We review some recent developments in Chern-Simons theory on a hyperbolic 3-manifold M with complex gauge group G. We focus on the case G=SL(N,C) and with M a knot complement. The main result presented in this note is the cluster partition function, a computational tool that uses cluster algebra techniques to evaluate the Chern-Simons path integral. We also review various applications and open questions regarding the cluster partition function and some of its relation with string theory.

The structural variations of multidomain proteins with flexible parts mediate many biological processes, and a structure ensemble can be determined by selecting a weighted combination of representative structures from a simulated structure pool, producing the best fit to experimental constraints such as interatomic distance. In this study, a hybrid structure-based and physics-based atomistic force field with an efficient sampling strategy is adopted to simulate a model di-domain protein against experimental paramagnetic relaxation enhancement (PRE) data that correspond to distance constraints. The molecular dynamics simulations produce a wide range of conformations depicted on a protein energy landscape. Subsequently, a conformational ensemble recovered with low-energy structures and the minimum-size restraint is identified in good agreement with experimental PRE rates, and the result is also supported by chemical shift perturbations and small-angle X-ray scattering data. It is illustrated that the regularizations of energy and ensemble-size prevent an arbitrary interpretation of protein conformations. Moreover, energy is found to serve as a critical control to refine the structure pool and prevent data overfitting, because the absence of energy regularization exposes ensemble construction to the noise from high-energy structures and causes a more ambiguous representation of protein conformations. Finally, we perform structure-ensemble optimizations with a topology-based structure pool, to enhance the understanding on the ensemble results from different sources of pool candidates.

Quantitative susceptibility mapping (QSM) uses the phase data in magnetic resonance signals to visualize a three-dimensional susceptibility distribution by solving the magnetic field to susceptibility inverse problem. Due to the presence of zeros of the integration kernel in the frequency domain, QSM is an ill-posed inverse problem. Although numerous regularization-based models have been proposed to overcome this problem, incompatibility in the field data, which leads to deterioration of the recovery, has not received enough attention. In this paper, we show that the data acquisition process of QSM inherently generates a harmonic incompatibility in the measured local field. Based on this discovery, we propose a novel regularization-based susceptibility reconstruction model with an additional sparsity-based regularization term on the harmonic incompatibility. Numerical experiments show that the proposed method achieves better performance than existing approaches.

Learning rate is arguably the most important hyper-parameter to tune when training a neural network. As manually setting right learning rate remains a cumbersome process, adaptive learning rate algorithms aim at automating such a process. Motivated by the success of the Barzilai–Borwein (BB) step-size method in many gradient descent methods for solving convex problems, this paper aims at investigating the potential of the BB method for training neural networks. With strong motivation from related convergence analysis, the BB method is generalized to adaptive learning rate of mini-batch gradient descent. The experiments showed that, in contrast to many existing methods, the proposed BB method is highly insensitive to initial learning rate, especially in terms of generalization performance. Also, the BB method showed its advantages on both learning speed and generalization performance over other available methods.

Finding the stationary states of a free energy functional is an important problem in phase field crystal (PFC) models. Many efforts have been devoted to designing numerical schemes with energy dissipation and mass conservation properties. However, most existing approaches are time-consuming due to the requirement of small effective step sizes. In this paper, we discretize the energy functional and propose efficient numerical algorithms for solving the constrained nonconvex minimization problem. A class of gradient-based approaches, which are the so-called adaptive accelerated Bregman proximal gradient (AA-BPG) methods, is proposed, and the convergence property is established without the global Lipschitz constant requirements. A practical Newton method is also designed to further accelerate the local convergence with convergence guarantee. One key feature of our algorithms is that the energy dissipation and mass conservation properties hold during the iteration process. Moreover, we develop a hybrid acceleration framework to accelerate the AA-BPG methods and most of the existing approaches through coupling with the practical Newton method. Extensive numerical experiments, including two three-dimensional periodic crystals in the Landau--Brazovskii (LB) model and a two-dimensional quasicrystal in the Lifshitz--Petrich (LP) model, demonstrate that our approaches have adaptive step sizes which lead to a significant acceleration over many existing methods when computing complex structures.