BRANCHES OF MATHEMATICS

The field of mathematics can be divided into pure mathematics and applied mathematics. Pure mathematics is concerned with the study of abstract mathematical properties and systems with no concern for application. Applied mathematics deals with solutions to problems that have practical apphcations. Engineers use both pure and applied mathematics in the solution of problems. 

There are many branches of mathematics, but all of the branches are sometimes grouped broadly into three areas algebra, geometry, and analysis. Each area has several subspecialties or branches. The branches of mathematics most 

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It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

A mathematician would classify the SCF equations as nonlinear equations. The term nonlinear has different meanings in different branches of mathematics. The branch of mathematics called chaos theory is the study of equations and systems of equations of this type. 

Arithmetic It is a branch of mathematics that deals with real numbers and computations with these numbers. 

The description of fuzzy, local density fragments is facilitated by the use of local coordinate systems, however, some compatibility conditions of such local coordinate systems must be fulfilled, reflecting the mutual relations of the fragments within the complete molecule. Manifold theory, topological manifolds, and in particular, differentiable manifolds [153-158], are the branches of mathematics dealing with the general properties of compatible local coordinate systems. 

When the change in a variable, say Ax, approaches zero it is called an infinitesimal. The branch of mathematics known as analysis, or the calculus,... 

An important part of AIM is the analysis of the electron density using the branch of mathematics called topology. Topology is the study of geometrical properties and spatial relations... 

Analytic geometry is a branch of mathematics in which geometry is described through the use of algebra. Rene Descartes (1596-1650) is credited for conceptualizing this mathematical discipline. Recalling the basics, we can express the points of a plane as a pair of numbers with x-axis and y-axis coordinates, designated by (x, y). Note that the x-axis coordinate is termed the abscissa , and the y-axis the ordinate . 

The mathematical apparatus for treating combinations of symmetry operations lies in the branch of mathematics known as group theory. A mathematical group behaves according to the following set of rules. A group is a set of elements and the operations that obey these rules. 

Statistics is a branch of mathematics that involves the study of data. Probability is the study of chance. This chapter will refresh your understanding of common statistical measures, graphs, and probability. Before proceeding to the lesson, take this ten-item Benchmark Quiz to see how much you remember about statistics and probability. These questions are similar to the type of questions that you will find on important tests. When you are finished, check the answer key carefully to assess your results. Your Benchmark Quiz analysis will help you determine how much time you need to spend on statistics and probability, and the specific areas in which you need the most careful review and practice. 

Clearly, however, electrons exist. And they must exist somewhere. To describe where that somewhere is, scientists used an idea from a branch of mathematics called statistics. Although you cannot talk about electrons in terms of certainties, you can talk about them in terms of probabilities. Schrodinger used a type of equation called a wave equation to define the probability of finding an atom s electrons at a particular point within the atom. There are many solutions to this wave equation, and each solution represents a particular wave function. Each wave function gives information about an electron s energy and location witbin an atom. Chemists call these wave functions orbitals. 

The purpose of the lectures was to discuss various properties of the Hilbert schemes of points on surfaces. Although it was not noticed until recently, the Hilbert schemes have relationship with many other branch of mathematics, such as topology, hyper-Kahler geometry, symplectic geometry, singularities, and representation theory. This is reflected to this note each chapter, which roughly corresponds to one lecture, discusses different topics. 

A branch of mathematics concerned with the study and applications of graphs and their uses in kinetic behavior of linked processes. Graph theory, especially when applied to the dynamic nature of networks, has provided... 

Sets are very general mathematical objects that are used in many branches of mathematics. Here the focus is on finite sets, that is, sets with a finite set of elements. A key concept in set theory is that of the universal set, U, sometimes called the universe of discourse, which is an unordered collection of n elements x1,x2, , xk,.. . , xn and is given by... 

Group theory is a branch of mathematics that describes the properties of an abstract model of phenomena that depend on symmetry. Despite its abstract tone, group theory provides practical techniques for making quantitative and verihable predictions about the behavior of atoms, molecules and solids. Once the basic ideas are clear, these techniques are easy to apply, requiring only simple arithmetic calculahons. 

The remaining errors in the data are usually described as random, their properties ultimately attributable to the nature of our physical world. Random errors do not lend themselves easily to quantitative correction. However, certain aspects of random error exhibit a consistency of behavior in repeated trials under the same experimental conditions, which allows more probable values of the data elements to be obtained by averaging processes. The behavior of random phenomena is common to all experimental data and has given rise to the well-known branch of mathematical analysis known as statistics. Statistical quantities, unfortunately, cannot be assigned definite values. They can only be discussed in terms of probabilities. Because (random) uncertainties exist in all experimentally measured quantities, a restoration with all the possible constraints applied cannot yield an exact solution. The best that may be obtained in practice is the solution that is most probable. Actually, whether an error is classified as systematic or random depends on the extent of our knowledge of the data and the influences on them. All unaccounted errors are generally classified as part of the random component. Further knowledge determines many errors to be systematic that were previously classified as random. 

Topology is a branch of mathematics investigating relations between objects and object s properties pertinent to continuous transformations of one object into another [8]. These transformations may involve considerable deformations of the objects. However, no cutting of them or gluing their points together are allowed by the transformations. Topological singularity of such molecules as those... 

The special branch of mathematics known as numerical analysis has assumed an added importance with the extensive use of digital computers. Since these calculators perform only the fundamental operations of arithmetic, it is necessary that all other mathematical operations be reduced to these terms. From a superficial viewpoint it might be concluded that such operations as differentiation and integration are inherently better suited to analog computers. This is not necessarily true, however, and depends upon the requirements of the particular problem at hand. 

As already evident from the previous section, symmetry properties of a molecule are of utmost importance in understanding its chemical and physical behaviour in general, and spectroscopy and photochemistry in particular. The selection rules which govern the transition between the energy states of atoms and molecules can be established from considerations of the behaviour of atoms or molecules under certain symmetry operations. For each type of symmetry, there is a group of operations and, therefore, they can be treated by group theory, a branch of mathematics. 

In this form the 125 letters contain little or no information, but they are very rich in entropy. Such considerations have led to the conclusion that information is a form of energy information has been called negative entropy. In fact, the branch of mathematics called information theory, which is basic to the programming logic of computers, is closely related to thermodynamic theory. Living organisms are highly ordered, nonrandom structures, immensely rich in information and thus entropy-poor. 

Space groups (like point groups) constitute a very pretty branch of mathematics, but that (presumably) is not why chemists study them. Space groups are important to a chemist because they are essential to solving and interpreting crystal structures. Therefore, we conclude this chapter with three topics that relate space group theory directly to the use of X-ray crystallography to obtain chemically useful information. 

The use of computers in science is widespread. Without powerful number-crunching facilities at his disposal, the modern scientist would be greatly handicapped, unable to perform the thousands or millions of calculations required to analyze his data or explore the implications of his favorite theory. He (or his assistant) thus requires at least some familiarity with computers, the programming of computers, and the methods which might be used by computers to solve his problems. An entire branch of mathematics, numerical analysis, exists to analyze the behavior of numerical algorithms. 

To deal with the problem in a rigorous fashion, we couch it within the branch of mathematics called graph theory. A graph G = (V,E) is a finite collection V of vertices and a finite collection E of edges. Each edge (v,w") consists of an unordered pair of distinct vertices. Each edge and each vertex may in addition have a label specifying certain information... 

Many other types of groups have been studied. They are of interest in geometry, differential equations, topology, and other branches of mathematics. In physics and chemistry, groups are used in the study of quantum mechanics molecular, crystal, and nuclear structure electrical circuits, etc. 

STOICHIOMETRY. The mathematics of chemical reactions and processes. It relates to all the quantitative aspects of chemical changes, both mass and energy Stoichiometry is based on file absolute laws of conversion of mass and of energy and on the chemical law of combining weights. This basis makes stoichiometry as exact as any other branch of mathematics. 

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