Forest-Like Permutations

Given a permutation , construct a graph G _π on the vertex set {1, 2,..., n} by joining i to j if (i) i < j and π(i) < π(j) and (ii) there is no k such that i < k < j and π(i) < π(k) < π(j). We say that π is forest-like if G _π is a forest. We first characterize forest-like permutations in terms of pattern avoidance, and then by a certain linear map being onto. Thanks to recent results of Woo and Yong, these show that forest-like permutations characterize Schubert varieties which are locally factorial. Thus forest-like permutations generalize smooth permutations (corresponding to smooth Schubert varieties). We compute the generating function of forest-like permutations. As in the smooth case, it turns out to be algebraic. We then adapt our method to count permutations for which G _π is a tree, or a path, and recover the known generating function of smooth permutations. Received March 27, 2006     

Asymptotic separation for independent trajectories of Markov processes

We say that n independent trajectories ξ₁(t),…,ξ _n (t) of a stochastic process ξ(t)on a metric space are asymptotically separated if, for some ɛ > 0, the distance between ξ _i (t _i ) and ξ _j (t _j ) is at least ɛ, for some indices i, j and for all large enough t ₁,…,t _n , with probability 1. We prove sufficient conitions for asymptotic separationin terms of the Green function and the transition function, for a wide class of Markov processes. In particular,if ξ is the diffusion on a Riemannian manifold generated by the Laplace operator Δ, and the heat kernel p(t, x, y) satisfies the inequality p(t, x, x) ≤ Ct ^−ν/2 then n trajectories of ξ are asymptotically separated provided . Moreover, if for some α∈(0, 2)then n trajectories of ξ^(α) are asymptotically separated, where ξ^(α) is the α-process generated by −(−Δ)^α/2. Received: 10 June 1999 / Revised version: 20 April 2000 / Published online: 14 December 2000 RID="*" ID="*" Supported by the EPSRC Research Fellowship B/94/AF/1782 RID="**" ID="**" Partially supported by the EPSRC Visiting Fellowship GR/M61573     

Exact laws for sums of ratios of order statistics from the Pareto distribution

Consider independent and identically distributed random variables {X _nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X _n(i) ≤ X _n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables R _ij = X _n(j)/X _n(i).     

The Johnson–Lindenstrauss Lemma Almost Characterizes Hilbert Space, But Not Quite

Let X be a normed space that satisfies the Johnson–Lindenstrauss lemma (J–L lemma, in short) in the sense that for any integer n and any x ₁,…,x _n ∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that ‖x _i −x _j ‖≤‖L(x _i )−L(x _j )‖≤O(1)⋅‖x _i −x _j ‖ for all i,j∈{1,…,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion 2^2O(log*n)2^{2^{O(\log^{*}n)}} . On the other hand, we show that there exists a normed space Y which satisfies the J–L lemma, but for every n, there exists an n-dimensional subspace E _n ⊆Y whose Euclidean distortion is at least 2^Ω(α(n)), where α is the inverse Ackermann function.     

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.)     

L p − Lq estimates for convolution operators with n-Dimensional singular measures

Let S ⊂ ℜⁿ⁺¹ be the graph of the function ϕ :[−1, 1] ⁿ → ℜ defined by ϕ (x ₁ , …, x_n) = ∑ _j=1 ⁿ |x_j|^αj, with1<α ₁ ≤ … ≤ α_n, let σ the Euclidean area measure on S. In this article we study the L^p − L^q boundedness of convolution operators with the singular Borel measure on Rⁿ⁺¹ given by μ (E)=σ (E ∩ S)     

Sudakov-type minoration for gaussian chaos processes

Consider a setA of symmetricn×n matricesa=(a _i,j) _i,j≤n . Consider an independent sequence (g _i) _i≤n of standard normal random variables, and letM=Esup_a∈A|Σ_i,j⪯na_i,jg_ig_j|. Denote byN ₂(A, α) (resp.N _t(A, α)) the smallest number of balls of radiusα for thel ₂ norm ofR ^{n 2} (resp. the operator norm) needed to coverA. Then for a universal constantK we haveα(logN ₂(A, α))^1/4≤KM. This inequality is best possible. We also show that forδ≥0, there exists a constantK(δ) such thatα(logN _t≤K(δ)M. Work partially supported by an N.S.F. grant.     

Exchangeable Gibbs partitions and Stirling triangles

For two collections of nonnegative and suitably normalized weights W = (W_j) and V = (V_n,k), a probability distribution on the set of partitions of the set {1, …, n} is defined by assigning to a generic partition {A_j, j ≤ k} the probability V_n,k , where |A_j| is the number of elements of A_j. We impose constraints on the weights by assuming that the resulting random partitions Π _n of [n] are consistent as n varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights W must be of a very special form depending on a single parameter α ∈ [− ∞, 1]. The case α = 1 is trivial, and for each value of α ≠ = 1 the set of possible V-weights is an infinite-dimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalized Stirling triangle. In particular, we show that the boundary is discrete for − ∞ ≤ α < 0 and continuous for 0 ≤ α < 1. For α ≤ 0 the extremes correspond to the members of the Ewens-Pitman family of random partitions indexed by (α,θ), while for 0 < α < 1 the extremes are obtained by conditioning an (α,θ)-partition on the asymptotics of the number of blocks of Π_n as n tends to infinity. Bibliography: 29 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 83–102.     

Regression of polynomial statistics on the sample mean and natural exponential families

A polynomial Q = Q(X ₁, …, X _n ) of degree m in independent identically distributed random variables with distribution function F is an unbiased estimator of a functional q(α ₁(F), …, α _m (F)), where q(u ₁, …, u _m ) is a polynomial in u ₁, …, u _m and α _j (F) is the jth moment of F (assuming the necessary moment of F exists). It is shown that the relation E(Q | X ₁ + … + X n) = 0 holds if and only if q(α ₁(θ), …, α _m (θ)) ≡ 0, where α _j (θ) is the jth moment of the natural exponential family generated by F. This result, based on the fact that X ₁ + … + X n is a complete sufficient statistic for a parameter θ in a sample from a natural exponential family of distributions F _θ(x) = ∫_−∞ ^x e ^θu−k(θ) dF(u), explains why the distributions appearing as solutions of regression problems are the same as solutions of problems for natural exponential families though, at the first glance, the latter seem unrelated to the former.     

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.     

OD-characterization of Almost Simple Groups Related to U3（5）

Let G be a finite group with order ｜G｜=p1^α1p2^α2……pk^αk, where p1 〈 p2 〈……〈 Pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph （or Gruenberg- Kegel graph） denoted .by г（G） （or GK（G））. This graph is constructed as follows： The vertex set of it is π（G） = {p1,p2,…,pk} and two vertices pi, pj with i≠j are adjacent by an edge （and we write pi - pj） if and only if G contains an element of order pipj. The degree deg（pi） of a vertex pj ∈π（G） is the number of edges incident on pi. We define D（G） ：= （deg（p1）, deg（p2）,..., deg（pk））, which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k non- isomorphic groups H such that ｜H｜ = ｜G｜ and D（H） = D（G）. Moreover, a 1-fold OD-characterizable group is simply called OD-characterizable. Let L ：= U3（5） be the projective special unitary group. In this paper, we classify groups with the same order and degree pattern as an almost simple group related to L. In fact, we obtain that L and L.2 are OD-characterizable; L.3 is 3-fold OD-characterizable; L.S3 is 6-fold OD-characterizable.     

Zeros of linear recurrence sequences

Let be an exponential polynomial over a field of zero characteristic. Assume that for each pair i,j with i≠j, α _i /α _j is not a root of unity. Define . We introduce a partition of into subsets (1≤i≤m), which induces a decomposition of f into , so that, for 1≤i≤m, , while for , the number either is transcendental or else is algebraic with not too small a height. Then we show that for all but at most solutions x∈ℤ of f(x)= 0, we have

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

Subsequences of normal sequences

In this paper, we characterize a set of indices τ={τ(0)<τ(1)<…} such that forany normal sequence (α(0), α(1),…) of a certain type, the subsequence (α(τ(0)), α(τ(1)),…) is a normal sequence of the same type. Assume that_n→∞. Then, we prove that τ has this property if and only if the 0–1 sequence (θ _τ (0), whereθ _τ (i)=1 or 0 according asi∈{τ(j);j=0, 1,…} or not, iscompletely deterministic in the sense of B. Weiss.     

Some results on the majorization theorem of connected graphs

Let π = (d ₁, d ₂, ..., d _n ) and π′ = (d′ ₁, d′ ₂, ..., d′ _n ) be two non-increasing degree sequences. We say π is majorizated by π′, denoted by π ⊲ π′, if and only if π ≠ π′, Σ _i=1 ⁿ d _i = Σ _i=1 ⁿ d′ _i , and Σ _i=1 ^j d _i ≤ Σ _i=1 ^j d′ _i for all j = 1, 2, ..., n. Weuse C _π to denote the class of connected graphs with degree sequence π. Let ρ(G) be the spectral radius, i.e., the largest eigenvalue of the adjacent matrix of G. In this paper, we extend the main results of [Liu, M. H., Liu, B. L., You, Z. F.: The majorization theorem of connected graphs. Linear Algebra Appl., 431(1), 553–557 (2009)] and [Bıyıkoğlu, T., Leydold, J.: Graphs with given degree sequence and maximal spectral radius. Electron. J. Combin., 15(1), R119 (2008)]. Moreover, we prove that if π and π′ are two different non-increasing degree sequences of unicyclic graphs with π ⊲ π′, G and G′ are the unicyclic graphs with the greatest spectral radii in C _π and C′ _π , respectively, then ρ(G) < ρ(G′).     

Multiplicative isometries on the Smirnov class

We show that T is a surjective multiplicative (but not necessarily linear) isometry from the Smirnov class on the open unit disk, the ball, or the polydisk onto itself, if and only if there exists a holomorphic automorphism Φ such that T(f)=f ○ Φ for every class element f or T(f) = [`(f<sup>°</sup> [`(j)] )]\overline {f^\circ \bar \varphi } for every class element f, where the automorphism Φ is a unitary transformation in the case of the ball and Φ(z ₁, ..., z _n ) = (l₁ z_i1 ,...,l_n z_in )(\lambda _1 z_{i_1 } ,...,\lambda _n z_{i_n } ) for |λ _j | = 1, 1 ≤ j ≤ n, and (i ₁; ..., i _n )is some permutation of the integers from 1through n in the case of the n-dimensional polydisk.     

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.     

Positive values of non-homogeneous indefinite quadratic forms of type (1, 4)

Let Г_r,n—r denote the infimum of all number Г > 0 such that for any real indefinite quadratic form inn variables of type (r, n—r), determinantD ≠ 0 and real numbers c₁; c₂,…, c_n, there exist integersx ₁,x₂,…,x_n satisfying 0 < Q(x₁+c₁,x₂ + c₂,…,x_n + c_n) ≤(Г|Z > |)^1/n. All the values of Г_r,n—r are known except for г_1,4. Earlier it was shown that 8 ≤Г_1,4 ≤16. Here we improve the upper bound to get Г_1,4 < 12.     

Varieties with at most quadratic growth

Let V be a variety of non-necessarily associative algebras over a field of characteristic zero. The growth of V is determined by the asymptotic behavior of the sequence of codimensions c _n (V), n = 1, 2, …, and here we study varieties of polynomial growth. Recently in [16], for any real number α, 3 < α < 4, a variety V was constructed satisfying C ₁ n ^α < c _n (V) < C ₂ n ^α , for some constants C ₁, C ₂. Motivated by this result here we try to classify all possible growth of varieties V such that c _n (V) < C _n ^α , with 0 < α < 2, for some constant C. We prove that if 0 < α < 1 then, for n large, c _n (V) ≤ 1, whereas if V is a commutative variety and 1 < α < 2, then lim _n→∞ log _n c _n (V) = 1 or c _n (V) ≤ 1 for n large enough.     

Asymptotic Behavior of the Maximum of Sums of I.I.D. Random Variables along Monotone Blocks

Let {X_i, Y_i}_i=1,2,... be an i.i.d. sequence of bivariate random vectors with P(Y₁ = y) = 0 for all y. Put M_n(j) = max_0≤k≤n-j (X_k+1 + ... X_k+j)I_k,j, where I_k,k+j = I{Y_k+1 < ⋯ < Y_k+j} denotes the indicator function for the event in brackets, 1 ≤ j ≤ n. Let L_n be the largest index l ≤ n for which I_k,k+l = 1 for some k = 0, 1, ..., n - l. The strong law of large numbers for “the maximal gain over the longest increasing runs,” i.e., for M_n(L_n) has been recently derived for the case where X₁ has a finite moment of order 3 + ε, ε > 0. Assuming that X₁ has a finite mean, we prove for any a = 0, 1, ..., that the s.l.l.n. for M(L_n - a) is equivalent to EX ₁ ^3+a I{X₁ > 0} < ∞. We derive also some new results for the a.s. asymptotics of L_n. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 179–189.