Papers

Recently, tensor-based physics-informed neural networks (T-PINNs) have received increasing attention. However, existing T-PINNs still face a fundamental challenge: they mainly rely on pre-specified low-rank tensor decompositions with manually tuned ranks, which limits their ability to capture the underlying structures of multivariate solution functions and hinders their practical deployment. To address this challenge, we propose a hierarchical rank-evolving (abbreviated as HRE) representation for multivariate functions, which endows us to faithfully capture the underlying structure of the targeted multivariate function accompanying with automatic rank determination. Concretely, in the hierarchical design of HRE representation, the target multivariate function is decomposed as a small-scale inner tensor with a set of univariate functions along each mode, where a customized tensor network decomposition can be readily deployed to capture the underlying structure of the small-scale inner tensor. In HRE representation, the crucial hyperparameters, ranks, can be adaptively revealed during the decomposition, freeing us from manual rank tuning and making HRE practically applicable to real-world problems. Besides, we build the HRE-PINNs correspondingly. Extensive numerical experiments, including high-dimensional static problems (Helmholtz equation and Poisson equation), nonlinear time-dependent problems (Klein-Gordon equation), and complex fluid-dynamics problems (flow mixing equation and Navier-Stokes equation), demonstrate that HRE-PINNs consistently outperform existing state-of-the-art approaches in terms of accuracy.

In this article, we propose a novel hyperbolic relaxation system for a general class of fourth-order nonlinear partial differential equations arising in the modelling of thin film flows or the phase separation of binary mixtures. The approximations are constructed to dissipate energies which recover the energy (Lyapunov) functional of the limit equation when the relaxation parameters vanish. Using the relative energy framework, we prove the convergence of weak entropy solutions of the relaxation system to smooth solutions of the limit equation. We validate our analysis with a series of numerical examples for thin film equations and Cahn-Hilliard equations.

Let $R$ be an excellent local domain. $R$ is said to be $NBIM$ if $Tor_{i}^{R}(R^{+}, k) = 0$ for some $i\geq d:=\dim(R)$. Bhatt, Iyengar, and Ma ask if equi-characteristic zero $NBIM$ rings are regular. If $R$ is of positive characteristic, Asgharzadeh and Mahdavi conjecture that $Ext^{i}_{R}(k,R^{\infty}) = 0$ for some $i>d$ implies that $R$ is regular. It is an open question whether $R^{+}$ and $R^{\infty}$ are $\mathfrak{m}$-adically idealwise separated in positive characteristic, a condition from the `local criterion of flatness'. These are analogues of Kunz's theorem and intimately related to the homological conjectures and singularities in algebraic geometry. We apply a result of Avramov, Hochster, Iyengar, and Yao on contracting endomorphisms to make progress on the first two. We observe that it implies toric $NBIM$ rings are regular and solves the conjecture for $F$-pure rings. These improvements are inaccessible by previous techniques and give new and simple proofs of earlier results. In mixed characteristic, we show several linked results for perfectoid-pure rings. We show the third statement when there is $R\rightarrow S$ finite and flat on the punctured spectrum and $S$ is regular, this uses Cohen-Macaulayness of $S^{+}$.

We show low-regularity well-posedness of the Korteweg-de Vries equation with spatially quasi-periodic initial data. To this end, we employ frequency-dependent time localization and a bilinear version of the Córdoba–Fefferman square function estimate to show a bilinear Strichartz estimate for quasi-periodic functions. The solutions are proved to preserve the Sobolev regularity of the initial data, which was not the case in earlier works. The argument extends to other dispersion relations and higher order nonlinearities.

We consider the magnetic Neumann Laplacian on bounded domains in $\mathbb{R}^2$ with outward peaks. The operator is associated with a large magnetic field depending on a parameter $\lambda$, and we investigate the behaviour of the eigenvalues as $\lambda$ tends to $+\infty$. We show that their asymptotic expansion is influenced by the sharpness $q$ and geometry of the peak, and that its main term is of order $\lambda^{\frac{2}{q+1}}$. This is an extension of previous works on magnetic Neumann Laplacians in smooth domains and domains with corners.

We develop an unfitted ghost finite element method for elliptic interface problems with discontinuous diffusion coefficients and apply it to the electrostatic simulation of low-dimensional semiconductor devices. The proposed approach is based on a fixed Cartesian grid and represents the geometry by level-set functions, avoiding mesh generation and remeshing even in the presence of interfaces. A snapping-back-to-grid strategy is used to control the small-cut-cell issue, while interface conditions are weakly enforced by a symmetric Nitsche formulation. The method is first validated on benchmark elliptic interface problems with different geometries and coefficient jumps, showing second-order accuracy for the solution and first-order accuracy for its gradient. As an application, we consider a graphene field-effect transistor described by a self-consistent drift-diffusion-Poisson model. The electrostatic potential is computed in a two-dimensional oxide/graphene/oxide structure, while charge transport in the graphene layer is modeled by one-dimensional bipolar drift-diffusion equations in the degenerate case and by including a field-dependent mobility model. The coupled nonlinear system is solved by a damped fixed-point iteration combined with a domain-decomposition treatment of the three-layer geometry. Numerical simulations reproduce transfer characteristics with a clear transition from an OFF state to an ON state and show the influence of the gate voltage on the two-dimensional electrostatic potential. The results also indicate that the effective thickness and discretization of the graphene layer affect the transverse potential profile, supporting the use of a full two-dimensional electrostatic description with explicit oxide/graphene interface conditions.

We study the discrepancy of balancing $n$ independent sample paths of fractional Brownian motion with Hurst exponent $H\in(0,1)$ on $[0,1]$, an infinite-dimensional analogue of balancing Gaussian vectors. We establish a phase transition at $H=1/2$: with high probability, the discrepancy is $\Omega(n^{1/2-H})$ and $\mathcal O(n^{1/2-H}(\log n)^{c(H)})$, where $c(H)=H+1/2$ if $H\geq 1/2$ and $c(H)=1/2$ otherwise. At the critical exponent $H=1/2$, we show that the discrepancy is $\Theta(1)$ with constant probability as $n\to\infty$. In this regime, we further characterize the geometry of the solution space by computing the expected number of local minima, establishing an overlap gap property near the existence threshold, and proving its absence at every diverging optimality threshold. We also give randomized polynomial-time algorithms that compute signings with discrepancy $\mathcal O(n^{1/2-H}\sqrt{\log n})$ for $H<1/2$, $\mathcal O((\log n)^{3/2})$ for $H=1/2$, and $\mathcal O(\sqrt{\log n})$ for $H>1/2$, with high probability. Our analysis combines a truncated balancing argument based on a wavelet representation of fractional Brownian motion with probabilistic methods.

We prove a Plancherel formula for a family of affine $\Psi$-bundles. As an application, we construct Fourier transforms and asymptotic Schwartz spaces for the family. We then prove the corresponding Poisson summation formula under suitable assumptions. The choice of affine $\Psi$-bundles we consider is motivated by applications to triple product $L$-functions explored in another paper of the authors and Leslie.

Computational solid mechanics relies on the discretisation of differential equations, requiring numerical differentiation and integration, often via quadrature. Recently, automatic differentiation (AD) has attracted interest as a tool to support these operations. This paper introduces the use of AD in nonlinear finite element analysis, focusing on stress updating, a key and repeatedly executed step in which stresses are computed from strain increments. Both implicit and explicit schemes require derivatives of constitutive models, which can be complex, error prone, and time consuming to derive analytically. We compare traditional analytical differentiation with automatic differentiation for implementing a hyperplastic Critical State model widely used in geotechnical engineering. Drained triaxial compression tests with varying overconsolidation ratios are simulated using both approaches. The results show that AD significantly simplifies the implementation of backward Euler stress integration by removing the need for manual derivation of derivatives. Despite this simplification, AD maintains high accuracy and robustness comparable to analytical approaches. The findings demonstrate that automatic differentiation can streamline the development of nonlinear material models, enabling efficient and reliable implementation of constitutive laws of arbitrary complexity. This opens the way for more flexible and maintainable computational mechanics codes.

Huybrechts conjectured that for every Brauer class $\alpha$ on a hyperkähler variety $X$ we have that $\mathop{\rm ind}(\alpha)\mid\mathop{\rm per}(\alpha)^{\dim(X)/2}$, strengthening the usual period-index conjecture. We show that this fails on certain hyperkähler fourfolds, both in type $\mathrm{K}3^{[2]}$ and $\mathrm{Kum}^2$.

We introduce the notion of R-full Schreier internal category, which is the monoid analogue of the notion of aspherical abelian groupoid. We associate with every R-full Schreier internal category a direction, which is a Schreier split extension with commutative and cancellative kernel. We show that this association is functorial and that such functor is a product preserving cofibration. Thanks to these properties, we equip the connected components of the fibres of such functor with canonical commutative monoid structures. Using the equivalence between Schreier internal categories and crossed semimodules of monoids, we describe these commutative monoids in terms of crossed Schreier extensions.

Using an argument of Jacob Lurie, we prove the existence of a E_2-lax monoidal structure on the Tate filtration coming from the standard cell structure on the infinite complex projective space. We then compare it to the filtration induced by the standard t-structure on spectra. In the last part of the paper, we construct a synthetic variant of the cellular Tate filtration and use it to compare Antieau's, Bhatt-Lurie's and Raksit's HKR filtrations on negative cyclic and periodic homology.

In previous work, the density of the martingale limit in Schröder branching processes was expressed as a convergent double power-law series with oscillatory terms. The proof relied on two assumptions, one of which imposed a geometric restriction on the critical angle of the Julia set of the offspring generating function near $1$. In this paper, we show that both assumptions can be removed. We derive explicit bounds on the expansion coefficients, which imply locally uniform convergence of the double series. The coefficients decay exponentially in one summation index, with the rate explicitly determined by the critical angle, and super-exponentially in the other. Finally, we investigate the behavior of the critical angle and describe regimes in which it approaches its minimum. This analysis shows how the geometry of the Julia set influences the magnitude of the oscillatory corrections in the left-tail expansion.

We present an analysis on the geometrical and physical nature of the $p_x + ip_y$ superconductor on the square lattice, with an emphasis on the topological phase transition at half-filling. We develop a local topological marker from specific Dirac points within the Brillouin zone, which is introduced via the addition of two one-dimensional (1D) $\mathbb{Z}$ $(\mathbb{Z}_2$) invariants defined on the Bloch sphere. We relate this topological marker to the electron spectral function integrated on frequency through the local momentum-resolved electron density, which may be measured via Angle Resolved Photoemission Spectroscopy (ARPES), and show that it remains well-protected including temperature effects. Integrating on a small area around a specific point in momentum space associated to the measure uncertainty, this also reveals the Van Hove logarithmic profile of the density of states in the derivative of the local marker while preserving the topological information. Topological transitions correspond to a protected semi-metal. We analyse the real space representation of this topological marker from correlation functions. We present physical responses such as the topological superfluid density.

Background: Over the last two decades, considerable effort in the applied mathematics community has been devoted to the development of polytopal methods, i.e., numerical methods for partial differential equations that support meshes more general than those used by traditional finite element methods. Despite their flexibility and attractive mathematical properties, polytopal methods have a steep implementation curve and, to date, no easily accessible open-source software framework has been available for their systematic implementation. Methods: In this article, we introduce poly_rust, a new software library implementing a domain-specific language (DSL) specifically designed for polytopal methods. poly_rust provides high-level constructs for expressing a wide range of such methods using a syntax that closely mirrors their mathematical formulation, thereby substantially lowering the implementation barrier for both newcomers and experienced users. Results: The library facilitates the rapid implementation, testing, and comparison of polytopal discretizations, making these methods more accessible to researchers, applied scientists, engineers, and students. In our vision, this will foster their dissemination beyond the numerical analysis community and encourage their adoption by practitioners in a broad range of application fields. It will also contribute to reproducibility of results, as DSL files are text files that can be easily made available, stored, and exchanged. Conclusions: poly_rust introduces, to our knowledge, the first DSL specifically tailored to polytopal methods. By combining mathematical expressiveness, ease of implementation, and an open-source framework, it provides a common platform for developing, sharing, reproducing, and comparing polytopal schemes, with the ambition of becoming a reference tool for the wider scientific community.

Multi-camera wireless perception requires a base station (BS) to maintain timely and reliable latent beliefs from distributed cameras under limited uplink resources. Conventional Age-of-Information (AoI) measures the age of the latest received update but not task-relevant latent content. We introduce Age-of-Latent (AoL) to quantify the freshness of each camera's latest decoded latent representation. A finite temporal window of integration (TWI) determines the task-commitment time and available uplink slots, creating a tradeoff among update opportunities, prediction duration, commit-time AoL, prediction reliability, and accumulated resource cost. Within this horizon, redundant or overlapping views enable correlated prediction, reducing reliance on the highest-cost communication and encoding configuration to maintain BS-side latent beliefs. We formulate a joint AoL-resource minimization problem coupling task-level TWI selection with slot-level encoder selection, scheduling, and NOMA power allocation under prediction-reliability constraints. We propose correlation-aware latent prediction for AoL minimization (CoLA), which uses Lyapunov optimization for task-level TWI selection based on AoL-resource cost and prediction uncertainty, and proximal policy optimization for slot-level resource control. Results on a warehouse multi-camera RF dataset show that CoLA adapts the TWI to camera-update availability and achieves the most favorable AoL-resource tradeoff among the benchmarks while maintaining prediction reliability, particularly under prolonged and severe camera outages.

Let $T$ be an exponentially stable strongly continuous semigroup on a Hilbert space $X$, $A$ its generator, $X_1$ the domain of $A$ with the norm $\|v\|_1 := \|Av\|$, and $C$ an admissible observation operator taking values in a Hilbert space $Y$. Russell and Weiss conjectured that if $(A,C)$ satisfies the infinite-dimensional Hautus test, then $(A,C)$ is exactly observable. After a counterexample was found their conjecture was modified to include the additional assumption that $T$ is similar to a contraction semigroup. We disprove this modified conjecture by constructing a counterexample in which $A$ is normal, $X$ has an orthonormal basis of eigenvectors of $A$ and $C: X_1 \to Y$ is Hilbert-Schmidt. Since $A$ is normal and $T$ is exponentially stable, it follows that $T$ is a contraction semigroup. In contrast, we prove that the Hautus test implies exact observability for self-adjoint $A$. Finally, we show that if $A$ is normal and $C:X_1 \to Y$ is compact, then the Hautus test implies that $A$ has compact resolvent. Consequently, for normal $A$ and finite-dimensional $Y$ the Hautus test implies exact observability.

In this paper, we study the boundedness properties of Erdélyi–Kober fractional integrals and Mellin fractional integrals on weighted Lebesgue spaces over $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion _{+}=$ $\left( 0,\infty \right) $. We establish sufficient conditions on different weight functions to ensure the boundedness of these operators between weighted integrable spaces. Our approach is mainly based on weighted Hardy-type inequalities, Hölder estimates, and suitable changes of variables associated with the multiplicative structure of the operators. We first investigate a class of Erdélyi–Kober type integral operators and derive weighted $L^{p}$-inequalities under appropriate assumptions on the weights. These results extend several classical inequalities related to Hardy operators and fractional integrals. We then consider Mellin fractional integral operators and obtain analogous boundedness results in weighted Lebesgue spaces. The obtained estimates reveal a close connection between Erd% élyi–Kober operators and Mellin-type fractional integrals within the framework of multiplicative harmonic analysis. The results presented in this paper provide a unified treatment of these fractional integral operators in weighted settings and generalize various previously known boundedness results. In particular, our conditions on the weights characterize the continuity of the operators on weighted $L^{p}$-spaces and illustrate the role played by the multiplicative structure of the underlying measure space.

A subgraph $H$ of a $k$-connected graph $G$ is called \emph{$k$-removable} if $G-E(H)$ remains $k$-connected. Halin proved that every $k$-connected graph $G$ with $\delta(G)\ge k+1$ has a $k$-removable edge. We extend this result from a single edge to matchings of any prescribed size by showing that, for positive integers $k$ and $m$, every $k$-connected graph $G$ with $\delta(G)\ge\max\{k+1,2m-2\}$ contains a $k$-removable matching of size $m$, unless $G\cong K_{2m-1}$, or $(k,m)=(1,2)$ and $G$ is a cycle. This confirms a conjecture of Li, Zhou, Fujita, and Mao. The minimum degree bound is sharp, and both exceptions are unavoidable. Consequently, $\max\{k+1,2m-1\}$ is the sharp minimum degree threshold guaranteeing such a matching without exceptions. The proof combines a prescribed-set strengthening of Halin's removable-edge theorem with an extremal analysis of maximum $k$-removable matchings.

This note aims to serve as an entry point to the literature on learning in games, a topic with significant theoretical appeal and a wide range of applications – from machine learning and data science to economics and beyond. Our presentation is structured around two complementary viewpoints: We first consider a single agent – the learner – engaged in a sequential decision process in an unknown, non-stationary, and possibly adversarial environment. We then examine what happens when the environment is shaped by the decisions of several interacting agents, not necessarily aware of each other's actions or goals, and all seeking to improve their individual rewards. In this general context, we examine a family of regularized learning policies based on best-responding to the past history of play, up to a regularization penalty intended to encourage exploration and prevent over-commitment to suboptimal choices. In the single-agent setting, we present some basic regret bounds for regularized learning in adversarial multi-armed bandits; in the multi-agent setting, we describe an ergodic equilibrium convergence result for zero-sum games in the spirit of classical results on fictitious play, as well as a "folk theorem" linking strategic and dynamic notions of stability – Nash equilibria and attracting points of regularized learning, respectively. We pay special attention to the information available to the players and, through a unified analysis framework, we study both oracle- and payoff-based (bandit) methods. Our goal is to provide a coherent and comprehensible – albeit, by necessity, not comprehensive – account of some recent ideas in the field, and to discuss their implications for the study of rationality.

Recent advancements in wireless endogenous security have explored leveraging the inherent randomness of wireless channels to enhance communication security, providing an effective alternative to traditional encryption methods. This paper proposes a wiretap coding scheme within the semantic communication framework, which leverages discrete semantic representations compatible with conventional digital modulation to jointly enhance communication security and reliability. We investigate two eavesdropping scenarios: (i) the eavesdropper employs a maximum a posteriori (MAP) decoder, and (ii) the eavesdropper has access to a decoder identical to that of the legitimate receiver. In the first scenario, we exploit mutual information as a metric to guide the design of an optimized coding strategy, minimizing information leakage while enhancing communication reliability. In the second scenario, considering the limitations of the eavesdropper's decoding capability, we employ generalized mutual information (GMI) to characterize recoverability under the prescribed decoding rule and guide reliability-aware code optimization.

In this paper, we introduce a nonlocal nonlinear Schrödinger (NLS) model on periodic domains, and establish its well-posedness, conservation laws, local limit, dispersion properties, and develop a structure-preserving asymptotically compatible Fourier collocation method. We prove that for a fixed nonlocal horizon $\delta$, the model is globally well posed and conserves mass and nonlocal energy, and the nonlocal NLS solution converges to the local NLS solution with order $O(\delta^2)$ under suitable regularity assumptions. We also derive the dispersion relation and group velocity for plane waves, establish their $O(\delta^2)$ local limits, and characterize their behavior at high frequencies. The numerical method combines Crank–Nicolson time stepping with Fourier collocation and preserves mass and energy. The existence, uniqueness, and convergence of the numerical solutions are proved. In particular, the error of the nonlocal NLS solution is $O(\tau^2+N^{s-r})$ in the $H^s$ norm, uniformly with respect to the horizon. Moreover, its total $H^s$-error relative to the local NLS solution is $O(\delta^2+\tau^2+N^{s-r})$ without any coupling condition among $\delta$, $\tau$, and $N$, which proves asymptotic compatibility of the proposed method. Numerical experiments in one, two, and three dimensions are presented to verify the theoretical accuracy and discrete conservation, confirm convergence under independent variation of horizon and discretization parameters, and show how the horizon and kernel affect dispersive wave propagation.

We study minimal noncommutative rings, that is noncommutative rings whose proper subrings and homomorphic images are all commutative. These rings were introduced by Bell and Danchev in order to test commutativity theorems. They raised the problems of describing all such rings in the finite and infinite cases. In the finite case, we give a classification into three pairwise disjoint classes, the first two of which are completely characterised. For the third class, we give a finite procedure which can produce any of (and only) the required rings. We also translate the problem into commutative algebra, in terms of finite local rings with small socle and a kind of cancellation property. Finally, we show that if an infinite minimal noncommutative ring exists, then it is a division algebra with very strange properties, and a counterexample to several longstanding conjectures.

A target moves between two sites according to a discrete-time Markov chain with a $2\times2$ transition matrix $M$. At each epoch one site is searched at positive cost, and a search may overlook a target that is present. Ross conjectured that an optimal policy is threshold in the posterior probability that the target is at site~1. MacPhee and Jordan proved the conjecture throughout the nonpositive-determinant ($\det M\le0$) regime and for part of the positive-determinant ($\det M>0$) regime, leaving the remaining cases open. We prove threshold optimality throughout the positive-determinant regime, completing Ross's conjecture for all parameter values.

We establish a uniform pole-subtracted limiting absorption principle for a fixed cluster of \(N\) disjoint three-dimensional high-contrast elastic resonators when the contrast tends to infinity and \(\omega=\delta^{1/2}\tau\) approaches the zero threshold. The exterior Dirichlet-to-Neumann map and a variational Grushin–Feshbach reduction give an exact decomposition of the cutoff resolvent into a uniformly bounded regular part and a finite-rank term governed by \[ \cM_\delta^\pm(\omega) = \delta K-\omega^2I_m \mp\ii\delta\omega\Gamma_0 +\mathcal O(\delta^2+\delta\omega^2). \] The cutoff-resolvent norm is uniformly equivalent to \(1+\|(\cM_\delta^\pm(\omega))^{-1}\|\); hence the finite-dimensional channel carries every loss of uniformity. An elastic optical identity factors the leading radiation matrix through one total-force map into \(\C^3\). Thus \(\rank\Gamma_0=3\) and \(\dim\Ker\Gamma_0=6N-3\). Compression to a static eigenspace of dimension \(r\) leaves at most three leading radiative channels. Simple bright poles have width \(O(\delta)\), whereas force-dark poles with a positive second radiation form have width \(O(\delta^2)\); the corresponding real-axis peaks have orders \(\delta^{-3/2}\) and \(\delta^{-5/2}\). We also treat multiple static eigenvalues, compute the spherical coefficients, and derive a conditional two-parameter crossover for a symmetry-broken dimer.

Quantum-classical coexistence networks (QCNs) share optical fiber between quantum and classical signals via wavelength-division multiplexing, offering a practical path to quantum communication over existing telecom infrastructure. However, co- and counter-propagating classical traffic introduce distinct depolarization noise, complicating channel characterization. We develop a tomography framework that infers per-link channel parameters of a QCN from end-to-end measurements alone. We first model each coexisting fiber by decomposing the signal evolution into photon loss, successful transmission, and three direction-dependent depolarization components. We then derive closed-form link-level estimators, and extend the approach to star-topology networks through a system of multiplicative equations across end-node pairs, together with a simple classical-signal-direction-switching protocol that resolves the remaining unknowns. On single-link experimental testbed data, we recover per-link depolarization probabilities accurately, with estimated process fidelities closely tracking the Bayesian-process-tomography baseline across multiple fiber lengths and wavelengths; residual gaps reflect the depolarization-only approximation. Absent a multi-link coexistence testbed, we validate the star-network estimators on emulated paths built from measured single-link channels. We further extend the framework in two directions: (i) a channel model that factorizes the coexisting fiber into a depolarizing-with-loss signal channel and a Raman-noise-injection channel on separate optical modes – a completely-positive, trace-preserving tensor product – whose link observables reduce exactly to our basic model; and (ii) a generalization to arbitrary topologies via a peeling algorithm (trees) and a least-squares estimator (meshes), validated by Monte-Carlo simulations on tree and cyclic-mesh networks.

We introduce orbital detection (OD), a framework for designing asymptotically optimal, low-complexity message passing (MP) receivers for digitally modulated multiple-input multiple-output (MIMO) systems, based on relaxing the discrete symbol prior into a mixed discrete-continuous density. The resulting orbital prior factors each symbol's distribution into a discrete radial component, supported on only the \(L << M\) amplitude rings of an arbitrary constellation \(\mathcal{M}\) of cardinality \(M = |\mathcal{M}|\), and a continuous, maximum-entropy phase density on each ring. This compresses the propagated posterior mean and variance losslessly into \(3L\) real scalars, and collapses the optimal \(\mathcal{O}(M)\)-complexity denoiser into a closed-form hierarchy whose per-symbol cost falls to \(\mathcal{O}(L)\) and ultimately \(\mathcal{O}(1)\): the orbital Bessel denoiser (OBD), its Bessel-free variant the orbital Gaussian denoiser (OGD), and the orbital phase denoiser (OPD), proved irreducible on the ring manifold. A Jacobi-Anger ladder recovers the exact detector with geometrically vanishing error. Five information-theoretic results follow. First, the OBD, OGD, and OPD share an identical leading-order state evolution (SE) fixed point. Second, the sole price is a change in the high-SNR error-decay law, from exponential to linear, which never hardens into an error floor. Third, for any underloaded system the induced rate loss vanishes exponentially in SNR, so every level is asymptotically capacity-achieving in the constellation-constrained sense, attaining \(\log_2 M\). Fourth, OD attains a minimum mean square error (MMSE) dimension \(d=1/2\), halfway between the \(d=0\) Bayes-optimal denoiser (BOD) and the \(d=1\) linear receiver. Fifth, a non-asymptotic optimal-transport bound in Wasserstein distance links constellation ring geometry directly to the achievable rate.

We construct two Stein structures on the same compact smooth four-manifold whose boundary contact forms are strictly identified and whose first Chern classes agree, but whose exact symplectic forms are not symplectomorphic. Moreover, no diffeomorphism makes the associated Weinstein structures homotopic. The examples are obtained by removing a tubular neighborhood of a smooth bicanonical curve from a fake projective plane and comparing the induced Stein structure with its conjugate. Their canonical Spinc structures differ by a nonzero class of order two. After the boundary forms are normalized using a common Boothby-Wang connection form, a hypothetical symplectomorphism preserves the oriented circle fiber and extends over the divisor cap, contradicting canonical-class rigidity. For the Weinstein statement, the required fiber-preservation is obtained from a monopole Floer grading asymmetry, using the Nelson-Weiler computation and Taubes's ECH-Seiberg-Witten correspondence. This gives an affirmative solution to a normalized version of the relative filling problem following Problem 4.96 in the K3 problem list.

We define Sobolev spaces associated with the Grushin operator with or without drift. We study some of their norm equivalences and establish embedding and algebra properties. We also prove the boundedness of the Riesz transforms on these spaces.

In this note we provide a simple explicit example of a discrete set $\Lambda\subset\mathbb{R}$ of unbounded density such that the Fourier transform of its counting measure $\widehat{\delta}_\Lambda$ is a tempered distribution of the form $c\delta_0'' + \mu$ where $\mu$ is a discrete measure supported by a union of three arithmetic progressions. This example is then used to construct a translational tiling of unbounded density of the real line by a function with compact support.