Stabilization of nonlinear systems with a dither control

In this paper a new approach is taken to analyze stabilization of a general nonlinear system with a dither input. Given the original system with a control, an autonomous relaxed system is constructed. It is shown that if the relaxed system is stable, then the original system with dither control would be stable in the finite time. An algorithm is given for constructing the dither control. The technique used here is general and does not have the limitations of the Dual Input Describing Function technique. Furthermore, in many cases it is possible to guarantee global contractive stability as well. Two examples are solved in detail using computer simulations for demonstration of the technique.     

The influence of couple-stresses on singular stress concentrations in elastic solids

Zusammenfassung Es werden im Rahmen einer linearisierten Theorie, welche Spannungsmomente berücksichtigt, singuläre Spannungskonzentrationen untersucht. Zu diesem Zweck wird mit Hilfe der exponentiellen Fouriertransformation für das Problem der Halbebene unter verteilter Oberflächenbelastung im ebenen Verschiebungszustand eine exakte Lösung aufgestellt. Diese Lösung wird anschliessend zum Studium des singulären Spannungsfeldes benützt, welches sich unter konzentrierter Normal-oder Tangentiallast sowie unter unstetiger Schubbelastung einstellt; sie wird auch zur Ermittlung der Spannungskonzentration benützt, welche von einem flachen Stempel herrührt. In jedem Falle werden die erhaltenen Singularitäten im einzelnen studiert und mit den entsprechenden Singularitäten der klassischen Elastizitätstheorie verglichen.

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This paper was prepared under Contral Nonr-220 (58) with the Office of Naval Research in Washington, D. C.

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The work underlying this paper was carried out in part while the second author was a Guggenheim Fellow at Keio University, where he held a Visiting Professorship.     

KKM—A Topological Approach For Trees

The Knaster–Kuratowski–Mazurkiewicz (KKM) theorem is a powerful tool in many areas of mathematics. In this paper we introduce a version of the KKM theorem for trees and use it to prove several combinatorial theorems.A 2-tree hypergraph is a family of nonempty subsets of T R (where T and R are trees), each of which has a connected intersection with T and with R. A homogeneous 2-tree hypergraph is a family of subsets of T each of which is the union of two connected sets.For each such hypergraph H we denote by (H) the minimal cardinality of a set intersecting all sets in the hypergraph and by (H) the maximal number of disjoint sets in it.In this paper we prove that in a 2-tree hypergraph (H)2(H) and in a homogeneous 2-tree hypergraph (H)3(H). This improves the result of Alon [3], that (H)8(H) in both cases.Similar results are proved for d-tree hypergraphs and homogeneous d-tree hypergraphs, which are defined in a similar way. All the results improve the results of Alon [3] and generalize the results of Kaiser [1] for intervals.     

Noncommutative Rigidity

Using very weak criteria for what may constitute a noncommutative geometry, I show that a pseudo-Riemannian manifold can only be smoothly deformed into noncommutative geometries if certain geometric obstructions vanish. These obstructions can be expressed as a system of partial differential equations relating the metric and the Poisson structure that describes the noncommutativity. I illustrate this by computing the obstructions for well known examples of noncommutative geometries and quantum groups. These rigid conditions may cast doubt on the idea of noncommutatively deformed space-time.     

Kinetics of phase transformation on a Bethe lattice in the presence of spin exchange

Kinetics of phase transformation on a Bethe lattice governed by single-spin-flip Glauber and spin-exchange Kawasaki dynamics is examined. For a general Glauber dynamics for which all processes (splitting and coagulation, growth and decay of clusters, as well as creation and annihilation of single-spin clusters) take place, the addition of the Kawasaki dynamics accelerates the transformation process without changing the qualitative behavior. In the growth-decay regime of the Glauber dynamics, regime in which the splitting and coagulation, and creation and annihilation processes due to single-spin flips are negligible, the Kawasaki dynamics strongly increases the fraction of transformed phase because of the splitting and coagulation of clusters induced by the spin-exchange processes. Acting alone, the Kawasaki dynamics leads to the growth of the clusters of each of the phases after the quenching of the temperature to a lower value. When the final temperature T(f) is smaller than a certain temperature T(f0), the average cluster radius grows linearly with time during both the initial and intermediate stages of the kinetic process, and diverges as log(2)(t(d)-t)(-1) when the time t approaches the value t(d) at which infinite clusters arise. It is shown that, among the various spin-exchange processes involved in Kawasaki dynamics, the main contribution is provided by those which decrease or increase the number of clusters by unity.     

Phase transformation in a lattice system in the presence of spin-exchange dynamics

A joint action of the Glauber single-spin-flip and the Kawasaki spin-exchange mechanisms upon the processes of phase transformation is examined in the framework of the one-dimensional kinetic Ising model. It is shown that the addition of the Kawasaki dynamics to that of Glauber accelerates the process of phase transformation in the initial stage, but slows it down in later stages. For the truncated form of Glauber dynamics, which excludes the processes of splitting and coagulation of clusters, the addition of the Kawasaki dynamics always accelerates the phase transformation process. Acting alone, the Kawasaki mechanism provides a cluster growth proportional to t(1/2) (where t is the time) in the initial stage and proportional to t(1/3) (Lifshitz-Slyozov-Wagner law) in the intermediate stage. In the final stage, a cluster size approaches exponentially its equilibrium value.