Direct Systems of Spherical Functions and Representations

Spherical representations and functions are the building blocks for harmonic analysis on riemannian symmetric spaces. In this paper we consider spherical functions and spherical representations related to certain infinite dimensional symmetric spaces $G_\infty/K_\infty = \varinjlim G_n/K_n$. We use the representation theoretic construction $\phi (x) =$ where $e$ is a $K_\infty$--fixed unit vector for $\pi$. Specifically, we look at representations $\pi_\infty = \varinjlim \pi_n$ of $G_\infty$ where $\pi_n$ is $K_n$--spherical, so the spherical representations $\pi_n$ and the corresponding spherical functions $\phi_n$ are related by $\phi_n(x) = <e_n, \pi_n(x)e_n>$ where $e_n$ is a $K_n$--fixed unit vector for $\pi_n$, and we consider the possibility of constructing a $K_\infty$--spherical function $\phi_\infty = \lim \phi_n$. We settle that matter by proving the equivalence of condtions (i) $\{e_n\}$ converges to a nonzero $K_\infty$--fixed vector $e$, and (ii) $G_\infty/K_\infty$ has finite symmetric space rank (equivalently, it is the Grassmann manifold of $p$--planes in \F^\infty where $p < \infty$ and \F is $\R$, \C or \H). In that finite rank case we also prove the functional equation $\phi(x)\phi(y) = \lim_{n\to \infty} \int_{K_n}\phi(xky)dk$ of Faraut and Olshanskii, which is their definition of spherical functions.Comment: 17 pages. New material added on the finite rank case

Photoinitiated oxidative addition of CF3I to gold(I) and facile aryl-CF3 reductive elimination.

Herein we report the mechanism of oxidative addition of CF3I to Au(I), and remarkably fast Caryl-CF3 bond reductive elimination from Au(III) cations. CF3I undergoes a fast, formal oxidative addition to R3PAuR (R = Cy, R = 3,5-F2-C6H4, 4-F-C6H4, C6H5, 4-Me-C6H4, 4-MeO-C6H4, Me; R = Ph, R = 4-F-C6H4, 4-Me-C6H4). When R = aryl, complexes of the type R3PAu(aryl)(CF3)I can be isolated and characterized. Mechanistic studies suggest that near-ultraviolet light (λmax = 313 nm) photoinitiates a radical chain reaction by exciting CF3I. Complexes supported by PPh3 undergo reversible phosphine dissociation at 110 °C to generate a three-coordinate intermediate that undergoes slow reductive elimination. These processes are quantitative and heavily favor Caryl-I reductive elimination over Caryl-CF3 reductive elimination. Silver-mediated halide abstraction from all complexes of the type R3PAu(aryl)(CF3)I results in quantitative formation of Ar-CF3 in less than 1 min at temperatures as low as -10 °C

Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric scattering transform on manifolds. Similar to the Euclidean scattering transform, the geometric scattering transform is based on a cascade of wavelet filters and pointwise nonlinearities. It is invariant to local isometries and stable to certain types of diffeomorphisms. Empirical results demonstrate its utility on several geometric learning tasks. Our results generalize the deformation stability and local translation invariance of Euclidean scattering, and demonstrate the importance of linking the used filter structures to the underlying geometry of the data.Comment: 35 pages; 3 figures; 2 tables; v3: Revisions based on reviewer comment

Discounting in the pigeon: Food-specified conditioned reinforcers

Discounting has been observed across a wide range of species, procedures, and reinforcers. One of the main differences in the literature is that studies with humans use money (conditioned reinforcer) as the reinforcer being discounted. This is in contrast to studies with non-human animals where the discounted outcomes are usually directly consumable like food and water (primary reinforcers). At present, there is no non-human animal equivalent to money that has been identified in the discounting literature. To further bridge the remaining gaps in the methodology, a common currency is needed to study conditioned reinforcers in the discounting paradigm. An adjusting-amount procedure was used to examine the discounting of conditioned reinforcers with pigeons. Pigeons made choices of smaller, more immediate amounts of tokens or larger, more delayed amounts of tokens and then exchanged the tokens for food. Pigeons discounted token reinforcers evidenced by the decrease in subjective value as delay increased. Further, tokens were discounted less steeply than real food reinforcers. The results indicate that pigeons can discount food-specific tokens and is an important first step towards developing a generalized token reinforcer for discounting procedures with non-human animals