34,028 research outputs found
A generalization of crossing families
For a set of points in the plane, a \emph{crossing family} is a set of line
segments, each joining two of the points, such that any two line segments
cross. We investigate the following generalization of crossing families: a
\emph{spoke set} is a set of lines drawn through a point set such that each
unbounded region of the induced line arrangement contains at least one point of
the point set. We show that every point set has a spoke set of size
. We also characterize the matchings obtained by selecting
exactly one point in each unbounded region and connecting every such point to
the point in the antipodal unbounded region.Comment: 14 pages, 10 figure
The untwisting number of a knot
The unknotting number of a knot is the minimum number of crossings one must
change to turn that knot into the unknot. The algebraic unknotting number is
the minimum number of crossing changes needed to transform a knot into an
Alexander polynomial-one knot. We work with a generalization of unknotting
number due to Mathieu-Domergue, which we call the untwisting number. The
untwisting number is the minimum number (over all diagrams of a knot) of right-
or left-handed twists on even numbers of strands of a knot, with half of the
strands oriented in each direction, necessary to transform that knot into the
unknot. We show that the algebraic untwisting number is equal to the algebraic
unknotting number. However, we also exhibit several families of knots for which
the difference between the unknotting and untwisting numbers is arbitrarily
large, even when we only allow twists on a fixed number of strands or fewer.Comment: 18 pages, 6 figures; to appear in Pacific J. Mat
Positive scalar curvature and higher-dimensional families of Seiberg-Witten equations
We introduce an invariant of tuples of commutative diffeomorphisms on a
4-manifold using families of Seiberg-Witten equations. This is a generalization
of Ruberman's invariant of diffeomorphisms defined using 1-parameter families
of Seiberg-Witten equations. Our invariant yields an application to the
homotopy groups of the space of positive scalar curvature metrics on a
4-manifold. We also study the extension problem for families of 4-manifolds
using our invariant.Comment: 22 pages, accepted for publication by the Journal of Topolog
Infinite families of crossing-critical graphs with prescribed average degree and crossing number
Siran constructed infinite families of k-crossing-critical graphs for every
k=>3 and Kochol constructed such families of simple graphs for every k=>2.
Richter and Thomassen argued that, for any given k>=1 and r>=6, there are only
finitely many simple k-crossing-critical graphs with minimum degree r. Salazar
observed that the same argument implies such a conclusion for simple
k-crossing-critical graphs of prescribed average degree r>6. He established
existence of infinite families of simple k-crossing-critical graphs with any
prescribed rational average degree r in [4,6) for infinitely many k and asked
about their existence for r in (3,4). The question was partially settled by
Pinontoan and Richter, who answered it positively for r in (7/2,4).
The present contribution uses two new constructions of crossing critical
simple graphs along with the one developed by Pinontoan and Richter to unify
these results and to answer Salazar's question by the following statement: for
every rational number r in (3,6) there exists an integer N_r, such that, for
any k>N_r, there exists an infinite family of simple 3-connected
crossing-critical graphs with average degree r and crossing number k. Moreover,
a universal lower bound on k applies for rational numbers in any closed
interval I in (3,6).Comment: 21 pages, 5 figure
Fifteen classes of solutions of the quantum two-state problem in terms of the confluent Heun function
We derive 15 classes of time-dependent two-state models solvable in terms of
the confluent Heun functions. These classes extend over all the known families
of three- and two-parametric models solvable in terms of the hypergeometric and
the confluent hypergeometric functions to more general four-parametric classes
involving three-parametric detuning modulation functions. In the case of
constant detuning the field configurations describe excitations of two-state
quantum systems by symmetric or asymmetric pulses of controllable width and
edge-steepness. The classes that provide constant detuning pulses of finite
area are identified and the factors controlling the corresponding pulse shapes
are discussed. The positions and the heights of the peaks are mostly defined by
two of the three parameters of the detuning modulation function, while the
pulse width is mainly controlled by the third one, the constant term. The
classes suggest numerous symmetric and asymmetric chirped pulses and a variety
of models with two crossings of the frequency resonance. We discuss the
excitation of a two-level atom by a pulse of Lorentzian shape with a detuning
providing one or two crossings of the resonance. We derive closed form
solutions for particular curves in the 3D space of the involved parameters
which compose the complete return spectrum of the considered two-state quantum
system
Logarithmic Kodaira-Akizuki-Nakano vanishing and Arakelov-Parshin boundedness for singular varieties
The article has two parts. The first part is devoted to proving a singular
version of the logarithmic Kodaira-Akizuki-Nakano vanishing theorem of Esnault
and Viehweg. This is then used to prove other vanishing theorems. In the second
part these vanishing theorems are used to prove an Arakelov-Parshin type
boundedness result for families of canonically polarized varieties with
rational Gorenstein singularities.Comment: 23 pages, typos/errors corrected, definition of strong ampleness is
made more general more typos/errors corrected, 6.4 made slightly more genera
Evolution for Khovanov polynomials for figure-eight-like family of knots
We look at how evolution method deforms, when one considers Khovanov
polynomials instead of Jones polynomials. We do this for the figure-eight-like
knots (also known as 'double braid' knots, see arXiv:1306.3197) -- a
two-parametric family of knots which "grows" from the figure-eight knot and
contains both two-strand torus knots and twist knots. We prove that parameter
space splits into four chambers, each with its own evolution, and two isolated
points. Remarkably, the evolution in the Khovanov case features an extra
eigenvalue, which drops out in the Jones (t -> -1) limit.Comment: 9 pages, add some missing reference
Generalization of Neron models of Green, Griffiths and Kerr
We explain some recent developments in the theory of Neron models for
families of Jacobians associated to variations of Hodge structures of weight
-1.Comment: 8 page
A determinantal formula for the Hilbert series of one-sided ladder determinantal rings
We give a formula that expresses the Hilbert series of one-sided ladder
determinantal rings, up to a trivial factor, in form of a determinant. This
allows the convenient computation of these Hilbert series. The formula follows
from a determinantal formula for a generating function for families of
nonintersecting lattice paths that stay inside a one-sided ladder-shaped
region, in which the paths are counted with respect to turns.Comment: 28 pages, AmS-LaTeX; an error in the definition of ladder
determinantal ring was correcte
The isotriviality of families of canonically-polarized manifolds over a special quasi-projective base
In this paper we prove that a smooth family of canonically polarized
manifolds parametrized by a special (in the sense of Campana) quasi-projective
variety is isotrivial.Comment: Shorter arguments in Sect. 3 plus a correction in the equality of
Cor. 3.2. To appear in Compositio Mathematic
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