Mathematical proofs

While one can argue intuitively that the average cost per share for Mary does not exceed that for John, we illustrate a mathematical proof using inequalities, The average cost per share for John is equal to... 

In recent years some theoretical results have seemed to defeat the basic principle of induction that no mathematical proofs on the validity of the model can be derived. More specifically, the universal approximation property has been proved for different sets of basis functions (Homik et al, 1989, for sigmoids Hartman et al, 1990, for Gaussians) in order to justify the bias of NN developers to these types of basis functions. This property basically establishes that, for every function, there exists a NN model that exhibits arbitrarily small generalization error. This property, however, should not be erroneously interpreted as a guarantee for small generalization error. Even though there might exist a NN that could... 

A mathematical proof of the inverse relationship starts from the maximum entropy conditions... 

The justification associated with a refinement can be formal or informal it could even simply say, Joe said this will work. The verification techniques can be applied in varying degrees of rigor, from casual inspection to mathematical proof. In between, there is the more cost-effective option of making refinements the focus of design reviews and basing systematically defined test code on the specifications (see Figure 6.43). 

As is evident from the above, both the physics invoked to derive the potential of Equation 7.31 and the numerical results presented show that Wx gives an accurate exchange potential for the excited states. When the proposal was initially made, there was no mathematical proof of the existence of a Kohn-Sham equation for excited states. It is only during the past few years that DFT of excited states [34-37], akin to its ground-state counterpart, is being developed. 

We are now in a position which allows us to give a microscopic analysis of electrophoresis. However, the detailed calculations rapidly become extremely complicated and we shall often be obliged to replace strict mathematical proofs by physical plausibility arguments. 

In the following an explicit mathematical proof is presented to show that symmetry numbers factors do not lead to isotope enrichment. That result should come as no surprise since the factor on the right hand side of Equation 4.118 can be identified as... 

If equal bond dipoles act in opposite directions in three-dimensional space, they counteract each other. A molecule with identical polar bonds that point in opposite directions is not polar. Figure 1.5 shows two examples, carbon dioxide and carbon tetrachloride. Carbon dioxide, CO2, has two polar C=0 bonds acting in opposite directions, so the molecule is non-polar. Carbon tetrachloride, CCI4, has four polar C—Cl bonds in a tetrahedral shape. You can prove mathematically that four identical dipoles, pointing toward the vertices of a tetrahedron, counteract each other exactly. (Note that this mathematical proof only applies if all four bonds are identical.) Therefore, carbon tetrachloride is also non-polar. 

I d still like to see mathematical proof that the "explosion" of water is not explained by ... 

It is not hard to understand why many metals favor an fee crystal structure there is no packing of hard spheres in space that creates a higher density than the fee structure. (A mathematical proof of this fact, known as the Kepler conjecture, has only been discovered in the past few years.) There is, however, one other packing that has exactly the same density as the fee packing, namely the hexagonal close-packed (hep) structure. As our third example of applying DFT to a periodic crystal structure, we will now consider the hep metals. 

We summarize results of relaxation analysis and describe the algorithm of approximation of steady state and relaxation in Section 4.3. After that, several examples of networks are analyzed. In Section 5 we illustrate the analysis of dominant systems on a simple example, the reversible triangle of reactions A2<- A3 A. This simplest example became very popular for the lumping analysis case study after the well-known work of Wei and Prater (1962). The most important mathematical proofs are presented in the appendices. 

Folger, T. (1990) Shuffling into hyperspace. Discover. 12(1) 66-64. (Mathematical proof of a perfect card shuffle.)... 

Identifying a president famous for a mathematical proof Naming a world conqueror who did a proof while in exile Pointing out some contemporary figures... 

When a scalar potential is set upon a transmission line, it speeds down the line at nearly light speed, revealing its vector nature. When it is set onto the middle of the transmission line, it speeds off in both directions simultaneously, revealing its bidirectional vector nature. In addition to this observation, there is rigorous mathematical proof as well. 

It is apparent that we can always get a set of 3 x 3 matrices, which form a representation of a given point group, by consideration of the effect that the symmetry operations of the point group have on a position vector. Why this works is shown pictorially in Fig. 5-3.2 for the a = a TCt operation of, T. The symmetry operation C8 on the position veotor p followed by a r on p produces a vector p which is coincidental with the one produced by the operation o on p. The matrices D(Ct), D(a r), X>(o ) then simply mirror what is being done to the point vector. The general mathematical proof that, if symmetry operations R, S, and T obey the relation SR — T, then the matrices D(R), D(S), and D(T), found as above, obey the relation... 

The traditional derivation of the Fokker-Planck equation (1.5) or (VIII. 1.1) is based on Kolmogorov s mathematical proof, which assumes infinitely many infinitely small jumps. In nature, however, all jumps are of some finite size. Consequently W is never a differential operator, but always of the type (V.1.1). Usually it also has a suitable expansion parameter and has the canonical form (X.2.3). If it then happens that (1.1) holds, the expansion leads to the nonlinear Fokker-Planck equation (1.5) as the lowest approximation. There is no justification for attributing a more fundamental meaning to Fokker-Planck and Langevin equations than in this approximate sense. 

Another particularly dangerous property of inverse MLR is that the quality of the model fit (expressed as RMSEE or r) must improve as the number of variables used in the model increases. A mathematical proof of this property will not be presented here, but it makes intuitive sense that the ability to explain changes in the Y-variable is improved as one has more X-variables to work with. This leads to the temptation to overfit the model, through the use of too many variables. If the number of variables is already sufficient for determining the Y-property in the presence of interfering effects, then the addition of more unnecessary variables only presents the opportunity to add more noise to the model and make the model more sensitive to unforeseen disturbances. A discussion on overfitting, as well as techniques for avoiding it, is provided in Section 8.3.7. 

Although lacking mathematical proof, I have nonetheless experienced that a soft-seated valve needs about 2.5 times less maintenance than a metal-seated valve This is logical considering the higher possible operating frequency of a... 

The rotational coordinates are Q 2 and Q 5. The rotational motion can be visualized by mapping the trough onto the surface of a 2D sphere the rotation is governed by the usual polar coordinate definitions, 6 and . This is also shown in equation (7) which has the usual form for a rotator with spherical harmonic solutions Ylm. The solutions will be written in the form I i//lo, hn ). For the high spin states case, it was found that l must be odd in order to obey the Pauli s exclusion principle and preserve the antisymmetric nature of the total wavefunctions at any point on the trough under symmetric operations [26]. In the current case, similar arguments show that l must be even. This is because the electronic basis is even under inversion and the whole vibronic wavefunction must also be even under inversion. A general mathematical proof can be found in Ref. [23],... 

It will be observed that some of the curves are concave upward, while others are concave downward. While the mathematical proof for this is not given, the physical explanation is not hard to find. In the case of the type 1 curves, the surface must approach a horizontal asymptote as the velocity is progressively slowed down owing to the increasing depth. Likewise, all curves that approach the normal or uniform depth line j... 

The final coup de grace against any alternative to the orthodox formulation was supposed to be delivered by John von Neumann (1932) with mathematical proof that dispersion-free states2 and hidden variables are impossible in quantum mechanics. He concluded [29] that ... 

The simple CSL model is directly applicable to the cubic crystal class. The lower symmetry of the other crystal classes necessitates the more sophisticated formalism known as the constrained coincidence site lattice, or CCSL (Chen and King, 1988). In this book we treat only cubic systems. Interestingly, whenever an even value is obtained for E in a cubic system, it will always be found that an additional lattice point lies in the center of the CSL unit cell. The true area ratio is then half the apparent value. This operation can always be applied in succession until an odd value is obtained thus, E is always odd in the cubic system. A rigorous mathematical proof of this would require that we invoke what is known as O-lattice theory (Bollman, 1967). The O-lattice takes into account all equivalence points between two neighboring crystal lattices. It includes as a subset not only coinciding lattice points (the CSL) but also all nonlattice sites of identical internal coordinates. However, expanding on that topic would take us well beyond the scope of this book. The interested reader is referred to Bhadeshia (1987) or Bollman (1970). 

In order to support this statement, Gibbs uses the picture of the mixing of a nondiffusive dye in a colorless solvent. He gives in addition a more mathematical proof, which is reproduced by Lorentz in a more transparent way. 

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