On the extremality of the polynomials used for obtaining the best known upper bounds for the kissing numbers

We prove that the polynomials used for obtaining the best known upper bounds for some kissing numbers (the maximum number of nonoverlapping unit spheres that can touch a unit sphere in n dimensions) are best between the polynomials of the same or lower degree. We give also some extremal polynomials we have obtained using a method proposed in [4]. The upper bounds obtained in this way are slightly better than these from [1]. However the improvements are not in the integer part for dimensionsn 18.     

The maximum asymptotic bias of S-estimates for regression over the neighborhoods defined by certain special capacities

The maximum asymptotic bias of an S-estimate for regression in the linear model is evaluated over the neighborhoods (called (c,γ)-neighborhoods) defined by certain special capacities, and its lower and upper bounds are derived. As special cases, the (c,γ)-neighborhoods include those in terms of -contamination, total variation distance and Rieder's (,δ)-contamination. It is shown that when the model distribution is normal and the (,δ)-contamination neighborhood is adopted, the lower and upper bounds of an S-estimate (including the LMS-estimate) based on a jump function coincide with the maximum asymptotic bias. The tables of the maximum asymptotic bias of the LMS-estimate are given. These results are an extension of the corresponding ones due to Martin et al. (Ann. Statist. 17 (1989) 1608), who used -contamination neighborhoods.     

Critical Points of Real Polynomials and Topology of Real Algebraic T-Surfaces

The paper is devoted to a special class of real polynomials, so-called T-polynomials, which arise in the combinatorial version of the Viro theorem. We study the relation between the numbers of real critical points of a given index of a T-polynomial and the combinatorics of lattice triangulations of Newton polytopes. We obtain upper bounds for the numbers of extrema and saddles of generic T-polynomials of a given degree in three variables, and derive from them upper bounds for Betti numbers of real algebraic surfaces in defined by T-polynomials. The latter upper bounds are stronger than the known upper bounds for arbitrary real algebraic surfaces in . Another result is the existence of an asymptotically maximal family of real polynomials of degree min three variables with 31m ³/36 + O(m ²) saddle points.     

Von Neumann Categories and Extended L2 Cohomology

In this paper we suggest a new general formalism for studying the L2 invariants of polyhedra and manifolds. First, we examine generality in which one may apply the construction of the extended Abelian category, which was earlier suggested by the author using the ideas of P. Freyd. This leads to the notions of a finite von Neumann category and a trace on such a category. Given a finite von Neumann category, we study the extended L2 homology and cohomology theories with values in the Abelian extension. Any trace on the initial category produces numerical invariants – the von Neumann dimension and the Novikov–Shubin numbers. Thus, we obtain the local versions of the Novikov–Shubin invariants, localized at different traces. In the Abelian case this localization can be made more geometric: we show that any torsion object determines a divisor – a closed subspace of the space of the parameters. The divisors of torsion objects together with the information produced by the local Novikov–Shubin invariants may be used to study multiplicities of intersections of algebraic and analytic varieties (we discuss here only simple examples demonstrating this possibility). We compute explicitly the divisors and the von Neumann dimensions of the extended L2 cohomology in the real analytic situation. We also give general formulae for the extended L2 cohomology of a mapping torus. Finally, we show how one can define a De Rham version of the extended cohomology and prove a De Rhamtype theorem.     

Variants of recognition problems for modular forms

We consider the problem of distinguishing two modular forms, or two elliptic curves, by looking at the coefficients of their L-functions for small primes (compared to their conductor). Using analytic methods based on large-sieve type inequalities we give various upper bounds on the number of forms having the first few coefficients equal to those of a fixed one. In addition, we consider similar questions of recognizing symmetric squares and CM forms from the behavior of small primes.Received: 30 August 2004     

Comparison theorem and estimates for transition probability densities of diffusion processes

We establish several comparison theorems for the transition probability density p _b (x,t,y) of Brownian motion with drift b, and deduce explicit, sharp lower and upper bounds for p _b (x,t,y) in terms of the norms of the vector field b. The main results are obtained through carefully estimating the mixed moments of Bessel processes. All constants are explicit in our lower and upper bounds, which is different from most of the previous estimates, and is important in many applications for example in statistical inferences for diffusion processes.Research partially supported by N.S.F. Grants DMS-0203823, and by Doctoral Program Fundation of the Ministry of Education of China, Grant No. 20020269015. Mathematics Subject Classification (2000):Primary: 60H10, 60H30; Secondary: 35K05     

Bracketing numbers for axis-parallel boxes and applications to geometric discrepancy

In the first part of this paper we derive lower bounds and constructive upper bounds for the bracketing numbers of anchored and unanchored axis-parallel boxes in the d$d$-dimensional unit cube.     

Performance of an ATM multiplexer queue in the fluid approximation using the Beneš approach

This paper describes a new method for evaluating the queue length distribution in an ATM multiplexer assuming the cell arrival process can be assimilated to a variable rate fluid input. The method is based on a result due initially to Bene allowing the analysis of queues with general input. Its extension to fluid input systems is considered here in the case of a superposition of on/off sources. We derive an upper bound on the complementary queue length distribution. The method is most easily applied in the case of Poisson burst arrivals (infinite sources model). In this case, we derive analytic expressions for the tail of the queue length distribution. A corrective factor is deduced to convert the upper bounds to good approximations. Numerical results justify the accuracy of the method and demonstrate the impact of certain traffic characteristics on queue performance.     

Christoffel functions,orthogonal polynomials,and Nevai's conjecture for Freud weights

We obtain upper and lower bounds for Christoffel functions for Freud weights by relatively new methods, including a new way to estimate discretization of potentials. We then deduce bounds for orthogonal polynomials on thereby largely resolving a 1976 conjecture of P. Nevai. For example, let W:=e ^–Q, whereQ: is even and continuous in, Q^" is continuous in (0, ) andQ ^'>0 in (0, ), while, for someA, B,

Inequality of Petrović and Giaccardi for Convex Function of Higher Order

In this paper we use Lidstone polynomials to prove further generalization of Giaccardi generalization of the well-known Petrovis inequality.     

Almost π-Lattices

In this paper we establish some conditions for an almost -domain to be a -domain. Next -lattices satisfying the union condition on primes are characterized. Using these results, some new characterizations are given for -rings.     

Combinatorial bounds on Hilbert functions of fat points in projective space

We study Hilbert functions of certain non-reduced schemes A supported at finite sets of points in , in particular, fat point schemes. We give combinatorially defined upper and lower bounds for the Hilbert function of A using nothing more than the multiplicities of the points and information about which subsets of the points are linearly dependent. When N=2, we give these bounds explicitly and we give a sufficient criterion for the upper and lower bounds to be equal. When this criterion is satisfied, we give both a simple formula for the Hilbert function and combinatorially defined upper and lower bounds on the graded Betti numbers for the ideal I_A defining A, generalizing results of Geramita et al. (2006) [16]. We obtain the exact Hilbert functions and graded Betti numbers for many families of examples, interesting combinatorially, geometrically, and algebraically. Our method works in any characteristic.     

Interpolation by Rational Functions with Nodes on the Unit Circle

From the Erds–Turán theorem, it is known that if f is a continuous function on and L _n (f, z) denotes the unique Laurent polynomial interpolating f at the (2 n + 1)th roots of unity, then Several years later, Walsh and Sharma produced similar result but taking into consideration a function analytic in and continuous on and making use of algebraic interpolating polynomials in the roots of unity.In this paper, the above results will be generalized in two directions. On the one hand, more general rational functions than polynomials or Laurent polynomials will be used as interpolants and, on the other hand, the interpolation points will be zeros of certain para-orthogonal functions with respect to a given measure on .     

Improvement of the Delsarte Bound for τ-Designs When It Is Not the Best Bound Possible

In this paper the problem for improvement of the Delsarte bound for -designs is investigated. Two main results are presented. Firstly, necessary and sufficient conditions for improving the bound are proved. We define test functions with the property that they are negative if and only if the Delsarte bound D(phmmat;, ) can be improved by linear programming. Then we investigate the infinite polynomial metric spaces and give exact intervals, when the Delsarte bound is not the best linear programming bound possible. Secondly, we derive a new bound for the infinite PMS. Analytical forms of the extremal polynomials of degree + 2 for non-antipodal PMS and of degree + 3 for antipodal PMS are given. The new bound is investigated in different asymptotical processes for infinite PMS. When and n grow simultaneously to infinity our bound is better than Delsarte bound.     

Petits discriminants des corps de nombres totalement imaginaires de degré 8

In this paper we show that there are, up to isomorphisms, exactly 15 totally imaginary eight-degree number fields having a discriminant smaller than 1656110. The proof of this result was obtained in the following manner: For each totally imaginary eight-degree number field with a discriminant smaller than 1656110, we constructed a polynomial, one root of which was a generator of the number field. In order to circumscribe the number of polynomials to be studied, we used, on the one hand, methods issuing from the geometry of numbers and on the other, the method developed by Odlyzko, Poitou, and Serre for the determination of lower bounds for discriminants.     

Connections between Interval and Unit Circle for Sobolev Orthogonal Polynomials. Strong Asymptotics on the Real Line

In this paper we show the connection between Sobolev orthogonal Laurent polynomials on the unit circle and Sobolev orthogonal polynomials on a bounded interval of the real line. As a consequence we deduce the strong outer asymptotics for Sobolev orthogonal polynomials with respect to the inner product

Improved Analytical Bounds for Gambler’s Ruin Probabilities

Given integer-valued wagers Feller (1968) has established upper and lower bounds on the probability of ruin, which often turn out to be very close to each other. However, the exact calculation of these bounds depends on the unique non-trivial positive root of the equation () = 1, where is the probability generating function for the wager. In the situation of incomplete information about the distribution of the wager, one is interested in bounds depending only on the first few moments of the wager. Ethier and Khoshnevisan (2002) derive bounds depending explicitly on the first four moments. However, these bounds do not make the best possible use of the available information. Based on the theory of s-convex extremal random variables among arithmetic and real random variables, a substantial improvement can be given. By fixed first four moments of the wager, the obtained new bounds are nearly perfect analytical approximations to the exact bounds of Feller.AMS 2000 Subject Classification: 60E15, 60G40, 91A60     

Integer valued polynomials over a number field

A number field is called a Pólya field if the module of integer valued polynomials over that field is generated by (f_i) _i=0 over the ring of integers, with deg(f_i)=i, i=0, 1, 2,... In this paper bounds on the class numbers and on the number of ramified primes in Pólya fields are derived.     

Integrating heuristic information into exact methods: The case of the vertex p-centre problem

We solve the vertex p-centre problem optimally using an exact method that considers both upper and lower bounds as part of its search engine. Tight upper bounds are generated quickly via an efficient three-level heuristic, which are then used to derive potential ‘lower bounds’ accordingly. These two pieces of information when used together make our chosen exact method more efficient at obtaining optimal solutions relatively quickly. The proposed implementation produced excellent results when tested on the OR Library data set. This integrated approach can be adopted for those exact methods that consider both upper and lower bounds within their search engine and hence provide a wider spectrum of applicability in other hard combinatorial problems.     

A geometric analysis of Renegar’s condition number,and its interplay with conic curvature

For a conic linear system of the form Ax ∈K, K a convex cone, several condition measures have been extensively studied in the last dozen years. Among these, Renegar’s condition number is arguably the most prominent for its relation to data perturbation, error bounds, problem geometry, and computational complexity of algorithms. Nonetheless, is a representation-dependent measure which is usually difficult to interpret and may lead to overly conservative bounds of computational complexity and/or geometric quantities associated with the set of feasible solutions. Herein we show that Renegar’s condition number is bounded from above and below by certain purely geometric quantities associated with A and K; furthermore our bounds highlight the role of the singular values of A and their relationship with the condition number. Moreover, by using the notion of conic curvature, we show how Renegar’s condition number can be used to provide both lower and upper bounds on the width of the set of feasible solutions. This complements the literature where only lower bounds have heretofore been developed.