Papers

We consider one-parameter families of random circle diffeomorphisms $g_{E,y}$ for which the unperturbed map $g_{0,\bar{0}}$ has a fixed point of order $2k$ and the dependence on the parameter $E$ is monotone. Under reasonable assumptions, we show that the rotation number $\rho(E)$ exhibits Lifshitz tail decay with exponent $-\frac{2k - 1}{2k}$, \[ \lim_{E \downarrow 0} \frac{\ln(-\ln(\rho(E) - \rho(0)))}{\ln(E)} = -\frac{2k-1}{2k}. \] The exponent is determined by the passage time through a parabolic bottleneck. A full rotation requires on the order of $E^{-\frac{2k - 1}{2k}}$ successive small perturbations, and the probability of such a streak decays exponentially as a function of its length. When $k=1$, the exponent is $-1/2$, and we recover as a corollary a purely dynamical proof of Lifshitz tail asymptotics at the spectral edges of the one-dimensional Anderson model.

We study the long-time behavior of a triangular system of Fisher–KPP type with $k$ interacting components, associated with a reducible multitype branching Brownian motion with $k$ types of particles. For this cascading system, we prove convergence in shape of each component to the minimal-speed Fisher–KPP traveling wave and determine the front asymptotics up to the constant order. This yields a PDE proof of Conjecture 1.2 from [4] on the convergence in distribution of the centered maximum particle in a cascading branching Brownian motion. We also derive asymptotic front-location estimates for such systems with general Fisher–KPP nonlinearities.

Graphs with bounded treewidth and bounded maximum degree are known to have tree-partitions of bounded width. What can be said if the bounded treewidth assumption is strengthened to bounded pathwidth? We prove that every graph with bounded pathwidth and bounded maximum degree has a tree-partition of bounded width, with the extra property that the underlying tree has bounded pathwidth. Moreover, we prove a lower bound showing that the bound on the pathwidth of the underlying tree is within a constant factor of optimal.

This paper studies major-minor linear-quadratic mean field games (MMLQMFGs) with erroneous initial information under a constrained observation structure. Each minor agent observes only its own state and the major agent's state, while the major agent observes its own state and the states of a subset of minor agents; neither side observes the mean field state directly. We show that the initial-information errors propagate linearly through the game dynamics and lead to explicit deviations in the major state, the actual mean field, and the agents' internally updated mean field states. Based on this structure, we formulate distributed error identification as a parameter-estimation problem from discrete-time local observations and construct maximum-likelihood estimators for unknown initial errors. We then propose an estimate-based strategy modification at an intermediate time by reconstructing the current mean field from the estimated errors and switching to the corresponding control law. We also characterize the resulting estimation errors and show that, in the present symmetric setting, the major agent's estimation precision depends on the number of observed minor agents but not on their identities. Numerical results illustrate the proposed method.

Starting from the axiomatic formulation of scale-space theory, we develop a viscosity-solution framework for multiscale image representations arising from degenerate elliptic-parabolic partial differential equations. Rather than introducing a new semigroup theory, we work within the standard viscosity-solution setting, using comparison principles to obtain well-posedness, uniqueness, and contraction in the supremum norm. This perspective is used to motivate a hybrid reconstruction operator in which a learned inverse map is followed by a nonlinear diffusion evolution. At the continuous level, the diffusion operator satisfies non-expansiveness, which provides stability for the reconstruction process; this framework is then evaluated on a CT-based mesothelioma classification task, where it attains an AUC of 0.875 with negligible variation across epochs, while the baseline model acquires AUC values from 0.49 to 0.80 without a clear convergence pattern. These observations are consistent with the stabilizing role suggested by the discussed viscosity theory.

Solutions to well-posed problems in continuum mechanics are continuously dependent upon prescribed boundary conditions. Because of this, variations in the enforcement of boundary data can impact the reliability of inversion techniques that rely on efficient and accurate forward models. To this end, it is necessary to understand how specific boundary implementation techniques can affect the performance of a given forward model. Our work focuses on the impact that key modeling decisions have on physics-informed neural network (PINN) solutions for initial boundary value problems in continuum mechanics. By interpolating boundary data over implicit boundary representations, we measure the performance of a physics-informed neural network across different configurations of soft and hard boundary enforcement. We target the problem of elastodynamic plane-strain and present a method of hard-enforcement of traction conditions over arbitrary, implicitly-defined, domain boundaries considering both first and second order formulations of the governing equations. We show that PINNs achieve a higher relative accuracy when solving the first-order plane strain problem and we observe a tradeoff between the final relative error and the total run time to complete training. This tradeoff is characterized by the number of hard and soft boundaries where, in the extremes, all soft-enforcement results in greater accuracy with a longer run time, while all hard-enforcement leads to lesser accuracy and a shorter run time.

The anthyphairetic process leads from a pair (a,b) of coprime positive integers to the pair (1,1) by successive subtractions of the smaller number from the bigger one. This process, which is a slow version of Euclid's algorithm applied to the pair (a,b), corresponds naturally to the process of successive blowups leading to the minimal embedded resolution of the plane curve defined by y^a - x^b = 0. This blowup process may be represented graphically by a special two-dimensional simplicial complex called a lotus. This allows to localize the various numbers appearing either during the anthyphairetic process or during the Euclidean algorithm at precise positions inside the lotus. In this introductory article, I recall first the construction of this lotus starting from the sequence of quotients generated by the Euclidean algorithm. I present then an alternative way of constructing it directly from the sequence of pairs of coprime integers generated by the anthyphairetic process, using what I call anthyphairetic rectangles. I conclude by explaining how to reconstruct from a lotus the corresponding sequence of pairs of coprime integers. This is a simple illustration of the way lotuses may serve as computational architectures.

FENE-type conformation-tensor models impose a finite-extensibility constraint that is absent from Oldroyd–B flow: the conformation tensor must satisfy $\CC\succ0$ and $\tr\CC<L^2$. Positive definiteness alone is therefore insufficient, since a numerical state can remain positive while crossing the singular trace barrier. Even a trace-preserving logarithmic parametrization is not enough by itself: high-order reconstruction can remain inside the finite-extensibility domain while injecting artificial FENE entropy. We develop and analyze a barrier-preserving entropy-compatible discretization for FENE-P type flows with polymer center-of-mass molecular diffusion and for trace-singular FENE-family closures with the same entropy structure. The method combines a trace-barrier free energy, a finite-extensibility logarithmic parametrization, a least-damping entropy-compatible barrier-log reconstruction, molecular diffusion paired with the barrier entropy variable, compatible quadrature for polymeric work, and a scaled FENE stress variable for the small-Weissenberg limit. For admissible discrete states we prove finite-extensibility preservation at entropy quadrature points, existence and bisection computability of the maximal entropy-admissible reconstruction parameter, a fully discrete free-energy inequality with relaxation and molecular-diffusion barrier dissipation, a quantitative AP stress closure, and a fixed-discretization Newtonian limit. A conditional relative-entropy estimate is derived on compact subsets of the finite-extensibility domain. Numerical diagnostics verify barrier preservation, entropy-compatible reconstruction, energy decay, AP closure, coupled velocity–pressure–stress accuracy, and high-Weissenberg robustness near the trace constraint.

Root-MUSIC is a spectral estimation algorithm that approximates the unknown signal frequencies by constructing a high-degree polynomial and finding a subset of roots which are closest to the complex unit circle. Previous works found asymptotic expectation formulas for the performance of Root-MUSIC under the implicit assumption that the aforementioned root selection criterion does not select extraneous roots – those which are unrelated to the correct parameters. This paper removes the need for this assumption by showing all extraneous roots lie outside an annulus of a certain thickness and therefore are not selected by the algorithm. This paper also provides sharp and non-asymptotic error bounds for the correct roots in terms of fundamental model parameters. All results hold under a natural separation condition on the correct signal frequencies and are applicable in both the single- and multi-snapshot models. Numerical simulations confirm our theory. The main mathematical insight of this paper is a geometric property of the Root-MUSIC polynomial: its correct roots are highly stable to noise while its extraneous roots must lie outside of an annulus.

Accurate channel state information (CSI) acquisition is critical for exploiting the spatial flexibility of fluid antenna systems (FASs). However, port selection and transmission optimization require CSI over a large number of candidate port positions, making direct port-wise estimation prohibitively costly in terms of pilot overhead. This paper addresses this challenge through geometry-structured channel reconstruction, which exploits the fact that the port-domain CSI can be parameterized by a small number of dominant propagation paths. We first establish fundamental mean square error (MSE) and normalized MSE (NMSE) benchmarks for both geometry-structured and unstructured channel reconstruction, providing analytical references for evaluating the intrinsic benefit of geometric modeling in conventional antenna systems and FASs. Motivated by the strong spatial correlation induced by densely distributed fluid antenna ports, we further propose a Bayesian reconstruction framework, termed geometry-structured expectation-maximization approximate message passing (GS-EM-AMP). The proposed algorithm incorporates geometric channel structure into the EM-AMP procedure and adaptively learns unknown statistical parameters from noisy observations. Numerical results demonstrate that GS-EM-AMP achieves near-bound reconstruction accuracy while maintaining strong robustness against steering-domain correlation, thereby offering an efficient and reliable solution for large-scale CSI acquisition in FASs.

Over the centuries, probability theory has grown from the calculus of games of chance into a central framework for reasoning under uncertainty. This article interprets that evolution not merely as a mathematical history, but as a transformation of rationality itself. From Pascal and Fermat's combinatorial symmetry to the inductive logic of Bayes and Laplace, from Poisson's statistics of events to Kolmogorov's axiomatic formalization, probability progressively incorporated uncertainty, time, and coherence into scientific judgment. This trajectory reaches a mature epistemological form in modern Bayesian inference, especially in Tarantola's view of probability as a logic of information, where prior knowledge and data are combined coherently. Yet this framework also exposes a limit: probability quantifies uncertainty about well-defined propositions, but does not by itself formalize the vagueness of the concepts used to describe them. The article therefore examines how rationality extends beyond probability. Fuzzy logic is presented as a rigorous language for graded meaning and qualitative judgment, while deep learning is analyzed as a distinct, powerful mode of prediction based on geometric interpolation and optimization rather than explicit inference. By situating probability, fuzzy logic, and deep learning in a common historical and epistemological perspective, the article clarifies their roles and limits. It argues that contemporary scientific rationality cannot be reduced to data-driven performance alone, but requires the explicit articulation of uncertainty, vagueness, and inference.

We develop a geometric convergence theory for neural-network optimization within the minimizing movement scheme (MMS) framework. Reformulating each neural MMS step as a minimization over the set of increments in a Hilbert space, we show that under a $C^2$ network with locally non-degenerate Jacobian this increment set is a boundaryless smooth embedded submanifold, on which a natural preconditioned (Gauss–Newton-type) gradient flow in parameter space induces exactly the Riemannian gradient flow. Under a strict interior-localization condition and an explicit data condition, the reached sublevel set is geodesically convex and the subproblem objective is geodesically strongly convex on it; both the continuous Riemannian gradient flow and its discrete companion via the exponential map converge linearly to the unique subproblem minimizer. Propagating finite-time inner-solver inexactness and neural-approximation error through the MMS iterations yields a uniform function-space tracking bound and an explicit trajectory budget, so the inexact neural iterates converge to an $O(\delta)$-neighborhood of the global minimum. Numerical experiments on nonlinear regression and a small-scale latent-diffusion testbed indicate that the Gauss–Newton-type inner solver achieves smaller trajectory errors with substantially fewer inner iterations than first-order baselines.

The Mycielskian is a standard construction studied in many an introductory graph theory course. It is natural to consider Mycielskians of cycles, some of the simplest of all graphs. This paper deals with the so-called “dimension” of such graphs. The dimension of a graph $G$ is the smallest positive integer $n$ such that there exists a one-to-one correspondence between the vertices of $G$ and some collection of points in $n$-dimensional Euclidean space such that if two vertices in $G$ are adjacent, then the distance between the corresponding points is $1$. In previous works, it had been proven that the dimension of the Mycielskian of a $k$-cycle is 3 when $k$ is $3$, $4$, or $5$, and 2 when $k=10$. In this paper, we answer the question completely. Namely, we show that the dimension is $3$ when $k\neq 10$, and $2$ when $k=10$.

We study regular two-dimensional free boundary minimal $T$-surfaces in the unit ball $\B^n$, with six faces, four triple junction arcs, and one tetrahedral vertex. A conformal minimal immersion of the standard flat $T$-cone, with the prescribed matching of junction traces, has the flat cone as its image. We also obtain rigidity criteria that do not require matching conformal parametrizations. In the disk-type class, two opposite planar faces suffice. More generally, it is enough that a natural junction invariant vanish on both junction edges of two opposite faces. For variations preserving the junction identifications and keeping the outer boundary on the sphere, we compute \[ \operatorname{ind}(TC)=n, \qquad \operatorname{nul}(TC)=3(n-2). \] The normal kernel consists exactly of the normal components of ambient rotations, and we compute the associated coupled harmonic Steklov spectrum. Every regular free boundary minimal $T$-surface with six faces has index at least $n$, with equality if and only if it is the flat $T$-cone; the same characterization holds for ambient variations. Finally, the flat cone is locally rigid, modulo orthogonal transformations, among sufficiently small compatible facewise $C^2$ free boundary minimal deformations.

We study the query complexity of sampling from high-dimensional Gaussian distributions using gradient information. In the standard oracle model, exact gradients expose only matrix-vector products with the precision matrix, leading to polynomial approximation barriers and a characteristic $\sqrt{\kappa}$ dependence on the known spectral range. We show that this barrier disappears when the sampler is allowed to query smoothed scores, namely gradients of the logarithms of the Gaussian-convolved densities. For a centered Gaussian target with precision matrix $\Lambda$, a smoothed-score query at noise level $\tau$ gives access to the resolvent $(\Lambda+\tau^{-1}I)^{-1}$. Combining geometrically spaced noise levels with sinc-quadrature rational approximation, we obtain a sampler with $q=O((\log\kappa+\log(e\sqrt d/\delta_{\rm TV}))\log(e\sqrt d/\delta_{\rm TV}))$ smoothed-score queries for total variation error $\delta_{\rm TV}$, improving the condition-number dependence from $\sqrt{\kappa}$ to logarithmic. We also study finite-bit gradient oracles. Using coordinatewise quantization of the transformed smoothed-score answers and a final dithering step, we obtain a sampling scheme whose total communicated gradient information is polylogarithmic in $\kappa$; in particular, for fixed dimension and accuracy, the bit complexity is $O(\log^2\kappa)$. To complement these upper bounds, we introduce a channel-synthesis, or reverse-Shannon, converse technique for sampling lower bounds. This converts total-variation simulation guarantees into communication requirements and yields an $\Omega(d\log\kappa)$ bit lower bound for $d\ge2$ and fixed total variation error below one, as $\kappa\to\infty$. Together, these results identify smoothed scores as a provably more informative oracle for sampling and give nearly matching upper and lower bounds for its finite-bit complexity.

Linear dimensionality reduction methods such as proper orthogonal decomposition (POD) make high-dimensional data amenable to analysis by identifying the principal components, or modes, that capture the most variance, or energy, in the data and constructing a low-dimensional representation in the subspace they span. Such linear methods struggle, however, for data with slowly decaying Kolmogorov $n$-widths, such as advection-dominated and turbulent flows, which require many modes for accurate reconstruction; moreover, energy-based truncation can discard low-energy modes needed to capture small-scale features. Recent nonlinear manifold methods using polynomial mappings with alternating or greedy mode selection achieve better reconstruction with fewer modes, but fix the form of the nonlinear mapping a priori, limiting expressivity. In contrast, neural network (NN) manifolds offer greater expressivity yet employ energy-based selection. We present SparseModesNet, a dimensionality reduction framework that employs linear encoding and nonlinear NN decoding. The decoder leverages LassoNet, a method enforcing hierarchical sparsity through a residual connection with a linear skip layer, to simultaneously select informative modes and learn a nonlinear mapping that minimizes reconstruction error. On benchmark advection-dominated and chaotic flows, SparseModesNet matches or exceeds state-of-the-art performance. For turbulent channel flow at friction Reynolds number $Re_\tau = 5200$, our method reduces reconstruction error by 51-78% compared to existing polynomial manifold methods while maintaining interpretability through physically meaningful mode selection.

We develop square-function estimates for Fourier multipliers whose local oscillation scale is \[ \rho(R)=\frac{R}{(\log R)^{\gamma-1}}, \qquad \gamma>1. \] This scale lies strictly between the dyadic scale and every fixed power-subdyadic scale at high frequency. For high-frequency symbols satisfying a localized Sobolev condition on balls of radius comparable to $\rho(R)$, we prove a pointwise square-function estimate and a weighted $L^2$ multiplier inequality. After adjoining a smooth compactly supported low-frequency part, we derive unweighted $L^p$ bounds. The weighted estimate is governed by a logarithmic geometric maximal operator which is strongly bounded above the critical $L^r$ threshold, satisfies weak type at the critical equality, and fails even weak type below it. As a model application, consider \[ L(\xi)=\frac12\log(e^2+|\xi|^2), \qquad m_{\gamma,\beta}(\xi)=L(\xi)^{-\beta}e^{iL(\xi)^\gamma}. \] For $p=2$, the associated multiplier is bounded on $L^2$ for every $\beta\geq0$. For $1<p<\infty$, $p\neq2$, it is bounded on $L^p$ under the sufficient condition \[ \beta> d(\gamma-1)\left|\frac12-\frac1p\right|. \] At the critical equality we obtain the corresponding Lorentz endpoint estimates.

We derive a multidimensional Stein's method for asymptotic independence in the case of a general target $\mu$ with a density, being invariant measure of a diffusion process. It allows us to give a general bound in Wasserstein distance between the law of a couple $(X, Y)$, where $X$ is a random variable, and $Y$ a random vector and $\mu \otimes \mathrm{Law}(Y)$. We focus in particular in the case where $X$ and $Y$ are differentiable in the Malliavin sense, by being function of a finite number of stochastic Wiener integrals.

Brauer graph algebras form a classical class of symmetric algebras with well-structured combinatorial properties and geometric models. Recently, they have been generalized to biserial fractional Brauer graph algebras, which can be regarded as a self-injective version of the classical Brauer graph algebras. In this paper, we show that the skew group algebras of biserial fractional Brauer graph algebras induced by the Nakayama automorphism are in fact skew-Brauer graph algebras. We then study two-term tilting complexes and Kauer moves for biserial fractional Brauer graph algebras. Moreover, we prove that a biserial fractional Brauer graph algebra is tilting-discrete if and only if its reduced form (which is a Brauer graph algebra) is tilting-discrete. Finally, we show that tilting-discrete biserial fractional Brauer graph algebras are closed under derived equivalence.

Block designs are combinatorial structures in which each pair of a set of varieties appears together in a fixed number of blocks. Complete graphs are graphs in which every pair of vertices are adjacent. We present some new constructions of block designs using complete graphs, including two infinite families of designs using edge sets of complete graphs.

Moment-based estimation is a theoretically attractive approach to parametric inference, especially when likelihood-based estimation is unavailable, misspecified, or computationally inconvenient. However, the moment equations involve sample averages, which makes moment-based estimation sensitive to outliers. We propose the SGR-GMM algorithm, a robust generalized method of moments (GMM) procedure that uses a spectral gradient reweighting (SGR) primitive to soft-reweight the per-observation gradients during the moment-matching optimization. Our analysis has three layers. First, for a fixed center, the SGR primitive is formulated as an entropy-regularized spectral game between a sample-weight player and a density-matrix player, which is analyzed using classical multiplicative-weights and matrix-multiplicative-weights regret bounds. Second, we establish explicit convergence radius and finite termination bound for the fixed-center updates in the SGR primitive. Third, we prove a local finite-sample parameter estimation error bound with explicit dependence on the contamination fraction, inlier gradient stability, local GMM identification strength, and optimization accuracy. We further specialize the SGR-GMM algorithm to obtain a robust diagonally-weighted GMM (DGMM) estimator for estimating heteroscedastic low-rank Gaussian mixtures observed under additive Gaussian noise and strong contamination. In the numerical experiments, the SGR primitive produces nearly-oracle gradient estimation and the robust DGMM specialization substantially improves over non-robust moment baselines. The code and data are available at https://github.com/liu-lzhang/sgr-gmm.

We prove precise almost sure lower path regularity results for a wide class of stochastic processes in all space dimensions $d\geq 1$. Examples include Gaussian processes, in particular, fractional Brownian motions with Hurst index $H\in (0,1)$, Rosenblatt processes, and solutions to stochastic differential equations driven by fractional Brownian motions with Hurst index $H\in (\frac{1}{4},1)$, all in arbitrary dimensions $d\ge 1$. Our key tool is a new continuity result for Riesz potentials of occupation measures, which we use as substitutes for local times.

In this paper, we study Kubo-Ando means of finite order over the real division algebras $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$. We establish an explicit bijection between Kubo-Ando means of order $n$ and normalized positive functions $f$ such that both $f$ and its transpose $f^{\top}(t)=tf(t^{-1})$ are matrix monotone of order $n$. We also prove that matrix monotonicity of a fixed order is independent of the underlying division algebra. We then investigate finite-order Kubo-Ando means under the canonical embeddings relating quaternionic, complex, and real positive definite matrices. These embeddings preserve functional calculus and the Loewner order, inducing natural correspondences between means on the original and embedded cones. We characterize their extensions to the ambient cones and study the geometry of the embeddings with respect to the Log-Euclidean metric. As an application, we obtain explicit affine formulas for Kubo-Ando means of order $2$ on $\mathscr{P}_{2}(\mathbb{D})$ and transfer them to the associated embedded cones of $4\times4$ positive definite matrices. In particular, we derive trace-determinant formulas for the geometric mean and show that they do not, in general, extend to the full ambient cones.

We compute, using matched asymptotic expansions, the long-time asymptotics of homoenergetic solutions to the nonlinear Boltzmann equation, in presence of a shear term, in the hyperbolic dominated regime, for homogeneous collision kernels for which there are infinitely many collisions as $\tau \to \infty$. The mass of the resulting solutions is concentrated in an increasing family of dyadic scales, something that makes the behaviour of these solutions very different from classical Maxwellian distributions.

We give a complete classification of the Jordan types occurring in the nilpotent commutator of a nilpotent matrix whose Jordan type is a hook partition. As a consequence, we also show that two partitions with the same generic commuting Jordan type need not commute with each other.

We study the relationship between the number of full twists in positive braid representations of satellite links and their companion links. We construct infinitely many satellite links that admit positive braid representations with arbitrarily many full twists, while their companion links do not admit any positive braid representation with more than one full twist. This exhibits an unexpected divergence between the braid-theoretic complexity of a satellite link and that of its companion.

We investigate the structure of the Earth co-orbital region at low eccentricity and low inclination using a semi-analytical model of the 1:1 mean-motion resonance. The dynamics of asteroids in co-orbital motion with the Earth is described through a resonant semi-secular Hamiltonian, allowing the classification of orbits into circulation, Trojan, horseshoe, and quasi-satellite states. By systematically exploring the phase space in the space of the orbital elements, we compute the fraction of each type of motion and quantify how different co-orbital states fill the Earth co-orbital region. We find that horseshoe orbits dominate the phase space, occupying more than half of the volume, followed by Trojan and circulating orbits, while quasi-satellites represent only a small fraction. The distribution of co-orbital states exhibits strong inhomogeneities, particularly as a function of the argument of perihelion, with clear concentration regions of horseshoe orbits associated with node-crossing geometries. We also study the short-term stability of this portion of phase space by means of the MEGNO indicator, and how the level of chaos differs between different co-orbital states. Finally, we discuss the implications of these results for the expected population of Earth co-orbitals and for planetary defence, showing that a large fraction of co-orbitals remains undiscovered.

We study the problem of determining a matrix whose $k$th multiplicative compound, with $k > 1$, is a prescribed matrix $M$. The cardinality of the set of matrices whose $k$th multiplicative compound equals $M$ is characterized in terms of $\rank(M)$. On the one hand, if $\rank(M)\le 1$, it is shown that there exist infinitely many such matrices for which a complete characterization is determined. On the other hand, if $\rank(M)>1$, then there exists a unique matrix – up to an overall sign – whose compound is $M$. An algorithm for finding a matrix whose compound equals $M$ is detailed, and its time complexity is analyzed.

We develop a structure-preserving computational framework for acoustic wave scattering by moving objects, comprising a new PML-domain-embedding model and a compatible numerical approximation. The model couples a perfectly matched layer (PML), used to truncate the acoustic wave equation, with a domain-embedding formulation that represents moving objects on a fixed computational domain. The resulting PML-domain-embedding (PML-DE) system enables moving-boundary scattering problems to be solved without remeshing. Using matched asymptotic expansions, we show that the diffuse-interface formulation converges to the corresponding sharpinterface system as the interface thickness tends to zero. We then construct an energy-dissipationrate-preserving finite-difference scheme for the PML-DE system. To improve computational efficiency, the scheme is combined with hierarchical local refinement informed by the moving-object location, the fixed PML region, and the evolving wave dynamics, all within the fixed computational domain. Numerical experiments demonstrate the accuracy of the computed scattering solutions, the effectiveness of the absorbing layer and object-embedding strategy, and the efficiency of the adaptive algorithm. The proposed framework provides a practical and robust computational approach for engineering applications involving complex acoustic wave-scattering problems.

We study the linear monodromy of codimension-one transversely affine foliations through multiplicative quotient stacks and logarithmic boundary topology. When the transverse affine structure admits a global presentation on $U$, a choice of multiplicative developing coordinate induces a morphism to $[\mathbb{C}^*/\Gamma]$, where $\Gamma\subset\mathbb{C}^*$ is the image of the linear monodromy. We prove that for infinite $\Gamma$ every strict semi-equivariant holomorphic self-map of $\mathbb{C}^*$ is a monomial, and classify the nonconstant strict semi-equivariant rational self-maps of $\mathbb{P}^1$ for arbitrary $\Gamma$. For a logarithmic connection along a simple normal crossings divisor, we describe the meridional characters and angular foliations on the boundary fibres of the Kato–Nakayama space, including the topology and closures of their leaves. We also describe the extension of the linear monodromy local system to this space and the effect of local Kummer base change on finite meridional monodromy.