Derivations and right ideals of algebras

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let ρ be a nonzero right ideal of R and let f(X₁,…,X_t) be a nonzero polynomial over K with constant term 0 such that μR≠0 for some coefficient μ of f(X₁,…,X_t). Suppose that d:R→R is a nonzero derivation. It is proved that if rankd(f(x₁,…,x_t))?m for all x₁,…,x_t∈ρ and for some positive integer m, then either ρ is generated by an idempotent of finite rank or d=ad(b) for some b∈End(V_D) of finite rank. In addition, if f(X₁,…,X_t) is multilinear, then b can be chosen such that rank(b)?2(6t+13)m+2.     

Construction of nonnegative symmetric matrices with given spectrum

Let σ = (λ₁, … , λ_n) be the spectrum of a nonnegative symmetric matrix A with the Perron eigenvalue λ₁, a diagonal entry c and let τ = (μ₁, … , μ_m) be the spectrum of a nonnegative symmetric matrix B with the Perron eigenvalue μ₁. We show how to construct a nonnegative symmetric matrix C with the spectrum

(λ₁+max{0,μ₁-c},λ₂,…,λ_n,μ₂,…,μ_m).     

On the corners of certain determinantal ranges

Let A be a complex n×n matrix and let SO(n) be the group of real orthogonal matrices of determinant one. Define Δ(A)={det(A°Q):Q∈SO(n)}, where ° denotes the Hadamard product of matrices. For a permutation σ on {1,…,n}, define It is shown that if the equation z_σ=det(A°Q) has in SO(n) only the obvious solutions (Q=(ε_iδ_σi,j),ε_i=±1 such that ε₁…ε_n=sgnσ), then the local shape of Δ(A) in a vicinity of z_σ resembles a truncated cone whose opening angle equals , where σ₁, σ₂ differ from σ by transpositions. This lends further credibility to the well known de Oliveira Marcus Conjecture (OMC) concerning the determinant of the sum of normal n×n matrices. We deduce the mentioned fact from a general result concerning multivariate power series and also use some elementary algebraic topology.     

Divisibility properties of power GCD matrices and power LCM matrices

Let a,b and n be positive integers and the set S={x₁,…,x_n} of n distinct positive integers be a divisor chain (i.e. there exists a permutation σ on {1,…,n} such that x_σ(1)|…|x_σ(n)). In this paper, we show that if a|b, then the ath power GCD matrix (S^a) having the ath power (x_i,x_j)^a of the greatest common divisor of x_i and x_j as its i,j-entry divides the bth power GCD matrix (S^b) in the ring M_n(Z) of n×n matrices over integers. We show also that if a?b and n?2, then the ath power GCD matrix (S^a) does not divide the bth power GCD matrix (S^b) in the ring M_n(Z). Similar results are also established for the power LCM matrices.     

Quasi sure quadratic variation of local times of smooth semimartingales

In this paper, we prove that the process of the quadratic variation of local times of smooth semimartingales can be constructed as the quasi sure limit of the form ∑_Δn(L_t^a_i+1n−L_t^a_in)², where Δ_n=(a_iⁿ,a_i+1ⁿ) is a sequence of subdivisions of [a,b], a_iⁿ=i(b−a)/2ⁿ+a, i=0,1,…,2ⁿ.     

Markov-Nikolskii type inequalities for exponential sums on finite intervals

Let Λn:={λ₀<λ₁<?<λn} be a set of real numbers. The collection of all linear combinations of eλ₀t,eλ₁t,…,eλnt over R will be denoted by

E(Λn):=span{eλ₀t,eλ₁t,…,eλnt}.     

Likelihood ratio orderings of spacings of heterogeneous exponential random variables

Let X₁,X₂,…,X_n be independent exponential random variables such that X_i has failure rate λ for i=1,…,p and X_j has failure rate λ^* for j=p+1,…,n, where p≥1 and q=n-p≥1. Denote by D_i:n(p,q)=X_i:n-X_i-1:n the ith spacing of the order statistics , where X_0:n≡0. It is shown that D_i:n(p,q)?_lrD_i+1:n(p,q) for i=1,…,n-1, and that if λ?λ^* then , and for i=1,…,n, where ?_lr denotes the likelihood ratio order. The main results are used to establish the dispersive orderings between spacings.     

Reduction of the c-numerical range to the classical numerical range

Let A be an n×n complex matrix and c=(c₁,c₂,…,c_n) a real n-tuple. The c-numerical range of A is defined as the set

     

On arithmetic and asymptotic properties of up-down numbers

Let σ=(σ₁,…,σ_N), where σ_i=±1, and let C(σ) denote the number of permutations π of 1,2,…,N+1, whose up-down signature sign(π(i+1)-π(i))=σ_i, for i=1,…,N. We prove that the set of all up-down numbers C(σ) can be expressed by a single universal polynomial Φ, whose coefficients are products of numbers from the Taylor series of the hyperbolic tangent function. We prove that Φ is a modified exponential, and deduce some remarkable congruence properties for the set of all numbers C(σ), for fixed N. We prove a concise upper bound for C(σ), which describes the asymptotic behaviour of the up-down function C(σ) in the limit C(σ)?(N+1)!.     

Thue-Morse at multiples of an integer

Let t=(tn)_n?0 be the classical Thue-Morse sequence defined by , where s₂ is the sum of the bits in the binary representation of n. It is well known that for any integer k?1 the frequency of the letter “1” in the subsequence t₀,tk,t2k,… is asymptotically 1/2. Here we prove that for any k there is an n?k+4 such that tkn=1. Moreover, we show that n can be chosen to have Hamming weight ?3. This is best in a twofold sense. First, there are infinitely many k such that tkn=1 implies that n has Hamming weight ?3. Second, we characterize all k where the minimal n equals k, k+1, k+2, k+3, or k+4. Finally, we present some results and conjectures for the generalized problem, where s₂ is replaced by sb for an arbitrary base b?2.     

Mean matrices and infinite divisibility

We consider matrices M with entries m_ij = m(λ_i, λ_j) where λ₁, … ,λ_n are positive numbers and m is a binary mean dominated by the geometric mean, and matrices W with entries w_ij = 1/m (λ_i, λ_j) where m is a binary mean that dominates the geometric mean. We show that these matrices are infinitely divisible for several much-studied classes of means.     

Simultaneous solutions of Sylvester equations and idempotent matrices separating the joint spectrum

We investigate simultaneous solutions of the matrix Sylvester equations A_iX-XB_i=C_i,i=1,2,…,k, where {A₁,…,A_k} and {B₁,…,B_k} are k-tuples of commuting matrices of order m×m and p×p, respectively. We show that the matrix Sylvester equations have a unique solution X for every compatible k-tuple of m×p matrices {C₁,…,C_k} if and only if the joint spectra σ(A₁,…,A_k) and σ(B₁,…,B_k) are disjoint. We discuss the connection between the simultaneous solutions of Sylvester equations and related questions about idempotent matrices separating disjoint subsets of the joint spectrum, spectral mapping for the differences of commuting k-tuples, and a characterization of the joint spectrum via simultaneous solutions of systems of linear equations.     

Construction of nonnegative matrices and the inverse eigenvalue problem

This article presents a technique for combining two matrices, an n?×?n matrix M and an m?×?m matrix B, with known spectra to create an (n?+?m???p)?×?(n?+?m???p) matrix N whose spectrum consists of the spectrum of the matrix M and m???p eigenvalues of the matrix B. Conditions are given when the matrix N obtained in this construction is nonnegative. Finally, these observations are used to obtain several results on how to construct a realizable list of n?+?1 complex numbers (λ₁,λ₂,λ₃,σ) from a given realizable list of n complex numbers (c ₁,c ₂,σ), where c ₁ is the Perron eigenvalue, c ₂ is a real number and σ is a list of n???2 complex numbers.     

Stochastic comparisons of order statistics from gamma distributions

Let (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n) be gamma random vectors with common shape parameter α(0<α?1) and scale parameters (λ₁,λ₂,…,λ_n), (μ₁,μ₂,…,μ_n), respectively. Let X₍₎=(X₍₁₎,X₍₂₎,…,X_(n)), Y₍₎=(Y₍₁₎,Y₍₂₎,…,Y_(n)) be the order statistics of (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n). Then (λ₁,λ₂,…,λ_n) majorizes (μ₁,μ₂,…,μ_n) implies that X₍₎ is stochastically larger than Y₍₎. However if the common shape parameter α>1, we can only compare the the first- and last-order statistics. Some earlier results on stochastically comparing proportional hazard functions are shown to be special cases of our results.     

A multiplicative inequality for vertex Folkman numbers

Let G be a graph and a₁,…,a_r be positive integers. The symbol G→(a₁,…,a_r) denotes that in every r-coloring of the vertex set V(G) there exists a monochromatic a_i-clique of color i for some i∈{1,…,r}. The vertex Folkman numbers F(a₁,…,a_r;q)=min{|V(G)|:G→(a₁,…,a_r) and K_q?G} are considered. Let a_i, b_i, c_i, i∈{1,…,r}, s, t be positive integers and c_i=a_ib_i, 1?a_i?s,1?b_i?t. Then we prove that

F(c₁,c₂,…,c_r;st+1)?F(a₁,a₂,…,a_r;s+1)F(b₁,b₂,…,b_r;t+1).     

Isospectral flow method for nonnegative inverse eigenvalue problem with prescribed structure

The nonnegative inverse eigenvalue problem is that given a family of complex numbers λ={λ₁,…,λ_n}, find a nonnegative matrix of order n with spectrum λ. This problem is difficult and remains unsolved partially. In this paper, we focus on its generalization that the reconstructed nonnegative matrices should have some prescribed entries. It is easy to see that this new problem will come back to the common nonnegative inverse eigenvalue problem if there is no constraint of the locations of entries. A numerical isospectral flow method which is developed by hybridizing the optimization theory and steepest descent method is used to study the reconstruction. Moreover, an error estimate of the numerical iteration for ordinary differential equations on the matrix manifold is presented. After that, a numerical method for the nonnegative symmetric inverse eigenvalue problem with prescribed entries and its error estimate are considered. Finally, the approaches are verified by the numerical test results.     

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.     

New upper bounds for Estrada index of bipartite graphs

Suppose G is a graph and λ₁,λ₂,…,λ_n are the eigenvalues of G. The Estrada index EE(G) of G is defined as the sum of e^λ_i, 1≤i≤n. In this paper some new upper bounds for the Estrada index of bipartite graphs are presented. We apply our result on a (4,6)-fullerene to improve our bound given in an earlier paper.     

Oscillating rim hook tableaux and colored matchings

We find a correspondence between oscillating m-rim hook tableaux and m-colored matchings, where m is a positive integer. An oscillating m  -rim hook tableau is defined as a sequence (λ⁰,λ¹,…,λ2n)$({\lambda }^0,{\lambda }^1,\dots ,{\lambda }^{2n})$ of Young diagrams starting with the empty shape and ending with the empty shape such that λi${\lambda }^i$ is obtained from λi−1${\lambda }^{i-1}$ by adding an m-rim hook or by deleting an m-rim hook. Our bijection relies on the generalized Schensted algorithm due to White. An oscillating 2-rim hook tableau is also called an oscillating domino tableau. When we restrict our attention to two column oscillating domino tableaux of length 2n  , we are led to a bijection between such tableaux and noncrossing 2-colored matchings on {1,2,…,2n}$\{1,2,\dots ,2n\}$, which are counted by the product CnC_n+1$C_n{C}_{n+1}$ of two consecutive Catalan numbers. A 2-colored matching is noncrossing if there are no two arcs of the same color that are intersecting. We show that oscillating domino tableaux with at most two columns are in one-to-one correspondence with Dyck path packings. A Dyck path packing of length 2n   is a pair (D,E)$(D,E)$, where D is a Dyck path of length 2n, and E is a dispersed Dyck path of length 2n that is weakly covered by D. So we deduce that Dyck path packings of length 2n   are counted by CnC_n+1$C_n{C}_{n+1}$.     

OMC for scalar multiples of unitaries

We establish the following case of the Determinantal Conjecture of Marcus [M. Marcus, Derivations, Plücker relations and the numerical range, Indiana Univ. Math. J. 22 (1973) 1137-1149] and de Oliveira [G.N. de Oliveira, Research problem: Normal matrices, Linear and Multilinear Algebra 12 (1982) 153-154]. Let A and B be unitary n × n matrices with prescribed eigenvalues a₁, … , a_n and b₁, … , b_n, respectively. Then for any scalars t and s