On the compactness of the set of invariant Einstein metrics

Let $M = G/H$ be a connected simply connected homogeneous manifold of a compact, not necessarily connected Lie group $G$. We will assume that the isotropy $H$-module $\mathfrak {g/h}$ has a simple spectrum, i.e. irreducible submodules are mutually non-equivalent. There exists a convex Newton polytope $N=N(G,H)$, which was used for the estimation of the number of isolated complex solutions of the algebraic Einstein equation for invariant metrics on $G/H$ (up to scaling). Using the moment map, we identify the space $\mathcal{M}_1$ of invariant Riemannian metrics of volume 1 on $G/H$ with the interior of this polytope $N$. We associate with a point ${x \in \partial N}$ of the boundary a homogeneous Riemannian space (in general, only local) and we extend the Einstein equation to $\bar{\mathcal{M}_1}= N$. As an application of the Aleksevsky--Kimel'fel'd theorem, we prove that all solutions of the Einstein equation associated with points of the boundary are locally Euclidean. We describe explicitly the set $T\subset \partial N$ of solutions at the boundary together with its natural triangulation. Investigating the compactification $\bar{\mathcal{M}_1}$ of $\mathcal{M}_1$, we get an algebraic proof of the deep result by B\"ohm, Wang and Ziller about the compactness of the set $\mathcal{E}_1 \subset \mathcal{M}_1$ of Einstein metrics. The original proof by B\"ohm, Wang and Ziller was based on a different approach and did not use the simplicity of the spectrum. In Appendix we consider the non-symmetric K\"ahler homogeneous spaces $G/H$ with the second Betti number $b_2=1$. We write the normalized volumes $2,6,20,82,344$ of the corresponding Newton polytopes and discuss the number of complex solutions of the algebraic Einstein equation and the finiteness problem.Comment: 25 pages, 4 figures. Some proofs, 3 references, and Appendix adde

High-throughput molecular imaging via deep-learning-enabled Raman spectroscopy.

Raman spectroscopy enables nondestructive, label-free imaging with unprecedented molecular contrast, but is limited by slow data acquisition, largely preventing high-throughput imaging applications. Here, we present a comprehensive framework for higher-throughput molecular imaging via deep-learning-enabled Raman spectroscopy, termed DeepeR, trained on a large data set of hyperspectral Raman images, with over 1.5 million spectra (400 h of acquisition) in total. We first perform denoising and reconstruction of low signal-to-noise ratio Raman molecular signatures via deep learning, with a 10× improvement in the mean-squared error over common Raman filtering methods. Next, we develop a neural network for robust 2-4× spatial super-resolution of hyperspectral Raman images that preserve molecular cellular information. Combining these approaches, we achieve Raman imaging speed-ups of up to 40-90×, enabling good-quality cellular imaging with a high-resolution, high signal-to-noise ratio in under 1 min. We further demonstrate Raman imaging speed-up of 160×, useful for lower resolution imaging applications such as the rapid screening of large areas or for spectral pathology. Finally, transfer learning is applied to extend DeepeR from cell to tissue-scale imaging. DeepeR provides a foundation that will enable a host of higher-throughput Raman spectroscopy and molecular imaging applications across biomedicine

The Bregman chord divergence

Distances are fundamental primitives whose choice significantly impacts the performances of algorithms in machine learning and signal processing. However selecting the most appropriate distance for a given task is an endeavor. Instead of testing one by one the entries of an ever-expanding dictionary of {\em ad hoc} distances, one rather prefers to consider parametric classes of distances that are exhaustively characterized by axioms derived from first principles. Bregman divergences are such a class. However fine-tuning a Bregman divergence is delicate since it requires to smoothly adjust a functional generator. In this work, we propose an extension of Bregman divergences called the Bregman chord divergences. This new class of distances does not require gradient calculations, uses two scalar parameters that can be easily tailored in applications, and generalizes asymptotically Bregman divergences.Comment: 10 page

Modeling concept drift: A probabilistic graphical model based approach

An often used approach for detecting and adapting to concept drift when doing classi cation is to treat the data as i.i.d. and use changes in classi cation accuracy as an indication of concept drift. In this paper, we take a different perspective and propose a framework, based on probabilistic graphical models, that explicitly represents concept drift using latent variables. To ensure effcient inference and learning, we resort to a variational Bayes inference scheme. As a proof of concept, we demonstrate and analyze the proposed framework using synthetic data sets as well as a real fi nancial data set from a Spanish bank

Self-avoiding walks and connective constants

The connective constant $\mu(G)$ of a quasi-transitive graph $G$ is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on $G$ from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph $G$. $\bullet$ We present upper and lower bounds for $\mu$ in terms of the vertex-degree and girth of a transitive graph. $\bullet$ We discuss the question of whether $\mu\ge\phi$ for transitive cubic graphs (where $\phi$ denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). $\bullet$ We present strict inequalities for the connective constants $\mu(G)$ of transitive graphs $G$, as $G$ varies. $\bullet$ As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator. $\bullet$ We describe so-called graph height functions within an account of "bridges" for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function. $\bullet$ A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions. $\bullet$ Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not. $\bullet$ The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with arXiv:1304.721

Do mutual funds have consistency in their performance?

Using a comprehensive data set of 714 Chinese mutual funds from 2004 to 2015, the study investigates these funds’ performance persistence by using the Capital Asset Pricing model, the Fama-French three-factor model and the Carhart Four-factor model. For persistence analysis, we categorize mutual funds into eight octiles based on their one year lagged performance and then observe their performance for the subsequent 12 months. We also apply Cross-Product Ratio technique to assess the performance persistence in these Chinese funds. The study finds no significant evidence of persis- tence in the performance of the mutual funds. Winner (loser) funds do not continue to be winner (loser) funds in the subsequent time period. These findings suggest that future performance of funds cannot be predicted based on their past performance.info:eu-repo/semantics/publishedVersio

The Effective Fragment Molecular Orbital Method for Fragments Connected by Covalent Bonds

We extend the effective fragment molecular orbital method (EFMO) into treating fragments connected by covalent bonds. The accuracy of EFMO is compared to FMO and conventional ab initio electronic structure methods for polypeptides including proteins. Errors in energy for RHF and MP2 are within 2 kcal/mol for neutral polypeptides and 6 kcal/mol for charged polypeptides similar to FMO but obtained two to five times faster. For proteins, the errors are also within a few kcal/mol of the FMO results. We developed both the RHF and MP2 gradient for EFMO. Compared to ab initio, the EFMO optimized structures had an RMSD of 0.40 and 0.44 {\AA} for RHF and MP2, respectively.Comment: Revised manuscrip