Celebrated Theorem

The proof that the sum of the squares of the dimensions of the irreducible representations is equal to the order of the group has three parts (1) intro, duction of the regular representation, (2) the Celebrated Theorem, (3) the final steps. 

The density functional methods were, in the views of many, legitimized by the introduction of the first Hohenberg-Kohn theorem [10]. The consequence of this celebrated theorem is that for a non-degenerate ground state and a given external potential, v(r), the electronic ground state energy can be expressed as... 

The above situation is the same as for the celebrated theorem of Kolmogorov-Arnold-Moser (KAM)—that is, the problem of small denominators. The convergence can be proved for sufficiently nonresonant combinations of the vibrational frequencies [31]. In other words, when tori of the vibrational motions on the NHIM Mq are sufficiently nonresonant, they survive under small perturbations. 

Onsager (1931) in his celebrated theorem on the reciprocal relations, was able to show that, as long as the forces and flows appearing in Eq. (13.4.1) are obtained in such a way that Eq. (13.3.11) is valid, and the forces are linearly independent, the phenomenological coefficients L t satisfy the relation... 

It is generally accepted that the derived weak statement of the problem and also its discretized form should possess the same conserved properties as the initial dynamical system. The celebrated theorem of Emmy Noether provides the basis for the systematic examination of the conserved properties of the weak formulation as well as of the discrete system. In the present paper we utilize this theorem in a weak sense, as it is more consistent with the whole setting. 

In 1977 OLSEN and DEGN [2] reported observations of nonperiodic behavior in an enz3rme system (peroxidase). Using a celebrated theorem of LI and YORKE [3] ("Period three implies chaos"), Olsen and Degn concluded that the observed nonperiodic behavior was chaos. Unfortunately, however, the Li and Yorke theorem says nothing about the measure of the parameter range of chaos the chaos predicted by the theorem could be of zero measure and hence unobservable. Thus the observation of period three does not necessarily imply chaos for a physical system. 

Bell s Theorem In a celebrated 1935 paper, Einstein, Podolsky and Rosen (EPR) [ein35] argued that quantum mechanics provides an essentially incomplete description of reality unless hidden variables exist. 

Perhaps the best starting point in a review of the nonequilibrium field, and certainly the work that most directly influenced the present theory, is Onsager s celebrated 1931 paper on the reciprocal relations [10]. This showed that the symmetry of the linear hydrodynamic transport matrix was a consequence of the time reversibility of Hamilton s equations of motion. This is an early example of the overlap between macroscopic thermodynamics and microscopic statistical mechanics. The consequences of time reversibility play an essential role in the present nonequilibrium theory, and in various fluctuation and work theorems to be discussed shortly. 

Equation (3.34) is the celebrated directionality theorem of Coulson which governs the bond angles for sp-hybridization. 

The celebrated Wigner-Eckart theorem states that the matrix elements of any tensor operator of an algebra G can be split into two pieces, a coupling coefficient and a piece that depends only on A that is,... 

In his classic paper on electric networks, G. Kirchhoff[38] (1847) implicitly established the celebrated Matrix-Tree-Theorem which, in modern terminology, expresses the complexity (i.e., the number of spanning trees) of any finite graph G as the determinant of a matrix which can easily be obtained from the adjacency matrix of G. Simple proofs were given by R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte [39] (1940), H. Trent [40] (1954), and H. Hutschenreuther [41] (1967) (for relations between the complexity and the spectrum of a graph see Ref. [36] pp 38, 39, 49, 50). 

The main properties of a polyhedral surface are undoubtedly related to the topology of the underlying manifold that it decorates. This topology can be completely described by two properties the Euler characteristic and the orientability of the surface. The former can easily be calculated from the celebrated Euler theorem [9] which states that the number of vertices, edges, and faces, denoted as V, E, and F respectively, obey the following mle ... 

This is but one form of the celebrated fluctuation-dissipation theorem wMch in this case, provides a link between the equilibrium fluctuations in capadtor voltage and which is a measure of the rate at which electrical... 

Duhem s theorem is another rule, similar to the phase rule, but less celebrated. It applies to closed systems for which the extensive state as well as the intensive state of the system is fixed. The state of such a system is said to be completely determined, and is characterized not only by the 2 + (JV—O-jt intensive phase-rule variables but also by the tt extensive variables represented by the masses (or mole numbers) of the phases. Thus the total number of variables is... 

Since the Density Functional Theory is about to celebrate its 30, h anniversary in 1994, the area has already been extensively reviewed and cross-examined. The theory itself seems to be quite well established, practical approximations that can be applied to real systems, however, are still the target for ongoing attack. The elegant route exists from the density to the energy that does not necessarily require the wavefunction. It has been paved by the famous Hochenberg-Kohn theorems introducing energy as a unique functional of density p(r)... 

In solid-state physics, theorists are dealing with many-electron systems where "many mean billions not just dozens as in molecular theories. This means that methods based on the electron density are much more widely used and much more intuitively appealing. Their constant efforts to develop such methods have been rewarded by a series of amazing theorems showing that it is possible to obtain the exact electron density without having recourse to the wavefunction. Naturally, these results have been taken up with some enthusiasm by workers in the field of molecular electronic structure. In this chapter the celebrated Kohn-Hohenberg-Sham approach is developed and its close relationship to the Hartree-Fock model is used to indicate how it can be implemented. The very different intuitions" of chemists and physicists about electronic structure generates some tensions in the interpretation of the results of these theories. 

In conclusion, we emphasise the following points (i) we have re-derived a previously obtained operator array formulation, which in its complex symmetric form permits a viable map of gravitational interactions within a combined quantum-classical structure (ii) the choice of representation allows the implementation of a global superposition principle valid both in the classical as well as the quantum domain (iii) the scope of the presentation has focused on obtaining well-known results of Einstein s theory of general relativity particularly in connection with the correct determination of the perihelion motion of the planet Mercury (iv) finally, we have obtained a surprising relation with Godel s celebrated incompleteness theorem. 

Pythagoras of Samos (c. 580-c. 500 bc) Greek philosopher and mathematician, who in about 520 bc went to Croton in Italy, where he founded an academy at which numbers and their mystical significance were studied. Pythagoras discovered irrational numbers and the celebrated Pythagoras theorem. 

Symmetry has taken us to a point where still quintic, quartic, and quadratic secular equations must be solved. However, a closer look at this equations shows that they can easily be solved. Apparently, a further symmetry principle is present, which leads to simple analytical solutions of the secular equations. Triphenylmethyl is an alternant hydrocarbon. In an alternant, atoms can be given two different colors in such a way that all bonds are between atoms of different colors hence, no atoms of the same color are adjacent. A graph with this property is bipartite, and its eigenvalue spectrum obeys the celebrated Coulson-Rushbrooke theorem [16]. 

As a special case, we get the celebrated revenue-equivalence theorem [100, 65], which states that the most popular auction formats, i.e. English, Dutch, first-price sealed-bid and second-price sealed-bid, all yield the same price on average in a single item allocation problem with symmetric agents. This is an immediate consequence because these auctions are all efficient in the simple private values model. 

Recall from Section 6.4 that the analogous equivalence relation, where ne and yoE are replaced by and y, is called the simple homotopy type, and that the celebrated Whitehead theorem implies that the simplicial complexes with the simple homotopy type of a point are precisely those that are contractible see Theorem 6.16. Therefore, the class of simplicial complexes that are NE-equivalent to a point relates to nonevasiveness in the same way as con-tractibility relates to collapsibility. Clearly, this means that this class should constitute an interesting object of study. 

Thus, the periodicity of the density of probabilities (3.80) was lowered at the level of eigen-function such as Eq. (3.88), regaining the celebrated Bloch theorem of Eq. (3.34), here in a generalized form the eigen-function of an electron in a periodic potential can be written as a product of a function carrying the potential periodicity and a basic exponential factor exp(/lx) . 

To complete this theoretical account we review also the celebrated Caldeira-Leggett quantum master equation and its relation to ZE. Moreover, we propose, for the first time, a physically transparent protocol to obtain the conventional ZE from the master equation in the diffusion limit. This novel protocol is universal cf. section 13.2.2), constructed based purely on the thermal dynamics consideration, validated by the central limiting theorem. 

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