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Godel and the "Theorem on Incompleteness", 2002.
Review of R. Rucker's discussion of the concept of infinity and how it relates to Godel's "Theorem on Incompleteness".
650 words (approx. 2.6 pages), 4 sources, AU$ 32.95
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Abstract
R. Rucker helps us better understand Godel's "Theorem on Incompleteness" by discussing infinity and whether it can be seen as a real entity. In his view, infinity can be seen as a tangible reality. He argues that it is quite possible that time may actually continue forever - and that is precisely what infinity is. Rucker also sees the possibility of the potential infinite divisibility of space into smaller and smaller pieces.
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Question of Mathematical Truth, 2002.
Examines the concept of mathematical truth and whether it really exists.
900 words (approx. 3.6 pages), 4 sources, AU$ 42.95
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Abstract
In this essay, I will discuss the question of mathematical truth and attempt to decide whether there can be such a thing as an "absolute fact."
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Statistical Analysis as a Function of Time, 2002.
This paper is an overview of the field of statistical analysis as a discipline, which is a function of time.
5,963 words (approx. 23.9 pages), 27 sources, APA, AU$ 171.95
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Abstract
This paper discusses statistical analysis as a dynamic form of study that evolves over time to meet developing needs and to exploit developing capabilities and technologies. The author points out that statistical analysis is the process through which data becomes knowledge and is a science to assist one in making decisions under conditions of uncertainty. The paper relates that the most appropriate logic bases for the discipline of statistical analysis in the contemporary period are rational, quantitative, positivist and causality.
Table of Contents
Introduction: Reflections on Statistics
Reviewing Statistical Analysis
Defining Statistical Analysis
Alternative Logic Bases for Statistical Analysis
Rational Model versus Naturalistic Model.
Quantitative Model versus Qualitative Model.
Positivist Model versus Normative Model.
Causality Model versus Plausibility Model
Exploratory Model versus Confirmatory Model.
Randomization Model.
Conclusion: Reviewing Statistical Analysis.
Examining the Classical Model of Statistical Analysis
Descriptive Statistical Analysis
Exploratory Statistical Analysis
Inferential Statistical Analysis
Probability Theory and Classical Statistical Analysis
Conclusion: Classical Statistical Analysis
From the Paper
"Descriptive statistical analysis describes the performance or activity of one group or class, without attempting to generalize about other groups or classes. Classification, description, and measurement are activities applicable to variables associated with social research. The classification of variables is based on an assumption that social units are comparable within the context of specific definitional criteria. A social researcher attempts to control variation through the classification of variables. The description of variables is an effort to assign some degree of uniqueness to each variable, in order to provide a basis for the establishment of relationships among variables. The measurement of the extent of the uniqueness of variables generates the quantitative indicators of the strength of the relationships between variables. The process of classification, description, and measurement facilitates the development of causal explanations for both regularities and variations in empirical phenomena. Comparisons are made according to the degree of differentiation of structure in data in relation to a common and less differentiated point of origin. Such comparability is dependent upon both the classification of the social unit and the dimension of that social unit that is being measured. The dimension is the variable being measured."
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Diophantus - The Father of Algebra, 2002.
Examining the life and works of mathematician Diophantus and why he was called the "Father of Algebra".
3,014 words (approx. 12.1 pages), 12 sources, MLA, AU$ 107.95
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Abstract
Diophantus is referred to more often than not as the "Father of Algebra", although algebra predated Diophantus. His contributions to the study of algebra, however, have led to this attribution. This paper reviews his life, his mathematics, his place in the history of mathematics and the relevance of his work in the 21st century. The review is presented in discussions of his life, his work, his place in mathematics history and the contemporary relevance of his contributions.
From the Paper
"Diophantus lived in the third century A.D. The best estimates of his birth and death years are 200 A.D. and 284 A.D. Other conjectures of these data range from 150 B.C. to 350 A.D. Exactly when he lived, however, is not nearly as relevant to contemporary society as is what he accomplished while he lived. What is generally agreed upon about Diophantus is that he was a normal man who married, had children, and lived a normal but scholarly life. Not all of his work has survived, at least not in a recorded form that may be attributed directly to him. That work which has survived and which can be directly attributed to him, however, has established him as mathematics theoretician of worthy note (Heath (Vol. I) 15-16)."
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The Babylonian Zero, 2002.
An examination on how the figure "zero" evolved during the Babylonian times.
2,176 words (approx. 8.7 pages), 17 sources, APA, AU$ 81.95
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Abstract
This paper begins by providing a history of the evolution of zero and discusses the origin of the symbol. It then discusses the origins of the concept of "zero" and how this was perceived differently by various ancient cultures such as the Egyptians, the Mayans and the Babylonians. It then focuses specifically on the "Babylonian zero" and how this differed in concept from other figures at the time. The paper includes several diagrams and pictures.
From the Paper
"The symbol zero evolved into its present form after quite a number of transformations. The idea of how the symbol was devised also harbors a few contradictory ideas. Opinions range from it being a dot originally, replaced by a circle with a dot in the center and then maturing to the current form, an oval shape that we all are familiar with. (Pearce, I., 2002). The Egyptian zero that evolved has also been equated with the hieroglyph for beauty, and that of the human windpipe, heart and lungs. (Williams, S. W., 2002)"
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Kinematic Geometry, 2002.
An examination of Horace Barlow's paper on ?Exploitation of Regularities in the Environment by the Brain?.
1,465 words (approx. 5.9 pages), 10 sources, MLA, AU$ 58.95
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Abstract
This paper explores the general perceptions of Horace Barlow, reflected in his paper ?The Exploitation of Regularities in the Environment by the Brain?, pertaining to the role of evolutionary internalized regularities, especially as they occur in theories of vision. The focus lies principally on issues relevant to the ecological validity of Shepard's kinematic geometry constraint in ordinary motion perception perspective. This paper also establishes the thought for two individual sets of assertions; perception of apparent motion modeled as kinematic geometry theory and internalization of the like.
From the Paper
"The limitations of kinematic geometry proposed in Barlow?s paper have been recognized, however kinematic geometry being a model for perception of apparent motion in my opinion is an idea that can expand into new dimensions. However internalization of kinematic geometry does project reservations about being a possibility. As indicated by Barlow, internalized principle of object observation gives way to the perception of apparent motion. The human brain?s support for a percept is purged from an external stimulus. Conforming to the putative universals are the preferred perceptual solutions. "
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Incorporating Fairness into Game Theory and Economics, 2002.
A referee report on Matthew Rabin's intention driven model of fairness.
2,340 words (approx. 9.4 pages), 6 sources, APA, AU$ 86.95
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Abstract
Matthew Rabin?s model of fairness is based on Geanakoplos, Pearce and Stacchetti's (1989) notion of ?psychological game?, in which payoffs depend on actions and on beliefs about actions. The paper describes how Rabin?s model shows how fairness expectations lead to different results than standard theory and demonstrates some general implications of fairness on game theory and economics. This paper contains a short description of Rabin?s model, gives some examples, propositions, proofs and critique.
From the Paper
"Suppose that (a1,a2) is a mutual-max outcome. Then both f1 and f2 must be nonnegative, thus reflect a positive regard for each other.
If each player chooses a strategy which maximizes both his own material well-being and the well-being of the other player this must maximize his own utility. In a case of mutual min outcome the f1 and f2 is non positive, thus, f~j(bj,ci)[1+fi(a1,bj)] is non negative. If each player is choosing a strategy which maximizes his own material well-being , this must maximize his utility."
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Encryption and Hash Algorithms, 2002.
A discussion of the differences between a code and a cipher.
2,025 words (approx. 8.1 pages), 7 sources, AU$ 86.95
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Abstract
Discusses differences between a code and a cipher. Requirements of each; how each works. History of encryption. Enigma machine of World War II. Pre-computer encrption. Development of computer program to encrypt data. Function of a "hash" (a number generated from text & smaller than the text itself). Privacy issues. Future of algorithms.
From the Paper
"Encryption and Hash Algorithms
Introduction
Stephen Levy (2001), reporting on the latest ?unbreakable code? begins his report by quoting Edgar Allan Poe. ?It may roundly be asserted that human ingenuity cannot concoct a cipher which human ingenuity cannot resolve? (Levy, 2001, 45). This article was selected to lead off this discussion of encryption because of two elements of confusion.
First, the headline read ?An Unbreakable Code?? and the article was about enciphering and deciphering, also called ?encryption? and ?decryption.? This is a common, and often-repeated mistake, one which can confuse the very field of study. A ?code? is not a ?cipher? anymore than a ?tennis ball? is a ?cabbage.?
A code is a..."
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Leonhard Euler, 1995.
Examines the life, career and major ideas of this 18th Century Swiss mathematician.
1,350 words (approx. 5.4 pages), 6 sources, AU$ 57.95
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From the Paper
"Leonhard Euler (1707-1783) published his first mathematical work in 1726, one year before Isaac Newton's death. Euler's enormous gifts and broad interests were ideally suited to this slot in history. In pure mathematics and mathematical physics, his work elaborated that of his predecessors, such as Newton and Leibniz, and exerted an enormous influence on those who followed him. Euler also systematized, standardized, and generally cleared the way for mathematical applications in numerous fields. In the course of his long and productive career, Euler "worthily united the ages of Newton and Gauss" (Morgan 133).
Euler was the most prolific mathematician in history. During his career, he published around 560 books and articles, and still left a backlog of over 300 works at his death. The St. Petersburg Academy did not finish publishing his "literary ..."
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Music and Mathematics, 1995.
Examines the inter-relations between music and mathematics. Discusses the theory and philosophy of music and focuses on the mathematical foundations of such composers as Mozart, Schoenberg, and Cage.
1,575 words (approx. 6.3 pages), 9 sources, AU$ 67.95
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From the Paper
"Music and mathematics are closely linked, and musical rhythm serves as an example of the practical use of different mathematical principles. It has recently been noted in fact that the mathematical regularity of certain music, such as that of Mozart, can be a spur to clearer thinking, at least for a short period of time after listening to a piece of music. Music has a psychological effect that is partly explained by its mathematical regularity, seen in the way music is divided into regular bars, beats, and different note lengths. Psychologists have discovered the importance of patterns in music and in aspects of human behavior. Music satisfies certain human needs for order and rhythm, and mathematics both explains and empowers this process.
Edward Rothstein writes about the relationship between music ..."
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Correlation and Regression Analysis, 1995.
This paper examines an application of the statistical procedures of correlation and regression analysis.
1,350 words (approx. 5.4 pages), 5 sources, AU$ 57.95
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From the Paper
"This research examines an application of the statistical procedures of correlation and regression analysis. The initial part of the examination describes correlation and regression procedures, and illustrates the use of the procedures in an application. Following the description and illustration, the accuracy and appropriateness of the application is discussed.
Description of the Procedure, and An Illustration of the Use of the Procedure in An Application
Correlation and regression procedures are described in this section. This description is followed by an illustration of the use of the procedures in an application."
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Sampling, 1995.
This paper examines an application of the statistical procedure of population sampling: Describes theory and techniques and assesses validity of application in population sampling.
1,350 words (approx. 5.4 pages), 5 sources, AU$ 57.95
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From the Paper
"This research examines an application of the statistical procedure of population sampling. The initial part of the examination describes sampling procedures, and illustrates the use of the procedures in an application. Following the description and illustration, the accuracy and appropriateness of the application is discussed.
Description of the Procedure, and An Illustration of the Use of the Procedure in An Application
Population sampling procedures are described in this section. This description is followed by an illustration of the use of the procedures in an application.
Description of Procedures ... "
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Normal Curve, 1994.
This paper defines and examines the normal distribution curve and its role in statistical analysis. Tables and graph.
1,350 words (approx. 5.4 pages), 6 sources, AU$ 57.95
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From the Paper
"Statisticians work with large masses of data. Before any conclusions can be drawn from such data, it must be condensed and arranged in a usable form. One of the most common ways to summarize and describe a mass of data is to arrange a frequency distribution table. These tables can then be graphed with the frequency scale on the y-axis and the interval being graphed on the x-axis. Above each interval a horizontal line is drawn which corresponds to the frequency of the interval, resulting in a stair-step histogram pattern. Connecting the midpoints of these class intervals produces a frequency polygon and an interval curve. Distribution curves which can be "folded" vertically so that the two halves of the curve are essentially the same are said to be bilaterally symmetrical. Perfectly symmetrical curves which have a bell shape are said to be normal curves, or Gaussian curve ... "
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Mathematics and Business, 1994.
This paper discusses that mathematics is at the core of understanding business and social sciences: Financial statements, supply and demand, forecasts, linear regression, equilibrium and elasticity.
1,350 words (approx. 5.4 pages), 5 sources, AU$ 57.95
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From the Paper
"Mathematics is at the core of understanding business and social sciences. Both disciplines make use of arithmetic, quantitative methods, statistics, linear regression and calculus as they seek to describe, predict and analyze the vast array of numerical data available in the fields. This research examines the application of math in these areas with a particular emphasis on math with regard to the supply and demand function.
Anyone selling a product or providing a service uses basic arithmetic to determine how much money they take in and how much money they pay out. When the expenses are less than the revenues, they make a profit. This simple accounting principle becomes more complex as the items associated with the various components increases in complexity. Revenues can be based on cash received, or they may be placed on accounts receivable."
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Evolution of Mathematics, 1993.
Ancient Greece to 1990s. Major figures & discoveries of mathematics. Looks at principles, calculus, physics, specialization and algebra. Compares the attitude differences between U.S and Japan.
3,600 words (approx. 14.4 pages), 16 sources, AU$ 154.95
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From the Paper
" The Evolution of Mathematics:
The American and Japanese Perspectives
Elementary forms of mathematics have probably been with man throughout his evolution. As human societies advanced, so too did mathematics. From the 1500s to the present, a long lineage of mathematicians have revolutionized the field. These men were often of European origin. Only in the last century has the United States and Japan emerged as dominant mathematical forces. At present, either of these nations could lead the field into the future.
The first systems of numeration were invented by the Greeks and the Romans (Struik, 1987, p. 80.81). Later, the Western merchant, Leonardo of Pisa, introduced the Hindu.Arabic system of numeration into Western Europe. Europeans came to accept these.."
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Learning Theories and Math, 1993.
A description of behaviorist, cognitive and humanistic approaches and the application to teaching math to children.
3,375 words (approx. 13.5 pages), 16 sources, AU$ 144.95
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From the Paper
"Application of Learning Theories in Early Childhood Mathematics
The major theories of learning which underlie curriculum planning in the schools are generally classified into three groups: behaviorism, cognitive development, and phenomenology or humanistic psychology. Each of these schools of thought arose from distinct philosophies and individuals who developed the theories within the philosophies. It is the purpose of this paper to discuss the major learning theories, the psychologists representing each group, the learning implications for each learning theory, and a representative mathematics curriculum for early childhood applying the various learning theories.
Behaviorism is the oldest learning theory, and it continues to be popular in the United States. Psychologists Thorndike..."
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