For Laplacians defined by measures on a bounded domain in R^n, we prove analogs of the classical eigenvalue estimates for the standard Laplacian: lower bound of sums of eigenvalues by Li and Yau, and gaps of consecutive eigenvalues by Payne, Polya and Weinberger. This work is motivated by the study of spectral gaps for Laplacians on fractals.
We study the heat kernel expansion of the Laplacian on n-forms defined on a subgraph of a directed complete graph. We derive two expressions for the subgraph heat kernel on 0-forms and compute the coefficients of the expansion. We also obtain the subgraph heat kernel of the Laplacian on 1-forms.
There are two important statements regarding the Trautman-Bondi mass at null infinity: one is the positivity, and the other
is the Bondi mass loss formula, which are both global in nature. The positivity of the quasi-local mass can potentially lead to a local description at null infinity. This is confirmed for the Vaidya spacetime in this note. We study the Wang-Yau quasi-local mass on surfaces of fixed size at the null infinity of the Vaidya spacetime. The optimal embedding equation is solved explicitly and the quasi-local mass is evaluated in terms of the mass aspect function of the Vaidya spacetime.
Very few previous studies have examined the forecast and delimiting of TV dramas evaluating indicators problem from the machine learning and statistic perspective. In this paper, we designed a series of web crawlers for collecting TV-drama-related indicators as raw data. The accurate prediction of the TV drama audience ratings and online views is achieved by the ARIMA model, RNNs, CLDNNs and RVM model. Statistical methods are applied to analyze and compare the TV ratings and the online views. Factor analysis is used to give a definition and calculation method of heat of TV dramas and rankings of the TV dramas based on heat. Finally, mixed CNNs is employed to predict heat of TV dramas using data of different dimensions. In this paper, web-crawler, traditional statistical method and the state-of-the art deep learning techniques are combined to give a basic application for predicting and ranking in TV drama industry.
For a fractal Schrodinger operator with a continuous potential and a coupling parameter, we obtain an analog of a semi-classical asymptotic formula for the number of bound states as the parameter tends to infinity. We also study Bohr's formula for Schrodinger operators on blowups of self-similar sets. For a Schrodinger operator defined by a fractal measure and a locally bounded potential that tends to infinity, we derive an analog of Bohr's formula under various assumptions. We demonstrate how these results can be applied to self-similar measures with overlaps, including the infinite Bernoulli convolution associated with the golden ratio, a family of convolutions of Cantor-type measures, and a family of measures that we call essentially of finite type.