Multidimensional models for analysing frequency modulated signals

In radio frequency (RF) applications, slowly varying signals often modulate the amplitude and frequency of fast carrier waves. Thus a numerical simulation of the differential algebraic equations (DAEs) modelling the electric circuit becomes tedious. Alternative models are required to achieve efficient simulations. A multivariate formulation of signals yields a suitable representation via decoupling the widely separated time scales. Consequently, the circuit's DAEs change into warped multirate partial DAEs. On the other hand, the transient behaviour of the circuit can also be approximated by a parameter-dependent DAE model including a multivariate structure. The properties of this alternative strategy are investigated. In particular, the two multidimensional approaches are compared with respect to the simulation of RF signals.     

Multirate models for simulating a colpitts oscillator

A model based on multirate partial differential algebraic equations yields an efficient numerical simulation of electric circuits in radio frequency applications. Considering frequency modulation, free parameters of the model are determined appropriately by a minimisation strategy. We apply the multirate approach to simulate a modified version of a Colpitts oscillator, which exhibits frequency modulation at widely separated time scales. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Transversality conditions for infinite horizon variational problems on time scales

We consider problems of the calculus of variations on unbounded time scales. We prove the validity of the Euler–Lagrange equation on time scales for infinite horizon problems, and a new transversality condition.     

Boundary value problems with global projection conditions

Parametrices of elliptic boundary value problems for differential operators belong to an algebra of pseudodifferential operators with the transmission property at the boundary. However, generically, smooth symbols on a manifold with boundary do not have this property, and several interesting applications require a corresponding more general calculus. We introduce here a new algebra of boundary value problems that contains Shapiro-Lopatinskij elliptic as well as global projection conditions; the latter ones are necessary, if an analogue of the Atiyah-Bott obstruction does not vanish. We show that every elliptic operator admits (up to a stabilisation) elliptic conditions of that kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. Moreover, we construct parametrices in the calculus.     

An application of time scales to economics

Economics is a discipline in which there appears to be many opportunities for applications of time scales. The time scales approach will not only unify the standard discrete and continuous models in economics, but also, for example, allows for payments which arrive at unequally spaced points in time. We present a dynamic optimization problem from economics, construct a time scales model, and apply calculus of variations to derive a solution. Time scale calculus would allow exploration of a variety of situations in economics.     

Leitmann’s direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales

The fundamental problem of the calculus of variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. This includes the discrete-time, the quantum, and the continuous/classical calculus of variations as particular cases. In this note we follow Leitmann’s direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale.     

On doubly warped product Finsler manifolds

In this paper, we introduce horizontal and vertical warped product Finsler manifolds. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemannian curvatures of doubly warped product Finsler manifold and its components, and consider the cases that this manifold is flat or has scalar flag curvature. We define the doubly warped Sasaki-Matsumoto metric for warped product manifolds and find a condition under which the horizontal and vertical tangent bundles are totally geodesic. We obtain some conditions under which a foliated manifold reduces to a Reinhart manifold. Finally, we study an almost complex structure on the tangent bundle of a doubly warped product Finsler manifold.     

Chern–Simons invariants and isometric immersions of warped products

In this paper, we calculate the Chern–Simons invariants on some 3-manifolds (e.g., Berger Sphere, warped product 3-manifolds) which obtain particular features in physics. We present the condition such that Berger sphere and warped product 3-manifolds are locally conformally flat. We also give a sufficient and necessary condition such that the warped product 3-manifolds can be isometrically immersed in \mathbbR⁴{\mathbb{R}^4} . The latter condition is different from those in the earlier works of others.     

Noether’s symmetry theorem for nabla problems of the calculus of variations

We prove a Noether-type symmetry theorem and a DuBois–Reymond necessary optimality condition for nabla problems of the calculus of variations on time scales.     

Chern–Simons invariants and isometric immersions of warped products

In this paper, we calculate the Chern–Simons invariants on some 3-manifolds (e.g., Berger Sphere, warped product 3-manifolds) which obtain particular features in physics. We present the condition such that Berger sphere and warped product 3-manifolds are locally conformally flat. We also give a sufficient and necessary condition such that the warped product 3-manifolds can be isometrically immersed in ${\mathbb{R}^4}$ . The latter condition is different from those in the earlier works of others.     

Poche de tourbillon pour Euler 2D dans un ouvert à bord

We consider Euler equations for an homogeneous incompressible non viscous fluid inside a smooth bounded domain of the plane. For an initial data of smooth vortex patch type, we obtain existence and uniqueness of a solution of the same type, locally in time if the initial patch is tangent to the boundary of the domain, and globally in time if the patch is far away from the boundary. We use pseudo-differential calculus to take care of the boundary condition. For the tangent limit case, we show that the velocity gradient of a vortex patch is Hölder continuous up to the boundary of the patch, using singular integrals. Our method provide also a result for several mutually tangent vortex patches in the plane.     

Polynomial chaos for multirate partial differential algebraic equations with random parameters

In radio frequency applications, a multivariate model yields an efficient representation of signals with amplitude modulation and/or frequency modulation. Periodic boundary value problems of multirate partial differential algebraic equations (MPDAEs) have to be solved to reproduce the quasiperiodic signals. Typically, technical parameters appear in the system, which may exhibit some uncertainty. Substitution by random variables results in a corresponding stochastic model. We apply the technique of the generalised polynomial chaos to obtain according solutions. A Galerkin approach yields larger coupled systems of MPDAEs. We analyse the properties of the coupled systems with respect to the original formulations. Thereby, we focus on the case of frequency modulation, since the case of amplitude modulation alone is straightforward.     

Linear Graininess Time Scales and Ladder Operators of Orthogonal Polynomials

In this paper we introduce linear graininess (LG) time scales. We further study orthogonal polynomials (OPs) with the weight function supported on LG time scales and derive the raising and lowering ladder operators by using the time scales calculus. We also derive a second order dynamic equation satisfied by these polynomials. The notion of an LG time scale encompasses the cases of the reals, the h-equidistant grid, the q-grid and, more general, a mixed (q, h)-grid. This allows a unified treatment of the ladder operators theory for classical OPs on these time scales. Moreover we will explain, why exclusively LG time scales provide the right framework for general OP theory.     

On imaging obstacles inside inhomogeneous media

We show that an obstacle inside a known inhomogeneous medium can be determined from measurements of the scattering amplitude at one frequency, without a priori knowledge of the boundary condition. We also show that an obstacle inside a known inhomogeneous anisotropic conducting medium can be determined from electrostatic current and voltage measurements on the boundary of a domain containing the obstacle. Moreover, two obstacles with boundary measurements which are merely comparable as operators must be identical. The first part of the paper gives an extension of the factorization method which may be of independent interest and also yields a new reconstruction procedure.     

Scattering by a highly oscillating surface

The boundary function method [A. B. Vasil'eva, V. F. Butuzov, and L. V. Kalachev, The boundary function method for singular perturbation problems, SIAM Studies in Applied Mathematics, Philadelphia, 1995] is used to build an asymptotic expansion at any order of accuracy of a scalar time‐harmonic wave scattered by a perfectly reflecting doubly periodic surface with oscillations at small and large scales. Error bounds are rigorously established, in particular in an optimal way on the relevant part of the field. It is also shown how the maximum principle can be used to design a homogenized surface whose reflected wave yields a first‐order approximation of the actual one. The theoretical derivations are illustrated by some numerical experiments, which in particular show that using the homogenized surface outperforms the usual approach consisting in setting an effective boundary condition on a flat boundary. Copyright © 2014 John Wiley & Sons, Ltd.     

Solving Boundary-Value Problems for Systems of Hyperbolic Conservation Laws with Rapidly Varying Coefficients

We study how boundary conditions affect the multiple-scale analysis of hyperbolic conservation laws with rapid spatial fluctuations. The most significant difficulty occurs when one has insufficient boundary conditions to solve consistency conditions. We show how to overcome this missing boundary condition difficulty for both linear and nonlinear problems through the recovery of boundary information. We introduce two methods for this recovery (multiple-scale analysis with a reduced set of scales, and a combination of Laplace transforms and multiple scales) and show that they are roughly equivalent. We also show that the recovered boundary information is likely to contain secular terms if the initial conditions are nonzero. However, for the linear problem, we demonstrate how to avoid these secular terms to construct a solution that is valid for all time. For nonlinear problems, we argue that physically relevant problems do not exhibit the missing boundary condition difficulty.     

Basic Calculus on Time Scales and some of its Applications

The study of dynamic systems on time scales not only unifies continuous and discrete processes, but also helps in revealing diversities in the corresponding results. In this paper we shall develop basic tools of calculus on time scales such as versions of Taylor’s formula, l’Hôspital’s rule, and Kneser’s theorem. Applications of these results in the study of asymptotic and oscillatory behavior of solutions of higher order equations on time scales are addressed. As a further application of Taylor’s formula, Abel-Gontscharoff interpolating polynomial on time scales is constructed and best possible error bounds are offered. We have also included notes at the end of each section which indicate further scope of the calculus developed in this paper.     

A Time Finite Element Method Based on the Generalized Variational Calculus

In this contribution, a new finite element method in the temporal domain is presented, in which the time step size is introduced as an additional variable. Thus, the variation of the time integral of the Lagrangean resulting from Hamilton's principle has to be carried out with respect to the rules of the generalized variational calculus. Apart from the usual time integral of the Euler‐Lagrange differential equations, the so‐called transversality condition is obtained as an additional result representing a time‐boundary term, which is used to obtain an optimal step size in the time domain.     

The Method of Singular Equations in Boundary Value Problems in Kinetic Theory

We develop a new effective method for solving boundary value problems in kinetic theory. The method permits solving boundary value problems for mirror and diffusive boundary conditions with an arbitrary accuracy and is based on the idea of reducing the original problem to two problems of which one has a diffusion boundary condition for the reflection of molecules from the wall and the other has a mirror boundary condition. We illustrate this method with two classical problems in kinetic theory: the Kramers problem (isothermal slip) and the thermal slip problem. We use the Bhatnagar-Gross-Krook equation (with a constant collision frequency) and the Williams equation (with a collision frequency proportional to the molecular velocity).__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 437–454, June, 2005.     

The logical representation of extensive games

Given an extensive formG, we associate with every choice an atomic sentence and with every information set a set of well-formed formulas (wffs) of prepositional calculus. The set of such wffs is denoted by Γ(G). Using the so-called topological semantics for propositional calculus (which differs from the standard one based on truth tables), we show that the extensive form yields a topological model of Γ(G), that is, every wff in Γ(G), is “true in G”. We also show that, within the standard truth-table semantics for propositional calculus, there is a one-to-one and onto correspondence between the set of plays ofG and the set of valuations that satisfy all the wffs in Γ(G).