Convergence for a class of delayed recurrent neural networks without M-matrix condition

In this paper, we investigate the asymptotic behavior of solutions to a class of recurrent neural network model with delays. Without assuming M-matrix condition, it is shown that every solution of the network tends to an equilibrium point as t→∞. Our results improve and extend some corresponding ones already known.     

Asymptotic constancy for a differential equation with multiple state-dependent delays

In this paper, we investigate the asymptotic behavior of solutions to a differential equation with multiple state-dependent delays. It is shown that every bounded solution of such an equation tends to a constant as t→∞. Our results improve and extend some corresponding ones already known.     

The determining equations for the nonclassical method of the nonlinear differential equation(s) with arbitrary order can be obtained through the compatibility

In this paper, firstly we show that the determining equations of the (1+1) dimension nonlinear differential equation with arbitrary order for the nonclassical method can be derived by the compatibility between the original equation and the invariant surface condition. Then we generalize this result to the system of the (m+1) dimension differential equations. The nonlinear Klein–Gordon equation, the (2+1)-dimensional Boussinesq equation and the generalized Nizhnik–Novikov–Veselov equation serve as examples illustrating this method.     

An iterative algorithm based on -proximal mappings for a system of generalized implicit variational inclusions in Banach spaces

In this paper, we give the notion of M-proximal mapping, an extension of P-proximal mapping given in [X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369–383], for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space and prove its existence and Lipschitz continuity. Further, we consider a system of generalized implicit variational inclusions in Banach spaces and show its equivalence with a system of implicit Wiener–Hopf equations using the concept of M-proximal mappings. Using this equivalence, we propose a new iterative algorithm for the system of generalized implicit variational inclusions. Furthermore, we prove the existence of solution of the system of generalized implicit variational inclusions and discuss the convergence and stability analysis of the iterative algorithm.     

-extension of Lie triple systems

In this paper, we introduce the notion of T^*-extension of a Lie triple system. Then we show that T^*-extension is compatible with nilpotency, solvability, and it preserves in certain sense the decomposition properties. In addition, we investigate the equivalence of T^*-extensions using cohomology. Finally, we show that every finite-dimensional nilpotent metrised Lie triple system over an algebraically closed field is the T^*-extension of an appropriate quotient system.     

Solutions to operator equations on Hilbert -modules

Let be a C^*-algebra, E,F and G be Hilbert -modules, , and . We generalize the Douglas theorem about the operator equation TX=T^′ from Hilbert space to Hilbert C^*-module. To the equation TX=T^′ and to the system of two equations TX=T^′ and XS=S^′, we get the forms of general solutions (in the case that there exists a solution), and give some sufficient and necessary conditions for the existence of solutions, and the existence of hermitian solutions and positive solutions (in the case G=E). In addition, the forms of general hermitian solution and general positive solution (in the case that there exists a solution and G=E) to the equation TX=T^′ are given too.     

Quantum -modules and generalized mirror transformations

In the previous paper [Hiroshi Iritani, Quantum D-modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (3) (2006) 577–622], the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c₁(M) of the tangent bundle is nef. In this paper, even when c₁(M) is not nef, we show that the equivariant Floer cohomology reconstructs the big quantum D-module under certain conditions on the ambient toric variety. The proof is based on a mirror theorem of Coates and Givental [T. Coates, A.B. Givental, Quantum Riemann — Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (1) (2007) 15–53]. The reconstruction procedure here gives a generalized mirror transformation first observed by Jinzenji in low degrees [Masao Jinzenji, On the quantum cohomology rings of general type projective hypersurfaces and generalized mirror transformation, Internat. J. Modern Phys. A 15 (11) (2000) 1557–1595; Masao Jinzenji, Co-ordinate change of Gauss–Manin system and generalized mirror transformation, Internat. J. Modern Phys. A 20 (10) (2005) 2131–2156].     

Anti-periodic solutions for a class of nonlinear th-order differential equations with delays

In this paper, we use the Leray–Schauder degree theory to establish new results on the existence and uniqueness of anti-periodic solutions for a class of nonlinear nth-order differential equations with delays of the form

x⁽ⁿ⁾(t)+f(t,x⁽ⁿ⁻¹⁾(t))+g(t,x(t−τ(t)))=e(t).

     

A new exclusion test for finding the global minimum

Exclusion algorithms have been used recently to find all solutions of a system of nonlinear equations or to find the global minimum of a function over a compact domain. These algorithms are based on a minimization condition that can be applied to each cell in the domain. In this paper, we consider Lipschitz functions of order α and give a new minimization condition for the exclusion algorithm. Furthermore, convergence and complexity results are presented for such algorithm.     

Stability and asymptotic stability of -methods for nonlinear stiff Volterra functional differential equations in Banach spaces

This paper is concerned with the stability and asymptotic stability of θ-methods for the initial value problems of nonlinear stiff Volterra functional differential equations in Banach spaces. A series of new stability and asymptotic stability results of θ-methods are obtained.     

Global Hopf bifurcation for differential equations with state-dependent delay

We develop a global Hopf bifurcation theory for a system of functional differential equations with state-dependent delay. The theory is based on an application of the homotopy invariance of S¹-equivariant degree using the formal linearization of the system at a stationary state. Our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state-dependent delay.     

Coincidence and common fixed points for hybrid strict contractions without the weakly commuting condition

In this work, we establish new coincidence and common fixed point theorems for hybrid strict contraction maps by dropping the assumption “f is T-weakly commuting” for a hybrid pair (f,T) of multivalued maps in Theorem 3.10 of T. Kamran (2004) [8]. As an application, an invariant approximation result is derived.     

A completeness result for the simply typed -calculus

In this paper, we define a realizability semantics for the simply typed λμ-calculus. We show that, if a term is typable, then it inhabits the interpretation of its type. This result serves to give characterizations of the computational behavior of some closed typed terms. We also prove a completeness result of our realizability semantics using a particular term model.     

Connected -tuple twin domination in de Bruijn and Kautz digraphs

For a digraph G, a k-tuple twin dominating set D of G for some fixed k≥1 is a set of vertices such that every vertex is adjacent to at least k vertices in D, and also every vertex is adjacent from at least k vertices in D. If the subgraph of G induced by D is strongly connected, then D is called a connected k-tuple twin dominating set of G. In this paper, we give constructions of minimal connected k-tuple twin dominating sets for de Bruijn digraphs and Kautz digraphs.     

Convexity of solutions and estimates for fully nonlinear elliptic equations

The starting point of this work is a paper by Alvarez, Lasry and Lions (1997) concerning the convexity and the partial convexity of solutions of fully nonlinear degenerate elliptic equations. We extend their results in two directions. First, we deal with possibly sublinear (but epi-pointed) solutions instead of 1-coercive ones; secondly, the partial convexity of C² solutions is extended to the class of continuous viscosity solutions. A third contribution of this paper concerns C^1,1 estimates for convex viscosity solutions of strictly elliptic nonlinear equations. To finish with, all the tools and techniques introduced here permit us to give a new proof of the Alexandroff estimate obtained by Trudinger (1988) and Caffarelli (1989).     

Cone-valued Lyapunov functions and stability for impulsive functional differential equations

The stability criteria in terms of two measures for impulsive functional differential equations are established via cone-valued Lyapunov functions and Razumikhin technique. The stability can be deduced from the (Q₀,Q)-stability of comparison impulsive differential equations. An example is given to illustrate the advantages of the results obtained.     

Noncommutative maximal ergodic theorems for positive contractions

Let M be a von Neumann algebra equipped with a normal semifinite faithful trace τ. Let T be a positive linear contraction on M such that τ○T?τ and such that the numerical range of T as an operator on L₂(M) is contained in a Stoltz region with vertex 1. We show that Junge and Xu's noncommutative Stein maximal ergodic inequality holds for the powers of T on Lp(M), 1<p?∞. We apply this result to obtain the noncommutative analogue of a recent result of Cohen concerning the iterates of the product of a finite number of conditional expectations.     

Travelling wave solutions for the nonlinear dispersion Drinfel’d–Sokolov () system

In this paper, travelling wave solutions for the nonlinear dispersion Drinfel’d–Sokolov system (called D(m,n) system) are studied by using the Weierstrass elliptic function method. As a result, more new exact travelling wave solutions to the D(m,n) system are obtained including not only all the known solutions found by Xie and Yan but also other more general solutions for different parameters m,n. Moreover, it is also shown that the D(m,1) system with linear dispersion possess compacton and solitary pattern solutions. Besides that, it should be pointed out that the approach is direct and easily carried out without the aid of mathematical software if compared with other traditional methods. We believe that the method can be widely applied to other similar types of nonlinear partial differential equations (PDEs) or systems in mathematical physics.     

On a signless Laplacian spectral characterization of -shape trees

Let M be an associated matrix of a graph G (the adjacency, Laplacian and signless Laplacian matrix). Two graphs are said to be cospectral with respect to M if they have the same M spectrum. A graph is said to be determined by M spectrum if there is no other non-isomorphic graph with the same spectrum with respect to M. It is shown that T-shape trees are determined by their Laplacian spectra. Moreover among them those are determined by their adjacency spectra are characterized. In this paper, we identify graphs which are cospectral to a given T-shape tree with respect to the signless Laplacian matrix. Subsequently, T-shape trees which are determined by their signless Laplacian spectra are identified.     

Existence of periodic solutions for a fourth-order -Laplacian equation with a deviating argument

In this paper, we study the existence of periodic solutions for a fourth-order p-Laplacian differential equation with a deviating argument as follows:

[φ_p(u^″(t))]^″+f(u^″(t))+g(u(t−τ(t)))=e(t).