12,613 research outputs found
The irrationality of some number theoretical series
We prove the irrationality of some factorial series. To do so we combine
methods from elementary and analytic number theory with methods from the theory
of uniform distribution
Quantum Probabilities as Behavioral Probabilities
We demonstrate that behavioral probabilities of human decision makers share
many common features with quantum probabilities. This does not imply that
humans are some quantum objects, but just shows that the mathematics of quantum
theory is applicable to the description of human decision making. The
applicability of quantum rules for describing decision making is connected with
the nontrivial process of making decisions in the case of composite prospects
under uncertainty. Such a process involves deliberations of a decision maker
when making a choice. In addition to the evaluation of the utilities of
considered prospects, real decision makers also appreciate their respective
attractiveness. Therefore, human choice is not based solely on the utility of
prospects, but includes the necessity of resolving the utility-attraction
duality. In order to justify that human consciousness really functions
similarly to the rules of quantum theory, we develop an approach defining human
behavioral probabilities as the probabilities determined by quantum rules. We
show that quantum behavioral probabilities of humans not merely explain
qualitatively how human decisions are made, but they predict quantitative
values of the behavioral probabilities. Analyzing a large set of empirical
data, we find good quantitative agreement between theoretical predictions and
observed experimental data.Comment: Latex file, 32 page
Diophantine approximation of Mahler numbers
Suppose that is a Mahler function and that is
in the radius of convergence of . In this paper, we consider the
approximation of by algebraic numbers. In particular, we prove that
cannot be a Liouville number. If is also regular, we show that
is either rational or transcendental, and in the latter case that
is an -number or a -number.Comment: 52 page
Some classical multiple orthogonal polynomials
Recently there has been a renewed interest in an extension of the notion of
orthogonal polynomials known as multiple orthogonal polynomials. This notion
comes from simultaneous rational approximation (Hermite-Pade approximation) of
a system of several functions. We describe seven families of multiple
orthogonal polynomials which have he same flavor as the very classical
orthogonal polynomials of Jacobi, Laguerre and Hermite. We also mention some
open research problems and some applications
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