Papers

Modern large language models (LLMs) can translate natural-language descriptions into operations research (OR) formulations. Post-training techniques including reinforcement learning and on-policy self-distillation have further improved this capability. However, three limitations remain in training LLMs for OR formulations. First, training commonly relies on synthetic formulations validated by human experts or stronger models, constraining scalable supervision. Second, credit assignment is either coarse or costly: outcome rewards score an entire trajectory without locating the responsible modeling decision, whereas process-level supervision requires an additional evaluator. Third, privileged self-distillation can induce style mismatch by using solver context unavailable at deployment. We find that a model can improve from solver-artifact feedback generated by its own rollouts, making self-distillation a practical, evaluator-free source of dense supervision. Therefore, we propose SOLID: Solver-Informed On-Policy LearnIng through Self-Distillation, a novel framework for self-improving OR language models without verified answers or external evaluators. SOLID executes candidate programs from multiple rollouts, clusters their objectives, and selects a majority-group artifact as a pseudo-reference. The model then performs updates using group-relative advantages and dense self-supervision signals. Across multiple OR benchmarks, SOLID improves solution accuracy for both general-purpose and OR-tuned models over outcome-only group-relative training. These results show that solver artifacts can support scalable self-improvement without trusted answers.

Consider the Dirichlet-type spaces on the bidisc defined by $$\mathcal{D}_{\beta}(\mathbb{D}^2)=\Bigg\{f(z_1,z_2)=\sum_{k,l}a_{kl}z_1^kz_2^l\in\mathcal{O}(\mathbb{D}^2): \sum_{k,l\ge 0}|a_{kl}|^2(k+l+1)^{\beta}<+\infty\Bigg\}.$$ Given $\beta_c\in(0,2],$ we construct a function $f$ that belongs to the Dirichlet-type space $\mathcal D_{2}(\mathbb{D}^2)$ of the bidisk and is cyclic in $\mathcal D_\beta(\mathbb{D}^2)$ if and only if $\beta\leq \beta_{c}.$ We also show that the critical index satisfies $\beta_c=2-\mathrm{dim}_H(\mathcal{Z}(f)\cap \mathbb{T}^2),$ where $\mathrm{dim_H}(\mathcal{Z}(f)\cap \mathbb{T}^2)$ is the Hausdorff dimension of the zero set of the function $f$ on the two-torus $\mathbb{T}^2.$

We study optimal boundary mixing of a nondiffusive scalar transported by a two-dimensional incompressible second-grade fluid. The control is the tangential traction in a Navier-slip boundary condition, and the principal objective is the terminal $(H^1(\Omega))'$ mix-norm, supplemented by a quadratic control cost and an optional enstrophy reward. The second-grade constitutive law introduces a spatially filtered acceleration and a generalized vorticity; after lifting the nonhomogeneous boundary data, both $\mathbf{g}$ and $\partial_t\mathbf{g}$ enter the state equation, where $\mathbf{g}$ is the boundary control variable. We prove global state well-posedness for a passive scalar and well-posedness on a common, control-independent local interval for an active scalar. We then establish existence of an optimal control, directional differentiability of the control-to-state map in the topology required by the terminal objective, and a weak backward adjoint formulation. A duality identity yields the first-order variational inequality. In the absence of the friction coefficient $\beta$ and enstrophy reward weight $\zeta$ (that is, $\beta=\zeta=0$), the additional adjoint regularity implies uniqueness of the optimal control for all sufficiently large control penalties $\gamma$.

We consider the problem of estimating a high-dimensional covariance matrix from a very limited number of samples. This problem is ubiquitous in computational fluid dynamics, where a small number of fluid snapshots must be used to construct a Gramian matrix determining a reduced-order model, as well as in computational geoscience, where a small ensemble of Earth system forecasts must be used to estimate the covariance matrix associated with the forecast uncertainty. It is common practice to regularize the small-sample covariance by imposing a "localization" structure that enforces a physically realistic correlation length scale, imposing a sparsity constraint, "shrinking" towards a prescribed target, or attenuating small correlations. We propose an alternate technique that regularizes the small-sample covariance by imposing hierarchical rank structure. Compared to regularization methods that assume sparsity such as spatial localization, hierarchical rank structure accommodates a wider range of covariance matrices, roughly corresponding to situations where long-range correlations vary more smoothly than short-range ones. It also results in a data-sparse matrix format that permits highly efficient matrix-vector products. We present theory and algorithms which show how to efficiently estimate a high-dimensional, hierarchically rank structured covariance matrix from limited samples. Through an error analysis and numerical experiments with a variety of model problems, we demonstrate that these techniques are effective at reducing sampling errors, and that in many cases they achieve smaller estimation error than conventional techniques.

Let $q$ be an odd prime power. We use complements of unions of subsets of $\mathbb F_{q^2}$ as locator sets and establish a sufficient condition under which a generalized Reed–Solomon (GRS) code is Hermitian self-orthogonal. Using cosets of multiplicative subgroups and sets with prescribed trace or norm values, we construct five families of Hermitian self-orthogonal GRS codes over $\mathbb F_{q^2}$. The Hermitian construction then yields five corresponding families of $q$-ary quantum maximum-distance-separable (MDS) codes. Under suitable parameter conditions, these quantum codes have minimum distances greater than $q/2+1$. By comparing codes of the same length, we give conditions under which our codes have strictly larger minimum distances than those obtainable from several previously known constructions based on trace maps, linear transformations, and cosets of multiplicative subgroups, either directly or via the propagation rule. We further show that such improvements occur for infinitely many values of $q$.

We prove a new asymptotic formula for the second moment of symmetric square L-functions in the weight aspect on average. This result implies that the associated L-function is non-vanishing at the central point for at least 76.69% of holomorphic Hecke cusp forms of bounded weight, improving the previous bound of 70.37%.

We prove that every trace-zero matrix $A\in M_n(\mathbb{C})$ admits a representation $A=BC-CB$ with $B,C\in M_n(\mathbb{C})$ and $\lVert B\rVert\lVert C\rVert\le K\lVert A\rVert$, where $K$ is an absolute constant independent of $n$, and $\lVert\cdot\rVert$ denotes the operator norm. For a fixed $t>0$, the proof splits according to whether $\lVert\operatorname{Re}(e^{\mathrm{i}\theta}A)\rVert_1\ge tn\lVert A\rVert$ holds for all $\theta\in\mathbb{R}$, where $\lVert\cdot\rVert_1$ denotes the trace norm. When this lower bound holds, we construct a commutator representation directly. Otherwise, the vector-selection theorem of Marcus, Spielman, and Srivastava yields smaller trace-zero compressions whose norms are small enough for the induction to close. We also construct an explicit family of zero-diagonal Hermitian unitaries that forces a lower bound of order $\sqrt{\log n}$ for $\lVert B\rVert\lVert C\rVert$ when either factor is required to be diagonal in the prescribed basis. The same family admits $\varepsilon$-pavings with fewer than $2\varepsilon^{-2}$ blocks and representations by two normal factors with optimal norm product $1/2$. This establishes a distinction between unrestricted commutator bounds and bounds under a prescribed diagonal restriction. The main results and their essential inputs are formalized in Lean 4 using Mathlib. The development also includes a formal derivation of the Kadison-Singer state-extension theorem from the same vector-selection theorem.

For any two links in the $3$-sphere, we provide a visual construction of a stable map $f$ from the $3$-sphere to the Euclidean plane such that $f$ has no cusp points, the set of definite (resp. indefinite) fold points of $f$ is isotopic to the first (resp. second) given link, and $f$ does not have a certain type of singular fibers. Moreover, we obtain a similar result for any $3$-manifold by setting the target space to the $2$-sphere.

Stein conjectured that for any integer $k\geq 2$, every oriented graph with minimum semidegree greater than $k/2$ contains every orientation of a path with $k$ edges. Recently, Chen, Hou and Zhou proved this conjecture to be true for any oriented path with two blocks, where a block of an oriented path is a maximal directed subpath within it. In this paper, we prove that every bipartite oriented graph with minimum semidegree at least $3k/8+2$ contains every oriented path with two blocks of length $k$ for $k\ge 2$. Moreover, in contrast to the general oriented setting, we highlight that the minimum semidegree threshold in the bipartite setting is closely related to the number of blocks.

Decentralized networks can learn collision-free periodic transmission schedules, but acquisition can consume much of a finite operating horizon. We characterize acquisition in a reservation protocol for a fixed population of $n$ nodes and a period of $\Gamma_n\ge n$ slots. Initially, each node attempts independently in each slot with probability proportional to $1/n$. A success reserves a slot that the node reuses every $\Gamma_n$ slots, retaining it after collisions. Nodes use only their own acknowledgments and small local memory. An exact finite-population Markov reduction, independent of the arrangement of reserved slots, characterizes the time until every node has a distinct reservation. As $n\to\infty$, mean acquisition takes $O(n^2)$ slots when $\Gamma_n=n$ and $O(n\log n)$ slots when $\Gamma_n\sim(1+\theta)n$, for fixed $\theta>0$. With a fixed number $\Gamma_n-n$ of spare slots, the standard deviation of acquisition time has the same asymptotic order as its mean. If that number diverges while $\Gamma_n/n$ remains bounded, the ratio of standard deviation to mean tends to zero. A diverging sublinear number of spare slots yields $o(n^2)$ mean acquisition while long-run utilization, the fraction $n/\Gamma_n$ of slots with successful transmissions, tends to one. Simulations validate these laws and compare acquisition and delivered packets per slot with existing access protocols.

This paper extends the convergence analysis of explicit exponential Runge–Kutta methods for linear parabolic problems $u'(t) + Au(t) = Bu(t)$, where $A$ generates an analytic semigroup and $B$ is relatively bounded with respect to $A$, from the third-order case to fourth-order schemes. By establishing the global error recursion relation and extending the defect-based analytical framework, we identify the terms responsible for stiff order reduction when $A$ and $B$ do not commute. Numerical experiments are performed on a non-commuting advection-diffusion problem to validate the theoretical results. Numerical tests using the classical four-stage, fourth-order schemes of Krogstad and Strehmel & Weiner exhibit an observed convergence order of approximately 2.75, which matches the theoretical prediction from the convergence analysis.

We prove that deciding global asymptotic stability for constrained finite-horizon linear model predictive control is undecidable. This holds at horizon one with identity state, input, and terminal weights, unique optimizers, and global feasibility. Separate reductions cover predicted-state boxes, hard input boxes, and quadratically softened input boxes. A fourth reduction fixes the state and input dimensions to three and six. Hence undecidability is not caused by long horizons, growing dimensions, failures of recursive feasibility, or nonuniqueness.

A nonlocal diffusion system describing the dynamics of multi-species populations and approximating the Shigesada-Kawasaki-Teramoto (SKT) model is analyzed in a bounded domain. The model consists of a system of integro-differential equations of Andreu-Mazón-Rossi-Toledo type, in which a nonlocal cross-diffusion operator acts on nonlinear diffusion potentials. We establish the global existence and uniqueness of strong solutions. The existence proof relies on suitable a priori estimates derived from an entropy inequality, whose proof requires the development of novel discrete chain-rule inequalities. As a by-product, these inequalities provide the basis for the construction of structure-preserving finite-volume schemes for the corresponding local SKT system. Uniqueness of solutions is established by means of a duality argument. Finally, one-dimensional numerical simulations illustrate the nonlocal-to-local limit and the qualitative behavior of solutions for superlinear and sublinear diffusion potentials.

Many variants of the multiple zeta values have been studied in recent years. There are a few among them that remain to be real numbers, such as the multiple mixed values defined by the authors as level-two generalizations, which include both Hoffman's multiple $t$-values and Kaneko–Tsumura's multiple $T$-values. A central question is when such a value descends to level one, that is, when it can be expressed as a $\Q$-linear combination of multiple zeta values. We call these values unramified. In this paper, we further consider the alternating version of the above variants and identify five families of unramified alternating multiple mixed values using the descent theory of Brown and Glanois. We conjecture that all unramified truly alternating multiple mixed values are given in this paper.

The Radon–Nikodym theorem for completely positive maps identifies the order interval below a map with positive contractions in the commutant of its minimal Stinespring representation. In this paper, we study the behavior of projection-valued Radon–Nikodym derivatives (sharp submaps) under composition. We prove that sharpness loss is determined by the multiplicative defect of the induced pullback map. In the finite-dimensional case, we show that this defect is determined by the orthogonal complement of the composite Kraus relation space, denoted $\mathcal{E}_{\Lambda,\Omega}$. A sharp submap remains sharp if and only if $\mathcal{E}_{\Lambda,\Omega}$ reduces the lifted Radon–Nikodym projection. Furthermore, we establish a tensor-factor criterion for the global preservation of sharpness and explicitly quantify the sharpness-loss defect.

This paper establishes a convergence theory for a continuous domain decomposition method for elliptic equations on manifolds. This method originated in \cite{lions2} in the setting of Euclidean domains and was later adapted and generalized to manifolds by \cite{qin_wang_wang}. Although its convergence was known, whether the convergence is geometric has remained open. We prove its geometric convergence and provide various estimates on the convergence rate.

Meta-learning has emerged as an effective paradigm for transferring knowledge across sequential bandit tasks. While substantial progress has been made for stochastic bandits and non-contextual adversarial bandits, meta-learning for adversarial linear contextual bandits (ALCBs) with random action sets remains largely unexplored. To address this problem, we propose Meta-LinEXP3, an online-within-online algorithm that constructs a predictable task-level prior from completed tasks to guide the inner LinEXP3 learner. For known context distributions, we develop a policy-centered estimator that achieves an intrinsic-dimension $\mathcal{O}(\sqrt{n})$ per-task regret bound. For unknown distributions, we introduce a past-only regularized moment estimator with an $\mathcal{O}(n^{2/3})$ leading regret term and explicit finite-sample error. We further establish a direct connection between prior accuracy and transfer regret, showing that increasingly accurate priors yield sublinear transfer-dependent regret across tasks. Experiments demonstrate the effectiveness of Meta-LinEXP3, including its application to structured hyperspectral tensor sampling.

We study fluctuation limit of a slow-fast system driven by $\alpha$-stable Lévy noise with $1<\alpha<2.$ The slow component is generated by an odd polynomial function $f(y):=y^q,$ while in the fast component, the drift is $g(y):=-|y|^p\operatorname{sgn}(y)$ for some $p>0.$ Although the noise is given, the fluctuation limit is either a stable process or a Brownian motion, depending on both $p$ and $\alpha.$ The critical line between stable limit and Brownian limit is $q+1-p=\alpha/2.$

In this paper, we derive the rotation-two-component $b$-family shallow-water wave system from the Green–Naghdi type equations incorporating the effects of the Coriolis force and constant vorticity. The system admits, among its special cases, the two-component Degasperis–Procesi (2DP) system as a reduction, thereby confirming its genuine role as a rotational extension of the $b$-family. We then focus on a specific reduction, the rotation-two-component Degasperis-Procesi system, whose particular interest lies in the fact that it combines two independent sources of difficulty: unlike the two-component Camassa-Holm type system, it possesses no conserved $H^1\times L^2$-type quantity, and in contrast to the standard 2DP system, it contains a higher-order nonlinear coupling term induced by the Coriolis force. To overcome this double difficulty, we introduce an auxiliary variable and establish a local $L^\infty$-estimate for $u$ without using any conservation law, based on which we derive two distinct blow-up criteria. When the Coriolis force vanishes, our results recover known criteria for the 2DP system and also provide new criteria for both the 2DP system and Degasperis-Procesi equation. The results extend naturally to the periodic setting and to relevant special cases of the rotational $b$-family system.

In this paper, we study metric bases and related metric properties for barycentric and partial matching subdivisions of the zero-divisor graph of $\mathbb Z_{pq}$, where $p$ and $q$ are distinct odd primes with $q>p$. We first recall the natural partition of the zero-divisor graph into the two prime classes and then give a detailed characterization of those subsets of $BS(\Gamma(\mathbb Z_{pq}))$ that form metric bases when $q\geq 2p-1$. The proof is expanded by separating the role of closed neighborhoods, rows of subdivision vertices, and forbidden twin configurations. We then investigate $M$-subdivision graphs obtained by subdividing selected edges of $\Gamma(\mathbb Z_{pq})$. In addition to the lower bounds for subdivisions of $p-3$ and $p-2$ edges, we prove an exact formula for matching subdivisions of arbitrary size $r$, $0\leq r\leq p-2$, namely $\dim(G_r)=p+q-r-4$. Several consequences are included to illustrate how a small matching subdivision can reduce the localization cost of the original zero-divisor network.

We study the Neumann boundary value problem $$ -\operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right) = \lambda a(x)g(u) \quad \text{in } \Omega, \qquad \frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\cdot \mathbf n = 0 \quad \text{on } \partial\Omega, $$ where $\Omega \subset \mathbb R^N$ is a bounded convex domain, $a\in L^\infty(\Omega)$ is an indefinite weight with $\int_\Omega a(x) \, dx< 0$, and $g$ is a nonlinearity. Under suitable assumptions on $g$, we prove the existence of two positive solutions for sufficiently large $\lambda>0$, using Szulkin's theory for nonsmooth functionals: a global minimizer $u_\lambda^{(l)}$ with negative energy, and a mountain-pass critical point $u_\lambda^{(s)}$ with positive energy. Furthermore, we study the asymptotic behaviour as $\lambda\to+\infty$ of both families of solutions. For the ground states we prove that, up to a subsequence, $u_\lambda^{(l)}\to u_\infty$ uniformly, where $u_\infty$ satisfies $\|\nabla u_\infty\|_{L^\infty(\Omega)}=1$ and is characterized by being a maximizer of $\int_\Omega a(x) G(u) \, dx$ and a minimizer of the kinetic energy among all such maximizers. In the model case $g(u)=|u|^{p-2}u$, $p>2$, we also show that the mountain-pass energy level satisfies the upper bound $c_\lambda=O(\lambda^{-2/(p-2)})$, and that $u_\lambda^{(s)}\to0$ in $C(\overline\Omega)$.

We study comparison inequalities for contact-process-type infection dynamics on $\mathbb Z^d$ with general local transmission mechanisms. The processes considered include both discrete-time oriented-percolation models and continuous-time contact processes in which infection events may transmit to random, possibly correlated sets of neighbouring sites. Our main results apply local-to-global comparison criteria beyond stochastic domination: if one local infection rule is more likely than another to hit every non-empty test set of neighbours, then the corresponding process has larger survival probabilities at all times. We compare classical independent-infection models with exchangeable fixed-budget, all-or-nothing, and burst-type infection mechanisms, deriving orderings for example of finite-time and infinite-time survival probabilities and stopping-time distributions. In continuous time, we further obtain consequences for critical survival thresholds of exchangeable infection laws. The results show that infection processes with identical or comparable mean-field infection intensity can nevertheless be rigorously ordered at the stochastic level once spatial geometry is taken into account. Additionally, it turns out that the classical continuous-time contact process dominates a variety of related continuous-time models, whereas its most natural discrete-time version, Bernoulli oriented percolation, is not dominant with respect to related discrete-time models.

Motivated by lifting problems for principal bundles, we study the following algebraic problem. Let $\smash{\widehat{G}}$ be a group with identity element $e$, let $G \leq \smash{\widehat{G}}$, and let $S$ be a strongly $G$-graded unital ring with principal component $R:=S_e$. We ask whether the given grading extends to a strong $\smash{\widehat{G}}$-grading without altering its $G$-homogeneous components. For every extension $\smash{\widehat{p}}$ of the Picard homomorphism of $S$, we construct a characteristic class in the relative third cohomology group \[ H^3_{\smash{\widehat{p}}}(\smash{\widehat{G}},G;\operatorname{U}(Z(R))), \] where $Z(R)$ denotes the center of $R$ and $\operatorname{U}(Z(R))$ its group of units. The~vanishing of this class is equivalent to the existence of a lift. When it vanishes, the relative second cohomology group \[ H^2_{\smash{\widehat{p}}}(\smash{\widehat{G}},G;\operatorname{U}(Z(R))) \] acts simply transitively on the equivalence classes of lifts with Picard homomorphism $\smash{\widehat{p}}$. We also prove that an $H$-grading is strong if and only if both its restriction to a normal subgroup $N \trianglelefteq H$ and its induced $H/N$-grading are strong. We illustrate the theory through a range of examples.

There are four covering numbers $\mathrm{c}_g^{\mathcal{G}}(H),\mathrm{c}_u^{\mathcal{G}}(H),\mathrm{c}_l^{\mathcal{G}}(H),\mathrm{c}_f^{\mathcal{G}}(H)$, each of which measures in a slightly different way how well the edges of a graph $H$ (called a host) can be covered with graphs of a class $\mathcal{G}$ (called a guest class). If we require the graphs of $\mathcal{G}$ to correspond to induced subgraphs of $H$, we obtain an induced variant $\mathrm{ic}_x^{\mathcal{G}}$ for each covering number $\mathrm{c}_x^{\mathcal{G}}$ which satisfies $\mathrm{c}_x^{\mathcal{G}}(H) \leq \mathrm{ic}_x^{\mathcal{G}}(H)$ for every graph $H$. Yet, in general $\mathrm{ic}_x^{\mathcal{G}}$ cannot be bounded in terms of $\mathrm{c}_x^{\mathcal{G}}$. If there exists for a guest class $\mathcal{G}$ and a host class $\mathcal{H}$ a function $f$ such that $\mathrm{ic}_x^{\mathcal{G}}(H) \leq f(\mathrm{c}_x^{\mathcal{G}}(H))$ for every graph $H \in \mathcal{H}$, we call $f$ a binding function. Within this work, we study for which structural properties of a guest class $\mathcal{G}$ and a host class $\mathcal{H}$ such binding functions exist. We consider guest classes $\mathcal{G}$ that are monotone, hereditary, component-closed or neither, and have bounded maximum average degree, bounded chromatic number or neither. The host classes $\mathcal{H}$ we consider have bounded treewidth, exclude some minor, have bounded maximum average degree, bounded chromatic number, or none of these properties. For $219$ out of the $240$ possible $3$-tuples of properties for $\mathcal{G}$ and $\mathcal{H}$ and covering numbers we either provide a binding function or an example where no such function exists. In particular, we show that such binding functions always exist for hereditary guest classes $\mathcal{G}$ of bounded maximum average degree for three of the four covering numbers, but may not for the fourth kind.

Lewis and Overton conjectured in 2008 that, for BFGS applied to the one-dimensional tilted absolute value with their Armijo–Wolfe line search, every nonterminating execution converges at a trial-normalized rate determined only by the tilt parameter and independent of the initial data. We prove that this initialization-independent rate assertion is false. In fact, for every tilt parameter in an explicit open interval, we construct two nonterminating executions with different sharp trial-normalized convergence rates. The proof reduces the line search to a scale-free state consisting of the iterate sign and the zero-crossing stepsize, and verifies two periodic state cycles by elementary rational inequalities. This yields an explicit family of counterexamples and shows that the asymptotic trial-normalized behavior of nonsmooth BFGS can depend essentially on the initialization even for this canonical one-dimensional model.

The celebrated theorems of Shilov, Arens–Royden, and Forster give direct descriptions of the first three integral cohomology groups of the Gelfand spectrum of a commutative complex Banach algebra. In his 1974 ICM address, Taylor asked whether the higher cohomology groups admit descriptions in terms of the underlying ring. We give a solution to this question in even degrees: The étale (aka Lichtenbaum) Chow group in every codimension is canonically isomorphic to the corresponding even integral cohomology group of the Gelfand spectrum.

We study compact Kähler varieties admitting int-amplified endomorphisms from the viewpoint of positivity of tangent sheaves. Our main result shows that the tangent sheaf of such a variety is weakly positively curved; in particular, it is pseudo-effective in a strong sense. This establishes a new link between complex dynamics and the positivity theory of tangent sheaves, and provides an alternative to equivariant MMP techniques in the compact Kähler setting. As an application, we prove a Kähler analogue of Yoshikawa's structure theorem: for a compact Kähler klt variety admitting an int-amplified endomorphism, after an equivariant quasi-étale cover, it admits an equivariant flat MRC fibration with irreducible fibres onto a complex torus; in the smooth case, the fibration is smooth whose periodic fibres are of Fano type. To this end, we prove the existence of minimal models in the case of numerical dimension zero canonical divisors for non-projective Kähler varieties. We further study rationally connected manifolds with pseudo-effective tangent bundle, focusing on their relation to Fano-type properties, almost homogeneity, and fundamental groups. In the proof, we also show that every surjective endomorphism of a compact klt Kähler variety lifts to a suitable maximally quasi-étale cover.

The purpose of this note is to show that the computation of the universal norms in Iwasawa cohomology for the $p$-adic cyclotomic extension implies an analogous result for any $p$-adic extension that contains the cyclotomic extension. In addition, we prove a twisted cyclotomic variant, which applies to all Lubin–Tate extensions.

We determine $2,540$ unknotting numbers of prime knots with at most $13$ crossings, that are unknown in the KnotInfo snapshot of 9 September 2026. The lower-bound calculations use Heegaard Floer correction-term obstructions and Greene's spanning-tree model. We also give explicit crossing-change constructions for upper bounds. For $959$ alternating knots with range $[2,3]$, we verify that no crossing change in a minimal diagram gives a knot of unknotting number one. We determine the unknotting numbers of all four knots in Brittenham and Hermiller's construction and thereby identify 13n3370 as an explicit counterexample to the original Bernhard-Jablan conjecture.

Fix integers $k,d\geq2$. Let $\mathrm{End}_d^k$ denote the parameter space of endomorphisms of $\mathbb{P}^k$ of algebraic degree $d$. We prove that there exists a dense Zariski open subset $U_{d,k}\subset\mathrm{End}_d^k$, defined over $\mathbb{Q}$, such that, for every $f\in U_{d,k}(\mathbb{C})$, every dominant rational self-map of $\mathbb{P}^k$ commuting with an iterate of $f$ is itself an iterate of $f$. The key intermediate result classifies dominant rational semiconjugacies to iterates of $f$ from normal projective varieties of dimension $k$. The proof uses finite-level monodromy and its action on the rooted preimage tree of a point outside the branch locus of an iterate of $f$. An analogous centralizer theorem holds generically for regular polynomial endomorphisms of $\mathbb{C}^k$. These results extend several theorems of Pakovich to higher dimensions. We give two applications. We first classify periodic dominant correspondences between a general endomorphism and an arbitrary endomorphism of degree at least two. We then show that, for a general $f\in\mathrm{End}_d^k$, an irreducible hypersurface of $\mathbb{P}^k$ is $f$-special in the sense of Ghioca–Tucker and DeMarco–Mavraki if and only if it is $f$-preperiodic.