exact solution of a 1d asymmetric exclusion model using a matrix formulation several recent works have shown that the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions can be solved exactly in the case of open boundaries here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices with this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules they obtain several explicit forms for these non commuting matrices which are in the general case infinite dimensional their approach allows exact expressions to be derived for the current and density profiles finally they discuss briefly two possible generalizations of their results the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles

the determinants of matrices with recursive entries abstract we prove a result concerning the evaluation of determinant of a matrix whose entries are given by certain linear inhomogeneous recurrence relations this result generalizes previous determinants evaluations due to krattenthaler neuwirth and the authors in c krattenthaler evaluations of some determinants of matrices related to the pascal triangle sem lothar combin 47 2002 19 article b47g e neuwirth treeway galton arrays and generalized pascal like determinants technical report university of vienna 2004 pp 1 16 a r moghaddamfar s m h pooya s navid salehy s nima salehy evaluating and generalizing certain determinants submitted for publication the main tool for proving our result is lu factorization of the recursive arrays introduced in c krattenthaler advanced determinant calculus a complement linear algebra appl 411 2005 68 166

nonzero bound on fiedler eigenvalue causes exponential growth of h infinity norm of vehicular platoon we consider platoons composed of identical vehicles and controlled in a distributed way that is each vehicle has its own onboard controller the regulation errors in spacing to the immediately preceeding and following vehicles are weighted differently by the onboard controller which thus implements an asymmetric bidirectional control scheme the weights can vary along the platoon we prove that such platoons have a nonzero uniform bound on the second smallest eigenvalue of the graph laplacian matrix the fiedler eigenvalue furthermore it is shown that existence of this bound always signals undesirable scaling properties of the platoon namely the h infinity norm of the transfer function of the platoon grows exponentially with the number of vehicles regardless of the controllers used hence the benefits of a uniform gap in the spectrum of a laplacian with an asymetric distributed controller are paid for by poor scaling as the number of vehicles grows

asymmetric simple exclusion process with open boundaries and askey wilson polynomials we study the one dimensional asymmetric simple exclusion process asep with open boundary conditions particles are injected and ejected at both boundaries it is clarified that the steady state of the model is intimately related to the askey wilson polynomials the partition function and the n point functions are obtained in the integral form with four boundary parameters the thermodynamic current is evaluated to confirm the conjectured phase diagram

mistuning based control design to improve closed loop stability margin of vehicular platoons we consider a decentralized bidirectional control of a platoon of n identical vehicles moving in a straight line the control objective is for each vehicle to maintain a constant velocity and inter vehicular separation using only the local information from itself and its two nearest neighbors each vehicle is modeled as a double integrator to aid the analysis we use continuous approximation to derive a partial differential equation pde approximation of the discrete platoon dynamics the pde model is used to explain the progressive loss of closed loop stability with increasing number of vehicles and to devise ways to combat this loss of stability if every vehicle uses the same controller we show that the least stable closed loop eigenvalue approaches zero as o 1 n2 in the limit of a large number n of vehicles we then show how to ameliorate this loss of stability by small amounts of mistuning i e changing the controller gains from their nominal values we prove that with arbitrary small amounts of mistuning the asymptotic behavior of the least stable closed loop eigenvalue can be improved to o 1 n all the conclusions drawn from analysis of the pde model are corroborated via numerical calculations of the state space platoon model

parabolic pde based multi agent formation control on a cylindrical surface abstractthis paper considers the modelling and control design of the multi agent systems in the 3 d space the communication graph of the agents is a mesh grid 2 d cylindrical surface different from most existing literatures where the agents are modelled by ordinary differential equations odes we treat the agents as a continuum in this paper more specifically we model the collective dynamics of the agents by two reaction advection diffusion 2 d partial differential equations pdes the pde states represent the agent positions and the equilibria correspond to possible formation manifolds these pdes can be open loop unstable and the boundary stabilisation problem of the pdes on the cylindrical surface is solved using the backstepping method an all explicit observer based output control scheme is constructed which is distributed in the sense that each agent only needs local information closed loop exponential stability in the l2 h1 and h2 spaces is proved for the controller designs numerical

one dimensional partially asymmetric simple exclusion process with open boundaries orthogonal polynomials approach the stationary state of the partially asymmetric simple exclusion process with open boundaries is reconsidered the so called matrix product ansatz is employed this enables us to construct the stationary state in the form of the product of the matrices d and e noticing the fact that the matrix c d e for the model is related to certain q orthogonal polynomials the model is analysed for a wide range of parameters the current and the correlation length are evaluated in the thermodynamic limit it turns out that the phase diagram for the correlation length is richer than that for the totally asymmetric case

diffusion and consensus on weakly connected directed graphs abstract let g be a weakly connected directed graph with asymmetric graph laplacian l consensus and diffusion are dual dynamical processes defined on g by x l x for consensus and p p l for diffusion in this survey we consider both these processes as well their discrete time analogues we define a basis of row vectors i i 1 k of the left null space of l and a basis of column vectors i i 1 k of the right null space of l in terms of the partition of g into strongly connected components this allows for complete characterization of the asymptotic behavior of both diffusion and consensus discrete and continuous in terms of these eigenvectors as an application of these ideas we present a treatment of the pagerank algorithm that is dual to the usual one we further show that the teleporting see below feature usually included in the algorithm is not strictly necessary together with 1 this is a complete and self contained overview of the asymptotics of consensus and diffusion on digraphs many of the ideas presented here can be found scattered in the literature though mostly outside mainstream mathematics and not always with complete proofs this paper seeks to remedy this by providing a compact and accessible survey

bethe ansatz and q operator for the open asep in this paper we look at the asymmetric simple exclusion process with open boundaries with a current counting deformation we construct a two parameter family of transfer matrices which commute with the deformed markov matrix of the system we show that these transfer matrices can be factorized into two commuting matrices with one parameter each which can be identified with baxter s q operator and that for certain values of the product of those parameters they decompose into a sum of two commuting matrices one of which is the usual one parameter transfer matrix for a given dimension of the auxiliary space using this we find the t q equation for the open asep and through functional bethe ansatz techniques we obtain an exact expression for the dominant eigenvalue of the deformed markov matrix

passivity based control for multi vehicle systems subject to string constraints in this paper we show how heterogeneous bidirectional vehicle strings can be modelled as port hamiltonian systems analysis of stability and string stability within this framework is straightforward and leads to a better understanding of the underlying problem nonlinear local control and additional integral action is introduced to design a suitable control law guaranteeing l 2 string stability of the system with respect to bounded disturbances

deviation bounds in multi agent systems described by undirected graphs the theory of port hamiltonian systems is used to derive upper bounds for the state deviations in multi agent systems described by undirected graphs pinned to a reference signal the upper bounds for the deviations in networks of first or second order agents respectively depend on the minimal eigenvalue of the extended laplacian of the system in networks of first order agents the deviations decay exponentially with a rate depending on the same minimal eigenvalue in case networks of second order systems meet specific design properties it can be shown that the deviations also decay exponentially with half the rate compared to first order systems

decentralized control of vehicle formations this paper investigates a method for decentralized stabilization of vehicle formations using techniques from algebraic graph theory the vehicles exchange information according to a pre specified communication digraph g a feedback control is designed using relative information between a vehicle and its in neighbors in g we prove that a necessary and sufficient condition for an appropriate decentralized linear stabilizing feedback to exist is that g has a rooted directed spanning tree we show the direct relationship between the rate of convergence to formation and the eigenvalues of the directed laplacian of g various special situations are discussed including symmetric communication graphs and formations with leaders several numerical simulations are used to illustrate the results

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

asymmetric exclusion model and weighted lattice paths we show that the known matrix representations of the stationary state algebra of the asymmetric simple exclusion process asep can be interpreted combinatorially as various weighted lattice paths this interpretation enables us to use the constant term method ctm and bijective combinatorial methods to express many forms of the asep normalization factor in terms of ballot numbers one particular lattice path representation shows that the coefficients in the recurrence relation for the asep correlation functions are also ballot numbers additionally the ctm has a strong combinatorial connection which leads to a new canonical lattice path representation and to the expansion which provides a uniform approach to computing the asymptotic behaviour in the various phases of the asep the path representations enable the asep normalization factor to be seen as the partition function of a more general polymer chain model having a two parameter interaction with a surface we show in the case 1 that the probability of finding a given number of particles in the stationary state can be expressed via non intersecting lattice paths and hence as a simple determinant

modeling complex systems by generalized factor analysis we propose a new modeling paradigm for large dimensional aggregates of stochastic systems by generalized factor analysis gfa models these models describe the data as the sum of a flocking plus an uncorrelated idiosyncratic component the flocking component describes a sort of collective orderly motion which admits a much simpler mathematical description than the whole ensemble while the idiosyncratic component describes weakly correlated noise we first discuss static gfa representations and characterize in a rigorous way the properties of the two components the extraction of the dynamic flocking component is discussed for time stationary linear systems and for a simple classes of separable random fields

multi agent deployment in 3 d via pde control this paper introduces a methodology for modelling analysis and control design of a large scale system of agents deployed in 3 d space the agents communication graph is a mesh grid disk 2 d topology in polar coordinates treating the agents as a continuum we model the agents collective dynamics by complex valued reaction diffusion 2 d partial differential equations pdes in polar coordinates whose states represent the position coordinates of the agents due to the reaction term in the pdes the agents can achieve a rich family of 2 d deployment manifolds in 3 d space which correspond to the pdes equilibrium as determined by the boundary conditions unfortunately many of these deployment surfaces are open loop unstable to stabilize them a heretofore open and challenging problem of pde stabilization by boundary control on a disk has been solved in this paper using a new class of explicit backstepping kernels that involve the poisson kernel a dual observer which is also explicit allows to estimate the positions of all the agents as needed in the leaders feedback by only measuring the position of their closest neighbors hence an all explicit control scheme is found which is distributed in the sense that each agent only needs local information closed loop exponential stability in the l 2 h 1 and h 2 spaces is proved for both full state and output feedback designs numerical simulations illustrate the proposed approach for 3 d deployment of discrete agents

crt based fully homomorphic encryption over the integers in 1978 rivest adleman and dertouzos introduced the basic concept of privacy homomorphism that allows computation on encrypted data without decryption it was an interesting work whose idea precedes the recent development of fully homomorphic encryption although actual example schemes proposed in the paper are all susceptible to simple known plaintext attacks in this paper we revisit one of their proposals in particular the third scheme which is based on the chinese remainder theorem and is ring homomorphic it is known that only a single pair of known plaintext ciphertext is needed to break this scheme however by exploiting the standard technique to insert an error to a message before encryption we can cope with this problem we present a secure modification of their proposal by showing that the proposed scheme is fully homomorphic and secure against the chosen plaintext attacks under the approximate gcd assumption and the sparse subset sum assumption when the message space is restricted to z 2 k interestingly the proposed scheme can be regarded as a generalization of the dghv scheme with larger plaintext space our scheme has o 5 ciphertext expansion overhead while the dghv has o 8 for the security parameter when restricted to the homomorphic encryption scheme with depth of o log the overhead is reduced to o our scheme can be used in applications requiring a large message space z q for log q o 4 or simd style operations on z q k for log q o k o 3 with o 5 ciphertext size as in the dghv

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

coherence in large scale networks dimension dependent limitations of local feedback we consider distributed consensus and vehicular formation control problems specifically we address the question of whether local feedback is sufficient to maintain coherence in large scale networks subject to stochastic disturbances we define macroscopic performance measures which are global quantities that capture the notion of coherence a notion of global order that quantifies how closely the formation resembles a solid object we consider how these measures scale asymptotically with network size in the topologies of regular lattices in 1 2 and higher dimensions with vehicular platoons corresponding to the 1 d case a common phenomenon appears where a higher spatial dimension implies a more favorable scaling of coherence measures with a dimensions of 3 being necessary to achieve coherence in consensus and vehicular formations under certain conditions in particular we show that it is impossible to have large coherent 1 d vehicular platoons with only local feedback we analyze these effects in terms of the underlying energetic modes of motion showing that they take the form of large temporal and spatial scales resulting in an accordion like motion of formations a conclusion can be drawn that in low spatial dimensions local feedback is unable to regulate large scale disturbances but it can in higher spatial dimensions this phenomenon is distinct from and unrelated to string instability issues which are commonly encountered in control problems for automated highways

leader enabled deployment onto planar curves a pde based approach we introduce an approach for stable deployment of agents onto families of planar curves namely 1 d formations in 2 d space the agents collective dynamics are modeled by the reaction advection diffusion class of partial differential equations pdes which is a broader class than the standard heat equation and generates a rich geometric family of deployment curves the pde models whose state is the position of the agents incorporate the agents feedback laws which are designed based on a spatial internal model principle namely the agents feedback laws allow the agents to deploy to a family of geometric curves that correspond to the model s equilibrium curves parameterized by the continuous agent identity 0 1 however many of these curves are open loop unstable stable deployment is ensured by leader feedback designed in a manner similar to the boundary control of pdes by discretizing the pde model with respect to we impose a fixed communication topology specifically a chain graph on the agents and obtain control laws that require communication with only an agent s nearest neighbors on the graph a pde based approach is also used to design observers to estimate the positions of all the agents which are needed in the leader s feedback by measuring only the position of the leader s nearest neighbor hence the leader uses only local information when employing output feedback

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

duality biorthogonal polynomials and multi matrix models the statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized christoffel darboux form constructed from sequences of biorthogonal polynomials for measures involving exponentials of a pair of polynomials v1 v2 in two different variables these kernels may be expressed in terms of finite dimensional windows spanned by finite subsequences having length equal to the degree of one or the other of the polynomials v1 v2 the vectors formed by such subsequences satisfy dual pairs of first order systems of linear differential equations with polynomial coefficients having rank equal to one of the degrees of v1 or v2 and degree equal to the other they also satisfy recursion relations connecting the consecutive windows and deformation equations determining how they change under variations in the coefficients of the polynomials v1 and v2 viewed as overdetermined systems of linear difference differential deformation equations these are shown to be compatible and hence to admit simultaneous fundamental systems of solutions the main result is the demonstration of a spectral duality property namely that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters

tableaux combinatorics for the asymmetric exclusion process and askey wilson polynomials introduced in the late 1960 s 33 43 the asymmetric exclusion process asep is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one dimensional lattice of n sites it has been cited as a model for traffic flow and protein synthesis in the most general form of the asep with open boundaries particles may enter and exit at the left with probabilities and and they may exit and enter at the right with probabilities and in the bulk the probability of hopping left is q times the probability of hopping right the first main result of this paper is a combinatorial formula for the stationary distribution of the asep with all parameters general in terms of a new class of tableaux which we call staircase tableaux this generalizes our previous work 14 15 for the asep with parameters 0 using our first result and also results of uchiyama sasamoto wadati 48 we derive our second main result a combinatorial formula for the moments of askey wilson polynomials since the early 1980 s there has been a great deal of work giving combinatorial formulas for moments of various other classical orthogonal polynomials e g hermite charlier laguerre meixner however this is the first such formula for the askey wilson polynomials which are at the top of the hierarchy of classical orthogonal polynomials

spectra of certain large tridiagonal matrices abstract we characterize the eigenvalues and eigenvectors of a class of complex valued tridiagonal n by n matrices subject to arbitrary boundary conditions i e with arbitrary elements on the first and last rows of the matrix for large n we show there are up to 4 eigenvalues the so called special eigenvalues whose behavior depends sensitively on the boundary conditions the other eigenvalues the so called regular eigenvalues vary very little as function of the boundary conditions for large n we determine the regular eigenvalues up to o n 2 and the special eigenvalues up to o n for some 0 1 the components of the eigenvectors are determined up to o n 1 the matrices we study have important applications throughout the sciences among the most common ones are arrays of linear dynamical systems with nearest neighbor coupling and discretizations of second order linear partial differential equations in both cases we give examples where specific choices of boundary conditions substantially influence leading eigenvalues and therefore the global dynamics of the system

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

information flow and its relation to stability of the motion of vehicles in a rigid formation in this note we consider the effect of information flow on the propagation of errors in spacing in a collection of vehicles trying to maintain a rigid formation during translational maneuvers the motion of each vehicle is described using a linear time invariant lti system we consider undirected and connected information flow graphs and assume that each vehicle can communicate with a maximum of q vehicles where q may vary with the size n of the collection we consider translational maneuver of a reference vehicle where its steady state velocity is different from its initial velocity in the absence of any disturbing forces acting on the vehicles during the maneuver it is desired that the collection be controlled in such a way that its motion asymptotically resembles that of a rigid body in the presence of bounded disturbing forces acting on the vehicles it is desired that the maximum deviation of the motion of the collection from that of a rigid body be bounded and be independent of the size of the collection we consider a decentralized feedback control scheme where the controller of each vehicle takes into account the aggregate errors in position and velocity from the vehicles with which it is in direct communication we assume that all vehicles start at their respective desired positions and velocities since the displacement of every vehicle at the end of the maneuver of the reference vehicle must be the same we show that the loop transfer function must have at least two poles at the origin we then show that if the loop transfer function has three or more poles at the origin then the motion of the collection is unstable that is its deviation from the rigid body motion is arbitrarily large if the size of the formation is sufficiently large if l is the number of poles of the transfer function relating the position of a vehicle with its control input we show that if q n n rarr0 as nrarrinfin then there is a low frequency sinusoidal disturbance of at most unit amplitude acting on each vehicle such that the maximum errors in spacing response increase at least as omega radicn q n l 1 a function p n is omega q n if there exists a nonzero constant c 0 and a n such that p n gesc q n for all n n a consequence of the results presented in this note is that the maximum errors in spacing and velocity of any vehicle can be made insensitive to the size of the collection only if there is at least one vehicle in the collection that communicates with at least omega n other vehicles in the collection

the computational complexity of simultaneous diophantine approximation problems simultaneous diophantine approximation in d dimensions deals with the approximation of a vector bf alpha alpha 1 cdots alpha d of d real numbers by vectors of rational numbers all having the same denominator this paper considers the computational complexity of algorithms to find good simultaneous approximations to a given vector bf alpha of d rational numbers we measure the goodness of an approximation using the sup norm we show that a result of h w lenstra jr produces polynomial time algorithms to find sup norm best approximations to a given vector bf alpha when the dimension d is fixed we show that a recent algorithm of a k lenstra h w lenstra jr and l lovasz to find short vectors in an integral lattice can be used to find a good approximation to a given vector bf alpha in d dimensions with a denominator q satisfying 1 leqq q leqq 2 d 2 n which is within a factor sqrt 5d 2 d 1 2 of the best approximation with denominato

string stability of interconnected systems introduces the notion of string stability of a countably infinite interconnection of a class of nonlinear systems intuitively string stability implies uniform boundedness of all the states of the interconnected system for all time if the initial states of the interconnected system are uniformly bounded it is well known that the input output gain of all the subsystems less than unity guarantees that the interconnected system is input output stable the authors derive sufficient weak coupling conditions which guarantee the asymptotic string stability of a class of interconnected systems under the same weak coupling conditions string stable interconnected systems remain string stable in the presence of small structural singular perturbations in the presence of parameter mismatch these weak coupling conditions ensure that the states of all the subsystems are all uniformly bounded when a gradient based parameter adaptation law is used and that the states of all the systems go to zero asymptotically

approximation error in pde based modelling of vehicular platoons we study the problem of how much error is introduced in approximating the dynamics of a large vehicular platoon by using a partial differential equation as was done in barooah mehta and hespanha barooah p mehta p g and hespanha j p 2009 mistuning based decentralised control of vehicular platoons for improved closed loop stability ieee transactions on automatic control 54 2100 2113 hao barooah and mehta hao h barooah p and mehta p g 2011 stability margin scaling laws of distributed formation control as a function of network structure ieee transactions on automatic control 56 923 929 in particular we examine the difference between the stability margins of the coupled ordinary differential equations ode model and its partial differential equation pde approximation which we call the approximation error the stability margin is defined as the absolute value of the real part of the least stable pole the pde model has proved useful in the design of distributed control schemes barooah et al 2009 hao et al 2011 it provides insight into the effect of gains of local controllers on the closed loop stability margin that is lacking in the coupled ode model here we show that the ratio of the approximation error to the stability margin is o 1 n where n is the number of vehicles thus the pde model is an accurate approximation of the coupled ode model when n is large numerical computations are provided to corroborate the analysis

nonequilibrium steady states of matrix product form a solver s guide we consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model among other things biological transport and traffic flow we begin with a broad overview of this class of driven diffusive systems which includes exclusion processes focusing on interesting physical properties such as shocks and phase transitions we then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form in addition to a gentle introduction to this matrix product approach how it works and how it relates to similar constructions that arise in other physical contexts we present a unified pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed we also review a number of more advanced topics including nonequilibrium free energy functionals the classification of exclusion processes involving multiple particle species existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled we conclude with a brief discussion of open problems for future research

bi orthogonal polynomial sequences and the asymmetric simple exclusion process we state the diffusion algebra equations of the stationary state of the three parameter and q asymmetric simple exclusion process as a linear functional acting on a tensor algebra from we construct a pair of sequences and of monic polynomials which are bi orthogonal that is they satisfy where is a scalar the uniqueness and existence of the pair of sequences arises from the determinant of the bi moment matrix whose elements satisfy a pair of q recurrence relations the determinant is evaluated using an ldu decomposition if the linear functional is represented as an inner product then the action of the polynomials qn on the boundary vector generate a basis whose orthogonal dual vectors are given by the action of pn on the dual boundary vector that is this basis gives the representation of the algebra which is associated with the al salam chihara polynomials obtained by sasamoto

automated traffic and the finite size resonance we investigate in detail what one might call the canonical automated traffic problem a long string of n 1 cars numbered from 0 to n moves along a one lane road in formation at a constant velocity and with a unit distance between successive cars each car monitors the relative velocity and position of only its neighboring cars this information is then fed back to its own engine which decelerates brakes or accelerates according to the information it receives the question is what happens when due to an external influence a traffic light turning green the zero th car the leader accelerates

transients of platoons with asymmetric and different laplacians abstract we consider an asymmetric control of platoons of identical vehicles with nearest neighbor interaction recent results show that if the vehicle uses different asymmetries for position and velocity errors the platoon has a short transient and low overshoots in this paper we investigate the properties of vehicles with friction to achieve consensus an integral part is added to the controller making the vehicle a third order system we show that the parameters can be chosen so that the platoon behaves as a wave equation with different wave velocities simulations suggest that our system has a better performance than other nearest neighbor scenarios moreover an optimization based procedure is used to find the controller properties

adding a single state memory optimally accelerates symmetric linear maps previous papers have proposed to add memory registers to the dynamics of discrete time linear systems in order to accelerate their convergence in particular it has been proved that adding one memory slot per agent allows faster convergence towards average consensus we here prove that this situation cannot be improved by adding more memory slots when the knowledge about the self adjoint linear map to be accelerated reduces to bounds on its extreme eigenvalues

flocks and formations given a large number the flock of moving physical objects we investigate physically reasonable mechanisms of influencing their orbits in such a way that they move along a prescribed course and in a prescribed and fixed configuration or in formation each agent is programmed to see the position and velocity of a certain number of others this flow of information from one agent to another defines a fixed directed loopless graph in which the agents are represented by the vertices this graph is called the communication graph to be able to fly in formation an agent tries to match the mean position and velocity of his neighbors his direct antecedents on the communication graph to his own this operation defines a directed laplacian on the communication graph a linear feedback is used to ensure stability of the coherent flight patterns we analyze in detail how the connectedness of the communication graph affects the coherence of the stable flight patterns and give a characterization of these stable flight patterns we do the same if in addition the flight of the flock is guided by one or more leaders finally we use this theory to develop some applications examples of these are flight guided by external controls flocks of flocks and some results about flocks whose formation is always oriented along the line of flight such as geese

stability margin scaling laws for distributed formation control as a function of network structure we describe a methodology for modeling analysis and distributed control design of a large vehicular formation whose information graph is a d dimensional lattice we derive asymptotic formulae for the closed loop stability margin based on a partial differential equation pde approximation of the formation we show that the exponent in the scaling law for the stability margin is influenced by the structure of the information graph and by the control architecture symmetric or asymmetric for a given fixed number of vehicles we show that the scaling law can be improved significantly by employing a higher dimensional information graph and or by introducing small asymmetry mistuning in the nominally symmetric proportional control gains we also provide a characterization of the error introduced by the pde approximation