Papers

Codes over non-unitary rings have been studied recently. In particular, codes over the commutative non-unitary ring $I_p$ (in the classification of Fine) of order $p^2$ where $p$ is a prime are being considered. For $p=2$ (resp. $p=3$), three categories of codes over $I_p$ have been studied: self-orthogonal codes, quasi self-dual codes, and self-dual codes over $I_p$. Using some related mass formulas and building-up constructions, classifications of these codes have been done up to the permutation equivalence (resp. the monomial equivalence) for certain small lengths. In this paper, we take the prime $p=5$ and consider the ring $I_5$. We introduce the notion of linear codes over $I_5$. We also define the same three categories of linear $I_5$-codes, study the structures of these $I_5$-codes and relate them to their associated residue and torsion codes. We classify the three categories of codes completely in lengths at most $4$ up to the monomial equivalence for a given type $\{ k_1 , k_2 \}$. Moreover, in the paper of Alahmadi et al. regarding the mass formula for self-orthogonal codes over $I_p$, mistakes in the classification of quasi self-dual codes over $I_5$ had been made such as incorrect automorphism group order of some codes or inconsistency with the mass formula for self-orthogonal codes over $I_p$ for length $n=2$ and type $\{ 1 , 0 \}$ and for length $n=3$ and type $\{ 1, 1 \}$. We correct and improve such results.

This paper provides two results that are useful in the study of the existence and the stability properties of almost periodic solutions for a given dynamical system. The obtained results are generalizations of recent results for periodic systems and are applied to the global entrainment problem in nonlinear time-invariant control systems. It is shown that local exponential stability for the unforced system and input-to-state stability with respect to small inputs can guarantee global entrainment to small almost periodic inputs. In this way, global entrainment is shown in Lotka-Volterra systems with a Volterra-Lyapunov stable interaction matrix. All results can be extended to the uniformly recurrent case.

We prove that heat flow generically produces Morse-minimal functions on round lens spaces. The result gives a rigorous explanation of a phenomenon first suggested by computational experiments: after high-frequency modes have been suppressed by diffusion, generic heat evolutions on lens spaces tend to settle into Morse functions with the smallest possible number of critical points. Precisely, for every lens space \(L(p,q)\) with \(p\geq2\), \(1\leq q\leq p/2\), and \((p,q)=1\), there is an open dense set of initial data \(f\in L^2(L(p,q),\mathbb R)\) such that the heat evolution \(e^{t\Delta}f\), where \(\Delta\) is taken in the non-positive sign convention, is, for all sufficiently large \(t\), a Morse function with exactly four critical points, of indices \(0,1,2,3\). The proof analyzes the asymptotic spectral expansion of the heat flow. In the most delicate case \(1<q<p/2\), the leading term is Morse–Bott with two critical circles, and the higher heat modes break these circles through a resonant Fourier mechanism. An arithmetic estimate shows that the fundamental reduced frequency dominates all higher multiples, and generic nonvanishing of the corresponding resonant coefficients gives the minimal critical-point count.

Jordan pairs and hermitian Jordan triples were discovered by mathematicians studying Jordan algebras, which describe the possible algebras of observables in quantum mechanics. We point out a striking correspondence between the doubly exceptional hermitian Jordan triple (the so-called “bi-Cayley” triple) and the structure of the Standard Model of particle physics. We also point out how ordinary quantum mechanics may be reformulated, and generalized, using hermitian Jordan triples.

We study Turán-type extremal problems for graphs containing a large $F$-sparse vertex set, meaning a vertex set whose induced subgraph contains few copies of $F$. For integers $r>s\ge 1$, we prove that if a $K_{r+1}$-free graph $G$ on $n$ vertices contains a set $M$ of size $m\ge \lceil sn/r\rceil$ such that $G[M]$ is $K_{s+1}$-free, then \[ e(G)\le m(n-m)+t_s(m)+t_{r-s}(n-m). \] We characterize the equality cases as the complete $r$-partite graphs whose vertex classes split into two balanced groups of total sizes $m$ and $n-m$, consisting of $s$ and $r-s$ classes, respectively. We also prove a color-critical extension for forbidden graphs that embed into a join of two edge-critical graphs, together with an asymptotic extension for general $H$-free graphs in which the prescribed large vertex set spans few copies of a fixed graph $F$ with $\chi(F)<\chi(H)$.

We present a least-squares weak Galerkin (LS-WG) finite element method for solving the Stokes equations on arbitrary polygonal and polyhedral meshes. By utilizing discrete weak derivatives on discontinuous polynomial spaces, the proposed framework naturally accommodates complex domain geometries and general partitions. Crucially, this least-squares formulation bypasses the traditional inf-sup (LBB) compatibility condition, transforming the standard indefinite saddle-point problem into an inherently symmetric and positive definite (SPD) discrete linear system. We establish the well-posedness of the numerical scheme and rigorously derive optimal-order error estimates in a custom discrete energy norm. Specifically, we prove convergence rates of $\mathcal{O}(h^k)$ for the discrete projection error and $\mathcal{O}(h^{k-1})$ for the global approximation error when employing polynomials of degree $k \ge 1$ for the velocity field and $k-1$ for the pressure. Extensive numerical experiments confirm these theoretical convergence rates, demonstrating the method's robustness, geometric flexibility, and overall efficiency.

Lawson showed that a non-totally geodesic Einstein minimal hypersurface in $\mathbb S^5$ is congruent to the Clifford hypersurface $\mathbb S^2(1/\sqrt2)\times \mathbb S^2(1/\sqrt2).$ It is also known, by work of Cartan and Ôtsuki, that a non-totally geodesic locally conformally flat minimal hypersurface in $\mathbb S^5$ is of Ôtsuki type, including the Clifford hypersurface $\mathbb S^1(1/2)\times \mathbb S^3(\sqrt3/2).$ In this paper we study closed minimal hypersurfaces $M$ in $\mathbb S^5$ satisfying $|W|^2=2|\mathring{\operatorname{Ric}}|^2,$ where $W$ is the Weyl tensor and $\mathring{\operatorname{Ric}}$ is the trace-free Ricci tensor. We call this the Euler-balanced condition. We prove that such a hypersurface is either totally geodesic or congruent to the Cartan minimal hypersurface.

We define analogs of superconformal algebras over an arbitrary commutative associative superalgebra. In the finitely generated case the universal central extensions of these algebras are finitely presented. In particular, we show that all known superconformal algebras are finitely presented.

Bayesian methods provide a natural framework for estimating a parameter in non-linear inverse problems and quantifying uncertainty in the estimation. However, when the forward model for such non-linear inverse problems is given by some Partial Differential Equation (PDE), Bayesian inference is typically carried out by resorting to MCMC methods. Since each MCMC iteration requires solving a PDE, these methods become computationally expensive and are often impractical for large-scale imaging problems. In this work, we develop a computationally efficient Bayesian framework for two such nonlinear imaging inverse problems: Quantitative Photoacoustic Tomography (QPAT) and Electrical Impedance Tomography (EIT). Building on a recently proposed two-stage pushforward methodology, we first formulate a Bayesian regression problem for an auxiliary variable whose posterior is available in closed form. This posterior is then pushed forward through a deterministic reconstruction map to obtain a posterior on the unknown parameter, avoiding MCMC sampling. We give a rigorous measure-theoretic justification to interpret the induced posterior as a Bayesian posterior and derive posterior contraction rates for both QPAT and EIT. Numerical results show that the proposed method provides accurate reconstructions and reliable uncertainty estimates at a arguably lower computational cost than standard Bayesian approaches.

We construct spatially local paracontrolled solutions on $\mathbb{R}^3$ for the stochastic wave–Klein–Gordon system \[ (\partial_t^2-c_{\mathrm{W}}^2\Delta)u=uv+\xi_{\mathrm{W}},\qquad (\partial_t^2-c_{\mathrm{K}}^2\Delta+m^2)v=uv+\xi_{\mathrm{K}}, \] with $c_{\mathrm{W}}\ne c_{\mathrm{K}}$. The independent forcings are white in time and stationary in space, generated by Fourier multipliers of respective orders $\beta_{\mathrm{W}},\beta_{\mathrm{K}}$, and we assume $\beta_*:=\max\{\beta_{\mathrm{W}},\beta_{\mathrm{K}}\}<1/8$. The main analytic step is the construction of the mixed random operators $I_a(w\prec\Psi_b)\circ\Psi_c$. At finite cutoff they decompose into a centered second Gaussian chaos and, when $b=c$, a deterministic covariance contraction. In the latter case the two propagation channels are different; the resulting low–high phase gap gives the diagonal shell bound $N^{-1+2\beta_b}$. The centered terms are controlled through four Schatten flattenings of a continuous-frequency kernel with two physical localizers. Weighted phase-layer estimates also construct the first Picard and cubic stochastic terms. A localized auxiliary system, a finite-cutoff Sobolev bootstrap, and finite propagation yield a compatible family of local solutions with compact-dependent measurable lifetimes. We prove uniqueness in the stated paracontrolled class, local Lipschitz dependence on the enhanced data and Cauchy data, independence of the localization pair, and convergence of the spectral approximations and nonlinear sources. Fixed cutoff profiles converge almost surely along the full sequence; arbitrary admissible cofinal cutoffs converge in probability up to the reference lifetime.

For nonlinear Fokker-Planck equations with mobility, the Wasserstein gradient flow structure is described by the generalized relative entropy as the energy functional and the modified Wasserstein metric $W_h$ as the associated metric structure. This work investigates the nonlinear effects induced by mobility and establishes the corresponding inequalities. For nonlinear diffusion, we establish a logarithmic Sobolev inequality, which yields the convergence rate of the free energy functional and the Talagrand inequality. By further exploiting the relationship between the weighted homogeneous Sobolev norm and the $W_h$ metric, we derive an HWI inequality relating the relative entropy, the $W_h$ metric, and the Fisher information. In the case of mobility dependent drift and linear diffusion, the convergence rate in the $W_h$ metric is also obtained by applying the Girsanov theorem.

A compact and high order HWENO scheme using ADER (Arbitrary high order using DERivatives) time discretization is developed for hyperbolic conservation laws on the triangular mesh, which is the extension of the work on the structured mesh (Luo et. al. (2024) \cite{luo2023}). The Lax-Wendroff procedure is employed to convert time derivatives to spatial derivatives. Thanks to this, the cell averages of the derivatives of the solution can be obtained by the time accurate solution as Gaussian points along the cell interfaces through the Green-Gauss theorem instead of by the evolution solution directly in the conventional HWENO methods. Comparing with the existing Runge-Kutta HWENO (RK-HWENO) method on the unstructured mesh (Zhao et. al. (2025) \cite{zhao2025}), the new method has the following advantages. Firstly, the RK-HWENO method must solve the additional equations for reconstructions and time advancing, which is avoided for the new method. Secondly, the HWENO reconstruction in the new method is performed once per time step and is different from the RK-HWENO method, in which the reconstruction is performed several times every time step. Because of these advantages the new method is more efficient than the RK-HWENO method with smaller numerical errors and less computational costs. Besides, comparing with the existing ADER-WENO methods \cite{dumbser20071,dumbser20072} under the same order of accuracy, the stencil of the new method is more compact since the both the function and its first derivative values are used in the reconstruction of the HWENO schemes. Numerical examples demonstrate that the new method can achieve the high order for smooth solutions both in space and time, keep non-oscillatory near discontinuities.

We develop a systematic approach to quantify $L^2$-relaxation times for non-reversible Markov processes based on the singular value gap of the generator introduced by Chatterjee. The inverse of the singular value gap is equivalent to the relaxation time of the time-averaged transition semigroup. We show that, moreover, the inverse singular value gap of the two-point motion is equivalent to the relaxation time of the non-averaged transition semigroup, and that it provides a lower bound on the usual spectral gap of the generator. We then introduce a method for deriving lower bounds on singular value gaps for Markov processes with degenerate noise that is based on the concept of a first- and second-order collapse of the generator. It follows ideas from hypocoercivity developed in a previous series of works but is simpler and more broadly applicable. In contrast to previous results, it includes settings with non-vanishing first-order collapse, and thus applies directly to Markov chains (in continuous time), but also to diffusion processes and piecewise-deterministic Markov processes. Our approach yields sharp upper and lower bounds for several classes of examples. First applications include the proof of a conjecture by Diaconis and Miclo on a square-root speed-up for lifted random walks on abelian groups, as well as bounds on relaxation times of switching flows perturbed by noise and of non-reversible diffusion processes.

Numerical integration is a cornerstone of various scientific computing applications, such as engineering simulations and model evidence computations in probabilistic machine learning. Bayesian Quadrature uses Gaussian process surrogates that explicitly encode structural assumptions about the integrand to obtain integral estimates with quantified uncertainty. These surrogates are predominantly based on stationary covariance functions, which results in model misspecification for integrands exhibiting nonstationary behavior. We tackle this issue through an adaptively growing, tree-based partition of the integration domain into local stationary models. Our method recombines the local integral estimates through a hierarchy of GP conditioning that reintroduces cross-subdomain correlations, while model selection criteria control the tree growth to avoid unnecessary partitioning. The resulting algorithm is simple, requires no MCMC, and adapts its evaluation budget to local integrand complexity. On benchmark integration problems and a model evidence computation for an epidemiological model, Hierarchical Bayesian Quadrature achieves substantial gains over standard Bayesian Quadrature on nonstationary integrands while matching its performance on stationary ones.

Variance reduction is indispensable in Byzantine-resilient decentralized stochastic optimization over multi-agent systems (MASs), as it effectively mitigates gradient noise to facilitate the resilient aggregation process. However, most existing Byzantine-resilient decentralized variance-reduced (VR) stochastic gradient algorithms rely on random data sampling, which proves inefficient in data-scarce yet high-dimensional tasks, for instance, image deblurring. This paper aims to pursue an alternative approach that achieves variance reduction via gradient sketching. We first formulate a class of structural risk minimization (SRM) problems over time-varying networks, where the local objectives are not necessarily decomposable and their gradients may be inaccessible. To solve the SRM problems in a decentralized manner, we incorporate a localized gradient-sketching technique into a preconditioned stochastic proximal gradient framework, proposing a novel gossip-based decentralized algorithm with variance reduction, dubbed Gossip-SEGA. To further enhance its resilience, we substitute the weighted-average aggregation in Gossip-SEGA with a norm-penalized approximation, thereby developing a Byzantine-resilient decentralized algorithm, namely RED-SEGA. Theoretically, we derive sufficient conditions for both consensus among reliable agents and (sub)linear convergence rate of RED-SEGA over time-varying networks in the presence of Byzantine agents with unknown identities. This is challenging, as the convergence and consensus errors must be bounded within a transformed variable-metric geometry, where the inherent complexities are further compounded by adversarial Byzantine agents and time-varying networks. The effectiveness and resilience of RED-SEGA are validated by solving a constrained least-squares problem and an image deblurring task under various Byzantine attacks.

We study sums of Kloosterman sums formed with a modular symbol. Employing Tauberian methods, we first give an estimate for a (Riesz) sum of Ramanujan sums formed with a modular symbol. We further define a zeta function that is analogous to the Selberg zeta function, establish its continuation to $\Re(s)>1/2$, give estimates for its growth and use this to prove a cancellation statement for sums of these twisted Kloosterman sums. We explain the connection of this construction to the eigenvalue 1/4 problem and formulate an analogue of Linnik's conjecture. Finally, we present numerical evidence that there is cancellation and also that the Kloosterman sums with a modular symbol are not correlated with classical Kloosterman sums.

We construct smoothly bounded pseudoconvex or convex domains in $\mathbb{C}^n$ whose weakly pseudoconvex loci realize prescribed closed sets.

We compute the Poisson derivations and the first Poisson cohomology group of uniparameter Poisson Nilpotent Algebras (PNA for short) under the hypothesis that Poisson-normal elements are Poisson-central. This applies in particular to certain Poisson algebras associated to Bott–Samelson varieties.

For an integer $L\ge2$, let $a=\lfloor L/2\rfloor$. Let $H(n,L)$ be the join of $K_a$ and an independent set of order $n-a$, with one extra edge in the independent set when $L$ is odd. We prove that, for fixed integers $q\ge4$ and $L>q$, and for all sufficiently large $n$, the graph $H(n,L)$ maximizes the number of copies of $C_q$ among all $n$-vertex graphs of circumference at most $L$. This settles a conjecture of Zhu, Győri, He, Lv, Salia and Xiao~[Bull. Lond. Math. Soc. 55 (2023)]. For even $q\ge6$, we also prove the boundary case $L=q$. We further determine the corresponding maximum when a long path is forbidden.

We study ideals in the semiring $\mathbb{N}$ of natural numbers, with a focus on those which are lost when extending from $\mathbb{N}$ to $\mathbb{Z}$. This leads to a new perspective on the classical theory of numerical semigroups, including the introduction of a natural multiplicative structure. We prove that unique factorization of ideals fails in $\mathbb{N}$ on several levels, introduce a handful of new tropical multiplicative invariants of numerical semigroups, characterize integral closures of ideals in terms of Newton polygons, and analyze the behavior of classical numerical semigroup invariants with respect to the product operation.

We study real interpolation for adapted sequence spaces with variable exponents. Let $(\Omega,\mathcal{F},\mathbb{P};(\mathcal{F}_n)_{n\geq 1})$ be a filtered complete probability space, let $p(\cdot)\in\mathcal{P}(\Omega)$, and let $0<q\leq\infty$ and $0<\theta<1$. We prove that \[ \left(L^{\mathrm{ad}}_{p(\cdot)},L^{\mathrm{ad}}_{\infty}\right)_{\theta,q} = L^{\mathrm{ad}}_{\widetilde{p}(\cdot),q}, \qquad \frac{1}{\widetilde{p}(\cdot)} = \frac{1-\theta}{p(\cdot)}, \] with equivalent quasi-norms. Here $L^{\mathrm{ad}}_{\widetilde{p}(\cdot),q}$ consists of adapted sequences $f=(f_n)_{n\geq 1}$ whose square function $\sigma(f)=\left(\sum_{n=1}^{\infty}|f_n|^2\right)^{1/2}$ belongs to the variable Lorentz space $L_{\widetilde{p}(\cdot),q}$. The proof uses a decomposition that preserves adaptedness and provides an upper estimate for the corresponding $K$-functional. No continuity condition on the variable exponent and no measurability relation between $p(\cdot)$ and the filtration are required.

A binary Coxeter code associated with a finite Coxeter system $(W,S)$ is an ${\mathbb F}_2$-linear span of indicators of standard cosets of a fixed rank. Coxeter codes, introduced in a recent paper by N. Coble and A. Barg, are a generalization of Reed–Muller codes which arise when $W={\mathbb Z}_2^m$ is the Coxeter group of type $mA_1$. In that paper, the authors proposed a conjectural value for the minimum distance of a general Coxeter code. This conjecture is proved in the present work. As a consequence, we obtain a Coxeter-theoretic generalization of Reed's majority-logic decoding algorithm for Reed–Muller codes.

This paper investigates inhomogeneous Poissonian pair correlation (PPC), its weak form, and equidistribution. We establish that weak inhomogeneous PPC does not imply inhomogeneous PPC. Furthermore, we construct a sequence satisfying weak inhomogeneous PPC that fails to be equidistributed, which stands in sharp contrast to the homogeneous case. Finally, we prove that for distinct $\gamma_1, \gamma_2 \in (0, \frac{1}{2}]$, weak $\gamma_1$-PPC does not imply weak $\gamma_2$-PPC, showing that different inhomogeneous parameters give rise to mutually independent notions of weak inhomogeneous PPC.

Let $\mu_0$ and $\mu_1$ be $\kappa_0$- and $\kappa_1$-strongly log-concave probability measures on $\R^n$, and let $(\mu_t)_{t\in[0,1]}$ be their quadratic Wasserstein geodesic. We prove the sharp Poincaré constant estimate \[ \sqrt{C_P(\mu_t)} \leq \frac{1-t}{\sqrt{\kappa_0}} + \frac{t}{\sqrt{\kappa_1}}. \] The coefficient is optimal for every $t,\kappa_0,\kappa_1$, and the result remains valid for extended-valued potentials without symmetry assumptions. Equality at an interior time holds exactly when the two endpoints have Gaussian factors in the same direction, with variances $\kappa_0^{-1}$ and $\kappa_1^{-1}$. All such directions form a maximal linear subspace. This gives an affirmative answer, without symmetry or parity assumptions, to a question of Aishwarya–Rotem. The proof develops a coupled Bochner method for the two endpoints. We solve a weighted Poisson equation involving the Hessian of the Brenier potential. From its solution we construct a Bochner couple, and separate estimates for its two fields are combined along the interpolation. No curvature bound is needed for the intermediate measures. Applications include centered Gaussian relative entropy, Gaussian Brunn–Minkowski inequalities with barycenter terms, and centered HWI, logarithmic Sobolev, and Talagrand estimates.

Probing and dual fixing are two powerful presolve techniques in mixed integer programming (MIP) solvers. Probing tentatively sets some binary variables to 0 or 1, applies linear constraint based domain propagation techniques to derive better variable bounds, and extracts useful information such as stronger variable implications and better global variable bounds. Dual fixing attempts to fix variables to lower or upper bounds while ensuring that at least one optimal solution is retained, as long as the problem was feasible. In this paper, we investigate how to combine the two approaches in MIP solvers to achieve a better performance. In particular, we first embed dual fixing into the probing framework, deriving more useful variables' implications for enhancing the capability of probing. Then, we develop an improved dual fixing technique where more variable fixings can be applied, and use the probing framework to detect the reductions. Computational results on the MIPLIB 2017 benchmark instances demonstrate the potential of the two proposed techniques in combining probing and dual fixing on the open-source MIP solver HiGHS.

The Lee-Yang theorem and its quantum extensions state that, for a broad class of Hamiltonians on any graph, the partition function's zeros in the complex magnetic field plane lie only on the imaginary axis. For these Hamiltonians, we prove that under a uniform Z-field of any strength h, the ground state has a spectral gap of at least h/4, independent of the system size and of the coupling strengths. The proof uses the zero-freeness of the partition function as given by Asano and Suzuki-Fisher to show exponential decay of the imaginary-time correlations for any product of Z-operators. Our result gives a polynomial-time quantum algorithm for computing the ground state energy of any Lee-Yang Hamiltonian.

We study sparse-array near-field beam focusing with spatial interference suppression, a problem arising in coherent satellite formations and other distributed non-terrestrial arrays. State-of-the-art designs solve it numerically through second-order cone programming (SOCP) with cutting-plane refinement, yet the achievable signal-to-interference ratio (SIR) and its link to classical adaptive beamforming have remained without an analytical characterization. We supply this characterization via a Lagrangian-dual analysis, obtaining three results. First, every optimal beamformer is a generalized matched filter against an effective spatial covariance induced by an optimal dual measure; this closed form recovers MVDR, LCMV, and SOCP-based focusing as special cases. Second, the dual measure has finite support of cardinality at most $M^2$ in general, sharpening to $M$ for uniform linear arrays ($M$ the number of array elements), which yields a finite-dimensional convergence certificate for cutting-plane methods. Third, a closed-form upper bound on the mean-SIR admits an asymptotic logarithmic scaling law in $M$ under near-collinear geometry, identifying array order, rather than the optimization algorithm, as the dominant performance factor. A Riemannian conjugate-gradient algorithm on the unit-torus manifold is developed for practical constant-modulus beamforming, and numerical results demonstrate that it closely approaches the derived performance limit.

We prove Bueno et al.'s conjectural formula for the rank of the modular multiplication matrix for every positive integer. The proof first applies the additive Fourier transform to split the row space by exact additive character order, and then applies the multiplicative character decomposition on unit groups to count the nonzero components. The same formula holds after deleting the zero row and zero column.

Roaming reactions, in which a dissociating fragment moves through a flat region of the potential surface rather than down the minimum-energy path, lie outside the assumptions of conventional transition state theory. The phase-space theory of roaming – unstable periodic orbits and their invariant manifolds organizing transport – has been developed for the Chesnavich model of $\mathrm{CH_4^+}\to\mathrm{CH_3^+}+\mathrm{H}$, which is cylindrically symmetric and reduces to two degrees of freedom (2-DoF). We construct and analyze a three-degree-of-freedom (3-DoF) extension. From the rigid-body formulation of Ezra and Wiggins, we break the symmetry with an azimuthal coupling respecting the three-fold ($C_3$) symmetry of the methyl fragment, obtaining a family $H_b$ whose planar reduction at $b=0$ is the 2-DoF model exactly and which is genuinely 3-DoF for $b>0$. This activates the out-of-plane degree of freedom at once: with the physical planar-top inertia ratio $I_z=2I_x$, arbitrarily weak coupling makes the periodic orbit on the roaming shelf transversely unstable, opening an escape route out of the reaction plane. Apart from a narrow elliptic window $0.58\lesssim b\lesssim0.63$, the instability persists across the range studied, changing type through a period-doubling at $b_c\approx0.63$. Because a periodic orbit cannot anchor a dividing surface in three degrees of freedom, we construct the objects that do – three three-dimensional normally hyperbolic invariant manifolds, one per transition state – at $b=0$, and prove that every compact interior piece of each persists for sufficiently small $b>0$. At $E=0.5\ \mathrm{kcal\,mol^{-1}}$ the coupling lowers the direct non-reactive fraction of a microcanonical ensemble of incoming trajectories by $0.032$ and raises the two roaming fractions by $0.040$; the effect decreases as the energy increases.

We propose a framework for controlling the collective dynamics of the time-delayed Kuramoto model based on a delay-free, higher-order approximation of the delayed interactions. By applying the Ott–Antonsen ansatz and the second-order averaging method to the resulting higher-order Kuramoto model, we obtain a one-dimensional reduced equation for the order parameter dynamics. Numerical simulations demonstrate that the higher-order approximation predicts the dynamics of the original delayed system more accurately than the conventional pairwise approximation and enables the realization of bistability and intermediate synchronization states. Our results demonstrate the effectiveness of higher-order interpretations of time delays for the control of oscillator networks with time-delayed interactions.