Singular Directions Research Articles

Propagation of singularities in many-body scattering

In this paper we describe the propagation of singularities of tempered distributional solutions u∈ S ′ of ( H− λ) u=0, λ>0, where H is a many-body Hamiltonian H= Δ+ V, Δ⩾0, V=∑ a V a , under the assumption that no subsystem has a bound state and that the two-body interactions V a are real-valued polyhomogeneous symbols of order −1 (e.g. Coulomb-type with the singularity at the origin removed). Here the term ‘singularity’ provides a microlocal description of the lack of decay at infinity. We use this result to prove that the wave front relation of the free-to-free S-matrix (which, under our assumptions, is all of the S-matrix) is given by the broken geodesic flow, broken at the ‘singular directions’, on S n−1 at time π. We also present a natural geometric generalization to asymptotically Euclidean spaces.

Academic contribution of professor chi tai chuang

In this paper we mainly introduce the academic contribution of Prof. Chi Tai Chuang about value distributions, normal families, singular directions and differential polynomials of meromorphic functions. For a long period of time, he obtained many important and interesting research results. For instance, R. NeverHnna's conjecture: Generalization of second fundamental theorem on meromorphic functions to small functions nearly was solved by him.

AFM analysis of sidewall formation in low angle incidence microchannel epitaxy of GaAs

In this paper we show a detailed AFM observation of GaAs epilayers grown on GaAs (0 0 1) substrate patterned with SiO 2 mask by low angle incidence microchannel epitaxy (LAIMCE). We have found that low index facets are formed as sidewalls of the epilayers grown in open window aligned along singular directions. We have also found that low index facets are formed as sidewalls for epilayers showing a zigzagging edge. In the case of the epilayers grown in an open window aligned 10° off [0 1 0], for which lateral growth is maximum, we have found round shape sidewalls and (0 0 1) terraces at the boundary of the sidewall and the top surface. We concluded that the lateral growth proceeds when the sidewall is rough, while it stops when low index facets are formed as sidewalls.

Observational Error Structures and the Value of Advanced Assimilation Techniques

Abstract The ability of four-dimensional variational (4DVAR) assimilation of data to reduce various observational error structures in a quasigeostrophic model is studied. It is found that 4DVAR with assimilation periods on the order of a week is very efficient at reducing error in phase space directions that have not amplified in the past, that is, those phase space directions that do not lie on the unstable manifold of the system. This is particularly true for observational errors that project in rapidly growing singular vector phase space directions. In general, long period 4DVAR changes the forecast error growth rates to rates similar to the leading Lyapunov exponents for the system. However, error structures that grow significantly faster than the leading Lyapunov vector and are not readily reduced by long period 4DVAR can be constructed by doing a singular vector decomposition in the subspace of growing backward Lyapunov vectors. This procedure is an approximation to calculating the singular vectors ...

Far–field asymptotics of the Green's tensor for a transversely isotropic solid

The dynamic time–harmonic Green's tensor for a transversely isotropic homogeneous linearly elastic solid is studied. First, the representation of the Green's tensor as an integral over the unit sphere is obtained. It consists of three parts: quasi–longitudinal (P), shear–horizontal (SH) and quasi–shear (SV). Then, an exact analytical solution for the SH part in terms of elementary functions is derived. The complete far–field asymptotic approximation of P and SV parts is obtained next, using the uniform stationary phase method. For the P wave it involves the leading term of the ray series since there is only one arrival of this wave. The wave surface for the SV wave contains conical points and cuspidal edges. The asymptotic description applicable near these singular directions is derived involving the Airy and Bessel functions. The directions close to the points of tangential contact of the SH and SV sheets of the wave surface are also treated. At the end of the paper numerical results in both frequency and time domain are presented. They show that the agreement between the outputs of the asymptotic and direct numerical codes is very good throughout all regions but the former can be orders of magnitude faster.

Grain boundary faceting and abnormal grain growth in nickel

A correlation between grain boundary faceting and abnormal grain growth has been observed in recrystallized polycrystalline Ni at varying annealing temperatures, with or without C added. Carburized Ni specimens deformed to 50 pct show faceted grain boundaries and abnormal grain growth when annealed at temperatures below 0.7 Tm, where Tm is the melting point of Ni in absolute scale. When annealed at or above 0.7 Tm, the grain boundaries are smoothly curved and, therefore, have a rough structure, and normal grain growth is observed. In the specimens annealed in vacuum without carburization, all grain boundaries are faceted at 0.55 Tm, and some of them become defaceted at higher temperatures. The specimens annealed in vacuum at temperatures between 0.55 and 0.95 Tm show abnormal grain growth. When the grain boundaries have a rough structure and are, therefore, nearly isotropic, normal grain growth is indeed expected, as shown by the simulation and analytical treatment. When all or a fraction of the grain boundaries are faceted, with the facet planes corresponding to the singular cusp directions in the variation of the boundary energy against the inclination angle, abnormal grain growth can occur either because some grain boundary junctions become immobile due to a torque effect, or the growth occurs by a step mechanism.

Feasible Trajectories in Task Space from a Singularity for a Nonredundant or Redundant Robot Manipulator

From a singularity with deficiency of rank 1, for any joint velocity the effector velocity in task space is zero along a special direction (the singular direction). However, any motion along the m -1 non singular directions can be produced. Thus, any feasible trajectories from the singularity and with nonzero initial velocity is orthogonal to the singular direction. The feasibility of an effector acceleration along the singular direction is studied. A necessary condition for existence of an effector acceleration along the singular direction is that the joint velocities have to appear explicitly in the expression of this acceleration. This condition is not sufficient, since the joint velocities are related to the effector velocity along the nonsingular directions as well. A simplified expression is found for the effector acceleration along the singular direction whose analysis allows one to characterize the feasible trajectories from the singularity with a nonzero initial acceleration. From some singularities, any path can be tracked. In other cases, the tangent and curvature to the path at the singular point are not arbitrary. In case any path cannot be tracked, it may be interesting to consider trajectories with zero initial acceleration and nonzero initial jerk. Some special examples are studied. A classification of singularities is displayed, and the set of all feasible trajectories is defined in the task space for each type of singularity Various examples are provided. Redundant and nonredundant robots are studied.

Study and resolution of singularities for a 6-DOF PUMA manipulator

Upon solving the inverse kinematics problem of robot manipulators, the inherent singularity problem should always be considered. When a manipulator is approaching a singular configuration, a certain degree of freedom will be lost such that there are no feasible solutions of the manipulator to move into this singular direction. In this paper, the singularities of a 6-DOF PUMA manipulator are analyzed in detail and all the corresponding singular directions in task space are clearly identified. In order to resolve this singularity problem, an approach denoted Singularity Isolation Plus Compact QP (SICQP) method is proposed. The SICQP method decomposes the work space into achievable and unachievable (i.e., singular) directions. Then, the exactness in the singular directions are released such that extra redundancy is provided to the achievable directions. Finally, the Compact QP method is applied to maintain the exactness in the achievable directions, and to minimize the tracking errors in the singular directions under the condition that feasible joint solutions must be obtained. In the end, some simulation results for PUMA manipulator are given to demonstrate the effectiveness of the SICQP method.

A Crack in Forever

Fiction Does this plot sound a bit familiar? Two exceptional young people fall passionately in love only to discover that one of them has contracted an incurable illness. In the new novel<i>A Crack in Forever</i>, the disease is AIDS, "an illness that will creep or crush, but either way, it heads in singular directions." Eric Moro is a handsome, brilliant, and vivacious medical student (is there any other kind?) who also works part-time as a nude model, saves the life of a swimmer by performing CPR, and is a whiz at dissection in the gross anatomy lab. He becomes involved in a torrid love affair with Alexandra Taylor, an attractive (of course) and gifted artist who later decides to become a skywriter and eventually enters law school. Unfortunately, Eric has unknowingly contracted AIDS from sharing an infected needle while trying heroin when he was 17 and dies at

Book Review: Theory of entire and meromorphic functions. Deficient and asymptotic values and singular directions

Book Review: Theory of entire and meromorphic functions. Deficient and asymptotic values and singular directions

Singular directions and differential polynomials of meromorphic functions

We prove the existance of a kind of singular directions concerning the differential polynomials\(f^{(k)} + \sum\limits_{j = 0}^{k - 1} {a_j f^{(j)} - af^n } \).

Singular directions in the configuration space of linear vibrating systems

Besides the familiar concept of principal directions of normal modes in the theory of oscillations of linear systems with constant coefficients, new notions are introduced: directions conjugate to the principal directions, and principal directions of forced vibrations. The basic properties of vibrations in these directions are established.

Singularities and perturbations for far-field caustics in the neighborhood of a crack tip

Adapting the principle of least time by Fermat, the farfield caustics around a crack tip were studied as a variational problem. The singular reflected rays were probed and classified using singularity theory; furthermore the singular directions were also defined.

Hayman directions of meromorphic functions

In this paper, we shall prove the existence of the singular directions related to Hayman's problems[1]. The results are as follows. Theorem I. Suppose that f(z) is a transcendental integral function in the finite plane, then there exists a direction H: argz= θ0 (0≤θ0>2π) such that for every positive e, every integer p(≠0, −1) and every finite complex number b(≠0), we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' \cdot \{ f\} ^p = b)} \right\} = + \infty $$ Theorem II. Suppose that f(z) is a transcendental integral function in the finite plane, then there exists a direction H:z= θ0 (0≤θ0>2π) such that for every positive e, every integrer p(≥3) and any finite complex numbers a(≠0) and b, we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' - a\{ f\} ^p = b)} \right\} = + \infty $$ Theorem III. Suppose that f(z) is a meromorphic function in the finite plane and satisfies the following condition $$\mathop {\lim }\limits_{r \to \infty } \frac{{T(r,f)}}{{(\log r)^3 }} = + \infty $$ then there exists a direction H:z= θ0 (0≤θ0>2π) such that for every positive e, every integer p(≥5) and every two finite complex numbers a(≠0) and b, we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' - a\{ f\} ^p = b)} \right\} = + \infty $$

Propagation of light along elliptic singular directions of absorbing gyrotropic crystals

Propagation of light along elliptic singular directions of absorbing gyrotropic crystals

Phonon focusing catastrophes

We present an elegant application of formal catastrope theory to phonon focusing phenomena in crystals. The standard analysis of phonon focusing breaks down near singular directions known as caustics. The theory we present, in addition to being numerically accurate, provides a qualitative picture of caustic surfaces which yields important new insights into the results of phonon propagation experiments. Supporting experimental evidence is also presented.

A Study of Open Orbits in Lead Using the Induced Torque Technique

Open orbits in single crystal lead have been studied using the induced torque method. [Formula: see text]-directed open orbits were observed for all magnetic field directions in the {111} planes, except at the [Formula: see text] directions. Both [Formula: see text]- and [Formula: see text]-directed open orbits were observed for magnetic field directions over a range of 30 ± 1/2° centred on [110]. The torque, with the magnetic field along the [Formula: see text] singular field directions, approached zero. It was also observed that regions of aperiodic open orbits occurred for magnetic field directions within a cone of angle 20° ± 2° centred about the [Formula: see text] directions. The dependence of torque amplitude on both the open orbit direction relative to the sweep plane of the magnetic field and the magnetic field strength was also studied.

On the Mathematical Characterization of the Breaking of Hadronic Internal Symmetry

The adjoint action of a semisimple complex Lie algebra G singles out some singular directions with special properties. The directions of breaking of the hadronic internal symmetry groups, e.g., SU(3) or SU(3) × SU(3), by the electromagnetic, semileptonic, and nonleptonic weak interactions, are of this type. They are Hermitian (Q,Y) or nilpotent (Cabibbo direction) elements in the corresponding complexified Lie algebra. Their mathematical characterization is provided by the adjoint action of G. In particular, they satisfy a generalized extremal property for every G−invariant polynomial.

Un método directo para imponer coacciones en cualquier dirección, en el análisis matricial de una estructura

When analysing a structure by matrix methods, it is common to come up against the problem of imposing restrictions in various directions, so that it is impossible to choose a system of Cartesian axes that will make the directions of these axes coincide with those along which restrictions are to be imposed. This work presents a general method for imposing restrictions in any direction, in a mathematically exact and direct form, modifying the stiffness matrix in the area corresponding to the nodes in which such restrictions are imposed, being the new stiffness matrix, equally symmetrical, and having singular directions.

Note on the Flow just behind Three-Dimensional Shock

The concise expressions which determine the derivatives of flow quantities just behind a steady three dimensional shock with respect to the space coordinates, are generally derived according to Thomas' procedure, under the assumption of regularity of shock surface. These expressions assume comparatively simple froms when we take the length of the lines of curvature as the parameters. As special cases, they are related to the flow just behind two-dimensional or axially symmetrical shock. The consistency relation is derived for axially symmetrical shock. Then it is is the same as shown that the distribution of singular shock directions for axially symmetrical shock that for two-dimensional shock.