Essey on MATHEMATICS and its importance in other Areas

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.