A category of kernels for equivariant factorizations and its implications for Hodge theory

Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesàro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.     

An exact duality theory for semidefinite programming and its complexity implications

In this paper, an exact dual is derived for Semidefinite Programming (SDP), for which strong duality properties hold without any regularity assumptions. Its main features are: (i) The new dual is an explicit semidefinite program with polynomially many variables and polynomial size coefficient bitlengths. (ii) If the primal is feasible, then it is bounded if and only if the dual is feasible. (iii) When the primal is feasible and bounded, then its optimum value equals that of the dual, or in other words, there is no duality gap. Further, the dual attains this common optimum value. (iv) It yields a precise theorem of the alternative for semidefinite inequality systems, i.e. a characterization of theinfeasibility of a semidefinite inequality in terms of thefeasibility of another polynomial size semidefinite inequality. The standard duality for linear programming satisfies all of the above features, but no such explicit gap-free dual program of polynomial size was previously known for SDP, without Slater-like conditions being assumed. The dual is then applied to derive certain complexity results for SDP. The decision problem of Semidefinite Feasibility (SDFP), which asks to determine if a given semidefinite inequality system is feasible, is the central problem of interest, he complexity of SDFP is unknown, but we show the following: (i) In the Turing machine model, the membership or nonmembership of SDFP in NP and Co-NP is simultaneous; hence SDFP is not NP-Complete unless NP=Co-NP. (ii) In the real number model of Blum, Shub and Smale, SDFP is in NP∩CoNP.     

A review on some contributions to perturbation theory, singular limits and well-posedness

In the beginning of the 1990s we devoted a sequence of papers to perturbation theory, singular limits and well-posedness problems. In particular, the strong well-posedness of the initial-boundary value problem for the compressible Euler equations was demonstrate for the first time. Our method also allowed singular limit results in the strong norm, even under assumptions weaker than the current ones in the literature (where the strong norm is not reached). It is worth noting that, until now, the above method and results have not been substantially improved. Hence an introduction to it still looks timely. Actually, in a forthcoming paper, by returning to this method, we improve (in a very substantial way) some important results recently appeared in the literature.     

A theta function identity and its implications

In this paper we prove a general theta function identity with four parameters by employing the complex variable theory of elliptic functions. This identity plays a central role for the cubic theta function identities. We use this identity to re-derive some important identities of Hirschhorn, Garvan and Borwein about cubic theta functions. We also prove some other cubic theta function identities. A new representation for is given. The proofs are self-contained and elementary.

     

Extreme value theory in some statistical analysis of genomic sequences

Because similarities in biological sequences often suggest similarities in structures and functions, profile searches using multiple alignment of families of related biological sequences provide useful hints for starting points for experimental investigations in molecular biology. Strategies are formulated for determining statistical significance of scores obtained by searching multiple alignment profiles with databanks, while accommodating for gaps in the profile. The methodology is validated with derivation of asymptotic distribution of the maximum of profile scores, even under weakly dependence conditions. Simulation studies show the proposed method is adequate for moderate sample sizes. The methodology is illustrated with an immunoglobulin protein domain study example.     

A general theory of some positive dependence notions

A general theory of concepts of positive dependence, which are weaker than association but stronger than orthant dependence, is developed. A random vector X is associated if and only if $\text{P(}\text{X}\text{∈ A ? B) ≥ P(}\text{X}\text{∈ A) P(}\text{X}\text{∈ B)}$ for all open upper sets A and B. By requiring the above inequality to hold only for some open upper sets A and B various notions of positive dependence which are weaker than association are obtained. First a general theory is given and then the results are specialized to some concepts of a particular interest. Various properties and interrelationships are derived and some applications are discussed.     

Percolation theory and some applications

The goal of the survey is to present mathematically rigorous results obtained in the recent years in the new mathematical discipline: percolation theory, which is on the border between the theory of random fields, stochastic geometry, and mathematical physics. Here not only classical percolation schemes are considered (bond and site problems on lattices) but also various generalizations that arose in connection with actual physical applications.Translated from Itogi Nauki i Tekhniki, Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika, Vol. 24, pp. 53–110, 1986.     

Asymptotic analysis of some nonlinear problems using Hopf-Cole transform and spectral theory

We consider initial boundary value problems for certain nonlinear scalar parabolic equations. A formula for the unique classical solution by Hopf-Cole transformations is obtained and the asymptotic behaviour of the solution as time goes to ∞ is studied.     

On some applications of graph theory,I

In a series of papers, of which the present one is Part I, it is shown that solutions to a variety of problems in distance geometry, potential theory and theory of metric spaces are provided by appropriate applications of graph theoretic results.     

On some applications of graph theory,I

In a series of papers, of which the present one is Part I, it is shown that solutions to a variety of problems in distance geometry, potential theory and theory of metric spaces are provided by appropriate applications of graph theoretic results.     

A new approach to the Morse-Conley theory and some applications

Summary We present a new approach to the Morse theory which is based on a generalization of the Conley index to non locally compact spaces. The variant of the Morse theory which we obtain seems suitable for the applications to nonlinear functionals analysis. Some applications are given here; they mainly concern the study of periodic solutions of second order Hamiltonian systems. Other applications are in some quoted papers.     

General multi-sum transformations and some implications

We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence \(\{g(k)\}\)), to be reduced to an infinite q-product times a single basic hypergeometric sum. Various applications are given, including summation formulae for some q orthogonal polynomials and various multi-sums that are expressible as infinite products.     

On the (non)-existence of some cubature formulas: gaps between a theory and its applications

An overview of the lower bounds for the number of points for integrals over the square and triangle is presented. This is compared with the number of points in known cubature formulae.     

About q-approximations and q-hulls over a Noetherian ring,some refinements of the Auslander-Bridger theory

We revisit the notion of q-approximations for a module over a Noetherian ring, originally due to Auslander and Bridger without a name and rediscovered later by Evans and Griffith. We introduce a somewhat symmetric notion of q-hull and provide existence theorems for both q-approximations and q-hulls. The main feature here are existence theorems and characterizations of minimal such ones when the ring is local. The q-hulls being close to the morphism obtained by Auslander and Bridger in their “approximation theorem”, we also obtain for the latter a minimal statement in the case when the ring is local.     

A duality theory for some non-convex functions of matrices

Abstract We study a special class of non-convex functions which appear in nonlinear elasticity, and we prove that they have a well-defined Legendre transform. Several examples are given, and an application to a nonlinear eigenvalue problem. Keywords: Duality, Legendre transform, Nonlinear elasticity Mathematics Subject Classification (2000): 05C38, 15A15, 05A15, 15A18     

Tolerance space theory and some applications

The paper deals with the notion of tolerance space (introduced by E. C. Zeeman, but discerned earlier by H. Poincaré), which formalizes the idea of resemblance. The category of tolerance spaces is described, their homology and homotopy theories developed. Applications include almost-fixed point theorems, almost-solution existence theorems for difference schemes and the three-channel principle (a general theorem on multichannel data transmission).