Harmonic Analysis and Random Schrödinger Operators

The present volume contains the Proceedings of the International Conference on Spectral Theory and Mathematical Physics held in Santiago de Chile in November 2014

Arteriovenous Blood Metabolomics: A Readout of Intra-Tissue Metabostasis.

The human circulatory system consists of arterial blood that delivers nutrients to tissues, and venous blood that removes the metabolic by-products. Although it is well established that arterial blood generally has higher concentrations of glucose and oxygen relative to venous blood, a comprehensive biochemical characterization of arteriovenous differences has not yet been reported. Here we apply cutting-edge, mass spectrometry-based metabolomic technologies to provide a global characterization of metabolites that vary in concentration between the arterial and venous blood of human patients. Global profiling of paired arterial and venous plasma from 20 healthy individuals, followed up by targeted analysis made it possible to measure subtle (<2 fold), yet highly statistically significant and physiologically important differences in water soluble human plasma metabolome. While we detected changes in lactic acid, alanine, glutamine, and glutamate as expected from skeletal muscle activity, a number of unanticipated metabolites were also determined to be significantly altered including Krebs cycle intermediates, amino acids that have not been previously implicated in transport, and a few oxidized fatty acids. This study provides the most comprehensive assessment of metabolic changes in the blood during circulation to date and suggests that such profiling approach may offer new insights into organ homeostasis and organ specific pathology

Localization via fractional moments for models on Z\mathbb{Z} with single-site potentials of finite support

One of the fundamental results in the theory of localization for discrete Schr\"odinger operators with random potentials is the exponential decay of Green's function and the absence of continuous spectrum. In this paper we provide a new variant of these results for one-dimensional alloy-type potentials with finitely supported sign-changing single-site potentials using the fractional moment method.Comment: LaTeX-file, 26 pages with 2 LaTeX figure

Scale-free uncertainty principles and Wegner estimates for random breather potentials

We present new scale-free quantitative unique continuation principles for Schr\"odinger operators. They apply to linear combinations of eigenfunctions corresponding to eigenvalues below a prescribed energy, and can be formulated as an uncertainty principle for spectral projectors. This extends recent results of Rojas-Molina & Veseli\'c, and Klein. We apply the scale-free unique continuation principle to obtain a Wegner estimate for a random Schr\"odinger operator of breather type. It holds for arbitrarily high energies. Schr\"odinger operators with random breather potentials have a non-linear dependence on random variables. We explain the challenges arising from this non-linear dependence

Localization criteria for Anderson models on locally finite graphs

We prove spectral and dynamical localization for Anderson models on locally finite graphs using the fractional moment method. Our theorems extend earlier results on localization for the Anderson model on \ZZ^d. We establish geometric assumptions for the underlying graph such that localization can be proven in the case of sufficiently large disorder

Adaptive estimation in circular functional linear models

We consider the problem of estimating the slope parameter in circular functional linear regression, where scalar responses Y1,...,Yn are modeled in dependence of 1-periodic, second order stationary random functions X1,...,Xn. We consider an orthogonal series estimator of the slope function, by replacing the first m theoretical coefficients of its development in the trigonometric basis by adequate estimators. Wepropose a model selection procedure for m in a set of admissible values, by defining a contrast function minimized by our estimator and a theoretical penalty function; this first step assumes the degree of ill posedness to be known. Then we generalize the procedure to a random set of admissible m's and a random penalty function. The resulting estimator is completely data driven and reaches automatically what is known to be the optimal minimax rate of convergence, in term of a general weighted L2-risk. This means that we provide adaptive estimators of both the slope function and its derivatives

Anderson localization for a class of models with a sign-indefinite single-site potential via fractional moment method

A technically convenient signature of Anderson localization is exponential decay of the fractional moments of the Green function within appropriate energy ranges. We consider a random Hamiltonian on a lattice whose randomness is generated by the sign-indefinite single-site potential, which is however sign-definite at the boundary of its support. For this class of Anderson operators we establish a finite-volume criterion which implies that above mentioned the fractional moment decay property holds. This constructive criterion is satisfied at typical perturbative regimes, e. g. at spectral boundaries which satisfy 'Lifshitz tail estimates' on the density of states and for sufficiently strong disorder. We also show how the fractional moment method facilitates the proof of exponential (spectral) localization for such random potentials.Comment: 29 pages, 1 figure, to appear in AH

Regularized Linear Inversion with Randomized Singular Value Decomposition

In this work, we develop efficient solvers for linear inverse problems based on randomized singular value decomposition (RSVD). This is achieved by combining RSVD with classical regularization methods, e.g., truncated singular value decomposition, Tikhonov regularization, and general Tikhonov regularization with a smoothness penalty. One distinct feature of the proposed approach is that it explicitly preserves the structure of the regularized solution in the sense that it always lies in the range of a certain adjoint operator. We provide error estimates between the approximation and the exact solution under canonical source condition, and interpret the approach in the lens of convex duality. Extensive numerical experiments are provided to illustrate the efficiency and accuracy of the approach.Comment: 20 pages, 4 figure

MetaboSearch: Tool for Mass-Based Metabolite Identification Using Multiple Databases

Searching metabolites against databases according to their masses is often the first step in metabolite identification for a mass spectrometry-based untargeted metabolomics study. Major metabolite databases include Human Metabolome DataBase (HMDB), Madison Metabolomics Consortium Database (MMCD), Metlin, and LIPID MAPS. Since each one of these databases covers only a fraction of the metabolome, integration of the search results from these databases is expected to yield a more comprehensive coverage. However, the manual combination of multiple search results is generally difficult when identification of hundreds of metabolites is desired. We have implemented a web-based software tool that enables simultaneous mass-based search against the four major databases, and the integration of the results. In addition, more complete chemical identifier information for the metabolites is retrieved by cross-referencing multiple databases. The search results are merged based on IUPAC International Chemical Identifier (InChI) keys. Besides a simple list of m/z values, the software can accept the ion annotation information as input for enhanced metabolite identification. The performance of the software is demonstrated on mass spectrometry data acquired in both positive and negative ionization modes. Compared with search results from individual databases, MetaboSearch provides better coverage of the metabolome and more complete chemical identifier information. Availability: The software tool is available at http://omics.georgetown.edu/MetaboSearch.html