Group, mathematical properties

A group is a set of abstract elements (members) that has specific mathematical properties. In general it is not necessary to specify the nature of the members of the group or the way in which they are related. However, in the applications of group theory of interest to physicists and chemists, the key word is symmetry. 

The standard deviation (denoted s or sd) is a measure of patient-to-patient variation. There are other potential measures but this quantity is used as it possesses a number of desirable mathematical properties and appropriately captures the overall amount of variability within the group of patients. 

The identity operation, designated as E, leaves the object unchanged. Although this operation may seem trivial, it is mathematically necessary in order to convey the mathematical properties of a group on the set of aU the synunetry operations apphcable to a given object. This point will be clarified later in this article. The reflection operation, designated as a, involves reflection of the object through a plane, known as... 

This development is appropriate only for saturated alkanes - - because there is no provision for atom identity, bond types, or number of hydrogen atoms in each skeletal group (except for the case of saturated hydrocarbons.) This branching index is essentially a mathematical property of graphs it does not adequately represent molecular graphs. Further, a single index would appear to be insufficient to relate to the wide variety of molecular properties, especially when biological and environmental properties are of interest. 

Note that the above discussion does not depend on knowing anything about the detailed functional form of wave functions. We do not have to solve the Schrodinger equation. The statements are based solely on the symmetry properties and the consequent mathematical properties of the symmetry group. 

The complete set of symmetry operations for the molecule together form a mathematical entity known as a group. The properties of such groups are interesting and will form the basis of the application of symmetry to molecular orbital problems. There are four rules that must be satisfied ... 

Consider the set of symmetry operations describing the symmetry of an actual object such as a molecule. Such a set of symmetry operations satisfies the properties of a group in the mathematical sense and is therefore called a synmietry point group. In most cases symmetry point groups contain a finite number of operations and are therefore finite groups. The properties of symmetry point groups will be considered in this section. 

Recent mathematical work suggests that—especially for nonlinear phenomena—certain geometric properties can be as important as accuracy and (linear) stability. It has long been known that the flows of Hamiltonian systems posess invariants and symmetries which describe the behavior of groups of nearby trajectories. Consider, for example, a two-dimensional Hamiltonian system such as the planar pendulum H = — cos(g)) or the... 

Other techniques that work well on small computers are based on the molecules topology or indices from graph theory. These fields of mathematics classify and quantify systems of interconnected points, which correspond well to atoms and bonds between them. Indices can be defined to quantify whether the system is linear or has many cyclic groups or cross links. Properties can be empirically fitted to these indices. Topological and group theory indices are also combined with group additivity techniques or used as QSPR descriptors. 

Dimensionless numbers are not the exclusive property of fluid mechanics but arise out of any situation describable by a mathematical equation. Some of the other important dimensionless groups used in engineering are Hsted in Table 2. 

The iatroduction of a plasticizer, which is a molecule of lower molecular weight than the resia, has the abiUty to impart a greater free volume per volume of material because there is an iucrease iu the proportion of end groups and the plasticizer has a glass-transition temperature, T, lower than that of the resia itself A detailed mathematical treatment (2) of this phenomenon can be carried out to explain the success of some plasticizers and the failure of others. Clearly, the use of a given plasticizer iu a certain appHcation is a compromise between the above ideas and physical properties such as volatiUty, compatibihty, high and low temperature performance, viscosity, etc. This choice is appHcation dependent, ie, there is no ideal plasticizer for every appHcation. 

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundaiy-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, separation of variables (not the classical concept) and the use of continuous transformation groups. The basic theoiy is available in Ames (see the references). 

I presented a group of subroutines—CORE, CHECKSTEP, STEPPER, SLOPER, GAUSS, and SWAPPER—that can be used to solve diverse theoretical problems in Earth system science. Together these subroutines can solve systems of coupled ordinary differential equations, systems that arise in the mathematical description of the history of environmental properties. The systems to be solved are described by subroutines EQUATIONS and SPECS. The systems need not be linear, as linearization is handled automatically by subroutine SLOPER. Subroutine CHECKSTEP ensures that the time steps are small enough to permit the linear approximation. Subroutine PRINTER simply preserves during the calculation whatever values will be needed for subsequent study. 

Molecular mechanics is a useful and reliable computational method for structure, energy, and other molecular properties. The mathematical basis for molecular models in MM3 has been described, along with the limitations of the method. One of the major difficulties associated with molecular mechanics, in general, and MM3 in particular is the lack of accurately parameterized diverse functional groups. This lack of diverse functional groups has severely limited the use of MM3 in pharmaceutical applications. 

A mathematical group, by definition, consists of a set of distinct elements G — E, A, B,C, D,..., endowed with a law of composition (such as multiplication, addition, or some other operation), such that the following properties are satisfied ... 

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