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EN
We consider the class of rational functions defined by the formula F(x, y) = φˉ¹(φ(x)φ(y)), where φ is a homographic function and we describe associative functions of the above form.
EN
We consider the class of rational functions defined by the formula F(x, y) = ϕ −1 (ϕ(x) + ϕ(y)), where ϕ is a homographic function and we describe all associative functions of the above form.
EN
We deal with the functional equation (so called addition formula) of the form f(x + y) = F(f(x),f(y)), where F is an associative rational function. The class of associative rational functions was described by A. Chéritat [1] and his work was followed by a paper of the author. For function F deﬁned by F(x,y) = ϕ−1(ϕ(x) + ϕ(y)), where ϕ is a homographic function, the addition formula is fulﬁlled by homographic type functions. We consider the class of the associative rational functions defined by formula F(u,v) =uv αuv + u + v, where α is a ﬁxed real numer.
4
Character sums and pair correlations
5
Rigidity of tame rational functions
EN
We introduce and establish some basic properties of the tame rational functions. The class of these functions contains all the rational functions with no recurrent critical points in their Julia sets. For tame non-exceptional functions we prove that the Lipschitz conjugacy, the same spectra of moduli of derivatives at periodic orbits and conformal conjugavcy are mutually equivalent. We prove also the following rigidity result: If h is a Borel measurable invertible map which conjugates two tame functions f and g a.e. and if h transports conformal measure m[sub f] to a measure equivalent to m[sub g] then h extends from a set of full measure m[sub f] to a conformal homeomorphism of neighbourhoods of respective Julia sets. This extends D. Sullivan's rigidity theorem for holomorphic expanding repellers. We provide also a few lines proof of E. Prado's theorem that two generalized polinomial-like maps at zero Teichmueller's distance are holomorphically conjugate.
EN
In the process of designing controllers for linear multivariable plants specially effective are algebraic methods which require from the transfer matrices of both, the plant and the controller to be presented in coprime fractional form with factorization carried on with respect to the ring of exponentially-stable, proper real-rational functions. The main objective of the paper is to show that this form of representation with simultaneous parametrization of all linear controllers that provide internal stability of the closed-loop system can be achieved in the simplest and most natural way by analysing the system shown in Fig. 3 - the so-called basic structure. Problems of choosing the parameter to meet some important design specifications, viz. a robust asymptotic tracking of the reference signal with disturbance and noise rejection are also considered and illustrated by two representative examples covering the area of continuous- and discrete-time systems.
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