Algebraic isomorphisms and strongly double triangle subspace lattices

Let D={{0},K,L,M,X} be a strongly double triangle subspace lattice on a non-zero complex reflexive Banach space X, which means that at least one of three sums K + L, L + M and M + K is closed. It is proved that a non-zero element S of AlgD is single in the sense that for any A,B∈AlgD, either AS = 0 or SB = 0 whenever ASB = 0, if and only if S is of rank two. We also show that every algebraic isomorphism between two strongly double triangle subspace lattice algebras is quasi-spatial.     

Kernels in quasi-transitive digraphs

Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively.A kernel N of D is an independent set of vertices such that for every w∈V(D)-N there exists an arc from w to N. A digraph is called quasi-transitive when (u,v)∈A(D) and (v,w)∈A(D) implies (u,w)∈A(D) or (w,u)∈A(D). This concept was introduced by Ghouilá-Houri [Caractérisation des graphes non orientés dont on peut orienter les arrêtes de maniere à obtenir le graphe d’ un relation d’ordre, C.R. Acad. Sci. Paris 254 (1962) 1370-1371] and has been studied by several authors. In this paper the following result is proved: Let D be a digraph. Suppose D=D₁∪D₂ where D_i is a quasi-transitive digraph which contains no asymmetrical infinite outward path (in D_i) for i∈{1,2}; and that every directed cycle of length 3 contained in D has at least two symmetrical arcs, then D has a kernel. All the conditions for the theorem are tight.     

Approximations for two variants of the Steiner tree problem in the Euclidean plane {\mathbb{R}^2}

Given n terminals in the Euclidean plane and a positive constant l, find a Steiner tree T interconnecting all terminals with the minimum total cost of Steiner points and a specific material used to construct all edges in T such that the Euclidean length of each edge in T is no more than l. In this paper, according to the cost b of each Steiner point and the different costs of some specific materials with the different lengths, we study two variants of the Steiner tree problem in the Euclidean plane as follows: (1) If a specific material to construct all edges in such a Steiner tree has its infinite length and the cost of per unit length of such a specific material used is c ₁, the objective is to minimize the total cost of the Steiner points and such a specific material used to construct all edges in T, i.e., ${{\rm min} \{b \cdot k_1+ c_1 \cdot \sum_{e \in T} w(e)\}}$ , where T is a Steiner tree constructed, k ₁ is the number of Steiner points and w(e) is the length of part cut from such a specific material to construct edge e in T, and we call this version as the minimum-cost Steiner points and edges problem (MCSPE, for short). (2) If a specific material to construct all edges in such a Steiner tree has its finite length L (l ≤ L) and the cost of per piece of such a specific material used is c ₂, the objective is to minimize the total cost of the Steiner points and the pieces of such a specific material used to construct all edges in T, i.e., ${{\rm min} \{b \cdot k_2+ c_2 \cdot k_3\}}$ , where T is a Steiner tree constructed, k ₂ is the number of Steiner points in T and k ₃ is the number of pieces of such a specific material used, and we call this version as the minimum-cost Steiner points and pieces of specific material problem (MCSPPSM, for short). These two variants of the Steiner tree problem are NP-hard with some applications in VLSI design, WDM optical networks and wireless communications. In this paper, we first design an approximation algorithm with performance ratio 3 for the MCSPE problem, and then present two approximation algorithms with performance ratios 4 and 3.236 for the MCSPPSM problem, respectively.     

New approximation-solvability of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators

In this paper, we consider and study a class of general nonlinear operator inclusion couples involving (A, η, m)-resolvent operators and relaxed cocoercive type operators in Hilbert spaces. We also construct a new perturbed iterative algorithm framework with errors and investigate variational graph convergence analysis for this algorithm framework in the context of solving the nonlinear operator inclusion couple along with some results on the resolvent operator corresponding to (A, η, m)-maximal monotonicity. The obtained results improve and generalize some well known results in recent literatures.     

Sharp distortion theorems for some subclass of close-to-convex functions

For real A, B such that −1 ? B < A ? 1, we denote by R(A,B) the class of functions, such that f′ ? (1 + Az)/(1 + Bz). Sharp distortion results for functions from R_A,B are obtained.     

Automatic Controls for Nonlinear Dynamical Systems with Lipschitzian Trajectories

We would like to investigate on the solution to the automatic control problem given by the differential equation y′(t) = f(t, y(t), w(t)) for a given initial function x in the initial domain D(x, ω, Y) for almost all t in the interval I, with controls given by w(t) = g(t, y(t), T(y)(t)), where T is a nonanticipating and Lipschitzian operator. The result will be generalized for a dynamical system y′(t) = f(t, y(t), T(y), u(t)).     

Considering the isolation set problem

Given a network, G = [N, E] the Isolation Set Problem (ISP) finds the set of arcs, D ⊆ E, that when removed will separate a predefined set of r distinguished nodes [2]. This involves eliminating connections from a specific set of nodes to the rest of a network. In our increasingly interconnected network-centric world, this might be isolating various units from Headquarters; isolating a portion of a computer network to disrupt communications or to quarantine a virus or some other form of cyber attack; or isolating a cell or sub-group in a terrorist or “dark” network, for example.     

Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights

We completely describe those positive Borel measures μ in the unit disc D such that the Bergman space Ap(w)⊂Lq(μ), 0<p,q<∞, where w belongs to a large class W of rapidly decreasing weights which includes the exponential weights , α>0, and some double exponential type weights.As an application of that result, we characterize the boundedness and the compactness of Tg:Ap(w)→Aq(w), 0<p,q<∞, w∈W, where Tg is the integration operator

     

Classification of small (0, 1) matrices

Denote by A_n the set of square (0, 1) matrices of order n. The set A_n, n ? 8, is partitioned into row/column permutation equivalence classes enabling derivation of various facts by simple counting. For example, the number of regular (0, 1) matrices of order 8 is 10160459763342013440. Let D_n, S_n denote the set of absolute determinant values and Smith normal forms of matrices from A_n. Denote by a_n the smallest integer not in D_n. The sets D₉ and S₉ are obtained; especially, a₉ = 103. The lower bounds for a_n, 10 ? n ? 19 (exceeding the known lower bound a_n ? 2f_n − 1, where f_n is nth Fibonacci number) are obtained. Row/permutation equivalence classes of A_n correspond to bipartite graphs with n black and n white vertices, and so the other applications of the classification are possible.     

The common Re-nnd and Re-pd solutions to the matrix equations AX = C and XB = D

In this article, we consider common Re-nnd and Re-pd solutions of the matrix equations AX = C and XB = D with respect to X, where A, B, C and D are given matrices. We give necessary and sufficient conditions for the existence of common Re-nnd and Re-pd solutions to the pair of the matrix equations and derive a representation of the common Re-nnd and Re-pd solutions to these two equations when they exist. The presented examples show the advantage of the proposed approach.     

Maps preserving the nilpotency of products of operators

Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ?:B(X)→B(X) is a surjective map such that A,B∈B(X) satisfy AB∈N(X) if and only if ?(A)?(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C?{0} such that one of the following holds:

(a)

There is a bijective bounded linear or conjugate-linear operator S:X→X such that ? has the form A?S[f(A)A]S^-1.

(b)

The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that ? has the form A ? S[f(A)A′]S⁻¹.

If X has dimension n with 3 ? n < ∞, and B(X) is identified with the algebra M_n of n × n complex matrices, then there exist a map f:M_n→C?{0}, a field automorphism ξ:C→C, and an invertible S ∈ M_n such that ? has one of the following forms:

     

Some sufficient conditions for the existence of kernels in infinite digraphs

A kernel N of a digraph D is an independent set of vertices of D such that for every w∈V(D)−N there exists an arc from w to N. If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. D is called a critical kernel imperfect digraph when D has no kernel but every proper induced subdigraph of D has a kernel. If F is a set of arcs of D, a semikernel modulo F of D is an independent set of vertices S of D such that for every z∈V(D)−S for which there exists an (S,z)-arc of D−F, there also exists an (z,S)-arc in D. In this work we show sufficient conditions for an infinite digraph to be a kernel perfect digraph, in terms of semikernel modulo F. As a consequence it is proved that symmetric infinite digraphs and bipartite infinite digraphs are kernel perfect digraphs. Also we give sufficient conditions for the following classes of infinite digraphs to be kernel perfect digraphs: transitive digraphs, quasi-transitive digraphs, right (or left)-pretransitive digraphs, the union of two right (or left)-pretransitive digraphs, the union of a right-pretransitive digraph with a left-pretransitive digraph, the union of two transitive digraphs, locally semicomplete digraphs and outward locally finite digraphs.     

On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers

In this paper, we have found upper and lower bounds for the spectral norms of r-circulant matrices in the forms A = C_r(F₀, F₁, …, F_n−1), B = C_r(L₀, L₁, …, L_n−1), and we have obtained some bounds for the spectral norms of Kronecker and Hadamard products of A and B matrices.     

Isospectral flows that preserve matrix structure

The matrix A = (a_ij) ∈ S_n is said to lie on a strict undirected graph G if a_ij = 0 (i ≠ j) whenever (i, j) is not in E(G). If S is skew-symmetric, the isospectral flow maintains the spectrum of A. We consider isospectral flows that maintain a matrix A(t) on a given graph G. We review known results for a graph G that is a (generalised) path, and construct isospectral flows for a (generalised) ring, and a star, and show how a flow may be constructed for a general graph. The analysis may be applied to the isospectral problem for a lumped-mass finite element model of an undamped vibrating system. In that context, it is important that the flow maintain other properties such as irreducibility or positivity, and we discuss whether they are maintained.     

A note on the perturbation of positive matrices by normal and unitary matrices

In a recent paper, Neumann and Sze considered for an n × n nonnegative matrix A, the minimization and maximization of ρ(A + S), the spectral radius of (A + S), as S ranges over all the doubly stochastic matrices. They showed that both extremal values are always attained at an n × n permutation matrix. As a permutation matrix is a particular case of a normal matrix whose spectral radius is 1, we consider here, for positive matrices A such that (A + N) is a nonnegative matrix, for all normal matrices N whose spectral radius is 1, the minimization and maximization problems of ρ(A + N) as N ranges over all such matrices. We show that the extremal values always occur at an n × n real unitary matrix. We compare our results with a less recent work of Han, Neumann, and Tastsomeros in which the maximum value of ρ(A + X) over all n × n real matrices X of Frobenius norm was sought.     

Unitarily invariant norms related to the numerical radius

We determine the maximum in the class of unitarily invariant norms ∥·∥ such that w(A) ? ∥A∥ for all n-square matrices A. Here w(A) denotes the numerical radius of A.     

On commutativity of two unary digraph operations: Subdividing and line-digraphing

For a digraph D, let L(D) and S(D) denote its line digraph and subdivision digraph, respectively. The motivation of this paper is to solve the digraph equation L(S(D))=S(L(D)). We show that L(S(D)) and S(L(D)) are cospectral if and only if D and L(D) have the same number of arcs. Further, we characterize the situation that L(S(D)) and S(L(D)) are isomorphic. Our approach introduces the new notion, the proper image D^* of a digraph D, and a new type of connectedness for digraphs. The concept D^* plays an important role in the main result of this paper. It is also useful in other aspects of the study of line digraphs. For example, L(D) is connected if and only if D^* is connected; L(D) is functional (contrafunctional) if and only if D^* is functional (contrafunctional). Some related results are also presented.     

On a question of A. Koldobsky

We construct an example of a non-convex star-shaped origin-symmetric body D⊂R³ such that its section function AD,ξ(t):=area(D∩{ξ^⊥+tξ}) is decreasing in t?0 for every fixed direction ξ∈S².     

Linear Complexity,k-Error Linear Complexity,and the Discrete Fourier Transform

Complexity measures for sequences of elements of a finite field play an important role in cryptology. We focus first on the linear complexity of periodic sequences. By means of the discrete Fourier transform, we determine the number of periodic sequences S with given prime period length N and linear complexity L_N, 0(S)=c as well as the expected value of the linear complexity of N-periodic sequences. Cryptographically strong sequences should not only have a large linear complexity, but also the change of a few terms should not cause a significant decrease of the linear complexity. This requirement leads to the concept of the k-error linear complexity L_N, k(S) of sequences S with period length N. For some k and c we determine the number of periodic sequences S with given period length N and L_N, k(S)=c. For prime N we establish a lower bound on the expected value of the k-error linear complexity.     

Inverse minimum cost flow problems under the weighted Hamming distance

Given a network N(V, A, u, c) and a feasible flow x⁰, an inverse minimum cost flow problem is to modify the cost vector as little as possible to make x⁰ form a minimum cost flow of the network. The modification can be measured by different norms. In this paper, we consider the inverse minimum cost flow problems, where the modification of the arcs is measured by the weighted Hamming distance. Both the sum-type and the bottleneck-type cases are considered. For the former, it is shown to be APX-hard due to the weighted feedback arc set problem. For the latter, we present a strongly polynomial algorithm which can be done in O(n · m²).