continuity analysis

This paper addresses the escape problem—determining whether iterates of a continuous map over the reals eventually leave a given closed Euclidean subset—within the bit-model of real computation. We present the first sound, robust, and partially complete decision algorithm: it always terminates on all instances stable under arbitrarily small functional perturbations, and its halting set is dense in the full parameter space. Our approach integrates real computability theory, robustness analysis, dynamical systems theory, and computability-theoretic techniques. The algorithm is applicable to affine linear systems and, conditionally on the Hyperbolicity Density Conjecture, to complex quadratic polynomials. As a consequence, it yields a novel proof strategy for the computability of the Mandelbrot set. Moreover, for both system classes, we achieve conditional decidability—substantially improving upon Hertling’s (2004) resolution of Penrose’s (1989) escape problem, which lacked robustness and density guarantees.

This study investigates the robustness of optimal transport maps when the target measure is contaminated while the reference measure remains fixed, with a focus on how the breakdown point depends on the transport cost function. By integrating optimal transport theory, Tukey depth analysis, and a general framework for convex cost functions, the authors establish that the breakdown point is independent of the specific choice of cost function and is instead determined solely by the Tukey depth of the reference measure. This result unifies and extends existing findings, providing the first complete characterization of the high breakdown point shared by all transport-based quantiles under a broad class of regular cost functions.

Existing methods for continuous Gromov–Wasserstein optimal transport (GWOT) implicitly rely on discretization and struggle to model parametric mappings between unknown continuous distributions, leading to fundamental deficiencies in theoretical modeling, numerical stability, and geometric fidelity. Method: We propose the first fully continuous GWOT framework—abandoning histogram approximations and grid-based constraints—and instead employ variational inference coupled with neural implicit mapping to learn differentiable, discretization-free parametric couplings. To ensure theoretical consistency with continuous OT, we integrate unbiased gradient estimation and adversarial distribution alignment. Contribution/Results: Experiments demonstrate that our method significantly outperforms state-of-the-art approaches in non-aligned, high-dimensional, and non-Euclidean manifold settings. It achieves superior mapping isometry and generalization on both synthetic and real-world data, establishing a differentiable and scalable theoretical and computational foundation for continuous GWOT.

This paper investigates the approximate controllability of the continuity equation under ReLU-type vector fields, aiming to construct a piecewise-constant time-varying control parameter θ = (w, a, b) that steers an initial density ρ_B arbitrarily close to a target density ρ_*. Using the relative entropy as the approximation error metric, the work establishes— for the first time—the approximability theory for ReLU vector fields in the relative entropy sense: arbitrary-precision approximation is achievable when the base and target distributions satisfy a relative tail-decay condition. It further provides an explicit upper bound on the number of control switches (i.e., segments of the piecewise-constant parameter), and characterizes the structure of the reachable set of the continuity equation under the relative entropy norm. This analysis furnishes a novel controllability-theoretic foundation and complexity guarantees for distribution transport via normalizing flows.

This paper addresses the lack of a unified topological foundation for robustness modeling in continuous quantum-valued metric spaces. Methodologically, it introduces, for the first time, a preorder-enriched Hausdorff–Smyth monad on such spaces, thereby naturally inducing a robust topology on the power set and proving that every topology is generated by some quantum-valued metric. The contributions are threefold: (1) establishing a unifying framework demonstrating equivalence between the generalized open-ball topology and the robust topology; (2) uniformly characterizing robustness under parameter perturbations via the monadic structure; and (3) providing a quantitatively rigorous mathematical foundation for imprecision and uncertainty in computational systems and physical models. Integrating quantum-valued metrics, uniformly continuous mappings, and monad theory, this work extends the categorical semantics of robustness formalization.

This work proposes a trajectory-restricted framework for linear convergence analysis that overcomes the conservatism of traditional first-order methods, whose guarantees often rely on global geometric conditions and worst-case constants. Instead of imposing regularity assumptions globally, our approach requires only local geometric properties—such as restricted Polyak–Łojasiewicz inequalities, error bounds, and quadratic growth—on the subset of the space actually traversed by the algorithm. We establish explicit relationships among the associated constants and show that, for piecewise polyhedral composite problems, once iterates enter a well-conditioned active manifold, convergence is governed by the restricted Hoffman constant of that manifold, yielding an improved effective condition number and faster local convergence. The results demonstrate that linear convergence fundamentally depends on the local geometry encountered along the algorithmic trajectory, rather than on global worst-case scenarios.

Existing approaches to persistent homology struggle to characterize the evolution, reorganization, and memory effects of homological features in parameter-dependent topological data. This work proposes a unified framework based on the zero modes of the combinatorial Hodge Laplacian to track homological feature evolution within a shared chain space. For the first time, it incorporates curvature and holonomy from differential geometry to capture local reorganization dynamics and cyclically accumulated memory, respectively. By transcending the representational limitations of traditional persistence diagrams, the method successfully identifies instability in feature tracking in time-varying point cloud experiments, distinguishes systems whose persistence diagrams are highly similar, and reveals higher-order cyclic memory structures that pairwise matching approaches fail to detect.

This work addresses the problem of deciding robust safety for dynamical systems governed by polynomial differential equations over bounded time horizons. By reducing δ-robust safety to the sound axiomatization of polynomial invariants, the authors construct the first complete logical proof system and integrate it with a computable algorithm to decide safety for arbitrary perturbation parameters δ. Innovatively leveraging subanalytic geometry, the approach enables inductive safety proofs and approximate decidability for general hybrid dynamical systems without requiring positive separation between initial and postconditions. This paper establishes, for the first time, a complete axiomatization framework for robust safety and provides an effective symbolic verification algorithm.

This work presents a formal verification in Isabelle/HOL of the global qualitative properties of the SIR epidemic model’s differential equations, including existence and uniqueness of solutions, non-negativity invariance, conservation of total population, monotonicity, and the threshold condition for infection growth. Building upon the Picard–Lindelöf flow framework from the Archive of Formal Proofs, the authors establish—for the first time—a complete formal bridge from local flows to global forward solutions and develop a reusable library of scalar compartment lemmas. By integrating the existence–uniqueness theorem, compactness arguments, conserved quantity analysis, and sign analysis of derivatives, all key properties are rigorously proved with no unproven assumptions, thereby providing a scalable foundation for the formal verification of epidemiological models.