Influence of anti-diagonals on the asymptotic stability for linear differential systems

Sufficient conditions are given for asymptotic stability of the linear differential system x′  =  B(t)x with B(t) being a 2  ×  2 matrix. All components of B(t) are not assumed to be positive. The matrix B(t) is naturally divisible into a diagonal matrix D(t) and an anti-diagonal matrix A(t). Our concern is to clarify a positive effect of the anti-diagonal part A(t)x on the asymptotic stability for the system x′  =  B(t)x.      

Influence of anti-diagonals on the asymptotic stability for linear differential systems

Sufficient conditions are given for asymptotic stability of the linear differential system x′  =  B(t)x with B(t) being a 2  ×  2 matrix. All components of B(t) are not assumed to be positive. The matrix B(t) is naturally divisible into a diagonal matrix D(t) and an anti-diagonal matrix A(t). Our concern is to clarify a positive effect of the anti-diagonal part A(t)x on the asymptotic stability for the system x′  =  B(t)x.     

Global optimal approximate solutions

Given non-void subsets A and B of a metric space and a non-self mapping T:A? B{T:A\longrightarrow B}, the equation T x = x does not necessarily possess a solution. Eventually, it is speculated to find an optimal approximate solution. In other words, if T x = x has no solution, one seeks an element x at which d(x, T x), a gauge for the error involved for an approximate solution, attains its minimum. Indeed, a best proximity point theorem is concerned with the determination of an element x, called a best proximity point of the mapping T, for which d(x, T x) assumes the least possible value d(A, B). By virtue of the fact that d(x, T x) ≥ d(A, B) for all x in A, a best proximity point minimizes the real valued function x? d(x, T x){x\longrightarrow d(x, T\,x)} globally and absolutely, and therefore a best proximity in essence serves as an ideal optimal approximate solution of the equation T x = x. The aim of this article is to establish a best proximity point theorem for generalized contractions, thereby producing optimal approximate solutions of certain fixed point equations. In addition to exploring the existence of a best proximity point for generalized contractions, an iterative algorithm is also presented to determine such an optimal approximate solution. Further, the best proximity point theorem obtained in this paper generalizes the well-known Banach’s contraction principle.     

On a criterion for the complete controllability of nonautonomous linear systems

We describe the controllability sets of linear nonautonomous systems ẋ = A(t)x + B(t)u, x ∈ ℝ ⁿ , u ∈ U ⊆ ℝ ^m , with entire matrix functions A(t) and B(t) and with a linear set U of control constraints. We derive a criterion for the complete controllability of these linear systems in terms of derivatives of the entire matrix functions A(t) and B(t) at zero. This complete controllability criterion is compared with the Kalman and Krasovskii criteria.     

Canonical form with respect to semiscalar equivalence for a matrix pencil with nonsingular first matrix

Polynomial n × n matrices A(x) and B(x) over a field \mathbbF \mathbb{F} are called semiscalar equivalent if there exist a nonsingular n × n matrix P over \mathbbF \mathbb{F} and an invertible n × n matrix Q(x) over \mathbbF \mathbb{F} [x] such that A(x) = PB(x)Q(x). We give a canonical form with respect to semiscalar equivalence for a matrix pencil A(x) = A _0x - A ₁, where A ₀ and A ₁ are n × n matrices over \mathbbF \mathbb{F} , and A ₀ is nonsingular.     

A Proof of the Jacobian Conjecture on Global Asymptotic Stability

Let ƒ∈C ¹ (R ¹, R ²), ƒ(0) = 0. The Jacobian Conjecture states that if for any x∈R ², the eigenvalues of the Jacobian matrix Dƒ(x) have negative real parts, then the zero solution of the differential equation x = ƒ(x) is globally asymptotically stable. In this paper we prove that the conjecture is true. This work is supported by the National Natural Science Foundation of China     

Global stabilization of nonlinear systems by quadratic Lyapunov functions

The system x = A (t, x)x + B(t, x)u, where A(t, x) and B(t, x) are, respectively, n × n and n × m (m<n) continuous matrices whose elements are uniformly bounded for t ≽ t ₀ and x ∈ ℝ ⁿ , is considered. It is assumed that the system has relative degree q = n - m + 1, and the determinant of the matrix composed of the last m rows of the matrix B(t, x) is bounded away from zero for t ≽ t ₀ and x ∈ ℝ ⁿ . A special quadratic Lyapunov function with constant positive definite coefficient matrix H depending only on the range of variation of the coefficients in the matrices A(t, x) and B(t, x) is constructed and applied to obtain a control u(t, x) =7n ~B⋆ (t, x)H depending on a scalar parameter 7n under which the system is globally asymptotically stable provided that it is closed. Here, ~B (t, x) is the scalar matrix obtained from the matrix B(t, x) by setting the first n - m rows to zero.     

Max-Min Problems on the Ranks and Inertias of the Matrix Expressions <Emphasis Type="Italic">A</Emphasis>−<Emphasis Type="Italic">BXC</Emphasis>±(<Emphasis Type="Italic">BXC</Emphasis>)<Superscript>∗</Superscript> with Applications

We introduce a simultaneous decomposition for a matrix triplet (A,B,C ^∗), where A=±A ^∗ and (⋅)^∗ denotes the conjugate transpose of a matrix, and use the simultaneous decomposition to solve some conjectures on the maximal and minimal values of the ranks of the matrix expressions A−BXC±(BXC)^∗ with respect to a variable matrix X. In addition, we give some explicit formulas for the maximal and minimal values of the inertia of the matrix expression A−BXC−(BXC)^∗ with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression D−CXC ^∗ subject to Hermitian solutions of a consistent matrix equation AXA ^∗=B, as well as the extremal ranks and inertias of the Hermitian Schur complement D−B ^∗ A ^∼ B with respect to a Hermitian generalized inverse A ^∼ of A. Various consequences of these extremal ranks and inertias are also presented in the paper.     

On pointwise and analytic similarity of matrices

LetA(ε) andB(ε) be complex valued matrices analytic in ε at the origin.A(ε)≈ _p B(ε) ifA(ε) is similar toB(ε) for any |ε|<r,A(ε)≈_a B(ε) ifB(ε)=T(ε)A(ε)T ⁻¹(ε) andT(ε) is analytic and |T(ε)|≠0 for |ε|<r! In this paper we find a necessary and sufficient conditions onA(ε) andB(ε) such thatA(ε)≈ _a B(ε) provided thatA(ε)≈ _p B(ε). This problem arises in study of certain ordinary differential equations singular with respect to a parameter ε in the origin and was first stated by Wasow. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024     

An algebraic invariant for Jordan automorphisms on β(<Emphasis Type="Italic">H</Emphasis>): The set of idempotents

Let H be an infinite dimensional complex Hilbert space. Denote by B(H) the algebra of all bounded linear operators on H, and by I(H) the set of all idempotents in B(H). Suppose that Φ is a surjective map from B(H) onto itself. If for every λ ∈ -1,1,2,3, and A, B ∈ B(H),A-λB ∈I(H) ⇔ Φ(A) -λΦ(B) ∈I(H, then Φ is a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that Φ(A) = TAT ^-1 for all A ∈ B(H), or Φ(A) = TA*T ^-1 for all A ∈ B(H); if, in addition, A-iB ∈I(H)⇔ Φ(A)-iΦ(B) ∈I(H), here i is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.     

An integrable equation with nonsmooth solitons

We present the bi-Hamiltonian structure and Lax pair of the equation ρ_t = bu_x+(1/2)[(u ² −u_x ² )ρ]_x, where ρ = u − u_xx and b = const, which guarantees its integrability in the Lax pair sense. We study nonsmooth soliton solutions of this equation and show that under the vanishing boundary condition u → 0 at the space and time infinities, the equation has both “W/M-shape” peaked soliton (peakon) and cusped soliton (cuspon) solutions.     

A new extended matrix KP hierarchy and its solutions

Starting from the matrix KP hierarchy and adding a new τ_B flow, we obtain a new extended matrix KP hierarchy and its Lax representation with the symmetry constraint on squared eigenfunctions taken into account. The new hierarchy contains two sets of times t_A and τ_B and also eigenfunctions and adjoint eigenfunctions as components. We propose a generalized dressing method for solving the extended matrix KP hierarchy and present some solutions. We study the soliton solutions of two types of (2+1)-dimensional AKNS equations with self-consistent sources and two types of Davey-Stewartson equations with selfconsistent sources.     

Generalized Bowen-Franks groups of integral matrices with the same zeta function

In this paper the relation between the zeta function of an integral matrix and its generalized Bowen-Franks groups is studied. Suppose that A and B are nonnegative integral matrices whose invertible part is diagonalizable over the field of complex numbers and A and B have the same zeta function. Then there is an integer m, which depends only on the zeta function, such that, for any prime q such that gcd(q,m)=1, for any g(x)∈Z[x] with g(0)=1, the q-Sylow subgroup of the generalized Bowen-Franks group BF_g(x)(A) and BF_g(x)(B) are the same. In particular, if m=1, then zeta function determines generalized Bowen-Franks groups.     

The long-step method of analytic centers for fractional problems

We develop a long-step surface-following version of the method of analytic centers for the fractional-linear problem min{t ₀ |t ₀ B(x) −A(x) εH, B(x) εK, x εG}, whereH is a closed convex domain,K is a convex cone contained in the recessive cone ofH, G is a convex domain andB(·),A(·) are affine mappings. Tracing a two-dimensional surface of analytic centers rather than the usual path of centers allows to skip the initial “centering” phase of the path-following scheme. The proposed long-step policy of tracing the surface fits the best known overall polynomial-time complexity bounds for the method and, at the same time, seems to be more attractive computationally than the short-step policy, which was previously the only one giving good complexity bounds. The research was partly supported by the Israeli-American Binational Science Foundation (BSF).     

Maximal subspaces for solutions of the second order abstract cauchy problem

For a continuous, increasing function ω: R → R \{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]-1u(t,x) is uniformly continues on R , and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A|z(A,ω) generates an O(ω(t))strongly continuous cosine operator function family.     

One sufficient condition for Hamiltonian graphs involving distances

In 1990 G. T. Chen proved that if G is a 2-connected graph of order n and 2|N(x) ∪ N(y)| + d(x) + d(y) ≥ 2n − 1 for each pair of nonadjacent vertices x, y ∈ V (G), then G is Hamiltonian. In this paper we prove that if G is a 2-connected graph of order n and 2|N(x) ∪ N(y)| + d(x)+d(y) ≥ 2n−1 for each pair of nonadjacent vertices x, y ∈ V (G) such that d(x, y) = 2, then G is Hamiltonian.     

Geometric computation of the numerical radius of a matrix

We use methods of geometric computing combined with hermitean matrix eigenvalue/eigenvector evaluations to find the numerical radius w(A) of a real or complex square matrix A simply, quickly, and accurately. The numerical radius w(A) is defined as the maximal distance of points in the field of values F(A) = { x^* A x | ||x||₂ = 1 }F(A) = \{ x^* A x \mid \|x\|_2 = 1 \} from zero in ℂ. Its value is an indicator of the transient behavior of the discrete dynamical system f _k + 1 = Af _k . We describe and test a MATLAB code for solving this optimization problem that can have multiple and even infinitely many solutions with maximal distance.     

Oscillation criteria related to integral averaging technique for linear matrix Hamiltonian systems

By employing a generalized Riccati technique and an integral averaging technique, new oscillation criteria are established for the linear matrix Hamiltonian system U′=A(x)U+B(x)V, V′=C(x)U−A∗(x)V under the assumption that all A(x), B(x)=B∗(x)>0, and C(x)=C∗(x) are n×n matrices of real-valued continuous functions on the interval I=[x₀,∞) (−∞<x₀). These criteria extend, improve, and unify a number of existing results and also handle cases not covered by known criteria. Several interesting examples that illustrate the importance of our results are included.     

Nonintegral criterion for oscillations of linear Hamiltonian matrix systems

In this paper we present nonintegral criteria for oscillation of linear Hamiltonian matrix system U^′=A(x)U+B(x)V, V^′=C(x)U−A^*(x)V under the hypothesis (H): A(x), B(x)=B^*(x)>0, and C(x)=C^*(x) are 2×2 matrices of real valued continuous functions on the interval I=[a,∞),(−∞<a). These criteria are conditions of algebraic type only. Our results are also useful for the detection of the oscillation of particular matrix differential systems.     

2-uniform congruences in majority algebras and a closure operator

A ternary term m(x, y, z) of an algebra is called a majority term if the algebra satisfies the identities m(x, x, y) = x, m(x, y, x) = x and m(y, x, x) = x. A congruence α of a finite algebra is called uniform if all of its blocks (i.e., classes) have the same number of elements. In particular, if all the α-blocks are two-element then α is said to be a 2-uniform congruence. If all congruences of A are uniform then A is said to be a uniform algebra. Answering a problem raised by Gr?tzer, Quackenbush and Schmidt [2], Kaarli [3] has recently proved that uniform finite lattices are congruence permutable. In connection with Kaarli’s result, our main theorem states that for every finite algebra A with a majority term any two 2-uniform congruences of A permute. Examples show that we can say neither “algebra” instead of “algebra with a majority term”, nor “3-uniform” instead of “2-uniform”. Given two nonempty sets A and B, each relation gives rise to a pair of closure operators, which are called the Galois closures on A and B induced by ρ. Galois closures play an important role in many parts of algebra, and they play the main role in formal concept analysis founded by Wille [4]. In order to prove our main theorem, we introduce a pair of smaller closure operators induced by ρ. These closure operators will hopefully find further applications in the future. Dedicated to the memory of Kazimierz Głazek Presented by E. T. Schmidt. Received November 29, 2005; accepted in final form May 23, 2006. This research was partially supported by the NFSR of Hungary (OTKA), grant no. T049433 and T037877.