The fundamental theorem of affine geometry in (L0)n(L^0)^n

Let $L^0$ be the algebra of equivalence classes of real valued random variables on a probability space. For each integer $n\geq 2$, we consider $(L^0)^n$--the $n$-ary Cartesian power of $L^0$--as a free $L^0$-module and establish the fundamental theorem of affine geometry in $(L^0)^n$: an injective map $T: (L^0)^n\to (L^0)^n$ which has local property and maps each $L^0$-line onto an $L^0$-line must be an $L^0$-affine linear map.Comment: 10 page

Energy-Efficient Offloading in Mobile Edge Computing with Edge-Cloud Collaboration

Multiple access mobile edge computing is an emerging technique to bring computation resources close to end mobile users. By deploying edge servers at WiFi access points or cellular base stations, the computation capabilities of mobile users can be extended. Existing works mostly assume the remote cloud server can be viewed as a special edge server or the edge servers are willing to cooperate, which is not practical. In this work, we propose an edge-cloud cooperative architecture where edge servers can rent for the remote cloud servers to expedite the computation of tasks from mobile users. With this architecture, the computation offloading problem is modeled as a mixed integer programming with delay constraints, which is NP-hard. The objective is to minimize the total energy consumption of mobile devices. We propose a greedy algorithm as well as a simulated annealing algorithm to effectively solve the problem. Extensive simulation results demonstrate that, the proposed greedy algorithm and simulated annealing algorithm can achieve the near optimal performance. On average, the proposed greedy algorithm can achieve the same application completing time budget performance of the Brute Force optional algorithm with only 31\% extra energy cost. The simulated annealing algorithm can achieve similar performance with the greedy algorithm.Comment: Accepted by the 18th International Conference on Algorithms and Architectures for Parallel Processing (ICA3PP 2018

Seeing Permeability From Images: Fast Prediction with Convolutional Neural Networks

Fast prediction of permeability directly from images enabled by image recognition neural networks is a novel pore-scale modeling method that has a great potential. This article presents a framework that includes (1) generation of porous media samples, (2) computation of permeability via fluid dynamics simulations, (3) training of convolutional neural networks (CNN) with simulated data, and (4) validations against simulations. Comparison of machine learning results and the ground truths suggests excellent predictive performance across a wide range of porosities and pore geometries, especially for those with dilated pores. Owning to such heterogeneity, the permeability cannot be estimated using the conventional Kozeny-Carman approach. Computational time was reduced by several orders of magnitude compared to fluid dynamic simulations. We found that, by including physical parameters that are known to affect permeability into the neural network, the physics-informed CNN generated better results than regular CNN, however improvements vary with implemented heterogeneity.Comment: 17 pages, 8 figure

Convergence Analysis for A Class of Iterative Methods for Solving Saddle Point Systems

Convergence analysis of a nested iterative scheme proposed by Bank,Welfert and Yserentant (BWY) ([Numer. Math., 666: 645-666, 1990]) for solving saddle point system is presented. It is shown that this scheme converges under weaker conditions: the contraction rate for solving the $(1,1)$ block matrix is bound by $(\sqrt{5}-1)/2$. Similar convergence result is also obtained for a class of inexact Uzawa method with even weaker contraction bound $\sqrt{2}/2$. Preconditioned generalized minimal residual method using BWY method as a preconditioner is shown to converge with realistic assumptions

A protocol to estimate the average fidelity of the bipartite system

In present work, we shall develope a protocol to estimate the average fidelity for the bipartite system. We show that the average fidelity should be known if the three measurable quantities, the average survive probability of the product state and the the average survive probablity of each subsystem, have been decided. Our protocol can be also applied to decide the selected element of the quantum proess matrix.Comment: 10 pages, 4 figure

ENGINE:Cost Effective Offloading in Mobile Edge Computing with Fog-Cloud Cooperation

Mobile Edge Computing (MEC) as an emerging paradigm utilizing cloudlet or fog nodes to extend remote cloud computing to the edge of the network, is foreseen as a key technology towards next generation wireless networks. By offloading computation intensive tasks from resource constrained mobile devices to fog nodes or the remote cloud, the energy of mobile devices can be saved and the computation capability can be enhanced. For fog nodes, they can rent the resource rich remote cloud to help them process incoming tasks from mobile devices. In this architecture, the benefit of short computation and computation delay of mobile devices can be fully exploited. However, existing studies mostly assume fog nodes possess unlimited computing capacity, which is not practical, especially when fog nodes are also energy constrained mobile devices. To provide incentive of fog nodes and reduce the computation cost of mobile devices, we provide a cost effective offloading scheme in mobile edge computing with the cooperation between fog nodes and the remote cloud with task dependency constraint. The mobile devices have limited budget and have to determine which task should be computed locally or sent to the fog. To address this issue, we first formulate the offloading problem as a task finish time inimization problem with given budgets of mobile devices, which is NP-hard. We then devise two more algorithms to study the network performance. Simulation results show that the proposed greedy algorithm can achieve the near optimal performance. On average, the Brute Force method and the greedy algorithm outperform the simulated annealing algorithm by about 28.13% on the application finish time.Comment: 10 pages, 9 figures, Technical Repor

Perturbative NNNN scattering in chiral effective field theory

Within the framework of chiral effective field theory, perturbative calculation for $NN$ scattering is carried out in partial waves with orbital angular momentum $L \geqslant 1$. The primary goal is to identify the lowest angular momenta at which perturbative treatment of chiral forces can apply. Results up to the order where the subleading two-pion exchange appears are shown. It is concluded that perturbation theory applies to all partial waves but ${}^1S_0$, ${}^3S_1 - {}^3D_1$, and ${}^3P_0$. Where it is applicable, perturbation theory with the delta-less chiral forces produces good agreement with the empirical phase shifts up to $k_\text{c.m.} \simeq 300$ MeV.Comment: Matches published version; suppression of TPEs accounted for at referee's suggestion; main conclusion unchanged, that most P waves and higher waves are perturbativ

Exact zero modes in a quantum compass chain under inhomogeneous transverse fields

We study the emergence of exact Majorana zero modes (EMZMs) in a one-dimensional quantum transverse compass model with both the nearest-neighbor interactions and transverse fields varying over space. By transforming the spin system into a quadratic Majorana-fermion model, we derive an exact formula for the number of the emergent EMZMs, which is found to depend on the partition nature of the lattice sites on which the magnetic fields vanish. We also derive explicit expressions for the wavefunctions of these EMZMs and show that they indeed depend on fine features of the foregoing partition of site indices. Based on the above rigorous results about the EMZMs, we provide an interpretation for the interesting dependence of the eigenstate-degeneracy on the transverse fields observed in prior literatures. As a special case, we employ a plane-wave ansatz to exactly solve an open compass chain with alternating nearest-neighbor interactions and staggered magnetic fields. Explicit forms of the canonical Majorana modes diagonalizing the model are given even for finite chains. We show that besides the possibly existing EMZMs, no almost Majorana zero modes exist unless the fields on both the two sublattices are turned off. Our results might shed light on the control of ground-state degeneracies by solely tuning the external fields in related systems.Comment: 17 pages, 4 figures; to appear in Physical Review

Topological Properties of Electrons in Honeycomb Lattice with Kekul\'{e} Hopping Textures

Honeycomb lattice can support electronic states exhibiting Dirac energy dispersion, with graphene as the icon. We propose to derive nontrivial topology by grouping six neighboring sites of honeycomb lattice into hexagons and enhancing the inter-hexagon hopping energies over the intra-hexagon ones. We reveal that this manipulation opens a gap in the energy dispersion and drives the system into a topological state. The nontrivial topology is characterized by the $\mathbb{Z}_2$ index associated with a pseudo time-reversal symmetry emerging from the $C_6$ symmetry of the Kekul\'{e} hopping texture, where the angular momentum of orbitals accommodated on the hexagonal "artificial atoms" behaves as the pseudospin. The size of topological gap is proportional to the hopping-integral difference, which can be larger than typical spin-orbit couplings by orders of magnitude and potentially renders topological electronic transports available at high temperatures.Comment: 7 pages, 7 figure

Scheme to Achieve Silicon Topological Photonics

We derive in the present work topological photonic states purely based on silicon, a conventional dielectric material, by deforming a honeycomb lattice of silicon cylinders into a triangular lattice of cylinder hexagons. The photonic topology is associated with a pseudo time reversal (TR) symmetry constituted by the TR symmetry respected in general by the Maxwell equations and the $C_6$ crystal symmetry upon design, which renders the Kramers doubling in the present photonic system with the role of pseudo spin played by the circular polarization of magnetic field in the transverse magnetic mode. We solve Maxwell equations, and demonstrate new photonic topology by revealing pseudo spin-resolved Berry curvatures of photonic bands and helical edge states characterized by Poynting vectors.Comment: 5 pages, 5 figure