On Complex Numbers in Higher Dimensions

The geometric approach to generalized complex and three-dimensional hyper-complex numbers and more general algebraic structures being based upon a general vector space structure and a geometric multiplication rule which was only recently developed is continued here in dimension four and above. To this end, the notions of geometric vector product and geometric exponential function are extended to arbitrary finite dimensions and some usual algebraic rules known from usual complex numbers are replaced with new ones. An application for the construction of directional probability distributions is presented.

Teaching Business Statistics: Some Useful Relationships

The purpose of this paper is to suggest an instructional approach in the introductory business statistics course that utilizes relationships between separately introduced topics. The paper will explore three “useful relationships” that can assist classroom instruction: (1) the relationship between the simple arithmetic mean, the weighted arithmetic mean, and the expected value of a discrete probability distribution; (2) the relationship between the use of the multiplication rule to calculate the joint probability associated with two events, use of tree diagrams, and the use of the binomial and hypergeometric distributions; and (3) the relationship between the geometric mean and the compound interest rate. Each discussion includes detailed examples of calculations to demonstrate the relationships.

Divergence & Probability Density : A Hypothesis

The earlier work by Datta et al ( 1 ) the note has been made with the algebraic properties of Probability ie P (A) +P(B )= P ( A*B ) . This is termed as multiplicative rule of Probability. By multiplication rule any number say a can be first diverged through a series of a ( pow ) x , Where x is an integer like 0 to n . REF :PROBABILITY & WAVEFUNCTION : A NOTE November 2019 DOI: 10.31219/osf.io/awk96

Curvature Spinors in Locally Inertial Frame and the Relations with Sedenion

In the 2-spinor formalism, the gravity can be dealt with curvature spinors with four spinor indices. Here we show a new effective method to express the components of curvature spinors in the rank-2 4 × 4 tensor representation for the gravity in a locally inertial frame. In the process we have developed a few manipulating techniques, through which the roles of each component of Riemann curvature tensor are revealed. We define a new algebra ‘sedon’, the structure of which is almost the same as sedenion except for the basis multiplication rule. Finally we also show that curvature spinors can be represented in the sedon form and observe the chiral structure in curvature spinors. A few applications of the sedon representation, which includes the quaternion form of differential Binanchi identity and hand-in-hand couplings of curvature spinors, are also presented.

About teaching students to methods of solving problems by combinator

This article reveals some aspects of the formation of skills to solve combinatorial problems when studying a school course in mathematics. It also considers methods for solving historical combinatorial problems, combinatorial problems and the rule of multiplication, developing skills for solving combinatorial problems, tasks on forming concepts, a tree of options, factorial, applying equations to equations and simplifying expressions, combinatorial problems for studying the concepts of permutations without repetitions, permutations with repetitions, placements without repetitions, placements with repetitions, combinations without repetitions, combinations with repetitions. In mathematics, there are many problems that require elements make available a different set, count the number of all possible combinations of elements formed by a certain rule. Such problems are called combinatorial, and the branch of mathematics involved in solving these problems is called combinatorics. Some combinatorial problems were solved in ancient China, and later in the Roman Empire. However, as an independent branch of mathematics, combinatorics took shape in Europe only in the 18th century. in connection with the development of probability theory. In ancient times, pebbles were often used to facilitate calculations. In this case, special attention was paid to the number of pebbles that could be laid out in the form of a regular figure. So square numbers appeared (1, 4, 16, 25, ...). In everyday life, we often face problems that have not one, but several different solutions. To make the right choice, it is very important not to miss any of them. To do this, iterate through all possible options. Such problems are called combinatorial. It turns out that the multiplication rule for three, four, etc. tests can be explained without going beyond the plane, using a geometric picture (model), which is called the tree of possible options. It, firstly, like any picture, is visual and, secondly, it allows you to take everything into account without missing anything.

A combined compromise solution (CoCoSo) method for multi-criteria decision-making problems

PurposeThe purpose of this paper is to discuss the advantage of a combinatory methodology presented in this study. The paper suggests that the comparison with results of previously developed methods is in high agreement.Design/methodology/approachThis paper introduces a combined compromise decision-making algorithm with the aid of some aggregation strategies. The authors have considered a distance measure, which originates from grey relational coefficient and targets to enhance the flexibility of the results. Hence, the weight of the alternatives is placed in the decision-making process with three equations. In the final stage, an aggregated multiplication rule is employed to release the ranking of the alternatives and end the decision process.FindingsThe authors described a real case of choosing logistics and transportation companies in France from a supply chain project. Some comparisons such as sensitivity analysis approach and comparing to other studies and methods provided to validate the performance of the proposed algorithm.Originality/valueThe algorithm has a unique structure among MCDM methods which is presented for the first time in this paper.

Auotomodeling rational mnemofunctions and their link to an analytical representation of distributions

Mnemofunctions of the form f(x/ε), where f is the proper rational function without singularities on the real line, are considered in this article. Such mnemofunctions are called automodeling rational mnemofunctions. They possess the following fine properties: asymptotic expansions in the space of distributions can be written in explicit form and the asymptotic expansion of the product of such mnemofunctions is uniquely determined by the expansions of multiplicands.Partial fraction decomposition of automodeling rational mnemofunctions generates the so-called sloped analytical representation of a distribution, i.e. the representation of a distribution by a jump of the boundary values of the functions analytical in upper and lower half-planes. Sloped analytical representation is similar to the classical Cauchy analytical representation, but its structure is more complicated. The multiplication rule of such representations is described in this article.

Students’ Combinatorial Thinking Processes in Solving Mathematics Problem

Combinatorial thinking is a way of thinking in solving combinatory problems. Combinatory problems are one of the difficult problems for students to solve. This study aims to analyses students’ combinatorial thinking processes in solving problems. Given two combinatory problems that consist of problems with multiplication rule and combination. The Problems were given to two 11th grade senior-high school students. The results obtained were that there was a tendency for male Participants to do the two different ways which are direct counting and using diagram. The female participants did the work with one way which is direct counting. On more complex issues, namely about combination, students' thinking models go through stages of set of outcomes. From this research, it is expected that combinatory material learning is emphasized on the discovery of formulas by students themselves inductively, especially deductively. So that in this case the students interpret the combinatory formula more.

Rational mnemofunctions on R

The subspace of rational distributions was considered it this paper. Distribution is called rational if it has analytical representation f = (f+, f-) where functions f+ and f- are proper rational functions. The embedding of the rational distributions subspace into the rational mnemofunctions algebra on was built by the mean of mapping Ra(f)=fε(x)=f+(x+iε)-f-(x-iε). A complete description of this algebra was given. Its generators were singled out; the multiplication rule of distributions in this algebra was formulated explicitly. Known cases when product of distributions is a distribution were analyzed by the terms of rational mnemofunctions theory. The conditions under which the product of arbitrary rational distributions is associated with a distribution were formulated.