Finite objectives

Eq. (87) really describes a needle crystal which, without noise, has no side branches. The corresponding star structure then cannot fill the space with constant density and the amount of material solidified in parabolic form increases with time, only fike rather than like P for a truly compact (initially finite) object in two dimensions. 

A finite object that exhibits the highest possible symmetry is the sphere, whose point group symbol Kb is derived form the German word Kugel. It has an infinite number of rotation axes of any order in every direction, all passing through the center, as well as an infinite number of ah planes each perpendicular to a Coo axis. Obviously, K is merely of theoretical interest as no chemical molecule can possess such symmetry. 

A single crystal, considered as a finite object, may possess a certain combination of point symmetry elements in different directions, and the symmetry operations derived from them constitute a group in the mathematical sense. The self-consistent set of symmetry elements possessed by a crystal is known as a crystal class (or crystallographic point group). Hessel showed in 1830 that there are thirty-two self-consistent combinations of symmetry elements n and n (n = 1,2,3,4, and 6), namely the thirty-two crystal classes, applicable to the description of the external forms of crystalline compounds. This important... 

Our discussion has centered on a single nonflow process, the expansion of a gas in a cylinder. The opposite process, compression of a gas in a cylinder, is described in exactly the same way. There are, however, many processes which are driven by other-than-mechanical forces. For example, heat flow occurs when a temperature difference exists, electricity flows under the influence of an electromotive force, and chemical reactions occur because a chemical potential exists. In general, a process is reversible when the net force driving it is only differential in size. Thus heat is transferred reversibly when it flows from a finite object at temperature T to another such object at temperature T - dT. 

Another classification is based on the presence or absence of translation in a symmetry element or operation. Symmetry elements containing a translational component, such as a simple translation, screw axis or glide plane, produce infinite numbers of symmetrically equivalent objects, and therefore, these are called infinite symmetry elements. For example, the lattice is infinite because of the presence of translations. All other symmetry elements that do not contain translations always produce a finite number of objects and they are called finite symmetry elements. Center of inversion, mirror plane, rotation and roto-inversion axes are all finite symmetry elements. Finite symmetry elements and operations are used to describe the symmetry of finite objects, e.g. molecules, clusters, polyhedra, crystal forms, unit cell shape, and any non-crystallographic finite objects, for example, the human body. Both finite and infinite symmetry elements are necessary to describe the symmetry of infinite or continuous structures, such as a crystal structure, two-dimensional wall patterns, and others. We will begin the detailed analysis of crystallographic symmetry from simpler finite symmetry elements, followed by the consideration of more complex infinite symmetry elements. 

Symmetry, in one or other of its aspects, is of interest in the arts, mathematics, and the sciences. The chemist is concerned with the symmetry of electron density distributions in atoms and molecules and hence with the symmetry of the molecules themselves. We shall be interested here in certain purely geometrical aspects of symmetry, namely, the symmetry of finite objects such as polyhedra and of repeating patterns. Inasmuch as these objects and patterns represent the arrangements of atoms in molecules or crystals they are an expression of the symmetries of the valence electron distributions of the component atoms. In the restricted sense in which we shall use the term, symmetry is concerned with the relations between the various parts of a body. If there is a particular relation between its parts the object is said to possess certain elements of symmetry. 

So we know prior mechanical elements of the telephone. The point I am trying to make is that it is far easier to trace the history of a finite object than it is to track concepts. We have countless clocks and precursors but who first thought of measuring time Show me a monument to his honor. Only through autobiography will we ever know who was inspired to do what with what idea. 

The division of crystals into crystal systems and crystal classes is based on the symmetry of the crystal as a finite object, or the symmetry of a single unit cell. In a unit cell all of the corners are equivalent points, since by translation along the axes the entire pattern can be... 

Recognition of non-Euclidean geometries creates the new problem of describing the position and motion of finite objects in curved space, a problem which does not occur in Euclidean spaces that extend uniformly to inhnity. An object such as the distance between two points. 

During exploitation, the polymer products suffer deformation and friction effects, and finally these ones are destroyed, as effect of gradual accumulation of some structural modifications that diminish their resistance to repeated deformations. Two are the most important reasons of finite objects deterioration 1) polymer ageing ... 

Mie theory is for spheres of finite size, and the coefficients are for a point of infinitesimal size. For a finite object, the total absorption is related to the volume around this infinitesimal point, and the... 

In this chapter the problem of elastic light scattering, i.e. interaction of electromagnetic waves with finite objects, is discussed. A detailed overview of one of the widely used methods for plasmonics, the discrete dipole approximation (DDA), is presented. This includes the theory of the DDA, practical recommendations for using available computer codes, and discussion of the DDA accuracy. 

Levin, L. A. (1976). Various measures of complexity for finite objects, (axiomatic description). Soviet Math, 17(522). 

From a computational point of view there is a natural distinction between the molecular and the materials perspective which lies chiefly in the treatment of the surroundings a molecule is treated as a finite object in space, while spatial periodicity is imposed on the material description in order to describe its extensive nature. Today, so-called density functional theory (DFT) methods prevail for ground state calculations of both molecules and materials comprised of up to several hundred atoms. For example, bond lengths can typically be calculated with standard DFT methodologies to accuracies within a few hundredths of an angstrom. Excitations can also be calculated with significant accuracy using an extension of DFT known as time-dependent DFT (TD-DFT). Optical excitations, calculated at the TD-DFT level, often match experiments with an accuracy of ca 0.1 eV for well-behaved systems. 

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