Existence of static solutions of the semilinear Maxwell equations

Abstract In this paper we study a model which describes the relation of the matter and the electromagnetic field from a unitarian standpoint in the spirit of the ideas of Born and Infeld. This model, introduced in [1], is based on a semilinear perturbation of the Maxwell equation (SME). The particles are described by the finite energy solitary waves of SME whose existence is due to the presence of the nonlinearity. In the magnetostatic case (i.e. when the electric field and the magnetic field does not depend on time) the semilinear Maxwell equations reduce to semilinear equation where “ ” is the curl operator, f′ is the gradient of a smooth function and is the gauge potential related to the magnetic field ( ). The presence of the curl operator causes (1) to be a strongly degenerate elliptic equation. The existence of a nontrivial finite energy solution of (1) having a kind of cylindrical symmetry is proved. The proof is carried out by using a variational approach based on two main ingredients: the Principle of symmetric criticality of Palais, which allows to avoid the difficulties due to the curl operator, and the concentration-compactness argument combined with a suitable minimization argument. Keywords: Maxwell equations, Natural constraint, Minimizing sequence Mathematics Subject Classification (2000): 35B40, 35B45, 92C15     

Bicomplex Quantum Mechanics: I. The Generalized Schrödinger Equation

We introduce the set of bicomplex numbers which is a commutative ring with zero divisors defined by where We present the conjugates and the moduli associated with the bicomplex numbers. Then we study the bicomplex Schr?dinger equation and found the continuity equations. The discrete symmetries of the system of equations describing the bicomplex Schr?dinger equation are obtained. Finally, we study the bicomplex Born formulas under the discrete symmetries. We obtain the standard Born’s formula for the class of bicomplex wave functions having a null hyperbolic angle.     

Identification of a time-dependent UV-photon source in an interstellar cloud

Abstract Consider an interstellar cloud that occupies the region , bounded by the known surface , and assume that the scattering cross section σ_s and the total cross section σ are also known. Then, we prove that it is possible to identify the source q=q(x,t) that produces UV-photons inside the cloud, provided that the UV-photon distribution function arriving at a location , far from the cloud, is measured at times , , ..., . Keywords: Photon transport, Semigroups and linear evolution equations, Inverse problems Mathematics Subject Classification (2000): 82A25, 82C70, 34K29, 65M32     

Criteria for the L p -dissipativity of systems of second order differential equations

Abstract  We give complete algebraic characterizations of the L^p-dissipativity of the Dirichlet problem for some systems of partial differential operators of the form , where are m× m matrices. First, we determine the sharp angle of dissipativity for a general scalar operator with complex coefficients. Next we prove that the two-dimensional elasticity operator is L^p-dissipative if and only if

The Inverse Problem for Perturbed Harmonic Oscillator on the Half-Line with a Dirichlet Boundary Condition

We consider the perturbed harmonic oscillator in , where is a real-valued potential. We prove that the mapping spectral data = {eigenvalues of T _D } {norming constants} is one-to-one and onto. The complete characterization of the set of spectral data which corresponds to is given. Dedicated to Vladimir Buslaev on the occasion of his 70th birthday Submitted: September 27, 2006. Accepted: January 9, 2007.     

A class number formula for the p-cyclotomic field

Let p be an odd prime number and . Let be the classical Stickelberger ideal of the group ring . Iwasawa [6] proved that the index equals the relative class number of . In [2], [4] we defined for each subgroup H of G a Stickelberger ideal of , and studied some of its properties. In this note, we prove that when mod 4 and [G : H] = 2, the index equals the quotient . Received: 13 January 2006     

Remarks on balance in generalized Tate cohomology

We consider two pairs of complete hereditary cotorsion theories on the category of left R-modules, such that We prove that for any left R-modules M, N and for any n ≧ 1, the generalized Tate cohomology modules can be computed either using a left of M and a left of M or using a right a right of N. Received: 17 December 2004     

Equidistribution of Rational Matrices in their Conjugacy Classes

Let G be a connected simply connected almost -simple algebraic group with non-compact and a cocompact congruence subgroup. For any homogeneous manifold of finite volume, and a , we show that the Hecke orbit T _a (x ₀ H) is equidistributed on as , provided H is a non-compact commutative reductive subgroup of G. As a corollary, we generalize the equidistribution result of Hecke points ([COU], [EO1]) to homogeneous spaces G/H. As a concrete application, we describe the equidistribution result in the rational matrices with a given characteristic polynomial. The second author partially supported by DMS 0333397. Received: May 2005 Revision: March 2006 Accepted: June 2006     

Representation Formulae for Solutions to Some Classes of Higher Order Systems and Related Liouville Theorems

Let m ≥ 1 be an integer and N > 2m. Let μ be a positive Radon measure on . We study necessary and sufficient conditions on possible distributional solutions of , that guarantee the validity of the representation formula a.e. on , where and c(2m) is a positive constant depending on m and N. Several consequences are derived. In particular we prove Liouville theorems for systems of higher order elliptic inequalities and weighted form of Hardy-Littlewood-Sobolev systems of integral equations. Received: March 2008     

Existence and Weyl’s law for spherical cusp forms

Let G be a split adjoint semisimple group over and a maximal compact subgroup. We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of . This proves a conjecture of Sarnak for -split groups, previously known only for the case G = PGL(n). The key idea amounts to a new type of simple trace formula. Received: April 2005 Revision: June 2006 Accepted: October 2006     

An inequality concerning the elasticity of Krull monoids with divisor class group $\mathbb{Z}_p$

Abstract  Let p be a prime integer and M a Krull monoid with divisor class group . We represent by S the set of nontrivial divisor classes of which contain prime divisors. We present a new inequality for the elasticity of M (denoted ρ (M)) which is dependent on the cardinality of S and argue that this inequality is the best possible. If M as above has | S| = 3, then it is known that , but for large p, not all the values in this containment set can be realized. For each | S| = 3, we produce a submonoid of such that

Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.     

Analysis of an Age Structured SEIRS Epidemic Model with Varying Total Population Size and Vaccination

This article focuses on the study of an age structured SEIRS epidemic model with a vaccinationprogram when the total population size is not kept at constant.We first give the explicit expression of thereproduction number R(ψ,(?)) in the presence of vaccine ((?) is the exponent of growth of total population),andshow that the infection-free steady state is linearly stable if R(ψ,(?))<1 and unstable if R(ψ,(?))>1,then weapply the theoretical results to vaccination policies to determine the optimal age or ages at which an individualshould be vaccinated.It is shown that the optimal strategy can be either one- or two-age strategies.     

On the geometry of metric measure spaces

We introduce and analyze lower (Ricci) curvature bounds  ⩾ K for metric measure spaces . Our definition is based on convexity properties of the relative entropy regarded as a function on the L ₂-Wasserstein space of probability measures on the metric space . Among others, we show that  ⩾ K implies estimates for the volume growth of concentric balls. For Riemannian manifolds,  ⩾ K if and only if  ⩾ K for all . The crucial point is that our lower curvature bounds are stable under an appropriate notion of D-convergence of metric measure spaces. We define a complete and separable length metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation. We also prove that the family of normalized metric measure spaces with doubling constant ⩽ C is closed under D-convergence. Moreover, the family of normalized metric measure spaces with doubling constant ⩽ C and diameter ⩽ L is compact under D-convergence.     

The Asymptotics of the Heat Semigroup for a General Bounded Domain with Mixed Boundary Conditions

Abstract Small-time asymptotics of the trace of the heat semigroup where {μ _ν } are the eigenvalues of the negative Laplacian in the (x ¹, x ²)-plane, is studied for a general bunded domain Ω with a smooth boundary ∂Ω, where a finite number of Dirichlet, Neumann and Robin boundary conditions, on the piecewise smooth parts Γ _i (i = 1, ..., n) of ∂Ω such that , are considered. Some geometrical properties associated with Ω are determined.     

A singular Moser-Trudinger embedding and its applications

Let Ω be a bounded domain in , we prove the singular Moser-Trudinger embedding: if and only if where and . We will also study the corresponding critical exponent problem.     

Some Categorical Algebraic Properties: Counter-Examples for Functor Categories

This work is a complement to the authors earlier papers, where it is shown that a functor category inherits from such properties as amalgamation, transferability and congruence extension if has either products or certain pushouts. A general scheme is given for constructing counter-examples which show that the latter condition on is essential. In particular, it is shown that the functor categories , , ( resp.) do not satisfy the amalgamation (congruence extension resp.) property in general. Moreover, one class of categories is described, where the condition of the existence of certain pushouts is not only sufficient, but also necessary for to preserve the considered properties of .Mathematics Subject Classifications (2000) 18A25, 18A32, 18B99, 08B26.Dali Zangurashvili: The support rendered by INTAS Grant 97 31961 is gratefully acknowledged.     

{\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pairs}}

Let R be a unital associative ring and two classes of left R-modules. In this paper we introduce the notion of a In analogy to classical cotorsion pairs as defined by Salce [10], a pair of subclasses and is called a if it is maximal with respect to the classes and the condition for all and Basic properties of are stated and several examples in the category of abelian groups are studied. Received: 17 March 2005     

Precise asymptotics in the Baum-Katz and davis law of large numbers for positively associated sequences

§ 1　 Introduction and resultsL et { X,Xi;i≥ 1} be a sequence of i.i.d.random variables,and set Sn= ni=1 Xi,n≥1.Hsu and Robbins[1 ] introduced the conceptof complete convergence.They together withErdos[2 ] proved n≥ 1 P(|Sn|≥εn) <∞ ,ε>0 (1)if and only if EX=0 and EX2 <∞ .L ater,Spitzer[3] proved n≥ 11n P(|Sn|≥εn) <∞ ,ε>0if and only if EX =0 and E|X|<∞ .More generally,it was shown by Baum and Katz[4 ]that,for 0 0 (…     

The Reduced Minimum Modulus in <Emphasis>C</Emphasis><Superscript>*</Superscript>-Algebras

We define the reduced minimum modulus of a nonzero element a in a unital C ^*-algebra by . We prove that . Applying this result to and its closed two side ideal , we get that dist , and for any if RR = 0, where and is the quotient homomorphism and . These results generalize corresponding results in Hilbert spaces.