Paper Example on Design and Analysis of Arrangement and Combinations

Published: 2021-07-08 20:35:51
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Middlebury College
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This work in design and analysis of arrangement and combinations is concerned with a new, combinatorial approach is the study of certain structures in sequence and number conversion algorithm. The approach is based on several invariants of combinatorial structures that are related to a formal group law, and which generalize classical invariants and the real number. There are three areas covered by the research, as explained below.

Our first objective is to use the theory of incidence arrangement and combinations to construct and study several sets of combination systems equipped with a group of isomorphism. These set of combinations are mapped onto individual systems arising in combinational mathematic (P Richard, Bjorner, Anders & 2010). The identification of projection maps as certain invariants of established systems is also part of the study such as the combinations geometry, which is considered in its right. Computational applications to formal group theory and combinatory permutations are also given.

Secondly the notion of permutation which relates to the act of arranging all the members of a set into some sequence or rearranging its elements, by the process called permuting. This differs from combinations that deal with the selection of some members of a set where the order is disregarded. There is also an anagram of a word which is a permutation of its letters. For instance, if the letters are already ordered in the original name; the anagram is a reordering of the letters. This study of permutations of finite sets is a topic in the field of combinatorics.

Thirdly, we focus area Algorithms, combinatorics and optimization (ACO) concerned with the study of discrete mathematical structures and how these mathematical structures can be maximized algorithmically to design fast optimization procedures. It will also include the study of graph theory, number theory, and partition theory. This focus area is in the combinatorial groups. It has many facets and a strong interdisciplinary character, spanning topics from fundamental mathematical questions to industrial collaborations. It has a rich mathematical theory that is interesting on its own and furthermore builds a fundamental basis for mathematical optimization algorithms. This allows students to create a strong background in mathematical optimization and to learn a variety of skills linked to mathematical modeling, the design and analysis of algorithms, and computational aspects (P Richard, Bjorner, Anders & et, al 2010).


Real number

Imaginary numbers


Graph theory

Number theory

Partition theory

Algebraic complex numbers

Applied mathematics

Pure mathematics

Enumerative combinatorics

Analytical combinatorics

Mathematical analysis

Taylor series

Power series

Differentiable function

Combinatorial Mathematics


This thesis deals with combinatorial geometry problems in functional analysis. Typically, a review deals with questions that have to do with functions with a particular emphasis on the internal structure involved. Combinatorial geometric analysis tackles similar problems differently. It turns out that it is sometimes preferable to think of a given function as a member of some larger geometric space which allows one draw on geometric methods and intuition to answer questions which had no apriority geometric flavor (P Richard, Bjorner, Anders & teal 2010). As it often happens in mathematics, although the original motivation for the development of electrical techniques was the usefulness in real analytic problems, the inherent richness and beauty of the geometry involve shifted the attention of many mathematicians to the study of those structures in their right. This is the approach of the present work.

The earliest study of complex mathematics is said to have been researched by Greek Mathematicians. They were doing research on the square roots of negative numbers. They tried to test several sets of numbers which would result in some volume errors. Then there was a discovery of algebraic solution by Italian mathematicians in the 16th century. They came up with the root of cubic and quartic polynomials. They also realized that this formula would provide a real solution to some problems by manipulating the square roots of negative numbers.

Complex mathematics can be defined as the set of figures which form the pair, for example, we can say (a, b) in a diagram like a complex plane. The plotting of these numbers in the graph is represented by "Re" as the real axis and "Im" as the imaginary axis in the complex plane. The real axis and imaginary axis should satisfy I squared is equals to -1.

In complex mathematics, complex numbers are represented in the form a + b. Letter a and b represent the real numbers while i is the imaginary unit in the equation of i squared is equals to -1. The concept of imaginary number can be imagined so many times in an equation, but sometimes we will find an entity that will match what we imagine (P Richard ,Bjorner, Anders & et,al 2010). Complex mathematics define a complex number whose real part of it is as zero as purely imaginary, whereas a complex number is sad to be a real number if the imaginary part is zero.

Additionally, to their use within mathematics, the complex number can be used in fields like electrical engineering, statistics, biology, economics, and physics. The Italian mathematicians introduced the concepts of real numbers called fictitious first. They were trying to find solutions of real numbers.

Complex mathematics consists of complex variables and theorem of function which are studied and interpreted to provide a real settlement of a problem. It is a branch of mathematics that investigates the system of complex numbers. It is applied in branches like applied mathematics, algebraic geometry, analytic combinatorics and number theory. Particularly in physics, it is needed in thermodynamics, hydrodynamics, and quantum mechanics. Also, complex mathematics is used in engineering fields like mechanical, nuclear aerospace and electrical engineering.

Taylor series consists of differential functions where complex numbers are summed to get the overall equation of the Taylor series. Complex mathematics should involve combinatorial problems. In these problems, one is required to provide a real solution following the right procedure of complex numbers. A real solution is achieved when the theoretical value is zero. A theorem in complex numbers state some general facts thus we are supposed to refer a particular theory when solving a problem. For example, when solving some problems from tailor series, we need to apply the properties of power series to expand the geometric theorem of a function in one complex number.

Combinatorics is a branch of mathematics where the operation of permutation, combination, and enumeration of an element is done. It has some characteristics and properties in mathematics activities. According to, discrete mathematics is a feature that incorporates a large subset of graph theory. Combinatorics also be referred to as combinatorial mathematics.

The most influential mathematician in this field is Richard P. He contributed a lot in this area. Enumerative Combinatorics is one of his works. In this section, he defines it as the most prestigious area which involves counting the total number of specific combinatorial items. An example of enumerative combinatorics is the Fibonacci number. Finally, after calculating permutation, objects are combined and later partitioned (P Richard ,Bjorner, Anders & et,al 2010). Probabilistic combinatorics is another field. The chance of the existence of certain object combinatorics evaluated. This is done by selecting the random discrete object, for example, random graph. This method is most effective in the application of graph theory. Finite is also a closely related area in this field of combinatorial mathematics.

The combinatory theory is the overall step in combinatorial mathematics. It is the study of various problems which are closely linked to integer partitions. Special functions are assigned to every object with the aim of finding expected value of that service. Partition theory is closely related to orthogonal polynomial and q-series. This method has an approach to the subject, an analysis is done on the same, and later statistics inference is arrived at through statistical mechanism.

The combination of mathematic ought to have some relevance in a mathematics class for the student to apply. They are required to understand the problem from the context of introduction, theorem, and its definition, proof or the solution and application in real life. A student needs to know the same problem can arise in real life, workplace, and in the society. We should spend more time in class to learn the formula and fundamental algorithms of questions to equip ourselves in real life in case a similar problem arose. Mathematics modeling is presented by the standard of practice in the common core math standard. It is in real sense emphasizing on mathematics teaching so as to benefits our students. Here is some relevance of a combination of mathematical research category: -

Providing context in a problem

If we plan to teach mathematics concepts, we also need to include application part of it, especially in work. It will help the student to develop the skills and fluency. For a good teacher to help his/her student to understand mathematics, putting it in context is important. This will contribute to spending less time in many similar problems and more time in the initial one. For example, the combination mathematics in a problem like 400 + 300 is in context than understanding a problem: "In June you paid $400, and in September you paid $300. When you're asked the total money paid in both months combined is more hectic that just understanding the context (P Richard ,Bjorner, Anders & et,al 2010).


Explaining a problem in details

Understanding life problems is not an easy task hence we need to provide a sustainable solution to a problem. It is done by providing a more than one question or multiple steps. For instance, in the above example, you need first to start with first June and September paying. In October you paid double the amount you paid in June. We are required to find total money paid in three months. Here a student needs to understand addition questions in depth and show some many skills of solving the problems (P Richard ,Bjorner, Anders & et,al 2010). We will not just simply add the two numbers but also able to comprehend the concept of doubling which means multiplication. Providing a deeper understanding of particular problems is important.

Providing resources and sample items

It involves realizing new ideas in mathematics problems and sharing with students. To thoroughly help the student, understand the problem a teacher is required to check out on the items which will help the one to acquire skills. For example, you need to encourage the student to connect mathematics problems and real life situations. Introducing videos in geometry lesson helps in inspiring the student to apply the knowledge in real life situations.

The combinatorial mathematical knowledge is needed to provide the basis for the understanding of real life problems. We are required to know the general knowledge involved in this branch of mathematics. Here we need to know discrete structures and its application in combinatorial mathematics. It includes a countable and infinite set of numbers which contain the enumerative combinatorics in a structure of any size (R Sriven.&R Paul,1996). When deciding whether some criteria is suitable or not, we need to select an object, construct its plane value and mak...

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