Three progressions in a foursome

Present paper envisages a novel approach to explore some real sequences by using the gamma function, arithmetic progression (AP) and geometric progression (GP). Particularly, applications of properties of both the arithmetic progression (AP) and geometric progression (GP) are focused to find out some real sequences which can be significantly useful in emerging fields of engineering science and technology. Real sequences play vital role in testing of convergence of infinite power series in real analysis and engineering mathematics. In this paper, several propositions pertaining to real sequences are explored and examined by way of presenting proofs. In addition to this, various numerical examples are also illustrated in order to emphasis the application aspect of real sequences explored herein. Finally, some significant conclusions are drawn for future scope and further findings with different versions of real sequences.

<p>The paper considers calculation methods of weight coefficients of local criteria for the person to make decision (PMD). Methods can be used by a decision-maker to form the integrated criteria in the form of the additive, multiplicative, minimax, nonlinear or combined convolution of local criteria, different in degree of importance, as well as to carry out a comparative assessment of the studied alternative options on their basis, range these options, and choose the best option among them.</p><p>The paper classifies the calculation methods of weight coefficients of criteria and distinguishes three groups, namely: methods based on the paired comparison of criteria, methods based on the analytical interrelation of indicators of criteria preference, and methods based on the formalistic approach. Among the methods based on paired comparison of criteria the following ones are distinguished: a classical method of paired comparison of criteria, and methods of paired comparison of criteria based on the fixed, floating, and exponential floating preferences of criteria. The last two methods of criteria comparison are respectively basic for the practical use of both a method of hierarchy analysis and a multiplicative method of hierarchy analysis. The paper considers in detail calculation methods of weight coefficients of criteria using analytical dependences of interrelation between the indicators of criteria importance based on an arithmetic and geometrical progression. The considered formal methods include the method of consecutive comparison of criteria known as a Cherchmen's method – Akoffa, method and a method of basic criterion.</p><p>It is shown that when using methods of paired comparison of criteria or methods based on the interrelation of weight coefficients of criteria obeyed to an analytical or geometrical progression with the strictly ranging criteria K K K ......K K 1 2 3 n-1 n , a difference in weight coefficients of the most important and least important criteria is strictly the fixed number of times  which depends on the used method and the number of the considered criteria n . So when using a classical method of paired comparison of criteria and a method based on the use of an arithmetic progression,   n, and when using a geometrical progression,   n , and it sharply increases with growing n.</p><p>To eliminate noted shortcomings in considered calculation methods of weight coefficients of criteria, methods of the adjusted assessment of criteria preference are developed and offered for a practical use: a method of paired comparison on the basis of the adjusted fixed preference of criteria; a method on the basis of the adjusted decreasing arithmetic progression, and a method on the basis of the adjusted decreasing geometrical progression.</p><p>The offered methods allow a decision-maker in case of strictly ranging criteria, as basic data, to set both the number of criteria, and the value - coefficient of superiority of the most important criterion in comparison with the least important criterion and have a pronounced practical focus.</p>

In the context of general education reform oriented toward the development of learners’ qualities and competencies, fostering mathematical problem-solving competence for students has become an essential requirement. This paper investigates the design and organization of problem-based situations in teaching the topic “Sequences – Arithmetic Progressions – Geometric Progressions” with the aim of developing mathematical problem-solving competence for Grade 11 students in accordance with the 2018 General Education Curriculum. Based on an analysis of theoretical perspectives on problem-based situations and mathematical problem-solving competence, the study clarifies the objectives and characteristics of the topic within the Grade 11 mathematics curriculum. A qualitative research approach is employed through document analysis, literature review, and examination of the current mathematics curriculum. The research proposes a process for designing and organizing problem-based situations in teaching sequences, arithmetic progressions, and geometric progressions, accompanied by a specific illustrative example. The findings provide practical references to support high school mathematics teachers in innovating teaching methods toward a competency-based approach.

Two well-studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression or in geometric progression. We study generalizations of this problem by varying the kinds of progressions to be avoided and the metrics used to evaluate the density of the resulting subsets. One can view a 3-term arithmetic progression as a sequence \(x, f_n(x), f_n(f_n(x))\), where \(f_n(x) = x + n\), n a nonzero integer. Thus, avoiding 3-term arithmetic progressions are equivalent to containing no three elements of the form \(x, f_n(x), f_n(f_n(x))\) with \(f_n \in \mathscr {F}_\mathrm{t}\), the set of integer translations. One can similarly construct related progressions using different families of functions. We investigate several such families, including geometric progressions (\(f_n(x) = nx\) with \(n > 1\) a natural number) and exponential progressions (\(f_n(x) = x^n\)). Progression-free sets are often constructed “greedily,” including every number so long as it is not in progression with any of the previous elements. Rankin characterized the greedy geometric-progression-free set in terms of the greedy arithmetic set. We characterize the greedy exponential set and prove that it has asymptotic density 1 and then discuss how the optimality of the greedy set depends on the family of functions used to define progressions. Traditionally, the size of a progression-free set is measured using the (upper) asymptotic density; however, we consider several different notions of density, including the uniform and exponential densities.

Voltage lift technique has been successfully employed in design of DC/DC converters, for example, four series Luo‐converters. However, the voltage transfer gain increases stage‐by‐stage in arithmetic progression. Super‐lift converters largely enhance the voltage transfer gain, which increases stage‐by‐stage in geometric progression. It effectively enhances the voltage transfer gain in power series. This study introduces a novel approach – modified super‐lift converters that implements the super‐lift operation as well, its voltage transfer gain increases stage‐by‐stage also in geometric progression.

Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2 k + p, where k is a positive integer and p is an odd prime. Erdős ever asked whether all these progressions can be obtained from covering congruences. In this paper, we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2 k +p, and give a quantitative form of Romanoff’s theorem on arithmetic progressions. As a corollary, we prove that the answer to the above Erdős problem is affirmative.

Si q≥2 est un nombre entier, on designe par Sq(n) la somme des chiffres en base q du nombre entier naturel n et par vq(n) sa valuation q-adique. L’objectif de cet article est d’etudier des sommes d’exponentielles de la forme ∑n≤xexp(2iπ(lmSq(n)+km′Sq(n+1)+θn)) afin d’en deduire certaines proprietes statistiques des nombres entiers n pour lesquels Sq(n) et Sq(n+1) appartiennent a des progressions arithmetiques donnees. Ceci permet d’etendre les resultats obtenus par Gelfond en 1968 et ceux obtenus par Mauduit-Sarkozy en 1996.

This is the first part of a two-part paper. Our paper was motivated by two classical papers: A paper of Sir Charles Wheatstone published in 1844 on representing certain powers of an integer as sums of arithmetic progressions and a paper of J. J. Sylvester published in 1882 for determining the number of ways a positive integer can be represented as the sum of a sequence of consecutive integers. There have been many attempts to extend Sylvester Theorem to the number of representations for an integer as the sums of different types of sequences, including sums of certain arithmetic progressions. In this part of the paper, we will make yet one more extension: We will describe a procedure for computing the number of ways a positive integer can be represented as the sums of all possible arithmetic progressions, together with an example to illustrate how this procedure can be carried out. In the process of doing this, we will also give an extension of Wheatstone’s work. In the second part of the paper, we will continue on the problems initiated by Wheatstone by studying certain relationships among the representations for different powers of an integer as sums of arithmetic progressions.

Introduction 1. Primes in arithmetic progressions 2. The main theorems II.Oscillation theorems for sifting functions 3. The statements 4. Proof of Theorem B2 5. Proof of Proposition 3.1 6. Proof of Proposition 3.2 7. Proof of Theorem C: preliminaries 8. Proof of Theorem C: completion 9. Proof of Theorem Bl 10.Proof of Theorem B3 III.Oscillation theorems for primes 11.Proof of Theorem Al 12. Proof of Theorem A2 13.Proofs of Theorem A3 and Proposition 2.1 I. INTRODUCTION 1. PRIMES IN ARITHMETIC PROGRESSIONSFor x ~ 2 real, q a positive integer, and a an integer coprime to q, let 0 and ~ be defined by (1.1) O(x; q, a) = E logp = qJZq) (1 +~(x; q, a)), p~x p=amodq

We introduce a new, elementary method for studying random differences in arithmetic progressions and convergence phenomena along random sequences of integers. We apply our method to obtain significant improvements on two results. The first improvement is the following. Let l be a positive integer and {u 1 ≥ u 2 ≥ · · · } be a decreasing sequence of probabilities satisfying u n · n 1/(l+1)→∞. Let R = R ω be the random sequence obtained by selecting the natural number n with probability u n . Then every set A of natural numbers with positive upper density contains an arithmetic progression a, a+r, a+2r,..., a+lr of length l + 1 with difference r ∈ R ω . The best previous result (by M. Christ and us) was the condition u n ·n 2−l+1 → ∞ with a logarithmic rate. The new bound is better when l ≥ 4. Our second improvement concerns almost everywhere convergence of double ergodic averages. We construct a (random) sequence {r 1 < r 2 < · · · } of positive integers such that r n /n 2−ϵ → ∞ for all ϵ > 0 and, for any measure preserving transformation T of a probability space, the averages $$\frac{1}{N}\sum\limits_{n < N} {{T^n}{F_1}\left( x \right)} {T^{{r_n}}}{F_2}\left( x \right)$$ converge for almost every x. Our best previous result was the growth rate r n /n (1+1/14)−ϵ → ∞ of the sequence {r n }.

On square-free arithmetic progressions in infinite words

Strongly sufficient sets and the distribution of arithmetic sequences in the [formula omitted] graph

Voltage lift technique has been successfully employed in design of DC/DC converters, for example, three series Luo‐converters. However, the output voltage increases in arithmetic progression. Super lift (SL) technique is the most significant contribution in power electronics, for example, four series SL converters. Their output voltage increases in geometric progression. This study introduces a novel approach – SL technique armed by split capacitors and split inductors that implements the output voltage increasing in higher geometric progression. It effectively enhances the voltage transfer gain in power series as well. From the authors experimental results, the power rates of these converters can be few kW.

Difference sets are not multiplicatively closed, Discrete Analysis 2016:17, 20pp. The famous sum-product problem of Erdős and Szemeredi asks the following. Let $A$ be a set of $n$ real numbers. Define the _sumset_ $A+A$ of $A$ to be the set $\{x+y:x,y\in A\}$ and the _product set_ $A.A$ to be the set $\{xy:x,y\in A\}$. Must at least one of $A+A$ and $A.A$ have size $n^{2-o(1)}$? Since neither set can have size greater than $n^2$, this is saying that the only way of making one of them a little smaller is to make the other one have almost maximal size. In particular, arithmetic progressions have very large product sets and geometric progressions have very large sumsets. Until recently the best known bound was $n^{4/3}$, up to logarithmic factors [3], though the exponent has now been very slightly improved [2]. More generally, the _sum-product phenomenon_ is a collection of statements that say that in various senses it is not possible for a set to be additively structured and multiplicatively structured at the same time. For example, an important result of Bourgain, Katz and Tao states that if $A$ is a subset of a finite field ${\mathbb F}_p$ and $p^{\delta} < |A| < p^{1-\delta}$, then one of $A+A$ and $A.A$ must have size at least $n^{1+\epsilon}$ for some $\epsilon$ that depends on $\delta$ only. An extra difficulty here is that finite fields can have nontrivial subfields, so somehow the proof has to use the fact that $\mathbb F_p$ does not. This paper gives an interesting twist on the sum-product phenomenon. The underlying philosophy is still that a set with additive structure cannot be too multiplicatively structured at the same time. However, the notion of additive structure is different. Instead of assuming that the set has a small sumset, Shkredov assumes that it is a difference set of another set. This is a genuinely different assumption, since if $A$ has a large difference set $D=A-A$, it is perfectly possible for $D+D$ to be much bigger than $D$. Nevertheless, when $A$ is a set of real numbers, it still implies, as the main result of the paper shows, that the product set $D.D$ has size at least $|D|^{1+c}$, where $c>0$ is an absolute constant. The same is true of the quotient set $D/D$. The paper also proves a result of a similar kind in the finite field $\mathbb F_p$ provided that $D$ is not too big. Since multiplicative subgroups have, by definition, very small product sets, this result implies in particular that a multiplicative subgroup $H$ of $\mathbb F_p$ that is not too large is not the difference set of any subset of $\mathbb F_p$. (The precise assumption required is that $H$ has order at most $p^{4/5-\epsilon}$.) [1] Jean Bourgain, Nets Katz and Terence Tao, _A sum-product estimate in finite fields, and applications_, GAFA, 14 (2004), 27-57, also available at [arxiv:0301343](https://arxiv.org/abs/math/0301343) [2] Sergei Konyagin and Ilya D, Shkredov, _On sum sets of sets, having small product set_, [arxiv:1503.05771](https://arxiv.org/abs/1503.05771) [3] Jozsef Solymosi, Bounding multiplicative energy by the sumset , Adv. in Math, 222 (2009), 402-408, also available at [arxiv:0806.1040](https://arxiv.org/abs/0806.1040) [Article image by Lee Coursey](https://www.flickr.com/photos/leeco/14514643440/in/photolist-o7BkP7-wbsvhS-87Eun-hFM6Pi-5wWnAE-ixoVt1-3uoZCv-rujdcL-nDrm5X-nASM8G-rrtCFn-7rZcJ-ciosiU-7g3EA-bmx166-hVkB6H-ptbTJY-7oA7X5-aXrfFV-bZ8qg1-mhLdxq-fDgqzF-7PNJ2e-ee44aR-efNP5W-rxKAnt-bW3rkY-4i25j5-eYtYcc-52bQM8-h7NLYq-eyVP1U-gipwap-4zibnz-ts19q8-7Sfrp9-qZ9Tgj-7q9giY-qxrbPZ-GJXD6y-5RT9F9-aAgwaC-dVCMcL-6y8utz-4uNzTu-a7WQPt-4RoPSW-8yyZHX-5Bcypu-o3vgdb)

“Multiply by adding”: Development of logarithmic-exponential covariational reasoning in high school students