We estimate the lattice sums arising in the context of the integer point counting in polyhedra.

In light of the rise of transition metal dichalcogenides as 2D semiconductors for device applications, band engineering becomes very important from an application point of view. In many of these materials, such as the canonical example of MoS$_{2}$, the semiconductor band gap depends on the layer number. It changes from indirect to direct as it evolves from a bulk semiconductor to a monolayer. Interestingly, it was predicted and experimentally confirmed that, by thinning the material from bulk to a bilayer, the indirect transition shows a strong blue-shift. Here, we present the results of scanning tunnelling spectroscopy measurements on MoS$_{2}$ that has been grown \textit{in situ} via molecular beam epitaxy on graphene on Ir(111) at thicknesses ranging from 1 to 5 layers. We find a drastic decrease of the band gap with increasing layer number, to values even below the band gap in bulk. We also observe that the pinning of the conduction band vanishes above 4 layers. Comparing our experimental data with density functional theory and \textit{GW} calculations indicates that an additional screening is introduced by the sample growth conditions.

It is often stated that a phase of standard, decelerated cosmological expansion is characterised by the absence of global event horizons, while a phase of accelerated expansion is associated with the absence of particle horizons. This is not necessarily true because such horizons, being non-local properties of the spacetime geometry, depend on the full (past and future) history of the given cosmological background. We provide examples of various different scenarios for the case in which the final asymptotic phase of standard expansion and decreasing curvature is connected, through a regular bounce, with an initial (and possibly infinitely extended in time) regime of growing curvature.

Sulfur is one of the most abundant elements in the Universe, yet the sulfur budget inferred from the observed sulfur-bearing molecules in dense cores is significantly lower than expected. Starless and pre-stellar cores represent the earliest stages of star formation and provide a laboratory for studying the physical and chemical processes that cause sulfur depletion. We aim to constrain sulfur chemistry in dense cores by measuring abundances of sulfur-bearing molecules and how they reflect core evolution and environmental effects. We observed nine cores in the Taurus Molecular Cloud, targeting 13 sulfur-bearing molecules, including CS, CCS, C$_3$S, OCS, SO, SO$_2$, H$_2$CS, and isotopologs. Molecular abundances and six abundance ratios were compared to three evolutionary tracers: H$_2$ column density, N$_2$D$^+$/N$_2$H$^+$, and the CO depletion factor. We also compared observations with 0D chemical models with different initial sulfur abundances. We find variations in abundances across cores. L1517B exhibits low abundances and a high depletion factor, whereas L1495B shows enhanced levels in oxygen-bearing species within the L1495 filament. Ratios tracing carbon- and oxygen-bearing species (CCS/$^{34}$SO and C$^{34}$S/$^{34}$SO) decrease with increasing H$_2$ column density and N$_2$D$^+$/N$_2$H$^+$ ratio. Other species and ratios show weak or no correlation with tracers. Models reproduce OCS, H$_2$CS, and HDCS reasonably well, but not all species simultaneously, especially between carbon- and oxygen-bearing molecules. The variations and lack of consistent correlations suggest that a single evolutionary parameter cannot describe sulfur chemistry and that the local environmental conditions strongly influence the observed abundances. Reproducing the full sample of sulfur-bearing molecules would require improved chemical networks and models that account for the core's physical structure.

Artificial general intelligence (AGI)--defined here as AI systems that match or exceed humans at most economically useful cognitive work--has moved from speculation to the centre of political and strategic debate. This paper examines three questions: how soon AGI might emerge, how it could reshape geopolitics, and whether Europe is adequately prepared. Drawing on empirical trends in AI capabilities, expert forecasting surveys, and policy analysis, we find that a plausible window for AGI emergence falls between 2030 and 2040, or potentially earlier, though substantial uncertainty remains. Our analysis of the geopolitical implications suggests that AGI could fundamentally alter the global distribution of economic and military power, intensify interstate competition, and strain existing governance frameworks. Assessing Europe's current positioning, we identify critical gaps: limited strategic awareness of frontier AI progress, structural weaknesses in compute infrastructure and talent retention, low rates of industrial AI adoption, and fragmented policy responses at both EU and Member State levels that do not match the potential scale of disruption.These findings point to a need for a coordinated European preparedness agenda. We outline policy options centred on building institutional capacity for AGI situational awareness, strengthening Europe's position in the AI value chain, and developing frameworks for international stability in an era of increasingly capable AI systems.

This study investigates the role of thermal boundary conditions on natural convection and entropy generation in non-Newtonian power-law fluids confined within a square cavity and a concentric cylindrical annulus. Steady, two-dimensional governing equations based on the incompressible power-law model and the Boussinesq approximation are solved using the Gridap.jl finite element framework. The numerical methodology is validated against benchmark solutions for both Newtonian and non-Newtonian convection, showing good agreement in terms of isotherm fields, streamlines, local Nusselt number distributions, and entropy generation. The effects of fluid rheology and heating mode are examined for shear-thinning, Newtonian, and shear-thickening fluids under uniform and non-uniform thermal boundary conditions. The results show that shear-thinning behavior enhances buoyancy-driven circulation, steepens thermal gradients, and increases heat transfer, whereas shear-thickening behavior suppresses convection and promotes conduction-dominated transport. Thermal boundary conditions are found to play an important role in controlling the intensity and spatial distribution of flow, heat transfer, and irreversibility. In both geometries, uniform heating produces stronger and more distributed convective structures, while non-uniform sinusoidal heating localizes thermal forcing and consistently reduces total entropy generation. An entropy analysis further reveals that viscous dissipation dominates irreversibility in shear-thinning fluids, whereas heat-transfer irreversibility becomes dominant as the power-law index increases. The study demonstrates that appropriate thermal boundary design, together with fluid rheology, provides an effective route for controlling heat transfer and minimizing thermodynamic losses in non-Newtonian convection systems. The source code and metadata are publicly available.

This study significantly advances multi-texture synthesis using Neural Cellular Automata (NCAs) by introducing a novel training methodology that enables robust self-regeneration of textures in damaged regions. This inherent healing mechanism, essential for dynamic and adaptive systems, extends beyond traditional computer graphics applications, highlighting the fundamental self-organizing properties of NCAs. Furthermore, we present a versatile grafting technique, enabling the seamless combination of distinct textures. This is achieved efficiently during the inference phase, without requiring specialized retraining, through precise initialization of the NCA's genome channels. Our findings demonstrate the generation of high-quality, complex textures with fluid transitions, showcasing a powerful and efficient paradigm for dynamic texture composition and self-repair in autonomous systems.

We investigate the orbital stability of black solitons for a broad class of quasilinear Schrödinger equations in one space dimension, with nonzero boundary conditions at infinity. Namely, our framework handles general defocusing semilinear nonlinearities and focusing or defocusing quasilinear nonlinearities. First, we establish sufficient conditions on the quasi-linear nonlinearities ensuring the existence of a local branch of finite-energy solitons parameterized by their speed. Within this branch, the black soliton, also called kink, corresponds to the stationary solution. Our main result is the orbital stability of the black soliton in the energy space, provided that the Vakhitov-Kolokolov (VK) slope condition holds; namely, that the derivative of the momentum with respect to the speed is negative at zero. Moreover, we derive an explicit formula for verifying this VK condition. The proof relies on the analysis of a carefully designed variational problem, which allows us to control the sup-norm of the evolution of a perturbation of the kink in terms of the energy and momentum, both of which are conserved by the flow. A delicate part of the argument is the analysis of minimizing sequences for this variational problem, since the infimum is not attained.

In this note, we show how to adapt Tao's slice rank method to extend the Ellenberg--Gijswijt theorem on cap sets to the problem of forbidding arithmetic progressions with restricted differences. In particular, we show that if $q$ is an odd prime power, there is $\varepsilon_q>0$ such that if $S \subseteq \mathbb{F}_q$ with $0 \in S$ and $|S|>(q+1)/2$ and $A \subseteq \mathbb{F}_q^n$ contains no three-term arithmetic progression whose common difference is in $S^n$, then $|A| \leq q^{(1-\varepsilon_q)n}$.

Thermodynamic geometry of two-dimensional fluids has been investigated using a square-well model as a prototype fluid. A comparison with the three-dimensional case is performed in the subcritical and supercritical domains of thermodynamic space. In the subcritical region, it is found that the R-crossing method has a narrower range of validity for two-dimensional fluids compared to three-dimensional ones. On the other hand, in the supercritical region, an analysis of different Widom lines, including the R Widom line, shows that for two-dimensional fluids these lines extend further into the supercritical region than their three-dimensional counterparts. A similar behavior is observed for the validity of the Clausius--Clapeyron equation in two-dimensional fluids.

This paper studies the zero-divisor graphs attached to several finite chain-ring families and computes the homological invariants of their edge ideals by using cochordal constructible systems. We begin with a general layered graph $C(q,L)$, whose vertices are arranged according to valuation layers and whose adjacency is governed by the single rule $k+\ell\ge L$, form some integers $k$ and $\ell$. This graph models the zero-divisor structure of a finite chain ring with residue field of order $q$ and nilpotency index $L$. We prove that $C(q,L)$ is cochordal, determine its type sequence, then correct and refine the Betti formula of its edge ideal [Dung and Vu, Cochordal zero divisor graphs and Betti numbers of their edge ideals, Comm. Algebra 54(2) (2026) 736--744]. The results are then specialized to the Gaussian quotient rings $\mathbb Z_{2^m}[i]$ and to the truncated polynomial rings $\mathbb Z_p[x]/(x^c)$. We compute projective dimension, regularity, independence number, height, Hilbert series, and Cohen--Macaulay behavior. The computations show that these quotient rings have $2$-linear resolutions, while Cohen--Macaulayness occurs only in the expected degenerate or complete-graph cases.

We consider inverse problems with linear forward models and Gaussian priors, but with unknown hyperparameters that may arise from the model, the noise, or the specification of the prior. We model this using a hierarchical Bayes framework resulting in a posterior distribution that is non-Gaussian, in general, and challenging to sample from. Consequently, we use an empirical Bayes framework for estimating the maximum a posteriori estimate of the hyperpameters by considering the marginalized posterior distribution. However, the optimization problem is also computationally challenging due to the need for repeated evaluation of log determinants. To address this issue, we propose a Majorization-Minimization with Monte Carlo approach, which we call M$^{3}$C, for hyperparameter estimation. Specifically, we replace the challenging optimization problem with a sequence of simpler ones by utilizing a majorization function (or majorant) for the log-determinant term, combined with a Monte Carlo estimator to approximate the majorant. We provide theoretical results, showing that under certain assumptions, the M$^{3}$C iterates converge with high probability to a critical point of the original cost function. A variety of numerical examples are provided from seismic tomography, super-resolution imaging, and contaminant source identification.

In this article, we investigate the combinatorial and algebraic properties of the lcm-lattice associated with the edge ideal of a hypergraph. Let $\H$ be a hypergraph, $I(\H)$ its corresponding edge ideal in a polynomial ring in $n$ variables, and $\mathrm{Icm}(I(\H))$ the associated lcm-lattice. We establish conditions under which the lcm-lattice of an edge ideal is Boolean, modular, or complemented. Furthermore, we extend these results to the case of the product of lcm-lattices in the complemented case. Additionally, we study the effects of polarization on the lcm-lattices of $I(\H)$ and its polarized ideal.

This paper introduces several techniques that improve the scalability of the deductive verification of data-level programs working on arrays and matrices. First of all, we introduce a technique to rewrite expressions with (nested) quantifiers, so suitable triggers can be generated for these expressions. We have proven this rewrite technique correct in a theorem prover. Second, we make reasoning about potentially overlapping arrays easier, by providing specification constructs to indicate and verify that two arrays are not aliases, or that they are immutable, so they can be modelled as mathematical sequences. All our techniques are implemented in the VerCors program verifier. We illustrate how our techniques improve scalability through a large number of experiments. Using our techniques on a set of typical GPU kernels, we achieve a reduction of verification time by, on average, a factor of 9, with outliers being up to 150 times faster. Additionally, applying these techniques to earlier experiments and an earlier case study of a radio telescope pipeline permitted the verification of results which were previously unobtainable and significantly reduced the verification time.

We develop the ARE method (Action-Rectification-Expansion), a structural framework for the organization of Leibniz terms in determinants through cyclic group actions and orbital decompositions. The symmetric group S_n is partitioned into (n-1)! disjoint orbits of size n under right composition by the cyclic group C_n. Each orbit admits a canonical representative and generates a family of determinant terms related by cyclic rotation. We prove explicit sign laws for orbital rotations, establish a rectification theorem transforming orbital polylines into parallel-line configurations through a single block permutation, and characterize companion orbitals through dihedral symmetries. The framework yields an exact reorganization of the Leibniz expansion preserving all n! terms while exposing hidden geometric and combinatorial structure. We further prove an impossibility theorem showing that no fixed-width direct extension of the classical Sarrus rule can capture all determinant terms for n >= 4. The method provides three equivalent visualizations: polylines, parallel rectified lines, and total-line representations. Deterministic orbital generation algorithms and computational verification against standard determinant methods are also presented. Although the approach does not reduce factorial complexity, it provides a systematic geometric and algebraic interpretation of determinant structure extending the conceptual spirit of Sarrus to arbitrary dimension.

A first dedicated search for pair production of new scalars predicted by the Inert Doublet Model is performed using proton-proton collisions. Data were collected with the CMS detector at the CERN LHC at $\sqrt{s}$ = 13 TeV and 13.6 TeV, corresponding to integrated luminosities of 138 fb$^{-1}$ and 35 fb$^{-1}$, respectively. Within this model, four additional scalar bosons (H, A, H$^+$, and H$^-$) are predicted. Through an additional discrete symmetry, the lightest new scalar, H, is stable, rendering it a viable dark matter candidate. These candidates can originate from quark-antiquark annihilation producing an offshell Z boson that decays to a pair of the new scalars. The target final state consists of exactly two opposite-charge same-flavour leptons (electrons or muons), with missing transverse momentum due to the stable neutral scalars, and very little hadronic activity. A parameterised neural network is used to separate the signal from the standard model background. No significant excess of events is observed. Exclusion limits at 95% confidence level are set on the production cross section of the two new neutral scalars, H and A, expressed in terms of their masses, $m_\mathrm{H}$ and $m_\mathrm{A}$, in the $m_\mathrm{H}$ vs. $m_\mathrm{A}$ plane. The observed (expected) exclusion region reaches $m_\mathrm{H}$ = 108 (106) GeV for $m_\mathrm{H}-m_\mathrm{A}$ = 78 (76) GeV and at $m_\mathrm{H}$ = 70 GeV, covers the range of $m_\mathrm{H}-m_\mathrm{A}$ = 40$-$90 (35$-$90) GeV.

This work presents a systematic study of the magnetocaloric effect in the nanostructured perovskite series $La_{0.6}Sr_{0.4}Fe_{1-x}Co_{x}O_3$ (x = 0, 0.2, 0.5, 0.8, and 1.0), synthesized by a pore-wetting method using polymeric membranes with pore diameters of 200 nm and 800 nm. All samples were calcined at 1000°C. Structural characterization was made by X-ray diffraction and confirmed the formation of a single-phase perovskite with distorted rhombohedral symmetry, without detectable secondary phases. We observed significant influence of substitution of Fe by Co on the morphology, as the analysis by scanning electron microscopy revealed a clear evolution from smaller to larger particles and from thin to thicker nanotubes, as the Co content increased. Magnetic measurements showed that the cationic substitution enhances ferromagnetic coupling, increasing both the saturation magnetization (MS) and the Curie temperature (TC). The magnetocaloric properties, determined through the Maxwell relations, exhibit a maximum entropy change of 1.13 J/(kg K) under an applied field of 3 T for the sample with x = 1. These results demonstrate that the combination of Co doping and controlled nanostructuring effectively optimizes the magnetocaloric response.

Let $G$ a bipartite graph with vertex bipartition $\{A,B\}$ and let $m=|E(G)|$. An $(A,B)$-uniformly ordered labeling of $G$ is a labeling $f\colon V\rightarrow [0,2m]$ which, among other conditions, requires that there exists $λ\in \mathbb N$ such that $f(a)\le λ$ and $f(b)>λ$ for all $a\in A$ and $b\in B$. The existence of such a labeling for $G$ implies the existence of a cyclic $G$-decomposition of $K_{2mx+1}$ for all positive integers $x$. In this paper, as a starting point, through this type of labeling we prove the existence of a cyclic $G$-decomposition in the case that $G$ is a cycle of even length with either one or two pendant paths of any length. Then, through a merging procedure, we are able to get this type of labeling for a specific class of bipartite graphs, which are obtained by iteratively adding an even cycle and a pendant path.

Using linearized elasticity as a convenient mechanical framework, we show that volumetric growth can be formulated as an optimization-driven process in which the growth tensor is determined implicitly by constrained optimization rather than prescribed through phenomenological evolution laws. At each incremental step, the displacement and growth fields satisfy equilibrium, mass-balance constraints, and an irreversibility condition enforcing accretive growth, while an objective functional encodes the driving mechanism of the process. Finite element discretization leads to a finite-dimensional constrained minimization problem in the growth variables alone and makes explicit the interpretation of the evolution as a projected gradient flow. Numerical examples illustrate the proposed framework.

ergodicity is an open-source Python library for computational work on stochastic dynamics, with particular emphasis on non-ergodicity, time-average behavior, heavy-tailed processes, and decision making under uncertainty. The package brings together three layers that are often split across ad hoc scripts: process definitions and simulators, analysis and fitting tools, and agent-based experimentation. This article documents the implemented software rather than presenting new stochastic theory. We describe the package architecture, the supported process families, the analysis workflow, and the practical boundaries of the current implementation. We also provide fully reproducible examples covering heavy-tailed ensemble spread, multiplicative Levy growth diagnostics, adaptive memory mean reversion, preasymptotic fluctuation analysis, and partial stochastic differential equation simulation. The package is positioned as an integration layer on top of the scientific Python stack, reducing the amount of glue code required to move from process specification to diagnostics and comparative experiments.

Understanding origin and evolution of cosmological magnetic fields requires knowledge of magnetic fields in different extragalactic environments. In this context, a powerful tool is the statistical analysis of the Faraday effect on the linear polarization of a sample of radio sources. This effect carries information about the magnetic fields in our Galaxy, extragalactic environments between the sources and the observer, and within the emitting radio source itself. An accurate disentangling of all these components is crucial to characterize magnetic fields in the LSS of the Universe. The significant amount of data delivered by new radio instruments enables the investigation of increasingly weak magnetic fields. However, a trustworthy characterization is only possible with advanced analysis techniques. In this work, we present a new algorithm capable of simultaneously disentangling the Faraday effect due to our Galaxy from extragalactic contributions, by properly taking into account the observing noise. The algorithm takes as an input a catalog of RM complemented by auxiliary information as, e.g., the redshift. We tested the algorithm with synthetic data to assess its performance and identify the range of Galactic magnetic field power spectrum slopes that allows us to properly disentangle Galactic and extragalactic terms. Furthermore, we tested the algorithm with synthetic catalogs, based on m- and cm-data currently available, corresponding to different observing setups, noise, and cuts in the absolute value of the Galactic latitude of the radio sources. Considering noise values and density of polarized sources consistent with existing catalogs, we demonstrated that the most robust results are obtained with sources with absolute Galactic latitude > 45deg, with inference of the extragalactic parameters at most within 5sigma, both for dispersion in Faraday rotation of ~1 and 10rad/m2.

We show that the cohomological invariant $r^\sharp$, introduced in [1] as a lower bound for the off-diagonal holonomy dimension of metric connections with totally skew torsion on product manifolds, predicts the behaviour of the torsion $3$-form under both dimensional reduction and Buscher T-duality. On a product $M = Σ_g \times M_2$ equipped with a product metric, when $r^\sharp = 0$ the parallel-form strata identify a flat circle factor $S^1_β\subset M_2$ via the de Rham splitting theorem, and the entire $H$-flux is converted into geometric flux under T-duality along $S^1_β$ (the parallel regime); when $r^\sharp = 1$, no such circle factor exists, and the $H$-flux survives T-duality along every flat circle factor as $H$-flux in the dual background (the transversely irreducible regime). When $M_2 = N \times T^k$ contains a torus factor, we prove that the Bouwknegt--Evslin--Mathai obstruction to successive T-dualities vanishes automatically for $H$-flux of pure bidegree $(2,1)$, that the resulting dualities are non-interfering and order-independent, and that $r^\sharp$ detects the \emph{irreducible kernel} of the $H$-flux: the component that survives T-duality along every flat circle factor and cannot be converted into geometric or non-geometric flux in any duality frame. This provides a metric refinement of topological T-duality: while the latter disregards the Riemannian metric entirely, $r^\sharp$ detects whether the cohomological coupling is aligned with the flat sub-factors identified by the Levi-Civita parallel-form strata.

We propose a variational framework for accretive surface growth driven by an optimality principle. Rather than prescribing a kinetic law, the configuration at each time step is obtained, within a time-discrete setting, as the solution of a constrained minimization problem. Growth is modeled as an irreversible surface deposition process subject to a global mass constraint, while the driving mechanism is encoded in an objective functional, here taken to be the structural mean compliance. The approach is illustrated on a linearly elastic cantilever beam whose cross-sectional height evolves through layered accretion, possibly involving prestrain and precurvature. Growth-induced residual stresses can alter the convexity of the compliance functional, leading to nonuniqueness and localization phenomena. We explore the possibility of adding a regularization term penalizing deviations from the previous-step configuration. Finally, through a formal limiting procedure, we derive from the time-discrete formulation a time-continuous limit in the form of a constrained gradient flow.

We present a fast time-domain simulator for optical cavities capable of reproducing non-linear dynamical regimes arising from ring-down effect during resonance crossings at high mirror velocities. The model is based on a recursive formulation of the intracavity electric field as a sum over round trips, preserving the cavity memory while maintaining high computational efficiency. The simulator is designed to achieve three main goals. First, the boundary conditions of the cavity can be modified at each simulation step, allowing arbitrary time-dependent variations of both mirror positions and input electric field. Second, the sampling frequency can be flexibly chosen by the user, however, it is internally adjusted before effectively executing the simulation to remain consistent with the cavity round-trip structure. Finally, high computational efficiency was obtained by avoiding the repeated evaluation of the full electric field history. The framework is validated through comparison with experimental data from the Virgo interferometer during a mechanical excitation experiment, showing good agreement in non-adiabatic regimes. Due to its efficiency and flexibility, the simulator provides a versatile tool for time-domain studies of optical resonators and future applications in real-time control and reinforcement-learning-based lock acquisition.

This paper establishes a sharp characterization of temporal decay rates for the incompressible Oldroyd-B model in a critical $L^p$ framework, covering the physically relevant and mathematically delicate case where both the fluid viscosity and the stress tensor damping are absent. We prove that an $L^2$-type condition on the low-frequencies part of the initial data $(u_0, τ_0)$ is almost both necessary and sufficient for obtaining optimal upper and lower bounds on the temporal decay of solutions in critical Besov spaces. A key contribution is a new decomposition of the stress tensor into its incompressible and compressible parts, combined with the introduction of an effective tensor to handle the loss of regularity in the high-frequencies velocity field. This is the first result to reveal such precise two-sided asymptotics for the incompressible Oldroyd-B model without viscosity or damping.

We assess the capability of recently developed foundational atomistic models (FAMs) to simulate iron alloys under the extreme pressures and temperatures of Earth's core. Static equations of state of hexagonal close-packed (hcp) and body-centered cubic (bcc) iron computed by 17 FAMs are benchmarked against ab initio calculations. Two representative models, MatterSim and MACE, are further evaluated for their ability to reproduce phonon spectra, liquid structure, and melting relations of iron at core conditions. While both models capture several key properties, MACE substantially overestimates the stability of bcc iron and fails to correctly describe the stability of hcp iron. Their performance is also examined for binary liquids, superionic phases, and a seven-component Fe-Ni-Si-S-O-H-C liquid. Although these FAMs were not explicitly trained on data from core conditions, they can reproduce several structural and dynamical properties across a wide range of compositions. However, none of the tested models consistently reproduces all first-principles benchmarks. By analyzing the origins of these discrepancies, we identify several limitations of current FAMs, particularly the lack of an explicit treatment of thermal electronic excitations, which significantly affect phase stability and thermodynamic properties under core conditions. We further discuss directions for improving FAMs to enable predictive simulations of core-forming materials under extreme conditions.

Research has become dependent on processing power and storage, one crucial aspect being data sharing. The Open Science Data Federation (OSDF) project aims to create a scientific global data distribution network based on the Pelican Platform. OSDF relies on the XRootD and Pelican projects. Nevertheless, OSDF must understand the XRootD limits under various configuration options, including transfer rate limits, proper buffer configuration, and storage type effect. We have thus executed a set of benchmarks to create a set of recommendations to share with the XRootD and Pelican teams. This work describes the tests and results performed using National Research Platform (NRP) hosts. The tests cover various file sizes and parallel streams and use clients from various distances from the server host. We also used several standalone clients (wget, curl, pelican) and the native HTCondor file transfer mechanisms. Applying the methodology creates a possibility to track how XRootD and the Pelican layer perform in different scenarios.

We show that the Cauchy problem associated with the parabolic-elliptic Keller-Segel model is locally ill-posed in $L^q(\mathbb{R}^n)$ for dimensions $n \in \{3,\dots,9\}$ and throughout the supercritical range $q\in [1,\frac{n}{2})$. The non-uniqueness is driven by an instability mechanism in self-similarity variables, in the spirit of the program proposed by Jia and Šverák for the three-dimensional Navier-Stokes equations.

We study Galois embedding problems arising from the 3-torsion of elliptic curves defined over $\mathbb{Q}$, extending the correspondence to all possible images of mod 3 Galois representations; namely, $\operatorname{GL}_2(\mathbb{F}_3),SD_{16},D_6,D_4$ and $C_2^2$. In the cyclotomic case, we show that solvability of these embedding problems is equivalent to the existence of infinitely many elliptic curves whose 3-division fields provide the corresponding solutions.

Recent studies have reported a linear correlation between the hard X-ray and high-energy neutrino luminosities of active galactic nuclei (AGN), suggesting a possible physical connection between these two messengers. In this work, we challenge this interpretation by demonstrating that the observed correlation may arise purely from selection effects. We analyze 10 years of IceCube public data for a sample of Seyfert galaxies and blazars from the \textit{Swift} BAT catalog. While our data reproduces the apparent $L_ν$--$L_X$ correlation for sources with mild (but not significant) neutrino evidence, we show through Monte Carlo simulations that the same correlation appears even when analyzing random sky positions with no astrophysical sources. The key issue is that TS-based source selection effectively restricts the neutrino flux to a narrow range (a factor of several), while the luminosity distance of the sample spans $\sim4$ orders of magnitude. This causes the luminosity $L = 4πD_L^2 F$ to be dominated by the distance term rather than intrinsic flux variations, creating an artificial correlation. While a robust flux correlation ($F_ν$--$F_X$) for high-significance sources may indicate a genuine physical link, our results demonstrate that a luminosity-luminosity correlation alone is insufficient to establish a physical relationship between neutrino and X-ray emission in AGN.