THE EXISTENCE OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH CHANGE OF SIGN

1 IntroductionIll tllis paper, we co1lsid(tr the fOllowillg elliptic prol)leInwllere n is a bOui1ded regular domai1l in RN(N > 3), N > p > 1, aild a: n - R is a Holdercontinuou8 function which chauges 8ign o11 fl so that the open setsandO-- = {z E Ola(z) < 0}are both nonemPty.We use the standard notations. Let II' lI, denote the norm of L'(o) for s21. DenoteW0lP(fl) the comPletion of C7(fl) under the norm IIuIl = llVuIl;, the norm of tlle dual spaceof WJlp(fl) will be denoted by Il…     

CLASSIFICATION OF SOLUTIONS FOR A CLASS OF SINGULAR INTEGRAL SYSTEM

In this paper, we consider the following integral system: u(x) = R n v q (y) | x y | nα dy, v(x) = R n u p (y) | x y | nμ dy, (0.1) where 0 < α, μ < n; p, q ≥ 1. Using the method of moving planes in an integral form which was recently introduced by Chen, Li, and Ou in [2, 4, 8], we show that all positive solutions of (0.1) are radially symmetric and decreasing with respect to some point under some general conditions of integrability. The results essentially improve and extend previously known results [4, 8].     

THE EXISTENCE OF MULTIPLE SOLUTIONS OF p-LAPLACIAN ELLIPTIC EQUATION

1 IntroductionIn this paper we consider the following quasilinear elliptic problemwhere -- A. 'u = --div(l V ulp--' 7 u), A 2 0 is a real parametcr, 0 < m < p -- 1 < q < oc. flis a bounded domain in RN(N 2 3).During the last decade HLaplaJce equatiolls have e11jOyed a growing attention. After theinitial works of Poliozeav[1], and of Brezis and Nirenberg[2], there has been great number ofcontributious to the study of that kind of problem (1.1)A(see [3-161). Recently, equationiuvolving th…     

THE JUMPING PHENOMENON OF THE DIMENSIONS OF COHOMOLOGY GROUPS OF TANGENT SHEAF

Let X be a compact complex manifold.Consider a small deformation π :X→B of X,the dimensions of the cohomology groups of tangent sheaf H q(X_t,T_X_t) may vary under this deformation.This article studies such phenomena by studying the obstructions to deform a class in H q(X,T X) with parameter t and gets a formula for the obstructions.     

CLASSIFICATION AND DIRECT DECOMPOSITIONS OF SOME BUTLER GROUPS OF COUNTABLE RANK

ABSTRACT

This paper is devoted to an extension of results concerning decompositions of particular almost completely decomposable groups of finite rank studied by Mader and the first author in 1-2 Mader, A. 2000. “Almost Completely Decomposable Abelian Groups. Algebra, Logic and Applications”. Vol. 13, Amsterdam: Gordon and Breach. Blagoveshchenskaya, E. and Mader, A. 1970. Decompositions of Almost Completely Decomposable Abelian Groups. Contemporary Mathematics, 171: 21–36.   to groups of infinite rank. Thus we obtain asystematic way to consider all decompositions of some fairly large class of torsion-free abelian groups. This will be illustrated discussing all pathological decompositions of Corner's[3] Corner, A.L.S. 1961. A Note on Rank and Decomposition of Torsion-free Abelian Groups. Proceedings Cambridge Philos. Soc., 57: 230–233. and 66, 239–240 [Google Scholar] group constructed 40 years ago.     

ITERATIONS OF SATISFACTION CLASSES AND MODELS OF PEANO ARITHMETIC

We consider iterations of satisfaction classes and apply them to construct expansions of models of Peano arithmetic to models of A|Δ+∑-AC. 1991 MSC: 03F35, 03C62.     

INTEGRABILITY OF GEODESIC FLOWS FOR METRICS ON SUBORBITS OF THE ADJOINT ORBITS OF COMPACT GROUPS

Let G/K be an orbit of the adjoint representation of a compact connected Lie group G, σ be an involutive automorphism of G and \( \tilde{G} \) be the Lie group of fixed points of σ. We find a sufficient condition for the complete integrability of the geodesic ow of the Riemannian metric on \( \tilde{G}/\left(\tilde{G}\cap K\right) \) which is induced by the bi-invariant Riemannian metric on \( \tilde{G} \). The integrals constructed here are real analytic functions, polynomial in momenta. It is checked that this sufficient condition holds when G is the unitary group U(n) and σ is its automorphism determined by the complex conjugation.     

A THEOREM ON THE CONVERGENCE OF SUMS OF INDEPENDENT RANDOM VARIABLES

1 IntroductionThroughout this paperl {X., n 2 1} is assumed to be a sequence of independent randomvariables. Fn denotes the distribution function of the partial surns S. = Z Xk. As is known,k = 1oothe series Z X. is said to be essentially convergent (a.s.) if there exists a sequence of constantsn = 1oo{b., n 2 1} such that Z (X. -- 6.) a.s. converges. When this happens we writeClearlyFn(x) = G.(x -- B.). (l.2)Our problem stems from an old conjecture in Probabilistic Number Theory sugges…     

ADIABATIC DECOMPOSITION OF THE -DETERMINANT OF THE DIRAC LAPLACIAN. I. THE CASE OF AN INVERTIBLE TANGENTIAL OPERATOR

ABSTRACT

We discuss the decomposition of the ζ-determinant of the square of the Dirac operator into the contributions coming from the different parts of the manifold. The result was announced in the Note Ref. [16]. The proof sketched in the Note was based on results of Brüning and Lesch (see Ref. [4]). In the meantime we have found another proof, more direct and elementary, and closer to the spirit of the original papers which initiated the study of the adiabatic decomposition of the spectral invariants (see Refs. [7] Douglas, R.G. and Wojciechowski, K.P. 1991. Adiabatic Limits of the η-Invariants. The Odd-dimensional Atiyah–Patodi–Singer Problem. Comm. Math. Phys., 142: 139–168. [Crossref], [Web of Science ®] , [Google Scholar] and [21] Singer, I.M. 1988. “The η-Invariant and the Index”. In Mathematical Aspects of String Theory Edited by: Yau, S.-T. pp. 239–258. Singapore: World Scientific Press.  [Google Scholar]). We discuss this proof in detail. We study the general case (non-invertible tangential operator) in forthcoming work (see Refs. [17] Park, J. and Wojciechowski, K.P. 2001. Scattering Theory and Adiabatic Decomposition of the ζ-Determinant of the Dirac Laplacian 01–02. IUPUI Preprint [Google Scholar] and [18] Park, J. and Wojciechowski, K.P. 2001. Adiabatic Decomposition of the ζ-Determinant of the Dirac Laplacian II. The Case of Non-invertible Tangential Operator In preparation [Google Scholar]). In the Appendix we present the computation of the cylinder contribution to the ζ-function of the Dirac Laplacian on a manifold with boundary, which we need in the main body of the paper. This computation is also used to show the vanishing result for the ζ-function on a manifold with boundary.     

USTIFICATION OF A TWO-DIMENSIONAL NONLINEAR SHELL MODEL OF KOITER'S TYPE

A two-dimensional nonlinear shell model"of Koiter's type"has recently been proposed by the first author. It is shown here that, according to two mutually exclusive sets of assumptions bearing on the associated manifold of admissible inextensional displacements, the leading term of a formal asymptotic expansion of the solution of this two-dimensional model, with the thickness as the"small" parameter, satisfies either the two-dimensional equations of a nonlinearly elastic "membrane" shell or those of a nonlinearly elastic "flexural" shell. These conclusions being identical to those recently drawn by B. Miara, then by V. Lods and B. Miara, for the leading term of a formal asymptotic expansion of the solution of the equations of three-dimensional nonlinear elasticity, again with the thickness as the "small" parameter, the nonlinear shell model of Koiter's type considered here is thus justified, at least formally.     

SOLUTIONS OF A SYSTEM OF FORCED BURGERS EQUATION IN TERMS OF GENERALIZED LAGUERRE POLYNOMIALS

In this article, we obtain explicit solutions of a linear PDE subject to a class of radial square integrable functions with a monotonically increasing weight function |x|n-1eβ|x|2/2,β≥ 0, x ∈ Rn. This linear PDE is obtained from a system of forced Burgers equation via the Cole-Hopf transformation. For any spatial dimension n 1, the solution is expressed in terms of a family of weighted generalized Laguerre polynomials. We also discuss the large time behaviour of the solution of the system of forced Burgers equation.     

一类矩阵对的广义特征值的扰动界限

关于矩阵特征值的扰动,下面的结果是熟知的:若A与C皆为n阶正规矩阵,它们的特征值分别为α_1,…,α_n与γ_1,…,γ_n,则据Wielandt-Hoffman定理,存在1,…,n的一个排列k_1,…,k_n,使得     

ON THE COADJOINT ORBITS OF MAXIMAL UNIPOTENT SUBGROUPS OF REDUCTIVE GROUPS

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \( \mathfrak{u} \) its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \( \mathfrak{u} \)* of \( \mathfrak{u} \). This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which has been used for G of rank at most 8, except E₈.When G is defined and split over the field of q elements, for q the power of a good prime for G, this algorithmic parametrization is used to calculate the number k(U(q); \( \mathfrak{u} \)*(q)) of coadjoint orbits of U(q) on \( \mathfrak{u} \)*(q). Since k(U(q), \( \mathfrak{u} \)*(q)) coincides with the number k(U(q)) of conjugacy classes in U(q), these calculations can be viewed as an extension of the results obtained in [11]. In each case considered here there is a polynomial h(t) with integer coefficients such that for every such q we have k(U(q)) = h(q). We also explain implications of our results for a parametrization of the irreducible complex characters of U(q).