Mathematical equivalence

The key term is which is the bond order between the atoms i and j. This parameter depends upon the number of bonds to the atom i the strength of the bond between i and j decreases as the number of bonds fo fhe atom i increases. The original bond-order potential [Abell 1985] is mathematically equivalent to the Finnis-Sinclair model if the bond order by is given by ... 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

This does not imply necessarily T y = T r = 0 or T s = 0, where is the Volta potential on the electrolyte surface. But for the subsequent analysis it is useful to notice that equation (7.15) is mathematically equivalent (in view of the general theoretical Equation (7.13) with the key experimental equation (7.11). 

At this point, we may proceed in one of two ways, which are mathematically equivalent. In the first procedure, we note that from the generating function (E.l) for Legendre polynomials Pi, equation (J.3) may be written as... 

The term Vmax refers to the maximum velocity obtained at infinite substrate concentration. VW is mathematically equivalent to the product of kC3) and the enzyme concentration ... 

Sorenson chose to write the concentration of hydrogen ions as a power of 10. For example, an aqueous solution that contained 5.00 x 10-2 M hydrogen ions (H+) would be denoted as containing the mathematically equivalent value of 10-130 M H+ ions. Sorenson designated the Ph or pH of the solution as the numerical value of the negative exponent of 10. Thus, a pH of 1.30 would be ascribed to the solution. In other words,... 

Bertsekas, D. (1994) Mathematical equivalence of the auction algorithm for assignment and the e-relaxation (preflow-push) method for min cost flow, in Large Scale Optimization. State-of-the-Art (ed. W.W. Hager), Papers Presented... 

Le Maguer and Yao (1995) presented a physical model of a plant storage tissue based on its cellular structure. The mathematical equivalent of this model was solved using a finite element-based computer method and incorporated shrinkage and different boundary conditions. The concept of volume average was used to express the concentration and absolute pressure in the intracellular volume, which is discontinuous in the tissue, as a... 

In natural waters, sinking rates (mathematically equivalent to the flow velocity for a stationary cell) are predicted to increase in proportion to the square of the cell radius according to a Stokes law dependency. For a spherical organism ... 

In order to tear a piece of paper into four equally wide strips, three tears must be made. One to tear the original paper in half and the other two to tear those halves in half again. A quartile is the mathematical equivalent of this to a range of ordered data. You should realize that the middle quartile (Q2 ) is, in effect, the median for the range. Similarly, the first quartile (Qi) is effectively the median of the lower half of the dataset and the third quartile (Q3) the median of the upper half. In the same way as for the median calculation, a quartile should be represented as the mean of two data points if it lies between them. 

We have found now an equation for the evolution of the density inside the cluster without any uncontrolled parameter, except for the dimensionless number C. Below we shall do two things. First, in Section V, we shall find the steady solutions for the density, that turns out to transform into a quite simple problem, mathematically equivalent to the equilibrium of self-gravitating atmosphere. Then, in Section VI we shall look at the possible existence of finite time singularities in the dynamical problem. 

The two pictures above (where U is viewed as acting on the wavefunction or acting on the Hamiltonian) are clearly mathematically equivalent. However, it is worth considering their physical equivalence in the language of canonical transformations. (A similar discussion of this issue may also be found in White [22].) In the first picture the Hamiltonian H, wavefunctions to and and transformation U are associated with particles defined by the operators c, Cj thus... 

Discussing physics with Bohr was sometimes quite exhausting. In 1925, while working at Bohr s institute, Heisenberg discovered quantum mechanics, the theory that superseded the old quantum theory that had been developed by Bohr and his colleagues. Then, in 1926, a theory that looked much different from Heisenberg s, but which turned out to be mathematically equivalent, was propounded by the Austrian physicist Erwin Schrodinger. 

In summary, microdosimetry is the study and quantification of the spatial and temporal distributions of absorbed energy in irradiated matter [15,17,22,23]. One makes a distinction between regional microdosimetry [the object of which is the study of micro-dosimetric distributions /(z)j and structural microdosimetry (a mathematically more advanced approach, which is concerned with characterizing the spatial distribution of individual energy deposition events, i.e., ionizations and/or excitations). Regional microdosimetry asserts that the effect is entirely determined by the amount of specific energy deposited in the relevant site (typically, a cell nucleus). The two kinds of microdosimetry, regional and structural, were shown to be in fact mathematically equivalent—once the sensitive site is judiciously determined [16]. 

It is always very useful to be able to predict at what level of external stress and in which directions the macroscopic yielding will occur under different loading geometry. Mathematically, the aim is to find functions of all stress components which reach their critical values equal to some material properties for all different test geometries. This is mathematically equivalent to derivation of some plastic instability conditions commonly termed as the yield criterion. Historically, the yield criteria derived for metals were appHed to polymers and, later, these criteria have been modified as the knowledge of the differences in deformation behavior of polymers compared to metals has been acquired [20,25,114,115]. 

Thus, the phase difference is related to the time integral of the electric field (the electric impulse) on the contour. At some time t the electric field (or its contour integral) can be zero, while the corresponding impulse (time integral) is non zero. What some might term action at a distance (from the magnetic field away from C) is mathematically equivalent to action from a previous time (when the electric field was present on C), assuming zero initial conditions. 

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