The Eigenvalues of Very Sparse Random Symmetric Matrices

Let W _n be an n × n random symmetric sparse matrix with independent identically distributed entries such that the values 1 and 0 are taken with probabilities p/n and 1-p/n, respectively; here is independent of n. We show that the limit of the expected spectral distribution functions of W _n has a discrete part. Moreover, the set of positive probability points is dense in (- +). In particular, the points , and 0 belong to this set.     

Selective Limit Theorems for Random Walks on Parabolic Biangle and Triangle Hypergroups

Let K be respectively the parabolic biangle and the triangle in and be a sequence in [0, +[ such that lim_p (p)=+. According to Koornwinder and Schwartz,⁽⁷⁾ for each there exist a convolution structure (*_(p)) such that (K, *_(p)) is a commutative hypergroup. Consider now a random walk on (K, *_(p)), assume that this random walk is stopped after j(p) steps. Then under certain conditions given below we prove that the random variables on K admit a selective limit theorems. The proofs depend on limit relations between the characters of these hypergroups and Laguerre polynomials that we give in this work.     

Interpolation Sequences for the Class of Functions of Finite η-Type Analytic in the Unit Disk

We establish conditions for the existence of a solution of the interpolation problem f( _n ) = b _n in the class of functions f analytic in the unit disk and such that

Approximation in the mean by bounded analytic functions

Let 1<p< and . LetC _q denote the Bessel capacity in the plane. Let be the set of homomorphisms ofH (G) such that (z)= and letNP denote the set of points in G for which is not a peak set forH (G). In this note, we show that ifC _q (NP)=0, thenH (G) is dense inL _a ^p (G), the Bergman space overG.Partially supported by NSF DMS-9401234     

Brownian Martingales and Some New Results on Interpolation of Hardy Spaces

Let H₁( ) be the usual Hardy space on . We show that the couple (H₁( ), L( ) is a Calderón couple. This result immediately follows from the following stronger one: Given any fH₁( ) +L( ) there exist two linear operators U and V satisfying the properties: (i) Uf=Nf (Nf being the non-tangential maximal function of f) and U is contractive from H₁( ) to L₁( ) and also from L( ) to L( ); (ii) V(Nf)=f, V is similtaneously bounded from L₁( ) to H₁( ) and from L( ) to L( ) and the norms of V on these spaces are controlled by a universal constant. We also have similar results on the couple (L_p( ), BMO ( )) for every 1相似文献   

Relations of Borel Type for Generalizations of Exponential Series

We prove that the condition is necessary and sufficient for the validity of the relation ln F() ln (, F), +, outside a certain set for every function from the class . Here, H(, f) is the class of series that converge for all 0 and have a form

Décroissance rapide de la distribution fλ

Let X be an open subset of ⁿ and (f₁, ...,f_p): X ^p be a holomorphic mapping. We prove that if (x⁰,0, ⁰) T^* × ^p does not belong to the characteristic variety of the _X []-module _X[]f, then there exists a conic neighborhood V × of (x⁰, ⁰) such the function is rapidely decreasing in | Im | for with Re bounded, for any (n,n)-form of class C with compact support in V. The following partial converse of this result is also established: if for all (n,n)-forms of class C with compact support in X, then .     

On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Let M _f(r) and _f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +.     

On Interpolation Sequences of One Class of Functions, Analytic in the Half-Plane

A new criterion of solvability of the interpolation problem f( _n )=b_n in the class of functions f, analytic in the right half-plane and such that there exists c ₁(0;+) such that |f(z)|c ₁exp((c₁|z|)) for all z , where is a positive increasing continuous differentiable function on [0;+), for which (t)+ as t+ and there exists c ₂(0;+) such that

Toeplitz operators with symbols from in the spaces

It is shown that the algebra of the multipliers of the space ^p (1<<) contains the closed subalgebra C_p+H _p , which coincides with the Douglas algebra C + H for =2. It is proved that a Toeplitz operator with symbol from C_p+H _p is Fredholm on ^p if and only if its symbol is invertible in C_p+H _p .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 124–128, 1987.The authors are grateful to V. I. Vasyunin for assistance.     

On convergence of types and processes in Euclidean space

Let be an Euclidean space; Y _n , Z, U random vectors in ; h _n , g _n affine transformations and let þ be a subgroup of the group G of all the in vertible affine transformations, closed relative to G. Suppose that g_n and where Z is nonsingular. The behaviour of _n = h _n g _n ^–1 as n is discussed first. The results are used then to prove that if for all t(0, ), where h _n þ and Z ₁ is nonsingular and nonsymmetric with respect to þ then H, for all t(0,) and is a continuous homomorphism of the multiplicative group of (0, ) into þ. The explicit forms of the possible are shown.     

Über Stabilität und Isoliertheit der Lösungen des Klassischen Plateauproblems

Zusammenfassung Satz 10.10 ist das Ergebnis dieser Arbeit. Wenn wir auch hier nicht zeigen können, daß alle Minimalflächen isolierte Lösungen des ihnen zugeordneten Plateau-problems sind, so haben wir doch in der Menge allerC -Minimalflächen eineC -offene undC -dichte Teilmenge, nämlich die strikt typischen Minimalflächen gefunden, die in der Tat diese Eigenschaft haben.Um von einer Minimalflächex, die von einer regulärenC -Kurveg berandet wird, zu entscheiden, ob sie strikt typisch ist, sind zwei Dinge notwendig:Man muß den komplexen Gradienten vonx berechnen und prüfen, ob die in ihr auftretenden Funktionen die verschiedenen Bedingungen 4.7, 5.1 und 7.7 erfüllt. Außerdem muß man prüfen, ob der Raum der Jacobifelder zux bezüglichg minimale Dimension im Sinne von [1] hat. Es wäre natürlich wünschenswert, daß gezeigt wird, daß fast alle Randkurveng:g: B ³ nur Lösungen des Plateau-problems haben, die ihrerseits strikt typische Minimalflächen sind.Dieses Problem jedoch kann man in einem Zusammenhang mit dem Sardschen Theorem sehen. In einem gewissen Sinne sind die strikt typischen Minimalflächen die regulären Punkte für die Zuordnung der Minimalflächen zu ihren Randkurven. In diesem Sinne wäre dann eine Randkurveg ein regulärer Wert für diese Abbildung , wenng nur strikt typische Minimalflächen berandet. Es ist aber ein zusätzliches Problem, aus der Existenz von vielen regulären Punkten für die Abbildung auf die Existenz vieler regulärer Werte von zu schließen.     

Generalization of some classical inequalities in the theory of orthogonal series

Let {X_i} _– be a sequence of random variables, E(X_i) 0. If 1, estimates for the -th moments can be derived from known estimates of the -th moment. Here we generalized the Men'shov-Rademacher inequality for =2 for orthonormal X_i, to the case 1 and dependent random variables. The Men'shov-Payley inequality >2 for orthonormal X_i) is generalized for >2 to general random variables. A theorem is also proved that contains both the Erdös -Stechkin theorem and Serfling's theorem withv > 2 for dependent random variables.Translated from Matematicheskie Zametki, Vol. 17, No. 2, pp. 219–230, February, 1975.This article was written while the author was working in the V. A. Steklov Mathematics Institute, Academy of Sciences of the USSR.     

The dirichlet problem for sub-elliptic second order equations

The C-regularity up to the boundary of solutions to the Dirichlet problem: is proved, using a comparison principle of L with a Hörmander's type operator X _j ^* X_j, where is a smooth bounded open subset of Rⁿ, and is a second-order degenerate elliptic operator with smooth coefficients, satisfying the so-called Fefferman-Phong's condition.     

On flag-transitive affine planes of orderq 3

Let be a translation plane of orderq ³,q an odd prime power, whose kern GF(q). Letl be the line at infinity of . LetG be a solvable collineation group of in the linear translation complement, which acts transitively onl , and letH be a maximal normal cyclic subgroup ofG. Then the restriction ofH onl acts semiregularly onl and {1, 2, 3, 6}, where is the restriction ofG onl (ifq –1(mod 3), then {1, 2}). Ifq {3, 5} and {1, 2}, then is determined completely, using a computer.     

Minimal Surfaces with an Elastic Boundary

LetD:= { C ³ (

Deformation of a submanifold in an Euclidean space with fixed Gauss image

In a previous paper [4] the present author studied a C mapping where M is an m-dimensional C manifold, I is some interval and for each tI the mapping is an immersion satisfying the following conditions, (i) The Gauss map associated with the immersion is regular. (ii) The Gauss image of the immersed submanifold is fixed against t for each point p of M. Such a mapping was called an admissible deformation. The purpose of the present paper is to give results obtained since then.     

The commutant of rationally cyclic subnormal operators and rational approximation

Let denote the set of analytic bounded point evaluations forR ^q (K, ). Assume that . In this paper, we first show that if is a finitely connected domain and if the evaluation map fromR ^q (K, )L () toH () is surjective, then | is absolutely continuous with respect to harmonic measure for . This generalizes Olin and Yang's corresponding result for polynomials and the proof we present here is simpler. We also provide an example that shows this absolute continuity property fails in general when is an infinitely connected domain. In the second part, we then offer a solution to a problem of Conway and Elias.     

A note on the optimal quadrature inH p

Summary The existence of optimal nodes with preassigned multiplicities is proved for the Hardy spacesH ^p (1<p<). This is then used to show that the exact order of convergence for the optimal qudrature formula withN nodes (including multiplicity) is where 1/p+1/q=1 and 1p.     

On the asymptotic and invariantσ-algebras of random walks on locally compact groups

Summary Consider a random walk of law on a locally compact second countable groupG. Let the starting measure be equivalent to the Haar measure and denote byQ the corresponding Markov measure on the space of pathsG . We study the relation between the spacesL (G , ^a ,Q) andL (G , ⁱ ,Q) where ^a and ⁱ stand for the asymptotic and invariant -algebras, respectively. We obtain a factorizationL (G , ^a ,Q) L (G , ⁱ ,Q)L (C) whereC is a cyclic group whose order (finite or infinite) coincides with the period of the Markov shift and is determined by the asymptotic behaviour of the convolution powers ⁿ.