Fano 3-folds with divisible anticanonical class

We show the nonvanishing of H ⁰(X,−K _X ) for any a Fano 3-fold X for which −K _X is a multiple of another Weil divisor in Cl(X). The main case we study is Fano 3-folds with Fano index 2: that is, 3-folds X with rank Pic(X)=1, -factorial terminal singularities and −K _X  = 2A for an ample Weil divisor A. We give a first classification of all possible Hilbert series of such polarised varieties (X,A) and deduce both the nonvanishing of H ⁰(X,−K _X ) and the sharp bound (−K _X )³≥ 8/165. We find the families that can be realised in codimension up to 4.     

Rates of uniform convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions

Assume that {Xn} is a strictly stationary β-mixing random sequence with the β-mixing coefficient βk = O(k-r), 0 < r ≤1. Yu (1994) obtained convergence rates of empirical processes of strictly stationary β-mixing random sequence indexed by bounded classes of functions. Here, a new truncation method is proposed and used to study the convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions. The research results show that if the envelope of the index class of functions is in Lp, p > 2 or p > 4, uniform convergence rates of empirical processes of strictly stationary β-mixing random sequence over the index classes can reach O((nr/(l+r)/logn)-1/2) or O((nr/(1+r)/ log n)-3/4) and that the Central Limit Theorem does not always hold for the empirical processes.``     

Maximal function characterizations of Hardy spaces on RD-spaces and their applications

Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a dimension n. For α∈ (0, ∞) denote by Hαp(X ), Hdp(X ), and H?,p(X ) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calder′on reproducing formula, it is shown that all these Hardy spaces coincide with Lp(X ) when p ∈ (1, ∞] a...     

Sum Complexes—a New Family of Hypertrees

A k-dimensional hypertree X is a k-dimensional complex on n vertices with a full (k−1)-dimensional skeleton and \binomn-1k\binom{n-1}{k} facets such that H _k (X;ℚ)=0. Here we introduce the following family of simplicial complexes. Let n,k be integers with k+1 and n relatively prime, and let A be a (k+1)-element subset of the cyclic group ℤ _n . The sum complex X _A is the pure k-dimensional complex on the vertex set ℤ _n whose facets are σ⊂ℤ _n such that |σ|=k+1 and ∑ _x∈σ x∈A. It is shown that if n is prime, then the complex X _A is a k-hypertree for every choice of A. On the other hand, for n prime, X _A is k-collapsible iff A is an arithmetic progression in ℤ _n .     

Equiarboreal graphs

A graphX is said to beequiarboreal if the number of spanning trees containing a specified edge inX is independent of the choice of edge. We prove that any graph which is a colour class in an association scheme (and thus any distance regular graph) is equiarboreal. We note that a connected equiarboreal graph withM edges andn vertices has edge-connectivity at leastM/(n−1).     

Almost completely decomposable groups with primary regulator quotients and their endomorphism rings

Let X be a block-rigid almost completely decomposable group of ring type with regulator A and p-primary regulator quotient X/A such that p ^l = exp X/A with natural l > 1. From the well-known fact p ^l End A ⊂ End X ⊂ End A it follows that End X = End X ∪ End A and p ^l End A = End X ∪ p ^l End A. Generalizing these, we determine the chain End X = ɛ _A ^(l) ⊂ ɛ _A ^(l−1) ⊂ ɛ _A ^(l−2) ⊂ ⋯ ⊂ ɛ _A ⁽¹⁾ ⊂ ɛ _A ⁽⁰⁾ = End A, satisfying p ^l−k ɛ _A ^(k) = End X ∪p ^l−k End A, and construct groups X _k ^′ and such that ɛ _A ^(k) = Hom , where k = 1, 2,..., l − 1. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 17–38, 2006.     

Affine cross-polytopes inscribed in a convex body

Let X be an affine cross-polytope, i.e., the convex hull of n segments A ₁ B ₁,…, A _n B _n in \mathbbRⁿ {\mathbb{R}^n} that have a common midpoint O and do not lie in a hyperplane. The affine flag F(X) of X is the chain O ∈ L ₁ ⊂⋯ ⊂ L _n = \mathbbRⁿ {\mathbb{R}^n} , where L _k is the k-dimensional affine hull of the segments A ₁ B ₁,…, A _k B _k , k ≤ n. It is proved that each convex body K ⊂ \mathbbRⁿ {\mathbb{R}^n} is circumscribed about an affine cross-polytope X such that the flag F(X) satisfies the following condition for each k ∈{2,…, n}:the (k−1)-planes of support at A _k and B _k to the body L _k ∩ K in the k-plane L _k are parallel to L _k −1.Each such X has volume at least V(K)/2 ^n(n−1)/2. Bibliography: 5 titles.     

The speed of convergence of a martingale

LetX _n, n≧0, be a martingale with respect to the σ-fieldsF _n and letB _n ² =Σ_1≧n E{(X ₁−X ₁₋₁)²|F ₁₋₁} It is known that ifB ₁ ² <∞ on some set Ω₀ thenX _∞=limX _n exists and is finite a.e. on Ω₀ We show that under suitable conditions there exists a constant ν<∞ for which lim supB _n ⁻¹ {log logB _n ² }^−1/2|X _∞−X _n−1 | ≦ √2(η+1). If “the fluctuations ofB _n are small” (in the sense of the Corollary) then ν=0 and the usual upper bound of a law of the iterated logrithm results. This upper bound is not necessarily achieved, though. Research supported in part by the NSF under Grant No. MCS 72-04534A04.     

On some Fano manifolds of large pseudoindex

In this paper we classify pairs (X,ℰ) with ℰ ample vector bundle of rank r on a smooth variety X of dimension n= 2r−1 such that K _X + det ℰ=? _x . Received: 7 April 2000     

On the existence of invariant probability measures

Let (X,A) be a measureable space andT:X →X a measurable mapping. Consider a family ℳ of probability measures onA which satisfies certain closure conditions. IfA ₀⊂A is a convergence class for ℳ such that, for everyA ∈A ₀, the sequence ((1/n) Σ _i ^=0/n−1 1 _A ∘T ⁱ) converges in distribution (with respect to some probability measurev ∈ ℳ), then there exists aT-invariant element in ℳ. In particular, for the special case of a topological spaceX and a continuous mappingT, sufficient conditions for the existence ofT-invariant Borel probability measures with additional regularity properties are obtained.     

Random markov processes and uniform martingales

It is shown here that a certain generalization of ann-step Markov chain is equivalent to the uniform convergence of the martingale {P(X ₀|X ₋₁ X ₋₂···X _−n)} _n=1 ^∞ . Ergodic and probabilistic properties of this process are explored. This work was initiated at SUNY/Albany.     

On products of unbounded operators

A symmetric operator X^ is attached to each operator X that leaves the domain of a given positive operator A invariant and makes the product AX symmetric. Some spectral properties of X^ are derived from those of X and, as a consequence, various conditions ensuring positivity of products of the form AX ₁ ... X _n are proved. The question of ^-complete positivity of the mapping p → Ap(X ₁,...,X _n) defined on complex polynomials in n variables is investigated. It is shown that the set ω is related to the McIntosh-Pryde joint spectrum of (X ₁,...,X _n) in case all the operators A, X ₁,...,X _n are bounded. Examples illustrating the theme of the paper are included. This revised version was published online in June 2006 with corrections to the Cover Date.     

On a characterization of the exponential distribution by spacings

Summary LetX be a non-negative random variable with probability distribution functionF. SupposeX _i,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(X _i+1,n−X_i,n) and (n−j)(X _j+1,n−X_j,n) for somei, j andn, (1≦i<j<n). The work was partly completed when the author was at the Dept. of Statistics, University of Brasilia, Brazil.     

Integral Equations and Operator Theory

A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λ XA. We characterize the set of extended eigenvalues, which we call extended point spectrum, for operators acting on finite dimensional spaces, finite rank operators, Jordan blocks, and C₀ contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. As an application, we show that the commutant of an operator A coincides with that of Aⁿ, n ≥ 2, n ∈ N if the extended point spectrum of A does not contain any n–th root of unity other than 1. The converse is also true if either A or A^* has trivial kernel.     

On the Cantor-Bendixson derivative, resolvable ranks, and perfect set theorems of A. H. Stone

A Borel derivative on the hyperspace 2 ^X of a compactumX is a Borel monotone mapD: 2 ^X →2 ^X . The derivative determines a Cantor-Bendixson type rank δ:2^X → ω₁ ∪ {∞} . We show that ifA⊂2 ^X is analytic andZ⊂A intersects stationary many layers δ₋₁({ξ}), then for almost all σ,F∩δ₋₁({ξ}) cannot be separated fromZ ∩∪ _a<ξ ^δ−1({a}) (and also fromZ ∩∪ _a>ξ ^δ−1({a}) by anyF _σ-set. Another main result involves a natural partial order on 2 ^X related to the derivative. The results are obtained in a general framework of “resolvable ranks” introduced in the paper. During our work on this paper the second author was a Visiting Professor at the Miami University, Ohio. This author would like to express his gratitude to the Department of Mathematics and Statistics for the hospitality.     

The nearness problems for symmetric centrosymmetric with a special submatrix constraint

We say that X=[x_ij]_i,j=1ⁿX=[x_{ij}]_{i,j=1}^n is symmetric centrosymmetric if x _ij  = x _ji and x _{n − j + 1,n − i + 1}, 1 ≤ i,j ≤ n. In this paper we present an efficient algorithm for minimizing ||AXA ^T  − B|| where ||·|| is the Frobenius norm, A ∈ ℝ ^m×n , B ∈ ℝ ^m×m and X ∈ ℝ ^n×n is symmetric centrosymmetric with a specified central submatrix [x _ij ] _{p ≤ i,j ≤ n − p} . Our algorithm produces a suitable X such that AXA ^T  = B in finitely many steps, if such an X exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim.     

Uniformity over primes of tamely ramified splittings

We fix a prime p and let f(X) vary over all monic integer polynomials of fixed degree n. Given any possible shape of a tamely ramified splitting of p in an extension of degree n, we prove that there exists a rational function φ(X)∈ℚ(X) such that the density of the monic integer polynomials f(X) for which the splitting of p has the given shape in ℚ[X]/f(X) is φ(p) (here reducible polynomials can be neglected). As a corollary, we prove that, for p≥n, the density of irreducible monic polynomials of degree n in ℤ _p [X] is the value at p of a rational function φ _n (X)∈ℚ(X). All rational functions involved are effectively computable. Received: 15 September 1998 / Revised version: 21 October 1999     

Rates of Convergence of Means of Euclidean Functionals

Let L be the Euclidean functional with p-th power-weighted edges. Examples include the sum of the p-th power-weighted lengths of the edges in minimal spanning trees, traveling salesman tours, and minimal matchings. Motivated by the works of Steele, Redmond and Yukich (Ann. Appl. Probab. 4, 1057–1073, 1994, Stoch. Process. Appl. 61, 289–304, 1996) have shown that for n i.i.d. sample points {X ₁,…,X _n } from [0,1] ^d , L({X ₁,…,X _n })/n ^(d−p)/d converges a.s. to a finite constant. Here we bound the rate of convergence of EL({X ₁,…,X _n })/n ^(d−p)/d . Y. Koo supported by the BK21 project of the Department of Mathematics, Sungkyunkwan University. S. Lee supported by the BK21 project of the Department of Mathematics, Yonsei University.     

The game for the speed of convergence in repeated games of incomplete information

We consider an infinitely repeated two-person zero-sum game with incomplete information on one side, in which the maximizer is the (more) informed player. Such games have value v _∞ (p) for all 0≤p≤1. The informed player can guarantee that all along the game the average payoff per stage will be greater than or equal to v _∞ (p) (and will converge from above to v _∞ (p) if the minimizer plays optimally). Thus there is a conflict of interest between the two players as to the speed of convergence of the average payoffs-to the value v _∞ (p). In the context of such repeated games, we define a game for the speed of convergence, denoted SG _∞ (p), and a value for this game. We prove that the value exists for games with the highest error term, i.e., games in which v _n (p)− v _∞ (p) is of the order of magnitude of . In that case the value of SG _∞ (p) is of the order of magnitude of . We then show a class of games for which the value does not exist. Given any infinite martingale 𝔛^∞={X _k }^∞ _k=1, one defines for each n : V _n (𝔛^∞) ≔E∑ⁿ _k=1 |X _k+1 − X _k|. For our first result we prove that for a uniformly bounded, infinite martingale 𝔛^∞, V _n (𝔛^∞) can be of the order of magnitude of n ^1/2−ε, for arbitrarily small ε>0. Received January 1999/Final version April 2002     

Record values and the exponential distribution

A sequence {X _n,n≧1} of independent and identically distributed random variables with continuous cumulative distribution functionF(x) is considered.X _j is a record value of this sequence ifX _j>max (X ₁, …,X _j−1). Let {X _L(n) n≧0} be the sequence of such record values. Some properties ofX _L(n) andX _L(n)−X_L(n−1) are studied when {X _n,n≧1} has the exponential distribution. Characterizations of the exponential distribution are given in terms of the sequence {X _L(n),n≧0} The work was partly completed when the author was at the Department of Statistics, University of Brasilia, Brazil.