Existence Of Such Singularities Research Articles

Singular Solutions of Elasticity Theory Problems for Ideal Contact Zones of Solids

Singular solutions of elasticity theory problems in the vicinity of the normal (at the right angle) intersection of the ideal contact surface of two elastic media with free, fixed or slip-hinge clamped surfaces were analytically investigated. The necessary and sufficient conditions of the existence of such singularities as well as basic relationships of the effect of elastic characteristics of contacting bodies on their order were established.

Observational constraints on finite scale factor singularities

We discuss the combined constraints on a Finite Scale Factor Singularity (FSF) universe evolution scenario, which come from the shift parameter ℛ, baryon acoustic oscillations (BAO) \U0001d49c, and from the type Ia supernovae.We show that observations allow existence of such singularities in the 2 × 109 years in future (at 1σ CL) which is much farther than a Sudden Future Singularity (SFS), and that at the present moment of the cosmic evolution, one cannot differentiate between cosmological scenario which allow finite scale factor singularities and the standard ΛCDM dark energy models. We also show that there is an allowed value of m = 2/3 within 1σ CL, which corresponds to a dust-filled Einstein-de-Sitter universe limit of the early time evolution and so it is pasted into a standard early-time scenario.

Anomaly-free singularities in the generalized Kohn variational method

We have carried out an analysis of singularities in Kohn variational calculations for low-energy (e+–H2) elastic scattering. Provided that a sufficiently accurate trial wavefunction is used, we argue that our implementation of the Kohn variational principle necessarily gives rise to singularities which are not spurious. We propose two approaches for optimizing a free parameter of the trial wavefunction in order to avoid anomalous behavior in scattering phase shift calculations, the first of which is based on the existence of such singularities. The second approach is a more conventional optimization of the generalized Kohn method. Close agreement is observed between the results of the two optimization schemes; further, they give results which are seen to be effectively equivalent to those obtained with the complex Kohn method. The advantage of the first optimization scheme is that it does not require an explicit solution of the Kohn equations to be found. We give examples of anomalies which cannot be avoided using either optimization scheme but show that it is possible to avoid these anomalies by considering variations in the nonlinear parameters of the trial function.

Obstacle to populating the string theory landscape

We construct domain walls and instantons in a class of models with coupled scalar fields, determining, in agreement with previous studies, that many such solutions contain naked timelike singularities. Vacuum bubble solutions of this type do not contain a region of true vacuum, obstructing the ability of eternal inflation to populate other vacua. We determine a criterion that potentials must satisfy to avoid the existence of such singularities and show that many domain wall solutions in type IIB string theory are singular.

Ambiguities in Powder Indexing: Conjunction of a Ternary and Binary Lattice Metric Singularity in the Cubic System.

A lattice metric singularity occurs when unit cells defining two (or more) lattices yield the identical set of unique calculated d-spacings. The existence of such singularities, therefore, has a practical and theoretical impact on the indexing of powder patterns. For example, in experimental practice an indexing program may find only the lower symmetry member of a singularity. Obviously, it is important to recognize such cases and know how to proceed. Recently, we described: a binary singularity involving a monoclinic and a rhombohedral lattice in a subcell-supercell relationship anda second type of singularity—a ternary singularity–in which two of the three lattices are in a derivative composite relationship. In this work, we describe a ternary lattice metric singularity involving a cubic P, a tetragonal P, and an orthorhombic C lattice. Furthermore, there is a binary singularity, involving a hexagonal P and orthorhombic P lattice, which is characterized by a set of unique d-spacings very close to that of the ternary singularity. The existence of such singularities is more common than once thought and requires a paradigm shift in experimental practice. In addition singularities provide opportunities in material design as they point to highly specialized lattices that may be associated with unusual physical properties.

Ambiguities in powder pattern indexing: A ternary lattice metric singularity

A lattice metric singularity occurs when unit cells defining two (or more) lattices yield the identical set of unique calculated d-spacings. The existence of such singularities, therefore, has a practical impact on the indexing of powder patterns. Lattice metric singularities often involve lattices that are in a derivative relationship one to another. A variety of types of singularities are possible depending on the number of different lattices involved (i.e., binary, ternary, quaternary), on the nature of the derivative lattice relationship (i.e., subcell/supercell, composite), on the Bravais type of each of the lattices, and on the the volume ratio(s) of primitive cells defining the lattices. In the laboratory, an encounter with a singularity can lead one into a trap; viz., the investigator using an indexing program, or by other means, may determine only one of the lattices with a high figure of merit. When this happens, it is critical to recognize that there exists more than one indexing solution. In a previous work, a binary singularity was described involving a monoclinic and a rhombohedral lattice. In the present work, we describe a second type of singularity—a ternary singularity—in which the two of the three lattices are in a derivative composite relationship.

Lattice metric singularities and their impact on the indexing of powder patterns

A lattice metric singularity occurs when unit cells defining two (or more) lattices yield the identical set of unique calculated d-spacings. The existence of such singularities, therefore, has a practical impact on the indexing of powder patterns. For example, when experimental data from ζ-LiBO2 were indexed, two solutions (a rhombohedral and a monoclinic lattice) with approximately the same figure of merit were found. These two lattices yield the same set of unique d-spacings even though they are characterized by different reduced cells with cell volumes in the ratio 2 to 1. From the indexing point of view, both answers are correct. A singularity of this type is common and not a mathematical rarity. In fact, any rhombohedral cell of this kind has a derivative monoclinic subcell, each of which gives the same set of unique calculated d-spacings. In actual cases like this, one can run into a trap. Due to experimental error and input parameters, an indexing program may determine only one of the cells with a high figure of merit. When this happens, it is critical to recognize that another solution exists, especially if one has determined the lower symmetry lattice.

Parallelism in MuPAD

S INGULAR is a computer algebra system designed to be a power-house for polynomial computations with special features for algebraic geometry and singularity theory.The development of S INGULAR started in 1986, motivated by the question of the existence of complete intersection singularities which are not quasi-homogeneous, but whose Poincare-complex is exact. The existence of such singularities could be shown in 1987 with an implementation of Mora's algorithm on an Atari in Modula-2 [11]. In 1990, S INGULAR was ported to Unix, and a first public-domain distribution followed. The code of S INGULAR was rewritten in C/C++. An interpreted user-level programming language and user-written libraries were added in 1993. With the release of version 1.0 in May 1997, many important features were added, including:• Polynomial factorization• Factorizing Buchberger algorithm• FGLM algorithm• Primary decomposition of ideals• MP links for general and efficient storage and inter-process communicationsThe goal of the poster is to discuss and demonstrate these new features of S INGULAR where, in combination with the posters on MP and MuPAD, special emphasize will be paid to S INGULAR 's MP interface and its resulting possibilities and problems.

Forbidden tangential orbit transfers between intersecting Keplerian orbits

The classical problem of tangential impulse transfer between coplanar Keplerian orbits is addressed. A completely analytic solution which does not rely on sequential calculation is obtained and this solution is used to demonstrate that certain choices of initial true anomaly can produce singularities in the parameters of the transfer orbit. A necessary and sufficient condition for the existence of such singularities is that the initial and final orbits intersect.

Random Nonlinear Wave Equations: Propagation of Singularities

We investigate the smoothness properties of the solutions of one-dimensional wave equations with nonlinear random forcing. We define singularities as anomalies in the local modulus of continuity of the solutions. We prove the existence of such singularities and their propagation along the characteristic curves. When the space variable is restricted to a bounded interval, we impose the Dirichlet boundary condition at the endpoints and we show how the singularities are reflected at the boundary.

Singularities of a variational partition function

By concrete examples it is shown that a variational approximation to the partition function (VPF) may have signularities of the phase transition types. Sufficient conditions for the existence of such singularities are given. One of these conditions is the presence of a continuous part in the spectrum of the underlying Hamiltonian.

Phonon Critical Points Reflected in Superconducting Tunneling Characteristics

Van Hove showed that there are mathematical singularities in the phonon distribution in frequency $g(\ensuremath{\omega})$ associated with stationary points in $\ensuremath{\omega}(q)$ versus $q$; $\frac{\mathrm{dg}}{d\ensuremath{\omega}}$ has at least two infinite discontinuities, and approaches $\ensuremath{-}\ensuremath{\infty}$ at the upper end of the spectrum. Additional weaker singularities have been discussed by Phillips. The current voltage characteristic of superconducting tunnel diodes exhibits structure related to the phonon spectrum; we investigate theoretically the nature of the mathematical singularities in the tunnel characteristic associated with the Van Hove and Phillips singularities, using the Eliashberg gap equation. Discontinuities and logarithmic singularities are predicted for the metal superconductor junction at $T=0$, and inverse square-root singularities as well in two-superconductor junctions. Analysis of Rowell's experimental curves for Pb and Sn in conjunction with neutron data on phonons in Pb provides definite evidence for the existence of such singularities and thus detailed confirmation of the theory.