tag:blogger.com,1999:blog-250222992012-02-14T19:57:45.048+11:00Reflections on mathematical research towards a doctorate of philosophy.ornoreply@blogger.comBlogger311100tag:blogger.com,1999:blog-25022299.post-720679079729032172006-11-15T17:19:00.000+11:002006-11-15T17:21:39.039+11:00

The scientist does nos study nature becaus it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.H. Poincare

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Man is not a circle with a single centre; he is an ellipse with two foci. Facts are one, ideas are the other.Victor Hugo, Les Miserables

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A representation is a formal systems for making explicit certain entities or types of information, together with a specification of how the system does this.

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A geometry, Klein said, is defined by a group of transformations, and investigates everything that is invariant under the transformations of this given group. Of symmetry one speaks with respect go a subgrup gamma of the total group.Physical occurrences happen not only in space but in space and time; the world is spread out not as a three- but as a four-dimensional continuum. The symmetry,

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For more than 130 years solitary waves where considered rare curiosities. Not until the 1960's did it become clear that such waves could form in the ocean as seismic waves of enormous wavelengths generated by shocks in the ocean's floor. Moreover, soliton waves are common in a great variety of other physical systems. In some circumstances it is almost impossible ot prevent them from forming.There

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The superficial similarity between the properties of solitons and of elementary particles is striking. Solitons may propagate without change of form. A solitons may be regarded as a local confinement of the energy of the wave field. When two solitons collide, each may come away with the same character as it had before the collision. When a soliton meets and antisoliton, both may be anhihilated.

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The progress of mathematics may be viewed as a movement from the infinite to the finite. At the start, the possibilities of a theory, for example the theory of enumeration, appear boundless. Rules for the enumeration of sets subject to various conditions appear to obey an indefinite variety of recursions and seem to lead to a bounty of generating functions. We are naively led to conjecture that

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A good teacher does not teach facts, he or she teaches enthusiasm, open-mindedness and values. Young people need encouragement. Left to themselves, they may not know hot to decide what is worthwhile. They may drop an original idea because they think someone else must have thought of it already. Students need to be taught to believe in themselves and not to give up. Gian-Carlo Rota

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Mathematics is the study of analogies between analogies. All science is. Scientists want to show that thins that don´t look alike are really the same. That is one of their innermost Freudian motivations. In fact, that is what we mean by understanding.

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The soliton fad is a catchy mixture of operator algebra and explicitlly solvable differential equations. How can anyone resist such temptation? So much for the good news. The bad news is that solitons seem unbudgingly one-dimensional, despite the insinuations of computer simulators. Chalk one up to one-dimensional physics.Gian-Carlo Rota

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Of all escapes from reality, mathematics is the most succesful ever. It is a fantasy that becomes all the more addictive because it works back to improve the same reality we are trying to evade. All other escapes -sex, drugs, hobbies, whatever- are ephemeral by comparison. The mathematician`s feeling of triumph, as he forces the world to obey the laws of his imagination has freely created, feeds

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If you look up at the sky on a clear cloudless night, you appear to see a hemispherical dome above you, punctuated by myriads of stars. In fact, you are realistically picturing the family of light rays that constitute the light cone centred at the event O that is occupied by your eye at the moment that you perceive the celestial scene.(Roger Penrose, The Road to Reality, 2005)

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Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independet reality.Minkowski

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It has been said that it is far easier to abstract a general mathematical theory from a single well-chosen example than it is to apply an existing abstract theory to a specific example.Peter Olver, 2000

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“Most current mathematical research, since the 60s, is devoted to fancysituations: it brings solutions which nobody understands to questions nobodyasked”(quoted from Bernard Beauzamy, “Real life Mathematics”, Irish Math. Soc. Bull. 48 (2002), 43-46).

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The mathematical models we devise to make sense of the world around and within us can only be approximations. Therefore, it seems reasonable that if they are to accurately reflect reality, the models themselves must be somewhat insensitive to perturbations. The attempts to give mathematical substance to these rather vague ideas have led to the concept of structural stability.The concept of

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The importance of coordinate transformations in the study of dynamical systems cannot be overestimated. For example, in the study of systems of linear constant coefficient ordinary differential equations, coordinate transformations allow one to decouple the system and hence reduce the system to a set of decoupled linear first-order equations which are easily solved. In the study of completely

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Watch this beautiful movie about black holes seen as waves. According to Einstein's math, when two massive black holes merge, all of space jiggles like a bowl of Jell-O as gravitational waves race out from the collision at light speed. Similar to ripples on a pond, gravitational waves are ripples in space and time, a four-dimensional concept that Einstein called spacetime.Black hole waves videoSo

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The Sato Theory clarifies the structure of the space of solutions, thetransformation groups on that space and the relation to linear equations (Laxequations.)(T. Tokihiro, 2004)

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A mechanical system which is in a state where its potential energy has anisolated minimum, is in a state of stable equilibrium.Lagrange

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Bacteria in our guts, insects in a field, trees in a forest and plants borne by air and water around the world are among many examples whose spatial distributions are important and interesting study. Mechanisms of dispersal include making small random moves, being carried along by air of water, and being attracted to certain areas by chemical signals, nutrients, light, etc. Patterns of organisms

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They say Australia is the number one seller of Easter Eggs and Bunnys. I'm at home, stranded in a long weekend, finding that the only amusements around here are expensive enough to unstabilize the economy. What a hack, it's a holiday. i wanted to go see some sightseeings but seems that a car is needed if one is to travel to another place than the city. Yes, I live in the suburbs, in Bundoora,

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"I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped- not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity assuming the form of a large solitary elevation,

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Just like Hilbert did state the most important mathematical problems for the 20th. century, Steven Smale goes in an attempt of doing the same for the 21st. century.I pay special attention to a few problems:Problem 8: Introduction of dynamics into economic theory:"Extend the mathematical model of general equilibrium theory to include price adjustments".Problem 10: The Closing Lemma.Problem 11: Is

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Turbulence:in poetry:http://www.poetrymagazines.org.uk/magazine/record.asp?id=10398in science:http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1103857186For the pleasure of the soul or for the study of the brain? chaotic dynamics, anyways

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So finally, after 3 weeks in Melbourne, my supervisor Reinout Quispel has arrived. The lines of investigation for my Masters thesis are still broad but all of them in differential -an now difference which seems to be a bit more complicated and exact- equations.I spend my time reading a couple interesting books: The Road to Reality and Mathematics in Nature; this books give me an overview of the

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This therefore is Mathematics:She reminds you of the invisible forms of the soul; She gives life to her own discoveries;She awakens the mind and purifies the intellect; She brings light to our intrinsic ideas; She abolishes oblivion and ignorance which are ours by birth. Proclus Diadochus, mathematician

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What if nothing ever really matters! What if we are just thinking about the time it takes to go from one place to another and forgetting that the time invested is what life is in its own. I like to think that there is a bright future awaiting for me, holding some of my existence so I cannot live entirely as I am right now. After a couple of years, I will have a MSc., and then I will be on my way

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So I’m here, in an island far from home. Human nature starts to make itself present. Mathematics is just a work or an occupation, just as any other could be; almost but not quite. It has its rewards in the simple fact of performing it. But doesn’t any occupation offer the same? The city is out there, far from La Trobe University, an hour by tram. These days I haven’t work much. Internet

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Tomorrow I move to my new apartment; the day after, my wife arrives to town. It’s been days of getting into study, trying to gasp the meaning of mathematical investigation. I realize that the fist step is to get enough tools as to be able to discover new ones. One has to dominate what is already there. I’m in the study of the book Nonlinear Differential Equations and Dynamical Systems by Verhulst

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I arrived to Melbourne on the morning of Tuesday 14th of March. I left my backpack at my temporary apartment and without delay I went to La Trobe University campus. Is there any point in writing all the details of a life already lived? Isn’t it better to move on to the next thing and just forget about what is done? No idea. But stubborn as I am, or as I intend to be, I will keep on with this

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