Papers

We study an indexed family of functions closely related to functions ana- lytic and with a real positive part in the right open half plane (the so-called positive functions) and to the functions analytic in the right open half-plane and bounded in modulus by one there (the so-called bounded functions). These two families are related by the Cayley transform. In the present paper we introduce an affine linear relation- ship between a subclass of positive functions and the family of bounded functions, and study the corresponding connections with passivity of linear systems, interpolation and operator models. This is therefore a multidisciplinary paper, with potential readers from engineering, linear system theory and operator theory, and some repetitions of known results are given to allow various audiences to read the work. Reproducing kernel Hilbert spaces of analytic functions are a key tool in the arguments. A special role is played by the de Branges-Rovnyak spaces associated to bounded functions, and we prove a related trace formula connecting an underlying pair of operators.

We first prove spectral convergence of the random feature method (RFM) for multidimensional targets in Sobolev, Gevrey, ultra-analytic, and bandlimited classes. The analysis establishes general high-probability approximation estimates in the interpolation scale generated by a kernel integral operator. On a single event determined only by the sampled features, one random space approximates every target in a prescribed source ball; moreover, for each target, a single coefficient vector defines an approximant that attains spectral accuracy simultaneously in all admissible error norms. For both regularity-adapted frequency distributions and uniform distributions on growing frequency windows, the resulting rates range from super-exponential to algebraic, depending on the regularity of the target. Second, we establish abstract error estimates for strong- and weak-form RFM discretizations, thereby converting the preceding approximation bounds into convergence estimates for multidimensional second-order elliptic boundary value and eigenvalue problems. Finally, for random feature matrices (RFMtxs), we prove super-exponential singular-value decay with Fourier features and exponential decay with $\tanh$ features, together with corresponding condition-number lower bounds. The analysis identifies a common mechanism: the same spectral approximation that yields high accuracy also drives severe ill-conditioning.

Let $K_{a,m}$ and $K_{b,n}$ be missing-digit Cantor sets with initial consecutive digit sets, and write $c_a=m/(a-1)$ and $c_b=n/(b-1)$. We prove that if $c_a+c_b\geq 1$, then the radial projection of $K_{a,m}\times K_{b,n}$ from every observer has nonempty interior; no multiplicative-independence assumption is needed for this implication. Conversely, when the bases are multiplicatively independent and $c_a+c_b<1$, we exhibit explicit unbounded open sets of observers for which the radial image is compact and nowhere dense. Rational observers with the same property are dense in each exterior corner region and occur arbitrarily close to the four corners of the product. Thus $c_a+c_b=1$ is the exact threshold for the all-observers interior property within the multiplicatively independent initial-block family. In particular, this supplies an explicit two-set counterexample to the nonempty-interior conclusions of two conjectures of Yu. We also establish a sufficient modular phase obstruction for affine translates of such sets. Combining it with a fixed-pin positive-measure theorem of Banaji and Yu yields consecutive-block division sets that are compact, perfect, of positive Lebesgue measure, and nowhere dense, with both factor dimensions tending to one. For the same family, Fourier $l^1$-dimension estimates and the incidence argument used in Yu's product theorem imply that the two self-products contain intervals for all sufficiently large $r$ and $s$, while the cross-division set remains nowhere dense.

In the linear time-invariant framework, passive systems are modeled by positive real functions. Dissipative systems can be modeled by the subset of quantita- tively Hyper-positive real functions, related through nested inclusions. This family was introduced and studied in our three previous works. Here, we further focus our attention on the proper subset of canonical Hyper-Positive functions. Although this family is "small", its exploration is well motivated: First, this set turns to be associated with absolute stability (the Lurie problem). Then, a systematic parametrization of all canonical Hyper-Positive functions is introduced. Moreover each canonical Hyper-Positive function can be viewed as an extreme point of the convex set of Hyper-Positive functions. Specifically a convex combination of canonical Hyper-Positive is Hyper-Positive, but not canonical. These observations hold in both frameworks: of analytic functions and of state-space realization arrays. Technically, some of the analysis is facilitated by employing Quadratic Matrix Inclusions of both, matrices and of matrix-valued rational functions.

We establish the existence and uniqueness theorem for the exterior Dirichlet problem for the special Lagrangian equation with prescribed asymptotic behavior at infinity, in both the viscosity setting for all the phases and classical setting for the critical and supercritical phases. These results generalize previous work by the second author by removing restrictive assumptions on the asymptotic matrix and improving the decay rate to the order $2-n$. We also solve the interior Dirichlet problem for the critical special Lagrangian equation and, as applications, all the above-mentioned corresponding problems for the three dimensional quadratic Hessian equation without any admissibility condition.

A new condition involving the complexity factor, a structure scalar arising from the orthogonal splitting of the Riemann tensor, is generated at the boundary of a spherically symmetric anisotropic fluid. This condition leads to the generation of a model, without invoking any further constraints on geometry or matter variables. The model generated is physically reasonable and suggests that there is a non-zero, positive minimum constraint on the complexity scalar value evaluated at the surface. This suggests that vanishing complexity is a globalized constraint which does not apply at the boundary, specifically for compact objects in the strong-gravity regime.

Let $g$ be a $C^\infty$ diffeomorphism in the isotopy class of a pseudo-Anosov homeomorphism $f$ such that $g$ and $f$ have the same topological entropy. In 1988, Handel proved that this implies the existence of a semiconjugacy $\pi$ from $g$ to $f$. He stated that, in general, there is at least one point $x$ such that $\pi^{-1}(x)$ is disconnected. We show that this is not the case: for every $x$, the set $\pi^{-1}(x)$ is the intersection of a nested sequence of closed topological disks, and hence is connected. We also prove that there is a unique $g$-invariant probability measure projecting to the measure of maximal entropy of $f$. This measure is entropy-maximizing, hyperbolic, and Bernoulli, and the semiconjugacy induces a metric isomorphism between the corresponding measure-preserving systems.

We consider a vegetation-rainfall-bushfire interaction model consisting of two partial differential equations describing the evolutions of bushfire intensity and water availability, together with an ordinary differential equation governing the vegetation density. The main feature of the model is that the ODE contains a nonlocal coefficient multiplying the vegetation density, which makes the mathematical analysis highly nontrivial. In this paper, we regard this nonlocal coefficient as the generator of a deformation of the metric structure of the underlying real Hilbert space. This interpretation enables us to formulate the ODE as a twisted gradient flow on state-dependent Hilbert spaces and to establish the existence of strong solutions to the associated initial-boundary value problem.

Predicting storm surge induced by landfalling tropical cyclones is crucial for flood mitigation and coastal risk management. Traditionally, physics-based numerical models simulate storm surge by solving the Navier–Stokes equations using numerical methods, but these simulations are computationally expensive. Generative models are promising for storm surge emulation because they can generate diverse realizations rather than producing a single deterministic prediction. However, their use for storm surge emulation remains largely unexplored. In this paper, we leverage diffusion models for storm surge surrogate modeling, combining a baseline prediction stage with conditional generation to provide a more interpretable modeling framework. We develop SurgeGen, a two-stage generative framework for generating storm surge scenarios conditioned on hypothetical storms with parameters defined in a continuous space. First, a baseline model produces a coarse estimate of the storm surge height. This estimate then conditions a diffusion model, which generates refined storm surge scenarios that better capture spatial patterns and variability. We demonstrate that our approach can generate realistic and diverse storm surge scenarios under conditions both within and outside the training distribution.

We extend Balmer's tensor-triangular geometry to arbitrary presentably symmetric monoidal $\infty$-categories $\mathcal V$ by introducing the atomic smashing frame. This construction is functorial in $\mathcal V$, recovers the classical Balmer spectrum in the stable compactly-rigidly generated case, and yields a formulation of the telescope conjecture for arbitrary $\mathcal V$. Using dualizable and locally rigid $\infty$-categories, we identify smashing ideals with locally rigid localizations and establish recollement results for smashing frames and atomic smashing frames. As a consequence, we obtain a recollement principle for the telescope conjecture. We also establish descent for the telescope conjecture along several natural classes of coverings. As a principal application to chromatic homotopy theory, we analyze the structure of the smashing frame of $\mathrm{Sp}$. We prove a prime decomposition in terms of the smashing frames of the $p$-local $\infty$-categories $\mathrm{Sp}_{(p)}$ and, at each prime, an embedding into the product of the smashing frames of the monochromatic layers $\mathrm{Sp}_{T(n)}$. In particular, the spatiality of the smashing frame of $\mathrm{Sp}$ reduces entirely to that of $\mathrm{Sp}_{T(n)}$. Beyond the stable setting, we characterize the telescope conjecture for $\infty$-topoi in terms of smashing fields, and for connective module categories and hypercomplete connective sheaves in terms of Pierce-type conditions. Finally, we introduce the Serre smashing frame. Over a connective $\mathbb{E}_\infty$-ring $R$, this frame sits between the atomic and usual smashing frames, providing an intermediate structural layer in the study of the telescope conjecture for the connective $R$-module $\infty$-category.

The classical Banach principle relies on contractive sequences being Cauchy. Gregori and Sapena (2002) established a fuzzy analogue but required this as an extra hypothesis, leaving it as an open problem. Gregori et al. (2020) later asked whether strictly fuzzy $\psi$-contractive sequences are necessarily Cauchy. We settle both problems negatively via a single counterexample: in a constructed GV-fuzzy metric space, there exists a sequence which is both GS-contractive and strictly fuzzy $\psi$-contractive, yet not Cauchy. This shows that contractivity never implies Cauchyness in general GV-fuzzy metric spaces, and additional structural conditions in fuzzy fixed point theorems are indispensable.

We refine the definition of eventually homological isomorphisms so that it becomes more natural and fruitful. We introduce singular equivalences of $n$-adjoint type which can be viewed as standard singular equivalences of different levels and are more convenient for reducing various homological conjectures and transferring homological properties. We develop the theories of singular equivalences of $n$-adjoint type for $n\ge 2$ and standard eventually homological isomorphisms between derived categories by giving their characterizations in terms of $\otimes$-(co)perfect complexes of bimodules, two constructions via opposite algebras and tensor product algebras, examples induced by homomorphisms of algebras and idempotents, and applications to the reduction of homological conjectures such as the finitistic dimension conjecture, Gorenstein symmetry conjecture and Auslander-Reiten conjecture, as well as to the preservation of homological properties such as Gorensteinness, the Fg condition, Happel's property and Han's property. Moreover, we show that essentially surjective standard eventually homological isomorphisms between derived categories correspond to standard singular equivalences with canonical left adjoint.

We propose a geometric parametrization of the filters in a single convolutional layer: the parameter is no longer an ordered family of filter vectors, but a fixed-dimensional subspace of the filter space. For one-dimensional finite-stride convolution, the filter-to-convolution-operator correspondence gives an injective linear map $\mathcal{C}:\mathcal{K}\to H$. This map sends filter subspaces in $\mathrm{Gr}(q,\mathcal{K})$ to operator subspaces in $\mathrm{Gr}(q,H)$; composing it with the Plücker embedding yields a projective parametrization $\Phi:\mathrm{Gr}(q,\mathcal{K})\to\mathbb{P}(\bigwedge^q H)$. Using $T_U\mathrm{Gr}(q,\mathcal{K})\cong\mathrm{Hom}(U,\mathcal{K}/U)$, we compute the differential of the induced Grassmannian map and show that the differential of $\Phi$ is injective at every point. We then use the vanishing equations for Plücker coordinates and standard affine coordinates on a Grassmannian to prove that $\mathrm{Gr}(q,\mathcal{C}(\mathcal{K}))\hookrightarrow\mathrm{Gr}(q,H)$ is a closed embedding, and hence that $\Phi$ is a closed embedding. Consequently, the parameter space is isomorphic to its projective image, the parametrization is finite and birational onto its image, every fiber is a singleton, and the resulting projective neural variety is smooth. For $k=4$ and $q=2$, we also use Singular to recover the image ideal and check its dimension, degree, chart rank, and smoothness. This computation illustrates, rather than replaces, the general proof. Finally, we discuss possible connections with filter redundancy and low-rank convolution, while distinguishing the proved geometric results from application proposals requiring numerical validation.

The contact process with avoidance is a generalization of the classical contact process (SIS epidemic) that introduces a mechanism for healthy individuals to avoid their infected neighbors. Let $G$ be a directed graph. At each time $t$, each vertex is either healthy or infected and each edge is either active or inactive. Each infected vertex infects each healthy neighbor across each active edge at rate $\lambda$ and recovers at rate $1$, while each active edge pointing from an infected vertex to a healthy vertex becomes inactive at rate $\alpha$. An inactive edge becomes active when its tail vertex recovers. This model has been previously studied on $\mathbb{Z}$, the $n$-cycle $\mathbb{Z}_n$, and the $n$-star graph; here we extend the study of this model to lattices $\mathbb{Z}^d$, $d \geq 2$. We show that for every $d \geq 2$ and fixed $\alpha > 0$, there exist constants $\lambda(\alpha,d)^-$ and $\lambda(\alpha,d)^+$ such that for all $\lambda < \lambda(\alpha,d)^-$ the infection dies out almost surely while for all $\lambda > \lambda(\alpha,d)^+$ the infection persists indefinitely with positive probability. Furthermore, we show that both $\lambda(\alpha,d)^-$ and $\lambda(\alpha,d)^+$ scale like $1/d$ as $d \rightarrow \infty$ and that there exists a constant $C(\alpha)$ such that when $\lambda > C(\alpha)/d$ the process has a nontrivial invariant measure for $d$ sufficiently large. Our methods and most of our results also apply to the SIRS model and a related model in which infected vertices enter an isolated state at rate $\alpha$ and transition from both isolated and infected to healthy at rate $1$.

For the canonical independent and identically distributed (IID) single-user multiple-input multiple-output (MIMO) channel with full channel state information at the transmitter (CSIT), the asymptotic capacity per receive antenna is a waterfilling integral over the Marčenko-Pastur law, whose shape depends only on the ratio of transmit to receive antennas. Closed forms are known for every ratio other than the square one, and there only above a finite signal-to-noise ratio (SNR). The square ratio behaves differently because the Marčenko-Pastur support reaches the origin. The waterfilling cutoff then stays strictly inside the support at every finite SNR, and the water level has been available only numerically. This letter closes that gap with a classical trigonometric parametrization of the square law, which removes the edge singularity and replaces the moving cutoff by a single angle. Both the SNR and the capacity become explicit functions of that angle, so the capacity curve is traced by sweeping the angle rather than by solving a scalar constraint at each operating point. The same parametrization delivers the low-SNR behavior in closed form, where the slope of capacity in SNR equals the upper edge of the limiting spectrum.

Let $\delta\in\mathbb{F}_{2^n}$ satisfy $\operatorname{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(\delta)=1$. We study the permutation behavior of $$ f(x) = \left(\frac{1}{x^2+x+\delta}\right)^{2^k}+x $$ over $\mathbb{F}_{2^n}$. Helleseth and Zinoviev proved that $f(x)$ is a permutation for $k=0,1$, and remarked that numerical evidence suggests that no other cases occur. In this paper, we confirm their assertion by proving that, for $0\leq k<n$, $f(x)$ is a permutation of $\mathbb{F}_{2^n}$ if and only if $k=0$ or $k=1$.

We construct a gauge-invariant BF representative of a conical defect in Jackiw–Teitelboim gravity, with its elliptic direction fixed by the reconstruction of the gravitational variables. In the gravitational sector, the BF observable reproduces the metric defect insertion and its distributional curvature source, and fixes the holonomy around the defect to an elliptic conjugacy class. For a single defect on the disk, the position integral combines with the residual \(\mathrm{PSL}(2,\mathbb R)\) quotient, leaving a fixed elliptic sector. The exact boundary quantization of this sector gives the elementary kernel and one-defect disk amplitude directly, without using a matrix model. The same local BF operator describes both sharp and blunt defects. Their difference appears only when several defect positions are integrated. BF source composition makes the fractional deficits additive. Requiring a fused source to remain in the gravitational cone sector gives the admissibility condition for defect collisions, while positivity of the hyperbolic area on each component gives the corresponding stability condition. These conditions coincide with those for Hassett weighted curves. Combining these BF results with the known Hassett compactification and conical Weil–Petersson geometry yields the fixed-order genus-zero contact terms and recovers the cluster expansion of deformed JT gravity. Thus sharp and blunt defects differ through the compactified geometry of their relative positions, rather than through the elementary BF operator.

Erdős et al. (JCT-B, 1989) conjectured that, for integers $r\ge 2$ and $\delta\ge 2$ with $3r-1\mid\delta$, every connected $K_{2r+1}$-free graph of order $n$ and minimum degree $\delta$ has diameter at most $ \frac{3r-1}{r}\cdot \frac{n}{\delta}+O(1)$. Czabarka et al. (JCT-B, 2021) later proposed the following generalization: for every $k\ge 3$ and $\delta\ge\left\lceil\frac{3k}{2}\right\rceil-1$, every connected $K_{k+1}$-free graph of order $n$ and minimum degree at least $\delta$ has diameter at most $(3-\frac{2}{k})\cdot\frac{n}{\delta}+O(1)$. We disprove the latter conjecture, including its $k$-colorable version, for every $k\ge 7$ and sufficiently large $\delta$. When $k=2r\ge 8$ and $3r-1\mid\delta$, our construction also disproves the conjecture of Erdős et al. (JCT-B, 1989).

This paper addresses infinite-dimensional Bayesian inference for inverse problem of partial differential equations with model parameters in infinite-dimensional Hilbert space. To effectively incorporate prior information, we propose a novel continuous normalizing flows based infinite-dimensional model. Specifically, by introducing a well-defined neural ordinary differential equation in infinite-dimensional space, a simple reference measure can be transformed into a more complex measure which encodes the prior information. A corresponding theoretical framework is established to ensure the well-posedness of our proposed Bayesian prior in infinite-dimensional space. We also provide training methods of the prior for two distinct data settings, along with two sampling algorithms for the resulting Bayesian posterior. The proposed framework is applied to three representative inverse problems: the simple smooth inverse problem, inverse scattering problem, and the inverse heat conduction problem. Numerical experiments support the theoretical analysis and demonstrate the efficiency of the proposed algorithms.

In this paper, we study the exterior overdetermined problems for the homogeneous k-Hessian equations $$\sigma_k(D^2u)=0\quad\text{in}~\mathbb{R}^n\setminus\overline{\Omega}$$ in three dimensional regimes. For $2\le k<n$ and smooth strictly star-shaped domain $\Omega\Subset\R^n$, we establish ball rigidity results in all three dimensional regimes. If $2\leq k<\frac{n}{2}$, the solution with boundary value $-1$ and limit zero at infinity is treated under strict $(k-1)$-convexity of $\partial\Omega$. If $k=\frac{n}{2}$, the solution with boundary value $0$ and logarithmic growth at infinity is considered under strict $(k-1)$-convexity of $\partial\Omega$; if $k>\frac{n}{2}$, the solution with boundary $1$ and fundamental power-growth at infinity is considered under strict $(k-1)$-convexity of $\partial\Omega$. In each case, for a positive constant $c$, $|Du|=c>0$ on $\partial\Omega$ forces $\Omega$ to be a Euclidean ball and determines the solution explicitly. The three arguments are organized by the single parameter $q=(n-k)/k$ and the scale function \[ F_q(t)=\frac{t^{1-q}-1}{1-q},\qquad F_1(t)=\log t. \] After normalization, $U=F_q(v)$ and $M_q[v]=vD^2v-qDv\otimes Dv$ satisfies $\sigma_k(M_q[v])=0$. A unified contact-point calculation gives the sharp bound $|Dv|\le b$;a boundary viscosity contact then yields $H_k\ge qbH_{k-1}$. The integral mechanism that closes the argument changes at the critical exponent: Rellich–Pohozaev identities are used for $q>1$, a scale-invariant Wronskian–Newton current for $q=1$, and a renormalized Newton–Jacobi mass for $q<1$. When $q\leq 1,$ we also obtain a global strict k-convex defining function by Minkowski gauge, this help us extend the exterior construction from convex domains to strictly star-shaped domains.

We study the singularity type and models of the Kähler–Ricci flow on compact manifolds constructed from the 1-parameter foliation of a circle-bundle over a product of Kähler–Einstein manifolds $N := N_1 \times \cdots \times N_r$, with metric constructed using the ansatz considered in \cite{DW2011}, \cite{WW} et. al. In the earlier work \cite{FT} by the authors, we considered the “two-bolt” case where both ends of the foliation close with the “bolt” $N$. In this article, we continue our work on the more subtle “nut-bolt” and “two-nut” cases. The former has one end of the interval closes with a nut-type collapse (i.e. $N' := N_2 \times \cdots \times N_r$) and the other with a bolt (i.e. $N$). The compactification $\widehat{M}$ is then a $\mathbb{CP}^{m+1}$-bundle over $N'$. The “two-nut” case is one that both ends close with nut-type collapses, necessarily two of the $N_i$'s must be $\mathbb{CP}^{m_0}$ and $\mathbb{CP}^{m_\ell}$, and the compactification $\widehat{M}$ is a $\mathbb{CP}^{m_0+m_\ell+1}$-bundle over $\prod_{k\geq 3}N_k$. We proved that in all “two-bolt”, “nut-bolt” and “two-nut” caess the singularity must be of Type I. Furthermore, we study the pointed Cheeger-Gromov limit of the rescaled and dilated sequence of the flow in all of three cases, and prove that the limit model must be $(\Sigma^{m+1}, g_\Sigma(t)) \times (\mathbb{C}^{k}, \textrm{flat})$ with $m, k \geq 0$, where $\Sigma$ is one of the following: $\mathbb{CP}^{m+1}$, $\textrm{Tot}(\mathcal{L}^{\oplus(m+1)})$, or a projectivization $\mathbb{P}\big(\mathcal{O}^{\oplus(m_0+1)} \oplus \mathcal{L}^{\oplus(m_\ell+1)}\big)$ with $m_0 + m_\ell = m$, and $\mathcal{L}$ is a line bundle over the product of some of the $N_1, \cdots, N_r$ factors. The metric $g_\Sigma(t)$ is a Kähler-Ricci shrinker satisfying the circle-bundle ansatz.

Let $P$ be a $d$-dimensional lattice polytope with Ehrhart polynomial $L_P(t)$. Motivated by the study of Ehrhart positivity and magic positivity, we investigate the Taylor coefficients $\mathsf{A}_j(P;k)$ in the shifted expansion $L_P(t)=\sum_{j=0}^{d}\mathsf{A}_j(P;k)(t-k)^j$ about a real center $k$. In this paper, we obtain the following four main results. (i) We give exact formulas for these coefficients in terms of the ordinary Ehrhart coefficients, the $h^*$-vector, elementary symmetric functions, and Stirling numbers. (ii) We denote by $\tau(P)$ and $\tau^+(P)$ the smallest nonnegative integral centers at which all Taylor coefficients are nonnegative and positive, respectively. If $s$ is the degree of the $h^*$-polynomial, then $0\leq\tau(P)\leq\tau^+(P)\leq\min\{\max\{0,s-1\},\lfloor\frac{d-1}{2}\rfloor\}$. As an application, we slightly improve an upper bound due to Beck, De Loera, Develin, Pfeifle, and Stanley. That is, every real root of $L_P(t)$ lies in $[-d,\lfloor\frac{d-1}{2}\rfloor)$. (iii) Let $\rho(P)$ be the smallest nonnegative real center such that the Taylor coefficients are nonnegative. If $\lambda_{\mathbb{R}}(f)$ denotes the largest real zero of $f(t)$, with value $-\infty$ when no such zero exists, then $\rho(P)=\max\{0,\max_{0\leq j<d}\lambda_{\mathbb{R}}\!(L_P^{(j)})\}$. (iv) We establish structural properties of the Taylor coefficients $\mathsf{A}_j(P;k)$, including derivative interlacing, palindromic reflection symmetries, and Laguerre and Newton inequalities. As a final note, these results provide a systematic partial answer to an open problem listed on the website of the American Institute of Mathematics.

Affine-invariant ensemble samplers are widely used in Bayesian applications. However, their quantitative convergence theory, in particular geometric ergodicity, remains a basic open question. We study the affine invariant ensemble Langevin dynamics, an interacting particle system that uses the empirical covariance of the whole ensemble as a preconditioner. While effective in practice, theoretical understanding of this method is not available beyond plain qualitative convergence in total variation, with a central difficulty being that the empirical covariance can become degenerate. For potentials with bounded Hessian that are strongly convex outside a ball, we prove geometric ergodicity using a novel Lyapunov function that combines the inverse covariance with a coercive exponential energy. We then show that directly applying the Euler–Maruyama scheme can diverge with positive probability, even for a one-dimensional Gaussian target. This motivates a covariance-trace regularization in continuous time. We prove geometric ergodicity of the regularized diffusion and, for sufficiently small step size, of its unadjusted Euler–Maruyama discretization. We also show that the invariant distributions of the discretization converge weakly to the product target distribution as the step size tends to zero. Lastly, we provide an explicit discretization that is both affine-invariant and geometrically ergodic, based on an adaptive step size clamping.

In this paper, we introduce generalized Hankel/Toeplitz matrices (GHM/GTM) and the associated generalized Vandermonde decomposition for nonuniform array signal processing and multi-dimensional super-resolution. The proposed framework was discovered from the study of resolution limit theory and extends the classical Hankel/Toeplitz structure by allowing substantially more flexible sampling geometries while preserving the underlying low-rank Vandermonde factorization. Through devising an optimal algorithm based on this GHM framework, we derive the state-of-the-art upper bound estimate for the computational resolution limit (CRL) of source-number detection in general $d$-dimensional super-resolution problems. For segmented sampling sets, whose geometry is closely related to sparse and distributed arrays, we establish deterministic lower bounds for the minimum singular values of the associated generalized Vandermonde matrices and derive corresponding stability and number-detection guarantees for multi-clump source configurations. To address the computational bottleneck of conventional multi-level Hankel constructions in high dimensions, we further introduce randomized GHM constructions whose matrix dimensions scale with the effective degrees of freedom rather than with the full tensor-product grid, together with deterministic recovery guarantees conditional on the realized Vandermonde factors. We also extend the framework to source localization by developing GHM-based MUSIC algorithms for nonuniform measurements, with stability characterized through the conditioning of the generalized Vandermonde factors. Numerical experiments on synthetic data demonstrate that the proposed GHM-based methods achieve competitive resolution and recovery accuracy while substantially reducing matrix size and computational cost, especially in high-dimensional settings.

Let $D=2^n$ and $M=3n+9\binom n2$. We study the right-invariant one-step-cliff metric $d_Q$ on $\operatorname{PU}(D)$, with unit penalty on Pauli weights one and two and penalty $Q$ on all higher weights. If $M/Q_D\to0$ and $MQ_D^{3/4}/D^2\to0$, then, for every fixed $0<x<\pi/\sqrt3$, \[ \mu_D\bigl(B_{Q_D}([I],x\sqrt{Q_D})\bigr)\le e^{-c_xD^2}. \] Here $\mu_D$ is normalized Haar measure. Thus the Haar-typical distance from the identity and the diameter are both asymptotic to $(\pi/\sqrt3)\sqrt{Q_D}$ throughout the window $M\ll Q_D\ll D^{8/3}M^{-4/3}$. Choosing $Q_D=\kappa D^{8/3}M^{-4/3}$ with sufficiently small fixed $\kappa>0$ yields a Haar-typical lower bound of order $D^{4/3}M^{-2/3}$ for the corresponding infinite-cliff Carnot–Carathéodory distance, outside an $e^{-\Omega(D^2)}$ exceptional set. The infinite-cliff diameter therefore has exponential lower rate at least $4/3$, disproving Brown's exponent-one conjecture. The same estimate gives a fixed-error no-ancilla two-qubit circuit lower bound of the same order.

More than twenty years ago, Baouendi and the first author proved that a holomorphic map between hyperquadrics of the same signature is either totally degenerate or has a nonvanishing normal derivative for its normal component. This established a CR analogue of the classical Hopf lemma in arbitrary codimension, in the absence of pseudoconvexity. They further conjectured that the same Hopf-type property holds for holomorphic maps between Levi-nondegenerate hypersurfaces of the same signature. In this paper, we provide a counterexample to this conjecture in full generality. We also prove the conjecture when the target hypersurface is a hyperquadric of any codimension, arguably the most important case for applications.

Motivated by extending the Morrison-Kawamata cone conjecture beyond Calabi-Yau varieties and Severi-Maehara type finiteness results to targets not necessarily of general type, we fix a smooth projective variety and study the finiteness of its log canonical models as klt boundaries vary. Our main result establishes finiteness for every smooth projective minimal surface. For each $\kappa\in\{-\infty,0,1\}$, we construct a smooth projective non-minimal surface of Kodaira dimension $\kappa$ with infinitely many log canonical models, showing that the minimality assumption cannot be omitted in general. We also investigate possible extensions of this finiteness result to higher dimensions.

In this paper, we mainly investigate $n$-cotorsion pairs in two types of abelian categories. Firstly, we construct left (right) $n$-cotorsion pairs in comma categories by improving some isomorphisms between homology groups and combining some classes of objects in comma categories. Secondly, based on known homological formulas over trivial ring extensions, we study some classes of modules satisfying certain conditions over such rings, and we establish left (right) $n$-cotorsion pairs over trivial ring extensions through these classes. Then corresponding conclusions over Morita rings with zero bimodule homomorphisms are directly presented. The hereditary property of left (right) $n$-cotorsion pairs is investigated in both cases. Finally, we apply main results obtained in comma categories and over trivial ring extensions to formal triangular matrix rings. At the same time, via these applications, we found that relevant conditions in known results have been improved and the known results have been elevated.

We establish two nonuniqueness phenomena on $\mathbb{CP}^1$. First, distinct normalized positive-mass model potentials can have the same non-pluripolar Monge–Ampère measure. By the contact-set formula, their zero-contact sets agree $\omega_{\mathrm{FS}}$-almost everywhere. Moreover, both have zero Lelong number at every point and identical multiplier ideal sheaves at every positive scale. Second, we construct distinct maximal test curves whose zero-contact sets agree pointwise at every parameter. Their inverse Legendre transforms are distinct bounded weak geodesic rays from zero with the same pointwise right initial tangent.

We use representation theory to calculate the $\pi$-electron energy levels of Buckminsterfullerene and other symmetrical conjugated systems according to the Hückel approximation.