arXiv subjects

Rafael Morales

Publications and source records attributed to Rafael Morales.

15 recordsLinked to original sources

arXiv2026

Multiplicative and Additive Finite Free Convolutions for q-Polynomials

We study $q$-analogs of finite free convolutions and their interaction with families of $q$-hypergeometric polynomials. First, we revisit the $q$-multiplicative finite free convolution, previously introduced in the literature, and show that it acts naturally on $q$-hypergeometric polynomials: the convolution of two such polynomials remains within the same class, with parameters obtained by concatenation. This observation provides a simple mechanism for constructing large families of $q$-hypergeometric polynomials whose zeros are real and whose logarithmic mesh is controlled. We illustrate it with an example of multiple little $q$-Jacobi polynomials of the first kind. A result of independent interest is also an alternative definition of the $q$-multiplicative convolution in terms of $q$-differential operators. Motivated by the additive finite free convolution, we introduce a $q$-additive finite free convolution and study its algebraic and analytic properties. Although this convolution does not preserve real-rootedness in general, we show that a natural modification involving a $q$-multiplicative convolution restores the preservation of real roots and interlacing for polynomials with bounded logarithmic mesh. Finally, we develop a systematic method to translate product identities of $q$-hypergeometric functions into convolution identities for $q$-hypergeometric polynomials. This approach yields several explicit formulas for $q$-additive convolutions and produces new families of real-rooted $q$-hypergeometric polynomials.

math.CA↗

arXiv2026

Predictions of Stellar Occultations by Haumea and the Event of 4 May 2026

Haumea is the third-largest of the five officially recognized dwarf planets and one of the four that reside in the trans-Neptunian region. It is among the most exotic bodies in the Solar System, with an exceptionally rapid rotation, a highly elongated triaxial shape, and a ring that orbits about three times more slowly than Haumea itself. Because of its large heliocentric distance, direct exploration by dedicated space missions is not feasible in the short term, so progress must rely on ground- and near-Earth facilities. Stellar occultations are among the most powerful tools to investigate trans-Neptunian objects. We present new predictions of stellar occultations by Haumea and its ring for stars down to Gaia G = 21, and assess their scientific potential, with special emphasis on the 4 May 2026 event. We computed occultation opportunities for the coming years and evaluated the 4 May 2026 geometry in detail, including Haumea's rotation phase, known 3D shape, pole orientation, and sky-plane motion, to estimate the expected shadow-path width. Because the target star has a very large Gaia RUWE, we also carried out a dedicated reliability analysis, including speckle observations. We identify eleven valuable events through 2030. For 4 May 2026, we derive an expected sky-plane shadow width of $2224 \pm 30$ km, substantially larger than conservative nominal assumptions and therefore highly favorable for observations. Speckle imaging reveals a companion at $\sim 0.12$ arcsec and $\Delta m \sim 3.1$; this companion is also expected to be occulted and shifts the nominal main-star path prediction on Earth by about 8 mas. These results confirm the strong scientific return expected from coordinated observations of upcoming Haumea occultations, especially the 4 May 2026 event, and provide an updated framework to improve constraints on Haumea's shape, density, ring properties, and environment.

astro-ph.EP↗

arXiv2025

Synchronous Rotation in the (120347) Salacia-Actaea System

We report on roughly 16 years of photometric monitoring of the transneptunian binary system (120347) Salacia-Actaea which provides significant evidence that Salacia and Actaea are tidally locked to the mutual orbital period in a fully synchronous configuration. The orbit of Actaea is updated, followed by a Lomb-Scargle periodogram analysis of the ground-based photometry which reveals a synodic period similar to the orbital period and a peak-to-peak lightcurve amplitude of {\delta}m = 0.0900 {\pm} 0.0036 mag (1{\sigma} uncertainty). Incorporating archival HST photometry that resolves each component, we argue that the periodicity in the unresolved data is driven by a longitudinally varying surface morphology on Salacia, and derive a sidereal rotation period that is within 1{\sigma} of the mutual orbital period. A rudimentary tidal evolution model is invoked that suggests synchronization occurred within 1.1 Gyr after Actaea was captured/formed.

astro-ph.EP↗

arXiv2025

Zeros of orthogonal little q-Jacobi polynomials: interlacing and monotonicity

We investigate the distribution of zeros of the little q-Jacobi polynomials and related q-hypergeometric families. We prove that the zeros of these orthogonal polynomials exhibit strong interlacing properties and obey natural monotonicity rules with respect to the parameters. A key tool in our approach is the logarithmic mesh, which quantifies the relative spacing of the positive real zeros and allows us to classify families of polynomials with prescribed interlacing patterns. Our results include new interlacing relations, monotonicity with respect to parameters, and structural decompositions in non-orthogonal regimes. Several classical families of q-hypergeometric polynomials, including q-Bessel and Stieltjes-Wigert polynomials, are treated as limit cases. The methods rely on a combination of classical orthogonality theory and q-difference equations.

math.CA↗

arXiv2025

Even Hypergeometric Polynomials and Finite Free Commutators

We study in detail the class of even polynomials and their behavior with respect to finite free convolutions. To this end, we use some specific hypergeometric polynomials and a variation of the rectangular finite free convolution to understand even real-rooted polynomials in terms of positive-rooted polynomials. Then, we study some classes of even polynomials that are of interest in finite free probability, such as even hypergeometric polynomials, symmetrizations, and finite free commutators. Specifically, we provide many new examples of these objects, involving classical families of special polynomials (such as Laguerre, Hermite, and Jacobi). Finally, we relate the limiting root distributions of sequences of even polynomials with the corresponding symmetric measures that arise in free probability.

math.CA↗

arXiv2024

Entropic Hetero-Associative Memory

The Entropic Associative Memory holds objects in a 2D relation or ``memory plane'' using a finite table as the medium. Memory objects are stored by reinforcing simultaneously the cells used by the cue, implementing a form of Hebb's learning rule. Stored objects are ``overlapped'' on the medium, hence the memory is indeterminate and has an entropy value at each state. The retrieval operation constructs an object from the cue and such indeterminate content. In this paper we present the extension to the hetero-associative case in which these properties are preserved. Pairs of hetero-associated objects, possibly of different domain and/or modalities, are held in a 4D relation. The memory retrieval operation selects a largely indeterminate 2D memory plane that is specific to the input cue; however, there is no cue left to retrieve an object from such latter plane. We propose three incremental methods to address such missing cue problem, which we call random, sample and test, and search and test. The model is assessed with composite recollections consisting of manuscripts digits and letters selected from the MNIST and the EMNIST corpora, respectively, such that cue digits retrieve their associated letters and vice versa. We show the memory performance and illustrate the memory retrieval operation using all three methods. The system shows promise for storing, recognizing and retrieving very large sets of object with very limited computing resources.

cs.CV↗

arXiv2024

Zeros of generalized hypergeometric polynomials via finite free convolution. Applications to multiple orthogonality

We address the problem of the weak asymptotic behavior of zeros of families of generalized hypergeometric polynomials as their degree tends to infinity. The main tool is the representation of such polynomials as a finite free convolution of simpler elements; this representation is preserved in the asymptotic regime, so we can formally write the limit zero distribution of these polynomials as a free convolution of explicitly computable measures. We derive a simple expression for the S-transform of the limit distribution, which turns out to be a rational function, and a representation of the Kampé de Fériet polynomials in terms of finite free convolutions. We apply these tools, as well as those from [arXiv:2309.10970], to the study of some well-known families of multiple orthogonal polynomials (Jacobi-Piñeiro and multiple Laguerre of the first and second kinds), obtaining results on their zeros, such as interlacing, monotonicity, and asymptotics.

math.CA↗

arXiv2024

Entropic associative memory for real world images

The entropic associative memory (EAM) is a computational model of natural memory incorporating some of its putative properties of being associative, distributed, declarative, abstractive and constructive. Previous experiments satisfactorily tested the model on structured, homogeneous and conventional data: images of manuscripts digits and letters, images of clothing, and phone representations. In this work we show that EAM appropriately stores, recognizes and retrieves complex and unconventional images of animals and vehicles. Additionally, the memory system generates meaningful retrieval association chains for such complex images. The retrieved objects can be seen as proper memories, associated recollections or products of imagination.

cs.LG↗

arXiv2024

Real roots of hypergeometric polynomials via finite free convolution

We examine two binary operations on the set of algebraic polynomials, known as multiplicative and additive finite free convolutions, specifically in the context of hypergeometric polynomials. We show that the representation of a hypergeometric polynomial as a finite free convolution of more elementary blocks, combined with the preservation of the real zeros and interlacing by the free convolutions, is an effective tool that allows us to analyze when all roots of a specific hypergeometric polynomial are real. Moreover, the known limit behavior of finite free convolutions allows us to write the asymptotic zero distribution of some hypergeometric polynomials as free convolutions of Marchenko-Pastur, reversed Marchenko-Pastur, and free beta laws, which has an independent interest within free probability.

math.CA↗

arXiv2024

Interlacing and monotonicity of zeros of Angelesco-Jacobi polynomials

Information about the behavior of zeros of classical families of multiple or Hermite-Padé orthogonal polynomials as functions of the intrinsic parameters of the family is scarce. We establish the interlacing properties of the zeros of Angelesco-Jacobi polynomials when one of the three main parameters is increased by 1, extending the work of dos Santos (2017). We also show their monotonicity with respect to (large values) of the parameter representing in the electrostatic model of the zeros the size of the positive charge fixed at the origin, as well as monotonicity with respect to the endpoint of the interval of orthogonality. These results are extended to zeros of multiple Jacobi-Laguerre and Laguerre-Hermite polynomials using asymptotic relations between these families.

math.CA↗

arXiv2023

Cluster formation in iterated Mean Field Games

We study a simple first-order mean field game in which the coupling with the mean field is only in the final time and gives an incentive for players to congregate. For a short enough time horizon, the equilibrium is unique. We consider the process of \emph{iterating} the game, taking the final population distribution as the initial distribution in the next iteration. Restricting to one dimension, we take this to be a model of coalition building for a population distributed over some spectrum of opinions. Our main result states that, given a final coupling of the form $G(x,m) = \int φ(x-z)\dif m(z)$ where $φ$ is a smooth, even, non-positive function of compact support, then as the number of iterations goes to infinity the population tends to cluster into discrete groups, which are spread out as a function of the size of the support of $φ$. We discuss the potential implications of this result for real-world opinion dynamics and political systems.

math.OC↗

arXiv2022

Entropic Associative Memory for Manuscript Symbols

Manuscript symbols can be stored, recognized and retrieved from an entropic digital memory that is associative and distributed but yet declarative; memory retrieval is a constructive operation, memory cues to objects not contained in the memory are rejected directly without search, and memory operations can be performed through parallel computations. Manuscript symbols, both letters and numerals, are represented in Associative Memory Registers that have an associated entropy. The memory recognition operation obeys an entropy trade-off between precision and recall, and the entropy level impacts on the quality of the objects recovered through the memory retrieval operation. The present proposal is contrasted in several dimensions with neural networks models of associative memory. We discuss the operational characteristics of the entropic associative memory for retrieving objects with both complete and incomplete information, such as severe occlusions. The experiments reported in this paper add evidence on the potential of this framework for developing practical applications and computational models of natural memory.

cs.LG↗

arXiv2020

An Entropic Associative Memory

Natural memories are associative, declarative and distributed. Symbolic computing memories resemble natural memories in their declarative character, and information can be stored and recovered explicitly; however, they lack the associative and distributed properties of natural memories. Sub-symbolic memories developed within the connectionist or artificial neural networks paradigm are associative and distributed, but are unable to express symbolic structure and information cannot be stored and retrieved explicitly; hence, they lack the declarative property. To address this dilemma, we use Relational-Indeterminate Computing to model associative memory registers that hold distributed representations of individual objects. This mode of computing has an intrinsic computing entropy which measures the indeterminacy of representations. This parameter determines the operational characteristics of the memory. Associative registers are embedded in an architecture that maps concrete images expressed in modality-specific buffers into abstract representations, and vice versa, and the memory system as a whole fulfills the three properties of natural memories. The system has been used to model a visual memory holding the representations of hand-written digits, and recognition and recall experiments show that there is a range of entropy values, not too low and not too high, in which associative memory registers have a satisfactory performance. The similarity between the cue and the object recovered in memory retrieve operations depends on the entropy of the memory register holding the representation of the corresponding object. The experiments were implemented in a simulation using a standard computer, but a parallel architecture may be built where the memory operations would take a very reduced number of computing steps.

cs.AI↗

arXiv2015

A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy

The theory of complex dimensions of fractal strings developed by Lapidus and van Frankenhuijsen has proven to be a powerful tool for the study of Minkowski measurability of fractal subsets of the real line. In a very general setting, the Minkowski measurability of such sets is characterized by the structure of corresponding complex dimensions. Also, this tool is particularly effective in the setting of self-similar fractal subsets of $\mathbb{R}$ which have been shown to be Minkowski measurable if and only if they are nonlattice. This paper features a survey on the pertinent results of Lapidus and van Frankenhuijsen and a preliminary extension of the theory of complex dimensions to subsets of Euclidean space, with an emphasis on self-similar sets that satisfy various separation conditions. This extension is developed in the context of box-counting measurability, an analog of Minkowski measurability, which is shown to be characterized by complex dimensions under certain mild conditions.

math-ph↗

arXiv2006

Asymmetric Reversal in Inhomogeneous Magnetic Heterostructures

Asymmetric magnetization reversal is an unusual phenomenon in antiferromagnet / ferromagnet (AF/FM) exchange biased bilayers. We investigated this phenomenon in a simple model system experimentally and by simulation assuming inhomogeneously distributed interfacial AF moments. The results suggest that the observed asymmetry originates from the intrinsic broken symmetry of the system, which results in local incomplete domain walls parallel to the interface in reversal to negative saturation of the FM. Magneto-optic Kerr effect unambiguously confirms such an asymmetric reversal and a depth-dependent FM domain wall in accord with the magnetometry and simulations.

cond-mat.mtrl-sci↗