Asymptotics for the Heat Kernel on H-Type Groups

We give sharp asymptotic estimates at infinity of all radial partial derivatives of the heat kernel on H-type groups. As an application, we give a new proof of the discreteness of the spectrum of some natural sub-Riemannian Ornstein-Uhlenbeck operators on these groups.Comment: 29 pages; submitte

Weighted sub-Laplacians on M\'etivier Groups: Essential Self-Adjointness and Spectrum

Let $G$ be a M\'etivier group and let $N$ be any homogeneous norm on $G$. For $\alpha>0$ denote by $w_\alpha$ the function $e^{-N^\alpha}$ and consider the weighted sub-Laplacian $\mathcal{L}^{w_\alpha}$ associated with the Dirichlet form $\phi \mapsto \int_{G} |\nabla_\mathcal{H}\phi(y)|^2 w_\alpha(y)\, dy$, where $\nabla_\mathcal{H}$ is the horizontal gradient on $G$. Consider $\mathcal{L}^{w_\alpha}$ with domain $C_c^\infty$. We prove that $\mathcal{L}^{w_\alpha}$ is essentially self-adjoint when $\alpha \geq 1$. For a particular $N$, which is the norm appearing in $\mathcal{L}$'s fundamental solution when $G$ is an H-type group, we prove that $\mathcal{L}^{w_\alpha}$ has purely discrete spectrum if and only if $\alpha>2$, thus proving a conjecture of J. Inglis.Comment: 15 pages; to appear on Proc. Amer. Math. So

Functional Calculus on Homogeneous Groups

In the first part of the thesis, we consider the following problem. Let G be a homogeneous group, and let (L_1,...,L_n) be a jointly hypoelliptic commutative finite family of formally self-adjoint, homogeneous, left-invariant differential operators without constant terms. Then, the operators L_j are essentially self-adjoint as operators on L^2(G) with domain C^infty_c(G), and their closures commute emph{as self-adjoint operators}. Therefore, one may consider the joint functional calculus associated with the family (L_1,...,L_n). More precisely, for every bounded Borel measurable function $m$ on $R^n$, the corresponding operator m(L_1,...,L_n) commutes with left translations, so that it admits a unique right convolution kernel K(m). The so-defined kernel transform K then maps S(R^n) continuously into S(G), and L^2(eta) isometrically into L^2(G) for some uniquely determined positive Radon measure eta on R^n; this latter property can be considered as an analogue of the Plancherel isomorphism. In addition, K maps L^1(eta) continuously into C_0(G), and this property can be considered as an analogue of the Riemann--Lebesgue lemma. We focus on the following properties of K: (RL) if K(m)in L^1(G), then m can be taken in C_0(R^n): this is again an analogue of the Riemann--Lebesgue lemma; (S) if K(m)in S(G), then m can be taken in S(R^n). We prove that properties (RL) and (S) are compatible with products, and we characterize the Rockland operators which satisfy property (S) when the underlying group G is abelian. We then consider the case of 2-step stratified groups, and families whose elements are either sub-Laplacians or vector fields of homogeneous degree 2. In this setting, we prove several sufficient conditions, as well as some necessary ones, for properties (RL) and (S); we even characterize them in some more specific settings. In addition, we study the case of general (that is, not necessarily homogeneous) sub-Laplacians on 2-step stratified groups, and prove that they always satisfy properties (RL) and (S). We also prove that, under some mild assumptions, a multiplier m can be taken so as to satisfy Mihlin--Hormander conditions of order infinity if and only if the corresponding kernel K(m) satisfies Calderon--Zygmund conditions of order infinity. In the second part of the thesis, we present some results which are joint work with T. Bruno. We fix the standard sub-Laplacian on an H-type group, and consider its heat kernel (p_s)_{s>0}. We provide sharp asymptotic estimates at $infty$ for basically all the derivatives of p_1. Because of the homogeneity of the family (p_s), these estimates can also be considered as short-time asymptotics

Progressive Time Delay to Teach High School Students with Intellectual Disability to Initiate Manding Siri® for Unknown Information

A multiple probe design across participants with intermittent generalization probes was used to evaluate (a) the effectiveness of progressive time delay to teach four high school students with intellectual disability to initiate using Siri® when asked an unknown question and (b) the generalized use of Siri® when asked questions from untrained communicative partners. Technology training occurred prior to baseline to teach all participants to use Siri®. Secondary data were collected on Siri’s® response and student engagement with the answer. Due to the 2020 Covid-19 pandemic and school closures, only one participant entered intervention and the study was unable to be completed. Implications for this study based on tier one data are discussed below

Introduction to the Milestones series

This new series will publish expository commentaries celebrating key contributions (Milestones) to our scholarly heritage

Boundedness of Bergman projectors on homogeneous Siegel domains

In this paper we study the boundedness of Bergman projectors on weighted Bergman spaces on homogeneous Siegel domains of Type II. As it appeared to be a natural approach in the special case of tube domains over irreducible symmetric cones, we study such boundedness on the scale of mixed-norm weighted Lebesgue spaces. The sharp range for the boundedness of such operators is essentially known only in the case of tube domains over Lorentz cones. In this paper we prove that the boundedness of such Bergman projectors is equivalent to variuos notions of atomic decomposition, duality, and characterization of boundary values of the mixed-norm weighted Bergman spaces, extending results moslty known only in the case of tube domains over irreducible symmetric cones. Some of our results are new even in the latter simpler context. We also study the simpler, but still quite interesting, case of the "positive" Bergman projectors, the integral operator in which the Bergman kernel is replaced by its absolute value. We provide a useful characterization which was previously known for tube domains.Comment: 34 pages, no figure

Feature-weighted categorized play across symmetric games

Experimental game theory studies the behavior of agents who face a stream of one-shot games as a form of learning. Most literature focuses on a single recurring identical game. This paper embeds single-game learning in a broader perspective, where learning can take place across similar games. We posit that agents categorize games into a few classes and tend to play the same action within a class. The agent’s categories are generated by combining game features (payoffs) and individual motives. An individual categorization is experience-based, and may change over time. We demonstrate our approach by testing a robust (parameter-free) model over a large body of independent experimental evidence over 2 × 2 symmetric games. The model provides a very good fit across games, performing remarkably better than standard learning models

On the Theory of Bergman Spaces on Homogeneous Siegel Domains

We consider mixed normed Bergman spaces on homogeneous Siegel domains. In the literature, two different approaches have been considered and several results seem difficult to be compared. In this paper we compare the results available in the literature and complete the existing ones in one of the two settings. The results we present are: natural inclusions, density, completeness, reproducing properties, sampling, atomic decomposition, duality, continuity of Bergman projectors, boundary values, transference.Comment: 31 pages, no figure