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Let \(H({\mathbb D})\) be the space of analytic functions on the unit disk \({\mathbb D}\). For \(p\in(1,\infty)\), \(\alpha\in(-2,\infty)\) and \(\beta\in[0,\infty)\), \(g\in H({\mathbb D})\) belongs to the space \(F(p,\alpha,\beta)\) if \[ \| g\|_{F(p,\alpha,\beta)}=\sup_{a\in D}\left(\int_\mathbb D| g'(z)| ^{p}(1-| z| ^{2})^{\alpha}(1-| \sigma_a(z)|^{2})^{\beta} \,dA(z)\right)^{\frac{1}{p}}<\infty, \] where \(\sigma_a(z)=(a-z)/(1-\overline az)\). The closed subspace \(F_0(p,\alpha,\beta)\) of \(F(p,\alpha,\beta)\) consists of those functions \(g\in F(p,\alpha,\beta)\) such that \[ \lim_{| a| \to1}\int_{\mathbb D}| g'(z)| ^{p}(1-| z| ^{2})^{\alpha}(1-| \sigma_a(z)| ^{2})^{\beta}\,dA(z)=0. \] These spaces were introduced in the dissertation of \textit{R. Zhao} [Ann. Acad. Sci. Fenn. Math., Diss. 105 (1996; Zbl 0851.30017)]. The \(R(p,\alpha,\beta)\)-spaces are defined in the following way. Let \(E_{k,j}\) be the pairwise disjoint set given by \[ E_{kj}=\left\{z\in \mathbb{D}:1-\frac{1}{2^{k}}\leq| z| <1-\frac{1}{2^{k+1}},\;\frac{\pi j}{2^{k+1}}\leq\arg z<\frac{\pi(j+1)}{2^{k+1}}\right\}, \] where \(k=0,1,2,\dots\) and \(j=0,1,2,\dots,2^{k+2}-1\), so that \[ \bigcup_{k=0}^{\infty}\bigcup_{j=0}^{2^{k+2}-1}E_{kj}=\mathbb{D}. \] For the sake of simplicity, we will rename these sets as \(E_m\), \(m\in\mathbb{N}\). More precisely, we denote \(m=j-1+\sum_{i=0}^{k}2^{i+1}\) so that \[ E_1=E_{0,0},\;\dots,\;E_4=E_{0,3},\;E_5=E_{1,0},\;\dots,\;E_{12}=E_{1,7},\;E_{13}=E_{2,0},\;\dots. \] Furthermore, let \(a_m\) denote the center of \(E_m\). Then \(R(p,\alpha,\beta)\) consists of those functions \(f\in H(\mathbb{D})\) for which \(f(z)=\sum_{m=1}^{\infty}f_m(z)\), where each \(f_m\in H(\mathbb{D})\) and \[ \sum_{m=1}^{\infty}\left(\int_\mathbb{D}| f_m(z)| ^{\frac{p}{p-1}}(1-| z| ^{2})^{-\frac{a}{p-1}}(1-| \sigma_{a_m}(z)| ^{2})^{-\frac{\beta}{p-1}}\,dA(z)\right)^{\frac{p-1}{p}}<\infty. \] The norm of \(R(p,\alpha,\beta)\) is given by \[ \| f\|_{R(p,\alpha,\beta)}=\inf\sum_{m=1}^{\infty}\left(\int_\mathbb{D}| f_m(z)| ^{\frac{p}{p-1}}(1-| z| ^{2})^{-\frac{a}{p-1}}(1-| \sigma_{a_m}(z)| ^{2})^{-\frac{\beta}{p-1}}\,dA(z)\right)^{\frac{p-1}{p}}, \] where the infimum is taken over all such representations of \(f\). Further, let the space \[ E(p,\alpha,\beta)=\left\{L\in F(p,\alpha,\beta)^{*}:L| _{(B_{F(p,\alpha,\beta)},co)}\;\text{is continuous}\right\} \] be the closed subspace of \(F(p,\alpha,\beta)^{*}\), that is, the dual of \(F(p,\alpha,\beta)\). The authors prove that \(F_0(p,\alpha,\beta)^{*}\) is isomorphic to \(R(p,\alpha,\beta)\), and that \(E(p,\alpha,\beta)\) is the unique isometric predual of \(F(p,\alpha,\beta)\) and that \(E(p,\alpha,\beta)\) is isomorphic to \(R(p,\alpha,\beta)\). Moreover, it is shown that the bidual of \(F_0(p,\alpha,\beta)\) is isometrically isomorphic to \(F(p,\alpha,\beta)\). Finally, the authors characterize the bounded multiplication operators on \(R(p,\alpha,\beta)\). Let \(A (\Delta)\) denote the collection of all functions analytic in the unit disk \( \Delta\), the following general family of function spaces was introduced and studied in great detail by the author in this paper. For \(0 < p < \infty\), \(- 2 < q < \infty\) and \(0 < s < \infty\), set \[ F(p, q,s) : = \Bigl \{f \in A (\Delta) : \sup_{a \in \Delta} I_a^{p,q,s} (f) < \infty \Bigr\} \] where \[ I_a^{p, q,s} (f) = \int_\Delta \bigl |f'(z) \bigr |^p \bigl( 1 - |z |^2 \bigr)^q \bigl( g(z,a) \bigr )^s dm (z) \] and \(g (z,a)\) is the Green's function of \(\Delta\) with logarithmic singularity at \(a \in \Delta\). The author proves some basic properties of this space and shows some relations between \(F(p,q,s)\) and several classical spaces, such as \(\alpha\)-Bloch space, BMOA, etc. By introducing the notion of the \(\alpha\)-Möbius invariance, it follows that \(F(p,q,s)\) is \({q + 2 \over p}\)-Möbius invariant and \(\alpha\)-Bloch spaces are maximal along certain \(\alpha\)-Möbius invariant spaces. This result generalizes a theorem for the Bloch space by \textit{L. A. Rubel} and \textit{R. M. Timoney} [Proc. Am. Math. Soc. 75, 45-49 (1979; Zbl 0405.46020)], and gives an answer to a question by \textit{K. Zhu} [Rocky Mt. J. Math. 23, No. 3, 1143-1177 (1993; Zbl 0787.30019)]. The estimate of the radial growth of the derivatives for the functions in \(F(p,q,s)\) is given. The author studies inclusion relations among the spaces \(F(p,q,s)\) and proves that all spaces \(F(p,q,s)\) are different for different \(s \in [0,1]\).

Let \(H({\mathbb D})\) be the space of analytic functions on the unit disk \({\mathbb D}\). For \(p\in(1,\infty)\), \(\alpha\in(-2,\infty)\) and \(\beta\in[0,\infty)\), \(g\in H({\mathbb D})\) belongs to the space \(F(p,\alpha,\beta)\) if \[ \| g\|_{F(p,\alpha,\beta)}=\sup_{a\in D}\left(\int_\mathbb D| g'(z)| ^{p}(1-| z| ^{2})^{\alpha}(1-| \sigma_a(z)|^{2})^{\beta} \,dA(z)\right)^{\frac{1}{p}}<\infty, \] where \(\sigma_a(z)=(a-z)/(1-\overline az)\). The closed subspace \(F_0(p,\alpha,\beta)\) of \(F(p,\alpha,\beta)\) consists of those functions \(g\in F(p,\alpha,\beta)\) such that \[ \lim_{| a| \to1}\int_{\mathbb D}| g'(z)| ^{p}(1-| z| ^{2})^{\alpha}(1-| \sigma_a(z)| ^{2})^{\beta}\,dA(z)=0. \] These spaces were introduced in the dissertation of \textit{R. Zhao} [Ann. Acad. Sci. Fenn. Math., Diss. 105 (1996; Zbl 0851.30017)]. The \(R(p,\alpha,\beta)\)-spaces are defined in the following way. Let \(E_{k,j}\) be the pairwise disjoint set given by \[ E_{kj}=\left\{z\in \mathbb{D}:1-\frac{1}{2^{k}}\leq| z| <1-\frac{1}{2^{k+1}},\;\frac{\pi j}{2^{k+1}}\leq\arg z<\frac{\pi(j+1)}{2^{k+1}}\right\}, \] where \(k=0,1,2,\dots\) and \(j=0,1,2,\dots,2^{k+2}-1\), so that \[ \bigcup_{k=0}^{\infty}\bigcup_{j=0}^{2^{k+2}-1}E_{kj}=\mathbb{D}. \] For the sake of simplicity, we will rename these sets as \(E_m\), \(m\in\mathbb{N}\). More precisely, we denote \(m=j-1+\sum_{i=0}^{k}2^{i+1}\) so that \[ E_1=E_{0,0},\;\dots,\;E_4=E_{0,3},\;E_5=E_{1,0},\;\dots,\;E_{12}=E_{1,7},\;E_{13}=E_{2,0},\;\dots. \] Furthermore, let \(a_m\) denote the center of \(E_m\). Then \(R(p,\alpha,\beta)\) consists of those functions \(f\in H(\mathbb{D})\) for which \(f(z)=\sum_{m=1}^{\infty}f_m(z)\), where each \(f_m\in H(\mathbb{D})\) and \[ \sum_{m=1}^{\infty}\left(\int_\mathbb{D}| f_m(z)| ^{\frac{p}{p-1}}(1-| z| ^{2})^{-\frac{a}{p-1}}(1-| \sigma_{a_m}(z)| ^{2})^{-\frac{\beta}{p-1}}\,dA(z)\right)^{\frac{p-1}{p}}<\infty. \] The norm of \(R(p,\alpha,\beta)\) is given by \[ \| f\|_{R(p,\alpha,\beta)}=\inf\sum_{m=1}^{\infty}\left(\int_\mathbb{D}| f_m(z)| ^{\frac{p}{p-1}}(1-| z| ^{2})^{-\frac{a}{p-1}}(1-| \sigma_{a_m}(z)| ^{2})^{-\frac{\beta}{p-1}}\,dA(z)\right)^{\frac{p-1}{p}}, \] where the infimum is taken over all such representations of \(f\). Further, let the space \[ E(p,\alpha,\beta)=\left\{L\in F(p,\alpha,\beta)^{*}:L| _{(B_{F(p,\alpha,\beta)},co)}\;\text{is continuous}\right\} \] be the closed subspace of \(F(p,\alpha,\beta)^{*}\), that is, the dual of \(F(p,\alpha,\beta)\). The authors prove that \(F_0(p,\alpha,\beta)^{*}\) is isomorphic to \(R(p,\alpha,\beta)\), and that \(E(p,\alpha,\beta)\) is the unique isometric predual of \(F(p,\alpha,\beta)\) and that \(E(p,\alpha,\beta)\) is isomorphic to \(R(p,\alpha,\beta)\). Moreover, it is shown that the bidual of \(F_0(p,\alpha,\beta)\) is isometrically isomorphic to \(F(p,\alpha,\beta)\). Finally, the authors characterize the bounded multiplication operators on \(R(p,\alpha,\beta)\). We study a pair of commuting difference operators arising from the elliptic \(C_2^{(1)}\)-face model. The operators, whose coefficients are expressed in terms of the Jacobi's elliptic theta function, act on the space of meromorphic functions on the weight space of the \(C_2\)-type simple Lie algebra. We show that the space of functions spanned by the level one characters of the affine Lie algebra \(\hat{\mathfrak {sp}}(4,\mathbb C)\) is invariant under the action of the difference operators

The paper aims to establish generic properties of collapse in the one-dimensional nonlinear Schrödinger equation with the self-focusing quintic nonlinearity, \[ i\psi_t + \psi_{xx} + |\psi|^4\psi = 0. \] It is commonly known that this equation has a family of soliton solutions generated by initial configuration \[ \psi_{\text{sol}} = (3/2)^{1/4}a[\cosh(a^2x/2)]^{-1/2}, \] with arbitrary scale parameter \(a\). The solitons are unstable because of the possibility of the critical collapse in this equation. In fact, an exact solution for a collapsing pulse is known in this case. Indeed, the equation maintains the pseudo-conformal invariance, which means that any solution \(\psi(x,t)\) can be transformed into a new solution, as follows: \[ \tilde{\psi}(x,t) = (a + bt)^{-1/2} \exp(\frac{ibx^2}{4(a + bt)})\psi(\frac{c + dt}{a + bt},\frac{x}{a + bt}), \] where the rows \((a,b)\) and \((c,d)\) form an \(SL(2)\) matrix with real elements. The application of this transformation to the soliton generates a new solution, which features the collapse at \(t = -a/b\), due to the presence of the singular factor, \((a + bt)^{-1/2}\). Nevertheless, it is known from the work of \textit{G. Perelman} [Ann. Henri Poincaré 2, No. 4, 605--673 (2001; Zbl 1007.35087)] and subsequent works that this exact solution represents a \textit{nongeneric} collapsing solution, the generic one featuring a different type of the near-collapse singular behavior, with the amplitude growing as \[ A \sim \sqrt{\frac{\ln|\ln(T-t)|}{T - t}} \] (\(T\) is the collapse time). The present work produces a rigorous proof of the fact that, while the nongeneric collapse regime is unstable (otherwise, it would be generic), one can identify a manifold of codimension 1 in a space of initial perturbations around the exact soliton, with some reasonably defined norm, such that the nongeneric collapse is stable on that manifold. The proof is based on a specially developed a priori estimate which allows one to split the initial perturbation into a dispersive (radiation) part, and one accounting for a modification of the soliton's parameters. Subsequently, a strong local estimate is obtained for the rate of the decay of the dispersive part. This analysis makes it possible to identify a restriction on the perturbation that maintains the nongeneric regime of the collapse, and thus determines the above-mentioned codimension-1 manifold on which this regime occurs. The author considers the nonlinear Schrödinger equation \[ i\psi_{t}=-\psi -\left|\psi\right|^{4}\psi \] with initial data \(\psi|_{t=0}=\varphi+\kappa\), where \(\kappa\) is small in suitable sense. It is shown that for a certain set of initial perturbations \(\kappa\) the solution \(\psi\) blows up in finite time. An asymptotic representation of the solution is obtained.

The paper aims to establish generic properties of collapse in the one-dimensional nonlinear Schrödinger equation with the self-focusing quintic nonlinearity, \[ i\psi_t + \psi_{xx} + |\psi|^4\psi = 0. \] It is commonly known that this equation has a family of soliton solutions generated by initial configuration \[ \psi_{\text{sol}} = (3/2)^{1/4}a[\cosh(a^2x/2)]^{-1/2}, \] with arbitrary scale parameter \(a\). The solitons are unstable because of the possibility of the critical collapse in this equation. In fact, an exact solution for a collapsing pulse is known in this case. Indeed, the equation maintains the pseudo-conformal invariance, which means that any solution \(\psi(x,t)\) can be transformed into a new solution, as follows: \[ \tilde{\psi}(x,t) = (a + bt)^{-1/2} \exp(\frac{ibx^2}{4(a + bt)})\psi(\frac{c + dt}{a + bt},\frac{x}{a + bt}), \] where the rows \((a,b)\) and \((c,d)\) form an \(SL(2)\) matrix with real elements. The application of this transformation to the soliton generates a new solution, which features the collapse at \(t = -a/b\), due to the presence of the singular factor, \((a + bt)^{-1/2}\). Nevertheless, it is known from the work of \textit{G. Perelman} [Ann. Henri Poincaré 2, No. 4, 605--673 (2001; Zbl 1007.35087)] and subsequent works that this exact solution represents a \textit{nongeneric} collapsing solution, the generic one featuring a different type of the near-collapse singular behavior, with the amplitude growing as \[ A \sim \sqrt{\frac{\ln|\ln(T-t)|}{T - t}} \] (\(T\) is the collapse time). The present work produces a rigorous proof of the fact that, while the nongeneric collapse regime is unstable (otherwise, it would be generic), one can identify a manifold of codimension 1 in a space of initial perturbations around the exact soliton, with some reasonably defined norm, such that the nongeneric collapse is stable on that manifold. The proof is based on a specially developed a priori estimate which allows one to split the initial perturbation into a dispersive (radiation) part, and one accounting for a modification of the soliton's parameters. Subsequently, a strong local estimate is obtained for the rate of the decay of the dispersive part. This analysis makes it possible to identify a restriction on the perturbation that maintains the nongeneric regime of the collapse, and thus determines the above-mentioned codimension-1 manifold on which this regime occurs. Full scale laboratory experiments on the dynamic behavior of a railway track and its constituents with respect to the coupling conditions to the ground are performed. Their effects on the frequency response functions of the track lead to special investigations on the wave propagation in the subsoil and the dynamic displacement fields of the rails and the sleepers by use of the holographic interfereometry. Moreover, wave propagation through the soil is observed when various inhomogenities are placed into the ground in order to isolate neighbouring constructions.

The paper studies the variance of the first-passage time \(\tau(0,x)\) from \(0\) to \(x\in\mathbb Z^d\) in the percolation model on the lattice \(\mathbb Z^d\), defined by \[ \tau(0,x)=\inf_{\gamma:0\rightarrow x}\sum_{e\in\gamma} t_e. \] In the above definition, \(\gamma=(v_0=0,e_0,v_1,\dots,e_N,v_N=x)\) is the lattice path that joins \(0\) to \(x\) and \((t_e)\) is a collection of independent random variables with common distribution \(\mu\) on the set of nearest-neighbor edges. For \(d=1\), \(\tau(0,x)\) becomes a sum of i.i.d.\ random variables and the variance of \(\tau(0,x)\) is of order \(\| x\|_1\), while for \(d\geq 2\), \(\tau(0,x)\) is the minimum over correlated sums of i.i.d.\ random variables. In the remarkable paper [Ann. Probab. 31, No. 4, 1970--1978 (2003; Zbl 1087.60070)], \textit{I. Benjamini} et al. proved that, in the case when the edge-weight distribution \(\mu\) is uniform on the set of two positive values (=\,weights) \(a\) and \(b\), the variance is sublinear in distance, i.e., \[ \operatorname{Var}\tau(0,x)\leq C(a,b)\frac{\| x\|_1}{\log\| x\|_1}, \] for all \(d\geq 2\). The present paper extends the above sublinear result to measures \(\mu\) obeying some moment condition. Let \(0 < a < b < \infty\), and for each edge e of \(\mathbb Z^d\) let \(\omega_e=a\) or \(\omega_e=b\), each with probability \(1/2\), independently. This induces a random metric \(\text{dist}_\omega\) on the vertices of \(\mathbb Z^d\), called first passage percolation. We prove that for \(d>1\), the distance \(\text{dist}_\omega(0,v)\) from the origin to a vertex \(v\), \(|v|>2\), has variance bounded by \(C|v|/\log|v|\), where \(C=C(a,b,d)\) is a constant which may only depend on a, b and d. Some related variants are also discussed.

The paper studies the variance of the first-passage time \(\tau(0,x)\) from \(0\) to \(x\in\mathbb Z^d\) in the percolation model on the lattice \(\mathbb Z^d\), defined by \[ \tau(0,x)=\inf_{\gamma:0\rightarrow x}\sum_{e\in\gamma} t_e. \] In the above definition, \(\gamma=(v_0=0,e_0,v_1,\dots,e_N,v_N=x)\) is the lattice path that joins \(0\) to \(x\) and \((t_e)\) is a collection of independent random variables with common distribution \(\mu\) on the set of nearest-neighbor edges. For \(d=1\), \(\tau(0,x)\) becomes a sum of i.i.d.\ random variables and the variance of \(\tau(0,x)\) is of order \(\| x\|_1\), while for \(d\geq 2\), \(\tau(0,x)\) is the minimum over correlated sums of i.i.d.\ random variables. In the remarkable paper [Ann. Probab. 31, No. 4, 1970--1978 (2003; Zbl 1087.60070)], \textit{I. Benjamini} et al. proved that, in the case when the edge-weight distribution \(\mu\) is uniform on the set of two positive values (=\,weights) \(a\) and \(b\), the variance is sublinear in distance, i.e., \[ \operatorname{Var}\tau(0,x)\leq C(a,b)\frac{\| x\|_1}{\log\| x\|_1}, \] for all \(d\geq 2\). The present paper extends the above sublinear result to measures \(\mu\) obeying some moment condition. The \(t\) copula and its properties are described with a focus on issues related to the dependence of extreme values. The Gaussian mixture representation of a multivariate \(t\) distribution is used as a starting point to construct two new copulas, the skewed \(t\) copula and the grouped \(t\) copula, which allow more heterogeneity in the modelling of dependent observations. Extreme value considerations are used to derive two further new copulas: the \(t\) extreme value copula is the limiting copula of componentwise maxima of \(t\) distributed random vectors; the \(t\) lower tail copula is the limiting copula of bivariate observations from a \(t\) distribution that are conditioned to lie below some joint threshold that is progressively lowered. Both these copulas may be approximated for practical purposes by simpler, better-known copulas, these being the Gumbel and Clayton copulas, respectively.

Just as the computation of the stable stems is one of the fundamental questions in algebraic topology, the homotopy groups of the \(C_2\)-equivariant and \(\mathbb{R}\)-motivic stable stems are fundamental to \(C_2\)-equivariant and \(\mathbb{R}\)-motivic homotopy theory. These computations also give additional information about the classical stable stems and therefore global results about the \(\mathbb{R}\)-motivic stable stems and the \(C_2\)-equivariant ones are certainly important. The paper under review provides a range of degrees, given by an inequality involving the stem and co-weight, in which the \(C_2\)-equivariant stable stems and the \(\mathbb{R}\)-motivic stable stems agree. This work extends a previous result of \textit{D. Dugger} and \textit{D. C. Isaksen} [Proc. Am. Math. Soc. 145, No. 8, 3617--3627 (2017; Zbl 1421.55011)] providing an isomorphism between the \(C_2\)-equivariant stable stems and \(\mathbb{R}\)-motivic stable stems in a smaller range. This result is used in a key way to study Mahowald invariants by Eva Belmont and Daniel C. Isaksen. The result is useful from a computational perspective and it is interesting in its own right because it provides qualitative information about the difference between the \(\mathbb{R}\)-motivic and \(C_2\)-equivariant stable stems. The paper concludes by describing why the main theorem is sharp in a sense the authors make precise. The proof improves on the method of Daniel Dugger and Daniel C. Isaksen using the respective cobar complexes by instead applying \(\mathbb{R}\)-motivic and \(C_2\)-equivariant \(\rho\)-Bockstein spectral sequences. This removes some of the extra data that appears in the cobar complexes, but does not survive to the \(E_2\)-pages of the respective Adams spectral sequences. The proof is therefore also interesting in its own right as an example of how the \(\rho\)-Bockstein spectral sequence can simplify computations of \(\mathbb{R}\)-motivic and \(C_2\)-equivariant homotopy groups. We establish an isomorphism between the stable homotopy groups \( \hat {\pi }^{\mathbb{R}}_{s,w}\) of the 2-completed \(\mathbb{R}\)-motivic sphere spectrum and the stable homotopy groups \( \hat {\pi }^{\mathbb{Z}/2}_{s,w}\) of the 2-completed \( \mathbb{Z}/2\)-equivariant sphere spectrum, valid in the range \( s \geq 3 w - 5\) or \( s \leq -1\).

Just as the computation of the stable stems is one of the fundamental questions in algebraic topology, the homotopy groups of the \(C_2\)-equivariant and \(\mathbb{R}\)-motivic stable stems are fundamental to \(C_2\)-equivariant and \(\mathbb{R}\)-motivic homotopy theory. These computations also give additional information about the classical stable stems and therefore global results about the \(\mathbb{R}\)-motivic stable stems and the \(C_2\)-equivariant ones are certainly important. The paper under review provides a range of degrees, given by an inequality involving the stem and co-weight, in which the \(C_2\)-equivariant stable stems and the \(\mathbb{R}\)-motivic stable stems agree. This work extends a previous result of \textit{D. Dugger} and \textit{D. C. Isaksen} [Proc. Am. Math. Soc. 145, No. 8, 3617--3627 (2017; Zbl 1421.55011)] providing an isomorphism between the \(C_2\)-equivariant stable stems and \(\mathbb{R}\)-motivic stable stems in a smaller range. This result is used in a key way to study Mahowald invariants by Eva Belmont and Daniel C. Isaksen. The result is useful from a computational perspective and it is interesting in its own right because it provides qualitative information about the difference between the \(\mathbb{R}\)-motivic and \(C_2\)-equivariant stable stems. The paper concludes by describing why the main theorem is sharp in a sense the authors make precise. The proof improves on the method of Daniel Dugger and Daniel C. Isaksen using the respective cobar complexes by instead applying \(\mathbb{R}\)-motivic and \(C_2\)-equivariant \(\rho\)-Bockstein spectral sequences. This removes some of the extra data that appears in the cobar complexes, but does not survive to the \(E_2\)-pages of the respective Adams spectral sequences. The proof is therefore also interesting in its own right as an example of how the \(\rho\)-Bockstein spectral sequence can simplify computations of \(\mathbb{R}\)-motivic and \(C_2\)-equivariant homotopy groups. The capacitated centred clustering problem (CCCP) consists of defining a set of clusters with limited capacity and maximum proper similarity per cluster. Each cluster is composed of individuals from whom we can compute a centre value and hence, determine a similarity measure. The clusters must cover the demands of their individuals. This problem can be applied to the design of garbage collection zones, defining salesmen areas, etc. In this work, we present two variations (\(p\)-CCCP and Generic CCCP) of this problem and their mathematical programming formulations. We first focus our attention on the Generic CCCP, and then we create a new procedure for \(p\)-CCCP. These problems being NP-HARD, we propose a two-phase polynomial heuristic algorithm. The first phase is a constructive phase for which we will propose two variants: the first technique uses known cluster procedures oriented by a log-polynomial geometric tree search, the other one uses unconstrained to constrained clustering. The second phase is a refinement of the variable neighbourhood search (VNS). We also show the results we have obtained for tests from the CCP literature, the design of garbage collection zones, and salesmen areas distribution using the approach implemented for the SISROT\(^\circledR\) system.

In the paper the properties of self-iterating Lie algebras introduced in [\textit{V. M. Petrogradsky}, J. Algebra. 302, No. 2, 881--886 (2006; Zbl 1109.17008)] are studied. Let \(K\) be a field of characteristic \(p>0\) \(R=K[t_i | i\in \mathbb {N}]/(t_i^p, i\in \mathbb{N})\) the truncated polynomial algebra over \(K\), \(v_1, v_2\in Der_KR\) \[ v_1=\partial _1{} +t{}_0(\partial _2{} +t{}_1(\partial _3{} + \ldots )), \] \[ v_2=\partial _2{}+ t{}_1(\partial _3{} _t{}_2(\partial _4{} _ {\ldots} )). \] The restricted Fibonacci algebra \(L=L_p(v_1, v_2)\) is a \(p\)-subalgebra in \(Der_KR\) generated by \(v_1, v_2\). For \(p=2,3,5\) it is proved that \(L\) has the Gelfand-Kirillov dimension equal to \(\ln p/\ln \lambda\) where \(\lambda =\frac{1+\sqrt 5}{2}.\) For the cases \(p=2,3\) the authors show that associative algebra \(A\) generated by \(v_1, v_2\) in \(End_KR\) has the Gelfand-Kirillov dimension \(2\ln p/\ln \lambda .\) It is proved that the restricted enveloping algebra \(u(L)\) is of intermediate growth. It is shown that for \(p=2,3\) \(L_p(v_1,v_2)\) has a nil-\(p\)-mapping, \(L_p(v_1,v_2)\), \(A\) and the augmentation ideal \(u_0(L)\) of \(u(L)\) are direct sums of two locally nilpotent subalgebras. The authors point out that the question whether \(u_0(L)\) is a nil-algebra remains open. The author presents an interesting example of a restricted Lie algebra \(L\) over a field \(K\) of characteristic~2. The algebra \(L\) is generated (as \(p\)-algebra) by two special derivations of the formal power series ring \(K[t_0,t_1,\dots]\). It is proved that \(L\) has a polynomial growth and a nil \(p\)-mapping. The Gelfand-Kirillov dimension of \(L\) is bounded from above by \(\log_\phi2\) where \(\phi\) is the golden ratio.

In the paper the properties of self-iterating Lie algebras introduced in [\textit{V. M. Petrogradsky}, J. Algebra. 302, No. 2, 881--886 (2006; Zbl 1109.17008)] are studied. Let \(K\) be a field of characteristic \(p>0\) \(R=K[t_i | i\in \mathbb {N}]/(t_i^p, i\in \mathbb{N})\) the truncated polynomial algebra over \(K\), \(v_1, v_2\in Der_KR\) \[ v_1=\partial _1{} +t{}_0(\partial _2{} +t{}_1(\partial _3{} + \ldots )), \] \[ v_2=\partial _2{}+ t{}_1(\partial _3{} _t{}_2(\partial _4{} _ {\ldots} )). \] The restricted Fibonacci algebra \(L=L_p(v_1, v_2)\) is a \(p\)-subalgebra in \(Der_KR\) generated by \(v_1, v_2\). For \(p=2,3,5\) it is proved that \(L\) has the Gelfand-Kirillov dimension equal to \(\ln p/\ln \lambda\) where \(\lambda =\frac{1+\sqrt 5}{2}.\) For the cases \(p=2,3\) the authors show that associative algebra \(A\) generated by \(v_1, v_2\) in \(End_KR\) has the Gelfand-Kirillov dimension \(2\ln p/\ln \lambda .\) It is proved that the restricted enveloping algebra \(u(L)\) is of intermediate growth. It is shown that for \(p=2,3\) \(L_p(v_1,v_2)\) has a nil-\(p\)-mapping, \(L_p(v_1,v_2)\), \(A\) and the augmentation ideal \(u_0(L)\) of \(u(L)\) are direct sums of two locally nilpotent subalgebras. The authors point out that the question whether \(u_0(L)\) is a nil-algebra remains open. This is a brief introduction to the theory of Enriques surfaces over arbitrary algebraically closed fields.

As part of the programme of understanding the classification of finite simple groups geometrically, one wishes to classify all flag-transitive \(P\)-geometries, that is geometries whose rank 2 part consists of the vertices and edges of the Petersen graph [see \textit{A. A. Ivanov} and the author, Eur. J. Comb. 10, No. 4, 347-361 (1989; Zbl 0709.20014)]. All known examples of these geometries are related to sporadic simple groups and it is hoped to classify the geometries as just these examples. Here it is shown that the universal closures of the amalgams \({\mathcal A}(Co_ 2)\) and \(\hat{\mathcal A}(Co_ 2)\) are isomorphic to \(Co_ 2\) and \(3^{23}\cdot Co_ 2\). The modulo involved in the latter extension is the 23-dimensional irreducible submodule of \(\Lambda/3\Lambda\), where \(\Lambda\) is the Leech lattice. The paper deals with a very interesting subject, namely with geometries of sporadic groups. The declared goal of this activity is construction of such a geometry which would be similar to the famous theory of buildings for the finite simple groups of Lie type. The paper under review contains a uniform construction of a class of geometries including geometries for five sporadic simple groups: the Mathieu groups \(M_{22}\) and \(M_{23}\), the Conway group \(Co_ 2\), the Janko group \(J_ 4\) and the Baby Monster \(F_ 2.\) A few words about the methods used in this paper. The geometries for the Mathieu groups \(M_{22}\) and \(M_{23}\) are constructed in purely geometric terms, more precisely in terms of the Steiner system S(5,8,24). For the larger groups an approach relying on the notion of representation of a geometry plays a crucial role. The \(Co_ 2\)-geometry is constructed in its ordinary representation of the Leech lattice \(\Lambda\) modulo \(2\Lambda\). The geometries for the groups \(J_ 4\) and \(F_ 2\) are constructed in ``nonabelian'' representations inside themselves. In addition the description problem for the universal representations of the constructed geometries is posed. This problem is solved for the examples related to the Mathieu groups. It should be mentioned that this paper is the second one in a series of papers [part I: the authors, Commun. Algebra 16, No.5, 925-953 (1988; Zbl 0639.20011)]. The aim of this series is to characterize the constructed geometries in terms of their diagrams. The authors have informed that they are close to the complete solution of the problem.

As part of the programme of understanding the classification of finite simple groups geometrically, one wishes to classify all flag-transitive \(P\)-geometries, that is geometries whose rank 2 part consists of the vertices and edges of the Petersen graph [see \textit{A. A. Ivanov} and the author, Eur. J. Comb. 10, No. 4, 347-361 (1989; Zbl 0709.20014)]. All known examples of these geometries are related to sporadic simple groups and it is hoped to classify the geometries as just these examples. Here it is shown that the universal closures of the amalgams \({\mathcal A}(Co_ 2)\) and \(\hat{\mathcal A}(Co_ 2)\) are isomorphic to \(Co_ 2\) and \(3^{23}\cdot Co_ 2\). The modulo involved in the latter extension is the 23-dimensional irreducible submodule of \(\Lambda/3\Lambda\), where \(\Lambda\) is the Leech lattice. The author explores a nontraditional approach to examining the problem of forecast uncertainty in extrapolative demographic models. It builds on prior research on stochastic time series forecast models, but diverges to examine their deterministic counterparts. The focus here is an examination of the structural integrity of the Lee-Carter method [\textit{L. R. Carter} and \textit{R. D. Lee} ``Modeling and Forecasting U.S. Sex Differentials in Mortality'', Int. Forecasting 8, 393-411 (1992)] applied to mortality forecasts. He investigates the nonlinear dynamics of the Lee-Carter method, particularly its sensitive dependence of the forecasts on the initial conditions of the model. He examines the Lee-Carter nonlinear demographic model, \(m_{x,t}= \exp(a_x+ b_xk_t+ e_{x,t})\), which is decomposed using SVD to derive a single time-varying linear index of mortality, \(k_t\). From a 90 year time series of \(k_t\), forty nine 40 year realizations are sampled. These realizations are modeled and estimated using Box-Jenkins techniques. The estimated parameters of these realizations and the first case of each of the samples are the initial conditions for the iterations of nonlinearized transformation of \(k_t\) to \(\exp (k_t)\). The terminal year for each of the 49 iterated series is 2065. The deterministic nonlinear dynamics of this system of 49 iterated series is investigated by testing its Lyapunov exponents as a nonparametric diagnostic of a one dimensional dynamical system. The exponents are all negative, indicating that chaos is not prevalent in this system. The nonexistencc of chaos suggests stability in the model and reaffirms the predictability of this one-dimensional map. Augmenting the iterations of the initial conditions with additive stochastic innovations, \(\{e_t, t>1\}\), creates a stochastic dynamical system of the form, \(k_t=k_{t-1} -c+\varphi flu+e_t\). Here, \(e_t\) is treated as a surrogate for some unanticipated time series event (e.g. an epidemic) that impacts the deterministic map. Gaussian white noise innovations do not move the iterations far from equilibrium and only for short time intervals. So, stepping the mean of the innovations by .01 produces stable Lyapunov exponents until the mean equnis .35 where some of the exponents are positive. At this point, deterministic chaos is evident, implying instability in the forecasts. The substantive implications of this instability are discussed.

Recall some definitions: 1. \textit{Hirzebruch surface} is a projective bundle of rank \(1\) over the projective line, i.e. one may take \(F_e:={\mathbb P}({\mathcal O}_{{\mathbb P}^1} \oplus {\mathcal O}_{{\mathbb P}^1}(-e))\) with \(e \geq 0\). 2. \textit{prioritary sheaf} (defined by \textit{A. Hirschowitz} and \textit{Y. Laszlo} [Math. Ann. 297, No. 1, 85--102 (1993; Zbl 0808.14007)]) on \({\mathbb P}^2\), is a coherent sheaf \(E\) on \({\mathbb P}^2\), torsionfree and with the property \(\text{Ext}^2(E,E(-1))=0\). This definition was extended naturally to sheaves on birationally ruled surfaces by Ch. Walter, who proved that the stack of prioritary torsion-free sheaves with fixed rank (\(\geq 2\)) and Chern classes \(c_1\), \(c_2\) is smooth, irreducible. 3. \textit{omalous bundles} (introduced by Donagi) on a complex variety (projective, smooth) are bundles with the determinant isomorphic to the anticanonical bundle and \newline \(c_2=c_2(\text{Tangent bundle})\). In this paper one proves ``unirationality of the stack of prioritary omalous bundles on Hirzebruch surfaces''. The main tool is the Buchdahl variant of the Beilinson spectral sequences, starting from the observation that the diagonal \(\Delta \subset X\times X\) for a Hirzebruch surface \(X\) is the zero locus of a global section in a rank \(2\) vector bundle on \(X \times X\). The main result of this paper is a new proof of the irreducibility of the moduli spaces of stable vector bundles over the projective plane. Until now this study was done in four separate cases: (1) \(c_ 1 = 0\), \(\text{rk} = 2\) (W. Barth (1977)); (2) \(c_ 1 = - 1\), \(\text{rk} = 2\) (K. Hulek (1979) und G. Ellingsrud, S. A. Stromme (1979)); (3) \(c_ 1 = 0\), \(\text{rk} \geq 3\) (K. Hulek (1980)), (4) the remaining cases (G. Ellingsrud (1983)). Here the irreducibility follows from the general result that ``le champ'' (in the sense of G. Laumon, L. Moret-Baily) of ``prioritaire'' sheaves with fixed Hilbert polynomial is irreducible.

Recall some definitions: 1. \textit{Hirzebruch surface} is a projective bundle of rank \(1\) over the projective line, i.e. one may take \(F_e:={\mathbb P}({\mathcal O}_{{\mathbb P}^1} \oplus {\mathcal O}_{{\mathbb P}^1}(-e))\) with \(e \geq 0\). 2. \textit{prioritary sheaf} (defined by \textit{A. Hirschowitz} and \textit{Y. Laszlo} [Math. Ann. 297, No. 1, 85--102 (1993; Zbl 0808.14007)]) on \({\mathbb P}^2\), is a coherent sheaf \(E\) on \({\mathbb P}^2\), torsionfree and with the property \(\text{Ext}^2(E,E(-1))=0\). This definition was extended naturally to sheaves on birationally ruled surfaces by Ch. Walter, who proved that the stack of prioritary torsion-free sheaves with fixed rank (\(\geq 2\)) and Chern classes \(c_1\), \(c_2\) is smooth, irreducible. 3. \textit{omalous bundles} (introduced by Donagi) on a complex variety (projective, smooth) are bundles with the determinant isomorphic to the anticanonical bundle and \newline \(c_2=c_2(\text{Tangent bundle})\). In this paper one proves ``unirationality of the stack of prioritary omalous bundles on Hirzebruch surfaces''. The main tool is the Buchdahl variant of the Beilinson spectral sequences, starting from the observation that the diagonal \(\Delta \subset X\times X\) for a Hirzebruch surface \(X\) is the zero locus of a global section in a rank \(2\) vector bundle on \(X \times X\). Codes with decomposable structure allow the use of multistage decoding procedures to achieve suboptimum bounded-distance error performance with reduced decoding complexity. This correspondence presents some new decomposable codes, including a class of distance-8 codes, that are constructed based on the \(|a+x|b+x|a+b+x|\) construction method. Some existing best codes are shown to be decomposable and hence can be decoded with multistage decoding.

Some Marcinkiewicz-Zygmund-Burkholder-Gundy inequalities are obtained for stopped random walks. [A further reference is the reviewer's book, Stopped random walks. Limit theorems and applications. (1988; Zbl 0634.60061), in particular Section I.5.] This clearly written book, useful for researcher and student and containing the author's research results of recent years, deals with the following part of random walk theory. Let \(X_ 1,X_ 2,..\). be i.i.d. random variables, \(S_ 0=0\), \(S_ n=S(n)=X_ 1+...+X_ n\), \(n\geq 1\), and let \(N_ t\), \(t\geq 0\), be positive integer random variables with \(N_ t\to \infty\) as \(t\to \infty\). Mostly, but not always, the \(N_ t\) are stopping times for the \(S_ n\)-process. The central part of the book is the study of limit theorems for \(S(N_ t)\) as \(t\to \infty:\) laws of large numbers and central limit theorems with convergence of moments, laws of iterated logarithm and uniform integrability. These theorems are applied in proving other limit theorems, e.g. in renewal theory on \({\mathbb{R}}\) and for \(M_ n=\max (S_ 0,...,S_ n).\) Chapter I gives general theorems on convergence of random variables \(Y(N_ t)\) with random index \(N_ t\), the Anscombe-Rényi theorem, convergence of moments in strong law and central limit theorem with deterministic and random index and existence of moments and uniform integrability of \(S(N_ t)\). Chapter II gives the fundamentals of renewal theory with \(X_ i\geq 0\) and of random walk theory. Chapter III is concerned with renewal theory on \({\mathbb{R}}\) for random walks with positive drift. Limit theorems of the types mentioned above are proved for the stopping time \(\nu_ t=\min \{n:\) \(S_ n>t\}\) and, where they make sense, for S(\(\nu\) (t)). The asymptotics of the renewal function and of \(S(\nu_ t)-t\), and their refinements, are dealt with. An important part here is played by ladder variable techniques. Some subjects of Chapter IV are: limit theorems for \(V(\tau_ t)\) where (U(n),V(n)) is a random walk in \({\mathbb{R}}^ 2 \)with E U(1)\(>0\) and \(\tau_ t=\min \{n:\) \(U(n)>t\}\), applications, limit theorems for \(M_ n\) by representing \(M_ n\) as a random sum of ladder steps, and first passage across general boundaries. Chapter V extends part of the previous results to functional limit theorems.

Some Marcinkiewicz-Zygmund-Burkholder-Gundy inequalities are obtained for stopped random walks. [A further reference is the reviewer's book, Stopped random walks. Limit theorems and applications. (1988; Zbl 0634.60061), in particular Section I.5.] A new method for the construction of a partial order on the set of multicriteria alter natives is presented. This method belongs to the family of verbal decision analysis methods and gives a more efficient means of problem solution. The method is based on psychologically valid operations for information elicitation from a decision maker: comparisons of two distances between the evaluations on the ordinal scales of two criteria. The information received from a decision maker is used for the construction of a binary relation between a pair of alternatives which yields s preference, indifference and incomparability relations. The notion of a method decisive power is introduced. The illustrative example is given.

This note is devoted to the topological classification problem and to the study of the itineraries for Lorenz maps on the interval. Extending results by \textit{J. H. Hubbard and C. T. Sparrow} [Commun. Pure Appl. Math. 43, 431-444 (1990; Zbl 0714.58041] the author first studies in detail Lorenz maps \(f\) with one discontinuity point \(c\), where \(\lim_{x\to c^-}f(x)=f(c^-)=1\), and \(\lim_{x\to c^+}f(x)=f(c^+)=0\), by introducing the associated kneading sequences. The results are then extended to the general case for Lorenz maps on the interval. The authors study the dynamics of Lorenz maps which are topologically expansive when regarded as maps on the cut up circle. These are maps \(f:I\to I\), \(I=[0,1]\) satisfying: (i) there exists \(c\in (0,1)\) such that f is continuous and strictly increasing on [0,c) and on (c,1], (ii) \(\lim_{x\uparrow c}f(x)=1\), \(\lim_{x\downarrow c}f(x)=0\), (iii) there exists \(\epsilon >0\) such that if \((x_ 0,x_ 1,...)\) and \((y_ 0,y_ 1,...)\) are distinct forward orbits then \(| x_ i-y_ i| \geq \epsilon\) for some i. Given a topologically expansive Lorenz map f, they define the kneading sequence \(k_ f(x)\) to be the sequence \(\epsilon_ 0,\epsilon_ 1,..\). of 0's and 1's with \(\epsilon_ i=0\) if \(f^ i(x)<c\) and \(\epsilon_ i=1\) if \(f^ i(x)>c\) and by appropriate limits if x is a preimage of c. The kneading invariant is defined by \(K(f)=(k_ f(0),k_ f(1))\). They prove that these maps are determined up to topological conjugacy by their kneading invariants and observe that this conjugacy is unique. This significantly extends the class of Lorenz maps for which this was previously known. Helpful diagrams illustrate the ideas. The authors also determine when a pair of sequences can be the kneading invariant for some topologically expansive Lorenz map. The results extend in a natural way to maps having multiple discontinuities.

This note is devoted to the topological classification problem and to the study of the itineraries for Lorenz maps on the interval. Extending results by \textit{J. H. Hubbard and C. T. Sparrow} [Commun. Pure Appl. Math. 43, 431-444 (1990; Zbl 0714.58041] the author first studies in detail Lorenz maps \(f\) with one discontinuity point \(c\), where \(\lim_{x\to c^-}f(x)=f(c^-)=1\), and \(\lim_{x\to c^+}f(x)=f(c^+)=0\), by introducing the associated kneading sequences. The results are then extended to the general case for Lorenz maps on the interval. Recent research reveals that pro-active real-time routing approaches that use stochastic knowledge about future requests can significantly improve solution quality compared to approaches that simply integrate new requests upon arrival. Many of these approaches assume that request arrivals on different days follow an identical pattern. Thus, they define and apply a single profile of past request days to anticipate future request arrivals. In many real-world applications, however, different days may follow different patterns. Moreover, the pattern of the current day may not be known beforehand, and may need to be identified in real-time during the day. In such cases, applying approaches that use a single profile is not promising. In this paper, we propose a new pro-active real-time routing approach that applies multiple profiles. These profiles are generated by grouping together days with a similar pattern of request arrivals. For each combination of identified profiles, stochastic knowledge about future request arrivals is derived in an offline step. During the day, the approach repeatedly evaluates characteristics of request arrivals and selects a suitable combination of profiles. The performance of the new approach is evaluated in computational experiments in direct comparison with a previous approach that applies only a single profile. Computational results show that the proposed approach significantly outperforms the previous one. We analyze further potential for improvement by comparing the approach with an omniscient variant that knows the actual pattern in advance. Based on the results, managerial implications that allow for a practical application of the new approach are provided.

Quasi-quadrics are introduced in [\textit{F. De Clerck, N. Hamilton, C. M. O'Keef} and \textit{T. Penttila}, Australas. J. Comb. 22, 151--166 (2000; Zbl 0970.51011)]. A set of points \(\mathcal H\) in \(PG(n,q^2)\) is called quasi-Hermitian variety, if it has the same intersection numbers with respect to hyperplanes as a non-singular Hermitian variety \(H(n-1,q^2)\) or as a cone with vertex a point and base a non-singular Hermitian variety \(H(n-2,q^2)\). The authors exhibit two constructions of quasi-Hermitian varieties which are not Hermitian varieties. A singular quasi-quadric is a point set \(\mathcal K\) in \(PG(n,q)\) which has the same numbers of points as a singular quadric \(Q\) and for which each intersection number with respect to hyperplanes is also an intersection number of \(Q\) with respect to hyperplanes. The authors prove the following theorem: Let \(\mathcal K\) be a set of \(q^3+q+1\) points in \(PG(4,q)\) such that every hyperplane intersects \(\mathcal K\) in \(q+1\), \(q^2+1\), or \(q^2+q+1\) points and designate a solid which intersects \(\mathcal K\) in \(q+1\) points as \textit{\((q+1)\)-solid}. If every \((q+1)\)-solid contains a line which meets \(\mathcal K\) in at least \(q\) points and if there are at least three \((q+1)\)-solids which intersect \(\mathcal K\) in a full line, then the set \(\mathcal K\) is a cone with vertex a point and base an ovoid. In a projective space \(\text{PG}(n,q)\), a quasi-quadric is a set of points that has the same intersection numbers with respect to hyperplanes as a non-degenerate quadric in that space. The importance of these quasi-quadrics lies in the fact that: (1) when \(n\) is odd, quasi-quadrics have two sizes of intersection with hyperplanes and these sets give rise to strongly regular graphs, two-weight codes, difference sets, SDP-designs, Reed-Muller codes and bent functions, and (2) when \(n\) is even, quasi-quadrics have three sizes of intersection with respect to hyperplanes, and certain sets of this type may be used to construct antipodal distance regular covers of complete graphs. The article collects many of the known results on quasi-quadrics and gives new geometric construction methods and theorems. The new construction methods, called pivotting, consist of replacing a subset of a quadric by an other set, leading to quasi-quadrics different from the original quadric.

Quasi-quadrics are introduced in [\textit{F. De Clerck, N. Hamilton, C. M. O'Keef} and \textit{T. Penttila}, Australas. J. Comb. 22, 151--166 (2000; Zbl 0970.51011)]. A set of points \(\mathcal H\) in \(PG(n,q^2)\) is called quasi-Hermitian variety, if it has the same intersection numbers with respect to hyperplanes as a non-singular Hermitian variety \(H(n-1,q^2)\) or as a cone with vertex a point and base a non-singular Hermitian variety \(H(n-2,q^2)\). The authors exhibit two constructions of quasi-Hermitian varieties which are not Hermitian varieties. A singular quasi-quadric is a point set \(\mathcal K\) in \(PG(n,q)\) which has the same numbers of points as a singular quadric \(Q\) and for which each intersection number with respect to hyperplanes is also an intersection number of \(Q\) with respect to hyperplanes. The authors prove the following theorem: Let \(\mathcal K\) be a set of \(q^3+q+1\) points in \(PG(4,q)\) such that every hyperplane intersects \(\mathcal K\) in \(q+1\), \(q^2+1\), or \(q^2+q+1\) points and designate a solid which intersects \(\mathcal K\) in \(q+1\) points as \textit{\((q+1)\)-solid}. If every \((q+1)\)-solid contains a line which meets \(\mathcal K\) in at least \(q\) points and if there are at least three \((q+1)\)-solids which intersect \(\mathcal K\) in a full line, then the set \(\mathcal K\) is a cone with vertex a point and base an ovoid. The Thurston-Bennequin invariant provides one notion of self-linking for any homologically trivial Legendrian curve in a contact three-manifold. Here we discuss related analytic notions of self-linking for Legendrian knots in \(\mathbb{R}^3\). Our definition is based upon reformulation of the elementary Gauss linking integral and is motivated by ideas from supersymmetric gauge theory. We recover the Thurston-Bennequin invariant as a special case.{ \copyright 2018 American Institute of Physics}

This is the fourth of a series of papers by the authors devoted to the study of the stability of general molecular systems in Thomas-Fermi or Hartree type models. In the preceding parts [for part III, cf. ibid., No. 3-4, 381-429 (1993; Zbl 0797.46053)] the authors proved that neutral molecules are stable if and only if one can bind complementing neutral subsystems which admit a full ground-state. This condition holds for Thomas-Fermi theories. But for the Hartree model it is restricted to a planar nuclear configuration. In this paper, the authors use an improved technique to extend the result to arbitrary neutral systems. The price they have to pay is the loss of precise information of the binding energy. Indeed, they prove that the interaction force between two neutral subsystems is attractive in some region, not necessarily at infinity. The paper depends heavily on the authors' previous papers. This work is the third of a series devoted to the study of the binding of atoms, molecules and ions and the stability of general molecular systems in the context of Hartree and Thomas-Fermi type theories. For part II, see the review above. It is proved, that neutral systems can be bound in the framework of Thomas-Fermi-von Weizsäcker and Thomas-Fermi-Dirac-von Weizäcker models and the same result is obtained for the Hartree model in the case of the two systems to bind are planar. Some estimates of the binding force are presented. The influence of the action of smearing the nucleii on the results is discussed. The systems with a slight positive excess charge are treated.

This is the fourth of a series of papers by the authors devoted to the study of the stability of general molecular systems in Thomas-Fermi or Hartree type models. In the preceding parts [for part III, cf. ibid., No. 3-4, 381-429 (1993; Zbl 0797.46053)] the authors proved that neutral molecules are stable if and only if one can bind complementing neutral subsystems which admit a full ground-state. This condition holds for Thomas-Fermi theories. But for the Hartree model it is restricted to a planar nuclear configuration. In this paper, the authors use an improved technique to extend the result to arbitrary neutral systems. The price they have to pay is the loss of precise information of the binding energy. Indeed, they prove that the interaction force between two neutral subsystems is attractive in some region, not necessarily at infinity. The paper depends heavily on the authors' previous papers. No review copy delivered.

The motion of vortex sheets with surface tension in three-dimensional Euler equations with vorticity is analyzed. In the paper, following the methodology of \textit{D. Coutand} and \textit{S. Shkoller} [J. Am. Math. Soc. 20, No. 3, 829--930 (2007; Zbl 1123.35038)], the well-posedness for short time of this problem is proved. The incompressible Euler equations for the motion of two fluids can be written as: \(\rho^{\pm}(u^{\pm}_t+\nabla_{u^{\pm}}u^{\pm})+ \nabla p^{\pm}=0\), in \(\Omega^{\pm}(t)\); \(\text{div}u^{\pm}=0\), in \(\Omega^{\pm}(t)\); \([p]_{\pm}=\sigma H\), on \(\Gamma(t)\); \([u\cdot n]_{\pm}=0\), on \(\Gamma(t)\); \(u^{-}\cdot n=0\), on \(\partial \Omega\); \(\Omega^{\pm}(0)=\Omega^{\pm}\), on \(\{t=0\}\); \(u^{\pm} (0)=u^{\pm}_0\), on \(\{t=0\}\times\Omega^{\pm}\), where the material interface \(\Gamma(t)\) moves with speed \(u^{+}(t)\cdot n(t)\), \(\rho^{+}\) and \(\rho^{-}\) are the densities of the two fluids occupying \(\Omega^{+}(t)\) and \(\Omega^{-}(t)\), respectively, \(H(t)\) is twice the mean curvature of \(\Gamma(t)\), \(\sigma>0\) is the surface tension parameter, and \(n(t)\) is outward unit normal on \(\partial\Omega^{+}(t)\). In Theorem 1.1 it is proved that under some assumptions on \(\Gamma(0)\), \(\partial\Omega\), and \(u_0\) there exists \(T>0\) and a solution of the above problem. This solution is unique. The paper deals with local existence and uniqueness of solutions in Sobolev spaces to free boundary incompressible Euler equations in vacuum: \[ \partial_t+\nabla_uu+\nabla p=0,\qquad \text{div}\;u=0\quad \text{in } Q, \] \[ p=\sigma H\text{ on}\;\partial Q,\quad \left(\partial_t+\nabla_u\right)|_{\partial Q}\in T(\partial Q),\quad u=u_0\text{ at }t=0, \quad Q(0)=\Omega, \] where \(Q(t)=\bigcup_{0\leq t\leq T} \{t\}\times\Omega(t)\), \(\Omega(t)\in \mathbb R_n\), \(n=2\) or 3, \(\partial Q(t)=\bigcup_{0\leq t\leq T} \{t\}\times\partial\Omega(t)\), \(\nabla_uu=u^j\partial u^i/\partial x_j\), the vector field \(u\) is the Eulerian or spatial velocity field defined on the time-dependent domain \(\Omega(t)\), \(p\) denotes the pressure function, \(H\) is twice the mean curvature of the boundary of the fluid \(\partial\Omega(t)\), and \(\sigma\) is the surface tension. The two following main theorems are proved. Theorem 1.1. Let \(\sigma>0\), \(\partial\Omega\) be of class \(H^{5.5}\), and \(u_0\in H^{4.5}(\Omega)\). Then, there exists \(T>0\) and a solution \(u(t)\), \(p(t)\), \(\Omega(t)\) of the problem with \(u\in L^\infty(0,T;H^{4.5}(\Omega(t)))\), \(p\in L^\infty(0,T;H^4(\Omega(t)))\). The solution is unique if \(u_0\in H^{5.5}(\Omega)\) and \(\partial\Omega\in H^{6.5}\). Theorem 1.2. Let \(\sigma=0\), \(\partial\Omega\) be of class \(H^3\), and \(u_0\in H^3(\Omega)\) and condition \(\nabla p\cdot n<0\) on \(\partial Q\) holds at \(t=0\). Then, there exists \(T>0\) and a unique solution \(u(t)\), \(p(t)\), \(\Omega(t)\) of the problem with \(u\in L^\infty(0,T;H^3(\Omega(t)))\), \(p\in L^\infty(0,T;H^{3.5}(\Omega(t)))\), and \(\partial\Omega(t)\in H^3\). In the proof, the Eulerian problem, set on the moving domain \(\Omega(t)\), is converted to a system on the fixed domain \(\Omega\), by using Lagrangian variables.

The motion of vortex sheets with surface tension in three-dimensional Euler equations with vorticity is analyzed. In the paper, following the methodology of \textit{D. Coutand} and \textit{S. Shkoller} [J. Am. Math. Soc. 20, No. 3, 829--930 (2007; Zbl 1123.35038)], the well-posedness for short time of this problem is proved. The incompressible Euler equations for the motion of two fluids can be written as: \(\rho^{\pm}(u^{\pm}_t+\nabla_{u^{\pm}}u^{\pm})+ \nabla p^{\pm}=0\), in \(\Omega^{\pm}(t)\); \(\text{div}u^{\pm}=0\), in \(\Omega^{\pm}(t)\); \([p]_{\pm}=\sigma H\), on \(\Gamma(t)\); \([u\cdot n]_{\pm}=0\), on \(\Gamma(t)\); \(u^{-}\cdot n=0\), on \(\partial \Omega\); \(\Omega^{\pm}(0)=\Omega^{\pm}\), on \(\{t=0\}\); \(u^{\pm} (0)=u^{\pm}_0\), on \(\{t=0\}\times\Omega^{\pm}\), where the material interface \(\Gamma(t)\) moves with speed \(u^{+}(t)\cdot n(t)\), \(\rho^{+}\) and \(\rho^{-}\) are the densities of the two fluids occupying \(\Omega^{+}(t)\) and \(\Omega^{-}(t)\), respectively, \(H(t)\) is twice the mean curvature of \(\Gamma(t)\), \(\sigma>0\) is the surface tension parameter, and \(n(t)\) is outward unit normal on \(\partial\Omega^{+}(t)\). In Theorem 1.1 it is proved that under some assumptions on \(\Gamma(0)\), \(\partial\Omega\), and \(u_0\) there exists \(T>0\) and a solution of the above problem. This solution is unique. We investigate finite groups \(G\) that possess a 2-cocycle for which all the complex irreducible projective representations have the same degree. We show that, provided the number of such inequivalent irreducible projective representations is relatively small compared to the order of \(G\), then the degree equality property is inherited by all the Sylow subgroups of \(G\).

My colleagues and I once created a mathematical model to explore the communication behavior of a mesh based network [\textit{Z.-S. Shen}, \textit{P. Drexel} and \textit{L. Urbach}, Math. Comput. Modelling 18, No. 12, 33-48 (1993; Zbl 0805.68013)], by using such mathematical and logic notations as first-order logic, linear algebraic, probabilistic, and various combinatorial expressions. Since these declarative statements can't be executed directly, later on, we had to ask a student to convert this mathematical model into an executable program in \(C\). After these many years, I still can remember the pain it took the student to understand what I meant by those ``weird'' expressions and the effort it took him to do the conversion. Apart from the quality of such a model, I wish at that time there were a better way for its construction and execution. Moreover, although I am not a professional modeler, I certainly hope to have some general understanding of the whole modeling process. Now, when I am reading this wonderful book by Hürlimann, I definitely wish it were available when I wrote that model. It has to be pointed out first that this book is not intended to hand over any magic solution to the users to build up their models. It is rather intended to help the modeling tool designers understand the whole modeling process better in order to develop more user friendly tools for the modeling practitioners. It is like a book written for the car designers so that we can have better cars to drive. As a consequence, it shows what could be a modeling language, and why, but not how to develop a model for a specific problem. This book consists of three parts: General concepts (Chapters 1-4), a discussion on modeling management systems (Chapters 5-7), and a proposal for such a system (Chapters 8-10). More specifically, after an introductory chapter, Chapter 2 defines the notion of modeling, particularly, mathematical modeling, and some of the related concepts. Many examples, old and new, simple and complicated, artistic and scientific, as well as some theoretical ``stuff'', are used to demonstrate and justify both the ends and the means of the modeling process, referred to as our ``power of abstraction''. Chapter 3 presents a modeling life cycle, beginning at specification, and ending at report writing, that summarizes various problems a modeler has to tackle. Chapter 4 contains an overview of different types and paradigms of mathematical models, such as analytical vs. simulation, linear vs. nonlinear, symbolic vs. numerical, etc.. It ends with a discussion of modeling uncertainty, by presenting various techniques and their inclusion in the respective model building process, which are well exemplified. Chapter 5 argues for, discusses the status quo of, and explores many issues related to, a general, uniform, modeling management system, i.e., a set of tools ``to provide computer-based assistance to the decision maker for creating, solving, manipulating, and integrating a variety of models'' [\textit{B. Shetty}, \textit{H. K. Bharagava} and \textit{R. Krishnan} (eds.), Model management in operations research, Annals of Operations Research 38, 1-4 (1992), Preface]. Chapter 6 presents a critical review of several frequently used approaches for modeling, such as spreadsheet, database, graphic modeling, CLP, algebraic languages, as well as some of the proposed general modeling frameworks, including a structured modeling, an embedded language approach, and a multi-view architecture. Chapter 7, ``the core chapter'', presents the author's own approach to the development of such a system, the LPL language [\((*)\)] (currently known as Logical Programming Language), with its implementation being discussed in the following Chapters 8 and 9. This language allows the modeler to specify the structure of the problem declaratively, while being able to express a solution algorithm procedurally. It also comes with an environment to support various steps of the aforementioned modeling life cycle. In the concluding Chapter 10, more than a dozen problems taken from various application areas are discussed, and the corresponding LPL codes given. This book ends up with almost twelve pages of references. I find the attached LPL WEB site [\((*)\)] [\url{http://diufpc03.unifr.ch/lpl/lpl.html}] not fancy, but quite useful. It providesa collection of LPL related papers and technical reports, including manuals, besides a free package of the LPL system. It also has an interactive feature. You can simply type in the LPL model for your problem, a built-in model solver will solve it, and display the results. Besides being rigorous, this book is quite readable. It is an ``essay'' after all. I definitely agree that besides stimulating modeling tool designers, this book can also be used as a textbook in a mathematical modeling course, perhaps accompanied with a LPL manual. Moreover, this book, particularly, the first part, maybe the second as well, is very informative, and should be interesting to more general audience. For example, now I know where to look for real life examples when I teach the modeling concepts in the beginning computer science courses. We define and implement a mathematical model for a general 2-d mesh system, which is arrays of processors with a bounded mesh architecture. As one of the simplest distributed architecture with fixed-connection, the 2-d mesh system has found many applications in computer sciences and engineering, particularly in computer communication. We use mathematical structures to characterize the mesh system and use \(C\) to have implemented an executable version of this model. In this paper, we will present the mathematical model itself, discuss some corresponding implementation issues and compare its behaviors with a simulator which we have been using to observe system behaviors.

My colleagues and I once created a mathematical model to explore the communication behavior of a mesh based network [\textit{Z.-S. Shen}, \textit{P. Drexel} and \textit{L. Urbach}, Math. Comput. Modelling 18, No. 12, 33-48 (1993; Zbl 0805.68013)], by using such mathematical and logic notations as first-order logic, linear algebraic, probabilistic, and various combinatorial expressions. Since these declarative statements can't be executed directly, later on, we had to ask a student to convert this mathematical model into an executable program in \(C\). After these many years, I still can remember the pain it took the student to understand what I meant by those ``weird'' expressions and the effort it took him to do the conversion. Apart from the quality of such a model, I wish at that time there were a better way for its construction and execution. Moreover, although I am not a professional modeler, I certainly hope to have some general understanding of the whole modeling process. Now, when I am reading this wonderful book by Hürlimann, I definitely wish it were available when I wrote that model. It has to be pointed out first that this book is not intended to hand over any magic solution to the users to build up their models. It is rather intended to help the modeling tool designers understand the whole modeling process better in order to develop more user friendly tools for the modeling practitioners. It is like a book written for the car designers so that we can have better cars to drive. As a consequence, it shows what could be a modeling language, and why, but not how to develop a model for a specific problem. This book consists of three parts: General concepts (Chapters 1-4), a discussion on modeling management systems (Chapters 5-7), and a proposal for such a system (Chapters 8-10). More specifically, after an introductory chapter, Chapter 2 defines the notion of modeling, particularly, mathematical modeling, and some of the related concepts. Many examples, old and new, simple and complicated, artistic and scientific, as well as some theoretical ``stuff'', are used to demonstrate and justify both the ends and the means of the modeling process, referred to as our ``power of abstraction''. Chapter 3 presents a modeling life cycle, beginning at specification, and ending at report writing, that summarizes various problems a modeler has to tackle. Chapter 4 contains an overview of different types and paradigms of mathematical models, such as analytical vs. simulation, linear vs. nonlinear, symbolic vs. numerical, etc.. It ends with a discussion of modeling uncertainty, by presenting various techniques and their inclusion in the respective model building process, which are well exemplified. Chapter 5 argues for, discusses the status quo of, and explores many issues related to, a general, uniform, modeling management system, i.e., a set of tools ``to provide computer-based assistance to the decision maker for creating, solving, manipulating, and integrating a variety of models'' [\textit{B. Shetty}, \textit{H. K. Bharagava} and \textit{R. Krishnan} (eds.), Model management in operations research, Annals of Operations Research 38, 1-4 (1992), Preface]. Chapter 6 presents a critical review of several frequently used approaches for modeling, such as spreadsheet, database, graphic modeling, CLP, algebraic languages, as well as some of the proposed general modeling frameworks, including a structured modeling, an embedded language approach, and a multi-view architecture. Chapter 7, ``the core chapter'', presents the author's own approach to the development of such a system, the LPL language [\((*)\)] (currently known as Logical Programming Language), with its implementation being discussed in the following Chapters 8 and 9. This language allows the modeler to specify the structure of the problem declaratively, while being able to express a solution algorithm procedurally. It also comes with an environment to support various steps of the aforementioned modeling life cycle. In the concluding Chapter 10, more than a dozen problems taken from various application areas are discussed, and the corresponding LPL codes given. This book ends up with almost twelve pages of references. I find the attached LPL WEB site [\((*)\)] [\url{http://diufpc03.unifr.ch/lpl/lpl.html}] not fancy, but quite useful. It providesa collection of LPL related papers and technical reports, including manuals, besides a free package of the LPL system. It also has an interactive feature. You can simply type in the LPL model for your problem, a built-in model solver will solve it, and display the results. Besides being rigorous, this book is quite readable. It is an ``essay'' after all. I definitely agree that besides stimulating modeling tool designers, this book can also be used as a textbook in a mathematical modeling course, perhaps accompanied with a LPL manual. Moreover, this book, particularly, the first part, maybe the second as well, is very informative, and should be interesting to more general audience. For example, now I know where to look for real life examples when I teach the modeling concepts in the beginning computer science courses. We present a survey of our recent achievements based on an asymptotic approximation method carried out by projection and averaging for initial value problems of infinite-dimensional difference equations with small parameters. Applications are given to delay integro-differential equations and to semilinear Schrödinger equations as well.

Let \(p\) be a prime, let \((K,\mathcal O,k)\) be a splitting \(p\)-modular system for the subgroups of a finite group \(G\), and let \(\mathcal OG\)-mod denote the category of finitely generated left \(\mathcal OG\)-modules. For a \(p\)-subgroup \(P\) of \(G\) and a subgroup \(H\) of \(G\) containing \(N_G(P)\), the Green correspondence can be viewed as an equivalence between a certain quotient category of a certain subcategory of \(\mathcal OG\)-mod and a certain quotient category of a certain subcategory of \(\mathcal OH\)-mod. In the paper under review, the authors investigate the behaviour of this equivalence with respect to the covering of blocks and to the covering of indecomposable modules, in the presence of a normal subgroup \(N\) of \(G\). One of their main results is a generalization of a theorem by \textit{M. E. Harris} and \textit{R. Knörr} [Commun. Algebra 13, 1213-1218 (1985; Zbl 0561.20002)]. The authors give a short and clear proof for the following useful theorem: Let \(N\) be a normal subgroup of \(G\) and \(b\) a block of \(N\) with defect group \(D\). Let \(H=N_ G(D)\) and let \(b_ 1\) be the Brauer correspondent of \(b\) in \(H\cap N\). Then the Brauer correspondence \(\beta \to \beta^ G\) is a defect group preserving one-to-one correspondence between the blocks of \(H\) covering \(b_ 1\) and the blocks of \(G\) covering \(b\). Corollary 4 deals with the case where \(D\) is a Sylow \(p\)-subgroup of \(N\). Then there is a natural one-to-one correspondence between the blocks of \(\bar H=H/O_{p'}(H\cap N)\) and the blocks of \(G\) covering the principal block of \(N\). The authors also remark in the introduction that E. C. Dade has another unpublished proof of their theorem.

Let \(p\) be a prime, let \((K,\mathcal O,k)\) be a splitting \(p\)-modular system for the subgroups of a finite group \(G\), and let \(\mathcal OG\)-mod denote the category of finitely generated left \(\mathcal OG\)-modules. For a \(p\)-subgroup \(P\) of \(G\) and a subgroup \(H\) of \(G\) containing \(N_G(P)\), the Green correspondence can be viewed as an equivalence between a certain quotient category of a certain subcategory of \(\mathcal OG\)-mod and a certain quotient category of a certain subcategory of \(\mathcal OH\)-mod. In the paper under review, the authors investigate the behaviour of this equivalence with respect to the covering of blocks and to the covering of indecomposable modules, in the presence of a normal subgroup \(N\) of \(G\). One of their main results is a generalization of a theorem by \textit{M. E. Harris} and \textit{R. Knörr} [Commun. Algebra 13, 1213-1218 (1985; Zbl 0561.20002)]. Quantified modal logic is notorious for being undecidable, with very few known decidable fragments such as the monodic ones. For instance, even the two-variable fragment over unary predicates is undecidable. In this paper, we study a particular fragment, namely the bundled fragment, where a first-order quantifier is always followed by a modality when occurring in the formula, inspired by the proposal by \textit{J. Wang} [In Proceedings Sixteenth Conference on Theoretical Aspects of Rationality and Knowledge, TARK 2017, Liverpool, UK, 24--26 July 2017, 515--534 (2017) \url{doi:10.4204/EPTCS.251.38}] in the context of non-standard epistemic logics of know-what, know-how, know-why, and so on.\par As always with quantified modal logics, it makes a significant difference whether the domain stays the same across possible worlds. In particular, we show that the predicate logic with the bundle \(\forall\square\) alone is undecidable over constant domain interpretations, even with only monadic predicates, whereas having the \(\exists\square\) bundle instead gives us a decidable logic. On the other hand, over increasing domain interpretations, we get decidability with both \(\forall\square\) and \(\exists\square\) bundles with unrestricted predicates, where we obtain tableau based procedures that run in PSPACE. We further show that the \(\exists\square\) bundle cannot distinguish between constant domain and variable domain interpretations.

The purpose of this paper is to study, in the presence of \(\Gamma\)-symmetry, the existence of the nonstationary periodic solutions \(x:\mathbb{R} \rightarrow V\) of the following autonomous Newtonian system \(\ddot{x}=- \nabla \varphi(x), x(0)=x(2\pi),\dot{x}(0)=\dot{x}(2\pi),\) where \(\varphi :V \rightarrow \mathbb{R}\) is a \(C^2\)-differentiable \(\Gamma\)-invariant function such that \((\nabla \varphi)^{-1}(0)=\{0\}\) and \(\nabla\varphi\) is asymptotically linear at infinity, i.e., there exists a symmetric \(\Gamma\)-equivariant linear map \(B:V\rightarrow V\) such that \(\nabla\varphi (x)=Bx+o(\|x\|)\) as \(\|x\| \rightarrow \infty\). Here \(\Gamma\) is a finite group, which is a symmetry group of certain regular polygon or polyhedra in \(\mathbb{R}^n\) acting on \(V=\mathbb{R}^n\) by permuting the coordinates of the vectors \(x\in V\). The authors obtain similar results as in the paper of \textit{H. Ruan} and \textit{S. Rybicki} [Nonlinear Anal., Theory Methods Appl. 68, No. 6, A, 1479--1516 (2008; Zbl 1154.47048)] for the above system allowing \(0\) and \(\infty\) to be isolated degenerate critical points of \(\varphi\). The main result is contained in Theorem 4.2, where the authors discuss the existence and nonexistence of certain maximal orbit types appearing in several degenerate cases. Computational examples are provided with \(\Gamma\) being the dihedral groups \(D_6,D_8,D_{10}\) and \(D_{12}\). The main purpose of this paper is to establish a relation between the so-called \textsl{primary} equivariant degree (a notion that can be described by a set of axioms and whose computation can be automatized) and the concept of equivariant degree for gradient maps. An application is provided to the problem of existence of nontrivial periodic solutions for a particular class of gradient systems in the presence of symmetries.

The purpose of this paper is to study, in the presence of \(\Gamma\)-symmetry, the existence of the nonstationary periodic solutions \(x:\mathbb{R} \rightarrow V\) of the following autonomous Newtonian system \(\ddot{x}=- \nabla \varphi(x), x(0)=x(2\pi),\dot{x}(0)=\dot{x}(2\pi),\) where \(\varphi :V \rightarrow \mathbb{R}\) is a \(C^2\)-differentiable \(\Gamma\)-invariant function such that \((\nabla \varphi)^{-1}(0)=\{0\}\) and \(\nabla\varphi\) is asymptotically linear at infinity, i.e., there exists a symmetric \(\Gamma\)-equivariant linear map \(B:V\rightarrow V\) such that \(\nabla\varphi (x)=Bx+o(\|x\|)\) as \(\|x\| \rightarrow \infty\). Here \(\Gamma\) is a finite group, which is a symmetry group of certain regular polygon or polyhedra in \(\mathbb{R}^n\) acting on \(V=\mathbb{R}^n\) by permuting the coordinates of the vectors \(x\in V\). The authors obtain similar results as in the paper of \textit{H. Ruan} and \textit{S. Rybicki} [Nonlinear Anal., Theory Methods Appl. 68, No. 6, A, 1479--1516 (2008; Zbl 1154.47048)] for the above system allowing \(0\) and \(\infty\) to be isolated degenerate critical points of \(\varphi\). The main result is contained in Theorem 4.2, where the authors discuss the existence and nonexistence of certain maximal orbit types appearing in several degenerate cases. Computational examples are provided with \(\Gamma\) being the dihedral groups \(D_6,D_8,D_{10}\) and \(D_{12}\). In this paper, we study the Moore-Penrose inverses of differences and products of projectors in a ring with involution. Some necessary and sufficient conditions for the existence of the Moore-Penrose inverse are given. Moreover, the expressions of the Moore-Penrose inverses of differences and products of projectors are presented.

The authors give a new representation for the heat propagator \(e^{-uL}\) corresponding to the sub-Laplacian \(L\) on the Heisenberg group \(\mathbb H=\mathbb C\times \mathbb R\). The representation is used to obtain estimates for the action of \(e^{-uL}\) in the scale of Sobolev type spaces. The approach is based on the use of the \(\tau\)-Weyl transform (\(\tau\) is the parameter corresponding to the differentiation in the \(\mathbb R\)-direction on \(\mathbb H\)) and the Fourier-Wigner transform studied in [\textit{A.\,Dasgupta} and \textit{M.\,W.\thinspace Wong}, Fields Institute Communications 52, 27--36 (2007; Zbl 1152.47034)]. The authors obtain a new formula for the inverse of the sub-Laplacian on the Heisenberg group. In addition, Sobolev spaces are introduced in order to estimate the images of such inverse operator. The main technique is based on the decomposition of the sub-Laplacian into a family of twisted Laplacians, and the use of \(\tau\)-Weyl transforms and \(\tau\)-Fourier-Wigner transforms of Hermite functions.

The authors give a new representation for the heat propagator \(e^{-uL}\) corresponding to the sub-Laplacian \(L\) on the Heisenberg group \(\mathbb H=\mathbb C\times \mathbb R\). The representation is used to obtain estimates for the action of \(e^{-uL}\) in the scale of Sobolev type spaces. The approach is based on the use of the \(\tau\)-Weyl transform (\(\tau\) is the parameter corresponding to the differentiation in the \(\mathbb R\)-direction on \(\mathbb H\)) and the Fourier-Wigner transform studied in [\textit{A.\,Dasgupta} and \textit{M.\,W.\thinspace Wong}, Fields Institute Communications 52, 27--36 (2007; Zbl 1152.47034)]. This research presents a numerical analysis of the normal field instability for an initially flat layer of ferrofluid under the influence of magnetic field. A coupling between the simplified lattice Boltzmann method and the self-correcting procedure is developed to capture the velocity field and magnetic field. The proposed method has the ability to simulate complex hedgehog and comb-like spike structures without using an additional magnetization equation. A single dipole permanent magnet is defined instead of multiple point dipoles which makes this method much simpler and more efficient compared to the numerical approaches available in the literature. A comparison between the simulation results and experimental findings is provided to verify the validity of our method. A criterion for the prediction of spikes is presented for uniform magnetic fields. This study also investigates the effects of different types of magnetic fields, their strengths, and the effect of surrounding non-magnetic fluid on the spike structures. Moreover, the description of magnetic field lines, distribution of magnetic flux density, and energy estimation are also provided in this work which gives a useful insight into the hydrodynamic as well as the magnetostatic behavior of ferrofluids.

To solve the algebraic equations arising from implicit methods applied to initial value problems in ordinary differential equations, an iterative method is usually invoked, and some variant of Newton's method is often used. To develop criteria for stopping the sequence of Newton's iterates, knowledge of its rate of convergence is required. For sufficient smoothness, this rate of convergence depends upon a power of the stepsize of integration, and for nonstiff problems, this dependence is well understood. Recently, \textit{J. L. M. van Dorsselaer} and \textit{M. N. Spijker} [IMA J. Numer. Anal. 14, No. 2, 183-209 (1994; Zbl 0799.65085)] isolated a special class of stiff problems for study, and showed for these problems how stiffness deteriorates the rates of convergence. The present authors continue this study through more detailed results for implicit linear multistep methods when used with a modified Newton method, and illustrate with a numerical simulation that the established rates of convergence can be observed for problems in the specified class. As well, corresponding results for the exact Newton method, and for global stopping errors are obtained. This paper deals with the numerical solution of stiff systems by implicit linear multistep methods and implicit Runge-Kutta methods. Employing Newton-type iterations, the stopping error is estimated and related to the discretization error. Numerical illustrations are given.

To solve the algebraic equations arising from implicit methods applied to initial value problems in ordinary differential equations, an iterative method is usually invoked, and some variant of Newton's method is often used. To develop criteria for stopping the sequence of Newton's iterates, knowledge of its rate of convergence is required. For sufficient smoothness, this rate of convergence depends upon a power of the stepsize of integration, and for nonstiff problems, this dependence is well understood. Recently, \textit{J. L. M. van Dorsselaer} and \textit{M. N. Spijker} [IMA J. Numer. Anal. 14, No. 2, 183-209 (1994; Zbl 0799.65085)] isolated a special class of stiff problems for study, and showed for these problems how stiffness deteriorates the rates of convergence. The present authors continue this study through more detailed results for implicit linear multistep methods when used with a modified Newton method, and illustrate with a numerical simulation that the established rates of convergence can be observed for problems in the specified class. As well, corresponding results for the exact Newton method, and for global stopping errors are obtained. We study several models of staircase polygons on the \(45^{\circ}\) rotated square lattice, which interact with an impenetrable surface while also being pushed towards or pulled away from the surface by a force. The surface interaction is governed by a fugacity \(a\) and the force by a fugacity \(y\). Staircase polygons are simplifications of more general self-avoiding polygons, a well-studied model of interacting ring polymers. For this simplified case we are able to exactly determine the limiting free energy in the full \(a-y\) plane, and demonstrate that staircase polygons exhibit four different phases, including a `mixed' adsorbed-ballistic phase.

In the paper [``A topological mean value theorem for the plane,'' Am. Math. Mon. 98, No.~2, 149--154 (1991; Zbl 0741.26003)], \textit{I. Rosenholtz} used the Jordan Curve Theorem for the Euclidean plane to prove the theorem of the title. The present authors prove a corresponding theorem which also holds for non-Jordan (differentiable, nonstop) curves \([a,b]\to \mathbb R^2\), and call it the weak topological mean value theorem. A smooth surface \(S\) in \(\mathbb R^3\) of class \(C^1\) is said to satisfy the weak topological mean value theorem if the corresponding theorem holds for differentiable nonstop curves in \(S\). The main result is that such a set \(S\) is necessarily an open subset of a plane. Suppose that point \(B\) is directly north of \(A\) in the plane. A path from \(A\) to \(B\) need not ever head directly north. E.g., it may consist of a northeast segment and a northwest segment meeting at right angles, or if it is required to be differentiable, it may start moving northeast, make a clockwise loop and cross itself moving northwest. In the latter case there is a point where the tangent line is parallel to the segment from \(A\) to \(B\), but is oppositely directed. The theorem to which the title refers asserts that if the path is differentiable and one-to-one, then there is some point where the tangent line points northward. The proof depends upon the Jordan Curve Theorem. Several interesting corollaries are drawn, one of which is an intermediate value theorem for derivative directions. A counterexample is given, however, to some natural conjectures about the connectedness of the set of velocity vectors and the set of speeds for a differentiable nonstop path which also has nonzero derivatives at the endpoints.

In the paper [``A topological mean value theorem for the plane,'' Am. Math. Mon. 98, No.~2, 149--154 (1991; Zbl 0741.26003)], \textit{I. Rosenholtz} used the Jordan Curve Theorem for the Euclidean plane to prove the theorem of the title. The present authors prove a corresponding theorem which also holds for non-Jordan (differentiable, nonstop) curves \([a,b]\to \mathbb R^2\), and call it the weak topological mean value theorem. A smooth surface \(S\) in \(\mathbb R^3\) of class \(C^1\) is said to satisfy the weak topological mean value theorem if the corresponding theorem holds for differentiable nonstop curves in \(S\). The main result is that such a set \(S\) is necessarily an open subset of a plane. In this paper, we prove that the Extended Courant Property fails to be true for certain smooth, strictly convex domains with Neumann boundary condition: there exists a linear combination of a second and a first Neumann eigenfunctions, with three nodal domains. For the proof, we revisit a deformation argument of \textit{D. Jerison} and \textit{N. Nadirashvili} [J. Am. Math. Soc. 13, No. 4, 741--772 (2000; Zbl 0948.35029)]. This argument being interesting in itself, we give full details. In particular, we carefully control the dependence of the constants on the geometry of our Lipschitz domains along the deformations.

In several cases there is a relation among the deformation theory of double covers of a variety embedded in a projective space (and namely the existence of small deformations which are embeddings), and the existence of certain double structures on it, see e.g [\textit{L.-Y. Fong}, J. Algebr. Geom. 2, No. 2, 295--307 (1993; Zbl 0788.14027)]. In the paper under review, the authors give some more evidence of this relation. Namely, they study canonical double covers of the projective plane or of an Hirzebruch surface, embedded by a complete linear system, showing that all their deformations preserve the canonical involution, and therefore the double covers do not have any small deformation whose canonical map is an embedding. On the other hand, they show that the mentioned rational surfaces do not have any canonically embedded double structure. The degenerations of smooth curves to a nonreduced one are studied. More precisely, the author considers a ribbon \(D\), i.e. a nonreduced one- dimensional scheme supported on a smooth curve \(D_{\text{red}}\) such that the ideal sheaf \(L\) of \(D_{\text{red}}\) in \(D\) is a line bundle on \(D_{\text{red}}\) with \(L^ 2=0\). The flat limit of a 1-parameter deformation of a smooth hyperelliptic curve (under some further assumptions) is proved to be a ribbon. Moreover all ribbons arise in that way. These results are applied to the theory of the Clifford index on ribbons.

In several cases there is a relation among the deformation theory of double covers of a variety embedded in a projective space (and namely the existence of small deformations which are embeddings), and the existence of certain double structures on it, see e.g [\textit{L.-Y. Fong}, J. Algebr. Geom. 2, No. 2, 295--307 (1993; Zbl 0788.14027)]. In the paper under review, the authors give some more evidence of this relation. Namely, they study canonical double covers of the projective plane or of an Hirzebruch surface, embedded by a complete linear system, showing that all their deformations preserve the canonical involution, and therefore the double covers do not have any small deformation whose canonical map is an embedding. On the other hand, they show that the mentioned rational surfaces do not have any canonically embedded double structure. We investigate the conformal geometry of spherically symmetric spacetimes in general without specifying the form of the matter distribution. The general conformal Killing symmetry is obtained subject to a number of integrability conditions. Previous results relating to static spacetimes are shown to be a special case of our solution. The general inheriting conformal symmetry vector, which maps fluid flow lines conformally onto fluid flow lines, is generated and the integrability conditions are shown to be satisfied. We show that there exists a hypersurface orthogonal conformal Killing vector in an exact solution of Einstein's equations for a relativistic fluid which is expanding, accelerating and shearing.

The authors give sufficient conditions in order that the following system of Voltera-Hammerstein nonlinear integral equations has a unique solution in \(L[0,\infty)\): \[ x(t)=w(t,x(t))+\mu\int_0^tm(t,s)g_i(s,x(s))\,ds+\lambda\int_0^\infty k(t,s)h_j(s,x(s))\,ds \] for all \(t\in [0,\infty)\), where \(w(t,x(t))\in L[0,\infty)\) is known, \(m(t,s)\), \(k(t,s)\), \(g_i(s,x(s))\) and \(h_j(s,x(s))\), \(i,j=1,2\) and \(i\neq j\) are real or complex valued functions that are measurable both in \(t\) and \(s\) on \([0,\infty)\) and \(\lambda,\mu\) are real or complex numbers. A Jungck-Rhoades common fixed point result for weakly compatible mappings from [\textit{G. Jungck} and \textit{B. E. Rhoades}, Fixed Point Theory 7, No. 2, 287--296 (2006; Zbl 1118.47045)] is used in the proof. We obtain two fixed point theorems for a class of operators called occasionally weakly compatible maps defined on a symmetric space. These results establish two of the most general fixed point theorems for four maps.

The authors give sufficient conditions in order that the following system of Voltera-Hammerstein nonlinear integral equations has a unique solution in \(L[0,\infty)\): \[ x(t)=w(t,x(t))+\mu\int_0^tm(t,s)g_i(s,x(s))\,ds+\lambda\int_0^\infty k(t,s)h_j(s,x(s))\,ds \] for all \(t\in [0,\infty)\), where \(w(t,x(t))\in L[0,\infty)\) is known, \(m(t,s)\), \(k(t,s)\), \(g_i(s,x(s))\) and \(h_j(s,x(s))\), \(i,j=1,2\) and \(i\neq j\) are real or complex valued functions that are measurable both in \(t\) and \(s\) on \([0,\infty)\) and \(\lambda,\mu\) are real or complex numbers. A Jungck-Rhoades common fixed point result for weakly compatible mappings from [\textit{G. Jungck} and \textit{B. E. Rhoades}, Fixed Point Theory 7, No. 2, 287--296 (2006; Zbl 1118.47045)] is used in the proof. This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions to the Chaffee-Infante equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained for different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the diffusion coefficient \(\lambda\) of the equation are investigated. In addition, all possible kink profile solitary wave solutions and approximate damped oscillatory solutions to the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on these studies, the main contribution in this paper is to reveal the diffusion effect on travelling wave solutions to the Chaffee-Infante equation.

The symmetrical monotone risk aversion is studied with and without assuming the rank-dependent expected utility model. The paper is a continuation of \textit{M. Abouda} and \textit{A. Chateauneuf} [Theory Decis. 52, 149--170 (2002; Zbl 1032.91049)]. It is analyzed the positivity of the bid-ask spreads in terms of the market-maker attitudes toward risk successively in the expected utility and the rank-dependence expected utility framework.

The symmetrical monotone risk aversion is studied with and without assuming the rank-dependent expected utility model. The paper is a continuation of \textit{M. Abouda} and \textit{A. Chateauneuf} [Theory Decis. 52, 149--170 (2002; Zbl 1032.91049)]. The minimization of operation costs for natural gas transport networks is studied. Based on a recently developed model hierarchy ranging from detailed models of instationary partial differential equations with temperature dependence to highly simplified algebraic equations, modeling and discretization error estimates are presented to control the overall error in an optimization method for stationary and isothermal gas flows. The error control is realized by switching to more detailed models or finer discretizations if necessary to guarantee that a prescribed model and discretization error tolerance is satisfied in the end. We prove convergence of the adaptively controlled optimization method and illustrate the new approach with numerical examples.

The aim is to prove the existence of global positive solutions of a weakly coupled system in \(\mathbb{R}^N\) \[ u_t = \delta \Delta u + v^p,\;v_t = \Delta v + u^q, \quad u(x,0) = u_0 (x) \geq 0,\;v(x,0) = v_0 (x) \geq 0 \] where \(\delta > 0\), \(pq > 1\), \(1 < \min (p,q) \leq \max (p,q) \leq N/(N - 2)\) and \(\max (\alpha, \beta) < N/2\), \(\alpha = (p + 1)/(pq - 1)\), \(\beta = (q + 1)/(pq - 1)\). By using ``hot spot'' change of variables the author investigates linearized stability of a semilinear parabolic system of equations. The above assumptions made on \(p,q\) enables the author to conclude that the trivial solution is stable and this is why global-in-time solutions with small initial data exist. The results of this brief survey paper are contained in part in the paper of the author, \textit{H. A. Levine} and \textit{Y. Uda} [Math. Methods Appl. Sci. 17, No. 10, 807-835 (1994; Zbl 0814.35046)]. We consider nonnegative solutions of the initial value problem in \(\mathbb{R}^ N\) for the system \(u_ t = \delta \Delta u + v^ p\), \(v_ t = \Delta v + u^ q\), where \(0 \leq \delta \leq 1\) and \(pq > 0\). We prove the following conditions. Suppose \(\min (p,q) \geq 1\) but \(pq > 1\). (a) If \(\delta = 0\) then \(u = v = 0\) is the only nonnegative global solution of the system. (b) If \(\delta > 0\), nonnegative nonglobal solutions always exist for suitable initial values. (c) If \(0 < \delta \leq 1\) and \(\max (\alpha, \beta) \geq N/2\), where \(q \alpha = \beta + 1\), \(p \beta = \alpha + 1\), then the conclusion of (a) holds. (d) If \(N > 2\), \(0 < \delta \leq 1\) and \(\max (\alpha, \beta) < (N-2)/2\), then global, nontrivial nonnegative solutions exist which belong to \(L^ \infty (\mathbb{R}^ N \times [0, \infty))\) and satisfy \(0 < u(x,t) \leq c | x |^{-2 \alpha}\) and \(0 < v(x,t) \leq c | x |^{-2 \beta}\) for large \(| x |\) for all \(t > 0\), where \(c\) depends only upon the initial data. (e) Suppose \(0 < \delta \leq 1\) and \(\max (\alpha, \beta) < N/2\). If \(N = 1,2\) or \(N > 2\) and \(\max (p,q) \leq N/(N-2)\), then global, nontrivial solutions exist which, after making the standard `hot spot' change of variables, belong to the weighted Hilbert space \(H^ 1(K)\) where \(K(x) \equiv \exp ({1 \over 4} | x |^ 2)\). They decay like \(e^{[\max (\alpha, \beta) - (N/2) + \varepsilon] t}\) for every \(\varepsilon > 0\). These solutions are classical solutions for \(t > 0\). (f) If \(\max (\alpha, \beta) < N/2\), then there are global nontrivial solutions which satisfy, in the hot spot variables, \[ \max (u,v) (x,t) \leq c(u_ 0, v_ 0) e^{-{1 \over 4} | x |^ 2} e^{[\max (\alpha, \beta) - (N/2) + \varepsilon]t}, \] where \(0 < \varepsilon = \varepsilon (u_ 0, v_ 0) < (N/2) - \max (\alpha, \beta)\). Suppose \(\min (p,q) \leq 1\). (g) If \(pq \leq 1\), all nonnegative solutions are global. Suppose \(\min (p,q) < 1\). (h) If \(pq > 1\) and \(\delta = 0\), then all nontrivial nonnegative maximal solutions are nonglobal. (i) If \(0 < \delta \leq 1\), \(pq > 1\) and \(\max (\alpha, \beta) \geq N/2\) all nontrivial nonnegative maximal solutions are nonglobal. (j) If \(0 < \delta \leq 1\), \(pq > 1\) and \(\max (\alpha, \beta) < N/2\), there are both global and nonglobal nonnegative solutions. We also indicate some extensions of these results to more general systems and to other geometries.

The aim is to prove the existence of global positive solutions of a weakly coupled system in \(\mathbb{R}^N\) \[ u_t = \delta \Delta u + v^p,\;v_t = \Delta v + u^q, \quad u(x,0) = u_0 (x) \geq 0,\;v(x,0) = v_0 (x) \geq 0 \] where \(\delta > 0\), \(pq > 1\), \(1 < \min (p,q) \leq \max (p,q) \leq N/(N - 2)\) and \(\max (\alpha, \beta) < N/2\), \(\alpha = (p + 1)/(pq - 1)\), \(\beta = (q + 1)/(pq - 1)\). By using ``hot spot'' change of variables the author investigates linearized stability of a semilinear parabolic system of equations. The above assumptions made on \(p,q\) enables the author to conclude that the trivial solution is stable and this is why global-in-time solutions with small initial data exist. The results of this brief survey paper are contained in part in the paper of the author, \textit{H. A. Levine} and \textit{Y. Uda} [Math. Methods Appl. Sci. 17, No. 10, 807-835 (1994; Zbl 0814.35046)]. We study Bojanov-Xu interpolation whose interpolation points are located on concentric circles in \(\mathbb{R}^2\). We prove that the integral means of the interpolation polynomial over a fixed circle or a fixed annulus are continuous functions of the radii of circles. We also give a distribution of the radii such that the integral means are convergent.

The authors study the inverse problem of determining the possibly anisotropic, conductivity \(\sigma\) of a bounded domain \(\Omega\subset\mathbb{R}^n\) \((n\geq 3)\) with Lipschitz boundary \(\partial\Omega\) when the Dirichlet-to-Neumann map is locally given on an open non-empty portion \(\Gamma\subset\partial\Omega\). More exactly, the unknown conductivities have the structure \(\sigma(x)= A(x,a(x))\), where \(A(x,t)\) is a known, matrix-valued function which verifies a condition of uniform ellipticity, as well as a condition of monotonicity with respect to the last variable \(t\), and \(a(x)\) is an unknown scalar function. The local Dirichlet-to-Neumann map \(\Lambda^\Gamma_\sigma\) is given, in a weak formulation, by the formula \[ \langle\Lambda^\Gamma_\sigma, \varphi\rangle= \int_\Omega \sigma\nabla_u\cdot\nabla\varphi, \] for any \(u,\varphi\in H^1(\Omega)\), \(u\) being a weak solution of the equation \(\text{div}(\sigma\nabla_u)= 0\) in \(\Omega\), \[ u|_{\partial\Omega},\varphi|_{\partial\Omega}\in H^{1/2}_{cs}(\Gamma):=\{f\in H^{1/2}(\partial\Omega); \text{supp\,}f\subset\Gamma\}. \] The authors improve their results of uniqueness and stability at the boundary obtained in [SIAM J. Math. Anal. 33, No.~1, 153--171 (2001; Zbl 0985.35104)] under the following aspects: {\parindent4mm \begin{itemize}\item[--] The uniqueness and stability estimates are adapted to the local Dirichlet-to-Neumann map. \item[--] The stability estimate at the boundary is obtained in the wider class of conductivities which are merely continuous in a neighborhood of some point at the boundary. \item[--] Analogous results are obtained when the local Dirichlet-to-Neumann map is replaced by the local Neumann-to-Dirichlet map. \end{itemize}} We prove results of uniqueness and stability at the boundary for the inverse problem of electrical impedance tomography in the presence of possibly anisotropic conductivities. We assume that the unknown conductivity has the form \(A=A(x,a(x))\), where \(a(x)\) is an unknown scalar function and \(A(x,t)\) is a given matrix-valued function. We also deduce results of uniqueness in the interior among conductivities A obtained by piecewise analytic perturbations of the scalar term \(a\).

The authors study the inverse problem of determining the possibly anisotropic, conductivity \(\sigma\) of a bounded domain \(\Omega\subset\mathbb{R}^n\) \((n\geq 3)\) with Lipschitz boundary \(\partial\Omega\) when the Dirichlet-to-Neumann map is locally given on an open non-empty portion \(\Gamma\subset\partial\Omega\). More exactly, the unknown conductivities have the structure \(\sigma(x)= A(x,a(x))\), where \(A(x,t)\) is a known, matrix-valued function which verifies a condition of uniform ellipticity, as well as a condition of monotonicity with respect to the last variable \(t\), and \(a(x)\) is an unknown scalar function. The local Dirichlet-to-Neumann map \(\Lambda^\Gamma_\sigma\) is given, in a weak formulation, by the formula \[ \langle\Lambda^\Gamma_\sigma, \varphi\rangle= \int_\Omega \sigma\nabla_u\cdot\nabla\varphi, \] for any \(u,\varphi\in H^1(\Omega)\), \(u\) being a weak solution of the equation \(\text{div}(\sigma\nabla_u)= 0\) in \(\Omega\), \[ u|_{\partial\Omega},\varphi|_{\partial\Omega}\in H^{1/2}_{cs}(\Gamma):=\{f\in H^{1/2}(\partial\Omega); \text{supp\,}f\subset\Gamma\}. \] The authors improve their results of uniqueness and stability at the boundary obtained in [SIAM J. Math. Anal. 33, No.~1, 153--171 (2001; Zbl 0985.35104)] under the following aspects: {\parindent4mm \begin{itemize}\item[--] The uniqueness and stability estimates are adapted to the local Dirichlet-to-Neumann map. \item[--] The stability estimate at the boundary is obtained in the wider class of conductivities which are merely continuous in a neighborhood of some point at the boundary. \item[--] Analogous results are obtained when the local Dirichlet-to-Neumann map is replaced by the local Neumann-to-Dirichlet map. \end{itemize}} A new method for computation of positive realizations of given transfer matrices of fractional linear continuous-time linear systems is proposed. Necessary and sufficient conditions for the existence of positive realizations of transfer matrices are given. A~procedure for computation of the positive realizations is proposed and illustrated by examples.

The authors consider nonlinear ordinary differential equations of the form \[ u''+ g(u)= p(t,u,u') \] satisfied by the functions \(u(t)\) defined on the interval \(0\leq t\leq 1\) under the boundary conditions \[ au(0)+ bu'(0)= A,\quad cu(1)+ du'(1)= B, \] and show that they have at least two solutions if the given functions \(g(x)\) and \(p(t,x,y)\) fulfill the following relations \[ \lim_{|x|\to\infty} [g(x)/x]= +\infty \] and \[ \exists k> 0\ni |p(t,x,y)|\leq k(1+ |x|+|y|),\quad\forall t\in [0,1],\quad (x,y)\in\mathbb{R}^2. \] Here, \(a\), \(b\), \(c\), \(d\), \(A\) and \(B\) refer to certain known real constants. Their proof consists of an application of a theorem due to \textit{A. Capietto}, \textit{J. Mawhin} and \textit{F. Zanolin} [Topol. Methods Nonlinear Anal. 6, No. 1, 175-188 (1995; Zbl 0849.34018)]. The authors continue their investigation of the solvability of boundary value problems associated with the second order differential equation (*) \(u' + f(u) = p(t,u,u')\), \(t \in [a,b]\), where no a priori bound for solutions may be proved. This fact requires a modification of the classical Leray-Schauder continuation technique. This modification was essentially established in the authors' paper published in [J. Differ. Equations 88, 347-395 (1990; Zbl 0718.34053)].

The authors consider nonlinear ordinary differential equations of the form \[ u''+ g(u)= p(t,u,u') \] satisfied by the functions \(u(t)\) defined on the interval \(0\leq t\leq 1\) under the boundary conditions \[ au(0)+ bu'(0)= A,\quad cu(1)+ du'(1)= B, \] and show that they have at least two solutions if the given functions \(g(x)\) and \(p(t,x,y)\) fulfill the following relations \[ \lim_{|x|\to\infty} [g(x)/x]= +\infty \] and \[ \exists k> 0\ni |p(t,x,y)|\leq k(1+ |x|+|y|),\quad\forall t\in [0,1],\quad (x,y)\in\mathbb{R}^2. \] Here, \(a\), \(b\), \(c\), \(d\), \(A\) and \(B\) refer to certain known real constants. Their proof consists of an application of a theorem due to \textit{A. Capietto}, \textit{J. Mawhin} and \textit{F. Zanolin} [Topol. Methods Nonlinear Anal. 6, No. 1, 175-188 (1995; Zbl 0849.34018)]. The papers for this special issue reveal the current areas of expert systems and artificial intelligence application research in the Pacific-Asian region. These 19 papers are selected from the most recent proceedings of the Japan/Korea Joint Conference on Expert Systems and its successor the 95 Pacific-Asia Conference on Expert Systems (PACES). The topics cover scheduling, design, diagnosis, integration with CIM information systems, machine learning and reasoning. (Autorreferat geändert) (Provider: Infodata)

The theory of institutions [\textit{J. Goguen} and \textit{R. Burstall}, J. Assoc. Comput. Mach. 39, No. 1, 95--146 (1992; Zbl 0799.68134)] is a categorical abstract model theory which formalizes the intuitive notion of logical system, including syntax, semantics, and the satisfaction between them. This paper studies definability within the theory of institutions, a version of abstract model theory that emerged in computing science studies of software specification and semantics. So, the authors generalize the concept of definability to arbitrary logics, formalized as institutions, and develop three general definability results. One generalizes the classical Beth theorem by relying on the interpolation properties of institution. Another relies on a meta Birkhoff axiomatizability property of the institution and constitutes a source for many new definability results, including definability in (fragments of) classical model theory. The third one gives a set of sufficient conditions for borrowing definability properties from another institution via an adequate encoding between institutions. The power of these general definability results is illustrated with several applications to (many-sorted) classical model theory and partial algebra, leading for example to definability results for (quasi-) varieties of models or partial algebras. Many other applications are expected for the multitude of logical systems formalized as institutions from computing science and logic. There is a population explosion among the logical systems used in computing science. Examples include first-order logic, equational logic, Horn-clause logic, higher-order logic, infinitary logic, dynamic logic, intuitionistic logic, order-sorted logic, and temporal logic; moreover, there is a tendency for each theorem prover to have its own idiosyncratic logical system. The concept of institution is introduced to formalize the informal notion of ``logical system''. The major requirement is that there is a satisfaction relation between models and sentences that is consistent under change of notation. Institutions enable abstracting away from syntactic and semantic detail when working on language structure ``in-the-large''; for example, we can define language features for building large structures from smaller ones, possibly involving parameters, without commitment to any particular logical system. This applies to both specification languages and programming languages. Institutions also have applications to such areas as database theory and the semantics of artificial and natural languages. A first main result of this paper says that any institution such that signatures (which define notation) can be glued together,also allows gluing together theories (which are just collections of sentences over a fixed signature). A second main result considers when theory structuring is preserved by institution morphisms. A third main result gives conditions under which it is sound to use a theorem prover for one institution on theories from another. A fourth main result shows how to extend institutions so that their theories may include, in addition to the original sentences, various kinds of constraint that are useful for defining abstract data types, including both ``data'' and ``hierarchy'' constraints. Further results show how to define institutions that allow sentences and constraints from two or more institutions. All our general results apply to such ``duplex'' and ``multiplex'' institutions.

The theory of institutions [\textit{J. Goguen} and \textit{R. Burstall}, J. Assoc. Comput. Mach. 39, No. 1, 95--146 (1992; Zbl 0799.68134)] is a categorical abstract model theory which formalizes the intuitive notion of logical system, including syntax, semantics, and the satisfaction between them. This paper studies definability within the theory of institutions, a version of abstract model theory that emerged in computing science studies of software specification and semantics. So, the authors generalize the concept of definability to arbitrary logics, formalized as institutions, and develop three general definability results. One generalizes the classical Beth theorem by relying on the interpolation properties of institution. Another relies on a meta Birkhoff axiomatizability property of the institution and constitutes a source for many new definability results, including definability in (fragments of) classical model theory. The third one gives a set of sufficient conditions for borrowing definability properties from another institution via an adequate encoding between institutions. The power of these general definability results is illustrated with several applications to (many-sorted) classical model theory and partial algebra, leading for example to definability results for (quasi-) varieties of models or partial algebras. Many other applications are expected for the multitude of logical systems formalized as institutions from computing science and logic. Since credit scoring was first applied in the 1940s the standard methodology has been to treat consumer lending decisions as binary classification problems, where the goal has been to make the best possible 'good/bad' classification of accounts on the basis of their eventual delinquency status. However, the real goal of commercial lending organizations is to forecast continuous financial measures such as contribution to profit, but there has been little research in this area. In this paper, continuous models of customer worth are compared to binary models of customer repayment behaviour. Empirical results show that while models of customer worth do not perform well in terms of classifying accounts by their good/bad status, they significantly outperform standard classification methodologies when ranking accounts based on their financial worth to lenders.

\textit{V. H. An} and \textit{H. H. Khoai} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 47, 117--126 (2018; Zbl 1413.30122)] considered the uniqueness of meromorphic functions when the derivative of some power of meromorphic functions share a set of elements with counting multiplicities. The authors of the paper use the notion of weighted sharing of sets to extend the results of V. H. An and H. H. Khoai. In the paper the authors consider uniqueness problem of meromorphic functions \(f\) and \(g\), when certain differential polynomials generated by them share a set. The main result of the paper is as follows: Let \(f\) and \(g\) be two nonconstant meromorphic functions, \(n, d, k\) be positive integers with \(\displaystyle n > 2k + \frac{2k + 8}{d}, d \geq 2\) and let \(S = \{a \in \mathbb{C}: a^{d} = 1\}\). If \((f^{n})^{(k)}\) and \((g^{n})^{(k)}\) share the set \(S\) with counting multiplicities, then one of the following two cases holds: (i) \(f = c_{1}e^{cz}\) and \(g = c_{2}e^{-cz}\) for three constants \(c_{1}, c_{2}\) and \(c\) such that \((-1)^{kd}(c_{1}c_{2})^{nd}(nc)^{2kd} = 1\); (ii) \(f = tg\) with \(t^{nd} = 1, t \in \mathbb{C}\).

\textit{V. H. An} and \textit{H. H. Khoai} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Comput. 47, 117--126 (2018; Zbl 1413.30122)] considered the uniqueness of meromorphic functions when the derivative of some power of meromorphic functions share a set of elements with counting multiplicities. The authors of the paper use the notion of weighted sharing of sets to extend the results of V. H. An and H. H. Khoai. Following Lutz (1995) we consider a problem to be weakly hard for a complexity class \(C\) if a nonnegligible part of the problems in \(C\) can be reduced to \(C\). For the exponential-time class \(E\) (the union of \(\text{DTIME}(2^{kn})\) for all \(k\)) Lutz formalized this concept by introducing a resource-bounded measure on this class and by saying that a subclass of \(E\) is negligible if it has measure \(0\in E\). In our talk we introduce and discuss some new weak hardness notions for \(E\), called \(E\)-nontriviality and strong \(E\)-nontriviality, which generalize Lutz's weak hardness notion for \(E\) and which are conceptually much simpler than Lutz's concept. Here a set \(A\) is \(E\)-nontrivial if the problems in \(E\) which can be reduced to \(A\) are not contained in a single level \(\text{DTIME}(2^{kn})\) of the hierarchy \(E\), and \(A\) is strongly \(E\)-nontrivial if, for any number \(k\) there is a problem in \(E\) which can be reduced to \(A\) and which is almost-everywhere \(2^{kn}\)-complex (i.e., bi-immune for the \(k\)-th level \(\text{DTIME}(2^{kn})\) of the hierarchy \(E\)). This work is joint work with Timur Bakibayev.

In the very interesting note under review, the authors provide a new constant in a Cwikel-Lieb-Rosenblum type inequality which estimates the number of negative eigenvalues of the Schrödinger operator involving the Heisenberg sub-Laplacian with a potential that is proportional to the characteristic function of a measurable set. More precisely, consider the Schrödinger operator \[ H=\Delta_{\mathbb{H}}-V \] with the Heisenberg sub-Laplacian \(\Delta_{\mathbb{H}}=-X^2-Y^2,\) \[ X=\dfrac{\partial}{\partial x}+\dfrac{1}{2}y\dfrac{\partial}{\partial t},\qquad Y=\dfrac{\partial}{\partial y}-\dfrac{1}{2}x\dfrac{\partial}{\partial t}, \] and a non-negative potential \(V.\) Set \(\lambda_1\leq\lambda_2\leq\cdots<0\) for the negative eigenvalues of \(H\) and let \(N(V)\) denote the number of such eigenvalues. Consider the potential \(V=c\chi_{\Omega}\) with a positive constant \(c\) on a given set \(\Omega\subset\mathbb{R}^3\) of finite measure. The main result proved states that \[ N(c\chi_{\Omega}) \leq \dfrac{c^2|\Omega|}{6}. \] The proof is based on the Birman--Schwinger principle and the Berezin-Lieb trace inequality, combined with the technique from [\textit{A. Laptev}, Oper. Theory, Adv. Appl. 78, 221--225 (1995; Zbl 0833.35032)]. We improve the Lieb constant in the Cwikel-Lieb-Rozenblum inequality for the number of bound states of Schrödinger operators whose potential equals the characteristic function of a measurable set. A similar idea also gives better constants in the Lieb-Thirring inequalities for the respective moments of eigenvalues.

In the very interesting note under review, the authors provide a new constant in a Cwikel-Lieb-Rosenblum type inequality which estimates the number of negative eigenvalues of the Schrödinger operator involving the Heisenberg sub-Laplacian with a potential that is proportional to the characteristic function of a measurable set. More precisely, consider the Schrödinger operator \[ H=\Delta_{\mathbb{H}}-V \] with the Heisenberg sub-Laplacian \(\Delta_{\mathbb{H}}=-X^2-Y^2,\) \[ X=\dfrac{\partial}{\partial x}+\dfrac{1}{2}y\dfrac{\partial}{\partial t},\qquad Y=\dfrac{\partial}{\partial y}-\dfrac{1}{2}x\dfrac{\partial}{\partial t}, \] and a non-negative potential \(V.\) Set \(\lambda_1\leq\lambda_2\leq\cdots<0\) for the negative eigenvalues of \(H\) and let \(N(V)\) denote the number of such eigenvalues. Consider the potential \(V=c\chi_{\Omega}\) with a positive constant \(c\) on a given set \(\Omega\subset\mathbb{R}^3\) of finite measure. The main result proved states that \[ N(c\chi_{\Omega}) \leq \dfrac{c^2|\Omega|}{6}. \] The proof is based on the Birman--Schwinger principle and the Berezin-Lieb trace inequality, combined with the technique from [\textit{A. Laptev}, Oper. Theory, Adv. Appl. 78, 221--225 (1995; Zbl 0833.35032)]. A nonstationary quasilinear problem of group pursuit is considered. Sufficient conditions for the unique solvability of this problem are obtained for some guaranteed time. The results are illustrated by an example model.

This paper deals with regularity properties of solutions of the equation \[ \psi_{tt}-t^{-4/3}\Delta\psi+{2\over t}\psi_t=f(x,t), \] with \(t>0\), \(x\in{\mathbb R}^n\) and \(f\) given. It is a sequel to a previous paper by the authors and \textit{T. Kinoshita} [J. Math. Phys. 51, No. 5, 052501, 18 p. (2010; Zbl 1310.58021)]. Letting \(\psi=u/t\), the authors reduce the problem to \[ u_{tt}-t^{-4/3}\Delta u=g(x,t), \] where \(g=tf\), and give an integral representation formula for solutions in terms of hypergeometric functions (that take an elementary form in this case), and solve a singular initial-value problem, pointing out the similarity with the Euler-Poisson-Darboux (EPD) equation to ``give a motivation for the choice of initial conditions.'' Uniqueness and regularity of the solution are also discussed. It is not clear whether the authors are aware that the equation for \(u\) is actually reducible to an inhomogeneous EPD equation (by introducing a new time variable \(T=t^{1/3}\)). Indeed, on page 3 or the previous paper, we read ``[t]o us, this approach seems to be more immediate than the one that uses the Euler-Poisson-Darboux equation.'' We consider the wave propagating in the Einstein and de Sitter space-time. The covariant d'Alembert's operator in the Einstein and de Sitter space-time belongs to the family of the non-Fuchsian partial differential operators. We introduce the initial value problem for this equation and give the explicit representation formulas for the solutions. We also show the \(L^p-L^q\) estimates for solutions.{ \copyright 2010 American Institute of Physics}

This paper deals with regularity properties of solutions of the equation \[ \psi_{tt}-t^{-4/3}\Delta\psi+{2\over t}\psi_t=f(x,t), \] with \(t>0\), \(x\in{\mathbb R}^n\) and \(f\) given. It is a sequel to a previous paper by the authors and \textit{T. Kinoshita} [J. Math. Phys. 51, No. 5, 052501, 18 p. (2010; Zbl 1310.58021)]. Letting \(\psi=u/t\), the authors reduce the problem to \[ u_{tt}-t^{-4/3}\Delta u=g(x,t), \] where \(g=tf\), and give an integral representation formula for solutions in terms of hypergeometric functions (that take an elementary form in this case), and solve a singular initial-value problem, pointing out the similarity with the Euler-Poisson-Darboux (EPD) equation to ``give a motivation for the choice of initial conditions.'' Uniqueness and regularity of the solution are also discussed. It is not clear whether the authors are aware that the equation for \(u\) is actually reducible to an inhomogeneous EPD equation (by introducing a new time variable \(T=t^{1/3}\)). Indeed, on page 3 or the previous paper, we read ``[t]o us, this approach seems to be more immediate than the one that uses the Euler-Poisson-Darboux equation.'' We propose a new framework to provide continuous services to users by a collection of mobile servers distributed over an interconnection network. We model those mobile servers as a subset of nodes, and assume that a user can receive the service if at least one adjacent node (including itself) plays the role of a server; i.e., we assume that the service could not be routed via the interconnection network. The main results obtained in this paper are summarized as follows: For the class of trees consisting of \(n\) nodes, \(\lfloor n/2\rfloor \) mobile servers are sometimes necessary and always sufficient to realize continuous services by the mobile servers, and for the class of Hamiltonian graphs with \(n\) nodes, \(\lceil (n+1)/3\rceil \) mobile servers are sometimes necessary and always sufficient.

The author studies the global well-posedness of the Cauchy problem for the equations of two-dimensional incompressible isotropic elastodynamics. A similar global existence result in the Lagrangian coordinates formulation was recently obtained by \textit{Z. Lei} [Commun. Pure Appl. Math. 69, No. 11, 2072--2106 (2016; Zbl 1351.35216)]. This paper aims at improvement of understanding the behavior of solutions, and also at presenting a different approach to study two-dimensional nonlinear wave equations in frequency space. By employing a modified energy method to derive the energy estimate and a Fourier transform method with a suitable choice of a \(Z\)-norm to derive a sharp \(L^\infty\)-estimate, the author establishes the global existence and the asymptotic behavior of smooth solutions for sufficiently small smooth initial data in the Eulerian coordinates formulation. We prove that for sufficiently small initial displacements in some weighted Sobolev space, the Cauchy problem of the systems of incompressible isotropic Hookean elastodynamics in two space dimensions admits a uniqueness global classical solution.

The author studies the global well-posedness of the Cauchy problem for the equations of two-dimensional incompressible isotropic elastodynamics. A similar global existence result in the Lagrangian coordinates formulation was recently obtained by \textit{Z. Lei} [Commun. Pure Appl. Math. 69, No. 11, 2072--2106 (2016; Zbl 1351.35216)]. This paper aims at improvement of understanding the behavior of solutions, and also at presenting a different approach to study two-dimensional nonlinear wave equations in frequency space. By employing a modified energy method to derive the energy estimate and a Fourier transform method with a suitable choice of a \(Z\)-norm to derive a sharp \(L^\infty\)-estimate, the author establishes the global existence and the asymptotic behavior of smooth solutions for sufficiently small smooth initial data in the Eulerian coordinates formulation. This monograph is devoted to study and design of planar robots mimicking the movements of snakes. Such kind of biological inspired robots are very interesting for science. Also they find important applications such as: inspection of industrial pipes and channels, performing dangerous tasks in hard-to-reach locations, etc. The complicated structure of the equations of motion for a snake robot moving on a planar surface makes the control problem challenging. Additionally, controlling snake robot is complicated by variable environment with uncertainty and disturbances causing variations in system parameters. To achieve efficient head-angle and velocity tracking with least computational complexity is a difficult problem. The aim of the book is to present efficient adaptive robust controlling method of snake robots. The book is divided into seven chapters. Chapter 1 introduces the state-of-art in the field of snake robotics, especially for dynamic modeling and the employed control methodologies. The following topics are presented: kinematics and dynamics of a snake robot, Euler-Lagrange equations, virtual holomonic constraint (VHC), body-shape control, serpenoid gait function, head-angle control, velocity control, robustness and multi-layered control. Chapters 2--5 present the following major research results: \begin{itemize} \item[1.] A sliding-mode control (SMC) used for efficient tracking of a planar snake robot moving in variable environment with bounded uncertainties. \item[2.] A time-delayed control (TDC) using an artificial delayed-based estimation to obtain satisfactory tracking efficiency of planar snake robots. \item[3.] An adaptive robust-time-delayed control (ARTDC) used to achieve improved tracking performance in presence of bounded estimation error. \item[4.] Differential flatness used to establish mapping between the output and input variables based on the serpenoid gait function. Flatness reduces order of systems that simplifies feedback controller design. \item[5.] The dynamic equations, obtained via Newton-Euler approach, to describe a snake robot's motion inside a channel. \end{itemize} Every chapter describes assumptions related to the considered model, description of control method used, stability analysis, plots presenting simulation results and detailed discussion. Summing up, the book reports the recent progress in methods used to control planar snake robots, achieved by the authors. The book is well--written. It can be recommended to researchers, engineers and graduate students who are interested in robotics, automatic control, optimal control problems and their effective solutions.

In the classification of A. M. Naveira for Riemannian almost-product manifolds [Rend. Mat. Appl., VII. Ser. 3, 577--592 (1983; Zbl 0538.53045) (see the previous review)], many of the classes correspond to foliated manifolds with special properties of its metric, such as bundle-like metrics, minimal foliations, totally geodesic foliations, etc. In the present paper, the author gives examples of the 36 classes, showing that these are not vacuous and also that the ten inclusion relations among them are strict. Nine examples are constructed explicitly, and the others are derived from them by means of products and conformal transformations. The basic examples are taken from the following manifolds: a) oriented real hypersurfaces \(M\) in \(\mathbb{C}^{n+1}\). If \(N\) is a unitary normal vector field on \(M\), and \(J\) the canonical complex structure on \(\mathbb{C}^{n+1}\), one defines the distribution on \(M\) generated by the vector field \(JN\). This distribution and its orthogonal define a Riemannian almost-product structure on \(M\); b) oriented real hypersurfaces \(M\) in \(\mathbb{H}^{n+1}\) \((\mathbb{H}= \) quaternion field). If \(J_1, J_2, J_3\) are the canonical almost-complex structures on \(\mathbb{H}^{n+1}\) such that \(J_1J_2 = J_3\), then a Riemannian almost-product structure on \(M\) is defined by means of the distribution generated by \(J_1N\), \(J_2N\), \(J_3N\), and its orthogonal; c) complex hypersurfaces \(M\) in \(\mathbb{H}^{n+1}\) with trivial normal vector bundle. Let \(N\), \(J_1N\) be a frame in the normal vector bundle of \(M\). Then one considers the distribution generated by \(J_2N\), \(J_3N\), and its orthogonal on \(M\). Let \(A\) be a Riemannian almost-product manifold with metric tensor \(<,>\) and almost-product tensor field \(P\). The author considers the symmetric 2-covariant tensor field \(\Phi\) defined by \(\Phi(\cdot,\cdot)=<P \cdot,\cdot>\). If \(\nabla\) denotes the Levi-Civita connection, then the covariant derivative \(\nabla \Phi\) satisfies \((\nabla \Phi)(M,N,Q)=(\nabla \Phi)(M,Q,N)=-(\nabla \Phi)(M,PN,PQ)\) for all vector fields \(M, N, Q\) on \(A\). Then he considers the vector space \(W\) of 3-covariant tensors satisfying the same identities as \(\nabla \Phi\), and by means of a study of the natural representation of the structure group \(O(p)\times O(q)\) on \(W\), he gives a classification of the Riemannian almost-product manifolds, obtaining 36 classes, including: i) a locally-product structure; ii) 7 types of totally geodesic foliations; iii) 18 types of foliations; iv) 4 types of antifoliations (Vidal spaces or almost-foliated manifolds, [see \textit{E. Vidal} and \textit{E. Vidal Costa}, J. Differ. Geom. 8, 297--304 (1973; Zbl 0284.53031]); v) 3 types with vertical distribution (the \((+1)\)-eigenbundle of \(P)\) of the so-called type \(D_1\); vi) 2 types with vertical distribution of the so-called type \(D_2\). The paper includes a geometrical interpretation of some of these classes, and in particular of the Vidal spaces. Following this line of research (which in turn is in the spirit of the Gray-Hervella classification of almost Hermitian manifolds), other papers have been written: \textit{O. Gil-Medrano} [Rend. Circ. Mat. Palermo, II. Ser. 32, 315--329 (1983; Zbl 0538.53043)] and \textit{A. Montesinos} [Mich. Math. J. 30, 31--36 (1983; Zbl 0538.53044)] (see the preceding two reviews) have completed the geometric interpretation, and \textit{V. Miquel} [Pac. J. Math. 111, 163--178 (1984; Zbl 0538.53046) (see the following review)] has given non trivial examples for each class.

In the classification of A. M. Naveira for Riemannian almost-product manifolds [Rend. Mat. Appl., VII. Ser. 3, 577--592 (1983; Zbl 0538.53045) (see the previous review)], many of the classes correspond to foliated manifolds with special properties of its metric, such as bundle-like metrics, minimal foliations, totally geodesic foliations, etc. In the present paper, the author gives examples of the 36 classes, showing that these are not vacuous and also that the ten inclusion relations among them are strict. Nine examples are constructed explicitly, and the others are derived from them by means of products and conformal transformations. The basic examples are taken from the following manifolds: a) oriented real hypersurfaces \(M\) in \(\mathbb{C}^{n+1}\). If \(N\) is a unitary normal vector field on \(M\), and \(J\) the canonical complex structure on \(\mathbb{C}^{n+1}\), one defines the distribution on \(M\) generated by the vector field \(JN\). This distribution and its orthogonal define a Riemannian almost-product structure on \(M\); b) oriented real hypersurfaces \(M\) in \(\mathbb{H}^{n+1}\) \((\mathbb{H}= \) quaternion field). If \(J_1, J_2, J_3\) are the canonical almost-complex structures on \(\mathbb{H}^{n+1}\) such that \(J_1J_2 = J_3\), then a Riemannian almost-product structure on \(M\) is defined by means of the distribution generated by \(J_1N\), \(J_2N\), \(J_3N\), and its orthogonal; c) complex hypersurfaces \(M\) in \(\mathbb{H}^{n+1}\) with trivial normal vector bundle. Let \(N\), \(J_1N\) be a frame in the normal vector bundle of \(M\). Then one considers the distribution generated by \(J_2N\), \(J_3N\), and its orthogonal on \(M\). The construction of an integrated numerical model is presented in this paper to deal with the interactions between vegetated surface and saturated subsurface flows. A numerical model is built by integrating the previously developed quasi-three-dimensional (Q3D) vegetated surface flow model with a two-dimensional (2D) saturated groundwater flow model. The vegetated surface flow model is constructed by coupling the explicit finite volume solution of 2D shallow water equations (SWEs) with the implicit finite difference solution of Navier-Stokes equations (NSEs) for vertical velocity distribution. The subsurface model is based on the explicit finite volume solution of 2D saturated groundwater flow equations (SGFEs). The ground and vegetated surface water interaction is achieved by introducing source-sink terms into the continuity equations. Two solutions are tightly coupled in a single code. The integrated model is applied to four test cases, and the results are satisfactory.

This paper deals with a continuous-time generalization of the secretary problem that was studied by \textit{F. T. Bruss} [J. Appl. Probab. 24, No.~4, 918--928 (1987; Zbl 0596.60046)].The authors study the problem within two frameworks. In the first case the objective is to stop on the best or on the second best object whilst in the second case the goal is to stop on the second best object. The optimal strategies and the probabilities of success are given. \textit{R. Cowan} and \textit{J. Zabczyk} [Teor. Veroyatn. Primen. 23, 606-614 (1978; Zbl 0396.62063)] have studied a continuous time generalization of the so-called secretary problem, where options arrive according to a homogeneous Poisson process of known intensity \(\lambda\). They give the complete strategy maximizing the probability of accepting the best option under the usual random arrival order and no-recall conditions. In this paper the solution is first extended to the case of an unknown intensity. It is then generalized to the case of an inhomogeneous Poisson process with intensity function \(\lambda\) (t) which is either supposed to be known or known up to a multiplicative constant.

This paper deals with a continuous-time generalization of the secretary problem that was studied by \textit{F. T. Bruss} [J. Appl. Probab. 24, No.~4, 918--928 (1987; Zbl 0596.60046)].The authors study the problem within two frameworks. In the first case the objective is to stop on the best or on the second best object whilst in the second case the goal is to stop on the second best object. The optimal strategies and the probabilities of success are given. Die betrachteten Gleichungen sind vom dritten und vierten Grade mit einer Unbekannten, oder Systeme von zwei Gleichungen mit zwei Unbekannten. Der Zweck ist die Hervorhebung des Vorteils, mehr, als dies gewöhnlich geschieht, von der Discriminante bei der Auflösung der Gleichungen Gebrauch zu machen, und die Betonung der Wichtigkeit, nach geometrischen Erläuterungen analytischer Methoden Umschau zu halten, wenn dies möglich ist.

The Gosset group of the title is a simple group, better known perhaps as \(O^+_8(2)\), or \(W(E_8)'/\{\pm 1\}\), where \(W(E_8)\) is the Weyl group of \(E_8\). It contains 17 classes of maximal subgroups. The aim of the paper is to describe in detail the correspondence between these subgroups and certain geometrical structures within the two 8-dimensional polytopes associated to the \(E_8\)-lattice. The corresponding exercise for \(E_7\) has already been carried out by the same author [ibid. 97, 83-89 (1993; Zbl 0802.05047)]. The Hesse simple group of order 1451520, which is the rotation group of the 7-dimensional polytope \(3_{21}\), is known to have eight classes of maximal subgroups. All but two of these classes can easily be described as stabilizers of certain geometric features of the \(3_{21}\). The two difficult subgroups \(U_ 3(3):2\) and \(L_ 2(8):3\), are described as stabilizers of two distinct Steiner \(S(2,4,28)\) systems whose 28 objects are the diameters of the \(3_{21}\), and whose tetrads are equatorial cubes of the \(3_{21}\).

The Gosset group of the title is a simple group, better known perhaps as \(O^+_8(2)\), or \(W(E_8)'/\{\pm 1\}\), where \(W(E_8)\) is the Weyl group of \(E_8\). It contains 17 classes of maximal subgroups. The aim of the paper is to describe in detail the correspondence between these subgroups and certain geometrical structures within the two 8-dimensional polytopes associated to the \(E_8\)-lattice. The corresponding exercise for \(E_7\) has already been carried out by the same author [ibid. 97, 83-89 (1993; Zbl 0802.05047)]. The purpose of this study is to describe the results obtained from a survey whose goal was to examine the combination of variables that have contributed to the success of a middle school math club. This is a case of a middle school in which the students are extremely successful in mathematics, and where the majority of the students voluntarily attend its math club. The results of the study show that the students have positive attitudes about mathematics and the club, and that some of the reasons that influenced them to attend the club were those of being with friends and eating donuts at the club. The results were similar for students of both genders and all grades. In addition, since positive attitudes are associated with higher levels of math achievement, such clubs have the potential to encourage students to enrol in additional mathematics classes while in high school, as well as pursuing mathematics related careers. (orig.)

The second author [\textit{F. Métayer}, ``Resolutions by polygraphs'', Theory Appl. Categ. 11, 148--184 (2003; Zbl 1020.18001)] defined homology using resolutions of \(\omega \)-categories via computads (polygraphs). The classical homology of a monoid \(M\) is defined in terms of resolutions of \(\mathbb{Z}\) by free \(\mathbb ZM\)-modules. The first author showed that the homologies coincide in the case where \(M\) is a group while the present paper proves the coincidence for a general monoid. An unfolding construction and path \(\omega \)-category are involved. The reviewer [\textit{R. H. Street}, J. Pure Appl. Algebra 8, 149-181 (1976; Zbl 0335.18005)] introduced the name computad (now 2-computad) for a directed graph equipped with 2-dimensional directed edges between directed paths in the graph. The name was chosen to suggest its relevance to formal computation as occurring in symbolic logic and computer science. Presentations of 2-categories by computads played a major role in the analysis of finite limits. The applicability of computads to 2-dimensional algebra was examined by \textit{J. Penon} [``Théories de catégories'', Diagrammes 2, 1-54 (1979; Zbl 0518.18010)]. In 1984, during the writing of ``The algebra of oriented simplexes'' [\textit{R. H. Street}, J. Pure Appl. Algebra 49, 283-335 (1987; Zbl 0661.18005)], the reviewer developed the inductive definition of \(n\)-computad for \(n\)-categories, yet found a way of expressing the requisite freeness property avoiding \(n\)-computads. The inductive definition appears in [\textit{A. J. Power}, Lect. Notes Math. 1488, 326-358 (1991; Zbl 0736.18004)]. In April 1985, after the reviewer's lecture on oriented simplexes, Samuel Eilenberg explained his work on higher rewriting systems and was excited by the connection to the lecture; over the next few years this was explored to some extent. More recently, the work of \textit{C. Squier, J. Groves, Y. Kobayashi}, and \textit{K. Brown} on rewriting systems in homological algebra was put into a higher categorical context by \textit{M. G. Citterio} [Theory Appl. Categ. 9, 92-105 (2001/02; Zbl 1018.18008)]. The very well written paper under review is a significant development in the area of non-additive homological algebra. The author introduces a notion of resolution of an \(n\)-category by a computad, closely related to a presentation, but instead of computad uses the term polygraph, as introduced by \textit{A. Burroni} [Theor. Comput. Sci. 115, 43-62 (1993; Zbl 0791.08004)]. The paper includes results on lifting morphisms to resolutions and the uniqueness of the liftings up to suitable homotopy.

The second author [\textit{F. Métayer}, ``Resolutions by polygraphs'', Theory Appl. Categ. 11, 148--184 (2003; Zbl 1020.18001)] defined homology using resolutions of \(\omega \)-categories via computads (polygraphs). The classical homology of a monoid \(M\) is defined in terms of resolutions of \(\mathbb{Z}\) by free \(\mathbb ZM\)-modules. The first author showed that the homologies coincide in the case where \(M\) is a group while the present paper proves the coincidence for a general monoid. An unfolding construction and path \(\omega \)-category are involved. In recent years, a variety of chaos-based digital image encryption algorithms have been suggested. Most of these algorithms implement permutations and diffusions at the pixel level by considering the pixel as the smallest (atomic) element of an image. In fact, a permutation at the bit level not only changes the position of the pixel but also alters its value. Here we propose an image cryptosystem employing the Arnold cat map for bit-level permutation and the logistic map for diffusion. Simulations have been carried out and analyzed in detail, demonstrating the superior security and high efficiency of our cryptosystem.

We present and study the concept of \(m\)-periodic Gorenstein objects relative to a pair \((\mathcal{A}, \mathcal{B})\) of classes of objects in an abelian category, as a generalization of \(m\)-strongly Gorenstein projective modules over associative rings. We prove several properties when \((\mathcal{A}, \mathcal{B})\) satisfies certain homological conditions, like for instance when \((\mathcal{A},\mathcal{B})\) is a GP-admissible pair. Connections to Gorenstein objects and Gorenstein homological dimensions relative to these pairs are also established. This paper generalizes the idea of the authors [in J. Pure Appl. Algebra 210, No. 2, 437-445 (2007; Zbl 1118.13014)]. Namely, we define and study a particular case of Gorenstein projective modules. We investigate some change of rings results for this new kind of modules. Examples over not necessarily Noetherian rings are given.

We present and study the concept of \(m\)-periodic Gorenstein objects relative to a pair \((\mathcal{A}, \mathcal{B})\) of classes of objects in an abelian category, as a generalization of \(m\)-strongly Gorenstein projective modules over associative rings. We prove several properties when \((\mathcal{A}, \mathcal{B})\) satisfies certain homological conditions, like for instance when \((\mathcal{A},\mathcal{B})\) is a GP-admissible pair. Connections to Gorenstein objects and Gorenstein homological dimensions relative to these pairs are also established. Methods are developed for structural analysis of metabolic networks expanding in size. Expansion proceeds in consecutive generations in which new reactions are attached to the network produced in the previous stage. Different rules are applied resulting in various modes of expansion. Expansion is performed on the set of glycolytic reactions as well as on a very large set of reactions taken from the KEGG database. It is shown that reactions and compounds strongly differ in the generation in which they are attached to the network allowing conclusions for the temporal order of the acquisition during network evolution. The expansion provides efficient tools for detecting new structural characteristics such as substrate-product relationships over long distances. (Copyright 2004 Japanese Society for Bioinformatics)

The concept of a full term of a given type was already introduced by \textit{K. Denecke} et al. [J. Autom. Lang. Comb. 6, No. 3, 253--262 (2001; Zbl 0993.68052)]. The set of these full terms forms a clone uder composition. Equipped with the superposition operation, this set forms an algebra satisfying the superassocoativity law. The freeness of this algebra is investigated using a generating set and a suitable homomorphism. Based on the theory of hypervarieities, the authors study hypersubstitutions and hyperidentities of the class of these algebras and detected solid varieties. For given depths of the terms \(s,t_1,\dots, t_n\) a formula is proved to calculate the depth of the composed term \(s(t_1,\dots, t_n)\) and if \(\sigma\) is a hypersubstitution and \(t\) is a term we derive a formula for the depth of \(\widehat\sigma[t]\).

The concept of a full term of a given type was already introduced by \textit{K. Denecke} et al. [J. Autom. Lang. Comb. 6, No. 3, 253--262 (2001; Zbl 0993.68052)]. The set of these full terms forms a clone uder composition. Equipped with the superposition operation, this set forms an algebra satisfying the superassocoativity law. The freeness of this algebra is investigated using a generating set and a suitable homomorphism. Based on the theory of hypervarieities, the authors study hypersubstitutions and hyperidentities of the class of these algebras and detected solid varieties. A novel finite element method for solving viscous Allen-Cahn equation with logarithmic potential is constructed in this paper. Firstly, for the logarithmic potential function, the domain of the function \(F(u)\) is extended from \((-1, 1)\) to \((-\infty, \infty)\) by regularization. Secondly, a new numerical scheme is proposed, that is, the mixed finite element method is used for discretization in the space and the second order scheme is adopted for discretization in the time. Then the unconditional energy stability of the numerical method is proved theoretically and the error estimates are analyzed in detail. Finally, some numerical examples are given to verify the effectiveness of the proposed method.

Let H be an Abelian semigroup, which is embeddable into a locally compact Abelian group. Let \(T_ t\) (t\(\in H)\), be a weakly continuous semigroup of contractions of the separable Banach space X. The author complements his result in Ergodic theory and related topics II, Proc. Conf., Georgenthal/GDR 1986, Teubner-Texte Math. 94, 108-112 (1987; Zbl 0637.22008) on equivalent conditions for \((T_ t)\) to be weakly mixing to zero, that is: for every \(x\in X\), \(x^*\in X^*\), there exists an invariant mean \(\ell\) so that \(\ell (| x^*,T,x|)=0.\) This generalises the well known result that a dynamical system is weakly mixing iff its two-fold product is ergodic. [For the entire collection see Zbl 0627.00018.] Let G be a locally compact abelian (additive) group, and let \(H\subset G\) be a semigroup with \(H\cup (-H)=G\). Furthermore, let \((T_ t)_{t\in H}\) be a weakly continuous contractive representation of H in a separable Banach space X. Given \(x\in X\), \((T_ t)_{t\in H}\) is called weakly mixing to zero in x if, for every x *\(\in X\) *, there exists an invariant mean \(\ell_{x\quad *,x}\) on H such that \(\ell_{x\quad *,x}(| <x\) \(*,T_.x>|)=0\). The note contains some characterizations of weak mixing to zero in x. Especially, under certain assumptions (e.g. in case X is reflexive) these results imply the following spectral mixing theorem: \((T_ t)_{t\in H}\) is weakly mixing to zero in x for all \(x\in X\) if and only if \(T\) \(*_ tx\) \(*=c(t)x\) *, \(| c(t)| =1\) (t\(\in H)\), implies x \(*=0\).

Let H be an Abelian semigroup, which is embeddable into a locally compact Abelian group. Let \(T_ t\) (t\(\in H)\), be a weakly continuous semigroup of contractions of the separable Banach space X. The author complements his result in Ergodic theory and related topics II, Proc. Conf., Georgenthal/GDR 1986, Teubner-Texte Math. 94, 108-112 (1987; Zbl 0637.22008) on equivalent conditions for \((T_ t)\) to be weakly mixing to zero, that is: for every \(x\in X\), \(x^*\in X^*\), there exists an invariant mean \(\ell\) so that \(\ell (| x^*,T,x|)=0.\) This generalises the well known result that a dynamical system is weakly mixing iff its two-fold product is ergodic. Common approximation tools return low-order approximations in the vicinities of singularities. Most prior works solve this problem for univariate functions. In this work we introduce a method for approximating non-smooth multivariate functions of the form \(f = g + r_+\) where \(g, r \in C^{M + 1}(\mathbb{R}^n)\) and the function \(r_+\) is defined by \[ r_+(y) = \begin{cases} r(y), & r(y) \geq 0 \\ 0, & r(y) < 0 \end{cases} , \qquad\forall y \in \mathbb{R}^n . \] Given scattered (or uniform) data points \(X \subset \mathbb{R}^n\), we investigate approximation by quasi-interpolation. We design a correction term, such that the corrected approximation achieves full approximation order on the entire domain. We also show that the correction term is the solution to a Moving Least Squares (MLS) problem, and as such can both be easily computed and is smooth. Last, we prove that the suggested method includes a high-order approximation to the locations of the singularities.

Let \(\Phi:\mathbb{R}\to [0,\infty)\) be a Young function such that \(\Phi\) is even, convex on \(\mathbb{R}\) and \(\Phi(0)=0\). Let \(\Omega\subset\mathbb{R}^n\) have finite Lebesgue measure and \(w\) be a weight on \(\Omega\), i.e., \(w\) is a positive locally integrable real function defined on \(\Omega\). The Orlicz (resp., weighted Orlicz) space \(L_\Phi(w)\) is induced by the modular \[ \rho(f) = \int_\Omega \Phi(f(x))\, dx \quad (\text{resp.,} \quad \rho(f,w) = \int_\Omega \Phi(f(x)) w(x)\, dx), \] and equipped with either the Luxemburg or Orlicz norm, which are equivalent. The definition can be formally generalized by replacing the Lebesgue measure by a general \(\sigma\)-finite measure space \((\Omega, \nu)\), where \(\Omega\) is an abstract set. Below there are samples of the main results which also hold true for general non-atomic \(\sigma\)-finite measure spaces. Theorem 1. Let \(\Phi\) be a Young function and for \(K>0\) put \(L_K(t) = \Phi(K\Phi^{-1}(t))\). Let \(S_K\) be the complementary function to \(L_K\) and assume that \(w\) is a weight function, bounded away from zero. Then \(L_\Phi(\chi_\Omega) = L_\Phi (w)\) if and only if there exists \(K>1\) such that \[ \int_\Omega S_K(w(x))\,dx < \infty. \] Theorem 2. Let \(w_j\), \(j=1,2\), be weight functions on \(\Omega\), \(w_j \geq 1\), and \(S_K\) be the function from Theorem 1 for some \(K>1\). Then \(L_\Phi(w_1) \hookrightarrow L_\Phi(w_2)\) if and only if \[ \int_\Omega S_k(w_2(x)/w_1(x)) w_1(x)\, dx < \infty. \] Theorem 3. For any Young function \(\Phi\), there exists an essentially unbounded weight \(w\) such that \(L_\Phi (\chi_\Omega) = L_\Phi(w)\) if and only if \[ \liminf_{t\to\infty} \Phi(Kt)/\Phi(t) =\infty \] for some \(K>1\). The applications of the above theorems allow to provide criteria for a composition operator to be continuous on \(L_\Phi(\Omega)\). This is an improvement and simplification of results in the literature. The main theorems are proved by using the techniques developed in the theory of Musiełak--Orlicz spaces. The authors consider composition operators \(C_{\tau}\) between Orlicz spaces \(L^{\phi}(X,\Sigma,\mu)\), generated by measurable and non-singular transformations \(\tau\) from \(\Omega\) into itself. They characterize the boundedness and compactness of composition operators between Orlicz spaces in terms of properties of the mapping \(\tau\), the function \(\phi\) and the measure space \((X,\Sigma,\mu)\). These results generalize earlier results known for \(L^p\)-spaces.

Let \(\Phi:\mathbb{R}\to [0,\infty)\) be a Young function such that \(\Phi\) is even, convex on \(\mathbb{R}\) and \(\Phi(0)=0\). Let \(\Omega\subset\mathbb{R}^n\) have finite Lebesgue measure and \(w\) be a weight on \(\Omega\), i.e., \(w\) is a positive locally integrable real function defined on \(\Omega\). The Orlicz (resp., weighted Orlicz) space \(L_\Phi(w)\) is induced by the modular \[ \rho(f) = \int_\Omega \Phi(f(x))\, dx \quad (\text{resp.,} \quad \rho(f,w) = \int_\Omega \Phi(f(x)) w(x)\, dx), \] and equipped with either the Luxemburg or Orlicz norm, which are equivalent. The definition can be formally generalized by replacing the Lebesgue measure by a general \(\sigma\)-finite measure space \((\Omega, \nu)\), where \(\Omega\) is an abstract set. Below there are samples of the main results which also hold true for general non-atomic \(\sigma\)-finite measure spaces. Theorem 1. Let \(\Phi\) be a Young function and for \(K>0\) put \(L_K(t) = \Phi(K\Phi^{-1}(t))\). Let \(S_K\) be the complementary function to \(L_K\) and assume that \(w\) is a weight function, bounded away from zero. Then \(L_\Phi(\chi_\Omega) = L_\Phi (w)\) if and only if there exists \(K>1\) such that \[ \int_\Omega S_K(w(x))\,dx < \infty. \] Theorem 2. Let \(w_j\), \(j=1,2\), be weight functions on \(\Omega\), \(w_j \geq 1\), and \(S_K\) be the function from Theorem 1 for some \(K>1\). Then \(L_\Phi(w_1) \hookrightarrow L_\Phi(w_2)\) if and only if \[ \int_\Omega S_k(w_2(x)/w_1(x)) w_1(x)\, dx < \infty. \] Theorem 3. For any Young function \(\Phi\), there exists an essentially unbounded weight \(w\) such that \(L_\Phi (\chi_\Omega) = L_\Phi(w)\) if and only if \[ \liminf_{t\to\infty} \Phi(Kt)/\Phi(t) =\infty \] for some \(K>1\). The applications of the above theorems allow to provide criteria for a composition operator to be continuous on \(L_\Phi(\Omega)\). This is an improvement and simplification of results in the literature. The main theorems are proved by using the techniques developed in the theory of Musiełak--Orlicz spaces. For a dynamic logic \(L\) we study dynamic logics \(L_ n\) for which programs allowed in formulas cannot use more than \(n\) variables. We prove that there exists a structure \(A\) of a finite signature such that for a wide class of dynamic logics \(L\) and for every natural number \(n\) the logic \(L_{n+1}\) is more expressive over \(A\) than \(L_ n\). This result is based on a construction of some canonical form for the formulas of \(L_ n\) over a free one-generated groupoid.

The authors study the quadratic congruential random number generator \(y_{n+1}= ay^ 2_ n+ by_ n+c \pmod m\), \(n\geq 0\), where \(a,b,c,y_ 0\in \mathbb{Z}_ m\) and \(m= p^ w\) is a prime power modulus with \(w\geq 3\). A sequence \((x_ n )_{n\geq 0}\) of quadratic congruential pseudorandom numbers in the interval \([0,1)\) is obtained by \(x_ n= y_ n/m\) for \(n\geq 0\). The main result of the paper is concerned with a discrepancy bound for the two-dimensional sequence \(x_ n= (x_ n, x_{n+1})\), \(0\leq n<m\). The authors prove for this discrepancy an explicit upper bound improving on an earlier one given in their paper [J. Comput. Appl. Math. 34, 243-249 (1991; Zbl 0731.11046)]. Knuth's quadratic congruential method [\textit{D. E. Knuth}, The Art of Computer Programming, Vol. 2 (1981; Zbl 0477.65002)] is analysed to examine the statistical independence of consecutive pairs in the pseudorandom stream produced. The basic congruence is \[ y_{n+1}\equiv ay^ 2_ n+by_ n+c(mod p^ m)\quad (n\geq 0) \] and a,b,c are to have optimal properties. Inequality results are obtained for the appropriately described discrepancy measure and this enables a contrast to be made with the performance of the linear congruential method mod \(p^ m\).

The authors study the quadratic congruential random number generator \(y_{n+1}= ay^ 2_ n+ by_ n+c \pmod m\), \(n\geq 0\), where \(a,b,c,y_ 0\in \mathbb{Z}_ m\) and \(m= p^ w\) is a prime power modulus with \(w\geq 3\). A sequence \((x_ n )_{n\geq 0}\) of quadratic congruential pseudorandom numbers in the interval \([0,1)\) is obtained by \(x_ n= y_ n/m\) for \(n\geq 0\). The main result of the paper is concerned with a discrepancy bound for the two-dimensional sequence \(x_ n= (x_ n, x_{n+1})\), \(0\leq n<m\). The authors prove for this discrepancy an explicit upper bound improving on an earlier one given in their paper [J. Comput. Appl. Math. 34, 243-249 (1991; Zbl 0731.11046)]. This paper is devoted to the computation of partial quotiens of decomposition of \(\frac{a_m}{N_m}\) into a continued fraction, where \({a_m}\) and \({N_m}\) are parameters of a parallelepipedal grid, which is an approximation of a quadratic algebraic grid by a rational grid. The results of this work allow us to implement fast algorithms for calculating the quality function of these rational approximations.

X\({}_ 1,...,X_ n\) are independent and identically distributed random p-dimensional vectors. The jth component of \(X_ i\) is assumed to have a marginal distribution which is symmetric about an unknown location parameter \(\theta_ j\). The problem is to estimate the vector \(\theta =(\theta_ 1,...,\theta_ p)\). If \(D=(D_ 1,...,D_ p)\) is the vector of estimators, the loss is \((D-\theta)Q(D-\theta)^ T\), where Q is a given positive definite matrix. This paper studies the asymptotic (as n increases) properties of the risk function when D is a class of M-estimators constructed by shrinking the estimators towards a given point using a James-Stein type of modification. M-estimators are described in \textit{P. J. Huber}, Robust statistics. (1981; Zbl 0536.62025). To my knowledge, the present monograph is the first systematic, booklength exposition of robust statistics. It gives a good foundation in robustness, but more to the theoretical than to the applied statistician. The treatment is theoretical, but the stress is on concepts rather than on mathematical completeness. The level of presentation is deliberately uneven: in some chapters single cases are treated with mathematical rigor; in others the results obtained in the simple cases are transferred by analogy to more complicated situations, where proofs are not always available. Also, selected numerical algorithms for computing robust estimates are described and, where possible, convergence proofs are given. The main point of this book is on estimation theory, only a few sections in Chapter 10 are related to test theory. Chapter 1 gives a general introduction and overview on the different concepts of robustness: qualitative, quantitative, influence curve, sensitivity curve, a.o. Chapter 2 contains an account of the formal mathematical background behind qualitative and quantitative robustness, e.g. weak topology, Lévy and Prohorov metrics, Fréchet and Gâteaux derivatives. Chapter 3 introduces and discusses the three basic types of estimates: M-, L-, and R-estimates. Chapter 4 treats the asymptotic minimax theory for estimating a location parameter. Chapter 5 deals with scale estimates and Chapter 6 with multiparameter problems, in particular joint estimation of location and scale. Chapter 7 considers the regression problem and investigates robust estimates of the parameters. Chapter 8 deals with robust covariance and correlation matrices, and Chapter 9 with the robustness of design. Chapter 10 gives exact finite sample results, especially in connection with some robust tests as mentioned above. Finally, in Chapter 11 miscellaneous topics are discussed.

X\({}_ 1,...,X_ n\) are independent and identically distributed random p-dimensional vectors. The jth component of \(X_ i\) is assumed to have a marginal distribution which is symmetric about an unknown location parameter \(\theta_ j\). The problem is to estimate the vector \(\theta =(\theta_ 1,...,\theta_ p)\). If \(D=(D_ 1,...,D_ p)\) is the vector of estimators, the loss is \((D-\theta)Q(D-\theta)^ T\), where Q is a given positive definite matrix. This paper studies the asymptotic (as n increases) properties of the risk function when D is a class of M-estimators constructed by shrinking the estimators towards a given point using a James-Stein type of modification. M-estimators are described in \textit{P. J. Huber}, Robust statistics. (1981; Zbl 0536.62025). [For the entire collection see Zbl 0523.00001.] Let T denote a rooted triangulation of the sphere, multiple edges allowed but not loops, with 2n faces. A triangulation is rooted when an edge is chosen, a direction is assigned to the edge and a left right orientation is chosen with respect to the directed edge. Let P(T,\(\lambda)\) denote the chromatic polynomial of the graph of T and let \(h_ n\) denote the sum of P(T,\(\lambda)\) over all rooted triangulations of the sphere with 2n faces. Starting from W. T. Tutte's nonlinear differential that the generating function, h(t), of the \(h_ n\) satisfies we show that for \(\lambda\geq 5\) or \(3(5/11)<\lambda<4\) the estimate \(h_ n\sim CR^{- n}/n^{5/2}\), \(n\to \infty\), \(\lambda\) fixed, is valid where \(R<1\) is the radius of convergence of h(t) and C is a positive constant defined explicitly in terms of R, \(\lambda\), and h'(R).

In [Ann. Inst. Fourier 35, No. 4, 163-174 (1985; Zbl 0564.46044)], \textit{J. Bourgain} proved that if \(\alpha\) is a real-valued function satisfying \(\alpha(t)/t\to 0\) at \(t\to 0\) and if \(f, f_1,\dots,f_n\in H^{\infty}(\mathbb T)\) are such that \(|f|\leq\alpha(|f_1|+\dots+|f_n|)\) on the unit disk \({\mathbb D}\), then \(f\) is in the closed ideal \(I\) generated by \(f_1,\dots,f_n\). In this paper the authors examine Bourgain's proof and extend it from Banach algebra \(H^{\infty}(\mathbb T)\) to arbitrary Douglas algebras. The essential difference in this case is that some estimates involving \({\overline{\partial}}\)-equations are no longer valid on \(D\) but rather on regions of \(D\) that are asymptotically close to the maximal ideal space of the Douglas algebra. In particular, an estimate for distance \(\text{dist}_A (f, I)\) from a function \(f\) in the Douglas algebra \(A\) to a finitely generated ideal \(I\) in \(A\) is obtained. Assume \(f_ 1,...,f_ N\) a finite set of functions in \(H^{\infty}(D)\), the space of bounded analytic functions on the open unit disc. We give a sufficient condition on a function f in \(H^{\infty}(D)\) to belong to the norm-closure of the ideal \(I(f_ 1,...,f_ N)\) generated by \(f_ 1,...,f_ N\), namely the property \[ | f(z)| \leq \alpha (| f_ 1(z)| +...+| f_ N(z)|)\quad for\quad z\in D \] for some function \(\alpha\) : \({\mathbb{R}}_+\to {\mathbb{R}}_+\) satisfying \(\lim_{t\to 0}\alpha (t)/t=0.\) The main feature in the proof is an improvement in the contour- construction appearing in L. Carleson's solution of the corona-problem. It is also shown that the property \[ | f(z)| \leq C\max_{1\leq j\leq N}| f_ j(z)| \quad for\quad z\in D \] for some constant C, does not necessary imply that f is in the closure of \(I(f_ 1,...,f_ N)\).

In [Ann. Inst. Fourier 35, No. 4, 163-174 (1985; Zbl 0564.46044)], \textit{J. Bourgain} proved that if \(\alpha\) is a real-valued function satisfying \(\alpha(t)/t\to 0\) at \(t\to 0\) and if \(f, f_1,\dots,f_n\in H^{\infty}(\mathbb T)\) are such that \(|f|\leq\alpha(|f_1|+\dots+|f_n|)\) on the unit disk \({\mathbb D}\), then \(f\) is in the closed ideal \(I\) generated by \(f_1,\dots,f_n\). In this paper the authors examine Bourgain's proof and extend it from Banach algebra \(H^{\infty}(\mathbb T)\) to arbitrary Douglas algebras. The essential difference in this case is that some estimates involving \({\overline{\partial}}\)-equations are no longer valid on \(D\) but rather on regions of \(D\) that are asymptotically close to the maximal ideal space of the Douglas algebra. In particular, an estimate for distance \(\text{dist}_A (f, I)\) from a function \(f\) in the Douglas algebra \(A\) to a finitely generated ideal \(I\) in \(A\) is obtained. Die Lösung der genannten Aufgabe der Elastostatik. Zwei Fälle - (1) die an beiden Enden frei gestützte Röhre und (2) die an einem Ende eingespannte Röhre.

It is proved that, if a homeomorphism of a separable locally compact metric space has a unique fixed point that is attracting or repelling, then its corresponding composition operator is cyclic. On the real line, it is shown that a composition operator is cyclic if and only if its symbol has at most one fixed point. Other results on the real line and the circle are discussed. This study is motivated by results of \textit{Z. Yin} [Int. J. Bifurcation Chaos Appl. Sci. Eng. 27, No. 6, Article ID 1750084, 12 p. (2017; Zbl 1370.37074)] who implicitly asked if the no-fixed-point condition is necessary for a composition operator to be cyclic on the real line. In this paper, the chaotic dynamics of composition operators on the space of real-valued continuous functions is investigated. It is proved that the hypercyclicity, topologically mixing property, Devaney chaos, frequent hypercyclicity and the specification property of the composition operator are equivalent to each other and are stronger than dense distributional chaos. Moreover, the composition operator \(C_\phi\) exhibits dense Li-Yorke chaos if and only if it is densely distributionally chaotic, if and only if the symbol \(\phi\) admits no fixed points. Finally, the long-time behaviors of the composition operator with affine symbol are classified in detail.

It is proved that, if a homeomorphism of a separable locally compact metric space has a unique fixed point that is attracting or repelling, then its corresponding composition operator is cyclic. On the real line, it is shown that a composition operator is cyclic if and only if its symbol has at most one fixed point. Other results on the real line and the circle are discussed. This study is motivated by results of \textit{Z. Yin} [Int. J. Bifurcation Chaos Appl. Sci. Eng. 27, No. 6, Article ID 1750084, 12 p. (2017; Zbl 1370.37074)] who implicitly asked if the no-fixed-point condition is necessary for a composition operator to be cyclic on the real line. This brief paper deals with realization and partial realization results for time-invariant systems with, in general, incommensurate delays. The results are obtained based upon the use of formal Laurent expansions whose coefficients are polynomial matrices of appropriate orders and which also integrated within truncated and infinite block Hankel matrices. The above mentioned polynomial matrices arise in a natural way from the transcendent equations associated with the delayed dynamics.

A ring with unity is called \textit{Baer (quasi-Baer)} if the left annihilator of each nonempty set (ideal) is generated by an idempotent element. The origins of the class of Baer rings evolved as an abstraction of the strictly algebraic properties of von Neumann algebras. This concept has been extended to nearrings. However in the classes of nearrings and rings without unity, the Baer concept splits into at least four distinct classes and at least eight classes for the quasi-Baer concept (see below). We investigate certain nearring and ring decompositions induced by Baer or quasi-Baer annihilator conditions. Examples are provided to illustrate and delimit our results. The present authors have already published several papers, too numerous to list here, some in collaboration with others, on annihilator conditions on polynomials and various types of Baer conditions on rings and near-rings. To describe the many results in this paper in detail would require a lot of space, so we will just give a flavour. A Baer ring is a ring with unity such that the right annihilator of every nonempty subset is generated as a right ideal by an idempotent. As often happens, the transfer of this definition and related ones to near-rings leads to several possibilities. Given a ring \(R\), the relationship between the Baer type properties of \(R\) and \(R[x]\) have been studied in part I [the authors, Commun. Algebra 29, No. 5, 2097--2112 (2001; Zbl 0990.16033)]. This has been taken further in this work by considering the situation in left near-rings \(N\) and the relationship between the various properties in a ring \(R\) and the near-ring \(R[x]\) of polynomials over \(R\), where the operations in \(R[x]\) are the usual addition and substitution. Several results linking the properties of the ring \(R\) and the near-ring \(R[x]\) are obtained, many similar to earlier ring results although the structure of \(R[x]\) arises from a very different operation to that in the ring of polynomials. There are also examples to show the limits of the results. There is much of interest here.

A ring with unity is called \textit{Baer (quasi-Baer)} if the left annihilator of each nonempty set (ideal) is generated by an idempotent element. The origins of the class of Baer rings evolved as an abstraction of the strictly algebraic properties of von Neumann algebras. This concept has been extended to nearrings. However in the classes of nearrings and rings without unity, the Baer concept splits into at least four distinct classes and at least eight classes for the quasi-Baer concept (see below). We investigate certain nearring and ring decompositions induced by Baer or quasi-Baer annihilator conditions. Examples are provided to illustrate and delimit our results. A two-dimensional model problem is considered for the potential \(\varphi\) in a human body of the current sources in the heart, which leads to a Cauchy problem for the 2D Laplace equation. This ill-posed problem is transformed into a first kind Fredholm integral equation. The numerical method is based on regularization by discretization, where a conjugate gradient iterative method is used. Numerical results of the solution of the model problem with perturbed initial data are presented.

Bielliptic and quasi-bielliptic surfaces form one of the four classes of minimal smooth projective surfaces of Kodaira dimension 0. In this article, we determine the automorphism group schemes of these surfaces over algebraically closed fields of arbitrary characteristic, generalizing work of \textit{C. Bennett} and \textit{R. Miranda} over the complex numbers [Rocky Mt. J. Math. 20, No. 1, 31--37 (1990; Zbl 0705.14042)]; we also find some cases that are missing from the classification of automorphism groups of bielliptic surfaces in characteristic 0. In this paper, the automorphism groups of the seven classes of the so- called hyperelliptic surfaces are calculated. Writing these as (E\(\times F)/G\), where E and F are elliptic curves and G is a finite group of translations of E acting on F not only as translations, covering space theory is then used to calculate the automorphisms. Letting M be the centralizer of G in Aut(E)\(\times Aut(F)\), it is then noted that in all cases M is generated by its E-translations, its F-translations, its E- automorphisms, and its F-automorphisms. Finally, two tables list the automorphism groups and generators for each.

Bielliptic and quasi-bielliptic surfaces form one of the four classes of minimal smooth projective surfaces of Kodaira dimension 0. In this article, we determine the automorphism group schemes of these surfaces over algebraically closed fields of arbitrary characteristic, generalizing work of \textit{C. Bennett} and \textit{R. Miranda} over the complex numbers [Rocky Mt. J. Math. 20, No. 1, 31--37 (1990; Zbl 0705.14042)]; we also find some cases that are missing from the classification of automorphism groups of bielliptic surfaces in characteristic 0. A useful challenge is to numerically solve differential algebraic equations (DAEs) such as \(M\dot{\mathbf u}= {\mathbf g}(t,{\mathbf u})\) for singular \(M\), or more generally \[ {\mathbf f}(t,{\mathbf u}\dot{\mathbf u})=\mathbf{0},\quad \text{where }M= {\partial{\mathbf f}\over\partial\dot{\mathbf u}}\text{ is singular}. \] Such DAEs are very close to a stiff ordinary differential equations (ODEs). Thus numerical integration methods for DAEs may be very similar to those for stiff ODEs. The most basic numerical scheme to integrate stiff ODEs such as \(\dot{\mathbf u}={\mathbf g}(t,{\mathbf u})\) is the implicit Euler scheme: \(({\mathbf u}_{k+1}-{\mathbf u}_k)/h= {\mathbf g}(t_{k+1},{\mathbf u}_{k_1})\) for time step \(h\). Although only first-order accurate, it is robust and eminently suitable for solving DAEs. In the hurly burly of computer arithmetic, round-off error may cause a DAE to look like a stiff ODE, possibly on ODE with extremely rapid growth. A numerical scheme for DAEs also must squash extremely rapid growth as well as it does rapid decay, just as the implicit Euler scheme does.

One considers the canonical fiber bundle \(\pi :{\mathbb R}^{n+1}\rightarrow {\mathbb R}^n\), the natural action of the Lie algebra of the Poincaré group \(P(1,n)\) in the module of vector fields over the fiber bundle and one determines the system of equations satisfied by the local coefficients of a \(P(1,n)\)-invariant connnection on the bundle. It comes out that the coefficients of a \(P(1,n)\)-invariant connection form geometric objects of the given representation of the group \(P(1,n)\) and one studies the relations between certain \(P(1,n)\)-invariant connections. For Part II, see ibid. 42, 81-87 (2002; Zbl 1030.53028) below. One considers the canonical fiber bundle \(\pi :{\mathbb R}^{n+1}\rightarrow {\mathbb R}^n\), the natural action of the Lie algebra of the Poincaré group \(P(1,n)\) in the module of vector fields over the fiber bundle and one determines the system of equations satisfied by the local coefficients of a \(P(1,n)\)-invariant connnection on the bundle. It comes out that the coefficients of a \(P(1,n)\)-invariant connection form geometric objects of the given representation of the group \(P(1,n)\) and one studies the relations between certain \(P(1,n)\)-invariant connections. For Part II, see ibid. 42, 81-87 (2002; Zbl 1030.53028) below.

One considers the canonical fiber bundle \(\pi :{\mathbb R}^{n+1}\rightarrow {\mathbb R}^n\), the natural action of the Lie algebra of the Poincaré group \(P(1,n)\) in the module of vector fields over the fiber bundle and one determines the system of equations satisfied by the local coefficients of a \(P(1,n)\)-invariant connnection on the bundle. It comes out that the coefficients of a \(P(1,n)\)-invariant connection form geometric objects of the given representation of the group \(P(1,n)\) and one studies the relations between certain \(P(1,n)\)-invariant connections. For Part II, see ibid. 42, 81-87 (2002; Zbl 1030.53028) below. A controller to rotate a rigid body between two successive orientations is designed. Particular features are the fact that it is based on the quaternion approach, known to provide singularity-free attitude description, and it is adaptive in the sense that it does not need specific knowledge of the inertia matrix. Global stability is proved analytically and tested in computer simulations.

All rings considered in this paper are commutative with identity. Let \(D\) be an integral domain with quotient field \(K,\) \(S\) be a multiplicatively closed subset of \(D\) and \(I\) a nonzero ideal of \(D.\) Following \textit{A. Hamed} and \textit{S. Hizem} [J. Pure Appl. Algebra 221, No. 11, 2869--2879 (2017; Zbl 1369.13015)], we say that \(I\) is \(S\)-\(v\)-principal if there are \(s\in S\) and \(a\in I_{v} \) such that \(sI\subset aD\) and \(D\) is called an \(S\)-GCD domain if each finitely generated nonzero ideal of \(D\) is \(S\)-\(v\)-principal. In this paper under review the author studies the transfer of the \(S\)-GCD property from the integral domain \(D\) to the ring of power series \(D[[X]].\) A nonzero fractional ideal of \(D\) is called \(S\)-\(v\)-invertible if there exists \(s\in S\) and a fractional ideal \(J\) of \(D\) such that \(sD\subset (IJ)_{v}\subset D.\) On the other hand, the ring \(D\) is called an \(S\)-\( H^{\ast }\)-domain if for any two integral \(S\)-\(v\)-invertible \(v\)-ideals \(I\) and \(J\) of \(D[[X]]\) such that \((IJ)_{0}\neq (0),\) there exists an \(s\in S\) such that \(s((IJ)_{0})_{v}\subset ((IJ)_{v})_{0}\) and \(s((IJ)_{v})_{0} \subset ((IJ)_{0})_{v},\) where \(I_{0}=\{f(0)\mid f\in I\}.\) An \(H^{\ast }\)-domain is an \(S\)-\(H^{\ast }\)-domain for \(S=\{1\}.\) First of all, the author investigates the relation between \(S\)-\(v\)-invertible \(v\)-ideals of \(D\) and those of \(D[[X]].\) He shows that if \(S\) is an anti-archimedean multiplicative subset of \(D\) and if \(D\) is an \(S\)-\( H^{\ast }\)-domain then for each \(S\)-\(v\)-invertible \(v\)-ideal \(I\) of \(D[[X]],\) there exists an \(S\)-\(v\)-invertible \(v\)-ideal \(L\) of \(D,\) an \(s\in S\) and a \( g\in K[[X]]\) such that \(sI\subset gL[[X]]\subset I.\) Based on this result, the author proves that if \(D\) is an integral domain such that \(D[[X]]\) is a \(TV\)-domain and \(S\) is an anti-archimedean multiplicative subset of \(D\) such that every finitely generated ideal of \( D[[X]]\) is \(S\)-\(v\)-invertible then \(D[[X]]\) is an \(S\)-GCD domain if and only if \(D\) is an \(S\)-GCD \(S\)-\(H^{\ast }\)-domain. As a consequence, a new characterization for a power series ring to be a GCD domain is given. More precisely, for a Krull domain \(D,\) the ring \(D[[X]]\) is a GCD domain if and only if \(D\) is a GCD, \(H^{\ast }\)-domain which implies in particular that for a UFD \(D,\) \(D[[X]]\) is a GCD domain if and only if \(D\) is an \(H^{\ast }\)-domain. Let's first recall some basic facts to understand the content of the paper. For an integral domain \(A\) and a nonzero fractional ideal \(I\) of \(A\), we put \(I_t=\bigcup\{J_v; J\) is a nonzero finitely generated fractional subideal of \(I\}\), where \(J_v=(J^{-1})^{-1}\). When \(I_t=I\) we say that \(I\) is a \(t\)-ideal and if \((II^{-1})_t=A\) then \(I\) is said to be \(t\)-invertible. The set \(T(A)\) of \(t\)-invertible fractional \(t\)-ideals of \(A\) is a group under the \(t\)-multiplication \(I*J=(IJ)_t\). The subset \(P(A)\) of nonzero principal fractional ideals of \(A\) is a subgroup of \(T(A)\). The quotient group \(Cl_t(A)=T(A)/P(A)\) is called the \(t\)-class group of \(A\). The authors studied the mapping \(\phi:Cl_t(A)\longrightarrow Cl_t(A[[X]])\) defined by \(\phi([I])=[(IA[[X]])_t]\), and extended old results of Claborn. To describe the content of the last part of the paper, let \(S\) be a multiplicative set of \(A\), not necessarily saturated. An ideal \(I\) of \(A\) is said to be \(S\)-principal if there exist \(s\in S\) and \(a\in I\) such that \(sI\subseteq aA\subseteq I\). The set denoted \(S-P(A)\) of the \(S\)-principal \(t\)-invertible \(t\)-ideals of \(A\) is a subgroup of \(T(A)\). Let \(S-Cl_t(A)=T(A)/S-P(A)\) be the quotient group. The authors generalize some classical results to \(S-Cl_t(A)\). The case when the isomorphism \(S-Cl_t(A)\simeq S-Cl_t(A[[X]])\) holds is investigated.

All rings considered in this paper are commutative with identity. Let \(D\) be an integral domain with quotient field \(K,\) \(S\) be a multiplicatively closed subset of \(D\) and \(I\) a nonzero ideal of \(D.\) Following \textit{A. Hamed} and \textit{S. Hizem} [J. Pure Appl. Algebra 221, No. 11, 2869--2879 (2017; Zbl 1369.13015)], we say that \(I\) is \(S\)-\(v\)-principal if there are \(s\in S\) and \(a\in I_{v} \) such that \(sI\subset aD\) and \(D\) is called an \(S\)-GCD domain if each finitely generated nonzero ideal of \(D\) is \(S\)-\(v\)-principal. In this paper under review the author studies the transfer of the \(S\)-GCD property from the integral domain \(D\) to the ring of power series \(D[[X]].\) A nonzero fractional ideal of \(D\) is called \(S\)-\(v\)-invertible if there exists \(s\in S\) and a fractional ideal \(J\) of \(D\) such that \(sD\subset (IJ)_{v}\subset D.\) On the other hand, the ring \(D\) is called an \(S\)-\( H^{\ast }\)-domain if for any two integral \(S\)-\(v\)-invertible \(v\)-ideals \(I\) and \(J\) of \(D[[X]]\) such that \((IJ)_{0}\neq (0),\) there exists an \(s\in S\) such that \(s((IJ)_{0})_{v}\subset ((IJ)_{v})_{0}\) and \(s((IJ)_{v})_{0} \subset ((IJ)_{0})_{v},\) where \(I_{0}=\{f(0)\mid f\in I\}.\) An \(H^{\ast }\)-domain is an \(S\)-\(H^{\ast }\)-domain for \(S=\{1\}.\) First of all, the author investigates the relation between \(S\)-\(v\)-invertible \(v\)-ideals of \(D\) and those of \(D[[X]].\) He shows that if \(S\) is an anti-archimedean multiplicative subset of \(D\) and if \(D\) is an \(S\)-\( H^{\ast }\)-domain then for each \(S\)-\(v\)-invertible \(v\)-ideal \(I\) of \(D[[X]],\) there exists an \(S\)-\(v\)-invertible \(v\)-ideal \(L\) of \(D,\) an \(s\in S\) and a \( g\in K[[X]]\) such that \(sI\subset gL[[X]]\subset I.\) Based on this result, the author proves that if \(D\) is an integral domain such that \(D[[X]]\) is a \(TV\)-domain and \(S\) is an anti-archimedean multiplicative subset of \(D\) such that every finitely generated ideal of \( D[[X]]\) is \(S\)-\(v\)-invertible then \(D[[X]]\) is an \(S\)-GCD domain if and only if \(D\) is an \(S\)-GCD \(S\)-\(H^{\ast }\)-domain. As a consequence, a new characterization for a power series ring to be a GCD domain is given. More precisely, for a Krull domain \(D,\) the ring \(D[[X]]\) is a GCD domain if and only if \(D\) is a GCD, \(H^{\ast }\)-domain which implies in particular that for a UFD \(D,\) \(D[[X]]\) is a GCD domain if and only if \(D\) is an \(H^{\ast }\)-domain. A proper vertex \(k\)-coloring of a graph \(G\) is called dynamic, if for every vertex \(v\) with degree at least 2, the neighbors of \(v\) receive at least two different colors. The smallest integer \(k\) such that \(G\) has a \(k\)-dynamic coloring is called the dynamic chromatic number of \(G\) and denoted by \(\chi_2(G)\). In this paper we study the dynamic chromatic number of graphs all of whose cycles have lengths divisible by \(\ell\), \(\ell\geq 2\). Let \(G\) be a graph and \(\ell\geq 3\) be a natural number. We prove that if the length of every cycle of \(G\) is divisible by \(\ell\) and \(G\) has no component isomorphic to \(C_5\), then \(\chi_2(G) \leq 4\). Also, it is shown that for every \(k\)-regular bipartite graph \(G\) \((k\geq 4)\), there is a \(4\)-dynamic coloring of \(G\) using \(2\) colors in each part.

The starting point for this research is the paper by \textit{Y. Lou} [J. Differ. Equations 223, No. 2, 400--426 (2006; Zbl 1097.35079)] dealing with the dynamics of a single species described by the logistic reaction-diffusion equation \[ \begin{aligned} u_{t} & =d\Delta u+u\left( m\left( x\right) -u\right) \quad \text{in }\Omega\times\mathbb{R}^{+},\\ \partial_{\nu}u & =0\quad\text{on }\partial\Omega\times\mathbb{R}^{+}, \end{aligned} \] where \(u\left( x,t\right) \) is the population density of a species at the location \(x\in\Omega\) and at the instant \(t>0,\) \(d\) is the dispersal rate of \(u,\) \(\Omega\) is a bounded domain in \(\mathbb{R}^{N},\) and \(m\left( x\right) \) represents both the local carrying capacity and the intrinsic growth rate of the species \(u.\) Lou made a striking observation that a population diffusing at any rate in an environment with spatially varying resources will reach a higher total equilibrium biomass in comparison with the population in an environment where total resources are distributed homogeneously, that is, \[ \int_{\Omega}\theta_{\mathrm{d}}>\int_{\Omega}m, \] where \(\theta_{\mathrm{d}}\) is a unique positive steady state. Arguing that the use of the same function \(m\left( x\right) \) for the representation of the local carrying capacity and the intrinsic growth rate is not realistic in ecology, the authors modify the logistic reaction term introducing two parameters, \(r(x)\) for intrinsic growth rate, and \(K(x)\) for carrying capacity, \[ r\left( x\right) u\left( 1-\frac{u}{K(x)}\right) . \] They establish the existence, uniqueness, and asymptotic stability of the positive steady state \(u_{\mathrm{d}}\) of a modified equation and explore its various properties paying a special attention to the case when the condition \[ \int_{\Omega}u_{\mathrm{d}}>\int_{\Omega}K \] holds. In the final part of the paper, the relevance of the results and their implications to population theory and conservation ecology are discussed. The paper is organized in two parts. The first one mainly deals with the total size of a population at equilibrium which satisfies a logistic type equation with diffusion in a spatially heterogeneous habitat which results in a nonconstant (and possibly taking negative values somewhere) intrinsic growth rate. There is only one positive equilibrium population distribution, which tends to the positive part of the intrinsic growth rate when the migration rate (the diffusion coefficient) tends to zero and to a homogeneous profile equal to the mean value of the intrinsic growth rate when the migration rate goes to infinity. The author proves that for any value of the migration rate, the total population is larger than the one corresponding to an infinite migration rate and moreover that, if the intrinsic growth rate is nonnegative, then the total population size is maximized for an intermediate value of the migration rate and minimized for values of the migration rate equal to zero and to infinity. These results are used in the second part which is devoted to the study of a Lotka-Volterra system for the competition of two species with diffusion in a heterogeneous habitat and the same intrinsic growth rate. The main results are related to the possibility of invasion of a species in logistic equilibrium by a very small population of the second species, that is, to the study of the linear stability of the ``semi-trivial'' equilibrium with zero population of the second species. The author considers the case of intra-specific competition larger than the inter-specific one (when coexistence is assured if mutation rate vanishes, i.e. the semi-trivial equilibrium is unstable and there is an attracting coexistence equilibrium) and proves the very interesting result that when diffusion (migration) is present, the invading species may go extinct for values of its mutation rate larger than the ones of the ``resident'' species (by proving that the semi-trivial equilibrium is stable). Moreover, if the intrinsic growth rate is nonnegative, the semi-trivial equilibrium may even be globally asymptotically stable proving that in this case there is no a coexistence steady state. The proofs are mainly based in the maximum principle, the characterization of the eigenvalues by the Rayleigh quotient, and regularity and compactness results from the theory of elliptic equations.

The starting point for this research is the paper by \textit{Y. Lou} [J. Differ. Equations 223, No. 2, 400--426 (2006; Zbl 1097.35079)] dealing with the dynamics of a single species described by the logistic reaction-diffusion equation \[ \begin{aligned} u_{t} & =d\Delta u+u\left( m\left( x\right) -u\right) \quad \text{in }\Omega\times\mathbb{R}^{+},\\ \partial_{\nu}u & =0\quad\text{on }\partial\Omega\times\mathbb{R}^{+}, \end{aligned} \] where \(u\left( x,t\right) \) is the population density of a species at the location \(x\in\Omega\) and at the instant \(t>0,\) \(d\) is the dispersal rate of \(u,\) \(\Omega\) is a bounded domain in \(\mathbb{R}^{N},\) and \(m\left( x\right) \) represents both the local carrying capacity and the intrinsic growth rate of the species \(u.\) Lou made a striking observation that a population diffusing at any rate in an environment with spatially varying resources will reach a higher total equilibrium biomass in comparison with the population in an environment where total resources are distributed homogeneously, that is, \[ \int_{\Omega}\theta_{\mathrm{d}}>\int_{\Omega}m, \] where \(\theta_{\mathrm{d}}\) is a unique positive steady state. Arguing that the use of the same function \(m\left( x\right) \) for the representation of the local carrying capacity and the intrinsic growth rate is not realistic in ecology, the authors modify the logistic reaction term introducing two parameters, \(r(x)\) for intrinsic growth rate, and \(K(x)\) for carrying capacity, \[ r\left( x\right) u\left( 1-\frac{u}{K(x)}\right) . \] They establish the existence, uniqueness, and asymptotic stability of the positive steady state \(u_{\mathrm{d}}\) of a modified equation and explore its various properties paying a special attention to the case when the condition \[ \int_{\Omega}u_{\mathrm{d}}>\int_{\Omega}K \] holds. In the final part of the paper, the relevance of the results and their implications to population theory and conservation ecology are discussed. Vorzüglicher Überblick über Umfang und Wesen der griechischen Mathematik und Astronomie, wobei auch dargelegt wird, was schon in der vorgriechischen Mathematik erreicht war und was griechischer Genius demgegenüber grundsätzlich Neues mit dem Aufbau des mathematischen Systems schuf. Bei den quadratischen Gleichungen sollte noch besonders betont werden, daß die griechische Lösungsmethode die genaue geometrische Übersetzung der babylonischen Formel darstellt. Man darf wohl annehmen, daß die arithmetische Lösungsmethode, die erst wieder bei \textit{Heron} nachweisbar ist, in der Zwischenzeit nicht verloren war; sie hatte eben in den wissenschaftlichen, d. h. geometrischen, Abhandlungen wie auch andere Teile der Logistik keinen Platz.

The article is dedicated to a wide variety of questions connected with amenability and paradoxical decompositions for pseudogroups and discrete metric spaces. The authors effectively demonstrate that paradoxical decompositions are associated with amenability and not with the property of ``exponential growth'' [compare with \textit{W. A. Deuber}, \textit{M. Simonowitz} and \textit{V. T. Sós}, Stud. Sci. Math. Hung. 30, 17-23 (1995; Zbl 0857.54030)]. It is proven that the existence of invariant means for pseudogroups is equivalent to Følner's condition. The authors define the Tarski number as the minimal number of pieces in paradoxical decompositions associated with nonamenable group actions and give estimates for this number. Let \(X\), \(Y\) be subsets of a metric space with metric \(d\). A mapping \(\varphi:X\to Y\) is called \(k\)-wobbling if \(\sup_{x\in X}d(\varphi(x),x)<k\). The sets \(X\), \(Y\) are wobbling equivalent if \(\varphi\) is a bijection. Such mappings were investigated by \textit{M. Laczkovich} [J. Reine Angew. Math. 404, 77-117 (1990; Zbl 0748.51017)] and using some of his results the authors prove that every Penrose tiling is equivalent to \(\tau\mathbb{Z}^2\) for some \(\tau\in\mathbb{R}\). Motivated by the Banach-Tarski paradoxical decomposition of the unit ball the authors say that a metric space \((M,d)\) is paradoxical if there is a decomposition \(M=M_1\dot\cup M_2\) such that \(M_1\), \(M_2\), \(M\) are pairwise wobbling equivalent. Using generalized Hall-Rado theorem about \((l_1,l_2)\)-matchings for countable locally finite bipartite graphs a characterization of paradoxical discrete countable spaces is given by a local test estimating size of neighbourhoods of finite sets. Considering graphs as metric spaces with the distance the length of the shortest path they prove that a locally finite infinite tree is paradoxical if and only if the lengths of chains with all vertices of degree 2 are bounded.

The article is dedicated to a wide variety of questions connected with amenability and paradoxical decompositions for pseudogroups and discrete metric spaces. The authors effectively demonstrate that paradoxical decompositions are associated with amenability and not with the property of ``exponential growth'' [compare with \textit{W. A. Deuber}, \textit{M. Simonowitz} and \textit{V. T. Sós}, Stud. Sci. Math. Hung. 30, 17-23 (1995; Zbl 0857.54030)]. It is proven that the existence of invariant means for pseudogroups is equivalent to Følner's condition. The authors define the Tarski number as the minimal number of pieces in paradoxical decompositions associated with nonamenable group actions and give estimates for this number. The holomorphic Siegel modular forms of degree two may be viewed as automorphic forms on the group \(\mathrm{GSp}(4)\). According to Langlands' principle of functoriality, one should be able to lift the automorphic representation so obtained to an automorphic representation on \(\mathrm{GL}(4)\) (or \(\mathrm{GL}(5)\)). This paper addresses the existence of these liftings, among others, for a holomorphic Siegel modular form \(F\) of degree two of full level that is not a Saito-Kurokawa lift from \(\mathrm{SL}(2)\). The proof is based on the converse theorem of \textit{J. W. Cogdell} and \textit{I. I. Piatetski-Shapiro} [Math. Res. Lett. 3, No. 1, 67--76 (1996; Zbl 0864.22009)]. It boils down to prove the niceness of Rankin-Selberg \(L\)-functions \(L(s, \pi \times \tau)\), where \(\pi\) is the automorphic representation generated by \(F\) and \(\tau\) stands for a cuspidal automorphic representation of \(\mathrm{GL}(2)\). Note that \(\pi\) is never generic, thus the Langlands-Shahidi method is not applicable here. One then makes use of \textit{M. Furusawa}'s integral representation for \(L(s, \pi \times \tau)\) [J. Reine Angew. Math. 438, 187--218 (1993; Zbl 0770.11025)], which unfolds to local integrals involving Whittaker functionals and Bessel functionals of \(\pi\). This requires a careful choice of local data and intricate computations. Also, a ``pull-back formula'' is applied to establish the entireness of \(L(s, \pi \times \tau)\). Apart from the existence of liftings, the authors also deduce results on the arithmeticity of critical \(L\)-values that seem out of reach of the techniques based on trace formula.

The problem ``which semigroup rings are rings with identity'' was raised a long time ago. In [Semigroup Forum 46, No. 1, 27-31 (1993; Zbl 0787.16024)], in order to investigate the existence of identity of an orthodox semigroup ring, \textit{F. Li} asked: for a ring \(R\) with identity and a regular semigroup \(S\), if \(RS\) is a ring with identity, is \(R\langle E(S)\rangle\) a ring with identity? The aim of this paper is to discuss this problem for FIC-semigroups and completely regular semigroups. The author proves that for an FIC-semigroup or a completely regular semigroup \(S\), if \(RS\) is a ring with identity, then \(R\langle E(S)\rangle\) is a ring with identity. \(E(S)\) denotes the set of idempotents of \(S\). Let \(S\) be an orthodox semigroup, and let \(E(S)\) be the band of idempotents of \(S\). For any \(x\in S\), put \(V(x) = \{y \in S\mid xyx = x\), \(yxy = y\}\). \(S\) is said to be \(N\)-perfect iff \(V(x)V(y) = V(yx)\) for all \(x,y \in D\). Assume that \(R\) is a ring with identity. The author gives a criterion for the existence of an identity in a contracted semigroup ring \(R_ 0[S]\) for \(S\) orthodox \(N\)-perfect. Assume that \(E(S)\) is a semilattice \(Y\) of rectangular bands \(E_ \alpha\) \((\alpha \in Y)\). Denote by \(M(Y)\) the set of all maximal elements of \(Y\) (under the usual semilattice order). If \(R_ 0[S]\) possesses an identity element then the following is true: (i) \(| E_ \alpha| = 1\) for every \(\alpha \in M(Y)\). So suppose that (i) holds; if this is the case then (Theorem 11) the following three conditions are equivalent: (ii) \(R_ 0[S]\) is a ring with identity, (iii) \(R_ 0[E(S)]\) is a ring with identity, (iv) \(R_ 0[Y]\) is a ring with identity. Thus the problem of the existence of an identity in a semigroup ring \(R_ 0[S]\) for \(S\) orthodox \(N\)-perfect is completely reduced to the same problem for a semilattice semigroup ring. The last problem was solved by Wenger: if \(Y\) is a semilattice then \(R_ 0[Y]\) is a ring with identity iff the set \(M(Y)\) is finite nonempty such that \(Y = Y\cdot M(Y)\). The author also poses an interesting problem which may be formulated as follows: let \(S\) be a regular semigroup such that the ring \(R_ 0[S]\) is a ring with identity \(1_{RS}\); is it true that \(\text{supp}(1_{RS}) \subset E(S)\)?

The problem ``which semigroup rings are rings with identity'' was raised a long time ago. In [Semigroup Forum 46, No. 1, 27-31 (1993; Zbl 0787.16024)], in order to investigate the existence of identity of an orthodox semigroup ring, \textit{F. Li} asked: for a ring \(R\) with identity and a regular semigroup \(S\), if \(RS\) is a ring with identity, is \(R\langle E(S)\rangle\) a ring with identity? The aim of this paper is to discuss this problem for FIC-semigroups and completely regular semigroups. The author proves that for an FIC-semigroup or a completely regular semigroup \(S\), if \(RS\) is a ring with identity, then \(R\langle E(S)\rangle\) is a ring with identity. \(E(S)\) denotes the set of idempotents of \(S\). Multiple hypothesis testing is a central topic in statistics. However, despite abundant research on the false discovery rate (FDR) and the corresponding Type-II error concept known as the false nondiscovery rate (FNR), we do not yet have a fine-grained understanding of the fundamental limits of multiple testing. The main contribution of this study is to derive a precise nonasymptotic trade-off between the FNR and FDR for a variant of the generalized Gaussian sequence model. Our approach is flexible enough to permit analyses of settings where the problem parameters vary with the number of hypotheses \(n\), including various sparse and dense regimes (with \(o(n)\) and \(O(n)\) signals). Moreover, we prove that the Benjamini-Hochberg and Barber-Candès algorithms are both rate-optimal up to constants across these regimes.

The authors present several inequalities for the trace of the product (in particular, for the power) of matrices. Related results can be found in [\textit{T. Ando}, \textit{F. Hiai} and \textit{K. Okubo}, Math. Inequal. Appl. 3, No. 3, 307--318 (2000; Zbl 0959.15015)]. The primary motivation of this paper is to consider whether the trace inequality \[ |\text{T{}r}(A^{p_1}B^{q_1}\dots A^{p_K}B^{q_K})|\leq\text{T{}r}(AB)\tag{*} \] is true for every \(p_i,q_i\geq 0\) with \(\sum_{i=1}^Kp_i=\sum_{i=1}^Kq_i=1\). It is also interesting to find cases where the opposite trace inequality \[ \text{T{}r}((A^{1/K}B^{1/K})^K)\leq|\text{T{}r}(A^{p_1}B^{q_1}\dots A^{p_K}B^{q_K})|\tag{**} \] holds, which is regarded as a far extended version of the Golden-Thompson inequality. These problems seem rather subtle, there are no counterexamples for the moment. In this paper, inequality (*) is proved under stronger assumptions that \(0\leq\sum_{i=1}^jq_i-\sum_{i=1}^jp_i\leq 1/2\) \((1\leq j\leq K-1)\) and \(0\leq\sum_{i=1}^jp_i-\sum_{i=1}^{j-1}q_i\leq 1/2\) \((1\leq j\leq K)\). But the inequality (**) is obtained only for some particular cases.

The authors present several inequalities for the trace of the product (in particular, for the power) of matrices. Related results can be found in [\textit{T. Ando}, \textit{F. Hiai} and \textit{K. Okubo}, Math. Inequal. Appl. 3, No. 3, 307--318 (2000; Zbl 0959.15015)]. Let \(k\) be an algebraically closed field of prime characteristic, Lie algebra \(\mathfrak{g}=so(5, k)\). In this paper, when \(p\)-character \(\chi\) is sub-regular nilpotent and has standard Levi form, the Loewy series of principle indecomposable \(\mathfrak{g}\)-modules is given.

Let \(n\) and \(e\) be integers greater than or equal to \(2\). Pappas and Rapoport conjectured that the subscheme \[\mathcal{N}_{n,e} = \{ A \in \mathrm{Mat}_{n \times n} \: : \: A^e = 0, \det(\lambda-A) = \lambda^n \}\] of the scheme \(\mathrm{Mat}_{n \times n}\) of \(n \times n\) matrices is reduced (Conjecture 5.8 of [\textit{G. Pappas} and \textit{M. Rapaport}, J. Algebraic Geom 12, 107--145 (2003; Zbl 1063.14029)]). They showed that this conjecture implies the flatness of the standard model of \(\mathrm{GL}_d\) in certain situations. The above conjecture is proved in full generality in this paper. The subject of the paper under review is to investigate some simply-defined schemes (\(M=M(\Lambda, \text{\textbf{r}}\), \(\widetilde M\), \(N\) etc.) in order to get information about more complicated objects, namely models (denoted by \(X\)) of Shimura varieties of endomorphism and level type over Spec \({\mathcal O}_E\), where \(E\) is the completion of the reflex field of \(X\) at a finite prime. Particularly, information about the phenomenon of possible non-flatness of \(X\) over Spec \({\mathcal O}_E\) is obtained. An example of such \(X\) is a Shimura variety associated to group \(G=\text{Res}_{E/F}\text{GL}(n)\) where \(E/F\) is a totally ramified extension of local fields. \(M=M(\Lambda, \text{\textbf{r}})\) is called a naive local model for the moduli problem corresponding to \(X\). \(M\) is closely related to \(X\), namely, it contains some closed subschemes (local models) which are locally for the étale topology isomorphic to \(X\). Schemes \(M(\Lambda, \text{\textbf{r}})\) are defined in terms of linear algebra. Namely, fix a totally ramified extension \(F/F_0\) of local fields, a vector space \(V\) over \(F\), an \({\mathcal O}_F\)-lattice \(\Lambda \subset V\) and a multidimension \(\text{\textbf{r}}=\{r_{\varphi}\}\) (here the \(\varphi\)'s run over all inclusions of \(F/F_0\) in \(\bar F_0\)) such that the reflex field of \(\text{\textbf{r}}\) is \(E\). \(M\) represents a functor that (roughly speaking) associates to an extension \(S/F_0\) the set of sublattices \(\mathcal F\) of \(\Lambda \otimes_{F_0}S\) such that the multidimension of \(\mathcal F\) is \(\text{\textbf{r}}\). Similarly, \(\widetilde M\) represents a functor associating to \(S\) the pair (\(\mathcal F\), a base of \(\mathcal F\)), and \(N(S)\) is a set of some \(r\times r\)-matrices over \(S\) (\(r=\sum_{\varphi}r_{\varphi}\)). There is a diagram \(M\overset{\pi}{\leftarrow} \widetilde M \overset{\phi}{\to}N\). Further, let \(M^{\text{loc}}\) be the closure of \(M\otimes_{{\mathcal O}_E}E\) in \(M\). The following theorems are proved: Theorem A. The morphism \(\phi\) is smooth. Theorem B. \(M^{\text{loc}}\) is normal, Cohen-Macaulay, its special fiber is reduced, normal with rational singularities. The existence of an inclusion of the special fiber of \(M\) to the affine Grassmannian permits to prove a theorem on its Schubert subvariety. The description of the resolution of the singularities of \(N\) over the Galois closure of \(F/F_0\) permits to calculate the complex of nearby cycles on \(M\) (together with the Gal(\(F_0\))-action) in terms of the constant sheaves on some varieties related to \(M\); the statement of this result is too large to be given here. The final result of the discussion about relations between \(X\) and \(M\) is the following: while \(X\) is not always flat over \({\mathcal O}_E\), it is conjectured that \(M^{\text{loc}}\) is always flat over \({\mathcal O}_E\).

Let \(n\) and \(e\) be integers greater than or equal to \(2\). Pappas and Rapoport conjectured that the subscheme \[\mathcal{N}_{n,e} = \{ A \in \mathrm{Mat}_{n \times n} \: : \: A^e = 0, \det(\lambda-A) = \lambda^n \}\] of the scheme \(\mathrm{Mat}_{n \times n}\) of \(n \times n\) matrices is reduced (Conjecture 5.8 of [\textit{G. Pappas} and \textit{M. Rapaport}, J. Algebraic Geom 12, 107--145 (2003; Zbl 1063.14029)]). They showed that this conjecture implies the flatness of the standard model of \(\mathrm{GL}_d\) in certain situations. The above conjecture is proved in full generality in this paper. [For the entire collection see Zbl 0742.00058.] We propose a new approach to the study of gene diffusion in a small isolated population. Our modeling is very different from those of Kolmogorov, Wright, Kimura, Feller, Bartlett and Malécot. This modeling takes into account some new biological features. For each generation we consider two successive steps, the population of young at the time of conception and the following population of adults at the time of reproduction. Consequently, we are able to separate the effect of genetic drift and mutation from the effect of zygotic selection. To get a better description, we have defined and used the notion of generalized genotypes that describe isoactive states of genes and constrained situations between genes. We make a strong and clear difference between conditional probabilities of alleles and unconditional or a priori probabilities of alleles. In previous classical models these probabilities have been confounded. We define and introduce two new types of mutation, one at the molecular level of nucleotide base sequence and the other at the level of phenotypic gene expression. Equations derived from this modeling can be solved and have nonlinear solutions. These solutions are complete and describe the evolution of genetical structure of the population from the beginning to the final equilibrium. Next, we have to extend these results mainly in two directions. One is to study the case of two or three linked loci and to introduce the notions of ``fitness'' for genotypes and of quantitative effects of alleles. The other is to modify this model to take into account the population demography, for instance the distribution of the numbers of offspring and the environmental drift of the number of each genotype among reproductive adults. Genetic drift and environmental drift are very different. Genetic drift is dependent on the type of mating, population size, and Mendelian segregation. Environmental drift is independent on population size only. Genetic drift modifies the population composition just at conception. Environmental drift modifies the population composition just before mating.