Change Is Possible
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Quran 5
"The Forbidden Rumi" and I Rumi is a great Muslim mystic and poet. His faith in God does not deter him from riding the winds of dangerous thinking and free expression. His most audacious poems were published in a collection called "The Forbidden Rumi". I have chosen one of them and interspersed its verses between the chapters of my book. I wanted to let some fresh air in my dry discussion of the Quranic text. You see, the fifth chapter of the Quran was the last one to have been composed. It discusses theological questions and lacks the poetic rhythm of the much shorter chapters of the early days. I am neither a poet nor a mystic. I am not even a Muslim. And yet I feel I have something in common with a Rumi. I do not let the requirements of Islamic tradition, the strict rules of modern scholarship and the taboos of our time constrict my freedom to say what I think and to explain my forbidden views. I do so however in a friendly manner and without any animosity. My study of the Quran is based on knowledge not on faith. My purpose is not to provide spiritual ammunition to the faith. It is much more prosaic. I propose a new way of reading the Quranic text. Instead of seeing in it a divine revelation, I see in it the work of a man called Muhammad. I do what is forbidden for a believer. But I recognize the believer's right to see in the Book the very words of God.
Time Changes of the Brownian Motion: Poincaré Inequality, Heat Kernel Estimate and Protodistance
Author: Jun Kigami
language: en
Publisher: American Mathematical Soc.
Release Date: 2019-06-10
In this paper, time changes of the Brownian motions on generalized Sierpinski carpets including n-dimensional cube [0,1]n are studied. Intuitively time change corresponds to alteration to density of the medium where the heat flows. In case of the Brownian motion on [0,1]n, density of the medium is homogeneous and represented by the Lebesgue measure. The author's study includes densities which are singular to the homogeneous one. He establishes a rich class of measures called measures having weak exponential decay. This class contains measures which are singular to the homogeneous one such as Liouville measures on [0,1]2 and self-similar measures. The author shows the existence of time changed process and associated jointly continuous heat kernel for this class of measures. Furthermore, he obtains diagonal lower and upper estimates of the heat kernel as time tends to 0. In particular, to express the principal part of the lower diagonal heat kernel estimate, he introduces “protodistance” associated with the density as a substitute of ordinary metric. If the density has the volume doubling property with respect to the Euclidean metric, the protodistance is shown to produce metrics under which upper off-diagonal sub-Gaussian heat kernel estimate and lower near diagonal heat kernel estimate will be shown.