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The authors present proofs of the results announced in their former paper [Russ. Acad. Sci., Dokl., Math. 46, No. 2, 279-281 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 326, No. 3, 414-416 (1992; Zbl 0792.32009)]. Let \(\Omega \subset \mathbb{C}^ n\) be a domain of holomorphy, \(n \geq 2\), and let \(\Gamma \subset \Omega\) be a smooth generic manifold of the form \(\Gamma=\{z \in \Omega:\rho_ 1(z)=\cdots=\rho_ k(z)=0\}\), where \(\rho_ 1,\dots,\rho_ k \in {\mathcal C}^ 3 (\Omega)\) \((1 \leq k \leq n- 1)\) and \(\partial \rho_ 1 \wedge \cdots \wedge \overline \partial \rho_ k \neq 0\) on \(\Gamma\). Fix \(z_ 0 \in \Gamma\) and let \(\Omega_ \Gamma: =(\Gamma + V_ 0) \cap \Omega\), where \(V_ 0\) is an open cone (with vertex at 0) contained in the normal space to \(\Gamma\) at \(z_ 0\). Assume that a compact set \(K \subset \Omega_ \Gamma\) is meromorphically \(p\)-convex, i.e. \(K=\widetilde K_ p\), where \(\widetilde K_ p:=\Omega \backslash \bigcup_{h \in {\mathcal O} (\Omega,\mathbb{C}^ p)} (\Omega \backslash h^{-1}(h(K)))\). The authors present conditions under which any CR-function \(f\) on \(\Gamma \backslash K\) of class \(L^ 1(\Gamma \backslash K\), loc) extends holomorphically to \(\Omega_ \Gamma \backslash K\). This is a generalization of their former results related to the case where \(K\) is holomorphically convex.

The authors present proofs of the results announced in their former paper [Russ. Acad. Sci., Dokl., Math. 46, No. 2, 279-281 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 326, No. 3, 414-416 (1992; Zbl 0792.32009)]. This paper is devoted to the well-posedness and long-time behavior of a stochastic Kirchhoff type suspension bridge equation with strong damping. The existence of the random attractor for a Kirchhoff type suspension bridge equation with white noise is established. Moreover, the upper semicontinuity of random attractors is also provided when the coefficient of random term approaches zero.

Let \(Y\) be a smooth projective surface defined over an algebraically closed field of characteristic \(\neq 2\). A nodal curve of \(Y\) is a smooth rational curve \(N\) such that \(K_YN=0\). An even set of nodal curves is a set \(N_1,\dots N_k\) of disjoint nodal curves such that the divisor \(N_1+\cdots+N_k\) is divisible by \(2\) in Pic\((Y)\). Equivalently, \(N_1,\dots N_k\) is an even set if there exists a double cover \(X\to Y\) branched precisely on the union of the \(N_i\). It is easy to show that the number of curves in an even set is divisible by \(4\). Let now \(V\) be a ruled surface and let \(C\) be a smooth fibre of the ruling of \(V\). Choose a point \(P\in V\), blow it up and then blow up the intersection point of the strict transform of \(C\) with the exceptional curve: the strict transforms of \(C\) and of the exceptional curve of the first blow up are a pair of disjoint nodal curves on the blown up surface. Performing this operation \(m\) times on different smooth fibres of the ruling of \(V\), one obtains \(m\) pairs of disjoint nodal curves such that any union of an even number of such pairs is an even set. \textit{I. Dolgachev, M. Mendes Lopes} and \textit{R. Pardini} [Compos. Math. 132, No. 3, 349--363 (2002; Zbl 1059.14050)] have shown that if \(Y\) is a rational surface containing \(N_1, \dots N_k\) disjoint nodal curves with an even subset of at least \(8\) nodal curves, then every even subset of the \(N_i\) arises as explained above. In the paper under consideration, the authors prove the same statement for a rational surface \(Y\) containing an even set of \(4\) nodal curves. In this situation the methods of [loc. cit.] cannot be applied, and they are replaced by a careful study of curves and adjoint systems on a rational surface. Let \(X\) be a projective rational surfaces with \(n(X)\) nodes, that arises from a birational map \(f:X'\to X\), where \(X'\) is smooth and \(f\) contracts \(n(X)\) disjoint nodal curves \(C_i\subset X'\). The Picard number \(\rho(X')\) is equal to the Picard number \(\rho(X)\) plus \(n(X)\). The authors consider the cases in which \(n(X)\) is big with respect to \(\rho(X')\), i.e. the cases in which \(\rho(X)=1\) or \(2\). In these situations, when the characteristic of the ground field is \(\neq 2\), the authors determine \(X'\) as a suitable blowing up of a minimal rational ruled surface. In the proof, they make use of the theory of codes: codes arise naturally in the previous picture by taking the subspace of Pic(\(X\))/2Pic(\(X\)) generated by the classes of the \(C_i\)'s. The result applies to the study of involutions on minimal surfaces of general type, having \(p_g=0\). Indeed, in this setting, the authors prove that no involutions exist on such a surface \(S\) when \(K_S^2=9\), and classify the possible surfaces with involution for the case \(K_S^2=8\).

Let \(Y\) be a smooth projective surface defined over an algebraically closed field of characteristic \(\neq 2\). A nodal curve of \(Y\) is a smooth rational curve \(N\) such that \(K_YN=0\). An even set of nodal curves is a set \(N_1,\dots N_k\) of disjoint nodal curves such that the divisor \(N_1+\cdots+N_k\) is divisible by \(2\) in Pic\((Y)\). Equivalently, \(N_1,\dots N_k\) is an even set if there exists a double cover \(X\to Y\) branched precisely on the union of the \(N_i\). It is easy to show that the number of curves in an even set is divisible by \(4\). Let now \(V\) be a ruled surface and let \(C\) be a smooth fibre of the ruling of \(V\). Choose a point \(P\in V\), blow it up and then blow up the intersection point of the strict transform of \(C\) with the exceptional curve: the strict transforms of \(C\) and of the exceptional curve of the first blow up are a pair of disjoint nodal curves on the blown up surface. Performing this operation \(m\) times on different smooth fibres of the ruling of \(V\), one obtains \(m\) pairs of disjoint nodal curves such that any union of an even number of such pairs is an even set. \textit{I. Dolgachev, M. Mendes Lopes} and \textit{R. Pardini} [Compos. Math. 132, No. 3, 349--363 (2002; Zbl 1059.14050)] have shown that if \(Y\) is a rational surface containing \(N_1, \dots N_k\) disjoint nodal curves with an even subset of at least \(8\) nodal curves, then every even subset of the \(N_i\) arises as explained above. In the paper under consideration, the authors prove the same statement for a rational surface \(Y\) containing an even set of \(4\) nodal curves. In this situation the methods of [loc. cit.] cannot be applied, and they are replaced by a careful study of curves and adjoint systems on a rational surface. The maximal operator plays the similar role as the summation operator in the sense of stability of operation. So, we can discuss ARMA processes in the maximal operation by the same way as in the summation operation. However, many papers already dealt with moving order statistics. In this paper, we discuss asymptotic behavior of maximal autoregressive (MAR) processes with the weight tending to 1.

The authors use the fixed point theorem of \textit{J. Brzdȩk} and \textit{K. Ciepliński} [Acta Math. Sci., Ser. B, Engl. Ed. 38, No. 2, 377--390 (2018; Zbl 1399.39063)] to prove hyperstability results for the Jensen equation in the setting of 2-Banach spaces. The aim of this article is to prove a fixed point theorem in 2-Banach spaces and show its applications to the Ulam stability of functional equations. The obtained stability results concern both some single variable equations and the most important functional equation in several variables, namely, the Cauchy equation. Moreover, a few corollaries corresponding to some known hyperstability outcomes are presented.

The authors use the fixed point theorem of \textit{J. Brzdȩk} and \textit{K. Ciepliński} [Acta Math. Sci., Ser. B, Engl. Ed. 38, No. 2, 377--390 (2018; Zbl 1399.39063)] to prove hyperstability results for the Jensen equation in the setting of 2-Banach spaces. Every chordal graph \( G\) admits a representation as the intersection graph of a family of subtrees of a tree. A classic way of finding such an intersection model is to look for a maximum spanning tree of the valuated clique graph of \(G\). Similar techniques have been applied to find intersection models of chordal graph subclasses as interval graphs and path graphs. In this work, we extend those methods to be applied beyond chordal graphs: we prove that a graph \(G\) can be represented as the intersection of a Helly separating family of graphs belonging to a given class if and only if there exists a spanning subgraph of the clique graph of \( G\) satisfying a particular condition. Moreover, such a spanning subgraph is characterized by its weight in the valuated clique graph of \(G\). The specific case of Helly circular-arc graphs is treated. We show that the canonical intersection models of those graphs correspond to the maximum spanning cycles of the valuated clique graph.

Soit \(v\) une solution de l'équation de Klein-Gordon quasilinéaire en dimension 1 d'espace \[ \square v+v= F(v,\partial_t v,\partial v,\partial_t \partial_x v,\partial^2_xv) \] à données de Cauchy régulières à support compact, de taille \(\varepsilon\to 0\), où \(F\) est une non-linéarité \({\mathcal C}^\infty\), à valeurs réelles, nulle su moins à l'ordre 2 à l'origine, affine en ses deux derniers arguments. On sait que \(v\) existe sur un intervalle de temps de longueur \(\geq e^{c/ \varepsilon^2} (c> 0)\) et qu'en général elle explose en temps fini de l'ordre de \(e^{c'/ \varepsilon^2}(c'>0)\). L'A. avait conjecturé [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16, No. 5, 563-591 (1999; Zbl 0937.35160)] une condition portant sur les termes quadratiques et cubiques de \(F\), nécessaire et suffisante pour l'existence globale en temps pour \(\varepsilon\) assez petit. Le premier des deux théorèmes principaux du présent article établit la suffisance de cette condition. La deuxième but de l'article est d'obtenir l'allure du comportement asymptotique lorsque \(t\to\infty\) des solutions sous les hypothèses ci-dessus. En particulier, les solutions n'ont pas en général un comportement de solution linéaire à l'infini. L'ingrédient de base dans la preuve des résultats est l'utilization des ``formes normales paradifférentielles''. Let \(u\) be a solution to the semilinear Klein-Gordon equation in one space dimension, \[ \partial^2_tu-\partial^2_x u+u=F(u, \partial_tu, \partial_xu, \partial_t \partial_xu, \partial^2_xu) \] where \(F\) is a quadratic nonlinearity and Cauchy data \(u(t=0)=\varepsilon u_0\), \(\partial_t u(t=0)= \varepsilon u\), are small in \(C_0^\infty\). The aim of the paper is to obtain an explicit lower bound of the existence time of the solution \((T)\): \[ \lim\inf_{\varepsilon\to 0}\varepsilon^2\log T_\varepsilon \geq A. \] This is an improvement of the known result for \(A=0\). The expression for \(A\) is directly computed from the Cauchy data and the given nonlinearity. The author first constructs the asymptotic solution and then derives the approximate solution. An interesting constraint on the nonlinearity (called null-condition) is obtained for which \(A\) is \(\infty\).

Soit \(v\) une solution de l'équation de Klein-Gordon quasilinéaire en dimension 1 d'espace \[ \square v+v= F(v,\partial_t v,\partial v,\partial_t \partial_x v,\partial^2_xv) \] à données de Cauchy régulières à support compact, de taille \(\varepsilon\to 0\), où \(F\) est une non-linéarité \({\mathcal C}^\infty\), à valeurs réelles, nulle su moins à l'ordre 2 à l'origine, affine en ses deux derniers arguments. On sait que \(v\) existe sur un intervalle de temps de longueur \(\geq e^{c/ \varepsilon^2} (c> 0)\) et qu'en général elle explose en temps fini de l'ordre de \(e^{c'/ \varepsilon^2}(c'>0)\). L'A. avait conjecturé [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16, No. 5, 563-591 (1999; Zbl 0937.35160)] une condition portant sur les termes quadratiques et cubiques de \(F\), nécessaire et suffisante pour l'existence globale en temps pour \(\varepsilon\) assez petit. Le premier des deux théorèmes principaux du présent article établit la suffisance de cette condition. La deuxième but de l'article est d'obtenir l'allure du comportement asymptotique lorsque \(t\to\infty\) des solutions sous les hypothèses ci-dessus. En particulier, les solutions n'ont pas en général un comportement de solution linéaire à l'infini. L'ingrédient de base dans la preuve des résultats est l'utilization des ``formes normales paradifférentielles''. <abstract> <p>In practice, objective condition may impose constraints on design region, which make it difficult to find the exact D-optimal design. In this paper, we propose a Multi-stage Differential Evolution (MDE) algorithm to find the global approximated D-optimal design in an experimental region with linear or nonlinear constraints. MDE algorithm is approved from Differential Evolution (DE) algorithm. It has low requirements for both feasible regions and initial values. In iteration, MDE algorithm pursues evolutionary equilibrium rather than convergence speed, so it can stably converge to the global D-optimal design instead of the local ones. The advantages of MDE algorithm in finding D-optimal design will be illustrated by examples.</p> </abstract>

The author develops some of the foundations of the theory of relatively hyperbolic groups as originally formulated by \textit{M. Gromov} in Chapter 8.6 of his article entitled Hyperbolic groups, [Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)]. Here is proved the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph. In the article is given a definition of a fine graph, i.e. it has only finitely many circuits of a given length containing any given edge. Moreover, there is shown how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form. Finally the author defines the boundary of a relatively hyperbolic group, and shows that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary. We should add that this very outstanding and valuable article was known since 1997 under the same title as a preprint of the author edited by the University of Southampton. [For the entire collection see Zbl 0626.00014.] The author gives three equivalent abstract definitions of (word) hyperbolic or negatively-curved groups. These groups include, but are not restricted to, the fundamental groups of closed negatively curved manifolds. The author shows how the curvature manifests itself in the combinatorics of the group. The author generalizes small-cancellation theory to this setting. The author exhibits numerous constructions which, when applied to hyperbolic groups, yield hyperbolic groups. The methods are general, powerful, and beautiful. The reader needs to beware that many of the proofs and statements, while correct in spirit, are incorrect in detail.

The author develops some of the foundations of the theory of relatively hyperbolic groups as originally formulated by \textit{M. Gromov} in Chapter 8.6 of his article entitled Hyperbolic groups, [Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015)]. Here is proved the equivalence of two definitions of this notion. One is essentially that of a group admitting a properly discontinuous geometrically finite action on a proper hyperbolic space. The other is that of a group which admits a cofinite action on a connected fine hyperbolic graph. In the article is given a definition of a fine graph, i.e. it has only finitely many circuits of a given length containing any given edge. Moreover, there is shown how a relatively hyperbolic group can be assumed to act on a proper hyperbolic space of a particular geometric form. Finally the author defines the boundary of a relatively hyperbolic group, and shows that the limit set of any geometrically finite action of the group is equivariantly homeomorphic to this boundary. We should add that this very outstanding and valuable article was known since 1997 under the same title as a preprint of the author edited by the University of Southampton. A study of Earth-Moon bi-impulsive trajectories is presented in this paper. The motion of the space vehicle is described by the classic planar circular restricted three-body problem. The velocity increments are computed through analytical expressions, which are derived from the development of the Jacobi Integral expression. To determine the trajectories, a new two-point boundary value problem (TPBVP) with prescribed value of Jacobi Integral is formulated. Internal and external trajectories are determined through the solution of this new TPBVP for several times of flight. A relation between the Jacobi Integral and the Kepler's energy at arrival is derived and several kinds of study are performed. Critical values of the Jacobi Integral, for which the Kepler's energy of the space vehicle on the arrival trajectory becomes negative, are calculated for several configurations of arrival at the low Moon orbit in both directions: clockwise and counterclockwise. Results show that the proposed method allows the estimation of the fuel consumption before solving the TPBVP, and it facilitates the determination of trajectories with large time of flight. However, increasing values of the time of flight are not necessarily related with the increase of the Jacobi Integral value, which means that the obtaining of new trajectories becomes more difficult as the Jacobi Integral increases. Moreover, the proposed method provides results to be used as initial guess for more complex models and for optimization algorithms in order to minimize the total fuel consumption. For this case, this paper presents an example where an internal trajectory with large time of flight is optimized considering the Sun's attraction.

If f is an integrable function on the real line \({\mathbb{R}}\) with respect to the measure \(dx/(1+x^ 2)\) and H stands for the Hilbert transform, then [f,H] is an operator defined by \[ [f,H]g(x)=H(fg)(x)-f(x)Hg(x)\quad (=\frac{1}{\pi}p.v.\int^{\infty}_{-\infty}\frac{f(y)-f(x)}{y- x}g(y)dy). \] The authors establish a characterization for the boundedness of [f,H] as an operator from the energy space \(E_{\alpha}\) with respect to the \(\alpha\)-Riesz kernel \(| x|^{\alpha -1}\), \(0<\alpha <1\), to itself; \(E_{\alpha}\) is the completion of \(C^{\infty}_ 0\)-functions on \({\mathbb{R}}\) with respect to the norm \(\| f\|_{\alpha}=(\int^{\infty}_{-\infty}| \xi |^{\alpha}| \hat f(\xi)|^ 2 d\xi)^{1/2},\) where \(\hat f\) is the Fourier transform of f: Theorem. [f,H]: \(E_{\alpha}\to E_{\alpha}\) is bounded if and only if \(f\in BMO_{\alpha}\). Here \(BMO_{\alpha}\) is the Banach space of functions \(f\in L^ 1(dx/(1+x^ 2))\), modulo constants, with the norm \(\| f\|_{BMO_{\alpha}}=\| \nabla f(x,y)|^ 2 y^{1- \alpha}\|^{1/2}_{Car,\alpha},\) where \(\nabla f(x,y)\) is the gradient of the Poisson extension of f to the upper half plane \({\mathbb{C}}_+\) and the number \(\| d\mu \|_{Car,\alpha}\), for a nonnegative (\(\alpha\)-Carleson) measure \(\mu\) on \({\mathbb{C}}_+\), is the minimum of constants B such that \(\iint_{\hat O}d\mu (x,y)\leq B Cap_{\alpha}(O)\) whenever \(O\subset {\mathbb{R}}\) is an open set. Here \(\hat O=\cup \{I\times (0,| I|):\) I component of \(\hat O\}\) and \(Cap_{\alpha}(\cdot)\) is the \(\alpha\)-Riesz capacity. The theorem leads to a result corresponding the fact that the dual of \(H^ 1\) is BMO; cf. \textit{C. Fefferman} and \textit{E. M. Stein} [Acta Math. 129, 137-193 (1972; Zbl 0257.46078)]. This paper is a major contribution to the study of \(H^p\) spaces, singular integrals, and harmonic analysis an \(\mathbb R^n\). Classically the theory of \(H^p\) spaces arose from analytic function theory. \(H^p\) was defined as the space of analytic functions in the upper half plane with boundary values in \(L^p(\mathbb R)\). More recently this definition has been generalised to \(\mathbb R^n\) by introducing generalised conjugate harmonic functions in \(\mathbb R_+^{n+1}=\{(x,t): x\in\mathbb R^n\), \(t>0\}\). The authors present several intrinsic descriptions of \(H^p\), of a real variable nature, not involving conjugate functions. These results greatly clarify the meaning of \(H^p\), as well as throwing new light on the behaviour of convolution operators an \(L^p\). The following is a summary of some of the main results. The first main result is the description of the dual of \(H^1\). \(H^1\) is the Banach space of all functions \(f\) in \(L^1(\mathbb R^n)\) such that \(R_jf\in L^1(\mathbb R^n)\), \(j=1,\dots,n\), where \(R_j\) is the \(j\)-th Riesz transform. (In terms of Fourier transforms, \((R_jf)^\wedge(y)= y_j\hat f(y)/|y|\). When \(n=1\) the definition says that the Hilbert transform of \(f\) is in \(L^1\), or equivalently \(\int f(x)\,dx=0\) and \(f=g+\bar h\) where \(g\) and \(h\) are in the classical ``analytic'' \(H^1\)). The authors prove that the dual of \(H'\) is the space of all functionals of the form \(\varphi\to\int f\varphi\) (suitably interpreted if \(f\varphi\not\in L^1\)), where \(\varphi\) is a function of bounded mean oscillation (BMO), which means that there is a constant \(C>0\) (depending an \(\varphi\)) such that \(\int_Q |f-f_Q|\leq C|Q|\) for any cube \(Q\) in \(\mathbb R^n\), where \(f_Q= |Q|^{-1}\int_Q f\). The proof of this remarkable and deep result depends an an inequality of Carleson and some clever manipulations of Littlewood Paley functions. The hard part is to show that a BMO function defines a bounded functional on \(H^1\). The essential difficulties are already present in the case \(n=1\). This duality leads to a new approach to convolution operators which brings out the usefulness of BMO as a substitute for \(L^\infty\). If \(T\) is a convolution operator (i.e. \(Tf= K*f\) for some distribution \(K\) an \(\mathbb R^n\)) mapping \(L^\infty\) into BMO, then the authors show, using the duality and a description of \(L^p\) related to BMO, that \(T\) maps \(H^1\) into \(H^1\), BMO into BMO, and \(L^p\) into \(L^p\) for \(1<p<\infty\). The condition that \(T\) map \(L^\infty\) into BMO is relatively easy to verify for singular integral operators of Calderón-Zygmund type. A more refined version of this result, involving Calderón's complex method of interpolation, enables the authors to prove new results on \(L^p\) multipliers. The authors then turn to \(H^p\) spaces for general \(p\) \((0<p<\infty)\). They define these first as spaces of harmonic functions an \(\mathbb R_+^{n+1}\), without reference to boundary values. Specifically, a harmonic function \(u_0\) is in \(H^p\) if there exist harmonic functions \(u_1,\dots,u_n\) satisfying \[ \partial u_j/\partial x_i=0, \quad \sum_{i=0}^n \partial u_i/\partial x_i \quad\text{and}\quad \sup_{t>0} \int_{\mathbb R^n}|u(x,t)|^p\,dx<\infty, \] where \(|u|^2= \sum_{i=0}^n |u_i|^2\). This definition is appropriate if \(p>(n-1)/n\). In general a more elaborate version (here omitted) is needed (the point is that \(|u|^p\) is subharmonic only if \(p\geq (n-1)/n\)). The main result is as follows: let \(u\) be harmonic in \(\mathbb R_+^{n+1}\) and define \(u^*(x)= \sup_t |u(x,t)|\). Then \(u\in H^p\) if and only if \(u^*\in L^p\). (For \(n=1\) this was proved by Burkholder, Gundy and Silverstein in 1971). Finally the authors consider boundary values. If \(u\in H^p\) then \(u(x,t)\to f(x)\) as \(t\to0\), in the distribution sense, where \(f\) is a tempered distribution on \(\mathbb R^n\). Denote the set of such \(f\) also by \(H^p\). Then \(H^p= L^p\) for \(p>1\) and for \(p=1\) this definition is consistent with the earlier one. The last result above characterises \(H^p\) in terms of Poisson integrals (as \(u\) is the Poisson integral of \(f\)). The authors show that the Poisson kernel can be replaced by any smooth approximate identity -- more precisely, fix a smooth function \(\varphi\) on \(\mathbb R^n\), decreasing rapidly at \(\infty\), with \(\int\varphi=1\). Put \(\varphi_t(x)= t^{-n} \varphi(x/t)\), and for any tempered distribution \(f\) write \(f^*(x)= \sup_{t>0} |\varphi_t*f(x)|\). Then (for \(0<p<\infty\)) \(f\in H^p\) if and only if \(f^*\in L^p\). (``Non-tangential'' versions of this and the preceding result are also given). This result implies for example that one could define \(H^p\) in terms of the wave equation rather than Laplace's and get the same space of functions on \(\mathbb R^n\). The paper concludes with a proof that certain singular integral operators map \(H^p\) to itself.

If f is an integrable function on the real line \({\mathbb{R}}\) with respect to the measure \(dx/(1+x^ 2)\) and H stands for the Hilbert transform, then [f,H] is an operator defined by \[ [f,H]g(x)=H(fg)(x)-f(x)Hg(x)\quad (=\frac{1}{\pi}p.v.\int^{\infty}_{-\infty}\frac{f(y)-f(x)}{y- x}g(y)dy). \] The authors establish a characterization for the boundedness of [f,H] as an operator from the energy space \(E_{\alpha}\) with respect to the \(\alpha\)-Riesz kernel \(| x|^{\alpha -1}\), \(0<\alpha <1\), to itself; \(E_{\alpha}\) is the completion of \(C^{\infty}_ 0\)-functions on \({\mathbb{R}}\) with respect to the norm \(\| f\|_{\alpha}=(\int^{\infty}_{-\infty}| \xi |^{\alpha}| \hat f(\xi)|^ 2 d\xi)^{1/2},\) where \(\hat f\) is the Fourier transform of f: Theorem. [f,H]: \(E_{\alpha}\to E_{\alpha}\) is bounded if and only if \(f\in BMO_{\alpha}\). Here \(BMO_{\alpha}\) is the Banach space of functions \(f\in L^ 1(dx/(1+x^ 2))\), modulo constants, with the norm \(\| f\|_{BMO_{\alpha}}=\| \nabla f(x,y)|^ 2 y^{1- \alpha}\|^{1/2}_{Car,\alpha},\) where \(\nabla f(x,y)\) is the gradient of the Poisson extension of f to the upper half plane \({\mathbb{C}}_+\) and the number \(\| d\mu \|_{Car,\alpha}\), for a nonnegative (\(\alpha\)-Carleson) measure \(\mu\) on \({\mathbb{C}}_+\), is the minimum of constants B such that \(\iint_{\hat O}d\mu (x,y)\leq B Cap_{\alpha}(O)\) whenever \(O\subset {\mathbb{R}}\) is an open set. Here \(\hat O=\cup \{I\times (0,| I|):\) I component of \(\hat O\}\) and \(Cap_{\alpha}(\cdot)\) is the \(\alpha\)-Riesz capacity. The theorem leads to a result corresponding the fact that the dual of \(H^ 1\) is BMO; cf. \textit{C. Fefferman} and \textit{E. M. Stein} [Acta Math. 129, 137-193 (1972; Zbl 0257.46078)]. The aim of this paper is to estimate parameters of generalized Pareto distribution based on generalized order statistics. Some non-Bayesian methods, such as MLE, bootstrap and unbiased estimators have been obtained to develop point and interval estimations. Bayesian estimations have also been derived under LSE and LINEX loss functions. To compare the performances of the employed methods, numerical results have been computed. To illustrate dependence and association properties of generalized order statistics, correlation coefficient and some informational measures in closed form have been obtained.

The notion of directional structures was introduced by E. Deák in 1964. This paper is a sequel of [ibid. 15, 45-61 (1980; Zbl 0558.54024)], where the author applies the theory of directional structures to obtain an internal characterization of subspaces of a Euclidean space in terms of the existence of an open subbase satisfying certain conditions. The main result is the following theorem: If a separable metrizable space X has a comparable complementary subbase S with inc \(S\leq n\) such that for each A,B\(\in S\), \(A\varsubsetneq B\Rightarrow \bar A\subset B\), then X can be topologically embedded in \({\mathbb{R}}^ n\). As the author writes ''this is the first part of a series intended to present some applications of the theory of directional structures to problems in general topology''. The author presents: (1) a generalization of de Groot's embedding theorem: a separable metrizable space can be topologically embedded into \(R^ n\) iff it has a comparable \(T_ 1\)- complementary subbase S with inc \(S\leq n\), where inc S means the supremum of the cardinalities of subcollections of S consisting of pairwise incomparable sets (the sets A,B\(\subset X\) are incomparable iff none of the sets \(A\cap B\), \(A\setminus B\), \(B\setminus A\) and \(X\setminus (A\cap B)\) is empty). (2) Some known characterizations of topological products of suborderable spaces as for instance: a space is homeomorphic to the product of compact orderable spaces iff it is \(T_ 1\)-space with comparable binary subbase.

The notion of directional structures was introduced by E. Deák in 1964. This paper is a sequel of [ibid. 15, 45-61 (1980; Zbl 0558.54024)], where the author applies the theory of directional structures to obtain an internal characterization of subspaces of a Euclidean space in terms of the existence of an open subbase satisfying certain conditions. The main result is the following theorem: If a separable metrizable space X has a comparable complementary subbase S with inc \(S\leq n\) such that for each A,B\(\in S\), \(A\varsubsetneq B\Rightarrow \bar A\subset B\), then X can be topologically embedded in \({\mathbb{R}}^ n\). Finite difference methods for solving one-dimensional Schrödinger eigenvalue problems on microcomputers have been developed by Killingbeck. In this paper the methods are generalized for applications to a wider class of potentials. Numerical examples are given for potentials of the form \(Ax^ 2+Bx^ 3+Cx^ 4\).

The paper is concerned with the Hasse principle and weak approximation for the class of varieties satisfying the Diophantine equation \[ P(t)={\mathbf N}_{K/k}(x_1,\dots,x_n)\neq 0, \tag{1} \] where \({\mathbf N}_{K/k}\) is a full norm form for an extension \(K/k\) of number fields and \(P(t)\) is a polynomial in \(k[t]\). A small number of results are known, and they mainly concern the cases where the polynomial \(P(t)\) splits completely over \(k\) with at most two roots. Other positive results have been obtained when \([K:k]\) is prime or the Galois group of the normal closure of \(K/k\) has cyclic Sylow subgroups. For this reason, the authors concentrate on the case where \(P(t)\) is irreducible of degree 2 and the extension \(K/k\) does not necessarily fall into the known cases. The main result of the paper can be stated as follows: Let \(P(t)\in{\mathbb Q}[t]\) be an irreducible quadratic polynomial and let \(K\) be a quartic extension of \({\mathbb Q}\) containing a root of \(P(t)\). Suppose that, for every valuation \(v\) of \({\mathbb Q}\), we are given a solution \((t_v,{\mathbf x}_v)\in {\mathbb Q}_v^5\) of (1). Let \(S\) be any finite set of valuations of \({\mathbb Q}\) and let \(\varepsilon>0\). Then there is a solution \((t,{\mathbf x})\in {\mathbb Q}^5\) of (1) such that \[ |t-t_v|_v<\varepsilon, \qquad |{\mathbf x} - {\mathbf x}_v|_v<\varepsilon, \] for every \(v\in S\). Thus the Hasse principle and the weak approximation hold for (1). The proof of the main theorem is based on analytic number theory, and in particular it is inspired by the paper [\textit{E. Fouvry} and \textit{H. Iwaniec}, Acta Arith. 79, No. 3, 249--287 (1997; Zbl 0881.11070)]. The authors establish an asymptotic formula which shows that of the primes of the form \(l^2+m^2\) not exceeding \(x\), counted according to multiplicity of representation, about the expected number have the number \(l\) also prime. In particular this number tends to \(\infty\) with \(x\). The treatment uses sieve ideas although, as is well known, sieve methods alone could not deliver such a result. Other ideas central to the treatment include an inequality of the large sieve type for numbers of the form \(\nu/d\) where \(\nu^2+1\equiv 0 \pmod d\), and estimates for bilinear forms over the Gaussian integers. Some of this work also provides some input into the later result of \textit{J. Friedlander} and \textit{H. Iwaniec} concerning prime values of \(m^2+n^4\), of which an announcement appears in [Proc. Natl. Acad. Sci. USA 94, No. 4, 1054-1058 (1997: Zbl 0870.11059)]. The authors devote some space to explaining the broader ideas of their approach as well as setting down the technical details. They also establish some easier related results in which primes are replaced by norms of ideals in abelian fields.

The paper is concerned with the Hasse principle and weak approximation for the class of varieties satisfying the Diophantine equation \[ P(t)={\mathbf N}_{K/k}(x_1,\dots,x_n)\neq 0, \tag{1} \] where \({\mathbf N}_{K/k}\) is a full norm form for an extension \(K/k\) of number fields and \(P(t)\) is a polynomial in \(k[t]\). A small number of results are known, and they mainly concern the cases where the polynomial \(P(t)\) splits completely over \(k\) with at most two roots. Other positive results have been obtained when \([K:k]\) is prime or the Galois group of the normal closure of \(K/k\) has cyclic Sylow subgroups. For this reason, the authors concentrate on the case where \(P(t)\) is irreducible of degree 2 and the extension \(K/k\) does not necessarily fall into the known cases. The main result of the paper can be stated as follows: Let \(P(t)\in{\mathbb Q}[t]\) be an irreducible quadratic polynomial and let \(K\) be a quartic extension of \({\mathbb Q}\) containing a root of \(P(t)\). Suppose that, for every valuation \(v\) of \({\mathbb Q}\), we are given a solution \((t_v,{\mathbf x}_v)\in {\mathbb Q}_v^5\) of (1). Let \(S\) be any finite set of valuations of \({\mathbb Q}\) and let \(\varepsilon>0\). Then there is a solution \((t,{\mathbf x})\in {\mathbb Q}^5\) of (1) such that \[ |t-t_v|_v<\varepsilon, \qquad |{\mathbf x} - {\mathbf x}_v|_v<\varepsilon, \] for every \(v\in S\). Thus the Hasse principle and the weak approximation hold for (1). The proof of the main theorem is based on analytic number theory, and in particular it is inspired by the paper [\textit{E. Fouvry} and \textit{H. Iwaniec}, Acta Arith. 79, No. 3, 249--287 (1997; Zbl 0881.11070)]. The modelling behaviour of 12--16-year-old pupils was studied on the basis of written reports about realistic mathematics projects. These were analysed by using a hypothetical project modelling route involving 16 actions. Application of this tool was useful in understanding the difficulties pupils have in carrying out the initial steps and the validation process.

The authors generalise their earlier results [Bull. Lond. Math. Soc. 40, No. 1, 129--138 (2008; Zbl 1171.32018)] on functions on curves to the case of maps between curves. Let \(F: (\mathcal{X},0)\to (\mathcal{Y},0)\) be a deformation of the map \(f: (X,0)\to (Y,0)\), such that \(X_t\setminus \{0\}\) and \(Y_t\setminus \{0\}\) are smooth. Then \(F\) is topologically trivial if and only if \(\mu(X_t)\), \(\mu(Y_t)\) and \(\deg(f_t)\) are constant, and \(F\) is Whitney equisingular if and only if in addition the multiplicities \(m(X_t)\) and \(m(Y_t)\) are constant. As an application some known equisingularity results for map germs are proved: an unfolding of a finitely determined map germ \(f: (\mathbb{C}^2,0)\to (\mathbb{C}^2,0)\) is Whitney equisingular if and only if the Milnor number of the discriminant curve \(\Delta_t\) is constant; an unfolding of a germ \(f: (\mathbb{C}^2,0)\to (\mathbb{C}^3,0)\) is topologically trivial if and only if the Milnor number of the double point curve \(D^2_t\) is constant. \((X,0) \subseteq ({\mathbb C}^n , 0)\) denotes a germ of a reduced space curve. For a finite function germ \(f:(X,0) \to ({\mathbb C} , 0)\), the Milnor number \(\mu (f)\) (in the sense of \textit{V. Goryunov} [J. Lond. Math. Soc., II. Ser. 61, 807--822 (2000; Zbl 0965.58030)], \textit{D. Mond} and \textit{D. van Straten} [J. Lond. Math. Soc., II. Ser. 63, No. 1, 177--187 (2001; Zbl 1017.32022)]) is an invariant with the following properties: (1) It is preserved under simultaneous deformation of \(f\) and \((X,0)\). (2) For smooth germs \((X,0) \), the number \(\mu (f)\) coincides with the usual Milnor number. If \((X,0) \) is smoothable, then \(\mu (f)\) is determined by the above conditions. The authors show: \(\mu (f) = \mu (X,0) + \text{deg}(f) -1\). This generalizes a result of the first author and Jorge Pérez. An algebraic proof of the general formula is given. The result is applied to study Whitney-equisingularity for a family \((X_t)\) of space curves. \((X_t)\) is shown to be Whitney-equisingular iff the first polar multiplicity of \((X_t)\) is constant. Furthermore, for a family \(f_t:X_t \to {\mathbb C}\) of functions on space curves, the following is obtained: (a) \(f_t\) is topologically trivial iff \(\mu (f_t)\) is constant. (b) \(f_t\) is Whitney-equisingular iff \(\mu (f_t)\) and the multiplicity \(m_0(X_t,0)\) are both constant.

The authors generalise their earlier results [Bull. Lond. Math. Soc. 40, No. 1, 129--138 (2008; Zbl 1171.32018)] on functions on curves to the case of maps between curves. Let \(F: (\mathcal{X},0)\to (\mathcal{Y},0)\) be a deformation of the map \(f: (X,0)\to (Y,0)\), such that \(X_t\setminus \{0\}\) and \(Y_t\setminus \{0\}\) are smooth. Then \(F\) is topologically trivial if and only if \(\mu(X_t)\), \(\mu(Y_t)\) and \(\deg(f_t)\) are constant, and \(F\) is Whitney equisingular if and only if in addition the multiplicities \(m(X_t)\) and \(m(Y_t)\) are constant. As an application some known equisingularity results for map germs are proved: an unfolding of a finitely determined map germ \(f: (\mathbb{C}^2,0)\to (\mathbb{C}^2,0)\) is Whitney equisingular if and only if the Milnor number of the discriminant curve \(\Delta_t\) is constant; an unfolding of a germ \(f: (\mathbb{C}^2,0)\to (\mathbb{C}^3,0)\) is topologically trivial if and only if the Milnor number of the double point curve \(D^2_t\) is constant. In this article, we consider the dynamics of \(N\) two-dimensional boson systems interacting through a pair potential \(N^{-1}V_a(x_i-x_j)\) where \(V_a(x)=a^{-2}V(x/a)\). It is well known that the Gross-Pitaevskii (GP) equation is a nonlinear Schrödinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if \(u_t\) solves the GP equation, then the family of \(k\)-particle density matrices \(\{\otimes_k u_t,k\geq 1\}\) solves the GP hierarchy. Denote by \(\psi_{N, t}\) the solution to the \(N\)-particle Schrödinger equation. Under the assumption that \(a=N^{-\varepsilon}\) for \(0<\varepsilon <3/4\), we prove that as \(N \to \infty\) the limit points of the \(k\)-particle density matrices of \(\psi_{N,t}\) are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by \(\int V(x)\text{d}x\).

Recall that a metric space \((X, d)\) is called ultrametric (or isosceles, or non-Archimedian) if \(d(x, z)\leq\max\{d(x, y),d(y, z)\}\) for all \(x,y,z\in X\). It was shown by \textit{B. Fichet} in [Classification and related methods of data analysis, Proc. 1st Conf. IFCS, Aachen/FRG 1987, 439--444 (1984; Zbl 0734.62002)] that every finite ultrametric space can be isometrically embedded into the Banach space \(l_p\) for each \(p\in[1,\infty]\). In this interesting paper it is shown that, in fact, every finite ultrametric space can be isometrically embedded into every infinite-dimensional Banach space. [For the entire collection see Zbl 0732.00020.] We deal with metric spaces of \(L_ p\)-type, i.e. spaces which are isometrically embeddable in an \(L_ p\)-space. We investigate their geometrical nature and show their link with the usual metric spaces occurring in data analysis. In particular, we take an interest in metric spaces of \(L_ 1\)-type which unify the classical structures of data analysis.

Recall that a metric space \((X, d)\) is called ultrametric (or isosceles, or non-Archimedian) if \(d(x, z)\leq\max\{d(x, y),d(y, z)\}\) for all \(x,y,z\in X\). It was shown by \textit{B. Fichet} in [Classification and related methods of data analysis, Proc. 1st Conf. IFCS, Aachen/FRG 1987, 439--444 (1984; Zbl 0734.62002)] that every finite ultrametric space can be isometrically embedded into the Banach space \(l_p\) for each \(p\in[1,\infty]\). In this interesting paper it is shown that, in fact, every finite ultrametric space can be isometrically embedded into every infinite-dimensional Banach space. For every measure \(\nu\), the integral \(I: f\to \int f d\mu\) is a linear functional on the set of real measurable functions. By the Daniell-Stone theorem, for every abstract intergral \(\Lambda: F \to \mathbb{R}\) on a Stone vector lattice \(F\) of real functions \(f:\Omega \to \mathbb{R}\) there is a measure \(\mu\) such that \(\int f d \mu = \Lambda(f)\) for all \(f \in F\). In this paper a computable version of this theorem is proven.

The author gives a description of the space of projective unitary representations of the orbifold fundamental group of a compact holomorphic orbifold \(X\) of dimension 2 in terms of the moduli space of stable parabolic bundles over \(X\). This allows the author to compute the cohomology of the \(SU(2)\)-character variety \({\mathcal R}(\Sigma)\) of any Seifert-fibered homology sphere. In particular, he shows that \(H^ k({\mathcal R}(\Sigma))=0\), for \(k\) odd. This extends to a complete computation of the cohomology of the character variety for Seifert fibrations which are torsion-free and of genus one and a partial computation for genus greater than one. Related results have been independently obtained by \textit{M. Furuta} and \textit{B. Steer} [Adv. Math. 96, No. 1, 38-102 (1992)] and \textit{S. Bauer} [Math. Ann. 290, No. 3, 509- 526 (1991; Zbl 0752.14035)]. The paper considers representation of Fuchsian groups of finite type with fixed weight at the parabolic elements within the concept of parabolic bundles introduced by Seshadri. The main aim of the paper is to give the explicit description of moduli spaces of parabolic bundles in the case of rank two bundles over the projective line. The algebro-geometric construction of the isomorphism of parabolic push arounds with the moduli space is used.

The author gives a description of the space of projective unitary representations of the orbifold fundamental group of a compact holomorphic orbifold \(X\) of dimension 2 in terms of the moduli space of stable parabolic bundles over \(X\). This allows the author to compute the cohomology of the \(SU(2)\)-character variety \({\mathcal R}(\Sigma)\) of any Seifert-fibered homology sphere. In particular, he shows that \(H^ k({\mathcal R}(\Sigma))=0\), for \(k\) odd. This extends to a complete computation of the cohomology of the character variety for Seifert fibrations which are torsion-free and of genus one and a partial computation for genus greater than one. Related results have been independently obtained by \textit{M. Furuta} and \textit{B. Steer} [Adv. Math. 96, No. 1, 38-102 (1992)] and \textit{S. Bauer} [Math. Ann. 290, No. 3, 509- 526 (1991; Zbl 0752.14035)]. The Maritime and Energy industry is facing rapid change with an increasing reliance on software embedded systems and integrated control and monitoring systems. From a practical stand point, challenges are related to increased system complexity, increasingly integrated sub-systems relying on Commercial-Of-The-Shelf software, longer supply chains for equipment and components delivered by different suppliers, and short duration for construction and commissioning of ships and offshore platforms. As a result, there is a lack of visibility into the architecture of systems, their design rationale, how subsystems/components were verified and integrated, and finally how systems were validated and certified with a particular focus on safety. In turn, this has hindered effective collaboration among stakeholders, including suppliers and system integrators. This talk will present a recent initiative, led by Simula Research Laboratory and Det Norske Veritas (DNV), Norway, to address the above problems. The general approach relies on model-centric processes, where models of the system specifications, architecture, and design properties, are used to support the documentation of architecture and design rationale, traceability among development artifacts, and guide safety analysis and testing, among other things. The project is focused on devising novel but scalable approaches to the long-standing model-driven development challenges.

In the moduli space \({\mathcal M}(\Sigma, d)\) of reduced curves in \(\mathbb{C}\mathbb{P}^2\) with given degree \(d\) and prescribed configuration of singularities, a couple of reduced curves \(C\) and \(C'\) is said to be a Zariski pair if \(C\) and \(C'\) have the same combinatoric and the pairs \((\mathbb{C}\mathbb{P}^2, C)\) and \((\mathbb{C}\mathbb{P}^2, C')\) are not homeomorphic. Given a pair of curves \((C,C')\in {\mathcal M}(\Sigma, d)\) with the same combinatorics , if their generic Alexander polynomials are different, then it is known that \((C,C')\) a Zariski pair. However, it may happen that these polynomials are the same although \((C,C')\) is a Zariski pair. In this case the pair is said to be Alexander equivalent. Various examples of such pairs (for reducible and irreducible curves) were given by many authors; in particular their existence in the case of irreducible curves of degree 6 was proved by \textit{A. Degtyarev} [J. Lond. Math. Soc., II. Ser. 78, No. 2, 329--351 (2008; Zbl 1158.14026)]: in the present paper the authors construct the first concrete example of an Alexander-equivalent Zariski pair dealing with irreducible sextics. The purpose of this paper is the topological study of the complement to irreducible sextics in the complex projective plane. The author makes a very effective use of the lattice structure of \(H_2(\tilde X)\) (the intersection lattice of the resolution of the double cover of a sextic curve \(C\) with simple singularities) in order to study properties of the fundamental group of the complement of \(C\) in \(\mathbb P^2\). In a recent paper, \textit{C. Eyral} and \textit{M. Oka} [J. Math. Soc. Japan 57, No. 1, 37--54 (2005; Zbl 1070.14031)] suggested the conjecture tha any irreducible sextic curve, which is not of torus type satisfies the following: its Alexander polynomial is trivial, and the fundamental group of its complement is abelian. The author proves the first part of Oka's conjecture for simple singularities and disproves the second part, both for simple singularities and non-simple singularities. Also, as a consequence of the techniques presented the author determines the rigid isotopy classes of sextics with simple singularities whose fundamental group factors through the dihedral group \(\mathbb D_{10}\) and \(\mathbb D_{14}\). This characterization brings up new examples of Alexander-equivalent Zariski pairs.

In the moduli space \({\mathcal M}(\Sigma, d)\) of reduced curves in \(\mathbb{C}\mathbb{P}^2\) with given degree \(d\) and prescribed configuration of singularities, a couple of reduced curves \(C\) and \(C'\) is said to be a Zariski pair if \(C\) and \(C'\) have the same combinatoric and the pairs \((\mathbb{C}\mathbb{P}^2, C)\) and \((\mathbb{C}\mathbb{P}^2, C')\) are not homeomorphic. Given a pair of curves \((C,C')\in {\mathcal M}(\Sigma, d)\) with the same combinatorics , if their generic Alexander polynomials are different, then it is known that \((C,C')\) a Zariski pair. However, it may happen that these polynomials are the same although \((C,C')\) is a Zariski pair. In this case the pair is said to be Alexander equivalent. Various examples of such pairs (for reducible and irreducible curves) were given by many authors; in particular their existence in the case of irreducible curves of degree 6 was proved by \textit{A. Degtyarev} [J. Lond. Math. Soc., II. Ser. 78, No. 2, 329--351 (2008; Zbl 1158.14026)]: in the present paper the authors construct the first concrete example of an Alexander-equivalent Zariski pair dealing with irreducible sextics. Proposals to improve the performance of TCP in high speed networks have been recently put forward. Examples of such proposals include High Speed TCP, Scalable TCP, and FAST. In contrast to the additive increase multiplicative decrease algorithm used in the standard TCP, Scalable TCP uses a Multiplicative Increase Multiplicative Decrease (MIMD) algorithm for the window size evolution. In this paper, we present a mathematical analysis of the MIMD congestion control algorithm in the presence of random losses. Random losses are typical to wireless networks but can also be used to model losses in wireline networks with a high bandwidth-delay product. Our approach is based on showing that the logarithm of the window size evolution has the same behaviour as the workload process in a standard G/G/1 queue. The Laplace-Stieltjes transform of the equivalent queue is then shown to directly provide the throughput of the congestion control algorithm and the higher moments of the window size. Using \(ns\)-2 simulations, we validate our findings using Scalable TCP.

Consider polynomials \[ F(x_1,\dots, x_r)= \sum^{n_1}_{t_1= 0}\cdots \sum^{n_r}_{t_r= 0} \alpha(t_1,\dots, t_r) x^{t_1}_1\cdots x^{r_r}_r \] with \(\alpha(0,\dots, 0)= 0\), and write \(\alpha\) for the vector of coefficients. Then the singular integral under consideration is \[ \int^\infty_{-\infty}\cdots \int^\infty_{-\infty}\,\Biggl|\, \int^1_0\cdots \int^1_0 \exp\{2\pi iF(x_1,\dots, x_r)\}\,dx_1\cdots dx_r\Biggr|^{2k} d\alpha, \] (see the book [Trigonometric sums in number theory and analysis, de Gruyter Expositions in Mathematics, 39. Walter de Gruyter, Berlin (2004; Zbl 1074.11043)] by \textit{G. I. Arkhipov}, \textit{V. N. Chubarikov} and \textit{A. A. Karatsuba}). The ``convergence exponent'' is the infimum of those real numbers \(\gamma\) for which the integral converges whenenver \(2k>\gamma\). For polynomials of other shapes, for example that in which the ranges for the \(t_i\) are replaced by the single condition \(t_1+\cdots+ t_r\leq n\), one gets other convergence exponent problems. In the present paper upper and lower bounds for convergence exponents are obtained, and in certain cases (unfortunately not including either of those mentioned above) the convergence exponent is identified. The first nine chapters of this book constitute a translation of the Russian original [Theory of multiple trigonometric sums (1987; Zbl 0638.10037)]. In the present English edition, three chapters have been added: 10. Estimates of multiple trigonometric sums with prime numbers, 11. Some applications of trigonometric sums and integrals, 12. Short Kloosterman sums. These contain expositions of more recent research and results.

Consider polynomials \[ F(x_1,\dots, x_r)= \sum^{n_1}_{t_1= 0}\cdots \sum^{n_r}_{t_r= 0} \alpha(t_1,\dots, t_r) x^{t_1}_1\cdots x^{r_r}_r \] with \(\alpha(0,\dots, 0)= 0\), and write \(\alpha\) for the vector of coefficients. Then the singular integral under consideration is \[ \int^\infty_{-\infty}\cdots \int^\infty_{-\infty}\,\Biggl|\, \int^1_0\cdots \int^1_0 \exp\{2\pi iF(x_1,\dots, x_r)\}\,dx_1\cdots dx_r\Biggr|^{2k} d\alpha, \] (see the book [Trigonometric sums in number theory and analysis, de Gruyter Expositions in Mathematics, 39. Walter de Gruyter, Berlin (2004; Zbl 1074.11043)] by \textit{G. I. Arkhipov}, \textit{V. N. Chubarikov} and \textit{A. A. Karatsuba}). The ``convergence exponent'' is the infimum of those real numbers \(\gamma\) for which the integral converges whenenver \(2k>\gamma\). For polynomials of other shapes, for example that in which the ranges for the \(t_i\) are replaced by the single condition \(t_1+\cdots+ t_r\leq n\), one gets other convergence exponent problems. In the present paper upper and lower bounds for convergence exponents are obtained, and in certain cases (unfortunately not including either of those mentioned above) the convergence exponent is identified. The authors complete some of their previous results concerning the reduction of bi-Hamiltonian manifolds [see Rend. Mat. Appl., VII. Ser. 10, No. 1, 11-37 (1990; Zbl 0719.58016)] and point out some of their applications to the integrable nonlinear evolution equations.

Let \(E\) be a ring spectrum that satisfies the standard assumptions for the construction and convergence of the \(E\)-based Adams spectral sequence, i.e. the corresponding spectral sequence converges to \(\pi_*(X)\) for \(E\)-complete spectra \(X\). In the present article the authors show that in this spectral sequence the presence of a vanishing line of fixed slope (at some term of the spectral sequence, with some intercept) is a generic property of \(X\), i.e. this property is passed on to retracts, and if two spectra in a cofibration sequence have this property, so does the third. The proof of this result is rather formal and actually applies in any category which satisfies the axioms of a stable homotopy category in the sense of \textit{M. Hovey, J. H. Palmieri} and \textit{N. P. Strickland} [Axiomatic stable homotopy theory, Mem. Am. Math. Soc. 610 (1997; Zbl 0881.55001)]. A stable homotopy category is, in this paper, defined to be a triangulated closed symmetric monoidal category with some structures which are adopted as axioms, keeping the homotopy category of spectra in mind. Starting with this definition, the authors construct cellular towers, Bousfield localization, Brown representability and so on in the category. They even prove analogous theorems to the nilpotence theorem and the thick subcategory theorem of Devinatz, Hopkins and Smith, while the situation is more general and the results are weaker. There are various examples that satisfy the axioms in other fields of mathematics, which naturally gives the applications there. This book consists of ten sections and two appendices. In the first two sections, the definition is given, and fundamental properties are studied including a construction of cellular towers. Sections 3 and 4 are devoted to Bousfield localization and Brown representability. Nilpotence and thick subcategory theorems are discussed in section 5. In section 6, the authors introduce a Noetherian category, which is simpler and behaves better than the homotopy category of spectra. In the next two sections, they state basic properties of special cases of connective and semisimple stable homotopy categories, which are similar to the homotopy category of spectra and is equivalent to the category of graded rational vector spaces, respectively. Section 9 contains several examples. This paper ends with some suggestions for further study in the last section. The appendices present triangulated categories and closed symmetric monoidal categories, which is useful for and reader unfamiliar with these categories.

Let \(E\) be a ring spectrum that satisfies the standard assumptions for the construction and convergence of the \(E\)-based Adams spectral sequence, i.e. the corresponding spectral sequence converges to \(\pi_*(X)\) for \(E\)-complete spectra \(X\). In the present article the authors show that in this spectral sequence the presence of a vanishing line of fixed slope (at some term of the spectral sequence, with some intercept) is a generic property of \(X\), i.e. this property is passed on to retracts, and if two spectra in a cofibration sequence have this property, so does the third. The proof of this result is rather formal and actually applies in any category which satisfies the axioms of a stable homotopy category in the sense of \textit{M. Hovey, J. H. Palmieri} and \textit{N. P. Strickland} [Axiomatic stable homotopy theory, Mem. Am. Math. Soc. 610 (1997; Zbl 0881.55001)]. A theorem of \textit{M. Boyle} and \textit{D. Handelman} [Ann. Math. (2) 133, No. 2, 249--316 (1991; Zbl 0735.15005)] gives necessary and sufficient conditions for an \(n\)-tuple of nonzero complex numbers to be the nonzero spectrum of some matrix with nonnegative entries, but is not constructive and puts no bound on the necessary dimension of the matrix. Working with polynomial matrices, we constructively reprove this theorem in a special case, with a bound on the size of the polynomial matrix required to realize a given polynomial.

A Riemannian manifold \((M,g)\) is \textit{semi-symmetric} if \(R\cdot R=0\) where the dot means that the curvature operator acts on \(R\) as a derivation. This was generalized by \textit{R. Deszcz} in [Bull. Soc. Math. Belg., Sér. A 44, No. 1, 1--34 (1992; Zbl 0808.53012)] to \(R\cdot R=L_R\bar Q(g,R)\) on the set \(U_R=\{x\in M:\bar Q(g,R)\neq 0\; \text{at}\; x\}\) where \(L_R\) is a function on \(U_R\) and \[ \bar Q(g,R)(X_1,X_2,X_3;X,Y)=(X\wedge Y)R(X_1,X_2)X_3 -R((X\wedge Y)X_1,X_2)X_3 \] \[ -R(X_1,(X\wedge Y)X_2)X_3-R(X_1,X_2)(X\wedge Y)X_3. \] Under this condition \((M,g)\) is said to be \textit{pseudo-symmetric}. A contact metric manifold is called an \(N(\kappa)\)-\textit{contact metric manifold} if the characteristic vector field \(\xi\) belongs to the \(\kappa\)-nullity distribution of the Riemannian manifold. The authors first study \(N(\kappa)\)-contact metric manifolds which are pseudo-symmetric or satisfy related conditions. Denoting the Ricci tensor by \(S\), the authors also study a \((2n+1)\)-dimensional, \(n>1\), non-flat, non-Sasakian, \(N(\kappa)\)-contact metric manifold \(M\) satisfying \(S\cdot R=0\) and prove that \(M\) is either locally isometric to \(E^{n+1}\times S^n(4)\) or to a particular \(N(1-{1\over n})\)-contact metric manifold. The author presents some results on pseudosymmetric manifolds, i.e. on semi-Riemannian manifolds which possess a pseudosymmetric Riemann- Christoffel curvature tensor. Various aspects of Riemannian geometry are studied: warped products, four-dimensional manifolds, geodesic mappings, conformal deformations of metrics, submanifolds and para-Kähler manifolds together with curvature conditions of pseudosymmetry type.

A Riemannian manifold \((M,g)\) is \textit{semi-symmetric} if \(R\cdot R=0\) where the dot means that the curvature operator acts on \(R\) as a derivation. This was generalized by \textit{R. Deszcz} in [Bull. Soc. Math. Belg., Sér. A 44, No. 1, 1--34 (1992; Zbl 0808.53012)] to \(R\cdot R=L_R\bar Q(g,R)\) on the set \(U_R=\{x\in M:\bar Q(g,R)\neq 0\; \text{at}\; x\}\) where \(L_R\) is a function on \(U_R\) and \[ \bar Q(g,R)(X_1,X_2,X_3;X,Y)=(X\wedge Y)R(X_1,X_2)X_3 -R((X\wedge Y)X_1,X_2)X_3 \] \[ -R(X_1,(X\wedge Y)X_2)X_3-R(X_1,X_2)(X\wedge Y)X_3. \] Under this condition \((M,g)\) is said to be \textit{pseudo-symmetric}. A contact metric manifold is called an \(N(\kappa)\)-\textit{contact metric manifold} if the characteristic vector field \(\xi\) belongs to the \(\kappa\)-nullity distribution of the Riemannian manifold. The authors first study \(N(\kappa)\)-contact metric manifolds which are pseudo-symmetric or satisfy related conditions. Denoting the Ricci tensor by \(S\), the authors also study a \((2n+1)\)-dimensional, \(n>1\), non-flat, non-Sasakian, \(N(\kappa)\)-contact metric manifold \(M\) satisfying \(S\cdot R=0\) and prove that \(M\) is either locally isometric to \(E^{n+1}\times S^n(4)\) or to a particular \(N(1-{1\over n})\)-contact metric manifold. This paper contains necessary and sufficient conditions for local invertibility of 6-node quadratic isoparametric finite element transformations associated with finite elements with two curved sides.

With the aid of the energy method [\textit{Y. Guo}, Invent. Math. 153, 593--630 (2003; Zbl 1029.82034)] the author proves the existence of global classical solution near the global Maxwellian \(\mu=e^{-|v|^2}\) for the Boltzmann equation \[ [\partial_t+v\cdot\nabla_x]F=Q[F,F], \quad F(0,x,v)=F_0(x,v) \] with respect to the spatially periodic function \(F(t,x,v)\), \(x=(x_1,x_2,x_3)\in [-\pi,\pi]^3=T^3\), \(v=(v_1,v_2,v_3)\in \mathbb R^3\). The Boltzmann collision operator \[ Q[F,G]=\int_{R^3\times S^2}|u-v|^{\gamma}[F(u')G(v')-F(u)G(v)]B(\theta)\,du\,d\omega, \] where \(-3<\gamma<0\), \(\omega\in S^2\) with the angular cutoff assumption \(0<B(\theta)\leq C|\cos\theta|\), \(\cos\theta=[u-v]\omega/|u-v|\), \(v'=v+[(u-v)\omega]\omega\), \(u'=u-[(u-v)\omega]\omega\), describes the collision between particles. Perhaps the most fundamental model for dynamics of dilute charged particles is described by the Vlasov-Maxwell-Boltzmann system, in which particles interact with themselves through collisions and with their self-consistent electromagnetic field. Despite its importance, no global in time solutions, weak or strong, have been constructed so far. It is shown in this article that any initially smooth, periodic small perturbation of a given global Maxwellian, which preserves the same mass, total momentum and reduced total energy, leads to a unique global in time classical solution for such a master system. The construction is based on a recent nonlinear energy method with a new a priori estimate for the dissipation: the linear collision operator \(L\), not its time integration, is positive definite for any solution \(f(t,x,v)\) with small amplitude to the Vlasov-Maxwell-Boltzmann system. As a by-product, such an estimate also yields an exponential decay for the simpler Vlasov-Poisson-Boltzmann system.

With the aid of the energy method [\textit{Y. Guo}, Invent. Math. 153, 593--630 (2003; Zbl 1029.82034)] the author proves the existence of global classical solution near the global Maxwellian \(\mu=e^{-|v|^2}\) for the Boltzmann equation \[ [\partial_t+v\cdot\nabla_x]F=Q[F,F], \quad F(0,x,v)=F_0(x,v) \] with respect to the spatially periodic function \(F(t,x,v)\), \(x=(x_1,x_2,x_3)\in [-\pi,\pi]^3=T^3\), \(v=(v_1,v_2,v_3)\in \mathbb R^3\). The Boltzmann collision operator \[ Q[F,G]=\int_{R^3\times S^2}|u-v|^{\gamma}[F(u')G(v')-F(u)G(v)]B(\theta)\,du\,d\omega, \] where \(-3<\gamma<0\), \(\omega\in S^2\) with the angular cutoff assumption \(0<B(\theta)\leq C|\cos\theta|\), \(\cos\theta=[u-v]\omega/|u-v|\), \(v'=v+[(u-v)\omega]\omega\), \(u'=u-[(u-v)\omega]\omega\), describes the collision between particles. [For the entire collection see Zbl 0547.00049.] An inverse spectral theory is presented for certain linear ordinary differential operators of arbitrary even order n which generalizes the Gel'fand-Levitan theory for Sturm-Liouville operators. We prove that the coefficients in these operators are uniquely determined by n-1 distinct spectral matrices. Our method of proof makes use of a transformation due to M. K. Fage which generalizes the Povzner-Levitan transformation for Sturm-Liouville operators.

Similar to the Euclidean 3-space, in the hyperbolic space \(\mathbb H^3\) of curvature \(-1\) for each \(H > 1\) there is a family \(\mathcal D_\tau\), \(\tau \in (-\infty, \tau_H]\), of \textit{Delaunay surfaces}, that is rotationally symmetric surfaces of constant mean curvature (CMC) equal to \(H\). For \(\tau \in (0, \tau_H)\) the CMC surface \(\mathcal D_\tau\) is embedded and is called \textit{unduloid}; for \(\tau \in (-\infty, 0)\) the surface \(\mathcal D_\tau\) is called \textit{nodoid} and it is not embedded. If \(\tau = \tau_H\), the surface is a Killing cylinder and if \(\tau = 0\) the surface is a hyperbolic sphere. For the Euclidean families of Delaunay surfaces it is known that unduloids are stable in the sense that the only global CMC deformations of an unduloid are unduloids. The same property holds for nodoids with \(\tau \in [\tau^H, 0)\), where the Delaunay parameter \(\tau^H\) depends only on \(H\). In [Contemp. Math. 314, 169--186 (2002; Zbl 1032.53002)] \textit{R. Mazzeo} and \textit{F. Pacard} proved that as \(\tau\) decreases to \(-\infty\), infinitely many new complete, cylindrically bounded CMC surfaces bifurcate from a nodoid \(\mathcal D_\tau\). In the paper under review the authors extend the result of Mazzeo-Pacard for nodoids of hyperbolic space \(\mathbb H^3\). Their main result is the following theorem. {Theorem 1.1} Let \(H > 1\). There exists \(l_H \in \mathbb N\) depending only on \(H\) such that for each \(l \in \mathbb N\), \(l \geqslant l_H\), there exists \(\tau_l < 0\) depending only on \(l\) and a smooth branch of surfaces in \(\mathbb H^3\) with CMC equal to \(H\) which bifurcates from the nodoid \(\mathcal D_{\tau_l}\). Moreover, if \(\gamma\) denotes the rotational axis of \(\mathcal D_{\tau_l}\), any element of this branch is a non-rotational surface invariant under the discrete group of symmetries generated by a rotation of angle \(2\pi/l\) with respect to \(\gamma\). As the principal tool for proving the main result the authors use the bifurcation theorem of Crandall and Rabinowitz. All complete, axially symmetric surfaces of constant mean curvature in \(\mathbb{R}^3\) lie in the one-prameter family \(D_\tau\) of Delaunay surfaces. The elements of this family which are embedded are called unduloids; all other elements, which correspond to parameter value \(\tau\in \mathbb{R}\), are immersed and are called nodoids. The unduloids are stable in the sense that the only global constant mean curvature deformations of them are to other elements of this Delaunay family. In the precisely written paper the authors prove that this same property is true for nodoids only when \(\tau\) is sufficiently close to zero. On the other hand, using methods of modem differential geometry, they show that as \(\tau\) decreases to \(-\infty\), infinitely many new families of complete, cylindrically bounded constant mean curvature surfaces bifurcate from this Delaunay family. Finally, they prove that the surfaces in these branches have only a discrete symmetry group. Unfortunately no figure completes the interesting work.

Similar to the Euclidean 3-space, in the hyperbolic space \(\mathbb H^3\) of curvature \(-1\) for each \(H > 1\) there is a family \(\mathcal D_\tau\), \(\tau \in (-\infty, \tau_H]\), of \textit{Delaunay surfaces}, that is rotationally symmetric surfaces of constant mean curvature (CMC) equal to \(H\). For \(\tau \in (0, \tau_H)\) the CMC surface \(\mathcal D_\tau\) is embedded and is called \textit{unduloid}; for \(\tau \in (-\infty, 0)\) the surface \(\mathcal D_\tau\) is called \textit{nodoid} and it is not embedded. If \(\tau = \tau_H\), the surface is a Killing cylinder and if \(\tau = 0\) the surface is a hyperbolic sphere. For the Euclidean families of Delaunay surfaces it is known that unduloids are stable in the sense that the only global CMC deformations of an unduloid are unduloids. The same property holds for nodoids with \(\tau \in [\tau^H, 0)\), where the Delaunay parameter \(\tau^H\) depends only on \(H\). In [Contemp. Math. 314, 169--186 (2002; Zbl 1032.53002)] \textit{R. Mazzeo} and \textit{F. Pacard} proved that as \(\tau\) decreases to \(-\infty\), infinitely many new complete, cylindrically bounded CMC surfaces bifurcate from a nodoid \(\mathcal D_\tau\). In the paper under review the authors extend the result of Mazzeo-Pacard for nodoids of hyperbolic space \(\mathbb H^3\). Their main result is the following theorem. {Theorem 1.1} Let \(H > 1\). There exists \(l_H \in \mathbb N\) depending only on \(H\) such that for each \(l \in \mathbb N\), \(l \geqslant l_H\), there exists \(\tau_l < 0\) depending only on \(l\) and a smooth branch of surfaces in \(\mathbb H^3\) with CMC equal to \(H\) which bifurcates from the nodoid \(\mathcal D_{\tau_l}\). Moreover, if \(\gamma\) denotes the rotational axis of \(\mathcal D_{\tau_l}\), any element of this branch is a non-rotational surface invariant under the discrete group of symmetries generated by a rotation of angle \(2\pi/l\) with respect to \(\gamma\). As the principal tool for proving the main result the authors use the bifurcation theorem of Crandall and Rabinowitz. The purpose of this article is to provide a number of basic mathematical models of enrollment trends at a college or university, specifically to model the effects of retention and recruitment. These models allow for adjustments in retention rates and recruitment goals. One can then see the immediate and long-range effects of various enrollment management strategies. (orig.)

In an earlier paper with the same title \textit{F. Colonius} and \textit{K. Kunisch} [J. Reine Angew. Math. 370, 1-29 (1986; Zbl 0584.34009)] have shown that the solutions to the output least squares problems of the parameter estimation in two point boundary value problems depend Hölder continuously on the observation and data describing the set of admissible parameters. In the present paper, the result is improved and Lipschitz continuous dependence is shown. Under the same assumption of a second- order sufficient optimality condition the new result is obtained by exploiting the special structure of the problem in deriving better bounds for the solutions and Lagrange multipliers. This paper studies parameter estimation problems for two point boundary value problems using the output least squares method. The notions of output least squares (OLS)-stability and output least squares stability under regularization (ROLS-stability) are introduced which allow to analyze continuous dependence of the (not necessarily unique) solution on the observation and the constraints. The proofs are based on perturbation theory of optimization problems in Banach spaces.

In an earlier paper with the same title \textit{F. Colonius} and \textit{K. Kunisch} [J. Reine Angew. Math. 370, 1-29 (1986; Zbl 0584.34009)] have shown that the solutions to the output least squares problems of the parameter estimation in two point boundary value problems depend Hölder continuously on the observation and data describing the set of admissible parameters. In the present paper, the result is improved and Lipschitz continuous dependence is shown. Under the same assumption of a second- order sufficient optimality condition the new result is obtained by exploiting the special structure of the problem in deriving better bounds for the solutions and Lagrange multipliers. The recent suggestion that late time quantum dynamics may be important for resolving cosmological issues associated with our observed universe requires a consideration of several subtle issues associated with quantum cosmology, as we describe here. The resolution of these issues will be important if we are to be able to properly ascribe probability measures associated with eternal inflation, and a string landscape.

Kamae and Mendès-France, A. Bertrand-Mathis, Bourgain and the reviewer have found connections between van der Corput's equidistribution theorem, difference sets, \(FC^+\)-sets (forcing continuity of positive measures) and Poincaré sets. Several of these results are systematized here and extended to means defined by different summability methods (the usual mean here corresponds to Cesàro summation) as well as to abstract groups replacing the unit circle. A new equivalent condition of the \(FC^+\) property is given with isometries. Quantitative versions are established for most of the cases. The paper is completed with a summary of previous results and ample references, to which I would like to add one, \textit{I. Kříž} [Graphs Comb. 3, 145-158 (1987; Zbl 0641.05044)]. The author regards k-uniform hypergraphs \(G=(Z,E)\), where Z is the set of all integers. G is called shift-invariant if \(M\in E\to \{x+a/a\in M\}\in E\) for any \(a\in Z\). \(X\subseteq Z\) is called independent in G, if 2 \(X\cap E=\emptyset.\) The author proves that the Bergelsons theorem that for every shift invariant k-uniform hypergraph for every independent set X in G, \(d(X)=\lim_{n\to \infty}| X\cap [-n,n]| /(2n+1)=0\) is not valid. May be \(\mu (G)=\sup \{d(X)/d(X)\) exists and X is independent in \(G\}\). For \(k=2\) the author find a shift-invariant graph with \(\mu (G)>1/2-\epsilon\) for an arbitrary \(\epsilon >0\). For \(k'>k\) one can easily construct a k'-uniform G' with the same properties for any \(k'>k\).

Kamae and Mendès-France, A. Bertrand-Mathis, Bourgain and the reviewer have found connections between van der Corput's equidistribution theorem, difference sets, \(FC^+\)-sets (forcing continuity of positive measures) and Poincaré sets. Several of these results are systematized here and extended to means defined by different summability methods (the usual mean here corresponds to Cesàro summation) as well as to abstract groups replacing the unit circle. A new equivalent condition of the \(FC^+\) property is given with isometries. Quantitative versions are established for most of the cases. The paper is completed with a summary of previous results and ample references, to which I would like to add one, \textit{I. Kříž} [Graphs Comb. 3, 145-158 (1987; Zbl 0641.05044)]. We propose a new procedure based on layered feed-forward neural networks for the identification of microplane material model parameters. Novelties are usage of the Latin hypercube sampling method for the generation of training sets, a systematic employment of stochastic sensitivity analysis and a genetic algorithm-based training of a neural network by an evolutionary algorithm. Advantages and disadvantages of this approach together with possible extensions are thoroughly discussed and analyzed.

Generalizing well known constructions in Riemannian geometry, the tangent bundle of a smooth manifold endowed with a Riemannian metric and a torsion free linear connection carries a natural almost Kähler structure, provided that the dual connection is torsion free, too. Continuing his earlier work, the author studies this construction in the case of statistical manifolds. Based on direct computations and a general criterion due to H.~Shima, he analyzes several examples, in which one actually obtains a family of appropriate connections and thus a family of almost Kähler structures. In each case, he characterizes the Kähler structures and the Kähler structures with constant holomorphic sectional curvature in the family. For part I, see Zbl 1111.53021. It is well known that a Riemannian metric on a manifold \(M\) induces an almost Kähler structure on the tangent bundle \(TM\). More generally an analogous construction can be carried out for a torsion free affine connection on a Riemannian manifold \((M,g)\) whose dual connection with respect to \(g\) is torsion free, too. Here the author applies this idea to statistical models endowed with the Fisher information as Riemannian metric and the so-called family of \(\alpha\)-connections. This is carried out explicitly for the manifolds of normal distributions and of discrete distributions on a finite set. It is shown that for special values of \(\alpha\) one actually obtains a Kähler structure with constant holomorphic sectional curvature.

Generalizing well known constructions in Riemannian geometry, the tangent bundle of a smooth manifold endowed with a Riemannian metric and a torsion free linear connection carries a natural almost Kähler structure, provided that the dual connection is torsion free, too. Continuing his earlier work, the author studies this construction in the case of statistical manifolds. Based on direct computations and a general criterion due to H.~Shima, he analyzes several examples, in which one actually obtains a family of appropriate connections and thus a family of almost Kähler structures. In each case, he characterizes the Kähler structures and the Kähler structures with constant holomorphic sectional curvature in the family. For part I, see Zbl 1111.53021. Three different localized representation methods and a manifold learning approach to face recognition are compared in terms of recognition accuracy. The techniques under investigation are (a) local nonnegative matrix factorization (LN MF); (b) independent component analysis (ICA); (c) NMF with sparse constraints (NMFsc); (d) locality-preserving projections (Laplacian faces). A systematic comparative analysis is conducted in terms of distance metric used, number of selected features, and sources of variability on AR and Olivetti face databases. Results indicate that the relative ranking of the methods is highly task-dependent, and the performances vary significantly upon the distance metric used.

The author continues topics from his former paper [Izv. Akad. Nauk Arm. SSR, 22, No.2, 193-199 (1987; Zbl 0624.30030)]. The paper under review deals with the following problem: Given a domain \(D:=K(0,1)\setminus \cup^{\infty}_{j=1}\overline{K(z_ j,r_ j)}\), where \(K(a,z):=\{z\in {\mathbb{C}}: | z-a| <r\}\), and the system of discs \(K_ j:=K(z_ j,r_ j)\), \(j\geq 1\), is such that (1) \(\bar K_ j\cap \bar K_ k=\emptyset\), \(j\neq k\), (2) the unit circle is the set of accumulation points of the sequence \((z_ j)^{\infty}_{j=1}\), find conditions (in terms of the \(z_ j' s\) and \(r_ j' s)\) under which for any function f holomorphic in D: if \(\lim_{| z| \to 1}f(z)=0\) then \(f\equiv 0\). Let \(D:=K(0,1)\setminus \cup^{\infty}_{j=1}\bar K_ j\), where \(K(a,r):=\{z\in {\mathbb{C}}:| z-a| <r\}\), \(K_ j+=K(z_ j,r_ j)\subset K(0,1)\), \(\bar K{}_ j\cap \bar K_ k=\emptyset\), \(j\neq k\), and each point of \(\partial K(0,1)\) is a limit point of the set \(\{z_ 1,z_ 2,...\}\). The author presents a sufficient condition for D to be an ``identity-domain'', i.e. if \(f\in {\mathcal O}(D)\) and f(z)\(\to 0\) as \(| z| \to 1\) then \(f\equiv 0\).

The author continues topics from his former paper [Izv. Akad. Nauk Arm. SSR, 22, No.2, 193-199 (1987; Zbl 0624.30030)]. The paper under review deals with the following problem: Given a domain \(D:=K(0,1)\setminus \cup^{\infty}_{j=1}\overline{K(z_ j,r_ j)}\), where \(K(a,z):=\{z\in {\mathbb{C}}: | z-a| <r\}\), and the system of discs \(K_ j:=K(z_ j,r_ j)\), \(j\geq 1\), is such that (1) \(\bar K_ j\cap \bar K_ k=\emptyset\), \(j\neq k\), (2) the unit circle is the set of accumulation points of the sequence \((z_ j)^{\infty}_{j=1}\), find conditions (in terms of the \(z_ j' s\) and \(r_ j' s)\) under which for any function f holomorphic in D: if \(\lim_{| z| \to 1}f(z)=0\) then \(f\equiv 0\). This paper gives a way to compute star products on a Poisson manifold. Let us recall that via the formality theorem, Kontsevich gave an explicit formula for star products on arbitrary Poisson manifolds in terms of graphs. However, there is no systematic way to compute the weights of the Konsevich graphs. In this paper, the author uses a derivative decomposition of the Hochschild differential complex and defines a generalized inverse of the Hochschild coboundary operator. This enables him to obtain a new systematic method to compute explicitely star products on Poisson manifolds. Finally, examples are discussed.

The authors consider a regularization method for solving a Volterra integral equation of the first kind. After discretization of the integral equation the classical Tikhonov regularization procedure is applied to compute the solution of the linear system. A modification of the method of \textit{S. Alliney} and \textit{F. Sgallari} [SIAM J. Appl. Math. 44, 627- 645 (1984; Zbl 0595.65140)] is also described. Three numerical examples are given in support of the presented method. The paper gives an inversion formula for the reconstruction of a function from projections which has been given previously by \textit{A. M. Cormack} [Representation of a function by its line integrals, with some radiological applications. I., J. Appl. Physics 34, 2722-2727 (1963); II., ibid. 35, 2908-2913 (1964)] and many others. The ill-posedness of this formula is discussed. No reconstructions have been done.

The authors consider a regularization method for solving a Volterra integral equation of the first kind. After discretization of the integral equation the classical Tikhonov regularization procedure is applied to compute the solution of the linear system. A modification of the method of \textit{S. Alliney} and \textit{F. Sgallari} [SIAM J. Appl. Math. 44, 627- 645 (1984; Zbl 0595.65140)] is also described. Three numerical examples are given in support of the presented method. Addendum to the author's paper in [J. Algebra 419, 34--51 (2014; Zbl 1380.17023)]

\textit{R. Askey} [IMA Vol. Math. Appl. 18, 151--158 (1989; Zbl 0694.33006)] introduced the \(q^{-1}\) Hermite-polynomials which has infinitely many orthogonality measures. The author provides algorithms to construct these measures using the properties of the theta-functions. These measures depend on an arbitrary number of parameters. [For the entire collection see Zbl 0687.00005.] If \(x\to ix\) in the recurrence relation for the continuous q-Hermite polynomials of Rogers, the resulting polynomials are orthogonal on the real line with respect to a positive measure when \(q>1\). Some explicit orthogonality relations are given here. The moment problem is indeterminate so there are many positive measures which can be used. One of these orthogonalities led to a new q-beta integral with four free parameters.

\textit{R. Askey} [IMA Vol. Math. Appl. 18, 151--158 (1989; Zbl 0694.33006)] introduced the \(q^{-1}\) Hermite-polynomials which has infinitely many orthogonality measures. The author provides algorithms to construct these measures using the properties of the theta-functions. These measures depend on an arbitrary number of parameters. The problem of stabilization and controllability for inhomogeneous multidimensional thermoelastic diffusion problem is considered for anisotropic material. By introducing a nonlinear feedback function on part of the boundary of the material, which is clamped along the rest of its boundary, we prove that the energy of the system decays to zero exponentially or polynomially. Both rates of decay are determined explicitly by the physical parameters. Via Russell's ``Controllability via Stabilizability'' principle, we prove that the considered system is partially controllable by a boundary function determined explicitly.

The function field of the hyperelliptic curve \(X_D: \;y^2=D(x)\) is \(K=\mathbb{F}_q (x, \sqrt{D(x)})\) over the finite field \(\mathbb{F}_q \) with odd \(q\) elements. The author determines \(\widehat {r}_2(D)\), the 2-rank of the Jacobian \(J_D(\mathbb{F}_q)\), i.e. the divisor class group with divisors of degree zero. The rank \(\widehat {r}_2(D)\) has a close relation to (and almost equals) the 2-rank \(r_2(D)\) of the ideal class group Pic(\(\mathcal{O}_D\)) of the maximal order \(\mathcal{O}_D\) of \(K\), which has been obtained by E. Artin and X. Zhang for imaginary and real \(K\) respectively in Artin's famous 1924 thesis and the reviewer's paper in [\textit{X. Zhang}, J. China Univ. Sci. Technol. 17, 425-431 (1987; Zbl 0643.12002)]. Here the author uses no arithmetic method, but starts with \(J_D[2](\mathbb{F}_q)=J_D[2]^G\) (the \(G\)-invariant elements), and reduces the actions of the Galois group \(G\) on \(J_D[2]\) to permutation-matrix multiplications. A point of order \(>2\) in \(J_D\) is proved to exist too. In this nice paper, the author studies ambiguous divisor classes in a quadratic extension of a function field over a finite field of characteristic not equal to 2. The author gives, among other things, a criterion for the parity of the class number. These results contained in the paper generalize those given in the thesis of E. Artin.

The function field of the hyperelliptic curve \(X_D: \;y^2=D(x)\) is \(K=\mathbb{F}_q (x, \sqrt{D(x)})\) over the finite field \(\mathbb{F}_q \) with odd \(q\) elements. The author determines \(\widehat {r}_2(D)\), the 2-rank of the Jacobian \(J_D(\mathbb{F}_q)\), i.e. the divisor class group with divisors of degree zero. The rank \(\widehat {r}_2(D)\) has a close relation to (and almost equals) the 2-rank \(r_2(D)\) of the ideal class group Pic(\(\mathcal{O}_D\)) of the maximal order \(\mathcal{O}_D\) of \(K\), which has been obtained by E. Artin and X. Zhang for imaginary and real \(K\) respectively in Artin's famous 1924 thesis and the reviewer's paper in [\textit{X. Zhang}, J. China Univ. Sci. Technol. 17, 425-431 (1987; Zbl 0643.12002)]. Here the author uses no arithmetic method, but starts with \(J_D[2](\mathbb{F}_q)=J_D[2]^G\) (the \(G\)-invariant elements), and reduces the actions of the Galois group \(G\) on \(J_D[2]\) to permutation-matrix multiplications. A point of order \(>2\) in \(J_D\) is proved to exist too. The continuous, conditionally cancellative t-subnorms that posses a continuous, additive generator are discussed. Conditions for a continuous, conditionally cancellative t-subnorm to have a continuous, additive generator are described. Constructions of corresponding additive generators are also shown.

Let \(R\) be a complete discrete valuation ring with finite residue field \({\mathbb F}_q\), and let \(r_n\) be the probability that a random monic polynomial over \(R\) of degree \(n\) splits into linear factors. The authors prove: (i) the recursion formula \[ r_n = \sum_{| d| =n} \prod_{0\leq i \leq q-1} q^{-\binom{d_1+1}{2}}r_{d_i}, \] where the sum is taken over all \(q\)-tuples \(d=(d_0,d_1.\dots,d_{q-1})\) of non-negative integers such that \(| d| =d_0+d_1+\dots+d_{q-1}=n\); (ii) the asymptotic formula \[ \log_q r_n = -{n^2\over 2(q-1)} - {1\over 2} n \log_q n + O(n), \] where the implied constant in the term \(O(n)\) depends on \(q\). Regarding this last term, they conjecture that it should have an oscillating behaviour, and they prove this conjecture for \(q=2\). Formulas for \(r_n\), and in fact much more general, were already found by recursive arguments in a series of papers by \textit{I. Del Corso} and \textit{R. Dvornicich} (see for instance [''Uniformity over primes of tamely ramified splittings'', Manuscr. Math. 101, No. 2, 239--266 (2000; Zbl 0962.11037)]). ``We fix a prime \(p\) and let \(f(X)\) vary over all monic integer polynomials of fixed degree \(n\). Given any possible shape of a tamely ramified splitting of \(p\) in an extension of degree \(n\), we prove that there exists a rational function \(\Phi(X)\in \mathbb Q(X)\) such that the density of the monic integer polynomials \(f(X)\) for which the splitting of \(p\) has the given shape in \(\mathbb Q[X]/(f(X))\) is \(\Phi(p)\) (here reducible polynomials can be neglected). As a corollary, we prove that, for \(p \geq n\), the density of irreducible monic polynomials of degree \(n\) in \(\mathbb Z_p[X]\) is the value at \(p\) of a rational function \(\Phi_n(X)\in\mathbb Q(X)\). All rational functions involved are effectively computable''.

Let \(R\) be a complete discrete valuation ring with finite residue field \({\mathbb F}_q\), and let \(r_n\) be the probability that a random monic polynomial over \(R\) of degree \(n\) splits into linear factors. The authors prove: (i) the recursion formula \[ r_n = \sum_{| d| =n} \prod_{0\leq i \leq q-1} q^{-\binom{d_1+1}{2}}r_{d_i}, \] where the sum is taken over all \(q\)-tuples \(d=(d_0,d_1.\dots,d_{q-1})\) of non-negative integers such that \(| d| =d_0+d_1+\dots+d_{q-1}=n\); (ii) the asymptotic formula \[ \log_q r_n = -{n^2\over 2(q-1)} - {1\over 2} n \log_q n + O(n), \] where the implied constant in the term \(O(n)\) depends on \(q\). Regarding this last term, they conjecture that it should have an oscillating behaviour, and they prove this conjecture for \(q=2\). Formulas for \(r_n\), and in fact much more general, were already found by recursive arguments in a series of papers by \textit{I. Del Corso} and \textit{R. Dvornicich} (see for instance [''Uniformity over primes of tamely ramified splittings'', Manuscr. Math. 101, No. 2, 239--266 (2000; Zbl 0962.11037)]). The extended Kalman filtering (EKF) algorithm instead of the error back-propagation (BP) algorithm is used to train artificial neural networks (ANNs) for chemical process modeling. The basic idea is, by modifying the EKF gain, to prevent overfitting or filtering divergence phenomenon caused by outliers in the training samples. The EKF-based ANNs training method proposed is also applied to estimate the conversion rate in the polyacrylonitrile production process. Numerical simulations show that the modified EKF algorithm is superior to the BP algorithm in resisting noise and outliers, as well as generalization.

A question of great interest is, as phrased in [\textit{V. Bangert} and \textit{P. Emmerich}, J. Differ. Geom. 94, No. 3, 367--385 (2013; Zbl 1278.53038)]: ``Suppose a complete Riemannian plane \(P\) satisfies the parallel axiom, i.e., for every geodesic \(c\) on \(P\) and every point \(p \in P\) not on \(c\) there exists a unique geodesic through \(p\) that does not intersect \(c\). Does this imply that \(P\) is isometric to the Euclidean plane?'' This paper answers the question in the affirmative under the additional hypothesis that the plane admits total curvature. The key lemma is that such a plane has total curvature zero. Under the hypothesis that the plane admits total curvature, it is possible to compute total curvature by exhaustion by closed subsets. The lemma is shown by first subdividing the plane into strips, then exhausting those strips by triangles. The parallel postulate controls the excess of the large triangles and hence, by Gauss-Bonnet, their total curvature. Note that this phrasing of Euclid's fifth postulate corresponds to Playfair's formulation. The original version states that if a line segment intersects two straight lines so that the two interior angles on the same side sum to less than \(\pi\), then the two lines must intersect on that side. This paper also shows that, with this version of the postulate, the question can be answered in the affirmative under an additional hypothesis; this time that the plane has no conjugate points. E. Hopf proved in 1943 that a Riemannian \(2\)-torus without conjugate points is flat. The authors use his method to study complete Riemannian metrics without conjugate points on the plane and the cylinder. The result in the case of the plane is the following: For every point \(p\) the area \(A_p(r)\) of the metric ball around \(p\) with radius \(r\) satisfies \[ \liminf_{r\to \infty}\frac{A_p(r)}{\pi r^2}\geq 1. \] Equality holds if and only if the metric is flat. For a cylinder it is shown that the metric is flat if both ends have subquadratic area growth. It is remarkable that both estimates are optimal. Previous results with stronger assumptions can be found in [\textit{V. Bangert} and \textit{P. Emmerich}, Commun. Anal. Geom. 19, No. 4, 773--806 (2011; Zbl 1260.53065)], \textit{K. Burns} and \textit{G. Knieper} J. Differ. Geom. 34, No. 3, 623--650 (1991; Zbl 0723.53024); \textit{H. Koehler}, Asian J. Math. 12, No. 1, 35--45 (2008; Zbl 1147.53034)].

A question of great interest is, as phrased in [\textit{V. Bangert} and \textit{P. Emmerich}, J. Differ. Geom. 94, No. 3, 367--385 (2013; Zbl 1278.53038)]: ``Suppose a complete Riemannian plane \(P\) satisfies the parallel axiom, i.e., for every geodesic \(c\) on \(P\) and every point \(p \in P\) not on \(c\) there exists a unique geodesic through \(p\) that does not intersect \(c\). Does this imply that \(P\) is isometric to the Euclidean plane?'' This paper answers the question in the affirmative under the additional hypothesis that the plane admits total curvature. The key lemma is that such a plane has total curvature zero. Under the hypothesis that the plane admits total curvature, it is possible to compute total curvature by exhaustion by closed subsets. The lemma is shown by first subdividing the plane into strips, then exhausting those strips by triangles. The parallel postulate controls the excess of the large triangles and hence, by Gauss-Bonnet, their total curvature. Note that this phrasing of Euclid's fifth postulate corresponds to Playfair's formulation. The original version states that if a line segment intersects two straight lines so that the two interior angles on the same side sum to less than \(\pi\), then the two lines must intersect on that side. This paper also shows that, with this version of the postulate, the question can be answered in the affirmative under an additional hypothesis; this time that the plane has no conjugate points. A coverage function \(f\) over a ground set \([m]\) is associated with a universe \(U\) of weighted elements and \(m\) sets \(A _{1},\dots ,A _{m } \subseteq U\), and for any \(T \subseteq [m]\), \(f(T)\) is defined as the total weight of the elements in the union \(\bigcup _{j \in T } A _{j }\). Coverage functions are an important special case of submodular functions, and arise in many applications, for instance as a class of utility functions of agents in combinatorial auctions. Set functions such as coverage functions often lack succinct representations, and in algorithmic applications, an access to a value oracle is assumed. In this paper, we ask whether one can test if a given oracle is that of a coverage function or not. We demonstrate an algorithm which makes \(O(m|U|)\) queries to an oracle of a coverage function and completely reconstructs it. This gives a polytime tester for succinct coverage functions for which \(|U|\) is polynomially bounded in \(m\). In contrast, we demonstrate a set function which is ``far'' from coverage, but requires \(2^{\tilde{\Theta}(m)}\) queries to distinguish it from the class of coverage functions.

The authors study on-line optimization for Markov control processes (MCPs) with finite states based on a single sample path [cf. \textit{X.-R. Cao} and \textit{H.-F. Chen}, IEEE Trans. Autom. Control 42, 1382-1393 (1997; Zbl 0889.93039)]. An on-line optimization algorithm for MCPs is proposed here based on the theory of Markov performance potentials and via estimating gradients of performance with respect to the policy parameters by simulating a single sample path. The convergence of the algorithm with probability one is proved under certain assumptions. Suppose \(\{X_t^{\{i\}}\}\) and \(\{\widetilde X_t^{\{j\}}\}, (t\geq 0)\) are two independent normalized Markov processes with the same state space \(Z^+\) and the same infinitesimal generator, but with different initial states \(i\) and \(j\) respectively. Two fundamental concepts, perturbation realization factors \((d_{ij})\) and performance potentials \((g_i)\), are proposed here; they are defined as: \(d_{ij} \triangleq E\{\int_0^{T\{i,j\}} [f(\widetilde X_t^{\{j\}}) -f(X_t^{\{i\}})] dt\}\) and \(d_{ij} =g_j-g_i\) (the reasonableness of it is well proved), where \(T\{i,j\} =\inf \{t:(X_t^{\{i\}}, \widetilde X_t^{\{j\}}) =(k,k)\), \(\forall k\in Z^+\}\) and \(f\) is a performance function. An easy approach is provided here for estimating the \(d_{ij}\)'s based on a single sample path. It is also shown under some minor conditions that the realization matrix \(D= [d_{ij}]\) and the performance potential \(g=[g_i]\) satisfy the Lyapunov equation and the Poisson equation respectively, and they can uniquely determine \(D\) and \(g\). These results provide a sound theoretical foundation for some algorithms for estimating performance sensitivities based on a single sample path of a Markov process.

The authors study on-line optimization for Markov control processes (MCPs) with finite states based on a single sample path [cf. \textit{X.-R. Cao} and \textit{H.-F. Chen}, IEEE Trans. Autom. Control 42, 1382-1393 (1997; Zbl 0889.93039)]. An on-line optimization algorithm for MCPs is proposed here based on the theory of Markov performance potentials and via estimating gradients of performance with respect to the policy parameters by simulating a single sample path. The convergence of the algorithm with probability one is proved under certain assumptions. In the present paper, we show the ill-posedness of the free boundary problem of the incompressible ideal magnetohydrodynamics (MHD) equations in two spatial dimensions for any positive vacuum permeability \(\mu_0\), in Sobolev spaces. The analysis is uniform for any \(\mu_0>0\).

This paper is concerned with Hopf algebras obtained by ``gauge'' transformations. Let \(A\) be a Hopf algebra, with comultiplication \(\Delta\), antipode \(\mathcal S\) and counit \(\varepsilon\). Let \(F\in A\otimes A\) be an invertible element satisfying \((\varepsilon \otimes 1) F = (1 \otimes \varepsilon) F = 1\). The algebra \(A\) together with \(\Delta^{F} = F\Delta F^{-1}\) and a certain \(\mathcal S^{F}\) is an example of a quasi-Hopf algebra, a notion introduced by \textit{V. G. Drinfel'd} [Leningr. Math. J. 1, No. 6, 1419-1457 (1990), translation from Algebra Anal. 1, No. 6, 114-148 (1989; Zbl 0718.16033)]. If in addition \(F_{23}(1 \otimes \Delta) F =F_{12}(\Delta \otimes 1) F\) then it is a genuine Hopf algebra. Let \(\mathfrak g\) be a complex Lie algebra, \(A\) the \(h\)-adic completion of the universal enveloping algebra of \({\mathfrak g}[[h]]\), \(F = \exp h f\) where \(f\) belongs to a commutative subalgebra \(\mathfrak c\). Then \(F\) satisfies the requirements above (in a topological setting) and hence one obtains new Hopf algebras (which are quantized enveloping algebras for a suitable Lie bialgebra structure on \(\mathfrak g\)) by twisting a universal enveloping algebra. The purpose of this paper is to show that the so-obtained Hopf algebras admit ``rational forms'' when \(\mathfrak g\) is simple and finite dimensional, and \(\mathfrak c\) is, say for simplicity, the Cartan subalgebra of \(\mathfrak g\). These rational forms are defined over a ring \(\mathbb{C}[h, \exp hu_{ij}]\) for some integers \(u_{ij}\). A link with generalized commutative (or braided) algebras is also established. The author defines the notion of quasi-Hopf algebra by weakening the coassociativity law for the coproduct in analogy to the weakening of the cocommutativity when passing from cocommutative to almost cocommutative Hopf algebras. A quasi-Hopf algebra is almost coassociative. The analogue of the R-matrix is an invertible element \(\Phi\) of \(A\otimes A\otimes A\), which is assumed to satisfy natural selfconsistency relations. The paper is organized as follows: {\S}1. Definition and elementary properties of quasi-Hopf algebras. The category mod A of left A-modules is discussed in case A denotes a quasi-bialgebra. There is an equivalence relation for quasi-bialgebras with different coproduct and different \(\Phi\), called ``twisting''. {\S}2. Quasi-Lie bialgebras. The author defines and studies the ``classical analogue'' or classical limit, the quasi-Lie bialgebras corresponding to quasi-Hopf quantized universal enveloping algebras. {\S}3. Quasitriangular, triangular, and coboundary quasi-Hopf algebras. In the case of quasitriangular, triangular, and coboundary quasi-Hopf algebras the category mod \(A\) is characterized by commutativity of some diagrams. The monoidal category mod \(A\) is symmetric or a tensor category if \(A\) is triangular. Quasitriangular quasi-Hopf quantized universal enveloping algebras are discussed together with their classical limits consisting of quasi-Lie bialgebras \({\mathfrak g}\) endowed with a symmetric \({\mathfrak g}\)-invariant element \(t\) of \({\mathfrak g}\otimes {\mathfrak g}\), \(t\) vanishes for triangular quasi-Hopf algebras. The classical limit of a coboundary quasi-Hopf quantized universal enveloping algebra is proved to be a quasi-Lie bialgebra \({\mathfrak g}\) endowed with a \({\mathfrak g}\)-invariant element \(\phi\) of \(\wedge^ 3{\mathfrak g}\). \(\phi\) vanishes in the triangular case. There is stated a one-to-one correspondence up to twisting between quasi-Hopf quantized universal enveloping algebras and the quasi-Lie bialgebras (\({\mathfrak g},t)\) endowed with a symmetric \({\mathfrak g}\)-invariant tensor \(t\), i.e. it is stated, that quasi-Lie algebras of that type can be quantized (Theorem 3.15). The idea of the proof is sketched in the introduction of the paper. The paper closes with a remark on the construction of link-invariants including R-matrix invariants for classical solutions of the quantum Yang-Baxter equation.

This paper is concerned with Hopf algebras obtained by ``gauge'' transformations. Let \(A\) be a Hopf algebra, with comultiplication \(\Delta\), antipode \(\mathcal S\) and counit \(\varepsilon\). Let \(F\in A\otimes A\) be an invertible element satisfying \((\varepsilon \otimes 1) F = (1 \otimes \varepsilon) F = 1\). The algebra \(A\) together with \(\Delta^{F} = F\Delta F^{-1}\) and a certain \(\mathcal S^{F}\) is an example of a quasi-Hopf algebra, a notion introduced by \textit{V. G. Drinfel'd} [Leningr. Math. J. 1, No. 6, 1419-1457 (1990), translation from Algebra Anal. 1, No. 6, 114-148 (1989; Zbl 0718.16033)]. If in addition \(F_{23}(1 \otimes \Delta) F =F_{12}(\Delta \otimes 1) F\) then it is a genuine Hopf algebra. Let \(\mathfrak g\) be a complex Lie algebra, \(A\) the \(h\)-adic completion of the universal enveloping algebra of \({\mathfrak g}[[h]]\), \(F = \exp h f\) where \(f\) belongs to a commutative subalgebra \(\mathfrak c\). Then \(F\) satisfies the requirements above (in a topological setting) and hence one obtains new Hopf algebras (which are quantized enveloping algebras for a suitable Lie bialgebra structure on \(\mathfrak g\)) by twisting a universal enveloping algebra. The purpose of this paper is to show that the so-obtained Hopf algebras admit ``rational forms'' when \(\mathfrak g\) is simple and finite dimensional, and \(\mathfrak c\) is, say for simplicity, the Cartan subalgebra of \(\mathfrak g\). These rational forms are defined over a ring \(\mathbb{C}[h, \exp hu_{ij}]\) for some integers \(u_{ij}\). A link with generalized commutative (or braided) algebras is also established. For the conjugate Hardy transform and for the Calderón transform, the BLO-norms are calculated and two-sided estimates of the norms in the BMO are obtained, most of which are exact.

The authors analyze the longtime behavior of solutions to the Keller-Segel-Stokes model \[ \begin{aligned} n_{t}+u\cdot \nabla n & =\Delta (n^m) -\nabla\cdot(n\chi(c)\nabla c),\cr c_{t}+u\cdot \nabla c & =\Delta c- nf(c),\cr u_t+\nabla p & =\eta \Delta u-n\nabla \phi,\cr \nabla\cdot u & =0, \end{aligned} \] in a bounded smooth domain \(\Omega\subset \mathbb{R}^2\), where \(n\) denotes the bacterial density, \(c\) the oxygen concentration and \(u\) the velocity field. Moreover, \(m>1\) is the adiabatic exponent, \(\chi\) the chemostatic sensitivity, \(f\) the oxygen consumption rate, \(p\) the pressure, \(\eta\) the viscosity, and \(\nabla \phi\) the gravitation force of the fluid. The following boundary conditions are assumed \[ \partial_\nu n|_{\partial \Omega}=\partial_\nu c|_{\partial \Omega}=u|_{\partial \Omega}=0, \] the initial densities \(n_0\), \(c_0\) are non-negative and \[ n_0\in L^\infty(\Omega),\qquad c_0\in W^{1,\infty}(\Omega),\qquad u_0\in D(A^\theta), \quad \theta>\frac{1}{2}, \] where \(A^\theta\) denotes the fractional power space associated with the Stokes operator. Under suitable assumptions on \(\chi, f\) and \(\phi\), the existence and boundedness of global weak solutions was previously shown in [\textit{Y. Tao}, and \textit{M. Winkler}, Discrete Contin. Dyn. Syst. 32, No. 5, 1901--1914 (2012; Zbl 1276.35105)]. The uniqueness of solutions remained open, and hence, the authors consider the trajectory attractor of the system, which is a suitable concept for problems without uniqueness. The existence of the trajectory attractor is proven, as well as the existence of a global attractor in a generalized sense. This paper deals with global existence of weak solutions to an initial-boundary value problem of a bio-fluid model obtained from coupling the Keller-Segel chemotaxis model with porous medium type diffusion to the incompressible Stokes system. The main result extends the one previously obtained by \textit{M. Di Francesco} et al. [Discrete Contin. Dyn. Syst. 28, No. 4, 1437--1453 (2010; Zbl 1276.35103)] regarding the exponent in the porous medium type diffusion. The proof combined entropy-like functionals and Moser-type iterations.

The authors analyze the longtime behavior of solutions to the Keller-Segel-Stokes model \[ \begin{aligned} n_{t}+u\cdot \nabla n & =\Delta (n^m) -\nabla\cdot(n\chi(c)\nabla c),\cr c_{t}+u\cdot \nabla c & =\Delta c- nf(c),\cr u_t+\nabla p & =\eta \Delta u-n\nabla \phi,\cr \nabla\cdot u & =0, \end{aligned} \] in a bounded smooth domain \(\Omega\subset \mathbb{R}^2\), where \(n\) denotes the bacterial density, \(c\) the oxygen concentration and \(u\) the velocity field. Moreover, \(m>1\) is the adiabatic exponent, \(\chi\) the chemostatic sensitivity, \(f\) the oxygen consumption rate, \(p\) the pressure, \(\eta\) the viscosity, and \(\nabla \phi\) the gravitation force of the fluid. The following boundary conditions are assumed \[ \partial_\nu n|_{\partial \Omega}=\partial_\nu c|_{\partial \Omega}=u|_{\partial \Omega}=0, \] the initial densities \(n_0\), \(c_0\) are non-negative and \[ n_0\in L^\infty(\Omega),\qquad c_0\in W^{1,\infty}(\Omega),\qquad u_0\in D(A^\theta), \quad \theta>\frac{1}{2}, \] where \(A^\theta\) denotes the fractional power space associated with the Stokes operator. Under suitable assumptions on \(\chi, f\) and \(\phi\), the existence and boundedness of global weak solutions was previously shown in [\textit{Y. Tao}, and \textit{M. Winkler}, Discrete Contin. Dyn. Syst. 32, No. 5, 1901--1914 (2012; Zbl 1276.35105)]. The uniqueness of solutions remained open, and hence, the authors consider the trajectory attractor of the system, which is a suitable concept for problems without uniqueness. The existence of the trajectory attractor is proven, as well as the existence of a global attractor in a generalized sense. In the book the methodology of artificial intelligence is applied in numerical optimization methods and in decision making procedures. Different cases of combined (hybrid) logic-computational methods for nonlinear programming problems, multicriterial optimization, global optimization of multi-extremal functions and combinatorial optimization are described. Some original decision making procedure with a finite number of alternatives is presented. The algorithms described in the book are given in the form of programs (in Prolog) and in the form of conversational systems.

The authors introduce almost compactness and near compactness in a smooth topological space [the third author, ibid. 48, No. 3, 371-375 (1992; Zbl 0783.54007)]\ and establish some of their simple properties. In 1986, \textit{R. Badard} introduced the concept of a smooth topological space. In this paper, we give some links between smooth topological spaces and the corresponding results in Chang's fuzzy topological spaces. The concepts of subspace, continuity and compactness are studied.

The authors introduce almost compactness and near compactness in a smooth topological space [the third author, ibid. 48, No. 3, 371-375 (1992; Zbl 0783.54007)]\ and establish some of their simple properties. The energy \(E(G)\) of a graph \(G\) is defined as the sum of the absolute values of its eigenvalues. A connected graph \(G\) of order \(n\) is said to be hypoenergetic if \(E(G)<n\). All connected hypoenergetic graphs with maximum degree \(\varDelta \leqslant 3\) have been characterized. In addition to the four (earlier known) hypoenergetic trees, we now show that complete bipartite graph \(K_{2,3}\) is the only hypoenergetic cycle-containing hypoenergetic graph. By this, the validity of a conjecture by \textit{S. Majstorović}, \textit{A. Klobučar}, and \textit{I. Gutman} [``Selected topics from the theory of graph energy: hypenergetic graphs,'' in: Applications of Graph Spectra, Math. Inst. Belgrade 65--105 (2009)] has been confirmed.

The relations \(\tilde {\mathcal L}\), \(\tilde {\mathcal R}\) are defined on a semigroup \(S\) with the set of idempotents \(E\) by \((\forall e \in E)(ae = a \Leftrightarrow be = b)\), \(a\tilde {\mathcal R}b\) iff \((\forall e \in E)(ea = a \Leftrightarrow eb = b)\), \(a,b \in S\). \(S\) is weakly abundant if each \(\tilde {\mathcal L}\)-class and each \(\tilde {\mathcal R}\)-class contains an idempotent. A weakly abundant semigroup satisfies the congruence condition if \(\tilde {\mathcal L}\) and \(\tilde {\mathcal R}\) are a right congruence and a left congruence respectively; a weakly abundant semigroup is called an \(E\)-abundant if it has the congruence condition. An \(E\)-abundant (weakly abundant) semigroup is said to be (weakly) left quasi-abundant if its set of idempotents forms a left quasi-normal band. Here, necessary and sufficient conditions for the set of idempotents of a weakly abundant semigroup to be a left quasi-normal band are proved and a description of left quasi-abundant semigroups in terms of weak spined products is presented. The relation \({\mathcal L}^*\) on a semigroup \(S\) is defined by \((a,b)\in{\mathcal L}^*\) iff \(a,b\) are in the Green relation \(\mathcal L\) in some over-semigroup of \(S\); relation \(\mathcal R^*\) is defined analogously. \(S\) is called abundant if every \({\mathcal L}^*\)-class and every \({\mathcal R}^*\)-class contains an idempotent. An abundant semigroup is called quasi-adequate if the idempotents form a sub-semigroup and is called left (right-) semi-perfect if the set of idempotents of \(S\) is a left (right) quasi-normal band. Properties of left semi-perfect abundant semigroups are investigated and their structure described.

The relations \(\tilde {\mathcal L}\), \(\tilde {\mathcal R}\) are defined on a semigroup \(S\) with the set of idempotents \(E\) by \((\forall e \in E)(ae = a \Leftrightarrow be = b)\), \(a\tilde {\mathcal R}b\) iff \((\forall e \in E)(ea = a \Leftrightarrow eb = b)\), \(a,b \in S\). \(S\) is weakly abundant if each \(\tilde {\mathcal L}\)-class and each \(\tilde {\mathcal R}\)-class contains an idempotent. A weakly abundant semigroup satisfies the congruence condition if \(\tilde {\mathcal L}\) and \(\tilde {\mathcal R}\) are a right congruence and a left congruence respectively; a weakly abundant semigroup is called an \(E\)-abundant if it has the congruence condition. An \(E\)-abundant (weakly abundant) semigroup is said to be (weakly) left quasi-abundant if its set of idempotents forms a left quasi-normal band. Here, necessary and sufficient conditions for the set of idempotents of a weakly abundant semigroup to be a left quasi-normal band are proved and a description of left quasi-abundant semigroups in terms of weak spined products is presented. We present an implementation of a linear-time approximation scheme for the traveling salesman problem on planar graphs with edge weights. We observe that the theoretical algorithm involves constants that are too large for practical use. Our implementation, which is not subject to the theoretical algorithm's guarantee, can quickly find good tours in very large planar graphs.

Let \(M\) be a closed orientable 3-manifold with a Heegaard splitting \(M=V \cup_F W\), and \(S\) a separating \(2\)-sphere \(S\) embedded in \(M\). We call \(S\) a Haken sphere if it intersects the splitting surface \(F\) only in an essential circle \(C\) in \(F\). Suppose genus\((F)=2\) and \(M\) is not prime. It is shown that, for every pair of two Haken spheres \(S, S'\) in \(M\), there is a sequence of Haken spheres \(S=S_0, S_1, \dots, S_n=S'\) such that \(S_k\) and \(S_{k+1}\) are \((i,j)\)-related as below, where \((i,j)=(1,1), (1,2), (2,1)\) or \((2,2)\). \(S \cap V\) (resp. \(S \cap W\)) is a disk \(D\) (resp. \(E\)) which separates \(V\) (resp. \(W\)) into two solid tori \(V_1\) and \(V_2\) (resp. \(W_1\) and \(W_2\)) with \(F \cap V_m = F \cap W_m\). \(V_i\) (resp. \(W_j\)) contains an essential non-separating disk \(\Delta\) (resp. \(\Sigma\)) disjoint from \(D\) (resp. \(E\)). When \(i \neq j\), a circle \(C''\) in \(F - (\partial \Delta \cup \partial \Sigma)\) can be extended to a Haken sphere \(S''\) if \(C''\) separates the circles \(\partial \Delta\) and \(\partial \Sigma\). Then we say that \(S\) and \(S''\) are \((i,j)\)-related. When \(i = j\), and \(V_i \cup W_i \cong S^2 \times S^1-\)(an open ball), we can take \(\Delta\) and \(\Sigma\) so that \(\partial \Delta = \partial \Sigma\). A circle \(C^*\) in \(F - \partial \Delta\) can be extended to a Haken sphere \(S^*\) if \(C^*\) separates \(F\) into two once-punctured tori. Then we say that \(S\) and \(S^*\) are \((i,i)\)-related. In the sequence, no pair of \(S_k\) and \(S_{k+1}\) are \((i,i)\)-related if \(M\) is a sum of two lens spaces. This case was studied by the first author in [Haken spheres in the connected sum of two lens spaces, preprint]. \textit{M. Scharleman} and \textit{A. Thompson} studied the case of \(M = S^3\) in [Proc. Lond. Math. Soc. (3) 87, No. 2, 523--544 (2003; Zbl 1047.57008)]. The authors consider knots \(K\) with an unknotting tunnel \(\gamma\). They investigate deeper a conjecture of Morimoto in which he stated that if a knot \(K\subset S^3\) has a single unknotting tunnel \(\gamma\), then \(\gamma\) can be moved to be level with respect to the natural height function on \(K\) given by a minimal presentation of \(K\). The proof of this conjecture is based on a ``thinning'' process of the \(1\)-complex \(K\cup\gamma\) which can simplify the presentation of this \(1\)-complex until the tunnel can be moved either to a level arc or a level circuit. In this paper the authors construct an obstruction \(\rho\in {\mathbb Q}_{\displaystyle /2{\mathbb Z}}\) to further useful motion of \(\gamma\) in the ``thinning'' process. When the knot is not a \(2\)-bridge knot the obstruction \(\rho\) can be defined in a way independent of the thin position, thereby \(\rho\) can be viewed as an invariant of the pair \((K,\gamma)\). This new invariant is easy to compute. Moreover when \(\rho\not = 1\) the tunnel can be isotoped onto a minimal genus Seifert surface, and then allows to prove a conjecture of H. Goda and M. Teragaito. The authors announce that in a forthcoming paper in Trans. Am. Math. Soc. they will focus on the case \(\rho = 1\).

Let \(M\) be a closed orientable 3-manifold with a Heegaard splitting \(M=V \cup_F W\), and \(S\) a separating \(2\)-sphere \(S\) embedded in \(M\). We call \(S\) a Haken sphere if it intersects the splitting surface \(F\) only in an essential circle \(C\) in \(F\). Suppose genus\((F)=2\) and \(M\) is not prime. It is shown that, for every pair of two Haken spheres \(S, S'\) in \(M\), there is a sequence of Haken spheres \(S=S_0, S_1, \dots, S_n=S'\) such that \(S_k\) and \(S_{k+1}\) are \((i,j)\)-related as below, where \((i,j)=(1,1), (1,2), (2,1)\) or \((2,2)\). \(S \cap V\) (resp. \(S \cap W\)) is a disk \(D\) (resp. \(E\)) which separates \(V\) (resp. \(W\)) into two solid tori \(V_1\) and \(V_2\) (resp. \(W_1\) and \(W_2\)) with \(F \cap V_m = F \cap W_m\). \(V_i\) (resp. \(W_j\)) contains an essential non-separating disk \(\Delta\) (resp. \(\Sigma\)) disjoint from \(D\) (resp. \(E\)). When \(i \neq j\), a circle \(C''\) in \(F - (\partial \Delta \cup \partial \Sigma)\) can be extended to a Haken sphere \(S''\) if \(C''\) separates the circles \(\partial \Delta\) and \(\partial \Sigma\). Then we say that \(S\) and \(S''\) are \((i,j)\)-related. When \(i = j\), and \(V_i \cup W_i \cong S^2 \times S^1-\)(an open ball), we can take \(\Delta\) and \(\Sigma\) so that \(\partial \Delta = \partial \Sigma\). A circle \(C^*\) in \(F - \partial \Delta\) can be extended to a Haken sphere \(S^*\) if \(C^*\) separates \(F\) into two once-punctured tori. Then we say that \(S\) and \(S^*\) are \((i,i)\)-related. In the sequence, no pair of \(S_k\) and \(S_{k+1}\) are \((i,i)\)-related if \(M\) is a sum of two lens spaces. This case was studied by the first author in [Haken spheres in the connected sum of two lens spaces, preprint]. \textit{M. Scharleman} and \textit{A. Thompson} studied the case of \(M = S^3\) in [Proc. Lond. Math. Soc. (3) 87, No. 2, 523--544 (2003; Zbl 1047.57008)]. A Total FETI (TFETI)-based domain decomposition algorithm with preconditioning by a natural coarse grid of the rigid body motions is adapted to the solution of multibody contact problems of elasticity in 2D and 3D and proved to be scalable. The algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition parameter and the discretization parameter is bounded. The analysis is based on the classical results by Farhat, Mandel, and Roux on scalability of FETI with a natural coarse grid for linear problems and on our development of optimal quadratic programming algorithms for bound and equality constrained problems. The algorithm preserves parallel scalability of the classical FETI method. Both theoretical results and numerical experiments indicate a high efficiency of our algorithm. In addition, its performance is illustrated on a real-world problem of analysis of the ball bearing.

\textit{T. Høholdt, J. van Lint} and \textit{R. Pellikaan} [V.S. Pless and W.C. Huffman (Eds), Handbook of Coding Theory, 1, 871--961 (1998; Zbl 0922.94015)] introduced the notions of order and weight functions over an algebra \(R\) defined over a finite field \(F_q\)\, to give a setting which can be used for the construction of algebraic geometric codes (AG codes) `without algebraic geometry'. As is well known the AG codes \(C(D,G)\)\, and their duals \(C(D,G)^*\)\, are based on a curve \({\mathcal X}\)\, over \(F_q\)\, and two rational divisors \(D, G\). The approach of Hohold, van Lint and Pellikaan allows to deal with one-point AG codes (when \(G=aP,\, P\in {\mathcal X}\)). In order to be able to give a similar construction for the case of two-points AG codes (\(G=a_1P_1+a_2P_2\)), \textit{C. Carvalho, C. Munuera, E. Silva} and \textit{F. Torres} [IEEE Trans. Inform. Theory 53 (5), 1919--1924 (2007)], introduce the notions of near order and near weight functions. In the present paper the authors characterize the algebras admitting two well-agreeing near weights. Section 2 of the paper recalls definitions and basic facts about weights and near weight over an algebra \(R\)\, and in particular the condition of well-agreement of two normal near weights \(\rho,\sigma\): \(\rho\) and \(\sigma\) agree well if \(\sharp(\mathbb{N}^2-H\))\, is finite and \({\mathcal U}_{\rho}\bigcap {\mathcal U}_{\sigma}=F_q\),\, where \({\mathcal U}_{\rho}=\{r\in R^*| \rho(r)\leq\rho(1) \}\bigcup\{0\}\) and \(H=\{(\rho(f),\sigma(f))| f\in R^*\}\). Section 3 shows that \(H\)\, is in fact a semigroup. The wanted characterization of the algebras \(R\)\, equipped with two well agreeing near weight is established in Sections 4 and 5. Section 4 proves that \(R\)\, should be a finitely generated integral domain over the finite field \(F_q\)\, (so \(R\)\, is an affine \(F_q\)-algebra) and that \(K\),\, the field of quotients of \(R\),\, is an algebraic function field of one variable over \(F_q\),\, while Section 5 shows (theorem 5.6) that there are two places \(P,Q\) of \(K\)\, such that \(\rho\) and \(\sigma\)\, are derived from the valuations associated to \(P\)\, and \(Q\). Then \(\bigcap_{S\in \mathbb{P}(K)-\{P,Q\}}\mathcal{O}_S \) is precisely the integral closure \(\bar{R}\)\, of \(R\)\, in \(K\). \{ For the entire collection see Zbl 0907.94001 \}. An algebraic geometry code is a generalization of the dual of Goppa's code [\textit{V. D. Goppa}, Sov. Math., Dokl. 24, 170-172 (1981); translation from Dokl. Akad. Nauk SSSR 259, 1289-1290 (1981; Zbl 0489.94014)], where the underlying curve is replaced by an algebraic set \(\mathcal X\) such that some theory of \(\mathcal X\) gives bounds on the dimension and the minimum distance of the code. In the paper under review the authors give an account of the construction and decoding of algebraic geometry codes. This is done in a self-contained and elementary way, where the key concept in their treatment is the notion of order function.

\textit{T. Høholdt, J. van Lint} and \textit{R. Pellikaan} [V.S. Pless and W.C. Huffman (Eds), Handbook of Coding Theory, 1, 871--961 (1998; Zbl 0922.94015)] introduced the notions of order and weight functions over an algebra \(R\) defined over a finite field \(F_q\)\, to give a setting which can be used for the construction of algebraic geometric codes (AG codes) `without algebraic geometry'. As is well known the AG codes \(C(D,G)\)\, and their duals \(C(D,G)^*\)\, are based on a curve \({\mathcal X}\)\, over \(F_q\)\, and two rational divisors \(D, G\). The approach of Hohold, van Lint and Pellikaan allows to deal with one-point AG codes (when \(G=aP,\, P\in {\mathcal X}\)). In order to be able to give a similar construction for the case of two-points AG codes (\(G=a_1P_1+a_2P_2\)), \textit{C. Carvalho, C. Munuera, E. Silva} and \textit{F. Torres} [IEEE Trans. Inform. Theory 53 (5), 1919--1924 (2007)], introduce the notions of near order and near weight functions. In the present paper the authors characterize the algebras admitting two well-agreeing near weights. Section 2 of the paper recalls definitions and basic facts about weights and near weight over an algebra \(R\)\, and in particular the condition of well-agreement of two normal near weights \(\rho,\sigma\): \(\rho\) and \(\sigma\) agree well if \(\sharp(\mathbb{N}^2-H\))\, is finite and \({\mathcal U}_{\rho}\bigcap {\mathcal U}_{\sigma}=F_q\),\, where \({\mathcal U}_{\rho}=\{r\in R^*| \rho(r)\leq\rho(1) \}\bigcup\{0\}\) and \(H=\{(\rho(f),\sigma(f))| f\in R^*\}\). Section 3 shows that \(H\)\, is in fact a semigroup. The wanted characterization of the algebras \(R\)\, equipped with two well agreeing near weight is established in Sections 4 and 5. Section 4 proves that \(R\)\, should be a finitely generated integral domain over the finite field \(F_q\)\, (so \(R\)\, is an affine \(F_q\)-algebra) and that \(K\),\, the field of quotients of \(R\),\, is an algebraic function field of one variable over \(F_q\),\, while Section 5 shows (theorem 5.6) that there are two places \(P,Q\) of \(K\)\, such that \(\rho\) and \(\sigma\)\, are derived from the valuations associated to \(P\)\, and \(Q\). Then \(\bigcap_{S\in \mathbb{P}(K)-\{P,Q\}}\mathcal{O}_S \) is precisely the integral closure \(\bar{R}\)\, of \(R\)\, in \(K\). In this paper, we investigate the abstract incomplete degenerate differential equations in locally convex spaces, associated with use of the modified Liuoville right-sided fractional derivatives [the author, Abstract Volterra integro-differential equations. Boca Raton, FL: CRC Press (2015; Zbl 1318.45004)]. The existence of solutions of abstract incomplete degenerate differential equations of second order is proved by considering the corresponding incomplete degenerate differential equations of order \(1/\gamma\;(0<\gamma<1/2)\) and using an approximation process when \(\gamma\to1/2-\).

Let \(\{X_{\mathbf j}; {\mathbf j}\in \mathbb{N}^d\}\), \(d>1\), be an i.i.d. random field of square integrable centered random elements in the separable Hilbert space \(\mathbb{H}\) and \(\xi_{\mathbf n}\), \({\mathbf n}\in \mathbb{N}^d,\) be the summation processes based on the collection of sets \([0,t_1]\times\cdots\times[0,t_d]\), \(0\leq t_i\leq 1\), \(i=1,\dots,d\). Let \(W\) be an \(\mathbb{H}\)-valued Brownian sheet with the same covariance operator as \(X_{\mathbf 1}\); let \(0<\alpha<1/2\) and \(p=(1/2-\alpha)^{-1}\). It is proved that \((n_1\cdots n_d)^{-1/2}\xi_{\mathbf n}\) converges in the Hölder space \(\mathrm{H}_{\alpha}^o(\mathbb{H})\) to \(W\) if and only if \(\sup_{t>0} t^pP(\| X_{\mathbf 1}\| >t)<\infty\), in contrast to the case \(d=1\) and \(\mathbb{H}=\mathbb{R}\) [see \textit{A.~Račkauskas} and \textit{Ch.~Suquet}, Teor. Jmovirn. Mat. Stat. 68, 115--124 (2003); translation in Theory Probab. Math. Stat. 68, 127--138 (2003; Zbl 1050.60040)]. Let \(X_1,X_2,\ldots\) be i.i.d. mean zero random variables, \(S(t)=\sum_{k\leq t}X_{k}\), \(t>0\), \(S(0)=0\), \(\xi_{n}(t)=S([nt])+(nt-[nt])X_{[nt]+1}\), \(t\in[0,1]\). The main result of this paper is the following: Assume that \(0<\alpha<1/2\) and \(p=1/(1/2-\alpha)\). Then \(n^{-1/2}\xi_{n} @>\text{D}>> W\) in the Hölder space \(H_{\alpha}^0[0,1]\) if and only if \(\lim_{t\to\infty}t^{p}\text{P}(| X_1|\geq t)=0\). Here \(W(t)\) is a standard Wiener process and \(@>\text{D}>>\) denotes the convergence in distribution. The obtained results are used to investigate the change point problem under epidemic alternative.

Let \(\{X_{\mathbf j}; {\mathbf j}\in \mathbb{N}^d\}\), \(d>1\), be an i.i.d. random field of square integrable centered random elements in the separable Hilbert space \(\mathbb{H}\) and \(\xi_{\mathbf n}\), \({\mathbf n}\in \mathbb{N}^d,\) be the summation processes based on the collection of sets \([0,t_1]\times\cdots\times[0,t_d]\), \(0\leq t_i\leq 1\), \(i=1,\dots,d\). Let \(W\) be an \(\mathbb{H}\)-valued Brownian sheet with the same covariance operator as \(X_{\mathbf 1}\); let \(0<\alpha<1/2\) and \(p=(1/2-\alpha)^{-1}\). It is proved that \((n_1\cdots n_d)^{-1/2}\xi_{\mathbf n}\) converges in the Hölder space \(\mathrm{H}_{\alpha}^o(\mathbb{H})\) to \(W\) if and only if \(\sup_{t>0} t^pP(\| X_{\mathbf 1}\| >t)<\infty\), in contrast to the case \(d=1\) and \(\mathbb{H}=\mathbb{R}\) [see \textit{A.~Račkauskas} and \textit{Ch.~Suquet}, Teor. Jmovirn. Mat. Stat. 68, 115--124 (2003); translation in Theory Probab. Math. Stat. 68, 127--138 (2003; Zbl 1050.60040)]. We introduce and investigate a weaker form of \(R\)-maps and \(\delta\)-continuous functions which is called almost \(\delta\)-semicontinuity. We obtain characterizations, basic properties and the relationship with other types of functions between topological spaces.

The paper studies two kinds of optimal Hermitian backward perturbations for the Hermitian eigenvalue problem which are obtained from different orthogonal decompositions of computed eigenvectors. It is shown that small residuals and almost orthogonality of the computed eigenvectors does not necessarily imply the smallness of all those optimal perturbations. This contradicts a result by \textit{S. Chandrasekharan} and \textit{I. C. F. Ipsen} [Numer. Math. 68, No. 2, 215-223 (1994; Zbl 0807.65034)] for the real symmetric eigenvalue problem. Bounds on the backward errors for the symmetric eigenvalue decomposition and the singular value decomposition in the two-norm and in the Frobenius norm are presented. When the computed eigenvectors have a small residual and are close to orthonormal then all backward errors tend to be small. Hence it does not matter how exactly a backward error is defined and how exactly residual and deviation from orthogonality are measured. The effects of the error bounds on implementation for eigenvectors and singular vectors are indicated. Finally, it is shown that the distance of an appropriately scaled matrix to its orthogonal QR factor is not much larger than its distance to the closest orthogonal matrix.

The paper studies two kinds of optimal Hermitian backward perturbations for the Hermitian eigenvalue problem which are obtained from different orthogonal decompositions of computed eigenvectors. It is shown that small residuals and almost orthogonality of the computed eigenvectors does not necessarily imply the smallness of all those optimal perturbations. This contradicts a result by \textit{S. Chandrasekharan} and \textit{I. C. F. Ipsen} [Numer. Math. 68, No. 2, 215-223 (1994; Zbl 0807.65034)] for the real symmetric eigenvalue problem. We present a novel way to implement a fingerprinting attack against Onion Routing anonymity systems such as Tor. Our attack is a realistic threat in the sense that it can be mounted by a single controller of entrance routers and furthermore require very few resources. The conventional fingerprinting attack based on incoming traffic does not work straightforwardly against Tor due to its multiplex and quantized nature of traffic. By contrast, our novel attack can degrade Tor's anonymity by a metric based on both incoming and outgoing packets. In addition, our method keeps the fingerprinting attack's advantage of being realistic in terms of the few required resources. Regarding evaluation, the effectiveness of our method is discussed in a comprehensive manner: experimentally and theoretically. In order to enhance further studies and show the significance of our idea, we also discuss methods for defending against our attack and other applications of our idea.

Let \(\lambda \in [ 0,1]\) and \(\mu >0\). The proximal average of the lower semicontinuous proper convex functions \(f_{0},f_{1}:\mathbb{R}^{d}\to]-\infty ,+\infty ]\) was defined by \textit{H. H. Bauschke, E. Matoušková} and \textit{S. Reich} [Nonlinear Anal., Theory Methods Appl., Ser. A 56, No. 5, 715--738 (2004; Zbl 1059.47060)] by \(\mathcal{P}_{\mu }(f_{0},f_{1};\lambda )(\xi )=\inf_{(1-\lambda)y_{0}+\lambda y_{1}=\xi }\{(1-\lambda )f_{0}(y_{0})+\lambda f_{1}(y_{1})+\frac{(1-\lambda )\lambda }{2\mu }||y_{0}-y_{1}||^{2}\}\). The authors prove that this function is separately convex in \(\mu \) and \(\lambda \), and give examples of convex quadratic functions \(f_{0}\) and \(f_{1}\) showing that it is not necessarily convex in any of the pairs \((\xi ,\lambda ),\) \((\lambda, \mu )\), \((\xi ,\mu )\) and \((f_{0},f_{1})\). They also propose some interpolation algorithms for plotting proximal averages, and present computational experience to show their efficiency in terms of computational time and image file size. The paper under review is a valuable and deep contribution to the convergence theory of certain sequences in Hilbert space. These sequences base on projection and proximal point methods. Herewith, this paper from functional analysis with its clear structure and thorough proofs is meaningful also for applied mathematics, especially, optimization theory in abstract spaces. This relation is reflected by the extension of projector classes and by the references. After preparations in terms of projector and mapping classes, as well as the classical J. von Neumann's and L. M. Bregman's results, two main lines of work on which the paper bases itself and which it continues, are as follows: (i) In the workshop ``Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications'', Haifa, 2000, \textit{H. Hundal} presented a hyperplane \(H\), a cone \(K\) and an initial point \(y_0\) in \(\ell_2\) such that the sequence of iterates of stepwise alternating projections on firstly \(H\) and secondly \(K\), weakly converges but not norm converges to a point in the intersection of \(H\) with \(K\). The authors extend this result to a counterexample on norm convergence of iterates given by averaged projections. (ii) Herewith, a question posed by \textit{S. Reich} becomes answered. In the paper, further counterexamples are presented in the line of research done by (iii) \textit{A. Genel} and \textit{J. Lindenstrauss}: firmly nonexpansive maps, (iv) \textit{O. Güler}: proximal point algorithms, and (v) \textit{Y. Censor} et al.: string-averaging projection methods. Finally, extensions to the Hilbert ball and Banach spaces are discussed, too. This paper with its rich and wide results may in the future serve for a deeper understanding of the numerical treatment of various problems from optimization, calculus of variations and optimal control.

Let \(\lambda \in [ 0,1]\) and \(\mu >0\). The proximal average of the lower semicontinuous proper convex functions \(f_{0},f_{1}:\mathbb{R}^{d}\to]-\infty ,+\infty ]\) was defined by \textit{H. H. Bauschke, E. Matoušková} and \textit{S. Reich} [Nonlinear Anal., Theory Methods Appl., Ser. A 56, No. 5, 715--738 (2004; Zbl 1059.47060)] by \(\mathcal{P}_{\mu }(f_{0},f_{1};\lambda )(\xi )=\inf_{(1-\lambda)y_{0}+\lambda y_{1}=\xi }\{(1-\lambda )f_{0}(y_{0})+\lambda f_{1}(y_{1})+\frac{(1-\lambda )\lambda }{2\mu }||y_{0}-y_{1}||^{2}\}\). The authors prove that this function is separately convex in \(\mu \) and \(\lambda \), and give examples of convex quadratic functions \(f_{0}\) and \(f_{1}\) showing that it is not necessarily convex in any of the pairs \((\xi ,\lambda ),\) \((\lambda, \mu )\), \((\xi ,\mu )\) and \((f_{0},f_{1})\). They also propose some interpolation algorithms for plotting proximal averages, and present computational experience to show their efficiency in terms of computational time and image file size. We consider difference approximations to the model hyperbolic equation \(u_ t=u_ x\) which compute each new value \(U(x,t+\Delta t)\) as a combination of the known values \(U(x-r\Delta x,t),...,U(x+s\Delta x,\Delta t)\). For such schemes we find the optimal order of accuracy: stability is possible for small \(\Delta\) t/\(\Delta\) x if and only if \(p\leq \min \{r+s,2r+2,2s\}\). A similar bound is established for implicit methods. In this case the most accurate schemes are based on Padé approximations P(z)/Q(z) to \(z^{\lambda}\) near \(z=1\), and we find an expression for the difference \(| Q|^ 2-| P|^ 2\); this allows us to test the von Neumann condition \(| P/Q| \leq 1\). We also determine the number of zeros of Q in the unit circle, which decides whether the implicit part is uniformly invertible.

A theorem on isomorphisms of graphs of lattices (dealing with proper cells) which was proved by the reviewer [Czech. Math. J. 35(110), 188-200 (1985; Zbl 0575.06004)] is generalized in the present paper for the case of isomorphisms of graphs of directed sets. For a long time, graph isomorphisms of distributive, modular and semimodular lattices, as well as of some partially ordered sets fulfilling certain covering conditions, have been investigated. In this paper, the author studies graph isomorphisms of locally finite lattices without assuming modularity. He deals with some types of cycles in lattice graphs, and correlates several properties of transposed intervals and direct and subdirect decompositions to those of certain graph isomorphisms.

A theorem on isomorphisms of graphs of lattices (dealing with proper cells) which was proved by the reviewer [Czech. Math. J. 35(110), 188-200 (1985; Zbl 0575.06004)] is generalized in the present paper for the case of isomorphisms of graphs of directed sets. Separable continuous images of ordered bicompacta are studied. It is proved in particular that a separable zero-dimensional image of ``two arrows'' is an ordered bicompactum.

The author [``Double implementation in Nash and strong Nash equilibria'', Soc. Choice Welfare 14, 439-447 (1997; Zbl 0881.90012)], considered a decision making problem where there is a set of options and a finite number of agents with preferences defined over the set of options, and obtained a necessary and sufficient condition for double implementation in Nash and strong Nash equilibria. One problem with the condition is that it is difficult to apply it directly to a given (social choice) correspondence. In the present paper the author provides an algorithm which helps to determine whether or not a correspondence satisfies the condition. We deal with the implementation problem where agents in some groups can cooperate (or form coalitions) and in some others cannot. We consider a situation where the planner does not know whether agents in a group can cooperate or not. We provide a necessary and sufficient condition for a correspondence to be implementable in such a situation.

The author [``Double implementation in Nash and strong Nash equilibria'', Soc. Choice Welfare 14, 439-447 (1997; Zbl 0881.90012)], considered a decision making problem where there is a set of options and a finite number of agents with preferences defined over the set of options, and obtained a necessary and sufficient condition for double implementation in Nash and strong Nash equilibria. One problem with the condition is that it is difficult to apply it directly to a given (social choice) correspondence. In the present paper the author provides an algorithm which helps to determine whether or not a correspondence satisfies the condition. No review copy delivered.

In a previous paper with the same title [Number Theory, Ulm/FRG 1987, Lect. Notes Math. 1380, 120--136 (1989; Zbl 0674.10032)], the mean value \[ I=\sum_{r=1}^R \int_0^V \biggl| \sum_M^{M'} d(m) g(m,v,y_r) e(f(m,v,y_r))\biggr|^2 \,dv \] was estimated. Here \(d(m)\) is the divisor function, the functions \(f\) and \(g\) satisfy certain conditions, and \(y_r\) runs over a well-spaced set of real numbers. In the present paper, the function \(f\) is supposed to be approximately of the form \(F(x/M)^aB(v,y)\), or of the form \(F(\log x) B(v,y)\), where \(F\) is a parameter, \(a\neq 0,1\), and B(v,y) is bounded. Under this assumption, the previous estimate for I can be refined and simplified. More precisely, the earlier estimate involved the factor \(\min (R^{1/2},(F/M)^{1/2})\) in one term, but now this factor can be removed. Because \(M\ll F\) by assumption, the result is indeed improved. Applications to Dirichlet polynomials and series are given. [For the entire collection see Zbl 0667.00007.] Let g(m,v,y), f(m,v,y) be real functions of m with the parameters v, y and let d(m) denote the number of divisors of m. Exponential sums of the type \[ S(M,M';v,y)=\sum^{M'}_{m=M}d(m) g(m,v,y) e^{2\pi if(m,v,y)} \] take a leading part in the theory of Riemann's zeta-function and in Dirichlet's divisor problem. It is proved an estimation for the mean square \[ \sum^{R}_{r=1}\int^{V}_{0}| S(M,M';v,y_ r)|^ 2 dv, \] where \(0<Y\leq y_ 1<...<y_ R\leq 2Y\) and \(y_{r+1}-y_ r\geq Y_ 0>0\). The result implies known mean value theorems for Riemann's zeta-function due to H. Iwaniec and for cusp form L-functions due to A. Good. New estimations are derived for the mean squares of the error term in the asymptotic formula for Dirichlet's divisor problem and of E(T) in \[ \int^{T}_{0}| \zeta (+it)|^ 2 dt=T(\log (T/2\pi)+2\gamma -1)+E(T). \] [For part II, see the following review.]

In a previous paper with the same title [Number Theory, Ulm/FRG 1987, Lect. Notes Math. 1380, 120--136 (1989; Zbl 0674.10032)], the mean value \[ I=\sum_{r=1}^R \int_0^V \biggl| \sum_M^{M'} d(m) g(m,v,y_r) e(f(m,v,y_r))\biggr|^2 \,dv \] was estimated. Here \(d(m)\) is the divisor function, the functions \(f\) and \(g\) satisfy certain conditions, and \(y_r\) runs over a well-spaced set of real numbers. In the present paper, the function \(f\) is supposed to be approximately of the form \(F(x/M)^aB(v,y)\), or of the form \(F(\log x) B(v,y)\), where \(F\) is a parameter, \(a\neq 0,1\), and B(v,y) is bounded. Under this assumption, the previous estimate for I can be refined and simplified. More precisely, the earlier estimate involved the factor \(\min (R^{1/2},(F/M)^{1/2})\) in one term, but now this factor can be removed. Because \(M\ll F\) by assumption, the result is indeed improved. Applications to Dirichlet polynomials and series are given. Consider the differential equation \[ x^{(2n)}(t) + \phi(t,x(t)) = 0, \quad t\in(0,1), \] with the homogeneous boundary conditions \(x^{(j)}(0)=x^{(j)}(1)=0\), \(j\in\{0,1,\ldots, r-1\}\). Using a previous result of the authors for nonlinear equations in partially ordered locally convex spaces, a combined method is obtained in order to solve numerically the above two-point boundary value problem. The combined method involves a modification of Newton's method and divided differences. A straightforward error estimation is an advantage of the method. A numerical example is also given.

For a given matrix polynomial \(P(\lambda )=\sum _{i=0}^mA_i\lambda ^i\), where the \(A_i\) are \(n\times n\) complex matrices, and for a given complex number \(\mu\) and integer \(\kappa \geq 2\), the authors determine the distance (suitably defined) from \(P(\lambda )\) to the set of matrix polynomials having \(\mu\) as an eigenvalue of geometric multiplicity at least \(\kappa\). They also obtain bounds for the distance from \(P(\lambda )\) to the set of matrix polynomials which have \(\mu\) as a multiple eigenvalue of any sort. This generalizes a result of \textit{A. N. Malyshev} [Numer. Math. 83, 443--454 (1999; Zbl 0972.15011)]. The author proves that the 2-norm distance from an \(n\times n\) matrix \(A\) to the matrices that have a multiple eigenvalue \(\lambda\) can be computed as the maximum with respect to \(\gamma\) of the second-lowest singular value of the \(2n\times 2n\) matrices with block-diagonal \(A-\lambda I\) in both entries and off-diagonal elements \(0\) and \(\gamma I\). Therefore, he can identify the 2-norm distance from \(A\) to the set of matrices with multiple eigenvalues -- and he is also able to give a constructive description of a minimizing matrix.

For a given matrix polynomial \(P(\lambda )=\sum _{i=0}^mA_i\lambda ^i\), where the \(A_i\) are \(n\times n\) complex matrices, and for a given complex number \(\mu\) and integer \(\kappa \geq 2\), the authors determine the distance (suitably defined) from \(P(\lambda )\) to the set of matrix polynomials having \(\mu\) as an eigenvalue of geometric multiplicity at least \(\kappa\). They also obtain bounds for the distance from \(P(\lambda )\) to the set of matrix polynomials which have \(\mu\) as a multiple eigenvalue of any sort. This generalizes a result of \textit{A. N. Malyshev} [Numer. Math. 83, 443--454 (1999; Zbl 0972.15011)]. Weak key analysis is a key issue in the design of chaotic ciphers. While most of the existing research focusing on the degradation of the chaotic sequences which causes weak keys, we point out that the parameters for which the chaotic sequences do not degrade are still possible to be weak keys. In this paper, we propose a new approach based on the rigorous statistical test to improve the weak key analysis. The weak keys of a specific chaotic cipher are investigated by using our method and a large number of new weak keys are detected. These results verify that our method is more effective. On the other hand, although statistical tests are now widely adopted to test the chaos-based bit sequences, there are few reports of analysis results on the weak keys or weak sequences of chaotic cipher. Thus our work may be helpful for current research on statistical tests of chaotic cipher.

A fundamental theorem (Wolfe's theorem) in geometric integration theory states that the space of flat \(m\)-forms, endowed with the flat norm, is isometric to the space of flat \(m\)-cochains. In the paper under review the authors generalize such a theorem to the setting of Sobolev differential forms and Sobolev cochains in \({\mathbb R}^n\). They use the results in a previous paper of the second and the third author [J. Math. Pures Appl. (9) 100, No. 6, 868--906 (2013; Zbl 1277.49051)] where a suitable framework of Sobolev cochains in complete metric measure spaces has been initiated. The main result is the following: Let \(1\leq m\leq n\) and \(1<q,p<\infty\). If \(p>n-m\) or \(q\leq \frac{pn}{n-p}\) then the space \(W_d^{q,p}({\mathbb R}^n,\wedge^m)\) (of \(L^q\)-integrable differential \(m\)-forms, \(\omega\), whose distributional exterior derivatives \(d\omega\) are \(L^p\)-integrable and endowed with the norm \[ ||\omega||_{q,p} = \max\{\left(\int||\omega(x)||^q dx \right)^{\frac1q},\left(\int||d\omega(x)||^p dx \right)^{\frac1p} \},) \] is isometrically isomorphic to the space \(W_{q,p}({\mathcal P}_m)\) (of equivalence classes of additive cochains with upper norm in \(L^q({\mathbb R}^n)\) and upper gradient in \(L^p({\mathbb R}^n)\), under the equivalence relation defined by \(X_1\sim X_2\) if \(||X_1-X_2||_{q,p} = 0\) where the Sobolev norm is based on a notion of upper gradient of a function defined in the cited paper). The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub)additive functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio-Kirchheim's theory of metric currents. The notion of weak differentiability we introduce is in analogy with Heinonen-Koskela's concept of upper gradients of functions. In one of the main results of our paper, we prove continuity estimates for cochains with \(p\)-integrable upper gradient in \(n\)-dimensional Lie groups endowed with a left-invariant Finsler metric. Our result generalizes the well-known Morrey-Sobolev inequality for Sobolev functions. Finally, we prove several results relating capacity and modulus to Hausdorff dimension.

A fundamental theorem (Wolfe's theorem) in geometric integration theory states that the space of flat \(m\)-forms, endowed with the flat norm, is isometric to the space of flat \(m\)-cochains. In the paper under review the authors generalize such a theorem to the setting of Sobolev differential forms and Sobolev cochains in \({\mathbb R}^n\). They use the results in a previous paper of the second and the third author [J. Math. Pures Appl. (9) 100, No. 6, 868--906 (2013; Zbl 1277.49051)] where a suitable framework of Sobolev cochains in complete metric measure spaces has been initiated. The main result is the following: Let \(1\leq m\leq n\) and \(1<q,p<\infty\). If \(p>n-m\) or \(q\leq \frac{pn}{n-p}\) then the space \(W_d^{q,p}({\mathbb R}^n,\wedge^m)\) (of \(L^q\)-integrable differential \(m\)-forms, \(\omega\), whose distributional exterior derivatives \(d\omega\) are \(L^p\)-integrable and endowed with the norm \[ ||\omega||_{q,p} = \max\{\left(\int||\omega(x)||^q dx \right)^{\frac1q},\left(\int||d\omega(x)||^p dx \right)^{\frac1p} \},) \] is isometrically isomorphic to the space \(W_{q,p}({\mathcal P}_m)\) (of equivalence classes of additive cochains with upper norm in \(L^q({\mathbb R}^n)\) and upper gradient in \(L^p({\mathbb R}^n)\), under the equivalence relation defined by \(X_1\sim X_2\) if \(||X_1-X_2||_{q,p} = 0\) where the Sobolev norm is based on a notion of upper gradient of a function defined in the cited paper). The estimation of random effects in frailty models is an important problem in survival analysis. Testing for the presence of random effects can be essential to improving model efficiency. Posterior consistency in dispersion parameters and coefficients of the frailty model was demonstrated in theory and simulations using the posterior induced by Cox's partial likelihood and simple priors. We also conducted simulation studies to test for the presence of random effects; the proposed method performed well in several simulations. Data analysis was also conducted. The proposed method is easily tractable and can be used to develop various methods for Bayesian inference in frailty models.

Let L be an ample line bundle on a complex projective smooth surface X. The spannedness and the very ampleness properties of the pluriadjoint bundles \((K_ X\otimes L)^{\otimes t}\), \(t\geq 2\), are investigated by using Reider's theorem [\textit{I. Reider}, Ann. Math., II. Ser. 127, No.2, 309-316 (1988; Zbl 0663.14010)]. If (X,L) is not in a special list and \((X',L')\) denotes the reduction of (X,L), then (1) \((K_{X'}\otimes L')^{\otimes 3}\) is very ample; (2) \((K_{X'}\otimes L')^{\otimes 2}\) is ample and spanned and is very ample unless X contains an irreducible curve satisfying some special conditions; (3) the map associated with \(\Gamma ((K_{X'}\otimes L')^{\otimes 2})\) has degree \(\leq 4\), equality implying that \(\chi\) (\({\mathcal O}_ X)=0\) and \(g(L)=2\). Let X be a smooth complex algebraic surface, L a nef divisor on X and \(K_ X\) the canonical divisor on X. In his main theorem the author constructs an effective divisor E depending on \(L^ 2\) and going through a base point of \(| K_ X+L|\) or pairs of points not separated by \(| K_ X+L|\) and gives all possible values of L.E and \(E^ 2\). He uses the notion of a divisor being in special position with respect to \(| K_ X+L|\) to get a decomposition of \(L=M+E\) due to Bogomolov [\textit{M. Reid}, Proc. Int. Symp. Algebraic Geometry, Kyoto 1977, 623-642 (1977; Zbl 0478.14003)]. Then he shows that his theorem implies: (a) results of Bombieri on the pluricanonical maps on surfaces of general type, (b) some results on surfaces with Kodaira dimension 0, due to Beauville, and (c) Sommese's and Van de Ven's results on the adjunction mapping.

Let L be an ample line bundle on a complex projective smooth surface X. The spannedness and the very ampleness properties of the pluriadjoint bundles \((K_ X\otimes L)^{\otimes t}\), \(t\geq 2\), are investigated by using Reider's theorem [\textit{I. Reider}, Ann. Math., II. Ser. 127, No.2, 309-316 (1988; Zbl 0663.14010)]. If (X,L) is not in a special list and \((X',L')\) denotes the reduction of (X,L), then (1) \((K_{X'}\otimes L')^{\otimes 3}\) is very ample; (2) \((K_{X'}\otimes L')^{\otimes 2}\) is ample and spanned and is very ample unless X contains an irreducible curve satisfying some special conditions; (3) the map associated with \(\Gamma ((K_{X'}\otimes L')^{\otimes 2})\) has degree \(\leq 4\), equality implying that \(\chi\) (\({\mathcal O}_ X)=0\) and \(g(L)=2\). Impulsive functional differential equations with finite delay are studied. The authors prove uniform asymptotic stability of the zero solution. They obtain some new Lyapunov functional in order to establish the obtained results. The paper generalizes some known results about the stability of impulsive functional differential equations.

In complete fuzzy metric spaces the author gives a fixed point theorem of Banach type, extending a result of \textit{V. Gregori} and \textit{A. Sapena} [Fuzzy Sets Syst. 125, No. 2, 245--252 (2002; Zbl 0995.54046)]. The authors have attempted to extend the Banach fixed point theorem to fuzzy contractive mappings on different types of complete fuzzy metric spaces. They have also introduced a uniform structure on the fuzzy metric space introduced in [\textit{A. George} and \textit{P. Veeramani}, ibid. 64, No. 3, 395-399 (1994; Zbl 0843.54014)]. The authors have proved fuzzy Banach contraction theorems in the sense of complete fuzzy metric space introduced in [\textit{A. George} and \textit{P. Veeramani}, loc. cit.] and in the sense of fuzzy completeness given by \textit{M. Grabiec} [ibid. 27, No. 3, 385-389 (1988; Zbl 0664.54032)]. They have also introduced a notion called fuzzy contractive sequence and observed that every fuzzy contractive sequence is G-Cauchy (definition of Cauchy sequence as given in Grabiec) and ask the question: Is a fuzzy contractive sequence a Cauchy sequence (in this case the definition of Cauchy sequence should be taken as given in [A. George and P. Veeramani, loc. cit.].

In complete fuzzy metric spaces the author gives a fixed point theorem of Banach type, extending a result of \textit{V. Gregori} and \textit{A. Sapena} [Fuzzy Sets Syst. 125, No. 2, 245--252 (2002; Zbl 0995.54046)]. This chapter presents the main concepts of morphological image processing. Mathematical morphology has application in diverse areas of image processing such as filtering, segmentation and pattern recognition, applied both to binary and gray-scale images. Section 4.2 addresses the basic binary morphological operations: erosion, dilation, opening and closing. We also present applications of the primary operators, paying particular attention to morphological reconstruction because of its importance and since it is still not widely known. In Sect. 4.3, the same concepts are extended to gray-scale images. Section 4.4 is devoted to watershed-based segmentation. There are many variants of the watershed transform. We introduce the watershed principles with real-world applications. The key to successful segmentation is the design of the marker to eliminate the over-segmentation problem. Finally, Sect. 4.5 presents the multi-scale watershed to segment brain structures from diffusion tensor imaging, a relatively recent imaging modality that is based on magnetic resonance.

In this interesting paper a bilevel programming problem is considered. The lower level problem is a one-parametric optimization problem. The concepts of optimistic and pessimistic strategies are introduced according to the assumptions on the reactions of the follower in front of the actions of the leader. The paper is restricted to the optimistic approach. Assuming that the lower level problem belongs to the generic class of one-parametric optimization problems introduced by \textit{H. Th. Jongen, P. Jonker} and \textit{F. Twilt} [J. Optim. Theory Appl. 48, 141-161 (1986; Zbl 0556.90086)], the authors develop an algorithm that provides a way to compute an optimistic solution to the bilevel programming problem. This algorithm is based on pathfollowing and jumps methods, under the global validity of the Mangasarian-Fromovitz constraint qualification at local minimizers of the lower level problem. The authors recognize that the practicability of the algorithmic procedure, based on maximal connected components of the parametric set of local minimizers of the lower level problem, seems to be restricted to relatively small problems. However, they propose two suggestions to deal with the time-consuming difficulty and the storage needs. We introduce generalized critical points and discuss their relationship with other concepts of critical points [resp., stationary points]. Generalized critical points play an important role in parametric optimization. Under generic regularity conditions, we study the set of generalized critical points, in particular, the change of the Morse index. We focus our attention to problems with equality constraints only and provide an indication of how the present theory can be extended to problems with inequality constraints as well.

In this interesting paper a bilevel programming problem is considered. The lower level problem is a one-parametric optimization problem. The concepts of optimistic and pessimistic strategies are introduced according to the assumptions on the reactions of the follower in front of the actions of the leader. The paper is restricted to the optimistic approach. Assuming that the lower level problem belongs to the generic class of one-parametric optimization problems introduced by \textit{H. Th. Jongen, P. Jonker} and \textit{F. Twilt} [J. Optim. Theory Appl. 48, 141-161 (1986; Zbl 0556.90086)], the authors develop an algorithm that provides a way to compute an optimistic solution to the bilevel programming problem. This algorithm is based on pathfollowing and jumps methods, under the global validity of the Mangasarian-Fromovitz constraint qualification at local minimizers of the lower level problem. The authors recognize that the practicability of the algorithmic procedure, based on maximal connected components of the parametric set of local minimizers of the lower level problem, seems to be restricted to relatively small problems. However, they propose two suggestions to deal with the time-consuming difficulty and the storage needs. The spatial component of ecological interactions plays an important role in shaping ecological communities. A crucial ecological question is how does habitat disturbance and fragmentation affect species persistence and diversity? In this paper, we develop a deterministic metapopulation model that takes into account a time-dependent patchy environment, thus our model and analysis take into account environmental changes. We investigate the effects that spatial variations have on persistence and coexistence of two competing species. In particular, we study the local behaviour of the model, and we provide a rigorous proof for the global analysis of our model. Also, we compare the results of the deterministic model with simulations of a stochastic version of the model.

In 1984, the reviewer gave a sharp upper bound for the sum \(g(T_n)\) of the distances between all ordered pairs of vertices in a strong tournament \(T_n\) of order \(n\) (see J. Graph Theory 8, 1-21 (1984; Zbl 0552.05048)). The result is strengthened by introducing the score \(s\) of some removable vertex \(v\) (i.e. \(T_n- v\) is strong): \(g(T_n)= \textstyle{{1\over 6}} (n+ 1) n(n- 1)+ 2n- 3- s(n- 1- s)\) and the extremal tournaments are characterized. This article has previously appeared in summary form [Graphs and other combinatorial topics. Proc. 3rd Czech. Symp., Prague 1982, Teubner-Texte Math. 59, 216-219 (1983; Zbl 0541.05056)].

In 1984, the reviewer gave a sharp upper bound for the sum \(g(T_n)\) of the distances between all ordered pairs of vertices in a strong tournament \(T_n\) of order \(n\) (see J. Graph Theory 8, 1-21 (1984; Zbl 0552.05048)). The result is strengthened by introducing the score \(s\) of some removable vertex \(v\) (i.e. \(T_n- v\) is strong): \(g(T_n)= \textstyle{{1\over 6}} (n+ 1) n(n- 1)+ 2n- 3- s(n- 1- s)\) and the extremal tournaments are characterized. Sorry, your browser doesn't support embedded PDFs, Download First Page

These lectures give an overview of the properties of Artin and elliptic \(L\)-functions. The underlying thesis of the lectures is that ``the connection between random matrices and \(L\)-functions is rooted ''in the so-called Sato-Tate laws''. In the case of Artin \(L\)-functions the author's description culminates in the proof of the Chebotarev density theorem. For elliptic \(L\)-functions one learns about the associated Galois representation, and about the Sato-Tate onjecture. Parallels are drawn between the Sato-Tate conjecture and the Chebotarev theorem. The final section looks at \(L\)-functions over function fields, giving a brief account of the work of \textit{N. M. Katz} and \textit{P. Sarnak} [Random matrices, Frobenius eigenvalues and monodromy, Providence, R. I.: Am. Math. Soc. (1999; Zbl 0958.11004)] This book has its origin -- as explicitly stated by the authors in its introduction -- in the remarkable discovery by \textit{A. M. Odlyzko} [The \(10^{20}\)th zero of the Riemann zeta-function and 70 millions of its neighbors, ATT Bell Laboratories, 1989; Math. Comput. 48, 273-308 (1987; Zbl 0615.10049)], carried out by means of numerical experiments, that the distribution of the spacings between successive (nontrivial) zeros of the Riemann zeta-function is empirically the same as the so-called GUE measure, which is a probability measure on \(\mathbb{R}\) that arises in random matrix theory. Odlyzko's numerical experiments had been, on its turn, inspired by work of \textit{H. L. Montgomery} [Analytic Number Theory, Proc. Symp. Pure Math. 24, St. Louis Univ. Missouri 1972, 181-193 (1973; Zbl 0268.10023)], who had determined the pair correlation distribution between the zeros in a restricted range, and had already noted the compatibility of his results with the GUE prediction. Recent new results have enforced the belief, explicitly conjectured in the book, that the distribution of the spacings between zeros of the Riemann zeta-function and also of quite general automorphic \(L\)-functions over \(\mathbb{Q}\) are, in fact, given by the GUE measure, satisfying the by now called Montgomery-Odlyzko law. The authors recognize that proving this in such generality, for arbitrary number fields, seems well beyond the range of existing techniques. In the book, they restrain this scope to the case of finite fields, namely the authors establish the Montgomery-Odlyzko law for wide classes of zeta and \(L\)-functions over finite fields. To fix ideas, the authors start by considering, already at the introduction, a special case, namely that of a finite field, \(\mathbb{F}_q\), and a proper, smooth, geometrically connected curve \(C/\mathbb{F}_q\), of genus \(g\) (the corresponding zeta function was introduced by E. Artin in his thesis). After reviewing in this example the defining of the normalized spacings between the zeros of the zeta function, they show that the spacing measure in this case is the probability measure which gives mass \(1/2g\) to each of the \(2g\) normalized spacings. Of course, the rest of the 420 page book is not that easy, but this example gives a clue to understand the essential phenomena, what is it all about, as the authors themselves like to remark. This pedagogical attitude is manifest throughout the book. They go on by recalling the definition of the GUE measure on \(\mathbb{R}\) (that is, the Wigner measure, for physicists): the limit for big \(N\) of the probability measure \(A\in U(N)\). Using the Kolmogorov-Smirnov discrepancy function, one is able to have a numerical measure of how close two probability measures in \(\mathbb{R}\) are. The generalized Sato-Tate conjecture follows, as well as a number of conjectures, in particular, involving the low-lying zeros of \(L\)-functions of elliptic curves over \(\mathbb{Q}\). In Chapter 1 we find statements of the main result in the book. It deals with the measures attached to spacings of eigenvalues and with the expected values of spacing measures. Three main theorems on the existence, universality and discrepancy for limits of expected values of spacing measures are given there, with some applications, corollaries, and an appendix on the continuity properties of the \(i\)th eigenvalue as a function on \(U(N)\). Chapter 2 deals with a reformulation of the main results, under a different viewpoint, and provides several discussions on the combinatorics of spacings of finitely many points on a line. Chapter 3 deals with reduction steps in proving the main theorems, while the next chapters are devoted, respectively, to test functions (Ch. 4), the Haar measure (Ch. 5), tail estimates, a determinant-trace inequality, and multi-eigenvalue location measures (Ch. 6), large \(N\) limits and Fredholm determinants (Ch. 7), the case of several variables, with corresponding large \(N\) scaling limits (Ch. 8), equidistribution, with two versions of Deligne's equidistribution theorem (Ch. 9), monodromy of families of curves and monodromy of some other families (Chs. 10, 11), GUE discrepancies in various families of curves, abelian varieties, hypersurfaces, and Kloosterman sums (Ch. 12), and finally, on the distribution of low-lying Frobenius eigenvalues in various families, of curves, abelian varieties, hypersurfaces, and Kloosterman sums, according to the measures corresponding to different groups of the \(G(N)\), \(USp\), and \(SO\) types, with a passage to the large \(N\) limit (Ch. 13). The book finishes with two appendices, on densities and large \(N\) limits, and on some graphs, how they were drawn and what they show, respectively. I've missed some more of these graphs, for different explicit examples, and also some specific applications (e.g. in physics, to honor Wigner's pioneering discoveries) of the results obtained in the book, but this is just a personal thought. To summarize, a very complete and useful reference on the subject, by two well known specialists in the field.

These lectures give an overview of the properties of Artin and elliptic \(L\)-functions. The underlying thesis of the lectures is that ``the connection between random matrices and \(L\)-functions is rooted ''in the so-called Sato-Tate laws''. In the case of Artin \(L\)-functions the author's description culminates in the proof of the Chebotarev density theorem. For elliptic \(L\)-functions one learns about the associated Galois representation, and about the Sato-Tate onjecture. Parallels are drawn between the Sato-Tate conjecture and the Chebotarev theorem. The final section looks at \(L\)-functions over function fields, giving a brief account of the work of \textit{N. M. Katz} and \textit{P. Sarnak} [Random matrices, Frobenius eigenvalues and monodromy, Providence, R. I.: Am. Math. Soc. (1999; Zbl 0958.11004)] Generally, partially shaded conditions frequently arise in large photovoltaic generation systems which in turn cause system losses. Therefore, arrival of multiple peak values is possible in the power-voltage characteristic curves of photovoltaic systems. Such kind of issues is resolved through the usage of novel control algorithms for multilevel dc link inverters. These algorithms depend on the arrangement of pulse width modulation and direct pulse width modulation system to be with MPPT performance based on fuzzy controller to control the multilevel DC link inverter. Under the partial shading conditions, the proposed algorithm is applied on single-phase seven-level inverter with isolated maximum power point tracking techniques. The maximum power recovery of photovoltaic sources is enabled by several maximum power point tracking (MPPT) techniques like modified perturb and observe (P\&O), incremental conductance and voltage hold P\&O. The spectral quality of the inverter output is observed through the simulation studies of MATLAB/Simulink working platform. Also, hardware results are verified with the help of ATMEGA8 microcontroller.

[For Part I, see Zbl 1013.46003 above.] This is the second part of the book dedicated to set-theoretic formalisms that allow us to use actual infinitely large and actual infinitesimal quantities. Applications of infinitesimal methods in topology, measure theory, optimization, and harmonic analysis are studied in detail. The contents of the second part of the book is as follows. Chapter 6 ``Technique of hyperapproximation'' (pp. 1--74) addresses the problem of approximating infinite-dimensional Banach spaces and operators between them by finite-dimensional spaces and finite-rank operators. Naturally, some infinitely large number plays the role of the dimension of such an approximate space. Chapter 7 ``Infinitesimals in harmonic analysis'' (pp. 75--182) provides the details of the nonstandard technique for `hyperapproximation' of locally compact abelian groups and Fourier transforms over them. Chapter 8 ``Exercises and unsolved problems'' (pp. 183--199) collects some exercises for drill and better understanding as well as several open questions whose complexity varies from nil to infinity. Appendix (pp. 200--205) contains a short essay on Boolean-valued models of set theory. The list of references (pp. 206--241) contains 533 items. The purpose of this book is to make new ideas in infinitesimal analysis related to Robinson's nonstandard analysis and Nelson's theory of internal sets more accessible. To this end, the authors start with presenting the semantic qualitative views of standard and nonstandard objects as well as the relevant apparatus at the ``naive'' level of rigor which is absolutely sufficient for effective applications without appealing to any logical formalism. They then give concise reference material pertaining to the modern axiomatic expositions of infinitesimal analysis within the classical Cantorian doctrine. They have found it appropriate to allot plenty of room to the ideological and historical facets of the topic, which has determined the plan and style of exposition. The contents is as follows. Chapter 1 ``Excursions in the history of calculus'' (pp. 1-11) contains the historical signposts alongside the qualitative motivation of the principles of infinitesimal analysis. Chapter 2 ``Naive foundations of infinitesimal analysis'' (pp. 12-43) contains a discussion of simplest implications of the principles of infinitesimal analysis for differential and integral calculus. This lays the ``naive'' foundation of infinitesimal analysis. Chapter 3 ``Set-theoretic formalism of infinitesimal analysis'' (pp. 44-115) gives formal details of the corresponding apparatus of nonstandard set theory. Chapter 4 ``Monads in general topology'' (pp. 116-165) and Chapter 5 ``Infinitesimals and subdifferentials'' (166-274) set forth the infinitesimal methods of general topology and subdifferential calculus. The list of references (275-309) contains 531 items. For Part II, see Zbl 1013.46004 below.

[For Part I, see Zbl 1013.46003 above.] This is the second part of the book dedicated to set-theoretic formalisms that allow us to use actual infinitely large and actual infinitesimal quantities. Applications of infinitesimal methods in topology, measure theory, optimization, and harmonic analysis are studied in detail. The contents of the second part of the book is as follows. Chapter 6 ``Technique of hyperapproximation'' (pp. 1--74) addresses the problem of approximating infinite-dimensional Banach spaces and operators between them by finite-dimensional spaces and finite-rank operators. Naturally, some infinitely large number plays the role of the dimension of such an approximate space. Chapter 7 ``Infinitesimals in harmonic analysis'' (pp. 75--182) provides the details of the nonstandard technique for `hyperapproximation' of locally compact abelian groups and Fourier transforms over them. Chapter 8 ``Exercises and unsolved problems'' (pp. 183--199) collects some exercises for drill and better understanding as well as several open questions whose complexity varies from nil to infinity. Appendix (pp. 200--205) contains a short essay on Boolean-valued models of set theory. The list of references (pp. 206--241) contains 533 items. The paper derives alternative formulae for the mean and variance for discrete probability distributions using cumulative and decumulative probabilities. The new formulae provide less complexity, in that instead of the standard weighted formulae around the x and \(x^ 2\) terms, no x term is used in finding the mean, and only x instead of \(x^ 2\) is used in the variance term. Examples illustrate this.

The main goal of the paper is to recover the gradient of the scalar conductivity defined on a bounded open set in \(\mathbb{R}^d\) from the Dirichlet to Neumann map from the \(p\)-Laplace equation. Given a bounded open set \(\Omega\in \mathbb{R}^d\) and a bounded conductivity \(\gamma>0\) on \(\overline{\Omega}\) consider the Dirichlet problem \[ \Delta_p^\gamma(u)=\operatorname{div}(\gamma (x)|\nabla u|^{p-2}\nabla u)=0\text{ in }\Omega;\quad u=v\text{ on }\partial\Omega. \] Given the knowledge of the DN map, we have to determine \(\nabla \gamma|_{\partial\Omega}\). For any boundary point the gradient is recovered by using Dirichlet data supported on an arbitrary small neighbourhood of the boundary point. The proof uses a Rellich type identity. There have been several boundary determination results in the case \(p=2\) (see, for example, [\textit{J. Sylvester} and \textit{G. Uhlmann}, Commun. Pure Appl. Math. 41, No. 2, 188--219 (1988; Zbl 0632.35074)]). The results of the present article are new for \(p\neq 2\). We extend a result of Kohn and Vogelius, showing that the boundary values of continuous isotropic conductivity can be recovered from voltage and current measurements at the boundary. Moreover, we prove sharp estimates to establish the continuous dependence of the boundary values of the conductivity on the voltage to current map. These estimates allow extension of the Kohn and Vogelius results to merely continuous conductivities.

The main goal of the paper is to recover the gradient of the scalar conductivity defined on a bounded open set in \(\mathbb{R}^d\) from the Dirichlet to Neumann map from the \(p\)-Laplace equation. Given a bounded open set \(\Omega\in \mathbb{R}^d\) and a bounded conductivity \(\gamma>0\) on \(\overline{\Omega}\) consider the Dirichlet problem \[ \Delta_p^\gamma(u)=\operatorname{div}(\gamma (x)|\nabla u|^{p-2}\nabla u)=0\text{ in }\Omega;\quad u=v\text{ on }\partial\Omega. \] Given the knowledge of the DN map, we have to determine \(\nabla \gamma|_{\partial\Omega}\). For any boundary point the gradient is recovered by using Dirichlet data supported on an arbitrary small neighbourhood of the boundary point. The proof uses a Rellich type identity. There have been several boundary determination results in the case \(p=2\) (see, for example, [\textit{J. Sylvester} and \textit{G. Uhlmann}, Commun. Pure Appl. Math. 41, No. 2, 188--219 (1988; Zbl 0632.35074)]). The results of the present article are new for \(p\neq 2\). A new meshless local Petrov-Galerkin (MLPG) method for solving two-dimensional frequency domain elastodynamic problems is proposed. Since the method utilizes, in its weak formulation, either the elastostatic or the frequency domain elastodynamic fundamental solution as test function, it is equivalent to the local boundary integral equation (LBIE) method. Nodal points spread over the analyzed domain are considered, and the moving least squares (MLS) interpolation scheme for the approximation of the interior and boundary variables is employed. Two integral equations suitable for the integral representation of the displacement fields in local sub-domains are used. The first utilizes the frequency domain fundamental solution, comprises only boundary integrals and exploits the elastodynamic companion solution, which is derived in the framework of the present work. The second equation makes use of the simple elastostatic fundamental solution, employs the elastostatic companion solution in order to get rid of tractions on the local boundaries and contains both boundary and volume integrals. On the global boundary, derivatives of the shape functions of the MLS approximation are avoided by considering displacements and tractions as independent variables. Direct numerical techniques for the accurate evaluation of both surface and volume integrals are employed and presented in detail. All the strongly singular integrals are computed directly through highly accurate integration techniques. Three representative numerical examples that demonstrate the accuracy of the proposed methodology are provided.

The article deals with variational methods for grid generation and construction of spatial mappings with prescribed properties. Such mappings are applied in many areas, examples of which are delivered in the text. The spatial mappings are treated with respect to the best-maximum principle, which allows to formulate an optimization problem for the class of quasi-isometric mappings. The optimization problem is discretized, its properties are delivered, and an algorithm based on continuation technique is proposed. The article contains examples of application in mesh regularization (untangling), morphing and surface flattening. The article is a progress work related to the previously published results by \textit{V. A. Garanzha} [Comput. Math. Math. Phys. 40, 1617--1637 (2000; Zbl 1004.65131)]. The author presents a grid generation method based on the minimization of a barrier functional with a feasible set consisting of quasi-isometric grids. The procedure for minimizing the deviation from isometry for a grid of given connectivity makes it possible to contract the feasible set into a small neighborhood of the optimal solution. A reliable and efficient technique for constructing a feasible solution is proposed, based on the penalty formulation and continuation technique.

The article deals with variational methods for grid generation and construction of spatial mappings with prescribed properties. Such mappings are applied in many areas, examples of which are delivered in the text. The spatial mappings are treated with respect to the best-maximum principle, which allows to formulate an optimization problem for the class of quasi-isometric mappings. The optimization problem is discretized, its properties are delivered, and an algorithm based on continuation technique is proposed. The article contains examples of application in mesh regularization (untangling), morphing and surface flattening. The article is a progress work related to the previously published results by \textit{V. A. Garanzha} [Comput. Math. Math. Phys. 40, 1617--1637 (2000; Zbl 1004.65131)]. We use the decomposition method for solving an equation of the form \(u= Nu\), where \(N\) is a nonlinear operator. We also give new formulae for the calculation of Adomian's polynomials for an operator and generalize these results for a multidimensional operator. These new results are then used for solving the Navier-Stokes equations.

Solutions \(u \in H^1(\mathbb{R}^N)\) of the nonlinear Schrödinger equation \[ -\Delta u + V(|y|)u = u^p, u>0\tag{1} \] are constructed when \(1<p<\frac{N+2}{N-2}\), \(N \ge 3\), and the radial potential \(V\) is positive, bounded and \[ V(|y|) =V_0 + \frac{a}{|y|^m} + O\left(\frac{1}{|y|^{m+\sigma}}\right) \text{ as }\: |y| \to \infty \] for some constants \(V_0, a,\sigma>0 \) and \(m> \max\{\frac{4}{p-1},2\}\). More precisely, let \(U\) be the radial solution of (1) when \(V =0\) and so that \(U(y) \to 0\) as \(|y|\to 0\). Define \(W_{r,h}(y)= \sum_{j=1}^{2k}U(y-x_j)\) where the \(2k\) points \(x_j\) are symmetrically chosen on the sphere \(y_1^2+y_2^2+y_3^2=r^2\) intersected with the two 2D planes \(y_3=\pm rh\), \(y_j=0, 4\le j \le N\). The parameters \(r,h>0\) are chosen in a small range dependent on \(k\). The main result is: For all \(k\) large enough there is a solution \(u_k\) of (1) of the form \(u_k = W_{r_k,h_k}+\omega_k\) where \(\omega_k\in H^1(\mathbb{R}^N)\) has certain symmetry properties and \[ \int_{\mathbb{R}^N}|\nabla \omega_k|^2+V|\omega_k|^2 \to 0 \text{ as } k \to \infty. \] This result is related to that of \textit{J. Wei} and \textit{S. Yan} [Calc. Var. Partial Differ. Equ. 37, No. 3--4, 423--439 (2010; Zbl 1189.35106)]. The latter result is roughly equivalent to taking \(h=0\) above. We consider the following nonlinear problem in \(\mathbb R^N\) \[ - \Delta u +V(|y|)u = u^p,\qquad u > 0 \quad \text{in }\mathbb R^N, \qquad u \in H^1(\mathbb R^N), \tag{1} \] where \(V(r)\) is a positive function, \(1<p <\frac{N+2}{N-2}\). We show that if \(V(r)\) has the following expansion: \[ V(r) = V_0+\frac {a}{r^m} + O\left(\frac {1}{r^{m+\theta}}\right),\quad \text{as } r\to +\infty, \] where \(a>0\), \(m>1\), \(\theta>0\), and \(V_0>0\) are some constants, then (1) has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.

Solutions \(u \in H^1(\mathbb{R}^N)\) of the nonlinear Schrödinger equation \[ -\Delta u + V(|y|)u = u^p, u>0\tag{1} \] are constructed when \(1<p<\frac{N+2}{N-2}\), \(N \ge 3\), and the radial potential \(V\) is positive, bounded and \[ V(|y|) =V_0 + \frac{a}{|y|^m} + O\left(\frac{1}{|y|^{m+\sigma}}\right) \text{ as }\: |y| \to \infty \] for some constants \(V_0, a,\sigma>0 \) and \(m> \max\{\frac{4}{p-1},2\}\). More precisely, let \(U\) be the radial solution of (1) when \(V =0\) and so that \(U(y) \to 0\) as \(|y|\to 0\). Define \(W_{r,h}(y)= \sum_{j=1}^{2k}U(y-x_j)\) where the \(2k\) points \(x_j\) are symmetrically chosen on the sphere \(y_1^2+y_2^2+y_3^2=r^2\) intersected with the two 2D planes \(y_3=\pm rh\), \(y_j=0, 4\le j \le N\). The parameters \(r,h>0\) are chosen in a small range dependent on \(k\). The main result is: For all \(k\) large enough there is a solution \(u_k\) of (1) of the form \(u_k = W_{r_k,h_k}+\omega_k\) where \(\omega_k\in H^1(\mathbb{R}^N)\) has certain symmetry properties and \[ \int_{\mathbb{R}^N}|\nabla \omega_k|^2+V|\omega_k|^2 \to 0 \text{ as } k \to \infty. \] This result is related to that of \textit{J. Wei} and \textit{S. Yan} [Calc. Var. Partial Differ. Equ. 37, No. 3--4, 423--439 (2010; Zbl 1189.35106)]. The latter result is roughly equivalent to taking \(h=0\) above. The articles of this volume will be reviewed individually.