اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدComputing the equilibrium measure of a system of intervals converging to a Cantor set
185
0
0.0
(
0
)
تأليف
Giorgio Mantica
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We describe a numerical technique to compute the equilibrium measure, in logarithmic potential theory, living on the attractor of Iterated Function Systems composed of one-dimensional affine maps. This measure is obtained as the limit of a sequence of equilibrium measures on finite unions of intervals. Although these latter are known analytically, their computation requires the evaluation of a number of integrals and the solution of a non-linear set of equations. We unveil the potential numerical dangers hiding in these problems and we propose detailed solutions to all of them. Convergence of the procedure is illustrated in specific examples and is gauged by computing the electrostatic potential.
قيم البحث
اقرأ أيضاً
We construct multidimensional Schrodinger operators with a spectrum that has no gaps at high energies and that is nowhere dense at low energies. This gives the first example for which this widely expected topological structure of the spectrum in the
class of uniformly recurrent Schrodinger operators, namely the coexistence of a half-line and a Cantor-type structure, can be confirmed. Our construction uses Schrodinger operators with separable potentials that decompose into one-dimensional potentials generated by the Fibonacci sequence and relies on the study of such operators via the trace map and the Fricke-Vogt invariant. To show that the spectrum contains a half-line, we prove an abstract Bethe--Sommerfeld criterion for sums of Cantor sets which may be of independent interest.
We study the orthogonal polynomials associated with the equilibrium measure, in logarithmic potential theory, living on the attractor of an Iterated Function System. We construct sequences of discrete measures, that converge weakly to the equilibrium
measure, and we compute their Jacobi matrices via standard procedures, suitably enhanced for the scope. Numerical estimates of the convergence rate to the limit Jacobi matrix are provided, that show stability and efficiency of the whole procedure. As a secondary result, we also compute Jacobi matrices of equilibrium measures on finite sets of intervals, and of balanced measures of Iterated Function Systems. These algorithms can reach large orders: we study the asymptotic behavior of the orthogonal polynomials and we show that they can be used to efficiently compute Greens functions and conformal mappings of interest in constructive function theory.
This paper is devoted to a new approach of the arithmetic of intervals. We present the set of intervals as a normed vector space. We define also a four-dimensional associative algebra whose product gives the product of intervals in any cases. This ap
proach allows to give a notion of divisibility and in some cases an euclidian division. We introduce differential calculus and give some applications.
In this paper we present the set of intervals as a normed vector space. We define also a four-dimensional associative algebra whose product gives the product of intervals in any cases. This approach allows to give a notion of divisibility and in some
cases an euclidian division. We introduce differential calculus and give some applications.
In this paper, we are going to discuss the following problem: Let $T$ be a fixed set in $mathbb{R}^n$. And let $S$ and $B$ he two subsets in $mathbb{R}^n$ such that for any $x$ in $S$, there exists an $r$ such that $x+ r T$ is a subset of $B$. How sm
all can be $B$ be if we know the size of $S$? Stein proved that for $n$ is greater than or equal to 3 and $T$ is a sphere centered at origin, then $S$ has positive measure implies $B$ has positive measure using spherical maximal operator. Later, Bourgain and Marstrand proved the similar result for $n =2$. And we found an example for why the result fails for $n=1$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
