Simultaneous approximation on two subsets of an open Riemann surface

Given functions f₁ and f₂ meromorphic, respectively, on subsets E₁ and E₂ of a Riemann surface R, we seek a function meromorphic on all of R, which simultaneously approximates f₁ on E₁ and f₂ on E₂.     

𝒞1 continuous discretization of gradient elasticity

For the modeling of size effects, gradient continua can be applied. In this contribution, strain gradients are used in a nonlinear hyperelastic material model. The resulting partial differential equations are solved numerically in terms of a Galerkin scheme requiring 𝒞¹ continuity of the shape functions. The performance of three 𝒞¹ continuous finite elements and the 𝒞¹ Natural Element method is compared. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cosine of angle and center of mass of an operator

We consider the notion of real center of mass and total center of mass of a bounded linear operator relative to another bounded linear operator and explore their relation with cosine and total cosine of a bounded linear operator acting on a complex Hilbert space. We give another proof of the Min-max equality and then generalize it using the notion of orthogonality of bounded linear operators. We also illustrate with examples an alternative method of calculating the antieigenvalues and total antieigenvalues for finite dimensional operators.     

On the convergence of the exponential multiplier method for convex programming

In this paper, we analyze the exponential method of multipliers for convex constrained minimization problems, which operates like the usual Augmented Lagrangian method, except that it uses an exponential penalty function in place of the usual quadratic. We also analyze a dual counterpart, the entropy minimization algorithm, which operates like the proximal minimization algorithm, except that it uses a logarithmic/entropy proximal term in place of a quadratic. We strengthen substantially the available convergence results for these methods, and we derive the convergence rate of these methods when applied to linear programs.Research supported by the National Science Foundation under Grant DDM-8903385, and the Army Research Office under Grant DAAL03-86-K-0171.     

Solving stochastic chemical kinetics by Metropolis-Hastings sampling

This study considers using Metropolis-Hastings algorithm for stochastic simulation of chemical reactions. The proposed method uses SSA (Stochastic Simulation Algorithm) distribution which is a standard method for solving well-stirred chemically reacting systems as a desired distribution. A new numerical solvers based on exponential form of exact and approximate solutions of CME (Chemical Master Equation) is employed for obtaining target and proposal distributions in Metropolis-Hastings algorithm to accelerate the accuracy of the tau-leap method. Samples generated by this technique have the same distribution as SSA and the histogram of samples show it''s convergence to SSA     

Generalized Derivatives for Solutions of Parametric Ordinary Differential Equations with Non-differentiable Right-Hand Sides

Sensitivity analysis provides useful information for equation-solving, optimization, and post-optimality analysis. However, obtaining useful sensitivity information for systems with nonsmooth dynamic systems embedded is a challenging task. In this article, for any locally Lipschitz continuous mapping between finite-dimensional Euclidean spaces, Nesterov’s lexicographic derivatives are shown to be elements of the plenary hull of the (Clarke) generalized Jacobian whenever they exist. It is argued that in applications, and in several established results in nonsmooth analysis, elements of the plenary hull of the generalized Jacobian of a locally Lipschitz continuous function are no less useful than elements of the generalized Jacobian itself. Directional derivatives and lexicographic derivatives of solutions of parametric ordinary differential equation (ODE) systems are expressed as the unique solutions of corresponding ODE systems, under Carathéodory-style assumptions. Hence, the scope of numerical methods for nonsmooth equation-solving and local optimization is extended to systems with nonsmooth parametric ODEs embedded.