Order Degeneracy Research Articles

Nonlinear Versions of Hyperbolic Problems for One Quasi-Linear Equation of Mixed Type

The Cauchy and Goursat problems for a quasi-linear second-order hyperbolic equation with admissible order and type degeneracy are considered. The conditions for nonexistence of a solution of the Cauchy problem are established. However, in conditions of the existence, the solution of the problem is constructed explicitly. In addition, nonlinear versions of the Goursat problem are examined and solutions of these problems are constructed in the explicit form. For the Cauchy problem, as well as for both of the characteristic problems, the structure of the domain of definition of the solution is described.

Goodness-of-fit tests for long memory moving average marginal density

This paper addresses the problem of fitting a known density to the marginal error density of a stationary long memory moving average process when its mean is known and unknown. In the case of unknown mean, when mean is estimated by the sample mean, the first order difference between the residual empirical and null distribution functions is known to be asymptotically degenerate at zero, and hence can not be used to fit a distribution up to an unknown mean. In this paper we show that by using a suitable class of estimators of the mean, this first order degeneracy does not occur. We also investigate the large sample behavior of tests based on an integrated square difference between kernel type error density estimators and the expected value of the error density estimator based on errors. The asymptotic null distributions of suitably standardized test statistics are shown to be chi-square with one degree of freedom in both cases of the known and unknown mean. In addition, we discuss the consistency and asymptotic power against local alternatives of the density estimator based test in the case of known mean. A finite sample simulation study of the test based on residual empirical process is also included.

Streamline topology near non-simple degenerate critical points in two-dimensional flow with symmetry about an axis

The local flow patterns and their bifurcations associated with non-simple degenerate critical points appearing away from boundaries are investigated under the symmetric condition about a straight line in two-dimensional incompressible flow. These flow patterns are determined via a bifurcation analysis of polynomial expansions of the streamfunction in the proximity of the degenerate critical points. The normal form transformation is used in order to construct a simple streamfunction family, which classifies all possible local streamline topologies for given order of degeneracy (degeneracies of order three and four are considered). The relation between local and global flow patterns is exemplified by a cavity flow.

Formula omitted]-theory for some elliptic and parabolic problems with first order degeneracy at the boundary

Let Ω be a smooth open bounded set in R N , let ϱ be the (smoothed in the interior) distance function from ∂ Ω, let ( a i j ) be a uniformly elliptic matrix with continuous entries in Ω and A the associated second order elliptic operator. Under suitable conditions, we prove that the operator L = − ϱ A + B , with B a first order operator with continuous coefficients, with Dirichlet boundary conditions, generates an analytic semigroup in L p ( Ω ) , 1 < p < ∞ , and in C ( Ω ¯ ) . In L p ( Ω ) we also give a precise description of the domain.

Kernel-based goodness-of-fit tests for copulas with fixed smoothing parameters

We study a test statistic based on the integrated squared difference between a kernel estimator of the copula density and a kernel smoothed estimator of the parametric copula density. We show for fixed smoothing parameters that the test is consistent and that the asymptotic properties are driven by a U -statistic of order 4 with degeneracy of order 1. For practical implementation we suggest to compute the critical values through a semiparametric bootstrap. Monte Carlo results show that the bootstrap procedure performs well in small samples. In particular, size and power are less sensitive to smoothing parameter choice than they are under the asymptotic approximation obtained for a vanishing bandwidth.

Kernel Based Goodness-of-Fit Test for Copulas with Fixed Smoothing Parameters

We study a test statistic on the integrated squared difference between a kernel estimator of the copula density and a kernel smoothed estimator of the parametric copula density. We show for fixed smoothing parameters that the test is consistent and that the asymptotic properties are driven by a U-statistic of order 4 with degeneracy of order 3. For practical implementation we suggest to compute the critical values through a semiparametric bootstrap. Monte Carlo results show that the bootstrap procedure performs well in small samples. In particular size and power are less sensitive to smoothing parameter choice than they are under the asymptotic approximation obtained for a vanishing bandwidth.

Kernel Based Goodness-of-Fit Tests for Copulas with Fixed Smoothing Parameters

We study a test statistic on the integrated squared difference between a kernel estimator of the copula density and a kernel smoothed estimator of the parametric copula density. We show for fixed smoothing parameters that the test is consistent and that the asymptotic properties are driven by a U-statistic of order 4 with degeneracy of order 3. For practical implementation we suggest to compute the critical values through a semiparametric bootstrap. Monte Carlo results show that the bootstrap procedure performs well in small samples. In particular size and power are less sensitive to smoothing parameter choice than they are under the asymptotic approximation obtained for a vanishing bandwidth.

Spectral Degeneracies in the Totally Asymmetric Exclusion Process

We study the spectrum of the Markov matrix of the totally asymmetric exclusion process (TASEP) on a one-dimensional periodic lattice at arbitrary filling. Although the system does not possess obvious symmetries except translation invariance, the spectrum presents many multiplets with degeneracies of high order when the size of the lattice and the number of particles obey some simple arithmetic rules. This behaviour is explained by a hidden symmetry property of the Bethe Ansatz. Assuming a one-to-one correspondence between the solutions of the Bethe equations and the eigenmodes of the Markov matrix, we derive combinatorial formulae for the orders of degeneracy and the number of multiplets. These results are confirmed by exact diagonalisations of small size systems. This unexpected structure of the TASEP spectrum suggests the existence of an underlying large invariance group.

Hidden symmetries in the asymmetric exclusion process

We present a spectral study of the evolution matrix of the totally asymmetricexclusion process on a ring at half filling. The natural symmetries (translation,charge conjugation combined with reflection) predict only twofold degeneracies.However, we have found that degeneracies of higher order also exist and, as thesystem size increases, higher and higher orders appear. These degeneracies becomegeneric in the limit of very large systems. This behaviour can be explained bythe Bethe ansatz and suggests the presence of hidden symmetries in the model.

P‐66: New Photo‐Alignment Technology Based on Chalcone Moieties: Molecular Design and Process Development

AbstractIn this paper, factors affecting the pretilt angle are investigated for our newly‐developed photo‐alignment materials. We believe that two factors are crucial for the pretilt angle generation. That is, surface tension of the alignment films, and the incident angle of the LPUV light which may affect the surface order degeneracy. In addition, our development efforts of the photo‐alignment materials will be shortly summarized.

Mechanism of Generation of Pretilt Angles on Polyimides with a Single Exposure to Polarized Ultraviolet Light

Behaviors of pretilt angles of liquid crystal (LC) cells were examined using polyimide (PI) films containing fluorine atoms by exposure to ultraviolet light (UV). Using a single obliquely polarized UV exposure, we obtained unidirectional LC alignment with any desired pretilt angles from 0° up to 90°. The pretilt angles depend on the concentration of fluorine atoms in the PI and the UV exposure time and angle. The results of dichroic ratio and capacitance measurements of the LC cells suggest that oblique polarized UV exposure breaks the surface order degeneracy of the LC molecules, generating uniform pretilt angles along the optic axis of the PI film.

Generation of Pretilt Angles on Polyimides with a Single Linearly Polarized UV Exposure

We designed polyimide (PI) films containing fluorine atoms for liquid crystal (LC) alignment using linearly polarized UV exposure. Using a single oblique polarized UV exposure we obtained unidirectional LC alignment with any desired pretilt angles from 0 ° up to 90 °. Pretilt angles depend on the concentration of fluorine atoms in PI and the UV exposure time and angle. Dichroic ratio and capacitance measurements of the LC cells suggest that oblique polarized UV exposure breaks the surface order degeneracy of the LC molecules, generating uniform pretilt angle along the optic axis of PI film.

Smooth Densities for Degenerate Stochastic Delay Equations with Hereditary Drift

We establish the existence of smooth densities for solutions of $\mathbf{R}^d$-valued stochastic hereditary differential systems of the form $dx(t) = H(t,x)dt + g(t,x(t - r))dW(t).$ In the above equation, $W$ is an $n$-dimensional Wiener process, $r$ is a positive time delay, $H$ is a nonanticipating functional defined on the space of paths in $\mathbf{R}^d$ and $g$ is an $n \times d$ matrix-valued function defined on $\lbrack 0,\infty) \times \mathbf{R}^d$, such that $gg^\ast$ has degeneracies of polynomial order on a hypersurface in $\mathbf{R}^d$. In the course of proving this result, we establish a very general criterion for the hypoellipticity of a class of degenerate parabolic second-order time-dependent differential operators with space-independent principal part.

Normalization and the detection of integrability: The generalized Van Der Waals potential

Deprit and Miller have conjectured that normalization of integrable Hamiltonians may produce normal forms exhibiting degenerate equilibria to very high order. Several examples in the class of coupled elliptic oscillators are known. In order to test the utility of normalization as a detector of integrability we normalize, to high order, a perturbed Keplerian system known to have several integrable limits; the generalized van der Waals Hamiltonian for a hydrogen atom. While the separable limits give rise to high order degeneracy we find a non-separable, integrable limit for which the normal form does not exhibit degeneracy. We conclude that normalization may, in certain cases, indicate integrability but is not guaranteed to uncover all integrable limits.

Models of Degenerate Fourier Integral Operators and Radon Transforms

Fourier integral operators whose Lagrangians project with singularities arise frequently in many areas of analysis and geometry [3], [5], [9], [10], [18], [20]. However, there are as yet few analytic tools available for their study, and even their simplest regularity properties with respect to the singularities of the Lagrangian are still obscure [4], [5], [13], [16]. In this paper we study a class of oscillatory integrals TA and Fourier integral operators R which can be expected to model the higher order singularities of the Lagrangian. They have homogeneous polynomial phases in two variables of order n, and indeed the case n = 3 is the model for Lagrangians which project with Whitney folds [3], [10], [11]. The main difficulty which sets the higher order degeneracy cases n > 4 apart from the the lower ones n = 2, 3 is that the critical varieties which arise there are usually not smooth manifolds. A systematic study of these oscillatory integrals was begun in [14]. The main idea in that work was to treat the critical varieties as smooth manifolds away from a lower-dimensional subvariety, and to keep track of the distance to this lower-dimensional subvariety. To achieve this we introduced a method of stationary phase which exhibited clearly the dependence on the distance between critical points. The method gave sharp bounds on the size of the kernel K(x, y) of TAT*, but no information on its phase and the resulting cancellations. It does not seem possible to refine this approach to the phase level, and this suggests looking instead for decompositions of the operators TA which can incorporate indirectly the required cancellations. The main goal of this paper is to introduce such decompositions. These decompositions, which we will try to describe momentarily, reflect the singular nature of the critical points, and are powerful enough to yield sharp bounds for the above oscillatory integrals and Radon transforms. Although the models under consideration are very

Fourier restriction theorems for degenerate curves

Fourier restriction theorems contain estimates of the formwhere σ is a measure on a smooth manifold M in ∝n. This paper is a continuation of [5], which considered this problem for certain degenerate curves in ∝n. Here estimates are obtained for all curves with degeneracies of finite order. References to previous work on this problem may be found in [5].

Bloch wave degeneracies in systematic high energy electron diffraction

For the systematic diffraction of high energy electrons by a thin crystalline slab, we study the accidental degeneracies of the Bloch waves excited in the specimen by the incident beam. It is shown that these Bloch waves are the eigenstates of a one dimensional band structure problem, and this is solved by wave matching methods. For a symmetric potential, the symmetry properties of the Bloch waves are discussed, and it is shown how accidental degeneracies of these waves can occur when the reflexion coefficient for waves incident on one unit cell of the one dimensional periodic potential vanishes. The form of the band structure and the Bloch waves in the neighbourhood of a degeneracy are derived by expanding the Kramers function in a Taylor series. It is then shown analytically how the degeneracy affects the diffracted waves emerging from the crystalline specimen (in particular, the Kikuchi pattern). To understand these effects fully, W. K. B. approximations for the Bloch waves are used to derive the Bloch wave excitations and the absorption coefficients. However, to predict the degeneracies themselves, it is shown that a different formula for the reflexion coefficient, due to Landauer, must be used. This formula shows how the critical voltage at which the Bloch waves degenerate depends on the form of the potential, and allows quick, accurate, computations of the critical voltages to be made. Also, a new higher order degeneracy is predicted for some of the systematic potentials of cadmium, lead and gold. Finally, to infer the potential in real space from measurements of critical voltages and several other quantities, we suggest an inversion scheme based on the Landauer formula for the reflexion coefficient. To a close approximation this potential is proportional to V 2 of the crystal charge density.

Phase Transition of a Hard-Core Lattice Gas. The Square Lattice with Nearest-Neighbor Exclusion

We consider an M×N periodic system of hard squares each of which prohibits the occupation of its nearest-neighbor sites by the other squares. An appropriate unitary transformation factors the matrix required in evaluating the grand canonical partition function into a direct sum of submatrices. Each submatrix belongs to a distinct irreducible representation of the dihedral group of order 2M. For an M×∞ system, only the submatrix belonging to the one-dimensional symmetric representation A1 must be considered. To investigate the possible existence of an order—disorder transition for a system infinite in both dimensions, detailed calculations are carried out with the M×∞ systems (M=2, 4, 6, ···, 18). All eigenvalues λi+ and eigenvectors of the submatrix are evaluated with eight-digit accuracy, and the various thermodynamic variables are expressed exactly in terms of the eigenvalues and eigenvectors. The result indicates that an order—disorder transition takes place continuously without any sharp break in a density (ρ)- vs-activity (z) plot, or a pressure (P)- vs-ρ plot. The transition point is characterized by the following set of numbers: zt=3.7966±0.0003, ρt/ρ0=0.73552±0.00001 (ρ0≡the close-packed density), and Pt/kT=0.7916±0.0001. Within numerical accuracy of the data, Pt/kT is equal to zt/(1+zt). The compressibility ρ−1dρ/dP appears to become infinite at the transition point, while the ratio λ1+/λ2+ of the two largest eigenvalues of the submatrix becomes unity. For a semi-infinite system in the neighborhood of zt, these two quantities exhibit, respectively, a maximum and a minimum given for large M by [(kTρ/ρ0)dρ/dP]max≃(0.0798±0.0001) lnM+(0.0910±0.0009) and [λ1+/λ2+]min≈exp[(6.25±0.04)/M]. We find that the activity z at [λ1+/λ2+]min converges rapidly to zt, thus providing a good means of locating the transition point. From the study of the eigenvalue spectrum over all densities, it appears that the transition point zt, the ordered (z&amp;gt;zt), and the disordered (z&amp;lt;zt) phases are, respectively, characterized by asymptotic degeneracy of order ∞, 2, and 0 in the modulus of the largest eigenvalue. The ordering of the hard-square lattice is investigated by introducing a radial distribution function g(l, ρ) (l is lattice spacing). The calculations of g(l, ρ) are carried out for the 4×4, 8×8, and 8×∞ systems. A long-range order parameter L(ρ) {≡liml→∞[g(l, ρ) −1], l: even} is introduced. For a two-dimensionally infinite system, all coefficients in the low-density expansion of L(ρ) vanish identically, while the first five terms of the high-density expansion of L(ρ) are computed. Therefore, the transition to an ordered state presumably takes place at density between 0 and ρ0.