Self-similar solutions for a convection-diffusion equation with absorption inR N

We prove the existence of a positive and smooth solution for the following semilinear elliptic problem: % MathType!End!2!1! for anya∈R ^N , 1<p<1+2/N andq=(p+1)/2. This solution decays exponentially as |x|→+∞. Moreover, if |a| is sufficiently small, this positive and rapidly decaying solution is unique. The existence of a positive, self-similar solution % MathType!End!2!1! follows for the following convection-diffusion equation with absorption: % MathType!End!2!1!. It is also a very singular solution. This solution decays as |x|→+∞ for anyt>0 fixed. Because of the nonvariational nature of the elliptic problem, a fixed point method is used for proving the existence result. The uniqueness is proved applying the Implicit Function Theorem. The work of the first author has been partially supported by Grant 1273/00003/88 of the University of the Basque Country. The work of the second author has been supported by Grant PB 86-0112-C02-00 of the Dirección General de Investigación Científica y Técnica.     

Exponential generating functions and complexity of Lie varieties

Suppose that % MathType!End!2!1! is a variety of Lie algebras, and letc _n( % MathType!End!2!1!) be the dimension of the linear span of all multilinear words onn distinct letters in the free algebraF( % MathType!End!2!1!,X) of the variety % MathType!End!2!1!. We consider an exponential generating function % MathType!End!2!1!, called the complexity function. The complexity function is an entire function of a complex variable provided the variety of Lie algebras is nontrivial. In this paper we introduce the notion of complexity for Lie varieties in terms of the growth of complexity functions; also we describe what the complexity means for the codimension growth of the variety. Our main goal is to specify the complexity of a product of two Lie varieties in terms of the complexities of multiplicands. The main observation here is thatC( % MathType!End!2!1!),z) behaves like a composition of three functionsC( % MathType!End!2!1!),z), exp(z), andC( % MathType!End!2!1!),z). Partially supported by grant RFFI 96-01-00146; the author is grateful to the University of Bielefeld for hospitality, where he was DAAD-fellow.     

Analytic Hausdorff gaps II: The density zero ideal

We prove two results about the quotient over the asymptotic density zero ideal. First, it is forcing equivalent to % MathType!End!2!1!, where % MathType!End!2!1! is the homogeneous probability measure algebra of characterc. Second, if it has analytic Hausdorff gaps, then they look considerably different from proviously known gaps of this form. Partially supported by NSERC.     

Many solutions for elliptic equations with critical exponents

Let Ω be an open bounded domain in ℝ^N(N ≥ 3) and . We are concerned with two kinds of critical elliptic problems. The first one is

The 2-length of the hughes subgroup

For a finite groupG and some prime powerp ⁿ , the -subgroup is defined by . Meixner proved that ifG is a finite solvable group and for somen≧1, then the Fitting length of is bounded by 4n. In the following note it is shown that the 2-length of is at mostn. This result cannot be derived from Meixner’s paper, since his result implies only that the 2-length is bounded by 2n.     

The two sides of a fourier-stieltjes transform and almost idempotent measures

For measuresμ on the circleT the quantities , need not be equal; it is shown, however, that they are continuous with respect to each other whenμ varies on bounded subsets ofM(T), the space of measures onT. It is also shown that measuresμ which areɛ-almost idempotent (i.e. ) are the sum of an idempotent measure and of a measureυ satisfying providedɛ is small enough (as a function of ‖μ‖).     

Almost orthogonal submatrices of an orthogonal matrix

Lett≥1 and letn, M be natural numbers,n<M. Leta=(a _i,j ) be ann xM matrix whose rows are orthonormal. Suppose that the ℓ₂-norms of the columns ofA are uniformly bounded. Namely, for allj Using majorizing measure estimates we prove that for every ε>0 there exists, a setI ⊃ {1,…,M} of cardinality at most such that the matrix , whereA _I =(a _i,j ) _j∈I , acts as a (1+ε)-isomorphism from ℓ ₂ ⁿ into . Research supported in part by a grant of the US-Israel BSF. Part of this research was performed when the author held a postdoctoral position at MSRI. Research at MSRI was supported in part by NSF grant DMS-9022140.     

On directional entropy functions

Given aZ ²-process, the measure theoretic directional entropy function,h( % MathType!End!2!1!), is defined on % MathType!End!2!1!. We relate the directional entropy of aZ ²-process to itsR ² suspension. We find a sufficient condition for the continuity of directional entropy function. In particular, this shows that the directional entropy is continuous for aZ ²-action generated by a cellular automaton; this finally answers a question of Milnor [Mil]. We show that the unit vectors whose directional entropy is zero form aG _δ subset ofS ¹. We study examples to investigate some properties of directional entropy functions. This research is supported in part by BSRI and KOSEF 95-0701-03-3.     

Asymptotics for Amitsur's Capelli-type polynomials and verbally prime PI-algebras

We consider associativePI-algebras over a field of characteristic zero. The main goal of the paper is to prove that the codimensions of a verbally prime algebra [11] are asymptotically equal to the codimensions of theT-ideal generated by some Amitsur's Capelli-type polynomialsE _M,L ^* [1]. We recall that two sequencesa _n,b _nare asymptotically equal, and we writea _n ≃b _n,if and only if lim _n→∞(a _n/b _n)=1.In this paper we prove that % MathType!End!2!1!, whereG is the Grassmann algebra. These results extend to all verbally primePI-algebras a theorem of A. Giambruno and M. Zaicev [9] giving the asymptotic equality % MathType!End!2!1! between the codimensions of the matrix algebraM _k(F) and the Capelli polynomials. The second author is partially supported by grants RFFI 04-01-00739a, E02-2.0-26.     

Existence of solutions for elliptic problems with critical Sobolev-Hardy exponents

Let Ω ⊂ ℝ ^N be a smooth bounded domain such that 0 ∈ Ω,N≥3, 0≤s<2,2^* (s)=2(N−s)/(N−2). We prove the existence of nontrival solutions for the singular critical problem with Dirichlet boundary condition on Ω for suitable positive parameters λ and μ. Corresponding author. This work is supported partly by the National Natural Science Foundation of China (No. 10171036) and the Natural Science Foundation of South-Central University For Nationalities (No. YZZ03001). The authors sincerely thank Prof. Daomin Cao (AMSS, Chinese Academy of Sciences) for helpful discussions and suggestions.     

On the riemann zeta-function and the divisor problem II

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If E ^*(t)=E(t)-2πΔ^*(t/2π) with , then we obtain

Branch rings, thinned rings, tree enveloping rings

We develop the theory of “branch algebras”, which are infinite-dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting on trees. In particular, for every field % MathType!End!2!1! we contruct a % MathType!End!2!1! which

Was sierpinski right? I

Aroused by Todorcevic’s breakthrough we prove here some complementary consistency results, mainly . We also get some generalization of his theorem to, e.g.,λ↛ forλ regular notω-Mahlo. The author thanks the NSF and N.S.E.R.C. for partially supporting the research. Preliminary versions of §3 and §2 were circulated in November ’84 and February ’85, respectively.     

Carleson measures and the heat equation

Let { }, where { } is the open unit disk on the complex plane { }. In G, we consider analytic solutions u(t, z) ({ }, { }) of the heat equation 2u_t=u_zz with initial data f(z)=u(0, z) belonging to the Fock space F, i.e., to the space of entire functions square summable with the weight e−|z|2.Conditions on a nonnegative measure μ on G are described under which for all f ∈ F we have { } Bibliography: 17 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 247, 1997, pp. 146–155. Translated by S. V. Kislyakov.     

The Balian-Low theorem and regularity of Gabor systems

For any positive real numbers A, B, and d satisfying the conditions , d>2, we construct a Gabor orthonormal basis for L²(ℝ), such that the generating function g∈L²(ℝ) satisfies the condition:∫_ℝ|g(x)|²(1+|x| ^A )/log ^d (2+|x|)dx < ∞ and .     

The Boggess-Polking extension theorem for<Emphasis Type="Italic">CR</Emphasis> functions on manifolds with corners

We give the “boundary version” of the Boggess-PolkingCR extension theorem. LetM andN be real generic submanifolds of ℂ ⁿ withN ⊂M and letV be a “wedge” inM with “edge”N and “profile” Σ ⊂T _NM in a neighborhood of a pointz _o.We identify in natural manner and assume that for a holomorphic vector fieldL tangent toM and verifying we have that the Levi form takes a value . Then we prove thatCR functions onV extend ∀_ω to a wedgeV ₁ “attached” toV in direction of a vector fieldiV such that |pr(iV(z ₀))−iv ₀| < ε (where pr is the projection pr:T _NX →T _MX | _N ).We then prove that when the Levi cone “relative to Σ”iZ _Σ = convex hull is open inT _MX, thenCR functions extend to a “full” wedge with edgeN (that is, with a profile which is an open cone ofT _NX). Finally, we prove that iff is defined in a couple of wedges ±V with profiles ±Σ such thatiZ _Σ =T _MX, and is continuous up toN, thenf is in fact holomorphic atz _o.     

Berger-Shaw's theorem forp-hyponormal operators

For an-multicyclicp-hyponormal operatorT, we shall show that |T|^2p –|T ^*|^2p belongs to the Schatten and that tr Area ((T)).     

Some existence and nonexistence theorems for compact graphs of constant mean curvature with boundary in parallel planes

Given H≥0 and bounded convex curves α₁, ...,⇌_n, α in the plane z=0 bounding domains D₁, …, D_n, D, respectively, with if i ∈ j and with D_i ⊂ D, we obtain several results proving the existence of a constanth depending only on H and on the geometry of the curves α_i, α such that the Dirichlet problem for the constant mean curvature H equation: where may accept or not a solution.     

Remainder estimates for the approximation numbers of weighted Hardy operators acting onL 2

We consider the weighted Hardy integral operatorT:L ²(a, b) →L ²(a, b), −∞≤a<b≤∞, defined by . In [EEH1] and [EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa _n(T) ofT. In this paper, we show that under suitable conditions onu andv, where ∥w∥_p=(∫ _a ^b |w(t)|^p dt)^1/p. Research supported by NSERC, grant A4021. Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic.     

Infinitely many solutions of dirichlet problem for <Emphasis Type="Italic">p</Emphasis>-mean curvature operator

The existence of infinitely many solutions of the following Dirichlet problem for p-mean curvature operator: is considered, where Θ is a bounded domain in R ⁿ (n>p>1) with smooth boundary ∂Θ. Under some natural conditions together with some conditions weaker than (AR) condition, we prove that the above problem has infinitely many solutions by a symmetric version of the Mountain Pass Theorem if . Supported by the National Natural Science Foundation of China (10171032) and the Guangdong Provincial Natural Science Foundation (011606).