Higher-order compact finite difference method for systems of reaction–diffusion equations

This paper is concerned with a compact finite difference method for solving systems of two-dimensional reaction–diffusion equations. This method has the accuracy of fourth-order in both space and time. The existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. Three monotone iterative algorithms are provided for solving the resulting discrete system efficiently, and the sequences of iterations converge monotonically to a unique solution of the system. A theoretical comparison result for the various monotone sequences is given. The convergence of the finite difference solution to the continuous solution is proved, and Richardson extrapolation is used to achieve fourth-order accuracy in time. An application is given to an enzyme–substrate reaction–diffusion problem, and some numerical results are presented to demonstrate the high efficiency and advantages of this new approach.     

Studying Baskakov–Durrmeyer operators and quasi-interpolants via special functions

We prove that the kernels of the Baskakov–Durrmeyer and the Szász–Mirakjan–Durrmeyer operators are completely monotonic functions. We establish a Bernstein type inequality for these operators and apply the results to the quasi-interpolants recently introduced by Abel. For the Baskakov–Durrmeyer quasi-interpolants, we give a representation as linear combinations of the original Baskakov–Durrmeyer operators and prove an estimate of Jackson–Favard type and a direct theorem in terms of an appropriate K-functional.     

Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization

In this paper, we introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the solutions of the variational inequality problem for two inverse-strongly monotone mappings. We introduce a new viscosity relaxed extragradient approximation method which is based on the so-called relaxed extragradient method and the viscosity approximation method. We show that the sequence converges strongly to a common element of the above three sets under some parametric controlling conditions. Moreover, using the above theorem, we can apply to finding solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. The results of this paper extended, improved and connected with the results of Ceng et al., [L.-C. Ceng, C.-Y. Wang, J.-C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Meth. Oper. Res. 67 (2008), 375–390], Plubtieng and Punpaeng, [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Appl. Math. Comput. 197 (2) (2008) 548–558] Su et al., [Y. Su, et al., An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. 69 (8) (2008) 2709–2719], Li and Song [Liwei Li, W. Song, A hybrid of the extragradient method and proximal point algorithm for inverse strongly monotone operators and maximal monotone operators in Banach spaces, Nonlinear Anal.: Hybrid Syst. 1 (3) (2007), 398-413] and many others.     

Sur des notions de n—categorie et n—groupoide non strictes via des ensembles multi-simpliciaux (On the notions of a nonstrict n–category and n–groupoid via multisimplicial sets)

In this paper we first give a simplicial approach to the definition of a nonstrict n–category that we call a n–nerve following the idea that a category could be interpreted as a simplicial set (its nerve). Then we prove that for n=2 our construction is equivalent to the usual nonstrict 2–category (bicategory). Next,we give a simplicial definition of a nonstrict n–groupoïd, and we associate to any topological space a n–groupoïd n (X) which generalises the famous Poincaré groupoïd 1 (X) and embodies the n–truncated homotopy type of . Conversely, we construct for each n–groupoïd a geometric realisation and we show that the functors geometric realisation and Poincaré n–groupoïd induce an equivalence between the category of n–groupoids and the category of n–truncated topological spaces, when we localise both categories by weak equivalence.     

Bowen–Franks Groups for Subshifts and Ext-Groups for C*-Algebras

We generalize the Bowen–Franks groups for topological Markov shifts to general subshifts as the Ext-groups for the associated C ^*-algebras. The generalized Bowen–Franks groups for subshifts are shown to be invariant under flow equivalence and, hence, invariant under topological conjugacy. They are regarded as the indices of Fredholm operators related to extensions of the associated C ^*-algebras so that they are described in terms of symbolic dynamical systems. In particular, the group for a sofic subshift is determined by the adjacency matrix of its left Krieger cover graph. The Bowen–Franks groups for some non sofic subshifts are calculated, proving that certain subshifts with the same topological entropy are not flow equivalent.     

Asymptotic behavior for a cellular replication and maturation model

We consider a nonlocal first order partial differential equation with time delay that models simultaneous cell replication and maturation processes. We establish a comparison principle and construct monotone sequences to show the existence and uniqueness of the solution to the equation. We then analyze the asymptotic behavior of the solution via upper–lower solution technique.     

Asymptotics of solutions to the Laplace–Beltrami equation on a rotation surface with a cusp

In this work we study an asymptotic behaviour of solutions to the Laplace–Beltrami operator on a rotation surface near a cuspidal point. To this end we use the WKB-approximation. This approach describes the asymptotic behaviour of the solution more explicitly than abstract theory for operators with operator-valued coefficients.     

Nonmonotone Levenberg–Marquardt Algorithms and Their Convergence Analysis

In this paper, two nonmonotone Levenberg–Marquardt algorithms for unconstrained nonlinear least-square problems with zero or small residual are presented. These algorithms allow the sequence of objective function values to be nonmonotone, which accelerates the iteration progress, especially in the case where the objective function is ill-conditioned. Some global convergence properties of the proposed algorithms are proved under mild conditions which exclude the requirement for the positive definiteness of the approximate Hessian T(x). Some stronger global convergence properties and the local superlinear convergence of the first algorithm are also proved. Finally, a set of numerical results is reported which shows that the proposed algorithms are promising and superior to the monotone Levenberg–Marquardt algorithm according to the numbers of gradient and function evaluations.     

Newton-Kantorovich Iterative Regularization for Nonlinear Ill-Posed Equations Involving Accretive Operators

The Newton-Kantorovich iterative regularization for nonlinear ill-posed equations involving monotone operators in Hilbert spaces is developed for the case of accretive operators in Banach spaces. An estimate for the convergence rates of the method is established.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 2, pp. 271–276, February, 2005.     

Abstract Cyclical Monotonicity and Monge Solutions for the General Monge–Kantorovich Problem

Abstract cyclical monotonicity is studied for a multivalued operator F : X L, where L R ^X . A criterion for F to be L-cyclically monotone is obtained and connections with the notions of L-convex function and of its L-subdifferentials are established. Applications are given to the general Monge–Kantorovich problem with fixed marginals. In particular, we show that in some cases the optimal measure is unique and generated by a unique (up to the a.e. equivalence) optimal solution (measure preserving map) for the corresponding Monge problem.     

Linear continuous operators for the Stieltjes moment problem in Gelfand–Shilov spaces

We obtain linear continuous operators providing a solution to the Stieltjes moment problem in the framework of Gelfand–Shilov spaces of rapidly decreasing smooth functions. The construction rests on an interpolation procedure due to R. Estrada for general rapidly decreasing smooth functions, and adapted by S.-Y. Chung, D. Kim and Y. Yeom to the case of Gelfand–Shilov spaces. It requires a linear continuous version of the so-called Borel–Ritt–Gevrey theorem in asymptotic theory.     

The unitary equivalence of linear operatops to integral operators

Criteria are established for the unitary equivalence of bounded linear operators to B-Carleman and bi-Carleman integral operators.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 601–610, April, 1976.     

On solvability of variational inequalities with discontinuous semimonotone operators

By using the method of monotone operators, a theorem on the existence of the solution with a special property is obtained for an elliptic variational inequality with discontinuous semimonotone operator; this theorem is then used to prove the existence of a semicorrect solution of a variational inequality with a differential semilinear high-order operator of elliptic type with a nonsymmetric linear part and discontinuous nonlinearity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 443–447, March, 1993.     

Extended Projection Methods for Monotone Variational Inequalities

In this paper, we prove that each monotone variational inequality is equivalent to a two-mapping variational inequality problem. On the basis of this fact, a new class of iterative methods for the solution of nonlinear monotone variational inequality problems is presented. The global convergence of the proposed methods is established under the monotonicity assumption. The conditions concerning the implementability of the algorithms are also discussed. The proposed methods have a close relationship to the Douglas–Rachford operator splitting method for monotone variational inequalities.     

Generalized intersection bodies

We study the structures of two types of generalizations of intersection-bodies and the problem of whether they are in fact equivalent. Intersection-bodies were introduced by Lutwak and played a key role in the solution of the Busemann–Petty problem. A natural geometric generalization of this problem considered by Zhang, led him to introduce one type of generalized intersection-bodies. A second type was introduced by Koldobsky, who studied a different analytic generalization of this problem. Koldobsky also studied the connection between these two types of bodies, and noted that an equivalence between these two notions would completely settle the unresolved cases in the generalized Busemann–Petty problem. We show that these classes share many identical structural properties, proving the same results using integral geometry techniques for Zhang's class and Fourier transform techniques for Koldobsky's class. Using a functional analytic approach, we give several surprising equivalent formulations for the equivalence problem, which reveal a deep connection to several fundamental problems in the integral geometry of the Grassmann manifold.     

On Convergence of One-Step Schemes for Weak Solutions of Quantum Stochastic Differential Equations

Several one-step schemes for computing weak solutions of Lipschitzian quantum stochastic differential equations (QSDE) driven by certain operator-valued stochastic processes associated with creation, annihilation and gauge operators of quantum field theory are introduced and studied. This is accomplished within the framework of the Hudson–Parthasarathy formulation of quantum stochastic calculus and subject to the matrix elements of solution being sufficiently differentiable. Results concerning convergence of these schemes in the topology of the locally convex space of solution are presented. It is shown that the Euler–Maruyama scheme,with respect to weak convergence criteria for Itô stochastic differential equation is a special case of Euler schemes in this framework. Numerical examples are given.     

Lie group classification of the generalized Lane–Emden equation

We carry out the Lie group classification of the generalized Lane–Emden equation xu^″+nu^′+xH(u)=0, which has many applications in mathematical physics and astrophysics. We show that the equation admits a three-dimensional equivalence Lie algebra. It is also shown that the principal Lie algebra, which in this case is trivial, has seven possible extensions. Three new cases arise for which the Lie point symmetry algebra is non-trivial. Comparison is then made of these cases with the Noether symmetry cases as well as the partial Noether operators.     

Regularization of Monotone Variational Inequalities with Mosco Approximations of the Constraint Sets

In this paper we study the convergence and stability in reflexive, smooth and strictly convex Banach spaces of a regularization method for variational inequalities with data perturbations. We prove that, when applied to perturbed variational inequalities with monotone, demiclosed, convex valued operators satisfying certain conditions of asymptotic growth, the regularization method we consider produces sequences which converge weakly to the minimal-norm solution of the original variational inequality, provided that the perturbed constraint sets converge to the constraint set of the original inequality in the sense of a modified form of Mosco convergence of order ≥1. If the underlying Banach space has the Kadeč–Klee property, then the sequence generated by that regularization method is strongly convergent. Mathematics Subject Classifications (2000) Primary: 47J0G, 47A52; secondary: 47H14, 47J20.     

The global attractor of the viscous Fornberg–Whitham equation

This paper aims to present a proof of the existence of the attractor for the one-dimensional viscous Fornberg–Whitham equation. In this paper, the global existence of solution to the viscous Fornberg–Whitham equation in L² under the periodic boundary conditions is studied. By using the time estimate of the Fornberg–Whitham equation, we get the compact and bounded absorbing set and the existence of the global attractor for the viscous Fornberg–Whitham equation.