The present book is the first Issue of a Series explaining various mathematical terms and concepts. It introduces the topics, definitions, main results and theorems avoiding proofs of the results. It may serve as a reference book. The first Issue consists of topics in Mathematical Analysis. The remaining ones covering topics of other mathematical disciplines will follow in the sequel. The contents are divided into Sections - numbered chapter wise. The discussion within the Sections is presented in the form of Definitions, Theorems, Notes and Examples. These subtitles within the Sections are numbered in decimal pattern. For instance, the equation number (c.s.e) refers to the eth equation in the sth section of Chapter c. When the number c coincides with the chapter at hand, it is dropped. Adequate references to the previously quoted results are made in the text avoiding their unnecessary repetition. For brevity, some set-theoretic notations and symbols are frequently used, e.g. the symbol means implies. The logarithm of a number to the exponential base e is denoted by ln. All the Latin mathematical symbols are normally italicized, while their Greek counterparts are in normal fonts."

The book offers a second course on Integral Calculus (of functions of real variables) for Graduate and Engineering students. Convergence (including uniform convergence) of improper integrals and various tests of Abel, Dirichlet and Weierstrass for the same are discussed. Improper integrals of quotient functions of various forms are evaluated. Integrations of continuous functions of 2 variables are discussed in relation with differential and integral properties of parameters in the functions. Eulerian integrals: Beta and Gamma functions, their transformation properties, relations connecting them, reflection and duplication formulae for the Gamma function and Frullani's integral are given. Double and triple integrals giving volume and areas of surfaces are discussed in the last 3 chapters. Numerous examples are solved illustrating the methods of change of order of integration. Dirichlet's integrals of 2nd, 3rd and pth orders are evaluated. Transfor- mations of integrals into 2 and 3-dimensional polar coordinates including Dirichlet's and Liouville's integrals are given. A short bibliography of the subject and an alphabetical index are added at the end.

Astrology has always been a fascinating subject to the mankind and the Cosmos ever remained a great mystery in spite of having been widely explored by the scientists. Taking birth in a traditional Vaishnavite family of India, I am also motivated towards this subject right since my childhood. Coming in contact with Mr. Yogendar Nath Dixit, a teacher of Mathematics at Allahabad (India) with profound knowledge of Indian astrology, in 1963, I learnt many more interesting characteristics of the subject from him during my long association with him. It was October 1995 when I was compelled to give a talk in an "International Conference on Integrated Systems of Medicine" organized at the Banaras Hindu University, Varanasi (India) by its organizers, I selected this topic and prepared a lecture (in English) with due consultations with many scholars of the subject. Varanasi has been well-known since ages for its scholars in astrology too. With my long expertise of mathematical training, I have tried to present here various aspects of astrology (with special reference to Indian astrology) in brief. Many explanations are given here in of mathematical tabular form that makes comprehension of the complex subject easier.

Infinitesimal transformations defining motions, affine motions, projective motions, conformal transformations and curvature collineations in various types of Finslerian spaces are discussed here. The notation and symbolism used is mainly based on 60] and author's works 24] - 42]. The present article offers an exposition of the axiomatic definition of tensors and their further developments from this very standpoint. Various types of tensor sand their examples have been included. A systematic study of manifolds endowed with a metric defined by the positive fourth-root of a 4th degree differential form was considered by P. Finsler in 1918, after whom such manifolds were eventually named. Thereafter, several geometers: E. Cartan, L. Berwald, J.A. Schouten, J. Douglas, W. Barthel, H. Rund, A. Lchnerowicz, A. Kawaguchi, H. Busemann, A. Moor, K. Takano, S.S. Chern, M.S. Knebelman etc. explored this domain extensively. The first treatise on the subject (in English) was published by Rund in 1959. Main aspects of the theory are presented here more elegantly and briefly.

Prayag is the ancient Sanskrit name of Allahabad - a city in North India, situated on the confluence of three rivers: the Ganges, Yamuna and Saraswati - the Goddess of Learning. Sant Tulsidas describes this Prayag as Tirthraj ("King of all Tirthas") in his classical epic Ram Charit Manas. When the sun crosses the line of Capricorn all the Tirthas, saints and other Rishis visit the Tirthraj Prayag. The University there can be regarded as the second oldest University in the entire Indian sub-continent. The first three Universities (of Calcutta, Bombay and Madras) were established at the same time in 1857 by a decree of the British Government. After three decades, the entire northern jurisdiction of the University of Calcutta was carved out in the name of the University of Allahabad on 23rd September, 1887. The author is privileged to have taught there for more than two decades. Learned teachers of mathematics at this University contributed a lot for its academic growth. Their names and their mathematical texts figure in this reportage. Some of them like Dr. Gorakh Prasad, Prof. B.N. Prasad and Prof. R.S. Mishra became legendary figures for their valuable services towards the subject.

The book comprises of 7 expository articles of different nature. The first one "Metrics of curved surfaces and spaces" deals with various possibilities of formulaefor measurement of distance between two infinitesimal points in different geometrical models especially Finslerian structures and their generalizations. "Differen- tial geometry: its past and future" describes briefly the development of the subject. Various techniques used therein are given. References, some standard books on "Finsler geometry and its applications" and recent "International conferences on Finsler geometry" held are also listed. "Physical field theories" characterizes Finslerian physics and Thermodynamic Finsler spaces. The smallest number which, on division by a, a + md, a + 2md leaves the remainders r, r + d, r + 2d respectively; where a, m, r and d are positive integers is discussed in the article on "Some properties of numbers." "Mathematics and computer science education in developing countries" and "Operations research" offer exposition of the subjects as indicated. Major events in the life of Professor Albert Einstein are presented in chronological order in the last article.

Summary 1. Projectively flat Finsler spaces were introduced by Douglas 2], and studied by Berwald 1] for 2-dimension. Okumura 11] gave certain properties of projectively flat non-Riemannian spaces admitting concircular and torse-forming vector fields in association with symmetric and recurrent properties of their curvature tensors. Meher 3] studied projective flatness in a Finsler space when Berwald's curvature tensor is recurrent. He derived relations connecting the curvature tensor and the recurrence vector. Pandey 13] derived a necessary and sufficient condition for the projective flatness of a Finsler space in terms of its isotropic property. Currently, projective flatness of Finsler spaces Fn, n > 2, is studied in association with their sectional curvature, symmetric and recurrent character of their curvature, and normal projective curvature tensor. The notation and symbolism used here are mainly based on the works 10] and 14]. 2. Concircular transformations, introduced by Yano 13] in Riemannian geometry, were extended by Takano 12] to affine geometry with recurrent curvature. Okumura 7] extended these to different types of Riemanna

The book, having eight chapters, offers one semester course (with four credit hours per week) on diverse topics usually included in the mathematics syllabus of the first year Engineering class of most Universities. The first chapter deals with the introductory topics in 2-dimensional coordinate geometry: straight lines and conic sections. A brief discussion of theory of equations is given in the second chapter. Vectors (mostly in 3-dimensional space) are introduced in the third chapter. Vector algebra and products of vectors are included. The next chapter deals with the matrices and determinants. Matrix algebra including multiplication of matrices is given. Further concepts such as rank of a matrix etc. are avoided. A comprehensive course on convergence of infinite series forms the subject mater of Chapter 5. Presuming that the students already had a first course on calculus only main concepts and results of differentiation and integration of functions of single variable have been given in the chapter 6. Power series, Maclaurin's and Taylor's expansions are explained in detail in the same chapter.

The present book offers a first course on Complex Analysis. It deals with the differential and integral properties of functions of a single complex variable. It includes conformal transformations and applications of complex analysis to potential theory and flow problems.The subject matter is presented here in five chapters. It starts with a historical note for the introduction and subsequent development of the subject right from the initial stage to the present modern complex analysis. The first chapter includes the introduction of complex numbers, their geometric representation, De Moivre's theorem, roots and logarithm of a complex number and real and imaginary parts of a complex function.The concepts of differential calculus for functions of a complex variable are outlined in the second chapter. It gives the concept of limit and differentiability of a function. Analytic properties and Cauchy-Riemann equations form the main subject matter of this chapter. The third chapter deals with the integration of functions of a complex variable.

The first chapter deals with the transformations (more precisely translation, reflection, rotation, dilation) in a plane while similar transformations in a 3-dimensional Euclidean space are dealt with in the second chapter. Non-linear, affine and locally affine transformations and rigid motions are included. Isometries are introduced in the third chapter. Chapter 4 offers a detailed account of product transformations which form an algebraic group. Infinitesimal transformations are introduced in Chapter 5. Cases when they may form a motion (preserving distances), translation (displacing each point through a fixed distance), conformal motion (preserving the angle between two directions), homothetic motion (a special case of conformal motion) and geodesic preserving transformations (called projective motions) are dealt therein. The last chapter defines a projective space in general and projective transformations: both singular and non-singular are studied. The chapter concludes with the projective transformations of a projective space onto itself, called collineations. Certain properties of collineations are studied.

The book has three chapters and an Appendix on different methods to calculate approximate areas of plain regions enclosed by curves. Tutorials, Tests and Examination Papers of different Universities where the author taught the course at the faculties of Engineering and Business Studies are provided. A short bibliography of books on the subject and alphabetical index of the topics covered are given in the end.The first chapter accounts the different methods for numerical solutions of ordinary differential equations. Picard's, Taylor's, Euler's, Runge-Kutta, Milne's and Adams-Bashforth's methods are given. The problems of curve fittings and spline fittings are explained in the second chapter. The 'Gaussian method of least squares' for the 'curve of best fit' is included. The concepts of Linear Programming are studied in the third chapter. Solution(s) of linear relations obtained analytically are also discussed by graphical methods. General problems of Linear Programming (LPP), canonical and standard forms of LPP and Simplex methods are discussed. Trapezoidal rule and Simpson's rules for approximate areas of plain regions are included in the Appendix

The book presents an introduction to mathematics starting right from the natural numbers. Covering a vast range of topics in Arithmetic, Algebra, Trigonometry, Coordinate Geometry, Theory of Equations, Sets, Matrices and Determinants, Solid Geometry, Calculus (both Differential and Integral), Analysis the book concludes with a chapter on Complex Numbers. Altogether, it has eleven chapters. Generally, the proofs are avoided but many results especially in Algebra, Trigonometry, Coordinate Geometry and Complex Numbers have been established. Fundamental theorems of Arithmetic, Algebra and Calculus are also given. The book is not intended to be a comprehensive text rather it offers a crash course of basic mathematics to the beginners and as a reference book to the advanced learners. Infinity ( ) is generally mistaken as a number or the logarithm of a negative number is just undefined concept at least to the beginners. These misconceptions are removed here. Almost all the authors claim r2 as the area of a circle of radius r (which, indeed, is zero) for circle having length (i.e. its circumference 2 r) only and is supposed to be of width zero. None takes pains to call r2 as the area of a circular (plane) region of radius r.