We introduce a new family of billiards which break time reversal symmetry in spite of having piece-wise straight trajectories. We show that our billiards preserve the ergodic and mixing properties of conventional billiards while they may turn into exponential the power law decay of correlations characteristic of Sinai type billiards. Such billiards can be implemented by squeezing the transverse...

We study expressive power of shallow and deep neural networks with piece-wise linear activation functions. We establish new rigorous upper and lower bounds for the network complexity in the setting of approximations in Sobolev spaces. In particular, we prove that deep ReLU networks more efficiently approximate smooth functions than shallow networks. In the case of approximations of 1D Lipschitz...

Thurston conjectured that a closed triangulated 3-manifold in which every edge has degree 5 or 6, and no two edges of degree 5 lie in a common 2-cell, has word-hyperbolic fundamental group. We establish Thurston's conjecture by proving that such a manifold admits a piece-wise Euclidean metric of non-positive curvature and the universal cover contains no isometrically embedded flat planes. The p...

Tempelmeier and Hilger (2015) study the stochastic dynamic lot sizing problem with multiple items and limited capacity. They propose a linear optimization formulation for the problem based on a piece-wise linear approximation of the nonlinear functions for the expected backorders and the expected inventory position. Our work builds on Tempelmeier and Hilger (2015). We correct an erroneous deriv...

In any problem involving images having scale-dependent structures, a key issue is the modeling of these multi-scale characteristics. Because multi-scale phenomena frequently possess nonstationary, piece-wise multi-model behaviour, the classic hidden Markov method can not perform well in modeling such complex images. In this paper we provide a new modeling approach to extend previous hierarchica...

A promising approach towards evaluating adaptive systems is to decompose the adaptation process and evaluate the system in a “piece-wise” manner. This paper presents a decomposition model that integrates two previous proposals. The main “stages” identified are: (a) collection of input data, (b) interpretation of the collected data, (c) modeling of the current state of the “world”, (d) deciding ...

Westudy the stability of an analog optimization circuit that solves quadratic programming (QP) problems. The circuit dynamics are modeled as a switched affine system. A piece-wise quadratic Lyapunov function and the KYP lemma are used to derive the stability criterion. The stability criterion characterizes the range of critical circuit parameters for which the QP circuit is globally exponential...

Building on previous results of the author this paper presents two new error estimates for the reachable set of an affine control system if only piece-wise constant admissible controls on a uniform mesh are used instead of all measurable admissible controls. It is natural to expect that the resulting " shrinkage " of the reachable set is of the order of the mesh size. In this paper it is proved...

In this paper a dipolar neural network designed for prediction of survival time is presented. The proposed method is based on minimization of piece-wise linear criterion functions. Basis exchange algorithms are exploited as a optimization technique. The method allows to estimate of discrete hazards as conditional probabilities of failure occurrence. The prediction capacity of neural network mod...