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

Functional Calculus on Non-Homogeneous Operators on Nilpotent Groups

We study the functional calculus associated with a hypoelliptic left-invariant differential operator $\mathcal{L}$ on a connected and simply connected nilpotent Lie group $G$ with the aid of the corresponding \emph{Rockland} operator $\mathcal{L}_0$ on the `local' contraction $G_0$ of $G$, as well as of the corresponding Rockland operator $\mathcal{L}_\infty$ on the `global' contraction $G_\infty$ of $G$. We provide asymptotic estimates of the Riesz potentials associated with $\mathcal{L}$ at $0$ and at $\infty$, as well as of the kernels associated with functions of $\mathcal{L}$ satisfying Mihlin conditions of every order. We also prove some Mihlin-H\"ormander multiplier theorems for $\mathcal{L}$ which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the `Plancherel measure' associated with $\mathcal{L}$ from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.Comment: 42 pages, no figure

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

Waiting times between orders and trades in double-auction markets

In this paper, the survival function of waiting times between orders and the corresponding trades in a double-auction market is studied both by means of experiments and of empirical data. It turns out that, already at the level of order durations, the survival function cannot be represented by a single exponential, thus ruling out the hypothesis of constant activity during trading. This fact has direct consequences for market microstructural models. They must include such a non-exponential behaviour to be realistic.Comment: 19 pages, 3 figures, paper presented at the WEHIA 2005, Colchester, U

Rap1B promotes VEGF-induced endothelial permeability and is required for dynamic regulation of the endothelial barrier

Vascular endothelial growth factor (VEGF), a key angiogenic and permeability factor, plays an important role in new blood vessel formation. However, abnormal VEGF-induced VEGFR2 signaling leads to hyperpermeability. We have shown previously that Rap1, best known for promoting cell adhesion and vessel stability, is a critical regulator of VEGFR2-mediated angiogenic and shear-stress EC responses. To determine the role of Rap1 role in endothelial barrier dynamics, we examined vascular permeability in EC-specific Rap1A- and Rap1B-knockout mice, cell-cell junction remodeling and EC monolayer resistivity in Rap1-deficient ECs under basal, inflammatory or elevated VEGF conditions. Deletion of either Rap1 isoform impaired de novo adherens junction (AJ) formation and recovery from LPS-induced barrier disruption in vivo However, only Rap1A deficiency increased permeability in ECs and lung vessels. Interestingly, Rap1B deficiency attenuated VEGF-induced permeability in vivo and AJ remodeling in vitro Therefore, only Rap1A is required for the maintenance of normal vascular integrity. Importantly, Rap1B is the primary isoform essential for normal VEGF-induced EC barrier dissolution. Deletion of either Rap1 isoform protected against hyper permeability in the STZ-induced diabetes model, suggesting clinical implications for targeting Rap1 in pathologies with VEGF-induced hyperpermeability

Positive Pluriharmonic Functions on Symmetric Siegel Domains

Given a symmetric Siegel domain $\mathscr D$ and a positive plurihamonic function $f$ on $\mathscr D$, we study the largest positive Radon measure $\mu$ on the Silov boundary $\mathrm b \mathscr D$ of $\mathscr D$ whose Poisson integral $\mathscr P \mu$ is $\leq f$. If $\mathscr D$ has no tubular irreducible factors of rank $\geq 2$, we show that $\mathscr P \mu$ is plurihamonic, and that $f-\mathscr P \mu$ is linear. As an application, we describe a possible analogue of the family of Clark measures associated with a holomorphic function from $\mathscr D$ into the unit disc in $\mathbb C$.Comment: 38 pages, no figure

Discrete event simulation in vehicle-to-vehicle transmission based on geometrical channel models

LAUREA MAGISTRALELa comunicazione veicolare: da una visione teorica delle tecnologie alla base, all'implementazione delle reti VANETs per la mobilità, fino alla simulazione ad eventi discreti della trasmissione veicolo-a-veicolo tramite l'utilizzo di un modello geometrico ad-hoc.Vehicular communication: from a theoretical overview about the background technologies, to the implementation of the VANETs networks for mobility, concluding with the discrete-event simulation under the vehicle-to-vehicle transmission using a geometric-based model created ad-hoc for the purpose

Clark measures associated with rational inner functions on bounded symmetric domains

Given a bounded symmetric domain in , we consider the Clark measures , , associated with a rational inner function from into the unit disc in . We show that , where is the dimension of the Šilov boundary of and is a suitable constant. Denoting with the closure of the space of holomorphic polynomials in , we characterize the for which when is a polydisc; we also provide some necessary and some sufficient conditions for general domain

Promoting vascular repair in the retina: can stem/progenitor cells help?

Since its first epidemic in the 1940s, retinopathy of prematurity (ROP) has been a challenging illness in neonatology. Higher than physiological oxygen levels impede the development of the immature retinal neuropil and vasculature. Current treatment regimens include cryotherapy, laser photocoagulation, and anti-VEGF agents. Unfortunately, none of these approaches can rescue the normal retinal vasculature, and each has significant safety concerns. The limitations of these approaches have led to new efforts to understand the pathological characteristics in each phase of ROP and to find a safer and more effective therapeutic approach. In the era of stem cell biology and with the need for new treatments for ROP, this review discusses the possible future use of unique populations of proangiogenic cells for therapeutic revascularization of the preterm retina