Papers

For $\beta \in \mathbb{R}$, we define the partition function $$ Z_\beta(X) = \sum_{i=1}^N \exp(\beta X_i) $$ of a centered Gaussian random vector $X=(X_1, \ldots, X_N)$ with $\mathbb{E}[X_i^2]= 1$. For $q \in \mathbb{R}$, we prove a complete phase transition for the generalized $L_q$-means of $Z_\beta(X)$ at $q=1$. The centered Gaussian vector with covariance matrix $\Delta_N$ obtained from a regular simplex configuration of unit vectors maximizes these means for $q<1$, and minimizes them for $q>1$. The two regimes are governed by different principles. For $q > 1$ {and $\beta\ne0$}, the moment functional is globally strongly convex on the entire set of correlation matrices, with an explicit modulus of convexity and a quantitative centroid-shape stability estimate. For $q<1$, the strategy is different. We prove a universal comparison for log-concave profiles of reverse Brascamp–Lieb type. Specializing this result to the Gumbel profile yields a Laplace-transform comparison between the partition functions.

In this article, we study the asymptotic expansion of the analytic torsion associated with positive Hermitian line bundles on projective spaces. We show that the expansion contains no terms of the form $k^{-i}\log k$ for $i\geq 1$. The proof combines the arithmetic Riemann-Roch theorem with an explicit computation of the arithmetic degree of spaces of global sections.

This paper studies the Cauchy problem for the stochastic Landau–Lifshitz–Baryakhtar equation on $\mathbb{R}^d$, $d=1,2,3$, subject to Stratonovich Gaussian perturbations. We establish a low-regularity well-posedness theory for initial data in $\mathbb{L}^2$, $\mathbb{H}^1$, and $\mathbb{H}^2$. For $\mathbb{L}^2$ initial data, we prove the existence and pathwise uniqueness of global very weak solutions in dimensions one and two, while in dimension three we construct martingale very weak solutions on every finite time interval. For arbitrary $\mathbb{H}^1$ initial data, we prove the existence and uniqueness of global pathwise weak solutions in all dimensions $d\leq3$. Furthermore, for $\mathbb{H}^2$ initial data, we establish the existence and uniqueness of global pathwise strong solutions in the same range of dimensions. The analysis is based on a frequency-truncation approximation scheme combined with stochastic compactness arguments in the whole-space setting. The key step in global continuation is to justify the stochastic effective-field balance at $\mathbb{H}^1$ regularity, where the drift is available only in a negative Sobolev space. Spatial mollification and passage to the limit yield the energy identity for local weak solutions. A coercive modification then gives global energy moments and excludes finite-time blow-up.

The Kromatic symmetric function (KSF) $\overline{X}_{G}$, introduced by Crew, Pechenik, and Spirkl (2026), is a $K$-theoretic analogue of the chromatic symmetric function (CSF) $X_G$. We study expansion formulas for the KSF in two different bases. First, we explore recursive ways to compute the KSF's expansion in the $K$-theoretic monomial symmetric function basis $\overline{\widetilde{m}}_\lambda$, generalizing formulas that were used by the second author and Samanta to show that the KSF distinguishes certain families of graphs that are not distinguished by the CSF. Second, we give new interpretations for the KSF's expansion in the $K$-theoretic power sum basis $\overline{p}_\lambda$, using an inclusion-exclusion approach like the one from Stanley (1995) instead of an acyclic orientation approach. Finally, we give $K$-analogues of Schmitt's and of Humpert and Martin's antipode formulas for the Hopf algebra of graphs, along with a Hopf algebra interpretation for both the $\overline{p}$-expansion and $\overline{\widetilde{m}}$-expansion of the KSF.

In a recent paper published in the Journal of Differential Equations (384, 2024), Cui and Zhao investigated an SIRS epidemic model with the nonlinear incidence rate $\frac{kI^p}{1+\alpha I^q}$, where $p>0$ and $q\geq0$ are arbitrary real numbers. They proved that the codimension of a Bogdanov-Takens bifurcation in this model is at most two, while the codimension of the Hopf bifurcation was left as an open problem. In this work, we investigate this problem while keeping $p$ and $q$ free throughout the analysis. A strategic nondimensionalization and parametrization remove the exponential dependence of the positive equilibrium and convert the generalized-Hopf conditions into algebraic focus equations coupled to explicit semialgebraic admissibility regions. Using algebraic focus reduction, strict semialgebraic admissibility, and validated interval computation, we rigorously prove the existence of admissible nondegenerate generalized Hopf points generating three, four, and five small-amplitude limit cycles. In particular, we rigorously certify a strict-interior codimension-five weak focus and a full-rank local unfolding, thereby proving that five small-amplitude limit cycles can bifurcate from a single positive equilibrium. We further derive exact algebraic obstructions to codimension six and find no admissible candidate in extensive continuation and projected-component searches. These results provide analytical and numerical evidence for the conjecture that five is maximal, but no codimension-six nonexistence theorem is claimed. The same parametrization also yields a simple proof that the Bogdanov-Takens codimension is at most two.

Nearly incompressible elasticity is often discretized by mixed methods to avoid locking, but the resulting saddle-point systems can be expensive to solve. We consider a weakly symmetric multipoint stress discretization whose stress and rotation unknowns can be eliminated independently over vertex interaction regions, leaving a symmetric positive definite system for cell-centered displacement only. Although smaller, the reduced system contains a parameter-dependent Schur complement that becomes difficult to precondition as the Lamé ratio increases. We show that its energy consists of a uniformly coercive shear part and a dominant semidefinite volumetric part. Standard cell-centered interpolation need not preserve the volumetric kernel and can introduce energy amplified near incompressibility. Motivated by this structure, we develop a two-grid preconditioner combining two complementary coarse subspaces: a conventional displacement subspace and a discrete-curl subspace lying exactly in the fine-grid volumetric kernel. Symmetric vertex-patch smoothing completes the displacement-only solver. The energy splitting and kernel-compatible decompositions yield a condition-number bound independent of the Lamé ratio for each fixed grid pair. Two- and three-dimensional experiments confirm the approximation properties of the locally eliminated discretization and show that the kernel correction prevents the deterioration of the displacement coarse subspace alone, also for discontinuous material coefficients.

Let $h\geq2$ be an integer and let $1/h<\theta\leq1/(h-1)$. In this paper, we prove that there exists a strongly minimal asymptotic basis $A$ of order $h$ such that $A(x)\asymp x^\theta$. This solves a problem posed by Nathanson in 1988.

We prove quantitative homogenization estimates for linear elliptic equations in nondivergence form with stationary, symmetric coefficients, finite range of dependence, and a deterministic upper ellipticity bound. No deterministic positive lower bound is imposed. We assume that the reciprocal of the infimum of the smallest eigenvalue on a unit ball has a finite moment of order $d$, together with the common continuity condition of Armstrong and Smart. Under these assumptions, we obtain algebraic probability bounds for finite-cell errors and for Dirichlet homogenization errors with nonzero sources. We construct quadratic correctors on the whole space, modulo affine functions, and prove first- and second-order large-scale regularity and the corresponding Liouville theorems. We also identify the effective matrix through the invariant density and quantify smooth spatial averages of the density and the weighted coefficients. The proof separates a unit-trace diffusion from its physical clock. A reverse Hölder estimate for the Green function of a stopped coarse process yields the integrability gain needed to control the clock at the critical moment. Applications include weighted gradient convergence and a finite-domain approximation of the effective matrix.

In this article, we study the monogenity of the number field generated by a root of an irreducible factor of the generalized $(k,t)$-Fibonacci and $(k,t)$-Lucas polynomial sequence. We use Dickson polynomials and their connection with cyclotomic fields to prove our main results. As special cases, we recover the monogenity results for the classical Fibonacci and Lucas polynomials and obtain corresponding monogenity results for Chebyshev polynomials of the first and second kinds.

The fixed-point fast sweeping methods with weighted essentially non-oscillatory (WENO) local solvers are a class of efficient and high-order accuracy numerical methods for solving steady-state solutions of hyperbolic conservation laws. However, with the classical WENO-JS local solver, the iteration residue of high-order fixed-point fast sweeping scheme often has difficulty to settle down to round-off errors. To achieve the full convergence in a fast sweeping method, the fixed-point fast sweeping methods with non-traditional WENO local solvers based on unequal-sized substencils were designed. However, the WENO schemes based on unequal-sized stencils are more complex and in general more expensive in computational costs than the classical WENO-JS schemes. In this paper, we go back to the classical WENO-JS local solver and develop a new fully convergent fifth-order fixed-point fast sweeping method for solving steady-state problems of hyperbolic conservation laws. Based on recent studies on the nonlinear weighting process around discontinuities of solution, we apply the technique of frozen weights for avoiding unnecessary adjustment of nonlinear weights in the WENO-JS local solver, which freezes the nonlinear weights once the residual sequence in the fast sweeping iterations has stabilized. Different from the existing work on frozen weights, we design a simple and robust approach to judge stabilization of iteration residues and determine the iteration step when the nonlinear weights are frozen in the fast sweeping method. Extensive numerical experiments on a wide range of challenging two-dimensional steady-state problems demonstrate that, unlike the previous fast sweeping method with the fifth-order WENO-JS local solver, the proposed new scheme consistently drives the iteration residues to round-off errors and achieves the full convergence.

Inspired by the Green-function monotonicity formulas of Colding, Minicozzi, and Manea, we establish weighted monotonicity formulas on closed weighted Riemannian manifolds satisfying ${\rm Ric}_f^m \ge (m-1)kg$, where $m>n\ge 3$ and $k>0$. Our results extend Manea's elliptic monotonicity framework to the setting of finite-dimensional Bakry-Émery curvature and provide a positive-curvature counterpart to the weighted formulas of Song-Wei-Wu. Starting from the Green function of $-\Delta_f+m(m-2)k/4$, we derive an exact weighted Bochner identity and prove three monotonicity formulas, together with a one-parameter family for $\beta\ge (m-2)/(m-1)$, under explicit pole-integrability conditions. We also treat the case of negative effective dimension $m<0$, for which the nonnegative square decomposition remains valid, while the change in the behavior near the Green pole leads to reversed area and volume monotonicity on finite sublevel sets. We conclude with comparisons and questions concerning entropy monotonicity formulas and positive-curvature RCD spaces.

Optimistic matrix mirror-prox (OMMP) computes $\epsilon$-approximate Nash equilibria in quantum zero-sum games with an $O(1/\varepsilon)$ average-iterate guarantee [arXiv:2311.10859]. We investigate whether this dependence on accuracy is tight and whether geometric last-iterate convergence can be guaranteed. We study these questions through explicit games with one qubit per player. First, we prove an $\Omega(1/\varepsilon)$ lower bound for the uniform-average output that includes the maximally mixed initial state, independently of the regularizer and step size. Second, we construct a fixed game on which optimistic gradient descent-ascent (OGDA), initialized at the maximally mixed state, has last-iterate Frobenius distance to equilibrium $\Theta(1/t)$ and duality gap $\Theta(1/t^3)$ for every sufficiently small fixed step size. A separate fixed game exhibits arbitrarily long delays in reducing the initial error by a constant factor across a family of initial states. Finally, we give a fixed game with a unique, strictly complementary equilibrium on which optimistic matrix multiplicative weights updates (OMMWU) converge only polynomially from the maximally mixed state for every fixed positive step size. The last-iterate Frobenius distance and quantum relative entropy from the equilibrium to the iterates decay as $\Theta(1/t)$, while the duality gap decays as $\Theta(1/t^2)$.

We prove the following two upper bounds for ordered Ramsey numbers: (1) Every ordered forest $F$ on $n$ vertices satisfies $R_{<}(F,F)=O(n^{1+\lceil\log\chi_{<}(F)\rceil})$. This in particular answers a question of Geneson, Holmes, Liu, Neidinger, Pehova and Wass. (2) There is a function $f$ such that, for every fixed ordered graph $H$ with maximum degree at most $\Delta$ and interval chromatic number at most $k$, it holds that $R_{<}(H,K_n)=O_H(n^{f(\Delta,k)})$.

A metric graph is of pseudocompact type if identifying parallel edges produces a tree. We give a constructive proof that, for such graphs, divisorial $d$-gonality is equivalent to the existence of a degree $d$ harmonic morphism to a tree. This mirrors the algebraic correspondence, for curves of compact type, between limit linear series of dimension one and admissible covers. We also deduce lifting results for positive-rank divisors, with a genus-preserving refinement when identifying parallel edges produces a path. Finally, we study the Brill-Noether theory in the path case.

We propose a holographic dual of the relative entropy of entanglement for a boundary tripartition $A:B:C$. Our proposal is that, at the leading order, \begin{align} E_R(\rho_{AB}) = \frac{1}{4G_N}\Big[\mathrm{Area}(\Gamma_{\min}) - \mathrm{Area}(\gamma_{AB})\Big] \notag \end{align} where $\Gamma_{\min}$ is the minimal bulk tripartition surface and $\gamma_{AB}$ is the minimal surface homologous to $AB$. For a general random tensor network, we prove the corresponding upper bound by constructing a separable state associated with $\Gamma_{\min}$ and evaluating its relative entropy. We also establish the matching lower bound rigorously for networks of one, two, and three Haar random tensors by bounding the maximal tripartite product-state overlap. In appropriate geometries, the minimal tripartition surface can form a nontrivial tri-junction resembling the Mercedes logo. The proof proceeds by sequential optimization over candidate product states, which has a natural geometric interpretation as a local search for the minimal tripartition surface.

We develop a Halpern-accelerated majorized alternating direction method of multipliers (ADMM) with possibly indefinite proximal terms for linearly constrained convex composite optimization. Majorization simplifies the subproblems but introduces a forward perturbation to the standard degenerate proximal point representation. We reformulate majorized ADMM as a perturbed degenerate proximal point method and characterize the fixed points of the induced mapping as Karush–Kuhn–Tucker (KKT) solutions. Under range compatibility and metric cocoercivity conditions, the mapping is nonexpansive in the seminorm induced by a possibly singular preconditioner. In finite dimensions, a verifiable block-matrix condition permits indefinite proximal regularization while retaining the required metric properties. A relaxed Halpern iteration then yields an $\mathcal O(1/k)$ fixed-point residual bound, sharp in the general nonexpansive setting, and an $\mathcal O(1/k)$ nonergodic KKT residual bound at the intermediate iterates. Numerical experiments illustrate the predicted residual decay and the benefits of majorization and indefinite proximal terms.

We address robust trajectory-planning problems with state constraints and with control/state-dependent uncertain inputs and measurements. Uncertain inputs and measurements satisfy quadratic inequalities with rank-2 indefinite multipliers; measurements depend on the uncertain-state realization and on the uncertain inputs. By means of Lagrangian duality, we derive finite nonsmooth constraints that guarantee robust satisfaction of the state constraints, yielding a formulation that can be addressed without bilevel solvers. We develop a two-step convex heuristic for the resulting nonconvex nonsmooth formulation, using a convex relaxation followed by a convex feasibility-recovery step. We test the two-step approach on a rendezvous scenario with state-dependent navigation uncertainty, and we reduce by a factor 13.8 the computation time with respect to the commercial solver Gurobi, while maintaining robust satisfaction of constraints.

We prove that for a supersingular irreducible symplectic variety admitting suitable lifting to characteristic zero has Tate Chow motive if it is of deformation type $K3^{[n]}$, OG6 (with Artin invariant $\neq$ 4), or OG10 (with Artin invariant $\neq$ 12), and has supersingular abelian Chow motive if it is of $\mathrm{Kum}^n$-type (with Artin invariant $\neq$ 3). In particular, any product of those irreducible symplectic varieties satisfies the supersingular Tate conjecture for the whole $\ell$-adic or crystalline cohomology ring.

Variational Quantum Algorithms (VQAs) rely heavily on classical optimization routines to navigate high-dimensional, noisy parameter landscapes. However, the evaluation of analytical gradients via the parameter-shift rule requires $\mathcal{O}(p)$ circuit executions, rendering this approach computationally prohibitive for deep circuits. While simultaneous perturbation methods provide an $\mathcal{O}(1)$ alternative, their gradient estimates suffer from severe $\mathcal{O}(p)$ spatial variance, which impedes convergence. To resolve this tension, we introduce the Cached Recycled Variance-Reduced Gradient (CRVG) by leveraging variance reduction techniques. We formulate two variants: a 3-Circuit (1-Sided) Non-Recursive CRVG and a Recursive CRVG. By introducing a center-point caching mechanism, the 3-Circuit CRVG mathematically recycles quantum measurements to reduce the inner-loop circuit complexity by 25% across both variants. We establish a theoretical separation between the two routing paths to achieve an $\epsilon^2$-stationary point: the non-recursive variant reaches a sample complexity of $\mathcal{O}(\max\{p/\epsilon^2, p^{2/3}/\epsilon^{10/3}\})$, while the recursive variant achieves an improved complexity of $\mathcal{O}(\max\{p/\epsilon^2, \sqrt{p}/\epsilon^3\})$. Empirically, extensive evaluations on MaxCut and Maximum Independent Set (MIS) benchmarks validate these findings. While standard simultaneous perturbation remains a robust baseline across varying depths, CRVG demonstrates targeted advantages in specific amortizable regimes, such as the non-recursive variant on EfficientSU2 circuits and both variants on depth-three QAOA, yielding superior energy minimums and tighter output variances. Ultimately, CRVG establishes a principled bias-throughput trade-off for scaling near-term quantum optimization.

We extend the definition of Khovanov-Lee homology to links in connected sums of interval bundles over surfaces and $S^1\times S^2$'s, and construct a Rasmussen-type invariant for links in these manifolds. As an application, we prove an inequality relating the Rasmussen-type invariant to the genus of surfaces with boundary in four-manifolds that are boundary connect sums of $D^2\times S^2$, $\mathbb{C} P^2\setminus B^4$, and $DTS^2$.

Recently Burniol Clotet proved that two transitive Anosov flows on 3-manifolds with equivalent strong unstable foliations must be conjugate up to time rescaling. We further explore the mechanism of Burniol Clotet's argument and treat the case of higher-dimensional Anosov flows and their isometric extensions by compact Lie groups.

Following the recent work of M. Womack \cite{Wo26} we consider the steady Ricci Yang-Mills solitons on 2d surfaces containing orbifold point $\mathbb{R}^2/\mathbb{Z}_p$. We show that there are a family of solitons depending parameter $\lambda$ which approaches to $\mathbb{Z}_p$-quotient of Hamilton cigar soliton as $\lambda \to (-2/p)^+$ and approaches to (after rescaling) $\mathbb{Z}_p$-quotient of round sphere as $\lambda \to \infty$. For any integer $q >p$ there is a $\lambda$ whose corresponding soliton metric is defined on football orbifold $S_{p,q}^2$.

Let $C_{k}^{(r)}$ be the $r$-uniform linear cycle on $k$ hyperedges. An $r$-graph is $C_{k}^{(r)}$-free if it contains no copy of $C_{k}^{(r)}$. Let $\operatorname{ex}_{r}(n,C_{k}^{(r)})$ denote the maximum number of hyperedges in an $n$-vertex $C_{k}^{(r)}$-free $r$-graph. Balogh, Narayanan and Skokan asked whether, for every pair of integers $r,k\ge 3$, the number of $C_k^{(r)}$-free $r$-graphs on $n$ labelled vertices is \[ 2^{(1+o(1))\operatorname{ex}_{r}(n,C_{k}^{(r)})}. \] While the analogous statement is known to fail for graphs ($r=2$), by a construction of Morris and Saxton, the general question remained open for hypergraphs. Very recently, Jiang and Longbrake answered the question affirmatively when $r\geq5$. In this paper, we completely resolve the problem by proving that the answer is affirmative for every pair of integers $r,k\geq3$. When $(r,k)\neq(3,3)$, we establish balanced supersaturation results for linear cycles and combine them with the hypergraph container method. For $(r,k)=(3,3)$, we instead decompose $3$-graphs according to their pair-codegrees, encode the subhypergraph formed by the hyperedges that contain a large codegree pair as a directed graph, and apply a multicolour entropy theorem.

A single-tone large-signal operating point of a nonlinear two-port is a periodic orbit of a periodically forced circuit. When the device has memory, the Floquet exponents of that orbit separate into fast (electrical) and slow (thermal and trapping) modes, and long-term memory is motion on the invariant manifold attached to the slow modes. Existence and uniqueness of that manifold follow from the parameterization method of Cabré, Fontich and de la Llave, applied to the stroboscopic map at the orbit; the manifold is the spectral submanifold of Haller and Ponsioen, without a small-forcing parameter. The manifold is a bundle over the circle of drive phase, its fiber dimension the number of slow exponents, and the dynamic X-parameter kernel of Verspecht et al. identifies its reduced dynamics from the transient after a step in drive amplitude. An exact reduced model has one memory state per slow exponent, the memoryless X-parameter surface is the fixed-point family of the reduced dynamics, and the envelope-domain model is the reduced dynamics driven by the envelope. The hypotheses are verified and the manifold constructed for a GaN HEMT compact model with thermal and trap memory: the trap's time constant at the operating point, $14 \mu$s, is set by the linearization, not by its $6$ ms emission time, and the trap's nonlinearity confines a polynomial representation of the manifold to a few millivolts of drive, so trap memory must be represented over the operating range, by tables or a fitted network, not by an expansion about the operating point.

A company's financial statements over many years must be assembled from many filings. Each filing may rename, combine, split or move its lines, and restates only the last two or three years to match. Filings provide no consistent, machine-readable mapping between successive statements, and neither a line's label nor its XBRL tag is fixed. We treat the reconstruction as an optimization problem. A tag that keeps its numbers from one filing to the next is taken to be the same line, and such runs are joined into strands before any edit is weighed. At each boundary between filings, the candidate edits that pass the check their kind requires are listed, each with the strand endpoints it would explain and a price for how much it asks the reader to believe beyond what the filings show. An integer program picks the cheapest set of the listed edits that accounts for every line exactly once, and the chosen edits trace each line through the filings. The result is a statement whose rows follow one line across every year, comparable within each filing's restated window and marked where a number was restated or derived, in any filing's layout, with each number cited to the filing that stated it and each declared total re-checked. On 200 companies drawn at random, 99.9% of the totals that can be checked add up. Among the 124 with a usable vendor comparison, the lines the method follows agree with the vendor's series on 96% of the cells both fill, reach 10% more cells than a tag alone and agree with the vendor on 90% of those. The method also yields a record of what each company changed in its reporting.

In this article, the problem of completing a tensor from some incomplete knowledge of its entries is treated by adopting a worst-case perspective, given the realistic assumption that the tensor's low-order ANOVA terms are dominant. We survey and leverage some recent all-purpose results from the field of Optimal Recovery to provide solutions on a theoretical level. But the accompanying constructions of optimal completion procedures, which often feature semidefinite programs, are not directly applicable in the tensor case due to the huge dimensions involved. To resolve the issue, we put forward a storage-friendly way to produce low-order ANOVA projections based on the fast Fourier transform (FFT), while exploiting the specificities of the completion problem to efficiently compute regularizers and extremal eigenvalues. Numerical experiments on synthetic tensors and real-world datasets demonstrate the accuracy and scalability of our FFT-based method.

This paper develops a model-free reinforcement learning (RL) algorithm based on the stochastic maximum principle for continuous-time stochastic control problems with continuous state and action spaces. For a parameterized Markovian policy, we establish the existence of the decoupling field for the adjoint backward stochastic differential equation, which allows the Hamiltonian gradient to be represented as a deterministic function of time and state. We then learn this Hamiltonian gradient directly from data and incorporate it into a policy gradient scheme with inexact gradients. We establish the convergence of the resulting policy gradient algorithm and establish the corresponding error estimates. Under suitable conditions on learning rates, exploration parameters, and approximation errors, the objective values converge and the $L^2$-norm of the policy gradient vanishes asymptotically. We further show that the proposed SMP-based policy gradient representation is equivalent to the existing continuous-time deterministic policy gradient representation based on the advantage-rate function. The effectiveness and efficiency of the proposed RL algorithm are demonstrated by numerical experiments.

We generalize Ren and Wang's incidence bound between points and lines in $\R^2$ \cite{RenWan23} to higher dimensions. We show how to use this incidence bound to improve the best known bound for Falconer's distance set problem in $\R^3$ and in $\R^4$. We show that if $d=3$ or $d=4$, and $E\subset \R^d$ is a Borel set of dimension $\dim_H(E) > d/2$, then \begin{equation*} \sup_{x\in E} \dim_H(\Delta_x(E)) \geq 2/3, \end{equation*} where $\Delta_x(E)$ is the pinned distance set of $E$ with respect to $x$. We also show how the incidence bound can be used to generalize the planar Furstenberg set bound, to sets in $\R^d$ that contain a $t$-dimensional set of hyperplanes, each of which contains an $s$-dimensional set of points, for any $d\ge 2$, $s \in (d-2, d-1]$ and $t \in (0, d]$.

We propose ABrA-GD (Adaptive Bregman Accelerated Gradient Descent), an adaptive algorithm for relatively smooth convex optimization. The algorithm is derived from a computable primal–dual certificate that guarantees progress by comparing the objective value with a lower bound on the minimum of a regularized objective. This certificate enables adaptation to both smoothness and geometry, using only the relative strong convexity constant as a problem-dependent input. We introduce the dual Bregman Length Distortion Factor (BLDF), which measures how dual Bregman lengths change under an anchor shift or rescaling. Under bounded local dual BLDF, ABrA-GD achieves an accelerated $O(1/k^2)$ rate for convex objectives and an accelerated linear rate for relatively strongly convex objectives.

Two recent constructions independently give bigraded commutative Hopf algebra structures on the homology of $\mathrm{GL}(\mathbb{Z})$ with rational Steinberg coefficients. One, by Ash–Miller–Patzt, uses the sharbly resolution. The other, by Brown–Chan–Galatius–Payne, uses the Quillen spectral sequence and rank-filtered maps on $BK(\mathbb{Z})$. We prove that the two structures coincide. Both are induced by join and parabolic restriction operations on Steinberg modules.