Some remarks on solvability and various indices for implicit differential equations

It has been shown [17,18,21] that the notion of index for DAEs (Differential Algebraic Equations), or more generally implicit differential equations, could be interpreted in the framework of the formal theory of PDEs. Such an approach has at least two decisive advantages: on the one hand, its definition is not restricted to a “state-space” formulation (order one systems), so that it may be computed on “natural” model equations coming from physics (which can be, for example, second or fourth order in mechanics, second order in electricity, etc.) and there is no need to destroy this natural way through a first order rewriting. On the other hand, this formal framework allows for a straightforward generalization of the index to the case of PDEs (either “ordinary” or “algebraic”). In the present work, we analyze several notions of index that appeared in the literature and give a simple interpretation of each of them in the same general framework and exhibit the links they have with each other, from the formal point of view. Namely, we shall revisit the notions of differential, perturbation, local, global indices and try to give some clarification on the solvability of DAEs, with examples on time-varying implicit linear DAEs. No algorithmic results will be given here (see [34,35] for computational issues) but it has to be said that the complexity of computing the index, whatever approach is taken, is that of differential elimination, which makes it a difficult problem. We show that in fact one essential concept for our approach is that of formal integrability for usual DAEs and that of involution for PDEs. We concentrate here on the first, for the sake of simplicity. Last, because of the huge amount of work on DAEs in the past two decades, we shall mainly mention the most recent results. This revised version was published online in June 2006 with corrections to the Cover Date.     

On eigenfunctions of the structures described by the “shallow-water” model on the plane

We propose a new method for solving the “shallow-water” equations. We show that from the equations of “shallow water” one obtains nonlinear Liouville-type equations, Helmholtz equations, etc. This allows one to construct eigenfunctions of various structures that appear in the flow in the two-dimensional case. We obtain exact and asymptotic solutions in an elliptic domain with singularities. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 17–32, 2006.     

A Newton’s method for perturbed second-order cone programs

We develop optimality conditions for the second-order cone program. Our optimality conditions are well-defined and smooth everywhere. We then reformulate the optimality conditions into several systems of equations. Starting from a solution to the original problem, the sequence generated by Newton’s method converges Q-quadratically to a solution of the perturbed problem under some assumptions. We globalize the algorithm by (1) extending the gradient descent method for differentiable optimization to minimizing continuous functions that are almost everywhere differentiable; (2) finding a directional derivative of the equations. Numerical examples confirm that our algorithm is good for “warm starting” second-order cone programs—in some cases, the solution of a perturbed instance is hit in two iterations. In the progress of our algorithm development, we also generalize the nonlinear complementarity function approach for two variables to several variables.     

Impedance Passive and Conservative Boundary Control Systems

The external Cayley transform is used for the conversion between the linear dynamical systems in scattering form and in impedance form. We use this transform to define a class of formal impedance conservative boundary control systems (colligations), without assuming a priori that the associated Cauchy problems are solvable. We give sufficient and necessary conditions when impedance conservative colligations are internally well-posed boundary nodes; i.e., when the associated Cauchy problems are solvable and governed by C ₀ semigroups. We define a “strong” variant of such colligations, and we show that “strong” impedance conservative boundary colligation is a slight generalization of the “abstract boundary space” construction for a symmetric operator in the Russian literature. Many aspects of the theory is illustated by examples involving the transmission line and the wave equations. Received: August 21, 2006. Accepted: October 22, 2006.     

Reduction of hugoniot-maslov chains for trajectories of solitary vortices of the “shallow water” equations to the hill equation

According to Maslov’s idea, many two-dimensional, quasilinear hyperbolic systems of partial differential equations admit only three types of singularities that are in general position and have the property of “structure self-similarity and stability.” Those are: shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is described by an infinite chain of ordinary differential equations (ODE) that generalize the well-known Hugoniot conditions for shock waves. After some reasonable closure of the chain for the case of solitary vortices in the “shallow water” equations, we obtain a nonlinear system of sixteen ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential. This means that, in some approximations, the trajectory of a solitary vortex can be described by the Hill equation. This result can be used to predict the trajectory of the vortex center if we know its observable part. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 47–66.     

The semantics of knowledge attributions

The basic idea of conversational contextualism is that knowledge attributions are context sensitive in that a given knowledge attribution may be true if made in one context but false if made in another, owing to differences in the attributors’ conversational contexts. Moreover, the context sensitivity involved is traced back to the context sensitivity of the word “know,” which, in turn, is commonly modelled on the case either of genuine indexicals such as “I” or “here” or of comparative adjectives such as “tall” or “rich.” But contextualism faces various problems. I argue that in order to solve these problems we need to look for another account of the context sensitivity involved in knowledge attributions and I sketch an alternative proposal.     

A Newton-like method for nonlinear system of equations

Tanabe (1988) proposed a variation of the classical Newton method for solving nonlinear systems of equations, the so-called Centered Newton method. His idea was based on a deviation of the Newton direction towards a variety called “Central Variety”. In this paper we prove that the Centered Newton method is locally convergent and we present a globally convergent method based on the centered direction used by Tanabe. We show the effectiveness of our proposal for solving nonlinear system of equations and compare with the Newton method with line search.     

A note on the regularity of reduced models obtained by nonlocal quasi-continuum-like approaches

The paper investigates model reduction techniques that are based on a nonlocal quasi-continuum-like approach. These techniques reduce a large optimization problem to either a system of nonlinear equations or another optimization problem that are expressed in a smaller number of degrees of freedom. The reduction is based on the observation that many of the components of the solution of the original optimization problem are well approximated by certain interpolation operators with respect to a restricted set of representative components. Under certain assumptions, the “optimize and interpolate” and the “interpolate and optimize” approaches result in a regular nonlinear equation and an optimization problem whose solutions are close to the solution of the original problem, respectively. The validity of these assumptions is investigated by using examples from potential-based and electronic structure-based calculations in Materials Science models. A methodology is presented for using quasi-continuum-like model reduction for real-space DFT computations in the absence of periodic boundary conditions. The methodology is illustrated using a basic Thomas–Fermi–Dirac case study.     

Introduction to the theory of functional differential equations and their applications. Group approach

In this work, we give an introduction to the theory of nonlinear functional differential equations of pointwise type on a finite interval, semi-axis, or axis. This approach is based on the formalism using group peculiarities of such differential equations. For the main boundary-value problem and the Euler-Lagrange boundary-value problem, we consider the existence and uniqueness of the solution, the continuous dependence of the solution on boundary-value and initial-value conditions, and the “roughness” of functional differential equations in the considered boundary-value problems. For functional differential equations of pointwise type we also investigate the pointwise completeness of the space of solutions for given boundary-value conditions, give an estimate of the rank for the space of solutions, describe types of degeneration for the space of solutions, and establish conditions for the “smoothness” of the solution. We propose the method of regular extension of the class of ordinary differential equations in the class of functional differential equations of pointwise type. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 8, Functional Differential Equations, 2004.     

Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem

The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities” argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the “acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization of the Godunov scheme.     

Mixing mixed-integer inequalities

Mixed-integer rounding (MIR) inequalities play a central role in the development of strong cutting planes for mixed-integer programs. In this paper, we investigate how known MIR inequalities can be combined in order to generate new strong valid inequalities.?Given a mixed-integer region S and a collection of valid “base” mixed-integer inequalities, we develop a procedure for generating new valid inequalities for S. The starting point of our procedure is to consider the MIR inequalities related with the base inequalities. For any subset of these MIR inequalities, we generate two new inequalities by combining or “mixing” them. We show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed-integer region associated with the base inequalities.?We discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed-integer programming problems. In particular, we present examples for production planning, capacitated facility location, capacitated network design, and multiple knapsack problems. We also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs. Finally we study some extensions of this mixing procedure. Received: April 1998 / Accepted: January 2001?Published online April 12, 2001     

Observations on the numerical stability of the Galerkin method

A simple factorization of the finite-dimensional Galerkin operators motivates a study of the numerical stability of a Galerkin procedure on the basis of its “potential stability” and the “conditioning” of its coordinate functions. Conditions sufficient for stability and conditions leading to instability are thereby identified. Numerical examples of stability and instability occurring in the application of the Galerkin method to boundary-integral equations arising in simple scattering problems are provided and discussed within this framework. Numerical instabilities reported by other authors are examined and explained from the same point of view. This revised version was published online in June 2006 with corrections to the Cover Date.     

A note on isomorphism and strategic equivalence of cooperative games

In this note, we will give several examples to illustrate that two essential games which are isomorphic are not necessarily S-equivalent when the cores of both games are “small” or empty. In other words, we show that whether two isomorphic games are S-equivalent can not be justified in terms of the “size” of the core.     

Semilocal Convergence Theorems for Newton’s Method Using Outer Inverses and Hypotheses on the Second Fréchet-Derivative

 We provide semilocal convergence theorems for Newton’s method in Banach space using outer or generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Fréchet-derivative. This way our Newton-Kantorovich hypotheses differ from earlier ones. Our results can be used to solve undetermined systems, nonlinear least squares problems and ill-posed nonlinear operator equations. We complete our study with some very simple examples to show that our results apply, where others fail.     

Reflected mean-field backward stochastic differential equations. Approximation and associated nonlinear PDEs

Mathematical mean-field approaches have been used in many fields, not only in Physics and Chemistry, but also recently in Finance, Economics, and Game Theory. In this paper we will study a new special mean-field problem in a purely probabilistic method, to characterize its limit which is the solution of mean-field backward stochastic differential equations (BSDEs) with reflections. On the other hand, we will prove that this type of reflected mean-field BSDEs can also be obtained as the limit equation of the mean-field BSDEs by penalization method. Finally, we give the probabilistic interpretation of the nonlinear and nonlocal partial differential equations with the obstacles by the solutions of reflected mean-field BSDEs.     

Homotopy classification of equivariant regular sections

In his paper [2], Bierstone proves the equivariant Gromov theorem which is an integrability theorem for “open regularity condition” of equivariant sections of a smooth G-fibre bundle under the assumption that all orbit bundles of base manifold are non-closed. Here, we prove the result without his assumption under a nice “open regularity condition” which we call “G-extensible”. One of the examples of “G-extensible condition” is given by notions of Thom-Boardman singularities.     

Stein domains and branched shadows of 4-manifolds

We provide sufficient conditions assuring that a suitably decorated 2-polyhedron can be thickened to a compact four-dimensional Stein domain. We also study a class of flat polyhedra in 4-manifolds and find conditions assuring that they admit Stein, compact neighborhoods. We base our calculations on Turaev’s shadows suitably “smoothed”; the conditions we find are purely algebraic and combinatorial. Applying our results, we provide examples of hyperbolic 3-manifolds admitting “many” positive and negative Stein fillable contact structures, and prove a four-dimensional analog of Oertel’s result on incompressibility of surfaces carried by branched polyhedra.      

Lie groupoids as generalized atlases

Starting with some motivating examples (classical atlases for a manifold, space of leaves of a foliation, group orbits), we propose to view a Lie groupoid as a generalized atlas for the “virtual structure” of its orbit space, the equivalence between atlases being here the smooth Morita equivalence. This “structure” keeps memory of the isotropy groups and of the smoothness as well. To take the smoothness into account, we claim that we can go very far by retaining just a few formal properties of embeddings and surmersions, yielding a very polymorphous unifying theory. We suggest further developments.     

Singular Cauchy problem for the equation of flow of thin viscous films with nonlinear convection

For multidimensional equations of flow of thin capillary films with nonlinear diffusion and convection, we prove the existence of a strong nonnegative generalized solution of the Cauchy problem with initial function in the form of a nonnegative Radon measure with compact support. We determine the exact upper estimate (global in time) for the rate of propagation of the support of this solution. The cases where the degeneracy of the equation corresponds to the conditions of “strong” and “weak” slip are analyzed separately. In particular, in the case of “ weak” slip, we establish the exact estimate of decrease in the L ²-norm of the gradient of solution. It is well known that this estimate is not true for the initial functions with noncompact supports. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 250–271, February, 2006.