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1. 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
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missing a key piece of middleware that is essential for the music application to run. Deployment estimation tool shows how
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Decide
Description: Real-time view of the installed base enables informed decisions to achieve maximum success.
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Perform
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May 21, 2011 - 12:00PM PT
Mobile Software: Driving
Innovation in the Multi-
Core Era
BY 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.
3. 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.
4. 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.
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. Trigonometry
From 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
7. 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
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 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
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 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
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, 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:
11. 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.
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 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
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, 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
14. 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
15. [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
16. 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
17. 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]La
w of
cosine
s
18. 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:
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d
it
]
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t
a
19. n
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e
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25. u
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i
:
List of Indian mathematicians
From 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 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 Jyeshtadeva, 1500–1610, Author
Madhava of Sangamagrama some elements of of Yuktibhāṣā, Madhava's Kerala school
Calculus hi Achyuta Pisharati, 1550–1621,
Parameshvara (1360–1455), discovered drk- Astronomer/mathematician, Madhava's Kerala school
ganita, a mode of astronomy based on Munishvara (17th century)
27. observations, Madhava's Kerala school Kamalakara (1657)
Nilakantha Somayaji,1444–1545 – Jagannatha Samrat (1730)
Mathematician and Astronomer, Madhava's 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
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media related
to: Mathematicians from
India
The Top Five All Time Mathematicians
School and Education - College
By: 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
30. 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.
31. 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
32. 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)
33. 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 ……
34. 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.
35. 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
36.
37. 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
38. 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:
3 3 3 3
1729 = 1 + 12 = 9 + 10
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 India
Aryabhata 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
39. 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
40. Bhaskara was an Indian
mathematician of the 7th
century
He was perhaps the first to conceive the
differential coefficient and differential calculus
Euclid
42. One of the oldest surviving fragments of
Euclid's Elements, found at Oxyrhynchus and
dated to circa AD 100
Pythagoras
43. 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
2 2 2
is, a + b = c .
44. 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
45. K.S.Chadrasekhara
n
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 Greece
46. In 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)
England
He 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
47. 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
48. 15 April 1707 - 18 September
1783 (aged 76) Basel, Switzerland
He 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
49. •
(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, electr
ostatics, astronomy and optics.
Brahmaguptha
50. 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
51. 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 water
Some important trigonometric results
attributed to Varahamihira
2 2
Sin x + cos x = 1
Dr.Calyampudi
Radhakrishna Rao
52. 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 interests
Robust estimation in univariate and
multivariate linear models: Current
investigation includes a new type of estimation
called Mu to cover estimates of parameters in
53. 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.
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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
54. obtained the 23rd root of the digit number '201'
mentally.For Shekuntala Devi Books click
sjgfvsayfgliafgilygvjswgbvIFwbgsBHGKWGBKhwsbgliWSBG.KUws
