In this paper, we prove that if the Frobenius traces agree at all but finitely many places, then two $l$-adic Galois representations, associated to rank-$2$ non-CM Drinfeld modules of generic characteristic, are isomorphic. As a generalization, we show that under a mild dimension hypothesis, the "Frobenius trace equality at all but finitely many places forces isomorphism" between two Galois representations over a local field of positive characteristic only holds under an absolute irreducibility assumption. Moreover, we formulate and prove a function field analogue of strong multiplicity one property for semisimple Galois representations over a local field of positive characteristic.

Decentralized Finance (DeFi) is a rapidly evolving segment of blockchain technology that enables a transformative approach to financial services through Web3 applications. By leveraging smart contracts, DeFi allows developers to build flexible and innovative financial instruments. Among the most prominent DeFi primitives by liquidity are decentralized exchange~(DEX) swap protocols~(such as Uniswap, Curve, and Balancer) that facilitate fast token-to-token exchanges. However, new exchange mechanisms also introduce new market inefficiencies that can be systematically exploited by arbitrageurs. This paper focuses on swap protocols based on the Automated Market Maker~(AMM), where the product of reserves is preserved as an invariant. We analyze the interaction between arbitrageurs and AMM liquidity pools and develop a mathematical model grounded in empirical pool configurations. Using this model, we derive bounds on the joint revenue of liquidity providers~(LPs) and arbitrageurs, propose a method to estimate the expected number of blocks until the occurrence of Impermanent Loss~(IL), and obtain a lower bound on the pool fee required to achieve a fixed target probability of staying in the Impermanent Gain (IG) zone within a block. The proposed framework extends existing LP risk-assessment methodologies by quantifying symbiotic profitability zones, providing a principled basis for fee selection that aligns LP-arbitrageur incentives and enhances market stability.

The long-standing tension in the Hubble constant $H_0$ has motivated extensive explorations of both new physics and observational systematics, for example, the late-time systematics in measuring the B-band absolute magnitude $M_B$ of type Ia supernovae, which is degenerated with $H_0$ via an intercept $-5a_B=M_B+5\lg (c/H_0/\mathrm{Mpc})+25$ in the linear relation $m_B=5\lg d_L(z)-5a_B$ between the apparent magnitude $m_B$ and logarithmic dimensionless luminosity distance $\lg d_L(z)$. Therefore, this intercept can be evaluated directly from pure observational quantities ($m_B$ and the redshift $z$) for a given model of $d_L(z)$ without knowing underlying systematics in $M_B$-$H_0$ degeneracy. Hence, the constancy of this intercept across different supernova datasets and different redshift bins within the same dataset for a given late-time model serves as a powerful diagnostic for disentangling late-time new physics from local supernova systematics. In this mini-review, we will show that: (1) there is a local $a_B$ tension in PantheonPlus around $z\sim0.01$, and the elimination of it leads to a $H_0$ measurement consistent with both SH0ES typical three-rung and first two-rung measurements; (2) there is a late-time $a_B$ tension in DES-Y5 around $z\sim0.1$, and the elimination of it largely reduces the preference for dynamical dark energy. We also update the late-time $a_B$-tension analysis for both DES-Y5 and DES-Dovekie supernovae, and find that this $a_B$ tension around $z\sim0.1$ is mainly driven by the inter-data tension between DES supernovae and DESI+Planck constraint, and the dynamical dark energy is preferred as a compromise of this tension. Finally, we briefly mention an interacting dark energy model that resolves this tension among DES, DESI, and Planck, and point out a crucial difference between the effective and apparent equations of state of dark energy.

Fast radio bursts (FRBs), thought to originate from magnetars, exhibit diverse polarization properties that constrain their emission physics and local magneto-ionic environments. The polarization position angle (PPA) is particularly sensitive to magnetic-field geometry in the emitting region and propagation effects in the magnetosphere and beyond. In hyper-active repeaters, PPAs are typically stable within bursts and over timescales of hours to days. Here, we present observations of the repeating source FRB~20240114A, which show significant burst-to-burst PPA variations. Using full-Stokes, high-time-resolution observations from the Nançay Radio Telescope (1.1--1.8\,GHz) and the Effelsberg 100-m telescope (1.3--1.5\,GHz) over $\sim1$~year, we measure rotation measures (RMs), polarization fractions, and time-resolved PPAs across 12 epochs. The RMs remain stable, and the emission is predominantly highly linearly polarized, with $\sim81\%$ of bursts showing $L/I > 0.8$, while circular polarization is weaker ($\sim16\%$ with $|V/I| > 0.1$). We find no evidence for Faraday conversion. The PPA exhibits rapid, stochastic variations from milliseconds to hours, spanning $\pm90^\circ$ during two active periods and $\pm50^\circ$ in a third. The distribution of PPA jumps shows that (1) there is no difference in the distribution of jumps on timescales shorter or longer than 1\,s; (2) positive and negative jumps are equally likely; and (3) a jump of $\pm90^\circ$, as expected from, e.g., orthogonal mode jumps, is not more common than any other value. This combination of stable RM, high linear polarization, and extreme PPA variability is not seen in other hyper-active repeaters. These results disfavor emission from a single fixed region and instead suggest multiple emission regions and/or strong magnetospheric and foreground propagation effects, such as plasma lensing.

Quantum control with restricted state access is central to near-term quantum devices, where full wave-function information is unavailable. We study this problem through multiqubit disentanglement scheduling from partial observations, where a controller receives only two-qubit reduced density matrices and selects which qubit pair to disentangle at each step. We introduce a modular hybrid quantum--classical policy framework consisting of classical preprocessing, a parameterized quantum circuit as a compact nonlinear latent block, and classical postprocessing for pair-selection probabilities. Benchmarking 4-, 5-, and 6-qubit tasks, we find that preprocessing is the dominant factor governing performance under reduced-state observations, while the quantum module provides a conditional compact representation whose utility depends on the input features and model budget. We further identify a performance--efficiency trade-off across policy families and find that increasing circuit width is generally more useful than increasing depth. These results provide practical design principles for hybrid policies in reduced-information quantum control.

For three-dimensional (3D) magnetic objects with linear size $L$ exceeding a few exchange lengths, the micromagnetic state exhibits pronounced informational sparsity: low-dimensional, high-gradient regions (e.g., domain walls) coexist with near-uniformly magnetized volumetric domains. Because standard micromagnetic simulation methods discretize the magnetization on near-uniform 3D grids with linear cell size $a$, they cannot take advantage of this sparsity. The computational problem scales as $\sim L^3$ and $\sim (1/a)^3$. In this Letter, we establish that direct tensor-train (TT) representations overcome these poor scalings by exploiting the spatial sparsity optimally, while preserving accuracy in a controlled way. Focusing on representative flux-closure configurations in soft-magnetic rectangular prisms, in the near-micrometer regime, we demonstrate that the parameter count of TT-compressed micromagnetic data scales approximately as $L^{1.8}$ and $(1/a)^{1.2}$. Hence the relative advantage over dense discretizations rapidly grows with the problem size and refinement level. These first results provide a strong motivation for future developments of micromagnetic solvers in TT format which could transcend the limitations of traditional simulators, with far reaching potential impacts on fundamental research and technology development.

Associating multiple sensory cues with a single experience or object is a fundamental process that improves object recognition and memory performance. However, neural mechanisms that bind sensory features during learning and augment memory expression are unknown. Here we demonstrate multisensory appetitive and aversive memory in Drosophila. Combining colours and odours improved memory performance, even when each sensory modality was tested alone. Temporal control of neuronal function revealed visually-selective mushroom body Kenyon Cells (KCs) to be required for enhancement of visual and olfactory memory recall after multisensory training. Synapse-level connectomics suggests that valence-relevant dopaminergic reinforcement could permit the KC-spanning serotonergic DPM neurons to bridge between previously modality-selective KC streams. Consistent with this model, DPM transmission is uniquely required during multisensory memory formation and for enhanced expression of olfactory memory afterwards. In addition, signalling via the DopR1 dopamine receptor is required in APL neurons, suggesting that reinforcing dopamine could locally release GABA-ergic inhibition to permit bridging microcircuits to function. Cross-modal binding thereby expands the KCs representing the olfactory memory engram into those representing the colour. We propose that broadening of the engram improves memory performance after multisensory learning and permits a single sensory feature to retrieve the memory of the multimodal experience.

We demonstrate a technique for spatially resolved temperature measurement utilizing Rydberg Doppler broadening thermometry. This method employs two focused laser beams arranged perpendicularly to excite laser-cooled atoms from the ground state to a Rydberg state via two photon absorption process. Temperature is obtained through the Doppler broadening of the spectral line. The perpendicular configuration allows for selective probing of a specific position within the atomic cloud, enabling localized temperature measurement. This technique, in principle, offers a temperature resolution on the order of \SI{}{\nano\kelvin}, attributed to the exceptionally narrow natural linewidth of the involved rubidium Rydberg transition line. Furthermore, the setup enables the measurement of position-velocity correlations within the cold atom ensemble. The velocity information is extracted through the Doppler shift, whereas the spatial information is inferred from the arrival time of ions detected by a channel electron multiplier detector. Using this technique, for the first time, we observed the spatial temperature gradient within the atomic cloud, with higher temperature in the wings and lower temperature in the center.

Despite the diversity and energy dependence of $γ$-ray pulse morphologies in Crab, Vela and Dragonfly, the phaseograms of these three canonical young pulsars can be organised within a single four-beam geometric template. Using \textit{Fermi} Large Area Telescope data, we fit the 60~MeV--3~GeV phaseograms with a mechanism-agnostic, geometry-first parametric model that incorporates phase-dependent Doppler shifts and constrains the three-dimensional locations and bulk motions of four emission sites. In each pulsar, the phaseogram admits a decomposition into two altitude-separated beam pairs. The lower-altitude pair is produced by plasma with bulk motion close to azimuthal corotation, sharpening the main peaks. The higher-altitude pair shows a radially outward bulk-motion component, suggestive of inertial effects in a toroidally dominated magnetic field, and contributes bridge/shoulder emission and ripple-like modulations overlapping the main peaks. As a posteriori, the lower-altitude pair is consistent with curvature-dominated outer-magnetospheric emission, while the higher-altitude pair is consistent with synchrotron-dominated emission from a current-sheet-like outflow. Higher-altitude site heights vary from $\simeq 0.7$ (Crab, $\approx 1$~kyr) to $\simeq 1.1$--$1.4$ light-cylinder radii (Vela and Dragonfly, $\approx 10$~kyr). This unified four-beam, observation-driven geometry maps an altitude-dependent azimuthal tilt of pulsed $γ$-ray emission, providing an observationally anchored framework amenable to systematic tests and readily extensible to other young pulsars.

Diffuse-interface (phase-field) models are widely used to describe multiphase mixtures and their interfacial dynamics. In multiphase settings, however, the constitutive closure should remain meaningful across different representations of the same mixture. Existing N-phase phase-field constructions commonly enforce reduction only when a phase is absent (restriction to a face of the Gibbs simplex), but do not address the natural requirement that physically identical phases can be merged without changing the governing equations. This requires characterizing thermodynamically admissible, mixture-aware constitutive closures that are consistent with merging identical phases at the PDE level. Here, we show that, under a small set of structural axioms, PDE-level reduction consistency uniquely fixes the admissible free-energy structure to an ideal-mixing contribution to an ideal-mixing contribution, a symmetric mean-field interaction term, and a constant-coefficient quadratic gradient penalty. yielding a thermodynamic closure that includes Maxwell--Stefan-type mobilities as a special case. The same requirement constrains the Onsager mobility matrix to a pairwise-exchange form with bilinear degeneracy in the volume fractions, yielding a thermodynamic closure that includes Maxwell--Stefan-type mobilities as a special case. These results provide a consistent closure for N-phase Navier--Stokes--Cahn--Hilliard mixture models and, in the bulk-only setting, for multiphase Maxwell--Stefan diffusion systems. Numerical experiments confirm the predicted mixture-aware reduction properties and illustrate the capabilities of the N-phase Navier--Stokes--Cahn--Hilliard framework in representative multiphase-flow computations.

A key task in AI practice is to assess potential impacts to prevent harm. Current AI tools assisting AI impact assessment have not been designed or evaluated for collaborative team brainstorming, and they do not capture the range of views in diverse teams. We studied how AI can support team brainstorming during AI impact assessment and made three contributions. First, we adapted two structured methods from strategic foresight and co-designed AI interventions for them in five in-person workshops with 28 participants in total. Second, we evaluated the interventions in ten in-person workshops with 54 participants, finding that AI improved impact assessment quality and brainstorming perceptions for a general-purpose AI use (a chatbot companion) but not for a specialised one (a kidney allocation application). Third, our findings result in broader design guidance for AI assistance in brainstorming: AI should only offer hints and not solutions during early ideation, initiating interaction only when participants face fixation or saturation; it should facilitate structuring ideas during convergence; leverage expertise to refine ideas; and overall, it should serve more in support of tedious brainstorming process tasks, rather than ideation that teams value to do themselves.

Electroweak radiative corrections form a crucial ingredient in modern precision calculations for particle processes at high-energy colliders such as the Large Hadron Collider. The salient features of electroweak corrections as well as currently used techniques and concepts for their calculation are reviewed. Recent progress in this enterprise is illustrated in a discussion of electroweak multi-gauge-boson production processes: massive di-boson production, vector-boson scattering, and massive tri-boson production.

This review provides an overview of thermal effects on hadron properties, focusing on the theoretical frameworks used to describe in-medium modifications of masses, decay widths, and spectral functions. We examine the application of finite-temperature quantum field theory -- specifically the imaginary-time formalism (ITF) -- to analyze both light- and heavy-hadron sectors. For light hadrons, we discuss the role of chiral symmetry restoration and the different definitions of thermal masses in effective field theories, like chiral perturbation theory. In the heavy-flavor sector, we review recent progress in describing open-heavy mesons and quarkonia using self-consistent unitarized approaches and nonrelativistic effective field theories. All these results are complemented by analyses of recent lattice-QCD calculations using the Euclidean formulation of QCD at finite temperature, relevant to extract screening masses and reconstructed spectral functions. Finally, we discuss the phenomenological impact of the thermal modifications on experimental observables in relativistic heavy-ion collisions, including numerical simulations, dilepton spectra, transport coefficients, and hadron femtoscopy. By combining phenomenological considerations with robust theoretical tools, this review provides a coherent picture of how thermal effects emerge in the hadronic phase and how they can be systematically studied within controlled frameworks. Ultimately, the discussion serves as a bridge between experimental observations in relativistic heavy-ion collisions and fundamental developments in finite-temperature QCD and effective field theories for hadronic systems.

The Baik-Ben Arous-Peche (BBP) transition sets fundamental limits for detecting low-rank structure in noisy high-dimensional data and underlies a wide range of spectral methods in many fields from physics to statistics and data sciences. In standard settings, this transition is continuous, implying that signal recovery emerges gradually above a sharp threshold. We show that BBP transitions can instead be discontinuous in very general settings and provide a full theory of this phenomenon. When the eigenvalue density vanishes faster than linearly at the spectral edge, the overlap between the leading eigenvector and the signal jumps discontinuously at the critical point. We study this mechanism in deformed Gaussian and reweighted Wishart ensembles. We analyze in detail the finite-size effects, which play a central and qualitatively new role in the discontinuous BBP transition. Unlike the continuous BBP transition, we establish the existence of an extended pre-critical region where informative eigenvectors emerge well before the asymptotic threshold. The main consequence-and difference from the continuous BBP transition-is that signal recovery can occur at significantly lower signal-to-noise ratio and it is accompanied by strong sample-to-sample variability. Our results show the relevance and the novelty of the discontinuous BBP transition, and highlight the practical implications for signal detection.

We investigate the lifetime of dynamical regimes under the impact of noise motivated by low-dimensional models of the atmosphere. One may expect that the inclusion of noise tends to make the system leave prescribed regions of the state space faster. However, for relevant systems with complexities ranging from phenomenological toy models to reduced models of atmospheric dynamics, this intuition has proven misleading. As long as the noise is sufficiently small, the noisy system stays in regimes of interest on average longer than its deterministic counterpart, an effect we call ``stochastic inertia''. This phenomenon has been observed through extensive numerical simulations for different noise levels. We propose a numerical technique for testing the occurrence of stochastic inertia, constructing, for any fixed noise level, a Markov chain on the set of points given by a sufficiently long trajectory of the system without noise. The method is shown to correctly predict the presence of stochastic inertia in simple systems, and its utility is demonstrated on a paradigm model of atmospheric dynamics.

We show that every minimally generically globally rigid graph in $\mathbb R^d$ which contains a subgraph isomorphic to $K_{d+2}$ is itself isomorphic to $K_{d+2}$, confirming a conjecture by Garamv{ö}lgyi, Jackson, and Jord{á}n. The proof is entirely generated by ChatGPT 5.5.

Type Ia supernova (SN Ia) standardisation parameters exhibit evidence for systematic variation across the host galaxy star-formation rate - stellar mass (SFR$-M_\star$) plane, motivating the incorporation of galaxy SFR information in cosmological inference. SFRs are commonly estimated via spectral energy distribution (SED) fitting with far-infrared (FIR) measurements to account for dust-obscured star formation. Such FIR coverage will, however, be limited for upcoming time-domain surveys such as the Rubin Observatory Legacy Survey of Space and Time (LSST), necessitating the use of alternative SFR tracers. Here, we reconstruct the SFR - $M_\star$ plane using 1.4 GHz diagnostics, to test the consistency of host classifications against FIR-constrained SED-based estimates. Within this plane, SN Ia host galaxies are divided into three regions: Region 1 (low-mass), Region 2 (high-mass star-forming) and Region 3 (high-mass passive). We find that ${\sim}84$ per cent of SN hosts retain identical region assignments when using radio versus FIR-constrained SED-derived SFRs. Measuring SN Ia nuisance parameters ($α,β, M$) within each subregion, we find consistent values between the two SFR - $M_\star$ plane reconstructions, indicating limited sensitivity to SFR estimator choice, with the largest deviations in Region 3 at ${\sim}1.1σ$. Across the three 1.4 GHz SFR - $M_\star$ subregions, we confirm the region-dependent variation in SN Ia standardisation parameters - particularly $β$ - reported in our earlier SED-based analysis. With near-complete radio coverage of the LSST footprint anticipated from current and forthcoming radio continuum surveys (e.g., Square Kilometre Array), radio SFR calibrations will become an increasingly useful and scalable approach to host galaxy classification, supporting the construction of robust SN Ia subsamples for precision cosmology.

The problem of determining whether a probabilistic program terminates almost surely (i.e.~with probability one) is undecidable, and actually $Π^0_2$-complete. For this reason, a growing literature has explored classes of programs for which this and related problems can be shown (semi-)decidable. In this work we consider the termination problem for the language of Probabilistic Higher-Order Recursion Schemes (PHORS). Using the weighted relational semantics of linear logic, we translate this problem into the computation of suitable generating functions associated with the program interpreted. This way, we establish the decidability of almost sure termination for a class of programs that extends Li et al.'s affine PHORS via a type discipline with bounded exponentials. To achieve this, we show that the generating functions for such programs are always algebraic, that is, solutions of polynomial equations, yielding an effective method to answer the termination problem.

In recent years, novel AI accelerators have emerged as promising alternatives to GPU for AI model training and inference tasks. One such accelerator, the Cerebras CS-3, achieves strong performance on large model training as well as scientific applications like molecular dynamics simulations. While dense compute workloads have been thoroughly explored for the CS-3, its potential for sparse workloads has not been fully examined. Applications requiring sparse linear algebra kernels, such as GNNs, linear solvers, and recommendation systems, could achieve good performance on a dataflow accelerator like the CS-3. In this work, we explore two key sparse linear algebra kernels, sparse-dense matrix multiplication (SpMM) and sampled dense-dense matrix multiplication (SDDMM), on the Cerebras CS-3. We propose low-level CS-3 kernel designs for these operations and optimize our designs to improve I/O performance, memory footprint, and scalability to large matrices. Our evaluation examines memory footprint and SpMM/SDDMM speedup relative to CPU. The evaluation suggests that the CS-3 can outperform CPU by 100$\times$ for SpMM with 90\% sparse matrices with performance improving as sparse matrix dimensionality increases. SDDMM on CS-3 can outperform CPU 20$\times$ for 90\% sparse matrices. We additionally find that as sparsity increases to beyond 99\%, the CS-3 suffers from performance degradation that makes it slower than CPU for SpMM.

Let $M$ be a smooth manifold without boundary and let $\mathcal{T}$ be a countable collection of manifolds with corners, each equipped with a smooth map to $M$. We show that the singular simplicial set $\mathrm{Sing}(M)$ of $M$ deformation retracts onto the simplicial subset $\mathrm{Sing}^{\mathcal{T}}\!(M)$ of smooth singular simplices that are transverse to every element of $\mathcal{T}$.

In this paper, we consider the Santa Claus problem in the CONGEST model. This NP-hard problem can be modeled as a bipartite graph of children and gifts where an edge indicates that a child desires a gift. Notably, each gift can have a different value. The goal is to assign the gifts to the children such that the least happy child is as happy as possible. Even though this is a well-studied problem in the sequential setting, we obtain the first results the distributed setting. In particular, we show that the complexity of computing an $\mathcal{O}(\log n/\log \log n)$-approximation is $\hat Θ(\sqrt n+D)$ rounds, where our $\widetildeΩ(\sqrt n+D)$-round lower bound is even stronger and holds for any approximation.

Starting with describing the Dirac equation in an unconventional manner, we introduce the notion of chirality and show how chiralities are mixed in the massive Dirac particles.Then we discuss the approximate chiral symmetry in QCD for the light quarks, i.e., u, d, and s quarks, proceeding to the introduction of the spontaneous symmetry breaking of chiral symmetry, the Wigner and Nambu-Goldstone realization of the symmetry, and the Nambu-Goldstone theorem. A remark is given on the U(1)$_A$ anomaly and its physical consequences.We stress the spectral degeneracies in the Wigner phase between the $σ$ -$π$ channels and between the $ρ$-$a_1$ channels, which would give definite evidence of the symmetry restoration.The chiral quark models are introduced of the Nambu-Jona-Lasinio type with the vector-axial vector interaction and/or the instanton-induced four-quark or six-quark interactions of a determinant form.A three-flavor linear $σ$ model with the determinant term is introduced, and some interesting results are expressed on the mass term of the $η'$ meson in the chiral limit.We also introduce a linear $σ$ model with parity-doublet nucleons incorporated. An intuitive account is given on how the chiral condensate may be reduced on the basis of Hellmann-Feynman theorem. A brief summary is given on some experiments and their implications on the possible restoration of the chiral symmetry, which include the pionic atoms, lepton-pair production in relativistic-heavy-ion collisions, an attempt to produce $η'$-mesonic nuclei.We mention the current active effort to construct the equation of state of nuclear and neutron-star matter incorporating the parity-doubling in the baryon sector and the occurrence of the restoration of chiral symmetry in the QCD matter.

We study the null junction conditions in (re-summed) quasi-topological gravity theories, showing that no null thin shells exist within the realms of standard distributional theory for the pure gravity regular black hole solutions we have analyzed. This implies that the usual derivation of the mass inflation instability, which makes use of null thin shells, is not applicable in these theories. The problem of stability of inner horizons of regular black holes in quasi-topological gravity is hence still open and must be addressed with a more refined analysis, which does not rely on thin shells or the vacuum condition.

Maximal Extractable Value (MEV) represents billions of dollars in extracted value that fundamentally shapes blockchain network dynamics and participant incentives. While research has focused on MEV extraction and mitigation, we lack systematic methods to attribute MEV opportunities to their on-chain origins. This paper formalizes the MEV opportunity attribution problem and introduces a systems framework for identifying which transactions create arbitrage opportunities and quantifying their contributions. We design and evaluate four attribution methods for atomic arbitrage on EVM-compatible networks: bot-data-driven, simulation-based, coefficient-based, and Shapley-based approaches. Through large-scale retrospective analysis spanning over one million blocks on Polygon, we demonstrate that the majority of atomic arbitrage opportunities can be traced to single source transactions, validating our central hypothesis about competitive MEV markets. We quantify a highly concentrated distribution of MEV creation, where a small subset of protocols generates most opportunities, and provide comparative analysis of method trade-offs in accuracy, cost, and scalability. Our findings offer insights for protocol designers reducing MEV leakage, validators optimizing transaction ordering, and analysts measuring ecosystem health through opportunity creation.

Viscoelastic rate-type fluid models constitute a fundamental framework for the mathematical description of complex materials exhibiting coupled elastic and viscous effects, with a wide range of applications in engineering, biomaterials, and medicine. In realistic regimes, thermal effects are essential and lead to strongly coupled systems in which heat conduction and temperature-dependent constitutive laws play a decisive role. In this paper, we develop a thermodynamically consistent model for heat-conducting viscoelastic rate-type fluids. We establish the existence of a global weak solution in the full three-dimensional setting. In contrast to the existing literature, no smallness, regularity, or structural restrictions on the initial data are imposed beyond natural energy and entropy bounds, and no additional regularising mechanisms such as artificial stress diffusion are required. The analysis is based on a weak-strong framework combining energy and entropy balances with compactness tools, allowing us to treat the full nonlinear coupling between the fluid velocity, the temperature, and the elastic stress.

We study the Liouville equation $\triangle u+e^{2u} =0$ in a Riemannian surface $(M, g)$ with nonnegative $Ricci$ curvature. Under some asymptotic lower bound assumptions, we classify all the solutions to this equation, meanwhile we obtain the rigidity results for the ambient manifold. Note that our assumptions are optimal in some sense and differ from the classical assumption of finite total curvature.

We derive a sharp upper bound on the residuals produced by the flexible GMRES (FGMRES) method. The bound shows that FGMRES exhibits two phases of convergence depending on the residual tolerance of the inner preconditioner. For small tolerances, the convergence of FGMRES is practically geometric with a constant rate throughout, while for looser tolerances the two-phase behavior becomes more pronounced. We also show that the derived bound cannot be improved and construct an example for which it becomes an equality.

We study semileptonic sum rules for $b \to c τ\overlineν$ transitions involving orbitally excited charm hadrons. Starting from the amplitude-level relation implied by the heavy quark symmetry, we construct sum rules relating these decays. We then examine deviations from the small-velocity limit. Our numerical analysis shows that the deviations generally increase once excited hadrons are involved, with tensor contributions often inducing sizable effects. At present, however, the relevant form factors are not yet sufficiently constrained. Further improvements in the hadronic inputs are essential for these sum rules to yield robust predictions for the corresponding lepton-universality ratios.

An important tool in the design of fault-tolerant graph data structures are $(L,f)$-replacement path coverings (RPCs). An RPC is a family $\mathcal{G}$ of subgraphs of a given graph $G$ such that, for every set $F$ of at most $f$ edges, there is a subfamily $\mathcal{G}_F \,{\subseteq}\, \mathcal{G}$ with the following properties. (1) No subgraph in $\mathcal{G}_F$ contains an edge of $F$. (2) For each pair of vertices $s,t$ that have a shortest path in $G-F$ with at most $L$ edges, one such path also exists in some subgraph in $\mathcal{G}_F$. The covering value of the RPC is the total number $|\mathcal{G}|$ of subgraphs. The query time is the time needed to compute the subfamily $\mathcal{G}_F$ given the set $F$. Weimann and Yuster [TALG'13] devised a randomized RPC with covering value $\widetilde{O}(fL^f)$ and query time $\widetilde{O}(f^2 L^f)$. This was derandomized by Karthik and Parter [TALG'24], who also reduced the query time to $\widetilde{O}(f^2 L)$. Their approach uses some heavy algebraic machinery involving error-correcting codes and an increased covering value of $O((cfL \log n)^{f+1})$ for some constant $c > 1$. We instead devise a much simpler derandomization via conditional expectations that lowers the covering value back to $\widetilde{O}(fL^{f+o(1)})$ and decreases the query time to $\widetilde{O}(f^{5/2}L^{o(1)})$, assuming $f = o(\log L)$. We also investigate the optimal covering value of any $(L,f)$-replacement path covering (deterministic or randomized) for different parameter ranges. We provide a new randomized construction as well as improving a known lower bound, also by Karthik and Parter. For example, for $f = o(\log L)$, we give an RPC with $\widetilde{O}( (L/f)^f L^{o(1)})$ subgraphs and show that this is tight up to the $L^{o(1)}$ term.

Living systems routinely consume energy to achieve motility, often using intricate biomolecular machinery. In this work, we show that active droplets can sustain indefinite self-propulsion of a spherical colloid in an otherwise homogeneous, isotropic, and autonomous environment. Our proposed minimal mechanism consists of phase-separating proteins, enzymes passivating them, and complementary enzymes anchored to the colloid surface that reactivate the proteins. This passivation-activation cycle gives rise to a symmetry breaking - nucleation and stabilization of a condensate near the colloid surface, which in turn exerts a repulsive force on the colloid. We numerically demonstrate that this mechanism can propel micron-sized colloids at speeds of up to a hundred microns per second. This propulsion mode is strongly resistant to Brownian fluctuations and external forces, suggesting that propulsion mechanisms based on biomolecular condensates may offer a complementary, motor-free route to biological transport.