of AdS/CFT, and provide evidence that either the dictionary must be exponentially hard to compute, or else the quantum Extended Church-Turing thesis must be false in quantum gravity.\par Our argument has its origins in a fundamental paradox in the AdS/CFT correspondence known as the wormhole growth paradox. The paradox is that the CFT is believed to be ``scrambling'' -- i.e. the expectation value of local operators equilibrates in polynomial time -- whereas the gravity theory is not, because the interiors of certain black holes known as ``wormholes'' do not equilibrate and instead their volume grows at a linear rate for at least an exponential amount of time. So what could be the CFT dual to wormhole volume? Susskind's proposed resolution was to equate the wormhole volume with the quantum circuit complexity of the CFT state. From a computer science perspective, circuit complexity seems like an unusual choice because it should be difficult to compute, in contrast to physical quantities such as wormhole volume.\par We show how to create pseudorandom quantum states in the CFT, thereby arguing that their quantum circuit complexity is not ``feelable'', in the sense that it cannot be approximated by any efficient experiment. This requires a specialized construction inspired by symmetric block ciphers such as DES and AES, since unfortunately existing constructions based on quantum-resistant one way functions cannot be used in the context of the wormhole growth paradox as only very restricted operations are allowed in the CFT. By contrast we argue that the wormhole volume is ``feelable'' in some general but non-physical sense. The duality between a ``feelable'' quantity and an ``unfeelable'' quantity implies that some aspect of this duality must have exponential complexity. More precisely, it implies that either the dictionary is exponentially complex, or else the quantum gravity theory is exponentially difficult to simulate on a quantum computer.\par While at first sight this might seem to justify the discomfort of complexity theorists with equating computational complexity with a physical quantity, a further examination of our arguments shows that any resolution of the wormhole growth paradox must equate wormhole volume to an ``unfeelable'' quantity, leading to the same conclusions. In other words this discomfort is an inevitable consequence of the paradox.

For each integer \(n \geq 2\), let \(\Omega^0_n\) denote the set of \(n \times n\) doubly stochastic matrices with all main diagonal entries equal to 0, and let \(J^0_n\) denote the matrix in \(\Omega^0_n\) with all off diagonal entries equal to \(1/(n-1)\). \textit{H. Minc} [Linear Multilinear Algebra 21, 109-148 (1987; Zbl 0621.15006)] conjectured that \(\text{per} A > \text{per} J^0_n\) when \(A \in \Omega^0_n\) with \(A \neq J^0_n\). In support of this conjecture, the author shows that the permanent function attains a strict local minimum on \(\Omega^0_n\) at \(J^0_n\). This paper is a continuation of the author's monograph [Permanents (1978; Zbl 0401.15005)] and his survey article [Linear Multilinear Algebra 12, 227-263 (1983; Zbl 0511.15002)]. In particular, it features S. Friedland's elegant proof of the Tverberg conjecture and the author's recent investigations on permanents of (0,1)-circulants. Moreover, the following topics are treated: permanents of doubly stochastic matrices and (1,-1)-matrices, the permanental dominance conjecture, various identities and inequalities involving permanents, and applications of the permanental polynomial to graph theory. Lists of the current status of old and new conjectures and problems are followed by a comprehensive annotated bibliography that contains (almost) every paper and book with material on permanents, published between 1982 and 1985 or awaiting publication at the time. This survey article shows that, in spite of the solution of the celebrated van der Waerden conjecture in 1981, the theory of permanents is still a lively and appealing field of research.

For each integer \(n \geq 2\), let \(\Omega^0_n\) denote the set of \(n \times n\) doubly stochastic matrices with all main diagonal entries equal to 0, and let \(J^0_n\) denote the matrix in \(\Omega^0_n\) with all off diagonal entries equal to \(1/(n-1)\). \textit{H. Minc} [Linear Multilinear Algebra 21, 109-148 (1987; Zbl 0621.15006)] conjectured that \(\text{per} A > \text{per} J^0_n\) when \(A \in \Omega^0_n\) with \(A \neq J^0_n\). In support of this conjecture, the author shows that the permanent function attains a strict local minimum on \(\Omega^0_n\) at \(J^0_n\). The authors obtained in a former paper several characterizations of \(\beta\)-continuity and showed that the notions of almost quasi-continuity and \(\beta\)-continuity are equivalent. Moreover, the same authors introduced the notion of weakly \(\beta\)-continuous functions and upper (lower) weakly \(\beta\)-continuous multifunctions. The purpose of the present paper is to continue the study of these notions. So the authors obtain many interesting characterizations and several properties concerning upper (lower) weakly \(\beta\)-continuous multifunctions. In conclusion, they examine under which conditions weak \(\beta\)-continuity, almost \(\beta\)-continuity, \(\beta\)-continuity and other forms of weak continuity coincide.

Let \(M\) be a finite-dimensional path-connected Riemannian manifold. The infinite-dimensional space of all smooth maps from the circle group \(S^1\) to \(M\) is the Fréchet manifold \(\Lambda M\) called the free loop space. The points in \(\Lambda M\) can be described as free loops in \(M\). \textit{M. F. Atiyah} [Colloq. Honneur L. Schwartz, Ec. Polytech. 1983, Vol. 1, Astérisque 131, 43-59 (1985; Zbl 0578.58039)] showed that there is a fundamental closed 2-form \(\omega\) on \(\Lambda M\) which, unlike the finite-dimensional case, can be degenerate at some points. In this paper, the author proves that the set of all points where \(\omega\) is nondegenerate is open. In modern physical theories such as string theory, the idea of representing particles not as points but rather as loops on some manifold \(M\) has resulted in the formulation of spinors on the loop space \(\Lambda M\) of \(M\). A string structure is defined as a lifting of the structural group to an \(S^1\)-central extension of the loop group. If \(\widetilde G\to \widetilde P\to X\) is a lifting of a principal Fréchet bundle \(G\to P\to X\) over a Fréchet manifold \(X\) and if \(S^1\to \widetilde G\to G\) is an \(S^1\)-central extension of \(G\), then the author shows that every connection on the principal bundle \(G\to P\to X\) together with a \(\widetilde G\)-invariant connection on \(S^1\to \widetilde P\to P\) defines a connection on \(\widetilde G\to \widetilde P\to X\), and, as a consequence of this result, that there exist connections on the string structure of \(\Lambda M\). Finally, the author defines Christoffel symbols on \(\Lambda M\) which are described in terms of the Christoffel symbols on \(M\) and compute the corresponding curvature. [For the entire collection see Zbl 0566.00010.] The Duistermaat-Heckman exact integration formula \[ \int_{M}e^{-tH} \omega^ n/n!=\int_{X}(e^{-tH(X)} e^{\omega}/\prod^{k}_{j=\quad 1}(tm_ j-i\alpha_ j)) \] for computation of oscillatory integrals is explained in the first part of the lecture. Then, following \textit{E. Witten} [J. Differ. Geom. 17, 661- 692 (1982; Zbl 0499.53056)], an application of this formula to the infinite-dimensional loop manifold \(M=Map(S',X)\) is heuristically discussed. It turns out that the Duistermaat-Heckmann formula for the infinite-dimensional loop space is reduced to the index theorem for the Dirac operator.

Let \(M\) be a finite-dimensional path-connected Riemannian manifold. The infinite-dimensional space of all smooth maps from the circle group \(S^1\) to \(M\) is the Fréchet manifold \(\Lambda M\) called the free loop space. The points in \(\Lambda M\) can be described as free loops in \(M\). \textit{M. F. Atiyah} [Colloq. Honneur L. Schwartz, Ec. Polytech. 1983, Vol. 1, Astérisque 131, 43-59 (1985; Zbl 0578.58039)] showed that there is a fundamental closed 2-form \(\omega\) on \(\Lambda M\) which, unlike the finite-dimensional case, can be degenerate at some points. In this paper, the author proves that the set of all points where \(\omega\) is nondegenerate is open. In modern physical theories such as string theory, the idea of representing particles not as points but rather as loops on some manifold \(M\) has resulted in the formulation of spinors on the loop space \(\Lambda M\) of \(M\). A string structure is defined as a lifting of the structural group to an \(S^1\)-central extension of the loop group. If \(\widetilde G\to \widetilde P\to X\) is a lifting of a principal Fréchet bundle \(G\to P\to X\) over a Fréchet manifold \(X\) and if \(S^1\to \widetilde G\to G\) is an \(S^1\)-central extension of \(G\), then the author shows that every connection on the principal bundle \(G\to P\to X\) together with a \(\widetilde G\)-invariant connection on \(S^1\to \widetilde P\to P\) defines a connection on \(\widetilde G\to \widetilde P\to X\), and, as a consequence of this result, that there exist connections on the string structure of \(\Lambda M\). Finally, the author defines Christoffel symbols on \(\Lambda M\) which are described in terms of the Christoffel symbols on \(M\) and compute the corresponding curvature. Representations of the quantum algebra \({\mathcal U}_q (\widehat{A}_N)\) are constructed on the space of \((N+1)\)-component anyons in \(\mathbb{R}\), extending analogous results on the lattice. Such representations can be obtained in terms of both fermionic or bosonic anyons, showing that the hard-core constraint is not necessary in the continuous case.

Let \(E\) be an elliptic curve defined over \(\mathbb Q\). Under the assumption that the Riemann hypothesis holds for \(L\)-function of elliptic curves, the author proves that the average analytic rank of elliptic curves over \(\mathbb Q\) is at most \(2\). It is also shown that the average analytic rank of elliptic curves in any family of quadratic twists is at most \(3/2\). More exactly, let \(E = E_{r,s}: y^2 = x^3 + rx + s\) be an elliptic curve (i.e. assume that \(\Delta_E = -16(4r^3 + 27s^2) \neq 0\)) and introduce the weight function \(w_T(E) = w_1(T^{-1/3} r) w_2(T^{-1/2} s)\), where \(w_1\) and \(w_2\) are infinitely differentiable nonnegative functions with compact support vanishing at \(0\). Write \(C = \{E_{r,s}: p^4 \mid r \Longrightarrow p^6 \nmid s\}\) and \(S(T) = \sum_{E \in C} w_T(E)\). Then \[ \frac 1{S(T)} \sum_{E \in C} w_T(E) r(E) \leq 2 + o(1) \] as \(T \longrightarrow \infty\), where \(r(E)\) denotes the analytic rank of \(E\). The proof uses an ``explicit formula'' due to \textit{A. Brumer} [Invent. Math. 109, 445--472 (1992; Zbl 0783.14019)] and the modularity of elliptic curves defined over \(\mathbb Q\). There has been much interest in the question of what to ``expect'' the rank of an elliptic curve over \(\mathbb{Q}\) to be. The traditional philosophy is that most elliptic curves would have as small of a rank as allowed by the sign of their functional equations, namely either 0 or 1. More recent work has uncovered families of elliptic curves with somewhat higher rank than this philosophy predicts. -- This paper studies the average analytic rank of elliptic curves over \(\mathbb{Q}\) (as given by their Hasse-Weil \(L\)- functions), modulo several standard conjectures. The elliptic curves are ordered, essentially by their Faltings' height. The paper shows that the average rank is then at most 2.3. The techniques of this paper are analytic making use of the explicit formulae. It assumes the Taniyama-Weil conjecture and a form of generalized Riemann hypothesis. For the results to apply to the ranks of the Mordell-Weil groups of the elliptic curves, one needs to assume the weak Birch and Swinnerton-Dyer conjecture as well. While these conjectures are substantial, they are generally expected to be true.

Let \(E\) be an elliptic curve defined over \(\mathbb Q\). Under the assumption that the Riemann hypothesis holds for \(L\)-function of elliptic curves, the author proves that the average analytic rank of elliptic curves over \(\mathbb Q\) is at most \(2\). It is also shown that the average analytic rank of elliptic curves in any family of quadratic twists is at most \(3/2\). More exactly, let \(E = E_{r,s}: y^2 = x^3 + rx + s\) be an elliptic curve (i.e. assume that \(\Delta_E = -16(4r^3 + 27s^2) \neq 0\)) and introduce the weight function \(w_T(E) = w_1(T^{-1/3} r) w_2(T^{-1/2} s)\), where \(w_1\) and \(w_2\) are infinitely differentiable nonnegative functions with compact support vanishing at \(0\). Write \(C = \{E_{r,s}: p^4 \mid r \Longrightarrow p^6 \nmid s\}\) and \(S(T) = \sum_{E \in C} w_T(E)\). Then \[ \frac 1{S(T)} \sum_{E \in C} w_T(E) r(E) \leq 2 + o(1) \] as \(T \longrightarrow \infty\), where \(r(E)\) denotes the analytic rank of \(E\). The proof uses an ``explicit formula'' due to \textit{A. Brumer} [Invent. Math. 109, 445--472 (1992; Zbl 0783.14019)] and the modularity of elliptic curves defined over \(\mathbb Q\). We study the bipartite von Neumann entropy of two particles interacting via a long-range scale-free potential \(V(r) \sim - g/r^{2}\) in three dimensions, close to the unbinding transition. The nature of the quantum phase transition changes from critical (\(-3/4 < g < 1/4\)) to first order \((g < - 3/4)\) with \(g = - 3/4\) as a multicritical point. Here, we show that the entanglement entropy has different behaviours for the critical and the first-order regimes. But there still exists an interesting multicritical scaling behaviour for all \(g \in ( - 2 < g < 1/4)\) controlled by the \(g = -3/4\) case.

Der Nachdruck des ersten ``modernen'' Werkes der mathematischen Kristallographie [(1891, JFM 23.0554.02)] ist aufs wärmste zu begrüßen. Das Werk zerfällt in zwei Teile: Das Ziel des ersten Abschnittes liegt in der Herleitung der 32 Kristallklassen. Um diese zu erhalten, wird die Symmetrie erläutert und das Operieren mit den Symmetrieelementen dargestellt. Der Gruppenbegriff gestattet die geeignete Formulierung der Ergebnisse. Diese Klassen beschreiben die makroskopische Gestalt der Kristalle, wobei der Autor zudem die daraus folgenden physikalischen Eigenschaften bespricht. Der zweite Abschnitt handelt von der inneren Struktur der Kristalle, für welche damals nur Hypothesen vorlagen. Die am weitesten führende war diejenige der Gitterstruktur. Mittels dieser gelangt der Autor zur Herleitung der 230 Raum- oder Bewegungsgruppen. Beide Teile sind auch heute noch lesenswert. Wir bemerken, daß der Autor übersehen hatte, daß nicht Hessel (1830/31), sondern Frankenheim die 32 Klassen erstmals herleitete. Wohl zitiert der Autor zwei Arbeiten dieses Forschers und weist darauf hin, daß dieser erstmals die sog. Bravais-Gitter aufstellte. Leider übersah er aber den 1826 in der Zeitschrift ISIS von Oken erschienenen Artikel ''Crystallonomische Aufsätze'', worin erstmals diese Klassen hergeleitet werden. Der Autor war ein Schüler von Felix Klein und von diesem angeregt, verfaßte er sein Werk im Sinne von dessen gruppentheoretischem Denken und in der noch heute gebrauchten Sprache. Zu Recht wird daher sein Werk als einer der beiden Klassiker der mathematischen Kristallographie bezeichnet. Der andere Klassiker wäre zweifelsohne sein Zeitgenosse E. S. von Fedorov, falls dieser seine Entdeckungen in lesbarer Form veröffentlicht hätte. Allein im Ural arbeitend, fehlte ihm die Schulung im geläufigen mathematischen Ausdruck. Dies hat wohl auch Chebychev veranlaßt, die Veröffentlichung der Arbeiten abzulehnen. Glücklicherweise wissen wir aus dem Briefwechsel von Schoenflies und Fedorov, daß nur eine enge Zusammenarbeit der beiden Forscher zur endgültigen und bereinigten Anzahl der 230 Gruppen führte. Allerdings scheint der Autor nicht bemerkt zu haben, daß der Begriff der symmorphen \((=\) arithmetischen) Klasse von Fedorov tiefer in die Struktur der Raumgruppen eingreift als derjenige der 32 Klassen, obschon er diese 73 Systeme auf Seite 598 erwähnt. Der Nachdruck ist von hervorragender Qualität und es ist zu hoffen, daß er heute, wo das Gebiet neue Aktualität genießt, viele Leser finden wird. Sie werden reich belohnt, insbesondere auch durch die Nennung von etwa 40 früheren Autoren und deren Werken, von Hauy, Delafosse, Frankenheim, und Hessel bis zu Fedorov und Barlow, was bezeugt, daß der Autor ein ausgezeichneter Kenner der Geschichte der Kristallographie war. Die krystallisirte Materie unterscheidet sich bekanntlich dadurch von den übrigen festen Körpern, dass ihr physikalisches Verhalten längs verschiedener Richtungen im allgemeinen verschieden ist. Nennt man alle Richtungen, in denen sich ein Krystall in jeder Beziehung gleichartig verhält, gleichwertige Richtungen, und denkt man sich von irgend einem Punkte \(O\) des Krystalles aus eine Gerade \(g\) und alle mit ihr gleichwertigen Geraden \(g_1,g_2,\dots\) gezogen, so ist ide Lage dieser \(N\) Geraden, wie die Erfahrung lehrt, durch bestimmte Symmetrieeigenschaften, wie Symmetrieaxen, Symmetrieebenen u. s. w., ausgezeichnet. Die Symmetrieeigenschaften der \(N\) Geraden \(g,g_1,g_2,\dots\) sind davon unabhängig, wie die Ausgangsrichtung \(g\) innerhalb der Krystallmasse angenommen wird; sie erhalten sich überdies während der wechselnden physikalichen Zustände, in denen sich der Krystall befinden kann. Diese Thatsache kann als das ``definirende Grundgesetz der krystallisirten Materie'' betrachtet werden und wird vom Verfasser zweckmässig als ``Symmetriegesetz'' bezeichnet. Die Thatsache, dass die Zahl der einzelnen Symmetrieeigenschaften, welche in den Krystallen vorkommen, nur eine geringe ist, und dass sie in wechselnder Verbindug die Gesamtsymmetrie eines jeden Krystalles constituiren, hat schon in früher Zeit zu Versuchen angeregt, selbständig solche Verbindungen der genannten Symmetrieelemente auszudenken, welche, wenn auch noch nicht beobachtet, so doch theoretisch möglich sind. Man kann aber die Symmetrieeigenschaften eines Krystalles nicht beliebig vorschreiben; vielmehr sind sie durch bestimmte geometrische Gesetze mit einander verbunden. Hier ist der Punkt, wo die mathematische Untersuchung einsetzt. Augenscheinlich handelt es sich darum, alle überhaupt möglichen Verbindungen von Symmetrieeigenschaften zu ermitteln, welche in den Krystallen auftreten können. Dabei ist zu berücksichtigen, dass die Zahl dieser Symmetrieeigenschaften für jeden Krystall nur eine begrenzte ist, und dass von Symmetrieaxen nur zwei-, drei-, vier- oder sechszählige auftreten. Wie der Marburger Krystallograph Hessel zuerst gezeigt hat, beträgt die Gesamtheit aller krystallographischen Symmetrieklassen 32. Ihrer Ableitung ist der erste Teil des Werkes gewidmet. Es werden zunächst diejenigen Krystallabteilungen ermittelt, welche nur Symmetrieaxen besitzen, und dann für jede von ihnen die ausserdem noch mit Summetrieebenen u. s. w. behafteten Klassen abgeleitet. Von hier aus ist es sodann leicht, diejenige Anordnung aller 323 Krystallklassen zu treffen, welche der gewöhnlichen Systematik entspricht. Lässt man, wie es der rein geometrischen Fragestellung entsprechen würde, beliebige \(n\)-zählige Symmetrieaxen zu, so ist die Zahl aller möglichen Symmetrieklassen unbegrenzt gross. Von ihnen stellen die 32 Krystallklassen diejenigen dar, deren Axen nur zwei-, drei, vier- oder sechszählig sind. Die Beschränkung auf derartige Axen kann als eine Folge des Gesetzes von den rationalen Indices betrachtet werden. Die deductive Ableitung der 32 möglichen Krystallklassen ist daher nur auf Grund ``zweier empirisch gewonnenen Gesetze'' möglich; das eine ist das fundamentale Symmetriegesetz, das andere ist das Gesetz der rationalen Indices. Die ``Structurtheorien'' sind hierin glücklicher; sie vermögen die Gesetze, welche die Symmetrie der Krystalle betreffen, ohne dass es nötig wäre, auf eines jener beiden Gesetze zu recurriren. Die Structurtheorien knüpfen bekanntlich an die fundamentale Hypothese an, dass die moleculare Eigenart der Krystalle ihren Ausdruck in der regelmässigen Anordnung der Krystallbausteine findet, und zwar prägt sich die Regelmässigkeit darin aus, dass alle diese Bausteine von gleicher Art sind, und dass jeder von ihnen von den benachbarten Bausteinen auf gleiche Weise umgeben ist. Diese Vorstellung ist, seitdem ihr Haüy, wenn auch in unvollkommener Weise, zuerst Ausdruck gegeben, ununterbrochen in Geltung geblieben. Der Verfasser beweist, dass sie in ungezwungener Weise das moleculare Verständnis der fundamentalen Gesetze der Krystallsubstanz vermittelt; er zeigt, dass das oberste Grundgesetz der krystallisirten Materie, nämlich das Symmetriegesetz, als eine an der Spitze der Theorie stehende Consequenz von principieller Bedeutung erscheint, dass sich auch die speciellen Gesetze, welche die Einteilung nach der Symmetrie betreffen, als natürliche Folgerungen der Ausgangshypothese ergeben, und dass sie endlich auch eine einfache Deutung des Gesetzes der rationalen Indices gestattet. Die auf keine andere Weise erklärbare Beschränkung auf die zwei-, drei, vier- un sechszähligen Symmetrieaxen, sowie die Einteilung in die 32 Klassen wird auf Grund dieser Hypothese überhaupt erst begreiflich. Die vorstehenden Bemerkungen lassen erkennen, welches die Aufgaben sind, die in der oben genannten Schrift zu lösen waren. Zunächst handelt es sich darum, die Natur eines regelmässigen Molekelhaufens genauer zu studiren und zu zeigen, dass ihm in dem nämlichen Sinn ein bestimmter Symmetriecharakter beizulegen ist, wie dem Krystall, welchen er darstellen soll. Die Einteilung aller Molekelhaufen nach der Symmetrie, sowie die Ableitung aller überhaupt existirenden regelmässigen Molekelhaufen ist die zweite Aufgabe, die zu behandeln war. Die Untersuchung gipfelt in dem Resultat, dass im ganzen 230 krystallographisch verwendbare regelmässige Molekelhaufen vorhanden sind, und dass sie rücksichtlich der Symmetrie in die nämlichen 32 Klassen zerfallen, zu welchen die von dem Symmetriegesetz und dem Gesetz der rationalen Indices ausgehende Deduction hinführt. Für jede dieser 32 Klassen sind sämtliche Molekelhaufen gleicher Symmetrie genau angegeben worden; die Aufgabe, ienen Krystall von bestimmter Symmetrie durch einen geeigneten Molekelhaufen darzustellen, kann daher an der Hand der genannten Schrift unmittelbar erledigt werden. Die grösste Zahl von Molekelhaufen verschiedener Art, welche einer einzelnen Krystallklasse entsprechen, beträgt 30; die bezügliche Krystallklasse ist die Holoedrie des rhombischen Systems. Hingegen tritt auch eine Krystallklasse auf, deren Symmetrie nur in einer einzigen Art von Molekelhaufen verkörpert ist, nämlich die mit dreizähliger Axe versehene Tetartoedrie des hexagonalen Systems. Es ist mithin die Möglichkeit gegeben, innerhalb jeder der 32 Krystallklassen wieder eine grössere oder geringere Zahl von Unterabteilungen zu statuiren, ja nach der Art des Molekelhaufens, durch welchen die Krystallsubstanz zu ersetzen ist. Der specifische Symmetriecharakter eines regelmässigen Molkelhaufens hängt augenscheinlich von zwei Factoren ab, nämlich von der räumlichen Anordnung der Molekeln und von ihrer Qualität, d. h. von ihrer Form, ihrer physikalischen Wirkungsweise, ihrer chemischen Beschaffenheit u. s. w. Hierüber sind die verschiedensten Annahmen möglich, und dementsprechend kann die Symmetrie eines Molekelhaufens auf mannigfaltige Weise begründet werden. Die Gesetze, welche hierfür massgebend sind, sind in der genannten Schrift in ausführlicher Weise abgeleitet worden. Im besondern wird erörtert, wie viele verschiedene Structurauffassungen auf Grund der Hypothese von der regelmässigen Anordnung der Molekeln überhaupt möglich sind, und in welhcem Verhältnis die einzelnen Structurauffassungen zu einander stehen. Unter allen Structurauffassungen giebt es zwei, welche als die beiden Extreme betrachtet werden können. Die eine drückt sich in der Bravais'schen Gittertheorie aus, die andere knüpft an Wiener und Sohncke an. Bravais nimmt bekanntlich an, dass die Schwerpunkte aller Molekeln ein Raumgitter bilden. Setzt man in jeden dieser Eckpunkte je eine gleichartige Molekel so ein, dass alle Molekeln parallel zu einander stehen, so erhält man einen Bravais'schen Molekelhaufen. Durch geeignete Wahl des Raumgitters und der Molkel kann man bewirken, dass die Symmetrie des zugehörigen Molekelhaufens mit der Symmetrie irgend einer der 32 Krystallklassen übereinstimmt. Die Raumgitter zerfallen nämlich rücksichtlich der Symmetrie in sieben Klassen, welche genau den sieben Krystallsystemen entsprechen. Soll nun für einen Krystall, der irgend einer Abteilung eines Krystallsystems angehört, die ihm entsprechende Structur hergestellt werden, so benutzt man ein Raumgitter, welches die Symmetrie des Krystallsystems besitzt, und stellt in jeden Gitterpunkt eine Molekel, deren Symmetrie genau mit derjenigen der bezüglichen Unterabteilung übereinstimmt. Die Bravais'sche Structurauffassung läuft daher darauf hinaus, den Molekeln dieselbe Symmetrie beizulegen, welche der Krystall besitzt; er stattet die kleinsten Teilchen genau mit denjenigen Eigenschaften aus, deren Vorkommen erklärt werden soll; ein Verfahren, das oftmals den ersten Versuch darstellt, um die physikalischen Eigenschaften unserem Verständnis näher zu bringen. Dem gegenüber sind Wiener und Sohncke davon ausgegangen, für die Erklärung der Symmetrie allein die Anordnung der individuellen Bausteine ins Auge zu fassen. Dieser Forderung kann, wie der Verfasser nachweist, im vollsten Umfange genügt werden; für jede der 32 Krystallklassen giebt es Molekelhaufen, welche, wie auch immer die Molekel beschaffen sein mag, die Symmetrie der bezügliche Krystallklasse besitzen. Diese Molekelhaufen sind gerade diejenigen 230, deren Existenz oben bereits angeführt wurde. Für die Structurauffassung, welche von ihnen ausgeht, unterliegt die Molekel weder nach Form noch Wirkungsweise irgend einer positiven Bestimmung. Die Möglichkeit, zwischen die beiden vorstehend genannten Structurtheorien noch eine Reihe anderer Structurauffassungen einzuordnen, beruht darauf, dass sich die Gesamtsymmetrie eines Krystalles im allgemeinen so in zwei Teile zerlegen lässt, dass einer von beiden der Molekel aufgeprägt wird, während sich der andere in der Anordnung, d. h. in der Art des Aufbaues darstellt. Jeder derartigen Zweiteilung der Krystallsymmetrie entspricht eine andere Structurvorstellung; je höher die Symmetrie eines Krystalles ist, um so mannigfaltiger ist daher im allgemeinen die Art, auf welche die Zweiteilung ausgeführt werden kann. Welche Structurauffassung in jedem speciellen Fall am zweckmässigsten zu Grunde gelegt wird, ist eine Frage, deren Entscheidung dem Krystallographen überlassen bleiben muss. Von mathematischer Seite könne es sich, wie der Verfasser bemerkt, nur darum handeln, die Gesamtheit aller überhaupt möglichen Structurauffassungen anzugeben und zu kennzeichnen.

Der Nachdruck des ersten ``modernen'' Werkes der mathematischen Kristallographie [(1891, JFM 23.0554.02)] ist aufs wärmste zu begrüßen. Das Werk zerfällt in zwei Teile: Das Ziel des ersten Abschnittes liegt in der Herleitung der 32 Kristallklassen. Um diese zu erhalten, wird die Symmetrie erläutert und das Operieren mit den Symmetrieelementen dargestellt. Der Gruppenbegriff gestattet die geeignete Formulierung der Ergebnisse. Diese Klassen beschreiben die makroskopische Gestalt der Kristalle, wobei der Autor zudem die daraus folgenden physikalischen Eigenschaften bespricht. Der zweite Abschnitt handelt von der inneren Struktur der Kristalle, für welche damals nur Hypothesen vorlagen. Die am weitesten führende war diejenige der Gitterstruktur. Mittels dieser gelangt der Autor zur Herleitung der 230 Raum- oder Bewegungsgruppen. Beide Teile sind auch heute noch lesenswert. Wir bemerken, daß der Autor übersehen hatte, daß nicht Hessel (1830/31), sondern Frankenheim die 32 Klassen erstmals herleitete. Wohl zitiert der Autor zwei Arbeiten dieses Forschers und weist darauf hin, daß dieser erstmals die sog. Bravais-Gitter aufstellte. Leider übersah er aber den 1826 in der Zeitschrift ISIS von Oken erschienenen Artikel ''Crystallonomische Aufsätze'', worin erstmals diese Klassen hergeleitet werden. Der Autor war ein Schüler von Felix Klein und von diesem angeregt, verfaßte er sein Werk im Sinne von dessen gruppentheoretischem Denken und in der noch heute gebrauchten Sprache. Zu Recht wird daher sein Werk als einer der beiden Klassiker der mathematischen Kristallographie bezeichnet. Der andere Klassiker wäre zweifelsohne sein Zeitgenosse E. S. von Fedorov, falls dieser seine Entdeckungen in lesbarer Form veröffentlicht hätte. Allein im Ural arbeitend, fehlte ihm die Schulung im geläufigen mathematischen Ausdruck. Dies hat wohl auch Chebychev veranlaßt, die Veröffentlichung der Arbeiten abzulehnen. Glücklicherweise wissen wir aus dem Briefwechsel von Schoenflies und Fedorov, daß nur eine enge Zusammenarbeit der beiden Forscher zur endgültigen und bereinigten Anzahl der 230 Gruppen führte. Allerdings scheint der Autor nicht bemerkt zu haben, daß der Begriff der symmorphen \((=\) arithmetischen) Klasse von Fedorov tiefer in die Struktur der Raumgruppen eingreift als derjenige der 32 Klassen, obschon er diese 73 Systeme auf Seite 598 erwähnt. Der Nachdruck ist von hervorragender Qualität und es ist zu hoffen, daß er heute, wo das Gebiet neue Aktualität genießt, viele Leser finden wird. Sie werden reich belohnt, insbesondere auch durch die Nennung von etwa 40 früheren Autoren und deren Werken, von Hauy, Delafosse, Frankenheim, und Hessel bis zu Fedorov und Barlow, was bezeugt, daß der Autor ein ausgezeichneter Kenner der Geschichte der Kristallographie war. The author is a well-known expert in history of science in antiquity and in the middle ages and in Sanskrit. The Royal Dutch Mathematical Society (KWG) had selected the field of history of mathematics for the Brouwer Prize in 2011, which was bestowed on the author. This article is based on the Brouwer Lecture which the author delivered on 14 April 2011 during the annual Dutch Mathematical Congress. The author discusses changing aims and current priorities in the history of mathematics, with special reference to comparisons between `Western' and `non-Western' traditions of mathematical knowledge. The author gives examples offering some remarkable results from late-medieval Indian mathematics that illustrate the creative tension between mathematical experiment and proof. For example, over 250 years before Newton's work, a Hindu astronomer named Madhava in South-West India had given the algorithm, in Sanskrit, for expansion of sine series which (when properly understood and translated) is exactly equivalent to the power series later found by Newton. The similarities in results found by Madhava and Newton and also in the basic structure of some of the tools they used are significant enough to have a relook on history of mathematics. However, few scholars outside mathematics have the requisite technical knowledge and interest in history of mathematics. This paper will be of interest to all those who have even little knowledge of mathematics and are interested in the pursuit of history of mathematics.

The book under review is the second, extended edition of the first printing, see [A Singular introduction to commutative algebra. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann. (Berlin): Springer. (2002; Zbl 1023.13001)], published in 2002. During these five years the computeralgebra system \textsc{Singular} (developed with a strong input of the authors of the book) became one of the most popular system for computations in commutative algebra and algebraic geometry. The actual version 3.0.4 is freely available under the GNU public license for various platforms via \texttt{http://www.singular.uni-kl.de}. In particular, during the last five years the development of \textsc{Singular} was supported by an international community that has created a large number of proceedures for some special computations available as libraries in \textsc{Singular}. Moreover, there is a kernel extension providing Gröbner bases algorithms and implementations for ideals and modules in non-commutative algebras with a certain condition for non-commutativity. Most of this was contributed by V. Lewandowsky. The second edition of the book reflects this rapid development of \textsc{Singular} in the following way: The Appendix C of the book, \textsc{Singular} - A short Introduction, concipated as a crash course of \textsc{Singular}'s programming language, is rewritten corresponding to the recent version. It covers more examples on how to write labraries and how to communicate with other systems, extending those of the first edition (Mathematica, Maple, MuPAD) by GAP and SAGE. A new CD is included containing all the examples of the book and most of the \textsc{Singular}-libaries. That means, the book does cover not only the theoretical background but also the actual version of the software with the examples in order to become familar with practical experiences. Moreover, the CD is completed with a lot of additional material, the manual (also available via the program), links to papers, other software resources and - of course - the sources and the binaries for \textsc{Singular}. The major extensions in the text are the following: (1) There is a new section of Chapter 1 ``Rings, Ideals and Standard Bases'' about non-commutative Gröbner bases. (2) Chapter 4 ``Primary Decomposition and Related Topics'' is extended by two new sections about characteristic and triangular sets with the corresponding decomposition algorithm. Characteristic sets are useful in order to compute the minimal associated prime ideals of an ideal. Triangular sets are a basic tool for the symbolic pre-processing to solve zero-dimensional systems of polynomial equations. (3) There is a new Appendix B ``Polynomial Factorization'' concerning univariate factorization over \(\mathbb F_p\) and \(\mathbb Q\) and algebraic extensions, as well as multivariate factorization over these fields and over the algebraic closure of \(\mathbb Q.\) In addition to what is said about the first edition the book is distinctive and highly useful in order to explore the beauties and difficulties of commutative algebra and algebraic geometry by computational and theoretical insights. It provides the theory in a clever way as well as all the requirements for practical experiments conceived for \textsc{Singular} but nevertheless there is no strict restriction in order to use the material with different computer algebra systems. It is highly recommended for all -- students and researchers -- who are interested in practical computations of their algebraic interests as well as for introductory research projects for students. In recent times the algorithmic and computational aspects of classical algebra became a separate research interest of the subject. For practical computations the most favorite computer algebra systems in commutative algebra with applications in algebraic geometry are \textit{CoCoA}, \textit{Macaulay2}, and \textit{Singular}. The computer algebra system \textit{Singular} was developed over the last two decades with a strong influence of the authors of the present book. It became a powerful tool for computations in particular for polynomial computations. The version 2.0.4 is freely available under the GNU public license for various platforms under \texttt{http://www.singular.uni-kl.de}. The book under review is completed by a CD-ROM that contains the previous version 2.0.3. The book is motivated by what the authors believe for the most useful way for studying commutative algebra with a view toward algebraic geometry and singularity theory. It grows out of several courses by the authors. The aim of the book is such an introduction to commutative algebra with a view towards to algorithmic aspects and computational practice. This introduction to commutative algebra is divided into seven chapters: 1. Rings, ideals, and standard bases, 2. Modules, 3. Noether normalization and applications, 4. Primary decomposition and related topics, 5. Hilbert function and dimension, 6. Complete local rings, 7. Homological algebra; and two appendices: A. Geometric background, B. \textit{Singular} - A short introduction. The first two chapters introduce the basics about rings, ideals and modules, emphasized for polynomial rings and their factor rings. This is illustrated how to use \textit{Singular} for computations. For further use (localization and singularity theory) the authors do not restrict to well-orderings for the computation of standard bases. -- The following four chapters are devoted to more advanced considerations: Chapter 3 describes Noether normalization as the cornerstone in the theory of affine algebras, theoretically as well as computationally. A highlight is the algorithm for the computation of the non-normal locus and the normalization of an affine ring, based on a criterion of Grauert and Remmert. The chapter ends with a section containing some of the larger procedures written in the \textit{Singular} programming language. Chapter 4 is devoted to primary decomposition and related topics. For the constructive approach the authors follow the ideas of \textit{P. Gianni}, \textit{B. Trager} and \textit{G. Zacharias} [see J. Symb. Comput. 6, 149-167 (1988; Zbl 0667.13008)]. The chapter 5 is concerned with the Hilbert functions and various related subjects. It culminates with a proof of the Jacobian criterion of affine algebras and its application for the computation of the singular locus. In chapter 6, the authors consider standard bases in power series rings. The basis for local analytic geometry is the fact that standard bases of ideals in power series rings can be computed if the ideal is generated by polynomials. -- Chapter 7 gives a short introduction to basic concepts of homological algebra. There are results about Koszul complexes, the proof of the Auslander-Buchsbaum formula and its application to a Cohen-Macaulay criterion. The first appendix provides an introduction to applications in algebraic geometry, in particular to elimination techniques and singularity theory. More relations might be found in the accompanying libraries of \textit{Singular}. The second appendix gives a crash course of the programming language of \textit{Singular}, data types, functions and control structures of the system, as well as of the procedures appearing in the libraries. Moreover, the authors show how \textit{Singular} might communicate with other systems (\textit{Maple, Mathematica, MuPAD}). In the introduction of the book, the authors describe how to use the book for various courses of different length and difficulties and seminars, using \textit{Singular} as the tool for explicit computations. Each chapter is completed by exercises that allow the reader to follow the theoretical as well as the computational aspects. The authors' most important new focus is the presentation of non-well orderings that allow them the computational approach for local commutative algebra. The accompanying CD-ROM also contains all the examples of the book. Finally one should mention that the book is not at all an introduction to \textit{Singular}. For that reason one might also consult the manual \textit{Singular} and the help files. In fact the book provides an introduction to commutative algebra from a computational point of view. So it might be helpful for students and other interested readers (familiar with computers) to explore the beauties and difficulties of commutative algebra by computational experiences. In this respect the book is one of the first samples of a new kind of textbooks in algebra.

The book under review is the second, extended edition of the first printing, see [A Singular introduction to commutative algebra. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann. (Berlin): Springer. (2002; Zbl 1023.13001)], published in 2002. During these five years the computeralgebra system \textsc{Singular} (developed with a strong input of the authors of the book) became one of the most popular system for computations in commutative algebra and algebraic geometry. The actual version 3.0.4 is freely available under the GNU public license for various platforms via \texttt{http://www.singular.uni-kl.de}. In particular, during the last five years the development of \textsc{Singular} was supported by an international community that has created a large number of proceedures for some special computations available as libraries in \textsc{Singular}. Moreover, there is a kernel extension providing Gröbner bases algorithms and implementations for ideals and modules in non-commutative algebras with a certain condition for non-commutativity. Most of this was contributed by V. Lewandowsky. The second edition of the book reflects this rapid development of \textsc{Singular} in the following way: The Appendix C of the book, \textsc{Singular} - A short Introduction, concipated as a crash course of \textsc{Singular}'s programming language, is rewritten corresponding to the recent version. It covers more examples on how to write labraries and how to communicate with other systems, extending those of the first edition (Mathematica, Maple, MuPAD) by GAP and SAGE. A new CD is included containing all the examples of the book and most of the \textsc{Singular}-libaries. That means, the book does cover not only the theoretical background but also the actual version of the software with the examples in order to become familar with practical experiences. Moreover, the CD is completed with a lot of additional material, the manual (also available via the program), links to papers, other software resources and - of course - the sources and the binaries for \textsc{Singular}. The major extensions in the text are the following: (1) There is a new section of Chapter 1 ``Rings, Ideals and Standard Bases'' about non-commutative Gröbner bases. (2) Chapter 4 ``Primary Decomposition and Related Topics'' is extended by two new sections about characteristic and triangular sets with the corresponding decomposition algorithm. Characteristic sets are useful in order to compute the minimal associated prime ideals of an ideal. Triangular sets are a basic tool for the symbolic pre-processing to solve zero-dimensional systems of polynomial equations. (3) There is a new Appendix B ``Polynomial Factorization'' concerning univariate factorization over \(\mathbb F_p\) and \(\mathbb Q\) and algebraic extensions, as well as multivariate factorization over these fields and over the algebraic closure of \(\mathbb Q.\) In addition to what is said about the first edition the book is distinctive and highly useful in order to explore the beauties and difficulties of commutative algebra and algebraic geometry by computational and theoretical insights. It provides the theory in a clever way as well as all the requirements for practical experiments conceived for \textsc{Singular} but nevertheless there is no strict restriction in order to use the material with different computer algebra systems. It is highly recommended for all -- students and researchers -- who are interested in practical computations of their algebraic interests as well as for introductory research projects for students. The articles of this volume will be reviewed individually.

The denominator of the rational function which is defined first as a formal power series with coefficients of local densities of the transforms between the two symmetric matrices, is computed in this paper. Let \(k\) be a \(\mathfrak p\)-adic number field, where \(\mathfrak p\) does not stand over 2, \(\mathcal O\) the ring of integers in \(k\), \(\pi\) a prime element of \(k\) and \(q\) the cardinal of \(\mathcal O/\mathfrak p\). For nondegenerate symmetric matrices \(A\) and \(B\) over \(\mathcal O\) of size \(m\) and \(n\), respectively, the local density \(\mu(B,A)\) is defined to be \[ \lim_{\ell \to \infty}q^{-\ln(m-(n+1)/2)}N_{\ell}(B,A), \] where \(N_{\ell}(B,A)\) is the number of elements \(T\) in \(M_{m,n}(\mathcal O/\mathfrak p^{\ell})\) satisfying \({}^tTAT\equiv B\pmod{\mathfrak p^\ell}\). Consider the formal power series \(P(B,A;X):=\sum^{\infty}_{r=0}\mu (\pi^rB,A)X^r\); it is absolutely convergent when \(| X|\) is sufficiently small and in fact, was proved to be a rational function by \textit{S. Böcherer} and \textit{F. Sato} [Comment. Math. Univ. St. Pauli 36, 53--86 (1987; Zbl 0629.10018)]. The author of this paper now has succeeded in computing its denominator; it is \[ (1-X) \prod^{n-1}_{i=1}(1-q^{(n-i)(n+i+1-m)}X^2). \] This paper proves the rationality of some formal power series, which are generating functions of some arithmetic objects over a p-adic number field. The authors obtain the rationality theorem for the following two power series: \[ P(S,T;X):=\sum^{\infty}_{r=0}\alpha_ p(S,p^ rT)X^ r\text{ and } Q(S,T;X):=\sum_{W}\alpha_ p(S,T[W])X^{ord_ p(\det (W))}, \] where S and T are regular half-integral symmetric matrices over \({\mathbb{Q}}_ p\) of size m and n \((m\geq n)\), respectively; \(\alpha_ p(S,T)\) denotes the p-adic local density of representations of T by S; the summation in the definition of Q is taken over \((M(n, {\mathbb{Z}}_ p)\cap GL(n, {\mathbb{Q}}_ p))/GL(n, {\mathbb{Z}}_ p).\) Moreover they compute the denominators of P and Q in the following two special cases: (I) m is even and S is unimodular (T is arbitrary), (II) T is anisotropic (S is arbitrary). The denominators of \(P(S,T;X)\) and \(Q(S,T;X)\) are given, respectively by \[ (1-X)\prod^{n-1}_{j=0}(1- p^{(n-j)(n+j+1-m)} X^ 2)\text{ and } \prod^{n}_{i=1}(1-p^{n+i- m} X)(1-p^{n-i} X). \]

The denominator of the rational function which is defined first as a formal power series with coefficients of local densities of the transforms between the two symmetric matrices, is computed in this paper. Let \(k\) be a \(\mathfrak p\)-adic number field, where \(\mathfrak p\) does not stand over 2, \(\mathcal O\) the ring of integers in \(k\), \(\pi\) a prime element of \(k\) and \(q\) the cardinal of \(\mathcal O/\mathfrak p\). For nondegenerate symmetric matrices \(A\) and \(B\) over \(\mathcal O\) of size \(m\) and \(n\), respectively, the local density \(\mu(B,A)\) is defined to be \[ \lim_{\ell \to \infty}q^{-\ln(m-(n+1)/2)}N_{\ell}(B,A), \] where \(N_{\ell}(B,A)\) is the number of elements \(T\) in \(M_{m,n}(\mathcal O/\mathfrak p^{\ell})\) satisfying \({}^tTAT\equiv B\pmod{\mathfrak p^\ell}\). Consider the formal power series \(P(B,A;X):=\sum^{\infty}_{r=0}\mu (\pi^rB,A)X^r\); it is absolutely convergent when \(| X|\) is sufficiently small and in fact, was proved to be a rational function by \textit{S. Böcherer} and \textit{F. Sato} [Comment. Math. Univ. St. Pauli 36, 53--86 (1987; Zbl 0629.10018)]. The author of this paper now has succeeded in computing its denominator; it is \[ (1-X) \prod^{n-1}_{i=1}(1-q^{(n-i)(n+i+1-m)}X^2). \] The authors assume that \(f\) is a continuous function defined on the unit ball of \(\mathbb R^{d }\), of the form \(f(x)=g(Ax)\), where \(A\) is a \(k\times d\) matrix and \(g\) is a function of \(k\) variables for \(k\ll d\). There are given a budget \(m\in \mathbb N\) of possible point evaluations \(f(x _{i })\), \(i=1,\dots ,m\), of \(f\), which are allowed to query in order to construct a uniform approximating function. Under certain smoothness and variation assumptions on the function \(g\), and an \(arbitrary\) choice of the matrix \(A\), the authors present in this paper {\parindent=4mm \begin{itemize} \item[1.] a sampling choice of the points \(\{x _{i }\}\) drawn at random for each function approximation; \item [2.] algorithms (algorithm 1 and algorithm 2) for computing the approximating function, whose complexity is at most polynomial in the dimension \(d\) and in the number \(m\) of points. \end{itemize}} Due to the arbitrariness of \(A\), the sampling points will be chosen according to suitable random distributions, and their results hold with overwhelming probability. Their approach uses tools taken from the compressed sensing framework, recent Chernoff bounds for sums of positive semidefinite matrices, and classical stability bounds for invariant subspaces of singular value decompositions.

These notes arose from a course on the index theorem, which was presented by the author at the International Centre for Theoretical Physics in Trieste in 1988. They contain a clearly written exposition on Clifford algebras and spinors, characteristic classes, elliptic complexes, and the index theorem. There are only a few proofs, but the paper contains a lot of carefully elaborated examples. The main reference for elliptic complexes and the index theorem is the author's book ``Invariance theory, the heat equation, and the Atiyah-Singer index theorem'' (1984; Zbl 0565.58035). Die vorliegende Monographie stellt eine stark erweiterte Fassung der Lecture Notes ''The index theorem and the heat equation'' des Verf. dar (1974; Zbl 0287.58006). Zentrales Thema ist die Behandlung des Atiyah- Singer-Indexsatzes mit den Methoden der Wärmeleitungsgleichung und der lokalen Differentialgeometrie. Grob gesprochen besteht die Idee darin, den Index eines elliptischen Komplexes (über einer kompakten (riemannschen) Mannigfaltigkeit M) als Integral über lokale Invarianten \(a_ n\) darzustellen, die mit Hilfe der Wärmeleitungskerne der (dem Komplex zugeordneten) Laplace-Operatoren über eine asymptotische Entwicklung gewonnen werden. Die \(a_ n\) erfüllen gewisse funktorielle Eigenschaften, die es erlauben, den Index (mittels der Invariantentheorie) in Beziehung zu charakteristischen Klassen, d.h. topologischen Invarianten der zugrundeliegenden Bündel zu setzen. Diesem Vorgehen entspricht der Aufbau des Buches, insbesondere der der ersten drei Kapitel. Das 1. Kapitel enthält eine klar verständliche Einführung in die Theorie der Pseudodifferentialoperatoren (über kompakten Mannigfaltigkeiten), insbesondere in die elliptische Theorie (elliptische komplexe Randwertprobleme) und deren funktionalanalytische Aspekte (Fredholm-Theorie), behandelt die Wärmeleitungsgleichung (für positive selbstadjungierte elliptische Operatoren) und bringt als zentrales Ergebnis die lokale Formel für den Index eines elliptischen Operators. Weiterhin werden speziellere Themen wie Lefschetzsche Fixpunktsätze und die \(\eta\)- und \(\zeta\)-Funktionen behandelt. Im 2. Kapitel wird sodann alles notwendige Material über charakteristische Klassen zusammengestellt. Ferner werden invariantentheoretische Charakterisierungen spezieller Klassen (z. B. der Pontryagin-Klasse und gemischter Klassen des Tangentialbündels etc.) angegeben, die stets durch eine Untersuchung einer geeigneten Teilmenge des Polynomrings in den \(g_{ij}\) und deren Ableitungen \(((g_{ij})\) sei dabei die riemannsche Metrik) gewonnen werden. Als Anwendung dieser Ideen wird ein Wärmeleitungsbeweis des Satzes von Gauß-Bonnet gegeben. Das 3. Kapitel stellt nun den eigentlichen Kern des Buches dar. Nach der Herleitung der jeweiligen Indexformeln für die vier klassischen elliptischen Komplexe (das sind: der de Rham-Komplex, der Signatur- Komplex, der Spin-Komplex (hier werden Ergebnisse über Spinoren, deren Darstellungen und Spinstrukturen auf Vektorbündeln eingesetzt) und der Dolbeault-Komplex) und einem Abstecher in die Kählersche Geometrie wird am Ende des Kapitels der Satz von Atiyah-Singer in seiner allgemeinen Form bewiesen. Die notwendigen Hilfsmittel aus der algebraischen Topologie (der Chern-Isomorphismus zwischen rationaler Kohomologie und K- Theorie; der Bottsche Periodizitätssatz) werden kurz in einem gesonderten Abschnitt zusammengestellt. Das weniger ausführlich geschriebene 4. Kapitel schließlich ist speziellen Themen gewidmet. So werden der de Rham-Komplex und der Satz von Gauß-Bonnet für berandete Mannigfaltigkeiten behandelt. Ferner wird die getwistete Index-Formel von Atiyah-Singer-Patodi (mit Hilfe der \(\eta\)-Invariante) und die Lefschetzsche Fixpunktformel für die vier klassischen Komplexe (s.o.) hergeleitet. Es folgen eine Anwendung der \(\eta\)-Invariante auf die K-Theorie sphärischer Raumformen, eine Untersuchung der Singerschen Vermutung über die Eulerform und lokale Formeln für die Invarianten der Wärmeleitungsgleichung. Den Abschluß bildet eine Anwendung des bisherigen Ergebnisses auf Fragen der Spektralgeometrie. Bis auf Anleihen aus der Invariantentheorie (ein Satz von Hermann Weyl in Kapitel 2) und der algebraischen Topologie (in Kapitel 3) und speziellere Verweise in Kapitel 4 ist die Darstellung in sich geschlossen.

These notes arose from a course on the index theorem, which was presented by the author at the International Centre for Theoretical Physics in Trieste in 1988. They contain a clearly written exposition on Clifford algebras and spinors, characteristic classes, elliptic complexes, and the index theorem. There are only a few proofs, but the paper contains a lot of carefully elaborated examples. The main reference for elliptic complexes and the index theorem is the author's book ``Invariance theory, the heat equation, and the Atiyah-Singer index theorem'' (1984; Zbl 0565.58035). If the open quantum system undergoes Markovian time evolution then its density operator (possibly time-dependent) is governed by Kossakowski-Lindblad equation (also known as GKSL master equation). For standard bilinear control systems (finite-dimensioanl and infinite-dimensional) with a switchable noise the Authors show that sufficient conditions under which reachable states (i.e.\ the semigroup orbit of initial density operator) fill the set of all states majorized by an initial state if the system undergo GKSL master equation. This result holds for finite-dimensional as well as (for the first time) for infinte-dimensional system.

With the standard notation for basic hypergeometric series, the authors prove in this paper the following identity (1) \[ \Big(\sum_{n=0}^{\infty}(-1)^n\,a^{n}\,q^{n(n-1)/2}\Big)\,\Big(\sum_{n=0}^{\infty}(-1)^n\,b^{n}\,q^{n(n-1)/2}\Big) =(q)_{\infty}(a)_{\infty}(b)_{\infty}\,\sum_{n=0}^{\infty}\frac{(abq^{n-1})_{n}\,q^{n}}{(q)_{n}(a)_{n}(b)_{n}}\,. \] Series of the form appearing in the left hand side of (1) are called partial theta functions. The authors deduce from (1) a generalized triple product identity proved by the second named author in [''Partial theta functions. I: Beyond the lost notebook.'' Proc. Lond. Math. Soc., 87, No.2, 363--395 (2003; Zbl 1089.05009)]. They also discuss the relationship of (1) to other theorems in \(q\)-hypergeometric series. It is shown how many of the partial theta function identities in Ramanujan's lost notebook can be generalized to infinite families of such identities. Key in our construction is the Bailey lemma and a new generalization of the Jacobi triple product identity. By computing residues around the poles of our identities we find a surprising connection between partial theta function identities and Garrett-Ismail-Stanton-type extensions of multisum Rogers-Ramanujan identities.

With the standard notation for basic hypergeometric series, the authors prove in this paper the following identity (1) \[ \Big(\sum_{n=0}^{\infty}(-1)^n\,a^{n}\,q^{n(n-1)/2}\Big)\,\Big(\sum_{n=0}^{\infty}(-1)^n\,b^{n}\,q^{n(n-1)/2}\Big) =(q)_{\infty}(a)_{\infty}(b)_{\infty}\,\sum_{n=0}^{\infty}\frac{(abq^{n-1})_{n}\,q^{n}}{(q)_{n}(a)_{n}(b)_{n}}\,. \] Series of the form appearing in the left hand side of (1) are called partial theta functions. The authors deduce from (1) a generalized triple product identity proved by the second named author in [''Partial theta functions. I: Beyond the lost notebook.'' Proc. Lond. Math. Soc., 87, No.2, 363--395 (2003; Zbl 1089.05009)]. They also discuss the relationship of (1) to other theorems in \(q\)-hypergeometric series. This note presents some straight-forward extensions of some results about certain oscillatory integrals with polynomial phases due to \textit{L. Corwin} and \textit{F. P. Greenleaf} [Commun. Pure Appl. Math. 31, 681-705 (1978) Zbl 0391.46033)].

This brief note gives important details on the mathematical innovations by A. B. Kempe and their influence on C. S. Peirce and Josiah Royce. While Kempe is mentioned in \textit{S. L. Pratt}'s article [Hist. Philos. Log. 28, No. 2, 133--150 (2007; Zbl 1134.03003)] reviewed below, Grattan-Guinness goes further and relates the peculiar apparatus of \(O\)- and \(F\)-relations at the center of Royce's logic to the details of an 1886 article by Kempe. This essay aims to explain and to defend the continuing relevance of Josiah Royce's (1855--1916) conception of logic. In line with other absolute idealists, like F. H. Bradley, Royce criticized the approach of Russell and Whitehead's \textit{Principia Mathematica} based on its taking for granted certain metaphysical notions like term and consequence. Instead, as Pratt explains, Royce aimed to develop a ``logic of order'' that included the means to generate the entities that other approaches to logic presupposed. For Royce this required a logic of the will whose primary instances were choices between incompatible courses of action (``all logic is the logic of the will'' (141)). Pratt goes on to argue that this seemingly more ambitious conception of logic can find useful application in contemporary debates about logical pluralism and the consequences of Gödel's incompleteness theorems.

This brief note gives important details on the mathematical innovations by A. B. Kempe and their influence on C. S. Peirce and Josiah Royce. While Kempe is mentioned in \textit{S. L. Pratt}'s article [Hist. Philos. Log. 28, No. 2, 133--150 (2007; Zbl 1134.03003)] reviewed below, Grattan-Guinness goes further and relates the peculiar apparatus of \(O\)- and \(F\)-relations at the center of Royce's logic to the details of an 1886 article by Kempe. A semiparametric model is used to examine the relationship between pollution and income for three non-point source pollutants. Statistical tests reject the quadratic specification in favor of the semiparametric model in all cases. However, the results do not support the inverted-U hypothesis for the pollution-income relationship.

The authors discuss the question of expressing an operator in an algebra of Hilbert space operators as a sum of commutators. They first note how the subject of commutators is related to derivations and describe the historical developements in the study of commutators. They then provide a survey of a number of important commutator results for special classes of operators. In the main new result here, they prove that, if \(\mathcal{R}\) denotes a von Neumann algebra of type \(\mathrm{II}_1\) and \(\mathscr{A}_f(\mathcal{R})\) is the algebra of all the (not necessarily bounded) operators affiliated with \(\mathcal{R}\), then the identity operator \(I\) in \(\mathcal{R}\) is the sum of two commutators in \(\mathscr{A}_f(\mathcal{R})\). This result leads to a negative answer to a conjecture made by the first two authors in [SIGMA, Symmetry Integrability Geom. Methods Appl. 10, Paper 009, 40 p. (2014; Zbl 1410.81008)] because it shows that there is a noncommutative polynomial in several variables with the property that whenever the variables are replaced by operators in \(\mathcal{R}\) the resulting operator has trace \(0\) but, replacing the variables with operators in \(\mathscr{A}_f(\mathcal{R})\), may yield a bounded operator with nonzero trace. We study some of the possibilities for formulating the Heisenberg relation of quantum mechanics in mathematical terms. In particular, we examine the framework discussed by Murray and von Neumann, the family (algebra) of operators affiliated with a finite factor (of infinite linear dimension).

The authors discuss the question of expressing an operator in an algebra of Hilbert space operators as a sum of commutators. They first note how the subject of commutators is related to derivations and describe the historical developements in the study of commutators. They then provide a survey of a number of important commutator results for special classes of operators. In the main new result here, they prove that, if \(\mathcal{R}\) denotes a von Neumann algebra of type \(\mathrm{II}_1\) and \(\mathscr{A}_f(\mathcal{R})\) is the algebra of all the (not necessarily bounded) operators affiliated with \(\mathcal{R}\), then the identity operator \(I\) in \(\mathcal{R}\) is the sum of two commutators in \(\mathscr{A}_f(\mathcal{R})\). This result leads to a negative answer to a conjecture made by the first two authors in [SIGMA, Symmetry Integrability Geom. Methods Appl. 10, Paper 009, 40 p. (2014; Zbl 1410.81008)] because it shows that there is a noncommutative polynomial in several variables with the property that whenever the variables are replaced by operators in \(\mathcal{R}\) the resulting operator has trace \(0\) but, replacing the variables with operators in \(\mathscr{A}_f(\mathcal{R})\), may yield a bounded operator with nonzero trace. We study the following inverse source problem for the wave operator \[ P_n:={\partial^2\over\partial t^2}- a^2\Delta_n\quad a>0, \] where \(\Delta_n\) is the Laplace operator, on the closed strip \(\overline G_T:= \{(x,t)\in\mathbb{R}^{n+1}: x\in\mathbb{R}^n\), \(0\leq t\leq T\}\), \(T>0\). Given any distribution \(v\in {\mathcal D}'(\mathbb{R}^{n+1})\) with \(\text{supp }v\subseteq \mathbb{R}^n\times(t>T)\), satisfying the wave equation \[ P_nv(x,t)= 0,\quad t>0,\quad\text{in }{\mathcal D}'(\mathbb{R}^n\times (t>T)), \] find a (source) distribution \(\nu\) with \(\text{supp }\nu\subseteq\overline G_T\) such that the wave potential \(E_n*\nu\), the convolution of the fundamental solution \(E_n\) of \(P_n\) and \(\nu\), satisfies the condition \(E_n*\nu(x,t)= v(x,t)\), \(t>T\). We deal with the solvability, the structure of the set of solutions, and the stability of the proposed problem.

The authors study the \(L^2\)-cohomology and \(L^2\)-signature for spaces with non-isolated conical singularities. These are called generalized Thom spaces in the paper, where arbitrary fibrations are used to construct the mapping cylinders and then the non-isolated conical singularities. The \(L^2\)-cohomology and \(L^2\)-signature for conical singularities were studied in the pioneering work of the first author in the 1980s. For the case of non-isolated singularities, the new issue is to identify the contribution from singular strata that now have positive dimensions. It turns out that the desired topological invariants from the singular strata were already created in the second author's previous work on the adiabatic limits of eta invariants associated with the fibrations [J. Am. Math. Soc. 4, No. 2, 265--321 (1991; Zbl 0736.58039)]. Consequently in the present paper, the authors are able to compute the \(L^2\)-signature of the singular spaces. In turn, and as an application, the \(L^2\)-signature formula gives a simpler proof of the adiabatic limit formula of eta invariants established in the afore-mentioned paper. The Dirac operator \(D_ x\) on the total space of fibration is treated when the metric along base direction is multiplied by a factor \(x\). The formula of adiabatic limit (as \(x\rightarrow 0\)) of the \(\eta(D_ x)\) is given where \(\eta\) is the Atiyah-Patodi-Singer invariant.

The authors study the \(L^2\)-cohomology and \(L^2\)-signature for spaces with non-isolated conical singularities. These are called generalized Thom spaces in the paper, where arbitrary fibrations are used to construct the mapping cylinders and then the non-isolated conical singularities. The \(L^2\)-cohomology and \(L^2\)-signature for conical singularities were studied in the pioneering work of the first author in the 1980s. For the case of non-isolated singularities, the new issue is to identify the contribution from singular strata that now have positive dimensions. It turns out that the desired topological invariants from the singular strata were already created in the second author's previous work on the adiabatic limits of eta invariants associated with the fibrations [J. Am. Math. Soc. 4, No. 2, 265--321 (1991; Zbl 0736.58039)]. Consequently in the present paper, the authors are able to compute the \(L^2\)-signature of the singular spaces. In turn, and as an application, the \(L^2\)-signature formula gives a simpler proof of the adiabatic limit formula of eta invariants established in the afore-mentioned paper. A new repetitive control method is presented for semi-active vibration control of machines, such as presses and forge hammers, that are subject to nearly periodic disturbances with known periods. The vibration of the machine is controlled by actuators consisting of a proof mass, a spring, and a damper with variable damping coefficients that are treated as control inputs. The system is approximated by a bilinear state equation involving products of state variables and control inputs. Using optimal control theory a control input vector is synthesized which minimizes an objective function that trades-off the quality of isolation against relative motions of the proof masses and accounts for inequality constraints on the control inputs. The optimal control law calls for a full state feedback and feedforward of a variable dependent on future disturbances. It is therefore proposed to estimate the system disturbance based on measurements from the most recent period and, assuming the disturbance to be repetitive, to use this estimate as a predicted disturbance for the consecutive period. An observer for simultaneous estimation of the state and the disturbance is developed. Although intended for a particular application, the repetitive control theory developed here is general and can be applied to other systems described by bilinear state equations.

The purpose of this paper is to study the relativistic Vlasov-Darwin system (RVDS), or its equivalent, called RVDG. The first theorem states that there exists a unique classical solution to the RVDG on a maximal interval, which satisfies certain conditions. Another theorem relates a global classical solution to the Cauchy problem of the RVDG to all solutions of a similar problem. The proofs use Fubinis theorem, mean value theorem, Cauchy sequences, Cauchy-Schwarz, Jensen, Poincaré, Hölder, Gronwall inequalities, and the Gronwall lemma. We show that a smooth, small enough Cauchy datum launches a unique classical solution of the relativistic Vlasov-Darwin (RVD) system globally in time. A similar result is claimed in [\textit{M. Seehafer}, Commun. Math. Sci. 6, No. 3, 749--764 (2008; Zbl 1157.35335)] following the work in [\textit{C. Pallard}, Int. Math. Res. Not. 2006, No. 15, Article ID 57191, 31 p. (2006; Zbl 1110.35095)]. Our proof does not require estimates derived from the conservation of the total energy, nor those previously given on the transversal component of the electric field. These estimates are crucial in the references cited above. Instead, we exploit the formulation of the RVD system in terms of the generalized space and momentum variables. By doing so, we produce a simple a priori estimate on the transversal component of the electric field. We widen the functional space required for the Cauchy datum to extend the solution globally in time, and we improve decay estimates given in [Seehafer, loc. cit.] on the electromagnetic field and its space derivatives. Our method extends the constructive proof presented in [\textit{G. Rein}, in: Handbook of differential equations: Evolutionary equations. Vol. III. Amsterdam: Elsevier/North-Holland. Handbook of Differential Equations, 383--476 (2007; Zbl 1193.35230)] to solve the Cauchy problem for the Vlasov-Poisson system with a small initial datum.

The purpose of this paper is to study the relativistic Vlasov-Darwin system (RVDS), or its equivalent, called RVDG. The first theorem states that there exists a unique classical solution to the RVDG on a maximal interval, which satisfies certain conditions. Another theorem relates a global classical solution to the Cauchy problem of the RVDG to all solutions of a similar problem. The proofs use Fubinis theorem, mean value theorem, Cauchy sequences, Cauchy-Schwarz, Jensen, Poincaré, Hölder, Gronwall inequalities, and the Gronwall lemma. This paper presents a novel method for encoding local binary descriptors for Visual Object Categorization (VOC). Nowadays, local binary descriptors, e.g. LBP and BRIEF, have become very popular in image matching tasks because of their fast computation and matching using binary bitstrings. However, the bottleneck of applying them in the domain of VOC lies in the high dimensional histograms produced by encoding these binary bitstrings into decimal codes. To solve this problem, we propose to encode local binary bitstrings directly by the Bag-of-Features (BoF) model with Hamming distance. The advantages of this approach are two-fold: (1) It solves the high dimensionality issue of the traditional binary bitstring encoding methods, making local binary descriptors more feasible for the task of VOC, especially when more bits are considered; (2) It is computationally efficient because the Hamming distance, which is very suitable for comparing bitstrings, is based on bitwise XOR operations that can be fast computed on modern CPUs. The proposed method is validated by applying on LBP feature for the purpose of VOC. The experimental results on the PASCAL VOC 2007 benchmark show that our approach effectively improves the recognition accuracy compared to the traditional LBP feature.

If \(T\) is a non-empty subset of the set \(\mathbb{N}\times \mathbb{N}\), where \(\mathbb{N}\) denotes the set of positive integers, \(\psi:T\times T\to \mathbb{N}\) is a map with the property that the equation \(\psi(x,y)=n\) has for every \(n\) finitely many solutions, then Lehmer's \(\psi\)-convolution of arithmetic functions \(f,g\) is defined by \(f\psi g=\sum_{\psi(x,y)=n}f(x)g(y)\). If the \(\psi\)-convolution is commutative and associative, then the set of all arithmetic functions forms a ring \(F\) under usual addition and the \(\psi\)-convolution as the product. The authors prove that \(F\) contains a unit element if and only if the map \(\psi\) is surjective. This has been known earlier only under the additional condition \(\psi(x,y)\geq\max\{x,y\}\) [the second author, Indian J. Pure Appl. Math. 20, 1184--1190 (1989; Zbl 0698.10004)]. It is shown moreover that if \( \psi\) preserves multiplicativity and is surjective, then the unit element is multiplicative. If F(x,y) is an integral-valued arithmetic function in two variables having the property that for every c the equation \(F(x,y)=c\) has only finitely many solutions, then Lehmer's product of two arithmetic functions f,g in one variable is defined by \(f*g(n)=\sum f(a)g(b)\), where the summation is taken over all pairs (a,b) of integers with \(F(a,b)=n\) [\textit{D. H. Lehmer}, Trans. Am. Math. Soc. 33, 945-957 (1931; Zbl 0003.10202); Am. Math. J. 53, 843-854 (1931; Zbl 0003.00303)]. The author shows that if F(x,y)\(\geq \max \{x,y\}\) and the set of all arithmetic functions with usual addition and Lehmer's product is a commutative ring then it has a unit element if and only if for every k the equation \(F(x,k)=k\) is solvable.

If \(T\) is a non-empty subset of the set \(\mathbb{N}\times \mathbb{N}\), where \(\mathbb{N}\) denotes the set of positive integers, \(\psi:T\times T\to \mathbb{N}\) is a map with the property that the equation \(\psi(x,y)=n\) has for every \(n\) finitely many solutions, then Lehmer's \(\psi\)-convolution of arithmetic functions \(f,g\) is defined by \(f\psi g=\sum_{\psi(x,y)=n}f(x)g(y)\). If the \(\psi\)-convolution is commutative and associative, then the set of all arithmetic functions forms a ring \(F\) under usual addition and the \(\psi\)-convolution as the product. The authors prove that \(F\) contains a unit element if and only if the map \(\psi\) is surjective. This has been known earlier only under the additional condition \(\psi(x,y)\geq\max\{x,y\}\) [the second author, Indian J. Pure Appl. Math. 20, 1184--1190 (1989; Zbl 0698.10004)]. It is shown moreover that if \( \psi\) preserves multiplicativity and is surjective, then the unit element is multiplicative. We describe an internet-based collaborative environment that protects geographically dispersed organizations of a critical infrastructure (e.g., financial institutions, telco providers) from coordinated cyber attacks. A specific instance of a collaborative environment for detecting malicious inter-domain port scans is introduced. This instance uses the open source Complex Event Processing (CEP) engine ESPER to correlate massive amounts of network traffic data exhibiting the evidence of those scans. The paper presents two inter-domain SYN port scan detection algorithms we designed, implemented in ESPER, and deployed on the collaborative environment; namely, Rank-based SYN (R-SYN) and Line Fitting. The paper shows the usefulness of the collaboration in terms of detection accuracy. Finally, it shows how Line Fitting can both achieve a higher detection accuracy with a smaller number of participants than R-SYN, and exhibit better detection latencies than R-SYN in the presence of low link bandwidths (i.e., less than 3Mbit/s) connecting the organizations to Esper.

The Dedekind function \(\Psi(n)=n\prod_{p| n}(1+1/p)\), where \(p\) denotes a prime, satisfies the asymptotic formula \[ \sum_{n\leq x}{n\over\Psi(n)}=\alpha x+E(x) \] where \(\alpha\) is a constant, and \(E(x)\ll(\log x)^{2/3}(\log\log x)^{4/3}\); see \textit{U. Balakrishnan} and \textit{Y.-F.~S. Pétermann} [Acta Arith. 75, 39--69 (1996; Zbl 0846.11054)]. The author shows that \[ \int_1^xE^2(t)\,dt= cx+O(x\varepsilon(x)), \] where \[ c={1\over12}\prod_p\Biggl(1-\biggl({1\over p^2}+{2\over p^3}\biggr)\biggl(1+{1\over p}\biggr)^{-2}\Biggr), \] and \(\varepsilon(x)= \exp(-A(\log x)^{3/5}(\log\log x)^{-1/5})\), with a certain positive constant~\(A\). The essentially analytic argument makes use of the zero-free region of the Riemann zeta-function. This paper explores the asymptotic formula for \(\sum_{n \leq x} a(n)\), for a class of arithmetic functions given as coefficients of the Dirichlet series in the title. Examples include \((\sigma (n)/n)^k\), \((\varphi (n)/n)^k\) and \((\sigma(n)/ \varphi (n))^k\), for any real \(k\). Both \(O\)- and \(\Omega\)-estimates for the error term are considered. The results obtained unify, and in many cases improve upon, those previously known.

The Dedekind function \(\Psi(n)=n\prod_{p| n}(1+1/p)\), where \(p\) denotes a prime, satisfies the asymptotic formula \[ \sum_{n\leq x}{n\over\Psi(n)}=\alpha x+E(x) \] where \(\alpha\) is a constant, and \(E(x)\ll(\log x)^{2/3}(\log\log x)^{4/3}\); see \textit{U. Balakrishnan} and \textit{Y.-F.~S. Pétermann} [Acta Arith. 75, 39--69 (1996; Zbl 0846.11054)]. The author shows that \[ \int_1^xE^2(t)\,dt= cx+O(x\varepsilon(x)), \] where \[ c={1\over12}\prod_p\Biggl(1-\biggl({1\over p^2}+{2\over p^3}\biggr)\biggl(1+{1\over p}\biggr)^{-2}\Biggr), \] and \(\varepsilon(x)= \exp(-A(\log x)^{3/5}(\log\log x)^{-1/5})\), with a certain positive constant~\(A\). The essentially analytic argument makes use of the zero-free region of the Riemann zeta-function. Several notions of sequential directional derivatives and sequential local approximations are introduced. Under (first-order) Hadamard differentiability assumptions of the data at the point of study, these concepts are utilized to analyze second-order necessary optimality conditions, which rely on given sequences, for local weak solutions in nonsmooth vector optimization problems with constraints. Some applications to minimax programming problems are also derived.

This book is devoted to deterministic methods for constrained global optimization - a subject which began to be studied more than two decades ago and for various reasons has received an intensive development in recent years (another book on this subject which has appeared recently is by \textit{P. M. Pardalos} and \textit{J. B. Rosen}, ``Constrained global optimization: Algorithms and applications'' (1987; Zbl 0638.90064)). The general problem discussed here is max\(\{\) f(x)\(|\) \(x\in X\), \(g_ i(x)\geq 0\) \((i=1,...,m)\}\), where X is a nonempty set in \(R^ n\), \(f,g_ i:\) \(X\to R\), such that a local maximum may not be a global one (so that conventional methods of local optimization cannot be used for locating a global maximum). The book consists of 10 chapters. The first three chapters deal with general questions such as Kuhn-Tucker-Lagrange optimality conditions and nonlinear duality (Chap. 1), convex and concave envelopes of functions (Chap. 2), and the basic approaches to solving nonconvex programming problems: direct and implicit enumeration, branch and bound, cuts (Chap. 3). The next four chapters discuss methods for solving the basic classes of constrained global optimization: maximizing a convex function over a polytope (Chap. 4), maximizing a convex function under reverse convex constraints (Chap. 5), continuous nonconvex programs (Chap. 6), nonconvex quadratic programming (Chap. 7), and the fixed charge problem (Chap. 8). The last two chapters are devoted to the reduction of constrained problems to unconstrained ones (Chap. 9) and the decomposition of nonconvex programs by Benders-Geoffrion's method (Chap. 10). By its systematic presentation and its well conceived structure the book may serve as a useful introduction to methods of nonconvex global optimization. Many results which were previously scattered in the journals are here given in a convenient form for the graduate student or the researcher. Also the book contains several unpublished results. However, the computational aspects of the algorithms are neglected; besides, only results published up to 1983 (in several parts, up to 1980) are discussed, so that the book is not an up-to-date account of the current state of research in nonconvex global optimization (this is particularly visible from the presentation of section 4.7 on Tuy-Zwart's method, Chap. 5 on problems with convex functions in the constraints and Chap. 7 on nonconvex quadratic programming). The book is divided into ten chapters. Chapter One, entitled Convex Sets and Functions, is devoted to mathematical preliminaries. Chapter Two has two main sections, one devoted to Kuhn-Tucker conditions, the other one to convex quadratic problems solvable in polynomial time. Algorithms based on Kuhn-Tucker conditions, and the approaches proposed by Shor, Khachiyan and Karmarkar are mentioned. Chapter Three deals with combinatorial optimization problems which can be formulated as nonconvex quadratic problems. The topics include linear and quadratic 0-1 programming, the quadratic and the 3-dimensional assignment problems, bilinear programming and the linear complementarity problems. Chapter Four ``Enumerative methods in Nonconvex Programming'', deals in the global concave minimization by ranking the extreme points, with the construction of linear underestimating functions, presents an algorithm (proposed by Manas) for the indefinite quadratic problem and mentions an algorithm by Zangwill for the concave cost network problem. Chapter Five is devoted to the cutting plane methods and Chapter Six to Branch and Bound methods. Chapter Seven is devoted to bilinear programming for nonconvex (and convex) quadratic problems. Chapter Eight is devoted to large scale problems with linear constraints and a quadratic objective function. Chapters Nine and Ten deal with the methods and the test problems for global indefinite quadratic programming problems. Each chapter contains some exercises and a substantial list of references; in addition to that, a bibliography of references for constrained global optimization is presented at the end of the book (237 titles).

This book is devoted to deterministic methods for constrained global optimization - a subject which began to be studied more than two decades ago and for various reasons has received an intensive development in recent years (another book on this subject which has appeared recently is by \textit{P. M. Pardalos} and \textit{J. B. Rosen}, ``Constrained global optimization: Algorithms and applications'' (1987; Zbl 0638.90064)). The general problem discussed here is max\(\{\) f(x)\(|\) \(x\in X\), \(g_ i(x)\geq 0\) \((i=1,...,m)\}\), where X is a nonempty set in \(R^ n\), \(f,g_ i:\) \(X\to R\), such that a local maximum may not be a global one (so that conventional methods of local optimization cannot be used for locating a global maximum). The book consists of 10 chapters. The first three chapters deal with general questions such as Kuhn-Tucker-Lagrange optimality conditions and nonlinear duality (Chap. 1), convex and concave envelopes of functions (Chap. 2), and the basic approaches to solving nonconvex programming problems: direct and implicit enumeration, branch and bound, cuts (Chap. 3). The next four chapters discuss methods for solving the basic classes of constrained global optimization: maximizing a convex function over a polytope (Chap. 4), maximizing a convex function under reverse convex constraints (Chap. 5), continuous nonconvex programs (Chap. 6), nonconvex quadratic programming (Chap. 7), and the fixed charge problem (Chap. 8). The last two chapters are devoted to the reduction of constrained problems to unconstrained ones (Chap. 9) and the decomposition of nonconvex programs by Benders-Geoffrion's method (Chap. 10). By its systematic presentation and its well conceived structure the book may serve as a useful introduction to methods of nonconvex global optimization. Many results which were previously scattered in the journals are here given in a convenient form for the graduate student or the researcher. Also the book contains several unpublished results. However, the computational aspects of the algorithms are neglected; besides, only results published up to 1983 (in several parts, up to 1980) are discussed, so that the book is not an up-to-date account of the current state of research in nonconvex global optimization (this is particularly visible from the presentation of section 4.7 on Tuy-Zwart's method, Chap. 5 on problems with convex functions in the constraints and Chap. 7 on nonconvex quadratic programming). The sparse nonlinear programming (SNP) is to minimize a general continuously differentiable function subject to sparsity, nonlinear equality and inequality constraints. We first define two restricted constraint qualifications and show how these constraint qualifications can be applied to obtain the decomposition properties of the Fréchet, Mordukhovich and Clarke normal cones to the sparsity constrained feasible set. Based on the decomposition properties of the normal cones, we then present and analyze three classes of Karush-Kuhn-Tucker (KKT) conditions for the SNP. At last, we establish the second-order necessary optimality condition and sufficient optimality condition for the SNP.

Let \(X\) be a non-empty set. A filter \(\mathcal{Q}\) consisting of reflexive relations on \(X\) is a quasi-uniformity on \(X\) if, for every \(Q\in\mathcal{Q}\), there is \(P\in \mathcal{Q}\) such that \(P\circ P\subseteq Q\). In this case, the pair \((X,\mathcal{Q})\) is called a quasi-uniform space. As shown, for example, by \textit{H.-P. A. Künzi} and \textit{C. Ryser} [Topol. Proc. 20, 161--183 (1995; Zbl 0876.54022)], a quasi-uniformity \(\mathcal{Q}\) on \(X\) induces on the family \(\mathcal{P}_0(X)\) of all non-empty subsets of \(X\) the quasi-uniformities \(\mathcal{Q}^+\), \(\mathcal{Q}^-\) and \(\mathcal{Q}^*\), called the upper Hausdorff quasi-uniformity, the lower Hausdorff quasi-uniformity, respectively, the Hausdorff (or Bourbaki) quasi-uniformity on \(\mathcal{P}_0(X)\) associated with \(\mathcal{Q}\). The authors investigate different properties of these three types of quasi-uniformities on \(\mathcal{P}_0(X)\), including the cases when \(X\) is a monoid, respectively, a conoid. One of the main results states that, if \((X,\mathcal{Q})\) is a quasi-uniform monoid, then so are \((\mathcal{P}_0(X),\mathcal{Q}^+)\), \((\mathcal{P}_0(X),\mathcal{Q}^-)\) and \((\mathcal{P}_0(X),\mathcal{Q}^*)\). As a consequence, one obtains that, in case \((X,\mathcal{Q})\) is a quasi-uniform conoid and \(\mathcal{P}_c(X)\) stands for the family of all non-empty convex subsets of \(X\), then \(\mathcal{Q}^+\), \(\mathcal{Q}^-\) and \(\mathcal{Q}^*\) induce quasi-uniformities on the conoid \(\mathcal{P}_c(X)\), denoted by \(\mathcal{Q}^+_c\), \(\mathcal{Q}^-_c\) and \(\mathcal{Q}^*_c\), respectively. Further results of the paper include properties of these three quasi-uniformities on \(\mathcal{P}_c(X)\). Different continuity properties of the scalar multiplication of the conoid \(\mathcal{P}_c(X)\) are also investigated. This paper initiates the systematic study of the preservation of quasi-uniform properties between a quasi-uniformity \({\mathcal U}\) on a set \(X\) and the Bourbaki quasi-uniformity \({\mathcal U}_*\) on the collection \({\mathcal P}_0 (X)\) of all nonempty subsets of \(X\). The authors prove that \(({\mathcal P}_0 (X), {\mathcal U}_*)\) is precompact (totally bounded) if, and only if, \((X, {\mathcal U})\) is precompact (totally bounded), and they give examples to show that the corresponding results hold neither for compactness nor hereditary precompactness. The principal result is an extension of the Isbell-Burdick Theorem: The Bourbaki quasi-uniformity \({\mathcal U}_*\) is right K-complete if, and only if, each stable filter on \((X, {\mathcal U})\) has a cluster point. As might be expected, along the way the authors provide a good many interesting results and examples concerning both right K-completeness and the related property that each stable filter has a cluster point.

Let \(X\) be a non-empty set. A filter \(\mathcal{Q}\) consisting of reflexive relations on \(X\) is a quasi-uniformity on \(X\) if, for every \(Q\in\mathcal{Q}\), there is \(P\in \mathcal{Q}\) such that \(P\circ P\subseteq Q\). In this case, the pair \((X,\mathcal{Q})\) is called a quasi-uniform space. As shown, for example, by \textit{H.-P. A. Künzi} and \textit{C. Ryser} [Topol. Proc. 20, 161--183 (1995; Zbl 0876.54022)], a quasi-uniformity \(\mathcal{Q}\) on \(X\) induces on the family \(\mathcal{P}_0(X)\) of all non-empty subsets of \(X\) the quasi-uniformities \(\mathcal{Q}^+\), \(\mathcal{Q}^-\) and \(\mathcal{Q}^*\), called the upper Hausdorff quasi-uniformity, the lower Hausdorff quasi-uniformity, respectively, the Hausdorff (or Bourbaki) quasi-uniformity on \(\mathcal{P}_0(X)\) associated with \(\mathcal{Q}\). The authors investigate different properties of these three types of quasi-uniformities on \(\mathcal{P}_0(X)\), including the cases when \(X\) is a monoid, respectively, a conoid. One of the main results states that, if \((X,\mathcal{Q})\) is a quasi-uniform monoid, then so are \((\mathcal{P}_0(X),\mathcal{Q}^+)\), \((\mathcal{P}_0(X),\mathcal{Q}^-)\) and \((\mathcal{P}_0(X),\mathcal{Q}^*)\). As a consequence, one obtains that, in case \((X,\mathcal{Q})\) is a quasi-uniform conoid and \(\mathcal{P}_c(X)\) stands for the family of all non-empty convex subsets of \(X\), then \(\mathcal{Q}^+\), \(\mathcal{Q}^-\) and \(\mathcal{Q}^*\) induce quasi-uniformities on the conoid \(\mathcal{P}_c(X)\), denoted by \(\mathcal{Q}^+_c\), \(\mathcal{Q}^-_c\) and \(\mathcal{Q}^*_c\), respectively. Further results of the paper include properties of these three quasi-uniformities on \(\mathcal{P}_c(X)\). Different continuity properties of the scalar multiplication of the conoid \(\mathcal{P}_c(X)\) are also investigated. This book is a student reference, review, supplemental learning, and example handbook (SRRSLEH) that mirrors the content of the author's popular textbook [Discrete mathematics with ducks. Boca Raton, FL: CRC Press (2012; Zbl 1250.05001)]. This handbook provides a review of key material, illustrative examples, and new problems with accompanying solutions that are helpful even for those using a traditional discrete mathematics textbook. Every chapter in SRRSLEH matches the corresponding chapter of DMwD. Chapters in SRRSLEH contain the following: -- a list of the notation introduced in the corresponding chapter, -- a list of definitions that students need to know from the corresponding chapter, -- theorems/facts of note appearing in the corresponding chapter, -- a list of proof techniques introduced, with templates and/or examples given for each one, -- a selection of examples from DMwD, written out formally and briefly rather than colloquially as in DMwD. A quick refresher for any discrete math student, this handbook enables students to find information easily and reminds them of the terms and results they should know during their course.

Grundlegend in der Theorie \(\omega\)-periodischer Lösungen des Differentialgleichungssystems (*) \(\dot x=f(t,x)\) (t\(\in {\mathbb R}\), \(x\in {\mathbb R}^ n)\) ist bekanntlich der Poincaré-Operator \(T_{\omega}(\xi)=\phi (\tau +\omega;\tau,\xi)\), der durch die Lösung \(\phi =\phi (t;\tau,\xi)\) von (*) mit \(\phi (\tau)=\xi\) erzeugt wird; so geben z.B. alle Fixpunkte \(\xi_{\omega}\) von \(T_{\omega}\) Anlaß zu \(\omega\)-periodischen Lösungen von (*) mit Anfangswert \(\xi_{\omega}\). Als hilfreich bei der Untersuchung des Operators \(T_{\omega}\) hat sich nun die ``Spiegelungsabbildung'' \(F(t,\xi)=\phi (- t;t,\xi)\) (t\(\in {\mathbb R}\), \(\xi \in {\mathbb R}^ n)\) erwiesen, die vom Autor im vorliegenden Büchlein systematisch untersucht wird. Die Kapitelüberschriften: Periodenverschiebungen; Definition und Eigenschaften der Spiegelungs\-abbildung; Konstruktion von Systemen aufgrund gegebener Spiegelungsabbildung; Systeme mit gleicher Spiegelungsabbildung; Spiegelungsabbildungen linearer Systeme; periodische Lösungen von Riccati-Gleichungen; Gleichungen mit linearer Spiegelungsabbildung; Einige weitere Eigenschaften der Spiegelungs\-abbildung; Spiegelungsabbildungen für Systeme mit kleinen Parametern. Das Buch ist interessant und klar geschrieben und stellt eine wertvolle Bereicherung jedes Kurses über die qualitative Theorie gewöhnlicher Differentialgleichungen dar, nicht zuletzt wegen der darin enthaltenen Übungsaufgaben. Eine erweiterte Fassung wurde 1986 publiziert: Minsk: Izdatel'stvo ``Universitetskoe'' vgl. das Referat im Zbl 0607.34038. Das vorliegende Büchlein ist im wesentlichen die gedruckte Version des im vorhergehenden Referat besprochenen getippten Vorlesungsskripts des Autors [Gomel': Gomel. Gos. Univ. (1985; Zbl 0607.34037)]. Zwei Paragraphen am Schluß wurden hinzugefügt, manche Übungs\-aufgaben abgeändert. Wo bekommt man schon bei uns ein solches Büchlein für weniger als einen Dollar?

Grundlegend in der Theorie \(\omega\)-periodischer Lösungen des Differentialgleichungssystems (*) \(\dot x=f(t,x)\) (t\(\in {\mathbb R}\), \(x\in {\mathbb R}^ n)\) ist bekanntlich der Poincaré-Operator \(T_{\omega}(\xi)=\phi (\tau +\omega;\tau,\xi)\), der durch die Lösung \(\phi =\phi (t;\tau,\xi)\) von (*) mit \(\phi (\tau)=\xi\) erzeugt wird; so geben z.B. alle Fixpunkte \(\xi_{\omega}\) von \(T_{\omega}\) Anlaß zu \(\omega\)-periodischen Lösungen von (*) mit Anfangswert \(\xi_{\omega}\). Als hilfreich bei der Untersuchung des Operators \(T_{\omega}\) hat sich nun die ``Spiegelungsabbildung'' \(F(t,\xi)=\phi (- t;t,\xi)\) (t\(\in {\mathbb R}\), \(\xi \in {\mathbb R}^ n)\) erwiesen, die vom Autor im vorliegenden Büchlein systematisch untersucht wird. Die Kapitelüberschriften: Periodenverschiebungen; Definition und Eigenschaften der Spiegelungs\-abbildung; Konstruktion von Systemen aufgrund gegebener Spiegelungsabbildung; Systeme mit gleicher Spiegelungsabbildung; Spiegelungsabbildungen linearer Systeme; periodische Lösungen von Riccati-Gleichungen; Gleichungen mit linearer Spiegelungsabbildung; Einige weitere Eigenschaften der Spiegelungs\-abbildung; Spiegelungsabbildungen für Systeme mit kleinen Parametern. Das Buch ist interessant und klar geschrieben und stellt eine wertvolle Bereicherung jedes Kurses über die qualitative Theorie gewöhnlicher Differentialgleichungen dar, nicht zuletzt wegen der darin enthaltenen Übungsaufgaben. Eine erweiterte Fassung wurde 1986 publiziert: Minsk: Izdatel'stvo ``Universitetskoe'' vgl. das Referat im Zbl 0607.34038. Let \(0<a\leq\infty \) and \(\Gamma:[0,a)\times R^ n\to Comp R^ n\) be a Caratheodory-type multifunction. The differential inclusion x'\(\in\Gamma (t,x)\) is said to have asymptotic equilibrium on the set \(A\subset [0,a)\times R^ n\) iff 1) for every point \((t_ 0,x_ 0)\in A\) the inclusion has the solution \(x(\cdot)\) such that \(x(t_ 0)=x_ 0\) and \(\lim_{t\to a}x(t)\) exists and 2) for every \((t_ 0,z)\in A\) there exists the solution \(x(\cdot)\) with \(\lim_{t\to a}x(t)=z\). It is proved that if \(\Gamma\) satisfies some estimate then the inclusion above has the asymptotic equilibrium on the set \(A=\{(t,x):| x|\leq r(t)\}\), where \(r(\cdot)\) is some increasing continuous function. Let \(\Delta\) (\(\cdot)\) be a matrix and \(\beta\) (\(\cdot)\) a positive scalar function of t. Using the asymptotic equilibrium on A sufficient conditions are found for the system (1) \(y'=B(t)y\), \(y(t_ 0)=y_ 0\), \((t_ 0,y_ 0)\in A\) and the perturbed system (2) \(x'\in B(t)x+\Gamma (t,x)\), \(x(t_ 0)=x_ 0\), \((t_ 0,x_ 0)\in A\) to be (\(\Delta\),\(\beta)\)-asymptotically equivalent on A in the sense of the equality \(\lim_{t\to a}|\Delta (t)(x(t)-y(t))| /\beta (t)=0.\) Coincidence of the globally attractive sets of (1) and (2) is obtained as a corollary.

The following properties are defined. Desargues property (DP). Any pair of non-collinear points has exactly four points collinear to both. Reye's property (RP). Any pair of non-collinear points has more than four points collinear to both. The authors prove from a geometrical point of view that if a partial linear space \(P\) of order two satisfies DP than \(P\) is isomorphic to \({\mathcal T}(\Omega)\), the geometry of 2-sets in a set \(\Omega\). They follow an algebraic approach. By a method of \textit{R. Brown} and \textit{S. P. Humphries} [Proc. Lond. Math. Soc., III. Ser. 52, 517-556 (1986; Zbl 0604.51008)] they define a quadratic form \(q\) on a vector space \(U\) of the universal representation of the space \(P\) and consider the quadratic space \(O(q)= \{a\in U\mid a\notin \text{Rad}(U)\), \(q(a)=1\}\). Then they prove that any connected reduced dual affine space with RP is a quadratic space. In addition they prove that for spaces \(P\) with DP and with \(|P|\) infinite there is only one symplectic representation. Es sei V ein endlichdimensionaler symplektischer Vektorraum über einem Körper K, so daß bezüglich der alternierenden Bilinearform f: (x,y)\(\to f(x,y)\) der Vektorraum V nicht total isotrop ist, d. h. sein Radikal rad (V) verschieden von V ausfällt. Mit \(Sp_ 0(V)\) sei die Gruppe derjenigen Isometrien von V bezeichnet, die die Identität auf rad (V) induzieren: Spezielle Beispiele solcher Isometrien sind die Transvektionen \(T^ k_ a: x\to x+kf(a,x)a,\) die mittels \(a\in V\) und \(k\in K\) definiert werden. Ist S eine Teilmenge von V, so sei Tv(S) diejenige Untergruppe von \(Sp_ 0(V)\), die durch die Transvektionen \(T^ k_ a\) für alle \(a\in S\) and \(k\in K\) erzeugt wird. Jeder Teilmenge S von V kann man einen Graphen G(S) zuweisen, dessen Ecken die Elemente von S sind; die Ecken a und \(b\in G(S)\) sind genau dann durch eine Kante verbunden, wenn f(a,b)\(\neq 0\) gilt. Die Verfasser zeigen, daß für eine Teilmenge S von \(V\setminus rad (V)\) die folgenden Bedingungen äquivalent sind: (i) Die Gruppe Tv(S) wirkt transitiv auf \(V\setminus rad V.\) (ii) Es ist \(Tv(S)=Sp_ 0(V)\). Aus jeder dieser beiden Bedingungen folgt stets die folgende Aussage (iii): S erzeugt V und der Graph G(S) ist zusammenhängend. Ein großer Teil der ersten Arbeit ist dem Nachweis gewidmet, daß aus (iii) sowohl (i) als auch (ii) folgt, sofern nur K verschieden von GF(2) ist. Für \(K=GF(2)\) ist dies nicht wahr und die Situation, der dann die zweite Arbeit gewidmet ist, ist wesentlich komplizierter. Immerhin bestimmt für \(K=GF(2)\) der Graph G(S), wobei S eine Basis von V ist, die Form f. Methodisch spielt in den beiden Arbeiten die t-Äquivalenz zwischen Teilmengen von V eine wichtige Rolle. Sie ist die gröbste Äquivalenzrelation, die von sogenannten elementaren t-Äquivalenzen erzeugt wird; dabei ist eine elementare t-Äquivalenz eine Surjektion s von \(S\subseteq V\) auf S'\(\subset V\), zu der Elemente a,b\(\in S\) und \(k\in K\) existieren, so daß \(s(x)=x\) für \(x\neq b\) und \(s(b)=T^ k_ a(b)\) gilt. Eine Basis P von V heißt vom orthogonalen Typ, wenn der Graph G(P) ein Baum ist und als Untergraphen das Diagramm der Liealgebra \(E_ 6\) enthält. Ist \(K=GF(2)\), so läßt sich zur symplektischen Form f und einer Basis P von V eine quadratische Form \(Q_ P\) assoziieren, so daß \(Q_ P(a)=1\) für alle \(a\in P\) gilt. Die Verfasser zeigen für V über GF(2) und eine Basis P vom orthogonalen Typ, daß zwei Elemente \(x,y\in V\setminus rad(V)\) genau dann zur selben Bahn von Tv(P) gehören, wenn \(Q_ P(x)=Q_ P(y)\) gilt. Ist P eine Basis von V über GF(2), die nicht vom orthogonalen Typ ist, so ist Tv(P) Erweiterung einer elementar abelschen 2-Gruppe durch eine symmetrische Gruppe; zu einer symmetrischen Gruppe ist Tv(P) genau dann isomorph, wenn P zu einer Teilmenge S t-äquivalent ist, so daß der Graph G(S) das Diagramm einer Liealgebra \(A_ n\) darstellt. Für den symplektischen Raum V über GF(2) gelte \(\dim [V/rad(V)]\geq 6;\) außerdem sei \(S=P\cup R\) eine Teilmenge von \(V\setminus rad(V),\) so daß P eine Basis bildet. Dann ist \(Tv(S)=Sp_ 0(V)\) genau dann, wenn die folgenden Bedingungen erfüllt sind: (a) Der Graph G(S) ist zusammenhängend und vom orthogonalen Typ. (b) Es gibt ein \(a\in R\) mit \(Q_ P(a)=0\).

The following properties are defined. Desargues property (DP). Any pair of non-collinear points has exactly four points collinear to both. Reye's property (RP). Any pair of non-collinear points has more than four points collinear to both. The authors prove from a geometrical point of view that if a partial linear space \(P\) of order two satisfies DP than \(P\) is isomorphic to \({\mathcal T}(\Omega)\), the geometry of 2-sets in a set \(\Omega\). They follow an algebraic approach. By a method of \textit{R. Brown} and \textit{S. P. Humphries} [Proc. Lond. Math. Soc., III. Ser. 52, 517-556 (1986; Zbl 0604.51008)] they define a quadratic form \(q\) on a vector space \(U\) of the universal representation of the space \(P\) and consider the quadratic space \(O(q)= \{a\in U\mid a\notin \text{Rad}(U)\), \(q(a)=1\}\). Then they prove that any connected reduced dual affine space with RP is a quadratic space. In addition they prove that for spaces \(P\) with DP and with \(|P|\) infinite there is only one symplectic representation. This paper discusses the stability of symplectic algorithms. There are several conclusions. First, symplectic Runge-Kutta methods and symplectic one-step methods with high-order derivative are unconditionally critically stable for Hamiltonian systems. However, only some of them are \(A\)-stable for non-Hamiltonian systems. Secondly, the hopscotch schemes are conditionally critically stable for Hamiltonian systems but their stability regions are only a segment of the imaginary axis for non- Hamiltonian systems. Finally, all linear symplectic multistep methods are conditionally critically stable except the trapezoid formula which is critically stable for Hamiltonian systems.