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In aerodynamics Bernoulli's principle is used to explain the pick up of an airplane wing in flight. (the Aerodynamic Development of the Formula One Car) wing is so constructed that air flows more quickly over its upper surface than its lower one, resulting in a reduction in pressure on the top surface when compared to the bottom. The resultant variation in pressure gives the pick up that maintains the aircraft in flight. If the wing is twisted overturned, the ensuing force is downwards. This gives details as to how racecars turn at such high speeds. The down force formed pushes the tyre into the road providing more control. In aerodynamics another vital feature is the pull or resistance acting on solid bodies moving through air. For instance, the propel force formed by the engine, must surmount the drag forces formed by the air flowing over an airplane. eorganizing the body…… [Read More]
Mathematician - Maria Gaetana Agnesi
Maria Gaetana Agnesi
Since the olden days, mathematics has been an area of study that has contributed much to diverse discoveries, inventions, and innovations of science and technology. Without mathematics, we will not experience the remarkable events of science, as well as the convenience that high technology brings to us. The academic mastery of mathematics is dominated by men, even up to these days. There are very few mathematician women who made a name in the field of mathematics. More especially in the past, social prejudices became a hindrance for women to master mathematics. At present, only three women captured success in the field of mathematics. They are Sonia Kovalevsky of Russia, Emmy Noether of Germany and U.S., and Maria Gaetana Agnesi of Italy (from Maria Agnesi and Her "Witch"). The following discussions in this paper is about Maria Gaetana Agnesi and her mathematics.…… [Read More]
The Jansenists were condemned by the pope in 1653 and 1713. Characteristic beliefs of the school included "the idea of the total sinfulness of humanity, predestination, and the need for Christians to rely upon a faith in God which cannot be validated through human reason. Jansenism often, but it continued to have a strong following among those who tended to reject papal authority, but not strong moral beliefs" ("Jansenism," About.com, 2008).
After his final conversion, Pascal moved to the Jansenist monastery in Port Royal. He had already convinced his younger sister to move to the nunnery in the same location. It was there he penned the work that would contain his famous wager, the famous Pensees. He continued to live at the monastery until his death in 1662, worn out, it was said, "from study and overwork," although later historians think that tuberculosis stomach cancer was the likely culprit (Ball…… [Read More]
" (assar, p.15) He had a wife and a young child by this time, and seemed to have a relatively stable if eccentric family and professional life. Then, the man, after a bout of mania became "frozen in a dreamlike state." (assar, p.19)
ash was treated for his dissociated states into paranoid schizophrenia with insulin therapy, drugs, shock therapy, and talk therapy, none of which seemed to help his condition. His wife at first stood by him, and then divorced him. The great mathematical genius that enabled ash to see patterns in behavior and numbers, and to construct predictable equations about human decision-making had dissolved into ravings about government agents, and nonsensical theorems.
After the failure of modern psychiatry and medicine to treat the mathematician, ash became "a phantom who haunted Princeton in the 1970s and 80s, scribbling on the blackboards and studying religious texts." (assar, p.19) Yet, while ash…… [Read More]
He however refused. Because of this, Polya could only return to his home country many years after the end of the war. Having taken wiss citizenship, Polya then married a wiss girl, tella Vera Weber, the daughter of a physics professor. He returned to Hungary only in 1967.
George Polya's professional life was as interesting as his personal pursuits. Before accepting an offer for an appointment in Frankfurt, Polya took time to travel to Paris in 1914, where he once again came into contact with a wide range of mathematicians.
Hurwitz influenced him greatly, and also held the chair of mathematics at the Eidgenssische Technische Hochschule Zurich. This mathematician arranged an appointment as Privatdozent for Polya at this institution, which the latter then accepted in favor of the Frankfurt appointment.
In addition to his teaching duties, Polya further pursued his passion for mathematics via his research efforts. He collaborated with…… [Read More]
In ancient China, the science of mathematics was subsumed under the larger practice of suan chu, or the "art of calculation." The Chinese are believed to be one of the first civilizations to develop and use the decimal numeral system. Their early mathematical studies have influenced science among neighboring Asian countries and beyond.
This paper examines the history of mathematical knowledge in China. It looks at the early Chinese achievements in the field of mathematics, including the decimal system, calculation of pi, the use of counting aids and the application of mathematical principles to everyday life. It also examines the influence of Indian and later, European mathematical knowledge into Chinese mathematics.
Unlike the ancient Greeks who prized knowledge for its own sake, much of the scientific studies conducted in ancient China were spurred by practical everyday needs. Because of its geographic location, China was prone to…… [Read More]
He was miserably poor. ... He opened his book and began to explain some of his discoveries. I saw quite at once that there was something out of the way; but my knowledge did not permit me to judge whether he talked sense or nonsense. ... I asked him what he wanted. He said he wanted a pittance to live on so that he might pursue his researches" (O'Connor & Robertson, 1)
As history returns to Ramanujan's ideas and finds accuracy in most of them, Rao's response would demonstrate the degree to which the young man's internal insights had somehow transcended those of the best math minds amongst his predecessors and contemporaries. So would this be demonstrated in his trigonometric principles, such as that which is commonly referred to as the Ramanujan Conjecture. This is stated as "an assertion on the size of the tau function, which has as generating…… [Read More]
Note the distinct similarities.
An examination of Escher's Circle Limit III can thus tell us much about distance in hyperbolic geometry. In both Escher's woodcut and the Poincare disk, the images showcased appear smaller as one's eye moves toward the edge of the circle. However, this is an illusion created by our traditional, Euclidean perceptions. Because of the way that distance is measured in a hyperbolic space, all of the objects shown in the circle are actually the same size. As we follow the backbones of the fish in Escher's representation, we can see, then, that the lines separating one fish from the next are actually all the same distance even though they appear to grow shorter. This is because, as already noted, the hyperbolic space stretches to infinity at its edges. There is no end. Therefore, the perception that the lines are getting smaller toward the edges is, in…… [Read More]
Agnes Meyer Driscoll
Like Yardley, Agnes Meyer Driscoll was born in 1889, and her most significant contribution was also made during World War I. Driscoll worked as a cryptanalyst for the Navy, and as such broke many Japanese naval coding systems. In addition, Driscoll developed many of the early machine systems. Apart from being significantly intelligent for any person of her time and age, Driscoll was also unusual in terms of her gender. Her interests led her to technical and scientific studies during her college career, which was not typical for women of the time (NA). When she enlisted in the United tates Navy during 1918, Driscoll was assigned to the Code and ignal section of Communications, where she remained as a leader in her field until 1949.
As mentioned above, Driscoll's work also involved remerging technology in terms of machine development. These were aimed not only at creating ciphers,…… [Read More]
Aristotle used mathematics in many of his other studies, as well. Another writer notes, "Aristotle used mathematics to try to 'see' the invisible patterns of sound that we recognize as music. Aristotle also used mathematics to try to describe the invisible structure of a dramatic performance" (Devlin 75-76). Aristotle used mathematics as a tool to enhance his other studies, and saw the value of creating and understanding theories of mathematics in everyday life and philosophy.
During his life, Aristotle also worked with theories developed by Eudoxus and others, and helped develop the theories of physics and some geometric theories, as well. Two authors quote Aristotle on mathematics. He writes, "These are in a way the converse of geometry. While geometry investigates physical lines but not qua physical, optics investigates mathematical lines, but qua physical, not qua mathematical" (O'Conner and obinson). He also commented on infinity, and did not believe that…… [Read More]
e. all loans. The same basic formulas using logarithms can be used to calculate the needed number of investments and/or the time period of investments at a given growth rate that will be needed in order to reach a target level of investments savings (Brown 2010). Both of these applications have very real implications for many individuals, whether they are trying to buy a home or planning for their retirement, as well as a n abundance of other issues related to personal banking. Logarithms are not only useful in highly technical scientific pursuits and investigations, then, but are directly applicable and necessary to situations that directly relate to and have an effect on people's daily lives.
What I found most interesting and surprising about the development of logarithms is that they are something that needed development in the first place. I suppose it is similar to having taken any invention…… [Read More]
Claude Shannon does not have the same name recognition as obert Oppenheimer, Albert Einstein, Alexander Bell, Bill Gates, or Doyle Brunson, but his work had an impact that rivaled each of these famous men. Shannon was a mathematician, an electrical engineer, and a cryptographer is famous in his field as the father of information theory. However, he also helped usher in the modern computer age, and used his mathematical knowledge to make money in Vegas playing blackjack, things that make him relevant to a modern society obsessed with computers and with gambling. In other words, Claude Shannon was a cool scientist before much of America realized that scientists could be cool.
Shannon always had tremendous promise as a scientist, and he realized that promise early in life. He was born April 30, 1916, and he spent much of his young life focused on attaining an education. He had an early…… [Read More]
To protect themselves, many Americans chose to avoid working with or becoming friends with those who immigrated. A lack of trust permeated everything that the Americans did in regards to the immigrants, at least with the men. This was not always true of the women, as they often got along together and shared the trials and difficulties of raising families. However, many men who owned shops and stores would not hire an immigrant laborer (Glazer, 1998).
They believed that immigrants took jobs away from people in the U.S., and they did not want to catch any diseases that these immigrants might have brought with them. The general attitude during this time period was that immigrants were so different from Americans that they could never mesh into one society, but that attitude has obviously changed, as today America is a mix of all kinds of people (Glazer, 1998; Sowell, 1997).
What…… [Read More]
Sublimation refers to this channeling of emotional intensity into creative work: to transform basic psychological or sexual urges into sublime revelations.
2. The collective unconscious is a term most commonly associated with the work of Carl Jung, a student of Freud's. Jung posited the existence of a grand database of human thought to which all persons have access. The idea that there is "nothing new under the sun" reflects the widespread belief in a collective unconscious. Common dreams, shared imagery, and similarity among world religions are extensions of the collective unconscious. The collective unconscious also serves as a wellspring of images, thoughts, sounds, and ideas that artists, musicians, and creative thinkers draw from during the creative process.
3. Archetypes are in fact part of the collective unconscious. Universal symbols or proto-ideas like "mother" or "father" are archetypal. Archetypes are what Plato referred to as the Forms. Jung deepened the theory…… [Read More]
Beautiful Mind" -- a Film
John Forbes Nash, Jr., an American Nobel Prize-winning mathematician, is such a notable individual that he is the subject of a book, a PBS documentary and a film. The film A Beautiful Mind (Crowe, et al. 2006) eliminates aspects of Nash's life and rewrites other aspects revealed in the book and documentary, possibly to make Nash a more sympathetic character for the audience. However, the film remains true to a consistent theme: in an individual's quest for satisfaction through self-fulfillment, the abnormal can also be the extraordinary.
The book and PBS documentary tell John Forbes Nash, Jr.'s story "from the outside looking in," immediately noting his abnormality in that he is a paranoid schizophrenic. The film takes a different approach, "from the inside looking out," so we experience the world as Nash experiences it and do not realize until half-way through the film that he…… [Read More]
Derivatives and Definite Integrals
Word Count (excluding title and works cited page): 628
Calculus pioneers of the seventeenth century such as Leibniz, ewton, Barrow, Fermat, Pascal, Cavelieri, and Wallis sought to find solutions to puzzling mathematical problems. Specifically, they expressed the functions for derivatives and definite integrals. Their areas of interest involved discussions on tangents, velocity and acceleration, maximums and minimums, and area. This introductory paper shall briefly introduce four specific questions related to these problems and the solutions that were sought.
In calculus, how a function changes in response to input is measured using a derivative. The derivative of a function is the result of mathematical differentiation. It measures the instantaneous rate of change of one certain quantity in relationship to another and is expressed as df (x)/dx. It can be interpreted geometrically as the slope of the curve of a mathematical function f (x) plotted as a function…… [Read More]
lives of Archimedes and Carl Friedrich Gauss, two of the greatest mathematicians of all time, through a point by point comparison of their childhood and education, mathematical contributions and the influence their work has on the science of mathematics.
Childhood and Education
Archimedes (287 BC to 212 BC) lived most of his life in Syracuse, Greece. This son of an astronomer and mathematician was born into a distinguished family and was able to comfortably devote his life to mathematical research.
Carl Friedrich Gauss (1777-1855) was born into a humble German family. His early mathematical promise marked him as a prodigy and eventually earned him admission to university.
Major Mathematical Ideas
The mathematical work of Archimedes centered on the theoretical, particularly geometry. His greatest mathematical contribution involved measuring areas and segments of plane and conic sections.
Gauss's work centered on number theory. Unlike Archimedes, Gauss also used ventured into applied mathematics…… [Read More]
decision to become a math major should not be taken lightly. ecent graduates are generally required to have a master's degree and the job market for mathematicians is competitive ("Mathematician: Summary" 2012). In 2010, there were only 3,100 positions for mathematicians in the U.S. And the need is expected to increase by only 16% between 2010 and 2020. Those who are able to secure a position as a mathematician generally work in federal agencies and in private science and engineering research companies. In 2010, the median salary for mathematicians was $47.78 per hour.
Given the competitive nature of the mathematician job market, math majors frequently augment their course of study with other course work or complete a double major ("Mathematicians: How to become a mathematician" 2012). For example, a math major who would like to secure a position in engineering research would benefit from engineering coursework or getting a second…… [Read More]
Pascal's Triangle [...] who really invented Pascal's Triangle. hile the mathematical formula known as "Pascal's Triangle" has long been attributed to its' namesake, Blaise Pascal, this is not really the case. The formula was simultaneously discovered centuries before Pascal by the Chinese and the Persians, so it seems, and it was even mentioned by Omar Khayyam centuries before Pascal's existence. hy has the formula been attributed to Pascal? There are no simple answers, but Pascal, one of the world's most famous mathematicians, was the first "modern" mathematician to realize the true potential of the formula and use it accordingly, and so, it still bears his name.
ho Invented Pascal's Triangle?
The mathematical formula known as "Pascal's Triangle" has long been attributed to the great mathematician and philosopher, Blaise Pascal, who lived in France during the 17th century. Pascal only lived to be thirty-nine years old, but during his lifetime, he…… [Read More]
Hypatia of Alexandria, daughter of Theon. Specifically, it will examine the life of Hypatia, especially her mathematical accomplishments. Hypatia was the first female mathematician that left a record that historians can trace. She was a philosopher, mathematician, and teacher who lived in Alexandria, Egypt from about 350 to 415 A.D. She was the daughter of Theon, a renowned mathematician and head of the library in Alexandria.
Historians do not agree on the year Hypatia was born. Some estimate it at around 355, while others place it as late as 370. What is known of Hypatia is that she was extremely influential in mathematics and philosophical thought. Hypatia was born in Alexandria and most historians believe she spent her entire life there. Some historians believe Hypatia studied mathematics in Athens, and then traveled through Europe (Coffin, 1998, p. 94), while others believe her father taught her most of what she knew…… [Read More]
The purpose of the paper is to develop a concept of the connection between mathematics and society from a historical perspective. Specifically, it will discuss the subject, what George Cantor accomplished for mathematics and what that did for society. George Cantor's set theory changed the way mathematicians of the time looked at their science, and he revolutionized the way the world looks at numbers.
George Cantor was a brilliant mathematician and philosopher who developed the modern mathematical idea of infinity, along with the idea of an infinite set of real numbers, called transfinite sets, or the "set theory." In addition, Cantor found that real numbers were not countable, while algebraic numbers were countable (Breen). Cantor's views were quite controversial when he first developed them in the late 1800s, and some mathematicians today question some of his hypothesis ("Transfinite Number"), however, his work is recognized as some of…… [Read More]
6). Pi is, therefore, on the level of philosophical discourse because many other mathematical problems elucidated by the ancients have since been solved. Arndt et al. claim that pi is "possibly the one topic within mathematics that has survived the longest," (6). Initial pi explorations may have been prehistoric. Ancient Egyptians and Mesopotamians later developed systems of writing and mathematics that enabled rigorous investigations into crucial problems. In 1650 BCE, ancient Egyptian scribe Ahmes recorded what are likely the first formulas for pi. The formulas are written on what is referred to as the Egyptian hind Papyrus (Eymard, Lafon & Wilson).
The Ahmes formulas relate the circle to the square, foreshadowing further investigations into pi by the Greeks. The Egyptians were therefore the first to record attempts to "square the circle," or relate the area of a square to that of a circle in search of a constant variable that…… [Read More]
Georg Cantor: A Genius Out of Time
If you open a textbook, in high school or college, in the first chapter you will be introduced to set theory and the theories of finite numbers, infinite numbers, and irrational numbers. The development of many theories of math took years upon years and the input of many mathematicians, as in the example of non-Euclidean geometry. This was the case with most math theories, however set theory was primarily the result of the work of one man, Georg Cantor. In his time, these hypotheses were considered greatly controversial by other mathematicians. However, now they are an integral part of the study of mathematics. Georg Cantor received more criticism than complement in his time and it eventually led him to mental illness. However, one must remember that many other things, once thought to be controversial are now considered to be fact. Georg Cantor should…… [Read More]
if, as Halmos suggests, math is a creative art then math must also be the handmaid of science.
Describing mathematics as a creative art helps students of math better understand the true roles of the mathematician. Numbers, while in many ways central to the art of math, do not comprise the whole lexicon of mathology. Mathematics does stem from "sheer pure intellectual curiosity," enabling students to perceive the world through new eyes (p. 379). Teaching mathematics can therefore be like teaching art. Some pupils will exhibit innate, almost supernatural talents and abilities and others struggle with the language and media unique to each subject.
Because mathematics integrates seamlessly with daily life, however, teachers can easily point out the ways mathematics underlies reality. Teaching mathematics from a multifaceted and creative perspective can enhance student learning, retention, and interest in one of the most…… [Read More]
Euclid's Fifth Postulate
Philosophical and Logical Problems Contained in Euclid's Fifth Postulate
Euclid gave the world much of the information it has on planar geometry in his five postulates. hile 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. 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. His fifth postulate states:
If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles."
There…… [Read More]
I've never "seen" a million dollars, but that doesn't mean it doesn't exist.
A couple of the other physics concepts can be difficult to comprehend, as well. For example, one concept is that things can exist in more than one space at a time, but people do not choose to see them, and so, when they look at them they disappear. This section of the film might turn away a lot of viewers, because much of the discussion may be over their heads and the might find it boring. These ideas are some of the most "out there" of the film, and the hardest for the mathematicians to really get across. The talk of what is real and what a person sees vs. what they remember was understandable, but many of the other concepts may just be too odd for people to wrap their heads around. For example, the atom…… [Read More]
He invented a planetary system, which consisted of spheres, the earth being still at the center, and twenty-seven concentric spheres rotating around the earth.
Actually, most of his accomplishments are difficult to explain at all to the nonprofessional, since they involve the complicated fields of math and astronomy. ut, for those who work in those areas, Eudoxus accomplishments are extraordinary. However, what his work does is make the work today so much easier. Those who labor in those fields know the practicalities, complexities, and almost impossibility of what Eudoxus did.
Eudoxus made important contributions to the theory of proportion, where he made a definition allowing possibly irrational lengths to be compared in a similar way to the method of cross multiplying used today. A major difficulty had arisen in mathematics by the time of Eudoxus, namely the fact that certain lengths were not comparable. The theory developed by Eudoxus is…… [Read More]
Euclid -- 323-285 B.C was a historical figure who taught at Alexandria in Egypt. There are three hypotheses revolving around Euclid's life. The first is that he wrote his magnum opus the Elements as also contributed a lot of other works. Another interesting hypothesis is that Euclid was a member of a group of mathematicians working at Alexandria with each one contributing to writing the 'complete works of Euclid', with the group engaged in writing books under the name of Euclid also after his demise. The third hypothesis is that Euclid of Megara lived roughly 100 years prior to Euclid of Alexandria. A team of mathematicians wrote the complete works of Euclid and took Euclid's name from Euclid of Megara. The proof surrounding the first hypothesis is significant that he wrote his magnum opus the Elements as also contributed a lot of other works. Scanty evidence is there that deny…… [Read More]
He made three main contributions to the theory of numbers: congruence theory, studies on the separation of the circle into equal parts, and theory of quadratic forms.
Algebra and Analysis
Up to Gauss's time, no one had been able to prove that every algebraic equation has at least one root. Gauss offered three proofs. And he modified the definition of a prime number.
We discussed briefly his assistance to astronomers in relocating Ceres. His success in this effort spurred him to develop the mathematical methods he used further. In 1809 his Theoria motus corporum coelestium used the method of least squares to determine the orbits of celestial bodies from observational data. In arguing his method, Gauss invented the Gaussian law of error, or, as we know it today, the normal distribution.
Mathematicians, for centuries, had been attempting to prove Euclid's postulate concerning parallels (the sum of…… [Read More]
" (Rizzoli, Oliverio, Montemanni and Gambardella, 2004)
According to Rizzoli, Oliverio, Montemanni and Gambardella objectives are that which "measure the fitness of a solution. They can be multiple and often they are also conflicting. The most common objective is the minimization of transportation costs as a function of the traveled distance or of the travel time; fixed costs associated with vehicles and drivers can be considered, and therefore the number of vehicles can also be minimized." (Rizzoli, Oliverio, Montemanni and Gambardella, 2004) Vehicle efficiency is another objective to consider and this is stated to be expressed as "the percentage of load capacity" and it is held that the higher the load capacity the better. The objective function is also used in representation of 'soft' constraints described as constraints "...which can be violated paying a penalty." (Rizzoli, Oliverio, Montemanni and Gambardella, 2004) oth independent variables and dependent variables are contained within…… [Read More]
Risk analysis? Why are these features important components of the Excel analysis process? If so, explain how, If not, explain why.
Base-case analysis is very much an art of Excel since the accountant / mathematician usually starts off with a "base case" set of assumptions and inputs. These are the accountant's best estimates, and he would use these to produce a set of expected results.
cenario testing, for instance, often involves these base-cases analyses.
In scenario testing, the mathematician sets up alternative situations, such as the ones formulated in the Table below ('inflation rate', 'earning rate', and 'salary growth'). Each situation consists of a base-case analysis of assumption and input and the whole model is then run through the Excel system.
Base case analysis is essential for business planning and risk management since one has to know one's base-case situation before structuring plans for one's business.
What-if analysis,…… [Read More]
Proof, a NOVA episode aired on PS [...] review the video, with a focus on what the video tells us about how people learn to do mathematics. Compare and contrast this with your own experiences with mathematics, particularly your approach toward learning about new mathematical problems and trying to solve them. "The Proof" is more than just a video about solving a complex mathematical problem. It is a story of determination, setting goals, and finding out that solutions come from many different places and ideas. You have to be open to new ideas when you try to solve anything, whether it is a complex mathematical problem, or a personal problem. The proof is really about keeping an open mind, and looking at all the angles of a problem.
The Proof" is an interesting look at one man's obsession with proving (or disproving) a theory (Fermat's Last Theorem), written…… [Read More]
'" (Molland 257) of course, this kind of thinking would eventually lead Dee to argue that "at length I perceived onely God (and by his good Angels) could satisfy my desire," and ultimately resulted in his extensive travels with the medium and alchemist Edward Kelley. Furthermore, this insistence on an astrological interpretation of cosmology directly influenced his other "scientific" works, something that is taken up in J. Peter Zetterberg's analysis of what he calls Dee's "hermetic geocentricity."
After discussing the somewhat limited commentary on Copernicus' theory of heliocentrism present in Dee's strictly scientific works, Zetterberg suggests that "to resolve the general ambiguity that surrounds the question of Dee's cosmological views it is necessary to leave his works on practical science and turn instead to his occult interests." In Monas hieroglyphica, the only work in which Dee "reveal[s] a cosmology," Zetterberg identifies a kind of hidden meaning Dee proposes to exist…… [Read More]
Considering this belief into account for the discussion of math and poetry, through the general observation and understanding, it is observed that math and poetry constitute the form of art also. The other instance of symmetry in math is proof. Math looks for an elegant proof above one which established the identical outcome through contradiction or examination of a lot of cases. The same thing is applicable in case of poetry. In case one is desirous of having a good poem, it is important that one must develop a style and follow the particular style across. Poetry as well as maths is two very distinct themes, nevertheless the same are found to be similar in formation and structure. Understanding math as a poem solves a lot of problem. When math is looked as a poem, it is observed that calculating is the same as a finding a pattern in the…… [Read More]
(Weisstein "Menger Sponge" 2009) It is created by dividing a cube into 27 equivalent cubes. This resembles a ubik's cube. The central cube is then removed. Each of the remaining cubes are then divided into 27 cubes each of which the central cube is removed. This process is continued to produce a Menger sponge, created by Austrian Mathematician Karl Menger. While 27 cubes result from the first iterative dividing of the starting cube, one can see that the problem becomes quickly insurmountable and needs the use of a computer. On the 6th interation, sixty-four million cubes are produced. The Hausfdorff dimensionality is 2.7268.
The figure shows a Menger sponge after the 4th interation
One of the best illustrations of a fractal, especially the ones that produce the complex looking diagrams is called the dragon curve. While there are programs that generate fractals, showing the results, the dragon curve…… [Read More]
He also has hallucinations about being followed by a federal agent, in keeping with his academic world where the government seeks on the one hand to employ mathematicians and scientists and on the other hand mistrusts them. Many of the encounters he has in his mind with this agent and others have the aura of a detective movie, showing that Nash is replaying films he has seen and that these serve as the inspiration for his visions. In a way, that serves as another pattern in his mind, linking what he saw in the theater with what he believes is happening to him. Nothing comes out of whole cloth but always comes from experience and is then reformed in a form it did not have in reality.
In this way, the film shows the viewer the kind of world experienced by the schizophrenic and why this world is disorienting and…… [Read More]
His work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth (Biography (http://en.wikipedia.org/wiki/Archimedes#Biography)."
One of today's industries that Archimedes is well-known for helping to develop and advance is the field of engineering. Engineering uses a significant amount of mathematics to develop and design countless elements of life, and industry. Archimedes has long since been credited with many of the concepts still used in the engineering field today.
At one point in his life a king commissioned him to design a very large ship that could be used for travel as well as a naval warship.
He did so and it was reputed to be an amazing vessel with garden decorations and a work out area.
Archimedes was well-known for developing concepts that were used much later in society. One of the many concepts that he developed and used during his…… [Read More]
Had this false belief not been perpetuated it may very well be Kepler who directly formulated the laws of gravity well before the time of Newton.
Kepler's second law, which is commonly referred to as the law of equal areas describes the speed at which any given planet will move while orbiting the sun. In his understanding and derivation of mathematical models to understand this process, Kepler noted that planets moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Again, this observation viewed through contemporary lenses makes the connection between the "hidden forces" of gravity as the evident driver, but Kepler concluded otherwise. He noted that if a line were drawn from the center of the planet to the center of the sun, such a line would sweep out the same area in equal period of time. His explanation for the…… [Read More]
First, math courses are required as part of college work in the pursuit of most degrees in the health care field. The level of required achievement is different, depending on the degree sought. For example, a student pursuing an LPN may take a semester or two of college algebra. A pre-med student is often required to take one or two semesters of calculus. A student pursuing a master's degree in health care administration will take courses in statistics, finance and accounting. The master's candidate can perhaps more easily see the relevance of the required math courses toward the future career. For the nursing student studying algebra or the pre-med student struggling through calculus, the correlation between academic study and actual practice may be unclear. They may wonder why they must undertake these courses, which seem to have little to do with the work in which they will eventually be engaged.…… [Read More]
A bit less far this time, certainly, but a bit none the less. He is always getting closer, but never makes up the lost ground completely.
Intuitively at least, it is clear that the human runner can quickly outdistance the tortoise (Papa-Grimaldi, 1996), but Zeno's paradox seems to defy this capability. In this regard, Cohen asks, "Achilles will certainly, with his celebrated speed, soon get very close behind the tortoise -- but why can't he, logically speaking at least, ever overtake the reptilian competitor?" (2002, p. 38). A similar paradox is presented by Zeno in "The Arrow," discussed further below.
Paradox of the Arrow. According to Dowden, this paradox concerns "a moving arrow [that] must occupy a space equal to itself at any moment. That is, at any moment it is at the place where it is. But places do not move. So, if at each moment, the arrow is…… [Read More]
It is almost as if Hawking wants science and religion to agree.
He also uses a sense of humor often times to get his point across. In UIAN, he uses visual jokes, written puns and several witticisms to get you in a light mood to keep going through the book and picking up the important ideas that are in there. His life work has been dissecting these questions and proposing answers and it seems important to him to get the reader and his listeners, students and followers excited with him.
Stephen illiam hawkings http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Hawking.html
Hawking's mother spent part of her life in dangerous places during orld ar II. His mother went to live in a safe town and gave birth to Stephen.
The family were soon back together living in Highgate, north London, where Stephen began his schooling.
In 1950 Stephen's father moved to the Institute for Medical Research in…… [Read More]
Beautiful Mind by Silvia Nasar: The Real Story Of Schizophrenia
For anyone who has seen the film A Beautiful Mind John Nash comes across as a man troubled by schizophrenia, yet able to achieve success in his life. hile his illness does cause him significant problems, he is still able to achieve greatness via his game theory, to manage a long-lasting relationship where his wife loves him unconditionally, to achieve social acceptance where his colleagues accept his condition, and to receive the ultimate career achievement in winning the Nobel prize. The film even shows Nash succeeding over his schizophrenia and become able to control it and cure himself. This depiction presents Nash's story as one full of positives where his struggle with schizophrenia and his life is seen in a romantic light. To see the real truth of schizophrenia, it is better to read Sylvia Nasar's biography of Nash titled…… [Read More]
They do not occupy space. Nevertheless, although the Form of a circle has never been seen -- -indeed, could never be seen -- -mathematicians and others do in fact know what a circle is. That they can define a circle is evidence that they know what it is. For Plato, therefore, the Form "circularity" exists, but not in the physical world of space and time. It exists as a changeless object in the world of Forms or Ideas, which can be known only by reason."
Forms have greater reality than objects in the physical world both because of their perfection and stability and because they are models of reality (Vincent, 2005). Circularity, squareness, and triangularity are all good examples of what Plato meant by Forms. An object in the physical world may be called a circle or a square or a triangle only to the extent that it resembles the…… [Read More]
Richard Le Gallienne, however, shows a Khayyam who may have liked to look at the stars but was far more contented to find peace and happiness in this world than in the next: "Look not above, there is no answer there; / Pray not, for no one listens to your prayer; / Near is as near to God as any Far, / and Here is just the same deceit as There." ith these verses, Khayyam represents a primarily atheistic vision of life, one in which no God exists except some deceitful spirit (an evil or devil?) that permeates life so fully that there is no escape from it. It is a dark and depressing vision, one that paints an even more complex of Khayyam.
In conclusion, Omar Khayyam was a brilliant mathematician and astronomer who also wrote verses that characterized slices of life. His style of writing was simple --…… [Read More]
The famous 1991 novel, Jurassic Park, is based on the subject of a wildlife preserve for dinosaurs. The renowned writer of this novel, Michael Crichton, hoisted the conventional phantom of the revivification of species that have been wiped out from the face of the earth by using conserving DNA samples ("Jurassic Park' 20 Years" C10). The uncontrolled genetic engineering produced outcomes that were not the concern of just the scientists in the novel but are the concern of the whole human civilization (Sharp 507).
Crichton was able to craft a vibrantly dramatic action-adventure story with the Jurassic Park that revolved around the ideas of gluttony and crookedness of science. In this vivid tale of Crichton, an affluent investor builds a theme park that was located on an island off the coast of Costa ica. The peculiar part of the tale is that the investor hires a scientist to…… [Read More]
Fans of science fiction are fond of recalling a remark by novelist Arthur C. Clarke, to the effect that any sufficiently advanced technology is indistinguishable from magic. I am currently typing these sentences onto a laptop, where I am also currently watching a grainy YouTube video of the legendary magician Harry Houdini, performing one of his legendary escapes -- from a straitjacket, in this case. Houdini is probably the most famous stage magician of the twentieth century, as witnessed by the fact that his name is familiar to my generation although he died almost a century ago. If Houdini were to suddenly reappear in front of me right now -- in the flesh, I mean, and not merely on YouTube -- how would I explain to him that the way in which all of this is taking place? To someone who has been dead for a century, the…… [Read More]
Sine, Cosine, And Tangent
When using trigonometric functions, the three sides of a right triangle (opposite, adjacent and hypotenuse) are identified in relation to a chosen angle. The trigonometric functions (sine, cosine, and tangent) are then defined in relation to the three sides of the right triangle.
The word "sine" comes from the Latin word "sinus," which means a bend or gulf, or the bosom of a garment. (Gelfand) The term was used as a translation for the Arabic word "jayb," the word for a sine that also meant the bosom of a garment, and which in turn comes from the Sanskrit word "jiva," which translates to bowstring.
Originally the word "sine" was applied to the line segment CD on a figure, which meant it was half the chord of twice the angle AO. A sine resembles a bowstring in this regard. The ratio of the sine CD to the…… [Read More]
' (Davidson; Lutz, 175) The target of such function is to better comprehend the manner varied circuits are combined during the meditation to generate the mental and behavioral variations which are indicated to prevail due to such experiences, incorporating the promotion of enhanced welfare. (Davidson; Lutz, 175)
Arnone, D; Schifano, F. Psychedelics in psychiatry. The British Journal of Psychiatry,
2006, vol. 188, no.3, pp: 88-89.
Aydin, K; Ucar, A; Oguz, K.K; Okur, O.O; Agayev, A; Unal, Z; Yilmaz, S; Ozturk, C.
Increased Gray Matter Density in the Parietal Cortex of Mathematicians: A Voxel-Based Morphometry Study. American Journal of Neuroradiology, November-December 2007, vol. 28, pp: 1859-1864.
Ball, Jeanne. Keeping your prefrontal cortex online: Neuroplasticity, stress and meditation. The Huffington Post, 11 August, 2000. p. 4.
Davidson, ichard J; Lutz, Antoine. Buddha's Brain: Neuroplasticity and Meditation.
IEEE Signal Processing Magazine, September, 2007, pp: 172-176.
Formica, Michael J. Mindfulness practice in everyday…… [Read More]
That could lead to all kinds of advertising insights," even though it could just be a coincidence (Baker 2008, p.3). The result could be a cutting-edge campaign for a rental car company stressing romance and danger, rather than safety and budget-conscious effectiveness.
No company wishes to miss out on the next new marketing trend, and all want to find the next 'hot' new connection between two seemingly discordant interests or types of behaviors. One idiosyncratic, accidental click or glance is meaningless, but being able to keep track of large amounts of data makes such decisions is significant for market researchers, as a consistent pattern can be drawn. And the blog world of online journalism is an even more willing source of "unfiltered immediacy" for marketers, who can track what types of blogs generate the largest amount of traffic for specific types of ads. Blogs are especially useful because marketers can…… [Read More]
He looks at thee methods: histoy (melding infomation about the divese geogaphical oigins of algeba with the poblems themselves), multiple epesentations (using notation, naative, geometic, gaphical, and othe epesentations togethe to build undestanding), and the object concept of function (teaching functions without genealizing about how taits of an individual elate to taits of a goup). The aticle seves to offe some inventive solutions to a common poblem in math education: How to make mateial elevant and compelling to a beadth of students.
Matinez, a.A. (2010). Tiangle sacifice to the gods. 1-11.
The aticle looks at Pythagoas, paticulaly the mythology suounding his life and his most famous discovey, the Pythagoean theoem. It calls into question the histoical evidence on which mathematics teaches base thei teaching of this theoy. The autho points out how vey little is known about Pythagoas and how he has been canonized by the math discipline because his…… [Read More]
" The point made by the poet is similar to the poem above. The reference to John,
The Father of our souls, shall be,
John tells us, doth not yet appear;
is a reference to the Book of Revelations, at the end of the Bible.
That despite the promises of an Eternal life for those who eschew sin, we are still frail and have the faults of people. We are still besought by sin and temptations and there's really no escape. People are people. No matter what we say or do, we find that life is not so simple. Consider this reference, which really refers to a person's frame of reference or "way of seeing."
Wise men are bad -- and good are fools,
This is a paradoxical statement: there is large gap between spirituality and reality. Those we consider wise or bad, might make decisions that are globally profound,…… [Read More]
ut science is about stepping stones: the creation of theories and hypothesis, and the testing of these hypotheses with empiricism. If these theories fail, then additional hypotheses have to be proposed. During the process of the testing these hypothesis, experimentalists will find evidence based that will enable to fine tuning of the hypothesis, and the process carries on. Indeed, most of quantum theory is hinged on the Uncertainty principle put forward by Werner Heisenberg. What apt that it be named the Uncertainty principle.
Eventually, one hopes that some consensus will come between those that support graduated equilibrium vs. phyletic gradualism in terms of evolution of species. Or a new theory will develop and come to the fore, if new fossil evidence comes to light. ut that does not mean that we subscribe to the watchmaker theory. William Paley, an eighteenth century moral theorist, philosopher and religious conservative, was perhaps the…… [Read More]
Moreover, his theories regarding the gravitation were supposed not to have been made possible without the attempts of his predecessors, as Galileo, to understand the world. Thus, Newton's luck may be put on the fact that he has lived in a period of discoveries, and, as he himself stated, he had seen further than other men, it is because he stood on the shoulders of giants.
All in all, Newton has been considered for almost 300 years to be the founding father of modern physical science, his discoveries being unprecedented, just as those in mathematical research. eing a polyvalent personality, he also studied chemistry, history and theology; his main method in all domains being the investigation of all forms and dimensions.
Cohen, I. ernard, The Newtonian Revolution, Cambridge, 1980, 546 pages;
Koyre, Alexandre, Newtonian Studies, Harvard U. Press, 1965, 673 pages;
Westfall, Richard S., Never at Rest: A iography…… [Read More]
William of Occam formulated the principle of Occam's Razor, which held that the simplest theory that matched all the known facts was the correct one. At the University of Paris, Jean Buridan questioned the physics of Aristotle and presaged the modern scientific ideas of Isaac Newton and Galileo concerning gravity, inertia and momentum when he wrote:
...after leaving the arm of the thrower, the projectile would be moved by an impetus given to it by the thrower and would continue to be moved as long as the impetus remained stronger than the resistance, and would be of infinite duration were it not diminished and corrupted by a contrary force resisting it or by something inclining it to a contrary motion (Glick, Livesay and Wallis 107)
Thomas Bradwardine and his colleagues at Oxford University also anticipated Newton and Galileo when they found that a body moving with constant velocity travels distance…… [Read More]
In the real world, most markets are far from fully competitive, labor-productivity within a country varies over time and full employment is just a dream in most capitalist economies. (Suranovic 1997)
Given the list of such 'unrealistic' assumptions made in the model, it is easy for us to dismiss the results of comparative advantage altogether or to accept it with a large dose of skepticism. But would such dismissal be justified? It is prudent to remember that almost all economic theories operate under a large number of 'unrealistic' assumptions that do not apply to the much more complex real world. In fact, we deliberately simplify the variables in economics in order to be able to construct models and to carry out analysis. Such simplification does not make all economic analysis redundant. It should also be remembered that comparative advantage and international trade would work to the benefit of all individuals…… [Read More]
But the view of Aristotle is more critical, rather than seeing the philosopher as a great prognosticator. Aristotle is presented as a great patriarch, occasionally overly venerated, as quite often his word was assumed to be 'gospel' during the heyday of the Catholic Church and scholasticism, although the website makes clear he should still be regarded as a worthy creator of the inklings of the modern scientific method.
The agenda of Philosophy Pages might also seem to be obvious, given the site's title, and its copious and helpful links to specific areas of Aristotelian philosophy, including Aristotle's doctrine of the universals, friendship, and other bibliographic sources. Although all of the data presented, however, is relatively accurate, it is extremely superficial and does not contain the copious amount of direct quotes from the philosopher's works -- far less even then the Encyclopedia page devoted to Aristotle. Also, the site is filled…… [Read More]
Questions about place and role of reason puzzle generations of philosophers as they are among the fundamental questions of philosophy. In case it appears that everything is planned and all events are mutually connected it may witness for the divine origin of the universe and man. Laplace proposed the theory of absolute determinism which stated that every process which took place in the universe had a reason so that the next or previous stage of this process could be predicted and described in the absolute form.
Determinism of Laplace had a lot of strong points at the time when he developed this theory. First of all Laplas was a mathematician and physicist and the principle of sufficient reason corresponded to all dynamic processes he studied: motion, oscillations, etc. This principle laid in the fundamentals of classic mechanics and was applied for any dynamical system on the hand with…… [Read More]
In addition, they both realize that stress can make his condition worse, and work to reduce stress on him. That would also be an important part of his treatment today. In the movie, it may have helped Nash that his imaginary "controller" wanted him to do things he could not agree with, such has harming his wife "because she knew too much." In the movie, the little girl appears and holds his hand, and then it dawns on him -- the little girl never ages. She can't be real. The controller can't be real. It is hard to know whether these events really happened in this way; his story is presented as a movie, and Nash's perceptions may have altered even the events that help him resist the draw of his hallucinations.
John Nash's story demonstrates also that hallucinations serve a purpose for the patient's personality. As a secret code…… [Read More]
As Kleiner & Movo*****z-Hadar show, the burden of proof lies with the mathematician eager to uncover some unknown universal law or theorem. His or her colleagues will, as the authors point out, be harangued and criticized because of a general resistance to new ways of thinking or profound revelations. Because of the difficulty in obtaining proof and subsequently communicating those proofs to the academic community, math remains one of the last bastions of reason in our society. Mathematics stimulates analysis and critical thinking, essential components of a good life. Philosophers use proofs too, such as to illustrate the existence or non-existence of God. Perhaps the most salient issue brought out by Kleiner & Movo*****z-Hadar in their article and the one most relevant to modern classrooms is their celebration of diversity of thought. While mathematics may occasionally manifest as speculation or even intuition, ultimately mathematicians demand proof: evidence, firmness, and rigorous…… [Read More]
operations can be described as the set of rules that mathematicians have established and agreed to adhere to in order to avoid mass confusion when simplifying mathematical equations or expressions. Actually, without this simple and significant order of operations, learning mathematics would not only be difficult but almost impossible. Therefore, order of operations is important in solving equations or expressions with several operations or in simplifying the equations or expressions. The main reason attributed to the ability of order of operations to contribute to simplifying expressions and equations is a standard that describes the order for the process through operations like addition, multiplication, subtraction, and division ("Order of Operations Lessons," n.d.).
During the process of developing the order of operations, mathematicians were very careful in order to promote the simplification of mathematical equations and expressions. If the order of operations is ignored, the most possible outcomes include the fact that…… [Read More]