Mathematics is a great Subject:
Mathematics is the
science and study of quantity ,structure ,space and change. There is debate
over whether mathematical objects such as numbers and points exist naturally or are human creations. The
mathematician Benjamin Peirce called mathematics "the science that draws necessary
conclusions". Albert Einstein, on the other hand, stated that "as far as the laws of
mathematics refer to reality, they are not certain; and as far as they are
certain, they do not refer to reality." The use of abstraction and logical reasoning, mathematics evolved from counting, calculation,measurement, and the systematic study of the shapes and motions of physical objects. Today, mathematics is used throughout
the world as an essential tool in many fields, including natural science, engineering, medicine, social sciences and many other subjects.
Role of Mathematics in other disciplines
Physical
Sciences In
mathematical physics, some basic axioms about mass, momentum, energy, force,
temperature, heat etc. are abstracted, from observations and physical
experiments and then the techniques of abstraction, generalisation and logical
deduction are used. It focuses on vector
spaces, matrix algebra, differential equations, integral equations, integral
transforms, infinite series, and complex variables. Its approach can be adapted
to applications in electromagnetism, classical mechanics, and quantum
mechanics.
Fluid Dynamics Many mathematicians and physicists
applied the basic laws of Newton to obtain mathematical models for solid and
fluid mechanics. This is one of the most widely applied areas of mathematics,
and is also 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.
Computational
Fluid Dynamics Computational
Fluid Dynamics is a discipline wherein we use computers to solve the Navier –
Stokes equations for specified initial and boundary condition for subsonic,
transonic and hypersonic flows. Many of our research workers use computers, but
usually these are used at the final stage when drastic simplifications have
already been made, partial differential equation have been reduced to ordinary
differential equations and those equations have even been solved.
Physical Oceanography Important fluid dynamics problem arise in physical
oceanography. Problems of waves, tides, cyclones flows in bays and estuaries,
the effect of efflux of pollutants from nuclear and other plants in sea water,
particularly on fish population in the ocean are important for study. From
defense point of view, the problem of under-water explosions, the flight of
torpedoes in water, the sailing of ships and submarines are also important.
Chemistry
Math is
extremely important in physical chemistry especially advanced topics such as
quantum or statistical mechanics. Quantum relies heavily on group theory and
linear algebra and requires knowledge of mathematical/physical topics such as
Hilbert spaces and Hamiltonian operators. Statistical mechanics relies heavily
on probability theory.
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, development of computer software for special biological and medical
problems, mathematical theory of epidemics, use of mathematical programming and
reliability theory in biosciences and mathematical problems in biomechanics,
bioengineering and bioelectronics.
In Population Dynamics, we study deterministic and stochastic models for growth of
population of micro-organisms and animals, subject to given laws of birth,
death, immigration and emigration. The models are in terms of differential
equations, difference equations, differential difference equations and integral
equations.
In Internal physiological Fluid Dynamics, we study flows of blood and other fluids
in the complicated network of cardiovascular and other systems. We also study
the flow of oxygen through lung airways and arteries to individual cells of the
human or animal body and the flow of synovial fluid in human joints.
In
External Physiological Fluid Dynamics we study the swimming of micro organisms and fish in water
and the flight of birds in air.
In Mathematical Ecology, we study the prey predator models and models where species
in geographical space are considered. Epidemic models for controlling epidemics
in plants and animals are considered and the various mathematical models pest
control is critically examined.
In Mathematical Genetics, we study the inheritance of genetic characteristics from
generation to generation and the method for genetically improving plant and
animal species. Decoding of the genetic code and research in genetic
engineering involve considerable mathematical modeling.
Mathematical theory of the Spread of
Epidemics determines the number of susceptible, infected and immune persons at
any time by solving systems of differential equations. The control of epidemics
subject to cost constraints involves the use of control theory and dynamic
programming. We have also to take account of the incubation period, the number
of carriers and stochastic phenomena. The probability generating function for
the stochastic case satisfies partial differential equations which cannot be
solved in the absence of sufficient boundary and initial conditions.
In Drug kinetics, we study the spread of drugs in the various compartments
of the human body. In mathematical models for cancer and other diseases, we
develop mathematical models for the study of the comparative effects of various
treatments.
Solid Biomechanics deals with the stress and strain in muscles and bones, with
fractures and injuries in skulls etc. and is very complex because of non
symmetrical shapes and the composite structures of these substances. This
involves solution of partial differential equations.
In Pollution Control Models, we study how to obtain maximum
reduction in pollution levels in air, water or noise with a given expenditure
or how to obtain a given reduction in pollution with minimum cost. Interesting
non- conventional mathematical programming problems arise here.
Social Sciences Disciplines such as 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.
Economics
In economic
theory and econometrics, a great deal of mathematical work is being done all
over the world. In econometrics, tools of matrices, probability and statistics
are used. 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.
Actuarial Science, Insurance and Finance Actuaries use mathematics and
statistics to make financial sense of the future, to analyze the project,
assess the financial risks involved, model the future financial outcomes and advice
the organization on the decisions to be made. Much of their work is on
pensions, ensuring funds stay solvent long into the future, when current
workers have retired. They also work in insurance, setting premiums to match
liabilities.
Mathematics is also used in many other areas of finance, from banking and
trading on the stock market, to producing economic forecasts and making
government policy.
Psychology and Archaeology Mathematics is even necessary in
many of the social sciences, such as psychology and archaeology. Archaeologists
use a variety of mathematical and statistical techniques to present the data
from archaeological surveys and try to distinguish patterns in their results
that shed light on past human behavior.
Mathematics
in Social Networks
Graph theory, text analysis, multidimensional scaling and cluster analysis, and
a variety of special models are some mathematical techniques used in analyzing
data on a variety of social networks.
Political Science In Mathematical Political Science, we analyze past election
results to see changes in voting patterns and the influence of various factors
on voting behavior, on switching of votes among political parties and
mathematical models for Conflict Resolution. Here we make use of Game
Theory.
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. Information theory, mathematical biology,
mathematical psychology etc. are all needed in the study of Linguistics.
Mathematics
in Music
Calculations are the root of all sorts of advancement in different disciplines.
The rhythm that we find in all music notes is the result of innumerable
permutations and combinations of SAPTSWAR.
Most of today's music is produced
using synthesizers and digital processors to correct pitch or add effects to
the sound. These tools are created by audio software engineers who work out
ways of manipulating the digital sound, by using a mathematical technique
called Fourier analysis.
Mathematics
in Art
“Mathematics and art are just two different languages that can be used to
express the same ideas the old Gothic Architecture is based on geometry. Even
the Egyptian Pyramids, the greatest feat of human architecture and engineering,
were based on mathematics.
Mathematics in management is a great challenge to imaginative minds. It is not meant
for the routine thinkers. 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.
Mathematics
in Engineering and Technology Mathematics has played an important role in the development
of mechanical, civil, aeronautical and chemical engineering through its
contributions to mechanics of rigid bodies, hydro-dynamics, aero-dynamics, heat
transfer, lubrication, turbulence, elasticity, etc..
The defense sector is an important employer of mathematicians; it needs people
who can design, build and operate planes and ships, and work on other advanced
technologies. It also needs clear-thinking and analytical strategists.
Mathematics in Computers An important area of applications of mathematics is in the
development of formal mathematical theories related to the development of
computer science. Now most applications of Mathematics to science and
technology today are via computers. The foundation of computer science is based
only on mathematics. It includes, logic, relations, functions, basic set
theory, count ability and counting arguments, proof techniques, mathematical
induction, graph theory, combinatory, discrete probability, recursion,
recurrence relations, and number theory, computer-oriented numerical analysis,
Operation Research techniques, modern management techniques like Simulation,
Monte Carlo program, Evaluation Research Technique, Critical Path Method,
Development of new computer languages, study of Artificial Intelligence,
Development of automata theory etc.
In Robotics Vision, computers built in the robots are trained to recognize
objects coming in their way through the pattern recognition programs built into
them. In manufacturing Robotics, the artificial arms and legs and other organs
have to be given the same degree of flexibility of rotation and motion as human
arms, legs and organs have. This requires special developments in mechanics.
