arXiv subjects

Pablo Carrasco

Publications and source records attributed to Pablo Carrasco.

5 recordsLinked to original sources

arXiv2026

A Characterization of $\alpha$-Convex Functions with Sharp Coefficient and Schwarzian Estimates

The class $M_\alpha$ of $\alpha$-convex functions, introduced by Mocanu in 1969, interpolates between starlike and convex functions. We prove a characterization of $M_\alpha$ that extends a theorem of Chuaqui, Duren, and Osgood from the convex case to the full class, and determine sharp values of $\beta$ for which $M_\alpha \subset C_\beta$ and $C_\beta \subset M_\alpha$. We also obtain a sharp Fekete--Szeg\H{o} inequality, bounds for the order and the Schwarzian norm, and an explicit formula for the Schwarzian norm of the $\alpha$-Koebe function for $\alpha = 1/n$, $n \in \mathbb{N}$, which we verify for $n \leq 9$ and conjecture to hold in general.

math.CV↗

arXiv2024

Tightening convex relaxations of trained neural networks: a unified approach for convex and S-shaped activations

The non-convex nature of trained neural networks has created significant obstacles in their incorporation into optimization models. In this context, Anderson et al. (2020) provided a framework to obtain the convex hull of the graph of a piecewise linear convex activation function composed with an affine function; this effectively convexifies activations such as the ReLU together with the affine transformation that precedes it. In this article, we contribute to this line of work by developing a recursive formula that yields a tight convexification for the composition of an activation with an affine function for a wide scope of activation functions, namely, convex or ``S-shaped". Our approach can be used to efficiently compute separating hyperplanes or determine that none exists in various settings, including non-polyhedral cases.

math.OC↗

arXiv2022

Schwarzian derivative for convex mappings of order alpha

The main purpose of this paper is to obtain sharp bounds of the norm of Schwarzian derivative for convex mappings of order $alpha$ in terms of the value of $f''(0)$, in particular, when this quantity is equal to zero. In addition, we obtain sharp bounds for distortion and growth for this mappings and we generalized the results obtained by Suita and Yamashita for this particular case.

math.CV↗

arXiv2021

Extended flexibility of Lyapunov exponents for Anosov diffeomorphisms

Bochi-Katok-Rodriguez Hertz proposed in [BKH21] a program on the flexibility of Lyapunov exponents for conservative Anosov diffeomorphisms, and obtained partial results in this direction. For conservative Anosov diffeomorphisms with strong hyperbolic properties we establish extended flexibility results for their Lyapunov exponents. We give examples of Anosov diffeomorphisms with the strong unstable exponent larger than the strong unstable exponent of the linear part. We also give examples of derived from Anosov diffeomorphisms with the metric entropy entropy larger than the entropy of the linear part. These results rely on a new type of deformation which goes beyond the previous Shub-Wilkinson and Baraviera-Bonatti techniques for conservative systems having some invariant directions. In order to estimate the Lyapunov exponents even after breaking the invariant bundles, we obtain an abstract result (Theorem 3.1) which gives bounds on exponents of some specific cocycles and which can be applied in various other settings. We also include various interesting comments in the appendices: our examples are $\mathcal{C}^2$ robust, the Lyapunov exponents are continuous (with respect to the map) even after breaking the invariant bundles, a similar construction can be obtained for the case of multiple eigenvalues.

math.DS↗

arXiv2016

Partially hyperbolic dynamics in dimension 3

Partial hyperbolicity appeared in the sixties as a natural generaliza- tion of hyperbolicity. In the last 20 years in this area there has been great activity. Here we survey the state of the art in some topics, focusing especially in partial hyperbolicity in dimension 3. The reason for this is not only that it is the smallest dimension in which non-degenerate partial hyperbolicity can occur, but also that the topology of 3-manifolds influences this dynamics in revealing ways.

math.DS↗