The Gleason theorem for the field of rational numbers and residue fields

Charges taking values in a fieldF and defined on orthomodular partially ordered sets (logics) of all projectors in some finite-dimensional linear space overF are considered. In the cases whereF is the field of rational numbers or a residue field, the Gleason representation , where is a linear operator, is proved.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 584–591, October, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01265.     

Which rational numbers are binding numbers?

The concept of the binding number of a graph was introduced by Woodall in 1973. in this paper we characterize the set F_n of all pairs (a, b) of integers such that there is a graph G with n vertices and binding number a/b that has a realizing set of b vertices.     

Unique expansions of real numbers

It was discovered some years ago that there exist non-integer real numbers q>1 for which only one sequence (ci) of integers ci∈[0,q) satisfies the equality . The set of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation.In this paper we consider for each fixed q>1 the set Uq of real numbers x having a unique representation of the form with integers ci belonging to [0,q). We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases q for which Uq is closed or even a Cantor set. We also study the set consisting of all sequences (ci) of integers ci∈[0,q) such that . We determine the numbers r>1 for which the map (defined on (1,∞)) is constant in a neighborhood of r and the numbers q>1 for which is a subshift or a subshift of finite type.     

Unique representation domains, II

Given a star operation ∗ of finite type, we call a domain R a ∗-unique representation domain (∗-URD) if each ∗-invertible ∗-ideal of R can be uniquely expressed as a ∗-product of pairwise ∗-comaximal ideals with prime radical. When ∗ is the t-operation we call the ∗-URD simply a URD. Any unique factorization domain is a URD. Generalizing and unifying results due to Zafrullah [M. Zafrullah, On unique representation domains, J. Nat. Sci. Math. 18 (1978) 19-29] and Brewer-Heinzer [J.W. Brewer, W.J. Heinzer, On decomposing ideals into products of comaximal ideals, Comm. Algebra 30 (2002) 5999-6010], we give conditions for a ∗-ideal to be a unique ∗-product of pairwise ∗-comaximal ideals with prime radical and characterize ∗-URD’s. We show that the class of URD’s includes rings of Krull type, the generalized Krull domains introduced by El Baghdadi and weakly Matlis domains whose t-spectrum is treed. We also study when the property of being a URD extends to some classes of overrings, such as polynomial extensions, rings of fractions and rings obtained by the D+XD_S[X] construction.     

Error-free computation with rational numbers

A method is described for doing error-free computation when the operands are rational numbers. A rational operanda/b is mapped onto the integer ¦a·b ^–1¦ _p and the arithmetic is performed inGF(p). A method is given for taking an integer result and finding its rational equivalent (the one which corresponds to the correct rational result).     

A method for exact computation with rational numbers

An easily programmed method is presented for solving N linear equations in N unknowns exactly for the rational answers, given that all coefficients and constants appearing in the equations are rational numbers. The rational answers are deduced from floating point approximations to the answers obtained by any of the standard solution algorithms. Criteria are given for determining for a particular set of equations the floating point precision needed.     

Diophantine representation of perfect numbers

There is explicitly given a polynomial (of degree 17 in 48 variables), the set of whose positive values (for natural values of the variables) is precisely the set of all perfect numbers. The construction of this polynomial is based on a new test for a number to be perfect, formulated in terms of the divisibility of binomial coefficients.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 78–89, 1979.     

Division in a binary representation for complex numbers

Computer operations involving complex numbers, essential in such applications as Fourier transforms or image processing, are normally performed in a ‘divide-and-conquer’ approach dealing separately with real and imaginary parts. A number of proposals have treated complex numbers as a single unit but all have foundered on the problem of the division process without which it is impossible to carry out all but the most basic arithmetic. This paper resurrects an early proposal to express complex numbers in a single ‘binary’ representation, reviews basic complex arithmetic and is able to provide a fail-safe procedure for obtaining the quotient of two complex numbers expressed in the representation. Thus, while an outstanding problem is solved, recourse is made only to readily accessible methods. A variety of extensions to the work requiring similar basic techniques are also identified. An interesting side-line is the occurrence of fractal structures, and the power of the ‘binary’ representation in analysing the structure is briefly discussed.     

A new infinite product representation for real numbers

We introduce a new algorithm that leads to a representation for any real number greater than one as an infinite product of rational numbers. Just as we can regard the Cantor product as being a product analogue of the series of Sylvester, this new product is analogous to the classical Engel representation for real numbers. The growth conditions satisfied by the digits in the product are likewise shown to correspond to those required for the Engel series. The representation for certain types of rational numbers via this algorithm is also considered.     

On the Hilbert 2-class field tower of some abelian 2-extensions over the field of rational numbers

It is well known by results of Golod and Shafarevich that the Hilbert 2-class field tower of any real quadratic number field, in which the discriminant is not a sum of two squares and divisible by eight primes, is infinite. The aim of this article is to extend this result to any real abelian 2-extension over the field of rational numbers. So using genus theory, units of biquadratic number fields and norm residue symbol, we prove that for every real abelian 2-extension over ? in which eight primes ramify and one of theses primes ≡ ?1 (mod 4), the Hilbert 2-class field tower is infinite.     

Estimates for polynomials in logarithms of some rational numbers

The Hermite-Pade approximations of the second type for algebra generated by a generalized Nikishin system of Markov functions corresponding to an infinite branching graph are investigated. Arithmetical applications of this construction are given. Namely, lower estimates for polynomials with integer coefficients in logarithms of some rational numbers are obtained. These estimates partially refine some known results obtained earlier by the Siegel method. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 179–194, 2005.     

Functional classical mechanics and rational numbers

The notion of microscopic state of the system at a given moment of time as a point in the phase space as well as a notion of trajectory is widely used in classical mechanics. However, it does not have an immediate physical meaning, since arbitrary real numbers are unobservable. This notion leads to the known paradoxes, such as the irreversibility problem. A “functional” formulation of classical mechanics is suggested. The physical meaning is attached in this formulation not to an individual trajectory but only to a “beam” of trajectories, or the distribution function on phase space. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values and there are corrections to the Newton trajectories. We give a construction of probability density function starting from the directly observable quantities, i.e., the results of measurements, which are rational numbers.     

A binomial product representation forp-adic numbers

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