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The objective of this paper is to construct fractional wavelet frames in L2(ℝ). A necessary condition and four sufficient conditions for fractional wavelet frames are given by virtue of fractional Fourier transform. The proposed inequalities generalize all the classical wavelet inequalities when θ = π/2. An example is presented at the end. [ABSTRACT FROM AUTHOR]

In this study, a hybrid method based on coupling discrete wavelet transforms (DWTs) and artificial neural network (ANN) for yield spread forecasting is proposed. The discrete wavelet transform (DWT) using five different wavelet families is applied to decompose the five different yield spreads constructed at shorter end, longer end, and policy relevant area of the yield curve to eliminate noise from them. The wavelet coefficients are then used as inputs into Levenberg-Marquardt (LM) ANN models to forecast the predictive power of each of these spreads for output growth. We find that the yield spreads constructed at the shorter end and policy relevant areas of the yield curve have a better predictive power to forecast the output growth, whereas the yield spreads, which are constructed at the longer end of the yield curve do not seem to have predictive information for output growth. These results provide the robustness to the earlier results. [ABSTRACT FROM AUTHOR]

In this paper, for any two positive integers p and q, we wish to introduce an alternative definition of relative (p, q)-th order of a meromorphic function with respect to another entire function which improves the earlier definition of relative (p, q)-th order of meromorphic function introduced by Banerjee and Jana [5]. Also in this paper we discuss some growth rates of composite entire and meromorphic functions on the basis of the improved definition of relative (p, q)-th order of meromorphic function. [ABSTRACT FROM AUTHOR]

‘Number rules the universe.’The Pythagoras‘If you wish to forsee the future of mathematics our course is to study the history and present conditions of the science.’Henri Poincaré‘The primary source (Urqell) of all mathematics are integers.’Hermann Minkowski 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 andk-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 theEuclid Elementsof 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. [ABSTRACT FROM PUBLISHER]

This paper deals with a brief history of the most remarkable Euler numberse, i and γ in mathematical sciences. Included are many properties of the constantse, i and γ 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. [ABSTRACT FROM PUBLISHER]

It is remarkable that a science (Probability) which began with consideration of games of chance, should have become the most important object of human knowledge.Probability has reference partly to our ignorance, partly to our knowledge. … The Theory of chances consists in reducing all events of the same kind to a certain number of cases equally possible, that is, such that we are equally undecided as to their existence; and determining the number of these cases which are favourable to the event sought. The ratio of that number to the number of all the possible cases is the measure of the probability . …P.S. LaplaceThe true logic of this world is to be found in theory of probability.James Clark Maxwell 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. [ABSTRACT FROM PUBLISHER]

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 arenot really constants, but they vary with cosmological time. Some concluding remarks with personal reminiscence are added in the end of the paper. [ABSTRACT FROM PUBLISHER]

In this paper, some algorithms for constructing tight wavelet frames on local fields using the unitary extension principles are suggested. We present a sufficient condition for finite number of functions to form a tight wavelet frame and establish general principles for constructing tight wavelet frames on local fields [ABSTRACT FROM AUTHOR]

The profound study of nature is the most fertile source of mathematical discoveries. Not only does this study, by offering a definite goal to research, have the advantage of excluding vague questions and futile calculations, but it is also a sure means of moulding analysis itself, and discerning those elements in it which it is still essential to know and which science ought to conserve. These fundamental elements are those which recur in all natural phenomena. Joseph Fourier pure mathematics enables us to discover the concepts and laws connecting them, which gives us the key to the understanding of the phenomena of nature. Albert Einstein 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 cn(t) in the Fourier expansion of a scalar field ψ(θ, t) satisfy equations of the simple harmonic motion. [ABSTRACT FROM AUTHOR]

Abstract: Using classical analytic techniques, the Wilker–Anglesio inequality and parameterized Wilker inequality for hyperbolic functions are proved. The main result is then applied to deriving a hyperbolic analogue of the Sándor–Bencze inequality. [Copyright &y& Elsevier]

Abstract: Let be a given real number, and , be two parameters, and let We determine the values which provide the best approximation: Furthermore, we establish a sharp Jordan’s inequality, and then apply it to improve the Yang Le inequality. [Copyright &y& Elsevier]

This paper deals with recent developments of Hilbert-type discrete and integral inequalities by introducing kernels, weight functions, and multiparameters. Included are numerous generalizations, extensions, and refinements of Hilbert-type inequalities involving many special functions such as beta, gamma, logarithm, trigonometric, hyper-bolic, Bernoulli's functions and Bernoulli's numbers, Euler's constant, zeta function, and hypergeometric functions with many applications. Special attention is given to many equivalent inequalities and to conditions under which the constant factors involved in inequalities are the best possible. Many particular cases of Hilbert-type inequalities are presented with numerous applications. A large number of major books and recent research papers published during 2009-2012 are included to stimulate new interest in future study and research. [ABSTRACT FROM AUTHOR]

This paper deals with a construction of wavelet frame packets on the positive half-line ℝ+ using the splitting trick for frames. It is shown that, as long as finitely many splitting steps are applied, the resulting sequence of functions is a frame of L 2(ℝ+). If the matrix M(ξ) associated with the splitting is unitary, then the splitting can be applied infinitely many times to prove the existence of frame with the frame bounds as shown in Theorem 3.3. [ABSTRACT FROM AUTHOR]

This paper deals with a construction of dyadic wavelet frames on a positive half-line + using the Walsh-Fourier transform. We prove necessary and sufficient conditions for the system [image omitted] to be a wavelet frame in L2(+). These conditions are found to be better than those of Daubechies [Ten Lecture on Wavelets, NSF-CBMS Regional Conferences in Applied Mathematics, Vol. 61, SIAM, Philadelphia, PA, 1992], Chui and Shi [Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal. 24 (1993), pp. 263-277], Casazza and Christenson [Weyl-Heisenberg frames for subspaces of L2(), Proc. Amer. Math. Soc. 129 (2001), pp. 145-154] and Christenson [Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bull. Amer. Math. Soc. 38 (2001), pp. 273-291; An Introduction to Frames and Riesz Bases, Birkhauser, Boston, 2003]. [ABSTRACT FROM AUTHOR]

'If you wish to foresee the future of mathematics our proper course is to study the history and present condition of the science.' Henri Poincare 'It is India that gave us the ingenious method of expressing all numbers by ten symbols, each symbol receiving a value of position, as well as an absolute value. We shall appreciate the grandeur of the achievement when we remember that it escaped the genius of Archimedes and Apollonius.' P.S. Laplace 'The Greeks were the first mathematicians who are still 'real' to us today. Oriental mathematics may be an interesting curiosity, but Greek mathematics is the real thing. The Greek first spoke of a language which modern mathematicians can understand.' G.H. Hardy 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with a Hilbert-type linear series operator and its norm. Several generalizations of the Hilbert-type inequalities whose kernels are symmetric and homogeneous of the first order are presented. Included are applications of some new extended Hilbert-type inequalities with the best possible constant factors and the equivalent forms are established. Also, the reverse forms and some particular forms are obtained as special cases of the results of this paper. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with a new generalization of the Garfunkel–Bankoff inequality by introducing a parameter, which relaxes the conditions of the Garfunkel–Bankoff inequality. As applications, an improved version of the Finsler–Hadwiger inequality is obtained. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with a generalization of L’Hôspital-type rules for monotonicity. The result is then used to establish a class of two-sided bounding inequalities for higher-order differentiable functions. As consequences, some refined upper and lower bounds for exponential function, logarithmic function, trigonometric functions and hyperbolic functions are derived. [Copyright &y& Elsevier]

Abstract: In this work, we first prove a generalized version of a parameterized Jordan-type inequality. We then use it to prove the generalized versions of Janous’s inequality and Tsintsifas’s inequality which reduce to two inequalities conjectured by Janous and Tsintsifas as special cases. [Copyright &y& Elsevier]

Abstract: In this work, we extend Jordan’s inequality to obtain a new type of inequality involving functions and their higher-order derivatives. The result is then used to obtain some higher accurate inequalities of Jordan type. [Copyright &y& Elsevier]

Abstract: A new generalized version of Aczél’s inequality is proved. This is a unified generalization of some known results. Moreover, the result is applied to the improvement of the well-known Bellman’s inequality. [Copyright &y& Elsevier]

Abstract: In this work, a weighted generalization of Rado’s inequality and Popoviciu’s inequality is given, from which some valuable inequalities of Rado–Popoviciu type are derived. Moreover, the result is used to obtain refinements of weighted mean value inequalities. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the third order two-point generalized right focal boundary value problem A few new results are given for the existence of at least one, two, three and infinitely many monotone positive solutions of the above boundary value problem by using the Krasnosel’skii fixed-point theorem in cones, and the Leggett–Williams fixed-point theorem. [Copyright &y& Elsevier]

'...the great primary waves of translation cross each other without change of any kind in the same manner as the small oscillations produced on the surface of a pool by a falling stone.' SCOTT RUSSELL '...as Sir Cyril Hinshelwood has observed... fluid dynamicists were divided into hydraulic engineers who observed things that could not be explained and mathematicians who explained things that could not be observed.' JAMES LIGHTHILL 'If you wish to foresee the future of mathematics our proper course is to study the history and present condition of the science.' HENRI POINCARÉ 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 Schrödinger (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. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode analysis. An extension of the higher modes of this work will be made in a subsequent paper. [Copyright &y& Elsevier]

Using the theory of majorization, new inequalities for convex sequences are proved. A necessary and sufficient condition for a convex sequence is established. This reveals an important relationship between the convex sequence and the majorized inequality. Finally, some applications of our new inequalities and some improvements of certain known results are presented. [Copyright &y& Elsevier]

The modern approach for wavelets imposes a Bayesian prior model on the wavelet coefficients to capture the sparseness of the wavelet expansion. The idea is to build flexible probability models for the marginal posterior densities of the wavelet coefficients. In this note, we derive exact expressions for two popular models for the marginal posterior density. We also illustrate the superior performance of these models over the standard models for wavelet coefficients. [ABSTRACT FROM AUTHOR]

This article deals with a study of some properties of the Mittag-Leffler functions including a definite integral and a recurrence relation. Several examples are given to illustrate the validity of theorems which have been proved in this work. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with a new generalized and sharp version of Jordan’s inequality which is a unified generalization of some known results. An application of our results to the improvement of the Yang Le inequality is provided in this paper. [Copyright &y& Elsevier]

This paper deals with the generalization and sharp versions of the Wolstenholme cyclic inequality and their applications. The inequalities of this paper improve and unify corresponding known results. Several interesting inequalities including the celebrated Ozeki inequality are obtained. Extensions of the Wolstenholme inequality for a complex polygon and the Wolstenholme inequality for a convex quadrilateral are derived. As example of applications, the well-known Erdös–Mordell inequality is improved in this paper. In addition, several extensions, unifications and refinements of Gueron–Shafrir’s inequalities and Mitrinović–Pecaric’s inequality are established. [Copyright &y& Elsevier]

Abstract: A new generalized and sharp version of Jordan’s inequality is proved and it is applied in the improvement of the Yang Le inequality. Moreover, a mistake in the proof of sharpening Jordan’s inequality due to Zhu [S.H. Wu, On generalizations and refinements of Jordan type inequality, Octogon Math. Mag. 12(1) (2004) 267–272] is corrected. [Copyright &y& Elsevier]

"As time goes on, it becomes increasely evident that rules that the mathematician finds interesting are the same as those that nature has chosen." PAUL DIRAC "... Some of the doors which these discoveries flung open led directly to new worlds whose full exploration has only become possible with the advent of high speed computing in the last twenty years." DAVID MUMFORD, CAROLINE SERIES and DAVID WRIGHT "Almost everyone using computers has experienced instances where computational results have sparked new insights." NORMAN J. ZABUSKY 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' 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' net, Indra' necklace and Indra' pearls which describe Klein' 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 surprisely very beautiful. [ABSTRACT FROM AUTHOR]

Abstract: Values of are determined for which there exist positive solutions of the fourth-order singular boundary value problems where , is parameter. For , we give criteria for the existence of eigenfunctions. [Copyright &y& Elsevier]

A study is made on the Hankel type integral transformations of generalized functions. An inversion theorem and a uniqueness theorem are proved. These transformations are then used to solve a class of partial differential equations involving the Bessel type differential operators. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, the existence of positive radially symmetric entire solutions for the equation Δmu=f(|x|,|∇u|)u−β (m⩾2) on Rn (n⩾3) is proved. Some properties of the solutions are obtained. The results of this paper may be regarded of results in [Math. Z. 67 (1957) 32; Arch. Math. 9 (1958) 308; Proc. Royal Soc. Edinburgh 82A (1979) 189; Hiroshima Math. J. (1989) 13]. [Copyright &y& Elsevier]

In the paper, the existence of positive radially symmetric entire solutions for a class of singular nonlinear polyharmonic equations on R2 is proved. Some properties are obtained. The results of this paper are generalizations of those in Refs. [Math. Z. 67 (1957) 32; Arch. Math. 9 (1958) 308; Proc. R. Soc. Edinburg 82A (1979) 189; Hiroshima Math. J. 17 (1989) 13]. [Copyright &y& Elsevier]

Kernel theorems for spaces of Cauchy ultradistribution, supported by an n-dimensional tube and cone of the product type, are investigated. [Copyright &y& Elsevier]

In this paper, by introducing three parameters A, B, and λ, and estimating the weight coefficient, we give a generalization of the extended Hardy–Hilbert''s inequality with a best possible constant factor, involving the β function. We also consider its equivalent inequality and the associated double series form. [Copyright &y& Elsevier]

An extension of Carlson''s inequality is made by using the Euler–Maclaurin summation formula. The integral analogues of this inequality are also presented. [Copyright &y& Elsevier]

A class of definite integrals ∫[sub a][sup b]{(b - x)(x - a)}[sup n-1]x[sup 1/2-α]dx is evaluated by means of explicit and recurrence formulas, where a, b > 0, n ∈ R and α ∈ R. [ABSTRACT FROM AUTHOR]

Presents a method for teaching inequality in Hilbert algebras. Introduction of the weight coefficient inequality; Improvement of the Hilbert equivalent form; Role of Hilbert inequality and equivalent form in the analysis.

Presents a historical development of wavelet transforms, Gabor transforms and their myriad of mathematical, physical, medical, biological and engineering concepts, while discussing the existence of Fourier transforms and the Hilbert space formalism in wavelet transform analysis. Factors associated with the use of the Fourier transform of a signal; Details on Gabor transforms; Historical concept of wavelet transforms.

A review of the article "A Brief Historical Development of Classical Mathematics Before the Renaissance," by Lokenath Debnath, which appeared in the journal "International Journal of Mathematical Education in Science and Technology," on July 2011, is presented.

The main objective of this paper is to study the continuous Bessel wavelet transformation and its inversion formula, and the Parseval relation using the theory of the Hankel convolution. A relation between the Bessel wavelet transformation and the Hankel–Hausdorff operator is established. [ABSTRACT FROM AUTHOR]