1. Orientation of the career of Mathematics 2. An aproximacion to the contents of the Plan of Studies 2.1 The basic mathematical disciplines 2.2. Perspective of the information 2.3. Perspective of the numerical analysis 2.4. Perspective of the operative investigation 2.5. Perspective of the statistics 2.6. Perspective of the economic mathematics |
The complex economy of the countries developed flexible mathematical demand, that have a basic training that allow them update permanently his knowledges, and that are capacitados to use, in the
Practical, the methods learnt and his intellectual maturity. Such fact has been ascertained by organisms like the Central Office of the Employment of Frankfurt, that ensures that the professional exits, in Germany, of the mathematical economists have improved, without exception, concerning the corresponding to the pure mathematicians. By this reason, the general purpose of the Plan of Studies of the Degree is to orient the mathematical studies in accordance with the requests of the market of work without incurring, in spite of this, in the error to provide an insufficient basic training. Of this form, the titled in Mathematics by the University of Alicante have to be in conditions to apply his knowledges to resolve successfully the problems "little nítidos" that they arise in any field of activity, but without closing the doors to the traditional exit of the Graduate in Mathematics in Spain, that is the education of the mathematics in the secondary education. By this reason, the Plan of Studies contains the asignatura optativa Didactic of the Mathematics of the Secondary Education.
It is interesting to happen magazine to the branches of economic activity where exists main demand of mathematicians. The experience of the University of Tréveris (in German Trier) is highly significant by what offers, likewise, the speciality of Economic Mathematics. The current situation of the employment between the mathematicians egresados of Trier has been analysed by means of one
Survey realised on the community of the titled between the years 1986 and 1991.
A 40%, roughly, of the titled polled work at present in the process electronic of data, realising tasks of development, sale and maintenance of the corresponding computer systems. Other activities realised by the mathematicians are the control of quality, the financial analysis, the investigation of markets (marketing), the planning of activities, the asesoría fiscal and in matter of pensions, the asesoría business and the estimate of risks in the sector of the insurances. By sectors of economic activity, those that provide more put of work to the mathematicians titled by Trier are the insurance (with a 21%), the industry, especially, the chemist-pharmaceutical (20%), the banking (19%) and the companies of consulting (13%).
As the report SIAM report on Mathematics in Industry (1998) (see the review of the same in the chapter 22), the sectors that absorb more graduates in Mathematical (M.S.) By North American universities are, apart from the academic, the one of services (finances, communications, transport) and of public administration, both with a 22% of the total, the consulting (engineering, programming), with a 18%, and the industry (electronic, computers, industry aerospacial), where works 12% of the titled.
To tenor of the reports presented in the Conference of Deans and Directors of Mathematics of Santiago of Compostela (February of 2000) and Barcelona (November of 2000), every time are more the universities that -looking for the extension of the market of work for his egresados- have opted for specialising his degrees in mathematics to some field of knowledge in which already have prestigious specialists, offering generally a double titulación. Like this, the Polytechnical universities of Catalonia, the one of Palm of Mallorca and the one of Alicante offer at present Mathematical- Telecommunications, Mathematical-Computer and Mathematical- Economy, respectively. Likewise, the representatives of the University of Cantabria and of Carlos III of Madrid informed of his intention to offer Mathematical-Economy, whereas the Autonomous University of Barcelona studies to offer the double titulación Mathematical-Physical. Similar initiatives are plantenado in other Spanish universities
The aim of the mathematics, born science does more than 4.000 years, is to analyse the reality through the mathematical structures and develop procedures that serve to apply the theoretical results to concrete problems. Of this way, the mathematics have turned into an important instrument for the remaining sciences and also for the technician.
The first mathematical developments, in Mesopotamia and Egypt, were related with the operations to explain (numerical systems ) and to measure (geodesy), from which formed the most ancient branches of the mathematics: the geometry, the astronomy, the arithmetic and the algebra. However, it is in Greece where fits to situate the birth of the mathematical science such and as today understand it. The "Elements" of Euclid, work written to the year 300 to. C., it is the first logical exhibition-deductiva of the arithmetic and the geometry developed until then. During the Half Age the Arabs conserved and perfected the Greek mathematics to the time that put in communication the Indian with Occident. To this it is necessary to add the decisive contribution of the Arab mathematicians to the birth of the algebra.
During the 17th century Newton and Leibnitz created, independently, the differential calculation and the integral calculation, that turned into useful tools for the sciences of the nature (firstly for the physics and, later, also for the chemistry, the biology and the medicine) and the technical disciplines (architecture and engineering). To ends of the 19th century attained order and relate of a systematic way some mathematical specialities with others, giving place to new disciplines more abstract, as the topology and the functional analysis, matters that will be entered more forward.
The society of the information in which today live is resulted of the simbiosis between the telecommunications and the computing. It has like base the ideas of Boole, founder around the middle of the 19th century of the mathematical logic, the mathematical model of Turing and the contributions of the eminent mathematical von Neumann and Wiener. The building of electronic computers increasingly powerful, from taken part of the 20th century, has allowed to pose and resolve new real problems arisen, frequently, further of the traditional fields of practical application of the mathematical -the sciences of the nature and the technical- as, for example, in the breast of the economic and social sciences. Of said problems occupy new mathematical disciplines, as the numerical analysis, the statistics, the operative investigation or the economic mathematics.
Like consequence of the complexity of the current mathematics, and of the economic orientation of the studies of Mathematics of the University of Alicante, seems necessary to explain to the student of Secondary Education -whose experience with the mathematics is limited to the linear algebra, to the geometry, to the mathematical analysis and to some notions of probability, always to elementary level- in what consists the Plan of Studies. With such end, have agrupado the asignaturas in six big areas whose objective and interconnections treat to expose of clear and succinct form.
The asignaturas of mathematical analysis, of algebra, of geometry and topology and of theory of the probability exert a basic paper in the Plan of Studies. These matters, that are very related some with others, treat to endow to the mathematical future of some indispensable and fundamental knowledges with sights to his back specialisation, to the time that contribute to develop in the student the capacity of abstraction and the logical thought.
The Mathematical Analysis I treat of the differential and integral calculation with real functions of real variable, whereas the Mathematical Analysis II and the Calculation Advanced (asignaturas of 2º course) develop the corresponding thematic about the scalar or vectorial functions of more than a variable and establish, likewise, the foundations for the study of the equations
Differential. Said equations, that study detalladamente in the signatura homónima of 3º, describe the relations between an unknown function (the incógnita of the equation) and his derivative, and appear with a lot of frequency in the formulation of models for scientific and technical problems, especially in those cases in which some magnitude evolves over time (dynamic models).
Paradoja Of Achilles and the turtle (Zenon of Elea, 490-430 To.C): Achilles pursues a turtle and, although it runs quite faster that this, never will be able to achieve it. The argument is as it follows: the turtle find initially in a point P; when Achilles arrives to P, then the turtle no longer find in P, but in´P ; when Achilles arrives to´P , she already is in a new point´´P ; and like this successively. Therefore, never it will achieve it. ¿Where it is the error in the reasoning?
All the asignaturas of mathematical analysis require the previous knowledge of the numerical groups and, in particular, of the real and complex numbers. Also they are indispensable some proposiciones about vectors, of matrices, determinantes and systems of linear equations. These matters study in Linear Algebra (1º) and in Theory of Matrices (2º), asignatura that it can consider also like an introduction to the Linear Operators no Negative (4º). The Algebra (of 4º course) effects a review advanced of the basic structures entered in Linear Algebra, and loans special attention to the possibility to resolve the algebraic equations by radical (Theory of Galois). The imaginary unit i was entered during the Renaissance so that it had solution the equation x^{2} = -1 that, obviously, lacks solution in the real field. Show that the complex numbers, that are the linear combinations of the real unit 1 and of the imaginary i, are not something unnatural or artistic, but that they are of big help in mathematics and allow to explain numerous natural phenomena and of the social sciences. They serve as it shows the following observations.
The fundamental theorem of the algebra establishes that all polynomial of degree n has exactly n roots in the field of the complex numbers (explaining each one of them so many times as it indicate his order of multiplicity). In other words looking for that a true algebraic equation particular (x^{2} = -1) had solution found us with that all they have them. To each linear differential equation ordinary with constant coefficients associates him an algebraic equation whose roots provide particular solutions of the differential equation. The solutions
Corresponding to real roots are exponential functions, whereas the corresponding to complex roots are products of exponential by breasts or cosenos. In relation with these functions suits to indicate that it exists an interesting relation between the exponential function (of complex variable) and the trigonometric functions: and^{ix} = cos x + i sen x. Substituting x by & pi in this formula, owed to Euler, obtains an expression in which appear the five more important numbers of the mathematics: and^{ix} + 1 = 0 (of her has said that it is the "most beautiful formula of the world"). The Mathematical Analysis IV (of 4º) occupy , between other things, of the study of the functions of complex variable. A lot of improper integrals (integrals of real functions on infinite intervals) and, for example,
,
Very important in probability (in relation with the normal distribution), are especially easy to calculate by means of the technicians of integration that use complex numbers (methods of waste). The Mathematical Analysis IV also occupy of the functional spaces arisen, at the beginning of the 20th century, by the need to develop general methods for the equations of the mathematical physics, of which only knew to resolve special cases. Al suppose that the solution of a linear differential equation ordinary is the sum of infinite addends of the form to_{}n x^{n} (that is to say, a seriede powers), the problem reduce to the resolution of a system of linear equations with infinite incógnitas (the coefficients to_{}n ). Such systems were resolved, with some atrevimiento, with help of the methods of the linear algebra. Proceeding of this way the mathematicians gave account that problems of distinct appearance could be reduced to a same model. Of this form, by abstraction, originated the fundamental principles of the functional analysis. The functions interpret now like elements -or points- of a true space that is endowed of an algebraic structure (generally of vectorial space) and of a topological structure (that it establishes a relation of vicinity between his elements), whereas the equations formulate , in this context, in terms of applications between such spaces.A lot of equations of the mathematical physics studied in Mathematical Analysis V (of 5º course) formulate by means of derivative partial (concerning the independent variables) of the unknown function and resolve , in occasions, decomposing the incógnita in series (sums innifinitas) of the independent variables. The most interesting series are the ones of powers (before mentioned) and the ones of Fourier, whose addends are of the form to_{} n cos nx + b_{n} sen nx. Given the periodic character of such sums (whose value repeat when adding to x multiples of %u03C0), the series of Fourier apply there where arise oscillating processes, as it occurs in the temporary series of economic nature, in electronics, in acoustics or in optics. The theoretical problems related with the convergence of the series of Fourier have impulsado fundamental advances in distinct fields of the mathematics and follow being considered today like problems very difficult.
Automatic diagnostic: The echography allows to register the vibration of each one of the membranes of the heart, providing a periodic curve and the one of the figure. A program of computer calculates the first terms of the successions {to_{}} n and {_{to}} n (coefficients of Fourier). In the case of the válvula mitral, are sufficient the coefficients to_{1}, to_{2},b_{1} and b_{1} to diagnose to the patient. This form of diagnostic diminishes costs in the sanitary system and, especially, avoid to the patient the risks and inherent annoyances to the proofs endoscópicas.
In Geometry and Topology I and in his Extension (asignaturas of 2º course) applies the linear algebra and the differential calculation to the study of the transformations in the flat and in the space, of the curves and of the surfaces (including conical and quadric). Also they provide an approximation to the general topological spaces (that they are, remember it, gifted groups of some structure of vicinity that allows to speak of limit) through the study of the metric spaces. Such spaces have a lot of common properties with the flat and the space of the intuition, and allow a geometrical treatment of important functional spaces (as the one of Hadamard, whose elements are the continuous functions in an enclosed interval and bounded die). The abstract versions of both matters (geometry in varieties and general topology) are the object of Geometry and Topology II (of 5º course).
The curves of the life: As Galileo, "the book of the nature" is written in mathematical language. The espiras of the shell of the ammonites increase his width in accordance with a constant factor. In words of Leonardo of Vinci, "these creatures expand his house and his ceiling gradually, in proportion, satisfied grows his body keeping hit to the sides of the shell". This is precisely the property that characterises to the espiral logarithmic discovered by Archimedes.
The Calculation of Probabilities and his Extension (both of 1º) allow to order the events associated with the random phenomena in accordance with the relative frequencies that can expect when repeating the corresponding experiment. This introduction to the calculation of probabilities bases the Statistics and his Extension (both of 3º), as well as Temporary Series (4º) and Methods Econométricos (5º). Besides, they facilitate the understanding of the abstract concepts of Theory of the Measure and Integration (optativa of 5º). These last theories no only possess innumerable applications to the mathematical analysis, but that base the theory of the probability and the mathematical statistics.
Paradoja Of Bertrand (1822-1900) ¿Which is the probability that a segment traced arbitrariamente inside a circle unit was of main length that the side of an equilateral triangle circumscribed in said circle?. If they consider parallel segments, then the result of the probability is a half, if they consider segments that split from the point P, the result of the probability is a third.
The computing pursues the replacement or simplification of the mental effort human in his more repetitive parts, freeing of this forms intellectual capacity for the creative production. Also it pretends to help us to gestionar big quantities of information, saving and selecting that that more interests us. Finally, it inhales to simplify the use of the machines, endowing to these of systems of interaction more amicable for the user. In consequence, the computing occupy of the register, transcription, utilisation and systematic manipulation of informations, as well as of the structures that take by base the theories of the information.
Although the computing is a very young science, his roots find , partly, in a remote past. It is very likely that the primitive man resorted to the fingers of the hands (and, when these were not sufficient, to the stones) to explain and effect incipientes arithmetical operations. The Egyptians formed montoncillos of 10 stones to explain, whereas in the Chinese of Confucio already used the abacus to explain and calculate, having proof of the use of the abacus in diverse Mediterranean civilisations and even in the pre-Columbian America. With the introduction, in Asia and in Europe, of the system of numbering of position (contrived in the Indian and popularised by the Arab) the arithmetical operations began to realise on the paper. The arrival of the logaritmos (Neper and Briggs) facilitated the approximate calculation of products, divisions, powers and roots, with the help of the corresponding tables. Shortly after, in the 17th century, would arise mechanical devices that facilitated the realisation of big sums (Pascal) and products (Leibnitz), but that they can not consider automatic machines by what required the intervention of the operator to the hour to effect the manoeuvres required by each operation and even the anotación of the intermediate steps. It did not occur the same with the machine of differences contrived, in full 19th century, by the also mathematical British Babbage to improve the precision of the calculations. Although it was a mechanical device, the machine of differences featured of devices of entrance, unit by heart, unit of control, arithmetical unit-logical and devices of exit. Exactly the same conception of the current computers, although the data and the instructions of the program were entered by means of cards perforadas (using with such end a loom of the period).
In 1944, the University of Harvard set up the first electromechanical computer, Mark I, whose programming required still the intervention of the operator. It did not occur like this with the first purely electronic computer of the history, the ENIAC, built in 1946 by the University of Pennsylvania, that was almost 200 faster times that Mark I. In 1947 could set up in Cambridge (England) the EDSAC, that is the first programmable electronic computer of the history. In his design had a leading paper John von Neumann, the mathematician more afamado of his period, member of the famous Institute of Studies Advanced of Princenton, author of valuable contributions to the functional analysis, to the economic mathematics, to the optimisation and also to the computing.
During the three last decades, the computing has transformed in a field of investigation of extraordinary dynamism, constituting in inseparable mate of the technology applied to our daily life. The availability of electronic computers allows, at present, realise a big quantity of operations with extraordinary rapidity.
The machine of Hollerith: The Office of the Census of the USA was forced by law to effect a census of the population each 10 years. A civil servant of said Office, Hollerith, conceived in 1886 a machine that allowed the swift treatment of the data stored in cards perforadas. In binary data (If/No), the step or no of the electrical current allowed to detect the answer. Said machine was used by all the North American Administration until taken part of the 20th century.
The word algorithm, and synonymous of a form to resolve abstract problems by means of some rule that repeat , proviene of the name of the Persian mathematician Alkwarizmi, whose book of text on the equations of second degree contributed to the origin of the algebra. The algorithms base the electronic calculation.
Two typical forms of computer application are, by a side, the sequential calculations of the numerical analysis, that drag quantities in iteraciones and processes of approximation, and, on the other hand, the treatment of big quantities of data to extract information.
A high percentage of the Gross Inner Product (GDP) in the states industrialised is the result of technical processes governed by sophisticated methods of planning without which is not possible to carry out any economic politics. Therefore, the elaboración of the information exerts a key paper in the planning of the company and in the organisation of the economic development. Of here the need to provide to the students of Mathematics a basic training in this field that enable them to use, in his professional future, the powerful instrumental computer available.
They are three the asignaturas of computer content in the First Cycle of the Plan of Studies, to know, Computing I (1º) and II (2º), and Logical of First Order (1º). This last asignatura, although oriented to the methods of automatic demonstration, has a high formative value to the explicitar the rules of the logical deduction. As asignaturas optativas of 2º cycle include Languages, Grammars and Automatons, Parallel Computation, Abstract Models of Calculation, Theory of the Complexity and Foundations of Artificial Intelligence.
They exist scarce real problems whose corresponding mathematical models can be resolved immediately. By general rule, the resolution of real problems demands a big volume of calculation. In fact, the invention of the computers, around the middle of the 20th century, made possible the mathematical treatment of some problems that before seemed inabordables. On the other hand, the generation of numbers aletorios allows to simulate, to low cost, the realisation of real experiments that require considerable investments and long (like this occurs in aeronaútica or and in physics of particles). The branch of the mathematics that occupy of such methods of calculation receives the name of analysis (or calculation) numerical.
There Are a lot of problems whose resolution requires a big volume of operations and for which the numerical analysis has developed special methods: the resolution of big systems of linear equations, the integration of functions, the differential equations or the determination of the roots of equations no linear. A lot of magnitudes of the financial mathematics (the annual tax equivalent of a credit, the internal tax of rentabilidad of a project of investment, etc.) They calculate resolving algebraic equations whose coefficients have a precise economic meaning. To centre the ideas referred us to continuation to the resolution of equations no linear.
The exact solution of an equation seldom can be obtained of explicit form, by means of an analytical method, but that, by the contrary, has to be approximated of iterative form. Treat of a process of successive approximations in which, in each step, improves the available approximate solution previously. In general it occurs that, after a lot of steps, will have to detain the calculations without having achieved the wished exact solution. The question that arises is: ¿how many steps will need , at most, to achieve an approximate solution with the precision wished?
Consider, for example, the consistent problem in determining the solution of the equation x = cos(x). We can proceed of the following way: it takes an initial value, say x=1'5, that will be the first approximation, x_{1}. It defines x_{2} = cos(x_{1}) and, in general, die x_{n}, define x_{n+1}= cos(x_{n}). Build like this a succession of real numbers that approximate increasingly to the solution looked for x*, but without achieving it never. Show that the succession generated satisfies the following property:
.
In other words the terms stop are approximations by excess of the root looked for, increasingly near to her, whereas the terms stop are approximations by defect. Therefore, this procedure provides an approximate solution with arbitrary precision %u03B5 (positive number arbitrariamente small). Detaining the process when the difference between two successive approximations are lower that %u03B5 will achieve a lower error that %u03B5.
The method described can seem extracted of the sleeve but has, in reality, an intuitive motivation quite simple.
Resolution of a no linear equation by successive approximations: Resolve the equation x = cos(x) can reformularse, geometrically, as the research of the intersection of the curve and=cos(x) with the straight and=x. The point of intersection can approximate building a "cloth of spider" that "acorrale" to the point, such and as it suggests the figure.
We observe, finally, that can be so many the equations to resolve simultaneously that only can address the problem by means of special methods and very efficient. For example, in the automatic control of the traffic of vehicles in the systems of real time, the system owe to be able to react, in fractions of second, to the sudden apparitions of obstacles providing a new route of trip.
The previous observations allow to define the aim of the numerical analysis as it follows: find methods of calculation that allow to resolve mathematical problems complexes of efficient form and with an admissible error predetermined.
Since the problems of calculation are present in almost all the fields of the mathematics, comprise easily that the numerical analysis was in narrow relation with many other disciplines, mainly with the linear algebra, with the optimisation, with the mathematical analysis and with the computing. Each course contains at least an asignatura of numerical analysis -Laboratory of Mathematical (1º), Extension of Laboratory of Mathematical (2º), Numerical Methods (3º), Methods of Mathematics Applied (4º) and Numerical Calculation (5º)-, to which it is necessary to add a wide offer of asignaturas optativas in the 2º cycle: Matrix Computation, Extension of Numerical Analysis, Laboratory of Symbolic Analysis and Extension of the Methods of Mathematics Applied.
Engineers, economists, political and military enfrentan with some frequency to situations in which have to choose between diverse alternative options, generally infinite, so as to extract the maximum utility of the available resources or, minimizar the damages if the circumstances are adverse. Like this, schedule the production of a company consists in assigning the hand of work, the machinery, the vehicles and the prime matter to each one of the activities to realise (for example, to each one of the lines of open production), and has to do so that the flow of box generated (the difference between income and expenses) was the possible elder. The ranchers decide the most convenient diet for each animal to his charge in function of his age and weight, and this diet has to guarantee the susbsistencia healthy, if the farm produces milk or eggs, whereas it has to minimizar the cost of the kilo of meat in channel if you treat of a farm of fatten. Análogamente, the services of emergency (parks of firemen, services of ambulances, etc.) They are situated minimizando the times of expect to the points of demand, the dangerous activities (nuclear head offices, centres of chemical experimentation, etc.) They realise minimizando the risks for the civil population, the military deployments effect minimizando the costs of transport, and like this successively.
In situations of decision like which finish to mention looks for the maximum or the minimum of a function, designated objective, that depends of the variable calls of decision (for example, the daily kilos of each class of think commercialised that will compose the diet of an animal) and that measures the utility or loss that associated to each possible decision. In general, such variables have to fulfil some requirements (owed to the budgetary limitations, to the minimum need of each one of the nutrients in the diet, etc.) That formulate by means of equations or inecuaciones.
The operative investigation uses the metodo scientific to address takes it of decisions, almost always concernientes to the optimum allocation of scarce resources, by means of the building and resolution of mathematical models. Such models are resolved by means of specific algorithms, of such form that the operative investigation is related with the economic sciences, by the nature of the problems that treats to resolve, with the computing, that facilitates the numerical resolution of the same, and with the remaining mathematical disciplines, that facilitate conceptual tools for the analysis of the models and allow the design of the algorithms. The denomination proviene of the mixed groups created in the armies allies during the II World-wide War for the optimum design of strategies of fight and of operations of supply.
The most known model of the operative investigation is the one of the linear programming, that looks for the optimisation of a linear objective function that is subject to some also linear restrictions. Geometrically, the problem consists in the research of a vértice optimum of between the finite vértices of a geometrical figure (a polyhedron of some dimension) that is the intersection of the semiespacios clear-cut by the restrictions. In the language of the economic sciences, this problem corresponds to a linear model of production that pursues to determine the level of optimum production for a company. They exist algorithms very efficient to resolve problems of linear programming. The most known of such methods is the simplex of Dantzig (1947), that allows to jump, in each step, of a vértice of the polyhedron until another better, until achieving, finally, a vértice optimum. The fact to select the best vértice between a finite group of them converts the problem of linear programming in a discreet problem. However, the discreet problems are not necessarily easy if they exist a lot of possible alternatives.
The Problem of the bridges of Königsberg (Leonhard Euler 1707-1783): ¿There Is a way that go through the seven bridges on the river Pregels, in Königsberg, and that cross each one of them exactly once? The answer depends of if we demanded, or no, the return to the point of game.
Often, in the models of optimisation, it is necessary to incorporate the exigencia that the solution have whole components (an incógnita can represent a true number of vehicles or of workers to which goes to encomendar true task). Such problems designate of combinatory optimisation, and can be very difficult to resolve, even using computers very powerful. The theory of the complexity treats to find a mathematical explanation to such computational difficulties.
Other important generalisations of the problems of linear programming present when it substitutes the linear objective function by general functions (no linear), or when the restrictions are not linear. The Greek mathematicians already posed problems of opimización no linear, as the isoperímetricos: ¿which is the flat figure of main area that can shut inside a perimeter of length given?
A problem of Fermat (1601-1665): Connect n points of the plane by means of a network of ways of minimum length. As Maupertius, the laws that figuran in the "book of the nature" (of the that spoke Galileo) formulate like problems of optimisation. In effect, the stable structures are those that provide a local minimum for some function (called function of potential). In the case of the soapy films supported in a frame given, said potential is, simply, the surface of the film (Plateau, 1801-1883). This type of surfaces "minimales" use a lot in architecture and engineering (the olympic complex of Munich is a good example). With help of two transparent plates and n pieces of wire of equal length can resolve the problem of Fermat. The figures To and B of up mustran the natural "solution" to the problem of Fermat with 4 and with 5 points.
The peculiarity of the problems of optimisation with objective function no linear or restrictions no linear consists in the possible existence of a lot of optimum local, that are those that allow to approximate the classical methods. By this reason have developed , during the last years, methods of global optimisation, that pretend to approximate an optimum solution of the problem. Such methods employ technical that often are of discreet or probabilistic nature. Many times such technicians are inspired by the nature (the evolution of the species like consequence of the genetic improvement) or of the technical (the temperate of the metals, that toughens them by means of successive warmings and coolings).
Multiplicity of local maxima in functions no linear: The figure shows the graphic of a function of two variables that take values between 0 and 10.
The Plan of Studies of the Degree in Mathematics contains an asignatura of Operative Investigation (in 2º course), that uses notions of Linear Algebra and of Mathematical Analysis II, and another devoted specifically to Optimisation (5º), whose foundations establish in Convex Analysis (of 3º). In these two asignaturas study systematically the problems of optimisation no linear.
Statistical means compilation, analysis and interpretation of data, and owe his name to his birth, like consequence of the need recaudatoria of the European states. It uses the statistics to realise censuses of population and to predict his future evolution. Also to examine the efficiency of the new medicines in the clinical studies, for the analysis and prediction of the natural catastrophes, in the design of agricultural experiments to find new species and abonos, to compare the images that the consumers have of different products that compete in a same market, in the controls of quality in the industrial manufacture or to predict the evolution of the economic magnitudes and, for example, the price of the dollar, the number of unemployed or the GDP. It can interest evaluate the time of curing of a medicine, predict the evolution in the short term of the number of unemployed or verify the influence of the interventions of the banks of broadcast in the market of capitals. The difficulty estriba in that the data of a survey, of a random experiment, of a clinical study or of an experiment any one are, in general, incomplete, very bulky and, besides, are used to to be contaminated by factors controlled or no controlled.
The statistics uses deterministic mathematical methods to analyse the data, and resort to the theory of the probability when you treat to obtain valid conclusions (infer) for all the population from the data contained in a sample (being able to measure the confidence that can have in the validity of such conclusions).
The level of life in the regions alicantinas: they Are many the appearances that inciden in the quality of life, as the level of income, the available comforts in the homes, the consumption of goods, the access to the educational and sanitary services, etc. The analysis of data allows, in these circumstances, build a scale, called level of life, that accumulates on an only axis all the available information (this technician of recucción of the dimension receives the technical name of analysis of main components). The scale of up was obtained by professors of the Department of Statistics and Operative Investigation of the University of Alicante, with data of 1985.
The asignatura of Analysis of Data (of 5º) is based in the geometry and the optimisation, but does not resort to the theory of the probability put that, in general, suppose available the relative data to all the population (and no of a random sample of the same). It does not occur the same with Statistics and his Extension (both of 3º, devoted the 2ª of them to the study of the linear models, that are intensivamente applied in sciences of the nature and in the technician), neither with Temporary Series and Prediction (of 4º) and Methods Econométricos (of 5º), asignaturas that also resort to the statistical inference.
As the classical definition, the central aim of the economy is the efficient allocation of scarce resources. The actors of the economy are the called economic agents, that can be individual (consumers, producing), or collective (groups of companies, families, institutions). The object of production or exchange between the economic agents are the commodities or services. The different agents produce, consume or exchange commodities following his own interest, inside a true group of actions that result them feasible. The big questions of the classical paradigm were two: (1) ¿they are compatible the individual actions of the different agents? And, (2) The results of the individual interaction, ¿are desirable from the social point of view?
The precise answer to them ask previous only was possible thanks to the development of suitable economic models (in the case mentioned, the models of general balance). A basic tool for the derivación of consequences in the classical models was the systematic employment of mathematical technicians. In the example of the models of general balance, the technical employees were convex analysis, theorems of fixed point, and mathematical programming.
The success of the employment of mathematical technicians in the models of general balance was the detonante that carried to the economists to pose new questions and to resolve them with mathematical tools, and, in a lot of cases, to create the suitable new technicians for the treatment of the new models. This interaction has carried to pose, the recent dates, new paradigms in the Economic Analysis, between which highlight the systematic studies of relations of conflict, the strategic behaviours, and the possibilities of manipulation of the asymmetries of information in favour of the better agents informed. The theory of games and his derivaciones, the problematic of the social interactions, the conflicts between justice and efficiency, are not more than examples of this new paradigm that has given origin to a mixed matter that in a lot of manuals know like Mathematical Economy.
An immediate consequence of the previous is the utilisation generalised of mathematics in almost all the economic disciplines, with main or lower degree of sophistication.
The bases of the economy study in two big matters: Microeconomía, that studies the behaviour individualizado of the economic agents and his interactions, and Macroeconomía, that analyses the behaviour of the big added economic.
As well as the Microeconomía has benefited to a large extent of tools like the previously indicated for his development, the Macroeconomía has used more extensivamente the analysis of data and the behaviour of the different statistical variables. The employment of statistical technicians in the economic modelling has carried to the birth of another important mixed matter: the Econometría. The employment of methods econométricos is particularly fruitful in the analysis of the financial variables and of the instruments of control of said variable, giving also origin to a new branch of the economy that today day entitle like Quantitative Finances.
On the other hand, the applications of this new form of analysis of the economic and social relations are extending (in each case with his specific peculiarities) to the study and systematisation of a lot of types of markets and/or matters: the economy of the transport, the economy of the health, the economy of the information, or the new "call economy", are not more than examples of new branches of study in which the cooperation between new economic models, mathematical tools, and suitable use of statistical technicians are giving clear answers to a lot of questions that, till lately, lacked all systematisation.
The Theory of Matrices (2º) enters elements of economic mathematics that are based in the linear algebra, being able to consider an introduction to the Linear Operators no Negative (4º). These tools are particularly useful in the models applied of general balance, in which the relations of production modelizan sectorialmente, following a diagram input-output.
The Economic Theory (3º) studies the bases of the classical models of general balance, and his main extensions. The Mathematical Methods for Economy (4º) study some mathematical technicians that have been born because of the needs of the economic models, as the functions multivaloradas, some results of fixed point and extensions, the bases of the Theory of Games, and his generalisations to the called abstract economies. The Theory of the Decision and his Extension (both of 5º course) study the most recent models of the Mathematical Economy: Problems of collective decision, arbitraje, theory of games with asymmetries of information, manipulation and mechanisms of allocation.. The Temporary Series and Prediction (of 4º) studies some models econométricos basic for the study of added and financial variables, and the Operative Investigation (of 2º course) provides the bases of the individual decision.
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