Cohen-Macaulay, shellable and unmixed clutters with a perfect matching of König type

Let C be a clutter with a perfect matching e₁,…,e_g of König type and let Δ_C be the Stanley-Reisner complex of the edge ideal of C. If all c-minors of C have a free vertex and C is unmixed, we show that Δ_C is pure shellable. We are able to describe, in combinatorial and algebraic terms, when Δ_C is pure. If C has no cycles of length 3 or 4, then it is shown that Δ_C is pure if and only if Δ_C is pure shellable (in this case e_i has a free vertex for all i), and that Δ_C is pure if and only if for any two edges f₁,f₂ of C and for any e_i, one has that f₁∩e_i⊂f₂∩e_i or f₂∩e_i⊂f₁∩e_i. It is also shown that this ordering condition implies that Δ_C is pure shellable, without any assumption on the cycles of C. Then we prove that complete admissible uniform clutters and their Alexander duals are unmixed. In addition, the edge ideals of complete admissible uniform clutters are facet ideals of shellable simplicial complexes, they are Cohen-Macaulay, and they have linear resolutions. Furthermore if C is admissible and complete, then C is unmixed. We characterize certain conditions that occur in a Cohen-Macaulay criterion for bipartite graphs of Herzog and Hibi, and extend some results of Faridi-on the structure of unmixed simplicial trees-to clutters with the König property without 3-cycles or 4-cycles.     

Balanced Leonard pairs

Let K denote a field, and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A : V → V and A^∗ : V → V that satisfy the following two conditions:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

Let (respectively v₀, v₁, … , v_d) denote a basis for V that satisfies (i) (respectively (ii)). For 0 ? i ? d, let a_i denote the coefficient of , when we write as a linear combination of , and let denote the coefficient of v_i, when we write A^∗v_i as a linear combination of v₀, v₁, … , v_d.In this paper we show a₀ = a_d if and only if . Moreover we show that for d ? 1 the following are equivalent; (i) a₀ = a_d and a₁ = a_d−1; (ii) and ; (iii) a_i = a_d−i and for 0 ? i ? d. These give a proof of a conjecture by the second author. We say A, A^∗ is balanced whenever a_i = a_d−i and for 0 ? i ? d. We say A,A^∗ is essentially bipartite (respectively essentially dual bipartite) whenever a_i (respectively ) is independent of i for 0 ? i ? d. Observe that if A, A^∗ is essentially bipartite or dual bipartite, then A, A^∗ is balanced. For d ≠ 2, we show that if A, A^∗ is balanced then A, A^∗ is essentially bipartite or dual bipartite.     

Depths and Cohen–Macaulay properties of path ideals

Given a tree T on n vertices, there is an associated ideal I   of R[_x1,…,_xn]$R[x_1,\dots ,x_n]$ generated by all paths of a fixed length ? of T  . We classify all trees for which R/I$R/I$ is Cohen–Macaulay, and we show that an ideal I whose generators correspond to any collection of subtrees of T satisfies the König property. Since the edge ideal of a simplicial tree has this form, this generalizes a result of Faridi. Moreover, every square-free monomial ideal can be represented (non-uniquely) as a subtree ideal of a graph, so this construction provides a new combinatorial tool for studying square-free monomial ideals.     

An eigenvalue localization theorem for pentadiagonal symmetric Toeplitz matrices

We express the eigenvalues of a pentadiagonal symmetric Toeplitz matrix as the zeros of explicitly given rational functions.     

Explicit inverse of a tridiagonal k−Toeplitz matrix

Summary We obtain explicit formulas for the entries of the inverse of a nonsingular and irreducible tridiagonal k–Toeplitz matrix A. The proof is based on results from the theory of orthogonal polynomials and it is shown that the entries of the inverse of such a matrix are given in terms of Chebyshev polynomials of the second kind. We also compute the characteristic polynomial of A which enables us to state some conditions for the existence of A^–1. Our results also extend known results for the case when the residue mod k of the order of A is equal to 0 or k–1 (Numer. Math., 10 (1967), pp. 153–161.).The work was supported by CMUC (Centro de Matemática da Universidade de Coimbra) and by Acção Integrada Luso-Espanhola E-6/03     

Fast Fourier-Galerkin methods for solving singular boundary integral equations: Numerical integration and precondition

We develop a fast fully discrete Fourier-Galerkin method for solving a class of singular boundary integral equations. We prove that the number of multiplications used in generating the compressed matrix is O(nlog³n), and the solution of the proposed method preserves the optimal convergence order O(n^−t), where n is the order of the Fourier basis functions used in the method and t denotes the degree of regularity of the exact solution. Moreover, we propose a preconditioning which ensures the numerical stability when solving the preconditioned linear system. Numerical examples are presented to confirm the theoretical estimates and to demonstrate the approximation accuracy and computational efficiency of the proposed algorithm.     

Multigraded Betti numbers of simplicial forests

We prove that multigraded Betti numbers of a simplicial forest are always either 0 or 1. Moreover a nonzero multidegree appears exactly in one homological degree in the resolution. Our work generalizes work of Bouchat [2] on edge ideals of graph trees.     

On the associated primes and the depth of the second power of squarefree monomial ideals

We present combinatorial characterizations for the associated primes of the second power of squarefree monomial ideals and criteria for this power to have positive depth or depth greater than one.     

The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem

A Cohen-Macaulay complex is said to be balanced of typea=(a ₁,a ₂, ...,a _s ) if its vertices can be colored usings colors so that every maximal face gets exactlya _i vertices of thei:th color. Forb=(b ₁,b ₂, ...,b _s ), 0≦b≦a, letf _b denote the number of faces havingb _i vertices of thei:th color. Our main result gives a characterization of thef-vectorsf=(f _b )_0≦b≦a′ or equivalently theh-vectors, which can arise in this way from balanced Cohen-Macaulay complexes. As part of the proof we establish a generalization of Macaulay’s compression theorem to colored multicomplexes. Finally, a combinatorial shifting technique for multicomplexes is used to give a new simple proof of the original Macaulay theorem and another closely related result. First and third authors partially supported by the National Science Foundation.     

Neighborhood Complexes of Stable Kneser Graphs

It is shown that the neighborhood complexes of a family of vertex critical subgraphs of Kneser graphs—the stable Kneser graphs introduced by L. Schrijver—are spheres up to homotopy. Furthermore, it is shown that the neighborhood complexes of a subclass of the stable Kneser graphs contain the boundaries of associahedra (simplicial complexes encoding triangulations of a polygon) as a strong deformation retract.* The first author was partially supported by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine. The second author was supported by the graduate school Algorithmische Diskrete Mathematik, which is funded by the Deutsche Forschungsgemeinschaft, grant GRK 219/3. The DAAD partially supported a stay at KTH, Stockholm, in December 1998, where this work was done: DAAD program AZ 313/S-PPP     

On the closed representation for the inverses of Hessenberg matrices

The general representation for the elements of the inverse of any Hessenberg matrix of finite order is here extended to the reduced case with a new proof. Those entries are given with proper Hessenbergians from the original matrix. It justifies both the use of linear recurrences of unbounded order for such computations on matrices of intermediate order, and some elementary properties of the inverse. These results are applied on the resolvent matrix associated to a finite Hessenberg matrix in standard form. Two examples on the unit disk are given.     

Embedded associated primes of powers of square-free monomial ideals

An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/I^t)=Ass(R/I) for all t≥1. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a square-free monomial ideal I is minimally not normally torsion-free then the least power t such that I^t has embedded primes is bigger than β₁, where β₁ is the monomial grade of I, which is equal to the matching number of the hypergraph H(I) associated to I. If, in addition, I fails to have the packing property, then embedded primes of I^t do occur when t=β₁+1. As an application, we investigate how these results relate to a conjecture of Conforti and Cornuéjols.     

On summation formulas due to Plana, Lindelöf and Abel, and related Gauss-christoffel rules, I

Three methods, old but not so well known, transform an infinite series into a complex integral over an infinite interval. Gauss quadrature rules are designed for each of them. Various questions concerning their construction and application are studied, theoretically or experimentally. They are so efficient that they should be considered for the development of software for special functions. Applications are made to slowly convergent alternating and positive series, to Fourier series, to the numerical analytic continuation of power series outside the circle of convergence, and to ill-conditioned power series.     

Strongly irreducible ideals of a commutative ring

An ideal I of a ring R is said to be strongly irreducible if for ideals J and K of R, the inclusion J∩K⊆I implies that either J⊆I or K⊆I. The relationship among the families of irreducible ideals, strongly irreducible ideals, and prime ideals of a commutative ring R is considered, and a characterization is given of the Noetherian rings which contain a non-prime strongly irreducible ideal.     

Multivariate Frobenius–Padé approximants: Properties and algorithms

The aim of this paper is to construct rational approximants for multivariate functions given by their expansion in an orthogonal polynomial system. This will be done by generalizing the concept of multivariate Padé approximation. After defining the multivariate Frobenius–Padé approximants, we will be interested in the two following problems: the first one is to develop recursive algorithms for the computation of the value of a sequence of approximants at a given point. The second one is to compute the coefficients of the numerator and denominator of the approximants by solving a linear system. For some particular cases we will obtain a displacement rank structure for the matrix of the system we have to solve. The case of a Tchebyshev expansion is considered in more detail.     

On the Gorenstein injective dimension and Bass formula

In this paper, we compare Krull dimension, Gorenstein injective dimension and injective dimension of a module in several cases. In fact, we establish some generalizations of the Bass formula. To this end, we generalize the Grothendieck non-vanishing theorem to a class of modules larger than finitely generated modules. Received: 21 May 2007     

A new type of Hermite matrix polynomial series

Conventional Hermite polynomials emerge in a great diversity of applications in mathematical physics, engineering, and related fields. However, in physical systems with higher degrees of freedom it will be of practical interest to extend the scalar Hermite functions to their matrix analogue. This work introduces various new generating functions for Hermite matrix polynomials and examines existence and convergence of their associated series expansion by using Mehler’s formula for the general matrix case. Moreover, we derive interesting new relations for even- and odd-power summation in the generating-function expansion containing Hermite matrix polynomials. Some new results for the scalar case are also presented.     

A characterization of Cohen-Macaulay modules and local cohomology

Let R be a commutative Noetherian ring, and let N be a non-zero finitely generated locally quasi-unmixed R-module. In this paper, the main result asserts that N is Cohen-Macaulay if and only if, for any N-proper ideal I of R generated by height_N I elements, the set of asymptotic primes of I with respect to N is equal to the set of presistent primes of I with respect to N. In addition, some applications about local cohomology are included. Received: 3 July 2005