Papers

A straightforward and computationally efficient indirect method based on STM sensitivity analysis is introduced for designing fuel-optimal low-thrust transfers under high-fidelity dynamics. Conventional indirect approaches require explicit expressions for the partial derivatives of the system dynamics to formulate the costate equations, making the derivation process complex for high-fidelity trajectory design. In this work, the costate equations are reformulated as ordinary differential equations involving only the state variables and their time derivatives. High-order dynamical effects are treated as black-box components, avoiding the need to derive partial derivatives of the system dynamics. A standard gradient-based or interior-point optimizer is used to determine the optimal costates and transfer parameters. The equivalence between the proposed method and conventional approaches is demonstrated through a classic Earth-Mars transfer scenario. An Earth-Mars transfer under high-fidelity dynamics is then presented, including perturbations from solar radiation pressure, solar J2 oblateness, Jupiter third-body gravity, and relativistic effects. Finally, the method is applied to a multiple-revolution Earth-Venus transfer under high-fidelity dynamics.

The LLM Jury, a Panel of LLM Evaluators (PoLL) reporting consensus scores, has become a practical alternative to single-judge LLM evaluation, yet its statistical behavior remains poorly understood. We formalize the LLM Jury under the Huber contamination model and show that PoLL incurs unbounded bias under any positive contamination, regardless of jury size, whenever a single judge fails in a biased, LLM-typical way (mode collapse, sycophancy, safety refusal). Framing jury consensus as classical robust mean estimation, we propose RoPoLL (Robust Panel of LLM-as-Judge), which preserves the PoLL panel but replaces the aggregation function with a robust mean estimator, instantiated with the geometric median (GM): tuning-free, with the optimal finite-sample breakdown point 1/2. A finite-sample error bound and a matching information-theoretic minimax lower bound agree on the parametric rate sigma*sqrt(d/N) and differ on the breakdown floor by a factor of sqrt(d), a statistical-computational gap that polynomial-time RoPoLL pays relative to the intractable Tukey halfspace median. Across 13 open-weight judges (4B-675B), three reward-model benchmarks, and four corruption regimes at rates up to 50%, RoPoLL dominates PoLL on every biased corruption type: by about 19% on cross-dimensional attacks at matched compute, and by orders of magnitude on heavy-tailed Byzantine adversaries. A 3-judge RoPoLL committee at 38B beats Mistral-Large-3 (675B) by 1.31x on HelpSteer-2 under 30% bimodal-random corruption, an 18x parameter advantage at better accuracy; a Noisy-GT control confirms the premium is paid against biased contamination, not benign imprecision.

Modern deep learning has been shown to operate using learning rates far larger than those justified by classical optimization theory. Most prior analyses of this phenomenon focus on deterministic gradient descent, leaving the stochastic setting largely unexplored. In this work, we provide sharp convergence guarantees for Stochastic Gradient Descent (SGD) applied to the multiclass cross-entropy loss, for both linear classifiers and two-layer neural networks. We show that the stochasticity of SGD may cause the dynamics to alternate between an edge-of-stability regime that is dominated by curvature-driven oscillations, and a stable regime in which the expected loss decreases at a controlled rate. Despite that, we prove that SGD stabilizes the dynamics, ensuring that the iterates return to stability in a fixed number of iterations and allowing convergence in the best-iterate sense even with large learning rates. Experiments validate our theoretical findings and illustrate the benefits of SGD in the large-stepsize regime.

For any conformal net, not necessarily rational, we construct the associated Cardy-type conformal field theory on the Minkowski spacetimes $(\mathbb R/2\pi\mathbb Z)\times\mathbb R$ for closed strings and $[0,\pi]\times\mathbb R$ for open strings within the framework of algebraic quantum field theory. In addition to verifying some of their basic properties, we prove three forms of Haag duality for multi-double-cones and boundary intervals, interpreted respectively as the Minkowskian versions of modular invariance, the Cardy consistency condition, and the Morita equivalence of boundary field algebras.

We introduce a discrete-energy Sobolev space $\mathcal{W}^{1,p}_{\mathscr V}(T)$ on Ahlfors regular snowtrees, a class of metric trees where every arc is a snowflake of the same type. Our main result shows that, for every partition $\mathscr V$ and every $1<p<\infty$, this discrete space coincides quantitatively with the Korevaar–Schoen space on $T$. This fact and the independence of the space on the particular partition used to define $\mathcal{W}^{1,p}_{\mathscr V}(T)$ are both novel even for the class of geodesic trees. We also determine the critical Korevaar-Schoen exponent for Ahlfors regular snowtrees and prove capacity attainment and upper estimates, which reveal the appropriate walk dimension needed for the corresponding probabilistic profile on these trees.

We study the compositionality of global dynamics through attractor lattices and order structures of recurrent dynamics in product and skew-product systems using Conley theory. For product systems, these structures can be characterized algebraically in terms of the structure of component systems, where we prove that the attractor lattice of the direct product of two flows is isomorphic to the coproduct of the attractor lattices of the component flows. We also consider fast-slow, skew-product systems that arise from singular perturbation of a parameterized dynamical system. These results provide a framework for decomposing global dynamics into lower-dimensional subsystems and suggest computational approaches for constructing Conley-Morse representations through composition.

Given two Banach spaces $X$ and $E$, one can associate a numerical invariant $\mathcal{CR}(X, E)$, called the coarse embeddability ratio, which provides a criterion for coarse and uniform embeddability. We compute the coarse embeddability ratio for several important classes of Banach spaces, using various tools from the nonlinear theory of Banach spaces. Finally, we find pairs of Banach spaces with arbitrarily large coarse embeddability ratio, resolving an open problem of Rosendal in the negative.

This work is about the shape optimization of long tubular objects in electromagnetic chirality (em-chirality). Em-chirality is a property of individual scattering objects or metamaterials describing their qualitatively different response to electromagnetic waves of opposite polarization handedness. The optimization is performed by a Newton-type iterative maximization of a regularized em-chirality measure with respect to the scatterer's shape. In this context, the differentiability of the object-to-far field operator map is analyzed rigorously, thereby extending previously known results on the domain derivative to the far field operator setting. Our optimal design algorithm is based on the electric field integral equation, which is employed both for the evaluation of scattered fields and for the computation of the domain derivative. Our implementation is done via the boundary element method. The numerical examples presented in this work yield strongly em-chiral scattering objects capable of exciting higher-order modes beyond the dipole regime with nonintuitive shapes that expand the known set of highly em-chiral scattering objects.

Let $\mathbb{H}_{n+1}$ denote a computable copy of the $(n+1)$-clique free universal homogeneous Henson graph, $G$ denote a finite subgraph of $\mathbb{H}_{n+1}$, and $k(G,n)$ denote the big Ramsey degree of $G$ in $\mathbb{H}_{n+1}$. We prove that for any computable coloring $\chi$ of the copies of $G$ in $\mathbb{H}_{n+1}$, there is a copy $\mathbb{H}'$ of $\mathbb{H}_{n+1}$ that is computable from $0^{(2\delta(G,n)-1)}$ in which $\chi$ takes no more than $k(G,n)$ colors, where $\delta(G,n)$ denotes the maximum number of levels of a diary for $G$ in $\mathbb{H}_{n+1}$ (this is a finite number). It follows that the statement, “Henson graphs have finite big Ramsey degrees," is provable in ACA$_0'$. Combining this with a recent result of Cholak, Dobrinen, and McCoy \cite{CDM} yields the equivalence of the statement with ACA$_0'$ over RCA$_0$.

We investigate four-dimensional electrostatic systems arising as spatial factors of static Einstein–Maxwell spacetimes with cosmological constant. Assuming that the electric field is everywhere collinear with the gradient of the lapse function, we prove that the harmonicity of one of the (anti-)self-dual components of the Weyl tensor imposes strong rigidity on the underlying geometry. More precisely, we show that the gradient of the lapse function is an eigenvector of the Ricci tensor and that the regular level sets of the lapse function are totally umbilic with constant mean curvature. As a consequence, the manifold is locally conformally flat and admits a local warped product structure with one-dimensional base and three-dimensional fiber of constant curvature.

We investigate matrices with entries in a number field such that some positive power has all its entries in the corresponding ring of integers. Our work generalizes previous results in several directions and we find applications to numerical semigroups.

Many measurements in physics determine a linear transformation from pairs of input and output vectors, where the transformation is known in advance to belong to a matrix Lie group, for example the Lorentz group for the four-momenta measured by two inertial observers, or the unitary group for the states produced by a quantum circuit. Among the classical groups, only the rotation and unitary groups have a well-known closed form solution, given by the Kabsch and Horn algorithms and by the polar decomposition. We extend our previously proposed Lie algebra method from the image of the matrix exponential to the entire Lie group. We derive the projection in closed form for the classical groups over the real and complex numbers, including the indefinite orthogonal and unitary, symplectic, spin, special Euclidean and similarity groups, and we give conditions under which the method is exact for noiseless data, with examples in which it fails otherwise and an extension by a data dependent involution which is exact on the whole group. For noisy data, we bound the error of the method to first order in the noise, and we show that, for the orthogonal and unitary groups, it agrees with the least squares solution over the group to first order when the input vectors are isotropic, but not otherwise. We apply the method to an eight-qubit quantum circuit and to the rigid registration of point clouds, and compare it with Gauss–Newton refinement and direct minimization. A Julia implementation is provided.

We prove a version of the Dvoretzky-Kiefer-Wolfowitz inequality for Markov chains with a regenerative structure. Suppose we have a regenerative Markov chain with stationary distribution $\pi$. Given a functional $\theta$ on the state space and a confidence level $1-\delta$, our result provides a uniform $1-\delta$ confidence band for the CDF of $\theta$ under $\pi$ based on the empirical CDF. By inversion, we get a $1-\delta$ confidence band for the quantile function of $\theta$ under $\pi$. Our bounds are fully explicit and nearly optimal. In addition, they are data-dependent in the following sense: in the formula for the width of the confidence band, the leading term can be computed directly from the sample path without any a priori information about the convergence rate of the chain. A convergence bound is required, but it contributes to the width of the confidence band only through a lower-order term. For this reason, our result is attractive for Markov chains whose convergence rate is much quicker in practice than what can be proved in theory. Data-dependent bounds of this type are called empirical concentration inequalities in the literature. Thus, our result is an empirical concentration inequality for the empirical CDF of $\theta$ given the sample path.

Consider a nonparametric regression model with one-dimensional covariates and a continuous regression function. Assume that the regression function from the left of the covariate support starts equal to zero and then changes at some unknown point. Our aim is to estimate this gradual change point. We define and compare various consistent estimators based on a new general optimization method in the case where the aim is to estimate the largest minimization point of some objective function. We discuss rates of convergence and estimating the regression function based on the gradual change structure. Bootstrap bias approximation is discussed. Further applications in a two sample case are considered, where two continuous regression functions first equal and then change at some point of interest.

This is a geometric retelling of Konyagin and Sevast'yanov's proof of Andrew's theorem, which is a tight upper bound on the number of vertices of a d-dimensional lattice polytope in terms of its volume.

We present $\texttt{bucket-graph-spprc}$ ($\texttt{bgspprc}$ for short), an open-source, header-only C++23 library for the shortest path problem with resource constraints (SPPRC), the pricing subproblem at the heart of branch-cut-and-price for vehicle routing and related problems. The library implements the bucket-graph labelling algorithm of Sadykov, Uchoa and Pessoa (2021), with bidirectional labelling, across-arc concatenation, bucket fixing and arc elimination, and a structure-of-arrays label store with SIMD-accelerated dominance. Its central design feature is a compile-time resource concept: a new SPPRC variant is added by implementing a fixed seven-function interface, and resources compose into a label state with no runtime dispatch, the state layout fixed at compile time. Five resources ship built in: time/capacity, ng-path elementarity relaxation, rank-1 cuts, cumulative cost, and pickup-and-delivery. In a reproducible, head-to-head comparison on shared public instances at an identical bound, $\texttt{bgspprc}$ outperforms PathWyse (Salani, Basso and Giuffrida, 2024), the main open-source comparator, by $1.3\times$–$2.35\times$ in shifted geometric mean (and by $1.3\times$–$2.3\times$ even when itself run single-threaded), and runs within $1.9\times$–$2.4\times$ of parallel pull labelling (Petersen and Spoorendonk, 2025), a different labelling technique for the same problem. The library, benchmark scripts, and pinned instances are publicly available.

We prove a Nekhoroshev type result for a time quasiperiodic perturbation of an integrable Hamiltonian system. More precisely, we assume that the integrable part is analytic and fulfills a generic nondegeneracy condition introduced by Nekhoroshev and called P-Steepness. We add a small perturbation which depends in a quasiperiodic way on time (with Diophantine frequency) and prove that – for times exponentially long with the inverse of the size $\varepsilon$ of the perturbation – the actions of the unperturbed system remain approximately constant. The proof is based on an extension to the time dependent case of the proof {of classical Nekhoroshev's theorem} given by Guzzo, Chierchia and Benettin, which however requires new ideas in order to deal with the more complex geometry of resonances of the time dependent case.

The number of trees is a central computational parameter in Random Forests: increasing it reduces finite-ensemble variability but increases training and prediction cost. Plateau-based tuning adapts this parameter through local comparisons of out-of-bag scores at a geometric triplet of tree counts. After the remaining hyperparameters have stabilized, however, the central triplet point need not converge to a deterministic value; instead, it fluctuates around a stationary regime. This paper develops a stationary-distribution theory for this process. The central ensemble size $B_t$ is modeled as a birth-death Markov chain on a geometric grid, and its stationary distribution is derived through local balance. Under a leading centered folded-normal approximation, equilibrium equations are obtained for the original update rule and a symmetric modified variant, implying that the stationary center $B_*=O(\varepsilon^{-2})$ as $\varepsilon\downarrow 0$. The stationary spread is also characterized. A local Gaussian approximation and a Fokker-Planck interpretation give grid-level variance constants. After conversion to the ensemble-size scale, $\sigma_{B,*}=O(\varepsilon^{-2})$, while the variance is $O(\varepsilon^{-4})$. The leading relative spread is independent of $\varepsilon$ and controlled by the scale factor and update rule. These results interpret plateau-based Random Forest tuning as a stochastic process rather than a deterministic stopping rule.

We study the squared singular value spectrum of a non-square product of independent real Gaussian matrices, equivalently the feature covariance spectrum of a deep linear neural network at initialization. Starting from the fixed-$m$ covariance diffusion previously obtained in the proportional depth-width limit, we record an equivalent matrix realization, describe its affine invariance, and derive the interacting diffusion satisfied by its eigenvalues. We then take a second limit, sending $m\to\infty$ on the accelerated spectral clock $\tau=mt$, which corresponds in this sequential construction to the relation $dm/n\to\bar\tau$. We establish convergence of the empirical spectral measure path to a deterministic mean-field limit and derive a closed Burgers equation for its $T$-transform. Together with the proportional depth-width limit, these results give a rigorous sequential route from the deep non-square Gaussian product to the free log-normal limit of its feature covariance spectrum; for more general initial laws, the transform yields a free multiplicative convolution form. We further analyze the support of the free log-normal law, give a fixed point iteration for numerical evaluation and a formal Marchenko–Pastur approximation at small time, and use the limiting spectrum to predict the risk in a toy random feature model.

We consider the finite-time optimal control of stochastic systems subject to a probabilistic constraint on the trajectories' safety. Such formulations are known as joint chance constrained optimal control problems. The common practice is to jointly minimise the expected cost of all trajectories, safe and unsafe. This leads to policies which invite constraint violations to exploit low-cost unsafe trajectories. When constraints represent states of critical failure, such behaviour is undesirable. We demonstrate that this behaviour can be overcome by only minimizing the expected cost of safe trajectories. The underlying rationale follows a practical intuition: In many applications, the cost incurred by unsafe trajectories is irrelevant (e.g., the battery usage of a crashed quadcopter), and one is usually interested in minimizing the cost of trajectories that are safe. We show that this problem can be cast as a constrained Markov Decision Process over an augmented state space. This allows solving it via dynamic programming. We derive bounds on the policies' safety under errors resulting from gridding approximations when the system's state space is continuous. Finally, we empirically compare dynamic programming as well as reinforcement learning solutions on a simulated 2D unicycle system in cluttered reach-avoid environments.

For a compactly supported probability measure $\mu$ on the $d$-dimensional space $\mathbb{R}^d$, the average distance problem asks us to minimize the average distance functional over all compact, connected, $\Sigma \subseteq \mathbb{R}^d$ satisfying the Hausdorff $1$-measure constraint $\mathcal{H}^1(\Sigma) \leq \ell$. This problem was first introduced in 2002 by Buttazzo, Oudet, and Stepanov to study optimal transport problems with free regions on which the transport cost vanishes, and has undergone a considerable amount of research since. Most recently, Kobayashi, Kim, and the author studied the structure of these regions using the barycentre field, a tool for studying the average distance functional introduced previously by Kobayashi, Hayase, and Kim. In this paper, we build upon this work to prove in much greater generality a topological description of minimizers of the average distance problem conjectured by Buttazzo, Oudet, and Stepanov. In particular, we prove this conjecture in all dimensions in the case originally studied by these authors.

When two initially entangled photons each undergo Compton scattering, the scattered electrons become correlated. However, the final reduced density matrix of one scattered pair is not influenced by the other scattered pair due to unitarity. Herein, we keep unitarity up to tree level for Compton double scattering and obtain different results than recent literature. The initial four particles, where the initial photons are entangled, are written as a superposition of two states with a relative phase. The final density matrix has two area divergences that are regularized with unitarity. The regularization procedure, i.e. solving for the roots of a polynomial that represents the probability for no scattering, suggests a novel definition of the scattering cross-section. Vieta's formulas relate these divergences to finite cross-sections. For an initial pure state, the formulas for the final density matrix and the correlation of final electronic polarizations are given. The correlation implies double scattering is analogous to Young's diffraction experiment. The two initial superposed states are the circular apertures while the Feynman amplitudes are the interfering complex light fields.

On the Kirby list, Akbulut poses the question of whether there exists a homology 3-sphere $Y$, other than $S^3$, with the following property: Any knot $K$, representing $0\in\pi_{1}(Y),$ which is slice in some contractible 4-manifold $X$ which $Y$ bounds, is already slice in $Y\times[0,1]$. In this paper, we make progress on this question by producing a class of deeply slice knots. We construct these knots by first specifying a pair $(X, K)$, where $X$ is a contractible 4-manifold with integral homology 3-sphere boundary and $K$ is slice in $X$. Then, we show the knot is deeply slice using concordance invariants from Heegaard Floer homology. We employ immersed curve techniques to compute these invariants.

In this paper, we attempt to enhance the theoretical understanding of convolutional neural networks (CNNs) as feature extractors in classification tasks by analyzing them through the lens of Cover's function-counting theory. Specifically, our focus lies on the notion of separation capacity, a combinatorial quantity derived from counting the number of realizable dichotomies (i.e., binary label assignments). Our contributions are threefold. First, we extend Cover's framework by establishing a conceptually insightful and practically useful formulation for the separation capacity. Second, leveraging this formulation, we identify the factors governing the separation capacity of feature extractors that employ a specific CNN architecture, so-called scattering networks, in terms of their network building blocks. Third, we provide practical insights for scattering network design.

In this paper, we prove that local polynomial convexity at the origin for the union of finitely many transverse totally real subspaces of maximal dimension is sufficient for Carleman approximation. Some new conditions are given for the polynomial convexity of the union of three transverse totally real planes in $\mathbb{C}^2$. We also provide a sufficient condition on the union of two Lipschitz graphs for Carleman approximation. Along the way, we provide sufficient conditions for union of two Lipschitz graphs to be polynomially convex. Finally, we find a family of surfaces in $\mathbb{C}^2$ with a hyperbolic complex point that allows Carleman approximation.

We establish a fundamental extension of Mercer's celebrated theorem by introducing a class of higher-order kernel operators acting on Sobolev spaces $H^k(\Theta)$, where $\Theta \subset \mathbb{R}^d$ is a bounded domain and $k\in\mathbb{N}_0$ corresponds to the order of weak differentiability. The spectral decomposition of these operators then yields Mercer-type expansions that are optimal in $H^k(\Theta\times\Theta)$. Notably, we derive from the embedding properties of Sobolev spaces, that for $k>d$, these expansions also converge uniformly without requiring the kernel to be positive definite. For positive definite kernels, we confirm the nuclearity of these higher-order operators and establish a significant refinement of Mercer's Theorem. These results lead to novel spectral representations of RKHS and have subtle implications for stochastic analysis. Applied to the covariance kernels of weakly differentiable random fields, our theory provides refined Karhunen-Loeve expansions that facilitate the simultaneous mean-square optimal approximation of both the process and its derivatives.

We present a complete machine-checked formalization of Dana Scott's landmark 1972 paper Continuous Lattices [Sco72], carried out in Lean 4 against mathlib and including the March 1972 Milner correction in [Sco72] (pp.~135–136). Scott's paper develops a model for \(\lambda\)-calculus from a topological starting point. He defines injective \(T_0\)-spaces – those with a strong extension property for continuous maps – and shows that they are exactly the continuous lattices: complete lattices whose Scott topology is determined by the order via the way-below relation (\(\ll\)). On this foundation he studies projections, retractions, products, function spaces, and inverse limits. The capstone (Theorem 4.4) constructs an inverse limit \(D_\infty\) of function-space approximants and proves \(D_\infty \cong [D_\infty \to D_\infty]\), yielding a purely mathematical model for Church's untyped \(\lambda\)-calculus. Our development formalizes 43 numbered results from Scott's Sections 1–4 (Propositions, Corollaries, Lemmas, and Theorems), each as a sorry-free Lean theorem, together with supporting infrastructure (step functions, the \(\Uparrow a\) basis of Scott opens, Milner's coarser-than-Scott hypothesis, the function-space tower, and the \(i_\infty\)/\(j_\infty\) pair). The formalization is classical (uses Classical.choice transitively) and follows Scott's proof dependency order. Where the Lean proof required choices not visible in the original – or where dead ends were encountered – we record detailed notes in Section 5. All proofs check with the standard footprint \(\texttt{[propext, Classical.choice, Quot.sound]}\). The formalization has been accepted into the Palomar registry.

We continue the study of Pascal-type residual constructions in projective four-space. Starting from two $k$-tuples of hyperplanes in $\mathbb P^4$ such that the $k$ diagonal intersection planes are contained in a hyperplane, one obtains a residual hypersurface of degree $k-1$ containing the remaining $k^2-k$ planes. In this work we consider the case $k=5$, where the twenty residual planes are contained in a quartic threefold. A balanced specialization of this construction is projectively equivalent to the celebrated Burkhardt quartic. In this model the twenty residual planes form one half of the forty Jacobi planes on the Burkhardt quartic. We reveal their incidence structure as governed by the directed complete graph on five vertices. The forty nodes naturally forced by these planes split as $30+10$, and the Burkhardt specialization adds five further nodes. We also write down the complementary twenty Jacobi planes explicitly and describe all forty Steiner hyperplanes in Pascal coordinates.

We derive sparse analogs of several Roth-type results, showing that they hold in $B_h$ sets of near-maximum size. It is shown that if a $B_h$ set is free of pairwise distinct solutions to a linear equation with more than $2h$ variables then it must be a constant factor smaller than the best-known upper bound on the size of any $B_h$ set. As a key input, it is established that extremal $B_h$ sets are Fourier pseudorandom. If the forbidden equation has a certain subdivision structure, an asymptotic saving is obtained. The case of Sidon sets ($h=2$) was previously studied by Conlon, Fox, Sudakov, and Zhao as well as Prendiville. When forbidding a non-translation-invariant equation $E$ from a Sidon set, it is shown that if $E$ has a zero-sum subcollection of at least five coefficients then the Sidon set must either be very small or generate a Cayley graph with bounded chromatic number. On the other hand, large Sidon sets are constructed that generate Cayley graphs with unbounded chromatic number and are also free of multiple equations with zero-sum subcollections of four coefficients. This can be viewed as a sparse analog of a result of Liu, Wu, Yang, and Zhang characterizing linear equations with vanishing chromatic threshold.

We consider the $L^2$-critical nonlinear Hartree equation in $\mathbb{R}^{1+4}$ and multisoliton solutions for which the trajectories are approximated to leading order by an $m$-body law. We obtain soliton clusters asymptotically following hyperbolic-parabolic trajectories of the corresponding $m$-body problem. By pseudo-conformal invariance, we then conclude finite-time collision blow-up with any number of clusters, each consisting of an arbitrary number of solitons, colliding simultaneously at distinct prescribed points.