Free Energy Wells and Overlap Gap Property in Sparse PCA

We study a variant of the sparse PCA (principal component analysis) problem in the “hard” regime, where the inference task is possible yet no polynomial-time algorithm is known to exist. Prior work, based on the low-degree likelihood ratio, has conjectured a precise expression for the best possible (subexponential) runtime throughout the hard regime. Following instead a statistical physics-inspired point of view, we show bounds on the depth of free energy wells for various Gibbs measures naturally associated to the problem. These free energy wells imply hitting time lower bounds that corroborate the low-degree conjecture: we show that a class of natural MCMC (Markov chain Monte Carlo) methods (with worst-case initialization) cannot solve sparse PCA with less than the conjectured runtime. These lower bounds apply to a wide range of values for two tuning parameters: temperature and sparsity misparametrization. Finally, we prove that the overlap gap property (OGP), a structural property that implies failure of certain local search algorithms, holds in a significant part of the hard regime. © 2022 Wiley Periodicals, Inc.     

A Reilly inequality for the first Steklov eigenvalue

Let M be a compact submanifold with boundary of a Euclidean space or a Sphere. In this paper, we derive an upper bound for the first non-zero eigenvalue p₁ of Steklov problem on M in terms of the r-th mean curvatures of its boundary ∂M. The upper bound obtained is sharp.     

Universal inequalities for the eigenvalues of a power of the Laplace operator

In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how one can use this abstract formulation both for giving different and simpler proofs for all the known results obtained for the eigenvalues of a power of the Laplace operator (i.e. the Dirichlet Laplacian, the clamped plate problem for the bilaplacian and more generally for the polyharmonic problem on a bounded Euclidean domain) and to obtain new ones. In a last paragraph, we derive new bounds for eigenvalues of any power of the Kohn Laplacian on the Heisenberg group.     

Laplacian eigenvalue functionals and metric deformations on compact manifolds

In this paper, we investigate critical points of the eigenvalues of the Laplace operator considered as functionals on the space of Riemannian metrics or a conformal class of metrics on a compact manifold. We introduce a natural notion of the critical metric of such a functional and obtain necessary and sufficient conditions for a metric to be critical. We derive specific consequences concerning possible locally maximizing metrics. We also characterize critical metrics of the ratio of two consecutive eigenvalues.     

Noncovalent Interactions of π Systems with Sulfur: The Atomic Chameleon of Molecular Recognition

The relative strength of noncovalent interactions between a thioether sulfur atom and various π systems in designed top pan molecular balances was determined by NMR spectroscopy. Compared to its oxygen counterpart, the sulfur atom displays a remarkable ability to interact with almost equal facility over the entire range of π systems studied, with the simple alkene emerging as the most powerful partner. With the exception of the O???heteroarene interaction, all noncovalent interactions of sulfur with π systems are favoured over oxygen.     

Triazole-linked dendro[60]fullerenes: modular synthesis via a ‘click’ reaction and acidity-dependent self-assembly on the surface

A series of Fréchet-type dendron functionalized [60]fullerene derivatives that bear a 1,2,3-triazole linkage group, referred to as triazole-linked dendro[60]fullerenes, were prepared via a modular synthetic protocol based on a Cu-catalyzed [3+2] cycloaddition (‘click’) reaction. Electronic properties of these dendro[60]fullerenes were investigated by UV-vis spectroscopy and cyclic voltammetry. Interfacial supramolecular self-assembly behavior of these dendro[60]fullerenes was studied using atomic force microscopy (AFM). The resulting self-assemblies showed different nanoscale packing geometries and morphologies on the surface, which are controllable by parameters such as the generation of dendron, the nature of peripheral functionalities, and the experimental conditions (e.g., acidity) applied. Correlations between molecular structure and self-assembling outcome were surveyed and discussed. The results of this study suggest a new avenue to gain better ‘bottom-up’ control over the self-assembly of dendrimer-fullerene hybrid materials in terms of shape and size.     

Weighing Matrices and String Sorting

In this paper we establish a fundamental link between the search for weighing matrices constructed from two circulants and the operation of sorting strings, an operation that has been studied extensively in computer science. In particular, we demonstrate that the search for weighing matrices constructed from two circulants using the power spectral density criterion and exploiting structural patterns for the locations of the zeros in candidate solutions, can be viewed as a string sorting problem together with a linear time algorithm to locate common strings in two sorted arrays. This allows us to bring into bear efficient algorithms from the string sorting literature. We also state and prove some new enhancements to the power spectral density criterion, that allow us to treat successfully the rounding error effect and speed up the algorithm. Finally, we use these ideas to find new weighing matrices of order 2n and weights 2n – 13, 2n – 17 constructed from two circulants.     

Evaluation of solid-phase microextraction desorption parameters for fast GC analysis of cocaine in coca leaves

By its simplicity and rapidity, solid-phase microextraction (SPME) appears as an interesting alternative for sample introduction in fast gas chromatography (fast GC). This combination depends on numerous parameters affecting the desorption step (i.e., the release of compounds from the SPME fiber coating to the GC column). In this study, different liner diameters, injection temperatures, and gas flow rates are evaluated to accelerate the thermal desorption process in the injection port. This process is followed with real-time direct coupling a split/splitless injector to a mass spectrometer by means of a short capillary. It is shown that an effective, quantitative, and rapid transfer of cocaine (COC) and cocaethylene (CE) is performed with a 0.75-mm i.d. liner, at 280 degrees C and 4 mL/min gas flow rate. The 7-microm polydimethylsiloxane (PDMS) coating is selected for combination with fast GC because the 100-microm PDMS fiber presents some limitations caused by fiber bleeding. Finally, the developed SPME-fast GC method is applied to perform in less than 5 min, the quantitation of COC extracted from coca leaves by focused microwave-assisted extraction. An amount of 7.6 +/- 0.5 mg of COC per gram of dry mass is found, which is in good agreement with previously published results.     

Adsorption of atrazine on soils: model study

The adsorption of the widely used herbicide atrazine onto three model inorganic soil components (silica gel, gamma-alumina, and calcite (CaCO(3)) was investigated in a series of batch experiments in which the aqueous phase equilibrated with the solid, under different solution conditions. Atrazine did not show discernible adsorption on gamma-alumina (theta=25 degrees C, 3.8相似文献   

The Effective Dynamics of the Volume Preserving Mean Curvature Flow

We consider the dynamics of small closed submanifolds (‘bubbles’) under the volume preserving mean curvature flow. We construct a map from (\(\text {n}+1\))-dimensional Euclidean space into a given (\(\text {n}+1\))-dimensional Riemannian manifold which characterizes the existence, stability and dynamics of constant mean curvature submanifolds. This is done in terms of a reduced area function on the Euclidean space, which is given constructively and can be computed perturbatively. This allows us to derive adiabatic and effective dynamics of the bubbles. The results can be mapped by rescaling to the dynamics of fixed size bubbles in almost Euclidean Riemannian manifolds.