Kind Of Degeneracy Research Articles

Anomalous Lamb waves in cubic crystals at a non-semisimple degeneracy of the fundamental matrix

An explicit equation is derived for the condition of non-semisimple degeneracy of the fundamental matrix and, hence, the occurrence of anomalous guided waves propagating along the crystallographic axes of a cubic crystal. It is revealed that the observed non-semisimple degeneracy appears at a certain phase velocity in any cubic crystal of type I, but this kind of degeneracy is not possible in cubic crystals of type II. The analysis is based on the Cauchy sextic formalism combined with the spectral analysis of the fundamental matrix.

Interactive and Robust Mesh Booleans

Boolean operations are among the most used paradigms to create and edit digital shapes. Despite being conceptually simple, the computation of mesh Booleans is notoriously challenging. Main issues come from numerical approximations that make the detection and processing of intersection points inconsistent and unreliable, exposing implementations based on floating point arithmetic to many kinds of degeneracy and failure. Numerical methods based on rational numbers or exact geometric predicates have the needed robustness guarantees, that are achieved at the cost of increased computation times that, as of today, has always restricted the use of robust mesh Booleans to offline applications. We introduce an algorithm for Boolean operations with robustness guarantees that is capable of operating at interactive frame rates on meshes with up to 200K triangles. We evaluate our tool thoroughly, considering not only interactive applications but also batch processing of large collections of meshes, processing of huge meshes containing millions of elements and variadic Booleans of hundreds of shapes altogether. In all these experiments, we consistently outperform prior robust floating point methods by at least one order of magnitude.

Distance between exceptional points and diabolic points and its implication for the response strength of non-Hermitian systems

Exceptional points are non-Hermitian degeneracies in open quantum and wave systems at which not only eigenenergies but also the corresponding eigenstates coalesce. This is in strong contrast to degeneracies known from conservative systems, so-called diabolic points, at which only eigenenergies degenerate. Here we connect these two kinds of degeneracies by introducing the concept of the distance of a given exceptional point in matrix space to the set of diabolic points. We prove that this distance determines an upper bound for the response strength of a non-Hermitian system with this exceptional point. A small distance therefore implies a weak spectral response to perturbations and a weak intensity response to excitations. This finding has profound consequences for physical realizations of exceptional points that rely on perturbing a diabolic point. Moreover, we exploit this concept to analyze the limitations of the spectral response strength in passive systems. A number of optical and photonics systems are investigated to illustrate the theory.

Shell-confined atom and plasma: Incidental degeneracy, metallic character, and information entropy

Shell confined atom can serve as a generalized model to explain both \emph{free} and \emph{confined} condition. In this scenario, an atom is trapped inside two concentric spheres of inner $(R_{a})$ and outer $(R_{b})$ radius. The choice of $R_{a}, R_{b}$ renders four different quantum mechanical systems. In hydrogenic atom, they are termed as (a) free hydrogen atom (FHA) (b) confined hydrogen atom (CHA) (c) shell-confined hydrogen atom (SCHA) (d) left-confined hydrogen atom (LCHA). By placing $R_{a}, R_{b}$ at the location of radial nodes of respective \emph{free} $n,\ell$ states, a new kind of degeneracy may arise. At a given $n$ of FHA, there exists $\frac{n(n+1)(n+2)}{6}$ number of iso-energic states with energy $-\frac{Z^{2}}{2n^{2}}$. Furthermore, within a given $n$, the individual contribution of each of these four potentials has also been enumerated. This incidental degeneracy concept is further explored and analyzed in certain well-known \emph{plasma} (Debye and exponential cosine screened) systems. Multipole oscillator strength, $f^{(k)}$, and polarizability, $\alpha^{(k)}$, are evaluated for (a)-(d) in some low-lying states $(k=1-4)$. In excited states, \emph{negative} polarizability is also observed. In this context, metallic behavior of H-like systems in SCHA is discussed and demonstrated. Additionally analytical closed-form expression of $f^{(k)}$ and $\alpha^{(k)}$ are reported for $1s,2s,2p,3d,4f,5g$ states of FHA. Finally, Shannon entropy and Onicescu {\color{red}information} energies are investigated in ground state in SCHA and LCHA in both position and momentum spaces. Much of the results are reported here for first time.

Homology Groups of Cubical Sets with Connections

Toward defining commutative cubes in all dimensions, Brown and Spencer introduced the notion of “connection” as a new kind of degeneracy. In this paper, for a cubical set with connections, we show that the connections generate an acyclic subcomplex of the chain complex of the cubical set. In particular, our results show that the homology groups of a cubical set with connections are independent of whether we normalize by the connections or we do not, that is, connections do not contribute to any nontrivial cycle in the homology groups of the cubical set.

Light sterile neutrinos and their implications on currently running long-baseline and neutrinoless double-beta decay experiments

Recent νe appearance data from the Mini Booster Neutrino Experiment support the excess of events reported by the Liquid Scintillator Neutrino Detector, which provides an indirect hint of the existence of eV-scale sterile neutrinos. As these sterile neutrinos can mix with the standard active neutrinos, in this paper we explore the effect of such active–sterile mixing on the determination of various oscillation parameters by the currently running long-baseline neutrino experiments T2K and NOνA. We find that the existence of sterile neutrinos can lead to new kinds of degeneracies among these parameters which would substantially deteriorate the mass hierarchy sensitivity of the NOνA experiment. We further notice that the inclusion of data from the T2K experiment helps in resolving the degeneracies. The impact of new CP-violating phases δ14 and δ34 on the maximal CP violation exclusion sensitivity for the NOνA experiment has also been illustrated. Finally, we discuss the implication of such light sterile neutrinos on neutrinoless double-beta decay processes in line with recent experimental results, as well as on the sensitivity reach of future experiments.

Constructive coupling effect of topological states and topological phase transitions in plumbene

Combining tight-binding (TB) models with first-principles calculations, we investigate electronic and topological properties of plumbene. Different from the other two-dimensional (2D) topologically nontrivial insulators in group IVA (from graphene to stanene), low-buckled plumbene is a topologically trivial insulator. The plumbene without spin-orbit coupling exhibits simultaneously two kinds of degeneracies, i.e., quadratic non-Dirac and linear Dirac band dispersions around the \Gamma and K/K' points, respectively. Our TB model calculations show that it is the coupling between the two topological states around the \Gamma and K/K' points that triggers the global topologically trivial property of plumbene. Quantum anomalous Hall effects with Chern numbers of 2 or -2 can be, however, achieved after an exchange field is introduced. When the plumbene is functionalized with ethynyl (PbC2H), quantum spin Hall effects appear due to the breaking of the coupling effect of the local topological states.

Pseudo-Hermitian Systems with P T $\mathcal {P}\mathcal {T}$ -Symmetry: Degeneracy and Krein Space

We show in the present paper that pseudo-Hermitian Hamiltonian systems with even PT-symmetry admit a degeneracy structure. This kind of degeneracy is expected traditionally in the odd PT-symmetric systems which is appropriate to the fermions as shown by Jones-Smith and Mathur [1] who extended PT-symmetric quantum mechanics to the case of odd time-reversal symmetry. We establish that the pseudo-Hermitian Hamiltonians with even PT-symmetry admit a degeneracy structure if the operator PT anticommutes with the metric operator {\eta} which is necessarily indefinite. We also show that the Krein space formulation of the Hilbert space is the convenient framework for the implementation of unbroken PT-symmetry. These general results are illustrated with great details for four-level pseudo-Hermitian Hamiltonian with even PT-symmetry.

Decay and persistence of spatial coherence during phonon-assisted relaxation in double quantum dots

We present a theoretical study of the evolution of spatial coherence during intraband relaxation between exciton states in a pair of vertically stacked semiconductor quantum dots coupled to acoustic phonons. We show that spatial coherence can be transferred to the ground state even in a system of uncoupled nonidentical quantum dots if a particular kind of degeneracy between the interlevel energy splittings is present. The phonon-assisted mechanism of coherence transfer leads to a dependence of the amount of the resulting coherence on the interdot distance and temperature. We analyze also the impact of carrier-phonon dynamics on a coupled system, where spatial coherence is present in the delocalized ground state.

Unique and Explicit Formulas for Green's Function in Three-Dimensional Anisotropic Linear Elasticity

Unique, explicit, and exact expressions for the first- and second-order derivatives of the three-dimensional Green's function for general anisotropic materials are presented in this paper. The derived expressions are based on a mixed complex-variable method and are obtained from the solution proposed by Ting and Lee (Ting and Lee, 1997,“The Three-Dimensional Elastostatic Green's Function for General Anisotropic Linear Elastic Solids,” Q. J. Mech. Appl. Math. 50, pp. 407–426) which has three valuable features. First, it is explicit in terms of Stroh's eigenvalues pα (α=1,2,3) on the oblique plane with normal coincident with the position vector; second, it remains well-defined when some Stroh's eigenvalues are equal (mathematical degeneracy) or nearly equal (quasi-mathematical degeneracy); and third, they are exact. Therefore, both new proposed solutions inherit these appealing features, being explicit in terms of Stroh's eigenvalues, simpler, unique, exact and valid independently of the kind of degeneracy involved, as opposed to previous approaches. A study of all possible degenerate cases validate the proposed scheme.

The existence of weak solutions to immiscible compressible two-phase flow in porous media: The case of fields with different rock-types

We study a model describing immiscible, compressible two-phase flow, such as water-gas, through heterogeneous porous media taking into account capillary and gravity effects. We will consider a domain made up of several zones with different characteristics: porosity, absolute permeability, relative permeabilities and capillary pressure curves. This process can be formulated as a coupled system of partial differential equations which includes a nonlinear parabolic pressure equation and a nonlinear degenerate diffusion-convection saturation equation. Moreover the transmission conditions are nonlinear and the saturation is discontinuous at interfaces separating different media. There are two kinds of degeneracy in the studied system: the first one is the degeneracy of the capillary diffusion term in the saturation equation, and the second one appears in the evolution term of the pressure equation. Under some realistic assumptions on the data, we show the existence of weak solutions with the help of an appropriate regularization and a time discretization. We use suitable test functions to obtain a priori estimates. We prove a new compactness result in order to pass to the limit in nonlinear terms. This passage to the limit is nontrivial due to the degeneracy of the system.

A nonlinear degenerate system modelling water-gas flows in porous media

The aim of this paper is to study a system modelling the flow of an incompressible phase (water) and a compressible phase (gas) in porous media. Two kinds of degeneracy appear for this problem: a dissipative term and an evolution term degenerate with respect to the saturation. Global weak solutions are established for the system by introducing several approximate models. The first one consists in obtaining a non-degenerate dissipative system. The second one is a time discretization method in order to overcome the degeneracy in the evolution term. At this step, the subproblem is a non- degenerate elliptic system which is strongly coupled and highly nonlinear. Then the Leray-Schauder fixed point theorem instead of a classical Schauder fixed point theorem is the key point to solve such a problem.

Propagation of singularities in the Cauchy problem for a class of degenerate hyperbolic operators

We consider second order degenerate hyperbolic Cauchy problems, the degeneracy coming either from low regularity (less than Lipschitz continuity) of the coefficients with respect to time, or from weak hyperbolicity. In the weakly hyperbolic case, we assume an intermediate condition between effective hyperbolicity and the Levi condition. We construct the fundamental solution and study the propagation of singularities using an unified approach to these different kinds of degeneracy.

Non-Hermitian polarizers: a biorthogonal analysis

The non-Hermitian operators of the ideal nonorthogonal multilayer optical polarizers are spectrally analyzed in the framework of skew-angular biorthonormal vector bases. It is shown that these polarizers correspond to skew projectors and their operators are generated by skew projectors, exactly as the canonical ideal polarizers correspond to Hermitian projectors. Thus the common feature of all the polarizers (Hermitian and non-Hermitian) is that their "nuclei" are (orthogonal or skew) projectors--the generating projectors. It is shown that if these nonorthogonal polarizers are looked upon as variable devices, two kinds of degeneracy may occur for suitable values of the inner parameter of the device: The corresponding operators may become normal (more precisely, Hermitian) or, on the contrary, very pathological--defective and singular. In the first case their eigenvectors and biorthogonal conjugate eigenvectors collapse into a unique pair of eigenvectors; in the second case their eigenvectors (as well as their biorthogonal conjugates) collapse into a single vector.

Bayesian Networks with Degenerate Gaussian Distributions

Bayesian networks compute marginal distributions through the “message passing” algorithm—a series of local calculations involving neighboring cliques of variables in a clique tree. In this work, these message passing computations are extended to the case of degenerate Gaussian potentials which are multivariate Gaussian densities that can have two different kinds of degeneracies corresponding to projections with zero variance and projections with infinite variance. The basic operations of the message passing algorithm, such as representing conditional distributions, extending potentials, and conditioning on observations, create or operate on potentials with various kinds of degeneracies thereby demonstrating the need for such methodology. Computer implementation of this scheme follows easily from the details within and some computational aspects are discussed. We also demonstrate an application of our methodology to automatic musical accompaniment.

Exchange Degeneracy of Relativistic Two-Particle Quantum States

The phenomenon of exchange degeneracy of 2-particle quantum states is studied in detail within the framework of Relativistic Schrodinger Theory (RST). In conventional quantum theory this kind of degeneracy refers to the circumstance that, under neglection of the interparticle interactions, symmetric and anti-symmetric 2-particle states have identical energy eigenvalues. However the analogous effect of RST degeneracy is rather related to the emergence of two types of mixtures (positive and negative) in connection with the vanishing or non-vanishing of certain components of the Hamiltonian (“exchange fields”). As a consequence, there arise two subcases of RST degeneracy: (i) mixture degeneracy through neglection of the exchange fields and (ii) exchange degeneracy through neglection of the mixture character of matter. The latter RST exchange degeneracy consists in the fact that the RST dynamics admits a certain set of pure-state solutions, as borderline case between positive and negative mixtures, and all these different solutions are generating the same physical situation, e.g., concerning mass eigenvalues and physical densities (of current and energy-momentum). The general results are exemplified by considering the 2-particle states for (scalar) Helium. Analogously as the conventional exchange degeneracy is broken (ortho- and para-Helium) by taking into account the interparticle interactions (e.g., Coulomb forces), the RST degeneracy is broken by simultaneously taking into account the mixture character of matter together with non-zero exchange fields.

Spectra of the Orr-Sommerfeld equation: the plane Poiseuille flow for the normal fluid revisited

We further investigate the strange spectra of the Orr-Sommerfeld operator using the plane Poiseuille flow as a basic stationary flow for normal fluids in the two-fluid system of helium II by a verified preconditioned complex-matrix solver. The strange spectra are composed of one pair of eigenvalues with the same phase speed (real part) but different amplification factors (imaginary part) corresponding to the specific Reynolds number and wavenumber we select. These kinds of degeneracy disappear for Reynolds number around 400, where the `drifting' of the complete spectra imposes much more complexity on the searching.

On the modeling of narrow gaps using the standard boundary element method.

Numerical methods based on the Helmholtz integral equation are well suited for solving acoustic scattering and diffraction problems at relatively low frequencies. However, it is well known that the standard method becomes degenerate if the objects that disturb the sound field are very thin. This paper makes use of a standard axisymmetric Helmholtz integral equation formulation and its boundary element method (BEM) implementation to study the behavior of the method on two test cases: a thin rigid disk of variable thickness and two rigid cylinders separated by a gap of variable width. Both problems give rise to the same kind of degeneracy in the method, and modified formulations have been proposed to overcome this difficulty. However, such techniques are better suited for the so-called thin-body problem than for the reciprocal narrow-gap problem, and only the first is usually dealt with in the literature. A simple integration technique that can extend the range of thicknesses/widths tractable by the otherwise unmodified standard formulation is presented and tested. This technique is valid for both cases. The modeling of acoustic transducers like sound intensity probes and condenser microphones has motivated this work, although the proposed technique has a wider range of applications.

Spatial shape-preserving interpolation using ν-splines

We present a global iterative algorithm for constructing spatial G 2‐continuous interpolating ν‐splines, which preserve the shape of the polygonal line that interpolates the given points. Furthermore, the algorithm can handle data exhibiting two kinds of degeneracy, namely, coplanar quadruples and collinear triplets of points. The convergence of the algorithm stems from the asymptotic properties of the curvature, torsion and Frenet frame of ν‐splines for large values of the tension parameters, which are thoroughly investigated and presented. The performance of our approach is tested on two data sets, one of synthetic nature and the other of industrial interest.

Phase-integral quantization of accidentally and intrinsically degenerate nonlinear systems

The dynamics of many nonlinearly coupled oscillator systems with internal resonances resembles the motion of a pendulum. For some other resonant systems the dynamics in the vicinity of the resonance cannot be mapped onto this “Standard Hamiltonian” of nonlinear theory. Often these two varieties, called “accidentally” and “intrinsically” degenerate systems respectively, can be converted into each other by changing external parameters. In this article, we introduce an alternative mapping (to the “Standard Map”) valid for both kinds of degeneracies by exploiting a connection between oscillator systems and asymmetric rotors. We apply this mapping to the phase-integral quantization to degenerate oscillator systems.