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For a fixed \(M>0\), the Barban-Davenport-Halberstam theorem says that \[ \sum_{q\leq Q}\sum_{\substack{a=1\\ (a,q)=1}}^q\Big(\psi(x;q,a)-\frac{x}{\varphi(q)}\Big)^2\ll xQ\log x \] if \(x(\log x)^{-M}\leq Q\leq x\), with \(\psi(x;q,a)=\sum_{p^m\leq x,~p^m\equiv a\pmod{q}}\log p\) and \(\varphi\) Euler's totient function. The outer sum is over the integers \(q\leq Q\), the inner sum over the primitive residue classes modulo \(q\). Let \(K\) be a number field that is normal over the rationals. The author, using a large sieve inequality, establishes an analogue of the Barban-Davenport-Halberstam theorem, where the sum is not over the primes \(p\) with \(p^m\leq x\), but over the prime ideals with \(m\)th power norm \(\leq x\) and \(\varphi(q)\) is replaced by the degree of the extension \(K(\zeta_q)\) over \(K\). Let \(K\) be a fixed number field, and assume that \(K\) is Galois over \(\mathbb{Q}\). Previously, the author showed [J. Number Theory 129, No. 11, 2735--2742 (2009; Zbl 1235.11090)] that when estimating the number of prime ideals with norm congruent to \(a\) modulo \(q\) via, the Chebotarëv Density Theorem, the mean square error in the approximation is small when averaging over all \(q\leq Q\) and all appropriate \(a\). In this article, the author replaces upper bound by an asymptotic formula. The proof is an adaption of Hooley's method for the classical case \(k=\mathbb{Q}\).

For a fixed \(M>0\), the Barban-Davenport-Halberstam theorem says that \[ \sum_{q\leq Q}\sum_{\substack{a=1\\ (a,q)=1}}^q\Big(\psi(x;q,a)-\frac{x}{\varphi(q)}\Big)^2\ll xQ\log x \] if \(x(\log x)^{-M}\leq Q\leq x\), with \(\psi(x;q,a)=\sum_{p^m\leq x,~p^m\equiv a\pmod{q}}\log p\) and \(\varphi\) Euler's totient function. The outer sum is over the integers \(q\leq Q\), the inner sum over the primitive residue classes modulo \(q\). Let \(K\) be a number field that is normal over the rationals. The author, using a large sieve inequality, establishes an analogue of the Barban-Davenport-Halberstam theorem, where the sum is not over the primes \(p\) with \(p^m\leq x\), but over the prime ideals with \(m\)th power norm \(\leq x\) and \(\varphi(q)\) is replaced by the degree of the extension \(K(\zeta_q)\) over \(K\). A subclass of affine transformations on the set of geometrical images of finite automata is investigated. The results about the characteristics and the form of these transformations are described.

Let \(f:M^n\to \mathbb R^q\) be a self-transverse immersion of a compact closed smooth manifold \(M^n\) into Euclidean space \(\mathbb R^q\). Then a point of \(\mathbb R^q\) is is an \(r\)-fold self-intersection point of \(f\) if it is the image under \(f\) of \(r\) distinct points of \(M^n\). The \(r\)-fold self-intersection point set \(S_r(f)\subset \mathbb R^q\) is an \(r\)-fold point manifold. For example, Boy's surface is a self-transverse immersion of the projective plane \(f:P^2\to \mathbb R^3\), so we have 2-fold and 3-fold self intersection points. If we concentrate on \(S_2(f)\), then we see that the 2-fold manifold \(S_2(f)\) is the image of an immersion of the circle \(S^1\subset P^2\) and \(S_3(f)\) is one point that is the image of an immersion of three points of \(S^1\). In general, the self-transversality of \(f\) implies that the \(r\)-fold manifold \(S_r(f)\subset \mathbb R^q\) is itself the image of an immersion of a compact manifold \(\Delta_r(f)\subset M^n\) of dimension \(rn-(r-1)q\). So self-transverse immersions produce \(r\)-fold point manifolds and recently mathematicians tried to classify immersions up to multiple point manifolds. This paper treats the case of immersions \(f:M^n\to \mathbb R^{2n-5}\), i.e. the case when the double point manifolds are 5-dimensional. Note that the cobordism class of 5-dimensional manifolds is generated by boundaries and the Dold manifold \(V^5\). The paper brings a proof that for \(n\) even the Dold manifold \(V^5\) cannot occur as a double point manifold of any immersion \(f:M^n\to \mathbb R^{2n-5}\), since by general position every immersion is regularly homotopic and so bordant to a self-transverse immersion. The paper [\textit{M. A. Asadi-Golmankhaneh} and \textit{P. J. Eccles}, J. Lond. Math. Soc., II. Ser. 62, No. 1, 278--290 (2000; Zbl 0959.57025)] contains a general approach to these problems, so that the present paper applies it. Let \(f:M^{n-k} \to\mathbb{R}^n\) be a self-transverse immersion of the closed smooth manifold \(M^{n-k}\) into the \(n\)-dimensional Euclidean space. For any positive integer \(r\) the \(r\)-fold intersection set \(I_r(f)\) is defined as the set of elements \(y\in f(M)\), such that there are distinct points \(x_1, \dots, x_r\in M\) and \(f(x_1)= \cdots= f(x_r)=y\). The self-transversality of \(f\) implies that \(I_r(f)\) it itself the image of an immersion of a manifold \(\Delta_r (f)\) of dimension \(n-kr\) in \(\mathbb{R}^n\). The bordism class of \(f\) corresponds to an element \(\alpha\in \pi^s_n MO(k)\). The authors consider the \(\mathbb{Z}_2\)-homology Hurewicz homomorphism \(h:\pi^s_n MO(k)\to H_n(QMO (k); \mathbb{Z}/2)\), where \(QMO(k)\) denotes the direct limit \(\lim\Omega^n \Sigma^n MO (k)\). They show that the manifold \(\Delta_r(f)\) is determined up to bordism by \(h (\alpha)\in H_n(QMO(k); \mathbb{Z}_2)\), and they prove a formula, which determines the normal Stiefel-Whitney classes of \(\Delta_r(f)\) by \(h(\alpha)\).

Let \(f:M^n\to \mathbb R^q\) be a self-transverse immersion of a compact closed smooth manifold \(M^n\) into Euclidean space \(\mathbb R^q\). Then a point of \(\mathbb R^q\) is is an \(r\)-fold self-intersection point of \(f\) if it is the image under \(f\) of \(r\) distinct points of \(M^n\). The \(r\)-fold self-intersection point set \(S_r(f)\subset \mathbb R^q\) is an \(r\)-fold point manifold. For example, Boy's surface is a self-transverse immersion of the projective plane \(f:P^2\to \mathbb R^3\), so we have 2-fold and 3-fold self intersection points. If we concentrate on \(S_2(f)\), then we see that the 2-fold manifold \(S_2(f)\) is the image of an immersion of the circle \(S^1\subset P^2\) and \(S_3(f)\) is one point that is the image of an immersion of three points of \(S^1\). In general, the self-transversality of \(f\) implies that the \(r\)-fold manifold \(S_r(f)\subset \mathbb R^q\) is itself the image of an immersion of a compact manifold \(\Delta_r(f)\subset M^n\) of dimension \(rn-(r-1)q\). So self-transverse immersions produce \(r\)-fold point manifolds and recently mathematicians tried to classify immersions up to multiple point manifolds. This paper treats the case of immersions \(f:M^n\to \mathbb R^{2n-5}\), i.e. the case when the double point manifolds are 5-dimensional. Note that the cobordism class of 5-dimensional manifolds is generated by boundaries and the Dold manifold \(V^5\). The paper brings a proof that for \(n\) even the Dold manifold \(V^5\) cannot occur as a double point manifold of any immersion \(f:M^n\to \mathbb R^{2n-5}\), since by general position every immersion is regularly homotopic and so bordant to a self-transverse immersion. The paper [\textit{M. A. Asadi-Golmankhaneh} and \textit{P. J. Eccles}, J. Lond. Math. Soc., II. Ser. 62, No. 1, 278--290 (2000; Zbl 0959.57025)] contains a general approach to these problems, so that the present paper applies it. We consider a class of singularly perturbed semilinear three-point boundary value problems. An accelerated uniformly convergent numerical method is constructed via the exponential fitted operator method using Richardson extrapolation techniques to solve the problem. To treat the semilinear term, we use quasi-linearization techniques. The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and \(\varepsilon \)-uniformly convergent for \(h \geq \varepsilon \), where the classical numerical methods fail to give a good result. It also improves the results of the methods existing in the literature. The method is shown to be second-order convergent independent of perturbation parameter \(\varepsilon \).

The authors present a new iteration method for systems of linear algebraic equations with a symmetric matrix of order \(m+n\) (\(m\geq n\)) consisting of one positive-definite block of order \(m\) and one zero block of order \(n\) in the main diagonal and of two mutually transposed off diagonal blocks of rank \(n\). This new method, called a generalized successive overrelaxation (SOR)-like method using Chebyshev semi-iteration (GSOR SI), is motivated as an improvement of the SOR-like method from \textit{G. H. Golub, X. Wu} and \textit{J.-Y. Yuan} [BIT 41, 71--85 (2001; Zbl 0991.65036)]. Convergence of the GSOR SI is proved and the asymptotic rate of convergence of the GSOR SI is shown to be greater than that of the SOR-like method under some assumptions concerning the preconditioner and the value of relaxation parameter. Two concrete examples illustrate that indeed the GSOR SI is more effective than the SOR-like method. The authors propose several successive overrelaxation (SOR)-like algorithms for solving augmented systems. A new approach is considered for solving the system arising from the finite element approximation to the Stokes equation. The null block in the coefficient matrix is replaced by some nonsingular matrix \(Q\) to get a special splitting. Application of the SOR method to such a splitting is called a SOR-like method. The optimal parameter for the SOR-like method and the behavior of its spectral radius are studied. For the special choise of the matrix \(Q\) some convergence results are given. Comparisons of the convergence rates and implementation of the proposed three algorithms are provided. Results from numerical experiments (flow in the unit square domain) are reported and discussed.

The authors present a new iteration method for systems of linear algebraic equations with a symmetric matrix of order \(m+n\) (\(m\geq n\)) consisting of one positive-definite block of order \(m\) and one zero block of order \(n\) in the main diagonal and of two mutually transposed off diagonal blocks of rank \(n\). This new method, called a generalized successive overrelaxation (SOR)-like method using Chebyshev semi-iteration (GSOR SI), is motivated as an improvement of the SOR-like method from \textit{G. H. Golub, X. Wu} and \textit{J.-Y. Yuan} [BIT 41, 71--85 (2001; Zbl 0991.65036)]. Convergence of the GSOR SI is proved and the asymptotic rate of convergence of the GSOR SI is shown to be greater than that of the SOR-like method under some assumptions concerning the preconditioner and the value of relaxation parameter. Two concrete examples illustrate that indeed the GSOR SI is more effective than the SOR-like method. Siehe Verf., Sc. Rep. Tôhoku Univ. I. 25 (1936), 257-280 (F. d. M. \(62_{\text{I}}\), 445).

Let \(\mathcal A\) be a pseudo-abelian tensor category. Any partition \(\lambda\) of an integer \(n\) gives a central idempotent \(d_{\lambda}\) in the group algebra \(Q\Sigma_n.\) Let \(S_{\lambda}(A)=d_{\lambda}(A^{\otimes n}).\) An object \(A\) of \(\mathcal A\) is called Schur-finite if there exists a partition \(\lambda\) such that \(S_{\lambda}(A)=0.\) An object \(A\) is called odd (resp. even) if \(S_{\lambda}(A)=0\) for \({\lambda}=(n)\) (resp. \({\lambda}=(1^n)\)). The object is called Kimura-finite if \(A=A_+ \oplus A_-\) with \(A_+\) even and \(A_-\) odd. Let \({\mathcal N}(A)\) denote the ideal of numerically trivial endomorphisms of an object \(A.\) It is known that \({\mathcal N}(A)\) is nilpotent if \(A\) is a Kimura-finite object [\textit{S.-I Kimura}, Math. Ann. 331, No. 1, 173--201 (2005; Zbl 1067.14006)]. Although the corresponding result is not true in general for Schur-finite objects, the authors prove that under some additional hypotheses on \(A\) (such objects are called special Schur-finite objects) it is true. As a consequence they prove that in the category of Chow motives, the Kimura-finitness, special Schur-finitness and the nilpotency of \(\text{CH}^{\text{ni}}(X^i\times X^i)_{\text{num}}\) for all \(i\) are all equivalent if \(X\) is a smooth projective variety satisfying the homological sign conjecture. Let \(A_{*}(X)\) denote the Chow group of \(*\)-dimensional cycles of an algebraic variety \(X\) with rational coefficients and \(A_{*}(X)_{\mathbb Z}\) be the corresponding Chow group with integer coefficients. For a curve \(C\), the degree \(0\) part of \(A_{0}(C)_{\mathbb Z}\) is represented by the Jacobian variety which is a finite dimensional object. This is no longer the case for the surfaces. \textit{D. Mumford} [J. Math. Kyoto Univ. 9, 195--204 (1968; Zbl 0184.46603)] proved that if \(S\) is a surface with \(p_{g}(S)>0\) then the group \(A_{0}(S)\) cannot be represented by a finite dimensional variety. The author suggests a new definition of ``finite dimensionality'' of Chow groups. Let \(K\) denote the kernel of the Albanese map \(\text{alb}: A_{0}(S)\rightarrow \mathbb Z\oplus \text{Alb}(S).\) Let \(S\) be the product of two smooth nonrational curves. Although \(K\) is infinite dimensional in the sense of Mumford the author proves the following Theorem 1.2. Assume that the base field is algebraically closed. Let \(c_{1},\dots c_{n}\) be cycles in the Albanese kernel in \(A_{0}(X),\) where \(X\) is a product of two curves. When \(n\) is larger than \(4p_{g}(x)\), then \(c_{1}\wedge\cdots\wedge c_{n}=0.\) Here \(c_{1}\wedge\cdots\wedge c_{n}=\sum_{\sigma\in S_n}\frac{{\text{sgn}}(\sigma)}{n!} c_{\sigma (1)}\times \dots \times c_{\sigma (n)}\in A_{*}(X\times \cdots \times X ).\) The theorem shows that there are finitely many ``linearly independent'' cycles in \(K.\) For a curve \(C\) the author proves the following theorems. Theorem 8.5. \noindent Let \(C\) be a smooth projective curve. For any \(n>0\), there exist \(0\)-cycles \(c_{1},\dots ,c_{n}\) in \(A_{*}(C)\) such that \(c_i\)'s have degree \(0\) and \(c_{1}\wedge\cdots\wedge c_{n}\neq 0\) Theorem 1.5. Let \(C\) be a smooth projective curve and \(n>2g(C).\) If \(c_{1},\dots ,c_{n}\in A_0 (C)\) are \(0\)-cycles with degree \(0\) then \(\text{Sym}( c_{1},\dots ,c_{n})=0.\) \noindent For algebraic cycles \(c_{1},\dots ,c_{n}\in A_* (X)\), \(\text{Sym}( c_{1},\dots ,c_{n})= \sum_{\sigma\in S_n}\frac{1}{n!} c_{\sigma (1)}\times \dots \times c_{\sigma (n)}\) holds.

Let \(\mathcal A\) be a pseudo-abelian tensor category. Any partition \(\lambda\) of an integer \(n\) gives a central idempotent \(d_{\lambda}\) in the group algebra \(Q\Sigma_n.\) Let \(S_{\lambda}(A)=d_{\lambda}(A^{\otimes n}).\) An object \(A\) of \(\mathcal A\) is called Schur-finite if there exists a partition \(\lambda\) such that \(S_{\lambda}(A)=0.\) An object \(A\) is called odd (resp. even) if \(S_{\lambda}(A)=0\) for \({\lambda}=(n)\) (resp. \({\lambda}=(1^n)\)). The object is called Kimura-finite if \(A=A_+ \oplus A_-\) with \(A_+\) even and \(A_-\) odd. Let \({\mathcal N}(A)\) denote the ideal of numerically trivial endomorphisms of an object \(A.\) It is known that \({\mathcal N}(A)\) is nilpotent if \(A\) is a Kimura-finite object [\textit{S.-I Kimura}, Math. Ann. 331, No. 1, 173--201 (2005; Zbl 1067.14006)]. Although the corresponding result is not true in general for Schur-finite objects, the authors prove that under some additional hypotheses on \(A\) (such objects are called special Schur-finite objects) it is true. As a consequence they prove that in the category of Chow motives, the Kimura-finitness, special Schur-finitness and the nilpotency of \(\text{CH}^{\text{ni}}(X^i\times X^i)_{\text{num}}\) for all \(i\) are all equivalent if \(X\) is a smooth projective variety satisfying the homological sign conjecture. The flow around two tandem circular cylinders was studied by a three-dimensional numerical simulation of the Navier-stokes equations at \(\text{Re}=220\). The improved virtual boundary method was applied to model the no-slip boundary condition of the cylinders. The results show that as the spacing ratio \(L/D\geq 4\), the three dimensionality occurs in the wake. When \(L/D\leq 3.5\) the wake keeps a two-dimensional state at the Reynolds number \(\text{Re}=220\). The critical spacing for the appearance of three-dimensional instability obtained is at the range \(3.5< L/D <4\), similar to the critical spacing found in two-dimensional case. Two sources of instability from upstream and downstream cylinder generate a complicated vortex structures in the wake, investigated by streamlines topology analysis in the streamwise plane. Many other interesting problems were also addressed in this paper.

The paper under review deals with the problem of separability. The authors review the existing tools for validating the assumption of separability for surface data (which is often made in situations where the covariance structure cannot be easily estimated) and, using a setup similar to [\textit{J. A. D. Aston} et al., Ann. Stat. 45, No. 4, 1431--1461 (2017; Zbl 1407.62147)], present an alternative test for the hypothesis of separability in hypersurface data. They derive an explicit expression for the minimal distance between the covariance operator and its approximation by a separable covariance operator (it turns out that this minimum vanishes if and only if the covariance operator is separable) and directly estimate the minimal distance (and not the covariance operator itself) from the available data. As a consequence the calculation of the test statistic does neither use an estimate of the full nonseparable covariance operator nor requires the specification of subspaces used for a projection. The authors also prove the consistency of the proposed estimator of the minimum distance and derive its asymptotic distribution under the null hypothesis \& alternative. These results are used to develop an asymptotic and a bootstrap test for the hypothesis of separability, which are consistent against all alternatives. Next, an investigation of the finite sample properties of the new tests (by means of a simulation study) and a comparison with some existing alternative tests for this problem are conducted. The assumption of separability of the covariance operator for a random image or hypersurface can be of substantial use in applications, especially in situations where the accurate estimation of the full covariance structure is unfeasible, either for computational reasons, or due to a small sample size. However, inferential tools to verify this assumption are somewhat lacking in high-dimensional or functional data analysis settings, where this assumption is most relevant. We propose here to test separability by focusing on \(K\)-dimensional projections of the difference between the covariance operator and a nonparametric separable approximation. The subspace we project onto is one generated by the eigenfunctions of the covariance operator estimated under the separability hypothesis, negating the need to ever estimate the full nonseparable covariance. We show that the rescaled difference of the sample covariance operator with its separable approximation is asymptotically Gaussian. As a by-product of this result, we derive asymptotically pivotal tests under Gaussian assumptions, and propose bootstrap methods for approximating the distribution of the test statistics. We probe the finite sample performance through simulations studies, and present an application to log-spectrogram images from a phonetic linguistics dataset.

The paper under review deals with the problem of separability. The authors review the existing tools for validating the assumption of separability for surface data (which is often made in situations where the covariance structure cannot be easily estimated) and, using a setup similar to [\textit{J. A. D. Aston} et al., Ann. Stat. 45, No. 4, 1431--1461 (2017; Zbl 1407.62147)], present an alternative test for the hypothesis of separability in hypersurface data. They derive an explicit expression for the minimal distance between the covariance operator and its approximation by a separable covariance operator (it turns out that this minimum vanishes if and only if the covariance operator is separable) and directly estimate the minimal distance (and not the covariance operator itself) from the available data. As a consequence the calculation of the test statistic does neither use an estimate of the full nonseparable covariance operator nor requires the specification of subspaces used for a projection. The authors also prove the consistency of the proposed estimator of the minimum distance and derive its asymptotic distribution under the null hypothesis \& alternative. These results are used to develop an asymptotic and a bootstrap test for the hypothesis of separability, which are consistent against all alternatives. Next, an investigation of the finite sample properties of the new tests (by means of a simulation study) and a comparison with some existing alternative tests for this problem are conducted. This paper studies the consensus problem of discrete-time single-integrator multi-agent systems (MASs) with relative-state-dependent (RSD) noises in possibly unbalanced switching networks. By using a time-varying quadratic Lyapunov function, we prove that the mean square and almost sure consensus of the single-integrator MASs with RSD noises can always be achieved by adopting a constant gain protocol if the switching network is uniformly strongly connected. We also give the statistic property of the convergent random variable with respect to the initial state of the system. Simulation results are given to verify the correctness of the consensus conclusion.

The classical theorem by Torelli says that a smooth complex algebraic curve is determined by the isomorphism class of its polarized Jacobian up to isomorphism. In this paper, the authors prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight system is generic. When the genus is at least two, using this result they also prove a Torelli theorem for the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of \textit{J. C. Hurtubise} [Duke Math. J. 83, No. 1, 19--50 (1996; Zbl 0857.58024); erratum ibid. 84, No. 3, 815 (1996)]. The paper is devoted to thorough analysis of algebraically integrable Hamiltonian systems (i.e., to complex integrable systems for which the compactified and desingularized joint level sets of the Hamiltonians are Abelian varieties with the linear structures given by the Hamiltonian [cf. \textit{P. van Moerbeke}, Proc. Symp. Pure Math. 49, Pt. 1, 107-131 (1989; Zbl 0688.70012)]), especially to the particular class of examples determined by some natural systems on the coadjoint orbits of a loop algebra of polynomials in one variable \(\lambda\) with values in a finite-dimensional semisimple Lie algebra [cf., e.g., \textit{M. Adler} and \textit{P. van Moerbeke}, Adv. Math. 38, 267-317 (1980; Zbl 0455.58017) and ibid., 318-379 (1980; Zbl 0455.58010)]. The reduced coadjoint orbit has a description as an open set of the union in the family of Jacobians, corresponding to a family of curves in the \((z,\lambda)\)-plane, and its symplectic geometry is related to that of the plane. The author deals with the following problems: Which integrable systems fit into the loop algebra framework, if not, are there other algebraic surfaces which can be invariantly associated to them, and classification of these surfaces. We cannot refer the deep results here. In rough terms, the vanishing of a certain invariant of the integrable system ensures a surface \(Q\) generalising the \((z,\lambda)\)-plane, in the loop case this \(Q\) is rational, and the classification is made according to whether certain curves \(S_h\) in \(Q\) (determined by the Lagrangian fibration of the integrable system) are hyperelliptic or not. Several examples related to recent work by M. R. Adams, J. Harnad, the author (coadjoint orbits), N.J. Hitchin (moduli spaces of stable \({\mathfrak {gl}}(r)\) pairs), E. K. Sklyanin (unusual brackets), S. Makai (\(K-3\) surfaces), and the genus two case are discussed. [The erratum concerns Proposition 3.5 of the paper].

The classical theorem by Torelli says that a smooth complex algebraic curve is determined by the isomorphism class of its polarized Jacobian up to isomorphism. In this paper, the authors prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight system is generic. When the genus is at least two, using this result they also prove a Torelli theorem for the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of \textit{J. C. Hurtubise} [Duke Math. J. 83, No. 1, 19--50 (1996; Zbl 0857.58024); erratum ibid. 84, No. 3, 815 (1996)]. The aim of this paper is to study the existence of periodic wave solutions of two-dimensional-nonlinear coupled Ginzburg-Landau equations with variable coefficients. The method of the auxiliary function and the homogeneous balance principle are adopted to achieve the aforesaid aim. The periodic wave solutions are obtained.

The idea of ``intuitionistic fuzzy sets'' as introduced by \textit{K. T. Atanassov} [Fuzzy Sets Syst. 20, 87--96 (1986; Zbl 0631.03040)] has been used by the authors for introducing `intuitionistic fuzzy' subgroups of a group \(G'\) (IFS in short) and `intuitionistic \(M\)-fuzzy subgroups' of an \(M\)-group \(G\). Some results relating to the two defined concepts have been developed; a theorem giving a necessary and sufficient condition for an IFS in an \(M\)-group \(G\) to be an intuitionistic \(M\)-fuzzy subgroup of \(G\) has been proved. Lastly, for two \(M\)-groups \(G\) and \(G'\) and an \(M\)-homomorphism \(f\) from \(G\) onto \(G\), it has been shown that \(f\) takes an intuitionistic \(M\)-fuzzy subgroup of \(G\) onto such a subgroup of \(G'\). By an intuitionistic fuzzy set (IFS) the author understands a pair of generalized characteristic functions and hence a generalization of a fuzzy set. These two functions for each point of the universe of discourse describe separately its degrees of membership and of nonmembership with respect to the IFS they characterize. These two degrees not necessarily add up to one. The motivation for introducing IFSs is explained with an example, set algebraic operations are introduced and some of their elementary properties given.

The idea of ``intuitionistic fuzzy sets'' as introduced by \textit{K. T. Atanassov} [Fuzzy Sets Syst. 20, 87--96 (1986; Zbl 0631.03040)] has been used by the authors for introducing `intuitionistic fuzzy' subgroups of a group \(G'\) (IFS in short) and `intuitionistic \(M\)-fuzzy subgroups' of an \(M\)-group \(G\). Some results relating to the two defined concepts have been developed; a theorem giving a necessary and sufficient condition for an IFS in an \(M\)-group \(G\) to be an intuitionistic \(M\)-fuzzy subgroup of \(G\) has been proved. Lastly, for two \(M\)-groups \(G\) and \(G'\) and an \(M\)-homomorphism \(f\) from \(G\) onto \(G\), it has been shown that \(f\) takes an intuitionistic \(M\)-fuzzy subgroup of \(G\) onto such a subgroup of \(G'\). We discuss various questions of the following kind: for a continuous map \(X \to Y\) from a compact metric space to a simplicial complex, can one guarantee the existence of a fiber large in the sense of Urysohn width? The \(d\)-width measures how well a space can be approximated by a \(d\)-dimensional complex. The results of this paper include the following. \begin{itemize} \item[(1)] Any piecewise linear map \(f: [0,1]^{m+2} \to Y^m\) from the unit euclidean \((m+2)\)-cube to an \(m\)-polyhedron must have a fiber of 1-width at least \(\frac{1}{2\beta m +m^2 + m + 1} \), where \(\beta = \sup_{y\in Y} rkH_1(f^{-1}(y))\) measures the topological complexity of the map. \item[(2)] There exists a piecewise smooth map \(X^{3m+1} \to \mathbb{R}^m\), with \(X\) a riemannian \((3m+1)\)-manifold of large \(3m\)-width, and with all fibers being topological \((2m+1)\)-balls of arbitrarily small \((m+1)\)-width. \end{itemize}

A semi-Markov process \({\mathcal J}\) with countable state space is introduced; it is assumed that a reward of \(\rho(i,\tau)\) accrues from remaining in state \(i\) for a length of time \(\tau\), thereby generalizing the standard linear reward structure. Sufficient conditions are given, in terms of \(\rho\) and the sojourn time distributions, for the expected total accrued reward by time \(t\) to be asymptotically linear in \(t\). Renewal theoretic methods are used, together with a formula, derived in the paper, relating the mean first passage times between two states for \({\mathcal J}\) and its dual process. The work continues that of the first author [ibid. 33, No. 4, 1011-1017 (1996; Zbl 0867.60079)]. Based on a semi-Markov process \(J(t)\), \(t\geq 0\), a reward process \(Z(t)\), \(t\geq 0\), is introduced where it is assumed that the reward function, \(\rho(k,x)\), is nonlinear; if the reward function is linear, i.e. \(\rho(k,x)= kx\), the reward process \(Z(t)\), \(t\geq 0\), becomes the classical one, which has been considered by many authors. An explicit formula for \({\mathbf E}(Z(t))\) is given in terms of the moments of the sojourn time distribution at \(t\), when the reward function is a polynomial.

A semi-Markov process \({\mathcal J}\) with countable state space is introduced; it is assumed that a reward of \(\rho(i,\tau)\) accrues from remaining in state \(i\) for a length of time \(\tau\), thereby generalizing the standard linear reward structure. Sufficient conditions are given, in terms of \(\rho\) and the sojourn time distributions, for the expected total accrued reward by time \(t\) to be asymptotically linear in \(t\). Renewal theoretic methods are used, together with a formula, derived in the paper, relating the mean first passage times between two states for \({\mathcal J}\) and its dual process. The work continues that of the first author [ibid. 33, No. 4, 1011-1017 (1996; Zbl 0867.60079)]. In this paper, pinning synchronization on directed network was considered. By analyzing, some general synchronization criteria on directed network were established. And then, we verified it on directed globally coupled network, directed scale-free network, and directed small-world network, respectively. The pinning nodes were selected, respectively, according to order of in-degrees and out-degrees. Through comparing and analyzing simulations, the optimal pinning scheme was found, and a practical principle was induced finally.{ \copyright 2011 American Institute of Physics}

Let \((X,\tau)\) be a topological space. A set \(S\subset X\) is regular closed if \(S= \operatorname {Cl(Int} (S))\). A set \(S\subset X\) is called \(A\)-set or \(A\)-open [\textit{J. Tong}, Acta Math. Hung. 48, 11-15 (1986; Zbl 0609.54012)] if \(S= U\cap W\), where \(U\) is an open set and \(W\) is a regular closed set. In this paper the author defines a new class of functions called \(A^*\)-continuous functions. A function \(f:(X,\tau)\to (Y,\sigma)\) is \(A^*\)-continuous at \(x\in X\) if for each open set \(V\) containing \(f(x)\), there exists an \(A\)-open set \(U\) in \(X\) containing \(x\) such that \(f(U)\subset V\). The function \(f\) is called \(A^*\)-continuous if it has this property for each point \(x\) in \(X\). Some characterizations of \(A^*\)-continuous functions are obtained. Also, new notions are introduced: \(A\)-\(T_2\) space, strongly \(A\)-closed graph, \(A\)-compact space, etc. Some properties of \(A^*\)-continuous functions are proved, for example, it is shown that the image of an \(A\)-compact set under \(A^*\)-continuity is compact. This paper introduces the following notions. A subset B of a topological space (X,\({\mathcal T})\) is defined to be an \({\mathcal A}\)-set if \(B=U-W\) where U is open and W is regular open. A function f: (X,\({\mathcal T})\to (Y,{\mathcal U})\) is called \({\mathcal A}\)-continuous if \(f^{-1}(V)\) is an \({\mathcal A}\)- set in X for each open set V in Y. \textit{O. Njåstad} [Pac. J. Math. 15, 961-970 (1965; Zbl 0137.419)] defined a subset S of (X,\({\mathcal T})\) to be an \(\alpha\)-set if \(S\subset int(cl(int S))\). \textit{A. S. Mashhour} et al [Acta Math. Hung. 41, 213-218 (1983; Zbl 0534.54006)] defined a function f: (X,\({\mathcal T})\to (Y,{\mathcal U})\) to be \(\alpha\)-continuous if \(f^{-1}(V)\) is an \(\alpha\)-set in X for each open set V in Y. The main result of this paper is that f is continuous if and only if it is \(\alpha\)-continuous and \({\mathcal A}\)-continuous.

Let \((X,\tau)\) be a topological space. A set \(S\subset X\) is regular closed if \(S= \operatorname {Cl(Int} (S))\). A set \(S\subset X\) is called \(A\)-set or \(A\)-open [\textit{J. Tong}, Acta Math. Hung. 48, 11-15 (1986; Zbl 0609.54012)] if \(S= U\cap W\), where \(U\) is an open set and \(W\) is a regular closed set. In this paper the author defines a new class of functions called \(A^*\)-continuous functions. A function \(f:(X,\tau)\to (Y,\sigma)\) is \(A^*\)-continuous at \(x\in X\) if for each open set \(V\) containing \(f(x)\), there exists an \(A\)-open set \(U\) in \(X\) containing \(x\) such that \(f(U)\subset V\). The function \(f\) is called \(A^*\)-continuous if it has this property for each point \(x\) in \(X\). Some characterizations of \(A^*\)-continuous functions are obtained. Also, new notions are introduced: \(A\)-\(T_2\) space, strongly \(A\)-closed graph, \(A\)-compact space, etc. Some properties of \(A^*\)-continuous functions are proved, for example, it is shown that the image of an \(A\)-compact set under \(A^*\)-continuity is compact. For the entire collection see Zbl 0571.00003.

Given a vertex operator algebra (VOA) \(V\) one may define chiral \(n\)-point functions at genus one following \textit{Y. Zhu} [J. Am. Math. Soc. 9, 237-302 (1996; Zbl 0854.17034)] and then use various sewing procedures to define such functions at higher genera. However, these procedures require a detailed description of the genus one \(n\)-point functions. In the paper under review the authors obtain explicit expressions for all genus one \(n\)-point functions in two important cases, namely when \(V\) is either a Heisenberg VOA (also called a free boson theory) or a lattice VOA, that is, a VOA associated to a positive-definite, even lattice. Roughly speaking, in a free boson theory the \(n\)-point functions are elliptic functions whose structure depends on certain combinatorial data determined by the states in \(V\) under consideration. In the case of a lattice VOA, the function is the product of two pieces, one determined by the Heisenberg sub-VOA of \(V\) and one which may be described in terms of the lattice itself and the genus one prime form. The paper concludes with a discussion of \(n\)-point functions from the point of view of their symmetry and elliptic properties. The author proves modular invariance of the characters of irreducible representations for a vertex operator algebra under certain finiteness conditions. In order to prove the result, an analogue of highest-weight theory in Lie algebras for vertex operator algebras is developed. Moreover, he proves that a vertex operator algebra allows a certain change-of-variables.

Given a vertex operator algebra (VOA) \(V\) one may define chiral \(n\)-point functions at genus one following \textit{Y. Zhu} [J. Am. Math. Soc. 9, 237-302 (1996; Zbl 0854.17034)] and then use various sewing procedures to define such functions at higher genera. However, these procedures require a detailed description of the genus one \(n\)-point functions. In the paper under review the authors obtain explicit expressions for all genus one \(n\)-point functions in two important cases, namely when \(V\) is either a Heisenberg VOA (also called a free boson theory) or a lattice VOA, that is, a VOA associated to a positive-definite, even lattice. Roughly speaking, in a free boson theory the \(n\)-point functions are elliptic functions whose structure depends on certain combinatorial data determined by the states in \(V\) under consideration. In the case of a lattice VOA, the function is the product of two pieces, one determined by the Heisenberg sub-VOA of \(V\) and one which may be described in terms of the lattice itself and the genus one prime form. The paper concludes with a discussion of \(n\)-point functions from the point of view of their symmetry and elliptic properties. We propose two classes of dynamic versions of the classical Erdős-Rényi graph: one in which the transition rates are governed by an external regime process, and one in which the transition rates are periodically resampled. For both models we consider the evolution of the number of edges present, with explicit results for the corresponding moments, functional central limit theorems and large deviations asymptotics.

In [Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Innsbruck: Univ. Innsbruck, Mathematisches Institut (Diss.) (1965; Zbl 1245.13020)] \textit{B. Buchberger} described an algorithm, known as Buchberger's algorithm, for the construction of Gröbner bases. In this paper, the authors present an analogue of this algorithm (see also the Knuth-Bendix algorithm) for the computation of standard bases in algebras of finite, numerical, and linear semialgebraic sets, as well as in the algebra of multisets. They refer to this algorithm as the replenishment algorithm and it can be used mainly to solve the membership problem in the mentioned kinds of algebras. Furthermore, they apply the replenishment algorithm to solve some algorithmic problems in the algebra of linear semialgebraic sets. See the review of the English translation in Zbl 1158.01307.

In [Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Innsbruck: Univ. Innsbruck, Mathematisches Institut (Diss.) (1965; Zbl 1245.13020)] \textit{B. Buchberger} described an algorithm, known as Buchberger's algorithm, for the construction of Gröbner bases. In this paper, the authors present an analogue of this algorithm (see also the Knuth-Bendix algorithm) for the computation of standard bases in algebras of finite, numerical, and linear semialgebraic sets, as well as in the algebra of multisets. They refer to this algorithm as the replenishment algorithm and it can be used mainly to solve the membership problem in the mentioned kinds of algebras. Furthermore, they apply the replenishment algorithm to solve some algorithmic problems in the algebra of linear semialgebraic sets. In this paper, we study viscosity approximations with \((\psi,\varphi)\)-weakly contractive mappings. We show that Moudafi's viscosity approximations follow from Browder and Halpern type convergence theorems. Our results generalize a number of convergence theorems including a strong convergence theorem of [\textit{Y. Song} and \textit{X. Liu}, Fixed Point Theory Appl. 2009, Article ID 824374, 13 p. (2009; Zbl 1167.65029)].

Given a positive integer \(n\), let \(X_n\) be the set \(\{1,2,\dots, n\}\). A partition \(\pi\) of \(X_n\) is said to have weight \(r\) if \(\pi\) has \(r\) distinct classes. The partition \(\pi\) and an \(r\)-element set \(A\) are said to be orthogonal if every class of \(\pi\) contains exactly one element of \(A\). An orthogonally labeled list for \(n\) and \(r\) is a sequence \(({\mathcal A},\pi)= A_1\), \(\pi_1\), \(A_2,\pi_2,\dots, A_{{n\choose r}}, \pi_{{n\choose r}}\) alternating between distinct \(r\)-sets \(A_i\) and distinct partitions \(\pi_i\) of weight \(r\), such that for \(i= 1,\dots,{n\choose r}- 1\), \(\pi_i\) is simultaneously orthogonal to \(A_i\) and \(A_{i+1}\), and \(\pi_{{n\choose r}}\) is simultaneously orthogonal to \(A_{{n\choose r}}\) and \(A_1\). A partition of the set \(X_n\) has type \[ \tau= d^{\mu(d_1)}_1 d^{\mu(d_2)}_2\cdots d^{\mu(d_k)}_k \] and weight \(r\) if it has \(\mu(d_i)\) classes of size \(d_i\), where \(d_1> d_2>\cdots> d_k\), \[ n= \sum^k_{i=1} d_i\mu(d_i)\quad\text{and}\quad r= \sum^k_{i=1}\mu(d_i). \] The partition type \(\tau\) is said to be exceptional if the number of distinct partitions of type \(\tau\) is less than \({n\choose r}\). Let \(G_{n,r}\) be the graph whose vertices are all the \(r\)-sets of \(X_n\), with two \(r\)-sets being adjacent if their intersection has exactly \(r-1\) elements. An orthogonally \(\tau\)-labeled list \(({\mathcal A},\pi)\) for which the set sequence \({\mathcal A}\) is a Hamiltonian cycle in \(G_{n,r}\) is called an orthogonally \(\tau\)-labeled Hamiltonian cycle. The Hamiltonian cycles of \(G_{n,r}\) are also called constant weight Gray codes. The partition type conjecture states that there exist orthogonally \(\tau\)-labeled Hamiltonian cycles for every non-exceptional partition type \(\tau\). This paper is mainly devoted to the proof of the partition type conjecture for a class of constant weight Gray codes. These results are applied in the theory of finite semigroups. The authors establish simple formulas for ranks and idempotent ranks of semigroups which are closed under conjugation by the permutations of \(X_n\). The authors continue their study on the orthogonal labeling of constant weight Gray codes, started in the paper reviewed above. The combinatorics and semigroup theory aspects are treated independently again. Among others, the authors generalize a known result on the rank and idempotent rank of the semigroup of transformations of \(\{1,2,\dots, n\}\) containing at most \(r\) elements in the image.

Given a positive integer \(n\), let \(X_n\) be the set \(\{1,2,\dots, n\}\). A partition \(\pi\) of \(X_n\) is said to have weight \(r\) if \(\pi\) has \(r\) distinct classes. The partition \(\pi\) and an \(r\)-element set \(A\) are said to be orthogonal if every class of \(\pi\) contains exactly one element of \(A\). An orthogonally labeled list for \(n\) and \(r\) is a sequence \(({\mathcal A},\pi)= A_1\), \(\pi_1\), \(A_2,\pi_2,\dots, A_{{n\choose r}}, \pi_{{n\choose r}}\) alternating between distinct \(r\)-sets \(A_i\) and distinct partitions \(\pi_i\) of weight \(r\), such that for \(i= 1,\dots,{n\choose r}- 1\), \(\pi_i\) is simultaneously orthogonal to \(A_i\) and \(A_{i+1}\), and \(\pi_{{n\choose r}}\) is simultaneously orthogonal to \(A_{{n\choose r}}\) and \(A_1\). A partition of the set \(X_n\) has type \[ \tau= d^{\mu(d_1)}_1 d^{\mu(d_2)}_2\cdots d^{\mu(d_k)}_k \] and weight \(r\) if it has \(\mu(d_i)\) classes of size \(d_i\), where \(d_1> d_2>\cdots> d_k\), \[ n= \sum^k_{i=1} d_i\mu(d_i)\quad\text{and}\quad r= \sum^k_{i=1}\mu(d_i). \] The partition type \(\tau\) is said to be exceptional if the number of distinct partitions of type \(\tau\) is less than \({n\choose r}\). Let \(G_{n,r}\) be the graph whose vertices are all the \(r\)-sets of \(X_n\), with two \(r\)-sets being adjacent if their intersection has exactly \(r-1\) elements. An orthogonally \(\tau\)-labeled list \(({\mathcal A},\pi)\) for which the set sequence \({\mathcal A}\) is a Hamiltonian cycle in \(G_{n,r}\) is called an orthogonally \(\tau\)-labeled Hamiltonian cycle. The Hamiltonian cycles of \(G_{n,r}\) are also called constant weight Gray codes. The partition type conjecture states that there exist orthogonally \(\tau\)-labeled Hamiltonian cycles for every non-exceptional partition type \(\tau\). This paper is mainly devoted to the proof of the partition type conjecture for a class of constant weight Gray codes. These results are applied in the theory of finite semigroups. The authors establish simple formulas for ranks and idempotent ranks of semigroups which are closed under conjugation by the permutations of \(X_n\). Verf. hat früher (Bull. math. Soc. Roumaine Sci. 39\(_{\text{II}}\) (1937), 71-85; F.~d.~M. 63\(_{\text{II}}\), 939) für beliebige, reelle, beschränkte Funktionen \(f(P)\) und beschränkte Integrationsbereiche \(\mathfrak{M}\) (im \(R_n\)) eine Definition des äußeren bzw. inneren (Lebesgueschen) Integrals \(\int\limits_{\mathfrak{M}}^{e} f\, dP\) bzw. \(\int\limits_{\mathfrak{M}}^{i} f\, dP\) gegeben. Jetzt werden die einfachsten Eigenschaften dieses Integrals ausführlich behandelt. Wir erwähnen: I. \(\int\limits_{\mathfrak{M}}^{e} f\, dP \leqq \sum\limits_{\varrho} \int\limits_{\mathfrak{M}_{\varrho}}^{e} f\, dP\), wenn \(\mathfrak{M}=\sum\limits_{\varrho} \mathfrak{M}_{\varrho}\) mit \(\mathfrak{M}_{\varrho}\mathfrak{M}_{\tau}=0\), \(\varrho \neq \tau\), wobei \(f \geqq 0\); II. Es sei \(f \geqq 0\) meßbar und \(f=\varphi+\psi\), wo \(\varphi\) und \(\psi\) beschränkt sind, ferner sei \(\mathfrak{M}\) beschränkt und meßbar, aber nicht Nullmenge. Falls dann \(\varphi\) und \(\psi\) ``von endlichem äußeren Maße'' sind, gilt \(\int\limits_{\mathfrak{M}}^{i}\varphi\,dP+\int\limits_{\mathfrak{M}}^{i}\psi\,dP \leqq \int\limits_{\mathfrak{M}} f\, dP \leqq \int\limits_{\mathfrak{M}}^{e}\varphi\,dP+\int\limits_{\mathfrak{M}}^{e}\psi\,dP\); III. Es sei \(\mathfrak{M}\) beschränkt und meßbar. Ferner seien die \(f_{\nu}(P)\), \(\nu=1,2,\ldots\), gleichmäßig beschränkt: \(|f_{\nu}(P)|<M\) für alle \(\nu\) und \(P\). Es existiere \(f(P)=\lim\limits_{\nu \to \infty}f_{\nu}(P)\) mit \(f(P) \geqq 0\), und es sei \(f(P)\) meßbar und von endlichem äußeren Maße. Dann gilt: \[ \varlimsup\limits_{\nu \to \infty} \left( \int\limits_{\mathfrak{M}}^{i} f_{\nu}\, dP \right) \leqq \int\limits_{\mathfrak{M}} f\, dP \leqq \varliminf\limits_{\nu \to \infty} \left[ \int\limits_{\mathfrak{M}}^{e} f_{\nu}\, dP + (M+M_{\nu}) \, g(f_{\nu}; \, \mathfrak{M}) \right], \] wo \(|f_{\nu}(P)|<M_{\nu}\) für alle \(P\) und wo \(g(f_{\nu}; \, \mathfrak{M})\) den ``Grad der Nichtmeßbarkeit von \(f_{\nu}\) auf \(\mathfrak{M}\)'' bezeichnet.

Let \(B_k\) be a basis for \(S_k(\Gamma(1))\) consisting of normalized Hecke eigenforms. For each \(f\in B_k\) and each primitive Dirichlet character \(\chi\), let \(L(f\otimes\chi, s)\) be the associated twisted \(L\)-function. This paper computes the mean square \[ \sum_{f\in B_k} w_f L(f\otimes\chi, \textstyle{{1\over 2}})^2, \] where \(w_f= \Gamma(k- 1)(4\pi)^{1-k}\| f\|^{-2}\), giving an asymptotic formula with error term \(O(D^3k^{-1/2}\log^4k)\), where \(D\) is the conductor of \(\chi\). More generally, mean values of \(L(f\otimes\chi_1,{1\over 2})L(f\otimes\chi_2,{1\over 2})\) are also considered. The first result is used to deduce that \[ \#\biggl\{f\in B_k: L\biggl(f\otimes\chi, {1\over 2}\biggr)\neq 0\biggr\}\gg|1+ \varepsilon_k(\chi)|^2 {D\over \phi(D)} {k\over(\log k)^2} \] for large enough \(k\). (Here \(\varepsilon_k(\chi)\) is the root number \(i^k\tau(\chi)^2/D\).) This holds uniformly for \(D\ll k^{1/6}(\log k)^{-2}\) if \(\chi\) is real, and for \(\log D\ll \log k)/ (\log\log k)\) for \(\chi\) complex. This non-vanishing result improves that of \textit{W. Kohnen} and \textit{W. Sengupta} [Acta Arith. 99, No. 1, 61--66 (2001; Zbl 0982.11026)], who only used a first moment. Let \(F_{2k}\) denote the set of normalized elliptic Hecke cusp forms of even weight \(2k\). Let \(L(f,D,s)\) be the twist of the Hecke \(L\)-series of \(f\in F_{2k}\) with the quadratic character \((\frac D*)\), where \(D\) is a fundamental discriminant. Then the authors show that \[ \sum_{f\in F_{2k}}L(f,D,k)\ll_{\varepsilon,D}k^{1+\varepsilon}\qquad(k\to\infty,(-1)^kD>0). \] Under the stronger assumption \(L(f,D,k)\ll_{\varepsilon,D}k^\varepsilon\) for some \(0 <\varepsilon <1\) they moreover derive \[ \#\{f\in F_{2k}\mid L(f, D, k)\neq 0\}\gg_{\varepsilon,D}k^{1-\varepsilon}/\log k. \]

Let \(B_k\) be a basis for \(S_k(\Gamma(1))\) consisting of normalized Hecke eigenforms. For each \(f\in B_k\) and each primitive Dirichlet character \(\chi\), let \(L(f\otimes\chi, s)\) be the associated twisted \(L\)-function. This paper computes the mean square \[ \sum_{f\in B_k} w_f L(f\otimes\chi, \textstyle{{1\over 2}})^2, \] where \(w_f= \Gamma(k- 1)(4\pi)^{1-k}\| f\|^{-2}\), giving an asymptotic formula with error term \(O(D^3k^{-1/2}\log^4k)\), where \(D\) is the conductor of \(\chi\). More generally, mean values of \(L(f\otimes\chi_1,{1\over 2})L(f\otimes\chi_2,{1\over 2})\) are also considered. The first result is used to deduce that \[ \#\biggl\{f\in B_k: L\biggl(f\otimes\chi, {1\over 2}\biggr)\neq 0\biggr\}\gg|1+ \varepsilon_k(\chi)|^2 {D\over \phi(D)} {k\over(\log k)^2} \] for large enough \(k\). (Here \(\varepsilon_k(\chi)\) is the root number \(i^k\tau(\chi)^2/D\).) This holds uniformly for \(D\ll k^{1/6}(\log k)^{-2}\) if \(\chi\) is real, and for \(\log D\ll \log k)/ (\log\log k)\) for \(\chi\) complex. This non-vanishing result improves that of \textit{W. Kohnen} and \textit{W. Sengupta} [Acta Arith. 99, No. 1, 61--66 (2001; Zbl 0982.11026)], who only used a first moment. [For the entire collection see Zbl 0509.00020.] The authors construct a Markov's process in \(R^ m\) so that its part in \(D_ 1\cup D_ 2\) is equivalent to a Wiener process with killing at the moment of the first hitting of S, where \(S=\{x=(x^ 1,x^ 2,...,x^ m)\in R^ m| x^ m=0\}\), \(D_ 1=\{x=(x^ 1,x^ 2,...,x^ m)\in R^ m| x^ m<0\}\), \(D_ 2=\{x=(x^ 1,x^ 2,...,x^ m)\in R^ m| x^ m>0\}\). To this aim, under the conditions: \(t>0\), \(x\in R^ m\) and \(\phi:R^ m\to R\) is a measurable and bounded function, they construct a function u(t,x,\(\phi)\) with the properties: a) the function u satisfies the heat equation \(\partial u/\partial t=\Delta u/2\) for \(t>0\), \(x\in D_ 1\cup D_ 2\), b) \(\lim_{t\downarrow 0}u(t,x,\phi)=\phi(x)\) for every continuity point x of the function \(\phi\), c) \[ q_ 1\frac{\partial u(t,x-0,\phi)}{\partial x^ m}+q_ 2\frac{\partial u(t,x+0,\phi)}{\partial x^ m}+\sum^{m}_{i,j=1}a^ i\frac{\partial u(t,x,\phi)}{\partial x^ i}+ \] \[ +\frac{1}{2}\sum^{m}_{i,j=1}b_{ij}\frac{\partial^ 2u(t,x,\phi)}{\partial x^ i\partial x^ j}=0, \] for \(t>0\), \(x\in S\), where \(q_ i,a_ i,b_{ij}\) are real constants and \(\partial u(t,x\pm 0,\phi)/\partial x^ m\) are one-sided derivatives of the function u(t,x,\(\phi)\) on the surface S, that is the limits of the derivatives \(\partial u(t,y,\phi)/\partial y^ m\), when y tends to \(x\in S\) along a non-tangential path belonging to \(D_ 2\) (sign ''\(+'')\) or to \(D_ 1\) (sign ''-''), respectively.

The article, written in Dutch, reflects the contents of a lecture given by the authors, with as listeners a general audience. It gives in easy words, the notions of complex numbers, its use and developments in a few recent years in the past, by people like Bombieri, Cardano, Tartaglia, Wantzel, Wessel, Hamilton, and one finds a few words about the influence of complex numbers in practice of today. One finds in the article portraits of Bombieri, Euler, Gauß, Hamilton, de Moivre, as well as a picture of the plaque on the bridge over the Royal Canal in Dublin remembering the invention by Hamilton regarding the quaternions. In short, also the content of the paper is certainly suited to enhance knowledge of the layman, as well as that of beginning students and pupils at secondary schools. The reviewer did read something (surprising at least to him as an unknown fact) remarkable in the paper. Namely, the quaternions were already discovered by Gauß in 1826, i.e. earlier than by Hamilton in 1843, but the publication thereoff came to light only in 1900! The reviewer took up the trouble to control that remark by means of Wikipedia sources. And indeed, see: [\textit{C. F. Gauss}, ``Mutationen des Raumes'', in: Carl Friedrich Gauss Werke. Achter Band. Königliche Gesellschaft der Wissenschaften. Berlin, Heidelberg: Springer. 357--362 (1900; \url{doi:10.1007/978-3-642-92474-3_67})] and \textit{S. L. Altmann}'s paper in [Math. Mag. 62, No. 5, 291--308 (1989; Zbl 0704.01009)]. The paper analyzes the logical status of Hamilton's quaternions with respect to the modern notions of vector and of rotation group. An article of 1840 by the French-Jewish Socialist banker Olinde Rodrigues (1794--1851) came nearer to the resolution of the rotation problem whereas ``in Hamilton's approach rotations become subservient to the algebra, which opens the door to a variety of misinterpretaions'' (301). The article is written from the standpoint of modern theory and does not fully discuss the intentions and influences of both Hamiltonian and Rodriguesian concepts. Nor does it give hints on possible interrelations and communications between both mathematicians. Reference to a former article in that same journal [\textit{B. L. van der Waerden}, ibid. 49, 227--234 (1976; Zbl 0348.01007)] would have been useful.

The article, written in Dutch, reflects the contents of a lecture given by the authors, with as listeners a general audience. It gives in easy words, the notions of complex numbers, its use and developments in a few recent years in the past, by people like Bombieri, Cardano, Tartaglia, Wantzel, Wessel, Hamilton, and one finds a few words about the influence of complex numbers in practice of today. One finds in the article portraits of Bombieri, Euler, Gauß, Hamilton, de Moivre, as well as a picture of the plaque on the bridge over the Royal Canal in Dublin remembering the invention by Hamilton regarding the quaternions. In short, also the content of the paper is certainly suited to enhance knowledge of the layman, as well as that of beginning students and pupils at secondary schools. The reviewer did read something (surprising at least to him as an unknown fact) remarkable in the paper. Namely, the quaternions were already discovered by Gauß in 1826, i.e. earlier than by Hamilton in 1843, but the publication thereoff came to light only in 1900! The reviewer took up the trouble to control that remark by means of Wikipedia sources. And indeed, see: [\textit{C. F. Gauss}, ``Mutationen des Raumes'', in: Carl Friedrich Gauss Werke. Achter Band. Königliche Gesellschaft der Wissenschaften. Berlin, Heidelberg: Springer. 357--362 (1900; \url{doi:10.1007/978-3-642-92474-3_67})] and \textit{S. L. Altmann}'s paper in [Math. Mag. 62, No. 5, 291--308 (1989; Zbl 0704.01009)]. In this paper, the existence of positive solutions of nonlinear operator equation \(u={\lambda}Tu\) are studied, where \({\lambda}>0\) is a parameter, \(T\) is a compact operator with sign-changing kernel. By using the Leray-Schauder degree theory, the existence of positive solutions of nonlinear operator equations with sign-changing kernel are obtained. The obtained abstract theorem can be applied easily, to illustrate this point the paper also introduces some applications.

A \(B_2[g]\) set \({\mathcal A}\) is a set of integers such that, for any integer \(n\), \(|\{(a_1,a_2),a_i\in{\mathcal A},\;a_1+a_2=n,\;a_1\leq a_2\}|\leq g\). Thus \(B_2[1]\) sets are Sidon sets. It was shown by \textit{J. Cilleruelo} [ibid. 89, 141--144 (2000; Zbl 0955.11005)] that if \({\mathcal A}\) is a \(B_2 [2]\) set in \(\{1,\dots,N\}\), then \(|{\mathcal A}|\leq(6N)^{1/2}+1\). The present author improves this result, showing that, for sufficiently large \(N\), \(|{\mathcal A}|\leq 2.363584\sqrt N\). A sequence \(A\) of integers is said to be a \(B_2[2]\) sequence if for every \(n\), the number of solutions of \(n=a+a', a\leq a', a\in A\) is at most two. Let \(F_2(N,2)=\max\{|A|:A\subset[1,N ]\), \(A\in B_2[2]\}\). In the present paper the author, using a new counting method, proves that \(F(N,2)\leq \sqrt {6N}+1\).

A \(B_2[g]\) set \({\mathcal A}\) is a set of integers such that, for any integer \(n\), \(|\{(a_1,a_2),a_i\in{\mathcal A},\;a_1+a_2=n,\;a_1\leq a_2\}|\leq g\). Thus \(B_2[1]\) sets are Sidon sets. It was shown by \textit{J. Cilleruelo} [ibid. 89, 141--144 (2000; Zbl 0955.11005)] that if \({\mathcal A}\) is a \(B_2 [2]\) set in \(\{1,\dots,N\}\), then \(|{\mathcal A}|\leq(6N)^{1/2}+1\). The present author improves this result, showing that, for sufficiently large \(N\), \(|{\mathcal A}|\leq 2.363584\sqrt N\). Let \(\mathbb F\) be a field and char \(\mathbb F=p>3\). Let \({\mathfrak g}\) denote the Lie superalgebra which is the tensor product of an exterior superalgebra and a finite-dimensional generalized Witt algebra or a special Lie algebra over \(\mathbb F\). In this paper it is proved that the standard filtration of the non-simple Lie superalgebra \(\mathfrak{g}\) is invariant under the automorphism group of \(\mathfrak{g}\) by means of investigating the ad nilpotent elements. Finally, the superalgebras of type \(W\) or \(S\) are classified by means of the parameters defining them.

The authors study the global asymptotic stability of the equilibrium for the damped superlinear oscillator \[ x''+a(t) \varphi_q(x') +\omega^2 x=0. \eqno(1) \] If \(\liminf_{t\to \infty} \int^{t+\gamma_0}_t a(s)\, ds = \gamma_0 >0\), where \(0<\gamma_0 < \frac{\pi}{\omega}\), the authors prove that the equilibrium of (1) is globally asymptotically stable if and only if \(\int^\infty_0 u(t)\, dt =-\infty\), where \(u(t)\) is the solution of \(u'+\omega^{q-2} a(t) \varphi_q(u) +1=0\) satisfying \(u(0)=0\). Explicit sufficient conditions and explicit necessary conditions are also given for the equilibrium of the damped superlinear oscillator to be globally attractive. The present paper is related to a work of \textit{L. Hatvani} and \textit{V. Totik} [Differ. Integral Equ. 6, No. 4, 835--848 (1993; Zbl 0777.34036)]. Conditions are given guaranteeing the property \(x(t) \to 0\), \(\dot x(t) \to 0\) \((t \to \infty)\) for every solution of the equation \(\ddot x+h(t) \dot x+k^ 2x=0\) \((t \geq 0, 0<k=\text{const.})\), where \(h\) is a nonnegative function. It is known that this property requires that in the average the damping coefficient \(h\) is not ``too small'' or ``too large''. In the first part we give a necessary and sufficient growth condition on \(h\), provided that \(h\) is not ``too small'' in some integral sense. Then, considering the case of small \(h\), we show that not only the size, but the distribution of the damping ``bumps'' is important. The main theorem takes into account both of them. Finally, we formulate theorems for the general case when \(h\) can be both small and large. It is pointed out that the conditions restricting \(h\) above and below are interdependent.

The authors study the global asymptotic stability of the equilibrium for the damped superlinear oscillator \[ x''+a(t) \varphi_q(x') +\omega^2 x=0. \eqno(1) \] If \(\liminf_{t\to \infty} \int^{t+\gamma_0}_t a(s)\, ds = \gamma_0 >0\), where \(0<\gamma_0 < \frac{\pi}{\omega}\), the authors prove that the equilibrium of (1) is globally asymptotically stable if and only if \(\int^\infty_0 u(t)\, dt =-\infty\), where \(u(t)\) is the solution of \(u'+\omega^{q-2} a(t) \varphi_q(u) +1=0\) satisfying \(u(0)=0\). Explicit sufficient conditions and explicit necessary conditions are also given for the equilibrium of the damped superlinear oscillator to be globally attractive. The present paper is related to a work of \textit{L. Hatvani} and \textit{V. Totik} [Differ. Integral Equ. 6, No. 4, 835--848 (1993; Zbl 0777.34036)]. The Wilcoxon two-sample nonparametric test is modified to analyze data with missing observations. The approach is to weigh each complete observation with its own estimated weight score, which is the inverse of the probability of completing the observations. The proposed modified method is applied to recently conducted clinical trial data. A small simulation study shows that the modified method is unbiased under the null hypothesis, while the Wilcoxon two sample test is biased when applied only to complete observations.

The purpose of this paper is to prove a theorem on functional equations of Hasse zeta functions of enveloping algebras of solvable Lie algebras, which was proposed by the author in her previous paper [cf. ibid. 72, No. 8, 187-188 (1996; Zbl 0868.11040)]. She proved there the theorem under certain conditions on \(p\)-mappings on the algebra. In this paper she succeeds in giving a proof of the theorem without any additional conditions. The theorem is the following. Let \(R\) be a finitely generated commutative ring over the integers \(\mathbb{Z}\). Let \({\mathfrak g}\) be a solvable Lie algebra over \(R\) which is free of finite rank \(n\) as an \(R\)-module, and let \(A\) be the universal enveloping algebra of \({\mathfrak g}\) over \(R\). Then we have \[ \zeta_A(s) =\zeta_R(s-n), \] where \(\zeta_A (s)\) and \(\zeta_R(s)\) are Hasse zeta functions of \(A\) and \(R\), respectively. In a preceding paper the author of this paper gave a generalization of Hasse's zeta function associated to a noncommutative ring defined over the integer ring \(\mathbb{Z}\). In this paper, she applies her theory to a universal enveloping algebra of a solvable Lie algebra. Her result is the following. Let \(R\) be a commutative finitely generated ring over \(\mathbb{Z}\). Let \({\mathfrak g}\) be a Lie algebra over \(R\) which is free of finite rank \(n\) as an \(R\)-module, and let \(A\) be a universal enveloping algebra of \({\mathfrak g}\). Assume that for each maximal ideal \({\mathfrak m}\) of \(R\), \({\mathfrak m}/{\mathfrak{gm}}\) is a completely solvable Lie algebra over \(R/{\mathfrak m}\) and has a \(p\)-mapping. Then we have that the Hasse zeta function \(\zeta_A(s)\) associated to \(A\) converges, and satisfies the functional equation \(\zeta_A(s)= \zeta_R(s-n)\).

The purpose of this paper is to prove a theorem on functional equations of Hasse zeta functions of enveloping algebras of solvable Lie algebras, which was proposed by the author in her previous paper [cf. ibid. 72, No. 8, 187-188 (1996; Zbl 0868.11040)]. She proved there the theorem under certain conditions on \(p\)-mappings on the algebra. In this paper she succeeds in giving a proof of the theorem without any additional conditions. The theorem is the following. Let \(R\) be a finitely generated commutative ring over the integers \(\mathbb{Z}\). Let \({\mathfrak g}\) be a solvable Lie algebra over \(R\) which is free of finite rank \(n\) as an \(R\)-module, and let \(A\) be the universal enveloping algebra of \({\mathfrak g}\) over \(R\). Then we have \[ \zeta_A(s) =\zeta_R(s-n), \] where \(\zeta_A (s)\) and \(\zeta_R(s)\) are Hasse zeta functions of \(A\) and \(R\), respectively. Im folgenden Beitrag wird versucht darzulegen, dass Primzahlen in der Mathematikausbildung, sowohl in der Schule als auch an der Universität, eine herausragende Rolle spielen können und sollen.

It is well known that the \(q\)-deformation of a rational number is a rational function in the parameter \(q\) with integer coefficients. Let \( [n]_q=\frac{1-q^n}{1-q}=1+q+q^2+\dots+q^{n-1}, \) \( [-n]_q=\frac{1-q^{-n}}{1-q}=-q^{-1}-q^{-2}-\dots- q^{-n} \) be the classical \(q\)-deformations of integers for any \(n \in \mathbb{Z}_{>0}\). The paper proves that for all \(x \in \mathbb{R},\) the \(q\)-deformations satisfy \( [x+1]_q=q[x]_q+1,\) \( [-1/x]_q=-q^{-1}/[x]_q\). The next main result: Let \(x=\frac{r\pm \sqrt{p}}{s}\) be a quadratic irrational. Its \(q\)-deformation \([x]_q\) satisfies the following (i) \([x]_q = \frac{\mathcal{R}\pm \sqrt{\mathcal{P}}}{\mathcal{S}}\) with \(\mathcal{R,P,S} \in \mathbb{Z}[q]\), and \(\mathcal{P}\) a palindrome; (ii) \([x]_q\) is solution of an equation \(\mathcal{A}X^2+ \mathcal{B}X+\mathcal{C}=0,\) with \(\mathcal{A,B,C} \in \mathbb{Z}[q]; \) (iii) there exists a matrix \(M_q \in \mathrm{GL}(2,\mathbb{Z}[q^{\pm 1}])\) such that \(M_q \cdot [x]_q=[x]_q \); (iv) \([x]_q\) has a periodic continued fraction expansion. Also they prove that the traces of the elements in \(\mathrm{PSL}_q (2,\mathbb{Z})\) are palindromic polynomials in \(\mathbb{Z}[q]\) with positive integer coefficients, modulo a multiplicative factor \(\pm q^N\). Illustrative examples of the given results are included in the work. We introduce a notion of \(q\)-deformed rational numbers and \(q\)-deformed continued fractions. A \(q\)-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the \(q\)-deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the \(q\)-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, \(q\)-deformation of the Farey graph, matrix presentations and \(q\)-continuants are given, as well as a relation to the Jones polynomial of rational knots.

It is well known that the \(q\)-deformation of a rational number is a rational function in the parameter \(q\) with integer coefficients. Let \( [n]_q=\frac{1-q^n}{1-q}=1+q+q^2+\dots+q^{n-1}, \) \( [-n]_q=\frac{1-q^{-n}}{1-q}=-q^{-1}-q^{-2}-\dots- q^{-n} \) be the classical \(q\)-deformations of integers for any \(n \in \mathbb{Z}_{>0}\). The paper proves that for all \(x \in \mathbb{R},\) the \(q\)-deformations satisfy \( [x+1]_q=q[x]_q+1,\) \( [-1/x]_q=-q^{-1}/[x]_q\). The next main result: Let \(x=\frac{r\pm \sqrt{p}}{s}\) be a quadratic irrational. Its \(q\)-deformation \([x]_q\) satisfies the following (i) \([x]_q = \frac{\mathcal{R}\pm \sqrt{\mathcal{P}}}{\mathcal{S}}\) with \(\mathcal{R,P,S} \in \mathbb{Z}[q]\), and \(\mathcal{P}\) a palindrome; (ii) \([x]_q\) is solution of an equation \(\mathcal{A}X^2+ \mathcal{B}X+\mathcal{C}=0,\) with \(\mathcal{A,B,C} \in \mathbb{Z}[q]; \) (iii) there exists a matrix \(M_q \in \mathrm{GL}(2,\mathbb{Z}[q^{\pm 1}])\) such that \(M_q \cdot [x]_q=[x]_q \); (iv) \([x]_q\) has a periodic continued fraction expansion. Also they prove that the traces of the elements in \(\mathrm{PSL}_q (2,\mathbb{Z})\) are palindromic polynomials in \(\mathbb{Z}[q]\) with positive integer coefficients, modulo a multiplicative factor \(\pm q^N\). Illustrative examples of the given results are included in the work. Ample psychological evidence suggests that people's learning behavior is often prone to a ``myside bias'' or ``irrational belief persistence'' in contrast to learning behavior exclusively based on objective data. In the context of Bayesian learning such a bias may result in diverging posterior beliefs and attitude polarization even if agents receive identical information. Such patterns cannot be explained by the standard model of rational Bayesian learning that implies convergent beliefs. Based on Choquet expected utility theory, we therefore develop formal models of Bayesian learning with psychological bias as alternatives to rational Bayesian learning. We derive conditions under which beliefs may diverge in the learning process despite the fact that all agents observe the same sample drawn from an i.i.d. process. Key to our approach is the description of ambiguous beliefs as neo-additive capacities [\textit{A. Chateauneuf, J. Eichberger} and \textit{S. Grant}, J. Econ. Theory 137, No.~1, 538--567 (2007; Zbl 1132.91420], which allows for a flexible and parsimonious parametrization of departures from additive probability measures.

Let \(\Omega\) be homogeneous of degree zero with mean value zero, \(P\) and \(Q\) real polynomials on \(\mathbb{R}^{n}\) with \(Q(0) = 0\) and \(\Omega \in B^{0,0}_{q}(S^{n-1})\) for some \(q > 1\). This note extends and improves a classical result of \textit{E. Stein} (see Theorem 5 in [Ann. Math. Stud. 112, 307--355 (1986; Zbl 0595.00015)]), to the following general form \[ \left| p.v. \int_{\mathbb{R}^{n}} e^{i(P(x)+1/Q(x))} \frac{\Omega(x/|x|)}{|x|^{n}} dx \right| \leq B, \] where \(B\) depends only on \(\| \Omega \|_{B^{0,0}_{q}(S^{n-1})}\), \(n\) and the degrees of \(P\) and \(Q\), but not on their coefficients. The articles of this volume will be reviewed individually under the abbreviation ''Beijing lectures in harmonic analysis, Ann. math. Stud. 112''.

Let \(\Omega\) be homogeneous of degree zero with mean value zero, \(P\) and \(Q\) real polynomials on \(\mathbb{R}^{n}\) with \(Q(0) = 0\) and \(\Omega \in B^{0,0}_{q}(S^{n-1})\) for some \(q > 1\). This note extends and improves a classical result of \textit{E. Stein} (see Theorem 5 in [Ann. Math. Stud. 112, 307--355 (1986; Zbl 0595.00015)]), to the following general form \[ \left| p.v. \int_{\mathbb{R}^{n}} e^{i(P(x)+1/Q(x))} \frac{\Omega(x/|x|)}{|x|^{n}} dx \right| \leq B, \] where \(B\) depends only on \(\| \Omega \|_{B^{0,0}_{q}(S^{n-1})}\), \(n\) and the degrees of \(P\) and \(Q\), but not on their coefficients. Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in \(C^\ast\)-algebras and Lie \(C^\ast\)-algebras and of derivations on non-Archimedean \(C^\ast\)-algebras and non-Archimedean Lie \(C^\ast\)-algebras for an \(m\)-variable additive functional equation.

Truth stressers, also known as modifiers in approximate reasoning applications of fuzzy logics, are modeled here as a kind of modalities. This generalizes an idea offered by \textit{P. Hájek} [Fuzzy Sets Syst. 124, No.~3, 329--333 (2001; Zbl 0997.03028)] for the particular truth stresser ``very true''. The authors extend the Hilbert-type calculus of monoidal t-norm logic MTL, as well as of suitable related mathematical fuzzy logics, with such operators, develop adequate algebraic semantics (the truth stressers correspond to kinds of interior operators in residuated lattices) and give also equivalent hypersequent calculi. And they get decidability results via the discussion of the finite embeddability property. The author considers a notion ``very true'' in the context of fuzzy logic. He builds a logic \(\text{BL}_{\text{vt}}\) on the ground of the basic fuzzy logic BL appending to its set of axioms three new axioms corresponding to the unary connective vt (very true). Then the author gives examples of particular vt operators in different t-norm logics, e.g., in Łukasiewicz many-valued logic, one can take \(\max(0,2x-1)\) to define the ``very true'' connective vt\((x)\), but in this case the axioms for ``very true'' fail to be tautologies for another t-norm logic, the product logic. Also a deduction theorem and a completeness theorem for \(\text{BL}_{\text{vt}}\) are given. The author points out that the question of standard completeness for the Łukasiewicz and the product logics with the vt operator and its additional axioms remains an open problem. I believe that the author's work presented in this paper is essential for the development of basic fuzzy logic and for the investigation of interconnections between different t-norm logics, but it doesn't seem quite new in the framework of many-valued logic to introduce an operator on the real interval \([0,1]\) which evaluates truth values in the way of ``very true''. A lot of people did it before. Nothing would change if one redid their work under the slogan of fuzziness.

Truth stressers, also known as modifiers in approximate reasoning applications of fuzzy logics, are modeled here as a kind of modalities. This generalizes an idea offered by \textit{P. Hájek} [Fuzzy Sets Syst. 124, No.~3, 329--333 (2001; Zbl 0997.03028)] for the particular truth stresser ``very true''. The authors extend the Hilbert-type calculus of monoidal t-norm logic MTL, as well as of suitable related mathematical fuzzy logics, with such operators, develop adequate algebraic semantics (the truth stressers correspond to kinds of interior operators in residuated lattices) and give also equivalent hypersequent calculi. And they get decidability results via the discussion of the finite embeddability property. We shall consider the general problem of causal propagation for spinor fields, focus attention in particular on the case constituted by ELKO fields and will show that the problem of causal propagation for ELKO fields is always solvable.

Let \(V\) be a nonsingular quadratic space over a field \(K\) of characteristic 2. An element of the orthogonal group is called a \textit{half turn} if it is a product of two distinct commuting reflections. The authors show that if \(n>4\), then every element \(\pi\) in the special orthogonal group \(\mathrm{SO}(V)\) is a product of an even number of half turns. If \(\dim B(\pi)=2k\), then the length of \(\pi\) with respect to the half turns is \(k\), \(k+1\), or \(k+2\). The case \(2K=K\) was done previously [\textit{E. W. Ellers}, J. Algebra 99, 275-294 (1986; Zbl 0587.20025)]. Let \(V\) be a vector space over a field of characteristic \(\neq 2\) and \(f\) be a symmetric bilinear form on \(V\). \(V\) is allowed to be of infinite dimension and \(f\) to be degenerate. Let \(R=V^{\perp}\). If \(g\in GL(V)\) then \(B(g)=Im(1-g)V\) and \(F(g)=Ker(1-g)V\). The group \(O(V,f)=\{g\in GL(V)\mid f(x,y)=f(gx,gy)\), \(\dim B(g)<\infty\) and \(F(g)\supset R\}\) is called the orthogonal group of \((V,f)\). Let \(O^+(V)=O(V,f)\cap SL(V)\). An element \(x\in O(V)\) is called a reflection if \(\dim B(x)=1\) and \(B(x)\not\subset R\). If \(z=xy\) where \(x,y\) are reflections and if \(B(x)\subset B(y)^{\perp}\) then \(z\) is called a half-turn. Thus a half-turn is an involution and the multiplicity of the eigenvalue \(-1\) is 2. The main result asserts that if \(g\in O^+(V)\) and \(2t=\dim B(g)+\dim(B(g)\cap R)\) then \(g\) is a product of \(t\), \(t+1\) or \(t+2\) half-turns. -- For a general picture of the problems similar to the one in question the author refers to his earlier article [Abh. Math. Semin. Univ. Hamb. 46, 97-127 (1977; Zbl 0367.50002)].

Let \(V\) be a nonsingular quadratic space over a field \(K\) of characteristic 2. An element of the orthogonal group is called a \textit{half turn} if it is a product of two distinct commuting reflections. The authors show that if \(n>4\), then every element \(\pi\) in the special orthogonal group \(\mathrm{SO}(V)\) is a product of an even number of half turns. If \(\dim B(\pi)=2k\), then the length of \(\pi\) with respect to the half turns is \(k\), \(k+1\), or \(k+2\). The case \(2K=K\) was done previously [\textit{E. W. Ellers}, J. Algebra 99, 275-294 (1986; Zbl 0587.20025)]. This paper presents the design, analysis and implementation of InterCondor system. The InterCondor system is an implementation of the concept of InterGrid. It uses Condor as a basic local Grid computing engine. It utilizes a series of Grid services, which including register service, data transfer service, task schedule service, security authentication service and status monitor service, to manage the resources such as remote sensing algorithms, remote sensing data, and computing resource under the management of Condor engine. We aim at integrating Grid Service data management, task schedule, and the computing power of Condor into remote sensing data processing and analysis to reduce the processing time of a huge amount of data and long-processing-time remote sensing task by algorithms issuance, data division, and the utilization of any computing resources unused on Internet.

The paper refers to a vivid discussion of whether nonlinear modifications of quantum mechanics yield nonlocal phenomena. The description is sketchy and basic definitions and conceptual support must be sought for elsewhere. In fact, to catch the argument of the author, one needs to have at hand section 3 of [\textit{I. Bialynicki-Birula} and \textit{J. Mycielski}, Ann. Phys. (NY) 100, 62-93 (1976) and a number of auxiliary references where an `essential nonlocality' is introduced. Examples are: \textit{J. Polchinski} [Phys. Rev. Lett. 66, 397-400 (1991; Zbl 0968.81504); Phys. Rev. A 58, 128 ff. (1998); Phys. Lett. A 143, 1 ff. (1990)]. Then one may try to analyze the general method for testing essential nonlocality of nonlinear modifications of quantum mechanics, according to the present author. The author shows that Weinberg's nonlinear quantum mechanics leads either to communication via Einstein-Podolsky-Rosen correlations, or to communications between branches of the wave function.

The paper refers to a vivid discussion of whether nonlinear modifications of quantum mechanics yield nonlocal phenomena. The description is sketchy and basic definitions and conceptual support must be sought for elsewhere. In fact, to catch the argument of the author, one needs to have at hand section 3 of [\textit{I. Bialynicki-Birula} and \textit{J. Mycielski}, Ann. Phys. (NY) 100, 62-93 (1976) and a number of auxiliary references where an `essential nonlocality' is introduced. Examples are: \textit{J. Polchinski} [Phys. Rev. Lett. 66, 397-400 (1991; Zbl 0968.81504); Phys. Rev. A 58, 128 ff. (1998); Phys. Lett. A 143, 1 ff. (1990)]. Then one may try to analyze the general method for testing essential nonlocality of nonlinear modifications of quantum mechanics, according to the present author. We address the problem of multi-task learning with no label correspondence among tasks. Learning multiple related tasks simultaneously, by exploiting their shared knowledge can improve the predictive performance on every task. We develop the multi-task Adaboost environment with Multi-Task Decision Trees as weak classifiers. We first adapt the well known decision tree learning to the multi-task setting. We revise the information gain rule for learning decision trees in the multi-task setting. We use this feature to develop a novel criterion for learning Multi-Task Decision Trees. The criterion guides the tree construction by learning the decision rules from data of different tasks, and representing different degrees of task relatedness. We then modify MT-Adaboost to combine Multi-task Decision Trees as weak learners. We experimentally validate the advantage of the new technique; we report results of experiments conducted on several multi-task datasets, including the Enron email set and Spam Filtering collection.

In [Math. Appl. 303, 153-157 (1994; Zbl 0822.17038)] the author showed that for noncommutative even part \(A_0\) of a simple unital associative superalgebra \(A=A_0\oplus A_1\) \((A_1\neq 0)\) with superinvolution \(*\) over a field of \(\text{char}\neq 2\) the sub-superalgebra \(H=H(A,*)\) of \(*\)-symmetric elements of the Jordan superalgebra \(A^+\) is a simple Jordan superalgebra and thus simple as a Jordan \(H\)-module. Here the author proves that if \(A_0\) is noncommutative and \(A\) is not a Clifford superalgebra over the center \(Z\) with \(A_0\) a quaternion algebra over \(Z\), then \(K=K(A,*)\) the sub-superalgebra of skew-symmetric elements of the Lie superalgebra \(A^-\) is again simple as a Jordan \(H\)-module. This also gives a characterization of the Jordan \(H\)-modules of \(A\). The classical Cohn's theorem describes the Jordan algebra of all reversible (symmetric) elements in the free associative algebra. This theorem has proved a keystone in the theory of Jordan algebras. The note under review shows that this theorem can be generalized in a natural way to Jordan superalgebras over a ring of scalars containing \(1/2\).

In [Math. Appl. 303, 153-157 (1994; Zbl 0822.17038)] the author showed that for noncommutative even part \(A_0\) of a simple unital associative superalgebra \(A=A_0\oplus A_1\) \((A_1\neq 0)\) with superinvolution \(*\) over a field of \(\text{char}\neq 2\) the sub-superalgebra \(H=H(A,*)\) of \(*\)-symmetric elements of the Jordan superalgebra \(A^+\) is a simple Jordan superalgebra and thus simple as a Jordan \(H\)-module. Here the author proves that if \(A_0\) is noncommutative and \(A\) is not a Clifford superalgebra over the center \(Z\) with \(A_0\) a quaternion algebra over \(Z\), then \(K=K(A,*)\) the sub-superalgebra of skew-symmetric elements of the Lie superalgebra \(A^-\) is again simple as a Jordan \(H\)-module. This also gives a characterization of the Jordan \(H\)-modules of \(A\). A new class of probability distributions, the so-called connected double truncated gamma distribution, is introduced. We show that using this class as the error distribution of a linear model leads to a generalized quantile regression model that combines desirable properties of both least-squares and quantile regression methods: robustness to outliers and differentiable loss function.

The present paper belongs to the very interesting field of arithmetic dynamical systems with focus on homoclinic points for nonexpansive automorphisms of compact abelian groups. Topics as the existence of nontrivial homoclinic points, expansiveness, entropy and adjoint automorphism (in the sense of \textit{M. Einsiedler} and \textit{K. Schmidt} [Monatsh. Math. 135, 203--220 (2002; Zbl 1007.22014)]) are connected in an ingenious way that yields useful informations about countable abelian group actions by automorphisms of compact abelian groups. Let \(\alpha: \mathbb{Z}^d\to \text{Aut}(X)\), \(\alpha({\mathbf n})= \alpha^{{\mathbf n}}\), be a homomorphism, \(X\) a compact Abelian group. \(\alpha\) is an expansive algebraic \(\mathbb{Z}^d\)-action if there exists open \(O\subset X\) (a neighborhood of the identity \(0_X\in X\)) with \(\bigcap_{{\mathbf n}\in \mathbb{Z}^d} \alpha^{-n}(O)= \{0_X\}\). If \(\alpha\) is a single expansive (and hence ergodic) automorphism of the \(m\)-torus \({\mathbf T}^m\), the adjoint of \(\alpha\) is the automorphism \(\alpha^*\) of \({\mathbf T}^m\) defined by the transpose of the matrix \(A\) defining \(\alpha\). The general notion of adjoint \(\alpha^*\) of an algebraic \(\mathbb{Z}^d\)-action is studied, and is shown to have completely positive entropy (although \(\alpha\) may not). \(\alpha\) and \(\alpha^*\) have the same entropy and \(\alpha\) is conjugate to \(\alpha^{***}\), but need not be conjugate to \(\alpha^{**}\) (reflexive), but those resulting from single automorphisms are always reflexive. Finally, a more general notion of adjoint action is mentioned for arbitrary expansive algebraic \(\mathbb{Z}^d\)-actions with not necessarily positive entropy.

The present paper belongs to the very interesting field of arithmetic dynamical systems with focus on homoclinic points for nonexpansive automorphisms of compact abelian groups. Topics as the existence of nontrivial homoclinic points, expansiveness, entropy and adjoint automorphism (in the sense of \textit{M. Einsiedler} and \textit{K. Schmidt} [Monatsh. Math. 135, 203--220 (2002; Zbl 1007.22014)]) are connected in an ingenious way that yields useful informations about countable abelian group actions by automorphisms of compact abelian groups. We present a scalable parallel algorithm for treating two-sided contact in a finite-element multi-physics code (ALE3D). This method assumes that proximity between the two sides changes continuously, and uses a local search to update proximity relations for each cycle. The evolving communication pattern is treated entirely by local, nearest-neighbor communication; there is no global communication. Opening and closing voids, overlapping and intersecting contact surfaces, and a number of other special features are reported.

It is shown that images of simple modules under projective functors in Bernstein-Gelfand-Gelfand category \(\mathcal O\) can be presented in the derived category by linear complexes of tilting modules. These complexes are also images of simple modules under projective functors in the Koszul dual of \(\mathcal O\). For part I cf. Pac. J. Math. 232, No. 2, 313-341 (2007; Zbl 1200.16012). Projective dimension of the structural modules in the Bernstein-Gelfand-Gelfand category \(\mathcal O\) is studied. The dimension is calculated for simple and (co)standard modules. It is observed that the extension algebra of Verma modules is \(\mathbb{Z}^2\)-graded. Some new classes of Koszul modules are found.

It is shown that images of simple modules under projective functors in Bernstein-Gelfand-Gelfand category \(\mathcal O\) can be presented in the derived category by linear complexes of tilting modules. These complexes are also images of simple modules under projective functors in the Koszul dual of \(\mathcal O\). For part I cf. Pac. J. Math. 232, No. 2, 313-341 (2007; Zbl 1200.16012). The Painlevé equations \((P1-P6)\) have numerous applications in modern mathematics and mathematical physics. The solutions of these equations, as they are meromorphic in the complex plane can be studied from the perspective of value distribution and growth theory, with such values as defect, deviation or multiplicity index estimated. The article is devoted to review of the basic features of transcendental meromorphic solutions of the equations \(P1\), \(P2\), \(P4\) and \(P34\), including the finiteness of order of growth, relatively regular distribution of values and limits for ramification indices. The article contains a new estimate of Petrenko's deficiency for solutions of equation \(P1\).

The paper deals with the focusing nonlinear Schrödinger equation \[ i q_t + q_{xx} - 2 \nu |q|^2 q = 0 \] with nonzero boundary conditions as \(x \to \pm \infty\). The solution of this equation by the inverse scattering transform was obtained in [\textit{G. Biondini} and \textit{G. Kovačič}, J. Math. Phys. 55, 031506 (2014; Zbl 1298.35187)]. In the present paper, the authors apply this method to the initial value problem with the following ``box-like'' initial condition: \(q(x, 0) = 1\) for \(|x| > L\) and \(q(x, 0) = b e^{i \alpha}\) for \(|x|<L\). The authors characterize the number and location of discrete eigenvalues for differents cases, when \(0 < b < 1\) and \(b > 1\). They also show that the limit of solutions, obtained by the inverse scattering transform, when the initial conditions are small deviations of the constant background, coincides with that of the direct linearization of the considered equation around the constant background, apart from the possible contribution of the discrete spectrum. Based on the results of the paper, the authors propose to use the inverse scattering transform to investigate the nonlinear stage of the modulation instability (the instability of a constant background with respect to long wavelength perturbations). Authors' abstract: The inverse scattering transform for the focusing nonlinear Schrödinger equation with non-zero boundary conditions at infinity is presented, including the determination of the analyticity of the scattering eigenfunctions, the introduction of the appropriate Riemann surface and uniformization variable, the symmetries, discrete spectrum, asymptotics, trace formulae and the so-called theta condition, and the formulation of the inverse problem in terms of a Riemann-Hilbert problem. In addition, the general behavior of the soliton solutions is discussed, as well as the reductions to all special cases previously discussed in the literature.

The paper deals with the focusing nonlinear Schrödinger equation \[ i q_t + q_{xx} - 2 \nu |q|^2 q = 0 \] with nonzero boundary conditions as \(x \to \pm \infty\). The solution of this equation by the inverse scattering transform was obtained in [\textit{G. Biondini} and \textit{G. Kovačič}, J. Math. Phys. 55, 031506 (2014; Zbl 1298.35187)]. In the present paper, the authors apply this method to the initial value problem with the following ``box-like'' initial condition: \(q(x, 0) = 1\) for \(|x| > L\) and \(q(x, 0) = b e^{i \alpha}\) for \(|x|<L\). The authors characterize the number and location of discrete eigenvalues for differents cases, when \(0 < b < 1\) and \(b > 1\). They also show that the limit of solutions, obtained by the inverse scattering transform, when the initial conditions are small deviations of the constant background, coincides with that of the direct linearization of the considered equation around the constant background, apart from the possible contribution of the discrete spectrum. Based on the results of the paper, the authors propose to use the inverse scattering transform to investigate the nonlinear stage of the modulation instability (the instability of a constant background with respect to long wavelength perturbations). \[ \begin{multlined} \cos ax = 1 - ax\sin bx - \frac{a(a-2b)}{2!}x^2\cos2bx\\ + \frac{a(a-3b)^2}{3!}x^3\sin3bx + \frac{a(a-4b)^2}{4!}x^4\cos4bx - \cdots,\end{multlined} \] \[ \begin{multlined} \sin ax = ax\cos bx - \frac{a(a-2b)}{2!}x^2\sin2bx - \frac{a(a- 3b)^2}{3!}x^3\cos3bx\\ + \frac{a(a-4b)^3}{4!}x^4\sin4bx +\cdots,\end{multlined} \] Beweis dieser Formeln von F. H. Peachell.

The paper under review deals with a problem in metric Diophantine approximation. To state the result of this paper we first introduce some notation. Let \(P_n\) denote the set of integer polynomials of degree at most \(n\) and let \(\Psi\) be a positive function. Let \(W\) be the set of all complex numbers \(z\) which satisfy the inequality \(|P(z)|<\Psi(H(P))\) for infinitely many integer polynomials \(P\in P_n\), where \(H(P)\) is the maximum of the modulus of the integer coefficients of \(P\). It is proved that, for \(n\geq 3\) the Lebesgue measure of the set \(W\) is zero if \[ \sum_{k=1}^\infty k^{n-2}\Psi^2(k)<\infty. \] The main novelty of this result is that the function \(\Psi\) is non-monotonic. The same result over the set of real numbers was established by \textit{V. Beresnevich} [Acta Arith. 117, No. 1, 71--80 (2005; Zbl 1201.11078)]. In 1932 \textit{K. Mahler} [J. Reine Angew. Math. 166, 118--136 (1931; Zbl 0003.15101, JFM 57.0242.03), ibid. 166, 137--150 (1932; Zbl 0003.38805, JFM 58.0207.01)] introduced a classification of real numbers \(x\) into the so-called classes of \(A, S, T,\) and \(U\) numbers according to the behavior of \(w_n(x)\) defined as the supremum of \(w > 0\) for which \(| P(x)| < H(P)^{-w}\) holds for infinitely many \(P\in P_n\). (Here \(P_n\) denotes the set of integral polynomials of degree \(\leq n\), and \(H(P)\) denotes the height of \(P\), i.e. the maximum of the absolute values of its coefficients.) By Minkowski's theorem on linear forms, one readily shows that \(w_n(x) > n\) for all \(x\in\mathbb R\). Mahler [Math. Ann. 106, 131--139 (1932; Zbl 0003.24602, JFM 58.0206.04)] proved that for almost all \(x\in\mathbb R\) (in the sense of Lebesgue measure) \(w_n(x)\leq 4n\), thus almost all \(x\in\mathbb R\) are in the \(S\)-class. Mahler has also conjectured that for almost all \(x\in\mathbb R\) one has the equality \(w_n(x)=n\). For about 30 years the progress in Mahler's problem was limited to \(n=2\) and 3 and to partial results for \(n > 3\). V. Sprindzhuk proved Mahler's conjecture in full. Let \(W_n(\Psi)\) be the set of \(x\in\mathbb R\) such that there are infinitely many \(P\in P_n\) satisfying \[ | P(x)| < \Psi(H(P)). \tag{1} \] \textit{A. Baker} [Proc. R. Soc. Lond., Ser. A 292, 92--104 (1966; Zbl 0146.06302)] has improved Sprindzhuk's theorem by showing that \(| W_n(\Psi)| = 0\) if \(\sum_{h=1}^\infty \Psi^{1/n}(h) < \infty\) and \(\Psi\) is monotonic. He also conjectured a stronger statement proved by \textit{V. I. Bernik} [Acta Arith. 53, 17--28 (1989; Zbl 0692.10042)] that \(| W_n(\Psi)| = 0\) if the sum \[ \sum_{h=1}^\infty h^{n-1}\Psi(h) \tag{2} \] converges and \(\Psi\) is monotonic. Later the author [Acta Arith. 90, No. 2, 97--112 (1999; Zbl 0937.11027)] has shown that \(| \mathbb R\setminus W_n(\Psi)| = 0\) if (2) diverges and \(\Psi\) is monotonic. Here the author proves the following: Theorem 1. Let \(\Psi: \mathbb R\to \mathbb R^+\) be an arbitrary function (not necessarily monotonic) such that the sum (2) converges. Then \(| W_n(\Psi)| = 0.\) The author remarks that Theorem 1 is no longer improvable as, by the result of his paper (loc. cit.), the convergence of (2) is crucial. Theorem 1 of this paper can be readily generalized for non-degenerate curves: Given a non-degenerate map \(f : I \to\mathbb R^n\) defined on an interval \(I\), for any function \(\Psi: \mathbb R\to\mathbb R^+\) such that the sum (2) converges for almost all \(x\in I\) the point \(f(x)\) is not \(\Psi\)-approximable.

The paper under review deals with a problem in metric Diophantine approximation. To state the result of this paper we first introduce some notation. Let \(P_n\) denote the set of integer polynomials of degree at most \(n\) and let \(\Psi\) be a positive function. Let \(W\) be the set of all complex numbers \(z\) which satisfy the inequality \(|P(z)|<\Psi(H(P))\) for infinitely many integer polynomials \(P\in P_n\), where \(H(P)\) is the maximum of the modulus of the integer coefficients of \(P\). It is proved that, for \(n\geq 3\) the Lebesgue measure of the set \(W\) is zero if \[ \sum_{k=1}^\infty k^{n-2}\Psi^2(k)<\infty. \] The main novelty of this result is that the function \(\Psi\) is non-monotonic. The same result over the set of real numbers was established by \textit{V. Beresnevich} [Acta Arith. 117, No. 1, 71--80 (2005; Zbl 1201.11078)]. Software Transactional Memory (STM) is one promising abstraction to simplify the task of writing highly parallel applications. Nonetheless, in workloads lacking enough parallelism, STM's optimistic approach to concurrency control can adversely degrade performance as transactions abort and restart often. In this paper, we describe a new scheduling-based solution to improve STM's performance in high-contention scenarios. Our Progressively Pessimistic Scheduler (ProPS) uses a fine-grained scheduling mechanism that controls the amount of concurrency in the system gradually as transactions abort and commit with success. Experimental results with the STMBench7 benchmark and the STAMP benchmark suite showed that current coarse-grained, conservative transaction schedulers are not suitable for workloads with long transactions, whereas ProPS is up to 40\% faster than all other scheduling alternatives.

Der Verf. entwickelt für ein von \textit{Th. P. Hill} [Am. Math. Mon. 90, 438-442 (1983; Zbl 0533.28004)] untersuchtes Teilungsproblem einen ``Algorithmus'', der zu einer ``gerechten'' Aufteilung eines einfach zusammenhängenden Gebietes D unter n Anrainern führt. Dazu wird zunächst bewiesen, daß - unter naheliegenden Voraussetzungen - D als Einheitskreis angenommen werden kann, wobei die Radien und konzentrischen Kreise von jedem der Anrainer mit dem Maß 0 bewertet werden. Die in Form einer Auktion ablaufende Verteilung führt nach endlich vielen Schritten zum Ziel. Das Problem, ein Gebiet unter n angrenzenden Ländern ''gerecht'' aufzuteilen, so daß jedes Land ein einzelnes angrenzendes Stück erhält, unterscheidet sich von dem bekannten Steinhausschen ''Kuchenverteilungsproblem'' dadurch, daß topologische Bedingungen zu beachten sind. Unter Zuhilfenahme eines Ergebnisses von \textit{L. E. Dubins} and \textit{E. H. Spanier} [Am. Math. Mon. 68, 1-17 (1961; Zbl 0108.316), p. 6] beweist der Verf. folgenden \(Satz:\) D,\(A_ 1,...,A_ n\) seien offene zusammenhängende Gebiete im \({\mathbb{R}}^ 2\), wobei alle \(A_ i\) an D ''angrenzen''. \(\mu_ 1,...,\mu_ n\) seien nichtatomare Wahrscheinlichkeitsmaße auf D und für n vorgegebene Zahlen \(p_ i\geq 0\) gelte \(\sum^{n}_{i=1}p_ i=1.\) Dann gibt es disjunkte offene zusammenhängende Teilmengen \(B_ 1,...,B_ n\), die jeweils an \(A_ 1,...,A_ n\) angrenzen, für die gilt: \(\mu_ i(B_ i)\geq p_ i\) und \(\overline{\cup_{1\leq i\leq n}B_ i}=D.\) Da es stets unendlich viele Lösungen des Verteilungsproblems gibt, erhebt sich die Frage nach vergleichsweise besseren bzw. ''optimalen'' Lösungen. Hierzu werden einige Optimalitätsbegriffe erörtert.

Der Verf. entwickelt für ein von \textit{Th. P. Hill} [Am. Math. Mon. 90, 438-442 (1983; Zbl 0533.28004)] untersuchtes Teilungsproblem einen ``Algorithmus'', der zu einer ``gerechten'' Aufteilung eines einfach zusammenhängenden Gebietes D unter n Anrainern führt. Dazu wird zunächst bewiesen, daß - unter naheliegenden Voraussetzungen - D als Einheitskreis angenommen werden kann, wobei die Radien und konzentrischen Kreise von jedem der Anrainer mit dem Maß 0 bewertet werden. Die in Form einer Auktion ablaufende Verteilung führt nach endlich vielen Schritten zum Ziel. This paper presents work on document retrieval for Italian carried out at ITC-irst. Two different approaches to information retrieval were investigated, one based on the Okapi weighting formula and one based on a statistical model. Development experiments were carried out using the Italian sample of the TREC-8 CLIR track. Performance evaluation was done on the Cross Language Evaluation Forum (CLEF) 2000 Italian monolingual track. The two methods achieved mean average precisions of 49.0\% and 47.5\%, respectively, which were the two best scores of their track.

The equations of the Föppl-Hencky small finite deflection theory are studied. Uniqueness of tensile solutions for annular membranes with given radial displacement at the inner edge and prescribed radial tensile stress (or radial displacement) at the outer edge is proved. The proof of both cases is based on Hopf's maximum principle. The paper supplies the earlier results obtained by \textit{H. J. Weinitschke} and the first author [ibid. 16, 135-147 (1986; Zbl 0585.73064)]. Axisymmetric deformations of annular membranes subjected to normal surface loads and radial edge loads or displacements are considered within the Föppl nonlinear membrane theory. When the inner edge \(r=a\) is free of radial traction, the solution of the annular membrane problem is shown to reduce to the solution for the circular membrane \((a=0)\). For nonvanishing traction at \(r=a\), the problem is reduced to a circular pseudo-membrane problem. For both cases, existence and uniqueness of tensile solutions of the annular membrane problem are proved, including a rigorous derivation of a stress concentration factor originally found by \textit{E. Schwerin} by formal methods [Z. Tech. Phys. 12, 651-659 (1929)].

The equations of the Föppl-Hencky small finite deflection theory are studied. Uniqueness of tensile solutions for annular membranes with given radial displacement at the inner edge and prescribed radial tensile stress (or radial displacement) at the outer edge is proved. The proof of both cases is based on Hopf's maximum principle. The paper supplies the earlier results obtained by \textit{H. J. Weinitschke} and the first author [ibid. 16, 135-147 (1986; Zbl 0585.73064)]. The Poincaré \(\delta\)-lemma (stable triviality of Spencer cohomology groups) for smooth algebras is proved.

Let \(RG\) be the group ring of the group \(G\) over the ring \(R\) with identity, and denote by \(R_0\) the centre of the ring \(R\), by \(C\) the centre of the group ring \(RG\) and by \(\overline G\) the group of inner automorphisms of the group ring \(RG\) induced by elements of the group \(G\). Descriptions of group rings \(RG\) which are Azumaya algebras are well-known. The main result of the authors is that the following statements are equivalent: (1) the group ring \(RG\) is a Hirata separable extension of the group ring \(RC\) and \(C\) is a direct summand of the group ring \(R_0G\) as a \(C\)-module; (2) the group ring \(R_0G\) is an Azumaya algebra; (3) the centre of the group \(G\) is of finite index and the order of the commutator subgroup of the group \(G\) is invertible in the ring \(R\). Moreover, a characterization is given of Azumaya \(C\)-algebras \(RG\) in terms of Hirata-separable extensions, and it is shown that the group ring \(RG\) is neither a Hirata-separable extension of the ring \(R\) nor a Galois extension of \((RG)^{\overline G}\) with Galois-group \(\overline G\). This paper is a continuation of a series of papers of the authors on Galois extensions [see, for instance, J. Algebra Number Theory, Adv. Appl. 2, No. 1, 21-26 (2009; Zbl 1196.16015)]. Let \(G\) be a group with center \(Z\), \(R\) a ring with unity, \(RG\) the group ring. The authors give a different proof of a result in the paper by \textit{F. R. DeMeyer, G. J. Janusz} [Trans. Am. Math. Soc. 279, 389-395 (1983; Zbl 0525.16002)], namely, that the group ring \(RG\) is an Azumaya algebra if and only if there exists a subgroup \(H\) with \(G=HZ\) such that the group ring \(RH\) is an Azumaya algebra. Moreover, another characterization of Azumaya group rings is given in terms of Hirata separable extensions. This paper is a continuation of a series of papers of the authors on Galois extensions the most recent of which is [Sci. Math. Jpn. 69, No. 3, 405-410 (2009; Zbl 1192.16041)].

Let \(RG\) be the group ring of the group \(G\) over the ring \(R\) with identity, and denote by \(R_0\) the centre of the ring \(R\), by \(C\) the centre of the group ring \(RG\) and by \(\overline G\) the group of inner automorphisms of the group ring \(RG\) induced by elements of the group \(G\). Descriptions of group rings \(RG\) which are Azumaya algebras are well-known. The main result of the authors is that the following statements are equivalent: (1) the group ring \(RG\) is a Hirata separable extension of the group ring \(RC\) and \(C\) is a direct summand of the group ring \(R_0G\) as a \(C\)-module; (2) the group ring \(R_0G\) is an Azumaya algebra; (3) the centre of the group \(G\) is of finite index and the order of the commutator subgroup of the group \(G\) is invertible in the ring \(R\). Moreover, a characterization is given of Azumaya \(C\)-algebras \(RG\) in terms of Hirata-separable extensions, and it is shown that the group ring \(RG\) is neither a Hirata-separable extension of the ring \(R\) nor a Galois extension of \((RG)^{\overline G}\) with Galois-group \(\overline G\). This paper is a continuation of a series of papers of the authors on Galois extensions [see, for instance, J. Algebra Number Theory, Adv. Appl. 2, No. 1, 21-26 (2009; Zbl 1196.16015)]. In recent years, the gap between theory and practice in quantum key distribution (QKD) has been significantly narrowed, particularly for QKD systems with arbitrarily flawed optical receivers. The status for QKD systems with imperfect light sources is however less satisfactory, in the sense that the resulting secure key rates are often overly dependent on the quality of state preparation. This is especially the case when the channel loss is high. Very recently, to overcome this limitation, Tamaki et al proposed a QKD protocol based on the so-called `rejected data analysis', and showed that its security -- in the limit of infinitely long keys -- is almost independent of any encoding flaw in the qubit space, being this protocol compatible with the decoy state method. Here, as a step towards practical QKD, we show that a similar conclusion is reached in the finite-key regime, even when the intensity of the light source is unstable. More concretely, we derive security bounds for a wide class of realistic light sources and show that the bounds are also efficient in the presence of high channel loss. Our results strongly suggest the feasibility of long distance provably secure communication with imperfect light sources.

In an earlier paper [ibid. 20, No. 4, 393-403 (1996; Zbl 0869.70010)], the authors have shown that the motion of charged test particles in an electromagnetic field can be derived from a gauge-invariant second-order Lagrangian which differs from the (gauge-dependent first-order) standard Lagrangian by a boundary term. The paper under review is devoted to the corresponding Hamiltonian theory. A characteristic feature of this approach is that, unlike in the standard formulation, the canonical momenta are gauge-invariant and have a clear physical meaning. The particle trajectories are constructed as the characteristic curves of a pre-symplectic form \(\Omega_N\) on the evolution space; this form \(\Omega_N\) is uniquely determined by the gauge-invariant Lagrangian via the Legendre transformation. New, gauge-independent, second order Lagrangian for the motion of classical, charged test particles is proposed. It differs from the standard, gauge-dependent, first order Lagrangian by boundary terms only. A new method of deriving equations of motion from field equations is developed. When applied to classical electrodynamics, this method enables us to obtain unambiguously the above second order Lagrangian from the general energy-momentum conservation principle.

In an earlier paper [ibid. 20, No. 4, 393-403 (1996; Zbl 0869.70010)], the authors have shown that the motion of charged test particles in an electromagnetic field can be derived from a gauge-invariant second-order Lagrangian which differs from the (gauge-dependent first-order) standard Lagrangian by a boundary term. The paper under review is devoted to the corresponding Hamiltonian theory. A characteristic feature of this approach is that, unlike in the standard formulation, the canonical momenta are gauge-invariant and have a clear physical meaning. The particle trajectories are constructed as the characteristic curves of a pre-symplectic form \(\Omega_N\) on the evolution space; this form \(\Omega_N\) is uniquely determined by the gauge-invariant Lagrangian via the Legendre transformation. Let \(\mathfrak M_{r} = \{E^{(j)}\}_{j=0}^{r}\) be a nested non-increasing system of sets \(E^{(j)} \subset {\mathbb R}\). A function \(\varphi \in C[0,\infty)\) belongs to \(\Phi_{k}\) if \(\varphi\) does not decrease, \(t^{-k} \varphi(t)\) does not increase on \((0,\infty)\), \(\varphi(0) = 0\), and \(\varphi(1) \not = 0\). The author proves that for each non-increasing system of sets \(\mathfrak M_{r}\) and each \(\varphi \in \Phi_{k}\), there exists a linear bounded operator \(A: W^{r} {\bar H}_{k}^{\varphi}(\mathfrak M_{r}) \rightarrow W^{r} H_{k}^{\varphi}({\mathbb R})\).

The authors apply finite element methods to elliptic problems on closed Riemannian manifolds and propose an approach different from constructing grids on manifolds (cf. [\textit{M. A. Olshanskii} and \textit{A. Reusken}, Numer. Math. 114, No. 3, 491--520 (2010; Zbl 1204.65136)]). The novelty of the approach is to avoid triangulations on curved manifolds by solving problems on Euclidean spaces. A numerical test for a problem on a 3-dimensional manifold is presented. A new finite element approach is introduced for the discretization of elliptic partial differential equations (PDEs) on surfaces. The finite element spaces are induced by triangulations of an `outer' domain. Applications are given to two-phase flow problems. The paper analyzes the two-dimensional case and shows that the (effective) spectral condition number of the diagonally scaled mass matrix and the diagonally scaled stiffness matrix behaves like \(h^{-3}|\log h|\) and \(h^{-2}|\log h|,\) respectively, where \(h\) is the meshsize of the outer triangulation. Examples are given.

The authors apply finite element methods to elliptic problems on closed Riemannian manifolds and propose an approach different from constructing grids on manifolds (cf. [\textit{M. A. Olshanskii} and \textit{A. Reusken}, Numer. Math. 114, No. 3, 491--520 (2010; Zbl 1204.65136)]). The novelty of the approach is to avoid triangulations on curved manifolds by solving problems on Euclidean spaces. A numerical test for a problem on a 3-dimensional manifold is presented. [For the entire collection see Zbl 0632.00020.] The author argues that: (1) Point-valued probabilities are a poor representation of uncertainty; (2) Interval-valued probabilities that result from knowledge of convex sets of distribution functions in reference classes, (properly) include Shafer's mass functions as a special case; (3) These probabilities yield a plausible non-monotonic form of inference (uncertain inference, inductive inference, statistical inference); (4) The proposed framework provides a very nearly classical decision theory. The author sketches the properties of interval-valued epistemic probability, and exhibits a structure for knowledge representation suitable for both uncertain inference from evidence and uncertain knowledge as a basis for decision.

This paper generalizes a limit theorem of \textit{G. Huisken} [ibid. 20, 237-266 (1984; Zbl 0556.53001)] and J. Urbas on the outward flow of hypersurfaces in \({\mathbb{R}}^{n+1}\) by mean curvature from the convex to the star shaped situation. Also, the (first) mean curvature is replaced by more general functions of the principal curvatures. The method is different and provides a fine piece of fruitful combination of differential geometry with PDE theory. Fairly intimate properties of hypersurface immersions like Codazzi tensors and their second covariant derivatives have to be applied to get the necessary uniform estimates for the parabolic flow equation. Let \(M_ 0\) be a smooth closed convex hypersurface with everywhere positive curvatures in Euclidean space \({\mathbb{R}}^{n+1}\). Suppose that \(M_ 0\) is smoothly deformed (that is, embedded into a smooth family \((M_ t\), \(t\geq 0)\) of hypersurfaces depending on a time parameter) such that, at every time t, the rate of change of \(M_ t\) in direction of the inward unit normal vector is equal to the (positive) mean curvature of the hypersurface at the point considered. The author shows that the parabolic evolution equation describing the problem has a smooth solution on a finite time interval \(0\leq t\leq T\), and the \(M_ t's\) converge to a point as \(t\to T\). Moreover, if the surfaces undergo suitable homotheties and the time parameter is transformed appropriately into a parameter \(\tilde t,\) \(0\leq \tilde t<\infty\), it is shown that the normalized surfaces converge to a sphere in the \(C^{\infty}\)-topology as \(\tilde t\to \infty\). As the author says, his approach is inspired by a paper of \textit{R. S. Hamilton} [J. Differ. Geom. 17, 255-306 (1982; Zbl 0504.53034)], and he can use many of the methods developed there. The case \(n=1\) was treated by \textit{M. E. Gage} [Invent. Math. 76, 357-364 (1984; Zbl 0542.53004)]. \{Reviewer's remark: A similar problem, related in spirit though not in the methods, was treated by \textit{W. J. Firey} [Mathematika 21, 1-11 (1974; Zbl 0311.52003)]. There the rate of change of the support function is proportional to the Gauss curvature.\}

This paper generalizes a limit theorem of \textit{G. Huisken} [ibid. 20, 237-266 (1984; Zbl 0556.53001)] and J. Urbas on the outward flow of hypersurfaces in \({\mathbb{R}}^{n+1}\) by mean curvature from the convex to the star shaped situation. Also, the (first) mean curvature is replaced by more general functions of the principal curvatures. The method is different and provides a fine piece of fruitful combination of differential geometry with PDE theory. Fairly intimate properties of hypersurface immersions like Codazzi tensors and their second covariant derivatives have to be applied to get the necessary uniform estimates for the parabolic flow equation. We present a model of the establishment and maintenance of communication between mobile agents. We assume that the agents move through a fixed environment modeled by a motion graph and are able to communicate if they are within distance at most \(d\) of each other. As the agents move randomly, we analyze the evolution in time of the connectivity between a set of \(w\) agents, asymptotically for a large number \(N\) of vertices, when \(w\) also grows large. The particular topologies of the environment we study here are the cycle and the toroidal grid.

A class of controlled branching processes with continuous time is introduced. It consists of discrete controlled branching processes indexed by renewal processes \(N\). At all time \(t \geq 0\), the size of the population is denoted by \(Y(t)\). At each jump time of \(N\), a number of individuals can be removed from the population, then all remaining individuals reproduce independently, then a certain number of new particles can be added, allowing to control the size of the population. Limit theorems, introduced in [\textit{I. del Pietro} et al., C. R. Acad. Bulg. Sci. 74, No. 3, 332--342 (2021; Zbl 1488.60197)], are proved for some critical controlled branching processes with continuous time. In this paper, a controlled branching process with continuous time is introduced and studied in the critical case. First, the authors define controlled branching processes with continuous time and with single and multiple control functions. Two limit theorems are stated in the cases when the mean of the renewal periods is either finite or infinite. The main result is the investigation of controlled branching processes with continuous time and multiple control functions. The case with three specific control functions is considered which is a controlled branching process with random migration. Finally, the regenerative controlled branching process with continuous time is studied.

A class of controlled branching processes with continuous time is introduced. It consists of discrete controlled branching processes indexed by renewal processes \(N\). At all time \(t \geq 0\), the size of the population is denoted by \(Y(t)\). At each jump time of \(N\), a number of individuals can be removed from the population, then all remaining individuals reproduce independently, then a certain number of new particles can be added, allowing to control the size of the population. Limit theorems, introduced in [\textit{I. del Pietro} et al., C. R. Acad. Bulg. Sci. 74, No. 3, 332--342 (2021; Zbl 1488.60197)], are proved for some critical controlled branching processes with continuous time. Necessary and sufficient globally optimal conditions - a matrix equation and a matrix inequality - are given for the existence of the optimal constant output feedback gain. Furthermore, it is shown that if the optimal output law \(L_ 0\) exists, it must be a solution derived from the corresponding optimal state-variable feedback problem, that is \(L_ 0C=K_ 0\), where \(K_ 0\) is the optimal state-variable feedback law. An example is given to show that a globally optimal output law may not be found even if the system is stabilizable by output feedback. These results are helpful for the understanding of the fundamental problems of output feedback and give the reasons for suboptimal approaches as adopted by \textit{W. S. Levine} and \textit{M. Athans} [IEEE Trans. Autom. Control AC-15, 44-48 (1970); see also Proc. 3rd Annual Princeton Conf. Inform Sci. Syst., 519-523 (1969; Zbl 0291.49015)] and other authors.

This second volume completes the first volume in covering the basic mathematics for courses in physics and engineering [Cambridge University Press 1995; Zbl 0840.00001]. As in the first volume, the book collects questions that have been posed to the author by his students in the course of many years offering tutorial for undergraduate physics students at Oxford University. Being an experimental physicist, most of the subjects are illustrated by examples and ways of dealing with questions of physics. The author insists on that the reader has to work through the problems at the end of each chapter in order to get the optimum benefit from the book. Mathematics comes as a natural way of dealing with problems from physics. Repeating what I wrote in the review of the first volume, this is a very interesting and useful book. It is a good support for students who want to see mathematics as more than a mere manipulation of definitions, theorems and formulae. This second volume is organized in eight chapters, in sequel to those of the first volume: 9. Integrals; 10. Vector operators; 11. Partial differential equations; 12. Fourier series; 13. Normal modes; 14. Waves; 15. Matrices; 16. Eigenvectors and eigenvalues. It also has an Appendix C, with a summary of the crucial equations for each chapter, and an Index. One of the most critical difficulties in current mathematics education, in all levels, is motivation. It seems that the traditional approaches to motivation, such as the importance of the subject and the beauty of its internal organization, are failing to attract students to mathematics. Enrolment in mathematics, even for science students, is declining all over the world. The usual pedagogical practice in mathematics seem to be a reason for this. Normally, the mathematics instructor goes to the classroom prepared to lecture on a topic and proceeds with the exposition. Students are explicitly or implicitly disencouraged to ask questions which deviate from the main course of the exposition. Many things that the student wanted to know about mathematics are never asked. The author, an experimental physicist, offers a tutorial for first year physics undergraduates of Oxford University, which ``supplement the conventional mathematical presentations with one that stressed explanations which are more meaningful to scientists.'' (p. xv). This book is a collection of questions that have been posed to the author by his students in the course of many years of this practice. Most of the questions are related to how does one translate physical facts, such as for example resonance, radioactive decay, gravity, into mathematical expressions. The book is addressed to first year undergraduates, hence the treatment is elementary and sometimes informal. The book has a Preface of four pages which, although this is not its objective, can be considered a most inspiring methodological proposal. First of all, it is feasible. It does not require any change in the regular way of conducting classes, no change of curricula or textbooks, which are some of the major obstacles to improve university mathematics education. The course simply runs in parallel to the regular courses, it does not matter how well or poorly it is conducted, as a supplementary activity. The proposal is very similar to a ``tailored'' pedagogy. The questions are organized in eight chapters and subsections dealing with: 1. Simultaneous equations; 2. Three-dimensional geometry; 3. Vectors; 4. Complex numbers; 5. Ordinary differential equations; 6. Partial derivatives; 7. Taylor series; 8. Lagrangian multipliers. Each chapter has a subsection on Problems. Two appendices give the basic techniques (20 pages) and useful formulae. An Index is very useful in providing quick reference to both interesting questions and mathematical topics. This is a very interesting and useful book. It may be a good support for students who see the mathematics taught in their classes as nothing more than a mere manipulation of definitions, theorems and formulae. A second volume is promised.

This second volume completes the first volume in covering the basic mathematics for courses in physics and engineering [Cambridge University Press 1995; Zbl 0840.00001]. As in the first volume, the book collects questions that have been posed to the author by his students in the course of many years offering tutorial for undergraduate physics students at Oxford University. Being an experimental physicist, most of the subjects are illustrated by examples and ways of dealing with questions of physics. The author insists on that the reader has to work through the problems at the end of each chapter in order to get the optimum benefit from the book. Mathematics comes as a natural way of dealing with problems from physics. Repeating what I wrote in the review of the first volume, this is a very interesting and useful book. It is a good support for students who want to see mathematics as more than a mere manipulation of definitions, theorems and formulae. This second volume is organized in eight chapters, in sequel to those of the first volume: 9. Integrals; 10. Vector operators; 11. Partial differential equations; 12. Fourier series; 13. Normal modes; 14. Waves; 15. Matrices; 16. Eigenvectors and eigenvalues. It also has an Appendix C, with a summary of the crucial equations for each chapter, and an Index. This article suggests a method of the determination of main parameters and dynamic characteristics of heat transfer crisis on a surface of fast heated wall. The new physical models describing process of transition from nucleate to film boiling are presented. The results of transient critical heat flux modeling are compared with the experimental data for saturated water pool boiling under atmospheric pressure.

The operator \(e^{it\Delta}\) is \(L^p(\mathbb{R}^n) \mapsto L^p(\mathbb{R}^n)\) bounded only when \(p = 2\), but the \(p\) range can be widened prior regularization. This type of phenomenon was already investigated, for example, in [\textit{T. A. Bui} et al., Rev. Mat. Iberoam. 36, No. 2, 455--484 (2020; Zbl 1448.35352)]. In the article under review, the authors consider mapping properties of \(e^{itL}\) for non-negative self-adjoint operators \(L\) in a metric space \((X, d, \mu)\) with doubling measure \(\mu\). Their main result is the inequality \[ \lVert e^{itL}(I + L)^{-\sigma_pn}f \rVert_{L^p(X)} \le C(I + \lvert t \rvert)^{\sigma_pn}\lVert f \rVert_{L^p(X)}, \qquad \text{for } \sigma_p := \bigg\lvert \frac{1}{2} - \frac{1}{p} \bigg\rvert, \tag{1}\] where \(p \in (p_0, p_0^\prime)\) with \(1\le p_0 < 2\), and \(e^{-tL}\) satisfies \[ \lVert \1_{B(x,t^{1/m})}e^{-tL}V_{t^{1/m}}^{\sigma_{p_0}} \1_{B(y,t^{1/m})} f \rVert_{L^2(X)} \le C \Big(1 + \frac{d(x,y)}{t^{1/m}}\Big)^{-n - \kappa} \lVert f \rVert_{L^{p_0}(X)}, \tag{2}\] where \(\kappa > [n/2] + 1\) and \(V_r(x) := \mu(B(x,r))\). The hypothesis (2) covers a wide class of operators. The authors develop a theory of Hardy spaces \(H_L^q(X)\) adapted to \(L\) and show that, by duality and interpolation, it suffices to prove (1) with \(H_L^q(X)\), for \(q < 1\), instead of \(L^p(X)\). To show that the operator is bounded in \(H^q_L(X)\), they prove new estimates for oscillatory multipliers to control off-diagonal terms. The proof uses several techniques, for example: \(L^2\) based estimates; a dyadic-like decomposition (amalgam blocks); and commutator estimates. \par Editorial note: The reviewer found some mistakes in the original paper [see \url{arXiv:2007.01469}] and contacted the authors. They uploaded an amended version of the paper in [\url{arXiv:2007.01469}]. The review above relates to the amended version. We prove an \(L^p\) estimate \[\| e^{-itL} \varphi(L)f \|_p \lesssim (1+|t|)^s \|f\|_p, \quad t\in \mathbb{R}, \quad s=n\Big|\frac{1}{2}-\frac{1}{p}\Big|\] for the Schrödinger group generated by a semibounded, self-adjoint operator \(L\) on a metric measure space \(\mathcal{X}\) of homogeneous type (where \(n\) is the doubling dimension of \(\mathcal{X})\). The assumptions on \(L\) are a mild \(L^{p_0}\to L^{p_0'}\) smoothing estimate and a mild \(L^2\to L^2\) off-diagonal estimate for the corresponding heat kernel \(e^{-tL}\). The estimate is uniform for \(\varphi\) varying in bounded sets of \(\mathscr{S}(\mathbb{R})\),or more generally of a suitable weighted Sobolev space. We also prove, under slightly stronger assumptions on \(L\), that the estimate extends to \[\|e^{-itL} \varphi(\theta L) f\|_p \lesssim (1+\theta^{-1}|t|)^s \|f\|_p, \quad \theta > 0, \quad t\in \mathbb{R},\] with uniformity also for \(\theta\) varying in bounded subsets of \((0,+\infty)\). For nonnegative operators uniformity holds for all \(\theta > 0\).

The operator \(e^{it\Delta}\) is \(L^p(\mathbb{R}^n) \mapsto L^p(\mathbb{R}^n)\) bounded only when \(p = 2\), but the \(p\) range can be widened prior regularization. This type of phenomenon was already investigated, for example, in [\textit{T. A. Bui} et al., Rev. Mat. Iberoam. 36, No. 2, 455--484 (2020; Zbl 1448.35352)]. In the article under review, the authors consider mapping properties of \(e^{itL}\) for non-negative self-adjoint operators \(L\) in a metric space \((X, d, \mu)\) with doubling measure \(\mu\). Their main result is the inequality \[ \lVert e^{itL}(I + L)^{-\sigma_pn}f \rVert_{L^p(X)} \le C(I + \lvert t \rvert)^{\sigma_pn}\lVert f \rVert_{L^p(X)}, \qquad \text{for } \sigma_p := \bigg\lvert \frac{1}{2} - \frac{1}{p} \bigg\rvert, \tag{1}\] where \(p \in (p_0, p_0^\prime)\) with \(1\le p_0 < 2\), and \(e^{-tL}\) satisfies \[ \lVert \1_{B(x,t^{1/m})}e^{-tL}V_{t^{1/m}}^{\sigma_{p_0}} \1_{B(y,t^{1/m})} f \rVert_{L^2(X)} \le C \Big(1 + \frac{d(x,y)}{t^{1/m}}\Big)^{-n - \kappa} \lVert f \rVert_{L^{p_0}(X)}, \tag{2}\] where \(\kappa > [n/2] + 1\) and \(V_r(x) := \mu(B(x,r))\). The hypothesis (2) covers a wide class of operators. The authors develop a theory of Hardy spaces \(H_L^q(X)\) adapted to \(L\) and show that, by duality and interpolation, it suffices to prove (1) with \(H_L^q(X)\), for \(q < 1\), instead of \(L^p(X)\). To show that the operator is bounded in \(H^q_L(X)\), they prove new estimates for oscillatory multipliers to control off-diagonal terms. The proof uses several techniques, for example: \(L^2\) based estimates; a dyadic-like decomposition (amalgam blocks); and commutator estimates. \par Editorial note: The reviewer found some mistakes in the original paper [see \url{arXiv:2007.01469}] and contacted the authors. They uploaded an amended version of the paper in [\url{arXiv:2007.01469}]. The review above relates to the amended version. We present a mathematical model for plane potential flow of inviscid incompressible fluid around and through a circular cylinder of porous material. The flow inside the cylider obeys Darcy's law. The flow problem is reduced to a nonlinear boundary value problem of Poincaré type, and this one in turn to a nonlinear infinite system of algebraic equations and to a related nonlinear integral equation.

The following review is a slight modification of the authors' abstract: Let \(\mathbf{I}=(I_1,I_2,\ldots,I_n)\) and \(\mathbf{I}'=(I_1',I_2',\ldots,I_n')\) be two vectors of independent Bernoulli random variables with respective parameters \((p_1,p_2,\ldots,p_n)\) and \((p_1',p_2',\ldots,p_n')\). Let \(X_1,X_2,\ldots,X_n\) be nonnegative exchangeable random variables that are independent of \(\mathbf{I}\) and \(\mathbf{I}'\). The authors show that if \((p_1,p_2,\ldots,p_n)\) is majorized by \((p_1',p_2',\ldots,p_n')\), then \((I_1'X_1,I_2'X_2,\ldots,I_n'X_n)\) is smaller than \((I_1X_1,I_2X_2,\ldots,I_nX_n)\) in the symmetric supermodular stochastic order. The comparison between \((I_1X_1,I_2X_2,\ldots,I_nX_n)\) and \((I_1'X_1,I_2'X_2,\ldots,I_n'X_n)\) in the multivariate symmetric usual stochastic order is also studied. Some results in this paper strengthen results of \textit{C. Ma} [J. Stat. Plann. Inference 84, No. 1--2, 11--25 (2000; Zbl 1131.60303)]. Linear combinations \(\sum_{i=1}^kb_iX_i\) and \(\sum_{i=1}^ka_iX_i\) of random variables \(X_1,\dots,X_k\) are ordered in the sense of the decreasing convex order and the Laplace order, where \((b_1,\dots,b_k)\) is majorized by \((a_1,\dots,a_k)\), when the underlying random variables are independent but possibly nonidentically distributed, and the joint density is arrangement increasing, respectively. Finite mixture distributions \(\sum_{i=1}^ka_iF_iX_i(x)\) and \(\sum_{i=1}^kb_iF_iX_i(x)\) are compared in the sense of the usual stochastic order, the convex order and higher-order stochastic dominance. The comparison between \(\sum_{i=1}^k b_iX_i\) and \(\sum_{i=1}^ka_iX_i\) is also studied for binary random variables \(I_{a_i}\), \(I_{b_i}\) \((i=1,\dots,k)\). Some applications in economics and reliability are described.

The following review is a slight modification of the authors' abstract: Let \(\mathbf{I}=(I_1,I_2,\ldots,I_n)\) and \(\mathbf{I}'=(I_1',I_2',\ldots,I_n')\) be two vectors of independent Bernoulli random variables with respective parameters \((p_1,p_2,\ldots,p_n)\) and \((p_1',p_2',\ldots,p_n')\). Let \(X_1,X_2,\ldots,X_n\) be nonnegative exchangeable random variables that are independent of \(\mathbf{I}\) and \(\mathbf{I}'\). The authors show that if \((p_1,p_2,\ldots,p_n)\) is majorized by \((p_1',p_2',\ldots,p_n')\), then \((I_1'X_1,I_2'X_2,\ldots,I_n'X_n)\) is smaller than \((I_1X_1,I_2X_2,\ldots,I_nX_n)\) in the symmetric supermodular stochastic order. The comparison between \((I_1X_1,I_2X_2,\ldots,I_nX_n)\) and \((I_1'X_1,I_2'X_2,\ldots,I_n'X_n)\) in the multivariate symmetric usual stochastic order is also studied. Some results in this paper strengthen results of \textit{C. Ma} [J. Stat. Plann. Inference 84, No. 1--2, 11--25 (2000; Zbl 1131.60303)]. We perform large-scale simulations of a two-dimensional restricted height conserved stochastic sandpile, focusing on particle diffusion and mobility, and spatial correlations. Quasistationary (QS) simulations yield the critical particle density to high precision \([p_c = 0.7112687(2)]\), and show that the diffusion constant scales in the same manner as the activity density, as found previously in the one-dimensional case. Short-time scaling is characterized by subdiffusive behavior (mean-square displacement \(\sim t^\gamma\) with \(\gamma < 1)\), which is easily understood as a consequence of the initial decay of activity, \( \rho (t) \sim t^{- \delta }\), with \(\gamma = 1 - \delta \). We verify that at criticality, the activity-activity correlation function \(C\left(r\right)\sim{{r}^{-\beta /{{\nu}_{\bot}}}} \), as expected at an absorbing-state phase transition. Our results for critical exponents are consistent with, and somewhat more precise than, predictions derived from the Langevin equation for stochastic sandpiles in two dimensions.

Let \(M\) be a compact smooth manifold, and \(\mathbb{T}:= \mathbb{R}/\mathbb{Z}\) or \(\{0\}\). A Tonelli Lagrangian will be a \(C^2\) real function \(L\) on \(\mathbb{T}\times TM\) such that: \(\bullet\) for each \((t,m)\in\mathbb{T}\times M\), \(L_{(t,m)}\) is convex on \(T_mM\) and \(\lim_{|\theta|\to\infty} L_{(t,m)}(\theta)/|\theta|= +\infty\); \(\bullet\) the flow \(\varphi\) \((\mathbb{R}\times \mathbb{T}\times TM\to \mathbb{T}\times TM)\) defined by the Euler-Lagrange equation \({d\over ds}L_\theta= L_m\) is complete; \(\bullet\) the previous properties hold as well for any \((L-u)\) instead of \(L\), \(u\) summing \(C^\infty(\mathbb{T}\times M,\mathbb{R})\). Let \(m(L)\) denote the set of \(\varphi\)-invariant probability measures \(\mu\) on \(\mathbb{T}\times TM\) which minimize the action \(\int_{\mathbb{T}\times TM} L\,d\mu\). Let \(A\) be a finite-dimensional space of Tonelli Lagrangians. Then the main theorem asserts that there exists a dense subset \(O\) of \(C^\infty(\mathbb{T}\times M,\mathbb{R})\), countable intersection of open subsets, such that \[ u\in O,\;L\in A\rightarrow\dim m(L-u)\leq \dim A; \] in other words, there exist at most \((1+\dim A)\) ergodic minimizing measures of \((L-u)\). This result generalizes a result of \textit{R. Mañé} [Nonlinearity 9, No. 2, 273--310 (1996; Zbl 0886.58037)], which corresponds to \(\dim A= 1\). The proof is mainly made in an abstract context. The authors notice finally that \(u\) can be replaced as well as by cohomology class of 1-forms on \(M\). It is proved here that minimizing measures of a Lagrangian flow are invariant and the Lagrangian is cohomologous to a constant on the support of their ergodic components. Moreover, it is shown that generic Lagrangians have a unique minimizing measure which is uniquely ergodic and is a limit of invariant probabilities supported on periodic orbits of the Lagrangian flows.

Let \(M\) be a compact smooth manifold, and \(\mathbb{T}:= \mathbb{R}/\mathbb{Z}\) or \(\{0\}\). A Tonelli Lagrangian will be a \(C^2\) real function \(L\) on \(\mathbb{T}\times TM\) such that: \(\bullet\) for each \((t,m)\in\mathbb{T}\times M\), \(L_{(t,m)}\) is convex on \(T_mM\) and \(\lim_{|\theta|\to\infty} L_{(t,m)}(\theta)/|\theta|= +\infty\); \(\bullet\) the flow \(\varphi\) \((\mathbb{R}\times \mathbb{T}\times TM\to \mathbb{T}\times TM)\) defined by the Euler-Lagrange equation \({d\over ds}L_\theta= L_m\) is complete; \(\bullet\) the previous properties hold as well for any \((L-u)\) instead of \(L\), \(u\) summing \(C^\infty(\mathbb{T}\times M,\mathbb{R})\). Let \(m(L)\) denote the set of \(\varphi\)-invariant probability measures \(\mu\) on \(\mathbb{T}\times TM\) which minimize the action \(\int_{\mathbb{T}\times TM} L\,d\mu\). Let \(A\) be a finite-dimensional space of Tonelli Lagrangians. Then the main theorem asserts that there exists a dense subset \(O\) of \(C^\infty(\mathbb{T}\times M,\mathbb{R})\), countable intersection of open subsets, such that \[ u\in O,\;L\in A\rightarrow\dim m(L-u)\leq \dim A; \] in other words, there exist at most \((1+\dim A)\) ergodic minimizing measures of \((L-u)\). This result generalizes a result of \textit{R. Mañé} [Nonlinearity 9, No. 2, 273--310 (1996; Zbl 0886.58037)], which corresponds to \(\dim A= 1\). The proof is mainly made in an abstract context. The authors notice finally that \(u\) can be replaced as well as by cohomology class of 1-forms on \(M\). We propose a practical design and implementation of network-adaptive high definition MPEG-2 video streaming combined with Cross-Layered channel Monitoring (CLM) over the IEEE 802.11a Wireless Local Area Network (WLAN). For wireless channel monitoring, we adopt a cross-layered approach, where an access point periodically measures lower layers such as Medium Access Control (MAC) and physical transmission information (e.g., MAC layer loss rate) and then sends the monitored information to the streaming server application. The adaptive streaming server with the CLM scheme reacts more quickly and efficiently to the fluctuating wireless channel than the end-to-end application-layer monitoring scheme. The streaming server dynamically performs priority-based frame dropping to adjust the sending rate according to the measured wireless channel condition. For this purpose, the proposed streaming system nicely provides frame-based prioritized packetization by using a real-time stream parsing module. Various evaluation results over an IEEE 802.11a WLAN testbed are provided to verify the intended Quality of Service adaptation capability. Experimental results showed that the proposed system can mitigate the quality degradation of video streaming due to the fluctuations of time-varying channel.

The paper is the continuation of the author's previous paper with the same title [ibid. 27, 33-43 (1987; Zbl 0645.46045)], in which non commutative analogues of Orlicz metric spaces of measurable functions are considered in von Neumann algebras set up. Let M be a von Neumann algebra with a faithful normal semifinite trace \(\tau\), \(C(M)\)- the *-algebra of all measurable operators affiliated with M. Given a continuous strictly monotone increasing function \(f\) on \([0,1)\) or \([0,\infty)\) with \(f(0)=0\) the author defines the non commutative functional space \[ L_ f=L_ f(M,\tau)=\{A\in C(M)| \exists l_ A>0:f(l_ A| A|)\in L_ 1\} \] and studies its metric and topological properties and the dual space \(L^*_ f\).

The paper is the continuation of the author's previous paper with the same title [ibid. 27, 33-43 (1987; Zbl 0645.46045)], in which non commutative analogues of Orlicz metric spaces of measurable functions are considered in von Neumann algebras set up. The conjugate gradient method is one of the most robust algorithms to solve large-scale monotone problems due to its limited memory requirements. However, in this article, we used the modified secant equation and proposed two optimal choices for the non-negative constant of the Hager-Zhang (HZ) conjugate gradient method by minimizing the upper bound of the condition number for the HZ search direction matrix. Two algorithms for solving large-scale non-linear monotone equations that incorporate the concept of projection method are provided. Based on monotone and Lipschitz continuous assumptions, we developed the global convergence of the methods. Computational results indicate that the proposed algorithms are effective and efficient.

The author deals with weak approximation of Itô type stochastic differential equations (i.e. approximation of functionals of the solution in its final stage). In an important part of the paper he studies the results of \textit{D. Talay}'s paper [RAIRO, Modélisation Math. Anal. Numér. 20, No. 1, 141-179 (1986; Zbl 0662.65129)]. He cites and proves results about properties of the solutions from Talay's paper, where he claims that one of Talay's smoothness assumptions had to be strengthened slightly. After recalling the first order weak convergence of the Euler and the Milshtein scheme, and after proving a sufficient criterion by Talay for second order weak convergence (which was slightly changed by the author), he proposes two discretization schemes which by virtue of this criterion have weak order two: One scheme is based on the stochastic Taylor expansion, the other is of Runge-Kutta type and was proposed by Rümelin. At the end, numerical tests of the discussed schemes are presented. We are interested in ``well'' discretizing a stochastic differential equation, in view to approximate numerically the expectations of a large class of functionals of the solution. Classical methods are not very efficient. \textit{G. N. Mil'stein} [Teor. Verojatn. Primen. 23, 414-419 (1978; Zbl 0391.60060)] has proposed a new method, and conjectured its rate of convergence. Here, we prove the announced result, and we introduce new schemes, which have the same rate of convergence as Milshtein's one, and permit to treat the multidimensional equations.

The author deals with weak approximation of Itô type stochastic differential equations (i.e. approximation of functionals of the solution in its final stage). In an important part of the paper he studies the results of \textit{D. Talay}'s paper [RAIRO, Modélisation Math. Anal. Numér. 20, No. 1, 141-179 (1986; Zbl 0662.65129)]. He cites and proves results about properties of the solutions from Talay's paper, where he claims that one of Talay's smoothness assumptions had to be strengthened slightly. After recalling the first order weak convergence of the Euler and the Milshtein scheme, and after proving a sufficient criterion by Talay for second order weak convergence (which was slightly changed by the author), he proposes two discretization schemes which by virtue of this criterion have weak order two: One scheme is based on the stochastic Taylor expansion, the other is of Runge-Kutta type and was proposed by Rümelin. At the end, numerical tests of the discussed schemes are presented. We consider the Bayesian lasso for regression, which can be interpreted as an \(L_{1}\) norm regularization based on a Bayesian approach when the Laplace or double-exponential prior distribution is placed on the regression coefficients. A crucial issue is an appropriate choice of the values of hyperparameters included in the prior distributions, which essentially control the sparsity in the estimated model. To choose the values of tuning parameters, we introduce a model selection criterion for evaluating a Bayesian predictive distribution for the Bayesian lasso. Numerical results are presented to illustrate the properties of our sparse Bayesian modeling procedure.

Let \(N_t(k)\), \(t\geq2\), be the maximum number of \(k\)-term arithmetic progressions of real numbers, any couple of which have \(t\) common elements. In the paper it is proved that \(N_2(t)=k(k-1)/2\) if \(k\) is a prime. Then the author proves that the same conclusion is true for \(k\geq10^{8000}\). The technique of proof (employing also a strong form of the theorem of \textit{R. Balasubramanian} and \textit{K. Soundararajan} [Acta Arith. 75, 1-38 (1996; Zbl 0853.11002)]) does not allow to lower this bound due to the lack of better bounds for primes \(n\) short intervals. For the case \(t\geq 3\) less precise lower and upper bounds for \(N_t(k)\) are proved. Let \(N\geq 5\) be an integer and \({\mathfrak A}= \{a_1, \dots, a_N\}\) a set of \(N\) integers, where \(a_1< \cdots< a_N\). A well-known conjecture of \textit{R. L. Graham} [Am. Math. Mon. 77, 775 (1970)] states that there exist \(a_i, a_j\in {\mathfrak A}\) with \(a_i/ (a_i, a_j)\geq N\). So far, this conjecture has been proved in a variety of special cases, last for all large \(N\) [see \textit{F. Y. Cheng} and \textit{C. Pomerance}, Rocky Mt. J. Math. 24, 961--975 (1994; Zbl 0823.11002)]. In the present paper the authors establish Graham's conjecture unconditionally, even in a stronger version: If \({\mathfrak A}= \{a_1, \dots, a_N\}\) is in addition so that \((a_1, \ldots, a_N)=1\) then there already exist \(a_i, a_j\in {\mathfrak A}\) with \(a_i/ (a_i, a_j)\geq N\). Moreover, the inequality is strict if both \({\mathfrak A}\) and \({\mathfrak A}^*\) are different from \(\{1, \dots, N\}\). Here, the `reciprocal set' \({\mathfrak A}^*\) is defined by \({\mathfrak A}^*= \{[a_1, \dots, a_N]/ a_1, \dots, [a_1, \dots, a_N]/ a_N\}\). A modification of the arguments leads to a more general result: If \({\mathfrak A}\) and \({\mathfrak B}\) are two \(N\) element sets then there exists \(a\in {\mathfrak A}\) and \(b\in {\mathfrak B}\) with \(\max (a/ (a,b), b/ (a, b))\geq N\). The inequality is strict unless \({\mathfrak A}= {\mathfrak B}= \{1, \dots, N\}\) or \({\mathfrak A}= {\mathfrak B}= \{1, \ldots, N\}^*\).

Let \(N_t(k)\), \(t\geq2\), be the maximum number of \(k\)-term arithmetic progressions of real numbers, any couple of which have \(t\) common elements. In the paper it is proved that \(N_2(t)=k(k-1)/2\) if \(k\) is a prime. Then the author proves that the same conclusion is true for \(k\geq10^{8000}\). The technique of proof (employing also a strong form of the theorem of \textit{R. Balasubramanian} and \textit{K. Soundararajan} [Acta Arith. 75, 1-38 (1996; Zbl 0853.11002)]) does not allow to lower this bound due to the lack of better bounds for primes \(n\) short intervals. For the case \(t\geq 3\) less precise lower and upper bounds for \(N_t(k)\) are proved. For the entire collection see Zbl 0746.00043.

Let \(\pmb k\) be a field and \(G_1\) and \(G_2\) be graphs on the index sets \([m]\) and \([n]\), respectively. The authors consider a polynomial ring \(P=\pmb k[x_1,\dots,x_{m+n-1}]\), a generic \(m\times n\) Hankel matrix \[ X=\begin{bmatrix} x_1&x_2&\cdots&x_n\\x_2&x_3&\cdots&x_{n+1}\\\vdots&\vdots&&\vdots\\ x_{m}&x_{m+1}&\cdots&x_{m+n-1}\end{bmatrix}, \] and the ideal \(I_{G_1,G_2}\) of \(P\) which is generated by all \(2\times 2\) minors of \(X\) of the form \[ \left|\begin{smallmatrix} X_{i,k}&X_{i,\ell}\\X_{j,k}&X_{j,\ell}\end{smallmatrix} \right|= \left|\begin{smallmatrix} x_{i+k-1}&x_{i+\ell-1}\\x_{j+k-1}&x_{j+\ell-1}\end{smallmatrix}\right|, \] where \(\{i,j\}\) is an edge in \(G_1\) and \(\{k,\ell\}\) is an edge in \(G_2\). It is shown that the ideal \(I_{G_1,G_2}\) is equal to an ideal \(I_G\) which has already been studied in [the first author et al., Bull. Iran. Math. Soc. 41, No. 4, 971--979 (2015; Zbl 1373.13021)]. Let \(X=\left( \begin{smallmatrix} x_1 & \dots & x_{n-1} & x_n\\ x_2 & \dots & x_n & x_{n+1}\\ \end{smallmatrix} \right) \) be the Hankel matrix of size \(2\times n\) and let \(G\) be a closed graph on the vertex set \([n]\). We study the binomial ideal \(I_G\subset K[x_1,\dots,x_{n+1}]\) which is generated by all the \(2\)-minors of \(X\) which correspond to the edges of \(G\). We show that \(I_G\) is Cohen-Macaulay. We find the minimal primes of \(I_G\) and show that \(I_G\) is a set theoretical complete intersection. Moreover, a sharp upper bound for the regularity of \(I_G\) is given.

Let \(\pmb k\) be a field and \(G_1\) and \(G_2\) be graphs on the index sets \([m]\) and \([n]\), respectively. The authors consider a polynomial ring \(P=\pmb k[x_1,\dots,x_{m+n-1}]\), a generic \(m\times n\) Hankel matrix \[ X=\begin{bmatrix} x_1&x_2&\cdots&x_n\\x_2&x_3&\cdots&x_{n+1}\\\vdots&\vdots&&\vdots\\ x_{m}&x_{m+1}&\cdots&x_{m+n-1}\end{bmatrix}, \] and the ideal \(I_{G_1,G_2}\) of \(P\) which is generated by all \(2\times 2\) minors of \(X\) of the form \[ \left|\begin{smallmatrix} X_{i,k}&X_{i,\ell}\\X_{j,k}&X_{j,\ell}\end{smallmatrix} \right|= \left|\begin{smallmatrix} x_{i+k-1}&x_{i+\ell-1}\\x_{j+k-1}&x_{j+\ell-1}\end{smallmatrix}\right|, \] where \(\{i,j\}\) is an edge in \(G_1\) and \(\{k,\ell\}\) is an edge in \(G_2\). It is shown that the ideal \(I_{G_1,G_2}\) is equal to an ideal \(I_G\) which has already been studied in [the first author et al., Bull. Iran. Math. Soc. 41, No. 4, 971--979 (2015; Zbl 1373.13021)]. Much of the theoretical work on strategic voting makes strong assumptions about what voters know about the voting situation. A strategizing voter is typically assumed to know how other voters will vote and to know the rules of the voting method. A growing body of literature explores strategic voting when there is uncertainty about how others will vote. In this paper, we study strategic voting when there is uncertainty about the voting method. We introduce three notions of manipulability for a set of voting methods: \textit{sure}, \textit{safe}, and \textit{expected} manipulability. With the help of a computer program, we identify voting scenarios in which uncertainty about the voting method may reduce or even eliminate a voter's incentive to misrepresent her preferences. Thus, it may be in the interest of an election designer who wishes to reduce strategic voting to leave voters uncertain about which of several reasonable voting methods will be used to determine the winners of an election.

\textit{J. H. Hubbard} and \textit{R. W. Oberste-Vorth} [NATO ASI Ser., Ser. C, Math. Phys. Sci. 464, 89--132 (1995; Zbl 0874.54037)] have given a characterization for the Julia set for the complex Hénon mappings. The authors of the article under review apply the Banach Contraction Principle to give a new, simple proof with a novel construction of function spaces. Moreover, further dependence of Julia sets on the parameters of Hénon mappings is obtained. [For Part I see Publ. Math., Inst. Hautes Étud. Sci. 79, 5-46 (1994; Zbl 0839.54029).] Let \(H: \mathbb{C}^2 \to\mathbb{C}^2\) be the Hénon mapping given by the formulae \(H(x,y)= (H_1(x,y), H_1(x,y))\), \(H_1(x,y) =p(x)-ay\), \(H_2(x,y)=x\). The key invariant sets are \(K_\pm\), the sets of bounded forward images, \(J_\pm= \partial K_\pm\). The paper describes the topological structure of these sets for \(p\) being a hyperbolic polynomial and \(|a|\) small, i.e. when \(H\) is a (small) perturbation of \(p\). The description involves projective and inductive limits of some objects defined by \(p\). The main result (Theorem 4.1) says that there is \(A\) such that if \(0<|a|<A\) then there exist homeomorphisms \(\Phi_-: \widehat \mathbb{C}_p \to J_-\) and \(\Phi_+: \check\mathbb{C}_p \to J_+\) such that \(H\circ \Phi_- =\Phi_-\circ \widehat p\) and \(H\circ \Phi_+ =\Phi_+ \circ\check p\), where \(\widehat\mathbb{C}_p\) denotes the projective limit \(\varprojlim (\mathbb{C},p)\) (that is the set \(\{(\dots, z_{-2}, z_{-1}, z_0): p(z_{-i-1}) =z_{-i}\) for \(i=0,1,2, \dots\})\), \(\widehat p(\dots, z_{-2},z_{-1}, z_0) = (\dots, z_{-1}, z_0, p(z_0))\) and \(\check \mathbb{C}_p\) is a suitable inductive limit for a mapping \(f\) defined by \(p\) and \(p'\) if \(p'\neq 0\), \(\check p\) is a corresponding bijection \(\check \mathbb{C}_p\to \check\mathbb{C}_p\).

\textit{J. H. Hubbard} and \textit{R. W. Oberste-Vorth} [NATO ASI Ser., Ser. C, Math. Phys. Sci. 464, 89--132 (1995; Zbl 0874.54037)] have given a characterization for the Julia set for the complex Hénon mappings. The authors of the article under review apply the Banach Contraction Principle to give a new, simple proof with a novel construction of function spaces. Moreover, further dependence of Julia sets on the parameters of Hénon mappings is obtained. We study the sequence of variational eigenvalues of a system of resonant type involving \(p\)- and \(q\)-Laplacians on \(\varOmega \subset \mathbb R^N\), with a coupling term depending on two parameters \(\alpha \) and \(\beta \) satisfying \(\alpha /p+\beta /q=1\). We show that the order of growth of the \(k\)th eigenvalue depends on \(\alpha +\beta , \lambda_k = O (k^{\frac {\alpha + \beta}{N}})\).

The first part of the paper deals with generalized differentiation for scalar functions: an attempt is made to unify well known differentiability notions previously introduced mainly in the convex and in the local Lipschitz case. The second part presents a necessary condition for local Pareto minima and an existence theorem that generalizes a previous result of the authors [see, e.g., Constrained optimization of infinite dimensional vector valued functions with application to infinite horizon integrals, Numer. Funct. Anal. Optimization (to appear; Zbl 0587.49009)]. The question of the existence of Pareto minima is studied. Some results are given in the general case of a cone with non-empty interior in a reflexive separable Banach space. An application to integral problems is given especially in connection with unbounded domain of integration. The convergence of minima of the truncated problem is also considered. For the case of a cone with empty interior the reader is referred to another paper of the authors [''Necessary conditions for local Pareto minima'', Boll. Unione Mat. Ital., VI. Ser., B (1986)].

The first part of the paper deals with generalized differentiation for scalar functions: an attempt is made to unify well known differentiability notions previously introduced mainly in the convex and in the local Lipschitz case. The second part presents a necessary condition for local Pareto minima and an existence theorem that generalizes a previous result of the authors [see, e.g., Constrained optimization of infinite dimensional vector valued functions with application to infinite horizon integrals, Numer. Funct. Anal. Optimization (to appear; Zbl 0587.49009)]. We introduce a new model of school choice with reserves in which asocial planner is constrained by a limited supply of reserve seats and tries to find the matching that is optimal according to a social welfare function. We construct the optimal distribution of reserves via a quartic-time dynamic programming algorithm. Due to the modular nature of the algorithm, the resulting mechanism is \textit{strategy-proof} for reserve-eligible students.

The totally asymmetric simple exclusion process (TASEP) on the one-dimensional lattice with the Bernoulli \(\rho\) measure as initial conditions, \(0< \rho <1\), is stationary in space and time. Let \(N_t(j)\) be the number of particles which have crossed the bond from \(j\) to \(j+1\) during the time span \([0,t]\). For \( j = (1-2p)t + 2w(\rho(1-\rho))^{1/3}t^{2/3}\) the authors prove that the fluctuations of \(N_t(j)\) for large \(t\) are of order \(t^{1/3}\) and determine the limiting distribution function \(F_w(s)\), which is a generalization of the GUE (Gaussian unitary ensemble) Tracy-Widom distribution. The family \(F_w(s)\) of distribution functions have been obtained before by \textit{J. Baik} and \textit{E. M. Rains} [J. Stat. Phys. 100, 523--541 (2000; Zbl 0976.82043)] in the context of the PNG model (polynuclear growth model) with boundary sources, which requires the asymptotics of a Riemann-Hilbert problem. In this work the authors arrive at \(F_w(s)\) through the asymptotics of a Fredholm determinant. \(F_w(s)\) is simply related to the scaling function for the space-time covariance of the stationary TASEP, equivalently to the asymptotic transition probability of a single second class particle. The purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by \textit{M. Prähofer} and \textit{H. Spohn} [Physica A 279, 342-352 (2000; Zbl 0976.82045)]. Depending on the strength of the sources, the limiting distribution functions are either the Tracy-Widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero. Moreover, we obtain transition functions between pairs of the above distribution functions in suitably scaled limits. There are also similar results for a discrete totally asymmetric exclusion process.

The totally asymmetric simple exclusion process (TASEP) on the one-dimensional lattice with the Bernoulli \(\rho\) measure as initial conditions, \(0< \rho <1\), is stationary in space and time. Let \(N_t(j)\) be the number of particles which have crossed the bond from \(j\) to \(j+1\) during the time span \([0,t]\). For \( j = (1-2p)t + 2w(\rho(1-\rho))^{1/3}t^{2/3}\) the authors prove that the fluctuations of \(N_t(j)\) for large \(t\) are of order \(t^{1/3}\) and determine the limiting distribution function \(F_w(s)\), which is a generalization of the GUE (Gaussian unitary ensemble) Tracy-Widom distribution. The family \(F_w(s)\) of distribution functions have been obtained before by \textit{J. Baik} and \textit{E. M. Rains} [J. Stat. Phys. 100, 523--541 (2000; Zbl 0976.82043)] in the context of the PNG model (polynuclear growth model) with boundary sources, which requires the asymptotics of a Riemann-Hilbert problem. In this work the authors arrive at \(F_w(s)\) through the asymptotics of a Fredholm determinant. \(F_w(s)\) is simply related to the scaling function for the space-time covariance of the stationary TASEP, equivalently to the asymptotic transition probability of a single second class particle. We obtain a stability result for the general linear equation in Hyers-Ulam sense.

The paper deals with the macroscopic mechanical behavior of composite materials. It treats materials with a microstructure consisting of either holes or rigid inclusions embedded in an elastic matrix. The approach follows that of the first author in a previous study (e.g., [\textit{O. P. Bruno}, Asymptotic Anal. 4, No. 4, 339-365 (1991; Zbl 0747.73035)]) on the conductivity of high-contrast mixtures. The effective elasticity tensor of a given composite is considered as a function of the elasticity tensor of the inclusions. Studying the analyticity of the effective tensor, the authors state that when the inclusions are well separated, there is a certain class of non-physical negative-definite values of the elastic moduli of the inclusions for which the effective tensor does not become singular. This result is used to establish representation formulae for the effective elasticity tensor. These formulae will, in turn, be useful in obtaining bounds for the elastic constants. More precise results concerning the singularity region of the effective tensor are given. The imply known homogenization theorems for domains with holes, and new results for domain with rigid inclusions. Bounds on the effective bulk and shear moduli of high-contrast composites are derived in the cases of well separated spherical or cylindrical holes or hard inclusions. We describe a construction, based on variational inequalities, which gives a hierarchy of upper and lower bounds (of odd orders \(2k+1\)), on the various effective moduli of random multiphase materials and polycrystals. The bounds of order \(2k+1\) on a given effective modulus can be explicitly evaluated if a truncated Taylor expansion of the given modulus is known to order \(2k+1\). Our calculations do not involve Green functions or \(n\)-point correlation functions, and they are very simple. We thus rederive known and obtain new sequences of bounds on the different effective moduli. We also describe a method that, for cell materials (i.e. materials in which cells of smaller and smaller length scales cover all space, with material properties assigned at random), permits one to calculate the truncated Taylor expansions that are needed for the explicit evaluation of the bounds. In connection with this, we show that the first coefficient in the low volume fraction expansion of any effective modulus of a cell material, coincides with the corresponding low volume fraction coefficient for an array of cells randomly distributed in a matrix.

The paper deals with the macroscopic mechanical behavior of composite materials. It treats materials with a microstructure consisting of either holes or rigid inclusions embedded in an elastic matrix. The approach follows that of the first author in a previous study (e.g., [\textit{O. P. Bruno}, Asymptotic Anal. 4, No. 4, 339-365 (1991; Zbl 0747.73035)]) on the conductivity of high-contrast mixtures. The effective elasticity tensor of a given composite is considered as a function of the elasticity tensor of the inclusions. Studying the analyticity of the effective tensor, the authors state that when the inclusions are well separated, there is a certain class of non-physical negative-definite values of the elastic moduli of the inclusions for which the effective tensor does not become singular. This result is used to establish representation formulae for the effective elasticity tensor. These formulae will, in turn, be useful in obtaining bounds for the elastic constants. More precise results concerning the singularity region of the effective tensor are given. The imply known homogenization theorems for domains with holes, and new results for domain with rigid inclusions. Bounds on the effective bulk and shear moduli of high-contrast composites are derived in the cases of well separated spherical or cylindrical holes or hard inclusions. Starting from the observation that one of the most successful methods for solving the Kohn-Sham equations for periodic systems - the plane-wave method - is a spectral method based on eigenfunction expansion, we formulate a spectral method designed towards solving the Kohn-Sham equations for clusters. This allows for efficient calculation of the electronic structure of clusters (and molecules) with high accuracy and systematic convergence properties without the need for any artificial periodicity. The basis functions in this method form a complete orthonormal set and are expressible in terms of spherical harmonics and spherical Bessel functions. Computation of the occupied eigenstates of the discretized Kohn-Sham Hamiltonian is carried out using a combination of preconditioned block eigensolvers and Chebyshev polynomial filter accelerated subspace iterations. Several algorithmic and computational aspects of the method, including computation of the electrostatics terms and parallelization are discussed. We have implemented these methods and algorithms into an efficient and reliable package called ClusterES (Cluster Electronic Structure). A variety of benchmark calculations employing local and non-local pseudopotentials are carried out using our package and the results are compared to the literature. Convergence properties of the basis set are discussed through numerical examples. Computations involving large systems that contain thousands of electrons are demonstrated to highlight the efficacy of our methodology. The use of our method to study clusters with arbitrary point group symmetries is briefly discussed.

The paper under review deals with the isotropic and spatially homogeneous Boltzmann equation for Bose-Einstein particles in the case of the hard sphere model, and tackles the problem of Bose-Einstein condensation in finite time. The main result states that if the initial energy distribution of particles has no mass at 0 but is singular enough near the origin, then condensation occurs at all positive times. In the first section, the author introduces the equation of interest, defines the notion of a distributional solution, recalls several results such as the existence of an equilibrium distributional solution and a critical temperature, as well as a result on the convergence to equilibrium, and finally raises the question of condensation in finite time. The latter is partially answered by the main theorem in Section 2, whereas Section 3 contains the proof. Implications of the result, in particular, the link between condensation and conservation of mass, are discussed in Section 4. In the appendix, it is proved that the definition of a distributional solution given in [\textit{X. Lu}, J. Stat. Phys. 116, No. 5--6, 1597--1649 (2004; Zbl 1097.82023)] is equivalent to the one in the present paper. The paper considers the spatially homogeneous Boltzmann equation for Bose-Einstein particles (BBE). In order to include the hard sphere model, the equation is studied in a weak form and its solutions (including initial data) are set in the class of isotropic positive Borel measures and therefore called isotropic distributional solutions. Stability of distributional solutions is established in the weak topology, global existence of distributional solutions that conserve the mass and energy is proved by weak convergence of approximate \(L^1\)-solutions, and moment production estimates for the distributional solutions are also obtained. As an application of the weak form of the BBE equation, it is shown that a Bose-Einstein distribution plus a Dirac \(\delta\)-function is an equilibrium solution to the BBE equation in the weak form if and only if it satisfies a low temperature condition and an exact ratio of the Bose-Einstein condensation.

The paper under review deals with the isotropic and spatially homogeneous Boltzmann equation for Bose-Einstein particles in the case of the hard sphere model, and tackles the problem of Bose-Einstein condensation in finite time. The main result states that if the initial energy distribution of particles has no mass at 0 but is singular enough near the origin, then condensation occurs at all positive times. In the first section, the author introduces the equation of interest, defines the notion of a distributional solution, recalls several results such as the existence of an equilibrium distributional solution and a critical temperature, as well as a result on the convergence to equilibrium, and finally raises the question of condensation in finite time. The latter is partially answered by the main theorem in Section 2, whereas Section 3 contains the proof. Implications of the result, in particular, the link between condensation and conservation of mass, are discussed in Section 4. In the appendix, it is proved that the definition of a distributional solution given in [\textit{X. Lu}, J. Stat. Phys. 116, No. 5--6, 1597--1649 (2004; Zbl 1097.82023)] is equivalent to the one in the present paper. Each open subspace of a weakly pseudocompact space is either weakly pseudocompact or locally compact Lindelöf. A topological sum is weakly pseudocompact if and only if (1) each summand is either weakly pseudocompact or locally compact Lindelöf and (2) the sum is either compact or not Lindelöf. If \(X\) is realcompact and the lattice of compactifications of \(X\) is a \(b\)-lattice or if \(X\) is a not Čech-complete \(G_\delta\)-diagonal space then \(X\) is not weakly pseudocompact. Weak pseudocompactness is neither an inverse invariant of perfect maps nor an invariant of open maps with compact fibers. There is a not pseudocompact space in which each zero set is weakly pseudocompact.

The authors develop a theory of quantum principal bundles over projective base spaces, and specifically over homogeneous projective varieties. They then study noncommutative principal bundles corresponding to a quotient \(G \rightarrow G/P\) of a semisimple group \(G\) by a parabolic subgroup \(P\) and a projective embedding of \(G/P\) coming from a section of a very ample line bundle. The classical description of principal bundles is given, then reformulated in terms of function algebras, which provides a Hopf-algebraic setting for quantum versions. In general, a \emph{quantum principal bundle}, or an \emph{\(H\)-principal bundle} for a Hopf algebra \(H\), is defined to be a sheaf \(\mathcal{F}\) of \(H\)-comodule algebras on a ringed space \((M,\mathcal{O}_M)\) equipped with an open covering \((U_i)\) such that \(\mathcal{F}(U_i)\) is a principal \(H\)-comodule algebra whose subalgebra of \(H\)-coinvariants coincides with \(\mathcal{O}_M(U_i)\). Quantum homogeneous projective varieties \(\tilde{\mathcal{O}}_q(G/P)\) corresponding to quantum sections of line bundles on \(G/P\) were constructed by \textit{N. Ciccoli} et al. as algebras of \(\mathcal{O}_q(P)\)-semi-coinvariants [J. Noncommut. Geom. 2, No. 4, 449--496 (2008; Zbl 1168.16024)]. Here the authors construct quantum principal bundles (\(\mathcal{O}_q(P)\)-principal bundles) over \(\tilde{\mathcal{O}}_q(G/P)\) corresponding to quantum sections. These bundles are worked out in detail for \(G = \operatorname{SL}_n({\mathbb C})\) and a specific \(P\) such that \(G/P \cong {\mathbb P}^{n-1}({\mathbb C})\). The final section of the paper develops a 2-cocycle twist deformation process for quantum principal bundles. Applications to the bundles worked out in the previous section give three new quantum principal bundles over \(\tilde{\mathcal{O}}_q( {\mathbb P}^{n-1}({\mathbb C}))\). Let \(G\) be a complex affine group which is a Poisson group, i.e., the function Hopf algebra \(O(G)\) has a Poisson bracket compatible with the Hopf algebra structure. Let \(H\) be a coisotropic subgroup of \(G\), i.e., the defining ideal of \(H\) is a Poisson subalgebra of \(O(G)\). Then \(G\to G/H\) is a surjective Poisson map, and a projective embedding of \(G/H\) is obtained via sections of a line bundle on \(G/H\). The main object of the paper is to quantize this set-up, where \(q\) is an invertible variable. A quantum coisotropic subgroup of \(O_q(G)\) is a \(C_q=C[q,q^{-1}]\)-coalgebra \(O_q(H)\) and a projection \(p\colon O_q(G)\to O_q(H)\) such that \(O_q(H)\) is a torsion-free \(C_q\)-module and \(p\) is a \(C_q\)-coalgebra map and an \(O_q(G)\)-module map. If, in addition, \(O_q(H)\) is a Hopf algebra, \(p\) is a Hopf algebra map, and a technical condition holds, then \(O_q(H)\) is called a quantum subgroup of \(O_q(G)\). When \(O_q(H)\) is a quantum coisotropic subgroup of \(O_q(G)\), the authors build a quantum deformation \(O_q(G/H)\) of the projective variety \(G/H\), which satisfies five conditions: (1) There is a one-dimensional corepresentation of \(O_q(H)\) which is a deformation of the corepresentation of \(O(H)\) corresponding to the character of \(H\) which defines the line bundle giving the projective embedding of \(G/H\); (2) \(O_q(G/H)\) is the ``semi-invariant'' functions in \(O_q(G)\) with respect to the given corepresentation of \(O_q(H)\); (3) \(O_q(G/H)\) is a graded subalgebra of \(O_q(G)\); (4) \(O_q(G/H)\) is a graded left coideal of \(O_q(G)\), so the comultiplication in \(O_q(G)\) induces a left \(O_q(G)\)-coaction, so that \(O_q(G/H)\) can be thought of as a quantum homogeneous space; (5) The semi-classical limit of \(O_q(G/H)\) is \(O(G/H)\) (embedded into \(O(G)\) as a graded subalgebra, left coideal and graded Poisson subalgebra). The quantum duality principle of Drinfeld was extended by the first and third authors to quasi-affine homogeneous spaces [Adv. Math. 199, No. 1, 104-135 (2006; Zbl 1137.58003)]. In the paper under review, this principle is extended to quantum projective homogeneous spaces. Finally they give the details of their quantization for the Grassmannians, i.e., \(G=SL_n\) and \(H\) is a parabolic subgroup. They sketch the quantization for quantum flag varieties.

The authors develop a theory of quantum principal bundles over projective base spaces, and specifically over homogeneous projective varieties. They then study noncommutative principal bundles corresponding to a quotient \(G \rightarrow G/P\) of a semisimple group \(G\) by a parabolic subgroup \(P\) and a projective embedding of \(G/P\) coming from a section of a very ample line bundle. The classical description of principal bundles is given, then reformulated in terms of function algebras, which provides a Hopf-algebraic setting for quantum versions. In general, a \emph{quantum principal bundle}, or an \emph{\(H\)-principal bundle} for a Hopf algebra \(H\), is defined to be a sheaf \(\mathcal{F}\) of \(H\)-comodule algebras on a ringed space \((M,\mathcal{O}_M)\) equipped with an open covering \((U_i)\) such that \(\mathcal{F}(U_i)\) is a principal \(H\)-comodule algebra whose subalgebra of \(H\)-coinvariants coincides with \(\mathcal{O}_M(U_i)\). Quantum homogeneous projective varieties \(\tilde{\mathcal{O}}_q(G/P)\) corresponding to quantum sections of line bundles on \(G/P\) were constructed by \textit{N. Ciccoli} et al. as algebras of \(\mathcal{O}_q(P)\)-semi-coinvariants [J. Noncommut. Geom. 2, No. 4, 449--496 (2008; Zbl 1168.16024)]. Here the authors construct quantum principal bundles (\(\mathcal{O}_q(P)\)-principal bundles) over \(\tilde{\mathcal{O}}_q(G/P)\) corresponding to quantum sections. These bundles are worked out in detail for \(G = \operatorname{SL}_n({\mathbb C})\) and a specific \(P\) such that \(G/P \cong {\mathbb P}^{n-1}({\mathbb C})\). The final section of the paper develops a 2-cocycle twist deformation process for quantum principal bundles. Applications to the bundles worked out in the previous section give three new quantum principal bundles over \(\tilde{\mathcal{O}}_q( {\mathbb P}^{n-1}({\mathbb C}))\). By direct numerical simulations of a chemical reaction-diffusion system coupled to a periodic external AC electric field with frequency equal to double frequency of the scroll wave rotation, we find that scroll rings resonate with the electric field and exhibit various dynamical behaviors, for example, their reversals, collapses, or growths, depending both on the initial phase of AC electric fields and on the initial phase of scroll rings. A kinematical model characterizing the drift velocity of the scroll rings along their radial directions as well as that of the scroll rings along their symmetry axes is proposed, which can effectively account for the numerical observations and predict the behaviors of the scroll rings. Besides, the existence of the equilibrium state of a scroll ring under the AC electric fields is predicted by the kinematical model and the predictions agree well with the simulations.{ \copyright 2018 American Institute of Physics}

The authors consider the moduli functor of curves with an action of a finite group and the moduli functor of Galois covers and show that both can be represented by closed subschemes of the Sato Grassmannian. More precisely, consider first curves with an action of a finite group \(G\), rigidified by the choice of a distinguished orbit, a formal trivialisation along this orbit and a group monomorphism of \(G\) into the group \(\text{QP}_r^m\) of \(r\times r\) matrices of quasipermutations with coefficients in the group of \(m\)th roots of unity, where \(r\) is the number of distinct points in the orbit and \(m={1\over r}|G|\). The authors prove that the corresponding moduli functor \({\mathcal M}^{\infty}_G(r)\) is represented by a closed subscheme of the Sato Grassmannian \(\text{Gr}(V)\), where \(V:=k((z_1)\times\ldots\times k((z_r))\), and give a characterisation of this subscheme. This generalises a previous result of the authors and \textit{J.~M.~Muñoz Porras} [Math. Ann. 327, No. 4, 606--639 (2003; Zbl 1056.14039)]. For the case of Galois covers, the moduli space is a subspace of the Hurwitz space previously studied by \textit{J.~M.~Muñoz Porras} and \textit{F. Plaza Martín} [Equations of Hurwitz schemes in the infinite Grassmannian, \texttt{math/0207091}]. The authors characterise its points as the points of the Hurwitz space whose stabilisers under the action of \(\text{QP}_r^m\) have order \(rm\). The representability of the corresponding functor by a subscheme of \(\text{Gr}(V)\) follows from this. The authors also construct the moduli spaces for Galois covers with fixed group and with fixed curve. The final section is concerned with explicit equations for the moduli spaces described above. During the past fifteen years, there have been several attempts to solve a Schottky-type problem for Prym varieties, that is to characterize the locus of Prym varieties inside the corresponding moduli space of polarized abelian varieties either geometrically or by certain equations in theta constants. Some partial results concerning this problem have been obtained, in the meantime, by Shiota, Taimanov, Li, Mulase, Plaza-Martín, and others using different approaches. The paper under review also points in this direction and has two main objectives. First, the authors generalize some previous results of Shiota and Plaza-Martín to the more general case of Prym varieties associated with curves admitting an automorphism of prime order. Then they give an explicit description of the equations defining the moduli space of curves with an automorphism of prime order as a subscheme of Sato's infinite Grassmannian. Using the formal approach developed by two of the authors [\textit{J. M. Muñoz Porras} and \textit{F. J. Plaza Martín}, Equations of Hurwitz schemes in the infinite Grassmannian, Preprint http://arxiv.org/abs/math/0207091] in order to characterize Hurwitz schemes in the framework of inifinite Grassmannians, and extending it to their new concept of formal Prym varieties, the authors establish an analogue of the classical Krichever map as well as an explicit characterization of formal Prym varieties as subvarieties of the the Sato Grassmannian. Finally, in the last section of the present paper, explicit equations of the moduli spaces of curves with automorphisms of prime order are derived within the same framework. The latter formal approach is based on the results and methods of another foregoing work of two of the authors, and being concerned with the equations defining the moduli spaces of pointed curves in the infinite Grassmannian [\textit{J. M. Muñoz Porras} and \textit{F. J. Plaza Martín}, J. Differ. Geom. 51, No. 3, 431--469 (1999; Zbl 1065.14512)].

The authors consider the moduli functor of curves with an action of a finite group and the moduli functor of Galois covers and show that both can be represented by closed subschemes of the Sato Grassmannian. More precisely, consider first curves with an action of a finite group \(G\), rigidified by the choice of a distinguished orbit, a formal trivialisation along this orbit and a group monomorphism of \(G\) into the group \(\text{QP}_r^m\) of \(r\times r\) matrices of quasipermutations with coefficients in the group of \(m\)th roots of unity, where \(r\) is the number of distinct points in the orbit and \(m={1\over r}|G|\). The authors prove that the corresponding moduli functor \({\mathcal M}^{\infty}_G(r)\) is represented by a closed subscheme of the Sato Grassmannian \(\text{Gr}(V)\), where \(V:=k((z_1)\times\ldots\times k((z_r))\), and give a characterisation of this subscheme. This generalises a previous result of the authors and \textit{J.~M.~Muñoz Porras} [Math. Ann. 327, No. 4, 606--639 (2003; Zbl 1056.14039)]. For the case of Galois covers, the moduli space is a subspace of the Hurwitz space previously studied by \textit{J.~M.~Muñoz Porras} and \textit{F. Plaza Martín} [Equations of Hurwitz schemes in the infinite Grassmannian, \texttt{math/0207091}]. The authors characterise its points as the points of the Hurwitz space whose stabilisers under the action of \(\text{QP}_r^m\) have order \(rm\). The representability of the corresponding functor by a subscheme of \(\text{Gr}(V)\) follows from this. The authors also construct the moduli spaces for Galois covers with fixed group and with fixed curve. The final section is concerned with explicit equations for the moduli spaces described above. In this paper, we consider the long term behavior of solutions to stochastic delay differential equations with additive noise. We first establish the existence of an invariant measure by regarding the stochastic equation as a perturbation of a deterministic equation. If the deterministic part is stable it is plausible to expect the existence of an invariant measure under some conditions on the perturbation. Different from traditional methods, a retarded Ornstein-Uhlenbeck process is introduced to the study of the existence of a continuous random dynamical system and tempered pullback attractors. In particular, we prove pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors. We also establish the upper semicontinuity of the random attractors when noise intensity approaches zero, and investigate the exponential stability of stationary solutions in the mean square sense and almost surely. It is found that our theoretical results and methods can be generalized to other stochastic delay differential equations, even with multiplicative noise. Finally, our theoretical results are illustrated by an applications to a stochastic Nicholson's blowflies model.

A complex manifold satisfies the ``density property'' if the integrable holomorphic vector fields generate a dense Lie subalgebra of the Lie algebra of all global holomorphic vector fields. This is an important property which has many implications and which was first verified for the affine complex space \({\mathbb C}^n\) by \textit{E. Andersén} and \textit{L. Lempert} [Invent. Math. 110, No. 2, 371--388 (1992; Zbl 0770.32015)]. The corresponding property for complex algebraic varieties is called ``algebraic density property''. Danilov-Gizatullin surfaces are two-dimensional complex affine algebraic varieties which can be characterized in a number of equivalent ways, e.g., as the complement of an ample section of a ruling in a Hirzebruch surface. In this article Donzelli proves that every Danilov-Gizatullin surface satisfies the algebraic density property. The paper deals with the problem of the structure of the group of holomorphic automorphisms \(\text{Aut}(\mathbb{C}^ n)\) of \(\mathbb{C}^ n\) (with the topology of locally uniform convergence). The authors consider two classes of holomorphic automorphisms of \(\mathbb{C}^ n\): 1. shears --- of the form \((z_ 1,\dots,z_ n)\mapsto(z_ 1,\dots,z_{n-1}\), \(f(z_ 1,\dots,z_{n-1})+z_ n)\), 2. overshears --- of the form \((z_ 1,\dots,z_ n)\mapsto(z_ 1,\dots,z_{n-1}, f(z_ 1,\dots,z_{n-1})+h(z_ 1,\dots,z_{n- 1})z_ n)\), where \(f, h\) are holomorphic functions on \(\mathbb{C}^{n-1}\), \(h\) nowhere 0. Let \(\text{Aut}_ 1(\mathbb{C}^ n)\) be the group of holomorphic automorphisms of \(\mathbb{C}^ n\) with Jacobi determinant 1. Denote by \(G(\mathbb{C}^ n)\) (resp. \(G_ 1(\mathbb{C}^ n))\) the subgroup of \(\text{Aut}(\mathbb{C}^ n)\) (resp. \(\text{Aut}_ 1(\mathbb{C}^ n))\) generated by overshears (resp. shears) i.e. all finite compositions of overshears (resp. shears). The first author in Complex Variables, Theory Appl. 14, No. 1-4, 223-235 (1990; Zbl 0705.58008) proved that for \(n\geq 2\) \(G_ 1(\mathbb{C}^ n)\) is dense in \(\text{Aut}_ 1(\mathbb{C}^ n)\) and that \(G_ 1(\mathbb{C}^ 2)\neq\text{Aut}_ 1(\mathbb{C}^ 2)\). In the paper the authors push further these results. Namely, they prove: 1. \(G_ 1(\mathbb{C}^ n)\neq\text{Aut}_ 1(\mathbb{C}^ n)\), 2. \(G(\mathbb{C}^ n)\neq\text{Aut}(\mathbb{C}^ n)\), 3. \(G(\mathbb{C}^ n)\) is dense in \(\text{Aut}(\mathbb{C}^ n)\).

A complex manifold satisfies the ``density property'' if the integrable holomorphic vector fields generate a dense Lie subalgebra of the Lie algebra of all global holomorphic vector fields. This is an important property which has many implications and which was first verified for the affine complex space \({\mathbb C}^n\) by \textit{E. Andersén} and \textit{L. Lempert} [Invent. Math. 110, No. 2, 371--388 (1992; Zbl 0770.32015)]. The corresponding property for complex algebraic varieties is called ``algebraic density property''. Danilov-Gizatullin surfaces are two-dimensional complex affine algebraic varieties which can be characterized in a number of equivalent ways, e.g., as the complement of an ample section of a ruling in a Hirzebruch surface. In this article Donzelli proves that every Danilov-Gizatullin surface satisfies the algebraic density property. Recently, linear codes constructed from defining sets have been studied widely and they have many applications. For an odd prime \(p\), let \(q=p^m\) for a positive integer \(m\) and \(\mathrm {Tr}_m\) the trace function from \(\mathbb {F}_q\) onto \(\mathbb {F}_p\). In this paper, for a positive integer \(t\), let \(D\subset \mathbb {F}^t_q\) and \(D=\{(x_1,x_2) \in (\mathbb {F}_q^{\ast})^2 : \mathrm{Tr}_{m}(x_1+x_2)=0\}\), we define a \(p\)-ary linear code \(\mathcal {C}_D\) by \[ \mathcal {C}_D=\left\{ \mathbf {c}(a_1,a_2) : (a_1,a_2)\in \mathbb F^2_q\right\} , \] where \[ \mathbf c(a_1,a_2)=\left( \mathrm{Tr}_m\left(a_1x^2_1 +a_2x^2_2\right)\right) _{(x_1,x_2)\in D}. \] We compute the weight enumerators of the punctured codes \(\mathcal {C}_D\).