It is well known that liveness properties cannot be proven using standard simulation arguments. This issue has been mitigated by extending standard notions of simulation for transition systems to fairness-preserving simulations for systems equipped with an additional fairness condition modeling liveness assumptions and/or liveness requirements. In the context of automated verification of finite-state systems, proofs by simulation are an appealing method as there exist efficient algorithms to find a simulation between two systems. However, applications of fair simulation to interactive verification have been much less studied. Perhaps one reason is that the definitions of fair simulation relations typically involve non-trivial nestings of inductive and coinductive relations, making them particularly difficult to use and to reason about. In this paper, we argue that in many cases, stronger notions of fair simulation involving more controlled alternations of fixed points are sufficient. Starting from known fair simulation techniques, we progressively build up a family of almost fair simulation relations for transition systems equipped with a Buechi fairness condition. The simulation relations we present can all be equipped with intuitive reasoning rules, leading to elegant deductive systems to prove fair trace inclusion. We mechanized our simulation relations and their associated deductive systems in the Rocq proof assistant, proved their soundness, and we demonstrate their use through a selection of examples.

We develop a theoretical and computational framework for estimating and correcting non-Abelian geometric distortions in time-bin photonic qudit processing when the relevant encoded object is a transported logical subspace rather than a collection of independent rays. In such settings, for example under mode mixing, multiplexed routing, or effective degeneracies, the geometric contribution is naturally matrix-valued and is described by a Wilczek-Zee holonomy on a rank-$m$ sub-bundle of the ambient Hilbert space. The framework generalises prior Abelian time-bin Pancharatnam-Berry feed-forward calibration, in which geometric distortions are represented by bin-resolved scalar phases, to the non-Abelian, matrix-valued case. We construct a gauge-covariant discrete estimator from overlap matrices between successive subspace frames: the polar factor of each overlap gives a unitary backward frame comparator, and the adjoint comparators compose to approximate the forward path-ordered exponential of the Wilczek-Zee connection. We prove gauge covariance under frame changes, polar optimality of the local comparator, consistency under partition refinement, and perturbative stability under well-conditioned overlap errors. We then formulate left- and right-acting feed-forward correction rules for removing the estimated holonomy from an effective logical operation. The work does not assume a device-specific transfer matrix, loss model, detector model, or experimental calibration pipeline; numerical studies use synthetic non-Abelian transport models to validate covariance, convergence, conditioning dependence, and correction fidelity.

We ask whether two classical time-machine geometries, the Ori (2005) compact-vacuum-core metric and the Ahmed (2018) four-dimensional generalisation of Misner space, remain admissible exact solutions when the gravitational sector is enlarged to the recently proposed $f(R,\mathcal{L}_{m},Φ,X)$ class, an extension of $f(R,\mathcal{L}_{m})$ that couples curvature, the matter Lagrangian density, a scalar field $Φ$, and its kinetic invariant $X = g^{μν}\nabla_μΦ\nabla_νΦ$. Working with the explicit model $f = R + \mathcal{L}_{m} + (λ/2)\,X$ and a vanishing scalar potential, we compute the curvature invariants, the modified field equations, and the effective stress-energy components produced by the harmonic scalar profile $Φ(x,y) = a(x^{2}-y^{2})/2$ in both backgrounds. The Ricci scalar vanishes for the Ori metric and obeys $R = e^{f}(f_{,xx}+f_{,yy})$ for the Ahmed metric; the kinetic invariant takes the explicit forms $X = a^{2}(x^{2}+y^{2})$ and $X = a^{2}e^{f}(x^{2}+y^{2})$, respectively. Both metrics solve the field equations of the modified theory with anisotropic matter sources, and the chronology-violating regions $g_{zz}<0$ (Ori) and $g_{ψψ}<0$ (Ahmed) survive the modification. Energy-density profiles measured by a closed-timelike-curve observer match those measured by a static observer outside the chronology horizon, so the additional scalar degree of freedom in $f(R,\mathcal{L}_{m},Φ,X)$ gravity does not enforce a chronology-protection mechanism in either background. The conclusion mirrors the parallel result for the Li time-machine and supplies a consistency test for scalar-extended modified gravity in non-globally-hyperbolic settings.

EuCd2As2 materials have two magnetic ordering states: antiferromagnetic (AFM) and ferromagnetic (FM) when their chemical tunability is utilized. While AFM-EuCd2As2 has a nonzero magnetoelectric response due to its symmetry breaking with spin configuration, FM-EuCd2As2 is an ideal candidate for studies of Weyl physics because of its minimum number of Weyl points with opposite chirality. In this article, we examine cryogenic low-frequency Raman spectroscopy of phonon modes in FM-EuCd2As2 crystals using circular polarization configurations, with support from density functional theory calculations, and investigate in-plane magneto-anisotropy by linear polarization configuration below the Curie temperature (Tc = 26 K). We attribute the anomalous enhancements in Raman intensities below the Curie temperature are due to spin-phonon coupling. Furthermore, we see that A-mode peaks can be distinguished by magneto-helical Raman spectroscopy through the magneto-optic effect and that the degree of circular polarization (DCP) of 12.5 meV peak reaches 60% at 4.2 K and becomes saturated. We also examine AFM-EuCd2As2 below Néel temperature (TN = 9 K) to compare with FM-EuCd2As2, but we hardly observe spin-phonon coupling and find negligible DCP values due to almost zero net magnetization. Our results contribute to the understanding of the phonon dynamics and the interplay between topology and magnetism in FM-EuCd2As2, through helical light and external magnetic fields. This lays the foundation for utilizing state-of-the-art Weyl systems for applications in thermoelectrics, phononic devices, and topological quantum computing.

We introduce a notion of proxy smallness for $t$-structures on triangulated categories associated to a Noetherian scheme. Specifically, the theory is developed in the presence of tensor actions. Consequently, our results yield a new characterization of schemes that are locally complete intersections in terms of $t$-structures, as well as a topological classification of preaisles on the bounded derived category of coherent sheaves.

The effectiveness of dimensionality reduction with quadratic manifolds hinges on the choice of a reduced basis and the associated quadratic correction terms. Existing approaches typically rely on subspaces spanned by the leading principal components of the training data. Although optimal for linear approximation, such bases are inherently suboptimal for quadratic manifold learning. Greedy basis-selection methods can significantly improve the representational capacity of quadratic manifolds by searching over a larger pool of candidate principal components, but the combinatorial cost limits the basis sizes that can be used in practice. This work proposes FastQM, an approach that treats the identification of an optimal quadratic approximation as a continuous optimization problem on the Stiefel manifold. By rotating the reduced basis within a candidate span of singular vectors, FastQM learns an ideal coordinate alignment tailored to quadratic manifold approximation. A feature-space formulation ensures that the optimization cost scales independently of the full state-space dimension. The efficacy of the proposed method is demonstrated on a turbulent airfoil-wake large-eddy simulation.

We develop the theory of lattice point counting on connected and simply connected nilpotent Lie groups of step-two, endowed with the parabolic type dilation and a family of homogeneous norms $ \mathcal{N}_{α,M}(x, t)=\left(|M_1x|^α+ |M_2t|^{α/ 2}\right)^{1 / α}$ adapted to the dilation structure, where $α>0$ and $M_1,M_2$ are invertible matrices. With appropriate notions of lattices, the domains to be counted are balls associated to these norms, and explicit counting discrepancy estimates are deduced for all possible dimensions and all $α>0$. The bounds are sharp when the group center is unidimensional and $α=2$, in certain rational sense. Our study also generalizes and even quantitively improves previous results on Heisenberg groups obtained by Garg--Nevo--Taylor \cite[\textit{Ann. Inst. Fourier}, 2015]{GNT15}: (i) In dimension $5$, the exponent of logarithmic factor is lowered from $2/3$ to ${1}/{3}$ if $α\in(3,4) $ or $α=1$; and the factor $\log ^{2/3} R $ is dropped if $α=4$ (i.e., the Cygan--Korányi norm case) or $α\in(2,3]$. (ii) In dimension $3$, the estimation is upgraded from $O_ε(R^{ 5/2+ε})$ to $O(R^{2}\log^{ 1/2} R)$ for $α=1$, and to $O(R^{{19}/{8}})$ for $α\in (1,2)$; and the factor $\log R$ is removed for $α>4$. Moreover, as a byproduct, we extend the lattice counting near Heisenberg spheres, recently considered by Campolongo--Taylor \cite[\textit{Matematica}, 2023]{CT23} and Srivastava--Taylor \cite[\textit{J. Fourier Anal. Appl.}, 2026]{ST26}, to the above step-two group setting with arbitrary dimensional group center, where some quantitive improvements are also attained. Our method relies upon Poisson's summation formulas, oscillatory integral estimates and asymptotic properties as well as recursion formulas of Bessel functions.

We prove a nonterminating well-poised basic $q$-Taylor expansion with complementary remainders for a two-basis infinite-product kernel implicitly proposed by the second author in \cite[Sec.~5]{Schlosser2008}. The well-poised parameter $c$ gives the rational $p=0$ basis, while the elliptic nome $p$ is a separate deformation; the infinite expansions treated here are specific to the basic case. We compute the two Taylor coefficient families and show that each one-family Taylor remainder tends to the complementary basis contribution. The proof uses the well-poised Cooper formula, Jackson's terminating ${}_8ϕ_7$ summation, Rogers' ${}_6ϕ_5$ summation, and theta interpolation, but not Bailey's nonterminating ${}_8ϕ_7$ summation, which is recovered as a consequence. We also record two quadratic one-family examples and discuss a multi-kernel outlook.

Core-collapse supernovae are the terminal explosions of massive stars. After successive phases of nuclear fusion proceeding up to silicon burning, these stars form an iron core that is supported by electron degeneracy pressure. The core eventually collapses to a proto-neutron star, and in most cases the outer layers of the star are ejected by a shock wave, with a kinetic energy of order $10^{51}\,\mathrm{erg}$. Neutrinos and multi-dimensional fluid flow play a key role in extracting energy from the collapsed core to drive the explosion. After adumbrating the astrophysical context of stellar evolution and transient observations, this chapter sketches the modern theory of neutrino-driven supernova explosions, and discusses the key role of nuclear physics and neutrino interaction rates in the supernova problem. It also outlines the role of neutrinos and gravitational waves as probes into the supernova core.

While Conformal Prediction (CP) has proven to be a powerful framework for uncertainty quantification, guaranteeing conditional coverage remains a central challenge. Although finite-sample, distribution-free conditional validity is known to be impossible without structural assumptions, we show that it is fundamentally equivalent to constructing a nonconformity score whose distribution is independent of the features. This theoretical characterization motivates PIT-CP, a new post-processing correction that maps any base nonconformity score to an approximately invariant one while preserving its geometry, interpretability, and marginal coverage. This perspective is particularly appealing in practice, since it may be neither economical nor time-effective to retrain a full generative model when a strong prediction-driven model already provides highly accurate point estimates. Our procedure reduces the problem to one-dimensional conditional density estimation on the induced score, rather than full conditional density estimation on the original outcome space. We show how to estimate this transform in practice and derive bounds on the conditional coverage gap, alongside volumetric and symmetric-difference bounds. We present known minimax-optimal conditional estimation techniques while also motivating the use of modern conditional density estimators, including Mixture Density Networks and Conditional Normalizing Flows. Finally, we empirically demonstrate on various datasets that our PIT-CP procedure matches or outperforms many state-of-the-art conformal prediction strategies with minimal effort and computational cost.

We study the pointwise convergence of solutions to the free Schrödinger equation with initial data in the Bessel potential spaces $L_s^p(\mathbb{R}^n)$. We establish new sufficient regularity indices for pointwise convergence across the full range $1 \leq p < \infty$, and demonstrate via counterexamples that these indices are sharp for all $1 \leq p \leq 2$ in one dimension, as well as for $p=1$ or $p$ large enough in higher dimensions. The proofs rely on the high-dimensional stationary phase method.

Gamma-ray bursts (GRBs) show diverse X-ray afterglow light-curves, including breaks and plateaus, whose physical origins remain debated. Previous claims linked high-energy ($E \ge 100$ MeV) detection to X-ray afterglow complexity or plateau incidence, but they were often based on small or heterogeneous samples. We present a large-scale, uniform, model-independent analysis of the complete Swift-XRT GRB afterglow catalog, including more than 1400 events. Our automated pipeline performs flare removal and segmented power-law fitting consistently across the sample. We find that both light-curve complexity and plateau incidence are strongly governed by the XRT observation start time, $t_{XRT}$. Apparent correlations between high-energy emission and X-ray morphology arise when $t_{XRT}$ is ignored, but vanish when the sample is stratified or controlled for this variable. X-ray complexity and plateaus are therefore not directly coupled to high-energy detectability, and early X-ray morphology is not predictive of high-energy emission. These results resolve conflicting claims in the literature and show that controlling for $t_{XRT}$ is essential in large-sample GRB studies. The automated pipeline provides a reproducible basis for future analyses of GRB afterglows from Swift and upcoming missions such as SVOM, Einstein Probe, and THESEUS.

Test Suite Minimization (TSM) reduces the size of test suites while preserving their fault detection capability. In black-box TSM, reduction is performed without relying on production-code instrumentation. While several black-box TSM approaches have explored metrics like test logs or test similarity, these often suffer from scalability and efficiency issues. Recently, change history has been explored as a lightweight and scalable indicator for guiding black-box TSM. However, existing approaches treat historical modifications uniformly, ignoring the temporal dynamics of software evolution where recently modified code tends to be more fault-prone. To address this limitation, we introduce temporal modeling into black-box TSM and propose Temporal Risk-driven Test Suite Minimization (TRTM). TRTM extracts modification history from version-control metadata and applies exponential temporal attenuation to weight changes based on recency, producing time-weighted class-level risk scores that reflect fault-proneness. Next, it determines dependencies between test cases and production classes by constructing static call graphs derived solely from test code, preserving the black-box setting. The risk scores of the classes exercised by each test case are then aggregated using statistical measures such as Average and Geometric Mean to compute a risk score for the test case. Finally, test cases with the highest risk scores are selected to construct the reduced suite. Evaluation on a large dataset containing 14 projects with 631 versions shows that TRTM consistently outperforms the state-of-the-art baseline, achieving a mean Accuracy of 0.72 (vs. 0.66) and Fault Detection Rate (FDR) of 0.75 (vs. 0.69), while also reducing execution time.

We study the facility location mechanism design problem where $n$ agents report their locations in Euclidean space, and the output is a single facility location. The cost function of each agent is the distance from the returned facility, and the objective is to minimize the social cost function (the sum of agent costs) in a strategyproof way. Our contributions: 1. Breaking the deterministic barrier. For $\mathbb{R}^2$, we give a random strategyproof mechanism (RR-CWM) achieving an expected approximation ratio of $\frac{4}π \approx 1.27$, which strictly improves upon the best deterministic strategyproof mechanism (which has a $\sqrt{2} \approx 1.41$ ratio). This closes the open problem of separating deterministic and random mechanisms for utilitarian facility location mechanism design in $\mathbb{R}^2$. For $\mathbb{R}^d$, we show that the expected approximation ratio of our mechanism is in $[1.41 - O(1/\sqrt{d}), 1.547]$. 2. Improved learning augmented mechanisms through randomization. We show our ideas can achieve better performance in the learning augmented setting in $\mathbb{R}^2$, where in addition to the input the mechanism also receives predictions. For the output prediction model of Agrawal et al. 2022 we show an improved expected consistency-robustness trade-off. Our results also imply improved performance for the input MAC predictions model of Barak et al. 2024. 3. The limitations of Random Dictators. We show a lower bound for the common mechanism class of GRD (Generalized Random Dictator) mechanisms, where only locations reported by the agents may be returned. We show that any GRD mechanism has a larger expected approximation ratio than our RR-CWM mechanism, as our lower bound for $\mathbb{R}^2$ is $\frac{4}π$ (matching the upper bound of RR-CWM, which is not a GRD mechanism). For $\mathbb{R}^d$, we show a lower bound of $\sqrt{2} - O(1/d)$.

The chord-power integrals $I_k$ are classical integral-geometric functionals of a planar convex body, obtained by integrating powers of the chord length against the kinematic measure on the space of lines meeting the body. We establish a single-expression closed form for $I_2$ on an arbitrary triangle, involving logarithms symmetric in the sides, and derive two analytic consequences: a power-sum series representation, and a sharp isoperimetric-type inequality with explicit constant involving $\ln 3$, attained uniquely by the equilateral triangle. The set $\{I_0, I_1, I_2\}$ identifies a triangle up to congruence, complementing J. Gates's algebraic recognition via $\{I_0, I_1, I_5\}$ with the minimal index set $\{0, 1, 2\}$.

The electric power supply for AI data centers is now the most significant bottleneck in the race toward Artificial General Intelligence, surpassing even the constraint of AI accelerator availability. To our knowledge, this paper is the first to describe the end-to-end power management process for a hyper-scale AI datacenter; from early power planning to accommodate next-generation accelerators 6--12 months before their general availability, to tuning power settings after large scale deployment, and finally to dynamic, runtime power management for evolving workloads. We present detailed power measurements for a 150 MW datacenter hosting a cluster of 83K GB200 GPUs. We share insights from building this state-of-the-art AI cluster. We hope this work encourages practitioners across the industry to share their own experiences as well.

Comparison geometry for Bakry-Émery Ricci curvature has been extensively developed by Wei-Wylie and others. Motivated by the weighted sectional curvature framework introduced by Wylie and further developed by Kennard-Wylie-Yeroshkin, we study radial comparison geometry on manifolds with density through a modified Hessian arising from this framework. Under nonnegative weighted sectional curvature together with suitable density control assumptions, we obtain a modified Hessian estimate for the radial function $u = \frac{1}{2}r^2$. From this estimate, we derive Hessian comparison, shape operator comparison, weighted Laplacian comparison, asymptotic radial volume density estimates, and polynomial weighted volume growth bounds. We introduce a normalized weighted radial volume density satisfying a monotonicity property analogous to the radial volume density monotonicity underlying Bishop-Gromov comparison. We also study rigidity phenomena associated with these comparison estimates. Equality in the Hessian comparison theorem yields radial conformal rigidity, while equality in the modified Hessian estimate forces the metric to have an exact metric cone structure.

Low-Gain Avalanche Diode (LGAD) sensors, offering timing resolutions of the order of tens of picoseconds, are being widely adopted in particle physics experiments and related applications. As these applications scale to large numbers of sensors with varying pixel geometries, conventional manual characterization techniques become inadequate for large-scale quality control. We present a modular probe card system for automated electrical characterization of pixelated LGAD sensors, consisting of a probe card, a switching board, precision measurement instruments, and control software. The system supports flexible pixel selection and measurement. Its performance is demonstrated through current-voltage (I-V) and capacitance-voltage (C-V) measurements of a $16 \times 16$ LGAD array. A rapid row-wise I-V scan of the full array is completed in approximately 20 minutes, while a pixel-by-pixel I-V scan from 0 to 300 V with a 1 V step requires about 340 minutes. The switching matrix introduces less than 1 nA of leakage current even in a conservative worst-case configuration, remaining small compared with the leakage current of a normal LGAD pixel. The modular architecture and automation capability make the system a practical and scalable solution for large-scale LGAD sensor quality control and distributed testing environments.

Urban decarbonization requires scaling rooftop solar across millions of fragmented producers, yet cities face a fundamental tension: energy data is easily manipulated, and economic incentives often reward speculation rather than actual infrastructure deployment. We present SolarChain, a platform that resolves both problems by anchoring digital accountability to the thermodynamic limits of solar energy conversion. Using real-time meteorological data, geospatial coordinates, and first-principles calculations of solar yield, the system establishes a hard physical boundary for every panel's maximum possible output; any reported generation exceeding this limit is automatically rejected before entering the shared ledger. This trustless verification enables a peer-to-peer marketplace with programmatic reward structures that continuously reinvest value into equipment maintenance and market liquidity, preventing the speculative hoarding that typically destabilizes blockchain-based marketplaces. When electricity is consumed, the corresponding digital credits are permanently retired in direct proportion to physical energy dissipation, creating an auditable one-to-one mapping between urban consumption and carbon accounting. Deployed across heterogeneous city nodes, the prototype demonstrates resilience against data injection attacks while lowering capital barriers for community-level solar expansion. Beyond energy, the framework offers a general model for coordinating economic activity with physical law in any domain where distributed infrastructure demands both data integrity and sustainable investment. We release the data and code as open-access on GitHub.

In this white paper, we summarize for the benefit of the wider research community on wireless communications, the two key results that we shared with the attendees of the 2026 IEEE Communication Theory Workshop in Azores, Portugal, about affine frequency division multiplexing (AFDM). Firstly, we show that in contrast to the wide perception by most researchers, AFDM can be implemented at marginal costs by means of a simple software upgrade (firmware patch) of conventional orthogonal frequency division multiplexing (OFDM), indicating that its adoption can potentially be achieved across a wide range of OFDM-based wireless infrastructure and systems. The most crucial relevance of this finding is that such an upgrade would enable, under the specific conditions of the corresponding systems and their applications, exploiting various advantageous features of AFDM, including robustness to doubly dispersive channels (i.e., to support high-mobility use-cases in 6G), inherent integrated sensing and communications (ISAC) compatibility (i.e., to support sensing use-cases in 802.11bf), and the straightforward introduction of low-complexity physical-layer security at the waveform level (as needed in next-generation IoT systems). Secondly, we also show that the same mathematical principles underpinning the aforementioned finding, also imply an inherent capability of AFDM to reap the full uncoded diversity of static linear time-invariant (LTI) channels, demonstrating that this simple upgrade taps into previously undiscovered strengths of multicarrier waveforms.

Irreversibility theorems -- such as the $A$-theorem -- establish a hierarchy among fixed points of the renormalization group flow. The strongest thesis of this type of theorems would be that there exists a scalar function $A$ (generally suggested by the topological Weyl anomaly) and a positive definite metric $G_{IJ}$ in the space of couplings such that the renormalization group flow satisfies a gradient equation, $\partial_I A= G_{IJ}β^J$, in which case $A$ is locally monotonic along the flow. In this paper we consider the long-range multiscalar $ϕ^4$ theory, a theory without a local energy-momentum tensor that is unitary in $d=2,3$ and that is believed to be conformally invariant at fixed points, and show that its renormalization group flow satisfies the gradient structure up to the third loop order in the coupling. We also show that $A$ and $G_{IJ}$ can be matched to the leading nontrivial order with the sphere free-energy $\tilde{F}$ and Zamolodchikov's metric $C_{IJ}$ of the corresponding conformal theory concentrating on the examples of the long-range vector $O(N)$ and hypercubic $H_N$ models. Our results imply a perturbative proof of the $\tilde{F}$-theorem at the leading nontrivial order. We conclude the paper discussing briefly whether this result should hold to the next orders in perturbation theory.

Trapped ions excited to high-lying electronic states combine strongly coupled collective vibrational and electronic degrees of freedom with long-ranged interparticle interactions. These ingredients enable the quantum simulation of biochemical processes, associated with the dynamics of excitons in non-perturbative parameter regimes. The key feature of such a quantum simulator are electronic-state-dependent molecular potential surfaces which can be strongly coupled. This allows to shed light on a variety of mechanisms underlying exciton transport. We illustrate this in a system of three trapped ions, which is amenable to an ab initio treatment. Given that ion traps can be routinely prepared with hundreds of ions, these quantum simulators can immediately realise scenarios which are inaccessible by current numerical methods.

Polymer-derived ceramics combine the thermal stability of ceramics with the versatile properties of carbon domains, but modeling their atomic-scale evolution during processing remains elusive due to the limitations of traditional computational methods. To address this issue, here we develop and apply a machine learning interatomic potential for silicon carbonitride-based (Si-C-N-H) systems, trained on a diversified database of over 9000 configurations -- including amorphous models, high-temperature states, surfaces, and crystal structure predictions -- to capture the full complexity of these materials. This potential enables large-scale molecular dynamics simulations of 8000-atom systems revealing the atomic-scale evolution of the polymer-derived ceramic during thermal treatment. A key result of this work is the occurrence of a phase separation where carbon domains progressively nucleate from the amorphous SiCN matrix during thermal processing, forming distinct graphene-like sheets while preserving the integrity of the ceramic network. The resulting models reproduce the experimental atomic pair distribution functions with exceptional fidelity, validating our approach and providing microscopic explanations for the material unique combination of ceramic and graphitic properties. In this process, defective 5- and/or 7-member carbon rings, mediate the transformation to stable 6-member aromatic structures. These findings offer new atomic-scale insights into the thermal stability and structural transformation pathways of polymer-derived ceramics, while our methodology opens avenues for studying complex amorphous systems with first-principles accuracy at experimentally relevant scales.

We propose a notion of scalar curvature lower bounds in a three-dimensional Riemannian manifold endowed with a $C^0$ metric based on the monotonicity of the Hawking mass along the inverse mean curvature flow. We present a stability theorem for continuous Riemannian metrics with nonnegative scalar curvature in such IMCF sense.

We consider the problem of computing sample points in each connected component of a semi-algebraic set defined by the non-vanishing or the positivity of an n-variate polynomial of degree d, with rational coefficients of bit size bounded by $τ$. Such a problem is a basic routine in effective real algebraic geometry, used in higher-level algorithms for solving polynomial systems over the reals and finds many applications in sciences. We design a probabilistic algorithm for solving this problem, which is based on reductions to different routines for solving zero-dimensional polynomial systems. It assumes that the input polynomial satisfies sufficiently generic properties (namely, smoothness of its defining hypersurface). This is done through the computations of critical points of well-chosen maps to capture the connected components of the semi-algebraic set under study. We derive a bit complexity estimate for the cost of this algorithm, which is, in terms of the B{é}zout bound d(d -1)^{n-1}, essentially cubic for obtaining parametrisations of the sought-for real points. Moreover, we also consider the case of obtaining rational approximations of those points, which are precise enough to lie in the same connected components as their exact counterparts, which yields a cost that is essentially quartic in the B{é}zout bound. In these complexity estimates, we take into account the degree structure of the input polynomial and its partial derivatives, allowing for a more refined bit complexity when the partial derivative of the input polynomial have degree lower than expected. We also analyse the probability of success of those algorithms. We report on practical experiments, benchmarking with random dense input polynomials as well as polynomials coming from applications, which were out of reach of the state-of-the-art implementations, and hence illustrate the practical efficiency of these new algorithms.

The compressible barotropic Navier--Stokes equations in vector-invariant form preserve the vorticity structure of the system and underlie modern atmospheric and ocean dynamical cores, yet no PDE theory has been developed for the compressible discrete system in this form. On a Delaunay--Voronoi mesh we prove via discrete exterior calculus, that every density-independent mass matrix with integration-by-parts-consistent divergence carries a sharp $\OO(h^2)$ energy residual of indeterminate sign that no operator choice can eliminate. This no-go theorem covers A-, B-, C-, D-, and quasi-B-grid staggerings. The density-weighted mass matrix is the unique algebraic remedy: it restores exact total energy while preserving the vector-invariant momentum equation, Lamb antisymmetry, and the topological conservation laws, at the cost of an $\OO(h^{r_\star})$ Kelvin defect matching the convergence rate. The residual is the cause of the Hollingsworth instability that has shaped vector-invariant dynamical-core design; the density-weighted construction removes it structurally. For the density-weighted~(DW) scheme on closed oriented Riemannian manifolds in $d = 2, 3$ we establish global well-posedness for $ν\ge 0$, convergence to smooth solutions uniformly in $ν$, and asymptotic preservation in the low-Mach limit; the density-free residual diverges as $\OO(M^{-1})$. Via a discrete Arnold energy-Casimir construction, exact discrete conservation forces Lyapunov stability around three classes of equilibria, excluding Hollingsworth instability: unconditional stability around hydrostatic and constant-flow stratified states, and conditional stability around sheared baroclinic states under a discrete Charney--Stern criterion. The DW scheme admits genuine baroclinic instability only when the continuum equations themselves do.

We investigate a geometric and distributional reinterpretation of Chatterjee's $ξ$-coefficient, which measures functional dependence between a response variable $Y$ and a predictor vector $\mathbf{X}$. For this purpose, we analyze the Markov product $(Y,Y')$, where $Y'$ is a copy of $Y$ that is conditionally independent of $Y$ given $\mathbf{X}$. Based on this construction, we introduce and study two dependence functions, denoted by $ϕ_{(Y,\mathbf{X})}$ and $κ_{(Y,\mathbf{X})}$. The proposed framework provides a geometric interpretation of the Markov product and extends Chatterjee's correlation coefficient to a richer and more interpretable object for the analysis of directed stochastic dependence. In particular, rather than only measuring how well $Y$ can be represented as a function of $\mathbf{X}$, the proposed dependence functions additionally quantify how strongly the corresponding Markov product is concentrated near the diagonal.

Predicting and controlling the transport of colloids in porous media is essential for applications ranging from contaminant remediation to drug delivery. In these complex environments, solute gradients are ubiquitous and could drive diffusiophoretic particle migration, yet their impact on macroscopic colloid dispersion remains poorly understood. Here we combine experiments and simulations to quantify how diffusiophoresis alters the spreading of a colloidal blob in a 2D ordered/disordered porous medium. A joint blob of colloids and salt at high concentration is introduced into a medium filled with salt at low concentration and advected by a background flow. Intuition suggests that when colloids are attracted toward or repelled from the solute-rich blob, dispersion should be suppressed or enhanced, respectively. Instead, we observe the opposite trend: longitudinal dispersion is enhanced in the attractive case, whereas dispersion is suppressed in the repulsive case. Numerical simulations reveal that this striking reversal arises from diffusiophoretic exchange of particles between slow and fast streamlines, which we capture using a minimal two-layer model of coupled fast and slow plug flows. Finally, we probe how geometric disorder in the medium modulates this mechanism. Our results demonstrate that diffusiophoresis can strongly modulate macroscopic dispersion of colloids in porous media with implications for transport in subsurface and biological environments.

Numerical simulations of compressible real-fluid flows are notoriously plagued by spurious pressure oscillations arising in regions of abrupt flow variations. As a possible remedy, several numerical formulations enforce the pressure equilibrium condition for the compressible Euler equations, typically at the cost of spoiling the correct conservation of total energy or by overspecifying the thermodynamical variables. This study proposes for the first time a numerical discretization procedure which is able to discretely preserve the full conservation of the linear invariants (mass, momentum and total energy) and to exactly enforce the pressure equilibrium condition. The method also preserves the conservation of kinetic energy by convection, and is based on the specification of nonlinear numerical fluxes for mass and internal energy which depend on the details of the equation of state. Both thermally perfect and real gases with an arbitrary equation of state are considered, and a simplified approximate pressure equilibrium preserving formulation with excellent performances is also proposed. The effectiveness of the novel formulations is assessed through a series of numerical simulations in supercritical and transcritical conditions with some of the most popular cubic equations of state.

Optimal maintenance policies play an important role in the reliability analysis of repairable systems. This paper examines a two-unit priority standby system with a repair facility, where the priority unit is subject to preventive maintenance. In the general case, we derive necessary and sufficient conditions under which maintenance increases the mean time to system failure. When the hazard rate of the priority unit has either a bathtub-shaped or an upside-down bathtub-shaped form, we establish conditions for the existence of both a threshold time, beyond which maintenance improves the mean time to system failure, and the optimal maintenance time. Expressions to find these quantities are provided in terms of the hazard rate and the mean residual life function. Furthermore, stochastic ordering techniques are used to compare two independent systems under this model, along with their respective threshold times and optimal maintenance times.