Limit behavior of global attractors for the complex Ginzburg–Landau equation on infinite lattices

In this work, the authors first show the existence of global attractors for the following lattice complex Ginzburg–Landau equation:

and for the following lattice Schrödinger equation:

Then they prove that the solutions of the lattice complex Ginzburg–Landau equation converge to that of the lattice Schrödinger equation as ε→0+. Also they prove the upper semicontinuity of as ε→0+ in the sense that .     

Approximate solution of the Fredholm integral equation of the first kind in a reproducing kernel Hilbert space

An approach for solving Fredholm integral equations of the first kind is proposed for in a reproducing kernel Hilbert space (RKHS). The interest in this problem is strongly motivated by applications to actual prospecting. In many applications one is puzzled by an ill-posed problem in space C[a,b] or L²[a,b], namely, measurements of the experimental data can result in unbounded errors of solutions of the equation. In this work, the representation of solutions for Fredholm integral equations of the first kind is obtained if there are solutions and the stability of solutions is discussed in RKHS. At the same time, a conclusion is obtained that approximate solutions are also stable with respect to _∞ or _L² in RKHS. A numerical experiment shows that the method given in the work is valid.     

Existence of an elliptic system involving Pucci operator

We study the existence of solutions for the nonlinear elliptic system

where Ω is a bounded domain, f₁ is superlinear and f₂ is sublinear at zero and infinity, h₁ and h₂ are perturbation terms. We will show that the system has at least two semi-trivial solutions (u,0), (0,v) and a nontrivial solution (u^*,v^*).     

Global stability of a rational difference equation

In this paper, we study the global stability of the difference equation , where the parameters a,a_i(0,∞) for i=0,…,k, x_-k,…, x_-1[0,∞) and x₀(0,∞). We prove that the unique positive equilibrium is globally asymptotically stable if and only if it is locally asymptotically. Also we provide sufficient condition for it to be globally asymptotically stable and our results solve the open problem proposed by Kulenović and Ladas (Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2002).     

Precise asymptotics for the first moment of the error variance estimator in linear models

Let σ² be the unknown error variance of a linear model and let be the estimator of σ² based on the residual sum of squares. In this work, we show the precise asymptotics in the law of the logarithm for the first moment of the error variance estimator.     

On periodic radical groups in which permutability is a transitive relation

A group G is said to be a -group if permutability is a transitive relation in the set of all subgroups of G. Our purpose in this paper is to study -groups in the class of periodic radical groups satisfying min-p for all primes p.     

Global existence and blow-up solutions and blow-up estimates for a non-local quasilinear degenerate parabolic system

This paper deals with p-Laplacian systems

with null Dirichlet boundary conditions in a smooth bounded domain ΩR^N, where p,q>1, , and a,b>0 are positive constants. We first get the non-existence result for a related elliptic systems of non-increasing positive solutions. Secondly by using this non-existence result, blow-up estimates for above p-Laplacian systems with the homogeneous Dirichlet boundary value conditions are obtained under Ω=B_R={xR^N:|x|<R}(R>0). Then under appropriate hypotheses, we establish local theory of the solutions and obtain that the solutions either exists globally or blow-up in finite time.     

Uniform asymptotic expansions of the Pollaczek polynomials

Two uniform asymptotic expansions are obtained for the Pollaczek polynomials P_n(cosθ;a,b). One is for , , in terms of elementary functions and in descending powers of . The other is for , in terms of a special function closely related to the modified parabolic cylinder functions, in descending powers of n. This interval contains a turning point and all possible zeros of P_n(cosθ) in θ(0,π/2].     

On oscillatory solutions of certain first order ordinary differential equations

By constructing a class of solutions to the integral inequality for t  t₀ large enough, where 0<A₁a(τ)A₂<+∞ and λ>1, that tend to zero as t→+∞ we address an open problem in the theory of nonlinear oscillations.     

Limit shape of convex lattice polygons with minimal perimeter

Let n be a positive integer and · any norm in . Denote by B the unit ball of · and the class of convex lattice polygons with n vertices and least ·-perimeter. We prove that after suitable normalization, all members of tend to a fixed convex body, as n→∞.     

Countable choice and compactness

We work in set-theory without choice ZF. Denoting by the countable axiom of choice, we show in that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak topology in ZF). We prove that this ball is (closely) convex-compact in the convex topology. Given a set I, a real number p1 (respectively p=0), and some closed subset F of [0,1]^I which is a bounded subset of ℓ^p(I), we show that (respectively DC, the axiom of Dependent Choices) implies the compactness of F.     

A survey on proper codes

The performance of a linear t-error correcting code over a q-ary symmetric memoryless channel with symbol error probability ε is characterized by the probability that a transmission error will remain undetected. This probability is a function of ε involving the code weight distribution and the weight distribution of the cosets of minimum weight at most t. When the undetectable error probability is an increasing function of ε, the code is called t-proper.

The paper presents sufficient conditions for t-properness and a list of codes known to be proper, many of which have been studied by these sufficient conditions. Special attention is paid to error detecting codes of interest in modern communication.     

On partitioning two matroids into common independent subsets

Let M₁ and M₂ be two matroids on the same ground set S. We conjecture that if there do not exist disjoint subsets A₁,A₂,…,A_k+1 of S, such that _isp_M1(A_i)≠Ø, and similarly for M₂, then S is partitioned into k sets, each independent in both M₁ and M₂. This is a possible generalization of König's edge-coloring theorem. We prove the conjecture for the case k=2 and for a regular case, in which both matroids have the same rank d, and S consists of k·d elements. Finally, we prove another special case related to a conjecture of Rota.     

On relations between 1-lines of Adams–Novikov spectral sequences modulo invariant prime ideals

We have a ring homomorphism Θ from the cohomology of the extended Morava stabilizer group G_n with coefficients in F[w^±1] to the cohomology of G_n+1 with coefficients in the graded field F((u_n))[u^±1]. In this note we study the behavior of Θ on H¹. Then it is shown that Θ is injective on H¹ for n1 and for all primes p.     

Discrete subspaces of countably tight compacta

Our main result is that the following cardinal arithmetic assumption, which is a slight weakening of GCH, “2^κ is a finite successor of κ for every cardinal κ”, implies that in any countably tight compactum X there is a discrete subspace D with . This yields a (consistent) confirmation of Alan Dow’s Conjecture 2 from [A. Dow, Closures of discrete sets in compact spaces, Studia Math. Sci. Hung. 42 (2005) 227–234].     

A priori error estimates for interior penalty discontinuous Galerkin method applied to nonlinear Sobolev equations

A discontinuous Galerkin method with interior penalties is presented for nonlinear Sobolev equations. A semi-discrete and a family of Fully-discrete time approximate scheme are formulated. These schemes can be symmetric or nonsymmetric. Hp-version error estimates are analyzed for these schemes. Just because of a damping term ·(b(u)u_t) included in Sobolev equation, which is the distinct character different from parabolic equation, special test functions are chosen to deal with this term. Finally, a priori L^∞(H¹) error estimate is derived for the semi-discrete time scheme and similarly, l^∞(H¹) and l²(H¹) for the Fully-discrete time schemes. These results also indicate that spatial rates in H¹ and time truncation errors in L² are optimal.     

On -differential graded algebras

We introduce the concept of N-differential graded algebras (N-dga), and study the moduli space of deformations of the differential of an N-dga. We prove that it is controlled by what we call the (M,N)-Maurer–Cartan equation.     

Fractal sets determined by dilation-stable processes

Let be a dilation-stable process on . We determine a Hausdorff measure function (a) such that the fractal set X[0,1]={X(t):0t1} has positive finite -measure. We also investigate the packing measure of X[0,1].     

Nonlinear maps preserving Lie products on factor von Neumann algebras

In this paper, we prove that every bijective map preserving Lie products from a factor von Neumann algebra into another factor von Neumann algebra is of the form A→ψ(A)+ξ(A), where is an additive isomorphism or the negative of an additive anti-isomorphism and is a map with ξ(AB-BA)=0 for all .     

Coclass and cohomology

Recently, it was proved by Leedham-Green and others that with a finite number of exceptions, every p-group of coclass r is a quotient of one of only a finite number of p-adic uniserial space groups. In this paper we use that structure to demonstrate that there are only finitely many isomorphism classes of cohomology rings of 2-groups of coclass r with coefficients in any fixed field k of characteristic 2. In addition, there is experimental evidence indicating that in many cases successive quotients of the uniserial space groups have isomorphic cohomology rings.