Representation dimension of m-replicated algebras

Let A be a finite-dimensional hereditary algebra over an algebraically closed field and A ^(m) be the m-replicated algebra of A. We prove that the representation dimension of A ^(m) is at most 3, and that the dominant dimension of A ^(m) is at least m.     

The Moore-Penrose inverse of a morphism with factorization

Given an m-by-n matrix A of rank r over a field with an involutory automorphism, it is well known that A has a Moore-Penrose inverse if and only if rank A¹A=r= rank AA¹. By use of the full-rank factorization theorem, this result may be restated in the category of finite matrices as follows: if (A₁, r, A₂) is an (epic, monic) factorization of A:m→n through r, then A has a Moore-Penrose inverse if and only if (A¹A₁, r, A₂) and (A₁, r, A₂A¹) are, respectively, (epic, monic) factorizations of A¹A:n→n and AA¹:m→m through r. This characterization of the existence of Moore-Penrose inverses is extended to arbitrary morphisms with (epic, monic) factorizations.     

Doubly stochastic matrices with some equal diagonal sums

Let A be doubly stochastic, and let τ₁,…,τ_m be m mutually disjoint zero diagonals in A, 1?m?n-1. E. T. H. Wang conjectured that if every diagonal in A disjoint from each τ_k (k=1,…,m) has a constant sum, then all entries in A off the m zero diagonals τ_k are equal to (n?m)^-1. Sinkhorn showed the conjecture to be correct. In this paper we generalize this result for arbitrary doubly stochastic zero patterns.     

Balancedness,nonemptiness of the core and the existence of a majority equilibrium

Consider a society with a finite number,n, of individuals who have to choose an alternative from a setA in them-dimensional Euclidean Space, R ^m . Assuming that the preference relation overA of every individual is convex and continuous,Greenberg [1979] showed that if the set of winning coalitions (i.e. those that have the veto power) consists of all coalitions with more thanmn/(m+1) individuals, then there exists an alternativea ^* ?A which is a majority equilibrium, or equivalently, that the core of the induced game is nonempty. In the present paper we strengthen this result by showing that this game not only has a nonempty core but is also balanced.     

Function algebras on which homomorphisms are point evaluations on sequences

In the study of the spectrum of a subalgebraA ofC(X), whereX is a completely regular Hausdorff space, a key question is, whether each homomorphism ?:A→R has the point evaluation property for sequences inA, that is whether, for each sequence (f _n ) inA, there exists a pointa inX such that ?(f _n )=f _n (a) for alln. In this paper it is proved that all algebras, which are closed under composition with functions inC ^∞ (R) and have a certain local property, have the point evaluation property for sequences. Such algebras are, for instance, the spaceC ^m (E) (m=0,1,...,∞) ofC ^m -functions on any real locally convex spaceE. This result yields in a trivial manner that each homomorphism ? onA is a point evaluation, ifX is Lindelöf or ifA contains a sequence which separates points inX. Further, also a well known result as well as some new ones are obtained as a consequence of the main theorem.     

Asymptotic expansions of the ordered spectrum of symmetric matrices

In this work, we build on ideas of Torki (2001 [6]) and show that if a symmetric matrix-valued map t?A(t) has a one-sided asymptotic expansion at t=0⁺ of order K then so does t?λ_m(A(t)), where λ_m is the mth largest eigenvalue. We derive formulas for computing the coefficients A₀,A₁,…,A_K in the asymptotic expansion. As an application of the approach we give a new proof of a classical result due to Kato (1976 [3]) about the one-sided analyticity of the ordered spectrum under analytic perturbations. Finally, as a demonstration of the derived formulas, we compute the first three terms in the asymptotic expansion of λ_m(A+tE) for any fixed symmetric matrices A and E.     

Universal terms of complexity three

If there is a function g such that g ^m mapsX/C bijectively ontoX, wherem is a positive integer and whereC is a finite subset ofX, thenm is a factor of |C|. This fact serves to establish the necessary and sufficient conditions for the wordA ^kB^mAⁿ to represent the successor and the predecessor functions on ε. These conditions entail that there is no nontrivial point universal word of complexity three.     

All spectral dominant norms are stable

A vector norm |·|on the space of n×n complex valued matrices is called stable if for some constant K&>0, not depending upon A or m, we have |A^m|?K|A|^m We show that such a norm is stable if and only if it dominates the spectralradius.     

Some lower bounds for the spectral radius of matrices using traces

Let A be an n×n matrix with eigenvalues λ₁,λ₂,…,λ_n, and let m be an integer satisfying rank(A)?m?n. If A is real, the best possible lower bound for its spectral radius in terms of m, trA and trA² is obtained. If A is any complex matrix, two lower bounds for are compared, and furthermore a new lower bound for the spectral radius is given only in terms of trA,trA²,‖A‖,‖A^∗A-AA^∗‖,n and m.     

Extremal Systems with Upper-Bounded Odd Intersections

In this paper we determine the maximum cardinality m of a family A= {A ₁,..., A _m} of subsets of a set of n elements with the property that for each A _i there are at most s A _j such that |A _i ∩ A _j | is odd (even). A tight upper bound is given in the case s < c(2 ^n,2/n). In the remaining cases we give an asymptotically tight upper bound. As an application we give a tight upper-bound for the cardinality of a family with even multi-intersection. Both results generalize a result of Berlekamp and Graver.     

Recursion relations for the extended Krylov subspace method

The evaluation of matrix functions of the form f(A)v, where A is a large sparse or structured symmetric matrix, f is a nonlinear function, and v is a vector, is frequently subdivided into two steps: first an orthonormal basis of an extended Krylov subspace of fairly small dimension is determined, and then a projection onto this subspace is evaluated by a method designed for small problems. This paper derives short recursion relations for orthonormal bases of extended Krylov subspaces of the type K^m,mi+1(A)=span{A^-m+1v,…,A^-1v,v,Av,…,A^miv}, m=1,2,3,…, with i a positive integer, and describes applications to the evaluation of matrix functions and the computation of rational Gauss quadrature rules.     

On D. Hägele’s approach to the Bessis-Moussa-Villani conjecture

The reformulation of the Bessis-Moussa-Villani (BMV) conjecture given by Lieb and Seiringer asserts that the coefficient α_m,k(A,B) of t^k in the polynomial Tr(A+tB)^m, with A,B positive semidefinite matrices, is nonnegative for all m,k. We propose a natural extension of a method of attack on this problem due to Hägele, and investigate for what values of m,k the method is successful, obtaining a complete determination when either m or k is odd.     

Preconditioned conjugate gradient method for generalized least squares problems

A variant of the preconditioned conjugate gradient method to solve generalized least squares problems is presented. If the problem is min (Ax − b)^TW⁻¹(Ax − b) with A ∈ R^m×n and W ∈ R^m×m symmetric and positive definite, the method needs only a preconditioner A₁ ∈ R^n×n, but not the inverse of matrix W or of any of its submatrices. Freund's comparison result for regular least squares problems is extended to generalized least squares problems. An error bound is also given.     

On averaging Frankl's conjecture for large union-closed-sets

Let F be a union-closed family of subsets of an m-element set A. Let n=|F|?2 and for a∈A let s(a) denote the number of sets in F that contain a. Frankl's conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element a∈A with n−2s(a)?0. Strengthening a result of Gao and Yu [W. Gao, H. Yu, Note on the union-closed sets conjecture, Ars Combin. 49 (1998) 280-288] we verify the conjecture for the particular case when m?3 and n?m²−²m/2. Moreover, for these “large” families F we prove an even stronger version via averaging. Namely, the sum of the n−2s(a), for all a∈A, is shown to be non-positive. Notice that this stronger version does not hold for all union-closed families; however we conjecture that it holds for a much wider class of families than considered here. Although the proof of the result is based on elementary lattice theory, the paper is self-contained and the reader is not assumed to be familiar with lattices.     

Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition

In this paper, first we introduce a new notion of commuting condition that φφ ₁ A = A φ ₁ φ between the shape operator A and the structure tensors φ and φ ₁ for real hypersurfaces in G ₂(? ^m+2). Suprisingly, real hypersurfaces of type (A), that is, a tube over a totally geodesic G ₂(? ^m+1) in complex two plane Grassmannians G ₂(? ^m+2) satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in G ₂(? ^m+2) satisfying the commuting condition. Finally we get a characterization of Type (A) in terms of such commuting condition φφ ₁ A = A φ ₁ φ.     

On the irreducibility of a certain class of Laguerre polynomials

R. Gow has investigated the problem of determining classical polynomials with Galois group A_m, the alternating group on m letters, in the case that m is even (odd m being previously handled in work of I. Schur). He showed that the generalized Laguerre polynomial L_m^(m)(x), defined below, has Galois group A_m provided m>2 is even and L_m^(m)(x) is irreducible (and obtained irreducibility in some cases). In this paper, we establish that L_m^(m)(x) is irreducible for almost all m (and, hence, has Galois group A_m for almost all even m).     

Existence of spectral well-behaved ∗-representations

There are several cases, where an m^∗-seminorm p is defined on a ∗-subalgebra of a given ∗-algebra A. This may lead to the construction of an unbounded ∗-representation of A. Such m^∗-seminorms are called unbounded. Given an unbounded m^∗-seminorm p of a ∗-algebra A, the concept of a p-spectral ∗-representation of A is introduced and studied in connection to well-behaved ∗-representations. More precisely, the existence of (p-) spectral well-behaved ∗-representations is investigated on ∗-algebras and locally convex ∗-algebras in terms of certain properties of Pták function, closely related to hermiticity and C^∗-spectrality of the ∗-subalgebras on which this function is defined. Various examples in diverse classes of locally convex algebras illuminate the elaborated results.     

A note on balancedness and nonemptiness of the core in voting games

Consider a society with a finite number,n, of individuals who have to choose an alternative from a setA in them-dimensional Euclidean space IR ^m . Assuming that the preference relation overA of every individual is convex and continuous, Greenberg (1979) showed some that if the set of winning coalitions (i.e. those that have the veto power) consists of all coalitions with more thanmn/m + 1 individuals the core of the induced game is nonempty. Greenberg and Weber (1984) have strengthened this result by showing that the induced game is in fact balanced. On the other hand Le Breton (1987), Schofield (1984a) and Strnad (1985) have generalized Greenberg's theorem to arbitrary voting games. The purpose of this note is to show that however the induced game is not generally balanced.     

Bijective linear maps on semimodules spanned by Boolean matrices of fixed rank

Let M_m,n(B) be the semimodule of all m×n Boolean matrices where B is the Boolean algebra with two elements. Let k be a positive integer such that 2?k?min(m,n). Let B(m,n,k) denote the subsemimodule of M_m,n(B) spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on B(m,n,k), then there exist permutation matrices P and Q such that T(A)=PAQ for all A∈B(m,n,k) or m=n and T(A)=PA^tQ for all A∈B(m,n,k). This result follows from a more general theorem we prove concerning the structure of linear mappings on B(m,n,k) that preserve both the weight of each matrix and rank one matrices of weight k². Here the weight of a Boolean matrix is the number of its nonzero entries.