Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications

This is the first installment in a series of papers aimed at generalizing symplectic capacities and homologies. The main purposes of this paper are to construct analogues of Ekeland-Hofer and Hofer-Zehnder symplectic capacities based on a class of Hamiltonian boundary value problems motivated by Clarke's and Ekeland's work, and to study generalizations of some important results about the original these two capacities (for example, the famous Weinstein conjecture, representation formula for $c_{\rm EH}$ and $c_{\rm HZ}$, a theorem by Evgeni Neduv, Brunn-Minkowski type inequality and Minkowski billiard trajectories proposed by Artstein-Avidan-Ostrover).Comment: Latex, 89 pages. Results in Section 1.6 are improved. Some typos are corrected. arXiv admin note: text overlap with arXiv:1903.0067

On the Regularizing Property of Stochastic Gradient Descent

Stochastic gradient descent is one of the most successful approaches for solving large-scale problems, especially in machine learning and statistics. At each iteration, it employs an unbiased estimator of the full gradient computed from one single randomly selected data point. Hence, it scales well with problem size and is very attractive for truly massive dataset, and holds significant potentials for solving large-scale inverse problems. In the recent literature of machine learning, it was empirically observed that when equipped with early stopping, it has regularizing property. In this work, we rigorously establish its regularizing property (under \textit{a priori} early stopping rule), and also prove convergence rates under the canonical sourcewise condition, for minimizing the quadratic functional for linear inverse problems. This is achieved by combining tools from classical regularization theory and stochastic analysis. Further, we analyze the preasymptotic weak and strong convergence behavior of the algorithm. The theoretical findings shed insights into the performance of the algorithm, and are complemented with illustrative numerical experiments.Comment: 22 pages, better presentatio

Improving the performance of SCTP Transport Protocol over wireless networks

[Abstract]: Stream Control Transmission Protocol(SCTP) is a reliable transport protocol combining the advantages of TCP and UDP. SCTP has many desirable features including multihoming, multistreaming, and partial data reliability. These features have made SCTP perform much more effectively in multimedia networking applications. They have also worked better in wireless environment which traditional transport protocols are ineffective and cumbersome. Before the transmission, an application using SCTP needs to establish an association between the client and the server. The establishment of association requires a number which will be used to create multiple streams. However, SCTP has not specified a method or suggested any ideas of determine the number. In our paper, we focus on the performance of SCTP protocol over the wireless networks. The ideas is to extend the SCTP with a process of determining an optimal number prior to the association establishing. We examine the modified SCTP on a simulated wireless networks, and the experiment results of simulation using NS2 have shown the modified SCTP is feasible and also demonstrated the modified SCTP’s superiority of performance over TCP and UDP over the wireless networks

From data towards knowledge: Revealing the architecture of signaling systems by unifying knowledge mining and data mining of systematic perturbation data

Genetic and pharmacological perturbation experiments, such as deleting a gene and monitoring gene expression responses, are powerful tools for studying cellular signal transduction pathways. However, it remains a challenge to automatically derive knowledge of a cellular signaling system at a conceptual level from systematic perturbation-response data. In this study, we explored a framework that unifies knowledge mining and data mining approaches towards the goal. The framework consists of the following automated processes: 1) applying an ontology-driven knowledge mining approach to identify functional modules among the genes responding to a perturbation in order to reveal potential signals affected by the perturbation; 2) applying a graph-based data mining approach to search for perturbations that affect a common signal with respect to a functional module, and 3) revealing the architecture of a signaling system organize signaling units into a hierarchy based on their relationships. Applying this framework to a compendium of yeast perturbation-response data, we have successfully recovered many well-known signal transduction pathways; in addition, our analysis have led to many hypotheses regarding the yeast signal transduction system; finally, our analysis automatically organized perturbed genes as a graph reflecting the architect of the yeast signaling system. Importantly, this framework transformed molecular findings from a gene level to a conceptual level, which readily can be translated into computable knowledge in the form of rules regarding the yeast signaling system, such as "if genes involved in MAPK signaling are perturbed, genes involved in pheromone responses will be differentially expressed"

A High Order Stochastic Asymptotic Preserving Scheme for Chemotaxis Kinetic Models with Random Inputs

In this paper, we develop a stochastic Asymptotic-Preserving (sAP) scheme for the kinetic chemotaxis system with random inputs, which will converge to the modified Keller-Segel model with random inputs in the diffusive regime. Based on the generalized Polynomial Chaos (gPC) approach, we design a high order stochastic Galerkin method using implicit-explicit (IMEX) Runge-Kutta (RK) time discretization with a macroscopic penalty term. The new schemes improve the parabolic CFL condition to a hyperbolic type when the mean free path is small, which shows significant efficiency especially in uncertainty quantification (UQ) with multi-scale problems. The stochastic Asymptotic-Preserving property will be shown asymptotically and verified numerically in several tests. Many other numerical tests are conducted to explore the effect of the randomness in the kinetic system, in the aim of providing more intuitions for the theoretic study of the chemotaxis models