اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدAsymptotic order of the geometric mean error for self-affine measures on Bedford-McMullen carpets
85
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Let $E$ be a Bedford-McMullen carpet associated with a set of affine mappings ${f_{ij}}_{(i,j)in G}$ and let $mu$ be the self-affine measure associated with ${f_{ij}}_{(i,j)in G}$ and a probability vector $(p_{ij})_{(i,j)in G}$. We study the asymptotics of the geometric mean error in the quantization for $mu$. Let $s_0$ be the Hausdorff dimension for $mu$. Assuming a separation condition for ${f_{ij}}_{(i,j)in G}$, we prove that the $n$th geometric error for $mu$ is of the same order as $n^{-1/s_0}$.
قيم البحث
اقرأ أيضاً
In this paper we consider the packing spectra for local dimension of Bernoulli measures supported on Bedford-McMullen carpets. We show that typically the packing dimension of the regular set is smaller than the packing dimension of the attractor. We
also consider a specific class of measures for which we are able to calculate the packing spectrum exactly and we show that the packing spectrum is discontinuous as a function on the space of Bernoulli measures.
A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincare inequalities. Our results yield ne
w examples of compact doubling metric measure spaces supporting Poincare inequalities: these examples have no manifold points, yet embed isometrically as subsets of Euclidean space.
A question related to some conjectures of Lutwak about the affine quermassintegrals of a convex body $K$ in ${mathbb R}^n$ asks whether for every convex body $K$ in ${mathbb R}^n$ and all $1leqslant kleqslant n$ $$Phi_{[k]}(K):={rm vol}_n(K)^{-frac{1
}{n}}left (int_{G_{n,k}}{rm vol}_k(P_F(K))^{-n},d u_{n,k}(F)right )^{-frac{1}{kn}}leqslant csqrt{n/k},$$ where $c>0$ is an absolute constant. We provide an affirmative answer for some broad classes of random polytopes. We also discuss upper bounds for $Phi_{[k]}(K)$ when $K=B_1^n$, the unit ball of $ell_1^n$, and explain how this special instance has implications for the case of a general unconditional convex body $K$.
We study the symmetrized noncommutative arithmetic geometric mean inequality introduced(AGM) by Recht and R{e} $$ |frac{(n-d)!}{n!}sumlimits_{{ j_1,...,j_d mbox{ different}} }A_{j_{1}}^*A_{j_{2}}^*...A_{j_{d}}^*A_{j_{d}}...A_{j_{2}}A_{j_{1}} | leq
C(d,n) |frac{1}{n} sum_{j=1}^n A_j^*A_j|^d .$$ Complementing the results from Recht and R{e}, we find upper bounds for C(d,n) under additional assumptions. Moreover, using free probability, we show that $C(d, n) > 1$, thereby disproving the most optimistic conjecture from Recht and R{e}.We also prove a deviation result for the symmetrized-AGM inequality which shows that the symmetric inequality almost holds for many classes of random matrices. Finally we apply our results to the incremental gradient method(IGM).
We present an alternative approach to some results of Koldobsky on measures of sections of symmetric convex bodies, which allows us to extend them to the not necessarily symmetric setting. We prove that if $K$ is a convex body in ${mathbb R}^n$ with
$0in {rm int}(K)$ and if $mu $ is a measure on ${mathbb R}^n$ with a locally integrable non-negative density $g$ on ${mathbb R}^n$, then begin{equation*}mu (K)leq left (csqrt{n-k}right )^kmax_{Fin G_{n,n-k}}mu (Kcap F)cdot |K|^{frac{k}{n}}end{equation*} for every $1leq kleq n-1$. Also, if $mu $ is even and log-concave, and if $K$ is a symmetric convex body in ${mathbb R}^n$ and $D$ is a compact subset of ${mathbb R}^n$ such that $mu (Kcap F)leq mu (Dcap F)$ for all $Fin G_{n,n-k}$, then begin{equation*}mu (K)leq left (ckL_{n-k}right )^{k}mu (D),end{equation*} where $L_s$ is the maximal isotropic constant of a convex body in ${mathbb R}^s$. Our method employs a generalized Blaschke-Petkantschin formula and estimates for the dual affine quermassintegrals.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
