Rational Numbers Research Articles

The ethics of research informed consent from the Kyrgyz perspective: A qualitative study.

To ensure informed consent is tailored to ethnic Asian communities, it is necessary to establish an ethical foundation that is relevant to the specific populations. We hypothesized that certain communitarian factors unique to traditional Kyrgyz culture may influence an individual's decision to participate in research. Guided by Seedhouse's (2005) Rational Field Theory, we conducted qualitative, in-depth interviews with cultural experts in Kyrgyzstan to identify the ethical foundations of decision-making for informed consent in Kyrgyz culture. The results indicate that Kyrgyz people have a distinctive decision-making style influenced by their nomadic culture and history, which values and prioritizes family integrity and reputation. These findings indicate that a multidimensional approach based on socio-cultural sensitivities is necessary to assess the appropriateness of consent procedures. We believe our results may have implications for revising the guidelines of local and regional research ethics committees in Kyrgyzstan and other Central Asian countries.

Rationality in four dimensions

By leveraging the physics of the Higgs branch, we argue that the conformal central charges a and c of an arbitrary 4D N=2 superconformal field theory (SCFT) are rational numbers. Our proof of the rationality of c is conditioned on a well-supported conjecture about how the Higgs branch of an SCFT is encoded in its protected chiral algebra. To establish the rationality of a, we further rely on a widely believed technical assumption on the high-temperature limit of the superconformal index. Published by the American Physical Society 2024

On the path homology of Cayley digraphs and covering digraphs

We develop a theory of covering digraphs, similar to the theory of covering spaces. By applying this theory to Cayley digraphs, we build a “bridge” between GLMY-theory and group homology theory, which helps to reduce path homology calculations to group homology computations. We show some cases where this approach allows us to fully express path homology in terms of group homology. To illustrate this method, we provide a path homology computation for the Cayley digraph of the additive group of rational numbers with a generating set consisting of inverses to factorials. The main tool in our work is a filtered simplicial set associated with a digraph, which we call the filtered nerve of a digraph.

On the generalized interlacing property for the zeros of Bessel functions

This paper investigates a generalized interlacing property between Bessel functions, particularly Jν and Jμ, where the difference |ν−μ| exceeds 2. This interlacing phenomenon is marked by a compensatory interaction with the zeros of Lommel polynomials, extending our understanding beyond the traditional |ν−μ|≤2 regime. The paper extends the generalized interlacing property to cylinder functions and the derivative of Bessel functions, as an application. It is also discussed that Siegel's extension of Bourget hypothesis to rational numbers of ν cannot be further improved to arbitrary real numbers.

Square Root Expressions and Irrational Numbers in Middle School Mathematics Textbooks Throughout the History of the Republic

Irrational numbers, a fundamental set that extends rational numbers to real numbers, have been a subject of limited study in the context of textbook analyses. This study, therefore, examines how square root expressions and irrational numbers have been addressed in middle school mathematics textbooks throughout the history of the Republic of Türkiye. The research was conducted using a comprehensive approach, with qualitative research methods and content analysis was applied to 37 mathematics textbooks (seven 5th grade, twelve 6th grade, nine 7th grade, and nine 8th grade) from 1932 to 2022. Data were collected from mathematics textbooks available at the Ferit Ragıp Tuncor Archive and Documentation Library affiliated with the Ministry of National Education. The results of this study indicate that changes in mathematics curricula are reflected in textbooks, and this reflection is directly influenced by the learning theory adopted in the curriculum. The study also provides insights into how irrational numbers are introduced in textbooks based on the sequence of topics, and recommendations are made based on the findings.

Promoting 5th Grade Students’ Spontaneous Focusing on Quantitative Relations

Spontaneous focusing on quantitative relations (SFOR) is related to the development of rational number knowledge and predicts future mathematical knowledge. However, scarce evidence exists on whether and how SFOR tendency can be supported with classroom instruction. In this quasi-experimental study, 306 5th grade students participated in one of two four-lesson interventions. We investigated the relative effectiveness of both interventions on students’ SFOR tendency and their knowledge of multiplicative relations and fractions. Students in the Spot the Relations intervention (n = 142) completed activities that supported their ability to recognize and describe multiplicative relations in their everyday surroundings. Students in the Fraction Estimation Game intervention (n = 164) trained with digital game-based number line estimation tasks that supported their fraction knowledge. The results revealed that students in the first group showed larger gains in their multiplicative relations knowledge and SFOR tendency, whereas those in the second group showed larger gains in their fraction knowledge from pre- to posttest. Although SFOR tendency was supported by both interventions over time, the effect was larger for Spot the Relations, which used instructional practices that supported students’ multiplicative knowledge and mathematical language and made multiplicative relations relevant and observable parts of the world.

ПАКЕТ GAP ДЛЯ РОЗЩЕПЛЕННЯ ЛІНІЙНИХ СИСТЕМ

There are a number of problems, the solution of which requires the breakdown of the initial system of equations using algebraic decoupling methods. This means reducing the matrix of coefficients to block-diagonal (or block-triangular) form by means of substitution of variables. The main computational tasks when using such methods are finding the centralizer of several matrices or compiling the algebra generated by these matrices. For calculations, it is convenient to use the GAP computer algebra system because the system itself is designed for discrete algebra calculations. The problem is that the GAP program does not support calculations with real numbers. For practical problems you can try to replace them (with some accuracy) by rational numbers. At the same time, the decision may turn out to be excessively cumbersome. On the other hand, the advantage of GAP is the complete absence of rounding errors.

A note on the two variable Artin's conjecture

In 1927, Artin conjectured that any integer a which is not −1 or a perfect square is a primitive root for a positive density of primes p. While this conjecture still remains open, there has been a lot of progress in last six decades. In 2000, Moree and Stevenhagen proposed what is known as the two variable Artin's conjecture and proved that for any multiplicatively independent rational numbers a and b, the set{p⩽x:p prime, mamodp∈〈b〉modp} has positive density under the Generalised Riemann Hypothesis for certain Dedekind zeta functions. While the infinitude of such primes is known, the only unconditional lower bound for the size of the above set is due to Ram Murty, Séguin and Stewart who in 2019 showed that for infinitely many pairs (a,b)#{p⩽x:p prime, mamodp∈〈b〉modp}≫xlog2⁡x. In this paper we improve this lower bound. In particular we show that given any three multiplicatively independent integers S={m1,m2,m3} such thatm1,m2,m3,−3m1m2,−3m2m3,−3m1m3,m1m2m3 are not squares, there exists a pair of elements a,b∈S such that#{p⩽x:p prime, mamodp∈〈b〉modp}≫xlog⁡log⁡xlog2⁡x. Further, under the assumption of a level of distribution greater than x23 in a theorem of Bombieri, Friedlander and Iwaniec (as modified by Heath-Brown), we prove the following conditional result. Given any two multiplicatively independent integers S={m1,m2} such thatm1,m2,−3m1m2 are not squares, there exists a pair of elements a,b∈{m1,m2,−3m1m2} such that#{p⩽x:p prime, mamodp∈〈b〉modp}≫xlog⁡log⁡xlog2⁡x.

Gabor frame multipliers and Parseval duals on the half real line

Recently, Gabor analysis on locally compact abelian (LCA) groups has become the focus of an active research. In practice, the time variable cannot be negative. The half real line [Formula: see text] is an LCA group under multiplication and the usual topology, with the Haar measure [Formula: see text]. This paper addresses Gabor frame multipliers and Parseval duals for [Formula: see text]. We introduce and characterize Gabor frame multipliers and Parseval Gabor frame multipliers based on Zak transform matrices. Our Zak transform matrix is essentially different from the conventional Zibulski–Zeevi matrix. It allows us to define Gabor frame generators by designing suitable matrix-valued functions of finite size. We also prove that an arbitrary Gabor frame [Formula: see text] admits a Parseval dual frame/tight dual frame whenever [Formula: see text] are rational numbers not greater than [Formula: see text].

Rotation number of 2-interval piecewise affine maps

AbstractWe study maps of the unit interval whose graph is made up of two increasing segments and which are injective in an extended sense. Such maps $$f_{\varvec{p}}$$ f p are parametrized by a quintuple $$\varvec{p}$$ p of real numbers satisfying inequations. Viewing $$f_{\varvec{p}}$$ f p as a circle map, we show that it has a rotation number $$\rho (f_{\varvec{p}})$$ ρ ( f p ) and we compute $$\rho (f_{\varvec{p}})$$ ρ ( f p ) as a function of $$\varvec{p}$$ p in terms of Hecke–Mahler series. As a corollary, we prove that $$\rho (f_{\varvec{p}})$$ ρ ( f p ) is a rational number when the components of $$\varvec{p}$$ p are algebraic numbers.

Rational numbers with small denominators in short intervals

We use bounds on bilinear forms with Kloosterman fractions and improve the error term in the asymptotic formula of Balazard and Martin (Bull Sci Math 187:Art. 103305, 2023) on the average value of the smallest denominators of rational numbers in short intervals.

GENERALISED QUADRATIC FORMS OVER TOTALLY REAL NUMBER FIELDS

Abstract We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version of the Hardy–Littlewood circle method over number fields.

Navigating Number Systems for Computational Precision: An Analytical Exploration

The main objective of this article is to study all numbers that have occurred so far in the theoretical discussions were real (or complex) numbers in the strict mathematical sense. That is, they were to be conceived as infinite decimal fractions, or as Dedekind cuts. For the purposes of computation such numbers have to be approximated by real numbers of a rather special type, such as terminating decimal fractions, or other rational numbers. The present article is devoted to a study of the number systems that can be used for the purpose of computation.

Unit fractions with shifted prime denominators

We prove that any positive rational number is the sum of distinct unit fractions with denominators in $\{p-1 : p\textrm { prime}\}$ . The same conclusion holds for the set $\{p-h : p\textrm { prime}\}$ for any $h\in \mathbb {Z}\backslash \{0\}$ , provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.

Root numbers of a family of elliptic curves and two applications

For each t∈Q∖{−1,0,1}, define an elliptic curve over Q by Et:y2=x(x+1)(x+t2).Using a formula for the root number W(Et) as a function of t and assuming some standard conjectures about ranks of elliptic curves, we determine (up to a set of density zero) the set of isomorphism classes of elliptic curves E/Q whose Mordell-Weil group contains Z×Z/2Z×Z/4Z, and the set of rational numbers that can be written as a product of the slopes of two rational right triangles.

Dynamic Tangled Derivative Logic of Metric Spaces

Dynamical systems are abstract models of interaction between space and time. They are often used in fields such as physics and engineering to understand complex processes, but due to their general nature, they have found applications for studying computational processes, interaction in multi-agent systems, machine learning algorithms and other computer science related phenomena. In the vast majority of applications, a dynamical system consists of the action of a continuous `transition function' on a metric space. In this work, we consider decidable formal systems for reasoning about such structures. Spatial logics can be traced back to the 1940's, but our work follows a more dynamic turn that these logics have taken due to two recent developments: the study of the topological mu-calculus, and the the integration of linear temporal logic with logics based on the Cantor derivative. In this paper, we combine dynamic topological logics based on the Cantor derivative and the `next point in time' operators with an expressively complete fixed point operator to produce a combination of the topological mu-calculus with linear temporal logic. We show that the resulting logics are decidable and have a natural axiomatisation. Moreover, we prove that these logics are complete for interpretations on the Cantor space, the rational numbers, and subspaces thereof.

Buckets of fun: Impacts of fraction ball activities on students' math-related emotions

Rational number learning can cause frustration and negative emotions for elementary school students. Fraction Ball, a play-based math intervention, allows students to actively learn rational numbers through engaging and interactive activities. Based on a cluster-randomized trial with 16 teachers and 360 students, our pre-registered analyses showed moderate positive impacts of Fraction Ball on overall students' self-reported math-related emotions. Exploratory analyses indicated that students with higher negative emotions at pretest showed larger experimental impacts on decreasing negative emotions at posttest. Finally, we found that Fraction Ball evidenced no trade-off between rational number learning and emotional outcomes at the classroom block level, indicating that positive learning gains in rational number skills were associated with increases in positive emotions and decreases in negative emotions.

Generalized Matrix Spectral Factorization with Symmetry and Construction of Quasi-Tight Framelets over Algebraic Number Fields

The rational field Q is highly desired in many applications. Algorithms using the rational number field Q algebraic number fields use only integer arithmetics and are easy to implement. Therefore, studying and designing systems and expansions with coefficients in Q or algebraic number fields is particularly interesting. This paper discusses constructing quasi-tight framelets with symmetry over an algebraic field. Compared to tight framelets, quasi-tight framelets have very similar structures but much more flexibility in construction. Several recent papers have explored the structure of quasi-tight framelets. The construction of symmetric quasi-tight framelets directly applies the generalized spectral factorization of 2×2 matrices of Laurent polynomials with specific symmetry structures. We adequately formulate the latter problem and establish the necessary and sufficient conditions for such a factorization over a general subfield F of C, including algebraic number fields as particular cases. Our proofs of the main results are constructive and thus serve as a guideline for construction. We provide several examples to demonstrate our main results.

Nearly computable real numbers

We call a sequence ( a n ) n of elements of a metric space nearly computably Cauchy if for every increasing computable function r : N → N the sequence ( d ( a r ( n + 1 ) , a r ( n ) ) ) n converges computably to 0. We show that there exists an increasing sequence of rational numbers that is nearly computably Cauchy and unbounded. Then we call a real number α nearly computable if there exists a computable sequence ( a n ) n of rational numbers that converges to α and is nearly computably Cauchy. It is clear that every computable real number is nearly computable, and it follows from a result by Downey and LaForte (Theoretical Computer Science 284 (2002) 539–555) that there exists a nearly computable and left-computable number that is not computable. We observe that the set of nearly computable real numbers is a real closed field and closed under computable real functions with open domain, but not closed under arbitrary computable real functions. Among other things we strengthen results by Hoyrup (Theory of Computing Systems 60 (2017) 396–420) and by Stephan and Wu (In New computational paradigms. First conference on computability in Europe, CiE 2005, Proceedings (2005) 461–469 Springer) by showing that any nearly computable real number that is not computable is weakly 1-generic (and, therefore, hyperimmune and not Martin-Löf random) and strongly Kurtz random (and, therefore, not K-trivial), and we strengthen a result by Downey and LaForte (Theoretical Computer Science 284 (2002) 539–555) by showing that no promptly simple set can be Turing reducible to a nearly computable real number.

ON THE SET OF KRONECKER NUMBERS

Abstract A positive even number is said to be a Maillet number if it can be written as the difference between two primes, and a Kronecker number if it can be written in infinitely many ways as the difference between two primes. It is believed that all even numbers are Kronecker numbers. We study the division and multiplication of Kronecker numbers and show that these numbers are rather abundant. We prove that there is a computable constant k and a set D consisting of at most 720 computable Maillet numbers such that, for any integer n, $kn$ can be expressed as a product of a Kronecker number and a Maillet number in D. We also prove that every positive rational number can be written as a ratio of two Kronecker numbers.