Berndt-Type Integrals of Order Three and Series Associated with Jacobi Elliptic Functions

In this paper, we first establish explicit evaluations of six classes of hyperbolic sums by special values of the Gamma function by using the tools of the Fourier series expansions and the Maclaurin series expansions of a few Jacobi elliptic functions developed in our previous paper. Then, using the method of contour integrations involving hyperbolic and trigonometric functions, we establish explicit evaluations of two families of Berndt-type integrals of order three by special values of the Gamma function. Furthermore, we present some interesting consequences and illustrative examples.Comment: 19 pages. arXiv admin note: text overlap with arXiv:2301.0821

Explicit Planning Helps Language Models in Logical Reasoning

Language models have been shown to perform remarkably well on a wide range of natural language processing tasks. In this paper, we propose a novel system that uses language models to perform multi-step logical reasoning. Our system incorporates explicit planning into its inference procedure, thus able to make more informed reasoning decisions at each step by looking ahead into their future effects. In our experiments, our full system significantly outperforms other competing systems. On a multiple-choice question answering task, our system performs competitively compared to GPT-3-davinci despite having only around 1.5B parameters. We conduct several ablation studies to demonstrate that explicit planning plays a crucial role in the system's performance

Kernel meets sieve: transformed hazards models with sparse longitudinal covariates

We study the transformed hazards model with time-dependent covariates observed intermittently for the censored outcome. Existing work assumes the availability of the whole trajectory of the time-dependent covariates, which is unrealistic. We propose to combine kernel-weighted log-likelihood and sieve maximum log-likelihood estimation to conduct statistical inference. The method is robust and easy to implement. We establish the asymptotic properties of the proposed estimator and contribute to a rigorous theoretical framework for general kernel-weighted sieve M-estimators. Numerical studies corroborate our theoretical results and show that the proposed method performs favorably over existing methods. Applying to a COVID-19 study in Wuhan illustrates the practical utility of our method

Statistical Guarantees of Generative Adversarial Networks for Distribution Estimation

Generative Adversarial Networks (GANs) have achieved great success in unsupervised learning. Despite the remarkable empirical performance, there are limited theoretical understandings on the statistical properties of GANs. This paper provides statistical guarantees of GANs for the estimation of data distributions which have densities in a H\"{o}lder space. Our main result shows that, if the generator and discriminator network architectures are properly chosen (universally for all distributions with H\"{o}lder densities), GANs are consistent estimators of the data distributions under strong discrepancy metrics, such as the Wasserstein distance. To our best knowledge, this is the first statistical theory of GANs for H\"{o}lder densities. In comparison with existing works, our theory requires minimum assumptions on data distributions. Our generator and discriminator networks utilize general weight matrices and the non-invertible ReLU activation function, while many existing works only apply to invertible weight matrices and invertible activation functions. In our analysis, we decompose the error into a statistical error and an approximation error by a new oracle inequality, which may be of independent interest

Adaptive Distribution Calibration for Few-Shot Learning with Hierarchical Optimal Transport

Few-shot classification aims to learn a classifier to recognize unseen classes during training, where the learned model can easily become over-fitted based on the biased distribution formed by only a few training examples. A recent solution to this problem is calibrating the distribution of these few sample classes by transferring statistics from the base classes with sufficient examples, where how to decide the transfer weights from base classes to novel classes is the key. However, principled approaches for learning the transfer weights have not been carefully studied. To this end, we propose a novel distribution calibration method by learning the adaptive weight matrix between novel samples and base classes, which is built upon a hierarchical Optimal Transport (H-OT) framework. By minimizing the high-level OT distance between novel samples and base classes, we can view the learned transport plan as the adaptive weight information for transferring the statistics of base classes. The learning of the cost function between a base class and novel class in the high-level OT leads to the introduction of the low-level OT, which considers the weights of all the data samples in the base class. Experimental results on standard benchmarks demonstrate that our proposed plug-and-play model outperforms competing approaches and owns desired cross-domain generalization ability, indicating the effectiveness of the learned adaptive weights

Numerical Analysis of Reinforced Concrete Piles under Blast Loads

Pile foundations are commonly used as foundation systems for high-rise buildings and bridges. This paper uses a fully coupled three dimensional numerical modelling procedure to study the performance of pile foundations subjected to ground shocks induced by surface explosions. The comprehensive numerical model includes the pile, surrounding soil, air and the explosive. Appropriate material models are incorporated and dynamic non-linear analysis is carried out using finite element techniques.  The soil in which the pile is buried could influence the blast performance of the pile. A parametric study is hence carried out to evaluate the effects of soil properties of density, friction angle, cohesion and Poisson’s ratio on the blast performance of the pile. It is found that density and cohesion of soil have significant effects on the deflection of the pile under blast loading. Poisson’s ratio has some effect, but effect of the soil friction angle is not very significant. The findings of this study will serve as a benchmark reference for future analysis and design of pile foundations to blast loading