Nonexistence of Slow Heteroclinic Travelling Waves for a Bistable Hamiltonian Lattice Model

The nonexistence of heteroclinic travelling waves in an atomistic model for martensitic phase transitions is the focus of this study. The elastic energy is assumed to be piecewise quadratic, with two wells representing two stable phases. We demonstrate that there is no travelling wave joining bounded strains in the different wells of this potential for a range of wave speeds significantly lower than the speed of sound. We achieve this using a profile-corrector method previously used to show existence of travelling waves for the same model at higher subsonic velocities.     

On the rotordynamic stability of two-pole synchronous electric machinery considering different load cases

This paper discusses stationary rotordynamics of synchronous electric machinery, considering different load cases. The model comprises the electrical operation in a rigid network and in an isolated condition. The mechanical part is modelled as a Laval Rotor (Jeffcott Rotor) with a noncircular shaft, accounting for both static- and dynamic rotor eccentricities. The results show, that the machine's electrical operation may influence the occurence of mechanical vibrations significantly and therefore demonstrate the importance of analysing the electromechanical interaction. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time decay for nonlinear wave equations in two space dimensions

The Cauchy Problem for the equation u_tt–u+|u|^p–1u=0 (x², t>0, >1) is studied. Smooth Cauchy data is prescribed, and no smallness condition is imposed. For >5, it is shown that the maximum amplitude of such a wave decays at the expected rate t^–1/2 as t. For 1+8<5, the maximum amplitude still decays, but at a slower rate. These results are then used to demonstrate the existence of the scattering operator when >_o, where _o is the root of the cubic equation ³-2²-7-8=0; thus _o4.15.Alfred P. Sloan Research Fellow     

System bandwidth and the existence of generalized shift-invariant frames

We consider the question whether, given a countable family of lattices ${({\mathrm{\Gamma }}_j)}_{j\in J}$ in a locally compact abelian group G, there exist functions ${(g_j)}_{j\in J}$ such that the resulting generalized shift-invariant system ${(g_j(??\gamma ))}_{j\in J,\gamma \in {\mathrm{\Gamma }}_j}$ is a tight frame of $L^2(G)$. This paper develops a new approach to the study of generalized shift-invariant system via almost periodic functions, based on a novel unconditional convergence property. From this theory, we derive characterizing relations for tight and dual frame generators, we introduce the system bandwidth as a measure of the total bandwidth a generalized shift-invariant system can carry, and we show that the so-called Calderón sum is uniformly bounded from below for generalized shift-invariant frames. Without the unconditional convergence property, we show, counter intuitively, that even orthonormal bases can have arbitrary small system bandwidth. Our results show that the question of existence of frame generators for a general lattice system is rather subtle and depends on analytical and algebraic properties of the lattice system.     

On a flow to untangle elastic knots

In this paper we present a method to untangle smooth knots by a gradient flow for a suitable energy. We show that the flow of smooth initial knots remains smooth for all time and approaches asymptotically an “optimal embedding” in its isotopy type. The method is to set up a gradient flow for the total energy of knots, which consists of bending energy and the Möbius energy of knots.     

Double-Pullback Transitions and Coalgebraic Loose Semantics for Graph Transformation Systems

The classical algebraic approach to graph transformation is a mathematical theory based on categorical techniques with several interesting applications in computer science. In this paper, a new semantics of graph transformation systems (in the algebraic, double-pushout (DPO) approach) is proposed in order to make them suitable for the specification of concurrent and reactive systems. Classically, a graph transformation system comes with a fixed behavioral interpretation. Firstly, all transformation steps are intended to be completely specified by the rules of the system, that is, there is an implicit frame condition: it is assumed that there is a complete control about the evolution of the system. Hence, the interaction between the system and its (possibly unknown) environment, which is essential in a reactive system, cannot be modeled explicitly. Secondly, each sequence of transformation steps represents a legal computation of the system, and this makes it difficult to model systems with control. The first issue is addressed by providing graph transformation rules with a loose semantics, allowing for unspecified effects which are interpreted as activities of the environment. This is formalized by the notion of double-pullback transitions, which replace (and generalize) the well-known double-pushout diagrams by allowing for spontaneous changes in the context of a rule application. Two characterizations of double-pullback transitions are provided: the first one describes them in terms of extended direct DPO derivations, and the second one as incomplete views of parallel or amalgamated derivations. The issue of constraining the behavior of a system to transformation sequences satisfying certain properties is addressed instead by introducing a general notion of logic of behavioral constraints, which includes instances like start graphs, application and consistency conditions, and temporal logic constraints. The loose semantics of a system with restricted behavior is defined as a category of coalgebras over a suitable functor. Such category has a final object which includes all finite and infinite transition sequences satisfying the constraints.