Papers

In the present paper, we show that, for an odd prime number $p$ and a nontrivial finite Galois extension $k$ of $\mathbb{Q}_{p}$, the $p$-adic representation of the absolute Galois group of $k$ determined by a Lubin-Tate formal group over the ring of integers of $k$ is not Aut-intrinsically Hodge-Tate [in the sense of Hoshi]. This settles the odd-degree cases left open in the previous works of Hoshi and the author and, together with the known even-degree case, completes the picture for finite Galois extensions of $\mathbb{Q}_{p}$ in the case where $p$ is odd. This exhibits a sharp contrast, from the viewpoint of anabelian geometry, between the $p$-adic cyclotomic character and other $p$-adic Lubin-Tate characters.

The convergence of large language models (LLMs) with 6G networks is fostering a paradigm of autonomous multi-agent cooperation, which in turn is expected to substantially increase east-west traffic. Although latent-space interaction mechanisms can enable more efficient collaboration than symbolic natural-language (NL) exchanges, prior work often abstracts away the associated communication overhead under practical wireless constraints. In embodied multi-agent settings, heterogeneous interaction media incur disparate inference and transmission costs, thereby inducing an inherent end-to-end (E2E) latency trade-off. To address this, we propose a joint design that integrates communication-media selection with wireless resource allocation. Through analytical characterization and simulation-based evaluation, we show that neither token-based transmission nor key-value (KV) cache-based transmission is uniformly optimal across operating regimes, as performance depends critically on system parameters such as available computational resources and channel conditions. Accordingly, we formulate a joint optimization problem aimed at minimizing the E2E latency of multi-agent collaboration and develop a low-complexity joint media selection and resource allocation (JMSRA) algorithm. Numerical results further confirm that, by adaptively coordinating the interaction media and bandwidth allocation over heterogeneous links, the proposed scheme achieves markedly reduced E2E latency relative to conventional NL-only and KV-cache-only baselines, enabling efficient and robust multi-agent collaboration in future wireless networks.

Neural operators learn mappings from function-dependent inputs to solutions, providing an effective framework for solving partial differential equations (PDEs). For time-dependent PDEs, existing methods typically perform long-horizon prediction through autoregressive rollout directly in high-dimensional physical field spaces, where each predicted state is recursively fed back as the input for the next step. Although effective for short-term prediction, this autoregressive rollout and the lack of continuous-time modeling lead to progressive error accumulation over long-horizon rollouts. In this work, we propose Autoregression-Free Neural Operators (AFNO), which map the time evolution of PDEs into a latent space and model continuous-time vector fields within it. AFNO uses flow matching to learn the latent vector field, thereby enabling continuous evolution over extended horizons, avoiding autoregressive rollout and capturing dynamics under varying parameter configurations through explicit conditioning on physical parameters. Theoretical analysis and extensive experiments on six PDEs demonstrate that AFNO improves long-horizon prediction stability and consistently reduces rollout errors compared with the baselines.

We study tautness properties of a Riemannian foliation by investigating a symmetric 2-tensor associated with the mean curvature of the foliation. As a consequence, we prove a tautness condition for Riemannian foliations on compact manifolds via the symmetric tautness tensor. Moreover, several applications of the aforementioned tensor are provided under various geometric conditions.

We introduce flag positroid pipe dreams (FPPs), whose role in the study of complete flag positroids is analogous to the role of Le-diagrams in the study of positroids. We develop the combinatorics of these diagrams and highlight some of their properties. FPPs are in bijection with intervals in the Bruhat order of the symmetric group, and the number of elbows in an FPP is the dimension of the corresponding Richardson cell in the decomposition of the nonnegative flag variety. We show how complete flag positroids can be built rank by rank via FPPs, and how the Le-diagrams of the positroid constituents of the flag can be obtained from the FPP via a simple standardization operation. Using partial FPPs, we give an alternative proof of a conjecture of Benedetti, Chavez, and Tamayo on the problem of characterizing elementary positroid quotients via cyclic shifts of decorated permutations, in the nonnegatively representable case. Our proof partially addresses a problem of Chen et al. regarding an explicit characterization of the cyclic shift operators purely in terms of decorated permutations. We show that the poset of nonnegatively representable elementary positroid quotients is self-dual. The maximal chains of this poset are in bijection with FPPs.

We extend Pólya's indicator diagram theory to encompass entire functions of order at most 1, allowing functions of maximal type. To do so, we introduce an extension of the complex plane in which indicator diagrams may be unbounded or even purely infinite. In this framework, we establish the Laplace and inverse Laplace transforms, and prove an analog of Pólya's theorem. As an application, we extend a characterization of the analytic continuability of power series. The resulting theory is illustrated by a range of classical examples of special functions.

The constrained $\ell_p^p/\ell_q^p$ ratio model is scale invariant and is therefore attractive for sparse signal recovery. However, its nonconvex, nonsmooth, and fractional structure makes a unified theoretical and algorithmic analysis challenging for $0<p\le1$ and $q>1$. This paper develops a unified framework for this general model, covering deterministic exact recovery, stable recovery for sparse and compressible signals, and convergence analysis of a fractional algorithm. We first establish two deterministic sufficient conditions for exact recovery: a local optimality criterion and a null-space condition ensuring uniform recovery. For the $\ell_1/\ell_q$ subfamily, this null-space condition is further converted into high-probability sample-complexity bounds for isotropic sub-Gaussian matrix. We then study noisy recovery. Under the $k$-sparsity assumption, we improve the RIP-based stable recovery theory by relaxing the required sufficient condition and deriving sharper reconstruction-error bounds. For compressible signals, we establish RIP–ROP based error estimates whose constants are independent of the ambient dimension, improving prior bounds with explicit dimension-dependent factors [1]. An RIP-only variant is also derived. On the algorithmic side, we propose a prox-linear Dinkelbach framework that directly handles the fractional structure of the constrained problem and prove its convergence. Numerical experiments demonstrate that suitable choices of $(p,q)$ are effective for high-dynamic-range sparse signals and coherent sensing matrices.

Lookahead-based acceleration methods, such as Nesterov's momentum, are widely used in optimization, but they often become unreliable in deep learning training mainly due to stochastic gradient noise and non-convex loss landscapes. In particular, standard lookahead relies on short-horizon update signals (e.g., differences between consecutive iterates), which are inherently noisy and can lead to unstable extrapolation directions. This work revisits Nesterov's acceleration from a trajectory perspective and argues that effective acceleration in deep learning should harness the low-frequency trends of optimization trajectories rather than extrapolating noisy one-step updates. Leveraging this insight, we propose EMA-Nesterov, a simple modification that replaces the standard Nesterov's lookahead direction with an exponential moving average (EMA) of parameter updates. This yields a stabilized lookahead direction that captures and harnesses the evolving trend of the training trajectory through a low-pass filter, while remaining adaptive to progressive changes via the geometric weighting structure of EMA. We show that EMA-Nesterov retains a theoretical accelerated convergence rate in convex problems that is analogous to Nesterov's accelerated gradient method. Furthermore, we provide empirical evidence on language model pre-training to verify that EMA-Nesterov is broadly applicable across a range of fine-tuned base optimizers, including Adam, SOAP, Muon, as well as complex optimizers that achieve state-of-the-art performance on optimization benchmarks (NanoGPT). Compared to prior lookahead methods, EMA-Nesterov achieves better performance by avoiding the instability of short-horizon lookahead and the non-adaptivity of long-horizon lookahead.

We observe that solving the Dirac equation for confined potentials using the generalized pseudospectral (GPS) method leads to deteriorating convergence of energy eigenvalues and highly oscillatory in wave functions as the confinement radius decreases. It is found that this issue stems from the first-order differentiation formulation employed in GPS method. Motivated by this insight, we adopt the kinetically balanced generalized pseudospectral method, which incorporates the kinetically-balanced condition into the GPS method. Numerical results demonstrate that the mono-kinetically-balanced generalized pseu dospectral (MKB-GPS) method yields converged energy eigenvalues and generates smooth, continuous wave functions. This is the first application of the MKB-GPS method to confined potentials, and its effectiveness is validated for small confinement radii.

We consider graph diffusion processes constructed from finite i.i.d. samples drawn from an unknown manifold embedded in ambient Euclidean space, where the graph affinity is defined by an ambient Gaussian kernel matrix. We show that the manifold heat semigroup $Q_t = e^{t\Delta}$ can be approximated directly by iterating the graph transition matrix $P$, under only low regularity assumptions on the test function $f$, including the case $f \in L^\infty$. We bound $\| P^n f - Q_t f \|$ in $\infty$-norm, with the operator application to $f$ properly defined, and we recover the classical graph-Laplacian pointwise rate $O(N^{-2/(d+6)})$ up to logarithmic factors, for diffusion times $t $ up to $O(1)$ and longer. The rate holds for in-sample error as well as out-of-sample generalization, where the estimator of $Q_t f$ at a new point is defined via kernel convolution. To handle non-uniform sampling densities on the manifold, we introduce a right-normalization of the graph transition matrix; under the assumption that the sampling density $p$ is $C^3$ and bounded away from zero, the same convergence rates hold. We numerically demonstrate the performance of the proposed estimator on simulated data.

We present an epidemiological model for vector-borne diseases that includes within-host viral load and antibody dynamics using structured transport equations. By incorporating the internal dynamics into the infected and recovered host compartments, the formulation introduces nonlinearities and nonlocalities. We establish analytical properties, including well-posedness and mass conservation, and characterize its characteristic curves. Furthermore, we derive a simplified Uniform Host Response (UHR) model featuring delay-type terms. For both the full and UHR frameworks, the basic reproduction number is determined and shown to serve as a threshold for the existence of an endemic equilibrium, and is related to the linear stability of the disease-free state. Finally, numerical experiments, parameterized specifically for Dengue fever, demonstrate how within-host mechanisms influence population-level epidemiological outcomes.

Motivated by a discrete inequality problem proposed by Duanyang Zhang as Problem 6 of the 2022 Spring NSMO, we prove a median version of Hardy's inequality. For a nonnegative function $f\in L^p(0,\infty)$, $p>1$, let $A(t)$ be the average of $f$ over $(0,t)$, and let $M(t)$ be the lower median of $f$ over $(0,t)$. We show that \[ \int_0^\infty |M(t)-A(t)|^p\,dt \leq 2^{1-p}\left(\frac p{p-1}\right)^p \int_0^\infty f(t)^p\,dt, \] and that the constant is best possible. The proof is based on a pointwise rearrangement estimate coming from the half-measure property of the median, followed by the classical Hardy inequality. A discrete form and its sharpness are also included.

We propose a quasi maximum likelihood estimation method for Bergomi-type stochastic volatility models with parametrized kernels, focusing on the estimation of the kernel parameters from high-frequency time-series observations of option prices. We first show that the cumulative forward variance, which can be reconstructed from option prices, solves an infinite-dimensional stochastic differential equation driven by a one-dimensional Brownian motion under the Bergomi-type model. To overcome this degeneracy, we introduce a nondegenerate proxy likelihood based on the Euler-Maruyama approximation and define an estimator through the associated estimating function. We establish consistency and asymptotic mixed normality of the proposed estimator under a regular class of kernels. Simulation studies and an empirical application to SPXW option data illustrate the finite-sample performance of the method and the practical relevance of the approach.

Neural network (NN) dynamics models and control policies achieve strong performance in robotics, but providing sound guarantees under uncertainty remains difficult, especially for closed-loop NN systems. Existing reachability tools provide formal over-approximations, yet are often non-differentiable, overly conservative, or too slow for modern learning and online planning pipelines. To address this, we present a parallelizable, differentiable reachability framework in JAX for continuous- and discrete-time systems with analytical and NN-based dynamics and controllers. Our framework combines Taylor-model flowpipe construction with CROWN-style linear bound propagation through a unified representation that preserves affine dependencies while supporting GPU-batched computation and automatic differentiation. Building on this reachability primitive, we develop (i) a certified training method that encourages reachability-friendly dynamics models and controllers, and (ii) a reachability-aware sampling-based MPC scheme with gradient-based refinement. Experiments on non-prehensile manipulation and quadrotor tasks, including hardware and higher-dimensional evaluations (up to 72D), demonstrate practical online planning while maintaining certified reachable-set over-approximations under bounded uncertainty.

We study the energy-critical inhomogeneous Hartree equation in space dimensions three and higher. Previous local well-posedness results left open the parameter regime where the inhomogeneity exponent is small and the Riesz potential exponent is either small or large. We establish local well-posedness for a new range of parameters, thereby substantially filling the remaining open parameter regime. In particular, our result completely resolves the remaining gap in dimensions $5$ and $6$.

In this article, we prove the strong monodromy conjecture for complex hyperplane arrangements by proving a conjecture of Budur, Musta\c t\u a and Teitler that $-n/d$ is a root of the $b$-function of an irreducible essential and central hyperplane arrangement $f$ of degree $d$ on $\mathbb C^n$.

We prove the soliton resolution conjecture for the Benjamin-Ono equation in the sharp well-posedness space $H^s\left(\mathbb{R}\right)$ for $s>-\frac12$. This extends soliton resolution to the sharp regularity, whereas previous results required weighted $H^{1,1}\left(\mathbb{R}\right)$ initial data. We first prove the soliton resolution conjecture in the sharp well-posedness space for the integrable equation. In particular, our work solves three important open problems, including the problem of embedded singular continuous spectrum, the extension of Gérard's explicit formula, and the inverse spectral problem for the Lax operator. A central ingredient of our work is a new modular operator-theoretic framework combining the closed quadratic-form theory of rough Lax operators, weighted limiting absorption, and distorted Fourier analysis, which yields a unified treatment of distributional spectral theory, continuous-spectrum scattering, and radiative asymptotics at the sharp well-posedness threshold. Another important ingredient is a Fourier-side regularity and stability result for the eigenfunctions of the rough Lax operator, established through a zero-frequency identity, resolvent commutator estimates, and an isolating-contour argument. This yields well-defined soliton centers that are stable under rough approximation. Our framework provides a broadly applicable framework for proving soliton resolution conjecture for integrable dispersive equations in low-regularity spaces.

The ferromagnetic Ising model on an $n\times n$ square lattice region $\Lambda$ with mixed boundary conditions can exhibit a phase transition as temperature varies. For this spin system, if we fix the spins on the top and bottom sides of the square to be $+$ and the left and right sides to be~$-$, a standard Peierls argument based on energy shows that below some critical temperature~$t_c$, any local Markov chain $\mathcal{M}$ requires time exponential in $n$ to mix. Spin glasses are magnetic alloys that generalize the Ising model by specifying the strength of nearest neighbor interactions on the lattice, including whether they are ferromagnetic or antiferromagnetic. Whenever a face of the lattice is bounded by an odd number of edges with ferromagnetic interactions, the face is considered frustrated because the local competing objectives cannot be simultaneously satisfied. We consider spin glasses with exactly four well-separated frustrated faces that are symmetric around the center of the lattice region under $90$ degree rotations. We show that local Markov chains require exponential time for all spin glasses in this class. This argument extends to the ferromagnetic Ising model with mixed boundary conditions described above, which behaves like spin glasses with frustrated faces on the boundary. The standard Peierls argument breaks down when the frustrated faces are on the interior of $\Lambda$ and yields weaker results when they are on the boundary of $\Lambda$ but not near the corners. We show that there is a universal temperature $T$ below which $\mathcal{M}$ will be slow for all spin glasses with four well-separated frustrated faces. Our argument shows that there is an exponentially small cut indicated by the free energy, carefully exploiting both entropy and energy to establish a small bottleneck in the state space to establish slow mixing.

We study length-filtered BFACF graphs of self-avoiding polygons in the simple cubic lattice, with merge heights recording the least length cap needed to connect configurations. We determine the complete minimal-layer merge trees for $4_1$ (four components at length 30 merging at 32) and $6_3$ (twelve components at length 40, two at 42, and one at 44). For one trefoil chirality, we certify six selected nonminimal components born at length 26 and merging with the minimal branch by length 28 or 30; these do not exhaust the full filtration. We characterize eventual connectivity of capped graphs by bounded birth levels of strong local-minimum components. A 64-edge Hamiltonian lattice unknot with no nonincreasing BFACF moves has exact merge height 66 with the elementary square, verified by an explicit move certificate. Repeated two-edge splicing produces frozen unknots of length $64m$ for every $m\ge1$. We also construct an 88-edge frozen trefoil by splicing the 64-edge unknot with a minimal 24-edge trefoil; iteration and reflection yield frozen trefoils of both chiralities with $64m+24$ edges. Hence the unknot and both trefoil chiralities have infinitely many nonminimal strong local-minimum births, and their capped BFACF graphs are not eventually connected. These results concern standard discrete BFACF moves, not continuous ropelength barriers; exact excess heights for the longer families and extensions to arbitrary knot types remain open.

Power flow feasibility assessment is computationally challenging for unbalanced three-phase distribution networks. This paper develops a vectorized semidefinite program (SDP) based on the bus injection model (BIM) and reformulates its dual as an exact-penalty problem, enabling us to develop a scalable three-cut proximal bundle method for feasibility assessment. The proposed bundle method is numerically over 400 times faster than MOSEK with less than 1/2000 of its memory; on the decomposed BIM-SDP, approximately 2 times faster with 75% less memory.

This paper investigates the performance of Newton's method, iterative Linear Quadratic Regulator (iLQR), and Differential Dynamic Programming (DDP) in solving discrete-time optimal control problems. We offer a unified perspective on these approaches, centered on the understanding that each method ultimately solves a sequence of quadratic programs. Building upon previous comparative works, this paper contributes additional mathematical explanations and results to the analysis. In particular, it is shown that iLQR is a principled Sequential Quadratic Programming (SQP) approach, rather than merely an approximation of DDP that neglects Hessian terms. This characteristic guarantees that iLQR will always produce a cost-descent direction and converge to an optimum, under some mild assumptions. In contrast, Newton's method and DDP lack these guarantees, especially when initialized far from an optimum. A series of numerical examples are presented to corroborate the mathematical reasoning and analysis developed in the paper.

In this work, we develop a new theory of multivariate V-filtration on D-modules along a simple normal crossing divisor and relate it with Sabbah's multi-filtration. We establish several new structural results and relate them with the Hodge filtration on free-monodromic local systems from geometric representation theory. As applications, we prove the Strong Monodromy Conjecture and its multivariate generalisation for hyperplane arrangements. Along the way, we confirm both the n/d-conjecture of Budur–Mustaţă–Teitler and its multivariate form due to Budur.

We add twists to a band of a genus-one Seifert surface, producing a knot $K(\ell,0)$. We prove $K$ and $K(\ell,0)$ have $S$-equivalent Seifert matrices if and only if the $(2,2)$-entry of the Seifert matrix vanishes and the sum of off-diagonal entries divides $\ell$. The necessity follows from the Alexander polynomial and a norm argument proving triviality of the $S$-equivalence subgroup $\mathcal{S}^+$ in the class group of binary quadratic forms (Aka–Feller–Miller–Wieser); sufficiency is an explicit $\Lambda_1$-operation. The Jones polynomial distinguishes the knots when $V(K)\neq1$, yielding infinite families of $S$-equivalent but inequivalent genus-one knots, illustrated by $9_{46}$. Also in this paper, we provide a partial answer for Problem~1.6 in Kirby's problem list (K3) and Problem~7.7 of Aka–Feller–Miller–Wieser.

This paper characterizes lattice-ordered algebras with the order-to-ring ideal property, i.e., those in which every order ideal is a ring ideal. Our main result shows that an Archimedean lattice-ordered algebra has this property if and only if it is an f-algebra admitting a nil-faithful submultiplicative lattice seminorm, where nil-faithful means that elements with zero seminorm are nilpotent. As a consequence, we prove that an Archimedean semiprime f-algebra has the order-to-ring ideal property if and only if it admits a submultiplicative lattice norm. These results correct and extend earlier incorrect work in the literature and yield a necessary and sufficient condition for all orthomorphisms on an Archimedean vector lattice to be central.

The concept of weakly regular residuated mappings was investigated for lattices recently by S. Radeleczki and L. Veres. We modify this concept for posets. We define so-called operator-residuated mappings in posets, show their important properties and point out differences between them and residuated mappings defined in the usual way. We modify the concept of a lattice homomorphism for posets where suprema need not exist and show its relation to residuated mappings and induced ideals. Finally, we present several examples of weakly regular residuated mappings in posets and show how such mappings may be constructed.

We introduce a user defined probabilistic polygonal representation for plane curves. Given a curve, we select vertices on the curve and connect consecutive vertices by line segments to obtain a polygonal approximation. Each segment is equipped with a user defined uncertainty parameter in the normal direction. This yields a collection of thin probabilistic geometric primitives that retain the geometrz of the underlying curve while extending it beyond the idealized deterministic one dimensional formulation. For each segment, we define a Random Variable that is uniform distributed in the tangent direction of the segment and Gaussian distributed in the normal direction of the segment. By matching the first and the second central moments, this construction induces a Gaussian component whose mean lies at the segment midpoint and whose covariance encodes both tangential and normal uncertainty. Combining the segment wise components with appropriate weights yields a Gaussian Mixture Model (GMM) representation of the user defined probabilistic polygonal representation of the plane curve. The proposed framework provides an analytically tractable probabilistic model that preserves local geometry, and uncertainty in the normal direction. It applies to smooth, closed, open, non regular, and self intersecting plane curves, allows adaptive discretization and varying uncertainty in the normal direction, and as a result supports uncertainty aware geometric modeling. Experiments on a collection of canonical plane curves show that the resulting GMM capture local tangent, local normal, and local arc length; resulting in the global shape of the underlying curves to be truthfully captured as well. The representation is particularly relevant for applications in uncertainty aware CAD and digital twins, probabilistic obstacle modeling in robotics, and probabilistic trajectory planning.

We propose Lie group embedded dynamical neural networks (LieEDNN) and the corresponding learning algorithms based on gradient descent and metric projection on smooth manifold, where we treat Lie group as an intrinsic representation for continuous symmetry of manifold geometry. Thereby we achieve learnable and stable dynamics on the underlying manifold for general Lie group, and we are able to utilize the powerful representation capability of Lie group such as SO(3) and SE(3) to solve real world engineering problems in areas such as robotics, graphics, and control. Two core challenges are: (i) General Lie groups are incompatible with addition arithmetic, which is necessary for neural network interactions. (ii) The dynamics evolve in the nonlinear representation space of special algebra rather than the normal Euclidean space, which violates the paradigm of common neural ODEs. To address these two challenges, we firstly introduce adjoint Lie group action on the Lie algebra, which induces a linear mapping and transfer to the block-wise structure of weight matrices, such that addition could operate on the Lie algebra as a vector space. Then we parameterize the Lie algebra and the adjoint action as linear transformation so that the architecture is aligned with neural network perceptrons. Explicitly, this embedding appears as block-wise manifold constraints on weights, and we develop algorithms to learn the equilibrium with stability guarantees of the temporal neural network dynamics. Experiments are implemented on a specific Lie group SE(3), with the application scenario of telescopic manipulators.

Competitive resource allocation problems over frequency and space can be formulated as minimax interaction between transmit power and worst-case interference. This formulation naturally arises in multi-operator low Earth orbit (LEO) satellite spectrum sharing, where transmissions from competing constellations interfere in real-time. Under Gaussian channels, the corresponding power-allocation problem admits a convex-concave formulation with a unique saddle point. Discrete constellations yield generally nonconvex mercury/water-filling formulations. In this paper we propose the adversarial water-filling (AWF) problem with corresponding theory and algorithms for these settings. In addition, we develop a domain-specific wireless foundation model for AWF to learn the AWF search dynamics. The architecture incorporates permutation-invariant channel representations, a constraint-aware graph neural network (GNN) with sparse message passing, and global latent variables capturing the low-dimensional water level implied by the AWF optimality. Through learned projected extragradient iterations, the model approximates stationary solutions of the constrained minimax problem arising under mercury/water-filling. We further establish projected-stationarity/Karush-Kuhn-Tucker consistency and conditional local convergence of the learned AWF dynamics under local regularity and stability conditions. Experiments demonstrate empirical generalization across unseen problem sizes, constraint structures, discrete constellations, and channel-weighted objectives, while achieving a median speedup exceeding one order of magnitude over Mirror-Prox on matched instances at comparable first-order solution quality. The related code can be found at https://github.com/convexsoft/AWF.

In what follows we generalize the notion of a complemented ring to rings that are not necessarily reduced. We then determine how our concepts fit in with other well-known classes of rings.

Convolutional neural receivers such as DeepRx outperform minimum mean-square error physical uplink shared channel detection on in distribution channel and waveform configurations, but their behavior under calibration drift when transmitter or channel parameters depart from the training envelope is poorly characterized.