We study nonlinear reactive transport in a layered porous medium separated by an $\varepsilon$-thin, highly heterogeneous fracture whose aperture and obstacle pattern vary periodically. Species transport in the bulk is governed by parabolic reaction--diffusion equations, coupled to a convection-diffusion-reaction problem in the fracture with nonlinear wall and obstacle reactions and Peclet number of order $O(\varepsilon^{-1})$. Via multiscale analysis as $\varepsilon \to 0$, when the fracture collapses to a flat interface, we derive a new type of homogenized model consisting of bulk diffusion--reaction equations coupled through nonlinear interface conditions and a first-order semilinear hyperbolic system on the interface. We prove well-posedness and regularity of the limit system, construct a multiscale approximation with boundary-layer correctors, and derive quantitative error estimates in suitable energy norms.

We introduce new logarithmic Hurwitz spaces $\mathcal{LH}^{\mathbb{Z}_{(p)}}_A$ and $\mathcal{LH}^{\mathbb{F}_{p}}_{A,Ξ}$ over $\mathbb{Z}_{(p)}$ and $\mathbb{F}_p$ respectively that in the mixed characteristic case can be considered as a compactification of the admissible cover stack parametrizing ramified covers of curves in characteristic $0$ of degree $p$ and in the equicharacteristic case compactify the space of separable maps between smooth curves of degree $p$. These Hurwitz spaces will carry a logarithmic structure and to emphasize that they are informative, we prove that in some first cases our Hurwitz spaces are log smooth. To achieve this, we introduce various Moduli spaces that parametrize Artin-Schreier covers and the locus of zeroes and poles of certain differential forms, show their smoothness and compute their dimension.

We use Hensel minimality, a non-Archimedean analog of o-minimality, to study several questions around transcendental number theory, unlikely intersections, and differential fields in a non-Archimedean setting. In particular, we focus on $p$-adic exponentiation and Tate uniformization on $\mathbb{C}_p$, which we show live in a Hensel minimal structure on $\mathbb{C}_p$. We start by constructing a large collection of derivations on Hensel minimal fields that respect definable functions, which we then apply to the $p$-adic Schanuel conjecture. We also study properties of local definability in analogy to work of Wilkie, and show that $p$-adic Schanuel implies a uniform version of itself. For Tate uniformization we show a strong closure property when blurring, and deduce that $\mathbb{C}_p$ with the blurred Tate uniformization is quasiminimal. Finally, we prove a result on $p$-adic density of likely intersections for powers of elliptic curves.

Large language models are increasingly integrated into academic writing workflows; however, the most widely used \LaTeX\ editors remain AI-peripheral -- offering compilation and collaboration, but no native intelligence. This separation forces researchers to leave their editing environment for AI assistance, fragmenting document context and interrupting writing flow. We present Bibby AI (trybibby.com), a native, AI-first \LaTeX\ editor that unifies the complete research writing lifecycle within a single interface. Bibby embeds an AI writing assistant, smart citation search, AI table and equation generation, an AI paper reviewer, abstract generator, literature review drafting, a deep research assistant, and real-time \LaTeX\ error detection and auto-fix -- all natively, without plugins or copy-paste workflows. We introduce LaTeXBench-500, a benchmark of 500 real-world compilation errors across six categories. Bibby achieves 91.4\% detection accuracy and 83.7\% one-click fix accuracy, outperforming Overleaf's native diagnostics (61.2\%) and OpenAI Prism (78.3 / 64.1\%) by large margins. Bibby demonstrates that a privacy-preserving, research-first AI editor can meaningfully accelerate every stage of academic manuscript preparation. We found that Bibby AI is a far superior alternative to overleaf latex and better than OpenAI Prism functionalities and AI.

The objective of automata learning is to reconstruct the implementation of a hidden automaton, to which only a teacher has access. The learner can ask certain kinds of queries to the teacher to gain more knowledge about the hidden automaton. The run-time of such a learning algorithm is then measured in the number of queries it takes until the hidden automaton is successfully reconstructed, which is usually parametric in the number of states of that hidden automaton. How can we prove such a run-time complexity of learning algorithms in a proof assistant if we do not have the hidden automaton and the number of states available? In the present paper, we solve this by considering learning algorithms themselves as generalized automata, more specifically as coalgebras. We introduce formal and yet compact definitions of what a learner and a teacher is, which make it easy to prove upper and lower bounds of different kinds of learning games in the proof assistant Agda. As a running example, we discuss the common number guessing game where a teacher thinks of a natural number and answers guesses by the learner with `correct', `too high', or `too low'. To demonstrate our framework, we formally prove in Agda that both the lower and upper bound on number of guesses by the learner is $\mathcal{O}(\log n)$, where $n$ is the teacher's secret number.

We define nonlocal predictability as how well one observer can predict another's measurement outcomes without classical communication, given full knowledge of the shared quantum state and measurement settings. The local bound on nonlocal predictability is defined as the maximum probability with which one observer can correctly predict the other's measurement outcome prior to measurement. We show that product states always meet this bound, while all pure entangled states and some classically correlated states can exceed it. This demonstrates a nonlocal phenomenon since the predictability of measurement outcomes increases after the measurement. Perfect nonlocal predictability for arbitrary projective measurements occurs only for maximally entangled states among all pure states, underscoring their special role. Comparing pure entangled states with their dephased versions, we find that dephasing on one subsystem can enhance nonlocal predictability for a broad class of states and measurements - a counterintuitive, noise-induced advantage that vanishes for maximally entangled states under any projective measurement.

A search for the decay $B^+\to D_s^+ K^+K^-$ has been reported by the LHCb Collaboration using $pp$ collision data corresponding to an integrated luminosity of $4.8\,\mathrm{fb}^{-1}$, collected at center-of-mass energies of 7, 8, and 13~TeV, in which no amplitude analysis of the $K^+K^-$ subsystem was performed. In this work, we study the resonant contributions to the decay $B^+\to D_s^+ K^+K^-$ within the perturbative QCD (PQCD) factorization framework. Contributions from the $S$-wave resonances $f_0(980)$, $f_0(1370)$, and $f_0(1500)$, the $P$-wave resonance $ϕ(1020)$, and the $D$-wave resonances $f_2(1270)$ and $f_2(1525)$ are taken into account. By introducing the corresponding two-meson distribution amplitudes for the $K^+K^-$ system, we perform a complete perturbative analysis of the quasi-two-body decays $B^+\to D_s^+(R\to)K^+K^-$, where $R$ denotes an intermediate resonance, and present the first PQCD predictions for the associated branching fractions. Using the narrow-width approximation, we further extract the branching fractions of the corresponding two-body decays $B^+\to D_s^+R$. Our results are consistent with the available experimental measurements and previous theoretical studies. Finally, we find that direct CP asymmetries vanish for these quasi-two-body decays within the Standard Model, so that any experimentally observed nonzero CP asymmetry would constitute a clear signal of physics beyond the Standard Model.

Many numerical studies validate droplet wall impact using only maximum spreading diameter, yet this metric alone cannot ensure correct droplet dynamics. We present a combined dynamic contact angle (DCA) model that merges the geometric accuracy of the generalized Hoffman-Voinov-Tanner law with the kinematic consistency of a Hoffman function-based approach, improving predictions of droplet spreading and receding. We simulate water-glycerol droplet impact on sapphire glass at Weber numbers 20 -- 250 and assess both contact angle formulations. Simulated radial velocity fields are processed in Python using SciPy and compared with Particle Image Velocimetry measurements in the longitudinal section of the spreading droplet. The Hoffman function-based model captures the main droplet kinematic trends and provides more consistent receding dynamics. The generalized Hoffman-Voinov-Tanner law matches the maximum spreading diameter within 7%. However, during receding, it shows a median absolute error in radial velocity up to three times higher than that of the Hoffman function-based solution. Average radial velocity and spreading velocity can differ from experimental trends even when maximum spreading is reproduced. These findings support validation combining geometric and kinematic metrics and motivate the combined model for predicting spreading and receding. Using the maximum spreading factor $β_{max}$ as the ratio of the maximum spreading diameter over the initial droplet diameter and the characteristic capillary number $Ca_{char}$ defined from the mean internal horizontal velocity at 300 micrometer above the substrate, we introduce a $(β_{max},\,Ca_{char})$ diagram to relate spreading characteristics to internal flow dynamics. We hypothesize that, given sufficient data, the contact-line geometry may be used to estimate internal kinematics.

Modulo sampling is a promising technology to preserve amplitude information that exceeds the observable range of analog-to-digital converters during the digitization of analog signals. Since conventional methods typically reconstruct the original signal by estimating the differences of the residual signal and computing their cumulative sum, each estimation error inevitably propagates through subsequent time samples. In this paper, to eliminate this error-propagation problem, we propose an algorithm that reconstructs the residual signal directly. The proposed method takes advantage of the high-frequency characteristics of the modulo samples and the sparsity of both the residual signal and its difference. Simulation results show that the proposed method reconstructs the original signal more accurately than a conventional method based on the differences of the residual signal.

The recent reconstruction of the Drosophila brain provides a neural network of unprecedented size and level of details. In this work, we study the geometrical properties of this system by applying network embedding techniques to the graph of synaptic connections. Since previous analysis have revealed an inhomogeneous degree distribution, we first employ a hyperbolic embedding approach that maps the neural network onto a point cloud in the two-dimensional hyperbolic space. In general, hyperbolic embedding methods exploit the exponentially growing volume of hyperbolic space with increasing distance from the origin, allowing for an approximately uniform spatial distribution of nodes even in scale-free, small-world networks. By evaluating multiple embedding quality metrics, we find that the network structure is well captured by the resulting two-dimensional hyperbolic embedding, and in fact is more congruent with this representation than with the original neuron coordinates in three-dimensional Euclidean space. In order to examine the network geometry in a broader context, we also apply the well-known Euclidean network embedding approach Node2vec, where the dimension of the embedding space, $d$ can be set arbitrarily. In 3 dimensions, the Euclidean embedding of the network yields lower quality scores compared to the original neuron coordinates. However, as a function of the embedding dimension the scores show an improving tendency, surpassing the level of the 2d hyperbolic embedding roughly at $d=16$, and reaching a maximum around $d=64$. Since network embeddings can serve as valuable inputs for a variety of downstream machine learning tasks, our results offer new perspectives on the structure and representation of this recently revealed and biologically significant neural network.

Preparing subradiant steady states of collectively emitting quantum two-level emitters (TLEs) is hindered by their dark, weakly interacting nature. Existing approaches rely on patterned driving, local control, or structured environments. We propose a platform-independent protocol based on projective measurements on a single TLE. For permutation-symmetric ensembles, a single measurement yields appreciable occupation of single-excitation subradiant steady states. For generic arrays, repeated measurements on one emitter drive the unmeasured TLEs into a nearly pure state with large overlap with the subradiant Dicke subspace.

In this work we investigate the computational complexity of the satisfiability problem of sub-fragments of the Bernays-Schoenfinkel class of first-order logic, also known as EPR (Effectively Propositional). While Bernays-Schoenfinkel is NEXPTIME-complete, we already can obtain fragments that are PSPACE-complete by restricting our clauses to DET-HORN or KROM. However such restrictions yield very different formulas to the canonical PSPACE-complete language of Quantified Boolean Formulas (QBF). This is despite Bernays-Schoenfinkel having a natural connection to an extension of QBF known as Dependency QBF. Our main contribution is the definition of a PSPACE-complete sub-fragment of Bernays-Schoenfinkel that extends from a translation of QBF, retains a similar two-player game evaluation for its semantics and can be restricted in various ways to obtain other complete problems, particularly those at different levels in the polynomial hierarchy. We use this definition to identify problems in the TPTP library that fall into this fragment and their level in the polynomial hierarchy.

Fast radio bursts (FRBs) are bright, energetic, radio pulses of extragalactic origin. A dichotomy has emerged in the population: some produce repeat bursts, while the majority do not. Most repeating sources only show rare repetitions, and none have been studied extensively over the wide bandwidths necessary to disentangle the physical processes that produce emission from distortions to bursts caused by intervening ionised gas. Here we present radio observations of the most active repeating source, FRB 20240114A. Using an ultrawideband receiving system, we have detected 5526 repetitions, revealing an extreme spectral and temporal variability in the burst emission. The bursts exhibit longer-term broadband variations in central emission frequency over multiple months, and narrowband bursts that have correlations in central frequencies on time scales of milliseconds to minutes. The spectral and temporal properties are consistent with the source undergoing magnification by foreground plasma lenses, potentially embedded in a turbulent circumsource medium. This extreme example highlights the role of plasma lenses in the observed properties of burst emission and can explain the diversity in activity and energetics of the entire FRB population.

We slightly extend the convex integration via staircase laminate toolbox recently developed by Kleiner, Müller, Székelyhidi, and Xie. As an example we revisit the proof by Astala-Faraco-Székelyhidi on optimal Meyers' regularity theory via this framework.

This paper extends the foundational work of Dollma \emph{et al}. on codes for ordered composite DNA sequences. We consider the general setting with an alphabet of size $q$ and a resolution parameter $k$, moving beyond the binary ($q=2$) case primarily studied previously. We investigate error-correcting codes for substitution errors and deletion errors under several channel models, including $(e_1,\ldots,e_k)$-composite error/deletion, $e$-composite error/deletion, and the newly introduced $t$-$(e_1,\ldots,e_t)$-composite error/deletion model. We first establish equivalence relations among families of composite-error correcting codes (CECCs) and among families of composite-deletion correcting codes (CDCCs). This significantly reduces the number of distinct error-parameter sets that require separate analysis. We then derive novel and general upper bounds on the sizes of CECCs using refined sphere-packing arguments and probabilistic methods. These bounds together cover all values of parameters $q$, $k$, $(e_1,\ldots,e_k)$ and $e$. In contrast, previous bounds were only established for $q=2$ and limited choices of $k$, $(e_1,\ldots,e_k)$ and $e$. For CDCCs, we generalize a known non-asymptotic upper bound for $(1,0,\ldots,0)$-CDCCs and then provide a cleaner asymptotic bound. On the constructive side, for any $q\ge2$, we propose $(1,0,\ldots,0)$-CDCCs, $1$-CDCCs and $t$-$(1,\ldots,1)$-CDCCs with near-optimal redundancies. These codes have efficient and systematic encoders. For substitution errors, we design the first explicit encoding and decoding algorithms for the binary $(1,0,\ldots,0)$-CECC constructed by Dollma \emph{et al}, and extend the approach to general $q$. Furthermore, we give an improved construction of binary $1$-CECCs, a construction of nonbinary $1$-CECCs, and a construction of $t$-$(1,\ldots,1)$-CECCs. These constructions are also systematic.

The calculation of thermal conductivity in insulating solids at temperatures below the Debye temperature is problematic, due to the breakdown of classical and semi-classical approaches. In this work, we present a fully quantum methodology to compute thermal conductivity based on Path Integral Monte Carlo (PIMC) simulations combined with the Green-Kubo linear response theory. The method is applied to crystalline argon modeled by a Lennard-Jones potential, a paradigmatic system where quantum effects strongly affect both thermodynamic and transport properties. From PIMC simulations, we obtain the temperature-dependent phonon frequencies, lifetimes, and specific heat. From the imaginary time correlations of the energy current, we extract the thermal transport coefficients based on a physically motivated prior. We show that the experimentally observed increase of the thermal conductivity at low temperatures cannot be explained within a standard Peierls-Boltzmann framework or quasi-harmonic approximation using phonon lifetimes alone. Instead, a distinct transport lifetime emerges from the analysis of heat-current correlations. Our results demonstrate that quantum Monte Carlo methods provide a robust, non-perturbative framework to investigate heat transport in insulating solids, beyond the limits of classical molecular dynamics and quasi-harmonic approximations.

Let $\mathcal{A}$ be an algebra, and let $\mathcal{A}^2 =$ span$\{ab : a, b \in \mathcal{A}\}$ be a subalgebra of $\mathcal{A}$. In this paper, we prove that if $\mathcal{A}^2$ has infinite codimension in $\mathcal{A}$ iff $\mathcal{A}$ has discontinuous square annihilation property (DSAP). In fact, in this case, the algebra $\mathcal{A}$ admits infinitely many non-equivalent algebra norms.

Traditional models of slow slip events (SSEs) often oversimplify fault geometry, yet imaging studies show that real subduction faults are segmented and complex. We investigate how fault interactions influence slip behavior using 3D quasi-dynamic earthquake sequence simulations of two parallel faults with uniform rate-weakening friction, accelerated with hierarchical matrices. Our results identify four slip regimes-periodic earthquakes, coexisting SSEs and earthquakes, only SSEs, and complex sequences-while a single planar fault under the same conditions produces only earthquakes. We quantify fault interaction using the maximum Coulomb stress induced on a target fault by unit, spatially uniform stress drop on a neighboring fault. Because the source stress drop is normalized, the metric depends only on geometry and is independent of friction and nucleation length, and it can be extended to arbitrary fault configurations. The occurrence of SSEs is confined to an intermediate range of interaction strength. We also reproduce the observed moment-duration scaling and show that it depends on event detection thresholds. These results demonstrate that complex fault geometry can naturally generate both slow and fast earthquakes through evolving traction heterogeneities.

In this paper we provide a framework for quantitative statements on distances and measures when studying algebraic varieties and morphisms of algebraic varieties over local fields. We will concentrate on local fields of the type $\mathbb{F}_\ell((t))$ and work uniformly with respect to finite extensions of $\mathbb{F}_\ell$. In this framework we prove analogues of standard results from local differential topology, including the implicit function theorem and study the behavior of smooth measures under push forward with respect to submersions.

We prove that the Hilbert scheme of the plane in positive characteristic admits an invertible top differential form. This implies certain integrability properties of the symmetric powers of the plane. This allows to define a function on the collection of monic polynomials over a local field which can be thought of as a variant of the inverse square root of the discriminant. In characteristic 0 it essentially coincides with this inverse square root, however in general it is quite different, and unlike this inverse square root, it is locally summable. In a sequel work [AGKS] we use this local summability in order to prove the positive characteristic analog of Harish-Chandra's local integrability theorem of characters of representations under certain conditions. The main results of this paper are known in characteristic zero. In fact a stronger result is known: there is a symplectic form on the Hilbert scheme of a plane.

A protostellar disk is threaded by a static magnetic field that is perpendicular to the disk-surface. The magnetic field acts to brake the protostellar disk and cause the disk material to move towards the protostar. General analytic equations are derived for the accretion speed, and mass accretion rate. Simplified analytic equations are also obtained for the disk energy dissipation, accretion timescale and the disk radial position plus disk surface density, as a function of time. In addition to providing physical insight, such equations might be useful as a check on computational models for protostar and protostellar disk formation.

The merger of a He white dwarf (WD) and a CO WD is the favored formation channel for R Coronae Borealis (RCB) stars. These stars exhibit ${^{16}}\mathrm{O}/{^{18}}\mathrm{O}$ ratios that are orders of magnitude lower than the solar value. However, it is not fully understood whether such low ${^{16}}\mathrm{O}/{^{18}}\mathrm{O}$ ratios can be achieved in WD merger remnants for the predicted lifetime of RCB stars of around $10^4\,\mathrm{years}$. In this work, we perform detailed nucleosynthesis calculations of a 3D magnetohydrodynamical simulation of a merger of a $0.3\,M_\odot$ He WD and a $0.6\,M_\odot$ CO WD for $4000\,\mathrm{s}$ at which point a steady state in temperature and density is reached. From this point, we follow several radial zones to study the long-term production of ${^{18}}\mathrm{O}$ and its variability throughout the burning region. We find that the asymmetric merger process leaves an imprint on the distribution of the abundances at the end of our hydrodynamic simulation. During the long-term evolution up to $100\,\mathrm{years}$, we observe ${^{16}}\mathrm{O}/{^{18}}\mathrm{O}$ ratios of order of unity, although the timescale on which ${^{18}}\mathrm{O}$ is destroyed again is highly location dependent. Importantly, our calculations suggest that in the outer layers of the burning shell, the dominant production channel is $^{14}\mathrm{C}(α,γ)^{18}\mathrm{O}$ instead of the commonly considered $^{14}\mathrm{N}(α,γ)^{18}\mathrm{F}(β^+)^{18}\mathrm{O}$ reaction, whereby the former can be sustained for longer periods of time. Furthermore, these outer regions do not reach the conditions necessary for fast $α$-captures in ${^{18}}\mathrm{O}$ to ${^{22}}\mathrm{Ne}$, thus being favorable to maintaining a low ${^{16}}\mathrm{O}/{^{18}}\mathrm{O}$ ratio.

We prove that the character of an irreducible cuspidal representation of $GL_n(\mathbb{F}_{\ell}((t)))$ is locally bounded up to a logarithmic factor by the orbital integral of a matrix coefficient of this representation. The characteristic $0$ analog of this result is part of the proof of the celebrated Harish-Chandra's integrability theorem. In a sequel work [AGKS] we use this result in order to prove a positive characteristic analog of Harish-Chandra's integrability theorem under some additional assumptions.

We consider a class of optimal control problems, with finite or infinite horizon, for a continuous-time Markov chain with finite state space. In this case, the control process affects the transition rates. We suppose that the controlled process can not be observed, and at any time the control actions are chosen based on the observation of a related stochastic process perturbed by an exogenous Brownian motion. We describe a construction of the controlled Markov chain, having stochastic transition rates adapted to the observation filtration. By a change of probability measure of Girsanov type, we introduce the so-called separated optimal control problem, where the state is the conditional (unnormalized) distribution of the controlled Markov chain and the observation process becomes a driving Brownian motion, and we prove the equivalence with the original control problem. The controlled equations for the separated problem are an instance of the Wonham filtering equations. Next we present an analysis of the separated problem: we characterize the value function as the unique viscosity solution to the dynamic programming equations (both in the parabolic and the elliptic case) we prove verifications theorems and a version of the stochastic maximum principle in the form of a necessary conditions for optimality.

In this paper, we investigate the enhancement of delocalization and coin-position entanglement in asymmetric discrete-time quantum walks (DTQWs). The asymmetry results from asymmetric coin operations, asymmetric initial states, and asymmetric polarization-dependent losses. By varying these asymmetry factors, the inverse participation ratio and entanglement entropy of the walker are numerically calculated for different coin and loss parameters, both for symmetric and asymmetric initial states. We then experimentally implement a 16-step asymmetric DTQW using a time-multiplexing fiber loop structure. By choosing an asymmetric initial state, both coin-position entanglement and delocalization are simultaneously enhanced under specific coin parameters. Moreover, we observe that with finite asymmetric polarization-dependent loss, the photon probability on the left side decreases significantly, while that on the right side increases and becomes more localized. Interestingly, under specific coin parameters, the entanglement and delocalization exhibit improved robustness against polarization-dependent loss. These results demonstrate that the DTQWs constitute an ideal platform for investigating photonic delocalization and hybrid entanglement.

Understanding why and how crystalline solids melt remains a central problem in condensed-matter physics. Dislocation loops are fundamental topological excitations that control the thermodynamic stability of crystals, yet their role in setting universal aspects of melting has remained unclear. Here we show, within dislocation-mediated melting theory, that the free-energy condition for loop proliferation leads to a universal ratio between the energy of a minimal dislocation loop and the thermal energy at melting. For minimal dislocation loops that begin to proliferate at the onset of melting, this ratio takes the purely geometric value $\mathcal{E}_* = E_{\rm loop}/(k_B T_m) \approx 25.1$, independent of elastic moduli and chemistry-dependent details. This result provides a microscopic explanation for recent empirical findings by Lunkenheimer \emph{et al.}, who identified a closely related universal energy scale $\approx 24.6$ from viscosity data. The same framework also rationalizes the empirical $2/3$ rule relating the glass-transition and melting temperatures.

The celebrated Harish-Chandra's integrability theorem states that the distributional character of an irreducible smooth representation of a p-adic group $G(F)$ is integrable, that is represented by an $L^1_{loc}(G(F))$ function. Here $F$ is a non-Archimedean local field of characteristic $0$ and $G$ is a reductive algebraic group defined over $F$. In this paper we focus on cuspidal representations of $GL_n(F)$ for a field $F$ of positive characteristic. We show that in this case the integrability holds under the hypothesis of existence of desingularization of (certain) algebraic varieties in positive characteristics. Furthermore, in the case $char(F)>n/2$ we establish the regularity of such characters unconditionally.

Let $\mathcal R_{n}$ be the set of all rational functions of the type $r(z) = f(z)/w(z)$, where $f(z)$ is a polynomial of degree at most $n$ and $w(z) = \prod_{j=1}^{n}(z-β_j)$, $|β_j|>1$ for $1\leq j\leq n$. In this work, we investigate the growth behavior of rational functions with prescribed poles by utilizing certain coefficients of the polynomial $f(z)$. The results obtained here not only refine and strengthen the findings of Rather et al. \cite{NS}, but also generalize recent growth estimates for polynomials due to Dhankhar and Kumar \cite{KD} to the broader setting of rational functions with fixed poles. Additionally, we establish corresponding results for such rational functions under suitable restrictions on their zeros.

Modern financial networks are highly connected and result in complex interdependencies of the involved institutions. In the prominent Eisenberg-Noe model, a fundamental aspect is clearing -- to determine the amount of assets available to each financial institution in the presence of potential defaults and bankruptcy. A clearing state represents a fixed point that satisfies a set of natural axioms. Existence can be established (even in broad generalizations of the model) using Tarski's theorem. While a maximal fixed point can be computed in polynomial time, the complexity of computing other fixed points is open. In this paper, we provide an efficient algorithm to compute a minimal fixed point that runs in strongly polynomial time. It applies in a broad generalization of the Eisenberg-Noe model with any monotone, piecewise-linear payment functions and default costs. Moreover, in this scenario we provide a polynomial-time algorithm to compute a maximal fixed point. For networks without default costs, we can efficiently decide the existence of fixed points in a given range. We also study claims trading, a local network adjustment to improve clearing, when networks are evaluated with minimal clearing. We provide an efficient algorithm to decide existence of Pareto-improving trades and compute optimal ones if they exist.

The next generation of mobile networks, 6G, is expected to enable data-driven services at unprecedented scale and complexity, with stringent requirements for trust, interoperability, and automation. Central to this vision is the ability to create, manage, and share high-quality datasets across distributed and heterogeneous environments. This paper presents the data architecture of the 6G-DALI project, which implements a federated dataspace and DataOps infrastructure to support secure, compliant, and scalable data sharing for AI-driven experimentation and service orchestration. Drawing from principles defined by GAIA-X and the International Data Spaces Association (IDSA), the architecture incorporates components such as federated identity management, policy-based data contracts, and automated data pipelines. We detail how the 6G-DALI architecture aligns with and extends GAIA-X and IDSA reference models to meet the unique demands of 6G networks, including low-latency edge processing, dynamic trust management, and cross-domain federation. A comparative analysis highlights both convergence points and necessary innovations.