Papers

Density-Driven Optimal Control (D2OC) provides an effective framework for steering multi-agent systems toward prescribed spatial distributions. However, existing D2OC formulations are primarily developed for Euclidean domains and do not directly account for intrinsic manifold geometry. This paper extends D2OC to second-order multi-agent systems evolving on Riemannian manifolds. The proposed Riemannian D2OC (R-D2OC) constructs a local distribution objective in the tangent space of each agent through logarithmic maps and uses its weighted center as the reference for a finite-horizon LQR. The resulting control is executed on the manifold through intrinsic second-order dynamics and parallel transport within a receding-horizon scheme. We establish local curvature-dependent bounds that quantify the approximation introduced by the tangent-space reduction and characterize the resulting target bias. Furthermore, we derive a conditional discrete-descent result showing that the closed-loop objective decreases when a local velocity-alignment condition is satisfied. Numerical simulations on a 3D ellipsoidal manifold demonstrate distribution-level control and empirically support the proposed approximation and descent results.

We prove a Pólya–Szegö inequality for functions defined on an $n$-dimensional submanifold $\Sigma$ of a complete noncompact Riemannian manifold with nonnegative sectional curvature. The associated rearrangement is a Schwarz rearrangement on $\mathbb R^n$, and the constant depends on the $L^n$-norm of the mean curvature of $\Sigma$ and an isoperimetric quantity obtained by Brendle. As applications, we derive Sobolev, Log-Sobolev, Hardy, and Gagliardo–Nirenberg inequalities on submanifolds of arbitrary codimension under a small total mean curvature assumption. In the critical Sobolev case, we obtain Moser–Trudinger inequalities on finite-volume submanifolds and exact growth inequalities on submanifolds with infinite volume. Under suitable assumptions, the Pólya–Szegö constant equals one; in this case, the critical constants in the inequalities coincide with the sharp Euclidean ones.

We prove Villani's conjecture on entropy production for Kac's walk with uniform collision angles. For every $N\geq2$, with total collision rate $N$, the optimal entropy production constant is $2/(N-1)$. We show that the known lower bound is sharp by constructing two families of smooth strictly positive probability densities, invariant under coordinate permutations and sign changes. The first consists of normalized products of inverse powers; the second is obtained by conditioning products of Gaussian mixtures on the energy sphere. With $N$ fixed, we compute the leading terms of entropy and entropy production. Successive parameter limits yield two independent proofs of sharpness.

Bowditch's theorem for hyperbolic groups establishes a fundamental correspondence between splittings over two-ended subgroups and the existence of local cut points in the Gromov boundary. While analogous results have been obtained for CAT(0) and relatively hyperbolic groups, no general theorem of this type exists for arbitrary finitely generated groups. The Morse boundary, introduced by Charney-Sultan and extended by Cordes, provides a quasi-isometry invariant boundary for any finitely generated group that naturally generalizes the Gromov boundary. In this paper, we prove that a splitting of a finitely generated group with connected Morse boundary over a two-ended Morse subgroup gives rise to a separating pair of points in the Morse boundary.

We prove that the quantum code distance is NP-hard to approximate within an additive error of $c N$, for some constant $c >0$, where $N$ is the number of qubits. Our reductions are deterministic. This improves the previous square-root additive gap to $\Omega(N)$ and resolves the explicitly stated linear-gap question of Kapshikar and Kundu. Our result holds for CSS codes with identical $X$- and $Z$-check spaces, and with a constant rate and constant relative distance. For every fixed $\lambda>1$, there is a constant $c>0$ such that hardness still holds even when every nonidentity stabilizer has weight greater than $\lambda$ times the quantum distance. We also improve the hardness gap of graph state distance on $N$ vertices of Grigorescu, Jha, and Samperton from cube-root to $\Omega(N)$, resolving their explicitly stated open question. Both hardness results are asymptotically optimal since both distances are at most $N$. Our graph state distance hardness result holds for balanced bipartite graphs with a binary adjacency matrix that is its own inverse (mod 2). Our main technique for both hardness bounds above is classical: we show how to convert any code $C$ of length $m$ into a self-dual code $A(C)$ of length $N=\Theta(m)$ while exactly doubling the original coset metric. The conversion is deterministic and efficient. We call it the metric self-dual completion of $C$. It comes with a linear embedding $\tau: \mathbb F_2^m \hookrightarrow \mathbb F_2^N$. The embedding doubles all Hamming distances between vectors in $\mathbb F_2^m$ and all pairwise distances between corresponding cosets. The embedding also guarantees that all codewords of $A(C)$ of weight at most $2m$ are exactly $\tau(C)$.

Density-Driven Optimal Control (D2OC) provides a principled approach to distributing multi-robot teams over non-uniform spatial distributions. Applying D2OC to nonholonomic robots, however, creates a gap between safety constraints imposed on a reference motion and the physical inputs that determine the actual robot motion. We address this issue by enforcing the safety constraint directly on the robot's physical inputs while preserving the density-driven coverage objective. The proposed framework combines D2OC with a control barrier function safety filter through a feedback-linearizing look-ahead point, allowing safety and actuator limits to be considered together during control. We further derive a safety margin that accounts for the look-ahead geometry, robot footprint, and motion during each control interval. Simulation results show that the proposed method maintains the required physical separation while achieving coverage performance comparable to a conventional reference-tracking approach, which can satisfy safety on the reference motion yet violate the corresponding physical clearance. Experiments on multiple nonholonomic robots in the Robotarium further demonstrate safe execution while driving the robots toward the desired spatial distribution. These results show that enforcing safety directly on the physical inputs can eliminate the mismatch between safety certification and physical robot motion in density-driven multi-robot coverage.

Let $M$ be the six-element commutative semigroup occurring as the common multiplicative reduct of the semirings $SR_6$ and $TR_6$. The closing paragraph of Shao, Ren, and Gao~\cite{ShaoRenGao2026} asks for the finite-basis and subvariety questions for the four remaining compatible additions on $M$. We answer these questions for the four isomorphism types $R_{01},R_{02},R_{11},R_{12}$. First, we classify all compatible additions on $M$: there are nine labelled additions and six isomorphism types, parametrized by $R_{ij}$ with $0\leq i\leq j\leq 2$. For each of the four new types we give a graph-theoretic criterion for every identity, an explicit infinite basis, and a proof of nonfinite basability. The generated varieties $\V(R_{01})$ and $\V(R_{02})$ have eleven subvarieties each, while $\V(R_{11})$ has sixty-six. The lattice $\Sub(\V(R_{12}))$ is countably infinite. Every identity in this variety reduces to a subset of twenty-five fixed identities together with two monotone path families $\gamma_n$ and $\gammaD_n$. This yields a canonical signature $(H,p,q)$, complete normal forms, explicit meet and join operations, and a formula for all covers. There are 153 fixed nodes, 43 one-parameter families, and 9 two-parameter families; exactly eighteen subvarieties are finitely based, and the unique limit subvariety is $\V(SR_6)$. The strong nonfinite-basis status of the four finite semirings remains open.

We study the log-concavity property of first Dirichlet eigenfunctions on domains in $\mathbb{CP}^2$. For every smooth $1$-convex domain $\Omega\subset\mathbb{CP}^2$, we prove the quantitative estimate \[ \nabla^2(-\log u) > \max\left\{\psi(s),\frac85\right\}g, \text{ where } \psi(|\grad f|^2) = \frac{s}{\sqrt{1+s}}-\log(1+s), \] for its first Dirichlet eigenfunction $u$. In particular, $u$ is strictly log-concave. As consequences, we obtain a uniform convexity estimate for the regular level sets of $u$ and the fundamental gap bound $\lambda_2-\lambda_1>46/5$.

We address a problem concerning cinema audiences: “A cinema has $n$ seats numbered from $1$ to $n$, and there are $n$ people with tickets numbered from $1$ to $n$. People enter the cinema in order. If someone has the ticket number $i$, they can choose seats whose numbers are multiples of $i$. They should exit the cinema if the permitted seats are occupied by previous audience members. In how many ways can they be seated under these conditions?" We give an algorithm to create the list of situations that meet these conditions. We also focus on finding the number of situations in two special cases: when exactly one seat is unoccupied whose total number is denoted by $\omega(n)$, and when all audiences $1, \ldots, n-1$ are seated, whose total number is denoted by $\psi(n)$. Giving the recursive formula $\psi(n)=1+\sum_{d|n, d\neq n}\psi(d)$ with the initial value $\psi(1)=1$, we provide an explicit formula for $\psi(p^\alpha q^\beta)$, where $p$ and $q$ are distinct prime numbers. Furthermore, we show that $\omega(n)=-n+\sum_{i=1}^n\psi(i)$.

The linearized buckling load of a stiffened panel depends on the stress stiffness its shell element assembles. We read it from exported operators against three truncations of one second variation. The classic pass of ANSYS SHELL181 carries a rotation-rotation block pairing the drilling freedom with the bending rotations and its perturbation pass does not; removing the block recovers the perturbation load factor to 0.02%. SHELL281 carries block and couplings in both passes. Abaqus S4 matches the critical mode of the complete second variation to 1.0000 on the translations and its load factor to 1.7%, against 17% and 34% for the other two forms. On an optimized panel under cantilevered shear a 20-node continuum lies 3% to 6% above that form, S4 and SHELL281, 11% and 23% below both SHELL181 passes and 25% above Abaqus S8R, at the finest meshes. On a conventionally stiffened panel the SHELL181 passes stand 1.0% and 3.5% above the complete form, 9% and 21% at half the rib pitch; under a shear flow, on cylinders, open beams and under axial compression the three forms coincide and no pass parts by more than 0.3%.

Dilution of precision is the standard way to describe navigation geometry and it does not apply in cislunar space: its derivation presumes a snapshot solution in which simultaneous measurements determine the state at one instant. A spacecraft tracked by inter-satellite ranging in the Earth-Moon three-body problem has no such solution: range is a function of position only, so the velocity columns of the measurement Jacobian vanish identically and the state is unobservable however many links are in view. The correct figure of merit is dynamical, because the three-body flow carries velocity information into later range measurements. We demonstrate this on four differential-corrected planar distant retrograde orbits, reporting the observability Gramian rather than a dilution of precision. Instantaneous range-only geometry has rank 2 of 4; the full state becomes observable only after a tracking arc of 2.09 h, the first sampled arc at full rank on a 0.26 h grid, and a square-root information filter recomputing the same quantity by a different route becomes well-posed at exactly that arc. Beyond it, conditioning improves monotonically with arc length, the Gramian condition number falling from 1.2e11 at 3 h to 1.4e5 at 48 h. Adding range rate lifts the instantaneous rank from 2 to 4, so a Doppler-capable link makes a snapshot solution exist where a range-only link cannot. Lifting the known-reference assumption, the joint problem of estimating a station and the satellites' states and clocks together is rank-deficient by exactly one without an external interferometric observable in all twenty realizations tested, and full rank with it; the deficiency persists only to four satellites. For the configurations studied, cislunar navigation requirements are better written as arc-length, conditioning and measurement-type budgets than as dilution-of-precision thresholds, which are undefined here.

We propose a computationally inexpensive algorithm to determine whether an opaque spherical body eclipses a luminous one, using only classical geometrical tools. We apply this method to the Sun-Earth-Moon system and predict solar eclipses, including their type, beginning, and duration, using the three-body problem. For predictions over a one-year period, the resulting errors are in the order of minutes.

For a positive integer $m$ and a real parameter $\alpha>-1$, consider the generalized Cesàro matrix on $\ell^2(\mathbb N_0)$ with entries $m(n-j+1)_{m-1}/(n+\alpha+1)_m$ for $j\le n$. We give a computer-assisted proof that this operator is hyponormal if and only if $\alpha\ge0$. The main algebraic ingredient is an explicit decomposition of an auxiliary defect operator into a positive diagonal operator and at most $m$ rank-one operators. Discrete Gram polynomials determine the signs and coefficients of these terms. For nonnegative parameters, weighted estimates reduce positivity to scalar inequalities. A uniform analytic estimate covers every $m\ge32768$, while finite rational certificates cover the remaining orders and entire parameter intervals, without parameter sampling or truncating an infinite operator. The proof also gives a uniform weighted lower bound for the self-commutator. For $-1<\alpha<0$, finite-support vectors give negative quadratic forms for every order.

We study online convex optimization on a bounded Euclidean domain when the learner receives only one subgradient at its prediction and knows neither the horizon, a gradient bound, nor the losses' strong-convexity parameters. We present a scale-invariant algorithm that couples projected adaptive gradient descent to one cone-constrained quadratic leader through a positive wealth process. The algorithm maintains $O(d)$ state, without a learning-rate grid or restarts. For a domain of diameter at most $D$, it guarantees regret at most $\sqrt{2}D(\sum_t\|\widehat g_t\|^2)^{1/2}+2D\max_t\|\widehat g_t\|$, with no logarithmic factor. A simultaneous quadratic-distance bound yields explicit improvements for unknown strong convexity, including a refinement governed by jumps in the observed gradient scale. For varying per-round curvature, the bound depends on the total shortfall below a reference level selected in hindsight. The proof uses a clipping-compatible surrogate and a potential involving the quadratic minimum, log wealth, and two scalar curvature statistics. The resulting strongly convex bound is not uniform at the classical logarithmic rate over all curvature-to-gradient ratios; the exact parameter dependence is stated. The contribution is a single-leader construction and its analysis, within an established literature on universal and scale-free online learning.

Despite the importance for interpretability, decision trees face severe scalability challenges. Existing global optimal methods are often limited by binary feature selection and shallow tree depths, whereas traditional heuristic approaches frequently sacrifice predictive accuracy. To overcome these limitations, this paper proposes a moving-horizon approximate branch-and-reduce method to train near-optimal deep classification trees on large-scale datasets with continuous features. Built on a hierarchical root-subtree optimization framework, the method solves the root-level problem via branch-and-reduce while approximating the induced subtree problem using greedy heuristics. Although the underlying framework is capable of guaranteeing global optimality, the approximation, which functions as a lookahead rollout in a reinforcement learning context, significantly boosts efficiency for deeper structures. A low-cost moving-horizon strategy is then employed to iteratively refine model accuracy. Extensive numerical results demonstrate that our method exceeds the testing accuracy of existing heuristic baselines while offering significantly greater scalability, in terms of both dataset size and tree depth, than global optimal solvers.

We study the nonlinear Hamilton–Jacobi system arising as the formal semi-classical limit of the chordal Belavin–Polyakov–Zamolodchikov equations with a common eigenvalue parameter $\lambda$. We determine the number of global real-valued solutions modulo additive constants. When $\lambda=0$, the number of solutions can be derived using the connection between the solutions and rational functions, due to Eremenko. We focus on the case when $\lambda\neq 0$ in this article. For $\lambda>0$, a Schwarzian correspondence relates the algebraic equations for the gradients to quasi-exponentials with prescribed critical points. Enumeration and transversality results of Mukhin, Tarasov, and Varchenko yield smooth real branches, and a Poisson bracket identity establishes their exactness. An energy bound rules out global real-valued solutions for $\lambda<0$.

In this paper, we study a one-to-one correspondence between $\epsilon$-unit generalized Killing spinors on a three-dimensional pseudo-Riemannian spin manifold $(M,g)$ and Lagrangian sections of an almost pseudo-Hermitian structure on its principal spin bundle.

We study torus-exterior replacement in four-manifolds, with the construction controlled by the peripheral homomorphism. Using torus exteriors arising from the work of Boyle and Kanenobu-Kazama, with explicitly controlled fundamental groups and peripheral systems, we construct smooth homotopy 4-spheres and manifolds homeomorphic to S^2 x S^2 and \#_3(S^2 x S^2). We also study direct gluings of two torus exteriors. A swap gluing of two one-null exteriors kills both meridians and gives a simply connected manifold with Euler characteristic \(4\) and signature \(0\), while a meridian-preserving gluing of doubly-null exteriors retains the complement groups as an amalgam over the meridian subgroup. Twisted peripheral gluings give cyclic and explicit finite nonabelian fundamental groups. We also consider the full Kanenobu-Kazama \(1\)-handle family and Litherland tori with full-rank peripheral subgroup; the latter have no null primitive boundary slope and lead naturally to amalgamated-product fundamental groups.

This 30 years old book is posted for free download by the generosity of Oxford University Press that granted permission to allow access to future generations of young scientists. A printed copy can be purchased from the publisher. The authors received for this book "The 1999 INFORMS Computing Society Prize for Research Excellence in the Interface Between Operations Research and Computer Science" from the Institute for Operations Research and the Management Sciences (INFORMS). For a book review by Janos D. Pinter go to: Journal of Global Optimization, 16: 107-108, 2000.

We develop inference for manifold-valued population minimizers when each observation belongs to a different participant and only locally private messages reach the analyst. The method releases randomized tangent gradients and combines them through Riemannian stochastic approximation and Polyak-Ruppert averaging. Directly inserting a private data surrogate into a nonlinear loss can shift its population target, whereas conditional centring of the released gradient preserves the first-order equation. We introduce symmetric-pair regression (SPR) to estimate the asymptotic variance from the same private messages used for point estimation, without holding out participants or requesting a second release. We prove the central limit theorem and consistency of the fully transcript-based sandwich covariance and intrinsic Wald region under local differential privacy. Simulations across various statistical problems and manifolds support the predicted decrease in estimation error and near-nominal coverage under moderate privacy. An application to NHANES anthropometric data illustrates private estimation of a leading body-size direction and its uncertainty.

We study the sharp $L^2$-Poincaré inequalities on Euclidean balls and with respect to the Gaussian measure, and we focus on their stability. In both settings, the sharp constant is the first nonzero eigenvalue of a self-adjoint operator and the set of optimizers is a finite dimensional linear space. Using the spectral decomposition, we establish sharp $L^2$-stability estimates and sharp gradient stability estimates, together with the stability of the stability inequalities in both norms, with explicit optimal constants and with the complete characterization of the equality cases. We then show that this process can be continued. The Poincaré deficit is equal to an infinite sum of $L^2$-distances to the increasing spaces of optimizers of the successive stability inequalities. It is also equal to an infinite sum of $L^2$-gradient distances to the same spaces. The coefficients of the first sum are the spectral gaps $\mu_j-\mu_{j-1}$, and the coefficients of the second sum are $\mu_1$ times the gaps of the reciprocal spectrum, $\mu_1\left(\mu_{j-1}^{-1}-\mu_j^{-1}\right)$. On the Euclidean ball, all the constants are expressed through the roots of Bessel functions, and we prove that the first two spectral levels are given by the degree one and the degree two Neumann modes in all dimensions, while the third level is given by the first radial mode when $N=2,3$ and by the degree three modes when $N\geq4$. In the Gaussian case, all the constants are rational numbers and the $L^2$-sum has all coefficients equal to $1$; we prove that this property characterizes the arithmetic spectrum.

We provide explicit bounds on the number of zeros for $\zeta$ in short intervals near the edge of the critical strip. Ultimately we show that even in short intervals a positive proportion of zeros will not be near the edge of the critical strip. For example with $h>100$ and $t>10^{100}$ the proportion of zeros of $\zeta$ with imaginary part in $[t-h,t+h]$ which have real part in $[\frac{1}{16},\frac{15}{16}]$ is at least $13\%$ and a positive proportion is already obtained at $t>10^{43}$.

Sub-fractional Brownian motion is self-similar and long-range dependent but has no stationary increments, so the increment covariance is not Toeplitz and no spectral density is available. We show that a complete finite-sample likelihood theory survives nonetheless. The model is a linear trend observed at $N$ equidistant times through a superposition of $m$ independent sub-fractional Brownian motions with known Hurst indices and a common scale. Nondegeneracy follows from realising the process as the even part of a two-sided fractional Brownian motion, and the maximum likelihood estimators of the trend and of the scale are explicit. The statistics on which inference rests are pivotal, their laws depending on the sample size alone, so intervals and tests of exact level are available at every $N\ge2$, together with complete sufficiency, minimum variance unbiasedness and attainment of the Cramér–Rao bound. An explicit variance bound gives strong consistency and asymptotic normality, and simulations confirm the exact coverage and the predicted effect of a misspecified Hurst vector.

A joint guidance and robust observer-based output-feedback synthesis framework is developed for discrete-time nonlinear systems subject to bounded process disturbances and measurement noise. The method co-designs a reference trajectory, time-varying Luenberger observer and observer-based linear output-feedback gains, and coupled ellipsoidal funnels that certify conditioned-invariant bounds on tracking and estimation errors. Nonlinear residuals are bounded by incremental quadratic constraints (QCs) parameterized by local Lipschitz constants, estimated via sampling approaches. The coupled stability conditions are formulated as linear matrix inequality (LMI) surrogates and solved using sequential convex programming with proximal regularization (prox-SCP) via semidefinite programming (SDP). Simulations of a unicycle model under persistent sensory noise and system disturbances demonstrate robust tracking and funnel-certified constraint satisfaction with respect to the jointly designed reference trajectory.

In this paper we study an equivalence between the existence of a Sobolev inequality without gain, \[\|\varphi\|_{L^p(v,\Omega)} \leq S(p,1) \| \sqrt{Q}\nabla \varphi\|_{L^p(\Omega)},\] that holds for smooth functions of compact support and the existence of a degenerate weak solution $(u,\nabla u)\in QH^{1,p}_0(v,\Omega)$ to a Dirichlet problem for the $p$-Laplacian with a zero order term: \begin{equation*} -v^{-1}\text{Div}(|\sqrt{Q}\nabla u|^{p-2}Q\nabla u)+F|u|^{p-2}u = |f|^{p-2}f - v^{-1}\text{Div}(v|g|^{p-2}g\mathbf{t}),\; x \in \Omega, \quad \text{and} \quad u =0, \; x \in \partial \Omega, \end{equation*} More precisely, we use the Sobolev inequality to prove the existence of a degenerate weak solution to this equation and then use the existence of such a solution to produce a Sobolev inequality. Moreover, we show that solutions are unique.

We compute the cohomology of the projective special linear Lie superalgebra $\mathfrak{psl}(n|n)$ over a field of characteristic zero in trivial and in adjoint coefficients in full detail through its associated Hochschild–Serre spectral sequences. Applications in physics exist for its real form, the projective special unitary Lie superalgebra $\mathfrak{psu}(n|n)$ — in particular, $\mathfrak{psu}(2,2|4)$ for AdS/CFT. We also find the cohomology of $\mathfrak{sl}(m|n)$ in trivial coefficients along the way and we point out a discrepancy with a previous statement in the literature. We present the explicit cohomology groups of low degree for $n\leqslant4$.

Consider the Monge-Kantorovich optimal transport problem between two Euclidean densities $\mu$ and $\nu$ and quadratic cost. The $\varepsilon$-Schrödinger bridge is the solution to the entropic regularized problem with regularization parameter $\varepsilon$. By taking the conditional expectation of the second coordinate given the first under this coupling, we obtain the $\varepsilon$-entropic Brenier map. As $\varepsilon$ goes down to zero, it is known that the entropic Brenier map converges to the quadratic cost optimal transport map between the two measures, i.e., the Brenier map. We show that, under some smoothness and log-concavity constraints on the marginals, the difference between the entropic Brenier map and the Brenier map is equal to $\varepsilon$ times one-half of the score function of the first marginal plus an error that is $o(\varepsilon)$ in $\mathbf{L}^2(\mu)$. This expansion holds irrespective of the second marginal $\nu$. The proof relies on an approximation of the Schrödinger bridge by a noisy version of the McCann interpolation. Under additional assumptions we show a second approximation to the Schrödinger bridge via so-called Mirror Langevin diffusions, which are Langevin diffusions on the Hessian manifold generated by the Brenier map. These two approximations, that appear quite different at first glance, are nonetheless shown to be closely connected. Our proofs utilize a random surface and a novel stochastic operation called the tangent Markov projection.

Semidefinite programming (SDP) certificates for feedback systems containing deep neural networks (NNs) typically scale with the total number of neurons, whereas small-gain tests are scalable but can be highly conservative. This paper develops a unified and scalable framework for incremental robust stability and performance analysis of feedback interconnections involving high-dimensional NNs and unmodeled dynamics. By combining a structured decomposition of the full-order SDP condition with scalable Lipschitz constant estimation algorithms, we derive reduced verification conditions that certify incremental convergence and incremental $\ell_2$-gain bounds. The dimensions of the resulting control-analysis linear matrix inequalities (LMIs) depend only on the widths of the last two network layers and are independent of network depth. The framework preserves the coupling between the plant and the NN, with the incremental small-gain condition recovered as a special case. To further reduce conservatism, we develop a multi-round alternating update scheme that iteratively refines the coupling variables while preserving scalability. Numerical experiments show that the proposed framework achieves state-of-the-art incremental $\ell_2$-gain bounds for large-scale NN feedback systems.

We study elastic knots obtained by minimizing the bending energy of a unit loop together with a small multiple of ropelength. We assume a generalized Fáry-Milnor inequality on the $C^1$-boundary of a knot class: every $H^2$-limit curve in the $C^1$-closure has total curvature at least $2\pi$ times the bridge index. Under this assumption we prove that every elastic knot in a one-component BB knot class, that is, a class whose bridge and braid indices coincide, is the round circle covered precisely that many times. The proof does not differentiate the nonsmooth ropelength functional. A minimal closed braid is contracted in the fibres of a solid torus, producing a family of comparison curves with quadratic bending-energy excess and thickness of linear order. This yields a lower thickness bound for the regularized minimizers. Variations on an intermediate scale then give free stationarity of the limiting curve for the normalized bending energy. Constant curvature and free stationarity reduce the classification to a linear distributional ordinary differential equation.

We study grazing closed trajectories of the billiard flow $\varphi^t$ on a compact surface $(\Sigma,g)$ with concave boundary. Under a non-degeneracy condition, which is satisfied for example when $(\Sigma,g)$ has negative sectional curvature, we compute the trace of a regularized pullback $E_{\epsilon}(\varphi^t)^* E_{\epsilon}$ near a closed trajectory grazing once, which is an analogue of the Atiyah–Bott–Guillemin trace formula for closed billiard trajectories.