Regular point, definition

A regular point of a function of n variables F(xj,..., x ) is defined as the point not being a critical point of this function. In other words, at a regular point, let it be for example the point x = 0, the function gradient (cf. definition (2.2)) does not vanish ... 

Short chains of amino acid residues are known as di-, tri-, tetrapeptide, and so on, but as the number of residues increases the general names oligopeptide and polypeptide are used. When the number of chains grow to hundreds, the name protein is used. There is no definite point at which the name polypeptide is dropped for protein. Twenty common amino acids appear regularly in peptides and proteins of all species. Each has a distinctive side chain (R in Figure 45.3) varying in size, charge, and chemical reactivity. 

The first definition is operational. It explicitly states criteria for deciding whether a player is a star. He is one who regularly scores an unusually large number of points or who in clever defensive acts, repetitiously and. To decide if a player is a star, you count his scoring or, alternatively, number the outstanding defensive plays he makes. 

Periodicity in space means that it repeats at regular intervals, known as the wavelength, A. Periodicity in time means that it moves past a fixed point at a steady rate characterised by the period r, which counts the crests passing per unit time. By definition, the velocity v = A/r. It is custom to use the reciprocals of wavelength 1/X — (k/2-ir) or 9, known as the wavenumber (k = wave vector) and 1/t — v, the frequency, or angular frequency u = 2itv. Since a sine or cosine (harmonic) wave repeats at intervals of 2n, it can be described in terms of the function... 

At the macroscopic level, a solid is a substance that has both a definite volume and a definite shape. At the microscopic level, solids may be one of two types amorphous or crystalline. Amorphous solids lack extensive ordering of the particles. There is a lack of regularity of the structure. There may be small regions of order separated by large areas of disordered particles. They resemble liquids more than solids in this characteristic. Amorphous solids have no distinct melting point. They simply become softer and softer as the temperature rises. Glass, rubber, and charcoal are examples of amorphous solids. 

The surface of a crystal is thus chemically unsaturated and the existence of a definite surface energy is due to the fact that the surface is unsaturated. Since the surface of a crystal consists of a regular lattice of orderly distributed atoms or ions the surface adhesional forces may be regarded as being distributed over the centres of these atoms (Haber, J.G.S.I. xxxiii. 50 Z it.f. Elektro-chem. XX. 521, 1914) although in all probability these points represent maxima in a continuous field. 

Line helices (chordal helices) are composed of dimensionless points. A chordal helix is uniform if all points are equivalent such a helix will have a circular cross-section and could be inscribed on a right circular cylinder. A nonuniform chordal helix is regular if it contains motifs repeated in a definite pattern and irregular if the motifs are not so repeated. For our purposes, chordal helices are useful mathematical abstractions. 

It has been explained that when testing mixture diagrams, factor space is usually a regular simplex with q-vertices in a q-1 dimension space. In such a case, the task of mathematical theory of experiments consists of determining in the given simplex the minimum possible number of points where the design points will be done and based on which coefficients of the polynomial that adequately describes system behavior will be determined. This problem, for the case when there are no limitations on ratios of individual components, as presented in the previous chapter, was solved by Scheffe in 1958 [5], However, a researcher may in practice often be faced with multicomponent mixtures where definite limitations are imposed on ratios of individual components ... 

The only way to avoid this is by strict analysis of the supply chain from the customer order to final product delivery. Definition of the optimized (theoretical) process and sequential work towards a high service level approach allow the identification of gaps, and of opportunities which might not always be the cheapest (ship versus train versus plane) but could be the most effective way to reduce capital costs and shorten planning scope - an important aspect, especially in volatile customer markets with long production processes on the (chemical) supplier side. As in the case of CIP, this needs clear parameters, KPIs, commitment from all players, and regular tracking. The most important parameters are the lead time for all products, optimal lot sizes, replenishment points, and safety inventories. 

The algorithm also ensures that a regular quadratic programme with a positive definite Hessian matrix is obtained at each step, provided that this is so for the initial point. Thus, although a "first-phase" procedure may sometimes be required to locate such a point, no rescue procedure is needed subsequently. 

Since the Flory interaction parameter, x> was derived by considering only interaction energies between the molecules, it should not contain any entropic contributions and Equation (2D-9) should yield the correct value for the Flory-/ parameter. Unfortunately, x contains not only enthalpic contributions from interaction energies, but also entropic contributions. The solubility parameter includes only interaction energies and by the definition of regular solutions does not include any excess entropy contributions. Blanks and Prausnitz (1964) point out that the Flory / parameter is best calculated from... 

Solids are classified as crystalline solids or amorphous solids. Crystalline solids, such as an ice cube or a sodium chloride crystal, have a definite melting point. Amorphous solids, such as a chocolate bar or glass, get softer and softer as the temperature is raised. The structures of crystalline solids feature regularly repeating arrangements of the constituent particles. The structure of amorphous solids is not regular, but something like that of liquids sometimes, amorphous solids are called supercooled liquids. ... 

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