Indian Mathematics

Some New results related to “On Direct & Inverse Systems in N-groups” were proved. Bhavanari Satyanarayana. & K. Syam Prasad "On Direct & Inverse Systems in N-groups", Indian J. Maths (BN Prasad Birth Commemoration Volume) 42 (2000) 183 - 192. (Zbl 1033.16021)

Śulbasūtra-s form a part of Kalpasūtra-s. They deal with construction of sacrificial altars and fire places. The altars and fire places are of varying geometrical shapes. Their shape, area, the number of bricks to be used for their construction are ordained in the Vedic texts  called Brāhmaņa-s. The bricks used for piling the 'citi' also have various shapes. Their shapes and even the vedic mantra-s to be uttered in accompaniment of piling are ordained in the Brāhmaņa-s. Verily Sulbasutra-s can be called the  the source book of Vedic Geometry.

Dans cet atelier, nous allons présenter brièvement un mathématicien indien hors du commun : Srinivasa Ramanujan, puis nous tirerons de son oeuvre quelques idées d'activités réalisables en classe de Terminale Scientifique. Ces activités, du domaine de la théorie des nombres, semblent en accord avec les tendances actuelles des programmes , qu'il s'agisse du retour de l'arithmétique dans l'enseignement de spécialité ou, moins spécifiquement, de la place croissante accordée à l'algorithmique.

Nephology is the study of clouds and cloud formation. Observations of clouds and their varied nature formed an important component of rainfall forecasting in ancient India. Scientific description of clouds is found in the  Ṛgveda II. 24. 4: “stone like, solid, down hanging and water-laden”. In the ṛk VII. 7. 4, Macdonell explains the word miha to mean mist. The Sāmaveda, began to make correct guesses about the height of clouds and was acquainted with the fact that clouds ascend and roam in the mountainous altitude also.
The Taittirīya Āranyaka classifies clouds to be of 7 types corresponding to the seven types of air current (or) wind in the atmosphere. In I. 10. 3 of the same text two more types are mentioned: (1) Śambara / Śāmbara and  (2) Bahusomagī. The latter is identified as ‘moving nimbus full of water’.
The Rāmāyaṇa classifies cloud into 3 types. Ascent of clouds in mountains and rainfall is stated in VI. 69/70. 124. It also discusses the process of cloud formation. The Mahābhārata informs us of the  4  types of clouds, different from the Rāmāyaṇa.
The phenomena of evaporation, cloud formation, classification of clouds and their relationship with winds are quite satisfactorily discussed in several Purāṇas and a full-fledged separate chapter has been devoted in them to these topics. The Vāyu, Brahmāṇḍa and Matsya purāṇas deserve mention in connection with clouds. According to Vāyu and Brahmāṇḍa there are 3 types of clouds. The Matsya Purāṇa furnishes still more elaborate and scientific information regarding clouds. The vv. 17-9 talk about 4 types of clouds.
The two Jyotiṣa texts, Bṛhat Samhitā and Mayūracitraka furnish a lot of details about clouds. They mention different clouds according to the capacity of rain they carry and astrologically predict their onset.
Meghamālā, a rare text, fitting to its name is a treatise on nephology. The first chapter of the text classifies clouds first as 4 in number and later 12 species of clouds  are enumerated. The chapter ten adds 10 more species to the aforesaid 12.
The Jambūdvīpaprajñapti, a jain text gives two classification of clouds. The first names 7 types of clouds and the second also enumerates 7 types. The Trilokasāra also a jain work enumerates 7 types of clouds.
The Buddhists refer to two general classification of clouds, Kālameha (or) monsoon clouds and Akālameha (or) storm clouds. The Samyutta Nikāya, a buddhist work tells about 5 types of clouds, while the Aṅguttara Nikāya mentions its own 4 categories of clouds.
The Tāntric Literature of Abhinavagupta named Tantrāloka is the first text classifying clouds according to their increasing height. 11 types of such clouds are listed against the modern classification recognising only 10 types.
The paper is an attempt to bring forth to light the scientific theories of Nephology as found in Indian tradition in comparison with the modern Meterology.

This note presents Virahāṅka's original proof of the sequence associated with prosody that is now known variously as the Virahāṅka-Fibonacci sequence, Fibonacci sequence, or just the Virahāṅka sequence. This sequence is also seen as the number of arrangements of beads in a necklace of a certain value, where each bead has the value of 1 or 2. This sequence was implicitly known as early as fourth to second century BCE in the work of Piṅgala, and was formally derived by Virahāṅka about 600 years before Fibonacci and, therefore, the last name is appropriate. The proof given here leads easily to the generalization of the Nārāyaṇa sequence.

Aryabhata I (b. 476 CE) enunciated the theory of diurnal rotation of the earth in his work Aryabhatiya. But scholars of astronomy like Varahamihira and Brahmagupta, who immediately followed Aryabhata, rebuked him for holding such a view. They vehemently attacked and refuted the theory. Their influence on the later scholars was so strong that even the followers of Aryabhata rejected the theory. For this, they misinterpreted  two passages of the text and changed (hampered) the text itself at two other instances.
(Some typographical errors that have crept into are regretted.)

This paper aims to examine the enunciation (uccāraṇa-kāla) time intervals for śrīvidyā pañcadaśī, a fifteen seed-syllable mantra (bīja-mantra) related to the homonymous śākta school Śrīvidyā or Traipuradarśaṇa. Following the indications provided in the Yoginīhṛdaya, with Dīpikā commentary by Amṛtānanda, and the Varivasyāraharasya by Bhāskararāya with Prakāśa auto-commentary, the research finds that these durations are not arbitrary at all but rather the result of a rigorous assessment. Moreover, the duration values suggest a specific conceptual goal that the mathematical rigor manifested by the authors seeks to fulfil: the progressive diminution of time intervals in order to achieve an atemporal dimension. The choice of the units of measurement itself is designed to meet this metaphysical and ritual need. By counting the intervals (both relative and overall) of mantra recitation, it is also possible to confirm the resonance nature of the sounds following nasalisations, sounds conceived by the authors as entirely independent of the reciter's phonatory activity.

P. P. Divakaran, in his significant work on Indian mathematics, 'Mathematics of India : Concepts, Methods, Connections', names this community as Nila School. This work of Divakaran is beautifully-written. Divakaran discusses what is ‘Indian’ in Indian mathematics. He takes us along with him for a voyage through the mathematico-cultural history of India. Starting from the Rgveda, the journey ends with the culmination of activities of Nila School.
Divakaran has proposed a hypothesis about the identity or the roots of Sangamagrama Madhava the founder and originator of the Nila School of Mathematics.

It is proposed here, in this paper,  to make some observations on his story about Sangamagrama Madhava.

This article draws attention to the need of mathematising the teachersof
mathematics. The paper questions and seek answers to the didactical
approaches that should be adopted to engage teachers in acts of thinking
mathematically.One of the proposed ways is by challenging teachers’
existing mathematical cognition in a constructive manner. The paper
further elaborates a task that was instrumental in setting up conditions
for thinking, reasoning and making conceptual connections.

Lilavati, as is well known,  is the most popular and  celebrated work on Indian mathematics.  The mathematics - the arithmetic and algebra - discussed therein  is outdated. But the philosophy of science  reflected in the work is very much relevant today.  The way in which concepts are looked upon, how they are conceived and develped; and the thought process involved are worth to be probed into.  In fact works like Lilavati assume significance from this angle  of view alone.

In same timespan , astronomer mathematician person likely to be astrophysicist & aeronautical engineer as both space & aircraft design & astronomy are closely related . So Lalla quoted in Vaimanika Shastra must be the same author of Shishya-Dhi-Vriddhi Tantra , treatise on astronomy .

This paper expounds very innovative results achieved between the mid-14th century and the beginning of the 16th century by Indian astronomers belonging to the so-called "Mādhava school". These results were in keeping with researches in trigonometry: they concern the calculation of the eight of the circumference of a circle. They not only expose an analog of the series expansion of arctan(1) usually known as the "Leibniz series", but also other analogs of series expansions, the convergence of which is much faster. These series expansions are derived from evaluations of the rests of the partial sums of the primordial series, by means of some convergents of generalized continued fractions. A justification and generalization of these results in modern terms is provided, which aims at restoring their full mathematical interest.

Keḷallūr Nīlakanṭha Somayājin (b. 1444 CE) is a prominent member of the mathematical tradition that flourished in Central Kerala during 13th to 18th centuries CE. This tradition is said to have originated thoughts on Calculus. Nīlakanṭha is well known as the author of Tantrasaṅgraha and Āryabhaṭīyabhāṣya.  But his works (all written in Sanskrit) are yet to be analysed from the angles of  the philosophy of Mathematics and conceptual history. The facts that his language resembles any of the classical works on Philosophy and that his works on Mathematics contain many quotations from Mīmāṃsā texts asserting that śāstra should essentially be rational are less noticed.
THIS PAPER, WITH FULL DOCUMENTATION SHOWS THAT HE HAD BEEN HOLDING A CRITICAL AND RATIONAL APPROACH TOWARDS THE DISCIPLINE OF ASTRONOMY.

Camp Zero part two held in Delhi, April4-6, 2017, proved to be a pleasant, all-in-the-family meeting of dedicated scholars. We are now to embark on the adventure of writing a monograph together, featuring first findings and insights yielded into the (philosophical) origins of the mathematical zero. This too in order to inspire and attract more scholars (and funds!) to ensure the future of the project.

In this paper, we introduce and investigate a subclass of analytic and biunivalent functions in the open unit disk U. Furthermore, we find upper bounds for the second and third coeffi cients for functions in this subclass. The results presented in this paper generalize and improve some recent works.

Newton (1643 - 1727) and Leibnitz (1646 - 1716) are regarded as the founders of the differential and the integral calculus. The notions of the integral calculus had however been understood in a rough way and applied to the determination of areas and volumes by the ancient Greeks. One thus finds the method of summation as envisaged in the integral calculus, from Archimedes to Kepler. Bhaskara (1114 – 1185) determines the area and volume of a sphere by exactly similar method. While Bhaskara thus had notions of the integral calculus as much as it was known in his time, he deserves special mention as the first mathematician ever who conceived of the differential calculus, and gave the first example of a differential coefficient. The principles of the differential calculus were conceived as the results of his astronomical considerations and can be found in Grahagaṇita, a treatise on the motion of planets, a third section of his great work Siddhānta Śiromaṇī.

It is a popular article. The gist : "Kelallur Nilakantha Somayajin (1444-1544 CE) is unparalleled in the Intellectual traditions of India. He is a staunch advocate  of the  fact that ज्यौतिष is a science which should be subjected to periodical revision and modification. He is the torch-bearer of the tradition upheld by Aryabhata and Vatasseri Paramesvara."

In a digraph D = (X, U) , not necessarily finite, an arc (x, y) ∈ U is reachable from a vertex u if there exists a directed walk W that originates from u and contains (x, y). A subset S ⊆ X is an arc-reaching set of D if for every arc (x, y) there exists a diwalk W originating at a vertex u ∈ S and containing (x, y). A minimal arc-reaching set is an arc-basis. S is a point-reaching set if for every vertex v there exists a diwalk W to v originating at a vertex u ∈ S. A minimal point-reaching set is a point-basis. We extend the results of Harary, Norman, and Cartwright on point-bases in finite digraphs to point- and arc-bases in infinite digraphs.

Determining and identifying different possible types of meters was the problem with which Vedic prosodists had to deal with. Those meters consist of varying combinations of Sanskrit syllables obtained by changing the long and short sounds within each syllable group. Each separate combination of such syllables is a potential poetic meter.
The effort of evolving an elegant system of classifying syllabic patterns led permanently to laying the foundation of the mathematics of permutations and combinations. Piṅgala's treatise on prosody (possibly 500-400 BC), Chandaḥśāstra, contains an enumeration of meters with fixed patterns of short and long syllables achieved by method which is considered as a first known binary system.

An elementary and self contained method is given for determining formulas involving the number of representations of an integer as a sum of two, four, six and eight squares or triangular numbers. Our method uses Ramanujan’s 1 1 summation formula and is based on K. Venkatachaliengar’s elementary approach to elliptic functions.

Popular attention has recently been captured by the results of the Bodleian Library's 2017 project of radiocarbon datingportions of the birch-bark fragments constituting what is known as the Bakhshālī Manuscript.  In this paper, we disagree with the interpretation of the findings announced by the Bodleian team. In particular, we argue that the earliest dated folio of this manuscript is unlikely to be the date of the whole text. Rather, the latest dateable folio is logically the date of the scribal activity. This fits well with past estimates of the date of the Bakhshālī Manuscript based on historical, philological and palaeographic arguments..  And we argue that the Bakhshālī Manuscript does include written zeros that function as arithmetical operators, i.e., as numbers in their own right, and not merely as place-holders, as asserted by the Bodleian team. Finally, we express regret that the Bodleian Library chose to announce scientific results without peer-review and through a p...

The geometry of nonholonomic bundle gerbes, provided with nonlinear connection structure, and nonholonomic gerbe modules is elaborated as the theory of Clifford modules on nonholonomic manifolds which positively fail to be spin. We explore an approach to such nonholonomic Dirac operators and derive the related Atiyah-Singer index formulas. There are considered certain applications in modern gravity and geometric mechanics of such Clifford-Lagrange/ Finsler gerbes and their realizations as nonholonomic Clifford and Riemann-Cartan modules.

This book is a cross-cultural reference volume of all attested numerical notation systems (graphic, non-phonetic systems for representing numbers), encompassing more than 100 such systems used over the past 5,500 years. Using a typology that defies progressive, unilinear evolutionary models of change, Stephen Chrisomalis identifies five basic types of numerical notation systems, using a cultural phylogenetic framework to show relationships between systems and to create a general theory of change in numerical systems. Numerical notation systems are primarily representational systems, not computational technologies. Cognitive factors that help explain how numerical systems change relate to general principles, such as conciseness or avoidance of ambiguity, which apply also to writing systems. The transformation and replacement of numerical notation systems relates to specific social, economic, and technological changes, such as the development of the printing press or the expansion of the global world-system.

We introduce C*-algebras over C_{\infty}(Q,C) as Banach-Kantorovich *-algebras over the algebra C_{\infty}(Q,C) of extended continuous complex-valued functions, defined on comeager subsets of Stonean compact Q, whose norm satisfies conditions similar to the axioms of C*-algebras, and show that such algebras can be uniquely up to a Q-C*-isomorphism represented by means of a continuous complete fiber bundle of C*-algebras over Q.

We provide an iterative process which converges strongly to a common fixed point of finite family of asymptoticallyk-strict pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.