A Theorem on Higher Bruhat Orders

We show that inclusion order and single-step inclusion coincide for higher Bruhat orders B(n,2) , i.e., . Received April 7, 1998.     

Unit Interval Orders of Open and Closed Intervals

A poset P = (X, ?) is a unit OC interval order if there exists a representation that assigns an open or closed real interval I(x) of unit length to each x ∈ P so that x ? y in P precisely when each point of I (x) is less than each point in I (y). In this paper we give a forbidden poset characterization of the class of unit OC interval orders and an efficient algorithm for recognizing the class. The algorithm takes a poset P as input and either produces a representation or returns a forbidden poset induced in P.     

Extrapolation for Higher Orders of Convergence

A new extrapolation procedure which encompasses Aitken's 2 processand extends to higher orders of convergence is presented. Thenew extrapolation yields information as to whether an algorithmis converging or diverging as well as a new approximation tothe solution point. The new extrapolation technique is comparedwith existing higher order extrapolation techniques. The valueof the new extrapolation is demonstrated analytically and empirically.     

A Note on the Homotopy Type of the Alexander Dual

We investigate the homotopy type of the Alexander dual of a simplicial complex. It is known that in general the homotopy type of \(K\) does not determine the homotopy type of its dual \(K^*\) . We construct for each finitely presented group \(G\) , a simply connected simplicial complex \(K\) such that \(\pi _1(K^*)=G\) and study sufficient conditions on \(K\) for \(K^*\) to have the homotopy type of a sphere. We extend the simplicial Alexander duality to the more general context of reduced lattices and relate this construction with Bier spheres using deleted joins of lattices. Finally we introduce an alternative dual, in the context of reduced lattices, with the same homotopy type as the Alexander dual but smaller and simpler to compute.     

A Point Process with Second Order MARKOV Dependent Intervals

The limiting behaviour of the point process with 2nd order MARKOV-dependent intervals is analysed through the use of “regeneration time”. The results for the case of periodic MARKOV-dependent intervals are briefly mentioned.     

Discrete Morse Theory and the Homotopy Type of Clique Graphs

We attach topological concepts to a simple graph by means of the simplicial complex of its complete subgraphs. Using Forman’s discrete Morse theory we show that the strong product of two graphs is homotopic to the topological product of the spaces of their complexes. As a consequence, we enlarge the class of clique divergent graphs known to be homotopy equivalent to all its iterated clique graphs.     

Chains in the Bruhat order

We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Littlewood-Richardson coefficients. In the second half of the paper, we study the classical flag manifold and discuss related combinatorial objects: flagged Schur polynomials, 312-avoiding permutations, generalized Gelfand-Tsetlin polytopes, the inverse Schubert-Kostka matrix, parking functions, and binary trees. A.P. was supported in part by National Science Foundation grant DMS-0201494 and by Alfred P. Sloan Foundation research fellowship. R.S. was supported in part by National Science Foundation grant DMS-9988459.     

高阶各向同性张量的搜求方法

基于单位张量是唯一带有根本性意义的二阶各向同性张量的结论,系统推演搜求高阶各向同性张量的具体方法.     

On Eigenvalue Intervals for Discrete Second Order Boundary Value Problems

In this paper, we consider discrete second order three-point boundary value problems. By exploring the properties of the associated Green's function and applying Guo-Krasnosel'skii's fixed point theorem, we show the existence of eigenvalue intervals.     

Homotopy Type of Complexes of Graph Homomorphisms between Cycles

In this paper we study the homotopy type of Hom(C_m,C_n), where C_k is the cyclic graph with k vertices. We enumerate connected components of Hom(C_m,C_n) and show that each such component is either homeomorphic to a point or homotopy equivalent to S¹. Moreover, we prove that Hom(C_m,L_n) is either empty or is homotopy equivalent to the union of two points, where L_n is an n-string, i.e., a tree with n vertices and no branching points.     

Tangent Bundle of a Manifold and its Homotopy Type

There is a homotopy equivalence :MM' between closed smooth manifoldsof an odd dimension such that *TM', TM are stably isomorphicbut not isomorphic to each other.     

The Second Lacuna in the Orders of Movement Groups of the Spaces U n

We prove the nonexistence of symmetrically linearly connected spaces of hyperplane elements admitting a movement group G _r with n ² - n + 2 r n² - 1, n 5.     

Pattern avoidance and the Bruhat order

The structure of order ideals in the Bruhat order for the symmetric group is elucidated via permutation patterns. The permutations with boolean principal order ideals are characterized. These form an order ideal which is a simplicial poset, and its rank generating function is computed. Moreover, the permutations whose principal order ideals have a form related to boolean posets are also completely described. It is determined when the set of permutations avoiding a particular set of patterns is an order ideal, and the rank generating functions of these ideals are computed. Finally, the Bruhat order in types B and D is studied, and the elements with boolean principal order ideals are characterized and enumerated by length.     

On the Upper Bound of Second Eigenvalues for Uniformly Elliptic Operators of any Orders

Abstract Let Ω R~m(m≥1)be a bounded domain with piecewise smooth boundary Ω.Let t and r bepositive integers with t>r+1. We consider the eigenvalue problems(1.1)and(12),and obtain Theorem 1and Theorem 2, which generalize the results in[1,2.5].