A Remark on Bernstein,Prokhorov, Bennett,Hoeffding, and Talagrand Inequalities

Let X ₁,..., X_n be independent random variables such that {X_j 1}=1 and E X _j=0 for all j. We prove an upper bound for the tail probabilities of the sum M _n=X₁+...+ X_n. Namely, we prove the inequality {M _nx} 3.7 {S_n x}, where S _n=₁+...+ _n is a sum of centered independent identically distributed Bernoulli random variables such that E S _n ² =ME M _n ² and {_k=1}=E S _n ² /(n+E S _n ² ) for all k (we call a random variable Bernoulli if it assumes at most two values). The inequality holds for x at which the survival function x{S _nx} has a jump down. For remaining x, the inequality still holds provided that we interpolate the function between the adjacent jump points linearly or log-linearly. If necessary, in order to estimate {S _nx} one can use special bounds for binomial probabilities. Up to the factor at most 2.375, the inequality is final. The inequality improves the classical Bernstein, Prokhorov, Bennett, Hoeffding, Talagrand, and other bounds.     

On rationalized H- and co-H-spaces with an appendix on decomposable H- and co-H-spaces

Let R be a subring of the rationals with 1/2, 1/3R; let S _R ⁿ denote the R-local n-sphere and define _R ⁿ :=S _R ⁿ for n odd, _R ⁿ :=S _R ⁿ for n>0 even. An H-space (resp. a 1-conn. co-H-space) is decomposable over R, if it is homotopy equivalent to a weak product of spaces _R ⁿ (resp. to a wedge of R-local spheres). We prove that, if E is grouplike decomposable of finite type over R, the functor [-,E] is determined on finite dim. complexes by the Hopf algebra M^*(E;R); here M^* denotes the unstable cohomotopy functor of H.J. Baues. If C is cogrouplike decomposable over R, the functor [C,-] is determined on 1-conn. R-local spaces by _*(C) as a cogroup in the category of M-Lie algebras. For R = the functor [-,E] is also determined by the Lie algebra _*(E) and [C,-] by the Berstein coalgebra associated to the comultiplication of C.     

Convolution equations in spaces of sequences with an exponential growth constraint

One describes the sets of the solutions of the convolution equations S*x=0 (on the set or on ₊={n:n0}) in the spaces of sequences of the type X=^X(, ), where. One proves that any 1-invariant subspace E,EX, coincides with KezS for some S and, after the Laplace transform can be represented in the form f·A(K_{(, )}), where K_{(, )}={z:kⁿ}_{n z} ^: }+{xX:x_k=0, k(, ), whose zeros do not accumulate to the circumference ¦¦=.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSP, Vol. 149, pp. 107–115, 1986.The author expresses his sincere gratitude to N. K. Nikol'skii for the formulation of the problem and for his interest in the paper.     

Diagonale Netze aus Schmieg- und Krümmungslinien I

In this paper we develop the theory of nets of curves in a regular C^r-2-surface Eⁿ (r1, n2) using the concept C^s-net (of curves); the term diagonal nets of curves defined by W. BPLASCHKE [2] in E² is generalized accordingly. A regular C^r-surface E³ (r2) of negative GAUSSian curvature is called a (C^r-)D_SK-surface if its asymptotic lines (S-lines) and lines of curvature (K-lines) locally form a pair of diagonal nets. For the C³-D_SK-surfaces a criterion is given and distinct categories are determined, in particular all those C³-D_SK-surfaces in which the S- and K-lines can be arranged as (curvilinear) kites, respectively parallelograms and their diagonals.

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Auszugsweise vorgetragen auf der Geometrietagung in Oberwolfach (1.10.l974).     

Spectral properties of Neumann Laplacian of horns

We study the Neumann Laplacian of unbounded regions in ⁿ with cusps at infinity so that the corresponding Dirichlet Laplacian has compact resolvent. Typical of our results is that of the region {(x, y)²x, y|<1} the Neumann Laplacian has absolutely continuous spectrum [0, ) of uniform multiplicity four and an infinity of eigenvaluesE _o<E ₁... and that for the region {(x, y)²y|1}, it has absolutely continuous spectrum [1/4, ) of uniform multiplicity 2 and an infinity of eigenvaluesE ₀=0<E ₁.... We use the Enss theory with a suitable asymptotic dynamics.The second author's research is partially funded under NSF grand number DMS-8801918     

A n 4 -polyhedra with free homology

An A _n ^k -polyhedron is a CW-complex which is (n–1)-connected and of dimension (n+k). We compute an algebraic category which is equivalent to the homotopy category of A _n ² -polyhedra with free homology (n3). This computation and a calculation of the -group _n+3(X) (n3) is used to obtain a complete algebraic homotopy invariant for A _n ⁴ -polyhedra with free homology (n3).As an application we compute the group of self-homotopy equivalences Aut(X) for a bouquet X=P².     

On discrete and continuous norms in Paley-Wiener spaces and consequences for exponential frames

It is well known that for certain sequences {t_n}_n the usual L^p norm ·_p in the Paley-Wiener space PW ^p is equivalent to the discrete norm f_p,{tn}:=( _n=– |f(t_n)|^p)^1/p for 1 p = < and f_,{tn}:=sup _n|f(t_n| for p=). We estimate f_p from above by Cf_p, _n and give an explicit value for C depending only on p, , and characteristic parameters of the sequence {t_n}_n. This includes an explicit lower frame bound in a famous theorem of Duffin and Schaeffer.     

Convergence of random products of compact contractions in Hilbert space

Let {T₁, ..., T_N} be a finite set of linear contraction mappings of a Hilbert space H into itself, and let r be a mapping from the natural numbers N to {1, ..., N}. One can form S_n=T_r(n)...T_r(1) which could be described as a random product of the T_i's. Roughly, the S_n converge strongly in the mean, but additional side conditions are necessary to ensure uniform, strong or weak convergence. We examine contractions with three such conditions. (W): x_n1, Tx_n1 implies (I-T)x_n0 weakly, (S): x_n1, Tx_n1 implies (I-T)x_n0 strongly, and (K): there exists a constant K>0 such that for all x, (I-T)x²K(x²–Tx²).We have three main results in the event that the T_i's are compact contractions. First, if r assumes each value infinitely often, then S_n converges uniformly to the projection Q on the subspace _i= ₁ ^N [x|T_ix=x]. Secondly we prove that for such compact contractions, the three conditions (W), (S), and (K) are equivalent. Finally if S=S(T₁, ..., T_N) denotes the algebraic semigroup generated by the T_i's, then there exists a fixed positive constant K such that each element in S satisfies (K) with that K.     

Some observations on the distribution of values of continued fractions

Summary There have been many studies of the values taken on by continued fractionsK(a _n /1) when its elements are all in a prescribed setE. The set of all values taken on is the limit regionV(E). It has been conjectured that the values inV(E), are taken on with varying probabilities even when the elementsa _n are uniformly distributed overE. In this article, we present the first concrete evidence that this is indeed so. We consider two types of element regions: (A)E is an interval on the real axis. Our best results are for intervals [–(1–), (1–)], 0 <1/2. (B)E is a disk in the complex plane defined byE={z:|z|(1–)}., 0<1/2.     

Numerical application of Euler's series transformation and its generalizations

Summary A nonlinear generalizationÊ _z of Euler's series transformation is compared with the (linear) Euler-Knopp transformationE _z and a twoparametric methodE . It is shown how to applyE orE _, to compute the valuef(z_o) of a functionf from the power series at 0 iff is holomorphic in a half plane or in the cut plane. BothE andE _, are superior toÊ _z . A compact recursive algorithm is given for computingE andE _,.     

Steiner minimal trees for regular polygons

Fifty years ago Jarnik and Kössler showed that a Steiner minimal tree for the vertices of a regularn-gon contains Steiner points for 3 n5 and contains no Steiner point forn=6 andn13. We complete the story by showing that the case for 7n12 is the same asn13. We also show that the set ofn equally spaced points yields the longest Steiner minimal tree among all sets ofn cocircular points on a given circle.     

Diophantine complexity

It is well known that any recursively enumerable set S of natural numbers can be represented by formulas of the following types a S x yx, z₁...Z_n (D (a,x,y,z₁,...,z_n)=0)¦(Davis normal form); aS z₁...z_n (E (a,z₁..., z_n)=E₂(a,z₁,...,z_n)) (exponential diophantine representation); and aS z₁... z_n, (D (a-, z₁),...)Z_n)=0) (diophantine representation). Each of these three representations yields a different measure for the complexity of S as a whole and the complexity of accepting individual members of S. A survey is presented of various results concerning such complexity measures, due to different authors.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeliniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 174, pp. 122–131, 1988.     

Knaster's problem on continuous maps of a sphere into Euclidean space

A survey of known results and additional new ones on Knaster's problem: on the standard sphere S^n–1Rⁿ find configurations of points A₁, , A_k, such that for any continuous map fS^n–1R^m one can find a rotation a of the sphere S^n–1 such that f(a(A₁)==f(a(A_k)) and some problems closely connected with it. We study the connection of Knaster's problem with equivariant mappings, with Dvoretsky's theorem on the existence of an almost spherical section of a multidimensional convex body, and we also study the set {a S0(n)f(a(A₁))==f(a(A_k))} of solutions of Knaster's problem for a fixed configuration of points A₁, , A_kS^n–1 and a map fS^n–1R^m in general position. Unsolved problems are posed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 169–178, 1987.     

On the convergence of orthogonal series to +∞

For any sequence of numbers _n0, _n=1 a _n ² =, a uniformly bounded orthonormal system of continuous functions _n(x) which is complete in L₂ (0, 1), and a sequence of numbers b_n(0< b_na_n) are constructed such that _n=1 ^Emphasis> b_nn(x)= everywhere on (0, 1).Translated from Matematicheskie Zametki, Vol. 11, No. 5, pp. 499–508, May, 1972.     

An Inequality for Tail Probabilities of Martingales with Bounded Differences

Let M _n =X₁+...+X_n be a martingale with bounded differences X_m=M_m-M_m-1 such that {|X_m| _m}=1 with some nonnegative _m. Write ²= ₁ ² + ... + _n ² . We prove the inequalities {M _nx}c(1-(x/)), {M _n x} 1- c(1- (-x/)) with a constant . The result yields sharp inequalities in some models related to the measure concentration phenomena.     

Ein Kontinuitätssatz für k-holomorphe Mengen

Let k denote a non-trivial non-archimedean complete valuated field and X an irreducible k-affinoid space. We discuss the Hartog's domain H^*:=(X×Eⁿ) (U×Eⁿ) where øUX is an affinoid subdomain, Eⁿ is the n-dimensional unit-polydisc over k and Eⁿ is the ringdomain of all z==(z₁,...,z_n)Eⁿ with some coordinate |z_i|=1. The main result is the non-archimedean version of Rothstein's extensiontheorem for analytic subvarieties: Every k-holomorphic subvariety AH^* whose every branch has dimension (dim X + 1) can be extended to a k-holomorphic subvariety X×Eⁿ such that every branch of has dimension (dim X + l).     

Convergence of diagonally stationary sequences in convex optimization

LetX be a real normed linear space,f, f ⁿ, n , be extended real-valued proper closed convex functions onX. A sequence {x _n} inX is called diagonally stationary for {f ⁿ} if for alln there existsx* _n f ⁿ (x _n) such that x* _{n *} 0. Such sequences arise in approximation methods for the problem of minimizingf. Some general convergence results based upon variational convergence theory and appropriate equi-well-posedness are presented.     

Projections on Lp-spaces of polyanalytic functions

The fundamental result: for an arbitrary bounded, simply connected domain in , the subspace L_n,m ^p() of the space L^p(, ) ( is the plane Lebesgue measure, p 1), consisting of the (m, n)-analytic functions in , is complemented in LP(, ) (a function f is said to be (m, n)-analytic if (^m+n/¯Z^mZⁿ)f=0 in ). Consequently, by virtue of a theorem of J. Lindenstrauss and A. Pelczyski, the space L_n,m ^P() is linearly homeomorphic to lP. In particular, for m=n=1 we obtain that the space of all harmonic L^P-functions in is complemented in L^P(, ). This result has been known earlier only for smooth domains.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 190, pp. 15–33, 1991.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).     

Wilson's functional equation on C

Summary We find the complete set of continuous solutionsf, g of Wilson's functional equation _{n = 0} ^{N – 1} f(x + wⁿy) = Nf(x)g(y), x, y C, given a primitiveN ^th rootw of unity.Disregarding the trivial solutionf = 0 andg any complex function, it is known thatg satisfies a version of d'Alembert's functional equation and so has the formg(z) = g (z) = N^–1 _{n = 0} ^{N – 1} E(wⁿz) for some C². HereE ₍₁, ₂)(x + iy) = exp( ₁x + ₂).For fixedg = g the space of solutionsf of Wilson's functional equation can be decomposed into theN isotypic subspaces for the action of Z _N on the continuous functions on C. We prove that ther ^th component, wherer {0, 1, ,N – 1}, of any solution satisfies the signed functional equation _{n = 0} ^{N – 1} f(x + wⁿy)w^nr = Ng(x)f(y), x, y C. We compute the solution spaces of each of these signed equations: They are 1-dimensional and spanned byz _{n = 0} ^{N – 1} w^nr E(wⁿz), except forg = 1 andr 0 where they are spanned by andz ^{N – r}. Adding the components we get the solution of Wilson's equation. Analogous results are obtained with the action ofZ _N on C replaced by that ofSO(2).The case ofg = 0 in the signed equations is special and solved separately both for Z _N andSO(2).