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Nikolai N. Vassiliev — 2019-12-28

ELLIPTIC INTEGRALS, FUNCTIONS, CURVES AND POLYNOMIALS

Semjon F. Adlaj — 2019-12-28

The extensive subject of elliptic integrals, functions and curves, being at the junction of analysis, algebra and geometry, has numerous applications in mechanics and physics. Two approaches to the study of elliptic functions have become classical, namely that of Jacobi and that of Weierstrass. Two separate chapters were devoted to these two approaches in the (well-known) course of modern analysis by Whittaker and Watson, without attempting to unite them [1, §§XX, XXII]. Also, two separate chapters are devoted to these two approaches in the latest version (1.0.22 on March 15, 2019) of the NIST Digital Library of Mathematical Functions [2, §§22, 23]. An wide-spread inculcation “explained” that the Weierstrass approach is more suitable for theoretical research, whereas the Jacobi elliptic functions are more common in applications. But, in fact, this dichotomy is artificial, and studying elliptic functions and curves may (and must) be combined in an algebraic approach, establishing a canonical “essential” elliptic function which linear fractional “symmetry” transformations acquire the simplest forms. Although such a natural and fundamental object to be (rightly) called the Galois essential elliptic function, was introduced only recently (already in our millennium), its use has quickly become fruitful, not only and not so much for the effective recovery of known results but also for achieving new calculations that once seemed too cumbersome to pursue. The methodological significance of this natural algebraic approach, which undoubtedly transcends back to the (revolutionary) contribution of Galois, is clearly manifested by its application to several fundamental problems of classical mechanics with the achievement of non-standard, capacious and highly efficient solutions.

APPROXIMATION OF AUTOMATA WITH RESPECT TO THE ANNIHILATION PREDICATE

Elena A. Tolkacheva — 2019-12-28

Automata are the most natural mathematical model for systems that may be in different states. The transfer of questions of approximability with respect to the annihilation predicate of the first kind from varieties of semigroups and polybasic structures to automata is connected with the fundamental property of the automaton — its recognizability. Considering a tribasic semigroup distributive algebra as a model of a semigroup automaton, we have shown that as a result of the transition to polybasic structures, the uniformity of the predicate of equality and the annihilation predicate of the first kind disappears. Consequently, the criteria for decidability of the problem of equality and annihilation predicates will be different. The approximability of automata by tribasic semigroup distributive algebras is associated with the algorithmic decidability of the corresponding problems. The question of the algorithmic decidability of the cancellation problem is of special interest. If there exists an algorithm that determines whether one word is zero for the other for any two words.

THE SQUARED ABSOLUTE VALUE OF (a +bi)+ (c +di)^(1/2^n )

Helmut Ruhland — 2019-12-28

The squared absolute value of A = (a + bi +(c +di)^0,5) can be expressed simply by AA = (a + bi +(c +di)^0,5)(a - bi +(c -di)^0,5), so by roots of complex numbers. Because this expression is real we are interested in an expression, if it exists, that uses only square roots of positive numbers. This expression with only roots of positive numbers is also given for the more general case with with 2^n-th roots.

USING CAPABILITIES OF RACKET TO IMPLEMENT A DOMAIN-SPECIFIC LANGUAGE

Ivan A. Dolgakov — 2019-12-28

An implementation of a domain-specific language, Landau, based on the Racket platform, is presented. Landau is a dynamical system specification language, used in an environment where parameters of the dynamical system must be determined from processing of observational data. That, in turn, requires efficient and accurate calculation of derivatives, which can be achieved with the automatic differentiation (AD) technique. Landau is a Turing incomplete statically typed language aimed to support code generation with AD. The Turing incompleteness provides the ability of analyzing the whole program workflow to figure out the chain of derivatives’ calculation. Landau has compile-time ranged for loops, if/else branching constructions, mutable variables and arrays. Landau can be compiled to both Racket and ANSI C. Landau implementation takes advantage of features that are unique to the Racket platform and make creation of DSLs more convenient than on other platform.

INFERRING TRIGGER NOISE FROM TRIGGER JITTER

Franсois Lamarche — 2019-12-28

We address the relationship between voltage noise at the input of a threshold comparator and random changes in the timing of the comparator output toggle timing (jitter). Models that go beyond the na¨ıve linear relationship are presented and shown to match experiment. They help reconcile predictions with verifiable device performance; more over, they allow an approximate measurement of the noise bandwidth without the help of high-frequency equipment. In the course our investigation, a PC/laptop toolset was constructed that lets a neophyte study noise via jitter. This toolset can also be used in education to teach the notion that a biased view of a random phenomenon can arise from triggering on the tails of its distribution.

THE CALCULATION OF THE PROBABILITY DENSITY IN PHASE SPACE OF A CHAOTIC SYSTEM ON THE EXAMPLE OF ROTATOR IN THE HARMONIC FIELD

Alexander V. Liapzev — 2019-12-28

The method of calculating the probability density distribution for chaotic systems described by the equations of classical nonlinear dynamics is proposed. Specific calculations are performed for the rotator in an external harmonic field. The results of calculation of probability density distribution in the cross section of the phase space are compared with the Poincare cross section for the chaotic attractor obtained by numerical solution of the dynamic equations. It is shown that the corresponding quantum problem for a rotator in an external harmonic field in the semiclassical limit leads to equations describing the distribution of probability density in the classical case.