Cohen and Lenstra introduced conjectures concerning the distribution of class numbers in quadratic fields, though many of these conjectures remain unproven. This paper investigates the 2-part of class groups in imaginary quadratic fields and examines their alignment with the Cohen-Lenstra heuristics. We provide detailed proofs of key theorems related to ideal decompositions and modular homomorphisms, and we explore the distribution of class groups of imaginary quadratic fields. Our analysis includes constructing imaginary quadratic fields with prescribed 2-class groups and discussing the implications of these findings on the Cohen-Lenstra conjecture.

We propose a unified framework for sparse stochastic optimal control in systems governed by time-varying dynamics and control-dependent diffusion—key features in robotic manipulators, energy systems, and other real-world applications. In contrast to conventional models assuming control-independent noise, our formulation explicitly captures the feedback between control effort and stochastic disturbances. This leads to a modified Hamilton–Jacobi–Bellman (HJB) equation that is nonconvex, nonsmooth, and nonmonotone. To address these challenges, we develop a relaxed viscosity solution framework that ensures analytical well-posedness. We also establish structural conditions under which sparse $L_0$-optimal control can be exactly recovered via its convex $L_1$ relaxation, leading to provable bang–off–bang strategies. Numerical simulations on robotic arms and photovoltaic–battery systems demonstrate the effectiveness of the proposed approach, showing superior control sparsity, robustness to noise, and safety guarantees under uncertainty. The results suggest our method offers a principled and scalable solution for real-time decision-making in noisy, resource-constrained environments.

Accurate traffic data imputation is essential for Intelligent Transportation Systems (ITS), particularly under complex missing patterns. This study presents the Adaptive and Truncated Schatten Norm Low-Rank Tensor Completion (LRTC-ATSN) model, which employs the flexible Schatten norm for low-rank approximation and introduces an adaptive truncation mechanism to suppress noise and retain core features. To address non-convex optimization, the model integrates the Adan algorithm with Nesterov momentum, achieving equilibrium between computational efficiency and recovery precision through dynamic parameter tuning. To evaluate performance, a framework was designed to simulate diverse real-world traffic scenarios with mixed missing patterns. Extensive experiments on datasets from Guangzhou and Seattle demonstrate that LRTC-ATSN outperforms existing methods, yielding 10.6% lower MAPE and 6.1% lower RMSE relative to the best baseline model. Even with 95.85% data loss, the model maintains high reliability. These results underscore LRTC-ATSN’s potential for enhancing ITS data robustness and applications across domains like finance and healthcare.

This paper proposes a scalable framework for sparse stochastic optimal control in time-varying systems with control-dependent diffusion. By relaxing the classical stochastic normality condition, we establish an approximate equivalence between ℓ₀ and ℓ₁ optimal control formulations under local or probabilistic regularity. The framework unifies sparsity, stochasticity, and safety by incorporating variational inequalities to handle state and control constraints. To overcome the curse of dimensionality, we develop efficient numerical solvers based on sparse grids, Tensor-Train decomposition, and neural residual networks. Numerical experiments on robotic and energy systems demonstrate substantial gains in control sparsity, safety compliance, and computational scalability, confirming the practicality and robustness of the proposed approach in high-dimensional, safety-critical settings.

Cohen and Lenstra introduced conjectures concerning the distribution of class numbers in quadratic fields, though many of these conjectures remain unproven. This paper investigates the 2-part of class groups in imaginary quadratic fields and examines their alignment with the Cohen–Lenstra heuristics. We provide detailed proofs of key theorems related to ideal decompositions and modular homomorphisms, and we explore the distribution of class groups of imaginary quadratic fields. Our analysis includes constructing imaginary quadratic fields with prescribed 2-class groups and discussing the implications of these findings on the Cohen–Lenstra conjecture.