Arithmetic Progressions Research Articles

Local edge (a, d) –antimagic coloring on sunflower, umbrella graph and its application

Suppose a graph G = (V, E) is a simple, connected and finite graph with vertex set V(G) and an edge set E(G). The local edge antimagic coloring is a combination of local antimagic labelling and edge coloring. A mapping f∶ V (G)→ {1, 2, ..., |V (G)|} is called local edge antimagic coloring if every two incident edges e_1and e_2, then the edge weights of e_1and e_2 maynot be the same, w(e_1) ≠ w(e_2), with e = uv ∈ G, w(e) = f(u)+ f(v) with the rule that the edges e are colored according to their weights, w_e. Local edge antimagic coloring has developed into local (a,d)-antimagic coloring. Local antimagic coloring is called local (a,d)-antimagic coloring if the set of edge weights forms an arithmetic sequence with a as an initial value and d as a difference value. The graphs used in this study are sunflower graphs and umbrella graphs. This research will also discuss one of the applications of local edge (a,d)-antimagic coloring, namely the design of the Sidoarjo line batik motif. The result show that χ_le(3,1) (Sf_n) = 3n and χ_le(3n/2,1) (U_(m,n) ) = m+1 . The local (a,d)-antimagic coloring is formed into a batik motif design with characteristics from the Sidoarjo region.

Arithmetic Sequences in Making Traditional Cast Nets in Lombok

Mathematical concepts can be found in the culture of the community, such as in daily activities and handicrafts. One type of handicraft is the pencar, a cast net (throw net) fishermen use to catch fish. This research aims to explore the arithmetic sequences involved in making a pencar. This research uses qualitative research with ethnography. The research was conducted in Marong Village, West Nusa Tenggara, Indonesia. Data were obtained by direct observation of the pencar and the process of making it, interviews with cast-net handycrafters, and literature studies on arithmetic sequences. The results of this study show that there are three arithmetic sequences in the pencar. The first sequence, the number of anak (meshes) between each anakan (widener) in the n-th widener row, is 3, 4, 5, and so on, formulated by Un = 3 + (n - 1). The second sequence, the number of anakan (widener) in the n-th widener row is 44, 45, 46, and so on, formulated by Un = 44 + (n - 1). The third sequence, the number of lubang (meshes) in the n-th anakan is 44, 90, 138, 188, and so on, formulated by Un = n^2 + 43n. The arithmetic sequences in pencar can be used as a problem context for culture-based learning in the arithmetic sequence topic.

The structure and number of Erdős covering systems

Introduced by Erdős in 1950, a covering system of the integers is a finite collection of arithmetic progressions whose union is the set \mathbb{Z} . Many beautiful questions and conjectures about covering systems have been posed over the past several decades, but until recently little was known about their properties. Most famously, the so-called minimum modulus problem of Erdős was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most 10^{16} . In this paper we answer another question of Erdős, asked in 1952, on the number of minimal covering systems. More precisely, we show that the number of minimal covering systems with exactly n elements is \exp \bigg( \bigg(\frac{4\sqrt{\tau}}{3} + o(1)\bigg) \frac{n^{3/2}}{(\log n)^{1/2}} \bigg) \quad \text{as $n \to \infty$, where} \quad \tau = \sum_{t = 1}^\infty \bigg( \log \frac{t+1}{t} \bigg)^2. En route to this counting result, we obtain a structural description of all covering systems that are close to optimal in an appropriate sense.

ANALISIS ETNOMATEMATIKA PADA AKTIVITAS MATEMATIKA BERBASIS BUDAYA PADA MASYARAKAT PESISIR DI TAMBAK GARAM

The purpose of this research is to explain how the mathematical activities in the salt ponds in Kebonangung Village are not realized by the coastal communities and students of the mathematical concepts related to ethnomathematics in mathematical activities in the salt ponds. This type of research is an ethnographic qualitative research method. Data obtained through interviews, observation, and documentation. The instrument in this study consisted of the main instrument, namely the researcher himself and auxiliary instruments in the form of interview guidelines, observation sheets, and documentation tools. The data validity technique used is method triangulation and source triangulation. The results of this study are salt ponds which have mathematical concepts including: (1). Rotation and circles that are on the windmill, (2). The rectangular shape found in the salt field, (3). Arithmetic sequences contained in the yield of each per tribe.

A pretentious proof of Linnik's estimate for primes in arithmetic progressions

AbstractIn the present paper, the author adopts a pretentious approach and recovers an estimate obtained by Linnik for the sums of the von Mangoldt function Λ on arithmetic progressions. It is the analogue of an estimate that Linnik established in his attempt to prove his celebrated theorem concerning the size of the smallest prime number of an arithmetic progression. Our work builds on ideas coming from the pretentious large sieve of Granville, Harper, and Soundararajan and it also borrows insights from the treatment of Koukoulopoulos on multiplicative functions with small averages.

Rankin-Selberg coefficients in large arithmetic progressions

Let $(\lambda_f(n))_{n\geq 1}$ be the Hecke eigenvalues of either a holomorphic Hecke eigencuspform or a Hecke-Maass cusp form $f$. We prove that, for any fixed $\eta>0$, under the Ramanujan-Petersson conjecture for $\rm GL_2$ Maass forms, the Rankin-Selberg coefficients $(\lambda_f(n)^2)_{n\geq 1}$ admit a level of distribution $\theta=2/5+1/260-\eta$ in arithmetic progressions.

Mathematical and Statistical Aspects of Estimating Small Oscillations Parameters in a Conservative Mechanical System Using Inaccurate Observations

This paper selects a set of reference points in the form of an arithmetic progression for planning an experiment to evaluate the parameters of systems of differential equations. This choice makes it possible to construct estimates of the parameters of a system of first-order differential equations based on the reversibility of the observation matrix, as well as estimates of the parameters of a system of second-order differential equations describing vibrations in a mechanical system by switching to a system of first-order differential equations. In turn, the reversibility of the observation matrix used in parameter estimation is established using the Vandermonde formula. A volumetric computational experiment has been carried out showing how, with an increase in the number of observations in the vicinity of reference points and with a decrease in the step of arithmetic progression, the accuracy of estimates of the parameters of the analyzed system increases. Among the estimated parameters, the most important are the oscillation frequencies of a conservative mechanical system, which establish its proximity to resonance, and therefore, determine the stability and reliability of the system.

A “pseudo-polynomial” algorithm for the Frobenius number and Gröbner basis

Given n⩾2 and a1,…,an∈N. Let S=〈a1,…,an〉 be a semigroup. The aim of this paper is to give an effective pseudo-polynomial algorithm on a1, which computes the Apéry set and the Frobenius number of S. We also find the Gröbner basis of the toric ideal defined by S, for the weighted degree reverse lexicographical order ≺w to x1,…,xn, without using Buchberger's algorithm. As an application we introduce and study some special classes of semigroups. Namely, when S is generated by generalized arithmetic progressions and generalized almost arithmetic progressions with the ratio a positive or a negative number. We determine symmetric and almost symmetric semigroups generated by a generalized arithmetic progression.

Two-Colorings of Normed Spaces without Long Monochromatic Unit Arithmetic Progressions

Given a natural , we construct a two-coloring of with the maximum metric satisfying the following. For any finite set of reals with diameter greater than such that the distance between any two consecutive points of does not exceed one, no isometric copy of is monochromatic. As a corollary, we prove that any normed space can be two-colored such that all sufficiently long unit arithmetic progressions contain points of both colors.

Implementation of “Kampus Mengajar" Programs to Improve Learning Motivation

This dedication is carried out to implement various activities of the 3rd Kampus Mengajar Program to motivate students to learn at Middle School 2 Pamarican. After observing the school, multiple problems were found, such as students' lack of enthusiasm during learning due to their lack of extrinsic motivation, students still needing help to read and count correctly, and teachers still using traditional learning media. This study used a qualitative descriptive method with data collection techniques through teacher interviews and school observations. This research was conducted at Middle School 2 Pamarican for four months, from March 2nd, 2022, to June 29th, 2022. This research was aimed at students in grades VII and VIII and was carried out directly. The results of this dedication activity include assisting in teaching, learning multiplication using jarimatika, operating numbers, arithmetic series, and the concept of 3 M with the GASING method, improving reading skills using five methods, learning to recite the Quran, and adapting technology aimed at increasing students' learning motivation. There were many obstacles and challenges during the implementation, especially when learning became limited, so teachers had to have high creativity to improve student learning outcomes.

A Bound for a Sum of Products of Two Characters and Its Application

Using the exponent pair method, a bound is derived for the sum ∑manb≤xχ1a(m)χ2b(n), where a,b are fixed positive integers, χ1,χ2 are primitive Dirichlet characters modulo q1 and q2, respectively, and χ1a,χ2b are not principal characters. As an application, an estimate for the error term in an asymptotic formula for the number of square-full integers simultaneously belonging to two arithmetic progressions is obtained.

A strengthening of Freiman's 3k−4$3k-4$ theorem

AbstractIn its usual form, Freiman's theorem states that if and are subsets of of size with small sumset (of size close to ), then they are very close to arithmetic progressions. Our aim in this paper is to strengthen this by allowing only a bounded number of possible summands from one of the sets. We show that if and are subsets of of size such that for any four‐element subset of the sumset has size not much more than , then already this implies that and are very close to arithmetic progressions.

Prime divisors of ℓ-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level ℓ

Let ℓ be any fixed prime number. We define the ℓ-Genocchi numbers by Gn:=ℓ(1−ℓn)Bn, with Bn the n-th Bernoulli number. They are integers. We introduce and study a variant of Kummer's notion of regularity of primes. We say that an odd prime p is ℓ-Genocchi irregular if it divides at least one of the ℓ-Genocchi numbers G2,G4,…,Gp−3, and ℓ-regular otherwise. With the help of techniques used in the study of Artin's primitive root conjecture, we give asymptotic estimates for the number of ℓ-Genocchi irregular primes in a prescribed arithmetic progression in case ℓ is odd. The case ℓ=2 was already dealt with by Hu et al. (2019) [14]. Using similar methods we study the prime factors of (1−ℓn)B2n/2n and (1+ℓn)B2n/2n. This allows us to estimate the number of primes p≤x for which there exist modulo p Ramanujan-style congruences between the Fourier coefficients of an Eisenstein series and some cusp form of prime level ℓ.

Distribution of moments of Hurwitz class numbers in arithmetic progressions and holomorphic projection

In this paper, we study moments of Hurwitz class numbers associated to imaginary quadratic orders restricted into fixed arithmetic progressions. In particular, we fix t t in an arithmetic progression t ≡ m ( mod M ) t\equiv m\ \, \left ( \operatorname {mod} \, M \right ) and consider the ratio of the 2 k 2k -th moment to the zeroeth moment for H ( 4 n − t 2 ) H(4n-t^2) as one varies n n . The special case n = p r n=p^r yields as a consequence asymptotic formulas for moments of the trace t ≡ m ( mod M ) t\equiv m\ \, \left ( \operatorname {mod} \, M \right ) of Frobenius on elliptic curves over finite fields with p r p^r elements.

DISTRIBUTION OF FROBENIUS ELEMENTS IN FAMILIES OF GALOIS EXTENSIONS

AbstractGiven a Galois extension $L/K$ of number fields, we describe fine distribution properties of Frobenius elements via invariants from representations of finite Galois groups and ramification theory. We exhibit explicit families of extensions in which we evaluate these invariants and deduce a detailed understanding and a precise description of the possible asymmetries. We establish a general bound on the generic fluctuations of the error term in the Chebotarev density theorem, which under GRH is sharper than the Murty–Murty–Saradha and Bellaïche refinements of the Lagarias–Odlyzko and Serre bounds, and which we believe is best possible (assuming simplicity, it is of the quality of Montgomery’s conjecture on primes in arithmetic progressions). Under GRH and a hypothesis on the multiplicities of zeros up to a certain height, we show that in certain families, these fluctuations are dominated by a constant lower order term. As an application of our ideas, we refine and generalize results of K. Murty and of Bellaïche, and we answer a question of Ng. In particular, in the case where $L/{\mathbb {Q}}$ is Galois and supersolvable, we prove a strong form of a conjecture of K. Murty on the unramified prime ideal of least norm in a given Frobenius set. The tools we use include the Rubinstein–Sarnak machinery based on limiting distributions and a blend of algebraic, analytic, representation theoretic, probabilistic and combinatorial techniques.

Bounds on bilinear forms with Kloosterman sums

AbstractWe prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by É. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, É. Fouvry, E. Kowalski, Ph. Michel and D. Milićević (2017), E. Kowalski, Ph. Michel and W. Sawin (2019, 2020) and I. E. Shparlinski (2019). These improvements rely on new estimates for Type II bilinear forms with incomplete Kloosterman sums. We also establish new estimates for bilinear forms with one variable from an arbitrary set by introducing techniques from additive combinatorics over prime fields. Some of these bounds have found a crucial application in the recent work of Wu (2020) on asymptotic formulas for the fourth moments of Dirichlet ‐functions. As new applications, an estimate for higher moments of averages of Kloosterman sums and the distribution of divisor function in a family of arithmetic progressions are also given.

On classification of sequences containing arbitrarily long arithmetic progressions

In this paper, we study the classification of sequences containing arbitrarily long arithmetic progressions. First, we deal with the question how the polynomial map [Formula: see text] can be extended so that it contains arbitrarily long arithmetic progressions. Under some growth conditions, we construct sequences which contain arbitrarily long arithmetic progressions. Also, we give a uniform and explicit arithmetic progression rank bound for a large class of sequences. Consequently, a dichotomy result is deduced on the finiteness of the arithmetic progression rank of certain sequences. Therefore, in this paper, we see a way to determine the finiteness of the arithmetic progression rank of various sequences satisfying some growth conditions.

Sifting for small primes from an arithmetic progression

In this work and its sister paper (Friedlander and Iwaniec (2023)), we give a new proof of the famous Linnik theorem bounding the least prime in an arithmetic progression. Using sieve machinery in both papers, we are able to dispense with the log-free zero density bounds and the repulsion property of exceptional zeros, two deep innovations begun by Linnik and relied on in earlier proofs.

On Gaussian decay rates of harmonic oscillators and equivalences of related Fourier uncertainty principles

We make progress on a question posed by Vemuri on the optimal Gaussian decay of harmonic oscillators, proving the original conjecture up to an arithmetic progression of times. The techniques used are a suitable translation of the problem at hand in terms of the free Schrödinger equation, the machinery developed in the work of Cowling, Escauriaza, Kenig, Ponce and Vega (2010), and a lemma which relates decay on average to pointwise decay. Such a lemma produces many more consequences in terms of equivalences of uncertainty principles. Complementing such results, we provide endpoint results in particular classes induced by certain Laplace transforms, both to the decay lemma and to the remaining cases of Vemuri’s conjecture, shedding light on the full endpoint question.

Deterministic Construction of Compressed Sensing Measurement Matrix with Arbitrary Sizes via QC-LDPC and Arithmetic Sequence Sets

It is of great significance to construct deterministic measurement matrices with good practical characteristics in Compressed Sensing (CS), including good reconstruction performance, low memory cost and low computing resources. Low-density-parity check (LDPC) codes and CS codes can be closely related. This paper presents a method of constructing compressed sensing measurement matrices based on quasi-cyclic (QC) LDPC codes and arithmetic sequence sets. The cyclic shift factor in each submatrix of QC-LDPC is determined by arithmetic sequence sets. Compared with random matrices, the proposed method has great advantages because it is generated based on a cyclic shift matrix, which requires less storage memory and lower computing resources. Because the restricted isometric property (RIP) is difficult to verify, mutual coherence and girth are used as computationally tractable indicators to evaluate the measurement matrix reconstruction performance. Compared with several typical matrices, the proposed measurement matrix has the minimum mutual coherence and superior reconstruction capability of CS signal according to one-dimensional (1D) signals and two-dimensional (2D) image simulation results. When the sampling rate is 0.2, the maximum (minimum) gain of peak signal-to-noise ratio (PSNR) and structural similarity index (SSIM) is up to 2.89 dB (0.33 dB) and 0.031 (0.016) while using 10 test images. Meanwhile, the reconstruction time is reduced by nearly half.