results for au:Cao_M in:math

- The risk of cascading blackouts greatly relies on failure probabilities of individual components in power grids. To quantify how component failure probabilities (CFP) influences blackout risk (BR), this paper proposes a sample-induced semi-analytic approach to characterize the relationship between CFP and BR. To this end, we first give a generic component failure probability function (CoFPF) to describe CFP with varying parameters or forms. Then the exact relationship between BR and CoFPFs is built on the abstract Markov-sequence model of cascading outages. Leveraging a set of samples generated by blackout simulations, we further establish a sample-induced semi-analytic mapping between the unbiased estimation of BR and CoFPFs. Finally, we derive an efficient algorithm that can directly calculate the unbiased estimation of BR when the CoFPFs change. Since no additional simulations are required, the algorithm is computationally scalable and efficient. Numerical experiments well confirm the theory and the algorithm.
- In this paper, the $k$ sample Behrens-Fisher problem is investigated in high dimensional setting. We propose a new test statistic and demonstrate that the proposed test is expected to have more powers than some existing test especially when sample sizes are unbalanced. We provide theoretical investigation as well as numerical studies on both sizes and powers of the proposed tests and existing test. Both theoretical comparison of the asymptotic power functions and numerical studies show that the proposed test tends to have more powers than existing test in many cases of unbalanced sample sizes.
- Roll stabilization (damping) is an important problem of ship motion control since excessive roll motion may cause motion sickness of human occupants and damage fragile cargo. This problem becomes especially non-trivial in situations where the same actuators are used for the vessel's roll and yaw stabilization. To keep the "trade-off" between the concurrent goals of accurate course steering and roll stabilization, an optimization problem is usually solved where the cost functional penalizes the roll angle, the steering error and the control effort. Since the vessel's motion is influenced by the uncertain wave disturbance, the optimal value of this functional and the optimal process are also uncertain. A standard approach, prevailing in the literature, is to approximate the wave disturbance by the "colored noise" with a known spectral density, reducing the optimization problem to conventional LQG control. In this paper, we propose a novel approach to optimal roll damping, approximating the disturbance by the polyharmonic signal with known frequencies yet uncertain amplitudes and phase shifts. For this class of external disturbances, an optimal universal controller (OUC) can be found. Unlike the LQG controller, minimizing only the expected value of the cost function, the OUC delivers the optimal solution for any uncertain parameters of the signal. The practical applicability of our algorithms is illustrated by numerical simulations.
- Some of the most interesting quantities associated with a factor graph are its marginals and its partition sum. For factor graphs \emphwithout cycles and moderate message update complexities, the sum-product algorithm (SPA) can be used to efficiently compute these quantities exactly. Moreover, for various classes of factor graphs \emphwith cycles, the SPA has been successfully applied to efficiently compute good approximations to these quantities. Note that in the case of factor graphs with cycles, the local functions are usually non-negative real-valued functions. In this paper we introduce a class of factor graphs, called double-edge factor graphs (DE-FGs), which allow local functions to be complex-valued and only require them, in some suitable sense, to be positive semi-definite. We discuss various properties of the SPA when running it on DE-FGs and we show promising numerical results for various example DE-FGs, some of which have connections to quantum information processing.
- In this paper, we propose a two-dimensional (2D) joint transmit array interpolation and beamspace design for planar array mono-static multiple-input-multiple-output (MIMO) radar for direction-of-arrival (DOA) estimation via tensor modeling. Our underlying idea is to map the transmit array to a desired array and suppress the transmit power outside the spatial sector of interest. In doing so, the signal-tonoise ratio is improved at the receive array. Then, we fold the received data along each dimension into a tensorial structure and apply tensor-based methods to obtain DOA estimates. In addition, we derive a close-form expression for DOA estimation bias caused by interpolation errors and argue for using a specially designed look-up table to compensate the bias. The corresponding Cramer-Rao Bound (CRB) is also derived. Simulation results are provided to show the performance of the proposed method and compare its performance to CRB.
- We consider the problem of transmitting classical information over a time-invariant channel with memory. A popular class of time-invariant channels with memory are finite-state-machine channels, where a \emphclassical state evolves over time and governs the relationship between the classical input and the classical output of the channel. For such channels, various techniques have been developed for estimating and bounding the information rate. In this paper we consider a class of time-invariant channels where a \emphquantum state evolves over time and governs the relationship between the classical input and the classical output of the channel. We propose algorithms for estimating and bounding the information rate of such channels. In particular, we discuss suitable graphical models for doing the relevant computations.
- Investigation of social influence dynamics requires mathematical models that are "simple" enough to admit rigorous analysis, and yet sufficiently "rich" to capture salient features of social groups. Thus, the mechanism of iterative opinion pooling from (DeGroot, 1974), which can explain the generation of consensus, was elaborated in (Friedkin and Johnsen, 1999) to take into account individuals' ongoing attachments to their initial opinions, or prejudices. The "anchorage" of individuals to their prejudices may disable reaching consensus and cause disagreement in a social influence network. Further elaboration of this model may be achieved by relaxing its restrictive assumption of a time-invariant influence network. During opinion dynamics on an issue, arcs of interpersonal influence may be added or subtracted from the network, and the influence weights assigned by an individual to his/her neighbors may alter. In this paper, we establish new important properties of the (Friedkin and Johnsen, 1999) opinion formation model, and also examine its extension to time-varying social influence networks.
- Recently the dynamics of signed networks, where the ties among the agents can be both positive (attractive) or negative (repulsive) have attracted substantial attention of the research community. Examples of such networks are models of opinion dynamics over signed graphs, recently introduced by Altafini (2012,2013) and extended to discrete-time case by Meng et al. (2014). It has been shown that under mild connectivity assumptions these protocols provide the convergence of opinions in absolute value, whereas their signs may differ. This "modulus consensus" may correspond to the polarization of the opinions (or bipartite consensus, including the usual consensus as a special case), or their convergence to zero. In this paper, we demonstrate that the phenomenon of modulus consensus in the discrete-time Altafini model is a manifestation of a more general and profound fact, regarding the solutions of a special recurrent inequality. Although such a recurrent inequality does not provide the uniqueness of a solution, it can be shown that, under some natural assumptions, each of its bounded solutions has a limit and, moreover, converges to consensus. A similar property has previously been established for special continuous-time differential inequalities (Proskurnikov, Cao, 2016). Besides analysis of signed networks, we link the consensus properties of recurrent inequalities to the convergence analysis of distributed optimization algorithms and the problems of Schur stability of substochastic matrices.
- This paper investigates the task assignment problem for multiple dispersed robots constrained by limited communication range. The robots are initially randomly distributed and need to visit several target locations while minimizing the total travel time. A centralized rendezvous-based algorithm is proposed, under which all the robots first move towards a rendezvous position until communication paths are established between every pair of robots either directly or through intermediate peers, and then one robot is chosen as the leader to make a centralized task assignment for the other robots. Furthermore, we propose a decentralized algorithm based on a single-traveling-salesman tour, which does not require all the robots to be connected through communication. We investigate the variation of the quality of the assignment solutions as the level of information sharing increases and as the communication range grows, respectively. The proposed algorithms are compared with a centralized algorithm with shared global information and a decentralized greedy algorithm respectively. Monte Carlo simulation results show the satisfying performance of the proposed algorithms.
- To understand the sophisticated control mechanisms of the human's endocrine system is a challenging task that is a crucial step towards precise medical treatment of many disfunctions and diseases. Although mathematical models describing the endocrine system as a whole are still elusive, recently some substantial progress has been made in analyzing theoretically its subsystems (or axes) that regulate production of specific hormones. Many of the relevant mathematical models are similar in structure to (or squarely based on) the celebrated Goodwin's oscillator. Such models are convenient to explain stable periodic oscillations at hormones' level by representing the corresponding endocrine regulation circuits as cyclic feedback systems. However, many real hormonal regulation mechanisms (in particular, testosterone regulation) are in fact known to have non-cyclic structures and involve multiple feedbacks; a Goodwin-type model thus represents only a part of such a complicated mechanism. In this paper, we examine a new mathematical model of hormonal regulation, obtained from the classical Goodwin's oscillator by introducing an additional negative feedback. Local stability properties of the proposed model are studied, and we show that the local instability of its unique equilibrium implies oscillatory behavior of almost all solutions. Furthermore, under additional restrictions we prove that almost all solutions converge to periodic ones.
- In this paper we propose an algorithm for path following control of the nonholonomic mobile robot based on the idea of the guiding vector field (GVF). The desired path may be an arbitrary smooth curve in its implicit form, that is, a level set of a predefined smooth function. Using this function and the robot's kinematic model, we design a GVF, whose integral curves converge to the trajectory. A nonlinear motion controller is then proposed which steers the robot along such an integral curve, bringing it to the desired path. We establish global convergence conditions for our algorithm and demonstrate its applicability and performance by experiments with real wheeled robots.
- Distributed algorithms of multi-agent coordination have attracted substantial attention from the research community; the simplest and most thoroughly studied of them are consensus protocols in the form of differential or difference equations over general time-varying weighted graphs. These graphs are usually characterized algebraically by their associated Laplacian matrices. Network algorithms with similar algebraic graph theoretic structures, called being of Laplacian-type in this paper, also arise in other related multi-agent control problems, such as aggregation and containment control, target surrounding, distributed optimization and modeling of opinion evolution in social groups. In spite of their similarities, each of such algorithms has often been studied using separate mathematical techniques. In this paper, a novel approach is offered, allowing a unified and elegant way to examine many Laplacian-type algorithms for multi-agent coordination. This approach is based on the analysis of some differential or difference inequalities that have to be satisfied by the some "outputs" of the agents (e.g. the distances to the desired set in aggregation problems). Although such inequalities may have many unbounded solutions, under natural graphic connectivity conditions all their bounded solutions converge (and even reach consensus), entailing the convergence of the corresponding distributed algorithms. In the theory of differential equations the absence of bounded non-convergent solutions is referred to as the equation's dichotomy. In this paper, we establish the dichotomy criteria of Laplacian-type differential and difference inequalities and show that these criteria enable one to extend a number of recent results, concerned with Laplacian-type algorithms for multi-agent coordination and modeling opinion formation in social groups.
- May 17 2016 math.CA arXiv:1605.04649v1It is well-known that the $L^p$ boundedness and weak $(1,1)$ estiamte $(\lambda>2)$ of the classical Littlewood-Paley $g_{\lambda}^{*}$-function was first studied by Stein, and the weak $(p,p)$ $(p>1)$ estimate was later given by Fefferman for $\lambda=2/p$. In this paper, we investigated the $L^p(\mu)$ boundedness of the non-homogeneous Littlewood-Paley $g_{\lambda,\mu}^{*}$-function with non-convolution type kernels and a power bounded measure $\mu$: $$ g_\lambda,\mu^*(f)(x) = \bigg(\iint_\mathbb R^n+1_+ \Big(\fractt + |x - y|\Big)^m \lambda |\theta_t^\mu f(y)|^2 \fracd\mu(y) dtt^m+1\bigg)^1/2,\ x ∈\mathbb R^n,\ \lambda > 1, $$ where $\theta_t^\mu f(y) = \int_{{\mathbb R}^n} s_t(y,z) f(z) d\mu(z)$, and $s_t$ is a non-convolution type kernel. Based on a big piece prior boundedness, we first gave a sufficient condition for the $L^p(\mu)$ boundedness of $g_{\lambda,\mu}^*$. This was done by means of the non-homogeneous good lambda method. Then, using the methods of dyadic analysis, we demonstrated a big piece global $Tb$ theorem. Finally, we obtaind a sufficient and necessary condition for $L^p(\mu)$ boundedness of $g_{\lambda,\mu}^*$-function. It is worth noting that our testing conditions are weak $(1,1)$ type with respect to measures.
- Recently it has been reported that range-measurement inconsistency, or equivalently mismatches in prescribed inter-agent distances, may prevent the popular gradient controllers from guiding rigid formations of mobile agents to converge to their desired shape, and even worse from standing still at any location. In this paper, instead of treating mismatches as the source of ill performance, we take them as design parameters and show that by introducing such a pair of parameters per distance constraint, distributed controller achieving simultaneously both formation and motion control can be designed that not only encompasses the popular gradient control, but more importantly allows us to achieve constant collective translation, rotation or their combination while guaranteeing asymptotically no distortion in the formation shape occurs. Such motion control results are then applied to (a) the alignment of formations orientations and (b) enclosing and tracking a moving target. Besides rigorous mathematical proof, experiments using mobile robots are demonstrated to show the satisfying performances of the proposed formation-motion distributed controller.
- Apr 26 2016 math.CA arXiv:1604.07037v1The main result of this paper is a bi-parameter $Tb$ theorem for Littlewood-Paley $g$-function, where $b$ is a tensor product of two pseudo-accretive functions. Instead of the doubling measure, we work with a product measure $\mu = \mu_n \times \mu_m$, where the measures $\mu_n$ and $\mu_m$ are only assumed to be upper doubling. The main techniques of the proof include a bi-parameter $b$-adapted Haar function decomposition and an averaging identity over good double Whitney regions. Moreover, the non-homogeneous analysis and probabilistic methods are used again.
- Apr 26 2016 math.CA arXiv:1604.06992v1Let $I_{\alpha}$ be the linear and $\mathcal{I}_{\alpha}$ be the bilinear fractional integral operators. In the linear setting, it is known that the two-weight inequality holds for the first order commutators of $I_{\alpha}$. But the method can't be used to obtain the two weighted norm inequality for the higher order commutators of $I_{\alpha}$. In this paper, we first give an alternative proof for the first order commutators of $I_{\alpha}$. This new approach allows us to consider the higher order commutators. This was done by showing that the commutator $[b,I_{\alpha}]$ can be represented as a finite linear combination of some paraproducts. Then, by using the Cauchy integral theorem, we show that the two-weight inequality holds for the higher order commutators of $I_{\alpha}$. In the bilinear setting, we present a dyadic proof for the characterization between $BMO$ and the boundedness of $[b,\mathcal{I}_{\alpha}]$. Moreover, some bilinear paraproducts are also treated in order to obtain the boundedness of $[b,\mathcal{I}_{\alpha}]$.
- This paper presents the analysis on the influence of distance mismatches on the standard gradient-based rigid formation control for second-order agents. It is shown that, similar to the first-order case as recently discussed in the literature, these mismatches introduce two undesired group behaviors: a distorted final shape and a steady-state motion of the group formation. We show that such undesired behaviors can be eliminated by combining the standard formation control law with distributed estimators. Finally, we show how the mismatches can be effectively employed as design parameters in order to control a combined translational and rotational motion of the formation.
- Feb 23 2016 math.CA arXiv:1602.06486v2This paper will be devoted to study the two-weight norm inequalities of the multilinear fractional maximal operator $\mathcal{M}_{\alpha}$ and the multilinear fractional integral operator $\mathcal{I}_{\alpha}$. The entropy conditions in the multilinear setting will be introduced and the entropy bounds for $\mathcal{M}_\alpha$ and $\mathcal{I}_\alpha$ will be given.
- In this paper, we investigated the boundedness of multilinear fractional strong maximal operator $\mathcal{M}_{\mathcal{R},\alpha}$ associated with rectangles or related to more general basis with multiple weights $A_{(\vec{p},q),\mathcal{R}}$. In the rectangles setting, we first gave an end-point estimate of $\mathcal{M}_{\mathcal{R},\alpha}$, which not only extended the famous linear result of Jessen, Marcinkiewicz and Zygmund, but also extended the multilinear result of Grafakos, Liu, Pérez and Torres ($\alpha=0$) to the case $0<\alpha<mn.$ Then, in one weight case, we gave several equivalent characterizations between $\mathcal{M}_{\mathcal{R},\alpha}$ and $A_{(\vec{p},q),\mathcal{R}}$, by applying a different approach from what we have used before. Moreover, a sufficient condition for the two weighted norm inequality of $\mathcal{M}_{\mathcal{R},\alpha}$ was presented and a version of vector-valued two weighted inequality for the strong maximal operator was established when $m=1$. In the general basis setting, we further studied the properties of the multiple weights $A_{(\vec{p},q),\mathcal{R}}$ conditions, including the equivalent characterizations and monotonic properties, which essentially extended one's previous understanding. Finally, a survey on multiple strong Muckenhoupt weights was given, which demonstrates the properties of multiple weights related to rectangles systematically.
- Let $m,n\ge 1$ and $g_{\lambda_1,\lambda_2}^*$ be the bi-parameter Littlewood-Paley $g_{\lambda}^{*}$-function defined by $$ g_\lambda_1,\lambda_2^*(f)(x)= \bigg(\iint_\R^m+1_+ \big(\fract_2t_2 + |x_2 - y_2|\big)^m \lambda_2 \iint_\R^n+1_+ \big(\fract_1t_1 + |x_1 - y_1|\big)^n \lambda_1|\theta_t_1,t_2 f(y_1,y_2)|^2 \fracdy_1 dt_1t_1^n+1 \fracdy_2 dt_2t_2^m+1 \bigg)^1/2, \lambda_1>1,\quad \lambda_2>1 $$ where $\theta_{t_1,t_2} f$ is a non-convolution kernel defined on $\mathbb{R}^{m+n}$. In this paper, we showed that the bi-parameter Littlewood-Paley function $g_{\lambda_1,\lambda_2}^*$ was bounded from $L^2(\R^{n+m})$ to $L^2(\R^{n+m})$. This was done by means of probabilistic methods and by using a new averaging identity over good double Whitney regions.
- Populations of flashing fireflies, claps of applauding audience, cells of cardiac and circadian pacemakers reach synchrony via event-triggered interactions, referred to as pulse couplings. Synchronization via pulse coupling is widely used in wireless sensor networks, providing clock synchronization with parsimonious packet exchanges. In spite of serious attention paid to networks of pulse coupled oscillators, there is a lack of mathematical results, addressing networks with general communication topologies and general phase-response curves of the oscillators. The most general results of this type (Wang et al., 2012, 2015) establish synchronization of oscillators with a delay-advance phase-response curve over strongly connected networks. In this paper we extend this result by relaxing the connectivity condition to the existence of a root node (or a directed spanning tree) in the graph. This condition is also necessary for synchronization.
- Sep 15 2015 math.OC arXiv:1509.03900v1In the set of stochastic, indecomposable, aperiodic (SIA) matrices, the class of stochastic Sarymsakov matrices is the largest known subset (i) that is closed under matrix multiplication and (ii) the infinitely long left-product of the elements from a compact subset converges to a rank-one matrix. In this paper, we show that a larger subset with these two properties can be derived by generalizing the standard definition for Sarymsakov matrices. The generalization is achieved either by introducing an "SIA index", whose value is one for Sarymsakov matrices, and then looking at those stochastic matrices with larger SIA indices, or by considering matrices that are not even SIA. Besides constructing a larger set, we give sufficient conditions for generalized Sarymsakov matrices so that their products converge to rank-one matrices. The new insight gained through studying generalized Sarymsakov matrices and their products has led to a new understanding of the existing results on consensus algorithms and will be helpful for the design of network coordination algorithms.
- Most of the distributed protocols for multi-agent consensus assume that the agents are mutually cooperative and "trustful," and so the couplings among the agents bring the values of their states closer. Opinion dynamics in social groups, however, require beyond these conventional models due to ubiquitous competition and distrust between some pairs of agents, which are usually characterized by repulsive couplings and may lead to clustering of the opinions. A simple yet insightful model of opinion dynamics with both attractive and repulsive couplings was proposed recently by C. Altafini, who examined first-order consensus algorithms over static signed graphs. This protocol establishes modulus consensus, where the opinions become the same in modulus but may differ in signs. In this paper, we extend the modulus consensus model to the case where the network topology is an arbitrary time-varying signed graph and prove reaching modulus consensus under mild sufficient conditions of uniform connectivity of the graph. For cut-balanced graphs, not only sufficient, but also necessary conditions for modulus consensus are given.
- In the recent paper by Hamadeh et al. (2012) an elegant analytic criterion for incremental output feedback passivity (iOFP) of cyclic feedback systems (CFS) has been reported, assuming that the constituent subsystems are incrementally output strictly passive (iOSP). This criterion was used to prove that a network of identical CFS can be synchronized under sufficiently strong linear diffusive coupling. A very important class of CFS consists of biological oscillators, named after Brian Goodwin and describing self-regulated chains of enzymatic reactions, where the product of each reaction catalyzes the next reaction, while the last product inhibits the first reaction in the chain. Goodwin's oscillators are used, in particular, to model the dynamics of genetic circadian pacemakers, hormonal cycles and some metabolic pathways. In this paper we point out that for Goodwin's oscillators, where the individual reactions have nonlinear (e.g. Mikhaelis-Menten) kinetics, the synchronization criterion, obtained by Hamadeh et al., cannot be directly applied. This criterion relies on the implicit assumption of the solution boundedness, dictated also by the chemical feasibility (the state variables stand for the concentrations of chemicals). Furthermore, to test the synchronization condition one needs to know an explicit bound for a solution, which generally cannot be guaranteed under linear coupling. At the same time, we show that these restrictions can be avoided for a nonlinear synchronization protocol, where the control inputs are "saturated" by a special nonlinear function (belonging to a wide class), which guarantees nonnegativity of the solutions and allows to get explicit ultimate bounds for them. We prove that oscillators synchronize under such a protocol, provided that the couplings are sufficiently strong.
- This paper studies synchronization of dynamical networks with event-based communication. Firstly, two estimators are introduced into each node, one to estimate its own state, and the other to estimate the average state of its neighbours. Then, with these two estimators, a distributed event-triggering rule (ETR) with a dwell time is designed such that the network achieves synchronization asymptotically with no Zeno behaviours. The designed ETR only depends on the information that each node can obtain, and thus can be implemented in a decentralized way.
- In this paper, we present a local $Tb$ theorem for the non-homogeneous Littlewood-Paley $g_{\lambda}^{*}$-function with non-convolution type kernels and upper power bound measure $\mu$. We show that, under the assumptions $\supp b_Q \subset Q$, $|\int_Q b_Q d\mu| \gtrsim \mu(Q)$ and $||b_Q||^p_{L^p(\mu)} \lesssim \mu(Q)$, the norm inequality $\big\| g_{\lambda}^{*}(f) \big\|_{L^p(\mu)} \lesssim \big\| f \big\|_{L^p(\mu)}$ holds if and only if the following testing condition holds : $$\sup_Q : cubes \ in \ \Rn \frac1\mu(Q)\int_Q \bigg(\int_0^\ell(Q) \int_\Rn \Big(\fractt+|x-y|\Big)^m\lambda|\theta_t(b_Q)(y,t)|^2 \fracd\mu(y) dtt^m+1\bigg)^p/2 d\mu(x) < ∞.$$ This is the first time to investigate $g_\lambda^*$-function in the simultaneous presence of three attributes : local, non-homogeneous and $L^p$-testing condition. It is important to note that the testing condition here is $L^p$ type with $p \in (1,2]$.
- Apr 30 2015 math.CA arXiv:1504.07850v2Let $n\ge 2$ and $g_{\lambda}^{*}$ be the well-known high dimensional Littlewood-Paley function which was defined and studied by E. M. Stein, \beginalign* g_\lambda^*(f)(x) =\bigg(\iint_\mathbb R^n+1_+ \Big(\fractt+|x-y|\Big)^n\lambda |∇P_tf(y,t)|^2 \fracdy dtt^n-1\bigg)^1/2, \ \quad \lambda > 1, \endalign* where $P_tf(y,t)=p_t*f(y)$, $p_t(y)=t^{-n}p(y/t)$ and $p(x) = (1+|x|^2)^{-(n+1)/2}$, $\nabla =(\frac{\partial}{\partial y_1},\ldots,\frac{\partial}{\partial y_n},\frac{\partial}{\partial t})$. In this paper, we give a characterization of two-weight norm inequality for $g_{\lambda}^{*}$-function. We show that, $\big\| g_{\lambda}^{*}(f \sigma) \big\|_{L^2(w)} \lesssim \big\| f \big\|_{L^2(\sigma)}$ if and only if the two-weight Muchenhoupt $A_2$ condition holds, and a testing condition holds : \beginalign* \sup_Q : cubes \ in \mathbb R^n \frac1\sigma(Q) \int_\mathbb R^n \iint_\widehatQ \Big(\fractt+|x-y|\Big)^n\lambda|∇P_t(\mathbf1_Q \sigma)(y,t)|^2 \fracw dx dtt^n-1 dy < ∞, \endalign* where $\widehat{Q}$ is the Carleson box over $Q$ and $(w, \sigma)$ is a pair of weights. We actually prove this characterization for $g_{\lambda}^{*}$-function associated with more general fractional Poisson kernel $p^\alpha(x) = (1+|x|^2)^{-{(n+\alpha)}/{2}}$. Moreover, the corresponding results for intrinsic $g_{\lambda}^*$-function are also presented.
- Apr 15 2015 math.CA arXiv:1504.03420v2In this paper, the multilinear fractional strong maximal operator $\mathcal{M}_{\mathcal{R},\alpha}$ associated with rectangles and corresponding multiple weights $A_{(\vec{p},q),\mathcal{R}}$ are introduced. Under the dyadic reverse doubling condition, a necessary and sufficient condition for two-weight inequalities is given. As consequences, we first obtain a necessary and sufficient condition for one-weight inequalities. Then, we give a new proof for the weighted estimates of multilinear fractional maximal operator $\mathcal{M}_\alpha$ associated with cubes and multilinear fractional integral operator $\mathcal{I}_{\alpha}$, which is quite different and simple from the proof known before.
- Jul 08 2014 math.DS physics.flu-dyn arXiv:1407.1579v1Modelling sediment transport in environmental turbulent fluids is a challenge. This article develops a sound model of the lateral transport of suspended sediment in environmental fluid flows such as floods and tsunamis. The model is systematically derived from a 3D turbulence model based on the Smagorinski large eddy closure. Embedding the physical dynamics into a family of problems and analysing linear dynamics of the system, centre manifold theory indicates the existence of slow manifold parametrised by macroscale variables. Computer algebra then constructs the slow manifold in terms of fluid depth, depth-averaged lateral velocities, and suspended sediment concentration. The model includes the effects of sediment erosion, advection, dispersion, and also the interactions between the sediment and turbulent fluid flow. Vertical distributions of the velocity and concentration in steady flow agree with the established experimental data. Numerical simulations of the suspended sediment under large waves show that the developed model predicts physically reasonable phenomena.
- May 29 2014 math.DS physics.flu-dyn arXiv:1405.7093v1The multiscale gap-tooth scheme uses a given microscale simulator of complicated physical processes to enable macroscale simulations by computing only only small sparse patches. This article develops the gap-tooth scheme to the case of nonlinear microscale simulations of thin fluid flow. The microscale simulator is derived by artificially assuming the fluid film flow having two artificial layers but no distinguishing physical feature. Centre manifold theory assures that there exists a slow manifold in the two-layer fluid film flow. Eigenvalue analysis confirms the stability of the microscale simulator. This article uses the gap-tooth scheme to simulate the two-layer fluid film flow. Coupling conditions are developed by approximating the values at the edges of patches by neighbouring macroscale values. Numerical eigenvalue analysis suggests that the gap-tooth scheme with the developed two-layer microscale simulator empowers feasible computation of large scale simulations of fluid film flows. We also implement numerical simulations of the fluid film flow by the gap-tooth scheme. Comparison between a gap-tooth simulation and a microscale simulation over the whole domain demonstrates that the gap-tooth scheme feasibly computes fluid film flow dynamics with computational savings.
- Apr 28 2014 math.DS arXiv:1404.6317v1The multiscale gap-tooth scheme is built from given microscale simulations of complicated physical processes to empower macroscale simulations. By coupling small patches of simulations over unsimulated physical gaps, large savings in computational time are possible. So far the gap-tooth scheme has been developed for dissipative systems, but wave systems are also of great interest. This article develops the gap-tooth scheme to the case of nonlinear microscale simulations of wave-like systems. Classic macroscale interpolation provides a generic coupling between patches that achieves arbitrarily high order consistency between the multiscale scheme and the underlying microscale dynamics. Eigen-analysis indicates that the resultant gap-tooth scheme empowers feasible computation of large scale simulations of wave-like dynamics with complicated underlying physics. As an pilot study, we implement numerical simulations of dam-breaking waves by the gap-tooth scheme. Comparison between a gap-tooth simulation, a microscale simulation over the whole domain, and some published experimental data on dam breaking, demonstrates that the gap-tooth scheme feasibly computes large scale wave-like dynamics with computational savings.
- Apr 16 2014 math.CA arXiv:1404.3819v1In this paper we study the gap probability problem in the Gaussian Unitary Ensembles of $n$ by $n$ matrices : The probability that the interval $J := (-a,a)$ is free of eigenvalues. In the works of Tracy and Widom, Adler and Van Moerbeke and Forrester and Witte on this subject, it has been shown that two Painleve type differential equations arise in this context. The first is the Jimbo-Miwa-Okomoto $\sigma-$form and the second is a particular Painleve IV. Using the ladder operator technique of orthogonal polynomials we derive three quantities associated with the gap probability, denoted by $\sigma_n(a)$, $R_n(a)$ and $r_n(a)$, and show that each one satisfying a second order, non-linear, differential equation as well as a second order, non-linear difference equation. In particular, in addition to providing an elementary derivation of the aforementioned $\sigma-$form and Painleve IV we show that the quantity $r_n(a)$ satisfies a particular case of Chazy's second degree second order differential equation. For the discrete equations we show that the quantity $r_n(a)$ satisfies a particular form of the modified discrete Painleve II equation obtained by Grammaticos and Ramani in the context of Backlund transformations. We also derive second order second degree difference equations for the quantities $R_n(a)$ and $\sigma_n(a)$.
- Feb 13 2014 math.OC arXiv:1402.2766v2In this paper, we study the discrete-time consensus problem over networks with antagonistic and cooperative interactions. Following the work by Altafini [IEEE Trans. Automatic Control, 58 (2013), pp. 935--946], by an antagonistic interaction between a pair of nodes updating their scalar states we mean one node receives the opposite of the state of the other and naturally by an cooperative interaction we mean the former receives the true state of the latter. Here the pairwise communication can be either unidirectional or bidirectional and the overall network topology graph may change with time. The concept of modulus consensus is introduced to characterize the scenario that the moduli of the node states reach a consensus. It is proved that modulus consensus is achieved if the switching interaction graph is uniformly jointly strongly connected for unidirectional communications, or infinitely jointly connected for bidirectional communications. We construct a counterexample to underscore the rather surprising fact that quasi-strong connectivity of the interaction graph, i.e., the graph contains a directed spanning tree, is not sufficient to guarantee modulus consensus even under fixed topologies. Finally, simulation results using a discrete-time Kuramoto model are given to illustrate the convergence results showing that the proposed framework is applicable to a class of networks with general nonlinear node dynamics.
- In this paper we propose an approach to the implementation of controllers with decentralized strategies triggering controller updates. We consider set-ups with a central node in charge of the computation of the control commands, and a set of not co-located sensors providing measurements to the controller node. The solution we propose does not require measurements from the sensors to be synchronized in time. The sensors in our proposal provide measurements in an aperiodic way triggered by local conditions. Furthermore, in the proposed implementation (most of) the communication between nodes requires only the exchange of one bit of information (per controller update), which could aid in reducing transmission delays and as a secondary effect result in fewer transmissions being triggered.
- Apr 08 2010 math.OC arXiv:1004.1095v1Motivated by applications in intelligent highway systems, the paper studies the problem of guiding mobile agents in a one-dimensional formation to their desired relative positions. Only coarse information is used which is communicated from a guidance system that monitors in real time the agents' motions. The desired relative positions are defined by the given distance constraints between the agents under which the overall formation is rigid in shape and thus admits locally a unique realization. It is shown that even when the guidance system can only transmit at most four bits of information to each agent, it is still possible to design control laws to guide the agents to their desired positions. We further delineate the thin set of initial conditions for which the proposed control law may fail using the example of a three-agent formation. Tools from non-smooth analysis are utilized for the convergence analysis.