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- 1. Home > Products + Services > What is MSMMobile Software Management encompasses a set of technologies and business processes that enable the managementof software assets in mobile devices throughout their lifecycle.As the pioneer of Mobile Software Management (MSM), Red Bend Software offers the industry’s leading software productsfor managing mobile software over the air. Together, Red Bend’s innovative and award-winning products create an end-to-end solution for independent, centralized and consistent control over managing all types of software and applications on anydevice, any platform, anytime.Red Bend offers the only holistic solution for gathering information from the device, analyzing that data, using it to formulatedecisions and then performing management actions on the device.InformDescription: On-device client software collects data from the device including software inventory, usage analytics, deviceconfiguration and settings, and network and device diagnostics.Real-World Example: vRapid Mobile® Software Management Client takes an inventory of what software and applicationsconsumers have installed and are using on their mobile phones. vDirect Mobile® Device Management Client transmitsthat data over the air using OMA DM standards.AnalyzeDescription: Back-end system processes, collates and presents data for analysis.Real-World Example: vRapid Mobile Software Management Center shows what percentage of consumers have not yetdownloaded the new music application that was launched two months ago. Analysis shows that most consumers aremissing a key piece of middleware that is essential for the music application to run. Deployment estimation tool shows howmany devices can be updated over the air, and how much bandwidth will be required for the full operation.DecideDescription: Real-time view of the installed base enables informed decisions to achieve maximum success.
- 2. Real-World Example: Service provider decides to offer a software update to bring the targeted devices up to the samebaseline.PerformDescription: On-device client software carries out those management decisions such as updating firmware over the air,installing or removing applications and changing device or network configuration.Real-World Example: A software update is deployed to those consumers with the missing middleware and the musicapplication in a single update package, so that the service provider can widen its addressable market for the musicapplication. The software update package is sent over the air to the vDirect Mobile device management client. vRapidMobile FOTA software performs the software update quickly, efficiently and reliably.May 21, 2011 - 12:00PM PTMobile Software: DrivingInnovation in the Multi-Core Era BY Rob Chandhok 6 CommentsMobile hardware is progressing at a blistering pace, but to deliver the type of user experiences enabledby awesome hardware software must keep pace. This goes beyond the need for innovations in OSes andapplications, to the underlying software that ties everything together.Mobile hardware is progressing at a blistering pace. Displays continue to increase in size, colorquality and resolution, while advancements such as glasses-free 3-D offer the promise of novel userexperiences. Processors are adding cores and clock speed faster than ever before, and 4G radios havebrought broadband data speeds to mobile devices. These unprecedented hardware innovations haveset the stage for a brave new world of mobile computing in which nearly anything is possible onhand-held devices. However, they account for only part of the equation.In order to deliver the type of user experiences enabled by these innovations software must keeppace – otherwise we will fall painfully short of capitalizing on the opportunities presented by thesehardware achievements. This goes beyond the need for innovations in OSes and applications, to theunderlying software that ties everything together. It’s the next great challenge faced by the mobileindustry.
- 3. Software as the Connective Tissue of the PhoneWhen it comes to mobile software, the importance of operating systems and applications is wellunderstood. The battle for smartphone OS market share evokes a feverish MLB pennant race, andthe fact that we’re all hopelessly addicted to Angry Birds proves that mobile apps have thoroughlypermeated the mainstream.Less understood, however, is the importance of the underlying software layer; the connective tissuethat ties hardware to software, such as optimizations between OS and chipset, performanceadvancements in web technology, and enhanced app performance. Without these efforts, gigahertz,cores and megabytes of RAM are nothing more than points on a spec sheet. In order to deliver thebest possible mobile experiences, hardware and software cannot be viewed separately. They areattached at the hip, and integrating them to work in perfect unison is the key to driving mobileinnovation forward.Immediate benefits of intelligent integration include better graphical frame rates in games, fasterweb page downloads and smoother rendering and scrolling. These are just a sampling of the userexperience improvements that will help mobile devices keep up with ever-increasing consumerexpectations.Innovating for the Future of the Mobile WebAll too often, the primary focus is on what the consumer wants today. It is our job to anticipate whatthe consumer will want tomorrow and innovate accordingly.While today’s consumers are still largely enamored with the simple inclusion of mobile browsers,tomorrow’s expectations will include desktop-level browser performance, Web pages and appsrunning on par with native apps and smooth HD multimedia streaming like the desktop equivalent.This is possible via complex but informed optimizations to the HTTP networking layer, HTML5browser core, and JavaScript engine. While powerful processors will strongly influence robust Webexperiences, the mobile software layer is significantly impacting how we get the most out of mobilehardware and continue to innovate on behalf of the consumer experience.While HTML5 will play an important role in the evolution of the mobile Web, it won’t come tofruition until mobile devices support the specification fully, from web and enterprise apps toentertainment and browsing. Forward-thinking developers making the transition to HTML5-based web apps stand to reap the benefits. The HTML5 family of standards runs faster, moreefficiently and with greater capabilities when the hardware and software have been tightlyintegrated.
- 4. The biggest remaining hurdle is ensuring that the same array of device capabilities, such as cameraaccess, is available to Web apps as their native counterparts. To this end, companies like Qualcommare enabling a rich set of device APIs within the browser so that Web apps have that same detailedcontrol and usage of the device’s hardware.Collaboration Is KeyThe mobile industry is built on partnerships within the diverse lines of business that make up theecosystem and we must continue to work closely together to make these advancements a reality —from ensuring common device APIs are defined, implemented, and utilized to working hand in handacross the mobile ecosystem to deliver web experiences that go beyond what we ever experienced ona PC. All stand to benefit greatly by software’s ongoing impact on mobile, and efficient collaborationwill expedite that process. Ultimately, intelligent and tight OS integration within the chip providestime to market advantages for OEMs who will see their devices running faster, smoother and moreefficiently.Enhancing mobile software is not a trickle down process. It starts with the seamless hardwareintegration and ends with developers bringing the experience to life. If we are serious about a futurewhere mobile phones are responsible for tasks currently held by computers we need to embrace therole of software in overall mobile performance and continue strongly supporting the softwaredevelopers that are driving innovation.Rob Chandhok is president of Qualcomm Internet Services and helps drive software strategy forQualcomm’s client and server platforms. He and other mobile industry thought leaders will bediscussing these topics and more June 1-2 at Uplinq 2011 in San Diego. His Twitter handleis@robchandhokThere are a massive range of mobile software for displaying OpenStreetMap or otherwise making use ofour geodata on phones and other mobile devices. Increasingly apps are allowing contribution toOpenStreetMap too. Because OpenStreetMap data is free and open for anyone and everyone, mobilesoftware is being developed by a wide range of people and companies, and OpenStreetMap data can beaccessed on almost any mobile platform.
- 5. Linux-based Access Linux ·Android ·DSLinux ·Familiar ·iPodLinux ·LiMo ·MeeGo (Moblin ·Maemo ·Qt Extended) ·Mobilinux ·Openmoko Linux ·OPhone ·SHR ·Qt Extended Improved ·Ubuntu Mobile ·webOS Other Bada ·BlackBerry OS ·BlackBerry Tablet OS ·GEOS ·iOS (iPhone) ·Nintendo DSi OS ·Nokia OS (S30 ·S40) ·Palm OS ·PSP OS · Symbian OS ·SavaJe ·Windows Mobile ·Windows Phone Related platforms BREW ·Java ME (FX Mobile)
- 6. TrigonometryFrom Wikipedia, the free encyclopedia"Trig" redirects here. For other uses, see Trig (disambiguation).The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints.Calculating the final position of the astronaut at the end of the arm requires repeated use of trigonometric functions of thoseangles. Trigonometry History Usage Functions Generalized Inverse functions Further reading Reference Identities Exact constants Trigonometric tables Laws and theorems Law of sines Law of cosines
- 7. Law of tangents Law of cotangents Pythagorean theorem Calculus Trigonometric substitution Integrals of functions Derivatives of functions Integrals of inverse functions V T ETrigonometry (from Greek trigōnon "triangle" + metron "measure"[1]) is a branch of mathematics thatstudies triangles and the relationships between their sides and the angles between these sides. Trigonometrydefines the trigonometric functions, which describe those relationships and have applicability to cyclicalphenomena, such as waves. The field evolved during the third century BC as a branch of geometry usedextensively for astronomical studies.[2] It is also the foundation of the practical art of surveying.Trigonometry basics are often taught in school either as a separate course or as part of a precalculus course.The trigonometric functions are pervasive in parts of pure mathematics and applied mathematics suchas Fourier analysis and the wave equation, which are in turn essential to many branches of science andtechnology. Spherical trigonometry studies triangles on spheres, surfaces of constant positive curvature,in elliptic geometry. It is fundamental to astronomy and navigation. Trigonometry on surfaces of negativecurvature is part of Hyperbolic geometry. Contents [hide] 1 History 2 Overview
- 8. o 2.1 Extending the definitions o 2.2 Mnemonics o 2.3 Calculating trigonometric functions 3 Applications of trigonometry 4 Standard identities 5 Angle transformation formulas 6 Common formulas o 6.1 Law of sines o 6.2 Law of cosines o 6.3 Law of tangents o 6.4 Eulers formula 7 See also 8 References o 8.1 Bibliography 9 External links[edit]HistoryMain article: History of trigonometryThe first trigonometric table was apparently compiled by Hipparchus, who is now consequently known as "the father oftrigonometry."[3]Sumerian astronomers introduced angle measure, using a division of circles into 360 degrees. [4] They and theirsuccessors the Babylonians studied the ratios of the sides of similar triangles and discovered some properties
- 9. of these ratios, but did not turn that into a systematic method for finding sides and angles of triangles.The ancient Nubians used a similar methodology.[5] The ancient Greeks transformed trigonometry into anordered science.[6]Classical Greek mathematicians (such as Euclid and Archimedes) studied the properties of chords andinscribed angles in circles, and proved theorems that are equivalent to modern trigonometric formulae,although they presented them geometrically rather than algebraically. Claudius Ptolemy expandeduponHipparchus Chords in a Circle in his Almagest.[7] The modern sine function was first defined in the SuryaSiddhanta, and its properties were further documented by the 5th century Indian mathematician andastronomer Aryabhata.[8] These Greek and Indian works were translated and expanded by medieval Islamicmathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, hadtabulated their values, and were applying them to problems in spherical geometry.[citation needed] At about thesame time, Chinese mathematicians developed trigonometry independently, although it was not a major field ofstudy for them. Knowledge of trigonometric functions and methods reached Europe via Latin translations of theworks of Persian and Arabic astronomers such as Al Battani and Nasir al-Din al-Tusi.[9] One of the earliestworks on trigonometry by a European mathematician is De Triangulis by the 15thcentury German mathematician Regiomontanus. Trigonometry was still so little known in 16th century Europethat Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explaining its basicconcepts.Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometrygrew to be a major branch of mathematics.[10]Bartholomaeus Pitiscus was the first to use the word, publishinghis Trigonometria in 1595.[11] Gemma Frisius described for the first time the method oftriangulation still usedtoday in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. Theworks of James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in thedevelopment of trigonometric series.[12] Also in the 18th century, Brook Taylor defined the general Taylorseries.[13][edit]Overview
- 10. In this right triangle: sin A = a/c; cos A = b/c;tan A = a/b.If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, becausethe three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees:they are complementary angles. The shape of a triangle is completely determined, except for similarity, by theangles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of thetriangle. If the length of one of the sides is known, the other two are determined. These ratios are given by thefollowing trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in theaccompanying figure: Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse. Cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse. Tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg. The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. Many English speakers find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA (see below underMnemonics). The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot), respectively:
- 11. The inverse functions are called the arcsine, arccosine, and arctangent,respectively. There are arithmetic relations between these functions, which areknown as trigonometric identities. The cosine, cotangent, and cosecant are sonamed because they are respectively the sine, tangent, and secant of thecomplementary angle abbreviated to "co-".With these functions one can answer virtually all questions about arbitrary trianglesby using the law of sines and the law of cosines. These laws can be used tocompute the remaining angles and sides of any triangle as soon as two sides andtheir included angle or two angles and a side or three sides are known. These lawsare useful in all branches of geometry, since every polygon may be described as afinite combination of triangles.[edit]Extending the definitionsFig. 1a - Sine and cosine of an angle θ defined using the unit circle.The above definitions apply to angles between 0 and 90 degrees (0 andπ/2 radians) only. Using the unit circle, one can extend them to all positive andnegative arguments (see trigonometric function). The trigonometric functionsare periodic, with a period of 360 degrees or 2π radians. That means their valuesrepeat at those intervals. The tangent and cotangent functions also have a shorterperiod, of 180 degrees or π radians.
- 12. The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus and infinite series. With these definitions the trigonometric functions can be defined for complex numbers. The complex exponential function is particularly useful. See Eulers and De Moivres formulas. Graphing process of y = sin(x) using a unit circle. Graphing process of y = tan(x) using a unit circle. Graphing process of y = csc(x) using a unit circle. [edit]Mnemonics Main article: Mnemonics in trigonometry A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters. For instance, a mnemonic for English speakers is SOH-CAH-TOA:Sine = Opposite ÷ HypotenuseCosine = Adjacent ÷ Hypotenuse
- 13. Tangent = Opposite ÷ Adjacent One way to remember the letters is to sound them out phonetically (i.e. "SOH-CAH-TOA", which is pronounced so-kə- tow-uh).[14] Another method is to expand the letters into a sentence, such as "Some Old Hippy Caught Another Hippy Trippin On Acid".[15] or "Some Old Houses, Cant Always Hide, Their Old Age" [edit]Calculating trigonometric functions Main article: Generating trigonometric tables Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how tointerpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions. Today scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan and sometimes cis) and their inverses. Most allow a choice of angle measurement methods: degrees, radians and, sometimes, grad.[citation needed] Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers have built-in instructions for calculating trigonometric functions.[citation needed] [edit]Applications of trigonometry
- 14. Sextants are used to measure the angle of the sun or stars with respectto the horizon. Using trigonometry and amarine chronometer, the positionof the ship can be determined from such measurements.Main article: Uses of trigonometryThere are an enormous number of uses of trigonometry andtrigonometric functions. For instance, the techniqueof triangulation is used in astronomy to measure the distance tonearby stars, in geography to measure distances betweenlandmarks, and in satellite navigation systems. The sine andcosine functions are fundamental to the theory of periodicfunctions such as those that describe sound and light waves.Fields that use trigonometry or trigonometric functionsinclude astronomy (especially for locating apparent positions ofcelestial objects, in which spherical trigonometry is essential) andhence navigation (on the oceans, in aircraft, and in space), musictheory, acoustics, optics, analysis of financialmarkets,electronics, probability theory, statistics, biology, medicalimaging (CATscans and ultrasound), pharmacy, chemistry, number theory (andhencecryptology), seismology, meteorology, oceanography,many physical sciences,land surveying and geodesy, architecture, phonetics, economics,electrical engineering, mechanical engineering, civilengineering, computergraphics, cartography, crystallography and game development.[edit]Standard identitiesIdentities are those equations that hold true for any value. [edit]Angle transformation formulas
- 15. [edit]CommonformulasTriangle withsides a,b,c andrespectively oppositeangles A,B,CCertain equationsinvolving trigonometricfunctions are true for allangles and are knownas trigonometricidentities. Someidentities equate anexpression to a differentexpression involving thesame angles. These are
- 16. listed in List oftrigonometric identities.Triangle identities thatrelate the sides andangles of a giventriangle are listed below.In the followingidentities, A, B and C are the angles of atriangleand a, b and c are thelengths of sides of thetriangle opposite therespective angles.[edit]Law of sinesThe law of sines (alsoknown as the "sinerule") for an arbitrarytriangle states: where R is the radius of the circumscribed circle of the triangle: Another law involving sines can be used to calculate the area of a
- 17. triangle. Giventwo sides andthe anglebetween thesides, the areaof the triangleis: All of the trigono metric functions of an angle θ can be constructed geometrical ly in terms of a unit circle centered at O. [edit]La w of cosine s
- 18. The lawofcosines (known asthe cosineformula, orthe "cosrule") is anextensionofthe Pythagoreantheorem toarbitrarytriangles: or equiv alentl y: [ e d it ] L a w o f t a
- 19. ngentsThe lawoftangents (alsoknownast
- 20. he"tanrule"): [ e d i t ] E u l e r s f o r m u l a
- 21. Eulersformula,whichstatesthat
- 22. ,producesthefollowinganalytical
- 23. identitiesforsine,cosine,andtan
- 24. gentintermsofeandtheimaginary
- 25. u n i t i :List of Indian mathematiciansFrom Wikipedia, the free encyclopedia This article needs attention from an expert in India. See the talk page for details. WikiProject India or the India Portal may be able to help recruit an expert. (June 2010)
- 26. Indian mathematician Komaravolu Chandrasekharan in Vienna, 1987.The chronology of spans from the Indus valley civilization and the Vedas to Modern times.Indian mathematicians have made a number of contributions to mathematics that have significantly influencedscientists and mathematicians in the modern era. These include place-value arithmetical notations, the ruler,the concept of zero, and most importantly, the Arabic-Hindu numerals predominantly used today and which canbe used in the future also. Contents [hide] 1 Classical 2 Medieval to Mughalperiod 3 Born in 1800s 4 Born in 1900s[edit]ClassicalPost-Vedic Sanskrit to Pala period mathematicians (5th c. BC to 11th c. AD) Aryabhata – Astronomer who gave accurate calculations for astronomical constants, 476AD-520AD Bhaskara I Brahmagupta – Helped bring the concept of zero into arithmetic (598 AD-670 AD) Mahavira Pavuluri Mallana – the first Telugu Mathematician Varahamihira Shridhara (between 650–850) – Gave a good rule for finding the volume of a sphere.[edit]Medieval to Mughal period13th century to 1800.13th century, Logician, mithila school Narayana Pandit Jyeshtadeva, 1500–1610, Author Madhava of Sangamagrama some elements of of Yuktibhāṣā, Madhavas Kerala school Calculus hi Achyuta Pisharati, 1550–1621, Parameshvara (1360–1455), discovered drk- Astronomer/mathematician, Madhavas Kerala school ganita, a mode of astronomy based on Munishvara (17th century)
- 27. observations, Madhavas Kerala school Kamalakara (1657) Nilakantha Somayaji,1444–1545 – Jagannatha Samrat (1730) Mathematician and Astronomer, Madhavas Kerala Srijan Gupta of Delhi (1997) school Mahendra Suri (14th century) Shankara Variyar (c. 1530) Raghunatha Siromani, (1475–1550), Logician, Navadvipa school[edit]Born in 1800s Ramchandra (1821–1880) Ganesh Prasad (1876–1935) Srinivasa Ramanujan (1887–1920) A. A. Krishnaswami Ayyangar (1892–1953)[edit]Born in 1900s Tirukkannapuram Vijayaraghavan (1902–1955) Dattaraya Ramchandra Kaprekar (1905–1986) Sarvadaman Chowla (1907–1995) Lakkoju Sanjeevaraya Sharma (1907–1998) Subrahmanyan Chandrasekhar (1910–1995) S. S. Shrikhande (born 1917) Harish-Chandra (1920–1983) Calyampudi Radhakrishna Rao (born 1920) Mathukumalli V. Subbarao (1921–2006) P. K. Srinivasan (1924–2005) Shreeram Shankar Abhyankar (born 1930) M. S. Narasimhan (born 1932) C. S. Seshadri (born 1932) K. S. S. Nambooripad (born 1935) Vinod Johri (born 1935) S. Ramanan (born 1937) C. P. Ramanujam (1938–1974) Shakuntala Devi (1939–present)
- 28. V. N. Bhat (1938–2009) S. R. Srinivasa Varadhan (born 1940) M. S. Raghunathan (born 1941) Gopal Prasad (born 1945) Vijay Kumar Patodi (1945–1976) S. G. Dani (born 1947) Raman Parimala (born 1948) Navin M. Singhi (born 1949) Narendra Karmarkar (born 1957) Manindra Agrawal (born 1966) Madhu Sudan (born 1966) Chandrashekhar Khare (born 1968) Manjul Bhargava (Indian origin American) (born 1974) Amit Garg (born 1978) Akshay Venkatesh (Indian origin Australian) (born 1981) Kannan Soundararajan (born 1982[citation needed]) Sucharit Sarkar (born 1983) L. Mahadevan Wikimedia Commons has media related to: Mathematicians from IndiaThe Top Five All Time MathematiciansSchool and Education - CollegeBy: AmarSingh 15-Oct-2010
- 29. ShareAryabhataAryabhatiya (499 A.D.), he made the fundamental advance in finding the lengths of chords of circles, by using the half chordrather than the full chord method used by Greeks. He gave the value of as 3.1416, claiming, for the first time, that it was anapproximation. He also gave methods for extracting square roots, summing arithmetic series, solving indeterminate equations ofthe type ax -by = c, and also gave what later came to be known as the table of Sines.BrahmaguptaHe was born in 598 A.D. His main work was Brahmasphutasiddhanta, which was a corrected version of old astronomical treatiseBrahmasiddhanta. This work was later translated into Arabic as Sind Hind.He gave the formula for the area of a cyclicquadrilateral . He was the first mathematician to treat algebra and arithmetic as two different branches of mathematics. He gavethe solution of the indeterminate equation Nx²+1 = y².BhaskaraBhaskaracharaya is the most well known ancient Indian mathematician. He was born in 1114 A.D. at Bijjada Bida (Bijapur,Karnataka) in the Sahyadari Hills.He was the first to declare that any number divided by zero is infinity and that the sum of anynumber and infinity is also infinity. He gave an example of what is now called "differential coefficient" and the basic idea ofwhat is now called "Rolles theorem".Srinivasa Aaiyangar RamanujanHe was born in a poor family at Erode in Tamil Nadu on December 22, 1887. His father, K. Srinivasa Iyengar worked as a clerkin a sari shop and his mother, Komalatammal was a housewife and also sang at a local temple.On 1 October 1892, Ramanujanwas enrolled at the local school. At the Kangayan Primary School, Ramanujan performed well. Just before the age of 10, inNovember 1897, he passed his primary examinations in English, Tamil, geography and arithmetic. With his scores, he finishedfirst in the distric .Largely self taught, he feasted on Loneys Trigonometry at the age of 13, and at the age of 15, his seniorfriends gave him Synopsis of Elementary Results in Pure and Applied Mathematics by George Carr.A few months earlier, he hadsent a letter to great mathematician G.H. Hardy, in which he mentioned 120 theorems and formulae. Ramanujan published 21papers, some in collaboration with Hardy. He died in Madras on April 26, 1920.D.R. KaprekarD.R. Kaprekar was born in 1905 in India.Kaprekar received his secondary school education in Thane and studied at FergussonCollege in Pune.In 1963, Kaprekar defined the property which has come to be known as self numbers. For example, 21 is not aself number, since it can be generated from 15: 15 + 1 + 5 = 21. But 20 is a self number, since it cannot be generated from anyother integer. Today his name is well-known and many other mathematicians have pursued the study of the properties hediscovered.India and the World of MathematicsIndia has a long tradition of pursuit of mathematics. Geometry had an important (practical)role to playin Vedic culture (circa 1000 BCE) and Pythagoras‟s theorem was known to mathematicians in India bythe eighth century before Christ. Astronomy went hand in hand with mathematics and not surprisingly
- 30. trigonometry was an area where India made considerable progress. The place value system with theuseof the zero seems to have been invented here in the early centuries of the Christian era. Scholarsfrom the middle east who came into contact with India absorbed the “Hindu methods” of calculationwith great enthusiasm and were to pass it on to Europe later. Al Biruni a 10th century scholar isfull of high praise for “Hindu mathematics” even while he is highly critical of several otheraspects of the culture of the subcontinent. In Kerala in the southwest of India, a mathematics schoolflourished for some 200 hundred years starting in the 14the century and the leadingmathematician of the School, Madhava, seems to have anticipated some of the essentialideas of Calculus. It is not clear if these discoveries had any influence on later developments inEurope. Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy.India has a long tradition of pursuit ofmathematics. Geometry had an important (practical)role to play in Vedic culture (circa 1000 BCE) and Pythagorass theorem was known tomathematicians in India by the eighth century before Christ. Astronomy went hand in hand withmathematics and not surprisingly trigonometry was an area where India made considerable progress. Theplace value system with the use of the zero seems to have been invented here in the early centuries of the Christian era. Scholars from the middle east who came into contact with India absorbed the "Hindu methods" of calculation with great enthusiasm and were to pass it on to Europe later. Al Biruni a 10th century scholar is fullof high praise for "Hindu mathematics" even while he is highly critical ofseveral other aspects ofthe culture of the subcontinent. In Kerala in the southwest of India, a mathematics school flourished for some 200 hundred years starting in the 14the century and the leading mathematician ofthe School, Madhava, seems to have anticipated some of the essential ideas of Calculus. It is not clear if these discoveries had any influence on later developments in Europe.The colonial periodAfter a dormant period of some two or three centuries, in the mid 19th century when the Britishcolonial administration set up universities along European lines, there was a a resumption of interestin mathematics in India – and it was now mathematics as pursued in the West. Many Britishacademics were involved in the promotion of mathematical activity in this country in those early daysand not surprisingly, the mathematical areas pursued in Britain were the ones Indians took to. In earlytwentieth century however Indian mathematicians were also becoming aware ofdevelopoments inmathematics elsewhere in Europe. Also Indianmathematicians were enrolling themselves in European(mostly British) universities and were geeting trained there.
- 31. In April 1907, a civil servant, V Ramaswamy Iyer who was also a mathematics enthusiast initiated theformation of the first mathematical (infact the first scientific) society in the Indian subcontinent. Thesociety called then The Indian Mathematics Club had at its inception only 20 members located indifferent cities (mainly Chennai, Mumbai and Pune); many among them, personalfriends of Ramaswamy Iyer, were not professionally involved with mathematics. The society startedsubscribing to some Mathematics journals and these volumes were circulated among the membership.In 1909 the society‟s name was changed to Indian Mathematical Society (IMS). It still functions todayunder that name and from that year has been publishing a journal, the “Journal of the IndianMathematical Society”. The first issue of the journal (published in 1910) has an article by one SeshuIyer on Green‟s Functions where he talks about Poincare‟s work on the subject; he says among otherthings that while the British treatment of the subject was largely from the point of view of physics theFrench, mainly Poincare, developed it from a purely mathematical standpoint. This is an indication thatIndian mathematicians were exploring mathematics beyond what the British had brought to them.I know of two great papers published in the Journal of the Indian Mathematical Society. The first is bySelberg on (what we now know as) the Selberg Trace formula. The second is Weil‟s paper on theclassification ofclassical semisimple groups over an arbitrary field of characteristic 0. There must beothers. Srīnivāsa Aiyangār Rāmānujan made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions.The first major mathematician from India to have had an international impact in the modern periodwas of course Srinivasa Ramanujan. And the decisive factor that was responsible for theflowering of that natural genius was the intervention of the towering Cambridge figure G H Hardy. Thefascinating story of Ramanujan is told admirably in Robert Kanigel‟s biography titled “The man whoknew infinity”. There are some very interesting essays about Ramanujan in a volume entitled“Ramanujan: Essays and Surveys” (an AMS publication, edited by Bruce Berndt and Robert A Rankin),among them the text of a talk given by Selberg about the influence Ramanujan‟s work had on him at aRamanujan Centenary Celebration in Mumbai (Bombay).Hardy never visited India but his mathematical influence in this country was pervasive. There wereother great mathematicians who had also some influence, but Hardy‟s was preeminent in the firstquarter and more of the 20th century. There were of course, as already mentioned, many visitorsmainly from Britain who spent considerable time in Indian universities. In the thirties and thereafter
- 32. one finds quite a few truly eminent names among the mathematicians visiting India for fairly longperiods. Some Jewish mathematicians sought refuge in India to escape the Nazi onslaught in Europe.Ramanuajan became an iconic figure and inspired many young men to take to research inmathematics. Research of good quality started emerging out of Universities of Calcutta and Madrasand a little later some others, notably Allahabad and Benares emerged as goodcentres of mathematical research. Here are some names that were recognised internationally in theirtime the first half of the twentieth century): S Mukhopadhyay (known for his “four vertex” theorem),Ganesh Prasad, Ananda Rau, B N Prasad, Vaidyanathaswamy, P C Vaidya, S S Pillai, P L Bhatnagar, TVijayaraghavan, S Minakshisundaram, S Chowla, K Chandrasekharan, D.D. Kosambi (see thisarticle for more details).Andre Weil spent 2 years (1930 – 32) at Aligarh Muslim University (Aligarh is a mid-sized town some100 kms south of Delhi) as Professor and Chair of the mathematics department. He was of course (at24) yet to achieve the kind of fame and standing that he did a little later in his career. In hisautobiography (Apprenticeship of a mathematician) he speaks fondly of his days there and of themany Indian friends he made during that sojourn. The English text of a talk about Indianmathematics of the thirties given in Moscow (in 1936) can be found in his collected works (springerVerlag 1980). The talk ends with “Nevertheless the intellectual potentialities of the Indian nation areunlimited, and not many years would perhaps be needed before India can take a worthy place in worldmathematics” Weil was again in India some 35 years after that to take part in international conferencein Algebraic Geometry in Mumbai (then Bombay). He went around the country renewing his oldacquaintances.Another frenchman who had a tremendous influence on mathematics in India was Rev Fr Racine. Hecame to India sent here by the Jesuit mission in 1934 to teach mathematics in St Joseph‟s College inThiruchi, a town in in the state of Tamilnadu. He was a student of E Cartan (and a friend of H Cartan).With that background he was able to influence students to pursue mathematical areas which occupiedcentre-stage in Europe rather than the traditional largely Cambridge inspired subjects. Many of hisstudents were to become distinguished mathematicians. Fr Racine stayed on in India till his death in1976 paying only infrequent short visits to France. Another big name who visited India several timesduring the late forties and later was M H Stone. On these visits he invariably spent substantialamounts of time in Chennai (then Madras). He spotted quite a few talented young men and helpedarrange for them to visit great institutions like the Institute in Princeton. Stone appears to haveenjoyed these visits immensely and he had many friends in India.Apart from the universities one other institution was promoting mathematical research vigorously –the Indian Statistical Institute. Many distinguished statisticians emerged from this institution in thefirst halfof the twentieth century: P. C. Mahalonobis, R. C. Bose, S. N. Roy, K. R. Nair, D. B. Lahiri, C.R. Rao to name some.The post independence (1947) scene (mainly about TIFR)
- 33. Tata Institute of Fundamental Research (TIFR)Two institutions, The Tata Institute of Fundamental Research (TIFR) in Mumbai (Bombay) and theIndian Statistical Institute (ISI) in Kolkata (Calcutta) have played a leading role in thepromotion ofmathematics in the country in the post-independence period. They both embarked onrunning graduate schools along the lines ofAmerican Universities and combining carefulrecruitment of young talent with rigorous training contributed immensely to thepromotionof mathematics. They had by the sixties established themselves as centres of excellence.Later in this page you will be able to read about ISI and some other institutions as well.TIFR made an unsuccessful effort to get Chern on its permanent faculty when he left China in 1949 –50: Chern opted to go to Chicago from where too he had an offer. In the fifties, a bid was made alsoto get Pjatetskii Shapiro but the necessary permission from the Soviet Union was not forthcoming.One of 20th century‟s greatest mathematicians, Carl Luidwig Siegel made four 3-month long visits toTIFR during the fifties and sixties. On each of these occasions he gave a course of lectures onadvanced topics and notes of these lectures were published by TIFR and are in demand to this day.During his stays in Mumbai, he invariably holidayed for a few days in Mahabaleshwar, a hill stationnear Mumbai.Laurent Schwarz too undertook several visits to TIFR and his lecture notes were again a wonderfulresource for graduate students. One year Schwarz deliberately planned his visit to Mumbai during themonsoon months (June – August), but sadly the rains failed that year. The monsoon when in fullswing is indeed a glorious experience on the West coast of India even if it does disrupt normal life nowand then. Schwarz however had better luck with his hobby of collecting butterflies.Over the last 5 decades TIFR has had a stream of many famous visitors Early (fifties and sixties of thelast century) visitors include J-P Kahane, B Malgrange, F Bruhat, J-L Koszul, J-L Lions, all French,Germans like Mass, Rademacher, Deuring and other Europeans, de Rham and Borel for example fromSwitzerland and Andreotti and Vesentini from Italy. All of them spent upward of two months inMumbai and gave lectures and the notes of these lectures again have enjoyed a great reputation.Long term visits of this kind by eminent mathematicians continues to this day, but there were alsovisitors at junior levels in large numbers at TIFR. TIFR has of course played host to many moreeminent names but for shorter durations like a week to one month: Atiyah, Deligne, Grauert, Gromov,Grothendieck, Hormander, Vaughn Jones, Ianaga, Ihara, Manin, Milnor, Mumford, Selberg, Serre,Smale, Stallings, Thom ……
- 34. A ninth century inscription from India - Bill CasselmanThere have been of course a large number of eminent visitors from America as well, some of whom(Bott, Browder, Stallings) spent extended periods oftime in the country. Two names stand out when itcomes to the amount oftime they have spent in this country: Armand Borel (he could also be classifiedas Swiss) and David Mumford. Borel‟s first visit was in 1961 and the visit fascinated him greatly. Hebecame a periodic visitor, Indian classical music being a big draw for him. He liked to be in Chennaiduring the “Music Season” – December – January. He was always accompanied by his wife Gaby.Mumford spent a whole academic year (with his family) in TIFR in 1967 – 68 and has been there forsome more extended visitsTIFR has also been regularly organising international meetings in diverse topics. The meetings startedin 1956 have been happening every four years. The topics are chosen to meet two criteria: theyshould be considered important by the mathematical community at large; secondlyloal mathematicians should have contributed to them significantly. Many of the visitors mentionedabove were participants in these meetings styled “International Colloquia”. Harish-ChandraIn the more recent past Robert Langlands spent an extended period of time in India. He was at TIFRmuch of this time but visited also Chennai, Bangalore, Delhi and Allahabad. Allahabad is where Harish-Chandra received his undergraduate education. There is now a research institute named after Harish-Chandra which too has hosted many eminent visitors from the west. Bill Casselman‟s interest in thehistory of mathematics has triggered some ofhis recent visits here (he had visited Mumbai for the firsttime some 30 years ago to participate in an international conference). His photograph of a ninthcentury inscription with the number 270 can be seen here. This is the oldest extant inscription where„0′ finds a place.
- 35. So India has been a happy (professional) destination for quite a fewmathematicians over the lasthundred years and more. As I said earlier, I have spoken mostly about visitors to one institution. Mylist is by no means exhaustive – I have left out the names of many of the visitors who went to Mumbaiin the late seventies and later.Institutions in the country other than TIFR and ISI too have played host to many mathematicians fromabroad even if they may not have done it on the scale of TIFR or ISI.The Indian mathematical community looks forward to welcoming their colleagues from abroad in largenumbers at the Congress. We do hope that many would spend some time beyond attending theCongress and will come back for longer visits later too. M S Raghunathan
- 36. Famous MathematiciansSrinivasa Ramanujan22 December 1887 -26 April 1920 (aged 32) Chetput, (Madras), India Ramanujan Number/Hardy-Ramanujan Number A common anecdote about Ramanujan relates to the number 1729. Hardy arrived at Ramanujans residence in a cab numbered 1729. Hardy commented that the number 1729
- 37. seemed to be uninteresting. Ramanujan is said to have stated on the spot that it wasactually a very interesting number mathematically, being the smallest natural numberrepresentable in two different ways as a sum of two cubes: 3 3 3 31729 = 1 + 12 = 9 + 1091 = 63 + (−5)3 = 43 + 33 (91 is divisor of 1729)Masahiko Fujiwara showed that 1729 is1 + 7 + 2 + 9 = 1919 × 91 = 1729Ramanujan Known for : Landau-Ramanujan Constant Mock theta functions Ramanujan Prime Ramanujans Sum Aryabhata 476 BC-550 BC India Aryabhata was the first astronomer to make an attempt at measuring the Earths circumference since Eratosthenes (circa 200 BC). Aryabhata accurately calculated the Earths circumference
- 38. as 24,835 miles, which was only 0.2% smallerthan the actual value of 24,902 miles. Thisapproximation remained the most accurate forover a thousand years.Statue of Aryabhata on the groundsof IUCAA,PuneBhaskara
- 39. Bhaskara was an Indianmathematician of the 7thcenturyHe was perhaps the first to conceive thedifferential coefficient and differential calculus Euclid
- 40. Born 300 BC Egypt "Father of Geometry"
- 41. One of the oldest surviving fragments ofEuclids Elements, found at Oxyrhynchus anddated to circa AD 100Pythagoras
- 42. Born 570 BC - 495 BC Samos Island GREEK Pythagoras has commonly been given credit fordiscovering the Pythagorean theorem, a theorem ingeometry that states that in a right-angled trianglethe area of the square on the hypotenuse (the sideopposite the right angle) is equal to the sum of theareas of the squares of the other two sides—that 2 2 2is, a + b = c .
- 43. For more details PPTClickHipparchos. born in Nicaea (now Iznik, Turkey)190 BC - 120BCHe developed trigonometry andconstructed trigonometric tables, and he hassolved several problems of spherical trigonometry.With his solar and lunar theories and histrigonometry, he may have been the first to developa reliable method to predict solar eclipses. Hisother reputed achievements include the discoveryof Earths precession, the compilation of the firstcomprehensive star catalog of the western world,and possibly the invention of the astrolabe, also ofthe armillary sphere, which he used during thecreation of much of the star catalogue
- 44. K.S.ChadrasekharanHe born in 21 November 1920 completed his highschool from Bapatla village in Guntur from AndhraPradesh. He completed his M.A. in mathematicsfrom thePresidency College, Chennai and a Ph.D.from the Department of Mathematics, University ofMadras in 1942. Fields Number TheoryThalesc.624 BC – c. 546 BC Greece
- 45. In mathematics, Thales used geometry to solveproblems such as calculating the height ofpyramids and the distance of ships from the shore.He is credited with the first use of deductivereasoning applied to geometry, by deriving fourcorollaries to Thales Theorem. As a result, he hasbeen hailed as the first true mathematician and isthe first known individual to whom a mathematicaldiscovery has been attributed. Also, Thales wasthe first person known to have studied electricity.John Wallis23 November 1616 - 28 October 1703 (aged 86)EnglandHe was an English mathematician who is givenpartial credit for the developmentofinfinitesimal calculus. Between 1643 and 1689he served aschief cryptographer for Parliament and, later,the royal court. He is also credited withintroducing the symbol ∞ for infinity. He
- 46. similarly used foran infinitesimal. Asteroid 31982 Johnwallis wasnamed after him.Rene Des Cartes • (31 March 1596 – 11 February 1650),He was a French philosopher,mathematician,scientist, and writer.Invented Analytic Geometry •Leonhard Paul Euler
- 47. 15 April 1707 - 18 September1783 (aged 76) Basel, SwitzerlandHe was apioneering Swissmathematician and physicist.He made important discoveries in field adiverseas infinitesimal calculus and graph theory. Healso introduced much of the modernmathematical terminology and notation,particularly for mathematical analysis, such asthe notion of a mathematical function. He isalso renowned for his work in mechanics, fluiddynamics, optics, and astronomy. •Johann CarlFriedrich Gauss
- 48. •(30 April 1777 – 23 February 1855) •“The Prince ofMathematicians”He wasa German mathematician and scientistwho contributed significantly to manyfields, including numbertheory, statistics, analysis,differentialgeometry, geodesy, geophysics, electrostatics, astronomy and optics.Brahmaguptha
- 49. Brahmagupta 598-668. The first IndianMathematician who framed theoperation of Zero. He separatedAlgebra and Arithmetic into twoseparate branches. He was the firstperson to calculate the length of theyear. He explained that the moon’sillumination can be computed by theangle it forms to the sunVarahamihira
- 50. Varahamihira 499-574 A.D. IndianAstronomer, Astrologer andMathematician, Calculated the distanceand positions of planets andresearched galaxies. Claimed thatplants and termites are the indicatorsof underground waterSome important trigonometric resultsattributed to Varahamihira 2 2Sin x + cos x = 1Dr.CalyampudiRadhakrishna Rao
- 51. He was born (10 September 1920) inHuvanna Hadagali, now in KarnatakaState, India. In 1941 he obtained an M.A. inmath from Andhra University in Waltair, AndhraPradesh, and in 1943 an M.A. in Statistics fromCalcutta University in Kolkata, West Bengal. Heworked at the Indian Statistical Institute untilmandatory retirement at age 60,Prof. Rao ranksamong the most notable statisticians of the lasthalf of the 20th century. He has received atleast 32 honorary doctorates from universitiesin 18 countries, and has been honored withmedals from countries worldwide, including theUnited States National Medal of Science. Hehas directly supervised more than 50 Ph.D.students who have in turn yielded more than350 Ph.D.’sSummary of research interestsRobust estimation in univariate andmultivariate linear models: Currentinvestigation includes a new type of estimationcalled Mu to cover estimates of parameters in
- 52. situations where M-estimation is not applicable,such as Oja’s median.Characterization of probability distributions: Ageneral solution of the integrated CauchyFunctional Equation is obtained.Matrix Algebra: Theory and applications ofantieigen values.Bootstrap: Bootstrap distributions underresampling schemes that ensure a certainnumber of distinct observations in each sampleare being investigated.-----------------------------------------------------------------------------------------------------------------------------------Shkuntaladevi -Guinness Book of world Records IndianwomenBorn on 4 November in 1939 at the city ofBangalore in Karnataka state India, ShakuntalaDevi is an outstanding calculating prodigy ofIndia.Shakuntala Devi is an outstandingcalculating prodigy of India. she solved themultiplication of two 13-digit numbers7,686,369,774,870 x 2,465,099,745,779 randomlypicked up by the computer department ofImperial College in London. And this, she did in28 seconds flat.This incident has been included on the 26thpage of the famous 1995 Guinness Book ofRecords.In the year 1977.Shakuntala Devi
- 53. obtained the 23rd root of the digit number 201 mentally.For Shekuntala Devi Books clicksjgfvsayfgliafgilygvjswgbvIFwbgsBHGKWGBKhwsbgliWSBG.KUws

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