Arithmetic Progressions Research Articles

Mathematical Critical Thinking Ability in Solving Mathematical Problems

This study aims to determine the mathematical critical thinking skills of class XII IPA 1 students at SMAN 5 Sidrap in solving mathematical problems in arithmetic sequences and series. The type of research used is descriptive research with a qualitative approach. In this study, there were 3 subjects, namely students with high, medium and low mathematical abilities. The instruments used in data collection were observation sheets, written tests and interview guidelines. The results showed that: (1) Students who met the Critical Thinking Level (CTL) 3 or critically were students who had high mathematical abilities. At this critical thinking level, students are able to formulate the main points of the problem, are able to reveal existing facts, are able to determine the theorems used and detect bias, Students are able to work on questions according to the initial plan, able to express their arguments clearly, able to re-examine answers and draw conclusions. (2) Students who meet CTL 2 or are quite critical are students who have moderate mathematical abilities. At this critical thinking level, students are able to formulate the main points of the problem, uncover existing facts, are able to determine the theorem used, are able to detect bias, are able to work on problems according to the initial plan, are able to express their arguments clearly, are less able to re-examine answers and draw a conclusion. (3) Students who meet CTL 0 or are not critical are students who have low mathematical abilities. At this critical thinking level, students have not been able to fulfill all indicators such as not being able to formulate the main points of the problem.

Bounds on the size of Progression-Free Sets in ℤ m n

Abstract In this note we give an overview of the currently known best lower and upper bounds on the size of a subset of ℤ n m avoiding k-term arithmetic progression. We will focus on the case when the length of the forbidden progression is 3. We also formulate some open questions.

Analysis of Students' Adaptive Reasoning Ability in Solving HOTS Problems Arithmetic Sequences and Series in Terms of Learning Style

This study describes students' adaptive reasoning abilities in solving HOTs type questions in Kolb's learning style. The method used to determine students' adaptive reasoning ability in solving HOTS (Higher Order Thinking Skill) uses qualitative research with a descriptive approach and type questions regarding learning styles (diverger, assimilator, converger, and accommodator types). The research subjects were thirty-five students of class XI. The results of data analysis and discussion can be concluded that students of class XI MIPS-3 SMA Negeri 4 Pasuruan have adaptive reasoning abilities in solving solutions to the HOTS type questions through the Polya step in terms of Kolb's learning style. Students who have adaptive reasoning ability solve the HOTS type problem solving through the Polya step in terms of Kolb's learning style. Students with a converger type of learning style can meet all indicators of adaptive reasoning ability (propose conjectures or conjectures, provide reasons or evidence about the truth of a statement, draw conclusions from an idea, check the validity of an argument, and find patterns from a mathematical problem) and all indicators in the Polya step (understand the problem, make a plan, implement the plan and re-examine the results).

Comminution sizes in sampling calculations

As was demonstrated by Gy empirically, then elucidated by Francois-Bongarcon, only the P95 size of comminuted material (i.e. the size of a screen rejecting 5% of the material in mass) is relevant to sampling theory variance formulas. In practice, when establishing sampling nomograms or using historic data for heterogeneity testing, the P95 is not always known and the material is not always preserved for P95 determinations by screening. A non-theoretical study of the experimental material previously used and published by Gy had suggested some unpublished, rule-of-thumb, order-of-magnitude correspondence between other Pxx sizes (e.g. P100, P80, P75, etc.) and P95, to be used as a last resort when no better determination is possible. This correspondence surprisingly followed an arithmetic progression. In this paper, various types of size distributions were researched and most of these rule-of-thumb formulas were reasonably confirmed, except for the correspondence between P95 and P100 (maximum size), which was therefore updated. The reasons for the arithmetic progression and its exception for P100 also became clearer in the process.

MISCONCEPTION ANALYSIS OF STUDENTS WITH IMPULSIVE-REFLECTIVE COGNITIVE STYLE IN SOLVING PATTERNS OF NUMBERS PROBLEMS

In the online learning process since March 2020, many students have experienced misconceptions in solving the questions given by the teacher, including mathematics learning activities. In online learning activities students are required to be able to understand the material quickly with all the limitations that students have so that misconceptions arise in students. This misconception can occur, one of which is influenced by differences in students' cognitive styles. This study aims to analyze students' misconceptions in solving problems related to number pattern material. The analysis was carried out on 1 subject with impulsive cognitive style and 1 subject with reflective cognitive style with the same learning outcomes. This type of research is descriptive research with a qualitative approach. Supporting instruments include the Matching Familiar Figure Test (MFFT) and a written test consisting of 9 multiple choice questions which include sub-materials of arithmetic number series, geometric number series, letter number series, and contextual questions related to PATTERNS OF NUMBERS with 4 answer choices. To analyze students' misconceptions, the Three Tier test method is used, namely the first tier consists of number pattern material questions in the form of multiple choice with 4 answer choices, the second tier is the column for students' reasons for giving answers, and the third tier is a column of students' confidence levels using the CRI method, and continue with the interview. The results showed that students with impulsive cognitive style experienced classificational, correlational, and theoretical misconceptions. Meanwhile, students with reflective cognitive style are correlation misconceptions and theoretical misconceptions. To anticipate the occurrence of misconceptions, teachers should provide variations in the learning process, so that students can focus on the learning process and teachers often check students' understanding of concepts.
 Keywords: Number pattern, Misconception, Cognitive Style, Three Tier, CRI

Analysis of determinant factors of life expectation age in East Nusa Tenggara Province

East Nusa Tenggara Province is one of the poorest provinces in Indonesia and various prevention programs are being intensified. The conditions of poverty experienced also have an impact on life expectancy. Recognizing this condition, the main objective to be achieved in this study is to determine the effect of the variable Education, Health services and GRDP on the Life Expectancy of the people in the Province of NTT. The analytical tool used is Panel Data Regression Analysis using secondary data published by the Central Statistics Agency during 2014-2018. Time series data was taken for an observation period of 5 years starting from 2014-2018 for the independent variables which include education, health facilities, and gross regional domestic product on the dependent variable life expectancy. So that from a total of 22 regencies/cities in the province of NTT multiplied by 5 years of observation, the number of units of analysis is 110 arithmetical series. The results showed that the variables of Education, Health services and GRDP either partially or simultaneously have a significant effect on life expectancy in East Nusa Tenggara Province.

Equidistribution of numerical semigroup gaps modulo m

For a positive integer m, a finite set of integers is said to be equidistributed modulo m if the set contains an equal number of elements in each congruence class modulo m. In this paper, we consider the problem of determining when the set of gaps of a numerical semigroup S is equidistributed modulo m. Of particular interest is the case when the nonzero elements of an Apéry set of S form an arithmetic sequence. We explicitly describe such numerical semigroups S and determine conditions for which the sets of gaps of these numerical semigroups are equidistributed modulo m.

Discrete vector-valued nonuniform Gabor frames

Gabor frames have interested many mathematicians and physicists due to their potential applications in time-frequency analysis, in particular, signal processing. A Gabor system is a collection of vectors which is obtained by applying modulation and shift operators to non-zero functions in signal spaces. In many applications, for example, signal processing related to Gabor systems, the corresponding shifts may not be uniform. That is, the set associated with shifts may not be a group under usual addition. We analyze discrete vector-valued nonuniform Gabor frames (DVNUG frames, in short) in discrete vector-valued nonuniform signal spaces, where the indexing set associated with shifts may not be a subgroup of real numbers under usual addition, but a spectrum which is based on the theory of spectral pairs. First, we give necessary and sufficient conditions for the existence of DVNUG Bessel sequences in discrete vector-valued nonuniform signal spaces in terms of Fourier transformations of the modulated window sequences. We provide a characterization of DVNUG frames in discrete vector-valued nonuniform signal spaces. It is shown that DVNUG frames are stable under small perturbation of window sequences associated with given discrete vector-valued nonuniform Gabor systems. We observed that the arithmetic mean sequences associated with window sequences of a given DVNUG frame collectively constitute a discrete nonuniform Gabor frame. Finally, we discuss an interplay between window sequences of DVNUG systems and their corresponding coordinates.

Integer Colorings with No Rainbow 3-Term Arithmetic Progression

In this paper, we study the rainbow Erdős-Rothschild problem with respect to 3-term arithmetic progressions. We obtain the asymptotic number of $r$-colorings of $[n]$ without rainbow 3-term arithmetic progressions, and we show that the typical colorings with this property are 2-colorings. We also prove that $[n]$ attains the maximum number of rainbow 3-term arithmetic progression-free $r$-colorings among all subsets of $[n]$. Moreover, the exact number of rainbow 3-term arithmetic progression-free $r$-colorings of $\mathbb{Z}_p$ is obtained, where $p$ is any prime and $\mathbb{Z}_p$ is the cyclic group of order $p$.

Sets Avoiding Six-Term Arithmetic Progressions in $\mathbb{Z}_6^{n}$ are Exponentially Small

We show that sets avoiding six-term arithmetic progressions in $\mathbb{Z}_6^n$ have size at most $5.709^n$. It is also pointed out that the “product construction” does not work in this setting; in particular we show that for the extremal sizes in small dimensions we have $r_6(\mathbb{Z}_6)=5$, $r_6(\mathbb{Z}_6^2)=25$, and $ 117\leq r_6(\mathbb{Z}_6^n)\leq 124$.

Topological characteristic factors and independence along arithmetic progressions

In 1994 Glasner studied the topological characteristic factor for minimal systems. It is shown that up to canonically defined proximal extensions, a characteristic family for τd= T × T2 × ⋯ × Td is the family of canonical PI flows of order d − 1. In this paper, we generalize Glasner’s work to the product system of finitely many minimal systems and give its relative version. As applications, we derive several applications related to independence pairs along arithmetic progressions and Δ-transitivity.

Students' Errors Analysis in Solving Problems in the Material of Sequences and Series based on Hadar's Criteria in terms of Students' Mathematical Understanding

This study aims to describe students' errors in solving problems in the material of Sequences and Series based on Hadar's criteria in terms of students' mathematical understanding. The type of research used is qualitative research with a descriptive approach. There are 6 research subjects, namely 2 students with high mathematical understanding, 2 students with moderate mathematical understanding, and 2 students with low mathematical understanding. The instrument used in data collection was an understanding test containing 7 questions, a series and series problem test containing 3 questions, as well as interview guidelines. The results showed that all subjects who worked on questions related to determining the P-th term in the series geometry and in working on problems related to determining the cattle population tend to make technical errors. Meanwhile, in working on problems related to determining the number of terms in an arithmetic sequence, the subject did not experience any errors in completing the answer except for the fifth subject and the sixth subject who did not answer at all.

The Goldbach Conjecture With Summands In Arithmetic Progressions

Abstract We prove that, for almost all $r \leq N^{1/2}/\log^{O(1)}N$, for any given $b_1 \ (\mathrm{mod}\ r)$ with $(b_1, r) = 1$, and for almost all $b_2 \ (\mathrm{mod}\ r)$ with $(b_2, r) = 1$, we have that almost all natural numbers $2n \leq N$ with $2n \equiv b_1 + b_2 \ (\mathrm{mod}\ r)$ can be written as the sum of two prime numbers $2n = p_1 + p_2$, where $p_1 \equiv b_1 \ (\mathrm{mod}\ r)$ and $p_2 \equiv b_2 \ (\mathrm{mod}\ r)$. This improves the previous result which required $r \leq N^{1/3}/\log^{O(1)}N$ instead of $r \leq N^{1/2}/\log^{O(1)}N$. We also improve some other results concerning variations of the problem.

Further results on Vanishing Coefficients in infinite products of the form (qb,qp−b;qp)∞3(qjb,q2p−jb;q2p)∞

This paper examines the phenomenon of coefficients that vanish in a class of infinite products related to those first examined by Hirschhorn, and later by Tang, Baruah and Kaur, and the first author of this paper. Several infinite families of vanishing coefficient phenomena are found. In particular, it is shown that if p is a prime in one of several arithmetic progressions modulo 24, then there exist integer values for j, v and w such that if b is any integer (or in same cases, any odd integer, or any even integer), then a result of the following type holds: If the sequence [Formula: see text] is defined by [Formula: see text] then [Formula: see text] for all n.

Regular handicap graphs of order n ≡ 4 (mod 8)

A handicap distance antimagic labeling of a graph G = ( V , E ) with n vertices is a bijection f : V → {1, 2, …, n } with the property that f ( x i )= i , the weight w ( x i ) is the sum of labels of all neighbors of x i , and the sequence of the weights w ( x 1 ), w ( x 2 ),…, w ( x n ) forms an increasing arithmetic progression. A graph G is a handicap distance antimagic graph if it allows a handicap distance antimagic labeling. We construct r -regular handicap distance antimagic graphs of order n ≡ 4 (mod 8) for all feasible values of r .

Integer colorings with no rainbow [formula omitted]-term arithmetic progression

In this paper, we study the rainbow Erdős–Rothschild problem with respect to k-term arithmetic progressions. For a set of positive integers S⊆[n], an r-coloring of S is rainbowk-AP-free if it contains no rainbow k-term arithmetic progression. Let gr,k(S) denote the number of rainbow k-AP-free r-colorings of S. For sufficiently large n and fixed integers r≥k≥3, we show that gr,k(S)<gr,k([n]) for any proper subset S⊂[n]. Further, we prove that limn→∞gr,k([n])/(k−1)n=rk−1. Our result is asymptotically best possible and implies that, almost all rainbow k-AP-free r-colorings of [n] use only k−1 colors.

Moments of the distribution of k$k$‐free numbers in short intervals and arithmetic progressions

We show estimates for the distribution of k $k$ -free numbers in short intervals and arithmetic progressions that, in certain ranges, agree with a conjecture by Montgomery.

Sylvester power and weighted sums on the Frobenius set in arithmetic progression

Let a1,a2,…,ak be positive integers with gcd(a1,a2,…,ak)=1. Frobenius number is the largest positive integer that is NOT representable in terms of a1,a2,…,ak. When k≥3, there is no explicit formula in general, but some formulae may exist for special sequences a1,a2,…,ak, including, those forming arithmetic progressions and their modifications. In this paper, we give formulae for the power and weighted sum of nonrepresentable positive integers. As applications, we show explicit expressions of these sums for a1,a2,…,ak forming arithmetic progressions.

A Szemerédi-type theorem for subsets of the unit cube

We investigate gaps of $n$-term arithmetic progressions $x, x+y, \ldots, x+(n-1)y$ inside a positive measure subset $A$ of the unit cube $[0,1]^d$. If lengths of their gaps $y$ are evaluated in the $\ell^p$-norm for any $p$ other than $1, 2, \ldots, n-1$, and $\infty$, and if the dimension $d$ is large enough, then we show that the numbers $\|y\|_{\ell^p}$ attain all values from an interval, the length of which depends only on $n$, $p$, $d$, and the measure of $A$. Known counterexamples prevent generalizations of this result to the remaining values of the exponent $p$. We also give an explicit bound for the length of the aforementioned interval. The proof makes the bound depend on the currently available bounds in Szemer\'{e}di's theorem on the integers, which are used as a black box. A key ingredient of the proof are power-type cancellation estimates for operators resembling the multilinear Hilbert transforms. As a byproduct of the approach we obtain a quantitative improvement of the corresponding (previously known) result for side lengths of $n$-dimensional cubes with vertices lying in a positive measure subset of $([0,1]^2)^n$.

Increasing sequence of topologies on the set of positive integers

In this paper we deal with topologies defined on the set of positive integers. We introduce a sequence of topologies Dm that are subtopologies of Golomb's topology D. The sum of all topologies Dm is Golomb's topology. We show also that all topologies Dm are Haudsdorff but not regular on N. Further, we characterize connectedness of arithmetic progressions in each topology Dm. An immediate consequence of these characterizations is local connectedness of the space (N,Dm) for each m∈N.