Transitivity and variational principles

We apply an order reasoning to mappings satisfying the triangle inequality. This general approach yields the Ekeland’s variational principle as one of the consequences. In addition we obtain an extension of the Brøndsted variational principle and of the Takahashi fixed point theorem.     

Discrete-analytical methods for the implementation of variational principles in environmental applications

A new method of constructing numerical schemes on the base of a variational principle for models including convection-diffusion operators is proposed. An original element is the use of analytical solutions of local adjoint problems formulated for the operators of convection-diffusion within the framework of the splitting technique. This results in numerical schemes which are absolutely stable, monotonic, transportive, and differentiable with respect to the state functions and parameters of the model. Artificial numerical diffusion is avoided due to the analytical solutions. The variational technique provides strong consistency between the numerical schemes of the main and adjoint problems. A theoretical study of the new class of schemes is given. The quality of the numerical approximations is demonstrated by an example of the non-linear Burgers equation. These new schemes enhance our variational methodology of environmental modelling. As one of the environmental applications, an inverse problem of risk assessment for Lake Baikal is presented.     

Error analysis of variational integrators of unconstrained Lagrangian systems

An error analysis of variational integrators is obtained, by blowing up the discrete variational principles, all of which have a singularity at zero time-step. Divisions by the time step lead to an order that is one less than observed in simulations, a deficit that is repaired with the help of a new past–future symmetry. G. W. Patrick is funded by the Natural Sciences and Engineering Reseach Council, Canada.     

Infinitesimal methods in control theory: Deterministic and stochastic

In Part I, methods of nonstandard analysis are applied to deterministic control theory, extending earlier work of the author. Results established include compactness of relaxed controls, continuity of solution and cost as functions of the controls, and existence of optimal controls. In Part II, the methods are extended to obtain similar results for partially observed stochastic control. Systems considered take the form:where the feedback control u depends on information from a digital read-out of the observation process y. The noise in the state equation is controlled along with the drift. Similar methods are applied to a Markov system in the final section.     

Hamilton-Jacobi theory over time scales and applications to linear-quadratic problems

In this paper we first derive the verification theorem for nonlinear optimal control problems over time scales. That is, we show that the value function is the only solution of the Hamilton-Jacobi equation, in which the minimum is attained at an optimal feedback controller. Applications to the linear-quadratic regulator problem (LQR problem) gives a feedback optimal controller form in terms of the solution of a generalized time scale Riccati equation, and that every optimal solution of the LQR problem must take that form. A connection of the newly obtained Riccati equation with the traditional one is established. Problems with shift in the state variable are also considered. As an important tool for the latter theory we obtain a new formula for the chain rule on time scales. Finally, the corresponding LQR problem with shift in the state variable is analyzed and the results are related to previous ones.     

Infinite dimensional groups and quantum field theory

This paper surveys recent work on representations of infinite dimensional groups and the connection with quantum field theory.     

Examples of asymptotic completeness in translation invariant systems with an unbounded number of particles

A tagged particle interacting with a finite but unbounded number of particles is considered. Some examples of asymptotic completeness are given which are uniform in the number of particles.     

σ-additivity in manuals and orthoalgebras

Notions of -additivity are introduced for orthoalgebras and for manuals. It is shown that the logic of a -additive manual is a -additive orthoalgebra, and that, conversely, every -additive orthoalgebra arises as such a logic. Using this theorem, it is shown that a given orthoalgebra admits at most one reasonable extension of its orthogonal sum operation to countable jointly orthogonal sets. It is also shown that every orthoalgebra can be embedded in a sigma-orthoalgebra and every orthomodular poset, in a -OMP.     

Archimedeanness and the MacNeille completion of pseudoeffect algebras and po-groups

Pseudoeffect (PE-) algebras are partial algebras differing from effect algebras in that they need not satisfy the commutativity assumption. PE-algebras typically arise from intervals of po-groups; this applies in particular to all those which satisfy a certain Riesz property.In this paper, we discuss the property of archimedeanness for PE-algebras on the one hand and for po-groups on the other hand. We prove that under the assumption of suphomogeneity, archimedeanness holds for a PE-algebra with the Riesz property if and only if it holds for its representing group. The algebra is in that case commutative. This result is established by using the technique of MacNeille completion. We give the exact condition for this completion to exist, and we clearly exhibit the role played by archimedeanness and by sup-homogeneity.     

A study of semiiterative methods for nonsymmetric systems of linear equations

Summary Given a nonsingular linear systemA x=b, a splittingA=M–N leads to the one-step iteration (1)x _m =T X _m–1 +c withT:=M ^–1N andc:=M ^–1 b. We investigate semiiterative methods (SIM's) with respect to (1), under the assumption that the eigenvalues ofT are contained in some compact set of , with 1. There exist SIM's which are optimal with respect to , but, except for some special sets , such optimal methods are not explicitly known in general. Using results about maximal convergence of polynomials and uniformly distributed nodes from approximation and function theory, we describe here SIM's which are asymptotically optimal with respect to . It is shown that Euler methods, extensively studied by Niethammer-Varga [NV], are special SIM's. Various algorithms for SIM's are also derived here. A 1-1 correspondence between Euler methods and SIM's, generated by generalized Faber polynomials, is further established here. This correspondence gives that asymptotically optimal Euler methods are quite near the optimal SIM's.Dedicated to Professor Karl Zeller (Universität Tübingen) on the occasion of his sixtieth birthday (December 28, 1984)     

On the Lagrangian form of the variational equations of Lagrangian dynamical systems

An extremal principle for obtaining the variational equations of a Lagrangian system is reviewed and formalized. Formalization is accomplished by relating the new Lagrangian function γ needed in such scheme to a prolongation of the original Lagrangian L. This formalization may be regarded as a necessary step before using the approach for stablishing nonintegrability of dynamical systems, or before applying it to analyse chaos-producing perturbations of integrable Lagrangian systems. The configuration manifold in which γ is defined is the double tangent bundle T(TQ) of the original configuration manifold Q modulo a flip mapping in such manifold. Our main result establishes that both the Euler–Lagrange equations and the corresponding variational equations of the original system can be viewed as the Lagrangian vector field associated with the composition of the first prolongation of the original Lagrangian with a flip mapping. Some applications of the approach to chaos and integrability issues are discussed.     

Mutational equations in metric spaces

This paper summarizes an extension of differential calculus to a mutational calculus for maps from one metric space to another. The simple idea is to replace half-lines allowing to define difference quotients of maps and their various limits in the case of vector space by transitions with which we can also define differential quotients of a map. Their various limits are called mutations of a map. Many results of differential calculus and set-valued analysis, including the Inverse Function Theorem, do not really rely on the linear structure and can be adapted to the nonlinear case of metric spaces and exploited. Furthermore, the concept of differential equation can be extended tomutational equation governing the evolution in metric spaces. Basic Theorems as the Nagumo Theorem, the Cauchy-Lipschitz Theorem, the Center Manifold Theorem and the second Lyapunov Method hold true for mutational equations.This work was motivated by evolution equations of tubes in visual servoing on one hand, mathematical morphology on the other, when the metric spaces are power spaces. This paper begins by listing some consequences of general theorems concerning mutational equations for tubes.     

Path-following and augmented Lagrangian methods for contact problems in linear elasticity

A certain regularization technique for contact problems leads to a family of problems that can be solved efficiently using infinite-dimensional semismooth Newton methods, or in this case equivalently, primal–dual active set strategies. We present two procedures that use a sequence of regularized problems to obtain the solution of the original contact problem: first-order augmented Lagrangian, and path-following methods. The first strategy is based on a multiplier-update, while path-following with respect to the regularization parameter uses theoretical results about the path-value function to increase the regularization parameter appropriately. Comprehensive numerical tests investigate the performance of the proposed strategies for both a 2D as well as a 3D contact problem.     

Supercategorification of quantum Kac–Moody algebras

We show that the quiver Hecke superalgebras and their cyclotomic quotients provide a supercategorification of quantum Kac–Moody algebras and their integrable highest weight modules.     

Majorizing functions and two-point Newton-type methods

The semi-local convergence of a Newton-type method used to solve nonlinear equations in a Banach space is studied. We also give, as two important applications, convergence analyses of two classes of two-point Newton-type methods including a method mentioned in [5] and the midpoint method studied in [1], [2] and [12]. Recently, interest has been shown in such methods [3] and [4].     

A mathematical flat integral realization and a large deviation result for the free Euclidean field

In this paper, we give a nonstandard construction of the free Euclidean field via S-white noise. This provides a flat integral realization of the free Euclidean field measure, which extends N. J. Cutland's flat integral representation of Wiener measure. Moreover, we show how a Cameron-Martin type formula for translations of the free field measure and a Schilder type large deviation principle for the scalar free field measure can be deduced from our nonstandard construction.SFB 237 Essen-Bochum-Düseldorf; BiBoS-Research Centre; CERFIM, Locarno, Switzerland.     

The limiting behavior of the value-function for variational problems arising in continuum mechanics

In this paper we study the limiting behavior of the value-function for one-dimensional second order variational problems arising in continuum mechanics. The study of this behavior is based on the relation between variational problems on bounded large intervals and a limiting problem on [0,∞)$[0,\infty )$.     

An extension of a logarithmic form of Cramér’s ruin theorem to some FARIMA and related processes

Cramér’s theorem provides an estimate for the tail probability of the maximum of a random walk with negative drift and increments having a moment generating function finite in a neighborhood of the origin. The class of (g,F)$(g,F)$-processes generalizes in a natural way random walks and fractional ARIMA models used in time series analysis. For those (g,F)$(g,F)$-processes with negative drift, we obtain a logarithmic estimate of the tail probability of their maximum, under conditions comparable to Cramér’s. Furthermore, we exhibit the most likely paths as well as the most likely behavior of the innovations leading to a large maximum.