Papers

This paper addresses the efficient solution of hierarchical Bayesian inverse problems with a high- or infinite-dimensional parameter field and a moderate number of hyperparameters. We focus on a class of problems in which the parameter-to-observable mapping is a linear PDE, so that, for fixed hyperparameters, the problem becomes conditionally Gaussian. Marginalizing the hyperparameters entails repeated evaluation of their marginal density, which in turn entails repeated large-scale log-determinant ratio computations and maximum a posteriori (MAP) estimations. To address these computational challenges, we introduce a scalable framework that relies on generalized low-rank approximations of the update from the prior to the posterior precision matrix. Though such a framework is state-of-the-art in non-hierarchical settings, in the case of nonlinear prior hyperparameters such as covariance length scales, a direct application to hierarchical problems is inefficient. We propose two amortized variants of this framework, comparing their theoretical properties and computational complexity to the direct method. In numerical experiments, we evaluate their performance on an advection-diffusion initial condition inverse problem in two and three spatial dimensions, with hyperparameters in both the prior and the noise covariance. We find that our amortized methods achieve a factor of 30-45 speedup for 100 marginal density evaluations relative to the direct method in the 3D problem.

Romik has presented a construction of a 1-dimensional dynamical system on the unit interval by developing an algorithm that returns the unique sequence of matrices associated with a positive primitive Pythagorean triple (in the sense of Barning), and projecting the map involved in this algorithm onto an appropriate 1-dimensional space via stereographic projection. Romik additionally computes the infinite, absolutely continuous invariant measure, and shows that the system is conservative and ergodic. Later, Cha et al. provided a method of calculating "Berggren trees", which are generalisations of the tree of positive primitive Pythagorean triples one may construct via Barning's theorem, except for different homogeneous quadratic equations in 3 variables. We present here a method of computing 1-dimensional dynamical systems induced from these Berggren trees following Romik's outline, and determine their absolutely continuous invariant measures by adapting the method of Keane.

We study strong Morita equivalence for crossed products of noncommutative solenoids by cyclic groups. Using their description as twisted group $\mathrm C^*$-algebras, we construct automorphisms induced by the natural linear action of $\mathrm{SL}_2(\mathbb Z[1/p])$ on $\mathbb Z[1/p]^2$, for every integer $p>1$. For odd primes, we use the Heisenberg–Weyl representation to construct compatible cyclic group actions on the equivalence bimodules relating solenoids with reciprocal parameters. This yields strong Morita equivalences between the corresponding crossed products by $\mathbb Z$ and, for finite-order matrices, by finite cyclic groups. The corresponding equivalence for flip crossed products for the action of $\mathbb Z/2\mathbb Z$ holds for every prime, including $2$. For prime $p$, we give representatives for the nontrivial finite subgroups of $\mathrm{SL}_2(\mathbb Z[1/p])$ and explicit arithmetic criteria for conjugacy among these representatives.

Parameters of interest in causal inference, such as treatment or policy effects, can often be expressed as linear functionals of an outcome regression function. Automatic debiased machine learning (AutoDML) is a unified framework for obtaining asymptotically normal estimators of such parameters, which requires estimation of both a regression function and a Riesz representer. Existing AutoDML neural network architectures, such as RieszNet and MADNet, use a shared intermediate covariate representation. However, it remains unclear whether this shared representation should be predictive of the Riesz representer or the outcome. We show that a shared representation of the covariates that preserves predictive power of the outcome while discarding information about the Riesz representer is asymptotically more efficient than the baseline AutoDML estimator that uses all covariates. Motivated by these results, we propose the outcome-adapted AutoDML estimator and establish its asymptotic behavior in a sample splitting framework. We provide a neural network implementation of the estimator that learns a sparse representation of the covariates that is predictive of the outcome but not predictive of the Riesz representer. We demonstrate the efficiency gains of our estimator over existing alternatives on synthetic data and achieve state-of-the-art estimation accuracy on the semi-synthetic IHDP benchmark dataset.

Accurate channel estimation in orthogonal frequency division multiplexing (OFDM) systems remains challenging when demodulation reference signal (DMRS) observations are sparse and noisy, and when DMRS configurations vary across deployment scenarios. This paper proposes DANCE (Diffusion-based Noise-Adaptive Null-space Channel Estimation), a diffusion-based channel estimator for OFDM systems. We formulate DMRS-aided channel estimation as a sparse linear inverse problem whose measurement operator is induced by the pilot pattern. The resulting range-null space decomposition separates the measurement-constrained range-space component from the unobserved null-space component, which is reconstructed through a learned diffusion prior. To avoid directly imposing noisy pilot samples as exact constraints, DANCE introduces a noise-adaptive posterior correction into the reverse diffusion process. The correction coefficient and the residual sampling variance are jointly calibrated according to the observation noise level, thereby reducing pilot-noise injection while retaining useful measurement information. We further design a conditional U-Net denoiser for complex-valued OFDM channel grids, where the real and imaginary components are represented as separate feature channels and downsampling is performed only along the subcarrier dimension. Simulations based on 5G NR tapped delay line (TDL) and clustered delay line (CDL) channel models show that DANCE achieves consistently lower normalized mean squared error (NMSE) than conventional estimators and diffusion-based posterior sampling methods under different signal-to-noise ratios, DMRS configurations, Doppler frequency shifts, and train-test distribution mismatches.

We consider the Multiscale Single-Index Model (MSIM), first introduced in \cite{oymak2021learning}, as a stylized model for hierarchical learning with scale separation. Each layer extracts a shared single-index feature at one physical scale and passes it to the next, thus defining a tractable setting in which to study how deep architectures learn multiscale representations. Under non-degeneracy and delocalization assumptions on the link function and planted features respectively, for fixed depth $K$ and local scale $d$, the first Wiener chaos of the target behaves as a perturbed spiked tensor, where the perturbation of order $d^{-1/2}$ comes from the non-linearity – revealing the MSIM as a natural non-linear analogue of the Tensor PCA model \cite{montanari2014statistical}. While this perturbative picture is sufficient to enable efficient spectral recovery based on Tensor unfolding (as already observed in \cite{oymak2021learning}), it is not precise enough for the analysis of backpropagation gradient-based methods. In this work, we address this limitation by performing a fine-grained analysis of the Wiener chaos using Edgeworth expansions. In the first chaos, this gives a finite-rank hierarchy at scales $d^{-q/2}$. In higher chaoses, balanced flattenings exhibit staircase singular-value plateaus of size $d^{-\rho/2}$ and multiplicity $d^{\rho}$ under a natural higher-chaos non-cancellation condition. Using this higher-chaos structure, and under an additional slow Hermite-energy tail condition, we first establish shallow-network approximation lower bounds, quantifying the benefit of depth in this model. Next, and most importantly, we prove that online SGD on the correlation objective, where all layers evolve in the same timescale, achieves $1 - o_d(1)$ recovery with $n = \widetilde{O}( d^{K-1})$ samples, recovering the same sample complexity as in the linear counterpart.

We prove a coarea-type result for scalar functions $f$ in fractional Sobolev spaces $W^{s, p} (\Omega)$ with $\Omega\subset \mathbb R^n$, $0<s<1$, and $1\leq p < \infty$. Our theorem shows that a.e. level set has zero Hausdorff $\mathcal{H}^{n-s}$ measure, where the level set $f^{-1} (y)$ is defined as the set all points at which $y$ is between the $\liminf$ and the $\limsup$ (as $r\downarrow 0$) of the averages of $f$ over the balls $B_r (y)$. A quite general construction of random series of wavelets shows also that with probability $1$ (many) level sets have indeed dimension $n-s$.

Computing the null space and null vectors of large-scale matrices is a fundamental task in numerical linear algebra and scientific computing. In this paper, an LSQR-based algorithm, termed LSQRNV, is proposed to compute a null vector of a large-scale rank-deficient matrix $A$ from an initial vector. The theoretical convergence properties of the algorithm are analyzed, demonstrating that it converges to a numerical null vector of $A$ at a rate dictated by its numerical condition number, and a rigorous accuracy bound is derived for the resulting approximation. By integrating a deflation technique with a tailored termination criterion, LSQRNV is extended to LSQRNS, which computes an orthonormal basis for the numerical null space of $A$ and explicitly determines its nullity. The aforementioned accuracy bound is rigorously generalized to the computed approximate numerical null space. Furthermore, with appropriate parameter settings, LSQRNV efficiently determines whether a large matrix is numerically rank-deficient or has full column rank. Numerical experiments corroborate the theoretical results, demonstrating the robustness, efficiency, and effectiveness of LSQRNS for large-scale null-space computations.

Learning dissipative dynamics from discrete observations is essential for reliable long-horizon prediction and physically meaningful parameter identification. For linearly damped Hamiltonian systems, the exact flow is generally not symplectic but conformally symplectic, contracting the canonical symplectic form by a scalar factor that reflects the net dissipation. We propose Conformal Symplectic Networks with damping identification (CSympNet-ID), a discrete-time map-learning framework that learns the one-step flow map directly from snapshot pairs while enforcing exact discrete conformal symplecticity by construction, without penalty terms or projection. The architecture composes an exact symplectic neural core with explicit diagonal scaling layers whose factors are parameterized exponentially by a scalar damping-rate parameter, thereby guaranteeing positivity and interpretability of the learned dissipation factor. We establish a scaling-conjugacy factorization for conformal symplectic maps and derive a pointwise-in-step density result for CSympNet-ID. We evaluate an irregular-step damped oscillator, a damped spring-mass chain, a damped nonlinear cubic oscillator, and additional high-dimensional extensions. CSympNet-ID gives the most favorable overall results among the compared models in the reported experiments, particularly in data-scarce regimes, target contraction-law recovery, and high-dimensional tests where unstructured baselines degrade rapidly.

Let $\mathbb{N}_0$ be the set of all nonnegative integers. For a nonempty set $\mathcal{A}\subseteq \mathbb{N}_0$ and integers $n,h\ge 2$, let $r_{h}(\mathcal{A},n)$ be the number of representations of $n$ as $a_1+\cdots+a_h$, where $a_1\le \cdots\le a_h$ and $a_i\in \mathcal{A}$ for $i=1,\cdots,h$. In 2016, Chen and Tang showed that, for any given distinct positive integers $u_1,\cdots,u_k$ and positive rational numbers $\alpha_1,\cdots,\alpha_k$ with $\alpha_1+\cdots+\alpha_k=1$, there are infinitely many sets $\mathcal{A}\subseteq \mathbb{N}_0$ such that $r_{h}(\mathcal{A},n)\ge 1$ for all nonnegative integers $n$ and the set of $n$ with $r_{h}(\mathcal{A},n)=u_i$ has density $\alpha_i$ for all integer $i=1,\cdots,k$. In this paper, we consider the irrational numbers $\alpha_i$ as well. As a main result, we prove that, for any nonnegative numbers $\alpha_0,\cdots,\alpha_m$ with $\alpha_0+\cdots+\alpha_m=1$, there are infinitely many sets $\mathcal{A}\subseteq \mathbb{N}_0$ such that the set of $n$ with $r_{2}(\mathcal{A},n)=i$ has density $\alpha_i$ for all integer $i=0,\cdots,m$. Other related results are also contained.

A BHEP test for the null hypotesis of multivariate normality on the basis of incomplete data is introduced. Estimators for the underlying unknown parameters in this situation are suggested. The test uses characteristic functions and circumvents the problem of singular covariance matrix estimates. As the sample size tends to infinity, an almost sure limit of the test statistic is obtained under the null hypothesis and under alternatives. The convergence in distribution under the null hypothesis is also proved. Critical values can be obtained using a bootstrap procedure. Simulation studies investigate size and power of the test and confirm the adequacy of the approach. A real data example demonstrates the application of the test.

This article investigates the application and analysis of two substructuring waveform relaxation algorithms namely Dirichlet-Neumann Waveform Relaxation (DNWR) and Neumann-Neumann Waveform Relaxation (NNWR) for solving hyperbolic partial differential equations (PDEs) with time delay. These equations are relevant in numerous physical and engineering contexts, such as wave propagation, biological processes, and control systems, where the system's dynamics are influenced by past states. The study emphasizes the stability, convergence, and computational efficiency of these non-overlapping domain decomposition methods when applied to such problems. Specifically, the DNWR and NNWR algorithms are analyzed using both Fourier and Laplace transforms in asymmetric domain decomposition to assess their capability to manage delayed terms in hyperbolic systems. Using Fourier analysis, we establish linear convergence estimate for the numerical errors. Laplace transform analysis enables a more in-depth study for characterizing finite-step convergence. Additionally, we derive the optimal parameters required to achieve finite step convergence in presence of heterogeneous spatial domain. Theoretical findings are complemented by numerical experiments, showcasing the methods' effectiveness in maintaining accuracy while reducing computational complexity. Additionally, the study explores potential extensions to more complex problems and diverse applications.

In this article, we prove a general criterion for obtaining robust transitivity of partially hyperbolic geodesic flows. As a consequence, we exhibit the first example of a $C^2$ open set of Riemannian metrics with conjugate points and transitive geodesic flow. In particular, such a set does not intersect the set of Riemannian metrics with Anosov geodesic flows.

This paper studies finite-horizon optimal stopping problems for semilinear stochastic evolution equations in real, separable Hilbert spaces, together with their associated parabolic variational inequalities. We prove continuous differentiability of the value function in infinite dimensions, thereby obtaining a smooth-fit principle for the corresponding optimal stopping problem. The analysis has two parts. First, we prove existence and uniqueness, in a suitable weighted class, of a mild solution to the variational inequality and identify it with the optimal stopping value function. We also establish local spatial Lipschitz continuity of the value function by probabilistic methods, without requiring any smoothing property of the underlying transition semigroup. Second, under a global regularizing assumption on this semigroup, we prove higher-order spatial regularity: the value function is continuously Fréchet differentiable and its gradient is locally Hölder continuous. The abstract results are then applied to a stochastic heat equation with additive noise.

We develop a new approach to cross-intersection problems in extremal set theory. The method builds on the iterative procedure introduced by Kupavskii and Zakharov (2024) and the $t$-cover method. It provides a flexible framework for deriving extremal and stability results for cross $t$-intersecting families. Our approach applies to a variety of combinatorial objects. As an application, we prove a product version of the seminal Erdős–Ko–Rado theorem for sufficiently spread set systems. Two families $\mathcal{F}$ and $\mathcal{G}$ of $k$-subsets of $[n]$ are called cross $t$-intersecting if $|F\cap G|\geq t$ for all $F\in\mathcal{F}$ and $G\in\mathcal{G}$. We determine the families maximizing $\min\{|\mathcal{F}|, |\mathcal{G}|\}$ for large $n$ and all $t\ge2$, generalizing results of Mörs (1985) and Füredi (1995) for cross $1$-intersecting families. We then determine the families maximizing $|\mathcal{F}||\mathcal{G}|$ under the condition $\max\{|\cap_{F\in\mathcal{F}}F|,|\cap_{G\in\mathcal{G}}G|\}<t$ for large $n$. This improves the bound obtained by Frankl and Wang (2024), and provides a characterization of extremal configurations. For a family $\mathcal{F}$ of subsets of $[n]$, we introduce its $t$-diversity $\gamma_t(\mathcal{F})$, defined as the minimum number of sets from $\mathcal{F}$ not containing a fixed $t$-subset. This serves as a natural generalization of the important notion of diversity for $t=1$. We obtain a stability result via $\gamma_t$, and determine the maximum of $\min\{\gamma_t(\mathcal{F}),\gamma_t(\mathcal{G})\}$ for cross $t$-intersecting families $\mathcal{F}$ and $\mathcal{G}$. These yield new results for $t$-intersecting families, including a stability theorem towards a conjecture of Ellis, Keller and Lifshitz (2019), which may also be regarded as a $t$-intersection version, for large $n$, of an influential theorem of Frankl (1987).

Let $G$ be a finite group and let $\tau = (H_1, \ldots, H_t)$ be a $t$-tuple of core-free subgroups of $G$. We say that $\tau$ is regular if $G$ contains elements $g_1, \ldots, g_t$ such that $\bigcap_i H_i^{g_i} = 1$, which is equivalent to the existence of a regular $G$-orbit on the Cartesian product $G/H_1 \times \cdots \times G/H_t$. Regular tuples were first investigated by Anagnostopoulou-Merkouri and Burness in a paper from 2024, partly motivated by the aim of seeking a natural generalisation of the classical and widely studied concept of a base for a transitive permutation group, which aligns with the special case where the $H_i$ are pairwise conjugate subgroups. In this paper, we focus on the case where $G$ is a finite almost simple classical group and each $H_i$ is a maximal subgroup contained in Aschbacher's collection $\mathcal{S}$ of irreducibly embedded subgroups. Our main theorem determines all the non-regular $t$-tuples of this form with $t \geqslant 2$, which extends earlier work by Burness, Guralnick and Saxl in the base size setting. In particular, we deduce that every pair of maximal subgroups in $\mathcal{S}$ is regular if $n \geqslant 15$, where $n$ is the dimension of the natural module for the socle of $G$, and this lower bound is best possible.

Since its introduction, the Keller–Segel model has become a cornerstone in the mathematical theory of chemotaxis and it has generated extensive analytical activity. Most studies consider homogeneous Neumann boundary conditions, which ensure mass conservation and simplify the qualitative analysis of solutions. To the best of the authors' knowledge, at present chemotaxis models incorporating boundary conditions that generate inward fluxes have only been studied in two recent papers, and we believe that this topic deserves and it may attract further mathematical attention. In this sense, in the present paper we investigate a two-species chemotaxis system with positive total flux. The model consists of two interacting populations, $u$ and $w$, coupled through elliptic/parabolic chemical signals $v$ and $z$, and subject to Robin-type boundary conditions allowing inward fluxes that depend on the product of the cellular and chemical densities. Unlike the classical conservative setting, the total mass is not preserved and it exhibits quadratic growth in time, exactly in line with one of the investigations above mentioned and dealing with a single-species taxis model. We show that, within the considered framework, standard logistic damping is not sufficient to compensate for the mass increase induced by the positive boundary flux. To restore control of the dynamics, stronger dissipative mechanisms involving gradient-dependent damping terms are required. Under suitable assumptions, we establish the global existence and boundedness of classical solutions in the presence of logistic-gradient damping.

Uniformly weighted divergence preferences (UWDP) introduced in Maccheroni et al. (2006) are an important class of risk-averse preferences that contain as a special case the monotone mean–variance utility. UWDP are characterised by the lowest expected value of an act in $L^\infty$ under an adversarially chosen probability measure combined with the divergence of this measure. Our main result provides an alternative, computationally friendlier formula, which establishes in full generality that UWDP are the translation-invariant hull of state-independent expected utility over $L^0$. Some consequences of the new representation are studied.

We present a surrogate modeling strategy for Stokes flows with liquid droplets suspended in a carrier fluid. Our approach is based on a multi-fidelity framework. At the lowest fidelity, droplets are treated as passive tracers, neglecting their influence on the ambient flow field. Building on this approximation, we derive a PDE that represents the current modeling error. This error equation is then solved approximately to correct the flow field and the procedure is iterated. Two fidelities are employed in an alternating fashion: Stokes flow in the absence of droplets and flow around a single droplet in free space. By systematically combining these models, the method captures droplet-flow, droplet-boundary, and droplet-droplet interactions. In this work, the framework is developed and validated for circular, non-deforming droplets in two spatial dimensions. The geometric self-similarity of the droplets allows us to construct an efficient offline-online strategy based on the reuse of precomputed single-droplet solutions. Extensions to deformable droplets are also discussed. Numerical experiments demonstrate the accuracy and efficiency of the proposed surrogate in a variety of tests, including scenarios with up to $10^4$ droplets. Notably, we show that the proposed surrogate achieves substantially reduced computational cost compared to fully resolved multi-fluid simulations with state-of-the-art software.

Functional laws of large numbers (FLLNs) describe the mean-field trajectory of epidemics on networks, but say nothing about the fluctuations around it. These fluctuations are governed by moments of the degree distribution not relevant at the level of the mean. A rigorous functional central limit theorem (FCLT) exists for the susceptible–infected ($SI$) process on configuration-model graphs, but no analogue exists for $SIS$, where recovery reintroduces vertices into the susceptible pool with partially known neighborhoods, breaking the clean neighborhood distribution the $SI$ derivation relies on. We develop a tractable variance approximation for Markovian $SIS$ on configuration-model graphs, combining Gleeson's approximate master equation (AME) framework with a van Kampen system-size expansion in the spirit of the $SI$ FCLT. We derive a closed drift and diffusion matrix for a reduced susceptible/$SI$-edge/$SS$-edge count vector and obtain the time-dependent covariance via the associated Langevin/Lyapunov equation. Validation against Gillespie simulation across Poisson, regular, and power-law networks shows close agreement, with deviations near the epidemic threshold and in strongly heterogeneous networks.

There has been recent interest in integer partitions whose parts satisfy parity restrictions: for example, those where all the odd parts are distinct, or those where all the even parts are larger than the odd parts. Often results about such partitions have been obtained by algebraic manipulation of generating functions. We show that a number of these identities can be proved in a bijective, and sometimes simpler, manner.

We study the cohomology of regular semisimple Hessenberg varieties associated with lollipop graphs as a module under the dot action. Using the natural basis introduced by Cho, Hong, and Lee, which we call the CHL basis, we establish structural properties of the dot action, including a result for classes satisfying \(i\)-decomposability. We also obtain an explicit elementary symmetric function expansion of the chromatic quasisymmetric functions of lollipop graphs in terms of \(h\)-admissible permutations and their associated partitions. Combining these geometric and combinatorial results, we construct a permutation module decomposition of the cohomology of the corresponding Hessenberg varieties, thereby proving a conjecture of Cho, Hong, and Lee for lollipop graphs.

We prove the existence of a global family of steady, doubly periodic gravity-capillary waves on Beltrami flows. This is the first rigorous existence result for genuinely three-dimensional inviscid surface waves, with or without vorticity, beyond the perturbative regime close to simple explicit solutions. The proof is based on reformulating the steady water wave problem as a bifurcation problem of the form `identity plus compact' and applying a global bifurcation argument in Hölder spaces. The main challenge is that the kernel of the linearisation at the bifurcation point is two-dimensional, and that both kernel elements are necessary to obtain genuinely three-dimensional solutions. Since this prevents the use of classical global bifurcation theory, we introduce a novel reformulation of the bifurcation problem using the parameterisation of a local family of solutions bifurcating from laminar flow. In this new parameter space, we apply a variation of analytic global bifurcation theory. Along the branch, we then prove a sharper blow-up alternative, namely blow-up of the surface gradient in $C^{0,\gamma}$.

We establish refined spectral asymptotics for the operator curl acting on a connected oriented closed Riemannian 3-manifold. We treat positive and negative eigenvalues separately and obtain explicit formulae for the first six global Weyl coefficients. With local Weyl coefficients we compute the first four coefficients as well as the sixth one, and we determine the fifth up to a universal constant. As a consequence, we prove that the eta function of curl (both in its local and global versions) is holomorphic in the complex half-plane $\operatorname{Re}s>-2$. Finally, under appropriate assumptions on the geodesic flow, we improve Bär's asymptotic formulae for the positive and negative counting functions, refining the remainder to $o(\lambda^2)$.

On a negatively curved closed manifold, there exists a well-defined map $\Psi^\bullet$ associating to every closed differential form a bounded cohomology class via integration over straight simplices. Classes in the image of this map, which, a priori, depend on the fixed family of straight simplices, are usually called De Rham classes, and constitute an interesting subspace of bounded cohomology. In this paper we prove that, in sufficiently high degrees, $\Psi^\bullet$ is a homomorphism of algebras, i.e., it sends the wedge product of closed differential forms to the cup product of the associated bounded cohomology classes. The degree in which $\Psi^\bullet$ starts to preserve products depends on the boundedness of Jacobians of straight simplices. For the barycentric straightening introduced by Besson, Courtois and Gallot, this happens for degrees $\ge 3$. As a corollary, the cup product of two De Rham classes vanishes, provided that its degree exceeds the dimension of the manifold (and the degrees of both classes are $\geq 3$). This result complements vanishing results for the cup product of De Rham classes due to Marasco and to Battista et al.

We study numerical semigroups $S_{a,b}(m)$ generated by the orbit of $m$ under the affine map $T_{a,b}(z)=az+b$, where $a\ge 2$, $m>1$, $\gcd(b,m)=1$, and $b\ge -(a-2)m-2$. This extends the usual affine-closed setting to feasible negative values of $b$. We write $A_i=(a^i-1)/(a-1)$ and let $n$ be the smallest positive integer such that $A_n\ge m$. We determine the minimal generators and give an explicit description of the Apéry set, obtaining homogeneity and formulas for the Frobenius number and the genus. We also study pseudo-Frobenius numbers via the induced Apéry parametrization and prove the sharp upper bound $\operatorname{t}(S_{a,b}(m))\le n-1$ for the type. We give a complete characterization of the symmetric members of the family in terms of the canonical representative of $m-1$. Finally, we exhibit a subfamily whose pseudo-Frobenius numbers form an arithmetic progression of length $n-1$; in particular, this subfamily attains the bound.

We prove Lubin's conjecture for height-one commuting pairs of formal power series defined over the ring of integers $\mathcal{O}$ of any finite extension of $\mathbb{Q}_p$. Using some $p$-adic Hodge theory, we show that such a pair is a pair of endomorphisms of a formal group defined over $\mathcal{O}$.

Pressure-driven tumor growth models describe the coupling between cell proliferation and mechanical pressure and naturally lead to moving free boundary problems. Their numerical approximation is challenging due to degenerate diffusion, pressure-dependent proliferation, and the stiffness of the pressure law \(p=n^\gamma\) for large \(\gamma\). In this paper, we propose a structure-preserving finite difference method for this class of pressure-driven tumor growth models with pressure-dependent proliferation. The method is derived from the Onsager variational principle. The key idea is to introduce a modified energy shifted by the homeostatic pressure, so that the growth term can be written in a dissipative form and incorporated together with the transport part into a unified Rayleighian formulation. This formulation leads to a time-discrete constrained minimization problem and a fully discrete scheme with explicit mobilities and an implicit pressure update. We prove that the scheme preserves nonnegativity and the homeostatic upper bound, satisfies a discrete modified energy dissipation law, and admits a fixed-grid stiff-pressure limiting structure. Numerical experiments in one and two spatial dimensions demonstrate the accuracy of the method, its convergence toward the Hele–Shaw limit for large \(\gamma\), and its ability to capture free boundary evolutions with topology changes.

Unknown linear time-varying (LTV) systems require the control policy to adapt from online closed-loop data as dynamics evolve. Existing methods usually update the policy by solving a one-shot optimization problem, which can be computationally demanding and sensitive to noisy model estimates. In this paper, we propose a policy gradient adaptive control (PGAC) method for LTV system control with unknown model parameters. Specifically, PGAC integrates online policy optimization into feedback by updating the state-feedback policy with one-step gradient descent of the linear quadratic regulator cost at each time instant. This incremental update is computationally light and naturally limits policy variations caused by noisy data. To explicitly compute the policy gradient online, we estimate local models from recent closed-loop trajectories using normalized sliding-window least-squares. We provide stability and convergence certificates of PGAC for two classes of LTV systems. For slowly time-varying systems, we prove that the closed-loop system achieves practical exponential stability without a dwell-time condition. For piecewise-constant LTV systems, we establish practical stability through a dwell-time contraction argument. We also provide average frozen-time optimality-gap bounds of the policy sequence for both classes. Finally, we validate the effectiveness of our method via numerical case studies of both LTV and nonlinear systems.

The spatial anisotropic Kepler problem comes from quantum mechanics, which models electron motion in semiconductors with donor impurities and depends on the anisotropic parameter $\beta\in(-1,+\infty)$. After reducing the system modulo rotational symmetry, we investigate periodic orbits in this two-degree-of-freedom setting. Combining an index comparison in \cite{HLOQS26}, the volume formula in \cite{CHHL23} and Franks' theorem, we prove that the system possesses infinitely many periodic orbits on any fixed compact and regular energy surface for $\beta\in(-1,0]$.