Rational Numbers Research Articles

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.

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 From approximately 1,200 published and unpublished study records, we identified 28 studies that met our inclusion criteria and coded the interventions for their instructional practices, intervention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].

Multivariate to bivariate reduction for noncommutative polynomial factorization

Based on Bergman's theorem, we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely,1.Given an n-variate noncommutative polynomial f∈F〈X〉 over a field F as an arithmetic circuit, computing a complete factorization of f into irreducible factors is deterministic polynomial-time reducible to factorization of a noncommutative bivariate polynomial g∈F〈x,y〉; the reduction transforms f into a circuit for g, and given a complete factorization of g, the reduction recovers a complete factorization of f in polynomial time.The reduction works both in the white-box and the black-box setting.2.We show over the field of rationals that bivariate linear matrix factorization problem for 4×4 matrices is at least as hard as factoring square-free integers and for 3×3 matrices it is in polynomial time.

Closure property and cognitive biases

In this note, we discuss two specific cognitive biases towards closure property as a learning obstacle in school mathematics in the context of number sets. These biases are related to rational and irrational numbers, two important and challenging issues from an educational point of view. The first bias concerns incorrect application of rules and properties of closure property in rational numbers to irrational numbers, and the second deals with the improper generalisation of the closure property in finite states to infinite states. These biases indicate the importance of teaching and learning the closure property.

On products of idempotents and nilpotents

This article studies the ring structure arising from products of idempotents and nilpotents. Thus the argument is concerned essentially with the one-sided IQNN property of rings. We first prove that if the $2$ by $2$ full matrix ring over a principal ideal domain $F$ of characteristic zero is right IQNN then $F$ contains infinitely many non-integer rational numbers; and that the concepts of right IQNN and right quasi-Abelian are independent of each other. We next introduce a ring property, called {\it right IAN}, as a generalization of both right IQNN and right quasi-Abelian; and provide several kinds of methods to construct right IAN rings. In the procedure, we also show that the right IQNN and right IAN do not go up to polynomial rings.

Rational Functions as Sums of Reciprocals of Polynomials

An algorithm is given to express a proper rational function (i.e., the degree of the denominator exceeds that of the numerator) over any field as an alternating sum of reciprocals of polynomials of increasing degree. This is analogous to existing algorithms for representing proper rational numbers.

String art with repeating decimals

Rational numbers are typically represented as decimals or fractions. This paper explores the artistic potential of these representations, focusing on converting decimal expansions of rational numbers into visual forms via string art. By dividing a circle into 10 equal parts numbered from zero to nine, repeating decimals can be transformed into visual paths defined by the sequence of digits. This approach was used to create a pair of artworks and several interactive GeoGebra animations. The visual appeal of the designs is complemented by a mathematical explanation using number theory results.

Experimental study on coherent structures by small particles suspended in high aspect-ratio (Γ=\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma =$$\\end{document} 2.5) thermocapillary liquid bridges of high Prandtl number

In time-dependent traveling-wave convection, it has been confirmed that small particles of low Stokes number (St≪\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{St}\\ll$$\\end{document} 1) form three-dimensional closed structures known as the particle accumulation structures (PASs). We focus on coherent structures formed in high aspect-ratio thermocapillary liquid bridges (Γ=\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma =$$\\end{document} 2.5), where oscillatory convection due to the so-called hydrothermal wave instability of mHTW=\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_{\ extrm{HTW}}=$$\\end{document} 1 stably emerges, and carry out experimental explorations into the volume ratio dependence of the emergence of coherent structures. Variation in the volume ratio induces the presence of coherent structures with different azimuthal wave numbers, that is, different spatial structures. Among the particles added to the liquid bridge, we perform spatio-temporal tracking of particles to quantify the correlation between the behavior of particles and the spatial structures of coherent structures with different rational wave numbers.

A simpler elementary proof that e is irrational

In this short note I present an elementary proof of irrationality for the number e, the base of the natural logarithm. It is simpler than other known proofs as it does not use comparisons with geometric series, nor Beukers' integrals, and it does not assume that e is a rational number from the beginning.

Deviation analysis in transient response of fractional-order systems: An elementary function-based lower bound

In this paper, transient response of commensurate fractional-order systems with non-zero initial conditions is investigated in the viewpoint of the existence of deviation from initial condition. In particular, firstly peak effect for a class of linear fractional order systems is inspected by presenting a lower bound for their responses. Such a lower bound is described by an elementary function, where the commensurate order of the system is considered as a rational number. Furthermore, it has been demonstrated that under particular circumstances the derived lower bound can be extended to apply for deviation analysis in response of a class of fractional-order nonlinear systems. Moreover, included are various illustrative examples intended to assess the applicability of the obtained lower bound in prediction of the deviation value and the time instance at which the peak effect occurs.

Mis-Out and Mis-In Examples: The Case of Rational Numbers

AbstractThis paper focuses on the definitions and the mis-out and mis-in examples of rational numbers that four prospective elementary teachers presented while working on rational number assignments. The participants were first asked to respond, individually, to an Individual Rational Number Assignment, consisting of items aiming at detecting their personal concept definitions of rational numbers and identifying the entities that they regarded as rational numbers. Then, to share their work with another prospective teacher, to identify similarities and differences in their responses, and to list issues that were raised during the individual or pair work, that they would like to discuss in class. The data exposed a tendency to provide one definition of rational numbers, to identify the term “rational” with “natural”, not to include a clarification that a rational number is a number, and a controversy regarding including (or not including) a statement that b ≠ 0 in the definition. Other observations related to a tendency not to categorize negative numbers (and perhaps also zero) as rational numbers and an inconsistency between their responses to the question “what is a rational number?” and their classification of examples of rational numbers. Recommendations for topics for discussion with prospective teachers, in light of the responses to the assignments, are suggested and methodological issues for considerations are proposed.