We present a model--independent reconstruction of the normalized dark energy density function, $X(z) \equiv \rho_{\mathrm{de}}(z)/\rho_{\mathrm{de}}(0)$, derived directly from the DES-SN5YR Type~Ia supernova sample. The analysis employs an inversion formalism that relates the derivative of the distance modulus, $\mu^{\prime}(z)$, to the expansion history, allowing the data to determine the shape of $X(z)$ without assuming a specific equation--of--state or dark energy density parameterization. A statistically optimized binning of the supernova sample (using 17 intervals following the Freedman--Diaconis criterion and 34 following Scott's rule) ensures a stable estimation of $\mu^{\prime}(z)$ and a controlled propagation of uncertainties throughout the inversion process. The resulting $X(z)$ remains statistically consistent with a constant value within one standard deviation across the entire redshift range, showing no significant evidence for an evolving dark energy component at present. In a direct comparison among $\Lambda$CDM, CPL, and the quadratic $X^2(z)$ parameterization -- where CPL and $X^2(z)$ each introduce two additional free parameters relative to $\Lambda$CDM -- the CPL model attains the best statistical agreement with the data, albeit only marginally and strictly within this restricted model set. These outcomes indicate that current observations are compatible with an almost constant dark energy density ($w \simeq -1$), while the inversion framework remains sensitive to subtle departures that forthcoming high--precision surveys could resolve.

Existing mathematical reasoning benchmarks are predominantly English only or translation-based, which can introduce semantic drift and mask languagespecific reasoning errors. To address this, we present AI4Math, a benchmark of 105 original university level math problems natively authored in Spanish. The dataset spans seven advanced domains (Algebra, Calculus, Geometry, Probability, Number Theory, Combinatorics, and Logic), and each problem is accompanied by a step by step human solution. We evaluate six large language models GPT 4o, GPT 4o mini, o3 mini, LLaMA 3.3 70B, DeepSeek R1 685B, and DeepSeek V3 685B under four configurations: zero shot and chain of thought, each in Spanish and English. The top models (o3 mini, DeepSeek R1 685B, DeepSeek V3 685B) achieve over 70% accuracy, whereas LLaMA 3.3 70B and GPT-4o mini remain below 40%. Most models show no significant performance drop between languages, with GPT 4o even performing better on Spanish problems in the zero shot setting. Geometry, Combinatorics, and Probability questions remain persistently challenging for all models. These results highlight the need for native-language benchmarks and domain-specific evaluations to reveal reasoning failures not captured by standard metrics.

We report observations of the ultra-high-energy gamma-ray source LHAASO J2108$+$5157, utilizing VERITAS, HAWC, Fermi-LAT, and XMM-Newton. VERITAS has collected $\sim$ 40 hours of data that we used to set ULs to the emission above 200 GeV. The HAWC data, collected over $\sim 2400$ days, reveal emission between 3 and 146 TeV, with a significance of $7.5~\sigma$, favoring an extended source model. The best-fit spectrum measured by HAWC is characterized by a simple power-law with a spectral index of $2.45\pm0.11_{stat}$. Fermi-LAT analysis finds a point source with a very soft spectrum in the LHAASO J2108+5157 region, consistent with the 4FGL-DR3 catalog results. The XMM-Newton analysis yields a null detection of the source in the 2 - 7 keV band. The broadband spectrum can be interpreted as a pulsar and a pulsar wind nebula system, where the GeV gamma-ray emission originates from an unidentified pulsar, and the X-ray and TeV emission is attributed to synchrotron radiation and inverse Compton scattering of electrons accelerated within a pulsar wind nebula. In this leptonic scenario, our X-ray upper limit provides a stringent constraint on the magnetic field, which is $\lesssim 1.5\ \mu$G.

In this paper we give a new generalization of token graphs. Given two integers $1\leq m \leq k$ and a graph $G$ we define the generalized token graph of the graph $G$, to be the graph $F_k^m(G)$ whose vertices correspond to configurations of $k$ indistinguishable tokens placed at distinct vertices of $G$, where two configurations are adjacent whenever one configuration can be reached from the other by moving $m$ tokens along $m$ edges of $G$. When $m=1$, the usual token graph $F_k(G)$ is recovered. We give sufficient and necessary conditions on the graph $G$ for $F_2^2(G)$ to be connected and we give sufficient and necessary conditions on the graph $G$ for $F_2^2(G)$ to be bipartite. We also analyze some properties of generalized token graphs, such as clique number, chromatic number, independence number and domination number. Finally, we conclude with an analysis of the automorphism group of the generalized token graph.

The article focuses on the urgent issue of femicide in Veracruz, Mexico, and the development of the MFM_FEM_VER_CP_2024 model, a mathematical framework designed to predict femicide risk using fuzzy logic. This model addresses the complexity and uncertainty inherent in gender based violence by formalizing risk factors such as coercive control, dehumanization, and the cycle of violence. These factors are mathematically modeled through membership functions that assess the degree of risk associated with various conditions, including personal relationships and specific acts of violence. The study enhances the original model by incorporating new rules and refining existing membership functions, which significantly improve the model predictive accuracy.

This study utilizes neural networks to evaluate the 2024 judicial reform in Mexico, a proposal designed to overhaul the judicial system by increasing transparency, judicial autonomy, and introducing the popular election of judges. The neural network model analyzes both converging and diverging factors that influence the reforms viability and public acceptance. Key areas of convergence include enhanced transparency and judicial autonomy, which are seen as improvements to the system. However, major points of divergence, such as the high costs of implementation and concerns about the legitimacy of electing judges, pose significant challenges. By integrating variables like transparency, decision quality, judicial independence, and implementation costs, the model predicts levels of public and professional acceptance of the reform. The neural networks multilayered structure allows for the modeling of complex relationships, offering predictive insights into how the reform may impact the Mexican judicial system. Initial findings suggest that while the reform could strengthen judicial autonomy, the risks of politicizing the judiciary and the financial burden it entails may reduce its overall acceptance. This research highlights the importance of using advanced AI tools to simulate public policy outcomes, providing valuable data to guide lawmakers in refining their proposals.

We show that the Lovelock type brane gravity is naturally holographic by providing a correspondence between bulk and surface terms that appear in the Lovelock-type brane gravity action functional. We prove the existence of relationships between the $\mathcal{L}_{\mbox{\tiny bulk}}$ and $\mathcal{L}_{\mbox{\tiny surf}}$ allowing $\mathcal{L}_{\mbox{\tiny surf}}$ to be determined completely by $\mathcal{L}_{\mbox{\tiny bulk}}$. In the same spirit, we provide relationships among the various conserved tensors that this theory possesses. We further comment briefly on the correspondence between geometric degrees of freedom in both bulk and surface space.

We explore the correspondence between the parallel surfaces framework, and the minimal surfaces framework, to uncover and apply new aspects of the geometrical and mechanical content behind the so-called Lovelock-type brane gravity (LBG). We show how this type of brane gravity emerges naturally from a Dirac-Nambu-Goto (DNG) action functional built up from the volume element associated with a world volume shifted a distance $\alpha$ along the normal vector of a germinal world volume, and provide all known geometric structures for such a theory. Our development highlights the dependence of the geometry for the displaced world volume on the fundamental forms, as well as on certain conserved tensors, defined on the outset world volume. Based on this, LBG represents a natural and elegant generalization of the DNG theory to higher dimensions. Moreover, our development allows for exploring disformal transformations in Lovelock brane gravity and analyzing their relations with scalar-tensor theories defined on the brane trajectory. Likewise, this geometrical correspondence would enable us to establish contact with tractable Hamiltonian approximations for this brane gravity theory, by exploiting the linkage with a DNG model, and thus start building a suitable quantum version.

We develop a covariant scheme to describe the dynamics of small perturbations on Lovelock type extended objects propagating in a flat Minkowski spacetime. The higher-dimensional analogue of the Jacobi equation in this theory becomes a wave type equation for a scalar field $\Phi$. Whithin this framework, we analyse the stability of membranes with a de Sitter geometry where we find that the Jacobi equation specializes to a Klein-Gordon (KG) equation for $\Phi$ possessing a tachyonic mass. This shows that, to some extent, these type of extended objects share the symmetries of the Dirac-Nambu-Goto (DNG) action which is by no means coincidental because the DNG model is the simplest included in this type of gravity.

We present a Born-Infeld type theory to describe the evolution of p-branes propagating in an N = (p+2)-dimensional Minkowski spacetime. The expansion of the BI-type volume element gives rise to the (p+1) Lovelock brane invariants associated with the worldvolume swept out by the brane. Contrary to the Lovelock theory in gravity, the number of Lovelock brane Lagrangians differs in this case, depending on the dimension of the worldvolume as a consequence that we consider the embedding functions, instead of the metric, as the field variables. This model depends on the intrinsic and the extrinsic geometries of the worldvolume and in consequence is a second-order theory as shown in the main text. A classically equivalent action is discussed and we comment on its Weyl invariance in any dimension which naturally requires the introduction of some auxiliary fields.

The importance of the working document is that it allows the analysis of information and cases associated with (SARS-CoV-2) COVID-19, based on the daily information generated by the Government of Mexico through the Secretariat of Health, responsible for the Epidemiological Surveillance System for Viral Respiratory Diseases (SVEERV). The information in the SVEERV is disseminated as open data, and the level of information is displayed at the municipal, state and national levels. On the other hand, the monitoring of the genomic surveillance of (SARS-CoV-2) COVID-19, through the identification of variants and mutations, is registered in the database of the Information System of the Global Initiative on Sharing All Influenza Data (GISAID) based in Germany. These two sources of information SVEERV and GISAID provide the information for the analysis of the impact of (SARS-CoV-2) COVID-19 on the population in Mexico. The first data source identifies information, at the national level, on patients according to age, sex, comorbidities and COVID-19 presence (SARS-CoV-2), among other characteristics. The data analysis is carried out by means of the design of an algorithm applying data mining techniques and methodology, to estimate the case fatality rate, positivity index and identify a typology according to the severity of the infection identified in patients who present a positive result. for (SARS-CoV-2) COVID-19. From the second data source, information is obtained worldwide on the new variants and mutations of COVID-19 (SARS-CoV-2), providing valuable information for timely genomic surveillance. This study analyzes the impact of (SARS-CoV-2) COVID-19 on the indigenous language-speaking population, it allows us to provide information, quickly and in a timely manner, to support the design of public policy on health.

The NAHU$^2$ project is a Franco-Mexican collaboration aimed at building the $\pi$-YALLI corpus adapted to machine learning, which will subsequently be used to develop computer resources for the Nahuatl language. Nahuatl is a language with few computational resources, even though it is a living language spoken by around 2 million people. We have decided to build $\pi$-YALLI, a corpus that will enable to carry out research on Nahuatl in order to develop Language Models (LM), whether dynamic or not, which will make it possible to in turn enable the development of Natural Language Processing (NLP) tools such as: a) a grapheme unifier, b) a word segmenter, c) a POS grammatical analyser, d) a content-based Automatic Text Summarization; and possibly, e) a translator translator (probabilistic or learning-based).

Some time ago has been studied mathematical models for biogas production due to its importance in the use of control and optimization of re\-new\-able resources and clean energy. In this paper we combine two algebraic methods to obtain solutions of Abel equation of first kind that arise from a mathematical model to biogas production formulated in France on 2001. The aim of this paper is obtain Liouvillian solutions of Abel's equations through Hamiltonian Algebrization. As an illustration, we present graphics of solutions for Abel equations and solutions for algebrized Abel equations.

We present a model--independent reconstruction of the normalized dark energy density function, $X(z) \equiv \rho_{\mathrm{de}}(z)/\rho_{\mathrm{de}}(0)$, derived directly from the DES-SN5YR Type~Ia supernova sample. The analysis employs an inversion formalism that relates the derivative of the distance modulus, $\mu^{\prime}(z)$, to the expansion history, allowing the data to determine the shape of $X(z)$ without assuming a specific equation--of--state or dark energy density parameterization. A statistically optimized binning of the supernova sample (using 17 intervals following the Freedman--Diaconis criterion and 34 following Scott's rule) ensures a stable estimation of $\mu^{\prime}(z)$ and a controlled propagation of uncertainties throughout the inversion process. The resulting $X(z)$ remains statistically consistent with a constant value within one standard deviation across the entire redshift range, showing no significant evidence for an evolving dark energy component at present. In a direct comparison among $\Lambda$CDM, CPL, and the quadratic $X^2(z)$ parameterization -- where CPL and $X^2(z)$ each introduce two additional free parameters relative to $\Lambda$CDM -- the CPL model attains the best statistical agreement with the data, albeit only marginally and strictly within this restricted model set. These outcomes indicate that current observations are compatible with an almost constant dark energy density ($w \simeq -1$), while the inversion framework remains sensitive to subtle departures that forthcoming high--precision surveys could resolve.

The importance of the working document is that it allows the analysis of the information and the status of cases associated with (SARS-CoV-2) COVID-19 as open data at the municipal, state and national level, with a daily record of patients, according to a age, sex, comorbidities, for the condition of (SARS-CoV-2) COVID-19 according to the following characteristics: a) Positive, b) Negative, c) Suspicious. Likewise, it presents information related to the identification of an outpatient and / or hospitalized patient, attending to their medical development, identifying: a) Recovered, b) Deaths and c) Active, in Phase 3 and Phase 4, in the five main population areas speaker of indigenous language in the State of Veracruz - Mexico. The data analysis is carried out through the application of a data mining algorithm, which provides the information, fast and timely, required for the estimation of Medical Care Scenarios of (SARS-CoV-2) COVID-19, as well as for know the impact on the indigenous language-speaking population in Mexico.

We develop a covariant scheme to describe the dynamics of small perturbations on Lovelock type extended objects propagating in a flat Minkowski spacetime. The higher-dimensional analogue of the Jacobi equation in this theory becomes a wave type equation for a scalar field $\Phi$. Whithin this framework, we analyse the stability of membranes with a de Sitter geometry where we find that the Jacobi equation specializes to a Klein-Gordon (KG) equation for $\Phi$ possessing a tachyonic mass. This shows that, to some extent, these type of extended objects share the symmetries of the Dirac-Nambu-Goto (DNG) action which is by no means coincidental because the DNG model is the simplest included in this type of gravity.

It is well known that the equation of state (EoS) of compact objects like neutron and quark stars is not determined despite there are several sophisticated models to describe it. From the electromagnetic observations, summarized in \cite{Lattimer01}, and the recent observation of gravitational waves from binary neutron star inspiral GW170817 \cite{Abbott2017_etal} and GW190425 \cite{Abbott2019}, it is possible to make an estimation of the range of masses and so constraint the mass of the neutron and quark stars, determining not only the best approximation for the EoS, but which kind of stars we would be observing. In this paper we explore several configurations of neutron stars assuming a simple polytropic equation of state, using a single layer model without crust. In particular, when the EoS depends on the mass rest density, $p=K \rho_{0}^{\Gamma}$, and when it depends on the energy density $p=K \rho^{\Gamma}$, considerable differences in the mass-radius relationships are found. On the other hand, we also explore quark stars models using the MIT bag EoS for different values of the vacuum energy density $B$.

Quantum chaos refers to signatures of classical chaos found in the quantum domain. Recently, it has become common to equate the exponential behavior of out-of-time order correlators (OTOCs) with quantum chaos. The quantum-classical correspondence between the OTOC exponential growth and chaos in the classical limit has indeed been corroborated theoretically for some systems and there are several projects to do the same experimentally. The Dicke model, in particular, which has a regular and a chaotic regime, is currently under intense investigation by experiments with trapped ions. We show, however, that for experimentally accessible parameters, OTOCs can grow exponentially also when the Dicke model is in the regular regime. The same holds for the Lipkin-Meshkov-Glick model, which is integrable and also experimentally realizable. The exponential behavior in these cases are due to unstable stationary points, not to chaos.

We present an exactly-solvable $p$-wave pairing model for two bosonic species. The model is solvable in any spatial dimension and shares some commonalities with the $p + ip$ Richardson-Gaudin fermionic model, such as a third order quantum phase transition. However, contrary to the fermionic case, in the bosonic model the transition separates a gapless fragmented singlet pair condensate from a pair Bose superfluid, and the exact eigenstate at the quantum critical point is a pair condensate analogous to the fermionic Moore-Read state.

Using coherent states as initial states, we investigate the quantum dynamics of the Lipkin-Meshkov-Glick (LMG) and Dicke models in the semi-classical limit. They are representative models of bounded systems with one- and two-degrees of freedom, respectively. The first model is integrable, while the second one has both regular and chaotic regimes. Our analysis is based on the survival probability. Within the regular regime, the energy distribution of the initial coherent states consists of quasi-harmonic sub-sequences of energies with Gaussian weights. This allows for the derivation of analytical expressions that accurately describe the entire evolution of the survival probability, from $t=0$ to the saturation of the dynamics. The evolution shows decaying oscillations with a rate that depends on the anharmonicity of the spectrum and, in the case of the Dicke model, on interference terms coming from the simultaneous excitation of its two-degrees of freedom. As we move away from the regular regime, the complexity of the survival probability is shown to be closely connected with the properties of the corresponding classical phase space. Our approach has broad applicability, since its central assumptions are not particular of the studied models.