Weyl’s Theorem for Algebraically Paranormal Operators

Let T be an algebraically paranormal operator acting on Hilbert space. We prove : (i) Weyls theorem holds for f(T) for every f $\in$ H((T)); (ii) a-Browders theorem holds for f(S) for every S $\prec$ T and f $\in$ H((S)); (iii) the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T.     

Pairs of Convex Bodies with Centrally Symmetric Intersections of Translates

For a pair of convex bodies $K$ and $K$ in $E^d$, the $d$-dimensional intersections $K \cap (x + K),$ $x \in E^d,$ are centrally symmetric if and only if $K$ and $K$ are represented as direct sums $K = R \oplus P$ and $K = R \oplus P$ such that: (i) $R$ is a compact convex set of some dimension $m$, $0 \le m \le d,$ and $R = z - R$ for a suitable vector $z \in E^d$, (ii) $P$ and $P$ are isothetic parallelotopes, both of dimension $d-m$.     

Poisson kernel is the unique approximate identity,asymptotically multiplicative on H∞

Let for anyf H(R), where (x): = ^–1(x^–1). Then (x) P (x + h) for some h R and > 0; P denotes the Poisson kernel.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 82–89, 1989.     

Solution of the Truncated Parabolic Moment Problem

Given real numbers with ₀₀ >0 , the truncated parabolic moment problem for entails finding necessary and sufficient conditions for the existence of a positive Borel measure , supported in the parabola p(x, y) = 0, such that We prove that admits a representing measure (as above) if and only if the associated moment matrix is positive semidefinite, recursively generated and has a column relation p(X, Y) = 0, and the algebraic variety () associated to satisfies card In this case, admits a rank -atomic (minimal) representing measure.Submitted: August 25, 2003     

Rate of decay to equilibrium in some semilinear parabolic equations

In this paper we prove, under various conditions, the so-called Lojasiewicz inequality $ \| E' (u + \varphi) \| \geq \gamma|E(u+\varphi) - E(\varphi)|^{1-\theta} $, where $ \theta \in (0,1/2] $, and > 0, while $ \| u \| $ is sufciently small and is a critical point of the energy functional E supposed to be only C⊃, instead of analytic in the classical settings. Here E can be for instance the energy associated to the semilinear heat equation $u_t = \Delta u - f(x,u) $ on a bounded domain $ \Omega \subset \mathbb{R}^N $. As a corollary of this inequality we give the rate of convergence of the solution u(t) to an equilibrium, and we exhibit examples showing that the given rate of convergence (which depends on the exponent and on the critical point through the nature of the kernel of the linear operator $ E' (\varphi)) $ is optimal.     

The Cauchy-Neumann problem for a viscous Hamilton-Jacobi equation

We study the weak solvability of viscous Hamilton-Jacobi equation: \,0,\,x\,\in\,\Omega,$" align="middle" border="0"> with Neumann boundary condition and irregular initial data ₀. The domain is a bounded open set and p > 0. The last part deals with the case a convex set and the initial data ₀ = in a open set D such that and      

Almost Integer Translates. Do Nice Generators Exist?

It has been shown earlier by the first author that for any nonzero perturbation of the integers $\lambda_n=n+o(1), \lambda_n\ne n$, there is a \textit{generator,} that is a function $\varphi\in L^2(\mathbf{R})$ such that the system of translates $\{\varphi(x-\lambda_n)\}$ is complete in $L^2(\mathbf{R})$. We ask if $\varphi$ can be chosen with fast decay. We prove that in general it cannot. On the other hand, if the perturbations are quasianalytically small, than it can, and this decay restriction is sharp. A certain class of complex measures which we call shrinkable is introduced, and it is shown that the zeros sets of such measures do dot admit generators with fast decay.     

Skorokhod decomposition of reflected diffusions on bounded Lipschitz domains with singular non-reflection part

Let be a compact set with interior G. Let L ¹ (G,dx), >0 dx-a.e. on G, and m:=dx. Let A=(a _ij ) be symmetric, and globally uniformly strictly elliptic on G. Let be such that ; f, , is closable in L ² (G,m) with closure ( ^r ,D( ^r )). The latter is fulfilled if satisfies the Hamza type condition, or _i L ¹ _loc (G,dx), 1id. Conservative, non-symmetric diffusion processes X _t related to the extension of a generalized Dirichlet form where satisfies are constructed and analyzed. If G is a bounded Lipschitz domain, H ^1,1 (G), and a _ij D( ^r ), a Skorokhod decomposition for X _t is given. This happens through a local time that is uniquely associated to the smooth measure 1_{{ Tr ()>0}} d, where Tr denotes the trace and the surface measure on G.This research has been financially supported by TMR grant HPMF-CT-2000-00942 of the European Union. Mathematics Subject Classification (2000): 60J60, 60J55, 31C15, 31C25, 35J25     

The conditions for the imbedding of the classes H k,R ω \tilde H_{k,R}^\omega

We find the necessary and sufficient conditions for the imbeddings in terms of the majorants and (R=L₂,C₂;l>k).Translated from Matematicheskie Zametki, Vol. 13, No. 2, pp. 169–178, February, 1973.     

On the Mean Value of the Index of Composition of an Integer

For each integer n 2, let be the index of composition of n, where . For convenience, we write (1)=(1)=1. We obtain sharp estimates for and , as well as for and . Finally we study the sum of running over shifted primes.Research supported in part by a grant from NSERC.Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA.     

Fine Properties of the Integrated Density of States and a Quantitative Separation Property of the Dirichlet Eigenvalues

We consider one-dimensional difference Schr?dinger equations with real analytic function V(x). Suppose V(x) is a small perturbation of a trigonometric polynomial V ₀(x) of degree k ₀, and assume positive Lyapunov exponents and Diophantine ω. We prove that the integrated density of states is H?lder continuous for any k > 0. Moreover, we show that is absolutely continuous for a.e. ω. Our approach is via finite volume bounds. I.e., we study the eigenvalues of the problem on a finite interval [1, N] with Dirichlet boundary conditions. Then the averaged number of these Dirichlet eigenvalues which fall into an interval , does not exceed , k > 0. Moreover, for , this averaged number does not exceed exp , for any . For the integrated density of states of the problem this implies that for any . To investigate the distribution of the Dirichlet eigenvalues of on a finite interval [1, N] we study the distribution of the zeros of the characteristic determinants with complexified phase x, and frozen ω, E. We prove equidistribution of these zeros in some annulus and show also that no more than 2k ₀ of them fall into any disk of radius exp. In addition, we obtain the lower bound (with δ > 0 arbitrary) for the separation of the eigenvalues of the Dirichlet eigenvalues over the interval [0, N]. This necessarily requires the removal of a small set of energies. Received: February 2006, Accepted: December 2007     

From Ginzburg-Landau to Gross-Pitaevskii

In this note we consider the Gross-Pitaevskii equation i _t ++(1–²)=0, where is a complex-valued function defined on ^N×, and study the following 2-parameters family of solitary waves: (x, t)=e ^it v(x ₁–ct, x), where and x denotes the vector of the last N–1 variables in ^N . We prove that every distribution solution , of the considered form, satisfies the following universal (and sharp) L -bound:

On context patterns associated with concept lattices

In this paper, we consider the following reconstruction problem: Given two ordered sets (G, ) and (M, ) representing join- and meet-irreducible elements, respectively together with three relationsJ,, onG×M modelling comparability (gm) and maximal noncomparability with respect tog (gm, butgm*) and with respect tom (gm, butgm*). We determine necessary and sufficient conditions for the existence of a finite latticeL and injections :GJ(L) and :MM(L) such that the given order relations and the abstract relations coincide with the one induced by the latticeL.     

Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with Zero Angles

Let C be the extended complex plane; G C a finite Jordan with 0 G; w= (z) the conformal mapping of G onto the disk normalized by . Let us set , and let be the generalized Bieberbach polynomial of degree n for the pair (G,0), which minimizes the integral in the class of all polynomials of degree not exceeding n with . In this paper we study the uniform convergence of the generalized Bieberbach polynomials with interior and exterior zero angles and determine its dependence on the properties of boundary arcs and the degree of their tangency.     

An open mapping theorem for measures

LetX andY be Hausdorff spaces and denote byM (X) andM (Y) the corresponding spaces of finite and non-negative Borel measures, endowed with the weak topology. A Borel map :XY induces the map :M (X)M (Y). We give necessary and sufficient conditions for to be open. In case of being a surjection between Suslin spaces, is open if and only if is.     

Cubic twists of <Emphasis Type="Italic">GL</Emphasis>(2) automorphic L-functions

Let and let be a cuspidal automorphic representation of . Consider the family of twisted L-functions L(s,) where ranges over the cubic Hecke characters of K. In this paper the mean value of this family of L-functions is computed; the result is consistent with the generalized Lindelöf hypothesis. From this mean value result a nonvanishing theorem is established: for given s there are infinitely many cubic twists such that the L-value at s is nonzero. At the center of the critical strip the number of such characters of norm less than X is . These results are obtained by introducing and studying three different families of weighted double Dirichlet series. These series are related by functional equations, some of which are obtained through the study of higher metaplectic Eisenstein series and the Hasse-Davenport relation. The authors establish the continuation of such series and then obtain their main result by Tauberian methods. Mathematics Subject Classification (1991) 11F66, 11F70, 11M41, 11N75, 22E55     

Interpolation theorems for a family of spanning subgraphs

Let G be a graph with order p, size q and component number . For each i between p – and q, let be the family of spanning i-edge subgraphs of G with exactly components. For an integer-valued graphical invariant if H H is an adjacent edge transformation (AET) implies |(H)-(H^')|1 then is said to be continuous with respect to AET. Similarly define the continuity of with respect to simple edge transformation (SET). Let M _j() and m _j() be the invariants defined by . It is proved that both M _p–() and m _p–(;) interpolate over , if is continuous with respect to AET, and that M _j() and m _j() interpolate over , if is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.     

Dirichlet and Bergman Spaces of Holomorphic Functions on the Unit Ball of ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let B denote the unit ball in ⁿ, n 1, and let and denote the volume measure and gradient with respect to the Bergman metric on B. In the paper we consider the weighted Dirichlet spaces , , and weighted Bergman spaces , , , of holomorphic functions f on B for which and respectively are finite, where and The main result of the paper is the following theorem.Theorem 1. Let f be holomorphic on B and .(a) If for some , then for all p, , with .(b) If for some p, , then for all with . Combining Theorem 1 with previous results of the author we also obtain the following.Theorem 2. Suppose is holomorphic in B. If for some p, , and , then . Conversely, if for some p, , then the series in * converges.     

On relations among moduli of continuity in Lorentz spaces

Estimates are obtained among moduli of continuity of functions in several variables that belong to various Lorentz spaces. The functions considered are periodic with period 1 in each variable. More exactly the following theorem is proved: If 0<,<(t) and (t) are so-called -functions such that _;>₁>1, and

A Class of Operators on L h 2 . II

Let D be the open unit disk in C, and L _h ² the space of quadratic integrable harmonic functions defined on D. Let be a function in L(D) with the property that (b) = lim _x b,xD (x) for all b D. Define the operator C on L _h ² as follows: C(f) = Q( · f), where Q is the orthogonal projection of L²(D) onto L _h ² . In this paper it is shown that if C is Fredholm, then is bounded away from zero on a neighborhood of D. Also, if C is compact, then |D 0, and the commutator ideal of (D) is K(D), where (D) denotes the norm closed subalgebra of the algebra of all bounded operators on L _h ² generated by , and K(D) is the ideal of compact operators on L _h ² . Finally, the spectrum of classes of operators defined on L _h ² is characterized.