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.     

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.     

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.     

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 .     

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).     

A Chebyshev type inequality for fuzzy integrals

In this paper, we prove a Chebyshev type inequality for fuzzy integrals. More precisely, we show that:

where μ is the Lebesgue measure on and f,g:[0,1]→[0,∞) are two continuous and strictly monotone functions, both increasing or both decreasing. Also, some examples and applications are presented.     

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.     

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→∞.     

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.     

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].     

The PI index of product graphs

The Padmakar–Ivan index of a graph G is the sum over all edges uv of G of number of edges which are not equidistant from u and v. In this work, an exact expression for the PI index of the Cartesian product of bipartite graphs is computed. Using this formula, the PI indices of C₄ nanotubes and nanotori are computed.     

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.     

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.     

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 .     

Mass formula for self-dual codes over Z

We give a mass formula for self-dual codes over Z_p², where p is an odd prime. Using the mass formula, we classify such codes of lengths up to n=8 over the ring Z₉, n=7 over Z₂₅ and n=6 over Z₄₉.     

Proximity problems on line segments spanned by points

Finding the closest or farthest line segment (line) from a point are fundamental proximity problems. Given a set S of n points in the plane and another point q, we present optimal O(nlogn) time, O(n) space algorithms for finding the closest and farthest line segments (lines) from q among those spanned by the points in S. We further show how to apply our techniques to find the minimum (maximum) area triangle with a vertex at q and the other two vertices in S{q} in optimal O(nlogn) time and O(n) space. Finally, we give an O(nlogn) time, O(n) space algorithm to find the kth closest line from q and show how to find the k closest lines from q in O(nlogn+k) time and O(n+k) space.     

Real hypersurfaces in complex projective space with recurrent structure Jacobi operator

We classify real hypersurfaces of complex projective space , m3, with -recurrent structure Jacobi operator and apply this result to prove the non-existence of such hypersurfaces with recurrent structure Jacobi operator.     

Covering point sets with two disjoint disks or squares

We study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks C_R and C_B with disjoint interiors such that the number of red points covered by C_R plus the number of blue points covered by C_B is maximized. We give an algorithm to solve this problem in O(n^8/3log²n) time, where n denotes the total number of points. We also show that the analogous problem of finding two axis-aligned unit squares S_R and S_B instead of unit disks can be solved in O(nlogn) time, which is optimal. If we do not restrict ourselves to axis-aligned squares, but require that both squares have a common orientation, we give a solution using O(n³logn) time.     

A non-normal topology generated by a two-point selection

We construct a two-point selection , where is the set of the irrational numbers, such that the space is not normal and it is not collectionwise Hausdorff either. Here, τ_f denotes the topology generated by the two-point selection f. This example answers a question posed by V. Gutev and T. Nogura. We also show that if is a two-point selection such that the topology τ_f has countable pseudocharacter, then τ_f is a Tychonoff topology.     

Jumps in cohomology and free group actions

A discrete group G has periodic cohomology over R if there is an element in a cohomology group cup product with which it induces an isomorphism in cohomology after a certain dimension. Adem and Smith showed that if , then this condition is equivalent to the existence of a finite dimensional free-G-CW-complex homotopy equivalent to a sphere. It has been conjectured by Olympia Talelli, that if G is also torsion-free then it must have finite cohomological dimension. In this paper we use the implied condition of jump cohomology over R to prove the conjecture for -groups and solvable groups. We also find necessary conditions for free and proper group actions on finite dimensional complexes homotopy equivalent to closed, orientable manifolds.