Symmetric block bases of sequences with large average growth

We show that if 0<ε≦1, 1≦p<2 andx ₁, …,x _n is a sequence of unit vectors in a normed spaceX such thatE ‖∑ _l ⁿ ε_i x _l‖≧n ^1/p, then one can find a block basisy ₁, …,y _m ofx ₁, …,x _n which is (1+ε)-symmetric and has cardinality at leastγn ^2/p-1(logn)⁻¹, where γ depends on ε only. Two examples are given which show that this bound is close to being best possible. The first is a sequencex ₁, …,x _n satisfying the above conditions with no 2-symmetric block basis of cardinality exceeding 2n ^2/p-1. This sequence is not linearly independent. The second example is a sequence which satisfies a lowerp-estimate but which has no 2-symmetric block basis of cardinality exceedingCn ^2/p-1(logn)^4/3, whereC is an absolute constant. This applies when 1≦p≦3/2. Finally, we obtain improvements of the lower bound when the spaceX containing the sequence satisfies certain type-condition. These results extend results of Amir and Milman in [1] and [2]. We include an appendix giving a simple counterexample to a question about norm-attaining operators.     

Three-Dimensional Polyhedra Can Be Described by Three Polynomial Inequalities

Bosse et al. conjectured that for every natural number d≥2 and every d-dimensional polytope P in ℝ ^d , there exist d polynomials p ₁(x),…,p _d (x) satisfying P={x∈ℝ ^d :p ₁(x)≥0,…,p _d (x)≥0}. We show that every three-dimensional polyhedron can be described by three polynomial inequalities, which confirms the conjecture for the case d=3 but also provides an analogous statement for the case of unbounded polyhedra. The proof of our result is constructive. Work supported by the German Research Foundation within the Research Unit 468 “Methods from Discrete Mathematics for the Synthesis and Control of Chemical Processes”.     

Test for properness using residue calculus

We will consider global problems in the ringK[X ₁, …,X _n] on the polynomials with coefficients in a subfieldK ofC. LetP=(P ₁, …,P _n):K ⁿ →K ⁿ be a polynomial map such that (P ₁,…,P _n) is a quasi-regular sequence generating a proper ideal, the main thing we do is to use the algebraic residues theory (as described in [5]) as a computational tool to give some result to test when a map (P ₁, …,P _n) is a proper map by computing a finite number of residue symbols.     

Bispectrality of Multivariable Racah–Wilson Polynomials

We construct a commutative algebra A_x{\mathcal{A}}_{x} of difference operators in ℝ ^p , depending on p+3 parameters, which is diagonalized by the multivariable Racah polynomials R _p (n;x) considered by Tratnik (J. Math. Phys. 32(9):2337–2342, 1991). It is shown that for specific values of the variables x=(x ₁,x ₂,…,x _p ) there is a hidden duality between n and x. Analytic continuation allows us to construct another commutative algebra A_n{\mathcal{A}}_{n} in the variables n=(n ₁,n ₂,…,n _p ) which is also diagonalized by R _p (n;x). Thus, R _p (n;x) solve a multivariable discrete bispectral problem in the sense of Duistermaat and Grünbaum (Commun. Math. Phys. 103(2):177–240, 1986). Since a change of the variables and the parameters in the Racah polynomials gives the multivariable Wilson polynomials (Tratnik in J. Math. Phys. 32(8):2065–2073, 1991), this change of variables and parameters in A_x{\mathcal{A}}_{x} and A_n{\mathcal{A}}_{n} leads to bispectral commutative algebras for the multivariable Wilson polynomials.     

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.     

Refined enumerations of alternating sign matrices: monotone (<Emphasis Type="Italic">d</Emphasis>,<Emphasis Type="Italic">m</Emphasis>)-trapezoids with prescribed top and bottom row

Monotone triangles are plane integer arrays of triangular shape with certain monotonicity conditions along rows and diagonals. Their significance is mainly due to the fact that they correspond to n×n alternating sign matrices when prescribing (1,2,…,n) as bottom row of the array. We define monotone (d,m)-trapezoids as monotone triangles with m rows where the d−1 top rows are removed. (These objects are also equivalent to certain partial alternating sign matrices.) It is known that the number of monotone triangles with bottom row (k ₁,…,k _n ) is given by a polynomial α(n;k ₁,…,k _n ) in the k _i ’s. The main purpose of this paper is to show that the number of monotone (d,m)-trapezoids with prescribed top and bottom row appears as a coefficient in the expansion of a specialisation of α(n;k ₁,…,k _n ) with respect to a certain polynomial basis. This settles a generalisation of a recent conjecture of Romik et al. (Adv. Math. 222:2004–2035, 2009). Among other things, the result is used to express the number of monotone triangles with bottom row (1,2,…,i−1,i+1,…,j−1,j+1,…,n) (which is, by the standard bijection, also the number of n×n alternating sign matrices with given top two rows) in terms of the number of n×n alternating sign matrices with prescribed top and bottom row, and, by a formula of Stroganov for the latter numbers, to provide an explicit formula for the first numbers. (A formula of this type was first derived by Karklinsky and Romik using the relation of alternating sign matrices to the six-vertex model.)     

相关序的进一步研究

Let (X1,X2,…,Xn) and (Y1,Y2,…Yn) be real random vectors with the same marginal distributions,if (X1,X2,…,Xn)≤c(Y1,Y2,…Yn), it is showed in this paper that ∑i=1^n Xi≤cx∑i=1^n Yi and max1≤k≤n∑i=1^k Xi≤icx max1≤k≤n∑i=1^k Yi hold. Based on this fact,a more general comparison theorem is obtained.     

Approximation of Projections of Random Vectors

Let X be a d-dimensional random vector and X _θ its projection onto the span of a set of orthonormal vectors {θ ₁,…,θ _k }. Conditions on the distribution of X are given such that if θ is chosen according to Haar measure on the Stiefel manifold, the bounded-Lipschitz distance from X _θ to a Gaussian distribution is concentrated at its expectation; furthermore, an explicit bound is given for the expected distance, in terms of d, k, and the distribution of X, allowing consideration not just of fixed k but of k growing with d. The results are applied in the setting of projection pursuit, showing that most k-dimensional projections of n data points in ℝ ^d are close to Gaussian, when n and d are large and k=clog (d) for a small constant c.     

An algorithmic approach to Ramanujan’s congruences

In this paper we present an algorithm that takes as input a generating function of the form $\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n}In this paper we present an algorithm that takes as input a generating function of the form ?_d|M?_n=1^￥(1-q^dn)^r_d=?_n=0^￥a(n)qⁿ\prod_{\delta|M}\prod_{n=1}^{\infty}(1-q^{\delta n})^{r_{\delta}}=\sum_{n=0}^{\infty}a(n)q^{n} and three positive integers m,t,p, and which returns true if a(mn+t) o 0 mod p,n 3 0a(mn+t)\equiv0\pmod{p},n\geq0, or false otherwise. Our method builds on work by Rademacher (Trans. Am. Math. Soc. 51(3):609–636, 1942), Kolberg (Math. Scand. 5:77–92, 1957), Sturm (Lecture Notes in Mathematics, pp. 275–280, Springer, Berlin/Heidelberg, 1987), Eichhorn and Ono (Proceedings for a Conference in Honor of Heini Halberstam, pp. 309–321, 1996).     

Return times, recurrence densities and entropy for actions of some discrete amenable groups

It is a theorem of Wyner and Ziv and Ornstein and Weiss that if one observes the initialk symbolsX ₀,…,X _k−1 of a typical realization of a finite valued ergodic process with entropyh, the waiting time until this sequence appears again in the same realization grows asymptotically like 2 ^hk [7, 12]. A similar result for random fields was obtained in [8]: in this case, one observes cubes in ℤ ^d instead of initial segments. In the present paper, we describe generalizations of this. We examine what happens when the set of possible return times is restricted. Fix an increasing sequence of sets of possible times {W _n } and defineR _k to be the firstn such thatX ₀,…,X _k−1 recurs at some time inW _n . It turns out that |W _{R k} | cannot drop below 2 ^hk asymptotically. We obtain conditions on the sequence {W _n } which ensure that |W _{R k} | is asymptotically equal to 2 ^hk . We consider also recurrence densities of initial blocks and derive a uniform Shannon-McMillan-Breiman theorem. Informally, ifU _k,n is the density of recurrences of the blockX ₀,…,X _k−1 inX _−n ,…,X _n , thenU _k,n grows at a rate of 2 ^hk , uniformly inn. We examine the conditions under which this is true when the recurrence times are again restricted to some sequence of sets {W _n }. The above questions are examined in the general context of finite-valued processes parametrized by discrete amenable groups. We show that many classes of groups have time-sequences {W _n } along which return times and recurrence densities behave as expected. An interesting feature here is that this can happen also when the time sequence lies in a small subgroup of the parameter group.     

On the theorem of M Golomb

Let X ₁, …, X _n be compact spaces and X = X ₁ × … × X _n . Consider the approximation of a function ƒ ∈ C(X) by sums g ₁(x ₁)+…+g _n (x _n ), where g _i ∈ C(X _i ), i = 1, …, n. In [8], Golomb obtained a formula for the error of this approximation in terms of measures constructed on special points of X, called ‘projection cycles’. However, his proof had a gap, which was pointed out by Marshall and O’Farrell [15]. But the question if the formula was correct, remained open. The purpose of the paper is to prove that Golomb’s formula holds in a stronger form.     

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.     

Convergence in law of partial sum processes in p-variation norm

Let X ₁, X ₂, … be a sequence of independent identically distributed real-valued random variables, S _n be the nth partial sum process S _n (t) ≔ X ₁ + ⋯ X _⌊tn⌋, t ∈ [0, 1], W be the standard Wiener process on [0, 1], and 2 < p < ∞. It is proved that n ^−1/2 S _n converges in law to σW as n → ∞ in p-variation norm if and only if EX ₁ = 0 and σ ² = EX ₁² < ∞. The result is applied to test the stability of a regression model. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-21/07     

On the strong law of large numbers for sequences of random variables without the independence condition

New sufficient conditions for the applicability of the strong law of large numbers to a sequence of dependent random variables X ₁, X ₂, …, with finite variances are established. No particular type of dependence between the random variables in the sequence is assumed. The statement of the theorem involves the classical condition Σ _n ^∞ (log₂ n)²/n ² < ∞, which appears in various theorems on the strong law of large numbers for sequences of random variables without the independence condition.     

Improved estimation of the generalized precision under the entropy loss

1.IntroductionInthisarticleweconsiderthepointestimationofthegeneralizedprecisionofamultivariatenormaldistributionwithanunknownmeanvector.TObespecific,letXI,'?XubelidobservationfromNc(~,E)wherebothpERPandZ>0arecompletelyunknown.Insteadoftheoriginaldatasetonecanreducetheproblembysufficiencyandlookonlyatnn(X,S),whereX~n--1ZXiandS~Z(Xi--X)(Xi--X)'.ItiswellknownthatXisi=1i~1mutuallyindependentofSandX~Nc(~,n--'Z),S~Wb(n--1,Z).ThelossfunctionweconsiderinthispaperistheentropylossL(6,IZ…     

On Chow and Brauer groups of a product of Mumford curves

Let C₁,···,C_d be Mumford curves defined over a finite extension of and let X=C₁×···×C_d. We shall show the following: (1) The cycle map CH₀(X)/n → H^2d(X, μ_n^⊗d) is injective for any non-zero integer n. (2) The kernel of the canonical map CH₀(X)→Hom(Br(X),) (defined by the Brauer-Manin pairing) coincides with the maximal divisible subgroup in CH₀(X).     

On the applicability conditions of the strong law of large numbers for sequences of independent random variables

We investigate relationship between Kolmogorov–s condition and Petrov–s condition in theorems on the strong law of large numbers for a sequence of independent random variables X ₁, X ₂, … with finite variances. The convergence (S _n − ES _n )/n → 0 holds a.s. (here, S _n = Σ _k=1 ⁿ X _k ), provided that Σ _n=1^∞ DX _n /n ² < ∞ (Kolmogorov’s condition) or DS _n = O(n ²/ψ(n)) for some positive non-decreasing function ψ(n) such that Σ1/(nψ(n)) < ∞ (Petrov’s condition). Kolmogorov’s condition is shown to follow from Petrov’s condition. Besides, under some additional restrictions, Petrov’s condition, in turn, follows from Kolmogorov’s condition.     

Asymptotic behavior of the ratio of tail probabilities of sum and maximum of independent random variables

For a fixed integer n ≥ 2, let X ₁ ,…, X _n be independent random variables (r.v.s) with distributions F ₁,…,F _n , respectively. Let Y be another random variable with distribution G belonging to the intersection of the longtailed distribution class and the O-subexponential distribution class. When each tail of F _i , i = 1,…,n, is asymptotically less than or equal to the tail of G, we derive asymptotic lower and upper bounds for the ratio of the tail probabilities of the sum X ₁ + ⋯ + X _n and Y. By taking different G’s, we obtain general forms of some existing results.     

Two-point concentration in random geometric graphs

A random geometric graph G _n is constructed by taking vertices X ₁,…,X _n ∈ℝ ^d at random (i.i.d. according to some probability distribution ν with a bounded density function) and including an edge between X _i and X _j if ‖X _i -X _j ‖ < r where r = r(n) > 0. We prove a conjecture of Penrose ([14]) stating that when r=r(n) is chosen such that nr ^d = o(lnn) then the probability distribution of the clique number ω(G _n ) becomes concentrated on two consecutive integers and we show that the same holds for a number of other graph parameters including the chromatic number χ(G _n ). The author was partially supported by EPSRC, the Department of Statistics, Bekkerla-Bastide fonds, Dr. Hendrik Muller’s Vaderlandsch fonds, and Prins Bernhard Cultuurfonds.     

On the stability of characterizations by an identical distribution property

Let X ₁, X ₂, … , X _n be i.i.d. random variables with common distribution F, and let b ₁, b ₂, … , b _n be real coefficients such that ∑ b _j ² = 1. We prove that F is close to the normal distribution in the Lévy metric whenever the distribution of the linear statistic ∑ b _j X _j is close to F.