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In this paper, we obtain some new Lyapunov-type inequalities for a class of even-order linear differential equations, the results are new and generalize and improve some early results in this field. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of nonlinear impulsive neutral partial functional differential equations with continuous distributed deviating arguments. For this class, we establish sufficient conditions for the H-oscillation of the solutions, using impulsive differential inequalities and an averaging technique with two different boundary conditions. We provide an example to illustrate the main result. [ABSTRACT FROM AUTHOR]

In this paper, we consider the singularly perturbed reaction-diffusion problem in one and two dimensions. The boundary value problem is decomposed into a first-order system to which a suitable weighted least-squares formulation is proposed. A robust, stable, and efficient approach is developed based on local discontinuous Galerkin (LDG) discretization for the weak form. Uniform error estimates are derived. Numerical examples are presented to illustrate the method and the theoretical results. Comparison studies are made between the proposed method and other methods. [ABSTRACT FROM AUTHOR]

In this paper we consider the convergence of the infinite dimensional version of the Kleinman–Newton algorithm for solving the algebraic Riccati operator equation associated with the linear quadratic regulator problem in a Hilbert space. We establish mesh independence for this algorithm and apply the result to systems governed by delay equations. Numerical examples are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

Judy and Motta developed a customizable electrodeposition process for fabrication of very small metal structures on a substrate. In this process, layers of metal of various shapes are placed on the substrate, then the substrate is inserted in an electroplating solution. Some of the metal layers have power applied to them, while the rest of the metal layers are not connected to the power initially. Metal ions in the plating solution start depositing on the powered layers and a surface grows from the powered layers. As the surface grows, it will touch metal layers that were initially unpowered, causing them to become powered and to start growing with the rest of the surface. The metal layers on the substrate are known as seed layer patterns, and different seed layer patterns can produce different shapes. This paper presents a mathematical model, a forward simulation method, and an inverse problem solution for the growth of a surface from a seed layer pattern. The model describes the surface evolution as uniform growth in the direction normal to the surface. This growth is simulated in two and three dimensions using the level set method. The inverse problem is to design a seed layer pattern that produces a desired surface shape. Some surface shapes are not attainable by any seed layer pattern. For smooth attainable shapes, we present a computational method that solves this inverse problem. [ABSTRACT FROM AUTHOR]

This paper studies robust stability of linear systems with polytopic uncertainty. New necessary and sufficient conditions for the existence of an affine parameter-dependent Lyapunov function assuring the Hurwitz or the Schur stability of a polytopic system are presented. These conditions are composed of a family of linear matrix inequality conditions of increasing precision. At each step, a set of linear matrix inequalities provides sufficient conditions for the existence of the affine parameter-dependent Lyapunov function, and necessity is asymptotically attained. Compared with the existing results in the literature, it is shown that the new stability conditions provide less conservative tests at each step. Numerical examples are given to illustrate the effectiveness of the new results. [ABSTRACT FROM AUTHOR]

In this paper, we study the solution of a certain non-smooth eigenvalue problem, using operator-splitting methods to solve an equivalent, constrained minimization problem. We present the Marchuk-Yanenko and Peaceman-Rachford schemes for solving the problem and compare their performance numerically on some model problems. The Peaceman-Rachford scheme turns out to be superior to the Marchuk-Yanenko scheme in terms of accuracy and computational efficiency. [ABSTRACT FROM AUTHOR]

In this paper a Galerkin method is used to construct series solutions of a homogeneous Sturm-Liouville problem defined on [0, &pie;]. The series constructed are shown to converge to a specified du Bois-Reymond function f in L² [0,&Pie;]. It is then shown that the series solutions can be made to converge uniformly to the specified du Bois-Reymond function when averaged by the Cesaro-one summability method. Therefore, in the Cesaro-one sense, every continuous function / o n [0, &Pie;] is the uniform limit of solutions of non-homogeneous Sturm-Liouville problems. [ABSTRACT FROM AUTHOR]

Traditional 'pencil and paper' derivations of the numerical differentiation formulae for f ′[x0] and f″[x0] have been done independently as if there was no connection among the two derivations. This new approach gives a parallel development of the formulae. It requires the solution of a 'linear system' that includes symbolic quantities as coefficients and constants. It is shown how the power of a computer algebra system such as Mathematica can be used to elegantly solve this linear system for f′[x0] and f″[x0]. The extension to derivations of higher order numerical differentiation formulas for the central, forward or backward differences are also presented. [ABSTRACT FROM AUTHOR]

In this paper we present the convergence analysis of iterative schemes for solving linear systems resulting from discretizing multidimensional linear second-order elliptic partial differential equations (PDEs) defined in a hyperparallelepiped ω and subject to Dirichlet boundary conditions on some faces of ω and Neumann on the others, using line cubic spline collocation (LCSC) methods. Specifically, we derive analytic expressions or obtain sharp bounds for the spectral radius of the corresponding Jacobi iteration matrix and from this we determine the convergence ranges and compute the optimal parameters for the extrapolated Jacobi and successive overrelaxation (SOR) methods. Experimental results are also presented. [ABSTRACT FROM AUTHOR]

In this paper linear periodic systems of difference equations are classified with respect to periodic similarity and kinematic similarity. Complete sets of invariants of periodic difference equations relative to such similarity transformations are given, and corresponding canonical forms are described. Also the irreducible periodic difference equations, i.e., those that cannot be reduced by such similarities to a nontrivial direct sum, are identified. [ABSTRACT FROM AUTHOR]

An exact line search method has been introduced by Benner and Byers [IEEE Trans. Automat. Control, 43 (1998), pp. 101–107] for solving continuous algebraic Riccati equations. The method is a modification of Newton's method. A convergence theory is established in that paper for the Newton-like method under the strong hypothesis of controllability, while the original Newton's method needs only the weaker hypothesis of stabilizability for its convergence theory. It is conjectured there that the controllability condition can be weakened to the stabilizability condition. In this article we prove that conjecture. [ABSTRACT FROM AUTHOR]

Numerical representations of choice functions allow the expression of a problem of choice as a problem of finding maxima of real-valued functions, which requires that less information be defined and which is easier to work with. In this paper, the existence of numerical representations of choice functions by imposing assumptions directly on the choice function is obtained. In particular, it is proved that (IIA) is a necessary and sufficient condition for a choice function to be representable, if the set of alternatives is countable, while the conjunction of (IIA) and a ‘continuity condition’ is sufficient to ensure it in separable topological spaces. [ABSTRACT FROM AUTHOR]

Abstract. In this paper, we obtain some applications of first order differential subordination and superordination results for some analytic functions defined by convolution. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, numerical solution of first-order systems of ordinary differential equations (ODEs) is considered and an error estimation method is proposed. By this method, the efficiency of pseudospectral method to solve the systems is determined and a modified pseudospectral method is presented. Furthermore, with providing some examples, the aforementioned cases are dealt with numerically. [Copyright &y& Elsevier]

Abstract: In this paper, nutrient-phytoplankton-zooplankton interaction with general uptake functions in which nutrient recycling is either instantaneous or delayed is considered. To account for higher predation, zooplankton''s death rate is modeled by a quadratic term instead of the usual linear function. Persistence conditions for each of the delayed and nondelayed models are derived. Numerical simulations with data from the existing literature are explored to compare the two models. It is demonstrated numerically that increasing zooplankton death rate can eliminate periodic solutions of the system in both the instantaneous and the delayed nutrient recycling models. However, the delayed nutrient recycling can actually stabilize the nutrient-plankton interaction. [Copyright &y& Elsevier]

In this study of two-step hybrid methods for second-order equations of the type y ″ = f ( y ), we apply P -series [Hairer, E., Lubich, C. and Wanner, G. (2002). Geometric Numerical Integration Structure-Preserving Algorithms for Ordinary Differential Equations . Springer Series in Computational Mathematics.] to formalise the approach of Chan [Chan, R. P. K. (2002). Two-step hybrid methods. Internal Publication .] to the order conditions, and present two characterizations of symmetry. Although order conditions can be obtained through the classical theory for the Nyström methods, it is of interest to derive particular simpler formulas for the class of two-step hybrid methods in order to facilitate the search for high-order methods. Moreover, the approach proves useful in analysing the symmetry of the hybrid methods. [ABSTRACT FROM AUTHOR]

In this paper, a mortar finite element method for parabolic problem is presented. Multigrid method is used for solving the resulting discrete system. It is shown that the multigrid method is optional, i.e., the convergence rate is independent of the mesh size L and the time step parameter Τ. [ABSTRACT FROM AUTHOR]

The spatiotemporal nonlinear Schrödinger equation with power-law nonlinearity in -symmetric potentials is investigated, and two families of analytical three-dimensional spatiotemporal structure solutions are obtained. The stability of these solutions is tested by the linear stability analysis and the direct numerical simulation. Results indicate that solutions are stable below some thresholds for the imaginary part of -symmetric potentials in the self-focusing medium, while they are always unstable for all parameters in the self-defocusing medium. Moreover, some dynamical properties of these solutions are discussed, such as the phase switch, power and transverse power-flow density. The span of phase switch gradually enlarges with the decrease of the competing parameter k in -symmetric potentials. The power and power-flow density are all positive, which implies that the power flow and exchange from the gain toward the loss domains in the cell. [ABSTRACT FROM AUTHOR]

The article discusses the stability conditions for solutions to impulsive equations based on a heterogeneous matrix-valued Lyapunov function. It notes the development of the method of matrix-valued Lyapunov functions which serve as a generalization of Lyapunov's direct method based on matrix-valued functions. It puts emphasis on the association of matrix-valued function elements to certain equations of the system being studied or to a composition of its subsystems. The impulsive differential equations and fuzzy equation demonstrating the principle comparison with heterogeneous Lyapunov function are also presented.

We formulate general principles on the existence of homeomorphic absolutely continuous on lines (ACL) solutions for the Beltrami equations with degeneration and derive from them a series of criteria and, in particular, a generalization and strengthening of the well-known Lehto existence theorem. Furthermore, we prove that in all these cases there exist the so-called strong ring solutions satisfying additional moduli conditions which play a great role in the research of various properties of such solutions. [ABSTRACT FROM AUTHOR]

We construct a function in AC0 that cannot be computed by a depth-2 majority circuit of size less than exp(ϴ(n1/5)). This solves an open problem due to Krause and Pudlák [Theoret. Comput. Sci., 174 (1997), pp. 137-156] and matches Allender's classic result [A note on the power of threshold circuits, in Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science (FOCS), Research Triangle Park, NC, 1989, pp. 580-584] that AC0 can be efficiently simulated by depth-3 majority circuits. To obtain our result, we develop a novel technique for proving lower bounds on communication complexity. This technique, the Degree/Discrepancy Theorem, is of independent interest. It translates lower bounds on the threshold degree of any Boolean function into upper bounds on the discrepancy of a related function. Upper bounds on the discrepancy, in turn, immediately imply lower bounds on communication and circuit size. In particular, we exhibit the first known function in AC0 with exponentially small discrepancy, exp(-Ω(n1/5)), thereby establishing the separations Σcc2... PPcc and Πcc2 ... PPcc in communication complexity. [ABSTRACT FROM AUTHOR]

We present new local anisotropic error estimates for the Lagrangian finite element interpolation. The bounds apply to affine equivalent elements and use information from directional derivatives of the function to interpolate along a set of adjacent edges. These new bounds do not require any geometric limitation but may vary, in some cases, with the node ordering. Several existing results are recovered from the new bounds. Examples compare the asymptotic behavior of the new and existing bounds when the diameter of the element goes to zero. For some elements with small or large angles, our new bound exhibits the same asymptotic behavior as the norm of the interpolation error while existing results do not have the correct asymptotic behavior. [ABSTRACT FROM AUTHOR]

We study the convergence of an adaptive interior penalty discontinuous Galerkin (IPDG) method for a two-dimensional model second order elliptic boundary value problem. Based on a residual-type a posteriori error estimator, we prove that after each refinement step of the adaptive scheme we achieve a guaranteed reduction of the global discretization error in the meshdependent energy norm associated with the IPDG method. In contrast to recent work on adaptive IPDG methods [O. Karakashian and F. Pascal, Convergence of Adaptive Discontinuous Galerkin Approximations of Second-order Elliptic Problems, preprint, University of Tennessee, Knoxville, TN, 2007], the convergence analysis does not require multiple interior nodes for refined elements of the triangulation. In fact, it will be shown that bisection of the elements is sufficient. The main ingredients of the proof of the error reduction property are the reliability and a perturbed discrete local efficiency of the estimator, a bulk criterion that takes care of a proper selection of edges and elements for refinement, and a perturbed Galerkin orthogonality property with respect to the energy inner product. The results of numerical experiments are given to illustrate the performance of the adaptive method. [ABSTRACT FROM AUTHOR]

Runge-Kutta-Nyström (RKN) methods provide a popular way to solve the initial value problem (IVP) for a system of ordinary differential equations (ODEs). Users of software are typically asked to specify a tolerance δ; that indicates in somewhat vague sense, the level of accuracy required. It is clearly important to understand the precise effect of changing δ; and to derive the strongest possible results about the behaviour of the global error that will not have regular behaviour unless an appropriate stepsize selection formula and standard error control policy are used. Faced with this situation sufficient conditions on an algorithm that guarantee such behaviour for the global error to be asympotatically linear in δ as δ → 0 , that were first derived by Stetter. Here we extend the analysis to cover a certain class of ODEs with low-order derivative discontinuities, and the class of ODEs with constant delays. We show that standard error control techniques will be successful if discontinuities are handled correctly and delay terms are calculated with sufficient accurate interpolants. It is perhaps surprising that several delay ODE algorithms that have been proposed do not use sufficiently accurate interpolants to guarantee asymptotic proportionality. Our theoretical results are illustrated numerically. [ABSTRACT FROM AUTHOR]

We discuss differential equations with nonlinear boundary conditions. We formulate sufficient conditions under which problems with deviating arguments have quasisolutions or solutions. To obtain the results, we apply the method of monotone iterations. [ABSTRACT FROM AUTHOR]

With reference to a multiobjective two-person nonzero-sum game, we define nondominated equilibrium solutions and provide a necessary and sufficient condition for a pair of mixed strategies to be a nondominated equilibrium solution. Using the necessary and sufficient condition, we formulate a mathematical programming problem yielding nondominated equilibrium solutions. We give a numerical example and demonstrate that nondominated equilibrium solutions can be obtained by solving the formulated mathematical programming problem. [ABSTRACT FROM AUTHOR]

The effects of truncating the (approximate) scattering transform in the D-bar reconstruction method for two-dimensional electrical impedance tomography are studied. The method is based on the uniqueness proof of Nachman [Ann. of Math. (2), 143 (1996), pp. 71-96] that applies to twice differentiable conductivities. However, the reconstruction algorithm has been successfully applied to experimental data, which can be characterized as piecewise smooth conductivities. The truncation is shown to stabilize the method against measurement noise and to have a smoothing effect on the reconstructed conductivity. Thus the truncation can be interpreted as regularization of the D-bar method. Numerical reconstructions are presented demonstrating that features of discontinuous high contrast conductivities can be recovered using the D-bar method. Further, a new connection between Calderón's linearization method and the D-bar method is established, and the two methods are compared numerically and analytically. [ABSTRACT FROM AUTHOR]

A mixed optimal model approximation is presented to obtain reduced-order models for truly fast descriptor systems. By a projection from truly fast descriptor systems to discrete-time systems, a mixed optimal model approximation for truly fast descriptor systems is transformed to a mixed optimal model approximation of the corresponding discrete-time systems. The structure of the fast descriptor systems is preserved in the model approximation procedure. The expression of the error and its gradient are given explicitly in terms of the solutions of certain Lyapunov equations. A numerical example is provided to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We consider a continuous function that changes its sign on an interval finitely many times and pose the problem of the approximation of this function by a polynomial that inherits its sign. For this approximation, we obtain (in the case where this is possible) Jackson-type estimates containing modified weighted moduli of smoothness of the Ditzian-Totik type. In some cases, constants in these estimates depend substantially on the location of points where the function changes its sign. We give examples of functions for which these constants are unimprovable. We also prove theorems that are analogous, in a certain sense, to inverse theorems of approximation without restrictions. [ABSTRACT FROM AUTHOR]

Sufficient conditions are derived for the controllability of nonlinear neutral evolution integrodifferential systems with infinite delay in a Banach space. The results are obtained by using the resolvent operators and the Schaefer fixed-point theorem. An example is given to illustrate the results. [ABSTRACT FROM AUTHOR]

We investigate conditions for zeros under which the Naftalevich-Tsuji product is a function of a bounded l-index analytic in the unit disk. [ABSTRACT FROM AUTHOR]

Estimation of the frequency and spatial dependent boundary traction vector from measured vibration responses in a vibrating structure is addressed. This problem, also referred to as the inverse problem, may in some circumstances be ill-conditioned. Here a technique to overcome the ill-conditioning is proposed. A subset of a set of available eigenmodes is chosen such that the problem becomes well-conditioned enough. It is shown that the ill-conditioning originates from the fact that not all eigenmodes are orthogonal over the surface where the traction vector is sought. Consequently, by choosing a set of eigenmodes orthogonal over the surface of interest, the problem becomes well-conditioned. The calculated traction vector is shown to converge to the true one in the sense of a L2-norm on the boundary of the body. The proposed technique is verified, using numerical simulation of measured responses, with good agreement. [ABSTRACT FROM AUTHOR]