Each Positive Rational Number Has the Form φ(m2)/φ(n2)

2020 ◽  
Vol 127 (9) ◽  
pp. 847-849
Author(s):  
Dmitry Krachun ◽  
Zhi-Wei Sun
Keyword(s):  
Rational Number ◽  
Positive Rational Number

scholarly journals Ramification theory and perfectoid spaces

2014 ◽  
Vol 150 (5) ◽  
pp. 798-834 ◽  
Author(s):  
Shin Hattori
Keyword(s):  
Positive Integer ◽  
Rational Number ◽  
Positive Rational Number ◽  
Ramification Theory ◽  
Local Monodromy ◽  
Residue Fields ◽  
Perfectoid Spaces ◽  
Residue Characteristic ◽  
The Absolute ◽  
Separable Extensions

AbstractLet $K_1$ and $K_2$ be complete discrete valuation fields of residue characteristic $p>0$. Let $\pi _{K_1}$ and $\pi _{K_2}$ be their uniformizers. Let $L_1/K_1$ and $L_2/K_2$ be finite extensions with compatible isomorphisms of rings $\mathcal{O}_{K_1}/(\pi _{K_1}^m)\, {\simeq }\, \mathcal{O}_{K_2}/(\pi _{K_2}^m)$ and $\mathcal{O}_{L_1}/(\pi _{K_1}^m)\, {\simeq }\, \mathcal{O}_{L_2}/(\pi _{K_2}^m)$ for some positive integer $m$ which is no more than the absolute ramification indices of $K_1$ and $K_2$. Let $j\leq m$ be a positive rational number. In this paper, we prove that the ramification of $L_1/K_1$ is bounded by $j$ if and only if the ramification of $L_2/K_2$ is bounded by $j$. As an application, we prove that the categories of finite separable extensions of $K_1$ and $K_2$ whose ramifications are bounded by $j$ are equivalent to each other, which generalizes a theorem of Deligne to the case of imperfect residue fields. We also show the compatibility of Scholl’s theory of higher fields of norms with the ramification theory of Abbes–Saito, and the integrality of small Artin and Swan conductors of $p$-adic representations with finite local monodromy.

scholarly journals Factorization invariants of the additive structure of exponential Puiseux semirings

2022 ◽  
Author(s):  
Harold Polo
Keyword(s):  
Rational Number ◽  
Arithmetic Progressions ◽  
Additive Structure ◽  
Positive Rational Number ◽  
Sets Of Lengths ◽  
Almost All

Exponential Puiseux semirings are additive submonoids of [Formula: see text] generated by almost all of the nonnegative powers of a positive rational number, and they are natural generalizations of rational cyclic semirings. In this paper, we investigate some of the factorization invariants of exponential Puiseux semirings and briefly explore the connections of these properties with semigroup-theoretical invariants. Specifically, we provide exact formulas to compute the catenary degrees of these monoids and show that minima and maxima of their sets of distances are always attained at Betti elements. Additionally, we prove that sets of lengths of atomic exponential Puiseux semirings are almost arithmetic progressions with a common bound, while unions of sets of lengths are arithmetic progressions. We conclude by providing various characterizations of the atomic exponential Puiseux semirings with finite omega functions; in particular, we completely describe them in terms of their presentations.

scholarly journals On computing the Lyapunov exponents of reversible cellular automata

2020 ◽  
Author(s):  
Johan Kopra
Keyword(s):  
Lyapunov Exponent ◽  
Cellular Automata ◽  
Lyapunov Exponents ◽  
Rational Number ◽  
Absolute Constant ◽  
Uniform Measure ◽  
Positive Rational Number ◽  
Reversible Cellular Automata

AbstractWe consider the problem of computing the Lyapunov exponents of reversible cellular automata (CA). We show that the class of reversible CA with right Lyapunov exponent 2 cannot be separated algorithmically from the class of reversible CA whose right Lyapunov exponents are at most $$2-\delta$$ 2 - δ for some absolute constant $$\delta >0$$ δ > 0 . Therefore there is no algorithm that, given as an input a description of an arbitrary reversible CA F and a positive rational number $$\epsilon >0$$ ϵ > 0 , outputs the Lyapunov exponents of F with accuracy $$\epsilon$$ ϵ . We also compute the average Lyapunov exponents (with respect to the uniform measure) of the reversible CA that perform multiplication by p in base pq for coprime $$p,q>1$$ p , q > 1 .

scholarly journals Sums of digits in q-ary expansions

2015 ◽  
Vol 11 (02) ◽  
pp. 593-611
Author(s):  
J. C. Saunders
Keyword(s):  
Rational Number ◽  
Explicit Construction ◽  
Positive Rational Number ◽  
Floor Function

Let sq(n) denote the sum of the digits of a number n expressed in base q. We study here the ratio [Formula: see text] for various values of q and α. In 1978, Kenneth B. Stolarsky showed that [Formula: see text] and that [Formula: see text] using an explicit construction. We show that for α = 2 and q ≥ 2, the above ratio can in fact be any positive rational number. We also study what happens when α is a rational number that is not an integer, terminating the resulting expression by using the floor function.

scholarly journals Positive Rational Number of the Form (Equation)

2020 ◽  
pp. 93-99
Author(s):  
Hongjian Li ◽  
Pingzhi Yuan ◽  
Hairong Bai
Keyword(s):  
Rational Number ◽  
Positive Rational Number ◽  
Form Equation ◽  
Positive Integers ◽  
Proper Representation

Let (Equation) and (Equation) be positive integers with (Equation) . In this paper, we show that every positive rational number can be written as the form (Equation) , where m,n∈N if and only if (Equation) or (Equation) . Moreover, if (Equation) , then the proper representation of such representation is unique.

scholarly journals Rational Catalan Combinatorics: The Associahedron

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Drew Armstrong ◽  
Brendon Rhoades ◽  
Nathan Williams
Keyword(s):  
Simplicial Complex ◽  
Rational Number ◽  
Catalan Number ◽  
Positive Rational Number ◽  
Dyck Paths ◽  
Narayana Numbers ◽  
International Audience ◽  
Positive Integers ◽  
Coprime Positive Integers ◽  
Product Formulas

International audience Each positive rational number $x>0$ can be written $\textbf{uniquely}$ as $x=a/(b-a)$ for coprime positive integers 0<$a$<$b$. We will identify $x$ with the pair $(a,b)$. In this extended abstract we use $\textit{rational Dyck paths}$ to define for each positive rational $x>0$ a simplicial complex $\mathsf{Ass} (x)=\mathsf{Ass} (a,b)$ called the $\textit{rational associahedron}$. It is a pure simplicial complex of dimension $a-2$, and its maximal faces are counted by the $\textit{rational Catalan number}$ $\mathsf{Cat} (x)=\mathsf{Cat}(a,b):=\frac{(a+b-1)! }{ a! b!}.$ The cases $(a,b)=(n,n+1)$ and $(a,b)=(n,kn+1)$ recover the classical associahedron and its Fuss-Catalan generalization studied by Athanasiadis-Tzanaki and Fomin-Reading. We prove that $\mathsf{Ass} (a,b)$ is shellable and give nice product formulas for its $h$-vector (the $\textit{rational Narayana numbers}$) and $f$-vector (the $\textit{rational Kirkman numbers}$). We define $\mathsf{Ass} (a,b)$ .

scholarly journals Soma iterada de algarismos de um número racional

2021 ◽  
Vol 43 ◽  
pp. e12
Author(s):  
Eudes Antonio Costa ◽  
Deyfila Da Silva Lima ◽  
Élis Gardel da Costa Mesquita ◽  
Keidna Cristiane Oliveira Souza
Keyword(s):  
Positive Integer ◽  
Rational Number ◽  
Positive Rational Number ◽  
Powers Of 2

The digital roots S* (x), of a n positive integer is the digit 0 ≤ b ≤ 9 obtained through an iterative digit sum process, where each iteration is obtained from the previous result so that only the b digit remains. For example, the iterated sum of 999999 is 9 because 9 + 9 + 9 + 9 + 9 + 9 = 54 and 5 + 4 = 9. The sum of the digits of a positive integer, and even the digital roots, is a recurring subject in mathematical competitions and has been addressed in several papers, for example in Ghannam (2012), Ismirli (2014) or Lin (2016). Here we extend the application Sast to a positive rational number x with finite decimal representation. We highlight the following result: given a rational number x, with finite decimal representation, and the sum of its digits is 9, so when divided x by powers of 2, the number resulting also has the sum of its digits 9. Fact that also occurs when the x number is divided by powers of 5. Similar results were found when the x digit sum is 3 or 6.

scholarly journals Rational Associahedra and Noncrossing Partitions

10.37236/3432 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Drew Armstrong ◽  
Brendon Rhoades ◽  
Nathan Williams
Keyword(s):  
Simplicial Complex ◽  
Rational Number ◽  
Perfect Matchings ◽  
Lattice Paths ◽  
Noncrossing Partitions ◽  
Positive Rational Number ◽  
Dyck Paths ◽  
Narayana Numbers ◽  
Positive Integers ◽  
Coprime Positive Integers

Each positive rational number $x>0$ can be written uniquely as $x=a/(b-a)$ for coprime positive integers $0<a<b$. We will identify $x$ with the pair $(a,b)$. In this paper we define for each positive rational $x>0$ a simplicial complex $\mathsf{Ass}(x)=\mathsf{Ass}(a,b)$ called the rational associahedron.  It is a pure simplicial complex of dimension $a-2$, and its maximal faces are counted by the rational Catalan number $$\mathsf{Cat}(x)=\mathsf{Cat}(a,b):=\frac{(a+b-1)!}{a!\,b!}.$$The cases $(a,b)=(n,n+1)$ and $(a,b)=(n,kn+1)$ recover the classical associahedron and its "Fuss-Catalan" generalization studied by Athanasiadis-Tzanaki and Fomin-Reading.  We prove that $\mathsf{Ass}(a,b)$ is shellable and give nice product formulas for its $h$-vector (the rational Narayana numbers) and $f$-vector (the rational Kirkman numbers).  We define $\mathsf{Ass}(a,b)$ via rational Dyck paths: lattice paths from $(0,0)$ to $(b,a)$ staying above the line $y = \frac{a}{b}x$.  We also use rational Dyck paths to define a rational generalization of noncrossing perfect matchings of $[2n]$.  In the case $(a,b) = (n, mn+1)$, our construction produces the noncrossing partitions of $[(m+1)n]$ in which each block has size $m+1$.

scholarly journals Explicit Values for Ramanujan's Theta Function ϕ(q)

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Bruce C Berndt ◽  
Örs Rebák
Keyword(s):  
Theta Function ◽  
Rational Number ◽  
Theta Functions ◽  
Positive Rational Number ◽  
Lost Notebook ◽  
Ramanujan's Lost Notebook ◽  
Ramanujan’S Theta Function ◽  
First Descriptions

This paper provides a survey of particular values of Ramanujan's theta function $\varphi(q)=\sum_{n=-\infty}^{\infty}q^{n^2}$, when $q=e^{-\pi\sqrt{n}}$, where $n$ is a positive rational number. First, descriptions of the tools used to evaluate theta functions are given. Second, classical values are briefly discussed. Third, certain values due to Ramanujan and later authors are given. Fourth, the methods that are used to determine these values are described. Lastly, an incomplete evaluation found in Ramanujan's lost notebook, but now completed and proved, is discussed with a sketch of its proof.

A poor person's approximation to π

2012 ◽  
Vol 96 (537) ◽  
pp. 408-414
Author(s):  
Daniel Shiu ◽  
Peter Shiu
Keyword(s):  
Real Number ◽  
Efficient Method ◽  
Rational Number ◽  
Existence And Uniqueness ◽  
Rational Numbers ◽  
Decimal Expansion ◽  
Arbitrary Length ◽  
Positive Rational Number ◽  
Computing Machine

Suppose that we have a computing machine that can only deal with the operations of addition, subtraction and multiplication, but not division, of integers. Can such a machine be used to find the decimal expansion of a given real number α to an arbitrary length? The answer is ‘yes’, at least in the sense that α is the limit of a sequence of rational numbers, and one can obtain the decimal expansion of a positive rational number a/b without division. For example, for any positive k, we try all non-negative c < b and d < 10ka, and see if 10ka = db + c. There will be success because all we are doing is reducing 10ka modulo b in a particularly moronic way. This guarantees the existence and uniqueness of d, and d/10k is then the desired expansion. Take, for example, = and k = 12; then, after a tedious search, we should find that, for c = 3, there is d = 230 769 230 769 because 3 × 1012 = 230769230769 × 13 + 3, so that = 0.230769230769 + × 10-12. Such a procedure of searching for c, d is hopelessly inefficient, of course, and a more efficient method is given in the next section.

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