Papers

We study the SCFT/VOA correspondence for the $(A_2,D_{3n-2})$ Argyres–Douglas theories with $n\geq2$. From the matching of the central charges and the equality of the Schur index with the vacuum supercharacter to all orders, we propose that the associated VOA is the logarithmic doublet algebra $\mathcal A(4n-2)$ of Feigin, Feigin, and Tipunin. For $n=2$, this correspondence was studied in detail by Buican and Nishinaka, and our proposal extends it to $n\geq3$. Using the Nahm sum expression for the vacuum supercharacter, we further propose a family of 3d $\mathcal N=2$ abelian Chern–Simons matter theories that flow to the 3d $\mathcal N=4$ SCFTs obtained by the R-twisted circle reduction, generalizing the $n=2$ construction. We determine the monopole superpotential and show that the Coulomb branch is $\mathbb C^2/\mathbb Z_2$ for every $n$. The superconformal index provides quantitative evidence for the proposed infrared fixed points, including the expected enhancement to $\mathcal N=4$ supersymmetry and the Higgs and Coulomb branch structure.

Let \(S\) be a smooth projective complex surface with irregularity \(q=h^{1,0}(S)\) and geometric genus \(g=h^{2,0}(S)\), and let \(S^{[n]}\) denote its Hilbert scheme of \(n\) points. We prove that, for every \(n\ge0\), the sequence \[ \left(h^{p,0}\bigl(S^{[n]}\bigr)\right)_{p=0}^{2n} \] is log-concave if and only if \(g\le\binom{q+1}{2}\). Moreover, log-concavity of all these sequences is already equivalent to log-concavity of the sequence for \(n=2\). We also prove that these sequences are unimodal for every \(n\ge0\) if and only if \(q\ge1\) or \(g=0\), and that this condition is already detected by the sequence for \(n=1\).

Let $p$ be a prime. We prove nontrivial bounds on short sums of Dirichlet characters mod $p$ evaluated at a class of polynomials, not necessarily homogeneous, in $n$ variables and of degree $k$. For large $n$, we further achieve nontrivial bounds for sums over boxes with side-lengths as short as $p^{1/(k-1)+\varepsilon}$, which breaks past the Burgess barrier of $p^{1/4+\varepsilon}$ as soon as $k\geq 6$. In the proof, we develop a new variation of the Burgess amplification method that reduces the problem to bounding additive character sums. This is the first case in which a Burgess-type method succeeds in the inhomogeneous setting.

We define the new basis of the Grothendieck group of unipotent representations in an exceptional Weyl group, making more precise and correcting results in previous papers.

We develop a global martingale optimal transport framework for joint S&P 500 (SPX)-VIX smile calibration across multiple maturities, without the conditional independence that Markovian stitching imposes between months. We prove that exact local and global feasibility are equivalent, yet the stitched laws, which carry earlier months only through the current SPX level, can form a strict subset of the global ones, so adjacent smiles cannot identify the dependence across months. We show that relative entropy against the standard Markov reference selects the stitched law, so this dependence must be supplied deliberately. We supply it through a finite-memory reference, in which a latent autoregressive state drives the VIX levels while every VIX marginal stays exact, and calibrate the resulting law on a junction tree whose size grows linearly with the number of maturities. On market data with up to nine SPX maturities, a law with memory and a law without it fit the same quotes but price VIX-conditioned SPX Asian claims differently: on every calendar, memory makes a claim on the maximum VIX cheaper and a claim on the minimum VIX dearer. Varying the memory strength on one calendar moves these differences steadily from zero, so the vanilla quotes, which fix the law of each date, do not determine prices that depend on the joint law across dates.

What are the transversality conditions for an optimal control problem when the boundary conditions are defined by differential equations? This seemingly bizarre question is motivated by trajectory optimization problems in the $N$-body system. The question, however, is more fundamental and goes beyond problems in astrodynamics to nonintegrable dynamical systems in general. The main contribution of this paper is the development of generic initial- and final-time transversality conditions for optimal control problems whose boundary conditions are defined in terms of differential equations with side conditions. The mathematical definition of differential boundary conditions are part of the foundations developed in this paper. To support the new fundamentals, the concept of coordinated/uncoordinated clock times and weak adjoint covectors are introduced. In the case of uncoordinated clock times, the new transversality conditions reveal that there exists a special situation where a weak adjoint covector is orthogonal to the vector field of the boundary differential equation. This condition is sharply different from the classical statement of orthogonality with respect to the endpoint manifold. The theorems developed in this paper are generic. An application of the theorems to several cases in the three-body problem are described in separate papers.

The classical technique of moving frames allows to effectively compute differential invariants of group actions. A standard observation is that the expressions of differential invariants contain radicals and more complicated algebraic functions, however the global Lie-Tresse theorem claims that for algebraic pseudogroup actions the algebra of differential invariants is generated by rational functions in jet variables. Here, we explain this apparent contradiction and illustrate it in a few well-elaborated examples. Constructively, our result gives a method to algorithmically compute global differential invariants.

We give deterministic FPTASes for two dense counting problems on which the known deterministic algorithms, based on zero-free interpolation, run in quasipolynomial time. For fixed $0<\gamma<1/2$ and $0<\theta\leq1$, the first approximates $\mathrm{haf}(A)$ for a symmetric matrix $A$ when its support graph $G$ has minimum degree at least $(1/2+\gamma)n$ and its nonzero entries lie in $[\theta,1]$. It also approximates permanents under the analogous bipartite condition, including full-support matrices in $[\theta,1]$. For fixed $\beta>0$ and $0<\kappa\leq1$, the second approximates the zero-field Ising partition function $Z(J)$ for zero-diagonal real symmetric matrices $J$ satisfying $\max_{i,j}|J_{ij}|\leq\beta/n$ and $\lambda_{\max}(J)\leq1-\kappa$. No separate lower-eigenvalue condition is imposed. We further prove $\log\mathrm{haf}(A)=h_A(G)-n/2+O_{\gamma,\theta}(1)$ and $Z(J)=2^n\det(I-J)^{-1/2}(1+O_{\beta,\kappa}(1/n))$. Here $h_A(G)$ is the maximum weighted fractional-matching entropy. For unweighted graphs, the first formula improves the Cuckler–Kahn error from $o(n)$ to $O_\gamma(1)$ on the fixed-margin class and extends it to weights in $[\theta,1]$. Both algorithms use a common Gaussian truncation principle. Each problem becomes an integral of a product of a fixed entire function over Gaussian coordinates, with possibly indefinite moment matrix entries of order $1/n$. Cancelling the linear term and exactly resumming the quadratic term leaves a coordinate remainder vanishing to order at least three. Complex dilation handles small supports. For large supports, we bound the recombined tail by a large-deviation rate that beats the entropy of the subsets. The truncation error is at most $(CR/n)^{R/2}+e^{-cn}$. This faster-than-geometric decay permits $R\log(en/R)=O(\log n+\log(1/\epsilon))$ and hence polynomial enumeration.

Let $\mathcal{B}$ be a braided finite tensor category and let $A$ be a simple commutative algebra in $\mathcal{B}$. We construct an order-reversing Galois connection between subalgebras of $A$ and tensor subcategories of $\mathcal{B}_A$. Let $\mathcal{B}'$ denote the Müger center and set $A':=A\cap\mathcal{B}'$. The closure operators are $B\mapsto\langle B,A'\rangle_{\mathrm{alg}}$ and $\mathcal{E}\mapsto\langle\mathcal{E},\mathcal{B}_A^{\mathrm{loc}}\rangle_{\otimes}$. Thus the closed subalgebras are those containing $A'$, while the closed tensor subcategories are those containing $\mathcal{B}_A^{\mathrm{loc}}$; equivalently, the fixed-point intervals $[A',A]_{\mathrm{alg}}$ and $[\mathcal{B}_A^{\mathrm{loc}},\mathcal{B}_A]_{\otimes}$ are anti-isomorphic as lattices. For a finite tensor category $\mathcal{C}$, the canonical algebra in $\mathcal{Z}(\mathcal{C})$ yields an anti-isomorphism between its subalgebras and tensor subcategories of $\mathcal{C}$. When $\mathcal{B}$ is nondegenerate, Frobenius extensions correspond to unimodular tensor subcategories. For a Hopf algebra in $\mathcal{B}$, the correspondence specializes to an order-preserving bijection between Hopf ideals and normal left coideal subalgebras.

We prove that generically on the basic stratum of split unitary Shimura varieties, the universal abelian variety with endomorphism structure and polarization has minimal automorphism group in a precise sense, except for a few degenerate cases. This is a direct analogue of Oort's conjecture on automorphisms of supersingular abelian varieties. On the way, we explicitly compute the generic automorphism group of the universal $p$-divisible group in a given basic isogeny class.

Let $G$ be a graph of order $n$, and let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of the positive and negative adjacency eigenvalues of $G$, respectively. Recently, Liu, Tang, and Zhang proved the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ of order $n$ satisfies $ \min\{s^+(G), s^-(G)\} \ge n-1. $ For positive square energy, we strengthen this result by showing that every $2$-connected graph $G$ of order $n$ which is not a cycle satisfies $s^+(G)\ge n$. The formally analogous assertion for $s^-$ is false: the complete graph $K_n$ satisfies $s^-(K_n)=n-1$. We prove a natural counterpart in the triangle-free class: every triangle-free $2$-connected noncycle $G$ satisfies $ \min\{s^+(G),s^-(G)\}>n. $ More generally, it is enough that some maximum-degree vertex of $G$ belongs to no triangle. Together with the exact square energies of cycles, this characterizes the triangle-free $2$-connected graphs for which $s^-(G)\ge n$; the only exceptions are the cycles $C_{4k+3}$ with $k\geq1$.

We develop a convergence theory for geometric multigrid with vertex-patch smoothers applied to cut finite element discretizations of the Poisson problem. The framework addresses non-inherited level forms and the mismatch between the physical and active domains. Using the discrete extension property, we prove two-level convergence bounds uniform in the mesh size and the cut geometry, and W-cycle bounds under an additional smallness assumption on the two-level rate. The numerical experiments intentionally use the stronger V-cycle, for which no convergence bound is claimed here. The convergence constants degrade with the degree $p$. Lowering the ghost penalty improves iteration counts. An aligned two-cell model exhibits a semidefiniteness threshold of order $p^{-2}$, whereas the visibility scale of a degree-$p$ cut mode decreases exponentially. Experiments at the model threshold reduce the iteration counts, but do not establish an assembled-operator threshold.

We consider a grand-canonical reduced BCS Hamiltonian on the full fermionic Fock space with an attractive pairing interaction supported in an energy shell around the Fermi surface. For fixed chemical potential, we prove that the exact finite-volume ground-state energy differs from the minimum of the corresponding finite-volume BCS functional by a volume-independent error. As a consequence, the BCS variational principle is exact for the ground-state energy density in the thermodynamic limit.

We investigate limits of semigroups of stochastic matrices defined by their invariant distribution. Given probability vectors $\gamma(\beta)$ depending on a parameter $\beta$, we introduce a notion of convergence as $\beta\to\infty$ for the corresponding semigroups of $\gamma(\beta)$-preserving stochastic matrices and investigate the structure of the resulting limit. In general, the limiting semigroup differs from the semigroup preserving the limiting distribution, showing that these two operations do not commute. We develop a general framework for such limiting semigroups and study in detail the case in which the invariant distributions are Gibbs vectors at the inverse temperature $\beta$. We show that the limiting semigroup consists of block-upper-triangular stochastic matrices subject to additional substochasticity constraints. We characterise and enumerate their extremal elements and determine the preorder on probability vectors induced by the action of the semigroup. The resulting notion of Block-Triangular majorisation interpolates between ordinary majorisation and upper triangular (aka unordered) majorisation. We show that it is completely characterised by a finite family of monotones and analyse the corresponding behaviour of Rényi $\alpha$-entropies as $\beta\to\infty$.

We introduce a class of Hall sets, which we call factor-parity Hall sets, whose elements split into good and bad brackets. We prove that the good brackets can be steered simultaneously and arbitrarily in small time, which yields sufficient conditions for the small-time local controllability of control-affine systems. This positive result draws on constructions of Kawski, Agrachev-Gamkrelidze and Krastanov. Conversely, we prove that each bad bracket generates an obstruction to controllability, hence a family of necessary conditions.

We prove a necessary condition conjecture ([27, Conjecture B]) for the maximal singularity of the Hilbert scheme of points in 3D for a large class of degrees, which in particular implies the 1978 Briançon-Iarrobino Conjecture for a tetrahedral degree, and vastly generalises the result in [19]. The novelty lies in introducing a new upper bound on the dimensions of the tangent spaces of the claimed non-maximally singular ideals, constructing canonical ideals that satisfy the claimed maximal singularity condition, providing a lower bound on the dimension of their tangent spaces, and finally comparing the two cases.

We propose a manifestly conformally invariant off-shell definition of the Weyl tensor in Galilean geometry. We also propose definitions for the electric and magnetic parts thereof. For the latter to vanish, observers with specific kinematical properties need to exist. While this is guaranteed by the Newton–Cartan equation, it might not be true for other Galilean invariant theories. Therefore, we propose that imposing the existence of observers for which the off-shell magnetic part is zero should be a necessary condition for a Galilean invariant theory to be called `Newtonian', in the spirit of the `Newtonian' condition, introduced by Trautman, in standard Newton–Cartan gravity. As a side result, we also show that there exists a unique Galilean boost-invariant connection that can be built from a Galilean structure and a choice of Coriolis field, even when the clock form is not closed. No extra structure is needed, contrasting with the standard approach used to construct boost-invariant connections which introduces a mass gauge field.

We study dimension-free differential estimates for Brenier maps under first-order perturbations of the two marginals. Caffarelli's contraction theorem yields such estimates when the source and target potentials satisfy a pointwise Hessian comparison. We show that the same conclusion remains true under arbitrary globally Lipschitz perturbations of both marginals, without additional assumptions. In particular, this answers the conjecture due to Fathi, Mikulincer, and Shenfeld. More precisely, let $d\ge1$, $L\ge0$, and let $B:\mathbb{R}^d\to\mathbb{R}$ be globally $L$-Lipschitz. We prove that the Brenier map from the standard Gaussian measure $\gamma_d$ to the probability measure proportional to $\mathrm{e}^{-B}\gamma_d$ has a globally Lipschitz representative whose bound depends only on $L$. The Brenier potential also belongs to $C^{1,1}(\mathbb{R}^d)$ and satisfies dimension-free two-sided Hessian bounds. Let $\mathfrak{C}(0,L)$ denote the upper-bound constant, then \[ \log\mathfrak{C}(0,L)=4L^2+\log L+\mathcal{O}(1) \qquad(L\to\infty), \] and a one-dimensional example with Brenier map $T_L$ satisfies $\log\operatorname{Lip}(T_L)\ge L^2/2$. We deduce this Gaussian estimate from an anisotropic two-marginal result. Let $V,W:\mathbb{R}^d\to\mathbb{R}$ be potentials, and let $Q,P$ be positive-definite matrices. Using the distributional curvature bounds $D^2V\preceq Q$ and $D^2W\succeq P$, we establish matrix Hessian bounds for Brenier maps between arbitrary globally Lipschitz perturbations of the reference marginals. Most importantly, the constants keep the directional geometry of the essential gradient ranges. For affine perturbations, our estimates recover the sharp noncommuting Caffarelli tensor. We further obtain pointwise displacement bounds for Gaussian perturbations and spectral Hessian estimates for compact Gaussian mixtures.

We partially answer an open problem, posed by Nick Higham, concerning the conditioning of solutions to Sylvester and Lyapunov equations. The question arises in the backward stability analysis of numerical algorithms for these equations. We first show that the solution to the Sylvester equation $AX-XB = C$ can be arbitrarily ill-conditioned even if $A, B, C$ and the Kronecker sum $I \otimes A - B^T \otimes I$ are all perfectly conditioned. We then derive general a priori bounds on the condition number of the solution, as well as bounds for the Sylvester equation when $A$ and $B$ are diagonalizable. We also provide lower bounds involving matrix exponentials and Zolotarev numbers. For the Lyapunov equation $AX+XA^T = -C$, we obtain upper bounds in two settings: (i) when $A$ is symmetric positive definite while $C$ is symmetric negative definite, and (ii) when $A$ is strictly dissipative and $C$ is symmetric positive definite.

We study heat content and spectral properties of subordinated sub-Laplacians on arbitrary Carnot groups. We consider restrictions of such operators to bounded open sets with zero Dirichlet boundary condition and study their spectral properties. In particular, for a large class of subordinators we give explicit eigenvalue estimates in terms of the subordinator and the eigenvalues of the sub-Laplacian. We also provide large-time asymptotics of the heat content for the subordinated sub-Laplacian in terms of its spectral gap. For fractional sub-Laplacians we prove short-time asymptotics for the corresponding heat content and relative heat content. Our approach combines semigroup methods, probabilistic techniques, geometric measure theory, and heat kernel estimates.

We consider an alternative notion of minimality for C*-dynamical systems, which we refer to as strong minimality. This notion coincides with the usual definition of minimality in the commutative setting, but diverges sharply from it in the noncommutative setting in a way which we argue addresses several fundamental defects in the usual definition. We demonstrate that strong minimality is deeply connected with Rørdam's notion of a C*-irreducible inclusion, giving a new perspective on the study of such inclusions. We provide various characterizations of this property, including for reduced crossed products, tensor products, and subhomogeneous C*-algebras.

Work of Gadish and Hainaut (after Petersen) models the compactly supported cohomology of a wedge of circles as a polynomial functor. We identify the coefficients of this functor, $\Phi[n,m]$, via a cobar construction of $\mathbf{FA}$-modules. This identification formally implies that these coefficients will arise in computations of graph homology, and we use this result to give examples of graph complexes whose homology may be embedded in $H_c^\ast(F(S^1\vee S^1,n))$. This includes the Payne-Willwacher marked graph complex in genus 2, for which we give a new, explicit decomposition in terms of $\mathbf{FA}$-modules. This allows us to describe $\mathsf{gr}_{11}H_c^{\ast}(\mathcal{M}_{2,n})$ as the cohomology of a complex of decorated trees and to show, for example, $\mathsf{gr}_{11}H_c^{n+1}(\mathcal{M}_{2,n})=0$.

The rate-optimal convergence rate of gradient descent (GD) with a fixed step-size is well known to be $\Theta(N^{-1})$ for $L$-Lipschitz smooth convex objectives in the prior art in convex optimization. Surprisingly, several recent works show that we can accelerate vanilla GD by applying a nonconstant, nonadaptive, deterministic step-size schedule. The best-known upper bounds so far in the non-anytime & anytime setups are $O(N^{-1.271})$ [Altschuler and Parrilo, 2025, Grimmer et al., 2023] and $O(N^{-1.119})$ [Zhang et al., 2025], respectively. On the other hand, the best reported lower bounds (or barriers) up to date in the non-anytime & anytime setups are $\Omega(N^{-1.635})$ and $\Omega(N^{-1.241})$ [Ye and Liu, 2026], respectively. We narrow these gaps by establishing stronger lower bounds for GD's convergence rate in both settings: $\Omega(N^{-1.450})$ for the non-anytime rate bound and $\Omega(N^{-1.184})$ for the anytime rate barrier.

The geometric morphology of deposited filaments can significantly influence the structural performance and stability of 3D concrete-printed (3DCP) structures. However, most finite element (FEM)-based approaches for buildability assessment represent printed layers as simplified rectangles, potentially limiting predictive accuracy. This study proposes a geometry-informed modelling framework that integrates the deep-learning-based filament shape prediction tool ShapeGen3DCP with a layer-activation FEM approach to investigate the effect of realistic filament geometries on buildability. The framework generates geometry-aware numerical models directly from material and process parameters, eliminating the need for experimental filament characterization or computationally intensive fluid-flow simulations. Validation against experimental data and a parametric study of rectilinear walls demonstrate that extrusion parameters and the resulting filament geometry can significantly influence buildability predictions. Realistic filament representations are particularly important for free-flow deposition, whereas layer-pressing strategies are less sensitive to geometric simplifications. Among the investigated representations, an elliptical approximation provides an effective balance between geometric fidelity and modelling simplicity. When rectangular representations are preferred to enable regular computational meshes for faster simulations, defining their dimensions based on volume conservation improves prediction reliability compared with calibrating them using either the maximum filament width or the interlayer contact width. Overall, the proposed methodology demonstrates the importance of incorporating filament geometry into 3DCP simulations and provides practical guidance for selecting efficient and accurate geometric representations for buildability assessment.

This paper gives a partial answer to a question of Dan Abramovich: consider a proper morphism $f : \mathcal{X} \to \mathcal{Z}$ with connected fibers, between smooth separated Deligne–Mumford stacks, which defines an isomorphism away from a smooth effective Cartier divisor $\mathcal{E} \subseteq \mathcal{X}$. Then, is $f$ a weighted blowup? We confirm that $f$ is an ordinary smooth blowup when $\mathcal{X}$ and $\mathcal{Z}$ are smooth separated schemes of finite type over $\mathbb{C}$, and when $f :\mathcal{X} \to \mathcal{Z}$ is a representable morphism of smooth separated Deligne–Mumford stacks. Further, we show that $f$ is a weighted blowup when $\mathcal{X}$ and $\mathcal{Z}$ are smooth separated Deligne–Mumford surfaces, i.e., $\dim \mathcal{X} = \dim \mathcal{Z} = 2$. As an application we determine when a reduction morphism between Hassett moduli stacks of weighted stable curves is given by a blowup along a smooth center.

Adaptive regularization is an effective means of improving the flexibility of classical variational reconstruction methods while retaining their interpretability and mathematical structure. In this work, we propose an approach for learning deep-reparameterized adaptive regularization maps (DREAM) for Total Variation (TV) and Total Generalized Variation (TGV) through algorithm unrolling in a zero-shot, self-supervised setting, requiring no access to large paired datasets. We first validate DREAM on a two-dimensional image-denoising problem and then apply it to large-scale dynamic MRI reconstruction. The resulting adaptive models are theoretically grounded, as they retain the convergence properties of the primal-dual algorithm used as the building block of the unrolled architecture. Our results show that the regularization parameter maps can be learned efficiently, requiring substantially fewer weight updates than state-of-the-art methods to achieve comparable performance. We further demonstrate that the CNN parametrization of the maps acts as an implicit prior and provides effective dimensionality reduction, outperforming direct optimization of the adaptive parameters. Across both applications, the performance gap between supervised and zero-shot training is smaller for DREAM than for the end-to-end deep-learning methods considered for comparison. Remarkably, in dynamic cardiac MRI reconstruction, zero-shot training matches the performance of the supervised model. Moreover, the learned adaptive maps are highly interpretable, providing a compelling alternative to purely deep-learning-based methods when reference data are difficult to obtain.

Transport models underpin the design and optimisation of chemical processes, yet their constitutive laws are typically hand-picked, their experimental initial conditions are assumed exact, and their computational cost restricts optimisation to a handful of process variables. Here we introduce a differentiable hybrid modelling framework that embeds neural networks within a GPU-accelerated population balance equation solver to discover constitutive laws and assimilate initial conditions directly from experimental crystallisation data. Using thirteen real experiments with multi-dimensional size and shape measurements, we show that treating measured initial conditions as exact yields a physically inconsistent rate law despite improved predictive accuracy, whereas assimilating them recovers a physically consistent, non-linear law and reduces trajectory error sevenfold relative to conventional modelling. End-to-end differentiability further enables gradient-based optimisation of high-dimensional operating profiles, identifying a 300-variable temperature profile that reduces crystal aspect ratio by 25% at constant mass, facilitating superior product attributes. Our framework augments, rather than replaces, mechanistic modelling, offering a general route to learning transport models from data and optimising processes with them.

Site-specific training can improve wireless receiver performance without increasing computational complexity. However, real-world results have so far focused on fully trainable neural receivers and single-layer transmissions. We study site-specific finetuning of three receiver architectures: fully trainable neural, model-driven neural, and model-based. We train and evaluate these receivers using new measurements from a standard-compliant 5G NR testbed at ETH Zurich with dual-layer uplink transmission, including measurement campaigns conducted more than six months apart. Our results show that site-specific finetuning (i) substantially improves fully trainable and model-driven neural receivers, while resulting in only marginal gains for the less tunable model-based receiver; (ii) enables a single neural receiver jointly finetuned for single- and dual-layer transmission to closely match receivers finetuned separately for each configuration; and (iii) remains effective across measurement campaigns separated by more than six months. We also investigate site-specific linear minimum mean-square error channel estimation using covariance matrices estimated from either synthetic channels or site-specific measurements. When combined with iterative detection and decoding, site-specific channel estimation achieves the lowest error rate observed in our datasets. Our finetuning code and measurement datasets are publicly available at https://github.com/IIP-Group/site_specific_training

On triangular meshes, we analyze local mesh refinement based on longest-edge bisection equipped with the serial longest-edge propagation-path closure. Ties are resolved by terminal priority: if the incoming shared edge is a longest edge of the neighboring triangle, the pair is declared terminal and that edge is bisected. For every adaptive mesh sequence $\mathcal{T}_0, \mathcal{T}_1, \ldots, \mathcal{T}_L$ with a sequence of marked sets $\mathcal{M}_0, \mathcal{M}_1, \ldots, \mathcal{M}_{L-1}$, we prove the cumulative closure estimate $$\#\mathcal{T}_L-\#\mathcal{T}_0 \lesssim\sum_{\ell=0}^{L-1}\#\mathcal{M}_\ell.$$ The proof derives single-mark locality from the uniform multiplicative gap between possible descendant diameters implied by finite similarity classes, and converts this locality into the cumulative estimate through a Binev–Dahmen–DeVore-type charging argument.

A basic property of holomorphic functions $f:D\to \mathbb{C}$ defined on domains $D$ of $\mathbb{C}$ is that $f$ is uniquely determined by its real part up to an additive constant. The same is true for slice regular functions defined on circular slice domains of the division algebra of quaternions or octonions. The aim of this paper is to extend the latter result to more general classes of algebras, including, among many others, the Clifford algebras $\mathbb{R}_{p,q}$ and the split octonions $\mathbb{S}\mathbb{O}$. New phenomena appear, as well as unexpected connections with graph theory and the theory of slice-Nash functions.