Ground State Solutions of Kirchhoff-type Fractional Dirichlet Problem with pp-Laplacian

We consider the Kirchhoff-type $p$-Laplacian Dirichlet problem containing the left and right fractional derivative operators. By using the Nehari method in critical point theory, we obtain the existence theorem of ground state solutions for such Dirichlet problem.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1607.0158

Infinitely Many Weak Solutions for Fractional Dirichlet Problem with pp-Laplacian

We focus on the study of $p$-Laplacian Dirichlet problem containing the left and right fractional derivative operators. By using the genus properties in critical point theory, we establish some new criteria to guarantee the existence of infinitely many weak solutions for the considered problem.Comment: 14 page

Existence and Multiplicity of Nontrivial Weak Solutions for Kirchhoff-type Fractional pp-Laplacian Equation

We discuss the Kirchhoff-type $p$-Laplacian Dirichlet problem containing the left and right fractional derivative operators. By using the mountain pass theorem and the genus properties in critical point theory, we get some new results on the existence and multiplicity of nontrivial weak solutions for such Dirichlet problem.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1605.0923

Sensitivity of parameter estimation near the exceptional point of a non-Hermitian system

The exceptional points of non-Hermitian systems, where $n$ different energy eigenstates merge into an identical one, have many intriguing properties that have no counterparts in Hermitian systems. In particular, the $\epsilon^{1/n}$ dependence of the energy level splitting on a perturbative parameter $\epsilon$ near an $n$-th order exceptional point stimulates the idea of metrology with arbitrarily high sensitivity, since the susceptibility $d\epsilon^{1/n}/d\epsilon$ diverges at the exceptional point. Here we theoretically study the sensitivity of parameter estimation near the exceptional points, using the exact formalism of quantum Fisher information. The quantum Fisher information formalism allows the highest sensitivity to be determined without specifying a specific measurement approach. We find that the exceptional point bears no dramatic enhancement of the sensitivity. Instead, the coalescence of the eigenstates exactly counteracts the eigenvalue susceptibility divergence and makes the sensitivity a smooth function of the perturbative parameter.Comment: 25 pages, 4 figure

Optimal Control of DERs in ADN under Spatial and Temporal Correlated Uncertainties

The control schemes of distributed energy resources (DERs) in active distribution networks (ADNs) are largely influenced by uncertainties. The uncertainties of DERs are complicated, containing spatial and temporal correlation, which makes it challenging to design proper control schemes, especially when there exist temporal-correlated units such as energy units (EUs). This paper provides an Ito process model to describe the characteristics of stochastic resources and EUs in a unified way, which makes it easy to evaluate the impacts of stochastic resources on temporal-correlated units. Based the moment form of the Ito process model, a moment optimization (MO) approach is provided to transform the stochastic control (SC) problem into an optimization problem with respect to the first-order and second-order moments of the system variables. The scale of MO is comparable to that of the corresponding deterministic control problem, which means that the computational efficiency of MO is much smaller than that of traditional approaches. Case studies also show that the proposed approach outperforms existing approaches in both the performance and computational efficiency, which means that the proposed approach has attractive potential for use in large-scale applications.Comment: 9 pages, 5 figure

Trivariate monomial complete intersections and plane partitions

We consider the homogeneous components U_r of the map on R = k[x,y,z]/(x^A, y^B, z^C) that multiplies by x + y + z. We prove a relationship between the Smith normal forms of submatrices of an arbitrary Toeplitz matrix using Schur polynomials, and use this to give a relationship between Smith normal form entries of U_r. We also give a bijective proof of an identity proven by J. Li and F. Zanello equating the determinant of the middle homogeneous component U_r when (A, B, C) = (a + b, a + c, b + c) to the number of plane partitions in an a by b by c box. Finally, we prove that, for certain vector subspaces of R, similar identities hold relating determinants to symmetry classes of plane partitions, in particular classes 3, 6, and 8.Comment: 21 pages, 15 figure

Learning to Navigate in Indoor Environments: from Memorizing to Reasoning

Autonomous navigation is an essential capability of smart mobility for mobile robots. Traditional methods must have the environment map to plan a collision-free path in workspace. Deep reinforcement learning (DRL) is a promising technique to realize the autonomous navigation task without a map, with which deep neural network can fit the mapping from observation to reasonable action through explorations. It should not only memorize the trained target, but more importantly, the planner can reason out the unseen goal. We proposed a new motion planner based on deep reinforcement learning that can arrive at new targets that have not been trained before in the indoor environment with RGB image and odometry only. The model has a structure of stacked Long Short-Term memory (LSTM). Finally, experiments were implemented in both simulated and real environments. The source code is available: https://github.com/marooncn/navbot

Quantification of the underlying mechanisms and relationship among cancer, metastasis and differentiation/development

Recurrence and metastasis have been regarded as two of the greatest obstacles for curing cancer. Cancer stem cell (CSC) have been found. They contribute to cancer development with the distinct feature of recurrence and resistance to the popular treatments such as drugs and chemotherapy. In addition, recent discoveries suggest that the epithelial mesenchymal transition (EMT) is an essential process in normal embryogenesis and tissue repair, which is a required step in cancer metastasis. Although there are many indications showing the connections between metastasis and stem cell, researches often studied them separately or at most bi-laterally, not in an integrated way. In this study, we aim at exploring the global mechanisms and interrelationship among cancer, development and metastasis which are currently poorly understood. To start, we constructed a core gene regulatory network motif which contain specific genes and microRNAs of CSC, EMT and cancer. We uncovered seven distinct states emerged from the underlying landscape. They are identified as Normal, Premalignant, Cancer, stem cell (SC), cancer stem cell (CSC), Lesion and Hyperlasia state. Given the biological definition of each state, we also discussed the metastasis ability of each state. We show how and which types of cells can be transformed to a cancer state and the connections among cancer, CSC and EMT. The barrier height and flux of the kinetic paths are explored to quantify how and which cells switch stochastically between the states. Our landscape model provides a quantitative way which reveals the global mechanisms of cancer, development and metastasis.Comment: 6 figures,9 page

Intrinsic ferromagnetism and quantum anomalous Hall effect in CoBr2 monolayer

The electronic, magnetic, and topological properties of CoBr2 monolayer are studied in the frame-work of the density-functional theory (DFT) combined with tight-binding (TB) modeling in terms of Wannier basis. Our DFT investigation and Monte Carlo simulation show that there exists intrinsic two-dimensional ferromagnetism in the CoBr2 monolayer thanks to large out-of-plane magnetocrystalline anisotropic energy. Our further study shows that the spin-orbits coupling makes it become a topologically nontrivial insulator with quantum anomalous Hall effect and topological Chern number C=4, and its edge states can be manipulated by changing the width of its nanoribbons and applying strains. The CoBr2 monolayer can be exfoliated from the layered CoBr2 bulk material because its exfoliation energy is between those of graphene and MoS2 monolayer and it is dynamically stable. These results make us believe that the CoBr2 monolayer can make a promising spintronic material for future high-performance devices

A physical mechanism of heterogeneity in stem cell, cancer and cancer stem cell

Heterogeneity is ubiquitous in stem cells (SC), cancer cells (CS), and cancer stem cells (CSC). SC and CSC heterogeneity is manifested as diverse sub-populations with self-renewing and unique regeneration capacity. Moreover, the CSC progeny possesses multiple plasticity and cancerous characteristics. Many studies have demonstrated that cancer heterogeneity is one of the greatest obstacle for therapy. This leads to the incomplete anti-cancer therapies and transitory efficacy. Furthermore, numerous micro-metastasis leads to the wide spread of the tumor cells across the body which is the beginning of metastasis. The epigenetic processes (DNA methylation or histone remodification etc.) can provide a source for certain heterogeneity. In this study, we develop a mathematical model to quantify the heterogeneity of SC, CSC and cancer taking both genetic and epigenetic effects into consideration. We uncovered the roles and physical mechanisms of heterogeneity from the three aspects (SC, CSC and cancer). In the adiabatic regime (relatively fast regulatory binding and effective coupling among genes), seven native states (SC, CSC, Cancer, Premalignant, Normal, Lesion and Hyperplasia) emerge. In non-adiabatic regime (relatively slow regulatory binding and effective weak coupling among genes), multiple meta-stable SC, CS, CSC and differentiated states emerged which can explain the origin of heterogeneity. In other words, the slow regulatory binding mimicking the epigenetics can give rise to heterogeneity. Elucidating the origin of heterogeneity and dynamical interrelationship between intra-tumoral cells has clear clinical significance in helping to understand the cellular basis of treatment response, therapeutic resistance, and tumor relapse.Comment: 7 pages, 2 figure