Magnetostatics of Magnetic Skyrmion Crystals

Magnetic skyrmion crystals are topological magnetic textures arising in the chiral ferromagnetic materials with Dzyaloshinskii-Moriya interaction. The magnetostatic fields generated by magnetic skyrmion crystals are first studied by micromagnetic simulations. For N\'eel-type skyrmion crystals, the fields will vanish on one side of the crystal plane, which depend on the helicity; while for Bloch-type skyrmion crystals, the fields will distribute over both sides, and are identical for the two helicities. These features and the symmetry relations of the magetostatic fields are understood from the magnetic scalar potential and magnetic vector potential of the hybridized triple-Q state. The possibility to construct magnetostatic field at nanoscale by stacking chiral ferromagnetic layers with magnetic skyrmion crystals is also discussed, which may have potential applications to trap and manipulate neutral atoms with magnetic moments.Comment: 5 pages, 2 figure

Mean-field backward stochastic differential equations on Markov chains

In this paper, we deal with a class of mean-field backward stochastic differential equations (BSDEs) related to finite state, continuous time Markov chains. We obtain the existence and uniqueness theorem and a comparison theorem for solutions of one-dimensional mean-field BSDEs under Lipschitz condition

Stochastic viscosity solution for stochastic PDIEs with nonlinear Neumann boundary condition

This paper is an attempt to extend the notion of viscosity solution to nonlinear stochastic partial differential integral equations with nonlinear Neumann boundary condition. Using the recently developed theory on generalized backward doubly stochastic differential equations driven by a L\'evy process, we prove the existence of the stochastic viscosity solution, and further extend the nonlinear Feynman-Kac formula.Comment: 16 pag

Control of Ultracold Atoms with a Chiral Ferromagnetic Film

We show that the magnetic field produced by a chiral ferromagnetic film can be applied to control ultracold atoms. The film will act as a magnetic mirror or a reflection grating for ultracold atoms when it is in the helical phase or the skyrmion crystal phase respectively. By applying a bias magnetic field and a time-dependent magnetic field, one-dimensional or two-dimensional magnetic lattices including honeycomb, Kagome, triangular types can be created to trap the ultracold atoms. We have also discussed the trapping height, potential barrier, trapping frequency, and Majorana loss rate for each lattice. Our results suggest that the chiral ferromagnetic film can be a platform to develop artificial quantum systems with ultracold atoms based on modern spintronics technologies.Comment: 9 pages, 6 figure

Non-smooth analysis method in optimal investment- a BSDE approach

In this paper, our aim is to investigate necessary conditions for optimal investment. We model the wealth process by Backward differential stochastic equations (shortly for BSDE) with or without constraints on wealth and portfolio process. The constraints can be very general thanks the non-smooth analysis method we adopted

Deep Competitive Pathway Networks

In the design of deep neural architectures, recent studies have demonstrated the benefits of grouping subnetworks into a larger network. For examples, the Inception architecture integrates multi-scale subnetworks and the residual network can be regarded that a residual unit combines a residual subnetwork with an identity shortcut. In this work, we embrace this observation and propose the Competitive Pathway Network (CoPaNet). The CoPaNet comprises a stack of competitive pathway units and each unit contains multiple parallel residual-type subnetworks followed by a max operation for feature competition. This mechanism enhances the model capability by learning a variety of features in subnetworks. The proposed strategy explicitly shows that the features propagate through pathways in various routing patterns, which is referred to as pathway encoding of category information. Moreover, the cross-block shortcut can be added to the CoPaNet to encourage feature reuse. We evaluated the proposed CoPaNet on four object recognition benchmarks: CIFAR-10, CIFAR-100, SVHN, and ImageNet. CoPaNet obtained the state-of-the-art or comparable results using similar amounts of parameters. The code of CoPaNet is available at: https://github.com/JiaRenChang/CoPaNet.Comment: To appear in ACML1

Reflected backward stochastic differential equations with jumps in time-dependent random convex domains

In this paper, we study a class of multi-dimensional reflected backward stochastic differential equations when the noise is driven by a Brownian motion and an independent Poisson point process, and when the solution is forced to stay in a time-dependent adapted and continuous convex domain ${\cal{D}}=\{D_t, t\in[0,T]\}$. We prove the existence an uniqueness of the solution, and we also show that the solution of such equations may be approximated by backward stochastic differential equations with jumps reflected in appropriately defined discretizations of $\cal{D}$, via a penalization method.Comment: 43 pages. arXiv admin note: text overlap with arXiv:1307.2124 by other author

Continuous dependence property of BSDE with constraints

In this paper, we study continuous properties of adapted solutions for backward stochastic differential equations with constraints (CBSDEs in short). Comparing with many existing literatures about this topic, our case is very general in the sense that constraints are formulated by general non-negative real functions. In general case, we proved a continuous property from below and a lower semi-continuous property of the minimal super-solution of CBSDE in its effective domain. Furthermore, in the special convex case, we obtained a continuous property with the help of convex analysis

Mean-field backward stochastic differential equations with subdifferrential operator and its applications

In this paper, we deal with a class of mean-field backward stochastic differential equations with subdifferrential operator corresponding to a lower semi-continuous convex function. By means of Yosida approximation, the existence and uniqueness of the solution is established. As an application, we give a probability interpretation for the viscosity solutions of a class of nonlocal parabolic variational inequalities

Spectral Learning for Supervised Topic Models

Supervised topic models simultaneously model the latent topic structure of large collections of documents and a response variable associated with each document. Existing inference methods are based on variational approximation or Monte Carlo sampling, which often suffers from the local minimum defect. Spectral methods have been applied to learn unsupervised topic models, such as latent Dirichlet allocation (LDA), with provable guarantees. This paper investigates the possibility of applying spectral methods to recover the parameters of supervised LDA (sLDA). We first present a two-stage spectral method, which recovers the parameters of LDA followed by a power update method to recover the regression model parameters. Then, we further present a single-phase spectral algorithm to jointly recover the topic distribution matrix as well as the regression weights. Our spectral algorithms are provably correct and computationally efficient. We prove a sample complexity bound for each algorithm and subsequently derive a sufficient condition for the identifiability of sLDA. Thorough experiments on synthetic and real-world datasets verify the theory and demonstrate the practical effectiveness of the spectral algorithms. In fact, our results on a large-scale review rating dataset demonstrate that our single-phase spectral algorithm alone gets comparable or even better performance than state-of-the-art methods, while previous work on spectral methods has rarely reported such promising performance