Prof.(Dr.) Z.K.Ansari,

Professor and Head,

Department of Applied Mathematics, JSS Academy of Technical Education,

NOIDA

Introduction

Mathematics, Science, Engineering, and Technology Education are central to the education process and have a profound effect on our Nation’s economic competitiveness and on the quality of life of its citizens.

Sciences, Engineering and Technology making it necessary to accept the challenge of promoting research in areas related to both theoretical and applied mathematics.

History

Mathematics is often presented as a large

collection of disparate facts to be

absorbed and used only with very specific

applications in mind.

Babylonians and Egyptians began using

arithmetic, algebra and geometry for

taxation and other financial calculations,

for building and construction, and for

astronomy.

Continue….. Pythagorean theorem provided great

intellectual interest to Babylonian scholars

of 2000 BC, who hunted for extraordinary

large multidigit numbers satisfying the

famous relation a2 + b2 = c2.

Ancient Chinese scholars took the joy to

create the first ‘Magic Square’. Between 600 BC and 300 BC the Ancient

Greeks began a systematic study of

mathematics in its own right with Greek

mathematics.

Cont…..Indian Mathematics

Vedas-1500BC and 800BC.

600BC ,Mathematics driven by the needs of the religion

and its demands for careful astronomical observations.

500 BC, mathematician Aryabhatta, developed the

theory of Trigonometry to help the astronomical

calculations.

He developed the method to calculate the square root.

Evaluated the value of π to a high degree of accuracy .

Two research centers established during this period.

One at Northeast India ( Kusumapura),headed by

Aryabhatta, and other at Ujjain headed the

mathematician Varahamihira.

Continue…

In the January 2006 issue of the Bulletin of the American Mathematical Society Mikhail B. Sevryuk mentioned that “The number of papers and books included in the Mathematical Reviews database since 1940 is now more than 1.9 million, and more than 75 thousand items are added to the database each year.

Prestigious award in mathematics is the Field Madel, established in 1936 and is often considered as equivalent to Nobel Prize.

Magic Square

Inspiration, pure and applied

mathematics

Today, all sciences suggest problems

studied by mathematicians, and many

problems arise within mathematics itself.

String theory, a still-developing scientific

theory which attempts to unify the four

fundamental forces of nature, continues to

inspire new mathematics.

Pure mathematics topics often turn out to

have applications, e.g. number theory in

cryptography.

Continue…. Several areas of applied mathematics

have merged with related traditions

outside of mathematics and become

disciplines in their own right, including

statistics ,operation research and

computer science.

Hundreds of specialized areas in

mathematics and latest Mathematics

Subject Classification runs to 46 pages.

String Theory Fundamental interactions, also called fundamental forces or

interactive forces, are modeled in fundamental physics as

patterns of relations in physical systems, evolving over time,

that appear not reducible to relations among entities more

basic. Four fundamental interactions are conventionally

recognized: gravitational, electromagnetic, and strong nuclear

and weak nuclear. Everyday phenomena of human experience

are mediated via gravitation and electromagnetism. The strong

interaction, synthesizing chemical elements via nuclear fusion

within stars, holds together the atom’s nucleus, and is released during an atomic bomb’s detonation. The weak interaction is involved in radioactive decay. (Speculations of a fifth force—perhaps an added gravitational effect—remain widely

disputed.)

Cryptography Cryptography is the practice and study of techniques

for secure communication in the presence of third parties. More generally, it is about constructing and analyzing protocols that overcome the influence of adversaries and which are related to various aspects in information security such as data confidentiality, data integrity, authentication, and non- repudiation . Modern cryptography intersects the disciplines of mathematics, computer science, and electrical engineering. Applications of cryptography include ATM cards, computer passwords, and electronic commerce.

Areas of Mathematics

Mathematics divided study of quantity,

structure, space, and change (i.e.

arithmetic ,algebra ,geometry and

analysis).

Subdivisions from the heart of

mathematics to other fields: to logic, to

set theory(foundation) to the empirical

mathematics of the various sciences

(applied mathematics), and more recently

to the rigorous study of uncertainty.

Quantity (Arithmetic)

Study of quantity starts with numbers ,

first the familiar natural numbers and

integers and arithmetical operations on

them. The deeper properties of integers

are studied in theory. The twin prime

conjecture and Goldbach’s conjecture are two unsolved problems in number theory.

Goldbach's conjecture

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and in all of mathematics. It states:

Every even integer greater than 2 can be expressed as the sum of two primes.

The conjecture has been shown to hold up through 4 × 1018 and is generally assumed to be true, but remains unproven despite considerable effort.

Goldbach's conjecture

Structure (Algebra) Mathematical objects, such as sets of

numbers and functions, exhibit internal

structure as a consequence of operation

that are defined on the set, e.g. number

theory studies properties of the set of

integers that can be expressed in terms of

arithmetic operation.

Groups ,rings ,fields and other abstract

systems; together such studies constitute

the domain of abstract algebra.

Cont….. Linear algebra, which is the general study

of vector spaces, and can be used to

model points in space. This is one

example of the phenomenon that the

originally unrelated areas of geometry

and algebra have very strong interactions

in modern mathematics.

Combinatorics Number Theory Group Theory

Combinatories

Graph Theory Order Theory Algebra

Space (Geometry)

The study of space originates with

geometry – in particular, Euclidean

geometry.

Differential geometry are the concepts of

fiber bundles and calculus on manifolds.

Quantity and space both play a role in

analytical geometry ,differential

geometry, and algebraic geometry.

Cont……. Pythagorean Theorem to include higher-

dimensional geometry, Non-Euclidean

geometries play a central role in general

relativity and topology.

Topology in all its many ramifications

may have been the greatest growth area in

20th century mathematics.

Geometry, Trigonometry, Differential Geometry ,Topology, Fractal

Geometry, Measure Theory

Topology

Topology is the mathematical study of shapes and spaces. It is an area of mathematics concerned with the properties of space that are preserved under continuous deformations including stretching and bending, but not tearing or gluing. This includes such properties as connectedness, continuity and boundary.

Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation. By the middle of the 20th century, topology had become a major branch of mathematics.

Topology

Cont…. Topology has many subfields.

General topology establishes the foundational aspects of topology and

investigates properties of topological spaces and investigates concepts

inherent to topological spaces. It includes point-set topology, which is the

foundational topology used in all other branches (including topics like

compactness and connectedness).

Algebraic topology tries to measure degrees of connectivity using algebraic

constructs such as homology and homotopy groups..

Differential topology is the field dealing with differentiable functions on

differentiable manifolds. It is closely related to differential geometry and

together they make up the geometric theory of differentiable manifolds.

Geometric topology primarily studies manifolds and their embeddings

(placements) in other manifolds. A particularly active area is low

dimensional topology, which studies manifolds of four or fewer dimensions.

This includes knot theory, the study of mathematical knots .

Change (Analysis)

Understanding and describing change is a

common theme in the natural sciences and

calculus was developed as a powerful tool to

investigate it.

Study of real numbers and functions of a

real variable is known as real analysis, with

complex analysis the equivalent field for the

complex numbers.

Relationships between a quantity and its rate

of change, studied as differential equations.

Calculus, Vector Calculus, Diff.Equation, Dynamical Systems, Chaos

theory, Complex Analysis

Applied Mathematics

Applied Mathematics concerns itself with

mathematical methods that are typically

used in science, engineering, business,

and industry.

Applied mathematics focuses on the

"formulation, study, and use of

mathematical models" in science

,engineering and other areas of

mathematical practice.

Statistics (Decision Sciences)

Applied mathematics has significant overlap

with the discipline of statistics, whose theory

is formulated mathematically, especially with

probability theory.

Minimizing the risk of a statistical action,

using a procedure in , e. g. parameter

estimation ,hypothesis testing etc.

Mathematical theory of statistics shares

concerns with other decision sciences-

operation research ,control theory etc.

Computational Mathematics

Computational Mathematics proposes and

studies methods for solving mathematical

problems that are typically too large for

human numerical capacity.

Numerical analysis and, scientific

computing also study non-analytic topics of

mathematical science, especially

algorithmic matrix and graph theory.

Main Disciplines and role of

mathematics

Physical Sciences-

- It is the branch of mathematical analysis that

emphasizes tools and techniques of particular

use to physicists and engineers.

- Vector spaces, matrix algebra, differential

equations, integral equations, integral

transforms, infinite series, and complex

variables are main tools.

Cont……. Fluid Dynamics

- Many mathematicians and physicists applied

the basic laws of Newton to obtain

mathematical models for solid and fluid

mechanics.

- Used in understanding volcanic eruptions,

flight, ocean currents.

- Civil and mechanical engineers still base

their models on this work, and numerical

analysis is one of their basic tools.

Cont…… Chemistry

-Math is extremely important in physical

chemistry especially advanced topics such

as quantum or statistical

mechanics(Probability theory).

- Quantum relies heavily on group theory

and linear algebra and requires knowledge

of mathematical/physical topics such as

Hilbert spaces and Hamiltonian operators.

Cont…

Biological sciences

-Biomathematics is a rich fertile field with

open, challenging and fascination problems

in the areas of mathematical genetics,

mathematical ecology, mathematical neuron-

physiology.

- Use of mathematical programming and

reliability theory in biosciences and

mathematical problems in biomechanics,

bioengineering and bioelectronics.

Cont…. Social Sciences

-Economics, sociology, psychology, and

linguistics all now make extensive use of

mathematical models, using the tools of

calculus, probability, and game theory,

network theory, often mixed with a

healthy dose of computing.

Cont….. Economics

- A great deal of mathematical thinking goes

in the task of national economic planning, and

a number of mathematical models for

planning have been developed.

-The models ,may be-stochastic / deterministic,

linear / non-linear, static / dynamic,

continuous/ discrete, microscopic/

macroscopic and all types of algebraic,

differential, difference and integral equations

arise for solution of these models. .

Cont…. Actuarial Science

-Actuaries use mathematics and statistics to

make financial sense of the future.

Mathematical Linguistics

-The concepts of structure and transformation

are as important for linguistic as they are for

mathematics. Development of machine

languages and comparison with natural and

artificial language require a high degree of

mathematical ability.

Cont…. Mathematics in Music

-Music scholars used mathematics to

understand musical scales, and some

composers have incorporated the Golden ratio

and Fibonacci numbers into their work.

Mathematics in Management

- Different Mathematical models are being

used to discuss management problems of

hospitals, public health, pollution, educational

planning and administration and similar other

problems of social decisions.

Cont….. Mathematics in Engineering and

Technology -Mechanical, civil, aeronautical and chemical

engineering involved with lots of

mathematics.

-Electrical/Electronics engineers through its

applications to information theory,

cybernetics, analysis and synthesis of

networks, automatic control systems, design

of digital computers etc using maximum

mathematics.

Cont…. Mathematics in Computers

-Most applications of Mathematics to science

and technology today are via computers.

- Operation Research techniques, modern

management Modeling and Simulation,

Monte Carlo program, Evaluation Research

Technique, Critical Path Method, Artificial

Intelligence, Development of automata

theory etc. are the some of the branches of

mathematics dealing with computers..

Some Other Discipline

Physical Oceanography

Psychology and Archaeology

Mathematics in Social Networks

Political Science

Mathematics in Art

And many more……..