Motor Imagery (MI) Electroencephalography (EEG) signals contain two crucial and complementary types of information: state information, which captures the global context of the task, and flow information, which captures fine-grained temporal dynamics. However, existing deep decoding models typically focus on only one of these information streams, resulting in unstable learning and sub-optimal performance. To address this, we propose the State-Flow Coordinated Network (StaFlowNet), a novel architecture that explicitly separates and coordinates state and flow information. We first employ a dual-branch design to extract the global state vector and temporal flow features separately. Critically, a novel state-modulated flow module is proposed to dynamically refine the learning of flow information. This modulated mechanism effectively integrates global context with fine-grained dynamics, thereby significantly enhancing task discriminability and decoding performance. Experiments on three public MI-EEG datasets demonstrate that StaFlowNet significantly outperforms state-of-the-art methods. Ablation studies further confirm that the state-modulated mechanism plays a crucial role in enhancing feature discriminability and overall performance.

We develop dual approaches for continuous-time stochastic control problems, enabling the computation of robust dual bounds in high-dimensional state and control spaces. Building on the dual formulation proposed in [L. C. G. Rogers, SIAM Journal on Control and Optimization, 46 (2007), pp. 1116--1132], we first formulate the inner optimization problem as a stochastic partial differential equation (SPDE); the expectation of its solution yields the dual bound. Curse-of-dimensionality-free methods are proposed based on the Pontryagin maximum principle and the generalized Hopf formula. In the process, we prove the generalized Hopf formula, first introduced as a conjecture in [Y. T. Chow, J. Darbon, S. Osher, and W. Yin, Journal of Computational Physics 387 (2019), pp. 376--409], under mild conditions. Numerical experiments demonstrate that our dual approaches effectively complement primal methods, including the deep BSDE method for solving high-dimensional PDEs and the deep actor-critic method in reinforcement learning.

We construct here global mild solutions in a critical setting for a class of transport-diffusion equations with a drift term that involves rough Calder{ó}n-Zygmund operators.

We investigate work extraction in open quantum batteries composed of interacting spin chains weakly coupled to engineered environments. Focusing on two- and four-qubit XX models initially prepared in thermal Gibbs states, we analyze how dissipation and dephasing, acting either locally or collectively, can generate and shape ergotropy during both transient and steady-state dynamics. By introducing a continuous interpolation between parallel and collective noise channels, we systematically characterize the impact of environmental structure on work extractability. We show that purely dissipative dynamics can activate finite ergotropy from completely passive thermal states, giving rise to temperature-dependent transient regimes where hotter initial states temporarily outperform colder ones in an ergotropic Mpemba-like fashion. In contrast, collective dissipation leads to steady states whose passivity crucially depends on the initial temperature and system size, a behavior we trace back to the emergence of non-trivial dark subspaces. Finally, we demonstrate that dephasing channels suppress both transient advantages and steady-state work extraction, highlighting the qualitative difference between dissipative and dephasing environments.

We propose a novel entropy flow on weighted graphs, which provides a principled framework that characterizes the evolution of probability distributions over graph structures while sharing geometric intuition with discrete Ricci flow. We provide its rigorous formulation, establish its fundamental theoretical properties, and prove the long-time existence and convergence of its solutions. To demonstrate its applicability, we employ entropy flow for community detection in real-world networks. Empirically, it achieves detection accuracy fully comparable to that of discrete Ricci flow. Crucially, by avoiding computations of optimal transport distances and shortest paths, our approach overcomes the fundamental computational bottleneck of Ollivier and Lin-Lu-Yau Ricci flows. As a result, entropy flow requires only $1.61\%$-$3.20\%$ of the computation time of Ricci flow. These results indicate that entropy flow provides a theoretically rigorous and computationally efficient framework for large-scale graph analysis.

Magnetism governs key properties of materials used in energy, data storage, and spintronic technologies, yet its complex coupling to lattice and electronic degrees of freedom challenges conventional first-principles approaches. We introduce an equivariant message-passing graph neural network that embeds atomic magnetic moments as explicit degrees of freedom, enabling the learning of magnetic interactions beyond collinear approximations. The model learns physically consistent and transferable representations of magnetic behaviour and can incorporate spin-orbit coupling, achieving near density-functional-theory accuracy with strong data efficiency across diverse magnetic systems by fine-tuning from a pre-trained model. Applications to structural transformations, finite-temperature magnetic phenomena, and materials screening for strongly spin-orbit coupled materials demonstrate transferable magnetic behaviour, establishing a practical foundation for data-driven, high-throughput discovery of complex magnetic materials.

We investigate propagation of generic waves on thin planar domain walls effectively described by the scalar DBI model. We pay a particular attention to the possibility of caustic (shock) formation - the process, which may lead to intensive particle emission by domain walls. It is demonstrated that no singularities arise in DBI in 2D flat spacetime in the hyperbolic case, if one starts from smooth initial conditions. Technically, this happens because the same family characteristics of the relevant PDE remain parallel at all the times, albeit not being straight lines generically. Crucially, characteristic curves cease to be parallel beyond the simplified setup of DBI in 2D flat spacetime. In particular, this is shown to be the case in $D>2$ for spherical waves, in an expanding Universe, and in the case of a minimal deformation of DBI necessary for avoiding the domain wall problem in cosmology. However, we prove that DBI remains shock free in the hyperbolic case in all these physically relevant situations. This strongly suggests that caustics can form on planar domain walls only due to the loss of hyperbolicity, and they have a cusp profile. We demonstrate, how the non-trivial structure of DBI characteristics beyond the 2D flat spacetime setup uncovered in this work can significantly affect cusp formation.

The pulsar timing array (PTA) is a powerful technique for detecting nanohertz gravitational wave backgrounds (GWBs). However, conventional PTAs lack sensitivity to parity violation in the GWB. In this work, we propose a dipole pulsar timing array system (dPTA). By deriving the overlap reduction functions (ORFs) from the cross-correlation of timing signals, we find that this system exhibits sensitivity to chiral GWBs in the nanohertz regime. Furthermore, through numerical calculations of its sensitivity curves, we demonstrate that the dPTA extends the detectable frequency range of PTAs for GWBs from the nanohertz to the microhertz regime.

We develop a theoretical description of quantum wave mixing (QWM) in a cascaded waveguide-QED system of two superconducting qubits, where the probe is driven by an external coherent tone and by the resonance fluorescence of a strongly driven source qubit. Starting from the field correlation functions of the source emission, we derive an effective master-equation treatment for the probe and identify the regime in which the incident fluorescence is characterized by anomalous correlations. When the coherent Rayleigh component of the source spectrum is suppressed, the probe equations of motion become equivalent to those for a qubit driven by a coherent tone and broadband squeezed light. This equivalence implies a selection rule for the peaks of the QWM spectrum, with a strong suppression of sidebands associated with processes involving an odd number of photons taken from the source field. Numerical simulations of the full cascaded two-qubit model for different ratios of radiative decay rates unambiguously confirm the participation of correlated photon pairs in QWM processes. The current research illustrates that the analysis of peak amplitudes can be used to probe photon statistics in the incident nonclassical field.

The Drazin index is a fundamental invariant in the analysis of singular matrices and their generalized inverses. While sharp results are available for block triangular matrices, the corresponding theory for anti-triangular block matrices is less developed. In this paper, we study matrices of the form \[ M=\begin{bmatrix} A & B \\ C & 0 \end{bmatrix}, \] under algebraic constraints on the blocks. Building on additive decompositions involving von Neumann inverses, we relate the Drazin index of $M$ to invariance properties of the index and minimal polynomial of expressions of the form $A^{2}A^{-}+I-AA^{-}$. This connection provides an effective mechanism to control the index of $M$ through suitable factorizations and associated block products. As a consequence, we derive explicit lower and upper bounds for $i(M)$ in terms of $i(A)$ and $i(BC)$, and characterize situations in which these bounds are attained. Under additional annihilation or orthogonality conditions on the blocks, we obtain closed-form representations for the Drazin inverse of $M$. Applications to adjacency matrices of directed graphs illustrate the sharpness of the bounds and the applicability of the results to structured matrices arising in graph-theoretic settings.

Collisions of lead nuclei at relativistic energies provide valuable insight into the properties of the quark gluon plasma formed in such collisions where the initial geometry and density profile play a crucial role in governing the subsequent evolution of the produced hot and dense fireball. The neutron skin thickness resulting from the difference between the neutron and proton density distributions in neutron rich lead nuclei plays an important role in nuclear structure studies. In this work we investigate the impact of neutron skin on the space time evolution of the fireball formed in Pb+Pb collisions at 2.76A TeV at the LHC and analyze how the presence of neutron skin affect bulk observables sensitive to the initial nuclear structure. The time evolution of initial profile along with the average $p_T$, particle spectra and anisotropic flow parameters are estimated to investigate the effect of neutron skin on these observables. The initial spatial anisotropy of the fireball is found to be affected by the neutron skin thickness significantly especially for the peripheral collisions. This leads to a substantial enhancement of the elliptic flow of hadrons with an even stronger effect observed for photons. In addition, the effect is found to be more pronounced for lower beam energy collisions of lead nuclei.

Alignment interactions in active matter are typically modeled as relaxational dynamics toward local consensus. In unbounded systems, this makes alignment effectively decoupled from local density and therefore unable to sustain self-confined collective motion without additional attractive forces. Here we show that this limitation can be overcome by extremal alignment rules in which the interaction neighborhood depends on the candidate orientation. For a broad class of candidate- dependent rules with pairwise additive utilities, the decision utility factorizes into the product of an average interaction score and the number of selected neighbors. This multiplicative structure couples orientational decisions to local density and thereby generates an effective cohesive bias without explicit cohesive forces. In metric models, however, the same mechanism drives collapse toward globally connected, effectively mean-field states that suppress spatial structure. We show that topological interactions regularize this tendency, stabilizing self-confined flocks of finite extent in open space. The resulting dynamics exhibits a rich dynamical phase diagram as a function of noise intensity and turning rate, including polarized flocks, swarms, and persistent swirling states. Our results identify candidate-dependent extremal alignment as a simple mechanism for generating cohesive, spatially structured active matter beyond the standard relaxational paradigm.

This paper primarily discusses the dynamical properties of a class of Lotka-Volterra models featuring the Allee effect and interspecific competition within the predator population. The constructed models employ Holling II and Holling I response functions for the predator, respectively.The existence of boundary equilibrium points under various parameter conditions and internal equilibrium points under specific parameter conditions is discussed. The equilibrium points of the system may be stable or unstable nodes, saddle points, saddle-nodes, or cusp points with a codimension of 2. The parameter conditions under which internal equilibrium points possess one zero eigenvalue and two non-zero eigenvalues, one zero eigenvalue and a pair of purely imaginary eigenvalues, or two zero eigenvalues and one non-zero eigenvalue are analyzed.

Deep stochastic state-space models enable Bayesian filtering in nonlinear, partially observed systems but typically assume a fixed latent structure. When this assumption is violated, parameter adaptation alone may result in persistent belief inconsistency. We introduce \emph{Cognitive Flexibility} (CF) as a representation-level operator that selects latent structures online via an innovation-based predictive score, while preserving the Bayesian filtering recursion. Structural mismatch is formalized as irreducible predictive inconsistency under fixed structure. The resulting belief--structure recursion is shown to be well posed, to exhibit a structural descent property, and to admit finite switching, with reduction to standard Bayesian filtering under correct specification. Experiments on latent-dynamics mismatch, observation-structure shifts, and well-specified regimes confirm that CF improves predictive accuracy under a mismatch while remaining non-intrusive when the model is correctly specified.

We study the property of hitting points for a class of $\mathbb{R}^d$-valued continuous Gaussian random fields on $\mathbb{R}^N$ with stationary increments, i.i.d. coordinates, and a regularly varying variance function $σ$ of index $0<H<1$. We first prove that if \[ \lim_{r\to 0^+} \frac{r^N}{σ^d\left(r\left( \log\log\frac{1}{r}\right)^{-1/N}\right)} = \infty, \] then every fixed point is polar (i.e., not hit almost surely). In general, this criterion may not be optimal in the critical dimension $d=N/H$. To aim for an optimal condition, we consider the specific case $σ(r) = r^H (\log(1/r))^γ$ and prove that, in the critical dimension $d=N/H$, points are polar if and only if $γ\le 1/d$, or equivalently in this specific case, \[ \int_{0^+} \frac{r^{N-1}}{σ^d(r)} dr = \infty. \] This integral condition is also necessary for points to be polar under general assumptions. Our main contribution lies in the proof of sufficiency of this condition in the specific case, where we extend a covering argument of Talagrand (1998) based on sojourn time estimates to obtain Hausdorff measure bounds and solve polarity of points in the critical dimension.

Interpretable AI can reveal physical principles governing intricate materials properties by uncovering explicit relationships between physical parameters and target properties. The sure-independence screening and sparsifying operator (SISSO) symbolic-regression approach identifies analytical expressions that correlate a target property with a small set of parameters, termed materials genes, selected from a large pool of candidates. However, multiple gene combinations can yield equally accurate SISSO models, with individual genes contributing with different weights. Here, we establish a derivative-based sensitivity analysis that resolves the non-uniqueness of symbolic-regression descriptions, enhances interpretability, thereby enabling deeper physical insight. This analysis reveals how distinct gene combinations encode equivalent information and identifies valence orbital radii, nuclear charges, and their products as the key quantities governing the equilibrium lattice constant of perovskites.

We investigate measures of distance and redshift in cosmological space-times that admit a shear-free foliation, which we henceforth refer to as `quasi-Newtonian'. Space expands isotropically in this description, and small-scale gravitational physics has a natural Newtonian limit, which makes it ideal for considering the physics of wide classes of cosmological models. By assuming that the energy-momentum tensor is dominated by rest-mass density, and that the 3-velocity of matter is small in the quasi-Newtonian frame, we derive fundamental results for kinematics and light propagation. Our results provide a new way of formulating general-relativistic cosmologies with non-perturbative structures in terms of quantities that can be understood from cosmological perturbation theory and post-Newtonian expansions, and allow us to quantify departures of observables from the predictions of Friedmann cosmology. It thereby provides a route to understanding inherently relativistic space-time structures, such as those that occur in Lemaître-Tolman-Bondi, Szekeres solutions, and Bianchi cosmologies in terms of Newtonian degrees of freedom. We illustrate our results using the degenerate Kasner solution as an example, and explain how our approach can be used to provide new insights into the current cosmological tensions.

This paper examines the influence of internal Gaussian noise on the performance of deep feedforward neural networks, focusing on the role of the noise injection stage relative to the activation function. Two scenarios are analyzed: noise introduced before and after the activation function, for both additive and multiplicative noise influence. The case of noise before activation function is similar to perturbations in the input channel of neuron, while the noise introduced after activation function is analogous to noise occurring either within the neuron itself or in its output channel. The types of noise and the method of their introduction were inspired by analog neural networks. The results show that the activation function acts as an effective nonlinear filter of noise. Networks with noise introduced before the activation function consistently achieve higher accuracy than those with noise applied after it, with additive noise being more effectively suppressed in this case. For noise introduced after the activation function, multiplicative noise is less detrimental than additive noise, and earlier hidden layers contribute more significantly to performance degradation due to cumulative noise amplification governed by the statistical properties of subsequent weight matrices. The study also demonstrates that pooling-based noise reduction is effective in both cases when noise is introduced before and after the activation function, consistently improving network performance.

In the last decades, energy-based models (EBMs) have become an important class of probabilistic models in which a component of the likelihood is intractable and therefore cannot be evaluated explicitly. Consequently, parameter estimation in EBMs is challenging for conventional inference methods. In this work, we provide a unified framework that connects noise contrastive estimation (NCE), reverse logistic regression (RLR), multiple importance sampling (MIS), and bridge sampling within the context of EBMs. We further show that these methods are equivalent under specific conditions. This unified perspective clarifies relationships among existing methods and enables the development of new estimators, with the potential to improve statistical and computational efficiency. Furthermore, this study helps elucidate the success of NCE in terms of its flexibility and robustness, while also identifying scenarios in which its performance can be further improved. Hence, rather than being a purely descriptive review, this work offers a unifying perspective and additional methodological contributions. The MATLAB code used in the numerical experiments is also made freely available to support the reproducibility of the results.

Picky eating in children can undermine dietary diversity and the development of healthy eating habits, while also creating recurring tension in family feeding routines. Prior interventions have explored food-centered designs, enhanced utensils, and mealtime interactive systems, but few position children as active participants in intervention processes that extend beyond single mealtime interactions. To better understand everyday responses to picky eating and child-acceptable intervention mechanisms, we conducted a formative study with caregivers and kindergarten teachers. Based on the resulting design considerations and iterative stakeholder review, we designed StoryEcho, a generative child-as-actor storytelling system for picky eating intervention. StoryEcho engages children outside mealtimes through personalized stories in which the child appears as a persistent story character and later shapes story development through real-world food-related behavior. The system combines non-mealtime story engagement, lightweight post-meal feedback, and behavior-informed story updates to support repeated intervention across everyday family routines. We evaluated StoryEcho in a between-group field study with 11 families of preschool children. Results provide preliminary evidence that StoryEcho can significantly increase children's willingness to approach and try target low-preference foods while reducing parental pressure around feeding. These findings suggest the promise of generative child-as-actor storytelling as a design approach for home-based behavior support that unfolds through recurring family routines.

Resilience in cyber-physical systems of systems (CPSoS) is often assessed using static indices or point-in-time metrics that do not adequately account for the temporal evolution of risk following a disruption. This paper formalizes resilience as a functional of the risk trajectory by modelling risk as a dynamic state variable. It is analytically shown that key resilience properties are structurally determined by maximum deviation (peak) and effective damping, and that cumulative risk exposure depends on their ratio. A simplified energy-dependent system illustrates the resulting differences in peak magnitude, recovery dynamics, and cumulative impact. The proposed approach links resilience assessment to stability properties of dynamic systems and provides a system-theoretically consistent foundation for the analysis of time-dependent resilience in CPSoS.

We consider the classic Multi-Armed Bandit setting to understand the exploration/exploitation tradeoffs made by different search heuristics. Since many search heuristics work by comparing different options (in evolutionary algorithms called "individuals"; in the Bandit literature called "arms"), we work with the "Dueling Bandits" setting. In each iteration, a comparison between different arms can be made; in the binary stochastic setting, each arm has a fixed winning probability against any other arm. A Condorcet winner is any arm that beats every other arm with a probability strictly higher than $1/2$. We show that evolutionary algorithms are rather bad at identifying the Condorcet winner: Even if the Condorcet winner beats every other arm with a probability $1-p$, the (1+1) EA, in its stationary distribution, chooses the Condorcet winner only with constant probability if $p=Ω(1/n)$. By contrast, we show that a simple EDA (based on the Max-Min Ant System with iteration-best update) will choose the Condorcet winner in its maintained distribution with probability $1-Θ(p)$. As a remedy for the (1+1) EA, we show how repeated duels can significantly boost the probability of the Condorcet winner in the stationary distribution.

We prove that the set of elements of a given finite order in the connected component $N_w$ of the normalizer $N_G(T)$ of a maximal torus $T$ of a semisimple group $G$ is either empty or a disjoint union of finitely many irreducible subvarieties $C_i$. The dimension of each $C_i$ equals the dimension of the subspace of fixed vectors for the action of the element $w$ of the Weyl group $W$ corresponding to the component $N_w$. Moreover, each $C_i$ is an orbit of the action of the torus $T$ on the component $N_w$ by conjugation.

The control of reflection and refraction at interfaces using engineered media is central to numerous optical technologies, with negative refraction and the suppression of backscattering representing two prominent research frontiers. In this work, we demonstrate that an effective negative refraction accompanied by an absence of backscattering can be realized at a moving spatiotemporal interface when temporal and spatial reflections occur concurrently. While such spatiotemporal co-reflection is prohibited in one-dimensional linear dispersive media, we show that it becomes permissible under oblique incidence within a specific range of traveling-wave modulation velocities. Leveraging this mechanism, we propose a spatiotemporal flat lens capable of nonreciprocal electromagnetic wave focusing. These findings provide a framework for developing advanced spatiotemporal metamaterials and time-varying metasurfaces.

Classical topological descriptors used in topological data analysis (TDA) are invariant under permutations of spatial axes and therefore cannot represent the loading direction, which is essential for modeling anisotropic mechanical response. Here, this limitation is addressed by introducing a \emph{direction-aware TDA framework} in which the compression axis is explicitly embedded into filtration functions used to compute both persistent homology and Euler characteristic profile descriptors. Across multiple porous-material datasets spanning a broad range of structural anisotropy, direction-aware descriptors yield higher predictive accuracy than their direction-agnostic counterparts, with performance gains that increase systematically with anisotropy. Notably, direction-aware descriptors remain competitive and often improve $R^2$ even for nominally isotropic ensembles, indicating sensitivity to mechanically relevant directional organization beyond bulk anisotropy measures. When used as inputs to gradient-boosted tree models, the proposed descriptors approach the accuracy of convolutional neural networks trained directly on voxelized structures while retaining a compact, transferable representation. The study considers multiple datasets spanning weak to strong anisotropy, enabling systematic validation of direction-aware topology across regimes. Overall, the results establish direction-aware TDA as a general route for linking porous structure to direction-dependent elastic properties and motivate its use in anisotropic materials modeling problems where a preferred direction naturally arises.

Multi-agent reinforcement learning serves as an effective tool for studying strategy adaptation in evolutionary games. Although prior work has integrated Q-learning with reputation mechanisms to promote cooperation, most existing algorithms adopt fixed exploration rates and overlook the influence of social context on exploratory behavior. In practice, individuals may adjust their willingness to explore based on their reputation and perceived social standing. To address this, we propose a Q-learning model that couples exploration rates with local reputation differences and incorporates asymmetric, state-dependent reputation updates. Our results show that each mechanism independently promotes cooperation, and their combination yields a reinforcing effect. The joint mechanism enhances cooperation by making ``high reputation--low exploration, low reputation--high exploration'', while adjusting reputation updates to amplify cooperative gains at low status and defection penalties at high status. This study thus offers insights into how social evaluation can shape learning behavior in complex environments.

Large language models (LLMs) have shown astonishing capability of generating software code, leading to its use to support developers in programming. Proposed tools have relied either on assistants for improved auto-complete or multi-agents, in which different model instances are orchestrated to perform parts of a problem to reach a complete solution. We argue that LLMs can enable a higher-level of abstraction, a new paradigm we called Test-Oriented Programming (TOP). Within this paradigm, developers only have to check test code generated based on natural language specifications, rather than focusing on production code, which could be delegated to the LLMs. To evaluate the feasibility of this proposal, we developed a proof-of-concept tool and used it to generate a small command-line program employing two different LLMs. We obtained promising results and identified challenges for the use of this paradigm for real projects.

Background: Composite endpoints in cardiovascular trials combine heterogeneous outcomes-mortality, nonfatal events, hospitalizations, and biomarkers-yet conventional analytical methods sacrifice information by targeting a single dimension. Cox time-to-first-event ignores post-first-event data; Win Ratio discards tied pairs; negative binomial regression treats death as noninformative censoring. Methods: We propose CWOT-CE: a Choquet integral-based composite endpoint analysis that encodes K = 6 outcome dimensions-survival, event-free time, AUC recurrent burden, last event time, biomarker, and alive status-and aggregates them through a non-additive fuzzy measure with pairwise interaction terms. The recurrent event process is represented as two complementary scalar summaries: the area under the cumulative count curve (AUC burden) and the last event time. Inference is via permutation test with exact finite-sample Type I error control and dual confidence interval by inversion. We conducted a simulation study comparing CWOT-CE against Cox TTFE, Win Ratio (WRrec), and WLW across 20 clinically motivated scenarios (1,000-5,000 replications). Results: Under the sharp null (5,000 replications), all methods maintained nominal Type I error (CWOT-CE: 4.8%, MCSE 0.3%). Across 17 non-null scenarios, CWOT-CE outperformed Cox TTFE in 15 (mean +28.8 pp), WLW in 14 (mean +27.2 pp), and Win Ratio in 10, with 5 ties and only 2 narrow losses (mean +5.6 pp). CWOT-CE showed particular advantages in high-correlation settings (+35.4 pp vs. WR), mortality-driven effects (+10.7 pp), and balanced multi-component effects (+10.1 pp). Shapley decomposition correctly identified effect-bearing components across all calibration scenarios. Conclusions: CWOT-CE with block recurrent encoding is broadly effective across clinically relevant scenarios while offering unique interpretive advantages through component attribution.

Consider a log canonical pair $(X,B)$ such that there is a Cartier divisor $D$ for which $T_X(-\log B) \otimes \mathcal O(D)$ is locally free and globally generated. Let $\mathcal F$ be a log canonical foliation of rank 1 on $X$. We prove that there exists a divisor $Γ$ such that $(X, Γ)$ is log canonical and $K_X + Γ\sim K_{\mathcal F} + D$. We then apply this result to prove several statements on the birational geometry of rank 1 log canonical foliations on log homogeneous varieties.

This study reconstructs the origin of a constant, here called $Ξ$ (Xi), as a primary scaling factor in Old Babylonian mathematics and astronomy. $Ξ$ arises from the practical necessity of precise measurements on the sky or a circle, through the harmonization of length-measure systems. The analysis of the Nippur measure (with its famous cubit) and the Gudea measure shows that $Ξ= 375/360$ represents the ratio of these established Old Babylonian measure systems. As a precision factor for circumference calculations, it remained in use until today. In Ptolemy's work, we find a slightly refined value of $Ξ= 377/360$. A further refinement of this coefficient led to our modern $π$, which still incorporates the two Old Babylonian components of a demonstrably two-stage calculation and refinement process. The accuracy increased by only 0.5\% compared to the first ratio. This factor, attested on several Old Babylonian cuneiform tablets including those from Susa, demonstrates the profound understanding of sexagesimal logic. The relative sexagesimal notation (60 = 1 = 1/60) enabled the universal application of $Ξ$ and its reciprocal for highly accurate calculations of arc-length on circular segments. This investigation leads ultimately to a surprising consequence: the modern radian measure is a direct descendant of this Old Babylonian coefficient.