Physical point of view

These conditions can be used sometimes for a reduction of the full geometry, but only if the problem is symmetrical or periodic from both the geometric and the physical points of view. 

These apparently paradoxical phenomena have been explained by Andronov 4 from the physical point of view, a major point being that at the two nonanalytic points impulsive inputs of energy occur which compensate for the continuous dissipation of energy on the analytic arcs of convergent spirals33 (compare with the theory of clocks of Section 6.9). 

Various in situ and ex situ methods have been used to determine the real surface area of solid electrodes. Each method10,15 32 67,73 74 218 is applicable to a limited number of electrochemical systems so that a universal method of surface area measurement is not available at present. On the other hand, a number of methods used in electrochemistry are not well founded from a physical point of view, and some of them are definitely questionable. In situ and ex situ methods used in electrochemistry have been recently reviewed by Trasatti and Petrii.73 A number of methods are listed in Table 3. 

From a physical point of view the change of the angular momentum per collision is... 

The model of non-correlated potential fluctuations is of special interest. First, it can be solved analytically, second, the assumption that subsequent values of orienting field are non-correlated is less constrained from the physical point of view. The theory allows for consideration of a rather general orienting field. When the spherical shape of the cell is distorted and its symmetry becomes axial, the anisotropic potential is characterized by the only given axis e. However, all the spherical harmonics built on this vector contribute to its expansion, not only the term of lowest order... 

From the physical point of view it is possible to suggest that the rate of bubble growth in micro-channel is determined by the following parameters ... 

Table 10.1 shows that only one stationary state is possible, when the meniscus is far enough from the outlet of the capillary. Contrary to this, when the capillary is filled, namely, the interface surface is near the outlet, and two and three stationary states are possible. From the physical point of view, these phenomena may be explained by the different contribution of the friction force due to the vapor. In the first case this force is dominant, whereas in the second its effect is negligible. 

From a physical point of view, the finite difference method is mostly based based on the further replacement of a continuous medium by its discrete model. Adopting those ideas, it is natural to require that the principal characteristics of a physical process should be in full force. Such characteristics are certainly conservation laws. Difference schemes, which express various conservation laws on grids, are said to be conservative or divergent. For conservative schemes the relevant conservative laws in the entire grid domain (integral conservative laws) do follow as an algebraic corollary to difference equations. 

As a rule, equations of gas dynamics are discontinuous. From a physical point of view it is fairly common to distinguish weak discontinuities relating to cutting waves and strong discontinuities relating to shock waves . For these reasons successive grid refinement can be made with caution when the accurate account of accuracy of numerical methods is performed. 

The gas motion near a disk spinning in an unconfined space in the absence of buoyancy, can be described in terms of a similar solution. Of course, the disk in a real reactor is confined, and since the disk is heated buoyancy can play a large role. However, it is possible to operate the reactor in ways that minimize the effects of buoyancy and confinement. In these regimes the species and temperature gradients normal to the surface are the same everywhere on the disk. From a physical point of view, this property leads to uniform deposition - an important objective in CVD reactors. From a mathematical point of view, this property leads to the similarity transformation that reduces a complex three-dimensional swirling flow to a relatively simple two-point boundary value problem. Once in boundary-value problem form, the computational models can readily incorporate complex chemical kinetics and molecular transport models. 

From the physical point of view it is obvious that there is a relationship between the distribution of density of a fluid and the geometric properties of the scalar field [/.To illustrate this we will proceed from the equation of motion... 

Model formulation. After the objective of modelling has been defined, a preliminary model is derived. At first, independent variables influencing the process performance (temperature, pressure, catalyst physical properties and activity, concentrations, impurities, type of solvent, etc.) must be identified based on the chemists knowledge about reactions involved and theories concerning organic and physical chemistry, mainly kinetics. Dependent variables (yields, selectivities, product properties) are defined. Although statistical models might be better from a physical point of view, in practice, deterministic models describe the vast majority of chemical processes sufficiently well. In principle model equations are derived based on the conservation law ... 

From the physical point of view there cannot exist, under equilibrium conditions, a measurable excess of charge in the bulk of an electrolyte solution. By electrostatic repulsion this charge would be dragged to the phase boundary where it would be the source of a strong electric field in the vicinity of the phase. This point will be discussed in Section 3.1.3. 

From a physical point of view, this new formulation includes exponential terms that are in agreement with the observed ab initio and experimental results. Moreover, it is easy to verify that the new expression converges to the classical one when r increases. That way, at long range, where the multipolar approximation is valid, the exponential part dies whereas, at short distances, the monopole-monopole interaction embodies a part of the penetration energy. Consequently, Emono-mono has the correct dependence at any range. 

For excitation of solutes with 0-0 transitions v0o>v (antiStokes spectral region of absorption), the situation is the opposite at the initial instant of time, the spectra are red-shifted as compared to the steady state spectra, Av1 (l)<0. In this case, the return of the spectrum to its normal position during configurational relaxation will lead to a blue shift with time. From the physical point of view, this means that the intermolecular energy excess, which the solvates possess before excitation, is partially converted into emitted energy leading to an increase in the radiation frequency with time. That is why the process may be called the up-relaxation of the fluorescence spectra. 

Finally, the third case corresponds to pumping of solutes with 0-0 transitions, Voo>v. In this case, the fluorescence spectrum, immediately after excitation, must be close to the steady state one and should not vary with time. From the physical point of view, this case corresponds to the situation, where solvates with a local field Rtl (corresponding to the equilibrium configuration in the excited state) are excited. 

These assumptions are partially different from those introduced in our previous model.10 In that work, in fact, in order to simplify the kinetic description, we assumed that all the steps involved in the formation of both the chain growth monomer CH2 and water (i.e., Equations 16.3 and 16.4a to 16.4e) were a series of irreversible and consecutive steps. Under this assumption, it was possible to describe the rate of the overall CO conversion process by means of a single rate equation. Nevertheless, from a physical point of view, this hypothesis implies that the surface concentration of the molecular adsorbed CO is nil, with the rate of formation of this species equal to the rate of consumption. However, recent in situ Fourier transform infrared (FT-IR) studies carried out on the same catalyst adopted in this work, at the typical reaction temperature and in an atmosphere composed by H2 and CO, revealed the presence of a significant amount of molecular CO adsorbed on the catalysts surface.17 For these reasons, in the present work, the hypothesis of the irreversible molecular CO adsorption has been removed. 

From the physical point of view the condition a < 3/2 and b < 3 can be interpreted as a criteria for a relativistic ideal gas to be degenerate. In other words, this means that if these conditions are satisfied, there exists a non-zero most probable velocity. 

As follows from the previous analysis for quasi and ordinary particles gases there exists a critical value of parameters a and b for which the least value of the distribution function for observable frequencies is observed. From the physical point of view this is in agreement with the absolute minimal realization of the most probable state. As in any equilibrium distribution, there is an unique most probable state which the system tends to achieve. In consequence we conclude that the observable temperature of the relic radiation corresponds to this state. Or, what is the same, the temperature of such radiation correspond to the temperature originated in the primary microwave cosmic background and the primitive quantum magnetic flow. 

Because of the basic requirements of collection and transport, sewer networks are normally dealt with from a physical point of view, i.e., the hydraulics and sewer solids transport processes have been focal points. From this point of view, new design and operational principles have been developed, to a great extent supported by numerical procedures and the ever-increasing capacity of computers. Under wet-weather conditions, the hydraulics and solids transport phenomena in a sewer play a major role, and the chemical and microbiological processes are typically of minor importance. Not surprisingly, interests devoted to urban drainage have focused on the physical behavior of the sewer. 

The solute-solvent system, from the physical point of view, is nothing but a system that can be decomposed in a determined collection of electrons and nuclei. In the many-body representation, in principle, solving the global time-dependent Schrodinger equation with appropriate boundary conditions would yield a complete description for all measurable properties [47], This equation requires a definition of the total Hamiltonian in coordinate representation H(r,X), where r is the position vector operator for all electrons in the sample, and X is the position vector operator of the nuclei. In molecular quantum mechanics, as it is used in this section, H(r,X) is the Coulomb Hamiltonian[46]. The global wave function A(r,X,t) is obtained as a solution of the equation ... 

From a physical point of view, system parameter values are never known precisely with the exception of zeros. They are fixed, for example, by the absence of physical connections between certain parts of a system. Also, in computing solutions, computers work with true zeros and fuzzy parameters. ... 

Because of the imprecise knowledge of nonzero system parameters, it is of interest to study system properties that rely on the internal connections of the process under study, and not on the specific numerical values of the system parameters. Among these system properties, structural observability makes the meaning of observability (in the usual sense) more complete from the physical point of view, because the real system involves parameters that are only approximately determined. Indeed, structural observability is a stronger property, as can be demonstrated following the proposition in Lin (1974). 

Neutron stars (NSs) are perhaps the most interesting astronomical objects from the physical point of view. They are associated with a variety of extreme phenomena and matter states for example, magnetic fields beyond the QED vacuum pair-creation limit, supranuclear densities, superfluidity, superconductivity, exotic condensates and deconfined quark matter, etc. 

Let us start by giving a brief introduction into the general method of constructing mixed phases by imposing the Gibbs conditions of equilibrium [23, 18]. From the physical point of view, the Gibbs conditions enforce the mechanical as well as chemical equilibrium between different components of a mixed phase. This is achieved by requiring that the pressure of different components inside the mixed phase are equal, and that the chemical potentials (p and ne) are the same across the whole mixed phase. For example, in relation... 

At least from the time of van Helmont on, the chemists when separating and describing gases usually examined the effect of the gases on animals and plants. Lavoisier (1777) understood that aerobic respiration is the process in which oxygen is consumed and carbon dioxide is produced, shortly after the discovery of oxygen (1771). However, it took about half a century to make aerobic respiration more comprehensible from a physical point of view (Joule 1843 Mayer 1845 and many others). 

Today, there an established software tool set does exist for the primary task, the calculation of modes and the description of field propagation. Approaches based on the finite element method (FEM) and finite differences (FD) are popular since long and can be applied to complex problems . The wave matching method, Green functions approaches, and many more schemes are used. But, as a matter of fact, the more dominant numerical methods are, the more the user has to scrutinize the results from the physical point of view. Recent mathematical methods, which can control accuracy absolutely - at least if the problem is well posed, help the design engineer with this. ... 

From the physics point of view, the system that we deal with here—a semiflexible polyelectrolyte that is packaged by protein complexes regularly spaced along its contour—is of a complexity that still allows the application of analytical and numerical models. For quantitative prediction of chromatin properties from such models, certain physical parameters must be known such as the dimensions of the nucleosomes and DNA, their surface charge, interactions, and mechanical flexibility. Current structural research on chromatin, oligonucleosomes, and DNA has brought us into a position where many such elementary physical parameters are known. Thus, our understanding of the components of the chromatin fiber is now at a level where predictions of physical properties of the fiber are possible and can be experimentally tested. 

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