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In this paper, finite element methods for the problems of wave propagation in a fluid-saturated porous medium are discussed. The medium is composed of a porous elastic solid (soil, rock, etc.) saturated by a compressible viscous fluid (oil, water, etc.), and the fluid may flow relatively to the solid. Biot's lowfrequency dynamic equations are chosen to describe the problems mentioned above, with stress-given boundary conditions, ABGs (Absorbing Boundary Conditions) on artificial boundaries and conditions on interfaces between the fluid-saturated porous medium and elastic solids. In the paper, a new kind of discrete ABCs is presented, and a discrete-time Galerkin method are utilized for obtaining approximate solutions. The numerical results show that they both are effective. Two dilatational waves (fast wave P1 and slow wave P2) and one rotational wave (S wave) are clearly visible in the figures of computational results, which coincide with theoretical analysis very well. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Cauchy problem for an elliptic equation in a plane domain. The problem is ill-posed. Using the method of quasi-reversibility, an approximation to the exact solution is given. Using Carleman's inequatily, we derive a sharp error estimate. [ABSTRACT FROM AUTHOR]

Some preconditioners for accelerating the classical iterative methods are given in Zhang et al. [Y. Zhang and T.Z. Huang, A class of optimal preconditioners and their applications, Proceedings of the Seventh International Conference on Matrix Theory and Its Applications in China, 2006. Y. Zhang, T.Z. Huang, and X.P. Liu, Modified iterative methods for nonnegative matrices and M-matrices linear systems, Comput. Math. Appl. 50 (2005), pp. 1587-1602. Y. Zhang, T.Z. Huang, X.P. Liu, A class of preconditioners based on the (I+S(α))-type preconditioning matrices for solving linear systems, Appl. Math. Comp. 189 (2007), pp. 1737-1748]. Another kind of preconditioners approximating the inverse of a symmetric positive definite matrix was given in Simons and Yao [G. Simons, Y. Yao, Approximating the inverse of a symmetric positive definite matrix, Linear Algebra Appl. 281 (1998), pp. 97-103]. Zhang et al. 's preconditioners and Simons and Yao's are generalized in this paper. These preconditioners are all of low construction cost, which all could be taken as approximate inverse of M-matrices. Numerical experiments of these preconditioners applied with Krylov subspace methods show the effectiveness and performance, which also show that the preconditioners proposed in this paper are better approximate inverse for M-matrices than Simons'. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the study begun in the previous paper with the same title. We analyze ER(2)16*+8(RP2n) and compute ER(2)*(RP16K+1), and use these to prove more nonimmersion theorems for RPn, including many in fairly low dimensions. In particular, we get 12 new nonimmersion results for RPn where n < 192, the range included in the tables Don Davis keeps. These complement the 10 already found in the first paper. [ABSTRACT FROM AUTHOR]

In previous work it was shown that the lower triangular generalized Hausdorff matrix Hαa with nonzero entries hnk= (n+α+1)-1, for a ≥ 0, is subnormal on Ι² if and only if α=0,1,2, …. For 0h : ..., is subnormal when .... In this paper we show that, when dj = Γ( j+1)Γ(h)/Γ( j+h), the square of the weights chosen above, then the corresponding operator Ch is bounded on Ι² for 0 < h < 3/2, that Hα is bounded on Ι² for all non-integer α < 0, and that Ch is closely related to Hh-1. This relationship leads to our main result that Ch is only subnormal when h = 1, when it corresponds to the original Cesaro operator with α = 0 and each dj = 1. [ABSTRACT FROM AUTHOR]

Abstract  The seminal paper by Kasner (Am. J. Math. 43, 217–221, 1921) is reprinted here, together with an editorial comment on its lasting scientific relevance, and a biography of the author. [ABSTRACT FROM AUTHOR]

The filter generator is an important building block in many stream ciphers. The generator consists of a linear feedback shift register of length n that generates an m-sequence of period 2′ - 1 filtered through a Boolean function of degree d that combines bits from the shift register and creates an output bit zt at any time t. The previous best attacks aimed at reconstructing the initial state from an observed keystream, have essentially reduced the problem to solving a nonlinear system of D = (Multiple line equation(s) cannot be represented in ASCII text) (i) equations in n unknowns using techniques based on linear algebra. This attack needs about D bits of keystream and the system can be solved in complexity O (Dω), where ω can be taken to be Strassen's reduction exponent ω = log2 (7) ≈ 2.807. This paper describes a new algorithm that recovers the initial state of most filter generators after observing O(D) keystream bits with complexity O((D - n)/2) ≈ O(D), after a pre-computation with complexity O(D(log2 D)³). [ABSTRACT FROM AUTHOR]

This paper gives a quantitative control of the order of zero of a weak solution to perturbations of the Baouendi-Grushin operator, which generalizes the result due to Aronszaijn, Krzywicki, and Szarski valid for elliptic operators in divergence form with Lipschitz continuous coefficients. [ABSTRACT FROM AUTHOR]

The present note is an addendum to our previous article ‘On a general Thue's equation.’ Corollary 2 in that paper provides a lower bound for the values of a polynomial with algebraic coefficients at integral points. Our purpose here is to prove a stronger version of it; also, we slightly simplify its proof, avoiding the use of Theorem 3 in the original paper. [ABSTRACT FROM AUTHOR]

This paper presents a quantitative discussion on the stability of the second-order periodic difference equation, which characterizes the discrete periodic time-varying system. The periodic parameter discrete system, which corresponds to Mathieu's equation in the continuous system, is represented by a second-order linear difference equation with a small parameter E in the varying term. The parameter E plays an important role in the determination of the stability. By applying McLachlan's method, the expression for the boundary between the stability and the instability can be determined analytically with regard to the parameters contained in the equation. For the case where the periodic parameter of the difference equation is represented as a sum of two even functions, the boundary between stability and instability is determined. The stability can be analyzed in a similar way for the case where the periodic parameter is represented as a sum of N even functions in Fourier series. [ABSTRACT FROM AUTHOR]

This paper reports on a design experiment conducted to explore the construction of meanings by 17 year old students, emerging from their interpretations and uses of algebraic like formalism. The students worked collaboratively in groups of two or three, using MoPiX, a constructionist computational environment with which they could create concrete entities in the form of models by using equations and animate them to link the equations' formalism to the produced visual representation. Our aim was to further study the ways in which the use of formalism in constructionist environments can create contexts for the emerging of mathematical meanings. Some illustrative examples of two groups of students' work indicate the potential of the activities and tools for expressing and reflecting on the mathematical nature of the available formalism. We particularly focused on the students' engagement in reification processes, i.e. making sense of structural aspects of equations, involved in conceptualising them as objects that underlie the behaviour of the respective models. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, based on minimization under constraints, yield a full branch of solutions, and extend earlier results (see [3,4,8]) where only a part of the branch was built. In dimension three, we also show that there are no travelling wave solutions of small energy. [ABSTRACT FROM AUTHOR]

The basic rules of elementary algebra can often appear beyond the grasp of many students. Even though most subjects, including calculus, prove to be more difficult, it is the simple rules of algebra that continue to be the “thorn in the side” of many mathematics students. In this paper we present a result intended to help students achieve a greater understanding of a familiar algebraic error. [ABSTRACT FROM AUTHOR]

The presented paper develops a new method for solving Einstein equations explicitly. Not only the solutions are new but also the mathematical framework of their construction which is given by a nonstandard function theory built over nonstandard algebras (in the following denoted as GAPs) and will be called here as “GAP Theory.” It is shown that the GAP theory succeeds in regions, where group theory fails: the group structure is too small to solve Einstein equations explicitly. [ABSTRACT FROM AUTHOR]

The article presents the first part of a paper about second-order rational difference equations. This study aims to demonstrate the results of the global character of solutions of a second-order rational difference equation. Moreover, a number of open problems and conjectures related to the topic will also be presented in the article.

In this paper, we present C∞ examples of degenerate hyperbolic and mixed type Monge-Ampère equations in the plane, which do not admit a local C3 solution. [ABSTRACT FROM AUTHOR]

Using the criterion of this paper, we solve the substitution problem and obtain an algorithm for determining the solvability of logical equations in the modal logic S4. α N . Another corollary of the criterion is the solvability of the corresponding quasiequational theory of the free modal algebra whose signature is enriched with constants for the free generators. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that with at most $$ O{\left( {N^{{\frac{{65}} {{66}}}} } \right)} $$ exceptions, all even positive integers up to N are expressible in the form $$ p^{2}_{2} + p^{3}_{3} + p^{4}_{4} + p^{5}_{5} $$ . This improves a recent result $$ O{\left( {N^{{\frac{{19193}} {{19200}} + \varepsilon }} } \right)} $$ due to C. Bauer. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, our main purpose is to establish the existence of positive bounded entire solutions of second order quasilinear elliptic equations under new conditions. The main results of the present paper are new and extend the previously known results. [Copyright &y& Elsevier]

This paper is devoted to studying the initial value problems of the nonlinear Kaup–Kupershmidt equations $$ \frac{{\partial u}} {{\partial t}} + a_{1} \frac{{u\partial ^{2} u}} {{\partial x^{2} }} + \beta \frac{{\partial ^{3} u}} {{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}} {{\partial x^{5} }} = 0,$$ ( x, t) ∈ R2, and $$ \frac{{\partial u}} {{\partial t}} + a_{2} \frac{{\partial u}} {{\partial x}}\frac{{\partial ^{2} u}} {{\partial x^{2} }} + \beta \frac{{\partial ^{3} u}} {{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}} {{\partial x^{5} }} = 0, $$ ( x, t) ∈ R2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup–Kupershmidt equations. The results show that a local solution exists if the initial function u0( x) ∈ Hs ( R), and s ≥ 5/4 for the first equation and s ≥ 301/108 for the second equation. [ABSTRACT FROM AUTHOR]

In this paper, we redefine the Fox torus homotopy groups and give a proof of the split exact sequence of these groups. Evaluation subgroups are defined and are related to the classical Gottlieb subgroups. With our constructions, we recover the Abe groups and prove some results of Gottlieb for the evaluation subgroups of Fox homotopy groups. We further generalize Fox groups and define a group τ = [Σ( V× W⋃ *), X] in which the generalized Whitehead product of Arkowitz is again a commutator. Finally, we show that the generalized Gottlieb group lies in the center of τ, thereby improving a result of Varadarajan. [ABSTRACT FROM AUTHOR]

In this paper we discuss local uniqueness, continuity, and differentiability properties of solutions of parameterized variational inequalities (generalized equations). To this end we use two types of techniques. One approach consists in formulating variational inequalities in a form of optimization problem based on regularized gap functions, and applying a general theory of perturbation analysis of parameterized optimization problems. Another approach is based on a theory of contingent (outer graphical) derivatives and some results about differentiability properties of metric projections. [ABSTRACT FROM AUTHOR]

We prove that the equation x2 − kxy + y2 + x = 0 with k ∊ N+ has an infinite number of positive integer solutions x and y if and only if k = 3. For k = 3 the quotient SHAPE="SOL" ALIGN="C" STYLE="S">xy is asymptotically equal to (3 + √5)/2 or (3 − √5)/2. Results of the paper are based on data obtained via Computer Algebra System (derive 5). Some derive procedures presented in the paper made it possible to discover interesting regularities concerning simple continued fractions of certain numbers. [Copyright &y& Elsevier]

We prove the boundary Harnack principle for ratios of solutions u/ v of non-divergence second order elliptic equations Lu = a D u + b D u = 0 in a bounded domain Ω ⊂ $ {\mathbb R} $. We assume that b ∈ L(Ω) and Ω is a twisted Hölder domain of order α ∈ (1/2, 1]. Based on this result, we derive the Hölder regularity of u/ v for uniform domains. Bibliography: 27 titles. [ABSTRACT FROM AUTHOR]

In this paper, Numerical solution of Partial Diferential-Algebraic Equa- tions(PDAEs) is considered by Multivariate Padè Approximations. We applied these method to one example. First Partial Diferential-Algebraic Equation(PDAE) has been converted to power series by two-dimensional diferential transformation,Then the numerical solution of equation was put into Multivariate Padè series form. Thus we obtained numerical solution of Partial Diferential-Algebraic Equa- tion(PDAE). [ABSTRACT FROM AUTHOR]

We take a process component as a pair of an interface and a behaviour. We study the composition of interacting process components in the setting of process algebra. We formalize the interfaces of interacting process components by means of an interface group. An interesting feature of the interface group is that it allows for distinguishing between expectations and promises in interfaces of process components. This distinction comes into play in case components with both client and server behaviour are involved. [ABSTRACT FROM AUTHOR]

In this paper, we describe the conditions, providing the existence of splicing of the measures λ , μ and v - finite, finitely additive and integral functions of sets, defined on the algebras of subsets B and C of some set Ω and the algebra U, based on the algebras B and C . Moreover, we show that the measure λ should be a common extension of the measures μ and v , and λ(B∩C)= λ(B)λ(C) for any B ε B, C ε C. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we obtained rich solutions for the discrete complex cubic Ginzburg–Landau equation by means of the extended tanh-function approach. These solutions include chirpless bright soliton, chirpless dark soliton, triangular function solutions and some solutions with alternating phases, and so on. Meanwhile, the range of parameters where some exact solution exists is given. [Copyright &y& Elsevier]

Abstract: In this paper we study the existence of time periodic solutions of a Cahn–Hilliard type equation based on Leray–Schauder’s fixed point theorem. [Copyright &y& Elsevier]

Abstract: In this paper, we show the existence of strong solutions for the suspension bridge equations. Furthermore, the existence of strong global attractors is investigated using a new semigroup scheme. [Copyright &y& Elsevier]

Abstract: In this paper, the existence of positive boundary blow-up weak solution for the quasilinear elliptic equation div(∣∇u∣ p−2∇u)= m(x)f(u) in a smooth bounded domain Ω ⊆ R N or global space Ω = R N are obtained under new conditions. Our proof is based on the method of sub and supersolutions. [Copyright &y& Elsevier]

Using the criterion of this paper, we solve the substitution problem and obtain an algorithm for determining the solvability of logical equations in the modal logic S4. α N . Another corollary of the criterion is the solvability of the corresponding quasiequational theory of the free modal algebra whose signature is enriched with constants for the free generators. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider Newton iteration methods for solving nonlinear equations on matrix Lie groups. Recently, Owren and Welfert have proposed a method where the original nonlinear equation is transformed into a nonlinear equation on the Lie algebra of the group via the exponential map, thus Newton iteration methods may be applied. Based on this we suggest two improved variants of Newton iteration algorithm. One is that the exponential map would be approximated by Cayley map and give a Cayley version Newton iteration method for solving nonlinear equations on quadratic Lie groups, then we show that, the proposed method converges quadratically; Another is a variant of Newton type method with accelerated convergence and the numerical tests reported seem to support that it converges with cubically. [Copyright &y& Elsevier]

A cell-centered finite volume method is proposed to approximate numerically the solution to the steady convection-diffusion equation on unstructured meshes of $d$-simplexes, where $d\geq 2$ is the spatial dimension. The method is formally second-order accurate by means of a piecewise linear reconstruction within each cell and at mesh vertices. An algorithm is provided to calculate nonnegative and bounded weights. Face gradients, required to discretize the diffusive fluxes, are defined by a nonlinear strategy that allows us to demonstrate the existence of a maximum principle. Finally, a set of numerical results documents the performance of the method in treating problems with internal layers and solutions with strong gradients. [ABSTRACT FROM AUTHOR]

Abstract: Let be a field and a finite directed multi-graph. The focus of this paper is to offer a complete description of all path algebras and admissible orders with the property that all of their finitely generated ideals have finite Gröbner bases and of those which contain a finitely generated ideal whose Gröbner bases are all infinite. [Copyright &y& Elsevier]

Abstract: In [E. Babolian, M.M. Hosseini, Reducing index, and pseudospectral methods for differential–algebraic equations, Appl. Math. Comput. 140 (2003) 77–90] a reducing index method has been proposed for some cases of semi-explicit DAEs (differential algebraic equations). In this paper, this method is generalized to more cases. Also, it is focused on Hessenberg index 2 systems and proposed reduction index method will be illustrated for this problem. The Hessenberg system and its obtained reduced index system are numerically solved through pseudospectral method. In addition, aforementioned methods will be considered by one example. [Copyright &y& Elsevier]

Abstract: In this paper we consider bisymmetric and centrosymmetric solutions to certain matrix equations over the real quaternion algebra . Necessary and sufficient conditions are obtained for the matrix equation AX = C and the following systems to have bisymmetric solutions, and the system to have centrosymmetric solutions. The expressions of such solutions of the matrix and the systems mentioned above are also given. Moreover a criterion for a quaternion matrix to be bisymmetric is established and some auxiliary results on other sets over are also mentioned. [Copyright &y& Elsevier]

The article presents some open problems and conjectures about some interesting types of difference equations. For various researchers investigated the local asymptotic stability of its unique prime period-2 solution, found its basin of attraction and determined the global stability character of the equilibrium of the equation.

Abstract: This paper deals with the higher-order Kirchhoff-type equation with nonlinear dissipationin a bounded domain, where m &#60; 1 is a positive integer, q, p, r &#60; 0 arepositive constants. We obtain that the solution exists globally if p ≤ r, while ifp > max{r, 2q}, then for any initial data with negative initial energy, the solution blowsup at finite time in L p +2 norm. [Copyright &y& Elsevier]

We define the topological degree for a class of operators and use it to find generalized zeros of operators. Using these results we get the existence of solutions for singular nonlinear elliptic equations. [ABSTRACT FROM AUTHOR]

The first author has proved in 1980 that the Bear-invariant of a group with respect to a given variety of groups commutes with direct limit. In this paper, we introduce the generalized Baer-invariant with respect to two varieties of groups and prove a similar results. Among other result, we construct a varietal covering group of direct limit of a given direct system of groups. [ABSTRACT FROM AUTHOR]

Discusses the theorem of finite-dimensional real associative division algebra according to F.G. Frobenius. History of the development of the Frobenius theorem; Ways to prove the Frobenius theorem in a algebraic way; Role of the Frobenius theorem in the U.S. algebra.

Abstract: In this paper, some new oscillation criteria are obtained for the first-order delay difference equation , where k,l1,l2,, ls are positive integer numbers, f is a function. Our results improve the known results in the literature. And some examples are given to demonstrate the advantage of our results. [Copyright &y& Elsevier]

In this paper, nonlinear dispersive variants of the generalized KdV equation are investigated. The compact and the noncompact physical structures are studied. Two general formulas for the focusing branch and the defocusing branch, that are of substantial interest, are formally derived. [Copyright &y& Elsevier]

Many characterizations of semipreconnected (=β-connected) spaces are obtained by Aho and Nieminen [2]. In this paper, we obtain further characterizations of semipreconnected spaces. We introduce β-separated and βs-connected sets and by using these sets we obtain several properties of β-connected subspaces. [ABSTRACT FROM AUTHOR]

In this paper we study some properties of automorphisms of C*-algebras satisfying certain equations. Among other results we show that if u, v are invertible elements of a C*-algebra A such that uvu-1 + cu-1vu = (1 + c)v, and uv*u-1 + cu-1v*u = (1 + c)v*, where c ∈ C with c ≠ -1 and if u is normal, then u commutes with v and v*. Further, if α, β are inner automorphisms of a W*-algebra M implemented by normal operators u and v, respectively, then α + α-1 = β + β-1 if and only if there is a central projection P ∈ M such that Puv and (1 - P)uv-1 are in the center of M and consequently M admits a decomposition via a central projection q = (1 - P) such that α = β on Mq and α = β-1 on M(1 - q). [ABSTRACT FROM AUTHOR]

The main purpose of this paper if to derive an explicit expression for Fourier-Chebyshev coefficient [multiple line equation(s) cannot be represented in ASCII text], which is initiatied by L. Gori and C.A. Micchelli. [ABSTRACT FROM AUTHOR]

In this paper, the analysis of the dynamic system for iterative improvement of a solution is discussed, and the new iterative improvement of a solution is proposed based on the dynamic system. We have proved that the new iterative improvement of the solution is convergent unconditionally. The numerical experiments illustrate that the new iterative improvement of the solution is more effective for an ill-conditional system of linear equations. [Copyright &y& Elsevier]

For transient thermal problems, discretizing in space by the finite element method yields a system of first-order ordinary differential equations which can be integrated in time by a finite difference based predictor-multicorrector scheme. At each time level the temperature field is obtained by solving a set of algebraic equations. In this paper the multigrid method is adopted for transient thermal problems. In order to model accurately the geometry of practical problems with various material properties, and to use the predictor-corrector scheme for the nonlinear transient analysis, the generalized multigrid approach, similar to that proposed in Reference 1, is employed. Some examples are included to illustrate the performance of this approach for nonlinear transient problems. [ABSTRACT FROM AUTHOR]

We show how the derivation of the Derrida–Lebowitz–Speer–Spohn equation can be prolonged to obtain a new equation, generalizing the models obtained in the paper by these authors. We then investigate its properties from both an analytical and numerical perspective. Specifically, a numerical method is presented to approximate solutions of the prolonged equation. Using this method, we investigate the relationship between the solutions of the prolonged equation and the Tracy–Widom GOE distribution. [ABSTRACT FROM AUTHOR]