Generating Certain Class of Real Sequences Using Gamma Function, Arithmetic and Geometric Progressions

<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.

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.

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.

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.

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.

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

Hecht discusses utopian socialism from its 2 main forerunners--Saint- Simon and Sismondi--to Lerous Proudhon and Fourier the great visionary utopians. Malthuss influence was particularly strong in France in the early to mid-1800s and even orthodox socialists were drawn into debate about his ideas. Some recognized the validity of Malthuss geometric and arithmetic progressions. They advocated prudence and morality and some followed Malthuss critique of the Poor Laws which they saw as contributing to idleness and high fertility. Some critics of Malthuss ideas emphasized the need to liberate the worker from exploitation in order to provide him with the means to support his family. Others violently objected to Malthuss notion of moral restraint in the support for chastity and late marriage or (like Fourier) in the advocacy of fertility control through free love refined eating habits and physical exercise. Many rejected Malthuss causal analysis of poverty and the remedies he proposed. Poverty was not caused by population outstripping resources. Neither was poverty inevitable nor was it caused by overpopulation. Instead overpopulation was caused by poverty and the inability of the working class to exercise foresight. Abstinence and restraint were no solutions to the population problem. The true remedy lay in the total reform of the social and economic system the socialization of property in the intervention of the state and in the organization of credit and work. A remedy was to be found in the growth of productive forces such as using machinery which would permit the quadrupling of production and at least a doubling of the population. Improvement in the standard of living would provide a minimum guaranteed income which would enable people to maintain their standard of living by restricting their fertility. Education would spread morality and foresight to the masses. Some utopians advocated contraception and expressed the outlines of a population policy that included family allowances and a graduated tax system to discourage large families. Most utopian socialists recognized that there were physical limits to growth; if their measures to control growth were advocated population would remain stationary.

In this paper we report the calculation applications interest of classic concepts of basic math, for example, fractions, decimal numbers, three rule in order functions, exponential, and geometric and arithmetic progression, that reveal themselves as effective tools in the application of interest calculation and percentages and resolving problems of the school environment, both for the student and the teacher. Introducing such concepts as the basis of the concepts of nancial mathematics to facilitate understanding and implementing nancial education which is an educational procedure that applies own methods, in which the teacher helps the student to build their autonomy to analyze and argue about nances, develop activities to help consumers to budget and manage your income, saving and investing. They are signicant elements and concepts so that citizens exercise activity, job and leisure, analyzing and arguing appeals before consumerism. By associating notions of economics with mathematics content, focusing on Basic Applied Mathematics in Financial Mathematics in teaching the intention is to show possibilities to improve the problem lies in the nancial outlook of students.

In writing the first book-length study of ancient Egyptian mathematics, Richard Gillings presents evidence that Egyptian achievements in this area are much more substantial than has been previously thought. He does so in a way that will interest not only historians of Egypt and of mathematics, but also people who simply like to manipulate numbers in novel ways. He examines all the extant sources, with particular attention to the most extensive of these--the Rhind Mathematical Papyrus, a collection of training exercises for scribes. This papyrus, besides dealing with the practical, commercial computations for which the Egyptians developed their mathematics, also includes a series of abstract numerical problems stated in a more general fashion.The mathematical operations used were extremely limited in number but were adaptable to a great many applications. The Egyptian number system was decimal, with digits sequentially arranged (much like our own, but reading right to left), allowing them to add and subtract with ease. They could multiply any number by two, and to accomplish more extended multiplications made use of a binary process, successively multiplying results by two and adding those partial products that led to the correct result. Division was done in a similar way. They could fully manipulate fractions, even though all of them (with one exception) were expressed in the unwieldy form of sumes of unit fractions--those having 1 as their numerator. (The exception was 2/3. The scribes recognized this as a very special quantity and took 2/3 of integral or fractional numbers whenever the change presented itself in the course of computation.) In expressing a rational quantity as a series of unit fractions, the scribes were generally able to choose a simple and direct solution from among the many--sometimes thousands--that are possible. Doing this without modern computers would seem quite as remarkable as building pyramids without modern machinery.The range of mathematical problems that were solved using these limited operational means is far wider than many historians of mathematics acknowledge. Gillings gives examples showing that the Egyptians were able, for example, to solve problems in direct and inverse proportion; to evaluate certain square roots; to introduce the concept of a harmonic mean between two numbers; to solve linear equations of the first degree, and two simultaneous equations, one of the second degree; to find the sum of terms of arithmetic and geometric progressions; to calculate the area of a circle and of cylindrical (possibly even spherical) surfaces; to calculate the volumes of truncated pyramids and cylindrical granaries; and to make use of rudimentary trigonometric functions in describing the slopes of pyramids. The Egyptian accomplishment that historians have tended to repeat uncritically, one after another, is one that Gillings can find no evidence to support: that the Egyptians knew the Pythagorean theorem, at least in the special case of the 3-4-5 right triangle.

The unique potential and different areas of applications of recurrent (synonymous: recursive) relationships in chemistry and chromatography are considered. Recurrent relations can be used in two forms: as functions of integer arguments, y(x+ 1) = ay(x) + b, and as functions of equidistant argument values, A(x+ Δx) = aA(x)+ b, Δx= const. The first form applies to all physicochemical properties of homologs in organic chemistry, because the number of carbon (and other) atoms in a molecule can be integer only. The second one applies to chemical variables depending on temperature, pressure, concentrations, etc., when the chemists should provide equal “steps” of their variations. Recurrent relations combine the properties of arithmetic and geometric progressions, which accounts for their unique approximation abilities. This was illustrated by approximating the number of isomers of alkanes, the boiling points of homologs (nonlinear dependencies), the melting points of homologs (alternation effects), the temperature dependence of the solubility of inorganic salts in water, and by revealing the anomalies of gas chromatographic retention indices and retention times in reversed-phase high performance liquid chromatography.

The article draws attention to the emergence of new operator operations through the reuse of simple mathematical operations and their application in everyday life, scientific and technical processes. An example of this idea is the fact that arithmetic and geometric progressions known from school mathematics arise from the repetition of simple operations. In addition, noting that the concept of a higher order derivative is a derivative of the multiple use of the derivative method, the final part of the article considers the original problem for a differential equation to determine an unknown function given by a higher order product. Of course, the solution to this problem requires a large number of solving problems from the student, subtle cognitive abilities. For example, a function with an upper limit of a variable integral, the application of the partial integration method to such functions, and other transformations require a great deal of mastery of explanation on the part of the teacher and considerable preparation in mastering the mastery of the student. Note that the purpose of this article is to deepen differential equations and expand the field of application in high school. Therefore, the main task is to describe the methods of integrating differential equations in the pedagogical and methodological base of school mathematics.

About Some Progressions

This paper, which professes to be a continuation of former researches on the same subject printed in the Transactions of the Royal Society, is divided into two chapters. In the first the author considers the nature of the law of those numbers in tables of mortality, which express the amount of persons living at the end of ages in regular arithmetical progression. He remarks that for short intervals the law approaches nearly to a decreasing geometrical progression, and that this must be the case whatever be the strict expression for the law of mortality, provided the intervals do not exceed certain limits. But he further remarks, that this property will be found to belong to very extensive portions of tables of mortality, and instances Deparcieux’s tables, where from the age of 25 to that of 45, the numbers living at the end of each year decrease very nearly in geometrical progression. Considering however the whole extent of such a table, it will be found that the ratio of this geometrical progression is not the same in all parts of the table. But before he enters on this consideration, the author draws some consequences from the hypothesis of a geometrical progression being the strict law of nature after a certain age. One of these is the equality of value of all life annuities commencing after that age. Another is, that the want of instances in history of persons living to very enormous ages (waving those of the patriarchs,) is no proof that such may not be the law of nature, as he shows by calculation, that out of 3,000,000 persons of 92, not more than one should on this supposition reach 192. This leads him to some general considerations on the causes of death, after which he resumes the consideration of the general law of the tables.