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A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector…
In this article, we consider inverse problems of determining a source term and a coefficient of a first-order partial differential equation and prove conditional stability estimates with minimum boundary observation data and relaxed…
We present quantum graphs with remarkably regular spectral characteristics. We call them {\it regular quantum graphs}. Although regular quantum graphs are strongly chaotic in the classical limit, their quantum spectra are explicitly…
There are numerous ways to represent real numbers. We may use, e.g., Cauchy sequences, Dedekind cuts, numerical base-10 expansions, numerical base-2 expansions and continued fractions. If we work with full Turing computability, all these…
In this note, we describe a family of particular algebraic, and nonquadratic, power series over an arbitrary finite field of characteristic 2, having a continued fraction expansion with all partial quotients of degree one. The main purpose…
We provide a non-trivial measure of irrationality for a class of Mahler numbers defined with infinite products which cover the Thue-Morse constant.
We investigate the topological structure of the decimal expansions of the three famous naturally occurring irrational numbers, $\pi$, $e$, and golden ratio, by explicitly calculating the diversity of the pair distributions of the ten digits…
We introduce the notions of quantum characteristic and quantum flatness for arbitrary rings. More generally, we develop the theory of quantum integers in a ring and show that the hypothesis of quantum flatness together with positive quantum…
We survay some nice result concerning the irrationals with a metric space point of view.Here is ofcourse nothing new may be or an expert in this field.
In order to prove irrationality of \sqrt{2} by using only decimal expansions (and not fractions), we develop in detail a model of real numbers based on infinite decimals and arithmetic operations with them.
Let $P$ be a prime polynomial in the variable $Y$ over a finite field and let $f$ be a quadratic irrational in the field of formal Laurant series in the variable $Y^{-1}$. We study the asymptotic properties of the degrees of the…
The properties of continued fractions whose partial quotients belong to a quadratic number field K are distinct from those of classical continued fractions. Unlike classical continued fractions, it is currently impossible to identify…
A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a…
We define a new condition number adapted to directionally uniform perturbations. The definitions and theorems can be applied to a large class of problems. We show the relation with the classical condition number, and study some interesting…
A discussion is given of the uncertainty principle in view of the introduction of a Gravitational Planck Constant. The need for such a gravitational constant is shown first. A reduced electromagnetic Planck constant and the analogous…
A criterion on the asymptotic stability of fractional-order systems with incomensurate orders is proposed in this paper. Existing methods always assume order parameters be rational numbers or the ratios of any two orders be rational…
A nonlinear partial differential equation is a nonlinear relationship between an unknown function and how it changes due to two or more input variables. A numerical method reduces such an equation to arithmetic for quick visualization, but…
In math.NT/0307308 we defined the irrationality base of an irrational number and, assuming a stronger hypothesis than the irrationality of Euler's constant, gave a conditional upper bound on its irrationality base. Here we develop the…
Consider the representation of a rational number in the form, associated with "centered" Euclidean algorithm. We prove a new formula for the limit distribution function for sequences of rationals with bounded sum of partial quotients.
Dynamical Systems theory generally deals with fixed point iterations of continuous functions. Computation by Turing machine although is a fixed point iteration but is not continuous. This specific category of fixed point iterations can only…