arXiv subjects

Antonio Acuaviva

Publications and source records attributed to Antonio Acuaviva.

24 records · Page 2Linked to original sources

arXiv2025

Factorizations and minimality of the Calkin Algebra norm for $C(K)$-spaces

For a scattered, locally compact Hausdorff space $K$, we prove that the essential norm on the Calkin algebra \break $\mathscr{B}(C_0(K))/\mathscr{K}(C_0(K))$ is a minimal algebra norm. The proof relies on establishing a quantitative factorization for the identity operator on $c_0$ through non-compact operators $T: C_0(K) \to X$, where $X$ is any Banach space that does not contain a copy of $\ell_1$ or whose dual unit ball is weak$^*$ sequentially compact. It follows that, for every ordinal $α$, the algebras $\mathscr{B}(C[0,α]))$ and $\mathscr{B}(C[0,α]))/\mathscr{K}(C[0,α]))$ have an unique algebra norm.

math.FA↗

arXiv2025

The class of Banach lattices is not primary

Building on a recent construction of Plebanek and Salguero-Alarcón, which solved the Complemented Subspace Problem for $C(K)$-spaces, and the subsequent work of De Hevia, Martínez-Cervantes, Salguero-Alarcón, and Tradacete solving the Complemented Subspace Problem for Banach lattices, we show that the class of Banach lattices is not primary. Specifically, we exhibit a compact Hausdorff space $L$ such that $C(L) \simeq X \oplus \tilde{X}$ and neither $X$ nor $\tilde{X}$ is isomorphic to a Banach lattice. In particular, it also follows that the class of $C(K)$-spaces is not primary.

math.FA↗

arXiv2024

Source-Seeking Problem with Robot Swarms

We present an algorithm to solve the problem of locating the source, or maxima, of a scalar field using a robot swarm. We demonstrate how the robot swarm determines its direction of movement to approach the source using only field intensity measurements taken by each robot. In contrast with the current literature, our algorithm accommodates a generic (non-degenerate) geometry for the swarm's formation. Additionally, we rigorously show the effectiveness of the algorithm even when the dynamics of the robots are complex, such as a unicycle with constant speed. Not requiring a strict geometry for the swarm significantly enhances its resilience. For example, this allows the swarm to change its size and formation in the presence of obstacles or other real-world factors, including the loss or addition of individuals to the swarm on the fly. For clarity, the article begins by presenting the algorithm for robots with free dynamics. In the second part, we demonstrate the algorithm's effectiveness even considering non-holonomic dynamics for the robots, using the vector field guidance paradigm. Finally, we verify and validate our algorithm with various numerical simulations.

cs.RO↗

arXiv2024

Resilient source seeking with robot swarms

We present a solution for locating the source, or maximum, of an unknown scalar field using a swarm of mobile robots. Unlike relying on the traditional gradient information, the swarm determines an ascending direction to approach the source with arbitrary precision. The ascending direction is calculated from measurements of the field strength at the robot locations and their relative positions concerning the centroid. Rather than focusing on individual robots, we focus the analysis on the density of robots per unit area to guarantee a more resilient swarm, i.e., the functionality remains even if individuals go missing or are misplaced during the mission. We reinforce the robustness of the algorithm by providing sufficient conditions for the swarm shape so that the ascending direction is almost parallel to the gradient. The swarm can respond to an unexpected environment by morphing its shape and exploiting the existence of multiple ascending directions. Finally, we validate our approach numerically with hundreds of robots. The fact that a large number of robots always calculate an ascending direction compensates for the loss of individuals and mitigates issues arising from the actuator and sensor noises.

cs.RO↗

arXiv2023

State Transfer in Complex Quantum Walks

Given a graph with Hermitian adjacency matrix $H$, perfect state transfer occurs from vertex $a$ to vertex $b$ if the $(b,a)$-entry of the unitary matrix $\exp(-iHt)$ has unit magnitude for some time $t$. This phenomenon is relevant for information transmission in quantum spin networks and is known to be monogamous under real symmetric matrices. We prove the following results: 1. For oriented graphs (whose nonzero weights are $\pm i$), the oriented $3$-cycle and the oriented edge are the only graphs where perfect state transfer occurs between every pair of vertices. This settles a conjecture of Cameron et al. On the other hand, we construct an infinite family of oriented graphs with perfect state transfer between any pair of vertices on a subset of size four. 2. There are infinite families of Hermitian graphs with one-way perfect state transfer, where perfect state transfer occurs without periodicity. In contrast, perfect state transfer implies periodicity whenever the adjacency matrix has algebraic entries (as shown by Godsil). 3. There are infinite families with non-monogamous pretty good state transfer in rooted graph products. In particular, we generalize known results on double stars (due to Fan and Godsil) and on paths with loops (due to Kempton, Lippner and Yau). The latter extends the experimental observation of quantum transport (made by Zimborás et al.) and shows non-monogamous pretty good state transfer can occur amongst distant vertices.

math.CO↗

arXiv2021

Equidistribution of zeros of some polynomials related to cyclic functions

In the study of the cyclicity of a function $f$ in reproducing kernel Hilbert spaces an important role is played by sequences of polynomials $\{p_n\}_{n\in \mathbb{N}}$ called \emph{optimal polynomial approximants} (o.p.a.). For many such spaces and when the functions $f$ generating those o.p.a. are polynomials without zeros inside the disk but with some zeros on its boundary, we find that the weakly asympotic distribution of the zeros of $1-p_nf$ is the uniform measure on the unit circle.

math.CV↗