How ∗ not ∗ to solve a Sudoku

We prove the NP-hardness of a consistency checking problem that arises in certain elimination strategies for solving Sudoku-type problems.     

Continuity of a Deformation in H<Superscript>1</Superscript> as a Function of Its Cauchy-Green Tensor in L<Superscript>1</Superscript>

Let $\Omega$ be a bounded Lipschitz domain in $\BBbR^n$. The Cauchy-Green, or metric, tensor field associated with a deformation of the set $\Omega$, i.e., a smooth-enough orientation-preserving mapping $\bTh\colon\Omega\to\BBbR^n$, is the $n\times n$ symmetric matrix field defined by $\bnabla\bTheta^T(x)\bnabla\bTheta(x)$ at each point $x\in\Omega$. We show that, under appropriate assumptions, the deformations depend continuously on their Cauchy-Green tensors, the topologies being those of the spaces $\bH^1(\Omega)$ for the deformations and $\bL^1(\Omega)$ for the Cauchy-Green tensors. When $n=3$ and $\Omega$ is viewed as a reference configuration of an elastic body, this result has potential applications to nonlinear three-dimensional elasticity, since the stored energy function of a hyperelastic material depends on the deformation gradient field $\bnabla\bTheta$ through the Cauchy-Green tensor.     

<Literal>smt</Literal>: a Matlab toolbox for structured matrices

The full exploitation of the structure of large scale algebraic problems is often crucial for their numerical solution. Matlab is a computational environment which supports sparse matrices, besides full ones, and allows one to add new types of variables (classes) and define the action of arithmetic operators and functions on them. The smt toolbox for Matlab introduces two new classes for circulant and Toeplitz matrices, and implements optimized storage and fast computational routines for them, transparently to the user. The toolbox, available in Netlib, is intended to be easily extensible, and provides a collection of test matrices and a function to compute three circulant preconditioners, to speed up iterative methods for linear systems. Moreover, it incorporates a simple device to add to the toolbox new routines for solving Toeplitz linear systems.     

Analysis of a FitzHugh–Nagumo–Rall model of a neuronal network

Pursuing an investigation started in (Math. Meth. Appl. Sci. 2007; 30 :681–706), we consider a generalization of the FitzHugh–Nagumo model for the propagation of impulses in a network of nerve fibres. To this aim, we consider a whole neuronal network that includes models for axons, somata, dendrites, and synapses (of both inhibitory and excitatory type). We investigate separately the linear part by means of sesquilinear forms, in order to obtain well posedness and some qualitative properties. Once they are obtained, we perturb the linear problem by a nonlinear term and we prove existence of local solutions. Qualitative properties with biological meaning are also investigated. Copyright © 2007 John Wiley & Sons, Ltd.     

Numerical bifurcation analysis of a ‘car‐following’ model

We study a system of ordinary differential equations describing a car‐following model for the motion of N car around a circular highway. All cars behave in the same way. The acceleration of each car is determined as a function of the headway (optimal velocity function). This model is known to have a solution with constant velocities and headways which, in a certain parameter regime, is stable and, varying the density of the cars, the loss of stability is generally due to a super‐ or subcritical Hopf bifurcation. Guided by analytical results, we numerically investigate the global bifurcation diagram for periodic solutions and obtain a complete picture of the dynamics of the model. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inequality for a trace on a unital <Emphasis Type="Italic">C</Emphasis>*-algebra

A new inequality for a trace on a unital C*-algebra is established. It is shown that the inequality obtained characterizes the traces in the class of all positive functionals on a unital C*-algebra. A new criterion for the commutativity of unital C*-algebras is proved.     

Three‐dimensional Stokes flow caused by a translating–rotating sphere bisected by a free surface bounding a semi‐infinite micropolar fluid

Explicit velocity and microrotation components and systematic calculation of hydrodynamic quasistatic drag and couple in terms of nondimensional coefficients are presented for the flow problem of an incompressible asymmetrical steady semi‐infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi‐infinite micropolar fluid. Two asymmetrical cases are considered for the motion of the sphere: parallel translation to the free surface and rotation about a diameter which is lying in the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. A linear slip, Basset‐type, boundary condition has been used. The variation of the resistance coefficients is studied numerically and plotted versus the micropolarity parameter and slip parameter. The two limiting cases of no‐slip and perfect slip are then recovered. Copyright © 2008 John Wiley & Sons, Ltd.     

Ott–Sudan–Ostrovskiy equation on a half-line

We consider the initial-boundary value problem for the Ott-Sudan-Ostrovskiy equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial-boundary value problem and the asymptotic behavior of solutions for large time.     

Stokes' first problem for a Newtonian fluid in a non‐Darcian porous half‐space using a Laguerre–Galerkin method

A Laguerre–Galerkin method is proposed and analysed for the Stokes' first problem of a Newtonian fluid in a non‐Darcian porous half‐space on a semi‐infinite interval. It is well known that Stokes' first problem has a jump discontinuity on boundary which is the main obstacle in numerical methods. By reformulating this equation with suitable functional transforms, it is shown that the Laguerre–Galerkin approximations are convergent on a semi‐infinite interval with spectral accuracy. An efficient and accurate algorithm based on the Laguerre–Galerkin approximations of the transformed equations is developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Lamé operators with finite monodromy—a combinatorial approach

We are describing Lamé differential operators with a full set of algebraic solutions. For each finite group G, we are describing the possible values of the degree parameter n such that the Lamé operator L_n has the projective monodromy group G. The main technical tool is the combinatorics associated to Belyi functions, ideas that we already used in (Rend. Sem. Mat. Univ. Padova 107 (2002) 191-208) for describing the case n=1. We also supply proofs to some finiteness properties conjectured by Baldassarri and by Dwork, and we work out an explicit formula for the number of essentially different Lamé equations when n=2. This approach can be generalized for arbitrary degree n (see (Counting Integral Lamé Equations by Means of Dessins d'Enfants, arXiv:math.CA/0311510) for n integer).     

Phasebehaviour of a 3D–Stell–Hemmer Liquid

Investigations for one– and two–dimensional systems have shown that the Stell–Hemmer potential can produce liquid–state anomalies. By means of a Monte Carlo simulation for a N; V; T–ensemble we calculate the phase behaviour and show that these anomalies disappear in three dimensions.     

A note on “Convergence of a Halpern-type iteration algorithm for a class of pseudo-contractive mappings”

In [C.O. Chidume, G. De Souza, Convergence of a Halpern-type iteration algorithm for a class of pseudocontractive mappings, Nonlinear Analysis (2007), doi:10.1016/j.na.2007.08.008], the authors proved a strong convergence result for strictly pseudo-contractive mappings using a Halpern-type iteration algorithm. However, the main result is not correct. In this note, we provide a counter-example to the theorem.     

Computational experiments with a lazy version of a <Emphasis Type="Italic">K</Emphasis> quickest simple path ranking algorithm

The quickest path problem is related to the classical shortest path problem, but its objective function concerns the transmission time of a given amount of data throughout a path, which involves both cost and capacity. The K-quickest simple paths problem generalises the latter, by looking for a given number K of simple paths in non-decreasing order of transmission time. Two categories of algorithms are known for ranking simple paths according to the transmission time. One is the adaptation of deviation algorithms for ranking shortest simple paths (Pascoal et al. in Comput. Oper. Res. 32(3):509–520, 2005; Rosen et al. in Comput. Oper. Res. 18(6):571–584, 1991), and another is based on ranking shortest simple paths in a sequence of networks with fixed capacity lower bounds (Chen in Inf. Process. Lett. 50:89–92, 1994), and afterwards selecting the K quickest ones. After reviewing the quickest path and the K-quickest simple paths problems we describe a recent algorithm for ranking quickest simple paths (Pascoal et al. in Ann. Oper. Res. 147(1):5–21, 2006). This is a lazy version of Chen’s algorithm, able to interchange the calculation of new simple paths and the output of each k-quickest simple path. Finally, the described algorithm is computationally compared to its former version, as well as to deviation algorithms.      

一类线性规划问题的随机中心路径算法

本文针对一类仅具不等式约束的线性规划问题提出了一种随机化的中心路径算法,并证明了算法的收敛性.初步的数值实验说明了算法的有效性.     

Comment on “On a stopped functional for a bidimensional process” by C. Makasu

We provide an elementary derivation of the result obtained by C. Makasu on a stopped functional for a bidimensional process.     

Rotation of a Neutron in the Coat of Helium-5 as a Classical Particle for a Relatively Large Value of the Hidden Parameter <Emphasis Type="Italic">t</Emphasis><Subscript>meas</Subscript>

Rotation of a neutron in the coat of helium-5 as a classical particle for a relatively large value of the hidden parameter (measurement time) t_meas = h/E_ms is considered. In consideration of the asymptotics as N → 0, equations for the mesoscopic energy E_ms are given. A model for the helium nucleus is introduced and the values of the mesoscopic parameters M_ms, and E_ms for helium-4 are calculated.     

Blow‐up phenomena for a pseudo‐parabolic equation

By the means of a differential inequality technique, we obtain a lower bound for blow‐up time if p and the initial value satisfy some conditions. Also, we establish a blow‐up criterion and an upper bound for blow‐up time under some conditions as well as a nonblow‐up and exponential decay under some other conditions. Copyright © 2014 John Wiley & Sons, Ltd.     

Dynamics and bifurcations of a modified Leslie‐Gower–type model considering a Beddington‐DeAngelis functional response

In this paper, a planar system of ordinary differential equations is considered, which is a modified Leslie‐Gower model, considering a Beddington‐DeAngelis functional response. It generates a complex dynamics of the predator‐prey interactions according to the associated parameters. From the system obtained, we characterize all the equilibria and its local behavior, and the existence of a trapping set is proved. We describe different types of bifurcations (such as Hopf, Bogdanov‐Takens, and homoclinic bifurcation), and the existence of limit cycles is shown. Analytic proofs are provided for all results. Ecological implications and a set of numerical simulations supporting the mathematical results are also presented.     

Flexural‐Gravity Wave Scattering by a Circular‐Arc‐Shaped Porous Plate

The interaction of flexural‐gravity waves with a thin circular‐arc‐shaped permeable plate submerged beneath the ice‐covered surface of water with uniform finite depth is considered under the assumption of linear theory. The problem is reduced to a second kind hypersingular integral equation for the potential difference across the plate which is solved approximately by an expansion–collocation method. Utilizing the solution, the reflection and the transmission coefficients and the hydrodynamic forces are evaluated numerically. The focus of the paper is to illustrate the effect of a porous curved plate submerged in finite depth water with an ice‐cover on the normally incident waves. Numerical results for a circular‐arc‐shaped plate for different configurations are derived and represented graphically. Also, by choosing an appropriate set of parameters, the known results for a circular‐arc‐shaped rigid plate submerged in deep water and a semicircular porous plate submerged in finite depth water with a free surface are recovered as special cases.