Papers

Slice regular functions are a generalization of holomorphic functions where alternative real $*$-algebras are considered instead of the field of complex numbers. For such functions we show that zero sets and function values on suitable subsets may be prescribed. As a consequence, we show that for any axially symmetric domain there exist slice regular functions which (due to the nature of its zero set) can not be extended to a larger such domain.

The Levin method transforms the evaluation of a highly oscillatory integral into the solution of a first-order linear ODE for a slowly varying auxiliary function. This ODE is typically approximated by collocation, after which the integral value is recovered from the auxiliary function at the endpoints. The present work develops a new extension of the Levin method for the summation of one-dimensional and multidimensional infinite oscillatory series. The summation problem is transformed into the solution of a functional equation involving transformed arguments of the unknown function. The resulting approach is particularly attractive in the multidimensional setting, where the range of existing numerical methods is relatively limited.

We study the problem of low-rank matrix recovery from linear measurements via the global nonconvex landscape of a low-rank factored formulation of the matrix LASSO (nuclear-norm–regularized least-squares). If the landscape is benign, that is, has no bad local optima, then practical and scalable algorithms can compute good statistical estimates. Previous state-of-the-art landscape guarantees have typically assumed that the linear measurement operator has the restricted isometry property, that is, the operator is approximately an isometry over all low-rank matrices. This is an unrealistic assumption for many applications; in particular, when the individual measurement matrices are themselves low-rank, we typically have poor upper isometry constants. To overcome this, we establish new guarantees of a benign landscape under a weaker isometry condition: rather than requiring upper isometry over all low-rank matrices, we only require it over the linear low-rank tangent space to the low-rank ground truth matrix. To illustrate the utility of this result, we apply it to the problem of matrix recovery from random rank-one linear measurements; via high-probability concentration bounds on the random measurement operator, we prove a novel landscape guarantee with statistically near-optimal sample complexity and recovery error.

We study bounded ratios and optimal bounding constants among the normalized coefficients of ternary Lorentzian polynomials. For every fixed $M$-convex support and in arbitrary degree, we give an explicit presentation of the bounded-ratio cone in terms of quadratic Hessian slices. We then express the optimal bounding constants through a variational formula combining local support functions with linear compatibility constraints between slices. For full support, we determine all compatibility relations in arbitrary degree; in degree three, this yields explicit optimal constants for every two-generator section. Finally, we compare the resulting Lorentzian bounds with those for volume polynomials and rank-three matroid basis profiles.

The Vlasov-Hartree system models a mixture of bosons and fermions interacting via Coulomb forces in which the bosons are described quantum-mechanically, while the fermions are described classically. For small initial data, we prove that the system exhibits modified scattering, i.e. the bosonic and fermionic subsystems each converge to solutions of the free Schrödinger and Vlasov equations respectively, but with coupled logarithmic corrections owing to the long-range nature of Coulomb interactions. We use a mixed Lagrangian and Fourier-based approach to derive explicit descriptions of the asymptotic dynamics of the solution and associated fields and Lagrangian trajectories. Notably, due to our purely Lagrangian analysis on the Vlasov side, we require no derivatives on the initial fermion density, the asymptotic fermionic profile is as regular as the initial data, and convergence occurs in almost the same space as the initial data.

Sloshing is an inevitable phenomenon in an ocean-going vessel that can have adverse effects. In this work, the mitigation of sloshing is investigated using multiple thin porous baffles of various configurations in a partially filled swaying tank. The boundary value problem is solved within the framework of a linearized potential flow theory using the scaled boundary finite element method (SBFEM). The flow through the thin porous baffles is assumed to follow Darcy's law. The computational domain is divided into a minimum number of subdomains due to the presence of porous baffles and to ensure star convexity. Higher-order polynomials are used along each subdomain edge to represent the unknown field, i.e., velocity potential. The developed numerical model is validated with the known results in the literature. Subsequently, various results, such as the amplification factor and the forces of the tank wall, are presented and discussed for the effect of configuration, porosity, slosh tank width, depth of baffle submergence and the space between adjacent baffles. From the parametric study, it is observed that top-mounted baffles enhance sloshing suppression by $50\%$ compared to bottom-mounted vertical baffles, considering all sloshing modes. Assessing the overall effectiveness, top-mounted convex baffle configuration emerges as the most efficient configuration for sloshing suppression, achieving a well-balanced reduction across all modes.

Finite training reduces the output signal-to-interference-plus-noise ratio (SINR) of an adaptive beamformer, and a natural question is how much of this loss is unavoidable. This paper determines this question, providing a beamforming counterpart of the Cramér–Rao bound in spectral estimation. An exact identity expresses the SINR loss as a bounded function of the error in the clairvoyant minimum-variance distortionless-response (MVDR) weight. It yields a local asymptotic minimax lower bound over all measurable data-dependent beamforming rules, including biased and irregular rules. The first-order coefficient is $\tr(\Mb\Jb_{\rm eff}^{-1})$, where $\Jb_{\rm eff}$ describes the information in the training data and $\Mb$ measures the sensitivity of the output SINR. Matching constructions determine this coefficient in two complex-Gaussian models. For an $N$-sensor uniform linear array with $K$ distinct point interferers and $2K+1\le N$, a data-driven split one-step beamformer attains the coefficient $C_\theta\le K$ at every interior scene of a fixed compact regular parameter set. For unrestricted covariance matrices, sample matrix inversion (SMI) attains the coefficient $N-1$ through the classical Reed–Mallett–Brennan law. The difference quantifies the first-order value of finite-source structure. Geometric formulas and numerical results describe the dependence on interference power and array geometry, and the finite-sample departure near a weak-source boundary.

Consider a parametric ODE control model. A function of the states and parameters is called observable if its value can in principle be reconstructed from input-output data. The observable functions form a field, called the observation field, represented naturally by a set of generators. Even when the model is not fully observable, this field captures the information still accessible from input-output data. We present an algorithm for computing a concise generating set for the observation field of a model with rational dynamics. The algorithm relies on two new results: one allows observable functions to be extracted from the coefficients of repeated Lie derivatives of the outputs, while the other reduces the required orders of differentiation by exploiting identifiable parameter combinations. We implement the resulting algorithm in StructuralIdentifiability$.$jl (https://github.com/SciML/StructuralIdentifiability.jl). For computational efficiency, we employ recent techniques for differential elimination and rational function field simplification. Using models from epidemiology, chemical kinetics, and cancer modeling, we show that the algorithm produces generators with domain-specific interpretations that can inform model analysis and development.

We prove that strong well-posedness is generic for McKean–Vlasov stochastic differential equations with jumps in a complete coefficient space that allows genuinely discontinuous state dependence. For every $T>0$, there exists a dense residual subset $\mathfrak R_T$, independent of the initial state, such that every coefficient $a\in\mathfrak R_T$ generates a unique strong solution for every deterministic initial condition $x\in\mathbb R^d$. Moreover, the solution map $x\mapsto X^{a,x}$ is continuous in $\mathcal S_T^2$, and every approximation by spatially Lipschitz coefficients converges locally uniformly in $x$ to the same canonical solution family. The coefficients are uniformly bounded, uniformly non-degenerate in the diffusion component, and Lipschitz continuous in the law with respect to the $2$-Wasserstein distance, while their state dependence may be merely measurable. The ambient space is obtained by completing a spatially Lipschitz core under a Krylov-stable metric based on local $L^{2(d+1)}$ coefficient errors. A stopped conditional Krylov–Melnikov occupation estimate converts convergence in this metric into occupation convergence along non-degenerate Itô–Lévy trajectories. The key probabilistic ingredient is a set-valued stability theorem: near a core coefficient, every strong solution of every nearby solvable equation remains close to the unique core solution, without any uniqueness assumption on the nearby equation. Together with closedness of the strong-solution relation, this permits a Baire-category argument on the completed coefficient space. We also give an intrinsic spatial-translation criterion showing that the completion contains natural classes with essential spatial discontinuities. A projective-limit argument yields the corresponding global-time generic well-posedness result.

Retrieval-augmented generation is usually formulated as a statistical information-retrieval problem. Graph-based variants add relational structure, but the mathematical status of that structure is often left underspecified. Three distinct questions tend to be conflated: which statements are certified as knowledge, which latent representations are useful for retrieval, and which multi-hop compositions are semantically admissible. We propose Proof-Carrying Algebraic-Geometric Retrieval (PAGR), a framework that separates these questions mathematically. Its symbolic layer is a many-sorted relational theory generated by a typed quiver, path equations, and positive Horn inclusions. A quiver representation assigns inner-product spaces to entity types and linear operators to relations. A cellular sheaf measures local-to-global consistency. Semiring provenance records derivations and supports machine-checkable certificates. The central principle is epistemic separation: learned geometry may rank and organize evidence, but cannot promote a hypothesis to certified ground truth. We show the certification criterion is invariant under arbitrary replacement of learned components. Further results include a conditional completeness bound, identification of the isometry group as the relevant symmetry for residual-based retrieval, a cohomological consistency diagnostic, and a bounded-bisimulation index for admissible-path expansion. PAGR is a mathematical architecture for separating where a system should look from what it is allowed to treat as knowledge.

In this note, we introduce a new comonotonic approximation for sums of lognormal random variables based on the famous Perron-Frobenius theorem.

We investigate positive radial solutions with an isolated nonremovable singularity for the semilinear elliptic equation \begin{align*} \Delta_{\mathbb{H}^N} u+\lambda u+u^p=0 \qquad\text{in }\mathbb{H}^N\setminus\{Q\}, \end{align*} where $N\geq 3$, $p>1$, $\lambda\le \frac{(N-1)^2}{4}$, and $Q\in\mathbb{H}^N$ is a prescribed pole. Our purpose is to describe how the locally Euclidean singular behavior near ${\{Q}\}$ interacts with the genuinely hyperbolic dynamics at infinity, and how this interaction changes across the Serrin and Sobolev critical exponents. For $\frac{N}{N-2}\le p<\frac{N+2}{N-2}$, we construct a family of positive radial singular solutions selecting the fast exponential mode at infinity. At the pole, these solutions exhibit the logarithmically corrected fundamental-solution profile when $p=\frac{N}{N-2}$, and the standard power-law profile when $\frac{N}{N-2}<p<\frac{N+2}{N-2}.$ At the conformally invariant pair $(p,\lambda)=(\frac{N+2}{N-2},\frac{N(N-2)}{4})$, we obtain an explicit singular solution and nonconstant Fowler-type solutions, and we classify all positive radial solutions with a nonremovable singularity via the positive periodic orbits of an associated autonomous equation. In the supercritical regime $p>\frac{N+2}{N-2}$ with $\lambda\le\frac{ N(N-2)}{4}$, we prove existence and uniqueness of the global positive radial singular solution, derive its two-term local asymptotic expansion near the pole, and establish a sharp trichotomy for its behavior at infinity.

This review develops a control-theoretic perspective on measurement-based feedback for continuously monitored open quantum systems, with the main analysis focused on finite-dimensional systems governed by diffusive stochastic master equations. We introduce the relevant state-space, invariant-subspace, and quantum non-demolition structures, interpret quantum filtering as nonlinear observer dynamics, and review open-loop asymptotics, filter stability, state-feedback stabilization, robustness, and reduced-order observer-based control. Particular emphasis is placed on a recurrence–contraction framework, in which Hamiltonian feedback removes non-target invariant obstructions while measurement-induced dynamics provide local exponential contraction. Although the detailed analysis is developed for finite-dimensional diffusive models, the underlying measurement–estimation–feedback architecture is relevant across a broad range of quantum platforms. We further discuss implementation challenges and open problems involving scalable estimation, sampling and delay, adaptation, hybrid and non-Markovian dynamics, practical stability, optimal control, and learning-based design. By organizing these developments around invariance, estimation, recurrence, and contraction, the review provides a tutorial bridge between measurement-based quantum feedback and nonlinear stochastic control.

We give a criterion for property $P_{\text{naive}}$ for subgroups of orientable big mapping class groups and apply it to mapping class groups of non-orientable infinite type surfaces through the orientation double cover.

Let $P_d$ be the convex hull of the permutations of $\{1,\ldots,d\}$ that avoid the patterns $132$ and $213$, and let $H_d(t)$ be its Ehrhart $h^*$-polynomial. We prove that all zeros of $H_d(t)$ are real and negative for $3\le d\le1000$ and for every $d\ge2^{62}$; for $d\le2$ the polynomial is constant. Expanding $H_d(t)$ in refined Eulerian polynomials does not suffice, since a coordinate is already negative at $d=7$. Let $A_{d,i}(t)$ enumerate all permutations of $\{1,\ldots,d\}$ with first entry $i$ by number of descents. For $c\ge0$, we prove that every nonnegative linear combination of $A_{d,1},\ldots,A_{d,d}$ and the adjacent differences $A_{d,r+1}-cA_{d,r}$, $2\le r\le d-2$, is real-rooted for all $d\ge4$ simultaneously if and only if $c\le2-\sqrt3$. This reduces real-rootedness to the positivity of explicit weighted tails of the coordinates. An exact certificate checks these inequalities for $d\le1000$, and coefficient estimates based on Darroch's theorem on the mode of a sum of Bernoulli variables prove them for $d\ge2^{62}$; the range $1001\le d<2^{62}$ remains open. Along the way we identify $P_d$, up to lattice equivalence, with twice a chain-poset permutahedron and with the Newton polytope of a discriminant, which transfers the known cube structure and volume of these models to $P_d$. We also show that the coefficient of $t^{d-1}$ in $H_d(t)$ counts strong tournament score sequences for $d\ge3$, confirming a conjecture of Rehberg in this range.

A second order finite difference scheme is analyzed for the nonlocal Cahn-Hilliard system with variable mobility and the singular Flory-Huggins logarithmic potential. The temporal discretization employs a modified Crank-Nicolson formulation, while the mobility is treated explicitly to guarantee ellipticity and reduce computational cost. To strictly enforce the positivity-preserving property at the discrete level, a nonlinear regularization term is added into the numerical scheme. Rigorous analysis shows that the scheme is uniquely solvable and stable under a modified numerical energy. The well-known analytical challenge associated with the variable mobility has to be handled carefully. To overcome this, a convergence framework is constructed based on two non-standard techniques: (1) a higher-order asymptotic expansion (extending up to the third order in time and fourth order in space) to retain a sufficiently order of accuracy; (2) a two-stage error strategy, wherein a rough error estimate first guarantees the discrete boundedness of the mobility and nonlinear terms, followed by a refined error analysis that yields the optimal convergence rate. Numerical experiments validate the theoretical results and demonstrate the robustness of the proposed scheme.

This paper studies a dynamical system approach for solving indefinite quadratic programming problems subject to linear constraints. We investigate the convergence of the trajectory generated by the system to a Karush-Kuhn-Tucker point of the quadratic programs. In addition, we derive an estimate for the distance between the trajectory and a solution of the problem. We further prove that the objective value is decreasing along the trajectory. An illustrative example and a numerical test are presented to demonstrate the behavior and performance of the proposed method.

We analyse symmetric interior-penalty discontinuous Galerkin approximations of elliptic stochastic partial differential equations driven by spatial Gaussian white noise on quasi-uniform polytopic meshes. The low regularity of the exact random field precludes the usual SIP consistency argument. We instead restrict the same continuum white-noise functional canonically to the discontinuous polynomial space and separate the error into stochastic forcing-restriction and deterministic discretisation components. For every fixed polynomial degree $k\geq1$, we prove the sharp two-sided estimate \[ \|Y-Y_h\|_{L^2(\Xi;L^2(\Omega))} \simeq h^{2-d/2}, \] giving rates $h$ in two dimensions and $h^{1/2}$ in three dimensions. A finite-dimensional lower bound establishes optimality. The analysis allows polytopic elements with arbitrarily many and arbitrarily small faces, including admissible non-convex elements. In two dimensions we also treat polygons with a single reentrant corner. Although the deterministic approximation exhibits an angle-dependent loss of regularity, the singular resolvent contribution is finite rank, and the sharp first-order stochastic rate is retained. The discontinuous structure yields an exact element-local sampler for the restricted white noise. Numerical experiments on irregular polygonal, non-convex polytopic and reentrant domains support the predicted convergence behaviour.

We generalize the one-dimensional infinite-order linear differential equation satisfied by the $p$-adic Hurwitz-type Euler zeta function (Abh. Math. Semin. Univ. Hambg. 91: 117–135, 2021) to $n$ variables. By encoding the differential operator as a convolution with a distribution kernel, we introduce partial zeta functions and partial operators indexed by subsets of $\{1,\dots,n\}$. A tensor product expansion of the distribution kernel is established, whose Möbius inversion via the binomial theorem yields the higher-dimensional analogue of the original equation in the form of an alternating-sum identity, reducing to the one-dimensional case when $n=1$. Convergence of the resulting series is confirmed by non-Archimedean estimates. As a second main contribution, we prove that, under the condition $|a|_p > 2p^{1/(p-1)}$, within the Banach space of bounded analytic functions on $\mathbb Z_p$, the shifted $p$-adic Hurwitz-type Euler zeta function is the unique solution to the one-dimensional equation. This uniqueness result establishes that the infinite-order $p$-adic differential equation admits at most one bounded analytic solution, and together with the existence theorem, it characterizes the shifted $p$-adic Hurwitz-type Euler zeta function uniquely. This sharply distinguishes it from the complex setting, where the operator series diverges on the Hurwitz zeta function, and the formal kernel fails to converge in the usual spaces of analytic test functions.

Let $T_n$ be the infinite $n$-regular tree, $n\ge3$. We prove that every finite connected vertex set $\Omega\subset T_n$ satisfies the sharp inequality \[ \sigma_1(\Omega)\le \frac{n}{(n-1)|\Omega|+1}. \] Equality holds if and only if $\Omega$ is a ball. Since \[ |\delta\Omega|=(n-2)|\Omega|+2, \] the result is equivalently a sharp upper bound at fixed external boundary cardinality, and hence a discrete Weinstock inequality on $T_n$.

Stochastic min-max optimization has attracted increasing attention due to its applications in modern machine learning, while existing theoretical studies mainly rely on the bounded variance assumption for stochastic gradients. Under heavy-tailed noise, where stochastic gradients only possess a finite $p$-th moment for $p\in(1,2]$, gradient clipping or normalization is commonly believed to be necessary to guarantee convergence. In this work, we revisit stochastic min-max optimization under heavy-tailed noise and provide a comprehensive theoretical study of stochastic gradient descent ascent (SGDA). We first show that vanilla SGDA, without any modification to its update rule, can converge under heavy-tailed noise in both nonconvex-strongly-concave (NC-SC) and nonconvex-concave (NC-C) settings, establishing the first convergence guarantees for SGDA in these regimes. Beyond unregularized problems, we further investigate regularized stochastic min-max optimization, where directly incorporating gradient normalization into proximal updates is nontrivial due to the incompatibility between normalization and proximal structures. We overcome this difficulty by developing new clipping-free algorithms, i.e., Stoc-TRGDAM and Stoc-TRGDmax, and they both can achieve the optimal dependence on the target accuracy without using gradient clipping.

We consider conforming $\Pone$ and lowest-order Crouzeix–Raviart (CR) approximations of the Dirichlet Poisson problem in two and three dimensions. We obtain computable a priori $L^2$-error bounds on anisotropic simplicial meshes without a global $H^2$-regularity assumption. For piecewise constant data, the Marini relation gives an equilibrated lowest-order Raviart–Thomas (RT) flux from the scalar CR solution. The square of the resulting Marini constant is the largest generalised eigenvalue of a problem involving only scalar conforming and CR matrices. Combined with the elementwise Poincaré constant, it controls both the conforming and CR errors. The CR $L^2$ estimate uses positivity of the discrete CR–conforming gap operator rather than a nonconforming Aubin–Nitsche argument. We also derive an exact decomposition of the Marini defect into the CR–conforming energy gap and an explicit geometric term with no aspect-ratio factor. On the algebraically graded L-shaped mesh family considered here, $q>3/2$ yields $\kappa_{M,h}=O(h)$ and computable $O(h^2)$ $L^2$-operator bounds.

For a compact set $K\subset\mathbb{C}$, let $\vartheta(K)$ be the infimum of the planar areas of the unit lemniscates of all monic polynomials with zeros in $K$, allowing arbitrary degree and repeated zeros. We prove that $\vartheta(K)=0$ whenever $\operatorname{cap}(K)=1$, with no regularity assumption on $K$. The proof uses a centered harmonic polynomial that is positive on all but a set of arbitrarily small area in the polynomial hull of $K$. A Fourier average of exterior harmonic measures realizes this polynomial as the logarithmic potential of a signed measure having bounded density with respect to the equilibrium measure. A positive perturbation and an $L^1$ approximation by empirical measures then produce the required polynomials. This extends the smooth-boundary result of Krishnapur, Lundberg, and Ramachandran to arbitrary compact sets of capacity one. Together with the capacity-greater-than-one theorem of Ghosh and Ramachandran and an elementary argument for unbounded sets, it follows that $\vartheta(F)=0$ for every closed infinite set $F\subset\mathbb{C}$ of transfinite diameter at least one, answering the vanishing question in Erdős Problem 1040.

In this paper, we investigate a class of magnetizable piezoelectric beam systems with two fractional infinite memory terms, whose fractional orders are denoted by \(\theta_1, \theta_2 \in [0,1]\). By introducing the Dafermos history variable, we construct an augmented state space and reformulate the system as an abstract evolution equation. The well-posedness of the system is established via semigroup theory. Through a frequency-domain analysis, we characterize the asymptotic behavior of the solutions. We show that the system is exponentially stable if \(\theta_1 = \theta_2 = 1\), and is polynomially stable in all other cases, with the explicit decay rate \(t^{-\frac{1}{2-2\theta_0}}\), where \(\theta_0 = \min\{\theta_1, \theta_2\}\). Furthermore, the optimality of the obtained decay rate is verified.

The Bott-Duffin theorem shows that any arbitrary positive real rational function of finite order, representing a passive network, can be represented by a finite number of positive value resistors, capacitors and inductors. It is shown here that if the positivity restriction is lifted, any rational function or polynomial of any order may be represented by a network of (possibly negative) inductors, capacitors and resistors. When there are no repeated roots this is a sum of simple sub-circuits each of which contains no more than four components. When there are repeated roots a more complex circuit is required, here featuring iterated Wheatstone bridges of special form.

The present study improves the qualitative analysis of backward stochastic variational dynamics on a general complete filtered probability space, considered in the spirit of Liang, Lyons and Qian (2011). Our primary objective is to overcome a substantial limitation in the study of Bensoussan, Li and Yam (2018), where the boundedness condition imposed on the multivalued subdifferential operator excludes standard obstacle-type constraints and indicator functions of convex sets. We prove the existence and uniqueness of a strong càdlàg solution under the natural assumption that the driving proper lower semicontinuous convex function is merely bounded from below by an affine/quadratic function. Furthermore, we incorporate an oblique reflection governed by a time-dependent, uniformly positive definite symmetric matrix, in the spirit of the pioneering results of Gassous, Răşcanu and Rotenstein (2012, 2015).

The goal of this paper is to study the dynamics of conformally symplectic Hamiltonian flows under the light of integrability. As the dynamics of conformally symplectic Hamiltonian flows are dissipative and differ fundamentally from their conservative counterpart we start by proposing several notions of integrability that are better suited to the problem. We will propose two notions of integrability: $C^1$-integrability and Hopf integrability, that depend on the existence of a global attractor and on its shape. Then our main theorem focuses on Tonelli Hamiltonians whose conformally symplectic flows do not have conjugate points. We prove that such flows are automatically Hopf integrable. The proof is geometric and studies the long time evolution of vertical subspaces under the flow. It also makes use of (discounted) weak KAM theory. We also establish several results about the asymptotic Maslov index for integrable conformally symplectic Hamiltonian flows. Finally, we describe some examples to illustrate differences between symplectic and conformally symplectic Hamiltonian flows and to illustrate the pertinence of our definitions of integrability.

We study area operators $\mathbb{A}_{\mu,l}$, $0<l<\infty$, induced by positive Borel measures on the right half-plane and acting on the Hardy spaces of Dirichlet series $\mathscr H^p$, $0<p<\infty$. We first disprove a conjecture proposed by the present authors in an earlier work by constructing a probability measure, valid for all $0<p,l<\infty$, for which the associated area operator is bounded although the measure fails the proposed Carleson conditions. We then characterize the boundedness and compactness of $\mathbb{A}_{\mu,l}$ on both $\mathscr H^p$ and the Hardy space $\mathscr H^p_0$ of Dirichlet series vanishing at $+\infty$, for every $0<p,l<\infty$, via integrability conditions for the associated cone integrals. In particular, the boundedness and compactness are equivalent for these operators. We also give direct proofs of compactness under vanishing Carleson measure and compact $H_{\mathrm i}^p$-Carleson embedding hypotheses. As an application, we establish an estimate for the essential norms of bounded Volterra operators. This leads to a complete characterization of compactness for Volterra operators on $\mathscr{H}^p$ and provides a different proof of compactness for those with Dirichlet series symbols in $\operatorname{VMOA}(\mathbb C_0)$.

Let $G$ and $H$ act commutatively and minimally on a compact metrizable space $X$. We prove that if the two subactions are point-distal, equivalently HPI, then the action generated by them is again point-distal. We also construct a common highly proximal extension on which both subactions are strictly HPI. As consequences, transitivity of mixed products and linear iterates upgrades to minimality in the point-distal category. We then compare the canonical structure of the two subactions with that of the joint action. The maximal highly proximal operations coincide, whereas the maximal joint isometric factor over a common factor is the meet of the two subaction-isometric factors. This gives a recursive description of the joint Furstenberg tower. Finally, we construct commuting minimal distal homeomorphisms of $\mathbb T^3$ whose canonical Furstenberg towers are different, showing that the meet formula can be strict.

Let $m\geq0$ and put $n=m+3$. We introduce a normalized positive Laurent class of decorated frieze patterns and prove a decorated analogue of the Conway-Coxeter classification theorem. Namely, such friezes of width $m$ are in canonical bijection with weighted triangulations of a convex $n$-gon, whose boundary edges are labeled by independent variables $y_1,\ldots,y_n$ and whose diagonals are labeled by $x_1,\ldots,x_m$. While a weighted Conway-Coxeter propagation algorithm constructs the frieze from a weighted triangulation, a main new ingredient is an explicit nonrecursive Laurent formula expressing each quiddity entry directly from the weighted triangles incident to the corresponding vertex. Conversely, specialization to $1$, Laurent positivity, and a decorated cutting-and-gluing procedure recover the weighted triangulation from the frieze.