61,437 research outputs found
Algebraic equations for constant width curves and Zindler curves
An explicit method to compute algebraic equations of curves of constant width and Zindler curves generated by a family of middle hedgehogs is given thanks to a property of Chebyshev polynomials. This extends the methodology used by Rabinowitz and Martinez-Maure in particular constant width curves to generate a full family of algebraic equations, both of curves of constant width and Zindler curves, defined by trigonometric polynomials as support functions
Algebraic Equations in State Condition
In this paper, we will prove that a problem deciding whether there is an
upper-triangular coordinate in which a character is not in the state of a
Hilbert point is NP-hard. This problem is related to the GIT-semistability of a
Hilbert point.Comment: -corrected some typos and added an example in the end of the fourth
sectio
Riemann-Hilbert problem for the small dispersion limit of the KdV equation and linear overdetermined systems of Euler-Poisson-Darboux type
We study the Cauchy problem for the Korteweg de Vries (KdV) equation with
small dispersion and with monotonically increasing initial data using the
Riemann-Hilbert (RH) approach. The solution of the Cauchy problem, in the zero
dispersion limit, is obtained using the steepest descent method for oscillatory
Riemann-Hilbert problems. The asymptotic solution is completely described by a
scalar function \g that satisfies a scalar RH problem and a set of algebraic
equations constrained by algebraic inequalities. The scalar function \g is
equivalent to the solution of the Lax-Levermore maximization problem. The
solution of the set of algebraic equations satisfies the Whitham equations. We
show that the scalar function \g and the Lax-Levermore maximizer can be
expressed as the solution of a linear overdetermined system of equations of
Euler-Poisson-Darboux type. We also show that the set of algebraic equations
and algebraic inequalities can be expressed in terms of the solution of a
different set of linear overdetermined systems of equations of
Euler-Poisson-Darboux type. Furthermore we show that the set of algebraic
equations is equivalent to the classical solution of the Whitham equations
expressed by the hodograph transformation.Comment: 32 pages, 1 figure, latex2
The Initial Value Problem For Maximally Non-Local Actions
We study the initial value problem for actions which contain non-trivial
functions of integrals of local functions of the dynamical variable. In
contrast to many other non-local actions, the classical solution set of these
systems is at most discretely enlarged, and may even be restricted, with
respect to that of a local theory. We show that the solutions are those of a
local theory whose (spacetime constant) parameters vary with the initial value
data according to algebraic equations. The various roots of these algebraic
equations can be plausibly interpreted in quantum mechanics as different
components of a multi-component wave function. It is also possible that the
consistency of these algebraic equations imposes constraints upon the initial
value data which appear miraculous from the context of a local theory.Comment: 8 pages, LaTeX 2 epsilo
Harmonic solutions to a class of differential-algebraic equations with separated variables
We study the set of T-periodic solutions of a class of T-periodically
perturbed Differential-Algebraic Equations with separated variables. Under
suitable hypotheses, these equations are equivalent to separated variables ODEs
on a manifold. By combining known results on Differential-Algebraic Equations,
with an argument about ODEs on manifolds, we obtain a global continuation
result for the T-periodic solutions to the considered equations. As an
application of our method, a multiplicity result is provided
Intercusp Geodesics and Cusp Shapes of Fully Augmented Links
We study the geometry of fully augmented link complements in by looking
at their link diagrams. We extend the method introduced by Thistlethwaite and
Tsvietkova to fully augmented links and define a system of algebraic equations
in terms of parameters coming from edges and crossings of the link diagrams.
Combining it with the work of Purcell, we show that the solutions to these
algebraic equations are related to the cusp shapes of fully augmented link
complements. As an application we use the cusp shapes to study the
commensurability classes of fully augmented links
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