### Article

## Topological regluing of rational functions

Regluing is a surgery that helps to build topological models for rational functions. It also has a holomorphic interpretation, with the °avor of in¯nite dimensional Thurston{TeichmÄuller theory. We will discuss a topological theory of regluing, and just trace a

direction, in which a holomorphic theory can develop.

In paper the authors consider creation of a sensitivity model for complex model of the interdependent electric, thermal and mechanical processes proceeding in an electronic equipment. The sensitivity model is formed by means of the principle of additional models creation. Use of complex model gives the chance to receive sensitivity functions of a process submodel variable of one physical nature to change of a process submodel parameter of other physical nature. It allows displaying actual parameters interrelation of heterogeneous physical processes within one design volume of electronic equipment.

Given n ≥ 1 and r ∈ [0, 1), we consider the set Rn, r of rational functions having at most n poles all outside of 1/rD, were D is the unit disc of the complex plane. We give an asymptotically sharp Bernstein-type inequality for functions in Rn, r in weighted Bergman spaces with “polynomially” decreasing weights. We also prove that this result can not be extended to weighted Bergman spaces with “super-polynomially” decreasing weights

We consider the set $R_n$ of rational functions of degree at most $n$ with no poles on the unit circle and its subclass $R_{n,r}$ consisting of rational functions without poles in the annulus $r<|z|<1/r$. We discuss an approach based on the model space theory which brings some integral representations for functions in $R_n$ and their derivatives. Using this approach we obtain $L^p$-analogs of several classical inequalities for rational functions including the inequalities by P. Borwein and T. Erdelyi, the Spijker Lemma and S.M. Nikolskii's inequalities. These inequalities are shown to be asymptotically sharp as $n$ tends to infinity and the poles of the rational functions approach the unit circle.

Authors are proved need of contact control methods application of electronic means elements when carrying out thermal diagnosing in the closed constructive volume. The method of the error calculation brought by the sensor at measurement of element temperature is described. In article topological thermal models of electronic component mounted on the printed circuit board and a pair of electronic component-sensor are presented and investigated in thermal modeling subsystem ASONIKA-T. This models are proposed to use in a new calculate method of error introduced by the sensor. The authors done the experimental check of the presented calculate method. The creation stages of thermal processes models are described in detail. In article is proved that the developed topological models and method fully meets requirements. The conclusion is drawn that their use in the process of electronic means thermal diagnostics will allow to increase the accuracy of temperature measurements due to compensation of the systematic error brought at measurement by the contact thermal sensor. As a result it will positively affect the reliability of definition of defective electronic components and the reliability of electronic means.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.