Home > Products + Services > What is MSM

Mobile Software Management encompasses a set of technologies and business processes that enable the management

of 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 products

for 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 any

device, any platform, anytime.

Red Bend offers the only holistic solution for gathering information from the device, analyzing that data, using it to formulate

decisions and then performing management actions on the device.

Inform

Description: On-device client software collects data from the device including software inventory, usage analytics, device

configuration and settings, and network and device diagnostics.

Real-World Example: vRapid Mobile® Software Management Client takes an inventory of what software and applications

consumers have installed and are using on their mobile phones. vDirect Mobile® Device Management Client transmits

that data over the air using OMA DM standards.Analyze

Description: 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 yet

downloaded the new music application that was launched two months ago. Analysis shows that most consumers are

missing a key piece of middleware that is essential for the music application to run. Deployment estimation tool shows how

many devices can be updated over the air, and how much bandwidth will be required for the full operation.Decide

Description: Real-time view of the installed base enables informed decisions to achieve maximum success.

Real-World Example: Service provider decides to offer a software update to bring the targeted devices up to the same

baseline.

Perform

Description: 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 music

application in a single update package, so that the service provider can widen its addressable market for the music

application. The software update package is sent over the air to the vDirect Mobile device management client. vRapid

Mobile FOTA software performs the software update quickly, efficiently and reliably. 

May 21, 2011 - 12:00PM PT

Mobile Software: Driving Innovation in the Multi-Core   Era

B Y  Rob Chandhok 6 Comments

Mobile hardware is progressing at a blistering pace, but to deliver the type of user

experiences enabled by awesome hardware software must keep pace. This goes beyond the

need for innovations in OSes and applications, to the underlying software that ties

everything together.

Mobile hardware is progressing at a blistering pace. Displays continue to increase in

size, color quality and resolution, while advancements such as glasses-free 3-D offer

the promise of novel user experiences. Processors are adding cores and clock speed

faster than ever before, and 4G radios have brought broadband data speeds to mobile

devices. These unprecedented hardware innovations have set the stage for a brave

new world of mobile computing in which nearly anything is possible on hand-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 keep pace – otherwise we will fall painfully short of capitalizing on the

opportunities presented by these hardware achievements. This goes beyond the need

for innovations in OSes and applications, to the underlying software that ties

everything together. It’s the next great challenge faced by the mobile industry.

Software as the Connective Tissue of the Phone

When it comes to mobile software, the importance of operating systems and

applications is well understood. The battle for smartphone OS market share evokes a

feverish MLB pennant race, and the fact that we’re all hopelessly addicted to Angry

Birds proves that mobile apps have thoroughly permeated the mainstream.

Less understood, however, is the importance of the underlying software layer; the

connective tissue that ties hardware to software, such as optimizations between OS

and chipset, performance advancements 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 the best possible mobile

experiences, hardware and software cannot be viewed separately. They are attached at

the hip, and integrating them to work in perfect unison is the key to driving mobile

innovation forward.

Immediate benefits of intelligent integration include better graphical frame rates in

games, faster web page downloads and smoother rendering and scrolling. These are

just a sampling of the user experience improvements that will help mobile devices

keep up with ever-increasing consumer expectations.

Innovating for the Future of the Mobile Web

All too often, the primary focus is on what the consumer wants today. It is our job to

anticipate what the 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 apps running 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, HTML5 browser core, and JavaScript

engine. While powerful processors will strongly influence robust Web experiences, the

mobile software layer is significantly impacting how we get the most out of mobile

hardware 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 to fruition until mobile devices support the specification fully, from web and

enterprise apps to entertainment 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, more efficiently and with greater capabilities when the

hardware and software have been tightly integrated.

The biggest remaining hurdle is ensuring that the same array of device capabilities,

such as camera access, is available to Web apps as their native counterparts. To this

end, companies like Qualcomm are enabling a rich set of device APIs within the

browser so that Web apps have that same detailed control and usage of the device’s

hardware.

Collaboration Is Key

The mobile industry is built on partnerships within the diverse lines of business that

make up the ecosystem 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 hand across the mobile ecosystem to

deliver web experiences that go beyond what we ever experienced on a PC. All stand

to benefit greatly by software’s ongoing impact on mobile, and efficient collaboration

will expedite that process. Ultimately, intelligent and tight OS integration within the

chip provides time to market advantages for OEMs who will see their devices running

faster, smoother and more efficiently.

Enhancing mobile software is not a trickle down process. It starts with the seamless

hardware integration and ends with developers bringing the experience to life. If we

are serious about a future where mobile phones are responsible for tasks currently

held by computers we need to embrace the role of software in overall mobile

performance and continue strongly supporting the software developers that are driving

innovation.

Rob Chandhok is president of Qualcomm Internet Services and helps drive software

strategy for Qualcomm’s client and server platforms. He and other mobile industry

thought leaders will be discussing these topics and more June 1-2 at Uplinq 2011 in

San Diego. His Twitter handle is@robchandhok

There are a massive range of mobile software for displaying OpenStreetMap or otherwise making use of

our geodata on phones and other mobile devices. Increasingly apps are allowing contribution to

OpenStreetMap too. Because OpenStreetMap data is free and open for anyone and everyone, mobile

software is being developed by a wide range of people and companies, and OpenStreetMap data can be

accessed on almost any mobile platform.

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)

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 those angles.

Trigonometry

History

Usage

Functions

Generalized

Inverse functions

Further reading

Reference

Identities

Exact constants

Trigonometric tables

Laws and theorems

Law of sines

Law of cosines

Law of tangents

Law of cotangents

Pythagorean theorem

Calculus

Trigonometric substitution

Integrals of functions

Derivatives of functions

Integrals of inverse functions

V

T

E

Trigonometry (from Greek trigōnon "triangle" + metron "measure"[1]) is a branch of mathematics that

studies triangles and the relationships between their sides and the angles between these sides. Trigonometry

defines the trigonometric functions, which describe those relationships and have applicability to cyclical

phenomena, such as waves. The field evolved during the third century BC as a branch of geometry used

extensively 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 such

as Fourier analysis and the wave equation, which are in turn essential to many branches of science and

technology. 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 negative

curvature is part of Hyperbolic geometry.

Contents

[hide]

1 History

2 Overview

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 Euler's formula

7 See also

8 References

o 8.1 Bibliography

9 External links

[edit]History

Main article: History of trigonometry

The first trigonometric table was apparently compiled by Hipparchus, who is now consequently known as "the father of

trigonometry."[3]

Sumerian astronomers introduced angle measure, using a division of circles into 360 degrees.[4] They and their

successors the Babylonians studied the ratios of the sides of similar triangles and discovered some properties

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 an

ordered science.[6]

Classical Greek mathematicians (such as Euclid and Archimedes) studied the properties of chords and

inscribed angles in circles, and proved theorems that are equivalent to modern trigonometric formulae,

although they presented them geometrically rather than algebraically. Claudius Ptolemy expanded

uponHipparchus' Chords in a Circle in his Almagest.[7] The modern sine function was first defined in the Surya

Siddhanta, and its properties were further documented by the 5th century Indian mathematician and

astronomer Aryabhata.[8] These Greek and Indian works were translated and expanded by medieval Islamic

mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had

tabulated their values, and were applying them to problems in spherical geometry.[citation needed] At about the same

time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study

for them. Knowledge of trigonometric functions and methods reached Europe via Latin translations of the works

of Persian and Arabic astronomers such as Al Battani and Nasir al-Din al-Tusi.[9] One of the earliest works on

trigonometry by a European mathematician is De Triangulis by the 15th

century German mathematician Regiomontanus. Trigonometry was still so little known in 16th century Europe

that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explaining its basic

concepts.

Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry

grew to be a major branch of mathematics.[10] Bartholomaeus Pitiscus  was the first to use the word, publishing

his Trigonometria in 1595.[11] Gemma Frisius described for the first time the method oftriangulation still used

today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The

works of James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the

development of trigonometric series.[12] Also in the 18th century, Brook Taylor defined the general Taylor series.

[13]

[edit]Overview

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, because

the 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 the

angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the

triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the

following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the

accompanying 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:

The inverse functions are called the arcsine, arccosine, and arctangent,

respectively. There are arithmetic relations between these functions, which are

known as trigonometric identities. The cosine, cotangent, and cosecant are so

named because they are respectively the sine, tangent, and secant of the

complementary angle abbreviated to "co-".

With these functions one can answer virtually all questions about arbitrary triangles

by using the law of sines and the law of cosines. These laws can be used to

compute the remaining angles and sides of any triangle as soon as two sides and

their included angle or two angles and a side or three sides are known. These laws

are useful in all branches of geometry, since every polygon may be described as a

finite combination of triangles.

[edit]Extending the definitions

Fig. 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 and

π/2 radians) only. Using the unit circle, one can extend them to all positive and

negative arguments (see trigonometric function). The trigonometric functions

are periodic, with a period of 360 degrees or 2π radians. That means their values

repeat at those intervals. The tangent and cotangent functions also have a shorter

period, of 180 degrees or π radians.

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 Euler's and De Moivre's 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 ÷ Hypotenuse

Cosine = Adjacent ÷ Hypotenuse

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, Can't 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

Sextants are used to measure the angle of the sun or stars with respect

to the horizon. Using trigonometry and amarine chronometer, the position

of the ship can be determined from such measurements.

Main article: Uses of trigonometry

There are an enormous number of uses of trigonometry and

trigonometric functions. For instance, the technique

of triangulation is used in astronomy to measure the distance to

nearby stars, in geography to measure distances between

landmarks, and in satellite navigation systems. The sine and

cosine functions are fundamental to the theory of periodic

functions such as those that describe sound and light waves.

Fields that use trigonometry or trigonometric functions

include astronomy (especially for locating apparent positions of

celestial objects, in which spherical trigonometry is essential) and

hence navigation (on the oceans, in aircraft, and in space), music

theory, acoustics, optics, analysis of financial

markets,electronics, probability theory, statistics, biology, medical

imaging (CAT

scans and ultrasound), pharmacy, chemistry, number theory (and

hencecryptology), seismology, meteorology, oceanography,

many physical sciences,

land surveying and geodesy, architecture, phonetics, economics, 

electrical engineering, mechanical engineering, civil

engineering, computer

graphics, cartography, crystallography and game development.

[edit]Standard identities

Identities are those equations that hold true for any value.

[edit]Angle transformation formulas

[edit]Common formulas

Triangle with

sides a,b,c and

respectively opposite

angles A,B,C

Certain equations

involving trigonometric

functions are true for all

angles and are known

as trigonometric

identities. Some

identities equate an

expression to a different

expression involving the

same angles. These are

listed in List of

trigonometric identities.

Triangle identities that

relate the sides and

angles of a given

triangle are listed below.

In the following

identities, A, B and C ar

e the angles of a

triangle

and a, b and c are the

lengths of sides of the

triangle opposite the

respective angles.

[edit]Law of sines

The law of sines (also

known as the "sine

rule") for an arbitrary

triangle 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

triangle. Given

two sides and

the angle

between the

sides, the area

of the triangle

is:

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]Law of cosines

The law

of

cosines (

known as

the cosine

formula, or

the "cos

rule") is an

extension

of

the Pythag

orean

theorem to

arbitrary

triangles:

or

equiv

alentl

y:

[

e

d

it

]

Law of tangents

T

h

e 

l

a

w

o

f

t

a

n

g

e

n

t

s 

(

a

ls

o

k

n

o

w

n

a

s

t

h

e

"t

a

n

r

u

l

e

")

:

[e

dit

]Euler's formula

Eul

er'

s

for

mu

la,

whi

ch

sta

tes

tha

t 

,

pro

du

ce

s

the

foll

owi

ng 

an

aly

tic

al i

de

ntit

ies

for

sin

e,

co

sin

e,

an

d

tan

ge

nt

in

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nd

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im

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uni

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)

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 influenced scientists 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 can be used in the future also.

Contents

[hide]

1 Classical

2 Medieval to Mughal

period

3 Born in 1800s

4 Born in 1900s

[edit]Classical

Post-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 period

13th century to 1800.13th century, Logician, mithila school

Narayana Pandit

Madhava of Sangamagrama some elements of

Calculus hi

Parameshvara (1360–1455), discovered drk-

ganita, a mode of astronomy based on

observations, Madhava's Kerala school

Nilakantha Somayaji ,1444–1545 –

Mathematician and Astronomer, Madhava's Kerala

school

Jyeshtadeva , 1500–1610, Author

of Yuktibhā ṣ ā , Madhava's Kerala school

Achyuta Pisharati , 1550–1621,

Astronomer/mathematician, Madhava's Kerala

school

Munishvara (17th century)

Kamalakara (1657)

Jagannatha Samrat (1730)

Srijan Gupta of Delhi (1997)

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)

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

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media related

to: Mathematicians from India

The Top Five All Time Mathematicians

School and Education - CollegeBy:  AmarSingh      15-Oct-2010

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Aryabhata

Aryabhatiya (499 A.D.), he made the fundamental advance in finding the lengths of chords of circles,

by using the half chord rather than the full chord method used by Greeks. He gave the value of as

3.1416, claiming, for the first time, that it was an approximation. He also gave methods for extracting

square roots, summing arithmetic series, solving indeterminate equations of the type ax -by = c, and

also gave what later came to be known as the table of Sines.

Brahmagupta

He was born in 598 A.D. His main work was Brahmasphutasiddhanta, which was a corrected version of

old astronomical treatise Brahmasiddhanta. This work was later translated into Arabic as Sind Hind.He

gave the formula for the area of a cyclic quadrilateral . He was the first mathematician to treat algebra

and arithmetic as two different branches of mathematics. He gave the solution of the indeterminate

equation Nx²+1 = y².

Bhaskara

Bhaskaracharaya 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 any number and infinity is also infinity. He gave an

example of what is now called "differential coefficient" and the basic idea of what is now called "Rolle's

theorem".

Srinivasa Aaiyangar Ramanujan

He was born in a poor family at Erode in Tamil Nadu on December 22, 1887. His father, K. Srinivasa

Iyengar worked as a clerk in a sari shop and his mother, Komalatammal was a housewife and also sang

at a local temple.On 1 October 1892, Ramanujan was enrolled at the local school. At the Kangayan

Primary School, Ramanujan performed well. Just before the age of 10, in November 1897, he passed

his primary examinations in English, Tamil, geography and arithmetic. With his scores, he finished first

in the distric .Largely self taught, he feasted on Loney's Trigonometry at the age of 13, and at the age

of 15, his senior friends gave him Synopsis of Elementary Results in Pure and Applied Mathematics by

George Carr.A few months earlier, he had sent a letter to great mathematician G.H. Hardy, in which he

mentioned 120 theorems and formulae. Ramanujan published 21 papers, some in collaboration with

Hardy. He died in Madras on April 26, 1920.

D.R. Kaprekar

D.R. Kaprekar was born in 1905 in India.Kaprekar received his secondary school education in Thane

and studied at Fergusson College in Pune.In 1963, Kaprekar defined the property which has come to be

known as self numbers. For example, 21 is not a self number, since it can be generated from 15: 15 +

1 + 5 = 21. But 20 is a self number, since it cannot be generated from any other integer. Today his

name is well-known and many other mathematicians have pursued the study of the properties he

discovered.

India and the World   of   Mathematics

India has a long tradition of pursuit of mathematics. Geometry had an important (practical)role to play

in Vedic culture (circa 1000 BCE) and Pythagoras’s theorem was known to mathematicians in India by

the eighth century before Christ. Astronomy went hand in hand with mathematics and not surprisingly

trigonometry was an area where India made considerable progress. The place value system with the

useof 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

full of high praise for “Hindu mathematics” even while he is highly critical of several other

aspects of the 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 of the 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.

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 Pythagoras's theorem was known

tomathematicians in India by the eighth century before Christ. Astronomy went hand in hand with

mathematics and not surprisingly trigonometry was an area where India made considerable progress. The

place 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 period

After a dormant period of some two or three centuries, in the mid 19th century when the British

colonial administration set up universities along European lines, there was a a resumption of interest in

mathematics in India – and it was now mathematics as pursued in the West. Many British academics

were involved in the promotion of mathematical activity in this country in those early days and not

surprisingly, the mathematical areas pursued in Britain were the ones Indians took to. In early

twentieth century however Indian mathematicians were also becoming aware ofdevelopoments in

mathematics elsewhere in Europe. Also Indianmathematicians were enrolling themselves in European

(mostly British) universities and were geeting trained there.

In April 1907, a civil servant, V Ramaswamy Iyer who was also a mathematics enthusiast initiated the

formation of the first mathematical (infact the first scientific) society in the Indian subcontinent. The

society called then The Indian Mathematics Club had at its inception only 20 members located in

different cities (mainly Chennai, Mumbai and Pune); many among them, personal

friends of Ramaswamy Iyer, were not professionally involved with mathematics. The society started

subscribing 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 today

under that name and from that year has been publishing a journal, the “Journal of the Indian

Mathematical Society”. The first issue of the journal (published in 1910) has an article by one Seshu

Iyer on Green’s Functions where he talks about Poincare’s work on the subject; he says among other

things that while the British treatment of the subject was largely from the point of view of physics the

French, mainly Poincare, developed it from a purely mathematical standpoint. This is an indication that

Indian 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 by

Selberg on (what we now know as) the Selberg Trace formula. The second is Weil’s paper on the

classification ofclassical semisimple groups over an arbitrary field of characteristic 0. There must be

others.

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 period

was of course Srinivasa Ramanujan. And the decisive factor that was responsible for the

flowering of that natural genius was the intervention of the towering Cambridge figure G H Hardy. The

fascinating story of Ramanujan is told admirably in Robert Kanigel’s biography titled “The man who

knew 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 a

Ramanujan Centenary Celebration in Mumbai (Bombay).

Hardy never visited India but his mathematical influence in this country was pervasive. There were

other great mathematicians who had also some influence, but Hardy’s was preeminent in the first

quarter and more of the 20th century. There were of course, as already mentioned, many visitors

mainly from Britain who spent considerable time in Indian universities. In the thirties and thereafter

one finds quite a few truly eminent names among the mathematicians visiting India for fairly long

periods. 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 in

mathematics. Research of good quality started emerging out of Universities of Calcutta and Madras

and a little later some others, notably Allahabad and Benares emerged as good

centres of mathematical research. Here are some names that were recognised internationally in their

time 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, T

Vijayaraghavan, S Minakshisundaram, S Chowla, K Chandrasekharan, D.D. Kosambi (see this

article for more details).

Andre Weil spent 2 years (1930 – 32) at Aligarh Muslim University (Aligarh is a mid-sized town some

100 kms south of Delhi) as Professor and Chair of the mathematics department. He was of course (at

24) yet to achieve the kind of fame and standing that he did a little later in his career. In his

autobiography (Apprenticeship of a mathematician) he speaks fondly of his days there and of the

many Indian friends he made during that sojourn. The English text of a talk about Indian

mathematics of the thirties given in Moscow (in 1936) can be found in his collected works (springer

Verlag 1980). The talk ends with “Nevertheless the intellectual potentialities of the Indian nation are

unlimited, and not many years would perhaps be needed before India can take a worthy place in world

mathematics” Weil was again in India some 35 years after that to take part in international conference

in Algebraic Geometry in Mumbai (then Bombay). He went around the country renewing his old

acquaintances.

Another frenchman who had a tremendous influence on mathematics in India was Rev Fr Racine. He

came to India sent here by the Jesuit mission in 1934 to teach mathematics in St Joseph’s College in

Thiruchi, 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 occupied

centre-stage in Europe rather than the traditional largely Cambridge inspired subjects. Many of his

students were to become distinguished mathematicians. Fr Racine stayed on in India till his death in

1976 paying only infrequent short visits to France. Another big name who visited India several times

during the late forties and later was M H Stone. On these visits he invariably spent substantial

amounts of time in Chennai (then Madras). He spotted quite a few talented young men and helped

arrange for them to visit great institutions like the Institute in Princeton. Stone appears to have

enjoyed 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 the first

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)

Tata Institute of Fundamental Research (TIFR)

Two institutions, The Tata Institute of Fundamental Research (TIFR) in Mumbai (Bombay) and the

Indian Statistical Institute (ISI) in Kolkata (Calcutta) have played a leading role in the

promotion ofmathematics in the country in the post-independence period. They both embarked on

running graduate schools along the lines ofAmerican Universities and combining careful

recruitment of young talent with rigorous training contributed immensely to the

promotionof 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 also

to 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 to

TIFR during the fifties and sixties. On each of these occasions he gave a course of lectures on

advanced 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 station

near Mumbai.

Laurent Schwarz too undertook several visits to TIFR and his lecture notes were again a wonderful

resource for graduate students. One year Schwarz deliberately planned his visit to Mumbai during the

monsoon months (June – August), but sadly the rains failed that year. The monsoon when in full swing

is indeed a glorious experience on the West coast of India even if it does disrupt normal life now and

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 the

last 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 from

Switzerland and Andreotti and Vesentini from Italy. All of them spent upward of two months in Mumbai

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 also visitors at

junior levels in large numbers at TIFR. TIFR has of course played host to many more eminent 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 ……

A ninth century inscription from India - Bill Casselman

There 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 it

comes to the amount oftime they have spent in this country: Armand Borel (he could also be classified

as Swiss) and David Mumford. Borel’s first visit was in 1961 and the visit fascinated him greatly. He

became a periodic visitor, Indian classical music being a big draw for him. He liked to be in Chennai

during 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 for

some more extended visits

TIFR has also been regularly organising international meetings in diverse topics. The meetings started

in 1956 have been happening every four years. The topics are chosen to meet two criteria: they should

be considered important by the mathematical community at large; secondly

loal mathematicians should have contributed to them significantly. Many of the visitors mentioned

above were participants in these meetings styled “International Colloquia”.

Harish-Chandra

In the more recent past Robert Langlands spent an extended period of time in India. He was at TIFR

much 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 the

history of mathematics has triggered some ofhis recent visits here (he had visited Mumbai for the first

time some 30 years ago to participate in an international conference). His photograph of a ninth

century inscription with the number 270 can be seen here. This is the oldest extant inscription where

‘0′ finds a place.

So India has been a happy (professional) destination for quite a fewmathematicians over the last

hundred years and more. As I said earlier, I have spoken mostly about visitors to one institution. My list

is by no means exhaustive – I have left out the names of many of the visitors who went to Mumbai in

the late seventies and later.

Institutions in the country other than TIFR and ISI too have played host to many mathematicians from

abroad 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 large

numbers at the Congress. We do hope that many would spend some time beyond attending the

Congress and will come back for longer visits later too.

M S Raghunathan

Famous Mathematicians

Srinivasa Ramanujan

22 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 Ramanujan's residence in a cab numbered 1729. Hardy commented that the number 1729

seemed to be uninteresting. Ramanujan is said to have stated on the spot that it was actually a very interesting number mathematically, being the smallest natural number representable in two different ways as a sum of two cubes:

1729 = 13 + 123 = 93 + 103

91 = 63 + (−5)3 = 43 + 33  (91 is divisor of 1729)

Masahiko Fujiwara showed that 1729 is

1 + 7 + 2 + 9 = 19

19 × 91 = 1729

Ramanujan Known for : Landau-Ramanujan Constant

                                            Mock theta functions

                                 Ramanujan Prime

                                 Ramanujan's Sum

Aryabhata

476 BC-550 BC IndiaAryabhata was the first astronomer to make an attempt at measuring the Earth's circumference since Eratosthenes (circa 200 BC). Aryabhata accurately calculated the Earth's circumference

as 24,835 miles, which was only 0.2% smaller than the actual value of 24,902 miles. This approximation remained the most accurate for over a thousand years.

Statue of Aryabhata on the grounds of IUCAA,Pune

Bhaskara

Bhaskara was an Indian mathematician of the 7th century He was perhaps the first to conceive the differential coefficient and differential calculus

      Euclid

Born 300 BC Egypt "Father of Geometry"

One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100

Pythagoras

Born 570 BC - 495 BC Samos Island GREEK     Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides—that is, a2 + b2 = c2.

For more details PPT Click 

Hipparchos.

 born in Nicaea (now Iznik, Turkey)190 BC - 120 BC He developed trigonometry and constructed trigonometric tables, and he has solved several problems of spherical trigonometry. With his solar and lunar theories and his trigonometry, he may have been the first to develop a reliable method to predict solar eclipses. His other reputed achievements include the discovery of Earth's precession, the compilation of the first comprehensive star catalog of the western world, and possibly the invention of the astrolabe, also of the armillary sphere, which he used during the creation of much of the star catalogue

K.S.Chadrasekharan

He born in 21 November 1920 completed his high school from Bapatla village in Guntur from Andhra Pradesh. He completed his M.A. in mathematics from thePresidency College, Chennai and a Ph.D. from the Department of Mathematics, University of Madras in 1942. Fields Number Theory

Thales

 c.624 BC – c. 546 BC GreeceIn mathematics, Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive

reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. As a result, he has been hailed as the first true mathematician and is the first known individual to whom a mathematical discovery has been attributed. Also, Thales was the first person known to have studied electricity.

John Wallis

23 November 1616 - 28 October 1703 (aged 86) EnglandHe was an English mathematician who is given partial credit for the development ofinfinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal court. He is also credited with

introducing the symbol ∞ for infinity. He

similarly used   for an infinitesimal. Asteroid 31982 Johnwallis was named 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

15 April 1707 - 18 September 1783 (aged 76) Basel, SwitzerlandHe was a pioneering Swissmathematician and physicist. He made important discoveries in field adiverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy.

•Johann Carl Friedrich Gauss

•(30 April 1777 – 23 February 1855)

•

“The Prince of Mathematicians”

He was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis,differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.

Brahmaguptha

Brahmagupta 598-668. The first Indian Mathematician who framed the operation of Zero. He separated Algebra and Arithmetic into two separate branches. He was the first person to calculate the length of the year. He explained that the moon’s illumination can be computed by the angle it forms to the sun

Varahamihira

Varahamihira 499-574 A.D. Indian Astronomer, Astrologer and Mathematician, Calculated the distance and positions of planets and researched galaxies. Claimed that plants and termites are the indicators of underground waterSome important trigonometric results attributed to Varahamihira  

                                                                    2             2Sin  x + cos x = 1

Dr.Calyampudi Radhakrishna Rao

He was born (10 September 1920) in Huvanna Hadagali, now in Karnataka State, India. In 1941 he obtained an M.A. in

math from Andhra University in Waltair, Andhra Pradesh, and in 1943 an M.A. in Statistics from Calcutta University in Kolkata, West Bengal. He worked at the Indian Statistical Institute until mandatory retirement at age 60,Prof. Rao ranks among the most notable statisticians of the last half of the 20th century. He has received at least 32 honorary doctorates from universities in 18 countries, and has been honored with medals from countries worldwide, including the United States National Medal of Science. He has directly supervised more than 50 Ph.D. students who have in turn yielded more than 350 Ph.D.’s

Summary of research interestsRobust estimation in univariate and multivariate linear models: Current investigation includes a new type of estimation called Mu to cover estimates of parameters in

situations where M-estimation is not applicable, such as Oja’s median.Characterization of probability distributions: A general solution of the integrated Cauchy Functional Equation is obtained.Matrix Algebra: Theory and applications of antieigen values.Bootstrap: Bootstrap distributions under resampling schemes that ensure a certain number of distinct observations in each sample are being investigated.-----------------------------------------------------------------------------------------------------------------------------------

Shkuntaladevi -Guinness Book of world Records Indian women

Born on 4 November in 1939 at the city of Bangalore in Karnataka state India, Shakuntala Devi is an outstanding calculating prodigy of India.Shakuntala Devi is an outstanding calculating prodigy of India. she  solved the multiplication of two 13-digit numbers 7,686,369,774,870 x 2,465,099,745,779 randomly picked up by the computer department of Imperial College in London. And this, she did in 28 seconds flat.This incident has been included on the 26th page of the famous 1995 Guinness Book of Records.In the year 1977.Shakuntala Devi obtained the 23rd root of the digit number '201' mentally.For Shekuntala Devi Books click

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