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We establish a compactness result for solutions of a certain class of hypoelliptic equations. This result allows us to show the existence of global weak solutions to the non-homogeneous Landau-Fermi-Dirac equation with Coulomb potential.

We compare ambient and outer Lipschitz geometry of Lipschitz normally embedded H\"older triangles in $\mathbb{R}^4$. In contrast to the case of $\mathbb{R}^3$ there are infinitely many equivalence classes. The equivalence classes are related to the so-called microknots.

The decays of the Z boson and CP-even or CP-odd scalar bosons into quark-antiquark pairs have been calculated at NLO in the framework of Implicit Regularization (IReg) , which operates strictly in the physical dimension and complies with the BPHZ procedure. The presence of the {\gamma}5 matrix is dealt without the need of gauge symmetry restoring counterterms and the Kinoshita-Lee-Nauenberg (KLN) theorem is verified. The results are compared to the ones obtained in the Dimensional Reduction scheme (DRed)., Comment: presented in the eQCD 2024 conference Acta Physica Polonica B Proceedings Supplement 17, 6-A14 (2024)

The Moderately Discontinuous Homology (MD-Homology, for short) was created recently in 2022 by Fern\'andez de Bobadilla at al. and it captures deep Lipschitz phenomena. However, to become a definitive powerful tool, it must be widely comprehended. In this paper, we investigate the MD-Homology of definable surface germs for the inner and outer metrics. We completely determine the MD-Homology of surfaces for the inner metric and we present a great variety of interesting MD-Homology of surfaces for the outer metric, for instance, we determine the MD-Homology of some bubbles, snake surfaces, and horns. Furthermore, we explicit the diversity of MD-Homology of surfaces for the outer metric in general, showing how hard it is to completely solve the outer classification problem. On the other hand, we show that, under specific conditions, the weakly outer Lipschitz equivalence determines completely the MD-Homology of surfaces for the outer metric, showing that these two subjects are quite related., Comment: 36 pages

We analyze the network of transactions among cities based on the electronic invoice database for the municipalities in the Cear\'a state, Brazil. It consists of approximately 3.7 billion records, registered during the period between the years 2016 to 2019. All the transactions are grouped in a unique dataset and represented as an asymmetrical adjacency matrix corresponding to a directed graph with connections weighted by the number of transactions among cities. Due to the large size of Cear\'a state, its unequal distribution of wealth, and spatially heterogeneous population, we initially determine communities of cities based on their mutual intensity of trades and verify to which extent their economic interests reflect a community cohesiveness. For the first task, we use the Infomap algorithm to detect the partition which provides the shortest description length and captures the optimal community structure of the network in terms of its associated flow dynamics. Surprisingly, the partition identified has five modules, whose two-dimensional geographical projections are all simply-connected domains. We proceed with the analysis of traded products by building bipartite structures represented in terms of adjacency matrices between municipalities and products, considering selling and buying. Using the revealed comparative advantage (RCA) concept, we define a non-monetary and binary activity index that can distinguish the RCA of a city in a class of goods or services as evidenced by trade flows. Finally, through the pairwise Maximum Entropy Model, we can associate to the largest communities their corresponding binary Ising-like Hamiltonian models. In an analogy with critical phenomena, our results reveal that each community operates at a "temperature" that is close to the corresponding "critical point", suggesting a high degree of "economic cohesiveness" in its trade network of cities., Comment: 29 pages, 10 figures

In this note, we present some results on the Mattei's conjecture, which asserts that the algebraic multiplicity of one-dimensional holomorphic foliations is a topological invariant. For instance, we establish a version of the A'Campo-L\^e Theorem adapted for foliations, which was originally formulated for hypersurfaces. More precisely, we show that, under certain conditions, the algebraic multiplicity equals to 1 of holomorphic one-dimensional foliations is a topological invariant. As a consequence, we prove that saddle-nodes are also topological invariant. Additionally, we address a fundamental result in the literature concerning Mattei's conjecture proved by Rosas-Bazan, which says that under the existence of a homeomorphism extending throughout a neighborhood of the exceptional divisor of the first blowing up, Mattei's conjecture has a positive answer. As a consequence of a characterization of the invariance of multiplicity proved here, we obtain that Mattei's conjecture has a positive answer if the homeomorphism extends locally in a small neighborhood where there is a separatrix not contained in the exceptional divisor of the first blowing up, but not necessarily extends throughout a neighborhood of whole exceptional divisor., Comment: 14 pages

Determining the carrying capacity of urban tourism destinations and public spaces is essential for sustainable management. This paper presents an online tool that calculates pedestrian carrying capacities for user-defined areas based on OpenStreetMap (OSM) data. The tool considers physical, real, and effective carrying capacities by incorporating parameters such as area per pedestrian, rotation factor, corrective factors, and management capacity. The carrying capacity calculator aids in balancing environmental, economic, social, and experiential factors to prevent overcrowding and preserve the quality of life for residents and visitors. This tool is particularly useful for tourism destination management, urban planning, and event management, ensuring positive visitor experiences and sustainable infrastructure development. We detail the implementation of the calculator, its underlying algorithm, and its application to the Santa Maria Maior parish in Lisbon, highlighting its effectiveness in managing urban tourism and public spaces.

Galaxies infalling into clusters undergo both star-formation quenching and morphological transformation due to environmental effects. We investigate these processes and their timescales using a local sample of 20,191 cluster and 11,674 field galaxies from SDSS. By analysing morphology as a function of distance from the star-formation main sequence, we show that environmental influence is especially pronounced for low-mass galaxies, which emerge from the green valley with early-type morphologies before their star formation is fully suppressed. Using the galaxies' positions in the clusters' Projected Phase Space, we examine the evolution of blue cloud, green valley, and red sequence fractions as a function of time since infall. Interestingly, the green valley fraction remains constant with time since infall, suggesting a balanced flow of galaxies in and out of this class. We estimate that galaxies less massive than $10^{10}\rm M_{\odot}$ spend approximately 0.4 Gyr in the green valley. By comparing quenched and early-type populations, we provide further evidence for the ``slow-then-rapid'' quenching model and suggest that it can also be applied to morphological transitions. Our results indicate that morphological transformation occurs at larger radii than complete star-formation quenching. About 75% of galaxies undergoing morphological transition in clusters are spirals evolving into S0s, suggesting that infalling galaxies retain their disks, while massive ellipticals are relics of early merger events. Finally, we show it takes approximately 2.5 and 1.2 Gyr after the delay-time ($\sim 3.8 {\rm Gyr}$) for the population of low mass galaxies in clusters to reach a 50% threshold in quenched and early-type fraction, respectively. These findings suggest morphological transition precedes full star formation quenching, with both processes possibly being causally linked., Comment: 14 pages, 10 figures

Strongly correlated electrons systems may exhibit a variety of interesting phenomena, for instance, superconductivity and pseudogap, as is the case of cuprates and pnictides. In strongly correlated electron systems, it is considered essential to understand, not only the nature of the pseudogap, but also the relationship between superconductivity and the pseudogap. In order to address this question, in the present work, we investigated a one-band Hubbard model treated by the Green's function method within an n-pole approximation. In the strongly correlated regime, antiferromagnetic correlations give rise to nearly flat band regions in the nodal points of the quasiparticle bands. As a consequence, a pseudogap emerges at the antinodal points of the Fermi surface. The obtained results indicate that the same antiferromagnetic correlations responsible for a pseudogap, can also favor superconductivity, providing an increase in the superconducting critical temperature Tc., Comment: 16 pages, 9 figures

We investigate the formation of particular patterns when light passes through glass. Experimentally, we use various glass plates, registering each reflected and transmitted outcome. Thus, we find the condition to form a common pattern, namely, randomly scratched plates produce the halos. We engineer other optical patterns by means of the specular holography method. Using the Defocusing Microscopy technique, we study the geometric properties of the glass, establishing the physical and mathematical model for a simulation. Our computational results agree well with the experimental ones, leading us to conclude that the pattern formation is governed by specular reflection on various points along the scratches. Our findings give an initial explanation for the pattern formation on glass, in particular for the halo design.

The burgeoning influence of Large Language Models (LLMs) in shaping public discourse and decision-making underscores the imperative to address inherent biases within these AI systems. In the wake of AI's expansive integration across sectors, addressing racial bias in LLMs has never been more critical. This paper introduces a novel framework called Comprehensive Bias Neutralization Framework (CBNF) which embodies an innovative approach to quantifying and mitigating biases within LLMs. Our framework combines the Large Language Model Bias Index (LLMBI) [Oketunji, A., Anas, M., Saina, D., (2023)] and Bias removaL with No Demographics (BLIND) [Orgad, H., Belinkov, Y. (2023)] methodologies to create a new metric called Bias Intelligence Quotient (BiQ)which detects, measures, and mitigates racial bias in LLMs without reliance on demographic annotations. By introducing a new metric called BiQ that enhances LLMBI with additional fairness metrics, CBNF offers a multi-dimensional metric for bias assessment, underscoring the necessity of a nuanced approach to fairness in AI [Mehrabi et al., 2021]. This paper presents a detailed analysis of Latimer AI (a language model incrementally trained on black history and culture) in comparison to ChatGPT 3.5, illustrating Latimer AI's efficacy in detecting racial, cultural, and gender biases through targeted training and refined bias mitigation strategies [Latimer & Bender, 2023]., Comment: 41 pages

We examine the subtleties of regularization schemes in four-dimensional space ($4S$), related in particular to the introduction of the $\gamma_5$ matrix. To illustrate we use a "Bumblebee" model featuring dynamically induced Lorentz symmetry violation. The analysis centers on how different regularization methods affect the solutions to the gap equation in this model. We highlight the resolution of ambiguities associated with the $\gamma_5$ matrix in ultraviolet divergent integrals by employing an enhanced Implicit Regularization (IREG) method. This method extends IREG to a quasi-four-dimensional space, $Q4S = 4S \oplus X$, drawing parallels with the consistent approach of Dimensional Reduction (DRED). Comparative analysis is conducted against results from the 't Hooft-Veltman regularization scheme, conventional IREG in strict $4S$, and sharp momentum cutoff techniques. Our results illustrate a scheme to compute $\gamma_5$ interactions in physical dimension of divergent amplitudes, confirming the approach in [1].

We explore the effects of Markovian bath coupling and initial position-momentum correlations on the coherence and purity of Gaussian quantum states. Our analysis focuses on the roles these factors play in the dynamics of quantum coherence, coherence lengths, and state purity. Our results reveal that initial position-momentum correlations have a remarkable impact on the quantum properties of the mixed state. These correlations lead to opposing behaviors in coherence and purity: as quantum coherence increases in response to stronger correlations, purity diminishes, and vice versa. This inverse relationship illustrates the phenomenon where, governed by these initial correlations, a state with greater mixing can display enhanced quantum coherence compared to a less mixed state. We also observe an unanticipated coherence freezing phenomenon, quantified by the relative entropy of coherence. Notably, this freezing is driven by initial position-momentum correlations, although the final frozen value is independent of these correlations.

For a foliation by CMC hypersurfaces on a complete Riemannian manifold $M^{n+1}$ with sectional curvature bounded from below by $-nK_0\leq 0$ and such that the mean curvature $H$ of the leaves of the foliation satisfies $|H|\geq \sqrt{K_0}$, under certain additional hypotheses, for instance, if $M$ is a compact (without boundary) or if ${\rm div} (\nabla_N N)=0$ where the gradient of the function ${\rm div} (N)$ is non zero, then we prove that $|H|\equiv \sqrt{K_0}$, where $N$ is a unit vector field orthogonal to the foliation. Moreover, in the the case that $M$ is compact, we prove also that all the leaves are totally umbilical. This gives, in particular, a generalization for the result proved by Barbosa, Kenmotsu and Oshikiri (1991), where was proved the above result in the case $K_0=0$ and $M$ compact. Under the same additional hypotheses as above, we obtain that for a foliation by CMC hypersurfaces on a complete Riemannian manifold $M$ with Ricci curvature bounded from below by $-nK_0\leq 0$, the mean curvature $H$ of the leaves of the foliation satisfies $|H|\leq \sqrt{K_0}$. This gives a positive partial answer to a conjecture due to Meeks III, P\'erez and Ros. We also provide some partial answers to several other problems., Comment: We have found a mistake in our proof. Our proof only works if we add some hypothesis on the foliation, for instance, if N(H) is constant on the leaves or div(\nabla_N N)=0 where the gradient of H is not zero. Thus, the conjectures are still open. We slightly rewrote the text and changed the title. Subsection 4.1 and Section 5 are new. 31 pages

In this paper, we prove metric analogues, in any dimension and in any co-dimension, of the famous Theorem of Mumford on smoothness of normal surfaces and the beautiful Theorem of Ramanujam that gives a topological characterization of $\mathbb{C}^2$ as an algebraic surface. For instance, we prove that a complex analytic set that is log-Lipschitz regular at 0 (i.e., a complex analytic set that has a neighbourhood of the origin which bi-log-Lipschitz homeomorphic to an Euclidean ball) must be smooth at 0. We prove even more, we prove that if a complex analytic set $X$ such that, for each $0<\alpha<1$, $(X,0)$ and $(\mathbb{R}^k,0)$ are bi-$\alpha$-H\"older homeomorphic, then $X$ must be smooth at 0. These results generalize the Lipschitz Regularity Theorem, which says that a Lipschitz regular complex analytic set must be smooth. Global versions of these results are also presented here and, in particular, we obtain a characterization of an affine linear subspace as a pure-dimensional entire complex analytic set., Comment: Typos and a mistake in Proposition 3.2 have been corrected. 23 pages

We show that two bi-Lipschitz equivalent Brieskorn-Pham hypersurfaces have the same multiplicities at $0$. Moreover we show that if two algebraic $(n-1)$-dimensional cones $P, R\subset\mathbb C^n$ with isolated singularities are homeomorphic, then they have the same degree., Comment: 5 pages

We investigate the role that dense environments have on the quenching of star formation and the transformation of morphology for a sample of galaxies selected from the Sloan Digital Sky Survey. We make a distinction between galaxies falling into groups $(13 \leq \log{(M_{\text{halo}}/M_{\odot})} < 14)$ and clusters $(\log{(M_{\text{halo}}/M_{\odot})} \geq 14)$, and compare to a large sample of field galaxies. Using galaxy position in projected phase space as a proxy for time since infall, we study how galaxy specific star formation rate (sSFR) and morphology, parameterized by the bulge-to-total light ratio (B/T), change over time. After controlling for stellar mass, we find clear trends of increasing quenched and elliptical fractions as functions of infall time for galaxies falling into both groups and clusters. The trends are strongest for low mass galaxies falling into clusters. By computing quenching and morphological transformation timescales, we find evidence that star formation quenching occurs faster than morphological transformation in both environments. Comparing field galaxies to recently infalling galaxies, we determine there is pre-processing of both star formation and morphology, with pre-processing affecting star formation rates more strongly. Our analysis favours quenching mechanisms that act quickly to suppress star formation, while other mechanisms that act on longer timescales transform morphology through bulge growth and disc fading., Comment: 14 pages, 12 figures, accepted by MNRAS

Learning to defer (L2D) aims to improve human-AI collaboration systems by learning how to defer decisions to humans when they are more likely to be correct than an ML classifier. Existing research in L2D overlooks key real-world aspects that impede its practical adoption, namely: i) neglecting cost-sensitive scenarios, where type I and type II errors have different costs; ii) requiring concurrent human predictions for every instance of the training dataset; and iii) not dealing with human work-capacity constraints. To address these issues, we propose the \textit{deferral under cost and capacity constraints framework} (DeCCaF). DeCCaF is a novel L2D approach, employing supervised learning to model the probability of human error under less restrictive data requirements (only one expert prediction per instance) and using constraint programming to globally minimize the error cost, subject to workload limitations. We test DeCCaF in a series of cost-sensitive fraud detection scenarios with different teams of 9 synthetic fraud analysts, with individual work-capacity constraints. The results demonstrate that our approach performs significantly better than the baselines in a wide array of scenarios, achieving an average $8.4\%$ reduction in the misclassification cost. The code used for the experiments is available at https://github.com/feedzai/deccaf