Papers

A continuous shape parameter enters both a radial density and the normalization of its transform coefficients. This coupling allows changes in shape to be compensated by changes in scale and exponential tilt. We study the coefficients b_k=e^a E[Y^k ]/[k!(d)_k ] for densities proportional to y^(d-1) exp(-{\alpha}y^q+e̱ta}y), with known integer q>=2, d,{\alpha}>0, and unrestricted real a,e̱ta}. For q=2, the first four coefficients determine all parameters globally, whereas three coefficients leave an analytic curve of solutions. We characterize the exact image of the two shape ratios and prove that two alternating corners decide whether an entire coefficient box is realizable and give its exact shape range. For every fixed q, the first q(q+1) coefficients give global identification and an everywhere full-rank map. A finite-difference criterion yields generic identification from 2q+1 coefficients, including every zero-tilt model against the full tilted family. For q=3, an exact rational certificate constructs two distinct shapes with identical first four coefficients and separated fifth coefficients. The results distinguish exact identification, realizability, and sensitivity for a parameter-dependent coefficient normalization.

We improve the shared-leg separation construction in Proposition 3.1 of OpenAI, *An Upper Bound of $9/4$ for the Matrix Multiplication Exponent* (October 2, 2026). If $M$ tensor blocks have a common first space and independent matched sectors on their other two spaces, then $M+n-1$ copies of their matched sum degenerate to the full direct sum of the blocks, each tensored with an $n$-dimensional dot product. For $n=M$, this reduces the number of source copies from $5M$ to $2M-1$. The construction uses polynomial interpolation and truncated local substitution, with no roots of unity or factorial denominators. It works over every field containing enough distinct elements. We verify that the remaining spectral and discrete-growth arguments of the cited preprint are independent of characteristic. Over an algebraic closure, the replacement therefore gives $\omega\leq9/4$ in every characteristic.

A tidal bore is a nonlinear step like surge propagating on the top of a river counter-current. The scientific literature reports that the passage of the tidal front may induce local flow reversals and, in some cases, a global reversal of the river current. Here, combining Particle Image Velocimetry (PIV) measurements and 2D numerical simulations based on a Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) method, we show that the flow classification of tidal bores must explicitly take into account the frame of reference. When the river flow does not reverse after the bore passing, this is a consequence of the perspective imposed by the background current (itself resulting from the previous tide and the natural river discharge toward the sea). Depending on the relative magnitudes of the river current and the velocity field induced by the bore, one may observe a global reversal, a local reversal (or both), or no reversal at all. When the tide is spatially inhomogeneous, with a tidal range that varies along the estuary rather than a simple step in water depth, transitions between bore types may occur during the bore passing.

A set of $m$ distinct positive integers is called a Diophantine $m$-tuple if the product of any two distinct elements, increased by one, is a perfect square. A Diophantine quadruple $\{a,b,c,d\}$ with $a<b<c<d$ is called regular if $$d=a+b+c+2abc+2\sqrt{(ab+1)(ac+1)(bc+1)}.$$ We prove that every irregular Diophantine quadruple satisfies $c<10^{338}$ and $d<10^{10^{24}}$.

The 2026 measles resurgence in Bangladesh motivates a spatial modelling framework that accounts for heterogeneous local transmission, vaccination activity, and interdivisional connectivity. We formulate a multipatch susceptible-vaccinated-infectious-recovered model for the eight administrative divisions of Bangladesh. Division-specific parameters are estimated from reported measles cases \cite{DGHS} and smoothed vaccination data using bounded nonlinear least squares, with uncertainty quantified through 1000 parametric bootstrap replicates. The fitted divisional systems are then connected through compartment-specific movement and cross-patch infection coefficients adjusted for connectivity and the source division's relative reproduction number. The fitted model captures the main division-specific epidemic patterns. Estimated local reproduction numbers range from \(1.1386\) in Khulna (95% CI:\(0.5703\)–\(1.3834\)) to \(4.3491\) in Dhaka (95% CI: \(1.4449\)–\(5.6307\)), while Chattogram also exhibits high transmission potential (\(\mathcal{R}_0^i=4.2275\); 95% CI: \(4.0019\)–\(4.3185\)). Baseline spatial coupling increases the network-wide epidemic peak from \(190.96\) to \(313.20\) infectious individuals and more than doubles cumulative infections from \(1086.42\) to \(2345.35\). Relative to baseline coupling, 50% and 80% restrictions reduce the epidemic peak by \(26.69\%\) and \(34.37\%\), and cumulative infections by \(25.07\%\) and \(41.48\%\), respectively. Conversely, a 50% increase in spatial coupling raises the peak by \(30.14\%\) and cumulative infections by \(23.54\%\). Although local transmission remains dominant in every division, cross-patch infection accounts for \(28.83\%\) of new infections in Rajshahi at the baseline epidemic peak. Source-sink analysis identifies Dhaka as the dominant cross-patch source and Chattogram as a smaller secondary source.

Let $K$ be an imaginary quadratic field with fundamental discriminant $D=-5n$, where $n>0$ is odd, and cyclic ideal class group $\text{Cl}(K)=\langle g\rangle\simeq C_{4M}$. Assume that the ramified prime class above $5$ is the unique involution $g^{2M}$. For the Kronecker eta invariant \[ F(A)=\frac{|\eta(z_A)|^2}{\sqrt{a_A}}, \] where $z_A$ and $a_A$ arise from a positive definite quadratic form representing the ideal class $A$, we show that the logarithms of the mirror ratios \[ R_j=\frac{F(g^{2M-j})}{F(g^j)},\qquad 0\le j<M, \] are related by an explicit discrete cosine transform to the odd class-character Hecke $L(1)$-values. A single level-$5$ generalized-Weber orbit realizes all of these mirrors. Combining Fourier nonvanishing, lower-power Weber class invariants and a ray-class cancellation argument, we prove \[ \mathbb{Q}(R_j)=H^{\langle c\rangle},\qquad R_j^{\sigma^{2M}}=R_j^{-1},\qquad \mathbb{Q}(R_j+R_j^{-1})=H^{\langle c,\sigma^{2M}\rangle}. \] Here $H$ is the Hilbert class field, $c$ denotes complex conjugation, and $\sigma$ corresponds to $g$ under the Artin map. Consequently \[ [\mathbb{Q}(R_j):\mathbb{Q}]=4M,\qquad [\mathbb{Q}(R_j+R_j^{-1}):\mathbb{Q}]=2M, \] and the latter degree becomes $M$ over the real genus field $\mathbb{Q}(\sqrt5)$. Thus the degree pattern $2^r,2^{r-1},2^{r-2}$ in the cyclic $2$-power hierarchy is a specialization of a uniform $C_{4M}$ theorem. We give exact low-class-number evaluations and certified computations at $D=-895,-3295,-8495$, separating respectively four, eight and sixteen non-genus Hecke $L(1)$-values and producing corresponding alternating Rogers-Ramanujan singular values. Classical mixed modular equations then give ordinary Rogers–Ramanujan companions at the same arguments.

The two 4-m class telescopes at the Devasthal site in India are complementary: the 4-m International Liquid Mirror Telescope (ILMT) is dedicated to the discovery of new transient phenomena, which can be followed up using the 3.6-m Devasthal Optical Telescope (DOT) for their optical characterisation. The ILMT continuously surveys a $22.3^\prime \times 22.3^\prime$ wide strip of the zenith sky. A dedicated transient detection and classification pipeline, PyLMT, based on optimised image subtraction and convolutional neural network (CNN) techniques, enables the discovery of numerous photometric transients with the ILMT. However, these transient discoveries still require spectroscopic confirmation to determine their types, for which spectra obtained with the DOT are essential. Here, we present a selection of photometric transients, specifically supernova candidates. We investigate their photometric and spectroscopic properties using ILMT light curves, supplemented with ZTF data and DOT spectroscopic observations.

Topological boundary states offer a platform for guided-wave propagation within bulk bandgaps. Their propagation characteristics depend not only on the spectral extent of the states but also on their dispersion within the operating bandwidth. A nearly linear dispersion provides more uniform group velocity across frequency, whereas a reduced slope enables slow-wave propagation. Despite advances in inverse design of prescribed dispersion relations and topological dispersion tuning, application-oriented characteristics of boundary-state dispersion have rarely been direct inverse-design objectives. Here, we develop a supercell-enabled, surrogate-assisted framework for the inverse design of boundary-state dispersion in a phononic crystal with pseudomagnetic fields. Area-preserving anisotropic deformation shifts the local Dirac points, and independent variation of 21 rows within the supercell provides distributed control over the boundary-state branch. A constraint-preserving surrogate reconstructs the dispersion from an initial frequency and positive frequency increments, with first- and second-difference losses introduced to capture local slope and curvature. Coupled with a genetic algorithm, the framework targets two prescribed-bandwidth design tasks: highly linear dispersion and slow-wave dispersion. For the former, finite-element verification yields relative group-velocity variations of 0.142% to 1.183% over prescribed bandwidths of 4% to 6%. For slow-wave designs, mean group velocities of 190 to 385 m/s are obtained over bandwidths of 0.2% to 0.8% while satisfying a 5% group-velocity variation limit. These results establish boundary-state dispersion as a direct inverse-design target beyond conventional bandwidth optimization, opening a flexible route to group-velocity control through application-oriented dispersion engineering over prescribed relative bandwidths.

As power systems shift toward high shares of wind and solar, long-duration storage becomes pivotal for balancing multi-day deficits and surpluses of uncertain supply and demand. Current dispatch strategies for long-duration storage either assign a fixed value to stored energy or extend the modeled horizon using a single forecast. Stochastic system dispatch with a multi-day horizon for long-duration storage remains unexplored, and existing stochastic dispatch only considers day-ahead uncertainty or Markovian uncertainty beyond one week of perfect foresight. Therefore, we benchmark stochastic multi-day dispatch against deterministic alternatives in a rolling-horizon setup. To emulate probabilistic multi-day forecasts, we combine diffusion with a semi-Markov model of weather regimes and a logistic growth model for meteorological forecast errors. At high renewable shares, stochastic operation reduces thermal backup generation and consistently eliminates loss of load. The benefits decrease with smaller shares of renewables, less long-duration storage, or higher forecast skill. Overall, our results suggest that systems with high shares of variable renewables can substantially benefit from probabilistic multi-day forecasts and stochastic dispatch.

We study disformal transformations of $\mathcal{N}$ scalar fields coupled to a Dirac fermion. The resulting axial coupling generically introduces higher derivatives in the scalar field equations of motion; for field-dependent free functions, these are absent if and only if the disformal transformation acts along a single field-space direction. We show that these higher derivatives do not signal an Ostrogradsky ghost. After the transformation, the fermion registers the derivatives of the matter-frame tetrad only through the axial current, so that every coupling to the derivatives of the scalar fields multiplies a fermion bilinear, which is Grassmann even and nilpotent. Such terms do not contribute to the body of the constraint matrix, and at each level of the Grassmann expansion the field equations are linear, with the principal part of the theory without fermions. Through a Hamiltonian analysis, we show that the number of degrees of freedom is that of the bosonic theory plus that of the fermion, both in the Einstein frame, for free functions depending on the scalar fields and their kinetic matrix, and for minimally coupled fermions in multi-field DHOST theories. The same mechanism applies to disformal transformations with higher derivatives of a single scalar field, for which fermions impose no condition beyond those required by bosonic matter.

We give an explicit construction proving that the exact stabilizer rank of $m$ copies of the qubit face-center magic state is $O(2^{3m/8})$. Two symmetry-related fifteen-term identities combine into a twenty-four-term decomposition at twelve copies. The balanced structure of this decomposition yields a substitution that adds eight copies at a cost of eight terms per existing term. The resulting bound $\gamma_T \leq 3/8$ improves the previously published exponent $\log_2(3)/4$.

Optical lattice clocks in space could improve satellite navigation systems and enable new tests of fundamental physics, including gravitational-wave detection and tests of relativity. In microgravity, atoms can tunnel to degenerate neighboring lattice sites, broadening the clock transition and limiting coherence times. We demonstrate Aubry-André localization in a bichromatic optical lattice formed by two magic-wavelength lasers, suppressing tunneling in a horizontal lattice clock emulating free fall. We recover a Fourier-limited clock Rabi spectrum and measure Ramsey coherence times of up to 2 s in shallow lattices, with up to 100-fold improvement over the delocalized regime. These results establish Aubry-André localization as a practical route to controlling tunneling in future space-based clocks, and open a path to operating terrestrial clocks at shallower lattice depths.

This paper addresses the control of autonomous underwater vehicles (AUVs) operating in dynamic environments, where wave-induced forces, modelling errors, and actuator limitations make reliable control challenging. Active disturbance rejection control (ADRC), together with an extended state observer (ESO), estimates lumped uncertainties online, simplifying control design using nominal models. However, existing stability analyses typically assume bounded control inputs and derivatives, which are difficult to guarantee under unknown disturbances and measurement noise. To bridge this theory–practice gap, this paper introduces an ADRC framework with active input saturation and post-saturation low-pass filtering. The saturation constrains the control magnitude, while the filter regularizes the saturated signal to ensure bounded input-rate variations. By explicitly enforcing these conditions, the proposed system guarantees semi-global practical asymptotic stability. High-fidelity simulations and experiments on an underwater robotic platform demonstrate robust control performance under severe environmental disturbances and actuator constraints. A supplementary video is available at https://youtu.be/AEFlZ69Yg00.

The Moran effect traditionally refers to the synchronization of population dynamics driven by shared environmental forcing rather than direct interactions among populations. Although extensively studied in ecology, the Moran effect has received limited attention in geography, and its applicability has yet to be fully explored. In this study, we build upon the classical Moran effect in population ecology as a conceptual foundation, developing a more comprehensive conceptual framework in a geographic context. We introduce the new concept Geographic Moran Effect (GME), defined as temporally coherent and synchronized variation across geographic entities induced by common external forcing, with the magnitude and potentially the direction of local responses varying according to a spatially varying sensitivity field. A two-step spatiotemporal factor framework is developed to identify and interpret the GME. The framework first tests for the possible presence of GME-type synchrony in spatiotemporal data and then examines the GME and its spatiotemporal evolution by investigating its structure through spatial and temporal dimensions separately. A synthetic experiment validates the framework's reliability, while an application to U.S. county-level presidential election data (2008-2024) demonstrates its ability to identify the GME and its spatial and temporal components. By redefining the Moran effect as a geographic theory and a distinct object of inference, this work provides a mechanism-oriented perspective on spatiotemporal analysis with broad applicability in geography, ecology, and public health, where explicitly identifying and interpreting GME-type synchrony is essential for understanding system-level dynamics.

A dynamic optimization workflow for linear gyrokinetic stability that couples the gyrokinetic code CGYRO with the libEnsemble framework for ensemble management has been developed. The workflow is designed to efficiently explore multi-dimensional plasma parameter spaces and to address the practical challenges of simulation-driven optimization in fusion plasma physics, including expensive objective evaluations, unavailable gradient information, non-smoothness arising from transitions between dominant instability branches, and simultaneous multi-core simulations on high-performance computing resources. Rather than optimizing the growth rate at a single wavenumber, the objective maximizes the growth rate-to-wavenumber ratio over the ion-scale spectrum. A derivative-free, model-based optimization algorithm employs manifold sampling to handle the nonsmooth composite structure of the objective function. The approach is applied to the optimization of local magnetic flux-surface shaping parameters, with an equilibrium baseline drawn from DIII-D negative triangularity experimental parameters. The workflow is general and extensible and provides a foundation for future optimization studies involving nonlinear transport objectives and higher-dimensional parameter spaces.

In humans, proprioception corresponds to the sense of self-movement or position, e.g. understanding the position of one's arm. This sense is often less well-understood and in popular understanding sometimes conflated with introspection. But, notably, proprioception is a separate sense, with dedicated receptors and verifiable outcomes. On the other hand though, language models cannot observe their own internal states during training (and for example answer the question which experts activated in a certain layer of an MoE) and so cannot accurately report on them. Yet, such information is, by virtue of being internal to the model, both accessible, and readily constructed into verifiable signals. We argue that this provides the basis for a digital analog of proprioception. As such, we train models through SFT and on-policy training to equip them with this ability. We show that modern models can be trained to understand a range of subtypes of proprioception, such as routing decision awareness, token entropy, decision depth or logit lens information, without affecting model capabilities. We then run these models through a broad evaluation of situational awareness, legibility, and epiphenomenality, to evaluate in what ways access to proprioception changes model behavior.

Learning management system traces are clustered to identify participation profiles, but unequal observation windows may cause profiles to reflect duration rather than participation alone. Using 16,448 complete learner–course sequences from 4,599 learners across 152 courses at a South African public university, we evaluated eight participation measures. All remained related to observation-window length even when expressed as rates or shares. Course-grouped adjustment reduced these relationships. Five retained measures were evaluated using K-means, Gaussian mixtures, Ward agglomerative clustering and K-medoids, with standardised measures and principal-component summaries. Evidence supported a broad two-profile grouping and a more detailed adjusted four-profile consensus: Gradually declining, Early-concentrated, Cyclical variable and Sustained active. Profile membership accounted for 2.87% of the variation in duration in the primary consensus, versus 38.18% using original measures. Mean assignment support was 0.739; 8.35% of assignments fell below 0.50, with greatest uncertainty in Sustained active. Two week-by-week sequence analyses agreed strongly with each other but only modestly with the four-profile consensus, indicating that the profiles combined multiple aspects of participation rather than distinct sequence patterns. Profiling should therefore assess duration effects, compare clustering methods and participation summaries, and report assignment consistency.

Financial Artificial Intelligence (AI) governance and model risk management (MRM) emphasize independent validation, transparency, and effective challenge of AI-generated computations. Text-to-SQL translates natural-language questions into Structured Query Language (SQL), but queries that agree on one database can still express different computations. We present VeriPIE, a neurosymbolic framework combining language-model generation with symbolic checks. It ensembles direct SQL generation with QuerySketch, a structured plan language checked by a backend implemented in Lean, a theorem prover and programming language. Plan diagnostics guide repair before SQL generation, while execution agreement and bounded searches for distinguishing databases inform candidate selection. These checks expose malformed plans and hidden disagreement without proving agreement with the user's intent. In single-run experiments on the complete Spider development set, VeriPIE achieves 72.05% test-suite accuracy (745/1,034), compared with DeepEye's 65.67% (679/1,034). Its two-query shortlist covers 78.43% (811 questions), compared with 74.66% for DIN-SQL (772) and 69.05% for DeepEye (714). On BIRD, VeriPIE reaches 68.90% execution accuracy and 73.73% two-query coverage. The central benefit is reviewable evidence about the generated computation: explicit plan checks, repair diagnostics, and concrete disagreements that support safer financial analysis.

In this paper, we present a comprehensive theoretical study of the electronic states in gapped graphene in the presence of a charge impurity and a magnetic field. We focus on the impurity-induced modification of the Landau levels under the influence of an attractive or repulsive Coulomb potential. Since analytical solutions of the Dirac equation are not available in this case, the study is carried out using the Wentzel-Kramers-Brillouin (WKB) approach. Special attention is paid to the screening of the Coulomb potential in the presence of metallic gates situated at some distances from graphene sheet, as is commonly the case in experiments. Using both analytical and numerical methods, we determine the energy spectrum as a function of the magnetic field strength and impurity charge for screened and unscreened Coulomb potentials. Our results reveal a significant dependence of the energy levels for sample-to-gate distances of less than 50 nm.

In this paper, we establish large deviations of invariant measures for fractional stochastic reaction-diffusion equations with polynomial reaction term driven by unbounded multiplicative noise on R^n. Under a dissipativity condition, we first prove the Freidlin-Wentzell large deviation principle of solution trajectories uniformly over all initial data in L^2(R^n). The key ingredient is a positive-time L^\infty-estimate for differences of controlled solutions which is independent of all initial data and hence yields uniform convergence of controlled stochastic solutions over the entire tphase space. Then using the uniform trajectory estimates, contraction and a second-moment estimate, we further prove the large deviations of invariant measures of the equation with a good quasipotential without introducing weighted spaces or establishing exponential tightness. The non-compactness of Sobolev embeddings in R^n is circumvented by uniform spatial tail-ends estimates. This is the first result on large deviations of invariant measures of the equation with multiplicative noise on unbounded domains.

Retrospective evaluation of early-warning models may not reflect information available when predictions are issued and outcomes become observable. We developed a time-synchronous prequential framework for online prediction of next-week student disengagement, operationalised as inactivity onset, from Moodle logs. Ordered by calendar time, models were updated only with available outcomes and remained unchanged for all predictions at each prediction cutoff; development, calibration and evaluation were chronologically separate. We compared stochastic gradient descent logistic regression (SGD-LR), online sequential extreme learning machine (OSELM) and echo state network (ESN) across predictive performance, fixed, frozen and rolling threshold policies, calendar-time variation and four behavioural feature sets. Held-out evaluation comprised 49,329 predictions with 37.68% inactivity-onset prevalence. Discrimination was similar (average precision 0.665-0.679; ROC-AUC 0.755-0.774). OSELM had the highest recall, SGD-LR the highest precision and specificity, and ESN the highest Matthews correlation coefficient and balanced accuracy. Lower thresholds increased recall and alert volume, whereas rolling adaptation did not improve MCC over frozen thresholds. Weekly performance and calibrated-probability distributions varied without sustained deterioration. During model selection, cumulative-and-temporal features achieved the highest individual-set AP, with modest gains from combining all feature groups. These findings indicate that streaming early-warning evaluation should consider calibration, threshold-dependent performance, alert burden and calendar-time variation alongside aggregate discrimination.

Peter Lax's 1968 discovery that certain nonlinear wave equations possess operator pairs (Lax pairs) opened the theory of completely integrable systems, yielding exact solutions, infinite conservation hierarchies, and perfectly elastic soliton collisions. When a nonlinear dispersive equation is provably non-integrable (admitting no Lax pair and only finitely many conservation laws), what mathematical structure governs its dynamics? The K(m,m) compacton equations, u_t + (u^m)_x + (u^m)_{xxx} = 0 for m >= 2, provide a testing ground: they support traveling waves with compact support, yet fail every known integrability test. We prove universal algebraic formulas determining the shape, width, and speed of every K(m,m) compacton. We develop a constraint-deficit mechanism explaining how the five independent conservation laws constrain emergent amplitudes and suppress the fundamental Fourier mode of the collision debris, while higher radiation modes remain unconstrained. The mechanism's conclusions rest on explicit hypotheses supported by numerical evidence. A phase-plane analysis reveals the compacton as a singular limit of smooth solitary waves through simultaneous regularity loss and support compactification. These results delineate intermediate territory between integrable order and generic nonlinear dynamics, organized by scaling universality and by the constraint deficit of a finite conservation structure.

As language-model-based coding agents evolve their programs and reuse code from one another, safety failures can acquire a history that extends beyond the agent in which they are detected. An unsafe change may originate in one agent, propagate through intermediate recipients, and survive subsequent revisions before an audit reveals it. Understanding this history is important for identifying the introducing change and distinguishing inherited behavior from independently occurring failures. We study origin attribution in self-evolving multi-agent coding systems: recovering the agent and program version that introduced an observed unsafe behavior into the target's code lineage, or determining that it arose locally. This task is challenging because rewriting obscures textual similarity, independent occurrences confound temporal ordering, and intermediate recipients can resemble the source. We propose a method that combines behavioral fingerprint matching with historical localization and recursive tracing. It compares candidate implementations beyond the unsafe outcome, retains evidence from multiple historical versions, and follows matching candidates upstream. On 158 queries across 85 controlled scenarios, the method achieves 96.8% exact attribution accuracy, compared with 77.2% for exhaustive fingerprint matching without recursion. Component ablations and evaluation on 34 additional queries using the same method and hyperparameters demonstrate the complementary roles of multiple representatives and recursive tracing. Further checks extend the evaluation to approval-boundary and resource-owner errors. Together, these results show how behavioral and historical evidence can recover the origins of unsafe behavior through code revision and cross-agent propagation.

Artificial intelligence (AI) increasingly participates in communication, decision-making, and collective action, making the social organisation of human-machine interaction central to Hybrid Intelligence (HI). This chapter reviews the behavioural sociology of humans and machines through an HI lens, examining how interaction and feedback shape the combination of human and artificial capabilities. Drawing on sociology, social psychology, human-computer interaction, organisational studies, collective intelligence, and science and technology studies, it traces processes from individual social perception to cooperation in hybrid groups and organisational and institutional arrangements. Particular attention is given to how machine participation alters familiar mechanisms: social categorisation attaches expectations to designed identities, reciprocity may involve unequal social obligations, and algorithmic recommendations acquire authority through organisational practices. The chapter distinguishes observable machine behaviour from human interpretations of intention, understanding, and moral responsibility. It also considers how behavioural effects can extend beyond direct human-machine encounters, while separating established findings from proposed longer-term consequences. It argues that the benefits of HI depend on how roles, incentives, reliance, and accountability are organised, and identifies priorities for empirical research on these evolving sociotechnical arrangements.

We consider the 1-D focusing cubic NLS with a smooth, spatially-dependent potential which decays rapidly as $x \rightarrow \pm \infty$. As is well known, under sufficient regularity conditions, solutions of the cubic NLS with such a spatially-dependent potential conserve both mass and energy. Additionally, in the absence of a potential, the 1-D focusing cubic NLS admits boosted soliton solutions. Furthermore, in the presence of a potential, the interaction of the potential with boosted soliton solutions to the free case is of active interest in the literature, particularly with respect to the dependence of this soliton-potential interaction on the velocity of the incident soliton. We adapt a finite-difference scheme to the setting of a smooth, rapidly-decaying spatially-dependent potential and show that this adaptation preserves the appropriate mass and energy invariants. Additionally, we use this algorithm to model soliton-potential interactions at a variety of incident soliton velocities.

We present the new CTEQ-JLab global QCD analysis, called CJ26, that simultaneously fits parton distribution functions of the proton and nuclear effects in deuteron targets. This work includes, for the first time in a global fit, the full set of JLab 6 GeV inclusive DIS measurements and the first published JLab 12 GeV data. The analysis focuses on the large-$x$ region, and carefully treats the interplay between higher-twist effects and off-shell corrections in low-energy proton and deuteron target data. We discuss their effect on quark and gluon distributions, and highlight the importance of quantifying the experimental correlated systematic uncertainty to achieve fits of the highest accuracy. We also outline how upcoming data may help unravel the nuclear and partonic dynamics in light nuclei. The extracted next-to-leading order parton distributions are available on the CJ web pages in LHAPDF format for general use, accompanied by leading- and higher-twist structure functions calculated with the simultaneously fitted PDFs, higher-twist corrections, and off-shell corrections. This write up clarifies several points that became clearer while finalizing the CJ26 paper and can be read as a companion to the latter.

Prescribed-time controllers drive the state to the origin at a time $T$ chosen in advance, with gains that grow without bound as $t$ approaches $T$. This paper shows the growth is a necessity and computes the exact price of avoiding it. The starting point is a simple estimate: any trajectory with $\|\dot z(t)\|\le L(t)\|z(t)\|$ satisfies $\|z(t)\|\ge\|z(0)\|e^{-\int_0^t L}$. From this: a gain profile $L$ supports exact prescribed-time convergence, over the unrestricted class of closed-loop vector fields admitting it, if and only if its integral over $[0,T)$ is infinite; so every design must either use an envelope of infinite integral or violate the linear growth bound at the origin, as finite-time and fixed-time designs do. Under a finite gain budget $\mathcal B=\int_0^T L$ the best worst-case contraction over the class is exactly $e^{-\mathcal B}$, a floor valid for every plant, and a finite-budget trajectory converges to a definite point away from the origin. A comparison lemma then carries the theory beyond linear growth: for envelopes $\|\dot z\|\le L(t)\,\omega(\|z\|)$, first arrival at the origin requires budget at least $\int_0^{\|z_0\|}du/\omega(u)$; arrival is impossible whenever this integral diverges, and when it is finite it is the exact threshold, attained inside the class. The classical settling expressions of finite-time control reappear as hitting-time floors, and the known fixed-time condition is the two-ended convergence of the same integral. For linear time-varying loops the results refine to logarithmic-norm budgets, corollaries of a known two-sided estimate, with the terminal error of gain freezing described exactly.

Machine learning surrogates of partial differential equations (PDEs) are commonly trained on data generated by numerical solvers. Therefore, the training data obey not the PDE but the modified equation of the numerical scheme, and the test loss, computed on data from the same solver, cannot distinguish a model that learned the physics from one that learned the discretization error. Here, we show that the difference is nevertheless visible in the trained model. We propose a suite of structure-preserving fidelity diagnostics, simple scores that measure whether a learned model respects one known property of the physics, such as a conserved quantity, a symmetry or a Hamiltonian structure, without access to the training data or the governing equation. Over five systems, the diagnostics track the error with respect to the true dynamics with a rank correlation of up to $0.94$, while the test loss is uncorrelated or anti-correlated with it. For the Burgers, Korteweg-de Vries and Navier-Stokes equations, the diagnostics separate numerical schemes that the test loss ranks as equivalent, including schemes of the same order. order.

Furusho, Hirose and Komiyama reformulated the pentagon equation using an involution called the flip, which led to a new characterisation of the graded Grothendieck-Teichmüller group $\mathsf{GRT}_1$. We show that the flip is a reflection in the cyclic structure of the operad of framed Drinfeld-Kohno Lie algebras, with the framings removed. Applying the reflection twice to the pentagon gives the two-cycle relation for every Drinfeld associator. This relation is precisely the condition for cyclic compatibility on the associativity morphism, giving a new proof that $\mathsf{GRT}_1$ acts by cyclic automorphisms of parenthesized ribbon chord diagrams.

Estimation error in short, correlated pricing panels is dominated by trajectory dispersion rather than sampling noise, creating an identification shortfall that only independently assigned variation can resolve. We propose a guided iteration methodology to decide how much variation to generate, where, and for how long: a cyclic price-perturbation scheme under a revenue-at-risk budget, framed as variance minimization over design-specific error covariances, analogous to portfolio selection. We establish four key results: (1) a closed-form minimum-variance combination rule; (2) proof that a convex cost of intensity yields an interior, spread-out allocation, whereas under a concave tolerable-volume-loss cost, concentrating intensity into fewer perturbations pays off only with heterogeneous candidates, priced-in fixed costs, or perturbations beyond the cost curve's inflection point; (3) a discounted recursion yielding a data-dependent exit timeline from the experimentation zone; and (4) an application to Dominick's retail data, in which about five designed independent units match the precision of approximately 700 passive equivalents in the anchor category, while a fast proxy across ten categories achieves a median 97% variance reduction relative to passive panels. Sixteen comparisons (eleven pre-specified) support the theory: greedy selection achieves 90-93% of the global optimum across more than fifty exhaustively verifiable instances, and the recursion reaches its predicted 90%-of-steady-state benchmark two cycles early, though a parametric bootstrap indicates that its delta-method interval understates the right-tail risk. Boundary conditions show that regularized covariance estimators extend plug-in noise tolerance from 29% to beyond 50%, submodularity holds up to a clear curvature threshold, and a shared shock imposes a fundamental precision floor.