Optical clocks have achieved remarkable estimated fractional frequency uncertainties reaching the $10^{-18}$ level and below, enabling applications in fundamental physics, general relativity, and geodesy. However, the challenge of verifying the international consistency of optical clocks remains critical as efforts intensify toward redefining the SI second based on an optical transition or transitions. We report on a two-month international clock comparison campaign involving seven optical clocks in four national metrology institutes (INRIM, LNE-OP, NPL, and PTB) connected via the optical fiber network established in Europe. The campaign resulted in optical frequency ratios with uncertainties ranging from $7.7\times10^{-18}$ to $6.1\times10^{-17}$. Among the results, the $^{171}$Yb$^+$(E3) clocks at NPL and PTB demonstrated agreement within an uncertainty of $7.7\times10^{-18}$, marking the first international verification of two independently developed optical clocks below one part in $10^{17}$. The operation of the $^{199}$Hg clock at LNE-OP (formerly LNE-SYRTE) resulted in frequency ratios with improved uncertainties with $^{171}$Yb$^+$(E3), $^{171}$Yb, and $^{87}$Sr optical clocks. These results provide input for the redefinition of the second and underscore how fiber-linked clock networks can advance metrology and scientific applications.

Clustering is a fundamental collective phenomenon in agent-based models (ABMs) of opinion dynamics. To study clustering in systems with co-evolving social and opinion variables, we derive stochastic partial differential equation (SPDE) models that describe the evolution of clusters on a reduced state space. We consider two settings: one in which opinions do not affect social interactions, and another one in which a feedback mechanism couples the two. Our approach extends reduced PDE modelling to a stochastic framework, which is essential for capturing long-term cluster behaviour. Numerical experiments demonstrate that the proposed reduced SPDEs substantially decrease computational cost compared to full-state SPDE models, such as the Dean-Kawasaki equation, while still accurately reproducing the clustering behaviour of the underlying ABM. As a result, these reduced models provide an efficient tool for studying systems with large populations, including those arising in the analysis of real-world data: in particular, we provide an application related to the large-scale General Social Survey (GSS), which comprises opinion and social data of the US population since 1972.

Many stochastic systems are built by wiring typed components together, but the wiring is often neither purely sequential nor type-homogeneous. This paper develops categorical semantics for such systems using ordered polycategories whose morphisms are Markov kernels. The basic operation is kernel slotwise composition, which connects one output slot of a many-output kernel to one input slot of another and marginalizes the internal wire. We prove its structural laws by assigning trace semantics to finite acyclic diagrams. We then introduce colored Markov polycategories, where objects and kernels carry colors and typed connections are realized by coherent interface kernels. This gives a colored kernel slotwise composition and trace semantics for typed stochastic diagrams. To describe systems whose structure changes, we co-index colored Markov polycategories and parameter spaces over an indexing category. Finally, for finite acyclic parameterized diagrams, we prove a diagrammatic differentiation result. The derivative of an expected scalar objective is obtained from local reverse-mode contributions at the parameterized vertices, with stochastic and deterministic kernels handled through admissible local gradient operators. The construction gives a typed, compositional language for finite acyclic stochastic systems and their parameter sensitivities.

We present a museum installation in a 180° dome theater, which gives the museum visitor the experience of conducting a symphony orchestra. We have pre-recorded a short music piece performed by a professional orchestra. This recording is played back in the dome with the visitor standing in the conductor's position. The visitor's gestures are captured with a vision-based skeleton tracker, steering the recording playback pace via a gesture recognition module that translates the gestures into a time control signal. This is sent to a playback module that plays the recording in the dome at the corresponding speed. The gesture recognition module is based on a hierarchical LSTM network, trained with recorded sequences of multiple conductors with different level of expertise conducting the same recording. The system is evaluated with a quantitative study of the estimated timing accuracy, a user study evaluating the musical realism and usability of the real-time control, and a field study to evaluate the performance of the entire system with real museum visitors.

We investigate the unique stationary measure of a positive recurrent reflecting Brownian motion in the upper half-plane, where the direction of reflection is constant on each half-axis. The Laplace transform of the stationary distribution is characterized by a functional equation, whose resolution is reduced to solving a discontinuous Riemann boundary value problem. By applying the Sokhotski-Plemelj formulas, we derive an explicit expression for the Laplace transform. Finally, we establish the local behavior of the stationary density at the origin and its asymptotics along the boundary axes using Tauberian theorems and asymptotic analysis.

A Killing tensor field on a Riemannian space corresponds to an integral of the geodesic flow polynomial in momenta. A Killing tensor field is called decomposable if it is a polynomial in Killing vector fields. In this paper, we first prove that the study of Killing tensor fields on symmetric spaces can be reduced to the case of compact irreducible ones. Then we introduce the class of top slot Killing tensor fields. We obtain an explicit and elegant description of such tensor fields and prove that the quadratic Killing tensor fields are spanned by the top-slot ones. We also show that quadratic Killing tensor fields on the quaternionic projective space and on the Cayley projective space are spanned by the indecomposable ones constructed in our earlier paper and the decomposable ones. This completes the classification of quadratic Killing tensor fields on Riemannian symmetric spaces of rank one.

Simultaneous multi-directional force measurement across all five digits is essential for studying hand coordination, compensatory forces, and myoelectric control, yet existing systems trade off digit coverage, force dimensionality, and anatomical adaptability. Reliable full-hand acquisition remains challenging because multi-axis calibration, hand-size adjustment, and consistent digit-specific force reconstruction are technically demanding. We present MyoKin3X, a customizable full-hand framework for simultaneous 3D force measurement of up to five digits providing robust and validated force reconstruction. It combines an anatomically versatile structure with five integrated 3D force sensors and a standalone software for synchronized electromyography and force acquisition. MyoKin3X provides in-place cross-calibration of all five sensors, single- and multi-digit maximal voluntary contraction recording, and automated coordinate transformation to digit-specific coordinate systems for standardized analysis across subjects and tasks. Calibration validation demonstrates high stability of the axis-specific calibration factors, with a mean coefficient of variation of 0.04% and maximum force error of +- 0.06N at 50N. It also shows effective inter-axis decoupling (mean crosstalk reduction: 92.71%; residual crosstalk below 0.02% for most axis pairs) and high predictive accuracy (R2 > 0.99 across sensors). The software includes four feedback modes: 1D ramps, fatigue protocols, 2D arbitrary target ramps, and 2D exploratory tasks. MyoKin3X therefore enables standardized full-hand force acquisition with validated measurement reliability, flexible protocol control, and real-time visualization for high-fidelity studies of hand motor control, muscle synergies, and human-machine interfacing.

We prove the existence of many non-trivial characteristic classes of smooth oriented bundles with fibre a product $ S^{n}\times S^{n} $ of odd-dimensional spheres. We do so by proving injectivity of the map from the ring of rational characteristic classes of oriented fibrations with fibre $ S^{n}\times S^{n} $; the latter is proven by Berglund--Zeman to be isomorphic to the group cohomology of the symmetric powers of the standard representation of a certain finite-index subgroup $ Γ$ of $ \mathrm{SL}_{2}(\mathbb{Z}) $. These characteristic classes of smooth bundles are not generalised Miller--Morita--Mumford classes, and they exist in arbitrarily large cohomological degrees. Inspired by an example given by Morita, we provide a collection of smooth oriented $ S^{n}\times S^{n} $-bundles, indexed by cyclic subgroups of $ Γ$, which detect any given non-zero characteristic class of such fibrations.

Quantum decoherence-the loss of quantum coherence due to interactions with an environment-plays a central role in quantum transport, and controlling this ubiquitous yet inevitable phenomenon is essential for practical quantum technologies. Despite its importance, the microscopic mechanisms of decoherence in infinite-size topological insulators remain poorly understood. Here, we develop a comprehensive theory that quantitatively investigates how quantum decoherence shapes the quantum spin Hall effect in macroscopic topological insulators, and reveal that decoherence-induced corrections scale quadratically with impurity density. Besides, we uncover a previously unidentified mechanism of the extrinsic spin Hall effect: a second-order skew-scattering process intrinsically tied to quantum decoherence-fundamentally distinct from, yet substantially stronger than, the conventional third-order skew-scattering mechanism. Furthermore, we predict a new scaling law in which the decoherence-induced spin Hall conductivity scales quadratically with the longitudinal conductivity, providing a clear experimental signature of decoherence effects. Our results establish the essential role of decoherence in quantum transport of topological insulators and reveal that macroscopic topological insulators offer a promising platform for next-generation spintronic applications.

Millimeter-wave (mmWave) communication depends on highly directional beamforming, while fast mobility, blockage, and rapid geometry changes in vehicle-to-everything (V2X) scenarios make beam tracking challenging. In cooperative multi-base-station (BS) systems, conventional hierarchical methods usually separate BS selection and beam selection, which may cause error propagation when beam states change abruptly. To address this issue, this paper proposes Cooperative Radio Sensing with Large Language Models (CRS-LLM), a cooperative beam prediction framework for next-step joint BS-beam prediction. CRS-LLM formulates beam tracking as a single classification problem over the joint BS-beam space, avoiding cascaded decision errors. To adapt channel state information (CSI) to large language models, a dual-view CSI tokenizer extracts frequency-domain and delay-domain channel features through a lightweight CNN front-end and temporal tokenization module. A truncated GPT-style backbone is then used for temporal modeling with parameter-efficient adaptation. In addition, a transition-aware switch-gated predictor combines a stable branch, a residual flip branch, and a low-rank transition prior to capture both smooth evolution and abrupt changes. Simulation results show that CRS-LLM outperforms CSI-Transformer, Hierarchical BS-Beam, and representative CNN- and recurrent-neural-network baselines in Top-1 accuracy and normalized beam gain under different SNR conditions, while also showing strong few-shot performance and promising zero-shot transferability.

In recent years, continuous-variable quantum key distribution (CV-QKD) has become a promising paradigm for enabling secure communication among multiple end users sharing the same telecommunication backbone. CV-QKD with reverse reconciliation naturally enables scalability from conventional point-to-point links to quantum access networks based on passive quantum broadcasting channels. Here, we report an experimental demonstration on an active $1:4$ multi-user CV quantum network (QN) in the finite-size regime. With $1.25\cdot10^9$ coherent states exchanged on each $11\text{km}$ quantum channel, the highest performance for secret key generation totaling $1.9\cdot10^{-1}$ bits/channel use. Furthermore, we investigate adaptable CV-QN protocols that comprehensively allow network operation in various security and key rates requirements of individual users. The results establish the practical security of CV-QN compatible with existing telecommunication for broad deployment, and allowing additional degree of freedom for connected end users in existing infrastructures.

The colonisation of extraterrestrial planets requires sustainable food production independent of Earth-based supplies. Due to the high costs and complicated logistics of food transport, in-situ cultivation will be essential. Growing plants directly in regolith offers a practical approach to achieve sustainable long-term human habitation beyond Earth. In this study, Laser-Induced Breakdown Spectroscopy (LIBS) technique was employed for bioimaging of broccoli (Brassica oleracea) and salad (Lactuca sativa) plants grown in Lunar regolith simulant and control substrate. For this purpose, the potential of the 2090 nm laser wavelength for bioimaging of plant tissue was studied compared to the conventional 1064 nm. The signal-to-noise ratio (SNR), total emissivity ($ε_{\mathrm{tot}}$), and Mg II / Mg I intensity ratio (ionisation degree) were all higher when using the 2090 nm laser wavelength compared to 1064 nm. These findings indicate that the 2090 nm laser produces a hotter and more efficiently ionised plasma, supporting its feasibility for bioimaging of plant tissues. Additionally, bioimaging with both laser wavelengths confirmed higher uptake of key plant nutrients such as magnesium (Mg) and calcium (Ca) from Lunar regolith simulant. These results support the potential of LIBS as a diagnostic tool for plant growth monitoring in extraterrestrial environments.

Singular theories, characterised by the presence of degeneracies in their Lagrangian or Hamiltonian descriptions, require the systematic implementation of constraints in order to obtain well-defined dynamics. While the symplectic framework provides the standard geometrical setting for conservative mechanical systems, those theories which exhibit dissipative effects are most appropriately discussed within the context of contact geometry. In this review, we present the geometrical structure underlying pre-symplectic and pre-contact manifolds, and develop the corresponding constraint algorithms that determine the admissible subset of phase space upon which consistent Hamiltonian evolution exists. We then close the discussion of each of the constraint algorithms with an example.

Second-order quantifier elimination is the problem of finding, given a formula with second-order quantifiers, a logically equivalent first-order formula. While such formulas are not computable in general, there are practical algorithms and subclasses with applications throughout computational logic. One of the most prominent algorithms for second-order quantifier elimination is the saturation-based SCAN algorithm. In this paper we show how the SCAN algorithm on clause sets can be extended to solve a more general problem: namely, finding a witness for the second-order quantifiers that results in a logically equivalent first-order formula. In addition, we provide a prototype implementation of the proposed method.

Primates exhibit a robust deviation from canonical allometric scaling: at fixed body mass, their lifespans exceed those of non-primate mammals by factors of two to three. A rhesus macaque (8 kg) lives 25-40 years, whereas a cat of similar mass rarely exceeds 18 years. This statistically significant clade-level excess cannot be explained by standard metabolic or ecological models. We provide a thermodynamic explanation within the Principle of Biological Time Equivalence (PBTE), where lifespan is determined by a finite cycle budget governed by entropy production. We show that primates reduce entropy production per physiological cycle through increased neural energy allocation. The neural power fraction acts as a control parameter, extending the effective lifetime cycle count. Three mechanisms, predictive regulation, enhanced repair, and behavioral buffering, jointly suppress dissipation. This yields a quantitative neuro-metabolic multiplier that explains primate longevity and provides testable predictions linking brain energetics, entropy production, and lifespan.

Concurrent advances across fields such as organoid technology, Microelectrode Arrays (MEAs), neuromorphic computing, and machine learning have given rise to a groundbreaking research paradigm: Synthetic Biological Intelligence (SBI). SBI refers to engineered systems in which living Biological Neural Networks (BNNs) are interfaced with hardware and software to perform task-oriented information processing in a closed loop. This cutting-edge technology, while still in its infancy, has the potential to deliver highly efficient performance across both computing capabilities and energy consumption. The early stage of this field underscores the need for reliable multi-scale and cross-domain interaction interfaces to support applications in robotics, biomedicine, signal processing, and neuroscience research. The hitherto lack of commercially available SBI platforms has slowed the development, as the conditions to produce a testbed are expensive and cumbersome. The introduction of standardized, platform- and cloud-integrated BNNs has been a crucial catalyst for the scientific community, improving the accessibility of SBI and leading the way to further developments. In this survey, we summarize the innovations that contributed to the emergence of SBI and the first testbed interfaces that enabled its embodiment. This work reframes SBI as a bio-digital interaction system and introduces a unified protocol across encoding, decoding, system engineering, and benchmarking.

In this paper, we study the $L^p$-boundedness of Stein's square function $\mathfrak{S}^α(\mathcal{L})$ associated with the sub-Laplacian $\mathcal{L}$ on Métivier group $G$. A key aspect of our result is that the smoothness condition is expressed in terms of the topological dimension $d$ of the underlying Métivier group $G$. Consequently, we also present several applications of the $L^p$-boundedness of $\mathfrak{S}^α(\mathcal{L})$. First, we provide an alternate proof of the sharp $L^p$-boundedness result for spectral multipliers on Métivier groups, recently obtained by Niedorf [Niedorf, Studia Math., 2025]. Next we prove $L^p$-boundedness of maximal spectral multipliers and consequently establish sharp $L^p$-boundedness result for the maximal Bochner-Riesz operator on Métivier groups, which also yields pointwise almost everywhere convergence of Bochner-Riesz means with smoothness parameter given in terms of the topological dimension of $G$. In case of Métivier groups our result improves upon the existing works of Mauceri-Meda [Mauceri, Meda, Rev. Mat. Iberoam., 1990] and Horwich-Martini [Horwich, Martini, J. Lond. Math. Soc., 2021]. Our result further imply the mixed norm regularity estimates for the solution of fractional Schrödinger equation on Métivier groups, where the regularity index is again expressed in terms of the topological dimension of $G$. Finally, we study the $L^{p_1}(G) \times L^{p_2}(G)$ to $L^p(G)$ boundedness of the bilinear Bochner-Riesz means and its maximal version, associated with the sub-Laplacian on Métivier group $G$. Our result improves upon the recent work of the author with Bagchi and Molla [Bagchi, Molla, Singh, J. Funct. Anal., 2026] in the range $2\leq p_1, p_2 <\infty$. In the same range, ......

In heterogeneous network systems such as ecological and social networks, structural stability depends on how connectivity changes under node removal, as different removal sequences can trigger distinct modes of systemic collapse. While robustness to random failures and targeted attacks has been extensively studied, most analyses have focused on connectivity loss or degree distribution, rather than on how scale-invariant organization emerges and evolves with system size. Here we examine how scale-free structure evolves under progressive degree-dependent node removal, systematically varying the hub-protection strength $θ$. Starting from scale-free networks, we apply the recently developed finite-size scaling (FSS) analysis to node-removed networks and compare the results with those from Kullback-Leibler (KL) divergence-based classification. We find that under random ($θ=0$) and hub-protecting removal ($θ>0$), the two criteria largely agree, whereas under hub-preferential removal ($θ<0$), networks may appear scale-free according to the KL criterion while failing the FSS test of scaling collapse. This discrepancy indicates that similarity to a reference degree distribution does not guarantee the persistence of scale-invariant organization across system sizes. The two diagnostics thus probe complementary aspects of network structure, and their joint use provides a more complete characterization of structural degradation.

A thermal spectrum is often taken as a signature that the emitted probe has reached detailed balance with the surrounding medium. We show that this interpretation is not generally valid by studying how the microscopic emission kernel determines the macroscopic spectrum. In $3+1$ dimensions, a simple thermal spectrum can be generated without probe thermalization when the relevant kernel belongs to a thermally degenerate class. A representative case is realized when the differential cross section depends on the scattering angle but carries no additional dependence on the Mandelstam variable $s$, as in low-energy Thomson scattering. Our results provide a kernel-based criterion for distinguishing genuine probe--medium exchange equilibrium from thermal spectra produced by the structure of the emission kernel itself.

We present a highly simplified nonlinear hydrodynamical model to emulate the main observed features of amplitude modulation (commonly known as Blazhko effect) in RR Lyrae stars. The model is based on the assumption that the periodic flow generated by the pulsation carries surplus helium in the ionization zones He I and II. Once this extra helium reaches a critical amount, a Rayleigh-Taylor-type instability leads to a back-flow of the surplus helium and the process starts over again, due to the continuing effect of pumping helium upward by the pulsation. This periodic variation of helium leads to various efficiency of radiation flux blocking in the helium ionization zone that shows up as a long-term periodic variation of the pulsation amplitude.

We consider a quasi-static nonlinear model in thermoviscoelasticity at a finite-strain setting in the Kelvin-Voigt rheology where both the elastic and viscous stress tensors comply with the principle of frame indifference under rotations. We refine the discretization schemes in [Badal-Friedrich-Kružík '23, Mielke-Roub\'ıček '20] by imposing frame indifference already at a time-discrete level. This is justified both analytically and numerically.

This paper studies data-driven approaches to the continuous-time linear quadratic regulator (LQR) problem based on two existing parameterizations, namely a closed-loop (CL) parameterization from behavioral system theory and an integral reinforcement learning (IRL) parameterization. The CL parameterization characterizes the closed-loop system via a matrix that satisfies equality constraints. While this parameterization has been extensively studied for discrete-time systems, we adapt key results to the continuous-time setting and develop a policy iteration (PI) scheme, derive a data-driven continuous-time algebraic Riccati equation (CARE), and introduce an alternative convex problem formulation. The IRL parameterization utilizes off-policy data to perform policy evaluation, which is then used for PI or value iteration. Within the IRL framework, we derive a policy gradient flow and propose convex reformulations of the LQR problem. Finally, we provide a unified treatment of these parameterizations that enables a systematic understanding of existing approaches and clarifies their structural relationships.

Context. The Large Array Survey Telescope (LAST) is a wide-field visual-band survey designed to explore the variable and transient sky with high cadence. Its raw data stream is automatically processed in near real time at the observatory site, producing science-quality images, catalogs, and transient alerts. Transient alerts are then reported to the Transient Name Server (TNS). Aims. The LAST pipeline comprises two major components: (i) processing and calibration of single images followed by coaddition of $20\times20$s exposures, producing single-image and coadded-image catalogs; and (ii) subtraction of coadded images from calibrated reference images followed by transient detection. In this work we present a detailed description and validation of the second component of the pipeline. Methods. Transient detection is based on the algorithm for proper image subtraction (ZOGY). We combine ZOGY subtraction with the Translient statistic for sub-pixel motion discrimination, together with a sequence of deterministic filtering steps, to produce a clean stream of transient candidates without the use of machine learning. Results. Using commissioning data, the pipeline achieves a preliminary $5σ$ limiting magnitude of $20.3$-$20.7$mag, a single-epoch transient detection efficiency of $\sim80$%, and a purity of $\gtrsim90$% at signal-to-noise ratio of $\geq7.5σ$.

The Circle Pattern Theorem characterizes the existence and rigidity of circle patterns with prescribed intersection angles on simplicial triangulations of closed surfaces. In this paper we extend the theorem to quasi-simplicial triangulations -- triangulations that may contain loops and multiple edges, but whose lifts to suitable finite coverings become simplicial. We prove that the curvature map on a quasi-simplicial triangulation has the same qualitative structure as in the simplicial case, provided the KAT inequalities are imposed on all subsets of the lifted vertex set. This generalization significantly widens the applicability of circle patterns: it accommodates economical one-vertex triangulations of higher genus surfaces, provides a natural framework for constructing circle patterns on orbifolds with prescribed cone angles, and resonates with the Delta complex structures routinely used in intrinsic geometry processing algorithms.

Coalition Logic is primarily concerned with what coalitions can achieve, whereas what coalitions cannot achieve -- their \emph{inability} -- has received comparatively little explicit attention. This asymmetry matters in artificial intelligence and safety-critical multi-agent systems, where one often needs to specify not merely what agents are instructed or disposed not to do, but what they are \emph{unable} to bring about. We develop a conservative extension of Coalition Logic with an explicit inability operator, interpreted as the negation of coalition ability. This operator is introduced as a conservative and formally tractable starting point for studying inability as a modal concept in its own right. We prove soundness, completeness, and conservativity over standard Coalition Logic, and analyse the resulting modal behaviour: anti-monotonicity with respect to coalition inclusion, contravariance with respect to goal strength, asymmetric interaction with conjunction and disjunction, failure of superadditivity, non-equivalence with opponent ability, and the connection between grand-coalition inability and systemic impossibility. Making this definable operator explicit reveals a systematic modal structure governing the limits of agency, and supports reasoning about constraints, negative capabilities, and impossibility in multi-agent systems.

In this paper, we investigate the existence of fixed-point-free automorphisms for finite-dimensional Lie algebras. By a result of Jacobson, a Lie algebra admitting a fixed-point-free automorphism is solvable. We prove that such a Lie algebra must be even strongly unimodular. We find a necessary and sufficient criterion such that a complex almost abelian Lie algebra admits a fixed-point-free automorphism. For complex filiform Lie algebras we show that the existence of a fixed-point-free automorphism is equivalent to not being characteristically nilpotent.

Modern large multicore systems often run multiple workloads that share CPUs under schedulers such as Linux CFS. To keep CPUs busy, these schedulers load-balance runnable work, causing each workload to execute on many cores. This weakens locality at the microarchitectural level: workloads lose reuse in caches, branch predictors, and prefetchers, and interfere more with one another - especially on chiplet-based systems, where spreading execution across cores also spreads it across LLC boundaries. A natural alternative is strict CPU partitioning, but hard partitions leave capacity idle when workloads do not fully use their reserved CPUs. We present Affinity Tailor, a userspace-guided kernel scheduling system built on a key insight: the kernel can preserve locality for workloads that share CPUs by treating demand-sized, topologically compact CPU sets as affinity hints rather than hard partitions. A userspace controller estimates each workload's CPU demand online and assigns a preferred CPU set sized to that demand, chosen to be as disjoint as possible from other workloads while spanning as few LLC domains as possible. The kernel then uses this set as an affinity hint, steering threads toward those CPUs while still allowing execution elsewhere when needed to preserve utilization. Deployed at Google, Affinity Tailor delivers geometric-mean per-CPU throughput gains of 12% on chiplet-based systems and 3% on non-chiplet systems over Linux CFS. Furthermore, faster execution reduces memory residency, yielding per-GB throughput gains of 3-7%. Our findings suggest that future schedulers should treat spatial locality as a first-class objective, even at the expense of work-conservation.

Salts are an integral part of the environment for living systems and, therefore, understanding their effects on proteins and other biomolecules is of fundamental interest. Small-angle X-ray scattering (SAXS) of protein solutions can provide valuable information on salt effects, but extracting this information has been a significant challenge. For example, SAXS data of bovine serum albumin (BSA) at various salt concentrations were fit to three different spherical models. Here we combined the newly developed FMAPIq approach with explicit-solvent all-atom molecular dynamics simulations to show that the complex effects of salt on the SAXS of BSA originate from the interplay of ions and hydration water, leading to a general picture of protein-ion-water interactions.

We study scale-invariant geometric quantities associated with embedded closed curves in Euclidean three-space, with an emphasis on their behavior under optimization within a fixed knot type. Given a Euclidean-invariant and scale-covariant size functional \(D\), we define the \(D\)-density of a curve \(γ\) by \(\len(γ)/D(γ)\), the \(D\)-compression radius by \(D(γ)/\Thi(γ)\), and the corresponding packing ratio as its reciprocal. For a single representative, ropelength factors as the product of the \(D\)-density and the \(D\)-compression radius. The main point is not this formal cancellation, but the separation it suggests after optimization: the density, compression, packing, and ropelength problems generally have different minimizing sequences. We develop this factorization framework for general scale-covariant size functionals. We prove the basic optimized inequality, give a criterion for equality after optimization, and compute the unknot case for the diameter and the minimal enclosing radius. We also prove polygonal approximation results for compression radii when \(D=\diam\) and when \(D=R_{\min}\), using standard convergence properties of polygonal thickness, and formulate the corresponding hypotheses for other \(L^p\)-type size functionals. Finally, we discuss relations with distortion, trunk, and supertrunk. The framework is intended as a structural companion to density-type invariants, rather than as an immediate source of stronger ropelength lower bounds. In particular, the optimized factorization by itself does not yield new ropelength bounds; such bounds require independent estimates for the density and compression factors.

Predictive simulation of electrochemical interfaces requires atomistic models that capture reactive bond rearrangements, long-range electrostatics, and charge distributions reflecting the electronic distinctness of electrode and electrolyte. Existing charge-aware machine-learned interatomic potentials (MLIPs) built on global charge equilibration (QEq) settle electrode and electrolyte at a common electrochemical potential, leaving no room for the interfacial gradient that the double layer requires and admitting spurious charge transfer between electronically disconnected regions. Per-fragment charge equilibration is the established remedy in classical molecular dynamics, but reliance on predefined molecular topology has confined it to non-reactive systems. We lift this restriction by making fragment identification itself a differentiable function of atomic geometry, yielding soft fragment-constrained charge equilibration (Soft-FQEq) -- a solver layer that restores fragment-resolved charge conservation in reactive MLIPs. The layer consumes four scalar MLP readouts from a shared atomic-feature network -- per-atom electronegativity, source charge, short-range energy, and a soft bond connectivity -- and returns equilibrated charges together with per-fragment chemical potentials. We implement Soft-FQEq as an extension of the hippynn framework on a HIP-NN feature network and train it on DFT energies, forces, and DDEC6 charges for IrO2/H2O/Na+/ClO4- interfaces. The trained model recovers a clear electrode-to-electrolyte gradient in the per-atom electrochemical potential. With the same trained weights but the fragment-constrained solver replaced by global QEq at inference, this gradient collapses to an essentially uniform profile, directly showing that the gradient cannot be sustained within global QEq while the fragment formulation recovers it.