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21.
For a knotted surface in -space, its generic projection into -space has branch points as its singularities, and its successive projection into -space has fold points and cusps as its singularities. In this paper, we show that for non-orientable knotted surfaces, the numbers of branch points and cusps can be minimized by isotopy.
22.
23.
Pankaj S. Joshi 《Pramana》2007,69(1):119-135
We consider here the genericity aspects of spacetime singularities that occur in cosmology and in gravitational collapse.
The singularity theorems (that predict the occurrence of singularities in general relativity) allow the singularities of gravitational
collapse to be either visible to external observers or covered by an event horizon of gravity. It is shown that the visible
singularities that develop as final states of spherical collapse are generic. Some consequences of this fact are discussed.
相似文献
24.
We study the structure and asymptotic behavior of the resolvent of elliptic cone pseudodifferential operators acting on weighted
Sobolev spaces over a compact manifold with boundary. We obtain an asymptotic expansion of the resolvent as the spectral parameter
tends to infinity, and use it to derive corresponding heat trace and zeta function expansions as well as an analytic index
formula.
相似文献
25.
Mitchell J. Feigenbaum 《Journal of statistical physics》1987,46(5-6):925-932
The grand canonical version of the spectrum of singularities formalism is presented, relying naturally upon certain Markov transition graphs. The structure of a graph is simply determined by the close return times of the dynamical system described. Thus, an intimate connection exists between the shape of the singularity curve and a small but interesting set of dynamical properties. 相似文献
26.
The nonlinear parametric programming problem is reformulated as a closed system of nonlinear equations so that numerical continuation
and bifurcation techniques can be used to investigate the dependence of the optimal solution on the system parameters. This
system, which is motivated by the Fritz John first-order necessary conditions, contains all Fritz John and all Karush-Kuhn-Tucker
points as well as local minima and maxima, saddle points, feasible and nonfeasible critical points. Necessary and sufficient
conditions for a singularity to occur in this system are characterized in terms of the loss of a complementarity condition,
the linear dependence of the gradients of the active constraints, and the singularity of the Hessian of the Lagrangian on
a tangent space. Any singularity can be placed in one of seven distinct classes depending upon which subset of these three
conditions hold true at a solution. For problems with one parameter, we analyze simple and multiple bifurcation of critical
points from a singularity arising from the loss of the complementarity condition, and then develop a set of conditions which
guarantees the unique persistence of a minimum through this singularity.
The research of this author was supported by National Science Foundation through NSF Grant
DMS-85-10201 and by the Air Force Office of Scientific Research through instrument number AFOSR-ISSA-85-00079. 相似文献
27.
28.
Laurent-Padé (Chebyshev) rational approximantsP
m
(w, w
−1)/Q
n
(w, w
−1) of Clenshaw-Lord type [2,1] are defined, such that the Laurent series ofP
m
/Q
n
matches that of a given functionf(w, w
−1) up to terms of orderw
±(m+n)
, based only on knowledge of the Laurent series coefficients off up to terms inw
±(m+n)
. This contrasts with the Maehly-type approximants [4,5] defined and computed in part I of this paper [6], where the Laurent
series ofP
m
matches that ofQ
n
f up to terms of orderw
±(m+n
), but based on knowledge of the series coefficients off up to terms inw
±(m+2n). The Clenshaw-Lord method is here extended to be applicable to Chebyshev polynomials of the 1st, 2nd, 3rd and 4th kinds and
corresponding rational approximants and Laurent series, and efficient systems of linear equations for the determination of
the Padé-Chebyshev coefficients are obtained in each case. Using the Laurent approach of Gragg and Johnson [4], approximations
are obtainable for allm≥0,n≥0. Numerical results are obtained for all four kinds of Chebyshev polynomials and Padé-Chebyshev approximants. Remarkably
similar results of formidable accuracy are obtained by both Maehly-type and Clenshaw-Lord type methods, thus validating the
use of either. 相似文献
29.
Heinz‐Jürgen Flad Reinhold Schneider Bert‐Wolfgang Schulze 《Mathematical Methods in the Applied Sciences》2008,31(18):2172-2201
We study the asymptotic regularity of solutions to Hartree–Fock (HF) equations for Coulomb systems. To deal with singular Coulomb potentials, Fock operators are discussed within the calculus of pseudo‐differential operators on conical manifolds. First, the non‐self‐consistent‐field case is considered, which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator, which guarantee ellipticity for the local part of the Fock operator on the open stretched cone ?+ × S2. This proves the existence of a parametrix with a corresponding smoothing remainder from which it follows, via a bootstrap argument, that the eigenfunctions of the Fock operator again satisfy asymptotic regularity conditions. Using a fixed‐point approach based on Cancès and Le Bris analysis of the level‐shifting algorithm, we show via another bootstrap argument that the corresponding self‐consistent‐field solutions to the HF equation have the same type of asymptotic regularity. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
30.
Xiaoxu Lian 《Optics Communications》2011,284(22):5253-5258
Young's interference experiment is regarded as a two-slit diffraction phenomenon, the polarization singularities in Young's two-slit configuration illuminated with two linearly, orthogonally polarized Gaussian vortex beams are studied. It is shown that generally, there exist L-lines (linearly polarization) and polarization singularities including C-points (circular polarization), S23 and S31 singularities even though the parameters of two beams are the same. The pair creation-annihilation and motion of polarization singularities take place upon propagation, or by varying a control parameter, such as the amplitude ratio of two beams or obscure ratio of slits etc. For a special case of the illumination with two linearly, orthogonally polarized Gaussian vortex-free beams, polarization singularities, in particular, C-points may occur if a parameter of two beams is not equal. 相似文献