A condition for isotopic approximation

In this note, we show that if two surfaces in are homeomorphic, then a simple and purely topological condition is sufficient to ensure the existence of an isotopy between them. When the surfaces are connected, the condition is merely that one surface is contained in some topological thickening of the other and separates the two boundary components of that thickening. The proof of this result is based on basic 3-manifold topology

HERA high Q2Q^2 events as indications of excited leptons with weak isotopic spin 3/2

The H1 and ZEUS anomalous events are interpreted as being due to the production and the decay of excited leptons $E$, which correspond to spin 1/2 resonances of the first generation lepton doublet ($\nu_e, e$) with W triplet. This assumption is supported by considering of Bethe-Salpeter equation in the ladder approximation with anomalous triple gauge boson vertex. The solution with weak isospin $I = 3/2$ is shown to exist for zero mass state, that means M_E$is small in comparison with TeV mass scale. The coupling of$E$with leptons and$W$is defined by the normalization condition. Calculation of the$E$width and the production cross-sections agrees with HERA data for value of the triple$W$coupling constant$\lambda \simeq 0.5$. Isotopic relations for different channels are presented as a tool for checking the interpretation.Comment: 8 pages, latex, no figure

Continuous rational maps into spheres

Let X be a compact nonsingular real algebraic variety. We prove that if a continuous map from X into the unit p-sphere is homotopic to a continuous rational map, then, under certain assumptions, it can be approximated in the compact-open topology by continuous rational maps. As a byproduct, we also obtain some results on approximation of smooth submanifolds by nonsingular subvarieties.Comment: To appear in Mathematische Zeitschrif

Shock-Wave Heating Model for Chondrule Formation: Prevention of Isotopic Fractionation

Chondrules are considered to have much information on dust particles and processes in the solar nebula. It is naturally expected that protoplanetary disks observed in present star forming regions have similar dust particles and processes, so study of chondrule formation may provide us great information on the formation of the planetary systems. Evaporation during chondrule melting may have resulted in depletion of volatile elements in chondrules. However, no evidence for a large degree of heavy-isotope enrichment has been reported in chondrules. In order to meet this observed constraint, the rapid heating rate at temperatures below the silicate solidus is required to suppress the isotopic fractionation. We have developed a new shock-wave heating model taking into account the radiative transfer of the dust thermal continuum emission and the line emission of gas molecules and calculated the thermal history of chondrules. We have found that optically-thin shock waves for the thermal continuum emission from dust particles can meet the rapid heating constraint, because the dust thermal emission does not keep the dust particles high temperature for a long time in the pre-shock region and dust particles are abruptly heated by the gas drag heating in the post-shock region. We have also derived the upper limit of optical depth of the pre-shock region using the radiative diffusion approximation, above which the rapid heating constraint is not satisfied. It is about 1 - 10.Comment: 58 pages, including 5 tables and 15 figures, accepted for publication in The Astrophysical Journa

Complete Subdivision Algorithms, II: Isotopic Meshing of Singular Algebraic Curves

Given a real valued function f(X,Y), a box region B_0 in R^2 and a positive epsilon, we want to compute an epsilon-isotopic polygonal approximation to the restriction of the curve S=f^{-1}(0)={p in R^2: f(p)=0} to B_0. We focus on subdivision algorithms because of their adaptive complexity and ease of implementation. Plantinga and Vegter gave a numerical subdivision algorithm that is exact when the curve S is bounded and non-singular. They used a computational model that relied only on function evaluation and interval arithmetic. We generalize their algorithm to any bounded (but possibly non-simply connected) region that does not contain singularities of S. With this generalization as a subroutine, we provide a method to detect isolated algebraic singularities and their branching degree. This appears to be the first complete purely numerical method to compute isotopic approximations of algebraic curves with isolated singularities

Isotopic Equivalence from Bezier Curve Subdivision

We prove that the control polygon of a Bezier curve B becomes homeomorphic and ambient isotopic to B via subdivision, and we provide closed-form formulas to compute the number of iterations to ensure these topological characteristics. We first show that the exterior angles of control polygons converge exponentially to zero under subdivision.Comment: arXiv admin note: substantial text overlap with arXiv:1211.035

Some conjectures on continuous rational maps into spheres

Recently continuous rational maps between real algebraic varieties have attracted the attention of several researchers. In this paper we continue the investigation of approximation properties of continuous rational maps with values in spheres. We propose a conjecture concerning such maps and show that it follows from certain classical conjectures involving transformation of compact smooth submanifolds of nonsingular real algebraic varieties onto subvarieties. Furthermore, we prove our conjecture in a special case and obtain several related results.Comment: arXiv admin note: text overlap with arXiv:1403.512

Predicting scattering properties of ultracold atoms: adiabatic accumulated phase method and mass scaling

Ultracold atoms are increasingly used for high precision experiments that can be utilized to extract accurate scattering properties. This calls for a stronger need to improve on the accuracy of interatomic potentials, and in particular the usually rather inaccurate inner-range potentials. A boundary condition for this inner range can be conveniently given via the accumulated phase method. However, in this approach one should satisfy two conditions, which are in principle conflicting, and the validity of these approximations comes under stress when higher precision is required. We show that a better compromise between the two is possible by allowing for an adiabatic change of the hyperfine mixing of singlet and triplet states for interatomic distances smaller than the separation radius. A mass scaling approach to relate accumulated phase parameters in a combined analysis of isotopically related atom pairs is described in detail and its accuracy is estimated, taking into account both Born-Oppenheimer and WKB breakdown. We demonstrate how numbers of singlet and triplet bound states follow from the mass scaling.Comment: 14 pages, 9 figure