Papers

We study the flip dynamics on perfect matchings of graphs, where a flip consists of replacing the edges of a perfect matching along an even cycle with the complementary alternating edges. In particular, we want to bound the minimum length of cycles such that any two perfect matchings are related by flips of such cycles. Given a finite graph $G$, we consider the families of decorated graphs obtained by replacing each vertex of $G$ with a decoration satisfying suitable conditions on the existence of perfect matchings in its subgraphs. We prove a general upper bound for this family. We then obtain stronger bounds for two families of decorations: clique decorations and Fisher decorations, the latter under the assumption that the underlying graph is planar. In both cases, the bounds are independent of the sizes of the decorations.

Parking functions correspond with preferences of $n$ cars which enter sequentially to park on a one-way street where (1) each car parks in the first available spot greater than or equal to its preference and (2) all cars successfully park. We generalize parking functions to parking completions: Here, we are given that some cars have already parked in a set of spots, which are indexed in a sequence $\mathbf{t}$. We then consider a preference list $\mathbf{c}$, where length of $\mathbf{t}$ + length of $\mathbf{c}$ = $n$. If all cars can park, we say that $\mathbf{c}$ is a parking completion. Adeniran et al. (2020) state an open problem which proposes a connection between the number of parking completions to the volumes of Pitman-Stanley polytopes by explicit computation on small values of $n$. In this paper, we provide a solution to this open problem by proving a theorem which explains that the set of parking completions is the set of $\mathbf{x}$-parking functions.

We provide a proof that the empirical spatial distribution estimator in $\mathbb R^d$ as well as the corresponding plug-in estimator of the spatial depth are uniformly $L^1$-consistent. The consistency rate only depends on the sample size $n$, not on the dimension $d$ or any tuning or regularization parameters. This is a rare property. The result of this note originates from a conversation with ChatGPT 5.4 Pro as part of some of our own earlier experiments on its mathematical reasoning capabilities.

The modular $b$-ary Gray code arranges fixed-length $b$-ary representations of intervals of natural numbers so that consecutive numbers differ in a single digit. Its usefulness, however, is tied to a fixed word length. We introduce, for every integer base $b\ge 2$, a variable-length Gray code $V_{b}$: a bijection from the natural numbers onto the set of all finite strings over the $b$-symbol alphabet $\{0,1,\dots,b-1\}$. The construction orders the natural numbers by codeword length into blocks and lists each block along the modular $b$-ary Gray code of the within-block offset, with its leading digit incremented modulo $b$. We prove four properties for arbitrary $b$. First, $V_{b}$ is a bijection, a complete code that assigns exactly one codeword to every finite string over the alphabet, including the empty string. Second, the Levenshtein (edit) distance between the codewords of two consecutive integers is always one. Third, codeword length is monotone non-decreasing in the encoded number. Fourth, the transition to codewords of length $k$ occurs exactly at $n=N_{k}=\sum_{i=0}^{k-1}b^{i}$, where $V_{b}(n)=1\,0^{k-1}$. The presented code, therefore, realizes a Hamiltonian enumeration under unit edit steps while retaining a near-optimal, self-adapting length profile. These properties suggest a use in machine learning: large language models routinely emit natural numbers as symbol strings, yet the positional notations they rely on are neither complete – strings with leading zeros are invalid or redundant – nor locally stable, since incrementing a number may rewrite many symbols at once. Because $V_{b}$ is a complete code and moves by a single edit between consecutive integers, it removes both obstacles and is a natural candidate representation for numeric tokens; we develop this argument and discuss the code's compression behavior relative to fixed-length Gray codes.

This paper illustrates the Pick-to-Learn methodology applied to the calibration of a Model Predictive Control policy. While developed around a specific example, the presentation is meant to highlight a methodology of broad applicability. The example concerns an aircraft traveling from an origin point to a destination point in the presence of uncertain crosswinds and a low-connectivity zone that should be avoided. The MPC policy is parameterized by two hyperparameters, which are selected from data by the P2L procedure. Starting from a dataset of 400 wind realizations, also called scenarios, P2L identifies a final compression set containing only two informative scenarios. The resulting MPC policy avoids the low-connectivity zone on all available scenarios and, according to the P2L theory, satisfies a probabilistic risk bound of $4.8\%$ at confidence level $1-10^{-5}$, where the risk is the probability of entering the low-connectivity zone in a future flight under a new wind realization not included in the sample.

Motivated by the celebrated Lonely Runner Conjecture, we study a variant in which the runners have time-dependent velocities. Let $n \geq 3$ runners start from the same point on the unit circle, where each runner $i\in[n]$ has a locally integrable velocity function $\nu_i\in L^1_{\mathrm{loc}}(\mathbb{R}_{>0})$. Assume that their velocities are strictly ordered almost everywhere and that the relative distance between every pair diverges. We prove that each of the slowest and fastest runners is at a distance strictly larger than $2^{-n+1}$ from every other runner at some time. Moreover, we show that the distance $2^{-n+1}$ is optimal. On the other hand, we construct examples in which every intermediate runner remains arbitrarily close to another runner at all times. As a consequence, we also obtain a sharp nonlinear analogue of a classical theorem of Schoenberg on billiard ball motion in the unit cube.

We establish asymptotic formulas for abelian extensions of global function fields ordered by conductor and subject to prescribed local conditions. Our proof combines harmonic analysis with a theory of frobenian functions over global function fields developed in this paper. We interpret our result via equidistribution on algebraic stacks.

We consider the $hp$-FEM applied to the Helmholtz scattering problem with wavenumber $k$, truncated with a perfectly-matched layer. The scatterer consists of a combination of Dirichlet, Neumann, and penetrable obstacles together with variable coefficients. Provided that the Helmholtz solution operator is polynomially bounded in $k$, all coefficients are piecewise smooth, all boundary surfaces are Gevrey and all coefficients restricted to boundary surfaces are Gevrey together with all their normal derivatives, we show that the $hp$-FEM is quasioptimal when $p\geq 1+\varepsilon \log k$ and $hk/p$ is sufficiently small; i.e., the $hp$-FEM does not suffer from the pollution effect. This result generalises the analogous results in both [Bernkopf, Chaumont-Frelet, Melenk 2025] (proved for piecewise analytic coefficients and analytic boundaries) and [Galkowski, Lafontaine, Spence, Wunsch 2024] (proved for smooth coefficients that are analytic near analytic obstacles) to a much larger class of scatterers.

An $n \times n$ matrix $W$ with exactly $w$ non-zero entries taken from the set of $k^{\rm th}$ complex roots of unity in each row and column satisfying $WW^{\ast} = wI_n$ is a complex generalised weighing matrix $CGW(n,w;k)$. We study such matrices through the centraliser algebras of monomial representations of finite groups. Using an exhaustive search over the linear characters of Schur covers, we classify, up to monomial equivalence, the complex generalised weighing matrices admitting a primitive group of rank at most five and degree at most $80$ acting by strong automorphisms, for coefficient orders $k \leq 6$, with partial results for larger degrees $100$. The census recovers known infinite families related to projective and affine finite geometries, describes infinite families related to Hamming schemes and settles the existence of some small open cases enumerated in the literature. We construct quantum error-correcting codes from the these matrices and determine their minimum distances exactly in all cases.

Let $d \geq 3$, \[ D \subsetneq \{ 0,1,\dots,d-1\}, \qquad |D| \geq 2, \ 0 \in D \] be a finite alphabet, and define the integer Cantor set \begin{align} \mathcal{C} := \mathcal{C}_{D} := \bigcup_{J \geq 0} \Big\{ \sum_{j =0}^J a_j d^j : a_j \in D \Big\}. \end{align} We prove that for any $\sigma$-finite measure-preserving system, $(X,\mu,T)$, and any $f \in L^p(X)$, $2\leq p<\infty$, the ergodic averages \begin{align} \frac{1}{|\mathcal{C}_N|} \sum_{n \in \mathcal{C}_N } f(T^n x), \qquad \mathcal{C}_N := \mathcal{C} \cap \{1,2,\dots,N \} \end{align} converge $\mu$-almost everywhere. By rescaling, this allows us to resolve the question of lacunary differentiation of Cantor measures at self-similar scales: if \begin{align} \mathcal{C}' := \Big\{ \sum_{j \geq 1} a_j d^{-j} : a_j \in D \Big\} \subset [0,1] \end{align} is a real-variable Cantor set, and $\nu$ denotes its natural measure, then we prove that \begin{align} \lim_{k \to \infty} \int f(x-d^{-k} t) \ d\nu(t) = f(x) \end{align} Lebesgue almost-everywhere for any $f \in L^2_{\text{loc}}(\mathbb{R})$.

The main purpose of this paper is to study the global solvability for a general class of quasilinear shifted wave equations on hyperbolic spaces, for smooth initial data with small amplitude. In contrast to the case of Euclidean spaces, when the space dimension is three, we do not need to assume structural conditions like the null conditions to ensure global existence. To achieve this, we establish the energy and local energy estimates for perturbed wave operators on $\mathbb{R}\times \mathbb{H}^n$. These estimates allow time-dependent metric perturbations and require only suitable smallness together with polynomial decay in the radial variable $r$. As a byproduct, for semilinear problems with power-type nonlinearities and radial data, we also obtain global solutions with low-regularity. In particular, we prove an analog of the radial Glassey conjecture on hyperbolic space.

We introduce the algebraic structure of an inverse brace, namely, a triple $(S,+,\circ)$ such that both $(S,+)$ and $(S,\circ)$ are inverse semigroups and the following identity holds $a\circ(b+c)=a\circ b-a+a\circ c$, for all $ a, b, c \in S$, where $-a$ denotes the inverse of $a \in S$, with respect to $+$. In particular, every weak brace is an inverse brace. We investigate the fundamental properties of inverse braces and analyze the relationship between additive and multiplicative idempotents, characterizing the condition under which an inverse brace is a weak brace. Our main results concern the connection with set-theoretic solutions to the Yang-Baxter equation. Specifically, we provide a class of inverse braces that yield solutions and give several examples. Finally, we introduce constructions of inverse braces via the matched product and the strong semilattice of inverse braces. We show that these constructions preserve the conditions required to produce solutions, thereby providing a systematic method for generating new examples.

In this work, we develop a geometry-split implicit-explicit (IMEX) framework for the compressible flow equations, wherein stiff regions are treated via an implicit hybridizable discontinuous Galerkin (HDG) method, while non-stiff regions are treated via an explicit discontinuous Galerkin (DG) method. Two implicit formulations are investigated: a mixed HDG method (HDG-MX) and a primal interior-penalty HDG method (HDG-IP). The spatial coupling between the implicit and explicit solutions is achieved in a conservative manner by appropriate interface conditions, while the temporal synchronization is maintained through the use of additive Runge-Kutta (ARK) schemes. We provide a detailed discussion on the computational performance of the resulting IMEX schemes. Verification and validation over a range of numerical experiments confirm that the proposed IMEX schemes achieve high-order accuracy in both space and time. Performance studies further indicate that the approach effectively alleviates geometry-induced stiffness and can provide speedups of up to approximately 50 relative to a fully explicit DG scheme, provided that the implicit region is chosen appropriately.

We study the inertial motion of incompressible continua within a geometric and variational framework, extending the classical Arnold-Ebin-Marsden theory from the group of volume-preserving diffeomorphisms of a fixed domain to configuration spaces of deformations with variable image. In the latter case, the lack of a group structure requires proving some results that are instead immediate in the classical setting. We show that the orientation-preserving deformations with suitable regularity constitute a Hilbert manifold and that volume-preserving deformations form a submanifold with tangent vectors that are mapped onto divergence-free vector fields by the Lagrangian-to-Eulerian-picture correspondence. In so doing, we also present the geometric structure corresponding to compressible continua. The kinetic energy of the continuum gives rise to both a Lagrangian action, from which the equations of inertial motion are deduced, and a metric, with geodesics that are identified precisely by inertial motions. While the inertial motion can coincide with a physical one only for incompressible perfect fluids, it can be used to provide a natural parametrization of the configuration manifold for generic continua. We obtain a general result of local-in-time existence of solutions for the geodesic flow equation and we present explicit examples showing that, for the same initial data, the geodesic followed by the continuum in the compressible case can leave the manifold of admissible deformations in finite time, while the corresponding incompressible geodesic exists for all times.

Gain-cell embedded dynamic random-access memory (GCRAM) offers dense and energy-efficient on-chip storage, but retention-time variations force frequent refresh operations to cover worst-case bits. Error-correction codes (ECCs) can alleviate this limitation by masking bit errors from weak cells and thereby reduce refresh cost. However, the trade-off between the additional access and logic energy introduced by ECCs and the power savings from longer refresh intervals is nontrivial, especially considering the wide range of available ECC options. To optimize overall power consumption, we propose an ECC selection method that combines a refresh-interval model with power analysis to identify the minimum-power ECC configurations under a given yield constraint. Across different memory bandwidths, activity factors, and read/write ratios, the evaluation results show that the best ECC option shifts from stronger codes in refresh-dominated operating regions to lower-overhead codes in access-dominated regions and achieves 46.8% to 94.8% reduction in total power relative to the no-ECC reference.

Let $D_n$ be the disjointness graph on the nonempty subsets of $[n]$, whose independent sets are exactly the intersecting families on $[n]$. We study the weighted independent-set polynomial $W(n)=\sum_F\prod_{S\in F}w(S)$, the sum running over these families, for the doubly exponential weight $w(S)=2^{2^{n-|S|}}-1$. The kernel-bearing (trivial) part $Z_\cap(n)$ is exact by inclusion-exclusion and satisfies $Z_\cap(n)\sim n\cdot 2^{3^{n-1}}$. For the kernel-free remainder we prove the exact prefactor $R(n)=(3/4+o(1))n\cdot 2^{3^{n-1}-2^{n-1}+2}$, whence $\log_2(Z_\cap(n)/R(n))=2^{n-1}-2+\log_2(4/3)+o(1)$, an additive $o(1)$, not merely a leading-order one. The engine is a second-level extremal theorem: among kernel-free maximal linked systems other than the $n$ one-flip stars, the largest weight exponent is $3^{n-1}-3\cdot 2^{n-2}+6$, a fixed gap $2^{n-2}-4$ below the maximum, with the extremisers classified exactly. None of this is special to the weight: for $w_B(S)=B^{B^{n-|S|}}-1$ with integer $B\ge 2$ the same stars dominate, the near-extremal families sit a gap $B^{n-2}-B^2$ below, and the prefactor is $1-B^{-B}$. The combinatorial input is the $p$-biased extremal problem for non-trivial intersecting families: $M_2(n,p)=p-pq^{n-1}+qp^{n-1}$ for all $n\ge 3$, $0<p\le 1/2$, $q=1-p$. This first level is essentially known: the extremal family is the Wheel coterie of Peleg and Wool, and at $p=1/Q$ the statement, with its maximiser classification, is the case $r=n$ of Borg's Hilton-Milner theorem for signed sets (2013). We give a short self-contained Erdős-Ko-Rado proof, uniform in real $p\in(0,1/2]$, whose layer-two rigidity feeds the second level. The novelty claimed lies at the second level and in the prefactor, where the classification cannot be read off the layer profile alone: at $n=5$ one profile carries two non-isomorphic types of extremisers.

For an explicitly represented finite empirical model, deciding whether the contextual fraction is strictly below one is NP-complete, while the standard exact linear program has one column for every global assignment. We identify a permutation-transport class in which this global problem collapses to a fixed-point calculation. Let a connected permutation gain graph act on a finite state set $O$, let $H \leq{ \rm Sym}(O)$ be its holonomy subgroup, let $F = {\rm Fix}(H)$, and let $p$ be an $H$-invariant root distribution. For the induced empirical model, \[ {\rm NCF}(e)=p(F),\qquad {\rm CF}(e)=1-p(F). \] Consequently, compatibility, $F$, and ${\rm CF}(e)$ are computable in $O(|O|(|V|+|E|))$ arithmetic and table operations. For every finite simple $2$-edge-connected graph, any deterministic exact algorithm in the explicit permutation-table query model requires at least $(|O|-1)|E|$ probes in the worst case, making the dependence on the input tables optimal up to constant factors. With a fixed spanning tree, chord insertions and deletions require $O(|O|)$ worst-case time, or time proportional to the moved-set representation, while compatibility and contextual-fraction queries take $O(1)$ time. Finally, for common-marginal realizable binary constraint languages, the support threshold ${\rm CF} < 1$ is polynomial-time equivalent to the associated finite-domain constraint-satisfaction problem and therefore inherits the Bulatov–Zhuk dichotomy. The results identify a query-optimal and dynamically maintainable tractability island inside the general contextual-fraction problem.

A foundational principle in nonlinear dynamics is that the structure of a dynamical system can be recovered from a small number of generic measurements or coordinates. We develop an analogous principle for the identification of dynamic models for time series with noise, which builds on previous identification results for noiseless dynamical systems. The noise is allowed to be non-iid, non-Gaussian, and dependent on the state. Our results cover noisily observed differential equations and discrete-time dynamical systems, as well as stochastic models with process noise. We illustrate the utility of this identification principle using a Lorenz-63 model and a Hénon map model, both with observational noise.

Let $P^+(n)$ denote the largest prime factor of $n$. One of Erdős and Turán's conjectures asserts that the asymptotic density of integers $n$ satisfying $P^+(n)<P^+(n+1)$ is 1/2. In this paper, we prove that this density is larger than 0.280, which improves the previous result 0.2017 by Lü and Wang (2025). We also prove that there exists a positive density of $n$ such that $P^+(n)<P^+(n+1)<x^{41/107+\varepsilon}$. Define $T_c(x):=\#\{p\leq x:P^+(p-1)\geq p^c\}$. For $1/2<c<1$, we also show that \begin{align*} \mathop{\lim \sup}_{x\rightarrow\infty}\frac{T_c(x)}{\pi(x)}\leq \min\left(-\frac{7}{2}\log c,\frac{1-\delta}{2c}\right), \end{align*} where $\delta=\delta(c)>0$.

We continue our study of quantum dynamics on a Lie group $G$, initiated in arXiv:2512.19840, by building the path integral that governs transition amplitudes in the Hilbert space $L^2(G)$. This relies on the proper handling of both a noncommutative momentum space and the presence of compact directions in $G$. We show that compactness can be handled through a sum over winding numbers in maximal tori of $G$, generalizing the similar sum commonly encountered for path integrals on a circle. As applications, we compute semiclassical approximations of propagators and partition functions of Euler-Arnold systems, up to (and including) two-loop order.

We study sharp estimates for the $p$-capacity on complete non-compact Riemannian manifolds under lower Ricci curvature bounds. First, we establish sharp comparison inequalities for the $p$-capacity of bounded smooth domains in manifolds satisfying $\operatorname{Ric}\ge -ng.$ The estimates are expressed in terms of the boundary mean curvature and correspond to natural warped-product model ends. We characterize all equality cases and show that equality forces the exterior region to be isometric to the corresponding warped product. We also obtain an analogous sharp estimate under nonnegative Ricci curvature, whose equality case is described by an asymptotically flat model end. Second, we investigate normalized lower bounds for the relative $p$-capacity of condensers. We introduce scale-invariant quantities involving the volume of the inner set and the diameter of the ambient domain, establish uniform positive lower bounds, and determine the optimal ranges of the normalization parameters.

We study the degeneration hierarchy of commutative, associative, finite-dimensional complex Artin algebras. Instead of studying degenerations in the Hilbert scheme, we introduce a singularity-theoretic notion of degeneration based on the correspondence between singularities of stable map germs and local algebras. This leads to a natural partially ordered set, the stable hierarchy, in which one algebra degenerates to another if nearby singularities realize the latter. Our first main result is that, in a wide range of dimensions, this hierarchy can be determined purely from symmetry data, namely from the automorphism groups of the algebras or singularities. The key tool is the theory of certain equivariant characteristic classes called Thom polynomials of multisingularities, established by Kazarian. Suitable substitutions into these polynomials completely characterize the hierarchy. As a consequence, the computation of degeneration posets becomes algorithmic in nature. In our second main result, we prove that our singularity-theoretic hierarchy extends the algebraic hierarchy obtained from deformation theory. While deformation theory requires the dimension (rank) to be fixed, our hierarchy generalizes this framework by comparing algebras of varying dimensions.

This work studies the mean-field limit of large networks of interacting stochastic leaky integrate-and-fire (LIF) neurons subject to short-term synaptic depression (STD). The macroscopic dynamics of this system is governed by a two-dimensional, non-linear McKean-Vlasov equation that couples the evolution of the neurons' membrane potentials with a synaptic depression variable. We investigate the long-time behavior of this limit system. To this end, we introduce an auxiliary linearized Markov process by freezing the interaction non-linearity to a constant. By exploiting the regeneration of the membrane potential at spike times, we are able to explicitly compute the conditional expectation of the synaptic depression variable, conditionally on the potential value, under the invariant measure of this two-dimensional linear process. This is a crucial ingredient to study time-dependent local perturbations thereof. As a consequence we are able to identify an analytic criterion guaranteeing the local stability of any invariant probability measure of the fully non-linear system. This stability criterion is formulated in terms of the zeros of the Laplace transform of a specific linear response function. Finally, we provide numerical examples demonstrating that the two-dimensional framework induces a richer spectrum of long-time dynamics than purely one-dimensional models. For example, synaptic depression can lead to low-frequency oscillations around a unique, unstable invariant measure where the oscillations are much slower than the neurons' firing rates.

A new approach for nonparametric estimation of the quantile density function (sparsity function) and its derivatives is suggested which is based on local polynomial estimation. The estimator has more advantageous properties at the boundaries than classical quantile density estimators. Asymptotic normality is shown and the bias, asymptotic variance as well as boundary properties are compared with other estimators.

We prove a conjecture of Montaldo, Oniciuc and Ratto concerning the nullity of a family of proper biharmonic maps from the flat two-torus to the round two-sphere. The proof reveals an unexpected connection between spectral geometry and arithmetic geometry. We show that the vanishing of a mixed Fourier eigenvalue produces a rational point on an explicitly defined affine quartic. By constructing an explicit polynomial isomorphism with an elliptic curve over $\Q$, the problem is reduced to the determination of a Mordell–Weil group. This yields a complete description of the rational points on the spectral curve and shows that none satisfies the positivity conditions required for a mixed Fourier mode. As a consequence, the mixed eigenvalues never vanish, confirming the Montaldo–Oniciuc–Ratto conjecture and proving that the nullity of every map in the family is equal to $5$.

By “skeletal homology” $K_{n}^{\varepsilon}(X)$ we mean the homology of the chain complex $S_{n}^{\varepsilon}(X)$ generated by skeletal $(n,\varepsilon )$-simplices, i.e.\ maps from the $0$-skeleton of the standard simplex into a metric space $X$ whose image has diameter less than $\varepsilon$. Goldfarb showed for finite metric spaces that this homology is isomorphic to the simplicial homology $H_{n}^{\Delta}(VR_{\varepsilon}(X))$ of the Vietoris–Rips complex. We give a new proof of this isomorphism for arbitrary metric spaces. We then develop a metric analogue of methods from singular homology by replacing the topological notion of “homotopic” with the metric notion of “close”. The key tool is a discrete prism construction, which shows that uniformly close functions with a common modulus of continuity induce the same homomorphism on skeletal homology. This leads to stability results for $K_{n}^{\varepsilon}(X)\cong H_{n}^{\Delta}(VR_{\varepsilon}(X))$, including broad extensions of the theorems of Hausmann and Latschev beyond the setting of Riemannian manifolds. Finally, repeated barycentric subdivision and restriction induce a natural homomorphism $\rho_{\varepsilon}:H_{n}% (X)\longrightarrow K_{n}^{\varepsilon}(X). $ We call its image the real homology $H_{n}^{\varepsilon}(X)$ at scale and define the corresponding phantom homology $P_{n}^{\varepsilon}(X) = K_{n}^{\varepsilon}% (X)/H_{n}^{\varepsilon}(X). $ We show that phantom homology can detect impending topological collapse, as illustrated by collapsing Riemannian manifolds such as Berger spheres.

We study moduli spaces and monodromy groups for surfaces fibered in conics over the projective line, with a view toward explicit presentations and arithmetic applications.

This study aims to develop a numerical homogenization method that can be applied to a heterogeneous stratified medium. Traditional scale transition methods are inadequate in capturing the essential gradient properties of some materials. Therefore, the focus of this work is to construct a homogenized model that considers the material property gradient. To achieve this, a two-step homogenization scheme is proposed. Firstly, the 3D model is decomposed into multiple 2D heterogeneous layers, and the behavior of each layer is estimated using a micro-mechanical model such as the Hashin-Shtrikman bounds. Secondly, a variational sum method is used to rebuild the behavior of the 3D environment. Finally, the method is applied to homogenize a thin plate with a porosity gradient.

The adaptive Bregman-Kaczmarz method recovers the exact, noise-free solution of a linear inverse problem even when every measurement it queries is corrupted, provided the corruption is fresh, independent and zero-mean. A block version was proposed for parallel hardware, but whether larger blocks actually converge faster was left open. We show the answer hinges on how the block is used: replacing the block sum with a block average collapses the analysis onto a single positive-semidefinite matrix, through which we prove that the guaranteed convergence improves monotonically with the batch size, with a total gain governed by the stable rank of the system matrix. We then address heterogeneous noise by introducing a noise-aware weighting that down-weights unreliable measurements, and prove it is strictly better than uniform weighting whenever the noise is not proportional to the row norms - a condition that essentially never holds in practice. The two improvements are compatible and their benefits combine. Finally, we show the adaptive step size interpolates automatically between a fast initial phase and a slowly vanishing tail that carries the error exactly to zero, and we explain how its hyperparameters can be estimated without knowing the true solution. Numerical experiments under sparse, heterogeneous noise illustrate these findings and confirm that the heuristic estimates produce effective step sizes.

In this article, we prove two types of results about the Frobenius dimension of associative algebras. First, we refine the known upper bound for the Frobenius dimension of a finite dimensional algebra in terms of its dimension as a vector space, and show that the only algebras reaching this bound are radical square zero algebras associated with single-vertex quivers. Second, we compute Frobenius dimension for low-dimensional algebras explicitly, and for truncated path algebras in terms of paths with no detours in their underlying quivers.