On the slowness of phase boundary motion in one space dimension

We study the limiting behavior of the solution of with a Neumann boundary condition or an appropriate Dirichlet condition. The analysis is based on “energy methods”. We assume that the initial data has a “transition layer structure”, i.e., u^? ≈ ±+M 1 except near finitely many transition points. We show that, in the limit as ? → 0, the solution maintains its transition layer structure, and the transition points move slower than any power of ?.     

Die Lösung der Prae-Maxwellschen Gleichungen mit Hilfe vollständiger Lösungssysteme

This paper deals with the Neumann problem of the pre-Maxwell partial differential equations for a vector field v defined in a region G ? R ³. We approximate its uniquely determined solution (integrability conditions assumed) uniformly on G by explicitly computable particular integrals and linear combinations of vector fields with a “fundamental” sequence of points .     

On asymmetric coverings and covering numbers

An asymmetric covering is a collection of special subsets S of an n‐set such that every subset T of the n‐set is contained in at least one special S with . In this paper we compute the smallest size of any for We also investigate “continuous” and “banded” versions of the problem. The latter involves the classical covering numbers , and we determine the following new values: , , , , and . We also find the number of non‐isomorphic minimal covering designs in several cases. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 218–228, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10022     

A refinement of Vietoris' inequality for sine polynomials

Let where In 1958, Vietoris proved that σ_n (x) is positive for all n ≥ 1 and x ∈ (0, π). We establish the following refinement. The inequalities hold for all natural numbers n and real numbers n ≥ 1 and x ∈ (0, π) if and only if      

On the existence of a rainbow 1‐factor in 1‐factorizations of K

Let be a 1‐factorization of the complete uniform hypergraph with and . We show that there exists a 1‐factor of whose edges belong to n different 1‐factors in . Such a 1‐factor is called a “rainbow” 1‐factor or an “orthogonal” 1‐factor. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 487–490, 2007     

Wave operator for the system of the Dirac–Klein–Gordon equations

We prove the existence of the wave operator for the system of the massive Dirac–Klein–Gordon equations in three space dimensions x∈ R ³ where the masses m, M>0. We prove that for the small final data , (?, ?)∈ H ^{2 + µ, 1} × H ^{1 + µ, 1}, with and , there exists a unique global solution for system (1) with the final state conditions Copyright © 2010 John Wiley & Sons, Ltd.     

The Khinchin Inequality for Generalized Rademacher Functions

We give a new proof of the Khinchin inequality for the sequence of k-Rademacher functions: We obtain constants which are independent of k. Although the constants are not best possible, they improve estimates of Floret and Matos [4] and they do have optimal dependence on p as p → ∞.     

Low‐curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes

We consider a class of fourth‐order nonlinear diffusion equations motivated by Tumblin and Turk's “low‐curvature image simplifiers” for image denoising and segmentation. The PDE for the image intensity u is of the form where g(s) = k²/(k² + s²) is a “curvature” threshold and λ denotes a fidelity‐matching parameter. We derive a priori bounds for Δu that allow us to prove global regularity of smooth solutions in one space dimension, and a geometric constraint for finite‐time singularities from smooth initial data in two space dimensions. This is in sharp contrast to the second‐order Perona‐Malik equation (an ill‐posed problem), on which the original LCIS method is modeled. The estimates also allow us to design a finite difference scheme that satisfies discrete versions of the estimates, in particular, a priori bounds on the smoothness estimator in both one and two space dimensions. We present computational results that show the effectiveness of such algorithms. Our results are connected to recent results for fourth‐order lubrication‐type equations and the design of positivity‐preserving schemes for such equations. This connection also has relevance for other related fourth‐order imaging equations. © 2004 Wiley Periodicals, Inc.     

Path homomorphisms,graph colorings,and boolean matrices

We investigate bounds on the chromatic number of a graph G derived from the nonexistence of homomorphisms from some path \begin{eqnarray*}\vec{P}\end{eqnarray*} into some orientation \begin{eqnarray*}\vec{G}\end{eqnarray*} of G. The condition is often efficiently verifiable using boolean matrix multiplications. However, the bound associated to a path \begin{eqnarray*}\vec{P}\end{eqnarray*} depends on the relation between the “algebraic length” and “derived algebraic length” of \begin{eqnarray*}\vec{P}\end{eqnarray*}. This suggests that paths yielding efficient bounds may be exponentially large with respect to G, and the corresponding heuristic may not be constructive. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 198–209, 2010     

Mixed boundary value problems for nonlinear elliptic equations in multidimensional non-smooth domains

The nonlinear elliptic equation is investigated. It is supposed that u fulfils a mixed boundary value condition and that Ω ? IRⁿ (n ≥ 3) has a piecewise smooth boundary. W^s,2 — regularity (s < 3/2) of u and L^p — properties of the first and the second derivatives of u are proven.     

Blowup for nonlinear wave equations describing boson stars

We consider the nonlinear wave equation modeling the dynamics of (pseudorelativistic) boson stars. For spherically symmetric initial data, u₀(x) ∈ C (?³), with negative energy, we prove blowup of u(t, x) in the H^1/2‐norm within a finite time. Physically this phenomenon describes the onset of “gravitational collapse” of a boson star. We also study blowup in external, spherically symmetric potentials, and we consider more general Hartree‐type nonlinearities. As an application, we exhibit instability of ground state solitary waves at rest if m = 0. © 2007 Wiley Periodicals, Inc.     

Tauberian theorem for m‐spherical transforms on the Heisenberg group

In this paper we prove a Tauberian type theorem for the space L ( H _n ). This theorem gives sufficient conditions for a L ( H _n ) submodule J ? L ( H _n ) to make up all of L ( H _n ). As a consequence of this theorem, we are able to improve previous results on the Pompeiu problem with moments on the Heisenberg group for the space L^∞( H _n ). In connection with the Pompeiu problem, given the vanishing of integrals ∫ z ^m L _g f ( z , 0) dσ ( z ) = 0 for all g ∈ H _n and i = 1, 2 for appropriate radii r₁ and r₂, we now have the (improved) conclusion f ≡ 0, where = · · · and form the standard basis for T^(0,1)( H _n ). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hyperbolic limit of the Jin‐Xin relaxation model

We consider the special Jin‐Xin relaxation model We assume that the initial data ( ) are sufficiently smooth and close to ( ) in L^∞ and have small total variation. Then we prove that there exists a solution ( ) with uniformly small total variation for all t ≥ 0, and this solution depends Lipschitz‐continuously in the L¹ norm with respect to time and the initial data. Letting , the solution converges to a unique limit, providing a relaxation limit solution to the quasi‐linear, nonconservative system These limit solutions generate a Lipschitz semigroup on a domain containing the functions with small total variation and close to . This is precisely the Riemann semigroup determined by the unique Riemann solver compatible with (0.1). © 2005 Wiley Periodicals, Inc.     

A generalization of the Oberwolfach problem

Let n > 1 be an integer and let a₂,a₃,…,a_n be nonnegative integers such that . Then can be factored into ‐factors, ‐factors,…, ‐factors, plus a 1‐factor. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 151–161, 2002     

The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters

Let X be an infinite set and let and denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by the Stone space of the Boolean algebra of all subsets of X. We show:

相似文献   

Derived Functors of Hom Relative to Flat Covers

We study the derived functors of Hom that are computed by using flat resolutions of Hom. These are denoted ⁿ. We compare these with the usual Extⁿ's and show that ¹⊂ Ext¹and indicate (using MacLane's terminology) why the class of associated short exact sequences is a proper class. When the ring is a Dedekind domain we classify the N such that ⁿ(–, N) = 0 and show that unlike the situation for other classically defined right derived functors of Hom, Hom is not balanced relative to the two classes of modules that make ¹ vanish.     

Eigenfunctions and Very Singular Similarity Solutions of Odd‐Order Nonlinear Dispersion PDEs: Toward a “Nonlinear Airy Function” and Others

Asymptotic properties of nonlinear dispersion equations (1) with fixed exponents n > 0 and p > n+ 1 , and their (2k+ 1) th‐order analogies are studied. The global in time similarity solutions, which lead to “nonlinear eigenfunctions” of the rescaled ordinary differential equations (ODEs), are constructed. The basic mathematical tools include a “homotopy‐deformation” approach, where the limit in the first equation in ( 1 ) turns out to be fruitful. At n= 0 the problem is reduced to the linear dispersion one: whose oscillatory fundamental solution via Airy’s classic function has been known since the nineteenth century. The corresponding Hermitian linear non‐self‐adjoint spectral theory giving a complete countable family of eigenfunctions was developed earlier in [ 1 ]. Various other nonlinear operator and numerical methods for ( 1 ) are also applied. As a key alternative, the “super‐nonlinear” limit , with the limit partial differential equation (PDE) admitting three almost “algebraically explicit” nonlinear eigenfunctions, is performed. For the second equation in ( 1 ), very singular similarity solutions (VSSs) are constructed. In particular, a “nonlinear bifurcation” phenomenon at critical values {p=p_l(n)}_l≥0 of the absorption exponents is discussed.     

On fans in multigraphs

We introduce a unifying framework for studying edge‐coloring problems on multigraphs. This is defined in terms of a rooted directed multigraph , which is naturally associated to the set of fans based at a given vertex u in a multigraph G. We call the “Fan Digraph.” We show that fans in G based at u are in one‐to‐one correspondence with directed trails in starting at the root of . We state and prove a central theorem about the fan digraph, which embodies many edge‐coloring results and expresses them at a higher level of abstraction. Using this result, we derive short proofs of classical theorems. We conclude with a new, generalized version of Vizing's Adjacency Lemma for multigraphs, which is stronger than all those known to the author. © 2005 Wiley Periodicals, Inc. J Graph Theory 51: 301–318, 2006     

Lower Estimates of the Norms of Extension Operators for Sobolev Spaces on the Halfline

It is proved that for an arbitrary extension operator its norm cannot be less than ε₀2^mm^—1/2p, ε₀ > 0. Previously only the upper estimates (≤ 8^m) for these norms were known.