Related papers: On absolutely normal and continued fraction normal…
This note examines the infinite divisibility of density-based transformations of normal random variables. We characterize a class of density-based transformations of normal variables which produces non-infinitely divisible distributions. We…
Consider the representation of a rational number as a continued fraction, associated with "odd" Euclidean algorithm. In this paper we prove certain properties for the limit distribution function for sequences of rationals with bounded sum…
Many systems exhibit a digit bias. For example, the first digit base 10 of the Fibonacci numbers, or of $2^n$, equals 1 not 10% or 11% of the time, as one would expect if all digits were equally likely, but about 30% of the time. This…
We study odd numbers through a straightforward indexing. We focus in particular on odd prime and composite numbers and their distribution. With a counting argument, we calculate the limit of two sums and compare their convergence rate.
We consider a family of integer sequences generated by nonlinear recurrences of the second order, which have the curious property that the terms of the sequence, and integer multiples of the ratios of successive terms (which are also…
In this paper we show how to construct several infinite families of polynomials $D(\bar{x},k)$, such that $\sqrt{D(\bar{x},k)}$ has a regular continued fraction expansion with arbitrarily long period, the length of this period being…
Many mathematical, man-made and natural systems exhibit a leading-digit bias, where a first digit (base 10) of 1 occurs not 11\% of the time, as one would expect if all digits were equally likely, but rather 30\%. This phenomenon is known…
Champernowne famously proved that the number $0.(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)...$ formed by concatenating all the integers one after another is normal base 10. We give a generalization of Champernowne's construction to various…
Many research works have concerned normality-preserving selection rules and operations on the sequence of digits of a given normal number that maintain or violate normality. This leads us to introduce rearrangement operations on finite…
We construct new families of completely regular codes by concatenation methods. By combining parity check matrices of cyclic Hamming codes, we obtain families of completely regular codes. In all cases, we compute the intersection array of…
Digital System Research has pioneered the mathematics and design for a new class of computing machine using residue numbers. Unlike prior art, the new breakthrough provides methods and apparatus for general purpose computation using several…
We prove that digital sequences modulo $m$ along squares are normal, which covers some prominent sequences like the sum of digits in base $q$ modulo $m$, the Rudin-Shapiro sequence and some generalizations. This gives, for any base, a class…
We believe we have made progress in the age-old problem of divisibility rules for integers. Universal divisibility rule is introduced for any divisor in any base number system. The divisibility criterion is written down explicitly as a…
We give a~detailed construction of the complete ordered field of real numbers by means of infinite decimal expansions. We prove that in the canonical encoding of decimals neither addition nor multiplication is {\em computable}, but that…
We formulate the problem of numerical analytic continuation in a way that lets us draw meaningful conclusions about properties of the spectral function based solely on the input data. Apart from ensuring consistency with the input data…
A real number $x$ is normal with respect to an integer base $b \geq 2$ if its digit expansion in this base is ``equitable'', in the sense that for $k \geq 1$, every ordered sequence of $k$ digits from $\{0, 1, \ldots, b-1\}$ occurs in the…
The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate…
If the odd and even parts of a continued fraction converge to different values, the continued fraction may or may not converge in the general sense. We prove a theorem which settles the question of general convergence for a wide class of…
In an unbounded plane, straight lines are used extensively for mathematical analysis. They are tools of convenience. However, those with high slope values become unbounded at a faster rate than the independent variable. So, straight lines,…
Natural numbers which are nontrivial multiples of some permutation of their base-$b$ digit representations are called permutiples. Specific cases include numbers which are multiples of cyclic permutations (cyclic numbers) and reversals of…