A statistical estimate of the structure of multi-extremal problems

The solution of ak-extremal problem is defined as the set of pairs (x _i ^* , _i),i = 1, ,k, where x _t ^* isi ^th local minimum and _i is the volume of the set of attraction of this minimum. A Bayesian estimate ofk and ( ₁ , , _k ) is constructed.This paper has been written while the author was a CNR visiting professor at the Institute of Mathematics of the Milano University.     

Some mildly wild circles in Sn arising from algebraic K-Theory

We study wild embeddings of S ¹ in S ⁿ which are tame in a sense introduced by Quinn. We show that if is a finitely presented group with H ₁()=H ₂()=0, then any finiteness obstruction K ₀() can be realized on the complement of such an embedded S ¹. We also realize trivially symmetric K _–1() obstructions on the complements of such embeddings. For trivially symmetric , the embeddings constructed are shown to be isotopy homogeneous.     

Foster's Formulas via Probability and the Kirchhoff Index

Building on a probabilistic proof of Fosters first formula given by Tetali (1994), we prove an elementary identity for the expected hitting times of an ergodic N-state Markov chain which yields as a corollary Fosters second formula for electrical networks, namely

A note on smoothed estimating functions

The kernel estimate of regression function in likelihood based models has been studied in Staniswalis (1989,J. Amer. Statist. Assoc.,84, 276–283). The notion of optimal estimation for the nonparametric kernel estimation of semimartingale intensity (t) is proposed. The goal is to arrive at a nonparametric estimate of ₀=(t ₀) for a fixed pointt ₀ [0, 1]. We consider the estimator that is a solution of the smoothed optimal estimating equation is the optimal estimating function as in Thavaneswaran and Thompson (1986,J. Appl. Probab.,23, 409–417).     

Detection of a signal of known shape in a multichannel system

We consider an n-channel signal detection system. Each of its channels may contain (or not contain) a signal. We assume that the signal is a function of known shape observed in the white Gaussian noise of level > 0. Let k be the number of channels containing signals. We study the asymptotically minimax variant of the detection problem as n depending on k and on the relation signal-noise in the channels. Bibliography: 9 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 88–112.This research was partially supported by the Russian Foundation for Basic Research, grants 00-01-00111, 00-15-96019, 02-01-00262, and 02-01-04001.Translated by Yu. I. Ingster and I. A. Suslina.     

Implicit elliptic differential equations

Let be a bounded domain in ⁿ (n 3) having a smooth boundary, letY be a closed, connected and locally connected subset of ^h , letf be a real-valued function defined on × ^h × ^nh ×Y, and letL be a linear, second-order elliptic operator. In this paper, the existence of strong solutionsu W ^2,p (, ^h ) W ₀ ^1,p (, ^h ) (n<p<+) to the implicit elliptic equationf(x, u, Du, Lu)=0, whereu=(u ₁,u ₂, ...,u _h ),Du=(Du ₁,Du ₂, ...,Du _h ) andLu=(Lu ₁,Lu ₂, ...,Lu _h ), is established. The abstract framework where the equation is studied is that of set-valued analysis.Dedicated to Professor G. Pulvirenti on the occasion of his sixtieth birthday     

A geometrical interpretation of the shape invariant for geodesic triangles in complex projective spaces

We show that the shape invariant of a triangle in the complex projective space P ⁿ , see [B], can be obtained by integrating the Kählerian form of P ⁿ over a domain parametrized by geodesics and bounded by a geodesic loop formed with sides of the triangle.The second author was supported by a grant from INDAM-Rome.     

Generalized Cheeger sets related to landslides

We study the maximization problem, among all subsets X of a given domain , of the quotient of the integral in X of a given function f by the integral on the boundary of X of another function g. This is a generalization of the well-known Cheeger problem corresponding to constant functions f,g. The non-constant case is motivated by applications to landslides modeling where the the supremum given by a variational blocking problem appears as a safety coefficient. We prove that this coefficient is equal to the supremum of the shape optimization problem formerly mentioned. For constant data, this amounts to studying the first eigenvalue of the 1-laplacian operator.We prove existence of optimal sets, and give some differential characterization of their internal boundary. We study their symmetry properties using the Steiner symmetrization. In dimension two, we give explicit solutions for data depending only on one variable.

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Résumé. Nous étudions le probléme de maximisation, parmi les ensembles X inclus dans un domaine fixé , du quotient de lintégrale dune fonction donnée f dans X par lintégrale dune autre fonction g sur le bord de X. Il sagit donc dune généralisation du célébre probléme de Cheeger (correspondant au cas f, g, constants). Le cas non-constant est motivé par des applications aux glissements de terrain, oú le supremum donné par un probléme variationnel de blocage, apparaít comme un coefficient de sreté. Nous démontrons que ce coefficient est égal á loptimum du probléme doptimisation de formes mentionné précédemment. Dans le cas de données constantes, cela revient á étudier la premiére valeur propre de lopérateur 1-laplacien.Nous démontrons lexistence densembles optimaux, et donnons une caractérisation différentielle de leur bord intérieur. Nous étudions leur symétrie á laide de la symétrisation de Steiner. En dimension deux, nous exhibons des solutions explicites dans le cas oú les données ne dépendent que dune variable.

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Received: 18 June 2004, Accepted: 12 July 2004, Published online: 10 December 2004Mathematics Subject Classification (2000): 49J40, 49Q10     

Differentiability of the minimal average action as a function of the rotation number

Let be a finite composition of exact twist diffeomorphisms. For any real number , letA() denote the minimal average action of -invariant measures with angular rotation number . We prove thatA() is differentiable at every irrational number and that for generic it is not differentiable at rational , thus verifying conjectures of S. Aubry. Moreover, we show that these results are valid for a variational principleh which satisfies the condition which we have called elsewhere (H). As a consequence, we generalize a result due to Bangert concerning geodesics on a two dimensional torus with an arbitrary, but sufficiently smooth metric.supported by NSF grant no. DMS-8806067.01 and a Guggenheim Fellowship.     

Canonical and Boundary Representations on a Hyperboloid of One Sheet

For the hyperboloid of one sheet X=G/H, G=SO₀(1,2), H=SO₀(1,1), canonical representations R _,, C, =0,1, are defined as the restrictions to G of representations of the overgroup =SO₀(2,2) associated with a cone. They act on the torus containing two copies of X as open G-orbits. We study boundary representations generated by R _,. For some , they contain Jordan blocks. The decomposition of R _, into irreducible constituents includes a finite number (depending on ) of irreducible parts of the boundary representations.     

Relative komplexe quotienten

Let XS be a holomorphic map, and let RX×SX be an equivalence relation. The restriction of R to the fibre ^–1(S) is denoted by R_s. The quotient X/R is called a relative complex quotient, if the quotient map XX/R is holomorphic over S. Two cases are studied: (C) All fibres of are locally R_s-separable (relative Cartan quotient); (R) All fibres of are holomorphically convex, and R_s is given by tke holomorphic functions on ^–1 (s) (relative Remmert quotient).     

Accelerated convergence for the Powell/Hestenes multiplier method

It is known that augmented Lagrangian or multiplier methods for solving constrained optimization problems can be interpreted as techniques for maximizing an augmented dual functionD _c(). For a constantc sufficiently large, by considering maximizing the augmented dual functionD _c() with respect to, it is shown that the Newton iteration for based on maximizingD _c() can be decomposed into taking a Powell/Hestenes iteration followed by a Newton-like correction. Superimposed on the original Powell/Hestenes method, a simple acceleration technique is devised to make use of information from the previous iteration. For problems with only one constraint, the acceleration technique is equivalent to replacing the second (Newton-like) part of the decomposition by a finite difference approximation. Numerical results are presented.     

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 a Parabolic Variational Inequality That Generalizes the Equation of Polytropic Filtration

We obtain conditions for the existence and uniqueness of a solution of a parabolic variational inequality that is a generalization of the equation of polytropic elastic filtration without initial conditions. The class of uniqueness of a solution of this problem consists of functions that increase not faster than e ^–t , > 0, as t –.     

Construction of ɛ-nets

LetS be a set ofn points in the plane and let be a real number, 0<<1. We give a deterministic algorithm, which in timeO(n ^–2 log(1/)+ ^–8) (resp.O(n ^–2 log(1/)+ ^–10) constructs an-netNS of sizeO((1/) (log(1/))²) for intersections ofS with double wedges (resp. triangles); this means that any double wedge (resp. triangle) containing more thatn points ofS contains a point ofN. This givesO(n logn) deterministic preprocessing for the simplex range-counting algorithm of Haussler and Welzl [HW] (in the plane).We also prove that given a setL ofn lines in the plane, we can cut the plane intoO( ^–2) triangles in such a way that no triangle is intersected by more thann lines ofL. We give a deterministic algorithm for this with running timeO(n ^–2 log(1/)). This has numerous applications in various computational geometry problems.     

On functions behaving like additive functions

Let 0 < 1. In the paper we consider the following inequality: |f(x + y) – f(x) – f(y)| min{|f(x + y)|, |f(x) + f(y)|}, wheref: R R. Solutions and continuous solutions of this inequality are investigated. They have similar properties as additive functions, e.g. if the solution is bounded above (below) on a set of positive inner Lebesgue measure then it is continuous. Some sufficient condition for this inequality is also given.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

A nonlinear equation for linear programming

We present a characterization of the normal optimal solution of the linear program given in canonical form max{c ^tx: Ax = b, x 0}. (P) We show thatx ^* is the optimal solution of (P), of minimal norm, if and only if there exists anR > 0 such that, for eachr R, we havex ^* = (rc – A^t_r)₊. Thus, we can findx ^* by solving the following equation for _r A(rc – A^t_r)₊ = b. Moreover,(1/r) _r then converges to a solution of the dual program.On leave from The University of Alberta, Edmonton, Canada. Research partially supported by the National Science and Engineering Research Council of Canada.     

On certain indestructibility of strong cardinals and a question of Hajnal

A model in which strongness of is indestructible under ⁺ -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number of elements of { |2 <}.     

The Smallest Convex Cover for Triangles of Perimeter Two

We find the unique smallest convex region in the plane that contains a congruent copy of every triangle of perimeter two. It is the triangle ABC with AB=2/3, B=60°, and BC1.00285.     

Estimates of Entire Functions of Exponential Type Less than π in Terms of Logarithmic Sums over Real Duffin and Schaeffer Sequences

We give uniform estimates of entire functions of exponential type less than having sufficiently small logarithmic sums over real sequences { _n } satisfying | _n –n|L and _n+1– _n for fixed positive constants L and . We thereby generalize results about logarithmic sums over the set of integers and so-called relatively h-dense sequences.