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

Proof of Two Conjectures of Brenti and Simion on Kazhdan-Lusztig Polynomials

We find an explicit formula for the Kazhdan-Lusztig polynomials P _ui,a,v _i of the symmetric group (n) where, for a, i, n such that 1 a i n, we denote by u _i,a = s _a s _a+1 ··· s _i–1 and by v _i the element of (n) obtained by inserting n in position i in any permutation of (n – 1) allowed to rise only in the first and in the last place. Our result implies, in particular, the validity of two conjectures of Brenti and Simion [4, Conjectures 4.2 and 4.3], and includes as a special case a result of Shapiro, Shapiro and Vainshtein [13, Theorem 1]. All the proofs are purely combinatorial and make no use of the geometry of the corresponding Schubert varieties.     

Weierstrass'' Theorem in Weighted Sobolev Spaces

We characterize the set of functions which can be approximated by polynomials with the following norm

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for a big class of weights w₀, w₁, …, w_k     

Reduced Words and Plane Partitions

Let w ₀ be the element of maximal length in thesymmetric group S _n , and let Red(w ₀) bethe set of all reduced words for w ₀. We prove the identity which generalizes Stanley's [20] formula forthe cardinality of Red(w ₀), and Macdonald's [11] formula .Our approach uses anobservation, based on a result by Wachs [21], that evaluation of certainspecializations of Schubert polynomials is essentially equivalent toenumeration of plane partitions whose parts are bounded from above. Thus,enumerative results for reduced words can be obtained from the correspondingstatements about plane partitions, and vice versa. In particular, identity(*) follows from Proctor's [14] formula for the number of planepartitions of a staircase shape, with bounded largest part.Similar results are obtained for other permutations and shapes;q-analogues are also given.     

Singularités génériques et quasi-résolutions des variétés de Schubert pour le groupe linéaire

We determine explicitly the irreducible components of the singular locus of any Schubert variety for being an algebraically closed field of arbitrary characteristic. We also describe the generic singularities along each of them.The case of covexillary Schubert varieties was solved in an earlier work of the author [Ann. Inst. Fourier 51 (2) (2001) 375]. Here, we first exhibit some irreducible components of the singular locus of X_w, by describing the generic singularity along each of them. Let Σ_w be the union of these components. As mentioned above, the equality is known for covexillary varieties, and we base our proof of the general case on this result. More precisely, we study the exceptional locus of certain quasi-resolutions of a non-covexillary Schubert variety X_w, and we relate the intersection of these loci to Σ_w. Then, by induction on the dimension, we can establish the equality.     

Conditioned Diffusions which are Brownian Bridges

Let X _t be a one-dimensional diffusion of the form dX _t=dB _t+(X _t)dt. Let Tbe a fixed positive number and let be the diffusion process which is X _t conditioned so that X ₀=X _T=x. If the drift is constant, i.e., , then the conditioned diffusion process is a Brownian bridge. In this paper, we show the converse is false. There is a two parameter family of nonlinear drifts with this property.     

Asymptotics for Lp-norms of Fourier series density estimators

LetX ₁, X₂, , be a sequence of independent, identically distributed bounded random variables with a smooth density functionf. We prove that is asymptotically normal, wheref _{m, n} is the Fourier series density estimator offandw is a nonnegative weight function.Communicated by Edward B. Saff.AMS classification: Primary 60F05, 60F25; Secondary 62G05.     

Whitney constants and approximation of -quasi-linear forms by -linear forms

We discuss some relations between Whitney constants w_m(B_X,Y) for bounded functions from, the unit ball of a real normed space X into another real normed space Y. In particular, we generalize a result of Tsar’kov that

to any n-dimensional X (here denotes linearized Whitney constant).     

Reinforced random walk

Summary Leta _i,i1, be a sequence of nonnegative numbers. Difine a nearest neighbor random motion =X ₀,X ₁, ... on the integers as follows. Initially the weight of each interval (i, i+1), i an integer, equals 1. If at timen an interval (i, i+1) has been crossed exactlyk times by the motion, its weight is . Given (X ₀,X ₁, ...,X _n)=(i₀, i₁, ..., i_n), the probability thatX _n+1 isi _n–1 ori _n+1 is proportional to the weights at timen of the intervals (i _n–1,i _n) and (i _n,ii_n+1). We prove that either visits all integers infinitely often a.s. or visits a finite number of integers, eventually oscillating between two adjacent integers, a.s., and that X _n /n=0 a.s. For much more general reinforcement schemes we proveP ( visits all integers infinitely often)+P ( has finite range)=1.Supported by a National Science Foundation Grant     

Some Combinatorial Properties of Schubert Polynomials

Schubert polynomials were introduced by Bernstein et al. and Demazure, and were extensively developed by Lascoux, Schützenberger, Macdonald, and others. We give an explicit combinatorial interpretation of the Schubert polynomial in terms of the reduced decompositions of the permutation w. Using this result, a variation of Schensted's correspondence due to Edelman and Greene allows one to associate in a natural way a certain set of tableaux with w, each tableau contributing a single term to . This correspondence leads to many problems and conjectures, whose interrelation is investigated. In Section 2 we consider permutations with no decreasing subsequence of length three (or 321-avoiding permutations). We show for such permutations that is a flag skew Schur function. In Section 3 we use this result to obtain some interesting properties of the rational function , where denotes a skew Schur function.Sara C. Billey: Supported by the National Physical Science Consortium. William Jockusch: Supported by an NSF Graduate Fellowship. Richard P. Stanley: Partially supported by NSF grants DMS-8901834 and DMS-9206374     

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.     

Leading coefficients of Kazhdan–Lusztig polynomials for Deodhar elements

We show that the leading coefficient of the Kazhdan–Lusztig polynomial P _x,w (q) known as μ(x,w) is always either 0 or 1 when w is a Deodhar element of a finite Weyl group. The Deodhar elements have previously been characterized using pattern avoidance in Billey and Warrington (J. Algebraic Combin. 13(2):111–136, [2001]) and Billey and Jones (Ann. Comb. [2008], to appear). In type A, these elements are precisely the 321-hexagon avoiding permutations. Using Deodhar’s algorithm (Deodhar in Geom. Dedicata 63(1):95–119, [1990]), we provide some combinatorial criteria to determine when μ(x,w)=1 for such permutations w. The author received support from NSF grants DMS-9983797 and DMS-0636297.     

Lower bounds for Kazhdan-Lusztig polynomials from patterns

Kazhdan-Lusztig polynomials P_x,w(q) play an important role in the study of Schubert varieties as well as the representation theory of semisimple Lie algebras. We give a lower bound for the values P_x,w(1) in terms of "patterns". A pattern for an element of a Weyl group is its image under a combinatorially defined map to a subgroup generated by reflections. This generalizes the classical definition of patterns in symmetric groups. This map corresponds geometrically to restriction to the fixed point set of an action of a one-dimensional torus on the flag variety of a semisimple group G. Our lower bound comes from applying a decomposition theorem for "hyperbolic localization" [Br] to this torus action. This gives a geometric explanation for the appearance of pattern avoidance in the study of singularities of Schubert varieties.     

Association Schemes of Quadratic Forms and Symmetric Bilinear Forms

Let X _n and Y _n be the sets of quadratic forms and symmetric bilinear forms on an n-dimensional vector space V over , respectively. The orbits of GL _n( ) on X _n × X _n define an association scheme Qua(n, q). The orbits of GL _n( ) on Y _n × Y _n also define an association scheme Sym(n, q). Our main results are: Qua(n, q) and Sym(n, q) are formally dual. When q is odd, Qua(n, q) and Sym(n, q) are isomorphic; Qua(n, q) and Sym(n, q) are primitive and self-dual. Next we assume that q is even. Qua(n, q) is imprimitive; when (n, q) (2,2), all subschemes of Qua(n, q) are trivial, i.e., of class one, and the quotient scheme is isomorphic to Alt(n, q), the association scheme of alternating forms on V. The dual statements hold for Sym(n, q).     

Strong consistency properties of nonparametric estimators for randomly censored data,I: The product-limit estimator

In reliability and survival-time studies one frequently encounters the followingrandom censorship model:X ₁,Y ₁,X ₂,Y ₂, is an independent sequence of nonnegative rv's, theX _n' s having common distributionF and theY _n' s having common distributionG, Z _n =min{X _n ,Y _n },T _n =I[X _n <-Y _n ]; ifX _n represents the (potential) time to death of then-th individual in the sample andY _n is his (potential) censoring time thenZ _n represents the actual observation time andT _n represents the type of observation (T _n =O is a censoring,T _n =1 is a death). One way to estimateF from the observationsZ ₁.T ₁,Z ₂,T ₂, (and without recourse to theX _n' s) is by means of theproduct limit estimator (Kaplan andMeier [6]). It is shown that a.s., uniformly on [0,T] ifH(T ^–)<1 wherel–H=(l–F) (l–G), uniformly onR if whereT _F =sup {x:F(x)<1}; rates of convergence are also established. These results are used in Part II of this study to establish strong consistency of some density and failure rate estimators based on .The third author's research was partly supported by National Research Council of Canada     

Limit theorems for the ratio of the empirical distribution function to the true distribution function

Summary We consider almost sure limit theorems for and where _n is the empirical distribution function of a random sample ofn uniform (0, 1) random variables anda _n 0. It is shown that (1) ifna _n /log₂ n then both and converge to 1 a.s.; (2) ifna _n /log₂ n=d>0 (d>1) then has an almost surely finite limit superior which is the solution of a certain transcendental equation; and (3) ifna _n /log₂ n0 then and have limit superior + almost surely. Similar results are established for the inverse function _n ^–1 .Supported by the National Science Foundation under MCS 77-02255     

Elementary symmetric polynomials of increasing order

Summary The asymptotic behaviour of elementary symmetric polynomials S _n ^(k) of order k, based on n independent and identically distributed random variables X ₁,..., X _n,is investigated for the case that both k and n get large. If , then the distribution function of a suitably normalised S _n ^(k) is shown to converge to a standard normal limit. The speed of this convergence to normality is of order , provided and certain natural moment assumptions are imposed. This order bound is sharp, and cannot be inferred from one of the existing Berry-Esseen bounds for U-statistics. If k at the rate n ^1/2 then a non-normal weak limit appears, provided the X _i's are positive and S _n ^(k) is standardised appropriately. On the other hand, if k at a rate faster than n ^1/2 then it is shown that for positive X _j'sthere exists no linear norming which causes S _n ^(k) to converge weakly to a nondegenerate weak limit.     

Study of some measures of dependence between order statistics and systems

Let be a random vector, and denote by X_1:n,X_2:n,…,X_n:n the corresponding order statistics. When X₁,X₂,…,X_n represent the lifetimes of n components in a system, the order statistic X_n−k+1:n represents the lifetime of a k-out-of-n system (i.e., a system which works when at least k components work). In this paper, we obtain some expressions for the Pearson’s correlation coefficient between X_i:n and X_j:n. We pay special attention to the case n=2, that is, to measure the dependence between the first and second failure in a two-component parallel system. We also obtain the Spearman’s rho and Kendall’s tau coefficients when the variables X₁,X₂,…,X_n are independent and identically distributed or when they jointly have an exchangeable distribution.