Nonwandering set of points of skew-product maps with base having closed set of periodic points

The aim of the present paper is to study the structure of the nonwandering set of points Ω() for the skew-product maps of the unit square , (x,y)→(f(x),g(x,y)), with base f having closed set of periodic points. For every and every point (x,y) with x periodic of period p_x by f and y not chain recurrent of F^p_x|_Ix, where , we prove that (x,y)Ω(F). On the other hand we construct a map with an isolated fixed point x₀ of f and y₀Ω(F|_Ix0) such that (x₀,y₀)Ω(F₀).     

Gradient estimates of Poisson equations on Riemannian manifolds and applications

For the reflected diffusion generated by on a connected and complete Riemannian manifold M with empty or convex boundary, we establish some sharp estimates of sup_xM|G|(x) of the Poisson equation in terms of the dimension, the diameter and the lower bound of curvature. Applications to transportation-information inequality, to Cheeger's isoperimetric inequality and to Gaussian concentration inequality are given. Several examples are provided.     

Existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function

This paper is concerned with the existence, multiplicity and stability of positive solutions of an indefinite weight boundary value problem

where aC[0,1] changes sign. The proof of our main result is based upon bifurcation techniques.     

Existence results for -point boundary value problem of second order ordinary differential equations

This study concerns the existence of positive solutions to the boundary value problemwhere ξ_i(0,1) with 0<ξ₁<ξ₂<<ξ_n-2<1, a_i, b_i[0,∞) with and . By applying the Krasnoselskii's fixed-point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of at least one positive solution or at least two positive solutions are established for the above general n-point boundary value problem.     

A new kind of graph coloring

A proper k-coloring C¹,C²,…,C^k of a graph G is called strong if, for every vertex uV(G), there exists an index i{1,2,…,k} such that u is adjacent to every vertex of Cⁱ. We consider classes of strongly k-colorable graphs and show that the recognition problem of is NP-complete for every k4, but it is polynomial-time solvable for k=3. We give a characterization of in terms of forbidden induced subgraphs. Finally, we solve the problem of uniqueness of a strong 3-coloring.     

A note on the signed edge domination number in graphs

Let G=(V(G),E(G)) be a graph. A function f:E(G)→{+1,−1} is called the signed edge domination function (SEDF) of G if ∑_e^′N[e]f(e^′)≥1 for every eE(G). The signed edge domination number of G is defined as is a SEDF of G}. Xu [Baogen Xu, Two classes of edge domination in graphs, Discrete Applied Mathematics 154 (2006) 1541–1546] researched on the edge domination in graphs and proved that for any graph G of order n(n≥4). In the article, he conjectured that: For any 2-connected graph G of order n(n≥2), . In this note, we present some counterexamples to the above conjecture and prove that there exists a family of k-connected graphs G_m,k with .     

Polynomial-time approximation schemes for piercing and covering with applications in wireless networks

Let be a set of disks of arbitrary radii in the plane, and let be a set of points. We study the following three problems: (i) Assuming contains the set of center points of disks in , find a minimum-cardinality subset of (if exists), such that each disk in is pierced by at least h points of , where h is a given constant. We call this problem minimum h-piercing. (ii) Assuming is such that for each there exists a point in whose distance from D's center is at most αr(D), where r(D) is D's radius and 0α<1 is a given constant, find a minimum-cardinality subset of , such that each disk in is pierced by at least one point of . We call this problem minimum discrete piercing with cores. (iii) Assuming is the set of center points of disks in , and that each covers at most l points of , where l is a constant, find a minimum-cardinality subset of , such that each point of is covered by at least one disk of . We call this problem minimum center covering. For each of these problems we present a constant-factor approximation algorithm (trivial for problem (iii)), followed by a polynomial-time approximation scheme. The polynomial-time approximation schemes are based on an adapted and extended version of Chan's [T.M. Chan, Polynomial-time approximation schemes for packing and piercing fat objects, J. Algorithms 46 (2003) 178–189] separator theorem. Our PTAS for problem (ii) enables one, in practical cases, to obtain a (1+ε)-approximation for minimum discrete piercing (i.e., for arbitrary ).     

Cut points in metric spaces

In this note, we will define topological and virtual cut points of finite metric spaces and show that, though their definitions seem to look rather distinct, they actually coincide. More specifically, let X denote a finite set, and let denote a metric defined on X. The tight span T(D) of D consists of all maps for which f(x)=sup_yX(xy−f(x)) holds for all xX. Define a map fT(D) to be a topological cut point of D if T(D)−{f} is disconnected, and define it to be a virtual cut point of D if there exists a bipartition (or split) of the support of f into two non-empty sets A and B such that ab=f(a)+f(b) holds for all points aA and bB. It will be shown that, for any given metric D, topological and virtual cut points actually coincide, i.e., a map fT(D) is a topological cut point of D if and only if it is a virtual cut point of D.     

A note on the duality mapping of a locally uniformly convex Banach space

Let X be a real locally uniformly convex Banach space with normalized duality mapping J:X→2^X*. The purpose of this note is to show that for every R>0 and every x₀X there exists a function , which is nondecreasing and such that (r)>0 for r>0,(0)=0 and

for all . Simply, it is shown that the necessity part of the proof of the original analogous necessary and sufficient condition of Prüß, for real uniformly convex Banach spaces, goes over equally well in the present setting. This is a natural setting for the study of many existence problems in accretive and monotone operator theories.     

Commuting traces and commutativity preserving maps on triangular algebras

Let A be a triangular algebra. The problem of describing the form of a bilinear map satisfying B(x,x)x=xB(x,x) for all xA is considered. As an application, commutativity preserving maps and Lie isomorphisms of certain triangular algebras (e.g., upper triangular matrix algebras and nest algebras) are determined.     

Varieties of nilpotent elements for simple Lie algebras I: Good primes

Let G be a simple algebraic group over k=C, or where p is good. Set g=LieG. Given rN and a faithful (restricted) representation , one can define a variety of nilpotent elements . In this paper we determine this variety when ρ is an irreducible representation of minimal dimension or the adjoint representation.     

Rigid realizations of graphs on small grids

A framework (G,p) is a straight line realization of a graph G=(V,E) in , given by a map . We prove that if (G,p) is an infinitesimally rigid framework then there is an infinitesimally rigid framework (G,q) for which the points q(v), vV(G), are distinct points of the k×k grid, where . We also show that such a framework on G can be constructed in O(|V|³) time.     

Detecting induced subgraphs

An s-graph is a graph with two kinds of edges: subdivisible edges and real edges. A realisation of an s-graph B is any graph obtained by subdividing subdivisible edges of B into paths of arbitrary length (at least one). Given an s-graph B, we study the decision problem Π_B whose instance is a graph G and question is “Does G contain a realisation of B as an induced subgraph?”. For several B’s, the complexity of Π_B is known and here we give the complexity for several more.Our NP-completeness proofs for Π_B’s rely on the NP-completeness proof of the following problem. Let be a set of graphs and d be an integer. Let be the problem whose instance is (G,x,y) where G is a graph whose maximum degree is at most d, with no induced subgraph in and x,yV(G) are two non-adjacent vertices of degree 2. The question is “Does G contain an induced cycle passing through x,y?”. Among several results, we prove that is NP-complete. We give a simple criterion on a connected graph H to decide whether is polynomial or NP-complete. The polynomial cases rely on the algorithm three-in-a-tree, due to Chudnovsky and Seymour.     

Simultaneous approximation and interpolation of increasing functions by increasing entire functions

We prove that, under suitable assumptions, an isomorphism g of dense subsets A,B of the real line can be taken to approximate a given increasing Cⁿ surjection f with the derivatives of g agreeing with those of f on a closed discrete set. For example, we have the following theorem. Let be a nondecreasing Cⁿ surjection. Let be a positive continuous function. Let be a closed discrete set on which f is strictly increasing. Let each of {A_i}, {B_i} be a sequence of pairwise disjoint countable dense subsets of such that for each and xE we have xA_i if and only if f(x)B_i. Then there is an entire function such that and the following properties hold.

(a) For all , Dg(x)>0.

(b) For k=0,…,n and all , |D^kf(x)−D^kg(x)|<ε(x).

(c) For k=0,…,n and all xE, D^kf(x)=D^kg(x).

(d) For each , g[A_i]=B_i.

This provides a version for increasing functions of a theorem of Hoischen. In earlier work, we proved that it is consistent that a similar theorem, omitting clause (c), holds when the sets A_i,B_i are of cardinality ₁ and have second category intersection with every interval. (See the introduction for the exact statement.) In this paper, we show how to incorporate clause (c) into the statement of the earlier theorem.

Regularity results for stable-like operators

For α[1,2) we consider operators of the form

and for α(0,1) we consider the same operator but where the f term is omitted. We prove, under appropriate conditions on A(x,h), that any solution u to will be in C^α+β if fC^β.     

Christoffel-type functions for -orthogonal polynomials for Freud weights

This paper gives upper and lower bounds of the Christoffel-type functions , for the m-orthogonal polynomials for a Freud weight W=e^-Q, which are given as follows. Let a_n=a_n(Q) be the nth Mhaskar–Rahmanov–Saff number, φ_n(x)=max{n^-2/3,1-|x|/a_n}, and d>0. Assume that QC(R) is even, , and for some A,B>1

Then for xR

and for |x|a_n(1+dn^-2/3)

     

Convergence of a Halpern-type iteration algorithm for a class of pseudo-contractive mappings

Let E be a real reflexive Banach space with uniformly Gâteaux differentiable norm. Let K be a nonempty bounded closed and convex subset of E. Let T:K→K be a strictly pseudo-contractive map and let L>0 denote its Lipschitz constant. Assume F(T){xK:Tx=x}≠0/ and let zF(T). Fix δ(0,1) and let δ^* be such that δ^*δL(0,1). Define , where δ_n(0,1) and limδ_n=0. Let {α_n} be a real sequence in (0,1) which satisfies the following conditions: . For arbitrary x₀,uK, define a sequence {x_n}K by x_n+1=α_nu+(1−α_n)S_nx_n. Then, {x_n} converges strongly to a fixed point of T.     

Equivalence relation groupoids associated with certain linearly ordered dimension groups

We consider linearly ordered, Archimedean dimension groups (G,G⁺,u) for which the group G/u is torsion-free. It will be shown that if, in addition, G/u is generated by a single element (i.e., ), then (G,G⁺,u) is isomorphic to for some irrational number τ(0,1). This amounts to an extension of related results where dimension groups for which G/u is torsion were considered. We will prove, in the case of the Fibonacci dimension group, that these results can be used to directly construct an equivalence relation groupoid whose C^*-algebra is the Fibonacci C^*-algebra.