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"Polygons"
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Uses cheese slices, pretzel sticks, a slice of bread, graph paper, a pencil, and more to introduce various polygons, flat shapes with varying numbers of straight sides.
Book
ON -DIVISIBLE GROUPS WITH SATURATED NEWTON POLYGONS
This paper concerns the classification of isogeny classes of
$p$
-divisible groups with saturated Newton polygons. Let
$S$
be a normal Noetherian scheme in positive characteristic
$p$
with a prime Weil divisor
$D$
. Let
${\\mathcal{X}}$
be a
$p$
-divisible group over
$S$
whose geometric fibers over
$S\\setminus D$
(resp. over
$D$
) have the same Newton polygon. Assume that the Newton polygon of
${\\mathcal{X}}_{D}$
is saturated in that of
${\\mathcal{X}}_{S\\setminus D}$
. Our main result (Corollary 1.1) says that
${\\mathcal{X}}$
is isogenous to a
$p$
-divisible group over
$S$
whose geometric fibers are all minimal. As an application, we give a geometric proof of the unpolarized analogue of Oort’s conjecture (Oort, J. Amer. Math. Soc.
17
(2) (2004), 267–296; 6.9).
Journal Article
Tits polygons
We introduce the notion of a Tits polygon, a generalization of the notion of a Moufang polygon, and show that Tits polygons arise in
a natural way from certain configurations of parabolic subgroups in an arbitrary spherical buildings satisfying the Moufang condition.
We establish numerous basic properties of Tits polygons and characterize a large class of Tits hexagons in terms of Jordan algebras. We
apply this classification to give a “rank
eBook
IS INFINITE DIMENSIONAL
Given a perfect valuation ring
$R$
of characteristic
$p$
that is complete with respect to a rank-1 nondiscrete valuation, we show that the ring
$\\mathbb{A}_{\\inf }$
of Witt vectors of
$R$
has infinite Krull dimension.
Journal Article
The study of circumsphere and insphere of a regular polyhedron
A regular polygon is a polygon that is both equilateral and equiangular. One of the properties of the regular polygon is has a circumcircle and an incircle. The analogy of a regular polygon on plane is a regular polyhedron in space, while the analogy of a circle on plane is a spehe in space. Relationship between regular polygon and circle on plane can be studied in space. The purpose of this paper is to show the existance and the characteristics of a circumsphere and an insphere of the regular polyhedron. The result show that each regular polyhedron has a circumsphere and an insphere. The center of the circumsphere of a regular polyhedron is intersection of the perpendicular bisector planes of the regular polyhedron and the center of the insphere of a regular polyhedron is intersection of the angle bisector planes of the regular polyhedron. The characteristics of a circumsphere and an insphere of a regular polyhedron are: 1) The center of a circumsphere and an insphere of a regular polyhedron is coincide; 2) The length of radius of a circumsphere and an insphere of a regular polyhedron meet the equation R=s2sin(180°2ev)sin(180°2en)cos(θ2);r=s2cot(180°2en)tan(θ2).
Journal Article
Characterizing and tuning exceptional points using Newton polygons
The study of non-Hermitian degeneracies—called exceptional points (EPs)—has become an exciting frontier at the crossroads of optics, photonics, acoustics, and quantum physics. Here, we introduce the Newton polygon method as a general algebraic framework for characterizing and tuning EPs. Newton polygons, first described by Isaac Newton, are conventionally used in algebraic geometry, with deep roots in various topics in modern mathematics. We propose and illustrate how the Newton polygon method can enable the prediction of higher-order EPs, using a recently experimentally realized optical system. Using the paradigmatic Hatano-Nelson model, we demonstrate how our method can predict the presence of the non-Hermitian skin effect. As further application of our framework, we show the presence of tunable EPs of various orders in
PT
-symmetric one-dimensional models. We further extend our method to study EPs in higher number of variables and demonstrate that it can reveal rich anisotropic behaviour around such degeneracies. Our work provides an analytic recipe to understand exceptional physics.
Journal Article
Best and Random Approximations with Generalized Disc–Polygons
We generalize the existing results on linear and spindle convexity to the so-called L-convexity. We consider the asymptotic behaviour of the distance between an L-convex disc K with sufficiently smooth boundary, and its approximating L-polygons, as the number of vertices tends to infinity. We consider two constructions: the best approximating inscribed and circumscribed L-polygons of K with respect to some metric; and random inscribed and circumscribed L-polygons of K, which are constructed from n i.i.d. random points chosen from the boundary of K. The asymptotic behaviour of the distance between K and the approximating L-polygons depend in both cases on the same, geometric limits. In order to prove the results for the circumscribed cases, we also introduce the notion of an L-convex duality.
Journal Article
CONSTANCY OF NEWTON POLYGONS OF -ISOCRYSTALS ON ABELIAN VARIETIES AND ISOTRIVIALITY OF FAMILIES OF CURVES
We prove constancy of Newton polygons of all convergent
$F$
-isocrystals on Abelian varieties over finite fields. Applying the constancy, we prove the isotriviality of proper smooth families of curves over Abelian varieties. More generally, we prove the isotriviality over projective smooth varieties on which any convergent
$F$
-isocrystal has constant Newton polygons.
Journal Article