Papers

We consider two sets of related models: initially, these are $SU(2)$ antiferromagnetic spin chains with $N$ sites of spin $S$, and the $O(3)$ nonlinear sigma model in two dimensions with topological coefficient $\Theta$ a multiple of $\pi$ (and later, the extensions of these with any semisimple Lie group symmetry). It is known that, in a continuum description, the low-energy behavior of the spin chain is given by the sigma model with $\Theta=2\pi S$. We study these models with $N$ odd and with antiperiodic (A) boundary condition (b.c.), respectively, which correspond. The A b.c. in the sigma model involves the $\mathbb{Z}_2$ inversion symmetry $\vec{n}\to-\vec{n}$, and amounts to a flux of a $\mathbb{Z}_2$ gauge field through a spacetime torus; summing over the two b.c.s for each direction would amount to gauging the $\mathbb{Z}_2$ inversion symmetry. We show directly that, if and only if $(-1)^{\Theta/\pi}=-1$, the gauging cannot be carried out; there is an 't Hooft anomaly. The partition function for the A b.c. exists, but is not gauge invariant; consequently, the sum over b.c.s cannot be made modular invariant. The gauged model would be a sigma model with target space $\mathbb{R}\mathbb{P}^2\cong \mathbb{S}^2/\mathbb{Z}_2$, and hence this model does not exist for $\Theta=\pi$ (mod $2\pi$). A related result is that, using semiclassical quantization, in the spin chain we obtain the known values of the ground-state crystal momentum, which at leading order depend only on $N$ modulo $4$ and $2S$ modulo $2$. For a large class of spin chains and associated sigma models we find similar results, but now $(-1)^{\Theta/\pi}$ is replaced by the value $\pm 1$ of the square of the time-reversal operator acting on a single spin, which is still determined by the coefficients of the topological terms, in a way that depends on the symmetry group.

The theory of connections is at the very core of differential geometry. Discovered by Levi-Civita and Christoffel and later studied by Cartan, Koszul, and others, connections appear in their most general form under the name of Ehresmann connections. An Ehresmann connection consists of a splitting of the tangent bundle of a submersion into the vertical sub-bundle and a given horizontal distribution. In this paper, we generalize Ehresmann connection to a categorical setting called tangent categories. Initially introduced by Rosický in 1984 and later generalized by Cockett and the first author in 2014, tangent categories provide a categorical framework to study geometry that extends well beyond smooth manifolds, including algebraic geometry and non-commutative geometry. In this paper we introduce and study Ehresmann connections in the context of tangent categories. We give various equivalent formulations in term of full and abstract connections and prove that they generalize Koszul connections. We also define parallel transport and curvature for such connections, and prove the structural equation and the Bianchi identity for the curvature.

Constrained simulation optimization (CSO) is a general framework for optimizing stochastic systems under performance constraints. It arises widely in practice where objective and constraint evaluations are available only through noisy simulation outputs. Compared with the unconstrained setting, the lack of accessible analytical gradients for simulation-based constraints makes it more challenging to develop efficient solution methods and establish non-asymptotic guarantees. To address this gap, we propose a novel single-loop algorithm, called min-max gradient search (MGS), which integrates a primal-dual framework with stochastic gradient estimators. Unlike conventional stochastic approximation methods based on gradient descent for solving simulation optimization problems, such as Zhou and Bhatnagar (2017) and Hu and Fu (2025), MGS performs alternating gradient descent and ascent on the primal and dual variables, which improves the objective while penalizing constraint violations. For the first time, we establish a finite-time convergence guarantee for single-loop CSO algorithms by showing that MGS converges to a stationary solution (a Karush-Kuhn-Tucker point under mild conditions) at a rate of $\tilde{O}(T^{-1/3})$, where $T$ is the number of iterations. Numerical experiments on a serial queuing system and a 2000-dimensional optimization problem demonstrate the superior performance and scalability of MGS.

We study $L^2$-averaged discrepancies of finite sequences of points in the torus $\mathbb{T}^d$ with respect to translated and dilated copies of convex bodies with non-smooth boundary. Under suitable anisotropic assumptions on the decay of the Fourier transform of the body, we prove matching lower and upper bounds for the averaged discrepancy, obtaining the rate $ N^{1 - \frac{d+1}{d^2+d-1}}$. This yields an intermediate regime between smooth convex bodies and polytopes and recovers the known exponent $2/5$ in dimension $d=2$. The argument relies on harmonic analysis techniques combined with averaging procedures adapted to the anisotropic setting. As an application, we analyze a class of convex bodies exhibiting mixed geometric features, including flat regions, curved parts, and edges.

In this article, a new type of multiple access, termed Environment-Division Multiple Access (EDMA), is introduced and its interaction with AI-native communication networks is illustrated. In particular, the key properties of EDMA, such as utilizing the features of wireless propagation environments, integrating advanced flexible antennas, and proactively reconfiguring propagation environments, are described. The article also illustrates two types of applications of AI tools to multiple access, namely AI-assisted EDMA and AI-native EDMA. Finally, open problems and important directions for future research in AI-assisted EDMA are discussed.

We introduce new contraction conditions based on classical inequality between arithmetic and geometric means. By incorporating an auxiliary semimetric $\delta$, we define arithmetic-geometric mean, multiplicative-type, and additive-type contractions. Connections between these types of contractions are found. Fixed point theorems are proved in the case of continuity of the above mentioned contractions. Under suitable regularity conditions on $\delta$ (such as being d-regular, strongly d-regular, or d-lower bounded) we obtain constructive corollaries. Various examples demonstrating our results are constructed. It is shown that with certain caveats fixed point theorem for additive-type mappings is equivalent to the fixed point theorem for perturbed metric spaces, which were recently introduced by M. Jleli and B. Samet.

We provide sufficient conditions for comparing $m$-generalized order statistics with respect to the increasing concave, increasing convex, and star-shaped stochastic orders. These conditions allow us to rank classical order statistics, selected censored type-II order statistics, and records. They depend on both the parameters of the generalized order statistics and the underlying distribution. Rather than assuming a specific parametric form, we adopt a nonparametric approach and assume some stochastic transform-ordered property, that is, some suitable shape condition. This framework encompasses many relevant classes of distributions that are related, via transform order, to the generalized and the negative generalized Pareto distribution.

We study the semilinear wave equation $ V(x) \partial_t^2 u + d(t) M(x,\nabla_x) u=\tilde{V}(x) \tilde{d}(t) |u|^{p-1}u$ on $ \mathbb{R}^N \times \mathbb{R}$ and show the existence of solutions which are localized in space and in time, called rogue waves, by means of variational methods. We introduce an energy functional on a suitable Hilbert space, and provide sufficient conditions on the coefficients $V, \tilde{V}, d, \tilde{d}$, the elliptic operator $M$ and $p>1$ for the existence of a critical point. Our approach is based on a detailed analysis of the wave type operator and in particular its spectral properties. Further regularity considerations show that critical points are weak solutions to our equation. Moreover, we provide examples of the coefficients and the elliptic operator which satisfy our assumptions.

In this paper we review some analytic results on the dynamics of the FPU system. In the first part of the paper, having in mind that the FPU Hamiltonian and the Toda Hamiltonian are close each other, we present some results on the action angle variables of the Toda system and deduce some stability properties for the dynamics of the FPU system. We first focus on the case of finitely many particles and then we study the limit $N\to\infty$. We present also some results on the continous limit of the Toda chain showing that it is well described by a couple of KdV equations. Then we study directly the dynamics of the function interpolating the FPU system and show that the dynamics is Hamiltonian and that the Hamiltonian is very close to a function of the first three Hamiltonians of the KdV hierarchy. In the second part of the paper we present some results valid in the thermodynamic limit, according to which the time autocorrelation functions of some suitably constructed observables decay slowly implying lower bounds on the thermalization times of the system.

We construct probabilistic strong solutions to the cubic Schrödinger equation on the three-dimensional ball with radial initial data, which is a significant improvement of a result by Bourgain–Bulut. These solutions lie in the supercritical regime with respect to the probabilistic scaling introduced by Deng–Nahmod–Yue. We achieve this result through gauge transformations that do not modify the equation, combined with a refined modulation analysis using random averaging operators.

We continue the study of fast functions, computable by linear-size circuits, that share useful properties of random functions. Motivated by cryptographic applications, we generalize and improve on previous results in this area, obtaining the following results: - For any constant $t$, we construct a fast $t$-wise independent hash function with algebraic degree $\log_2 t$ (over $\mathbb F_2$), simultaneously optimizing both asymptotic circuit size and degree. - We simplify and improve a recent construction (ITCS 2026) of a family of fast codes with fast duals, both meeting the Gilbert-Varshamov bound. Unlike the previous construction, our construction has negligible failure probability, can accommodate general fields and rates, supports a systematic encoding, and admits fast universal encoders. - We strengthen the above to support stronger random-like properties, such as optimal combinatorial list-decoding. This is achieved by constructing, for any constant $t$, a family of fast linear functions that map any $t$ linearly independent inputs to uniform and statistically independent outputs. Prior to our work, this was only known for $t=1$. We demonstrate the usefulness of the above results to cryptography. This includes the first nontrivial protocols for perfectly secure multiparty computation whose circuit complexity scales linearly with the number of parties, as well as protocols for computing encrypted matrix-vector products with optimal asymptotic circuit complexity.

Estimating the logarithm of the determinant of large sparse symmetric positive definite matrices is an important problem in numerical linear algebra, machine learning, Gaussian processes, and uncertainty quantification. We propose a range-deflated stochastic Lanczos quadrature method, termed DSLQ, for matrix-free log-determinant estimation inspired by Hutch ++. The method constructs a randomized approximation space directly from products with a scaled version of the original matrix, and uses the associated orthogonal projector to decompose the trace of matrix logarithm into projected and complementary contributions. Both contributions are evaluated through the Gauss-Lanczos quadrature, avoiding explicit formation of the matrix logarithm or a low-rank matrix-function surrogate. Since the approximation space is generated from the matrix rather than from the matrix logarithm, the standard Hutch++ approximation bound does not apply directly. We therefore derive an error analysis specific to the DSLQ construction, quantifying the residual induced by the matrix-generated subspace and propagating this effect through the stochastic residual estimator and the Lanczos quadrature approximations. Extensive numerical experiments on both large sparse synthetic matrices ( Gaussian Markov Random fields, Bayesian inverse problem.) and large-scale real-world matrices, demonstrate that our method substantially reduces computational time while retaining competitive accuracy, thereby confirming its effectiveness, scalability, and favorable accuracy-cost trade-off for large-scale log-determinant estimation.

For smooth nonconvex-strongly-concave minimax problems, existing second-order methods share a common double-loop structure where the inner maximization is solved to sufficiently high accuracy before each second-order step. First-order methods are often adopted in the inner loops for scalability, but they also undermine the condition-insensitivity of second-order methods, limiting these methods to instances with mild conditioning. To resolve this issue, we propose a novel single-loop framework based on an equivalent regularized minimization reformulation of the original problem. By deriving a new adaptive cubic-quadratic majorization to dynamically absorb the non-Lipschitz components of the reformulated Hessian, we establish a regularized Newton method with robust theoretical guarantees across multiple settings. For deterministic problems, our single-loop method matches the $\mathcal{O}(\varepsilon^{-1.5})$ global iteration complexity of the existing double-loop second-order methods, while automatically achieving a local superlinear rate that is unavailable in existing works due to the inner-loop bottleneck. For the stochastic setting, we achieve $\mathcal{O}(\varepsilon^{-3})$ gradient and $\mathcal{O}(\varepsilon^{-2})$ Hessian complexities by integrating a recursive variance reduction, strictly improving those of the double-loop methods by $\mathcal{O}(\varepsilon^{-0.5})$ factors. In both deterministic and stochastic experiments, our methods significantly outperform the benchmarks, offering substantial speedups over the double-loop methods even under mildly conditioned instances. As a byproduct of our analysis, we close a gap in stochastic second-order methods for nonconvex minimization, where the best known result contains a nontrivial technical issue.

We study the distribution of cokernels of Haar-random matrices over the $p$-adic integers with prescribed zero patterns, motivated by the Cohen–Lenstra heuristics. A central feature of our approach is that the asymptotic cokernel distribution is governed by the reductions modulo $p$ of these matrices, viewed as random matrices over the finite field $\mathbb{F}_p$. For several families of support patterns arising from stair-shaped zero regions, including general stair-shaped patterns, band matrices, and matrices with two symmetric stair-shaped zero regions, we show that convergence of the cokernel distribution to the Cohen–Lenstra distribution is equivalent to an asymptotic nonsingularity condition over $\mathbb{F}_p$. We further propose a conjecture for general support patterns and give examples showing that analogous rank-$r$ criteria fail for $r\ge 1$.

Nikiforov conjectured that, for every fixed $k\ge2$ and all sufficiently large $n$, the unique $n$-vertex $C_{2k+2}$-free graph with maximum adjacency spectral radius is $S^+_{n,k}$, where $S_{n,k}=K_k\vee\overline K_{n-k}$ and $S^+_{n,k}$ is obtained from $S_{n,k}$ by adding one edge inside the independent part. Cioabă, Desai and Tait proved this conjecture for $n\ge k^{O(k)}$. Later, Li and Ning raised the problem of determining the optimal exponent $\gamma=\gamma(k)$ such that the same conclusion holds for $n\ge \Omega(k^{\gamma(k)})$. We prove a stronger uniform theorem for Nikiforov's matrices $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$. More precisely, for every $\epsilon>0$ there are constants $C_\epsilon$ and $k_\epsilon$ such that for all $0\le\alpha\le1-\epsilon$, $k\ge k_\epsilon$ and $n\ge C_\epsilon k$, every $n$-vertex $C_{2k+2}$-free graph $G$ satisfies $\rho_\alpha(G)\le\rho_\alpha(S^+_{n,k})$, with equality if and only if $G\cong S^+_{n,k}$. In particular, the case $\alpha=0$ answers the problem of Li and Ning, and the $A_\alpha$-spectral even-cycle threshold is linear in $k$, uniformly for all $\alpha$ bounded away from $1$. Our proof introduces a weighted rooted Erdős–Gallai type path lemma, which may be of independent interest in Perron-vector methods for spectral extremal graph problems. The same method also yields asymptotically tight $A_\alpha$-spectral bounds for two local forbidden-subgraph families, namely $(K_1\vee P_\ell)$-free graphs and $F_s$-free graphs, where $F_s$ denotes the friendship graph.

We present a differentiable framework for end-to-end mutual information (MI) optimization over linear Gaussian directed acyclic graphs (DAGs). The framework targets network-wide design under global constraints, such as a total transmit power budget, and covers MIMO precoding, amplify-and-forward relays, RIS-aided channels, and branching/merging topologies within a common linear Gaussian model. Its core ingredient is a K-recursion that analytically propagates all node-pair covariances along the DAG in topological order, including non-adjacent cross-covariances that are necessary for correctly handling branching and merging paths. The resulting covariances yield a closed-form log-determinant expression for the end-to-end MI as a smooth function of the controllable factors. Complex-valued reverse-mode automatic differentiation on this K-recursion then returns the exact Wirtinger gradient at every controllable factor in a single backward sweep, and projected gradient ascent (PGA) is used to maximize the MI under the global constraints. Because no closed-form gradient expression per topology is required, the same topology-agnostic implementation applies to any linear Gaussian DAG. A single topology-agnostic implementation is applied to four representative DAG classes: single-link MIMO, a diamond DAG, a two-hop AF relay, and input-covariance shaping. The same implementation reaches the classical water-filling optimum in the settings where it is available and yields MI improvements in non-single-link topologies without using topology-specific gradient formulas. A further experiment on a multi-layer Gaussian network (11 nodes, 5 layers) illustrates applicability to nontrivial multi-layer topologies for which no closed-form gradient is available.

We disprove the visibility conjecture in the plane and prove the sharp upper bound for the almost-sure dimension of visible parts. Precisely, let $K \subset \mathbb{R}^{2}$ be a compact set. For $\sigma \in S^{1}$, let $\mathrm{Vis}_{\sigma}(K) \subset K$ be the visible part of $K$ in direction $\sigma$. We prove that $\operatorname{dim}_{\mathrm{H}} \mathrm{Vis}_{\sigma}(K) \leq \tfrac{3}{2}$ for $\mathcal{H}^{1}$ almost every $\sigma \in S^{1}$. This is sharp: we construct a compact set $K \subset \mathbb{R}^{2}$ such that $\operatorname{dim}_{\mathrm{H}} \mathrm{Vis}_{\sigma}(K)\geq \tfrac{3}{2}$ for all $\sigma \in S^1$.

Let $(T(n),\Omega)$ be the covering of the generalized Kronecker quiver $K(n)$, where $\Omega$ is a bipartite orientation. Given a regular Auslander–Reiten component $\cD$ of $\modd(T(n),\Omega)$, we introduce two invariants: the width $\cW(\cD)$ and the number of flow modules $b(\cD)$. We show that $\cW(\cD)\geq \frac{b(\cD)+1}{2}$. In particular, we get $\{\cW(\cD)| \cD \text{ is a regular component} \}=\mathbb{N}$.

This paper explores stability dichotomies of thermoelastic Timoshenko-Boltzmann systems with hereditary memory, comparing Fourier's parabolic (no thermal relaxation) and Cattaneo's hyperbolic (with thermal relaxation) heat conduction via semigroup theory and spectral analysis. The Fourier-semigroup achieves exponential stability through the $\delta$-condition plus a specific parameter condition, while sole $\delta$-condition ensures $1/2$-order polynomial stability. The Cattaneo-semigroup retains this polynomial stability but requires a thermal relaxation-included modified parameter condition for exponential stability. As thermal relaxation time approaches zero, Cattaneo's modified parameter converges to Fourier's, formalizing their asymptotic connection. Results show thermal relaxation alters exponential criteria but preserves polynomial decay, supporting stability analysis under time-dependent thermal loading in structural mechanics.

We investigate monomial ideals satisfying the non-pure dual exchange property, a notion introduced in connection with componentwise polymatroidal ideals. Our contributions are twofold. First, we show that in two variables, every integrally closed monomial ideal satisfies this property; as a consequence, we characterize polymatroidal ideals in two variables. Second, for strongly stable (Borel) ideals in three variables, we establish a practical criterion: it suffices to verify the defining condition only for the Borel generators, and this verification reduces to simple inequalities involving the degrees in the second and third variables.

Zhou et al. (2025) proposed a distance Laplacian analog of Brouwer's conjecture on partial sums of Laplacian eigenvalues, asserting that for any connected graph $G$, $\sum_{i=1}^r \partial_i^L(G)\le W(G)+\binom{r+2}{3},$ where $\partial_i^L(G)$ are the eigenvalues of the distance Laplacian matrix and $W(G)$ is the Wiener index. We prove this inequality for three broad classes of graphs, thereby improving and extending existing results. First, we prove that all connected graphs of diameter at most $D$ satisfy the inequality once the order $n$ satisfies $n\ge\lceil\frac49(D+1)^3\rceil$. Second, we show that the inequality holds for every diameter-$2$ graph with the only exceptions being $K_{1,3}$ at $r=2$ and $K_{1,4}$ at $r=3$. Third, we prove that if the maximum degree is $\Delta(G)=n-k$, then the inequality holds for all $n\ge N(k)$, where $N(2)=10$ and $N(k)=\lceil 5(k-1)^{3/2}\rceil$ for $k\ge 3$. Our proofs rely on decomposing the distance Laplacian matrix into Laplacian matrices of auxiliary graphs whose edges are vertex pairs at distance at least a prescribed value, together with classical eigenvalue inequalities.

We give a variational formulation of the continuous adjoint Kutta condition for two-dimensional subcritical potential flow, with emphasis on the Kutta condition and the role of the wake. We show that the adjoint Kutta condition can be imposed by a penalty term evaluated at the trailing edge, with the corresponding Lagrange multiplier determined by stationarity of the Lagrangian with respect to circulation, and that a wake treatment is not required. Some of the implications of these results for adjoint consistency are also briefly discussed.

The present paper is devoted to studying derivations and local derivations on the Euclidean Lie algebras $\mathfrak{e}(n)$. We give a complete desrciption of the derivation algebra of $\mathfrak{e}(n)$ for $n\geq 4$. Furthermore, we prove that every local derivations on the $\mathfrak{e}(n)$ is a derivation.

We study finite adaptability for decision-making under uncertainty, where a small set of candidate solutions is prepared in advance and the best response is selected after uncertainty is realized. While existing methods have made significant progress on exact formulations, scalability remains a persistent challenge due to (i) the combinatorial nature of assigning decisions to uncertainty realizations, and (ii) the joint optimization of uncertainty set partition and subsequent decisions. We propose a framework that makes the partition of the uncertainty set explicit and uses polyhedral partitions as the basis for policy design. Under mild regularity conditions and for general risk measures, we show that such policies converge to the optimal fully adjustable policy as the number of regions increases. Building on this result, we develop a parametric partition framework that allows flexible policy design with tractable reformulations for both robust and stochastic finite adaptability problems. To improve scalability, we introduce an approximate-learn-parallel framework that integrates partition learning with parallel optimization while preserving solution robustness. Computational experiments on classical testbeds in both robust and stochastic settings show that the proposed method scales to larger instances and yields competitive policy performance.

In this paper we utilize a convex integration scheme to construct non-trivial weak solutions to the $k$-generalized KdV equation which lie in $$ \bigcap_{\epsilon > 0} C_t^0 L_x^{k-\epsilon}([0,1] \times \mathbb{T}) $$ and, when $k \ge 3$, it may also be chosen in \[ \bigcap_{\epsilon >0} C_t^0 H_x^{\frac{1}{2} - \frac{1}{k} - \epsilon}([0,1] \times \mathbb{T}) \] attaining identically $0$ initial data. Since our solutions do not lie in $C_t^0 L_x^k$, this requires introducing a new notion of weak solution, which is in fact stronger than the classical notion of a weak solution when the nonlinearity is integrable. This result shows that a necessary condition for unconditional uniqueness for $k$-gKdV is that the nonlinearity lies in $C_t^0L^1_x$. In the case of KdV this is in fact also sufficient.

Let $W(n)$ be the finite-dimensional simple Lie superalgebra of fundamental type in the Cartan type series of Kac's classification result \cite{Kac77} over an algebraically closed field of characteristic $0$. Let $\mathbf{g}$ be the graded-zero part of $W(n)$ which is isomorphic to $\mathfrak{gl}(n)$. In the first part of this paper, following the basic idea of taking the “minimal" parabolic subalgebra $\mathsf{P}$ as a working platform in \cite{DSY} we introduce the Whittaker category $\mscrw$ for representations of $W(n)$ associated with a nilpotent element $e$ in $\mathbf{g}_0$ and with $W(n)_{-1}$. This Whittaker category turns out to be close to the classical Whittaker category McDowell and Miličić-Soergel studied in \cite{Mc} and \cite{MS}, respectively (or see \cite{Back}). We finally classify the simple objects in $\mscrw$. In the second part, we introduce the finite $W$-algebra associated with $e$, we then establish a generalized Skryabin's equivalence between the representation category of the finite $W$-superalgebra and the category $\mscrw'$ of so-called weakened Whittaker modules over $W(n)$. Here $\mscrw'$ naturally contains $\mscrw$ as a full subcategory.

We study the dynamics of planar sets under area-preserving crystalline curvature flow. We prove under mild assumptions on $\phi$ that the flat flow solution from regular initial data coincides with a classical ODE evolution, extending the results of \cite{Almgren1995} to the area-preserving setting. We also show that for arbitrary initial data, the flat flow converges exponentially in time to a disjoint union of Wulff shapes, and under a non-bubbling assumption, the flow eventually becomes regular. Both of these results are novel for area-preserving crystalline flow of general sets, i.e. without assuming geometric properties such as convexity or star-shapedness. A key ingredient of independent interest is that planar almost-minimizers are Lipschitz $\phi$-regular, which we prove by exploiting a sharp minimality estimate for distinguished line segments, as opposed to the excess decay argument given in \cite{Ambrosio2002regularity}. The novelty of our approach lies in the application of $\phi$-minimal barriers for energy competition arguments, both for the geometric rigidity of the discretized flat flow and for the regularity of almost-minimizers.

In this paper we describe a communication-strategy study for multi-GPU three-dimensional finite-difference time-domain computation with convolutional perfectly matched layer boundary conditions using CUDA. The metrics used to determine the most effective implementation include runtime, throughput in millions of output points per second, strong-scaling efficiency, CPML overhead, host-staged versus direct GPU-to-GPU exchange speedup, and enlarged-ghost speedup. On a single NVIDIA Quadro RTX 6000 GPU, the CPML implementation sustains 2,889–3,290 million output points per second with less than 1% boundary-layer overhead, providing the single-GPU baseline for the multi-GPU study. The results show that direct GPU-to-GPU peer exchange is the dominant optimization with a 2.46–2.76$\times$ speedup over host-staged exchange, while enlarged ghost regions give only modest benefits because the reduced communication frequency is partly offset by redundant computation and additional memory traffic. On NVIDIA Quadro RTX 8000 GPUs, the implementation gives up to a 1.51$\times$ speedup on two GPUs for the tested strong-scaling cases, while four GPUs enable larger grids that approach or exceed single-GPU memory capacity.

Let $d_r(n)$ be the maximum, over all binary phylogenetic trees with $n$ leaves, of the minimum number of $r$-state characters required to define the tree. Semple and Steel proved that $d_r(n)\geq\lceil(n-3)/(r-1)\rceil$, and Bordewich and Semple proved that equality holds for each fixed $r$ and all sufficiently large $n$. We study the corresponding threshold $n_r$, the least $N$ for which equality holds for every $n\geq N$. The Bordewich–Semple construction yields an explicit polynomial upper bound of order $O(r^5)$ for this threshold. We prove the near-linear estimate \[ 3r+1\leq n_r\leq \ceil{64(r-1)\log_2(r+1)}+3\qquad(r\geq4). \] The proof constructs, for every binary phylogenetic tree with $m=n-3$ internal edges, a linked quartet certificate whose conflict graph has maximum degree at most $16\lceil\log_2(m+2)\rceil+4$. Equitable coloring then packs the certificate into exactly $\lceil m/(r-1)\rceil$ $r$-state characters once $m\geq\lceil64(r-1)\log_2(r+1)\rceil$. We also include the lower bound $n_r\geq3r+1$, obtained from the snowflake obstruction, and state the natural conjecture that this lower bound is the exact threshold for all $r\geq4$. The conjectural endpoint is consistent with the known small-state thresholds: $n_4=13$ and $n_5=16$, while the cases $r=2,3$ are also explicitly classified.

This paper presents a successive convexification framework for trajectory optimization under continuous-time Signal Temporal Logic (CT-STL) specifications. The framework employs generalized mean-based robustness (GMSR), a smooth and exact parameterization of discrete-time STL, as a logical building block for constructing differentiable CT-STL constraints in optimal control. It is integrated with time-dilation for free-final-time problems, finite-dimensional control parameterization, multiple-shooting discretization of the dynamics, and a convergence-guaranteed sequential convex programming method, prox-convex, to solve the nonconvex program. The main CT-STL realization embeds temporal aggregation into augmented continuous-time dynamics. This augmentation-based construction is largely transcription-independent, can be incorporated into existing optimal-control pipelines with minimal structural changes, and enables smooth CT-STL parameterizations with accuracy controlled by a user-selected regularization parameter. We also discuss a complementary dense-time realization that evaluates CT-STL formulas directly on the integration subnodes used for dynamics discretization, yielding a smooth and exact parameterization on the numerical trajectory representation, up to the accuracy of the integration scheme. The proposed GMSR-based formulations mitigate the locality and gradient-masking behavior of standard quantitative semantics and therefore provide a favorable landscape for gradient-based trajectory optimization. The framework is demonstrated through trajectory-optimization examples for a double-integrator system with continuous-time \always{}, \eventually{}, and \until{} specifications, and a 6-DoF quadrotor flight problem with combined \always{}, \implication{}, and \eventually{}-type specifications. The implementation is available at https://github.com/UW-ACL/TrajOpt_CT-STL.