Indian Mathematics Research Articles

Glimpses of contributions of some top Indian mathematicians: A review article

Indian mathematics has its deep roots in the Vedas, different from what is known as Vedic Mathematics. Vedic age gave rise to a new era of progress in the field of Science, Technology and Development. The Hindu Scripture Vedas is synonymous with all kinds of original source of knowledge and intellectual wisdom in the universe leading to modern knowlege in modern mathematics. Indian mathematicians made tremendous contributions to the entire world of Mathematics and Science. Decimal number system as well as the invention of zero (0) are among the greatest contributions of Indian mathematicians. The theory of trigonometry, Mathematical Modelling, Algebra, algorithm, modern arithmetic, sine and cosine functions leading to modern trigonometry, Diophantine equation, square root, cube root, negative numbers are also developed by Indian mathematicians. In this review article, the work of some of the renowned Indian mathematicians from Indus Valley civilization and the Vedas to modern times are covered in short with the hope that it may reveal hidden fundamental mathematical ideas as basic ideal tools which may usefully motivate for further research work in every domain of mathematical sciences, natural and applied sciences, engineering and social sciences. Moreover, there are many more remarkable Indian mathematicians who contributed to the origin of mathematical sciences. They have made several general contributions to mathematics that have significantly influenced scientists and mathematicians in the modern times.

HISTORICAL AND EDUCATIONAL ASPECTS OF DISCOVERIES AND INVENTIONS THAT REVOLUTIONIZED MANKIND

This article presents discoveries and inventions from different periods of mankind, which played an important role in social and technological evolution. Thus, from the period of the Ancient World (prehistory - 400 AD), we present: the appearance of stone tools (which occurred in East Africa and belongs to the first hominids); pottery (appeared in 10500 BC); the development of metallurgy (began in the Middle East, around 6500 BC); the invention of the ox-drawn plow (which occurred around 4000 BC); the construction of the first pyramid in Egypt (2600 BC); the development of iron processing (as part of the development of metallurgy, it occurred around 1400 BC); modernization of papermaking technology (attributed to Tsai Lun, China, around 105 AD); Another historical period that we analyzed in terms of discoveries and innovations that revolutionized humanity was the Middle Ages (400 - 1500). Thus, from this period we presented the following discoveries and inventions: the discovery of the number zero (occurred in 520 and belongs to Indian mathematicians); woodcut printing (appeared in sixth century China); the first printed newspaper (year 700); the development of algebra (it belongs to the Greek mathematician Diophantos, 3rd century AD); gunpowder (it was discovered around 850); the establishment of the University of Bologna (made in 1088); The last period approached was the current one. From this period we presented the following discoveries: magnetism - a new form of electricity; devices controlled only by hand gestures.

Rich History of Indian Mathematics

Rich History of Indian Mathematics

Ramanujan's Beautiful Integrals

Throughout his entire mathematical life, Ramanujan loved to evaluate definite integrals. One can find them in his problems submitted to the Journal of the Indian Mathematical Society, notebooks, Quarterly Reports to the University of Madras, letters to Hardy, published papers and the Lost Notebook. His evaluations are often surprising, beautiful, elegant, and useful in other mathematical contexts. He also discovered general methods for evaluating and approximating integrals. A survey of Ramanujan's contributions to the evaluation of integrals is given, with examples provided from each of the above-mentioned sources.

On the Indian tradition of mathematical names: A scientific analysis

The current paper explores the potential interlink between names of individuals in a society and its collective social consciousness, particularly with reference to the pervasive occurrence of the ‘mathematical names’ in the current Hindu society spanning the Indian subcontinent and beyond. Initially, an attempt is made to put things into mathematical perspective by drawing a quick sketch of some of the stellar achievements of the Indian mathematicians. Under the six broad categories of geometry, trigonometry, numeration, arithmetic, algebra, and mathematics in the Vedic tradition, a concise simple description of these subdivisions is presented, underlining selected mathematical concepts and terms, sometimes by producing the textual references. We hypothesize that such terms permeate as names in the current Hindu society, reflecting the impressions of the tremendously rich mathematical heritage left by the stalwart Hindu mathematicians. Accordingly, an attempt is made to juxtapose these terms with the names current in the Indian Hindu society. By employing an extensive dataset of university student names in India and the directories of Facebook and LinkedIn, we produce both qualitative and quantitative evidence of the presence of such names in the Indian subcontinent. Our hypothesis has also been examined by taking surveys of people bearing these mathematical names, as well as by documenting the ‘conscious procedures’ that go behind the naming of a Hindu Indian child. In trying to investigate if such a phenomenon is unique to the Indian tradition, a stark contrast with the ‘names in mathematics’ as prevalent in the post-renaissance Europe is presented, as cultural roots of mathematics are explored. Evidently, the large magnitude and the span of such names substantiates the presence of these names as the extant remains of the colossal impact of multifarious mathematical traditions existing in India. Interestingly, the present research also brings to the fore, certain unseen facets of the Indian Hindu society as regards the education of mathematics to women – through an indirect exploration of their names. We also show that the pervasive occurrence of these names is not merely the result of semantic chance events, but must denote the richness of the Indian mathematical legacy. By presenting some cross-cultural comparisons, we bring about the specific uniqueness of Indian mathematical and scientific traditions that led to the pervasiveness of ‘mathematical names’ in India vis-a-vis other cultures. Finally, an attempt is made to clarify some subtle points on the associations between mathematics and religion in India and other cultures of the world. It is sincerely hoped that the present study may shed light on the cultural roots of mathematics and may furnish a new dimension in the study of mathematics, culture, and civilizations across the world.

JYOTPATTI

<p>Bhaskaracharya ,(Bhaskar II, 1114- 1185 AD), was one of the great mathematicians in India. His text “<em>Sindhant Shiromani</em>” (SS) was treated as the base of the further research oriented results by most of all the mathematicians after him.<br> SS contains of two parts:, <em>Goladhyaya and Grahaganit</em>. Jyotpatti is the last chapter in <em>Goladhyaya. Jyotpatti</em> consists of 25 <em>Shloka</em> (stanzas), all in Sanskrit language. It is a general impression that SS contains <em>Lilavat</em>i and <em>Beejganit</em> also, but that is not so.<br> <em>Jyotpatti</em> deals with trigonometry. This was a milestone in developing geometry in India. <em>Jya</em> means sine and <em>Utapatti</em> means creation. Hence the name <em>Jyotpatti ( jya + upapatti</em>). <em>Jya and Kojya( or kotijya)</em> stand for the Rsine and Rcosine ratios respectively. The trigonometry developed by Bhaskara II is based on a circle of radius R, and not on a right-angled triangle as taught in the schools.<br> After defining <em>Jya, Kotijya and Utkrama</em> (<em>verse jya) </em>etc, Bhaskara obtains these ratios for the standards angles of 30,45, 60, 36 and 28,(all in degrees) by inscribing a regular polygon in a circle of radius R. Bhaskara called these angles as <em>Panchajyaka. </em>Not only this, Bhaskara developed these results for addition and subtraction of two angles. This result was further developed for the similar results, for the multiple angles. Bhaskara compares <em>jya and kotijya</em> with the longitude –latitude of earth and those with lateral threads of a cloth.<br> Contents in Jyotpatti (Only a few mentioned here)<br> (1) R jya 45 =R, and other similar R jya values. (All in degrees)<br> (2) R jya 36 = 0.5878 approx.<br> (3) Sn = side of a regular polygon of n sides = D sin (π/n), D is the diameter of circle in which polygon is inscribed.<br> (4) Derivation of formulae for sin (θ + ϕ) and cosine (θ + ϕ) called as <em>samas bhavana</em> and <em>antar bhavana</em>.<br> (5) Concept of derivatives, that is, δ(sin θ) = (cos θ) δθ etc. Which is Rolle’s Theorem.<br> Indian mathematicians developed trigonometry in different way than that of western mathematicians. Though <em>Jyotpatti</em> is a small text, it is a landmark in development of ancient and medieval trigonometry.</p>

Mathematics and Map Drawing in the Eighteenth Century

We consider the mathematical theory of geographical maps, with an emphasis on the eighteenth century works of Euler, Lagrange and Delisle. This period is characterized by the frequent use of maps that are no more obtained by the stereographic projection or its variations, but by much more general maps from the sphere to the plane. More especially, the characteristics of the desired geographical maps were formulated in terms of an appropriate choice of the images of the parallels and meridians, and the mathematical properties required by the map concern the distortion of the maps restricted to these lines. The paper also contains some notes on the general use of mathematical methods in cartography in Greek Antiquity and in the modern period, and on the mutual influence of the two fields, mathematics and geography. The final version of this paper will appear in Ganita Bharati (Indian mathematics).

Remembering Ramanujan

Mathematicians like Guido Grandi, Ernesto Cesaro and others found novel way of assigning finite sum to divergent series. This created a new scope of understanding leading to analytic continuation of real valued functions. One among such methods was called ``Ramanujan Summation'' proposed by Indian Mathematician Srinivasa Ramanujan. In this paper, I try to highlight how Ramanujan could have possibly arrived at those values by looking through his notebook jottings and extending further to provide Geometrical meaning behind those values obtained by him. Finally, I provide a novel way to arrive at the general formula obtained by Ramanujan regarding his summation of zeta function.

Secure and Efficient Application of Manet Using RSA Using Vedic Method Combine With Visual Cryptography and Identity Based Cryptography Technique

Security and rapid transmission of data could be very important a part of military devices. Propose technique present the stable, green and rapid way to send pictures by using the Identity Based Cryptography and Visual Cryptography. In this application Identity Based Cryptography is used with Visual cryptography. In Identity primarily based cryptography the RSA Cryptosystem is used to generate public and private key by using Ancient Indian Mathematics for fast mathematical calculation. RSA is the most secure and general algorithm. Implementation of it by using the Vedic multiplication could be very green in time period of area, speed compared to its cutting-edge mathematics implementations. The regenerations of public and private key are adopted to make the system greater steady from diverse attacks.

Conceptualisation of Ancient Indian Concept/Representation of Numbers through Word Numerals and Alphabetic Numerals Existent till Today

This article is an effort to reveal that – It was ancient India which guided the world how to represent systematically the numbers in word forms as well as in alphabets maintaining ascendancy of the alphabets which is still in use by the beginners in the study of Arithmetic. It was the Ancient Indian Mathematicians who probably used first and developed these numeral systems. They used word numerals and alphabetic numerals system for their astronomical studies. The aim of this article is to present an account on the evolution and development of word numeral systems and alphabetic numeral systems from ancient to the modern period.

Implementation of 64 Bit Complex Floating-Point Multiplier on FPGA using Vedic Mathematics Sutra- Urdhva Tiryagbhyam

Multipliers play crucial role in present days in the area of digital signal processing and in communication systems applications. The entire system performance depends on speed area and power of the multipliers. In our paper, we developed a 64x64 bit complex floating-point multiplier with 64bit IEEE 754 format multipliers having less delay. Vedic multiplier of ripple carry adder based is suggested for mantissa multiplication in IEEE 754 format. Suggested Vedic multiplier uses historic Vedic Indian mathematics sutra called Urdhva-Tiryagbhyam for Vedic multiplication. The architecture Proposed for 64x64 bit complex floating-point multiplier is in Xilinx ISE 14.2 FPGA navigator in Verilog HDL. Eventually, the outcomes of the suggested multiplier will differentiate with traditional booth multiplier and array multiplier which represents clearly that complex multiplication using suggested architecture gives less delay, power and low area.

The role of history in learning and teaching mathematics: A personal perspective

This is a slightly revised version of the Presidential address (General) delivered at the 84th Annual Conference of the Indian Mathematical Society held at Jammu, India during November 2018.

Ancient Indian Mathematical Evolution since Counting

Ancient Indian Mathematical Evolution since Counting

Recent results on vector bundles, principal bundles and related topics

This is a short survey of the interesting work in the theory of vector bundles (with various augmenting structures), principal bundles and related topics by Indian mathematicians in the last decade. The focus is on topics in which Indian mathematicians have made major contributions.

Geometric group theory and hyperbolic geometry: Recent contributions from Indian mathematicians

Geometric group theory emerged as a distinct branch of mathematics through the seminal work of Gromov [25] in 1987 and since then it has been a very active area of research intermingling with many other fields of mathematics. This paper is a survey of contributions made in the past decade in Geometric group theory by the Indian mathematicians.

A survey of symplectic and contact topology

In this article, we give a brief survey of major historical developments in the field of Contact and Symplectic Geometry. This field has grown into an area in its own right due to rapid progress seen in the last five decades. The community of Indian mathematicians working on this field is small but steadily growing. The contribution from Indian mathematicians to this field is noted in the article.

Analytic number theory in India during 2001–2010

In this article we summarize the contribution of Indian mathematicians to analytic number theory during 2001–2010.

Finite element methods: Research in India over the last decade

This article is a summary of major contributions of Indian mathematicians to the mathematical aspects of the finite element method in the last one decade: 2008–2017. We briefly trace out the historical origins of the topic in India and abroad. A section on the method itself is included so that this review is accessible to anybody with a background in partial differential equations and numerical techniques for solving it.

Some recent results in the area of non-complete algebraic varieties

We will briefly describe the basic theory of non-complete algebraic varieties developed by Japanese algebraic geometers and some of the main contributions of Indian mathematicians in this area during the last ten years.

Theoretical developments in the study of partial differential equations

In this article we review some of the main contributions of Indian mathematicians in the theoretical analysis of partial differential equations in the last decade.