Homogeneous equation, definition

This definition can be expressed in an alternative form by noting that the equations (29.1) can be considered as r homogeneous equations in the variables. .. M. The condition that these equations are linearly independent is that, from the table of coefficients (called a matrix)... 

Our simple definition of morphisms for affine algebraic sets does not work for projective algebraic sets. The trouble is that it automatically implied that the morphism will extend to a morphism of the ambient affine space. There is no analogous fact in the projective case. Look at the case of Example D. Let Ei be the conic with homogeneous equation... 

Actually Eq. (50) is a homogeneous equation so that a more general definition would be that (r, E.Q) is proportional to (rather than "is") the total number of neutrons.In the present work we assume a normalization that sets the proportionality constant to unity. Moreover, all the adjoint functions considered in this section could, by a proper normalization, be referred to the same detector. 

This reaction is said to be homogeneous if it occurs within a single phase. For the time being, we are concerned only with reactions that take place in the gas phase or in a single liquid phase. These reactions are said to be elementary if they result from a single interaction (i.e., a collision) between the molecules appearing on the left-hand side of Equation (1.7). The rate at which collisions occur between A and B molecules should be proportional to their concentrations, a and b. Not all collisions cause a reaction, but at constant environmental conditions (e.g., temperature) some definite fraction should react. Thus, we expect... 

Neal and Nader [260] considered diffusion in homogeneous isotropic medium composed of randomly placed impermeable spherical particles. They solved steady-state diffusion problems in a unit cell consisting of a spherical particle placed in a concentric shell and the exterior of the unit cell modeled as a homogeneous media characterized by one parameter, the porosity. By equating the fluxes in the unit cell and at the exterior and applying the definition of porosity, they obtained... 

As done below for two examples, expressions can also be derived for the scalar variance starting from the model equations. For the homogeneous flow under consideration, micromixing controls the variance decay rate, and thus y can be chosen to agree with a particular model for the scalar dissipation rate. For inhomogeneous flows, the definitions of G and M(n) must be modified to avoid spurious dissipation (Fox 1998). We will discuss the extension of the model to inhomogeneous flows after looking at two simple examples. 

Sigma (a) bonds Sigma bonds have the orbital overlap on a line drawn between the two nuclei, simple cubic unit cell The simple cubic unit cell has particles located at the corners of a simple cube, single displacement (replacement) reactions Single displacement reactions are reactions in which atoms of an element replace the atoms of another element in a compound, solid A solid is a state of matter that has both a definite shape and a definite volume, solubility product constant (/ p) The solubility product constant is the equilibrium constant associated with sparingly soluble salts and is the product of the ionic concentrations, each one raised to the power of the coefficient in the balanced chemical equation, solute The solute is the component of the solution that is there in smallest amount, solution A solution is defined as a homogeneous mixture composed of solvent and one or more solutes. 

Equation 2.63 is valid for any homogeneous or heterogeneous reaction. The only difference is in the definition of activities. For a species in a perfect gas-phase mixture a = pi/p°, where pi is the partial pressure of species i andp° is the standard pressure (1 bar). For a real gas-phase mixture a =f/p°, where is the fugacity of i. The fugacity concept was developed for the same reason as the activity to extend to real gases the formalism used to describe perfect gas mixtures. In the low total pressure limit (p -> 0), fi = pi. 

Two important ways in which heterogeneously catalyzed reactions differ from homogeneous counterparts are the definition of the rate constant k and the form of its dependence on temperature T. The heterogeneous rate equation relates the rate of decline of the concentration (or partial pressure) c of a reactant to the fraction / of the catalytic surface area that it covers when adsorbed. Thus, for a first-order reaction,... 

After an introductory chapter we review in Chap. 2 the classical definition of stress, strain and modulus and summarize the commonly used solutions of the equations of elasticity. In Chap. 3 we show how these classical solutions are applied to various test methods and comment on the problems imposed by specimen size, shape and alignment and also by the methods by which loads are applied. In Chap. 4 we discuss non-homogeneous materials and die theories relating to them, pressing die analogies with composites and the value of the concept of the representative volume element (RVE). Chapter 5 is devoted to a discussion of the RVE for crystalline and non-crystalline polymers and scale effects in testing. In Chap. 6 we discuss the methods so far available for calculating the elastic properties of polymers and the relevance of scale effects in this context. 

Hence, equilibrium constants of homogeneous electron-transfer reactions between (A) and B are evidently connected to the differences in reduction potentials of A and B. This connection reflects a definite physical phenomenon. Namely, if two redox systems are in the same solution, they react with each other until a unitary electric potential is reached. For the transfer of only one electron at room temperature, the following simplified equation can be employed ... 

As indicated above, studying the reaction at constant volume in a single homogeneous phase greatly simplifies the analysis of the results (see also Chapter 4). There are, however, many industrial processes carried out in flow or stirred reactors in which the volume cannot be assumed constant. In such cases, the rigorous definition of the rate equations in terms of the extent of reaction must be used. 

However the obtained system of the equations remains still too difficult for analyzing. For definiteness the (6,0) SWNT has been studied (N=6). To facilitate the solution of the equation system (8) we shall research first of all the case when only one oscillation mode (k=0) is induced. Obviously, the case corresponds to the oscillations, which are homogeneous along SWNT perimeter. For further simplification of the equation system we shall suppose that (pG = (p G, y/a = f/. Using Lorenz s invariance property for running... 

Equation (3) has the same form as one of Gibbs s fundamental equations for a homogeneous phase, and owing to this formal similarity the term surface phase is often used. It must be remembered, however, that the surface phase is not physically of the same definiteness as an ordinary phase, with a precise location in space neither do the quantities c , if, mf refer to the total amounts of energy, entropy, or material components present in the surface region as it exists physioally they are surface excesses , or the amounts by which the actual system exceeds the idealized system in these quantities. Care must be taken not to confuse the exact mathematical expression, surface phase , with the physical concept of the surface layer or surface film. 

The definition of a partial molar property, Eq. (11.2), provides the me-for calculation of partial properties from solution-property data. Implicit in definition is a second, equally important, equation that allows the calculation solution properties from knowledge of the partial properties. The derivation this second equation starts with the observation that the thermodynamic propertl of a homogeneous phase are functions of temperature, pressure, and the numb of moles of the individual species which comprise the phase. For thermodyna property M we may therefore write... 

Frequency-dependent properties are in fact usually defined by a time-dependent generalization of this equation. For example, when a periodic homogeneous electric field is applied, then the time development of defines the frequency-dependent properties. In the limit of a time-independent field the same equation then serves as a definition of frequency-independent properties. 

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