The Local Frobenius Theorem

On Gibbs Measures and Spectra of Ruelle Transfer Operators

Canadian Mathematical Bulletin ◽

10.4153/cmb-2016-073-2 ◽

2017 ◽

Vol 60 (2) ◽

pp. 411-421

Author(s):

Luchezar Stoyanov

Keyword(s):

Spectral Radius ◽

Spectral Properties ◽

Gibbs Measures ◽

Transfer Operator ◽

Frobenius Theorem ◽

Transfer Operators

AbstractWe prove a comprehensive version of the Ruelle–Perron–Frobenius Theorem with explicit estimates of the spectral radius of the Ruelle transfer operator and various other quantities related to spectral properties of this operator. The novelty here is that the Hölder constant of the function generating the operator appears only polynomially, not exponentially as in previously known estimates.

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On the Stability in Bonnet's Theorem of the Surface Theory

Georgian Mathematical Journal ◽

10.1515/gmj.2007.543 ◽

2007 ◽

Vol 14 (3) ◽

pp. 543-564

Author(s):

Yuri G. Reshetnyak

Keyword(s):

Differential Operators ◽

Sobolev Class ◽

Integrability Condition ◽

Complete Integrability ◽

Integral Representations ◽

Completely Integrable Systems ◽

Frobenius Theorem ◽

Fundamental Tensor ◽

The Stability ◽

Codazzi Equations

Abstract In the space , 𝑛-dimensional surfaces are considered having the parametrizations which are functions of the Sobolev class with 𝑝 > 𝑛. The first and the second fundamental tensor are defined. The Peterson–Codazzi equations for such functions are understood in some generalized sense. It is proved that if the first and the second fundamental tensor of one surface are close to the first and, respectively, to the second fundamental tensor of the other surface, then these surfaces will be close up to the motion of the space . A difference between the fundamental tensors and the nearness of the surfaces are measured with the help of suitable 𝑊-norms. The proofs are based on a generalization of Frobenius' theorem about completely integrable systems of the differential equations which was proved by Yu. E. Borovskiĭ. The integral representations of functions by differential operators with complete integrability condition are used, which were elaborated by the author in his other works.

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Transversals in permutation groups and factorisations of complete graphs

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700020323 ◽

2002 ◽

Vol 65 (2) ◽

pp. 277-288 ◽

Cited By ~ 2

Author(s):

Gil Kaplan ◽

Arieh Lev

Keyword(s):

Permutation Group ◽

Directed Graph ◽

Sufficient Conditions ◽

Combinatorial Problems ◽

Permutation Groups ◽

Transitive Permutation Group ◽

Complete Graphs ◽

Frobenius Theorem ◽

Disjoint Cycles ◽

Finite Set

Let G be a transitive permutation group acting on a finite set of order n. We discuss certain types of transversals for a point stabiliser A in G: free transversals and global transversals. We give sufficient conditions for the existence of such transversals, and show the connection between these transversals and combinatorial problems of decomposing the complete directed graph into edge disjoint cycles. In particular, we classify all the inner-transitive Oberwolfach factorisations of the complete directed graph. We mention also a connection to Frobenius theorem.

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