On the multiplicity and the Cohen-Macaulayness of fiber cones of graded algebras

Let A be a Noetherian local ring with the maximal ideal m and an m-primary ideal J. Let S=?_n≥0S_n be a finitely generated standard graded algebra over A. Set S⁺=?_n>0S_n. Denote by F_J(S)=?_n≥0→(S_n/JS_n) the fiber cone of S with respect to J. The paper characterizes the multiplicity and the Cohen-Macaulayness of F_J(S) in terms of minimal reductions of S⁺.     

Completion of a partial integral matrix to a unimodular matrix

We first characterize submatrices of a unimodular integral matrix. We then prove that if n entries of an n × n partial integral matrix are prescribed and these n entries do not constitute a row or a column, then this matrix can be completed to a unimodular matrix. Consequently an n × n partial integral matrix with n − 1 prescribed entries can always be completed to a unimodular matrix.     

On the Courant-Fischer theory for Krein spaces

Let J=I_r⊕-I_n-r,0<r<n. An n×n complex matrix A is said to be J-Hermitian if JA=A^∗J. An extension of the classical theory of Courant and Fischer on the Rayleigh ratio of Hermitian matrices is stated for J-Hermitian matrices. Applications to the theory of small oscilations of a mechanical system are presented.     

On the Picard bundle

Fix a holomorphic line bundle ξ over a compact connected Riemann surface X of genus g, with g?2, and also fix an integer r such that degree(ξ)>r(2g−1). Let Mξ(r) denote the moduli space of stable vector bundles over X of rank r and determinant ξ. The Fourier-Mukai transform, with respect to a Poincaré line bundle on X×J(X), of any F∈Mξ(r) is a stable vector bundle on J(X). This gives an injective map of Mξ(r) in a moduli space associated to J(X). If g=2, then Mξ(r) becomes a Lagrangian subscheme.     

Ideals of Some Matrix Rings

For a commutative ring K the conception of a strongly maximal ideal J was introduced by Kuzucuoglu and Levchuk in 2000. Denote by R_n(K,J) the ring of all n×n-matrices over K with elements from J on and above the main diagonal. Recent results on ideals of the ring R_n(K,J) for this case, ideals of the associated Lie ring and normal subgroups of the adjoint group are considered in this paper. Also ideals of R_n(K,J) for the case of an arbitrary associative ring K with the identity are investigated.     

Required nonzero patterns for nonsingular sign regular matrices

An n×n real matrix is called sign regular if, for each k(1?k?n), all its minors of order k have the same nonstrict sign. The zero entries which can appear in a nonsingular sign regular matrix depend on its signature because the signature can imply that certain entries are necessarily nonzero. The patterns for the required nonzero entries of nonsingular sign regular matrices are analyzed.     

An algorithm for unimodular completion over Laurent polynomial rings

We present a new and simple algorithm for completion of unimodular vectors with entries in a multivariate Laurent polynomial ring over an infinite field K. More precisely, given n?3 and a unimodular vector V=^t(v₁,…,v_n)∈Rⁿ (that is, such that 〈v₁,…,v_n〉=R), the algorithm computes a matrix M in M_n(R) whose determinant is a monomial such that MV=^t(1,0,…,0), and thus M^-1 is a completion of V to an invertible matrix.     

On the completion of a partial integral matrix to a unimodular matrix

We prove that if a partial integral matrix has a free diagonal then this matrix can be completed to a unimodular matrix. Such a condition is necessary in a general sense. Consequently if an n × n (n ? 2) partial integral matrix has 2n − 3 prescribed entries and any n entries of these do not constitute a row or a column then it can be completed to a unimodular matrix. This improves a recent result of Zhan.     

Unitary equivalence to a complex symmetric matrix: An algorithm

We present a necessary and sufficient condition for a 3×3 matrix to be unitarily equivalent to a symmetric matrix with complex entries, and an algorithm whereby an arbitrary 3×3 matrix can be tested. This test generalizes to a necessary and sufficient condition that applies to almost every n×n matrix. The test is constructive in that it explicitly exhibits the unitary equivalence to a complex symmetric matrix.     

Number theory of matrix semigroups

Unlike factorization theory of commutative semigroups which are well-studied, very little literature exists describing factorization properties in noncommutative semigroups. Perhaps the most ubiquitous noncommutative semigroups are semigroups of square matrices and this article investigates the factorization properties within certain subsemigroups of M_n(Z), the semigroup of n×n matrices with integer entries. Certain important invariants are calculated to give a sense of how unique or non-unique factorization is in each of these semigroups.     

Properties of chains of prime ideals in an amalgamated algebra along an ideal

Let f:A→B be a ring homomorphism and let J be an ideal of B. In this paper, we study the amalgamation of A with B along J with respect to f (denoted by A?^fJ), a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced and studied by D’Anna and Fontana in 2007, and other classical constructions (such as the A+XB[X], the A+XB?X? and the D+M constructions). In particular, we completely describe the prime spectrum of the amalgamated duplication and we give bounds for its Krull dimension.     

Circular law, extreme singular values and potential theory

Consider the empirical spectral distribution of complex random n×n matrix whose entries are independent and identically distributed random variables with mean zero and variance 1/n. In this paper, via applying potential theory in the complex plane and analyzing extreme singular values, we prove that this distribution converges, with probability one, to the uniform distribution over the unit disk in the complex plane, i.e. the well known circular law, under the finite fourth moment assumption on matrix elements.     

The spectrum is continuous on the set of quasi-n-hyponormal operators

In this paper it is shown that the spectrum σ, a set-valued function, is continuous when the function is restricted to the set of all ‘quasi-n-hyponormal’ operators acting on an infinite-dimensional separable Hilbert space, where a quasi-n-hyponormal operator is defined to be unitarily equivalent to an n×n upper triangular operator matrix whose diagonal entries are hyponormal operators.     

Subnormal structure of non-stable unitary groups over rings

Let R be a commutative ring with identity in which 2 is invertible. Let H denote a subgroup of the unitary group U(2n,R,Λ) with n≥4. H is normalized by EU(2n,J,Γ^J) for some form ideal (J,Γ^J) of the form ring (R,Λ). The purpose of the paper is to prove that H satisfies a “sandwich” property, i.e. there exists a form ideal (I,Γ^I) such that

EU(2n,IJ⁸Γ^J,Γ)⊆H⊆CU(2n,I,Γ^I).     

A combinatorial problem associated with nonograms

Associated with an m × n matrix with entries 0 or 1 are the m-vector of row sums and n-vector of column sums. In this article we study the set of all pairs of these row and column sums for fixed m and n. In particular, we give an algorithm for finding all such pairs for a given m and n.     

Filtrations,rees rings,and ideal transforms

Let I be an ideal, and let f = {K_n|n ≥ 0 } be a filtration of the Noetherian ring R, such that Iⁿ ⊆ K_n for all n ≥ 0. We study when the Rees ring $\text{R}$(f) is either finite or integral over the Rees ring $\text{R}$(I), for two types of filtrations f which have recently drawn interest. If I and J are ideals in R, and if m(n) is the least power of J such that (Iⁿ : J^{m(n) + 1}), we show that the function m(n) is eventually non-decreasing. For J regular, we characterize when it is eventually constant.     

Deformations of the generalised Picard bundle

Let X be a nonsingular algebraic curve of genus g?3, and let M_ξ denote the moduli space of stable vector bundles of rank n?2 and degree d with fixed determinant ξ over X such that n and d are coprime. We assume that if g=3 then n?4 and if g=4 then n?3, and suppose further that n₀, d₀ are integers such that n₀?1 and nd₀+n₀d>nn₀(2g-2). Let E be a semistable vector bundle over X of rank n₀ and degree d₀. The generalised Picard bundle W_ξ(E) is by definition the vector bundle over M_ξ defined by the direct image where U_ξ is a universal vector bundle over X×M_ξ. We obtain an inversion formula allowing us to recover E from W_ξ(E) and show that the space of infinitesimal deformations of W_ξ(E) is isomorphic to H¹(X,End(E)). This construction gives a locally complete family of vector bundles over M_ξ parametrised by the moduli space M(n₀,d₀) of stable bundles of rank n₀ and degree d₀ over X. If (n₀,d₀)=1 and W_ξ(E) is stable for all E∈M(n₀,d₀), the construction determines an isomorphism from M(n₀,d₀) to a connected component M⁰ of a moduli space of stable sheaves over M_ξ. This applies in particular when n₀=1, in which case M⁰ is isomorphic to the Jacobian J of X as a polarised variety. The paper as a whole is a generalisation of results of Kempf and Mukai on Picard bundles over J, and is also related to a paper of Tyurin on the geometry of moduli of vector bundles.     

On the numbers of positive entries of reducible nonnegative matrices

Let RM(n,d) be the class {A∣A is an n×n reducible nonnegative matrix and the greatest common divisor of the lengths of all cycles in D(A) is d}, where D(A) is the associated digraph of A. In this paper we determine the set of numbers of positive entries of A for A∈RM(n,d). We also characterize the reducible nonnegative matrices with the maximum and minimum numbers of positive entries.     

Sparse anti-magic squares and vertex-magic labelings of bipartite graphs

A sparse anti-magic square is an n×n array whose non-zero entries are the consecutive integers 1,…,m for some m?n² and whose row-sums and column-sums form a set of consecutive integers. We derive some basic properties of these arrays and provide constructions for several infinite families of them. Our main interest in these arrays is their application to constructing vertex-magic labelings for bipartite graphs.     

The intricacy of avoiding arrays is 2

Let A be any n×n array on the symbols [n], with at most one symbol in each cell. An n×n Latin square L avoids A if all entries in L differ from the corresponding entries in A. If A is split into two arrays B and C in a special way, there are Latin squares L_B and L_C avoiding B and C, respectively. In other words, the intricacy of avoiding arrays is 2, the number of arrays into which A has to be split.