Let MD,E,F=
be an upper triangular operator matrix on the Hilbert space H1⊕H2⊕H3. In this paper, we obtain necessary and sufficient conditions of σ*(MD,E,F)=σ*(A) ∪ σ*(B) ∪ σ*(C) for every D ∈ B(H2, H1), E ∈ B(H3, H1) and F ∈ B(H3, H2), in terms of the spectral properties of diagonal entries A, B and C in MD,E,F, where σ* is the point spectrum, the residual spectrum and the continuous spectrum. Moreover, we construct some examples illustrating our main results.
Trench gave the definition of the (R,Sσ)-commutative matrix in[Characterization and properties of (R,Sσ)-commutative matrices, Linear Algebra Appl., 2012, 436:4261-4278]. This paper discusses the general structure of (R,Sσ)-commutative matrix, and the least squares problem and the best approximation problem of linear equations AX=B, YA=D in the set of (R,Sσ)-commutative matrices for given matrix X, Y, B, D. Then we analysis the expressions of the least-square commutative solution and the best approximate solution of (R,Sσ)-commutative matrix in detail. At the same time, the necessary and sufficient conditions for the existence of the (R,Sσ)-commutative solution are analyzed when the linear matrix equations is consistent.
The aim of this paper is to define and study the R-matrix of a bimonad distributive law. Assume that F and G are bimonads, we give necessary and sufficient conditions for C(F,G)(φ), the category of F, G-bimodules, to be a braided monoidal category.
A sequence {Pi}i∈N of covers of a topological space X is called a pointstar network for X if the family {st(x, Pi)}i∈N is a network at x in X for each x ∈ X. The main purpose of this paper is to characterize the spaces which has a point-star network consisting of cs-finite cs-coverings and express their as certain images of metric spaces. It is proved that the following are equivalent for a topological space X when the property P of set families satisfies some suitable conditions:
(1) X has a point-star network consisting of cs-coverings with property P.
(2) X has a point-star network consisting of sn-coverings with property P.
(3) X is a Cauchy sn-symmetric space with a σ-P cs-network.
(4) X is a Cauchy sn-symmetric space with a σ-P sn-network.
(5) X is a sequence-covering, π and σ-P-image of a metrizable space.
(6) X is a 1-sequence-covering, compact and σ-P-image of a metrizable space.
The above and some related work contain the study of locally finite or point-finite families as a special case, expand the research from bases to cs-networks, and enrich the idea of mutual classification on mappings and spaces.
For the following p-Kirchhoff type functional
we prove the existence and uniqueness of global minimum or mountain pass type critical points on the Lp-normalized manifold Sc:={u∈W1,p(Rn):∫Rn|u|pdx=cp}. We show that these critical points indeed are optimizers of a certain Gagliardo-Nirenberg inequality. Especially, when p ∈ (1, 2], they are unique up to translations. We extend some known results for the case of p=2 in previous papers.
We introduce the notion of estimation entropy for semigroup action systems and give some properties. Furthermore, we investigate the concept of Δ-weakly mixing sets for semigroup actions and give a characterisation of Δ-weakly mixing sets by Δ-Xiong chaotic sets.
The main purpose of this paper is using the analytic methods and the properties of character sums to study the computational problem of one kind hybrid power mean involving the character sums of polynomials and Dirichlet L-functions modulo p, an odd prime, and give some exact computational formulas for them.
Pointwise preimage entropy is similar to topological entropy but, in general, their properties are not completely coincident such as additivity of under Cartesian product. In this paper we show that the description of the additivity of pointwise preimage entropy of the torus maps under Cartesian product given by myself extends to the case of the maps of compact nilmanifolds.
In this paper, the initial boundary value problem for some nonlinear higherorder Kirchhoff-type equation with damping and source terms in a bounded domain is studied. We prove the existence of global solutions for this problem by constructing a stable set and establish the energy decay estimate by applying a difference inequality due to Nakao. Meanwhile, under the condition of the positive initial energy, it is shown that the solution blows up in the finite time and the lifespan estimate of solution is also given.
Suppose X, Y are reflexive strictly convex Banach spaces. Let A ∈ B(X, Y) be a bounded linear operator with R(A) closed. T=EAF ∈ B(X, Y) is a multiplicative perturbation of A. In this paper, we investigate the multiplicative perturbations of the metric generalized inverse of A. We present the upper bound of ||TM-AM under the condition that the range R(A) and the null space N(A) are uniformly strong unique of order α and order β, respectively. As an application, we consider the multiplicative perturbation of metric generalized inverse in Lp space.
This paper studies the Menon-Sury's identity with both Dirichlet characters and additive characters, and we shall give the explicit formula of the following sum
gcd(a1-1,…,as-1,b1,…,br,n)χ1(a1)…χs(as)λ1(b1)…λr(br),
where n is a positive integer, s, r are nonnegative integers, Zn* is the group of units of the ring Zn=Z/nZ, gcd(,) represents the greatest common divisor, χi (1 ≤ i ≤ s) are Dirichlet characters mod n with conductors di, λj (1 ≤ j ≤ r) are additive characters of Zn. From the point of view of Fourier analysis on finite Abelian groups, our result presents the explicit expression of Fourier coefficients of the function f (a1,…,as,b1,…,br)=gcd (a1-1,…,as-1,b1,…,br,n) on the Abelian group (Zn*)s×(Zn)r.
