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Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for multivariate multipoint evaluation over any finite field \(\mathbb {F}\) that outputs the evaluations of an m-variate polynomial of degree less than d in each variable at N points in time, \(\begin{equation*} (d^m+N)^{1+o(1)}\cdot {{\sf poly}}(m,d,\log |\mathbb {F}|), \end{equation*}\) for all \(m\in \mathbb {N}\) and all sufficiently large \(d\in \mathbb {N}\). A previous work of Kedlaya and Umans (FOCS 2008 and SICOMP 2011) achieved the same time complexity when the number of variables m is at most \(d^{o(1)}\) and had left the problem of removing this condition as an open problem. A recent work of Bhargava, Ghosh, Kumar, and Mohapatra (STOC 2022) answered this question when the underlying field is not too large and has characteristic less than \(d^{o(1)}\). In this work, we remove this constraint on the number of variables over all finite fields, thereby answering the question of Kedlaya and Umans over all finite fields. Our algorithm relies on a non-trivial combination of ideas from three seemingly different previously known algorithms for multivariate multipoint evaluation, namely the algorithms of Kedlaya and Umans, that of Björklund, Kaski, and Williams (IPEC 2017 and Algorithmica 2019), and that of Bhargava, Ghosh, Kumar, and Mohapatra, together with a result of Bombieri and Vinogradov from analytic number theory about the distribution of primes in an arithmetic progression. We also present a second algorithm for multivariate multipoint evaluation that is completely elementary and, in particular, avoids the use of the Bombieri–Vinogradov theorem. However, it requires a mild assumption that the field size is bounded by an exponential tower in d of bounded height. More specifically, our second algorithm solves the multivariate multipoint evaluation problem over a finite field \(\mathbb {F}\) in time, \(\begin{equation*} (d^m+N)^{1+o(1)}\cdot {{\sf poly}}(m,d,\log |\mathbb {F}|), \end{equation*}\) for all \(m\in \mathbb {N}\) and all sufficiently large \(d\in \mathbb {N}\) , provided that the size of the finite field \(\mathbb {F}\) is at most \((\exp (\exp (\exp (\cdots (\exp (d)))))\) , where the height of this tower of exponentials is fixed. [ABSTRACT FROM AUTHOR]

We give a purely combinatorial proof for the infinitary van der Waerden's theorem. [ABSTRACT FROM AUTHOR]

The global impact of the SARS-CoV-2 pandemic has underscored the need for a deeper understanding of viral evolution to anticipate new viruses or variants. Genetic recombination is a fundamental mechanism in viral evolution, yet it remains poorly understood. In this study, we conducted a comprehensive research on the genetic regions associated with genetic recombination features in SARS-CoV-2. With this aim, we implemented a two-phase transfer learning approach using genomic spectrograms of complete SARS-CoV-2 sequences. In the first phase, we utilized a pre-trained VGG-16 model with genomic spectrograms of HIV-1, and in the second phase, we applied HIV-1 VGG-16 model to SARS-CoV-2 spectrograms. The identification of key recombination hot zones was achieved using the Grad-CAM interpretability tool, and the results were analyzed by mathematical and image processing techniques. Our findings unequivocally identify the SARS-CoV-2 Spike protein (S protein) as the pivotal region in the genetic recombination feature. For non-recombinant sequences, the relevant frequencies clustered around 1/6 and 1/12. In recombinant sequences, the sharp prominence of the main hot zone in the Spike protein prominently indicated a frequency of 1/6. These findings suggest that in the arithmetic series, every 6 nucleotides (two triplets) in S may encode crucial information, potentially concealing essential details about viral characteristics, in this case, recombinant feature of a SARS-CoV-2 genetic sequence. This insight further underscores the potential presence of multifaceted information within the genome, including mathematical signatures that define an organism's unique attributes. [ABSTRACT FROM AUTHOR]

The Hurwitz zeta-function ζ (s , α) , s = σ + i t , with parameter 0 < α ⩽ 1 is a generalization of the Riemann zeta-function ζ (s) ( ζ (s , 1) = ζ (s) ) and was introduced at the end of the 19th century. The function ζ (s , α) plays an important role in investigations of the distribution of prime numbers in arithmetic progression and has applications in special function theory, algebraic number theory, dynamical system theory, other fields of mathematics, and even physics. The function ζ (s , α) is the main example of zeta-functions without Euler's product (except for the cases α = 1 , α = 1 / 2 ), and its value distribution is governed by arithmetical properties of α. For the majority of zeta-functions, ζ (s , α) for some α is universal, i.e., its shifts ζ (s + i τ , α) , τ ∈ R , approximate every analytic function defined in the strip { s : 1 / 2 < σ < 1 } . For needs of effectivization of the universality property for ζ (s , α) , the interval for τ must be as short as possible, and this can be achieved by using the mean square estimate for ζ (σ + i t , α) in short intervals. In this paper, we obtain the bound O (H) for that mean square over the interval [ T − H , T + H ] , with T 27 / 82 ⩽ H ⩽ T σ and 1 / 2 < σ ⩽ 7 / 12 . This is the first result on the mean square for ζ (s , α) in short intervals. In forthcoming papers, this estimate will be applied for proof of universality for ζ (s , α) and other zeta-functions in short intervals. [ABSTRACT FROM AUTHOR]

Calculations about compound interest serve as the basis of most financial decisions; therefore, it is imperative to explore what mathematical knowledge people need to correctly calculate simple and compound interest. The aim of this study is to analyze the relationship between college students' competence in calculating simple and compound interest and their understanding of the arithmetic and geometric progressions. It is also pointed out whether the results vary according to gender. Population proportion tests are carried out, and gender proportion differences are considered for inferential analysis. The dichotomous Probit model was used for correlation analysis. Results demonstrate that 59.8% of students know how to formulate a whole-number succession, and only 30.9% in the case of fractional numbers. Less than 50% of students can calculate compound interest, but 76.7% can calculate simple interest. There is no significant difference between men and women. The results show a positive relationship between male students' competence in calculating compound interest with the possibility to correctly formulate a geometric succession. Findings can be useful for mathematics teaching strategy design and its applications in finance contexts with the purpose of training students to be better at finance decision making. [ABSTRACT FROM AUTHOR]

Accurate estimation of design floods is necessary for developing effective flood‐management strategies. Climate change (CC) studies on floods generally consider alterations in mean runoff using ensembles compared to a base period. In this study, we examined the plausibility and implications of applying individual climate model‐generated flows versus their ensembles to estimate peak floods (magnitude and timing of occurrence), using Budhigandaki River Basin of Nepal as a case study. Annual maximum one‐day floods were derived for four future climate scenario projections (cold‐dry, cold‐wet, warm‐wet, and warm‐dry) from simulated daily flow series. Future floods of six return periods estimated for the individual climate scenarios were compared with their "Ensemble" (combiner for the ensemble series is the arithmetic mean of daily floods), "Average," and 'Baseline." Results showed that magnitudes of the flood peaks are such that those estimated using "Ensemble" < "Average" < individual series. We conclude that ensemble series should not be used for flood estimation because of the averaging effect. Designers should consider at the least the "Average" instead of the "Ensemble" series while designing climate‐resilient flood structures. Furthermore, the occurrences of flood peaks are likely to be confined within the monsoon season for the "Ensemble" but spread out in the other months for the individual climate scenarios. This could have direct implications on the availability and mobilization of resources as well as the need for a year‐round operational early warning system for flood risk management. [ABSTRACT FROM AUTHOR]

Communication among hard-of-hearing individuals presents challenges, and to facilitate communication, sign language is preferred. Many people in the deaf and hard-of-hearing communities struggle to understand sign language due to their lack of sign-mode knowledge. Contemporary researchers utilize glove and vision-based approaches to capture hand movement and analyze communication; most researchers use vision-based techniques to identify disabled people's communication because the glove-based approach causes individuals to feel uncomfortable. However, the glove solution successfully identifies motion and hand dexterity, even though it only recognizes the numbers, words, and letters being communicated, failing to identify sentences. Therefore, artificial intelligence (AI) is integrated with the sign language prediction system to identify disabled people's sentence-based communication. Here, wearable glove-related sign language information is utilized to analyze the recognition system's efficiency. The collected inputs are processed using progression learning deep convolutional neural networks (PLD-CNNs). The technique known as progression learning processes sentences by dividing them into words, creating a training dataset. The model assists in efforts to understand sign language sentences. A memetic optimization algorithm is used to calibrate network performance, minimizing recognition optimization problems. This process maximizes convergence speed and reduces translation difficulties, enhancing the overall learning process. The created system is developed using the MATLAB (R2021b) tool, and its proficiency is evaluated using performance metrics. The experimental findings illustrate that the proposed system works by recognizing sign language movements with excellent precision, recall, accuracy, and F1 scores, rendering it a powerful tool in the detection of gestures in general and sign-based sentences in particular. [ABSTRACT FROM AUTHOR]

In this article, we study the distribution of index of Farey fractions by providing asymptotic formulas for moments of index of Farey fractions twisted by Dirichlet characters for Farey fractions with B -free denominators. Additionally, we reconsider the square-free case earlier done in Alkan et al. (Ramanujan J 16(2):131–161, 2008), and obtain new results for moments of indices with square-free denominators. We also obtain higher level correlation measures of the index function, generalizing earlier known results on two level correlations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Assuming a widely believed hypothesis concerning the least prime in an arithmetic progression, we show that polynomials of degree less than n over a finite field Fq with q elements can be multiplied in time O(n logq log(n logq)), uniformly in q. Under the same hypothesis, we show how to multiply two n-bit integers in time O(n logn); this algorithm is somewhat simpler than the unconditional algorithm from the companion paper [22]. Our results hold in the Turing machine model with a finite number of tapes. [ABSTRACT FROM AUTHOR]

Switching power converters bring a desired output magnitude using semiconductor devices. These switching converters are generally classified as voltage lift and super lift. The conventional lift converters such as buck, boost and buck–boost come under the category of voltage lift technique, in which the output voltage increases in arithmetic progression. The other lift approach that increases the output voltage in the geometric progression is the super-lift method. Nowadays super-lift converters find their role in various applications, because of their high gain and low ripple. The converter taken for analysis is a modified negative output super-lift Luo converter (MNOSLC). Current mode control (CMC) is employed to maintain a stable DC voltage at the load side. This is investigated using different types of controllers such as PI, fuzzy, sliding mode controller (SMC) and fuzzy SMC. The performance of these controllers is compared and it is found that fuzzy SMC results in better line and load regulation. A prototype is developed that validates the simulation results. [ABSTRACT FROM AUTHOR]

This document provides a summary of a special issue of the journal Mathematics, which features research papers in the field of algebraic, analytic, and computational number theory. The papers cover a wide range of topics, including generalized Lucas pseudoprimality, Fibonacci numbers, piecewise linear difference equations, properties of Euler's function, regularities in ordered semihypergroups, zero-density results for automorphic functions, properties of Fibonacci finite operator quaternions, and more. The authors of the special issue believe that the techniques used in these papers have the potential to yield new results in these areas of mathematics. [Extracted from the article]

The aggregation operators are important tools in the process of information aggregation. The aim of this paper is to investigate arithmetic aggregation operators under the type2 picture fuzzy environment with the help of new operations on type-2 picture fuzzy numbers. The type-2 picture fuzzy set is an extended version of Cuong’s picture fuzzy set, which not only considers the degree of acceptance or rejection, but also takes into account the neutral degree and invalid degree during the analysis. Under these environments, by using the new operations on type-2 picture fuzzy numbers, several series of arithmetic aggregation operators for type-2 picture fuzzy sets are proposed, namely type-2 picture fuzzy weighted average operators, type-2 picture fuzzy ordered weighted average operators, and type-2 picture fuzzy hybrid average operators, along with their desirable properties. Furthermore, a decision-making approach based on these operators is also presented to make the evaluation result more objective. Finally, an illustrative example is provided to demonstrate the practicality and reasonableness of the proposed method. [ABSTRACT FROM AUTHOR]

We study covering sumsets of a prime field based on its multiplicative structure. By developing various sufficient analytic and algebraic criteria for their existence, it is shown that covering sumsets arise in two main families, namely in the form of complementary sumsets and in the form of double sumsets. In each case, the abundance of covering sumsets is supported by providing asymptotically growing lower bounds on their number which in turn point out a rich array of fruitful connections to seemingly unrelated topics such as the Titchmarsh divisor problem, Mersenne primes, Fermat quotients, partitions into cycles, quadratic reciprocity, Gauss and Jacobi sums, and density results in class field theory resulting from Chebotarev's theorem. Moreover, representations of an element taken from a prime field, in terms of the sums in a covering sumset, furnish us with new formulas for the class numbers of quadratic fields, Bernoulli numbers and Bernoulli polynomials. In this way, curious tendencies among the number of representations are discovered over half intervals. Lastly, our findings show in different circumstances that the summands of a covering sumset can seldom form an arithmetic progression, thereby indicating a tension between additive and multiplicative structures in a prime field. [ABSTRACT FROM AUTHOR]

Research in learning media development is driven by the common problem of student disengagement, which hinders their understanding of the material. The specific problem addressed in this research is students' lack of attention and comprehension while studying arithmetic sequences and series. The purpose of this study is to document the process and outcomes of developing interactive learning media using Articulate Storyline specifically for the material on sequences and arithmetic series. The research follows the Research and Development (R&D) approach, employing the Thiagarajan model (Model 4-D) consisting of four stages: 1) the definition stage (define), 2) the design stage (design), 3) the development stage (development), and 4) the dissemination stage (disseminate). The research was conducted with a sample of 31 students from class X B at SMA Tunas Luhur. The results of the media validity test showed a score of 4.82, indicating high validity. Furthermore, the practicality and effectiveness tests conducted with both small and large groups achieved a perfect score of 100% in all aspects. This research indicates that the interactive learning media for arithmetic sequences and series based on Articulate Storyline meet the criteria of validity, practicality, and effectiveness.

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This paper establishes the linear algebra of the (r, β)-Stirling matrix. Along the way, this paper derives various identities, such as its factorization and relationship to the Pascal matrix and the Stirling matrix of the second kind. Additionally, this paper develops a natural extension of the Vandermonde matrix, which can be used to study and evaluate successive power sums of arithmetic progressions. [ABSTRACT FROM AUTHOR]

For given positive integers a 1 , a 2 , ⋯ , a k with gcd (a 1 , a 2 , ⋯ , a k) = 1 , the denumerant d (n) = d (n ; a 1 , a 2 , ⋯ , a k) is the number of nonnegative solutions (x 1 , x 2 , ⋯ , x k) of the linear equation a 1 x 1 + a 2 x 2 + ⋯ + a k x k = n for a positive integer n. For a given nonnegative integer p, let S p = S p (a 1 , a 2 , ⋯ , a k) be the set of all nonnegative integer n's such that d (n) > p . In this paper, by introducing the p-numerical semigroup, where the set N 0 \ S p is finite, we give explicit formulas of the p-Frobenius number, which is the maximum of the set N 0 \ S p , and related values for the triple of arithmetic progressions. The main aim is to determine the elements of the p-Apéry set. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Sylvester's theorem states that every number can be decomposed into a sum of consecutive positive integers except powers of 2. In a way, this theorem characterizes the partitions of a number as a sum of consecutive integers. The first generalization we propose of the theorem characterizes the partitions of a number as a sum of arithmetic progressions with positive terms. In addition to synthesizing and rediscovering known results, the method we propose allows us to state a second generalization and characterize the partitions of a number into parts whose differences between consecutive parts form an arithmetic progression. To achieve this, we will analyze the set of divisors in arithmetics that modify the usual definition of the multiplication operation between two integers. As we will see, symmetries arise in the set of divisors based on two parameters: t 1 , being even or odd, and t 2 , congruent to 0, 1, or 2 (mod 3). This approach also leads to a unique representation result of the same nature as Sylvester's theorem, i.e., a power of 3 cannot be represented as a sum of three or more terms of a positive integer sequence such that the differences between consecutive terms are consecutive integers. [ABSTRACT FROM AUTHOR]

Using the exponent pair method, a bound is derived for the sum ∑ m a n b ≤ x χ 1 a (m) χ 2 b (n) , where a , b are fixed positive integers, χ 1 , χ 2 are primitive Dirichlet characters modulo q 1 and q 2 , respectively, and χ 1 a , χ 2 b 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. [ABSTRACT FROM AUTHOR]

In this paper, we explore the existence of m-term arithmetic progressions in F q n with a given common difference whose terms are all primitive elements, and at least one of them is normal. We obtain asymptotic results for m ≥ 4 and concrete results for m ∈ { 2 , 3 } , where the complete list of exceptions when the common difference belongs to F q is obtained. The proofs combine character sums, sieve estimates and computational arguments using the software SageMath. [ABSTRACT FROM AUTHOR]

In this paper, we present an algorithm on vanishing coefficients with arithmetic progressions in the sum of two generalized eta-quotients by using the theory of modular forms, and utilize our algorithm to prove five conjectures on matching coefficients in series expansions of certain q -products and their reciprocals. One of which was provided by Baruah and Das, and the others were found by Schlosser. For instance, we prove that for any n ≥ 0 , λ 12 (10 n + r) = λ 12 ′ (10 n + r − 6) , r ∈ { 7 , 9 } , where the sequences { λ 12 (n) } n ≥ 0 and { λ 12 ′ (n) } n ≥ 0 are defined by ∑ n = 0 ∞ λ 12 (n) q n = 1 R (q) R (q 2) R (q 4) R (q 8) = (∑ n = 0 ∞ λ 12 ′ (n) q n) − 1 , and where R (q) is the Rogers–Ramanujan continued fraction, which has the following celebrated product representation: R (q) = ∏ n = 0 ∞ (1 − q 5 n + 1) (1 − q 5 n + 4) (1 − q 5 n + 2) (1 − q 5 n + 3). Finally, we find that this phenomenon also exists in other infinite q -series expansions. [ABSTRACT FROM AUTHOR]

Let...) be a smooth function supported on [1, 2] with derivatives bounded by... and d3(n) be the number of ways to write n as a product of three factors. We get the asymptotic formula for the nonlinear exponential sum.... [ABSTRACT FROM AUTHOR]

For integers m and n, we study the problem of finding good lower bounds for the size of progression-free sets in (Z m n , +) . Let r k (Z m n) denote the maximal size of a subset of Z m n without arithmetic progressions of length k and let P - (m) denote the least prime factor of m. We construct explicit progression-free sets and obtain the following improved lower bounds for r k (Z m n) : If k ≥ 5 is odd and P - (m) ≥ (k + 2) / 2 , then r k (Z m n) ≫ m , k ⌊ k - 1 k + 1 m + 1 ⌋ n n ⌊ k - 1 k + 1 m ⌋ / 2 . If k ≥ 4 is even, P - (m) ≥ k and m ≡ - 1 mod k , then r k (Z m n) ≫ m , k ⌊ k - 2 k m + 2 ⌋ n n ⌊ k - 2 k m + 1 ⌋ / 2 . Moreover, we give some further improved lower bounds on r k (Z p n) for primes p ≤ 31 and progression lengths 4 ≤ k ≤ 8 . [ABSTRACT FROM AUTHOR]

In this paper, we shall establish a hierarchy of functional equations (as a G-function hierarchy) by unifying zeta-functions that satisfy the Hecke functional equation and those corresponding to Maass forms in the framework of the ramified functional equation with (essentially) two gamma factors through the Fourier–Whittaker expansion. This unifies the theory of Epstein zeta-functions and zeta-functions associated to Maass forms and in a sense gives a method of construction of Maass forms. In the long term, this is a remote consequence of generalizing to an arithmetic progression through perturbed Dirichlet series. [ABSTRACT FROM AUTHOR]

We reformulate the fourth-order equation of the Moore–Gibson–Thompson (MGT) type to a fractional semilinear fourth-order equation with structural damping and a time-nonlocal nonlinearity. The solution blow-up for this problem is established by the test function method. First, we recall some definitions and elementary properties of the fractional derivatives, and then we study the absence of global weak solutions. [ABSTRACT FROM AUTHOR]

A graph G(Z, D) with vertex set Z is called an integer distance graph if its edge set is obtained by joining two elements of Z by an edge whenever their absolute difference is a member of D. When D = P or D ⊆ P where P is the set of all prime numbers then we call it a prime distance graph. After establishing the chromatic number of G(Z, P) as four, Eggleton has classified the collection of graphs as belonging to class i if the chromatic number of G(Z, D) is i. The problem of characterizing the family of graphs belonging to class i when D is of any given size is open for the past few decades. As coloring a prime distance graph is equivalent to producing a prime distance labeling for vertices of G, we have succeeded in giving a prime distance labeling for certain class of all graphs considered here. We have proved that if D = {2, 3, 5, 7, 7th prime, 10th prime, 13th prime, 16th prime, (7 + Ps-1 j=1 4 ×3j)th prime, . . ., (4 + Ps j=1 4 × 3j)th prime for any s ∈ N}, then there exists a prime distance graph with distance set D in class 4 and if D = {2, 3, 5, 4th prime, 6th prime, 8th prime, (4 + Ps-1 j=1 3 × 2j)th prime, . . ., (2 + Ps j=1 3 × 2j)th prime for any s ∈ N} then there exists a prime distance graphs with distance set D in class 3. Further, we have also obtained some more interesting results that are either general or existential such as a) If D is a specific sequence of integers in arithmetic progression then there exist a prime distance graph with distance set D, b) If G is any prime distance graph in class i for 1 ≤ i ≤ 4 then G × K2 is also a prime distance graph in the respective class i, c) A countable union of disjoint copies of prime distance graph is again a prime distance graph, d) The Middle/Total graph of a path on n vertices is a prime distance graph. In addition we also provide a new different proof for establishing a fact that all cycles are prime distance graph. [ABSTRACT FROM AUTHOR]

In the spirit of the landmark work of Granville and Soundararajan, we further develop the theory of limitations to equidistribution in arithmetic progressions for a general class of nonnegative arithmetical functions. Our framework can be applied to numerous specific functions of interest to obtain new results. In particular, we obtain average results on irregularities in equidistribution for sequences of Beatty primes, integers free of small primes, as well as the divisor function supported on integers free of small primes. We also obtain limitations to equidistribution of primes in short arithmetic progressions. [ABSTRACT FROM AUTHOR]

For any prime p and t|(p - 1), let A be the multiplicative subgroup of Z* p of order t, and 2A = A + A, a union of A cosets of A, together with f0g in case t is even. For fixed n, we characterize all A for which A = n. For n = 1; 2; 3; 4 we provide, with proof, a complete list of all such groups, while for n = 5 to 10 we make a conjecture based on our data. A has maximal doubling if λA = min(k, [t/2]). We show A has maximal doubling if t < log3 p. Finally, we find all groups A contained in an arithmetic progression of length at most 3/2 |A|, generalizing a result of Chowla, Mann and Straus. [ABSTRACT FROM AUTHOR]

Given a family F = {A1;: :: ;As} of subsets of Zn, define ΔF to be the multiset of all (cyclic) distances dist(x; y), where {x; y} - Ai, x 6= y, for some i 2 {1; ... s}. Taking inspiration from a Euclidean distance problem of Erd}os, we say that F is Erd}os-deep if the multiplicities of distances that occur in F are precisely 1; 2; ... k - 1 for some integer k. In the case s = 1, it is known that a modular arithmetic progression in Zn achieves this property (under mild conditions); conversely, arithmetic progressions are the only such sets, except for one sporadic case when n = 6. Here, we consider in detail the case s = 2. In particular, we classify Erd}os-deep pairs fA1;A2g when each Ai is an arithmetic progression in Zn. We also give a construction of a much wider class of Erd}os-deep families {A1,...}Asg when s is a square integer. [ABSTRACT FROM AUTHOR]

Recently, a relation between Schreier-type sets and TurÁn graphs was discovered. In this note, we give a combinatorial proof and obtain a generalization of the relation. Specifically, for p, q ≥ 1, let Aq := {F ⊂ N: |F| = 1 or F is an arithmetic progression with difference q} and Sr(n, p, q) := #{F ⊂ {1,..., n}: p min F ⊂ |F| and F ⊂ Aq}. We show that Sr(n, p, q) = T(n + 1, pq + 1, q), where T(...) is the number of edges of a modification of Tur'an graphs. We also prove that Sr(n, p, q) is the partial sum of certain sequences. [ABSTRACT FROM AUTHOR]

Let f be a normalized primitive holomorphic cusp form of even integral weight k for the full modular group Γ = SL(2, Z). Denote by λf (n) the nth normalized Fourier coefficient of f. In this paper, we are interested in an asymptotic formula for the sum X n≤x n -¡ l(mod q) λ 2j f (n), where j ≥ 5 is any positive integer, and q is a prime with (l, q) = 1. [ABSTRACT FROM AUTHOR]

The Lerch zeta function is defined by a Dirichlet series depending on two fixed parameters. In the paper, we consider the approximation of analytic functions by discrete shifts of the Lerch zeta function, and we prove that, for arbitrary parameters and a step of arithmetic progression, there is a closed non-empty subset of the space of analytic functions defined in the critical strip such that its functions can be approximated by discrete shifts of the Lerch zeta function. The set of those shifts is infinite, and it has a positive density. For the proof, the weak convergence of probability measures in the space of analytic functions is applied. [ABSTRACT FROM AUTHOR]

The inter-annual stable carbon isotope ratio (δ13C) of three tree-ring cores of P. euphratica (Populus euphratica Oliv.) was determined from Ejina Oasis in Northwest China. A robust and representative δ13C chronology is generated from the three δ13C series using an arithmetic mean method. After eliminating the influence of the δ13C from elevated atmospheric carbon dioxide (CO2) concentration, we obtained a carbon isotopic discrimination (Δ13C) chronology. According to the significant correlation between the tree-ring Δ13C and instrumental data, we reconstructed the mean maximum temperature anomalies from previous December to current September (TDS) for the period 1901–2011. The reconstruction explained 43.6% of the variance over the calibration period. Three high-temperature periods (1929–1965, 1972–1974, and 1992–2006) and three low-temperature periods (1906–1926, 1966–1968, and 1975–1991) were found in the reconstructed series. Comparisons between the reconstructed TDS and the observed mean temperature from previous December to current September in Anxi meteorological station and the temperature index in north-central China demonstrated the reconstructed TDS has the advantage of reliability and stability. The significant spatial correlation declared that the reconstruction has a broad spatial representation and can represent the temperature variation characteristics in a wide geographical area. In addition, we found that the area of Ejina Oasis is smaller (larger) when the mean maximum temperature is higher (lower), which may be due to a conjunction effect of natural and anthropogenic activities. Significant periodicities and correlations suggested that the TDS variations in Ejina Oasis were regulated by solar radiation and atmospheric circulations at the interannual and interdecadal time scales. [ABSTRACT FROM AUTHOR]

We examine a few mathematical characteristics of Rogers–Ramanujan type identities as a follow-up work. Recently authors interpreted Rogers–Ramanujan type identities combinatorially using signed color partitions. In the present study, we discovered several congruences for the coefficients of powers of q that are in arithmetic progressions modulo powers of 2 and 3. [ABSTRACT FROM AUTHOR]

This paper reports on the formalization in Mizar system [1], [2] of ten selected problems from W. Sierpinski's book "250 Problems in Elementary Number Theory" [7] (see [6] for details of this concrete dataset). This article is devoted mainly to arithmetic progressions: problems 52, 54, 55, 56, 60, 64, 70, 71, and 73 belong to the chapter "Arithmetic Progressions", and problem 50 is from "Relatively Prime Numbers". [ABSTRACT FROM AUTHOR]

We show that there is a red-blue colouring of [N] with no blue 3-term arithmetic progression and no red arithmetic progression of length eC(log N)3/4 (log log N)1/4. Consequently, the two-colour van der Waerden number w(3,k) is bounded below by kb(k), where b(k) = c(log k/log log k)1/3. Previously it had been speculated, supported by data, that w(3,k) = O(k²). [ABSTRACT FROM AUTHOR]

In this note we show a simple formula for the coefficients of the polynomial associated with the sums of powers of the terms of an arbitrary arithmetic progression. This formula consists of a double sum involving only ordinary binomial coefficients and binomial powers. Arguably, this is the simplest formula that can probably be found for the said coefficients. Furthermore, we give an explicit formula for the Bernoulli polynomials involving the Stirling numbers of the first and second kind. [ABSTRACT FROM AUTHOR]

For any graph G = (V, E), the order and size of G are p and q. A bijection l from V (G) to {1, 2,.., p} is called (a, d) -edge local antimagic labeling if for any two adjacent edges are not received the same edge-weight (color) and the set of all edge-weights are formed an arithmetic progression {a, a+d, a+2d,..., a+(c-1)d}, for some integers a, d > 0 and c is the number of distinct colors used in the proper coloring. An edge-weight (color) w(uv) is the sum of two end vertices labels, w(uv) = f(u) + f(v), uv E(G). The (a, d) -edge local antimagic coloring number is the least color (edge-weight) used in any (a, d) -edge local antimagic labeling. In the present study, we introduce a new type of labeling and a parameter, also we obtain the (a, d) -edge local antimagic coloring number for paths and wheel graph Wn, n = 3, 4, 5. Moreover, we obtain an upper bound of the (a, d) -edge local antimagic coloring number for wheel Wn, n = 6. [ABSTRACT FROM AUTHOR]

We briefly discuss a congruence relation of the subsequence of the Fibonacci m-step numbers. Then we use the obtained result to establish many identities regarding the Tribonacci numbers mod q with indices in arithmetic progression and connect them with the existing results. Finally, we discuss arbitrary linear recurrence sequences. [ABSTRACT FROM AUTHOR]

In this paper, using a deep result on the existence of primitive divisors of Lehmer numbers due to Y. Bilu, G. Hanrot and P. M. Voutier, we first give an explicit formula for all positive integer solutions of the Diophantine equation (x-d)²+x²+(x+d)² = yn (*), when n is an odd prime and d = pr, p > 3, a prime. So this improves the results of the papers of A. Koutsianas and V. Patel and A. Koutsianas. Secondly, under the assumption of our first result, we prove that (*) has at most one solution (x, y). Next, for a general d, we prove the following two results: (i) if every odd prime divisor q of d satisfies q ≢ ±1 (mod 2n), then (*) has only the solution (x, y, d, n) = (21, 11, 2, 3), and (ii) if n > 228000 and d > 8√2, then all solutions (x, y) of (*) satisfy yn < 23/2d³. [ABSTRACT FROM AUTHOR]

The appearance of (k, r)-integers in an arithmetic progression by using the exponent pair method is studied. Moreover, the result to r-free integers in the arithmetic progression is deduced. [ABSTRACT FROM AUTHOR]

Finite-state dimension, introduced by Dai, Lathrop, Lutz and Mayordomo quantifies the information rate in an infinite sequence as measured by finite-state automata. In this paper, we define a relative version of finite-state dimension. The finite-state relative dimension dim F S Y (X) of a sequence X relative to Y is the finite-state dimension of X measured using the class of finite-state gamblers with oracle access to Y. We show its mathematical robustness by equivalently characterizing this notion using the relative block entropy rate of X conditioned on Y. We derive inequalities relating the dimension of a sequence to the relative dimension of its subsequences along any arithmetic progression (A. P.). These enable us to obtain a strengthening of Wall's Theorem on the normality of A. P. subsequences of a normal sequence, in terms of relative dimension. In contrast to the original theorem, this stronger version has an exact converse yielding a new characterization of normality. We also obtain finite-state analogues of van Lambalgen's theorem on the symmetry of relative normality. [ABSTRACT FROM AUTHOR]

Featured Application: An author-level metric for evaluating a researcher's cumulative scholarly output is presented which can be effectively utilized by abstract and citation databases and search engines. As the academic world yields an ever-increasing research output in terms of journal papers, conference proceedings, and books, the rating of published works and authors becomes imperative. All the big citation databases and search engines are currently using cumulative output indices, such as h-index, i10-index, and g-index, which do not consider the number of co-authors or the researcher's sequence number in the authors list of a publication. In this context, the article presents a novel computational approach for evaluating a researcher's scholarly output by taking into account the total number of co-authors, the sequence number of the researcher in the authors list, and the number of citations received per year by an article. Arithmetic progression is applied to quantify the credit for each co-author of a publication. The respective credits of a researcher are then accumulated for all their publications to obtain the rating. The method yields a truer value of the researcher's impact in terms of their scholarly activities. A global implementation of the metric presented in this work will curb the unethical practice of including the names of non-contributing researchers in the authors list and expecting reciprocity in return. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a simple graph and H be a subgraph of G. Then G admits an H-covering, if every edge in E(G) belongs to at least one subgraph of G that is isomorphic to H. An (a, d)-H-antimagic total labeling of G is a bijection f: V (G) ∪ E(G) → {1, 2, 3, ..., |V (G)| + |E(G)|} such that for all subgraphs H0 of G isomorphic to H, the H0 weights w(H0) = P v∈V (H0) f (v) + P e∈E(H0) f (e) constitute an arithmetic progression {a, a + d, a + 2d, ..., a + (n - 1)d}, where a and d are positive integers and n is the number of subgraphs of G isomorphic to H. The labeling f is called a super (a, d)-H-antimagic total labeling if f (V (G)) = {1, 2, 3, ..., |V (G)|}. In [9], authors have posed an open problem to characterize the super (a, d)-G+e-antimagic total labeling of the graph Gu[Sn], where n ≥ 3 and 4 ≤ d ≤ p+q+2. In this paper, a partial solution to this problem is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we prove several identities involving linear combinations of convolutions of the generalized Fibonacci and Lucas sequences. Our results apply more generally to broader classes of second-order linearly recurrent sequences with constant coefficients. As a consequence, we obtain as special cases many identities relating exactly four sequences amongst the Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, and Jacobsthal-Lucas number sequences. We make use of algebraic arguments to establish our results, frequently employing the Binet-like formulas and generating functions of the corresponding sequences. Finally, our identities above may be extended so that they include only terms whose subscripts belong to a given arithmetic progression of the non-negative integers. [ABSTRACT FROM AUTHOR]

We study an open question at the interplay between the classical and the dynamical Mordell-Lang conjectures in positive characteristic. Let K be an algebraically closed field of positive characteristic, let G be a finitely generated subgroup of the multiplicative group of K, and let X be an (irreducible) quasiprojective variety defined over K. We consider K-valued sequences of the form an : = f(φn(x0)), where φ : X → X and f : X → P¹ are rational maps defined over K and x0 ∈ X is a point whose forward orbit avoids the indeterminacy loci of φ and f. We show that the set of n for which an ∈ G is a finite union of arithmetic progressions along with a set of Banach density zero. [ABSTRACT FROM AUTHOR]

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(xi) = i, the weight w(xi) is the sum of labels of all neighbors of xi, and the sequence of the weights w(x1),w(x2), . . ., w(xn) 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. [ABSTRACT FROM AUTHOR]