results for au:Wang_Z in:math

- We study a quartic matrix model with partition function $Z=\int d\ M\exp{\rm Tr}\ (-\Delta M^2-\frac{\lambda}{4}M^4)$. The integral is over the space of Hermitian $(\Lambda+1)\times(\Lambda+1)$ matrices, the matrix $\Delta$, which is not a multiple of the identity matrix, encodes the dynamics and $\lambda>0$ is a scalar coupling constant. We proved that the logarithm of the partition function is the Borel sum of the perturbation series, hence is a well defined analytic function of the coupling constant in certain analytic domain of $\lambda$, by using the multi-scale loop vertex expansions. All the non-planar graphs generated in the perturbation expansions have been taken care of on the same footing as the planar ones. This model is derived from the self-dual $\phi^4$ theory on the 2 dimensional Moyal space, also called the 2 dimensional Grosse-Wulkenhaar model. This would also be the first fully constructed matrix model which is non-trivial and not solvable.
- May 16 2018 math.PR arXiv:1805.05526v1In this paper we prove a new strong uniqueness result and a weak existence result for possibly \it degenerate multidimensional stochastic differential equations with Sobolev diffusion coefficients and rough drifts. In particular, examples with Hölder diffusion coefficients are provided to show our results.
- We study novel invariants of modular categories that are beyond the modular data, with an eye towards a simple set of complete invariants for modular categories. Our focus is on the $W$-matrix $-$the quantum invariant of a colored framed Whitehead link from the associated TQFT of a modular category. We prove that the $W$-matrix and the set of punctured $S$-matrices are strictly beyond the modular data $(S,T)$. Whether or not the triple $(S,T,W)$ constitutes a complete invariant of modular categories remains an open question.
- We propose an encoding for topological quantum computing with mapping class group representations, which is obviously not universal without leakage. We are interested in whether or not some generalized version of the Knill-Gottesmann theorem holds for such a quantum computing scheme, and how to make universal gate sets with supplemental primitives. As a first step, we prove that for abelian anyons, all gates from mapping class group representations are generalized Clifford gates, and this theorem does not hold for the Fibonacci anyon.
- May 15 2018 math.OC arXiv:1805.04815v1Series FACTS devices are one of the key enablers for very high penetration of renewables due to their capabilities in continuously controlling power flows on transmission lines. This paper proposes a bilevel optimization model to optimally locate variable series reactor (VSR) and phase shifting transformer (PST) in the transmission network considering high penetration of wind power. The upper level problem seeks to minimize the \textcolorblackinvestment cost on series FACTS, the cost of wind power curtailment and possible load shedding. The lower level problems capture the market clearing under different operating scenarios. Due to the poor scalability of $B\theta$ formulation, the \textslshift factor structure of FACTS allocation is derived. A customized reformulation and decomposition algorithm is designed and implemented to solve the proposed bilevel model with binary variables in both upper and lower levels. Detailed numerical results based on 118-bus system demonstrate the efficient performance of the proposed planning model and the important role of series FACTS for facilitating the integration of wind power.
- May 09 2018 math.CV arXiv:1805.02906v1Let $\Omega$ be an internal chord-arc Jordan domain and $\varphi:\mathbb S\rightarrow\partial\Omega$ be a homeomorphism. We show that $\varphi$ has finite dyadic energy if and only if $\varphi$ has a diffeomorphic extension $h: \mathbb D\rightarrow \Omega$ which has finite energy.
- The aim of this paper is to establish properties of the solutions to the $\alpha$-harmonic equations: $\Delta_{\alpha}(f(z))=\partial{z}[(1-{|{z}|}^{2})^{-\alpha} \overline{\partial}{z}f](z)=g(z)$, where $g:\overline{\mathbb{ID}}\rightarrow\mathbb{C}$ is a continuous function and $\overline{\mathbb{D}}$ denotes the closure of the unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$. We obtain Schwarz type and Schwarz-Pick type inequalities for the solutions to the $\alpha$-harmonic equation. In particular, for $g\equiv 0$, the solutions to the above equation are called $\alpha$-harmonic functions. We determine the necessary and sufficient conditions for an analytic function $\psi$ to have the property that $f\circ\psi$ is $\alpha$-harmonic function for any $\alpha$-harmonic function $f$. Furthermore, we discuss the Bergman-type spaces on $\alpha$-harmonic functions.
- Apr 24 2018 math.FA arXiv:1804.08102v1In this paper, we prove that all doubling measures on the unit disk $\mathbb{D}$ are Carleson measures for the standard Dirichlet space $\mathcal{D}$. The proof has three ingredients. The first one is a characterization of Carleson measures which holds true for general reproducing kernel Hilbert spaces. The second one is another new equivalent condition for Carleson measures, which holds true only for the standard Dirichlet space. The third one is an application of the dyadic method to our settings.
- This paper seeks to extend the theory of composition operators on analytic functional Hilbert spaces from analytic symbols to quasiconformal ones. The focus is the boundedness but operator-theoretic questions are discussed as well. In particular, we present a thorough analysis of $L^p$-estimates of a class of singular integral operators $P_\varphi$ associated with a quasiconformal mapping $\varphi$.
- Apr 11 2018 math.AP arXiv:1804.03319v2This paper is concerned with the uniqueness of solutions to the following nonlocal semi-linear elliptic equation \beginequation\labelellip\tag$\ast$ ∆u-\beta u+\lambda\frace^u\int_\Omegae^u=0~\mathrmin~\Omega, \endequation where $\Omega$ is a bounded domain in $\mathbb{R}^2$ and $\beta, \lambda$ are positive parameters. The above equation arises as the stationary problem of the well-known classical Keller-Segel model describing chemotaxis. For equation \eqrefellip with Neumann boundary condition, we establish an integral inequality and prove that the solution of (\refellip) is unique if $0<\lambda \leq 8\pi$ and $u$ satisfies some symmetric properties. While for \eqrefellip with Dirichlet boundary condition, the same uniqueness result is obtained without symmetric condition by a different approach inspired by some recent works [19,21]. As an application of the uniqueness results, we prove that the radially symmetric solution of the classical Keller-Segel system with subcritical mass subject to Neumann boundary conditions will converge to the unique constant equilibrium as time tends to infinity if $\Omega$ is a disc in two dimensions. As far as we know, this is the first result that asserts the exact asymptotic behavior of solutions to the classical Keller-Segel system with subcritical mass in two dimensions.
- Apr 10 2018 math.CO arXiv:1804.02802v2In the cops and robber games played on a simple graph $G$, Aigner and Fromme's lemma states that one cop can guard a shortest path in the sense that the robber cannot enter this path without getting caught after finitely many steps. In this paper, we extend Aigner and Fromme's lemma to cover a larger family of graphs and give metric characterizations of these graphs. In particular, we show that a generalization of block graphs, namely vertebrate graphs, are 1-guardable. We use this result to give the cop number of some special class of multi-layer generalized Peterson graphs.
- Apr 10 2018 math.CO arXiv:1804.02578v1We show a cyclic permutation analogue of Erdős-Szekeres Theorem. In particular, we show that every cyclic permutation of length $(k-2)(m-2)+2$ has either an increasing cyclic sub-permutation of length $k$ or a decreasing cyclic sub-permutation of length $m$. We also show that the result is tight.
- Apr 03 2018 math.OC arXiv:1804.00106v1This paper investigates the problem on the minimum ellipsoid containing the intersection of multiple ellipsoids, which has been extensively applied to information science, target tracking and data fusion etc. There are three major relaxation methods involving SDP relaxation, S-procedure relaxation and bounding ellipsoid relaxation, which are derived by different ideas or viewpoints. However, it is unclear for the interrelationships among these methods. This paper reveals the equivalence among the three relaxation methods by three stages. Firstly, the SDP relaxation method can be equivalently simplified to a decoupled SDP relaxation method. Secondly, the equivalence between the SDP relaxation method and the S-procedure relaxation method can be obtained by rigorous analysis. Thirdly, we establish the equivalence between the decoupled SDP relaxation method and the bounding ellipsoid relaxation method. Therefore, the three relaxation methods are unified through the decoupled SDP relaxation method. By analysis of the computational complexity, the decoupled SDP relaxation method has the least computational burden among the three methods. The above results are helpful for the research of set-membership filter and distributed estimation fusion. Finally, the performance of each method is evaluated by some typical numerical examples in information fusion and filtering.
- Mar 28 2018 math.CA arXiv:1803.09500v1We extend some one parameter theorems on fractional integrals due to Sawyer and Wheeden to the two parameter setting using a recent iteration method of Tanaka and Yabuta.
- We consider minimizers of the following mass critical Hartree minimization problem: \[ e_\lambda(N):=\underset{u∈H^1(R^d),\,\|u\|^2_2=N}\inf E_\lambda(u),\,\ d\ge 3, \]where the Hartree energy functional $E_\lambda(u)$ is defined by \[ E_\lambda(u):=\int_R ^d|∇u(x)|^2dx+\lambda \int_R ^dg(x)u^2(x)dx-\frac12 \int_R ^d\int_R ^d \fracu^2(x)u^2(y)|x-y|^2dxdy,\,\ \lambda>0,\]and the steep potential $g(x)$ satisfies $0=g(0)=\inf _{R^d}g(x)\le g(x)\le 1$ and $1-g(x)\in L^{\frac{d}{2}}(R^d)$. We prove that there exists a constant $N^*>0$, independent of $\lambda g(x)$, such that if $N\ge N^*$, then $e_\lambda(N)$ does not admit minimizers for any $\lambda >0$; if $0<N<N^*$, then there exists a constant $\lambda ^*(N)>0$ such that $e_\lambda(N)$ admits minimizers for any $\lambda >\lambda ^*(N)$, and $e_\lambda(N)$ does not admit minimizers for $0<\lambda <\lambda ^*(N)$. For any given $0<N<N^*$, the limit behavior of positive minimizers for $e_\lambda(N)$ is also studied as $\lambda\to\infty$, where the mass concentrates at the bottom of $g(x)$.
- Mar 26 2018 math.CO arXiv:1803.08641v1The dimension of a partially-ordered set (poset), introduced by Dushnik and Miller (1941), has been studied extensively in the literature. Recently, Ueckerdt (2016) proposed a variation called local dimension which makes use of partial linear extensions. While local dimension is bounded above by dimension, they can be arbitrarily far apart as the dimension of the standard example is $n$ while its local dimension is only $3$. Hiraguchi (1955) proved that the maximum dimension of a poset of order $n$ is $n/2$. However, we find a very different result for local dimension, proving a bound of $\Theta(n/\log n)$. This follows from connections with covering graphs using difference graphs which are bipartite graphs whose vertices in a single class have nested neighborhoods. We also prove that the local dimension of the $n$-dimensional Boolean lattice is $\Omega(n/\log n)$ and make progress toward resolving a version of the removable pair conjecture for local dimension.
- This paper investigates noncoherent detection in a two-way relay channel operated with physical layer network coding (PNC), assuming FSK modulation and short-packet transmissions. For noncoherent detection, the detector has access to the magnitude but not the phase of the received signal. For conventional communication in which a receiver receives the signal from a transmitter only, the phase does not affect the magnitude, hence the performance of the noncoherent detector is independent of the phase. PNC, however, is a multiuser system in which a receiver receives signals from multiple transmitters simultaneously. The relative phase of the signals from different transmitters affects the received signal magnitude through constructive-destructive interference. In particular, for good performance, the noncoherent detector in PNC must take into account the influence of the relative phase on the signal magnitude. Building on this observation, this paper delves into the fundamentals of PNC noncoherent detector design. To avoid excessive overhead, we do away from preambles. We show how the relative phase can be deduced directly from the magnitudes of the received data symbols. Numerical results show that our detector performs nearly as well as a "fictitious" optimal detector that has perfect knowledge of the channel gains and relative phase.
- This paper proposes an approach to address the voltage stability assessment (VSA) considering N-1 contingency. The approach leverages the sensitivity based Thevenin index (STI) which involves evaluating the Jacobian matrix at current operating condition. Since the N-1 contingency case is hypothetical, there is no information regarding the operating condition after a foreseen contingency. The proposed approach first estimates the post-contingency operating point as well as possible PV-PQ transitions based on the current operating point. Then the STI for each contingency can be predicted using the estimated operating condition. Numerical results based on IEEE 14-bus system demonstrate the accuracy of the proposed approach in predicting the voltage stability margin under contingency. Moreover, the on-line implementation of the proposed approach is promising since it only involves solving several linear equations.
- With the increased complexity of power systems due to the integration of smart grid technologies and renewable energy resources, more frequent changes have been introduced to system status, and the traditional serial mode of state estimation algorithm cannot well meet the restrict time-constrained requirement for the future dynamic power grid, even with advanced computer hardware. To guarantee the grid reliability and minimize the impacts caused by system status fluctuations, a fast, even SCADA-rate, state estimator is urgently needed. In this paper, a graph based power system modeling is firstly explored and a graph computing based state estimation is proposed to speed up its performance. The power system is represented by a graph, which is a collection of vertices and edges, and the measurements are attributes of vertices and edges. Each vertex can independently implement local computation, like formulations of the node-based H matrix, gain matrix and righthand-side (RHS) vector, only with the information on its connected edges and neighboring vertices. Then, by taking advantages of graph database, these node-based data are conveniently collected and stored in the compressed sparse row (CSR) format avoiding the complexity and heaviness introduced by the sparse matrices. With communications and synchronization, centralized computation of solving the weighted least square (WLS) state estimation is completed with hierarchical parallel computing. The proposed strategy is implemented on a graph database platform. The testing results of IEEE 14-bus, IEEE 118-bus systems and a provincial system in China verify the accuracy and high-performance of the proposed methodology.
- Mar 05 2018 math.OC arXiv:1803.00660v1As microgrids have advanced from early prototypes to relatively mature technologies, converting data center integrated commercial buildings to microgrids provides economic, reliability and resiliency enhancements for the building owners. Thus, microgrid design and economically sizing distributed energy resources (DER) are becoming more demanding to gain widespread microgrids commercial viability. In this paper, an optimal DER sizing formulation for a hybrid AC/DC microgrid configuration has been proposed to leverage all benefits that AC or DC microgrid could solely contribute. Energy storage (ES), photovoltaics (PV) and power electronics devices are coordinately sized for economic grid-connected and reliable islanded operations. Time-of-use (TOU) energy usages charges and peak demand charges are explicitly modeled to achieve maximum level of cost savings. Numerical results obtained from a real commercial building load demonstrate the benefits of the proposed approach and the importance of jointly sizing DER for the grid-connected and islanded modes.
- The alternating gradient descent (AGD) is a simple but popular algorithm which has been applied to problems in optimization, machine learning, data ming, and signal processing, etc. The algorithm updates two blocks of variables in an alternating manner, in which a gradient step is taken on one block, while keeping the remaining block fixed. When the objective function is nonconvex, it is well-known the AGD converges to the first-order stationary solution with a global sublinear rate. In this paper, we show that a variant of AGD-type algorithms will not be trapped by "bad" stationary solutions such as saddle points and local maximum points. In particular, we consider a smooth unconstrained optimization problem, and propose a perturbed AGD (PA-GD) which converges (with high probability) to the set of second-order stationary solutions (SS2) with a global sublinear rate. To the best of our knowledge, this is the first alternating type algorithm which takes $\mathcal{O}(\text{polylog}(d)/\epsilon^{7/3})$ iterations to achieve SS2 with high probability [where polylog$(d)$ is polynomial of the logarithm of dimension $d$ of the problem].
- Feb 27 2018 math.CO arXiv:1802.08918v1For any $t\geq 1$ and an edge-colored multigraph $G$, we show that $G$ has $t$ color-disjoint rainbow spanning trees if and only if for any partition $P$ of $V(G)$, there are at least $t(|P|-1)$ distinct colors occurring in the crossing edges of $P$. Our theorem generalizes two previous results: Nash-Williams-Tutte theorem and Schrijver's theorem. As an application, we resolve a conjecture of Jahanbekam and West: $r(n,t)={n-2 \choose 2}+t$ whenever $n\geq 2t+2 \geq 6$. Here $r(n,t)$ is the maximum number of colors in an edge-coloring of $K_n$ not having $t$ edge-disjoint rainbow spanning trees.
- Feb 27 2018 math.OC arXiv:1802.08720v1It is a major task to develop effective strategies for defending the power system against deliberate attacks. It is critical to comprehensively consider the human-related and environmental risks and uncertainties, which is missing in existing literature. This paper considers the load demand uncertainties and wind generation uncertainties in addition to the interactive attacker/defender behaviors. Specifically, a defender-attacker-nature-operator model is proposed, which incorporates the attack/defense interaction, the corrective re-dispatch of the operator, the coordination between the attack strategy and the stochastic nature of load demands and wind generations. The Column-and-Constraint Generation (C&CG) algorithm is adopted for solving the proposed model by decomposing the proposed model into a master problem and a sub-problem. Simulations are performed using MATLAB and CPLEX on a modified IEEE RTS79 system. The simulation results verify the validity of the proposed model.
- Feb 22 2018 math.NA arXiv:1802.07682v1In this article, we propose an implicit finite difference scheme for a two-dimensional parabolic stochastic partial differential equation (SPDE) of Zakai type. The scheme is based on a Milstein approximation to the stochastic integral and an alternating direction implicit (ADI) discretisation of the elliptic term. We prove its mean-square stability and convergence in $L_2$ of first order in time and second order in space, by Fourier analysis, in the presence of Dirac initial data. Numerical tests confirm these findings empirically.
- The popular cubic regularization (CR) method converges with first- and second-order optimality guarantee for nonconvex optimization, but encounters a high sample complexity issue for solving large-scale problems. Various sub-sampling variants of CR have been proposed to improve the sample complexity.In this paper, we propose a stochastic variance-reduced cubic-regularized (SVRC) Newton's method under both sampling with and without replacement schemes. We characterize the per-iteration sample complexity bounds which guarantee the same rate of convergence as that of CR for nonconvex optimization. Furthermore, our method achieves a total Hessian sample complexity of $\mathcal{O}(N^{8/11} \epsilon^{-3/2})$ and $\mathcal{O}(N^{3/4}\epsilon^{-3/2})$ respectively under sampling without and with replacement, which improve that of CR as well as other sub-sampling variant methods via the variance reduction scheme. Our result also suggests that sampling without replacement yields lower sample complexity than that of sampling with replacement. We further compare the practical performance of SVRC with other cubic regularization methods via experiments.
- The farthest point map sends a point in a compact metric space to the set of points farthest from it. We focus on the case when this metric space is a convex centrally symmetric polyhedron, so that we can compose the farthest point map with the antipodal map. The purpose of this work is to study the properties of this composition of the two maps above. We showed that: 1. this map is piecewise biquadratic; 2. it has no generalized periodic points; 3. its limit point set coincides with its generalized fixed point set; 4. each of its orbit converges; 5. its limit set is contained in a finite union of hyperbolas. We will define some of these terminologies in the article.
- Feb 19 2018 math.NA arXiv:1802.05743v1A first-order, Monte Carlo ensemble method has been recently introduced for solving parabolic equations with random coefficients in [26], which is a natural synthesis of the ensemble-based, Monte Carlo sampling algorithm and the ensemble-based, first-order time stepping scheme. With the introduction of an ensemble average of the diffusion function, this algorithm leads to a single discrete system with multiple right-hand sides for a group of realizations, which could be solved more efficiently than a sequence of linear systems. In this paper, we pursue in the same direction and develop a new multilevel Monte Carlo ensemble method for solving random parabolic partial differential equations. Comparing with the approach in [26], this method possesses a high-order accuracy in time and further reduces the computational cost by using the multilevel Monte Carlo method. Rigorous numerical analysis shows the method achieves the optimal rate of convergence. Several numerical experiments are presented to illustrate the theoretical results.
- Feb 15 2018 math.CO arXiv:1802.04930v1Given a graph $G$ and a positive integer $k$, the \emphGallai-Ramsey number is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) triangle or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds on the Gallai-Ramsey numbers for books $B_{m} = K_{2} + \overline{K_{m}}$ and prove sharp results for $m \leq 5$.
- Feb 15 2018 physics.comp-ph math.NA arXiv:1802.04961v1In this paper, a unified gas kinetic scheme for multiphase dilute gas-particle system is proposed. The UGKS multiphase (UGKS-M) is a finite volume method, which captures flow physics in the regimes from collisionless multispecies transport to the two-fluid hydrodynamic Navier-Stokes (NS) solution with the variation of Knudsen number, and from granular flow regime to dusty gas dynamics with the variation of Stokes number. The main reason for preserving the multiscale nature in UGKS-M is mainly coming from the direct modeling of the flow physics in the scales of discrete cell size and time step, where the ratio of the time step over the particle collision time determines flow behavior in different regimes. For the particle phase, the integral solution of the kinetic equation is used in the construction of the numerical flux, which takes into account the particle transport, collision, and acceleration. The gas phase, which is assumed to be in the continuum flow regime, evolves numerically by the gas kinetic scheme (GKS), which is a subset of the UGKS for the Navier-Stokes (NS) solutions. The interaction between the gas and particle phase is calculated based on a velocity space mapping method, which solves accurately the kinetic acceleration process. The stability of UGKS-M is determined by the CFL condition only. With the inclusion of the material temperature evolution equation of solid particles, the UGKS-M conserves the total mass, momentum, and energy for the whole multiphase system. In the numerical tests, the UGKS-M shows good multiscale property in capturing the particle trajectory crossing (PTC), particle wall reflecting phenomena, and vortex-induced segregation of inertial particles under different Stokes numbers. The scheme is also applied to simulate shock induced fluidization problem and the simulation results agree well with experimental measurement.
- Feb 15 2018 math.RT arXiv:1802.05197v1Let $\mathcal{X}$ be a class of left $R$-modules, $\mathcal{Y}$ be a class of right $R$-modules. In this paper, we introduce and study Gorenstein $(\mathcal{X}, \mathcal{Y})$-flat modules as a common generalization of some known modules such as Gorenstein flat modules \citeEJT93, Gorenstein $n$-flat modules \citeSUU14, Gorenstein $\mathcal{B}$-flat modules \citeEIP17, Gorenstein AC-flat modules \citeBEI17, $\Omega$-Gorenstein flat modules \citeEJ00 and so on. We show that the class of all Gorenstein $(\mathcal{X}, \mathcal{Y})$-flat modules have a strong stability. In particular, when $(\mathcal{X}, \mathcal{Y})$ is a perfect (symmetric) duality pair, we give some functorial descriptions of Gorenstein $(\mathcal{X}, \mathcal{Y})$-flat dimension, and construct a hereditary abelian model structure on $R$-Mod whose cofibrant objects are exactly the Gorenstein $(\mathcal{X}, \mathcal{Y})$-flat modules. These results unify the corresponding results of the aforementioned modules.
- In this paper, local linear estimators are adapted for the unknown infinitesimal coefficients associated with continuous-time asset return model with jumps, which can correct the bias automatically due to their simple bias representation. The integrated diffusion models with jumps, especially infinite activity jumps are mainly investigated. In addition, under mild conditions, the weak consistency and asymptotic normality is provided through the conditional Lindeberg theorem. Furthermore, our method presents advantages in bias correction through simulation whether jumps belong to the finite activity case or infinite activity case. Finally, the estimators are illustrated empirically through the returns for stock index under five-minute high sampling frequency for real application.
- Microgrids consisting of multiple distributed energy resources (DERs) provide a promising solution to integrate renewable energies, e.g., solar photovoltaic (PV) systems. Hybrid AC/DC microgrids leverage the merits of both AC and DC power systems. In this paper, a control strategy for islanded multi-bus hybrid microgrids is proposed based on the Finite-Control-Set Model Predictive Control (FCS-MPC) technologies. The control loops are expedited by predicting the future states and determining the optimal control action before switching signals are sent. The proposed algorithm eliminates the needs of PI, PWM, and droop components, and offers 1) accurate PV maximum power point tracking (MPPT) and battery charging/discharging control, 2) DC and multiple AC bus voltage/frequency regulation, 3) a precise power sharing scheme among DERs without voltage or frequency deviation, and 4) a unified MPC design flow for hybrid microgrids. Multiple case studies are carried out, which verify the satisfactory performance of the proposed method.
- Motivated by the quasi-local mass problem in general relativity, we study the rigidity of isometric immersions with the same mean curvature into a warped product space. As a corollary of our main result, two star-shaped hypersurfaces in a spatial Schwarzschild or AdS-Schwarzschild manifold with nonzero mass differ only by a rotation if they are isometric and have the same mean curvature. We also give similar results if the mean curvature condition is replaced by an $\sigma_2$-curvature condition.
- Large eddy simulation (LES) has been increasingly used to tackle vortex-dominated turbulent flows. In LES, the quality of the simulation results hinges upon the quality of the numerical discretizations in both space and time. It is in this context we perform a Fourier analysis of several popular methods in LES including the discontinuous Galerkin (DG), finite difference (FD), and compact difference (CD) methods. We begin by reviewing the semi-discrete versions of all methods under-consideration, followed by a fully-discrete analysis with explicit Runge-Kutta (RK) time integration schemes. In this regard, we are able to unravel the true dispersion/dissipation behavior of DG and Runge-Kutta DG (RKDG) schemes for the entire wavenumber range. The physical-mode is verified to be a good approximation for the asymptotic behavior of these DG schemes in the low wavenumber range. After that, we proceed to compare the DG, FD, and CD methods in dispersion and dissipation properties. Numerical tests are conducted using the linear advection equation to verify the analysis. In comparing different methods, it is found that the overall numerical dissipation strongly depends on the time step. Compact difference (CD) and central finite difference (FD) schemes, in some particular settings, can have more numerical dissipation than the DG scheme with an upwind flux. This claim is then verified through a numerical test using the Burgers' equation.
- Motivated by mobile edge computing and wireless data centers, we study a wireless distributed computing framework where the distributed nodes exchange information over a wireless interference network. Our framework follows the structure of MapReduce. This framework consists of Map, Shuffle, and Reduce phases, where Map and Reduce are computation phases and Shuffle is a data transmission phase. In our setting, we assume that the transmission is operated over a wireless interference network. We demonstrate that, by duplicating the computation work at a cluster of distributed nodes in the Map phase, one can reduce the amount of transmission load required for the Shuffle phase. In this work, we characterize the fundamental tradeoff between computation load and communication load, under the assumption of one-shot linear schemes. The proposed scheme is based on side information cancellation and zero-forcing, and we prove that it is optimal in terms of computation-communication tradeoff. The proposed scheme outperforms the naive TDMA scheme with single node transmission at a time, as well as the coded TDMA scheme that allows coding across data, in terms of the computation-communication tradeoff.
- In this paper, an energy harvesting scheme for a multi-user multiple-input-multiple-output (MIMO) secrecy channel with artificial noise (AN) transmission is investigated. Joint optimization of the transmit beamforming matrix, the AN covariance matrix, and the power splitting ratio is conducted to minimize the transmit power under the target secrecy rate, the total transmit power, and the harvested energy constraints. The original problem is shown to be non-convex, which is tackled by a two-layer decomposition approach. The inner layer problem is solved through semi-definite relaxation, and the outer problem is shown to be a single-variable optimization that can be solved by one-dimensional (1-D) line search. To reduce computational complexity, a sequential parametric convex approximation (SPCA) method is proposed to find a near-optimal solution. Furthermore, tightness of the relaxation for the 1-D search method is validated by showing that the optimal solution of the relaxed problem is rank-one. Simulation results demonstrate that the proposed SPCA method achieves the same performance as the scheme based on 1-D search method but with much lower complexity.
- We study the problem of centralized exact repair of multiple failures in distributed storage. We describe constructions that achieve a new set of interior points under exact repair. The constructions build upon the layered code construction by Tian et al., designed for exact repair of single failure. We firstly improve upon the layered construction for general system parameters. Then, we extend the improved construction to support the repair of multiple failures, with varying number of helpers. In particular, we prove the optimality of one point on the functional repair tradeoff of multiple failures for some parameters. Finally, considering minimum bandwidth cooperative repair (MBCR) codes as centralized repair codes, we determine explicitly the best achievable region obtained by space-sharing among all known points, including the MBCR point.
- In this paper, we study a multi-user multiple-input-multiple-output secrecy simultaneous wireless information and power transfer (SWIPT) channel which consists of one transmitter, one cooperative jammer (CJ), multiple energy receivers (potential eavesdroppers, ERs), and multiple co-located receivers (CRs). We exploit the dual of artificial noise (AN) generation for facilitating efficient wireless energy transfer and secure transmission. Our aim is to maximize the minimum harvested energy among ERs and CRs subject to secrecy rate constraints for each CR and total transmit power constraint. By incorporating norm-bounded channel uncertainty model, we propose a iterative algorithm based on sequential parametric convex approximation to find a near-optimal solution. Finally, simulation results are presented to validate the performance of the proposed algorithm outperforms that of the conventional AN-aided scheme and CJ-aided scheme.
- Using non-commutative differential forms, we construct a complex called singular Hochschild cochain complex for any associative algebra over a field. The cohomology of this complex is isomorphic to the Tate-Hochschild cohomology in the sense of Buchweitz. By a natural action of the cellular chain operad of the spineless cacti operad, introduced by R. Kaufmann, on the singular Hochschild cochain complex, we provide a proof of the Deligne's conjecture for this complex. More concretely, the complex is an algebra over the (dg) operad of chains of the little $2$-discs operad. By this action, we also obtain that the singular Hochschild cochain complex has a $B$-infinity algebra structure and its cohomology ring is a Gerstenhaber algebra. Inspired by the original definition of Tate cohomology for finite groups, we define a generalized Tate-Hochschild complex with the Hochschild chains in negative degrees and the Hochschild cochains in non-negative degrees. There is a natural embedding of this complex into the singular Hochschild cochain complex. In the case of a self-injective algebra, this embedding becomes a quasi-isomorphism. In particular, for a symmetric algebra, this allows us to show that the Tate-Hochschild cohomology ring, equipped with the Gerstenhaber algebra structure, is a Batalin-Vilkovisky algebra.
- In this paper, we prove the existence of the free boundary minimal hypersurface of least area in compact manifolds with boundary. Such hypersurface can be viewed as the ground state of the volume spectrum introduced by Gromov. Moreover, we characterize the orientation and Morse index of them.
- Jan 23 2018 math.AP arXiv:1801.07062v1The flux limited Keller-Segel (FLKS) system is a macroscopic model describing bacteria motion by chemotaxis which takes into account saturation of the velocity. The hyper-bolic form and some special parabolic forms have been derived from kinetic equations describing the run and tumble process for bacterial motion. The FLKS model also has the advantage that traveling pulse solutions exist as observed experimentally. It has attracted the attention of many authors recently. We design and prove a general derivation of the FLKS departing from a kinetic model under stiffness assumption of the chemotactic response and rescaling the kinetic equation according to this stiffness parameter. Unlike the classical Keller-Segel system, solutions of the FLKS system do not blow-up in finite or infinite time. Then we investigate the existence of radially symmetric steady state and long time behaviour of this flux limited Keller-Segel system.
- An exponential time-integrator scheme of second-order accuracy based on the predictor-corrector methodology, denoted PCEXP, is developed to solve multi-dimensional nonlinear partial differential equations pertaining to fluid dynamics. The effective and efficient implementation of PCEXP is realized by means of the Krylov method. The linear stability and truncation error are analyzed through a one-dimensional model equation. The proposed PCEXP scheme is applied to the Euler equations discretized with a discontinuous Galerkin method in both two and three dimensions. The effectiveness and efficiency of the PCEXP scheme are demonstrated for both steady and unsteady inviscid flows. The accuracy and efficiency of the PCEXP scheme are verified and validated through comparisons with the explicit third-order total variation diminishing Runge-Kutta scheme (TVDRK3), the implicit backward Euler (BE) and the implicit second-order backward difference formula (BDF2). For unsteady flows, the PCEXP scheme generates a temporal error much smaller than the BDF2 scheme does, while maintaining the expected acceleration at the same time. Moreover, the PCEXP scheme is also shown to achieve the computational efficiency comparable to the implicit schemes for steady flows.
- The private search problem is introduced, where a dataset comprised of $L$ i.i.d. records is replicated across $N$ non-colluding servers, each record takes values uniformly from an alphabet of size $K$, and a user wishes to search for all records that match a privately chosen value, without revealing any information about the chosen value to any individual server. The capacity of private search is the maximum number of bits of desired information that can be retrieved per bit of download. The asymptotic (large $K$) capacity of private search is shown to be $1-1/N$, even as the scope of private search is further generalized to allow approximate (OR) search over a number of realizations that grows with $K$. The results are based on the asymptotic behavior of a new converse bound for private information retrieval with arbitrarily dependent messages.
- Jan 16 2018 math.QA arXiv:1801.04296v1Acyclic anyon models are non-abelian anyon models for which thermal anyon errors can be corrected. In this note, we characterize acyclic anyon models and raise the question if the restriction to acyclic anyon models is a deficiency of the current protocol or could it be intrinsically related to the computational power of non-abelian anyons. We also obtain general results on acyclic anyon models and find new acyclic anyon models such as $SO(8)_2$ and untwisted Dijkgraaf-Witten theories of nilpotent finite groups.
- In this paper, a secure spatial modulation (SM) system with artificial noise (AN)-aided is investigated. To achieve higher secrecy rate (SR) in such a system, two high-performance schemes of transmit antenna selection (TAS), leakage-based and maximum secrecy rate (Max-SR), are proposed and a generalized Euclidean distance-optimized antenna selection (EDAS) method is designed. From simulation results and analysis, the four TAS schemes have an decreasing order: Max-SR, leakage-based, generalized EDAS, and random (conventional), in terms of SR performance. However, the proposed Max-SR method requires the exhaustive search to achieve the optimal SR performance, thus its complexity is extremely high as the number of antennas tends to medium and large scale. The proposed leakage-based method approaches the Max-SR method with much lower complexity. Thus, it achieves a good balance between complexity and SR performance. In terms of bit error rate (BER), their performances are in an increasing order: random, leakage-based, Max-SR, and generalized EDAS.
- Decoupled fractional Laplacian wave equation can describe the seismic wave propagation in attenuating media. Fourier pseudospectral implementations, which solve the equation in spatial frequency domain, are the only existing methods for solving the equation. For the earth media with curved boundaries, the pseudospectral methods could be less attractive to handle the irregular computational domains. In the paper, we propose a radial basis function collocation method that can easily tackle the irregular domain problems. Unlike the pseudospectral methods, the proposed method solves the equation in physical variable domain. The directional fractional Laplacian is chosen from varied definitions of fractional Laplacian. Particularly, the vector Grünwald-Letnikov formula is employed to approximate fractional directional derivative of radial basis function. The convergence and stability of the method are numerically investigated by using the synthetic solution and the long-time simulations, respectively. The method's flexibility is studied by considering homogeneous and multi-layer media having regular and irregular geometric boundaries.
- Jan 03 2018 math.OC arXiv:1801.00172v1The Variable Series Reactors (VSRs) can efficiently control the power flow through the adjustment of the line reactance. When they are appropriately allocated in the power network, the transmission congestion and generation cost can be reduced. This paper proposes a planning model to optimally allocate VSRs considering AC constraints and multi-scenarios including base case and contingencies. The planning model is originally a non-convex large scale mixed integer nonlinear program (MINLP), which is generally intractable. The proposed Benders approach decomposes the MINLP model into a mixed integer linear program (MILP) master problem and a number of nonlinear subproblems. Numerical case studies based on IEEE 118-bus demonstrate the high performance of the proposed approach.
- Dec 29 2017 math.OC arXiv:1712.09443v2This paper proposes a dynamic game-based maintenance scheduling mechanism for the asset owners of the natural gas grid and the power grid by using a bilevel approach. In the upper level, the asset owners of the natural gas grid and the power grid schedule maintenance to maximize their own revenues. This level is modeled as a dynamic game problem, which is solved by the backward induction algorithm. In the lower level, the independent system operator (ISO) dispatches the system to minimize the loss of power load and natural gas load in consideration of the system operating conditions under maintenance plans from the asset owners in the upper level. This is modeled as a mixed integer linear programming problem. For the model of the natural gas grid, a piecewise linear approximation associated with the big-M approach is used to transform the original nonlinear model into the mixed integer linear model. Numerical tests on a 6-bus system with a 4-node gas grid show the effectiveness of the proposed model.
- Dec 25 2017 math.OC arXiv:1712.08528v1In the past extensive researches have been conducted on demand side management (DSM) program which aims at reducing peak loads and saving electricity cost. In this paper, we propose a framework to study decentralized household demand side management in a residential distribution network which consists of multiple smart homes with schedulable electrical appliances and some rooftop photovoltaic generation units. Each smart home makes individual appliance scheduling to optimize the electric energy cost according to the day-ahead forecast of electricity prices and its willingness for convenience sacrifice. Using the developed simulation model, we examine the performance of decentralized household DSM and study their impacts on the distribution network operation and renewable integration, in terms of utilization efficiency of rooftop PV generation, overall voltage deviation, real power loss, and possible reverse power flows.
- Let $(X,\omega)$ be a compact Hermitian manifold of complex dimension $n$. In this article, we first survey recent progress towards Grauert-Riemenschneider type criterions. Secondly, we give a simplified proof of Boucksom's conjecture given by the author under the assumption that the Hermitian metric $\omega$ satisfies $\partial\overline{\partial}\omega^l=$ for all $l$, i.e., if $T$ is a closed positive current on $X$ such that $\int_XT_{ac}^n>0$, then the class $\{T\}$ is big and $X$ is Kähler. Finally, as an easy observation, we point out that Nguyen's result can be generalized as follows: if $\partial\overline{\partial}\omega=0$, and $T$ is a closed positive current with analytic singularities, such that $\int_XT^n_{ac}>0$, then the class $\{T\}$ is big and $X$ is Kähler.
- Existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying "true" statistical models. To address this issue, we take a first step towards taming model misspecification by studying the high-dimensional sparse phase retrieval problem with misspecified link functions. In particular, we propose a simple variant of the thresholded Wirtinger flow algorithm that, given a proper initialization, linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions. We further provide extensive numerical experiments to support our theoretical findings.
- Dec 19 2017 math.CO arXiv:1712.06499v1We investigate the rigidity of the Hopf algebra of quasi-symmetric functions with respect to monomial, fundamental and quasi-symmetric Schur basis respectively. It is showed that the linear map induced by sending $M_{\alpha}$ to $M_{\alpha^r}$ is the unique nontrivial graded algebra automorphism that takes the monomial basis into itself, but it does not preserve the comultiplication. We also show that the linear map induced by sending $F_{\alpha}$ to $F_{\alpha^c}$ is the unique nontrivial graded coalgebra automorphism preserving the fundamental basis. However, there are no nontrivial graded Hopf algebra automorphisms that preserve the quasi-symmetric Schur basis.
- Dec 13 2017 math.OC arXiv:1712.04419v1Nowadays common practice in deploying photovoltaic distributed generations (PVDGs) is customer-based installation in the distribution network. Increasing level of PVDG applications and expedite approval by utilities have raised concern about the negative impacts of PVDG installations on the distribution network operations such as reverse power flows and undesirable voltage fluctuations. One potential solutions is to optimize the siting and sizing of these distributed renewable generation resources. This paper presents a comparative study on both optimal and randomized installation of PVDGs with the latter modeling real life customer-based renewable integration. The proposed models examine and compare the impacts of PVDG installation on distribution network operation. Numerical simulations have been performed on a local distribution network model with realistic load profiles, GIS information, local solar insolation, and feeder and voltage settings. It is found that when the distribution system has a medium penetration ratio optimal PVDG installations may introduce essential improvements in terms of voltage deviation and energy loss reduction than randomized installation. However, if the penetration ratio is very low or extremely high there will be not significant difference between the two.
- Dec 12 2017 math.CO arXiv:1712.03247v1For any $r\geq 2$ and $k\geq 3$, the $r$-color size-Ramsey number $\hat R(\mathcal{G},r)$ of a $k$-uniform hypergraph $\mathcal{G}$ is the smallest integer $m$ such that there exists a $k$-uniform hypergraph $\mathcal{H}$ on $m$ edges such that any coloring of the edges of $\mathcal{H}$ with $r$ colors yields a monochromatic copy of $\mathcal{G}$. Let $\mathcal{P}_{n,k-1}^{(k)}$ denote the $k$-uniform tight path on $n$ vertices. Dudek, Fleur, Mubayi and Rődl showed that the size-Ramsey number of tight paths $\hat R(\mathcal{P}_{n,k-1}^{(k)}, 2) = O(n^{k-1-\alpha} (\log n)^{1+\alpha})$ where $\alpha = \frac{k-2}{\binom{k-1}{2}+1}$. In this paper, we improve their bound by showing that $\hat R(\mathcal{P}_{n,k-1}^{(k)}, r) = O(r^k (n\log n)^{k/2})$ for all $k\geq 3$ and $r\geq 2$.
- In this paper, we focus on studying central limit theorems for functionals of some specific stationary random processes. In classical probability theory, it is well-known that for non-linear functionals of stationary Gaussian sequences, we can get a central-limit result via Hermite polynomials and the diagram formula for cumulants. In this paper, the main result is an analogous central limit theorem, in a free probability setting, for real-valued functionals of a stationary semicircular sequence with long-range dependence, namely the correlation function of the underlying time series tends to zero as the lag goes to infinity.
- Dec 08 2017 math.OC arXiv:1712.02396v1This paper presents an anomaly detection method using a hybrid observer -- which consists of a discrete state observer and a continuous state observer. We focus our attention on anomalies caused by intelligent attacks, which may bypass existing anomaly detection methods because neither the event sequence nor the observed residuals appear to be anomalous. Based on the relation between the continuous and discrete variables, we define three conflict types and give the conditions under which the detection of the anomalies is guaranteed. We call this method conflict-driven anomaly detection. The effectiveness of this method is demonstrated mathematically and illustrated on a Train-Gate (TG) system.
- Let $(X,\omega)$ be a Hermitian manifold and let $(E,h^E)$, $(F,h^F)$ be two Hermitian holomorphic line bundle over $X$. Suppose that the maximal rank of the Chern curvature $c(E)$ of $E$ is $r$, and the kernel of $c(E)$ is foliated, i.e. there is a foliation $Y$ of $X$, of complex codimension $r$, such that the tangent space of the leaf at each point $x\in X$ is contained in the kernel of $c(E)$. In this paper, local versions of Demailly-Bouche's holomorphic Morse inequalities (which give asymptotic bounds for cohomology groups $H^{q}(X,E^k\otimes F^l)$ as $k,l,k/l\rightarrow \infty$) are presented. The local version holds on any Hermitian manifold regardless of compactness and completeness. The proof is a variation of Berman's method to derive holomorphic Morse inequalities on compact complex manifolds with boundary.
- Nov 28 2017 math.NA arXiv:1711.09392v1In this paper we study the problem of computing the effective diffusivity for a particle moving in chaotic and stochastic flows. In addition we numerically investigate the residual diffusion phenomenon in chaotic advection. The residual diffusion refers to the non-zero effective (homogenized) diffusion in the limit of zero molecular diffusion as a result of chaotic mixing of the streamlines. In this limit traditional numerical methods typically fail since the solutions of the advection-diffusion equation develop sharp gradients. Instead of solving the Fokker-Planck equation in the Eulerian formulation, we compute the motion of particles in the Lagrangian formulation, which is modelled by stochastic differential equations (SDEs). We propose a new numerical integrator based on a stochastic splitting method to solve the corresponding SDEs in which the deterministic subproblem is symplectic preserving while the random subproblem can be viewed as a perturbation. We provide rigorous error analysis for the new numerical integrator using the backward error analysis technique and show that our method outperforms standard Euler-based integrators. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several typical chaotic and stochastic flow problems of physical interests.
- The full configuration interaction quantum Monte Carlo method in the lens of inexact power iterationNov 28 2017 math.NA physics.comp-ph arXiv:1711.09153v2In this paper, we propose a general analysis framework for inexact power iteration, which can be used to efficiently solve high dimensional eigenvalue problems arising from quantum many-body problems. Under the proposed framework, we establish the convergence theorems for several recently proposed randomized algorithms, including the full configuration interaction quantum Monte Carlo (FCIQMC) and the fast randomized iteration (FRI). The analysis is consistent with numerical experiments for physical systems such as Hubbard model and small chemical molecules. We also compare the algorithms both in convergence analysis and numerical results.
- We investigate, in the Luttinger model with fixed box potential, the time evolution of an inhomogeneous state prepared as a localized fermion added to the noninteracting ground state. We proved that, if the state is evolved with the interacting Hamiltonian, the averaged density has two peaks moving in opposite directions, with a constant but renormalized velocity. We also proved that a dynamical `Landau quasi-particle weight' appears in the oscillating part of the averaged density, asymptotically vanishing with large time. The results are proved with the Mattis-Lieb diagonalization method. A simpler proof with the exact Bosonization formulas is also provided.
- Nov 22 2017 math.OC arXiv:1711.07853v2In this paper, we develop semidefinite programming (SDP) models aimed at solving optimal power flow (OPF) problems in distribution systems. We propose two models: the symmetrical SDP model which modifies the existing BFM-SDP model. Then based on the symmetrical SDP model, we develop a voltage regulation model that solves OPF problems with binding voltage constraints. To evaluate the accuracy of our proposed OPF models, we rely on OpenDSS, a power flow solver, to generate power flow solutions as the benchmarks. Comprehensive case studies are conducted showing our SDP models have better numerical stability and yield more accurate results than existing approaches
- This paper divides into two parts. Let $(X,\omega)$ be a compact Hermitian manifold. Firstly, if the Hermitian metric $\omega$ satisfies the assumption that $\partial\overline{\partial}\omega^k=0$ for all $k$, we generalize the volume of the cohomology class in the Kähler setting to the Hermitian setting, and prove that the volume is always finite and the Grauert-Riemenschneider type criterion holds true, which is a partial answer to a conjecture posed by Boucksom. Secondly, we observe that if the anticanonical bundle $K^{-1}_X$ is nef, then for any $\varepsilon>0$, there is a smooth function $\phi_\varepsilon$ on $X$ such that $\omega_\varepsilon:=\omega+i\partial\overline{\partial}\phi_\varepsilon>0$ and Ricci$(\omega_\varepsilon)\geq-\varepsilon\omega_\varepsilon$. Furthermore, if $\omega$ satisfies the assumption as above, we prove that for a Harder-Narasimhan filtration of $T_X$ with respect to $\omega$, the slopes $\mu_\omega(\mathcal{F}_i/\mathcal{F}_{i-1})\geq 0$ for all $i$, which generalizes a result of Cao which plays a very important role in his studying of the structures of Kähler manifolds.
- In this paper, we provide a set of definitions upon which one can prove in a rigorous way most of the main results achieved in the pattern of zeros classification of fractional quantum Hall states.
- Nov 17 2017 math.NA arXiv:1711.06038v1The steady-state Poisson-Nernst-Planck (ssPNP) equations are an effective model for the description of ionic transport in ion channels. It is observed that an ion channel exhibits voltage-dependent switching between open and closed states. Different conductance states of a channel imply that the ssPNP equations probably have multiple solutions with different level of currents. We propose numerical approaches to study multiple solutions to the ssPNP equations with multiple ionic species. To find complete current-voltage (I-V ) and current-concentration (I-C) curves, we reformulate the ssPNP equations into four different boundary value problems (BVPs). Numerical continuation approaches are developed to provide good initial guesses for iteratively solving algebraic equations resulting from discretization. Numerical continuations on V , I, and boundary concentrations result in S-shaped and double S-shaped (I-V and I-C) curves for the ssPNP equations with multiple species of ions. There are five solutions to the ssPNP equations with five ionic species, when an applied voltage is given in certain intervals. Remarkably, the current through ion channels responds hysteretically to varying applied voltages and boundary concentrations, showing a memory effect. In addition, we propose a useful computational approach to locate turning points of an I-V curve. With obtained locations, we are able to determine critical threshold values for hysteresis to occur and the interval for V in which the ssPNP equations have multiple solutions. Our numerical results indicate that the developed numerical approaches have a promising potential in studying hysteretic conductance states of ion channels.
- Hilbert-Huang transform (HHT) has drawn great attention in power system analysis due to its capability to deal with dynamic signal and provide instantaneous characteristics such as frequency, damping, and amplitudes. However, its shortcomings, including mode mixing and end effects, are as significant as its advantages. A preliminary result of an extended Kalman filter (EKF) method to enhance HHT and hopefully to overcome these disadvantages is presented in this paper. The proposal first removes dynamic DC components in signals using empirical mode decomposition. Then an EKF model is applied to extract instant coefficients. Numerical results using simulated and real-world low-frequency oscillation data suggest the proposal can help to overcome the mode mixing and end effects with a properly chosen number of modes.
- Nov 08 2017 math.NA arXiv:1711.02199v1The paper is concerned with overlapping domain decomposition and exponential time differencing for the diffusion equation discretized in space by cell-centered finite differences. Two localized exponential time differencing methods are proposed to solve the fully discrete problem: the first method is based on Schwarz iteration applied at each time step and involves solving stationary problems in the subdomains at each iteration, while the second method is based on the Schwarz waveform relaxation algorithm in which time-dependent subdomain problems are solved at each iteration. The convergence of the associated iterative solutions to the corresponding fully discrete multidomain solution and to the exact semi-discrete solution is rigorously proved. Numerical experiments are carried out to confirm theoretical results and to compare the performance of the two methods.
- We construct an explicit projective bimodule resolution for the Leavitt path algebra of a row-finite quiver. We prove that the Leavitt path algebra of a row-countable quiver has Hochschild cohomolgical dimension at most one, that is, it is quasi-free in the sense of Cuntz-Quillen. The construction of the resolution relies on an explicit derivation of the Leavitt path algebra.
- Safely serving the school transportation demand with the minimum number of buses is one of the highest financial goals of school transportation directors. To achieve that objective, a good and efficient way to solve the routing and scheduling problem is required. Due to the growth of the computing power, the spotlight has been shed on solving the combined problem of the school bus routing and scheduling problem. We show that an integrated multi-school bus routing and scheduling can be formulated with the help of trip compatibility. A novel decomposition algorithm is proposed to solve the integrated model. The merit of this integrated model and the decomposition method is that with the consideration of the trip compatibility, the interrelationship between the routing and scheduling sub-problems will not be lost in the process of decomposition. Results show the proposed decomposed problem could provide the solutions using the same number of buses as the integrated model in much shorter time (as little as 0.6%) and that the proposed method can save up to 26% number of buses from existing research.
- School bus planning is usually divided into routing and scheduling due to the complexity of solving them concurrently. However, the separation between these two steps may lead to worse solutions with higher overall costs than that from solving them together. When finding the minimal number of trips in the routing problem, neglecting the importance of trip compatibility may increase the number of buses actually needed in the scheduling problem. This paper proposes a new formulation for the multi-school homogeneous fleet routing problem that maximizes trip compatibility while minimizing total travel time. This incorporates the trip compatibility for the scheduling problem in the routing problem. Since the problem is inherently just a routing problem, finding a good solution is not cumbersome. To compare the performance of the model with traditional routing problems, we generate eight mid-size data sets. Through importing the generated trips of the routing problems into the bus scheduling (blocking) problem, it is shown that the proposed model uses up to 13% fewer buses than the common traditional routing models.
- We survey some recent work on topological quantum computation with gapped boundaries and boundary defects and list some open problems.