The determinant of the sum of two normal matrices with prescribed eigenvalues

Given complex numbers α₁,...,α_n, β₁,...,β_n, what can we say about the determinant of A+B, where A (B) is an n×n normal matrix with eigenvalues α₁,...,α_n (β₁,...,β_n)? Some partial answers are offered to this question.     

Continuous semigroups on ordered Banach spaces

Let ($\text{B}$, $\text{B}$₊, ∥ · ∥) denote a Banach space $\text{B}$, ordered by a proper norm-closed convex cone $\text{B}$₊, with a Riesz norm ∥ · ∥, and define the canonical half-norm N associated with $\text{B}$₊ by

$\text{N(a)=}\text{inf}\text{{∥a+b∥;b?B}{}_{\mathrm{+}}\text{}}$

. The analogs of the Hille-Yosida and Feller-Miyadera-Phillips theorems characterizing the generators H of C₀- or C₀¹-semigroups S = {S_t}_{t ? 0} of positive operators, i.e., operators such that S_t$\text{B}$₊?$\text{B}$₊, are proved. In these theorems conditions of norm-dissipativity, e.g.,

$\text{∥(I + αH) a ∥ ? ∥ a ∥, α > 0, a ? D(H)}$

are replaced by N-dissipativity, i.e.,

$\text{N((I + αH)a) ? N(a), α > 0, a ? D(H)}$

.     

Conformally invariant tensors of an almost Hermitian manifold associated with the holomorphic curvature tensor

We consider an almost Hermitian manifold and apply the conformal change of metric to its holomorphic curvature tensor. In such a way we find that the generalized Bochner curvature tensor can be expressed as a linear combination of B ₁, B ₂, and B ₃ such that (6.4) holds. Each of the tensors B ₁, B ₂, B ₃ is conformally invariant and satisfies the condition (1.2) of K?hler type.     

Simultaneous quasidiagonalization of complex matrices

Let

be the complex algebra generated by a pair of n × n Hermitian matrices A, B. A recent result of Watters states that A, B are simultaneously unitarily quasidiagonalizable [i.e., A and B are simultaneously unitarily similar to direct sums C₁⊕…⊕C_t,D₁⊕…⊕D_t for some t, where C_i, D_i are k_i × k_i and k_i?2(1?i?t)] if and only if [p(A, B), A]² and [p(A, B), B]² belong to the center of

for all polynomials p(x, y) in the noncommuting variables x, y. In this paper, we obtain a finite set of conditions which works. In particular we show that if A, B are positive semidefinite, then A, B are simultaneously quasidiagonalizable if (and only if) [A, B]², [A², B]² and [A, B²]² commute with A, B.     

Upper bounds for p-divisibility of sets of Bernoulli numbers

Kummer's criterion asserts that a prime p divides the class number of the pield of pth roots of unity if and only if p divides the numerator of at least one of the Bernoulli numbers B₂, B₄,…,B_p?3. We partition the set {B₂,B₄,…,B_p?3} into certain “divisions” and prove that up to a higher order term at most one-half of the B_2k in a given division can be divisible by p.     

On rational invariants of a set of symmetric matrices

Some identities resulting from the Cayley-Hamilton theorem are derived. Some applications include: (a) for k = 1,2,…,n ? 1 a condition is found for a pair (A,B) of symmetric operators acting in Euclidean n-space to have common invariant k-subspace (provided that A does not have multiple eigenvalues); (b) it is shown that the field of rational invariants of (A,B) is isomorphic to a subfield of a rational function field with n(n+3)/2 generators consisting of elements symmetric with respect to the permutaion group P_n; (c) it is shown that any rational invariant of (g+2) symmetric operators A,B,C₁,C₂,…, C_g can be expressed as a rational function of invariants of one or two operators that are taken for pairs (A,B), (A,C₂),…, (A,C_g, (A,B+C₁), (A,B+C₂),…,(A,B+C_g).     

A non-steady flow of liquid in a porous pipe with variable permeability

LetB denote the infinitesimal operator of a strongly continuous semigroup S(t), with resolvent R_λ, on Banach space L. We define related operators P and V so that λR_λf = Pf + λVf + o(λ), as λ → 0⁺. For α, η > 0 and possibly unbounded, linear operator A, we let U_{α, η}(t) represent a strongly continuous semigroup generated by αA + ηB. We show that under appropriate simultaneous convergence of α and η, U_{α, η}(t) converges strongly to a strongly continous semigroup U(t), having infinitesimal operator characterized through PA(VA)^rf where r =min{j ? 0, PA(VA)^j ≠ 0}. We apply the abstract perturbation theorem to a singular perturbation initial-value problem, of Tihonov-type, for a non-linear system of ordinary differential equations.     

Note on Besicovitch’s theorem on the possible values of upper and lower derivatives

LetB ₁, ..., B _k be Busemann-Feller and regular differential bases composed of intervals of the corresponding dimensions. It is proved that if B ₁, ...,B _k satisfy a certain condition (called the completeness condition), then, for their Cartesian product B ₁ × ... × B _k , an analog of Besicovitch’s theorem on the possible values of strong upper and lower derivatives is valid.     

A criterion of diagonalizability of a pair of matrices over the ring of principal ideals by common row and separate column transformations

We establish that a pair A, B, of nonsingular matrices over a commutative domain R of principal ideals can be reduced to their canonical diagonal forms D ^A and D ^B by the common transformation of rows and separate transformations of columns. This means that there exist invertible matrices U, V _A, and V _B over R such that UAV _a=D^A and UAV _B=D^B if and only if the matrices B _*A and D _* ^B D^A where B _* 0 is the matrix adjoint to B, are equivalent.     

The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil

Given n-square Hermitian matrices A,B, let A_i,B_i denote the principal (n?1)- square submatrices of A,B, respectively, obtained by deleting row i and column i. Let μ, λ be independent indeterminates. The first main result of this paper is the characterization (for fixed i) of the polynomials representable as det(μA_i?λB_i) in terms of the polynomial det(μA?λB) and the elementary divisors, minimal indices, and inertial signatures of the pencil μA?λB. This result contains, as a special case, the classical interlacing relationship governing the eigenvalues of a principal sub- matrix of a Hermitian matrix. The second main result is the determination of the number of different values of i to which the characterization just described can be simultaneously applied.     

A note on Stewart's theorem for definite matrix pairs

Let A and B be n×n Hermitian matrices. The matrix pair (A, B) is called definite pair and the corresponding eigenvalue problem βAx = αBx is definite if c(A, B) ≡ inf_{6x6= 1}{|^H(A+iB)x|} > 0. In this note we develop a uniform upper bound for differences of corresponding eigenvalues of two definite pairs and so improve a result which is obtained by G.W. Stewart [2]. Moreover, we prove that this upper bound is a projective metric in the set of n × n definite pairs.     

Collinear subsets of lattice point sequences—An analog of Szemerédi's theorem

Szemerédi's theorem states that given any positive number B and natural number k, there is a number n(k, B) such that if n ? n(k, B) and 0 < a₁ < … < a_n is a sequence of integers with a_n ? Bn, then some k of the a_i form an arithmetic progression. We prove that given any B and k, there is a number m(k, B) such that if m ? m(k, B) and u₀, u₁, …, u_m is a sequence of plane lattice points with ∑_i=1^m…u_i ? u_i?1… ? Bm, then some k of the u_i are collinear. Our result, while similar to Szemerédi's theorem, does not appear to imply it, nor does Szemerédi's theorem appear to imply our result.     

On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn

Let B₁, B₂, ... be a sequence of independent, identically distributed random variables, letX₀ be a random variable that is independent ofB_n forn?1, let ρ be a constant such that 0<ρ<1 and letX₁,X₂, ... be another sequence of random variables that are defined recursively by the relationshipsX_n=ρX_n-1+B_n. It can be shown that the sequence of random variablesX₁,X₂, ... converges in law to a random variableX if and only ifE[log⁺¦B₁¦]<∞. In this paper we let {B(t):0≦t<∞} be a stochastic process with independent, homogeneous increments and define another stochastic process {X(t):0?t<∞} that stands in the same relationship to the stochastic process {B(t):0?t<∞} as the sequence of random variablesX₁,X₂,...stands toB₁,B₂,.... It is shown thatX(t) converges in law to a random variableX ast →+∞ if and only ifE[log⁺¦B(1)¦]<∞ in which caseX has a distribution function of class L. Several other related results are obtained. The main analytical tool used to obtain these results is a theorem of Lukacs concerning characteristic functions of certain stochastic integrals.     

A new fuzzy compactness defined by fuzzy nets

Let (Ω, $\text{A}$, μ) be a probability space and let $\text{B}$ be a subsigma algebra of $\text{A}$. Let A= L_∞Ω, $\text{A}$, μ , let A= L_∞Ω, $\text{B}$, μ, and let f?A. It is shown that best L_∞-approximations of f by elements of B comprise an interval in B; that is, there exists $\text{f}\text{,}\text{f}\text{?B}$ such that a function g?B is a best L_∞-approximation to f if and only if $\text{f}\text{? g ?}\text{f}$ a.e. on Ω. The difference, $\text{f}\text{? f}$, of $\text{f}$ and f is completely characterized in terms of special sets that have been developed in [2]. Then it is established that the best best L_∞-approximation, f_{$\text{B}$,∞}, to f by elements of B is the average of $\text{f}$ and $\text{f}$, where the function f_{$\text{B}$,∞} is defined by f_{$\text{B}$,∞}(ω) lim_{p → ∞}f_$\text{B}$,P(ξ) and f_$\text{B}$,P denotes the best L_p-approximation to f elements of L_p(Ω, $\text{B}$, μ).     

On some properties of base-matroids

Let M=(E,F) be a rank-n matroid on a set E and B one of its bases. A closed set θ⊆E is saturated with respect to B, or B-saturated, when |θ∩B|=r(θ), where r(θ) is the rank of θ.The collection of subsets I of E such that |I∩θ|?r(θ), for every closed B-saturated set θ, turns out to be the family of independent sets of a new matroid on E, called base-matroid and denoted by M_B. In this paper we prove some properties of M_B, in particular that it satisfies the base-axiom of a matroid.Moreover, we determine a characterization of the matroids M which are isomorphic to M_B for every base B of M.Finally, we prove that the poset of the closed B-saturated sets ordered by inclusion is isomorphic to the Boolean lattice B_n.     

The determinants of certain matrices arising from the Boolean lattice

Let B be the Boolean lattice on an n-set with B=?B_i the rank decomposition. Let M(n,i) be the incidence matrix between B_i and B_n−i. We obtain a recursive formula for the determinant of the matrix M(n,i).     

Factorizing multivariate time series operators

Motivated by problems occurring in the empirical identification and modelling of a n-dimensional ARMA time series X(t) we study the possibility of obtaining a factorization (I + a₁B + … + a_pB^p) X(t) = [Π_i=1^p (I ? α_iB)] X(t), where B is the backward shift operator. Using a result in [3] we conclude that as in the univariate case such a factorization always exists, but unlike the univariate case in general the factorization is not unique for given a₁, a₂,…, a_p. In fact the number of possibilities is limited upwards by $\text{(np)!}\text{(n!)}{}^{\mathrm{p}}$, there being cases, however, where this maximum is not reached. Implications for the existence and possible use of transformations which removes nonstationarity (or almost nonstationarity) of X(t) are mentioned.     

Semi Laguerre Planes

A Dembowski semi-plane is a semi-plane obtained from a projective plane by Dembowski's method [1]. A semi Laguerre plane is an incidence structure J = (P, B₁ ∪ B₂, I) for which: (a) every element of P is incident with one element of B₁, (b) an element of B₁ and an element of B₂ are incident with at most one common element of P, (c) each residual space of J (with respect to B₁) is a Dembowski semi-plane, (d) B₂ ≠ ? and each element of B₂ is incident with at least 4 elements of P. We prove that all semi Laguerre planes are substructures of Laguerre planes or special Laguerre planes (in the sense of Thas, Willems [3], [4]). Therefore, these incidence structures are related to optimal codes ([5], [6]).     

Boundedness for pseudodifferential operators on multivariate α-modulation spaces

The α-modulation spaces M ^s,α _p,q (R ^d ), α∈[0,1], form a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that a pseudodifferential operator σ(x,D) with symbol in the Hörmander class S ^b _ρ,0 extends to a bounded operator σ(x,D):M ^s,α _p,q (R ^d )→M ^s-b,α _p,q (R ^d ) provided 0≤α≤ρ≤1, and 1<p,q<∞. The result extends the well-known result that pseudodifferential operators with symbol in the class S ^b _1,0 maps the Besov space B ^s _p,q (R ^d ) into B ^s-b _p,q (R ^d ).