Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ballB⊂C ⁿ with its relative logarithmic capacity inC ⁿ with respect to the same ballB. An analogous comparison inequality for Borel subsets of euclidean balls of any generic real subspace ofC ⁿ is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of plurisubharmonic lemniscates associated to the Lelong class of plurisubharmonic functions of logarithmic singularities at infinity onC ⁿ as well as the Cegrell class of plurisubharmonic functions of bounded Monge-Ampère mass on a hyperconvex domain Ω⊂(C ⁿ . Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of plurisubharmonic functions. This work was partially supported by the programmes PARS MI 07 and AI.MA 180.     

Discrete isoperimetric and Poincaré-type inequalities

We study some discrete isoperimetric and Poincaré-type inequalities for product probability measures μ ⁿ on the discrete cube {0, 1} ⁿ and on the lattice Z ⁿ . In particular we prove sharp lower estimates for the product measures of boundaries of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions μ on Z which satisfy these inequalities on Z ⁿ . The class of these distributions can be described by a certain class of monotone transforms of the two-sided exponential measure. A similar characterization of distributions on R which satisfy Poincaré inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes. Received: 30 April 1997 / Revised version: 5 June 1998     

Diffeomorphic types of complements of nice point arrangements in CP<Superscript><Emphasis Type="Italic">l</Emphasis></Superscript>

We use a new method to study arrangement in CP ^l , define a class of nice point arrangements and show that if two nice point arrangements have the same combinatorics, then their complements are diffeomorphic to each other. In particular, the moduli space of nice point arrangements with same combinatorics in CP ^l is connected. It generalizes the result on point arrangements in CP ³ to point arrangements in CP ^l for any l.     

The decidability of some classes of Stone algebras

Iterating the triple construction applied consecutively to n Boolean algebras, we introduce two finitely axiomatizable subclasses SAⁱ_n{{\bf SA}^{\rm i}_n} and SA^s_n{{\bf SA}^{\rm s}_n} of the class SA _n of all Stone algebras of degree n with all the structure homomorphisms in their P-product representation injective or surjective, respectively. Then the class of all Post algebras of degree n is definitionally equivalent to the intersection SAⁱ_n ?SA^s_n{{\bf SA}^{\rm i}_{n} \cap {\bf SA}^{\rm s}_{n}}. We show that for each n ≥ 2 the class SAⁱ_n{{\bf SA}^{\rm i}_n} is hereditarily undecidable while SA^s_n{{\bf SA}^{\rm s}_{n}} is decidable. As a consequence we obtain several (un)decidability results for various axiomatic classes of Stone algebras: among them the decidability of the class of all Stone algebras of degree n which are dually pseudocomplemented and form a dual Stone algebra under the operation of dual pseudocomplement, and undecidability of the class of all Stone algebras with Boolean dense set. On the other hand, the class of all finite members in SA _n is decidable.     

Lipschitz classes on local fields

The Lipschitz class Lipαon a local field K is defined in this note,and the equivalent relationship between the Lipschitz class Lipαand the Holder type space C~α(K)is proved.Then,those important characteristics on the Euclidean space R~n and the local field K are compared,so that one may interpret the essential differences between the analyses on R~n and K.Finally,the Cantor type fractal functionθ(x)is showed in the Lipschitz class Lip(m,K),m＜(ln 2/ln 3).     

Fully copositive matrices

The class of fully copositive (C ₀ ^f ) matrices introduced in [G.S.R. Murthy, T. Parthasarathy, SIAM Journal on Matrix Analysis and Applications 16 (4) (1995) 1268–1286] is a subclass of fully semimonotone matrices and contains the class of positive semidefinite matrices. It is shown that fully copositive matrices within the class ofQ ₀-matrices areP ₀-matrices. As a corollary of this main result, we establish that a bisymmetricQ ₀-matrix is positive semidefinite if, and only if, it is fully copositive. Another important result of the paper is a constructive characterization ofQ ₀-matrices within the class ofC ₀ ^f . While establishing this characterization, it will be shown that Graves's principal pivoting method of solving Linear Complementarity Problems (LCPs) with positive semidefinite matrices is also applicable toC ₀ ^f Q ₀ class. As a byproduct of this characterization, we observe that aC ₀ ^f -matrix is inQ ₀ if, and only if, it is completelyQ ₀. Also, from Aganagic and Cottle's [M. Aganagic, R.W. Cottle, Mathematical Programming 37 (1987) 223–231] result, it is observed that LCPs arising fromC ₀ ^f Q ₀ class can be processed by Lemke's algorithm. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Corresponding author.     

A Character Theoretic Condition for F(G) > 1

ABSTRACT

If G is a nontrivial finite group with the property that χ (1) ² divides bi| G | for all irreducible (complex) characters χ , then G has a nontrivial normal abelian subgroup.

     

An optimal multivariate spline method of recovery of twice differentiable functions

A continuous quadratic polynomial spline of several variables is constructed. It solves the optimal recovery problem studied by V.F. Babenko, S.V. Borodachov, and D.S. Skorokhodov for the class of functions defined on a convex polytope in R ^d , whose second derivatives in any direction are uniformly bounded, and for a periodic analogue of this class. The information consists of the values and gradients of the function at some finite set of nodes in R ^d .     

Partial regularity of mass-minimizing rectifiable sections

Let B be a fiber bundle with compact fiber F over a compact Riemannian n-manifold M ⁿ. There is a natural Riemannian metric on the total space B consistent with the metric on M. With respect to that metric, the volume of a rectifiable section σ: M → B is the mass of the image σ(M) as a rectifiable n-current in B. Theorem 1. For any homology class of sections of B, there is a mass-minimizing rectifiable current T representing that homology class which is the graph of a C¹ section on an open dense subset of M. Mathematics Subject Classifications (2000): 49F20, 49F22, 49F10, 58A25, 53C42, 53C65.     

Boolean Metric Spaces and Boolean Algebraic Varieties

Abstract

The concepts of Boolean metric space and convex combination are used to characterize polynomial maps A ⁿ ?→?A ^m in a class of commutative Von Neumann regular rings including p-rings, that we have called CFG-rings. In those rings, the study of the category of algebraic varieties (i.e., sets of solutions to a finite number of polynomial equations with polynomial maps as morphisms) is equivalent to the study of a class of Boolean metric spaces, that we call here CFG-spaces.     

Bahadur representation of nonparametric M-estimators for spatial processes

Under some mild conditions, we establish a strong Bahadur representation of a general class of nonparametric local linear M-estimators for mixing processes on a random field. If the socalled optimal bandwidth hn = O（｜n｜^-1/5）, n ∈ Z^d, is chosen, then the remainder rates in the Bahadur representation for the local M-estimators of the regression function and its derivative are of order O（｜n｜^-4/5 log ｜n｜）. Moreover, we derive some asymptotic properties for the nonparametric local linear M-estimators as applications of our result.     

Lacunary Series in Weighted Hyperholomorphic B p,q (G) Spaces

The goal of this article is two-fold. First, we consider a class of hyperholomorphic functions, the so called B ^{p, q} (G) space in ?³. Then, we use the B ^{p, q} (G) space to characterize the hyperholomorphic α-Bloch space. Second, we obtain characterizations of the weighted hyperholomorphic B ^{p, q} (G)-functions by the coefficients of certain lacunary series expansions in Clifford Analysis.     

Weingarten surfaces and nonlinear partial differential equations

The sine-Gordon equation has been known for a long time as the equation satisfied by the angle between the two asymptotic lines on a surface inR ³ with constant Gauss curvature –1. In this paper, we consider the following question: Does any other soliton equation have a similar geometric interpretation? A method for finding all the equations that have such an interpretation using Weingarten surfaces inR ³ is given. It is proved that the sine-Gordon equation is the only partial differential equation describing a class of Weingarten surfaces inR ³ and having a geometricso(3)-scattering system. Moreover, it is shown that the elliptic Liouville equation and the elliptic sinh-Gordon equation are the only partial differential equations describing classes of Weingarten surfaces inR ³ and having geometricso(3,C)-scattering systems.     

Singular sets of convex bodies and surfaces with generalized curvatures

In this paper we consider generalized surfaces with curvature measures and we study the properties of those k-dimensional subsets Σ ^k of such surfaces where the curvatures have positive density with respect to k-dimensional Hausdorff measure. Special attention is given to boundaries of convex bodies inR ³. We introduce a class of convex sets whose curvatures live only on integer dimension sets. For such convex sets we consider integral functionals depending on the curvature and the area ofK and on the curvature andH ^k of Σ ^k .     

Totally real embeddings of the torus intoC 2

In this short note we combine a construction of Viro and a result of Eliashberg and Harlamov to prove that there exist smooth totally real embeddings of the torus intoC ² which are isotopic but not so within the class of totally real surfaces. We also show how Viro's construction can be used to define an isotopy invariant for a certain class of complex curves inC P ².     

On t-Structures and Tilting Theory

Abstract

Let A be a right coherent associative ring with unit. We introduce the notion of coendofinite complex and we associate to such a complex a t-structure in D ^b (mod A). We give conditions for the heart of that t-structure to be a module category. We also give some applications in connection with derived equivalent rings and tilting theory. In particular for a tilting module over a finite dimensional k-algebra, we get a reformulation of Brenner-Butler's theorem in terms of t-structures.     

On finite retract lattices of monounary algebras

For a monounary algebra (A, f) we denote R ^∅(A, f) the system of all retracts (together with the empty set) of (A, f) ordered by inclusion. This system forms a lattice. We prove that if (A, f) is a connected monounary algebra and R ^∅(A, f) is finite, then this lattice contains no diamond. Next distributivity of R ^∅(A, f) is studied. We find a representation of a certain class of finite distributive lattices as retract lattices of monounary algebras.     

Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators

We consider quite general h-pseudodifferential operators on R ⁿ with small random perturbations and show that in the limit h → 0 the eigenvalues are distributed according to a Weyl law with a probabality that tends to 1. The first author has previously obtained a similar result in dimension 1. Our class of perturbations is different.