Degeneracies in a quantum system can be systematic or accidental in nature.
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These degeneracies are connected to the existence of bound orbits in classical Physics.
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Compactification of moduli spaces generally require allowing certain degeneracies – for example, allowing certain singularities or reducible varieties. This is notably used in the Deligne–Mumford compactification of the moduli space of algebraic curves.
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The JTE is usually associated with degeneracies that are well localised in space, like those occurring in a small molecule or associated to an isolated transition metal complex. However, in many periodic high-symmetry solid-state systems, like perovskites, some crystalline sites allow for electronic degeneracy giving rise under adequate compositions to lattices of JT-active centers. This can produce a cooperative JTE, where global distortions of the crystal occur due to local degeneracies. In order to determine the final electronic and geometric structure of a cooperative JT system, it is necessary to take into account both the local distortions and the interaction between the different sites, which will take such form necessary to minimise the global energy of the crystal.
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Degenerate primers are widely used and extremely useful in the field of microbial ecology. They allow for the amplification of genes from thus far uncultivated microorganisms or allow the recovery of genes from organisms where genomic information is not available. Usually, degenerate primers are designed by aligning gene sequencing found in GenBank. Differences among sequences are accounted for by using IUPAC degeneracies for individual bases.
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Zhegalkin (also Žegalkin, Gégalkine or Shegalkin) polynomials form one of many possible representations of the operations of Boolean algebra. Introduced by the Russian mathematician Ivan Ivanovich Zhegalkin in 1927, they are the polynomial ring over the integers modulo 2. The resulting degeneracies of modular arithmetic result in Zhegalkin polynomials being simpler than ordinary polynomials, requiring neither coefficients nor exponents. Coefficients are redundant because 1 is the only nonzero coefficient.
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Nevertheless, also twofold degeneracies continue to be important. Among larger systems, a focus in the literature has been on benzene and its radical cation, as well as on their halo (especially fluoro) derivatives. Already in the early 1980s, a wealth of information emerged from the detailed analysis of experimental emission spectra of 1,3,5- trifluoro- and hexafluoro (and chloro) benzene radical cations. The Jahn-Teller effect in the 1,3,5-trifluoro benzene radical cation is discussed in Section 13.4.
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The set of all operators which commute with the Hamiltonian of a quantum system are said to form the symmetry group of the Hamiltonian. The commutators of the generators of this group determine the algebra of the group. An n-dimensional representation of the Symmetry group preserves the multiplication table of the symmetry operators. The possible degeneracies of the Hamiltonian with a particular symmetry group are given by the dimensionalities of the irreducible representations of the group.
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It is a type of degeneracy resulting from some special features of the system or the functional form of the potential under consideration, and is related possibly to a hidden dynamical symmetry in the system. It also results in conserved quantities, which are often not easy to identify. Accidental symmetries lead to these additional degeneracies in the discrete energy spectrum. An accidental degeneracy can be due to the fact that the group of the Hamiltonian is not complete.
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However, these effects become properties of the hadron wave functions themselves using the front form. This also eliminates the many orders of magnitude conflict between the measured cosmological constant and quantum field theory. Some aspects of vacuum structure in light-front quantization can be analyzed by studying properties of massive states. In particular, by studying the appearance of degeneracies among the lowest massive states, one can determine the critical coupling strength associated with spontaneous symmetry breaking.
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In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler to) the rest of the class, and the term degeneracy is the condition of being a degenerate case. The definitions of many classes of composite or structured objects often implicitly include inequalities. For example, the angles and the side lengths of a triangle are supposed to be positive. The limiting cases, where one or several of these inequalities become equalities, are degeneracies.
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Among Rellich's most important mathematical contributions are his work in the perturbation theory of linear operators on Hilbert spaces: he studied the dependence of the spectral family E_\varepsilon(\lambda) of a self-adjoint operator A_\varepsilon on the parameter \varepsilon. Although the origins and applications of the problem are in quantum mechanics, Rellich's approach was completely abstract. Rellich successfully worked on many partial differential equations with degeneracies. For instance, he showed that in the elliptic case, the Monge-Ampère differential equation, while not necessarily uniquely soluble, can have at most two solutions.
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Here singlet and triplet degeneracies originate not from the whole system but from the two identical particles in it. The half- integer spin value distinguishes trions from excitons in many phenomena; for example, energy states of trions, but not excitons, are split in an applied magnetic field. Trion states were predicted theoretically in 1958; they were observed experimentally in 1993 in CdTe/Cd1−xZnxTe quantum wells, and later in various other optically excited semiconductor structures. There are experimental proofs of their existence in nanotubes supported by theoretical studies.
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Crystal field theory (CFT) describes the breaking of degeneracies of electron orbital states, usually d or f orbitals, due to a static electric field produced by a surrounding charge distribution (anion neighbors). This theory has been used to describe various spectroscopies of transition metal coordination complexes, in particular optical spectra (colors). CFT successfully accounts for some magnetic properties, colors, hydration enthalpies, and spinel structures of transition metal complexes, but it does not attempt to describe bonding. CFT was developed by physicists Hans Bethe and John Hasbrouck van Vleck in the 1930s.
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In 1982, Hamilton published his formulation of Nash's reasoning, casting the theorem into the setting of tame Fréchet spaces; Nash's fundamental use of restricting the Fourier transform to regularize functions was abstracted by Hamilton to the setting of exponentially decreasing sequences in Banach spaces. His formulation has been widely quoted and used in the subsequent time. He used it himself to prove a general existence and uniqueness theorem for geometric evolution equations; the standard implicit function theorem does not often apply in such settings due to the degeneracies introduced by invariance under the action of the diffeomorphism group. In particular, the well-posedness of the Ricci flow follows from Hamilton's general result.
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It is important to note that many cooperative JT systems would be expected to be metals from band theory as, to produce them, a degenerate orbital has to be partially filled and the associated band would be metallic. However, under the perturbation of the symmetry-breaking distortion associated to the cooperative JTE, the degeneracies in the electronic structure are destroyed and the ground state of these systems is often found to be insulating (see e.g.). In many important cases like the parent compound for colossal magnetoresistance perovskites, LaMnO3, an increase of temperature leads to disorder in the distortions which lowers the band splitting due to the cooperative JTE, thus triggering a metal- insulator transition.
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Thermally induced spin crossover is due to the higher electronic degeneracies of the LS form and lower vibrational frequencies of the HS form, thus increasing the entropy. The Raman spectrum of an iron(II) complex in the HS and LS state, emphasizing the changes in the M-L vibrational modes, where a shift from 2114 cm−1 to 2070 cm−1 corresponds to changes in the stretching vibrational modes of the thiocyanate ligand from a LS state to a HS state, respectively. SCO behavior can be followed with UV-vis spectroscopy. In some cases, the absorption bands obscured due to the high intensity absorption bands caused by the Metal-to-Ligand Charge Transfer (MLCT) absorption bands.
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The result that every simple planar graph can be drawn with straight line edges is called Fáry's theorem.For the relation between Tutte's and Fáry's theorem, and the history of rediscovery of Fáry's theorem, see . The Tutte spring theorem proves this for 3-connected planar graphs, but the result is true more generally for planar graphs regardless of connectivity. Using the Tutte spring system for a graph that is not 3-connected may result in degeneracies, in which subgraphs of the given graph collapse onto a point or a line segment; however, an arbitrary planar graph may be drawn using the Tutte embedding by adding extra edges to make it 3-connected, drawing the resulting 3-connected graph, and then removing the extra edges.
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Challenges associated with 3D shape-based similarity queries With the skeleton modeling 3D retrieval method, figuring out an efficient way to index 3D shape descriptors is very challenging because 3D shape indexing has very strict criteria. The 3D models must be quick to compute, concise to store, easy to index, invariant under similarity transformations, insensitive to noise and small extra features, robust to arbitrary topological degeneracies, and discriminating of shape differences at many scales. 3D search and retrieval with multimodal support challenges In order to make the 3D search interface simple enough for novice users who know little on 3D retrieval input source requirements, a multimodal retrieval system, which can take various types of input sources and provide robust query results, is necessary. So far, only a few approaches have been proposed.
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Full CI problems including several million up to a few billion determinants are possible using current algorithms. Because full CI results are exact within the space spanned by the orbital basis set, they are invaluable in benchmarking approximate quantum chemical methods. This is particularly important in cases such as bond-breaking reactions, diradicals, and first-row transition metals, where electronic near-degeneracies can invalidate the approximations inherent in many standard methods such as Hartree-Fock theory, multireference configuration interaction, finite-order Møller-Plesset perturbation theory, and coupled cluster theory. Although fewer N-electron functions are required if one employs a basis of spin-adapted functions (Ŝ2 eigenfunctions), the most efficient full CI programs employ a Slater determinant basis because this allows for the very rapid evaluation of coupling coefficients using string-based techniques advanced by Nicholas C. Handy in 1980.
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The non-existence of Gliese 581f was accepted relatively quickly: it was shown that the radial velocity variations that led to the claimed discovery of Gliese 581f were instead associated with the stellar activity cycle rather than an orbiting planet. Nevertheless, the existence of planet g remained controversial: Vogt responded in the media that he stood by the discovery and questions arose as to whether the effect was due to the assumption of circular rather than eccentric orbits or the statistical methods used. Bayesian analysis found no clear evidence for a fifth planetary signal in the combined HIRES/HARPS data set, though other studies led to the conclusion that the data did support the existence of planet g, albeit with strong degeneracies in the parameters as a result of the first eccentric harmonic with the outer planet Gliese 581d. Using the assumption that the noise present in the data was correlated (red noise rather than white noise), Roman Baluev called into question not only the existence of planet g, but Gliese 581d as well, suggesting there were only three planets (Gliese 581b, c, and e) present.
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