On Convolutions and Linear Combinations of Pseudo-Isotropic Distributions

Let E be a Banach space, and let E* be its dual. For understanding the main results of this paper it is enough to consider E= ⁿ . A symmetric random vector X taking values in E is called pseudo-isotropic if all its one-dimensional projections have identical distributions up to a scale parameter, i.e., for every E* there exists a positive constant c() such that (, X) has the same distribution as c() X ₀, where X ₀ is a fixed nondegenerate symmetric random variable. The function c defines a quasi-norm on E*. Symmetric Gaussian random vectors and symmetric stable random vectors are the best known examples of pseudo-isotropic vectors. Another well known example is a family of elliptically contoured vectors which are defined as pseudo-isotropic with the quasi-norm c being a norm given by an inner product on E*. We show that if X and Y are independent, pseudo-isotropic and such that X+Y is also pseudo-isotropic, then either X and Y are both symmetric -stable, for some (0, 2], or they define the same quasi-norm c on E*. The result seems to be especially natural when restricted to elliptically contoured random vectors, namely: if X and Y are symmetric, elliptically contoured and such that X+Y is also elliptically contoured, then either X and Y are both symmetric Gaussian, or their densities have the same level curves. However, even in this simpler form, this theorem has not been proven earlier. Our proof is based upon investigation of the following functional equation:

The Groups H3(F(X)/FM) and CH2(X) for Generic Splitting Varieties of Quadratic Forms

Let F be a field of characteristic different from 2 and be a quadratic form over F. Let X be an arbitrary projective homogeneous generic splitting variety of . For example, we can take X to be equal to the variety X_,m of totally isotropic m-dimensional subspaces of V, where V is the quadratic space corresponding to and < dim V. In this paper, we study the groups CH²(X) and H³(F(X)/F) = ker(H ³(F) H ³(F(X))). One of the main results of this paper claims that the group Tors CH²(X) is always zero or isomorphic to . In many cases we prove that Tors CH²(X) = 0 and compute the group H ³(F(X)/F) completely. As an application of the main results, we give a criterion of motivic equivalence of eight-dimensional forms except for the case where the Schur indices of their Clifford algebras equal 4.     

The lattice points of ann-dimensional tetrahedron

We show that the number of orderedm-tuples of points on the integer lattice, inside or on then-dimensional tetrahedron bounded by the hyperplanesX ₁=0,X ₂=0, ...,X _n=0 andw ₁ X ₁+w ₂ X _n+...+w _n X_n=X, with the property that, for eachj, no more thank such points have non-zerojth ordinate, is asymptotically

A note on the product of random elements of a semigroup

LetX=(X ₀,X ₁, ...) be a Markov chain on the discrete semigroupS. X is assumed to have one essential classC such thatCK, whereK is the kernel ofS. We study the processY=(Y ₀,Y ₁,...) whereY _n =X ₀ X ₁ ...X _n using the auxiliary process which is a Markov chain onS×S. The essential classes and the limiting distribution of theZ-chain are determined. (These results were obtained earlier byH. Muthsam, Mh. Math.76, 43–54 (1972). However, his proofs contained an error restricting the validity of his results.Supported in part by the Danish Ministry of Education and the Toroch Ellida Ljungbergs fond.     

The Descent Monomials and a Basis for the Diagonally Symmetric Polynomials

Let R(X) = Q[x ₁, x ₂, ..., x _n] be the ring of polynomials in the variables X = {x ₁, x ₂, ..., x _n} and R*(X) denote the quotient of R(X) by the ideal generated by the elementary symmetric functions. Given a S _n, we let g In the late 1970s I. Gessel conjectured that these monomials, called the descent monomials, are a basis for R*(X). Actually, this result was known to Steinberg [10]. A. Garsia showed how it could be derived from the theory of Stanley-Reisner Rings [3]. Now let R(X, Y) denote the ring of polynomials in the variables X = {x ₁, x ₂, ..., x _n} and Y = {y ₁, y ₂, ..., y _n}. The diagonal action of S _n on polynomial P(X, Y) is defined as Let R (X, Y) be the subring of R(X, Y) which is invariant under the diagonal action. Let R *(X, Y) denote the quotient of R (X, Y) by the ideal generated by the elementary symmetric functions in X and the elementary symmetric functions in Y. Recently, A. Garsia in [4] and V. Reiner in [8] showed that a collection of polynomials closely related to the descent monomials are a basis for R *(X, Y). In this paper, the author gives elementary proofs of both theorems by constructing algorithms that show how to expand elements of R*(X) and R *(X, Y) in terms of their respective bases.     

Strong uniform consistency of nonparametric regression function estimates

Let (X, Y) be a ^dx-valued random vector and let r(t)=E(Y/X=t) be the regression function of Y on X that has to be estimated from a sample (X _i, Y_i), i=1,..., n. We establish conditions ensuring that an estimate of the form

A Borel extension approach to weakly compact operators on C 0(T)

Let X be a quasicomplete locally convex Hausdorff space. Let T be a locally compact Hausdorff space and let C ₀(T) = is continuous and vanishes at infinity} be endowed with the supremum norm. Starting with the Borel extension theorem for X-valued -additive Baire measures on T, an alternative proof is given to obtain all the characterizations given in [13] for a continuous linear map to be weakly compact.     

Entropy for Random Partitions and Its Applications

Asymptotic properties of partitions of the unit interval are studied through the entropy for random partition

Non-intersection exponents for Brownian paths

Summary LetX andY be independent 3-dimensional Brownian motions,X(0)=(0,0,0),Y(0)=(1,0,0) and letp _r =P(X[0,r] Y[0,r]=). Then the non-intersection exponent exists and is equal to a similar non-intersection exponent for random walks. Analogous results hold inR ² and for more than 2 paths.Supported in part by NSF grant DMS 8702620Supported by NSF grant DMS 8702879 and an Alfred P. Sloan Research Fellowship     

The Jacobian conjecture,the <Emphasis Type="Italic">d</Emphasis>-inversion approximation and its natural boundary

Let F ? \mathbbC[ X, Y ]² F \in \mathbb{C}{\left[ {X,\,Y} \right]^2} be an étale map of degree deg F = d. An étale map G ? \mathbbC[ X,Y ]² G \in \mathbb{C}{\left[ {X,Y} \right]^2} is called a d-inverse approximation of F if deg G ≤ d and F ◦ G =(X + A(X, Y), Y + B(X, Y)) and G ◦ F =(X + C(X, Y), Y + D(X, Y)), where the orders of the four polynomials A, B, C, and D are greater than d. It is a well-known result that every \mathbbC² {\mathbb{C}^2} -automorphism F of degree d has a d-inverse approximation, namely, F ⁻¹. In this paper, we prove that if F is a counterexample of degree d to the two-dimensional Jacobian conjecture, then F has no d-inverse approximation. We also give few consequences of this result. Bibliography: 18 titles.     

Markov Processes with Equal Capacities

Let X and be transient standard Markov processes in weak duality with respect to a -finite measure m. Let (Y, , ) be a second dual pair with the same state space E as (X, , m). Let Cap ^X and Cap ^Y be the 0-order capacities associated with (X, , m) and (Y, , ), and let V and denote the potential kernels for Y and . Assume that singletons are polar with respect to both X and Y, and that semipolar sets are of capacity zero for both dual pairs. We show that if Cap ^X (B)=Cap ^Y (B) for every Borel subset of E then there is a strictly increasing continuous additive functional D=(D _t) _t0 of (X, , m) such that

Coconvex Pointwise Approximation

Assume that a function f C[–1, 1] changes its convexity at a finite collection Y := {y ₁, ... y _s} of s points y _i (–1, 1). For each n > N(Y), we construct an algebraic polynomial P _n of degree n that is coconvex with f, i.e., it changes its convexity at the same points y _i as f and

Ramified coverings over a complete Kähler space

Let X and Y be reduced complex spaces with countable topology. Let be a locally semi-finite holomorphic map such that the analytic set is nowhere dense in X. If Y is complete Kähler, then we prove that X is also complete Kähler. Especially if is a (not necessarily finitely sheeted) ramified covering over a complete Kähler space Y, then X is also complete Kähler. Received: 2 August 2002     

Asymptotic behavior of a class of stochastic semigroups in the Bernoulli scheme

A family of subalgebras describing the space of complex-valued 2×2 matrices is selected. In this space, a stochastic semigroupY _n =X _n X _–1 ...X ₁,n= , is considered, where {X_i, i= } are independent equally distributed random matrices taking two values. For a stochastic semigroupY _n , whose phase space belongs to one of the subalgebras, the index of exponential growth is determined explicitly.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1580–1584, November, 1993.     

Central limit theorems for nonlinear functionals of stationary Gaussian processes

Let X=(X _t ,t) be a stationary Gaussian process on (, ,P), letH(X) be the Hilbert space of variables inL ² (,P) which are measurable with respect toX, and let (U _s ,s) be the associated family of time-shift operators. We sayYH(X) (withE(Y)=0) satisfies the functional central limit theorem or FCLT [respectively, the central limit theorem of CLT if in [respectively,], where

Distance-2-Matchings of Random Graphs

In this paper we study the maximal size of a distance-2-matching in a random graph G _n;M , i.e., the probability space consisting of subgraphs of the complete graph over n vertices, K _n , having exactly M edges and uniform probability measure. A distance-2-matching in a graph Y, M ₂, is a set of Y-edges with the property that for any two elements every pair of their 4 incident vertices has Y-distance 2. Let M₂(Y) be the maximal size of a distance-2-matching in Y. Our main results are the derivation of a lower bound for M₂(Y) and a sharp concentration result for the random variable AMS Subject Classification: 05C80, 05C70.     

Real Regulators on Milnor Complexes

Let X/_C be a projective algebraic manifold. In the first part of this paper, we construct a natural real regulator on the cohomology of a Gersten–Milnor complex, and where appropriate, compare it to the Beilinson regulator into real Deligne cohomology. The second part is devoted to a calculation on curves. In particular, we arrive at an elementary new proof of the known nontriviality of the regulator on CH²(X,2), for a general elliptic curve X. Further, if X P ² is a general curve of degree 3, then for any open Riemann surface U contained in the complement , we arrive at a nontrivial regulator calculation on CH²(^U,2). We conclude this paper with an application to points of finite order on a curve X.     

A Limit Result for U-Statistics of Binary Variables

Define , where is a symmetric U-type statistic, H _k() is the Hermite polynomial of degree k, and {X, X _n, n1} are independent identically distributed binary random variables with Pr(X{–1, 1}})=1. We show that according as EX=0 or EX0, respectively.     

Smoothness Properties of Semiflows for Differential Equations with State-Dependent Delays

Differential equations with state-dependent delay can often be written as (t)=f(x_t) with a continuously differentiable map f from an open subset of the space C¹=C¹([-h,0], {}^n), {h>0}, into {}^n. In a previous paper we proved that under two mild additional conditions the set is a continuously differentiable n-codimensional submanifold of C ¹, on which the solutions define a continuous semiflow F with continuously differentiable solution operators F_t=F(t,·), t 0. Here we show that under slightly stronger conditions the semiflow F is continuously differentiable on the subset of its domain given by {t> h}. This yields, among others, Poincaré return maps on transversals to periodic orbits. All hypotheses hold for an example which is based on Newton's law and models automatic position control by echo.     

A Lower Bound for the Sectional Genus of Quasi-Polarized Surfaces

Let (X,L) be a quasi-polarized variety, i.e. X is a smooth projective variety over the complex numbers and L is a nef and big divisor on X. Then we conjecture that g(L) = q(X), whereg(L) is the sectional genus of L and . In this paper, we treat the case . First we prove that this conjecture is true for , and we classify (X,L) withg(L)=q(X), where is the Kodaira dimension of X. Next we study some special cases of .