Ramsey Theory Problems over the Integers: Avoiding Generalized Progressions

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.

We study the equation (x−4r)3+(x−3r)3+(x−2r)3+(x−r)3+x3+(x+r)3+(x+2r)3+(x+3r)3+(x+4r)3=yp, which is a natural continuation of previous works carried out by A. Argáez-García and the fourth author (perfect powers that are sums of cubes of a three, five and seven term arithmetic progression). Under the assumptions 0<r≤106, p≥5 a prime and gcd(x,r)=1, we show that solutions must satisfy xy=0. Moreover, we study the equation for prime exponents 2 and 3 in greater detail. Under the assumptions r>0 a positive integer and gcd(x,r)=1 we show that there are infinitely many solutions for p=2 and p=3 via explicit constructions using integral points on elliptic curves. We use an amalgamation of methods in computational and algebraic number theory to overcome the increased computational challenge. Most notable is a significant computational efficiency obtained through appealing to Bilu, Hanrot and Voutier’s Primitive Divisor Theorem and the method of Chabauty, as well as employing a Thue equation solver earlier on.

In their valuable book Board games around the world . Bell and Cornelius discuss possible investigations arising from the rather complicated medieval game of Rithmomachia . One such investigation gives rise to the following problem: Find all sets of four positive integers in which three of the four are in arithmetic progression, another three are in geometric progression, and another three are in harmonic progression (that is to say, their reciprocals are in arithmetic progression). The authors give two such sets, and state without proof that they are the only ones, apart from multiplication throughout by a common factor. The proof that this is so is an interesting investigation in itself, and opens up some further mathematics.

We analyze the uniform asymptotics of the equilibrium value (as a function of initial state) in the case when its payoffs are averaged with respect to a density that depends on some scale parameter and this parameter tends to zero; for example, the Cesaro and Abel averages as payoffs for the uniform and the exponential densities, respectively. We also investigate the robustness of this asymptotics of the equilibrium value with respect to the choice of distribution when its scale parameter is small enough. We establish the class of densities such that the existence of the asymptotics of the equilibrium value for some density guarantees the same asymptotics for a piecewise-continuous density; in particular, this class includes the uniform, exponential, and rational densities. By reducing the general n-person dynamic games to mappings that assigns to each payoff its corresponding equilibrium value, we gain an ability to consider dynamic games in continuous and discrete time, both in deterministic and stochastic settings.

It is shown that there is a subsetS of integers containing no (k+1)-term arithmetic progression such that if the elements ofS are arbitrarily colored (any number of colors),S will contain ak-term arithmetic progression for which all of its terms have the same color, or all have distinct colors.

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is shown to be no more than $\lfloor{(4 n-1)(n-1)/18}\rfloor$. The number of $k$-term arithmetic progressions that lie within an $n$-point subset of the line is shown to be at most $(n-r)(n+r-k+1)/(2 k-2)$, where $r$ is the remainder when $n$ is divided by $k-1$. This upper bound is achieved when the $n$ points themselves form an arithmetic progression, but for some values of $k$ and $n$, it can also be achieved for other configurations of the $n$ points, and a full classification of such optimal configurations is given. These results are achieved using a new general method based on ordering relations.

Ideals of nowhere dense sets in some topologies on positive integers

Szemeredi’s theorem says that relatively dense sets contain arithmetic progressions. The purpose of this chapter is to present a result of Leth from Leth (Proc Am Math Soc 134:1579–1589, 2006) which shows that certain sparse sets contain “near” arithmetic progressions. We then detail the connection between the aforementioned theorem of Leth and the Erdős-Turan conjecture.

We examine the ideals of nowhere dense sets in three topologies on the set of positive integers, namely Furstenberg’s, Rizza’s and the common division topology. We mainly concentrate on inclusions between these ideals, we present a diagram showing these and we explore all possible inclusions between them. We present a formula for the closure of a set in the common division topology. We answer a question posed by Kwela and Nowik (Topol Appl. 248:149–163, 2018) by constructing a set in {{mathcal {I}}}_G {setminus } ({{mathcal {I}}}_K cup {{mathcal {I}}}_F). Therefore, the main diagram of comparison between the ideals of nowhere dense sets in various topologies from the article by M. Kwela and A. Nowik is completed.

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.

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

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.