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

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.

Factorization invariants of the additive structure of exponential Puiseux semirings

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.

Soma iterada de algarismos de um número racional

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.

On computing the Lyapunov exponents of reversible cellular automata

We 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 .

Positive Rational Number of the Form (Equation)

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.

Ramanujan’s function k(τ)=r(τ)r 2(2τ) and its modularity

We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

Sums of digits in q-ary expansions

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.

Ramification theory and perfectoid spaces

Let $K_1$ 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.

Rational Associahedra and Noncrossing Partitions

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 $m+1$.