Fraction Representation Research Articles

A renormalization scheme for semi-regular continued fractions

AbstractIn this article we study a renormalization scheme with which we find all semi-regular continued fractions of a number in a natural way. We define two maps, $$\hat{T}_{slow}$$ T ^ slow and $$\hat{T}_{fast}$$ T ^ fast : these maps are defined for $$(x,y)\in [0,1]$$ ( x , y ) ∈ [ 0 , 1 ] , where x is the number for which a semi- regular continued fraction representation is developed by $$\hat{T}_{slow}$$ T ^ slow according to the parameter y. The set of all possible semi-regular continued fraction representations of x are bijectively constructed as the parameter y varies. The map $$\hat{T}_{fast}$$ T ^ fast is a “sped up" version of the map $$\hat{T}_{slow}$$ T ^ slow , and we show that $$\hat{T}_{fast}$$ T ^ fast is ergodic with respect to a probability measure which is mutually absolutely continuous with Lebesgue measure. In contrast, $$\hat{T}_{slow}$$ T ^ slow preserves no such measure, but does preserve an infinite, $$\sigma $$ σ -finite measure mutually absolutely continuous with Lebesgue measure. Furthermore, we generate a sequence of substitutions which generate a symbolic coding of the orbit of y under rotation by x modulo one. In the last section we highlight how our scheme can be used to generate semi-regular continued fractions explicitly for specific continued fraction algorithms such as Nakada’s $$\alpha $$ α -continued fractions.

PROSES KOGNITIF SISWA SEKOLAH DASAR DALAM MEMAHAMI MATERI PECAHAN DITINJAU DARI STIFIN FRAMEWORK: THINKING DAN FEELING

AbstractThe understanding of students, their learning methods, and cognitive characteristics are important aspects of classroom learning management. Fractional math problems require reasoning and mathematical logic, so to complete the gap in previous research, namely how types of intelligence form patterns of student thought processes, this study examines the level of student understanding and problem-solving strategies for fractions based on STIFin intelligence characteristics. This research uses a mixed-method approach in data collection. The research instrument uses the Fraction Sense Test (FST) and interviews and STIFin tests. The study involved 16 fourth-grade students of SD 47 Kota Jambi. The results showed that students with the thinking intelligence type scored the highest compared to sensing, intuiting, and feeling. Furthermore, there are four strategies students use in developing problem-solving strategies for fractions: (1) comparing fractions using other simple benchmark fractions like ½, (2) comparing fractions based on the denominator and numerator, (3) creating a fraction representation in the form of flat shapes to understand and compare fractions.

Correction: Continued fraction representation of the generating function of Bernoulli polynomials

Correction: Continued fraction representation of the generating function of Bernoulli polynomials

Solving the membership problem for certain subgroups of [formula omitted]

For positive integers u and v, let Lu=[10u1] and Rv=[1v01]. Let Gu,v be the group generated by Lu and Rv. In a previous paper, the authors determined a characterization of matrices M=[abcd] in Gu,v when u,v≥3 in terms of the short continued fraction representation of b/d. We extend this result to the case where uv≥5. Additionally, we compute [Gu,v:Gu,v] where Gu,v={[1+uvn1vn2un31+uvn4]∈SL2(Z):(n1,n2,n3,n4)∈Z4} for u,v≥1, extending a result of Chorna, Geller, and Shpilrain and solving the membership problem for Gu,v with u,v≥1.

P System with Fractional Reduction

Membrane computing is a branch of natural computing, which is a new computational model abstracted from the study of the function and structure of living biological cells. The study of numerical computation based on membrane computation has received increasing attention in recent years, where maximum parallelism in the execution of evolutionary rules plays an important role in improving the efficiency of numerical computation. Numbers in numerical computation are usually represented as decimals or fractions, and this paper investigates the fundamental problem in fraction representation and operations—fraction simplification. By improving the parallelization of two traditional fractional reduction algorithms, we design the corresponding fractional reduction class cells P System Π1 and P System Π2. Combining these two P Systems, this paper designs P System Π3. The feasibility and effectiveness of the P System designed in this paper are verified experimentally with the simulation software UPSimulator, and the characteristics and application scenarios of the three P Systems are analyzed.

Polynomial continued fractions for exp(π)

ABSTRACT We present two (inequivalent) polynomial continued fraction representations of the number with all their elements in ; no such representation was seemingly known before. More generally, a similar result for is obtained for every such that . The proof uses a classical polynomial continued fraction representation of , for and , of which we present a new proof that enables us to obtain the exact rate of convergence of the convergents of the continued fraction for . We also deduce some consequences of arithmetic interest concerning the elements of certain polynomial continued fraction representations of the (transcendental) Gel'fond-Schneider numbers , where and , where is the field of algebraic numbers, embedded into .

ON CERTAIN CONTINUED FRACTIONS INVOLVING BASIC BILATERAL HYPERGEOMETRIC FUNCTION 2Ψ2

In this paper we establish certain continued fractions representation for the ratio of two 2Ψ2’s. We also discuss certain continued fraction representation for the ratio of two 3Ψ2’s with two bases and 4Ψ3’s with one base.

Statistically characterized subgroups of the circle (II): Continued fractions

In this note, we continue the investigation of the new version of characterized subgroups of the circle group T, namely, “statistically characterized subgroups” (shortly, “s-characterized subgroups”) recently introduced in [12]. We primarily investigate these subgroups for sequences arising out of continued fraction representation of irrational numbers α in line of [22] and [20] (followed by [3–5]) comparing their main results for this new notion and show that these Borel subgroups are strictly larger in size (so nontrivial) than the corresponding characterized subgroups, always having cardinality c and containing the subgroup 〈α〉 and in the process answer the Open Question 6.4 [12] which had asked about the size of the s-characterized subgroup for the celebrated Fibonacci sequence.

Continued fraction representation of the generating function of Bernoulli polynomials

Continued fraction and quasi-reciprocal continued fraction expansions of the generating function of Bernoulli numbers have been obtained. The convergence and uniform convergence of continued fraction expansions have been proved. Representations of the generating function of Bernoulli polynomials in the form of the product of three continued fractions, as well as the product of three quasi-reciprocal continued fractions, have been found.

Algorithms for the Minimal Rational Fraction Representation of Sequences Revisited

Algorithms for the Minimal Rational Fraction Representation of Sequences Revisited

Continued fraction representation of the generating function of Bernoulli polynomials

Continued fraction and quasi-reciprocal continued fraction expansions of the generating function of Bernoulli numbers have been obtained. The convergence and uniform convergence of continued fraction expansions have been proved. Representations of the generating function of Bernoulli polynomials in the form of the product of three continued fractions, as well as the product of three quasi-reciprocal continued fractions, have been found.

On Generalized Mock Theta Functions of Tenth Order

In this paper we have given 3 independent variable generalization of tenth order mock theta functions and have found some relations between generalized mock theta functions of different order. We have given a continued fraction representation of generalized tenth order mock theta functions. We have also given the q-integral representation and multibasic expansion for these functions. Further by specializing the parameters we have connected mock theta functions with continued fraction of Ramanujan.
 HIGHLIGHTS
 
 Mock theta functions are last gift to mathematics by Ramanujan. These functions are also called mysterious functions as the exact meaning of order of mock theta functions is mystery for mathematicians
 The generalization of tenth order mock theta functions connects them with different order mock theta functions and with Ramanujan’s continued fractions
 The specialization of parameters of generalized mock theta functions gives the famous Ramanujan’s continued fraction

An Authentic Learning Approach to Assist the Computational Thinking in Mathematics Learning for Elementary School

Programme for International Student Assessment (PISA) 2018 report revealed that in mathematics the ability of Indonesian students to solve mathematical problems is still very low. The Trends in International Mathematics and Science Study (TIMSS) strengthened the PISA report by mentioning that in general, Indonesian students' ability in solving fraction problems is low. In this study, We tried to improve students' ability in solving fractions by using computational thinking assisted by an authentic learning approach. A system was utilized to help the students in learning fractions. Totally twenty-two fifth-grade students were assigned in this study. While they learned fractions using the system, they needed to capture an object around them to make a representation of the fraction problem. Moreover, by using the system, they also need to find the best and fastest way to solve a fraction problem. The result showed that their learning achievements improved especially in fraction representation, fraction operation, linguistics, and creativity. And multiple regression results showed that students’ activity can predict the students’ learning achievement. The more active the learn fraction using the system, the higher score they get. In the final, the result of this study contributes essential implications along with a conclusion and suggestion for future research.

The relation between non-symbolic magnitude representation and symbolic fraction representation

The relation between non-symbolic magnitude representation and symbolic fraction representation

Mode spectrum characteristics and onset of the low-shear MHD stability regime

Equilibria with extended regions of weak magnetic shear, including some tokamak scenarios and stellarators, can be susceptible to pressure-driven internal MHD instabilities even though there is no mode rational surface in the plasma. Nonresonant modes, in particular, can have properties that are unattractive for confinement, including displacing substantial volumes of the plasma and leading to more efficient pressure gradient flattening in the nonlinear regime. The onset and linear properties of the low shear stability regime are examined using the initial-value, extended-MHD code M3D-C1. For monotonic q-profiles, we demonstrate a clear correlation between the convergents associated with the continued fraction representation of q0 and the spectrum of unstable modes. Nonresonant modes are observed to be destabilized preferentially to any other resonant instability with the same toroidal mode number when n > 1. Using the observed connection between the spectrum associated with q0 and the overall equilibrium stability properties, we suggest a technique for reducing the uncertainty on both q0 and magnetic shear in the core region, obtained either via measurement or through the analysis and reconstruction of experimental results.

A Comparative Analysis of Fraction Addition and Subtraction Contents in the Mathematics Textbooks in the U.S. and South Korea

Developing textbooks of optimal quality is crucial for enriching the students’ learning and understanding. This study examined fraction addition and subtraction problems in the U.S. and South Korean mathematics textbooks according to the types of denominators. In particular, we investigated Everyday Mathematics (EM) and South Korean mathematics (KM) textbooks revised in 2015 and developed an analytic framework encompassing horizontal and vertical dimensions to examine the learning opportunities presented to students by the textbooks. We assessed their topic sequence and frequency of fraction addition and subtraction contents with regard to the former and examined their contextual features, cognitive demands, and mathematical activities with regard to the latter. We observed that EM provided inadequate learning opportunities for fraction subtraction problems, representation problems, and high-order cognitive abilities. However, KM provided more even learning opportunities for fraction addition and subtraction problems, various contextual features, and high and low thinking skills. Moreover, we found that EM emphasized understanding and resolving activities, whereas KM underscored exploring and explaining activities. The findings of this study suggested updating fraction addition and subtraction contents in the U.S., South Korea, and other countries by considering horizontal and vertical dimensions.

Lanczos recursion on a quantum computer for the Green's function and ground state

A state-preserving quantum counting algorithm is used to obtain coefficients of a Lanczos recursion from a single ground state wavefunction on the quantum computer. This is used to compute the continued fraction representation of an interacting Green's function for use in condensed matter, particle physics, and other areas. The wavefunction does not need to be re-prepared at each iteration. The quantum algorithm represents an exponential reduction in memory over known classical methods. An extension of the method to determining the ground state is also discussed.

Predicting nonresonant pressure-driven MHD modes in equilibria with low magnetic shear

Nonresonant internal modes can be difficult to anticipate as there is no resonant surface in the plasma. However, equilibria that are unstable to multiple nonresonant magnetohydrodynamic (MHD) modes may be more prone to global loss of confinement since these instabilities generate spatially extended linear displacements, potentially enhancing magnetic field line chaos via nonlinear interactions. Here, we successfully predict the unstable nonresonant pressure-driven modes for equilibria with zero shear in the plasma core, irrational q on axis, and a central pressure gradient, which is consistent with pre-crash profiles in sawtoothing tokamak plasmas in the large-aspect-ratio limit. A criterion for identifying nonresonant modes most likely to be unstable is developed from the convergents of the continued fraction representation of q0. A higher-order analysis of the standard Energy Principle reveals the conditions under which these modes are expected to dominate. Linear growth rate spectra, as a function of toroidal mode number (up to n = 30), calculated using the initial-value extended-MHD code, M3D-C1, recover the characteristic dependence observed for ideal infernal modes. Nonresonant modes have also been invoked in some ideal sawtooth crash models. This work provides a mechanism to predict the mode numbers of infernal modes and, potentially, the width of some post-sawtooth-crash profiles.

CONTINUED FRACTIONS, INTERMEDIATE FRACTIONS AND THEIR RELATION TO THE BEST APPROXIMATIONS

The subject of this paper is the current state of art in theory of continued fractions, intermediate fractions and their relation to the best rational approximations of the first and second kind. The paper provides an overview of the some well known and even some new properties of continued fractions, and the various terms associated with them. In addition to intermediate fractions, paper considers the fine intermediate fractions and gave some statements to position these fractions in the continued fraction representation of numbers.

IN SEARCH OF STRATEGIES USED BY PRIMARY SCHOOL PUPILS FOR DEVELOPING FRACTION SENSE

Purpose – Most literature has focused solely on either knowledge about number sense or understanding of fractions. To fill the research gap, this study examined pupils’ abilities in both number sense and fractions. In particular, it investigated Year 4 and Year 5 pupils’ use of strategies in developing their fraction sense. Methodology – This study adopted a descriptive research design, utilising a mixed approach in data collection. An instrument called the Fraction Sense Test (FST) and a clinical interview were used to collect data. The FST comprised 3 strands: fraction concept, fraction representation and effect of operation. A two-stage cluster sampling method was employed to select 396 Year 4 and Year 5 pupils. The sampling involved random selection of the primary schools in the first stage, followed by pupils within the selected schools in the second stage. In addition to descriptive statistics, content analysis of interview transcripts was conducted to identify the presence of concepts and strategies applied among the pupils. Findings – The study found that the pupils scored lowest in effect of operation. It was also revealed that there were four strategies which helped the pupils to develop fraction sense, namely (1) comparing fractions using benchmark fractions of common fractions such as ½, ¼, zero and 1, (2) understanding denominators to determine the size of equal parts, (3) comparing fractions using unit fraction, and (4) applying the strategies in (1) and (2) to manipulate fractions in effect of operation. Significance – The findings provide useful input to facilitate the development of fraction sense ability.