Topic

# Degeneracy (mathematics)

About: Degeneracy (mathematics) is a research topic. Over the lifetime, 2984 publications have been published within this topic receiving 53767 citations.

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TL;DR: In this paper, it was shown that if the total electronic state of orbital and spin motion is degenerate, then a non-linear configuration of the molecule will be unstable unless the degeneracy is the special twofold one (discussed by Kramers 1930) which can occur only when the molecule contains an odd number of electrons.

Abstract: In a previous paper (Jahn and Teller 1937) the following theorem was established: A configuration of a polyatomic molecule for an electronic state having orbital degeneracy cannot be stable with respect to all displacements of the nuclei unless in the original configuration the nuclei all lie on a straight line. The proof given of this theorem took no account of the electronic spin, and in the present paper the justification of this is investigated. An extension of the theorem to cover additional degeneracy arising from the spin is established, which shows that if the total electronic state of orbital and spin motion is degenerate, then a non-linear configuration of the molecule will be unstable unless the degeneracy is the special twofold one (discussed by Kramers 1930) which can occur only when the molecule contains an odd number of electrons. The additional instability caused by the spin degeneracy alone, however, is shown to be very small and its effect for all practical purposes negligible. The possibility of spin forces stabilizing a non-linear configuration which is unstable owing to orbital degeneracy is also investigated, and it is shown that this is not possible except perhaps for molecules containing heavy atoms for which the spin forces are large. Thus whilst a symmetrical nuclear configuration in a degenerate orbital state might under exceptional circumstances be rendered stable by spin forces, it is not possible for the spin-orbit interaction to cause instability of an orbitally stable state. 1—General theorem for molecules with spin Just as before we must see how the symmetry of the molecular framework determines whether the energy of a degenerate electronic state with spin depends linearly upon nuclear displacements. This is again determined by the existence of non-vanishing perturbation matrix elements which are linear in the nuclear displacements. These matrix elements are integrals involving the electronic wave functions with spin and the nuclear displacements, and we deduce as before from their transformation properties whether for a given molecular symmetry they can be different from zero.

2,539 citations

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TL;DR: In this article, it is shown that the quantum field trajectories are of paramount importance, which trajectories connect seemingly degenerate vacua thereby eliminating the degeneracy. But the authors focus on a special mechanism which is responsible for the restoration of broken gauge symmetry.

1,388 citations

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TL;DR: In this article, the local C(1 + Alpha) nature of weak solutions of elliptic equations of the type (1.1) in the introduction under the degeneracy (or singularity) assumptions (A sub 1)-(A sub 3).

Abstract: : It is demonstrated the local C(1 + Alpha) nature of weak solutions of elliptic equations of the type (1.1) in the introduction under the degeneracy (or singularity) assumptions (A sub 1)-(A sub 3).

1,106 citations

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TL;DR: In this paper, it was shown that the Haldane phase is characterized by a double degeneracy of the entanglement spectrum, which cannot be lifted unless either a phase boundary to another, topologically trivial, phase is crossed, or the symmetry is broken.

Abstract: We show that the Haldane phase of $S=1$ chains is characterized by a double degeneracy of the entanglement spectrum. The degeneracy is protected by a set of symmetries (either the dihedral group of $\ensuremath{\pi}$ rotations about two orthogonal axes, time-reversal symmetry, or bond centered inversion symmetry), and cannot be lifted unless either a phase boundary to another, ``topologically trivial,'' phase is crossed, or the symmetry is broken. More generally, these results offer a scheme to classify gapped phases of one-dimensional systems. Physically, the degeneracy of the entanglement spectrum can be observed by adiabatically weakening a bond to zero, which leaves the two disconnected halves of the system in a finitely entangled state.

876 citations

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TL;DR: An edit-distance algorithm for shock graphs that finds the optimal deformation path in polynomial time is employed and gives intuitive correspondences for a variety of shapes and is robust in the presence of a wide range of visual transformations.

Abstract: This paper presents a novel framework for the recognition of objects based on their silhouettes. The main idea is to measure the distance between two shapes as the minimum extent of deformation necessary for one shape to match the other. Since the space of deformations is very high-dimensional, three steps are taken to make the search practical: 1) define an equivalence class for shapes based on shock-graph topology, 2) define an equivalence class for deformation paths based on shock-graph transitions, and 3) avoid complexity-increasing deformation paths by moving toward shock-graph degeneracy. Despite these steps, which tremendously reduce the search requirement, there still remain numerous deformation paths to consider. To that end, we employ an edit-distance algorithm for shock graphs that finds the optimal deformation path in polynomial time. The proposed approach gives intuitive correspondences for a variety of shapes and is robust in the presence of a wide range of visual transformations. The recognition rates on two distinct databases of 99 and 216 shapes each indicate highly successful within category matches (100 percent in top three matches), which render the framework potentially usable in a range of shape-based recognition applications.

773 citations