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## Volume and entropy in abstract analytic number theory and thermodynamics

We develop the recent research [1] and introduce the notions of volume and entropy in abstract analytic number theory. The introduction of negative numbers in the generalized partition problem, together with the meaning of such a generalization in some applications of the theory, is discussed.

A vocationally-oriented course on Physics with medico-biological direction is proposed in this book. The matters of application of physical methods for substances investigation are considered here in detail.

The book may ne helpful for students of medical and pharmaceutical institutes of higher education and colleges, chemical, biological and pharmaceutical faculties of universities, as well as for students of core classes and teachers of medico-biological specialties.

In this paper we propose a new machine learning concept called randomized machine learning, in which model parameters are assumed random and data are assumed to contain random errors. Distinction of this approach from "classical" machine learning is that optimal estimation deals with the probability density functions of random parameters and the "worst" probability density of random data errors. As the optimality criterion of estimation, randomized machine learning employs the generalized information entropy maximized on a set described by the system of empirical balances. We apply this approach to text classification and dynamic regression problems. The results illustrate capabilities of the approach.

Properties of Erdos measure and the invariant Erdos measure for the golden ratio and all values of the Bernoulli parameter are studies. It is proved that a shift on the two-sided Fibonacci compact set with invariant Erdos measure is isomorphic to the integral automorphism for a Bernoulli shift with countable alphabet. An effective algorithm for calculating the entropy of an invariant Erdos measure is proposed. It is shown that, for certain values of the Bernulli parameter, the algorithm gives the Hausdorff dimension of an Erdos measure to 15 decimal places.

Properties of Erdos measure and the invariant Erdos measure for the golden ratio and all values of the Bernoulli parameter are studies. It is proved that a shift on the two-sided Fibonacci compact set with invariant Erdos measure is isomorphic to the integral automorphism for a Bernoulli shift with countable alphabet.

The present volume is the fourth issue of the Almanac series entitled ‘Evolution’. Thus, one can maintain that our Almanac, which has actually turned into a Yearbook, has succeeded (see below).

The title of the present volume is ‘From Big Bang to Nanorobots’. In this way we demonstrate that all phases of megaevolution and Big History are covered in the articles of the present Yearbook. Several articles also present forecasts about possible future developments.

We single out the main features of the mathematical theory of noble gases. It is proved that the points of degeneracy of the Bose gas fractal dimension in momentum space coincide with the critical points of noble gases, while the jumps of the critical indices and the Maxwell rule are related to tunnel quantization in thermodynamics. We consider semiclassical methods for tunnel quantization in thermodynamics as well as those for second and ultrasecond quantization (the creation and annihilation operators for pairs of particles). Each noble gas is associated with a new critical point of the limit negative pressure. The negative pressure is equivalent to covering the (P,Z)- diagram by the second sheet.

The formula for calculating the entropy and the Hausdorff dimension of an invariant Erdos measure for the pseudogolden ratio and all values Bernoulli parameter is obtained. This formula make possible calculating the entropy and the Hausdorff dimension with high accuracy.