Elliptic semiplanes and group divisible designs with orthogonal resolutions

In this paper we prove that there exists an elliptic semiplaneS(v, k, m) withk –m 2 if and only if there exists a group divisible design GDD _k ((k –m)(k – 1);k –m; 0, 1) withm pairwise orthogonal resolutions. As an example of this theorem, we construct an elliptic semiplaneW(45, 7, 3) and show thatW is isomorphic to the elliptic semiplaneS(45, 7, 3) given by R. D. Baker.     

Finiteness of Ricci flatN=2 supersymmetric σ-models

We study the ultraviolet behavior of two dimensional supersymmetric non-linear -models with target space an arbitrary Kähler manifoldM, so that the models areN=2 supersymmetric. We point out that these models have an additional fermionic axial symmetry if and only if the metric onM is Ricci flat. We show that the preservation of this symmetry in perturbation theory implies that both bare and renormalized metrics onM are Ricci flat. Combining this result with the constraint ofN=2 supersymmetry requiring that all counter-terms to the metric beyond one-loop order be cohomologically trivial, we argue thatN=2 models defined on Ricci flat Kähler manifolds are on-shell ultraviolet finite to all orders of perturbation theory.     

Periodic behavior of the stochastic Brusselator in the mean-field limit

Summary We prove a propagation of chaos result for the mean-field limit of a model for a trimolecular chemical reaction called Brusselator. Then we show that the pair of nonlinear (i.e. law-dependent) stochastic differential equations describing the evolution of the concentration of the molecules at a given site in the mean field limit has a solution with a periodic law (in t). Finally we identify the limit and establish a central limit theorem for the periodic law in the case where the noise tends to zero.Part of this work was performed while on leave at the Department of Mathematics and Statistics, Carleton University, Ottawa, Canada and supported by NSERC operating grants of M. Csörgö and D. Dawson     

Renewal theorem for a class of stationary sequences

Summary A renewal theorem is obtained for stationary sequences of the form _n=(...,X _n-1,X _n,X _n+1...), whereX _n, , are i.i.d. r.v.s. valued in a Polish space. This class of processes is sufficiently broad to encompass functionals of recurrent Markov chains, functionals of stationary Gaussian processes, and functionals of one-dimensional Gibbs states. The theorem is proved by a new coupling construction.Research supported by the National Science Foundation     

How large can a hereditarily separable or hereditarily Lindelöf space be?

The main result of this paper is that ifV satisfies GCH andω ₁<λ<μ are arbitrary regular cardinals, then in some cardinal preserving forcing extensionW ofV we have λ^<λ=λ=2 ^N 0,μ=2 ^λ and there are a hereditarily separableX ?2 ^λ with |X|= \(2^{\aleph _0 } \) =μ and a hereditarily LindelöfY ?2 ^μ withw(Y)= \(2^{2^\aleph 0} \) =μ. So far similar results have only been obtained under the assumption of CH.     

Observables on hypergraphs

Observables on hypergraphs are described by event-valued measures. We first distinguish between finitely additive observables and countably additive ones. We then study the spectrum, compatibility, and functions of observables. Next a relationship between observables and certain functionals on the set of measures M(H) of a hypergraph H is established. We characterize hypergraphs for which every linear functional on M(H) is determined by an observable. We define the concept of an effect and show that observables are related to effect-valued measures. Finally, we define operational transformations from M(H) to itself and show that they can be described as a certain combination of effects.On leave from University of Berne, Institute of Mathematical Statistics, Sidlerstrasse 5, CH-3012 Berne, Switzerland.     

An explicit formula for the discussion of stability in autonomous Hamiltonian systems

The well known theorem of Arnold and Moser gives (in most cases) an answer to the question of the stability of the equilibria of autonomous Hamiltonian systems which can be reduced to two degrees of freedom. To apply their theorem, the Hamiltonian has to be transformed into its Birkhoff normal form of order two. In this paper this transformation is reduced to the multiplication of polynomials, which can easily be performed algebraically on a computer. The conical precessions of a dynamically symmetric satellite in a circular orbit serve as an example.

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Zusammenfassung Der bekannte Satz von Arnold und Moser liefert (auer in wenigen Ausnahmefällen) eine Antwort auf die Frage nach der Stabilität von Gleichgewichtslagen autonomer Hamiltonscher Systeme, die sich auf zwei Freiheitsgrade reduzieren lassen. Damit dieser Satz angewendet werden kann, mu die Hamiltonfunktion auf ihre Birkhoff-Normalform der Ordnung zwei transformiert werden. Diese Transformation wird hier auf Multiplikationen von Polynomen reduziert, die einfach, in symbolischer Weise, auf dem Computer ausgeführt werden können. Als Beispiel dienen die konischen Präzessionen eines dynamisch symmetrischen Satelliten auf einer Kreisbahn.

     

Complete integrability of the supersymmetric model (cos φ)2

Complete integrability of the supersymmetric, two-dimensional sine-Gordon model of field theory within the framework of the Hamiltonian interpretation of the method of the inverse problem is proved. The classical r-matrix of the model is computed, and its equivalence to the r-matrix the Grassmann Thirring model is established. Variables of creation-annihilation type are constructed, and the spectrum of elementary excitations of the system is obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 120, pp. 122–135, 1982.