For a self-similar measure in d-dimensional Euclidean space with overlaps but satisfies the so-called bounded measure type condition introduced by Tang and the authors, we set up a framework for deriving a closed formula for the Lq-spectrum of the measure for nonnegative q. The framework allows us to include iterated function systems that have different contraction ratios and those in higher dimension. For self-similar measures with overlaps, closed formulas for the Lq-spectrum have only been obtained earlier for measures satisfying Strichartz second-order identities. We illustrate how to use our results to prove the differentiability of the Lq-spectrum, obtain the multifractal dimension spectrum, and compute the Hausdorff dimension of the measure.

We study free scalar field theory on a graph, which gives rise to a modified version of discrete Green’s function on a graph studied in [8]. We show that this gives rise to a graph invariant, which is closely related to the 2-dimensional Weisfeiler-Lehman algorithm for graph isomorphism testing. We then consider the same theory over the integers, which leads to the consideration of certain quadratic forms over the integers as initiated in [14], associated to the graphs. The quadratic form represented by the combinatorial Laplacian respects a well-behaved wedge sum of graphs, and appears to capture important graph properties reg

We study orthogonal polynomials with respect to self-similar measures, focusing on the class of infinite Bernoulli convolutions, which are defined by iterated function systems with overlaps, especially those defined by the Pisot, Garsia, and Salem numbers. By using an algorithm of Mantica, we obtain graphs of the coefficients of the 3-term recursion relation defining the orthogonal polynomials. We use these graphs to predict whether the singular infinite Bernoulli convolutions belong to the Nevai class. Based on our numerical results, we conjecture that all infinite Bernoulli Convolutions with contraction ratios greater than or equal to 1/2 belong to Nevai's class, regardless of the probability weights assigned to the self-similar measures.

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.

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.