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This paper is written to commemorate the centennial anniversary of the Mathematical Association of America. It deals with a short history of different kinds of natural numbers including triangular, square, pentagonal, hexagonal and "k"-gonal numbers, and their simple properties and their geometrical representations. Included are Euclid's and Pythagorean's main contributions to elementary number theory with the main contents of the "Euclid Elements" of the 13-volume masterpiece of mathematical work. This is followed by Euler's new discovery of the additive number theory based on partitions of numbers. Special attention is given to many examples, Euler's theorems on partitions of numbers with geometrical representations of Ferrers' graphs, Young's diagrams, Lagrange's four-square theorem and the celebrated Waring problem. Included are Euler's generating functions for the partitions of numbers, Euler's pentagonal number theorem, Gauss' triangular and square number theorems and the Jacobi triple product identity. Applications of the theory of partitions of numbers to different statistics such as the Bose-Einstein, Fermi-Dirac, Gentile, and Maxwell-Boltzmann statistics are briefly discussed. Special attention is given to pedagogical information through historical approach to number theory so that students and teachers at the school, college and university levels can become familiar with the basic concepts of partitions of numbers, partition functions and their modern applications, and can pursue advanced study and research in analytical and computational number theory.

This paper deals with a brief history of the most remarkable Euler numbers "e,"?"i"?and?"?" in mathematical sciences. Included are many properties of the constants "e,"?"i"?and?"?" and their applications in algebra, geometry, physics, chemistry, ecology, business and industry. Special attention is given to the growth and decay phenomena in many real-world problems including stability and instability of their solutions. Some specific and modern applications of logarithms, complex numbers and complex exponential functions to electrical circuits and mechanical systems are presented with examples. Included are the use of complex numbers and complex functions in the description and analysis of chaos and fractals with the aid of modern computer technology. In addition, the phasor method is described with examples of applications in engineering science. The major focus of this paper is to provide basic information through historical approach to mathematics teaching and learning of the fundamental knowledge and skills required for students and teachers at all levels so that they can understand the concepts of mathematics, and mathematics education in science and technology.

This paper deals with a brief history of probability theory and its applications to Jacob Bernoulli's famous law of large numbers and theory of errors in observations or measurements. Included are the major contributions of Jacob Bernoulli and Laplace. It is written to pay the tricentennial tribute to Jacob Bernoulli, since the year 2013 marks the tricentennial anniversary of Bernoulli's law of large numbers since its posthumous publication in 1713. Special attention is given to Bayes' celebrated theorem and the famous controversy between the Bayesian and frequentism approaches to probability and statistics. This paper is also written to pay a special tribute to Thomas Bayes since the year 2013 marks the 250th anniversary of Bayes' celebrated work in probability and statistics, since its posthumous publication in 1763. This is followed by a short review of the modern axiomatic theory of probability first created by A.N. Kolmogorov in 1933.

This paper deals with a short biography of Paul Dirac, his first celebrated work on quantum mechanics, his first formal systematic use of the Dirac delta function and his famous work on quantum electrodynamics and quantum statistics. Included are his first discovery of the Dirac relativistic wave equation, existence of positron and the intrinsic spin and helicity of electrons. Special attention is given to Dirac's original visionary work on the existence of the magnetic monopole, and on his Large Number Hypothesis that led to the conclusion that physical quantities universally considered as constant of nature are "not really constants," but they vary with cosmological time. Some concluding remarks with personal reminiscence are added in the end of the paper.

This article deals with a brief biographical sketch of Joseph Fourier, his first celebrated work on analytical theory of heat, his first great discovery of Fourier series and Fourier transforms. Included is a historical development of Fourier series and Fourier transforms with their properties, importance and applications. Special emphasis is made to his splendid research contributions to mathematical physics, pure and applied mathematics and his unprecedented public service accomplishments in the history of France. This is followed by historical comments about the significant and major impact of Fourier analysis on mathematical physics, probability and mathematical statistics, mathematical economics and many areas of pure and applied mathematics including geometry, harmonic analysis, signal analysis, wave propagation and wavelet analysis. Special attention is also given to the Fourier integral formula, Brownian motion and stochastic processes and many examples of applications including isoparametric inequality, everywhere continuous but nowhere differentiable functions, Heisenberg uncertainty principle, Dirichlets' theorem on primes in arithmetic progression, the Poisson summation formula and solutions of wave and diffusion equations. It is also shown that Fourier coefficients c[subscript n](t) in the Fourier expansion of a scalar field psi(theta, t) satisfy equations of the simple harmonic motion.

This article deals with a brief history of Fibonacci's life and career. It includes Fibonacci's major mathematical discoveries to establish that he was undoubtedly one of the most brilliant mathematicians of the Medieval Period. Special attention is given to the Fibonacci numbers, the golden number and the Lucas numbers and their fundamental properties with enlightening examples. A large number of exciting applications of these numbers to mathematical, physical, biological and engineering sciences is included. It also contains a wide variety of material accessible to college and even high school mathematics students and teachers at all levels. Included is also mathematical information that puts the professionals and prospective mathematical scientists at the forefront of current research. (Contains 8 figures.)

This article deals with a short history of mathematics and mathematical scientists during the ancient and medieval periods. Included are some major developments of the ancient, Indian, Arabic, Egyptian, Greek and medieval mathematics and their significant impact on the Renaissance mathematics. Special attention is given to many results, theorems, generalizations, and new discoveries of arithmetic, algebra, number theory, geometry and astronomy during the above periods. A number of exciting applications of the above areas is discussed in some detail. It also contains a wide variety of important material accessible to college and even high school students and teachers at all levels. Included also is mathematical information that puts the professionals and prospective mathematical scientists at the forefront of current research. (Contains 3 figures.)

This article is essentially devoted to a brief historical introduction to Euler's formula for polyhedra, topology, theory of graphs and networks with many examples from the real-world. Celebrated Konigsberg seven-bridge problem and some of the basic properties of graphs and networks for some understanding of the macroscopic behaviour of real physical systems are included. We also mention some important and modern applications of graph theory or network problems from transportation to telecommunications. Graphs or networks are effectively used as powerful tools in industrial, electrical and civil engineering, communication networks in the planning of business and industry. Graph theory and combinatorics can be used to understand the changes that occur in many large and complex scientific, technical and medical systems. With the advent of fast large computers and the ubiquitous Internet consisting of a very large network of computers, large-scale complex optimization problems can be modelled in terms of graphs or networks and then solved by algorithms available in graph theory. Many large and more complex combinatorial problems dealing with the possible arrangements of situations of various kinds, and computing the number and properties of such arrangements can be formulated in terms of networks. The Knight's tour problem, Hamilton's tour problem, problem of magic squares, the Euler Graeco-Latin squares problem and their modern developments in the twentieth century are also included. (Contains 15 figures.)

This tricentennial tribute commemorates Euler's major contributions to mathematical and physical sciences. A brief biographical sketch is presented with his major contributions to certain selected areas of number theory, differential and integral calculus, differential equations, solid and fluid mechanics, topology and graph theory, infinite products, infinite series, elliptic integrals, calculus of variations and the Fourier coefficients. This short article is written to provide a definite impression of Euler's personal and professional life as well as of the range, power and depth of his unique contributions to mathematical and physical sciences. By any standards, his legacy of complete works entitled "Opera Omnia" of nearly 80 volumes will remain forever as a remarkable example of his scholarly work in the history of mathematics and sciences. (Contains 4 figures.)

This paper deals with a brief introduction to major remarkable discoveries of the "soliton" and the "inverse scattering transform" in the 1960s. The discovery of the soliton (or the solitary waves) began with the famous physical experiments of the Scottish Engineer and Naval Architect John Scott Russell in the Glasgow-Edinburgh Canal in 1834. The main objective of this paper is to introduce Scott Russell's vision to the reader. This was followed by the famous mathematical derivation of the Korteweg-de Vries (KdV) equation in 1895 for the propagation of the great solitary wave of Scott Russell in one direction on the free surface of water in a shallow canal. In 1965, Norman Zabusky and Martin Kruskal discovered the existence of solitons and the interaction of solitons from their computer experiments. In 1967, C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura developed an ingenious method for finding the exact solution of the KdV equation. This paper is also concerned with several fundamental nonlinear partial differential equations including the Korteweg and de Vries (KdV) equations, the nonlinear Schrodinger (NLS) equation, the Sine-Gordon (SG) equation and the Toda lattice equation. Thus, the soliton and the inverse scattering transform are now regarded as the major remarkable discoveries in mathematical sciences of the second half of the twentieth century. A brief comment is made on the discovery of compacton in the 1990s and physical interaction of two or more compactons. A compacton is a soliton with a finite compact support with "no" oscillatory trail. Interestingly, compactons can describe the intrinsic localized modes in anharmonic crystals. Concluding remarks are made on the recent developments of nonlinear wave phenomena and their applications to a wide variety of problems. (Contains 7 figures.)

This paper deals with a brief historical introduction to fractals, fractal dimension and fractal geometry. Many fractals including the Cantor fractal, the Koch fractal, the Minkowski fractal, the Mandelbrot and Given fractal are described to illustrate self-similar geometrical figures. This is followed by the discovery of dynamical systems and iterative mappings with the remarkable examples of the Julia fractals and the Mandelbrot fractals. Fractals, chaos, bifurcations and Hausdorff dimension seem to be essential elements of fractal geometry. Some applications of fractals to fracture mechanics and turbulence are briefly discussed. Included are Mumford's new computer aided explorations to visualize Kleinian groups of maps of the complex plane of fractional linear transformations. This leads to a family of extremely beautiful symmetrical fractal structures including Indra's net, Indra's necklace and Indra's pearls which describe Klein's mathematical vision. It is shown that modern computer technology and computer graphical analysis have the remarkable ability and power to generate totally new and unexpected complex fractal structures that are surprisingly very beautiful.

This paper deals with a brief historical introduction to fractional calculus with basic ideas, definitions and results of fractional order integration and differentiation of arbitrary order.