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A map that is defined on the set of all symmetric matrices over a field is rank-one nonincreasing if it maps the matrices of rank one to matrices of rank at most one. In the case of the fields with two or three elements nonstandard examples of such additive maps are known to exist. In this paper, all these maps are characterized. Moreover, the existence of nonstandard maps is understood in a geometric sense since they exist precisely in the two cases, where the whole n-dimensional vector space over can be written as a union of two hyperbolic/parabolic quadrics of low indices. This geometric interpretation enables us to realize that these weird examples represent just a tip of an iceberg that appears in an analogous problem on symmetric tensors, where nonstandard maps exist also over larger fields. A few open problems related to the multilinear analog and to homogeneous polynomials are posed. [ABSTRACT FROM AUTHOR]

Current practice in the use of the method of fundamental solutions (MFS) for inverse Stefan problems typically involves setting the source and collocation points at some distance, h,  from the boundaries of the domain in which the solution is required, and then varying their number, , so that the obtained solution fulfils a desired tolerance, Tol, when a random noise level is added to the boundary conditions. This leads to an open question: can h and be chosen simultaneously so that is minimized, thereby leading to a lower computational expense in the solution of the inverse problem? Here, we develop a novel, simple and practical algorithm to help answer this question. The algorithm is used to study the effect of Tol and on both h and . Its effectiveness is shown through three test problems and numerical experiments show promising results: for example, even with as high as 5% and Tol as low as , we are able to find satisfactory solutions for as low as 8. [ABSTRACT FROM AUTHOR]

This article presents a novel algorithm for topology optimization using an orthotropic material model. Based on the virtual work principle, mathematical formulations for effective orthotropic material properties of an element containing two materials are derived. An algorithm is developed for structural topology optimization using four orthotropic material properties, instead of one density or area ratio, in each element as design variables. As an illustrative example, minimum compliance problems for linear and nonlinear structures are solved using the present algorithm in conjunction with the moving iso-surface threshold method. The present numerical results reveal that: (1) chequerboards and single-node connections are not present even without filtering; (2) final topologies do not contain large grey areas even using a unity penalty factor; and (3) the well-known numerical issues caused by low-density material when considering geometric nonlinearity are resolved by eliminating low-density elements in finite element analyses. [ABSTRACT FROM AUTHOR]

In recent literature there are an increasing number of papers where the forbidden sets (FSs) of difference equations are computed. We review and complete different attempts to describe the FS and propose new perspectives for further research and a list of open problems in this field. [ABSTRACT FROM AUTHOR]

In this article, a repairable system with age-dependent failure type and minimal repair based on a cumulative repair-cost limit policy is studied, where the information of entire repair-cost history is adopted to decide whether the system is repaired or replaced. As the failures occur, the system has two failure types: (i) a Type-I failure (minor) type that is rectified by a minimal repair, and (ii) a Type-II failure (catastrophic) type that calls for a replacement. We consider a bivariate replacement policy, denoted by (n,T), in which the system is replaced at life ageT, or at then-th Type-I failure, or at thekth Type-I failure (k < nand due to a minor failure at which the accumulated repair cost exceeds the pre-determined limit), or at the first Type-II failure, whichever occurs first. The optimal minimum-cost replacement policy (n,T)* is derived analytically in terms of its existence and uniqueness. Several classical models in maintenance literature could be regard as special cases of the presented model. Finally, a numerical example is given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The article presents information on a student research project that was performed to enable students to investigate an abstract algebra class through modular arithmetic. It further depicts the mathematical palette by topping idempotents off with modular arithmetic and a series of exercises determining for which n there are more than two idempotents. Several numerical calculations are also presented to elaborate this.

New additional simple conditions of Dirichlet type are proposed for the numerical uniqueness of the solution of elastostatic axisymmetric problems formulated in terms of Boussinesq and Timpe potential functions, as particular cases of the Papkovich–Neuber representation. Applications to hollow cylinder, subjected to general axisymmetric load and displacement boundary conditions, are shown and numerically solved by the network method. [ABSTRACT FROM PUBLISHER]

The lattice Boltzmann Peierls Callaway (LBPC) method is a recent development of the versatile lattice Boltzmann formalism aimed at a numerical experiment on mesoscale thermal transport in a multiphase phonon gas. Two aspects of mesoscopic thermal transport are discussed: the finite phonon mean free path and the interface thermal resistance. Based on the phonon momentum screening length measured in the LBPC computational apparatus, the validity of the Umklapp collision relaxation time in the Callaway collision operator is examined quantitatively. The discrete nature of the spatio-temporal domain in the LBPC method, along with the linear approximation of the exponential screening mechanism in the Callaway operator, reveals a large discrepancy between the effective phonon mean free path and the analytic phonon mean free path when the relaxation time is small. The link bounce back interface phonon collision rule is used to realize the interface thermal resistance between phonon gases with dissimilar dispersion relations. Consistent with the Callaway collision operator for the bulk phonon dynamics, the interface phonon collision process is regarded as a linear relaxation mechanism toward the local pseudo-equilibrium phonon distribution uniquely defined by the energy conservation principle. The interface thermal resistance is linearly proportional to the relaxation time of the proposed phonon interface collision rule. [ABSTRACT FROM AUTHOR]

It is well known that convection-diffusion equations suffer from the most difficult problems to gain a stable and accurate solution. Sometimes, the convection terms may cause oscillatory behavior of solutions. In this article, the streamline upwind Petrov-Galerkin (SUPG) scheme is applied to eliminate overshoots and undershoots produced by the convection term in the meshless local Petrov-Galerkin (MLPG) method. The accuracy and stability of the present method are discussed using two test cases. The results of the present method are compared with results of other upwind schemes and the finite volume method (FVM) using a high-order upwind scheme. Our analysis shows that the present method, with its simple implementation, can give excellent results for convection-dominated problems. [ABSTRACT FROM PUBLISHER]

The critical region of the hierarchical reference theory (HRT) is investigated both by analytical and numerical methods. This is closely related to a recent work by one of us where the HRT was unified with another accurate theory, the self-consistent Ornstein–Zernike theory (SCOZA). It was found that somehow the properties of HRT alone would be closely related to the critical properties of the unified problem and vice versa. Our investigation is facilitated by the discovery of a situation where HRT and SCOZA give identical results. For the more general HRT situation we then find that an additional intermediate term appears, and the generalized scaling of SCOZA is replaced with standard scaling. Further we find that the numerical results can be accurately expressed by simple analytic expressions in the critical region. With an interplay between the leading scaling function and subleading contributions, simple rational numbers are found for the critical indices. [ABSTRACT FROM AUTHOR]

The new originally capital general solution of determining the prototype filter function as the response that satisfies the specifications of all pole low-pass continual time filter functions of odd and even order is presented in this article. In this article, two new classes of filter functions are proposed using orthogonal and orthonormal Jacobi polynomials. The approximation problem of filter function was solved mathematically, most directly applying the summed Christoffel–Darboux formula for the orthogonal polynomials. The starting point in solving the approximation problem is a direct application of the Christoffel–Darboux formula for the initial set of continual Jacobi orthogonal polynomials in the finite interval in full respect to the weighting function with two free real parameters. General solution of the filter functions is obtained in a compact explicit form, which is shown to enable generation the Jacobi filter functions in a simple way by choosing the numerical values of the free real parameters. For particular specifications of free parameters, the proposed solution is used with the same criterion of approximation to generate the appropriate particular filter functions as are: the Gegenbauer, Legendre and Chebyshev filter functions of the first and second kind as well. The examples of proposed filter functions of even and odd order are illustrated and compared with classical solutions. [ABSTRACT FROM AUTHOR]

For a vertically polarized line source, in the context of HF (3-30 MHz) ground wave propagation over a curved rough sea surface in the presence of islands, this paper adapts the FB-SA (Forward-Backward Spectral Acceleration) method to compute the attenuation function over sea-land (island)-sea mixed paths for different shapes and heights of the islands. The rigorous FB-SA numerical method is based on the method of moments and was originally developed for scattering from rough surfaces and is especially efficient in solving a huge problem, which is required to predict the ground wave propagation over a long surface. In addition, for zero island height, this method is compared with an analytical approach expressed from a (residue) series, in which the roots of a differential equation, depending on the Airy function, must be calculated. In addition, from an intuitive approach and from the work of Barrick and Ishimaru, this analytical approach is extended to include the sea roughness and then validated from the FB-SA. [ABSTRACT FROM AUTHOR]

In diffusion transport, the flux at a point is typically modeled in terms of the local gradient of a potential. When heterogeneities are present, this local model can break down and it may be more appropriate to model the diffusion flux as a weighted sum of gradients present throughout the domain. Here a discrete nonlocal flux model-consistent with control-volume implementations-is developed. This scheme is referred to as the control-volume weighted flux scheme (CVWFS). The key component is the modeling of the diffusion flux at a given control-volume face in terms of a weighted sum of gradients at that face and at faces up- and downstream. Criteria for choosing the weights are proposed. This results in numerical solution schemes in which the coefficient matrix is diagonally dominant, has positive off-diagonal elements, and zero row sums. For a particular power-law weighting scheme it is shown how the CVWFS is related to the definition of the Caputo fractional derivative and the one-shift Grunwald approximation of the Riemann-Liouville fractional derivative. On developing transients and boundary condition treatments, the accuracy and suitability of the CVWFS scheme is demonstrated by solving a number of problems governed by Caputo fractional diffusion equations. [ABSTRACT FROM AUTHOR]

We investigate an application of the method of fundamental solutions (MFS) to heat conduction in two-dimensional bodies, where the thermal diffusivity is piecewise constant. We extend the MFS proposed in Johansson and Lesnic [A method of fundamental solutions for transient heat conduction, Eng. Anal. Bound. Elem. 32 (2008), pp. 697-703] for one-dimensional heat conduction with the sources placed outside the space domain of interest, to the two-dimensional setting. Theoretical properties of the method, as well as numerical investigations, are included, showing that accurate results can be obtained efficiently with small computational cost. [ABSTRACT FROM AUTHOR]

We present a flux-based wave decomposition (FBWD) scheme for numerical computation of isentropic two-fluid two-phase flows. We use Tait's equation of state for both the gas and the liquid phases. Extension of the scheme to the second order is made by the MUSCL-Hancock approach. The solutions of several two-phase shock tube problems resulting from the present scheme are compared with the solutions obtained by the earlier HLL scheme. Investigation of the effect of the interfacial pressure models and the drag coefficient are also included. It is concluded that the present FBWD scheme is robust and accurate for computation of isentropic two-fluid models. [ABSTRACT FROM AUTHOR]

This article deals with the peristaltic flow of a Sisko fluid in an endoscope. The inner tube of the endoscope is fixed, while outer tube is flexible. Continuity and momentum equations are utilized in the mathematical analysis and obtained both the analytical and numerical solutions. The analytical solution has been found by the homotopy analysis method and the numerical solutions are carried out by the shooting technique. The comparison of both the solutions are presented. And the quantitative behaviours of the solutions are discussed. [ABSTRACT FROM AUTHOR]

The radial basis functions (RBFs) method is employed to handle a class of multi-dimensional parabolic inverse problems. Because they are not modelled by classical parabolic initial-boundary value problems, theoretical behaviour and numerical approximation of these problems have been active areas of research. Based on the idea of RBF approximation, a fast and highly accurate meshless method is developed for solving an inverse problem with a control parameter. Moreover, with the meshless property, it can be used to handle multi-dimensional parabolic inverse problems defined on very complicated geometries. [ABSTRACT FROM AUTHOR]

The dominant method of small molecule quantum chemistry over the last twenty years is CCSD(T). Despite this success, RHF-based CCSD(T) fails for systems away from equilibrium. Work over the last ten years has lead to modifications of CCSD(T) that improve the description of bond breaking. These new methods include ΛCCSD(T), CCSD(2)T, CCSD(2) and CR-CC(2,3), which are new perturbative corrections to single-reference CCSD. We present a unified derivation of these methods and compare them at the level of formal theory and computational accuracy. None of the methods is clearly superior, although formal considerations favour ΛCCSD(T) and computational accuracy for the systems considered favours CR-CC(2,3). [ABSTRACT FROM AUTHOR]

This article reviews the developments made primarily by the author in mathematical theories and algorithms for systematic many-body descriptions of correlated electrons and anharmonic lattice vibrations in polymers and molecular crystals. It also documents underlying size-extensive electron-correlation methods and their computerized implementations as well as a few pivotal applications. [ABSTRACT FROM AUTHOR]

The optimized virtual orbital space (OVOS) method is described from the perspective of several developments that have occurred since the method was originally introduced 25 years ago during the author's postdoctoral stay in the research group of Rod Bartlett. We describe the OVOS approach in the context of the restricted Hartree-Fock (RHF), unrestricted HF (UHF), and multi-configuration self-consistent field (MCSCF) methods used to generate the reference zero-order wave function for the OVOS calculations. [ABSTRACT FROM AUTHOR]

The relationship between standard single-reference coupled-cluster (CC) and quadratic configurations interaction (QCI) methods is reviewed from the viewpoint of both past and recent developments in the methodology of post-Hartree-Fock approaches to the many-electron correlation problem, emphasising their advantages and shortcomings on the basis of many-body perturbation theory (MBPT) analysis, the required computational effort, and their practical exploitations. [ABSTRACT FROM AUTHOR]

We improve in three ways the results of [6] that establish the acoustic limit for DiPerna-Lions solutions of Boltzmann equation. First, we enlarge the class of collision kernels treated to that found in [13], thereby treating all classical collision kernels to which the DiPerna-Lions theory applies. Second, we improve the scaling of the kinetic density fluctuations with Knudsen number from O(εm) for some [image omitted] to [image omitted]. Third, we extend the results from periodic domains to bounded domains with a Maxwell reflection boundary condition, deriving the impermeable boundary condition for the acoustic system. [ABSTRACT FROM AUTHOR]

In this paper, we present a class of non-conventional two-step explicit methods for solving the second-order initial value problems (IVPs) with periodic or oscillatory solutions. The methods have second algebraic order, a large interval of periodicity and high phase-lag order. For multi-dimensional problems, we give the vector form of the methods with the aid of a special vector operation. Some numerical results are reported to illustrate the efficiency of our methods. [ABSTRACT FROM AUTHOR]

We introduce a new variational formulation of the transport equation based on a mixed form of the transport equation. We prove some theoretical results about existence and uniqueness of the solution of this abstract problem by using desirable functional spaces. We then apply a mixed and hybrid finite element method to discretize the transport equation by introducing appropriate basis functions. We present some numerical results to illustrate the efficiency of our method. [ABSTRACT FROM AUTHOR]

An implementation of MNDO-type semiempirical methods is described, which include analytic second derivatives and interface to ONIOM hybrid models. Analytic second derivatives are much more efficient than numerical derivatives and have very low cost even for systems containing hundreds of atoms. ONIOM calculations using a combination of semiempirical and molecular mechanics methods are practical, even for second derivative calculations having hundreds of atoms in the quantum model system and thousands of atoms in the full system. [ABSTRACT FROM AUTHOR]

This aritcle describes a simple alternative unified method of solving nonlinear regular perturbation problems. The procedure is based upon the manipulation of Taylor's approximation for the expansion of the nonlinear term in the perturbed equation. An essential feature of this technique is the relative simplicity used and the associated unified computational procedure that is employed. As such it should be of interest to teachers of applied mathematics courses, particularly those courses which include perturbation methods. One of the merits of this approach is that it leads on naturally to a scheme based on Taylor's expansions and, consequently, allows the regular perturbation method to be introduced into a course much earlier than is currently common. The method is illustrated by implementing it to four perturbation problems, including two algebraic equations and two initial-value problems. The new approach is compared and contrasted with the traditional perturbation scheme in order to demonstrate its relative merit. [ABSTRACT FROM AUTHOR]

The article presents an elementary proof of the random walk hitting time theorem. This theorem states that a left-continuous random walk on (Z) that start at a nonnegative integer (k) wherein the probability that hits the origin (n) is equal to k/n. It discusses the hitting time theorem which is equivalent to the ballot theorem that dates back to Whitford and Bertrand. It presents equations and solutions to prove these theorems. It also presents minor modification or assumptions in the hitting time theorem.

In this paper, we consider the initial-boundary value problem of quasi-linear wave equation outside of star-shaped obstacle in three space dimensions. We study the general case that the nonlinear term may explicitly depend on the unknown function itself. By some new weighted space-time L2 estimates of the linear wave equation for the unknown function itself, we obtained sharp lower bounds of the lifespan [image omitted] for the quasi-linear initial boundary value problem. [ABSTRACT FROM AUTHOR]

In this article, we shall be concerned with the existence of solutions for the strongly non-linear boundary value problem: [image omitted] where A is an elliptic operator of finite order defined from an anisotropic Sobolev space of order m to its dual, g is a Caratheodory function satisfying essentially a sign condition on u with no growth restrictions and f belongs to L1. [ABSTRACT FROM AUTHOR]

The homogenization of a binary structure made of very small particles of general shape is performed when the diffusivity is finite and when the size of the particles has a critical value with respect to a rarefaction number. The limit problem involves an auxiliary function which can be interpreted as the solution of a problem of cellular type, thus filling the gap with classical methods of multiple scales. [ABSTRACT FROM AUTHOR]

We consider a Finite Element Method (F.E.M.) implementation for the asymptotic partial decomposition. The advantage of this approach is an important reduction of the number of nodes. The convergence is proved for some model problems. Finally the relation with the "mixed formulation" is discussed. [ABSTRACT FROM AUTHOR]

In this paper we prove a Hölder and Lipschitz stability estimates of determining the residual stress by a single pair of observations from a part of the lateral boundary or from the whole boundary. These estimates imply first uniqueness results for determination of residual stress from few boundary measurements. [ABSTRACT FROM AUTHOR]

We study the stationary problem of a viscous, incompressible Navier-Stokes fluid flowing through a flexible tube with thickness. The behavior of the elastic walls of the tube is described by the equations of nonlinear elasticity for a St.Venant-Kirchhoff material. For smooth enough applied exterior forces we prove the existence of a solution to the coupled problem. [ABSTRACT FROM AUTHOR]

In this article, general existence theorems are presented for a quasilinear system We obtain some existence theorems by a simple application of the Schauder fixed-point theorem and degree theory. We do not require conditions of the nonlinearity f ,&ThickSpace; g at zero or at infinity, and we do not need upper bounds for p ,&ThickSpace; q involving the dimension n . We study the case where problem ( P ) is not of variational type. [ABSTRACT FROM AUTHOR]

The spatial statistics formalism is applied to electromagnetic fields analysis. Fields are considered as realizations of a random function. Their spatial structure is studied by a method known as variographic analysis. To infer unknown values of the fields, an interpolation method called kriging is then used. It is shown how kriging can be performed on experimental or numerical data to speed up the fields estimation process. [ABSTRACT FROM AUTHOR]

Numerical solution based on similarity reductions for partial differential equations used to get the numerical scheme for the regularized long wave (RLW) equation. The similarity reductions for RLW equation are obtained locally on subdomains defined by the classical three-point stencil. The ordinary differential equation, which deduced from the similarity reduction can be linearized, integrated analytically and then obtain the solution. This approch eliminates the difficulties associated with boundary conditions for the similarity reduction over the whole solution domain. Numerical results are obtained for test problem. The computed results using our scheme confirm the accuracy of our scheme. [ABSTRACT FROM AUTHOR]

In the present investigation, numerical methods are developed for approximate solution of stochastic boundary-value problems. In particular, shooting methods are examined for numerically solving systems of Stratonovich boundary-value problems. It is proved that these methods accurately approximate the solutions of stochastic boundary-value problems. An error analysis of these methods is performed. Computational simulations are given. [ABSTRACT FROM AUTHOR]

A method is presented for the efficient solution of the inverse geometric problem of detection of subsurface cavities and flaws using thermographic techniques. Here, a superposition of clusters of sources/sinks with a boundary-element solution of the forward problem offers a numerical scheme that does not require remeshing of the interior geometry as the inverse problem is solved iteratively to detect the flaw or cavity. The approach offers tremendous advantage in reducing the computational burden involved in remeshing and presents a promising technique for three-dimensional applications. [ABSTRACT FROM AUTHOR]

In this article we give three characterizations of real self-adjoint operators associated with the singular Sturm-Liouville expressions The first characterization is based on the construction of extensions of the domain of the minimal operator associated with ℓ, the second is based on the behavior of certain functions in their domain near the endpoints and the third is based on the boundary conditions satisfied by such extensions. [ABSTRACT FROM AUTHOR]