Degeneracy Space Research Articles

Mechanism creation in tensegrity structures by cellular morphogenesis

Tensegrity structures are self-equilibrated statically and kinematically indeterminate structures. Cellular morphogenesis represents a generative process for the composition of complex tensegrity structures based on elementary cells. This article discusses the mechanism creation by the integration of mobility conditions in the generation of tensegrity structures using cellular morphogenesis. The creation of finite and infinitesimal mechanisms in tensegrity structures is described by the adhesion of cells sharing less than d nodes (d being the dimension of the workspace) and by the fusion of cells with the removal of two edges. Parametric descriptions of the infinitesimal displacements in the case of trivial and finite mechanisms are derived by the analysis of rigid assemblies corresponding to the rigid parts of the structure. In addition, an interpretation of the degeneracy of tensegrity structures in the case of self-stressable mechanisms is also presented. Analytical solutions of the degeneracy space for configurations resulting from adhesion and the fusion of two cells with the removal of two edges are described using symbolic calculations on the rigidity matrices of tensegrity structures. Although the study focuses on selected configurations and arrangements, the generalization of the ideas and findings included in this paper can lead to the generation of tensegrity structures with predefined static as well as kinematic properties, thus further enabling the application of tensegrity systems in science and engineering.

Chain of Dirac spectrum loops of nodes in crossed magnetic and electric fields

New semimetal systems along with Dirac and Weyl semimetals contain compounds, in which the energy of electron excitations vanishes not at nodes but on lines. A higher dimension of the degeneracy space changes many physical properties. We consider a chain of loops consisting of Dirac spectrum nodes in nonsymmorphic crystalline compounds placed in external mutually perpendicular magnetic and electric fields. An exact solution for the spectrum is obtained under the assumption of particle-hole symmetry. An analysis of this spectrum shows the existence of a line of critical values of the magnetic and electric fields, at which a quantum phase transition to a gapless state occurs. The use of the obtained spectrum allows also predicting a number of new oscillation and resonance effects in the field of magneto-optical phenomena.

Homotopy-Theoretic Study &Atomic-Scale Observation of Vortex Domains in Hexagonal Manganites.

Essential structural properties of the non-trivial “string-wall-bounded” topological defects in hexagonal manganites are studied through homotopy group theory and spherical aberration-corrected scanning transmission electron microscopy. The appearance of a “string-wall-bounded” configuration in RMnO3 is shown to be strongly linked with the transformation of the degeneracy space. The defect core regions (~50 Å) mainly adopt the continuous U(1) symmetry of the high-temperature phase, which is essential for the formation and proliferation of vortices. Direct visualization of vortex strings at atomic scale provides insight into the mechanisms and macro-behavior of topological defects in crystalline materials.

Characterization of the non-uniqueness of used nuclear fuel burnup signatures through a Mesh-Adaptive Direct Search

The use of passive gamma and neutron signatures from fission indicators is a common means of estimating used fuel burnup, enrichment, and cooling time. However, while characteristic fission product signatures such as 134Cs, 137Cs, 154Eu, and others are generally reliable estimators for used fuel burnup within the context where the assembly initial enrichment and the discharge time are known, in the absence of initial enrichment and/or cooling time information (such as when applying NDA measurements in a safeguards/verification context), these fission product indicators no longer yield a unique solution for assembly enrichment, burnup, and cooling time after discharge. Through the use of a new Mesh-Adaptive Direct Search (MADS) algorithm, it is possible to directly probe the shape of this “degeneracy space” characteristic of individual nuclides (and combinations thereof), both as a function of constrained parameters (such as the assembly irradiation history) and unconstrained parameters (e.g., the cooling time before measurement and the measurement precision for particular indicator nuclides). In doing so, this affords the identification of potential means of narrowing the uncertainty space of potential assembly enrichment, burnup, and cooling time combinations, thereby bounding estimates of assembly plutonium content. In particular, combinations of gamma-emitting nuclides with distinct half-lives (e.g., 134Cs with 137Cs and 154Eu) in conjunction with gross neutron counting (via 244Cm) are able to reasonably constrain the degeneracy space of possible solutions to a space small enough to perform useful discrimination and verification of fuel assemblies based on their irradiation history.

Shifted cosmological parameter and shifted dust matter in a two-phase tachyonic field universe

We propose a model of the evolution of the tachyonic scalar field over two phases in the universe. The field components do not interact in phase I, while in the subsequent phase II, they change flavours due to relative suppression of the radiation contribution. In phase II, we allow them to interact mutually with time-independent perturbation in their equations of state, as Shifted Cosmological Parameter (SCP) and Shifted Dust Matter (SDM). We determine the solutions of their scaling with the cosmic redshift in both phases. We further suggest the normalized Hubble function diagnostic, which, together with the low- and high-redshift $H(z)$ data and the concordance values of the present density parameters from the CMBR, BAO statistics etc., constrains the strength of interaction, by imposing the viable conditions to break degeneracy in 3-parameter $(\gamma, \varepsilon, \dot{\phi}^2)$ space. The range of redshifts $(z=0.1$ to $z=1.75)$ is chosen to highlight the role of interaction during structure formation, and it may lead to a future analysis of power spectrum in this model \emph{vis a vis} Warm Dark Matter (WDM) or $\Lambda$CDM models. We further calculate the influence of interaction in determining the age of the universe at the present epoch, within the degeneracy space of model parameters.

Zero-curvature transonic states and one-component surface waves in anisotropic elastic media

The existence of one-component surface waves requires a degeneracy in the Stroh sextic equation. An extraordinary zero-curvature transonic state, a point on the slowness surface where both the curvature and its first derivative equal zero, will yield a triple degeneracy in the Stroh equation. Relationship between extraordinary zero-curvature transonic states and one-component surface waves is investigated showing that they are linked via a space of degeneracy associated with the Stroh equation. Moreover, some generalized subsonic surface waves containing generalized Stroh eigenvectors are also found along the space of degeneracy.

Tetra-radical and ionic S 1/S 0 conical intersections of cyclobutadiene

We have located two conical intersections between the first singlet excited (S 1) and singlet ground (S 0) states of cyclobutadiene (CBD) using the complete active space self-consistent field (CASSCF) method. One is the ionic-structure S 1/S 0 conical intersection (CI ionic), which was located by carrying out a minimum-energy-path calculation from the Franck–Condon point of the HOMO to LUMO double-electron excited state, and the other is the tetra-radical S 1/S 0 conical intersection (CI tetra), which was located by exploring the S 1/S 0 degeneracy space. While CI ionic is only involved in the automerization of CBD, CI tetra is involved in not only the automerization but also the criss–cross reaction. It is possible for one of the highest constrained compounds, tetrahedrane, to be produced if S 1 excited CBD undergoes a transition to the S 0 state via the tetra-radical S 1/S 0 conical intersection. In this paper, we discuss the possibility that unsubstituted tetrahedrane can be produced by irradiating CBD.

Acceleration of the Z to E photoisomerization of penta-2,4-dieniminium by hydrogen out-of-plane motion: theoretical study on a model system of retinal protonated Schiff base

We report the result of comparison between two reaction coordinates [on the potential energy surface of the first excited state (S(1))] produced by CASSCF and these energies recalculated by MRMP2 in the Z to E photoisomerization of penta-2,4-dieniminium (PDI) as the minimal model of the retinal protonated Schiff base (RPSB). One coordinate is the S(1) state minimum-energy-path (MEP) in mass-weighted coordinates from the S(1) vertically excited point, where a strong hydrogen-out-of plane (HOOP) motion is not exhibited. The energy profile of the S(1) MEP at the MRMP2//CASSCF level shows a barrier for the rotation around the reactive C-C and hits the S(1)/S(0) degeneracy space where the central C-C-C-C dihedral angle is distorted by 65 degrees . The other coordinate is an S(1) coordinate obtained by the relaxed scan strategy. The relaxed coordinate along the central C-C-C-C dihedral angle, which we call the HOOP coordinate, shows strong HOOP motion. According to the MRMP2//CASSCF calculation, there is no barrier on the HOOP coordinate. Furthermore, the S(1) to S(0) transition may be possible without the large skeletal deformation by HOOP motion because the HOOP coordinate encounters the S(1)/S(0) degeneracy space where the central C-C-C-C dihedral angle is distorted by only 40 degrees . Consequently, if PDI is a suitable model molecule for the RPSB as often assumed, the 11-cis to all-trans photoisomerization is predicted to be accelerated by the HOOP motion.

Revisiting the S1/S0 Degeneracy Space along the Exocyclic Methylene Twist Motion of Fulvene through a Two-Step Procedure

We have characterized the degeneracy space (DS) between the ground (S0) state and the first excited (S1) state along the exocyclic methylene twist motion of fulvene, using our calculation strategy, i.e., a two-step procedure with CASSCF. The origin of the "cancellation error" on locating degeneracy points under geometrical constraints is analyzed, leading to a method to assess adequacy of the strategy. According to our estimation, these S1/S0 DPs are optimized for energy within 2.0 × 10(-)(3) Eh Å(-)(1) (the value of root-mean-square). From the obtained S1/S0 DS, we provide some information about the exocyclic methylene rotation by 180(◦).

Extraordinary degeneracy and space of degeneracy in transversely isotropic elastic media

A large variety of degeneracy exist in both static and dynamic regimes in anisotropic elastic media. Extraordinary degeneracies occur mostly in the static regime and they are of great importance in understanding the existence of the so-called space of degeneracy in the dynamic regime. This paper deals with the interconnection between degeneracies in the static and dynamic regimes. Both the modified Lekhnitskii formalism and the Stroh formalism are employed in exploring the degeneracies in the transversely isotropic media. It is demonstrated that the space of degeneracy can be regarded as an extension of the static degeneracy, and the development of the static extraordinary degeneracy into the dynamic regime is via a number of spaces of degeneracy of various types. The static degeneracy can also span a space of degeneracy.

Helimagnetism in XY models: Domain walls, frustrations, fractional vortices, and phase transitions.

We consider in this paper a model for two-dimensional XY helimagnets, which was proposed in Pimpinelli et al. [J. Phys. Condens. Matter 3, 4693 (1991)]. In this model the usual spin waves and vortices coexist with chirality degrees of freedom in such a way that the degeneracy space of the order parameter is O(2)\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$. In Pimpinelli et al. the low-temperature behavior of the model was addressed, and an infinite sequence of first-order phase transitions was shown to take place, due to the spin-wave coupling between domains with opposite chirality. Here we investigate the same model at higher temperatures, where vortices in the spin system and kinks on domain walls are thermally excited. The kinks in domain walls are associated with noninteger (fractional) vortices. This is presumably a common feature also in conventional helimagnets, where the ${\mathit{Z}}_{2}$ chirality is due to the twofold handedness of helices. We also discuss the applicability of this scenario to the helimagnetic compound ${\mathrm{BaCo}}_{2}$(${\mathrm{AsO}}_{4}$${)}_{2}$.

Existence of one component surface waves in anisotropic elastic media

The search for one component surface waves has been connected with the space of simple reflection and the space of degeneracy. We demonstrate that the existence of the one component surface wave requires degeneracy in the Stroh eigenvalue problem and that the degeneracy can be either semisimple or non-semisimple. Relating the one component surface wave to the two component surface wave, we find that the one component surface wave results from a confluence between the secluded supersonic surface wave and the space of degeneracy. A sequence of theorems concerning the existence of one component supersonic surface waves is established.

Defects in liquid crystals: homotopy theory and experimental studies

The fundamental concepts of the homotopy theory of defects in liquid crystals and the results of experimental studies in this field are presented. The concepts of degeneracy space, homotopy groups, and topological charge, which are used for classifying the topologically stable inhomogeneous distributions in different liquid-crystalline phases are examined (uni and biaxial nematics, cholesterics, smectics, and columnar phases). Experimental data are given for the different mesophases on the structure and properties of dislocations, disclinations, point defects in the volume (hedgehogs) and on the surface of the medium (boojums), monopoles, domain formations, and solitons. Special attention is paid to the results of studies of defects in closed volumes (spherical drops, cylindrical capillaries), and to the connection between the topological charges of these defects and the character of the orientation of the molecules of the liquid crystal at the surface. A set of fundamentally new effects that can occur in studying the topological properties of defects in liquid crystals is discussed.

The topology of non-uniform media in condensed matter physics

Abstract Methods of algebraic topology have been employed recently to classify defects and non-singular textures of condensed matter systems, and to describe two-defect processes. In this article a systematic review is presented of these methods and their applications. The non-uniform media are characterized by fields valued in a space of degeneracy, whose topological properties are investigated. Ways to represent such a space are reported. An introduction is given to the necessary mathematical tools, viz. the homotopy groups and exact sequences thereof. Combinations, entanglements and transformations of singularities are discussed. The connection between the homotopic defect classification and the description of many-defect systems is elaborated. The limitations and necessary modifications of the method for systems of broken translational symmetry are examined.

The structure of the solution set for the Yang-Mills equations

AbstractThe solution set for the (sourceless) Yang-Mills equations on a spacetime with compact Cauchy surface is a smooth manifold (i.e. the equations are linearization stable) except at solutions that are symmetric. At such symmetric solutions, the structure is described by a homogeneous quadratic form. The degeneracy space for this form is tangent to a manifold of symmetric solutions. Symmetry breaking occurs for perturbations in the nondegenerate directions of the quadratic form. The terms ‘symmetry’ and ‘stability’ in the present work are compared to these terms as used elsewhere in the literature.

The 2:1 anisotropic oscillator, separation of variables and symmetry group in Bargmann space

We present a detailed analysis of the separation of variables for the time-dependent Schrödinger equation for the anisotropic oscillator with a 2:1 frequency ratio. This reduces essentially to the time-independent one, where the known separability in Cartesian and parabolic coordinates applies. The eigenvalue problem in parabolic coordinates is a multiparameter one which is solved in a simple manner by transforming the system to Bargmann’s Hilbert space. There, the degeneracy space appears as a subspace of homogeneous polynomials which admit unique representations of a solvable symmetry algebra s3 in terms of first order operators. These representations, as well as their conjugate representations, are then integrated to indecomposable finite-dimensional nonunitary representations of the corresponding group S3. It is then shown that the two separable coordinate systems correspond to precisely the two orbits of the factor algebra s3/u (1) [u (1) generated by the Hamiltonian] under the adjoint action of the group. We derive some special function identitites for the new polynomials which occur in parabolic coordinates. The action of S3 induces a nonlinear canocical transformation in phase space which leaves the Hamiltonian invariant. We discuss the differences with previous works which present su (2) as the algebra responsible for the degeneracy of the two-dimensional anisotropic oscillator.

Symmetry of a Hydrogen Atom in a Weak Magnetic Field

The degeneracy problem of a hydrogen atom in a uniform weak magnetic field $H$ is considered from the standpoint of the invariance group that the system obeys. Evidently the degeneracy space has a constant $z$ component of the angular momentum. Under such a classification, the dynamical invariance of the problem is shown to be an isomorph to the Lie group of the linear transformation of the straight line.

The Maslov-Morse index of indefinite metrics

We compute the Maslov-Morse index of geodesics on a manifold with indefinite metric. It is shown that the multiplicity of conjugate points in the sense of Maslov is equal to the signature of a quadratic form obtained by restricting the metric to the space of degeneracy for the projection of the tangent space onto the Lagrangian manifold, if this latter spans the principal axes of the metric.

NDA Approvals

The use of passive gamma and neutron signatures from fission indicators is a common means of estimating used fuel burnup, enrichment, and cooling time. However, while characteristic fission product signatures such as 134Cs, 137Cs, 154Eu, and others are generally reliable estimators for used fuel burnup within the context where the assembly initial enrichment and the discharge time are known, in the absence of initial enrichment and/or cooling time information (such as when applying NDA measurements in a safeguards/verification context), these fission product indicators no longer yield a unique solution for assembly enrichment, burnup, and cooling time after discharge. Through the use of a new Mesh-Adaptive Direct Search (MADS) algorithm, it is possible to directly probe the shape of this “degeneracy space” characteristic of individual nuclides (and combinations thereof), both as a function of constrained parameters (such as the assembly irradiation history) and unconstrained parameters (e.g., the cooling time before measurement and the measurement precision for particular indicator nuclides). In doing so, this affords the identification of potential means of narrowing the uncertainty space of potential assembly enrichment, burnup, and cooling time combinations, thereby bounding estimates of assembly plutonium content. In particular, combinations of gamma-emitting nuclides with distinct half-lives (e.g., 134Cs with 137Cs and 154Eu) in conjunction with gross neutron counting (via 244Cm) are able to reasonably constrain the degeneracy space of possible solutions to a space small enough to perform useful discrimination and verification of fuel assemblies based on their irradiation history.

NDA Approvals

The use of passive gamma and neutron signatures from fission indicators is a common means of estimating used fuel burnup, enrichment, and cooling time. However, while characteristic fission product signatures such as 134Cs, 137Cs, 154Eu, and others are generally reliable estimators for used fuel burnup within the context where the assembly initial enrichment and the discharge time are known, in the absence of initial enrichment and/or cooling time information (such as when applying NDA measurements in a safeguards/verification context), these fission product indicators no longer yield a unique solution for assembly enrichment, burnup, and cooling time after discharge. Through the use of a new Mesh-Adaptive Direct Search (MADS) algorithm, it is possible to directly probe the shape of this “degeneracy space” characteristic of individual nuclides (and combinations thereof), both as a function of constrained parameters (such as the assembly irradiation history) and unconstrained parameters (e.g., the cooling time before measurement and the measurement precision for particular indicator nuclides). In doing so, this affords the identification of potential means of narrowing the uncertainty space of potential assembly enrichment, burnup, and cooling time combinations, thereby bounding estimates of assembly plutonium content. In particular, combinations of gamma-emitting nuclides with distinct half-lives (e.g., 134Cs with 137Cs and 154Eu) in conjunction with gross neutron counting (via 244Cm) are able to reasonably constrain the degeneracy space of possible solutions to a space small enough to perform useful discrimination and verification of fuel assemblies based on their irradiation history.