arXiv subjects

Antonio Montalban

Publications and source records attributed to Antonio Montalban.

8 recordsLinked to original sources

arXiv2024

The spark of synchronization in heterogeneous networks of chaotic maps

We investigate the emergence of synchronization in heterogeneous networks of chaotic maps. Our findings reveal that a small cluster of highly connected maps is responsible for triggering the spark of synchronization. After the spark, the synchronized cluster grows in size and progressively moves to less connected maps, eventually reaching a cluster that may remain synchronized over time. We explore how the shape of the network's degree distribution affects the onset of synchronization and derive an expression based on the network's construction that determines the expected time for a network to synchronize. Understanding how the network structure affects the spark of synchronization is particularly important for the control and design of more robust systems that require some level of coherence between a subset of units for better functioning. Numerical simulations in finite-sized networks are consistent with this analysis.

nlin.AO↗

arXiv2023

Emergence of chaotic cluster synchronization in heterogeneous networks

Many real-world complex systems rely on cluster synchronization to function properly. A cluster of nodes exhibits synchronous behavior while others behave erratically. Predicting the emergence of these clusters and understanding the mechanism behind their structure and variation in response to parameter change is a daunting task in networks that lack symmetry. We unravel the mechanism for the emergence of cluster synchronization in heterogeneous random networks. We develop a heterogeneous mean field approximation together with a self-consistent theory to determine the onset and stability of the cluster. Our analysis shows that cluster synchronization occurs in a wide variety of heterogeneous networks, node dynamics, and coupling functions. The results could lead to a new understanding of the dynamical behavior of networks ranging from neural to social.

nlin.AO↗

arXiv2015

A mathematical analysis of the evolutionary benefits of sexual reproduction

The question as to why most higher organisms reproduce sexually has remained open despite extensive research, and has been called "the queen of problems in evolutionary biology". Theories dating back to Weismann have suggested that the key must lie in the creation of increased variability in offspring, causing enhanced response to selection. Rigorously quantifying the effects of assorted mechanisms which might lead to such increased variability, and establishing that these beneficial effects outweigh the immediate costs of sexual reproduction has, however, proved problematic. Here we introduce an approach which does not focus on particular mechanisms influencing factors such as the fixation of beneficial mutants or the ability of populations to deal with deleterious mutations, but rather tracks the entire distribution of a population of genotypes as it moves across vast fitness landscapes. In this setting simulations now show sex robustly outperforming asex across a broad spectrum of finite or infinite population models. Concentrating on the additive infinite populations model, we are able to give a rigorous mathematical proof establishing that sexual reproduction acts as a more efficient optimiser of mean fitness, thereby solving the problem for this model. Some of the key features of this analysis carry through to the finite populations case.

q-bio.PE↗

arXiv2014

Computable structures in generic extensions

In this paper, we investigate connections between structures present in every generic extension of the universe $V$ and computability theory. We introduce the notion of {\em generic Muchnik reducibility} that can be used to to compare the complexity of uncountable structures; we establish basic properties of this reducibility, and study it in the context of {\em generic presentability}, the existence of a copy of the structure in every extension by a given forcing. We show that every forcing notion making $ω_2$ countable generically presents some countable structure with no copy in the ground model; and that every structure generically presentble by a forcing notion that does not make $ω_2$ countable has a copy in the ground model. We also show that any countable structure $\mathcal{A}$ that is generically presentable by a forcing notion not collapsing $ω_1$ has a countable copy in $V$, as does any structure $\mathcal{B}$ generically Muchnik reducible to a structure $\mathcal{A}$ of cardinality $\aleph_1$. The former positive result yields a new proof of Harrington's result that counterexamples to Vaught's conjecture have models of power $\aleph_1$ with Scott rank arbitrarily high below $ω_2$. Finally, we show that a rigid structure with copies in all generic extensions by a given forcing has a copy already in the ground model.

math.LO↗

arXiv2013

A computability theoretic equivalent to Vaught's conjecture

We prove that, for every theory $T$ which is given by an ${\mathcal L}_{ω_1,ω}$ sentence, $T$ has less than $2^{\aleph_0}$ many countable models if and only if we have that, for every $X\in 2^ω$ on a cone of Turing degrees, every $X$-hyperarithmetic model of $T$ has an $X$-computable copy. We also find a concrete description, relative to some oracle, of the Turing-degree spectra of all the models of a counterexample to Vaught's conjecture.

math.LO↗

arXiv2011

Relative to any non-hyperarithmetic set

We prove that there is a structure, indeed a linear ordering, whose degree spectrum is the set of all non-hyperarithmetic degrees. We also show that degree spectra can distinguish measure from category.

math.LO↗

arXiv2011

A fixed point for the jump operator on structures

Assuming that $0^#$ exists, we prove that there is a structure that can effectively interpret its own jump. In particular, we get a structure $\mathcal A$ such that \[ Sp({\mathcal A}) = \{{\bf x}':{\bf x}\in Sp ({\mathcal A})\}, \] where $Sp ({\mathcal A})$ is the set of Turing degrees which compute a copy of $\mathcal A$. It turns out that, more interesting than the result itself, is its unexpected complexity. We prove that higher-order arithmetic, which is the union of full $n$th-order arithmetic for all $n$, cannot prove the existence of such a structure.

math.LO↗

arXiv2008

K-trivials are NCR

We show that for every K-trivial real X, there is no representation of a continuous probability measure m such that X is 1-random relative to m.

math.LO↗