Related papers: Rotation number of primitive vector sequences
If p is a prime, then the numbers 1, 2, ..., p-1 form a group under multiplication modulo p. A number g that generates this group is called a primitive root of p; i.e., g is such that every number between 1 and p-1 can be written as a power…
We provide necessary and sufficient conditions on the existence of common hypercyclic vectors for multiples of the backward shift operator along sparse powers. Our main result strongly generalizes corresponding results which concern the…
We examine linear sums of primitive roots and their inverses in finite fields. In particular, we refine a result by Li and Han, and show that every $p> 13$ has a pair of primitive roots $a$ and $b$ such that $a+ b$ and $a^{-1} + b^{-1}$ are…
Let $X$ be a closed invariant subset of the half--open annulus $\mathbb{A}$ such that $\mathbb{A} \setminus X$ is homeomorphic to $\mathbb{A}$. We prove that either the rotation number of all forward semi--orbits of accessible points of $X$…
Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…
Assuming a suitable rotation of the Universe as a whole, we can attribute the rotation of galaxies to it. Using Lagrangian Mechanics we then can find the equations governing the rotation of a galaxy. We find in this way the azimuthal…
The distribution of differences of consecutive members of sequences of primes is investigated. A quantitative measure for oscillations among these differences is the curvature of the sequence. If the sequence is not too sparse, then sharp…
By using the matrix formulation of the two-step approach to distributions of patterns in random sequences, recurrence and explicit formulas for the generating functions of successions in random permutations of arbitrary multisets are…
The way a field transforms under rotations determines its statistics--as is easy to see for scalar, Dirac, and vector fields.
We describe all weights which are appropriated for the unitarization of linear representations of primitive partially ordered sets of finite type.
This article investigates integer sequences that partition the sequence into blocks of various lengths - irregular arrays. The main result of the article is explicit formulas for numbering of irregular arrays. A generalization of Cantor…
In this work I look at the distribution of primes by calculation of an infinite number of intersections. For this I use the set of all numbers which are not elements of a certain times table in each case. I am able to show that it exists a…
We define a transformation that associates certain exponential moment sequences with ordinary moment sequences in a natural way. The ingredients of this transformation are series reversion, the Sumudu transform (a variant of the Laplace…
In this paper, closed formulas for the eigenvectors of a particular class of matrices generated by generalized permutation matrices, named generalized circulant matrices, are presented.
Let $\mathcal{D}$ be a Hermitian symmetric space of tube type, and let $S$ be its Shilov boundary. We give a realization of the universal covering $\widetilde{S}$ of $S$. Then we describe on $\widetilde{S}$ a primitive for the generalized…
We present a topological recursion formula for calculating the intersection numbers defined on the moduli space of open Riemann surfaces. The spectral curve is $x = \frac{1}{2}y^2$, the same as spectral curve used to calculate intersection…
We present a general framework for calculating the Volterra-type convolution of polynomials from an arbitrary polynomial sequence $\{P_k(x)\}_{k \geqslant 0}$ with $\deg P_k(x) = k$. Based on this framework, series representations for the…
Modulo a prime number, we define semi-primitive roots as the square of primitive roots. We present a method for calculating primitive roots from quadratic residues, including semi-primitive roots. We then present progressions that generate…
The denominator of the Hilbert series of a finitely generated R-module M does not always divide the denominator of the Hilbert series of R. For this reason, we define the universal denominator. The universal denominator of a module M is the…
Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix $H$ to a unitary analogue. In the first type the unitary matrix is $e^{i(\lambda\II - H)}$ where $\lambda$ is the…