Precontact manifolds extend contact geometry by weakening the maximal non-integrability condition of the defining $1$-form. We clarify the geometric foundations of this structure by studying general pairs of a $1$-form and a $2$-form under mild regularity conditions. We characterize them through their class, analyse the role of distinguished vector fields, such as Reeb or Liouville fields, and study other associated geometrical objects. Precontact structures are then treated as the special case of pairs formed by a nowhere-vanishing $1$-form and its exterior derivative. We also define Hamiltonian dynamics on precontact manifolds. Several examples are presented to illustrate the theory.

The classical Erdős-Kac theorem states that for $n$ chosen uniformly at random from $1, \dots, N$, the random variable $(ω(n) - \log\log N)/\sqrt{\log\log N}$ converges in distribution to the standard Gaussian as $N$ tends to infinity. Banks and Shparlinski showed that this Gaussian convergence holds for any Beatty sequence $\lfloorαn + β\rfloor$ in place of $n$. Continuing in this spirit, Crnčević, Hernández, Rizk, Sereesuchart and Tao considered the joint distribution of $ω(n)$ and $ω(\lfloorαn\rfloor)$, which they showed to be asymptotically independent for irrational values of $α$. Generalising both results, we show that for any positive integer $k$, real numbers $α_1, \dots, α_k > 0$ and $β_1, \dots, β_k$, where $α_i/α_j$ is irrational for $i\neq j$, the joint distribution of $(ω(\lfloorα_in + β_i\rfloor) - \log\log N)/\sqrt{\log\log N}$ converges to the $k$-dimensional standard Gaussian. We next discuss quantitative bounds on the rate of convergence which do not depend on the values taken by the relevant parameters. Banks and Shparlinski remarked that such quantitative bounds may be given for a single Beatty sequence $\lfloorαn + β\rfloor$ under Diophantine type assumptions on $α$. We show that such assumptions are in fact unnecessary. Specifically, for any real numbers $α> 0, β$, we show that the Kolmogorov distance between the random variable $(ω(\lfloorαn + β\rfloor) - \log\log N)/\sqrt{\log\log N}$ and the standard Gaussian is bounded above by $O(\log\log\log N/\sqrt{\log\log N})$ as $N$ tends to infinity. On the other hand, we show that universal quantitative bounds of this kind do not exist for higher-degree generalised polynomials or for the joint convergence for multiple Beatty sequences.

We present a simple explicit family $\mathcal{B}$ of $910$ $6$-subsets of $[60]=\{1,\dots,60\}$ such that every $6$-subset $S\subset[60]$ intersects at least one block $B\in\mathcal{B}$ in at least three elements, i.e.\ $|S\cap B|\ge 3$. Equivalently, $\mathcal{B}$ is a covering (dominating set) of the Johnson graph $J(60,6)$ with covering radius $3$ in the Johnson metric. The construction is purely combinatorial, based on a fixed split of $[60]$ into two halves, a pairing of each half, and a pigeonhole argument. We also record a crude counting lower bound and a straightforward generalization to $[2m]$ (with $m$ even).

We use admissible covers to characterize irreducible stable curves that are $(d,h)$-elliptic, that is, that are limits of smooth curves admiting finite maps of degree-$d$ to smooth curves of genus $h\geq 1$.

Low Earth orbit (LEO) satellite systems experience significant Doppler effects due to high mobility. While Doppler shifts can be largely compensated, residual frequency uncertainty induces a structured form of channel uncertainty that can limit achievable rates. We model this effect using a block-fading channel of the form $ \mathbf{H} = \mathbf{F} + s \mathbf{G} $, where $s$ is an unknown scalar random parameter. We first study this model in a general $N\times N$ MIMO setting. For this channel, we derive achievable rate lower bounds based on explicit transmission schemes and capacity upper bounds using a duality approach. We study Gaussian signaling and propose a practical superposition scheme with subspace alignment (SN) and successive interference cancellation, where a coarse-layer stream serves as an implicit pilot for decoding refined-layer data. We characterize asymptotic capacity in the near-coherent and high-SNR regimes, and show via Doppler-OFDM simulations that the proposed SN scheme achieves near-optimal rates with low complexity.

We study conformal tractor bundles from an extrinsic viewpoint, relating them to codimension two spacelike immersions into Lorentzian manifolds. We show that, at least locally, every Riemannian conformal structure admits a natural realization of its normal conformal tractor bundle as the pullback of the tangent bundle of a suitably constructed Lorentzian ambient space. Finally, we reformulate the classical equations characterizing parallel sections of the normal conformal tractor bundle in this extrinsic setting, showing that they can be expressed entirely in terms of the geometry of the associated spacelike immersion. This extrinsic perspective provides additional geometric insight into parallel standard tractors and conformal holonomy.

This paper addresses openness, density and structural stability conditions of one-parameter families of 3D piecewise smooth vector fields (PSVFs) defined around typical singularities. Our treatment is local and the switching set, $M$, is a $2D$ surface embedded in $\mathbb{R}^3$. In short, we analyze the robustness and normal forms of certain codimension one singularities that occur in PSVFs. The main machinery used in this paper involves the theory of contact between a vector field and $M$, Bifurcation Theory and the Topology of Manifolds. Our main result states robust mathematical statements resembling the classical Kupka-Smale Theorem in the sense that we establish the openness and density of a large class of PSVFs presenting generic and quasi-generic singularities. Due to the lack of uniqueness of certain solutions associated with PSVFs, we employ Filippov's theory as the basis of our approach throughout the paper.

We show that the Baumslag-Solitar group $BS(1,2)$ is undistorted in the Lodha-Moore group $G_0$ using an explicit lower bound for the word length of $G_0$. We also show that Thompson's group $F$ is distorted in $G_0$.

We provide characterizations of continuous eigenvalues for minimal symbolic dynamical systems described by $S$-adic structures satisfying natural mild conditions, such as recognizability and primitiveness. Under the additional assumptions of finite alphabet rank or decisiveness of the directive sequence, these characterizations are expressed in terms of associated sequences of local coboundaries. We emphasize the role of combinatorics in the study of continuous eigenvalues through the interplay between coboundaries and extension graphs, and we give several types of sufficient conditions for the nonexistence of trivial letter-coboundaries. As further results, we apply coboundaries in the context of bounded discrepancy, and in particular we obtain a simple characterization of letter-balance for primitive substitutive subshifts. Moreover, we recover a result of Tijdeman on the minimal factor complexity of transitive subshifts with rationally independent letter frequencies. Finally, we use linear-algebraic duality to refine known descriptions of the possible values of eigenvalues in terms of measures of bases.

We prove that resonances of the Schrödinger operator with compactly supported potential can contain arbitrary subset of the angle $\{z: -\text{Im} z > C |\text{Re} z|\}$ that satisfies Blaschke condition. We also establish sufficient conditions for the subsets of wider domains.

We study the estimation of quadratic Sobolev-type integral functionals of an unknown density on the unit sphere. The functional is defined through fractional powers of the Laplace--Beltrami operator and provides a global measure of smoothness and spectral energy. Our approach relies on spherical needlet frames, which yield a localized multiscale decomposition while preserving tight frame properties in the natural square-integrable function space on the sphere. We construct unbiased estimators of suitably truncated versions of the functional and derive sharp oracle risk bounds through an explicit bias--variance analysis. When the smoothness of the density is unknown, we propose a Lepski-type data-driven selection of the resolution level. The resulting adaptive estimator achieves minimax-optimal rates over Sobolev classes, without resorting to nonlinear or sparsity-based methods.

We study the committee selection problem in the canonical impartial culture model with a large number of voters and an even larger candidate set. Here, each voter independently reports a uniformly random preference order over the candidates. For a fixed committee size $k$, we ask when a committee can collectively beat every candidate outside the committee by a prescribed majority level $α$. We focus on two natural notions of collective dominance, $α$-winning and $α$-dominating sets, and we identify sharp threshold phenomena for both of them using probabilistic methods, duality arguments, and rounding techniques. We first consider $α$-winning sets. A set $S$ of $k$ candidates is $α$-winning if, for every outside candidate $a \notin S$, at least an $α$-fraction of voters rank some member of $S$ above $a$. We show a sharp threshold at \[ α_{\mathrm{win}}^\star = 1 - \frac{1}{k}. \] Specifically, an $α$-winning set of size $k$ exists with high probability when $α< α_{\mathrm{win}}^\star$, and is unlikely to exist when $α> α_{\mathrm{win}}^\star$. We then study the stronger notion of $α$-dominating sets. A set $S$ of $k$ candidates is $α$-dominating if, for every outside candidate $a \notin S$, there exists a single committee member $b \in S$ such that at least an $α$-fraction of voters prefer $b$ to $a$. Here we establish an analogous sharp threshold at \[ α_{\mathrm{dom}}^\star = \frac{1}{2} - \frac{1}{2k}. \] As a corollary, our analysis yields an impossibility result for $α$-dominating sets: for every $k$ and every $α> α_{\mathrm{dom}}^\star = 1 / 2 - 1 / (2k)$, there exist preference profiles that admit no $α$-dominating set of size $k$. This corollary improves the best previously known bounds for all $k \geq 2$.

We propose a combinatorial bandit formulation to opportunistically trigger sleep modes in gNode-B (gNB) small cells (SCs), followed by a cell range expansion (CRE)-based load balancing procedure. This is implemented by ensuring that the fifth generation (5G) quality of service identifier (5QI)-requirements of user equipments (UEs) are maintained. The key challenge is the fact that while deactivating a given SC gNB reduces its own consumption, it may increase the load on neighboring gNBs and the macro gNB (coverage cell), impacting the overall energy efficiency. This phenomenon is accurately characterized by modeling the dynamic cell load that jointly takes into account the location of the UEs, their relative locations to all the SCs, and their data demands. We experimentally show that the proposed combinatorial upper confidence bound (CUCB) followed by the load balancer outperforms not only the naive strategies like arbitrarily keeping all the SCs on, but also other state-of-the-art reinforcement learning solutions. The proposed algorithm can be implemented as open-radio access network (O-RAN) near-real-time (NRT) RAN intelligent controller (RIC) xApps.

We study mean field equations with singular sources on a compact Riemann surface with boundary $(Σ,g)$, subject to homogeneous Neumann boundary conditions: \[ -Δ_g v = ρ\left( \frac{V e^{v}}{\int_ΣV e^{v}\, d v_g} - \frac{1}{|Σ|_g}\right) - \sum_{ξ\in Q} \frac{\varrho(ξ)}{2}γ(ξ) \left(δ_ξ- \dfrac{1}{|Σ|_g}\right) \text{in }Σ; \qquad \partial_{ν_g} v = 0 \text{ on }\partialΣ. \] Here, $V$ is a smooth positive function, $ρ$ is a non-negative parameter, $Q\subsetΣ$ is a finite set of prescribed singular points, and the singular weights satisfy $γ(ξ)\in(-1,+\infty)\setminus(\mathbb{N}\cup\{0\})$. The coefficients are given by $\varrho(ξ)=8π$ for $ξ\inΣ\setminus\partialΣ$ and $\varrho(ξ)=4π$ for $ξ\in\partialΣ$. We construct blow-up solutions in the non-quantized singular regime, including purely singular and mixed singular-regular blow-up cases, with parameters approaching resonant values. The construction is achieved via a Lyapunov-Schmidt reduction under suitable stability assumptions. Key words: Singular mean field equations, Blow-up phenomena, Lyapunov-Schmidt reduction, Riemann surfaces with boundary

Certain integrable hierarchies appearing in random matrix theory, enumerative geometry, and conformal field theory are governed by Virasoro/$W$-algebra constraints and their $W$-representations.Motivated by the Gaussian Hermitian $β$-ensemble and recent studies of superintegrable partition function hierarchies, we build an explicit bridge from symmetric group class algebras to bosonic Fock spaces and further to geometry. On the algebraic side, we decompose the transposition class sum into cut and join channels and recover the classical cut-and-join operator on the ring of symmetric functions. On the geometric side, we use the Grojnowski-Nakajima Fock space identification to realize the ladder operator $E_1=[W_0,p_1]$ as the Hecke correspondence on $\mathrm{Hilb}_n(\mathbb C^2)$, and we interpret the cubic generator $W_0$ as a normal ordered triple incidence correspondence. We then explain how the $β$-deformed cubic generator $W_0^{(β)}$ arises from the Ward identities/Virasoro constraints of the Gaussian $β$-ensemble via a background charge parametrization, clarifying its conformal field theoretic meaning. Finally, using the Grojnowski-Nakajima Heisenberg-Fock isomorphism $Φ_{\mathrm{Hilb}}:Λ\xrightarrow{\sim}\bigoplus_{n\ge0}H_T^*(\Hilb^n(\mathbb C^2))$, we transport the resulting commutator hierarchy to Hilbert schemes, where $E_1$ is realised by the Hecke correspondence (adding one point) and the diagonal correction terms are computed by equivariant localization from the $T$-weights of the tangent bundle $T\Hilb^n(\mathbb C^2)$ and the tautological bundle $\mathcal V$. This provides a geometric realization framework that unifies $β$-deformed integrable structures and offers new tools for studying quiver gauge theory partition functions.

We prove an almost minimal R=T theorem for self-dual Galois representations with coefficients in a finite field satisfying a property called rigid. We also prove the rigidity property for a large family of residual Galois representations attached to regular algebraic self-dual representations. Our theorem is based on a Taylor--Wiles patching argument for G-valued Galois representation, where G equals GO(2m) or GSp(2m).

We present a class of potentials $q \colon \mathbb{R}^{n} \to (0,\infty)$ that implies the weighted Schrödinger semigroup $φ^{-1}\mathrm{e}^{-tH}φ$ to map a weighted Lebesgue function space $\mathrm{L}_μ^{1}(\mathbb{R}^{n})$ into a weighted Lebesgue function space $\mathrm{L}_μ^{2}(\mathbb{R}^{n})$ continously at every time $t>0$ by Logarithmic Sobolev inequalities for $H=-Δ+ q(x)$ with it's strictly positive ground state $φ\colon \mathbb{R}^{n} \to (0,\infty)$. We use the self-adjointness of $\mathrm{e}^{-tH}$ in $\mathrm{L}^{2}(\mathbb{R}^{n})$ to infer an intrinsic ultracontractivity, i.e. $\forall t>0 \ \exists C_{t} > 0 \ : \ \left| \mathrm{e}^{-tH} u (x) \right| \ \leq \ C_{t} φ(x) \| u \|_{2}$ for every $u \in \mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ almost everywhere in $\mathbb{R}^{n}$.

We prove the existence of strong and weak solutions to the semilinear wave equation with coefficients depending both on time and space variables, with continuous nonlinearity satisfying the sign condition. The uniqueness is proven under slightly more restrictive assumptions. Furthermore, the results obtained in abstract setting are illustrated on practical examples.

The Koopman operator and its data-driven approximations, such as extended dynamic mode decomposition (EDMD), are widely used for analysing, modelling, and controlling nonlinear dynamical systems. However, when the true Koopman eigenfunctions cannot be identified from finite data, multi-step predictions may suffer from structural inaccuracies and systematic bias. To address this issue, we analyse the first and second moments of the multi-step prediction residual. By decomposing the residual into contributions from the one-step approximation error and the propagation of accumulated inaccuracies, we derive a closed-form expression characterising these effects. This analysis enables the development of a novel and computationally efficient algorithm that enforces unbiasedness and reduces variance in the resulting predictor. The proposed method is validated in numerical simulations, showing improved uncertainty properties compared to standard EDMD. These results lay the foundation for uncertainty-aware and unbiased Koopman-based prediction frameworks that can be extended to controlled and stochastic systems.

The explicit description and computation of functions defined on Riemann surfaces of various genera depends on the choice of language: while the Jacobi parametrization is widely known and used, the Schottky uniformization has been proven to provide an alternative approach, useful in particular for (but not limited to) numerical calculations. Despite capturing the geometry of the Riemann surface completely, the two languages are subject to rather different sets of auxiliary symmetries. In this article we translate and compare the symplectic transformations inherent in the Jacobi parametrization to the freedom in choosing Möbius transformations generating the Schottky group for the Schottky uniformization. Our results are aimed at transferring functional relations expressed in the Schottky language to the Jacobi language and vice versa. An immediate application would be the efficient numerical evaluation of special functions in a physics context by favorably tuning the Schottky cover leading to quicker convergence.

The wave speed of a stochastic wave equation driven by Riesz noise on the unbounded multidimensional spatial domain is estimated based on discrete measurements. Central limit theorems for second-order variations of the observations in space, time, and space-time are established. Under general assumptions on the spatial and temporal sampling frequencies, the resulting method-of-moments estimators are asymptotically normally distributed. The covariance structure of the discrete increments admits a closed-form representation involving two different Fejér-type kernels, enabling a precise analysis of the interplay between spatial and temporal contributions.

In the first part of this article we present a growth condition on the potential $q$ in the Schrödinger operator $H=-Δ+ q(x)$ in $\mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ that implies Rosen inequalities for the ground state $φ$ of $H$, i.e. $\forall \varepsilon > 0 \exists γ(\varepsilon) > 0 \ : \ - \ln\left( φ(x) \right) \leq \varepsilon q(x) + γ(\varepsilon)$. While these inequalities are not particularly interesting in themselves, they offer Logarithmic Sobolev inequalities which are absolutely essential to prove an intrinsic ultracontractivity of the associated Schrödinger semigroup $\mathrm{e}^{-tH}$, i.e. $\forall t>0 \exists C_{t} > 0 \ : \ \left| \mathrm{e}^{-tH} u (x) \right| \ \leq \ C_{t} φ(x) \| u \|_{2}$ holds for every $u \in \mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ almost everywhere in $\mathbb{R}^{n}$ which we prove in the second part of this article. For proving Rosen inequalities we focus on solving a radial Schrödinger inequality and use Agmon's version of the comparison principle and Young's inequality for increasing functions. We follow the classic method proving intrinsic ultracontractivity of $\mathrm{e}^{-tH}$ by using weighted Sobolev function spaces, weighted Schrödinger semigroups and Logarithmic Sobolev inequalities.

We construct an application, which takes as input a simple path and a possibly infinite collection of loops, and outputs a continuous path by adding the loops chronologically to the simple path as the simple path encounters them. By studying the regularity properties of this application and using lattice discretisations, we prove that chronologically adding the loops from a Brownian loop soup encountered by an independent radial SLE$_2$ path produces a continuous path which has the law of a planar Brownian motion. This resolves a conjecture of Lawler and Werner. This construction produces a coupling between SLE$_2$ and Brownian motion, and we further show that this joint law is the scaling limit of the loop-erased random walk and the random walk itself. The arguments are robust and can be applied for instance in the off-critical setup, where the scaling limit of loop-erased random walk is Makarov and Smirnov's massive SLE$_2$.

Let $M$ be a compact Riemannian manifold without boundary, with $L^2$-normalized Laplace-Beltrami eigenfunctions $\{e_j\}_j$, which satisfy $Δ_g e_j = -λ_j^2 e_j$. We study the following inner product of eigenfunctions \[ \langle e_{i_1} e_{i_2} \ldots e_{i_k}, e_{i_{k+1}} \rangle = \int e_{i_1} e_{i_2}\ldots e_{i_k} \overline{e_{i_{k+1}}} \, dV. \] We show that, after a mild averaging in the frequency variables, the main $\ell^2$-concentration of this inner product is determined by the measure of a set of configurations of $(k+1)$-gons whose side lengths are the frequencies $λ_{i_1}, λ_{i_2}, \dots, λ_{i_{k+1}}$. We prove that a rapidly vanishing proportion of this mass lies in the regime where $λ_{i_1}, λ_{i_2}, \dots, λ_{i_{k+1}}$ cannot occur as the side lengths of any $(k+1)$-gon.

Studying Courant algebroids on dg ringed manifolds, we observe that the associated Roytenberg-Weinstein $L_\infty$ algebra admits a local structure reminiscent of a shifted contact structure. On a dg ringed manifold with an $n$-orientation, its symplectification produces a sheaf of $(2-n)$-shifted symplectic formal moduli problems, which we call the Courant contact model. This construction can be interpreted as a ($\mathbb{Z}/2\mathbb{Z}$-graded) theory in the Batalin-Vilkovisky formalism whenever $n$ is odd. After developing the procedure of reduction and extension of scalars, we show how twisted backgrounds in type I supergravity naturally lead to Courant algebroids over the Dolbeault complex. Specialising to the case of a Calabi-Yau fivefold, we show that the Courant contact model for that Courant algebroid is equivalent to a central extension of minimal type I BCOV theory. Inspired by this, we extend the conjecture of Costello and Li and place it within the setting of generalized geometry, conjecturing a description of the BV formulation of type I supergravity in general twisted backgrounds.

We establish rigorously the equivalence between classical moment closure and a nonlinear variational approximation of the Fokker-Planck equation for dilute polymeric flow in the linearized Hookean spring chain setting. The variational formulation is based on the Dirac-Frankel principle applied to a Gaussian approximation manifold endowed with the Fisher-Rao information metric. We show that the invariance of this manifold under the linear configurational dynamics yields an exact evolution for the macroscopic conformation tensor, recovering the classical diffusive Oldroyd-B closure. While the equivalence only holds in the linearized setting, the associated variational framework provides an abstract error representation and a starting point for the systematic construction of reduced approximation schemes for polymeric flows with nonlinear forcing laws.

We are interested in analyzing the preservation of bifurcations in a class of piecewise smooth vector fields with a nonregular switching set under a smoothing process that approximates them by smooth vector fields. We examine cases in which the codimension is either preserved or altered, as well as whether the generic nature of the bifurcation is maintained.

We consider expectations of the form $E [tr h_1(X_1^N)... tr h_r(X_r^N)]$, where $X_i^N$ are self-adjoint polynomials in various independent classical random matrices and $h_i$ are smooth test function and obtain a large $N$ expansion of these quantities, building on the framework of polynomial approximation and Bernstein-type inequalities recently developed by Chen, Garza-Vargas, Tropp, and van Handel. As applications of the above, we prove the higher-order asymptotic vanishing of cumulants for smooth linear statistics, establish a Central Limit Theorem, and demonstrate the existence of formal asymptotic expansions for the free energy and observables of matrix integrals with smooth potentials.

In the current work we present a spectral analysis of the additive and multiplicative Schwarz methods within the framework of domain decomposition techniques, by investigating the spectral properties of these classical Schwarz preconditioning matrix-sequences, with emphasis on their convergence behavior and on the effect of transmission operators. In particular, after a general presentation of various options, we focus on restricted variants of the Schwarz methods aimed at improving parallel efficiency, while preserving their convergence features. In order to rigorously describe and analyze the convergence behavior, we employ the theory of generalized locally Toeplitz (GLT) sequences, which provides a robust framework for studying the asymptotic spectral distribution of the discretized operators arising from Schwarz iterations. By associating each operator sequence with the appropriate GLT symbol, we derive explicit expressions for the GLT symbols of the convergence factors, for both additive and multiplicative Schwarz methods. The GLT-based spectral approach offers a unified and systematic understanding of how the spectrum evolves with mesh refinement and overlap size (in the algebraic case). Our analysis not only deepens the theoretical understanding of classical Schwarz methods, but also establishes a foundation for examining future restricted or hybrid Schwarz variants using symbolic spectral tools. These results enable the prediction of the remarkable efficiency of block Jacobi/Gauss--Seidel and block additive/multiplicative Schwarz preconditioners for GLT sequences, as further illustrated through a wide choice of numerical experiments.

By employing harmonic analysis techniques, we derive weak-type Caffarelli-Kohn-Nirenberg inequalities under natural parameter conditions. A key feature of these weak-type versions is that they remain valid even at critical parameter values where the classical inequalities fail. As an important corollary, we obtain weak-type Hardy inequalities that hold true even in the critical dimension \(d = p\). The methods developed here are sufficiently flexible to handle homogeneous, non-homogeneous and anisotropic weights, providing a unified approach to various endpoint cases in interpolation theory.