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Register a new userHomomorphisms from Functional Equations: The Goldie Equation, III
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In this the sequel to arXiv1910.05816, we derive a necessary and sufficient condition characterizing which real-valued continuous solutions of a multivariate Goldie functional equation express homomorphy between the multivariate Popa groups defined and characterized in the earlier work. This enables us to deduce that all (real-valued) continuous solutions are homomorphisms between such groups. We use this result also to characterize as Popa homomorphisms smooth solutions of a related more general equation, also of Levi-Civita type. A key result here (Theorem 2) on purely radial behaviour is generalized in arXiv2105.07794 to a Banach-algebra setting involving radial tilting behaviour.
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In this sequel to arXiv1407.4089 by the second author, we extend to multi-dimensional (or infinite-dimensional) settings the Goldie equation arising in the general regular variation of `General regular variation, Popa groups and quantifier weakening, J. Math. Anal. Appl. 483 (2020) 123610, 31 pp. (arXiv1901.05996). The theory focusses on extension of the treatment there of Popa groups, permitting a characterization of Popa homomorphisms (in two complementary theorems, 4A and 4B below). This in turn enables a characterization of the (real-valued) solutions of the multivariate Goldie equation, to be presented in the further sequel arXiv1910.05817. The Popa groups here contribute to a structure theorem describing Banach-algebra value
The aim of this note is to characterize all pairs of sufficiently smooth functions for which the mean value in the Cauchy Mean Value Theorem is taken at a point which has a well-determined position in the interval. As an application of this result, a partial answer is given to a question posed by Sahoo and Riedel.
We are concerned below with the characterization in a unital commutative real Banach algebra $mathbb{A}$ of continuous solutions of the Go{l}k{a}b-Schinzel functional equation (below), the general Popa groups they generate and the associated Goldie functional equation. This yields general structure theorems involving both linear and exponential homogeneity in $mathbb{A}$ for both these functional equations and also explict forms, in terms of the recently developed theory of multi-Popa groups [BinO3,4], both for the ring $C[0,1]$ and for the case of $mathbb{R}^{d}$ with componentwise product, clarifying the context of recent developments in [RooSW]. The case $mathbb{A}=mathbb{C}$ provides a new viewpoint on continuous complex-valued solutions of the primary equation by distinguishing analytic from real-analytic ones.
We present a detailed probabilistic and structural analysis of the set of weighted homomorphisms from the discrete torus $mathbb{Z}_m^n$, where $m$ is even, to any fixed graph: we show that the corresponding probability distribution on such homomorphisms is close to a distribution defined constructively as a certain random perturbation of some dominant phase. This has several consequences, including solutions (in a strong form) to conjectures of Engbers and Galvin and a conjecture of Kahn and Park. Special cases include sharp asymptotics for the number of independent sets and the number of proper $q$-colourings of $mathbb{Z}_m^n$ (so in particular, the discrete hypercube). We give further applications to the study of height functions and (generalised) rank functions on the discrete hypercube and disprove a conjecture of Kahn and Lawrenz. For the proof we combine methods from statistical physics, entropy and graph containers and exploit isoperimetric and algebraic properties of the torus.
We consider discrete-time dynamical systems with a linear relaxation dynamics that are driven by deterministic chaotic forces. By perturbative expansion in a small time scale parameter, we derive from the Perron-Frobenius equation the corrections to ordinary Fokker-Planck equations in leading order of the time scale separation parameter. We present analytic solutions to the equations for the example of driving forces generated by N-th order Chebychev maps. The leading order corrections are universal for N larger or equal to 4 but different for N=2 and N=3. We also study diffusively coupled Chebychev maps as driving forces, where strong correlations may prevent convergence to Gaussian limit behavior.
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