اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA Computer Verification of a Conjecture About Erdos-Mordell Curve
134
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this paper we consider Erdos-Mordell inequality and its extension in the plane of triangle to the Erdos-Mordell curve. Algebraic equation of this curve is derived, and using modern computer tools in mathematics, we verified one conjecture that relates to Erdos-Mordell curve.
قيم البحث
اقرأ أيضاً
The Serpinsky-Knopp curve is characterized as the only curve (up to isometry) that maps a unit segment onto a triangle of a unit area, so for any pair of points in the segment, the square of the distance between their images does not exceed four times the distance between them.
We prove a special case of a dynamical analogue of the classical Mordell-Lang conjecture. In particular, let $phi$ be a rational function with no superattracting periodic points other than exceptional points. If the coefficients of $phi$ are algebrai
c, we show that the orbit of a point outside the union of proper preperiodic subvarieties of $(bP^1)^g$ has only finite intersection with any curve contained in $(bP^1)^g$. We also show that our result holds for indecomposable polynomials $phi$ with coefficients in $bC$. Our proof uses results from $p$-adic dynamics together with an integrality argument. The extension to polynomials defined over $bC$ uses the method of specializations coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of $(phi,phi)$ on $bA^2$.
In this paper, we prove the Uniform Mordell-Lang Conjecture for subvarieties in abelian varieties. As a byproduct, we prove the Uniform Bogomolov Conjecture for subvarieties in abelian varieties.
Very recently, E. H. Lieb and J. P. Solovej stated a conjecture about the constant of embedding between two Bergman spaces of the upper-half plane. A question in relation with a Werhl-type entropy inequality for the affine $AX+B$ group. More precisel
y, that for any holomorphic function $F$ on the upper-half plane $Pi^+$, $$int_{Pi^+}|F(x+iy)|^{2s}y^{2s-2}dxdyle frac{pi^{1-s}}{(2s-1)2^{2s-2}}left(int_{Pi^+}|F(x+iy)|^2 dxdyright)^s $$ for $sge 1$, and the constant $frac{pi^{1-s}}{(2s-1)2^{2s-2}}$ is sharp. We prove differently that the above holds whenever $s$ is an integer and we prove that it holds when $srightarrowinfty$. We also prove that when restricted to powers of the Bergman kernel, the conjecture holds. We next study the case where $s$ is close to $1.$ Hereafter, we transfer the conjecture to the unit disc where we show that the conjecture holds when restricted to analytic monomials. Finally, we overview the bounds we obtain in our attempts to prove the conjecture.
A finite subset of a Euclidean space is called an $s$-distance set if there exist exactly $s$ values of the Euclidean distances between two distinct points in the set. In this paper, we prove that the maximum cardinality among all 5-distance sets in
$mathbb{R}^3$ is 20, and every $5$-distance set in $mathbb{R}^3$ with $20$ points is similar to the vertex set of a regular dodecahedron.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
