325 research outputs found
Construction of -energy and associated energy measures on the Sierpi\'{n}ski carpet
We establish the existence of a scaling limit of discrete
-energies on the graphs approximating the planar Sierpi\'{n}ski carpet for
, where is
the Ahlfors regular conformal dimension of the Sierpi\'{n}ski carpet.
Furthermore, the function space defined as the collection of
functions with finite -energies is shown to be a reflexive and separable
Banach space that is dense in the set of continuous functions with respect to
the supremum norm. In particular, recovers the
canonical regular Dirichlet form constructed by Barlow and Bass or Kusuoka and
Zhou. We also provide -energy measures associated with the
constructed -energy and investigate its basic properties like
self-similarity and chain rule.Comment: 80 pages, 9 figures; fixed several typos and some mistakes, edited
some proofs. Sections 4 and 5 from the previous version have been merged into
one section. Theorem 2.20 and subsection 6.3 are ne
Sierpiński carpet上のp-エネルギーと対応するエネルギー測度の構成
京都大学新制・課程博士博士(情報学)甲第24262号情博第806号京都大学大学院情報学研究科先端数理科学専攻(主査)教授 木上 淳, 教授 磯 祐介, 准教授 白石 大典学位規則第4条第1項該当Doctor of InformaticsKyoto UniversityDFA
First-order Sobolev spaces, self-similar energies and energy measures on the Sierpi\'{n}ski carpet
We construct and investigate -Sobolev space, -energy, and the
corresponding -energy measures on the planar Sierpi\'{n}ski carpet for all
. Our method is based on the idea of Kusuoka and Zhou
[Probab. Theory Related Fields (1992), no. 2, 169--196], where
Brownian motion (the case ) on self-similar sets including the planar
Sierpi\'{n}ski carpet were constructed. Similar to this earlier work, we use a
sequence of discrete graph approximations and the corresponding discrete
-energies to define the Sobolev space and -energies. However, we need a
new approach to ensure that our -Sobolev space has a dense set of
continuous functions when is less than the Ahlfors regular conformal
dimension. The new ingredients are the use of Loewner type estimates on
combinatorial modulus to obtain Poincar\'e inequality and elliptic Harnack
inequality on a sequence of approximating graphs. An important feature of our
Sobolev space is the self-similarity of our -energy, which allows us to
define corresponding -energy measures on the planar Sierpi\'{n}ski carpet.
We show that our Sobolev space can also be viewed as a Korevaar-Schoen type
space. We apply our results to the attainment problem for Ahlfors regular
conformal dimension of the Sierpi\'{n}ski carpet. In particular, we show that
if the Ahlfors regular conformal dimension, say , is
attained, then any optimal measure which attains should
be comparable with the -energy measure of some function in
our -Sobolev space up to a multiplicative constant.
In this case, we also prove that the Newton-Sobolev space corresponding to any
optimal measure and metric can be identified as our self-similar -Sobolev space.Comment: 162 pages 4 figures; Several updates, sections reorganized, and a new
section added; Comments welcom
Photon polarization entanglement induced by biexciton: experimental evidence for violation of Bell's inequality
We have investigated the polarization entanglement between photon pairs
generated from a biexciton in a CuCl single crystal via resonant hyper
parametric scattering. The pulses of a high repetition pump are seen to provide
improved statistical accuracy and the ability to test Bell's inequality. Our
results clearly violate the inequality and thus manifest the quantum
entanglement and nonlocality of the photon pairs. We also analyzed the quantum
state of our photon pairs using quantum state tomography.Comment: 4 pages, 5 figure
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