Detachment waves in frictional contact: I — A two-mass system

This work studies a mathematical model for the motion of a two mass–spring–damper system with friction, and has a two-fold purpose. The first is to start a mathematical study of the way detachment or slip waves, when the system transits from stick to slip motion, propagate in discrete and continuous systems. The introduction of friction changes the problems into systems of differential set-valued inclusions, which are mathematically interesting, but rather complex. The second is to use such a ‘simple’ system as an example of a very general existence and uniqueness theorem for systems described by set-valued pseudomonotone operators, Kuttler and Shillor (1999) and Andrews et al. (2020).     

Optimal control of a nonconvex perturbed sweeping process

The paper concerns optimal control of discontinuous differential inclusions of the normal cone type governed by a generalized version of the Moreau sweeping process with control functions acting in both nonconvex moving sets and additive perturbations. This is a new class of optimal control problems in comparison with previously considered counterparts where the controlled sweeping sets are described by convex polyhedra. Besides a theoretical interest, a major motivation for our study of such challenging optimal control problems with intrinsic state constraints comes from the application to the crowd motion model in a practically adequate planar setting with nonconvex but prox-regular sweeping sets. Based on a constructive discrete approximation approach and advanced tools of first-order and second-order variational analysis and generalized differentiation, we establish the strong convergence of discrete optimal solutions and derive a complete set of necessary optimality conditions for discrete-time and continuous-time sweeping control systems that are expressed entirely via the problem data.     

集值逆上鞅的收敛定理及应用

研究了取值于Ｂａｎａｃｈ空间的集值逆上鞅的收敛性，给出了集值逆上鞅在集列Ｗｉｊｓｍａｎ收敛，弱收敛及Ｋｕｒａｔｏｗｓｋｉ－Ｍｏｓｃｏ收敛意义下的收敛定理，并给出了它们在连续参数集值鞅中的应用。     

A note on badly approximabe sets in projective space

Recently, Ghosh and Haynes (J Reine Angew Math 712:39–50, 2016) proved a Khintchine-type result for the problem of Diophantine approximation in certain projective spaces. In this note we complement their result by observing that a Jarník-type result also holds for ‘badly approximable’ points in real projective space. In particular, we prove that the natural analogue in projective space of the classical set of badly approximable numbers has full Hausdorff dimension when intersected with certain compact subsets of real projective space. Furthermore, we also establish an analogue of Khintchine’s theorem for convergence relating to ‘intrinsic’ approximation of points in these compact sets.     

Inverse stability and convergence of difference approximations for boundary value problems for differential inclusions

A general stability inequality is proved. It will be used to show the discrete weak convergence of sequences of approximate solutions obtained bydifference approximations for boundary value problems for differential inclusions.     

A note on the convergence of discretized dynamic iteration

In this note we prove convergence results, including error estimates, for the dynamic iteration scheme where the forward Euler and backward Euler method are used to compute the iterates. The proofs are interesting in that they are exact analogues of the proof for the continuous case, using discrete versions of Gronwall's inequality.     

ARMS: an algebraic recursive multilevel solver for general sparse linear systems

This paper presents a general preconditioning method based on a multilevel partial elimination approach. The basic step in constructing the preconditioner is to separate the initial points into two parts. The first part consists of ‘block’ independent sets, or ‘aggregates’. Unknowns of two different aggregates have no coupling between them, but those in the same aggregate may be coupled. The nodes not in the first part constitute what might be called the ‘coarse’ set. It is natural to call the nodes in the first part ‘fine’ nodes. The idea of the methods is to form the Schur complement related to the coarse set. This leads to a natural block LU factorization which can be used as a preconditioner for the system. This system is then solved recursively using as preconditioner the factorization that could be obtained from the next level. Iterations between levels are allowed. One interesting aspect of the method is that it provides a common framework for many other techniques. Numerical experiments are reported which indicate that the method can be fairly robust. Copyright © 2002 John Wiley & Sons, Ltd.     

Classroom note: A simple proof of an interesting Fibonacci generalization

It is reasonably well known that the ratios of consecutive terms of a Fibonacci series converge to the golden ratio. This note presents a simple, complete proof of an interesting generalization of this result to a whole family of ‘precious metal ratios’.     

Lower convergence of approximate solutions to vector quasi-variational problems

In this article, various types of approximate solutions for vector quasi-variational problems in Banach spaces are introduced. Motivated by [M.B. Lignola, J. Morgan, On convergence results for weak efficiency in vector optimization problems with equilibrium constraints, J. Optim. Theor. Appl. 133 (2007), pp. 117–121] and in line with the results obtained in optimization, game theory and scalar variational inequalities, our aim is to investigate lower convergence properties (in the sense of Painlevé–Kuratowski) for such approximate solution sets in the presence of perturbations on the data. Sufficient conditions are obtained for the lower convergence of ‘strict approximate’ solution sets but counterexamples show that, in general, the other types of solutions do not lower converge. Moreover, we prove that any exact solution to the limit problem can be obtained as the limit of a sequence of approximate solutions to the perturbed problems.     

An efficient algorithm for solving the discrete minisum problem

For a simple nonsmooth minimization problem, the discrete minisum problem, an efficient hybrid method is presented. This method consists of an ‘inner algorithm’ (Newton method) for solving the necessary optimality conditions and a gradient-type ‘outer algorithm’. By this way we combine the large convergence area of the gradient technique with the fast final convergence of the Newton method.     

A numerical scheme for a class of sweeping processes

The aim of this paper is to study a whole class of first order differential inclusions, which fit into the framework of perturbed sweeping process by uniformly prox-regular sets. After obtaining well-posedness results, we propose a numerical scheme based on a prediction-correction algorithm and we prove its convergence. Finally we apply these results to a problem coming from the modelling of crowd motion.     

New Korn-type inequalities and regularity of solutions to linear elliptic systems and anisotropic variational problems involving the trace-free part of the symmetric gradient

The aim of? this note is to investigate a regularity theory for minimizers of energies whose density depends on the trace-free part of the symmetric gradient, where integrands of anisotropic growth are considered. An adequate coercive inequality guarantees the existence of minimizers of such energies in suitable Sobolev classes. Moreover, various other Korn-type inequalities are shown, which can be used to prove the smoothness of weak solutions to linear elliptic systems involving the trace-free part of the symmetric gradient. In particular, Campanato-type estimates for solutions to such systems are established so that all tools are available to prove the interior regularity of minimizers of energies depending on the trace-free part of the symmetric gradient.     

Discrete uniform distribution for a sum of additive functions

We consider the limit distribution of values of a sum of sets of strongly additive arithmetic functions with shifted argument. We obtain sufficient and necessary conditions for a weak convergence of distributions of that sum to the discrete uniform law. The case where those functions take values 0 or 1 on primes is studied.     

Fixed point theorems for fuzzy numbers

In this paper we study the problem of the existence of fixed points for fuzzy maps. We first define uniform structures on sets of fuzzy subsets and describe the ‘closed convergence’ and ‘myope convergence’ structures which have properties close to the corresponding traditional ones. We then give two ways in order to obtain fixed point results. We finally give applications of these results, particularly to fuzzy Markov processes.     

Discrete approximations to optimization of neutral functional differential inclusions

The paper deals with the convergence of discrete approximations to optimization problems governed by neutral functional differential inclusions. The discrete approximations through Euler finite difference are constructed and the convergence of discrete approximations is proved.     

Spectral decomposition theorem in equicontinuous nonautonomous discrete dynamical systems

In this paper, we define the notion of weak chain recurrence and study properties of weak chain recurrent sets in a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a compact metric space. Our main result is the Smale’s spectral decomposition theorem in an equicontinuous nonautonomous discrete dynamical system.     

Numerical study of linear and nonlinear string vibrations by means of physical discretization

To solve partial differential equations numerically, discretization of the continuous model is required and may be achieved either mathematically or physically. This paper illustrates how physical discretization of a continuous string may be accomplished by employing discrete model theory which has as its essential substance Newtonian mechanics.Typical examples of wave motion in discretized ‘linear’ and ‘non-linear’ strings are discussed. They include the transverse vibrations of a string after having been subjected to a given initial displacement, reflection and superposition of wave pulses in the string, and resonance of the string when coupled to a harmonic vibrator. The equations that arise after application of discrete model theory to these problems, describe the subsequent motion of the string, and are solved numerically by computer. In all cases the results obtained for the discrete linear string agree remarkably well with those for the corresponding continuous physical string. The stability of the solutions obtained by discretization are also investigated.     

A discrete location model with fuzzy accessibility measures

In crisply defined discrete location problems, a number of facilities are to be located at specific points within an area, according to precisely quantified criteria. However in many location problems, especially those associated with social policies, non-crisply defined criteria are used such as, how ‘near’ or ‘accessible’ a facility is, or how ‘important’ certain issues are, etc. In these cases a fuzzy sets approach is more appropriate.This paper presents an application of the set partitioning (set covering with equality constraints) type of integer programming formulation to a discrete location problem with fuzzy accessibility criteria. The solution method suggested uses the symmetry of the objectives and the constraints introduced by Bellman and Zadeh.     

Asymptotic structural characteristics of fuzzy measure and their applications

In fuzzy measure theory, as Sugeno's fuzzy measures lose additivity in general, the concept ‘almost’, which is well known in classical measure theory, splits into two different concepts, ‘almost’ and ‘pseudo-almost’. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called ‘waxing’ and ‘waning’, and to introduce some new concepts, such as ‘autocontinuity’, ‘converse-autocontinuity’ and ‘pseudo-autocontinuity’. These concepts describe some asymptotic structural characteristics of a fuzzy measure.In this paper, by means of the asymptotic structural characteristics of fuzzy measure, we also give four forms of generalization for both Egoroff's theorem, Riesz's theorem and Lebesgue's theorem respectively, and prove the almost everywhere (pseudo-almost everywhere) convergence theorem, the convergence in measure (pseudo-in measure) theorem of the sequence of fuzzy integrals. In the last two theorems, the employed conditions are not only sufficient, but also necessary.     

Behaviour of mushy regions under the action of a volumetric heat source

In this paper we consider solutions to Stefan problems in spatial dimensions N ? 1. We find the necessary conditions on the heat source for the appearance of a ‘mushy region’ (i.e. a region where the temperature coincides identically with the temperature of the change of phase) inside a purely liquid (or solid) phase. For sources depending on energy such conditions are connected only with the local behaviour of the source near the energy level corresponding to the beginning of the change or phase. Both weak and smooth solutions are considered; in the latter case the behaviour of the solution at the free boundary is investigated in detail.