Equações algébricas, máxima absorção ressonante em espectroscopia Mössbauer, problemas diretos e inversos em espalhamento de partículas e problema inverso em dinâmica. - Jesse Moreira Oliveira (2014)
In Chapter 1, begins the study of algebraic equations, more specifically of cubic equations. The chapter begins with a brief historical introduction which illustrates the challenges faced by mathematicians to reach a solution. Shows up, then the most famous method of resolution, which is assigned to the mathematician GirolanoCardano and is known as a method of Cardano. However, we highlight that in using this method there is the appearance of some strange roots that must be carefully analyzed. The objective of this chapter is to show an analytic form, quickly and efficiently, to calculate the real roots of such equations. With this goal in mind, and the fact that appear strange roots in Cardano's method, is shown following the chapter alternatives of solving using complex numbers and hyperbolic functions which are illustrated by an example solved. Well, but the reader until reading this paragraph might be thinking we're presenting a thesis of mathematics, however, it is good to emphasize that such work is a thesis of chemistry. The origin of the algebraic equations study was motivated by Titrations, i.e. the area of chemistry that investigates the titrations. Shows up in some sections of this chapter application of cubic equations in titration weak acid with strong base and vice versa: titrations that naturally lead to cubic equations, whose resolutions are often taught roughly by textbooks. The purpose of these sections is to show a way to treat the problem of titling without approximations. Show in graphic form, some results obtained for titrations in certain concentrations, compared with results obtained by a famous text book. Another motivation for the study of cubic equations, originated in physical chemistry, more specifically the problem of real gases. It is known that the greatness which measures the greater or lesser degree of ideality of a gas is the compressibility factor, which is simply the ratio of the volume of the real gas and ideal gas. The real gas is often described by the theory of van Der Waals, which leads to an equation for calculating the cubic volume. Again apply the analytical methods for solving cubic equations, and is an important equation that allows us to obtain the compressibility factor for a real gas: something that is usually done only for numerical calculations in text books. The fundamental physical arguments to the description of the resulting resonance line of transitions of quantum States induced by electromagnetic radiation are often not sufficiently well understood by novice students in natural sciences or non-specialists in spectroscopy. Textbooks in general tend to treat the problem on the basis of complex theoretical formalisms, inaccessible to all those who are not sufficiently familiar with the arguments advanced mathematical and physical of quantum mechanics. In Chapter 2, algebraic developments are presented more straightforward and simpler with the intent and offer an alternative didactics and serve as a source of reference to readers with varying degrees of theoretical domain in advanced physics. The central point is to show that the resonance line width of gamma radiation is conceptually twice the natural line width for energy related to the nuclear State excited of the transition. The algebraic arguments are used in order to offer a direct pathway and deductive, intentionally more readily achievable than those now found in the scientific literature. This case study was chosen from the particular perspective of physical principle corresponding to the nuclear spectroscopy Mössbauer. Generally speaking the problems of applied mathematics can be classified into direct and Inverse Problems Problems. Such classification is based, in General, in cause-and-effect relationships. A direct problem as one that seeks to determine the effect from a cause known or observed, for example, you kick a ball with a force (known), F, and are interested in finding out the range x. the inverse problem consists in determining the cause which led to right effect. Example, knowing the range, x, of the ball, it is intended to determine the force, F, with which the ball was kicked. Inverse problem-solving frequently used mathematical techniques more refined, since such problems often take into account the experimental error (example, the reach of the ball is not accurate since it is measured with an instrument) and measures that can make you not have unique solution or even not show solution (for example, kicks in different angles and forces, can take the ball to the same scope). Problems of this nature are classified as badly placed. In Chapter 3, a brief introduction to inverse problems. We decided to in this chapter by introducing the theme, showing some historical problems and its applications. The choice of apesentados problems was made based on the mathematical difficulty of resolution, that is, had-if you care to choose, problems whose resolution could be accompanied by any reader with simple math knowledge. Shows the mathematical representation of a direct and inverse problem and is one of the methods of solving such problems (namely, least squares). He concludes this chapter, showing an important quote demonstrates the importance of studying such problems. In chemistry an important example of inverse problem occurs in kinetics. On kinetic, knowing the speed of reactions (effect), seeks to investigate the rate constants (cause). Such study philosophy is exactly the same as found in the dynamics of populations, as well as in the dynamics of growth of living organisms. Motivated by the similarity with the problem of chemical kinetics Chapter 4 comes to show the inverse problem of growth of fish. Shows the von Bertalanfy differential equation for growth of fish and its resolution. The following sections of chapter illustrate the resolution of inverse problem, namely the problem of the determination of the causes (of growth) from experimental data. The data were supplied by a company that specializes in fish and refer to Thai tilapia (Oreochromisniloticus). There are a few articles in the literature showing the resolution of this problem with the method of least squares (M.Q.), however, it was noticed some difficulty of convergence of the method, yielding inconsistent results. Thus, it was decided to introduce a small damping factor in the method, i.e. snapped using a more robust method, known as Levenberg-Marquardt (Boms). Shows up in chapter tackling the problem with this method, since the whole resolution was made with a computer program written in Matlab language, by our group. Robustness of tests were conducted to check Boms method your behavior in misplaced problems, it was decided to add a lot of noise (errors) to data and parse the result, which is illustrated in one of the figures of the chapter. Both the Bom as M.Q. method uses inversion of matrices in its Constitution and therefore as a consequence are susceptible to experimental errors. In this way, we decided to attack the problem of dynamics of fish by a technique out of matrix inversion. It was decided to use artificial neural networks, being M.L.P architecture networks and Hopfield our chosen. Wrote computer programs for solving this problem with the networks mentioned, gettingfit at the same time, two mathematical models to data, something that is unheard of in this type of work. Chapter 5 shows the inverse problem of the scattering angle. The inversion methodology Firsovwas used to retrieve the potential energy function for interacting helium-helium (the choice of the system was motivated by the presence of a potential that describes with precision) from angles of scattering. Using a combination of simulated data accurate for large scattering angles and a Lennard-Jones potential for small angles it was possible to recover the potential short-range in excellent agreement with theoretical results. Scattering angle errors ranged from 1% to 10% for the collision energy of 2 x 10-3 eV to 2 eV. The inverted Potential was obtained with an accuracy of 2% to 8%. This chapter explores the possibility of using the approach of low collision energies Firsov, unlike previous works on the subject. The method proved robust, stable against experimental errors and very easy to be implemented numerically. Chapter 6 shows the inverse problem of quantum scattering within the Born approximation. The formalism of quantum mechanics, we obtain the scattering amplitude function whose square module is the differential cross section (important parameter in Atomic Collisions). It is shown in the chapter which, with certain approximations (Born Approximation), this function has a direct relationship with the potential energy function and, therefore, knowing that function (which measures how the interaction between the particles) can determine the scattering amplitude and, consequently, the differential cross section. The problem of determining the amplitude of scattering (effect) from the potential energy (cause) is what we call direct problem of quantum scattering of particles. This problem is resolved in this chapter, by a computer program, written by our group to the prototype system H-H (the choice of the system was motivated due to the presence of a potential that describes with great precision the system). The inverse problem is equivalent to determining the potential energy function (cause) from the scattering amplitude (effect), which is obtained from the experimental data of cross section. Non-linear Inverse problems with analytical solutions are rare occurrences, however, using Fourier integrals achieved analytical solution to the inverse problem of quantum scattering of particles. It has been proven that this inversion analytical process is able to recover the potential as was done for the Yukawa potential (the choice of this potential was attributed to be a short-range potential, as required by Born approximation, with applications in chemistry). Shows the numerical resolution of inverse problem, using Fourier integrals, by a simple algorithm. The resolution was made using a computer program, written in Matlab language, capable of integrating the integral equations, taking other actions and, finally, to solve the inverse problem. Shows up at the end of the chapter the results of inverse problem. You can see the robustness of the method, because it is able to recover all the potential range. The errors shown in the process of reversal if the integrator used numerical error (rule of the rectangle with 500 rectangles) and the limits of integration, because it is an improper integral with infinite upper bound which was numerically approximated by a large number (40).