Hausdorff Distance between Norm Balls and their Linear Maps

We consider the problem of computing the (two-sided) Hausdorff distance between the unit $\ell_{p_{1}}$ and $\ell_{p_{2}}$ norm balls in finite dimensional Euclidean space for $1 < p_1 < p_2 \leq \infty$, and derive a closed-form formula for the same. We also derive a closed-form formula for the Hausdorff distance between the $k_1$ and $k_2$ unit $D$-norm balls, which are certain polyhedral norm balls in $d$ dimensions for $1 \leq k_1 < k_2 \leq d$. When two different $\ell_p$ norm balls are transformed via a common linear map, we obtain several estimates for the Hausdorff distance between the resulting convex sets. These estimates upper bound the Hausdorff distance or its expectation, depending on whether the linear map is arbitrary or random. We then generalize the developments for the Hausdorff distance between two set-valued integrals obtained by applying a parametric family of linear maps to different $\ell_p$ unit norm balls, and then taking the Minkowski sums of the resulting sets in a limiting sense. To illustrate an application, we show that the problem of computing the Hausdorff distance between the reach sets of a linear dynamical system with different unit norm ball-valued input uncertainties, reduces to this set-valued integral setting

An equivalent condition to the Jensen inequality for the generalized Sugeno integral.

For the classical Jensen inequality of convex functions, i.e., [Formula: see text] an equivalent condition is proved in the framework of the generalized Sugeno integral. Also, the necessary and sufficient conditions for the validity of the discrete form of the Jensen inequality for the generalized Sugeno integral are given

Space-time extensions II

The global extendibility of smooth causal geodesically incomplete spacetimes is investigated. Denote by $\gamma$ one of the incomplete non-extendible causal geodesics of a causal geodesically incomplete spacetime $(M,g_{ab})$. First, it is shown that it is always possible to select a synchronised family of causal geodesics $\Gamma$ and an open neighbourhood $\mathcal{U}$ of a final segment of $\gamma$ in $M$ such that $\mathcal{U}$ is comprised by members of $\Gamma$, and suitable local coordinates can be defined everywhere on $\mathcal{U}$ provided that $\gamma$ does not terminate either on a tidal force tensor singularity or on a topological singularity. It is also shown that if, in addition, the spacetime, $(M,g_{ab})$, is globally hyperbolic, and the components of the curvature tensor, and its covariant derivatives up to order $k-1$ are bounded on $\mathcal{U}$, and also the line integrals of the components of the $k^{th}$-order covariant derivatives are finite along the members of $\Gamma$---where all the components are meant to be registered with respect to a synchronised frame field on $\mathcal{U}$---then there exists a $C^{k-}$ extension $\Phi: (M,g_{ab}) \rightarrow (\widehat{M},\widehat{g}_{ab})$ so that for each $\bar\gamma\in\Gamma$, which is inextendible in $(M,g_{ab})$, the image, $\Phi\circ\bar\gamma$, is extendible in $(\widehat{M},\widehat{g}_{ab})$. Finally, it is also proved that whenever $\gamma$ does terminate on a topological singularity $(M,g_{ab})$ cannot be generic.Comment: 42 pages, no figures, small changes to match the published versio

From elephant to goldfish (and back): memory in stochastic Volterra processes

We propose a new theoretical framework that exploits convolution kernels to transform a Volterra path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. This transformation is achieved by embedding a Markovian "memory process" within the dynamics of the non-Markovian process. We discuss existence and path-wise regularity of solutions for the stochastic Volterra equations introduced and we provide a financial application to volatility modeling. We also propose a numerical scheme for simulating the processes. The numerical scheme exhibits a strong convergence rate of 1/2, which is independent of the roughness parameter of the volatility process. This is a significant improvement compared to Euler schemes used in similar models. We propose a new theoretical framework that exploits convolution kernels to transform a Volterra path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. This transformation is achieved by embedding a Markovian "memory process" (the goldfish) within the dynamics of the non-Markovian process (the elephant). Most notably, it is also possible to go back, i.e., the transformation is reversible. We discuss existence and path-wise regularity of solutions for the stochastic Volterra equations introduced and we propose a numerical scheme for simulating the processes, which exhibits a remarkable convergence rate of $1/2$. In particular, in the fractional kernel case, the strong convergence rate is independent of the roughness parameter, which is a positive novelty in contrast with what happens in the available Euler schemes in the literature in rough volatility models

Hölder-Besov boundedness for periodic pseudo-differential operators

In this work we give Holder-Besov estimates for periodic Fourier multipliers. We present a class of bounded pseudo-differential operators on periodic Besov spaces with symbols of limited regularity

Riesz transforms, Cauchy-Riemann systems and amalgam Hardy spaces

In this paper we study Hardy spaces $\mathcal{H}^{p,q}(\mathbb{R}^d)$, $0<p,q<\infty$, modeled over amalgam spaces $(L^p,\ell^q)(\mathbb{R}^d)$. We characterize $\mathcal{H}^{p,q}(\mathbb{R}^d)$ by using first order classical Riesz transforms and compositions of first order Riesz transforms depending on the values of the exponents $p$ and $q$. Also, we describe the distributions in $\mathcal{H}^{p,q}(\mathbb{R}^d)$ as the boundary values of solutions of harmonic and caloric Cauchy-Riemann systems. We remark that caloric Cauchy-Riemann systems involve fractional derivative in the time variable. Finally we characterize the functions in $L^2(\mathbb{R}^d) \cap \mathcal{H}^{p,q}(\mathbb{R}^d)$ by means of Fourier multipliers $m_\theta$ with symbol $\theta(\cdot/|\cdot|)$, where $\theta \in C^\infty(\mathbb{S}^{d-1})$ and $\mathbb{S}^{d-1}$ denotes the unit sphere in $\mathbb{R}^d$.Comment: 24 page

Analysis of models for quantum transport of electrons in graphene layers

We present and analyze two mathematical models for the self consistent quantum transport of electrons in a graphene layer. We treat two situations. First, when the particles can move in all the plane \RR^2, the model takes the form of a system of massless Dirac equations coupled together by a selfconsistent potential, which is the trace in the plane of the graphene of the 3D Poisson potential associated to surface densities. In this case, we prove local in time existence and uniqueness of a solution in H^s(\RR^2), for $s > 3/8$ which includes in particular the energy space H^{1/2}(\RR^2). The main tools that enable to reach $s\in (3/8,1/2)$ are the dispersive Strichartz estimates that we generalized here for mixed quantum states. Second, we consider a situation where the particles are constrained in a regular bounded domain $\Omega$. In order to take into account Dirichlet boundary conditions which are not compatible with the Dirac Hamiltonian $H_{0}$, we propose a different model built on a modified Hamiltonian displaying the same energy band diagram as $H_{0}$ near the Dirac points. The well-posedness of the system in this case is proved in $H^s_{A}$, the domain of the fractional order Dirichlet Laplacian operator, for $1/2\leq s<5/2$