semigroup generator calculus

This work proposes a unified mathematical framework grounded in dynamic information flow for constructing structurally rigorous models of future prediction. By integrating filtering theory, regular conditional probabilities, Markov semigroups, infinitesimal generators, and multiple information geometries—including Hilbert, Fisher–Rao, and Wasserstein—the approach conceptualizes prediction as the construction of conditional distributions governed by informational, geometric, and modeling constraints. The framework elucidates deep connections among classical results such as the tower property and semigroup laws, as well as Itô’s formula and backward equations. Explicit transition laws, spectral decompositions, term structures, and asymptotic behaviors are derived within canonical models like Ornstein–Uhlenbeck and Cox–Ingersoll–Ross, thereby establishing a compact mathematical mapping from idealized theoretical constructs to empirical forecasting.

This work proposes a novel Toeplitz filtering framework for accurately estimating the spectral properties of linear evolution operators—such as Koopman or transfer operators—from equation-free equilibrium trajectory data. By introducing Toeplitz structure into spectral estimation and incorporating structural priors like self-adjointness or skew-symmetry on the infinitesimal generator, the method enables efficient recovery of eigenvalues, eigenfunctions, and spectral measures. Coupled with a primal-dual statistical learning algorithm, the framework achieves both statistical consistency and computational efficiency. Numerical experiments demonstrate that the approach precisely reconstructs fine-grained spectral structures in both deterministic and chaotic dynamical systems—features often missed by conventional data-driven techniques.

This paper systematically investigates the algebraic structure and computational properties of finite semigroupoids—i.e., categories without identities—in automata theory. Addressing the key issue that associativity and type consistency are logically independent, it introduces a rigorous framework distinguishing strict versus lax homomorphisms for arrow-typed semigroupoids and establishes the first systematic enumeration method. Using relational and declarative programming techniques, the work enables abstract generation of partial composition tables, automated homomorphism checking, and construction of minimal transformation representations. Contributions include: (1) the first open-source tool supporting enumeration, homomorphism analysis, and representation reduction for semigroupoids; (2) verification of existence and essential distinctions among multiple type structures; and (3) strengthened algebraic characterization of typed computation, providing a novel foundation for categorical semantics and formal verification.

This study addresses the computational inefficiency in enumerating congruences on finite inverse semigroups. Methodologically, it introduces a novel algorithm integrating computational group theory, automata theory, and the algebraic structure of inverse semigroups: it constructs a state-transition automaton from generating pairs, leverages group-action orbit decompositions and the normal series property of inverse semigroups to efficiently enumerate and verify congruences, and—uniquely—systematically unifies these three theoretical frameworks to enable algebraically guided pruning and parallelization. Experiments demonstrate that the algorithm achieves speedups of 2–4 orders of magnitude over existing methods and successfully computes congruences for inverse semigroups of order exceeding 1,000, substantially extending the feasible problem scale. The primary contribution is the establishment of a cross-theoretical framework for congruence computation on inverse semigroups, providing a new paradigm for automated algebraic analysis.

This study addresses ambiguity in both the initial distribution and transition mechanism of continuous-time Markov processes. To jointly model these two sources of uncertainty, we introduce the novel concept of *imprecise Markov semigroups*. We establish geometric and topological ergodicity criteria—applicable to Euclidean spaces, Riemannian manifolds, and general measurable spaces—and rigorously prove sufficient conditions for exponential decay of uncertainty over time. Our methodology integrates convex analysis, operator semigroup theory, differential geometry, and measure theory. This work constitutes the first extension of classical ergodicity theory to settings with imprecise probabilistic specifications. The resulting framework provides a verifiable theoretical foundation and a unified analytical toolset for robust machine learning and uncertainty-aware visual modeling.

This work addresses the lack of a unified convergence analysis framework for population-based optimization algorithms, which hinders systematic comparison and generalization. The authors propose an operator calculus framework that models diverse algorithms as compositions of three fundamental operators—mutation, selection, and recombination—acting on probability measures. By leveraging mean-field limits, they derive a continuous-time transport-reaction-jump partial differential equation governing the algorithmic dynamics. Building upon operator semigroup theory and functional analysis on spaces of probability measures, they develop a modular Lyapunov method that enables dissipativity verification operator by operator. Under explicit stability and regularity conditions, they establish exponential decay of both a state-space Lyapunov functional and the search error, thereby providing a unified guarantee of exponential convergence for a broad class of distributed optimization algorithms.

This work addresses the efficient approximation of the Koopman operator for nonlinear dynamical systems by proposing a novel construction of observables based on the continuous wavelet transform. It establishes, for the first time, a rigorous proof that these wavelet-based observables serve as eigenfunctions of the Koopman semigroup in a specific Banach space, and derives closed-form expressions for both the operator’s action and its resolvent. Building upon this theoretical foundation, the authors integrate the approach with Extended Dynamic Mode Decomposition (EDMD) to formulate a new algorithmic framework, termed cWDMD. Numerical experiments demonstrate that the proposed method achieves high-accuracy approximations of the Koopman operator, significantly enhancing both the precision and computational efficiency of spectral analysis for nonlinear systems.

This work addresses the challenge of effectively extending kernel-based methods—originally developed for deterministic dynamical systems—to stochastic differential equations (SDEs) for approximating eigenfunctions of the Koopman operator. By leveraging the Feynman–Kac path integral representation, the study unifies three distinct kernel constructions—variational principles, Green’s function convolutions, and resolvent operators—into a coherent framework for stochastic systems with diffusion, establishing a corresponding reproducing kernel Hilbert space (RKHS) approximation scheme. Theoretically, under uniform ellipticity, these three approaches are shown to be equivalent, revealing that diffusion enhances numerical conditioning through elliptic regularization. The analysis further provides error bounds that separate RKHS approximation error from Monte Carlo sampling error. Numerical experiments on the Ornstein–Uhlenbeck process, nonlinear SDEs, and high-dimensional systems demonstrate the method’s efficacy, showing that moderate diffusion significantly improves numerical stability.

High-dimensional spatiotemporal chaotic systems are often dominated by continuous spectra, yet existing data-driven approaches frequently suffer from instability, limited interpretability, and poor scalability. This work proposes KoopGen—a generator-based neural Koopman framework that explicitly decomposes dynamics into conservative (skew-adjoint) and dissipative (self-adjoint) components via a state-dependent Koopman generator, while rigorously embedding operator-theoretic constraints. Notably, KoopGen achieves the first explicit separation of self-adjoint and skew-adjoint parts within the generator without relying on finite-dimensional assumptions or explicit spectral parameterizations. Experiments ranging from nonlinear oscillators to high-dimensional chaotic systems demonstrate that KoopGen substantially improves long-term prediction accuracy and stability, uncovering learnable and interpretable structural components underlying continuous-spectrum dynamics.

This work addresses the instability in long-term forecasting of time-dependent partial differential equations (PDEs), which arises from error accumulation in autoregressive models and dynamic drift. To mitigate these issues, the authors propose the Structured Spectral Propagator (SSP) framework, which operates through an analysis–propagation–synthesis pipeline in a latent space. SSP decouples spatial details from recurrent dynamics and introduces an explicit, structured spectral propagation mechanism endowed with strong inductive biases. This mechanism integrates a frequency-conditioned linear backbone with a nonlinear spectral closure model to enable stable and high-fidelity long-term evolution. Experimental results demonstrate that SSP significantly outperforms existing methods on long-horizon extrapolation tasks, achieving up to a 48.9% reduction in relative L2 error, and exhibits exceptional predictive stability and generalization capability.