Elementary equivalent pairs of algebras associated with sets

The elementary equivalence of two full relation algebras, partition lattices or function monoids are shown to be equivalent to the second order equivalence of the cardinalities of the corresponding sets. This is shown to be related to elementary equivalence of permutation groups and ordinals. Infinite function monoids are shown to be ultrauniversal.Presented by Walter Taylor.The work of the second author was supported by a grant from the University of Cape Town Research Committee, and by the Topology Research Group from the University of Cape Town and the South African Council for Scientific and Industrial Research.     

On the convergence of a fourth order evolution equation to the Allen-Cahn equation

We construct special sequences of solutions to a fourth order nonlinear parabolic equation of Cahn-Hilliard/Allen-Cahn type, converging to the second order Allen-Cahn equation. We consider the evolution equation without boundary, as well as the stationary case on domains with Dirichlet boundary conditions. The proofs exploit the equivalence of the fourth order equation with a system of two second order elliptic equations with “good signs”.     

两类方程Cauchy问题经典解的等价性

对于给出的一类二阶线性双曲型方程,通过未知变量替换,将其化为一阶对称双曲型方程组.可以证明这个一阶对称双曲型方程组与原来的二阶线性双曲型方程的Cauchy问题的经典解在某种意义下是等价的.     

Matrix representations of fourth order boundary value problems with finite spectrum

We show that a class of regular self-adjoint fourth order boundary value problems is equivalent to a certain class of matrix problems. Equivalent here means that they have exactly the same eigenvalues. Such an equivalence was previously known only in the second order case.     

A second order <Emphasis Type="Italic">η</Emphasis>-approximation method for constrained optimization problems involving second order invex functions

A new approach for obtaining the second order sufficient conditions for non-linear mathematical programming problems which makes use of second order derivative is presented. In the so-called second order η-approximation method, an optimization problem associated with the original nonlinear programming problem is constructed that involves a second order η-approximation of both the objective function and the constraint function constituting the original problem. The equivalence between the nonlinear original mathematical programming problem and its associated second orderη-approximated optimization problem is established under second order invexity assumption imposed on the functions constituting the original optimization problem.     

Classification of scalar third order ordinary differential equations linearizable via generalized contact transformations

Whereas Lie had linearized scalar second order ordinary differential equations (ODEs) by point transformations, and later Chern had extended to the third order by using contact transformation, till recently no work had been done for higher order (or systems) of ODEs. Lie had found a unique class defined by the number of infinitesimal symmetry generators but the more general ODEs were not so classified. Recently, classifications of higher order and systems of ODEs were provided. In this paper we relate contact symmetries of scalar ODEs with point symmetries of reduced systems. We define a new type of transformation that builds upon this relation and obtain equivalence classes of scalar third order ODEs linearizable via these transformations. Four equivalence classes of such equations are seen to exist.     

Identification of heat-conduction coefficient via method of variational imbedding

The inverse parabolic problem of coefficient identification from over-posed data is embedded into a fourth order in space and second order in time elliptic boundary value problem. The latter is well posed for redundant data at boundaries. The equivalence of the two problems is demonstrated. A difference scheme of splitting-type is employed, and featuring examples are elaborated numerically.     

Floquet theory for second order linear homogeneous difference equations

In this paper we provide a version of the Floquet’s theorem to be applied to any second order difference equations with quasi-periodic coefficients. To do this we extend to second order linear difference equations with quasi-periodic coefficients, the known equivalence between the Chebyshev equations and the second order linear difference equations with constant coefficients. So, any second order linear difference equations with quasi-periodic coefficients is essentially equivalent to a Chebyshev equation, whose parameter only depends on the values of the quasi-periodic coefficients and can be determined by a non-linear recurrence. Moreover, we solve this recurrence and obtaining a closed expression for this parameter. As a by-product we also obtain a Floquet’s type result; that is, the necessary and sufficient condition for the equation has quasi-periodic solutions.     

PRECONDITIONING MIXED ELEMENT METHOD FOR NONSELFADJOINT AND INDEFINITE SECOND ORDER ELLIPTIC PROBLEMS*

In this paper, an equivalence between mixed element method and nonconforming element method for nonselfad joint and indefinite second order elliptic problems is established without using any bubble functions. It is proved that the H~1-condition number of preconditioned operator B_h~(-1)A_h is uniformly bounded and its B_h-singular values cluster in a positive finite interval, where A_h is the equivalent nonconforming element discretization of nonselfad joint and indefinite second order elliptic operator A, B_h is usual noncon forming element discretization of selfadjoint and positive definite second order elliptic operator B. Finally a simple V-cycle multigrid implementation of B_h~(-1) is given.     

On Commutative and Nonassociative Syntactic Calculi and Categorial Grammars

Two axiomatizations of the nonassociative and commutative Lambek syntactic calculus are given and their equivalence is proved. The first axiomatization employs Permutation as the only structural rule, the second one, with no Permutation rule, employs only unidirectional types. It is also shown that in the case of the Ajdukiewicz calculus an analogous equivalence is valid only in the case of a restricted set of formulas. Unidirectional axiomatizations are employed in order to establish the generative power of categorial grammars based on the nonassociative and commutative Lambek calculus with product. Those grammars produce CF-languages of finite degree generated by CF-grammars closed with respect to permutations.     

Oscillation and other properties of linear impulsive and nonimpulsive delay equations

Many of the oscillation results for linear impulsive equations were justified by the following scheme. First, the equivalence of the oscillation of the impulsive equation and some specially constructed nonimpulsive equation was established; further, on the base of well-known results for the nonimpulsive case, the oscillation of the impulsive equation was analyzed. In the present paper, we prove the “oscillation equivalence” result for a linear impulsive equation with a distributed delay and discuss the possibility to expand this approach to the other properties of impulsive equations, for example, stability and asymptotic behavior. In addition, a linear impulsive equation of the second order is considered.     

On asymptotic equivalence of linear and quasilinear difference equations

The asymptotic equivalence of systems of difference equations of linear and quasilinear type is investigated. The first result on the asymptotic equivalence of linear systems is a discrete analog of an improved version of the Levinson's well-known theorem on asymptotic equivalence of linear differential equations, while the second one providing conditions for asymptotic equivalence of linear and quasilinear systems is related to that of Yakubovich in differential equations case.     

Non‐standard analysis in ACA0 and Riemann mapping theorem

This research is motivated by the program of reverse mathematics and non‐standard arguments in second‐order arithmetic. Within a weak subsystem of second‐order arithmetic ACA₀, we investigate some aspects of non‐standard analysis related to sequential compactness. Then, using arguments of non‐standard analysis, we show the equivalence of the Riemann mapping theorem and ACA₀ over WKL₀. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Second order analysis for the optimal control of parabolic equations under control and final state constraints

We consider the optimal control of a semilinear parabolic equation with pointwise bound constraints on the control and finitely many integral constraints on the final state. Using the standard Robinson’s constraint qualification, we provide a second order necessary condition over a set of strictly critical directions. The main feature of this result is that the qualification condition needed for the second order analysis is the same as for classical finite-dimensional problems and does not imply the uniqueness of the Lagrange multiplier. We establish also a second order sufficient optimality condition which implies, for problems with a quadratic Hamiltonian, the equivalence between solutions satisfying the quadratic growth property in the L ¹ and \(L^{\infty }\) topologies.     

Restrictions of continuous functions

Let α = {α₁, ...,α_k} be a finite multiset of non-negative real numbers. Consider the sequence of all positive integer multiples of all α _i ’s, and note the multiplicity of each term in this sequence. This sequence of multiplicities is the resonance sequence generated by {α ₁, ...,α_k}. Two multisets are combinatiorially equivalent if they generate the same resonance sequence. The paper is devoted to the classification of multisets up to combinatorial equivalence. We show that the problem of combinatorial equivalence of multisets is closely related to the problem of classification of systems of second order ordinary differential equations up to focal equivalence.     

Asymptotic forms of positive solutions of quasilinear ordinary differential equations with singular nonlinearities

In this paper we consider positive solutions of second order quasilinear ordinary differential equations with singular nonlinearities. We obtain asymptotic equivalence theorems for asymptotically superlinear solutions and decaying solutions. By using these theorems, exact asymptotic forms of such solutions are determined. Furthermore, we can establish the uniqueness of decaying solutions as an application of our results.     

Equivalence of Entropy and Renormalized Solutions of Anisotropic Elliptic Problem in Unbounded Domains with Measure Data

We consider a class of anisotropic elliptic equations of second order with variable exponents of non-linearity where a special Radon measure is used as the right-hand side. We establish uniqueness of entropy and renormalized solutions of the Dirichlet problem in anisotropic Sobolev spaces with variable exponents of non-linearity for arbitrary domains and certain other their properties. In addition, we prove the equivalence of entropy and renormalized solutions of the problem under consideration.     

在仿射等价类中找具有好的密码学性质的布尔函数

The Boolean functions in an affine equivalence class are of the same algebraic degree and nonlinearity, but may satisfy different order of correlation immunity and propagation criterion. A method is presented in this paper to find Boolean functions with higher order correlation immunity or satisfying higher order propagation criterion in an affine equivalence class. 8 AES s-box functions are not better Boolean functions in their affine equivalence class.     

First order theory of permutation groups

We solve the problem of the elementary equivalence (definability) of the permutation groups over cardinals ℵ_α. We show that it suffices to solve the problem of elementary equivalence (definability) for the ordinals α in certain second order logic, and this is reduced to the case of α < (2^ℵ ₀)⁺. We solve a problem of Mycielski and McKenzie on embedding of free groups in permutation groups, and discuss some weak second-order quantifiers. An erratum to this article is available at .