Unions and Intersections of Families of Lp-Spaces

Let L^Ψ and E^Ψ be the ORLICZ space and the space of finite elements respectively, on a measure space (Ω, Σ, μ), and let T ? (0, ∞). It is proved that if inf {p: p ? T} ? T or sup {p: p ? T} ? T and μ is an infinite atomless measure, then there is no ORLICZ function Ψ such that: \documentclass{article}\pagestyle{empty}\begin{document}$ L^\varphi = Lin\mathop { \cup L^p }\limits_{p\varepsilon T} $\end{document} or \documentclass{article}\pagestyle{empty}\begin{document}$ E^\varphi = Lin\mathop { \cup L^p }\limits_{p\varepsilon T} $\end{document} and moreover, there is no ORLICZ function Ψ such that: \documentclass{article}\pagestyle{empty}\begin{document}$ L^\varphi = Lin\mathop { \cap L^p }\limits_{p\varepsilon T} $\end{document} or \documentclass{article}\pagestyle{empty}\begin{document}$ E^\varphi = Lin\mathop { \cap L^p }\limits_{p\varepsilon T} $\end{document}.     

The color space of a graph

If k is a prime power, and G is a graph with n vertices, then a k‐coloring of G may be considered as a vector in $\mathbb{GF}$(k)ⁿ. We prove that the subspace of $\mathbb{GF}$(3)ⁿ spanned by all 3‐colorings of a planar triangle‐free graph with n vertices has dimension n. In particular, any such graph has at least n − 1 nonequivalent 3‐colorings, and the addition of any edge or any vertex of degree 3 results in a 3‐colorable graph. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 234–245, 2000     

Inversion of the Radon transform with incomplete data

The Radon transform R(p, θ), θ∈S^n?1, p∈?¹, of a compactly supported function f(x) with support in a ball B_a of radius a centred at the origin is given for all $ \theta \in \mathop {S^{n - 1} }\limits^\tilde $, where $ \mathop {S^{n - 1} }\limits^\tilde $ is an open set on S^n?1, and all p∈(? ∞, ∞), n≥2. An approximate formula is given to calculate f(x) from the given data.     

Periodic problem for a nonlinear‐damped wave equation on the boundary

In this work we investigate the existence of periodic solutions in t for the following problem: We employ elliptic regularization and monotone method. We consider $\mbox{\boldmath{$\Omega$}}\mbox{\boldmath{$\subset$}}{\mathbb{R}}^{{{n}}} \ (n\geqslant 1)$ an open bounded set that has regular boundary Γ and Q=Ω ×(0,T), T>0, a cylinder of ${\mathbb{R}}^{n+1}$ with lateral boundary Σ = Γ × (0,T). Copyright © 2009 John Wiley & Sons, Ltd.     

Compact complex threefolds which are Kähler outside a smooth rational curve

Let M be a compact complex manifold containing an irreducible curve C such that M — C is Kähler; in this paper we study the link between some cohomological properties of C and the obstructions to the existence of a Kähler metric on the whole of M. In particular we get that, if M is not Kähler, then C is a $ \left({\partial + \bar \partial} \right) $ –exact current, or there exists a positive current S of bidimension (1, 1) such that $ \partial \bar \partial S = 0,\,\chi _C S = 0 $ and S + C is $ \left({\partial + \bar \partial} \right) $ –exact. If C is a smooth rational curve, more precise results are given in connection with the normal bundle N_C|M.     

A Self‐Dual Polar Factorization for Vector Fields

We show that any nondegenerate vector field u in \begin{align*}L^{\infty}(\Omega, \mathbb{R}^N)\end{align*}, where Ω is a bounded domain in \begin{align*}\mathbb{R}^N\end{align*}, can be written as \begin{align*}u(x)= \nabla_1 H(S(x), x)\quad {\text for a.e.\ x \in \Omega}\end{align*}}, where S is a measure‐preserving point transformation on Ω such that \begin{align*}S^2=I\end{align*} a.e. (an involution), and \begin{align*}H: \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}\end{align*} is a globally Lipschitz antisymmetric convex‐concave Hamiltonian. Moreover, u is a monotone map if and only if S can be taken to be the identity, which suggests that our result is a self‐dual version of Brenier's polar decomposition for the vector field as \begin{align*}u(x)=\nabla \phi (S(x))\end{align*}, where ? is convex and S is a measure‐preserving transformation. We also describe how our polar decomposition can be reformulated as a (self‐dual) mass transport problem. © 2012 Wiley Periodicals, Inc.     

What is the smallest number of dicycles in a dicycle decomposition of an eulerian digraph?

A digraph D with n vertices is said to be decomposable into a set S of dicycles if every arc of D is contained in exactly one member of S. Counterexamples are given to the following conjectures which are generalizations of three well-known conjectures by G. Hajós, P. Erd?s, and P.J. Kelly: (1) [B. Jackson] Every eulerian-oriented graph is decomposable into at most \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{n}{2} $\end{document} dicycles. (2) [W. Bienia & H. Meyniel] Every eulerian digraph is decomposable into at most n dicycles. Certain observations lead us to make three other conjectures: (a) Every eulerian-oriented graph is decomposable into at most \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{2n}}{3} $\end{document} dicycles. (b) Every symmetric digraph with n > 1 is decomposable into at most 2n – 3 dicycles. (c) Every eulerian digraph with n > 1 is decomposable into at most \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{8n}}{3} $\end{document} – 3 dicycles.     

The PENROSE Transform for Bundles Non-Trivial on the General Line

Let L₀ be a fixed projective line in CP ³ and let M ? C ⁴ be the complexified MINKOWSKI space interpreted as the manifold of all projective lines L ? CP ³ with L ∩ L ₀ ?? Ø. Let D ? M , D ′ ? CP ³/ L ₀ be open sets such that \documentclass{article}\pagestyle{empty}\begin{document}$ D' = \mathop \cup \limits_{L \in D} $\end{document}. Under certain topological conditions on D, R. S. WARD'S PENROSE transform sets up an 1–1 correspondence between holomorphic vector bundles over D ′ trivial over each L ? D and holomorphic connections with anti-self-dual curvature over D (anti-self-dual YANG-MILLIS fields). In the present paper WARD'S construction is generalized to holomorphic vector bundles E over D′ satisfying the condition that \documentclass{article}\pagestyle{empty}\begin{document}$ E|_L \cong E|_{\tilde L} $\end{document} for all \documentclass{article}\pagestyle{empty}\begin{document}$ L,\tilde L \in D $\end{document}.     

On the number of subtrees for almost all graphs

In this article it is shown that the number of common edges of two random subtrees of K_n having r and s vertices, respectively, has a Poisson distribution with expectation 2λμ if $\mathop {\lim }\limits_{n \to \infty } r/n = \lambda$ and $\mathop {\lim }\limits_{n \to \infty } s/n = \mu$. Also, some estimations of the number of subtrees for almost all graphs are made by using Chebycheff's inequality. © 1994 John Wiley & Sons, Inc.     

The Structure of the Spectrum of Fourth-Order Differential Operators with Random Coefficients

In this paper it is proved that under certain conditions on the coefficients the random operator H = $H = \frac{1}{{r\left( {x_t } \right)}}\left[ {\frac{{d^2 }}{{dt^2 }}\left( {\frac{1}{{p\left( {x_t } \right)}}\frac{{d^2 }}{{dt^2 }}} \right) + q\left( {x_t } \right)} \right], \in R^1$ being a stationary, ergodic Markov process with compact State space K, has almost surely pure point spectrum and exponentially decreasing eigenfunctions. The method used here can be extended to operators corresponding to certain matrix Sturm-Liouville problems.     

Inexpensive d‐dimensional matchings

Suppose that independent U(0, 1) weights are assigned to the ${d\choose 2}n^{2}$ edges of the complete d‐partite graph with n vertices in each of the d maximal independent sets. Then the expected weight of the minimum‐weight perfect d‐dimensional matching is at least $\frac{3}{16}n^{1-(2/d)}$. We describe a randomized algorithm that finds a perfect d‐dimensional matching whose expected weight is at most 5d³n^1?(2/d)+d¹⁵ for all d≥3 and n≥1. © 2002 John Wiley & Sons, Inc. Random Struct. Alg., 20, 50–58, 2002     

Integral Operators in Sobolev Spaces on Domains with Boundary

Properties of integral operators with weak singularities arc investigated. It is assumed that G ? ?ⁿ is a bounded domain. The boundary δG should be smooth concerning the Sobolev trace theorem. It will be proved that the integral operators $\int {_G \frac{{f\left(\Theta \right)}}{{x - y|^{n - 1} }}u\left(\nu \right)d\partial G_\nu }$ and $ \int {_{\partial G} \frac{{f\left(\Theta \right)}}{{|x - y|^{n - 1} }}u\left(y \right)d\partial G_y }$ maps W_p^k(G) into W_p^k+1(G) and W_p^k?1(G) into W_p^k/p(G), respectively, and are bounded. Here θ ∈ S ? ?ⁿ, where S is the unit sphere. Furthermore, f possesses bounded first order derivatives and is bounded on S. Then applications to first order systems are discussed.     

The phase transition in the evolution of random digraphs

Let $ \mathop {\rm D}\limits^ \to $(n, M) denote a digraph chosen at random from the family of all digraphs on n vertices with M arcs. We shall prove that if M/n ≤ c < 1 and ω(n) → ∞, then with probability tending to 1 as n → ∞ all components of $ \mathop {\rm D}\limits^ \to $(n, M) are smaller than ω(n), whereas when M/n ≥ c > 1 the largest component of $ \mathop {\rm D}\limits^ \to $(n, M) is of the order n with probability 1 - o(1).     

Compactness Conditions for the Green Operator in Lp(ℝ) Corresponding to a General Sturm–Liouville Operator

We consider the equation ℝ, where , for ℝ, (ℝ), (ℝ), (ℝ), (ℝ) := C(ℝ)). We give necessary and sufficient conditions under which, regardless of , the following statements hold simultaneously: I) For any (ℝ) Equation (0.1) has a unique solution (ℝ) where $\int ^{\infty}_{-\infty}$ ℝ. II) The operator (ℝ) → (ℝ) is compact. Here is the Green function corresponding to (0.1). This result is applied to study some properties of the spectrum of the Sturm–Liouville operator.     

Nullstellensatz effectif et Conjecture de Serre (Théorème de Quillen-Suslin) pour le Calcul Formel

Let k be an arbitrary field, X₁,….,X_n indeterminates over k and F₁…, F₃ ε ∈ k[X₁…,X_n] polynomials of maximal degree $ d: = \mathop {\max }\limits_{1 \le i \le a} \deg $ (F_i). We give an elementary proof of the following effective Nullstellensatz: Assume that F₁,…,F have no common zero in the algebraic closure of k. Then there exist polynomials P₁…, P₃ ε ∈ k[X₁…,X_n] such that $ 1: = \mathop \Sigma \limits_{1 \le i \le a} $ P_iF_i and This result has many applications in Computer Algebra. To exemplify this, we give an effective quantitative and algorithmic version of the Quillen-Suslin Theorem baaed on our effective Nullstellensatz.     

Infinitely Many Positive Solutions to Some Scalar Field Equations with Nonsymmetric Coefficients

In this paper the equation $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}} - \Delta u + a(x)u = |u|^{p - 1} u\;{\rm in }\;{\R}^N $ is considered, when N ≥ 2, p > 1, and $p < {{N + 2} \over {N - 2}}$ if N ≥ 3. Assuming that the potential a(x) is a positive function belonging to $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}L_{{\rm loc}}^{N/2} ({\R}^N )$ such that a(x) → a_∞ > 0 as |x|→∞ and satisfies slow decay assumptions but does not need to fulfill any symmetry property, the existence of infinitely many positive solutions, by purely variational methods, is proved. The shape of the solutions is described as is, and furthermore, their asymptotic behavior when $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}|a(x) - a_\infty |_{L_{{\rm loc}}^{N/2} ({\R}^N )} \to 0$ . © 2012 Wiley Periodicals, Inc.     

A Note on Bounds for the Supremum Metric for Discrete Random Variables

For positive or negative integer-valued random variables X and Y with finite second moments the inequality sup \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop {\sup }\limits_n |\Pr \{ X \le n\} - \Pr \{ Y \le n\} |\, \le \,|EX - EY| + \frac{1}{2}(EX(EX - 1) + (EY(Y - 1)) $\end{document} is established by elementary manipulation, and shown to be tight. Use of generating functions and an inversion formula yields the larger bound with 1/2 replaced by 2/π.     

A mixed problem for electrostatic potential in semiconductors

In this article we deal with the solution in Ω ? R ² of the quasi linear equation ?Δu = f(x, y, u(x, y)) subject to mixed boundary data and representing Gauss' law in a semiconductor device, where u and f are, respectively, the electrostatic potential and the space charge density after a suitable scaling. In the following we consider the associated variational problem of finding in a suitable subspace of H¹(Ω) the minimum of the functional $ J(u)\, = \,\int {_\Omega } (\frac{1}{2}\left| {\nabla u\left| {^2 \, - \,{\cal F}(x,y,u)\,d\Omega,} \right.} \right. $, where $ {\cal F}(x,y,u)\, = \,\int f (x,y,\xi)\,d\xi, $ and we prove existence and uniqueness of a weak solution according to the technique of Convex Analysis. The numerical study is then carried on by a piecewise linear finite element approximation, which is proved to converge in the H¹-norm to the exact solution of the variational problem; some numerical examples are also included. © 1994 John Wiley & Sons, Inc.     

A Weak Convergence Theorem for Functionals of Sums of Martingale Differences

For each n, let (S_nk), 1 ≦ k ≦ k_n, be a mean zero square — integrable martingale adapted to increasing s?-fields (b_nk), 0 ≦ k ≦ k_n, and let (b_nk), 0 ≦ k ≦ k_n, be a system of random variables such that b_n0 = 0 < b_n1 <…< b = 1 and such that b_nk is b_{n, k?1} measurable for each k. We present sufficient conditions under which \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{i = 0}^{k_n - 1} {f_n (b_{ni,\;} S_{ni})\;(S_{n,i + 1} \; - \;S_{ni})\; \to \int\limits_0^1 {f(t,\;W(t))\;d{\rm W(t)}} } $\end{document} as n → ∞, where {W(t) : 0 ≦ t ≦ 1} is a standard WIENER process.     

Classification of edge‐transitive rose window graphs

Given natural numbers n?3 and 1?a, r?n?1, the rose window graph R_n(a, r) is a quartic graph with vertex set $\{{{x}}_{{i}}|{{i}}\in {\mathbb{Z}}_{{n}}\} \cup \{{{y}}_{{i}}|{{i}}\in{\mathbb{Z}}_{{n}}\}Given natural numbers n?3 and 1?a, r?n?1, the rose window graph R_n(a, r) is a quartic graph with vertex set $\{{{x}}_{{i}}|{{i}}\in {\mathbb{Z}}_{{n}}\} \cup \{{{y}}_{{i}}|{{i}}\in{\mathbb{Z}}_{{n}}\}$ and edge set $\{\{{{x}}_{{i}},{{x}}_{{{i+1}}}\} \mid {{i}}\in {\mathbb{Z}}_n \} \cup \{\{{{y}}_{{{i}}},{{y}}_{{{i+r}}}\}\mid {{i}} \in{\mathbb{Z}}_{{n}}\}\cup \{\{{{x}}_{{{i}}},{{y}}_{{{i}}}\} \mid {{i}}\in {\mathbb{Z}}_{{{n}}}\}\cup \{\{{{x}}_{{{i+a}}},{{y}}_{{{i}}}\} \mid{{i}} \in {\mathbb{Z}}_{{{n}}}\}$. In this article a complete classification of edge‐transitive rose window graphs is given, thus solving one of the three open problems about these graphs posed by Steve Wilson in 2001. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 216–231, 2010