Thursday, July 29, 2010

Role of Mathematics in other disciplines

A famous Jain Mathematician, Ācārya Mahāvira (9th century) writes that-“Bahubhirvi pralāpaih kim trailokye sacarācare, Yatkimcidvastu tatsarva ganitena binā nahi”. The verse shows the importance of Mathematics as-“What is good of saying much in vain? Whatever there is in all three worlds, which are possessed of moving and non-moving being all that indeed cannot exist as apart from Mathematics”.

Here are some main disciplines in which Role of Mathematics is widely accepted.

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, generalization and logical deduction are used. It is the branch of mathematical analysis that emphasizes tools and techniques of particular use to physicists and engineers. 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.

In mathematics, a particle is a point-like, perfectly rigid, solid object. Particle mechanics deals with the results of subjecting particles to forces. It includes celestial mechanics — the study of the motion of celestial objects. The role of Space Dynamics is very important in mechanics. Here we have to consider the trajectories which are time-optimal i.e. which take the least time in going from one point to another and in which the object starts and reaches the destination with zero velocity. Similarly we may have to consider energy- optimal trajectories. We have also to consider the internal and external ballistic of rocket and the path of inter continental ballistic missiles.

Fluid Dynamics

Understanding the conditions that result in avalanches, and developing ways to predict when they might occur, uses an area of mathematics called fluid mechanics. 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. In the 19th century, phenomena involving heat, electricity, and magnetism were successfully modeled; and in the 20th century, relativistic mechanics, quantum mechanics, and other theoretical constructs were created to extend and improve the applicability of earlier ideas. One of the most widespread numerical analysis techniques for working with such models involves approximating a complex, continuous surface, structure, or process by a finite number of simple elements, known as the finite element method (FEM). This technique was developed by the American engineer Harold Martin and others to help the Boeing Company analyze stress forces on new jet wing designs in the 1950s. FEM is widely used in stress analysis, heat transfer, fluid flow, and torsion analysis.
Fluid Dynamics is also very important in Atmospheric Sciences, in dynamic meteorology and weather prediction. Another use in the study of diffusion of pollutants in the atmosphere e.g. to find out what proportion of pollutants emitted from chimneys or refineries reach hospitals and other buildings. It is also needed for the study of effect of leakages of poisonous gases.

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.

Other fields of chemistry also use a significant amount of math. For example, most modern IR and NMR spectroscopy machines use the Fourier transform to obtain spectra. Even biochemistry has important topics which rely heavily on math, such as binding theory and kinetics.
Even Pharmaceutical companies require teams of mathematicians to work on clinical data about the effectiveness or dangers of new drugs. Pure scientific research in chemistry and biology also needs mathematicians, particularly those with higher degrees in computer science, to help develop models of complicated processes.

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.

Mathematical and computational methods have been able to complement experimental structural biology by adding the motion to molecular structure. These techniques have been able to bring molecules to life in a most realistic manner, reproducing experimental data of a wide range of structural, energetic and kinetic properties. Mathematical models have played, and will continue to play, an important role in cellular biology. A major goal of cell biology is to understand the cascade of events that controls the response of cells to external legends (hormones, transport proteins, antigens, etc.). Mathematical modeling has also made an enormous impact on neuroscience. . Three-dimensional topology and two-dimensional differential geometry are two additional areas of mathematics when it interacts with biology. Its application is also very important to cellular and molecular biology in the area of structural biology. This area is at the interface of three disciplines: biology, mathematics and physics.

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.

The models may be stochastic or deterministic, linear or non-linear, static or dynamic, continuous or discrete, microscopic or macroscopic and all types of algebraic, differential, difference and integral equations arise for the solution of these models. At a later stage more sophisticated models for international economies, for predicting the results of various economic policies and for optimizing the results are developed.
Another important subject for economics is Game theory. The whole economic situation is regarded as a game between consumers, distributors, and producers, each group trying to optimize its profits. The subject tries to develop optimal strategies for each group and the equilibrium values of games.

Actuarial Science, Insurance and Finance

Actuaries use mathematics and statistics to make financial sense of the future. For example, if an organization is embarking on a large project, an actuary may analyze the project, assess the financial risks involved, model the future financial outcomes and advise 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. Statistical measures are used during excavation to monitor which pits are most successful and decide on further excavation. Finds are analyzed using statistical and numerical methods to spot patterns in the way the archaeological record changes over time, and geographically within a site and across the country. Archaeologists also use statistics to test the reliability of their interpretations.

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 has had a great influence on research in literature. In deciding whether a given poem or essay could have been written by a particular poet or author, we can compare all the characteristics of the given composition with the characteristics of the poet or other works of the author with the help of a computer.

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. Music theorists often use mathematics to understand musical structure and communicate new ways of hearing music. This has led to musical applications of set theory, abstract algebra, and number theory. Music scholars have also used mathematics to understand musical scales, and some composers have incorporated the Golden ratio and Fibonacci numbers into their work.
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. This is part of the area of digital signal processing (DSP) which has many other applications including speech recognition, image enhancement and data compression.

Mathematics in Art

"Mathematics and art are just two different languages that can be used to express the same ideas." It is considered that the universe is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures. The old Goethic Architecture is based on geometry. Even the Egyptian Pyramids, the greatest feat of human architecture and engineering, were based on mathematics. Artists who strive and seek to study nature must therefore first fully understand mathematics. On the other hand, mathematicians have sought to interpret and analyze art though the lens of geometry and rationality. This branch of mathematics studies the nature of geometric objects by allowing them to distort and change. An area that benefits most from the visual approach is topology.

Moreover the study of origami and mathematics can be classified as topology, although some feel that it is more closely aligned with combinatorics, or, more specifically, graph theory. Huzita's axioms are one important contribution to this field of study.

Mathematics in Management

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. In order to apply mathematics to management, one must know the mathematical techniques and the conditions under which these techniques are applicable. In addition, one must also understand the situations under which these can be applied. In all the problems of management, the basic problem is the maximization or minimization of some objective function, subject to the constraints in available resources in manpower and materials. Thus OR techniques is the most powerful mathematical tool in the field of Management.

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.. It has become of great interest to electrical engineers through its applications to information theory, cybernetics, analysis and synthesis of networks, automatic control systems, design of digital computers etc. The new mathematical sciences of magneto-hydrodynamics and plasma dynamics are used for making flow meters, magneto-hydrodynamic generates and for experiments in controlled nuclear fusion.

It is well known that most of the technological processes in industry are described effectively by using mathematical frame work. This frame work is then subsequently used to analyze and comprehend advantages and disadvantages in adopting efficient and novel methodologies in these processes, resulting into the introduction of Mathematical Technology.

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, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, 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.

All mathematical processes of use in applications are being rapidly converted into computer package algorithms. There are computer packages for solution of linear and non linear equations, inversions of matrices, solution of ordinary and partial differential equations, for linear, non linear and dynamic programming techniques, for combinatorial problems and for graph enumeration and even for symbolic differentiation and integration.

Cryptography is the practice and study of hiding information. In modern times cryptography is considered a branch of both mathematics and computer science and is affiliated closely with information theory, computer security and engineering. Cryptography is used in applications present in technologically advanced societies; examples include the security of ATM cards, computer passwords and electronic commerce, which all depend on cryptography. It is the mathematics behind cryptography that has enabled the e-commerce revolution and information age.

Pattern Recognition is concerned with training computers to recognize pattern in noisy and complex situations. e.g. in recognizing signatures on bank cheques, in remote sensing 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.

Computerized Tomography uses the important break through in reconstruction of images of brain and objects from the knowledge of the proportions of photons observed along different lines sent through the object. These proportions can be expressed as line integrals of a function.

Fractals Geometry enable us to design models of irregular objects like clouds, coast lines, lightening turbulence etc. and this uses a combination of probability theory, mathematics and computers. This shows that mathematics can enable us to handle apparently irregular patterns as much as it can enable us to study regular patterns.

In Computer Graphics we find the virtual landscapes and things within them are three-dimensional mathematical objects, and these objects behave and interact according to the equations for the rules of physics that apply within the game. These rules might cover gravity, speed and force, and even stop your character falling through a solid floor but allow them to sink in quicksand. This type of mathematics is used in computer graphics for movies, and mathematics plays an important part in many areas of IT, including programming, designing hardware and project management.

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