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Cryptography, 2002.
An insight into the use of cryptography in data security.
724 words (approx. 2.9 pages), 3 sources, MLA, AU$ 31.95
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Abstract
This paper analyzes cryptography, the encryption or transformation of data into some unreadable form in order to ensure privacy by keeping the information hidden from anyone for whom it is not intended. It provides a brief overview of cryptography, discusses methods of encryption and description and examines cryptographic protection in Microsoft Windows 2000 as an example of cryptography utilization.
From the Paper
"Cryptography is the study of mathematical techniques related to aspects of information security such as confidentiality, data integrity, entity authentication, and data origin authentication. It is defined as the science of protecting data. Cryptographic mechanisms help organizations provide a complete suite of security services. The fundamental goal of cryptography is to adequately address systems and information security in the prevention and detection of cheating and malicious activities."
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Mathematics Curriculum Review, 2002.
A comprehensive analysis of the problems in the elementary school's mathematics curriculum.
3,545 words (approx. 14.2 pages), 6 sources, APA, AU$ 120.95
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Abstract
This paper examines the question of how to reverse the trend of lack of educational progress, specifically in the world of mathematics. This is considered through an evaluation of three elementary schools' stated mathematics curriculum, and how they compare to the standards of the National Council of Teachers of Mathematics published standards. The process of this evaluation is a point by point comparison between the NCTM standards and the printed curriculum guidelines for these schools. Specific points which are supportive, and which may fail to reach the guidelines are identified and discussed for each school. The purpose of this evaluation is not to approve or reject these curricula, but rather to identify specific applications which can be either improved through change, or strengthened by building upon existing positive initiatives.
Introduction
Discussion of the NCTM Standards
West New York Public Schools, West MY
Bogota Public Schools, Bogota, NJ
North Bergen Public School System, North Bergen, NJ
Bibliography
From the Paper
"According to national statistics, the mathematical educational progress of American elementary students has failed to keep progress with the rest of the world. This stinging indictment of the educational system of the most technologically advanced culture in the world has caused a serious evaluation of the standards and goals of the elementary system. According to the National Council of Teachers of Mathematics, there are knowledgeable teachers in the system. The teaching staff has adequate support and resources. In a society which depends daily on mathematics, there is opportunity for students to learn and apply math principles and facts. There also is an abundance of access to technology to support the educational process. Finally, if students are considering careers, those in math related fields, such as engineering, financial planning, accounting and many others are some of the highest paying positions in our current job market."
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Isaac Newton, 2002.
A look at the scientific discoveries of Isaac Newton.
606 words (approx. 2.4 pages), 3 sources, APA, AU$ 26.95
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Abstract
This paper begins by providing a brief biographical overview of Isaac Newton, from his birth in England in 1642 to his groundbreaking scientific theories and discoveries. The paper covers Newton's scientific achievements, starting with the fact that he established a unified theory of approach to modern science. It discusses his discoveries relating to the white light, the telescope and to the field of optics in general. The paper also covers Newton's mathematical achievements in the form of calculus and his most famous discovery of all - gravity.
From the Paper
"Newton?s discoveries in optics were offset by his even more groundbreaking discoveries in pure mathematics and the science of mechanics. One of the most important modern mathematical tools ?The Integral Calculus? was the brainchild of Newton. It need not be mentioned that without this mathematical tool the progress that the scientific community achieved in many disciplines would have been significantly delayed. However Newton?s discoveries in the field of mechanics outweigh all his other accomplishments. Though Galileo had already discovered the first law of motion his theory was based on the movement of objects without any external influence or attraction between them. Newton?s three laws of motion explained the hitherto inexplicable behavior of all physical bodies in motion. Still more astounding was Newton?s discovery of gravity. All these four laws put together explained the mechanical motion of all earthly and heavenly bodies. Newton not only proposed these laws but also ratified them by using the integral calculus."
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Euclid's Fifth Postulate, 2002.
A paper which discusses the philosophical and logical problems contained in Euclid's 'Fifth Postulate' on planar geometry.
1,622 words (approx. 6.5 pages), 3 sources, APA, AU$ 63.95
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Abstract
Euclid gave the world much of the information it has on planar geometry in his five postulates. The paper shows that while the first four are relatively easy to understand, the fifth one is very difficult in relation to the others. It is this fifth postulate that many people feel can never be proven. The paper discusses how there are those that say it is simply incorrect, those that say it's both true and false and others that say there is no possible way to prove it, and Euclid himself may have realized that the task was impossible. The author of the paper surmizes that if someday the fifth postulate is proven to be either true or false, and the decision is agreed upon, then it could change the way mathematics are done and the way geometry is looked at.
From the Paper
"Theoretically it would be possible for the lines to move toward one another so slowly, because of the low degree of angle, that they take a huge amount of space to come together at the end. But is it possible to have such a slight angle that the lines are almost parallel? They would be so close to parallel at that point that the impression that they are drawing closer together wouldn't be noticed unless they were looked at over miles at one time. That must be possible, but they still must meet somewhere in infinity.
Perhaps Euclid was right and the lines do meet somewhere, but the angles can be so minute that the lines go on almost to infinity, and we don't have the capabilities to calculate just how far that is yet. Perhaps Euclid is wrong and lines will go on into infinity still never touching, but only being a hair's width apart. Mathematicians may never know, since they haven't discovered any way to prove Euclid's fifth postulate by now."
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Archimedes and Carl Friedrich Gauss, 2002.
A comparison and contrast of two brilliant mathematicians, Archimedes and Carl Friedrich Gauss.
1,541 words (approx. 6.2 pages), 6 sources, MLA, AU$ 61.95
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Abstract
This paper discusses the lives of Archimedes and Carl Friedrich Gauss, two of the greatest mathematicians of all time. The paper provides a point by point comparison of their childhood and education, outlines each of their mathematical contributions and examines the influence their work continues to have on the science of mathematics.
From the Paper
"Far more details survive about the life of Archimedes than about any other ancient scientist, but scholars disagree on which details are fact and which are anecdotal. The most famous Archimedes story centers on how he determined the proportion of gold and silver in a crown made for Hieron through measuring water displacement. Since he supposedly made the discovery while in the bathtub, the excited Archimedes ran naked through the streets of Syracuse shouting ?Eureka!? (Muir 20)."
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Existence and Uniqueness Theorem, 2002.
A study of the relationship and accuracy of the existence and uniqueness theorems.
1,835 words (approx. 7.3 pages), 3 sources, MLA, AU$ 71.95
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Abstract
This paper aims to provide information on generalization of existence and uniqueness theorem for ordinary differential equations (ODE). This contains mathematical definitions, notations, symbolisms, formulas, graphics, and equations of the theorems mentioned. A few examples were also provided to illustrate and prove the definitions of the theorems.
From the Paper
The fundamental theorem of Existence and Uniqueness answer the questions does a solution exists and is it unique. If at least one solution can be determined for a given problem, a solution to that problem is said to exist. Frequently, mathematicians seek to prove the existence of solutions (by means of a so-called existence theorem) and then investigate their uniqueness (by means of a so-called uniqueness theorem). The solutions to an ordinary differential equation (frequently abbreviated as ODE) satisfy the existence and uniqueness properties.
The foundation of the theory of differential equations is the theorem on the existence of solutions for the initial value problem
x = f ( x ), x ( t o ) = x o
for a function x: R - Rn and f: Rn - Rn The main tool that we will use in developing our theory is the reformation of the differential equation as an integral equation. Suppose that both x(t) and f(x) are continuous functions. Then we can formally integrate both sides with respect to t to obtain:
x ( t ) = x o + ....
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Isaac Newton, 2002.
This paper discusses the life and work of Isaac Newton.
600 words (approx. 2.4 pages), 3 sources, MLA, AU$ 26.95
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Abstract
This paper discusses the life and work of Isaac Newton and how his laws and discoveries have ensured that his name is imprinted in the history of science. The author illustrates how Newton is not only one of the greatest scientists but also one of the most influential scientific personalities.
From the Paper
"Isaac Newton was the greatest and the most influential scientist of all times. Born in Woolsthrope, England on a Christmas day in 1642 Newton was a bright child with an incredible mechanical aptitude. Newton entered the Cambridge University when he was eighteen years of age and soon he mastered the science and mathematical concepts of his time and went on to continue his independent research. It was during this period that Newton laid the foundation for the subsequent discoveries that were to revolutionize the scientific world. Newton was conferred the honorable Fellow of Royal Society of London in 1671."
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"The Life of Isaac Newton" by Richard Westfall, 2002.
This paper is a review of "The Life of Isaac Newton" by Richard Westfall, a detailed portrait of the English mathematician, physical scientist, and theologian.
1,565 words (approx. 6.3 pages), 2 sources, AU$ 62.95
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Abstract
This paper describes the book, "Life of Isaac Newton" by Richard Westfall, which tells chronologically the life of a solitary scholar, Trinity College professor, government administrator and elder statesman of the English scientific community by showing his accomplishments and human weakness. This paper tells the story of the "apple" and points out that Newton may have gotten the idea when he was young but it took many years for him to develop his theories.
From the Paper
"For a number of years, Newton did not publish anything and seemed to immerse himself in the study of chemistry and its "occultist" neighbor, alchemy. Avoiding the more mystical areas of the science, there is no doubt he was searching for both knowledge as well as gold . Newton also was delving into some dangerous theological areas, doubting the existence of the Trinity and attributing it to a corruption of the true earlier Christian religion. Despite holding these beliefs until his death, he successfully kept them a secret, and even managed to be appointed to the Lucasian chair of Trinity College without having to take the usual step of taking on the holy orders. He kept his then-heretical religious beliefs a secret until his deathbed, when he refused to take his final communion "
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Cryptography, 2002.
An overview of the science of cryptography - the creation of a pattern by switching letters around.
2,770 words (approx. 11.1 pages), 6 sources, APA, AU$ 100.95
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Abstract
Kids decoder rings in cereal boxes, the puzzles in the comic pages of the daily newspapers and high-tech encryption all have something in common, they are all variations of cryptography. The paper shows how, ever since the early days of civilization, people have been trying to encode massages to keep secrets from falling into the hands of the wrong person. Today the science and math of cryptography go way beyond switching letters around according to a certain pattern, but if a person remembers that the basic idea is the same, cryptography can be a fascinating endeavor into math, science, and even into language itself. This paper reviews the history of cryptography and the many things encryption has been used for in the past. It then looks at how encryption is used in modern times and for what purposes. The paper explains cryptography from a mathematical point of view, following the development of encryption and cryptography mathematically. Finally, it looks at the future of this science.
From the Paper
"One of the most important developments came in the form of the Wheel Cipher. The Wheel Cipher was created by Thomas Jefferson, possibly with the help of Dr. Robert Patterson, a mathematician at the University of Pennsylvania. In 1913, Captain Parket Hitt reinvented the Wheel Cipher in strip form. This lead to the creation M-138 -A, used in World War II. Just a few years later in 1916, Major Joseph O. Mauborgne ut Hitt?s strip cipher back into the wheel form, strengthened the alphabet construction, and produced the device that would lead to the M-94 cipher device. These devices, along with encryption courtesy of the Navajo people, helped the allies defeat Germany, Japan, and Italy in World War II."
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Object Oriented Hypermedia Design Model, 2002.
A brief overview of the Object Oriented Hypermedia design model and the four-step process involved in its development.
2,480 words (approx. 9.9 pages), 8 sources, MLA, AU$ 91.95
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Abstract
The Object Oriented Hypermedia Design Model uses an object oriented framework to allow a concise description of complex information items, and allow the specification of complex navigation patterns and interface transformations. This paper provides an explanation for each step in the process and discusses. The past, present and future business uses of the model.
From the Paper
"A well-designed application is important because business owners understand that how a website functions will either create repeat customers or discourage customers from visiting the site. It is essential that a website is easy to navigate and that it functions in an efficient manner. It is also important for a business to be able to correct problems with the system quickly, which will prevent the loss of customers and profits. As a result of the demands that are placed on business to have an efficient website a precise software production process is needed. (Abrah?o, Fons, Pastor 2000, 2) The OOHDM process provides the stability needed to accommodate an e-commerce site."
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Karl Gauss, 2002.
An examination of the many theories developed by Karl Gauss, a famous mathematician, (1777-1855).
1,221 words (approx. 4.9 pages), 3 sources, MLA, AU$ 50.95
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Abstract
This paper looks at the life and work of Karl Gauss. It examines his theory on Plate Tectonics, the theory of Motion of Heavenly Bodies and several other theories that were developed during his lifetime. The writer first briefly gives a bio of Gauss and then attempts to explain the theories in laymen's terms.
From the Paper
"There are many well known mathematicians from history whose work is well known and position widely recognised. However, there are also many lesser known mathematicians that have also made equally valuable contributions. Karl Friedrich Gauss is one of these, and as such is a worthwhile individual to study. Gauss developed many ideas and theories which are still in use today. He is best known for his theory of plate tectonics and his work entitled ?Theoria Motus Corporum Coelestium? ; Theory of the Motion of Heavenly Bodies in 1809. With Wilhelm E. Weber; a physicist he also developed a theory concerning geomagnetism. Much of his work is still used today, including work in the fields of physics, astronomy, and his statistical theories are even used in software algorithms. In this we see man who has made large contributions to the world of mathematics and related disciplines (Schaaf, 1964)."
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The History and Development of Calculus, 2002.
A study of the origins of mathematics and the growth of calculus.
1,825 words (approx. 7.3 pages), 7 sources, MLA, AU$ 71.95
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Abstract
This paper presents a detailed examination of the history of calculus. The writer takes the reader on an exploratory path through the origins of mathematics and then on to the history of calculus. The people who are credited with its invention as well as the forms that it took are all included in the discussion.
From the Paper
"The history of mathematics is one in which the topic follows the actual subject. Mathematics are taught by building on foundational blocks. Each block is taught and mastered and when that is completed the next block is introduced. The origin and history of mathematics follows the same path. The history of calculus is perhaps the most interesting of the mathematical techniques. The history and origin of calculus is founded in philosophy as well as science and it is one of the most fascinating of the mathematical theories and practices."
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Georg Cantor: A Genius Out of Time, 2002.
A review of the life and work of the mathematician Georg Cantor.
2,755 words (approx. 11.0 pages), 4 sources, MLA, AU$ 100.95
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Abstract
This paper is a biographical description of the work of Georg Cantor and his work in the development of set theory. In his time, these hypotheses were considered greatly controversial by other mathematicians. However, now they are an integral part of the study of mathematics.
From the Paper
"Georg attended several private schools in Frankfurt, and in 1859, entered the distinguished Grossherzoglich Hessiche Provinzialrealschule in Darmstadt. He left this institution in 1860 with high recommendations in mathematics. His father discouraged the study of math due to the fact that he wished him to become an engineer, a job that paid considerably more than mathematics. He originally attended Grossherzogliche Hoehere Gewerbeschule (Grand-Ducal Higher Polytechnic, later changed to Technische Hochschule) at Darmstadt following his father?s wishes and studying Engineering. Later, when Georg convinced his father that his heart was truly in math, his father relented and he began the study of Mathematics in 1862 (Johnson, 1997). "
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Fermat?s Last Theorem, 2001.
This paper takes a look at this mathematical theorem and how it has fascinated mathematicians for hundreds of years.
650 words (approx. 2.6 pages), 5 sources, MLA, AU$ 28.95
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Abstract
This paper briefly gives a background of Pierre de Fermat and states this famous theorem - FLT. It looks at a few working examples of problems related to the theorem and how mathematicians think that they have finally solved them.
From the Paper
"Pierre de Fermat was born near Montauban in 1601. He was born in a family reared by a leather-merchant who was his father and was educated at home. He was essentially a lawyer and was an amateur mathematician. Throughout his life, Fermat published only one mathematical paper, which was written anonymously and appeared as an appendix to a book. He died in 1655. (Ball) Fermat?s Last Theorem (FLT) has been one of the most fascinating theorems in mathematics. This theorem has been one the great, unsolved problems in this field for three hundred and fifty some years. Some experts believe, however, that the problem has been solved."
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Manipulatives in Mathematics Curriculum, 2006.
This paper discusses mathematics education in early education programs.
875 words (approx. 3.5 pages), 5 sources, APA, AU$ 38.95
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Abstract
In this article, the writer explains that manipulatives are defined as materials that are physically handled by students in order to help them see actual examples of mathematical principles at work. The writer notes that manipulatives are incorporated into curriculum with the aim of helping the student understand mathematics, rather than increasing efficiency in calculation. The writer maintains that manipulatives are very useful especially in early education. The writer notes that there is a wide array of math manipulatives on the Internet. Some may be bought while others can be enjoyed for free on the web. The writer provides examples and pictures and discusses how it would be possible to use them in teaching children.
Outline:
What are Manipulatives?
References
From the Paper
"Manipulatives are incorporated into curriculum with the aim of helping the student understand mathematics, rather than increasing efficiency in calculation. Manipulatives are very useful especially in early education. Moreover, its use is not exclusive to teachers and schools, parents who would choose to help their children with school lessons can also employ them to help their children understand math concepts. Most students dislike math because they think it is very complicated. This prejudice towards this subject result to poor performance of students in math subjects. The development of this negative mind set on the subject may have started when in their childhood. Traditional ways of teaching may have bored them and cause them to dislike the subject, which they will carry to adulthood. That is why it is important that at a young age, kids should learn to enjoy math. And the use of manipulatives can help them enjoy and appreciate it. Manipulatives come in colorful packages that attract children, their interactive design also allows children to play with them as they learn. There is a wide array of math manipulatives in the Internet."
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Archimedes' Stomachion: A Perpetually Recombining Mystery, 2008.
A discussion of the geometric puzzle - Archimedes' Stomachion.
1,674 words (approx. 6.7 pages), 3 sources, APA, AU$ 66.95
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Abstract
This paper discusses all aspects of the ever complex geometric puzzle - Archimedes' Stomachion. It explains that the Stomachion is a collection of fourteen differently shaped geometrical objects, most self-evidently organizing to form a cube and for this reason, the object or set of objects is often also given the name Loculus of Archimedes, meaning Archimedes' Box. The paper points out that the object is one rife with mystery, from its historical obscurity to its continued rendering of insight and revelation, the Stomachion is an object that is increasingly yielding of questions as much as of answers. The paper also examines the Stomachion's history and its current relevance to mathematics and scientific culture. The paper concludes that there remains any number of aspects of this puzzle that are as yet uncovered and owing both to the condition of the parchment upon which Archimedes' original ideas are expressed and to the unfurling complexity of the puzzle, it remains uncertain what additional implications were either perceived or intended by Archimedes.
From the Paper
"Composing his ideas during the 2nd century BCE, Archimedes is not generally believed to have invented the puzzle in question so much as channeled its implications into a discourse on its mathematical suggestions, which have since proven increasingly extensive. Nothing of Archimedes' investigation here was spoken of for roughly 2000 years, with a parchment communicating through several centuries of mathematical discourse gradually becoming obscured under the oppressive thumb of religious revisionism. What is today known as the Archimedes palimpsest is a document beneath a document, with the former a descendent of Archimedes' preponderance on the subject of the riddle at hand. Its existence was unknown until 1907, when Danish philologist Johan Ludvig Heiberg discovered the forgery untouched by any interested parties for several hundred years in a monastery in Constantinople. (Wikipedia, 1) Using a magnifying glass in accordance with the technology available at the time, he could make out only small fractions of a text divulging the existences of the Stomachion, with only smatterings of information accessible regarding its meaning or purposes."
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