Papers

The rapid growth of resident space objects is increasing the complexity of space situational awareness sensor tasking, challenging classical optimization methods as they allocate finite, heterogeneous, and distributed sensing resources across ever-larger catalogues. Existing deep reinforcement learning approaches show promise in reduced settings, but fixed-dimensional state and action representations limit their ability to scale to large, dynamic catalogues and distributed sensing networks. We introduce VISTA (Variable-Entity Intelligent Sensor Tasking Architecture), a scalable deep reinforcement learning architecture for persistent uncertainty-driven catalogue maintenance across variable object populations and sensor configurations. VISTA combines physics- and mission-informed top-K retrieval with entity-centric attention, recurrent memory, and pointer-based action decoding, thereby keeping each agent's observation and action spaces independent of catalogue size. We evaluate VISTA across different scenarios, from fixed-size single-sensor benchmarks to large-scale space-based tasking and heterogeneous cooperative sensing. With 30 orbiting targets, VISTA recovers the catalogue 31.2% faster than the fixed-dimensional recurrent baseline. In the large-scale regime, VISTA reduces five-hour uncertainty by 97.5% relative to the strongest classical reference and by 99.3% relative to the recurrent learner. Zero-shot tests up to 20,000 objects reveal near-linear relations between sensing capacity, catalogue size, and recovery horizon. Learned policies also exhibit sensor modality adaptation and generalization to population and initial-uncertainty shifts. Together, these results demonstrate that VISTA provides a scalable framework for adaptive space situational awareness sensor tasking across large, distributed networks of heterogeneous ground- and space-based sensors.

We describe the first Galactic-scale extreme-precision radial velocity (EPRV) survey and outline our goals of measuring Galactic accelerations and deriving constraints on the dark matter distribution in our Galaxy. We are targeting quiet subgiant stars within a kiloparsec in radius of the Sun and about 2 kiloparsec in vertical height relative to the Galactic mid-plane. The observations described here were carried out using the Echelle SPectrograph for Rocky Exoplanets and Stable Spectroscopic Observations (ESPRESSO) at the Very Large Telescope (VLT), between 2021-2025. We began by operating at a typical RV precision of $\sim$ 70 cm/s to select quiet stars from stellar activity indicators, and are now operating at $\sim$ 50 cm/s. We selected stars with low stellar jitter (determined from their RV scatter as a function of epoch) with the goal that they yield a measurement of the Galactic acceleration on approximately a decade timescale. We refer to this as our “gold" sample. We also identify stars with possible companions (either brown dwarfs or planetary companions) or hitherto unknown stellar binaries that we have culled from our survey. For a few cases, we present initial constraints on the companion mass and orbital period. We also present the photospheric and fundamental stellar parameters for our “gold" sample of sub-giant stars. This first Galactic-scale EPRV survey provides a pathfinder for next-generation surveys with extremely large telescopes, opening the possibility of directly mapping the gravitational acceleration field across the Milky Way and potentially beyond.

In this short note we prove a rigidity result for orientable minimal submanifolds in the round sphere of the lowest non-trivial Morse index. In particular, we show that the only closed and connected $(2m-2)$-dimensional orientable minimal submanifolds in $\mathbb{S}^{2m}$ of index less than or equal to $2m+1$ are the totally geodesic spheres.

We prove that a smooth, regular, asymptotically flat stationary vacuum black hole has a nonextremal Kerr exterior if no null geodesic orthogonal to its stationary Killing field is trapped, establishing the rigidity conjecture of Alexakis, Ionescu, and Klainerman. In the rotating case, we start with the rotational Killing field constructed near the bifurcation sphere and study a maximal domain carrying its complete circle action. The main difficulty is to recover boundary regularity and the curvature condition needed for local Killing extension. We derive uniform estimates along complete rotational orbits from nontrapping and combine them with convexity along short null geodesics. These estimates show that a nonempty remaining boundary in the ergoregion is a timelike photon surface for the stationary quotient metric. Its null geodesics lift to spacetime null geodesics orthogonal to the stationary field. The vacuum circularity identities give their continuation through contact with the ergosurface, keeping their stationary projections in a compact set. This contradicts nontrapping and yields the global rotational symmetry required by black hole uniqueness.

We study homological Bogomolov multipliers and finitary Bogomolov multipliers of infinite groups, with emphasis on polycyclic groups. We obtain structural descriptions for torsion-free nilpotent groups of class two, semidirect products, wreath products, and infinite simple groups. The class-two case reveals a connection with rational points on linear sections of Grassmannians, explaining an arithmetic obstruction to a general algorithm. On the positive side, we give a deterministic algorithm for virtually abelian polycyclic groups and an algorithm for finitary Bogomolov multipliers of arbitrary polycyclic groups, both implemented in GAP.

Reservoirs with the same mean relaxation rate can be indistinguishable at the level of average energy transport while producing different fluctuations. We study a bosonic mode coupled to hot and cold Poisson streams of thermal ancillas through finite beam-splitter collisions. The averaged evolution is a compound-Poisson semigroup generated by finite Gaussian event channels. Its nonequilibrium steady state is an exact mixture of thermal states governed by a random affine fixed point. Consequently, the mean-occupation dynamics and bath-resolved mean heat currents coincide with those of the matched continuous Lindblad reservoir, whereas higher correlations retain the collision strength. For equal collision transmissivities, we derive an exact superthermal bunching law controlled by the temperature contrast and collision strength. We also obtain the asymptotic rates of the first two heat cumulants for an ideal stationary event-resolved two-point-measurement record and separate local one-collision contributions from temporal correlations. Fock-space diagonalization and Monte Carlo trajectories validate the formulas and connect full resets to the weak-collision Gaussian limit.

Deriving upper bounds on code size from existing bounds is a classical approach in coding theory, dating back to the seminal results of Elias, Bassalygo, and Levenshtein. We study a framework encompassing the Bassalygo–Elias and Levenshtein inequalities for binary and nonbinary (constant-weight) codes and provides certain generalizations. The asymptotic cost of transferring a bound between different symbol compositions is expressed in terms of mutual information, yielding an information-theoretic optimal transport formulation. We determine the optimal permutation-transport cost between arbitrary compositions in terms of their least common majorant in the majorization order. Specializing to symmetric constant-weight compositions yields explicit transport profiles. As a byproduct, we establish unimodality of the asymptotic constant-weight rate as a function of the relative weight. We prove that the resulting closure operators are idempotent and that applying transport before outer Bassalygo–Elias averaging leaves the unrestricted bound obtained from the same input unchanged. We also establish necessary and sufficient conditions for an upper bound to be a fixed point of the closure operator. Since the asymptotic rate function is an upper bound for itself and it is a fixed point, we conclude the Schur concavity of the constant-composition rate function. Finally, we survey existing upper bounds for binary and nonbinary constant-weight and constant-composition codes, combine them into optimized envelopes within the transport framework, and obtain improved theoretical and numerical bounds.

On 7 and 8 September 2026 programmatic search produced singularities: a forced Navier-Stokes singularity at every fixed viscosity, statements (C) and (D) of the Clay problem, and two Euler singularities. The unforced problem, statements (A) and (B), stands open, and a search for it needs a target. This paper fixes one: the positive defect problem, that a Leray-Hopf solution from smooth data on the periodic cube loses energy on a finite window, at fixed viscosity, beyond what viscosity removes. A positive defect implies blowup and so a negative answer to statement (B); the converse is not known. The paper proves the target equivalent to a floor on the energy flux through the Littlewood-Paley shells, the Fourier-side form of the coarse-grained flux of the Onsager theory of turbulence, averaged over the window; states necessary conditions on a candidate: a singular time of Type II in velocity, energy concentrating on a set of zero length, a pressure outside L^2, a velocity outside the Onsager-critical class L^3_t B^{1/3}_{3,c_0}, an obstruction to collapse onto a fixed steady Euler profile; and states what cannot certify one: no finite computation witnesses a Galerkin-uniform ceiling, and selection and forcing return the question to a positive defect. A pseudo-spectral search at 128^3 and 256^3 shows which condition of the reduction binds: the fine-shell flux floor holds to within 1 to 6 percent of the ceiling for a third of a turnover time, and fails in scale at the Kolmogorov wavenumber, so a candidate must differ from generic turbulence in the depth of its cascade, not in its timing. Every implication not marked otherwise is a theorem in Lean 4 over Mathlib; the library contains no Navier-Stokes object, and the equation enters only through hypotheses.

In this paper we find a sufficient condition on an Ahlfors–David regular metric measure space under which it admits a partition into parts of prescribed measures and optimal (up to a constant) diameters. The proof uses the construction of dyadic cubes. The process is algorithmic: the pieces are cut out one by one via a filling procedure on the tree of dyadic cubes. We introduce the notion of spaces which admit a connected dyadic cube decomposition and prove that they admit a partition of the kind described above. We then develop several techniques to obtain such spaces and show some natural examples of this type.

We construct a bar-type operation for the affine Hecke algebra $\Ha$ attached to a symmetrizable Kac–Moody group. Our main tool is a geometric bimodule defined using the equivariant $K$-theory of a suitable version of the Steinberg variety for this Kac–Moody group. Using elements of this bimodule corresponding to standard and costandard Hodge $D$-modules on the Kashiwara flag variety, we define a bar involution on $\mathcal{H}_{W}^a$ by relating the left and right actions of $\mathcal{H}_{W}^a$ on this bimodule. A central issue is that the resulting formulas for the bar operation do not preserve the algebra itself and naturally produce infinite expansions. To address this, we introduce certain completions of $\mathcal{H}_{W}^a$ on which the bar operation is well-defined. For this construction, we prove a number of combinatorial finiteness results for products in $\mathcal{H}_{W}^a$. In affine type, we further analyze the level-zero part of the algebra and obtain an explicit combinatorial formula for a bar operation on a suitable localization of Cherednik's double affine Hecke algebra. In this case the bar operation itself is not an involution but we also show that this bar operation becomes an involution when the lattice parameter $q$ is set to $1$.

The Voyager spacecraft have been detecting plasma oscillations of a few kHz for over 40 years. However, there is still no widely accepted hypothesis for the astrophysical source of the oscillations. Having entered the local interstellar medium, these detections have persisted, adding to the challenge of identifying the astrophysical source. Neutron stars were one of the earliest hypothesized candidates of the electromagnetic radiation responsible for the detected plasma oscillations, but no theoretical mechanism could be identified to explain the frequency of the oscillations. We explore the recently developed theory of co-production of electromagnetic radiation in gravitational wave backgrounds as a possible source of these low-frequency oscillations and investigate whether neutron stars could be the astrophysical source.

In this paper we study the Borel summability and quantum modularity of the free energy computed by refined topological recursion on the Weber-type refined spectral curve. On the one hand, we verify expected properties, namely, Stokes rays come in a pair and the Borel-Laplace sum of the free energy coincides with the Barnes double gamma function. Somewhat unexpectedly, on the other hand, we prove that the generating function of the Stokes constants along each ray is a Jacobi quantum modular function. Our proof suggests that quantum modularity may be a general feature of refined topological recursion, applicable beyond the Weber-type refined spectral curve.

Manipulation policies must know which objects matter and where they are, yet the pretrained backbones that current robot foundation models build on, from language in vision-language-action models (VLAs) to video generation in world-action models (WAMs), do not directly require this metric grounding, leaving it to be learned implicitly from robot demonstrations. We propose Grounded Action Models (GAMs), a new paradigm of robot foundation models built with 3D grounding. GAM can be conditioned using language, points, or box prompts, which are first transformed into a shared object-centric representation of the selected objects. This representation captures target-focused visual features and metric object geometry, which is mixed with robot state history through a multi-stream transformer to predict action chunks. Although GAMs can be run autonomously, they can also serve as a low-level controller that a high-level planner controls using its various input modalities, allowing for long-horizon and memory-dependent manipulation. On RoboTwin 2.0, GAM achieves an average success rate of 55.3% across 50 tasks (vs. 52.0% for Spatial Forcing), including 47.6% under scene randomization (vs. 30.4% for Abot-M0), with its action policy trained only on clean-scene demonstrations. On LIBERO-PRO, it achieves a state-of-the-art average success rate of 61% (vs. 53% for $\pi_{0.5}$) across 16 perturbation settings, with the largest gains when targets are relocated or newly designated. On two real robots, GAM retains 17/20 successes under visual shift on a bimanual YAM versus 4/20 for $\pi_{0.5}$, while its composition with a Molmo2 planner on a Franka achieves 64.7% ID and 49.8% OOD step completion on long-horizon and memory-dependent tasks.

The target trial framework is increasingly used to define the causal estimand for an observational analysis. However, designing time-to-event analyses that align with the causal estimand is challenging in pregnancy because the timescale on which outcomes are often defined, "gestational age", differs from the timescale on which outcomes are measured, "time since study entry". We argue that neither timescale is "best" but that each aligns with a different estimand. In this article, we use the target trial framework to define the causal estimands corresponding to each timescale. We consider a (hypothetical) target trial comparing randomization to 17-alpha hydroxyprogesterone caproate (17-OHPC) versus placebo between 16-20 weeks of gestation to prevent delivery before 37 weeks' gestation, whether by miscarriage, stillbirth, or live birth. We define the estimand of time-to-event analyses on the "time since study entry" timescale as the effect of randomization to 17-OHPC versus placebo between 16-20 weeks of gestation on delivery before 37 weeks'. In contrast, the causal estimand of time-to-event analyses using the "gestational age" timescale corresponds to the effect had, counter to fact, all participants been randomized at one gestational age (e.g., 16 weeks). We review specific challenges to estimation and possible solutions for each timescale.

In this paper, we examine how intentionally designing for belonging impacts outcomes in a technology program serving a population of youth who have not had safe and supported experiences with technology - youth impacted by foster care. We reflect on the design of an internship program which engaged two foster-involved youth in creative self-expression and co-design with technology. We first outline how we designed the program to foster belonging, then analyze intern and facilitator reflections to understand how the program impacted the interns' experience. We highlight how the internship design affected youth's inclusion and overall engagement; self-expression; exploration; and self-efficacy. We conclude with a discussion of design recommendations for technologists and service providers aiming to engage youth in expressive and co-design technology experiences, particularly youth impacted by foster care.

Multimodal Large Language Models (MLLMs) commonly reuse visual encoders pretrained with CLIP, although the features of these ViTs are ultimately consumed by autoregressive LLMs. We refer to this mismatch as the semantic-interface gap and introduce MedMLIP, a framework that pretrains the visual encoder through report generation with a frozen LLM, while employing Local Relational Distillation (LRD) to preserve relationships among visual patches to avoid visual collapse. We pretrain MedMLIP on IU-Xray and Open-PMC-300K and evaluate the resulting encoders on VQA-RAD and SLAKE. Only the ViT is transferred, while the guiding LLM and projector are replaced, allowing us to assess cross-LLM transferability. Our cross-LLM transfer experiments demonstrate the value of pretraining visual encoders for their autoregressive LLM interface while trying to preserve more fine-grained visual information. Code and the pretrained model are available at https://github.com/SkyCol/MedMLIP

In agentic coding, developers shift from implementing changes themselves to specifying intent, evaluating the agent's work, and making approval decisions. However, delegating implementation can introduce cognitive debt that erodes project comprehension over time, constraining developers' ability to provide oversight. In this work, we investigate the role of persistent shared representations in supporting project comprehension and oversight of coding agents. We present Vibe-GUIDE, an agentic coding interface built around a structural, live, manipulable, and adaptive graph representation organized by functional modules. We evaluated our interface in a randomized between-subjects study comparing how 16 developers completed three cumulative coding tasks using our interface or a Chat-only baseline. We found that Vibe-GUIDE can support project comprehension and sustained task performance while keeping developers cognitively involved in oversight. These findings show how persistent shared representations can complement natural-language interaction and help developers maintain the understanding needed to oversee agent-generated changes as projects evolve.

Let $d \geq 1$ be fixed and let \[ D(n,d) := \sum_{j=0}^{d} \binom{n-1}{j}. \] We show that if $x^{(1)}, \dots, x^{(m)}$ are independent uniform points of $\{\pm 1\}^n$ then uniformly for $m \leq D(n,d)$, there exists a constant $C_d > 0$ such that \[ \mathbb{P}((x^{(1)})^{\otimes d}, \dots, (x^{(m)})^{\otimes d} \text{ are linearly independent}) = 1 - O_d\left(\frac{\log^{C_d} n}{n^{1/2}} \right). \] This achieves the exact dimensional threshold and answers a question asked by Baldi and Vershynin. We discuss applications of the result to the semidefinite relaxation of the cut-polytope and to matrix factorization.

Mixed Graphical Models (MGMs) provide a flexible framework for structure learning from heterogeneous data by treating sets of both continuous and discrete. Bayesian inference for MGMs remains challenging due to the combinatorial complexity of graph space exploration and posterior computation. In this paper, we model discrete components through latent Gaussian variables and consider two likelihood specifications: a copula-based ranked likelihood, yielding the copula-MGM, and a probit formulation based on cut-off points, yielding the probit-MGM. We then propose the Mixed Graph WWA, a class of MCMC methods for posterior simulation in Bayesian MGMs. Building upon the WWA algorithm, we develop two specialized algorithms: copula-WWA for copula-MGMs and probit-WWA for probit-MGMs. Both methods exploit the latent Gaussian representations to perform posterior inference through a Gibbs sampling scheme that alternates between latent-variable augmentation and graph-structure updates. Through extensive simulation studies we demonstrate that the proposed methods achieve graph recovery accuracy comparable to or better than existing approaches, including copula-BD MCMC and probit-BD MCMC based on the Birth-Death MCMC methodology, while maintaining efficient posterior exploration and favorable effective sample size per unit computational time. We further illustrate the practical utility of our approach through an application to the PAM$50$ breast cancer gene expression dataset, where the inferred MGMs reveal meaningful dependencies between gene expression profiles and cancer subtypes. These results highlight the effectiveness of the Mixed Graph WWA method as a scalable and principled tool for Bayesian structure learning in MGMs.

Excessive force may damage tissue and increase the risk of anastomotic leakage in robotic colorectal surgery. Although the da Vinci 5 provides force sensing, this capability is unavailable on earlier da Vinci systems and many other surgical robotic platforms. In this work, we present a vision-based pipeline for estimating 3D interaction forces from soft-tissue deformation in stereo endoscopic video. We dynamically reconstruct the tissue point cloud in an object-centered coordinate frame, track tissue points with geometric constraints, and predict the 3D force vector with a neural network. We progressively evaluate the pipeline on rubber-glove phantoms, ex vivo porcine colons, and in vivo colorectal surgical video sequences. Under varying tissue orientations and positions within the endoscopic view, as well as different camera viewpoints, the proposed method achieves average root mean square error (RMSEs) of 0.77 N and 1.30 N on the phantom and porcine colon, respectively. Compared with the camera-frame representation, the object-centered representation reduces average RMSE by 51.3% and 56.7%, while geometry-constrained tracking reduces RMSE by 19.8% and 25.3% compared with CoTracker. We further qualitatively demonstrate the feasibility of vision-based force estimation on an in vivo colorectal surgical sequence, as a step toward clinical translation of vision-based, sensorless force estimation.

We prove two main algorithmic results in spectral discrepancy. First, we give a deterministic polynomial-time rounding theorem for rational positive semidefinite matrices of arbitrary rank. The algorithm starts from any rational fractional signing and assigns one sign per original matrix. Its discrepancy is less than $3.37\,\|\sum_i \mathrm{Tr}(A_i)A_i\|^{1/2}$. This yields Kadison–Singer half-partitions with error below $1.69\sqrt{\varepsilon}$, as well as deterministic graph signings that control signed adjacency and signed degrees simultaneously. The proof builds on the spectral-potential method of Ezeunala and Jiang (2026) and introduces a new way to choose rounding directions. We prove polynomial bit complexity for the rounding procedure. Second, we give a Las Vegas algorithm for the Bilu–Linial signing problem on an arbitrary prescribed graph. If $G$ has $n$ vertices and maximum degree $\Delta\ge3$, the algorithm terminates almost surely. It uses fewer than $100n^3$ insertion attempts in expectation and returns a signing with $\|A_s\|<2\sqrt{2(\Delta-1)}$. For bipartite graphs its one-sided form gives the sharp universal bound $\|A_s\|<2\sqrt{\Delta-1}$. The algorithm builds the signing by inserting vertices and recursively deleting and restoring neighbors after rejected insertions. In the analysis, the $\sqrt2$ gap to the Bilu–Linial conjecture comes from a factor of two in the bound for vertex deletions in the two-sided case. On a $d$-regular bipartite Ramanujan base the same signing produces a Ramanujan $2$-lift of that prescribed base.

This article argues that a central development in the history of computing is the mathematical evolution of addressing. During the three decades separating Kateryna Yushchenko's Address Programming Language (1958) from Lenore Mullin's A Mathematics of Arrays (1988), addressing evolved from a machine-level mechanism for locating data into a formal theory of indexing, reduction, and machine realization, passing through Kenneth Iverson's algebra of arrays and Philip Abrams' recognition of shape as an architectural resource. Mullin's theory establishes indexing itself, via the function {\psi}, as the primitive from which array operations compose, reduce to a Denotational Normal Form, transform into an Operational Normal Form, and realize on hardware through dimension lifting. Historical claims are sourced against primary material, with inferences flagged as such; mathematical claims are stated as identities; performance claims are marked validated, theoretically established, or provisional. The result links programming languages, array mathematics, and computer architecture in one account.

Disaster-risk-reduction archives describe hazard events in prose that databases such as EM-DAT (Delforge et al., 2025) cannot ingest directly. We present an LLM pipeline that generates candidate georeferenced event records using a controlled hazard vocabulary and fixed schema, retaining evidence for review. Applied to 10,000 documents from PreventionWeb, the knowledge hub managed by UNDRR, it produced 3,572 records from 1,913 documents across 24 hazard types and resolved 81% of location mentions to OpenStreetMap geometries. On 171 human-positive document windows from a stratified 217-document reference set, GPT-5 achieved 86.0% pooled attribute $F_1$, versus 44.2% for the spaCy-gazetteer baseline. Evaluation pools hazard families, location strings, and event years within documents, without assessing their assignment to individual events. GPT-5.4 ranked highest among ten LLMs (86.6% $F_1$). Verbatim evidence occurrence was 72.0% for GPT-5 and 47.2% for GPT-5.4, measuring textual traceability without establishing attribute support. We report production failure modes and automated label and location-rule compliance checks. Prompts, schema, and outputs will be released for adaptation to national reporting archives.

The first almost-linear time maximum and minimum cost flow algorithm of Chen-Kyng-Liu-Peng-Probst Gutenberg-Sachdeva (FOCS 2022), reduced these flow objectives to a sequence of min-ratio cycle problems. Solving this core primitive requires approximately minimizing the ratio of a linear gradient term and an undirected length term. In Chen-Kyng-Liu-Peng-Probst Gutenberg-Sachdeva (FOCS 2022) and the subsequent work of Chen-Kyng-Liu-Meierhans-Probst Gutenberg (STOC 2024), intricate data structures were given to solve the min-ratio problem. We show that such a cycle can be extracted directly from the dynamic distance oracle of Kyng-Meierhans-Probst Gutenberg (STOC 2024) using linearity. This simplifies previous algorithms that relied on multiple additional steps to extract the cycle, and can be seen as evidence that solving the min-ratio cycle problem really is all about distances. As a result, we obtain a faster primal maxflow and min-cost flow solver that also extends to incremental graphs.

We study the stability of nonlinear systems subject to both slow and fast time variations. Both can have discontinuities, covering switched systems as a special case. The fast variation is assumed to be periodic; thus, we rely on averaging to construct an average system. Importantly, the equilibrium of the average system depends on the slow variation and is assumed to be exponentially stable when this slow input is frozen. Using perturbation and Lyapunov analyses, we establish a practical stability result showing that the system state remains within a neighborhood of the moving equilibrium when the total variation (flows and jumps) of the slow variation is appropriately bounded and the fast input varies sufficiently fast. The result is illustrated via a nonlinear switched system with slow-fast switching and a mode-dependent equilibrium.

The continuity equation serves as a fundamental principle for mass transport in continuous media. While its mathematical structure mirrors that of probability transport and the Liouville equation, an informational interpretation of macroscopic fluid flow is less commonly explored in engineering contexts. This paper treats fluid density as a spatial probability density function, modeling macroscopic motion as the continuous transport of uncertainty. I derive the temporal evolution of Shannon entropy under general flow conditions, establishing how entropy generation depends on macroscopic compressibility and microscopic diffusion. Microscopic diffusion is shown to act as a strictly positive entropy source governed exactly by local Fisher Information. Furthermore, the framework yields an explicit algebraic scaling law characterizing the equilibrium thickness of compressive mixing layers at a local Peclet number of unity. The theoretical model is computationally validated through finite-difference simulations of one-dimensional canonical flows and advection-diffusion within a three-dimensional, spatially varying Arnold-Beltrami-Childress (ABC) flow, yielding a mean relative error of $0.001$. The framework provides an analytical perspective on thermodynamic irreversibility, with potential applications to turbulence modeling, thermal entropy generation in heat exchangers, in-cylinder mixing in internal combustion engines, and aerodynamic flows.

Sequential recommendation must operate under long-tailed item distributions and popularity-driven concentration, often forcing practitioners to trade short-list accuracy against long-tail exposure. In this work, we study feature decorrelation as a mechanism for shaping representation geometry in dot-product sequential recommenders, and analyze how this, in turn, affects popularity-driven concentration. We propose a decorrelation-regularized training framework that augments next-item prediction with an auxiliary redundancy-reduction term, and instantiate it with BT-SR, which uses the Barlow Twins objective. To form label-consistent positive pairs without synthetic corruptions, we pair user histories that share the same next-item target. Beyond accuracy, we provide a geometric analysis showing how decorrelation suppresses shared low-rank directions in the user representation space that can give popular items a global scoring advantage, and we introduce a bucket-based alignment concentration metric to quantify this effect. Experiments on five public benchmarks show that BT-SR consistently improves next-item ranking quality, while the decorrelation strength acts as a simple control knob that reallocates accuracy across head and tail items, enabling accuracy-exposure trade-offs. Our analysis also reveals that the impact on head-vs-tail exposure differs across datasets, reflecting interactions between decorrelation and data temporal structure.

Constrained optimization is central to many engineering systems in which decisions must satisfy strict safety and operational requirements, especially in real-time settings with limited computational budgets. In such scenarios, optimization algorithms are often terminated before full convergence, making *anytime feasibility* essential for safe deployment. Existing methods that guarantee feasibility at every iterate typically rely on exact gradient information, an assumption that is often violated in practice due to measurement noise, stochastic approximations, or model mismatch. We develop an anytime-feasible first-order method for nonlinear constrained optimization under norm-bounded errors in the objective and constraint gradients. The method computes a robust search direction by solving a second-order cone program and selects a step size through safeguarded backtracking. Assuming exact function evaluations and a strictly feasible initialization, the method preserves strict feasibility and guarantees sufficient objective decrease whenever the computed search direction is nonzero. We establish a uniform positive lower bound on the accepted step sizes, an O(1/K) bound on the average squared search direction norm, and convergence of the search directions to zero. We also show that a zero search direction at a strictly feasible point certifies approximate first-order stationarity. We validate the proposed method on a multi-agent navigation task in cluttered environments and show that it maintains collision-free trajectories despite noisy gradient information.

Let $F$ be a totally real Galois number field of degree $g\leq5$ and let $\ell$ be an odd prime. Let $E/F$ be an elliptic curve with an $F$-rational point of order $\ell$. Under explicit arithmetic and local hypotheses, together with an explicit non-vanishing condition modulo $\ell$, we prove that there are $\gg_{F,E,\ell}\frac{X^{1/(2g)}}{\log X}$ totally negative square classes $d\in F^\times/(F^\times)^2$ with $\left|\mathrm{N}_{F/\mathbb{Q}}\bigl(D(F(\sqrt{d})/F)\bigr)\right|<X $ for which $\operatorname{Sel}_\ell(E^d,F)$ is trivial. Such twists have rank zero and trivial $\ell$-part of the Shafarevich–Tate group. The condition is verifiable by a finite computation, which we carry out in an example. We lift the method of James and Ono from $\mathbb{Q}$ to the totally real setting, combining a theorem of Morrow, which relates the Selmer group of such a twist to the class group of the CM extension $F(\sqrt{d})$, with an indivisibility theorem of Takai for relative class numbers. Takai twists only by quadratic Hecke characters, whereas Morrow's conditions are local; we extend the twisting argument to primitive quadratic residue-class characters to make the two results compatible.

We employ equation-driven, physics-constrained deep learning to solve the fixed-boundary Grad-Shafranov (GS) equilibrium problem, constructing axisymmetric magnetohydrodynamic equilibria with tokamak-relevant characteristics. Equilibria across linear (Solov'ev) and nonlinear profile functions are constructed using Physics-Informed Kolmogorov-Arnold Networks (KANs) that approximate GS solutions while satisfying appropriate boundary conditions. A highly nonlinear pressure profile recreating high-confinement mode phenomenology, such as pressure pedestals and significant bootstrap current components, is also considered. To enable efficient convergence, guided training schemes are employed, specifically homotopy-based continuation curriculum learning and transfer learning via pretrained networks. While computing nonlinear equilibria employing standard Multi-Layer Perceptrons under unguided physics-informed training remains an elusive or computationally inefficient task, our framework overcomes this limitation. Specifically, we demonstrate that the combination of three key elements, namely KAN architecture, guided training schemes, and the self-scaled Broyden optimization method, enables stable, efficient, and accurate equilibrium computation with simultaneous profile parameter identification in view of equilibrium constraints.