Application of Mathematics

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Published on February 23, 2014

Author: KarthikMuraliIyer

Source: slideshare.net

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This was an Inter Collegiate and a State Level Contest named SIGMA '08. Won a special prize for this paper. This research emphasized on how simple concepts of Mathematics helps into constructing complex mathematical models for space programming and their individual importance in real time applications.

Application of Mathematics A short research on the application of a few selected mathematical concepts, what do they signify in the world of numerical science and a case study of a single project titled “Global Precipitation Measurement” that encompasses the amalgamation of all the concepts considered for this research. Karthik Murali 9/1/08 Numerical Methods

Application of Mathematics INTRODUCTION We keep the broad definition here, that mathematics includes all the related areas which touch on quantitative, geometric, and logical themes. This includes Statistics, Computer Science, Logic, Applied Mathematics, and other fields which are frequently considered distinct from mathematics, as well as fields which study the study of mathematics i.e. History of Mathematics, Mathematics Education and so on. We draw the line only at experimental sciences, philosophy, and computer applications. Personal perspectives vary widely, of course. A fairly standard definition is the one in the Columbia Encyclopedia (5th edition):"Mathematics is deductive study of numbers, geometry, and various abstract constructs, or structures. The latter often arise from analytical models in the empirical sciences, but may emerge from purely mathematical considerations." Some definitions of mathematics heard from others:  That which mathematicians do.  The study of well-defined things.  The study of statements of the form "P implies Q".  The science of patterns (Keith Devlin) Contrary to common perception, mathematics does not consist of ‘crunching numbers’ or ‘solving equations’. As we shall see there are branches of mathematics concerned with setting up equations, or analyzing their solutions, and there are parts of mathematics devoted to creating methods for doing computations. But there are also parts of mathematics which have nothing at all to do with numbers or equations.

ORIGIN OF MATHEMATICS The word ‘mathematics’ (Greek: μαθηματικά or mathēmatiká) comes from the Greek word μάθημα (máthēma) which means learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times. Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), in Latin ars mathematica, meant the mathematical art. APPLIED MATHEMATICS Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the sciences, business, and other areas. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a role. Most experiments, surveys and observational studies require the informed use of statistics. (Many statisticians, however, do not consider themselves to be mathematicians, but rather part of an allied group.) Numerical analysis investigates computational methods for efficiently solving a broad range of mathematical problems that are typically too large for human numerical capacity; it includes the study of rounding errors or other sources of error in computation. Mathematical Physics Mathematical Fluid Dynamics Numerical Analysis Optimization Probability Statistics Financial Mathematics Game Theory

MATHEMATICAL PHYSICS Mathematical physics is the scientific discipline concerned with ‘the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories.’ It can be seen as underpinning both theoretical physics and computational physics. MATHEMATICAL FLUID DYNAMICS Fluid mechanics is the study of the physics of continuous materials which take the shape of their container. Like any mathematical model of the real world, fluid mechanics makes some basic assumptions about the materials being studied. These assumptions are turned into equations that must be satisfied if the assumptions are to hold true. For example, consider an incompressible fluid in three dimensions. The assumption that mass is conserved means that for any fixed closed surface (such as a sphere) the rate of mass passing from outside to inside the surface must be the same as rate of mass passing the other way. (Alternatively, the mass inside remains constant, as does the mass outside). This can be turned into an integral equation over the surface. NUMERICAL ANALYSIS Numerical analysis is the study of algorithms or the problems of continuous mathematics as distinguished from discrete mathematics.

OPTIMIZATIONS In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. Many real-world and theoretical problems may be modeled in a general framework. The branch of applied mathematics and numerical analysis that is concerned with the development of deterministic algorithms that are capable of guaranteeing convergence in finite time to the actual optimal solution of a non-convex problem is called global optimization. PROBABILITY Probability is the likelihood or chance that something is the case or will happen. Probability theory is used extensively in areas such as statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems. Two major applications of probability theory in everyday life are in risk assessment and in trade commodity market. Governments typically apply probabilistic methods in environmental regulation where it is called ‘pathway analysis’, often measuring well being using methods that are stochastic in nature, and choosing projects to undertake based on statistical analyses of their probable effect on the population as a whole.

It is not correct to say that statistics are involved in the modeling itself, as typically the assessments of risk are one-time and thus require more fundamental probability models, e.g. "the probability of another 9/11". A law of small numbers tends to apply to all such choices and perception of the effect of such choices, which makes probability, measures a political matter. STATISTICS Statistics is a mathematical science pertaining to the collection, analysis, interpretation or explanation, and presentation of data. It is applicable to a wide variety of academic disciplines, from the natural and social sciences to the humanities. Statistics is also used for making informed decisions in government and business. Statistical methods can be used to summarize or describe a collection of data; this is called descriptive statistics. In addition, patterns in the data may be modeled in a way that accounts for randomness and uncertainty in the observations, and then used to draw inferences about the process or population being studied; this is called inferential statistics. Both descriptive and inferential statistics comprise applied statistics. There is also a discipline called mathematical statistics, which is concerned with the theoretical basis of the subject. The word statistics is also the plural of statistic (singular), which refers to the result of applying a statistical algorithm to a set of data, as in economic statistics, crime statistics, etc.

MATHEMATICAL FINANCE Mathematical finance is the branch of applied mathematics concerned with the financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive, and extend, the mathematical or numerical models suggested by financial economics. Thus, for example, while a financial economist might study the structural reasons why a company may have a certain share price, a financial mathematician may take the share price as a given, and attempt to use stochastic calculus to obtain the fair value of derivatives of the stock. In terms of practice, mathematical finance also overlaps heavily with the fields of financial engineering and computational finance. Many universities around the world now offer degree and research programs in mathematical finance. GAME THEORY Game theory is a branch of applied mathematics that is often used in the context of economics. It studies strategic interactions between agents. In strategic games, agents choose strategies that will maximize their return, given the strategies the other agents choose. The essential feature is that it provides a formal modeling approach to social situations in which decision makers interact with other agents. Game theory extends the simpler optimization approach developed in neoclassical economics. These were some of the applications of Mathematics in the real world. Mathematics is an essential part of our everyday lives. It is very much vivid that without mathematics, many of the world’s complex problems would not have turned so easy.

CASE STUDY: - Global Precipitation Measurement [GPM] Goddard NASA Some of the concepts of applied mathematics that are discussed above are put to use or we could say, are implemented in the GPM Project taken up by NASA. Mathematical Physics, Mathematical Fluid Dynamics, Numerical Analysis, Optimization, Probability & Statistics are some of the few that we have seen so far. Mathematics is still an unexplored area. There is still so much to be revealed. Global Precipitation is a project initialized for the betterment of mankind. Indirectly, Mathematics too, becomes a factor that is very clearly helping the Earth in many ways. CONCLUSION:Mathematics is a powerful subject with the context and constraints of its own. Its applications are being developed on such a large scale that it will nearly take light years to study all of them. Mathematics is seriously a boon to mankind as it is indirectly and unknowingly applied in most of the complex problems; the world faces. Mathematics has no boundaries and will always hold its importance till eternity.

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