Isochronicity for a class of reversible systems

This paper deals with the relation between isochronicity and first integral for a class of reversible systems: , , which associates to the first integral of the form H(x,y)=F(x)y²+G(x). Two necessary and sufficient conditions are given to characterize isochronicity for these systems. Moreover, we apply these results to show that there exists a class of polynomial reversible systems of degree n with isochronous center for any n.     

On the algebraic structure of Abelian integrals for a kind of perturbed cubic Hamiltonian systems

The finite generators of Abelian integral are obtained, where Γh is a family of closed ovals defined by H(x,y)=x²+y²+ax⁴+bx²y²+cy⁴=h, h∈Σ, ac(4ac−b²)≠0, Σ=(0,h₁) is the open interval on which Γh is defined, f(x,y), g(x,y) are real polynomials in x and y with degree 2n+1 (n?2). And an upper bound of the number of zeros of Abelian integral I(h) is given by its algebraic structure for a special case a>0, b=0, c=1.     

Images of locally finite derivations of polynomial algebras in two variables

In this paper we show that the image of any locally finite k-derivation of the polynomial algebra k[x,y] in two variables over a field k of characteristic zero is a Mathieu subspace. We also show that the two-dimensional Jacobian conjecture is equivalent to the statement that the image of every k-derivation D of k[x,y] such that and is a Mathieu subspace of k[x,y].     

Positive integral operators in unbounded domains

We study positive integral operators in with continuous kernel k(x,y). We show that if the operator is compact and Hilbert-Schmidt. If in addition k(x,x)→0 as |x|→∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and is trace class. Replacing the first assumption by the stronger then and the bilinear series converges also in L¹. Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.     

Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial

Using the theory of elliptic curves, we show that the class number h(−p) of the field appears in the count of certain factors of the Legendre polynomials , where p is a prime >3 and m has the form (p−e)/k, with k=2,3 or 4 and . As part of the proof we explicitly compute the Hasse invariant of the Hessian curve y²+αxy+y=x³ and find an elementary expression for the supersingular polynomial ss_p(x) whose roots are the supersingular j-invariants of elliptic curves in characteristic p. As a corollary we show that the class number h(−p) also shows up in the factorization of certain Jacobi polynomials.     

The third order Melnikov function of a quadratic center under quadratic perturbations

We study quadratic perturbations of the integrable system , where H=(x²+y²)/2. We prove that the first three Melnikov functions associated to the perturbed system give rise at most to three limit cycles.     

Mapping properties for oscillatory integrals in d-dimensions

For aj,bj?1, j=1,2,…,d, we prove that the operator maps into itself for , where , and k(x,y)=φ(x,y)eig(x,y), φ(x,y) satisfies (1.2) (e.g. φ(x,y)=|x−y|^iτ,τ real) and the phase g(x,y)=xa⋅yb. We study operators with more general phases and for these operators we require that aj,bj>1, j=1,2,…,d, or al=bl?1 for some l∈{1,2,…,d}.     

On the spanning fan-connectivity of graphs

Let G be a graph. The connectivity of G, κ(G), is the maximum integer k such that there exists a k-container between any two different vertices. A k-container of G between u and v, C_k(u,v), is a set of k-internally-disjoint paths between u and v. A spanning container is a container that spans V(G). A graph G is k^∗-connected if there exists a spanning k-container between any two different vertices. The spanning connectivity of G, κ^∗(G), is the maximum integer k such that G is w^∗-connected for 1≤w≤k if G is 1^∗-connected.Let x be a vertex in G and let U={y₁,y₂,…,y_k} be a subset of V(G) where x is not in U. A spanningk−(x,U)-fan, F_k(x,U), is a set of internally-disjoint paths {P₁,P₂,…,P_k} such that P_i is a path connecting x to y_i for 1≤i≤k and . A graph G is k^∗-fan-connected (or -connected) if there exists a spanning F_k(x,U)-fan for every choice of x and U with |U|=k and x∉U. The spanning fan-connectivity of a graph G, , is defined as the largest integer k such that G is -connected for 1≤w≤k if G is -connected.In this paper, some relationship between κ(G), κ^∗(G), and are discussed. Moreover, some sufficient conditions for a graph to be -connected are presented. Furthermore, we introduce the concept of a spanning pipeline-connectivity and discuss some sufficient conditions for a graph to be k^∗-pipeline-connected.     

Classification of solutions of a critical Hardy-Sobolev operator

In this article we classify all positive finite energy solutions of the equation in Rⁿ where and a point x∈Rⁿ is denoted as x=(y,z)∈R^k×R^n-k. As a consequence we obtain the best constant and extremals of a related Hardy-Sobolev inequality.     

Spike solutions of a nonlinear Schrödinger equation with degenerate potential

We study the existence of positive solutions to the elliptic equation ε²Δu(x,y)−V(y)u(x,y)+f(u(x,y))=0 for (x,y) in an unbounded domain subject to the boundary condition u=0 whenever is nonempty. Our potential V depends only on the y variable and is a bounded or unbounded domain which may coincide with . The positive parameter ε is tending to zero and our solutions u_ε concentrate along minimum points of the unbounded manifold of critical points of V.     

The Diametric Theorem in Hamming Spaces—Optimal Anticodes

For a Hamming space (ⁿ_α, d_H), the set ofn-length words over the alphabet _α = {0, 1,…,α − 1} endowed with the distanced_H, which for two wordsxⁿ = (x₁,…,x_n),yⁿ = (y₁,…,y_n) ∈ ⁿ_αcounts the number of different components, we determine the maximal cardinality of subsets with a prescribed diameterdor, in another language, anticodes with distanced. We refer to the result as the diametric theorem.In a sense anticodes are dual to codes, which have a prescribedlowerbound on the pairwise distance. It is a hopeless task to determine their maximal sizes exactly.We find it remarkable that the diametric theorem (for arbitrary α) can be derived from our recent complete intersection theorem, which can be viewed as a diametric theorem (for α = 2) in the restricted case, where alln-length words considered have exactlykones.     

Equitable colorings of Kronecker products of graphs

For a positive integer k, a graph G is equitably k-colorable if there is a mapping f:V(G)→{1,2,…,k} such that f(x)≠f(y) whenever xy∈E(G) and ||f⁻¹(i)|−|f⁻¹(j)||≤1 for 1≤i<j≤k. The equitable chromatic number of a graph G, denoted by χ₌(G), is the minimum k such that G is equitably k-colorable. The equitable chromatic threshold of a graph G, denoted by , is the minimum t such that G is equitably k-colorable for k≥t. The current paper studies equitable chromatic numbers of Kronecker products of graphs. In particular, we give exact values or upper bounds on χ₌(G×H) and when G and H are complete graphs, bipartite graphs, paths or cycles.     

Distance-two labelings of digraphs

For positive integers j?k, an L(j,k)-labeling of a digraph D is a function f from V(D) into the set of nonnegative integers such that |f(x)-f(y)|?j if x is adjacent to y in D and |f(x)-f(y)|?k if x is of distance two to y in D. Elements of the image of f are called labels. The L(j,k)-labeling problem is to determine the -number of a digraph D, which is the minimum of the maximum label used in an L(j,k)-labeling of D. This paper studies -numbers of digraphs. In particular, we determine -numbers of digraphs whose longest dipath is of length at most 2, and -numbers of ditrees having dipaths of length 4. We also give bounds for -numbers of bipartite digraphs whose longest dipath is of length 3. Finally, we present a linear-time algorithm for determining -numbers of ditrees whose longest dipath is of length 3.     

Galois closure of essentially finite morphisms

Let X be a reduced connected k-scheme pointed at a rational point x∈X(k). By using tannakian techniques we construct the Galois closure of an essentially finite k-morphism f:Y→X satisfying the condition H⁰(Y,O_Y)=k; this Galois closure is a torsor dominating f by an X-morphism and universal for this property. Moreover, we show that is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over Y is still an essentially finite vector bundle over X. We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor f:Y→X under a finite group scheme satisfying the condition H⁰(Y,O_Y)=k, Y has a fundamental group scheme π₁(Y,y) fitting in a short exact sequence with π₁(X,x).     

Positive definite matrices and differentiable reproducing kernel inequalities

Let I⊆R be a interval and be a reproducing kernel on I. By the Moore-Aronszajn theorem, every finite matrix k(xi,xj) is positive semidefinite. We show that, as a direct algebraic consequence, if k(x,y) is appropriately differentiable it satisfies a 2-parameter family of differential inequalities of which the classical diagonal dominance is the order 0 case. An application of these inequalities to kernels of positive integral operators yields optimal Sobolev norm bounds.