Papers

Let $\mathscr{G}_{n,k}$ denote the family of all connected graphs of order $n$ with matching number $k$. Liu, Lou, and Trevisan~(Linear Algebra Appl., 2026) posed the following problem: Determine the spectrally minimal graphs in $\mathscr{G}_{n,k}$. In this paper we prove that for every graph $G \in \mathscr{G}_{n,k}$, $ \rho(G) \ge \sqrt{\frac{n + 2k - 3}{k}}, $ and we completely characterize the extremal graphs when $k \mid (n-3)$. As applications, we establish $\rho(G) + k \ge 3\sqrt[3]{n/4}$ for $k \ge 2$, settling the asymptotic order of $\rho + k$ as $\Theta(n^{1/3})$ – strictly smaller than the $\Theta(\sqrt{n})$ order suggested by the disproved Aouchiche–Hansen conjecture.

A significant result by Lickorish and Wallace shows that every closed, orientable 3-manifold can be obtained from a Dehn surgery on one link in 3-sphere. As links and Dehn surgeries vary vastly in the universe, a question arises: how can we describe their properties in vague? We introduce a counting model on links and Dehn surgeries, and prove that under this model, (1) a randon link is hyperbolic; (2) a random 3-manifold is hyperbolic.

In this paper, we study last-iterate convergence of stochastic first-order methods for constrained smooth convex–concave minimax optimization under the standard bounded-variance stochastic oracle. A fundamental challenge is that the last iterates of vanilla stochastic extragradient (S-EG) and stochastic optimistic gradient descent–ascent (S-OGDA) may fail to converge in the presence of stochastic gradient noise, even for simple bilinear problems. To overcome this difficulty, we introduce a simple perturbation framework that regularizes the original convex–concave problem into a strongly convex–strongly concave one. Applying S-EG and S-OGDA to the perturbed problem yields two simple single-loop methods, referred to as perturbed S-EG (PS-EG) and perturbed S-OGDA (PS-OGDA). We establish last-iterate convergence by first deriving convergence in terms of the squared distance to the saddle point of the perturbed problem and then translating this estimate into guarantees for the restricted primal–dual gap. Based on this framework, we establish two types of convergence guarantees. When the optimization horizon is known a priori, both PS-EG and PS-OGDA achieve an $\mathcal{O}(T^{-1/4})$ last-iterate convergence rate for the restricted primal–dual gap, which coincides with the standard primal–dual gap on compact feasible domains. When the optimization horizon is unknown, we develop an anytime variant based on diminishing perturbations and diminishing stepsizes. For general closed convex feasible sets, both PS-EG and PS-OGDA achieve an $\mathcal{O}(T^{-1/5})$ last-iterate convergence rate for the restricted primal–dual gap. Furthermore, in the unconstrained setting, PS-EG admits a sharper $\mathcal{O}(T^{-1/4})$ anytime convergence rate in terms of the gradient norm.

We investigate the existence of normalized solutions to the gradient-type Schrödinger system \begin{equation*} \begin{cases} -\Delta u+ V_1(x)u+\lambda u= uv^2 & \text{ in } \Omega,\\ -\Delta v+ V_2(x)v+\lambda v= u^2v & \text{ in } \Omega %\frac{\partial u}{\partial \nu}=\frac{\partial v}{\partial \nu}=0 \, & \text{ on } \partial \Omega \end{cases} \end{equation*} subject to the mass constraint $\int_{\Omega}\left(|u|^2+|v|^2 \right)dx=a>0$ and Neumann boundary conditions, where $\Omega\subset \mathbb{R}^3$ is a smooth bounded domain, each $V_i$ is continuous, and $\lambda$ is a Lagrange multiplier. Applying a minimax principle that incorporates Morse index information, we establish the existence of nontrivial normalized solutions of mountain pass type. The proof is based on a refined blow-up analysis adapted to such gradient-type systems, together with new Liouville-type theorems for finite Morse index solutions of the associated limit systems in $\mathbb{R}^3$ and $\mathbb{R}^3_+$.

A positive integer $n$ is said to be nondeficient if $\sigma(n) \geq 2n$. Letting the positive divisors of a positive integer $n$ be written as $1 = d_0 < d_1 < \cdots < d_k < d_{k+1} = n$, and letting $\mathcal{S}$ denote a set of integers, if there exist values $\lambda_j \in \mathcal{S}$ such that $1 + \sum_{j=1}^{k} \lambda_j d_j = n$, then $n$ is said to be an $\mathcal{S}$-perfect number. Ross, in 2024, introduced the study of $\mathcal{S}$-perfect numbers, and concluded with two conjectures that each concern both $\{ -1, 1 \}$-perfect numbers and nondeficient numbers. We disprove both of these conjectures.

This paper provides the study of the homogenization of the 3D non-homogeneous incompressible heat-conducting fluid in perforated domains with holes of subcritical size, where the viscosity and the heat conductivity coefficient are assumed to depend on the temperature. The diameter of the holes is of order $\varepsilon^{\alpha} \ (\alpha>3)$, where $\varepsilon > 0$ is a small parameter that measures the mutual distance between the holes. We prove that as $\varepsilon\to 0$, the limit behavior of velocity, density and temperature is governed by the original system in homogeneous domain without holes.

We prove an $\mathrm{MSO}_2$ zero-one law for a very sparse Erdős-Rényi graph after pruning by component order. Let $p_n=c_n/n$, where $c_n\to0$, and delete every component of order less than $f(n)$, where $f(n)\to\infty$. If \[ f(n)\bigl(\log f(n)+\log(1/c_n)\bigr)=o(\log n), \] then the resulting graph satisfies a zero-one law for $\mathrm{MSO}_2$, with quantification over sets of vertices and sets of edges. The proof combines uniform component counts, an MSO Feferman-Vaught decomposition for disjoint unions, and semilinearity of the order spectra of MSO-definable classes of finite trees. We also show that the term $f(n)\log f(n)$ cannot simply be omitted: star components can occur at first-order-visible Poisson thresholds. We further establish first-order limit laws for bond percolation on the discrete torus $T_L^d$. In the two-sided subpolynomial regime, pruning below a sufficiently slow threshold yields a zero-one law. For the unpruned model in either one-sided polynomial regime, the reciprocal exponents $\alpha=1/k$ are precisely the critical scales. At such a scale, an extended limit of $N p_N^k$ or $N q_N^k$ equal to $0$ or $\infty$ gives a zero-one law; a positive finite limit gives a convergence law but not a zero-one law; and the absence of an extended limit gives failure of convergence. Finally, $\mathrm{MSO}_1$ already detects the parity of the torus side length through bipartiteness, producing a natural obstruction to monadic convergence in a near-deterministic regime.

We study distributed experimental design with multiple candidate experiments, where local sites possess only partial information and transmit design recommendations to a fusion center. Unlike centralized design, in which the experiment that maximizes expected information gain can be selected directly, distributed design requires combining heterogeneous and potentially conflicting local recommendations. Formulating as a multi-class Bayes fusion problem, centralized oracle design is treated as an unknown label and each site is characterized by a local recommendation mechanism. The proposed fusion rule minimizes posterior expected information regret, rather than merely maximizing the number of local votes or the posterior probability (MAP) of the oracle label. This distinction is essential because different incorrect experimental choices may incur different losses in information gain. We show that majority vote is optimal only under restrictive symmetry assumptions and can otherwise be strictly suboptimal. Regret-weighted multi-class Chernoff bounds are derived to identify the pairwise separations governing distributed design performance. Numerical studies identify two operational regimes: MAP is effective when oracle-label accuracy and information regret are aligned, while regret-weighted Bayes fusion reduces information loss when the most probable oracle label is not the lowest-regret decision.

Noguchi introduced the notion of Wronskian curvature positivity in his Second Main Theorem and asked for further examples to which his theorem applies. We give a complete answer: a complex manifold admits a smooth connection satisfying this condition if and only if it admits a holomorphic projective connection. For a fixed torsion-free connection, the condition is equivalent to the holomorphicity of its projective class.

Holographic multiple-input multiple-output (HMIMO) has emerged as a promising technology for future wireless systems by enabling continuous electromagnetic (EM) field control over large apertures. However, user-side EM exposure has become an increasingly important concern in large-scale array systems. This paper addresses this issue by developing a multiuser uplink HMIMO model, where a physically consistent specific absorption rate (SAR) model is established to quantify the EM exposure. On this basis, a spectral efficiency (SE) maximization problem is addressed by developing a modified iterative water-filling algorithm. Simulation results demonstrate that the proposed algorithm effectively improves the system SE while satisfying the SAR constraints.

Tumor evolution is shaped by cell division, cell death, competition, and constraints imposed by the local microenvironment. Because these dynamics are usually not observed directly, phylogenetic trees inferred from somatic variation in sampled tumor cells can provide an indirect record of the population history that produced the sample. In this paper, we examine whether the distribution of inferred internal branching times exhibits signatures that depend on the underlying tumor growth regime. Specifically, we study the distribution of internal branching times in continuous-time birth-death models of tumor evolution. Exponentially growing populations exhibit a unimodal distribution of internal branching times, with the mode located near the root. In contrast, logistic growth, which models expansion constrained by carrying capacity, yields a substantially more intricate genealogical structure: the distribution of branching times undergoes a systematic transition as the time elapsed since tumor initiation increases. Specifically, this progression shifts from an expansion-dominated phase, through an intermediate early-recent bimodal phase, to a final recent-dominated phase. Extensive simulations of reconstructed tumor genealogies support these theoretical findings.

Krauth and Mézard predicted in 1989 that the storage capacity of the Ising perceptron at zero margin is an explicit constant $\alpha_\star\approx0.8330786$. Let $M_N$ be the largest number of random patterns that can be stored by an $N$-dimensional Ising perceptron. We give a computer-assisted proof that \[ \frac{M_N}{N}\xrightarrow{\mathbb P}\alpha_\star, \qquad \alpha_\star\in[0.833078599,0.833078600]. \] Previous work established matching conditional lower and upper bounds, subject respectively to a one-variable global sign condition of Ding–Sun and a two-variable global sign condition of Huang. We rigorously verify both conditions using Arb ball arithmetic. For Huang's condition, a moment-coordinate reparametrization compresses the unbounded parameter plane onto a compact convex body. Convex duality and certified adaptive sweeps control its bulk, while a ray-concavity argument treats the degenerate maximizer. We also re-establish the shared parameter rectangle and verify the full Ding–Sun condition, including its curvature and endpoint requirements. Combining these verifications with the existing sharp-threshold and universality theorems proves the result for Gaussian disorder and for every fixed i.i.d.\ mean-zero, unit-variance subgaussian disorder law, including Bernoulli disorder. The complete verification programs, certificates, and raw records accompany the paper.

We identify a two-dimensional time-reversal-invariant topological metallic phase on a bilayer Lieb lattice, characterized by a quantized layer–resolved pseudo-spin Chern number. Without the orbital-angular-momentum-dependent (OAM-dependent) coupling, the system gives rise to a time-reversal-invariant topological semimetal with a zero indirect gap and quantized pseudo-spin Chern number. Opposite-sign intralayer OAM-dependent coupling immediately converts the zero-indirect-gap semimetal into a metal, in which the global spectrum is metallic while the layer–resolved pseudo-spin Chern number remains well defined as long as the direct gap at each crystal momentum and the pseudo-spin gap remain open. The model also exhibits asymmetric boundary states: in the semimetallic regime, one edge hosts perfectly flat bands, whereas the opposite edge supports gapless counter-propagating modes forming a one-dimensional Dirac cone. An edge-localized interlayer coupling gaps only the counter-propagating edge states, leaving the flat-band edge essentially intact, while intralayer OAM-dependent coupling bends the exact flat band into a dispersive boundary mode without affecting the gapped Dirac edge. These results open a route toward the controlled engineering of layer–resolved topological gapless phases in synthetic and quantum materials.

This paper develops a model-free reinforcement learning framework for continuous–time extended mean field control problems, where both the dynamics and reward may depend on the joint distribution of states and controls. We adopt deterministic feedback policies, under which the state–action distribution is induced directly as a push–forward of the state law. This avoids optimization over stochastic kernels and bypasses key limitations of existing approaches in extended mean field settings. We first establish a model–free sensitivity formula for parameterized McKean–Vlasov dynamics and use it to derive a deterministic policy gradient formula expressed through an advantage–rate function on the Wasserstein space. We then refine this formula by introducing local value and advantage–rate representations that depend on the state, action, and joint state–action distribution, yielding a policy gradient that includes both action derivatives and measure–derivative terms with respect to the control distribution. These characterizations lead to a martingale–based learning principle and motivate a continuous–time deep deterministic policy gradient algorithm combining particle approximations, measure–dependent neural networks, temporal–difference learning, and exploration in either action or parameter space. Numerical experiments on stochastic Cucker–Smale consensus control and optimal liquidation with trade crowding demonstrate the efficiency, stability, and robustness of the proposed method, including problems with explicit dependence on the control distribution.

The Smith-Ward problem about matrix ranges, posed in the 1980s, was recently resolved in the negative by Marcel Scherer (arXiv:2607.04274) by obtaining a three-dimensional operator system $\mathcal{S} \subseteq M_4(C_r^*(\mathbb{F}_2))$ without the lifting property. However, this operator system must be exact. In this paper we show that, for every finitely generated $C^*$-algebra $\mathcal{A}$ without the local lifting property (LLP), there exists a three-dimensional operator system $\mathcal{S} \subseteq M_{n+2}(\mathcal{A})$ without the lifting property (LP), thus generalizing Scherer's result and eliminating the reliance on $\text{Ext}(\mathcal{A})$ not being a group. In particular, we prove that whenever $\mathcal{T}$ is a finite-dimensional operator system without the LP, then $M_{n+2}(C_u^*(\mathcal{T}))$ contains a $3$-dimensional operator system without the LP for some $n \leq 2(\dim(\mathcal{T})-1)$. In this way, we yield a plethora of counterexamples to the Smith-Ward problem. In particular, unlike Scherer's example, these three-dimensional operator systems fail both the LP and exactness. We also prove the existence of a three-dimensional operator system that detects nuclearity for unital $C^*$-algebras, strengthening previous work of Kavruk (J. Funct. Anal., volume 269, 2015).

We give a tight single-change covering design with $v=26$ and $k=6$. This answers Problem 1 of the Nineteenth British Combinatorial Conference, which asked whether such a design exists with block size greater than $5$. We also describe the satisfiability search that found the design, including negative search results at the smallest admissible order $v=21$.

Graded $n$-absorbing ideals generalize graded prime ideals by extending absorption properties to products of $(n+1)$ homogeneous elements. We study several generalizations of graded prime ideals, including graded $n$-absorbing, graded weakly $n$-absorbing, graded strongly $n$-absorbing, and graded $n$-absorbing primary ideals, as well as related graded $n$-absorbing subgroups. Our primary result establishes a combinatorial model for graded $n$-absorbing principal monomial ideals in polynomial rings with the standard grading. By identifying principal monomial ideals with exponential vectors in $\mathbb{N}^{m}$, we show that the graded $n$-absorbing principal monomial ideals correspond precisely to lattice points in the simplex $\{\alpha \in \mathbb{N}^{m}: \vert \alpha \vert \leq n\}.$ Consequently, the Hasse diagram of principal monomial ideals is realized as the 1-skeleton of the Cayley graph of $\mathbb{N}^{m}$, yielding a geometric and combinatorial interpretation of graded $n$-absorption.

Autonomous-vehicle motion planners must resolve conflicts among safety, regulation, comfort, and efficiency in real time while exposing those decisions for audit. We present W-SQP, a weighted tiered-slack nonlinear model predictive controller (NMPC) that compiles nine driving-rule families into a four-tier shared-slack nonlinear program solved online with CasADi and IPOPT; the name denotes the weighted quadratic slack penalty, not a sequential-quadratic-programming solver. Strongly separated tier penalties bias residual violations toward lower-priority rules while leaving actuation bounds hard. The controller replans from its executed state at $10$\,Hz and records per-rule residuals on every cycle. A $90$\,ms solver-time limit returns an anytime iterate that is projected through the vehicle dynamics before execution; median and maximum observed wall-clock solve times were $28$ and $104$\,ms. We evaluate W-SQP in closed loop on 150 Waymo Open Motion Dataset scenarios in Waymax against reactive and proposal-and-select baselines, and introduce a log-independent protocol that separates safety and regulatory compliance from resemblance to the recorded human trajectory. Under this protocol, W-SQP shows no systematic group-level deficit relative to expert replay on the log-independent safety and regulatory rules, with several localized regressions in the hardest, highest-divergence scenarios. The results characterize W-SQP as an auditable, priority-biased, anytime-capable NMPC prototype rather than a hard-real-time or formally safe controller.

Lattice point counting in gauge balls on the Heisenberg group $\mathbb{H}^q$ is a non-commutative analogue of the Euclidean multidimensional sphere problem, initiated by Garg, Nevo and Taylor \cite[Ann. Inst. Fourier, 2015]{GNT15}. The case of particular interest is when the gauge is taken as the Cygan–Korányi norm and the error term reads: $$\mathcal{E}_q(t)=\#\left(\mathbb{Z}^{2 q+1} \cap \mathcal{B}_t\right)-\operatorname{vol}(\mathcal{B}_1) \, t^{2 q+2},$$ with $\mathcal{B}_t=\{(v,w)\in\mathbb{H}^q: (|v|^4 + w^2)^{1/4} \le t \}$, which is closely related to the Gauss circle problem. When $q\ge3$, Gath \cite[Ann. Sc. Norm. Super. Pisa Cl. Sci., 2022]{Gat22} improved upon \cite[]{GNT15} by showing that $ |\mathcal{E}_q(t)|\lesssim t^{2q-1+ 1/3}$ and proposed the conjecture that the optimal order should be $2q-1$. In this paper, through Landau's formula and the $5,6$-th Derivative Tests of van der Corput, we arrive at that $|\mathcal{E}_q(t)| \lesssim t^{2 q-1 + 241/753} $ for any $ q \geq 4$, and recover the bound of Gath for $q=3$ up to a logarithmic factor. This, via a simpler method, provides the first progress towards Gath's conjecture.

Using elementary $q$-series manipulations, we establish a positivity property for the tails of the Jacobi theta series. Specifically, for integers $k\ge 1$ and $n\ge 0$, define \[ \sum_{n\ge0}\sum_{m\in\mathbb{Z}}J_{k,n}(m)z^m q^{n} = \frac{(-1)^k q^{-\binom{k+1}{2}}}{(z)_{\infty}(q/z)_\infty} \sum_{j\ge k}(-1)^jq^{\binom{j+1}{2}}z^{-j}(1-z^{2j+1}), \] where $(a)_\infty:=\prod_{n\ge0}(1-aq^n)$ denotes the $q$-shifted factorial. We prove that for all integers $k\ge 1$ and $n\ge 0$, the coefficients $J_{k,n}(m)$ are positive for all integers $-(k+n)\le m\le k+n$.

Mixed precision Runge–Kutta methods reduce the cost of the expensive implicit solves in diagonally implicit Runge–Kutta (DIRK) schemes by evaluating them in low precision, while retaining the accuracy of the scheme for larger time steps. The accuracy lost to the low-precision perturbation can be recovered through inexpensive explicit corrections; however, these corrections have an adverse impact on stability. Recently proposed stabilized corrections remedy this by applying a stabilization matrix to the correction step, but their runtime cost has not previously been quantified. In this work, we present a numerical study of the runtime performance of these stabilized corrections. Using spectral semi-discretizations of two nonlinear partial differential equations, the inviscid Burgers' equation and the porous medium equation, we compare uncorrected mixed precision DIRK methods against explicitly corrected and stabilized variants across half, single, double, and quadruple precision pairings, for SDIRK methods of orders two through four. We report convergence, runtime, and speedups, and show that the stabilized corrections improve the accuracy of the mixed precision schemes while preserving substantial runtime savings. All experiments were performed on an Intel Xeon Platinum 8480+ CPU with Julia version 1.11.4.

Reduced-order models, such as latent dynamics models, are becoming mainstream for accelerating simulations for parameterized physical systems governed by nonlinear conservation laws. However, most existing latent dynamics frameworks suffer from two important limitations: they do not provide uncertainty estimates for model predictions, and they do not guarantee adherence to the underlying conservation laws. While these challenges have been addressed separately in prior work, a unified framework that simultaneously provides uncertainty quantification and exact conservation-law preservation remains largely unexplored. In this work, we develop a variational latent neural field framework that integrates Gaussian process-inspired surrogates, enabling estimation of predictive confidence for both in-distribution and out-of-distribution parameter regimes. Three variants of the framework are considered: IRS-UQ, PI-IRS-UQ, and ECLEIRS-UQ, corresponding to unconstrained, physics-informed, and conservation-structure-preserving formulations, respectively. Exact conservation-structure preservation is achieved by embedding the solution dynamics within a conservation-law manifold through a space-time divergence-free representation of the solution-flux field. We demonstrate the applicability of the framework through three numerical experiments: 1) 1-D advection, 2) 2-D Euler and 3) 2-D shallow water equations in parameterized settings. Numerical experiments demonstrate that the proposed approach provides accurate predictions together with uncertainty estimates, while remaining robust to sparse and noisy training data. Comparisons between the proposed three approaches show that conservation-structure preserving latent representations improve robustness to degraded training data while maintaining competitive predictive accuracy and uncertainty quantification capability.

Let $k>\ell\ge 1$ be integers. For a family $\mathcal{F}$ of $k$-uniform hypergraphs ($k$-graphs), the $\ell$-degree Turán density $\gamma^{(k)}_\ell(\mathcal{F})$ of $\mathcal{F}$ is defined as the asymptotic maximum of the normalized minimum $\ell$-degree over all $\mathcal{F}$-free $k$-graphs. In this paper, we prove that for all $k>\ell>k/2$, every rational number $\alpha\in[0,1)$ can be realized as the $\ell$-degree Turán density $\gamma^{(k)}_\ell(\mathcal{F})$ for some finite family $\mathcal{F}$ of $k$-graphs. Furthermore, for any $k>\ell\ge 1$, we construct an explicit infinite sequence of values realized by single forbidden $k$-graphs: for each integer $q\ge2$, there exists a $k$-graph $F$ such that $\gamma^{(k)}_\ell(F)=1-q^{\ell-k}$. We also establish a strengthened non-principality property for $\ell$-degree Turán densities: for all $k>\ell>1$, there exist two $k$-graphs $F_1$ and $F_2$ satisfying $0< \gamma^{(k)}_{\ell}(\{F_1,F_2\})<\min\{ \gamma^{(k)}_\ell(F_1),\gamma^{(k)}_\ell(F_2)\}$.

In this paper, we study the initial value problem of the sixth-order Boussinesq equation with quadratic and cubic nonlinearities in arbitrary spatial dimensions. First, by using the Fourier restriction norm method and a high-low frequency decomposition, we establish the nonlinear smoothing for this equation, namely, the integral form of the solution to the Duhamel formulation enjoys higher regularity than its linear counterpart. Finally, by using the nonlinear smoothing, we establish the uniform convergence of the integral term and its spatial decay for each fixed $t$.

We prove a rigidity theorem for $\pi_*$-étale $\mathbb E_k$-algebras over an $\mathbb E_{k+1}$-ring spectrum: the category of $\pi_*$-étale extensions of an $\mathbb E_k$-algebra is identified with the ordinary category of étale Dirac algebras over its graded homotopy Dirac ring. The proof develops a relative Goerss-Hopkins type obstruction theory in synthetic spectra, including an $I$-complete version. As an application, the completed obstruction theory constructs the $I_n$-complete $\mathbb E_3$-$MU_{(p)}$-algebra realization of the Lubin-Tate theory, hence an $\mathbb E_4$-orientation $MU_{(p)}\to E_n$.

In arXiv:2511.13287, we derived exact formulas for generating functions counting the number of $n$-ary words avoiding strictly increasing subwords of length $k$, and provided applications in probability theory as well as the continuous limit as $n\to\infty$. We also conjectured several corresponding formulas for the case where the “strictly” requirement is dropped. In this paper, we prove those formulas.

This work studies the inverse problem of recovering the relative magnitudes of the tension, bending, and bearing loads acting on a crack from its stress-intensity-factor profile along the crack front, using the public SIFBench finite-element data. The central claim is not forensic load recovery on field cases, but a rigorous characterization of when the combined load is identifiable at all, together with an estimator that returns calibrated uncertainty precisely in the regimes where it is not. For a known geometry the forward map from loads to profile is exactly linear, and identifiability reduces to a single geometric question: whether the three elementary load profiles are linearly independent as functions along the front. When they are nearly dependent, many different load combinations produce almost the same profile and the inverse problem is illposed; the analysis shows that the degree of ill-posedness is controlled by an intrinsic stability margin, not by the conditioning number alone. A single crack-front operator serves both as a structured forward surrogate and as the differentiable map required by a simplex-constrained, set-valued inverse estimator. On the SIFBench corner-crack scenario the empirical behaviour matches the theory: the typical geometry is well posed while a sizable minority is genuinely ill-posed, so a point estimate is reliable on the majority and provably uninformative on the rest. Validation is on controlled synthetic noise; no real fracture cases are used or claimed.

Let $I_2$ be the commutative non-unitary ring of order $4$ arising in the classification of Fine. In this paper, we investigate cyclic codes over $I_2$ through their associated residue and torsion codes over $\mathbb{F}_2$. We introduce the notions of twisted and untwisted cyclic codes and characterize cyclicity in terms of a compatibility condition involving the twist map and the cyclic shift. Connections between cyclic codes over $I_2$ and binary quasi-cyclic codes are established via Gray maps. In particular, we show that the Gray image of a cyclic code over $I_2$ is a binary quasi-cyclic code of index $2$. We also study duality properties of cyclic codes over $I_2$ and prove that the dual of a cyclic code is again cyclic. Finally, we classify permutation inequivalent cyclic codes over $I_2$ for lengths $n \le 7$ and determine various structural properties of these codes.

A homeomorphism of an interval is a positive bump if its support is a single open interval on which it moves every point to the right. Choosing a fundamental domain $[m,b(m))$ for the action of a positive bump $b$ on its support splits the remainder of the support into two intervals, called the feet of $b$. A finite set of positive bumps is geometrically fast if fundamental domains can be chosen so that all the resulting feet are pairwise disjoint. The crossing graph of such a set has the bumps as its vertices, two bumps being adjacent whenever their supports overlap but are not nested, and the set is irreducible if its crossing graph is connected. We prove that for every $n\geq 2$, every group generated by an irreducible geometrically fast set of $n$ positive bumps is isomorphic to the $n$-ary Thompson group $F_n$. This answers the strong version of a problem posed by Brin and Zaremsky (Oberwolfach Rep. 15 (2018)).

Standard learning rate schedules such as cosine annealing are tied to a fixed training horizon, limiting their ability to accommodate post hoc horizon extension. Warmup-stable-decay (WSD) partially addresses this issue by maintaining a long constant-rate phase before a short linear cooldown, allowing training to resume from a pre-decay checkpoint. However, its peak learning rate is still tuned based on the original training horizon and can become suboptimal when training is extended. Motivated by stochastic convex optimization, we propose WSqD (Warmup with Square-root base and linear Decay), a learning rate schedule that replaces WSD's constant stable phase with a shifted inverse-square-root base while retaining the final linear cooldown. In the stochastic convex setting, WSqD provably attains the minimax-optimal $O(1/\sqrt{T})$ last-iterate convergence rate. Importantly, its base learning rate schedule is horizon-independent, and the training horizon is needed only to determine when to begin the final cooldown. Empirically, on language-model pretraining using the SlimPajama corpus, WSqD matches or outperforms carefully tuned WSD and other baselines across multiple training horizons while reusing a single peak learning rate.