On Conservative Confidence Intervals

The subject of the paper – (conservative) confidence intervals – originates in applications to auditing. Auditors are interested in upper confidence bounds for an unknown mean for all sample sizes n. The samples are drawn from populations such that often only a few observations are nonzero. The conditional distribution of an observation given that it is nonzero usually has a very irregular shape. However, it can be assumed that observations are bounded. We propose a way to reduce the problem to inequalities for tail probabilities of certain relevant statistics. Note that a traditional approach involving limit theorems forces to impose additional conditions on regularity of samples and leads to approximate or asymptotic bounds. In the case of , as a statistic we can use sample mean, say , and we have to use Hoeffding [7] inequalities, since currently they are the best available. This leads to upper confidence bounds for which are of (asymptotic) size at most in the case of risk =0.05, where is the unknown standard deviation. We have , where is the bound in a model with normally distributed observations. It seems that the bound is very robust and can be improved replacing Hoeffding's inequalities by more refined ones. The commonly used Stringer bound (it is still not known whether it is an upper confidence bound) is of asymptotic size c with equality only for Bernoulli distributions, and the ratio c / can be arbitrary large already for rather simple distributions. Our bounds can involve a priori information (professional judgment of an auditor) of type ₀ or/and ₀, which leads to improvements. Most of the results also hold for sampling without replacement from finite populations. The i.i.d. condition can be replaced by a martingale-type dependence assumption. Finally, the results can be extended to the noni.i.d. case and for settings with several samples.     

Convergence theorems for set-valued amarts and uniform amarts

( ) , — , - . .     

The Darboux-Type Theorems for ∂-Symplectic and ∂-Contact Structures

A -symplectic structure on a complex manifold M of complex dimension2n is given by a smooth -closed (2, 0)-form such that ⁿ is nonvanishing. We prove that a version of the Darboux theorem isvalid for such a structure: locally can be represented as _i=1 ⁿ f _i f _{n + i} for appropriate smooth complex valuedfunctions f ₁, ..., f _2n . We also present a contact version of this theorem.     

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.     

Newton polyhedra and the Bergman kernel

The purpose of this paper is to study singularities of the Bergman kernel at the boundary for pseudoconvex domains of finite type from the viewpoint of the theory of singularities. Under some assumptions on a domain in ⁿ⁺¹ , the Bergman kernel B(z) of takes the form near a boundary point p: where (w,) is some polar coordinates on a nontangential cone with apex at p and means the distance from the boundary. Here admits some asymptotic expansion with respect to the variables ^{1/ m} and log(1/) as 0 on . The values of d _F >0, m _F ₊ and m are determined by geometrical properties of the Newton polyhedron of defining functions of domains and the limit of as 0 on is a positive constant depending only on the Newton principal part of the defining function. Analogous results are obtained in the case of the Szegö kernel. Mathematics Subject Classification (2000):32A25, 32A36, 32T25, 14M25.     

Further results on the covering radii of the Reed-Muller codes

LetR(r, m) by therth order Reed-Muller code of length2 ^m , and let (r, m) be its covering radius. We obtain the following new results on the covering radius ofR(r, m): 1. (r+1,m+2) 2(r, m)+2 if 0rm–2. This improves the successive use of the known inequalities (r+1,m+2)2(r+1,m+1) and (r+1,m+1) (r, m).2.(2, 7)44. Previously best known upper bound for (2, 7) was 46. 3. The covering radius ofR(1,m) inR(m–1,m) is the same as the covering radius ofR(1,m) inR(m–2,m) form4.     

Perturbation of the poles of the scattering matrix under variation of the boundary condition

The behavior of the poles z_n(), n=1,2,... of the scattering matrix of the operatorl u=–u(x), x , (u/n)+(x)u|=0 as 0 is considered. It is proved that |z_n()–z_n|=0(^(1/2)qn), where q_n is the order of the pole of the scattering matrix for the operator ₀u=–u, u/=0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 183–191, 1981.     

Conservative weightings and ear-decompositions of graphs

A subsetJ of edges of a connected undirected graphG=(V, E) is called ajoin if |CJ||C|/2 for every circuitC ofG. Answering a question of P. Solé and Th. Zaslavsky, we derive a min-max formula for the maximum cardinality of a joint ofG. Namely, =(+|V|–1)/2 where denotes the minimum number of edges whose contraction leaves a factor-critical graph.To study these parameters we introduce a new decomposition ofG, interesting for its own sake, whose building blocks are factor-critical graphs and matching-covered bipartite graphs. We prove that the length of such a decomposition is always and show how an optimal join can be constructed as the union of perfect matchings in the building blocks. The proof relies on the Gallai-Edmonds structure theorem and gives rise to a polynomial time algorithm to construct the optima in question.     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

Über die Häufigkeit großer Differenzen konsekutiver Primzahlen

The following theorem is going to be proved. Letp _m be them-th prime and putd _m :=p _m+1–p _m . LetN(,T), 1/21,T3. denote the number of zeros =+i of the Riemann zeta function which fulfill and ||T. Letc2 andh0 be constants such thatN(,T)T ^c(1–) (logT) ^h holds true uniformly in 1/21. Let >0 be given. Then there is some constantK>0 such that      

Asymptotic behaviour of Gaussian random fields

Summary Let X={X(t), t ^N} be a centred Gaussian random field with covariance X(t)X(s)=r(t–s) continuous on ^N×^N and r(0)=1. Let (t,s)=((X(t)–X(s)) ²)^1/2; (t,s) is a pseudometric on ^N. Assume X is -separable. Let D ₁ be the unit cube in ^N and for 0<k, D _k= {x^N: k ^–1 xD₁}, Z(k)=sup{X(t),tD _k}. If X is sample continuous and ¦r(t)¦ =o(1/log¦t¦) as ¦t¦8 then Z(k)-(2Nlogk) ^1/20 as k a.s.     

On Fourier multipliers between Besov spaces with 0 <p o ≤ min{1,p 1}

, 0<0 min{1,p1}.     

Probabilistic Methods for Decomposition Dimension of Graphs

In a graph G, the distance from an edge e to a set FE(G) is the vertex distance from e to F in the line graph L(G). For a decomposition of E(G) into k sets, the distance vector of e is the k-tuple of distances from e to these sets. The decomposition dimension dec(G) of G is the smallest k such that G has a decomposition into k sets so that the distance vectors of the edges are distinct. For the complete graph K _n and the k-dimensional hypercube Q _k , we prove that (2–o(1))lgndec(K _n )(3.2+o(1))lgn and k/lgk dec(Q _k ) (3.17+o(1))k/lgk. The upper bounds use probabilistic methods directly or indirectly. We also prove that random graphs with edge probability p such that p n ^1– for some positive constant have decomposition dimension (lnn) with high probability. Acknowledgments.The authors thank Noga Alon for clarifying and strengthening the results in Sections 3 and 4. Thanks also go to a referee for repeated careful readings and suggestions.AMS classifications: 05C12, 05C35, 05D05, 05D40     

Positivity and Stability for One-Sided Coupled Operator Matrices

Many evolutionary systems can be described by an abstract Cauchy problem governed by an operator matrix. Assuming this problem to be one-sided coupled and well-posed we study in this paper the positivity and the stability of the associated matrix semigroup. The abstract results are illustrated by several examples.     

Lipschitz and darbo conditions for the superposition operator in ideal spaces

Summary In this paper we give necessary and sufficient conditions for the superposition operator Fx(s)=f(s, x(s)) to satisfy a Lipschitz condition Fx₁ - Fx₂kx₁ - x₂ or a Darbo condition (FN)k(N) in ideal spaces of measurable functions, where is the Hausdorff measure of noncompactness. Moreover, we characterize a large class of spaces in which the above mentioned two conditions are equivalent.

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Sunto In questo lavoro diamo delle condizioni necessarie e sufficienti perchè l'operatore di sovrapposizione Fx(s)=f (s, x(s)) soddisfi alla condizione di Lipschitz Fx₁–Fx₂ kx₁–x₂ o quella di Darbo (FN)k(N) in spazi ideali di funzioni misurabili, ove è la misura di non compattezza di Hausdorff. Inoltre, caratterizziamo un'ampia classe di spazi in cui le suddette due condizioni sono equivalenti.

     

Diagonals of positive semigroups

LetE be a Dedekind complete complex Banach lattice and letD denote the diagonal projection from the spaceL _r (E) onto the centerZ(E) ofE. Let {T(t)} _t0 be a positive strongly continuous semigroup of linear operators with generatorA. The first main result is that if the spectral bounds(A) equals to zero, then the functionD(T(t)) is a center valuedp-function. The second main result is that if for >0 the diagonalD(R(, A)) of the resolvent operatorR(, A) is strictly positive, then (D(R(, A))) ^–1 is a center valued Bernstein function. As an application of these results it follows that the order limit lim₀D(R(,A)) exists inZ(E) and equals the order limit lim _m D((R(, A)) ^m ) for any >0.     

Four-dimensional compact projective planes with doubly transitive action on the fixed pencil

We consider a four-dimensional compact projective plane =( , ) whose collineation group is six-dimensional and solvable with a nilradical N isomorphic to Nil × R, where Nil denotes the three-dimensional, simply connected, non-Abelian, nilpotent Lie group. We assume that fixes a flag pW, acts transitively on _p \{W}, and fixes no point in the set W{p}. We study the actions of and N on and on the pencil _p \{W}, in the case that does not contain a three-dimensional elation group. In the special situation that acts doubly transitively on _p {W}, we will determine all possible planes . There are exactly two series of such planes.     

Probabilities of moderate deviations

Let X₁, X₂, ... be a sequence of independent, identically distributed random variables, and let. The rate of convergence of probabilities of the form andis studied for any > ₀ and some r and ₀. Moreover, necessary and sufficient conditions are given that the relations be satisfied uniformly with respect to x in the region 0 x clog n, where and c are some positive constants, and. Local limit theorems are also presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituts im. V. A. Steklova AN SSSR, Vol. 85, pp. 6–16, 1979.     

On the distribution of sequences connected with digit-representation

Let N=e(s)b(s)+e(s–1)b(s–1)+...+e(1)b(1)+e(0) be the digit representation of the integer N to base b or to -scale, that is with respect to the best approximation denominators of an irrational number . Let f:N₀ Z with f(0)=0 be an arbitrary function and r(0),r(1),... be an arbitrary sequence of integers and F(N):=f(e(s))r(s)+...+f(e (1))r(1)+f(e(0))r(0). Conditions for the uniform distribution modulo one of the sequence {F(N)x}_NN, x are given.     

Perron-frobenius theory for Banach spaces with a hyperbolic cone

If X is a real Banach space, then the inequality x defines so-called hyperbolic cone in E=X. We develop a relevant version of Perron-Frobenius-Krein-Rutman theory.