Topics

1

History of Mathematics / Biography: Arthur Cayley

« on: August 05, 2013, 01:40:45 PM »

ARTHUR CAYLEY (1821–1895) was an English mathematician who was instrumental in developing the theory of matrices. He was the first to use a single symbol such as A to represent a matrix, thereby introducing the idea that a matrix is a single entity rather than just a collection of numbers. Cayley practiced law until the age of 42, but his primary interest from adolescence was mathematics, and he published almost 200 articles on the subject in his spare time. In 1863 he accepted a professorship in mathematics at Cambridge, where he taught until his death. Cayley’s work on matrices was of purely theoretical interest in his day, but in the 20th century many of his results found application in physics, the social sciences, business, and other fields. One of the most common uses of matrices today is in computers, where matrices are employed for data storage, error correction, image manipulation, and many other purposes. These applications have made matrix algebra more useful than ever.

2

Applied Maths / Mathematics and the Modern World 4. Unbreakable Codes

« on: August 03, 2013, 12:30:06 PM »

If you read spy novels, you know about secret codes and how the hero “breaks” the code. Today secret codes have a much more common use. Most of the information that is stored on computers is coded to prevent unauthorized use. For example, your banking records, medical records, and school records are coded. Many cellular and cordless phones code the signal carrying your voice so that no one can listen in. Fortunately, because of recent advances in mathematics, today’s codes are “unbreakable.” Modern codes are based on a simple principle: Factoring is much harder than multiplying. For example, try multiplying 78 and 93; now try factoring 9991. It takes a long time to factor 9991 because it is a product of two primes 97 and 103, so to factor it, we have to find one of these primes. Now imagine trying to factor a number N that is the product of two primes p and q, each about 200 digits long. Even the fastest computers would take many millions of years to factor such a number! But the same computer would take less than a second to multiply two such numbers. This fact was used by Ron Rivest, Adi Shamir, and Leonard Adleman in the 1970s to devise the RSA code. Their code uses an extremely large number to encode a message but requires us to know its factors to decode it. As you can see, such a code is practically unbreakable. The RSA code is an example of a “public key encryption” code. In such codes, anyone can code a message using a publicly known procedure based on N, but to decode the message, they must know p and q, the factors of N. When the RSA code was developed, it was thought that a carefully selected 80-digit number would provide an unbreakable code. But interestingly, recent advances in the study of factoring have made much larger numbers necessary.

Reference

J. Stewart, L. Redlin, S. Watson Algebra and Trigonometry 3rd ed. Brooks/Cole 2012

3

Applied Maths / Mathematics and the Modern World 3. Automotive Design

« on: August 03, 2013, 12:08:50 PM »

Automotive Design

Computer-aided design (CAD) has completely changed the way in which car companies design and manufacture cars. Before the 1980s automotive engineers would build a full-scale “nuts and bolts” model of a proposed new car; this was really the only way to tell whether the design was feasible. Today automotive engineers build a mathematical model, one that exists only in the memory of a computer. The model incorporates all the main design features of the car. Certain polynomial curves, called splines, are used in shaping the body of the car. The resulting “mathematical car” can be tested for structural stability, handling, aerodynamics, suspension response, and more. All this testing is done before a prototype is built. As you can imagine, CAD saves car manufacturers millions of dollars each year. More importantly, CAD gives automotive engineers far more flexibility in design; desired changes can be created and tested within seconds. With the help of computer graphics, designers can see how good the “mathematical car” looks before they build the real one. Moreover, the mathematical car can be viewed from any perspective; it can be moved, rotated, or seen from the inside. These manipulations of the car on the computer monitor translate mathematically into solving large systems of linear equations.

4

Applied Maths / Mathematics and the Modern World 2. Computers

« on: May 18, 2013, 03:00:22 PM »

For centuries machines have been designed to perform specific tasks. For example, a washing machine washes clothes, a weaving machine weaves cloth, an adding machine adds numbers, and so on. The computer has changed all that.
     The computer is a machine that does nothing—until it is given instructions on what to do. So your computer can play games, draw pictures, or calculate π to a million decimal places; it all depends on what program (or instructions) you give the computer. The computer can do all this because it is able to accept instructions and logically change those instructions based on incoming data. This versatility makes computers useful in nearly every aspect of human endeavor.
     The idea of a computer was described theoretically in the 1940's by the mathematician Allan Turing in what he called a universal machine. In 1945 the mathematician John Von Neumann, extending Turing’s ideas, built one of the first electronic computers.
          Mathematicians continue to develop new theoretical bases for the design of computers. The heart of the computer is the “chip,” which is capable of processing logical instructions. To get an idea of the chip’s complexity, consider that the Pentium chip has over 3.5 million logic circuits!

Alan Turing (1912-1954)

John von Neumann with the stored-program computer at the Institute for Advanced Study, Princeton, New Jersey, in 1945.

Further reading

Saba Fatema Alan Turing: The Master of Code Breaking http://forum.daffodilvarsity.edu.bd/index.php?topic=9702.msg38548#msg38548

5

Applied Maths / Mathematics and the Modern World Changing Words, Sound, and Pictures into Number

« on: May 15, 2013, 05:02:05 PM »

Pictures, sound, and text are routinely transmitted from one place to another via the Internet, fax machines, or modems. How can such things be transmitted through telephone wires? The key to doing this is to change them into numbers or bits (the digits 0 or 1). It’s easy to see how to change text to numbers. For example, we could use the correspondence

A = 00000001, B = 00000010, C = 00000011, D = 00000100, E = 00000101,

and so on. The word “BED” then becomes

000000100000010100000100

.
By reading the digits in groups of eight, it is possible to translate this number back to the word “BED.”

     Changing sound to bits is more complicated. A sound wave can be graphed on an oscilloscope or a computer. The graph is then broken down mathematically into simpler components corresponding to the different frequencies of the original sound. (A branch of mathematics called Fourier analysis is used here.) The intensity of each component is a number, and the original sound can be reconstructed from these numbers. For example, music is stored on a CD as a sequence of bits; it may look like 101010001010010100101010 10000010 11110101000101011.... (One second of music requires 1.5 million bits!) The CD player reconstructs the music from the numbers on the CD.

    Changing pictures into numbers involves expressing the color and brightness of each dot (or pixel) into a number. This is done very efficiently using a branch of mathematics called wavelet theory. The FBI uses wavelets as a compact way to store the millions of fingerprints they need on file.

6

Mathematics / Algebra

« on: May 14, 2013, 06:20:28 PM »

Algebra

Algebra is a branch of mathematics that uses variables to solve equations. When solving an algebraic problem, at least one variable will be unknown. Using the numbers and expressions that are given, the unknown variable(s) can be determined.

Early Algebra

The history of algebra began in ancient Egypt and Babylon. The Rhind Papyrus, which dates to 1650 B.C.E., provides insight into the types of problems being solved at that time.

Portion of Rhind Papyrus

The Babylonians are credited with solving the first quadratic equation. Clay tablets that date to between 1800 and 1600 B.C.E. have been found that show evidence of a procedure similar to the quadratic equation. The Babylonians were also the first people to solve indeterminate equations, in which more than one variable is unknown. The Greek mathematician Diophantus continued the tradition of the ancient Egyptians and Babylonians into the common era. Diophantus is considered the “father of algebra,” and he eventually furthered the discipline with his book Arithmetica. In the book he gives many solutions to very difficult indeterminate equations. It is important to note that, when solving equations, Diophantus was satisfied with any positive number whether it was a whole number or not. By the ninth century, an Egyptian mathematician, Abu Kamil, had stated and proved the basic laws and identities of algebra. In addition, he had solved many problems that were very complicated for his time.

Medieval Algebra

During medieval times, Islamic mathematicians made great strides in algebra. They were able to discuss high powers of an unknown variable and could work out basic algebraic polynomials. All of this was done without using modern symbolism. In addition, Islamic mathematicians also demonstrated knowledge of the binomial theorem.

Modern Algebra

An important development in algebra was the introduction of symbols for the unknown in the sixteenth century. As a result of the introduction of symbols, Book III of La géometrie by René Descartes strongly resembles a modern algebra text. Descartes’s most significant contribution to algebra was his development of analytical algebra. Analytical algebra reduces the solution of geometric problems to a series of algebraic ones. In 1799, German mathematician Carl Friedrich Gauss was able to prove Descartes’s theory that every polynomial equation has at least one root in the complex plane. Following Gauss’s discovery, the focus of algebra began to shift from polynomial equations to studying the structure of abstract mathematical systems. The study of the quaternion became extensive during this period. The study of algebra went on to become more interdisciplinary as people realized that the fundamental principles could be applied to many different disciplines. Today, algebra continues to be a branch of mathematics that people apply to a wide range of topics.

René Descartes (1596-1650)

Carl Friedrich Gauss (1777-1855)

Current Status of Algebra

Today, algebra is an important day-to-day tool; it is not something that is only used in a math course. Algebra can be applied to all types of real-world situations. For example, algebra can be used to figure out how many right answers a person must get on a test to achieve a certain grade. If it is known what percent each question is worth, and what grade is desired, then the unknown variable is how many right answers one must get to reach the desired grade. Not only is algebra used by people all the time in routine activities, but many professions also use algebra just as often. When companies figure out budgets, algebra is used. When stores order products, they use algebra. These are just two examples, but there are countless others. Just as algebra has progressed in the past, it will continue to do so in the future. As it is applied to more disciplines, it will continue to evolve to better suit peoples’ needs. Although algebra may be something not everyone enjoys, it is one branch of mathematics that is impossible to ignore.

7

Science Discussion Forum / Neils Henrik Abel Father of Group theory

« on: April 23, 2013, 05:01:24 PM »

Niels Henrik Abel (1802–1829) Norwegian Algebra Born on August 5, 1802.

Niels Abel might have been one of the great mathematicians of the 19th century had he not died of tuberculosis at age 26. He is remembered, and honored, in mathematics for putting an end to the three-century-long search for a solution by radicals of the quintic equation. His theoretical work in the topics of group theory and algebra paved the way for continued significant research in these areas.
          Abel’s short life was dominated by poverty, chiefly due to the severe economic hardships his homeland of Norway endured after the Napoleonic wars, exacerbated by difficult family circumstances. A schoolteacher, thankfully, recognized Abel’s talent for mathematics as a young student and introduced him to the works of Leonhard Euler, Joseph-Louis Lagrange, and other great mathematicians. He also helped raise money to have Abel attend university and continue his studies. Abel entered the University of Christiania in the city of Christiania (present-day Oslo), Norway, in 1821.
          During his final year of study, Abel began working on the solution of quintic equations (fifth-degree polynomial equations) by radicals. Although scholars for a long time knew general formulae for solving for quadratic, cubic, and quartic equations using nothing more than basic arithmetical operations on the coefficients that appear in the equation, no one had yet found a similar formula for solving quintics. In 1822 Abel believed he had produced one. He shared the details of his method with the Danish mathematician Ferdinand Degen in hopes of having the work published by the Royal Society of Copenhagen. Degen had trouble following the logic behind Abel’s approach and asked for a numerical illustration of his method. While trying to produce a numerical example, Abel found an error in his paper that eventually led him to understand the reason why general solutions to fifth- and higher-degree equations are impossible. Abel published this phenomenal discovery in 1825 in a self-published pamphlet “Mémoire sur les équations algébriques où on démontre l’impossibilité de la résolution de l’équation générale ducinquième degré” (Memoir on the impossibility of algebraic solutions to the general equations of the fifth degree), which he later presented as a series of seven papers in the newly established Journal for Pure and Applied Mathematics (commonly known as Crelle’s Journal for its German founder August Leopold Crelle). At first, reaction to this work was slow, but as the reputation of the journal grew, more and more scholars took note of the paper, and news of Abel’s accomplishment began to spread across Europe. A few years later Abel was honored with a professorship at the University of Berlin. Unfortunately, Abel had contracted tuberculosis by this time, and he died on April 6, 1829, a few days before receiving the letter of notification.
          In 1830 the Paris Academy awarded Abel, posthumously, the Grand Prix for his outstanding work. Although Abel did not write in terms of the modern-day concepts of group theory, mathematicians call groups satisfying the commutative property “Abelian groups” in his honor. In 2002, on the bicentenary of his birth, the Norwegian Academy of Science and Letters created a new mathematics prize, the Abel Prize, similar to the Nobel Prize, to be awarded annually.
          Research in the field of commutative algebra continues today using the approach developed by Abel during his short life. His influence on the development of Abstract Algebra is truly significant.

Related post
http://forum.daffodilvarsity.edu.bd/index.php?topic=9210.0

8

Science Discussion Forum / Professor Stephen W. Hawking The Man of Singularity

« on: April 20, 2013, 08:09:58 PM »

Professor Stephen W. Hawking (at home)

Stephen Hawking is a theoretical physicist recognized for his groundbreaking scientific work concerning the relationship between black holes and the beginning of the universe. He studies the basic physical laws governing the universe in an effort to understand how the universe began. Hawking uses complex mathematical models to explore his ideas and develop his scientific theories. He believes anyone can understand the basic ideas of his research and endeavors to share his excitement about science with anyone who is interested, regardless of academic background. He is the author of several books written for nonscientists explaining the concepts of his research on black holes and the universe. His best-selling book A Brief History of Time, first published in 1988, sold over 9 million copies as of 2010 and is published in more than 30 languages. His personal story is also a human interest story. At the age of 21, while attending Cambridge University, Hawking was diagnosed with amyotrophic lateral sclerosis (ALS), also known as “motor neuron disease.” This degenerative disease affects voluntary muscle coordination. ALS does not affect brain function and Hawking is able to continue working in spite of the disabling effects of the disease. His confinement to a motorized wheelchair, his use of a computer-generated voice synthesizer, and his many appearances in the media have made Hawking one of the most recognized scientists around the world.
         

       

            Born January 8, 1942, in Oxford, England, Hawking realized at a young age that he wanted to study science. He attended Oxford University and planned to study mathematics, but chose to study physics because the university did not have a program in mathematics. After completing his studies at Oxford, Hawking earned a Ph.D. in cosmology from Cambridge University. While working as a research associate at Cambridge, Hawking became interested in the study of black holes and the history of the universe. In his dissertation, Hawking theoretically proved that the universe began as a single point of infinite density, known as a “singularity.” As a theoretical physicist, Hawking relies on mathematical models to describe and build scientific theory, which can then be supported or refuted through observation. In particular, Hawking applies known mathematics to study particular objects in the universe, called “black holes,” and to study the universe itself. His work concerning the origins of the universe is built from the mathematical model of the general theory of relativity. This theory, developed by Albert Einstein and published in 1915, describes gravity in terms of its geometric relationship to space and time. Using the mathematics of general relativity, Hawking demonstrated that the equations imply the universe had a beginning as a singularity. This moment at the beginning of the universe is known as the “Big Bang.” However, theoretical physicists such as Hawking have been unable to determine the precise conditions that enable the Big Bang to occur because the mathematics of time and space become undefined at the point of a singularity. Singularities also occur when stars collapse under their own gravitational force and become black holes. Applying the mathematics of general relativity, Hawking also demonstrated that time should come to an end inside a black hole, although the equations are again undefined at the point of a singularity. His research into the structure of black holes further led to his development of a theoretical model concerning radiation emitted from black holes, which is known as “Hawking radiation.” Hawking states he has always been intrigued by life’s big questions and wants to find scientific answers to those questions. His extensive work in the mathematical exploration of black holes and the structure of the universe has led to profound insights in the fields of theoretical physics and cosmology. Among his many academic honors and awards, he held the prestigious Lucasian Chair of Mathematics at Cambridge University from 1979 to 2009, a post once held by Isaac Newton. He continues to work toward his goal of achieving a complete understanding of the universe and why it exists as it does. He and other theoretical physicists are searching for mathematical models to combine quantum mechanics (the study of subatomic particles) and general relativity. He claims that he does not particularly enjoy working with complex mathematical equations because he does not find them intuitive. Rather, he thinks about his ideas geometrically by envisioning mental pictures and visual images. It is these mental pictures and images he uses to try to convey his theoretical ideas. Since writing A Brief History of Time, he has continued his efforts to share his ideas and has written several books for the nonscientist, including Black Holes and Baby Universes and Other Essays and The Universe in a Nutshell.

          He had a guest appearance on an episode of the television series ‘Star Trek: The Next Generation’, playing poker with Data, Albert Einstein, and Isaac Newton in the episode ‘Descent, Part I’. The animated television series ‘The Simpsons’ has occasionally featured him in episodes.

Useful WebPages
Stephen Hawking official page
http://www.hawking.org.uk/
Stephen Hawking – Wikipedia
http://en.wikipedia.org/wiki/Stephen_Hawking
Science Kids – Stephen Hawking
http://www.sciencekids.co.nz/sciencefacts/scientists/stephenhawking.html

Further Reading
Hawking, S. Black Holes and Baby Universes and Other Essays. New York: Bantam, 1993.
Hawking, S. A Brief History of Time. New York: Bantam, 1988.
Hawking, S. God Created the Integers: The Mathematical Breakthroughs That Changed History. New York: Running Press, 2007.
Hawking, S. The Universe in a Nutshell. New York: Bantam, 2001.

9

Science Discussion Forum / Professor Dr. Jamal Nazrul Islam (1939-2013)

« on: April 07, 2013, 06:18:50 PM »

LIFE SKETCH

Prof. Dr. Jamal Nazrul Islam

Early Life and Education
Professor Islam was born on 24 February 1939 in Jhenaidah, British India (what is now Bangladesh). He studied at Chittagong Collegiate School, going on to Lawrence College in Marit, West Pakistan where he passed the Senior Cambridge and Senior Cambridge Higher Senior Cambridge exams. He received a Bachelor of science BSc degree from St. Xavier's College, Calcutta St. Xavier's College at the University of Calcutta. Islam took his mathematical tripos and PhD from Trinity College, Cambridge in 1964, followed by a DSc in 1982.

When J.N. Islam got admitted to Chittagong Collegiate School, he was doubly  promoted for his extraordinary result in the admission test. Islam was the only person who studied Mathematics in the Higher Senior Cambridge level which is now GCE Advanced Level in the whole Pakistan. He used to say that “I never used calculator”.

Career At a Glance
In his lifetime Dr. Islam has worked in various institutions over the world.
 
*A post-doctoral fellow of the Department of Physics and Astronomy, University of Maryland, College Park, Maryland (1963-1965).
*Staff member, Institute of Theoretical Astronomy, University of Cambridge, (1967-1971).
*Visiting appointments at the Institute for Advanced Study, Princeton, New Jersey, (1968,1973, 1984).
*Visiting associate at the California Institute of Technology (famously known as CALTECH), (1971-1972).
*Senior research associate in the Department of Astronomy, University of Washington, (1972-1973).
*Lecturer in Applied Mathematics, King’s College London, (1973-1974).
*Fellow of science research council at University College, Cardiff, UK, (1975-1978).
*Lecturer and Reader at the City University, London, UK, (1978-1984).
*Professor and Head of the Department of Mathematics, University of Chittagong
*Professor Emeritus at the University of Chittagong, Bangladesh.
*Member of University Grants Commission (UGC) of Bangladesh.

Research Interests
His research areas include Applied Mathematics, Theoretical Physics, Mathematical Physics, theory of Gravitation, General Relativity, Mathematical Cosmology and Quantum Field Theory. Dr. Islam authored/coauthored/edited more than 50 scientific articles, books and some popular articles published in various prestigious scientific journals. Besides this the books he has written in Bangla are no less important,however. Particularly noteworthy are ''Black Hole'', published from the Bangla Academy, “The Mother Tongue, Scientific Research and other Articles” from Rahat Publishers and “Art, Literature and Society” from Siraj Publishers. The latter two are compilations.
He has participated and organized in several international conferences, seminars, symposia, workshops and meetings both in Bangladesh and in many countries of the world.

Selected Publications
Books Authored/Coauthored/Edited
1. Islam, J.N. (1983): The Ultimate Fate of the Universe. Cambridge University Press, Cambridge, England. ISBN 978-0-521-11312-0. (Digital print version published in 2009).
2. Bonnor, W.B., Islam, J.N., MacCallum, M.A.H. (eds.)(1983): Classical General Relativity: Proceedings of the Conference on Classical (Non-Quantum) General Relativity, Cambridge University Press, Cambridge, England. ISBN 0-521-26747-1.
3. Islam, J.N. (1985): Rotating Fields in General Relativity, Cambridge University Press, Cambridge, England. ISBN 978-0-521-11311-3]. (Digital print version published in 2009).
4. Islam, J.N. (1992, 2nd edition 2001): An Introduction to Mathematical Cosmology, Cambridge University Press, Cambridge, England. ISBN 0-521-49973-9.

Scholarly Articles
1. Islam, J.N.: Modified Mandelstam Representation for Heavy Particles. J. Math. Phys. 3, 1098-1106 (1962). http://dx.doi.org/10.1063/1.1703852
2. Islam, J.N.: Acnodes and Cusps and the Mandelstam Representation. J. Math. Phys. 4, 872-878 (1963). http://dx.doi.org/10.1063/1.1704012
3. Islam, J.N., Kim, Y.S.: Analytic Property of Three-Body Unitarity Integral. Phys. Rev. 138, B1222–B1229 (1965). http://link.aps.org/doi/10.1103/PhysRev.138.B1222
4. Islam, J.N.: Leading Landau Curves of a Class of Feynman Diagrams. J. Math. Phys. 7, 652-660 (1966). http://dx.doi.org/10.1063/1.1704978
5. Islam, J.N.: Green Function Formulation of the Dirac Field in Curved Space. Proc. R. Soc. Lond. A 294, 437-448 (1966). http://dx.doi.org/10.1098/rspa.1966.0217
6. Islam, J.N.: Field Equations in the Neighbourhood of a Particle in a Conformal Theory of Gravitation. Proc. R. Soc. Lond. A 306, 487-501 (1968). http://dx.doi.org/10.1098/rspa.1968.0163
7. Islam, J.N.: Field Equations in the Neighbourhood of a Particle in a Conformal Theory of Gravitation. II. Proc. R. Soc. Lond. A 313, 71-82 (1969). http://dx.doi.org/10.1098/rspa.1969.0180
8. Islam, J.N.: Some general relativistic inequalities for a star in hydrostatic equilibrium. Mon. Not. R. Astron. Soc. 145, 21-29 (1969)]. http://adsabs.harvard.edu/abs/1969MNRAS.145...21I
9. Islam, J.N.: Some general relativistic inequalities for a star in hydrostatic equilibrium-II. Mon. Not. R. Astron. Soc. 147, 377-386 (1970). http://adsabs.harvard.edu/abs/1970MNRAS.147..377I
10. Islam, J.N.: A class of approximate exterior rotating solutions of Einstein's equations. Math. Proc. Camb. Phil. Soc. 79, 161-166 (1976). http://dx.doi.org/10.1017/S0305004100052178
11. Islam, J.N.: Possible Ultimate Fate of the Universe. Quart. J. R. Astron. Soc. 18, 3-8 (1977). http://adsabs.harvard.edu/abs/1977QJRAS..18....3I
12. Islam, J.N.: On the static field in general relativity. Math. Proc. Camb. Phil. Soc. 81, 485-496 (1977). http://dx.doi.org/10.1017/S0305004100053548
13. Islam, J.N.: On the static field in general relativity: II. Math. Proc. Camb. Phil. Soc. 83, 299-306 (1978). http://dx.doi.org/10.1017/S0305004100054566
14. Islam, J.N.: On the stationary axisymmetric Einstein-Maxwell equations. Gen. Relativ. Gravit. 9, 687-690 (1978). http://link.springer.com/article/10.1007%2FBF00760140
15. Islam, J.N.: A Class of Exact Interior Solutions of the Einstein-Maxwell Equations. Proc. R. Soc. Lond. A 353, 523-531 (1977). http://dx.doi.org/10.1098/rspa.1977.0048
16. Islam, J.N.: The Ultimate Fate of the Universe. Sky & Telescope 57, 13-18 (1979).
17. Islam, J.N.: The long-term future of the universe. Vistas in Astronomy 23, 265–277 (1979). http://dx.doi.org/10.1016/0083-6656(79)90014-X
18. Islam, J.N.: Recently Found Solution of Einstein's Equations. Phys. Rev. Lett. 43, 601-602 (1979) http://dx.doi.org/10.1103/PhysRevLett.43.601
19. Islam, J.N.: The Far Future of the Universe. Endeavour, 8, 32-34 (1984). http://dx.doi.org/10.1016/0160-9327(84)90127-3
20. Islam, J.N.: On Rotating Charged Dust in General Relativity. Proc. R. Soc. Lond. A 362, 329-340 (1978). http://dx.doi.org/10.1098/rspa.1978.0136
21. Islam, J.N.: On Rotating Charged Dust in General Relativity. II. Proc. R. Soc. Lond. A 367, 71-280 (1979). http://dx.doi.org/10.1098/rspa.1979.0087
22. Islam, J.N.: On Rotating Charged Dust in General Relativity. III. Proc. R. Soc. Lond. A 372, 111-115 (1980). http://dx.doi.org/10.1098/rspa.1980.0104
23. Islam, J.N.: On rotating charged dust in general relativity. IV. Proc. R. Soc. Lond. A 385, 189-205 (1983). http://dx.doi.org/10.1098/rspa.1983.0010
24. Islam, J.N.: On Rotating Charged Dust in General Relativity. V. Proc. R. Soc. Lond. A 389, 291-298 (1983). http://dx.doi.org/10.1098/rspa.1983.0110
25. Islam, J.N., Schutz, B.F.: Motion of primordial black holes in the early universe and their likely distribution today. Gen. Relativ. Gravit. 12, 881-893 (1980). http://link.springer.com/article/10.1007%2FBF00757357
26. Islam, J.N.: The cosmological constant and classical tests of general relativity. Phys. Lett. A 97, 239–241 (1983). http://dx.doi.org/10.1016/0375-9601(83)90756-9
27. Boachie, L. A., Islam, J.N.: On a certain solution of the Einstein-Maxwell equations. Phys. Lett. A 93, 321–322 (1983). http://dx.doi.org/10.1016/0375-9601(83)90107-X
28. Islam, J.N.: Closed form for Van Stockum interior solution of Einstein's equations. Phys. Lett. A 94, 421–423 (1983). http://dx.doi.org/10.1016/0375-9601(83)90843-5
29. Islam, J.N., Bergh, N. V. d., Wils, P.: General solutions for axisymmetric differentially rotating charged dust with vanishing Lorentz force. Class. Quant. Grav. 1, 705-714 (1984).
30. Islam, J.N.: On Yang-Mills Theory in the Temporal Gauge. Proc. R. Soc. Lond. A 421, 279-301 (1989). http://dx.doi.org/10.1098/rspa.1989.0012
31. Islam, J.N.: Schrödinger Functional Equation for Yang-Mills Theory. Prog. Theor. Phys. 89, 161-185 (1993) http://ptp.oxfordjournals.org/content/89/1/161.abstract, http://ptp.ipap.jp/link?PTP/89/161/
32. Islam, J.N.: The Schrödinger equation in quantum field theory. Found. Phys. 24, 593-630 (1994). http://link.springer.com/article/10.1007%2FBF02054667
33. Azad, A.K., Islam, J.N.: Cosmological constant in the Bianchi type-I-modified Brans-Dicke cosmology. Pramana J. Phys. 60, 21-27 (2003). http://link.springer.com/article/10.1007%2FBF02705065
34. Islam, M.A., Islam, J.N.: Anharmonic solution of Schrödinger time-independent equation. Pramana J. Phys. 77, 243-261 (2011). http://link.springer.com/article/10.1007/s12043-011-0101-8
35. Firoz, K.A., Moon, Y.-J., Park, S.-H., Kudela, K., Islam, J.N., Dorman, L. I.: On the Possible Mechanisms of Two Ground-level Enhancement Events. ApJ 743, 190 (18pp) (2011). http://dx.doi.org/10.1088/0004-637X/743/2/190
36. Panna, N., Islam, J.N.: Construction of an Exact Solution of Time-Dependent Ginzburg-Landau Equations by Standard Integral for Front Propagation in Superconductors. Science Journal Of Mathematics and Statistics, Article ID sjms-101, 4 Pages, (2012). http://www.sjpub.org/sjms/abstract/sjms-101
37. Panna, N., Islam, J.N.: Construction of an exact solution of time-dependent Ginzburg-Landau equations and determination of the superconducting-normal interface propagation speed in superconductors. Interaction of Lasers with Atoms, Molecules and Clusters University of Hyderabad, Hyderabad 9-12 January 2012. (Forthcoming)

Death
Dr. Islam died on 16th March 2013 in Chittagong, Bangladesh. With his death we have lost a creative, thoughtful and active member of the relativity-and-gravitation community.

Reference
For more details one can see the following reference
1. http://en.wikipedia.org/wiki/Jamal_Nazrul_Islam (References are compiled by the present author)

10

Science Discussion Forum / Higgs boson finally discovered?

« on: July 05, 2012, 05:48:35 PM »

Most people think they know what mass is, but they understand only part of the story. For instance, an elephant is clearly bulkier and weighs more than an ant. Even in the absence of gravity, the elephant would have greater mass—it would be harder to push and set in motion. Obviously the elephant is more massive because it is made of many more atoms than the ant is, but what determines the masses of the individual atoms? What about the elementary particles that make up the atoms—what determines their masses? Indeed, why do they even have mass?

Unlike protons and neutrons, truly elementary particles—such as quarks and electrons—are not made up of smaller pieces. The explanation of how they acquire their rest masses gets to the very heart of the problem of the origin of mass. The account proposed by contemporary theoretical physics is that fundamental particle masses arise from interactions with the Higgs field. A key player in physicists’ tentative theories about mass is a new kind of field that permeates all of reality, called the Higgs field. Elementary particle masses are thought to come about from the interaction with the Higgs field. If the Higgs field exists, theory demands that it have an associated particle, the Higgs boson. In short, this Higgs particle becomes the source of all mass. Using particle accelerators (giant machines such as LHC), scientists are now hunting for the Higgs. Scientists are studying Higgs particle for not only it is important but it's existence is directly related to our existence in this universe.

The Higgs boson is named after Peter Ware Higgs (born 29th may 1929), a British theoretical physicist, who was one of six authors in the 1960s who wrote the ground-breaking papers covering what is now known as the Higgs mechanism and described the related Higgs field and boson.

On 4 July 2012, the two main experiments at the LHC (Large Hadron Collider), ATLAS (A Toroidal LHC ApparatuS) and CMS (Compact Muon Solenoid), both reported independently the confirmed existence of a previously unknown particle with a mass of about 125 GeV/c² (about 133 proton masses, on the order of 10^-25 kg), which is "consistent with the Higgs boson" and widely believed to be the Higgs boson. They acknowledged that further work would be needed to confirm that it is indeed the Higgs boson and not some other previously unknown particle (meaning that it has the theoretically predicted properties of the Higgs boson) and, if so, to determine which version of the Standard Model it best supports.

One possible signature of a Higgs boson from a simulated proton–proton collision. It decays almost immediately into two jets of hadrons and two electrons, visible as lines.

Finally found (Image: Thomas McCauley/Lucas Taylor/CERN/CMS Collaboration)

There's a 5-in-10 million chance that this is a fluke. That was enough for physicists to declare that the Higgs boson – the world's most-wanted particle – has been discovered. Rapturous applause, whistles and cheers filled the auditorium at CERN, near Geneva, Switzerland.

Almost 50 years after its existence was first predicted, the breakthrough means that the standard model of particle physics, which explains all known particles and the forces that act upon them, is now complete.

A Higgs boson with a mass of around 125 to 126 gigaelectronvolts (GeV) was seen separately by the twin CMS and ATLAS detectors at the LHC, each with a confidence level of 5 sigma, or standard deviations, the heads of the experiments announced today at CERN.

Even by particle physicists' strict standards, that's statistically significant enough to count as a particle discovery.
"I think we have it," said Rolf Heuer, director general of CERN, as he concluded a hotly-anticipated seminar.

It's elementary
The Higgs boson gives all elementary particles mass, allowing for the existence of matter. It is the fundamental unit, or quantum, of the Higgs field, an all pervading entity that all particles must pass through. Some, like the photon, slip through unhindered – they are massless. Others, though, must struggle like a fly trapped in treacle. The Higgs and its field are required by the standard model, but had never been conclusively detected before today's report.

The seminar was full of emotion and elation. An emotional Peter Higgs, who postulated the boson that is his namesake in 1964, wiped a tear from his eye as the teams finished their presentations in the Cern auditorium. "I would like to add my congratulations to everyone involved in this achievement," he added later. "It's really an incredible thing that it's happened in my lifetime." 

"This is an important result and should earn Peter Higgs the Nobel Prize," Professor Stephen Hawking told BBC News.

There was also some humour: "It is very nice of the standard model boson to be at that mass," says Gianotti. "Because of that mass we can measure it. Thanks nature."
The physicists were a little reticent to call the discovery a "Higgs boson", preferring to call it the discovery of a "new boson".

That's because they don't yet know its properties – and so can't confirm how similar it is to the Higgs of the standard model. "It's the beginning of a long journey to investigate all the properties of this particle," says Heuer.
One property that needs to be investigated is the particle's spin: the standard model says it should have a value of zero; a more exotic boson would give a value of two. Oliver Buchmueller of CMS says the LHC should be able to determine the new boson's spin by the end of 2012.

We know that the standard model is not complete – it does not contain dark matter or gravity, for a start – so a non-standard model Higgs could be very exciting.
"Everyone is not just excited about the discovery, but about the prospects this discovery offers to the field," says Heuer.

References
Paul Rincon Higgs boson-like particle discovery claimed at LHC, BBC News Science and Environment, 4 July 2012. Link: http://www.bbc.co.uk/news/world-18702455.

Gordon Kane The Mysteries of Mass, Published in The Frontiers of Physics, Scientific American Special Edition, 2005

11

Science Discussion Forum / Probability and the Birthday problem Part-III

« on: June 02, 2012, 07:30:06 PM »

INTERESTING FACTS
We note that the same phenomenon pertains to deaths, for example, the probability that at least two former presidents of the United States share a birth month and day and the probability that at least two former presidents share a death day and month is the same. From the table (See Probability and the Birthday problem Part-I) for n = 44 presidents, the probability is 0.9328. Which shows that, there is about 93.28% chance that the former any two or more US Presidents share their birthday or death day.

James Polk (11th US President)

Warren G Harding (29th US President)

Millard Fillmore (13th US President)

William Howard Taft (27th US President)

John Adams (2nd US President)

Thomas Jefferson (3rd US President)

James Monroe (5th US President)

James Polk and Warren Harding were born on November 2; Milliard Fillmore and William Howard Taft died on March 8. John Adams, Thomas Jefferson, and James Monroe all died on July 4 (Independence Day). Of course, these presidents’ birth and death events do not prove the phenomenon.

References

Woolfson, M. M. Every day Probability and Statistics, Imperial College Press, 2008.
Deep, R. Probability and Statistics, Academic Press, 2006.
Blinder, S. M. Guide to Essential Math A Review for Physics Chemistry and Engineering Students, Academic Press, 2008.

To know more on the birth and death date/month/year of the US Presidents the following website would be helpful to the readers
http://en.wikipedia.org/wiki/List_of_Presidents_of_the_United_States_by_date_of_birth#Born_on_the_same_date
http://en.wikipedia.org/wiki/List_of_Presidents_of_the_United_States_by_date_of_death#Died_on_the_same_day_in_the_same_year

To see the previous parts of this topic please visit
Probability and the Birthday problem Part-I  http://forum.daffodilvarsity.edu.bd/index.php/topic,9321.0.html
Probability and the Birthday problem Part-II http://forum.daffodilvarsity.edu.bd/index.php/topic,9322.0.html

12

Science Discussion Forum / Probability and the Birthday problem Part-II

« on: June 02, 2012, 07:15:03 PM »

In my previous post Probability and the Birthday problem Part-I I asked-

What is the probability (or chance) that I share my birthday with any two or more of my students (in a section)?

The answer is given in this part.

In the following table I present the chances (in percentage) of sharing my birthday.

References

Woolfson, M. M. Every day Probability and Statistics, Imperial College Press, 2008.
Deep, R. Probability and Statistics, Academic Press, 2006.
Blinder, S. M. Guide to Essential Math A Review for Physics Chemistry and Engineering Students, Academic Press, 2008.

To see the previous part of this topic please visit
Probability and the Birthday problem Part-I  http://forum.daffodilvarsity.edu.bd/index.php/topic,9321.0.html

13

Science Discussion Forum / Probability and the Birthday problem Part-I

« on: June 02, 2012, 06:47:49 PM »

THE BIRTHDAY PROBLEM

In this current semester I take the course Mathematics-I in the department of EEE. There are two sections. In each the number of students are 43 and 42.
Now one day I ask myself what is the probability for sharing my birthday to any other students?
Or, I can ask what is the probability that any two or more students share the same birthday?

It is a possibly surprising fact that if you have 23 students in your class or at a party, there is a better than 50% chance that two people share the same birthday (same month and day, not necessarily in the same year).

Let us solve this problem. If we ignore the complication of leap-days there are 365 days on which a birthday can occur. A moment of thought tells us that if there are 366 people in a room then it is certain (probability = 1) that at least one pair of them must share a birthday (think about it!!). If the number of people were reduced to 365 or 364, then while it is not certain that at least one pair share a birthday, our intuition tells us that the probability of this must be very close to 1 — indeed we should be astonished if no two people shared a birthday with such numbers. If the number of people were reduced further then, clearly, the probability would reduce but would still remain high down to 200 people or so.
Let us now start at the other end with two people in a room. The probability that they have the same birthday is low and is easy to calculate. One way of thinking about it is to consider that one of them tells you his birthday. Then you ask the other one for his birthday. There are 365 possible answers he can give but only one of them will match the birthday given by the first person. Hence the probability of the birthdays being the same is 1/365.
The other possible way to look at this problem is to first ask the question “In how many possible ways can two people have birthdays without any restriction?” Since there are 365 possibilities for each of them then the answer to this is 365×365—every date in the calendar for one combined with every date in the calendar for the other. Now, we are going to ask a question that is not, perhaps, the obvious one to ask — “In how many ways can the two people have different birthdays?”.Well, the first one has 365 possibilities but once his birthday is fixed then, for the birthdays to be different, the second has only 364 possibilities so the answer is 365 × 364. Now, we can find the probability that they have different birthdays as

p_diff = (Number of ways of having different birthdays)/(Total number of ways of having birthdays)
= (365 × 364)/(365 × 365)
= 364/365
= 0.997260
≈ 99.72%

Now, either the two individuals have the same birthday or they do not and these are the only possibilities and they are mutually exclusive so that

p_diff + p_same = 1,
or, p_same = 1 − p_diff = 1 – 364/365 = 1/365

the answer obtained previously. The reason we have introduced this apparently convoluted way of getting the answer is because it better lends itself to dealing with the problem of when there are three or more people in the room. Now consider three people in a room. The number of ways they can have birthdays, without any restriction, is

365 × 365 × 365.

The number of ways they can have different birthdays is found by noting that the first person can have a birthday in 365 ways, for the second to be different from the first there are 364 ways and for the third to be different from the other two there are 363 ways. Hence the total number of ways they can have different birthdays is

365×364×363.

Hence the probability that they have different birthdays is
p_diff = number of ways birthdays can be different number of ways they can have birthdays

= (365 × 364 × 363)/(365 × 365 × 365) = 0.9917958.

If the birthdays are not all different then at least two of them must be the same and the probability of this is

p_{not diff} = 1 − p_diff = 0.0082042.

We should now be able to see by extending the principle that the probability that at least two people out of five people in a room would share a birthday is

p_{not diff} = 1 – (365 × 364 × 363 × 362 × 361)/ (365 × 365 × 365 × 365 × 365) = 0.027136.

The advantage of calculating the probability in the way that we do, by finding the probability that all birthdays are different and then subtracting from 1, is now evident. Suppose that, we tried to directly find the probability that two or more people in a set of five shared a birthday. We would have to consider one pair the same and the others all different, three with the same birthday with the other two different, two pairs with the same birthday, and many other combinations. All birthdays being different is a single outcome, the probability of which is comparatively easy to find.
Now, we ask the question that is really the basis of this section—“How many people must there be in a room for there to be a probability greater than 0.5 (or, 50% chance) that at least two of them would share a birthday?” This can be done with a calculator. We work out the following product term-by-term until the answer we get is less than 0.5.

(365/365) × (364/365) × (363/365) × (362/365) × (361/365) × (360/365) × • • •

The value of n for which the answer first falls below 0.5 is when the probability of all birthdays being different is less than 0.5 and hence when there is a probability greater than 0.5 that two or more people have the same birthday. This is the number of people that fulfills our requirement. What is your intuitive assessment of n at this point? In the following table, we give the probability of having at least two people with the same birthday with increasing n. We see that the probability just exceeds 0.5 when 23 individuals are present — a result that most people find surprisingly low.

There is greater than 50% chance that two of these will share a birthday!

References
Woolfson, M. M. Every day Probability and Statistics, Imperial College Press, 2008.
Deep, R. Probability and Statistics, Academic Press, 2006.
Blinder, S. M. Guide to Essential Math A Review for Physics Chemistry and Engineering Students, Academic Press, 2008.

14

Science Discussion Forum / The Giants of Calculus

« on: May 31, 2012, 04:01:14 PM »

WHAT IS CALCULUS?
Calculus was invented as a tool for solving problems. Prior to the development of calculus, there were a variety of different problems that could not be addressed using the mathematics that was available. For example, scientists did not know how to measure the speed of an object when that speed was changing over time. Also, a more effective method was desired for finding the area of a region that did not have straight edges. Geometry, algebra, and trigonometry, which were well understood, did not provide the necessary tools to adequately address these problems.
             At the time in which calculus was developed, automobiles had not been invented. However, automobiles are an example of how calculus may be used to describe motion. When the driver pushes on the accelerator of a car, the speed of that car increases. The rate at which the car is moving, or the velocity, increases with respect to time. When the driver steps on the brakes, the speed of the car decreases. The velocity decreases with respect to time.
             As a driver continues to press on the accelerator of a car, the velocity of that car continues to increase. Acceleration is a concept that is used to describe how velocity changes over time. Velocity and acceleration are measured using a fundamental concept of calculus that is called the derivative.
             Derivatives can be used to describe the motion of many different objects. For example, derivatives have been used to describe the orbits of the planets and the descent of space shuttles. Derivatives are also used in a variety of different fields. Electrical engineers use derivatives to describe the change in current within an electric circuit. Economists use derivatives to describe the profits and losses of a given business.
             Derivatives along with the concept of a tangent line can be used to find the maximum or minimum value for a given situation. For example, a business person may wish to determine how to maximize profit and minimize expense. Astronomers also use derivatives and the concept of a tangent line to find the maximum or minimum distance of Earth from the Sun.
             The derivative is closely related to another important concept in calculus, the integral. The integral, much like the derivative, has many applications. For example, physicists use the integral to describe the compression of a spring. Engineers use the integral to find the center of mass or the point at which an object balances. Mathematicians use the integral to find the areas of surfaces, the lengths of curves, and the volumes of solids.
             The basic concepts that underlie the integral can be described using two other mathematical concepts that are important to the study of calculus are area and limit. Many students know that finding the area of a rectangle requires multiplying the base of the rectangle by the height of the rectangle. Finding the area of a shape that does not have all straight edges is more difficult.

Area under a curve f(x)

Area approximated by 4 rectangles

Area approximated by 10 rectangles

Area approximated by 20 rectangles

            An interesting relationship in calculus is that the derivative and the integral are inverse processes. Much like subtraction reverses addition; differentiation (finding the derivative) reverses integration. The reverse of this statement, integration reverses differentiation, is also true. This relationship between derivatives and integrals is referred to as the "Fundamental Theorem of Calculus". The Fundamental Theorem of Calculus allows integrals to be used in motion problems and derivatives to be used in area problems.

WHO INVENTED CALCULUS?
Pinpointing who invented calculus is a difficult task. The current content that comprises calculus has been the result of the efforts of numerous scientists. These scientists have come from a variety of different scientific backgrounds and represent many nations and both genders. History, however, typically recognizes the contributions of two scientists as having laid the foundations for modern calculus: Gottfried Wilhelm Leibniz (1646-1716) and Sir Isaac Newton (1642-1727).
              Leibniz was born in Leipzig, Germany, and had a Ph.D. in law from the University of Altdorf. He had no formal training in mathematics. Leibniz taught himself mathematics by reading papers and journals. Newton was born in Woolsthorpe, England. He received his master's degree in mathematics from the University of Cambridge.
             The question of who invented calculus was debated throughout Leibniz's and Newton's lives. Most scientists on the continent of Europe credited Leibniz as the inventor of calculus, whereas most scientists in England credited Newton as the inventor of calculus. History suggests that both of these men independently discovered the Fundamental Theorem of Calculus, which describes the relationship between derivatives and integrals.
             The contributions of Leibniz and Newton have often been separated based on their area of concentration. Leibniz was primarily interested in examining methods for finding the area beneath a curve and extending these methods to the examination of volumes. This led him to detailed investigations of the integral concept.
Leibniz is also credited for creating a notation for the integral, ∫. The integral symbol looks like an elongated S. Because finding the area under a curve requires summing rectangles, Leibniz used the integral sign to indicate the summing process. Leibniz is also credited for developing d notation for finding a derivative. Both of these symbols are still used in calculus today.
              Newton was interested in the study of fluxions. Fluxions refers to methods that are used to describe how things change over time. As discussed earlier, the motion of an object often changes over time and can be described using derivatives. Today, the study of fluxions is referred to as the study of calculus. Newton is also credited with finding many different applications of calculus to the physical world.
             It is important to note that the ideas of Leibniz and Newton had built upon the ideas of many other scientists, including Kepler, Galileo, Cavalieri, Fermat, Descartes, Torricelli, Barrow, Gregory, and Huygens. Also, calculus continued to advance due to the efforts of the scientists who followed. These individuals included the Bernoulli brothers, L'Hôpital, Euler, Lagrange, Cauchy, Cantor, and Peano. In fact, the current formulation of the limit concept is credited to Louis Cauchy. Cauchy's definition of the limit concept appeared in a textbook in 1821, almost 100 years after the deaths of Leibniz and Newton.

Pierre de Fermat (1601-1665)

Joseph-Louis Lagrange (1736-1813)

Gottfried Wilhelm Leibniz (1646-1716)

Augustine Louis Cauchy (1789-1857)

Leonhard Euler (1707-1783)

Who Uses Calculus Today?
Calculus is used in a broad range of fields for a variety of purposes. Advancements have been made and continue to be made in the fields of medicine, research, and education that are supported by the methods of calculus. Everyone experiences the benefits of calculus in their daily lives. These benefits include the availability of television, the radio, the telephone, and the World Wide Web.
            Calculus is also used in the design and construction of houses, buildings, bridges, and computers. A background in calculus is required in a number of different careers, including physics, chemistry, engineering, computer science, education, and business. Because calculus is important to so many different fields, it is considered to be an important subject of study for both high school and college students.
            It is unlikely the Leibniz or Newton could have predicted the broad impact that their discovery would eventually have on the world around them.

Based on the article by Barbara M. Moskal published in
Brandenberger, B. M. Jr. (editor in chief) Mathematics Volume 1 Macmillan Science Reference, Thomson/Gale, 2002.

Also See
Hall, A. R. Philosophers at War The Quarrel between Newton and Leibniz, Cambridge University Press, 2002.
Edwards, C. H. Jr. The Historical Development of the Calculus, Springer-Verlag, New York, 1979.
Bardi, J. S. Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time,
Thunder's Mouth Press, New York, 2006

15

Science Discussion Forum / Leonhard Euler The Mathematician & the Master of us All

« on: May 30, 2012, 12:58:47 PM »

Leonhard Euler (1707–1783) is a name well known in many academic fields: philosophy, hydrodynamics (the dynamics of moving fluids), astronomy, optics, and physics. His true fame comes, however, through his prolific work in pure mathematics. He produced more scholarly work in mathematics than have most other mathematicians. His abilities were so great that his contemporaries called him “Analysis Incarnate” (in human form of analysis). A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."
           

Portrait by Johann Georg Brucker (1756)

         

                                                                                                                                  A 1753 portrait by Emanuel Handmann. This portrayal suggests
                                                                                                                         problems of the right eyelid, and possible strabismus. The left eye, which here appears healthy, was later affected by a cataract.

             
             Euler (pronounced “oiler”) was born in Switzerland in 1707. He had originally intended to follow the career path of his father, who was a Calvinist clergyman. Euler studied theology and Hebrew at the University of Basel.
            Johann Bernoulli, however, tutored Euler in mathematics on the side. Euler’s facility in the subject was so great that his father, an amateur mathematician himself, soon favored the decision of his son to pursue mathematics rather than join the clergy.
            Euler first taught at the Academy of Sciences in St. Petersburg, Russia, in 1727. He married and eventually became the father of thirteen children. His children provided him with great joy, and children were often playing
in the room or sitting on his lap while Euler worked. It was in Russia that he lost sight in one eye after working for three days to solve a mathematics problem that Academy members urgently needed but had predicted would take months to solve.
            Euler was a very productive writer, completing five hundred books and papers in his lifetime and having four hundred more published posthumously. The Swiss edition of his complete works is contained in seventy four volumes. He wrote Introductio in Analysin Infinitorum in 1748. This book introduces much of the material that is found in modern algebra and trigonometry textbooks.
            Euler wrote the first treatment of differential calculus in 1755 in Institutiones Calculi Differentialis and in 1770 explored determinate and indeterminate algebra in Anleitung zur Algebra. Three-dimensional surfaces and conic sections were also extensively treated in his writings.
            Euler introduced many of the important mathematical symbols that are now in standard usage, such as Σ (for summation), π (the ratio of the circumference of a circle to its diameter), f (x) (function notation), e (the base of a natural logarithm), and i (square root of negative one). He was the first to develop the calculus of variations. One of the more notable equations that he developed was e^iθ = cos θ + i sin θ, which shows that exponential and trigonometric functions are related. Another important equation he developed establishes a relationship among five of the most significant numbers, e^iπ +1 = 0.
           

Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. In the middle, it shows his polyhedral formula V-E+F = 2

Old Swiss 10 Franc banknote honoring Euler

                                                                                         1957 stamp of the former Soviet Union commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.

            All of Euler’s work was not strictly academic, however. He enjoyed solving practical problems such as the famous “seven bridges of Königsberg” problem that led to Euler circuits and paths. He even performed calculations simply for their own sake.
            Euler later went to Berlin to become the director of Mathematics at the Academy of Science under Frederick the Great and to enjoy a more free political climate. However, Euler was viewed as being rather unsophisticated, and Frederick referred to him as a “mathematical Cyclops.” He returned to Russia when a more liberal leader, Catherine the Great, came to rule.
            By 1766, Euler was completely blind but continued to dictate his work to his secretary and his children. His last words, uttered as he suffered a fatal stroke in 1783, imitated his work in eloquence and simplicity: “I die.”

References
Brandenberger, Barry M. Jr. (editor in chief) Mathematics The Macmillan Science Library Volume 1-4, Macmillan Reference USA, Thomson/Gale, 2002.
Dunham, W. Euler The Master of Us All, Mathematical Association of America, 1999.
Nahin, P. J. Dr. Euler's Fabulous Formula Cures Many Mathematical Ills with a new preface by the author, Princeton University Press, 2006.
Tent, M. B. Leonhard Euler and the Bernoullis Mathematicians from Basel, A K Peters, 2009.
Suisky, D. Euler as Physicist, Springer, 2008.