1,453 research outputs found
On polynomially integrable Birkhoff billiards on surfaces of constant curvature
We present a solution of the algebraic version of Birkhoff Conjecture on
integrable billiards. Namely we show that every polynomially integrable real
bounded convex planar billiard with smooth boundary is an ellipse. We extend
this result to billiards with piecewise-smooth and not necessarily convex
boundary on arbitrary two-dimensional surface of constant curvature: plane,
sphere, Lobachevsky (hyperbolic) plane; each of them being modeled as a plane
or a (pseudo-) sphere in equipped with appropriate quadratic
form. Namely, we show that a billiard is polynomially integrable, if and only
if its boundary is a union of confocal conical arcs and appropriate geodesic
segments. We also present a complexification of these results. These are joint
results of Mikhail Bialy, Andrey Mironov and the author. The proof is split
into two parts. The first part is given by Bialy and Mironov in their two joint
papers. They considered the tautological projection of the boundary to
and studied its orthogonal-polar dual curve, which is piecewise
algebraic, by S.V.Bolotin's theorem. By their arguments and another Bolotin's
theorem, it suffices to show that each non-linear complex irreducible component
of the dual curve is a conic. They have proved that all its singularities and
inflection points (if any) lie in the projectivized zero locus of the
corresponding quadratic form on . The present paper provides the
second part of the proof: we show that each above irreducible component is a
conic and finish the solution of the Algebraic Birkhoff Conjecture in constant
curvature.Comment: To appear in the Journal of the European Mathematical Society (JEMS),
69 pages, 2 figures. A shorter proof of Theorem 4.24. Minor precisions and
misprint correction
Convexity preserving interpolatory subdivision with conic precision
The paper is concerned with the problem of shape preserving interpolatory
subdivision. For arbitrarily spaced, planar input data an efficient non-linear
subdivision algorithm is presented that results in limit curves,
reproduces conic sections and respects the convexity properties of the initial
data. Significant numerical examples illustrate the effectiveness of the
proposed method
Discrete Riemann Surfaces and the Ising model
We define a new theory of discrete Riemann surfaces and present its basic
results. The key idea is to consider not only a cellular decomposition of a
surface, but the union with its dual. Discrete holomorphy is defined by a
straightforward discretisation of the Cauchy-Riemann equation. A lot of
classical results in Riemann theory have a discrete counterpart, Hodge star,
harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of
holomorphic forms with prescribed holonomies. Giving a geometrical meaning to
the construction on a Riemann surface, we define a notion of criticality on
which we prove a continuous limit theorem. We investigate its connection with
criticality in the Ising model. We set up a Dirac equation on a discrete
universal spin structure and we prove that the existence of a Dirac spinor is
equivalent to criticality
Dynamics of a family of piecewise-linear area-preserving plane maps II. Invariant circles
This paper studies the behavior under iteration of the maps
T_{ab}(x,y)=(F_{ab}(x)-y,x) of the plane R^2, in which F_{ab}(x)=ax if x>=0 and
bx if x<0. The orbits under iteration correspond to solutions of the nonlinear
difference equation x_{n+2}= 1/2(a-b)|x_{n+1}| + 1/2(a+b)x_{n+1} - x_n. This
family of maps has the parameter space (a,b)\in R^2. These maps are
area-preserving homeomorphisms of R^s that map rays from the origin into rays
from the origin. This paper shows the existence of special parameter values
where T_{ab} has every nonzero orbit an invariant circle with irrational
rotation number, and these invariant circles are piecewise unions of arcs of
conic sections. Numerical experiments suggest the possible existence of many
other parameter values having invariant circles.Comment: v2 corresponds to second half of old part I; 27 pages latex, 9 ps
figure files. Current part I is math.DS/0301294, part III is math.DS/0505103;
v3 reflects prior work of Beardon, Bullett and Rippo
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