Prof.(Dr.) Z.K.Ansari PDF/Mathematics-An... · 2020-01-15 · &RQW« ,QGLDQ0DWKHPDWLFV y...
Transcript of Prof.(Dr.) Z.K.Ansari PDF/Mathematics-An... · 2020-01-15 · &RQW« ,QGLDQ0DWKHPDWLFV y...
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Prof.(Dr.) Z.K.Ansari,
Professor and Head,
Department of Applied Mathematics, JSS Academy of Technical Education,
NOIDA
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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.
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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.
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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.
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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.
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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.
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Magic Square
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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.
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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.
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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.)
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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.
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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.
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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.
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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.
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Goldbach's conjecture
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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.
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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.
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Combinatorics Number Theory Group Theory
Combinatories
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Graph Theory Order Theory Algebra
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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.
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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.
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Geometry, Trigonometry, Differential Geometry ,Topology, Fractal
Geometry, Measure Theory
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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.
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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 .
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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.
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Calculus, Vector Calculus, Diff.Equation, Dynamical Systems, Chaos
theory, Complex Analysis
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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. .
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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.
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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.
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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.
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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..
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Some Other Discipline
Physical Oceanography
Psychology and Archaeology
Mathematics in Social Networks
Political Science
Mathematics in Art
And many more……..
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