Papers

The manuscript establishes sharp bound of the third-order Toeplitz determinant for a subclass of starlike mappings defined on the unit ball in a complex Banach space and on bounded starlike circular domains in $\mathbb{C}^n.$

Response-adaptive randomization (RAR) in clinical trials aims to improve ethical and statistical efficiency by dynamically allocating patients to treatments based on observed outcomes. While RAR based on a target optimal allocation have been extensively studied for two-arms settings, their extension to multi-treatment experiments ($K \geq 2$) remains theoretically fragmented, with most existing methods focusing on specific algorithms or restricted target allocations. In this paper, we introduce a unified framework for response-adaptive targeting, the $\alpha$-Rebalancing Targeting Strategies ($\alpha$RTS), which generalize the ERADE two-armed strategy of Hu et al. [2009]. We prove that all designs in this family share fundamental asymptotic properties: strong consistency, asymptotic normality of allocation proportions and treatment effect estimators, and asymptotic efficiency. To address sparse target regimes (where some treatments are asymptotically eliminated), we further propose $\alpha$RTS with Forced Exploration, a variant that guarantees infinite sampling for all treatments while preserving the asymptotic guarantees. Extensive simulations illustrate the finite-sample behavior of $\alpha$RTS variants in a 3-armed context, highlighting in particular the critical role of forced exploration in sparse settings.

We investigate the relationship between volumetric and surface growth within a recently proposed optimization-driven framework for linearly elastic solids. In this approach, growth is not prescribed through an evolution law; instead, the growth distribution is determined as the solution of a constrained optimization problem. Focusing on processes driven by the minimization of the work performed by external loads in one-dimensional and axisymmetric settings, we derive explicit analytical solutions for the resulting growth distributions. Although growth is initially formulated as a volumetric process through a distributed growth strain tensor, we show that the optimal growth distributions are singular and concentrate on boundaries or internal interfaces. These results provide a variational mechanism through which, under certain conditions, surface growth is selected as the optimal realization of volumetric growth.

This study presents a high-order compact finite difference gas-kinetic scheme (FD-CGKS) that introduces a novel spatial discretization strategy for the efficient implementation of space-time coupled high-order schemes on structured grids. A conservative nonlinear compact discretization is achieved by formulating numerical fluxes from physical fluxes at both nodal and interfacial locations. To simplify the multidimensional spatial reconstruction required for the GKS flux evaluation, we propose a dual-grid approach that updates conservative variables on both a primary grid and an identical dual grid, offset by half the mesh spacing. By leveraging the time-accurate interface solutions from the gas-kinetic evolution model, the scheme explicitly updates averaged spatial derivatives between virtual interfaces, naturally enabling compact high-order reconstruction. Furthermore, a nonlinear GENO method is incorporated to capture flow discontinuities with high resolution and robustness, effectively suppressing spurious oscillations. The proposed framework, which also offers new perspectives for designing schemes based on space-time decoupled Riemann solvers, is systematically validated. Comprehensive benchmark computations of inviscid and viscous flows demonstrate the scheme's high accuracy in resolving a wide spectrum of flow features, from smooth multiscale structures to strong shock discontinuities.

The tacnode process is a universal determinantal point process arising in non-intersecting particle systems and random tiling models. In this paper, we study the generating function for the counting functions of the tacnode process on a union of $m$ intervals, $m\in\mathbb{N}^{+}$. Our first result provides an integral representation for the $m$-point generating function in terms of the Hamiltonian governing a system of $8m+4$ coupled differential equations. Combined with several differential identities for this Hamiltonian, the representation yields the large gap asymptotics, up to and including the constant term. As further applications, we obtain asymptotic formulae for the expectations, variances, and covariances of the counting functions, and establish a central limit theorem for their joint fluctuations. These results extend the previously known $1$-point theory for the tacnode process to the multi-interval setting with multiple discontinuities.

In this work we establish that the recently introduced fractional Sobolev spaces based on the Riesz fractional gradient of Musielak-Orlicz functions by one of the authors, coincide with the space of Bessel potentials of functions on such generalized Orlicz setting. Moreover, we identify them as complex interpolation spaces, and exploiting the well known properties for interpolation of operators we obtain several structural properties for those spaces.

The increasing penetration of distributed energy resources (DERs) is transforming low-voltage (LV) networks into active systems, including energy communities, whose generation, storage, and energy exchange activities require enhanced coordination with upstream medium-voltage (MV) networks. In the proposed Stackelberg structure, a local market operator aggregates multiple LV communities and acts as a single leader, determining DER operations and boundary energy exchanges, while the MV network serves as the follower, ensuring efficient system feasibility through an economic dispatch that includes both conventional and utility-scale PV generation. The proposed bottom-up coordination scheme concentrates discrete DER scheduling at the LV level while the MV level retains a convex continuous formulation, enabling an exact single-level reformulation via the Karush-Kuhn-Tucker (KKT) conditions. In addition, a distributed coordination algorithm that combines Lagrangian Dual Decomposition (LDD) with the Alternating Direction Method of Multipliers (ADMM) is developed to coordinate LV communities in parallel while preserving data confidentiality. The framework is validated using the IEEE 33-bus system at the MV level and six European 206-bus LV test feeders. Results indicate that the LDD-ADMM algorithm closely matches the exact reformulation, with an average relative deviation of 1.7e-4, with deviations confined to periods of scarcity for the cheap resource. Furthermore, leaders' decisions can induce operating conditions that increase followers' costs relative to their independently optimal dispatch, a pattern reinforced by comparison with a feasibility-based single-level relaxation that satisfies the required energy exchanges but fails to achieve a cost-efficient allocation of MV resources.

We consider piecewise linear polygonal chains connecting two given points $A, B \in \mathbb{R}^3$ and consisting of exactly $n+1$ segments (i.e., having $n$ turning points). The absolute value of the turning angle at each interior point is bounded by a given number $\varphi \in (0,\pi)$. Under the condition $n\varphi \leq \pi$, we describe the set to which all interior vertices of such a polygonal chain belong (Theorem 1). It is proved that for any point $B^{(1)}$ from this set, there exists a polygonal chain with the specified parameters (Lemma 1). Based on these results, we obtain an explicit formula describing the set of all admissible sequences $(B^{(1)}, \ldots, B^{(n)})$ of the angular points of the polygonal chain. The obtained description can serve as a basis for constructing algorithms to enumerate admissible polygonal chains and to solve optimization problems for an objective function that accounts for the cost of traversing the segments and the cost of turns.

This paper develops a learning-based boundary control framework for stabilizing a parabolic equation defined on time-varying spatial domain. Although the partial differential equation (PDE) backstepping method provides a systematic theoretical framework for such moving-boundary systems, its real-time implementation is hindered by the need to repeatedly solve time-varying kernel PDEs on evolving domains. To overcome this limitation, we first formulate the time-varying backstepping design as an operator that maps the moving-boundary trajectory to the corresponding backstepping kernel. By mapping the time-varying domain of the backstepping kernel equation onto a fixed reference domain, we establish the continuous dependence of the kernel on the moving-boundary trajectory, which provides the theoretical basis for approximating the backstepping design operator by a neural operator. Based on the approximate kernel operator, we construct the corresponding boundary feedback controller to stabilize the system. It is shown that the closed-loop system admits an exponential decay estimate on any prescribed finite time interval. For numerical implementation, DeepONet is employed to learn the time-varying kernel operator from offline-generated numerical kernel solutions and is subsequently deployed online to generate the required time-varying kernels without repeatedly solving the kernel PDE. Numerical benchmarks demonstrate that the proposed neural-operator-based implementation bypasses repeated online solution of the time-varying kernel PDE, achieves a significant acceleration of close to three orders of magnitude compared with conventional numerical kernel solvers, and thus enables real-time stabilization of the system on time-varying spatial domain.

We study local stationary solutions of finite-horizon discrete-time Pontryagin systems near a steady extremal. Suppose that the stationarity equation for the control is regular, the reduced state–costate map is hyperbolic, and the endpoint conditions satisfy a scaled transversality condition with respect to the stable and unstable subspaces. Then the linearized boundary-value problem admits an inverse whose Green estimate is uniform in the horizon. The Green kernel separates interior decay from the two reflections induced by the endpoint conditions. For $x_0=x_{\rm in}$ and $p_T=r_x(x_T,y)$, a contraction argument in a weighted norm proves existence and uniqueness in a neighborhood independent of $T$, together with uniform Lipschitz estimates and a pointwise quadratic remainder. We also derive an explicit admissible data radius and an a posteriori criterion for existence and local uniqueness near an approximate trajectory. For these graph boundary conditions, a one-sided Green estimate shows that a perturbation of the terminal reward changes the initial control and the gradient with respect to the initial state of the stationary objective by $O(e^{-\alpha_{\rm ter} T})$ for every $\alpha_{\rm ter}$ below the dichotomy rate. For linear-quadratic systems with invertible $A$, stabilizable $(A,B)$, $Q\succ0$, $R\succ0$, and a nonpositive terminal Hessian, a symplectic graph condition verifies the assumptions, and the finite-horizon Riccati matrix and initial feedback gain converge at rate $O(e^{-2\gamma T})$. Numerical experiments verify the certificates and the predicted decay rates.

Discretization-aware LES yields an exact expression for the discrete target flux in finite-volume LES by recognizing that a coarse finite difference is a top-hat-filtered exact derivative (the "filter-swap" property). That argument is purely spatial; here we observe that the forward-Euler time difference is itself a (one-sided) top-hat-filtered exact time derivative, and repeat the construction in space-time. The resulting exact discrete flux decomposition extends the spatial one with a single temporal term: a flux-quadrature error that shrinks with the quadrature order of the time integrator. In a Burgers experiment this term grows with the CFL number while the spatial terms do not, and a Smagorinsky closure augmented with its leading order - a Lax-Wendroff-type diffusion - stays accurate at coarse time steps where space-only closures degrade.

We study the large–\(d\) dynamics of the mass–deformed bosonic \(\mathrm{BFSS}_{d+1}\) matrix quantum mechanics using a Hubbard–Stratonovich localization of the Yang–Mills interaction. After integrating out the matrix coordinates, the theory reduces to a holonomy–dependent effective action for an auxiliary adjoint kernel. We introduce a commuting–symmetric saddle and its maximally symmetric specialization, in which the interaction is encoded in a single dynamically generated mass shift \(k_0\). The resulting large–\(d\) description is a gauged matrix harmonic oscillator with self–consistent frequency \(s^2=m+k_0\), fixed by a gap equation. We analyze the low–temperature \(X\)-space physics, the holonomy effective action, the Yang–Mills observable, and the associated phase structure. We then identify a correlated double–scaling limit in which \(d\to\infty\), \(m\to\infty\), and \(\kappa=m^{3/2}/d\) is held fixed. In this limit the Yang–Mills interaction and the explicit mass deformation remain parametrically balanced: the theory interpolates between the commutator–dominated BFSS regime and the mass–dominated Gaussian regime. The double–scaled theory exhibits two complementary large–\(d\) regimes. At low temperature, the enhanced gap pushes the deconfinement scale upward and opens a parametrically large uniform–holonomy region, where the bulk dynamics behaves as weakly coupled \(\mathrm{BFSS}_2\)–type gauged harmonic–oscillator sectors. At the same time, the high–temperature branch reveals an overlap window in which the Gaussian description remains self–consistent while the commutator contribution per matrix pair is parametrically suppressed. The resulting dynamics is therefore \(\mathrm{BFSS}_2\)–like in its enlarged uniform–holonomy sector and IKKT–like in its almost–commuting matrix behavior.

A dynamical system is said to be surjunctive if every injective endomorphism of the system is surjective and it is said to be injunctive if every surjective endomorphism is injective. An endomorphism of a dynamical system is called pre-injective if its restriction to every homoclinicity class of the phase space is injective. One says that a dynamical system has the Moore property if every surjective endomorphism of the system is pre-injective and that it has the Myhill property if every pre-injective endomorphism is surjective. We give characterisations of surjunctivity and injunctivity for $\Z$-subshifts of finite type in terms of their irreducible components and their Cantor-Bendixson decomposition. We also prove that a $\Z$-subshift of finite type is surjunctive if and only if it has the Moore property and that every injunctive $\Z$-subshift of finite type is surjunctive. This implies in particular that a $\Z$-subshift of finite type has the Moore property whenever it has the Myhill property.

Hypergraph polytopes (or nestohedra) form a broad class of polytopes obtained by truncating faces of a simplex according to a hypergraph. In earlier work, the authors constructed q-tridendriform algebras on the set of faces of certain families of hypergraph polytopes, including associahedra and permutohedra. The well-definedness of these structures relied on a connectedness property on the hypergraphs involved, called strictness. Nevertheless, notable examples of hypergraph polytopes such as cyclohedra fell outside this setting. We introduce a new connectedness condition, called anti-strictness, which goes opposite to strictness and captures a different class of hypergraph polytopes, including associahedra, permutohedra and cyclohedra. Our main result produces natural (-1)-tridendriform algebras in the anti-strict framework, which match previously introduced tridendriform algebras in the overlap of the two frameworks, thereby extending the range of hypergraph polytopes admitting such algebraic structures.

We study the optimal placement and tuning of a small number of pointwise viscous dampers for a vibrating string. Starting from a finite element discretization of the damped wave equation, the system is transformed into a first-order phase-space formulation, which enables a unified Lyapunov trace framework. Three optimization criteria are considered: average total energy, average total displacement, and energy for a fixed initial state. For all criteria, explicit gradient formulas with respect to damper positions and viscosities are derived, requiring only one primal and one dual Lyapunov solve. Due to the strong non-convexity of the problem, a simple heuristic based on an explicit single-damper formula is proposed to generate effective initial guesses. Numerical examples illustrate the influence of spectral selection and discretization on the optimal damping configuration.

The problem of computing the volume of tubes in riemannian manifolds goes back to Weyl and Hotelling. Here we find explicit Taylor series for the volume of a tube in a Lie group equipped with a bi-invariant metric. The coefficients are smooth valuations, given by the convolution powers of the surface area valuation. We show that the tube coefficients can be naturally described as the unique valuations given by universal formulas through the formalism of differential graded Lie and Gerstenhaber algebras; in fact, they are generated by the gauge action on the Maurer–Cartan cone in the free differential graded Lie algebra on one generator. Moreover, we introduce a new convolution product on the corresponding free Gerstenhaber algebra which is compatible with the convolution of valuations and differential forms. To complete the picture, we show that a Lie group – not necessarily connected – admits a smooth bi-invariant valuation, beyond the Euler characteristic and the Haar measure, if and only if it admits a bi-invariant riemannian metric.

We explore the numerical solution of large-scale non-autonomous Differential Riccati Equations (DREs). While we assume to discretize the differential operator using a Backward Differentiation Formula (BDF) of order s, we solve the generalized Algebraic Riccati Equation (gARE) resulting at each time step by different state-of-the-art methods. In particular, we compare the performance of the inexact Newton- Kleinman method with line search and the low-rank RADI iteration, considering for both methods two different initialization strategies: zero initialization and warm-start. A comprehensive panel of numerical results illustrate the potential and limitations of these methods when employed within a numerical pipeline for the solution of DREs, rather than for the isolated solution of a single gARE, as commonly considered in the existing literature.

In this paper, we study the information-theoretic characterization of simultaneous signalling and control over channels modeled by partially observable Markov decision processes (POMDPs). The problem is formulated as an optimization over randomized control strategies that maximize the directed information from actions to observations, subject to an average-cost constraint. We derive a novel dynamic programming framework in which the state is defined on the space of conditional probability distributions, leading to a high-level “meta” dynamic program. Specifically, we show that two coupled information states, namely, the posterior distribution of the system state and a distribution over such posteriors, satisfy Markov recursions and provide sufficient statistics for optimal control. This structure enables the decomposition of optimal strategies into separated randomized policies that depend only on these information states. Our results establish necessary and sufficient conditions for optimality and unify classical stochastic control and information-theoretic formulations. In particular, we show that in the absence of signalling, the proposed framework reduces to the standard dynamic programming equations for POMDPs. The developed approach provides a principled foundation for analyzing and designing control systems with intrinsic information constraints.

Let $H$ be a reproducing kernel Hilbert space over the unit disk $\mathbb{D}$, where analytic monomials span a dense subset. Given $\mathcal{N} \subseteq\mathbb{Z}_+$ and $\Lambda \subseteq \mathbb{D}$ we say that $(\Lambda,\mathcal{N})$ is a uniqueness pair for $H$ if $\Lambda$ is a uniqueness set for the subspace of $H$ spanned by $\{z^n:\;n\in\mathcal{N}\}$. We examine uniqueness pairs in the Dirichlet-type spaces $\mathbb{D}_\alpha$, $0\leq\alpha\leq1$. We prove two complementary results. First, if $\mathcal{N}$ contains sufficiently long finite arithmetic progressions with fixed gap size, then no sequence $\Lambda$ tending sufficiently rapidly to the boundary forms a uniqueness pair with $\mathcal{N}$. Second, if $\mathcal{N}$ satisfies a suitable arithmetic sparsity condition then one can construct uniqueness pairs $(\Lambda,\mathcal{N})$ with the points of $\Lambda$ tending to the boundary arbitrarily fast.

We prove Sobolev interpolation inequalities on extension domains that have a form reminiscent of the corresponding whole-space inequalities. This form is crucial in certain applications, which we discuss as well. The technical key ingredient is the notion of a Lebesgue $W^{1,p}$-extension domain, which we introduce here, and our proof that, for $1<p<\infty$, any $W^{1,p}$-extension domain is a Lebesgue $W^{1,p}$-extension domain.

We prove approximate tensorization for hypercontractivity and logarithmic-Sobolev constants for a class of primitive reversible quantum Markov semigroups satisfying the positive off-diagonal scaling (PODS) condition. This class includes qubit examples and generalized depolarizing semigroups with respect to full-rank states in arbitrary finite dimensions. For any such semigroup $(\Phi_t)_{t\ge 0}$ and every tensor power $n$, we show that the log-Sobolev constant of the product semigroup $\Phi_t^{\otimes n}$ is at least $2/(3\ln 2)\approx 0.96$ times the log-Sobolev constant of the single-site semigroup $\Phi_t$, independently of $n$ and the local dimension $d$. The proof first establishes an exact tensorization of the $(q,2)$-hypercontractive inequality for integer $q$, in particular $q=3$, and then extends the estimate to all real $q>2$ by complex interpolation; the standard implication from hypercontractivity to logarithmic-Sobolev inequalities yields the stated almost tensorization result. In the qubit case, we further prove exact tensorization for primitive reversible PODS semigroups and obtain sharp $(q,2)$-hypercontractivity estimates for generalized qubit depolarizing channels.

This paper considers sequential gray-box optimization where the objective function is given as the composition of a loss function and a parametric model. Crucially, the parameters of the model are unknown and need to be iteratively estimated from noisy observations of the model outputs. This problem setup generalizes the parametric black-box optimization problem known as (contextual) stochastic linear bandit. To address the sequential gray-box optimization problem, we propose a structure-exploiting method that leverages known problem structure given in terms of the loss function and an a priori set of admissible parameters. The method is based on the principle of optimism in the face of uncertainty and trades off exploration and exploitation by minimizing a lower confidence bound on the true objective function. We provide a detailed regret analysis of the novel method, improving on state-of-the-art results for the special case of linear stochastic bandits due to the use of a recently published bound for the parameter confidence sets arising in multi-output linear least-squares estimation. Numerical examples illustrate the superior performance of structure-exploiting methods compared to structure-agnostic approaches.

The Hyperfibonacci sequence of the $r$th generation is defined recursively as a generalization of Fibonacci numbers, where each term is obtained by summing the terms of the Hyperfibonacci sequence of the preceding generation. We introduce the transformation matrix for Hyperfibonacci numbers, which enables us to determine the next term in a given generation. We explore the algebraic structure of that matrix, and its power of $n$, similarity transformations between these matrices and their Jordan canonical forms. Finally, we analyze the powers of these matrices using their Jordan forms, obtaining compact and elegant formulas for expressing $r$-generation Hyperfibonacci numbers in terms of Fibonacci numbers.

We give a constructive proof for existence of absolutely continuous curves connecting each pair $\mu,\nu \in \mathrm{PL}_\infty^p(\mathbb{R}^n)$, for every $1\leq p\leq \infty$, where $(\mathrm{PL}_\infty^p(\mathbb{R}^n),\mathfrak{d}_\infty^p)$ is the complete metric space of absolutely continuous measures with density in $L^p(\mathbb{R}^n)$ and bounded support introduced in [1].

We study duality for monoids in an infinite-dimensional setting that are invariant under the action of the infinite symmetric group Sym. Our main result is an equivariant Minkowski–Weyl theorem for monoids. More precisely, we analyze the evolution of dual monoids along stabilizing Sym-invariant chains and describe the eventual behavior of their equivariant Hilbert bases. In addition, we develop a systematic study of structural properties of dual symmetric monoids, including a characterization of the duals of positive and non-positive monoids.

Let (R,m) be a Noetherian local ring and M1,...,Mn finitely generated R-modules. Set M is the direct sum of Mi. The main purpose of this paper is to extend the results on the sequential Cohen-Macaulayness of the direct sum of modules (Taniguchi et al, 2018), the sequential generalized Cohen-Macaulayness of the direct sum of modules (Cuong and Nhan, 2003) We first describe the largest submodule of M of dimension less than dimM by using the largest submodule so f its component modules. Then we give a necessary and sufficient condition for M being 1-Cohen-Macaulay. The purpose of this paper is to characterize the sequential 1-Cohen-Macaulayness of the direct sum M. We show that M is sequentially 1-Cohen-Macaulay if and only if Mi is sequentially 1-Cohen-Macaulay for all i <=n. We provide an example to clarify the results. We employ inductive methods as well as the dimension filtration of a finitely generated module.

Lossless data compression remains central to computer science, with direct impact on storage, communication bandwidth, computational cost, and energy consumption. It is also closely related to Algorithmic Information Theory, where compressibility provides an operational measure of structure and non-randomness. This paper presents the 2026 Algorithmic Information Theory Data Compression Challenge, a benchmark for evaluating general-purpose lossless compressors under realistic constraints. Submissions were encouraged to use arithmetic or range coding, limited to at most 8 GB of memory, and required to include a decompressor no larger than 1 MB. The benchmark comprised sixteen heterogeneous files, split into public training and hidden testing datasets. In total, 117 valid submitted compressors were evaluated alongside established reference compressors using compression ratio, compression and decompression time, Weissman score, and Pareto-frontier analysis. The results show that performance depends strongly on the optimization criterion: fast compressors achieved the best speed-oriented scores, whereas modelling-intensive compressors produced smaller outputs at higher computational cost. A Normalized Compression Distance analysis further revealed clusters of related submissions and distinguished incremental variants from more independent implementations. Selected submissions were described for their methodological novelty or competitive performance and further tested on four large external datasets, where several achieved competitive or superior results relative to established compressors. Overall, the challenge confirms the importance of probabilistic modelling, hidden testing, and external datasets for assessing compression performance and generalization. Benchmark resources, leaderboard data, binaries, and selected source code are publicly available at https://aitdcc.github.io.

Blind recognition of polar-coded transmissions is an important task in non-cooperative wireless forensics and security-oriented signal analysis. When the code length is known or has been estimated, recovering the frozen/information bit-position pattern is a key step in identifying the underlying polar-code structure and enabling subsequent information recovery from intercepted observations. In this paper, blind recognition of polar codes is investigated from a hypothesis-testing perspective under the successive cancellation (SC)-based synthetic bit-channel representation. First, under an ideal SC-consistent condition, we formulate position-wise recognition as a binary hypothesis test between frozen-position and information-position models, which provides a theoretical benchmark for analyzing their intrinsic distinguishability. Second, we show that the adopted soft recognition metric admits an exact shifted log-likelihood-ratio interpretation. This justifies ln 2 as the neutral threshold under equal priors and costs, while unequal priors or costs lead to the corresponding Bayesian threshold shift. Third, under the ideal SC-consistent model and this neutral setting, we derive upper and lower bounds on the position-wise and sequence-level recognition error probabilities with multiple independent observations. The resulting overlap coefficient is further related to the classical Bhattacharyya parameter, establishing an interpretable link between blind-recognition difficulty and polar synthetic-channel reliability. Simulation results show that the derived bounds characterize the recognition performance under the ideal SC-consistent model and capture the effects of code length, the number of intercepted observations, and SNR. Further paired comparisons in the tested settings indicate that the SC-consistent recursion provides a good sequence-level match to the realistic SC-recursive procedure.

We develop an ABP approach to Sobolev and Michael-Simon type inequalities under volume noncollapsing assumptions. The main new observation is a refinement of Brendle's contact-set argument: the ABP image contains the full geodesic ball centered at the minimum point of the Neumann potential, with radius equal to the ABP parameter. This allows one to use lower bounds for the volumes of geodesic balls, either at a fixed scale or under prescribed volume-growth assumptions, rather than positive asymptotic volume ratio. The central application is a Michael-Simon type inequality for immersed submanifolds of ambient manifolds with nonnegative sectional curvature and volume noncollapsing. The resulting inequality contains a lower-order term determined by the noncollapsing scale and applies to submanifolds with controlled mean curvature. In the intrinsic case, the same method gives an ABP proof of Varopoulos' $L^{1}$-Sobolev inequality with lower-order term, identifying the optimal constant in front of the gradient term, as well as explicit lower bounds for the isoperimetric profile in terms of lower bounds on the volumes of geodesic balls. Further geometric applications include Topping-type diameter estimates for submanifolds involving the $L^{n-1}$-norm of the mean curvature and various heat kernel and spectral estimates in the minimal case.

We study monoids in an infinite-dimensional setting that are invariant under the action of the infinite symmetric group Sym. Our main result establishes a local–global principle characterizing equivariant finite generation for arbitrary Sym-invariant monoids, extending earlier results that required additional assumptions. We further analyze local–global phenomena for other fundamental properties, including positivity, normality, seminormality, and simplicity. In addition, we obtain structural results for symmetric monoids, including characterizations of positivity and non-positivity, a description of their groups of units, and explicit formulas for the ranks of local symmetric monoids and stabilizing Sym-invariant chains.