We show that the semi-classical analysis of generic Euclidean path integrals necessarily requires complexification of the action and measure, and consideration of complex saddle solutions. We demonstrate that complex saddle points have a natural interpretation in terms of the Picard-Lefschetz theory. Motivated in part by the semi-classical expansion of QCD with adjoint matter on R3 ⇥ S1, we study quantum-mechanical systems with bosonic and fermionic (Grassmann) degrees of freedom with harmonic degenerate minima, as well as (related) purely bosonic systems with harmonic non-degenerate minima. We find exact finite action non-BPS bounce and bion solutions to the holomorphic Newton equations. We find not only real solutions, but also complex solution with non-trivial monodromy, and finally complex multi-valued and singular solutions. Complex bions are necessary for obtaining the correct non-perturbative structure of these models. In the supersymmetric limit the complex solutions govern the ground state properties, and their contribution to the semiclassical expansion is necessary to obtain consistency with the supersymmetry algebra. The multi-valuedness of the action is either related to the hidden topological angle or to the resurgent cancellation of ambiguities. We also show that in the approximate multi-instanton description the integration over the complex quasi-zero mode thimble produces the most salient features of the exact solutions. While exact complex saddles are more difficult to construct in quantum field theory, the relation to the approximate thimble construction suggests that such solutions may be underlying some remarkable features of approximate bion saddles in quantum field theories.

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 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.

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

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.