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From Wikipedia

Mathematical model

Note: The term model has a different meaning inmodel theory, a branch of mathematical logic. An artifact which is used to illustrate a mathematical idea may also be called a mathematical model, and this usage is the reverse of the sense explained below.

A mathematical model is a description of a system using mathematical language. The process of developing a mathematical model is termed mathematical modelling (also written modeling). Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines (e.g. computer science, artificial intelligence), but also in the social sciences (such as economics, psychology, sociology and political science); physicists, engineers, statisticians, operations research analysts and economists use mathematical models most extensively.

Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures.

Examples of mathematical models

  • Population Growth. A simple (though approximate) model of population growth is theMalthusian growth model. A slightly more realistic and largely used population growth model is the logistic function, and its extensions.
  • Model of a particle in a potential-field. In this model we consider a particle as being a point of mass which describes a trajectory in space which is modeled by a function giving its coordinates in space as a function of time. The potential field is given by a function V : R3→ R and the trajectory is a solution of the differential equation
m \frac{d^2}{dt^2} x(t) = - \operatorname{grad} \left( V \right) (x(t)).
Note this model assumes the particle is a point mass, which is certainly known to be false in many cases in which we use this model; for example, as a model of planetary motion.
  • Model of rational behavior for a consumer. In this model we assume a consumer faces a choice of n commodities labeled 1,2,...,n each with a market price p1, p2,..., pn. The consumer is assumed to have a cardinal utility function U (cardinal in the sense that it assigns numerical values to utilities), depending on the amounts of commodities x1, x2,..., xn consumed. The model further assumes that the consumer has a budget M which is used to purchase a vector x1, x2,..., xn in such a way as to maximize U(x1, x2,..., xn). The problem of rational behavior in this model then becomes an optimization problem, that is:
\max U(x_1,x_2,\ldots, x_n)
subject to:
\sum_{i=1}^n p_i x_i \leq M.
x_{i} \geq 0 \; \; \; \forall i \in \{1, 2, \ldots, n \}
This model has been used in general equilibrium theory, particularly to show existence and Pareto efficiency of economic equilibria. However, the fact that this particular formulation assigns numerical values to levels of satisfaction is the source of criticism (and even ridicule). However, it is not an essential ingredient of the theory and again this is an idealization.

Modelling requires selecting and identifying relevant aspects of a situation in the real world.

Background

Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis, engineers can build a descriptive model of the system as a hypothesis of how the system could work, or try to estimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can try out different control approaches in simulations.

A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. The values of the variables can be practically anything; real or integer numbers, boolean values or strings, for example. The variables represent some properties of the system, for example, measured system outputs often in the form of signals, timing data, counters, and event occurrence (yes/no). The actual model is the set of functions that describe the relations between the different variables.

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From Encyclopedia

Mathematics

The invention and ideas of many mathematicians and scientists led to the development of the computer, which today is used for mathematical teaching purposes in the kindergarten to college level classrooms. With its ability to process vast amounts of facts and figures and to solve problems at extremely high speeds, the computer is a valuable asset to solve the complex math-laden research problems of the sciences as well as problems in business and industry. Major applications of computers in the mathematical sciences include their use in mathematical biology, where math is applied to a discipline such as medicine, making use of laboratory animal experiments as surrogates for a human biological system. Mathematical computer programs take the data drawn from the animal study and extrapolate it to fit the human system. Then, mathematical theory answers the question of how far these data can be transformed yet still preserve similarity between species. Mathematical ecology tries to understand the patterns of nature as society increasingly faces shortages in energy and depletion of its limited resources. Computers can also be programmed to develop premium tables for life insurance companies, to examine the likely effects of air pollution on forest productivity, and to simulate mathematical model outcomes that are used to predict areas of disease outbreaks. Mathematical geography computer programs model flows of goods, people, and ideas over space so that commodity exchange, transportation, and population migration patterns can be studied. Large-scale computers are used in mathematical physics to solve equations that were previously intractable, and for problems involving a third dimension, numerous computer graphics packages display three-dimensional spatial surfaces. A byproduct of the advent of computers is the ability to use this tool to investigate nonlinear methods. As a result, the stability of our solar system has been checked for millions of years to come. In the information age, information needs to be stored, processed, and retrieved in various forms. The field of cryptography is loaded with computer science and mathematics complementing each other to ensure the confidentiality of information transmitted over telephone lines and computer networks. Encoding and decoding operations are computationally intense. Once a message is coded, its security may hinge on the inability of an intruder to solve the mathematical riddle of finding the prime factors of a large number. Economical encoding is required in high-resolution television because of the enormous amount of information. Data compression techniques are initially mathematical concepts before becoming electromagnetic signals that emerge as a picture on the TV screen. Mathematical application software routines that solve equations, perform computations, or analyze experimental data are often found in area-specific subroutine libraries which are written most often in Fortran or C. In order to minimize inconsistencies across different computers, the Institute of Electrical and Electronics Engineers (IEEE) standard is met to govern the precision of numbers with decimal positions. The basic configuration of mathematics learning in the classroom is the usage of stand-alone personal computers or shared software on networked microcomputers. The computer is valued for its ability to aid students to make connections between the verbal word problem, its symbolic form such as a function, and its graphic form. These multiple representations usually appear simultaneously on the computer screen. For home and school use, public-domain mathematical software and shareware are readily available on the Internet and there is a gamut of proprietary software written that spans the breadth and depth of the mathematical branches (arithmetic, algebra, geometry, trigonometry, elementary functions, calculus, numerical analysis, numerical partial differential equations, number theory, modern algebra, probability and statistics, modeling, complex variables, etc.). Often software is developed for a definitive mathematical maturity level. In lieu of graphics packages, spreadsheets are useful for plotting data and are most useful when teaching arithmetic and geometric progressions. Mathematics, the science of patterns, is a way of looking at the world in terms of entities that do not exist in the physical world (the numbers, points, lines and planes, functions, geometric figures——all pure abstractions of the mind) so the mathematician looks to the mathematical proof to explain the physical world. Several attempts have been made to develop theorem-proving technology on computers. However, most of these systems are far too advanced for high school use. Nevertheless, the non-mathematician, with the use of computer graphics, can appreciate the sets of Gaston Julia and Benoit B. Mandelbrot for their artistic beauty. To conclude, an intriguing application of mathematics to the computer world lies at the heart of the computer itself, its microprocessor. This chip is essentially a complex array of patterns of propositional logic (p and q, p or q, p implies q, not p, etc.) etched into silicon . see also Data Visualization; Decision Support Systems; Interactive Systems; Physics. Patricia S. Wehman Devlin, Keith. Mathematics: The Science of Patterns. New York: Scientific American Library, 1997. Sangalli, Arturo. The Importance of Being Fuzzy and Other Insights from the Border between Math and Computers. Princeton, NJ: Princeton University Press, 1998.

mathematics

mathematics deductive study of numbers, geometry, and various abstract constructs, or structures; the latter often "abstract" the features common to several models derived from the empirical, or applied, sciences, although many emerge from purely mathematical or logical considerations. Mathematics is very broadly divided into foundations, algebra, analysis, geometry, and applied mathematics, which includes theoretical computer science. Branches of Mathematics Foundations The term foundations is used to refer to the formulation and analysis of the language, axioms, and logical methods on which all of mathematics rests (see logic ; symbolic logic ). The scope and complexity of modern mathematics requires a very fine analysis of the formal language in which meaningful mathematical statements may be formulated and perhaps be proved true or false. Most apparent mathematical contradictions have been shown to derive from an imprecise and inconsistent use of language. A basic task is to furnish a set of axioms effectively free of contradictions and at the same time rich enough to constitute a deductive source for all of modern mathematics. The modern axiom schemes proposed for this purpose are all couched within the theory of sets , originated by Georg Cantor, which now constitutes a universal mathematical language. Algebra Historically, algebra is the study of solutions of one or several algebraic equations, involving the polynomial functions of one or several variables. The case where all the polynomials have degree one (systems of linear equations) leads to linear algebra. The case of a single equation, in which one studies the roots of one polynomial, leads to field theory and to the so-called Galois theory. The general case of several equations of high degree leads to algebraic geometry, so named because the sets of solutions of such systems are often studied by geometric methods. Modern algebraists have increasingly abstracted and axiomatized the structures and patterns of argument encountered not only in the theory of equations, but in mathematics generally. Examples of these structures include groups (first witnessed in relation to symmetry properties of the roots of a polynomial and now ubiquitous throughout mathematics), rings (of which the integers, or whole numbers, constitute a basic example), and fields (of which the rational, real, and complex numbers are examples). Some of the concepts of modern algebra have found their way into elementary mathematics education in the so-called new mathematics. Some important abstractions recently introduced in algebra are the notions of category and functor, which grew out of so-called homological algebra. Arithmetic and number theory , which are concerned with special properties of the integers—e.g., unique factorization, primes, equations with integer coefficients (Diophantine equations), and congruences—are also a part of algebra. Analytic number theory, however, also applies the nonalgebraic methods of analysis to such problems. Analysis The essential ingredient of analysis is the use of infinite processes, involving passage to a limit . For example, the area of a circle may be computed as the limiting value of the areas of inscribed regular polygons as the number of sides of the polygons increases indefinitely. The basic branch of analysis is the calculus . The general problem of measuring lengths, areas, volumes, and other quantities as limits by means of approximating polygonal figures leads to the integral calculus. The differential calculus arises similarly from the problem of finding the tangent line to a curve at a point. Other branches of analysis result from the application of the concepts and methods of the calculus to various mathematical entities. For example, vector analysis is the calculus of functions whose variables are vectors. Here various types of derivatives and integrals may be introduced. They lead, among other things, to the theory of differential and integral equations, in which the unknowns are functions rather than numbers, as in algebraic equations. Differential equations are often the most natural way in which to express the laws governing the behavior of various physical systems. Calculus is one of the most powerful and supple tools of mathematics. Its applications, both in pure mathematics and in virtually every scientific domain, are manifold. Geometry The shape, size, and other properties of figures and the nature of space are in the province of geometry. Euclidean geometry is concerned with the axiomatic study of polygons, conic sections, spheres, polyhedra, and related geometric objects in two and three dimensions—in particular, with the relations of congruence and of similarity between such objects. The unsuccessful attempt to prove the "parallel postulate" from the other axioms of Euclid led in the 19th cent. to the discovery of two different types of non-Euclidean geometry . The 20th cent. has seen an enormous development of topology , which is the study of very general geometric objects, called topological spaces, with respect to relations that are much weaker than congruence and similarity. Other branches of geometry include algebraic geometry and differential geometry , in which the methods of analysis are brought to bear on geometric problems. These fields are now in a vigorous state of development. Applied Mathematics The term applied mathematics loosely designates a wide range of studies with significant current use in the empirical sciences. It includes numerical methods and computer science, which seeks concrete solutions, sometimes approximate, to explicit mathematical problems (e.g., differential equations, large systems of linear equations). It has a major use in technology for modeling and simulation. For example, the huge wind tunnels , formerly used to test expensive prototypes of airplanes, have all but disappeared. The entire design and testing process is now largely carried out by computer simulation, using mathematically tailored software. It also includes mathematical physics, which now strongly interacts with all of the central areas of mathematics. In addition, probability theory and mathematical statistics are often considered parts of applied mathematics. The distinction between pure and applied mathematics is now becoming less significant. Development of Mathematics The earliest records of mathematics show it arising in response to practical needs in agriculture, business, and industry. In Egypt and Mesopotamia, where evidence dates from the 2d and 3d millennia BC, it was used for surveying and mensuration; estimates of the value of π ( pi ) are found in both locations. There is some evidence of similar developments in India and China during this same period, but few records have survived. This early mathematics is generally empirical, arrived at by trial and error as the best available means for obtaining results, with no proofs given. However, it is now known that the Babylonians were aware of the necessity of proofs prior to the Greeks, who had been presumed the originators of this important step. Greek Contributions A profound change occurred in the nature and approach to mathematics with the contributions of the Greeks. The earlier (Hellenic) period is represented by Thales (6th cent. BC), Pythagoras , Plato , and Aristotle , and by the schools associated with them. The Pythagorean theorem, known earlier in Mesopotamia, was discovered by the Greeks during this period. During the Golden Age (5th cent. BC), Hippocrates of Chios made the beginnings of an axiomatic approach to geometry and Zeno of Elea proposed his famous paradoxes concerning the infinite and the infinitesimal, raising questions about the nature of and relationships among points, lines, and numbers. The discovery through geometry of irrational numbers, such as 2 , also dates from this period. Eudoxus of Cnidus (4th cent. BC) resolved certain of the problems by proposing alternative methods to those involving infinitesimals; he is known for his work on geometric proportions and for his exhaustion theory for determining areas and volumes. The later (Hellenis


From Yahoo Answers

Question:The importance of mathematical modeling/simulations in the experimenting process? Do you see any advantages/disadvantages using this type of "experimenting" over the experimental work? Even if you could give me some good web sites to look at that will help me better understand mathematical modeling and simulations.

Answers:Disadvantage: First of all, any model is not the real system. There is never a perfect model. Results could be influenced by model innacuracies. Advantages: A lot cheaper than running experiments on the real thing, where experiments may not even be possible, or they may just be impractical. For instance, suppose the experiment invlved economic impact of using a certain production process over a 3 year period. It wouldn't be practical to run 3 year experiments, but simulation experiments on a model of the system could be done with nearly instantaneous results.

Question:I am doing this assignment,and i have to get the advantages and disadvantages of mathematics...but i already have the advantages,,all i need is the disadvantages of mathematics.I really need help

Answers:It wears on your brain.

Question:A direct answer OR where can i get it on the internet(website)? An alternative answer can be a list of the advantages of mathematical modelling.

Answers:All approaches in the sciences use mathematical modeling. Newton's calculus is mathematical model for gravitation effects. Einstein's model is better and could make predictions which Newton's couldn't (as well). Other part - modelling is much cheaper than carrying out physical experiments in many cases, and in many cases only modeling is possible -- black holes for example

Question:A plague of locusts attacked an area of fruit trees covering 200 hectares. The insects move devouring it at a constant rate of 3.28 hectares per day. Find a mathematical model that determines the number of hectares that are invade in x days. Describes another model of the number of hectares not attacked yet after x days. How many hectares of locusts have eaten after 7 days? How quickly wipe out the entire area?

Answers:number of hectares that are invaded in x days: 3.28x Describes another model of the number of hectares not attacked yet after x days: 200 - 3.28x How many hectares of locusts have eaten after 7 days? 3.28 * 7 How quickly wipe out the entire area? 200/3.28

From Youtube

Mathematical Modelling :

GG Bridge -mathematical modeling :mathematical modeling of the golden gate bridge cables by Dr.Kember