101,989 research outputs found
Generalized Mean Curvature Flow in Carnot Groups
No full textIn this paper we study the generalized mean curvature flow of sets in the sub-Riemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting in [4] and [12]. We establish two special cases of the comparison principle, existence, uniqueness and basic geometric properties of the flow
Regularity for a class of quasilinear degenerate parabolic equations in the Heisenberg group
No full textWe extend to the parabolic setting some of the ideas originated with Xiao Zhong's proof in [31] of the Hölder regularity of p-harmonic functions in the Heisenberg group Hn. Given a number p ≥ 2, in this paper we establish the C1 smoothness of weak solutions of a class of quasilinear PDE in Hn modeled on the equation
Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups H^n
No full textWe prove that Lipschitz intrinsic graphs in the Heisenberg groups , with n > 1, which are vanishing viscosity solutions of the minimal surface equation, are smooth and satisfy the PDE in a strong sense
Regularity of non-characteristic minimal graphs in the Heisenberg group
No full textMinimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits.
Our main results are a-priori estimates on the solutions of the approximating Riemannian PDE and the ensuing C∞ regularity of the sub-Riemannian minimal surface along its Legendrian foliation
System identification through Lipschitz regularized deep neural networks
No full textIn this paper we use neural networks to learn governing equations from data. Specifically we reconstruct the right-hand side of a system of ODEs x ̇(t)=f(t,x(t)) directly from observed uniformly time-sampled data using a neural network. In contrast with other neural network-based approaches to this problem, we add a Lipschitz regularization term to our loss function. In the synthetic examples we observed empirically that this regularization results in a smoother approximating function and better generalization properties when compared with non-regularized models, both on trajectory and non-trajectory data, especially in presence of noise. In contrast with sparse regression approaches, since neural networks are universal approximators, we do not need any prior knowledge on the ODE system. Since the model is applied component wise, it can handle systems of any dimension, making it usable for real-world data
Uniform Gaussian Bounds for Subelliptic Heat Kernels and an Application to the Total Variation Flow of Graphs over Carnot Groups
No full textIn this paper we study heat kernels associated with a Carnot
group G, endowed with a family of collapsing left-invariant
Riemannian metrics σε which converge in the Gromov-
Hausdorff sense to a sub-Riemannian structure on G as ε→
0. The main new contribution are Gaussian-type bounds on
the heat kernel for the σε metrics which are stable as ε→0
and extend the previous time-independent estimates in [16].
As an application we study well posedness of the total variation
flow of graph surfaces over a bounded domain in a step
two Carnot group (G; σε ). We establish interior and boundary
gradient estimates, and develop a Schauder theory which are
stable as ε → 0. As a consequence we obtain long time
existence of smooth solutions of the sub-Riemannian flow
(ε = 0), which in turn yield sub-Riemannian minimal surfaces
as t → ∞
Robust Neural Network Approach to System Identification in the High-Noise Regime
No full textWe present a new algorithm for learning unknown governing equations from trajectory data, using a family of neural networks. Given samples of solutions to an unknown dynamical system we approximate the forcing term using a family of neural networks. We express the equation in integral form and use Euler method to predict the solution at every successive time step using at each iteration a different neural network as a prior for f. This procedure yields M-1 time-independent networks, where M is the number of time steps at which the solution is observed. Finally, we obtain a single function which approximates the forcing term by neural network interpolation. Unlike our earlier work, where we numerically computed the derivatives of data, and used them as target in a Lipschitz regularized neural network to approximate f, our new method avoids numerical differentiations, which are unstable in presence of noise. We test the new algorithm on multiple examples in a high-noise setting. We empirically show that generalization and recovery of the governing equation improve by adding a Lipschitz regularization term in our loss function and that this method improves our previous one especially in the high-noise regime, when numerical differentiation provides low quality target data. Finally, we compare our results with other state of the art methods for system identification
Le armi di achille: note sull'aggressività in psicoterapia psicoanalitica
No full textCos'è che spinge l'uomo ad esistere? E' semplicemente la preoccupazione per la propria sicurezza e sopravvivenza? Vogliamo credere che il bambino fin dalla più tenera età ricerchi solamente una base sicura, e che non sia altrettanto desideroso di "imparare a tremare"? L'intento di questo libro non è quello di "conoscere", bensì quello di "capire" l'aggressività salendo verso i piani alti della coscienza, dove l'aggressività è rappresentata attraverso l'arte, la religione, la mitologia, l'estetica, ecc
The alpha-latrotoxin mutant LTXN4C enhances spontaneous and evoked transmitter release in CA3 pyramidal neurons
No full textAlpha-latrotoxin (LTX) stimulates vesicular exocytosis by at least two mechanisms that include (1) receptor binding-stimulation and (2) membrane pore formation. Here, we use the toxin mutant LTX(N4C) to selectively study the receptor-mediated actions of LTX. LTX(N4C) binds to both LTX receptors (latrophilin and neurexin) and greatly enhances the frequency of spontaneous and miniature EPSCs recorded from CA3 pyramidal neurons in hippocampal slice cultures. The effect of LTX(N4C) is reversible and is not attenuated by La3+ that is known to block LTX pores. On the other hand, LTX(N4C) action, which requires extracellular Ca2+, is inhibited by thapsigargin, a drug depleting intracellular Ca2+ stores, by 2-aminoethoxydiphenyl borate, a blocker of inositol(1,4,5)-trisphosphate-induced Ca2+ release, and by U73122, a phospholipase C inhibitor. Furthermore, measurements using a fluorescent Ca2+ indicator directly demonstrate that LTX(N4C) increases presynaptic, but not dendritic, free Ca2+ concentration; this Ca2+ rise is blocked by thapsigargin, suggesting, together with electrophysiological data, that the receptor-mediated action of LTX(N4C) involves mobilization of Ca2+ from intracellular stores. Finally, in contrast to wild-type LTX, which inhibits evoked synaptic transmission probably attributable to pore formation, LTX(N4C) actually potentiates synaptic currents elicited by electrical stimulation of afferent fibers. We suggest that the mutant LTX(N4C), lacking the ionophore-like activity of wild-type LTX, activates a presynaptic receptor and stimulates Ca2+ release from intracellular stores, leading to the enhancement of synaptic vesicle exocytosis
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