almost equitable partitions and new necessary conditions for network controllability abstract in this paper we consider the controllability problem for multi agent networked control systems the main results of the paper are new graph theoretic necessary conditions for controllability involving almost equitable graph vertex partitions we generalize the known results on the role of graph symmetries and uncontrollability to weighted digraphs with multiple leaders and we also consider the broadcasted control scenario our results show that the internal structure of communities in a graph can induce obstructions to controllability that cannot be characterized by symmetry arguments alone and that in some cases depend on the number theoretic properties of the communities we show via examples that our results can be used to account for a large portion of uncontrollable inducing leader selections that could not have otherwise been accounted for using symmetry results

an efficient sixth order solution for anisotropic poisson equation with completed richardson extrapolation and multiscale multigrid method we present an efficient numerical method for anisotropic poisson equations the sixth order accuracy is achieved through applying completed richardson extrapolation on two fourth order solutions computed from different scale grids with unequal mesh size discretization theoretical analysis is conducted to demonstrate that the richardson extrapolation is able to obtain a sixth order solution by removing the leading truncation error terms of the fourth order solution from grid with unequal mesh sizes the gain in efficiency is obtained through adopting partial semi coarsening multigrid method to solve the resulting linear systems and multiscale multigrid computation to speed up the whole solution numerical experiments are conducted to verify the accuracy and efficiency of the proposed method and the results are compared with the existing fourth order methods for solving 2d and 3d anisotropic poisson equations

a new eighth order implicit finite difference method to solve the three dimensional helmholtz equation abstract this paper introduces a novel method for deducing high order finite difference schemes for the three dimensional helmholtz equation p 2 p f the new method can reach an eighth order of accuracy it is based on an implicit formulation by computing simultaneously the unknown variable and its corresponding derivatives the scheme is obtained from taylor series expansion and plane wave theory analysis and it is constructed from a few modifications to the standard centered finite differences a classical dispersion analysis and the error between the numerical wavenumber and the exact wavenumber is given numerical experiments are presented to verify the feasibility and accuracy of the proposed methods for a broad range of values moreover the numerical method is capable to solve with great precision the monotone and oscillatory helmholtz problem with variable values thus the proposed formulation results in an attractive method easy to implement with high order of accuracy but nearly the same computational cost as those of an explicit formulation

high order compact scheme with multigrid local mesh refinement procedure for convection diffusion problems we derive a new fourth order compact finite difference scheme which allows different meshsize in different coordinate directions for the two dimensional convection diffusion equation a multilevel local mesh refinement strategy is used to deal with the local singularity problem a corresponding multilevel multigrid method is designed to solve the resulting sparse linear system numerical experiments are conducted to show that the local mesh refinement strategy works well with the high order compact discretization scheme to recover high order accuracy for the computed solution our solution method is also shown to be effective and robust with respect to the level of mesh refinement and the anisotropy of the problems

continuous time quantum walk on integer lattices and homogeneous trees this paper is concerned with the continuous time quantum walk on d and infinite homogeneous trees by using the generating function method we compute the limit of the average probability distribution for the general isotropic walk on and for nearest neighbor walks on d and infinite homogeneous trees in addition we compute the asymptotic approximation for the probability of the return to zero at time t in all these cases

solution of high order compact discretized 3d elliptic partial differential equations by an accelerated multigrid method abstract a robust and efficient numerical solver for three dimensional linear elliptic partial differential equations without cross derivative terms but variable coefficients is presented the proposed algorithm is based on a general meshsize fourth order compact finite difference scheme and an accelerated geometric multigrid solver the former not only increases the accuracy of the discretization but also does not affect the bandwidth of the common second order finite difference scheme which is relevant for distributed memory parallel implementations of the multigrid preconditioner the acceleration technique on the other hand allows for a dramatic reduction of the number of cycles and smoothing steps required to achieve convergence furthermore it is found that depending on the coefficients of the elliptic partial differential equation the multigrid preconditioned solver can be the only way to obtain a reliable numerical solution or attain convergence

fast and high accuracy multigrid solution of the three dimensional poisson equation we employ a fourth order compact finite difference scheme fos with the multigrid algorithm to solve the three dimensional poisson equation we test the influence of different orderings of the grid space and different grid transfer operators on the convergence and efficiency of our high accuracy algorithm fourier smoothing analysis is conducted to show that fos has a smaller smoothing factor than the traditional second order central difference scheme cds a new method of fourier smoothing analysis is proposed for the partially decoupled red black gauss seidel relaxation with fos numerical results are given to compare the computed accuracy and the computational efficiency of fos with multigrid against cds with multigrid

controllability of multi agent systems with periodically switching topologies and switching leaders abstractthis paper considers controllability of multi agent systems with periodically switching topologies and switching leaders the concept of m periodic controllability is proposed and a criterion for m periodic controllability is established the effect of the duration of subsystems on controllability is analysed by utilising a property of analytic functions in addition the influence of switching periods on controllability is investigated and an algorithm is proposed to search for the fewest periods to ensure controllability a necessary condition for m periodic controllability is obtained from the perspective of eigenvectors of the subsystems laplacian matrices for a system with switching leaders it is proved that switching leader controllability is equivalent to multiple leader controllability furthermore both the switching order and the tenure of agents being leaders have no effect on the controllability some examples are provided to illustrate the theoretical results

a cubic b spline semi analytical algorithm for simulation of 3d steady state convection diffusion reaction problems abstract in this work a new cubic b spline based semi analytical algorithm is presented for solving 3d anisotropic convection diffusion reaction cdr problems in the inhomogeneous medium the mathematical model is expressed by the quasi linear second order elliptic partial differential equations epde with mixed derivatives and variable coefficients the final approximation is obtained as a sum of the rough primary solution and the modified spline interpolants with free parameters the primary solution mathematically satisfies boundary conditions thus the free parameters of interpolants are chosen to satisfy the governing equation in the solution domain the numerical examples demonstrate the high accuracy of the proposed method in solving 3d cdr problems in single and multi connected domains

racah type expressions for the 6j coefficients of the orthosymplectic superalgebra osp 1 2 using generating functions various expressions for the 6j coefficients of the osp 1 2 superalgebra are derived some of these expressions bear a close resemblance to the racah formula for the su 2 6j coefficients as a consequence it is shown that the osp 1 2 6j coefficients exhibit regge symmetries

on the controllability properties of circulant networks this paper examines the controllability of a group of first order agents adopting a weighted consensus type coordination protocol over a circulant network specifically it is shown that a circulant network with laplacian eigenvalues of maximum algebraic multiplicity q is controllable from q nodes our approach leverages on the cauchy binet formula which in conjunction with the popov belevitch hautus test leads to new insights on structural aspects of network controllability

controllability of heterogeneous multi agent systems under directed and weighted topology abstractthis paper considers the controllability problem for both continuous and discrete time linear heterogeneous multi agent systems with directed and weighted communication topology first two kinds of neighbour based control protocols based on the distributed protocol of first order and second order multi agent systems are proposed under which it is shown that a heterogeneous multi agent system is controllable if the underlying communication topology is controllable then under special leader selection the result shows that the controllability of a heterogeneous multi agent system is solely decided by its communication topology graph furthermore some necessary and or sufficient conditions are derived for controllability of communication topology from algebraic and graphical perspectives finally simulation examples are presented to demonstrate the effectiveness of the theoretical results

a transformation free hoc scheme and multigrid method for solving the 3d poisson equation on nonuniform grids a high order compact hoc difference scheme is proposed to solve the three dimensional 3d poisson equation on nonuniform orthogonal cartesian grids involving no coordinate transformation from the physical space to the computational space theoretically the proposed scheme has third to fourth order accuracy its fourth order accuracy is achieved under uniform grid settings then a multigrid method is developed to solve the linear system arising from this hoc difference scheme and the corresponding multigrid restriction and interpolation operators are constructed using the volume law numerical experiments are conducted to show the computed accuracy of the hoc scheme and the computational efficiency of the multigrid method

a new method to deduce high order compact difference schemes for two dimensional poisson equation abstract this paper introduces a novel method for deducing high order compact difference schemes for the two dimensional 2d poisson equation like finite volume method a dual partition is introduced combining simpson integral formula and parabolic interpolation a family of fourth order and sixth order compact difference schemes are obtained based on three different types of dual partitions moreover several new fourth order compact schemes are gained and numerical experiments are shown two of them are much better than almost any other fourth order schemes which have been presented in others work the outline for the nonlinear poisson equation is also given numerical experiments are presented to verify the feasibility of this new method and the high accuracy of these fourth order and sixth order compact difference schemes

quadratic stabilisability of multi agent systems under switching topologies this paper addresses the stabilisability of multi agent systems mass under switching topologies necessary and or sufficient conditions are presented in terms of graph topology these conditions explicitly reveal how the intrinsic dynamics of the agents the communication topology and the external control input affect stabilisability jointly with the appropriate selection of some agents to which the external inputs are applied and the suitable design of neighbour interaction rules via a switching topology an mas is proved to be stabilisable even if so is not for each of uncertain subsystem in addition a method is proposed to constructively design a switching rule for mass with norm bounded time varying uncertainties the switching rules designed via this method do not rely on uncertainties and the switched mas is quadratically stabilisable via decentralised external self feedback for all uncertainties with respect to applications of the stabilisability results the formation control and the cooper

on the relationship between continuous and discrete time quantum walk quantum walk is one of the main tools for quantum algorithms defined by analogy to classical random walk a quantum walk is a time homogeneous quantum process on a graph both random and quantum walks can be defined either in continuous or discrete time but whereas a continuous time random walk can be obtained as the limit of a sequence of discrete time random walks the two types of quantum walk appear fundamentally different owing to the need for extra degrees of freedom in the discrete time case

laplacian graph eigenvectors abstract if g is a graph its laplacian is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix the main thrust of the present article is to prove several laplacian eigenvector principles which in certain cases can be used to deduce the effect on the spectrum of contracting adding or deleting edges and or of coalescing vertices one application is the construction of two isospectral graphs on 11 vertices having different degree sequences only one of which is bipartite and only one of which is decomposable

multigrid method and fourth order compact difference discretization scheme with unequal meshsizes for 3d poisson equation a fourth order compact difference discretization scheme with unequal meshsizes in different coordinate directions is employed to solve a three dimensional 3d poisson equation on a cubic domain two multgrid methods are developed to solve the resulting sparse linear systems one is to use the full coarsening multigrid method with plane gauss seidel relaxation which uses line gauss seidel relaxation to compute each planewise solution the other is to construct a partial semi coarsening multigrid method with the traditional point or plane gauss seidel relaxations numerical experiments are conducted to test the computed accuracy of the fourth order compact difference scheme and the computational efficiency of the multigrid methods with the fourth order compact difference scheme

dynamics of continuous time quantum walks in restricted geometries we study quantum transport on finite discrete structures and we model the process by means of continuous time quantum walks a direct and effective comparison between quantum and classical walks can be attained based on the average displacement of the walker as a function of time indeed a fast growth of the average displacement can be advantageously exploited to build up efficient search algorithms by means of analytical and numerical investigations we show that the finiteness and the inhomogeneity of the substrate jointly weaken the quantum walk performance we further highlight the interplay between the quantum walk dynamics and the underlying topology by studying the temporal evolution of the transfer probability distribution and the lower bound of long time averages

performance and design of cycles in consensus networks abstract this work explores the role of cycles in consensus seeking networks for analysis and synthesis purposes cycles are critical for many reasons including improving the convergence rate of the system resilience to link failures and the overall performance of the system the focus of this work examines how cycles impact the h 2 performance of consensus networks a first contribution shows that the addition of cycles always improves the performance of the system we provide an analytic characterization of how the addition of edges improves the performance and show that it is related to the inverse of the cycle lengths and the number of shared edges between independent cycles these results are then used to consider the design of consensus networks in this direction we present an l 1 relaxation method that leads to a convex program for adding a fixed number of edges to a consensus networks we also demonstrate how this relaxation can be used to embed additional performance criteria such as maximization of the algebraic connectivity of the graph

laplacian controllability classes for threshold graphs abstract let g be a graph on n vertices with laplacian matrix l and let b be a binary vector of length n the pair l b is controllable if the smallest l invariant subspace containing b is of dimension n the graph g is called essentially controllable if l b is controllable for every b ker l completely uncontrollable if l b is uncontrollable for every b and conditionally controllable if it is neither essentially controllable nor completely uncontrollable in this paper we completely characterize the graph controllability classes for threshold graphs we first observe that the class of threshold graphs contains no essentially controllable graph we prove that a threshold graph is completely uncontrollable if and only if its laplacian matrix has a repeated eigenvalue in the process we fully characterize the set of conditionally controllable threshold graphs

zero forcing sets and controllability of dynamical systems defined on graphs in this technical note controllability of systems defined on graphs is discussed we consider the problem of controllability of the network for a family of matrices carrying the structure of an underlying directed graph a one to one correspondence between the set of leaders rendering the network controllable and zero forcing sets is established to illustrate the proposed results special cases including path cycle and complete graphs are discussed moreover as shown for graphs with a tree structure the proposed results of the present technical note together with the existing results on the zero forcing sets lead to a minimal leader selection scheme in particular cases

controllability of the multi agent system modeled by the threshold graph with one repeated degree abstract the controllability issue of multi agent systems modeled by a special class of graphs is studied suppose the modeling graph is simple connected and driven by only one controller the author shows that the system is controllable if the graph has exactly two vertices with the same degree and either one of these two vertices is selected to receive the control input this result is extended to the case of the threshold graph that has only one repeated degree specifically it is shown that if the threshold graph has only one repeated degree and its multiplicity is m then m 1 is the minimum number of the controllers required to make the system controllable in addition the necessary and sufficient condition that characterizes the binary control vectors to ensure the controllability of the system is derived

representations of osp 2 1 and the metaplectic representation shift operator techniques are used to treat the irreducible representations of the superalgebra osp 2 1 apart from obtaining the well known gradestar dispin representations which arise when the even part is the compact su 2 algebra the case when the star conditions on the even part are those satisfied by the noncompact su 1 1 algebra is also treated in this case no gradestar representations arise and the star representations are found to consist of the direct sum of two discrete series representations of su 1 1 one of these representations can be realized in terms of functions of a single complex variable and turns out to be a simple example of a metaplectic representation

error scaling laws for linear optimal estimation from relative measurements in this paper we study the problem of estimating vector valued variables from noisy ldquorelativerdquo measurements which arises in sensor network applications the problem can be posed in terms of a graph whose nodes correspond to variables and edges to noisy measurements of the difference between two variables the optimal minimum variance linear unbiased estimate of the node variables with an arbitrary variable as the reference is considered this paper investigates how the variance of the estimation error of a node variable grows with the distance of the node to the reference node a classification of graphs namely dense or sparse in rd 1 les d les 3 is established that determines this growth rate in particular if a graph is dense in 1 d 2 d or 3 d a node variable s estimation error is upper bounded by a linear logarithmic or bounded function of distance from the reference corresponding lower bounds are obtained if the graph is sparse in 1 d 2 d and 3 d these results show that naive measures of graph density such as node degree are inadequate predictors of the estimation error being true for the optimal linear unbiased estimate these scaling laws determine algorithm independent limits on the estimation accuracy achievable in large graphs

observability through a matrix weighted graph observability of an array of identical linear time invariant systems with incommensurable output matrices is studied where an array is called observable when identically zero relative outputs imply synchronized solutions for the individual systems it is shown that the observability of an array is equivalent to the connectivity of its interconnection graph whose edges are assigned matrix weights moreover to better understand the relative behavior of distant units pairwise observability that concerns with the synchronization of a certain pair of individual systems in the array is studied this milder version of observability is shown to be closely related to certain connectivity properties of the interconnection graph as well pairwise observability is also analyzed using the circuit theoretic tool effective conductance the observability of a certain pair of units is proved to be equivalent to the nonsingularity of the matrix valued effective conductance between the associated pair of nodes of a resistive network with matrix valued parameters whose node admittance matrix is the laplacian of the array s interconnection graph

nodes selection strategy in cooperative tracking problem in this paper a new optimization problem is addressed for node selection that has application potentials in input output switches for sensors in control system design and leader determination in social networks the purpose of the addressed problem is to develop a strategy for selecting a subset of nodes as controlled nodes in order to minimize certain objective function consisting of the convergence speed and the energy of control action over a finite time horizon for networks with fixed controlled nodes an upper bound of the objective function is obtained which is shown to be convex and independent of the time horizon for networks with switched controlled nodes a greedy algorithm is proposed to reduce the computation complexity resulting from the length of the time horizon where the nodes selection is carried out over divided small time intervals the cost gap is also analyzed between the strategy of optimizing over the whole time horizon and the strategy of optimizing over the small intervals finally the proposed nodes selection strategy is validated through simulations and two regions are found in which the number of optimal controlled nodes is determined

upper and lower bounds for controllable subspaces of networks of diffusively coupled agents this technical note studies the controllability of diffusively coupled networks where some agents called leaders are under the influence of external control inputs first we consider networks where agents have general linear dynamics then we turn our attention to infer network controllability from its underlying graph topology to do this we consider networks with agents having single integrator dynamics for such networks we provide lower and upper bounds for the controllable subspaces in terms of the distance partitions and the maximal almost equitable partitions respectively we also provide an algorithm for computing the maximal almost equitable partition for a given graph and a set of leaders

comparison of second and fourth order discretizations for multigrid poisson solvers we combine a compact high order difference approximation with multigrid v cycle algorithm to solve the two dimensional poisson equation with dirichlet boundary conditions this scheme along with several different orderings of grid space and projection operators is compared with the five point formula to show the dramatic improvement in computed accuracy on serial and vector machines

strong structural controllability of systems on colored graphs this article deals with strong structural controllability of leader follower networks the system matrix defining the network dynamics is a pattern matrix in which a priori given entries are equal to zero while the remaining entries take nonzero values these nonzero entries correspond to edges in the network graph the network is called strongly structurally controllable if for all choices of real values for the nonzero entries in the pattern matrix the system is controllable in the classical sense the novelty of this article is that we consider the situation that prespecified nonzero entries in the system s pattern matrix are constrained to take identical nonzero values these constraints can be caused by symmetry properties or physical constraints on the network restricting the system matrices to those satisfying these constraints yields a new notion of strong structural controllability the aim of this article is to establish graph theoretic conditions for this more general property of strong structural controllability

on achieving size independent stability margin of vehicular lattice formations with distributed control we study the stability margin of a vehicular formation with distributed control in which the control at each vehicle only depends on the information from its neighbors in an information graph we consider a d dimensional lattice as information graph of which the 1 d platoon is a special case the stability margin is measured by the real part of the least stable eigenvalue of the closed loop state matrix which quantifies the rate of decay of initial errors in 1 it was shown that with symmetric control in which two neighbors put equal weight on information received from each other the stability margin of a 1 d vehicular platoon decays to 0 as 0 1 n2 where n is the number of vehicles moreover a perturbation analysis was used to show that with vanishingly small amount of asymmetry in the control gains the stability margin scaling can be improved to 0 1 n in this technical note we show that with judicious choice of nonvanishing asymmetry in control the stability margin of the closed loop can be bounded away from zero uniformly in n asymmetry in control gains thus makes the control architecture highly scalable the results are also generalized to d dimensional lattice information graphs that were studied in 2 and the correspondingly stronger conclusions than those derived in 2 are obtained in addition we show that the size independent stability margin can be achieved with relative position and relative velocity rprv feedback as well as relative position and absolute velocity rpav feedback while the analysis in 1 2 was only for the rpav case

constructing a controllable graph under edge constraints abstract this paper is concerned with the problem of constructing a controllable graph subject to some practical edge constraints specifically suppose the total amount of vertices and the upper bounds on the graph diameter or on the vertex degree are given we consider the problem of exploring a class of feasible graphs that satisfy the constraints using the hybrid of a path graph and an antiregular graph we propose a simple and systematic method to generate a class of controllable graphs whose diameters or degrees cover the full possible ranges the method to select the control vector to ensure the controllability of the combined graph is also proposed numerical examples are provided to demonstrate our results

unconditionally stable finite difference scheme and iterative solution of 2d microscale heat transport equation abstract a two dimensional time dependent heat transport equation at the microscale is derived a second order finite difference scheme in both time and space is introduced and the unconditional stability of the finite difference scheme is proved a computational procedure is designed to solve the discretized linear system at each time step by using a preconditioned conjugate gradient method numerical results are presented to validate the accuracy of the finite difference scheme and the efficiency of the proposed computational procedure

brief paper finite time multi agent deployment a nonlinear pde motion planning approach the systematic flatness based motion planning using formal power series and suitable summability methods is considered for the finite time deployment of multi agent systems into planar formation profiles along predefined spatial temporal paths thereby a distributed parameter setting is proposed where the collective leader follower agent dynamics is modeled by two boundary controlled nonlinear time varying pdes governing the motion of an agent continuum in the plane the discretization of the pde model directly induces a decentralized communication and interconnection structure for the multi agent system which is required to achieve the desired spatial temporal paths and deployment formations

controllability of multiagent networks with antagonistic interactions this paper addresses the controllability of a class of antagonistic multiagent networks with both positive and negative edges all the agents of the multiagent network run a consensus algorithm using a signed laplacian based on the generalized equitable partition we propose a graph theoretic characterization of an upper bound on the controllable subspace then we provide a necessary condition for the controllability of the system and give an algorithm to compute the partition furthermore we prove that for a structurally balanced network the controllability is equivalent to that of the corresponding all positive network if the leaders are chosen from the same vertex set several examples are given to illustrate these results

wigner operator and racah operator of the lie superalgebra osp 1 2 starting with the wigner eckart theorem about the osp 1 2 irreducible tensor j phys a math gen 20 5423 1987 the osp 1 2 wigner operator and racah operator are studied their general definitions orthogonality properties and coupling laws are given and the connections between them and the corresponding operators of the so 3 algebra are established the results obtained in this paper are a natural development of the authors theory on the osp 1 2 irreducible tensor

the group with grassmann structure uosp 1 2 the finite dimensional representations of the lie superalgebraosp 1 2 and the group with grassmann structureosp 1 2 have been studied the explicit expression of the projection operator of the superalgebraosp 1 2 has been found the operator permits an arbitrary finite dimensional representation to be expanded in the components multiple to the irreducible ones the clebsch gordan coefficients for the tensor product of two arbitrary irreducible representations have been obtained the matrix elements of the irreducible representations of the groupuosp 1 2 the analoque of the compact form of the grouposp 1 2 are studied the explicit form of these matrix elements the differential equations satisfied by them and the integral of their product have been found

regularity controllability observability of an nds with descriptor form subsystems and generalized lfts abstract this paper investigates regularity controllability and observability for a networked dynamic system nds with its subsystems being described in a descriptor form and system matrices of each subsystem being represented by a generalized linear fractional transformation glft of its parameters except a well posedness condition no any other constraints are put on either parameters or connections of a subsystem based on the kronecker canonical form kcf of a matrix pencil some matrix rank based necessary and sufficient conditions are established respectively for the regularity and complete controllability observability of the nds in which the associated matrix depends affinely on both subsystem parameters and subsystem connections these conditions keep the property that all the involved numerical computations are performed on each subsystem independently which is attractive in the analysis and synthesis of a large scale nds moreover some explicit and easily checkable requirements are derived for subsystem dynamics parameters with which a completely controllable observable nds can be constructed more easily

racah wigner calculus for the super rotation algebra i the symmetry properties and the pseudo orthogonality relations of the super rotation clebsh gordan coefficients for the tensor product of two irreducible representations of the super rotation algebra are derived the symmetric super rotation 3 j symbol and the symmetric and invariant super rotation 6 j symbol are defined their basic properties are described and their relations to the usual 3 j and 6 j symbols are given

controllability of discrete time multi agent systems with directed topology and input delay this paper investigates the controllability of first order and second order discrete time multi agent systems with directed topology and input delay the problem is studied in the leader follower framework where a number of agents are selected to be leaders and serve as control inputs to all other agents sufficient and necessary conditions are derived for the controllability of first order discrete time multi agent systems with sampling period and feedback gain satisfying certain conditions it is proved under three different protocols that the controllability of second order discrete time multi agent systems is equivalent to that of a pair of submatrices of laplacian matrix in addition the controllability of both first order and second order discrete time multi agent systems with input delay is shown through some transformations to be equivalent to that of the transformed non delayed systems finally some simulation examples are given to illustrate the theoretical results

controllability of system dynamics on networks quantum walks and random walks abstract we consider a model of system dynamics on networks consisting of a certain number of stations nodes connected by edges at every time step a certain quantity leaves the stations and gets re distributed among the neighboring stations preserving an appropriate norm one has the control at every station on the relative quantity which is sent to the various neighbors controllability is achieved when every possible distribution of the given quantity in a natural set can be obtained this general model includes in particular classical and quantum walks on graphs the case of quantum walks is treated in detail and it is proved that controllability is achieved if and only if the underlying graph is not bipartite this extends to general graphs a result previously proved in the literature for the case where the underlying graph is regular then extensions are presented to general systems modeling system dynamics on networks

controllability and observability of network of networks via cartesian products the paper presents a system theoretic analysis framework for a network of networks formed from smaller factor networks via graph cartesian products we provide a compositional framework for extending the controllability and observability of the factor networks to that of the composite network of networks we then delve into the effectiveness of designing control and estimation algorithms for the composite network via its symmetry and gramian structure an example demonstrating the usefulness of our results in the context of social networks with a cartesian product structure is then presented

calculation of primitive 6 j symbols the 6 j symbols of a group are independent of the subgroup chain chosen to define the basis states the authors present an improved algorithm for calculating the primitive 6 j symbols for a compact group with a faithful irrep by recursive building up using only the kronecker product rules and the general properties of 6 j previously one has sometimes needed to search for the useful equations by systematically trying all equations which involve unknown 6 j the authors show that the primitive 6 j may all be easily solved in terms of a subclass the core 6 j they discuss how the core 6 j can usually be solved proving that the method is complete for so3 the authors conjecture that the algorithm is complete for all groups

controllability and observability of grid graphs via reduction and symmetries in this paper we investigate the controllability and observability properties of a family of linear dynamical systems whose structure is induced by the laplacian of a grid graph this analysis is motivated by several applications in network control and estimation quantum computation and discretization of partial differential equations specifically we characterize the structure of the grid eigenvectors by means of suitable decompositions of the graph for each eigenvalue based on its multiplicity and on suitable symmetries of the corresponding eigenvectors we provide necessary and sufficient conditions to characterize all and only the nodes from which the induced dynamical system is controllable observable we discuss the proposed criteria and show through suitable examples how such criteria reduce the complexity of the controllability respectively observability analysis of the grid

flatness based trajectory planning for diffusion reaction systems in a parallelepipedon a spectral approach spectral analysis is considered for the flatness based solution of the trajectory planning problem for a boundary controlled diffusion reaction system defined on a 1 r dimensional parallelepipedon by exploiting the riesz spectral properties of the system operator it is shown that a suitable reformulation of the resolvent operator allows a systematic introduction of a basic output which yields a parametrization of both the system state and the boundary input in terms of differential operators of infinite order their convergence is verified for both infinite dimensional and finite dimensional actuator configurations by restricting the basic output to certain gevrey classes involving non analytic functions with this a systematic approach is introduced for basic output trajectory assignment and feedforward tracking control towards the realization of finite time transitions between stationary profiles

brief paper leader localization in multi agent systems subject to failure a graph theoretic approach in this paper structural controllability of a leader follower multi agent system with multiple leaders is studied a graphical condition for structural controllability based on the information flow graph of the system is provided the notions of p link and q agent controllability in a multi leader setting are then introduced which provide quantitative measures for the controllability of a system subject to failure in the agents and communication links the problem of leader localization is introduced which is concerned with finding the minimum number of agents whose selection as leaders results in a p link or q agent controllable network polynomial time algorithms are subsequently presented to solve the problem for both cases of undirected and directed information flow graphs

disturbance decoupling problem for multi agent systems a graph topological approach abstract this paper studies the disturbance decoupling problem for multi agent systems with single integrator dynamics and a directed communication graph we are interested in topological conditions that imply the disturbance decoupling of the network and more generally guarantee the existence of a state feedback rendering the system disturbance decoupled in particular we will develop a class of graph partitions which can be described as a topological translation of controlled invariant subspaces in the context of dynamical networks then we will derive sufficient conditions in terms of graph partitions such that the network is disturbance decoupled as well as conditions guaranteeing solvability of the disturbance decoupling problem the proposed results are illustrated by a numerical example

on the derivation of highest order compact finite difference schemes for the one and two dimensional poisson equation with dirichlet boundary conditions the primary aim of this paper is to answer the question what are the highest order five or nine point compact finite difference schemes to answer this question we present several simple derivations of finite difference schemes for the one and two dimensional poisson equation on uniform quasi uniform and nonuniform face to face hyperrectangular grids and directly prove the existence or nonexistence of their highest order local accuracies our derivations are unique in that we do not make any initial assumptions on stencil symmetries or weights for the one dimensional problem the derivation using the three point stencil on both uniform and nonuniform grids yields a scheme with arbitrarily high order local accuracy however for the two dimensional problem the derivation using the corresponding five point stencil on uniform and quasi uniform grids yields a scheme with at most second order local accuracy and on nonuniform grids yields at most first order local accuracy when expanding the five poin

flatness based boundary control of a class of quasilinear parabolic distributed parameter systems motion planning and control of a boundary controlled quasilinear parabolic partial differential equation in one spatial variable is considered the approach relies on a flatness property of the system namely that the system solution can be differentially parameterized in terms of a flat output which in the case considered is a boundary value such a parameterization allows straightforward motion planning and computation of a control law the approach is based on power series in the spatial variable and the convergence of these series is ensured by choosing the flat output to be a nonanalytic smooth function of appropriate gevrey class

multigrid method and fourth order compact scheme for 2d poisson equation with unequal mesh size discretization abstract a fourth order compact difference scheme with unequal mesh sizes in different coordinate directions is employed to discretize a two dimensional poisson equation in a rectangular domain multigrid methods using a partial semicoarsening strategy and line gauss seidel relaxation are designed to solve the resulting sparse linear systems numerical experiments are conducted to test the accuracy of the fourth order compact difference scheme and to compare it with the standard second order difference scheme convergence behavior of the partial semicoarsening and line gauss seidel relaxation multigrid methods is examined experimentally

on the reachability and observability of path and cycle graphs in this technical note we investigate the reachability and observability properties of a network system running a laplacian based average consensus algorithm when the communication graph is a path or a cycle specifically we provide necessary and sufficient conditions based on simple rules from number theory to characterize all and only the nodes from which the network system is reachable respectively observable interesting immediate corollaries of our results are i a path graph is reachable observable from any single node if and only if the number of nodes of the graph is a power of two n 2i i n and ii a cycle is reachable observable from any pair of nodes if and only if n is a prime number for any set of control observation nodes we provide a closed form expression for the unreachable unobservable eigenvalues and for the eigenvectors of the unreachable unobservable subsystem

algebraic structure of tensor superoperators for the super rotation algebra ii it is shown that the sets of tensor superoperators for the super rotation algebra can be used to build explicit bases for the representations of several superalgebras the representations built in this way are the fundamental representations of the special linear superalgebras sl 2j 1 2j and of the orthosymplectic superalgebras osp 2j 1 2j and the 4j 1 dimensional representations of osp 1 2 and sl 1 2 stavraki superalgebras it is shown that the chain osp 1 2 osp 2j 1 2j or osp 2j 2j 1 explains the existence of a series of nontrival zeros for the super rotation 6 j symbol sr6 j symbols

on tensor operators of the lie superalgebra osp 1 2 the author gives a complete theory on tensor operators of the osp 1 2 algebra including a new definition of the irreducible tensor a coupling law of two irreducible tensors the wigner eckart theorem and a series of methods of calculating reduced matrix elements etc one can see the overall application of the osp 1 2 racah coefficients

on the controllability and observability of networked dynamic systems some necessary and sufficient conditions are obtained for the controllability and observability of a networked system with linear time invariant lti dynamics the topology of this system is fixed but arbitrary and every subsystem is permitted to have different dynamics these conditions essentially depend only on transmission zeros of every subsystem and the subsystem connection matrix which makes them attractive in the analysis and synthesis of a large scale networked system as an application these conditions are utilized to characterize systems whose steady estimation accuracy with the distributed predictor of zhou 2013 is equal to that of the lumped kalman filter it has been made clear that to guarantee this equivalence the steady update gain matrix of the kalman filter must be block diagonal

racah coefficients of the lie superalgebra osp 1 2 using the indefinite metric the author studies the coupling of three irreps of the osp 1 2 algebra and gives the general definition orthogonality conditions symmetry properties and calculating formulae of the racah coefficients of the osp 1 2 algebra eight kinds of racah coefficients exist for the osp 1 2 algebra and they are all expressed in terms of the racah coefficients of the so 3 algebra

zero forcing number constrained matchings and strong structural controllability abstract the zero forcing number is a graph invariant introduced to study the minimum rank of the graph in 2008 aazami proved the np hardness of computing the zero forcing number of a simple undirected graph we complete this np hardness result by showing that the non equivalent problem of computing the zero forcing number of a directed graph allowing loops is also np hard the rest of the paper is devoted to the strong controllability of a networked system this kind of controllability takes into account only the structure of the interconnection graph but not the interconnection strengths along the edges we provide a necessary and sufficient condition in terms of zero forcing sets for the strong controllability of a system whose underlying graph is a directed graph allowing loops moreover we explain how our result differs from a recent related result discovered by monshizadeh et al finally we show how to solve the problem of finding efficiently a minimum size input set for the strong controllability of a self damped system with a tree structure

parametrization of the linear zeros of 6j coefficients the linear zeros of 6j coefficients are fully parametrized apart from a multiplicative factor in terms of four integers

high order compact scheme for the steady stream function vorticity equations a higher order compact scheme that is o h4 on the nine point 2 d stencil is formulated for the steady stream function vorticity form of the navier stokes equations the resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software we also show that special treatment of the wall boundary conditions is required the method is tested on representative model problems and compares very favourably with other schemes in the literature

controllability improvement for multi agent systems leader selection and weight adjustment abstractfor an uncontrollable system adding leaders and adjusting edge weights are two methods to improve controllability in this paper the controllability of multi agent systems under directed topologies is studied especially on the leader selection problem and the weight adjustment problem for a multi agent system necessary and sufficient algebraic conditions for controllability with fewest leaders are proposed to improve controllability by adjusting edge weights the system is supposed to be structurally controllable which holds if and only if the communication topology contains a spanning tree it is also proved that the number of the fewest edges needed to be assigned on new weights equals the rank deficiency of controllability matrix in addition a leader selection algorithm and a weight adjustment algorithm are presented simulation examples are provided to illustrate the theoretical results

haar wavelet collocation method for three dimensional elliptic partial differential equations a new collocation method based on haar wavelet is presented for numerical solution of three dimensional elliptic partial differential equations with dirichlet boundary conditions an important advantage of the method is that it can be applied to both linear as well as nonlinear problems the algorithm based on this new method is simple and can be easily implemented in any programming language experimental rates of convergence of the proposed method are calculated which are in agreement with theoretical results the proposed method is applied to several benchmark problems from the literature including linear and nonlinear elliptic problems as well as systems of elliptic partial differential equations the numerical experiments confirm the accuracy and diverse applicability of the method

a rbf based technique for 3d convection diffusion reaction problems in an anisotropic inhomogeneous medium abstract we present a rbf based semi analytical technique for solving 3d convection diffusion reaction cdr equations to model transport in an anisotropic inhomogeneous medium the mathematical model is expressed as the boundary value problem for elliptic partial differential equation epde main feature of the presented technique is the separately satisfaction of the conditions on the boundary of the domain and the epde inside to be more precise we transform the original epde to the equation with homogeneous boundary condition bc and seek the approximate solution as a sum of the modified rbfs mrbfs the mrbfs satisfy the homogeneous bc of the problem so any linear combination also satisfies the homogeneous bc the rbfs of three types are used in the framework of the method the multiquadric mq rbf the gaussian rbf and the conical one the coefficients of the linear combination are determined so that it satisfies the governing equation of the epde ten numerical examples demonstrate the high effectiveness of the presented technique in solving 3d cdr problems in single and double connected domains

a general meshsize fourth order compact difference discretization scheme for 3d poisson equation a fourth order compact difference scheme with unrestricted general meshsizes in different coordinate directions is derived to discretize three dimensional poisson equation on a regular cubic domain the difference scheme derivation procedure makes use of the symbolic representation of the finite difference schemes and is easier to understand in such complex three dimensional manipulations we use a preconditioned conjugate gradient method to solve the resulting sparse linear systems and verify the formal order of convergence of the derived fourth order finite difference scheme

containment control in mobile networks in this paper the problem of driving a collection of mobile robots to a given target destination is studied in particular we are interested in achieving this transfer in an orderly manner so as to ensure that the agents remain in the convex polytope spanned by the leader agents while the remaining agents only employ local interaction rules to this aim we exploit the theory of partial difference equations and propose hybrid control schemes based on stop go rules for the leader agents non zenoness liveness and convergence of the resulting system are also analyzed

graph controllability classes for the laplacian leader follower dynamics in this paper we consider the problem of obtaining graph theoretic characterizations of controllability for the laplacian based leader follower dynamics our developments rely on the notion of graph controllability classes namely the classes of essentially controllable completely uncontrollable and conditionally controllable graphs in addition to the topology of the underlying graph the controllability classes rely on the specification of the control vectors our particular focus is on the set of binary control vectors the choice of binary control vectors is naturally adapted to the laplacian dynamics as it captures the case when the controller is unable to distinguish between the followers and moreover controllability properties are invariant under binary complements we prove that the class of essentially controllable graphs is a strict subset of the class of asymmetric graphs and provide numerical results that suggests that the ratio of essentially controllable graphs to asymmetric graphs increases as the number of vertices increases although graph symmetries play an important role in graph theoretic characterizations of controllability we provide an explicit class of asymmetric graphs that are completely uncontrollable namely the class of block graphs of steiner triple systems we prove that for graphs on four and five vertices a repeated laplacian eigenvalue is a necessary condition for complete uncontrollability but however show through explicit examples that for eight and nine vertices a repeated eigenvalue is not necessary for complete uncontrollability for the case of conditional controllability we give an easily checkable necessary condition that identifies a class of binary control vectors that result in a two dimensional controllable subspace several constructive examples demonstrate our results

a fourth order kernel free boundary integral method for implicitly defined surfaces in three space dimensions abstract the kernel free boundary integral kfbi method is a finite difference version of the traditional boundary integral method for elliptic and parabolic partial differential equations on complex domains it evaluates boundary or volume integrals involved in the solution of boundary integral equations bies by solving equivalent but simple interface problems on regular grids so that the integral kernel or green function is never needed or computed this is the essential difference of the kfbi method from the traditional ones it takes advantage of the well conditioning property of discrete bies so that the number of krylov subspace iterations is essentially independent of discretization parameter or system dimension this paper presents a fourth order kernel free boundary integral method for second order elliptic partial differential equations on complex domains in three space dimensions whose boundaries are given by implicitly defined surfaces it represents the domain boundary and discretizes data on it by intersection points of the surface with cartesian grid lines the approach has a variety of advantages the current work solves simple interface problems with corrected 27 point compact finite difference schemes and calculates the discrete equations with a fast fourier transform based elliptic solver numerical examples show that the proposed method is efficient as well as accurate

algorithms for the polynomial zeros of degree 2 of the 3 j and the 6 j coefficients simple algorithms which are based on the principle of factorisation of an integer are proposed to generate the polynomial zeros of degree 2 of the 3 j and 6 j coefficients

controllability of multi agent systems with directed and weighted signed networks abstract this paper studies the controllability of multi agent systems with signed networks which are represented by directed weighted signed graphs the adjacency weights of network depict the property of interactions positive weight means cooperative interaction while negative weight antagonistic interaction first by using switching equivalent transformation the controllability of three kinds of signed networks is studied structurally balanced anti balanced and strictly unbalanced it is shown that the controllability of the structurally balanced network is equivalent to that of the associated underlying network then a graph theoretic necessary condition for controllability is given by virtue of almost equitable partition of directed weighted signed networks furthermore some necessary and sufficient conditions are given for the controllability of generic linear multi agent systems in addition several examples are presented to illustrate the theoretical results

non stationary quantum walks on the cycle we consider quantum walks on the cycle in the non stationary case where the coin operation is allowed to change at each time step we characterize in algebraic terms the set of possible state transfers and prove that as opposed to the stationary case the associate probability distribution may converge to a uniform distribution among the nodes of the associated graph

high order compact finite difference schemes for a system of third order boundary value problem abstract high order compact finite difference schemes are devised for general third order differential equations raised in the study of obstacle boundary value problems these high order compact schemes are constructed by a modification without additional workload in computation of the second order scheme developed in noor and al said 2002 10 the convergence rates of these schemes are shown to be third and fourth order respectively numerical experiments show that the proposed schemes are of high accuracy and are efficient

analysis of quantum walks with time varying coin on d dimensional lattices in this paper we present a study of discrete time quantum walks whose underlying graph is a d dimensional lattice the dynamical behavior of these systems is of current interest because of their applications in quantum information theory as tools to design quantum algorithms we assume that at each step of the walk evolution the coin transformation is allowed to change so that we can use it as a control variable to drive the evolution in a desired manner we give an exact description of the possible evolutions and of the set of possible states that can be achieved with such a system in particular we show that it is possible to go from a state where there is probability 1 for the walker to be found in a vertex to a state where all the vertices have equal probability we also prove a number of properties of the set of admissible states in terms of the number of steps needed to obtain them we provide explicit algorithms for state transfer in low dimensional cases as well as results that allow to reduce algorithms on two dimensional lattices to algorithms on the one dimensional lattice the cycle

quadratic zeros of racah 6j coefficients a geometrical approach it is shown that the projective symmetries of the polynomial of quadratic 6j coefficients form the symmetrical group s6 nonlinear rational symmetries of are found partial parametrizations of the zeros of are presented

protocols design and uncontrollable topologies construction for multi agent networks this note investigates the controllability issues of multi agent systems where each node contains generic linear dynamics first a neighbor based control protocol is proposed under which it is shown that the controllability of a multi agent system is solely decided by its communication topology structure we then further consider the effects of communication topology on the controllability from a graph theory point of view by explicitly constructing topologies the construction exhibits a partition of the designed graph with identified leaders under which the system is shown to be uncontrollable