Exact independent

A convenient analogy for understanding latent variables is reconstructing the spectrum of a mixture from the spectra of the pure chemicals contained in the mixture. The spectra of these pure chemicals would be the latent variables of the measured spectrum because they are not directly accessible in the spectrum of the mixture. However, PCs are not necessarily the spectra of the pure chemicals in the mixtures representing the samples. PCs represent whatever independent phenomena affect the spectra of the samples composing the calibration set. If one sample constituent varies entirely independently of everything else, and this constituent has a spectrum of its own, then one of the PCs will indeed represent the spectrum of that constituent. It is most unusual for any one constituent to vary in a manner that is exactly independent of any other. There is inevitably some correlation between the various constituents in a set of specimens, and any PC will represent the sum of the effects of these correlated constituents. Even if full independence is accomplished, there is dependence in that the sum of all constituents must equal 100%. Consequently, the PC representing that source of independent variability will look like the difference between the constituent of interest and all the other constituents in the samples. The spectrum of the constituent considered could be extracted mathematically, but the PCs will not look exactly like the spectrum of the pure constituent. 

Equation (2.16) is exact independent of the strength or the shape of the light pulse. 

The simple van t Hoff equation (Eq. 28) is not completely correct if the complex formation results in a change of the specific heat capacity ACp, in which case neither AH nor AS are exactly independent of temperature. A more precise form of the van t Hoff equation is [13] ... 

In this table any calculated e is about 90% of the exact independent-pair contributions, and n is the number of valence pairs (i. e. of bonds). 

Applications of quantum mechanics to chemistry invariably deal with systems (atoms and molecules) that contain more than one particle. Apart from the hydrogen atom, the stationary-state energies caimot be calculated exactly, and compromises must be made in order to estimate them. Perhaps the most useful and widely used approximation in chemistry is the independent-particle approximation, which can take several fomis. Conuiion to all of these is the assumption that the Hamiltonian operator for a system consisting of n particles is approximated by tlie sum... 

Defining EJh + oij, replacing v /(-co) by v r(0), since the difference is only a phase factor, which exactly cancels in the bra and ket, and assuming that the electric field vector is time independent, we find... 

It is important to recognize that thennodynamic laws are generalizations of experimental observations on systems of macroscopic size for such bulk systems the equations are exact (at least within the limits of the best experimental precision). The validity and applicability of the relations are independent of the correchiess of any model of molecular behaviour adduced to explain them. Moreover, the usefiilness of thennodynamic relations depends cmcially on measurability, unless an experimenter can keep the constraints on a system and its surroundings under control, the measurements may be worthless. 

The work depends on the detailed path, so Dn is an inexact differential as symbolized by the capitalization. (There is no established convention about tliis symbolism some books—and all mathematicians—use the same symbol for all differentials some use 6 for an inexact differential others use a bar tln-ough the d still others—as in this article—use D.) The difference between an exact and an inexact differential is crucial in thennodynamics. In general, the integral of a differential depends on the path taken from the initial to the final state. Flowever, for some special but important cases, the integral is independent of the path then and only then can one write... 

One may now consider how changes can be made in a system across an adiabatic wall. The first law of thermodynamics can now be stated as another generalization of experimental observation, but in an unfamiliar form the M/ork required to transform an adiabatic (thermally insulated) system, from a completely specified initial state to a completely specifiedfinal state is independent of the source of the work (mechanical, electrical, etc.) and independent of the nature of the adiabatic path. This is exactly what Joule observed the same amount of work, mechanical or electrical, was always required to bring an adiabatically enclosed volume of water from one temperature 0 to another 02. 

If the adiabatic work is independent of the path, it is the integral of an exact differential and suffices to define a change in a function of the state of the system, the energy U. (Some themiodynamicists call this the internal energy , so as to exclude any kinetic energy of the motion of the system as a whole.)... 

The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if... 

The nth virial coefficient = < is independent of the temperature. It is tempting to assume that the pressure of hard spheres in tln-ee dimensions is given by a similar expression, with d replaced by the excluded volume b, but this is clearly an approximation as shown by our previous discussion of the virial series for hard spheres. This is the excluded volume correction used in van der Waals equation, which is discussed next. Other ID models have been solved exactly in [14, 15 and 16]. ... 

The most conunon choice for a reference system is one with hard cores (e.g. hard spheres or hard spheroidal particles) whose equilibrium properties are necessarily independent of temperature. Although exact results are lacking in tluee dimensions, excellent approximations for the free energy and pair correlation fiinctions of hard spheres are now available to make the calculations feasible. 

This is the quasi-chemical approximation introduced by Fowler and Guggenlieim [98] which treats the nearest-neighbour pairs of sites, and not the sites themselves, as independent. It is exact in one dimension. The critical temperature in this approximation is... 

The preferable theoretical tools for the description of dynamical processes in systems of a few atoms are certainly quantum mechanical calculations. There is a large arsenal of powerful, well established methods for quantum mechanical computations of processes such as photoexcitation, photodissociation, inelastic scattering and reactive collisions for systems having, in the present state-of-the-art, up to three or four atoms, typically. " Both time-dependent and time-independent numerically exact algorithms are available for many of the processes, so in cases where potential surfaces of good accuracy are available, excellent quantitative agreement with experiment is generally obtained. In addition to the full quantum-mechanical methods, sophisticated semiclassical approximations have been developed that for many cases are essentially of near-quantitative accuracy and certainly at a level sufficient for the interpretation of most experiments.These methods also are com-... 

From the derivation of the method (4) it is obvious that the scheme is exact for constant-coefficient linear problems (3). Like the Verlet scheme, it is also time-reversible. For the special case A = 0 it reduces to the Verlet scheme. It is shown in [13] that the method has an 0 At ) error bound over finite time intervals for systems with bounded energy. In contrast to the Verlet scheme, this error bound is independent of the size of the eigenvalues Afc of A. 

The discovery of nbozymes (Section 28 11) in the late 1970s and early 1980s by Sidney Altman of Yale University and Thomas Cech of the University of Colorado placed the RNA World idea on a more solid footing Altman and Cech independently discovered that RNA can catalyze the formation and cleavage of phosphodiester bonds—exactly the kinds of bonds that unite individual ribonucleotides in RNA That plus the recent discovery that ribosomal RNA cat alyzes the addition of ammo acids to the growing peptide chain in protein biosynthesis takes care of the most serious deficiencies in the RNA World model by providing precedents for the catalysis of biologi cal processes by RNA... 

The first thing to realize about scattering by liquids is that individual molecules can no longer be viewed as independent scatterers. If a liquid were perfectly uniform in density at the molecular level, its molecules could always be paired in such a way that the light scattered by each member of a pair would be exactly out of phase with the other, resulting in destructive interference. No net scattering results in this case. The second thing to realize, however, is that density is not perfectly uniform at the molecular level. 

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