an algorithm for simplex tableau reduction the push to pull solution strategy the simplex algorithm requires artificial variables for solving linear programs which lack primal feasibility at the origin point we present a new general purpose solution algorithm called push to pull which obviates the use of artificial variables the algorithm consists of preliminaries for setting up the initialization followed by two main phases the push phase develops a basic variable set bvs which may or may not be feasible unlike simplex and dual simplex this approach starts with an incomplete bvs initially and then variables are brought into the basis one by one if the bvs is complete but the optimality condition is not satisfied then push phase pushes until this condition is satisfied using the rules similar to the ordinary simplex since the proposed strategic solution process pushes towards an optimal solution it may generate an infeasible bvs the pull phase pulls the solution back to feasibility using pivoting rules similar to the dual simplex method all phases use the usual gauss pivoting row operation and it is shown to terminate successfully or indicates unboundedness or infeasibility of the problem a computer implementation which further reduces the size of simplex tableau to the dimensions of the original decision variables is provided this enhances considerably the storage space and the computational complexity of the proposed solution algorithm illustrative numerical examples are also presented

structure of the sets of mutually unbiased bases with cyclic symmetry mutually unbiased bases that can be cyclically generated by a single unitary operator are of special interest for they can be readily implemented in practice we show that for a system of qubits finding such a generator can be cast as the problem of finding a symmetric matrix over the field 2 equipped with an irreducible characteristic polynomial of a given fibonacci index the entanglement structure of the resulting complete sets is determined by two additive matrices of the same size

multi stage robust chinese remainder theorem it is well known that the traditional chinese remainder theorem crt is not robust in the sense that a small error in a remainder may cause a large reconstruction error a robust crt was recently proposed for a special case when the greatest common divisor gcd of all the moduli is more than 1 and the remaining integers factorized by the gcd are co prime it basically says that the reconstruction error is upper bounded by the remainder error level tau if tau is smaller than a quarter of the gcd of all the moduli in this paper we consider the robust reconstruction problem for a general set of moduli we first present a necessary and sufficient condition on the remainder errors with a general set of moduli and also a corresponding robust reconstruction method this can be thought of as a single stage robust crt we then propose a two stage robust crt by grouping the moduli into several groups as follows first the single stage robust crt is applied to each group then with these robust reconstructions from all the groups the single stage robust crt is applied again across the groups this is easily generalized to multi stage robust crt with this two stage robust crt the robust reconstruction holds even when the remainder error level tau is above the quarter of the gcd of all the moduli and an algorithm on how to group a set of moduli for a better reconstruction robustness is proposed in some special cases

an inequality sorting algorithm for a class of linear programming problems abstract the linear programming problem lpp considered here is maximize b t x subject to the inequalities c a x d where a is an m n real matrix of rank r m the vectors c and d are real and d 0 and the lpp is feasible with bounded optimal solutions an algorithm that normalizes the inequalities arranges them in an ascending order on the right most rm quantities and then chooses r linearly independent inequalities is described it computes a solution of the lpp with these r inequalities that the solution determines which inequalities are not satisfied on the left most lm inequalities the unsatisfied inequalities are then negated and the algorithm is executed for the second time this process is continued until all the inequalities are satisfied the algorithm is in practice amazingly fast often direct with computational complexity o mn 2 an application of this algorithm to a general 1pp is discussed

consensus of high order multi agent systems with large input and communication delays we study in this paper the consensus problem for multi agent systems with agents characterized by high order linear systems with time delays in both the communication network and inputs provided that the open loop dynamics of the agents is not exponentially unstable but may be polynomially unstable and the communication topology contains a directed spanning tree a truncated predictor feedback approach is established to solve the consensus problem it is shown that if the delays are constant and exactly known the consensus problems can be solved by both full state feedback and observer based output feedback protocols for arbitrarily large yet bounded delays if it is further assumed that the open loop dynamics of the agents only contains zero eigenvalues the delays are allowed to be time varying and unknown numerical examples are worked out to illustrate the effectiveness of the proposed approaches

distributed tracking control for linear multiagent systems with a leader of bounded unknown input this technical note considers the distributed tracking control problem of multiagent systems with general linear dynamics and a leader whose control input is nonzero and not available to any follower based on the relative states of neighboring agents two distributed discontinuous controllers with respectively static and adaptive coupling gains are designed for each follower to ensure that the states of the followers converge to the state of the leader if the interaction graph among the followers is undirected the leader has directed paths to all followers and the leader s control input is bounded a sufficient condition for the existence of the distributed controllers is that each agent is stabilizable simulation examples are given to illustrate the theoretical results

distributed output tracking of high order nonlinear multi agent systems with unstable linearization abstract this paper addresses the problem of distributed output tracking for a class of high order nonlinear multi agent systems whose linearization parts may have unstable modes for the case where the graph topology is directed and the leader is the neighbor of only a small portion of followers distributed tracking control laws are designed by using the algebra graph theory it is shown that all the states of the closed loop system are bounded and the tracking errors can be tuned to be arbitrarily small finally the design procedure is applied to underactuated unstable mechanical multi agent systems from which the efficiency of the tracking controllers is demonstrated

multicomplementary operators via finite fourier transform a complete set of d 1 mutually unbiased bases exists in a hilbert space of dimension d whenever d is a power of a prime we discuss a simple construction of d 1 disjoint classes each one having d 1 commuting operators such that the corresponding eigenstates form sets of unbiased bases such a construction works properly for prime dimension we investigate an alternative construction in which the real numbers that label the classes are replaced by a finite field having d elements one of these classes is diagonal and can be mapped to cyclic operators by means of the finite fourier transform which allows one to understand complementarity in a similar way as for the position momentum pair in standard quantum mechanics the relevant examples of two and three qubits and two qutrits are discussed in detail

o n3 noniterative heuristic algorithm for linear programs with error free implementation we present an o n 3 noniterative heuristic algorithm and describe an implementation of p adic arithmetic to solve the linear programming problems error free an optimality test as well as numerical examples depict the scope of such an algorithm in the few cases where it did not lead to the optimal solution it did provide a nonnegative solution of the system ax b which could be used as the basic feasible solution for any one of the widely used simplex methods say the revised simplex algorithm without the need to include artificial variables in ax b for consistency check thus the algorithm could be at least a pre processor for any simplex algorithm and avoids the enhancement of dimensions of the constraint matrix a and consequent additional space computational complexities and errors

conditions for synchronizability in arrays of coupled linear systems synchronization in arrays of coupled continuous time linear systems is studied sufficiency of certain conditions for the existence of a synchronizing feedback law are analyzed it is shown that for neutrally stable systems that are detectable from their outputs a linear feedback law exists under which any number of coupled systems synchronize provided that the directed weighted graph describing the interconnection is fixed and connected an algorithm generating one such feedback law as well as the trajectory that the solutions converge to are presented it is also shown that for critically unstable systems detectability is not sufficient whereas full state coupling is for the existence of a linear feedback law that is synchronizing for all coupling configurations described by a connected graph

consensus of linear multi agent systems with reduced order observer based protocols this paper considers the consensus problems for both continuous and discrete time linear multi agent systems with directed communication topologies distributed reduced order observer based consensus protocols are proposed based on the relative outputs of neighboring agents a multi step algorithm is presented to construct a reduced order protocol under which a continuous time multi agent system whose communication topology contains a directed spanning tree can reach consensus this algorithm is further modified to achieve consensus with a prescribed convergence rate these two algorithms have a favorable decoupling property in light of the modified algebraic riccati equation an algorithm is then given to construct a reduced order protocol for the discrete time case

consensus of multi agent systems based on sampled data control this article studies the consensus problem in directed networks assuming that each agent is with double integrator dynamics and only obtains the measurements of its positions relative to its neighbours at sampling instants we propose a protocol based on sampled data control and derive an equivalent characterisation of the solvability of the consensus problem under this protocol in virtue of this equivalent characterisation we further consider two cases fixed topology and switching topology for the first case we present a set of sampling periods and feedback coefficients which ensure that the protocol can solve a consensus problem for the second case we derive sufficient conditions for the protocol to solve a consensus problem under arbitrary switching signals and under a class of switching signals respectively finally simulations are provided to illustrate the effectiveness of the theoretical results

complete sets of cyclic mutually unbiased bases in even prime power dimensions we present a construction method for complete sets of cyclic mutually unbiased bases mubs in hilbert spaces of even prime power dimensions in comparison to the usual complete sets of mubs complete cyclic sets possess the additional property of being generated by a single unitary operator the construction method is based on the idea of obtaining a partition of multi qubit pauli operators into maximal commuting sets of operators with the help of a suitable element of the clifford group as a consequence we explicitly obtain complete sets of cyclic mubs generated by a single element of the clifford group in dimensions 2 m for m 1 2 24

distributed robust control of linear multi agent systems with parameter uncertainties this article considers the distributed robust control problems of uncertain linear multi agent systems with undirected communication topologies it is assumed that the agents have identical nominal dynamics while subject to different norm bounded parameter uncertainties leading to weakly heterogeneous multi agent systems distributed controllers are designed for both continuous and discrete time multi agent systems based on the relative states of neighbouring agents and a subset of absolute states of the agents it is shown for both the continuous and discrete time cases that the distributed robust control problems under such controllers in the sense of quadratic stability are equivalent to the h control problems of a set of decoupled linear systems having the same dimensions as a single agent a two step algorithm is presented to construct the distributed controller for the continuous time case which does not involve any conservatism and meanwhile decouples the feedback gain design from the commun

distributed coordination for second order multi agent systems with nonlinear dynamics using only relative position measurements in this paper we study the distributed coordination for second order multi agent systems with intrinsic nonlinear dynamics under an undirected graph that characterizes the interaction among the agents or followers both the leaderless consensus problem and the coordinated tracking problem with a dynamic leader are considered by introducing a distributed filter for each agent or follower the proposed control algorithms use only relative position measurements in the absence of communication in the special case without the intrinsic nonlinear dynamics i e for multi agent systems with double integrator dynamics we further derive a necessary and sufficient condition for the leaderless consensus problem under a general directed graph finally simulation examples are provided to show the effectiveness of the proposed control algorithms

synchronization reachable topology and synchronization of discrete time linear multi agent systems in this note we consider the synchronization problem for discrete time linear multi agent systems with time varying network topology to describe a new joint connectivity condition for synchronization the concept of synchronization reachable topology is introduced under the conditions that agent dynamics is neutrally stable and network topology is synchronization reachable a synchronization and an exponential synchronization result are obtained using paracontraction theory the two results generalize in a natural way the classical results for the consensus of first order discrete time multi agent systems finally the effectiveness of the results is illustrated by a simulation example

computing a matrix symmetrizer exactly using modified multiple modulus residue arithmetic a symmetric solution x satisfying the matrix equation xa atx is called a symmetrizer of the matrix a a general algorithm to compute a matrix symmetrizer is obtained a new multiple modulus residue arithmetic called floating point modular arithmetic is described and implemented on the algorithm to compute an error free matrix symmetrizer

brief paper synchronization in networks of identical linear systems the paper investigates the synchronization of a network of identical linear state space models under a possibly time varying and directed interconnection structure the main result is the construction of a dynamic output feedback coupling that achieves synchronization if the decoupled systems have no exponentially unstable mode and if the communication graph is uniformly connected the result can be interpreted as a generalization of classical consensus algorithms stronger conditions are shown to be sufficient but to some extent also necessary to ensure synchronization with the diffusive static output coupling often considered in the literature

open problems in computational linear algebra in the recent past a few important open problems such as those involving developing polynomial time iterative algorithms for solving linear programs in both integer and real number models and for testing and generating primes have been solved there are still several problems we have not found solutions over decades centuries problems such as those involving i devising a deterministic noniterative polynomial time algorithm for linear programs ii deciding a priori all required fail proof prime bases for errorfree computations for linear systems and linear optimization iii determining computational complexities in some deterministic algorithms iv designing algorithms with the lowest possible bound of complexity for matrix multiplications v developing a polynomial time deterministic algorithm for computing the error bounds in an error free computation vi verification of the solution of some nonlinear optimization problems in polynomial time vii obtaining the error bounds in the solution of some linear nonlinear problems when solved probabilistically are open problems the solution of these open problems has the potential to revolutionize the whole area of computational mathematics and science as well as super grid computing we present here some of these open problems precisely along with the related discussions

exact and robust reconstructions of integer vectors based on multidimensional chinese remainder theorem md crt the robust chinese remainder theorem crt has been recently proposed for robustly reconstructing a large nonnegative integer from erroneous remainders it has found many applications in signal processing including phase unwrapping and frequency estimation under sub nyquist sampling motivated by the applications in multidimensional md signal processing in this article we propose the md crt and robust md crt for integer vectors specifically by rephrasing the abstract crt for rings in number theoretic terms we first derive the md crt for integer vectors with respect to a general set of integer matrix moduli which provides an algorithm to uniquely reconstruct an integer vector from its remainders if it is in the fundamental parallelepiped of the lattice generated by a least common right multiple of all the moduli for some special forms of moduli we present explicit reconstruction formulae moreover we derive the robust md crt for integer vectors when the remaining integer matrices of all the moduli left divided by their greatest common left divisor gcld are pairwise commutative and coprime two different reconstruction algorithms are proposed and accordingly two different conditions on the remainder error bound for the reconstruction robustness are obtained which are related to a quarter of the minimum distance of the lattice generated by the gcld of all the moduli or the smith normal form of the gcld

optimum statistical estimates in conditions of ambiguity an ambiguity problem arises if a distance is measured by observing phase shift of signals when the range of the measured distance is greater than the wavelength of the signals used this ambiguity generally can be resolved by pumping extra information into the system a classical estimation problem is the obtained at the expense of reduced autonomy of the estimator due to the demand for external information not inherent to the system the interval lambda of the unambiguous estimates can be considerably extended using observations of several different wavelengths lambda sub i this yields lambda as the least common multiple of all lambda sub i a method of developing optimum estimation rules for this case is introduced optimum estimation rules for some typical cases with gaussian observation errors are also developed

decentralised consensus for multiple lagrangian systems based on event triggered strategy abstractthis paper considers the decentralised event triggered consensus problem for multi agent systems with lagrangian dynamics under undirected graphs first a distributed leaderless and event triggered consensus control algorithm is presented based on the definition of generalised positions and velocities for all agents there is only one triggering function for both the generalised positions and velocities and no zeno behaviour exhibited under the proposed consensus strategy second an adaptive event triggered consensus control algorithm is proposed for such multi agent systems with unknown constant parameters third based on sliding mode method an event triggered consensus control algorithm is considered for the case with external disturbance finally simulation results are given to illustrate the theoretical results

sampled data discrete time coordination algorithms for double integrator dynamics under dynamic directed interaction in this article we study two sampled data based discrete time coordination algorithms for multi vehicle systems with double integrator dynamics under dynamic directed interaction for both algorithms we derive sufficient conditions on the interaction graph the damping gain and the sampling period to guarantee coordination by using the property of infinity products of stochastic matrices when the conditions on the damping gain and the sampling period are satisfied the first algorithm guarantees coordination on positions with a zero final velocity if the interaction graph has a directed spanning tree jointly while the second algorithm guarantees coordination on positions with a constant final velocity if the interaction graph has a directed spanning tree at each time interval simulation results are presented to show the effectiveness of the theoretical results

a robust generalized chinese remainder theorem for two integers a generalized chinese remainder theorem crt for multiple integers from residue sets has been studied recently where the correspondence between the remainders and the integers in each residue set modulo several moduli is not known a robust crt has also been proposed lately to robustly reconstruct a single integer from its erroneous remainders in this paper we consider the reconstruction problem of two integers from their residue sets where the remainders not only are out of order but also may have errors we prove that two integers can be robustly reconstructed if their remainder errors are less than m 8 where m is the greatest common divisor of all the moduli we also propose an efficient reconstruction algorithm finally we present some simulations to verify the efficiency of the proposed algorithm this paper is motivated from and has applications in the determination of multiple frequencies from multiple undersampled waveforms

an iterative algorithm for solving a finite dimensional linear operator equation t x f with applications abstract this paper proposes an iterative algorithm for solving a general finite dimensional linear operator equation t x f and demonstrates that it will get the exact solution within a finite number of iteration steps this algorithm unifies all the iterative methods in huang 2008 3 peng 2005 7 and peng and peng 2006 8 and provides an iterative method for solving an inverse problem related to hermitian generalized hamiltonian matrices 2 12

kalman filtering with intermittent observations motivated by navigation and tracking applications within sensor networks we consider the problem of performing kalman filtering with intermittent observations when data travel along unreliable communication channels in a large wireless multihop sensor network the effect of communication delays and loss of information in the control loop cannot be neglected we address this problem starting from the discrete kalman filtering formulation and modeling the arrival of the observation as a random process we study the statistical convergence properties of the estimation error covariance showing the existence of a critical value for the arrival rate of the observations beyond which a transition to an unbounded state error covariance occurs we also give upper and lower bounds on this expected state error covariance

computing matrix symmetrizers part 2 new methods using eigendata and linear means a comparison abstract over any field f every square matrix a can be factored into the product of two symmetric matrices as a s 1 s 2 with s i s i t f n n and either factor can be chosen nonsingular as was discovered by frobenius in 1910 frobenius symmetric matrix factorization has been lying almost dormant for a century the first successful method for computing matrix symmetrizers i e symmetric matrices s such that sa is symmetric was inspired by an iterative linear systems algorithm of huang and nong 2010 in 2013 29 30 the resulting iterative algorithm has solved this computational problem over r and c but at high computational cost this paper develops and tests another linear equations solver as well as eigen and principal vector or schur normal form based algorithms for solving the matrix symmetrizer problem numerically four new eigendata based algorithms use respectively svd based principal vector chain constructions gram schmidt orthogonalization techniques the arnoldi method or the schur normal form of a in their formulations they are helped by datta s 1973 method that symmetrizes unreduced hessenberg matrices directly the eigendata based methods work well and quickly for generic matrices a and create well conditioned matrix symmetrizers through eigenvector dyad accumulation but all of the eigen based methods have differing deficiencies with matrices a that have ill conditioned or complicated eigen structures with nontrivial jordan normal forms our symmetrizer studies for matrices with ill conditioned eigensystems lead to two open problems of matrix optimization

taussky s theorem symmetrizability and modal analysis revisited this paper is concerned with symmetrization and diagonalization of real matrices and their implications for the dynamics of linear second order systems governed by equations of motion having asymmetric coefficient matrices results in the light of taussky s theorem are presented the connection of the symmetrizers with the eigenvalue problem is brought out an alternative proof of taussky s theorem for real matrices is presented diagonalization of two real symmetric but not necessarily positive definite matrices is discussed in the context of undamped non gyroscopic systems a commutator of two matrices with respect to a given third matrix is defined this commutator is found to play an interesting role in deciding simultaneous diagonalizability of two or three matrices errors in a few previously known results are brought out pseudo conservative systems are studied and their connection with the so called symmetrizable systems is critically examined results for modal analysis of general non conservative systems are presented illustrative examples are given

a computer implementation of the push and pull algorithm and its computational comparison with lp simplex method the simplex algorithm requires artificial variables for solving linear programs which lack primal feasibility at the origin point we present a new general purpose solution algorithm called push and pull which obviates the use of artificial variables the algorithm consists of preliminaries for setting up the initialization followed by two main phases the push phase develops a basic variable set bvs which may or may not be feasible unlike simplex and dual simplex this approach starts with an incomplete bvs initially and then variables are brought into the basis one by one if the bvs is complete but the optimality condition is not satisfied then push phase pushes until this condition is satisfied using the rules similar to the ordinary simplex since the proposed strategic solution process pushes towards an optimal solution it may generate an infeasible bvs the pull phase pulls the solution back to feasibility using pivoting rules similar to the dual simplex method all phases use the usual gauss pivoting row operation and it is shown to terminate successfully or indicates unboundedness or infeasibility of the problem a computer implementation which enables the user to run either push and pull or ordinary simplex algorithms is provided the fully coded version of the algorithm is available from the authors upon request a comparison analysis to test the efficiency of push and pull algorithm comparing to ordinary simplex is accomplished illustrative numerical examples are also presented

an algorithm for linear programming which requires o m n n 2 m n 1 5 n l arithmetic operations we present an algorithm for linear programming which requires o m n n 2 m n 1 5 n l arithmetic operations wherem is the number of constraints andn is the number of variables each operation is performed to a precision of o l bits l is bounded by the number of bits in the input the worst case running time of the algorithm is better than that of karmarkar s algorithm by a factor of sqrt m n

distributed and truncated reduced order observer based output feedback consensus of multi agent systems this technical note is concerned with output feedback consensus of both continuous time and discrete time multi agent systems mass characterized by high order linear systems with directed communication topologies distributed reduced order observer based protocols are established by only using the relative outputs and inputs of neighboring agents it is shown that consensusability by the proposed output feedback protocols is equivalent to the consensusability by state feedback protocol under the condition that the open loop dynamics of the mass is not exponentially unstable a truncated reduced order observer based protocol utilizing only the relative outputs of neighboring agents is also established a numerical example is given to illustrate the effectiveness of the proposed approaches

a robust chinese remainder theorem with its applications in frequency estimation from undersampled waveforms the chinese remainder theorem crt allows to reconstruct a large integer from its remainders modulo several moduli in this paper we propose a robust reconstruction algorithm called robust crt when the remainders have errors we show that using the proposed robust crt the reconstruction error is upper bounded by the maximal remainder error range named remainder error bound if the remainder error bound is less than one quarter of the greatest common divisor gcd of all the moduli we then apply the robust crt to estimate frequencies when the signal waveforms are undersampled multiple times it shows that with the robust crt the sampling frequencies can be significantly reduced

a closed form robust chinese remainder theorem and its performance analysis chinese remainder theorem crt reconstructs an integer from its multiple remainders that is well known not robust in the sense that a small error in a remainder may cause a large error in the reconstruction a robust crt has been recently proposed when all the moduli have a common factor and the robust crt is a searching based algorithm and no closed from is given in this paper a closed form robust crt is proposed and a necessary and sufficient condition on the remainder errors for the closed form robust crt to hold is obtained furthermore its performance analysis is given it is shown that the reason for the robustness is from the remainder differential process in both searching based and our proposed closed form robust crt algorithms which does no exist in the traditional crt we also propose an improved version of the closed form robust crt finally we compare the performances of the traditional crt the searching based robust crt and our proposed closed form robust crt and its improved version algorithms in terms of both theoretical analysis and numerical simulations the results demonstrate that the proposed closed form robust crt its improved version has the best performance has the same performance but much simpler form than the searching based robust crt

distance estimation using wrapped phase measurements in noise measuring a distance using phase measurements is a common practice in many areas of engineering almost inevitably these measurements are accompanied by noise and are always subject to ambiguity resulting from the phase of modulo 2 pi in the presence of phase ambiguity where for instance the unknown distance is far longer than the wavelength of the signal carrying the phase measurement the distance cannot be uniquely determined one way to resolve this phase ambiguity is to measure the signal phase at multiple frequencies converting the phase ambiguity problem into one of solving a family of diophantine equations typically under some reasonable assumptions the diophantine problems can be solved using the chinese reminder theorem as documented in the literature however the existing algorithms can experience significant computational overhead for a given application because an exhaustive search is required in this paper a novel method addressing the phase ambiguity issue using lattice theoretic ideas is proposed and a closed form algorithm is presented for the estimation of the number of wavelengths in the unknown distance using the phase measurements taken at multiple wavelengths the algorithm is extremely efficient as the diophantine equations are solved without searching the unknown distance can then be estimated via a maximum likelihood method using the unwrapped phase measurement a statistical bound of the measurement noise which ensures that the number of whole wavelengths in the unknown distance can be found with a probability close to unity is derived the robustness efficiency and estimation accuracy of the proposed method are demonstrated by the simulated results presented

optimal estimates of two common remainders for a robust generalized chinese remainder theorem estimation of multiple common remainders from a sequence of erroneous residue sets is an important step for the robust generalized chinese remainder theorem crt this paper considers the problem of how to estimate the two common remainders from their residue sets modulo a set of moduli to measure the errors properly under the modular operation we introduce two type circular distances based on these circular distances two estimation methods are proposed by properly grouping the erroneous remainders into two ordered clusters both of the two methods perform better with lower computational complexities than the existing methods in this paper theoretical analysis as well as analytical results for the two proposed methods are obtained for the first method the two optimal estimates are proved to be in a finite set with no more than l candidates where l is the number of the given moduli the second method have closed forms simulation results show that the two proposed methods have nearly the same performance these optimal estimates can improve the performance of the robust generalized crt significantly

consensus seeking of high order dynamic multi agent systems with fixed and switching topologies this article studies a consensus problem for a group of high order dynamic agents with fixed and switching topologies two linear consensus protocols which only depend on the agent s own information and its neighbours partial information are proposed respectively for the group with and without communication time delays for the case of fixed topology and zero communication time delay a necessary and sufficient condition for the consensus is established the parameter design of the protocol is also discussed under switching topology the consensus is investigated for two cases the case of zero communication time delay and the case of nonuniform communication time delays for each case a sufficient condition for the consensus is provided moreover for a group of double triple integrator agents with fixed topology and uniform constant time delay some necessary and sufficient conditions for the consensus are given finally numerical simulations are worked out to illustrate the effectiveness of the

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

a variation on karmarkar s algorithm for solving linear programming problems the algorithm described here is a variation on karmarkar s algorithm for linear programming it has several advantages over karmarkar s original algorithm in the first place it applies to the standard form of a linear programming problem and produces a monotone decreasing sequence of values of the objective function the minimum value of the objective function does not have to be known in advance secondly in the absence of degeneracy the algorithm converges to an optimal basic feasible solution with the nonbasic variables converging monotonically to zero this makes it possible to identify an optimal basis before the algorithm converges

distributed algorithms for reaching consensus on general functions this paper presents analysis and design results for distributed consensus algorithms in multi agent networks we consider continuous consensus functions of the initial state of the network agents under mild smoothness assumptions we obtain necessary and sufficient conditions characterizing any algorithm that asymptotically achieves consensus this characterization is the building block to obtain various design results for networks with weighted directed interconnection topologies we first identify a class of smooth functions for which one can synthesize in a systematic way distributed algorithms that achieve consensus we apply this result to the family of weighted power mean functions and characterize the exponential convergence properties of the resulting algorithms we establish the validity of these results for scenarios with switching interconnection topologies finally we conclude with two discontinuous distributed algorithms that achieve respectively max and min consensus in finite time

all mutually unbiased product bases in dimension 6 all mutually unbiased bases in dimension 6 consisting of product states only are constructed several continuous families of pairs and two triples of mutually unbiased product bases are found to exist but no quadruple the exhaustive classification leads to a proof that a complete set of seven mutually unbiased bases if it exists cannot contain a triple of mutually unbiased product bases

about mutually unbiased bases in even and odd prime power dimensions mutually unbiased bases generalize the x y and z qubit bases they possess numerous applications in quantum information science it is well known that in prime power dimensions n pm with p prime and m a positive integer there exists a maximal set of n 1 mutually unbiased bases in the present paper we derive an explicit expression for those bases in terms of the operations of the associated finite field galois division ring of n elements this expression is shown to be equivalent to the expressions previously obtained by ivanovic 1981 j phys a math gen 14 3241 in odd prime dimensions and wootters and fields 1989 ann phys 191 363 in odd prime power dimensions in even prime power dimensions we derive a new explicit expression for the mutually unbiased bases the new ingredients of our approach are basically the following we provide a simple expression of the generalized pauli group in terms of the additive characters of the field and we derive an exact groupal composition law between the elements of the commuting subsets of the generalized pauli group renormalized by a well chosen phase factor

graph laplacian based leaderless consensus for multi agent systems in t s form abstract the graph laplacian based leaderless consensus protocol design problem for nonlinear multi agent systems mas in takagi sugeno form is highly challenging the main reason is that the similarity transformation of the mas dynamics in an abbreviated form with the kronecker product is quite restricted we provide a positive answer to this problem

generation of bases with definite factorization for an n qubit system and mutually unbiased sets construction we propose a systematic procedure to construct all the possible bases with a definite factorization structure eigenstates of n commuting monomials constructed as products of pauli operators for an n qubit system as well as the possibility of collecting them into mutually unbiased sets we also discuss an algorithm for the determination of basis separability and propose a criteria for complementarity between such bases the results are applied to generate non isomorphic complete sets of mutually unbiased bases

maximal entanglement collective coordinates and tracking the king maximal entangled states provide a basis to two d dimensional particles in hilbert space d prime 2 the maximally entangled states forming this basis are uniquely related to product states in the collective center of mass and relative coordinates these states are associated underpinned with lines of finite geometry whose constituent points are associated with product states carrying mutual unbiased bases labels this representation is shown to be convenient for the study of the mean king problem and a variant thereof termed tracking the king which proves to be a novel quantum communication channel the main topics and notations used are reviewed in an attempt to keep the paper self contained

maximum likelihood estimation based robust chinese remainder theorem for real numbers and its fast algorithm robust chinese remainder theorem crt has been recently investigated for both integers and real numbers where the folding integers are accurately recovered from erroneous remainders in this paper we consider the crt problem for real numbers with noisy remainders that follow wrapped gaussian distributions we propose the maximum likelihood estimation mle based crt when the remainder noises may not necessarily have the same variances furthermore we present a fast algorithm for the mle based crt algorithm that only needs to search for the solution among l elements where l is the number of remainders then a necessary and sufficient condition on the remainder errors for the mle crt to be robust is obtained which is weaker than the existing result finally we compare the performances of the newly proposed algorithm and the existing algorithm in terms of both theoretical analysis and numerical simulations the results demonstrate that the proposed algorithm not only has a better performance especially when the remainders have different error levels variances but also has a much lower computational complexity

towards robustness in residue number systems the problem of robustly reconstructing a large number from its erroneous remainders with respect to several moduli namely the robust remaindering problem may occur in many applications including phase unwrapping frequency detection from several undersampled waveforms wireless sensor networks etc assuming that the dynamic range of the large number is the maximal possible one i e the least common multiple lcm of all the moduli a method called robust chinese remainder theorem for solving the robust remaindering problem has been recently proposed in this paper by relaxing the assumption that the dynamic range is fixed to be the lcm of all the moduli a tradeoff between the dynamic range and the robustness bound for two modular systems is studied it basically says that a decrease in the dynamic range may lead to an increase of the robustness bound we first obtain a general condition on the remainder errors and derive the exact dynamic range with a closed form formula for the robustness to hold we then propose simple closed form reconstruction algorithms furthermore the newly obtained two modular results are applied to the robust reconstruction for multimodular systems and generalized to real numbers finally some simulations are carried out to verify our proposed theoretical results

solving linear programming problems exactly a linear programming problem in an inequality form having a bounded solution is solved error free using an algorithm that sorts the inequalities removes the redundant ones and uses the p adic arithmetic c elsevier science inc 1997

cyclic mutually unbiased bases fibonacci polynomials and wiedemann s conjecture we relate the construction of a complete set of cyclic mutually unbiased bases i e mutually unbiased bases generated by a single unitary operator in power of two dimensions to the problem of finding a symmetric matrix over f2 with an irreducible characteristic polynomial that has a given fibonacci index for dimensions of the form 22k we present a solution that shows an analogy to an open conjecture of wiedemann in finite field theory finally we discuss the equivalence of mutually unbiased bases