Condensable algebras in 2+1d non-chiral topological orders characterize gapped boundary conditions and interfaces. Applied to the Symmetry Topological Field Theory, they allow classification of symmetric gapped phases and impose sharp constraints on possible phase transitions. A condensable algebra is specified not only by its underlying set of anyons, which end on the boundary or interface, but also by its algebra structure. We introduce the concept of twin condensable algebras, which have the same anyon decomposition, but inequivalent algebra structure. We revisit the classification of condensable algebras in $\mathcal{Z}(\text{Vec}_G^ω)$, i.e. in group-theoretical topological orders for finite groups $G$ with anomaly $ω$. In this context we are able to identify twin algebras that arise from different mechanisms, such as subgroup data, SPT cocycles, and symmetry actions. In particular, we construct infinite families of examples of twins from so-called Gassmann triples, and exhibit cases in which the reduced topological orders are inequivalent despite having identical anyon content. Physically, twin algebras describe distinct symmetric phases that have isomorphic spaces of ground states, but inequivalent order parameters. Such twin phases never exhibit relative spontaneous symmetry breaking, and can be used to construct phase transitions without hidden symmetry breaking, which are intrinsically beyond Landau transitions.

We introduce the concept of twin phases for a symmetry $\mathcal{S}$, defined as inequivalent phases, whose order parameters are part of the same generalized charge under $\mathcal{S}$. Stable, direct transitions between such twin phases are never spontaneous-symmetry-breaking transitions, even after (partially) gauging the initial symmetry $\mathcal{S}$: they are phase transitions without hidden symmetry breaking. We illustrate this with an (anomalous) finite group symmetry in 1+1d, which exhibits such intrinsically beyond Landau transitions.

Empirical Bayes analyses routinely model noisy measurements of latent parameters as normal, justifying this by an informal appeal to the central limit theorem (CLT). This paper puts this heuristic appeal on firmer analytical grounds. We show that the denoising regret of the nonparametric maximum likelihood estimator (NPMLE) and related sieve methods is controlled by the rate attained under exact normality, plus a term reflecting the quality of the CLT approximation. The CLT need only hold marginally for each coordinate, and moreover only on average, without needing high-dimensional normal approximations. We identify two asymptotic regimes in which the normal approximation is adequate and the empirical Bayesian prior remains informative, and we show that our guarantees are robust to dependence and to variance estimation.

Communication costs are a major bottleneck in distributed learning and first-order optimization. A common approach to alleviate this issue is to compress the gradient information exchanged between agents. However, such compression typically degrades the convergence guarantees of gradient-based methods. Error feedback mechanisms provide a simple and computationally cheap remedy for this issue, but numerous variants have been proposed, and their relative performance remains poorly understood. This paper provides tight convergence analyses for two of the main error-feedback algorithms from the literature, the classic Error Feedback method (EF) and Error Feedback 21 (EF21), by identifying optimal step-size choices and constructing optimal Lyapunov functions tailored to each method. The results hold independently of the number of agents and recover the known best guarantees possible in the single-agent regime.

The crucial feature of resurgence theory is the ambiguity of non-perturbative behavior, reflected either in the different choices of integration contours or in the existence of several solutions to Ward identities. This is well illustrated by considering exactly solvable models, of which the prominent example is Chern-Simons theory. Its important chapter, which should have a direct generalization to arbitrary Yang-Mills, is the consideration of Wilson averages in the double-scaling limit of large representation and small coupling. For historical reasons, we call it a Kashaev limit. It possesses a natural interpretation in terms of quasiclassical/WKB approximation, which is, however, somewhat peculiar and thus sheds new light on the old story. The crucial point is the appearance of Alexander polynomials $Δ$ in two seemingly opposite roles: the classical $A$-polynomials have common roots with $Δ$, while Jones polynomials tend to $Δ^{-1}$ in the perturbative expansion. The consistency is provided by the peculiar form of the quantum $A$-polynomial, and the resolution of the puzzle is the co-existence of two different branches (phases) in the quasiclassical limit -- with non-trivial and with vanishing classical actions. The first leads to classical $A$-polynomials and hyperbolic volumes, the second -- to inverse Alexanders.

We consider axisymmetric no-swirl solutions to the three-dimensional incompressible Euler equations, with initial velocity in $C^{1,α}\cap L^2$, where $α\in\left[\frac{1}{3},1\right)$. Motivated by Shkoller's Lagrangian clock-and-driver framework for finite-time blow-up below the $C^{1,\frac{1}{3}}$ threshold, we introduce coherent Lagrangian classes of initial data and conditional solutions for which the same clock mechanism yields a supercritical/critical barrier when $α\geq\frac{1}{3}$, with a genuinely depleted barrier for $α>\frac{1}{3}$ and an exponential bound at the critical endpoint $α=\frac{1}{3}$. In the general case, we formulate a matrix-clock criterion in terms of the smallest singular value of the deformation gradient and show that, under cusp-tail, Dini coherence, near-field compatibility, and bounded transverse-distortion hypotheses, this singular value cannot collapse in finite time. In the on-axis case, the criterion reduces to the scalar clock inequality $\displaystyle \dot{J}(t)\gtrsim -B(t)J(t)-CJ(t)^{3α}$, which rules out Shkoller-type clock collapse for $α\geq\frac{1}{3}$. These results do not enlarge the known Lorentz-space global regularity classes. Rather, they in particular identify the supercritical Lagrangian obstruction dual to Shkoller's subcritical blow-up mechanism in the case $α>\frac{1}{3}$.

We study the global evolution problem for the Einstein equations under T2 symmetry on T3, allowing vacuum, scalar-field, and compressible-fluid matter models, governed by a general equation of state including isothermal and polytropic fluids. Under this symmetry, we obtain the first non-perturbative, global existence and stability theory with connection coefficients being merely square-integrable, which allows both impulsive gravitational waves and shock waves. In areal gauge, we introduce new fluid and geometric variables and reformulate the Einstein-Euler system as a first-order system of nonlinear balance laws with constraints and an entropy structure. The resulting formulation exhibits hyperbolicity, null forms, entropy currents, div-curl structure, maximum principles, and spacetime estimates. This leads to a notion of tame Einstein-Euler flow for which the essential geometric and fluid variables are square-integrable (finite energy), and the secondary variables are absolutely continuous (or, more generally, of bounded variation). In this non-perturbative and weak regularity setting, the equations remain meaningful even when the Weyl curvature concentrates into Dirac masses along timelike hypersurfaces, and the Ricci curvature remains only integrable. Our main results are a global existence theorem for areal foliations, a nonlinear stability theorem for well-prepared initial data, and a nonlinear instability theorem for geometrically oscillatory data, the latter producing measure corrections to the stress energy tensor. In the future-contracting regime, the areal foliation reaches a geometric singularity where the volume of T3 spatial slices degenerates to zero. The areal function reaches zero generically in the non-vacuum Gowdy-symmetric and vacuum torus-symmetric cases. In the future-expanding regime, the areal foliation is complete.

This paper develops a functional theory for multi-target detection, where a compactly supported signal is recovered from a single noisy observation containing many unknown translations of the signal. Our formulation allows continuous, off-grid translations and correlated stationary Gaussian process noise, extending beyond the discrete, grid-aligned, white-noise models common in prior work. We analyze two uninitialized recovery algorithms based on autocorrelation analysis; in particular, both algorithms first estimate the signal's bispectrum via a debiased third-order empirical autocorrelation. The signal is then recovered from the estimated bispectrum using either a functional frequency marching scheme or a Kotlarski-type deconvolution formula. For both algorithms, we prove non-asymptotic recovery guarantees for compactly supported signals without bandlimiting assumptions. The resulting error bounds depend on the smoothness of the signal and the accuracy of bispectrum estimation, with the latter governed by the noise characteristics and the number of signal occurrences. Numerical experiments validate our theory and demonstrate accurate recovery in low-SNR regimes.

In this article, a one-off option for temporary service rate control for the M/M/1 queue is considered. After taking this option, during a single exponentially distributed period, two service rates are available for use. Once service rate control is lost, the system continues with a fixed service rate $μ$. The objective is to minimise the sum of holding costs and service costs. We approximate the expected total saved cost by taking the one-off option, depending on the starting state or starting distribution. Using the Value Iteration algorithm with $M$-uniform geometric recurrence, we present methods to approximate the expected total saved future cost, as well as the expected total saved discounted future cost. Furthermore, we obtain theoretical results on the structure of optimal policies and strong Blackwell optimality. The paper is concluded by numerically applying the methods to various instances of the model.

Our purpose is to revisit the screw theory in light of the affine tensor formalism, introducing the co-momentum and momentum tensors. Our target-applications of the Euler-Poincaré equation are problems of mechanics such as the motion of the rigid body or the statics and the dynamics of Cosserat arches, in relation to the concept of momentum tensor. Using the framework of Ehresmann connections on the principal bundle of affine frames, we show that the Euler-Poincaré equation means that the momentum tensor is parallel-transported.

We consider the Boolean model of continuum percolation, where points are placed in $\mathbb{R}^d$ by a Poisson point process and pairs of points with distance at most 1 are connected by an edge. The time constant is the limiting ratio of the chemical distance (i.e. graph distance) to the Euclidean distance for pairs of distant connected points. Yao, Chen, and Guo established the existence of a time constant in the supercritical regime. We show that above the critical intensity, the time constant is a Lipschitz continuous function of the intensity. The proof adapts a recent argument of Can, Nakajima, and Nguyen to the continuous setting.

We introduce a stochastic interacting particle system in separable Hilbert spaces together with its associated mean-field formulation. The model is shown to retain the characteristic consensus-driven structure of classical Consensus-Based Optimization, while accounting for the analytical challenges of infinite-dimensional dynamics. We establish well-posedness of the proposed dynamics and analyze the associated consensus mechanism. Furthermore, we derive convergence guarantees under suitable assumptions on the objective functional, showing concentration of the dynamics toward the minimizer in the long-time regime. This extends the applicability of the method to a broad class of infinite-dimensional optimization problems. In addition, we study the corresponding finite-particle system relevant for numerical implementation and propose a practical algorithm.

We present a general framework for the high-precision numerical evaluation of multivariate hypergeometric functions defined as solutions of holonomic systems of partial differential equations. Our approach adapts and extends methods originally developed for multi-loop Feynman integrals to the setting of hypergeometric functions of many variables. In particular, we construct Pfaffian systems for arbitrary multivariate hypergeometric functions by applying the Laporta reduction algorithm to suitable systems of differential relations. Next, we construct a numerical scheme based on the Frobenius method, which allows us to compute local power-series solutions with controlled precision and to transport them along prescribed paths in the space of variables. A central part of the paper is devoted to a systematic analysis of multivaluedness and branch structure: we show how the Frobenius method can be used to access different Riemann sheets in a controlled way and to track changes of the solution under analytic continuation around singular loci.

GenEO (`Generalised Eigenvalue problems on the Overlap') is a method for constructing coarse spaces used in the preconditioning of iterative solvers for discrete PDEs. This method combines a (small) number of modes of local PDE eigenproblems to obtain a global coarse space. A coarse solve is then combined with local solves of the global PDE to obtain the preconditioner. A substantial theory for GenEO has been developed for the case when the local elgenproblems are positive semi-definite. This has been applied mostly to positive definite global PDEs, but also recently extended to the case of convection--diffusion--reaction problems, which may be neither self-adjoint, nor positive definite. However, when the global problem is highly indefinite, coarse spaces built from positive semi-definite local eigenproblems fail to be robust in practice. In this paper we consider highly indefinite global PDE problems, characterised by a large parameter $k$ (allowing also highly variable coefficients), and we develop a new spectral coarse space built from solving eigenvalue problems based on \textit{local copies of the global problem}. We put no constraint on the diameters of the local domains, thus allowing the local eigenvalue problems to be indefinite. The new method (which we call $H_k$-GenEO) is seen to be much more robust as $k$ increases than methods based on positive semi-definite eigenproblems. We provide sufficient conditions for robustness of the preconditioned GMRES iterative method, in terms of the tolerance of the local eigenproblems and the size of the subdomains for the local PDE solves. In practice the method is observed to be robust with respect to $k$ under even weaker conditions on the local eigenproblem tolerance. The experiments also suggest the method can be resilient to high variation in PDE coefficients.

Microwave linear analog computers (MiLACs) have recently emerged to enable high-performance and efficient beamforming in the analog domain. In this paper, we introduce a dual-functionality framework for MiLAC-aided transceivers. Beyond analog-domain precoding/combining (active beamforming), a MiLAC and its antenna array can simultaneously act as a reconfigurable intelligent surface (RIS) (passive beamforming). This allows the MiLAC to execute beamforming for transmission/reception while reflecting external incident signals. We provide an optimal reconfiguration strategy for this dual-functional MiLAC, and characterize the fundamental limits on the trade-off between active and passive rate, namely the capacity region bounds and the sum-rate capacity.

Dynamical systems reconstruction (DSR) aims to learn surrogate models that capture the dynamics underlying time-series data. Reliably deploying these surrogates requires uncertainty estimates consistent with the learned dynamics. We expose a dynamic-probabilistic consistency (DPC) gap: the pursuit of finite-horizon probabilistic objectives can degrade dynamics or decouple predictive uncertainty from the local tangent dynamics it ought to reflect. We isolate three mechanisms behind this gap: core collapse, noise masking, and blind uncertainty. Specifically, we show that open-loop Gaussian rollout objectives can penalize Jacobian-generated covariance growth in chaotic systems, encouraging optimization shortcuts that weaken physical expansion or decouple uncertainty from it. To mitigate this gap, we propose KAFFEE (Kalman-Aware Framework For Ergodic Emulation), a differentiable extended Kalman filter-based training framework that evaluates likelihood on local predictive residuals (innovations) while transporting covariance through learned local Jacobians. On stochastic hyperchaotic Lorenz-96, KAFFEE reduces the identified failure modes, improves reconstruction of dynamical invariants relative to open-loop objectives, and maintains competitive predictive scores. We further show that the DPC gap appears when probabilistically adapting a DSR foundation model across 13 chaotic systems, where KAFFEE enables in-context Bayesian filtering while largely preserving zero-shot dynamics.

We introduce weak and strong poset Ramsey-Turán numbers for $t$-chains in host poset families, focusing on the Boolean lattice family $\mathcal{B}=\{B_n:n\ge 1\}$. For any poset $P$, we show $\operatorname{RT}(\mathcal{B};n,P,l,t)\le \operatorname{RT}^{\sharp}(\mathcal{B};n,P,l,t)$, with equality when $P$ is a chain. In particular, for $t=1$, $\operatorname{RT}(\mathcal{B};n,C_k,l)=\operatorname{RT}^{\sharp}(\mathcal{B};n,C_k,l)=(k-1)(l-1)$. We also give universal upper bounds for both versions. For fixed $k,l,t$ with $\min\{l-1,k-1\}\ge 1$, we prove $\operatorname{RT}^{\sharp}(\mathcal{B};n,A_k,l,t)=Θ(n^t)$. More generally, for every non-chain poset $P$, the strong number is $Θ(n^t)$ for fixed $l,t$. Finally, if $h(P)=r>t$ and $l(n)=\lfloor M_n^β\rfloor$ with $0<β\le α<1$, then both weak and strong versions admit lower bounds of order $Ω\!\left(2^{βn}n^{-β/2}\right)$.

Orbital transfers in multi-body systems are often studied as isolated trajectory design problems, making it difficult to identify the larger transport structure connecting families of periodic orbits, including which families act as hubs, gateways, relays, or persistently difficult-to-access regions. This work introduces a reachable-set-based framework for constructing orbital networks in the circular restricted three-body problem. Finite-$ΔV$ and finite-time-of-flight reachable-set overlaps are used to infer accessibility relationships between representative periodic orbit families on a common Jacobi energy manifold and to assemble these relationships into a weighted orbital network. Applied to the Earth-Moon system, the resulting network reveals distinct accessibility regimes in which direct reachability, graph connectedness, and feasible multileg closure emerge separately. The analysis identifies multi-orbiter cycler orbits as the dominant hub, gateway, and relay families, with the (3,2)-cycler dominating across much of the sampled budget plane and the short-period (1,1)-cycler dominating in the low-time-of-flight regime, while the stable 2:1 resonant orbit remains persistently difficult to access. Although the maximum-budget network is nearly complete in a binary sense, its weighted accessibility remains strongly non-uniform. Selected proxy-supported connections are refined into concrete trajectories through differential correction, with corrected transfer costs remaining below the proxy estimates in all tested cases. Together, the results demonstrate how reachable-set overlap geometry can expose large-scale transport structure in nonlinear gravitational systems without requiring exhaustive pairwise trajectory optimization.

It has been conjectured that generic smooth solutions of the two-dimensional Euler equation exhibit linear growth of vorticity gradients. We prove an elementary arbitrary-background perturbation principle in the odd symmetric setting. More precisely, for any compactly supported nonnegative function in the half-plane, one can find an arbitrarily small smooth nonnegative perturbation whose associated solution undergoes linear-in-time filamentation in the quadrant. The main ingredients are the lower bound of the center of mass given by Iftimie-Sideris-Gamblin, and the velocity estimate for the sparse part to capture those slowly moving particles.

A hyperbolic structure on a group $G$ is a (not necessarily properly discontinuous) cobounded action of $G$ on a Gromov hyperbolic space, considered up to coarsely $G$-equivariant quasi-isometry. We show that for groups $G$ acting geometrically and positively on a horocyclic product $X\bowtie Y$, all hyperbolic structures on $G$ come from the two actions on the factors. The ingredients of the proof include Malcev rigidity and a new ("vertical") boundary of horocyclic products. We also give the first example of a group acting geometrically on a horocyclic product of two mixed millefeuille spaces.

The polar-angle marginal of a centred bivariate Gaussian distribution, obtained after integrating out the radial coordinate, gives the two-dimensional angular central Gaussian (ACG) distribution of Tyler. While its trigonometric and vector-valued moments have been studied in detail, to our knowledge there are no explicit closed-form expressions for the \emph{linear} moments $\mathbf{E}[θ]$ and $\mathbf{E}[θ^{2}]$ on the natural domain $θ\in\left]-π/2,π/2\right[$. Here \textit{linear} refers to the ordinary moments $\intθ^{k}f(θ)\,dθ$ of the angle regarded as a real-valued variable, in contrast to the circular (trigonometric) moments $\mathbf{E}[e^{ikθ}]$ customary in directional statistics. We provide such expressions: the mean is a simple arctangent of the parameters, while the second moment is given by the real part of a dilogarithm. The derivation, based on a contour integration around the branch cut of $\arctan z$, is elementary. These quantities naturally arise in physics, where $θ$ is interpreted as a real-valued phase rather than a circular variable.

Maritime weather routing must optimize route geometry under dynamic wind-wave conditions, obstacle constraints, and fixed-arrival requirements. We present Bézier Evolve and Refine Strategy (\name{}), a two-stage framework that combines global evolutionary search (CMA-ES) with local variational refinement (FMS). Routes are parametrized as Bézier curves and evaluated with dense along-path sampling, enabling smooth trajectories while preserving practical feasibility constraints and accounting for mid-segment effects. We evaluate \name{} on synthetic benchmarks designed to stress seven operational criteria: continuity, obstacle avoidance, dynamic adaptation, flexible objective design, constant-load feasibility, just-in-time arrival, and local optimality. Across these tests, \name{} matches or improves published baselines while maintaining robust convergence under challenging flow fields and land geometries. We then validate the method on real ocean data using hourly ERA5 forcing over 366 daily departures in 2024 for two trans-oceanic corridors (Atlantic and Pacific), with a physics-based model of an 88~m cargo vessel with optional rigid wingsails. In real-ocean experiments, route optimization alone reduces mean propulsive energy by 23--59\% versus great-circle baselines of the same propulsion mode. Combined with wind-assisted propulsion, total savings reach up to 75\%. These results show that \name{} provides a practical and scalable foundation for just-in-time, energy-efficient weather routing in maritime decarbonization workflows.

Satellite communications are envisioned as a key enabler for ubiquitous coverage in future 6G networks, yet the broadcast nature renders them vulnerable to eavesdropping, especially given the long-distance transmissions and associated high uncertainties. In this paper, we propose the physical layer security enhancement for multi-beam satellite communications with the assistance of an aerial reconfigurable intelligent surface (ARIS). Considering the high dynamics and uncertainties of channels, we characterize the channel distribution with moment-based ambiguity sets. Accordingly, a distributionally robust secrecy rate optimization is formulated through joint design of transmit and reflection beamforming. We then introduce a conditional value-at-risk-based reformulation to convert the probabilistic constraints into deterministic forms. An alternating optimization framework is subsequently employed to iteratively update the transmit and reflective beamforming vectors until convergence. Simulation results demonstrate that the proposed distributionally robust scheme significantly enhances secrecy performance, and maintains reliable performance across various channel error distributions.

This paper was triggered by recent results on the maximal `average distance between consecutive points' of uniformly distributed sequences (u.f.d.s.). Here we address a generalized version of this question, consider pairwise maximal/minimal/total distances in blocks/segments of $ d \geq 2 $ consecutive points of u.f.d.s., and derive sharp upper bounds for all three aggregations. Our main idea of proof consists in, firstly, adding degrees of freedom, secondly, translating the resulting problem to a solvable optimization problem over the compact family of $ d $-stochastic measures, and, thirdly, showing that the obtained bounds are also sharp bounds for the original problem.

In the finite-dimensional setting, it is known that STFT phase retrieval is always possible when the window's ambiguity function does not vanish. However, it is not known how many zeros are allowed in the ambiguity function for the window still to allow phase retrieval. In order to tackle this problem, we first consider a two-window approach where the second window equals the Fourier transform of the first window. This allows us to apply the uncertainty principle in order to obtain sufficient conditions for phase retrieval. Using the relation between STFT phase retrieval and ambiguity sampling, we can prove sufficient conditions for the single-window phase retrieval problem, showing that only approximately eight ninths of the entries of the window's ambiguity function (and only three quarters in prime dimensions) are required to be nonzero.

We propose the \emph{sketched landing method}, a randomized variant of the landing method for optimization under orthogonality constraints. Each landing step consists of the sum of a \emph{normal} component, which reduces infeasibility, and a \emph{tangent} component, which decreases the objective function. Our main contribution is the introduction of low-dimensional random \emph{sketch matrices} to reduce the computational cost of these directions. We consider both dense (Gaussian) and sparse (subsampling) sketch matrices, and show how they reduce the per-iteration cost while preserving convergence guarantees in expectation.

We study the Hausdorff dimension spectrum of asymptotic approximation rates of complex Diophantine approximation and that of the asymptotic average excursion time of cusp excursions on the Bianchi orbifold $\mathbb{H}^3/\operatorname{PSL}(2,\mathbb {Z}[i])$ via a unified approach using the Hurwitz map. In particular, we construct a conformal graph directed system (CGDS) for the Hurwitz map and show that the Lyapunov exponent of the Hurwitz CGDS simultaneously captures the asymptotic approximation rate and the the asymptotic average excursion time. Applying the multifractal analysis of Lyapunov exponents for this system, we obtain a formula and real-analyticity for the Hausdorff dimension spectrum functions.

We propose a new family of polynomial approximation kernels for approximating nonnegative polynomials on the hypercube $[-1,1]^n$. Our Kernels produce polynomial sums-of-squares of degree $r$, achieving an $O(\log^3 r/r^2)$ error in the $\ell_1$-norm of the coefficients. This improves on the known error bound $O(1/r)$ from the literature. As a corollary, we obtain an improved convergence rate for the Lasserre hierarchy for polynomial optimization on the hypercube, again improving a known rate by Baldi and Slot from $O(1/r)$ to $O(\log^3 r/r^2)$.

We study the space of conformally covariant $n$-point functions of spinning operators using methods from invariant theory, commutative algebra, and combinatorics. We show that the rational part of any such function can be expressed in terms of the basic building blocks introduced by Costa, Penedones, Poland, Rychkov, thereby providing a rigorous proof of a result that is widely used in the physics literature. We reformulate the problem of enumeration of $n$-point structures in terms of counting lattice points in fractional matching polytopes, and compute these counts using vector partition functions, Hilbert functions, and Kostka numbers. We show that all algebraic relations between the building blocks follow from Gram constraints and compute the number of algebraically independent building blocks. For three-point functions, we derive closed counting formulas for arbitrary integer spins, both with and without Bose symmetry, and discuss a necessary and sufficient condition for the partial conservation operator to lift to a differential operator written in terms of the building blocks. We provide code that generates a basis of three-point structures satisfying these constraints for given values of spins and scaling dimensions.

We show that every pair of 2-torsion elements in the Tate-Shafarevich group of an elliptic curve are visible in the same abelian surface. This was previously only known for a single 2-torsion element. Our result explains some of the original observations on visibility in the paper of Cremona and Mazur.