Papers

Threshold analyses of quantum error-correcting codes are well established for stochastic error models, in which errors occur randomly with given probabilities. However, errors in actual devices can also be coherent, such as unwanted $Z$ rotations due to imperfect control, which are not captured by stochastic error models. For the surface code, numerical studies have indicated threshold behavior even under coherent errors, but a rigorous proof of threshold existence is lacking. Here we prove that a positive threshold for coherent $Z$-rotation errors exists for quantum low-density parity-check codes with a bounded number of logical qubits, including the surface code and other topological codes. Specifically, we show that the maximum-likelihood Pauli recovery suppresses the entanglement infidelity exponentially in the code distance up to a prefactor linear in the number of physical qubits whenever the rotation angles lie below a constant that is independent of the code size. The proof combines Fourier analysis to retain the interference among the amplitudes of coherent errors with the cluster expansion of abstract polymer models. Our results expand the theoretical foundation of quantum error correction and offer a statistical-mechanical description of quantum error correction beyond stochastic errors.

In this paper we construct a two parameter family of generalized Gaussian fields which interpolates between the Gaussian Free Field, conditioned Gaussian Free Fields, and Gaussian Free Fields with independent noise. Depending on the values of the parameters, we characterize these generalized Gaussian fields as partial conditionings of a Gaussian Free Field with perturbations of its first two Fourier modes. We show that this family arises as the fluctuations of the height function associated with the corner process of a two parameter variant of Wigner random matrices.

It is known that, for ordinary Hölder multifunctions with 0 < {\alpha} < 1, Hölder regularity alone does not guarantee the existence of a Hölder selection, and in general even a continuous selection may fail to exist. We ask how this situation changes when the parameter is measured through a Stieltjes clock. For 0 < {\alpha} < 1, we characterize the clocks for which every compact-valued g-Hölder multifunction admits a g-Hölder selection. The determining condition is uniform disconnectedness of the clock image Im(g). For left-continuous and nondecreasing clocks, this condition is equivalent to uniform jump dominance: every positive clock increment contains a jump carrying a fixed positive fraction of that increment. Under this condition, a selection can be chosen through any prescribed point of the graph, with explicit control of its Hölder regularity. Conversely, when the condition fails, there exists a compact-valued g-Hölder multifunction with no g-Hölder selection. The results show that the selection property is governed not simply by the presence of jumps, but by how their sizes are distributed across scales.

The quantitative success of relativistic viscous hydrodynamics in describing the short-lived quark–gluon plasma raises a fundamental question: how does far-from-equilibrium QCD matter approach hydrodynamic behavior so rapidly? Microscopic kinetic-theory studies have related this onset to attractor dynamics, but typically assume transverse homogeneity. We relax this assumption by introducing gradient modes with finite transverse wave number $k$, extending the analysis of arXiv:2212.00820. These couple different spherical harmonics of the momentum distribution, and the resulting dynamics is controlled by the competition among the expansion rate $1/\tau$, the collision rate $1/\tau_R$, and the gradient scale $k$. At early times, longitudinal expansion suppresses this coupling, and different spherical harmonic sectors follow their homogeneous attractors. We find that, at later times and sufficiently small $k$, this gradient coupling drives the system toward a global attractor manifold spanned by hydrodynamic sound and shear modes, on a timescale $\tau_D$ that depends on both the azimuthal harmonic $m$ and $k\tau_R$. For sufficiently large $k\tau_R$, the spectral gap closes: perturbations retain finite damping rates and therefore equilibrate, but no longer follow an isolated hydrodynamic attractor or admit a reduced description involving only a few hydrodynamic modes.

We consider a certain pair of families of Picard rank 1 Calabi–Yau threefolds, that have appeared earlier in mathematical literature in unrelated contexts: the family $\mathcal{X}$ of degree 33 threefolds in $G(2,6)$ (constructed by Miura) and the family $\mathcal{Y}$ of arithmetically Gorenstein degree 21 threefolds in $\mathbb{P}^8$ (constructed by Schenck–Stillman–Yuan). After establishing a natural geometric correspondence between their general members, we go on to show that any pair of corresponding threefolds in these families satisfy certain classical dualities. We moreover discover that their geometries may be related by a mathematical gauged linear sigma model, using which we prove that they are derived equivalent. This settles a conjecture of Miura, who predicted the existence of non-trivial Fourier–Mukai partners to members of $\mathcal{X}$, based on a study of its mirror moduli. This conjecture was also formulated later by Gerhardus–Jockers, in a physical context. In fact, starting from the other family $\mathcal{Y}$, we conjecturally arrive at the same mirror moduli. Thus, such a pair is a new, and quite possibly the last, addition to the small list of deformation families of non-birational, double-mirror Calabi–Yau threefolds having Picard rank 1.

In this work, we implement at the quantum level the gauge-fixing conditions that relate the homogeneous SU(2) gauge theory of Ashtekar variables, describing the classical cosmological sector, to the usual minisuperspace formulation. We impose the so-called divergence constraint, which fixes the gauge to the homogeneous one and, at the classical level, recovers a finite-dimensional phase space, together with the diagonal constraint, which imposes the diagonality of the metric and has already been extensively studied in the literature. We construct the corresponding quantum operators for general cosmological models and provide a suitable regularization in terms of holonomies and fluxes. We then show that a special class of homogeneous spin-network states describing the Bianchi I and Bianchi IX models satisfies the quantum gauge-fixing constraints, using techniques developed in Reduced Quantum Loop Gravity. This provides a quantum-level link between the cosmological theory with full SU(2) gauge symmetry and standard Loop Quantum Cosmology, by reason of the effective Abelian structure of the selected states.

Can one construct a graph $G$ on the set of integer points ${\mathbb Z}^2$ in the plane such that the length of the shortest path between any two vertices of $G$ differs from their Euclidean distance by at most an absolute constant? This question of Benjamini, Erd\H os, Kleiner, Kozma, Schramm, and the first-named author has been open for a long time. We give an affirmative answer to a weaker form of this question, based on the following geometric statement, which is of independent interest. There exists a constant $c>0$ such that for every $i=1,2,\ldots,$ every $\rho$-fat plane convex set $S$ can be cut into $2^i$ convex pieces of equal area, each of which is at least $c\rho$-fat. (A convex set is $\rho$-fat if the ratio of its inradius to its circumradius is at least $\rho$.) We prove that there exists an (unweighted) spanning subgraph $G$ of an enlarged copy of ${\mathbb Z}^2$ such that, for every pair of vertices at Euclidean distance $d$, their shortest-path distance in $G$ lies between $d-O(1)$ and $d+o(d^{5/6})$. The same bound can be achieved by a planar graph with vertex set ${\mathbb Z}^2$, in which every edge joins two vertices at Euclidean distance at most 2.

We study efficient algorithms for realizing the first-order oracle complexity of optimization of $G$-Lipschitz convex functions with respect to the $\ell_q$-norm over an $\ell_p$-ball of radius $R$, where $1\leq p,q\leq\infty$. For $p<q$, we obtain error $\widetilde{O}_{p,q}(GR/T^{1/p-(1/q-1/2)_{+}})$ after $T$ oracle queries, attaining the lower bounds in a COLT 2015 open problem (Guz15b) with efficient algorithms. In particular, the rate is $\widetilde{O}(GR/T)$ for Euclidean Lipschitzness with an $\ell_1$-ball domain of radius $R$ ($p=1,q=2$). Our solution consists of reducing convex Lipschitz optimization to the chasing nested convex sets problem (BBE+20) in sublevel sets of an evolving maximum of affine lower bounds, that is, the bundle (LNN95): at each query we either find a point with low function value or we produce a deep cut in the current sublevel of the bundle that we chase. The dichotomy between stability of selectors and forced movement by deep cuts bounds the number of iterations of the algorithm nearly optimally. For nested convex subsets of the $\ell_p$ ball $R B_p^d$, we introduce novel stable selectors whose movement is nearly optimal in high dimensions. A Monte Carlo average of the proposed selector achieves near-optimal rates with high probability and can be implemented for our optimization algorithm in polynomial time in the real-arithmetic model.

Episodic test-time adaptation resets a frozen segmenter to source weights $M_0$ on each case and adapts for a fixed step count. A fixed horizon conflates a cohort-level question, how far to adapt, with an irreducibly per-case one, whether this case should be adapted at all. Cohort means hide that decision: on cross-vendor cardiac MRI the mean $\Delta$Dice from adaptation is statistically indistinguishable from zero while 58.7% of cases are individually made worse. We quantify this harm as harmful accepted area (HA), the harmful fraction of the edited area a controller deploys. Held-out tuning gives a stronger baseline than a fixed horizon, but the budget it selects transfers on neither of the two main medical benchmarks, and no global budget can condition on the case. We show that prediction fragmentation—the disagreement geometry between $M_0$ and the adapted mask $M_k$—predicts HA with no labels or extra backward passes at decision time, comparably on three benchmarks (Spearman $\rho$ 0.50–0.60), at a quarter of gradient-norm's latency. A case-level router built on it cuts HA from 0.228 to 0.139 on a benchmark that took no part in its design, with the design frozen and only cut-points recalibrated there. On the cardiac benchmark the design was selected on, the router cuts HA from 0.129 to 0.013 at matched Dice and 1.10 deployed updates, against the retrospective-best budget found post hoc on evaluation labels, and reduces that 58.7% to 20.0%, an upper bound we quantify. Where the retained cases are not net-helped (as on prostate), the router still cuts HA but concedes accuracy, a boundary we report. Thresholds are fit once on a labeled split disjoint from evaluation; decisions use no labels or gradients. The template ports across architecture and domain (nnU-Net$\to$SegFormer, Cityscapes$\to$ACDC) with coordinate, thresholds and per-bucket actions instantiated per domain.

We consider a large-scale incentive allocation problem where the entire trade-off curve between budget and profit has to be maintained approximately at all times. The application originally comes from assigning coupons to users of ride-sharing apps, where each user can have a limit on the number of coupons assigned to them. We consider a more general form, where the coupons for each user form a matroid, and the set of coupons assigned to each user must be an independent set. We show the entire trade-off curve can be maintained approximately in near real time.

The study of entanglement inevitably leads to the concept of regularization, where measures are evaluated on infinitely many copies of a state. Single-letter formulas try to make these asymptotic entanglement quantities accessible through a calculation on one copy of a state, but are rarely available. We introduce a decreasing hierarchy of computable upper bounds on the asymptotic relative entropy of entanglement with respect to positive-partial-transpose (PPT) states. The regularization of every fixed hierarchy level equals this asymptotic quantity. Additivity at any level therefore yields a single-letter formula, even when the usual one-copy relative entropy is nonadditive. We establish such additivity at the first nontrivial level for two broad multiparameter families, both containing all Werner states. The upper-bound construction also extends to sandwiched Renyi divergences. The hierarchy motivates $k$-multinegative states, a generalization of binegative states and the associated $k$-multinegativity, which gives explicit upper bounds on both the asymptotic relative entropy and exact PPT entanglement cost. We construct states which are $k$-multinegative at arbitrarily large depths $k$ that separate consecutive levels of the previously introduced entanglement-cost hierarchy, disproving its conjectured finite collapse. These results provide state-dependent single-letter formulas while identifying a limitation of universal finite-level characterizations.

In recent work, McNamara and Wang demonstrated that the failure of factorization for a gravitational path integral (GPI) can be understood as an incompleteness of its spectrum of states by constructing a unitary quantum field theory (QFT) into which the GPI embeds. We examine this construction from an algebraic point of view, and use it to verify a conjecture of the author concerning the existence and uniqueness of a unitary completion for the GPI by means of the theory of conditional expectations. We explore how the unitary QFT can be regarded as a single coherent theory in which the quantitative signatures of ensemble averaging emerge purely from projecting to the underlying, smooth GPI. The conditional expectation relating the two theories may therefore be interpreted as a filter in the sense of Liu. The factorizing theory differs from a more standard QFT in the sense that it is reducible, meaning the Hilbert space associated with the empty set is non-trivial. We explain how the structure of reducible QFTs are generically encoded in weak Hopf algebraic symmetries and demonstrate how the weak structure allows for generalized $\alpha$-sectors and half wormholes to coexist within a single Hilbert space.

We study prophet inequalities under intersections of $q$ partition matroids, where an online algorithm irrevocably selects elements with independent nonnegative values drawn from known distributions and revealed in an adversarial order. We prove an $\Omega(q/\log q)$ lower bound on the competitive ratio. Together with the known $O(q)$ upper bounds, this resolves, up to a logarithmic factor, the optimal dependence on $q$, an open question posed by Correa, Cristi, Fielbaum, Pollner, and Weinberg (IPCO 2022) and Saxena, Velusamy, and Weinberg (ITCS 2023). Our construction also yields an $\Omega(d/\log d)$ lower bound for $d$-single-minded auctions, where buyers request fixed bundles of at most $d$ unit-capacity items. Our construction and analysis build on the big-decisions-first framework of Rubinstein and Singla (STOC 2026). Previous lower bounds for this problem hold even for i.i.d. Bernoulli values, whereas our instance uses nonidentical distributions. For i.i.d. Bernoulli values, we prove a $q^{1-o(1)}$ lower bound, improving the previous $q^{1/2+\Omega(1/\log\log q)}$ of Saxena, Velusamy, and Weinber (ITCS 2023). To this end, we construct $p^p$ pairwise qualitatively independent partitions into $p$ blocks on a ground set of only $p^{2+o(1)}$ points.

On the occasion of Gerard 't Hooft's 80th birthday, I present my personal recollections of a scientific meeting held in the Orsay Campus of the University of Paris in January 1972. In this meeting Gerard gave for the first time a detailed account of his groundbreaking work on the renormalization properties of Yang-Mills theories, both in the massless phase as well as in the phase with spontaneous symmetry breaking.

Mission-level autonomy for disaster response, search and rescue, and tactical UGV operations requires repeated path and mission planning as terrain conditions, agent types, and objectives change. Pixel-grid search is costly for repeated kilometer-scale queries, while semantic abstractions must maintain valid costs and connectivity as conditions change. We present HOPHY (Hierarchical Off-Road Planning using Hypergraphs), a reusable hierarchical terrain representation that organizes map-scale terrain into geometrically connected semantic regions (GSNodes), connectivity-preserving critical regions (Coarse Regions), and typed hyperedges for terrain, agent, and weather context. Hyperedge intersections select affected regions and incident edges for state updates without rebuilding the hierarchy. Across real off-road maps spanning kilometer-scale areas, HOPHY achieves 100% planning success and less than 0.01% median cost deviation from the oracle (pixel A*), with substantially lower query and replanning latency than the evaluated pixel and abstraction baselines. Applied to a multi-robot task-allocation (MRTA) problem, these gains reduce total computation by 79x over pixel A* and 7.2x over the fastest abstraction baseline, with mission makespan comparable to pixel A*. Finally, we demonstrate HOPHY on a physical Clearpath Jackal that successfully executes a 1.5-km, eight-task mission across mixed-surface outdoor terrain and a blockage-triggered replanned route.

Detecting phases of matter in general relies on identifying the correct order parameter - a task that remains notoriously difficult for unknown transitions and traditionally is guided by physical intuition and educated guess. Neural networks have recently offered an alternative route by locating phase transitions in known models without any a priori physical knowledge. Yet these approaches remain black boxes and only identify phases without elucidating their properties. Moreover, they often struggle when confronted with realistic, noisy experimental data, which constitute the ultimate testbed for automated methods in physics. Here, we bridge these perspectives by introducing TetrisCNN, a convolutional architecture with parallel branches of differently shaped filters, reminiscent of Tetris blocks, that learns sparse, interpretable latent representations directly in terms of spin correlators. Applied to experimental snapshots of two-dimensional Ising and XY quantum simulators measured in multiple bases, the network not only detects phase transitions and crossovers but also expresses its latent representation and decision boundaries as symbolic formulas built from experimentally measurable spin correlators. This framework opens the way to integrating interpretable neural networks with quantum simulators to uncover and understand new phases of matter.

We study the free minimization problem for a nonlinear, nonlocal functional associated with Robin-type nonlocal exterior conditions depending on a positive parameter $\alpha$. We prove that, for every $\alpha>0$, the minimizer exists, is unique, and satisfies suitable decay properties at infinity. We also investigate the regularity of the maps $\alpha \mapsto u_\alpha$ and $\alpha \mapsto E_\alpha$, where~$u_\alpha$ denotes the minimizer and $E_\alpha$ the corresponding energy. Finally, we derive asymptotic expansions of the energy as $\alpha\to+\infty$ and as $\alpha\to0^+$. The latter regime requires distinguishing between two cases, depending on whether the forcing term has zero mass. In one case, the limiting energy possesses a minimizer and $E_\alpha$ converges to its energy, but in the other case $E_\alpha$ diverges to $-\infty$.

Mapping and monitoring aquatic environments can benefit from hybrid aerial-amphibious drones able to combine flight and water-surface navigation within the same mission. This paper presents a PX4 firmware extension for such platforms, introducing manual and autonomous marine navigation modes integrated with the standard PX4 mission pipeline and QGroundControl interface. The proposed framework preserves existing flight functionalities and safety mechanisms while enabling unified planning and execution of hybrid aerial-marine missions with differentiated aerial and marine waypoints. Simulated case studies validate the implementation and demonstrate stable surface navigation under calm and wavy conditions.

Godsil, Roberson, Šámal and Severini conjectured that the quantum chromatic number of the categorical product of two graphs equals the minimum of the quantum chromatic numbers of the factors. We disprove this conjecture: we construct explicit finite graphs $G,H$ with \[ \chi(G\times H) \leq 1538 < 1539 = \min(\chi_q(G),\chi_q(H)).\]The graphs are obtained from Zhu's counterexample to Hedetniemi's conjecture by using a base graph for which the Lovász theta number of the complement, and not only the fractional chromatic number, is large. The lower bound for the first factor is the theta bound. For the second factor we adapt Zhu's argument to projections that do not commute: the step that fixes the colors of a clique is replaced by identities between operators. Both lower bounds hold for colorings by projections in an arbitrary nonzero unital $C^*$-algebra. Hence the conjecture also fails for the spatial, approximate, commuting-operator and $C^*$-algebraic variants of the quantum chromatic number. We also give smaller counterexamples certified by exact integer data. The graph constructions, the certificates and the counterexample statements in the projective formulation are formalized in Lean~4.

We show that manifolds admitting Zoll families of minimal hypersurfaces are diffeomorphic to $\mathbb{S}^{n}$ or to $\mathbb{R}\mathbb{P}^{n}$, and we characterize their Zoll families in both cases. Then, we prove that a metric on $\mathbb{R}\mathbb{P}^3$ is surface Zoll if and only if its projective widths are all equal. Our last result asserts that a closed surface with the first four widths equal to $2\pi$ is isometric to $\mathbb{R}\mathbb{P}^2$ with its constant curvature one metric.

Multiple pendulum with branching is proposed as a convenient platform to study energy transfer between different modes. The antiphase oscillation mode is localized to the "child" links, which makes it easy to prepare initial conditions without exciting the antiphase mode. A manageable expression for the energy transfer is derived theoretically and evaluated with experimental data.

We study first-order black-box convex optimization over an $\ell_p$-ball for objectives Lipschitz in the $\ell_q$-norm, solving in the affirmative the nonsmooth version of the COLT open question (Guz15b) on whether the geometry of a smaller feasible set ($p < q$) can improve convergence rates in convex optimization, and matching prior lower bounds up to logarithmic factors. Our rates include $\widetilde{O}(1/T)$ for convex Euclidean-Lipschitz optimization over the $\ell_1$-ball, improving on the $O(1/\sqrt{T})$ classical rate under general assumptions. The key technical device is a new online learning game, where the comparator is evaluated using the maximum of affine losses observed so far. We bound the value of this game above and below in terms of a combinatorial online learning quantity: the sequential fat-shattering dimension, which we characterize for the $\ell_p / \ell_q$ case. Our results generally apply when the feasible set $\mathcal{X}$ and the set of possible subgradients $\mathcal{H}$ admit a type of minimax theorem, advancing on a fundamental question by Sridharan (Sri12, Section 10.1.2, Q3). As a geometric consequence of our analysis, of independent interest, we obtain estimates for the expected distance of a convex hull of samples to their mean in several Banach geometries, a version of the celebrated Wendel's theorem (Wen62), but quantitative and for bounded general distributions as opposed to centrally symmetric ones.

Scaling optical input/output for co-packaged optics is limited by the fiber-to-chip interface. Conventional edge coupling offers low loss and broad bandwidth but confines channels to a single row along the chip facet. On the other hand, surface couplers are either narrowband or difficult to fabricate with high yield and, importantly, compete with back-end metal routing. In this work, we introduce volumetric edge coupling, in which the chip edge is structured in three dimensions so that optical coupling can occur over a two-dimensional region rather than along a single line, providing a route toward interfacing multiple waveguides with multiple cores of a multicore fiber while preserving optical access from the chip perimeter. We investigate a single-channel realization based on total-internal-reflection (TIR)-mediated evanescent coupling through the 54.7$^\circ$ sidewall of a KOH-etched silicon cavity, with the coupling profile shaped by a wedged buried-oxide ridge. Finite-difference time-domain simulations predict a peak coupling efficiency of 88% (-0.56 dB) at 1550 nm and a 1-dB bandwidth of 86.45 nm spanning the C-band. The design further exhibits a vertical alignment tolerance of approximately $\pm$2 $\mu$m, weak sensitivity to transverse offsets up to 5 $\mu$m, and a $\pm$0.8$^\circ$ 1-dB angular tolerance. The reflected optical field also provides a potential alignment signal, offering a path toward reduced active alignment overhead in future multicore implementations.

We study the large-$N$ asymptotics of a constrained spherical integral for non-Hermitian random matrices, in which the norms and mutual scalar product of two vectors are fixed. In the delocalized regime, the asymptotics are governed by the non-Hermitian transforms $\mathcal R_1$ and $\mathcal R_2$. At saddle-point level, the constrained integral exhibits a transition to a localized regime controlled by the largest singular value of a shifted matrix and by the overlap of its associated left and right singular vectors. Motivated by the Hermitian spherical-integral mechanism and by the Coulomb-gas picture, we formulate conjectures for one-eigenvalue large deviations and boundary fluctuations. Throughout, our analysis is carried out in the spirit of mathematical physics.

Large language models are increasingly used in healthcare communication, yet most evaluations emphasize response quality while assuming that the user's concern has been interpreted correctly. We introduce HerHealthEval, a controlled evaluation framework for multilingual understanding of women's-health communication. For each clinical case, HerHealthEval provides matched versions in English, French, and Modern Standard Arabic using six communicative forms: canonical, clinical, layperson, indirect or hedged, emotionally concerned, and deliberately under-specified. The first five express the same underlying concern and retain the same clinical information, whereas the under-specified form intentionally omits relevant details to test whether the model recognizes that clarification is needed. We evaluate a multilingual instruction model and QLoRA-adapted variants on concern classification, risk calibration, clarification behavior, parse compliance, and cross-form consistency. Results reveal that aggregate accuracy and consistency can conceal safety-relevant failures. A multilingual adaptation model reaches 0.994 under-triage in French and Arabic under language-asymmetric risk supervision. A controlled re-adaptation using source-derived, language-invariant risk labels reduces under-triage to 0.572 and 0.558, respectively. These findings show that robust multilingual healthcare evaluation requires explicit testing of register variation, uncertainty handling, and the provenance and invariance of adaptation labels.

We revisit the analysis of Bayesian convergence to the truth under finite additivity in a recent paper by Nielsen (J. Philos. Logic, 2021, doi:10.1007/s10992-020-09569-2) Its principal theorem proves that the posteriors of a probability function converge to the truth almost uniformly if and only if the function has two properties: an approximation property, and so-called countable additivity on conditional hitting times. We show that the two properties are necessary but not sufficient, so that the theorem is false, and we locate the error in its published proof. We construct a merely finitely additive probability function that has both properties and whose posteriors converge to the truth almost surely but not almost uniformly. Three of the paper's four remaining theorems, and its corollary, lose their published proofs as well, two with the failed implication and two to a separate defect that we also identify. Two of the four results we reprove and one we leave undecided; the last is the corollary, which our counterexample does not refute, and which we establish for a family including the counterexample and leave open in general. We also show that almost-sure convergence of posteriors to the truth for every event does not characterize countable additivity. We close with what a repaired characterization of almost-uniform convergence would have to add.

We prove that solution maps of the Vlasov–Poisson equation are discontinuous at the zero initial datum in $X_p=L_x^pL_v^\infty$ for every $1\leq p<\infty$, in $\mathbb{R}^{d} \times \mathbb{R}^{d}$ with $d \le 3$, and for both attractive and repulsive interactions. For every $T>0$, the trajectory of the solution in $L^\infty([0,T];X_p)$ is discontinuous at zero, and for every sufficiently small fixed $t>0$, the fixed-time map with values in $X_p$ is also discontinuous at zero. The counterexamples are smooth, nonnegative, bounded by one, and supported in a common compact subset of phase space. Their mass and initial $X_p$ norm tend to zero, whereas their solution norm at the observation time is at least one. The construction places narrow spatial packets on a lattice. Free transport allows a different velocity to select a packet at each spatial point, while a smallness estimate on the field ensures that the nonlinear characteristics are close to that of the free transport. For the same families, the phase-space $L^q$ norms converge to zero uniformly in time for every $1\leq q<\infty$.

We investigate martingale properties of heat dissipated by generic non-equilibrium overdamped Langevin dynamics with multiple dimensions and transport coefficients that may depend explicitly on time and/or space. We find quantitative criteria that establish that heat dissipation may exhibit transient martingale (time-conserved), submartingale (time increasing), and supermartingale (time decreasing) properties in Langevin dynamics with drift coefficients with spatial non-linear dependencies. Furthermore, we reveal a tight link between the heat statistics at stopping (e.g. first-passage) times with that of a functional quantifying the phase-space contraction in the system dynamics. The theoretical results are used to predict the statistical properties of the heat dissipated by a spherical microscopic particle subject to gravity and double-layer forces and hydrodynamically interacting with a rigid wall, whose validity is confirmed by numerical simulations. These results extend the scope of stochastic energetics by revealing a non-trivial relation between heat fluctuations' statistical properties and the non-linear features of non-equilibrium processes.

Scalable machine learning in challenging underwater environments is strongly limited by the lack of labeled real-world training data. This data is often expensive and laborious to gather, making large-scale real-world data challenging to gather and curate. However, simulated data can help close the gap, enabling many learning-based tasks for underwater perception. In this work, we extend OceanSim, an IsaacSim-based underwater perception simulator, with a Synthetic Data Generation (SDG) pipeline for training models to be used in underwater scenarios. The proposed pipeline enables users to generate large, automatically labeled, photorealistic datasets with configurable scene appearance, structure, and sensor settings. We evaluate the pipeline on a real-world sea urchin detection task and study how different forms of synthetic scene variation affect sim-to-real performance. Based on these experiments, we discuss findings on our results, main limitations of the current pipeline and identify future directions for improving underwater rendering fidelity, scene diversity, and the evaluation of sim-to-real generalization. The open-source code can be found at https://github.com/umfieldrobotics/OceanSim.

In 2020 M. Mucciconi and L. Petrov introduced a long-range deformation of the quantum open non-relativistic Toda system. We prove the integrability of the deformed Toda system by constructing a $2 \times 2$ Lax operator, which produces the commutative family of differential operators containing the Hamiltonian of the deformed Toda system. Moreover, we show that the same integrable deformation exists on both classical and quantum levels and can be applied to both non-relativistic and relativistic Toda systems. For the open non-relativistic deformed Toda systems we also present an $n \times n$ Lax matrix and prove that it produces the same family of Hamiltonians. We also show how to obtain the van Diejen-type deformed Toda system. Lastly, we show that on the quantum level the algebraic Bethe ansatz technique can be applied to the deformed Toda system.