Narrow Results

We study perturbations of a class of analytic two-dimensional autonomous systems with perturbations depending periodically on time; for instance one can imagine a periodically driven or forced system with one degree of freedom. In the first part of the paper, we revisit a problem considered by Chow and Hale on the existence of subharmonic solutions. In the analytic setting, under more general (weaker) conditions on the perturbation, we prove their results on the bifurcation curves dividing the region of non-existence from the region of existence of subharmonic solutions. In particular our results apply also when one has degeneracy to the first order — i.e. when the subharmonic Melnikov function is identically constant. Moreover we can deal as well with the case in which degeneracy persists to arbitrarily high orders, in the sense that suitable generalizations to higher orders of the subharmonic Melnikov function are also identically constant. The bifurcation curves consist of four branches joining continuously at the origin, where each of them can have a singularity (although generically they do not). The branches can form a cusp at the origin: we say in this case that the curves are degenerate as the corresponding tangent lines coincide. The method we use is completely different from that of Chow and Hale, and it is essentially based on the proof of convergence of the perturbation theory. It also allows us to treat the Melnikov theory in degenerate cases in which the subharmonic Melnikov function is either identically vanishing or has a zero which is not simple. This is investigated at length in the second part of the paper. When the subharmonic Melnikov function has a non-simple zero, we consider explicitly the case where there exist subharmonic solutions, which, although not analytic, still admit a convergent fractional series in the perturbation parameter.

Information of sensitivity analysis, in a linear programming problem, is usually more important than the optimal solution itself. However, traditional sensitivity analysis, which perturbs exactly one coefficient and then determines the range preserving the optimality of the current optimal base, is impractical for the assignment problem. An optimal basic solution of the assignment problem is inherently degenerate, so it may be that the optimal base has changed but the optimal assignment remains unchanged. Furthermore, elements of a column (or row) in a cost matrix of assignment problem are usually closely related and change simultaneously, not uniquely. This paper focuses on two kinds of sensitivity analyses for the assignment problem. One is to determine the sensitivity range, over which the current optimal assignment or all the optimal assignments remain optimal, while perturbing the elements of one column (or row) in a cost matrix of the assignment problem simultaneously but dependently. The other is to perturb elements of one column (or row) in a cost matrix of the assignment problem simultaneously but independently. Numerical illustrations are presented to demonstrate that the approaches are useful in practice.

We investigate how the data from various future neutrino oscillation experiments will constrain the physics parameters for a three active neutrino mixing model. The investigations properly account for the degeneracies and ambiguities associated with the phenomenology as well as estimates of experimental measurement errors. Combinations of various reactor measurements with the expected J-PARC (T2K) and NuMI offaxis (Nova) data, both with and without the increased flux associated with proton driver upgrades, are considered. The studies show how combinations of reactor and offaxis data can resolve degeneracies (e.g. the θ₂₃ degeneracy) and give more precise information on the oscillation parameters. A primary purpose of this investigation is to establish the parameter space regions where CP violation can be discovered and where the mass hierarchy can be determined. We find that, even with augmented flux from proton drivers, such measurements demand that sin² 2θ₁₃ be fairly large and in the range where it is measurable by reactor experiments.

In a model that supports both Abelian (Abrikosov–Nielsen–Olesen) and non-Abelian strings we analyze the parameter space to find examples in which these strings not only coexist but are degenerate in tension. We prove that both solutions are locally stable, i.e. there are no negative modes in the string background. The tension degeneracy is achieved at the classical level and is expected to be lifted by quantum corrections. The setup of the model, analogous to that of Witten's superconducting cosmic strings, had been extended to include non-Abelian strings previously [M. Shifman, Phys. Rev. D87, 025025 (2013), arXiv:1212.4823 [hep-th]].

In this work, we report the effect of the spin–orbit (SO) interaction on the band structure of TlInSe₂. Calculation was performed by implementing density functional theory (DFT) method. Our results show that SO interaction is significant for two high symmetrical point ($\mathrm{\Gamma }$, $T$) and line ($\kappa$, $A$) of Brillouin zone in the band structure and negligible changes was observed in the bands near the Fermi level. The maximum SO splitting is $\sim$0.9 eV.

In this work, we illustrate the use of eigenfunction method (EFM) in quantum mechanics by taking a specific example namely the splitting of energy levels under perturbation. Initially, the symmetry group of the Hamiltonian is taken to be $C_{4v}$ which under the action of a perturbing potential changes to a lower symmetry namely $C_{2v}$. We have constructed an irreducible basis using the EFM for both the symmetries and show the simplifications that ensure the calculation of the expectation value of the perturbing Hamiltonian.

By virtue of the entangled state representation, we show that a new function space for Landau state vectors can be constructed. As a result, the magnetic translation and degeneracy of some Landau states can be analyzed in a direct and transparent manner.

In this paper, the Bremsstrahlung emission and re-absorption mechanisms are studied mainly through Inverse Bremsstrahlung and Compton Scattering. The Radiation Specific Power is calculated numerically assuming the suitable forms of Energy Distribution Function in plasma conditions. The calculation of Spectral Emission shows that, the Bremsstrahlung emission is strongly forward and backward peak relative to electron direction in overdense and high temperature plasma. Finally, some of the conditions for dominant of the re-absorption mechanism are explained.

We perform an analytical study of the Hopf bifurcations and their degeneracies in Chua's equation. In the case of the equilibrium at the origin only codimension-two Hopf bifurcations appear. However, for the nontrivial equilibria we prove the existence of codimension-three Hopf bifurcations. Numerical results are in strong agreement with the analytical ones.

In this work, we consider some degeneracies of homoclinic and heteroclinic connections organized by the triple-zero degeneracy, in Chua's equation. This allows us to numerically study the homoclinic-heteroclinic transition exhibited by the curve of Takens–Bogdanov bifurcations as it passes through the triple-zero degeneracy. Several codimension-two degenerate homoclinic and heteroclinic connections organized by the triple-zero bifurcation are involved in this transitional homoclinic-heteroclinic mechanism. In particular, we point out that the existence of a curve of T-points and a curve of Belyakov points (its equilibrium passes from saddle to saddle-focus) is necessary for this process to occur. Closed curves of homoclinic connections with different pulses are also found.

When a flow suffers a discontinuity in its vector field at some switching surface, the flow can cross through or slide along the surface. Sliding along the switching surface can be understood as the flow along an invariant manifold inside a switching layer. It turns out that the usual method for finding sliding modes — the Filippov convex combination or Utkin equivalent control — results in a degeneracy in the switching layer whenever the flow is tangent to the switching surface from both sides. We derive the general result and analyze the simplest case here, where the flow curves parabolically on either side of the switching surface (the so-called fold–fold or two-fold singularities). The result is a set of zeros of the fast switching flow inside the layer, which is structurally unstable to perturbation by terms nonlinear in the switching parameter, terms such as ${(\mathrm{\text{sign}}x)}^2$ [where the superscript does mean “squared”]. We provide structurally stable forms, and show that in this form the layer system is equivalent to a generic singularity of a two timescale system. Finally we show that the same degeneracy arises when a discontinuity is smoothed using standard regularization methods.

The taxonomy of flattened Moebius strips (FMS) is reexamined in order to systematize the basis for its development. An FMS is broadly characterized by its twist and its direction of traverse. All values of twist can be realized by combining elementary FMS configurations in a process called fusion but the result is degenerate; a multiplicity of configurations can exist with the same value of twist. The development of degeneracy is discussed in terms of several structural factors and two principles, conservation of twist and continuity of traverse. The principles implicate a corresponding pair of constructs, a process of symbolic convolution, and the inner product of symbolic vectors. Combining constructs and structural factors leads to a systematically developed taxonomy in terms of twist categories assembled from permutation groups. Taxonomical structure is also graphically revealed by the geometry of an expository edifice that validates the convolution process while displaying the products of fusion. A formulation that combines some of the algebraic precepts of Quantum Mechanics with the primitive combinatorics and degeneracies inherent to the FMS genus is developed. The potential for further investigation and application is also discussed. An appendix outlines the planar extension of the fusion concept and another summarizes a related application of convolution.

In this paper, the generalized Dirac oscillator with $\kappa$-Poincaré algebra is structured by replacing the momentum operator p with $p-im\omega \beta V(r)r$ in $\kappa$-deformation Dirac equation. The deformed radial equation is derived for this model. Particularly, by solving the deformed radial equation, the wave functions and energy spectra which depend on deformation parameter $𝜀$ have been obtained for these quantum systems with $V(r)$ being a Yukawa-type potential, inverse-square-type singular potential and central fraction power singular potential in two-dimensional space, respectively. The results show that the deformation parameter $𝜀$ can lead to decreasing of energy levels for the above quantum systems. At the same time, the degeneracy of energy spectra has been discussed and the corresponding conditions of degeneracy have been given for each case.

Algebraic theory of coding is one of modern fields of applications of algebra. This theory uses matrix algebra intensively. This paper is devoted to an application of Kronecker's matrix forms of presentations of the genetic code for algebraic analysis of a basic scheme of degeneracy of the genetic code. Similar matrix forms are utilized in the theory of signal processing and encoding. The Kronecker family of the genetic matrices is investigated, which is based on the genetic matrix [C A; U G], where C, A, U, G are the letters of the genetic alphabet. This matrix in the third Kronecker power is the (8*8)-matrix, which contains all 64 genetic triplets in a strict order with a natural binary numeration of the triplets by numbers from 0 to 63. Peculiarities of the basic scheme of degeneracy of the genetic code are reflected in the symmetrical black-and-white mosaic of this genetic (8*8)-matrix. This mosaic matrix is connected algorithmically with Hadamard matrices unexpectedly, which are famous in the theory of signal processing and encoding, spectral analysis, quantum mechanics and quantum computers. Furthermore, many kinds of cyclic permutations of genetic elements lead to reconstruction of initial Hadamard matrices into new Hadamard matrices unexpectedly. This demonstrates that matrix algebra is one of promising instruments and of adequate languages in bioinformatics and algebraic biology.

In this paper, we are concerned with the following elliptic problems which are related to the well-known Caffarelli–Kohn–Nirenberg inequalities:

Nickel phthalocyanines (NiPc) substituted by trifluorosulfonyl or trimethylsilylethynyl groups have been prepared and characterized using NMR and electronic absorption spectroscopy. In particular, when trifluorosulfonyl groups are introduced to two, so-called, α-benzo positions of one benzene ring of the Pc skeleton, the Q-band splits similarly to the Q-bands of metal-free phthalocyanines. The splitting width of the Q-bands can be interpreted as due mainly to splitting of the LUMOs by the substituents, although splitting between the HOMO-1 and HOMO-2 is also effective to a lesser extent.

In this paper we present a novel 2-D graphical representation of DNA sequences in the first quadrant Cartesian coordinate system. This representation has been mathematically proven to be a zero length of circuit, i.e., without any degeneracy. A practicable formula based upon an iterative comparison method has been developed to calculate the frequencies of nucleic acid bases (A, T, G, and C) from the starting point to any considered point. Furthermore, this new graphical representation, in comparison with our previously published method, may have a unique application in representing the universal genetic code.

We introduce and study an inductively defined analogue $f_{\mathrm{\text{IND}}}()$ of any increasing graph invariant $f()$. An invariant $f()$ is increasing if $f(H)\le f(G)$ whenever $H$ is an induced subgraph of $G$. This inductive analogue simultaneously generalizes and unifies known notions like degeneracy, inductive independence number, etc., into a single generic notion. For any given increasing $f()$, this gets us several new invariants and many of which are also increasing. It is also shown that $f_{\mathrm{\text{IND}}}()$ is the minimum (over all orderings) of a value associated with each ordering.

We also explore the possibility of computing $f_{\mathrm{\text{IND}}}()$ (and a corresponding optimal vertex ordering) and identify some pairs $({𝒞},f())$ for which $f_{\mathrm{\text{IND}}}()$ can be computed efficiently for members of ${𝒞}$. In particular, it includes graphs of bounded $f_{\mathrm{\text{IND}}}()$ values. Some specific examples (like the class of chordal graphs) have already been studied extensively.

We further extend this new notion by (i) allowing vertex weighted graphs, (ii) allowing $f()$ to take values from a totally ordered universe with a minimum and (iii) allowing the consideration of $r$-neighborhoods for arbitrary but fixed $r\ge 1$. Such a generalization is employed in designing efficient approximations of some graph optimization problems. Precisely, we obtain efficient algorithms (by generalizing the known algorithm of Ye and Borodin [Y. Ye and A. Borodin, Elimination graphs, ACM Trans. Algorithms  8(2) (2012) 1–23] for special cases) for approximating optimal weighted induced ${𝒫}$-subgraphs and optimal ${𝒫}$-colorings (for hereditary ${𝒫}$’s) within multiplicative factors of (essentially) $k$ and $k/(m-1),$ respectively, where $k$ denotes the inductive analogue (as defined in this work) of optimal size of an unweighted induced ${𝒫}$-subgraph of the input and $m$ is the minimum size of a forbidden induced subgraph of ${𝒫}$. Our results generalize the previous result on efficiently approximating maximum independent sets and minimum colorings on graphs of bounded inductive independence number to optimal ${𝒫}$-subgraphs and ${𝒫}$-colorings for arbitrary hereditary classes ${𝒫}$. As a corollary, it is also shown that any maximal ${𝒫}$-subgraph approximates an optimal solution within a factor of $k+1$ for unweighted graphs, where $k$ is maximum size of any induced ${𝒫}$-subgraph in any local neighborhood $N_G(u)$.

In the 2 × 2 matrices representing retarders and ideal polarizers, the eigenvectors are orthogonal. An example of the opposite case, where eigenvectors collapse onto one, is matrices M representing crystal plates sandwiched between a crossed polarizer and analyser. For these familiar combinations, M² = 0, so black sandwiches can be regarded as square roots of zero. Black sandwiches illustrate physics associated with degeneracies of non-Hermitian matrices.