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(Math) Calculus: 2006 Abel Prize (Harmonic Analysis)

 
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adedios
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PostPosted: Sun Mar 26, 2006 9:21 am    Post subject: (Math) Calculus: 2006 Abel Prize (Harmonic Analysis) Reply with quote






American Mathematical Society
23 March 2006

Lennart Carleson wins 2006 Abel Prize
Prize of almost one million dollars

The Norwegian Academy of Science and Letters has awarded the 2006 Abel (AH-bell) Prize to Lennart Carleson, Royal Institute of Technology, Sweden. The prize amount is over $900,000 and in mathematics is comparable to the Nobel Prize in other sciences. The Abel Committee citation says: "Carleson's work has forever altered our view of analysis. Not only did he prove extremely hard theorems, but the methods he introduced to prove them have turned out to be as important as the theorems themselves."
King Harald of Norway will present the Abel Prize to Lennart Carleson at an award ceremony in Oslo 23 May.

Carleson's mathematical work: Harmonic analysis and dynamical systems

Harmonic analysis
Harmonic analysis is a vast extension of the mathematics underlying ordinary calculus. It has applications to real world phenomena ranging from the firing of neurons in the brain to quantum computers, and is based on concepts students see in trigonometry. Many phenomena, ranging from the vibrations of violin strings to the propagation of heat through a metal bar, can be viewed as sums of the simple wave patterns called sines and cosines. Such summations are now called Fourier series. One of Carleson's many triumphs was settling a conjecture that had remained unsolved for over 150 years. He showed that every continuous function (one with a connected graph) is equal to the sum of its Fourier series except perhaps at some negligible points. (He actually proved a more general result, that a broader class of functions, called square-integrable, equal the sum of their Fourier series except perhaps at those points.)

Dynamical Systems
Dynamical systems are mathematical models that seek to describe the behavior of large classes of phenomena, such as those observed in weather, financial markets, and many biological systems, from fluctuations in fish populations to epidemiology. Even the simplest dynamical systems can be surprisingly complex. With Michael Benedicks, Carleson studied a dynamical system first proposed in 1976 by the astronomer Michel Hénon, which exhibits the intricacies of weather dynamics and turbulence. This system was generally believed to have a so-called strange attractor, drawn in beautiful detail by computer graphics tools, but poorly understood mathematically. In a great tour de force, Benedicks and Carleson provided the first proof of the existence of this strange attractor in 1991, which has opened the way to a systematic study of this class of dynamical systems.

Biography of Lennart Carleson
Born March 18, 1928 in Stockholm, Carleson received his PhD from Uppsala University in 1950. He was director of the Mittag-Leffler Institute (Stockholm) from 1968 to 1984, building it into one of the prestigious research centers in the world, and was president of the International Mathematical Union (IMU) from 1978 to 1982. During his term as president, he succeeded in getting the People's Republic of China represented in the IMU and was instrumental in the creation of the Nevanlinna Prize, which rewards young computer scientists.

The Abel Prize
The Niels Henrik Abel Memorial Fund, established in 2002, awards the Abel Prize for outstanding scientific work in the field of mathematics. The Norwegian Academy of Science and Letters appoints an international Abel Prize Committee to select a laureate from among the nominees and administers the annual award of nearly 6 million Norwegian Kroner (currently about $910,000). Previous Abel winners are Jean-Pierre Serre (2003), Isadore M. Singer and Sir Michael Atiyah (2004), and Peter Lax (2005). See www.abelprisen.no.

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Questions to explore further this topic:

Tutorials on pre-calculus topics in math

http://archives.math.utk.edu/v.....index.html
http://www.bymath.com/index.html
http://www.jimloy.com/math/math.htm

Tutorials on calculus

http://www.karlscalculus.org/calculus.html

Limits and continuity
http://archives.math.utk.edu/v.....index.html

Derivatives
http://archives.math.utk.edu/v.....index.html

Applications of differentiation
http://archives.math.utk.edu/v.....index.html

Integration
http://archives.math.utk.edu/v.....index.html

Applications of integration
http://archives.math.utk.edu/v.....index.html

Sequences and series
http://archives.math.utk.edu/v.....index.html

What is harmonic analysis?

http://www.math.yale.edu/~mmm82/hrmwav.htm
http://www.math.lsu.edu/grad/harmgrp.html

An online text on harmonic analysis

http://math.ut.ee/~toomas_l/harmonic_analysis/

What is Fourier analysis?

http://www.astro-med.com/knowledge/fourier.html

Fourier analysis of waves

http://hyperphysics.phy-astr.g.....geowv.html

What is a Fourier series?

http://en.wikipedia.org/wiki/Fourier_series
http://www.kettering.edu/~drus.....urier.html
http://physics.nad.ru/Physics/English/harm_txt.htm
http://physics.nad.ru/Physics/English/harm_ref.htm

What are wavelets?

http://www.amara.com/current/wavelet.html
http://www.amara.com/IEEEwave/IEEEwavelet.html
http://users.rowan.edu/~polika.....orial.html

What is complex analysis?

http://www.maths.mq.edu.au/~wc.....lnica.html

Applications of harmonic analysis in nonlinear approximation and image processing

http://www.math.ac.vn/conferen.....cohen1.pdf

Harmonic analysis in acoustics

http://www.phon.ucl.ac.uk/cour.....eek1-4.pdf

Harmonic analysis on AC electrical circuits

http://www.ibiblio.org/obp/ele...../AC_7.html

Signal processing in chemical analysis

http://www.wam.umd.edu/~toh/sp.....essing.pdf

Harmonic analysis of tides

http://www.math.sunysb.edu/~tony/tides/index.html'

GAMES

http://www.bbc.co.uk/education.....wheel.html
http://www.ababasoft.com/kids/index.html


Last edited by adedios on Sat Jan 27, 2007 3:13 pm; edited 1 time in total
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adedios
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PostPosted: Wed Aug 23, 2006 1:28 pm    Post subject: Math and aftermath Reply with quote

Math and aftermath
STAR SCIENCE By Eusebio L. Koh, Ph.D.
The Philippine STAR 08/24/2006

Many years ago while taking an electrical engineering class at the University of the Philippines in Diliman, I was thoroughly impressed by my professor, Bartolome Blanco, who solved a differential equation with initial conditions by some sort of a "trick." He called his trick the Laplace transform.

With thirty-some bright-eyed classmates agog at learning new ideas, I was wondering if he was pulling our legs. Ever quick on the uptake, Prof. Blanco said, "If any of you wish to impugn what I taught here, you can check out my solution by plugging it into the equation or you can run to the Math Department to see what was going on."

Wow! Some of us had brushed elbows with the simplest differential equations, never heard of transforms, except in reference to converting high voltage to low voltage in AC circuits. And "impugn"? I had read that word somewhere in a long forgotten book but that was the first time I heard it used. Now you understand why I was impressed with my "Prof" and had visions of following in his footsteps.

So much so that I jumped on the opportunity to teach engineering at UP upon my graduation, when Dean C. Ortigas offered me an instructorship. At the same time, I took more math classes on the side: a class on differential equations here, a class on advanced engineering math there. It certainly gave me a step up when UP later sent me abroad for graduate studies.

Now fast forward to what I ended up doing. My doctoral dissertation at the State University of New York at Stony Brook was on the Hankel transformation of generalized functions with applications to sorts of differential equations. This is really a mouthful and I am going too far ahead of myself. With your indulgence, let me start from scratch.

1. Mathematics is the study of relations and their properties. A relation is a correspondence between two sets of objects. These objects can be anything: numbers, figures, sets, or even relations, too. When the correspondence assigns a unique object from the second set to each element of the first set, we call that relation a function. A function is usually restricted to a relation between sets of numbers. If the relation is between sets of functions, we call it an operator. If the relation is from a set of functions to a set of numbers, we call it a functional. Here are some examples.

Functions: linear, quadratic, rational, exponential, trigonometric, special.

Operators: differential operator - d/dx, integral operator- º…dx.

Functionals: definite integrals, Dirac delta.

2. Calculus is a branch of math dealing with change. The rate of change of a function f is its derivative f’ and the process of finding the derivative is called differentiation. Because our world is always in a state of flux, processes, natural or otherwise, can be described by equations involving derivatives. These are called differential equations, which are creatures more intractable than algebraic equations. Their solutions are functions while the latter has numbers for solutions. For example:

Algebraic equation: x square +4x+3=0 is solved by x = -1 or x = -3.

Differential equation: y"+4y’+3y=0 is solved by y=a[exp(-x)]+b[exp(-3x)], a & b constants.

The last equation describes a simple damped spring mass system or a simple electric circuit with a battery, a capacitor and a resistance and the solution is a combination of two decaying exponential functions. The solution is found by the observation that an exponential function replicates itself on differentiation. Thus the task is reduced to solving an algebraic equation. More complicated equations are solved by such techniques as infinite series, integral transformation, Fourier series, special functions and numerical methods and computer codes.

3. An integral transformation is an operator that takes a set of functions and maps it into a set of transforms via integration. The most famous is the Laplace transform given by F(s) = L[f(t)] = ºexp(-st)f(t)dt with the important property L[f’(t)] = sL[f(t)] — f(0). Thus when f(0) = 0, differentiation corresponds to multiplication by s in the transform domain. The Laplace transform has been used to justify Heaviside’s calculus used by engineers. The term exp(-st) is called the kernel of the transform. By changing the kernel as well as the range of integration, other integral transforms were developed and used for solving other equations. The Hankel transform replaces exp(-st) with Ãst Jµ(st) where Jµ(st) is Bessel function of order µ. This transform is suitable for differential equations with variable coefficients that arise in physical problems with axial symmetry. Other transforms go by such names as Fourier, Weierstrass, Stieltjes, Meijer, Hilbert, Jacobi, convolution, etc., each with its suitability for certain differential operators. The definitions, properties and applications are enough to fill several books.

4. In 1926, Dirac introduced his d-function as d(x)=0 when x­0, ºf(x) d(a-x)dx=f(a) and used it in quantum mechanics. This is not a function in the classical sense. To be sure there were other improper functions such as Hadamard’s pseudofunction pf x, Cauchy’s pv of a divergent integral, derivatives of Dirac- dthat have appeared in some physicists’ works. In the 1950s, Laurent Schwartz came up with his Theory of Distributions to explain all these as continuous linear functionals on test functions with compact supports. In the ‘60s, the Russians Gelfand and Shilov came up with their four-volume treatise on Generalized Functions not drastically different from Schwartz.

5. In the ’60s and ’70s, Armen Zemanian and others developed various integral transformations of generalized functions (gfs). Zemanian and I developed the generalized Hankel transform under some restrictions. Later, I went farther than our work going into representations of Hankel transforms, transforms of negative orders, transforms in higher dimensions. I later developed the generalized Meijer transform (with my student M. Ali), Hankel Transforms of Banach-space-valued gfs (with my student C. K. Li).

6. There are two methods of extending the integral transformation to generalized functions. The Adjoint Method uses a Parseval—type relation to define the transform of a gf F as the application of F on the transform of a suitable test function. Schwartz used this method for the Fourier transform, Gelfand and Shilov for the Hilbert transform, Zemanian for the Hankel and other transforms. What’s involved is the construction of two test function spaces such that one is the transform of the other. Details can be found in their books.

The other method defines the transform as the application of the generalized function F on the kernel as an element of a test function space. This necessitates the construction of a suitable complete semi-normed space that contains the kernel. Zemanian used this method for the Laplace transform and I did it for the Hankel transform. Details may be found in Zemanian’s book and Brychkov and Prudnikov (1998).

The University of the Philippines is a great source of inspiration for our youth despite its lack of material resources. It is, however, rich in a faculty that is a fountain of wisdom and knowledge. I credit Prof. Blanco in igniting a spark to my career. But there were many excellent professors I had. I would be remiss not to acknowledge Professor Josefina Constantino who instilled in me a love of the English language. From the time I became Professor Emeritus in 1999, I have published short stories, poems and essays online and in print. In these writings my math background seems to creep in.

American poet Adelaide Crapsey invented a verse form in the 1920s called cinquain, which is a poem of five iambic lines of two, four, six, eight and two syllables. There are no restrictions on rhyme but there is a central theme in the cinquain. In 1927, Angela M. Gloria published three cinquains in the Philippine Herald. Here are some unpublished cinquains of mine on scientists:

Newton

The man

Felt the apple

Dropped on his head with such

Gravity, he couldn’t help yell,

"Aha!"

Einstein

He did

Tinker with his

Relatives; speed boggles

The mind as energy and mass

Equate.

Archimedes

He ran

Out in the buff

Hollering "Eureka!"

For he had found that buoyant force

Pushed up.

Leibniz

Symbols

He used for thoughts,

Logic was the richer,

And calculus was brought to the

Limit. * * *
Dr. Eusebio (Seb) L. Koh has been a professor emeritus of mathematics at the University of Regina since 1999. He has taught mechanical engineering and mathematics in the Philippines, the United States, Canada, Germany and Saudi Arabia. He holds master’s degrees from Purdue University and the University of Birmingham in England and a Ph.D. from SUNY, Stony Brook. He has published over 50 refereed research papers in mathematics and sundry articles in engineering. He has given lectures and conference seminars in the Philippines, other Asian countries, North America and Europe. Since his retirement from university teaching, he has published short stories and poems in Our Own Voice, The Best Philippine Short Stories and the Prairie Messenger, the Catholic weekly in Saskatchewan. Since 1993, he writes a column for Filipino Journal, a Philippine semi-monthly in Winnipeg. When not writing, he is into chess, tennis, crossword and Sudoku puzzles, volunteer church work, Knights of Columbus or learning to cook.
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PostPosted: Mon Aug 13, 2007 11:02 am    Post subject: Indians predated Newton 'discovery' by 250 years Reply with quote

University of Manchester
13 August 2007

Indians predated Newton 'discovery' by 250 years

A little known school of scholars in southwest India discovered one of the founding principles of modern mathematics hundreds of years before Newton – according to new research.

Dr George Gheverghese Joseph from The University of Manchester says the ‘Kerala School’ identified the ‘infinite series ’- one of the basic components of calculus - in about 1350.

The discovery is currently - and wrongly - attributed in books to Sir Isaac Newton and Gottfried Leibnitz at the end of the seventeenth centuries.

The team from the Universities of Manchester and Exeter reveal the Kerala School also discovered what amounted to the Pi series and used it to calculate Pi correct to 9, 10 and later 17 decimal places.

And there is strong circumstantial evidence that the Indians passed on their discoveries to mathematically knowledgeable Jesuit missionaries who visited India during the fifteenth century.

That knowledge, they argue, may have eventually been passed on to Newton himself.

Dr Joseph made the revelations while trawling through obscure Indian papers for a yet to be published third edition of his best selling book ‘The Crest of the Peacock: the Non-European Roots of Mathematics’ by Princeton University Press.

He said: “The beginnings of modern maths is usually seen as a European achievement but the discoveries in medieval India between the fourteenth and sixteenth centuries have been ignored or forgotten.

“The brilliance of Newton’s work at the end of the seventeenth century stands undiminished – especially when it came to the algorithms of calculus.

“But other names from the Kerala School, notably Madhava and Nilakantha, should stand shoulder to shoulder with him as they discovered the other great component of calculus- infinite series.

“There were many reasons why the contribution of the Kerala school has not been acknowledged - a prime reason is neglect of scientific ideas emanating from the Non-European world - a legacy of European colonialism and beyond.

“But there is also little knowledge of the medieval form of the local language of Kerala, Malayalam, in which some of most seminal texts, such as the Yuktibhasa, from much of the documentation of this remarkable mathematics is written.

He added: “For some unfathomable reasons, the standard of evidence required to claim transmission of knowledge from East to West is greater than the standard of evidence required to knowledge from West to East.

“Certainly it’s hard to imagine that the West would abandon a 500-year-old tradition of importing knowledge and books from India and the Islamic world.

“But we’ve found evidence which goes far beyond that: for example, there was plenty of opportunity to collect the information as European Jesuits were present in the area at that time.

“They were learned with a strong background in maths and were well versed in the local languages.

“And there was strong motivation: Pope Gregory XIII set up a committee to look into modernising the Julian calendar.

“On the committee was the German Jesuit astronomer/mathematician Clavius who repeatedly requested information on how people constructed calendars in other parts of the world. The Kerala School was undoubtedly a leading light in this area.

“Similarly there was a rising need for better navigational methods including keeping accurate time on voyages of exploration and large prizes were offered to mathematicians who specialised in astronomy.

“Again, there were many such requests for information across the world from leading Jesuit researchers in Europe. Kerala mathematicians were hugely skilled in this area.”
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