Elliptic gradient constraint problem

We study the existence and regularity of gradient constraint problem. It arises in elastoplasticity and finance. First, we consider linear double obstacle problem which comes from viscosity solution to Hamilton–Jacobi equation and find the solution has C^1,α regularity by estimating Campanato-type integral oscillation. Then, by perturbation method and fixed point theorem in C^1,α space, we prove the existence of C^1,α solution.     

Computable analysis of a boundary‐value problem for the generalized KdV–Burgers equation

In this paper, we investigate the computability of the solution operator of the generalized KdV‐Burgers equation with initial‐boundary value problem. Here, the solution operator is a nonlinear map H^{3m ? 1}(R⁺) × H^m(0,T)→C([0,T];H^{3m ? 1}(R⁺)) from the initial‐boundary value data to the solution of the equation. By a technique that is widely used for the study of nonlinear dispersive equation, and using the type 2 theory of effectivity as computable model, we prove that the solution map is Turing computable, for any integer m ≥ 2, and computable real number T > 0. Copyright © 2014 John Wiley & Sons, Ltd.     

A Description of the Global Attractor for a Class of Reaction – Diffusion Systems with Periodic Solutions

We examine the autonomous reaction–diffusion system with Dirichlet boundary conditions on (0, 1), where α, β are real, α > 0, and g is C¹ and satisfies some conditions which we need in order to prove the existence of solutions. We construct a solution of (RD) for every initial value in L²((0, 1)) × L²((0, 1)), we show that this solution is uniquely determined and that the solution has C^∞–smooth representatives for all positive t. We determine the long time behaviour of each solution. In particular, we show that each solution of (RD) tends either to the zero solution or to a periodic orbit. We construct all periodic orbits and show that their number is always finite. It turns out that the global attractor is a finite union of subsets of L² × L², which are finite–dimensional manifolds, and the dynamics in these sets can be described completely.     

Error Estimates for Barycentric Finite Volumes Combined with Nonconforming Finite Elements Applied to Nonlinear Convection-Diffusion Problems

The subject of the paper is the derivation of error estimates for the combined finite volume-finite element method used for the numerical solution of nonstationary nonlinear convection-diffusion problems. Here we analyze the combination of barycentric finite volumes associated with sides of triangulation with the piecewise linear nonconforming Crouzeix-Raviart finite elements. Under some assumptions on the regularity of the exact solution, the L ²(L ²) and L ²(H ¹) error estimates are established. At the end of the paper, some computational results are presented demonstrating the application of the method to the solution of viscous gas flow.     

Nonhomogeneous Neumann problem for the Poisson equation in domains with an external cusp

In this work we analyze the existence and regularity of the solution of a nonhomogeneous Neumann problem for the Poisson equation in a plane domain Ω with an external cusp. In order to prove that there exists a unique solution in H¹(Ω) using the Lax–Milgram theorem we need to apply a trace theorem. Since Ω is not a Lipschitz domain, the standard trace theorem for H¹(Ω) does not apply, in fact the restriction of H¹(Ω) functions is not necessarily in L²(∂Ω). So, we introduce a trace theorem by using weighted Sobolev norms in Ω. Under appropriate assumptions we prove that the solution of our problem is in H²(Ω) and we obtain an a priori estimate for the second derivatives of the solution.     

Superconvergent interpolants for collocation methods applied to Volterra integro-differential equations with delay

Standard software based on the collocation method for differential equations delivers a continuous approximation (called the collocation solution) which augments the high order discrete approximate solution that is provided at mesh points. This continuous approximation is less accurate than the discrete approximation. For ‘non-standard’ Volterra integro-differential equations with constant delay, that often arise in modeling predator-prey systems in Ecology, the collocation solution is C ⁰ continuous. The accuracy is O(h ^s+1) at off-mesh points and O(h ^2s ) at mesh points where s is the number of Gauss points used per subinterval and h refers to the stepsize. We will show how to construct C ¹ interpolants with an accuracy at off-mesh points and mesh points of the same order (2s). This implies that even for coarse mesh selections we achieve an accurate and smooth approximate solution. Specific schemes are presented for s=2, 3, and numerical results demonstrate the effectiveness of the new interpolants.     

The solution of the equation AX + BX ⋆ = 0

We give a complete solution of the matrix equation AX?+?BX ^??=?0, where A, B?∈?? ^m×n are two given matrices, X?∈?? ^n×n is an unknown matrix, and ? denotes the transpose or the conjugate transpose. We provide a closed formula for the dimension of the solution space of the equation in terms of the Kronecker canonical form of the matrix pencil A?+?λB, and we also provide an expression for the solution X in terms of this canonical form, together with two invertible matrices leading A?+?λB to the canonical form by strict equivalence.     

Qualitative Properties of Saddle-Shaped Solutions to Bistable Diffusion Equations

We consider the elliptic equation ? Δu = f(u) in the whole ?^2m , where f is of bistable type. It is known that there exists a saddle-shaped solution in ?^2m . This is a solution which changes sign in ?^2m and vanishes only on the Simons cone 𝒞 = {(x ¹, x ²) ∈ ? ^m × ? ^m : |x ¹| = |x ²|}. It is also known that these solutions are unstable in dimensions 2 and 4.

In this article we establish that when 2m = 6 every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution.

These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established.     

L^p- L^q decay estimates of solutions to Cauchy problems of thermoviscoelastic systems

L^p- L^q decay estimate of solution to Cauchy problem of a linear thermoviscoelastic system is studied. By using a diagonalization argument of frequency analysis, the coupled system will be decoupled micrologically. Then with the help of the information of characteristic roots for the coefficient matrix of the system, L^p- L^q decay estimate of parabolic type of solution to the Cauchy problem is obtained.     

Large time behavior for the incompressible magnetohydrodynamic equations in half‐spaces

The weighted L^r‐asymptotic behavior of the strong solution and its first‐order spacial derivatives to the incompressible magnetohydrodynamic (MHD) equations is established in a half‐space. Further, the L^∞‐decay rates of the second‐order spatial derivatives of the strong solution are derived by using the Stokes solution formula and employing a decomposition for the nonlinear terms in MHD equations. Copyright © 2014 John Wiley & Sons, Ltd.     

指数Diophantine方程4~x+b~y=(b+4)~2

本文研究了指数Diophantine方程4~x+b~y=(b+4)~2的解.设b1是给定的正奇数,运用有关指数Diophantine方程的已知结果以及有关Pell方程的Stormer定理的推广,证明了方程4~x+b~y=(b+4)~2仅有正整数解(x,y,z)=(1,1,1).     

Symmetry Properties of Positive Solutions of Parabolic Equations on ℝ N : II. Entire Solutions

We consider nonautonomous quasilinear parabolic equations satisfying certain symmetry conditions. We prove that each positive bounded solution u on ? ^N  × (?∞, T) decaying to zero at spatial infinity uniformly with respect to time is radially symmetric around some origin in ? ^N . The origin depends on the solution but is independent of time. We also consider the linearized equation along u and prove that each bounded (positive or not) solution is a linear combination of a radially symmetric solution and (nonsymmetric) spatial derivatives of u. Theorems on reflectional symmetry are also given.     

A conflict between sequential rationality and consistency principles

It is shown that no solution concept that selects sequentially rational (perfect, proper, persistent, or members of some stable set of) equilibria satisfies the following consistency property. Suppose that in every solution of the game G, player i's action is a, and denote by G ^a the game in which player i is restricted to choose a. Then some player j≠i has an action c that is used with positive probability in both some solution of G and some solution of G ^a. This result illustrates a conflict between a mild consistency condition and sequential rationality. Received: January 2001/Final version: April 2002     

Error analysis of the finite-strip method for parabolic equations

The finite-strip method (FSM) is a hybrid technique which combines spectral and finite-element methods. Finite-element approximations are made for each mode of a finite Fourier series expansion. The Galerkin formulated method is set apart from other weighted-residual techniques by the selection of two types of basis functions, a piecewise linear interpolating function and a trigonometric function. The efficiency of the FSM is due in part to the orthogonality of the complex exponential basis: The linear system which results from the weak formulation is decoupled into several smaller systems, each of which may be solved independently. An error analysis for the FSM applied to time-dependent, parabolic partial differential equations indicates the numerical solution error is O(h² + M^?r). M represents the Fourier truncation mode number and h represents the finite-element grid mesh. The exponent r ≥ 2 increases with the exact solution smoothness in the respective dimension. This error estimate is verified computationally. Extending the result to the finite-layer method, where a two-dimensional trigonometric basis is used, the numerical solution error is O(h² + M^?r + N^?q). The N and q represent the trucation mode number and degree of exact solution smoothness in the additional dimension. © 1993 John Wiley & Sons, Inc.     

An inner product and orthonormalization inl p -spaces and solution of infinite systems of infinite linear equations

Let the rows of an infinite square matrixM be elements ofl ^p -space (p>1) andX be an infinite column vector of unknowns andC an infinite column vector of real numbers. To our knowledge the solvability ofMX=C has nowhere been satisfactorily studied in the literature. This is also true of Riesz’classical work [2]. A reason for this is that not until recently [1] an appropriate inner product and the corresponding orthonormalization forp≠2 has been introduced. In this paper, based on [1], it is shown thatMX=C has a solution which is an element ofl ^q if and only if upon our process of orthonormalization of the rows ofM the system yields an equivalent systemAX=K where the rows ofA form an orthonormal sequence (in our sense) of elements ofl ^p andK becomes an element ofl ^q withp ⁻¹+q ⁻¹=1. A solution is then given byX=(A ^(q) (AA ^(q) )⁻¹)K whereA ^(q) is ourq-transpose ofA. This solution is the solution of the minimall ^q -norm. Otherwise, the obvious dual concept of the best approximating solution inl ^q -norm is introduced and obtained. 1980 Mathematics Subject Classification: Primary 46C10, Secondary 15A06     

Perturbation analysis of the matrix equation

Consider the nonlinear matrix equation X-A^*X^-pA=Q with 0<p1. This paper shows that there exists a unique positive definite solution to the equation. A perturbation bound and the backward error of an approximate solution to this solution is evaluated. We also obtain explicit expressions of the condition number for the unique positive definite solution. The theoretical results are illustrated by numerical examples.     

On Interacting Systems of Hilbert-Space-Valued Diffusions

A nonlinear Hilbert-space-valued stochastic differential equation where L ^-1 (L being the generator of the evolution semigroup) is not nuclear is investigated in this paper. Under the assumption of nuclearity of L ^-1 , the existence of a unique solution lying in the Hilbert space H has been shown by Dawson in an early paper. When L ^-1 is not nuclear, a solution in most cases lies not in H but in a larger Hilbert, Banach, or nuclear space. Part of the motivation of this paper is to prove under suitable conditions that a unique strong solution can still be found to lie in the space H itself. Uniqueness of the weak solution is proved without moment assumptions on the initial random variable. A second problem considered is the asymptotic behavior of the sequence of empirical measures determined by the solutions of an interacting system of H -valued diffusions. It is shown that the sequence converges in probability to the unique solution Λ ₀ of the martingale problem posed by the corresponding McKean—Vlasov equation. Accepted 4 April 1996     

Lower Bounds for Catalan's Equation

This paper begins with a short historical survey on Catalan's equation, namely x^p-y^q=1, where p andq are prime numbers and x, y are non-zero rational integers. It is conjectured that the only solution is the trivial solution 3²-2³=1. We prove that there is no non-trivial solution with p orq smaller than 30000. The tools to reach such a result are presented. A crucial role is played by a recent estimate of linear forms in two logarithms obtained by Laurent, Mignotte and Nestrenko. The criteria used are also quite recent. We give information on the enormous amount of computation needed for the verification.     

Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

Integer knapsack problems with set-up weights

The Integer Knapsack Problem with Set-up Weights (IKPSW) is a generalization of the classical Integer Knapsack Problem (IKP), where each item type has a set-up weight that is added to the knapsack if any copies of the item type are in the knapsack solution. The k-item IKPSW (kIKPSW) is also considered, where a cardinality constraint imposes a value k on the total number of items in the knapsack solution. IKPSW and kIKPSW have applications in the area of aviation security. This paper provides dynamic programming algorithms for each problem that produce optimal solutions in pseudo-polynomial time. Moreover, four heuristics are presented that provide approximate solutions to IKPSW and kIKPSW. For each problem, a Greedy heuristic is presented that produces solutions within a factor of 1/2 of the optimal solution value, and a fully polynomial time approximation scheme (FPTAS) is presented that produces solutions within a factor of ε of the optimal solution value. The FPTAS for IKPSW has time and space requirements of O(nlog n+n/ε ²+1/ε ³) and O(1/ε ²), respectively, and the FPTAS for kIKPSW has time and space requirements of O(kn ²/ε ³) and O(k/ε ²), respectively.