Semialgebraic sets and the Łojasiewicz-Siciak condition

The paper contains a full geometric characterization of compact semialgebraic sets in C satisfying the ?ojasiewicz-Siciak condition. The ?ojasiewicz-Siciak condition is a certain estimate for the Siciak extremal function. In a previous paper, we gave a sufficient criterion for a compact, connected, and semialgebraic set in C to satisfy this condition. In the present paper, we remove completely the connectedness assumption and prove that the aforementioned sufficient condition is also necessary. Moreover, we obtain some new results concerning the ?ojasiewicz-Siciak condition in C^N. For example, we prove that if K₁,...,K_p are compact, nonpluripolar, and pairwise disjoint subsets of C^N, each satisfying the ?ojasiewicz-Siciak condition, and K:= K₁?· · ·?K_p is polynomially convex, then K satisfies this condition as well.     

Graphic sequences and split graphs

The split graph K _rVK _s on r+s vertices is denoted by S _r,s. A graphic sequence π = (d ₁, d ₂, ···, d _n) is said to be potentially S _r,s-graphic if there is a realization of π containing S _r,s as a subgraph. In this paper, a simple sufficient condition for π to be potentially S _r,s-graphic is obtained, which extends an analogous condition for p to be potentially K _r+1-graphic due to Yin and Li (Discrete Math. 301 (2005) 218–227). As an application of this condition, we further determine the values of σ(S _r,s, n) for n ≥ 3r + 3s - 1.     

Cofiniteness and finiteness of local cohomology modules over regular local rings

Let (R,m) be a commutative Noetherian regular local ring of dimension d and I be a proper ideal of R such that mAss _R (R/I) = Assh _R (_I). It is shown that the R- module H^ht(I) _I (R) is I-cofinite if and only if cd(I,R) = ht(I). Also we present a sufficient condition under which this condition the R-module H ⁱ _I (R) is finitely generated if and only if it vanishes.     

On <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-factorization of complete bipartite multigraphs

Let λK _m,n be a complete bipartite multigraph with two partite sets having m and n vertices, respectively. A K _p,q -factorization of λK _m,n is a set of edge-disjoint K _p,q -factors of λK _m,n which partition the set of edges of λK _m,n . When p = 1 and q is a prime number, Wang, in his paper [On K _1,q -factorization of complete bipartite graph, Discrete Math., 126: (1994), 359-364], investigated the K _1,q -factorization of K _m,n and gave a sufficient condition for such a factorization to exist. In papers [K _1,k -factorization of complete bipartite graphs, Discrete Math., 259: 301-306 (2002),; K _p,q -factorization of complete bipartite graphs, Sci. China Ser. A-Math., 47: (2004), 473-479], Du and Wang extended Wang’s result to the case that p and q are any positive integers. In this paper, we give a sufficient condition for λK _m,n to have a K _p,q -factorization. As a special case, it is shown that the necessary condition for the K _p,q -factorization of λK _m,n is always sufficient when p : q = k : (k + 1) for any positive integer k.     

On reliability analysis of consecutive k-out-of-n systems with arbitrarily dependent components

In this paper, we consider the linear and circular consecutive k-out-of-n systems consisting of arbitrarily dependent components. Under the condition that at least n?r+1 components (r ≤ n) of the system are working at time t, we study the reliability properties of the residual lifetime of such systems. Also, we present some stochastic ordering properties of residual lifetime of consecutive k-out-of-n systems. In the following, we investigate the inactivity time of the component with lifetime T_r:n at the system level for the consecutive k-out-of-n systems under the condition that the system is not working at time t > 0, and obtain some stochastic properties of this conditional random variable.     

Coercivity results of a modified <Emphasis Type="Italic">Q</Emphasis><Subscript>1</Subscript>-finite volume element scheme for anisotropic diffusion problems

In this paper, we study a so-called modified Q ₁-finite volume element scheme that is obtained by employing the trapezoidal rule to approximate the line integrals in the classical Q ₁-finite volume element method. A necessary and sufficient condition is obtained for the positive definiteness of a certain element stiffness matrix. Based on this result, a sufficient condition is suggested to guarantee the coercivity of the scheme on arbitrary convex quadrilateral meshes. When the diffusion tensor is an identity matrix, this sufficient condition reduces to a geometric one, covering some standard meshes, such as the traditional h ^1+γ -parallelogram meshes and some trapezoidal meshes. More interesting is that, this sufficient condition has explicit expression, by which one can easily judge on any diffusion tensor and any mesh with any mesh size h > 0. The H ¹ error estimate of the modified Q ₁-finite volume element scheme is obtained without the traditional h ^1+γ -parallelogram assumption. Some numerical experiments are carried out to validate the theoretical analysis.     

Existence of Semilinear Differential Equations with Nonlocal Initial Conditions

In this paper we give the existence of mild solutions for semilinear Cauchy problems u′（t） = Au（t） ＋f（t, u（t））, t ∈ I, a.e. with nonlocal initial condition u（O） = g（u） ＋uo when the map g loses compactness in Banach spaces.     

Cauchy problem for the Korteweg-de Vries equation in the case of a nonsmooth unbounded initial function

In the strip П = (?1, 0) × ?, we establish the existence of solutions of the Cauchy problem for the Korteweg-de Vries equation u _t + u _xxx + uu _x = 0 with initial condition either 1) u(?1, x) = ?xθ(x), or 2) u(?1, x) = ?xθ(?x), where θ is the Heaviside function. The solutions constructed in this paper are infinitely smooth for t ∈ (?1, 0) and rapidly decreasing as x → +∞. For the case of the first initial condition, we also establish uniqueness in a certain class. Similar special solutions of the KdV equation arise in the study of the asymptotic behavior with respect to small dispersion of the solutions of certain model problems in a neighborhood of lines of weak discontinuity.     

Global hypoellipticity for first-order operators on closed smooth manifolds

The main goal of this paper is to address global hypoellipticity issues for the class of first-order pseudo-differential operators L = D_t + C(t, x,D_x), where (t, x) ∈ T × M, T is the one-dimensional torus, M is a closed manifold, and C(t, x,D_x) is a first-order pseudo-differential operator on M, smoothly depending on the periodic variable t. In the case of separation of variables, when C(t, x,D_x) = a(t)p(x,D_x) + ib(t)q(x,D_x), we give necessary and sufficient conditions for the global hypoellipticity of L. In particular, we show that the famous (P) condition of Nirenberg-Treves is neither necessary nor sufficient to guarantee the global hypoellipticity of L.     

Lower bounds for polynomials of many variables

A polynomial P(ξ) = P(ξ₁,..., ξ _n ) is said to be almost hypoelliptic if all its derivatives D ^ν P(ξ) can be estimated from above by P(ξ) (see [16]). By a theorem of Seidenberg-Tarski it follows that for each polynomial P(ξ) satisfying the condition P(ξ) > 0 for all ξ ∈ R ⁿ , there exist numbers σ > 0 and T ∈ R¹ such that P(ξ) ≥ σ(1 + |ξ|) ^T for all ξ ∈ R ⁿ . The greatest of numbers T satisfying this condition, denoted by ST(P), is called Seidenberg-Tarski number of polynomial P. It is known that if, in addition, P ∈ I _n , that is, |P(ξ)| → ∞ as |ξ| → ∞, then T = T(P) > 0. In this paper, for a class of almost hypoelliptic polynomials of n (≥ 2) variables we find a sufficient condition for ST(P) ≥ 1. Moreover, in the case n = 2, we prove that ST(P) ≥ 1 for any almost hypoelliptic polynomial P ∈ I₂.     

Disjoint cliques in claw-free graphs

A graph is said to be claw-free if it does not contain an induced subgraph isomorphic to K_1,3. Let s and k be two integers with 0 ≤ s ≤ k and let G be a claw-free graph of order n. In this paper, we investigate clique partition problems in claw-free graphs. It is proved that if n ≥ 3s+4(k?s) and d(x)+d(y) ≥ n?2s+2k+1 for any pair of non-adjacent vertices x, y of G, then G contains s disjoint K₃s and k ? s disjoint K₄s such that all of them are disjoint. Moreover, the degree condition is sharp in some cases.     

Continuity of topological entropy for perturbation of time-one maps of hyperbolic flows

In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type FP_n over a profinite ring R, analogous to the Bieri–Eckmann criteria for abstract groups. We use these to prove that the class of groups of type FP_n is closed under extensions, quotients by subgroups of type FP_n, proper amalgamated free products and proper HNN-extensions, for each n. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type FP_∞ over all profinite R. For any class C of finite groups closed under subgroups, quotients and extensions, we also construct pro-C groups of type FP_n but not of type FP_n+1 over Z_? for each n. Finally, we show that the natural analogue of the usual condition measuring when pro-p groups are of type FP_n fails for general profinite groups, answering in the negative the profinite analogue of a question of Kropholler.     

Criterion for the Solvability of the Weighted Cauchy Problem for an Abstract Euler–Poisson–Darboux Equation

In a Banach space E, we consider the abstract Euler–Poisson–Darboux equation u″(t) + kt^?1u′(t) = Au(t) on the half-line. (Here k ∈ ? is a parameter, and A is a closed linear operator with dense domain on E.) We obtain a necessary and sufficient condition for the solvability of the Cauchy problem u(0) = 0, lim _t→0+t ^k u′(t) = u₁, k < 0, for this equation. The condition is stated in terms of an estimate for the norms of the fractional power of the resolvent of A and its derivatives. We introduce the operator Bessel function with negative index and study its properties.     

Almost everywhere divergence of lacunary subsequences of partial sums of fourier series

If an increasing sequence {n _m } of positive integers and a modulus of continuity ω satisfy the condition Σ _m=1 ^∞ ω(1/n _m )/m < ∞, then it is known that the subsequence of partial sums \(S_{n_m } \left( {f,x} \right)\) converges almost everywhere to f(x) for any function f ∈ H ₁ ^ω . We show that this sufficient convergence condition is close to a necessary condition for a lacunary sequence {n _m }.     

Boundary value problem for one evolution equation

The aim of the paper is to investigate the boundary value problem of the evolution equation Lu = K (x,t) _ut - Δu + a (x,t) u = f (x,t). The characteristic property of this type of equations is the failure of the Petrovski’s “A” condition when coefficients are constant [1]. In this case, Cauchy problem is incorrect in the sense of Hadamard. Hence in this paper, the space, guaranteeing the correctness of the boundary value problem in the sense of Hadamard, is selected by adding some additional conditions to the coefficients of the equation.     

Integral averaging technique for oscillation of damped half-linear oscillators

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator (a(t)?_p(x′))′ + b(t)?_p(x′) + c(t)?_p(x) = 0, where ?p(x) = |x|^p?1 sgn x for x ∈ ? and p > 1. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if p ≠ 2 is presented.     

On the boundedness of generalized solutions of higher-order nonlinear elliptic equations with data from an Orlicz–Zygmund class

In the present paper, a 2mth-order quasilinear divergence equation is considered under the condition that its coefficients satisfy the Carathéodory condition and the standard conditions of growth and coercivity in the Sobolev space W^m,p(Ω), Ω ? Rⁿ, p > 1. It is proved that an arbitrary generalized (in the sense of distributions) solution u ∈ W₀^m,p (Ω) of this equation is bounded if m ≥ 2, n = mp, and the right-hand side of this equation belongs to the Orlicz–Zygmund space L(log L)^n?1(Ω).     

Relative Gorenstein Dimensions

In the last years (Gorenstein) homological dimensions relative to a semidualizing module C have been subject of several works as interesting extensions of (Gorenstein) homological dimensions. In this paper, we extend to the noncommutative case the concepts of G _C -projective module and dimension, weakening the condition of C being semidualizing as well. We prove that indeed they share the principal properties of the classical ones and relate this new dimension with the classical Gorenstein projective dimension of a module. The dual concepts of G _C -injective modules and dimension are also treated. Finally, we show some interesting interactions between the class of G _C -projective modules and the Bass class associated to C on one side, and the class of G\({_{C^{\vee}}}\) -injective modules (C ^∨ = Hom _R (C, E) where E is an injective cogenerator in R-Mod) and the Auslander class associated to C in the other.     

On local equivalence of the majorant of partial sums and Paley function of Franklin series

In this paper we prove that the majorant of partial sums and the Paley function of Franklin series have equivalent norms in the space L _p (I), p > 0, provided that the “peak” intervals of Franklin functions with non-vanishing coefficients lie in I. Examples of series emphasizing that this condition is essential are also given.