Rational Numbers Research Articles

Reordered Computable Numbers

AbstractA real number is called left-computable if there exists a computable increasing sequence of rational numbers converging to it. In this article we are investigating a proper subset of the left-computable numbers. We say that a real number x is reordered computable if there exist a computable function $$f :\mathbb {N} \rightarrow \mathbb {N}$$ f : N → N with $$\sum _{k=0}^{\infty } 2^{-f(k)} = x$$ ∑ k = 0 ∞ 2 - f ( k ) = x and a bijective function $$\sigma :\mathbb {N} \rightarrow \mathbb {N}$$ σ : N → N such that the rearranged series $$\sum _{k=0}^{\infty } 2^{-f(\sigma (k))}$$ ∑ k = 0 ∞ 2 - f ( σ ( k ) ) converges computably. In this article we will give some examples and counterexamples for reordered computable numbers and we will show that these numbers are closed under addition, multiplication and the Solovay reduction. Finally, we will also present a density theorem for reordered computable numbers.

Irrationality and transcendence of infinite series of rational terms

The aim of this paper is to prove two results concerning the irrationality and transcendence of a certain class of infinite series which consist of rational numbers and converge very quickly.

Notes on Characterizations of 2d Rational SCFTs: Algebraicity, Mirror Symmetry, and Complex Multiplication

AbstractS. Gukov and C. Vafa proposed a characterization of rational superconformal field theories (SCFTs) in dimensions with Ricci‐flat Kähler target spaces in terms of the Hodge structure of the target space, extending an earlier observation by G. Moore. The idea is refined, and a conjectural statement on necessary and sufficient conditions for such SCFTs to be rational is obtained, which is indeed proven to be true in the case the target space is . In the refined statement, the algebraicity of the geometric data of the target space turns out to be essential, and the Strominger–Yau–Zaslow fibration in the mirror correspondence also plays a vital role.

Birational Automorphism Groups of Severi–Brauer Surfaces Over the Field of Rational Numbers

Birational Automorphism Groups of Severi–Brauer Surfaces Over the Field of Rational Numbers

Decompositions of the positive real numbers into disjoint sets closed under addition and multiplication

The general problem studied in this paper is to decompose the positive elements R+ of a totally ordered ring R into the disjoint union of sets that are closed under addition and multiplication. We mainly investigate the case when R is a subfield of the reals R. We prove that for any n∈N the positive real numbers can be decomposed into n disjoint pieces which are also closed under addition and multiplication. We construct infinitely many different n-decompositions (n∈N) for all fields containing at least two algebraically independent transcendental numbers. We characterise all possible decompositions into two pieces for fields of transcendence degree one over Q. Further, we prove that the positive elements of real algebraic extensions of the rational numbers are indecomposable into finitely many pieces.

Stripe Embedding: Efficient Maps with Exact Numeric Computation

We consider the fundamental problem of injectively mapping a surface mesh with disk topology onto a boundary constrained convex domain. We start from the basic observation that mapping a strip of triangles onto a rectangular shape always yields a valid embedding, if the vertices that bound the strip are sorted coherently along the sides of the rectangle. Based on this intuition, we propose a straightforward algorithm, called Stripe Embedding, that operates by decomposing the input mesh into a set of triangle strips and then embeds each strip into the target domain by means of linear interpolation between two previously embedded vertices. Thanks to its simplicity, Stripe Embedding is extremely efficient and permits to switch to an exact implementation without almost increasing its running times. Stripe Embedding is up to three orders of magnitude faster than the Tutte embedding for same numerical model and, even when implemented with costly rational numbers, it is faster than any floating point implementation of prior methods at any scale.

A note on the large-c conformal block asymptotics and α-heavy operators

We consider α-heavy conformal operators in CFT2 which dimensions grow as h=O(cα) with α being non-negative rational number and conjecture that the large-c asymptotics of the respective 4-point Virasoro conformal block is exponentiated similar to the standard case of α=1. It is shown that the leading exponent is given by a Puiseux polynomial which is a linear combination of power functions in the central charge with fractional powers decreasing from α to 0 according to some pattern. Our analysis is limited by considering the first six explicit coefficients of the Virasoro block function in the coordinate. For simplicity, external primary operators are chosen to be of equal conformal dimensions that, therefore, includes the case of the vacuum conformal block. The consideration is also extended to the 4-point W3 conformal block of four semi-degenerate operators, in which case the exponentiation hypothesis works the same way. Here, only the first three block coefficients can be treated analytically.

Enumerating the Rationals

Summary We introduce a new binary tree that enumerates the rational numbers in the interval ( 0 , 1 ) and provide historical context.

An interesting class of exact solutions to Binet's equation, with references to Newton's and Einstein's theories of gravity

Abstract To adequately describe the motion of a particle in a central force field, one has to specify the apsidal angle (α). The particle's trajectory can only be truly periodic (closed), if α/π is a rational number. For example, F∼1/r^2 then α=π, and the trajectory is an ellipse. This is the simplest case when the trajectory equation (Binet's equation) is exactly solvable. In this article, bounded motion in a two-component central potential field V(r)=-a_1/r-a_2/r^2 will be examined and illustrated. This case is more complex, but the corresponding Binet's equation is still exactly solvable. Moreover, the perturbational solution to the relativistic trajectory equation is, in fact, the exact solution for a force field of the same type. It means that the relativistic force F∼1/r^4 is formally replaced with an effective F∼1/r^3 force. As will be shown, this differently interpretable solution gives a very accurate prediction to the anomalous shift of Mercury's perihelion.

Galois actions on Tate modules of Abelian varieties with semi-stable reduction

Let p be a rational prime number, let K denote a finite extension of Qp, K‾ some fixed algebraic closure of K. Let GK be the absolute Galois group of K and let IK⊂GK be its inertial subgroup. Let A be an Abelian variety defined over K, with semi-stable reduction. In this note, we give a criterion for which Vp(A)IK=0, where Vp(A) is the p-adic Tate module associated to A.

Universal quadratic forms and Northcott property of infinite number fields

AbstractWe show that if a universal quadratic form exists over an infinite degree, totally real extension of the field of rationals , then the set of totally positive integers in the extension does not have the Northcott property. In particular, this implies that no universal form exists over the compositum of all totally real Galois fields of a fixed prime degree over . Further, by considering the existence of infinitely many square classes of totally positive units, we show that no classical universal form exists over the compositum of all such fields of degree (for each fixed odd integer ).

Identifying Key Instructional Practices in Rational Number Interventions: A Systematic Review and Meta-Analysis

Impacts of rational number interventions among U.S. students in Grades 3 through 9 with mathematics difficulties are examined using a systematic review and meta-analysis. The primary goal of the meta-analysis was to identify instructional practices that are key drivers of student impacts. From approximately 1,200 published and unpublished study records, we identified 28 studies that met our inclusion criteria and coded the interventions for their instructional practices, intervention characteristics, and study design features. The random-effects mean effect size across all 28 studies (90 effect sizes) was 0.68 ( SE = 0.08, p < .001, 95% confidence interval [CI]: [0.51, 0.85]). The 95% prediction interval was −0.36 to 1.8. Using meta-regression techniques, we found the teaching of mathematical language ( β = 0.50) and the use of the number line ( β = 0.47) during intervention to be significantly associated with positive impacts when adjusted for controls. We discuss implications for intervention practice and study limitations along with the challenges of examining complex, multifaceted interventions.

Moment evolution equations for rational random dynamical systems: an increment decomposition method

Abstract Statistical moments are commonly used tools for exploring the ensemble behavior in gene regulation and population dynamics, where the rational vector fields are particularly ubiquitous, but how to efficiently derive the corresponding moment evolution equations was not much involved. Traditional derivation methods rely on fractional reduction and Itô formula, but it may become extremely complicated if the vector field is described by multivariate fractional polynomials. To resolve this issue, we present a novel incremental decomposition method, by which the rational vector field is divided into two parts: (proper) fractional polynomials and non-fractional polynomials. For the non-fractional polynomial part, we deduce the variation rate of a statistical moment by the Itô formula, but for the fractional polynomial part we acquire the corresponding variation rate by a relation analogous to that between the moment generating function and the distinct statistical moments. As application of the novel technique, the resultant infinite-dimensional moment systems associated with two typical examples are truncated with the schemes of derivative matching closure and the Gaussian moment closure. By comparing the lower-order statistical moments obtained from the closed moment systems with the counterparts obtained from direct simulation, the correctness of the proposed technique is verified. The present study is significant in facilitating the development of moment dynamics towards more complex systems.

Infinite temperature spin dc conductivity of the spin-1/2 XXZ chain

Abstract Using the Bethe ansatz method and the thermodynamic Bethe ansatz equations for the higher-spin integrable XXZ chain, the regular zero frequency contribution to the spin current correlation (spin dc conductivity) is analyzed for the spin-1/2 XXZ chain with the anisotropy 0 ⩽ Δ < 1 . In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies and exactly evaluate the spin dc conductivity at zero magnetic field, which is proportional to the inverse temperature β at the β → 0 limit. We find that the high temperature proportionality constant σ 0 reg of the spin dc conductivity is discontinuous at all rational numbers of the anisotropy parameter p 0 = π / cos − 1 ⁡ Δ ⩾ 2 with the gap increasing larger than the second power of growing magnetization on one quasi-particle. The isotropic Δ = 1 point is exceptional. Close to this point, σ 0 reg slowly increases in proportion to the first power of the one quasi-particle magnetization. On the other hand, σ 0 reg is proportional to the second power of the same when p 0 approaches irrational numbers.

Translation lengths in crossing and contact graphs of quasi-median graphs

Given a quasi-median graph X , the crossing graph \Delta X and the contact graph \Gamma X are natural hyperbolic models of X . In this article, we show that the asymptotic translation length in \Delta X or \Gamma X of an isometry of X is always rational. Moreover, if X is hyperbolic, these rational numbers can be written with denominators bounded above uniformly; this is not true in full generality. Finally, we show that, if the quasi-median graph X is constructible in some sense, then there exists an algorithm computing the translation length of every computable isometry. Our results encompass contact graphs in CAT(0) cube complexes and extension graphs of right-angled Artin groups.

Diophantine approximation by rational numbers of certain parity types

AbstractFor a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd, and odd/even rational numbers. We study algorithms to find best approximations by rational numbers of given parities and compare these algorithms with continued fraction expansions.

Gamification of Al-Khawarizmi Number Basis Module for Gifted and Talented Muslim Students

There are a few bright and intelligent students who have a poor level of achievement in mathematics. They think that mathematics is a boring subject and not important in daily life. A special module named the Al-Khawarizmi Number Basis Module will be used as an intervention step for students who need this guidance and support to achieve excellence in mathematics, increase their level of understanding of professional mathematics, and give encouragement to Muslim students in cultivating a mathematical life. The main objective of this research is to develop a game card for gifted and talented Muslim students at Kolej PERMATA Insan, Universiti Sains Islam Malaysia. This module helps students enhance their understanding and achievement in mathematics. This module contains three main topics, namely rational numbers, factors and multiples, and standard form. The effectiveness of this module needs to be evaluated and analyzed carefully. This way, the module can improve the level of achievement and understanding of gifted and talented Muslim students, which can then be shared with students that are in mainstream schools. In completing this module, students are also tested with a game card called Find and Match: Al-Khawarizmi Robot of Number (Riz and Ron). Then, students will undergo a pre-test and post-test to assess their level of achievement. The implementation of this module can increase understanding and strengthen the professional knowledge of mathematics among gifted and talented Muslim students and encourage students to cultivate mathematical knowledge throughout life.

Transabdominal Ultrasound of the Stomach in Patients with Functional Dyspepsia: A Review.

Background/Aim: Dyspepsia is a very common condition worldwide and has an immense impact on quality of life. Functional dyspepsia (FD) is defined by dyspeptic symptoms with the absence of any structural abnormality that can explain the cause. Ultrasonography (US) is a non-invasive imaging modality that can be applied to assess gastric function. The aim of this review paper is to assess how ultrasonography can sort out patients with dyspepsia and help diagnose functional dyspepsia. Methods: Using the keywords "functional dyspepsia" and "ultrasonography", the PubMed database was screened for publications on the use of ultrasonography to identify functional dyspepsia. Afterward, two screening processes were performed to narrow the articles down to a rational number. Results: A total of 169 articles were obtained from the literature search, and 31 of these were included after the screening process. Ultrasonography was capable of identifying functional dyspepsia pathophysiology using both two-dimensional (2D) and three-dimensional (3D) ultrasound. Conclusions: We conclude that ultrasonography is a non-invasive, safe, low-cost, and widely accessible method that can help diagnose functional dyspepsia through the exclusion of organic dyspepsia and assessing FD pathophysiology. Incorporation of ultrasound in the work-up of patients with functional dyspepsia allows for a sound diagnostic approach and can further improve patient management.

Yukawa couplings at infinite distance and swampland towers in chiral theories

We study limits of vanishing Yukawa couplings of 4d chiral matter fields in Quantum Gravity, using as a laboratory type IIA orientifolds with D6-branes. In these theories chiral fermions arise at brane intersections, where an infinite tower of charged particles dubbed gonions are localised. We show that in the limit Y → 0 some of these towers become asymptotically massless, while at the same time the kinetic term of some chiral fields becomes singular and at least two extra dimensions decompactify. For limits parametrised by a large complex structure saxion u, Yukawa couplings have a behaviour of the form Y ~ 1/ur, with r some positive rational number. Moreover, in this limit some of the gauge couplings associated to the Yukawa vanish. The lightest gonion scales are of order mgon ~ gsMP with s > 1, verifying the magnetic WGC with room to spare and with no need of its tower/sublattice versions. We also show how this behaviour can be understood in the context of the emergence of kinetic terms in Quantum Gravity. All these results may be very relevant for phenomenology, given the fact that some of the Yukawa couplings in the Standard Model are very small.

Combining Precision Boosting with LP Iterative Refinement for Exact Linear Optimization

This article studies a combination of the two state-of-the-art algorithms for the exact solution of linear programs (LPs) over the rational numbers in practice, that is, without any roundoff errors or numerical tolerances. By integrating the method of precision boosting inside an LP iterative refinement loop, the combined algorithm is able to leverage the strengths of both methods: the speed of LP iterative refinement, in particular, in the majority of cases when a double-precision floating-point solver is able to compute approximate solutions with small errors, and the robustness of precision boosting whenever extended levels of precision become necessary. We compare the practical performance of the resulting algorithm with both pure methods on a large set of LPs and mixed-integer programs (MIPs). The results show that the combined algorithm solves more instances than a pure LP iterative refinement approach while being faster than pure precision boosting. When embedded in an exact branch-and-cut framework for MIPs, the combined algorithm is able to reduce the number of failed calls to the exact LP solver to zero while maintaining the speed of the pure LP iterative refinement approach. History: Accepted by Antonio Frangioni, Area Editor for Design and Analysis of Algorithms: Continuous. Funding: The work for this article has been conducted within the Research Campus Modal funded by the German Federal Ministry of Education and Research (BMBF) [Grants 05M14ZAM and 05M20ZBM].