The Stochastic Solution to a Cauchy Problem for Degenerate Parabolic Equations

We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally H\"older continuous with exponent $\delta\in (0, 1]$, the stochastic solution, which represents the price of a European option, is shown to be a classical solution to the Cauchy problem. This improves the standard requirement $\delta\ge 1/2$. Uniqueness results, including a Feynman-Kac formula and a comparison theorem, are established without assuming the usual linear growth condition on the diffusion coefficient. When the stochastic solution is not smooth, it is characterized as the limit of an approximating smooth stochastic solutions. In deriving the main results, we discover a new, probabilistic proof of Kotani's criterion for martingality of a one-dimensional diffusion in natural scale.Comment: Keywords: local martingales, local stochastic solutions, degenerate Cauchy problems, Feynman-Kac formula, necessary and sufficient condition for uniqueness, comparison principl

Effects of N(2080)3/2−N(2080){3/2}^- and N(2270)3/2−N(2270)3/2^- molecules on K∗ΣK^\ast \Sigma photoproduction

In the present work, we re-analyze the available data for $\gamma p\to K^{\ast +}\Sigma^0$ and $\gamma p \to K^{\ast 0}\Sigma^+$ by considering the contributions from the $N(2080){3/2}^-$ and $N(2270)3/2^-$ molecules instead of any nucleon resonances in the $s$ channel, where the $N(2080)3/2^-$ was proposed to be a $K^\ast \Sigma$ molecule as the strange partner of the $P_c^+(4457)$ hadronic molecular state, and the $N(2270)3/2^-$ was assumed to be a $K^*\Sigma^*$ molecule as the strange partner of the $\bar{D}^\ast \Sigma^\ast_c$ bound states that are predicated as members in the same heavy-quark spin symmetry multiplet as the $P_c$ states. It turns out that all the available cross-section data can be well reproduced, indicating that the molecular structures of the possible $N(2080){3/2}^-$ and $N(2270)3/2^-$ states are compatible with the available data for $K^\ast\Sigma$ photoproduction reactions. Further analysis shows that for both $\gamma p\to K^{\ast +}\Sigma^0$ and $\gamma p \to K^{\ast 0}\Sigma^+$ reactions, the $N(2080){3/2}^-$ exchange provides dominant contributions to the cross-sections in the near-threshold energy region, and significant contributions from the $N(2270)3/2^-$ exchange to the cross-sections in the higher energy region are also found. Predictions of the beam asymmetry $\Sigma$, target asymmetry $T$, and recoil baryon asymmetry $P$ are presented and compared with those from our previous work. Measurements of the data on these observables are called on to further constrain the reaction mechanisms of $K^\ast\Sigma$ photoproduction reactions and to verify the molecular scenario of the $N(2080){3/2}^-$ and $N(2270)3/2^-$ states.Comment: 10 pages, 8 figure

A Review of Dynamic Wireless Power Transfer for In‐Motion Electric Vehicles

Dynamic wireless power transfer system (DWPT) in urban area ensures an uninterrupted power supply for electric vehicles (EVs), extending or even providing an infinite driving range with significantly reduced battery capacity. The underground power supply network also saves more space and hence is important in urban areas. It must be noted that the railways have become an indispensable form of public transportation to reduce pollution and traffic congestion. In recent years, there has been a consistent increase in the number of high‐speed railways in major cities of China, thereby improving accessibility. Wireless power transfer for train is safer and more robust when compared with conductive power transfer through pantograph mounted on the trains. Direct contact is subject to wear and tear; in particular, the average speed of modern trains has been increasing. When the pressure of pantograph is not sufficient, arcs, variations of the current, and even interruption in power supply may occur. This chapter provides a review of the latest research and development of dynamic wireless power transfer for urban EV and electric train (ET). The following key technology issues have been discussed: (1) power rails and pickups, (2) segmentations and power supply schemes, (3) circuit topologies and dynamic impedance matching, (4) control strategies, and (5) electromagnetic interference

Learning to Weight Samples for Dynamic Early-exiting Networks

Early exiting is an effective paradigm for improving the inference efficiency of deep networks. By constructing classifiers with varying resource demands (the exits), such networks allow easy samples to be output at early exits, removing the need for executing deeper layers. While existing works mainly focus on the architectural design of multi-exit networks, the training strategies for such models are largely left unexplored. The current state-of-the-art models treat all samples the same during training. However, the early-exiting behavior during testing has been ignored, leading to a gap between training and testing. In this paper, we propose to bridge this gap by sample weighting. Intuitively, easy samples, which generally exit early in the network during inference, should contribute more to training early classifiers. The training of hard samples (mostly exit from deeper layers), however, should be emphasized by the late classifiers. Our work proposes to adopt a weight prediction network to weight the loss of different training samples at each exit. This weight prediction network and the backbone model are jointly optimized under a meta-learning framework with a novel optimization objective. By bringing the adaptive behavior during inference into the training phase, we show that the proposed weighting mechanism consistently improves the trade-off between classification accuracy and inference efficiency. Code is available at https://github.com/LeapLabTHU/L2W-DEN.Comment: ECCV 202

A Bayesian measurement error model for two-channel cell-based RNAi data with replicates

RNA interference (RNAi) is an endogenous cellular process in which small double-stranded RNAs lead to the destruction of mRNAs with complementary nucleoside sequence. With the production of RNAi libraries, large-scale RNAi screening in human cells can be conducted to identify unknown genes involved in a biological pathway. One challenge researchers face is how to deal with the multiple testing issue and the related false positive rate (FDR) and false negative rate (FNR). This paper proposes a Bayesian hierarchical measurement error model for the analysis of data from a two-channel RNAi high-throughput experiment with replicates, in which both the activity of a particular biological pathway and cell viability are monitored and the goal is to identify short hair-pin RNAs (shRNAs) that affect the pathway activity without affecting cell activity. Simulation studies demonstrate the flexibility and robustness of the Bayesian method and the benefits of having replicates in the experiment. This method is illustrated through analyzing the data from a RNAi high-throughput screening that searches for cellular factors affecting HCV replication without affecting cell viability; comparisons of the results from this HCV study and some of those reported in the literature are included.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS496 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org