Alternating Sign Matrices and Some Deformations of Weyl's Denominator Formulas

An alternating sign matrix is a square matrix whose entries are 1, 0, or –1, and which satisfies certain conditions. Permutation matrices are alternating sign matrices. In this paper, we use the (generalized) Littlewood's formulas to expand the products and 2 as sums indexed by sets of alternating sign matrices invariant under a 180° rotation. If we put t = 1, these expansion formulas reduce to the Weyl's denominator formulas for the root systems of type B _n and C _n. A similar deformation of the denominator formula for type D _n is also given.     

Another construction for large sets of Kirkman triple systems

A large set of Kirkman triple systems of order v, denoted by LKTS(v), is a collection , where every is a KTS(v) and all form a partition of all triples on X. In this article, we give a new construction for LKTS(6v + 3) via OLKTS(2v + 1) with a special property and obtain new results for LKTS, that is there exists an LKTS(3v) for , where p, q ≥ 0, r _i , s _j ≥ 1, q _i is a prime power and mod 12.      

Small bound for birational automorphism groups of algebraic varieties (with an Appendix by Yujiro Kawamata)

We give an effective upper bound of |Bir(X)| for the birational automorphism group of an irregular n-fold (with n = 3) of general type in terms of the volume V = V(X) under an “albanese smoothness and simplicity” condition. To be precise, . An optimum linear bound is obtained for those threefolds with non-maximal albanese dimension. For all n ≥ 3, a bound is obtained when alb _X is generically finite, alb(X) is smooth and Alb(X) is simple. The author is supported by an Academic Research Fund of NUS.     

An Extension of the Admissibility-Type Conditions for the Exponential Dichotomy of C 0-Semigroups

In the present paper we obtain a sufficient condition for the exponential dichotomy of a strongly continuous, one-parameter semigroup , in terms of the admissibility of the pair . It is already known the equivalence between the -admissibility condition and and the hyperbolicity of a C ₀-semigroup , when we assume a priori that the kernel of the dichotomic projector (denoted here by X ₂) is T(t)-invariant and is an invertible operator. We succeed to prove in this paper that the admissibility of the pair still implies the existence of an exponential dichotomy for a C ₀-semigroup even in the general case where the kernel of the dichotomic projector, X ₂, is not assumed to be T(t)-invariant.      

Dirac’s Type Sufficient Conditions for Hamiltonicity and Pancyclicity

Let G = (V, E) be a any simple, undirected graph on n ≥ 3 vertices with the degree sequence . We consider the class of graphs satisfying the condition where , is a positive integer. It is known that is hamiltonian if θ ≤ δ. In this paper,

Exact Distribution of the Local Score for Markovian Sequences

Let be a sequence of letters taken in a finite alphabet Θ. Let be a scoring function and the corresponding score sequence where X _i = s(A _i ). The local score is defined as follows: . We provide the exact distribution of the local score in random sequences in several models. We will first consider a Markov model on the score sequence , and then on the letter sequence . The exact P-value of the local score obtained with both models are compared thanks to several datasets. They are also compared with previous results using the independent model.     

Orbits of the hyperoctahedral group as Euclidean designs

The hyperoctahedral group H in n dimensions (the Weyl group of Lie type B _n ) is the subgroup of the orthogonal group generated by all transpositions of coordinates and reflections with respect to coordinate hyperplanes.With e ₁ , ..., e _n denoting the standard basis vectors of ⁿ and letting x _k = e ₁ + ··· + e _k (k = 1, 2, ..., n), the set

Notes on estimating inverse-Gaussian and gamma subordinators under high-frequency sampling

We study joint efficient estimation of two parameters dominating either the inverse-Gaussian or gamma subordinator, based on discrete observations sampled at satisfying as . Under the condition that as we have two kinds of optimal rates, and . Moreover, as in estimation of diffusion coefficient of a Wiener process the -consistent component of the estimator is effectively workable even when T _n does not tend to infinity. Simulation experiments are given under several h _n ’s behaviors.     

A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ) ^m u  =  u ^q in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .      

On the VC-dimension of uniform hypergraphs

Let be a k-uniform hypergraph on [n] where k−1 is a power of some prime p and n≥ n ₀(k). Our main result says that if , then there exists E ₀∊ such that {E∩ E ₀: E∊ } contains all subsets of E ₀. This improves a longstanding bound of due to Frankl and Pach [7].Research supported in part by NSF grants DMS-0400812 and an Alfred P. Sloan Research Fellowship.Research supported in part by NSA grant H98230-05-1-0079. Part of this research was done while working at University of Illinois at Chicago.     

The Spectrum of a Random Geometric Graph is Concentrated

Consider n points, x ₁,... , x _n , distributed uniformly in [0, 1] ^d . Form a graph by connecting two points x _i and x _j if . This gives a random geometric graph, , which is connected for appropriate r(n). We show that the spectral measure of the transition matrix of the simple random walk on is concentrated, and in fact converges to that of the graph on the deterministic grid.      

A remark on the conjecture of Erdös, Faber and Lovász

The celebrated Erd?s, Faber and Lovász Conjecture may be stated as follows: Any linear hypergraph on ν points has chromatic index at most ν. We show that the conjecture is equivalent to the following assumption: For any graph , where ν(G) denotes the linear intersection number and χ(G) denotes the chromatic number of G. As we will see for any graph G = (V, E), where denotes the complement of G. Hence, at least G or fulfills the conjecture.      

Higher integrability of the gradient for vectorial minimizers of decomposable variational integrals

We consider local minimizers of variational integrals , where F is of anisotropic (p, q)-growth with exponents . If F is in a certain sense decomposable, we show that the dimensionless restriction together with the local boundedness of u implies local integrability of for all exponents . More precisely, the initial exponents for the integrability of the partial derivatives can be increased by two, at least locally. If n = 2, then we use these facts to prove -regularity of u for any exponents .     

Minimizing total completion time for UET tasks with release time and outtree precedence constraints

Brucker et al. (Math Methods Oper Res 56: 407–412, 2003) have given an O(n ²)-time algorithm for the problems , outtree and , outtree . In this note, we show that their algorithm admits an O(n log n)-time implementation.     

On hyperbolicity cones associated with elementary symmetric polynomials

Elementary symmetric polynomials can be thought of as derivative polynomials of . Their associated hyperbolicity cones give a natural sequence of relaxations for . We establish a recursive structure for these cones, namely, that the coordinate projections of these cones are themselves hyperbolicity cones associated with elementary symmetric polynomials. As a consequence of this recursion, we give an alternative characterization of these cones, and give an algebraic characterization for one particular dual cone associated with together with its self-concordant barrier functional.     

Harmonic Functions for a Class of Integro-differential Operators

We consider the operator defined on functions by

Analysis of degenerate elliptic operators of Grušin type

We analyse degenerate, second-order, elliptic operators H in divergence form on L ₂(R ⁿ  × R ^m ). We assume the coefficients are real symmetric and a ₁ H _δ  ≥ H ≥ a ₂ H _δ for some a ₁, a ₂ > 0 where

Association schemes from the action of PGL(2, q) fixing a nonsingular conic in PG(2, q)

The group PGL(2,q) has an embedding into PGL(3,q) such that it acts as the group fixing a nonsingular conic in PG(2,q). This action affords a coherent configuration (q) on the set (q) of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions ₊(q) and ₋(q) of (q) to the set ₊(q) of secant (hyperbolic) lines and to the set ₋(q) of exterior (elliptic) lines, respectively, are both association schemes; moreover, we show that the elliptic scheme ₋(q) is pseudocyclic.We further show that the coherent configurations (q ²) with q even allow certain fusions. These provide a 4-class fusion of the hyperbolic scheme ₊(q ²), and 3-class fusions and 2-class fusions (strongly regular graphs) of both schemes ₊(q ²) and ₋(q ²). The fusion results for the hyperbolic case are known, but our approach here as well as our results in the elliptic case are new.     

Variational integrals of splitting-type: higher integrability under general growth conditions

Besides other things we prove that if , , locally minimizes the energy