Papers

This paper studies a partially observed stochastic optimal control problem for jump-diffusion states observed through multivariate counting processes with state-dependent intensities. Because the observation jumps carry information about the latent state through their intensities, the variational analysis involves coupled state and likelihood-ratio perturbations. By introducing a reference probability measure and augmenting the state with the counting-process likelihood ratio, we establish a stochastic maximum principle whose first-order necessary condition is expressed as a conditional Hamiltonian stationarity relation with respect to the observation filtration. Finally, an LQ example is used to examine the compatibility of a linear adjoint ansatz and to illustrate numerically the coupled partial-information stationarity system.

Inverse Optimal Control (IOC) aims to recover the cost function that explains observed trajectories as solutions of an optimal control problem. Classical IOC formulations rely on bilevel optimization, which repeatedly solves a nested optimal control problem and quickly becomes computationally prohibitive for realistic systems. Recent projection-based approaches offer a promising alternative but suffer from numerical instability when solved with gradient-based methods due to violations of standard constraint qualifications. In this paper, we show that these difficulties stem from the geometric structure of the IOC feasible set. We demonstrate that the set of trajectories satisfying the optimality conditions naturally forms a manifold and reformulate IOC as an optimization problem on this manifold. Based on this insight, we propose a Riemannian Inverse Optimal Control (RIOC) method that projects observed trajectories onto the manifold of optimal solutions while preserving feasibility by construction. Experiments on real human arm trajectories show that the proposed method achieves comparable or better reconstruction accuracy than classical bilevel IOC while reducing computation time by about a factor of four. These results highlight the potential of geometric optimization methods to improve the scalability and reliability of IOC for robotics and human motion analysis.

Groundwater models are routinely non-identifiable: sparse subsurface observations leave many combinations of parameters, states, and boundary conditions equally consistent with the available data. Existing uncertainty quantification (UQ) methods address this by exploring a finite set of model realizations, but incomplete exploration can systematically underestimate the true range of admissible solutions. We propose a fundamentally different approach based on Optimization-based Bound Tightening (OBBT), which represents uncertainty directly as intervals and tightens them by extremizing variables over a constraint system encoding physical laws and observations. This yields guaranteed outer bounds on all uncertain variables without sampling, side-stepping the exploration problem entirely. To apply OBBT to groundwater flow, we discretize Darcy's law using a finite-volume scheme and handle the resulting bilinear terms through McCormick relaxations. We show that these relaxations can break the sign coupling between fluxes and head gradients, permitting non-physical rotational flow and failing to provide sufficient information for effective bound tightening. We identify flow sign prescription and irrotationality constraints as effective remedies and characterize their respective strengths and limitations. We demonstrate the framework on three numerical examples - a 1D steady-state model, a 2D steady-state model across four experimental configurations, and a 2D transient model on a hexagonal grid - and discuss computational performance, scalability, and directions for future research. OBBT offers a conservative, deterministic, and physically grounded alternative to ensemble-based UQ, with natural connections to null space theory and data assimilation.

In this paper, we develop an abstract theory of derivatives for Banach spaces based on objects that we call bidual assignments. This framework encompasses both the Semadeni derivative and the recently introduced Semadeni–Pełczyński derivative. More generally, suitable ideals of subsets of dual spaces give rise to a broad family of derivatives within this setting. We establish direct-sum and tensor-product formulas for these derivatives, showing that they behave naturally with respect to direct sums and injective tensor products. We then obtain explicit descriptions of derivatives associated with compact trees and study the corresponding questions for finite products of compact lines. As the main application, we establish classification results for spaces of the form $C(K^n,X)$. In particular, for uncountable ordinals $\alpha$ and $\beta$, an integer $n\geq1$, and Banach spaces $X$ satisfying suitable rigidity assumptions, we prove that \[ C([0,\alpha]^n,X)\sim C([0,\beta]^n,X) \quad\text{if and only if}\quad C([0,\alpha])\sim C([0,\beta]). \] This extends Kislyakov's classification of the spaces $C([0,\alpha])$ and its vector-valued extension due to Galego to finite powers of ordinal intervals.

These are lecture notes for the PCMI 2024 Graduate Summer School for the mini-workshop on motivic explorations in enumerative geometry. Motivic homotopy theory allows to do enumerative geometry over an arbitrary field, which leads to additional arithmetic and geometric information. The goal of the mini-workshop is to explain why and how this works. We will also provide a toolbox for solving enumerative geometry problems in this setting, including the use of tropical geometry. We start with two classical examples in enumerative geometry, namely Bezout's theorem and the count of lines on a smooth cubic surface. We then explain how to solve these problems, first over the complex and real numbers, and then over an arbitrary field, using the A1-degree from motivic homotopy theory. Then we introduce tropical geometry, more precisely we focus on tropical plane curves and show how they can be used to prove Bezout's theorem for curves over an arbitrary field. Finally, we discuss tropical correspondence theorems.

The Euler Characteristic Curve (ECC) records the Euler characteristic of a linearly embedded cell complex as a function of filtration height in a given direction, and the Euler Characteristic Transform (ECT) is the injective shape descriptor obtained by collecting ECCs over many directions. How the ECT is encoded for a neural network is itself an inductive bias, conventionally fixed by discretizing each ECC. We introduce a continuous encoding: for each direction and each vertex it records the net Euler-characteristic change attributed to that vertex, producing a per-direction token sequence that a small transformer maps to a feature vector. We separate the resulting pipeline into two stages on orthogonal axes: an ECC encoder that acts within each direction, mapping its curve to a fixed-length vector, and an ECT representation that acts across directions, aggregating the per-direction vectors into one. We study six ECT representation architectures spanning a range of inductive biases, from a structure-agnostic feedforward baseline to convolutional and complex-valued models that preserve equivariance under planar rotations. Across six classification benchmarks covering point clouds, graphs, cubical complexes, and meshes, the continuous encoding improves accuracy on all six datasets, and control experiments attribute the gain to the tokenization itself rather than to the added transformer capacity. The representation architecture matters less than the encoding, and the payoff from its inductive biases depends on the encoding: a feedforward network performs best under continuous encoding but is less robust under discretization than convolutional architectures.

We study Euler characteristics of tropical subvarieties of tropical abelian varieties. We prove that every H-regular subvariety, locally modeled on tropicalizations of sufficiently well-behaved very affine varieties, has nonnegative signed Euler characteristic. This gives a tropical analogue of a theorem of Green-Lazarsfeld for subvarieties of complex abelian varieties. The main input is a local vanishing theorem for H-regular tropical fans, which also yields a Lefschetz-type theorem for affine H-regular tropical varieties. We further show that the signed Euler characteristic inequality fails for general tropical subvarieties of tropical abelian varieties, and we construct a 3-dimensional tropical fan whose link is not homotopy equivalent to a bouquet of 2-spheres.

The Dyn-Farkhi conjecture asserts that the square of the Hausdorff distance from a compact set to its convex hull is subadditive with respect to Minkowski addition. The conjecture is elementary in dimension 1, was recently proved by Meyer in dimension 2, and was disproved in dimensions $n\geq3$ by Fradelizi, Madiman, Marsiglietti, and Zvavitch. The symmetric case $A=B$, however, remained open. We show that the conjecture already fails in this restricted setting. More precisely, for every $n\geq3$, we construct a compact set $A\subset\mathbb{R}^n$ such that $$d(A(k))=d(A)>0$$ for every $1\leq k\leq n-1$, where $d(X)$ is the Hausdorff distance from $X$ to its convex hull and $A(k):=\frac1k (A+\dots+A)$ is the $k$-fold iterated Minkowski average of $A$. We also prove that the threshold $k=n$ is sharp: for every nonempty compact $A\subset\mathbb{R}^n$ with $n\geq 2$, we have $$d(A(n))\leq \left(1-\frac{n-1}{n(2n-1)}\right)d(A).$$

In this paper, we combine a data-driven system representation with a framework to systematically construct (approximate) solutions to nonlinear optimal control problems. By immersing the unknown dynamics into an extended state space, solutions are characterised via purely data-dependent algebraic conditions. This allows us to design dynamic state-feedback controllers with local stability and performance guarantees for unknown nonlinear, input-affine systems directly using data, without explicitly identifying the dynamics.

Moonshine is an autonomous agent whose central objective is to generate mathematical conjectures. Its core capability is to extract structure from classical problems, distill new concepts, and formulate conjectures of mathematical significance. Rather than treating the solution of a single proposition as its endpoint, Moonshine builds an extensible theoretical framework through conjecture generation, bridge building, and obstacle identification. This article uses Moonshine's exploration of the Jacobian conjecture as an example. It shows how the central logic of whether local nondegeneracy can force global injectivity is transferred to one-hidden-layer affine-ridge sigmoid networks. This leads to the formulation of the Neural Jacobian Conjecture (NJC): if such a network has strictly positive Jacobian determinant on the whole space, then it must be globally injective. By invoking GPT-5.5-pro and DeepSeek-V4-pro separately, Moonshine obtained independent complete proofs for the case \(N=n+1\). In addition, with the assistance of ChatGPT through interactive use of its web interface with GPT-5.5-pro, a geometric-topological proof was developed. These results provide preliminary evidence for the plausibility of the conjecture. The general higher-width case \(N\ge n+2\), however, remains unresolved and is left for further investigation. This work illustrates Moonshine's ability to autonomously generate meaningful mathematical problems and make rigorous progress on them.

We formulate a nonlinear geometric theory of visco-anelasticity that unifies viscoelastic and anelastic responses within a single thermodynamic framework. At each material point, the total deformation gradient is multiplicatively decomposed into elastic, viscous, and anelastic distortions, thereby generalizing the Bilby-Kröner-Lee decomposition to visco-anelasticity. The theory explicitly incorporates the material metric, which encodes the evolving natural configuration of the solid, the transformed structural tensors, and provides a consistent formulation of the constitutive equations, the balance laws, the thermodynamic potentials, and the kinetic equations. The first and second laws of thermodynamics are systematically applied to derive the constitutive and evolution equations without invoking observer invariance. Anisotropy is treated in full generality through structural tensors. As illustrative examples, we specialize the general framework to isotropic and transversely isotropic visco-anelastic solids. Two examples within the class of universal deformations, which admit closed or partially closed form solutions, show how the proposed framework can be used to model the coupled viscous and anelastic response of incompressible anisotropic solids with distributed eigenstrains and the associated residual stresses. This geometric framework unifies nonlinear viscoelasticity and anelasticity by coupling time-dependent and eigenstrain-driven effects within a single, fully consistent geometric formulation. In particular, the proposed framework clarifies the geometric structure of the elastic, viscous, and anelastic distortions and resolves ambiguities associated with intermediate configurations in existing formulations of nonlinear viscoelasticity and viscoplasticity.

In this paper, we study some refined properties of uniformly quasi-greedy bases. In particular, we characterize the stability of uniformly quasi-greediness under scalings with bounded quotient and show that unconditionality is sufficient to guarantee this property. For the "isometric" case, i.e., when the uniformly quasi-greedy constant is $1$, we prove that 1-uniformly quasi-greediness implies disjointness if the underlying norm is strictly monotone. We also introduce a stronger notion of quasi-greediness by requiring uniform order boundedness of the greedy sums over greedy orderings. The characterization and properties of such bases are derived along the lines of uniformly quasi-greedy bases.

We study a one-sided degenerate Cahn-Hilliard equation with anisotropic traction flux, arising as a reduced continuum description of mechanically biased cell interactions in digit-forming organoids. The equation combines a one-sided degenerate mobility with a density-weighted anisotropic higher-order transport term. This traction term is not generated by the variational derivative of the Cahn-Hilliard energy and therefore produces sign-indefinite contributions in the energy balance. For nonnegative initial data, we prove the global-in-time existence of nonnegative weak solutions. The proof combines an energy estimate for the diffusive flux with a mobility-matched entropy method adapted to the vacuum degeneracy. A key point is that the entropy variable cancels the mobility, turning the anisotropic traction contribution into a coercive first-order term in the entropy inequality, while the energy estimate supplies a weighted control of the diffusive flux.

In this work, we review and revisit the generating function techniques that provide exact analytical solutions for aggregation and fragmentation equations across several physical regimes including spontaneous and collisional shattering. For discrete coagulation-fragmentation equations with size-independent rates under monodisperse initial conditions, we show the derivation of several explicit closed-form solutions. We also briefly report the exact solutions for continuous, three-particle, $D$-particle collisions and two-component generalizations. Source-driven aggregation yields steady distributions featuring a universal $s^{-3/2}$ power-law decay and a cutoff mass scaling $s_{*} \sim t^{2}$. We also remind that exploitation of the low-rank representations of the aggregation kernel allows to organize efficient modelling the aggregation kinetics and extend this idea showing a simple way for the fast accounting of the spontaneous framentation terms. We exploit our numerical algorithm and study the numerical solutions for multiple models of spontaneous fragmentation and find a new example of temporal oscillations for the closed aggregation-fragmentation process.

Let $\Gamma = (G, \sigma)$ be a signed graph and $v$ a non-isolated vertex of $\Gamma$. Let $\Gamma-v$ denote the graph obtained by deleting the vertex $v$ together with all signed edges incident to it from $\Gamma$, and $d_{\Gamma}(v)$ the degree of $v$ in $\Gamma$. In this paper, we prove that the largest eigenvalue $\lambda_1(\Gamma)$ of $\Gamma$ satisfies \[ \lambda_1(\Gamma) \le \sqrt{\lambda_1^2(\Gamma - v) + 2d_\Gamma(v) - 1}, \] and we also present a refined version of this bound. Moreover, we characterize the extremal signed graphs achieving equality when $\Gamma$ is connected and $d_\Gamma(v)\ge 2$, which are switching equivalent to the balanced complete signed graph.

Secure aggregation is a vital component for mitigating gradient leakage in federated learning, but its communication cost conventionally scales with the gradient dimension. This becomes prohibitive for large models and even more pronounced in decentralized federated learning with limited bandwidth and unreliable nodes. Top-K gradient sparsification is an effective approach to reduce communication by transmitting only a few entries of the full gradient, while maintaining competitive model accuracy. Nevertheless, the top-K entries selected by each user are unpredictable and vary across users, which poses a challenge for efficient sparse secure aggregation. This paper studies information-theoretic secure aggregation with top-K sparsification in decentralized federated learning under user dropouts and user collusion. We propose a communication-efficient sparse secure aggregation scheme that offloads dimension-dependent overhead to an offline phase and protects private gradients using random masks and permutations. Experimental results demonstrate that our scheme preserves accuracy comparable to full-gradient aggregation even with only 1% gradient sparsification, while substantially reducing the communication cost.

We study Stein fillings of planar contact 3-manifolds under certain constraints on their relative trisections. We partially classify the diffeomorphism types of such fillings admitting relative trisections with genus at most 2.

Positive structures in Lie groups with respect to a subset $\Theta$ of the set of positive roots provide a generalization of Lusztig's total positivity in split real Lie groups to the setting of general real semisimple Lie groups. In [GW25] Lie groups G admitting a positive structure were classified and many key properties of the unipotent positive semigroups were established. In this article we focus on the positive semigroup in G. We establish key geometric properties of elements in the positive semigroup. Further we introduce corresponding positive and non-negative parts of flag varieties and determine the topology of the non-negative parts of flag varieties in many cases. For symplectic flag varieties we provide explicit descriptions of the positive and non-negative flag varieties.

Two (pseudo-)Riemannian metrics are said to be geodesically equivalent if they share the same geodesics considered as unparametrized curves. In 1865, E. Beltrami posed the problem of describing all geodesically equivalent metric pairs locally. At generic points, this problem was solved by Dini in the Riemannian setting and later extended to the pseudo-Riemannian case by Bolsinov, Matveev, and Pucacco. In the present paper, we solve the Beltrami problem in dimension two by providing a complete local classification of geodesically equivalent pseudo-Riemannian metrics in a neighbourhood of a singular point. This yields a full solution of the problem in dimension two.

In many dynamical systems applications, one seeks representative trajectories summarizing families of periodic orbits arising from parameter variation or uncertainty. While the Fréchet mean provides a natural notion of averaging in nonlinear metric spaces, its application to periodic trajectories is complicated by the interplay between geometric shape and temporal dynamics. Inspired by ideas from shape analysis, we develop a framework for computing Fréchet means of periodic orbits by representing trajectories as closed curves and introducing a metric structure on a quotient space that accounts for circular phase shifts. Within this framework, we establish the existence of empirical Fréchet means in the resulting infinite-dimensional quotient space. A central finding is that the resulting Fréchet mean depends strongly on the chosen parametrization: time parametrization preserves dynamical information, whereas arc length parametrization emphasizes geometric structure. To reconcile these viewpoints, we propose a decoupled approach that computes a geometric Fréchet mean using arc length parametrization and subsequently reconstructs representative dynamics through harmonic averaging of aligned speed profiles. We further introduce curvature- and medoid-based diagnostic measures that quantify the representativeness of the resulting mean and identify situations in which averaging produces geometric artifacts or fails to capture heterogeneous ensembles. Numerical experiments for the Van der Pol oscillator, the Rosenzweig-MacArthur predator-prey model, and the Morris-Lecar neuronal model demonstrate that the proposed methodology yields robust geometric summaries together with consistent averaged dynamics. The framework provides a principled approach for constructing representative periodic trajectories and assessing their validity in uncertainty quantification and dynamical systems.

The Fermat principle of least time determines the path of a single ray of light given its initial and final positions and the medium electromagnetic properties. However, a single ray is not a measurable physical object. We derive here an upgraded variational principle for the geometric optics limit of Maxwell equations in an arbitrary medium, based on measuring the intensity of the wave on two planes. The principle provides the complete Fresnel rays bundle connecting associated points at the first and second planes. One of the applications of the present theory is to use the reciprocity between Fresnel rays and phase normals to determine the phase of an electromagnetic wave from two intensity measurements.

We prove the uniformization theorem for domains in the Riemann sphere via the Kobayashi metric. Our proof does not rely on the existence of the elliptic modular function and instead uses the characterization of complete Kobayashi hyperbolicity for domains in the sphere.

We investigate the long-time asymptotics of a full soliton gas for the Camassa–Holm equation. The analysis starts from a pure-soliton Riemann–Hilbert (RH) problem with \(2N\) poles and two distinct types of residue conditions. We prove that, as \(N\to\infty\), this discrete RH problem converges to a limiting soliton gas RH problem whose jump matrix contains two nonzero reflection coefficients. In this sense, the limiting problem gives a full soliton gas model for the Camassa–Holm equation, in contrast to the previously studied half soliton gas models, whose jump matrices involve only one nonzero reflection coefficient. The limiting RH problem is analyzed by the Deift–Zhou nonlinear steepest descent method. The presence of two nonzero reflection coefficients requires two different types of triangular factorizations of the jump matrix and leads to a more delicate \(g\)-function mechanism. The main difficulty lies in the construction of suitable \(g\)-functions adapted to the Camassa–Holm phase, together with the precise control of their behavior near the distinguished point \(k=i/2\) and at infinity. Depending on the location of the spectral endpoints \(\eta_1\) and \(\eta_2\), different \(g\)-function mechanisms arise. In this paper, we focus on Case I and derive the long-time asymptotic formulas in three elliptic-wave regions of the self-similar plane. In each region, the leading term is given by a finite-gap elliptic function, while in the central region the first correction is of order \(\mathcal O(t^{-1/2})\) and involves parabolic cylinder functions.

We show that q-Gaussian $C^*$-algebras for $q \in (-1, 1)$ have the Dixmier averaging property, and hence are simple with a unique trace. We argue by combining rapid decay and spectral gap estimates for the commutators with the generators, which are obtained from free probability.

Max-policy iteration is an approach to computing precise numeric program invariants by successive attempts at resolving maximum operators and reduction to mathematical optimization. Mathematical optimization, though, may be expensive. Here, we show, for max-policy iteration on systems of equations over integers as well as over floating point numbers, that mathematical optimization can be replaced by plain value iteration – which is still guaranteed to terminate. As an application, a precise bound analysis for integer or floating point variables is obtained, avoiding widening operators altogether. We also consider max-policy iteration over the rational numbers, where the right-hand sides are maxima of minima of affine combinations of unknowns. We propose min-policy iteration as an alternative to linear programming for solving the optimization problems posed by max-policy iteration. We prove that max-min policy iteration is guaranteed to return the least solution for bounded systems. We also show how to extend this algorithm to unbounded systems, and how to construct certificates of soundness as well as of optimality of the computed results.

We consider the problem of disjointness of rescalings $(\varphi_{\kappa t})_{t\in \mathbb{R}}$, $\kappa\in\mathbb{R}$ of a flow $(\varphi_{t})_{t\in \mathbb{R}}$ in the context of smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces. We show that, when the genus of the surface is $g\ge 2$, almost every locally Hamiltonian flow with 2g-2 non-degenerate simple saddles is such that any distinct two rational rescalings $(\varphi_{\kappa t})_{t\in \mathbb{R}}$ and $(\varphi_{\kappa' t})_{t\in \mathbb{R}}$ with $\kappa=p/q$ and $\kappa'=p'/q'$ of different absolute values, are disjoint. Previous results on disjointness of rescalings were available only for rescalings for locally Hamiltonian flows and their special flow representations in genus one. The result is proved using a criterion for disjointness based on the study of the distribution of Birkhoff sums of a special representation and in particular estimates on their exponential tails decay. A key novel geometric ingredient in the proof is the existence of a sequence of rigidity times which display what we call bounded-type rigidity, so that a large set of points comes back in time $q$ with distance $O(1/q)$. To produce such bounded-type rigidity times we exploit a particular way of degeneration of a translation surface to a flat torus for which the vertical flow has bounded-type rotation number.

This paper introduces a restricted Chen-Nagano variational principle for the Einstein-Hilbert functional on compact Riemannian manifolds. Instead of considering arbitrary symmetric variations of the metric, we restrict the variational problem to an infinite-dimensional subspace determined by the Chen-Nagano gauge constraint. We derive the corresponding restricted Euler-Lagrange equations and obtain a novel structural characterization of critical metrics. The resulting criticality condition is expressed by the equation $E_g = B_g^{*}(\theta) + c\,g$ which may be regarded as a restricted counterpart to the classical Einstein equation. Furthermore, we demonstrate that this variational framework naturally leads to generalized Ricci almost soliton structures and, in the gradient case, to gradient Ricci almost solitons. Several global rigidity consequences of this restricted principle are also established.

We prove a phase space criterion for dynamical Amrein-Berthier uncertainty principles. The abstract result says that, for a Fourier integral operator $A\in FIO(\chi)$ associated with a tame canonical transformation $\chi$, the localized operator $\mathbf{1}_E A\mathbf{1}_F$ is compact on $L^2(\mathbb {R}^d)$ whenever $\chi$ satisfies a vertical non refocusing condition: high frequency covectors issued from a spatially localized region cannot return to a vertical direction over the observation region. In the linear symplectic case this condition is equivalent to the familiar nondegeneracy $\det B\neq0$ of the upper right block of the symplectic matrix. We apply this compactness theorem to Schrödinger propagators for Yajima–type Hamiltonians, including quadratic electric and linear magnetic growth, and obtain two–time Amrein–Berthier inequalities for compact localization sets at all nonrefocusing times. The result extends the compactness mechanism behind the dynamical Amrein-Berthier principle to a genuinely microlocal setting.

We study the approximation of certain renormalized conserved quantities for the NLS hierarchy with nonzero boundary conditions, in the long-wave regime, by the energies of the KdV hierarchy. We extend this to all $n \in \mathbb{N}$ by proving an approximation result for the transmission coefficient of the Lax operator of the NLS hierarchy, which is a Dirac operator in the nonrelativistic regime, by the transmission coefficient of a Schrödinger operator, which is the Lax operator of the KdV hierarchy. This yields a formal approximation result between the hierarchies, which we quantify using energy methods and previously established well-posedness results.

We prove dynamical Amrein-Berthier uncertainty principles for fractional Schrödinger flows. For the free Hamiltonian $H=(-\Delta)^\alpha$ on $L^2(\mathbb{R}^n)$, with $\alpha>\frac{1}{2}$, we show that two–time localization on finite measure sets $E,F$ forces the quantitative estimate \begin{equation*} \|u(t)\|_{L^{2}}\lesssim_{E,F,T,n,\alpha} \|u(0)\|_{L^{2}(E^{c})} + \|u(T)\|_{L^{2}(F^{c})}, \qquad T\neq0,\ t\in \mathbb{R} \end{equation*} for $u(t)=e^{-itH}u(0)$ at every time. The threshold $\alpha>\frac{1}{2}$ is tied to the stationary phase structure of the fractional kernel. If $\alpha\ge1$ the sets can be arbitrary finite measure sets; if $\frac{1}{2}<\alpha<1$ we impose the finiteness of a natural interaction energy \begin{equation*} \textstyle \mathcal{I}_{\gamma}(E,F) = \int_{\mathbb{R}^n \times \mathbb{R}^n} \mathbf{1}_{F}(x)|x-y|^{2\gamma}\mathbf{1}_{E}(y)\,dx\,dy<\infty, \qquad \gamma = \frac{n(1-\alpha)}{2 \alpha-1} \end{equation*} of the pair $(E,F)$, essentially equivalent to a sufficiently fast joint decay of the measure of the sets at infinity. In particular, compact support at two distinct times is impossible for a nonzero solution. We also prove corresponding results for one dimensional fractional Hamiltonians $(-\partial_x^2+V)^\alpha$ under weighted scattering assumptions, and for higher order Hamiltonians $(-\Delta)^m+V$ for suitable classes of decaying potentials $V$.