Papers

We establish maximal concentration bounds for the iterates generated by stochastic approximation algorithms with general step sizes, where the noise has a finite-state Markovian component plus a Martingale-difference component. When the Martingale-difference noise is bounded, we show that the tail of the error can be sub-Gaussian, sub-Weibull, or something lighter than any Pareto but heavier than any Weibull, depending on the step size sequence and on whether the random operator is almost surely contractive, almost surely non-expansive, or expansive with positive probability. Our analysis relies on a novel Lyapunov function involving the moment-generating function of the solution to a Poisson equation, together with an auxiliary projected algorithm. We complement the upper bounds with worst-case examples showing that qualitatively sharper bounds are impossible. We further study the case of unbounded Martingale-difference noise when the average operator is contractive, and the step sizes are of order $1/k$. In this setting, we show that if the random operator is almost surely non-expansive, then the error tail is at most three times heavier than the noise tail, whereas if the random operator is expansive with positive probability, then the error may have substantially heavier tails. These results are obtained through a novel black-box truncation argument that reduces the unbounded-noise setting to the bounded-noise case.

Most value-based and actor–critic reinforcement learning methods rely on Bellman-style recursions, yet these recursions collapse under non-exponential discounting common in human preferences and survival processes. We show the breakdown is structural: exponential discounting sits at a fragile intersection of multiplicativity and time homogeneity, and violating either property breaks standard dynamic programming. To overcome this, we propose Pontryagin-Guided Direct Policy Optimization (PG-DPO), a variational framework that abandons recursion and couples the Pontryagin Maximum Principle with Monte Carlo rollouts via an Adjoint-MC projection enforcing pointwise Hamiltonian maximization. Across multi-dimensional hyperbolic and survival-discount benchmarks, PG-DPO improves accuracy and stability where equation-driven solvers and critic-based baselines diverge.

This paper focuses on the estimation of partially observed branching processes. First, the estimators from a frequentist perspective proposed in the literature are reviewed. The main objective of this paper is to present computational tools based on sequential Monte Carlo methods to perform Bayesian inference for these processes. In particular, the Liu-West particle filter is applied to perform Bayesian estimation of the parameters of interest for an epidemic model fitted by a partially observed branching process. As application, the example given in [8] is revisited and extended.

We give an affirmative full-range solution to Gaunt's 2019 Open Problem~2.10. The problem asks whether, for every \(\nu>-1/2\) and \(0<\gamma<1\), the reciprocal-power integral \(\int_0^x e^{-\gamma t}I_\nu(t)t^{-\nu}\,\dd t\) is bounded by a constant multiple of \(e^{-\gamma x}I_{\nu+1}(x)x^{-\nu}\), uniformly for all \(x>0\). Earlier exponential-tilt estimates proved such endpoint majorants only under an additional smallness condition on \(\gamma\). We prove the estimate throughout the natural range \(0<\gamma<1\), with an explicit admissible constant. More generally, if \(\mu>-1\), \(q>-1\), \(0<\gamma<1\), and \(w(x)x^{-q}\) is nondecreasing on \((0,\infty)\), then for every \(\theta\in(\gamma,1)\), \(\int_0^x e^{-\gamma t}w(t)t^{-\mu}I_\mu(t)\,\dd t\) is controlled by an explicit multiple of \(e^{-\gamma x}w(x)x^{-\mu}I_{\mu+1}(x)\). The case \(w\equiv1\), \(q=0\), and \(\mu=\nu\) resolves Gaunt's problem. The weighted theorem also yields shifted-order and moment estimates, applies to approximate power weights and monotone regularly varying amplitudes, and provides two-sided estimates under a reversed comparison. We further analyze the sharp power-weighted quotient via endpoint expansions, a stationary equation, and parameter monotonicity.

Let \(\nu_p(G)\) be the number of Sylow \(p\)-subgroups of a finite group \(G\), let \(\sigma_p(G)\) be their common order, and set \[ \gamma(G)=\int_0^1\sum_{p\in\pi(G)}\nu_p(G)x^{\sigma_p(G)}\,dx =\sum_{p\in\pi(G)}\frac{\nu_p(G)}{\sigma_p(G)+1}. \] A recent conjectural extension of the simple-group theorem for this invariant asserted that a nonsolvable finite group has \(\gamma(G)=9/2\) precisely when \(G\cong A_5\). We disprove this assertion by a direct and verifiable construction. More generally, we prove an exact direct-product compensation formula for \(A_5\) with an arbitrary nilpotent factor. The formula reduces the equality \(\gamma(A_5\times N)=9/2\) to a finite Egyptian-fraction equation in the orders of the Sylow subgroups of \(N\). Taking \(N=\C_2\times\C_7\times\C_{11}\times\C_{13}\times\C_{17}\times\C_{19}\times\C_{29}\times\C_{71}\times\C_{83}\), the loss in the \(2\)-Sylow contribution is exactly compensated by the new normal Sylow subgroups. Consequently \(G=A_5\times N\) is nonsolvable, is not isomorphic to \(A_5\), has solvable radical \(N\), and nevertheless satisfies \(\gamma(G)=9/2\). Several further explicit compensation certificates are also recorded.

Let $M$ be a cancellative commutative monoid and call a submonoid $S$ of $M$ an undermonoid if $\G(S)=\G(M)$ inside the Grothendieck group of $M$. Gotti and Li asked whether the finite factorization property is hereditary once it is known on all undermonoids: if every undermonoid of $M$ is a finite factorization monoid, must every submonoid of $M$ be a finite factorization monoid? We give an affirmative answer. Equivalently, for every cancellative commutative monoid $M$, the following two conditions coincide: every submonoid of $M$ is an FFM, and every undermonoid of $M$ is an FFM. The proof isolates a fixed length $\ell$ and an infinite set of length-$\ell$ factorizations of one element $b$. In the non-group case, a divisor-complement ideal $I=\{m\in M:m\nmid_M b\}$ enlarges the bad submonoid to a bad undermonoid while preserving the chosen length-$\ell$ factorizations. In the group case, a maximality argument over submonoids for which these factorizations survive is combined with a two-sided perturbation $S\mapsto S+\Nzero(2b+u)$. The key point is that the perturbation creates no new units and does not split any atom occurring in the fixed factorizations. This yields an undermonoid with infinitely many factorizations of $b$, contradicting the hypothesis.

The domination problem is a well-studied problem in graph theory. In this paper, we study two natural variants: the hop domination problem and the $2$-step domination problem. Let $G$ be a graph with vertex set $V$ and edge set $E$. For a graph $G$, a subset $S \subseteq V(G)$ is called an hop dominating set if every vertex not in $S$ lies at distance of exactly $2$ from at least one vertex in $S$. For $v\in V(G)$, let $N(v,2)$ denote the set of vertices in $V(G)$ that are at distance exactly $2$ from $v$. For a graph $G$, a subset $S \subseteq V(G)$ is called an \emph{$2$-step dominating set} if every vertex $v\in V(G)$ lies at a distance of exactly $2$ from at least one vertex in $S$. The Hop Domination (HD) problem and the \textsc{$2$-Step Domination} ($2$SD) problems ask whether a graph contains a hop domination set or a $2$-step domination set of size at most $k$, respectively. We study the computational complexity of these problems, and show that both are NP-complete, even when restricted to $d$-regular graphs for every $d\geq 3$, claw-free graphs and also unit disk graphs.

We develop a regularity and compactness theory for stable capillary minimal hypersurfaces in the half-space $\mathbb{H}^{n+1}$ with contact angle $\theta \in (0,\pi)$ and dimension $n \geq 2$. One key analytic ingredient is a capillary differential Schoen inequality, which allows us to establish a boundary sheeting theorem in the spirit of Bellettini. The other ingredient is a refined classification of stable capillary minimal cones, and we show that for any contact angle $\theta\in(0,\pi)$, the stable capillary minimal hypercone in $\mathbb H^5$ with an isolated singularity must be flat. As a consequence, for any $n\leq4$ and $\theta\in(0,\pi)$, we obtain the Bernstein theorem for embedded complete stable capillary minimal hypersurfaces in $\mathbb H^{n+1}$ with Euclidean area growth.

Bilevel optimization problems arise in many applications where decisions must account for the optimal response of another system, such as in game-theoretic settings. However, these problems are notoriously challenging, as even linear bilevel programs are strongly NP-hard. In this work, we consider bilevel optimization with a known upper-level objective and an unknown, potentially time-varying lower-level response, accessible only through noisy zeroth-order observations. We propose W-SparQ-BL, a Bayesian optimization framework that models the lower-level mapping using multi-output Gaussian processes and enables efficient optimization under uncertainty. Our approach leverages a sparse, observation-based approximation to control the effect of noise and temporal variability, while requiring only limited access to additional information over time. We establish regularity results linking the lower-level response to standard RKHS assumptions for common kernels, including Matérn and squared exponential. We prove that W-SparQ-BL achieves sublinear dynamic regret in both stationary and time-varying settings. Experiments on representative time-varying game-theoretic problems demonstrate the effectiveness of our approach.

Let $\mathscr{C}$ be a reduced $0$-Auslander extriangulated category. Motivated by Pan–Zhu silting reduction for such categories, we introduce the notion of (signed) presilting sequences in $\mathscr{C}$ and establish a bijection between (signed) presilting sequences in $\mathscr{C}$ and (signed) $\tau$-exceptional sequences over $\Lambda = \text{End}_{\mathscr{C}}(P)$, where $P$ is a projective generator of $\mathscr{C}$. This correspondence provides a new perspective on the Buan–Marsh bijection between signed $\tau$-exceptional sequences and ordered support $\tau$-rigid objects. Furthermore, we introduce a new category $\mathfrak{M}(\mathscr{C})$, called the $\tau$-cluster morphism category of $\mathscr{C}$, whose objects are certain extension-closed subcategories of $\mathscr{C}$ and whose morphisms are described in terms of signed presilting sequences. As an application, we recover the $\tau$-cluster morphism category of $\Lambda$ from $\mathfrak{M}(\mathscr{C})$.

Bergeron, Garsia, Haiman, and Tesler (1999) introduced two Schur positivity conjectures concerning the action of the $\nabla$ operator on modified Hall–Littlewood polynomials. Although the proofs of the celebrated compositional shuffle conjecture and its rational generalization, along with various symmetric function identities, have largely resolved these conjectures in the affirmative, we provide an elegant alternative proof driven by $q$-series techniques. Through this approach, we derive explicit Schur function expansions for two-column partitions and extend our results to arbitrary positive powers $\nabla^k$.

In this paper, we study a particular conservative standard map in complex dimension 2. In this example, Siegel disks can be visualized and analyzed numerically as to the smoothness of their boundaries. We formulate and numerically support some conjectures.

In Section 6 of the paper “Fraïssé's Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition”, we applied the methods developed in earlier sections to show that a certain reduct of the generic permutation has finite big Ramsey degrees. Unfortunately, this reduct was incorrectly identified as the generic partial order. We are grateful to Jan Hubička for bringing this error to our attention. In this note we correct the statements that rely on this misidentification and demonstrate that the reduct in question is in fact the generic 2-dimensional partial order. We emphasize that the arguments presented in Section 6 remain valid, with the sole exception of the Claim in the proof of Theorem 6.4, whose role was to (incorrectly) identify the reduct of the generic permutation as the generic partial order. This correction has an unexpected positive consequence. Rather than reproving a well-known result whose existing proof is already notably elegant, this note demonstrates that our general framework can be used to establish that a previously unexplored class of generic relational structures has finite big Ramsey degrees. This observation opens a potentially new direction for further research in the thriving area of big Ramsey combinatorics. In the addendum, we combine a recent result by Oudrar and Pouzet with our analysis of finite big Ramsey degrees for structures admitting finite monomorphic decomposition to characterize the existence of finite Big Ramsey degrees for all countable relational structures whose language has a linear order and age has polynomial growth.

Let $r,\ell\geq2$ be integers. Given $r$-graphs $G$ and $F_1,\dots,F_\ell$, we write $G\to(F_1,\dots,F_\ell)$ if every $\ell$-edge-coloring of $G$ yields a monochromatic copy of $F_i$ in the $i$th color for some $1\leq i\leq\ell$, otherwise we write $G\not\to(F_1,\dots,F_\ell)$. The Ramsey number $R(F_1,\dots,F_\ell)$ is the minimum number of vertices in an $r$-graph $G$ satisfying $G\to(F_1,\dots,F_\ell)$. In this note we prove that for any integers $t_1\geq\dots\geq t_\ell>r$, there exists an $r$-graph $G$ such that $G\not\to(K^{(r)}_{t_1},\dots,K^{(r)}_{t_\ell})$ but $G\to(K^{(r)}_s,K^{(r)}_{t_\ell-1})$, where $s=R(K^{(r)}_{t_1},\dots,K^{(r)}_{t_\ell})-1$. This extends recent work by Mendonça, Miralaei, and Mota, who established the statement for $r=2$.

Berggren and Popović introduced quadratic-permutation-polynomial interleaved Zadoff–Chu sequences and, from exhaustive data, conjectured that all normalized QPP-interleaved Zadoff–Chu sequences are inequivalent to ordinary Zadoff–Chu sequences precisely for prime-power lengths $N=p^n$ with $p>3$ and $n>1$. We give an exact local arithmetic criterion. For a normalized QPP $\pi_{a,b}(k)=ak^2+bk\pmod N$, the interleaved sequence is equivalent, under the standard five CAZAC-preserving operations, to a Zadoff–Chu sequence if and only if, for every prime power $p^\alpha\Vert N$, the valuation of $a$ satisfies \[ \nu_p(a)\ge \begin{cases} 0, & p=2,\ \alpha=1,\\ \alpha-1, & p=2,\ \alpha\ge2,\\ \alpha-1, & p=3,\\ \alpha, & p>3. \end{cases} \] The proof is based on a third finite-difference invariant of the lifted Zadoff–Chu phase, namely \[ \Delta^3\bigl((ak^2+bk+\varepsilon_N+2q)(ak^2+bk)\bigr) =12a(2ak+3a+b). \] As a consequence, the conjectured prime-power boundary is not correct: the exact non-vacuous condition for all nonzero normalized QPPs to be inequivalent to Zadoff–Chu sequences is that $N$ is odd, $9\nmid N$, and $p^2\mid N$ for at least one prime $p\ge5$. In particular, $N=75=3\cdot5^2$ is the smallest non-prime-power counterexample to the conjectured “only if” direction. A second corollary records the corresponding statement for irreducible QPPs.

A group is self-simulable if all its computable actions admit SFT covers, which means roughly that they can be implemented with finitely many tiling constraints. We prove that a graph product of infinite finitely-generated groups is self-simulable if and only if its defining graph has no disconnecting clique consisting of amenable groups. In particular, a right-angled Artin group (a.k.a.\ a graph group) is self-simulable if and only if the defining graph has no disconnecting clique. As an application, we obtain that a graph product of infinite finitely-generated groups splits (algebraically, or in a certain geometric sense) over an amenable subgroup if and only if the graph has a disconnecting clique consisting of amenable groups.

Let $f$ and $g$ be two Hénon-Sibony maps of $\mathbb{C}^k$. We show that if they have the same forward Julia set, then they share a common iterate, thereby extending Lamy's results from dimension 2.

We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be evaluated with little computational effort for a given eigenpair, assuming the matrix perturbations are measured by the spectral or Frobenius norm. We also show how symmetric perturbations can be exploited in the analysis. By means of two numerical experiments we demonstrate that problems incorporating eigenvector nonlinearities potentially need to be treated with additional care, when compared to the linear or eigenvalue-nonlinear theory.

Voloshyn introduced rational Weyl group elements in connection with rational normal forms on complex reductive groups and conjectured that their number in type $D_r$, for odd $r$, is $2^r-1$. We prove a stronger structural statement. For every odd integer $r\geq 5$, the rational elements of $W(D_r)$ are precisely the longest element $w_0$ and two explicitly described signed cyclic elements $c_I,d_I$ for each non-empty subset $I\subseteq\{1,\ldots,r-1\}$. Consequently, the rationality graph $\Gamma(D_r)$ is obtained by gluing two explicitly labelled subset-toggle graphs at $w_0$; it has $2^r-1$ vertices, and its only vertices of valency one are $c_{\{1\}}$ and $d_{\{1\}}$. The proof combines a two-level acyclic description of the root-poset graphs $\Gamma_{c_I}$ with a rigidity theorem for simple left multiplications of the signed cyclic family. A self-contained simply-laced reflection-preservation lemma and a terminal-layer argument provide the descent step, while every forbidden one-step move from the family is excluded by an explicit loop or two-cycle.

An important step in the proof of the Herman invariant tori conjecture was the introduction of a normal form with poles along the resonance loci, replacing the Birkhoff normal form, which we call the Hamiltonian normal form. This paper is extracted from previous versions (arXiv:1206.1245, arXiv:1909.06053) and aims to present this Hamiltonian normal form in its simplest form. It is expected that, not only theoretically but also in numerical computations, it will provide better approximations than the standard Birkhoff normal form.

Let $n,$ $d,$ and $r$ be integers such that $0 \leq r \leq d \leq n,$ and let $P \subset \mathbb R^n$ be a $d$-dimensional convex lattice polytope. In this article, we prove that the ratio of the $r$-dimensional volume of a normal-sized miniature of $P$ to that of $P$ is given by $1:\binom{d+r+1}{r},$ which generalizes the author's previous results on the volumes of the unit hypercube and lattice simplices in the case where $r = d = n.$ This theorem is proven by establishing that the number of horizontal miniatures of $P$ with resolution $t$ is a polynomial of degree $d+1$ in $t$ whose leading coefficient is $\mathrm{vol}\,P/(d+1),$ which is derived from Ehrhart theory.

We introduce the extra slow Tamari lattices, a new family of lattices defined on faithfully balanced tableaux. These tableaux arise naturally from the representation theory of type \( A \) quivers, and our construction extends the classical Tamari lattice and the slow Tamari lattice. We explicitly describe meets and joins in the extra slow Tamari lattices, and then prove that they are lattices. We then show that they are semidistributive, trim, polygonal, and congruence uniform. Their join-irreducible elements are described in terms of a three-color analogue of the positive roots of type \( A \), which leads to descriptions of their spines and congruence lattices. We also obtain several enumerative results for the extra slow Tamari lattices and their associated structures. Finally, we derive new structural and enumerative results for the slow Tamari lattices.

We revisit the instability properties of the recovery of the absorption coefficient for the radiative transfer equation in the diffusive regime. To this end, we develop a rather robust framework building on [Koch-Rüland-Salo, 2021] which allows us to deal with nonlinear critical stability transition phenomena. In particular, this permits us to consider rather general geometries based on the identification of compression properties of the forward operator. Given the albedo operator as the measurement data, we show that in the regime of vanishing Knudsen number there is a transition from Hölder to logarithmic stability in the inverse problem for the radiative transfer equation. As a central ingredient, we rely on suitable a priori estimates for the radiative transfer equation which we deduce by building on the strategy from [Demattè-Velázquez, 2025].

We study single-loop gradient-flow dynamics for nested optimization, where the outer variable evolves while auxiliary variables track the inner solution map. While existing analyses typically rely on problem- and condition-specific Lyapunov constructions, we propose, to our knowledge, the first unified Lyapunov template for continuous-time analysis that covers minimax, bilevel via a lifted penalty formulation, and min–min–max. Our proof is modular, built from reusable lemmas that yield a unified characterization of time-scale separation. This characterization bridges regimes from strong convexity/concavity to mere convexity through an error-bound condition, and produces explicit closed-form thresholds that avoid the coupled ratio conditions common in discrete-time analyses. We further compare the penalty dynamics with the ideal hyper-gradient flow, derive a finite-time tracking bound, and discuss an Euler one-step analogue; hypercleaning diagnostics show that the predicted relative time-scale regions remain visible under stable forward-Euler discretization.

We consider the asymptotics of sums of the form $$ \frac1{F_n^\sigma} \sum_{m = 1}^{F_n-1} \frac{f(m/F_n)}{\left|{\sin(\pi m/F_n)}\right|^\sigma} \frac{f(F_{n-1}m/F_n)}{\left|{\sin(\pi F_{n-1}m/F_n)}\right|^\sigma} $$ where $(F_n)_{n \in \mathbb N} = (1, 1, 2, 3, 5, 8, 13, \dots)$ are the Fibonacci numbers. Such sums appear, for example, in the context of discrepancy theory and numerical integration methods reformulated as energy minimization problems. We show that for parameters $\sigma > 1$ and a large class of functions $f$ the above sum behaves asymptotically like $$ C n + D + O\left((1-\varepsilon)^{n}\right) $$ for some constants $C$ and $D$. These constants can be given via infinite series connected to the Dedekind zeta function over the algebraic number field $\mathbb Q(\sqrt5)$. In special cases we even observe simple closed-form expressions for such sums as above, explicitly proving that $$ \sum_{m=1}^{F_n-1} \frac1{\sin(\pi m/F_n)^2} \frac1{\sin(\pi F_{n-1} m/F_n)^2} = \frac{4n}{75} F_{2n} - \frac{17}{225}F_n^2 - (-1)^n \frac2{15} - \frac19. $$

We present the Matlab toolbox MacaulayLab, which implements numerical linear algebra algorithms for solving multivariate polynomial systems and rectangular multiparameter eigenvalue problems. Its structure and functionality are the result of several years of research and algorithmic development. We demonstrate how the software works and compare its performance with other software packages, such as PNLA, PHCpack, and MultiParEig. Some core features of MacaulayLab are the fact that it solves two key problems via one common approach, works independently of the chosen polynomial basis and monomial order, and is capable of dealing with positive-dimensional solution sets at infinity. The toolbox (including its future updates) and a large collection of test problems are freely available online.

We study two new parameters for finite posets motivated by the problem of efficiently determining the set of successors of a given element. A plane map of a poset $P=(X,\leq)$ is an injective mapping of $X$ into the Cartesian plane $\mathbb{R}^2$. Given two different points $a$ and $b$ in the plane, we say that $b$ dominates $a$ if $a<b$ coordinatewise. We say that an element $x$ of $P$ is tight in a plane map $\mu$ if the following holds: $x<y$ in $P$ if and only if $\mu(y)$ dominates $\mu(x)$. Note that, by definition, every 2-dimensional poset admits a map such that every element of the poset is tight. For any poset $P$, we define the mapability of $P$, $\mathrm{dmap}(P)$, to be the maximum number of elements that are tight in a single map, and we define the atlas thickness of $P$, $\mathrm{at}(P)$, to be the size of the smallest collection of maps such that every element is tight in at least one map of the collection. We relate these parameters to the classical notions of dimension and width: for every poset $P$, we show that $\mathrm{dim}(P) \le 2\mathrm{at}(P) \le \mathrm{width}(P)+1$. On the other hand, there exists a sequence of posets $(P_n)_{n \ge 1}$ such that the atlas thickness of $P_n$ is doubly exponential in the dimension of $P_n$. On the computational side, we prove that it is NP-complete, for a given poset $P$, to compute the mapability of $P$ and to decide whether $\mathrm{at}(P) \le 2$. In contrast to the latter, we show that computing the mapability of a poset is fixed-parameter tractable with respect to the natural parameter.

This paper presents a Markov-chain-based method for the early-phase analysis and design of hybrid spare-management architectures for large-scale satellite constellations.} The hybrid strategy combines two channels: an indirect path that stages spares in parking orbits via heavy launch for later transfer to constellation planes, and a direct path that delivers spares to in-plane orbits using small launch vehicles. {To assess the long-run viability of such concepts of operations, satellite failure and replenishment processes are modeled as a Markov chain:} the indirect channel follows a periodic-review reorder-point/order-quantity policy, while the direct channel uses a standard reorder-point/order-quantity policy. These coupled chains yield a periodic steady state over the right ascension of the ascending node cycle via fixed-point iteration, and the stationary distributions provide rigorous cost and resilience metrics. By directly modeling the stochastic, multi-echelon dynamics governed by orbital mechanics, our framework avoids the aggregation assumptions of prior works and remains valid across a wider operating domain. We also introduce an approximate analysis that preserves delay statistics while significantly reducing model size. Building on this fast, accurate analysis, we formulate a cost minimization problem with resilience constraints and solve it using a genetic algorithm. The framework is channel-neutral; the optimization autonomously selects the preferred path and roles. {A case study validates the analysis against Monte Carlo simulations and demonstrates the practical value of the framework in identifying the conditions under which the hybrid policy outperforms pure strategies.

Communication is a major bottleneck in distributed learning, especially in large-scale settings and in federated learning environments with slow links. Three standard ways to reduce this cost are communication compression, local training, and communication-computation overlap. Methods that combine these ingredients are used in practice and have been found to be effective for large-scale training, but there is little theory for methods that combine all three. We study a heterogeneous-compute setting in which different workers may take different numbers of local steps, and we propose LOSCAR-SGD, a Local SGD method that communicates only a sparse subset of model coordinates and continues optimizing while communication is in flight. A key ingredient is a delay-corrected merge rule that incorporates delayed synchronized information without discarding the progress made during the overlap phase. We give convergence guarantees for smooth non-convex objectives and show how sparsity, overlap, and worker heterogeneity affect the rate. To the best of our knowledge, this is the first theory for this combination of ingredients. Experiments further show that communication-computation overlap reduces training time and that the delay-corrected merge outperforms naive overwriting.

For a weighted tree, the Lin–Lu–Yau Ricci curvature admits an explicit formula in terms of the edge weights. Consequently, the constant-curvature equation is equivalent to an eigenvalue problem for an edge-indexed Ricci matrix $R_T$. Building on the spectral characterization of discrete Einstein metrics on trees, we classify all finite trees whose discrete Einstein metric has positive curvature, equivalently all trees satisfying $\lambda_{\max}(R_T)<0$. For caterpillars with spine order $m\ge 12$, this occurs precisely for the endpoint families $T_m(a,0,\ldots,0,b)$ with $1\le a,b\le 3$ and $(a,b)\ne(3,3)$. The remaining cases $3\le m\le 11$ are settled by an exact finite verification using rational characteristic polynomials and Sturm root counts. We also determine the zero level set $\lambda_{\max}(R_T)=0$: among caterpillars, it consists of the stable family $(3,0,\ldots,0,3)$ together with nine exceptional short-spine caterpillars, while $S_3^2$ is the unique non-caterpillar zero example.