Papers

We study the robustness of the $F$-test in random design linear models, and reach a somewhat nuanced conclusion. On the positive side, one of our main results is that the size of the test is close to its nominal level as soon as either the distribution of the normalised error vector is close to uniform on the unit sphere, or the design matrix, after applying a suitable column space-preserving orthogonalisation scheme, is close to being uniformly distributed. This provides a sense in which the $F$-test is doubly robust. Our conclusion is reached by establishing a Kolmogorov to Wasserstein distance Hölder continuity property controlling the departure of the $F$-statistic from its notional $F$-distribution under the null. Writing $n$, $p$ and $p_0$ for the sample size and the dimensions of the full and null models respectively, we prove that the Hölder exponent is $1/3$ when $\min(p-p_0,n-p) = 1$ and $1/2$ when $\min(p-p_0,n-p) \geq 2$. On the other hand, these exponents are relatively small and cannot be improved in general, revealing that the size of the test may depart from its nominal level quite quickly as we move away from settings where the test is exact. In some cases, our conclusions may be improved by working with a local Kolmogorov distance that focuses on discrepancies between distribution functions in the right tail.

Using the theory of von Neumann algebras, Vaughan Jones showed abstractly that there is a bridge between holomorphic cusp forms and random matrices. This bridge is supplied by a distinguished function he called a tracelike vector. He posed the following problem: construct a tracelike vector to make this bridge explicit. We carry out Jones' vision for the full modular group. We construct holomorphic tracelike vectors for every finite-covolume Fuchsian group throughout the permissible Bergman range. For the modular group at the critical parameter $s=13$, we use the resulting unit tracelike vector to assign explicit group coefficients to cusp form multiplication operators, and form positive finite random matrices from a shared Haar-distributed unitary matrix. Almost surely, their normalised traces recover all joint operator moments, first as the matrix size tends to infinity, and then as the group-word cutoff tends to infinity. We also identify the limiting mixed second moment with a series of Petersson norms that involve Maass-Shimura operators of cusp forms. The exact nature of our use of AI is detailed in the introduction.

We study the simultaneous low Mach number limit of a compressible Navier–Stokes–Euler two-phase system in $\mathbb R^3$, in which the pressure terms in both phases are scaled by $\varepsilon^{-2}$. For sufficiently small $H^3$-perturbations around the constant equilibrium, we establish the global well-posedness of the scaled compressible system and derive global energy-dissipation estimates that are uniform with respect to the Mach number $\varepsilon$. A key feature of the analysis is the degenerate dissipation structure: viscosity acts only on the Navier–Stokes phase, while the drag coupling transfers dissipation to the inviscid Euler phase through the relaxation mode. We then prove the global well-posedness and large-time decay of the limiting incompressible two-phase system, and show that the relative velocity $u-\omega$ decays faster than the full velocity pair. Finally, for the well-prepared initial data, we introduce pressure-corrected acoustic variables to remove the singular pressure mismatch and establish a global-in-time $H^2$-error. As a result, the scaled compressible solutions converge uniformly in time to the corresponding solution of the limiting incompressible system.

Let $B$ and $D$ be both algebras and coalgebras in a braided monoidal category $\mathcal{C}$. In the literature, there are equivalent conditions for $(B,D)$ to form a generalized smash biproduct (or cross product) denoted by $B\times D$, which were mainly given by Bespalov and Drabant in 1999 as well as by Bulacu, Caenepeel and Torrecillas in 2013. This paper is devoted to constructing another biproduct $D\times B^\ast$ when $B$ has a left dual object $B^\ast$ in $\mathcal{C}$. As results, we show that $B\times D$ is left smash over $D\times B^\ast$, and establish a specific isomorphism ${}_{B\times D}\mathfrak{YD}(\mathcal{C})^{B\times D}\cong{}_{D\times B^\ast}\mathfrak{YD}(\mathcal{C})^{D\times B^\ast}$ between the categories of the (left-right) Yetter-Drinfeld modules in $\mathcal{C}$. The constructions and results are provided in three levels: (1) $B\times D$ is an algebra and a coalgebra; (2) $B\times D$ is a bialgebra; (3) $B\times D$ is a Hopf algebra.

How much unknown process noise can a bounded experiment tolerate while still establishing that the plant can be stabilized? We answer this question for scalar discrete-time systems with bounded inputs, exact state measurements, and a disturbance-energy budget proportional to the experiment length. We distinguish individual stabilizability of every consistent model, stabilization by one common gain, and a common quadratic certificate. We determine their exact noise ceilings for every drift. The last two requirements coincide; individual stabilizability generally tolerates more noise. Bounded periodic inputs approach the ceilings, while adversarial disturbances prevent their attainment. We also determine the long-horizon limits. For unstable plants, a short experiment can establish individual stabilizability at noise levels where every sufficiently long experiment fails. The same obstruction yields upper bounds for multidimensional systems with real eigenvalues. Input design may use plant knowledge, but certification uses only the recorded data and noise bound. The results therefore provide fundamental robustness benchmarks for unknown-plant designs.

We consider a strongly dispersive Boussinesq-type system arising as a model for the propagation of weakly nonlinear water waves. The system is characterized by a nonlinearity parameter $0<\epsilon\le 1$ and a shallow water parameter $0<\mu\le 1$. In the Boussinesq regime $\mu \sim \epsilon$, Saut, Wang and Xu \cite{SWX2017} used a hyperbolic approach to prove the well-posedness of the system on a time scale of order $1/\epsilon$, in both one and two dimensions, under a non-cavitation condition. In this paper, we establish well-posedness on a time scale of order $ ( \mu^{ 1/4} \epsilon^{-1} h_0 )^{4/3}$ in one dimension and of order $(\mu^{ 1/4}\epsilon^{-1} h_0)^{2-}$ in two dimensions, assuming a non-cavitation condition with a parameter $h_0>0$. Our results show quantitatively how the lifespan depends on the balance between nonlinearity and dispersion in the regimes $\epsilon\lesssim \mu$ and $\mu\ll\epsilon$. They also make explicit how the existence time deteriorates as $h_0\to0$, that is, as the wave approaches cavitation. Our proofs combine dispersive and Strichartz estimates with energy estimates.

Routing unknown quantum information across distributed communication networks requires autonomous, measurement-free mechanisms to prevent wave-function collapse. Szegedy quantum walks provide a mechanism for spatial state transport. The conventional walks on unweighted graphs suffer from severe back-reflection, spatial dispersion, and channel crosstalk. In this paper, we introduce a deterministic topological quantum routing architecture based on coined Szegedy quantum walks on edge-duplicated signed graphs. By treating edge signs as localized phase shifts within a balanced coin reflection, we enforce an exact zero back-scattering condition across routing nodes. We demonstrate deterministic Perfect State Transfer (PST) with unit fidelity at exact arrival times across fundamental archetypes, including the signed dumbbell switch $D_{2m,0,2n}$ and scalable glued binary trees. Furthermore, we analyse routing performance under realistic open-system amplitude and phase damping noise channels.

Let $q$ be an odd prime power and $m\geq 1$. Let $Q$ be a nondegenerate quadratic form on $\mathbb F_q^{2m}$. For every sphere $S_j=\{Q=j\}$ with $j\in \mathbb F_q^\times$, we prove the sharp extension estimate $R_{S_j}^*(2\to r)\lesssim_m1$ for $r\ge 2(m+1)/m$, uniformly in $q$, $Q$, and $j$. As an application, we show that if $E\subseteq\mathbb F_q^{2m}$ satisfies $|E|/q^{m+1/3}\to\infty$, then almost every pin $y\in E$ determines $(1-o(1))q$ values of $Q(x-y)$. The proof combines two arithmetic Hecke operator estimates with an induction in the dimension, a centered sphere–cone estimate, and an orthogonal decomposition.

In this paper, we determine the set of Euler classes of all taut foliations for a family of closed graph manifolds. As a consequence, for any closed aspherical graph manifold $Y$, every integral class in the closed dual Thurston unit ball can be virtually realized as the Euler class of some taut foliation. In particular, $Y$ virtually admits a taut foliation with vanishing Euler class. We present examples of graph manifolds with arbitrarily large first Betti number, which admit no taut foliations with vanishing real Euler classes. This shows that the virtual requirement is necessary. We also provide examples of graph manifolds with arbitrarily large first Betti number, for which every taut foliation has vanishing real Euler class, but none has vanishing integral Euler class. This illustrates the subtle difference between real Euler class zero and integral Euler class zero.

We consider the 653 equational laws of order up to 4 for an associative binary operation, and all of their conjunctions. We determine that there are only 456 equivalence classes of such conjunctions (associative equational theories), and find all implications between them. The conjunction operation makes this set of theories into a semi-lattice. These results are formalized in Lean. This is a semigroup analogue of the Equational Theories Project, but extended to conjunctions of equations.

Let $Z$ be a compact Kähler manifold endowed with a Hamiltonian action of a compact connected Lie group $U$, and let $G\subset U^{\mathbb C}$ be a real compatible subgroup. Using the gradient map $\mu_{\mathfrak p}$ associated with the momentum map, we establish a Calabi-Matsushima type decomposition for the Lie algebra of the stabilizer $G_z$ at a critical point of $f\circ\mu_{\mathfrak p}$, where $f$ is the restriction to $\mathfrak p$ of a suitable $Ad_U$-invariant strictly convex function on $i\mathfrak u$. More precisely, we show that the isotropy algebra decomposes into eigenspaces corresponding to nonnegative eigenvalues of an adjoint endomorphism, with zero eigenspace given by its reductive part. This extends the classical decomposition for the complexified action of $U^{\mathbb C}$ to the real reductive setting. As an application, we prove that the identity component of the compact stabilizer at critical points of $f\circ\mu_{\mathfrak p}$ is a maximal compact subgroup of the identity component of the $G$-stabilizer.

We construct exact representations of finite element functions of arbitrary polynomial degree on simplicial meshes in any spatial dimension by two-hidden-layer multilayer perceptrons (MLPs). The first hidden layer uses $\mathrm{ReLU}^0+\mathrm{ReLU}$, and the second uses the degree-dependent activation $\mathrm{ReLU}^k$. The neural networks can realize both zero-skeleton and prescribed-skeleton representatives of discontinuous finite element functions, where the former vanish on the mesh skeleton and the latter admit independently prescribed polynomial data on each relatively open lower-dimensional mesh subsimplex; in particular, continuous finite element functions can be represented pointwise on the closed domain. In the continuous case, a half-open decomposition reduces the second-hidden-layer width by assigning each subsimplex to a single incident element. The number of nonzero parameters grows linearly with the number of elements for fixed spatial dimension, polynomial degree, and output dimension, and it is of the same order as the number of global degrees of freedom of the finite element space in the continuous case, provided that the mesh is shape-regular. All parameters in the neural network can be explicitly computed via the information of the mesh and finite element function without training.

We study the width required for a randomly initialized hidden layer of a neural network to achieve rank lifting. Namely, given a dataset $X \in \mathbb{R}^{m \times d}$ of $m$, $d$-dimensional input vectors separated by an angle of at least $\theta$, we consider the random feature matrix $\sigma(XR)$, where $R$ is standard Gaussian. For positively homogeneous nonpolynomial activations, which include sign, Heaviside, ReLU, and ReLU powers among others, we prove that $$n \gtrsim \frac{1}{\theta}\max\left\{m,\log\left(\frac{1}{\delta}\right)\right\}$$ neurons suffice for $\sigma(XR)$ to have full row rank $m$ with probability at least $1-\delta$. This dimension-free bound exponentially improves the previous general-dimensional guarantee for sign features (Drago et al., 2026) and is essentially tight. The proof shows that one random feature column escapes every proper subspace of $\mathbb{R}^m$ with probability $\Omega(\theta)$, using a coupling of nearby Gaussian directions and a local crossing of the induced hyperplane arrangement. We also study stable rank lifting, where the goal is to establish a quantitative analogue of exact rank lifting, i.e., a lower bound on the smallest eigenvalue of the empirical feature Gram matrix in high-probability. Our analysis unifies and generalizes stable rank guarantees for all $q$-homogeneous non-polynomial activations following prior work in Panigrahi et al. (2020) and Song (2026). In particular, we combine a diagonally dominant Taylor tail of the population kernel with truncation and matrix concentration, to show that for positively homogeneous nonpolynomial activations, stable rank lifting is achieved at width $$n \gtrsim C^q \frac{m}{\theta^{2q+1}} \log^{2q+\frac{1}{2}}\left(\frac{m}{\theta}\right) \log\left(\frac{m}{\delta}\right),$$ where $q$ is the degree of the activation and $C > 0$ is some universal constant.

Landmark shape spaces arise from configurations of $n$ distinct landmarks with rigid motions factored out and scale normalized. When equipped with sufficiently regular metrics inherited from right-invariant metrics on diffeomorphism groups, the shape spaces inherit geodesic completeness from geodesics of diffeomorphisms. We now answer the corresponding stochastic question if the Riemannian Brownian motion on landmark shape spaces exists for all time and hence rule out initially distinct landmarks colliding when following a Brownian flow. We show that with general classes of metrics, including degenerate metrics with infinitesimal rigid motions in their null space, landmark shape spaces are stochastically complete, thus making the use of Riemannian Brownian motion for modelling shape stochasticity in applied fields well-founded.

Policy mirror descent (PMD) enjoys fast convergence in regularized Markov decision processes (MDPs), but existing guarantees often rely on exact or increasingly accurate policy evaluation. We analyze PMD coupled with a persistent critic advanced by one temporal-difference (TD) update. For finite discounted MDPs, we establish global linear convergence in value for exact coordinate-wise Bellman updates, with any positive constant actor stepsize and arbitrary finite critic initialization. The proof combines a resolvent-based auxiliary distribution with a decaying Bellman-violation correction and a potential weighted by inverse coordinate weights. We then study stochastic TD-PMD with general strongly convex mirror maps under a single off-policy Markov trajectory. With suitably chosen constant stepsizes and a finite-batch TD update, the method achieves an expected value gap of $\epsilon$ after $\widetilde{O}(1/((1-\gamma)^5 \widetilde{\sigma}_b \epsilon))$ transitions. The stochastic analysis relies on the trajectory-wise Lipschitz continuity of the regularizer, derived from uniform bounds on vertex Bregman divergences, together with a visitation-weighted resolvent estimate for signed critic-error propagation that yields an inverse-linear dependence on behavior coverage $\widetilde{\sigma}_b$. In contrast to many prior guarantees for regularized policy optimization, our sample-complexity guarantee holds without trajectory resets, generative-model access, or nested policy-evaluation loops. Numerical results are consistent with the theoretical convergence analysis.

We define a family of random sets of points on the sphere $\mathbb S^2$, the Pearl ensemble, depending on several parameters. The expected value of the logarithmic energy can be computed exactly up to terms of order $O(\sqrt{N}\log N)$, with $N$ the number of points. For properly chosen values of the parameters, the coefficient of the linear term in the expansion of the logarithmic energy can be taken as close as desired to the value $(1-\log 3)/2\sim -0.0493061\ldots$. Among the explicit constructions currently known to us for which the logarithmic energy expansion has been rigorously determined up to the linear term, the Pearl ensemble yields the smallest coefficient.

For every fixed integer $d\geq2$, we prove that the finite graphs receiving a continuous homomorphism from the standard Schreier graph $F(2^{\mathbb Z^d})$ form a $\Sigma^0_1$-complete set. This extends a theorem of Gao, Jackson, Krohne and Seward when $d=2$. For the positive part of the reduction, we show that if a graph $H$ satisfies a certain property then there is a continuous graph homomorphism from $F(2^{\mathbb Z^d})$ to $H$, in particular, the complete graph $K_4$ satisfies this property. This extends a theorem of Gao and Jackson. We also prove that $\big (\mathcal H_d\big )$ is a strictly decreasing sequence.

We study convergent and overconvergent isocrystals that become trivial after pullback along a proper surjective morphism. On a geometrically unibranch variety over an algebraically closed field, every such object is already trivialized by a finite étale cover. Over a perfect field, this shows that a proper cover giving geometric triviality can be replaced by a finite étale cover over the ground field, and implies finiteness of the geometric monodromy group. For proper geometrically unibranch varieties, finite geometric monodromy is also sufficient. In the overconvergent case, a dominant morphism can replace the given proper cover. A nodal curve shows why the geometrically unibranch hypothesis is needed.

At what rate does the von Neumann entropy of an ensemble of quantum states change under Hamiltonian evolution of its constituents? Bravyi proposed the small incremental mixing conjecture controlling the mixing rate of a binary ensemble $\{(1\!-\!p,\rho_1),(p,\rho_2)\}$ by $c\,\|{H}\|h_{2}(p)$ (with the binary entropy $h_{2}$). The proof of Bravyi's conjecture was subsequently reduced to the matrix inequality $\|[A,\log B]\|_{1}\!\leq\! c\,h_{2}(p)$ for positive trace-class operators $A\!\leq\! B$ with $\text{Tr} A\!=\!p$ and $\text{Tr} B\!=\!1$ on separable Hilbert spaces by Mariën, Audenaert, Van Acoleyen, and Verstraete, and they conjectured the optimal constant to be $c\!=\!1$. Here, I prove the latter conjecture using an exact integral representation of the commutator $[A,\log B]$ obtained from the operator layer cake theorem, due to Cheng and Liu. The result gives a sharp dimension-independent limit on how rapidly unitary evolution of one component can change the entropy of a binary quantum ensemble. It follows that mixing rates satisfy small incremental mixing with the optimal constant, and entangling rates of bipartite Hamiltonians are bounded by $(2\log d\!+\!1)\|H\|$. Moreover, the result corrects a conjecture by Lieb and Vershynina for mixing rates of general ensembles.

In this paper the incomparability of certain Rahm entropy maximal functions introduced by Rahm and Orlicz maximal functions that arise in quantitative endpoint estimates for Calderón-Zygmund operators is established. Also the dual Muckenhoupt-Wheeden two-weight estimates introduced by Lerner, Ombrosi and Pérez are revisited, improving the possible choices for the maximal function in the left hand side of the inequality, via a transference principle from two-weight endpoint inequalities.

Fall Armyworm (FAW) poses a serious threat to maize production in many regions due to its aggressive feeding habits and rapid development cycle. In this study, we developed and analyzed a stage-structured mathematical model to evaluate the impact of different FAW larval instars on maize dynamics during the vegetative and reproductive stages. Analytical results indicate that the two models have unique and positively bounded solutions for all time $t\geq 0$ and admit four equilibrium points: the trivial, non-trivial, maize extinction and coexistence equilibria. The behavior of the model was studied using stability analysis to find conditions under which FAW dies out or continues to spread. Furthermore, sensitivity analysis and numerical simulations were conducted to examine how key parameters affect FAW population dynamics and maize. Numerical simulations of the model in both stages indicate that there is high destruction of maize plants in both vegetative and reproductive stages of maize production due to increased egg production and larval population density. The extensive damage caused by large populations of eggs, larvae and adult moths motivated an extension of the two models to include intervention measures such as traditional methods like handpicking and chemical pesticides. Numerical results indicate that these control strategies significantly suppressed the FAW population with a resultant increase in the maize plant population towards its maximum capacity during the vegetative and reproductive stages, respectively.

We derive a method of analytic function expansion using barycentric rational interpolation, in which an arbitrary analytic weight function defines the interpolation and nodes. Truncating these expansions yields successive approximations, and the derivation of the expansions gives the approximation error as a byproduct. We demonstrate the method with cosine and the Bessel functions of the first kind for several choices of weight functions, and we examine the accuracy and convergence of the approximate interpolations as a function of the weight and the number of nodes used in the interpolation.

We establish nonasymptotic spectral comparison results for matrices whose entries are non-linear functions of an iid random field. Under an exponential decay assumption on an $\mathbb L^\infty$-coupling coefficient, we derive high-probability bounds for the Hausdorff distance between their spectra and those of Gaussian matrices with matching covariance structures. We further derive a comparison with a covariance-matched free model and derive corresponding noncommutative Khintchine-type bounds. The results apply, in particular, to matrices whose entries are nonlinear transformations of causal linear processes, Volterra-type processes, and neural networks. The proof combines finite-memory approximation, block decomposition, Gaussian interpolation, and resolvent estimates.

Distinguishing periodic from chaotic dynamics in a time series is a fundamental challenge in both physics and engineering. Yet, end-to-end learned architectures must discover both a representation and a decision boundary from data, at substantial cost. We introduce TopTimeNet, which decouples these tasks: a fixed, non-learned stage extracts a $42$-dimensional geometric and topological descriptor from Takens delay embeddings and persistent homology, and a lightweight learnable stage performs classification. On a benchmark of $49$ nonlinear dynamical systems, a $1{,}638$-parameter configuration matches the mean accuracy of one with $33\times$ more trainable parameters. Additionally, this approach delivers mean accuracy comparable to convolutional neural networks and surpasses the average performance of converged Transformer models, while requiring three to four orders of magnitude fewer trainable parameters. Robustness also depends sharply on where noise is introduced: TopTimeNet degrades gracefully under perturbations to its precomputed features, but degrades sharply when noise is introduced into the raw signal and the full feature-extraction pipeline is recomputed, showing that robustness to perturbations of the precomputed features does not imply robustness of the complete raw-signal-to-prediction pipeline. These results show that decoupling fixed geometric and topological feature construction from a lightweight discriminative stage can achieve comparable classification accuracy with substantially fewer trainable parameters.

We prove that a compact immersed disk of constant mean curvature in $\mathbb S^3$ or in $\mathbb H^3$ whose boundary is mapped diffeomorphically onto a circle is an embedded totally umbilical disk.

In this paper, we establish symmetry results for solutions of overdetermined problems arising in optimal transportation. These problems involve the Monge-Ampère equation with a Dirichlet boundary condition $u=0$ on $\partial \Omega$ and the natural boundary condition $Du(\Omega)=\Omega^{*}$. We show that symmetry holds when the target is the unit ball $B$. For more general target domains, including the cases $\Omega^*=\Omega$ and arbitrary $\Omega^{*}$, symmetry is retained under an additional volume constraint on $\Omega$ and a boundary condition on $|Du|$. Finally, we introduce a curvature-type overdetermined problem for the Monge-Ampère equation and obtain ellipsoidal or spherical symmetry under an additional integral normalization condition. Our proofs rely on distinct techniques in different contexts: optimal transport and convex analysis for some cases, integral identities and isoperimetric inequalities for others, and P-function methods for the remaining cases. As a byproduct, in the $\tau=2$ case, a new maximum principle for the $P$-function $\phi(x)=\sum_{k,l=1}^{n}{\frac{\partial{S_{\tau}(D^2{u})}}{\partial{u_{kl}}}u_{k}u_{l}}-2\binom{n-1}{\tau-1}\int_{0}^{u}{f^{\frac{\tau}{n}}(t)\,dt}$ is established for arbitrary positive and nondecreasing $f$, which is of independent interest.

With the rapid development of the low-altitude economy, low-altitude operations are booming, where complex missions require collaborative efforts among multiple heterogeneous low-altitude aircrafts (LAAs). Specifically, different LAAs assume distinct roles: some for sensing, some for communication, some for computing, and others for mission execution, together forming a sensing-communication-computing-control (SC3) closed loop, akin to a reflex arc. To enable efficient coordination in such multi-LAA swarms, we introduce the concept of operational-capability entropy (OCE) to quantify the effective work capability of operational LAAs. Accordingly, by jointly considering heterogeneous OCE and channel conditions among LAAs, we formulate the power allocation problem with the goal of minimizing the linear quadratic regulator (LQR) cost, which serves as a metric for mission efficiency. The resulting complex optimization problem is decomposed into two convex subproblems that are solved iteratively, with closed-form solutions derived for each. Simulation results demonstrate that the proposed mission-oriented adaptive power allocation scheme significantly outperforms traditional ones.

When $G$ is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of $G$. In this paper, we prove a similar theorem for certain subclasses of rank 2 affine MV polytopes. For the Kac-Moody group $\widehat{SL_2}$, an affine MV polytopes splits into three subpolytopes: a lower, a middle and an upper polytope. The lower polytopes are natural generalizations of finite-type polytopes with highest vertex labelled by an arbitrary Weyl element. We extend the known results from this subclass of finite-type MV polytopes to the case of lower affine MV polytopes of rank 2. We prove that for an element $w$ of the affine Weyl group, the class of lower affine MV polytopes with highest vertex $w$ are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by $w^{-1}$. To do this, we describe the BZ data of a rank 2 affine MV polytope and show that certain generalized minor functions satisfy the conditions of a lower polytope. As any upper polytope is a reflection of some lower polytope, a analogous result will hold for the class of upper affine MV polytopes of rank 2.

We study the reversible reaction $1+2\rightleftarrows3+4$ in a dilute gas described by a Boltzmann system. We prove that the outgoing flux of one product species determines the forward collision kernel throughout the domain on a prescribed admissible set of relative velocities and outgoing directions. The elastic collision laws are known, and the reverse kernel is fixed by a prescribed reciprocity relation. The measurements use two synchronized incoming packets with fixed relative velocity and a common velocity tending to infinity. Under bounded angularly integrated collision rates, the short transit time allows us to isolate the contribution of one forward reaction, with estimates uniform as the packets concentrate. Suitable tests of the product flux yield the spatial line integrals of the kernel and a Fourier reconstruction, without differentiating the nonlinear boundary map. We also estimate the effects of finite beam speed, packet width, detector resolution, and measurement noise on the recovered line integrals.

We introduce notions of non-commutative Fano varieties and non-commutative minifolds, and study the latter in dimension three. We prove that the Euler matrices of full exceptional collections generating geometric helices on these minifolds form five mutation classes, distinguished by their Euler degrees \(22,40,54,64,72.\) The first four classes are realized by (commutative) Fano threefolds \(X_{22},V_5,Q^3,\mathbb P^3,\) respectively. The fifth is realized by the non-commutative family \(M_{72}\) and cannot be realized by a full exceptional collection on a smooth projective threefold.