660 research outputs found
Quasi-classical approximation in vortex filament dynamics. Integrable systems, gradient catastrophe and flutter
Quasiclassical approximation in the intrinsic description of the vortex
filament dynamics is discussed. Within this approximation the governing
equations are given by elliptic system of quasi-linear PDEs of the first order.
Dispersionless Da Rios system and dispersionless Hirota equation are among
them. They describe motion of vortex filament with slow varying curvature and
torsion without or with axial flow. Gradient catastrophe for governing
equations is studied. It is shown that geometrically this catastrophe manifests
as a fast oscillation of a filament curve around the rectifying plane which
resembles the flutter of airfoils. Analytically it is the elliptic umbilic
singularity in the terminology of the catastrophe theory. It is demonstrated
that its double scaling regularization is governed by the Painleve' I equation.Comment: 25 pages, 5 figures, minor typos correcte
The Blaschke conjecture and great circle fibrations of spheres
We construct an explicit diffeomorphism taking any fibration of a sphere by
great circles into the Hopf fibration, using elementary geometry--indeed the
diffeomorphism is a local (differential) invariant, algebraic in derivatives.Comment: 61 pages, 8 figures, corrected errors in the published versio
Global Energy Matching Method for Atomistic-to-Continuum Modeling of Self-Assembling Biopolymer Aggregates
This paper studies mathematical models of biopolymer supramolecular aggregates that are formed by the self-assembly of single monomers. We develop a new multiscale numerical approach to model the structural properties of such aggregates. This theoretical approach establishes micro-macro relations between the geometrical and mechanical properties of the monomers and supramolecular aggregates. Most atomistic-to-continuum methods are constrained by a crystalline order or a periodic setting and therefore cannot be directly applied to modeling of soft matter. By contrast, the energy matching method developed in this paper does not require crystalline order and, therefore, can be applied to general microstructures with strongly variable spatial correlations. In this paper we use this method to compute the shape and the bending stiffness of their supramolecular aggregates from known chiral and amphiphilic properties of the short chain peptide monomers. Numerical implementation of our approach demonstrates consistency with results obtained by molecular dynamics simulations
General-affine invariants of plane curves and space curves
We present a fundamental theory of curves in the affine plane and the affine
space, equipped with the general-affine groups and ,
respectively. We define general-affine length parameter and curvatures and show
how such invariants determine the curve up to general-affine motions. We then
study the extremal problem of the general-affine length functional and derive a
variational formula. We give several examples of curves and also discuss some
relations with equiaffine treatment and projective treatment of curves.Comment: 51 pages, 4 figures, to appear in Czechoslovak Mathematical Journal,
version2: typos are fixe
Conicoid Mirrors
The first order equation relating object and image location for a mirror of
arbitrary conic-sectional shape is derived. It is also shown that the parabolic
reflecting surface is the only one free of aberration and only in the limiting
case of distant sources.Comment: 9 page
Element sets for high-order Poincar\'e mapping of perturbed Keplerian motion
The propagation and Poincar\'e mapping of perturbed Keplerian motion is a key
topic in celestial mechanics and astrodynamics, e.g. to study the stability of
orbits or design bounded relative trajectories. The high-order transfer map
(HOTM) method enables efficient mapping of perturbed Keplerian orbits over many
revolutions. For this, the method uses the high-order Taylor expansion of a
Poincar\'e or stroboscopic map, which is accurate close to the expansion point.
In this paper, we investigate the performance of the HOTM method using
different element sets for building the high-order map. The element sets
investigated are the classical orbital elements, modified equinoctial elements,
Hill variables, cylindrical coordinates and Deprit's ideal elements. The
performances of the different coordinate sets are tested by comparing the
accuracy and efficiency of mapping low-Earth and highly-elliptical orbits
perturbed by with numerical propagation. The accuracy of HOTM depends
strongly on the choice of elements and type of orbit. A new set of elements is
introduced that enables extremely accurate mapping of the state, even for high
eccentricities and higher-order zonal perturbations. Finally, the high-order
map is shown to be very useful for the determination and study of fixed points
and centre manifolds of Poincar\'e maps.Comment: Pre-print of journal articl
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