## The powers of all subsystems are positive odd rational numbers and depend on the switching signals. 모든 하위 시스템의 전력은 양의 홀수 유리수이며 스위칭 신호에 따라 다릅니다.

Adaptive quantised control for switched uncertain nonlinear systems with unknown control directions

## This paper studies the problem of finite-time control for switched nonlinear systems (SNSs), where the powers of chained integrators associated with individual subsystems can be different positive odd rational numbers from each other. 이 논문은 개별 서브시스템과 관련된 연쇄 적분기의 전력이 서로 다른 양의 홀수 유리수일 수 있는 스위치 비선형 시스템(SNS)에 대한 유한 시간 제어 문제를 연구합니다.

Finite-time H∞ control for switched nonlinear systems

## Difficulties of high school entrants in understanding rational numbers This paper presents an analysis of the resolution of a question about rational numbers, posed by students entering high school. 고등학생의 유리수 이해 어려움 이 논문은 고등학교에 입학하는 학생들이 제기한 유리수 문제의 해결에 대한 분석을 제시한다.

Sobre dificuldades de ingressantes no ensino médio na compreensão de números racionais

## We argue that (a) the distinction of equivalence gives a unified framework of equal fractions that has not previously been described in the literature; (b) a conceptual understanding of both fraction equivalences is integral to understanding rational numbers; and (c) knowledge of both conceptions of equivalence is important for developing a conceptual understanding of fraction arithmetic. 우리는 (a) 등가의 구별이 이전에 문헌에서 설명되지 않은 등분의 통일된 프레임워크를 제공한다고 주장합니다. (b) 두 분수 등가물에 대한 개념적 이해는 유리수를 이해하는 데 필수적입니다. (c) 두 등가 개념에 대한 지식은 분수 산술의 개념적 이해를 발전시키는 데 중요합니다.

Two conceptions of fraction equivalence

## , with n being the element from the set of all positive rational numbers. nan

Multiphase separation model for binary mixed micelles

## One such example is the set of rational numbers Q. 그러한 예 중 하나는 유리수 Q의 집합입니다.

Banach Spaces and Hilbert Spaces

## The irreducibility of P G ( x ) over rational numbers Q has a close relationship with the automorphism group, reconstruction and controllability of a graph. 유리수 Q에 대한 PG( x )의 기약성은 그래프의 automorphism 그룹, 재구성 및 제어 가능성과 밀접한 관계가 있습니다.

Note on graphs with irreducible characteristic polynomials

## We present a survey of 67 oxygen-rich stars from 7 to 1 mm, in their rotational transitions from J = 1 → 0 to J = 5 → 4, for vibrational numbers v from 0 to 6 in the three main SiO isotopologs. 우리는 3개의 주요 SiO 동위 원소에서 0에서 6까지의 진동 수 v에 대해 J = 1 → 0에서 J = 5 → 4로의 회전 전이에서 7에서 1mm의 산소가 풍부한 별 67개에 대한 조사를 제시합니다.

SiO, 29SiO, and 30SiO Emission from 67 Oxygen-rich Stars: A Survey of 61 Maser Lines from 7 to 1 mm

## We perform calculations for the states of the total angular momentum L=0-4 with the complete set of vibrational numbers v=0-10. nan

Variational calculations of the H2+ and HD+ rovibrational energies

## be a sequence of rational numbers whose mth divided difference is integer-valued. m번째로 나눈 차이가 정수 값인 유리수의 시퀀스입니다.

Functions with integer-valued divided differences

## We present a uniform description of sets of m linear forms in n variables over the field of rational numbers whose computation requires m(n − 1) additions. 계산에 m(n - 1) 덧셈이 필요한 유리수 필드에 대해 n 변수의 m 선형 형식 집합에 대한 균일한 설명을 제시합니다.

Sets of Linear Forms Which Are Hard to Compute

## The present article is devoted to representations of rational numbers in terms sign-variable Cantor expansions. 현재 기사는 부호 변수 칸토어 확장의 관점에서 유리수 표현에 전념합니다.

Rational numbers represented by sign-variable Cantor series

## 29 (1984), 101–108] concerning rational approximation to fractal sets by rational numbers inside and outside the set in question. 29 (1984), 101–108] 문제의 집합 내부와 외부의 유리수에 의한 프랙탈 집합에 대한 합리적 근사에 관한 것입니다.

On intrinsic and extrinsic rational approximation to Cantor sets

## For irrational numbers, the Brillouin zone extends to infinity. 무리수의 경우 Brillouin 영역은 무한대로 확장됩니다.

Calculating the polarization in bipartite lattice models: Application to an extended Su-Schrieffer-Heeger model

## Cauchy sequences, Dedekind cuts, base-10 expansions and continued fractions are examples of well-known representations of irrational numbers. 코시 수열, 데데킨트 컷, 10진법 확장 및 연속 분수는 무리수의 잘 알려진 표현의 예입니다.

Computable irrational numbers with representations of surprising complexity

## In this article, we introduce for the first time the radiation properties of a spiral shape that is obtained from the work on irrational numbers by Theodorus (465 BC–398 BC) of Cyrene (presently near Shahhat, Libya). 이 기사에서 우리는 Cyrene(현재 리비아 샤하트 근처)의 Theodorus(기원전 465-398년)의 무리수에 대한 연구에서 얻은 나선 모양의 복사 특성을 처음으로 소개합니다.

The Theodorus of Cyrene Spiral Geometry and Its Application to Antenna Design

10.29069/FORSCIENCE.2021V9N1.E878

## Difficulties of high school entrants in understanding rational numbers This paper presents an analysis of the resolution of a question about rational numbers, posed by students entering high school. 고등학생의 유리수 이해 어려움 이 논문은 고등학교에 입학하는 학생들이 제기한 유리수 문제의 해결에 대한 분석을 제시한다.

Sobre dificuldades de ingressantes no ensino médio na compreensão de números racionais

## For a polynomial $f(x)\in\mathbb{Q}[x]$ and rational numbers c, u, we put $f_c(x)\coloneqq f(x)+c$ , and consider the Zsigmondy set $\calZ(f_c,u)$ associated to the sequence $\{f_c^n(u)-u\}_{n\geq 1}$ , see Definition 1. 다항식 $f(x)\in\mathbb{Q}[x]$ 및 유리수 c, u에 대해 $f_c(x)\coloneqq f(x)+c$ 를 넣고 Zsigmondy 집합 $\를 고려합니다.$\{f_c^n(u)-u\}_{n\geq 1}$시퀀스와 연관된 calZ(f_c,u)$, 정의 1 참조.

Primitive prime divisors in the critical orbits of one-parameter families of rational polynomials

10.1016/B978-0-12-817414-2.00009-9

## We are able to conceive irrational numbers, idealized geometrical shapes, abstract topological properties, and so on without ever perceiving them. 무리수, 이상화된 기하학적 모양, 추상적인 위상적 성질 등은 우리가 인지하지 못한 채 상상할 수 있습니다.

Cerebral underpinning of advanced mathematical activity

## These include natural numbers, integers, rational numbers, as well as irrational and real numbers. 여기에는 자연수, 정수, 유리수, 무리수 및 실수가 포함됩니다.

Back to High School—Algebra Revisited

10.23925/1983-3156.2021V23I1P683-712

## Keywords: Rational numbers, Fractions, Meaning of fractions, Learning, Exploratory approach. 키워드: 유리수, 분수, 분수의 의미, 학습, 탐색적 접근.

Quando As Frações Não São Apenas Partes de Um Todo…!When fractions are not just parts of a whole…

## Our main result is an explicit evaluation of the Mahler measure of PD as a convergent series whose each term is given in terms of rational numbers, multinomial coefficients, and the L-norm of the vector of coefficients of PD. 우리의 주요 결과는 각 항이 유리수, 다항 계수 및 PD 계수 벡터의 L-노름으로 제공되는 수렴 계열로서 PD의 말러 측정을 명시적으로 평가한 것입니다.

Evaluating the Mahler measure of linear forms via the Kronecker limit formula on complex projective space

## The theory of dessins d’enfants on compact Riemann surfaces, which are bipartite maps on compact orientable surfaces, are combinatorial objects used to study branched covers between compact Riemann surfaces and the absolute Galois group of the field of rational numbers. 조밀한 방향성 표면의 이분 맵인 조밀한 리만 표면의 데신 덴팡 이론은 조밀한 리만 표면과 유리수 분야의 절대 갈루아 그룹 사이의 분기된 덮개를 연구하는 데 사용되는 조합 객체입니다.

Dessins D’enfants and Some Holomorphic Structures on the Loch Ness Monster

## Unlike most of the literature, we make no sparsity assumption on $\beta^*$, but instead adopt a different regularization: In the noiseless setting, we assume $\beta^*$ consists of entries, which are either rational numbers with a common denominator $Q\in\mathbb{Z}^+$ (referred to as $Q$-rationality); or irrational numbers supported on a rationally independent set of bounded cardinality, known to learner; collectively called as the mixed-support assumption. 대부분의 문헌과 달리 $\beta^*$에 대한 희소성 가정을 하지 않고 대신 다른 정규화를 채택합니다. 무소음 설정에서 $\beta^*$가 항목으로 구성되어 있다고 가정합니다. 공통 분모 $Q\in\mathbb{Z}^+$($Q$-합리성이라고 함); 또는 학습자에게 알려진 합리적으로 독립적인 경계 카디널리티 세트에서 지원되는 무리수; 혼합 지원 가정이라고 통칭합니다.

Inference in High-Dimensional Linear Regression via Lattice Basis Reduction and Integer Relation Detection

## The powers of all subsystems are positive odd rational numbers and depend on the switching signals. 모든 하위 시스템의 전력은 양의 홀수 유리수이며 스위칭 신호에 따라 다릅니다.

Adaptive quantised control for switched uncertain nonlinear systems with unknown control directions

## The main objective of this article is to explain the positive influence of a new and innovative teaching methodology, applied to the mathematical operation of Rational Numbers (RN) of students; called Didactic Transposition (DT). 이 기사의 주요 목적은 학생들의 유리수(RN)의 수학적 연산에 적용된 새롭고 혁신적인 교수법의 긍정적인 영향을 설명하는 것입니다. DT(Didactic Transposition)라고 합니다.

Metodología moderna con influencia psicométrica para mejorar la comprensión de la parte operativa de los números fraccionarios

## The rational and irrational numbers present little research in basic education and, when they are approached, they are usually presented with a focus on privileging only operational, finite, infinite and exact aspects, which limits the approach and understanding of this important theme in the teaching of Mathematics. 합리적이고 비합리적인 숫자는 기초 교육에서 거의 연구를 나타내지 않으며, 접근할 때 일반적으로 조작적, 유한, 무한 및 정확한 측면에만 특권을 부여하는 데 초점을 맞춰 제시되며, 이는 교육에서 이 중요한 주제에 대한 접근 및 이해를 제한합니다. 수학.

Frações contínuas para a resolução da equação de Pell

## The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. 유리수에 대한 일반 혼합 정수 프로그램의 반올림 오류 없는 솔루션에 대한 마지막 이정표는 2013년 Cook, Koch, Steffy 및 Wolter가 발표한 하이브리드 정밀도 분기 제한 알고리즘이었습니다.

A Computational Status Update for Exact Rational Mixed Integer Programming

## While one might ultimately be more interested in solutions over, say, the field of rational numbers, the problem is far more tractable over finite fields, and local-global principles such as the Birch-Swinnerton-Dyer conjecture establish strong, albeit subtle, relationships between the two cases. 궁극적으로 유리수 필드에 대한 솔루션에 더 관심이 있을 수 있지만 문제는 유한 필드에 대해 훨씬 다루기 쉬우며 Birch-Swinnerton-Dyer 추측과 같은 로컬 글로벌 원칙은 미묘하지만 강력하고 관계를 설정합니다. 두 경우 사이.

The Weil Conjectures

## We prove new upper bounds on the number of representations of rational numbers mn as a sum of four unit fractions, giving five different regions, depending on the size of m in terms of n. 유리수 mn의 표현 수에 대한 새로운 상한을 4개의 단위 분수의 합으로 증명하여 n에 대한 m의 크기에 따라 5개의 다른 영역을 제공합니다.

Sums of four and more unit fractions and approximate parametrizations

## In particular, the authors prove the “Alien Intruder Theorem” guaranteeing the existence of a model of R# including the rational numbers in which each rational acts as a nonstandard natural number. 특히 저자들은 각각의 유리수가 비표준 자연수로 작용하는 유리수를 포함하는 R# 모델의 존재를 보장하는 "외계인 침입자 정리"를 증명한다.

Alien Intruders in Relevant Arithmetic

10.23925/1983-3156.2021V23I1P632-654

## The objective of this article is to identify signs of learning by students in a 4 th -grade class of elementary school about the meaning of the quotient of rational numbers, based on the analysis of 46 protocols related on two activities with this subject. 이 글의 목적은 초등학교 4학년 학생들이 이 주제와 관련된 두 가지 활동과 관련된 46개의 프로토콜을 분석하여 유리수의 몫의 의미에 대한 학습 징후를 식별하는 것입니다.

Indícios de aprendizagens de alunos de 4º ano sobre os números racionais envolvendo o significado quociente

## This paper studies the problem of finite-time control for switched nonlinear systems (SNSs), where the powers of chained integrators associated with individual subsystems can be different positive odd rational numbers from each other. 이 논문은 개별 서브시스템과 관련된 연쇄 적분기의 전력이 서로 다른 양의 홀수 유리수일 수 있는 스위치 비선형 시스템(SNS)에 대한 유한 시간 제어 문제를 연구합니다.

Finite-time H∞ control for switched nonlinear systems

## The present article is devoted to representations of rational numbers in terms of sign variable Cantor expansions. 현재 기사는 부호 변수 Cantor 확장의 관점에서 유리수의 표현에 전념합니다.

Rational numbers defined in terms of certain generalized series

## Consider the set of irreducible denominators of the rational numbers representable by finite continued fractions all of whose partial quotients belong to some finite alphabet. 부분 몫이 모두 일부 유한 알파벳에 속하는 유한 연속 분수로 나타낼 수 있는 유리수의 기약 분모 집합을 고려하십시오.

A strengthening of the Bourgain-Kontorovich method: three new theorems

## As to Patterns and Algebra, all 21 teacher participants mentioned that the topics that hard to be applied with modern technologies were irrational numbers, solving problems in linear equations, operation on polynomials and inequalities, and fundamental operations on algebraic expressions. 패턴과 대수에 관해서는 21명의 교사 참가자 모두 현대 기술에서 적용하기 어려운 주제가 무리수, 선형 방정식 문제 풀기, 다항식 및 부등식에 대한 연산, 대수식에 대한 기본 연산이라고 언급했습니다.

Modern Mathematics Applications: Solutions to Challenges Encountered in Teaching Spiral Progression in Mathematics 7

## A Puiseux monoid is an additive submonoid of the nonnegative rational numbers. Puiseux 모노이드는 음이 아닌 유리수의 가산 서브모노이드입니다.

When Is a Puiseux Monoid Atomic?

## We argue that (a) the distinction of equivalence gives a unified framework of equal fractions that has not previously been described in the literature; (b) a conceptual understanding of both fraction equivalences is integral to understanding rational numbers; and (c) knowledge of both conceptions of equivalence is important for developing a conceptual understanding of fraction arithmetic. 우리는 (a) 등가의 구별이 이전에 문헌에서 설명되지 않은 등분의 통일된 프레임워크를 제공한다고 주장합니다. (b) 두 분수 등가물에 대한 개념적 이해는 유리수를 이해하는 데 필수적입니다. (c) 두 등가 개념에 대한 지식은 분수 산술의 개념적 이해를 발전시키는 데 중요합니다.

Two conceptions of fraction equivalence

## We obtain infinite product formulas for certain combinations of gamma functions, which include irrational numbers such as √ 2 as well as some nested radicals. √ 2와 같은 무리수와 일부 중첩된 라디칼을 포함하는 감마 함수의 특정 조합에 대해 무한 곱 공식을 얻습니다.

Euler’s reflection formula, infinite product formulas, and the correspondence principle of quantum mechanics

## Multiple notations pose a challenge for learners but could also present an opportunity, in that cross-notation knowledge could help learners to achieve a better understanding of rational numbers than could easily be achieved from within-notation knowledge alone. 다중 표기법은 학습자에게 도전이 될 수 있지만 교차 표기법 지식이 학습자가 표기법 내 지식만으로 쉽게 달성할 수 있는 것보다 유리수에 대한 더 나은 이해를 달성하는 데 도움이 될 수 있다는 점에서 기회를 제공할 수도 있습니다.

Cross-notation knowledge of fractions and decimals.

## This study investigated Pre-Service Teachers’ mastery level, achievements differences, and correlation of their procedural knowledge and conceptual knowledge of rational numbers. 본 연구에서는 예비교사의 숙련도, 성취도 차이, 유리수에 대한 절차지식과 개념지식 간의 상관관계를 조사하였다.

Pre-Service Teachers’ Conceptual and Procedural Knowledge of Rational Numbers in E. P. College of Education, Bimbilla, Ghana

## Previous work has established a dichotomy for Unions of Conjunctive Queries (UCQ) when the probabilities are arbitrary rational numbers, showing that, for each query, its complexity is either in polynomial time or \#P-hard. 이전 작업에서는 확률이 임의의 유리수일 때 UCQ(Unions of Conjunctive Queries)에 대한 이분법을 확립했으며, 이는 각 쿼리의 복잡성이 다항식 시간 또는 \#P-hard임을 보여줍니다.

A Dichotomy for the Generalized Model Counting Problem for Unions of Conjunctive Queries

10.1093/OSO/9780198852650.003.0013

## This chapter describes how to multiply and divide, albeit approximately, by some of the world’s most famous irrational numbers, such as π‎, Euler’s number e, 2, 3, both of which occur frequently in the study of triangles, and the Golden Ratio, also sometimes called the Divine proportion. 이 장에서는 삼각형 연구에서 자주 발생하는 π‎, 오일러 수 e, 2, 3과 같은 세계에서 가장 유명한 무리수와 황금비를 곱하고 나누는 방법에 대해 설명합니다. , 때로는 신성한 비율이라고도 합니다.

Multiplying and Dividing Irrationally

## 7), but also non-natural rational numbers (e. 7) 뿐만 아니라 비자연적 유리수(e.

The Number Sense Represents (Rational) Numbers.

## One such example is the set of rational numbers Q. 그러한 예 중 하나는 유리수 Q의 집합입니다.

Banach Spaces and Hilbert Spaces

## As a consequence, we show that the class of bounded-error affine languages remains the same when the AfAs are restricted to use rational numbers only. 결과적으로, 우리는 AfA가 유리수만 사용하도록 제한될 때 경계 오류 affine 언어의 클래스가 동일하게 유지된다는 것을 보여줍니다.

Computational limitations of affine automata and generalized affine automata

## The theory of Frobenius groups with abelian Frobenius kernel largely reduces to algebraic number theory and indeed to a tractable part of algebraic number theory: the study of unramified primes in abelian extensions of the field of rational numbers [2]. Abelian Frobenius 커널이 있는 Frobenius 그룹의 이론은 대수적 수 이론으로, 실제로는 대수적 수 이론의 다루기 쉬운 부분으로 축소됩니다.

Many Frobenius complements have even order

## Hence, all real numbers of the form ∛x cannot be considered as if those are points on the ‘Real Line’; and there are no points on the ‘Real Line’ corresponding to the irrational numbers of the form ∛x. 따라서 ∛x 형식의 모든 실수는 '실선'의 점인 것처럼 간주될 수 없습니다. ∛x 형식의 무리수에 해당하는 '실선'에는 점이 없습니다.

Real Line – An Incomplete Number System

## We present a sieve result for rational numbers as an analogue of that in the Kubilius model. 유리수에 대한 체 결과를 Kubilius 모델의 유사체로 제시합니다.

Sieving the Rationals

## org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>–adic continued fraction algorithm proving that it stops in a finite number of steps when processes rational numbers, solving a problem left open in a paper by Browkin [Math. org/1998/Math/MathML" alttext="p"> <mml:의미론> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:의미론> </mml:수학> </inline-formula> – 유리수를 처리할 때 유한한 수의 단계에서 멈춘다는 것을 증명하는 연속 분수 알고리즘으로 Browkin[Math.

10.22108/IJGT.2021.128359.1686

## ‎We explicitly describe the set‎ ‎of all {it isomorphism types} of irreducible representations of $G$‎ ‎over the field of complex numbers $mathbb{C}$ as well as the‎ ‎isomorphism types over the field of rational numbers $mathbb{Q}$‎. ‎복소수 $mathbb{C}$의 필드에 대한 $G$의 기약 표현의 모든 {it isomorphism types}의 집합과 유리수 필드에 대한 ‎ ‎동형 유형을 명시적으로 설명합니다. $mathbb{Q}$‎.

Rational and Quasi-Permutation Representations of Holomorphs of Cyclic $p$-Groups

## We propose a modular forms-based model for counting -number fields having the same local properties as the splitting field of the mod p-Galois representation associated with an elliptic curve over the rational numbers. 유리수에 대한 타원 곡선과 관련된 mod p-Galois 표현의 분할 필드와 동일한 로컬 속성을 갖는 숫자 필드를 계산하기 위한 모듈식 형식 기반 모델을 제안합니다.

On Bhargava’s Heuristics for GL2(𝔽 p )-Number Fields and the Number of Elliptic Curves of Bounded Conductor

## In the earlier work, the SEL series expansion for any real number is constructed and characterizations of rational numbers by using such expansion are established. 초기 연구에서는 임의의 실수에 대한 SEL 급수 전개를 구성하고 이러한 전개를 사용하여 유리수의 특성을 확립하였다.

SEL Egyptian fraction expansion and characterizations of rational numbers

## In this paper, we study associated PB splines with local knot vectors that are arbitrarily distributed in $$([0,1]\cap {\mathbb {Q}})^d, d=1,2$$ ( [ 0 , 1 ] ∩ Q ) d , d = 1 , 2 , where $${\mathbb {Q}}$$ Q is the set of rational numbers. 이 논문에서는 $$([0,1]\cap {\mathbb {Q}})^d, d=1,2$$ ( [ 0 , 1 ] ∩ Q ) d , d = 1 , 2 , 여기서 $${\mathbb {Q}}$$ Q 는 유리수의 집합입니다.

On linear independence of linear and bilinear point-based splines

## By the primitive element theorem, for all but finitely many rational numbers r we have $$F={{\mathbb {Q}}}(\alpha +r\beta )$$. 원시 요소 정리에 따르면 유한하게 많은 유리수 r에 대해 $$F={{\mathbb {Q}}}(\alpha +r\beta )$$가 있습니다.

An effective version of the primitive element theorem

## In finite type arithmetic, the real numbers are represented by rapidly converging Cauchy sequences of rational numbers. 유한 유형 산술에서 실수는 유리수의 코시 수열을 빠르게 수렴하여 표현됩니다.

The abstract type of the real numbers

## Integers can be regarded as an integral domain, the rational numbers can be regard as a fractional domain. 정수는 정수 영역으로 간주될 수 있고 유리수는 분수 영역으로 간주될 수 있습니다.

The Generalization and Proof of “Square Root of 2 Is Not a Rational Number” on the Integral Domain

## Ancient records substantiate that more than 5000 years back Vedic Ascetics were successful in approximating these numbers in terms of rational numbers and used these approximations for ritual sacrifices, they also indicated clearly that these numbers are incommensurable. 고대 기록에 따르면 5000년 이상 전에 베다 고행자들은 이 숫자를 합리적인 숫자로 근사하는 데 성공했으며 의식 희생을 위해 이 근사치를 사용했으며 이 숫자는 비교할 수 없다는 점을 분명히 밝혔습니다.

Origin of Irrational Numbers and Their Approximations

## The ultimate objective is to understand the concept of fractions as rational numbers (p divided by q), that can be placed on a number line (Gunderson et al. 궁극적인 목표는 분수의 개념을 유리수(p 나누기 q)로 이해하는 것이며, 이는 숫자 라인에 표시될 수 있습니다(Gunderson et al.

Designing a Mixed Reality Extension for an Educational Board Game on Fractions

## If the arguments of functions sin x and cos x, expressed in radians, are equal to x = r 2 \pi, where r are rational numbers, then the values of the functions are algebraic numbers. 라디안으로 표시되는 함수 sin x 및 cos x의 인수가 x = r 2 \pi(여기서 r은 유리수)인 경우 함수의 값은 대수입니다.

Algebraic values of sines and cosines and their arguments

## In this paper, we study the computational applications of the Levi-Civita field whose elements are functions from the additive abelian group of rational numbers to the real numbers field, with left-finite support. nan

On computational applications of the Levi-Civita field

## The irreducibility of P G ( x ) over rational numbers Q has a close relationship with the automorphism group, reconstruction and controllability of a graph. 유리수 Q에 대한 PG( x )의 기약성은 그래프의 automorphism 그룹, 재구성 및 제어 가능성과 밀접한 관계가 있습니다.

Note on graphs with irreducible characteristic polynomials

## Using the corresponding result for polynomial automata, we are able to prove that the ZERONESS problem for weighted alternating automata with the rational numbers as weights is decidable.

A Nivat Theorem for Weighted Alternating Automata over Commutative Semirings

## An introduction to irrational numbers is done using basic trigonometry, generally taught in pre-university courses.

Teaching Irrational Numbers Through Trigonometry

## Perhaps the most severe is the limited expressiveness in that degrees of belief are restricted to constant rational numbers, which makes it impossible to express arbitrary belief distributions.

Reasoning about Beliefs and Meta-Beliefs by Regression in an Expressive Probabilistic Action Logic

## This works is concerned with the finite-time optimal stabilization problem for a class of switched non-strict-feedback nonlinear systems whose powers are possibly different positive odd rational numbers in the sense the powers of each subsystem might differ from others.

Global finite-time stabilization of switched high-order rational power nonlinear systems

## This article intends to review quasirandom sequences, especially the Faure sequence to introduce a new version of scrambled of this sequence based on irrational numbers, as follows to prove the success of this version of the random number sequence generator and use it in future calculations.

The New Scramble for Faure Sequence Based on Irrational Numbers

## It follows from Hilbert's Irreducibility Theorem that for most rational numbers $c$ the specialized polynomial $P(c,x)$ has Galois group isomorphic to $G$ and factors in the same way as $P$.

Galois groups over rational function fields and Explicit Hilbert Irreducibility

## Then we derive explicit expression using some Dedekind eta functions when the Bloch vector $$\mathbf {k}$$ k are some rational numbers.

On the regular part of the Bloch Green’s function for the Laplacian: analytical formula and critical points

10.22405/2226-8383-2021-22-1-118-132

## Using tropical functions, we prove that the degrees of the variables of the above monomial can be represented as quadratic polynomials in the order index of the element of the Somos sequence, whose free terms are periodic sequences of rational numbers.

Тропические последовательности, ассоциированные с последовательностями Сомоса

## be a sequence of rational numbers whose mth divided difference is integer-valued. m번째로 나눈 차이가 정수 값인 유리수의 시퀀스입니다.

Functions with integer-valued divided differences

10.22405/2226-8383-2021-22-2-104-120

## In addition, we consider the representations of natural numbers based on the denominators of partial convergents of the continued fraction expansions of irrational numbers. nan

Об аналоге задачи Гельфонда для обобщенных разложений Цеккендорфа

## The rational numbers are countable.

Listing the Positive Rationals

## We then outline a method to solve the problem of approximating scales with frequency ratios generated by rational numbers with small numerators and denominators, via equal temperament.

Subsets of Scales in Compositions Constructed by Similarity

## We apply our results to the Gauss map and obtain new precise asymptotics in the theorem of Lévy on the regular continued fraction expansion of irrational numbers in (0, 1).

Precise asymptotics on the Birkhoff sums for dynamical systems

## We study rational periodic points of polynomial fd,c(x) = x d + c over the field of rational numbers, where d is an integer greater than 2 and c 6= −1.

Rational Periodic Points of xd + c and Fermat-Catalan Equations

## Students often show difficulties in understanding rational numbers.

Profiles in understanding operations with rational numbers

## Here, h1,…,hr are algebraic numbers linearly independent over the field of rational numbers.

Joint Universality of the Zeta-Functions of Cusp Forms

## From such sets A , B , sparse linear systems over the rational numbers arise.

Unique sums and differences in finite Abelian groups

## For all Eichler orders with a same squarefree level in a definite quaternion algebra over the field of rational numbers, we prove that a weighted sum of Jacobi theta series associated to these orders is a Jacobi Eisenstein series.

Eichler orders and Jacobi forms of squarefree level

## We then derive an asymptotically optimal mechanism under a minor technical assumption: we assume the agents’ valuations are rational numbers with bounded denominators.

An asymptotically optimal VCG redistribution mechanism for the public project problem

10.1088/1757-899X/1047/1/012154

## Orthogonal coding is analogous to convolutional coding over the field of rational numbers, allows to provide the required transmission quality at lower energy costs.

Interference immunity in channels with random phase and fading when using orthogonal coding and frequency modulation

## Rational numbers, such as fractions, decimals and percentages, are a persistent challenge in the mathematics curriculum.

Probing the neural basis rational numbers: the role of inhibitory control and magnitude representations

## In particular, we prove that the roots and the quotient of such series are irrational numbers.

An irrationality result for a recursive construction

## ABSTRACT This note provides a short, self-contained proof of the famous fact that any countable metric space without isolated points is homeomorphic to the space of rational numbers.

Countable Metric Spaces Without Isolated Points

## This, applied to the case where $K$ is the maximal algebraic extension of the field $\mathbb Q$ of rational numbers in the field $\mathbb Q _{p}$ of $p$-adic numbers, for a given prime number $p$, proves the existence of an algebraic extension $E _{p}$ of $\mathbb Q$, such that dim$(E _{p}) \le 1$, $E _{p}$ has a Henselian valuation with a residue field of characteristic $p$, and $E _{p}$ is not a $C _{1}$-field.

Fields of dimension one algebraic over a global or local field need not be of type C1

## We present a survey of 67 oxygen-rich stars from 7 to 1 mm, in their rotational transitions from J = 1 → 0 to J = 5 → 4, for vibrational numbers v from 0 to 6 in the three main SiO isotopologs. 우리는 3개의 주요 SiO 동위 원소에서 0에서 6까지의 진동 수 v에 대해 J = 1 → 0에서 J = 5 → 4로의 회전 전이에서 7에서 1mm의 산소가 풍부한 별 67개에 대한 조사를 제시합니다.

SiO, 29SiO, and 30SiO Emission from 67 Oxygen-rich Stars: A Survey of 61 Maser Lines from 7 to 1 mm

## These lattices are important in the theory of cluster algebras and their rank generating functions can be used to define q-analogues of rational numbers. nan

On a rank-unimodality conjecture of Morier-Genoud and Ovsienko

## Moreover, the GB hierarchy directly follows the distribution of rational numbers that is represented by the modified Farey diagram, which represents the 3D atomic structure of the symmetrical tilt GBs in an fcc crystal.

3D arrangement of atomic polyhedra in tilt grain boundaries

## This paper addresses the adaptive finite-time consensus tracking problem for high-order nonlinear multi-agent systems (MASs) with powers of positive odd rational numbers under prescribed performance.

Adaptive neural network finite-time tracking control for a class of high-order nonlinear multi-agent systems with powers of positive odd rational numbers and prescribed performance

## ” After noting that Plato was interested in adding irrational numbers and must have added √ 2 + √3, Popper writes: “It seems a plausible hypothesis that Plato knew of [ √ 2 + √3 ≈ π], but was unable to prove whether or not is was a strict inequality or only an approximation.

Plato’s Approximation of Pi

## In particular, we devise a superadditive, computable sequence of rational numbers so that the associated limit value in the sense of Fekete’s lemma is not a computable number.

On the Effectiveness of Fekete’s Lemma in Information Theory

## In particular, we obtain an expressive numerical p-admissible concrete domain based on the rational numbers.

An Algebraic View on p-Admissible Concrete Domains for Lightweight Description Logics

## The study was based on the theory of constructivism in a bid to understand whether learners’ transition from whole numbers to rational numbers enabled them to deal with the more complex concept of fractions.

Grade 9 learners’ understanding of fraction concepts: Equality of fractions, numerator and denominator

## For the rational numbers which are very common in smart cities, we design a coding technology to realize the secure homomorphic computation of them.

Secure Data Set Operation Protocols for Outsourced Cloud Data to Protect User Privacy in Smart City

## Irrational numbers didn’t have reoccurrence sequence but this paper calculated a sequence with respect to Quantum Perspective Model.

Can Irrational Numbers (Such as Square Root of the Number Five) Be Reached by Analysis of Genetic Sequences?

## 16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if \begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.

Bernoulliness of $[T,\text{Id}]$ when $T$ is an irrational rotation: towards an explicit isomorphism

## In particular, it is known that some nonzero integral powers of the Gauss sums in this case are in quadratic fields over the field of rational numbers.

Powers of Gauss sums in quadratic fields