Papers

Many nonsmooth, nonconvex objectives in learning and signal recovery are $\rho$-weakly convex. We minimize such a function when the weak-convexity parameter $\rho$ is unknown and only stochastic candidate and function-difference information is available. We propose the Adaptive Prox-Guided Scheme (APS), a single-trial framework that adapts the proximal parameter online and bidirectionally through a noisy descent test. APS attains a high-probability $O(\varepsilon^{-2})$ iteration bound for Moreau-envelope stationarity. The result allows biased, heavy-tailed function-difference estimates, while the candidate oracle need only be sufficiently accurate with constant probability when the proximal parameter is small and may be arbitrary otherwise. APS does not need to identify which case applies. A model-based extension covers adaptive proximal-point, prox-linear, and proximal-gradient methods. When the oracles are exact, the same analysis yields a deterministic $O(\varepsilon^{-2})$ bound and, for the proximal-point instance, an $\varepsilon$-subgradient stationary point.

Let $p$ be a prime. For every field $F$ of characteristic $p$ we exhibit pairs of non-isomorphic finite-dimensional $p$-nilpotent restricted Lie algebras $L$ and $H$ over $F$ whose restricted enveloping algebras $u(L)$ and $u(H)$ are isomorphic as $F$-algebras. Such pairs exist in every dimension at least $p+5$, with $\dim L'=p$ and $\dim H'=p+1$. Thus, the restricted isomorphism problem has a negative answer over every field of positive characteristic, even for $p$-nilpotent algebras over perfect fields.

In this paper, a novel optimal soft-landing guidance law with inequality approach-angle path constraints is analytically derived. The proposed guidance law prevents ground collision and enables approach-angle control by constraining the optimal trajectory to remain within a convex inverted pyramid originating at the landing point. A 3D point-mass linear kinematic model in a constant gravitational field is employed, together with a quadratic control-effort cost and terminal constraints on position and velocity. Analytical open-loop and closed-loop solutions, together with the optimal final time, are derived using Pontryagin's Minimum Principle and the optimality conditions at the transitions between unconstrained and constrained arcs. It is additionally shown that the optimal final time decreases when the path constraints become active. The resulting guidance law is continuous, piecewise linear in time, and nonlinear in the states in closed-loop. When a constraint becomes active, the controller cancels the gravitational component normal to the constraint, causing the trajectory to evolve along the constraint surface. The proposed guidance law is evaluated in simulations under various initial conditions, demonstrating accurate landing performance and consistent satisfaction of the path constraints.

For a finite poset $\mathcal{P}$, its height $h(\mathcal{P})$ is the number of cover relations in its longest chain. When $\mathcal{P}$ is a lattice $\mathcal{L}$, we label its elements $x$ with $h(x_\downarrow) = h([\hat{0},x])$ and its cover relations $x \lessdot y$ with $h(y_\downarrow) - h(x_\downarrow)$. When a lattice $\mathcal{L}'$ extends $\mathcal{L}$, $h(x_\downarrow)_\mathcal{L} \leq h(x_\downarrow)_{\mathcal{L}'}$. We study lattices $\mathcal{L}$ and $\mathcal{L}'$ such that $h(x_\downarrow)_\mathcal{L} = h(x_\downarrow)_{\mathcal{L}'}$. Cover relations labeled $1$ in $\mathcal{L}$ induce a poset that we call the (long) skeletal poset $\mathrm{SK}(\mathcal{L})$. Its Hasse diagram is the largest spanning subgraph that the Hasse diagrams of $\mathcal{L}$ and $\mathcal{L}'$ have in common. An example of lattices $\mathcal{L}$ and $\mathcal{L}'$ is the alt-Tamari lattices introduced by Chenevière, where every alt-Tamari lattice $\mathrm{alt}\text{-}\mathrm{Tam}_n$ extends the Tamari lattice $\mathrm{Tam}_n$/refines the Dyck lattice $\mathrm{Dyck}_n$ such that $h(x_\downarrow)_{\mathrm{Tam}_n} = h(x_\downarrow)_{\mathrm{alt}\text{-}\mathrm{Tam}_n}$. We study $\mathrm{SK}(\mathrm{Tam}_n)$ with another poset we introduce. We enumerate intervals in these posets. For a well-chosen distributive lattice, we introduce its altitude lattices, which generalize the alt-Tamari lattices $\mathrm{alt}\text{-}\mathrm{Tam}_n$. Altitude lattices within a family have the same number of linear intervals. They are related to each other via extensions, refinements, and embeddings of some skeletal posets. For a poset $\mathcal{P}$ with $\hat{0}$, we define its Kneser graphs $KG(k) := (V(k),E)$, where $V(k) := \{x: h(x_\downarrow) = k, 1 \leq k \leq h(\mathcal{P})\}$ and $E := \{(x,y): x_\downarrow \cap y_\downarrow =\hat{0}\}$. We give some observations about them in a reconstruction setting.

Generalizing the construction of two-block group algebra (2BGA) codes, we introduce a family of two-block quantum LDPC codes constructed using the action of a group on the cosets of its subgroup. This replaces the regular group actions of the earlier two-block constructions and significantly expands the search space, yielding new quantum LDPC codes outside the 2BGA family. Through a computer search, we identify several new quantum LDPC codes, including weight-6 codes with parameters $[[48,8,6]]$, $[[96,8,10]]$, and $[[224,12,16]]$, as well as weight-8 codes with parameters $[[84,16,8]]$, $[[112,16,10]]$, $[[128,16,12]]$, and $[[168,16,15]]$. Furthermore, we introduce a maximally packed syndrome extraction schedule of depth $w+2$, including initialization and measurement steps, for any code with a maximum stabilizer weight of $w$ from our family. Under a standard circuit-level noise model, our codes, when decoded using BP-OSD, perform competitively with BB codes, achieving thresholds of $\approx0.65\%$ for the weight-6 family and $\approx0.35\%$ for the weight-8 family. Finally, we introduce a group-theoretic framework to generate sequences of graph-based covers of 2BGA codes, recovering and extending recent results on code constructions of this type.

We study sparse semi-continuous tensor data with excess zeros, heavy right tails, and slice-specific dispersion. Such features arise naturally in monetary-valued multi-way data, such as international trade, where most exporter–importer–product–year cells are zero while positive values are continuous and highly variable. To model these data, we propose a Bayesian hierarchical tensor factorization model that places a low-rank CP structure on a latent Poisson rate tensor and couples it with a conditional Gamma model for positive outcomes, with rate parameters that can vary across slices within a mode. The model therefore separates the occurrence and magnitude of positive observations while borrowing strength across all tensor dimensions through a shared low-rank latent structure. To scale posterior inference to large arrays, we develop a hybrid variational–Monte Carlo algorithm that combines efficient coordinate ascent updates with a partially collapsed augmented-data sampler. Applied to approximately 60 million trade flows, the method surfaces multiway dependence across exporters, importers, products, and years that is difficult to recover from gravity-type or pairwise network analyses, which do not jointly model the product and temporal dimensions.

We develop Hamiltonian dynamics-based algorithms for smooth convex optimization that achieve accelerated rates of convergence. By exploiting contraction of averaged Hamiltonian flow trajectories rather than requiring contraction at trajectory endpoints, we show that Hamiltonian dynamics-based optimization methods admit deterministic and accelerated convergence guarantees, extending prior work that is limited to quadratic objectives or holds only in expectation. We analyze an idealized continuous-time algorithm and derive practical discrete-time implementations with optimal first-order complexity, thereby establishing Hamiltonian dynamics as a useful algorithmic primitive for deterministic accelerated convex optimization.

We investigate the image of polynomials multilinear in graded variables evaluated on the algebra of upper triangular matrices endowed with a group grading. We show that, in general, such an image need not be a vector subspace. However, under the additional assumption that the identity component of the grading is commutative, we prove that the image is always a vector subspace. We further investigate the image of polynomials evaluated on inverse and direct limits of algebras. As a consequence, we prove that the image of a polynomial evaluated on a direct limit of upper triangular matrix algebras whose identity component is commutative is always a vector subspace.

Starting from a spectral problem posed in a Clifford algebra with $d$ generators and Euclidean signature, we study an integrable, coupled system of PDEs that can be viewed as a vector perturbation of the Camassa–Holm equation with residual orthogonal symmetry. In the two-component case $d=2$, we show that the travelling wave solutions correspond to a Liouville integrable Hamiltonian system with two degrees of freedom, making use of a reciprocal transformation linking the coupled PDEs to a symmetry of the Hirota–Satsuma system. We also present a symmetry classification of all integrable two-component perturbations of Camassa–Holm, and find that besides the $d=2$ system analyzed here, the coupled 2CH system studied by Olver and Rosenau (as well as by Chen, Liu and Zhang, and Falqui), and equations related to either of those systems by Miura transformations, we also obtain a new system that (to the best of our knowledge) has not been reported previously. For the case of an arbitrary number of components $d$, we additionally investigate the short-pulse (high-frequency) regime, in which the limiting dynamics are governed by a vector-valued Hunter-Saxton type system. Furthermore, we provide a detailed analysis of the corresponding measure-valued (weak) solutions associated with this system.

We investigate the nonlinear dynamics of a single-component plasma confined in a time-dependent harmonic trap regarding aspects of symmetry and invariant functions. The system is described as a fluid in an isentropic adiabatic regime by a system of partial differential equations. A convenient change of variables, with a Gaussian ansatz for the number density distribution, allows a consistent mathematical description in terms of ordinary differential equations, from which we follow up with an analysis concerning the corresponding differential operators algebraic structure and Noether symmetries in specific physical regimes. For each studied case, proper invariants are identified. The obtained conserved quantities capture an interplay between the internal plasma dynamics and the time modulation of the trap, resulting in a sharp restriction for the system evolution.

Analog in-memory computing enables highly parallel matrix-vector multiplications with reduced data movement, but the resulting operations are noisy, quantized, and affected by device- and circuit-level non-idealities. This paper studies approximate inverse preconditioning for Krylov subspace methods in a hybrid analog-digital architecture that combines a conventional digital processor with an analog in-memory computing (AIMC) unit. The digital host performs sparse products with the coefficient matrix and the precision-sensitive Krylov operations, while the AIMC unit performs preconditioner applications through analog crossbar matrix-vector multiplications. Since the realized preconditioner is inexact and application-dependent, the outer iteration is formulated as the flexible GMRES method. We show that analog execution changes the usual preconditioner design problem in the sense that a stronger digital preconditioner may be less effective after analog scaling, write noise, input/output perturbations, quantization, and clipping are taken into account. We use exact block inverses and sparse approximate inverses as reference constructions and develop a noise-matching criterion for Monte Carlo approximate inverse (MCAI) construction. We also study damping and nesting to improve robustness. Numerical experiments, in which analog matrix-vector products are simulated rather than executed on hardware, show that analog-aware choices of block size, damping, MCAI construction accuracy, and nesting are important for robust convergence.

We prove that target-avoidance sets for Manneville–Pomeau maps are strong winning for Schmidt's game. More precisely, for the class of nonuniformly expanding interval maps considered here, there exists a single parameter $\alpha>0$ such that for every target $p\in[0,1]$, the set of points whose forward orbit does not accumulate on $p$ is $\alpha$-strong winning. The proof induces on the uniformly expanding region $[r_1,1]$. The resulting first-return map has infinitely many branches, so we approximate it by finite-branch expanding maps, apply a theorem of Hu–Li–Yu to those finite approximants, and then transfer the resulting strategies first to the induced map and then to the original Manneville–Pomeau map.

Farrow structures based on linear–phase FIR subfilters provide an efficient realization of variable fractional–delay (VFD) filters with reduced implementation complexity. While the all–linear–phase configuration admits a decoupled least–squares (LS) formulation with an analytical solution, this decoupling fails when branches of mixed types, linear–phase and general FIR, are required, as occurs when a group–delay constraint is imposed. This letter presents a unified LS design for Farrow structures via a nullspace parameterization of the per–branch symmetry constraints, yielding an analytical solution that accommodates arbitrary per–branch types. Numerical results demonstrate that the proposed framework satisfies group–delay constraints that the all linear–phase approach cannot meet, while substantially reducing the number of free parameters relative to the unconstrained general FIR baseline.

In this paper, we consider Newton-Puiseux parametrizations of irreducible curves in (n+1)-space, n greater or equal to 1, within a fixed semigroup under the action of Mather's group A. We establish criteria for eliminating parameters while preserving the Newton-Puiseux form, extending known results for plane curves.

This paper revisits the strong duality in the linear quadratic regulator (LQR) for continuous-time and discrete-time systems, and explores its interconnection with typical assumptions and the uniqueness of primal-dual solutions. Using a linear operator $\Psi$, we formulate a common nonconvex LQR problem that captures both time domains. We then derive its Lagrange dual problem and establish the strong duality via a rank-constrained tight semidefinite program (SDP) relaxation. Further, we show that the primal-dual optimal solutions to the SDP relaxation, after dropping the rank constraint, recover the classical algebraic Riccati equations and optimal feedback gains in a constructive manner. The dual derivation and strong duality analysis rely on mild standard assumptions and exploit the properties of the linear operator and its adjoint, revealing a structural symmetry between the two time domains.

Let $G$ be a finite group and let $\F$ be a finite field. We prove that any finite-dimensional $G$-graded associative algebra $A$ over $\F$ has a finite basis for its $G$-graded polynomial identities.

The end space of an infinite graph arises naturally in many contexts as an important invariant and an interesting construction. It compactifies a locally finite graph and Diestel shows how to extend the end space to a larger space, called the tangle space, which is able to compactify any infinite graph. In both ends and tangles, it is the vertex-connectivity structure of the graph that is being studied. If we switch our attention to edge-connectivity, we can analogously define edge-ends. There is a space known as the edge-direction space which turns out to play an analogous role as the tangle space in its relationship with the end space: the edge-direction space provides a larger compact space in which the not necessarily compact space of edge-ends lives in. In this paper, we make this analogy precise, providing a natural edge analogue definition of tangles and proving they result in exactly the edge-directions. We also describe a combinatorial construction of certain Boolean algebras which give rise, via Stone duality, to the tangle and the edge-direction spaces. Finally, we pursue functorial definitions of the combinatorial constructions used in the paper, inspired by the famously functorial nature of Stone duality and by previous work by one of the authors and colleagues on trying to functorialize the end space construction. We hope our work will provide foundation and inspiration for further work on infinite graph theory that makes ample use of category theory and powerful algebraic constructions such as Boolean algebras and Stone duality.

Let $K$ be a $1$-separated set of diameter at most $R-1$, and let $\ell_m$ denote a collection of $m$ points on a line, with consecutive points of distance $1$ apart. Conlon and Fox (2019) demonstrated a coloring of $n$-dimensional Euclidean space avoiding red congruent copies of $\ell_2$ and blue congruent copies of $K$ for $|K| > 10000^n\log_2 R$. We show here a stronger bound, that in fact $|K| > (6.79 + o(1))^n\log R$ suffices for arbitrary $1$-separated $K$, while the improvement $|K| > (5 + o(1))^n\log R$ holds in many cases, including when $K = \ell_m$, or more generally when $K$ is contained in a low-dimensional affine subspace. We also make a special study of the case when $n=2$, demonstrating a two-coloring of two-dimensional Euclidean space avoiding red copies of $\ell_2$ and blue copies of $\ell_{6330}$. This latter result addresses a question of Erdős and Graham.

The main thrust of these notes is 3-fold: (1) An analysis of certain $K(\pi,1)$'s that arise from the connections between configuration spaces, braid groups, and mapping class groups, (2) a function space interpretation of these results, and (3) a homological analysis of the cohomology of some of these groups for genus zero, one, and two surfaces possibly with marked points, as well as the cohomology of certain associated function spaces. An example of the type of results given here is an analysis of the space k particles moving on a punctured torus up to equivalence by the natural $SL(2,\mathbb{Z})$ action.

In this paper we consider minimal surfaces in hyperbolic space of arbitrary codimension that are critical for renormalized area. We give two criteria which imply that $Y$ must be a totally geodesic disk. One can be viewed as a 'gap' result for the renormalized area.

For an amenable unimodular random rooted network, we show that the presence of a positive point mass in the expected spectral measure implies that, with positive probability, there exists an eigenfunction with finite support.

Graph neural network architectures based on the graph Laplacian approximate the Laplace-Beltrami operator, thus limiting their application to isotropic operators. As a nonlinear alternative to the Laplace-Beltrami operator, we consider estimates of the Finsler Laplacian on point clouds sampled from a manifold. We prove that these discrete estimates converge to the true operator on the manifold as the number of point samples grows. Moreover, we show that this operator can be expressed as a graph neural network layer, which we use to define a family of Finslerian graph neural networks constrained to express Finsler geometry. We show that Finslerian graph neural networks recover the geometry underlying nonlinear diffusion equations in practice.

We give a short proof of the quantitative isoperimetric inequality. Our argument is based on a notion of quantitative calibrations which induce a natural distance controlling both the Fraenkel asymmetry and the tilt excess. The proof of our key result which can be viewed as a nonlinear, geometric version of Fuglede's result in $BV$ is direct and self-contained. In particular, we do not make any use of regularity theory for almost minimizers.

We investigate the thermal partition and one-point functions of the three-dimensional conformal field theory dual to a massive interacting scalar field in AdS$_4$. Using thermal inversion formulas, we determine the asymptotic behaviour of the spectral density and OPE coefficients involving heavy operators at fixed spin. We first analyse these CFT data for the generalised free field, corresponding to the non-interacting bulk theory. Then we compute the first-order perturbative corrections induced by the cubic and quartic bulk interactions. The thermal observables considered here probe a sector associated with operators of large dimension and, in the bulk description, a regime dominated by states with large particle number. This regime remains comparatively unexplored even in generalised free field theory. Remarkably, the asymptotic formulas obtained from thermal inversion remain quantitatively accurate far from the asymptotic regime, describing CFT data reliably already at intermediate conformal weights. Our results show that this feature survives the inclusion of bulk interactions and provide new analytic control over heavy-state data in conformal field theories.

Organizations and researchers show increasing interest in using large language models (LLMs) in place of human participants in A/B tests, in the hope of experimenting faster and at lower cost. We study when a treatment effect estimated on LLM outcomes can recover the effect for the human population of interest. Distributional equivalence between LLM and human outcomes would make any standard estimator valid but is unrealistic. We therefore develop a statistical framework that adapts surrogate endpoint theory to LLMs, showing that calibrating LLM outcomes to human outcomes identifies the average treatment effect under surrogacy and comparability conditions that are jointly weaker than distributional equivalence. We present a falsification test for surrogacy and a bound on the worst-case bias from limited overlap between the LLM and human samples. We further show that the stochasticity inherent to LLMs can weaken surrogacy for identification while also introducing bias and variance during estimation, but that using an average over multiple LLM draws per unit as the surrogate mitigates these issues. Simulations validate the results, and an empirical application to the Upworthy Research Archive dataset shows that raw LLM outputs recover only 39% of the human treatment effect while nonparametric calibration closes the gap. A central takeaway is that A/B testing on LLM responses is correct only by assumption, whereas A/B testing on humans is correct by design, and that the required assumptions are hardest to justify precisely where LLMs promise the greatest benefit. We discuss the choice of LLM, prompting, and temperature as design variables, the compounded challenge posed by long-term outcomes, and how to size human pilot studies for validation.

We formulate and analyze a sex- and stage-structured model for Anopheles dynamics under the sterile insect technique (SIT), motivated by the need for tools robust to insecticide resistance and outdoor transmission. The model tracks aquatic stages, adult males, unmated females, and females mated with wild or sterile males; includes egg-laying capacity and larval competition; and uses a refractory period followed by density-dependent mate search. The resulting Holling type-II mating term generates a mate-finding Allee effect. After establishing well-posedness, we prove that this Allee effect makes the mosquito-free equilibrium locally stable for all admissible parameters and globally asymptotically stable when a quick-mate-search reproduction number $R_0^q$ is below one. When $R_0^q>1$, habitat capacity is large, and larval competition is weak, two positive equilibria arise through a saddle-node bifurcation: a stable natural equilibrium and an unstable Allee equilibrium separating persistence from extinction. For a reduced model, a Goh-Volterra Lyapunov functional estimates the persistence basin. We then show how constant and population-responsive sterile-male releases reshape this bistability. Sufficiently large releases annihilate the positive equilibria in a second saddle-node bifurcation, while a sufficiently large constant release drives local elimination from every admissible initial state. Thus SIT need only push the population across the Allee separatrix, after which mate-finding failure can complete extinction. In a free-horizon optimization framework with an Allee-threshold stopping rule, a hybrid release strategy reduces the sterile-male requirement by about $5\%$ relative to the best constant-only strategy and $39\%$ relative to the best population-responsive-only strategy. These results recast the Allee effect as a control lever for vector suppression.

Over-the-air federated learning (FL) leverages the superposition property of multiple-access channels to enable communication-efficient distributed model training. Existing integrated sensing, communication, and computation (ISCC)-enabled over-the-air FL systems typically require dedicated resources for the sensing module, inevitably compromising FL performance due to resource competition. In this paper, we propose a sensing-native over-the-air FL framework that explores built-in distributed wireless sensing capability with zero overhead per model aggregation. Specifically, the high-dimensional local gradient signals possessing favorable autocorrelation property are concurrently leveraged for target distance estimation, while the gradient statistics already required for over-the-air FL serve as a ready-made gateway to deliver locally-sensed results to the edge server for cooperative localization. To combat inter-device interference, channel fading, and communication noise, we put forth a robust trilateration-based target positioning method building upon an efficient matched-filtering-based distance estimation. Then, by explicitly characterizing the impact of imperfect model aggregation and noisy gradient-statistics transmission on the sensing-native over-the-air FL convergence, we develop a statistics-aware communication-learning co-design approach. We first derive the closed-form optimal power budgets allocated to local gradients and their statistics, based on which an efficient successive convex approximation method is proposed for receiver beamforming optimization. Simulation results show that the proposed framework simultaneously achieves superior learning and sensing performance compared to representative baselines.

We study the differential inclusion $\text{d}z_t\in F(z_t)\text{d}x_t$, with initial condition $z_0=\xi$, where $F$ is a nonconvex-valued multifunction, and $x$ a path of bounded $q$-variation, for some $1\leqslant q<2$, extending the work of Bailleul, Brault, and Coutin (2020). We obtain existence of local and global solutions to this inclusion without assuming $F$ bounded. If $z(\xi,x)$ denotes such a solution, we obtain measurability of $z$ with respect to $x$ and $\xi$. To establish this, we introduce a Skorokhod-type distance and prove that Young integration is continuous with respect to it. By the way, we prove that a compact-valued $\gamma$-Hölder map $F$ has, for any $p>1/\gamma$ and $\xi\in F(0)$, a selection $f(\xi)$ of bounded $p$-variation, started at $\xi$, such that $f$ is measurable in $\xi$.

In this paper, we present the numerical analysis of an elliptic eigenvalue problem in which the essential boundary condition is imposed weakly by means of the Nitsche method. The resulting discrete eigenvalue problem is studied within the framework of compact operator theory. We prove norm convergence of the discrete solution operator and derive error estimates for the eigenvalues and eigenfunctions, with rates depending on the chosen Nitsche variant. In addition, we develop an a posteriori error analysis and propose a residual-based estimator suitable for adaptive refinement. Several numerical experiments are presented to assess the convergence, stability and robustness of the method, including the influence of the Nitsche stabilization parameter and the performance of the adaptive strategy.

We give the first polynomial-time constant-factor approximation of the Gromov-Hausdorff distance d_GH between finite point sets in the Euclidean plane; in fixed Euclidean dimension such an approximation was previously known only on the line (Majhi, Vitter and Wenk, 2024). Global alignment cannot succeed: the classical dimension drop defeats alignment by isometries, a multiplicity gap defeats alignment by bijections, and a reflection barrier defeats sorting under any single global reflection pattern. The algorithm is therefore local. Guessing the images of one diameter pair pins every point's longitudinal coordinate to within O(d_GH). Heights are read in windows whose height spread is at most a fixed multiple of their length, where a chain argument makes every compatible match local in the plane. One reflection sign per window is then chosen by 2-SAT; at the right frame and guess, any solution yields a correspondence of distortion O(d_GH). For the bijective relative of d_GH, half the least additive distortion over bijections, the same scheme reduces the planar problem to a matching question that we leave open.