Papers

Procrustes-Wasserstein alignment jointly estimates a matching and rotation without supplied correspondences, but alternating minimization can stop at suboptimal solutions. Rubix solves the equally weighted planar problem globally under squared Euclidean loss. Each matching $\sigma$ of two centered $n$-point sets defines a complex correlation $z_\sigma=\sum_i\bar x_i y_{\sigma(i)}$. Their convex hull is the permutation polygon: supporting vertices give optimal matchings at fixed rotations, and the farthest vertex gives the global alignment. We prove the sharp bound of $n(n-1)$ vertices for $n\ge2$, answering Rote's rotation-assignment open problem. In exact arithmetic, assignment queries recover the polygon in $\mathcal O(n^5)$ operations. Assignment-based bounds extend the approach to three-dimensional rotations and partial matching at a supplied translation through branch-and-bound. On timed MPEG-7 shape pairs, Rubix attains every numerical reference value in 12 ms on average, 50 times faster than a rotation grid at the same accuracy. Its distances improve gravity-aligned matching of real 3D scans, shape retrieval and noisy crystal classification over alternating minimization.

Let $W$ be an algebra with involution over a field of characteristic zero. We develop a theory of $W$-polynomial identities with involution for finite-dimensional $(W,*)$-algebras $A$, using the multiplier algebra with involution of $A$. We prove that the $(W,*)$-exponent of $A$ exists and coincides with the ordinary $*$-exponent. We then consider a four-dimensional subalgebra $M$ of the algebra of $4\times4$ upper triangular matrices, endowed with the reflection involution, and study two non-equivalent $(W,*)$-algebra structures on it. For both structures, we determine the corresponding $T_W^*$-ideals of $W$-polynomial identities with involution and compute the $(W,*)$-codimension sequences explicitly; for one of them, we also determine the complete $(W,*)$-cocharacter sequence. Finally, we prove that these two $W$-algebras with involution generate distinct varieties of almost polynomial growth.

Let $\mathcal{G}(G,A,\alpha)$ be a discrete 2-group, given as the {categorical} extension of an ordinary group $G$ by an abelian group $A$. We define principal $\mathcal{G}$-bundles over a manifold in terms of Čech data over a good open cover. In the case of the manifold $S^2$, we calculate this bicategory explicitly, and identify its 2-skeleton. We compare with the holonomy bicategory $\mathrm{Fun}(*/\!\!/ \pi_{\le 2}(S^2), * /\!\!/ \mathcal{G})$, and provide a concrete equivalence between the two.

The regionally proximal and equicontinuous structure relations are fundamental relations in topological dynamics that capture the equicontinuous behavior in a topological dynamical system and its factors. For minimal actions of abelian groups, these relations are known to be equivalence relations and are known to coincide. In this paper, we generalize these facts to semigroup actions: for minimal actions of commutative semigroups, the regionally proximal and equicontinuous structure relations are equivalence relations and the two coincide. We also develop the machinery of natural extensions for commutative semigroup actions that act by surjections, concluding that the maximal equicontinuous factor of a minimal action of a commutative semigroup is the same as the maximal equicontinuous factor of the group action into which it embeds. We formally verify all of the results in this paper in Lean. The main results are verified in a Palomar submission, and we link to a Github repository containing code for the complete verification.

Models can be retrained as new data arrive, but deploying every new version risks replacing a good policy with a worse one. We study how to plan policy updates (i.e., meta-policy) before future candidates are trained, balancing the benefits of improvement against the risk of performance regression. Our offline meta-policy maximizes expected cumulative value subject to a budget on the expected number of updates that perform worse than the policies they replace. We estimate the value and risk of possible switches from historical learning trajectories, represent an update schedule as a path in a directed acyclic graph, and select a schedule using dynamic programming. A leading-order analysis identifies the signal-to-noise ratio of policy improvement as a key driver of update frequency, waiting times, and risk allocation: clearer improvements support earlier, more frequent updates, while noisier improvements call for longer waits or greater risk expenditure. Their asymptotic rates also reveal a diminishing marginal cost of achieving greater safety over time. Experiments on synthetic and clinical trial data illustrate the performance–risk tradeoff and compare our method with alternative baselines.

Rényi entropy estimation has been comprehensively investigated by Acharya, Orlitsky, Suresh and Tyagi (SODA 2015; IEEE Trans. Inf. Theory 2017) and consequent works, whereas only the sample complexity of Rényi entropy estimation of integer order has been settled. In this paper, we settle the sample complexity of Rényi entropy estimation of noninteger order, thereby completing the complexity picture of Rényi entropy estimation. Specifically, we show that for any noninteger $\alpha > 0$, it is sufficient and necessary to use \[ \Theta\!\left(\frac{d^{\max\{1/\alpha,1\}}}{\varepsilon^{1/\alpha}\log(d)} + \frac{d^{|1-1/\alpha|}}{\varepsilon^2}\right) \] samples to estimate the Rényi entropy of order $\alpha$ of an unknown discrete distribution over an alphabet of size $d$ to within additive error $\varepsilon$. For the upper bound, we reduce the bias using a refined polynomial approximation estimator for large probabilities. For the lower bound, we employ a different hard instance equipped with a new moment matching construction. The constructive moment matching has constant bounded high-order moments, while attaining a fixed ratio between the $\alpha$-th moments, which is of independent interest.

How large of a minimum degree does an $n$-vertex graph need before we are sure that it contains a perfect matching? Dirac's theorem states that a graph on an even number of vertices in which each vertex has degree at least $n/2$ has this property. In this short expository note, intended to be used in the classroom, we discuss how this statement generalizes to hypergraphs. In particular, we highlight an elegant connection between fractional perfect matchings in hypergraphs and a probabilistic inequality about nonnegative random variables, which was conjectured by Feige. We also present a very short self-contained proof of Feige's conjecture.

In this paper, we introduce the crystallization $C(SO_0(2n+1)/SO_0(2n-1))$ of the $C^*$-algebra $C(SO_q(2n+1)/SO_q(2n-1))$. We show that every irreducible representation of $C(SO_0(2n+1)/SO_0(2n-1))$ arises as the limit, as $q \to 0^+$, of a family of irreducible representations of $C(SO_q(2n+1)/SO_q(2n-1))$. As a consequence, we prove that $C(SO_0(2n+1)/SO_0(2n-1))$ is the universal $C^*$-algebra generated by a finite set of generators and relations. We then construct a chain of unital essential short exact sequences of $C^*$-algebras, enabling us to compute the $K$-groups of $C(SO_0(2n+1)/SO_0(2n-1))$ and determine explicit generators for them. Using the theory of homogeneous $C^*$-extensions, we prove that $C(SO_0(2n+1)/SO_0(2n-1))$ is isomorphic to $C(SO_q(2n+1)/SO_q(2n-1))$ for every $q \in (0,1)$.

We construct a family of metric outer measures whose critical exponents form a packing spectrum interpolating between Hausdorff and packing dimensions. The spectrum is countably stable, bi-Lipschitz invariant, and locally Lipschitz continuous in the parameter below the packing endpoint, where a jump may occur. For nonempty compact sets, we prove a variational formula in terms of windowed local mass exponents of probability measures. We establish product inequalities pairing the packing spectrum with upper intermediate dimensions and obtain a converse characterization by products with compact sets. We characterize all attainable profiles by monotonicity, continuity below the packing endpoint, and a sharp inequality for the upper right Dini derivative. Every admissible profile is realized by a compact binary digit set whose uniform digit measure attains the variational supremum at every parameter below the packing endpoint and whose complementary digit set attains the product supremum at every parameter.

We extend the Nair–Tenenbaum estimates for short sums of nonnegative arithmetic functions at polynomial values to mixed moments involving Kloosterman sums. Combining these estimates with lower bounds obtained by refining the method of Fouvry and Michel, we prove that, for every fixed nonzero integer \(a\) and fixed \(z,\nu>0\), \[ \sum_{n\leq x}z^{\omega(n)} |\operatorname{Kl}(a;n)|^\nu \asymp_{a,z,\nu} x(\log x)^{z\mathfrak{s}(\nu)-1}, \] where \(\omega(n)\) counts distinct prime divisors and \(\mathfrak{s}(\nu)\) is the \(\nu\)-th absolute moment of the Sato–Tate measure. We also obtain upper bounds for mixed absolute moments of Hecke eigenvalues and Kloosterman sums, determine the frequency of large Kloosterman values on a logarithmic scale, and establish the order of magnitude predicted by Li and Sarnak for the diagonal second moment of classical Kloosterman sums. We obtain matching bounds for the smoothed spectral counting variance on the modular surface in a logarithmic range and the lower bound \(\int_T^{2T}S(t)^2\,dt\gg T^2/\log T\), where \(S(t)\) is the remainder in Weyl's law.

The heavy-ball method can accelerate convergence near the minimizer, but it may also enter persistent oscillations on general strongly convex objectives. How should it be initialized to reliably realize this local advantage? We study the initialization neighborhood in which convergence can be guaranteed using only curvature and regularity bounds. This neighborhood shrinks as momentum approaches one; for Lipschitz-continuous Hessians, its radius is of the same order as damping. Matching upper and lower bounds show that this initialization requirement cannot be relaxed uniformly over the function class. We then derive a startup rule based on the current gradient: a finite gradient-descent warm-up enters the guaranteed region before switching to heavy-ball. With parameters optimal for strongly convex quadratics, the subsequent iterates satisfy an accelerated local asymptotic rate bound. Code is available at https://anonymous.4open.science/r/HeavyBall-811D/.

Feature caching accelerates diffusion sampling by replacing expensive network evaluations with predictions from previously computed activations. However, forecasts based only on past features cannot directly incorporate changes in the current denoising state. We investigate whether inexpensive, freshly computed features can serve as observations for correcting these predictions. We introduce an observation-corrected Koopman framework for accelerating frozen diffusion models. Using calibration trajectories, we identify finite-dimensional, time-dependent Koopman approximations that jointly describe the increments of shallow and deep network features. During accelerated sampling, these operators predict the evolution of expensive deep features, while innovations in the observed shallow features correct the predicted state. Periodic full evaluations refresh the observer, and all generative-model parameters remain unchanged. This formulation enables controlled comparisons of temporal prediction and observation correction. Across three 10,000-image runs per dataset, our method reduces paired Inception-feature MSE by $19.9\%$ on CIFAR-10 and $11.9\%$ on a ten-class ImageNet subset relative to channelwise affine prediction under the same four-partial-step schedule. Matched ablations attribute additional reductions of $4.54\%$ and $4.67\%$ to observation correction. The observer achieves $1.89\times$ and $1.85\times$ measured speedups over DDIM-50, supporting improved reference-sampler fidelity without retraining the denoiser.

In the recent work of Bergeron-Charollois-García \cite{BCG23}, a conjectural analytic expression for elliptic units for complex cubic fields was proposed: namely, as the value of a smoothed quotient of elliptic Gamma functions. The complex number depended on the conductor ideal, ray class group element, smoothing ideal, and a `torsion point' $h$. If this analytic function is to fit into the framework of explicit class field theory, then, as in the case of torsion points on CM elliptic curves, our choice of $h$ should only depend on its congruence class modulo some lattice $L$. This conjecture has been supported by much numerical evidence. In this paper, we use the cocycle relations satisfied by the smoothed elliptic Gamma function to prove that the construction is indeed independent of $h$.

In the upper half-space model of the hyperbolic space $\mathbb{H}^{n+1}$, a translation hypersurface of type I is a graph $x_{n+1}=f_1(x_1)+\cdots+f_n(x_n)$ and a translation hypersurface of type II is a graph $x_n=f_1(x_1)+\cdots+f_{n-1}(x_{n-1})+g(x_{n+1})$. We classify the minimal translation hypersurfaces of both types for every $n\geq 2$. We obtain these results as particular cases of a classification of the singular minimal translation hypersurfaces of $\mathbb{R}^{n+1}$ for an arbitrary exponent $\alpha\neq 0$, with respect to a direction orthogonal (type I) or parallel (type II) to the hyperplane over which the hypersurface is a graph. The minimal translation hypersurfaces of type I are exactly the parabolic cylinders over a one-parameter family of arches, which contradicts a nonexistence theorem in the literature. The minimal translation hypersurfaces of type II are the parabolic cylinders, the vertical totally geodesic hyperplanes and, when $n\geq 3$, the vertical cylinders over the Scherk surface.

We study decentralized adaptive sensing, where multiple agents choose measurements from evolving local beliefs while exchanging information over a communication graph. We ask whether the measurements actually selected by an adaptive policy have collected enough evidence to distinguish the true target from every plausible alternative. We develop a pathwise certificate based on the Rényi–Chernoff information accumulated along the realized sensing trajectory. It yields nonasymptotic MAP-error bounds and an anytime, network-wide stopping rule for arbitrary history-dependent sensing policies, while separating accumulated statistical information from a bounded network-mixing transient. Linear growth of the information against the least-resolved competitor implies exponential decay of MAP and squared-localization error. A classical pairwise KL converse, specialized to the adaptive decentralized transcript, shows that insufficient information on any pair prevents a positive uniform error exponent, confirming the hardest competitor as a fundamental bottleneck. Across policies, graph topologies, sensor profiles, and seeds, the worst-competitor score correlates more strongly with localization speed than an average-pair proxy in both 1D ($r=0.89$ versus $0.40$) and structured 2D sensing ($r=0.77$ versus $0.48$). Our results provide a practical way to certify and diagnose adaptive multi-agent sensing systems using the evidence they actually collect.

We prove estimates for Gromov's Long Neck Foliated Conjecture in the spin cases, which involves leafwise scalar curvature. The same methods yield a quantitative version of a theorem of Su–Wang–Zhang and a Llarull-type theorem for closed foliated manifolds. A refinement, which involves mean curvature and partially generalizes a theorem of Cecchini–Zeidler, is also obtained.

It is well known that the compressible Euler equations with planar shock initial data may admit infinitely many entropy weak solutions, as shown by Chiodaroli, De Lellis and Kreml [Comm. Pure Appl. Math. (2015)]. In this paper, we introduce a suitable notion of global-in-time weak solutions to the two-dimensional (2D) compressible Navier-Stokes equations with density-dependent viscosities in $\mathbb{R} \times \mathbb{T}$. Within this solution framework, we establish the first vanishing viscosity limit selection criterion for the physically relevant planar shock solution to the 2D compressible Euler equations. More precisely, we prove the uniqueness and stability of the 2D Euler planar shock solution within a class of inviscid limits of Navier-Stokes solutions emanating from near-planar initial data while the shock direction perturbation can be arbitrarily large. Starting from the relative $\kappa$-entropy for 2D compressible Navier-Stokes equations in $\mathbb{T}^2$ by Bresch, Noble and Vila [ESAIM Proc. Surveys (2017)], we first establish the relative $\frac12$-entropy between 2D weak solution of compressible Navier-Stokes equations in $\mathbb{R} \times \mathbb{T}$ and one-dimensional (1D) strong solution for averaged initial data in $\mathbb{R}$. Then we achieve the desired vanishing viscosity limit selection criterion by successfully bridging the fundamental structural gap between 2D Eulerian formulation of the original Navier-Stokes system in the relative $\frac12$-entropy and 1D vanishing viscosity limit to a shock solution of Euler equations in Lagrangian coordinates by Kang and Vasseur [Invent. Math. (2021)]. The framework developed herein can be applied to the three-dimensional limit selection criterion in $\mathbb{R} \times \mathbb{T}^2$, as well as to more complex planar Riemann solutions and interacting wave patterns.

We show that zero is a characterising slope for the untwisted Whitehead double of any knot $K$. More generally, this holds for all $n$-twisted Whitehead doubles of $K$, with the assumption that $n\ne pq$ if $K$ is a non-trivial $(p,q)$-torus knot or $(p,q)$-cable. In particular, zero is a characterising slope for all twist knots.

It was known that the Gaussian measures with monomial weights satisfy the sharp pointwise curvature-dimension condition $CD(1, \infty)$, and consequently they allow us to derive the sharp constant $1$ for the Poincaré inequality only with partial monomial weights. In this paper, we first establish the Gaussian measures with full monomial weights satisfy the integrated curvature-dimension condition $ICD(2, \infty)$ which is sufficient for us to prove the sharp Poincaré inequalities with optimal constant $2$ and the stability of the Brascamp–Lieb inequality for full monomial Gaussian measures. Several of these results are extended to homogeneous Gaussian measures on convex cones. Using the sharp Poincaré inequality and the hypercontractivity, we obtain an improved Beckner inequality that recover the sharp Poincaré and logarithmic Sobolev constants at the endpoints. By developing a Laguerre spectral decomposition of the associated Ornstein–Uhlenbeck type generator, we derive an exact identity for the Poincaré deficit and obtain sharp gradient stability for the Poincaré inequality with full monomial weights. We further establish $L^2$ and weighted gradient stability estimates for the Brascamp–Lieb inequality with explicit constants when all monomial exponents exceed $1$. The $L^2$ estimate extends to log-concave homogeneous weights under a subharmonicity assumption. Finally, we obtain an improved integrated curvature-dimension bound under a scale-invariant Hessian condition and identify the sharp constant for the Poincaré inequality in the radial class.

We construct coupled Dirac-harmonic maps from manifolds of the form $S^1\times M$ where $M$ is a spin manifold equipped with a pair of harmonic spinors satisfying a norm condition. This method produces coupled Dirac-harmonic maps from closed manifolds.

For a normed space $E$, we denote by $E_w$ and $E'_{w^*}$ the space $E$ and its dual $E'$ endowed with the weak topology $w$ and the weak$^*$ topology $w^*$, respectively. It is well known that for a Banach space $E$, both spaces $E_w$ and $E'_{w^*}$ are Fréchet–Urysohn topological spaces if and only if $E$ is finite-dimensional. This statement remains true for all linear subspaces of $E_w$ and $E'_{w^*}$. Moreover, for any normed space $E$, every Fréchet–Urysohn subgroup of $E_w$ is locally finite-dimensional; that is, it has an open subgroup contained in a finite-dimensional linear subspace. Similarly, for any Banach space $E$, every Fréchet–Urysohn subgroup of $E'_{w^*}$ is locally finite-dimensional. Analogous results are obtained for the free locally convex space $L(X)$, its weak version $L_p(X)$, and the free topological vector space $V(X)$ over a Tychonoff space~$X$. The same conclusions hold for Baire subgroups and Baire linear subspaces of $E_w$, $E'_{w^*}$, $L(X)$, $L_p(X)$, and $V(X)$. We emphasize that even for metrizable subgroups, these results are new and nontrivial. In fact, all results are established for the much wider family of $\kappa$-Fréchet–Urysohn subgroups of topological vector spaces. The class of $\kappa$-Fréchet–Urysohn spaces is the natural framework in which the proofs of our results work.

We study learning optimal policies in average-reward multichain Markov decision processes (MDPs), where the optimal gain may depend on the initial state and recurrence structures vary across policies, creating challenges for reinforcement learning (RL) methods. We propose an asynchronous value-iteration-based RL algorithm that requires no model knowledge beyond the MDP's transition graph and leverages Bather's decomposition to hierarchically partition the state space into communicating subsystems and transient states. This decomposition induces a recasting of the global decision problem into structured subproblems, which our algorithm exploits. We show that the algorithm converges to the optimal gain and produces gain-optimal policies after finite time. Building on this base algorithm, we develop two further algorithms: one approximately solves the multichain average optimality equations to obtain near gain-optimal policies, and another targets near bias-optimality by approximating the optimal bias function and solving an induced average-reward multichain MDP using the base algorithm. We provide almost-sure convergence guarantees for all three algorithms and empirically compare their tradeoffs, showing that the latter two also consistently improve transient performance relative to the base algorithm. To our knowledge, these are the first essentially model-free average-reward RL algorithms for general multichain MDPs without reductions to discounted problems.

We establish a Bochner formula for double forms in terms of the curvature operator of the second kind. As an application, we prove that a complete Riemannian manifold of dimension $n \ge 4$ with harmonic Weyl tensor and $\frac{3(n-1)}{4}$-nonnegative curvature operator of the second kind is either globally conformally equivalent to a space of positive constant curvature or is isometric to a flat manifold. We prove vanishing theorems for the Lichnérowicz Laplacian $\Delta$ on $(p,q)$ double forms. These generalize recent results of Nienhaus-Petersen-Wink \cite{NPW23} and Dai-Fu-Lu-Yang \cite{DF24,DFY24,FL1,FLD}.

Given a manifold $M$ with an asymptotically cylindrical end, we construct the parametrix for the shifted Laplace operator $\Delta_g + 1$ on the blown-up $b$-double space $M_b^2$. We will review the geometry of manifold with a cylindrical end and construct the inverse directly via separation of variables, which motivates the parametrix construction. Next, we review the mapping properties and show how to get elliptic regularity from the mapping properties of the parametrix. Through this explicit example, we hope this paper provides another reference for the theory of $b$-operators introduced by Melrose \cite{Mel93}.

Conventional wireless communication systems employ antennas with fixed radiation patterns. Enhancing system performance by increasing the number of antennas would push the limits of cost and size for communication devices. To overcome this issue, novel antenna technologies are urgently desired to break through the fundamental performance bottleneck and engineering limits. Pixel antennas provide a general framework for highly reconfigurable antennas, which can flexibly adjust radiation patterns via controlling variable lumped elements placed across adjacent pixels, unlocking novel degrees of freedom (DoF) in the beamspace domain. Recently, an emerging antenna coding technique links wireless system performance to the antenna configurations. This enables the use of pixel antennas to enhance system performance by leveraging the DoF from the antenna side. In this article, we provide a comprehensive overview on antenna coding by pixel antennas. Specifically, based on the pixel antenna model, the concept of antenna coding is firstly introduced, followed by a discussion on the classifications of antenna coding. Two cases for channel gain and multiplexing gain enhancement are highlighted to show the effectiveness of antenna coding in wireless communications. Besides, various optimization approaches for antenna coding design are investigated. Finally, the challenges and opportunities are outlined, providing guidance for future works.

Over the past few decades, several notions of graph curvature have been introduced. Among them, Ollivier curvature, defined through optimal transport, and its Lin–Lu–Yau variant provide notions of curvature for graphs which retain several features of their counterpart in Riemannian geometry. A classical problem in this context is to understand the global structure of positively curved spaces. Lu and Wang proved that there are only finitely many positively curved planar graphs of minimum degree at least 3. In this paper, we study two directions motivated by this result. We first show that for fixed $2\leq s\leq t$, there are only finitely many positively curved $K_{s,t}$-minor-free graphs with minimum degree at least $s$. In particular, this generalizes the result of Lu and Wang to graphs embeddable on any fixed surface. We then give a complete classification of positively curved 5-connected plane triangulations. In particular, we show that there are only five examples, on 12, 14, 15, 16, and 17 vertices.

For a finite group \(G\), let \(N(G)\) be the simplicial complex of subsets that do not generate \(G\). We separate a general Frattini-topological reduction from the structure special to finite \(p\)-groups. For every finite noncyclic group, the non-cone Frattini core is homotopy equivalent to the order complex of the proper part of the subgroup lattice of \(G/\Phi(G)\). For a finite \(p\)-group this quotient is an elementary abelian vector space, and the core becomes the non-spanning complex of a uniform parallel extension of \(\mathrm{PG}(r-1,p)\), where \(r=d(G)\) and \(q=|\Phi(G)|\). We use this geometry to determine the exact simplicial isomorphism data and full simplicial automorphism group, identify top homology equivariantly with the appropriate restriction of the Steinberg module, and derive modular consequences. We also give an explicit supportwise formula for every multigraded Betti number and compare it with a closed single-sum formula for the \(\mathbb Z\)-graded Betti numbers. Known matroidal and building-theoretic inputs are stated separately from the group-specific consequences. The resulting framework also yields the homotopy type, depth, regularity, projective dimension, face enumeration, and the complete Cohen–Macaulay classification.

For a digraph $D$, let $\alpha_2(D)$ be the largest size of a vertex set containing no directed $2$-cycle. Let $f_2(a)$ be the least positive integer $k$ such that every $k$-strongly connected digraph $D$ with $\alpha_2(D)\le a$ has a Hamilton cycle. Jackson and Ordaz conjectured that $f_2(a)\le a+1$. Towards this conjecture, we establish the near-linear bound $f_2(a)=O\!\left(\frac{a(\log a)^4}{(\log\log a)^2}\right),$ and hence $f_2(a)=O_\varepsilon(a^{1+\varepsilon})$ for every fixed $\varepsilon>0$. We also disprove the pancyclicity conjecture of Jackson and Ordaz that every digraph $D$ with $\kappa(D)\ge\alpha_2(D)+1$ contains a directed cycle of every length from $2$ to $|V(D)|$.

We extend the BDF2-type finite-element integrator proposed in [M. Aldé, M. Feischl, D. Praetorius; BDF2-type integrator for Landau-Lifshitz-Gilbert equation in micromagnetics: unconditional weak convergence to weak solutions. (Math. Comp., 2026, doi:10.1090/mcom/4210)] to the Landau-Lifshitz-Gilbert (LLG) equation with lower-order effective-field contributions and current-induced non-conservative torques arising in micromagnetic applications. To incorporate these additional terms while retaining a linear solve at each time step, we employ an implicit-explicit (IMEX) time splitting. Our main theorem proves unconditional weak convergence of the fully discrete approximations towards weak solutions of the LLG equation. The analysis removes the mild but artificial CFL-type restrictions required at the first and final time steps in [Aldé M., Feischl M., Praetorius D.; Math. Comp., 2026]. Furthermore, in the presence of the Dzyaloshinskii-Moriya interaction (DMI), the convergence is likewise unconditional and does not require the CFL-type restriction imposed in [G. Hrkac, C.-M. Pfeiler, D. Praetorius, M. Ruggeri, A. Segatti, and B. Stiftner. Convergent tangent plane integrators for the simulation of chiral magnetic skyrmion dynamics. (Adv. Comput. Math., 2019. doi: 10.1007/s10444- 019-09667-z)]. Numerical experiments support the theoretical findings.

We give an explicit description of the orbits of the split real form $G_{2(2)}$ on the complex flag manifolds of $G_2$ and study their induced invariant CR structures. In particular we study their Levi forms and show that all possible cases in the simple homogeneous setting occur, namely Levi nondegeneracy, $k$-nondegeneracy for $2\leq k\leq 3$, and holomorphic degeneracy. Moreover, following the classical approach to orbits on flag manifolds, we give equations for these homogeneous CR manifolds.