Berger and Coburn proposed an endpoint boundedness criterion for Toeplitz operators on the Bargmann--Fock space in which the decisive quantity is the heat transform of the symbol at the borderline time $t=\tfrac14$, the time naturally singled out by the Weyl calculus under the Bargmann transform. We show that this criterion fails for general measurable symbols in every complex dimension $n\ge 1$. Concretely, we construct a measurable symbol $g\in L^2(\mathbb C^n,dμ)$ such that $gk_a\in L^2(dμ)$ for every normalized reproducing kernel $k_a$, and the associated Toeplitz form extends to a bounded operator on $H^2(\mathbb C^n,dμ)$, but the heat transform $g^{(1/4)}$ is unbounded on $\mathbb C^n$. The example is obtained by summing translated bounded "blocks" whose Toeplitz norms are summable while their $t=\tfrac14$ heat profiles have fixed size. The blocks are produced by combining a Hilbert--Schmidt estimate for Weyl quantization with the Bargmann correspondence between Weyl and Toeplitz operators.

We construct examples of minimal and uniquely ergodic systems realizing all possible behaviors in the interplay of measurable and topological nilfactors. To build such examples, we adapt an idea that stems from Furstenberg's construction of a minimal but not uniquely ergodic system on $\mathbb{T}^2$.

We show that, under very general hypotheses, topological quantum field theories (TQFTs) cannot detect homotopy spheres bounding parallelisable manifolds, such as Milnor's exotic 7-dimensional sphere. The result holds for a wide variety of target categories (or $(\infty,n)$-categories) and arbitrary tangential structures. An appendix contains results on the mapping class groups of (stably-) framed manifolds that may be of independent interest.

Orthogonal Time Frequency Space (OTFS) modulation offers robust performance in high-mobility scenarios by transforming time-varying channels into the delay-Doppler (DD) domain. However, in high-mobility environment such as emerging 5G Non-Terrestrial Networks (NTN), the extreme orbital velocities of Low Earth Orbit (LEO) satellites frequently cause the physical Doppler shifts to exceed the fundamental grid range. This Doppler ambiguity induces severe model mismatch and renders traditional MLE channel estimators ineffective. To address this challenge, this paper proposes a novel low-complexity pilot-aided Doppler ambiguity detection and compensation framework. We first mathematically derive the OTFS input-output relationship in the presence of aliasing, revealing that Doppler ambiguity manifests itself as a distinct phase rotation along the delay dimension. Leveraging this insight, we developed a two-stage estimator that utilizes pairwise phase differences between pilot symbols to identify the integer ambiguity, followed by a refined Maximum Likelihood Estimation (MLE) for channel recovery. We investigate two pilot arrangements, Embedded Pilot with Guard Zone (EP-GZ) and Data-Surrounded Pilot (DSP), to analyze the trade-off between interference suppression and spectral efficiency. Simulation results demonstrate that the proposed scheme effectively eliminates the error floor caused by ambiguity, achieving Bit Error Rate (BER) and Normalized Mean Square Error (NMSE) performance comparable to the exhaustive search benchmark while maintaining a computational complexity similar to standard MLE.

The construction of Maximum Distance Profile (MDP) convolutional codes in general requires the use of very large finite fields. In contrast convolutional codes with optimal column distances maximize the column distances for a given arbitrary finite field. In this paper, we present a construction of such convolutional codes. In addition, we prove that for the considered parameters the codes that we constructed are the only ones achieving optimal column distances. The structure of the presented convolutional codes with optimal column distances is strongly related to first order Reed-Muller block codes and we leverage this fact to develop a reduced complexity version of the Viterbi algorithm for these codes.

Extending Katz-Sarnak heuristics, Ballini-Lombardo-Verzobio [BLV25] conjectures a limiting distribution as $p \to \infty$ for $\# A_g(\mathbb F_p,t)$, the number of $g$-dimensional PPAVs over $\mathbb F_p$ with trace $t$, as a product of natural local factors $v_\ell(t)$ for non-archimedean places $\ell$ and the Sato-Tate measure $\text{ST}_g$ corresponding to $\infty$. We prove that their conjecture is true for all $g$. As a consequence, we obtain analogous results on the distribution of curves of genus $2$ and $3$, answering questions of Bergström-Howe-García-Ritzenthaler [BHLR24] and [BLV25].

We consider a wireless network comprising multiple singleantenna repeaters that amplify and instantaneously re-transmit received signals in a full-duplex (FD) communication setting. Specifically, we study a massive multiple-input multiple output base station that simultaneously serves multiple uplink (UL) and downlink (DL) user equipment (UE) over the same frequency band. The focus is on the problem of repeater weight optimization at each active repeater to maximize the sum of the weighted minimum spectral efficiencies (SEs) for both UL and DL UEs. The resulting non-convex optimization problem is tackled using a successive convex approximation technique. To demonstrate the effectiveness of the proposed approach, we evaluate its performance against benchmark systems with and without repeater assistance. The optimized FD design achieves SE improvements of up to 4-fold and 2.5-fold compared to its half-duplex counterpart.

Using torus fibrations over K3 orbisurfaces, we construct new smooth solutions to the $G_2$ Hull-Strominger system. These manifolds arise as total spaces of principal $T^3$ (orbi)bundles over singular K3 surfaces. Our construction is based on the choice of three divisors on a singular K3 surface that are primitive with respect to a particular Kählermetric. The stable bundle is obtained via an adaptation of the Serre construction to the singular setting.

The effective sample size quantifies the amount of independent information contained in a dataset, accounting for redundancy due to correlation between observations. While widely used in geostatistics for scalar data, its extension to functional spatial data has remained largely unexplored. In this work, we introduce a novel definition of the effective sample size for functional geostatistical data, employing the trace-covariogram as a measure of correlation, and show that it retains the intuitive properties of the classical scalar ESS. We illustrate the behavior of this measure using a functional autoregressive process, demonstrating how serial dependence and the allocation of variability across eigen-directions influence the resulting functional ESS. Finally, the approach is applied to a real meteorological dataset of geometric vertical velocities over a portion of the Earth, showing how the method can quantify redundancy and determine the effective number of independent curves in functional spatial datasets.

This work introduces MADS-PIP, an efficient framework that integrates a penalty-interior point strategy into the mesh adaptive direct search (MADS) algorithm for solving nonsmooth blackbox optimization problems with general inequality and equality constraints. Inequality constraints are partitioned into two subsets: one treated via a logarithmic barrier applied to an aggregated interior constraint violation, and the other handled through an exterior quadratic penalty. All equality constraints are treated by the exterior penalty. A merit function defines a sequence of unconstrained subproblems, which are solved approximately using MADS, while a carefully designed update rule drives the penalty-barrier parameter to zero. In the nonsmooth setting, we establish convergence results ensuring feasibility for general constraints as well as Clarke stationarity for inequality-constrained problems. Computational experiments on both analytical test sets and challenging blackbox problems demonstrate that the proposed MADS-PIP algorithm is competitive with, and often outperforms, MADS with the progressive barrier strategy, particularly in the presence of equality constraints.

We prove that the motive of the punctual Quot scheme $\mathrm{Quot}^d(\mathscr O^{\oplus r}_{\mathbb A^n})_0$ stabilises, when $n \to \infty$, to $[\mathrm{Gr}(d-1,\infty)]\cdot \sum_{i=0}^{r-1}\mathbb L^{di}$. We similarly show that the Poincaré polynomial of the Quot scheme $ \mathrm{Quot}^d(\mathscr O^{\oplus r}_{\mathbb A^n})$ stabilises and we compute the limit in terms of the infinite Grassmannian. Finally, we prove that the motive of the nested Hilbert scheme stabilises to the motive of the infinite flag variety and we compute the cohomology ring in the limit. These results provide affirmative evidence to a question of Pandharipande concerning the cohomology of Quot schemes on $\mathbb A^\infty$.

For any $ν\in(\frac 37,\frac12)$, we prove the existence of an $H^1$ solution $u$ of the mass critical generalized Korteweg-de Vries equation on the time interval $(0,T_0]$, for some $T_0>0$, which blows up at the time $t=0$ and at the point $x=0$ with the rate $\|\partial_x u (t,x)\|_{L^2} \approx t^{-ν}$. Such a blowup rate is associated to a blowup residue of the form $r_α(x)= x^{α-\frac 12}$ for $x>0$ close to the blowup point, where $α=\frac{3ν-1}{2-4ν}$. The condition $ν\in(\frac37,\frac12)$ is equivalent to $α>1$, which corresponds to the full range for which the residue $r_α$ belongs to $H^1$. Such blowup at a finite point is in contrast with all the blowup solutions constructed for this equation, except the one constructed previously by the authors corresponding to the special value $ν=\frac 25$. Finally, we present some open problems regarding the blowup phenomenon for the mass critical gKdV equation.

We propose a systematic framework for constructing geometric integrators for Hamiltonian systems on Jacobi manifolds. By combining Poissonization of Jacobi structures with homogeneous symplectic bi-realizations, Jacobi dynamics are lifted to homogeneous Poisson Hamiltonian systems, enabling the construction of structure-preserving Jacobi Hamiltonian integrators. The resulting schemes are constructed explicitly and applied to a range of examples, including contact Hamiltonian systems and classical models. Numerical experiments highlight their qualitative advantages over standard integrators, including better preservation of geometric structure and improved long-time behavior.

We consider the stochastic heat equation on a compact smooth Riemannian manifold without boundary satisfying \begin{equation*} \partial_tu(t,x)=\frac{1}{2}Δ_Mu(t,x)+σ(t,x,u)\dot{W}(t,x),\quad (t,x)\in\mathbb{R}_+\times M, \end{equation*} where $\dot{W}$ is a centered Gaussian noise that is white in time and colored in space. Assuming that $σ$ is Lipschitz in $u$ and uniformly bounded, we estimate small ball probabilities for the solution $u$ when $u(0,x)\equiv 0$.

For $π$ a finitely presented group, Hausmann and Weinberger defined $q(π) \in \mathbb Z$ to be the minimum Euler characteristic over all closed, oriented $4$-manifolds with fundamental group $π$. This short note establishes that this minimum value in general differs depending on whether one minimizes over topological manifolds or only those admitting a smooth structure.

Identifying the effects of mechanical ventilation strategies and protocols in critical care requires analyzing data from heterogeneous patient-ventilator systems within the context of the clinical decision-making environment. This research develops a framework to help understand the consequences of mechanical ventilation (MV) and adjunct care decisions on patient outcome from observations of critical care patients receiving MV. Developing an understanding of and improving critical care respiratory management requires the analysis of existing secondary-use clinical data to generate hypotheses about advantageous variations and adaptations of current care. This work introduces a perspective of the joint patient-ventilator-care systems (so-called J6) to develop a scalable method for analyzing data and trajectories of these complex systems. To that end, breath behaviors are analyzed using evolutionary game theory (EGT), which generates the necessary quantitative precursors for deeper analysis through probabilistic and stochastic machinery such as reinforcement learning. This result is one step along the pathway toward MV optimization and personalization. The EGT-based process is analytically validated on synthetic data to reveal potential caveats before proceeding to real-world ICU data applications that expose complexities of the data-generating process J6. The discussion includes potential developments toward a state transition model for the simulating effects of MV decision using empirical and game-theoretic elements.

Structured perturbation results for invariant subspaces of $Δ$-Hermitian and Hamiltonian matrices are provided. The invariant subspaces under consideration are associated with the eigenvalues perturbed from a single defective eigenvalue. The results show how the original eigenvectors and generalized eigenvectors are involved in composing such perturbed invariant subspaces and eigenvectors.

We prove a triangular decomposition theorem for the lower crystal lattice $\mathcal{O}_{t}^{A_{0}}(G)$ of the quantized function algebra $\mathcal{O}_{t}(G)$, where $G$ is a connected simply connected complex Lie group with Lie algebra $\mathfrak {g}$ of type $A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$, $E_{6}$ or $E_{7}$. As a consequence, we prove the inclusion $\mathcal{O}_{t}^{A_{0}}(G)\subseteq\mathcal{O}_{t}^{A_{0}}(K)$ conjectured by Matassa \& Yuncken in these cases. We also give a precise definition of the specialization map used by Matassa \& Yuncken, which helps simplify their description of the crystallized algebra. This allows us to prove that the crystallized algebra $C(K_{0})$ is a compact quantum semigroup for the above mentioned cases, thus extending an earlier result for type $A_{n}$ compact quantum groups. As another consequence of the triangular decomposition, we prove that the notions of crystallized quantized function algebra given by Matassa \& Yuncken coincide with that of Giri \& Pal in the type $A_{n}$ case.

The $R$-hulloid, in the Euclidean space $\mathbb{R}^3$, of the set of vertices $V$ of a tetrahedron $T$ is the minimal closed set containing $V$ such that its complement is the union of open balls of radius $R$. When $R$ is greater than the circumradius of $T$, the boundary of the $R$-hulloid consists of $V$ and possibly of four spherical subsets of well defined spheres of radius $R$ through the vertices of $T$. The existence of a value $R^*$ such that these subsets collapse into a point $O^*$, in the interior of $T$, is investigated; in such a case $O^*$ belongs to four spheres of radius $R^*$, each one through three vertices of $T$ and not containing the fourth one. As a consequence, the range of $ρ$ such that $V$ is a $ρ$-body is described completely. This work generalizes to dimension three previous results, proved in the planar case and related to the three circles Johnson's Theorem.

We consider the localisation of the 2-category of diffeological groupoids at weak equivalences from the perspective of anafunctors, and with this language, prove that the localisation of the 2-category of Lie groupoids is an essentially full sub-bicategory of that of diffeological groupoids. In particular, we solve the open problem affirmatively of whether two Lie groupoids that are diffeologically Morita equivalent are Morita equivalent in the usual Lie sense.

We develop a theory of additive group actions on affine ind-schemes through a purely algebraic and topological framework. Affine ind-schemes are described via complete, second-countable, linearly topologized rings, and actions of the additive group are encoded by restricted exponential homomorphisms. We introduce the notion of a topologically integrable derivation, a continuous derivation whose formal exponential converges in the sense of restricted power series, and show that this notion provides the correct extension of locally nilpotent derivations to the infinite-dimensional setting. Our first main result establishes a one-to-one correspondence between topologically integrable derivations and additive group actions on affine ind-schemes, extending the classical correspondence for affine varieties. We then investigate the structure of such actions admitting a slice. In this context, we prove an ind-scheme analog of the classical slice theorem: if an additive group action admits a slice, then the underlying affine ind-scheme is equivariantly isomorphic to a product with the affine line, and the action is given by translation on the second factor. Several examples illustrate the necessity of the topological hypotheses and highlight phenomena absent in the finite-type case.

A class of continuous controllers termed Robust Integral of the Signum of the Error (RISE) have been published over the last decade as a means to yield asymptotic convergence of the tracking error for classes of nonlinear systems that are subject to exogenous disturbances and/or modeling uncertainties. The development of this class of controllers relies on a property related to the integral of the signum of an error signal. A proof for this property is not available in previous literature. The stability of some RISE controllers is analyzed using differential inclusions. Such results rely on the hypothesis that a set of points is Lebesgue negligible. This paper states and proves two lemmas related to the properties.

Gaussian process regression uses data measured at sensor locations to reconstruct a spatially dependent function with quantified uncertainty. However, if only a limited number of sensors can be deployed, it is important to determine how to optimally place the sensors to minimize a measure of the uncertainty in the reconstruction. We consider the Bayesian D-optimal criterion to determine the optimal sensor locations by choosing sensors from a candidate set of sensors. Since this is an NP-hard problem, our approach models sensor placement as a column subset selection problem (CSSP) on the covariance matrix, computed using the kernel function on the candidate sensor points. We propose an algorithm that uses the Golub-Klema-Stewart framework (GKS) to select sensors and provide an analysis of lower bounds on the D-optimality of these sensor placements. To reduce the computational cost in the GKS step, we propose and analyze algorithms for the D-optimal sensor placements using Nyström approximations on the covariance matrix. Moreover, we propose several algorithms that select sensors via Nyström approximation of the covariance matrix, utilizing the randomized Nyström approximation, random pivoted Cholesky and greedy pivoted Cholesky. We demonstrate the performance of our method on two applications: thin liquid film dynamics and sea surface temperature.

We construct five irreducible divisors in the moduli space of complex cubic fourfolds parametrising smooth cubic fourfolds that contain highly singular hyperplane sections. We prove that each is not a Noether-Lefschetz (or Hassett) divisor, utilising the computational method developed by Addington-Auel.

We recall that a parking function of length $n+1$ is said to be prime if removing any instance of 1 yields a parking function of length $n$. In this article, we study prime parking functions from multiple lenses. We derive an explicit formula for the average value of the total displacement of prime parking functions. We present a formula for the displacement-enumerator of prime parking functions that involves a sum over Łukasiewicz paths. We describe the one-to-one correspondence between parking functions and labeledŁukasiewicz paths via Dyck paths. We introduce the concept of $\ell$-forward differences and use this as a vehicle for examining ties, ascents, and descents in prime parking functions. We establish a link between Schur functions corresponding to the partition $(i,1^{n-i})$ and fundamental quasisymmetric functions indexed by prime parking function tie sets of size $n-i.$

In this note we discuss the Efimov effect emerging in a three-particle quantum system with zero-range interactions. In particular, we consider two non-interacting identical bosons plus a different lighter particle such that the interaction between a boson and the light particle is resonant. We also assume the validity of the Born-Oppenheimer approximation. Under these conditions, we show that the three-particle system exhibits infinitely many negative eigenvalues which accumulate at zero and satisfy the universal geometrical law characterising the Efimov effect. The result we find is a generalisation of previous results recently obtained in [13, 24].

In this paper, we discuss k-quasi-m-isometric completion problem of unilateral weighted shifts and composition operators on directed graphs with one circuit and more than one branching vertex.

We study the quantum Pais-Uhlenbeck oscillator at the resonant (equal-frequency) point, where the dynamics becomes non-diagonalisable and the conventional Fock-space construction collapses. At the classical level, the degenerate system admits more than one Hamiltonian formulation generating the same equations of motion, leading to a nontrivial quantisation ambiguity. Working first in the ghostly two-dimensional Hamiltonian formulation, we construct differential intertwiners that generate a spectrum-generating algebra acting on the generalised eigenspaces of the Hamiltonian. This algebra organises the generalised eigenvectors into finite Jordan chains and closes into a hidden $su(2)$ Lie algebra that exists only at resonance. We then show that quantising a classically equivalent Hamiltonian yields a radically different quantum theory, with a fully diagonalisable spectrum and genuine degeneracies. Our results demonstrate that the resonant Pais-Uhlenbeck oscillator provides a concrete example in which classically equivalent Hamiltonians define inequivalent quantum theories.

We deal with the numerical solution of the time-dependent partial differential equations using the adaptive space-time discontinuous Galerkin (DG) method. The discretization leads to a nonlinear algebraic system at each time level, the size of the system is varying due to mesh adaptation. A Newton-like iterative solver leads to a sequence of linear algebraic systems which are solved by GMRES solver with a domain decomposition preconditioner. Particularly, we consider additive and hybrid two-level Schwarz preconditioners which are efficient and easy to implement for DG discretization. We study the convergence of the linear solver in dependence on the number of subdomains and the number of element of the coarse grid. We propose a simplified cost model measuring the computational costs in terms of floating-point operations, the speed of computation, and the wall-clock time for communications among computer cores. Moreover, the cost model serves as a base of the presented adaptive domain decomposition method which chooses the number of subdomains and the number of element of the coarse grid in order to minimize the computational costs. The efficiency of the proposed technique is demonstrated by two benchmark problems of compressible flow simulations.

In this paper, we establish an angle duality and a gap principle for convex combinations of incomplete polynomials, extending two results of Ge and Gonek in [IMRN, 2024].Our approach is geometric: we introduce an ``angle gain'' mechanism for points inside the lune region and quantify how moving away from the unit circle forces a definite increase in the relevant angle functional.This yields a robust lower bound that is uniform under convex mixing and leads to the desired separation phenomenon.The main difficulty is that incomplete polynomials and their convex combinations may have highly nonuniform root distributions on the unit circle, so classical convex-hull type constraints are too coarse; one must instead control the local geometry of chords and boundary arcs and relate it to critical-point behavior through sharp trigonometric inequalities.